Fluid Mechanics, Fourth Edition
Founders of Modern Fluid Dynamics
Ludwig Prandtl
(1875–1953)
G. I. Taylor
(1886–1975)
(Biographical sketches of Prandtl and Taylor are given in Appendix C.)
Photograph of Ludwig Prandtl is reprinted with permission from the Annual Review of Fluid
Mechanics, Vol. 19, Copyright 1987 by Annual Reviews www.AnnualReviews.org.
Photograph of Geoffrey Ingram Taylor at age 69 in his laboratory reprinted with permission
from the AIP Emilio Segrè Visual Archieves. Copyright, American Institute of Physics, 2000.
Fluid Mechanics
Fourth Edition
Pijush K. Kundu
Oceanographic Center
Nova Southeastern University
Dania, Florida
Ira M. Cohen
Department of Mechanical Engineering and
Applied Mechanics
University of Pennsylvania
Philadelphia, Pennsylvania
with contributions by P. S. Ayyaswamy and H. H. Hu
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Notices
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Library of Congress Cataloging-in-Publication Data
Kundu, Pijush K.
Fluid mechanics / Pijush K. Kundu, Ira M. Cohen. – 4th ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-12-373735-9 (alk. paper) /
1. Fluid mechanics. I. Cohen, Ira M. II. Title.
QA901.K86 2008
620.1’06–dc22
2007042765
Reprinted January 2010 – ISBN: 978-0-12-381399-2
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1
The fourth edition is dedicated to the memory of Pijush K. Kundu and also to my
wife Linda and daughters Susan and Nancy who have greatly enriched my life.
“Everything should be made as simple as possible,
but not simpler.”
—Albert Einstein
“If nature were not beautiful, it would not be worth studying it.
And life would not be worth living.”
—Henry Poincaré
In memory of Pijush Kundu
Pijush Kanti Kundu was born in Calcutta,
India, on October 31, 1941. He received a
B.S. degree in Mechanical Engineering in
1963 from Shibpur Engineering College of
Calcutta University, earned an M.S. degree
in Engineering from Roorkee University in
1965, and was a lecturer in Mechanical Engineering at the Indian Institute of Technology
in Delhi from 1965 to 1968. Pijush came to
the United States in 1968, as a doctoral student at Penn State University. With Dr. John
L. Lumley as his advisor, he studied instabilities of viscoelastic fluids, receiving his doctorate in 1972. He began his lifelong interest in
oceanography soon after his graduation, working as Research Associate in Oceanography at Oregon State University from 1968 until 1972. After spending a year at
the University de Oriente in Venezuela, he joined the faculty of the Oceanographic
Center of Nova Southeastern University, where he remained until his death in 1994.
During his career, Pijush contributed to a number of sub-disciplines in physical
oceanography, most notably in the fields of coastal dynamics, mixed-layer physics,
internal waves, and Indian-Ocean dynamics. He was a skilled data analyst, and, in
this regard, one of his accomplishments was to introduce the “empirical orthogonal
eigenfunction” statistical technique to the oceanographic community.
I arrived at Nova Southeastern University shortly after Pijush, and he and I worked
closely together thereafter. I was immediately impressed with the clarity of his scientific thinking and his thoroughness. His most impressive and obvious quality, though,
was his love of science, which pervaded all his activities. Some time after we met,
Pijush opened a drawer in a desk in his home office, showing me drafts of several
chapters to a book he had always wanted to write. A decade later, this manuscript
became the first edition of “Fluid Mechanics,” the culmination of his lifelong dream;
which he dedicated to the memory of his mother, and to his wife Shikha, daughter
Tonushree, and son Joydip.
Julian P. McCreary, Jr.,
University of Hawaii
Contents
New in This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Preface to Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Preface to Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Preface to First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
Author’s Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
Chapter 1
Introduction
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Units of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solids, Liquids, and Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continuum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fluid Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Classical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Static Equilibrium of a Compressible Medium . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
3
4
5
8
9
12
16
18
22
24
24
Chapter 2
Cartesian Tensors
1.
2.
3.
4.
5.
Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotation of Axes: Formal Definition of a Vector . . . . . . . . . . . . . . . . . . . . .
Multiplication of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Second-Order Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contraction and Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
26
29
30
32
vii
viii
Contents
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Force on a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kronecker Delta and Alternating Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cross Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Operator ∇: Gradient, Divergence, and Curl . . . . . . . . . . . . . . . . . . . . . . . .
Symmetric and Antisymmetric Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eigenvalues and Eigenvectors of a Symmetric Tensor . . . . . . . . . . . . . . . .
Gauss’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boldface vs Indicial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
36
37
38
38
40
41
44
47
49
49
50
51
51
Chapter 3
Kinematics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lagrangian and Eulerian Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eulerian and Lagrangian Descriptions: The Particle Derivative . . . . . . . .
Streamline, Path Line, and Streak Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference Frame and Streamline Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shear Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vorticity and Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Relative Motion near a Point: Principal Axes . . . . . . . . . . . . . . . . . . . . . . . .
Kinematic Considerations of Parallel Shear Flows . . . . . . . . . . . . . . . . . . .
Kinematic Considerations of Vortex Flows . . . . . . . . . . . . . . . . . . . . . . . . . .
One-, Two-, and Three-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . .
The Streamfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
54
55
57
59
60
61
62
64
67
68
71
73
75
77
79
Chapter 4
Conservation Laws
1.
2.
3.
4.
5.
6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time Derivatives of Volume Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Streamfunctions: Revisited and Generalized . . . . . . . . . . . . . . . . . . . . . . . .
Origin of Forces in Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stress at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
82
84
87
88
90
ix
Contents
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Momentum Principle for a Fixed Volume . . . . . . . . . . . . . . . . . . . . . . . . . . .
Angular Momentum Principle for a Fixed Volume . . . . . . . . . . . . . . . . . . .
Constitutive Equation for Newtonian Fluid . . . . . . . . . . . . . . . . . . . . . . . . . .
Navier–Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mechanical Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
First Law of Thermodynamics: Thermal Energy Equation . . . . . . . . . . . .
Second Law of Thermodynamics: Entropy Production . . . . . . . . . . . . . . .
Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Applications of Bernoulli’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
93
98
100
104
105
111
115
116
118
122
124
129
134
136
137
Chapter 5
Vorticity Dynamics
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vortex Lines and Vortex Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Role of Viscosity in Rotational and Irrotational Vortices . . . . . . . . . . . . . .
Kelvin’s Circulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vorticity Equation in a Nonrotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity Induced by a Vortex Filament: Law of Biot and Savart . . . . . . .
Vorticity Equation in a Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interaction of Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vortex Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
140
141
144
149
151
152
157
161
161
163
163
Chapter 6
Irrotational Flow
1.
2.
3.
4.
5.
6.
7.
8.
Relevance of Irrotational Flow Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity Potential: Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow at a Wall Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Irrotational Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow past a Half-Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
165
167
169
171
173
174
174
175
x
Contents
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Flow past a Circular Cylinder without Circulation . . . . . . . . . . . . . . . . . . .
Flow past a Circular Cylinder with Circulation . . . . . . . . . . . . . . . . . . . . . .
Forces on a Two-Dimensional Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Source near a Wall: Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow around an Elliptic Cylinder with Circulation . . . . . . . . . . . . . . . . . . .
Uniqueness of Irrotational Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical Solution of Plane Irrotational Flow. . . . . . . . . . . . . . . . . . . . . . .
Axisymmetric Irrotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Streamfunction and Velocity Potential for Axisymmetric Flow . . . . . . . .
Simple Examples of Axisymmetric Flows . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow around a Streamlined Body of Revolution . . . . . . . . . . . . . . . . . . . . .
Flow around an Arbitrary Body of Revolution . . . . . . . . . . . . . . . . . . . . . . .
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
178
180
184
189
190
192
194
195
201
203
205
206
208
209
209
212
212
Chapter 7
Gravity Waves
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wave Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Surface Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Some Features of Surface Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . .
Approximations for Deep and Shallow Water . . . . . . . . . . . . . . . . . . . . . . .
Influence of Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Group Velocity and Energy Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Group Velocity and Wave Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonlinear Steepening in a Nondispersive Medium . . . . . . . . . . . . . . . . . . .
Hydraulic Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finite Amplitude Waves of Unchanging Form in a
Dispersive Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stokes’ Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Waves at a Density Interface between Infinitely Deep Fluids . . . . . . . . . .
Waves in a Finite Layer Overlying an Infinitely Deep Fluid . . . . . . . . . . .
Shallow Layer Overlying an Infinitely Deep Fluid . . . . . . . . . . . . . . . . . . .
Equations of Motion for a Continuously Stratified Fluid . . . . . . . . . . . . . .
Internal Waves in a Continuously Stratified Fluid . . . . . . . . . . . . . . . . . . . .
Dispersion of Internal Waves in a Stratified Fluid . . . . . . . . . . . . . . . . . . . .
Energy Considerations of Internal Waves in a Stratified Fluid . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
214
214
216
219
223
229
234
237
238
242
246
248
250
253
255
259
262
263
267
270
272
276
277
xi
Contents
Chapter 8
Dynamic Similarity
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Nondimensional Parameters Determined from Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Dimensional Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Buckingham’s Pi Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Nondimensional Parameters and Dynamic Similarity . . . . . . . . . . . . . . . .
6. Comments on Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7. Significance of Common Nondimensional Parameters. . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
279
280
284
285
287
290
292
294
294
294
Chapter 9
Laminar Flow
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analogy between Heat and Vorticity Diffusion . . . . . . . . . . . . . . . . . . . . . .
Pressure Change Due to Dynamic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .
Steady Flow between Parallel Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Steady Flow in a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Steady Flow between Concentric Cylinders . . . . . . . . . . . . . . . . . . . . . . . . .
Impulsively Started Plate: Similarity Solutions . . . . . . . . . . . . . . . . . . . . . .
Diffusion of a Vortex Sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Decay of a Line Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow Due to an Oscillating Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
High and Low Reynolds Number Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creeping Flow around a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonuniformity of Stokes’ Solution and Oseen’s Improvement . . . . . . . .
Hele-Shaw Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
295
297
297
298
302
303
306
313
315
317
320
322
327
332
334
335
337
337
Chapter 10
Boundary Layers and Related Topics
1.
2.
3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary Layer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Different Measures of Boundary Layer Thickness . . . . . . . . . . . . . . . . . . .
Boundary Layer on a Flat Plate with a Sink at the Leading Edge:
Closed Form Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
340
340
346
348
xii
Contents
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Boundary Layer on a Flat Plate: Blasius Solution . . . . . . . . . . . . . . . . . . .
von Karman Momentum Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Effect of Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Description of Flow past a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . .
Description of Flow past a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamics of Sports Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two-Dimensional Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Secondary Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perturbation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Example of a Regular Perturbation Problem . . . . . . . . . . . . . . . . . . . . .
An Example of a Singular Perturbation Problem . . . . . . . . . . . . . . . . . . . .
Decay of a Laminar Shear Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
352
362
364
366
368
375
376
381
388
389
394
396
401
407
409
410
Chapter 11
Computational Fluid Dynamics
1.
2.
3.
4.
5.
6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Incompressible Viscous Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Three Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
411
413
418
426
440
461
463
464
Chapter 12
Instability
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Method of Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thermal Instability: The Bénard Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
Double-Diffusive Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Centrifugal Instability: Taylor Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kelvin–Helmholtz Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instability of Continuously Stratified Parallel Flows. . . . . . . . . . . . . . . . . .
Squire’s Theorem and Orr–Sommerfeld Equation . . . . . . . . . . . . . . . . . . . .
Inviscid Stability of Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Some Results of Parallel Viscous Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental Verification of Boundary Layer Instability . . . . . . . . . . . . .
Comments on Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
467
469
470
482
486
493
500
507
510
514
520
522
xiii
Contents
13. Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14. Deterministic Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
523
525
533
535
Chapter 13
Turbulence
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Historical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correlations and Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Averaged Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kinetic Energy Budget of Mean Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kinetic Energy Budget of Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . .
Turbulence Production and Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectrum of Turbulence in Inertial Subrange . . . . . . . . . . . . . . . . . . . . . . . .
Wall-Free Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wall-Bounded Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eddy Viscosity and Mixing Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coherent Structures in a Wall Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Turbulence in a Stratified Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Taylor’s Theory of Turbulent Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
537
539
541
543
547
554
556
559
562
564
570
580
584
586
591
598
598
600
601
Chapter 14
Geophysical Fluid Dynamics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vertical Variation of Density in Atmosphere and Ocean . . . . . . . . . . . . . .
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Approximate Equations for a Thin Layer on a Rotating Sphere . . . . . . . .
Geostrophic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ekman Layer at a Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ekman Layer on a Rigid Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shallow-Water Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Normal Modes in a Continuously Stratified Layer . . . . . . . . . . . . . . . . . . .
High- and Low-Frequency Regimes in Shallow-Water
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11. Gravity Waves with Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12. Kelvin Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
603
605
607
610
613
617
622
625
628
634
636
639
xiv
Contents
13. Potential Vorticity Conservation in Shallow-Water
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14. Internal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15. Rossby Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16. Barotropic Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17. Baroclinic Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18. Geostrophic Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
644
647
657
663
665
673
676
677
Chapter 15
Aerodynamics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Aircraft and Its Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Airfoil Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Forces on an Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kutta Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generation of Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conformal Transformation for Generating Airfoil Shape . . . . . . . . . . . . .
Lift of Zhukhovsky Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wing of Finite Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lifting Line Theory of Prandtl and Lanchester . . . . . . . . . . . . . . . . . . . . . .
Results for Elliptic Circulation Distribution . . . . . . . . . . . . . . . . . . . . . . . . .
Lift and Drag Characteristics of Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Propulsive Mechanisms of Fish and Birds . . . . . . . . . . . . . . . . . . . . . . . . . .
Sailing against the Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
679
680
683
684
684
687
688
692
695
697
701
704
706
708
709
711
711
Chapter 16
Compressible Flow
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Basic Equations for One-Dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . .
Stagnation and Sonic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Area–Velocity Relations in One-Dimensional Isentropic Flow . . . . . . . .
Normal Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Operation of Nozzles at Different Back Pressures . . . . . . . . . . . . . . . . . . . .
Effects of Friction and Heating in Constant-Area Ducts . . . . . . . . . . . . . .
Mach Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Oblique Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Expansion and Compression in Supersonic Flow . . . . . . . . . . . . . . . . . . . .
713
717
721
724
729
733
741
747
750
752
756
xv
Contents
12. Thin Airfoil Theory in Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
758
761
763
763
Chapter 17
Introduction to Biofluid Mechanics
1.
2.
3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Circulatory System in the Human Body . . . . . . . . . . . . . . . . . . . . . . . .
Modelling of Flow in Blood Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction to the Fluid Mechanics of Plants . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
765
766
782
831
837
838
838
Appendix A
Some Properties of Common Fluids
A1.
A2.
A3.
A4.
Useful Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of Pure Water at Atmospheric Pressure . . . . . . . . . . . . . . . . . . .
Properties of Dry Air at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . .
Properties of Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
841
842
842
843
Appendix B
Curvilinear Coordinates
B1. Cylindrical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B2. Plane Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B3. Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
845
847
847
Appendix C
Founders of Modern Fluid Dynamics
Ludwig Prandtl (1875–1953) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geoffrey Ingram Taylor (1886–1975) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
851
852
853
Appendix D
Visual Resources
855
Index
857
New in This Book
We are pleased to include a free copy of the DVD Multimedia Fluid Mechanics,
2/e, with this copy of Fluid Mechanics, Fourth Edition. You will find it in a plastic
sleeve on the inside back cover of the book. If you are purchasing a used copy, be
aware that the DVD might have been removed by a previous owner.
Inspired by the reception of the first edition, the objectives in Multimedia Fluid
Mechanics, 2/e, remain to exploit the moving image and interactivity of multi–
media to improve the teaching and learning of fluid mechanics in all disciplines by
illustrating fundamental phenomena and conveying fascinating fluid flows for
generations to come.
The completely new edition on the DVD includes the following:
•
Twice the coverage with new modules on turbulence, control volumes,
interfacial phenomena, and similarity and scaling
•
Four times the number of fluids videos, now more than 800
•
Now more than 20 Virtual labs and simulations
•
Dozens of new interactive demonstrations and animations
Additional new features:
xvi
•
Improved navigation via side bars that provide rapid overviews of
modules and guided browsing
•
Media libraries for each chapter that give a snapshot of videos, each with
descriptive labels
•
Ability to create movie playlists, which are invaluable in teaching
•
Higher-resolution graphics, with full or part screen viewing options
•
Operates on either a PC or a Mac OSX
Preface
Fluid mechanics has a vast scope and touches every aspect of our lives. Just look at
the contents of the 39 volumes of the Annual Review of Fluid Mechanics (1969–2007)
for validation of that statement. We cover only a tiny fraction of that scope in this
book.
This Fourth Edition continues to evolve due to the kindness of readers and users
who write to me suggesting corrections. Specifically, Roger Berlind of Columbia
University is responsible for the revisions to the Thermal Wind subsection of
Chapter 14. Howard Hu has revised, streamlined, and updated his chapter on Computational Fluid Dynamics, and P. S. Ayyaswamy has contributed a new chapter (17)
on Introduction to Biofluid Mechanics. It is an excellently written contribution
and unique in that its level is appropriate for this book. It is between the advanced
treatises and overly simplified treatments available elsewhere. I have tried to update
much of the remaining material, particularly on turbulence, where so many new
papers appear each year. Reference to the collection of the National Committee for
Fluid Mechanics Films, now available for viewing via the Internet, is made in a new
Appendix D. These films may be old but remain an excellent resource for visualization
of flows.
On a more personal note, the bladder cancer (transitional cell carcinoma) diagnosed in the Fall of 2003, and attacked by a sequence of surgeries and a regimen of
chemotherapy, was never completely killed and grew back to visibility in the Fall
of 2006. The minimum visible spot on MRI can contain 70,000 TCC. A new and
harsher regimen of chemotherapy was prescribed through the Spring of 2007, during
the period when the updates were prepared. The irony of the fact that all the chemical
poisons infused into my veins are fluids is not lost on me. Because of the fatigue,
I accomplished less than I had hoped. Since radiologists cannot distinguish viable
living TCC from those that have been killed and remain in place, the only means of
discerning living cancer is to image again and see if there is new growth. The image
in early June confirmed the message that my body had already sent me: the cancer
was growing back and causing pain. Radiation was tried for a while to shrink the
painful tumor but that was unsuccessful. In late July a new regimen of chemotherapy
began, to last perhaps through the end of the year and beyond. The initial response
was positive and, although I have come to realize that my condition is incurable, I was
ever hopeful. After three infusions, however, the toxic effects of the chemotherapy
were more destructive than the aggressiveness of the cancer. I was left with no lung
capacity, no muscle strength, and was very ill and weak when page proofs came back
for checking. I remain so at the time of this writing.
xvii
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Preface
I appreciate all the help provided by Ms. Susan Waddington in the preparation
of the final stages of this manuscript.
I am very grateful to my family, friends, and colleagues for their support throughout this ordeal.
Ira M. Cohen
Preface to Third Edition
This edition provided me with the opportunity to include (almost) all of the additional
material I had intended for the Second Edition but had to sacrifice because of the
crush of time. It also provided me with an opportunity to rewrite and improve the
presentation of material on jets in Chapter 10. In addition, Professor Howard Hu
greatly expanded his CFD chapter. The expansion of the treatment of surface tension
is due to the urging of Professor E. F. “Charlie” Hasselbrink of the University of
Michigan.
I am grateful to Mr. Karthik Mukundakrishnan for computations of boundary
layer problems, to Mr. Andrew Perrin for numerous suggestions for improvement
and some computations, and to Mr. Din-Chih Hwang for sharing his latest results
on the decay of a laminar shear layer. The expertise of Ms. Maryeileen Banford in
preparing new figures was invaluable and is especially appreciated.
The page proofs of the text were read between my second and third surgeries
for stage 3 bladder cancer. The book is scheduled to be released in the middle of my
regimen of chemotherapy. My family, especially my wife Linda and two daughters
(both of whom are cancer survivors), have been immensely supportive during this
very difficult time. I am also very grateful for the comfort provided by my many
colleagues and friends.
Ira M. Cohen
xix
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Preface to Second Edition
My involvement with Pijush Kundu’s Fluid Mechanics first began in April 1991 with
a letter from him asking me to consider his book for adoption in the first year graduate
course I had been teaching for 25 years. That started a correspondence and, in fact,
I did adopt the book for the following academic year. The correspondence related
to improving the book by enhancing or clarifying various points. I would not have
taken the time to do that if I hadn’t thought this was the best book at the first-year
graduate level. By the end of that year we were already discussing a second edition
and whether I would have a role in it. By early 1992, however, it was clear that I
had a crushing administrative burden at the University of Pennsylvania and could not
undertake any time-consuming projects for the next several years. My wife and I met
Pijush and Shikha for the first time in December 1992. They were a charming, erudite,
sophisticated couple with two brilliant children. We immediately felt a bond of warmth
and friendship with them. Shikha was a teacher like my wife so the four of us had a
great deal in common. A couple of years later we were shocked to hear that Pijush had
died suddenly and unexpectedly. It saddened me greatly because I had been looking
forward to working with Pijush on the second edition after my term as department
chairman ended in mid-1997. For the next year and a half, however, serious family
health problems detoured any plans. Discussions on this edition resumed in July of
1999 and were concluded in the Spring of 2000 when my work really started. This
book remains the principal work product of Pijush K. Kundu, especially the lengthy
chapters on Gravity Waves, Instability, and Geophysical Fluid Dynamics, his areas of
expertise. I have added new material to all of the other chapters, often providing an
alternative point of view. Specifically, vector field derivatives have been generalized,
as have been streamfunctions. Additional material has been added to the chapters on
laminar flows and boundary layers. The treatment of one-dimensional gasdynamics
has been extended. More problems have been added to most chapters. Professor
Howard H. Hu, a recognized expert in computational fluid dynamics, graciously
provided an entirely new chapter, Chapter 11, thereby providing the student with an
entree into this exploding new field. Both finite difference and finite element methods
are introduced and a detailed worked-out example of each is provided.
I have been a student of fluid mechanics since 1954 when I entered college to
study aeronautical engineering. I have been teaching fluid mechanics since 1963 when
I joined the Brown University faculty, and I have been teaching a course corresponding
to this book since moving to the University of Pennsylvania in 1966. I am most grateful
to two of my own teachers, Professor Wallace D. Hayes (1918–2001), who expressed
fluid mechanics in the clearest way I have ever seen, and Professor Martin D. Kruskal,
whose use of mathematics to solve difficult physical problems was developed to a
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Preface to Second Edition
high art form and reminds me of a Vivaldi trumpet concerto. His codification of rules
of applied limit processes into the principles of “Asymptotology” remains with me
today as a way to view problems. I am grateful also to countless students who asked
questions, forcing me to rethink many points.
The editors at Academic Press, Gregory Franklin and Marsha Filion (assistant)
have been very supportive of my efforts and have tried to light a fire under me. Since
this edition was completed, I found that there is even more new and original material
I would like to add. But, alas, that will have to wait for the next edition. The new figures
and modifications of old figures were done by Maryeileen Banford with occasional
assistance from the school’s software expert, Paul W. Shaffer. I greatly appreciate
their job well done.
Ira M. Cohen
Preface to First Edition
This book is a basic introduction to the subject of fluid mechanics and is intended for
undergraduate and beginning graduate students of science and engineering. There is
enough material in the book for at least two courses. No previous knowledge of the
subject is assumed, and much of the text is suitable in a first course on the subject. On
the other hand, a selection of the advanced topics could be used in a second course.
I have not tried to indicate which sections should be considered advanced; the choice
often depends on the teacher, the university, and the field of study. Particular effort
has been made to make the presentation clear and accurate and at the same time easy
enough for students. Mathematically rigorous approaches have been avoided in favor
of the physically revealing ones.
A survey of the available texts revealed the need for a book with a balanced
view, dealing with currently relevant topics, and at the same time easy enough for
students. The available texts can perhaps be divided into three broad groups. One
type, written primarily for applied mathematicians, deals mostly with classical topics
such as irrotational and laminar flows, in which analytical solutions are possible.
A second group of books emphasizes engineering applications, concentrating on
flows in such systems as ducts, open channels, and airfoils. A third type of text is
narrowly focused toward applications to large-scale geophysical systems, omitting
small-scale processes which are equally applicable to geophysical systems as well as
laboratory-scale phenomena. Several of these geophysical fluid dynamics texts are
also written primarily for researchers and are therefore rather difficult for students.
I have tried to adopt a balanced view and to deal in a simple way with the basic ideas
relevant to both engineering and geophysical fluid dynamics.
However, I have taken a rather cautious attitude toward mixing engineering and
geophysical fluid dynamics, generally separating them in different chapters. Although
the basic principles are the same, the large-scale geophysical flows are so dominated
by the effects of the Coriolis force that their characteristics can be quite different
from those of laboratory-scale flows. It is for this reason that most effects of planetary
rotation are discussed in a separate chapter, although the concept of the Coriolis force
is introduced earlier in the book. The effects of density stratification, on the other hand,
are discussed in several chapters, since they can be important in both geophysical and
laboratory-scale flows.
The choice of material is always a personal one. In my effort to select topics,
however, I have been careful not to be guided strongly by my own research interests.
The material selected is what I believe to be of the most interest in a book on general
fluid mechanics. It includes topics of special interest to geophysicists (for example,
the chapters on Gravity Waves and Geophysical Fluid Dynamics) and to engineers
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Preface to First Edition
(for example, the chapters on Aerodynamics and Compressible Flow). There are also
chapters of common interest, such as the first five chapters, and those on Boundary
Layers, Instability, and Turbulence. Some of the material is now available only in
specialized monographs; such material is presented here in simple form, perhaps
sacrificing some formal mathematical rigor.
Throughout the book the convenience of tensor algebra has been exploited freely.
My experience is that many students feel uncomfortable with tensor notation in the
beginning, especially with the permutation symbol εijk . After a while, however, they
like it. In any case, following an introductory chapter, the second chapter of the book
explains the fundamentals of Cartesian Tensors. The next three chapters deal with
standard and introductory material on Kinematics, Conservation Laws, and Vorticity
Dynamics. Most of the material here is suitable for presentation to geophysicists as
well as engineers.
In much of the rest of the book the teacher is expected to select topics that are
suitable for his or her particular audience. Chapter 6 discusses Irrotational Flow; this
material is rather classical but is still useful for two reasons. First, some of the results
are used in later chapters, especially the one on Aerodynamics. Second, most of the
ideas are applicable in the study of other potential fields, such as heat conduction
and electrostatics. Chapter 7 discusses Gravity Waves in homogeneous and stratified
fluids; the emphasis is on linear analysis, although brief discussions of nonlinear
effects such as hydraulic jump, Stokes’s drift, and soliton are given.
After a discussion of Dynamic Similarity in Chapter 8, the study of viscous flow
starts with Chapter 9, which discusses Laminar Flow. The material is standard, but
the concept and analysis of similarity solutions are explained in detail. In Chapter 10
on Boundary Layers, the central idea has been introduced intuitively at first. Only
after a thorough physical discussion has the boundary layer been explained as a singular perturbation problem. I ask the indulgence of my colleagues for including the
peripheral section on the dynamics of sports balls but promise that most students
will listen with interest and ask a lot of questions. Instability of flows is discussed at
some length in Chapter 12. The emphasis is on linear analysis, but some discussion
of “chaos” is given in order to point out how deterministic nonlinear systems can lead
to irregular solutions. Fully developed three-dimensional Turbulence is discussed in
Chapter 13. In addition to standard engineering topics such as wall-bounded shear
flows, the theory of turbulent dispersion of particles is discussed because of its geophysical importance. Some effects of stratification are also discussed here, but the
short section discussing the elementary ideas of two-dimensional geostrophic turbulence is deferred to Chapter 14. I believe that much of the material in Chapters 8–13
will be of general interest, but some selection of topics is necessary here for teaching
specialized groups of students.
The remaining three chapters deal with more specialized applications in geophysics and engineering. Chapter 14 on Geophysical Fluid Dynamics emphasizes
the linear analysis of certain geophysically important wave systems. However, elements of barotropic and baroclinic instabilities and geostrophic turbulence are also
included. Chapter 15 on Aerodynamics emphasizes the application of potential theory to flow around lift-generating profiles; an elementary discussion of finite-wing
theory is also given. The material is standard, and I do not claim much originality or
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Preface to First Edition
innovation, although I think the reader may be especially interested in the discussions
of propulsive mechanisms of fish, birds, and sailboats and the material on the historic
controversy between Prandtl and Lanchester. Chapter 16 on Compressible Flow also
contains standard topics, available in most engineering texts. This chapter is included
with the belief that all fluid dynamicists should have some familiarity with such topics
as shock waves and expansion fans. Besides, very similar phenomena also occur in
other nondispersive systems such as gravity waves in shallow water.
The appendices contain conversion factors, properties of water and air, equations
in curvilinear coordinates, and short bibliographical sketches of Founders of Modern
Fluid Dynamics. In selecting the names in the list of founders, my aim was to come
up with a very short list of historic figures who made truly fundamental contributions.
It became clear that the choice of Prandtl and G. I. Taylor was the only one that would
avoid all controversy.
Some problems in the basic chapters are worked out in the text, in order to
illustrate the application of the basic principles. In a first course, undergraduate engineering students may need more practice and help than offered in the book; in that
case the teacher may have to select additional problems from other books. Difficult
problems have been deliberately omitted from the end-of-chapter exercises. It is my
experience that the more difficult exercises need a lot of clarification and hints (the
degree of which depends on the students’ background), and they are therefore better
designed by the teacher. In many cases answers or hints are provided for the exercises.
Acknowledgments
I would like to record here my gratitude to those who made the writing of this book
possible. My teachers Professor Shankar Lal and Professor John Lumley fostered my
interest in fluid mechanics and quietly inspired me with their brilliance; Professor
Lumley also reviewed Chapter 13. My colleague Julian McCreary provided support,
encouragement, and careful comments on Chapters 7, 12, and 14. Richard Thomson’s
cheerful voice over the telephone was a constant reassurance that professional science
can make some people happy, not simply competitive; I am also grateful to him for
reviewing Chapters 4 and 15. Joseph Pedlosky gave very valuable comments on
Chapter 14, in addition to warning me against too broad a presentation. John Allen
allowed me to use his lecture notes on perturbation techniques. Yasushi Fukamachi,
Hyong Lee, and Kevin Kohler commented on several chapters and constantly pointed
out things that may not have been clear to the students. Stan Middleman and Elizabeth
Mickaily were especially diligent in checking my solutions to the examples and
end-of-chapter problems. Terry Thompson constantly got me out of trouble with my
personal computer. Kathy Maxson drafted the figures. Chuck Arthur and Bill LaDue,
my editors at Academic Press, created a delightful atmosphere during the course of
writing and production of the book.
Lastly, I am grateful to Amjad Khan, the late Amir Khan, and the late Omkarnath
Thakur for their music, which made working after midnight no chore at all. I recommend listening to them if anybody wants to write a book!
Pijush K. Kundu
This page intentionally left blank
Author’s Notes
Both indicial and boldface notations are used to indicate vectors and tensors. The
comma notation to represent spatial derivatives (for example, A,i for ∂A/∂xi ) is used
in only two sections of the book (Sections 5.6 and 13.7), when the algebra became
cumbersome otherwise. Equal to by definition is denoted by ≡; for example, the
ratio of specific heats is introduced as γ ≡ Cp /Cv . Nearly equal to is written as ≃,
proportional to is written as ∝, and of the order is written as ∼.
Plane polar coordinates are denoted by (r, θ), cylindrical polar coordinates are
denoted by either (R, ϕ, x) or (r, θ, x), and spherical polar coordinates are denoted by
(r, θ, ϕ) (see Figure 3.1). The velocity components in the three Cartesian directions
(x, y, z) are indicated by (u, v, w). In geophysical situations the z-axis points upward.
In some cases equations are referred to by a descriptive name rather than a number
(for example, “the x-momentum equation shows that . . . ”). Those equations and/or
results deemed especially important have been indicated by a box.
A list of literature cited and supplemental reading is provided at the end of most
chapters. The list has been deliberately kept short and includes only those sources that
serve one of the following three purposes: (1) it is a reference the student is likely to
find useful, at a level not too different from that of this book; (2) it is a reference that
has influenced the author’s writing or from which a figure is reproduced; and (3) it
is an important work done after 1950. In currently active fields, reference has been
made to more recent review papers where the student can find additional references
to the important work in the field.
Fluid mechanics forces us fully to understand the underlying physics. This is
because the results we obtain often defy our intuition. The following examples support
these contentions:
1. Infinitesmally small causes can have large effects (d’Alembert’s paradox).
2. Symmetric problems may have nonsymmetric solutions (von Karman vortex
street).
3. Friction can make the flow go faster and cool the flow (subsonic adiabatic flow
in a constant area duct).
4. Roughening the surface of a body can decrease its drag (transition from laminar
to turbulent boundary layer separation).
5. Adding heat to a flow may lower its temperature. Removing heat from a flow
may raise its temperature (1-dimensional adiabatic flow in a range of subsonic
Mach number).
6. Friction can destabilize a previously stable flow (Orr-Sommerfeld stability
analysis for a boundary layer profile without inflection point).
xxvii
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Author’s Notes
7. Without friction, birds could not fly and fish could not swim (Kutta condition
requires viscosity).
8. The best and most accurate visualization of streamlines in an inviscid (infinite
Reynolds number) flow is in a Hele-Shaw apparatus for creeping highly viscous
flow (near zero Reynolds number).
Every one of these counterintuitive effects will be treated and discussed in
this text.
This second edition also contains additional material on streamfunctions, boundary conditions, viscous flows, boundary layers, jets, and compressible flows. Most
important, there is an entirely new chapter on computational fluid dynamics that introduces the student to the various techniques for numerically integrating the equations
governing fluid motions. Hopefully the introduction is sufficient that the reader can
follow up with specialized texts for a more comprehensive understanding.
An historical survey of fluid mechanics from the time of Archimedes (ca.
250 B.C.E.) to approximately 1900 is provided in the Eleventh Edition of
The Encyclopædia Britannica (1910) in Vol. XIV (under “Hydromechanics,”
pp. 115–135). I am grateful to Professor Herman Gluck (Professor of Mathematics at the University of Pennsylvania) for sending me this article. Hydrostatics and
classical (constant density) potential flows are reviewed in considerable depth. Great
detail is given in the solution of problems that are now considered obscure and arcane
with credit to authors long forgotten. The theory of slow viscous motion developed by
Stokes and others is not mentioned. The concept of the boundary layer for high-speed
motion of a viscous fluid was apparently too recent for its importance to have been
realized.
IMC
Chapter 1
Introduction
Fluid Mechanics . . . . . . . . . . . . . . . . . . . 1
Units of Measurement . . . . . . . . . . . . . . 2
Solids, Liquids, and Gases . . . . . . . . . 3
Continuum Hypothesis . . . . . . . . . . . . . 4
Transport Phenomena . . . . . . . . . . . . . 5
Surface Tension . . . . . . . . . . . . . . . . . . . . 8
Fluid Statics . . . . . . . . . . . . . . . . . . . . . . . 9
Example 1.1. . . . . . . . . . . . . . . . . . . . . . 12
8. Classical Thermodynamics . . . . . . . . 12
First Law of Thermodynamics . . . . 13
Equations of State . . . . . . . . . . . . . . . . 13
Specific Heats . . . . . . . . . . . . . . . . . . . . 14
1.
2.
3.
4.
5.
6.
7.
Second Law of Thermodynamics .
T dS Relations . . . . . . . . . . . . . . . . . . .
Speed of Sound. . . . . . . . . . . . . . . . . .
Thermal Expansion Coefficient . .
9. Perfect Gas . . . . . . . . . . . . . . . . . . . . . .
10. Static Equilibrium of
a Compressible Medium . . . . . . . . . .
Potential Temperature and
Density . . . . . . . . . . . . . . . . . . . . . . .
Scale Height of the Atmosphere . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . .
15
15
16
16
16
18
20
22
22
24
24
1. Fluid Mechanics
Fluid mechanics deals with the flow of fluids. Its study is important to physicists,
whose main interest is in understanding phenomena. They may, for example, be
interested in learning what causes the various types of wave phenomena in the atmosphere and in the ocean, why a layer of fluid heated from below breaks up into cellular
patterns, why a tennis ball hit with “top spin” dips rather sharply, how fish swim, and
how birds fly. The study of fluid mechanics is just as important to engineers, whose
main interest is in the applications of fluid mechanics to solve industrial problems.
Aerospace engineers may be interested in designing airplanes that have low resistance and, at the same time, high “lift” force to support the weight of the plane. Civil
engineers may be interested in designing irrigation canals, dams, and water supply
systems. Pollution control engineers may be interested in saving our planet from the
constant dumping of industrial sewage into the atmosphere and the ocean. Mechanical engineers may be interested in designing turbines, heat exchangers, and fluid
couplings. Chemical engineers may be interested in designing efficient devices to
mix industrial chemicals. The objectives of physicists and engineers, however, are
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50001-0
1
2
Introduction
not quite separable because the engineers need to understand and the physicists need
to be motivated through applications.
Fluid mechanics, like the study of any other branch of science, needs mathematical analyses as well as experimentation. The analytical approaches help in finding the
solutions to certain idealized and simplified problems, and in understanding the unity
behind apparently dissimilar phenomena. Needless to say, drastic simplifications are
frequently necessary because of the complexity of real phenomena. A good understanding of mathematical techniques is definitely helpful here, although it is probably
fair to say that some of the greatest theoretical contributions have come from the
people who depended rather strongly on their unusual physical intuition, some sort
of a “vision” by which they were able to distinguish between what is relevant and
what is not. Chess player, Bobby Fischer (appearing on the television program “The
Johnny Carson Show,” about 1979), once compared a good chess player and a great
one in the following manner: When a good chess player looks at a chess board, he
thinks of 20 possible moves; he analyzes all of them and picks the one that he likes.
A great chess player, on the other hand, analyzes only two or three possible moves;
his unusual intuition (part of which must have grown from experience) allows him
immediately to rule out a large number of moves without going through an apparent
logical analysis. Ludwig Prandtl, one of the founders of modern fluid mechanics,
first conceived the idea of a boundary layer based solely on physical intuition. His
knowledge of mathematics was rather limited, as his famous student von Karman
(1954, page 50) testifies. Interestingly, the boundary layer technique has now become
one of the most powerful methods in applied mathematics!
As in other fields, our mathematical ability is too limited to tackle the complex
problems of real fluid flows. Whether we are primarily interested either in understanding the physics or in the applications, we must depend heavily on experimental
observations to test our analyses and develop insights into the nature of the phenomenon. Fluid dynamicists cannot afford to think like pure mathematicians. The
well-known English pure mathematician G. H. Hardy once described applied mathematics as a form of “glorified plumbing” (G. I. Taylor, 1974). It is frightening to
imagine what Hardy would have said of experimental sciences!
This book is an introduction to fluid mechanics, and is aimed at both
physicists and engineers. While the emphasis is on understanding the elementary
concepts involved, applications to the various engineering fields have been discussed
so as to motivate the reader whose main interest is to solve industrial problems. Needless to say, the reader will not get complete satisfaction even after reading the entire
book. It is more likely that he or she will have more questions about the nature of
fluid flows than before studying this book. The purpose of the book, however, will be
well served if the reader is more curious and interested in fluid flows.
2. Units of Measurement
For mechanical systems, the units of all physical variables can be expressed in terms
of the units of four basic variables, namely, length, mass, time, and temperature.
In this book the international system of units (Système international d’ unités) and
commonly referred to as SI units, will be used most of the time. The basic units
3
3. Solids, Liquids, and Gases
TABLE 1.1
Quantity
SI Units
Name of unit
Symbol
Equivalent
meter
kilogram
second
kelvin
hertz
newton
pascal
joule
watt
m
kg
s
K
Hz
N
Pa
J
W
s−1
kg m s−2
N m−2
Nm
J s−1
Length
Mass
Time
Temperature
Frequency
Force
Pressure
Energy
Power
TABLE 1.2
Common Prefixes
Prefix
Symbol
Multiple
Mega
Kilo
Deci
Centi
Milli
Micro
M
k
d
c
m
µ
106
103
10−1
10−2
10−3
10−6
of this system are meter for length, kilogram for mass, second for time, and kelvin
for temperature. The units for other variables can be derived from these basic units.
Some of the common variables used in fluid mechanics, and their SI units, are listed
in Table 1.1. Some useful conversion factors between different systems of units are
listed in Section A1 in Appendix A.
To avoid very large or very small numerical values, prefixes are used to indicate
multiples of the units given in Table 1.1. Some of the common prefixes are listed in
Table 1.2.
Strict adherence to the SI system is sometimes cumbersome and will be abandoned in favor of common usage where it best serves the purpose of simplifying
things. For example, temperatures will be frequently quoted in degrees Celsius (◦ C),
which is related to kelvin (K) by the relation ◦ C = K − 273.15. However, the old
English system of units (foot, pound, ◦ F) will not be used, although engineers in the
United States are still using it.
3. Solids, Liquids, and Gases
Most substances can be described as existing in two states—solid and fluid. An
element of solid has a preferred shape, to which it relaxes when the external forces
on it are withdrawn. In contrast, a fluid does not have any preferred shape. Consider
a rectangular element of solid ABCD (Figure 1.1a). Under the action of a shear force
F the element assumes the shape ABC′ D′ . If the solid is perfectly elastic, it goes
back to its preferred shape ABCD when F is withdrawn. In contrast, a fluid deforms
4
Introduction
Figure 1.1 Deformation of solid and fluid elements: (a) solid; and (b) fluid.
continuously under the action of a shear force, however small. Thus, the element of
the fluid ABCD confined between parallel plates (Figure 1.1b) deforms to shapes
such as ABC′ D′ and ABC′′ D′′ as long as the force F is maintained on the upper plate.
Therefore, we say that a fluid flows.
The qualification “however small” in the forementioned description of a fluid is
significant. This is because most solids also deform continuously if the shear stress
exceeds a certain limiting value, corresponding to the “yield point” of the solid. A
solid in such a state is known as “plastic.” In fact, the distinction between solids and
fluids can be hazy at times. Substances like paints, jelly, pitch, polymer solutions, and
biological substances (for example, egg white) simultaneously display the characteristics of both solids and fluids. If we say that an elastic solid has “perfect memory”
(because it always relaxes back to its preferred shape) and that an ordinary viscous
fluid has zero memory, then substances like egg white can be called viscoelastic
because they have “partial memory.”
Although solidsand fluidsbehaveverydifferently when subjectedto shearstresses,
they behave similarly under the action of compressive normal stresses. However,
whereas a solid can support both tensile and compressive normal stresses, a fluid
usuallysupportsonlycompression(pressure)stresses.(Someliquidscansupportasmall
amount of tensile stress, the amount depending on the degree of molecular cohesion.)
Fluids again may be divided into two classes, liquids and gases. A gas always
expands and occupies the entire volume of any container. In contrast, the volume of a
liquid does not change very much, so that it cannot completely fill a large container;
in a gravitational field a free surface forms that separates the liquid from its vapor.
4. Continuum Hypothesis
A fluid, or any other substance for that matter, is composed of a large number of
molecules in constant motion and undergoing collisions with each other. Matter is
therefore discontinuous or discrete at microscopic scales. In principle, it is possible to
study the mechanics of a fluid by studying the motion of the molecules themselves, as
is done in kinetic theory or statistical mechanics. However, we are generally interested
in the gross behavior of the fluid, that is, in the average manifestation of the molecular
motion. For example, forces are exerted on the boundaries of a container due to the
constant bombardment of the molecules; the statistical average of this force per unit
5. Transport Phenomena
area is called pressure, a macroscopic property. So long as we are not interested in the
mechanism of the origin of pressure, we can ignore the molecular motion and think
of pressure as simply “force per unit area.”
It is thus possible to ignore the discrete molecular structure of matter and replace
it by a continuous distribution, called a continuum. For the continuum or macroscopic
approach to be valid, the size of the flow system (characterized, for example, by the
size of the body around which flow is taking place) must be much larger than the mean
free path of the molecules. For ordinary cases, however, this is not a great restriction,
since the mean free path is usually very small. For example, the mean free path for
standard atmospheric air is ≈5 × 10−8 m. In special situations, however, the mean
free path of the molecules can be quite large and the continuum approach breaks
down. In the upper altitudes of the atmosphere, for example, the mean free path of
the molecules may be of the order of a meter, a kinetic theory approach is necessary
for studying the dynamics of these rarefied gases.
5. Transport Phenomena
Consider a surface area AB within a mixture of two gases, say nitrogen and oxygen
(Figure 1.2), and assume that the concentration C of nitrogen (kilograms of nitrogen
per cubic meter of mixture) varies across AB. Random migration of molecules across
AB in both directions will result in a net flux of nitrogen across AB, from the region
of higher C toward the region of lower C. Experiments show that, to a good approximation, the flux of one constituent in a mixture is proportional to its concentration
Figure 1.2
Mass flux qm due to concentration variation C(y) across AB.
5
6
Introduction
gradient and it is given by
qm = −km ∇C.
(1.1)
Here the vector qm is the mass flux (kg m−2 s−1 ) of the constituent, ∇C is the concentration gradient of that constituent, and km is a constant of proportionality that
depends on the particular pair of constituents in the mixture and the thermodynamic
state. For example, km for diffusion of nitrogen in a mixture with oxygen is different
than km for diffusion of nitrogen in a mixture with carbon dioxide. The linear relation (1.1) for mass diffusion is generally known as Fick’s law. Relations like these
are based on empirical evidence, and are called phenomenological laws. Statistical
mechanics can sometimes be used to derive such laws, but only for simple situations.
The analogous relation for heat transport due to temperature gradient is Fourier’s
law and it is given by
q = −k∇T ,
(1.2)
where q is the heat flux (J m−2 s−1 ), ∇T is the temperature gradient, and k is the
thermal conductivity of the material.
Next, consider the effect of velocity gradient du/dy (Figure 1.3). It is clear that
the macroscopic fluid velocity u will tend to become uniform due to the random
Figure 1.3 Shear stress τ on surface AB. Diffusion tends to decrease velocity gradients, so that the
continuous line tends toward the dashed line.
7
5. Transport Phenomena
motion of the molecules, because of intermolecular collisions and the consequent
exchange of molecular momentum. Imagine two railroad trains traveling on parallel
tracks at different speeds, and workers shoveling coal from one train to the other. On
the average, the impact of particles of coal going from the slower to the faster train will
tend to slow down the faster train, and similarly the coal going from the faster to the
slower train will tend to speed up the latter. The net effect is a tendency to equalize the
speeds of the two trains. An analogous process takes place in the fluid flow problem
of Figure 1.3. The velocity distribution here tends toward the dashed line, which can
be described by saying that the x-momentum (determined by its “concentration” u)
is being transferred downward. Such a momentum flux is equivalent to the existence
of a shear stress in the fluid, just as the drag experienced by the two trains results
from the momentum exchange through the transfer of coal particles. The fluid above
AB tends to push the fluid underneath forward, whereas the fluid below AB tends
to drag the upper fluid backward. Experiments show that the magnitude of the shear
stress τ along a surface such as AB is, to a good approximation, related to the velocity
gradient by the linear relation
τ =µ
du
,
dy
(1.3)
which is called Newton’s law of friction. Here the constant of proportionality µ
(whose unit is kg m−1 s−1 ) is known as the dynamic viscosity, which is a strong
function of temperature
T . For ideal gases the random thermal speed is roughly
√
proportional
to
T
;
the
momentum
transport, and consequently µ, also vary approx√
imately as T . For liquids, on the other hand, the shear stress is caused more by the
intermolecular cohesive forces than by the thermal motion of the molecules. These
cohesive forces, and consequently µ for a liquid, decrease with temperature.
Although the shear stress is proportional to µ, we will see in Chapter 4 that the
tendency of a fluid to diffuse velocity gradients is determined by the quantity
ν≡
µ
,
ρ
(1.4)
where ρ is the density (kg/m3 ) of the fluid. The unit of ν is m2 /s, which does not
involve the unit of mass. Consequently, ν is frequently called the kinematic viscosity.
Two points should be noticed in the linear transport laws equations (1.1), (1.2),
and (1.3). First, only the first derivative of some generalized “concentration” C appears
on the right-hand side. This is because the transport is carried out by molecular processes, in which the length scales (say, the mean free path) are too small to feel the curvature of the C-profile. Second, the nonlinear terms involving higher powers of ∇C do
not appear. Although this is only expected for small magnitudes of ∇C, experiments
show that such linear relations are very accurate for most practical values of ∇C.
It should be noted here that we have written the transport law for momentum
far less precisely than the transport laws for mass and heat. This is because we
have not developed the language to write this law with precision. The transported
quantities in (1.1) and (1.2) are scalars (namely, mass and heat, respectively), and the
8
Introduction
corresponding fluxes are vectors. In contrast, the transported quantity in (1.3) is itself
a vector, and the corresponding flux is a “tensor.” The precise form of (1.3) will be
presented in Chapter 4, after the concept of tensors is explained in Chapter 2. For now,
we have avoided complications by writing the transport law for only one component
of momentum, using scalar notation.
6. Surface Tension
A density discontinuity exists whenever two immiscible fluids are in contact, for
example at the interface between water and air. The interface in this case is found
to behave as if it were under tension. Such an interface behaves like a stretched
membrane, such as the surface of a balloon or of a soap bubble. This is why drops of
liquid in air or gas bubbles in water tend to be spherical in shape. The origin of such
tension in an interface is due to the intermolecular attractive forces. Imagine a liquid
drop surrounded by a gas. Near the interface, all the liquid molecules are trying to
pull the molecules on the interface inward. The net effect of these attractive forces is
for the interface to contract. The magnitude of the tensile force per unit length of a
line on the interface is called surface tension σ , which has the unit N/m. The value
of σ depends on the pair of fluids in contact and the temperature.
An important consequence of surface tension is that it gives rise to a pressure
jump across the interface whenever it is curved. Consider a spherical interface having
a radius of curvature R (Figure 1.4a). If pi and po are the pressures on the two sides
of the interface, then a force balance gives
σ (2π R) = (pi − po )πR 2 ,
from which the pressure jump is found to be
pi − po =
2σ
,
R
(1.5)
Figure 1.4 (a) Section of a spherical droplet, showing surface tension forces. (b) An interface with radii
of curvatures R1 and R2 along two orthogonal directions.
9
7. Fluid Statics
showing that the pressure on the concave side is higher. The pressure jump, however,
is small unless R is quite small.
Equation (1.5) holds only if the surface is spherical. The curvature of a general
surface can be specified by the radii of curvature along two orthogonal directions,
say, R1 and R2 (Figure 1.4b). A similar analysis shows that the pressure jump across
the interface is given by
pi − p o = σ
1
1
+
R1
R2
,
which agrees with equation (1.5) if R1 = R2 .
It is well known that the free surface of a liquid in a narrow tube rises above
the surrounding level due to the influence of surface tension. This is demonstrated in
Example 1.1. Narrow tubes are called capillary tubes (from Latin capillus, meaning
“hair”). Because of this phenomenon the whole group of phenomena that arise from
surface tension effects is called capillarity. A more complete discussion of surface
tension is presented at the end of Chapter 4 (Section 19) as part of an expanded section
on boundary conditions.
7. Fluid Statics
The magnitude of the force per unit area in a static fluid is called the pressure. (More
care is needed to define the pressure in a moving medium, and this will be done in
Chapter 4.) Sometimes the ordinary pressure is called the absolute pressure, in order
to distinguish it from the gauge pressure, which is defined as the absolute pressure
minus the atmospheric pressure:
pgauge = p − patm .
The value of the atmospheric pressure is
patm = 101.3 kPa = 1.013 bar,
where 1 bar = 105 Pa. The atmospheric pressure is therefore approximately 1 bar.
In a fluid at rest, the tangential viscous stresses are absent and the only force
between adjacent surfaces is normal to the surface. We shall now demonstrate that
in such a case the surface force per unit area (“pressure”) is equal in all directions.
Consider a small triangular volume of fluid (Figure 1.5) of unit thickness normal to
the paper, and let p1 , p2 , and p3 be the pressures on the three faces. The z-axis is
taken vertically upward. The only forces acting on the element are the pressure forces
normal to the faces and the weight of the element. Because there is no acceleration
of the element in the x direction, a balance of forces in that direction gives
(p1 ds) sin θ − p3 dz = 0.
10
Introduction
Figure 1.5 Demonstration that p1 = p2 = p3 in a static fluid.
Because dz = ds sin θ, the foregoing gives p1 = p3 . A balance of forces in the
vertical direction gives
−(p1 ds) cos θ + p2 dx − 21 ρg dx dz = 0.
As ds cos θ = dx, this gives
p2 − p1 − 21 ρg dz = 0.
As the triangular element is shrunk to a point, the gravity force term drops out, giving
p1 = p2 . Thus, at a point in a static fluid, we have
p1 = p2 = p3 ,
(1.6)
so that the force per unit area is independent of the angular orientation of the surface.
The pressure is therefore a scalar quantity.
We now proceed to determine the spatial distribution of pressure in a static fluid.
Consider an infinitesimal cube of sides dx, dy, and dz, with the z-axis vertically
upward (Figure 1.6). A balance of forces in the x direction shows that the pressures
on the two sides perpendicular to the x-axis are equal. A similar result holds in the
y direction, so that
∂p
∂p
=
= 0.
∂x
∂y
(1.7)
This fact is expressed by Pascal’s law, which states that all points in a resting fluid
medium (and connected by the same fluid) are at the same pressure if they are at the
same depth. For example, the pressure at points F and G in Figure 1.7 are the same.
A vertical equilibrium of the element in Figure 1.6 requires that
p dx dy − (p + dp) dx dy − ρg dx dy dz = 0,
11
7. Fluid Statics
Figure 1.6
Fluid element at rest.
Figure 1.7
Rise of a liquid in a narrow tube (Example 1.1).
which simplifies to
dp
= −ρg.
dz
(1.8)
This shows that the pressure in a static fluid decreases with height. For a fluid of
uniform density, equation (1.8) can be integrated to give
p = p0 − ρgz,
(1.9)
where p0 is the pressure at z = 0. Equation (1.9) is the well-known result of hydrostatics, and shows that the pressure in a liquid decreases linearly with height. It implies
that the pressure rise at a depth h below the free surface of a liquid is equal to ρgh,
which is the weight of a column of liquid of height h and unit cross section.
12
Introduction
Example 1.1. With reference to Figure 1.7, show that the rise of a liquid in a narrow
tube of radius R is given by
h=
2σ sin α
,
ρgR
where σ is the surface tension and α is the “contact” angle.
Solution. Since the free surface is concave upward and exposed to the atmosphere, the pressure just below the interface at point E is below atmospheric. The
pressure then increases linearly along EF. At F the pressure again equals the atmospheric pressure, since F is at the same level as G where the pressure is atmospheric.
The pressure forces on faces AB and CD therefore balance each other. Vertical equilibrium of the element ABCD then requires that the weight of the element balances
the vertical component of the surface tension force, so that
σ (2π R) sin α = ρgh(π R 2 ),
which gives the required result.
8. Classical Thermodynamics
Classical thermodynamics is the study of equilibrium states of matter, in which the
properties are assumed uniform in space and time. The reader is assumed to be familiar
with the basic concepts of this subject. Here we give a review of the main ideas and
the most commonly used relations in this book.
A thermodynamic system is a quantity of matter separated from the surroundings
by a flexible boundary through which the system exchanges heat and work, but no
mass. A system in the equilibrium state is free of currents, such as those generated
by stirring a fluid or by sudden heating. After a change has taken place, the currents
die out and the system returns to equilibrium conditions, when the properties of the
system (such as pressure and temperature) can once again be defined.
This definition, however, is not possible in fluid flows, and the question arises as
to whether the relations derived in classical thermodynamics are applicable to fluids
in constant motion. Experiments show that the results of classical thermodynamics
do hold in most fluid flows if the changes along the motion are slow compared to a
relaxation time. The relaxation time is defined as the time taken by the material to
adjust to a new state, and the material undergoes this adjustment through molecular
collisions. The relaxation time is very small under ordinary conditions, since only
a few molecular collisions are needed for the adjustment. The relations of classical
thermodynamics are therefore applicable to most fluid flows.
The basic laws of classical thermodynamics are empirical, and cannot be proved.
Another way of viewing this is to say that these principles are so basic that they
cannot be derived from anything more basic. They essentially establish certain basic
definitions, upon which the subject is built. The first law of thermodynamics can be
regarded as a principle that defines the internal energy of a system, and the second
law can be regarded as the principle that defines the entropy of a system.
13
8. Classical Thermodynamics
First Law of Thermodynamics
The first law of thermodynamics states that the energy of a system is conserved. It
states that
Q+W =
e,
(1.10)
where Q is the heat added to the system, W is the work done on the system, and e
is the increase of internal energy of the system. All quantities in equation (1.10) may
be regarded as those referring to unit mass of the system. (In thermodynamics texts it
is customary to denote quantities per unit mass by lowercase letters, and those for the
entire system by uppercase letters. This will not be done here.) The internal energy
(also called “thermal energy”) is a manifestation of the random molecular motion of
the constituents. In fluid flows, the kinetic energy of the macroscopic motion has to be
included in the term e in equation (1.10) in order that the principle of conservation of
energy is satisfied. For developing the relations of classical thermodynamics, however,
we shall only include the “thermal energy” in the term e.
It is important to realize the difference between heat and internal energy. Heat and
work are forms of energy in transition, which appear at the boundary of the system
and are not contained within the matter. In contrast, the internal energy resides within
the matter. If two equilibrium states 1 and 2 of a system are known, then Q and W
depend on the process or path followed by the system in going from state 1 to state 2.
The change e = e2 − e1 , in contrast, does not depend on the path. In short, e is a
thermodynamic property and is a function of the thermodynamic state of the system.
Thermodynamic properties are called state functions, in contrast to heat and work,
which are path functions.
Frictionless quasi-static processes, carried out at an extremely slow rate so that
the system is at all times in equilibrium with the surroundings, are called reversible
processes. The most common type of reversible work in fluid flows is by the expansion
or contraction of the boundaries of the fluid element. Let v = 1/ρ be the specific
volume, that is, the volume per unit mass. Then the work done by the body per unit
mass in an infinitesimal reversible process is −pdv, where dv is the increase of v.
The first law (equation (1.10)) for a reversible process then becomes
de = dQ − pdv,
(1.11)
provided that Q is also reversible.
Note that irreversible forms of work, such as that done by turning a paddle wheel,
are excluded from equation (1.11).
Equations of State
In simple systems composed of a single component only, the specification of two
independent properties completely determines the state of the system. We can write
relations such as
p = p(v, T ) (thermal equation of state),
e = e(p, T )
(caloric equation of state).
(1.12)
14
Introduction
Such relations are called equations of state. For more complicated systems composed
of more than one component, the specification of two properties is not enough to
completely determine the state. For example, for sea water containing dissolved salt,
the density is a function of the three variables, salinity, temperature, and pressure.
Specific Heats
Before we define the specific heats of a substance, we define a thermodynamic property called enthalpy as
h ≡ e + pv.
(1.13)
This property will be quite useful in our study of compressible fluid flows.
For single-component systems, the specific heats at constant pressure and constant volume are defined as
∂h
Cp ≡
,
(1.14)
∂T p
∂e
.
(1.15)
Cv ≡
∂T v
Here, equation (1.14) means that we regard h as a function of p and T , and find the
partial derivative of h with respect to T , keeping p constant. Equation (1.15) has an
analogous interpretation. It is important to note that the specific heats as defined are
thermodynamic properties, because they are defined in terms of other properties of
the system. That is, we can determine Cp and Cv when two other properties of the
system (say, p and T ) are given.
For certain processes common in fluid flows, the heat exchange can be related
to the specific heats. Consider a reversible process in which the work done is given
by p dv, so that the first law of thermodynamics has the form of equation (1.11).
Dividing by the change of temperature, it follows that the heat transferred per unit
mass per unit temperature change in a constant volume process is
dQ
∂e
=
= Cv .
dT v
∂T v
This shows that Cv dT represents the heat transfer per unit mass in a reversible
constant volume process, in which the only type of work done is of the pdv type.
It is misleading to define Cv = (dQ/dT )v without any restrictions imposed, as the
temperature of a constant-volume system can increase without heat transfer, say, by
turning a paddle wheel.
In a similar manner, the heat transferred at constant pressure during a reversible
process is given by
dQ
∂h
=
= Cp .
dT p
∂T p
15
8. Classical Thermodynamics
Second Law of Thermodynamics
The second law of thermodynamics imposes restriction on the direction in which
real processes can proceed. Its implications are discussed in Chapter 4. Some consequences of this law are the following:
(i) There must exist a thermodynamic property S, known as entropy, whose
change between states 1 and 2 is given by
S2 − S1 =
2
1
dQrev
,
T
(1.16)
where the integral is taken along any reversible process between the two states.
(ii) For an arbitrary process between 1 and 2, the entropy change is
S2 − S1
1
2
dQ
T
(Clausius-Duhem),
which states that the entropy of an isolated system (dQ = 0) can only increase.
Such increases are caused by frictional and mixing phenomena.
(iii) Molecular transport coefficients such as viscosity µ and thermal conductivity
k must be positive. Otherwise, spontaneous “unmixing” would occur and lead
to a decrease of entropy of an isolated system.
T dS Relations
Two common relations are useful in calculating the entropy changes during a process.
For a reversible process, the entropy change is given by
T dS = dQ.
(1.17)
On substituting into (1.11), we obtain
T dS = de + p dv
T dS = dh − v dp
(Gibbs),
(1.18)
where the second form is obtained by using dh = d(e + pv) = de + p dv +
v dp. It is interesting that the “T dS relations” in equations (1.18) are also valid for
irreversible (frictional) processes, although the relations (1.11) and (1.17), from which
equations (1.18) is derived, are true for reversible processes only. This is because
equations (1.18) are relations between thermodynamic state functions alone and are
therefore true for any process. The association of T dS with heat and −pdv with
work does not hold for irreversible processes. Consider paddle wheel work done at
constant volume so that de = T dS is the element of work done.
16
Introduction
Speed of Sound
In a compressible medium, infinitesimal changes in density or pressure propagate
through the medium at a finite speed. In Chapter 16, we shall prove that the square
of this speed is given by
∂p
2
c =
,
(1.19)
∂ρ s
where the subscript “s” signifies that the derivative is taken at constant entropy. As
sound is composed of small density perturbations, it also propagates at speed c. For
incompressible fluids ρ is independent of p, and therefore c = ∞.
Thermal Expansion Coefficient
In a system whose density is a function of temperature, we define the thermal expansion coefficient
1 ∂ρ
α≡−
,
(1.20)
ρ ∂T p
where the subscript “p” signifies that the partial derivative is taken at constant pressure.
The expansion coefficient will appear frequently in our studies of nonisothermal
systems.
9. Perfect Gas
A relation defining one state function of a gas in terms of two others is called an
equation of state. A perfect gas is defined as one that obeys the thermal equation of
state
p = ρRT ,
(1.21)
where p is the pressure, ρ is the density, T is the absolute temperature, and R is the
gas constant. The value of the gas constant depends on the molecular mass m of the
gas according to
R=
Ru
,
m
(1.22)
where
Ru = 8314.36 J kmol−1 K−1
is the universal gas constant. For example, the molecular mass for dry air is
m = 28.966 kg/kmol, for which equation (1.22) gives
R = 287 J kg−1 K−1
for dry air.
17
9. Perfect Gas
Equation (1.21) can be derived from the kinetic theory of gases if the attractive
forces between the molecules are negligible. At ordinary temperatures and pressures
most gases can be taken as perfect.
The gas constant is related to the specific heats of the gas through the relation
R = Cp − Cv ,
(1.23)
where Cp is the specific heat at constant pressure and Cv is the specific heat at constant
volume. In general, Cp and Cv of a gas, including those of a perfect gas, increase with
temperature. The ratio of specific heats of a gas
γ ≡
Cp
,
Cv
(1.24)
is an important quantity. For air at ordinary temperatures, γ = 1.4 and C p =
1005 J kg−1 K−1 .
It can be shown that assertion (1.21) is equivalent to
e = e(T )
h = h(T )
and conversely, so that the internal energy and enthalpy of a perfect gas can only be
functions of temperature alone. See Exercise 7.
A process is called adiabatic if it takes place without the addition of heat. A
process is called isentropic if it is adiabatic and frictionless, for then the entropy of
the fluid does not change. From equation (1.18) it is easy to show that the isentropic
flow of a perfect gas with constant specific heats obeys the relation
p
= const.
ργ
(isentropic)
(1.25)
Using the equation of state p = ρRT , it follows that the temperature and density
change during an isentropic process from state 1 to state 2 according to
T1
=
T2
p1
p2
(γ −1)/γ
and
ρ1
=
ρ2
p1
p2
1/γ
(isentropic)
(1.26)
See Exercise 8. For a perfect gas, simple expressions can be found for several
useful thermodynamic properties such as the speed of sound and the thermal expansion
coefficient. Using the equation of state p = ρRT , the speed of sound (1.19) becomes
c=
γ RT ,
(1.27)
where equation (1.25) has been used. This shows that the speed of sound increases
as the square root of the temperature. Likewise, the use of p = ρRT shows that the
18
Introduction
thermal expansion coefficient (1.20) is
α=
1
,
T
(1.28)
10. Static Equilibrium of a Compressible Medium
In an incompressible fluid in which the density is not a function of pressure, there is
a simple criterion for determining the stability of the medium in the static state. The
criterion is that the medium is stable if the density decreases upward, for then a particle
displaced upward would find itself at a level where the density of the surrounding
fluid is lower, and so the particle would be forced back toward its original level. In
the opposite case in which the density increases upward, a displaced particle would
continue to move farther away from its original position, resulting in instability. The
medium is in neutral equilibrium if the density is uniform.
For a compressible medium the preceding criterion for determining the stability
does not hold. We shall now show that in this case it is not the density but the entropy
that is constant with height in the neutral state. For simplicity we shall consider the
case of an atmosphere that obeys the equation of state for a perfect gas. The pressure
decreases with height according to
dp
= −ρg.
dz
A particle displaced upward would expand adiabatically because of the decrease of
the pressure with height. Its original density ρ0 and original temperature T0 would
therefore decrease to ρ and T according to the isentropic relations
T
=
T0
p
p0
(γ −1)/γ
and
ρ
=
ρ0
p
p0
1/γ
,
(1.29)
where γ = Cp /Cv , and the subscript 0 denotes the original state at some height z0 ,
where p0 > p (Figure 1.8). It is clear that the displaced particle would be forced back
toward the original level if the new density is larger than that of the surrounding air
at the new level. Now if the properties of the surrounding air also happen to vary
with height in such a way that the entropy is uniform with height, then the displaced
particle would constantly find itself in a region where the density is the same as that
of itself. Therefore, a neutral atmosphere is one in which p, ρ, and T decrease in
such a way that the entropy is constant with height. A neutrally stable atmosphere is
therefore also called an isentropic or adiabatic atmosphere. It follows that a statically
stable atmosphere is one in which the density decreases with height faster than in an
adiabatic atmosphere.
19
10. Static Equilibrium of a Compressible Medium
Figure 1.8
Adiabatic expansion of a fluid particle displaced upward in a compressible medium.
It is easy to determine the rate of decrease of temperature in an adiabatic atmosphere. Taking the logarithm of equation (1.29), we obtain
ln Ta − ln T0 =
γ −1
[ln pa − ln p0 ],
γ
where we are using the subscript “a” to denote an adiabatic atmosphere. A differentiation with respect to z gives
1 dTa
γ − 1 1 dpa
=
.
Ta dz
γ pa dz
Using the perfect gas law p = ρRT , Cp − Cv = R, and the hydrostatic rule
dp/dz = −ρg, we obtain
g
dTa
≡ Ŵa = −
dz
Cp
(1.30)
where Ŵ ≡ dT /dz is the temperature gradient; Ŵa = −g/Cp is called the adiabatic
temperature gradient and is the largest rate at which the temperature can decrease
with height without causing instability. For air at normal temperatures and pressures,
the temperature of a neutral atmosphere decreases with height at the rate of g/Cp ≃
10 ◦ C/km. Meteorologists call vertical temperature gradients the “lapse rate,” so that
in their terminology the adiabatic lapse rate is 10 ◦ C/km.
Figure 1.9a shows a typical distribution of temperature in the atmosphere. The
lower part has been drawn with a slope nearly equal to the adiabatic temperature gradient because the mixing processes near the ground tend to form a neutral atmosphere,
with its entropy “well mixed” (that is, uniform) with height. Observations show that
the neutral atmosphere is “capped” by a layer in which the temperature increases with
height, signifying a very stable situation. Meteorologists call this an inversion, because
the temperature gradient changes sign here. Much of the atmospheric turbulence and
mixing processes cannot penetrate this very stable layer. Above this inversion layer the
20
Introduction
Figure 1.9 Vertical variation of the (a) actual and (b) potential temperature in the atmosphere. Thin
straight lines represent temperatures for a neutral atmosphere.
temperature decreases again, but less rapidly than near the ground, which corresponds
to stability. It is clear that an isothermal atmosphere (a vertical line in Figure 1.9a) is
quite stable.
Potential Temperature and Density
The foregoing discussion of static stability of a compressible atmosphere can be
expressed in terms of the concept of potential temperature, which is generally denoted
by θ . Suppose the pressure and temperature of a fluid particle at a certain height are
p and T . Now if we take the particle adiabatically to a standard pressure ps (say,
the sea level pressure, nearly equal to 100 kPa), then the temperature θ attained by
the particle is called its potential temperature. Using equation (1.26), it follows that the
actual temperature T and the potential temperature θ are related by
T =θ
p
ps
(γ −1)/γ
.
(1.31)
Taking the logarithm and differentiating, we obtain
1 dT
1 dθ
γ − 1 1 dp
=
+
.
T dz
θ dz
γ p dz
Substituting dp/dz = −ρg and p = ρRT , we obtain
T dθ
dT
g
d
=
+
=
(T − Ta ) = Ŵ − Ŵa .
θ dz
dz
Cp
dz
(1.32)
Now if the temperature decreases at a rate Ŵ = Ŵa , then the potential temperature θ
(and therefore the entropy) is uniform with height. It follows that the stability of the
21
10. Static Equilibrium of a Compressible Medium
atmosphere is determined according to
dθ
>0
dz
dθ
=0
dz
dθ
<0
dz
(stable),
(neutral),
(1.33)
(unstable).
This is shown in Figure 1.9b. It is the gradient of potential temperature that determines
the stability of a column of gas, not the gradient of the actual temperature. However,
the difference between the two is negligible for laboratory-scale phenomena. For
example, over a height of 10 cm the compressibility effects result in a decrease of
temperature in the air by only 10 cm × (10 ◦ C/km) = 10−3 ◦ C.
Instead of using the potential temperature, one can use the concept of potential
density ρθ , defined as the density attained by a fluid particle if taken isentropically to
a standard pressure ps . Using equation (1.26), the actual and potential densities are
related by
1/γ
p
.
(1.34)
ρ = ρθ
ps
Multiplying equations (1.31) and (1.34), and using p = ρRT , we obtain
θρθ = ps /R = const. Taking the logarithm and differentiating, we obtain
−
1 dρθ
1 dθ
=
.
ρθ dz
θ dz
(1.35)
The medium is stable, neutral, or unstable depending upon whether dρθ /dz is negative, zero, or positive, respectively.
Compressibility effects are also important in the deep ocean. In the ocean the
density depends not only on the temperature and pressure, but also on the salinity,
defined as kilograms of salt per kilogram of water. (The salinity of sea water is ≈3%.)
Here, one defines the potential density as the density attained if a particle is taken to
a reference pressure isentropically and at constant salinity. The potential density thus
defined must decrease with height in stable conditions. Oceanographers automatically
account for the compressibility of sea water by converting their density measurements
at any depth to the sea level pressure, which serves as the reference pressure.
From (1.32), the temperature of a dry neutrally stable atmosphere decreases
upward at a rate dTa /dz = −g/Cp due to the decrease of pressure with height and
the compressibility of the medium. Static stability of the atmosphere is determined
by whether the actual temperature gradient dT /dz is slower or faster than dTa /dz. To
determine the static stability of the ocean, it is more convenient to formulate the criterion in terms of density. The plan is to compare the density gradient of the actual static
state with that of a neutrally stable reference state (denoted here by the subscript “a”).
22
Introduction
The pressure of the reference state decreases vertically as
dpa
= −ρa g.
dz
(1.36)
In the ocean the speed of sound c is defined by c2 = ∂p/∂ρ, where the partial derivative
is taken at constant values of entropy and salinity. In the reference state these variables
are uniform, so that dpa = c2 dρa . Therefore, the density in the neutrally stable state
varies due to the compressibility effect at a rate
dρa
ρg
1 dpa
1
= 2
= 2 (−ρa g) = − 2 ,
dz
c dz
c
c
(1.37)
where the subscript “a” on ρ has been dropped because ρa is nearly equal to the actual
density ρ.
The static stability of the ocean is determined by the sign of the potential density
gradient
dρpot
dρ
dρa
dρ
ρg
=
−
=
+ 2.
dz
dz
dz
dz
c
(1.38)
The medium is statically stable if the potential density gradient is negative, and
so on. For a perfect gas, it can be shown that equations (1.30) and (1.38) are
equivalent.
Scale Height of the Atmosphere
Expressions for pressure distribution and “thickness” of the atmosphere can be
obtained by assuming that they are isothermal. This is a good assumption in the
lower 70 km of the atmosphere, where the absolute temperature remains within 15%
of 250 K. The hydrostatic distribution is
dp
pg
= −ρg = −
.
dz
RT
Integration gives
p = p0 e−gz/RT ,
where p0 is the pressure at z = 0. The pressure therefore falls to e−1 of its surface
value in a height RT /g. The quantity RT /g, called the scale height, is a good measure
of the thickness of the atmosphere. For an average atmospheric temperature of T =
250 K, the scale height is RT /g = 7.3 km.
Exercises
1. Estimate the height to which water at 20 ◦ C will rise in a capillary glass
tube 3 mm in diameter exposed to the atmosphere. For water in contact with glass the
wetting angle is nearly 90◦ . At 20 ◦ C and water-air combination, σ = 0.073 N/m.
(Answer: h = 0.99 cm.)
23
Exercises
2. Consider the viscous flow in a channel of width 2b. The channel is aligned
in the x direction, and the velocity at a distance y from the centerline is given by the
parabolic distribution
y2
u(y) = U0 1 − 2 .
b
In terms of the viscosity µ, calculate the shear stress at a distance of y = b/2.
3. Figure 1.10 shows a manometer, which is a U-shaped tube containing mercury of density ρm . Manometers are used as pressure measuring devices. If the fluid
in the tank A has a pressure p and density ρ, then show that the gauge pressure in the
tank is
p − patm = ρm gh − ρga.
Note that the last term on the right-hand side is negligible if ρ ≪ ρm . (Hint: Equate
the pressures at X and Y .)
4. A cylinder contains 2 kg of air at 50 ◦ C and a pressure of 3 bars. The air is
compressed until its pressure rises to 8 bars. What is the initial volume? Find the final
volume for both isothermal compression and isentropic compression.
5. Assume that the temperature of the atmosphere varies with height z as
T = T0 + Kz.
Show that the pressure varies with height as
p = p0
T0
T0 + Kz
where g is gravity and R is the gas constant.
Figure 1.10 A mercury manometer.
g/KR
,
24
Introduction
6. Suppose the atmospheric temperature varies according to
T = 15 − 0.001z
where T is in degrees Celsius and height z is in meters. Is this atmosphere stable?
7. Prove that if e(T , v) = e(T ) only and if h(T , p) = h(T ) only, then the
(thermal) equation of state is equation (1.21) or pv = kT .
8. For a reversible adiabatic process in a perfect gas with constant specific
heats, derive equations (1.25) and (1.26) starting from equation (1.18).
9. Consider a heat insulated enclosure that is separated into two compartments
of volumes V1 and V2 , containing perfect gases with pressures and temperatures of
p1 , p2 , and T1 , T2 , respectively. The compartments are separated by an impermeable membrane that conducts heat (but not mass). Calculate the final steady-state
temperature assuming each of the gases has constant specific heats.
10. Consider the initial state of an enclosure with two compartments as described
in Exercise 9. At t = 0, the membrane is broken and the gases are mixed. Calculate
the final temperature.
11. A heavy piston of weight W is dropped onto a thermally insulated cylinder
of cross-sectional area A containing a perfect gas of constant specific heats, and
initially having the external pressure p1 , temperature T1 , and volume V1 . After some
oscillations, the piston reaches an equilibrium position L meters below the equilibrium
position of a weightless piston. Find L. Is there an entropy increase?
Literature Cited
Taylor, G. I. (1974). The interaction between experiment and theory in fluid mechanics. Annual Review of
Fluid Mechanics 6: 1–16.
Von Karman, T. (1954). Aerodynamics, New York: McGraw-Hill.
Supplemental Reading
Batchelor, G. K. (1967). “An Introduction to Fluid Dynamics,” London: Cambridge University Press,
(A detailed discussion of classical thermodynamics, kinetic theory of gases, surface tension effects,
and transport phenomena is given.)
Hatsopoulos, G. N. and J. H. Keenan (1981). Principles of General Thermodynamics. Melbourne, FL:
Krieger Publishing Co. (This is a good text on thermodynamics.)
Prandtl, L. and O. G. Tietjens (1934). Fundamentals of Hydro- and Aeromechanics, New York: Dover
Publications. (A clear and simple discussion of potential and adiabatic temperature gradients is given.)
Chapter 2
Cartesian Tensors
1. Scalars and Vectors . . . . . . . . . . . . . . 25
2. Rotation of Axes: Formal Definition
of a Vector . . . . . . . . . . . . . . . . . . . . . . . 26
3. Multiplication of Matrices . . . . . . . . 29
4. Second-Order Tensor . . . . . . . . . . . . 30
5. Contraction and Multiplication . . 32
6. Force on a Surface . . . . . . . . . . . . . . . 33
Example 2.1 . . . . . . . . . . . . . . . . . . . . 35
7. Kronecker Delta and Alternating
Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
8. Dot Product . . . . . . . . . . . . . . . . . . . . . 37
9. Cross Product . . . . . . . . . . . . . . . . . . . 38
10. Operator ∇: Gradient, Divergence,
and Curl . . . . . . . . . . . . . . . . . . . . . . . . 38
11. Symmetric and Antisymmetric
Tensors . . . . . . . . . . . . . . . . . . . . . . . . . .
12. Eigenvalues and Eigenvectors of a
Symmetric Tensor . . . . . . . . . . . . . . . .
Example 2.2 . . . . . . . . . . . . . . . . . . . .
13. Gauss’ Theorem . . . . . . . . . . . . . . . . .
Example 2.3 . . . . . . . . . . . . . . . . . . . .
14. Stokes’ Theorem . . . . . . . . . . . . . . . . .
Example 2.4 . . . . . . . . . . . . . . . . . . . .
15. Comma Notation . . . . . . . . . . . . . . . .
16. Boldface vs Indicial Notation. . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . .
40
41
42
44
45
47
48
49
49
50
51
51
1. Scalars and Vectors
In fluid mechanics we need to deal with quantities of various complexities. Some
of these are defined by only one component and are called scalars, some others are
defined by three components and are called vectors, and certain other variables called
tensors need as many as nine components for a complete description. We shall assume
that the reader is familiar with a certain amount of algebra and calculus of vectors.
The concept and manipulation of tensors is the subject of this chapter.
A scalar is any quantity that is completely specified by a magnitude only, along
with its unit. It is independent of the coordinate system. Examples of scalars are
temperature and density of the fluid. A vector is any quantity that has a magnitude
and a direction, and can be completely described by its components along three
specified coordinate directions. A vector is usually denoted by a boldface symbol,
for example, x for position and u for velocity. We can take a Cartesian coordinate
system x1 , x2 , x3 , with unit vectors a1 , a2 , and a3 in the three mutually perpendicular
directions (Figure 2.1). (In texts on vector analysis, the unit vectors are usually denoted
by i, j, and k. We cannot use this simple notation here because we shall use ij k to
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50002-2
25
26
Cartesian Tensors
Figure 2.1 Position vector OP and its three Cartesian components (x1 , x2 , x3 ). The three unit vectors
are a1 , a2 , and a3 .
denote components of a vector.) Then the position vector is written as
x = a1 x1 + a2 x2 + a3 x3 ,
where (x1 , x2 , x3 ) are the components of x along the coordinate directions. (The
superscripts on the unit vectors a do not denote the components of a vector; the a’s
are vectors themselves.) Instead of writing all three components explicitly, we can
indicate the three Cartesian components of a vector by an index that takes all possible
values of 1, 2, and 3. For example, the components of the position vector can be
denoted by xi , where i takes all of its possible values, namely, 1, 2, and 3. To obey the
laws of algebra that we shall present, the components of a vector should be written
as a column. For example,
x1
x = x2 .
x3
In matrix algebra, one defines the transpose as the matrix obtained by interchanging
rows and columns. For example, the transpose of a column matrix x is the row matrix
xT = [x1
x2
x3 ].
2. Rotation of Axes: Formal Definition of a Vector
A vector can be formally defined as any quantity whose components change similarly
to the components of a position vector under the rotation of the coordinate system.
27
2. Rotation of Axes: Formal Definition of a Vector
Figure 2.2
Rotation of coordinate system O 1 2 3 to O 1′ 2′ 3′ .
Let x1 x2 x3 be the original axes, and x1′ x2′ x3′ be the rotated system (Figure 2.2). The
components of the position vector x in the original and rotated systems are denoted
by xi and xi′ , respectively. The cosine of the angle between the old i and new j axes
is represented by Cij . Here, the first index of the C matrix refers to the old axes,
and the second index of C refers to the new axes. It is apparent that Cij = Cj i . A
little geometry shows that the components in the rotated system are related to the
components in the original system by
xj′ = x1 C1j + x2 C2j + x3 C3j =
3
xi Cij .
(2.1)
i=1
For simplicity, we shall verify the validity of equation (2.1) in two dimensions only.
Referring to Figure 2.3, let αij be the angle between old i and new j axes, so that
Cij = cos αij . Then
x1′ = OD = OC + AB = x1 cos α11 + x2 sin α11 .
(2.2)
As α11 = 90◦ − α21 , we have sin α11 = cos α21 = C21 . Equation (2.2) then becomes
x1′ = x1 C11 + x2 C21 =
2
xi Ci1 .
i=1
In a similar manner
x2′ = PD = PB − DB = x2 cos α11 − x1 sin α11 .
(2.3)
28
Cartesian Tensors
Figure 2.3 Rotation of a coordinate system in two dimensions.
As α11 = α22 = α12 − 90◦ (Figure 2.3), this becomes
x2′ = x2 cos α22 + x1 cos α12 =
2
xi Ci2 .
(2.4)
i=1
In two dimensions, equation (2.1) reduces to equation (2.3) for j = 1, and to equation (2.4) for j = 2. This completes our verification of equation (2.1).
Note that the index i appears twice in the same term on the right-hand side of
equation (2.1), and a summation is carried out over all values of this repeated index.
This type of summation over repeated indices appears frequently in tensor notation.
A convention is therefore adopted that, whenever an index occurs twice in a term, a
summation over the repeated index is implied, although no summation sign is explicitly
written. This is frequently called the Einstein summation convention. Equation (2.1)
is then simply written as
(2.5)
xj′ = xi Cij ,
where a summation over i is understood on the right-hand side.
The free index on both sides of equation (2.5) is j , and i is the repeated or dummy
index. Obviously any letter (other than j ) can be used as the dummy index without
changing the meaning of this equation. For example, equation (2.5) can be written
equivalently as
xi Cij = xk Ckj = xm Cmj = · · · ,
because they all mean xj′ = C1j x1 + C2j x2 + C3j x3 . Likewise, any letter can also
be used for the free index, as long as the same free index is used on both sides of
the equation. For example, denoting the free index by i and the summed index by k,
equation (2.5) can be written as
xi′ = xk Cki .
(2.6)
29
3. Multiplication of Matrices
This is because the set of three equations represented by equation (2.5) corresponding
to all values of j is the same set of equations represented by equation (2.6) for all
values of i.
It is easy to show that the components of x in the old coordinate system are
related to those in the rotated system by
xj = Cj i xi′ .
(2.7)
Note that the indicial positions on the right-hand side of this relation are different
from those in equation (2.5), because the first index of C is summed in equation (2.5),
whereas the second index of C is summed in equation (2.7).
We can now formally define a Cartesian vector as any quantity that transforms like
a position vector under the rotation of the coordinate system. Therefore, by analogy
with equation (2.5), u is a vector if its components transform as
u′j = ui Cij .
(2.8)
3. Multiplication of Matrices
In this chapter we shall generally follow the convention that 3 × 3 matrices are represented by uppercase letters, and column vectors are represented by lowercase letters.
(An exception will be the use of lowercase τ for the stress matrix.) Let A and B be
two 3 × 3 matrices. The product of A and B is defined as the matrix P whose elements
are related to those of A and B by
Pij =
3
Aik Bkj ,
k=1
or, using the summation convention
Pij = Aik Bkj .
(2.9)
P = A • B.
(2.10)
Symbolically, this is written as
A single dot between A and B is included in equation (2.10) to signify that a single
index is summed on the right-hand side of equation (2.9). The important thing to note
in equation (2.9) is that the elements are summed over the inner or adjacent index k.
It is sometimes useful to write equation (2.9) as
Pij = Aik Bkj = (A • B)ij ,
where the last term is to be read as the “ij -element of the product of matrices A
and B.”
30
Cartesian Tensors
In explicit form, equation (2.9) is written as
A11 A12 A13
B11 B12 B13
P11 P12 P13
P21 P22 P23 = A21 A22 A23 B21 B22 B23 (2.11)
B31 B32 B33
P31 P32 P33
A31 A32 A33
Note that equation (2.9) signifies that the ij -element of P is determined by multiplying
the elements in the i-row of A and the j -column of B, and summing. For example,
P12 = A11 B12 + A12 B22 + A13 B32 .
This is indicated by the dotted lines in equation (2.11). It is clear that we can define
the product A • B only if the number of columns of A equals the number of rows of B.
Equation (2.9) can be used to determine the product of a 3 × 3 matrix and a
vector, if the vector is written as a column. For example, equation (2.6) can be written
T x , which is now of the form of equation (2.9) because the summed index
as xi′ = Cik
k
k is adjacent. In matrix form equation (2.6) can therefore be written as
T
x1
C11 C12 C13
x1′
′
x2 = C21 C22 C23 x2 .
x3
C31 C32 C33
x3′
Symbolically, the preceding is
x′ = CT • x,
whereas equation (2.7) is
x = C • x′ .
4. Second-Order Tensor
We have seen that scalars can be represented by a single number, and a Cartesian
vector can be represented by three numbers. There are other quantities, however, that
need more than three components for a complete description. For example, the stress
(equal to force per unit area) at a point in a material needs nine components for a
complete specification because two directions (and, therefore, two free indices) are
involved in its description. One direction specifies the orientation of the surface on
which the stress is being sought, and the other specifies the direction of the force on
that surface. For example, the j -component of the force on a surface whose outward
normal points in the i-direction is denoted by τij . (Here, we are departing from the
convention followed in the rest of the chapter, namely, that tensors are represented by
uppercase letters. It is customary to denote the stress tensor by the lowercase τ .) The
first index of τij denotes the direction of the normal, and the second index denotes
the direction in which the force is being projected.
This is shown in Figure 2.4, which gives the normal and shear stresses on an
infinitesimal cube whose surfaces are parallel to the coordinate planes. The stresses
31
4. Second-Order Tensor
Figure 2.4 Stress field at a point. Positive normal and shear stresses are shown. For clarity, the stresses
on faces FBCG and CDHG are not labeled.
are positive if they are directed as in this figure. The sign convention is that, on a
surface whose outward normal points in the positive direction of a coordinate axis,
the normal and shear stresses are positive if they point in the positive direction of
the axes. For example, on the surface ABCD, whose outward normal points in the
positive x2 direction, the positive stresses τ21 , τ22 , and τ23 point toward the x1 , x2
and x3 directions, respectively. (Clearly, the normal stresses are positive if they are
tensile and negative if they are compressive.) On the opposite face EFGH the stress
components have the same value as on ABCD, but their directions are reversed. This
is because Figure 2.4 shows the stresses at a point. The cube shown is supposed to be
of “zero” size, so that the faces ABCD and EFGH are just opposite faces of a plane
perpendicular to the x2 -axis. That is why the stresses on the opposite faces are equal
and opposite.
Recall that a vector u can be completely specified by the three components ui
(where i = 1, 2, 3). We say “completely specified” because the components of u in
any direction other than the original axes can be found from equation (2.8). Similarly,
the state of stress at a point can be completely specified by the nine components τij
(where i, j = 1, 2, 3), which can be written as the matrix
τ11 τ12 τ13
τ = τ21 τ22 τ23 .
τ31 τ32 τ33
The specification of the preceding nine components of the stress on surfaces parallel
to the coordinate axes completely determines the state of stress at a point, because
32
Cartesian Tensors
the stresses on any arbitrary plane can then be determined. To find the stresses on any
arbitrary surface, we shall consider a rotated coordinate system x1′ x2′ x3′ one of whose
axes is perpendicular to the given surface. It can be shown by a force balance on a
tetrahedron element (see, e.g., Sommerfeld (1964), page 59) that the components of
τ in the rotated coordinate system are
′ =C C τ .
τmn
im j n ij
(2.12)
Note the similarity between the transformation rule equation (2.8) for a vector, and the
rule equation (2.12). In equation (2.8) the first index of C is summed, while its second
index is free. The rule equation (2.12) is identical, except that this happens twice. A
quantity that obeys the transformation rule equation (2.12) is called a second-order
tensor.
The transformation rule equation (2.12) can be expressed as a matrix product.
Rewrite equation (2.12) as
′
T
= Cmi
τij Cj n ,
τmn
which, with adjacent dummy indices, represents the matrix product
τ′ = CT • τ • C.
This says that the tensor τ in the rotated frame is found by multiplying C by τ and
then multiplying the product by CT .
The concepts of tensor and matrix are not quite the same. A matrix is any arrangement of elements, written as an array. The elements of a matrix represent the components of a tensor only if they obey the transformation rule equation (2.12).
Tensors can be of any order. In fact, a scalar can be considered a tensor of zero
order, and a vector can be regarded as a tensor of first order. The number of free
indices correspond to the order of the tensor. For example, A is a fourth-order tensor
if it has four free indices, and the associated 81 components change under the rotation
of the coordinate system according to
A′mnpq = Cim Cj n Ckp Clq Aij kl .
(2.13)
Tensors of various orders arise in fluid mechanics. Some of the most frequently
used are the stress tensor τij and the velocity gradient tensor ∂ui /∂xj . It can be shown
that the nine products ui vj formed from the components of the two vectors u and
v also transform according to equation (2.12), and therefore form a second-order
tensor. In addition, certain “isotropic” tensors are also frequently used; these will be
discussed in Section 7.
5. Contraction and Multiplication
When the two indices of a tensor are equated, and a summation is performed over
this repeated index, the process is called contraction. An example is
33
6. Force on a Surface
Ajj = A11 + A22 + A33 ,
which is the sum of the diagonal terms. Clearly, Ajj is a scalar and therefore independent of the coordinate system. In other words, Ajj is an invariant. (There are
three independent invariants of a second-order tensor, and Ajj is one of them; see
Exercise 5.)
Higher-order tensors can be formed by multiplying lower tensors. If u and v are
vectors, then the nine components ui vj form a second-order tensor. Similarly, if A
and B are two second-order tensors, then the 81 numbers defined by Pij kl ≡ Aij Bkl
transform according to equation (2.13), and therefore form a fourth-order tensor.
Lower-order tensors can be obtained by performing contraction on these multiplied forms. The four contractions of Aij Bkl are
Aij Bki = Bki Aij = (B • A)kj ,
Aij Bik = ATj i Bik = (AT • B)j k ,
Aij Bkj = Aij BjTk = (A • BT )ik ,
(2.14)
Aij Bj k = (A • B)ik .
All four products in the preceding are second-order tensors. Note in equation (2.14)
how the terms have been rearranged until the summed index is adjacent, at which
point they can be written as a product of matrices.
The contracted product of a second-order tensor A and a vector u is a vector. The
two possibilities are
Aij uj = (A • u)i ,
Aij ui = ATj i ui = (AT • u)j .
The doubly contracted product of two second-order tensors A and B is a scalar. The
two possibilities are Aij Bj i (which can be written as A : B in boldface notation) and
Aij Bij (which can be written as A : BT ).
6. Force on a Surface
A surface area has a magnitude and an orientation, and therefore should be treated as
a vector. The orientation of the surface is conveniently specified by the direction of
a unit vector normal to the surface. If dA is the magnitude of an element of surface
and n is the unit vector normal to the surface, then the surface area can be written as
the vector
dA = n dA.
Suppose the nine components of the stress tensor with respect to a given set of
Cartesian coordinates are given, and we want to find the force per unit area on a
surface of given orientation n (Figure 2.5). One way of determining this is to take
a rotated coordinate system, and use equation (2.12) to find the normal and shear
stresses on the given surface. An alternative method is described in what follows.
34
Cartesian Tensors
Figure 2.5 Force f per unit area on a surface element whose outward normal is n.
Figure 2.6
(a) Stresses on surfaces of a two-dimensional element; (b) balance of forces on element ABC.
For simplicity, consider a two-dimensional case, for which the known stress
components with respect to a coordinate system x1 x2 are shown in Figure 2.6a. We
want to find the force on the face AC, whose outward normal n is known (Figure 2.6b).
Consider the balance of forces on a triangular element ABC, with sides AB = dx2 ,
BC = dx1 , and AC = ds; the thickness of the element in the x3 direction is unity. If
F is the force on the face AC, then a balance of forces in the x1 direction gives the
component of F in that direction as
F1 = τ11 dx2 + τ21 dx1 .
35
6. Force on a Surface
Dividing by ds, and denoting the force per unit area as f = F/ds, we obtain
dx2
dx1
F1
= τ11
+ τ21
ds
ds
ds
= τ11 cos θ1 + τ21 cos θ2 = τ11 n1 + τ21 n2 ,
f1 =
where n1 = cos θ1 and n2 = cos θ2 because the magnitude of n is unity (Figure 2.6b).
Using the summation convention, the foregoing can be written as f1 = τj 1 nj , where
j is summed over 1 and 2. A similar balance of forces in the x2 direction gives
f2 = τj 2 nj . Generalizing to three dimensions, it is clear that
fi = τj i nj .
Because the stress tensor is symmetric (which will be proved in the next chapter),
that is, τij = τj i , the foregoing relation can be written in boldface notation as
f = n • τ.
(2.15)
Therefore, the contracted or “inner” product of the stress tensor τ and the unit outward
vector n gives the force per unit area on a surface. Equation (2.15) is analogous to
un = u • n, where un is the component of the vector u along unit normal n; however,
whereas un is a scalar, f in equation (2.15) is a vector.
Example 2.1. Consider a two-dimensional parallel flow through a channel. Take
x1 , x2 as the coordinate system, with x1 parallel to the flow. The viscous stress tensor
at a point in the flow has the form
0 a
τ=
,
a 0
where the constant a is positive in one half of the channel, and negative in the other
half. Find the magnitude and direction of force per unit area on an element whose
outward normal points at 30◦ to the direction of flow.
Solution by using equation (2.15): Because the magnitude of n is 1 and it points
at 30◦ to the x1 axis (Figure 2.7), we have
√
3/2
n=
.
1/2
The force per unit area is therefore
√
0 a
3/2
f =τ•n=
a 0
1/2
=
√a/2
3 a/2
The magnitude of f is
f = (f12 + f22 )1/2 = |a|.
=
f1
f2
.
36
Cartesian Tensors
Figure 2.7 Determination of force on an area element (Example 2.1).
If θ is the angle of f with the x1 axis, then
√
3 a
f2
=
and
sin θ =
f
2 |a|
cos θ =
1 a
f1
=
.
f
2 |a|
Thus θ = 60◦ if a is positive (in which case both sin θ and cos θ are positive), and
θ = 240◦ if a is negative (in which case both sin θ and cos θ are negative).
Solution by using equation (2.12): Take a rotated coordinate system x1′ , x2′ ,
with x1′ axis coinciding with n (Figure 2.7). Using equation (2.12), the components
of the stress tensor in the rotated frame are
′
= C11 C21 τ12 + C21 C11 τ21 =
τ11
′
= C11 C22 τ12 + C21 C12 τ21 =
τ12
√
3
2
√
3
2
√
1
1 3
a
+
2
2 2 a
√
3
11
2 a − 2 2a
=
=
√
3
2 a,
1
2 a.
√
The normal stress is therefore 3 a/2, and the shear stress is a/2. This gives a
magnitude a and a direction 60◦ or 240◦ depending on the sign of a.
7. Kronecker Delta and Alternating Tensor
The Kronecker delta is defined as
δij =
1
0
which is written in the matrix form as
if i = j
,
if i = j
1 0 0
δ = 0 1 0 .
0 0 1
(2.16)
37
8. Dot Product
The most common use of the Kronecker delta is in the following operation: If we
have a term in which one of the indices of δij is repeated, then it simply replaces the
dummy index by the other index of δij . Consider
δij uj = δi1 u1 + δi2 u2 + δi3 u3 .
The right-hand side is u1 when i = 1, u2 when i = 2, and u3 when i = 3. Therefore
δij uj = ui .
(2.17)
From its definition it is clear that δij is an isotropic tensor in the sense that its
components are unchanged by a rotation of the frame of reference, that is, δij′ = δij .
Isotropic tensors can be of various orders. There is no isotropic tensor of first order,
and δij is the only isotropic tensor of second order. There is also only one isotropic
tensor of third order. It is called the alternating tensor or permutation symbol, and is
defined as
1 if ij k = 123, 231, or 312 (cyclic order),
0 if any two indices are equal,
εij k =
(2.18)
−1 if ij k = 321, 213, or 132 (anticyclic order).
From the definition, it is clear that an index on εij k can be moved two places (either
to the right or to the left) without changing its value. For example, εij k = εj ki where
i has been moved two places to the right, and εij k = εkij where k has been moved
two places to the left. For a movement of one place, however, the sign is reversed.
For example, εij k = −εikj where j has been moved one place to the right.
A very frequently used relation is the epsilon delta relation
εij k εklm = δil δj m − δim δj l .
(2.19)
The reader can verify the validity of this relationship by taking some values for ij lm.
Equation (2.19) is easy to remember by noting the following two points: (1) The
adjacent index k is summed; and (2) the first two indices on the right-hand side,
namely, i and l, are the first index of εij k and the first free index of εklm . The remaining
indices on the right-hand side then follow immediately.
8. Dot Product
The dot product of two vectors u and v is defined as the scalar
u • v = v • u = u1 v1 + u2 v2 + u3 v3 = ui vi .
It is easy to show that u • v = uv cos θ , where u and v are the magnitudes and θ is the
angle between the vectors. The dot product is therefore the magnitude of one vector
times the component of the other in the direction of the first. Clearly, the dot product
u • v is equal to the sum of the diagonal terms of the tensor ui vj .
38
Cartesian Tensors
9. Cross Product
The cross product between two vectors u and v is defined as the vector w whose
magnitude is uv sin θ, where θ is the angle between u and v, and whose direction is
perpendicular to the plane of u and v such that u, v, and w form a right-handed system.
Clearly, u × v = −v × u, and the unit vectors obey the cyclic rule a1 × a2 = a3 . It
is easy to show that
u × v = (u2 v3 − u3 v2 )a1 + (u3 v1 − u1 v3 )a2 + (u1 v2 − u2 v1 )a3 ,
(2.20)
which can be written as the symbolic determinant
1 2 3
a a a
u × v = u1 u2 u3 .
v1 v2 v3
In indicial notation, the k-component of u × v can be written as
(u × v)k = εij k ui vj = εkij ui vj .
(2.21)
As a check, for k = 1 the nonzero terms in the double sum in equation (2.21) result
from i = 2, j = 3, and from i = 3, j = 2. This follows from the definition
equation (2.18) that the permutation symbol is zero if any two indices are equal. Then
equation (2.21) gives
(u × v)1 = εij 1 ui vj = ε231 u2 v3 + ε321 u3 v2 = u2 v3 − u3 v2 ,
which agrees with equation (2.20). Note that the second form of equation (2.21) is
obtained from the first by moving the index k two places to the left; see the remark
below equation (2.18).
10. Operator ∇: Gradient, Divergence, and Curl
The vector operator “del”1 is defined symbolically by
∇ ≡ a1
∂
∂
∂
∂
+ a2
+ a3
= ai
.
∂x1
∂x2
∂x3
∂xi
(2.22)
When operating on a scalar function of position φ, it generates the vector
∇φ = ai
∂φ
,
∂xi
1 The inverted Greek delta is called a “nabla” (ναβλα). The origin of the word is from the Hebrew
(pronounced navel), which means lyre, an ancient harp-like stringed instrument. It was on this
instrument that the boy, David, entertained King Saul (Samuel II) and it is mentioned repeatedly
in Psalms as a musical instrument to use in the praise of God.
39
10. Operator ∇: Gradient, Divergence, and Curl
whose i-component is
(∇φ)i =
∂φ
.
∂xi
The vector ∇φ is called the gradient of φ. It is clear that ∇φ is perpendicular to the
φ = constant lines and gives the magnitude and direction of the maximum spatial rate
of change of φ (Figure 2.8). The rate of change in any other direction n is given by
∂φ
= (∇φ) • n.
∂n
The divergence of a vector field u is defined as the scalar
∇•u≡
∂ui
∂u1
∂u2
∂u3
=
+
+
.
∂xi
∂x1
∂x2
∂x3
(2.23)
So far, we have defined the operations of the gradient of a scalar and the divergence of a vector. We can, however, generalize these operations. For example, we can
define the divergence of a second-order tensor τ as the vector whose i-component is
(∇ • τ)i =
∂τij
.
∂xj
It is evident that the divergence operation decreases the order of the tensor by one.
In contrast, the gradient operation increases the order of a tensor by one, changing
Figure 2.8
Lines of constant φ and the gradient vector ∇φ.
40
Cartesian Tensors
a zero-order tensor to a first-order tensor, and a first-order tensor to a second-order
tensor.
The curl of a vector field u is defined as the vector ∇ × u, whose i-component
can be written as (using equations (2.21) and (2.22))
(∇ × u)i = εij k
∂uk
.
∂xj
(2.24)
The three components of the vector ∇ × u can easily be found from the right-hand
side of equation (2.24). For the i = 1 component, the nonzero terms in the double
sum in equation (2.24) result from j = 2, k = 3, and from j = 3, k = 2. The three
components of ∇ × u are finally found as
∂u2
∂u3
−
∂x2
∂x3
,
∂u1
∂u3
−
∂x3
∂x1
,
and
∂u2
∂u1
−
∂x1
∂x2
.
(2.25)
A vector field u is called solenoidal if ∇ • u = 0, and irrotational if ∇ × u = 0. The
word “solenoidal” refers to the fact that the magnetic induction B always satisfies
∇ • B = 0. This is because of the absence of magnetic monopoles. The reason for the
word “irrotational” will be clear in the next chapter.
11. Symmetric and Antisymmetric Tensors
A tensor B is called symmetric in the indices i and j if the components do not change
when i and j are interchanged, that is, if Bij = Bj i . The matrix of a second-order
tensor is therefore symmetric about the diagonal and made up of only six distinct
components. On the other hand, a tensor is called antisymmetric if Bij = −Bj i . An
antisymmetric tensor must have zero diagonal terms, and the off-diagonal terms must
be mirror images; it is therefore made up of only three distinct components. Any
tensor can be represented as the sum of a symmetric part and an antisymmetric part.
For if we write
Bij = 21 (Bij + Bj i ) + 21 (Bij − Bj i )
then the operation of interchanging i and j does not change the first term, but changes
the sign of the second term. Therefore, (Bij + Bj i )/2 is called the symmetric part of
Bij , and (Bij − Bj i )/2 is called the antisymmetric part of Bij .
Every vector can be associated with an antisymmetric tensor, and vice versa. For
example, we can associate the vector
ω1
ω = ω2 ,
ω3
with an antisymmetric tensor defined by
41
12. Eigenvalues and Eigenvectors of a Symmetric Tensor
0 −ω3 ω2
0 −ω1 ,
R ≡ ω3
−ω2 ω1
0
(2.26)
where the two are related as
Rij = −εij k ωk
(2.27)
ωk = − 21 εij k Rij .
As a check, equation (2.27) gives R11 = 0 and R12 = −ε123 ω3 = −ω3 , which is in
agreement with equation (2.26). (In Chapter 3 we shall call R the “rotation” tensor
corresponding to the “vorticity” vector ω.)
A very frequently occurring operation is the doubly contracted product of a
symmetric tensor τ and any tensor B. The doubly contracted product is defined as
P ≡ τij Bij = τij (Sij + Aij ),
where S and A are the symmetric and antisymmetric parts of B, given by
Sij ≡
1
(Bij + Bj i ) and
2
Aij ≡
1
(Bij − Bj i ).
2
Then
P = τij Sij + τij Aij
= τij Sj i − τij Aj i
= τj i Sj i − τj i Aj i
= τij Sij − τij Aij
(2.28)
because Sij = Sj i and Aij = −Aj i ,
because τij = τj i ,
interchanging dummy indices.
(2.29)
Comparing the two forms of equations (2.28) and (2.29), we see that τij Aij = 0, so
that
τij Bij =
1
τij (Bij + Bj i ).
2
The important rule we have proved is that the doubly contracted product of a symmetric
tensor τ with any tensor B equals τ times the symmetric part of B. In the process,
we have also shown that the doubly contracted product of a symmetric tensor and an
antisymmetric tensor is zero. This is analogous to the result that the definite integral
over an even (symmetric) interval of the product of a symmetric and an antisymmetric
function is zero.
12. Eigenvalues and Eigenvectors of a Symmetric Tensor
The reader is assumed to be familiar with the concepts of eigenvalues and eigenvectors
of a matrix, and only a brief review of the main results is given here. Suppose τ is a
42
Cartesian Tensors
symmetric tensor with real elements, for example, the stress tensor. Then the following
facts can be proved:
(1) There are three real eigenvalues λk (k = 1, 2, 3), which may or may not be all
distinct. (The superscripted λk does not denote the k-component of a vector.)
The eigenvalues satisfy the third-degree equation
det |τij − λδij | = 0,
which can be solved for λ1 , λ2 , and λ3 .
(2) The three eigenvectors bk corresponding to distinct eigenvalues λk are mutually orthogonal. These are frequently called the principal axes of τ. Each b is
found by solving a set of three equations
(τij − λδij ) bj = 0,
where the superscript k on λ and b has been omitted.
(3) If the coordinate system is rotated so as to coincide with the eigenvectors, then
τ has a diagonal form with elements λk . That is,
λ1 0 0
τ′ = 0 λ2 0
0 0 λ3
in the coordinate system of the eigenvectors.
(4) The elements τij change as the coordinate system is rotated, but they cannot be
larger than the largest λ or smaller than the smallest λ. That is, the eigenvalues
are the extremum values of τij .
Example 2.2.
The strain rate tensor E is related to the velocity vector u by
∂uj
1 ∂ui
Eij =
.
+
2 ∂xj
∂xi
For a two-dimensional parallel flow
u=
u1 (x2 )
0
,
show how E is diagonalized in the frame of reference coinciding with the principal
axes.
Solution: For the given velocity profile u1 (x2 ), it is evident that E11 = E22 = 0,
and E12 = E21 = 21 (du1 /dx2 ) = Ŵ. The strain rate tensor in the unrotated coordinate
system is therefore
0 Ŵ
E=
.
Ŵ 0
12. Eigenvalues and Eigenvectors of a Symmetric Tensor
Figure 2.9 Original coordinate system O x1 x2 and rotated coordinate system O x1′ x2′ coinciding with
the eigenvectors (Example 2.2).
The eigenvalues are given by
−λ Ŵ
= 0,
det |Eij − λδij | =
Ŵ −λ
whose solutions are λ1 = Ŵ and λ2 = −Ŵ. The first eigenvector b1 is given by
1
b
b11
0 Ŵ
1
= λ1
,
Ŵ 0
b21
b21
√
whose solution is b11 = b21 = 1/ 2, thus normalizing the magnitude to unity. The
√
√
first eigenvector is therefore b1 = [1/ 2, √
1/ 2],√writing it in a row. The second
eigenvector is similarly found as b2 = [−1/ 2, 1/ 2]. The eigenvectors are shown
in Figure 2.9. The direction cosine matrix of the original and the rotated coordinate
system is therefore
√1
− √1
2
2
C=
,
√1
2
√1
2
which represents rotation of the coordinate system by 45◦ . Using the transformation
rule (2.12), the components of E in the rotated system are found as follows:
′
= Ci1 Cj 2 Eij = C11 C22 E12 + C21 C12 E21
E12
1 1
1 1
= √ √ Ŵ− √ √ Ŵ=0
2 2
2 2
43
44
Cartesian Tensors
′
E21
=0
′
E11
= Ci1 Cj 1 Eij = C11 C21 E12 + C21 C11 E21 = Ŵ
′
E22
= Ci2 Cj 2 Eij = C12 C22 E12 + C22 C12 E21 = −Ŵ
(Instead of using equation (2.12), all the components of E in the rotated system can be
found by carrying out the matrix product CT • E • C.) The matrix of E in the rotated
frame is therefore
Ŵ 0
′
.
E =
0 −Ŵ
The foregoing matrix contains only diagonal terms. It will be shown in the next
chapter that it represents a linear stretching at a rate Ŵ along one principal axis, and a
linear compression at a rate −Ŵ along the other; there are no shear strains along the
principal axes.
13. Gauss’ Theorem
This very useful theorem relates a volume integral to a surface integral. Let V be a
volume bounded by a closed surface A. Consider an infinitesimal surface element
dA, whose outward unit normal is n (Figure 2.10). The vector n dA has a magnitude
dA and direction n, and we shall write dA to mean the same thing. Let Q(x) be a
scalar, vector, or tensor field of any order. Gauss’ theorem states that
V
∂Q
dV =
∂xi
Figure 2.10 Illustration of Gauss’ theorem.
A
dAi Q.
(2.30)
45
13. Gauss’ Theorem
The most common form of Gauss’ theorem is when Q is a vector, in which case the
theorem is
∂Qi
dV = dAi Qi ,
A
V ∂xi
which is called the divergence theorem. In vector notation, the divergence theorem is
•
∇ Q dV = dA • Q.
A
V
Physically, it states that the volume integral of the divergence of Q is equal to the
surface integral of the outflux of Q. Alternatively, equation (2.30), when considered
in its limiting form for an infintesmal volume, can define a generalized field derivative
of Q by the expression
1
DQ = lim
dAi Q.
(2.31)
V →0 V A
This includes the gradient, divergence, and curl of any scalar, vector, or tensor Q.
Moreover, by regarding equation (2.31) as a definition, the recipes for the computation
of the vector field derivatives may be obtained in any coordinate system. For a tensor
Q of any order, equation (2.31) as written defines the gradient. For a tensor of order
one (vector) or higher, the divergence is defined by using a dot (scalar) product under
the integral
1
dA • Q,
(2.32)
div Q = lim
V →0 V A
and the curl is defined by using a cross (vector) product under the integral
1
V →0 V
curl Q = lim
A
dA × Q.
(2.33)
In equations (2.31), (2.32), and (2.33), A is the closed surface bounding the volume V.
Example 2.3. Obtain the recipe for the divergence of a vector Q(x) in cylindrical polar coordinates from the integral definition equation (2.32). Compare with
Appendix B.1.
Solution: Consider an elemental volume bounded by the surfaces R − R/2,
R + R/2, θ − θ/2, θ + θ/2, x − x/2 and x + x/2. The volume
enclosed
V is R θ R x. We wish to calculate div Q = lim V→0 1V A dA • Q at the
central point R, θ , x by integrating the net outward flux through the bounding surface
A of V:
Q = iR QR (R, θ, x) + iθ Qθ (R, θ, x) + ix Qx (R, θ, x).
46
Cartesian Tensors
In evaluating the surface integrals, we can show that in the limit taken, each of the
six surface integrals may be approximated by the product of the value at the center
of the surface and the surface area. This is shown by Taylor expanding each of the
scalar products in the two variables of each surface, carrying out the integrations, and
applying the limits. The result is
R
R
, θ, x R +
θ x
R→0
R θ R x
2
2
θ →0
x→0
R
R
, θ, x R −
θ x
− QR R −
2
2
x
x
R θ R − Qx R, θ, x −
R θ R
+ Qx R, θ, x +
2
2
θ
θ
+ Q R, θ +
, x • iθ − iR
R x
2
2
θ
θ
, x • − i θ − iR
R x ,
+ Q R, θ −
2
2
1
div Q = lim
QR R +
where an additional complication arises because the normals to the two planes θ ±
θ/2 are not antiparallel:
Q R, θ ±
θ
θ
θ
, x = QR R, θ ±
, x iR R, θ ±
,x
2
2
2
θ
θ
, x iθ R, θ ±
,x
+ Qθ R, θ ±
2
2
θ
+ Qx R, θ ±
, x ix .
2
Now we can show that
iR θ ±
θ
2
θ
iθ (θ),
= iR (θ) ±
2
iθ θ ±
θ
2
= iθ (θ) ∓
θ
iR (θ).
2
Evaluating the last pair of surface integrals explicitly,
R
R
, θ, x
R+
θ x
R→0
R θ R x
2
2
θ →0
x→0
R
R
, θ, x
R−
θ x
− QR R −
2
2
div Q = lim
1
QR R +
47
14. Stokes’ Theorem
x
x
+ Qx R, θ, x +
− Qx R, θ, x −
R θ R
2
2
θ
θ
θ
θ
+ QR R, θ +
,x
− QR R, θ +
,x
R x
2
2
2
2
θ
θ
+ Qθ R, θ +
, x − Qθ R, θ −
,x R x
2
2
θ
θ
θ
θ
,x
− QR R, θ −
,x
R x ,
− QR R, θ −
2
2
2
2
where terms of second order in the increments have been neglected as they will vanish
in the limits. Carrying out the limits, we obtain
div Q =
1 ∂Qθ
∂Qx
1 ∂
(RQR ) +
+
.
R ∂R
R ∂θ
∂x
Here, the physical interpretation of the divergence as the net outward flux of a vector
field per unit volume has been made apparent by its evaluation through the integral
definition.
This level of detail is required to obtain the gradient correctly in these coordinates.
14. Stokes’ Theorem
Stokes’ theorem relates a surface integral over an open surface to a line integral
around the boundary curve. Consider an open surface A whose bounding curve is C
(Figure 2.11). Choose one side of the surface to be the outside. Let ds be an element of
the bounding curve whose magnitude is the length of the element and whose direction
is that of the tangent. The positive sense of the tangent is such that, when seen from
the “outside” of the surface in the direction of the tangent, the interior is on the left.
Then the theorem states that
A
(∇ × u) • dA =
u • ds,
(2.34)
C
which signifies that the surface integral of the curl of a vector field u is equal to the
line integral of u along the bounding curve.
The line integral of a vector u around a closed curve C (as in Figure 2.11)
is called the “circulation of u about C.” This can be used to define the curl of a
vector through the limit of the circulation integral bounding an infinitesmal surface
as follows:
1
u • ds,
(2.35)
n • curl u = lim
A→0 A C
48
Cartesian Tensors
Figure 2.11 Illustration of Stokes’ theorem.
where n is a unit vector normal to the local tangent plane of A. The advantage of the
integral definitions of the field derivatives is that they may be applied regardless of
the coordinate system.
Example 2.4. Obtain the recipe for the curl of a vector u(x) in Cartesian coordinates
from the integral definition given by equation (2.35).
Solution: This is obtained by considering rectangular contours in three perpendicular planes intersecting at the point (x, y, z). First, consider the elemental rectangle
in the x = const. plane. The central point in this plane has coordinates (x, y, z) and
the area is y z. It may be shown by careful integration of a Taylor expansion of
the integrand that the integral along each line segment may be represented by the
product of the integrand at the center of the segment and the length of the segment
with attention paid to the direction of integration ds. Thus we obtain
y
y
1
(curl u)x = lim
, z − uz x, y −
,z
z
uz x, y +
y→0
y z
2
2
z→0
z
z
1
uy x, y, z −
− uy x, y, z +
y .
+
y z
2
2
Taking the limits,
(curl u)x =
∂uy
∂uz
−
.
∂y
∂z
49
16. Boldface vs Indicial Notation
Similarly, integrating around the elemental rectangles in the other two planes
∂uz
∂ux
−
,
∂z
∂x
∂uy
∂ux
−
.
(curl u)z =
∂x
∂y
(curl u)y =
15. Comma Notation
Sometimes it is convenient to introduce the notation
A,i ≡
∂A
,
∂xi
(2.36)
where A is a tensor of any order. In this notation, therefore, the comma denotes a
spatial derivative. For example, the divergence and curl of a vector u can be written,
respectively, as
∂ui
= ui,i ,
∂xi
∂uk
(∇ × u)i = εij k
= εij k uk,j .
∂xj
∇•u=
This notation has the advantages of economy and that all subscripts are written on
one line. Another advantage is that variables such as ui,j “look like” tensors, which
they are, in fact. Its disadvantage is that it takes a while to get used to it, and that
the comma has to be written clearly in order to avoid confusion with other indices
in a term. The comma notation has been used in the book only in two sections, in
instances where otherwise the algebra became cumbersome.
16. Boldface vs Indicial Notation
The reader will have noticed that we have been using both boldface and indicial notations. Sometimes the boldface notation is loosely called “vector” or dyadic notation,
while the indicial notation is called “tensor” notation. (Although there is no reason
why vectors cannot be written in indicial notation!). The advantage of the boldface
form is that the physical meaning of the terms is generally clearer, and there are no
cumbersome subscripts. Its disadvantages are that algebraic manipulations are difficult, the ordering of terms becomes important because A • B is not the same as
B • A, and one has to remember formulas for triple products such as u × (v × w) and
u • (v × w). In addition, there are other problems, for example, the order or rank of
a tensor is not clear if one simply calls it A, and sometimes confusion may arise in
products such as A • B where it is not immediately clear which index is summed. To
add to the confusion, the singly contracted product A • B is frequently written as AB
in books on matrix algebra, whereas in several other fields AB usually stands for the
uncontracted fourth-order tensor with elements Aij Bkl .
50
Cartesian Tensors
The indicial notation avoids all the problems mentioned in the preceding. The
algebraic manipulations are especially simple. The ordering of terms is unnecessary because Aij Bkl means the same thing as Bkl Aij . In this notation we deal with
components only, which are scalars. Another major advantage is that one does not
have to remember formulas except for the product εij k εklm , which is given by equation (2.19). The disadvantage of the indicial notation is that the physical meaning of a
term becomes clear only after an examination of the indices. A second disadvantage
is that the cross product involves the introduction of the cumbersome εij k . This, however, can frequently be avoided by writing the i-component of the vector product of u
and v as (u × v)i using a mixture of boldface and indicial notations. In this book we
shall use boldface, indicial and mixed notations in order to take advantage of each. As
the reader might have guessed, the algebraic manipulations will be performed mostly
in the indicial notation, sometimes using the comma notation.
Exercises
1. Using indicial notation, show that
a × (b × c) = (a • c)b − (a • b)c.
[Hint: Call d ≡ b × c. Then (a × d)m = εpqm ap dq = εpqm ap εij q bi cj . Using
equation (2.19), show that (a × d)m = (a • c)bm − (a • b)cm .]
2. Show that the condition for the vectors a, b, and c to be coplanar is
εij k ai bj ck = 0.
3. Prove the following relationships:
δij δij = 3
εpqr εpqr = 6
εpqi εpqj = 2δij .
4. Show that
C • CT = CT • C = δ,
where C is the direction cosine matrix and δ is the matrix of the Kronecker delta.
Any matrix obeying such a relationship is called an orthogonal matrix because it
represents transformation of one set of orthogonal axes into another.
5. Show that for a second-order tensor A, the following three quantities are
invariant under the rotation of axes:
I1 = Aii
A A
I2 = 11 12
A21 A22
I3 = det(Aij ).
A22 A23
+
A32 A33
A11 A13
+
A31 A33
51
Supplemental Reading
[Hint: Use the result of Exercise 4 and the transformation rule (2.12) to show that
I1′ = A′ii = Aii = I1 . Then show that Aij Aj i and Aij Aj k Aki are also invariants. In
fact, all contracted scalars of the form Aij Aj k · · · Ami are invariants. Finally, verify
that
I2 = 21 [I12 − Aij Aj i ]
I3 = Aij Aj k Aki − I1 Aij Aj i + I2 Aii .
Because the right-hand sides are invariant, so are I2 and I3 .]
6. If u and v are vectors, show that the products ui vj obey the transformation
rule (2.12), and therefore represent a second-order tensor.
7. Show that δij is an isotropic tensor. That is, show that δij′ = δij under rotation
of the coordinate system. [Hint: Use the transformation rule (2.12) and the results of
Exercise 4.]
8. Obtain the recipe for the gradient of a scalar function in cylindrical polar
coordinates from the integral definition.
9. Obtain the recipe for the divergence of a vector in spherical polar coordinates
from the integral definition.
10. Prove that div(curl u) = 0 for any vector u regardless of the coordinate
system. [Hint: use the vector integral theorems.]
11. Prove that curl(grad φ) = 0 for any single-valued scalar φ regardless of the
coordinate system. [Hint: use Stokes’ theorem.]
Literature Cited
Sommerfeld, A. (1964). Mechanics of Deformable Bodies, New York: Academic Press. (Chapter 1 contains
brief but useful coverage of Cartesian tensors.)
Supplemental Reading
Aris, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, NJ:
Prentice-Hall. (This book gives a clear and easy treatment of tensors in Cartesian and non-Cartesian
coordinates, with applications to fluid mechanics.)
Prager, W. (1961). Introduction to Mechanics of Continua, New York: Dover Publications. (Chapters 1
and 2 contain brief but useful coverage of Cartesian tensors.)
This page intentionally left blank
Chapter 3
Kinematics
1. Introduction . . . . . . . . . . . . . . . . . . . . . 53
2. Lagrangian and Eulerian
Specifications . . . . . . . . . . . . . . . . . . . . 54
3. Eulerian and Lagrangian Descriptions:
The Particle Derivative . . . . . . . . . . . 55
4. Streamline, Path Line, and Streak
Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5. Reference Frame and Streamline
Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6. Linear Strain Rate . . . . . . . . . . . . . . . 60
7. Shear Strain Rate . . . . . . . . . . . . . . . . 61
8. Vorticity and Circulation . . . . . . . . . 62
9. Relative Motion near a Point:
Principal Axes . . . . . . . . . . . . . . . . . . . 64
10. Kinematic Considerations of
Parallel Shear Flows . . . . . . . . . . . . 67
11. Kinematic Considerations of Vortex
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Solid-Body Rotation . . . . . . . . . . . . 68
Irrotational Vortex . . . . . . . . . . . . . . 70
Rankine Vortex . . . . . . . . . . . . . . . . . 71
12. One-, Two-, and Three-Dimensional
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 71
13. The Streamfunction . . . . . . . . . . . . . 73
14. Polar Coordinates . . . . . . . . . . . . . . 75
Exercises . . . . . . . . . . . . . . . . . . . . . . . 77
Supplemental Reading . . . . . . . . . . 79
1. Introduction
Kinematics is the branch of mechanics that deals with quantities involving space and
time only. It treats variables such as displacement, velocity, acceleration, deformation,
and rotation of fluid elements without referring to the forces responsible for such a
motion. Kinematics therefore essentially describes the “appearance” of a motion.
Some important kinematical concepts are described in this chapter. The forces are
considered when one deals with the dynamics of the motion, which will be discussed
in later chapters.
A few remarks should be made about the notation used in this chapter and
throughout the rest of the book. The convention followed in Chapter 2, namely,
that vectors are denoted by lowercase letters and higher-order tensors are denoted
by uppercase letters, is no longer followed. Henceforth, the number of subscripts
will specify the order of a tensor. The Cartesian coordinate directions are denoted
by (x, y, z), and the corresponding velocity components are denoted by (u, v, w).
When using tensor expressions, the Cartesian directions are denoted alternatively
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50003-4
53
54
Kinematics
z
Figure 3.1 Plane, cylindrical, and spherical polar coordinates: (a) plane polar; (b) cylindrical polar;
(c) spherical polar coordinates.
by (x1 , x2 , x3 ), with the corresponding velocity components (u1 , u2 , u3 ). Plane
polar coordinates are denoted by (r, θ), with ur and uθ the corresponding velocity
components (Figure 3.1a). Cylindrical polar coordinates are denoted by (R, ϕ, x),
with (uR , uϕ , ux ) the corresponding velocity components (Figure 3.1b). Spherical polar coordinates are denoted by (r, θ, ϕ), with (ur , uθ , uϕ ) the corresponding
velocity components (Figure 3.1c). The method of conversion from Cartesian to
plane polar coordinates is illustrated in Section 14 of this chapter.
2. Lagrangian and Eulerian Specifications
There are two ways of describing a fluid motion. In the Lagrangian description, one
essentially follows the history of individual fluid particles (Figure 3.2). Consequently,
the two independent variables are taken as time and a label for fluid particles. The label
can conveniently be taken as the position vector a of the particle at some reference time
t = 0. In this description, any flow variable F is expressed as F (a, t). In particular,
the position vector is written as r = r(a, t), which represents the location at t of a
particle whose position was a at t = 0.
In the Eulerian description, one concentrates on what happens at a spatial point
r′ , so that the independent variables are taken as r′ and t ′ . (Here the primes are meant
to distinguish Lagrangian dependent variables from Eulerian independent variables.)
Flow variables are written, for example, as F (r′ , t ′ ).
55
3. Eulerian and Lagrangian Descriptions: The Particle Derivative
z
u
particle
path
r (t)
r (0) 5 a
y
x
Figure 3.2 Particle—Lagrangian description. Independent variables: (a, t); dependent variables: r(a, t),
u = (∂r/∂t)a , ρ = ρ(a, t), and so on.
The velocity and acceleration of a fluid particle in the Lagrangian description are
simply the partial time derivatives
u = ∂r/∂t, acceleration a = ∂u/∂t = ∂ 2 r/∂t 2
(3.1)
as the particle identity is kept constant during the differentiation. In the Eulerian
description, however, the partial derivative ∂/∂t ′ gives only the local rate of change
at a point r′ and is not the total rate of change as seen by the fluid particle. Additional
terms are needed to form derivatives following a particle in the Eulerian description,
as explained in the next section.
The Eulerian specification is used in most problems of fluid flows. The
Lagrangian description is used occasionally when we are interested in finding particle
paths of fixed identity; examples can be found in Chapters 7 and 13.
3. Eulerian and Lagrangian Descriptions: The Particle
Derivative
Classical mechanics has two alternative descriptions: the field description (Eulerian)
and the particle description (Lagrangian), associated with two of the great European
mathematical physicists of the eighteenth century [Leonhard Euler (1707–1783) and
Joseph Louis, Comte de Lagrange (1736–1813)]. Most of this book is written in
the field description (Figure 3.3) but it is frequently very useful to express a particle derivative in the field description. Thus we wish to compare and relate the two
descriptions.
Consider any fluid property F (r′ , t ′ ) = F (a, t) at the same position and time
in the two descriptions. F may be a scalar, vector, or tensor property. We seek to
express (∂F /∂t)a , which is the rate of change of F as seen by an observer on the
fixed particle labeled by coordinate a = r(0) at t = 0, in field variables. That is,
56
Kinematics
z9
(x9, y9, z9, t9)
u
r
y9
x9
Figure 3.3 Field—Eulerian description. Independent variables: (x ′ , y ′ , z′ , t ′ ); dependent variables:
u(r′ , t ′ ), ρ(r′ , t), and so on.
we ask what combination of r′ , t ′ field derivatives corresponds to (∂F /∂t)a ? We do
our calculation at r′ = r and t ′ = t so we are at the same point and time in the two
descriptions. Thus
F (a, t) = F [r(a, t), t] = F (r′ , t ′ ).
(3.2)
Differentiating, taking care to differentiate dependent variables with respect to independent variables, and using the chain rule,
[∂F (a, t)/∂t]a = (∂F /∂t ′ )r′ (∂t ′ /∂t) + (∂F /∂r′ )t ′ • (∂r′ /∂r) • (∂r/∂t)a .
(3.3)
Now ∂t ′ /∂t is simply the ratio of time scales used in the two descriptions. We take this
equal to 1 by measuring the time in the same units (say seconds). Here ∂r′ /∂r is the
transformation matrix between the two coordinate systems. If r′ and r are not rotated or
stretched with respect to each other, but with parallel axes and with lengths measured
in the same units (say meters), then ∂r′ /∂r = I, the unit matrix, with elements δij .
Since (∂r/∂t)a = u, we have the result
(∂F /∂t)a = ∂F /∂t ′ + (∇ ′ F ) • u ≡ DF /Dt.
(3.4)
The total rate of change D/Dt is generally called the material derivative (also called
the substantial derivative, or particle derivative) to emphasize the fact that the derivative is taken following a fluid element. It is made of two parts: ∂F /∂t is the local
rate of change of F at a given point, and is zero for steady flows. The second part
ui ∂F /∂xi is called the advective derivative, because it is the change in F as a result
of advection of the particle from one location to another where the value of F is different. (In this book, the movement of fluid from place to place is called “advection.”
57
4. Streamline, Path Line, and Streak Line
Figure 3.4
Streamline coordinates (s, n).
Engineering texts generally call it “convection.” However, we shall reserve the term
convection to describe heat transport by fluid movements.)
In vector notation, equation (3.4) is written as
DF
∂F
=
+ u • ∇F.
Dt
∂t
(3.5)
The scalar product u • ∇F is the magnitude of u times the component of ∇F in the
direction of u. It is customary to denote the magnitude of the velocity vector u by q.
Equation (3.5) can then be written in scalar notation as
DF
∂F
∂F
=
+q
,
Dt
∂t
∂s
(3.6)
where the “streamline coordinate” s points along the local direction of u (Figure 3.4).
4. Streamline, Path Line, and Streak Line
At an instant of time, there is at every point a velocity vector with a definite direction.
The instantaneous curves that are everywhere tangent to the direction field are called
the streamlines of flow. For unsteady flows the streamline pattern changes with time.
Let ds = (dx, dy, dz) be an element of arc length along a streamline (Figure 3.5),
and let u = (u, v, w) be the local velocity vector. Then by definition
dx
dy
dz
=
=
,
u
v
w
(3.7)
along a streamline. If the velocity components are known as a function of time, then
equation (3.7) can be integrated to find the equation of the streamline. It is easy to
show that equation (3.7) corresponds to u × ds = 0. All streamlines passing through
58
Kinematics
Figure 3.5 Streamline.
Figure 3.6 Streamtube.
any closed curve C at some time form a tube, which is called a streamtube (Figure 3.6).
No fluid can cross the streamtube because the velocity vector is tangent to this surface.
In experimental fluid mechanics, the concept of path line is important. The path
line is the trajectory of a fluid particle of fixed identity over a period of time. The
path line or particle path is represented as in Section 2 by r = r(a, t) where a is
the location of the particle at the reference time, say t = 0. Then u = ∂r/∂t for
fixed particle. Assuming a nonzero Jacobian determinant, we may invert to obtain
the reference particle location at t = 0, a = a(r, t). Path lines and streamlines are
identical in a steady flow, but not in an unsteady flow. Consider the flow around a
body moving from right to left in a fluid that is stationary at an infinite distance from
the body (Figure 3.7). The flow pattern observed by a stationary observer (that is,
5. Reference Frame and Streamline Pattern
Figure 3.7
Several streamlines and a path line due to a moving body.
an observer stationary with respect to the undisturbed fluid) changes with time, so
that to the observer this is an unsteady flow. The streamlines in front of and behind
the body are essentially directed forward as the body pushes forward, and those
on the two sides are directed laterally. The path line (shown dashed in Figure 3.7)
of the particle that is now at point P therefore loops outward and forward again as the
body passes by.
The streamlines and path lines of Figure 3.7 can be visualized in an experiment
by suspending aluminum or other reflecting materials on the fluid surface, illuminated
by a source of light. Suppose that the entire fluid is covered with such particles, and a
brief time exposure is made. The photograph then shows short dashes, which indicate
the instantaneous directions of particle movement. Smooth curves drawn through
these dashes constitute the instantaneous streamlines. Now suppose that only a few
particles are introduced, and that they are photographed with the shutter open for a
long time. Then the photograph shows the paths of a few individual particles, that is,
their path lines.
A streak line is another concept in flow visualization experiments. It is defined as
the current location of all fluid particles that have passed through a fixed spatial point
at a succession of previous times. It is determined by injecting dye or smoke at a fixed
point for an interval of time. Suppose a particle on a streak line passes the location of
dye ξ at a time τ ≤ t. Then the equation of the streak line is x = x[a(ξ , τ ), t]. See
Aris (1962), Chapter 4 for more details. In steady flow the streamlines, path lines,
and streak lines all coincide.
5. Reference Frame and Streamline Pattern
A flow that is steady in one reference frame is not necessarily so in another. Consider
the flow past a ship moving at a steady velocity U, with the frame of reference (that
is, the observer) attached to the river bank (Figure 3.8a). To this observer the local
flow characteristics appear to change with time, and thus appear to be unsteady. If,
59
60
Kinematics
Figure 3.8 Flow past a ship with respect to two observers: (a) observer on river bank; (b) observer on ship.
on the other hand, the observer is standing on the ship, the flow pattern is steady
(Figure 3.8b). The steady flow pattern can be obtained from the unsteady pattern of
Figure 3.8a by superposing on the latter a velocity U to the right. This causes the
ship to come to a halt and the river to move with velocity U at infinity. It follows that
any velocity vector u in Figure 3.8b is obtained by adding the corresponding velocity
vector u′ of Figure 3.8a and the free stream velocity vector U.
6. Linear Strain Rate
A study of the dynamics of fluid flows involves determination of the forces on an
element, which depend on the amount and nature of its deformation, or strain. The
deformation of a fluid is similar to that of a solid, where one defines normal strain as
the change in length per unit length of a linear element, and shear strain as change
of a 90◦ angle. Analogous quantities are defined in a fluid flow, the basic difference
being that one defines strain rates in a fluid because it continues to deform.
Consider first the linear or normal strain rate of a fluid element in the x1 direction
(Figure 3.9). The rate of change of length per unit length is
1 D
1 A′ B ′ − AB
(δx1 ) =
δx1 Dt
dt
AB
∂u1
∂u1
1 1
δx1 +
δx1 dt − δx1 =
.
=
dt δx1
∂x1
∂x1
The material derivative symbol D/Dt has been used because we have implicitly
followed a fluid particle. In general, the linear strain rate in the α direction is
∂uα
,
∂xα
(3.8)
where no summation over the repeated index α is implied. Greek symbols such as α
and β are commonly used when the summation convention is violated.
The sum of the linear strain rates in the three mutually orthogonal directions
gives the rate of change of volume per unit volume, called the volumetric strain rate
(also called the bulk strain rate). To see this, consider a fluid element of sides δx1 ,
61
7. Shear Strain Rate
Figure 3.9
Linear strain rate. Here, A′ B ′ = AB + BB ′ − AA′ .
δx2 , and δx3 . Defining δ ᐂ ≡ δx1 δx2 δx3 , the volumetric strain rate is
1 D
D
1
(δ ᐂ) =
(δx1 δx2 δx3 ),
δ ᐂ Dt
δx1 δx2 δx3 Dt
1 D
1 D
1 D
(δx1 ) +
(δx2 ) +
(δx3 ),
=
δx1 Dt
δx2 Dt
δx3 Dt
that is,
∂u1
∂u2
∂u3
∂ui
1 D
(δ ᐂ) =
+
+
=
.
δ ᐂ Dt
∂x1
∂x2
∂x3
∂xi
(3.9)
The quantity ∂ui /∂xi is the sum of the diagonal terms of the velocity gradient
tensor ∂ui /∂xj . As a scalar, it is invariant with respect to rotation of coordinates.
Equation (3.9) will be used later in deriving the law of conservation of mass.
7. Shear Strain Rate
In addition to undergoing normal strain rates, a fluid element may also simply deform
in shape. The shear strain rate of an element is defined as the rate of decrease of the
angle formed by two mutually perpendicular lines on the element. The shear strain so
calculated depends on the orientation of the line pair. Figure 3.10 shows the position
of an element with sides parallel to the coordinate axes at time t, and its subsequent
position at t + dt. The rate of shear strain is
dα + dβ
∂u1
∂u2
1
1
1
=
δx2 dt +
δx1 dt
dt
dt δx2 ∂x2
δx1 ∂x1
=
∂u1
∂u2
+
.
∂x2
∂x1
(3.10)
An examination of equations (3.8) and (3.10) shows that we can describe the
deformation of a fluid element in terms of the strain rate tensor
1
eij ≡
2
∂uj
∂ui
.
+
∂xj
∂xi
(3.11)
62
Kinematics
Figure 3.10 Deformation of a fluid element. Here, dα = C A / C B; a similar expression represents dβ.
The diagonal terms of e are the normal strain rates given in (3.8), and the off-diagonal
terms are half the shear strain rates given in (3.10). Obviously the strain rate tensor
is symmetric as eij = ej i .
8. Vorticity and Circulation
Fluid lines oriented along different directions rotate by different amounts. To define
the rotation rate unambiguously, two mutually perpendicular lines are taken, and the
average rotation rate of the two lines is calculated; it is easy to show that this average
is independent of the orientation of the line pair. To avoid the appearance of certain
factors of 2 in the final expressions, it is generally customary to deal with twice the
angular velocity, which is called the vorticity of the element.
Consider the two perpendicular line elements of Figure 3.10. The angular velocities of line elements about the x3 axis are dβ/dt and −dα/dt, so that the average is
1
2 (−dα/dt + dβ/dt). The vorticity of the element about the x3 axis is therefore twice
this average, as given by
∂u1
∂u2
1
1
1
−
δx2 dt +
δx1 dt
ω3 =
dt δx2
∂x2
δx1 ∂x1
∂u1
∂u2
−
.
=
∂x1
∂x2
From the definition of curl of a vector (see equations 2.24 and 2.25), it follows that
the vorticity vector of a fluid element is related to the velocity vector by
ω=∇×u
or
ωi = εij k
∂uk
,
∂xj
(3.12)
63
8. Vorticity and Circulation
whose components are
ω1 =
∂u2
∂u3
−
,
∂x2
∂x3
ω2 =
∂u1
∂u3
−
,
∂x3
∂x1
ω3 =
∂u2
∂u1
−
.
∂x1
∂x2
(3.13)
A fluid motion is called irrotational if ω = 0, which would require
∂uj
∂ui
=
∂xj
∂xi
i = j.
(3.14)
In irrotational flows, the velocity vector can be written as the gradient of a scalar
function φ(x, t). This is because the assumption
ui ≡
∂φ
,
∂xi
(3.15)
satisfies the condition of irrotationality (3.14).
Related to the concept of vorticity is the concept of circulation. The circulation Ŵ
around a closed contour C (Figure 3.11) is defined as the line integral of the tangential
component of velocity and is given by
Ŵ≡
u • ds,
(3.16)
C
where ds is an element of contour, and the loop through the integral sign signifies that
the contour is closed. The loop will be omitted frequently because it is understood
that such line integrals are taken along closed contours called circuits. Then Stokes’
theorem (Chapter 2, Section 14) states that
•
(3.17)
u ds = (curl u) • dA
C
Figure 3.11 Circulation around contour C.
A
64
Kinematics
which says that the line integral of u around a closed curve C is equal to the “flux” of
curl u through an arbitrary surface A bounded by C. (The word “flux” is generally used
to mean the integral of a vector field normal to a surface. [See equation (2.32), where
the integral written is the net outward flux of the vector field Q.]) Using the definitions
of vorticity and circulation, Stokes’ theorem, equation (3.17), can be written as
Ŵ=
ω • dA.
(3.18)
A
Thus, the circulation around a closed curve is equal to the surface integral of the
vorticity, which we can call the flux of vorticity. Equivalently, the vorticity at a point
equals the circulation per unit area. That follows directly from the definition of curl
as the limit of the circulation integral. (See equation (2.35) of Chapter 2.)
9. Relative Motion near a Point: Principal Axes
The preceding two sections have shown that fluid particles deform and rotate. In this
section we shall formally show that the relative motion between two neighboring
points can be written as the sum of the motion due to local rotation, plus the motion
due to local deformation.
Let u(x, t) be the velocity at point O (position vector x), and let u + du be
the velocity at the same time at a neighboring point P (position vector x + dx; see
Figure 3.12). The relative velocity at time t is given by
Figure 3.12 Velocity vectors at two neighboring points O and P.
65
9. Relative Motion near a Point: Principal Axes
∂ui
dxj ,
∂xj
(3.19)
∂u1
∂u1
∂u1
dx1 +
dx2 +
dx3 .
∂x1
∂x2
∂x3
(3.20)
dui =
which stands for three relations such as
du1 =
The term ∂ui /∂xj in equation (3.19) is the velocity gradient tensor. It can be decomposed into symmetric and antisymmetric parts as follows:
1
∂ui
=
∂xj
2
∂uj
∂ui
+
∂xj
∂xi
+
1
2
∂uj
∂ui
−
∂xj
∂xi
,
(3.21)
which can be written as
∂ui
1
= eij + rij ,
∂xj
2
(3.22)
where eij is the strain rate tensor defined in equation (3.11), and
rij ≡
∂uj
∂ui
−
,
∂xj
∂xi
(3.23)
is called the rotation tensor. As rij is antisymmetric, its diagonal terms are zero
and the off-diagonal terms are equal and opposite. It therefore has three independent
elements, namely, r13 , r21 , and r32 . Comparing equations (3.13) and (3.22), we can
see that r21 = ω3 , r32 = ω1 , and r13 = ω2 . Thus the rotation tensor can be written in
terms of the components of the vorticity vector as
0 −ω3 ω2
r = ω3 0 −ω1 .
−ω2 ω1 0
(3.24)
Each antisymmetric tensor in fact can be associated with a vector as discussed in
Chapter 2, Section 11. In the present case, the rotation tensor can be written in terms
of the vorticity vector as
rij = −εij k ωk .
(3.25)
This can be verified by taking various components of equation (3.24) and comparing
them with equation (3.23). For example, equation (3.24) gives r12 = −ε12k ωk =
−ε123 ω3 = −ω3 , which agrees with equation (3.23). Equation (3.24) also appeared
as equation (2.27).
Substitution of equations (3.21) and (3.24) into equation (3.19) gives
dui = eij dxj − 21 εij k ωk dxj ,
66
Kinematics
which can be written as
dui = eij dxj + 21 (ω × dx)i .
(3.26)
In the preceding, we have noted that εij k ωk dxj is the i-component of the cross product
−ω × dx. (See the definition of cross product in equation (2.21).) The meaning of the
second term in equation (3.25) is evident. We know that the velocity at a distance x
from the axis of rotation of a body rotating rigidly at angular velocity is × x. The
second term in equation (3.25) therefore represents the relative velocity at point P due
to rotation of the element at angular velocity ω/2. (Recall that the angular velocity is
half the vorticity ω.)
The first term in equation (3.25) is the relative velocity due only to deformation
of the element. The deformation becomes particularly simple in a coordinate system coinciding with the principal axes of the strain rate tensor. The components of e
change as the coordinate system is rotated. For a particular orientation of the coordinate system, a symmetric tensor has only diagonal components; these are called the
principal axes of the tensor (see Chapter 2, Section 12 and Example 2.2). Denoting
the variables in the principal coordinate system by an overbar (Figure 3.13), the first
part of equation (3.25) can be written as the matrix product
Figure 3.13 Deformation of a spherical fluid element into an ellipsoid.
67
10. Kinematic Considerations of Parallel Shear Flows
ē11 0 0
d x̄1
d ū = ē • d x̄ = 0 ē22 0 d x̄2 .
d x̄3
0 0 ē33
(3.27)
Here, ē11 , ē22 , and ē33 are the diagonal components of e in the principal coordinate
system and are called the eigenvalues of e. The three components of equation (3.26)
are
d ū1 = ē11 d x̄1
d ū2 = ē22 d x̄2
d ū3 = ē33 d x̄3 .
(3.28)
Consider the significance of the first of equations (3.27), namely, d ū1 =ē11 d x̄1 (Figure 3.13). If ē11 is positive, then this equation shows that point P is moving away
from O in the x̄1 direction at a rate proportional to the distance d x̄1 . Considering all points on the surface of a sphere, the movement of P in the x̄1 direction
is therefore the maximum when P coincides with M (where d x̄1 is the maximum)
and is zero when P coincides with N. (In Figure 3.13 we have illustrated a case
where ē11 >0 and ē22 <0; the deformation in the x3 direction cannot, of course, be
shown in this figure.) In a small interval of time, a spherical fluid element around
O therefore becomes an ellipsoid whose axes are the principal axes of the strain
tensor e.
Summary: The relative velocity in the neighborhood of a point can be divided
into two parts. One part is due to the angular velocity of the element, and the other
part is due to deformation. A spherical element deforms to an ellipsoid whose axes
coincide with the principal axes of the local strain rate tensor.
10. Kinematic Considerations of Parallel Shear Flows
In this section we shall consider the rotation and deformation of fluid elements in the
parallel shear flow u = [u1 (x2 ), 0, 0] shown in Figure 3.14. Let us denote the velocity
gradient by γ (x2 ) ≡ du1 /dx2 . From equation (3.13), the only nonzero component
of vorticity is ω3 = −γ . In Figure 3.13, the angular velocity of line element AB is
−γ , and that of BC is zero, giving −γ /2 as the overall angular velocity (half the
Figure 3.14 Deformation of elements in a parallel shear flow. The element is stretched along the principal
axis x̄1 and compressed along the principal axis x̄2 .
68
Kinematics
vorticity). The average value does not depend on which two mutually perpendicular
elements in the x1 x2 -plane are chosen to compute it.
In contrast, the components of strain rate do depend on the orientation of the
element. From equation (3.11), the strain rate tensor of an element such as ABCD,
with the sides parallel to the x1 x2 -axes, is
0
e = 21 γ
0
1
2γ
0
0 0 ,
0 0
which shows that there are only off-diagonal elements of e. Therefore, the element
ABCD undergoes shear, but no normal strain. As discussed in Chapter 2, Section 12
and Example 2.2, a symmetric tensor with zero diagonal elements can be diagonalized
by rotating the coordinate system through 45◦ . It is shown there that, along these
principal axes (denoted by an overbar in Figure 3.14), the strain rate tensor is
1
0 0
2γ
ē = 0 − 21 γ 0 ,
0
0 0
so that there is a linear extension rate of ē11 = γ /2, a linear compression rate of
ē22 = −γ /2, and no shear. This can be understood physically by examining the
deformation of an element PQRS oriented at 45◦ , which deforms to P′ Q′ R′ S′ . It is
clear that the side PS elongates and the side PQ contracts, but the angles between the
sides of the element remain 90◦ . In a small time interval, a small spherical element in
this flow would become an ellipsoid oriented at 45◦ to the x1 x2 -coordinate system.
Summarizing, the element ABCD in a parallel shear flow undergoes only shear
but no normal strain, whereas the element PQRS undergoes only normal but no shear
strain. Both of these elements rotate at the same angular velocity.
11. Kinematic Considerations of Vortex Flows
Flows in circular paths are called vortex flows, some basic forms of which are described
in what follows.
Solid-Body Rotation
Consider first the case in which the velocity is proportional to the radius of the streamlines. Such a flow can be generated by steadily rotating a cylindrical tank containing
a viscous fluid and waiting until the transients die out. Using polar coordinates (r, θ),
the velocity in such a flow is
uθ = ω0 r
ur = 0,
(3.29)
where ω0 is a constant equal to the angular velocity of revolution of each particle
about the origin (Figure 3.15). We shall see shortly that ω0 is also equal to the angular
69
11. Kinematic Considerations of Vortex Flows
Figure 3.15 Solid-body rotation. Fluid elements are spinning about their own centers while they revolve
around the origin. There is no deformation of the elements.
speed of rotation of each particle about its own center. The vorticity components of
a fluid element in polar coordinates are given in Appendix B. The component about
the z-axis is
ωz =
1 ∂ur
1 ∂
(ruθ ) −
= 2ω0 ,
r ∂r
r ∂θ
(3.30)
where we have used the velocity distribution equation (3.28). This shows that the
angular velocity of each fluid element about its own center is a constant and equal
to ω0 . This is evident in Figure 3.15, which shows the location of element ABCD at
two successive times. It is seen that the two mutually perpendicular fluid lines AD
and AB both rotate counterclockwise (about the center of the element) with speed ω0 .
The time period for one rotation of the particle about its own center equals the time
period for one revolution around the origin. It is also clear that the deformation of the
fluid elements in this flow is zero, as each fluid particle retains its location relative
to other particles. A flow defined by uθ = ω0 r is called a solid-body rotation as the
fluid elements behave as in a rigid, rotating solid.
The circulation around a circuit of radius r in this flow is
2π
uθ r dθ = 2π ruθ = 2π r 2 ω0 ,
(3.31)
Ŵ = u • ds =
0
which shows that circulation equals vorticity 2ω0 times area. It is easy to show
(Exercise 12) that this is true of any contour in the fluid, regardless of whether or
not it contains the center.
70
Kinematics
Irrotational Vortex
Circular streamlines, however, do not imply that a flow should have vorticity
everywhere. Consider the flow around circular paths in which the velocity vector
is tangential and is inversely proportional to the radius of the streamline. That is,
C
ur = 0.
r
Using equation (3.29), the vorticity at any point in the flow is
(3.32)
uθ =
0
.
r
This shows that the vorticity is zero everywhere except at the origin, where it cannot
be determined from this expression. However, the vorticity at the origin can be determined by considering the circulation around a circuit enclosing the origin. Around a
contour of radius r, the circulation is
2π
Ŵ=
uθ r dθ = 2π C.
ωz =
0
This shows that Ŵ is constant, independent of the radius. (Compare this with the case
of solid-body rotation, for which equation (3.30) shows that Ŵ is proportional to r 2 .)
In fact, the circulation around a circuit of any shape that encloses the origin is 2π C.
Now consider the implication of Stokes’ theorem
Ŵ=
ω • dA,
(3.33)
A
for a contour enclosing the origin. The left-hand side of equation (3.32) is nonzero,
which implies that ω must be nonzero somewhere within the area enclosed by the
contour. Because Ŵ in this flow is independent of r, we can shrink the contour without
altering the left-hand side of equation (3.32). In the limit the area approaches zero, so
that the vorticity at the origin must be infinite in order that ω • δA may have a finite
nonzero limit at the origin. We have therefore demonstrated that the flow represented
by uθ = C/r is irrotational everywhere except at the origin, where the vorticity is
infinite. Such a flow is called an irrotational or potential vortex.
Although the circulation around a circuit containing the origin in an irrotational
vortex is nonzero, that around a circuit not containing the origin is zero. The circulation
around any such contour ABCD (Figure 3.16) is
•
•
•
ŴABCD =
u • ds.
u ds +
u ds +
u ds +
AB
BC
CD
DA
Because the line integrals of u • ds around BC and DA are zero, we obtain
ŴABCD = −uθ r θ + (uθ + uθ )(r + r) θ = 0,
where we have noted that the line integral along AB is negative because u and ds
are oppositely directed, and we have used uθ r = const. A zero circulation around
ABCD is expected because of Stokes’ theorem, and the fact that vorticity vanishes
everywhere within ABCD.
12. One-, Two-, and Three-Dimensional Flows
Figure 3.16 Irrotational vortex. Vorticity of a fluid element is infinite at the origin and zero everywhere else.
Rankine Vortex
Real vortices, such as a bathtub vortex or an atmospheric cyclone, have a core
that rotates nearly like a solid body and an approximately irrotational far field
(Figure 3.17a). A rotational core must exist because the tangential velocity in an
irrotational vortex has an infinite velocity jump at the origin. An idealization of such
a behavior is called the Rankine vortex, in which the vorticity is assumed uniform
within a core of radius R and zero outside the core (Figure 3.17b).
12. One-, Two-, and Three-Dimensional Flows
A truly one-dimensional flow is one in which all flow characteristics vary in one
direction only. Few real flows are strictly one dimensional. Consider the flow in a
conduit (Figure 3.18a). The flow characteristics here vary both along the direction
of flow and over the cross section. However, for some purposes, the analysis can
be simplified by assuming that the flow variables are uniform over the cross section
(Figure 3.18b). Such a simplification is called a one-dimensional approximation, and
is satisfactory if one is interested in the overall effects at a cross section.
A two-dimensional or plane flow is one in which the variation of flow characteristics occurs in two Cartesian directions only. The flow past a cylinder of arbitrary
cross section and infinite length is an example of plane flow. (Note that in this context
the word “cylinder” is used for describing any body whose shape is invariant along the
length of the body. It can have an arbitrary cross section. A cylinder with a circular
cross section is a special case. Sometimes, however, the word “cylinder” is used to
describe circular cylinders only.)
Around bodies of revolution, the flow variables are identical in planes containing
the axis of the body. Using cylindrical polar coordinates (R, ϕ, x), with x along the
axis of the body, only two coordinates (R and x) are necessary to describe motion
71
72
Kinematics
Figure 3.17 Velocity and vorticity distributions in a real vortex and a Rankine vortex: (a) real vortex;
(b) Rankine vortex.
Figure 3.18 Flow through a conduit and its one-dimensional approximation: (a) real flow; (b)
one-dimensional approximation.
73
13. The Streamfunction
(see Figure 6.27). The flow could therefore be called “two dimensional” (although not
plane), but it is customary to describe such motions as three-dimensional axisymmetric
flows.
13. The Streamfunction
The description of incompressible two-dimensional flows can be considerably simplified by defining a function that satisfies the law of conservation of mass for such
flows. Although the conservation laws are derived in the following chapter, a simple
and alternative derivation of the mass conservation equation is given here. We proceed
from the volumetric strain rate given in (3.9), namely,
∂ui
1 D
(δ ᐂ) =
.
δ ᐂ Dt
∂xi
The D/Dt signifies that a specific fluid particle is followed, so that the volume of a
particle is inversely proportional to its density. Substituting δ ᐂ ∝ ρ −1 , we obtain
−
∂ui
1 Dρ
.
=
ρ Dt
∂xi
(3.34)
This is called the continuity equation because it assumes that the fluid flow has no
voids in it; the name is somewhat misleading because all laws of continuum mechanics
make this assumption.
The density of fluid particles does not change appreciably along the fluid path
under certain conditions, the most important of which is that the flow speed should be
small compared with the speed of sound in the medium. This is called the Boussinesq
approximation and is discussed in more detail in Chapter 4, Section 18. The condition
holds in most flows of liquids, and in flows of gases in which the speeds are less than
about 100 m/s. In these flows ρ −1 Dρ/Dt is much less than any of the derivatives in
∂ui /∂xi , under which condition the continuity equation (steady or unsteady) becomes
∂ui
= 0.
∂xi
In many cases the continuity equation consists of two terms only, say
∂u ∂v
+
= 0.
∂x
∂y
(3.35)
This happens if w is not a function of z. A plane flow with w = 0 is the most
common example of such two-dimensional flows. If a function ψ(x, y, t) is now
defined such that
∂ψ
,
u≡
∂y
(3.36)
∂ψ
v≡−
,
∂x
74
Kinematics
then equation (3.34) is automatically satisfied. Therefore, a streamfunction ψ can be
defined whenever equation (3.34) is valid. (A similar streamfunction can be defined
for incompressible axisymmetric flows in which the continuity equation involves R
and x coordinates only; for compressible flows a streamfunction can be defined if the
motion is two dimensional and steady (Exercise 2).)
The streamlines of the flow are given by
dy
dx
=
.
u
v
(3.37)
Substitution of equation (3.35) into equation (3.36) shows
∂ψ
∂ψ
dx +
dy = 0,
∂x
∂y
which says that dψ = 0 along a streamline. The instantaneous streamlines in a flow
are therefore given by the curves ψ = const., a different value of the constant giving
a different streamline (Figure 3.19).
Consider an arbitrary line element dx = (dx, dy) in the flow of Figure 3.19.
Here we have shown a case in which both dx and dy are positive. The volume rate
of flow across such a line element is
v dx + (−u) dy = −
∂ψ
∂ψ
dx −
dy = −dψ,
∂x
∂y
showing that the volume flow rate between a pair of streamlines is numerically equal
to the difference in their ψ values. The sign of ψ is such that, facing the direction
of motion, ψ increases to the left. This can also be seen from the definition equation
Figure 3.19 Flow through a pair of streamlines.
75
14. Polar Coordinates
(3.35), according to which the derivative of ψ in a certain direction gives the velocity
component in a direction 90◦ clockwise from the direction of differentiation. This
requires that ψ in Figure 3.19 must increase downward if the flow is from right
to left.
One purpose of defining a streamfunction is to be able to plot streamlines. A more
theoretical reason, however, is that it decreases the number of simultaneous equations
to be solved. For example, it will be shown in Chapter 10 that the momentum and
mass conservation equations for viscous flows near a planar solid boundary are given,
respectively, by
u
∂u
∂u
∂ 2u
+v
= ν 2,
∂x
∂y
∂y
∂u ∂v
+
= 0.
∂x
∂y
(3.38)
(3.39)
The pair of simultaneous equations in u and v can be combined into a single equation
by defining a streamfunction, when the momentum equation (3.37) becomes
∂ψ ∂ 2 ψ
∂ 3ψ
∂ψ ∂ 2 ψ
=
ν
.
−
∂y ∂x ∂y
∂x ∂y 2
∂y 3
We now have a single unknown function and a single differential equation. The
continuity equation (3.38) has been satisfied automatically.
Summarizing, a streamfunction can be defined whenever the continuity equation
consists of two terms. The flow can otherwise be completely general, for example, it
can be rotational, viscous, and so on. The lines ψ=C are the instantaneous streamlines,
and the flow rate between two streamlines equals dψ. This concept will be generalized
following our derivation of mass conservation in Chapter 4, Section 3.
14. Polar Coordinates
It is sometimes easier to work with polar coordinates, especially in problems involving circular boundaries. In fact, we often select a coordinate system to conform to
the shape of the body (boundary). It is customary to consult a reference source for
expressions of various quantities in non-Cartesian coordinates, and this practice is
perfectly satisfactory. However, it is good to know how an equation can be transformed from Cartesian into other coordinates. Here, we shall illustrate the procedure
by transforming the Laplace equation
∇ 2ψ =
∂ 2ψ
∂ 2ψ
+
,
∂x 2
∂y 2
to plane polar coordinates.
Cartesian and polar coordinates are related by
x = r cos θ
y = r sin θ
θ = tan−1 (y/x),
r = x2 + y2.
(3.40)
76
Kinematics
Let us first determine the polar velocity components in terms of the streamfunction.
Because ψ = f (x, y), and x and y are themselves functions of r and θ , the chain
rule of partial differentiation gives
∂x
∂y
∂ψ
∂ψ
∂ψ
=
+
.
∂r θ
∂x y ∂r θ
∂y x ∂r θ
Omitting parentheses and subscripts, we obtain
∂ψ
∂ψ
∂ψ
=
cos θ +
sin θ = −v cos θ + u sin θ.
∂r
∂x
∂y
(3.41)
Figure 3.20 shows that uθ = v cos θ − u sin θ, so that equation (3.40) implies ∂ψ/∂r
= −uθ . Similarly, we can show that ∂ψ/∂θ = rur . Therefore, the polar velocity
components are related to the streamfunction by
ur =
1 ∂ψ
,
r ∂θ
uθ = −
∂ψ
.
∂r
This is in agreement with our previous observation that the derivative of ψ gives the
velocity component in a direction 90◦ clockwise from the direction of differentiation.
Now let us write the Laplace equation in polar coordinates. The chain rule gives
∂ψ ∂r
∂ψ ∂θ
∂ψ
sin θ ∂ψ
∂ψ
=
+
= cos θ
−
.
∂x
∂r ∂x
∂θ ∂x
∂r
r ∂θ
Figure 3.20 Relation of velocity components in Cartesian and plane polar coordinates.
77
Exercise
Differentiating this with respect to x, and following a similar rule, we obtain
∂
sin θ ∂ψ
sin θ ∂ψ
∂ψ
sin θ ∂
∂ψ
∂ 2ψ
= cos θ
−
−
cos θ
−
cos θ
.
∂r
∂r
r ∂θ
r ∂θ
∂r
r ∂θ
∂x 2
(3.42)
In a similar manner,
cos θ ∂ψ
cos θ ∂ψ
∂ψ
cos θ ∂
∂ψ
∂
∂ 2ψ
+
+
sin
θ
+
sin
θ
.
=
sin
θ
∂r
∂r
r ∂θ
r ∂θ
∂r
r ∂θ
∂y 2
(3.43)
The addition of equations (3.41) and (3.42) leads to
∂ 2ψ
1 ∂
∂ 2ψ
+
=
2
r ∂r
∂x
∂y 2
r
∂ψ
∂r
+
1 ∂ 2ψ
= 0,
r 2 ∂θ 2
which completes the transformation.
Exercises
1. A two-dimensional steady flow has velocity components
u=y
v = x.
Show that the streamlines are rectangular hyperbolas
x 2 − y 2 = const.
Sketch the flow pattern, and convince yourself that it represents an irrotational flow
in a 90◦ corner.
2. Consider a steady axisymmetric flow of a compressible fluid. The equation
of continuity in cylindrical coordinates (R, ϕ, x) is
∂
∂
(ρRuR ) +
(ρRux ) = 0.
∂R
∂x
Show how we can define a streamfunction so that the equation of continuity is satisfied
automatically.
3. If a velocity field is given by u = ay, compute the circulation around a circle
of radius r = 1 about the origin. Check the result by using Stokes’ theorem.
4. Consider a plane Couette flow of a viscous fluid confined between two flat
plates at a distance b apart (see Figure 9.4c). At steady state the velocity distribution is
u = Uy/b
v = w = 0,
78
Kinematics
where the upper plate at y = b is moving parallel to itself at speed U , and the lower
plate is held stationary. Find the rate of linear strain, the rate of shear strain, and
vorticity. Show that the streamfunction is given by
ψ=
Uy 2
+ const.
2b
5. Show that the vorticity for a plane flow on the xy-plane is given by
2
∂ 2ψ
∂ ψ
ωz = −
+
.
∂x 2
∂y 2
Using this expression, find the vorticity for the flow in Exercise 4.
6. The velocity components in an unsteady plane flow are given by
u=
x
1+t
and
v=
2y
.
2+t
Describe the path lines and the streamlines. Note that path lines are found by following
the motion of each particle, that is, by solving the differential equations
dx/dt = u(x, t)
and
dy/dt = v(x, t),
subject to x = x0 at t = 0.
7. Determine an expression for ψ for a Rankine vortex (Figure 3.17b), assuming
that uθ = U at r = R.
8. Take a plane polar element of fluid of dimensions dr and r dθ . Evaluate the
right-hand side of Stokes’ theorem
ω • dA = u • ds,
and thereby show that the expression for vorticity in polar coordinates is
1 ∂
∂ur
ωz =
(ruθ ) −
.
r ∂r
∂θ
Also, find the expressions for ωr and ωθ in polar coordinates in a similar manner.
9. The velocity field of a certain flow is given by
u = 2xy 2 + 2xz2 ,
v = x 2 y,
w = x 2 z.
Consider the fluid region inside a spherical volume x 2 + y 2 + z2 = a 2 . Verify the
validity of Gauss’ theorem
∇ • u dV = u • dA,
by integrating over the sphere.
79
Exercise
10. Show that the vorticity field for any flow satisfies
∇ • ω = 0.
11. A flow field on the xy-plane has the velocity components
u = 3x + y
v = 2x − 3y.
Show that the circulation around the circle (x − 1)2 + (y − 6)2 = 4 is 4π .
12. Consider the solid-body rotation
uθ = ω0 r
ur = 0.
Take a polar element of dimension r dθ and dr, and verify that the circulation is
vorticity times area. (In Section 11 we performed such a verification for a circular
element surrounding the origin.)
13. Using the indicial notation (and without using any vector identity) show that
the acceleration of a fluid particle is given by
1 2
∂u
+∇
q + ω × u,
a=
∂t
2
where q is the magnitude of velocity u and ω is the vorticity.
14. The definition of the streamfunction in vector notation is
u = −k × ∇ψ,
where k is a unit vector perpendicular to the plane of flow. Verify that the vector
definition is equivalent to equations (3.35).
Supplemental Reading
Aris, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, NJ:
Prentice-Hall. (The distinctions among streamlines, path lines, and streak lines in unsteady flows are
explained; with examples.)
Prandtl, L. and O. C. Tietjens (1934). Fundamentals of Hydro- and Aeromechanics, New York: Dover
Publications. (Chapter V contains a simple but useful treatment of kinematics.)
Prandtl, L. and O. G. Tietjens (1934). Applied Hydro- and Aeromechanics, New York: Dover Publications.
(This volume contains classic photographs from Prandtl’s laboratory.)
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Chapter 4
Conservation Laws
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2. Time Derivatives of Volume
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
General Case . . . . . . . . . . . . . . . . . . . . . . . . . 82
Fixed Volume . . . . . . . . . . . . . . . . . . . . . . . . 83
Material Volume. . . . . . . . . . . . . . . . . . . . . . 84
3. Conservation of Mass . . . . . . . . . . . . . . . . . 84
4. Streamfunctions: Revisited and
Generalized . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5. Origin of Forces in Fluid . . . . . . . . . . . . . . 88
6. Stress at a Point . . . . . . . . . . . . . . . . . . . . . . 90
7. Conservation of Momentum . . . . . . . . . . . 92
8. Momentum Principle for a Fixed
Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 95
9. Angular Momentum Principle for a
Fixed Volume . . . . . . . . . . . . . . . . . . . . . . . . . 98
Example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 99
10. Constitutive Equation for Newtonian
Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Non-Newtonian Fluids . . . . . . . . . . . . . . 103
11. Navier–Stokes Equation . . . . . . . . . . . . . 104
Comments on the Viscous Term . . . . . . 105
12. Rotating Frame . . . . . . . . . . . . . . . . . . . . . 105
Effect of Centrifugal Force . . . . . . . . . . 109
Effect of Coriolis Force . . . . . . . . . . . . . . 109
13. Mechanical Energy Equation . . . . . . . . 111
Concept of Deformation Work and
Viscous Dissipation . . . . . . . . . . . . . . . 112
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50004-6
14.
15.
16.
17.
18.
19.
Equation in Terms of Potential
Energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equation for a Fixed Region . . . . . . . . .
First Law of Thermodynamics:
Thermal Energy Equation . . . . . . . . . . .
Second Law of Thermodynamics:
Entropy Production . . . . . . . . . . . . . . . . .
Bernoulli Equation . . . . . . . . . . . . . . . . . .
Steady Flow . . . . . . . . . . . . . . . . . . . . . . . .
Unsteady Irrotational Flow . . . . . . . . . .
Energy Bernoulli Equation . . . . . . . . . .
Applications of Bernoulli’s
Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pitot Tube . . . . . . . . . . . . . . . . . . . . . . . . . .
Orifice in a Tank . . . . . . . . . . . . . . . . . . . .
Boussinesq Approximation. . . . . . . . . . .
Continuity Equation . . . . . . . . . . . . . . . .
Momentum Equation . . . . . . . . . . . . . . . .
Heat Equation . . . . . . . . . . . . . . . . . . . . . .
Boundary Conditions . . . . . . . . . . . . . . . .
Boundary Condition at a moving,
deforming surface . . . . . . . . . . . . . . . . .
Surface tension revisited:
generalized discussion . . . . . . . . . . . . .
Example 4.3 . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . .
113
114
115
116
118
119
121
121
122
122
123
124
125
126
127
129
130
130
133
134
136
137
81
82
Conservation Laws
1. Introduction
All fluid mechanics is based on the conservation laws for mass, momentum, and
energy. These laws can be stated in the differential form, applicable at a point. They
can also be stated in the integral form, applicable to an extended region. In the integral
form, the expressions of the laws depend on whether they relate to a volume fixed in
space, or to a material volume, which consists of the same fluid particles and whose
bounding surface moves with the fluid. Both types of volumes will be considered
in this chapter; a fixed region will be denoted by V and a material volume will be
denoted by ᐂ. In engineering literature a fixed region is called a control volume,
whose surfaces are called control surfaces.
The integral and differential forms can be derived from each other. As we shall
see, during the derivation surface integrals frequently need to be converted to volume
integrals (or vice versa) by means of the divergence theorem of Gauss
∂F
dAi F,
(4.1)
dV =
V ∂xi
A
where F (x, t) is a tensor of any rank (including vectors and scalars), V is either a
fixed volume or a material volume, and A is its boundary surface. Gauss’ theorem
was presented in Section 2.13.
2. Time Derivatives of Volume Integrals
In deriving the conservation laws, one frequently faces the problem of finding the
time derivative of integrals such as
d
F dV ,
dt V (t)
where F (x, t) is a tensor of any order, and V (t) is any region, which may be fixed or
move with the fluid. The d/dt sign (in contrast to ∂/∂t) has been written because only
a function of time remains after performing the integration in space. The different
possibilities are discussed in what follows.
General Case
Consider the general case in which V (t) is neither a fixed volume nor a material
volume. The surfaces of the volume are moving, but not with the local fluid velocity. The rule for differentiating an integral becomes clear at once if we consider a
one-dimensional (1D) analogy. In books on calculus,
b
b(t)
d
∂F
db
da
dx +
F (b, t) −
F (a, t).
(4.2)
F (x, t) dx =
dt x=a(t)
dt
dt
a ∂t
This is called the Leibniz theorem, and shows how to differentiate an integral whose
integrand F as well as the limits of integration are functions of the variable with
respect to which we are differentiating. A graphical illustration of the three terms on
83
2. Time Derivatives of Volume Integrals
Figure 4.1
Graphical illustration of Leibniz’s theorem.
the right-hand sideof the Leibniz theorem is shown in Figure 4.1. The continuous line
shows the integral F dx at time t, and the dashed line shows the integral at time t +dt.
The first term on the right-hand side in equation (4.2) is the integral of ∂F /∂t over the
region, the second term is due to the gain of F at the outer boundary moving at a rate
db/dt, and the third term is due to the loss of F at the inner boundary moving at da/dt.
Generalizing the Leibniz theorem, we write
d
dt
V (t)
F (x, t) dV =
V (t)
∂F
dV +
∂t
A(t)
dA • uA F,
(4.3)
where uA is the velocity of the boundary and A(t) is the surface of V (t). The surface
integral in equation (4.3) accounts for both “inlets” and “outlets,” so that separate
terms as in equation (4.2) are not necessary.
Fixed Volume
For a fixed volume we have uA = 0, for which equation (4.3) becomes
d
dt
V
F (x, t) dV =
V
∂F
dV ,
∂t
(4.4)
which shows that the time derivative can be simply taken inside the integral sign if
the boundary is fixed. This merely reflects the fact that the “limit of integration” V is
not a function of time in this case.
84
Conservation Laws
Material Volume
For a material volume ᐂ(t) the surfaces move with the fluid, so that uA = u, where
u is the fluid velocity. Then equation (4.3) becomes
D
Dt
ᐂ
F (x, t) d ᐂ =
ᐂ
∂F
dᐂ +
∂t
A
dA • uF.
(4.5)
This is sometimes called the Reynolds transport theorem. Although not necessary,
we have used the D/Dt symbol here to emphasize that we are following a material
volume.
Another form of the transport theorem is derived by using the mass conservation
relation equation (3.32) derived in the last chapter. Using Gauss’ theorem, the transport
theorem equation (4.5) becomes
D
∂F
∂
F dᐂ =
+
(F uj ) d ᐂ.
Dt ᐂ
∂t
∂xj
ᐂ
Now define a new function f such that F ≡ ρf , where ρ is the fluid density. Then
the preceding becomes
∂
∂(ρf )
D
+
(ρf uj ) d ᐂ
ρf d ᐂ =
Dt
∂t
∂xj
∂f
∂f
∂ρ
∂
ρ
=
d ᐂ.
+f
+f
(ρuj ) + ρuj
∂t
∂t
∂xj
∂xj
Using the continuity equation
∂
∂ρ
+
(ρuj ) = 0.
∂t
∂xj
we finally obtain
D
Dt
ᐂ
ρf d ᐂ =
ᐂ
ρ
Df
d ᐂ.
Dt
(4.6)
Notice that the D/Dt operates only on f on the right-hand side, although ρ is variable.
Applications of this rule can be found in Sections 7 and 14.
3. Conservation of Mass
The differential form of the law of conservation of mass was derived in Chapter 3,
Section 13 from a consideration of the volumetric rate of strain of a particle. In this
chapter we shall adopt an alternative approach. We shall first state the principle in
an integral form for a fixed region and then deduce the differential form. Consider a
85
3. Conservation of Mass
Figure 4.2
Mass conservation of a volume fixed in space.
volume fixed in space (Figure 4.2). The rate of increase of mass inside it is the volume
integral
d
∂ρ
ρ dV =
dV .
dt V
V ∂t
The time derivative has been taken inside the integral on the right-hand side because
the volume is fixed and equation (4.4) applies. Now the rate of mass flow out of the
volume is the surface integral
ρu • dA,
A
because ρu • dA is the outward flux through an area element dA. (Throughout the
book, we shall write dA for n dA, where n is the unit outward normal to the surface.
Vector dA therefore has a magnitude dA and a direction along the outward normal.)
The law of conservation of mass states that the rate of increase of mass within a fixed
volume must equal the rate of inflow through the boundaries. Therefore,
∂ρ
(4.7)
dV = − ρu • dA,
V ∂t
A
which is the integral form of the law for a volume fixed in space.
The differential form can be obtained by transforming the surface integral on the
right-hand side of equation (4.7) to a volume integral by means of the divergence
theorem, which gives
∇ • (ρu) dV .
ρu • dA =
A
V
86
Conservation Laws
Equation (4.7) then becomes
V
∂ρ
+ ∇ • (ρu) dV = 0.
∂t
The forementioned relation holds for any volume, which can be possible only if the
integrand vanishes at every point. (If the integrand did not vanish at every point, then
we could choose a small volume around that point and obtain a nonzero integral.)
This requires
∂ρ
+ ∇ • (ρu) = 0,
∂t
(4.8)
which is called the continuity equation and expresses the differential form of the
principle of conservation of mass.
The equation can be written in several other forms. Rewriting the divergence
term in equation (4.8) as
∂
∂ρ
∂ui
(ρui ) = ρ
+ ui
,
∂xi
∂xi
∂xi
the equation of continuity becomes
1 Dρ
+ ∇ • u = 0.
ρ Dt
(4.9)
The derivative Dρ/Dt is the rate of change of density following a fluid particle; it
can be nonzero because of changes in pressure, temperature, or composition (such
as salinity in sea water). A fluid is usually called incompressible if its density does
not change with pressure. Liquids are almost incompressible. Although gases are
compressible, for speeds 100 m/s (that is, for Mach numbers <0.3) the fractional
change of absolute pressure in the flow is small. In this and several other cases
the density changes in the flow are also small. The neglect of ρ −1 Dρ/Dt in the
continuity equation is part of a series of simplifications grouped under the Boussinesq
approximation, discussed in Section 18. In such a case the continuity equation (4.9)
reduces to the incompressible form
∇ • u = 0,
whether or not the flow is steady.
(4.10)
87
4. Streamfunctions: Revisited and Generalized
4. Streamfunctions: Revisited and Generalized
Consider the steady-state form of mass conservation from equation (4.8),
∇ · (ρu) = 0.
(4.11)
In Exercise 10 of Chapter 2 we showed that the divergence of the curl of any vector
field is identically zero. Thus we can represent the mass flow vector as the curl of a
vector potential
ρu = ∇ × ,
(4.12)
where we can write = χ ∇ψ + ∇φ in terms of three scalar functions. We are
concerned with the mass flux field ρu = ∇χ × ∇ψ because the curl of any gradient
is identically zero (Chapter 2, Exercise 11). The gradients of the surfaces χ = const.
and ψ = const. are in the directions of the surface normals. Thus the cross product is
perpendicular to both normals and must lie simultaneously in both surfaces χ = const.
and ψ = const. Thus streamlines are the intersections of the two surfaces, called
streamsurfaces or streamfunctions in a three-dimensional (3D) flow. Consider an
edge view of two members of each of the families of the two streamfunctions χ = a,
χ = b, ψ = c, ψ = d. The intersections shown as darkened dots in Figure 4.3 are
the streamlines coming out of the paper. We calculate the mass per time through a
surface A bounded by the four streamfunctions with element dA having n out of the
paper. By Stokes’ theorem,
· ds = (χ∇ψ + ∇φ) · ds
ρu · dA = (∇ × ) · dA =
ṁ =
C
C
A
A
χ dψ = b(d − c) + a(c − d) = (b − a)(d − c).
= (χ dψ + dφ) =
C
C
Here we have used the vector identity ∇φ • ds = dφ and recognized that integration
around a closed path of a single-valued function results in zero. The mass per time
through a surface bounded by adjacent members of the two families of streamfunctions is just the product of the differences of the numerical values of the respective
streamfunctions. As a very simple special case, consider flow in a z = constant plane
Figure 4.3 Edge view of two members of each of two families of streamfunctions. Contour C is the
boundary of surface area A : C = ∂A.
88
Conservation Laws
(described by x and y coordinates). Because all the streamlines lie in z = constant
planes, z is a streamfunction. Define χ = −z, where the sign is chosen to obey the
usual convention. Then ∇χ = −k (unit vector in the z direction), and
ρu = −k × ∇ψ;
ρu = ∂ψ/∂y,
ρv = −∂ψ/∂x,
in conformity with Chapter 3, Exercise 14.
Similarly, in cyclindrical polar coordinates as shown in Figure 3.1, flows, symmetric with respect to rotation about the x-axis, that is, those for which ∂/∂φ = 0,
have streamlines in φ = constant planes (through the x-axis). For those axisymmetric
flows, χ = −φ is one streamfunction:
1
ρu = − iφ × ∇ψ,
R
then gives ρRux = ∂ψ/∂R, ρRuR = −∂ψ/∂x. We note here that if the density may
be taken as a constant, mass conservation reduces to ∇ • u = 0 (steady or not) and
the entire preceding discussion follows for u rather than ρu with the interpretation of
streamfunction in terms of volumetric rather than mass flux.
5. Origin of Forces in Fluid
Before we can proceed further with the conservation laws, it is necessary to classify
the various types of forces on a fluid mass. The forces acting on a fluid element can
be divided conveniently into three classes, namely, body forces, surface forces, and
line forces. These are described as follows:
(1) Body forces: Body forces are those that arise from “action at a distance,” without physical contact. They result from the medium being placed in a certain
force field, which can be gravitational, magnetic, electrostatic, or electromagnetic in origin. They are distributed throughout the mass of the fluid and are
proportional to the mass. Body forces are expressed either per unit mass or per
unit volume. In this book, the body force per unit mass will be denoted by g.
Body forces can be conservative or nonconservative. Conservative body
forces are those that can be expressed as the gradient of a potential function:
g = −∇,
(4.13)
where is called the force potential. All forces directed centrally from a source
are conservative. Gravity, electrostatic and magnetic forces are conservative.
For example, the gravity force can be written as the gradient of the potential
function
= gz,
5. Origin of Forces in Fluid
where g is the acceleration due to gravity and z points vertically upward. To
verify this, equation (4.13) gives
∂
∂
∂
+j
+k
(gz) = −kg,
g = −∇(gz) = − i
∂x
∂y
∂z
which is the gravity force per unit mass. (Here we have changed our usual
convention for unit vectors and used the more standard form.) The negative
sign in front of kg ensures that g is downward, along the negative z direction.
The expression = gz also shows that the force potential equals the potential
energy per unit mass. Forces satisfying equation (4.13) are called “conservative” because the resulting motion conserves the sum of kinetic and potential
energies, if there are no dissipative processes. Conservative forces also satisfy
the property that the work done is independent of the path.
(2) Surface forces: Surface forces are those that are exerted on an area element by
the surroundings through direct contact. They are proportional to the extent
of the area and are conveniently expressed per unit of area. Surface forces can
be resolved into components normal and tangential to the area. Consider an
element of area dA in a fluid (Figure 4.4). The force dF on the element can
be resolved into a component dFn normal to the area and a component dFs
tangential to the area. The normal and shear stress on the element are defined,
respectively as,
dFs
dFn
τs ≡
.
τn ≡
dA
dA
These are scalar definitions of stress components. Note that the component of
force tangential to the surface is a two-dimensional (2D) vector in the surface.
The state of stress at a point is, in fact, specified by a stress tensor, which has
nine components. This was explained in Section 2.4 and is again discussed in
the following section.
(3) Line forces: Surface tension forces are called line forces because they act along
a line (Figure 1.4) and have a magnitude proportional to the extent of the line.
They appear at the interface between a liquid and a gas, or at the interface
between two immiscible liquids. Surface tension forces do not appear directly
in the equations of motion, but enter only in the boundary conditions.
Figure 4.4
Normal and shear forces on an area element.
89
90
Conservation Laws
6. Stress at a Point
It was explained in Chapter 2, Section 4 that the stress at a point can be completely
specified by the nine components of the stress tensor τ. Consider an infinitesimal rectangular parallelepiped with faces perpendicular to the coordinate axes (Figure 4.5).
On each face there is a normal stress and a shear stress, which can be further resolved
into two components in the directions of the axes. The figure shows the directions of
positive stresses on four of the six faces; those on the remaining two faces are omitted
for clarity. The first index of τij indicates the direction of the normal to the surface on
which the stress is considered, and the second index indicates the direction in which the
stress acts. The diagonal elements τ11 , τ22 , and τ33 of the stress matrix are the normal
stresses, and the off-diagonal elements are the tangential or shear stresses. Although
a cube is shown, the figure really shows the stresses on four of the six orthogonal
planes passing through a point; the cube may be imagined to shrink to a point.
We shall now prove that the stress tensor is symmetric. Consider the torque on
an element about a centroid axis parallel to x3 (Figure 4.6). This torque is generated
only by the shear stresses in the x1 x2 -plane and is (assuming dx3 = 1)
1 ∂τ12
dx1
dx1
1 ∂τ12
dx1 dx2
+ τ12 −
dx1 dx2
T = τ12 +
2 ∂x1
2
2 ∂x1
2
1 ∂τ21
1 ∂τ21
dx2
dx2
− τ21 +
dx2 dx1
− τ21 −
dx2 dx1
.
2 ∂x2
2
2 ∂x2
2
Figure 4.5 Stress at a point. For clarity, components on only four of the six faces are shown.
91
6. Stress at a Point
Figure 4.6
Torque on an element.
After canceling terms, this gives
T = (τ12 − τ21 ) dx1 dx2 .
The rotational equilibrium of the element requires that T = I ω̇3 , where ω̇3 is the
angular acceleration of the element and I is its moment of inertia. For the rectangular element considered, it is easy to show that I = dx1 dx2 (dx12 + dx22 )ρ/12. The
rotational equilibrium then requires
(τ12 − τ21 ) dx1 dx2 =
that is,
τ12 − τ21 =
ρ
dx1 dx2 (dx12 + dx22 ) ω̇3 ,
12
ρ
(dx12 + dx22 ) ω̇3 .
12
As dx1 and dx2 go to zero, the preceding condition can be satisfied only if τ12 = τ21 .
In general,
τij = τj i .
(4.14)
See Exercise 3 at the end of the chapter.
The stress tensor is therefore symmetric and has only six independent components. The symmetry, however, is violated if there are “body couples” proportional to
the mass of the fluid element, such as those exerted by an electric field on polarized
fluid molecules. Antisymmetric stresses must be included in such fluids.
92
Conservation Laws
7. Conservation of Momentum
In this section the law of conservation of momentum will be expressed in the differential form directly by applying Newton’s law of motion to an infinitesimal fluid
element. We shall then show how the differential form could be derived by starting
from an integral form of Newton’s law.
Consider the motion of the infinitesimal fluid element shown in Figure 4.7.
Newton’s law requires that the net force on the element must equal mass times the
acceleration of the element. The sum of the surface forces in the x1 direction equals
∂τ11 dx1
∂τ11 dx1
− τ11 +
dx2 dx3
τ11 +
∂x1 2
∂x1 2
∂τ21 dx2
∂τ21 dx2
− τ21 +
dx1 dx3
+ τ21 +
∂x2 2
∂x2 2
∂τ31 dx3
∂τ31 dx3
− τ31 +
dx1 dx2 ,
+ τ31 +
∂x3 2
∂x3 2
which simplifies to
∂τ21
∂τ31
∂τ11
+
+
∂x1
∂x2
∂x3
dx1 dx2 dx3 =
∂τj 1
d ᐂ,
∂xj
Figure 4.7 Surface stresses on an element moving with the flow. Only stresses in the x1 direction are
labeled.
93
8. Momentum Principle for a Fixed Volume
where d ᐂ is the volume of the element. Generalizing, the i-component of the surface
force per unit volume of the element is
∂τij
,
∂xj
where we have used the symmetry property τij = τj i . Let g be the body force per
unit mass, so that ρg is the body force per unit volume. Then Newton’s law gives
ρ
∂τij
Dui
= ρgi +
.
Dt
∂xj
(4.15)
This is the equation of motion relating acceleration to the net force at a point and
holds for any continuum, solid or fluid, no matter how the stress tensor τij is related
to the deformation field. Equation (4.15) is sometimes called Cauchy’s equation of
motion.
We shall now deduce Cauchy’s equation starting from an integral statement of
Newton’s law for a material volume ᐂ. In this case we do not have to consider the
internal stresses within the fluid, but only the surface forces at the boundary of the
volume (along with body forces). It was shown in Chapter 2, Section 6 that the surface
force per unit area is n • τ, where n is the unit outward normal. The surface force on an
area element dA is therefore dA • τ. Newton’s law for a material volume ᐂ requires
that the rate of change of its momentum equals the sum of body forces throughout
the volume, plus the surface forces at the boundary. Therefore
D
Dui
τij dAj ,
(4.16)
ρgi d ᐂ +
ρui d ᐂ =
dᐂ =
ρ
Dt ᐂ
Dt
A
ᐂ
ᐂ
where equations (4.6) and (4.14) have been used. Transforming the surface integral
to a volume integral, equation (4.16) becomes
∂τij
Dui
ρ
d ᐂ = 0.
− ρgi −
Dt
∂xj
As this holds for any volume, the integrand must vanish at every point and therefore
equation (4.15) must hold. We have therefore derived the differential form of the
equation of motion, starting from an integral form.
8. Momentum Principle for a Fixed Volume
In the preceding section the momentum principle was applied to a material volume
of finite size and this led to equation (4.16). In this section the form of the law will be
derived for a fixed region in space. It is easy to do this by starting from the differential
form (4.15) and integrating over a fixed volume V . Adding ui times the continuity
equation
94
Conservation Laws
∂
∂ρ
+
(ρuj ) = 0,
∂t
∂xj
to the left-hand side of equation (4.15), we obtain
∂τij
∂
∂
(ρui ) +
(ρui uj ) = ρgi +
.
∂t
∂xj
∂xj
(4.17)
Each term of equation (4.17) is now integrated over a fixed region V . The time
derivative term gives
d
dMi
∂(ρui )
dV =
ρui dV =
,
(4.18)
∂t
dt V
dt
V
where
Mi ≡
ρui dV ,
V
is the momentum of the fluid inside the volume. The volume integral of the second
term in equation (4.17) becomes, after applying Gauss’ theorem,
∂
ρui uj dAj ≡ Ṁiout ,
(4.19)
(ρui uj ) dV =
V ∂xj
A
where Ṁiout is the net rate of outflux of i-momentum. (Here ρuj dAj is the mass
outflux through an area element dA on the boundary. Outflux of momentum is defined
as the outflux of mass times the velocity.) The volume integral of the third term in
equation (4.17) is simply
(4.20)
ρgi dV = Fbi ,
where Fb is the net body force acting over the entire volume. The volume integral of
the fourth term in equation (4.17) gives, after applying Gauss’ theorem,
∂τij
τij dAj ≡ Fsi ,
(4.21)
dV =
A
V ∂xj
where Fs is the net surface force at the boundary of V . If we define F = Fb + Fs
as the sum of all forces, then the volume integral of equation (4.17) finally
gives
F=
dM
out
+ Ṁ ,
dt
where equations (4.18)–(4.21) have been used.
(4.22)
95
8. Momentum Principle for a Fixed Volume
Equation (4.22) is the law of conservation of momentum for a fixed volume. It
states that the net force on a fixed volume equals the rate of change of momentum
within the volume, plus the net outflux of momentum through the surfaces. The
equation has three independent components, where the x-component is
Fx =
dMx
+ Ṁxout .
dt
The momentum principle (frequently called the momentum theorem) has wide application, especially in engineering. An example is given in what follows. More illustrations can be found throughout the book, for example, in Chapter 9, Section 4,
Chapter 10, Section 11, Chapter 13, Section 10, and Chapter 16, Sections 2 and 3.
Example 4.1. Consider an experiment in which the drag on a 2D body immersed
in a steady incompressible flow can be determined from measurement of the velocity
distributions far upstream and downstream of the body (Figure 4.8). Velocity far
upstream is the uniform flow U∞ , and that in the wake of the body is measured to be
u(y), which is less than U∞ due to the drag of the body. Find the drag force D per
unit length of the body.
Solution: The wake velocity u(y) is less than U∞ due to the drag forces exerted
by the body on the fluid. To analyze the flow, take a fixed volume shown by the dashed
lines in Figure 4.8. It consists of the rectangular region PQRS and has a hole in the
center coinciding with the surface of the body. The sides PQ and SR are chosen far
enough from the body so that the pressure nearly equals the undisturbed pressure p∞ .
The side QR at which the velocity profile is measured is also at a far enough distance
for the streamlines to be nearly parallel; the pressure variation across the wake is
therefore small, so that it is nearly equal to the undisturbed pressure p∞ . The surface
forces on PQRS therefore cancel out, and the only force acting at the boundary of the
chosen fixed volume is D, the force exerted by the body at the central hole.
For steady flow, the x-component of the momentum principle (4.22) reduces to
D = Ṁ out ,
Figure 4.8
Momentum balance of flow over a body (Example 4.1).
(4.23)
96
Conservation Laws
where Ṁ out is the net outflow rate of x-momentum through the boundaries of the
region. There is no flow of momentum through the central hole in Figure 4.8. Outflow
rates of x-momentum through PS and QR are
Ṁ PS = −
Ṁ QR =
b
−b
2
U∞ (ρU∞ dy) = −2bρU∞
,
b
−b
u(ρu dy) = ρ
b
u2 dy.
(4.24)
(4.25)
−b
An important point is that there is an outflow of mass and x-momentum through PQ
and SR. A mass flux through PQ and SR is required because the velocity across QR
is less than that across PS. Conservation of mass requires that the inflow through
PS, equal to 2bρU∞ , must balance the outflows through PQ, SR, and QR. This
gives
b
PQ
SR
u dy,
2bρU∞ = ṁ + ṁ + ρ
−b
where ṁPQ and ṁSR are the outflow rates of mass through the sides. The mass balance
can be written as
ṁPQ + ṁSR = ρ
b
−b
(U∞ − u) dy.
Outflow rate of x-momentum through PQ and SR is therefore
Ṁ PQ + Ṁ SR = ρU∞
b
−b
(U∞ − u) dy,
(4.26)
because the x-directional velocity at these surfaces is nearly U∞ . Combining equations (4.22)–(4.26) gives a net outflow of x-momentum of:
Ṁ out = Ṁ PS + Ṁ QR + Ṁ PQ + Ṁ SR = −ρ
b
−b
u(U∞ − u) dy.
The momentum balance (4.23) now shows that the body exerts a force on the fluid in
the negative x direction of magnitude
D=ρ
b
−b
u(U∞ − u) dy,
which can be evaluated from the measured velocity profile.
A more general way of obtaining the force on a body immersed in a flow is by using
the Euler momentum integral, which we derive in what follows. We must assume that
the flow is steady and body forces are absent. Then integrating (4.17) over a fixed
volume gives
97
8. Momentum Principle for a Fixed Volume
V
∇ · (ρuu − τ)dV =
A
(ρuu − τ) · dA,
(4.27)
where A is the closed surface bounding V . This volume V contains only fluid particles.
Imagine a body immersed in a flow and surround that body with a closed surface. We
seek to calculate the force on the body by an integral over a possibly distant surface.
In order to apply (4.27), A must bound a volume containing only fluid particles. This
is accomplished by considering A to be composed of three parts (see Figure 4.9),
A = A 1 + A2 + A3 .
Here A1 is the outer surface, A2 is wrapped around the body like a tight-fitting rubber
glove with dA2 pointing outwards from the fluid volume and, therefore, into the body,
and A3 is the connection surface between the outer A1 and the inner A2 . Now
(ρuu − τ) · dA3 → 0
as A3 → 0,
A3
because it may be taken as the bounding surface of an evanescent thread. On the
•
surface
of a solid body, u dA2 = 0 because no mass enters or leaves the surface.
Here A2 τ · dA2 is the force the body exerts on the fluid from our definition of τ.
Then the force the fluid exerts on the body is
τ · dA2 = −
(ρuu − τ) · dA1 .
(4.28)
FB = −
A2
A1
Using similar arguments, mass conservation can be written in the form
ρu · dA1 = 0.
(4.29)
A1
Equations (4.28) and (4.29) can be used to solve Example 4.1. Of course, the same
final result is obtained when τ ≈ constant pressure on all of A1 , ρ = constant, and
the x component of u = U∞ i on segments PQ and SR of A1 .
Figure 4.9
Surfaces of integration for the Euler momentum integral.
98
Conservation Laws
9. Angular Momentum Principle for a Fixed Volume
In mechanics of solids it is shown that
T=
dH
,
dt
(4.30)
where T is the torque of all external forces on the body about any chosen axis, and
dH/dt is the rate of change of angular momentum of the body about the same axis.
The angular momentum is defined as the “moment of momentum,” that is
H ≡ r × u dm,
where dm is an element of mass, and r is the position vector from the chosen axis
(Figure 4.10). The angular momentum principle is not a separate law, but can be
derived from Newton’s law by performing a cross product with r. It can be shown
that equation (4.30) also holds for a material volume in a fluid. When equation (4.30)
is transformed to apply to a fixed volume, the result is
T=
dH
+ Ḣout ,
dt
where
T=
H=
Ḣout =
A
V
A
r × (τ · dA) +
V
r × (ρg dV ),
r × (ρu dV ),
r × [(ρu · dA)u].
Figure 4.10 Definition sketch for angular momentum theorem.
(4.31)
9. Angular Momentum Principle for a Fixed Volume
Here T represents the sum of torques due to surface and body forces, τ • dA is
the surface force on a boundary element, and ρgdV is the body force acting on
an interior element. Vector H represents the angular momentum of fluid inside the
fixed volume because ρudV is the momentum of a volume element. Finally, Ḣout
is the rate of outflow of angular momentum through the boundary, ρu • dA is the
mass flow rate, and (ρu • dA)u is the momentum outflow rate through a boundary
element dA.
The angular momentum principle (4.31) is analogous to the linear momentum
principle (4.22), and is very useful in investigating rotating fluid systems such as
turbomachines, fluid couplings, and even lawn sprinklers.
Example 4.2. Consider a lawn sprinkler as shown in Figure 4.11. The area of the
nozzle exit is A, and the jet velocity is U . Find the torque required to hold the rotor
stationary.
Solution: Select a stationary volume V shown by the dashed lines. Pressure
everywhere on the control surface is atmospheric, and there is no net moment due
to the pressure forces. The control surface cuts through the vertical support and the
torque T exerted by the support on the sprinkler arm is the only torque acting on V .
Apply the angular momentum balance
T = Ḣzout .
Let ṁ = ρAU be the mass flux through each nozzle. As the angular momentum is
the moment of momentum, we obtain
Ḣzout = (ṁU cos α)a + (ṁU cos α)a = 2aρAU 2 cos α.
Therefore, the torque required to hold the rotor stationary is
T = 2aρAU 2 cos α.
Figure 4.11 Lawn sprinkler (Example 4.2).
99
100
Conservation Laws
When the sprinkler is rotating at a steady state, this torque is balanced by both air
resistance and mechanical friction.
10. Constitutive Equation for Newtonian Fluid
The relation between the stress and deformation in a continuum is called a constitutive
equation. An equation that linearly relates the stress to the rate of strain in a fluid
medium is examined in this section.
In a fluid at rest there are only normal components of stress on a surface, and
the stress does not depend on the orientation of the surface. In other words, the stress
tensor is isotropic or spherically symmetric. An isotropic tensor is defined as one
whose components do not change under a rotation of the coordinate system (see
Chapter 2, Section 7). The only second-order isotropic tensor is the Kronecker delta
1 0 0
δ = 0 1 0 .
0 0 1
Any isotropic second-order tensor must be proportional to δ. Therefore, because the
stress in a static fluid is isotropic, it must be of the form
τij = −pδij ,
(4.32)
where p is the thermodynamic pressure related to ρ and T by an equation of state
(e.g., the thermodynamic pressure for a perfect gas is p = ρRT ). A negative sign is
introduced in equation (4.32) because the normal components of τ are regarded as
positive if they indicate tension rather than compression.
A moving fluid develops additional components of stress due to viscosity. The
diagonal terms of τ now become unequal, and shear stresses develop. For a moving
fluid we can split the stress into a part −pδij that would exist if it were at rest and a
part σij due to the fluid motion alone:
τij = −pδij + σij .
(4.33)
We shall assume that p appearing in equation (4.33) is still the thermodynamic pressure. The assumption, however, is not on a very firm footing because thermodynamic
quantities are defined for equilibrium states, whereas a moving fluid undergoing diffusive fluxes is generally not in equilibrium. Such departures from thermodynamic
equilibrium are, however, expected to be unimportant if the relaxation (or adjustment)
time of the molecules is small compared to the time scale of the flow, as discussed in
Chapter 1, Section 8.
The nonisotropic part σ, called the deviatoric stress tensor, is related to the
velocity gradients ∂ui /∂xj . The velocity gradient tensor can be decomposed into
symmetric and antisymmetric parts:
∂uj
∂uj
1 ∂ui
1 ∂ui
∂ui
+
.
=
+
−
∂xj
2 ∂xj
∂xi
2 ∂xj
∂xi
101
10. Constitutive Equation for Newtonian Fluid
The antisymmetric part represents fluid rotation without deformation, and cannot by
itself generate stress. The stresses must be generated by the strain rate tensor
eij ≡
1
2
∂uj
∂ui
+
∂xj
∂xi
,
alone. We shall assume a linear relation of the type
σij = Kij mn emn ,
(4.34)
where Kij mn is a fourth-order tensor having 81 components that depend on the thermodynamic state of the medium. Equation (4.34) simply means that each stress component is linearly related to all nine components of eij ; altogether 81 constants are
therefore needed to completely describe the relationship.
It will now be shown that only two of the 81 elements of Kij mn survive if it
is assumed that the medium is isotropic and that the stress tensor is symmetric. An
isotropic medium has no directional preference, which means that the stress–strain
relationship is independent of rotation of the coordinate system. This is only possible
if Kij mn is an isotropic tensor. It is shown in books on tensor analysis (e.g., see Aris
(1962), pp. 30–33) that all isotropic tensors of even order are made up of products of
δij , and that a fourth-order isotropic tensor must have the form
Kij mn = λδij δmn + µδim δj n + γ δin δj m ,
(4.35)
where λ, µ, and γ are scalars that depend on the local thermodynamic state. As σij
is a symmetric tensor, equation (4.34) requires that Kij mn also must be symmetric in
i and j . This is consistent with equation (4.35) only if
γ = µ.
(4.36)
Only two constants µ and λ, of the original 81, have therefore survived under the
restrictions of material isotropy and stress symmetry. Substitution of equation (4.35)
into the constitutive equation (4.34) gives
σij = 2µeij + λemm δij ,
where emm = ∇ · u is the volumetric strain rate (explained in Chapter 3, Section 6).
The complete stress tensor (4.33) then becomes
τij = −pδij + 2µeij + λemm δij .
(4.37)
The two scalar constants µ and λ can be further related as follows. Setting i = j ,
summing over the repeated index, and noting that δii = 3, we obtain
τii = −3p + (2µ + 3λ) emm ,
102
Conservation Laws
from which the pressure is found to be
1
p = − τii +
3
2
µ + λ ∇ · u.
3
(4.38)
Now the diagonal terms of eij in a flow may be unequal. In such a case the stress
tensor τij can have unequal diagonal terms because of the presence of the term
proportional to µ in equation (4.37). We can therefore take the average of the
diagonal terms of τ and define a mean pressure (as opposed to thermodynamic
pressure p) as
1
(4.39)
p̄ ≡ − τii .
3
Substitution into equation (4.38) gives
2
µ + λ ∇ · u.
p − p̄ =
3
(4.40)
For a completely incompressible fluid we can only define a mechanical or mean
pressure, because there is no equation of state to determine a thermodynamic pressure.
(In fact, the absolute pressure in an incompressible fluid is indeterminate, and only
its gradients can be determined from the equations of motion.) The λ-term in the
constitutive equation (4.37) drops out because emm = ∇ ·u = 0, and no consideration
of equation (4.40) is necessary. For incompressible fluids, the constitutive equation
(4.37) takes the simple form
τij = −pδij + 2µeij
(incompressible),
(4.41)
where p can only be interpreted as the mean pressure. For a compressible fluid, on
the other hand, a thermodynamic pressure can be defined, and it seems that p and p̄
can be different. In fact, equation (4.40) relates this difference to the rate of expansion
through the proportionality constant κ = λ + 2µ/3, which is called the coefficient
of bulk viscosity. In principle, κ is a measurable quantity; however, extremely large
values of Dρ/Dt are necessary in order to make any measurement, such as within
shock waves. Moreover, measurements are inconclusive about the nature of κ. For
many applications the Stokes assumption
2
λ + µ = 0,
3
(4.42)
is found to be sufficiently accurate, and can also be supported from the kinetic theory of
monatomic gases. Interesting historical aspects of the Stokes assumption 3λ+2µ = 0
can be found in Truesdell (1952).
To gain additional insight into the distinction between thermodynamic pressure
and the mean of the normal stresses, consider a system inside a cylinder in which a
piston may be moved in or out to do work. The first law of thermodynamics may be
written in general terms as de = dw + dQ = −p̄dv + dQ = −pdv + T dS, where
103
10. Constitutive Equation for Newtonian Fluid
the last equality is written in terms of state functions. Then T dS − dQ = (p − p̄)dv.
The Clausius-Duhem inequality (see under equation 1.16) tells us T dS − dQ ≥ 0
for any process and, consequently, (p − p̄)dv ≥ 0. Thus, for an expansion, dv > 0,
so p > p̄, and conversely for a compression. Equation (4.40) is:
2
2
1 Dρ
2
1 Dv
1
p − p̄ =
µ + λ ∇ ·u = −
µ+λ
=
µ+λ
,
v= .
3
3
ρ Dt
3
v Dt
ρ
Further, we require (2/3)µ + λ > 0 to satisfy the Clausius-Duhem inequality statement of the second law.
With the assumption κ = 0, the constitutive equation (4.37) reduces to
τij = − p + 23 µ∇ · u δij + 2µeij
(4.43)
This linear relation between τ and e is consistent with Newton’s definition of viscosity
coefficient in a simple parallel flow u(y), for which equation (4.43) gives a shear stress
of τ = µ(du/dy). Consequently, a fluid obeying equation (4.43) is called a Newtonian
fluid. The fluid property µ in equation (4.43) can depend on the local thermodynamic
state alone.
The nondiagonal terms of equation (4.43) are easy to understand. They are of the
type
∂u2
∂u1
τ12 = µ
,
+
∂x2
∂x1
which relates the shear stress to the strain rate. The diagonal terms are more difficult
to understand. For example, equation (4.43) gives
1 ∂ui
∂u1
τ11 = −p + 2µ −
,
+
3 ∂xi
∂x1
which means that the normal viscous stress on a plane normal to the x1 -axis is proportional to the difference between the extension rate in the x1 direction and the average
expansion rate at the point. Therefore, only those extension rates different from the
average will generate normal viscous stress.
Non-Newtonian Fluids
The linear Newtonian friction law is expected to hold for small rates of strain because
higher powers of e are neglected. However, for common fluids such as air and water
the linear relationship is found to be surprisingly accurate for most applications. Some
liquids important in the chemical industry, on the other hand, display non-Newtonian
behavior at moderate rates of strain. These include: (1) solutions containing polymer
molecules, which have very large molecular weights and form long chains coiled
together in spongy ball-like shapes that deform under shear; and (2) emulsions and
slurries containing suspended particles, two examples of which are blood and water
104
Conservation Laws
containing clay. These liquids violate Newtonian behavior in several ways—for example, shear stress is a nonlinear function of the local strain rate. It depends not only on
the local strain rate, but also on its history. Such a “memory” effect gives the fluid an
elastic property, in addition to its viscous property. Most non-Newtonian fluids are
therefore viscoelastic. Only Newtonian fluids will be considered in this book.
11. Navier–Stokes Equation
The equation of motion for a Newtonian fluid is obtained by substituting the constitutive equation (4.43) into Cauchy’s equation (4.15) to obtain
ρ
2
∂p
∂
Dui
2µeij − µ(∇ · u)δij ,
=−
+ ρgi +
Dt
∂xi
∂xj
3
(4.44)
where we have noted that (∂p/∂xj )δij = ∂p/∂xi . Equation (4.44) is a general
form of the Navier–Stokes equation. Viscosity µ in this equation can be a function of the thermodynamic state, and indeed µ for most fluids displays a rather
strong dependence on temperature, decreasing with T for liquids and increasing
with T for gases. However, if the temperature differences are small within the
fluid, then µ can be taken outside the derivative in equation (4.44), which then
reduces to
ρ
∂eij
∂p
2µ ∂
Dui
=−
+ ρgi + 2µ
−
(∇ · u)
Dt
∂xi
∂xj
3 ∂xi
1 ∂
∂p
2
+ ρgi + µ ∇ ui +
(∇ · u) ,
=−
∂xi
3 ∂xi
where
∇ 2 ui ≡
∂ 2 ui
∂ 2 ui
∂ 2 ui
∂ 2 ui
+
+
,
=
∂xj ∂xj
∂x12
∂x22
∂x32
is the Laplacian of ui . For incompressible fluids ∇ · u = 0, and using vector notation,
the Navier–Stokes equation reduces to
ρ
Du
= −∇p + ρg + µ ∇ 2 u.
Dt
(incompressible)
(4.45)
If viscous effects are negligible, which is generally found to be true far from boundaries of the flow field, we obtain the Euler equation
ρ
Du
= −∇p + ρg.
Dt
(4.46)
105
12. Rotating Frame
Comments on the Viscous Term
For an incompressible fluid, equation (4.41) shows that the viscous stress at a point is
σij = µ
∂uj
∂ui
+
∂xj
∂xi
,
(4.47)
which shows that σ depends only on the deformation rate of a fluid element at a point,
and not on the rotation rate (∂ui /∂xj − ∂uj /∂xi ). We have built this property into the
Newtonian constitutive equation, based on the fact that in a solid-body rotation (that
is a flow in which the tangential velocity is proportional to the radius) the particles do
not deform or “slide” past each other, and therefore they do not cause viscous stress.
However, consider the net viscous force per unit volume at a point, given by
∂σij
∂
=µ
Fi =
∂xj
∂xj
∂uj
∂ui
+
∂xj
∂xi
=µ
∂ 2 ui
= −µ(∇ × ω)i ,
∂xj ∂xj
(4.48)
where we have used the relation
(∇ × ω)i = εij k
∂
∂ωk
= εij k
∂xj
∂xj
= (δim δj n − δin δj m )
=−
∂un
εkmn
∂xm
∂ 2 uj
∂ 2 un
∂ 2 ui
=
−
∂xj ∂xm
∂xj ∂xi
∂xj ∂xj
∂ 2 ui
.
∂xj ∂xj
In the preceding derivation the “epsilon delta relation,” given by equation (2.19),
has been used. Relation (4.48) can cause some confusion because it seems to show
that the net viscous force depends on vorticity, whereas equation (4.47) shows that
viscous stress depends only on strain rate and is independent of local vorticity. The
apparent paradox is explained by realizing that the net viscous force is given by either
the spatial derivative of vorticity or the spatial derivative of deformation rate; both
forms are shown in equation (4.48). The net viscous force vanishes when ω is uniform
everywhere (as in solid-body rotation), in which case the incompressibility condition
requires that the deformation is zero everywhere as well.
12. Rotating Frame
The equations of motion given in Section 7 are valid in an inertial or “fixed” frame of
reference. Although such a frame of reference cannot be defined precisely, experience
shows that these laws are accurate enough in a frame of reference stationary with
respect to “distant stars.” In geophysical applications, however, we naturally measure
positions and velocities with respect to a frame of reference fixed on the surface of the
earth, which rotates with respect to an inertial frame. In this section we shall derive
the equations of motion in a rotating frame of reference. Similar derivations are also
given by Batchelor (1967), Pedlosky (1987), and Holton (1979).
106
Conservation Laws
Figure 4.12 Coordinate frame (x1 , x2 , x3 ) rotating at angular velocity with respect to a fixed frame
(X1 , X2 , X3 ).
Consider (Figure 4.12) a frame of reference (x1 , x2 , x3 ) rotating at a uniform
angular velocity with respect to a fixed frame (X1 , X2 , X3 ). Any vector P is represented in the rotating frame by
P = P1 i1 + P2 i2 + P3 i3 .
To a fixed observer the directions of the rotating unit vectors i1 , i2 , and i3 change with
time. To this observer the time derivative of P is
d
dP
= (P1 i1 + P2 i2 + P3 i3 )
dt F
dt
dP2
dP3
di1
di2
di3
dP1
+ i2
+ i3
+ P1
+ P2
+ P3
.
= i1
dt
dt
dt
dt
dt
dt
To the rotating observer, the rate of change of P is the sum of the first three terms,
so that
di2
di3
di1
dP
dP
+ P2
+ P3
.
(4.49)
=
+ P1
dt F
dt R
dt
dt
dt
Now each unit vector i traces a cone with a radius of sin α, where α is a constant
angle (Figure 4.13). The magnitude of the change of i in time dt is |di| = sin α dθ ,
which is the length traveled by the tip of i. The magnitude of the rate of change
is therefore (di/dt) = sin α (dθ/dt) = sin α, and the direction of the rate of
change is perpendicular to the (, i)-plane. Thus di/dt = × i for any rotating
unit vector i. The sum of the last three terms in equation (4.49) is then P1 × i1
107
12. Rotating Frame
Figure 4.13 Rotation of a unit vector.
+P2 × i2 + P3 × i3 = × P. Equation (4.49) then becomes
dP
dP
=
+ × P,
dt F
dt R
(4.50)
which relates the rates of change of the vector P as seen by the two observers.
Application of rule (4.50) to the position vector r relates the velocities as
u F = u R + × r.
(4.51)
Applying rule (4.50) on u F , we obtain
du F
du F
=
+ × u F,
dt F
dt R
which becomes, upon using equation (4.51),
d
du F
= (u R + × r) R + × (u R + × r)
dt
dt
du R
dr
=
+×
+ × u R + × ( × r).
dt R
dt R
This shows that the accelerations in the two frames are related as
a F = a R + 2 × u R + × ( × r),
˙ = 0,
(4.52)
The last term in equation (4.52) can be written in terms of the vector R drawn perpendicularly to the axis of rotation (Figure 4.14). Clearly, × r = × R. Using the
108
Conservation Laws
Figure 4.14 Centripetal acceleration.
vector identity A × (B × C) = (A • C)B − (A · B)C, the last term of equation (4.52)
becomes
× ( × R) = −( · )R = −2 R,
where we have set · R = 0. Equation (4.52) then becomes
a F = a + 2 × u − 2 R,
(4.53)
where the subscript “R” has been dropped with the understanding that velocity u and
acceleration a are measured in a rotating frame of reference. Equation (4.53) states
that the “true” or inertial acceleration equals the acceleration measured in a rotating
system, plus the Coriolis acceleration 2 × u and the centripetal acceleration −2 R.
Therefore, Coriolis and centripetal accelerations have to be considered if we are
measuring quantities in a rotating frame of reference. Substituting equation (4.53) in
equation (4.45), the equation of motion in a rotating frame of reference becomes
Du
1
= − ∇p + ν∇ 2 u + (gn + 2 R) − 2 × u,
Dt
ρ
(4.54)
where we have taken the Coriolis and centripetal acceleration terms to the right-hand
side (now signifying Coriolis and centrifugal forces), and added a subscript on g to
mean that it is the body force per unit mass due to (Newtonian) gravitational attractive
forces alone.
109
12. Rotating Frame
Figure 4.15 Effective gravity g and equipotential surface.
Effect of Centrifugal Force
The additional apparent force 2 R can be added to the Newtonian gravity gn to
define an effective gravity force g = gn + 2 R (Figure 4.15). The Newtonian gravity
would be uniform over the earth’s surface, and be centrally directed, if the earth were
spherically symmetric and homogeneous. However, the earth is really an ellipsoid with
the equatorial diameter 42 km larger than the polar diameter. In addition, the existence
of the centrifugal force makes the effective gravity less at the equator than at the poles,
where 2 R is zero. In terms of the effective gravity, equation (4.54) becomes
1
Du
= − ∇p + ν∇ 2 u + g − 2 × u.
Dt
ρ
(4.55)
The Newtonian gravity can be written as the gradient of a scalar potential function.
It is easy to see that the centrifugal force can also be written in the same manner.
From Definition (2.22), it is clear that the gradient of a spatial direction is the unit
vector in that direction (e.g., ∇x = ix ), so that ∇(R 2 /2) = Ri R = R. Therefore,
2 R = ∇(2 R 2 /2), and the centrifugal potential is −2 R 2 /2. The effective gravity
can therefore be written as g = −∇, where is now the potential due to the
Newtonian gravity, plus the centrifugal potential. The equipotential surfaces (shown
by the dashed lines in Figure 4.15) are now perpendicular to the effective gravity. The
average sea level is one of these equipotential surfaces. We can then write = gz,
where z is measured perpendicular to an equipotential surface, and g is the effective
acceleration due to gravity.
Effect of Coriolis Force
The angular velocity vector points out of the ground in the northern hemisphere.
The Coriolis force −2 × u therefore tends to deflect a particle to the right of its
110
Conservation Laws
Figure 4.16 Deflection of a particle due to the Coriolis force.
direction of travel in the northern hemisphere (Figure 4.16) and to the left in the
southern hemisphere.
Imagine a projectile shot horizontally from the north pole with speed u. The
Coriolis force 2u constantly acts perpendicular to its path and therefore does not
change the speed u of the projectile. The forward distance traveled in time t is ut, and
the deflection is ut 2 . The angular deflection is ut 2 /ut = t, which is the earth’s
rotation in time t. This demonstrates that the projectile in fact travels in a straight
line if observed from the inertial outer space; its apparent deflection is merely due to
the rotation of the earth underneath it. Observers on earth need an imaginary force
to account for the apparent deflection. A clear physical explanation of the Coriolis
force, with applications to mechanics, is given by Stommel and Moore (1989).
It is the Coriolis force that is responsible for the wind circulation patterns around
centers of high and low pressure in the earth’s atmosphere. Fluid flows from regions
of higher pressure to regions of lower pressure, as (4.55) indicates acceleration of
a fluid particle in a direction opposite the pressure gradient. Imagine a cylindrical
polar coordinate system, as defined in Appendix B1, with the x-axis normal (outwards) to the local tangent plane to the earth’s surface and the origin at the center
of the “high” or “low.” If it is a high pressure zone, uR is outwards (positive) since
flow is away from the center of high pressure. Then the Coriolis acceleration, the
last term of (4.55), becomes −2 × u = −z ur = −uθ is in the −θ direction
(in the Northern hemisphere), or clockwise as viewed from above. On the other
hand, flow is inwards toward the center of a low pressure zone, which reverses the
direction of ur and, therefore, uθ is counter-clockwise. In the Southern hemisphere,
the direction of z is reversed so that the circulation patterns described above are
reversed.
111
13. Mechanical Energy Equation
Although the effects of a rotating frame will be commented on occasionally in
this and subsequent chapters, most of the discussions involving Coriolis forces are
given in Chapter 14, which deals with geophysical fluid dynamics.
13. Mechanical Energy Equation
An equation for kinetic energy of the fluid can be obtained by finding the scalar
product of the momentum equation and the velocity vector. The kinetic energy equation is therefore not a separate principle, and is not the same as the first law of
thermodynamics. We shall derive several forms of the equation in this section. The
Coriolis force, which is perpendicular to the velocity vector, does not contribute to
any of the energy equations. The equation of motion is
ρ
∂τij
Dui
= ρgi +
.
Dt
∂xj
Multiplying by ui (and, of course, summing over i), we obtain
∂τij
D 1 2
u = ρui gi + ui
,
ρ
Dt 2 i
∂xj
(4.56)
where, for the sake of notational simplicity, we have written u2i for ui ui = u21 +u22 +u23 .
A summation over i is therefore implied in u2i , although no repeated index is explicitly
written. Equation (4.56) is the simplest as well as most revealing mechanical energy
equation. Recall from Section 7 that the resultant imbalance of the surface forces at a
point is ∇ · τ, per unit volume. Equation (4.56) therefore says that the rate of increase
of kinetic energy at a point equals the sum of the rate of work done by body force g
and the rate of work done by the net surface force ∇ · τ per unit volume.
Other forms of the mechanical energy equation are obtained by combining
equation (4.56) with the continuity equation in various ways. For example, u2i /2
times the continuity equation is
∂
1 2 ∂ρ
u
+
(ρuj ) = 0,
2 i ∂t
∂xj
which, when added to equation (4.56), gives
∂τij
1 2
∂
∂ 1 2
uj ρui = ρui gi + ui
ρui +
.
∂t 2
∂xj
2
∂xj
Using vector notation, and defining E ≡ u2i /2 as the kinetic energy per unit volume,
this becomes
∂E
+ ∇ · (uE) = ρu · g + u · (∇ · τ).
∂t
(4.57)
The second term is in the form of divergence of kinetic energy flux uE. Such flux
divergence terms frequently arise in energy balances and can be interpreted as the net
112
Conservation Laws
loss at a point due to divergence of a flux. For example, if the source terms on the
right-hand side of equation (4.57) are zero, then the local E will increase with time if
∇·(uE) is negative. Flux divergence terms are also called transport terms because they
transfer quantities from one region to another without making a net contribution over
the entire field. When integrated over the entire volume, their contribution vanishes if
there are no sources at the boundaries. For example, Gauss’ theorem transforms the
volume integral of ∇ · (uE) as
Eu · dA,
∇ · (uE) dV =
A
V
which vanishes if the flux uE is zero at the boundaries.
Concept of Deformation Work and Viscous Dissipation
Another useful form of the kinetic energy equation will now be derived by examining
how kinetic energy can be lost to internal energy by deformation of fluid elements.
In equation (4.56) the term ui (∂τij /∂xj ) is velocity times the net force imbalance
at a point due to differences of stress on opposite faces of an element; the net force
accelerates the local fluid and increases its kinetic energy. However, this is not the
total rate of work done by stress on the element, and the remaining part goes into
deforming the element without accelerating it. The total rate of work done by surface
forces on a fluid element must be ∂(τij ui )/∂xj , because this can be transformed to
a surface integral of τij ui over the element. (Here τij dAj is the force on an area
element, and τij ui dAj is the scalar product of force and velocity. The total rate of
work done by surface forces is therefore the surface integral of τij ui .) The total work
rate per volume at a point can be split up into two components:
∂
(ui τij ) =
∂xj
total work
(rate/volume)
τij
∂ui
∂xj
+
deformation
work
(rate/volume)
ui
∂τij
.
∂xj
increase
of KE
(rate/volume)
We have seen from equation (4.56) that the last term in the preceding equation results
in an increase of kinetic energy of the element. Therefore, the rest of the work rate
per volume represented by τij (∂ui /∂xj ) can only deform the element and increase
its internal energy.
The deformation work rate can be rewritten using the symmetry of the stress tensor. In Chapter 2, Section 11 it was shown that the contracted product of a symmetric
tensor and an antisymmetric tensor is zero. The product τij (∂ui /∂xj ) is therefore
equal to τij times the symmetric part of ∂ui /∂xj , namely eij . Thus
Deformation work rate per volume = τij
∂ui
= τij eij .
∂xj
On substituting the Newtonian constitutive equation
2
τij = −pδij + 2µeij − µ(∇ · u)δij ,
3
(4.58)
113
13. Mechanical Energy Equation
relation (4.58) becomes
2
Deformation work = −p(∇ · u) + 2µeij eij − µ(∇ · u)2 ,
3
where we have used eij δij = eii = ∇ · u. Denoting the viscous term by φ, we obtain
where
Deformation work (rate per volume) = −p(∇ · u) + φ,
(4.59)
2
2
1
2
φ ≡ 2µeij eij − µ(∇ · u) = 2µ eij − (∇ · u)δij .
3
3
(4.60)
The validity of the last term in equation (4.60) can easily be verified by completing
the square (Exercise 5).
In order to write the energy equation in terms of φ, we first rewrite equation (4.56)
in the form
∂
D 1 2
ui = ρgi ui +
(ui τij ) − τij eij ,
(4.61)
ρ
Dt 2
∂xj
where we have used τij (∂ui /∂xj ) = τij eij . Using equation (4.59) to rewrite the
deformation work rate per volume, equation (4.61) becomes
∂
D 1 2
u = ρg · u +
(ui τij ) + p(∇ · u) − φ
ρ
Dt 2 i
∂xj
(4.62)
rate of work by
total rate of
rate of work
rate of
body force
work by τ
by volume
expansion
viscous
dissipation
It will be shown in Section 14 that the last two terms in the preceding equation
(representing pressure and viscous contributions to the rate of deformation work)
also appear in the internal energy equation but with their signs changed. The term
p(∇ · u) can be of either sign, and converts mechanical to internal energy, or vice
versa, by volume changes. The viscous term φ is always positive and represents a
rate of loss of mechanical energy and a gain of internal energy due to deformation of
the element. The term τij eij = p(∇ · u) − φ represents the total deformation work
rate per volume; the part p(∇ · u) is the reversible conversion to internal energy by
volume changes, and the part φ is the irreversible conversion to internal energy due
to viscous effects.
The quantity φ defined in equation (4.60) is proportional to µ and represents
the rate of viscous dissipation of kinetic energy per unit volume. Equation (4.60)
shows that it is proportional to the square of velocity gradients and is therefore more
important in regions of high shear. The resulting heat could appear as a hot lubricant in
a bearing, or as burning of the surface of a spacecraft on reentry into the atmosphere.
Equation in Terms of Potential Energy
So far we have considered kinetic energy as the only form of mechanical energy. In
doing so we have found that the effects of gravity appear as work done on a fluid
114
Conservation Laws
particle, as equation (4.62) shows. However, the rate of work done by body forces can
be taken to the left-hand side of the mechanical energy equations and be interpreted
as changes in the potential energy. Let the body force be represented as the gradient
of a scalar potential = gz, so that
ui gi = −ui
∂
D
(gz) = − (gz),
∂xi
Dt
where we have used ∂(gz)/∂t = 0, because z and t are independent. Equation (4.62)
then becomes
∂
D 1 2
ui + gz =
(ui τij ) + p(∇ • u) − φ,
ρ
Dt 2
∂xj
in which the function = gz clearly has the significance of potential energy per unit
mass. (This identification is possible only for conservative body forces for which a
potential may be written.)
Equation for a Fixed Region
An integral form of the mechanical energy equation can be derived by integrating
the differential form over either a fixed volume or a material volume. The procedure
is illustrated here for a fixed volume. We start with equation (4.62), but write the
left-hand side as given in equation (4.57). This gives (in mixed notation)
∂
∂
∂E
+
(ui E) = ρg • u +
(ui τij ) + p(∇ • u) − φ,
∂t
∂xi
∂xj
where E = ρu2i /2 is the kinetic energy per unit volume. Integrate each term of the
foregoing equation over the fixed volume V . The second and fourth terms are in
the flux divergence form, so that their volume integrals can be changed to surface
integrals by Gauss’ theorem. This gives
d
E dV + Eu • dA
dt
rate of change
of KE
=
rate of outflow
across
boundary
ρg u dV
•
rate of work
by body
force
+
ui τij dAj +
rate of work
by surface
force
p(∇ u) dV −
•
rate of work
by volume
expansion
φ dV
rate of viscous
dissipation
(4.63)
where each term is a time rate of change. The description of each term in equation (4.63) is obvious. The fourth term represents rate of work done by forces at the
boundary, because τij dAj is the force in the i direction and ui τij dAj is the scalar
product of the force with the velocity vector.
The energy considerations discussed in this section may at first seem too
“theoretical.” However, they are very useful in understanding the physics of fluid
115
14. First Law of Thermodynamics: Thermal Energy Equation
flows. The concepts presented here will be especially useful in our discussions of
turbulent flows (Chapter 13) and wave motions (Chapter 7). It is suggested that the
reader work out Exercise 11 at this point in order to acquire a better understanding of
the equations in this section.
14. First Law of Thermodynamics: Thermal Energy Equation
The mechanical energy equation presented in the preceding section is derived from
the momentum equation and is not a separate principle. In flows with temperature
variations we need an independent equation; this is provided by the first law of thermodynamics. Let q be the heat flux vector (per unit area), and e the internal energy
per unit mass; for a perfect gas e = CV T , where CV is the specific heat at constant
volume (assumed constant). The sum (e + u2i /2) can be called the “stored” energy
per unit mass. The first law of thermodynamics is most easily stated for a material
volume. It says that the rate of change of stored energy equals the sum of rate of work
done and rate of heat addition to a material volume. That is,
D
qi dAi .
(4.64)
τij ui dAj −
ρgi ui d ᐂ +
ρ e + 21 u2i d ᐂ =
Dt ᐂ
A
A
ᐂ
Note that work done by body forces has to be included on the right-hand side if
potential energy is not included on the left-hand side, as in equations (4.62)–(4.64).
(This is clear from the discussion of the preceding section and can also be understood
as follows. Imagine a situation where the surface integrals in equation (4.64) are zero,
and also that e is uniform everywhere. Then a rising fluid particle (u • g < 0), which is
constantly pulled down by gravity, must undergo a decrease of kinetic energy. This is
consistent with equation (4.64).) The negative sign is needed on the heat transfer term,
because the direction of dA is along the outward normal to the area, and therefore
q • dA represents the rate of heat outflow.
To derive a differential form, all terms need to be expressed in the form of volume
integrals. The left-hand side can be written as
D
D
1
1
ρ
ρ e + u2i d ᐂ =
e + u2i d ᐂ,
Dt ᐂ
2
2
ᐂ Dt
where equation (4.6) has been used. Converting the two surface integral terms into
volume integrals, equation (4.64) finally gives
∂qi
∂
1 2
D
(τij ui ) −
.
(4.65)
e + ui = ρgi ui +
ρ
Dt
2
∂xj
∂xi
This is the first law of thermodynamics in the differential form, which has both
mechanical and thermal energy terms in it. A thermal energy equation is obtained if
the mechanical energy equation (4.62) is subtracted from it. This gives the thermal
energy equation (commonly called the heat equation)
ρ
De
= −∇ • q − p(∇ • u) + φ,
Dt
(4.66)
116
Conservation Laws
which says that internal energy increases because of convergence of heat, volume
compression, and heating due to viscous dissipation. Note that the last two terms in
equation (4.66) also appear in mechanical energy equation (4.62) with their signs
reversed.
The thermal energy equation can be simplified under the Boussinesq approximation, which applies under several restrictions including that in which the flow speeds
are small compared to the speed of sound and in which the temperature differences
in the flow are small. This is discussed in Section 18. It is shown there that, under
these restrictions, heating due to the viscous dissipation term is negligible in equation (4.66), and that the term −p(∇ • u) can be combined with the left-hand side of
equation (4.66) to give (for a perfect gas)
ρCp
DT
= −∇ • q.
Dt
(4.67)
If the heat flux obeys the Fourier law
q = −k∇T ,
then, if k = const., equation (4.67) simplifies to:
DT
= κ∇ 2 T .
Dt
(4.68)
where κ ≡ k/ρCp is the thermal diffusivity, stated in m2 /s and which is the same as
that of the momentum diffusivity ν.
The viscous heating term φ may be negligible in the thermal energy equation (4.66) if flow speeds are low compared with the sound speed, but not in the
mechanical energy equation (4.62). In fact, there must be a sink of mechanical energy
so that a steady state can be maintained in the presence of the various types of forcing.
15. Second Law of Thermodynamics: Entropy Production
The second law of thermodynamics essentially says that real phenomena can only
proceed in a direction in which the “disorder” of an isolated system increases. Disorder of a system is a measure of the degree of uniformity of macroscopic properties in
the system, which is the same as the degree of randomness in the molecular arrangements that generate these properties. In this connection, disorder, uniformity, and
randomness have essentially the same meaning. For analogy, a tray containing red
balls on one side and white balls on the other has more order than in an arrangement
in which the balls are mixed together. A real phenomenon must therefore proceed in
a direction in which such orderly arrangements decrease because of “mixing.” Consider two possible states of an isolated fluid system, one in which there are nonuniformities of temperature and velocity and the other in which these properties are
uniform. Both of these states have the same internal energy. Can the system spontaneously go from the state in which its properties are uniform to one in which they are
117
15. Second Law of Thermodynamics: Entropy Production
nonuniform? The second law asserts that it cannot, based on experience. Natural
processes, therefore, tend to cause mixing due to transport of heat, momentum, and
mass.
A consequence of the second law is that there must exist a property called
entropy, which is related to other thermodynamic properties of the medium. In
addition, the second law says that the entropy of an isolated system can only
increase; entropy is therefore a measure of disorder or randomness of a system.
Let S be the entropy per unit mass. It is shown in Chapter 1, Section 8 that the
change of entropy is related to the changes of internal energy e and specific volume
v (= 1/ρ) by
T dS = de + p dv = de −
p
dρ.
ρ2
The rate of change of entropy following a fluid particle is therefore
T
DS
De
p Dρ
=
− 2
.
Dt
Dt
ρ Dt
(4.69)
Inserting the internal energy equation (see equation (4.66))
ρ
De
= −∇ • q − p(∇ • u) + φ,
Dt
and the continuity equation
Dρ
= −ρ(∇ • u),
Dt
the entropy production equation (4.69) becomes
ρ
DS
1 ∂qi
φ
=−
+
Dt
T ∂xi
T
∂ qi
qi ∂T
φ
=−
− 2
+ .
∂xi T
T
T ∂xi
Using Fourier’s law of heat conduction, this becomes
∂T 2 φ
k
∂ qi
DS
+ .
+ 2
=−
ρ
Dt
∂xi T
∂xi
T
T
The first term on the right-hand side, which has the form (heat gain)/T, is the entropy
gain due to reversible heat transfer because this term does not involve heat conductivity. The last two terms, which are proportional to the square of temperature and
velocity gradients, represent the entropy production due to heat conduction and viscous generation of heat. The second law of thermodynamics requires that the entropy
production due to irreversible phenomena should be positive, so that
µ, k > 0.
118
Conservation Laws
An explicit appeal to the second law of thermodynamics is therefore not required in
most analyses of fluid flows because it has already been satisfied by taking positive
values for the molecular coefficients of viscosity and thermal conductivity.
If the flow is inviscid and nonheat conducting, entropy is preserved along the
particle paths.
16. Bernoulli Equation
Various conservation laws for mass, momentum, energy, and entropy were presented in the preceding sections. The well-known Bernoulli equation is not a separate
law, but is derived from the momentum equation for inviscid flows, namely, the Euler
equation (4.46):
∂ui
∂
1 ∂p
∂ui
+ uj
=−
(gz) −
,
∂t
∂xj
∂xi
ρ ∂xi
where we have assumed that gravity g = −∇(gz) is the only body force. The advective
acceleration can be expressed in terms of vorticity as follows:
∂uj
∂uj
∂ui
1
∂
∂ui
+ uj
uj
= uj
−
= uj rij +
uj uj
∂xj
∂xj
∂xi
∂xi
∂xi 2
1 2
1 2
∂
∂
= −uj εij k ωk +
(4.70)
q = −(u × ω)i +
q ,
∂xi 2
∂xi 2
where we have used rij = −εij k ωk (see equation 3.23), and used the customary
notation
q 2 = u2j = twice kinetic energy.
Then the Euler equation becomes
1 2
1 ∂p
∂
∂
∂ui
+
(gz) = (u × ω)i .
+
q +
∂t
∂xi 2
ρ ∂xi
∂xi
(4.71)
Now assume that ρ is a function of p only. A flow in which ρ = ρ(p) is called
a barotropic flow, of which isothermal and isentropic (p/ρ γ = constant) flows are
special cases. For such a flow we can write
dp
∂
1 ∂p
=
,
(4.72)
ρ ∂xi
∂xi
ρ
where dp/ρ is a perfect differential, and therefore the integral does not depend on
the path of integration. To show this, note that
x
x0
dp
=
ρ
x
x0
1 dp
dρ =
ρ dρ
x
x0
dP
dρ = P (x) − P (x0 ),
dρ
(4.73)
119
16. Bernoulli Equation
where x is the “field point,” x0 is any arbitrary reference point in the flow, and we
have defined the following function of ρ alone:
dP
1 dp
≡
.
dρ
ρ dρ
(4.74)
The gradient of equation (4.73) gives
x
∂
dp
∂P
dP ∂p
1 ∂p
=
=
=
,
∂xi x0 ρ
∂xi
dp ∂xi
ρ ∂xi
where equation (4.74) has been used. The preceding equation is identical to equation (4.72).
Using equation (4.72), the Euler equation (4.71) becomes
∂ui
dp
∂ 1 2
+
q +
+ gz = (u × ω)i .
∂t
∂xi 2
ρ
Defining the Bernoulli function
1
B ≡ q2 +
2
1
dp
+ gz = q 2 + P + gz,
ρ
2
(4.75)
the Euler equation becomes (using vector notation)
∂u
+ ∇B = u × ω.
∂t
(4.76)
Bernoulli equations are integrals of the conservation laws and have wide applicability
as shown by the examples that follow. Important deductions can be made from the
preceding equation by considering two special cases, namely a steady flow (rotational or irrotational) and an unsteady irrotational flow. These are described in what
follows.
Steady Flow
In this case equation (4.76) reduces to
∇B = u × ω.
(4.77)
The left-hand side is a vector normal to the surface B = constant, whereas the
right-hand side is a vector perpendicular to both u and ω (Figure 4.17). It follows
that surfaces of constant B must contain the streamlines and vortex lines. Thus, an
inviscid, steady, barotropic flow satisfies
1 2
q +
2
dp
+ gz = constant along streamlines and vortex lines
ρ
(4.78)
120
Conservation Laws
Figure 4.17 Bernoulli’s theorem. Note that the streamlines and vortex lines can be at an arbitrary angle.
Figure 4.18 Flow over a solid object. Flow outside the boundary layer is irrotational.
which is called Bernoulli’s equation. If, in addition, the flow is irrotational (ω = 0),
then equation (4.72) shows that
dp
1 2
q +
+ gz = constant everywhere.
(4.79)
2
ρ
It may be shown that a sufficient condition for the existence of the surfaces containing streamlines and vortex lines is that the flow be barotropic. Incidentally, these
are called Lamb surfaces in honor of the distinguished English applied mathematician and hydrodynamicist, Horace Lamb. In a general, that is, nonbarotropic flow, a
path composed of streamline and vortex line segments can be drawn between any two
points in a flow field. Then equation (4.78) is valid with the proviso that the integral be
evaluated on the specific path chosen. As written, equation (4.78) requires the restrictions that the flow be steady, inviscid, and have only gravity (or other conservative)
body forces acting upon it. Irrotational flows are studied in Chapter 6. We shall note
only the important point here that, in a nonrotating frame of reference, barotropic
irrotational flows remain irrotational if viscous effects are negligible. Consider the
flow around a solid object, say an airfoil (Figure 4.18). The flow is irrotational at all
points outside the thin viscous layer close to the surface of the body. This is because a
particle P on a streamline outside the viscous layer started from some point S, where
the flow is uniform and consequently irrotational. The Bernoulli equation (4.79) is
therefore satisfied everywhere outside the viscous layer in this example.
121
16. Bernoulli Equation
Unsteady Irrotational Flow
An unsteady form of Bernoulli’s equation can be derived only if the flow is irrotational.
For irrotational flows the velocity vector can be written as the gradient of a scalar
potential φ (called velocity potential):
u ≡ ∇φ.
(4.80)
The validity of equation (4.80) can be checked by noting that it automatically satisfies
the conditions of irrotationality
∂uj
∂ui
=
∂xj
∂xi
i = j.
On inserting equation (4.80) into equation (4.76), we obtain
dp
1
∂φ
+ q2 +
+ gz = 0,
∇
∂t
2
ρ
that is
1
∂φ
+ q2 +
∂t
2
dp
+ gz = F (t),
ρ
(4.81)
where the integrating function F (t) is independent of location. This form of the
Bernoulli equation will be used in studying irrotational wave motions in Chapter 7.
Energy Bernoulli Equation
Return to equation (4.65) in the steady state with neither heat conduction nor viscous
stresses. Then τij = −pδij and equation (4.65) becomes
ρui
∂
∂
(e + q 2 /2) = ρui gi −
(ρui p/ρ).
∂xi
∂xi
If the body force per unit mass gi is conservative, say gravity, then gi = −(∂/∂xi )(gz),
which is the gradient of a scalar potential. In addition, from mass conservation,
∂(ρui )/∂xi = 0 and thus
p q2
∂
e+ +
+ gz = 0.
(4.82)
ρui
∂xi
ρ
2
From equation (1.13), h = e + p/ρ. Equation (4.82) now states that gradients of
B ′ = h + q 2 /2 + gz must be normal to the local streamline direction ui . Then
B ′ = h + q 2 /2 + gz is a constant on streamlines. We showed in the previous section
that inviscid, non-heat conducting flows are isentropic (S is conserved along particle
paths), and in equation (1.18) we had the relation dp/ρ = dh when S = constant.
Thus the path integral dp/ρ becomes a function h of the endpoints only if, in
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Conservation Laws
the momentum Bernoulli equation, both heat conduction and viscous stresses may
be neglected. This latter form from the energy equation becomes very useful for
high-speed gas flows to show the interplay between kinetic energy and internal energy
or enthalpy or temperature along a streamline.
17. Applications of Bernoulli’s Equation
Application of Bernoulli’s equation will now be illustrated for some simple flows.
Pitot Tube
Consider first a simple device to measure the local velocity in a fluid stream by
inserting a narrow bent tube (Figure 4.19). This is called a pitot tube, after the French
mathematician Henri Pitot (1695–1771), who used a bent glass tube to measure the
velocity of the river Seine. Consider two points 1 and 2 at the same level, point 1 being
away from the tube and point 2 being immediately in front of the open end where the
fluid velocity is zero. Friction is negligible along a streamline through 1 and 2, so that
Bernoulli’s equation (4.78) gives
u2
u2
p1
p2
p2
+ 1 =
+ 2 =
,
ρ
2
ρ
2
ρ
from which the velocity is found to be
u1 =
2(p2 − p1 )/ρ.
Figure 4.19 Pitot tube for measuring velocity in a duct.
123
17. Applications of Bernoulli’s Equation
Pressures at the two points are found from the hydrostatic balance
p1 = ρgh1
and
p2 = ρgh2 ,
so that the velocity can be found from
u1 =
2g(h2 − h1 ).
Because it is assumed that the fluid density is very much greater than that of the
atmosphere to which the tubes are exposed, the pressures at the tops of the two fluid
columns are assumed to be the same. They will actually differ by ρatm g(h2 − h1 ).
Use of the hydrostatic approximation above station 1 is valid when the streamlines
are straight and parallel between station 1 and the upper wall. In working out this
problem, the fluid density also has been taken to be a constant.
The pressure p2 measured by a pitot tube is called “stagnation pressure,” which
is larger than the local static pressure. Even when there is no pitot tube to measure
the stagnation pressure, it is customary to refer to the local value of the quantity
(p + ρu2 /2) as the local stagnation pressure, defined as the pressure that would be
reached if the local flow is imagined to slow down to zero velocity frictionlessly. The
quantity ρu2 /2 is sometimes called the dynamic pressure; stagnation pressure is the
sum of static and dynamic pressures.
Orifice in a Tank
As another application of Bernoulli’s equation, consider the flow through an orifice
or opening in a tank (Figure 4.20). The flow is slightly unsteady due to lowering of
the water level in the tank, but this effect is small if the tank area is large as compared
to the orifice area. Viscous effects are negligible everywhere away from the walls of
the tank. All streamlines can be traced back to the free surface in the tank, where they
have the same value of the Bernoulli constant B = q 2 /2 + p/ρ + gz. It follows that
the flow is irrotational, and B is constant throughout the flow.
We want to apply Bernoulli’s equation between a point at the free surface in
the tank and a point in the jet. However, the conditions right at the opening (section
A in Figure 4.20) are not simple because the pressure is not uniform across the jet.
Although pressure has the atmospheric value everywhere on the free surface of the jet
(neglecting small surface tension effects), it is not equal to the atmospheric pressure
inside the jet at this section. The streamlines at the orifice are curved, which requires
that pressure must vary across the width of the jet in order to balance the centrifugal
force. The pressure distribution across the orifice (section A) is shown in Figure 4.20.
However, the streamlines in the jet become parallel at a short distance away from the
orifice (section C in Figure 4.20), where the jet area is smaller than the orifice area.
The pressure across section C is uniform and equal to the atmospheric value because
it has that value at the surface of the jet.
124
Conservation Laws
Figure 4.20 Flow through a sharp-edged orifice. Pressure has the atmospheric value everywhere across
section CC; its distribution across orifice AA is indicated.
Application of Bernoulli’s equation between a point on the free surface in the
tank and a point at C gives
patm
patm
u2
+ gh =
+ ,
ρ
ρ
2
from which the jet velocity is found as
u=
2gh,
which simply states that the loss of potential energy equals the gain of kinetic energy.
The mass flow rate is
ṁ = ρAc u = ρAc 2gh,
where Ac is the area of the jet at C. For orifices having a sharp edge, Ac has been
found to be ≈62% of the orifice area.
If the orifice happens to have a well-rounded opening (Figure 4.21), then the jet
does not contract. The streamlines right at the exit are then parallel, and the pressure
at the exit is uniform and
√ equal to the atmospheric pressure. Consequently the mass
flow rate is simply ρA 2gh, where A equals the orifice area.
18. Boussinesq Approximation
For flows satisfying certain conditions, Boussinesq in 1903 suggested that the density
changes in the fluid can be neglected except in the gravity term where ρ is multiplied
by g. This approximation also treats the other properties of the fluid (such as µ, k, Cp )
as constants. A formal justification, and the conditions under which the Boussinesq
approximation holds, is given in Spiegel and Veronis (1960). Here we shall discuss the
125
18. Boussinesq Approximation
Figure 4.21 Flow through a rounded orifice.
basis of the approximation in a somewhat intuitive manner and examine the resulting
simplifications of the equations of motion.
Continuity Equation
The Boussinesq approximation replaces the continuity equation
1 Dρ
+ ∇ • u = 0,
ρ Dt
(4.83)
∇ • u = 0.
(4.84)
by the incompressible form
However, this does not mean that the density is regarded as constant along the direction
of motion, but simply that the magnitude of ρ −1 (Dρ/Dt) is small in comparison to
the magnitudes of the velocity gradients in ∇ • u. We can immediately think of several
situations where the density variations cannot be neglected as such. The first situation
is a steady flow with large Mach numbers (defined as U/c, where U is a typical
measure of the flow speed and c is the speed of sound in the medium). At large Mach
numbers the compressibility effects are large, because the large pressure changes
cause large density changes. It is shown in Chapter 16 that compressibility effects
are negligible in flows in which the Mach number is <0.3. A typical value of c for
126
Conservation Laws
air at ordinary temperatures is 350 m/s, so that the assumption is good for speeds
<100 m/s. For water c = 1470 m/s, but the speeds normally achievable in liquids
are much smaller than this value and therefore the incompressibility assumption is
very good in liquids.
A second situation in which the compressibility effects are important is unsteady
flows. The waves would propagate at infinite speed if the density variations are
neglected.
A third situation in which the compressibility effects are important occurs when
the vertical scale of the flow is so large that the hydrostatic pressure variations cause
large changes in density. In a hydrostatic field the vertical scale in which the density
changes become important is of order c2 /g ∼ 10 km for air. (This length agrees with
the e-folding height RT /g of an “isothermal atmosphere,” because c2 = γ RT ; see
Chapter 1, Section 10.) The Boussinesq approximation therefore requires that the
vertical scale of the flow be L ≪ c2 /g.
In the three situations mentioned the medium is regarded as “compressible,” in
which the density depends strongly on pressure. Now suppose the compressibility
effects are small, so that the density changes are caused by temperature changes
alone, as in a thermal convection problem. In this case the Boussinesq approximation
applies when the temperature variations in the flow are small. Assume that ρ changes
with T according to
δρ
= −αδT ,
ρ
where α = −ρ −1 (∂ρ/∂T )p is the thermal expansion coefficient. For a perfect gas
α = 1/T ∼ 3 × 10−3 K −1 and for typical liquids α ∼ 5 × 10−4 K−1 . With a temperature difference in the fluid of 10 ◦ C, the variation of density can be only a few
percent at most. It turns out that ρ −1 (Dρ/Dt) can also be no larger than a few
percent of the velocity gradients in ∇ • u. To see this, assume that the flow field is
characterized by a length scale L, a velocity scale U , and a temperature scale δT .
By this we mean that the velocity varies by U and the temperature varies by δT , in
a distance of order L. The ratio of the magnitudes of the two terms in the continuity
equation is
(U/ρ)(δρ/L)
δρ
(1/ρ)u(∂ρ/∂x)
(1/ρ)(Dρ/Dt)
∼
=
= αδT ≪ 1,
∼
∇•u
∂u/∂x
U/L
ρ
which allows us to replace continuity equation (4.83) by its incompressible
form (4.84).
Momentum Equation
Because of the incompressible continuity equation ∇ • u = 0, the stress tensor is
given by equation (4.41). From equation (4.45), the equation of motion is then
ρ
Du
= −∇p + ρg + µ∇ 2 u.
Dt
(4.85)
127
18. Boussinesq Approximation
Consider a hypothetical static reference state in which the density is ρ0 everywhere and
the pressure is p0 (z), so that ∇p0 = ρ0 g. Subtracting this state from equation (4.85)
and writing p = p0 + p ′ and ρ = ρ0 + ρ ′ , we obtain
ρ
Du
= −∇p ′ + ρ ′ g + µ∇ 2 u.
Dt
(4.86)
Dividing by ρ0 , we obtain
1
ρ′
ρ ′ Du
= − ∇p ′ + g + ν∇ 2 u,
1+
ρ0 Dt
ρ0
ρ0
where ν = µ/ρ0 . The ratio ρ ′ /ρ0 appears in both the inertia and the buoyancy terms.
For small values of ρ ′ /ρ0 , the density variations generate only a small correction to
the inertia term and can be neglected. However, the buoyancy term ρ ′ g/ρ0 is very
important and cannot be neglected. For example, it is these density variations that
drive the convective motion when a layer of fluid is heated. The magnitude of ρ ′ g/ρ0
is therefore of the same order as the vertical acceleration ∂w/∂t or the viscous term
ν∇ 2 w. We conclude that the density variations are negligible the momentum equation,
except when ρ is multiplied by g.
Heat Equation
From equation (4.66), the thermal energy equation is
ρ
De
= −∇ • q − p(∇ • u) + φ.
Dt
(4.87)
Although the continuity equation is approximately ∇ • u = 0, an important point is
that the volume expansion term p(∇ • u) is not negligible compared to other dominant
terms of equation (4.87); only for incompressible liquids is p(∇ • u) negligible in
equation (4.87). We have
p ∂ρ
DT
DT
p Dρ
•
≃
= −pα
.
−p∇ u =
ρ Dt
ρ ∂T p Dt
Dt
Assuming a perfect gas, for which p = ρRT , Cp − Cv = R and α = 1/T , the
foregoing estimate becomes
−p∇ • u = −ρRT α
DT
DT
= −ρ(Cp − Cv )
.
Dt
Dt
Equation (4.87) then becomes
ρCp
DT
= −∇ • q + φ,
Dt
(4.88)
where we used e = Cv T for a perfect gas. Note that we would have gotten Cv
(instead of Cp ) on the left-hand side of equation (4.88) if we had dropped ∇ • u in
equation (4.87).
128
Conservation Laws
Now we show that the heating due to viscous dissipation of energy is negligible under the restrictions underlying the Boussinesq approximation. Comparing
the magnitudes of viscous heating with the left-hand side of equation (4.88), we
obtain
2µeij eij
µU 2 /L2
ν U
φ
∼
∼
=
.
ρCp (DT /Dt)
ρCp uj (∂T /∂xj )
ρ0 Cp U δT /L
Cp δT L
In typical situations this is extremely small (∼ 10−7 ). Neglecting φ, and assuming
Fourier’s law of heat conduction
q = −k∇T ,
the heat equation (4.88) finally reduces to (if k = const.)
DT
= κ∇ 2 T ,
Dt
where κ ≡ k/ρCp is the thermal diffusivity.
Summary: The Boussinesq approximation applies if the Mach number of the
flow is small, propagation of sound or shock waves is not considered, the vertical scale of the flow is not too large, and the temperature differences in the
fluid are small. Then the density can be treated as a constant in both the continuity and the momentum equations, except in the gravity term. Properties of
the fluid such as µ, k, and Cp are also assumed constant in this approximation.
Omitting Coriolis forces, the set of equations corresponding to the Boussinesq
approximation is
∇•u=0
Du
1 ∂p
=−
+ ν∇ 2 u
Dt
ρ0 ∂x
1 ∂p
Dv
=−
+ ν∇ 2 v
Dt
ρ0 ∂y
Dw
1 ∂p ρg
=−
−
+ ν∇ 2 w
Dt
ρ0 ∂z
ρ0
DT
= κ∇ 2 T
Dt
ρ = ρ0 [1 − α(T − T0 )],
(4.89)
where the z-axis is taken upward. The constant ρ0 is a reference density corresponding to a reference temperature T0 , which can be taken to be the mean temperature
in the flow or the temperature at a boundary. Applications of the Boussinesq set
can be found in several places throughout the book, for example, in the problems of
wave propagation in a density-stratified medium, thermal instability, turbulence in a
stratified medium, and geophysical fluid dynamics.
19. Boundary Conditions
19. Boundary Conditions
The differential equations we have derived for the conservation laws are subject to
boundary conditions in order to properly formulate any problem. Specifically, the
Navier-Stokes equations are of a form that requires the velocity vector to be given on
all surfaces bounding the flow domain.
If we are solving for an external flow, that is, a flow over some body, we must
specify the velocity vector and the thermodynamic state on a closed distant surface.
On a solid boundary or at the interface between two immiscible liquids, conditions
may be derived from the three basic conservation laws as follows.
In Figure 4.22, a “pillbox” is drawn through the interface surface separating
medium 1 (fluid) from medium 2 (solid or liquid immiscible with fluid 1). Here dA1
and dA2 are elements of the end face areas in medium 1 and medium 2, respectively,
locally tangent to the interface, and separated from each other by a distance l. Now
apply the conservation laws to the volume defined by the pillbox. Next, let l → 0,
keeping A1 and A2 in the different media. As l → 0, all volume integrals → 0 and
the integral over the side area, which is proportional to l, tends to zero as well. Define
a unit vector n, normal to the interface at the pillbox and pointed into medium 1.
Mass conservation gives ρ1 u1 · n = ρ2 u2 · n at each point on the interface as the
end face area becomes small. (Here we assume that the coordinates are fixed to the
interface, that is, the interface is at rest. Later in this section we show the modifications
necessary when the interface is moving.)
If medium 2 is a solid, then u2 = 0 there. If medium 1 and medium 2 are immiscible liquids, no mass flows across the boundary surface. In either case, u1 · n = 0
on the boundary. The same procedure applied to the integral form of the momentum
equation (4.16) gives the result that the force/area on the surface, ni τij is continuous
across the interface if surface tension is neglected. If surface tension is included, a
jump in pressure in the direction normal to the interface must be added; see Chapter 1,
Section 6 and the discussion later in this section.
Applying the integral form of energy conservation (4.64) to a pillbox of infinitesimal height l gives the result ni qi is continuous across the interface, or explicity,
k1 (∂T1 /∂n) = k2 (∂T2 /∂n) at the interface surface. The heat flux must be continuous
at the interface; it cannot store heat.
Two more boundary conditions are required to completely specify a problem
and these are not consequences of any conservation law. These boundary conditions
are: no slip of a viscous fluid is permitted at a solid boundary v1 · t = 0; and no
Figure 4.22 Interface between two media; evaluation of boundary conditions.
129
130
Conservation Laws
temperature jump is permitted at the boundary T1 = T2 . Here t is a unit vector
tangent to the boundary.
Known violations of the no-slip boundary condition occur for superfluid helium
at or below 2.17◦ K, which has an immeasurable small (essentially zero) viscosity. On
the other hand, the appearance of slip is created when water or water-based fluids flow
over finely textured “superhydrophobic” (strongly water repellent) coated surfaces.
This is described by Gogte et al. (2005). Surface textures must be much smaller than
the capillary length for water and were typically about 10µm . The fluid did not slip
on the protrusions but did not penetrate the valleys because of the surface tension,
giving the appearance of slip. Both slip and temperature jump are known to occur
in highly rarefied gases, where the mean distances between intermolecular collisions
become of the order of the length scales of interest in the problem. The details are
closely related to the manner of gas-surface interaction of momentum and energy.
A recent review was given by McCormick (2005).
Boundary condition at a moving, deforming surface
Consider a surface in space that may be moving or deforming in some arbitrary way.
Examples may be flexible solid boundaries, the interface between two immiscible
liquids, or a moving shock wave, as described in Chapter 16. The first two examples
do not permit mass flow across the interface, whereas the third does. Such a surface can be defined and its motion described in inertial coordinates by the equation
f (x, y, z, t) = 0. We often must treat problems in which boundary conditions must
be satisfied on such a moving, deforming interface. Let the velocity of a point that
remains on the surface be us . An observer that remains on the surface always sees
f = 0, so for that observer,
df/dt = ∂f/∂t + us • ∇f = 0 on f = 0.
(4.90)
A fluid particle has velocity u. If no fluid flows across f = 0, then u • ∇f =
us • ∇f = −∂f/∂t. Thus the condition that there be no mass flow across the surface
becomes,
(4.91)
∂f/∂t + u • ∇f ≡ Df/Dt = 0 on f = 0.
If there is mass flow across the surface, it is proportional to the relative velocity
between the fluid and the surface, (ur )n = u • n − us • n, where n = ∇f/|∇f |.
(ur )n = u • ∇f/|∇f | + [1/|∇f |][∂f/∂t] = [1/|∇f |]Df/Dt.
(4.92)
Thus the mass flow rate across the surface (per unit surface area) is represented by
[ρ/|∇f |]Df/Dt on f = 0.
(4.93)
Again, if no mass flows across the surface, the requirement is Df/Dt = 0 on f = 0.
Surface tension revisited: generalized discussion
As we discussed in Section 1.6 (p. 8), attractive intermolecular forces dominate
in a liquid, whereas in a gas repulsive forces are larger. However, as a liquid-gas
131
19. Boundary Conditions
phase boundary is approached from the liquid side, these attractive forces are not
felt equally because there are many fewer liquid phase molecules near the phase
boundary. Thus there tends to be an unbalanced attraction to the interior of the
liquid of the molecules on the phase boundary. This is called “surface tension”
and its manifestation is a pressure increment across a curved interface. A somewhat more detailed description is provided in texts on physicochemical hydrodynamics. Two excellent sources are Probstein (1994, Chapter 10) and Levich (1962,
Chapter VII).
H. Lamb, Hydrodynamics (6th Edition, p. 456) writes, “Since the condition of
stable equilibrium is that the free energy be a minimum, the surface tends to contract
as much as is consistent with the other conditions of the problem.” Thus we are led
to introduce the Helmoltz free energy (per unit mass) via
F = e − T S,
(4.94)
where the notation is consistent with that used in Section 1.8. If the free energy
is a minimum, then the system is in a state of stable equilibrium. F is called the
thermodynamic potential at constant volume [E. Fermi, T hermodynamics, p. 80].
For a reversible, isothermal change, the work done on the system is the gain in total
free energy F ,
dF = de − TdS − SdT,
(4.95)
where the last term is zero for an isothermal change. Then, from (1.18), dF = −pdv =
work done on system. (These relations suggest that surface tension decreases with
increasing temperature.)
For an interface of area = A, separating two media of densities ρ1 and ρ2 , with
volumes V1 and V2 , respectively, and with a surface tension coefficient σ (corresponding to free energy per unit area), the total (Helmholtz) free energy of the system can
be written as
F = ρ1V1F1 + ρ2 V2 F2 + Aσ.
(4.96)
If σ > 0, then the two media (fluids) are immiscible; on the other hand, if σ < 0,
corresponding to surface compression, then the two fluids mix freely. In the following,
we shall assume that σ = const. Flows driven by surface tension gradients are called
Marangoni flows and are not discussed here. Our discussion will follow that given by
G. K. Batchelor, An Introduction to Fluid Dynamics, pp. 61ff.
We wish to determine the shape of a boundary between two stationary fluids
compatible with mechanical equilibrium. Let the equation of the interface surface be
given by f (x, y, z) = 0 = z − ζ (x, y). Align the coordinates so that ζ (0, 0) = 0,
∂ζ /∂x|0,0 = 0, ∂ζ /∂y|0,0 = 0. See Figure 4.23. A normal to this surface is obtained
by forming the gradient, n = ∇[z − ζ (x, y)] = k − i∂ζ /∂x − j∂ζ /∂y. The (x, y, z)
components of n are (−∂ζ /∂x, −∂ζ /∂y, 1). Now the tensile forces on the bounding
132
Conservation Laws
z
dr
surface z 2 (x, y) 5 0
0
y
x
Figure 4.23 Geometry of equilibrium interface with surface tension.
line of the surface are obtained from the line integral
= σ dr × n
= σ (i dx + j dy + k dz) × (k − i∂ζ /∂x − j∂ζ /∂y)
= σ [−k(∂ζ /∂y)dx − jdx + k(∂ζ /∂x)dy + idy − j(∂ζ /∂x)dz + i(∂ζ /∂y)dz].
This integral is carried out over a contour C, which bounds the area A. Let that contour
C be in a z = const. plane so that dz = 0 on C. Then note that
(idy − jdx) = −k × (idx + jdy) = −k × dr = 0.
Then the tensile force acting on the bounding line C of the surface A
= kσ [−(∂ζ /∂y)dx + (∂ζ /∂x)dy].
Now use Stokes’ theorem in the form
F • dr = (∇ × F) • dA, where here F = −(∂ζ /∂y)i + (∂ζ /∂x)j. Then
A
C=∂A
∇ × F = (∂Fy /∂x − ∂Fx /∂y)k = (∂ 2 ζ /∂x 2 + ∂ 2 ζ /∂y 2 )k, and
σ
C=∂A
[(−∂ζ /∂y)dx + (∂ζ /∂x)dy] = σ
A
(∂ 2 ζ /∂x 2 + ∂ 2 ζ /∂y 2 )dAz .
(4.97)
We had expanded in a small neighborhood of the origin so the force per surface
area is the last integrand = σ (∂ 2 ζ /∂x 2 + ∂ 2 ζ /∂y 2 )0,0 , and this is interpreted as a
pressure difference across the surface. The curvature of the surface in the y = 0 plane
= [∂ 2 ζ /∂x 2 ][1 + (∂ζ /∂x)2 ]−3/2 . Since this is evaluated at (0,0) where ∂ζ /∂x = 0,
133
19. Boundary Conditions
the curvature reduces to ∂ 2 ζ /∂x 2 ≡ 1/R1 (defining R1 ). Similarly, the curvature in
the x = 0 plane at (0,0) is ∂ 2 ζ /∂y 2 ≡ 1/R2 (defining R2 ). Thus we say
p = σ (1/R1 + 1/R2 ),
(4.98)
where the pressure is greater on the side with the center of curvature of the interface.
Batchelor (loc. cit., p. 64) writes “An unbounded surface with a constant sum of the
principal curvatures is spherical, and this must be the equilibrium shape of the surface.
This result also follows from the fact that in a state of (stable) equilibrium the energy
of the surface must be a minimum consistent with a given value of the volume of the
drop or bubble, and the sphere is the shape which has the least surface area for a given
volume.” The original source of this analysis is Lord Rayleigh (J. W. Strutt), “On the
Theory of Surface Forces,” Phil. Mag. (Ser. 5), Vol. 30, pp. 285–298, 456–475 (1890).
For an air bubble in water, gravity is an important factor for bubbles of millimeter
size, as we shall see here. The hydrostatic pressure for a liquid is obtained from
pL + ρgz = const., where z is measured positively upwards from the free surface
and g is downwards. Thus for a gas bubble beneath the free surface,
pG = pL + σ (1/R1 + 1/R2 ) = const. − ρgz + σ (1/R1 + 1/R2 ).
Gravity and surface tension are of the same order in effect over a length scale
(σ/ρg)1/2 . For an air bubble in water at 288 ◦ K, this scale = [7.35 × 10−2 N/m/
(9.81 m/s2 × 103 kg/m3 )]1/2 = 2.74 × 10−3 m.
Example 4.3. Calculation of the shape of the free surface of a liquid adjoining an
infinite vertical plane wall. With reference to Figure 4.24, as defined above, 1/R1 =
[∂ 2 ζ /∂x 2 ][1 + (∂ζ /∂x)2 ]−3/2 = 0, and 1/R2 = [∂ 2 ζ /∂y 2 ][1 + (∂ζ /∂y)2 ]−3/2 .
At the free surface, ρgζ − σ/R2 = const. As y → ∞, ζ → 0, and R2 → ∞, so
const. = 0. Then ρgζ /σ − ζ ′′ /(1 + ζ ′2 )3/2 = 0.
Multiply by the integrating factor ζ ′ and integrate. We obtain (ρg/2σ )ζ 2 + (1 +
′2
−1/2
ζ )
= C. Evaluate C as y → ∞, ζ → 0, ζ ′ → 0. Then C = 1. We look at
y = 0, z = ζ = h to find h. The slope at the wall, ζ ′ = tan(θ + π/2) = − cot θ.
Then 1 + ζ ′2 = 1 + cot 2 θ = csc2 θ . Thus we now have (ρg/2σ )h2 = 1 − 1/ csc θ
Z
Gas
interface z 5 (x, y)
Solid h
u
Liquid
y
Figure 4.24 Free surface of a liquid adjoining a vertical plane wall.
134
Conservation Laws
= 1−sin θ, so that h2 = (2σ/ρg)(1−sin θ). Finally we seek to integrate to obtain the
shape of the interface. Squaring and rearranging the result above, the differential equation we must solve may be written as 1+(dζ /dy)2 = [1−(ρg/2σ )ζ 2 ]−2 . Solving for
the slope and taking the negative square root (since the slope is negative for positive y),
dζ /dy = −{1 − [1 − (ρg/2σ )ζ 2 ]2 }1/2 [1 − (ρg/2σ )ζ 2 ]−1 .
Define σ/ρg = d 2 , ζ /d = η. Rewriting the equation in terms of y/d and η, and
separating variables,
2(1 − η2 /2)η−1 (4 − η2 )−1/2 dη = d(y/d).
The integrand on the left is simplified by partial fractions and the constant of integration is evaluated at y = 0 when η = h/d. Finally
cosh−1 (2d/ζ ) − (4 − ζ 2 /d 2 )1/2 − cosh−1 (2d/ h) + (4 − h2 /d 2 )1/2 = y/d
gives the shape of the interface in terms of y(ζ ).
Analysis of surface tension effects results in the appearance of additional dimensionless parameters in which surface tension is compared with other effects such
as viscous stresses, body forces such as gravity, and inertia. These are defined in
Chapter 8.
Exercises
1. Let a one-dimensional velocity field be u = u(x, t), with v = 0 and
w = 0. The density varies as ρ = ρ0 (2 − cos ωt). Find an expression for u(x, t)
if u(0, t) = U .
2. In Section 3 we derived the continuity equation (4.8) by starting from the integral form of the law of conservation of mass for a fixed region. Derive equation (4.8)
by starting from an integral form for a material volume. [Hint: Formulate the principle
for a material volume and then use equation (4.5).]
3. Consider conservation of angular momentum derived from the angular
momentum principle by the word statement: Rate of increase of angular momentum
in volume V = net influx of angular momentum across the bounding surface A of V
+ torques due to surface forces + torques due to body forces. Here, the only torques
are due to the same forces that appear in (linear) momentum conservation. The possibilities for body torques and couple stresses have been neglected. The torques due to
the surface forces are manipulated asfollows. The torque about a point O due to the
element of surface force τmk dAm is ǫij k xj τmk dAm , where x is the position vector
from O to the element dA. Using Gauss’ theorem, we write this as a volume integral,
∂xj
∂τmk
∂
dV
(xj τmk )dV = εij k
τmk + xj
εij k
∂xm
∂xm
∂x m
V
V
∂τmk
= εij k
dV ,
τj k + x j
∂xm
V
135
Exercises
where we have used ∂xj /∂xm = δj m . The second term is V x × ∇ · τ dV and
combines
with the remaining terms in the conservation
of angular momentum to give
x×(Linear
Momentum:
equation
(4.17))
dV
=
V
V ǫij k τj k dV . Since the left-hand
side = 0 for any volume V , we conclude that εij k τkj = 0, which leads to τij = τj i .
4. Near the end of Section 7 we derived the equation of motion (4.15) by starting
from an integral form for a material volume. Derive equation (4.15) by starting from
the integral statement for a fixed region, given by equation (4.22).
5. Verify the validity of the second form of the viscous dissipation given in
equation (4.60). [Hint: Complete the square and use δij δij = δii = 3.]
6. A rectangular tank is placed on wheels and is given a constant horizontal
acceleration a. Show that, at steady state, the angle made by the free surface with the
horizontal is given by tan θ = a/g.
7. A jet of water with a diameter of 8 cm and a speed of 25 m/s impinges normally
on a large stationary flat plate. Find the force required to hold the plate stationary.
Compare the average pressure on the plate with the stagnation pressure if the plate is
20 times the area of the jet.
8. Show that the thrust developed by a stationary rocket motor is F = ρAU 2 +
A(p − patm ), where patm is the atmospheric pressure, and p, ρ, A, and U are,
respectively, the pressure, density, area, and velocity of the fluid at the nozzle exit.
9. Consider the propeller of an airplane moving with a velocity U1 . Take a
reference frame in which the air is moving and the propeller [disk] is stationary. Then
the effect of the propeller is to accelerate the fluid from the upstream value U1 to
the downstream value U2 > U1 . Assuming incompressibility, show that the thrust
developed by the propeller is given by
F =
ρA 2
(U2 − U12 ),
2
where A is the projected area of the propeller and ρ is the density (assumed constant).
Show also that the velocity of the fluid at the plane of the propeller is the average value
U = (U1 + U2 )/2. [Hint: The flow can be idealized by a pressure jump, of magnitude
p = F /A right at the location of the propeller. Also apply Bernoulli’s equation
between a section far upstream and a section immediately upstream of the propeller.
Also apply the Bernoulli equation between a section immediately downstream of the
propeller and a section far downstream. This will show that p = ρ(U22 − U12 )/2.]
10. A hemispherical vessel of radius R has a small rounded orifice of area A at
the bottom. Show that the time required to lower the level from h1 to h2 is given by
2π
1 5/2
2
5/2
3/2
3/2
t= √
h1 − h2
.
−
R h1 − h2
5
A 2g 3
11. Consider an incompressible planar Couette flow, which is the flow between
two parallel plates separated by a distance b. The upper plate is moving parallel to
136
Conservation Laws
itself at speed U , and the lower plate is stationary. Let the x-axis lie on the lower plate.
All flow fields are independent of x. Show that the pressure distribution is hydrostatic
and that the solution of the Navier–Stokes equation is
u(y) =
Uy
.
b
Write the expressions for the stress and strain rate tensors, and show that the viscous
dissipation per unit volume is φ = µU 2 /b2 .
Take a rectangular control volume for which the two horizontal surfaces coincide
with the walls and the two vertical surfaces are perpendicular to the flow. Evaluate
every term of energy equation (4.63) for this control volume, and show that the balance
is between the viscous dissipation and the work done in moving the upper surface.
12. The components of a mass flow vector ρu are ρu = 4x 2 y, ρv = xyz,
ρw = yz2 . Compute the net outflow through the closed surface formed by the planes
x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.
(a) Integrate over the closed surface.
(b) Integrate over the volume bounded by that surface.
13. Prove that the velocity field given by ur = 0, uθ = k/(2π r) can have only
two possible values of the circulation. They are (a) Ŵ = 0 for any path not enclosing
the origin, and (b) Ŵ = k for any path enclosing the origin.
14. Water flows through a pipe in a gravitational field as shown in the accompanying figure. Neglect the effects of viscosity and surface tension. Solve the appropriate
conservation equations for the variation of the cross-sectional area of the fluid column
A(z) after the water has left the pipe at z = 0. The velocity of the fluid at z = 0 is
uniform at v0 and the cross-sectional area is A0 .
15. Redo the solution for the “orifice in a tank” problem allowing for the fact
that in Fig. 4.20, h = h(t). How long does the tank take to empty?
Literature Cited
Aris, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, NJ:
Prentice-Hall. (The basic equations of motion and the various forms of the Reynolds transport theorem
are derived and discussed.)
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press. (This
contains an excellent and authoritative treatment of the basic equations.)
Supplemental Reading
Fermi, E. (1956). Thermodynamics, New York: Dover Publications, Inc.
Gogte, S. P. Vorobieff, R. Truesdell, A. Mammoli, F. van Swol, P. Shah, and C. J. Brinker (2005). “Effective
slip on textured superhydrophobic surfaces.” Phys. Fluids 17: 051701.
Lamb, H. (1945). Hydrodynamics, Sixth Edition, New York: Dover Publications, Inc.
Levich, V. G. (1962). Physicochemical Hydrodynamics, Second Edition, Englewood Cliffs, NJ:
Prentice-Hall, Chapter VII.
Lord Rayleigh (J. W. Strutt) (1890). “On the Theory of Surface Forces.” Phil. Mag. (Ser. 5), 30: 285–298,
456–475.
McCormick, N. J. (2005). “Gas-surface accomodation coefficients from viscous slip and temperature jump
coefficients.” Phys. Fluids 17: 107104.
Holton, J. R. (1979). An Introduction to Dynamic Meteorology, New York: Academic Press.
Pedlosky, J. (1987). Geophysical Fluid Dynamics, New York: Springer-Verlag.
Probstein, R. F. (1994). Physicochemical Hydrodynamics, Second Edition, New York: John Wiley & Sons,
Chapter 19.
Spiegel, E. A. and G. Veronis (1960). On the Boussinesq approximation for a compressible fluid. Astrophysical Journal 131: 442–447.
Stommel H. M. and D. W. Moore (1989) An Introduction to the Coriolis Force. New York: Columbia
University Press.
Truesdell, C. A. (1952). Stokes’ principle of viscosity. Journal of Rational Mechanics and Analysis 1:
228–231.
Supplemental Reading
Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability, London: Oxford University Press.
(This is a good source to learn the basic equations in a brief and simple way.)
Dussan V., E. B. (1979). “On the Spreading of Liquids on Solid Surfaces: Static and Dynamic Contact
Lines.” Annual Rev. of Fluid Mech. 11, 371–400.
Levich, V. G. and V. S. Krylov (1969). “Surface Tension Driven Phenomena.” Annual Rev. of Fluid Mech.
1, 293–316.
137
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Chapter 5
Vorticity Dynamics
1. Introduction . . . . . . . . . . . . . . . . . . . . . 139
2. Vortex Lines and Vortex
Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3. Role of Viscosity in Rotational and
Irrotational Vortices . . . . . . . . . . . . . 141
Solid-Body Rotation . . . . . . . . . . . . . 141
Irrotational Vortex . . . . . . . . . . . . . . . 142
Discussion. . . . . . . . . . . . . . . . . . . . . . . 143
4. Kelvin’s Circulation Theorem . . . . 144
Discussion of Kelvin’s Theorem . . 147
Helmholtz Vortex Theorems. . . . . . 149
5. Vorticity Equation in a Nonrotating
Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6. Velocity Induced by a Vortex Filament:
Law of Biot and Savart . . . . . . . . . . 151
7. Vorticity Equation in a Rotating
Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Meaning of (ω • ∇)u . . . . . . . . . . . . . 155
Meaning of 2( • ∇)u . . . . . . . . . . . . 156
8. Interaction of Vortices . . . . . . . . . . . . 157
9. Vortex Sheet . . . . . . . . . . . . . . . . . . . . . 161
Exercises . . . . . . . . . . . . . . . . . . . . . . . . 161
Literature Cited . . . . . . . . . . . . . . . . . 163
Supplemental Reading . . . . . . . . . . . 163
1. Introduction
Motion in circular streamlines is called vortex motion. The presence of closed streamlines does not necessarily mean that the fluid particles are rotating about their own
centers, and we may have rotational as well as irrotational vortices depending on
whether the fluid particles have vorticity or not. The two basic vortex flows are the
solid-body rotation
1
uθ = ωr,
(5.1)
2
and the irrotational vortex
uθ =
Ŵ
.
2π r
(5.2)
These are discussed in Chapter 3, Section 11, where also, the angular velocity in the
solid-body rotation was denoted by ω0 = ω/2. Moreover, the vorticity of an element
is everywhere equal to ω for the solid-body rotation represented by equation (5.1), so
that the circulation around any contour is ω times the area enclosed by the contour.
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50005-8
139
140
Vorticity Dynamics
In contrast, the flow represented by equation (5.2) is irrotational everywhere except
at the origin, where the vorticity is infinite. All the vorticity of this flow is therefore
concentrated on a line coinciding with the vortex axis. Circulation around any circuit
not enclosing the origin is therefore zero, and that enclosing the origin is Ŵ. An
irrotational vortex is therefore called a line vortex. Some aspects of the dynamics of
flows with vorticity are examined in this chapter.
2. Vortex Lines and Vortex Tubes
A vortex line is a curve in the fluid such that its tangent at any point gives the direction
of the local vorticity. A vortex line is therefore related to the vorticity vector the same
way a streamline is related to the velocity vector. If ωx , ωy , and ωz are the Cartesian
components of the vorticity vector ω, then the orientation of a vortex line satisfies the
equations
dy
dz
dx
=
=
,
ωx
ωy
ωz
(5.3)
which is analogous to equation (3.7) for a streamline. In an irrotational vortex, the
only vortex line in the flow field is the axis of the vortex. In a solid-body rotation, all
lines perpendicular to the plane of flow are vortex lines.
Vortex lines passing through any closed curve form a tubular surface, which is
called a vortex tube. Just as streamlines bound a streamtube, a group of vortex lines
bound a vortex tube (Figure 5.1). The circulation around a narrow vortex tube is
dŴ = ω • dA, which is similar to the expression for the rate of flow dQ = u • dA
through a narrow streamtube. The strength of a vortex tube is defined as the circulation
around a closed circuit taken on the surface of the tube and embracing it just once.
From Stokes’ theorem it follows that the strength of a vortex tube is equal to the mean
vorticity times its cross-sectional area.
Figure 5.1 Analogy between streamtube and vortex tube.
141
3. Role of Viscosity in Rotational and Irrotational Vortices
3. Role of Viscosity in Rotational and Irrotational Vortices
The role of viscosity in the two basic types of vortex flows, namely the solid-body
rotation and the irrotational vortex, is examined in this section. Assuming incompressible flow, we shall see that in one of these flows the viscous terms in the momentum
equation drop out, although the viscous stress and dissipation of energy are nonzero.
The two flows are examined separately in what follows.
Solid-Body Rotation
As discussed in Chapter 3, fluid elements in a solid-body rotation do not deform.
Because viscous stresses are proportional to deformation rate, they are zero in this
flow. This can be demonstrated by using the expression for viscous stress in polar
coordinates:
1 ∂ur
∂ uθ
= 0,
+r
σrθ = µ
r ∂θ
∂r r
where we have substituted uθ = ωr/2 and ur = 0. We can therefore apply the inviscid
Euler equations, which in polar coordinates simplify to
−ρ
u2θ
∂p
=−
r
∂r
∂p
0=−
− ρg.
∂z
(5.4)
The pressure difference between two neighboring points is therefore
dp =
∂p
∂p
1
dr +
dz = ρrω2 dr − ρg dz,
∂r
∂z
4
where uθ = ωr/2 has been used. Integration between any two points 1 and 2 gives
p2 − p1 =
1 2 2
ρω (r2 − r12 ) − ρg(z2 − z1 ).
8
(5.5)
Surfaces of constant pressure are given by
z2 − z1 =
1 2
(ω /g)(r22 − r12 ),
8
which are paraboloids of revolution (Figure 5.2).
The important point to note is that viscous stresses are absent in this flow. (The
viscous stresses, however, are important during the transient period of initiating the
motion, say by steadily rotating a tank containing a viscous fluid at rest.) In terms of
velocity, equation (5.5) can be written as
1
1
p2 − ρu2θ 2 + ρgz2 = p1 − ρu2θ 1 + ρgz1 ,
2
2
142
Vorticity Dynamics
Figure 5.2 Constant pressure surfaces in a solid-body rotation generated in a rotating tank containing
liquid.
which shows that the Bernoulli function B = u2θ /2 + gz + p/ρ is not constant for
points on different streamlines. This is expected of inviscid rotational flows.
Irrotational Vortex
In an irrotational vortex represented by
uθ =
Ŵ
,
2π r
the viscous stress is
σrθ
1 ∂ur
µŴ
∂ uθ
=µ
= − 2,
+r
r ∂θ
∂r r
πr
which is nonzero everywhere. This is because fluid elements do undergo deformation
in such a flow, as discussed in Chapter 3. However, the interesting point is that the net
viscous force on an element again vanishes, just as in the case of solid body rotation.
In an incompressible flow, the net viscous force per unit volume is related to vorticity
by (see equation 4.48)
∂σij
= −µ(∇ × ω)i ,
∂xj
(5.6)
which is zero for irrotational flows. The viscous forces on the surfaces of an element
cancel out, leaving a zero resultant. The equations of motion therefore reduce to
the inviscid Euler equations, although viscous stresses are nonzero everywhere. The
3. Role of Viscosity in Rotational and Irrotational Vortices
pressure distribution can therefore be found from the inviscid set (5.4), giving
dp =
ρŴ 2
dr − ρg dz,
4π 2 r 3
where we have used uθ = Ŵ/(2πr). Integration between any two points gives
ρ
p2 − p1 = − (u2θ 2 − u2θ 1 ) − ρg(z2 − z1 ),
2
which implies
u2
u2
p1
p2
+ θ 1 + gz1 =
+ θ 2 + gz2 .
ρ
2
ρ
2
This shows that Bernoulli’s equation is applicable between any two points in the flow
field and not necessarily along the same streamline, as would be expected of inviscid
irrotational flows. Surfaces of constant pressure are given by
u2θ 1
u2θ 2
1
Ŵ2
1
z2 − z1 =
−
=
− 2 ,
2g
2g
8π 2 g r12
r2
which are hyperboloids of revolution of the second degree (Figure 5.3). Flow is
singular at the origin, where there is an infinite velocity discontinuity. Consequently,
a real vortex such as that found in the atmosphere or in a bathtub necessarily has a
rotational core (of radius R, say) in the center where the velocity distribution can be
idealized by uθ = ωr/2. Outside the core the flow is nearly irrotational and can be
idealized by uθ = ωR 2 /2r; here we have chosen the value of circulation such that uθ
is continuous at r = R (see Figure 3.17b). The strength of such a vortex is given by
Ŵ = (vorticity)(core area) = πωR 2 .
One way of generating an irrotational vortex is by rotating a solid circular cylinder
in an infinite viscous fluid (see Figure 9.7). It is shown in Chapter 9, Section 6 that
the steady solution of the Navier–Stokes equations satisfying the no-slip boundary
condition (uθ = ωR/2 at r = R) is
uθ =
ωR 2
2r
r R,
where R is the radius of the cylinder and ω/2 is its constant angular velocity; see
equation (9.15). This flow does not have any singularity in the entire field and is
irrotational everywhere. Viscous stresses are present, and the resulting viscous dissipation of kinetic energy is exactly compensated by the work done at the surface of
the cylinder. However, there is no net viscous force at any point in the steady state.
Discussion
The examples given in this section suggest that irrotationality does not imply the
absence of viscous stresses. In fact, they must always be present in irrotational flows
143
144
Vorticity Dynamics
Figure 5.3 Irrotational vortex in a liquid.
of real fluids, simply because the fluid elements deform in such a flow. However
the net viscous force vanishes if ω = 0, as can be seen in equation (5.6). We have
also given an example, namely that of solid-body rotation, in which there is uniform
vorticity and no viscous stress at all. However, this is the only example in which
rotation can take place without viscous effects, because equation (5.6) implies that
the net force is zero in a rotational flow if ω is uniform everywhere. Except for this
example, fluid rotation is accomplished by viscous effects. Indeed, we shall see later
in this chapter that viscosity is a primary agent for vorticity generation.
4. Kelvin’s Circulation Theorem
Several theorems of vortex motion in an inviscid fluid were published by Helmholtz
in 1858. He discovered these by analogy with electrodynamics. Inspired by this work,
Kelvin in 1868 introduced the idea of circulation and proved the following theorem:
In an inviscid, barotropic flow with conservative body forces, the circulation around
a closed curve moving with the fluid remains constant with time, if the motion is
observed from a nonrotating frame. The theorem can be restated in simple terms as
follows: At an instant of time take any closed contour C and locate the new position
of C by following the motion of all of its fluid elements. Kelvin’s circulation theorem
states that the circulations around the two locations of C are the same. In other words,
DŴ
= 0,
Dt
(5.7)
145
4. Kelvin’s Circulation Theorem
Figure 5.4
Proof of Kelvin’s circulation theorem.
where D/Dt has been used to emphasize that the circulation is calculated around a
material contour moving with the fluid.
To prove Kelvin’s theorem, the rate of change of circulation is found as
DŴ
D
Dui
D
ui dxi =
=
dxi + ui
(dxi ),
(5.8)
Dt
Dt C
Dt
C Dt
C
where dx is the separation between two points on C (Figure 5.4). Using the momentum
equation
Dui
1 ∂p
1
=−
+ gi + σij, j ,
Dt
ρ ∂xi
ρ
where σij is the deviatoric stress tensor (equation (4.33)). The first integral in equation (5.8) becomes
Dui
1 ∂p
1
dxi = −
dxi + gi dxi +
σij, j dxi
Dt
ρ ∂xi
ρ
=−
dp
+
ρ
gi dxi +
1
σij, j dxi ,
ρ
where we have noted that dp = ∇p • dx is the difference in pressure between two
neighboring points. Equation (5.8) then becomes
DŴ
dp
D
1
•
•
•
= g dx −
+
(∇ σ) dx + ui
(dxi ).
(5.9)
Dt
ρ
ρ
Dt
C
C
C
Each term of equation (5.9) will now be shown to be zero. Let the body force be
conservative, so that g = −∇, where is the force potential or potential energy
146
Vorticity Dynamics
per unit mass. Then the line integral of g along a fluid line AB is
B
g • dx = −
B
∇ • dx = −
d = A − B .
A
A
A
B
When the integral is taken around the closed fluid line, points A and B coincide,
showing that the first integral on the right-hand side of equation (5.9) is zero.
Now assume that the flow is barotropic, which means that density is a function
of pressure alone. Incompressible and isentropic (p/ρ γ = constant for a perfect gas)
flows are examples of barotropic flows. In such a case we can write ρ −1 as some
function of p, and we choose to write this in the form of the derivative ρ −1 ≡ dP /dp.
Then the integral of dp/ρ between any two points A and B can be evaluated, giving
A
B
dp
=
ρ
B
A
dP
dp = PB − PA .
dp
The integral around a closed contour is therefore zero.
If viscous stresses can be neglected for those particles making up contour C, then
the integral of the deviatoric stress tensor is zero. To show that the last integral in
equation (5.9) vanishes, note that the velocity at point x + dx on C is given by
u + du =
D
Dx
D
(x + dx) =
+
(dx),
Dt
Dt
Dt
so that
du =
D
(dx),
Dt
The last term in equation (5.9) then becomes
C
ui
D
(dxi ) =
Dt
C
ui dui =
C
d
1 2
2 ui
= 0.
This completes the proof of Kelvin’s theorem.
We see that the three agents that can create or destroy vorticity in a flow are
nonconservative body forces, nonbarotropic pressure-density relations, and viscous
stresses. An example of each follows. A Coriolis force in a rotating coordinate system
generates the “bathtub vortex” when a filled tank, initially as rest on the earth’s
surface, is drained. Heating from below in a gravitational field creates a buoyant force
generating an upward plume. Cooling from above and mass conservation require that
the motion be in cyclic rolls so that vorticity is created. Viscous stresses create vorticity
in the neighborhood of a boundary where the no-slip condition is maintained. A short
distance away from the boundary, the tangential velocity may be large. Then, because
there are large gradients transverse to the flow, vorticity is created.
4. Kelvin’s Circulation Theorem
Discussion of Kelvin’s Theorem
Because circulation is the surface integral of vorticity, Kelvin’s theorem essentially
shows that irrotational flows remain irrotational if the four restrictions are satisfied:
(1) Inviscid flow: In deriving the theorem, the inviscid Euler equation has been
used, but only along the contour C itself. This means that circulation is preserved if there are no net viscous forces along the path followed by C. If C
moves into viscous regions such as boundary layers along solid surfaces, then
the circulation changes. The presence of viscous effects causes a diffusion of
vorticity into or out of a fluid circuit, and consequently changes the circulation.
(2) Conservative body forces: Conservative body forces such as gravity act through
the center of mass of a fluid particle and therefore do not tend to rotate it.
(3) Barotropic flow: The third restriction on the validity of Kelvin’s theorem is that
density must be a function of pressure only. A homogeneous incompressible
liquid for which ρ is constant everywhere and an isentropic flow of a perfect
gas for which p/ρ γ is constant are examples of barotropic flows. Flows that are
not barotropic are called baroclinic. Consider fluid elements in barotropic and
baroclinic flows (Figure 5.5). For the barotropic element, lines of constant p are
parallel to lines of constant ρ, which implies that the resultant pressure forces
pass through the center of mass of the element. For the baroclinic element, the
Figure 5.5 Mechanism of vorticity generation in baroclinic flow, showing that the net pressure force does
not pass through the center of mass G. The radially inward arrows indicate pressure forces on an element.
147
148
Vorticity Dynamics
lines of constant p and ρ are not parallel. The net pressure force does not pass
through the center of mass, and the resulting torque changes the vorticity and
circulation.
As an example of the generation of vorticity in a baroclinic flow, consider a
gas at rest in a gravitational field. Let the gas be heated locally, say by chemical
action (such as explosion of a bomb) or by a simple heater (Figure 5.6). The
gas expands and rises upward. The flow is baroclinic because density here is
also a function of temperature. A doughnut-shaped ring-vortex (similar to the
smoke ring from a cigarette) forms and rises upward. (In a bomb explosion, a
mushroom-shaped cloud occupies the central hole of such a ring.) Consider a
closed fluid circuit ABCD when the gas is at rest; the circulation around it is
then zero. If the region near AB is heated, the circuit assumes the new location
A′ B′ CD after an interval of time; circulation around it is nonzero because
u • dx along A′ B′ is nonzero. The circulation around a material circuit has
therefore changed, solely due to the baroclinicity of the flow. This is one of
the reasons why geophysical flows, which are dominated by baroclinicity,
are full of vorticity. It should be noted that no restriction is placed on the
compressibility of the fluid, and Kelvin’s theorem is valid for incompressible
as well as compressible fluids.
(4) Nonrotating frame: Motion observed with respect to a rotating frame of reference can develop vorticity and circulation by mechanisms not considered in
our demonstration of Kelvin’s theorem. Effects of a rotating frame of reference
are considered in Section 7.
Under the four restrictions mentioned in the foregoing, Kelvin’s theorem essentially
states that irrotational flows remain irrotational at all times.
Figure 5.6 Local heating of a gas, illustrating vorticity generation on baroclinic flow.
5. Vorticity Equation in a Nonrotating Frame
Helmholtz Vortex Theorems
Under the same four restrictions, Helmholtz proved the following theorems on vortex
motion:
(1) Vortex lines move with the fluid.
(2) Strength of a vortex tube, that is the circulation, is constant along its length.
(3) A vortex tube cannot end within the fluid. It must either end at a solid boundary
or form a closed loop (a “vortex ring”).
(4) Strength of a vortex tube remains constant in time.
Here, we shall prove only the first theorem, which essentially says that fluid
particles that at any time are part of a vortex line always belong to the same vortex line.
To prove this result, consider an area S, bounded by a curve, lying on the surface of a
vortex tube without embracing it (Figure 5.7). As the vorticity vectors are everywhere
lying on the area element S, it follows that the circulation around the edge of S is
zero. After an interval of time, the same fluid particles form a new surface, say S′ .
According to Kelvin’s theorem, the circulation around S′ must also be zero. As this is
true for any S, the component of vorticity normal to every element of S′ must vanish,
demonstrating that S′ must lie on the surface of the vortex tube. Thus, vortex tubes
move with the fluid. Applying this result to an infinitesimally thin vortex tube, we get
the Helmholtz vortex theorem that vortex lines move with the fluid. A different proof
may be found in Sommerfeld (Mechanics of Deformable Bodies, pp. 130–132).
5. Vorticity Equation in a Nonrotating Frame
An equation governing the vorticity in a fixed frame of reference is derived in this
section. The fluid density is assumed to be constant, so that the flow is barotropic.
Viscous effects are retained. Effects of nonbarotropic behavior and a rotating frame
Figure 5.7
Proof of Helmholtz’s vortex theorem.
149
150
Vorticity Dynamics
of reference are considered in the following section. The derivation given here uses
vector notation, so that we have to use several vector identities, including those for
triple products of vectors. Readers not willing to accept the use of such vector identities
can omit this section and move on to the next one, where the algebra is worked out
in tensor notation without using such identities.
Vorticity is defined as
ω ≡ ∇ × u.
Because the divergence of a curl vanishes, vorticity for any flow must satisfy
∇ • ω = 0.
(5.10)
An equation for rate of change of vorticity is obtained by taking the curl of the equation
of motion. We shall see that pressure and gravity are eliminated during this operation.
In symbolic form, we want to perform the operation
1
∂u
+ u • ∇u = − ∇p + ∇ + ν∇ 2 u ,
(5.11)
∇×
∂t
ρ
where
is the body force potential. Using the vector identity
1
1
u • ∇u = (∇ × u) × u + ∇(u • u) = ω × u + ∇q 2 ,
2
2
and noting that the curl of a gradient vanishes, (5.11) gives
∂ω
+ ∇ × (ω × u) = ν∇ 2 ω,
∂t
(5.12)
where we have also used the identity ∇ × ∇ 2 u = ∇ 2 (∇ × u) in rewriting the viscous
term. The second term in equation (5.12) can be written as
∇ × (ω × u) = (u • ∇)ω − (ω • ∇)u,
where we have used the vector identity
∇ × (A × B) = A∇ • B + (B • ∇)A − B∇ • A − (A • ∇)B,
and that ∇ • u = 0 and ∇ • ω = 0. Equation (5.12) then becomes
Dω
= (ω • ∇)u + ν∇ 2 ω.
Dt
(5.13)
This is the equation governing rate of change of vorticity in a fluid with constant
ρ and conservative body forces. The term ν∇ 2 ω represents the rate of change of ω
due to diffusion of vorticity in the same way that ν∇ 2 u represents acceleration due
to diffusion of momentum. The term (ω • ∇)u represents rate of change of vorticity
due to stretching and tilting of vortex lines. This important mechanism of vorticity
151
6. Velocity Induced by a Vortex Filament: Law ofBiot and Savart
generation is discussed further near the end of Section 7, to which the reader can
proceed if the rest of that section is not of interest. Note that pressure and gravity
terms do not appear in the vorticity equation, as these forces act through the center
of mass of an element and therefore generate no torque.
6. Velocity Induced by a Vortex Filament: Law of
Biot and Savart
It is often useful to be able to calculate the velocity induced by a vortex filament with
arbitrary orientation in space. This result is used in thin airfoil theory. We shall derive
the velocity induced by a vortex filament for a constant density flow. (What actually
is required is a solenoidal velocity field.) We start with the definition of vorticity,
ω ≡ ∇ × u. Take the curl of this equation to obtain
∇ × ω = ∇ × (∇ × u) = ∇(∇ • u) − ∇ 2 u.
We shall asume that mass conservation can be written as ∇ • u = 0, (for example, if
ρ = const) and solve the vector Poisson equation for u in terms of ω. The Poisson
equation in the form ∇ 2 φ = −ρ(r)/ε leads to the solution expressed as φ(r) =
(4π ε)−1 V ′ ρ(r ′ )|r − r ′ |−1 dV ′ where the integration is over all of V ′ (r ′ ) space.
Using this form for each component of vorticity, we obtain for u,
u = (4π)−1
(∇ ′ × ω)|r − r ′ |−1 dV ′
(5.14)
V′
We take V ′ to be a small cylinder wrapped around the vortex line C through the point
r′ . See Figure 5.8. Equation (5.14) can be rewritten in general as
u = (4π)−1
{∇ ′ × [ω/|r − r ′ |] − [∇ ′ |r − r ′ |−1 ] × ω}dV ′
(5.15)
V′
We use the divergence theorem on the first integral in the form
A=∂V dA × F. Then (5.15) becomes
u = (4π)
−1
′
′
dA × ω/|r − r | +
A′ =∂V ′
V (∇
× F)dV =
dV ′ (∇ ′ |r − r ′ |) × ω/|r − r ′ |2
(5.16)
V′
Now shrink V ′ and A′ = ∂V ′ to surround the vortex line segment in the neighborhood
of r′ . On the two end faces of A′ , dA′ ||ω so dA′ × ω = 0. Since, ∇ • ω = 0, ω
is constant along a vortex line, so A′ dA′ × ω = ( A′ dA′ ) × ω = 0 and
sides
A′sides
sides
dA′ = 0 because the generatrix of A′sides is a closed curve. For the second
integral, dV ′ = dA′ • dl, where dA′ is an element of end face area and dl is arc length
along the vortex line. Now, by Stokes’ theorem, end ω • dA′ = C=∂A′ u • ds = Ŵ,
where Ŵ is the circulation around the vortex line C and ds is an element of arc
152
Vorticity Dynamics
argument point
r
r2r9
Γ
C
ω
parameter
point
r9
Figure 5.8 Geometry for derivation of Law of Biot and Savart.
length on the generatrix of A′ . Then ωdA′ • dl = Ŵdl since ω is parallel to dl. Now
∇ ′ |r − r ′ | = −1r−r′ (unit vector), so (5.16) reduces to u = −(4π )−1 C (1r−r′ /|r −
r ′ |2 ) × (Ŵdl) for any length of vortex line C. For a small segment of vortex line dl,
du = (Ŵ/4π )[dl × 1r−r′ /|r − r ′ |2 ]
(5.17)
is an expression of the Law of Biot and Savart.
7. Vorticity Equation in a Rotating Frame
A vorticity equation was derived in Section 5 for a fluid of uniform density in a
fixed frame of reference. We shall now generalize this derivation to include a rotating frame of reference and nonbarotropic fluids. The flow, however, will be assumed
nearly incompressible in the Boussinesq sense, so that the continuity equation is
approximately ∇ • u = 0. We shall also use tensor notation and not assume any
vector identity. Algebraic manipulations are cleaner if we adopt the comma notation introduced in Chapter 2, Section 15, namely, that a comma stands for a spatial
derivative:
A,i ≡
∂A
.
∂xi
A little practice may be necessary to feel comfortable with this notation, but it is very
convenient.
We first show that the divergence of ω is zero. From the definition ω = ∇ × u,
we obtain
ωi,i = (εinq uq,n ),i = εinq uq,ni .
In the last term, εinq is antisymmetric in i and n, whereas the derivative uq,ni is
symmetric in i and n. As the contracted product of a symmetric and an antisymmetric
153
7. Vorticity Equation in a Rotating Frame
tensor is zero, it follows that
ωi,i = 0
∇•ω=0
or
(5.18)
which shows that the vorticity field is nondivergent (solenoidal), even for compressible and unsteady flows.
The continuity and momentum equations for a nearly incompressible flow in
rotating coordinates are
ui,i = 0,
∂ui
+ uj ui,j + 2εij k
∂t
j uk
(5.19)
1
= − p,i + gi + νui,jj ,
ρ
(5.20)
where is the angular velocity of the coordinate system and gi is the effective gravity
(including centrifugal acceleration); see equation (4.55). The advective acceleration
can be written as
uj ui,j = uj (ui,j − uj,i ) + uj uj,i
1
= −uj εij k ωk + (uj uj ),i
2
1 2
= −(u × ω)i + (uj ),i ,
2
(5.21)
where we have used the relation
εij k ωk = εij k (εkmn un,m )
= (δim δj n − δin δj m ) un,m = uj,i − ui,j .
(5.22)
The viscous diffusion term can be written as
νui,jj = ν(ui,j − uj,i ),j + νuj,ij = −νεij k ωk,j ,
(5.23)
where we have used equation (5.22) and the fact that uj,ij = 0 because of the
continuity equation (5.19). Relation (5.22) says that ν∇ 2 u = −ν∇ × ω, which we
have used several times before (e.g., see equation (4.48)). Because × u = −u × ,
the Coriolis term in equation (5.20) can be written as
2εij k
j uk
= −2εij k
k uj .
(5.24)
Substituting equations (5.21), (5.23), and (5.24) into equation (5.20), we obtain
∂ui
+
∂t
1 2
u +
2 j
− εij k uj (ωk + 2
,i
where we have also assumed g = −∇.
k)
1
= − p,i − νεij k ωk,j ,
ρ
(5.25)
154
Vorticity Dynamics
Equation (5.25) is another form of the Navier–Stokes equation, and the vorticity
equation is obtained by taking its curl. Since ωn = εnqi ui,q , it is clear that we need
to operate on (5.25) by εnqi ( ),q . This gives
1 2
∂
(εnqi ui,q ) + εnqi
u +
− εnqi εij k [uj (ωk + 2 k )],q
∂t
2 j
,iq
1
= −εnqi
− νεnqi εij k ωk,j q .
p,i
(5.26)
ρ
,q
The second term on the left-hand side vanishes on noticing that εnqi is antisymmetric
in q and i, whereas the derivative (u2j /2 + ),iq is symmetric in q and i. The third
term on the left-hand side of (5.26) can be written as
−εnqi εij k [uj (ωk + 2
k )],q
= −[un (ωk + 2
= −(δnj δqk − δnk δqj )[uj (ωk + 2
k )],k
= −un (ωk,k + 2
+ [uj (ωn + 2
k,k ) − un,k (ωk
= −un (0 + 0) − un,k (ωk + 2
= −un,j (ωj + 2
j ) + uj
+2
k )],q
n )],j
k ) + uj (ωn
k ) + uj (ωn
+2
+2
n ),j
n ),j
(5.27)
ωn,j ,
where we have used ui,i = 0, ωi,i = 0 and the fact that the derivatives of are zero.
The first term on the right-hand side of equation (5.26) can be written as follows:
1
1
1
−εnqi
p,i
= − εnqi p,iq + 2 εnqi ρ,q p,i
ρ
ρ
ρ
,q
=0+
1
[∇ρ × ∇p]n ,
ρ2
(5.28)
which involves the n-component of the vector ∇ρ × ∇p. The viscous term in equation (5.26) can be written as
−νεnqi εij k ωk,j q = −ν(δnj δqk − δnk δqj )ωk,j q
= −νωk,nk + νωn,jj = νωn,jj .
(5.29)
If we use equations (5.27)–(5.29), vorticity equation (5.26) becomes
∂ωn
= un,j (ωj + 2
∂t
j ) − uj ωn,j
+
1
[∇ρ × ∇p]n + νωn,jj .
ρ2
Changing the free index from n to i, this becomes
Dωi
= (ωj + 2
Dt
j )ui,j
+
1
[∇ρ × ∇p]i + νωi,jj .
ρ2
In vector notation it is written as
Dω
1
= (ω + 2) • ∇u + 2 ∇ρ × ∇p + ν∇ 2 ω.
Dt
ρ
(5.30)
155
7. Vorticity Equation in a Rotating Frame
This is the vorticity equation for a nearly incompressible (that is, Boussinesq) fluid
in rotating coordinates. Here u and ω are, respectively, the (relative) velocity and
vorticity observed in a frame of reference rotating at angular velocity . As vorticity
is defined as twice the angular velocity, 2 is the planetary vorticity and (ω + 2)
is the absolute vorticity of the fluid, measured in an inertial frame. In a nonrotating
frame, the vorticity equation is obtained from equation (5.30) by setting to zero
and interpreting u and ω as the absolute velocity and vorticity, respectively.
The left-hand side of equation (5.30) represents the rate of change of relative
vorticity following a fluid particle. The last term ν∇ 2 ω represents the rate of change
of ω due to molecular diffusion of vorticity, in the same way that ν∇ 2 u represents
acceleration due to diffusion of velocity. The second term on the right-hand side is
the rate of generation of vorticity due to baroclinicity of the flow, as discussed in
Section 4. In a barotropic flow, density is a function of pressure alone, so ∇ρ and ∇p
are parallel vectors. The first term on the right-hand side of equation (5.30) plays a
crucial role in the dynamics of vorticity; it is discussed in more detail in what follows.
Meaning of (ω • ∇)u
To examine the significance of this term, take a natural coordinate system with s
along a vortex line, n away from the center of curvature, and m along the third normal
(Figure 5.9). Then
∂
∂
∂u
∂
•
•
+ in
+ im
(5.31)
u=ω
(ω ∇)u = ω
is
∂s
∂n
∂m
∂s
where we have used ω • in = ω • im = 0, and ω • is = ω (the magnitude of ω). Equation (5.31) shows that (ω • ∇) u equals the magnitude of ω times the derivative of
u in the direction of ω. The quantity ω(∂u/∂s) is a vector and has the components ω(∂us /∂s), ω(∂un /∂s), and ω(∂um /∂s). Among these, ∂us /∂s represents the
increase of us along the vortex line s, that is, the stretching of vortex lines. On the
other hand, ∂un /∂s and ∂um /∂s represent the change of the normal velocity components along s and, therefore, the rate of turning or tilting of vortex lines about the m
and n axes, respectively.
To see the effect of these terms more clearly, let us write equation (5.30) and
suppress all terms except (ω • ∇)u on the right-hand side, giving
Dω
∂u
= (ω • ∇)u = ω
(barotropic, inviscid, nonrotating)
Dt
∂s
whose components are
Dωs
∂us
=ω
,
Dt
∂s
Dωn
∂un
=ω
,
Dt
∂s
and
Dωm
∂um
=ω
.
Dt
∂s
(5.32)
The first equation of (5.32) shows that the vorticity along s changes due to stretching of
vortex lines, reflecting the principle of conservation of angular momentum. Stretching
decreases the moment of inertia of fluid elements that constitute a vortex line, resulting
in an increase of their angular speed. Vortex stretching plays an especially crucial role
in the dynamics of turbulent and geophysical flows The second and third equations
156
Vorticity Dynamics
Figure 5.9 Coordinate system aligned with vorticity vector.
of (5.32) show how vorticity along n and m change due to tilting of vortex lines.
For example, in Figure 5.9, the turning of the vorticity vector ω toward the n-axis
will generate a vorticity component along n. The vortex stretching and tilting term
(ω • ∇) u is absent in two-dimensional flows, in which ω is perpendicular to the plane
of flow.
Meaning of 2( • ∇) u
Orienting the z-axis along the direction of , this term becomes 2( • ∇)u =
2 (∂u/∂z). Suppressing all other terms in equation (5.30), we obtain
Dω
∂u
=2
Dt
∂z
(barotropic, inviscid, two-dimensional)
whose components are
∂w
Dωz
=2
,
Dt
∂z
Dωx
∂u
=2
,
Dt
∂z
and
Dωy
∂v
=2
.
Dt
∂z
This shows that stretching of fluid lines in the z direction increases ωz , whereas a
tilting of vertical lines changes the relative vorticity along the x and y directions.
Note that merely a stretching or turning of vertical fluid lines is required for this
mechanism to operate, in contrast to (ω • ∇) u where a stretching or turning of vortex
lines is needed. This is because vertical fluid lines contain “planetary vorticity” 2.
A vertically stretching fluid column tends to acquire positive ωz , and a vertically
shrinking fluid column tends to acquire negative ωz (Figure 5.10). For this reason
large-scale geophysical flows are almost always full of vorticity, and the change of
due to the presence of planetary vorticity 2 is a central feature of geophysical fluid
dynamics.
We conclude this section by writing down Kelvin’s circulation theorem in a
rotating frame of reference. It is easy to show that (Exercise 5) the circulation theorem
157
8. Interaction of Vortices
e
Figure 5.10 Generation of relative vorticity due to stretching of fluid columns parallel to planetary
vorticity 2. A fluid column acquires ωz (in the same sense as ) by moving from location A to location B.
is modified to
DŴa
=0
Dt
(5.33)
where
Ŵa ≡
(ω + 2) dA = Ŵ + 2
•
• dA.
A
A
Here, Ŵa is circulation due to the absolute vorticity (ω + 2) and differs from Ŵ by
the “amount” of planetary vorticity intersected by A.
8. Interaction of Vortices
Vortices placed close to one another can mutually interact, and generate interesting
motions. To examine such interactions, we shall idealize each vortex by a concentrated
line. A real vortex, with a core within which vorticity is distributed, can be idealized
by a concentrated vortex line with a strength equal to the average vorticity in the core
times the core area. Motion outside the core is assumed irrotational, and therefore
inviscid. It will be shown in the next chapter that irrotational motion of a constant
density fluid is governed by the linear Laplace equation. The principle of superposition
therefore holds, and the flow at a point can be obtained by adding the contribution
of all vortices in the field. To determine the mutual interaction of line vortices, the
important principle to keep in mind is the Helmholtz vortex theorem, which says that
vortex lines move with the flow.
Consider the interaction of two vortices of strengths Ŵ1 and Ŵ2 , with both Ŵ1
and Ŵ2 positive (that is, counterclockwise vorticity). Let h = h1 + h2 be the distance
between the vortices (Figure 5.11). Then the velocity at point 2 due to vortex Ŵ1 is
directed upward, and equals
V1 =
Ŵ1
.
2π h
158
Vorticity Dynamics
Figure 5.11 Interaction of line vortices of the same sign.
Similarly, the velocity at point 1 due to vortex Ŵ2 is downward, and equals
V2 =
Ŵ2
.
2π h
The vortex pair therefore rotates counterclockwise around the “center of gravity” G,
which is stationary.
Now suppose that the two vortices have the same circulation of magnitude Ŵ, but
an opposite sense of rotation (Figure 5.12). Then the velocity of each vortex at the
location of the other is Ŵ/(2π h) and is directed in the same sense. The entire system
therefore translates at a speed Ŵ/(2π h) relative to the fluid. A pair of counter-rotating
vortices can be set up by stroking the paddle of a boat, or by briefly moving the blade
of a knife in a bucket of water (Figure 5.13). After the paddle or knife is withdrawn,
the vortices do not remain stationary but continue to move under the action of the
velocity induced by the other vortex.
The behavior of a single vortex near a wall can be found by superposing two
vortices of equal and opposite strength. The technique involved is called the method
of images, which has wide applications in irrotational flow, heat conduction, and
electromagnetism. It is clear that the inviscid flow pattern due to vortex A at distance
h from a wall can be obtained by eliminating the wall and introducing instead a vortex
of equal strength and opposite sense at “image point” B (Figure 5.14). The velocity at
any point P on the wall, made up of VA due to the real vortex and VB due to the image
vortex, is then parallel to the wall. The wall is therefore a streamline, and the inviscid
boundary condition of zero normal velocity across a solid wall is satisfied. Because
of the flow induced by the image vortex, vortex A moves with speed Ŵ/(4π h) parallel
to the wall. For this reason, vortices in the example of Figure 5.13 move apart along
the boundary on reaching the side of the vessel.
Now consider the interaction of two doughnut-shaped vortex rings (such as smoke
rings) of equal and opposite circulation (Figure 5.15a). According to the method of
images, the flow field for a single ring near a wall is identical to the flow of two rings
8. Interaction of Vortices
Figure 5.12 Interaction of line vortices of opposite spin, but of the same magnitude. Here Ŵ refers to the
magnitude of circulation.
Figure 5.13 Top view of a vortex pair generated by moving the blade of a knife in a bucket of water.
Positions at three instances of time 1, 2, and 3 are shown. (After Lighthill (1986).)
of opposite circulations. The translational motion of each element of the ring is caused
by the induced velocity of each element of the same ring, plus the induced velocity
of each element of the other vortex. In the figure, the motion at A is the resultant of
VB , VC , and VD , and this resultant has components parallel to and toward the wall.
Consequently, the vortex ring increases in diameter and moves toward the wall with
a speed that decreases monotonically (Figure 5.15b).
Finally, consider the interaction of two vortex rings of equal magnitude and
similar sense of rotation. It is left to the reader (Exercise 6) to show that they should
both translate in the same direction, but the one in front increases in radius and
therefore slows down in its translational speed, while the rear vortex contracts and
translates faster. This continues until the smaller ring passes through the larger one,
at which point the roles of the two vortices are reversed. The two vortices can pass
through each other forever in an ideal fluid. Further discussion of this intriguing
problem can be found in Sommerfeld (1964, p. 161).
159
160
Vorticity Dynamics
Figure 5.14 Line vortex A near a wall and its image B.
Figure 5.15 (a) Torus or doughnut-shaped vortex ring near a wall and its image. A section through the
middle of the ring is shown. (b) Trajectory of vortex ring, showing that it widens while its translational
velocity toward the wall decreases.
161
9. Vortex Sheet
Figure 5.16 Vortex sheet.
9. Vortex Sheet
Consider an infinite number of infinitely long vortex filaments, placed side by side on a
surface AB (Figure 5.16). Such a surface is called a vortex sheet. If the vortex filaments
all rotate clockwise, then the tangential velocity immediately above AB is to the right,
while that immediately below AB is to the left. Thus, a discontinuity of tangential
velocity exists across a vortex sheet. If the vortex filaments are not infinitesimally
thin, then the vortex sheet has a finite thickness, and the velocity change is spread out.
In Figure 5.16, consider the circulation around a circuit of dimensions dn and
ds. The normal velocity component v is continuous across the sheet (v = 0 if the
sheet does not move normal to itself ), while the tangential component u experiences
a sudden jump. If u1 and u2 are the tangential velocities on the two sides, then
dŴ = u2 ds + v dn − u1 ds − v dn = (u2 − u1 ) ds,
Therefore the circulation per unit length, called the strength of a vortex sheet,
equals the jump in tangential velocity:
γ ≡
dŴ
= u2 − u 1 .
ds
The concept of a vortex sheet will be especially useful in discussing the flow over
aircraft wings (Chapter 15).
Exercises
1. A closed cylindrical tank 4 m high and 2 m in diameter contains water to a
depth of 3 m. When the cylinder is rotated at a constant angular velocity of 40 rad/s,
show that nearly 0.71 m2 of the bottom surface of the tank is uncovered. [Hint: The free
surface is in the form of a paraboloid. For a point on the free surface, let h be the
height above the (imaginary) vertex of the paraboloid and r be the local radius of the
paraboloid. From Section 3 we have h = ω02 r 2 /2g, where ω0 is the angular velocity
of the tank. Apply this equation to the two points where the paraboloid cuts the top
and bottom surfaces of the tank.]
162
Vorticity Dynamics
2. A tornado can be idealized as a Rankine vortex with a core of diameter 30 m.
The gauge pressure at a radius of 15 m is −2000 N/m2 (that is, the absolute pressure
is 2000 N/m2 below atmospheric). (a) Show that the circulation around any circuit
surrounding the core is 5485 m2 /s. [Hint: Apply the Bernoulli equation between
infinity and the edge of the core.] (b) Such a tornado is moving at a linear speed
of 25 m/s relative to the ground. Find the time required for the gauge pressure to
drop from −500 to −2000 N/m2 . Neglect compressibility effects and assume an air
temperature of 25 ◦ C. (Note that the tornado causes a sudden decrease of the local
atmospheric pressure. The damage to structures is often caused by the resulting excess
pressure on the inside of the walls, which can cause a house to explode.)
3. The velocity field of a flow in cylindrical coordinates (R, ϕ, x) is
uR = 0
uϕ = aRx
ux = 0
where a is a constant. (a) Show that the vorticity components are
ωϕ = 0
ωR = −aR
ωx = 2ax
(b) Verify that ∇ • ω = 0. (c) Sketch the streamlines and vortex lines in an Rx-plane.
Show that the vortex lines are given by xR 2 = constant.
4. Consider the flow in a 90◦ angle, confined by the walls θ = 0 and θ = 90◦ .
Consider a vortex line passing through (x, y), and oriented parallel to the z-axis.
Show that the vortex path is given by
1
1
+ 2 = constant.
x2
y
[Hint: Convince yourself that we need three image vortices at points (−x, −y),
(−x, y) and (x, −y). What are their senses of rotation? The path lines are given
by dx/dt = u and dy/dt = v, where u and v are the velocity components at the
location of the vortex. Show that dy/dx = v/u = −y 3 /x 3 , an integration of which
gives the result.]
5. Start with the equations of motion in the rotating coordinates, and prove
Kelvin’s circulation theorem
D
(Ŵa ) = 0
Dt
where
Ŵa =
(ω + 2) • dA
Literature Cited
Assume that the flow is inviscid and barotropic and that the body forces are conservative. Explain the result physically.
6. Consider the interaction of two vortex rings of equal strength and similar
sense of rotation. Argue that they go through each other, as described near the end of
Section 8.
7. A constant density irrotational flow in a rectangular torus has a circulation
Ŵ and volumetric flow rate Q. The inner radius is r1 , the outer radius is r2 , and the
height is h. Compute the total kinetic energy of this flow in terms of only ρ, Ŵ, and Q.
8. Consider a cylindrical tank of radius R filled with a viscous fluid spinning
steadily about its axis with constant angular velocity . Assume that the flow is in
a steady state. (a) Find A ω · dA where A is a horizontal plane surface through the
fluid normal to the axis of rotation and bounded by the wall of the tank. (b) The tank
then stops spinning. Find again the value of A ω · dA.
9. In Figure 5.11, locate point G.
Literature Cited
Lighthill, M. J. (1986). An Informal Introduction to Theoretical Fluid Mechanics, Oxford, England: Clarendon Press.
Sommerfeld, A. (1964). Mechanics of Deformable Bodies, New York: Academic Press. (This book contains
a good discussion of the interaction of vortices.)
Supplemental Reading
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.
Pedlosky, J. (1987). Geophysical Fluid Dynamics, New York: Springer-Verlag. (This book discusses the
vorticity dynamics in rotating coordinates, with application to geophysical systems.)
Prandtl, L. and O. G. Tietjens (1934). Fundamentals of Hydro- and Aeromechanics, New York: Dover
Publications. (This book contains a good discussion of the interaction of vortices.)
163
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Chapter 6
Irrotational Flow
1. Relevance of Irrotational Flow
Theory . . . . . . . . . . . . . . . . . . . . . . . . .
2. Velocity Potential: Laplace
Equation . . . . . . . . . . . . . . . . . . . . . . .
3. Application of Complex
Variables . . . . . . . . . . . . . . . . . . . . . . .
4. Flow at a Wall Angle . . . . . . . . . . .
5. Sources and Sinks . . . . . . . . . . . . . .
6. Irrotational Vortex. . . . . . . . . . . . . .
7. Doublet . . . . . . . . . . . . . . . . . . . . . . . .
8. Flow past a Half-Body . . . . . . . . .
9. Flow past a Circular Cylinder
without Circulation . . . . . . . . . . . . .
10. Flow past a Circular Cylinder
with Circulation . . . . . . . . . . . . . . . .
11. Forces on a Two-Dimensional
Body . . . . . . . . . . . . . . . . . . . . . . . . . . .
Blasius Theorem . . . . . . . . . . . . . . . .
Kutta–Zhukhovsky
Lift Theorem . . . . . . . . . . . . . . . . .
Unsteady Flow . . . . . . . . . . . . . . . . .
12. Source near a Wall: Method of
Images . . . . . . . . . . . . . . . . . . . . . . . . .
13. Conformal Mapping . . . . . . . . . . . .
14. Flow around an Elliptic Cylinder
with Circulation . . . . . . . . . . . . . . . .
165
167
169
171
173
174
174
175
178
180
184
184
185
188
189
190
192
15. Uniqueness of Irrotational Flows. 194
16. Numerical Solution of Plane
Irrotational Flow . . . . . . . . . . . . . . . 195
Finite Difference Form of the
Laplace Equation . . . . . . . . . . . . 196
Simple Iteration Technique . . . . . 198
Example 6.1 . . . . . . . . . . . . . . . . . . . 199
17. Axisymmetric Irrotational
Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
18. Streamfunction and Velocity
Potential for Axisymmetric
Flow . . . . . . . . . . . . . . . . . . . . . . . . . 203
19. Simple Examples of Axisymmetric
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Uniform Flow . . . . . . . . . . . . . . . . . . 205
Point Source . . . . . . . . . . . . . . . . . . . 205
Doublet . . . . . . . . . . . . . . . . . . . . . . . . 205
Flow around a Sphere . . . . . . . . . . 206
20. Flow around a Streamlined
Body of Revolution . . . . . . . . . . . . . . 206
21. Flow around an Arbitrary Body
of Revolution . . . . . . . . . . . . . . . . . . . 208
22. Concluding Remarks . . . . . . . . . . . . 209
Exercises . . . . . . . . . . . . . . . . . . . . . . . 209
Literature Cited . . . . . . . . . . . . . . . . 212
Supplemental Reading . . . . . . . . . . 212
1. Relevance of Irrotational Flow Theory
The vorticity equation given in the preceding chapter implies that the irrotational flow
(such as the one starting from rest) of a barotropic fluid observed in a nonrotating
frame remains irrotational if the fluid viscosity is identically zero and any body forces
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50006-X
165
166
Irrotational Flow
are conservative. Such an ideal flow has a nonzero tangential velocity at a solid surface
(Figure 6.1a). In contrast, a real fluid with a nonzero ν must satisfy a no-slip boundary
condition. It can be expected that viscous effects in a real flow will be confined to
thin layers close to solid surfaces if the fluid viscosity is small. We shall see later
that the viscous layers are thin not just when the viscosity is small, but when a
non-dimensional quantity Re = U L/ν, called the Reynolds number, is much larger
than 1. (Here, U is a scale of variation of velocity in a length scale L.) The thickness
of such boundary layers, within which viscous diffusion of vorticity is important,
approaches zero as Re → ∞ (Figure 6.1b). In such a case, the vorticity equation
implies that fluid elements starting from rest, or from any other irrotational region,
remain irrotational unless they move into these boundary layers. The flow field can
therefore be divided into an “outer region” where the flow is inviscid and irrotational
and an “inner region” where viscous diffusion of vorticity is important. The outer flow
can be approximately predicted by ignoring the existence of the thin boundary layer
and applying irrotational flow theory around the solid object. Once the outer problem
is determined, viscous flow equations within the boundary layer can be solved and
matched to the outer solution.
An important exception in which this method would not work is where the solid
object has such a shape that the boundary layer separates from the surface, giving rise
to eddies in the wake (Figure 6.2). In this case viscous effects are not confined to thin
layers around solid surfaces, and the real flow in the limit Re → ∞ is quite different
Figure 6.1 Comparison of a completely irrotational flow and a high Reynolds number flow: (a) ideal
flow with ν = 0; (b) flow at high Re.
Figure 6.2 Examples of flow separation. Upstream of the point of separation, irrotational flow theory is
a good approximation of the real flow.
167
2. Velocity Potential: Laplace Equation
from the ideal flow (ν = 0). Ahead of the point of separation, however, irrotational
flow theory is still a good approximation of the real flow (Figure 6.2).
Irrotational flow patterns around bodies of various shapes is the subject of this
chapter. Motion will be assumed inviscid and incompressible. Most of the examples
given are from two-dimensional plane flows, although some examples of axisymmetric flows are also given later in the chapter. Both Cartesian (x, y) and polar (r, θ)
coordinates are used for plane flows.
2. Velocity Potential: Laplace Equation
The two-dimensional incompressible continuity equation
∂u ∂v
+
= 0,
∂x
∂y
(6.1)
guarantees the existence of a stream function ψ, from which the velocity components
can be derived as
∂ψ
∂ψ
u≡
v≡−
.
(6.2)
∂y
∂x
Likewise, the condition of irrotationality
∂u
∂v
−
= 0,
∂x
∂y
(6.3)
guarantees the existence of another scalar function φ, called the velocity potential,
which is related to the velocity components by
u≡
∂φ
∂x
and
v≡
∂φ
.
∂y
(6.4)
Because a velocity potential must exist in all irrotational flows, such flows are frequently called potential flows. Equations (6.2) and (6.4) imply that the derivative
of ψ gives the velocity component in a direction 90◦ clockwise from the direction
of differentiation, whereas the derivative of φ gives the velocity component in the
direction of differentiation. Comparing equations (6.2) and (6.4) we obtain
∂φ
∂ψ
=
∂x
∂y
Cauchy–Riemann conditions
(6.5)
∂φ
∂ψ
=−
∂y
∂x
from which one of the functions can be determined if the other is known. Equipotential lines (on which φ is constant) and streamlines are orthogonal, as equation (6.5)
implies that
∂ψ
∂φ
∂ψ
∂φ ∂ψ
∂φ ∂ψ
∂φ
•
•
+j
+j
+
= 0.
∇φ ∇ψ = i
i
=
∂x
∂y
∂x
∂y
∂x ∂x
∂y ∂y
This demonstration fails at stagnation points where the velocity is zero.
168
Irrotational Flow
The streamfunction and velocity potential satisfy the Laplace equations
∇ 2φ =
∂ 2φ
∂ 2φ
+ 2 = 0,
2
∂x
∂y
(6.6)
∇ 2ψ =
∂ 2ψ
∂ 2ψ
+
= 0,
2
∂x
∂y 2
(6.7)
as can be seen by cross differentiating equation (6.5). Equation (6.7) holds for
two-dimensional flows only, because a single streamfunction is insufficient for
three-dimensional flows. As we showed in Chapter 4, Section 4, two streamfunctions
are required to describe three-dimensional steady flows (or, if density may be regarded
as constant, three-dimensional unsteady flows). However, a velocity potential φ can be
defined in three-dimensional irrotational flows, because u = ∇φ identically satisfies
the irrotationality condition ∇ × u = 0. A three-dimensional potential flow satisfies
the three-dimensional version of ∇ 2 φ = 0.
A function satisfying the Laplace equation is sometimes called a harmonic function. The Laplace equation is encountered not only in potential flows, but also in heat
conduction, elasticity, magnetism, and electricity. Therefore, solutions in one field of
study can be found from a known analogous solution in another field. In this manner,
an extensive collection of solutions of the Laplace equation have become known. The
Laplace equation is of a type that is called elliptic. It can be shown that solutions
of elliptic equations are smooth and do not have discontinuities, except for certain
singular points on the boundary of the region. In contrast, hyperbolic equations such
as the wave equation can have discontinuous “wavefronts” in the middle of a region.
The boundary conditions normally encountered in irrotational flows are of the
following types:
(1) Condition on solid surface—Component of fluid velocity normal to a solid
surface must equal the velocity of the boundary normal to itself, ensuring that
fluid does not penetrate a solid boundary. For a stationary body, the condition is
∂φ
=0
∂n
or
∂ψ
=0
∂s
(6.8)
where s is direction along the surface, and n is normal to the surface.
(2) Condition at infinity—For the typical case of a body immersed in a uniform
stream flowing in the x direction with speed U , the condition is
∂φ
=U
∂x
or
∂ψ
=U
∂y
(6.9)
However, solving the Laplace equation subject to boundary conditions of the
type of equations (6.8) and (6.9) is not easy. Historically, irrotational flow theory
was developed by finding a function that satisfies the Laplace equation and then
determining what boundary conditions are satisfied by that function. As the Laplace
equation is linear, superposition of known harmonic functions gives another harmonic
function satisfying a new set of boundary conditions. A rich collection of solutions
has thereby emerged. We shall adopt this “inverse” approach of studying irrotational
169
3. Application of Complex Variables
flows in this chapter; numerical methods of finding a solution under given boundary
conditions are illustrated in Sections 16 and 21.
After a solution of the Laplace equation has been obtained, the velocity components are then determined by taking derivatives of φ or ψ. Finally, the pressure
distribution is determined by applying the Bernoulli equation
p + 21 ρq 2 = const.,
between any two points in the flow field; here q is the magnitude of velocity. Thus,
a solution of the nonlinear equation of motion (the Euler equation) is obtained in
irrotational flows in a much simpler manner.
For quick reference, the important equations in polar coordinates are listed in the
following:
1 ∂uθ
1 ∂
(rur ) +
=0
r ∂r
r ∂θ
(continuity),
(6.10)
1 ∂ur
1 ∂
(ruθ ) −
=0
r ∂r
r ∂θ
(irrotationality),
(6.11)
ur =
∂φ
1 ∂ψ
=
,
∂r
r ∂θ
(6.12)
uθ =
1 ∂φ
∂ψ
=−
,
r ∂θ
∂r
(6.13)
∇2φ =
1 ∂
r ∂r
1 ∂ 2φ
= 0,
r 2 ∂θ 2
(6.14)
∇2ψ =
1 ∂
r ∂r
1 ∂ 2ψ
∂ψ
r
+ 2 2 = 0,
∂r
r ∂θ
(6.15)
r
∂φ
∂r
+
3. Application of Complex Variables
In this chapter z will denote the complex variable
z ≡ x + iy = r eiθ ,
(6.16)
√
where i = −1, (x, y) are the Cartesian coordinates, and (r, θ) are the polar coordinates. In the Cartesian form the complex number z represents a point in the xy-plane
whose real axis is x and imaginary axis is y (Figure 6.3). In the polar form, z represents the position vector 0z, whose magnitude is r = (x 2 + y 2 )1/2 and whose angle
with the x-axis is tan−1 (y/x). The product of two complex numbers z1 and z2 is
z1 z2 = r1 r2 ei(θ1 +θ2 ) .
Therefore, the process of multiplying a complex number z1 by another complex
number z2 can be regarded as an operation that “stretches” the magnitude from r1 to
r1 r2 and increases the argument from θ1 to θ1 + θ2 .
170
Irrotational Flow
Figure 6.3 Complex z-plane.
When x and y are regarded as variables, the complex quantity z = x + iy is
called a complex variable. Suppose we define another complex variable w whose real
and imaginary parts are φ and ψ:
w ≡ φ + iψ.
(6.17)
If φ and ψ are functions of x and y, then so is w. It is shown in the theory of complex
variables that w is a function of the combination x + iy = z, and in particular has
a finite and “unique derivative” dw/dz when its real and imaginary parts satisfy the
pair of relations, equation (6.5), which are called Cauchy–Riemann conditions. Here
the derivative dw/dz is regarded as unique if the value of δw/δz does not depend on
the orientation of the differential δz as it approaches zero. A single-valued function
w = f (z) is called an analytic function of a complex variable z in a region if a finite
dw/dz exists everywhere within the region. Points where w or dw/dz is zero or
infinite are called singularities, at which constant φ and constant ψ lines are not
orthogonal. For example, w = ln z and w = 1/z are analytic everywhere except at
the singular point z = 0, where the Cauchy–Riemann conditions are not satisfied.
The combination w = φ + iψ is called complex potential for a flow. Because
the velocity potential and stream function satisfy equation (6.5), and the real and
imaginary parts of any function of a complex variable w(z) = φ + iψ also satisfy
equation (6.5), it follows that any analytic function of z represents the complex potential of some two-dimensional flow. The derivative dw/dz is an important quantity in
the description of irrotational flows. By definition
dw
δw
= lim
.
δz→0 δz
dz
As the derivative is independent of the orientation of δz in the xy-plane, we may take
δz parallel to the x-axis, leading to
dw
∂w
∂
δw
= lim
=
=
(φ + iψ),
δx→0 δx
dz
∂x
∂x
171
4. Flow at a Wall Angle
which implies
dw
= u − iv.
dz
(6.18)
It is easy to show that taking δz parallel to the y-axis leads to an identical result. The
derivative dw/dz is therefore a complex quantity whose real and imaginary parts give
Cartesian components of the local velocity; dw/dz is therefore called the complex velocity. If the local velocity vector has a magnitude q and an angle α with the x-axis, then
dw
= qe−iα .
dz
(6.19)
It may be considered remarkable that any twice differentiable function w(z), z = x+iy
is an identical solution to Laplace’s equation in the plane (x, y). A general function of
the two variables (x, y) may be written as f (z, z∗ ) where z∗ = x − iy is the complex conjugate of z. It is the very special case when f (z, z∗ ) = w(z) alone that we
consider here.
As Laplace’s equation is linear, solutions may be superposed. That is, the sums
of elemental solutions are also solutions. Thus, as we shall see, flows over specific
shapes may be solved in this way.
4. Flow at a Wall Angle
Consider the complex potential
w = Azn
(n
1
),
2
(6.20)
where A is a real constant. If r and θ represent the polar coordinates in the z-plane,
then
w = A(reiθ )n = Ar n (cos nθ + i sin nθ),
giving
φ = Ar n cos nθ
ψ = Ar n sin nθ.
(6.21)
For a given n, lines of constant ψ can be plotted. Equation (6.21) shows that ψ = 0
for all values of r on lines θ = 0 and θ = π/n. As any streamline, including the
ψ = 0 line, can be regarded as a rigid boundary in the z-plane, it is apparent that
equation (6.20) is the complex potential for flow between two plane boundaries of
included angle α = π/n. Figure 6.4 shows the flow patterns for various values of n.
Flow within a certain sector of the z-plane only is shown; that within other sectors
can be found by symmetry. It is clear that the walls form an angle larger than 180 ◦
for n < 1 and an angle smaller than 180 ◦ for n > 1. The complex velocity in terms
of α = π/n is
Aπ (π −α)/α
dw
= nAzn−1 =
z
,
dz
α
172
Irrotational Flow
y
h
-m
m
U
a
a
xs
x
xs
Figure 6.4 Irrotational flow at a wall angle. Equipotential lines are dashed.
Figure 6.5 Stagnation flow represented by w = Az2 .
which shows that at the origin dw/dz = 0 for α < π , and dw/dz = ∞ for α > π .
Thus, the corner is a stagnation point for flow in a wall angle smaller than 180 ◦ ;
in contrast, it is a point of infinite velocity for wall angles larger than 180 ◦ . In both
cases the origin is a singular point.
The pattern for n = 1/2 corresponds to flow around a semi-infinite plate. When
n = 2, the pattern represents flow in a region bounded by perpendicular walls. By
including the field within the second quadrant of the z-plane, it is clear that n = 2
also represents the flow impinging against a flat wall (Figure 6.5). The streamlines
and equipotential lines are all rectangular hyperbolas. This is called a stagnation flow
because it represents flow in the neighborhood of the stagnation point of a blunt body.
Real flows near a sharp change in wall slope are somewhat different than those
shown in Figure 6.4. For n < 1 the irrotational flow velocity is infinite at the origin, implying that the boundary streamline (ψ = 0) accelerates before reaching
173
5. Sources and Sinks
this point and decelerates after it. Bernoulli’s equation implies that the pressure
force downstream of the corner is “adverse” or against the flow. It will be shown
in Chapter 10 that an adverse pressure gradient causes separation of flow and generation of stationary eddies. A real flow in a corner with an included angle larger than
180 ◦ would therefore separate at the corner (see the right panel of Figure 6.2).
5. Sources and Sinks
Consider the complex potential
w=
m
m
ln z =
ln (reiθ ).
2π
2π
(6.22)
m
θ,
2π
(6.23)
The real and imaginary parts are
φ=
m
ln r
2π
ψ=
from which the velocity components are found as
ur =
m
2π r
uθ = 0.
(6.24)
This clearly represents a radial flow from a two-dimensional line source at the origin,
with a volume flow rate per unit depth of m (Figure 6.6). The flow represents a line
sink if m is negative. For a source situated at z = a, the complex potential is
w=
Figure 6.6
Plane source.
m
ln (z − a).
2π
(6.25)
174
Irrotational Flow
Figure 6.7 Plane irrotational vortex.
6. Irrotational Vortex
The complex potential
w=−
iŴ
ln z.
2π
(6.26)
represents a line vortex of counterclockwise circulation Ŵ. Its real and imaginary
parts are
Ŵ
Ŵ
φ=
θ
ψ =−
ln r,
(6.27)1
2π
2π
from which the velocity components are found to be
ur = 0
uθ =
Ŵ
.
2π r
(6.28)
The flow pattern is shown in Figure 6.7.
7. Doublet
A doublet or dipole is obtained by allowing a source and a sink of equal strength
to approach each other in such a way that their strengths increase as the separation
1 The argument of transcendental functions such as the logarithm must always be dimensionless. Thus a
constant must be added to ψ in equation (6.27) to put the logarithm in proper form. This is done explicitly
when we are solving a problem as in Section 10 in what follows.
175
8. Flow past a Half-Body
distance goes to zero, and that the product tends to a finite limit. The complex potential
for a source-sink pair on the x-axis, with the source at x = −ε and the sink at x = ε, is
m
m
m
z+ε
w=
ln (z + ε) −
ln (z − ε) =
ln
,
2π
2π
2π
z−ε
m
2ε
mε
ln 1 +
+ ··· ≃
.
≃
2π
z
πz
Defining the limit of mε/π as ε → 0 to be µ, the preceding equation becomes
w=
µ
µ
= e−iθ ,
z
r
(6.29)
whose real and imaginary parts are
φ=
x2
µx
+ y2
ψ =−
x2
µy
.
+ y2
(6.30)
The expression for ψ in the preceding can be rearranged in the form
µ 2
µ 2
x + y+
.
=
2ψ
2ψ
2
The streamlines, represented by ψ = const., are therefore circles whose centers lie
on the y-axis and are tangent to the x-axis at the origin (Figure 6.8). Direction of
flow at the origin is along the negative x-axis (pointing outward from the source of
the limiting source-sink pair), which is called the axis of the doublet. It is easy to
show that (Exercise 1) the doublet flow equation (6.29) can be equivalently defined
by superposing a clockwise vortex of strength −Ŵ on the y-axis at y = ε, and a
counterclockwise vortex of strength Ŵ at y = −ε.
The complex potentials for concentrated source, vortex, and doublet are all singular at the origin. It will be shown in the following sections that several interesting
flow patterns can be obtained by superposing a uniform flow on these concentrated
singularities.
8. Flow past a Half-Body
An interesting flow results from superposition of a source and a uniform stream. The
complex potential for a uniform flow of strength U is w = U z, which follows from
integrating the relation dw/dz = u − iv. Adding to that, the complex potential for a
source at the origin of strength m, we obtain,
w = Uz +
m
ln z,
2π
whose imaginary part is
ψ = U r sin θ +
m
θ.
2π
(6.31)
(6.32)
176
Irrotational Flow
Figure 6.8 Plane doublet.
From equations (6.12) and (6.13) it is clear that there must be a stagnation point to
the left of the source (S in Figure 6.9), where the uniform stream cancels the velocity
of flow from the source. If the polar coordinate of the stagnation point is (a, π ), then
cancellation of velocity requires
U−
m
= 0,
2π a
giving
a=
m
.
2π U
(This result can also be found by finding dw/dz and setting it to zero.) The value of
the streamfunction at the stagnation point is therefore
ψs = U r sin θ +
m
m
m
θ = U a sin π +
π= .
2π
2π
2
177
8. Flow past a Half-Body
Figure 6.9
ψ = m/2.
Irrotational flow past a two-dimensional half-body. The boundary streamline is given by
The equation of the streamline passing through the stagnation point is obtained by
setting ψ = ψs = m/2, giving
U r sin θ +
m
m
θ= .
2π
2
(6.33)
A plot of this streamline is shown in Figure 6.9. It is a semi-infinite body with a
smooth nose, generally called a half-body. The stagnation streamline divides the field
into a region external to the body and a region internal to it. The internal flow consists
entirely of fluid emanating from the source, and the external region contains the
originally uniform flow. The half-body resembles several practical shapes, such as
the front part of a bridge pier or an airfoil; the upper half of the flow resembles the
flow over a cliff or a side contraction in a wide channel.
The half-width of the body is found to be
h = r sin θ =
m(π − θ)
,
2π U
where equation (6.33) has been used. The half-width tends to hmax = m/2U as θ → 0
(Figure 6.9). (This result can also be obtained by noting that mass flux from the source
is contained entirely within the half-body, requiring the balance m = (2hmax )U at
a large downstream distance where u = U .)
The pressure distribution can be found from Bernoulli’s equation
1
1
p + ρq 2 = p∞ + ρU 2 .
2
2
A convenient way of representing pressure is through the nondimensional excess
pressure (called pressure coefficient)
Cp ≡
p − p∞
1
2
2 ρU
=1−
q2
.
U2
178
Irrotational Flow
Figure 6.10 Pressure distribution in irrotational flow over a half-body. Pressure excess near the nose is
indicated by ⊕ and pressure deficit elsewhere is indicated by ⊖.
A plot of Cp on the surface of the half-body is given in Figure 6.10, which shows
that there is pressure excess near the nose of the body and a pressure deficit beyond
it. It is easy to show by integrating p over the surface that the net pressure force is
zero (Exercise 2).
9. Flow past a Circular Cylinder without Circulation
Combination of a uniform stream and a doublet with its axis directed against the
stream gives the irrotational flow over a circular cylinder. The complex potential for
this combination is
a2
µ
,
(6.34)
w = Uz + = U z +
z
z
where a ≡
√
µ/U . The real and imaginary parts of w give
a2
cos θ
φ=U r+
r
a2
sin θ.
ψ =U r−
r
(6.35)
It is seen that ψ = 0 at r = a for all values of θ, showing that the streamline
ψ = 0 represents a circular cylinder of radius a. The streamline pattern is shown in
Figure 6.11. Flow inside the circle has no influence on that outside the circle. Velocity
components are
∂φ
a2
= U 1 − 2 cos θ.
ur =
∂r
r
1 ∂φ
a2
= −U 1 + 2 sin θ,
uθ =
r ∂θ
r
179
9. Flow past a Circular Cylinder without Circulation
Figure 6.11 Irrotational flow past a circular cylinder without circulation.
from which the flow speed on the surface of the cylinder is found as
q|r = a = |uθ |r = a = 2U sin θ,
(6.36)
where what is meant is the positive value of sin θ . This shows that there are stagnation
points on the surface, whose polar coordinates are (a, 0) and (a, π). The flow reaches
a maximum velocity of 2 U at the top and bottom of the cylinder.
Pressure distribution on the surface of the cylinder is given by
Cp =
p − p∞
1
2
2 ρU
=1−
q2
= 1 − 4 sin2 θ.
U2
Surface distribution of pressure is shown by the continuous line in Figure 6.12. The
symmetry of the distribution shows that there is no net pressure drag. In fact, a general
result of irrotational flow theory is that a steadily moving body experiences no drag.
This result is at variance with observations and is sometimes known as d’Alembert’s
paradox. The existence of tangential stress, or “skin friction,” is not the only reason
for the discrepancy. For blunt bodies, the major part of the drag comes from separation
of the flow from sides and the resulting generation of eddies. The surface pressure in
the wake is smaller than that predicted by irrotational flow theory (Figure 6.12),
resulting in a pressure drag. These facts will be discussed in further detail in
Chapter 10.
The flow due to a cylinder moving steadily through a fluid appears unsteady to
an observer at rest with respect to the fluid at infinity. This flow can be obtained by
superposing a uniform stream along the negative x direction to the flow shown in
Figure 6.11. The resulting instantaneous flow pattern is simply that of a doublet, as
is clear from the decomposition shown in Figure 6.13.
180
Irrotational Flow
Figure 6.12 Comparison of irrotational and observed pressure distributions over a circular cylinder. The
observed distribution changes with the Reynolds number Re; a typical behavior at high Re is indicated by
the dashed line.
Figure 6.13 Decomposition of irrotational flow pattern due to a moving cylinder.
10. Flow past a Circular Cylinder with Circulation
It was seen in the last section that there is no net force on a circular cylinder in steady
irrotational flow without circulation. It will now be shown that a lateral force, akin
to a lift force on an airfoil, results when circulation is introduced into the flow. If
a clockwise line vortex of circulation −Ŵ is added to the irrotational flow around
a circular cylinder, the complex potential becomes
iŴ
a2
ln (z/a),
(6.37)
+
w =U z+
z
2π
181
10. Flow past a Circular Cylinder with Circulation
whose imaginary part is
Ŵ
a2
ln (r/a),
sin θ +
ψ =U r−
r
2π
(6.38)
where we have added to w the term −(iŴ/2π ) ln a so that the argument of the logarithm is dimensionless, as it must be always.
Figure 6.14 shows the resulting streamline pattern for various values of Ŵ. The
close streamline spacing and higher velocity on top of the cylinder is due to the
addition of velocity fields of the clockwise vortex and the uniform stream. In contrast,
the smaller velocities at the bottom of the cylinder are a result of the vortex field
counteracting the uniform stream. Bernoulli’s equation consequently implies a higher
pressure below the cylinder and an upward “lift” force.
The tangential velocity component at any point in the flow is
uθ = −
Ŵ
a2
∂ψ
= −U 1 + 2 sin θ −
.
∂r
2π r
r
Figure 6.14 Irrotational flow past a circular cylinder for different values of circulation. Point S represents
the stagnation point.
182
Irrotational Flow
At the surface of the cylinder, velocity is entirely tangential and is given by
uθ | r = a = −2 U sin θ −
which vanishes if
sin θ = −
Ŵ
,
2π a
(6.39)
Ŵ
.
4π aU
(6.40)
For Ŵ < 4πaU , two values of θ satisfy equation (6.40), implying that there are two
stagnation points on the surface. The stagnation points progressively move down as
Ŵ increases (Figure 6.14) and coalesce at Ŵ = 4π aU . For Ŵ > 4π aU , the stagnation
point moves out into the flow along the y-axis. The radial distance of the stagnation
point in this case is found from
a2
Ŵ
= 0.
uθ |θ =−π/2 = U 1 + 2 −
2π r
r
This gives
r=
1
[Ŵ ± Ŵ 2 − (4π aU )2 ],
4π U
one root of which is r > a; the other root corresponds to a stagnation point inside the
cylinder.
Pressure is found from the Bernoulli equation
p + ρq 2 /2 = p∞ + ρU 2 /2.
Using equation (6.39), the surface pressure is found to be
pr = a
Ŵ
= p∞ + 21 ρ U 2 − −2U sin θ −
2π a
2
.
(6.41)
The symmetry of flow about the y-axis implies that the pressure force on the cylinder
has no component along the x-axis. The pressure force along the y-axis, called the
“lift” force in aerodynamics, is (Figure 6.15)
L=−
2π
0
pr = a sin θ a dθ.
Substituting equation (6.41), and carrying out the integral, we finally obtain
L = ρU Ŵ,
where we have used
0
2π
sin θ dθ =
0
2π
sin3 θ dθ = 0.
(6.42)
10. Flow past a Circular Cylinder with Circulation
Figure 6.15 Calculation of pressure force on a circular cylinder.
It is shown in the following section that equation (6.42) holds for irrotational flows
around any two-dimensional shape, not just circular cylinders. The result that lift force
is proportional to circulation is of fundamental importance in aerodynamics. Relation
equation (6.42) was proved independently by the German mathematician, Wilhelm
Kutta (1902), and the Russian aerodynamist, Nikolai Zhukhovsky (1906); it is called the
Kutta–Zhukhovsky lift theorem. (Older western texts transliterated Zhukhovsky’s name
as Joukowsky.) The interesting question of how certain two-dimensional shapes, such
as an airfoil, develop circulation when placed in a stream is discussed in chapter 15. It
will be shown there that fluid viscosity is responsible for the development of circulation.
The magnitude of circulation, however, is independent of viscosity, and depends on flow
speed U and the shape and “attitude” of the body.
For a circular cylinder, however, the only way to develop circulation is by rotating
it in a flow stream. Although viscous effects are important in this case, the observed
pattern for large values of cylinder rotation displays a striking similarity to the ideal
flow pattern for Ŵ > 4πaU ; see Figure 3.25 in the book by Prandtl (1952). For
lower rates of cylinder rotation, the retarded flow in the boundary layer is not able
to overcome the adverse pressure gradient behind the cylinder, leading to separation;
the real flow is therefore rather unlike the irrotational pattern. However, even in the
presence of separation, observed speeds are higher on the upper surface of the cylinder,
implying a lift force.
A second reason for generating lift on a rotating cylinder is the asymmetry generated due to delay of separation on the upper surface of the cylinder. The resulting
asymmetry generates a lift force. The contribution of this mechanism is small for
two-dimensional objects such as the circular cylinder, but it is the only mechanism
for side forces experienced by spinning three-dimensional objects such as soccer,
tennis and golf balls. The interesting question of why spinning balls follow curved
paths is discussed in Chapter 10, Section 9. The lateral force experienced by rotating
bodies is called the Magnus effect.
The nonuniqueness of solution for two-dimensional potential flows should be
noted in the example we have considered in this section. It is apparent that solutions
for various values of Ŵ all satisfy the same boundary condition on the solid surface
183
184
Irrotational Flow
(namely, no normal flow) and at infinity (namely, u = U ), and there is no way to
determine the solution simply from the boundary conditions. A general result is that
solutions of the Laplace equation in a multiply connected region are nonunique. This
is explained further in Section 15.
11. Forces on a Two-Dimensional Body
In the preceding section we demonstrated that the drag on a circular cylinder is zero
and the lift equals L = ρU Ŵ. We shall now demonstrate that these results are valid
for cylindrical shapes of arbitrary cross section. (The word “cylinder” refers to any
plane two-dimensional body, not just to those with circular cross sections.)
Blasius Theorem
Consider a general cylindrical body, and let D and L be the x and y components of
the force exerted on it by the surrounding fluid; we refer to D as “drag” and L as
“lift.” Because only normal pressures are exerted in inviscid flows, the forces on a
surface element dz are (Figure 6.16)
dD = −p dy,
dL = p dx.
We form the complex quantity
dD − i dL = −p dy − ip dx = −ip dz∗ ,
where an asterisk denotes the complex conjugate. The total force on the body is
therefore given by
(6.43)
D − iL = −i p dz∗ ,
C
Figure 6.16 Forces exerted on an element of a body.
185
11. Forces on a Two-Dimensional Body
where C denotes a counterclockwise contour coinciding with the body surface.
Neglecting gravity, the pressure is given by the Bernoulli equation
1
p∞ + ρU 2 = p + 21 ρ(u2 + v 2 ) = p + 21 ρ(u + iv)(u − iv).
2
Substituting for p in equation (6.43), we obtain
D − iL = −i [p∞ + 21 ρU 2 − 21 ρ(u + iv)(u − iv)] dz∗ ,
(6.44)
C
Now the integral of the constant term (p∞ + 21 ρU 2 ) around a closed contour is zero.
Also, on the body surface the velocity vector and the surface element dz are parallel
(Figure 6.16), so that
u + iv = u2 + v 2 eiθ ,
dz = |dz| eiθ .
The product (u + iv) dz∗ is therefore real, and we can equate it to its complex
conjugate:
(u + iv) dz∗ = (u − iv) dz.
Equation (6.44) then becomes
i
D − iL = ρ
2
C
dw
dz
2
dz,
(6.45)
where we have introduced the complex velocity dw/dz = u − iv. Equation (6.45)
is called the Blasius theorem, and applies to any plane steady irrotational flow. The
integral need not be carried out along the contour of the body because the theory
of complex variables shows that any contour surrounding the body can be chosen,
provided that there are no singularities between the body and the contour chosen.
Kutta–Zhukhovsky Lift Theorem
We now apply the Blasius theorem to a steady flow around an arbitrary cylindrical body,
around which there is a clockwise circulation Ŵ. The velocity at infinity has a magnitude U and is directed along the x-axis. The flow can be considered a superposition
of a uniform stream and a set of singularities such as vortex, doublet, source, and sink.
As there are no singularities outside the body, we shall take the contour C in
the Blasius theorem at a very large distance from the body. From large distances, all
singularities appear to be located near the origin z = 0. The complex potential is then
of the form
w = Uz +
iŴ
µ
m
ln z +
ln z + + · · · .
2π
2π
z
The first term represents a uniform flow, the second term represents a source, the third
term represents a clockwise vortex, and the fourth term represents a doublet. Because
186
Irrotational Flow
the body contour is closed, the mass efflux of the sources must be absorbed by the
sinks. It follows that the sum of the strength of the sources and sinks is zero, thus we
should set m = 0. The Blasius theorem, equation (6.45), then becomes
2
iρ
µ
iŴ
D − iL =
U+
(6.46)
− 2 + · · · dz.
2
2π z z
To carry out the contour integral in equation (6.46), we simply have to find the
coefficient of the term proportional to 1/z in the integrand. The coefficient of 1/z in
a power series expansion for f (z) is called the residue of f (z) at z = 0. It is shown
in complex variable theory that the contour integral of a function f (z) around the
contour C is 2πi times the sum of the residues at the singularities within C:
f (z) dz = 2π i[sum of residues].
C
The residue of the integrand in equation (6.46) is easy to find. Clearly the term µ/z2
does not contribute to the residue. Completing the square (U + iŴ/2π z)2 , we see that
the coefficient of 1/z is iŴ U/π . This gives
iŴU
iρ
D − iL =
2π i
,
2
π
which shows that
D = 0,
L = ρU Ŵ.
(6.47)
The first of these equations states that there is no drag experienced by a body in
steady two-dimensional irrotational flow. The second equation shows that there is a
lift force L = ρU Ŵ perpendicular to the stream, experienced by a two-dimensional
body of arbitrary cross section. This result is called the Kutta–Zhukhovsky lift theorem, which was demonstrated in the preceding section for flow around a circular
cylinder. The result will play a fundamental role in our study of flow around airfoil
shapes (Chapter 15). We shall see that the circulation developed by an airfoil is nearly
proportional to U , so that the lift is nearly proportional to U 2 .
The following points can also be demonstrated. First, irrotational flow over a
finite three-dimensional object has no circulation, and there can be no net force on
the body in steady state. Second, in an unsteady flow a force is required to push a body,
essentially because a mass of fluid has to be accelerated from rest.
Let us redrive the Kutta–Zhukhovsky lift theorem from considerations of vector
calculus without reference to complex variables. From equations (4.28) and (4.33),
for steady flow with no body forces, and with I the dyadic equivalent of the Kronecker
delta δij
(ρuu + pI − σ) · dA1 .
FB = −
A1
187
11. Forces on a Two-Dimensional Body
Assuming an inviscid fluid, σ = 0. Now additionally assume a two-dimensional
constant density flow that is uniform at infinity u = U ix . Then, from Bernoulli’s
theorem, p + ρq 2 /2 = p∞ + ρq 2 /2 = p0 , so p = p0 − ρq 2 /2. Referring to
Figure 6.17, for two-dimensional flow dA1 = ds×iz dz, where here z is the coordinate
out of the paper. We will carry out the integration over a unit depth in z so that the
result for FB will be force per unit depth (in z).
With r = xix + yiy , dr = dxix + dyiy = ds, dA1 = ds × iz · 1 = −iy dx + ix dy.
Now let u = U ix + u′ , where u′ → 0 as r → ∞ at least as fast as 1/r. Substituting
for uu and q 2 in the integral for FB , we find
FB = − ρ
A1
{U U ix ix + U ix (u′ ix + v ′ iy ) + (u′ ix + v ′ iy )ix U
+ u′ u′ + (ix ix + iy iy )[p0 /ρ − U 2 /2 − U u′
− (u′2 + v ′2 )/2]} · (−iy dx + ix dy).
Let r → ∞ so that the contour C is far from the body. The constant terms U 2 ,
p0 /ρ, −U 2 /2 integrate to zero around the closed path. The quadratic terms u′ u′ ,
(u′2 + v ′2 )/2 1/r 2 as r → ∞ and the perimeter of the contour increases only
as r. Thus the quadratic terms → 0 as r → ∞. Separating the force into x and y
components,
FB = −ix ρU [(u′ dy − v ′ dx) + (u′ dy − u′ dy)] − iy ρU (v ′ dy + u′ dx).
c
c
We note that the first integrand is u′ · ds × iz , and that we may add the constant
U ix to each of the integrands because the integration of a constant velocity over a
Figure 6.17 Domain of integration for the Kutta–Zhukhovsky theorem.
188
Irrotational Flow
closed contour or surface will result in zero force. The integrals for the force then
become
FB = −ix ρU
(U ix + u′ ) · dA1 − iy ρU (U ix + u′ ) · ds.
c
At
The first integral is zero by equation (4.29) (as a consequence of mass conservation for constant density flow) and the second is the circulation Ŵ by definition.
Thus,
FB = −iy ρU Ŵ
(force/unit depth),
where Ŵ is positive in the counterclockwise sense. We see that there is no force
component in the direction of motion (drag) under the assumptions necessary for
the derivation (steady, inviscid, no body forces, constant density, two-dimensional,
uniform at infinity) that were believed to be valid to a reasonable approximation for
a wide variety of flows. Thus it was labeled a paradox—d’Alembert’s paradox (Jean
Le Rond d’Alembert, 16 November 1717–29 October 1783).
Unsteady Flow
The Euler momentum integral [(4.28)] can be extended to unsteady flows as follows.
The extension may have some utility for constant density irrotational flows with
moving boundaries; thus it is derived here.
Integrating (4.17) over a fixed volume V bounded by a surface A (A = ∂V )
containing within it only fluid particles, we obtain
•
ρuu dA +
τ • dA
d/dt ρu dV = −
V
A = ∂V
A = ∂V
where body forces g have been neglected, and the divergence theorem has been used.
Because the immersed body cannot be part of V , we take A = A1 + A2 + A3 , as
shown in Figure 4.9. Here A1 is a “distant” surface, A2 is the body surface, and A3
is the connection between A1 and A2 that we allow to vanish. We identified the force
on the immersed body as
FB = − τ • dA2
A2
Then,
FB = −
A1
(ρuu − τ ) • dA1 −
A2
ρuu • dA2 − d/dt
ρu dV
(6.48)
V
If the flow is unsteady because of a moving boundary (A2 ), then u • dA2 = 0, as we
showed at the end of Section 4.19. If the body surface is described by f (x, y, z, t) = 0,
then the condition that no mass of fluid with local velocity u flow across the boundary
is (4.92): Df/Dt = ∂f/∂t + u • ∇f = 0. Since ∇f is normal to the boundary
189
12. Source near a Wall: Method of Images
(as is dA2 ), u • ∇f = −∂f/∂t on f = 0. Thus u • dA2 is in general = 0 on the body
surface. Equation (6.48) may be simplified if the density ρ = const. and if viscous
effects can be neglected in the flow. Then, by Kelvin’s theorem the flow is circulation
preserving. If it is initially irrotational, it will remain so. With ∇ ×u = 0, u = ∇φ and
ρ = const., the last integral in (6.48) can be transformed by the divergence theorem
d/dt ρu dV = ρd/dt ∇φdV = ρd/dt
φI • dA
V
V
A = ∂V
With A = A1 + A2 + A3 and A3 → 0, the A1 and A2 integrals can be combined
with the first two integrals in (6.48) to yield
•
FB = − (ρuu + pI + ρI∂φ∂t) dA1 − (ρuu + ρI∂φ/∂t) • dA2 , (6.49)
A1
A2
where τ = −pI + σ and σ = 0 with the neglect of viscosity. The Bernoulli equation
for unsteady irrotational flow [(4.81)], ρ∂φ/∂t + p + ρu2 /2 = 0, where the function
of integration F (t) has been absorbed in the φ, can be used if desired to achieve a
slightly different form.
12. Source near a Wall: Method of Images
The method of images is a way of determining a flow field due to one or more
singularities near a wall. It was introduced in Chapter 5, Section 8, where vortices
near a wall were examined. We found that the flow due to a line vortex near a wall can
be found by omitting the wall and introducing instead a vortex of opposite strength
at the “image point.” The combination generates a straight streamline at the location
of the wall, thereby satisfying the boundary condition.
Another example of this technique is given here, namely, the flow due to a line
source at a distance a from a straight wall. This flow can be simulated by introducing
an image source of the same strength and sign, so that the complex potential is
m
m
m
ln (z − a) +
ln (z + a) −
ln a 2 ,
2π
2π
2π
m
m
ln (x 2 − y 2 − a 2 + i2xy) −
ln a 2 .
=
2π
2π
w=
(6.50)
We know that the logarithm of any complex quantity ζ = |ζ | exp (iθ) can be written
as ln ζ = ln |ζ | + iθ. The imaginary part of equation (6.50) is therefore
ψ=
m
2xy
,
tan−1 2
2π
x − y 2 − a2
from which the equation of streamlines is found as
2π ψ
2
2
= a2.
x − y − 2xy cot
m
190
Irrotational Flow
Figure 6.18 Irrotational flow due to two equal sources.
The streamline pattern is shown in Figure 6.18. The x and y axes form part of the
streamline pattern, with the origin as a stagnation point. It is clear that the complex
potential equation (6.48) represents three interesting flow situations:
(1) flow due to two equal sources (entire Figure 6.18);
(2) flow due to a source near a plane wall (right half of Figure 6.18); and
(3) flow through a narrow slit in a right-angled wall (first quadrant of Figure 6.18).
13. Conformal Mapping
We shall now introduce a method by which complex flow patterns can be transformed
into simple ones using a technique known as conformal mapping in complex variable
theory. Consider the functional relationship w = f (z), which maps a point in the
w-plane to a point in the z-plane, and vice versa. We shall prove that infinitesimal
figures in the two planes preserve their geometric similarity if w = f (z) is analytic.
Let lines Cz and Cz′ in the z-plane be transformations of the curves Cw and Cw′ in the
w-plane, respectively (Figure 6.19). Let δz, δ ′ z, δw, and δ ′ w be infinitesimal elements
along the curves as shown. The four elements are related by
δw =
dw
δz,
dz
(6.51)
δ′w =
dw ′
δ z.
dz
(6.52)
If w = f (z) is analytic, then dw/dz is independent of orientation of the elements,
and therefore has the same value in equation (6.51) and (6.52). These two equations
191
13. Conformal Mapping
Figure 6.19 Preservation of geometric similarity of small elements in conformal mapping.
then imply that the elements δz and δ ′ z are rotated by the same amount (equal to the
argument of dw/dz) to obtain the elements δw and δ ′ w. It follows that
α = β,
which demonstrates that infinitesimal figures in the two planes are geometrically
similar. The demonstration fails at singular points at which dw/dz is either zero or
infinite. Because dw/dz is a function of z, the amount of magnification and rotation
that an element δz undergoes during transformation from the z-plane to the w-plane
varies. Consequently, large figures become distorted during the transformation.
In application of conformal mapping, we always choose a rectangular grid in the
w-plane consisting of constant φ and ψ lines (Figure 6.20). In other words, we define
φ and ψ to be the real and imaginary parts of w:
w = φ + iψ.
The rectangular net in the w-plane represents a uniform flow in this plane. The constant φ and ψ lines are transformed into certain curves in the z-plane through the
transformation w = f (z). The pattern in the z-plane is the physical pattern under
investigation, and the images of constant φ and ψ lines in the z-plane form the equipotential lines and streamlines, respectively, of the desired flow. We say that w = f (z)
transforms a uniform flow in the w-plane into the desired flow in the z-plane. In fact,
all the preceding flow patterns studied through the transformation w = f (z) can be
interpreted this way.
If the physical pattern under investigation is too complicated, we may introduce
intermediate transformations in going from the w-plane to the z-plane. For example,
the transformation w = ln (sin z) can be broken into
w = ln ζ
ζ = sin z.
Velocity components in the z-plane are given by
u − iv =
dw
dw dζ
1
=
= cos z = cot z.
dz
dζ dz
ζ
192
Irrotational Flow
Figure 6.20 Flow patterns in the w-plane and the z-plane.
A simple example of conformal mapping is given immediately below as a special case
of the flow discussed in Section 4 above. Consider the transformation, w = φ + iψ =
z2 = x 2 − y 2 + 2ixy. Streamlines are ψ = const = 2xy, rectangular hyperbolae.
See one quadrant of the flow depicted in Fig. 6.5. Uniform flow in the w-plane has
been mapped onto flow in a right hand corner in the z-plane by this transformation.
A more involved example is shown in the next section. Additional applications are
discussed in Chapter 15.
14. Flow around an Elliptic Cylinder with Circulation
We shall briefly illustrate the method of conformal mapping by considering a transformation that has important applications in airfoil theory. Consider the following
transformation:
z=ζ+
b2
,
ζ
(6.53)
relating z and ζ planes. We shall now show that a circle of radius b centered at the
origin of the ζ -plane transforms into a straight line on the real axis of the z-plane. To
prove this, consider a point ζ = b exp (iθ) on the circle (Figure 6.21), for which the
corresponding point in the z-plane is
z = beiθ + be−iθ = 2b cos θ.
As θ varies from 0 to π , z goes along the x-axis from 2b to −2b. As θ varies from π
to 2π , z goes from −2b to 2b. The circle of radius b in the ζ -plane is thus transformed
into a straight line of length 4b in the z-plane. It is clear that the region outside
the circle in ζ -plane is mapped into the entire z-plane. It can be shown that the
region inside the circle is also transformed into the entire z-plane. This, however, is
193
14. Flow around an Elliptic Cylinder with Circulation
Figure 6.21 Transformation of a circle into an ellipse by means of the Zhukhovsky transformation
z = ζ + b2 /ζ .
of no concern to us because we shall not consider the interior of the circle in the
ζ -plane.
Now consider a circle of radius a > b in the ζ -plane (Figure 6.21). Points
ζ = a exp (iθ) on this circle are transformed to
z = a eiθ +
b2 −iθ
e ,
a
(6.54)
which traces out an ellipse for various values of θ. This becomes clear by elimination
of θ in equation (6.54), giving
y2
x2
+
= 1.
2
2
(a + b /a)
(a − b2 /a)2
(6.55)
For various values of a > b, equation (6.55) represents a family of ellipses in the
z-plane, with foci at x = ± 2b.
The flow around one of these ellipses (in the z-plane) can be determined by
first finding the flow around a circle of radius a in the ζ -plane, and then using
the transformation equation (6.53) to go to the z-plane. To be specific, suppose the
desired flow in the z-plane is that of flow around an elliptic cylinder with clockwise
circulation Ŵ, which is placed in a stream moving at U . The corresponding flow in
the ζ -plane is that of flow with the same circulation around a circular cylinder of
radius a placed in a stream of the same strength U for which the complex potential
is (see equation (6.37))
iŴ
iŴ
a2
ln ζ −
ln a.
(6.56)
+
w=U ζ+
ζ
2π
2π
The complex potential w(z) in the z-plane can be found by substituting the inverse
of equation (6.53), namely,
ζ =
1
1
z + (z2 − 4b2 )1/2 ,
2
2
(6.57)
194
Irrotational Flow
into equation (6.56). (Note that the negative root, which falls inside the cylinder, has
been excluded from equation (6.57).) Instead of finding the complex velocity in the
z-plane by directly differentiating w(z), it is easier to find it as
u − iv =
dw
dw dζ
=
.
dz
dζ dz
The resulting flow around an elliptic cylinder with circulation is qualitatively quite
similar to that around a circular cylinder as shown in Figure 6.14.
15. Uniqueness of Irrotational Flows
In Section 10 we saw that plane irrotational flow over a cylindrical object is nonunique.
In particular, flows with any amount of circulation satisfy the same boundary
conditions on the body and at infinity. With such an example in mind, we are ready
to make certain general statements concerning solutions of the Laplace equation. We
shall see that the topology of the region of flow has a great influence on the uniqueness
of the solution.
Before we can make these statements, we need to define certain terms. A reducible
circuit is any closed curve (lying wholly in the flow field) that can be reduced to a
point by continuous deformation without ever cutting through the boundaries of the
flow field. We say that a region is singly connected if every closed circuit in the region
is reducible. For example, the region of flow around a finite body of revolution is
reducible (Figure 6.22a). In contrast, the flow field over a cylindrical object of infinite
length is multiply connected because certain circuits (such as C1 in Figure 6.22b) are
reducible while others (such as C2 ) are not reducible.
To see why solutions are nonunique in a multiply connected region, consider the
two circuits C1 and C2 in Figure 6.22b. The vorticity everywhere within C1 is zero,
thus Stokes’ theorem requires that the circulation around it must vanish. In contrast,
the circulation around C2 can have any strength Ŵ. That is,
u • dx = Ŵ,
(6.58)
C2
Figure 6.22 Singly connected and multiply connected regions: (a) singly connected; (b) multiply
connected.
16. Numerical Solution of Plane Irrotational Flow
where the loop around the integral sign has been introduced to emphasize that the
circuit C2 is closed. As the right-hand side of equation (6.58) is nonzero, it follows
that u • dx is not a “perfect differential,” which means that the line integral between
any two points depends on the path followed (u • dx is called a perfect differential
if it can be expressed as the differential of a function, say as u • dx = df . In that
case the line integral around a closed circuit must vanish). In Figure 6.22b, the line
integrals between P and Q are the same for paths 1 and 2, but not the same for paths 1
and 3. The solution is therefore nonunique, as was physically evident from the whole
family of irrotational flows shown in Figure 6.14.
In singly connected regions, circulation around every circuit is zero, and the solution of ∇ 2 φ = 0 is unique when values of φ are specified at the boundaries (the
Dirichlet problem). When normal derivatives of φ are specified at the boundary (the
Neumann problem), as in the fluid flow problems studied here, the solution is unique
within an arbitrary additive constant. Because the arbitrary constant is of no consequence, we shall say that the solution of the irrotational flow in a singly connected
region is unique. (Note also that the solution depends only on the instantaneous
boundary conditions; the differential equation ∇ 2 φ = 0 is independent of t.)
Summary: Irrotational flow around a plane two-dimensional object is nonunique because it allows an arbitrary amount of circulation. Irrotational flow around
a finite three-dimensional object is unique because there is no circulation.
In Sections 4 and 5 of Chapter 5 we learned that vorticity is solenoidal (∇ ·ω = 0),
or that vortex lines cannot begin or end anywhere in the fluid. Here we have learned
that a circulation in a two dimensional flow results in a force normal to an oncoming
stream. This is used to simulate lifting flow over a wing by the following artifice,
discussed in more detail in our chapter on Aerodynamics. Since Stokes’ theorem tells
us that the circulation about a closed contour is equal to the flux of vorticity through
any surface bounded by that contour, the circulation about a thin airfoil section is
simulated by a continuous row of vortices (a vortex sheet) along the centerline of
a wing cross-section (the mean camber line of an airfoil). For a (real) finite wing,
these vortices must bend downstream to form trailing vortices and terminate in starting
vortices (far downstream), always forming closed loops. Although the wing may be
a finite three dimensional shape, the contour cannot cut any of the vortex lines without
changing the circulation about the contour. Generally, the circulation about a wing
does vary in the spanwise direction, being a maximum at the root or centerline and
tending to zero at the wingtips.
Additional boundary conditions that the mean camber line be a streamline and
that a real trailing edge be a stagnation point serve to render the circulation distribution
unique.
16. Numerical Solution of Plane Irrotational Flow
Exact solutions can be obtained only for flows with simple geometries, and approximate methods of solution become necessary for practical flow problems. One of these
approximate methods is that of building up a flow by superposing a distribution of
sources and sinks; this method is illustrated in Section 21 for axisymmetric flows.
195
196
Irrotational Flow
Another method is to apply perturbation techniques by assuming that the body is thin.
A third method is to solve the Laplace equation numerically. In this section we shall
illustrate the numerical method in its simplest form. No attempt is made here to use
the most efficient method. It is hoped that the reader will have an opportunity to learn
numerical methods that are becoming increasingly important in the applied sciences
in a separate study. See Chapter 11 for introductory material on several important
techniques of computational fluid dynamics.
Finite Difference Form of the Laplace Equation
In finite difference techniques we divide the flow field into a system of grid points,
and approximate the derivatives by taking differences between values at adjacent grid
points. Let the coordinates of a point be represented by
x = i x
(i = 1, 2, . . . ,),
y = j y
(j = 1, 2, . . . ,).
Here, x and y are the dimensions of a grid box, and the integers i and j are the
indices associated with a grid point (Figure 6.23). The value of a variable ψ(x, y)
can be represented as
ψ(x, y) = ψ(i x, j y) ≡ ψi,j ,
Figure 6.23 Adjacent grid boxes in a numerical calculation.
197
16. Numerical Solution of Plane Irrotational Flow
where ψi,j is the value of ψ at the grid point (i, j ). In finite difference form, the first
derivatives of ψ are approximated as
∂ψ
1
≃
ψi+ 1 ,j − ψi− 1 ,j ,
2
2
∂x i,j
x
∂ψ
∂y
i,j
≃
1
ψi,j + 1 − ψi,j − 1 .
2
2
y
The quantities on the right-hand side (such as ψi+1/2,j ) are half-way between the
grid points and therefore undefined. However, this would not be a difficulty in the
present problem because the Laplace equation does not involve first derivatives. Both
derivatives are written as first-order centered differences.
The finite difference form of ∂ 2 ψ/∂x 2 is
∂ 2ψ
∂x 2
i,j
1
≃
x
∂ψ
∂x
i+ 21 ,j
−
∂ψ
∂x
i− 21 ,j
,
1
1
1
(ψi+1,j − ψi,j ) −
(ψi,j − ψi−1,j ) ,
x x
x
1
[ψi+1,j − 2ψi,j + ψi−1,j ].
=
x 2
≃
(6.59)
Similarly,
∂ 2ψ
∂y 2
i,j
≃
1
[ψi,j +1 − 2ψi,j + ψi,j −1 ]
y 2
(6.60)
Using equations (6.59) and (6.60), the Laplace equation for the streamfunction in a
plane two-dimensional flow
∂ 2ψ
∂ 2ψ
+
= 0,
∂x 2
∂y 2
has a finite difference representation
1
1
[ψi+1,j − 2ψi,j + ψi−1,j ] +
[ψi,j +1 − 2ψi,j + ψi,j −1 ] = 0.
2
x
y 2
Taking x = y, for simplicity, this reduces to
ψi,j =
1
[ψi−1,j + ψi+1,j + ψi,j −1 + ψi,j +1 ],
4
(6.61)
which shows that ψ satisfies the Laplace equation if its value at a grid point equals
the average of the values at the four surrounding points.
198
Irrotational Flow
Simple Iteration Technique
We shall now illustrate a simple method of solution of equation (6.61) when the values
of ψ are given in a simple geometry. Assume the rectangular region of Figure 6.24,
in which the flow field is divided into 16 grid points. Of these, the values of ψ are
known at the 12 boundary points indicated by open circles. The values of ψ at the
four interior points indicated by solid circles are unknown. For these interior points,
the use of equation (6.61) gives
ψ2,2 =
1
4
B +ψ
B
ψ1,2
3,2 + ψ2,1 + ψ2,3 ,
ψ3,2 =
1
4
B + ψB + ψ
ψ2,2 + ψ4,2
3,3 ,
3,1
ψ2,3 =
1
4
B
ψ1,3
ψ3,3 =
1
4
B +ψ
B
ψ2,3 + ψ4,3
3,2 + ψ3,4 .
B
+ ψ3,3 + ψ2,2 + ψ2,4
(6.62)
,
Figure 6.24 Network of grid points in a rectangular region. Boundary points with known values are
indicated by open circles. The four interior points with unknown values are indicated by solid circles.
16. Numerical Solution of Plane Irrotational Flow
In the preceding equations, the known boundary values have been indicated by a
superscript “B.” Equation set (6.62) represents four linear algebraic equations in four
unknowns and is therefore solvable.
In practice, however, the flow field is likely to have a large number of grid points,
and the solution of such a large number of simultaneous algebraic equations can only
be performed using a computer. One of the simplest techniques of solving such a
set is the iteration method. In this a solution is initially assumed and then gradually
improved and updated until equation (6.61) is satisfied at every point. Suppose the
values of ψ at the four unknown points of Figure 6.24 are initially taken as zero.
Using equation (6.62), the first estimate of ψ2,2 can be computed as
ψ2,2 =
1
B
ψ B + 0 + ψ2,1
+0 .
4 1,2
The old zero value for ψ2,2 is now replaced by the preceding value. The first estimate
for the next grid point is then obtained as
ψ3,2 =
1
B
B
ψ2,2 + ψ4,2
+ ψ3,1
+0 ,
4
where the updated value of ψ2,2 has been used on the right-hand side. In this manner,
we can sweep over the entire region in a systematic manner, always using the latest
available value at the point. Once the first estimate at every point has been obtained,
we can sweep over the entire region once again in a similar manner. The process is
continued until the values of ψi,j do not change appreciably between two successive
sweeps. The iteration process has now “converged.”
The foregoing scheme is particularly suitable for implementation using a computer, whereby it is easy to replace old values at a point as soon as a new value
is available. In practice, a more efficient technique, for example, the successive
over-relaxation method, will be used in a large calculation. The purpose here is not to
describe the most efficient technique, but the one which is simplest to illustrate. The
following example should make the method clear.
Example 6.1. Figure 6.25 shows a contraction in a channel through which the flow
rate per unit depth is 5 m2 /s. The velocity is uniform and parallel across the inlet and
outlet sections. Find the flow field.
Solution: Although the region of flow is plane two-dimensional, it is clearly
singly connected. This is because the flow field interior to a boundary is desired, so
that every fluid circuit can be reduced to a point. The problem therefore has a unique
solution, which we shall determine numerically.
We know that the difference in ψ values is equal to the flow rate between two
streamlines. If we take ψ = 0 at the bottom wall, then we must have ψ = 5 m2 /s at the
top wall. We divide the field into a system of grid points shown, with x = y = 1m.
Because ψ/y (= u) is given to be uniform across the inlet and the outlet, we must
have ψ = 1 m2 /s at the inlet and ψ = 5/3 = 1.67 m2 /s at the outlet. The
resulting values of ψ at the boundary points are indicated in Figure 6.25.
199
200
Irrotational Flow
Figure 6.25 Grid pattern for irrotational flow through a contraction (Example 16). The boundary values
of ψ are indicated on the outside. The values of i,j for some grid points are indicated on the inside.
The FORTRAN code for solving the problem is as follows:
DIMENSION S(10, 6)
DO 10 I = 1, 6
10 S(I, 1) = 0.
DO 20 J = 2, 3
20 S(6, J) = 0.
Set ψ = 0 on top and bottom walls
DO 30 I = 7, 10
30 S(I, 3) = 0.
DO 40 I = 1, 10
40 S(I, 6) = 5.
DO 50 J = 2, 6
Set ψ at inlet
50 S(1, J) = J - 1.
DO 60 J = 4, 6
Set ψ at outlet
60 S(10, J) = (J - 3) * (5. / 3.)
DO 100 N = 1, 20
DO 70 I = 2, 5
DO 70 J = 2, 5
70 S(I, J) = (S(I, J + 1) + S(I, J-1) + S(I + 1, J) + S(I - 1, J)) / 4.
DO 80 I = 6, 9
DO 80 J = 4, 5
80 S(I, J) = (S(I, J + 1) + S(I, J - 1) + S(I + 1, J) + S(I - 1, J)) / 4.
100 CONTINUE
PRINT 1, ((S(I, J), I = 1, 10), J = 1, 6)
1 FORMAT (' ', 10 E 12.4)
END
17. Axisymmetric Irrotational Flow
Figure 6.26 Numerical solution of Example 6.1.
Here, S denotes the stream function ψ. The code first sets the boundary values.
The iteration is performed in the N loop. In practice, iterations will not be performed
arbitrarily 20 times. Instead the convergence of the iteration process will be checked,
and the process is continued until some reasonable criterion (such as less than 1%
change at every point) is met. These improvements are easy to implement, and the
code is left in its simplest form.
The values of ψ at the grid points after 50 iterations, and the corresponding
streamlines, are shown in Figure 6.26.
It is a usual practice to iterate until successive iterates change only by a prescribed
small amount. The solution is then said to have “converged.” However, a caution is
in order. To be sure a solution has been obtained, all of the terms in the equation
must be calculated and the satisfaction of the equation by the “solution” must be
verified.
17. Axisymmetric Irrotational Flow
Several examples of irrotational flow around plane two-dimensional bodies were
given in the preceding sections. We used Cartesian (x, y) and plane polar (r, θ)
coordinates, and found that the problem involved the solution of the Laplace equation in φ or ψ with specified boundary conditions. We found that a very powerful tool in the analysis was the method of complex variables, including conformal
transformation.
Two streamfunctions are required to describe a fully three-dimensional
flow (Chapter 4, Section 4), although a velocity potential (which satisfies the
three-dimensional version of ∇ 2 φ = 0) can be defined if the flow is irrotational.
201
202
Irrotational Flow
If, however, the flow is symmetrical about axis, one of the streamfunctions is known
because all streamlines must lie in planes passing through the axis of symmetry. In
cylindrical polar coordinates, one streamfunction, say, χ , may be taken as χ = −ϕ.
In spherical polar coordinates (see Figure 6.27), the choice χ = −ϕ is also appropriate if all streamlines are in ϕ = const. planes through the axis of symmetry. Then
ρu = ∇χ × ∇ψ. We shall see that the streamfunction for these axisymmetric flows
does not satisfy the Laplace equation (and consequently the method of complex variables is not applicable) and the lines of constant φ and ψ are not orthogonal. Some
simple examples of axisymmetric irrotational flows around bodies of revolution, such
as spheres and airships, will be given in the rest of this chapter.
In axisymmetric flow problems, it is convenient to work with both cylindrical
and spherical polar coordinates, often going from one set to the other in the same
problem. In this chapter cylindrical coordinates will be denoted by (R, ϕ, x), and
spherical coordinates by (r, θ, ϕ). These are illustrated in Figure 6.27a, from which
their relation to Cartesian coordinates is seen to be
cylindrical
spherical
x=x
y = R cos ϕ
z = R sin ϕ
x = r cos θ
y = r sin θ cos ϕ
z = r sin θ sin ϕ
(6.63)
Note that r is the distance from the origin, whereas R is the radial distance from
the x-axis. The bodies of revolution will have their axes coinciding with the x-axis
(Figure 6.27b). The resulting flow pattern is independent of the azimuthal coordinate
ϕ, and is identical in all planes containing the x-axis. Further, the velocity component
uϕ is zero.
Important expressions for curvilinear coordinates are listed in Appendix B. For
axisymmetric flows, several relevant expressions are presented in the following for
quick reference.
Figure 6.27 (a) Cylindrical and spherical coordinates; (b) axisymmetric flow. In Fig. 6.27, the coordinate axes are not aligned according to the conventional definitions. Specifically in (a), the polar axis
from which θ is measured is usually taken to be the z-axis and ϕ is measured from the x-axis. In
(b), the axis of symmetry is usually taken to be the z-axis and the angle θ or ϕ is measured from the
x-axis.
203
18. Streamfunction and Velocity Potential for Axisymmetric Flow
Continuity equation:
∂ux
1 ∂
+
(RuR ) = 0
∂x
R ∂R
1 ∂ 2
1 ∂
(r ur ) +
(uθ sin θ ) = 0
r ∂r
sin θ ∂θ
(cylindrical)
(6.64)
(spherical)
(6.65)
Laplace equation:
∂φ
∂ 2φ
1 ∂
2
R
+ 2 = 0 (cylindrical)
∇ φ=
R ∂R
∂R
∂x
∂
1 ∂
∂φ
1
∂φ
∇2φ = 2
r2
+ 2
sin θ
=0
∂r
∂θ
r ∂r
r sin θ ∂θ
(6.66)
(spherical)
(6.67)
Vorticity:
∂ux
∂uR
−
(cylindrical)
∂x
∂R
1 ∂
∂ur
ωϕ =
(ruθ ) −
(spherical)
r ∂r
∂θ
ωϕ =
(6.68)
(6.69)
18. Streamfunction and Velocity Potential for
Axisymmetric Flow
A streamfunction can be defined for axisymmetric flows because the continuity equation involves two terms only. In cylindrical coordinates, the continuity equation can
be written as
∂
∂
(Rux ) +
(RuR ) = 0
∂x
∂R
(6.70)
which is satisfied by u = −∇ϕ × ∇ψ, yielding
ux ≡
1 ∂ψ
R ∂R
(cylindrical),
(6.71)
1 ∂ψ
uR ≡ −
.
R ∂x
The axisymmetric stream function is sometimes called the Stokes streamfunction. It
has units of m3 /s, in contrast to the streamfunction for plane flow, which has units of
m2 /s. Due to the symmetry of flow about the x-axis, constant ψ surfaces are surfaces
of revolution. Consider two streamsurfaces described by constant values of ψ and
ψ + dψ (Figure 6.28). The volumetric flow rate through the annular space is
∂ψ
∂ψ
dQ = −uR (2πR dx) + ux (2πR dR) = 2π
dx +
dR = 2π dψ,
∂x
∂R
204
Irrotational Flow
Figure 6.28 Axisymmetric streamfunction. The volume flow rate through two streamsurfaces is 2π ψ.
where equation (6.71) has been used. The form dψ = dQ/2π shows that the difference in ψ values is the flow rate between two concentric streamsurfaces per unit radian
angle around the axis. This is consistent with the extended discussion of streamfunctions in Chapter 4, Section 4. The factor of 2π is absent in plane two-dimensional
flows, where dψ = dQ is the flow rate per unit depth. The sign convention is the same
as for plane flows, namely, that ψ increases toward the left if we look downstream.
If the flow is also irrotational, then
ωϕ =
∂ux
∂uR
−
= 0.
∂x
∂R
(6.72)
On substituting equation (6.71) into equation (6.72), we obtain
1 ∂ψ
∂ 2ψ
∂ 2ψ
−
= 0,
+
R ∂R
∂R 2
∂x 2
(6.73)
which is different from the Laplace equation (6.66) satisfied by φ. It is easy to show
that lines of constant φ and ψ are not orthogonal. This is a basic difference between
axisymmetric and plane flows.
In spherical coordinates, the streamfunction is defined as u = −∇ϕ × ∇ψ,
yielding
1 ∂ψ
ur = 2
r sin θ ∂θ
(spherical),
(6.74)
1 ∂ψ
,
uθ = −
r sin θ ∂r
which satisfies the axisymmetric continuity equation (6.65).
205
19. Simple Examples of Axisymmetric Flows
The velocity potential for axisymmetric flow is defined as
cylindrical
spherical
uR =
∂φ
∂R
ur =
∂φ
∂R
ux =
∂φ
∂x
uθ =
1 ∂φ
r ∂θ
(6.75)
which satisfies the condition of irrotationality in a plane containing the x-axis.
19. Simple Examples of Axisymmetric Flows
Axisymmetric irrotational flows can be developed in the same manner as plane flows,
except that complex variables cannot be used. Several elementary flows are reviewed
briefly in this section, and some practical flows are treated in the following sections.
Uniform Flow
For a uniform flow U parallel to the x-axis, the velocity potential and streamfunction
are
cylindrical
spherical
φ = Ux
φ = U r cos θ
ψ = 21 U R 2
(6.76)
ψ = 21 U r 2 sin2 θ
These expressions can be verified by using equations (6.71), (6.74), and (6.75). Equipotential surfaces are planes normal to the x-axis, and streamsurfaces are coaxial
tubes.
Point Source
For a point source of strength Q (m3 /s), the velocity is ur = Q/4π r 2 . It is easy to
show (Exercise 6) that in polar coordinates
φ=−
Q
4πr
ψ =−
Q
cos θ.
4π
(6.77)
Equipotential surfaces are spherical shells, and streamsurfaces are conical surfaces
on which θ = const.
Doublet
For the limiting combination of a source–sink pair, with vanishing separation and
large strength, it can be shown (Exercise 7) that
φ=
m
cos θ
r2
ψ =−
m 2
sin θ,
r
(6.78)
where m is the strength of the doublet, directed along the negative x-axis. Streamlines
in an axial plane are qualitatively similar to those shown in Figure 6.8, except that
they are no longer circles.
206
Irrotational Flow
Figure 6.29 Irrotational flow past a sphere.
Flow around a Sphere
Irrotational flow around a sphere can be generated by the superposition of a uniform
stream and an axisymmetric doublet opposing the stream. The stream function is
ψ =−
m 2
1
sin θ + U r 2 sin2 θ.
r
2
(6.79)
This shows that ψ = 0 for θ = 0 or π (any r), or for r = (2m/U )1/3 (any θ).
Thus all of the x-axis and the spherical surface of radius a = (2m/U )1/3 form the
streamsurface ψ = 0. Streamlines of the flow are shown in Figure 6.29. In terms of
the radius of the sphere, velocity components are found from equation (6.79) as
a 3
1 ∂ψ
cos θ,
=U 1−
ur = 2
r
r sin θ ∂θ
(6.80)
1 ∂ψ
1 a 3
uθ = −
sin θ.
= −U 1 +
r sin θ ∂r
2 r
The pressure coefficient on the surface is
Cp =
p − p∞
1
2
2 ρU
=1−
uθ
U
2
=1−
9 2
sin θ,
4
(6.81)
which is symmetrical, again demonstrating zero drag in steady irrotational flows.
20. Flow around a Streamlined Body of Revolution
As in plane flows, the motion around a closed body of revolution can be generated
by superposition of a source and a sink of equal strength on a uniform stream. The
closed surface becomes “streamlined” (that is, has a gradually tapering tail) if, for
example, the sink is distributed over a finite length. Consider Figure 6.30, where there
is a point source Q (m3 /s) at the origin O, and a line sink distributed on the x-axis
207
20. Flow around a Streamlined Body of Revolution
Figure 6.30 Irrotational flow past a streamlined body generated by a point source at O and a distributed
line sink from O to A.
from O to A. Let the volume absorbed per unit length of the line sink be k (m2 /s).
An elemental length dξ of the sink can be regarded as a point sink of strength k dξ ,
for which the streamfunction at any point P is [see equation (6.77)]
dψsink =
k dξ
cos α.
4π
The total streamfunction at P due to the entire line sink from O to A is
a
k
ψsink =
cos α dξ.
4π 0
(6.82)
The integral can be evaluated by noting that x − ξ = R cot α. This gives dξ =
R dα/ sin2 α because x and R remain constant as we go along the sink. The streamfunction of the line sink is therefore
α1
k
R
kR α1 d(sin α)
ψsink =
cos α 2 dα =
,
4π θ
4π θ
sin α
sin2 α
k
1
1
kR
=
−
(r − r1 ).
(6.83)
=
4π sin θ
sin α1
4π
To obtain a closed body, we must adjust the strengths so that the efflux from the source
is absorbed by the sink, that is, Q = ak. Then the streamfunction at any point P due
to the superposition of a point source of strength Q, a distributed line sink of strength
k = Q/a, and a uniform stream of velocity U along the x-axis, is
ψ =−
Q
Q
1
cos θ +
(r − r1 ) + U r 2 sin2 θ.
4π
4π a
2
(6.84)
208
Irrotational Flow
A plot of the steady streamline pattern is shown in the bottom half of Figure 6.30,
in which the top half shows instantaneous streamlines in a frame of reference at rest
with the fluid at infinity.
Here we have assumed that the strength of the line sink is uniform along its
length. Other interesting streamlines can be generated by assuming that the strength
k(ξ ) is nonuniform.
21. Flow around an Arbitrary Body of Revolution
So far, in this chapter we have been assuming certain distributions of singularities, and
determining what body shape results when the distribution is superposed on a uniform
stream. The flow around a body of given shape can be simulated by superposing a
uniform stream on a series of sources and sinks of unknown strength distributed on a
line coinciding with the axis of the body. The strengths of the sources and sinks are then
so adjusted that, when combined with a given uniform flow, a closed streamsurface
coincides with the given body. The calculation is done numerically using a computer.
Let the body length L be divided into N equal segments of length ξ , and let kn
be the strength (m2 /s) of one of these line sources, which may be positive or negative
(Figure 6.31). Then the streamfunction at any “body point” m due to the line source
n is, using equation (6.83),
ψmn = −
kn m
rn−1 − rnm ,
4π
where the negative sign is introduced because equation (6.83) is for a sink. When
combined with a uniform stream, the streamfunction at m due to all N line sources is
ψm = −
N
kn m
2
rn−1 − rnm + 21 U Rm
.
4π
n=1
Figure 6.31 Flow around an arbitrary axisymmetric shape generated by superposition of a series of line
sources.
209
Exercise
Setting ψm = 0 for all N values of m, we obtain a set of N linear algebraic equations
in N unknowns kn (n = 1, 2, . . . , N), which can be solved by the iteration technique
described in Section 16 or some other matrix inversion routine.
22. Concluding Remarks
The theory of potential flow has reached a highly developed stage during the last
250 years because of the efforts of theoretical physicists such as Euler, Bernoulli,
D’Alembert, Lagrange, Stokes, Helmholtz, Kirchhoff, and Kelvin. The special interest in the subject has resulted from the applicability of potential theory to other fields
such as heat conduction, elasticity, and electromagnetism. When applied to fluid flows,
however, the theory resulted in the prediction of zero drag on a body at variance with
observations. Meanwhile, the theory of viscous flow was developed during the middle of the Nineteenth Century, after the Navier–Stokes equations were formulated.
The viscous solutions generally applied either to very slow flows where the nonlinear
advection terms in the equations of motion were negligible, or to flows in which the
advective terms were identically zero (such as the viscous flow through a straight
pipe). The viscous solutions were highly rotational, and it was not clear where the
irrotational flow theory was applicable and why. This was left for Prandtl to explain,
as will be shown in Chapter 10.
It is probably fair to say that the theory of irrotational flow does not occupy the
center stage in fluid mechanics any longer, although it did so in the past. However,
the subject is still quite useful in several fields, especially in aerodynamics. We shall
see in Chapter 10 that the pressure distribution around streamlined bodies can still be
predicted with a fair degree of accuracy from the irrotational flow theory. In Chapter 15
we shall see that the lift of an airfoil is due to the development of circulation around
it, and the magnitude of the lift agrees with the Kutta–Zhukhovsky lift theorem. The
technique of conformal mapping will also be essential in our study of flow around
airfoil shapes.
Exercises
1. In Section 7, the doublet potential
w = µ/z,
was derived by combining a source and a sink on the x-axis. Show that the same
potential can also be obtained by superposing a clockwise vortex of circulation −Ŵ
on the y-axis at y = ε, and a counterclockwise vortex of circulation Ŵ at y = −ε,
and letting ε → 0.
2. By integrating pressure, show that the drag on a plane half-body (Section 8)
is zero.
3. Graphically generate the streamline pattern for a plane half-body in the following manner. Take a source of strength m = 200 m2 /s and a uniform stream U =
10 m/s. Draw radial streamlines from the source at equal intervals of θ = π/10,
210
Irrotational Flow
with the corresponding streamfunction interval
ψsource =
m
θ = 10 m2 /s.
2π
Now draw streamlines of the uniform flow with the same interval, that is,
ψstream = U y = 10 m2 /s.
This requires y = 1 m, which you can plot assuming a linear scale of 1 cm = 1 m.
Now connect points of equal ψ = ψsource + ψstream . (Most students enjoy doing this
exercise!)
4. Take a plane source of strength m at point (−a, 0), a plane sink of equal
strength at (a, 0), and superpose a uniform stream U directed along the x-axis. Show
that there are two stagnation points located on the x-axis at points
±a
m
+1
π aU
1/2
.
Show that the streamline passing through the stagnation points is given by ψ = 0.
Verify that the line ψ = 0 represents a closed oval-shaped body, whose maximum
width h is given by the solution of the equation
πUh
h = a cot
.
m
The body generated by the superposition of a uniform stream and a source–sink pair is
called a Rankine body. It becomes a circular cylinder as the source–sink pair approach
each other.
5. A two-dimensional potential vortex with clockwise circulation Ŵ is located at
point (0, a) above a flat plate. The plate coincides with the x-axis. A uniform stream
U directed along the x-axis flows over the vortex. Sketch the flow pattern and show
that it represents the flow over an oval-shaped body. [Hint: Introduce the image vortex
and locate the two stagnation points on the x-axis.]
If the pressure at x = ±∞ is p∞ , and that below the plate is also p∞ , then show
that the pressure at any point on the plate is given by
p∞ − p =
ρŴ 2 a 2
ρU Ŵa
.
−
2π 2 (x 2 + a 2 )2
π(x 2 + a 2 )
Show that the total upward force on the plate is
F =
ρŴ 2
− ρU Ŵ.
4π a
6. Consider a point source of strength Q (m3 /s). Argue that the velocity components in spherical coordinates are uθ = 0 and ur = Q/4π r 2 and that the velocity
211
Exercises
potential and streamfunction must be of the form φ = φ(r) and ψ = ψ(θ). Integrating
the velocity, show that φ = −Q/4πr and ψ = −Q cos θ/4π .
7. Consider a point doublet obtained as the limiting combination of a point
source and a point sink as the separation goes to zero. (See Section 7 for its two
dimensional counterpart.) Show that the velocity potential and streamfunction in
spherical coordinates are φ = m cos θ/r 2 and ψ = −m sin2 θ/r, where m is the
limiting value of Q δs/4π , with Q as the source strength and δs as the separation.
8. A solid hemisphere of radius a is lying on a flat plate. A uniform stream
U is flowing over it. Assuming irrotational flow, show that the density of the material
must be
33 U 2
,
ρh ρ 1 +
64 ag
to keep it on the plate.
9. Consider the plane flow around a circular cylinder. Use the Blasius theorem
equation (6.45) to show that the drag is zero and the lift is L = ρU Ŵ. (In Section 10,
we derived these results by integrating the pressure.)
10. There is a point source of strength Q (m3 /s) at the origin, and a uniform
line sink of strength k = Q/a extending from x = 0 to x = a. The two are combined
with a uniform stream U parallel to the x-axis. Show that the combination represents
the flow past a closed surface of revolution of airship shape, whose total length is the
difference of the roots of
x2
a2
x
Q
.
±1 =
a
4π U a 2
11. Using a computer, determine the surface contour of an axisymmetric
half-body formed by a line source of strength k (m2 /s) distributed uniformly along
the x-axis from x = 0 to x = a and a uniform stream. Note that the nose is more
pointed than that formed by the combination of a point source and a uniform stream.
By a mass balance (see√
Section 8), show that the far downstream asymptotic radius
of the half-body is r = ak/π U .
12. For the flow described by equation (6.30) and sketched in Figure 6.8, show
for µ > 0 that u < 0 for y < x and u > 0 for y > x. Also, show that v < 0 in the
first quadrant and v > 0 in the second quadrant.
13. A hurricane is blowing over a long “Quonset hut,” that is, a long half-circular
cylindrical cross-section building, 6 m in diameter. If the velocity far upstream is
U∞ = 40 m/s and p∞ = 1.003 × 105 N/m, ρ∞ = 1.23 kg/m3 , find the force per
unit depth on the building, assuming the pressure inside is p∞ .
14. In a two-dimensional constant density potential flow, a source of strength m
is located a meters above an infinite plane. Find the velocity on the plane, the pressure
on the plane, and the reaction force on the plane.
212
Irrotational Flow
15. Consider a two-dimensional, constant density potential flow over a circular
cylinder of radius r = a with axis coincident with a right angle corner, as shown in
the figure below. Solve for the streamfunction and velocity components.
c
5
co
n
st
y
a
x
Literature Cited
Prandtl, L. (1952). Essentials of Fluid Dynamics, New York: Hafner Publishing.
Supplemental Reading
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.
Milne-Thompson, L. M. (1962). Theoretical Hydrodynamics, London: Macmillan Press.
Shames, I. H. (1962). Mechanics of Fluids, New York: McGraw-Hill.
Vallentine, H. R. (1967). Applied Hydrodynamics, New York: Plenum Press.
Chapter 7
Gravity Waves
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
The Wave Equation. . . . . . . . . . . . . . . . . .
Wave Parameters . . . . . . . . . . . . . . . . . . . .
Surface Gravity Waves . . . . . . . . . . . . . . .
Formulation of the Problem . . . . . . . . .
Solution of the Problem . . . . . . . . . . . . .
Some Features of Surface Gravity
Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pressure Change Due to Wave
Motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
Particle Path and Streamline . . . . . . . .
Energy Considerations . . . . . . . . . . . . . .
Approximations for Deep and
Shallow Water . . . . . . . . . . . . . . . . . . . . . .
Deep-Water Approximation . . . . . . . . . .
Shallow-Water Approximation . . . . . . .
Wave Refraction in Shallow Water . . .
Influence of Surface Tension . . . . . . . . .
Standing Waves . . . . . . . . . . . . . . . . . . . . .
Group Velocity and Energy Flux . . . . .
Group Velocity and Wave
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . .
Physical Motivation . . . . . . . . . . . . . . . . .
Layer of Constant Depth . . . . . . . . . . . .
Layer of Variable Depth H (x) . . . . . . .
Nonlinear Steepening in a
Nondispersive Medium . . . . . . . . . . . . . .
Hydraulic Jump . . . . . . . . . . . . . . . . . . . . .
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50007-1
214
214
216
219
219
221
223
224
224
227
229
230
231
233
234
237
238
242
242
243
244
246
248
13. Finite Amplitude Waves of
Unchanging Form in a Dispersive
Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Finite Amplitude Waves in Deep Water:
The Stokes Wave . . . . . . . . . . . . . . . . . . 251
Finite Amplitude Waves in Fairly
Shallow Water: Solitons . . . . . . . . . . . 252
14. Stokes’ Drift . . . . . . . . . . . . . . . . . . . . . . . . . 253
15. Waves at a Density Interface
between Infinitely Deep Fluids . . . . . . . 255
16. Waves in a Finite Layer Overlying
an Infinitely Deep Fluid . . . . . . . . . . . . . 259
Barotropic or Surface Mode . . . . . . . . . . 261
Baroclinic or Internal Mode . . . . . . . . . . 261
17. Shallow Layer Overlying an
Infinitely Deep Fluid. . . . . . . . . . . . . . . . . 262
18. Equations of Motion for a
Continuously Stratified Fluid . . . . . . . . 263
19. Internal Waves in a Continuously
Stratified Fluid . . . . . . . . . . . . . . . . . . . . . . 267
The w = 0 Limit . . . . . . . . . . . . . . . . . . . . 269
20. Dispersion of Internal Waves in a
Stratified Fluid . . . . . . . . . . . . . . . . . . . . . . 270
21. Energy Considerations of Internal
Waves in a Stratified Fluid . . . . . . . . . . . 272
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Literature Cited . . . . . . . . . . . . . . . . . . . . . 277
213
214
Gravity Waves
1. Introduction
It is perhaps not an overstatement to say that wave motion is the most basic feature
of all physical phenomena. Waves are the means by which information is transmitted
between two points in space and time, without movement of the medium across the
two points. The energy and phase of some disturbance travels during a wave motion,
but motion of the matter is generally small. Waves are generated due to the existence of
some kind of “restoring force” that tends to bring the system back to its undisturbed
state, and of some kind of “inertia” that causes the system to overshoot after the
system has returned to the undisturbed state. One type of wave motion is generated
when the restoring forces are due to the compressibility or elasticity of the material
medium, which can be a solid, liquid, or gas. The resulting wave motion, in which the
particles move to and fro in the direction of wave propagation, is called a compression
wave, elastic wave, or pressure wave. The small-amplitude variety of these is called
a “sound wave.” Another common wave motion, and the one we are most familiar
with from everyday experience, is the one that occurs at the free surface of a liquid,
with gravity playing the role of the restoring force. These are called surface gravity
waves. Gravity waves, however, can also exist at the interface between two fluids of
different density, in which case they are called internal gravity waves. The particle
motion in gravity waves can have components both along and perpendicular to the
direction of propagation, as we shall see.
In this chapter, we shall examine some basic features of wave motion and illustrate
them with gravity waves because these are the easiest to comprehend physically. The
wave frequency will be assumed much larger than the Coriolis frequency, in which
case the wave motion is unaffected by the earth’s rotation. Waves affected by planetary
rotation will be considered in Chapter 14. Wave motion due to compressibility effects
will be considered in Chapter 16. Unless specified otherwise, we shall assume that the
waves have small amplitude, in which case the governing equation becomes linear.
2. The Wave Equation
Many simple “nondispersive” (to be defined later) wave motions of small amplitude
obey the wave equation
∂ 2η
= c2 ∇ 2 η,
(7.1)
∂t 2
which is a linear partial differential equation of the hyperbolic type. Here η is any
type of disturbance, for example the displacement of the free surface in a liquid,
variation of density in a compressible medium, or displacement of a stretched string
or membrane. The meaning of parameter c will become clear shortly. Waves traveling
only in the x direction are described by
2
∂ 2η
2∂ η
=
c
,
∂t 2
∂x 2
which has a general solution of the form
η = f (x − ct) + g (x + ct) ,
(7.2)
(7.3)
215
2. The Wave Equation
where f and g are arbitrary functions. Equation (7.3), called d’Alembert’s solution,
signifies that any arbitrary function of the combination (x±ct) is a solution of the wave
equation; this can be verified by substitution of equation (7.3) into equation (7.2). It
is easy to see that f (x − ct) represents a wave propagating in the positive x direction
with speed c, whereas g(x + ct) propagates in the negative x direction at speed c.
Figure 7.1 shows a plot of f (x − ct) at t = 0. At a later time t, the distance x needs to
be larger for the same value of (x − ct). Consequently, f (x − ct) has the same shape
as f (x), except displaced by an amount ct along the x-axis. Therefore, the speed of
propagation of wave shape f (x − ct) along the positive x-axis is c.
As an example of solution of the wave equation, assume initial conditions in the
form
∂η
η(x, 0) = F (x) and
(x, 0) = G(x),
(7.4)
∂t
Then equation (7.3) requires that
f (x) + g(x) = F (x)
and
which gives the solution
1
1 x
f (x) =
F (x) −
G(ξ ) dξ ,
2
c x0
−
dg
1
df
+
= G(x),
dx
dx
c
1 x
1
F (x) +
G(ξ ) dξ ,
2
c x0
(7.5)
The case of zero initial velocity [G(x) = 0] is interesting. It corresponds to an initial
displacement of the surface into an arbitrary profile F (x), which is then left alone.
In this case equation (7.5) reduces to f (x) = g(x) = F (x)/2, so that solution (7.5)
becomes
η = 21 F (x − ct) + 21 F (x + ct),
(7.6)
g(x) =
The nature of this solution is illustrated in Figure 7.2. It is apparent that half the initial
disturbance propagates to the right and the other half propagates to the left. Widths
of the two components are equal to the width of the initial disturbance. Note that
boundary conditions have not been considered in arriving at equation (7.6). Instead,
the boundaries have been assumed to be so far away that the reflected waves do not
return to the region of interest.
Figure 7.1
Profiles of f (x − ct) at two times.
216
Gravity Waves
Figure 7.2 Wave profiles at three times. The initial profile is F (x) and the initial velocity is assumed to
be zero. Half the initial disturbance propagates to the right and the other half propagates to the left.
3. Wave Parameters
According to Fourier’s principle, any arbitrary disturbance can be decomposed into
sinusoidal wave components of different wavelengths and amplitudes. Consequently,
it is important to study sinusoidal waves of the form
2π
η = a sin
(x − ct) .
(7.7)
λ
The argument 2π(x − ct)/λ is called the phase of the wave, and points of constant
phase are those where the waveform has the same value, say a crest or trough. Since η
varies between ±a, a is called the amplitude of the wave. The parameter λ is called the
wavelength because the value of η in equation (7.7) does not change if x is changed
by ±λ. Instead of using λ, it is more common to use the wavenumber defined as
k≡
2π
,
λ
(7.8)
which is the number of complete waves in a length 2π . It can be regarded as the
“spatial frequency” (rad/m). The waveform equation (7.7) can then be written as
η = a sin k(x − ct).
(7.9)
217
3. Wave Parameters
The period T of a wave is the time required for the condition at a point to repeat itself,
and must equal the time required for the wave to travel one wavelength:
T =
λ
.
c
(7.10)
The number of oscillations at a point per unit time is the frequency, given by
ν=
1
.
T
(7.11)
Clearly c = λν. The quantity
ω = 2π ν = kc,
(7.12)
is called the circular frequency; it is also called the “radian frequency” because it is
the rate of change of phase (in radians) per unit time. The speed of propagation of the
waveform is related to k and ω by
c=
ω
,
k
(7.13)
which is called the phase speed, as it is the rate at which the “phase” of the wave
(crests and troughs) propagates. We shall see that the phase speed may not be the
speed at which the envelope of a group of waves propagates. In terms of ω and k, the
waveform equation (7.7) is written as
η = a sin(kx − ωt).
(7.14)
So far we have been considering waves propagating in the x direction only. For
three-dimensional waves of sinusoidal shape, equation (7.14) is generalized to
η = a sin(kx + ly + mz − ωt) = a sin(K • x − ωt),
(7.15)
where K = (k, l, m) is a vector, called the wavenumber vector, whose magnitude is
given by
K 2 = k 2 + l 2 + m2 .
(7.16)
It is easy to see that the wavelength of equation (7.15) is
λ=
2π
,
K
(7.17)
which is illustrated in Figure 7.3 in two dimensions. The magnitude of phase velocity
is c = ω/K, and the direction of propagation is that of K. We can therefore write the
phase velocity as the vector
ωK
c=
,
(7.18)
KK
where K/K represents the unit vector in the direction of K.
218
Gravity Waves
Figure 7.3 Wave propagating in the xy-plane. The inset shows how the components cx and cy are added
to give the resultant c.
From Figure 7.3, it is also clear that the phase speeds (that is, the speeds of
propagation of lines of constant phase) in the three Cartesian directions are
ω
ω
ω
cy =
cz = .
(7.19)
cx =
k
l
m
The preceding shows that the components cx , cy , and cz are each larger than the
resultant c = ω/K. It is clear that the components of the phase velocity vector c do
not obey the rule of vector addition. The method of obtaining c from the components
cx and cy is illustrated at the top of Figure 7.3. The peculiarity of such an addition
rule for the phase velocity vector merely reflects the fact that phase lines appear to
propagate faster along directions not coinciding with the direction of propagation,
say the x and y directions in Figure 7.3. In contrast, the components of the “group
velocity” vector cg do obey the usual vector addition rule, as we shall see later.
We have assumed that the waves exist without a mean flow. If the waves are
superposed on a uniform mean flow U, then the observed phase speed is
c0 = c + U.
A dot product of the forementioned with the wavenumber vector K, and the use of
equation (7.18), gives
ω0 = ω + U • K,
(7.20)
where ω0 is the observed frequency at a fixed point, and ω is the intrinsic frequency
measured by an observer moving with the mean flow. It is apparent that the frequency
4. Surface Gravity Waves
of a wave is Doppler shifted by an amount U • K due to the mean flow. Equation (7.20)
is easy to understand by considering a situation in which the intrinsic frequency ω is
zero and the flow pattern has a periodicity in the x direction of wavelength 2π/k. If
this sinusoidal pattern is translated in the x direction at speed U , then the observed
frequency at a fixed point is ω0 = U k.
The effects of mean flow on frequency will not be considered further in this
chapter. Consequently, the involved frequencies should be interpreted as the intrinsic
frequency.
4. Surface Gravity Waves
In this section we shall discuss gravity waves at the free surface of a sea of liquid of
uniform depth H , which may be large or small compared to the wavelength λ. We
shall assume that the amplitude a of oscillation of the free surface is small, in the sense
that both a/λ and a/H are much smaller than one. The condition a/λ ≪ 1 implies
that the slope of the sea surface is small, and the condition a/H ≪ 1 implies that the
instantaneous depth does not differ significantly from the undisturbed depth. These
conditions allow us to linearize the problem. The frequency of the waves is assumed
large compared to the Coriolis frequency, so that the waves are unaffected by the
earth’s rotation. Here, we shall neglect surface tension; in water its effect is limited
to wavelengths <7 cm, as discussed in Section 7. The fluid is assumed to have small
viscosity, so that viscous effects are confined to boundary layers and do not affect the
wave propagation significantly. The motion is assumed to be generated from rest, say,
by wind action or by dropping a stone. According to Kelvin’s circulation theorem,
the resulting motion is irrotational, ignoring viscous effects, Coriolis forces, and
stratification (density variation).
Formulation of the Problem
Consider a case where the waves propagate in the x direction only, and that the
motion is two dimensional in the xz-plane (Figure 7.4). Let the vertical coordinate z
Figure 7.4
Wave nomenclature.
219
220
Gravity Waves
be measured upward from the undisturbed free surface. The free surface displacement
is η(x, t). Because the motion is irrotational, a velocity potential φ can be defined
such that
∂φ
∂φ
w=
.
(7.21)
u=
∂x
∂z
Substitution into the continuity equation
∂u ∂w
+
= 0,
∂x
∂z
(7.22)
∂ 2φ
∂ 2φ
+ 2 = 0.
2
∂x
∂z
(7.23)
gives the Laplace equation
Boundary conditions are to be satisfied at the free surface and at the bottom. The
condition at the bottom is zero normal velocity, that is
w=
∂φ
=0
∂z
at
z = −H.
(7.24)
At the free surface, a kinematic boundary condition is that the fluid particle never
leaves the surface, that is
Dη
= wη at z = η,
Dt
where D/Dt = ∂/∂t + u(∂/∂x), and wη is the vertical component of fluid velocity
at the free surface. This boundary condition is the specialization of that discussed in
Chapter 4.19 to zero mass flow across the wave surface. The forementioned condition
can be written as
∂η
∂φ
∂η
=
.
(7.25)
+ u
∂t
∂x z=η
∂z z=η
For small-amplitude waves both u and ∂η/∂x are small, so that the quadratic term
u(∂η/∂x) is one order smaller than other terms in equation (7.25), which then simplifies to
∂η
∂φ
=
,
(7.26)
∂t
∂z z=η
We can simplify this condition still further by arguing that the right-hand side can be
evaluated at z = 0 rather than at the free surface. To justify this, expand ∂φ/∂z in a
Taylor series around z = 0:
∂φ
∂φ
∂ 2 φ
∂φ
·
·
·
≃
+
.
=
+
η
∂z z=η
∂z z=0
∂z z=0
∂z2 z=0
Therefore, to the first order of accuracy desired here, ∂φ/∂z in equation (7.26) can
be evaluated at z = 0. We then have
∂η
∂φ
=
at z = 0.
(7.27)
∂t
∂z
221
4. Surface Gravity Waves
The error involved in approximating equation (7.26) by (7.27) is explained again later
in this section.
In addition to the kinematic condition at the surface, there is a dynamic condition
that the pressure just below the free surface is always equal to the ambient pressure,
with surface tension neglected. Taking the ambient pressure to be zero, the condition is
p=0
z = η.
at
(7.28)
Equation (7.28) follows from the boundary condition on τ • n, which is continuous
across an interface as established in Chapter 4, Section 19. As before, we shall simplify
this condition for small-amplitude waves. Since the motion is irrotational, Bernoulli’s
equation (see equation (4.81))
∂φ
1
p
+ (u2 + w 2 ) + + gz = F (t),
∂t
2
ρ
(7.29)
is applicable. Here, the function F (t) can be absorbed in ∂φ/∂t by redefining φ.
Neglecting the nonlinear term (u2 + w 2 ) for small-amplitude waves, the linearized
form of the unsteady Bernoulli equation is
p
∂φ
+ + gz = 0.
∂t
ρ
(7.30)
Substitution into the surface boundary condition (7.28) gives
∂φ
+ gη = 0
∂t
at
z = η.
(7.31)
As before, for small-amplitude waves, the term ∂φ/∂t can be evaluated at z = 0
rather than at z = η to give
∂φ
= −gη
∂t
at
z = 0.
(7.32)
Solution of the Problem
Recapitulating, we have to solve
∂ 2φ
∂ 2φ
+ 2 = 0.
2
∂x
∂z
(7.22)
subject to the conditions
∂φ
=0
∂z
at
z = −H,
(7.24)
∂φ
∂η
=
∂z
∂t
at
z = 0,
(7.27)
∂φ
= −gη
∂t
at
z = 0.
(7.32)
222
Gravity Waves
In order to apply the boundary conditions, we need to assume a form for η(x, t). The
simplest case is that of a sinusoidal component with wavenumber k and frequency ω,
for which
η = a cos(kx − ωt).
(7.33)
One motivation for studying sinusoidal waves is that small-amplitude waves on a water
surface become roughly sinusoidal some time after their generation (unless the water
depth is very shallow). This is due to the phenomenon of wave dispersion discussed
in Section 10. A second, and stronger, motivation is that an arbitrary disturbance
can be decomposed into various sinusoidal components by Fourier analysis, and the
response of the system to an arbitrary small disturbance is the sum of the responses
to the various sinusoidal components.
For a cosine dependence of η on (kx − ωt), conditions (7.27) and (7.32) show
that φ must be a sine function of (kx − ωt). Consequently, we assume a separable
solution of the Laplace equation in the form
φ = f (z) sin(kx − ωt),
(7.34)
where f (z) and ω(k) are to be determined. Substitution of equation (7.34) into the
Laplace equation (7.22) gives
d 2f
− k 2 f = 0,
dz2
whose general solution is
f (z) = Aekz + Be−kz .
The velocity potential is then
φ = (Aekz + Be−kz ) sin(kx − ωt).
(7.35)
The constants A and B are now determined from the boundary conditions (7.24) and
(7.27). Condition (7.24) gives
B = Ae−2kH .
(7.36)
Before applying condition (7.27) in the linearized form, let us explore what would
happen if we applied it at z = η. From (7.35) we get
∂φ
= k(Aekη − Be−kη ) sin(kx − ωt),
∂z z=η
Here we can set e kη ≃ e −kη ≃ 1 if kη ≪ 1, valid for small slope of the free surface.
This is effectively what we are doing by applying the surface boundary conditions
equations (7.27) and (7.32) at z = 0 (instead of at z = η), which we justified
previously by a Taylor series expansion.
Substitution of equations (7.33) and (7.35) into the surface velocity condition
(7.27) gives
k(A − B) = aω.
(7.37)
223
5. Some Features of Surface Gravity Waves
The constants A and B can now be determined from equations (7.36) and (7.37) as
A=
aω
k(1 − e−2kH )
B=
aω e−2kH
.
k(1 − e−2kH )
The velocity potential (7.35) then becomes
φ=
aω cosh k(z + H )
sin(kx − ωt),
k
sinh kH
(7.38)
from which the velocity components are found as
cosh k(z + H )
cos(kx − ωt),
sinh kH
sinh k(z + H )
w = aω
sin(kx − ωt).
sinh kH
u = aω
(7.39)
We have solved the Laplace equation using kinematic boundary conditions alone.
This is typical of irrotational flows. In the last chapter we saw that the equation of
motion, or its integral, the Bernoulli equation, is brought into play only to find the
pressure distribution, after the problem has been solved from kinematic considerations
alone. In the present case, we shall find that application of the dynamic free surface
condition (7.32) gives a relation between k and ω.
Substitution of equations (7.33) and (7.38) into (7.32) gives the desired relation
ω = gk tanh kH ,
(7.40)
The phase speed c = ω/k is related to the wave size by
c=
g
tanh kH =
k
2π H
gλ
tanh
,
2π
λ
(7.41)
This shows that the speed of propagation of a wave component depends on its
wavenumber. Waves for which c is a function of k are called dispersive because
waves of different lengths, propagating at different speeds, “disperse” or separate.
(Dispersion is a word borrowed from optics, where it signifies separation of different
colors due to the speed of light in a medium depending on the wavelength.) A relation
such as equation (7.40), giving ω as a function of k, is called a dispersion relation
because it expresses the nature of the dispersive process. Wave dispersion is a fundamental process in many physical phenomena; its implications in gravity waves are
discussed in Sections 9 and 10.
5. Some Features of Surface Gravity Waves
Several features of surface gravity waves are discussed in this section. In particular, we
shall examine the nature of pressure change, particle motion, and the energy flow due
to a sinusoidal propagating wave. The water depth H is arbitrary; simplifications that
result from assuming the depth to be shallow or deep are discussed in the next section.
224
Gravity Waves
Pressure Change Due to Wave Motion
It is sometimes possible to measure wave parameters by placing pressure sensors at
the bottom or at some other suitable depth. One would therefore like to know how deep
the pressure fluctuations penetrate into the water. Pressure is given by the linearized
Bernoulli equation
∂φ
p
+ + gz = 0.
∂t
ρ
If we define
p′ ≡ p + ρgz,
(7.42)
as the perturbation pressure, that is, the pressure change from the undisturbed pressure
of −ρgz, then Bernoulli’s equation gives
p ′ = −ρ
∂φ
.
∂t
(7.43)
On substituting equation (7.38), we obtain
ρaω2 cosh k(z + H )
cos(kx − ωt),
k
sinh kH
which, on using the dispersion relation (7.40), becomes
p′ =
p ′ = ρga
cosh k(z + H )
cos(kx − ωt).
cosh kH
(7.44a)
(7.44b)
The perturbation pressure therefore decays into the water column, and whether it
could be detected by a sensor depends on the magnitude of the water depth in relation
to the wavelength. This is discussed further in Section 6.
Particle Path and Streamline
To examine particle orbits, we obviously need to use Lagrangian coordinates. (See
Chapter 3, Section 2 for a discussion of the Lagrangian description.) Let (x0 +ξ, z0 + ζ )
be the coordinates of a fluid particle whose rest position is (x0 , z0 ), as shown in Figure 7.5. We can use (x0 , z0 ) as a “tag” for particle identification, and write ξ(x0 , z0 , t)
and ζ (x0 , z0 , t) in the Lagrangian form. Then the velocity components are given by
∂ξ
,
∂t
∂ζ
w=
,
∂t
u =
(7.45)
where the partial derivative symbol is used because the particle identity (x0 , z0 ) is
kept fixed in the time derivatives. For small-amplitude waves, the particle excursion
(ξ, ζ ) is small, and the velocity of a particle along its path is nearly equal to the
fluid velocity at the mean position (x0 , z0 ) at that instant, given by equation (7.39).
225
5. Some Features of Surface Gravity Waves
Figure 7.5
Orbit of a fluid particle whose mean position is (x0 , z0 ).
Therefore, equation (7.45) gives
∂ξ
cosh k(z0 + H )
= aω
cos(kx0 − ωt),
∂t
sinh kH
∂ζ
sinh k(z0 + H )
= aω
sin(kx0 − ωt).
∂t
sinh kH
Integrating in time, we obtain
ξ = −a
cosh k(z0 + H )
sin(kx0 − ωt),
sinh kH
sinh k(z0 + H )
ζ = a
cos(kx0 − ωt).
sinh kH
Elimination of (kx0 − ωt) gives
sinh k(z0 + H ) 2
cosh k(z0 + H ) 2
2
2
ξ
+ζ
a
= 1,
a
sinh kH
sinh kH
(7.46)
(7.47)
which represents ellipses. Both the semimajor axis, a cosh[k(z0 + H )]/sinh kH and
the semiminor axis, a sinh[k(z0 + H )]/sinh kH decrease with depth, the minor axis
vanishing at z0 = −H (Figure 7.6b). The distance between foci remains constant
with depth. Equation (7.46) shows that the phase of the motion (that is, the argument
of the sinusoidal term) is independent of z0 . Fluid particles in any vertical column are
therefore in phase. That is, if one of them is at the top of its orbit, then all particles at
the same x0 are at the top of their orbits.
To find the streamline pattern, we need to determine the streamfunction ψ, related
to the velocity components by
∂ψ
cosh k(z + H )
= u = aω
cos(kx − ωt),
∂z
sinh kH
(7.48)
sinh k(z + H )
∂ψ
= −w = −aω
sin(kx − ωt),
∂x
sinh kH
(7.49)
226
Gravity Waves
Figure 7.6 Particle orbits of wave motion in deep, intermediate and shallow seas.
where equation (7.39) has been introduced. Integrating equation (7.48) with respect
to z, we obtain
ψ=
aω sinh k(z + H )
cos(kx − ωt) + F (x, t),
k
sinh kH
where F (x, t) is an arbitrary function of integration. Similarly, integration of equation (7.49) with respect to x gives
ψ=
aω sinh k(z + H )
cos(kx − ωt) + G(z, t),
k
sinh kH
where G(z, t) is another arbitrary function. Equating the two expressions for ψ we
see that F = G = function of time only; this can be set to zero if we regard ψ as due
to wave motion only, so that ψ = 0 when a = 0. Therefore
ψ=
aω sinh k(z + H )
cos(kx − ωt).
k
sinh kH
(7.50)
Let us examine the streamline structure at a particular time, say, t = 0, when
ψ ∝ sinh k(z + H ) cos kx.
It is clear that ψ = 0 at z = −H , so that the bottom wall is a part of the ψ = 0
streamline. However, ψ is also zero at kx = ±π/2, ±3π/2, . . . for any z. At these
5. Some Features of Surface Gravity Waves
Figure 7.7
Instantaneous streamline pattern in a surface gravity wave propagating to the right.
values of kx, equation (7.33) shows that η vanishes. The resulting streamline pattern
is shown in Figure 7.7. It is seen that the velocity is in the direction of propagation
(and horizontal ) at all depths below the crests, and opposite to the direction of
propagation at all depths below troughs.
Energy Considerations
Surface gravity waves possess kinetic energy due to motion of the fluid and potential
energy due to deformation of the free surface. Kinetic energy per unit horizontal area
is found by integrating over the depth and averaging over a wavelength:
λ 0
ρ
Ek =
(u2 + w 2 ) dz dx.
2λ 0 −H
Here the z-integral is taken up to z = 0, because the integral up to z = η gives a
higher-order term. Substitution of the velocity components from equation (7.39) gives
λ
0
ρω2
1
2
2
Ek =
cosh2 k(z + H ) dz
a
cos
(kx
−
ωt)
dx
2 sinh2 kH λ 0
−H
0
1 λ 2 2
sinh2 k(z + H ) dz .
(7.51)
a sin (kx − ωt) dx
+
λ 0
−H
In terms of free surface displacement η, the x-integrals in equation (7.51) can be
written as
1 λ 2
1 λ 2 2
a cos2 (kx − ωt) dx =
a sin (kx − ωt) dx
λ 0
λ 0
1 λ 2
η dx = η2 ,
=
λ 0
227
228
Gravity Waves
where η2 is the mean square displacement. The z-integrals in equation (7.51) are easy
to evaluate by expressing the hyperbolic functions in terms of exponentials. Using
the dispersion relation (7.40), equation (7.51) finally becomes
Ek = 21 ρgη2 ,
(7.52)
which is the kinetic energy of the wave motion per unit horizontal area.
Consider next the potential energy of the wave system, defined as the work done
to deform a horizontal free surface into the disturbed state. It is therefore equal to the
difference of potential energies of the system in the disturbed and undisturbed states.
As the potential energy of an element in the fluid (per unit length in y) is ρgz dx dz
(Figure 7.8), the potential energy of the wave system per unit horizontal area is
ρg λ η
ρg λ 0
Ep =
z dz dx −
z dz dx,
λ 0 −H
λ 0 −H
ρg λ η
ρg λ 2
=
z dz dx =
η dx.
(7.53)
λ 0 0
2λ 0
(An easier way to arrive at the expression for Ep is to note that the potential energy
increase due to wave motion equals the work done in raising column A in Figure 7.8
to the location of column B, and integrating over half the wavelength. This is because
an interchange of A and B over half a wavelength automatically forms a complete
wavelength of the deformed surface. The mass of column A is ρη dx, and the center of gravity is raised by η when A is taken to B. This agrees with the last form
in equation (7.53).) Equation (7.53) can be written in terms of the mean square
displacement as
(7.54)
Ep = 21 ρgη2 .
Comparison of equation (7.52) and equation (7.54) shows that the average kinetic
and potential energies are equal. This is called the principle of equipartition of energy
Figure 7.8 Calculation of potential energy of a fluid column.
229
6. Approximations for Deep and Shallow Water
and is valid in conservative dynamical systems undergoing small oscillations that are
unaffected by planetary rotation. However, it is not valid when Coriolis forces are
included, as will be seen in Chapter 14. The total wave energy in the water column
per unit horizontal area is
E = Ep + Ek = ρgη2 = 21 ρga 2 ,
(7.55)
where the last form in terms of the amplitude a is valid if η is assumed sinusoidal,
since the average of cos2 x over a wavelength is 1/2.
Next, consider the rate of transmission of energy due to a single sinusoidal component of wavenumber k. The energy flux across the vertical plane x = 0 is the
pressure work done by the fluid in the region x < 0 on the fluid in the region x > 0.
Per unit length of crest, the time average energy flux is (writing p as the sum of a
perturbation p′ and a background pressure −ρgz)
0
0
0
F =
z dz
p ′ u dz − ρgu
pu dz =
−H
0
−H
=
−H
p ′ u dz ,
(7.56)
−H
where denotes an average over a wave period; we have used the fact that u = 0.
Substituting for p′ from equation (7.44a) and u from equation (7.39), equation (7.56)
becomes
0
ρa 2 ω3
F = cos2 (kx − ωt)
cosh2 k(z + H ) dz.
k sinh2 kH −H
The time average of cos2 (kx − ωt) is 1/2. The z-integral can be carried out by writing
it in terms of exponentials. This finally gives
c
2kH
2
1
F = 2 ρga
1+
.
(7.57)
2
sinh 2kH
The first factor is the wave energy given in equation (7.55). Therefore, the second
factor must be the speed of propagation of wave energy of component k, called the
group speed. This is discussed in Sections 9 and 10.
6. Approximations for Deep and Shallow Water
The analysis in the preceding section is applicable whatever the magnitude of λ
is in relation to the water depth H . Interesting simplifications result for H /λ ≪ 1
(shallow water) and H /λ ≫ 1 (deep water). The expression for phase speed is given
by equation (7.41), namely,
gλ
2π H
tanh
.
(7.41)
c=
2π
λ
230
Gravity Waves
Approximations are now derived under two limiting conditions in which
equation (7.41) takes simple forms.
Deep-Water Approximation
We know that tanh x → 1 for x → ∞ (Figure 7.9). However, x need not be very
large for this approximation to be valid, because tanh x = 0.94138 for x = 1.75. It
follows that, with 3% accuracy, equation (7.41) can be approximated by
gλ
g
=
,
(7.58)
c=
2π
k
for H > 0.28λ (corresponding to kH > 1.75). Waves are therefore classified as
deep-water waves if the depth is more than 28% of the wavelength. Equation (7.58)
shows that longer waves in deep water propagate faster. This feature has interesting
consequences and is discussed further in Sections 9 and 10.
A dominant period of wind-generated surface gravity waves in the ocean is ≈10 s,
for which the dispersion relation (7.40) shows that the dominant wavelength is 150 m.
The water depth on a typical continental shelf is ≈100 m, and in the open ocean it
is about ≈4 km. It follows that the dominant wind waves in the ocean, even over the
continental shelf, act as deep-water waves and do not feel the effects of the ocean
Figure 7.9 Behavior of hyperbolic functions.
6. Approximations for Deep and Shallow Water
bottom until they arrive near the beach. This is not true of gravity waves generated by
tidal forces and earthquakes; these may have wavelengths of hundreds of kilometers.
In the preceding section we said that particle orbits in small-amplitude gravity
waves describe ellipses given by equation (7.47). For H > 0.28λ, the semimajor and
semiminor axes of these ellipses each become nearly equal to aekz . This follows from
the approximation (valid for kH > 1.75)
sinh k(z + H )
cosh k(z + H )
≃
≃ ekz .
sinh kH
sinh kH
(The various approximations for hyperbolic functions used in this section can easily be
verified by writing them in terms of exponentials.) Therefore, for deep-water waves,
particle orbits described by equation (7.46) simplify to
ξ = −a ekz0 sin(kx0 − ωt)
ζ = a ekz0 cos(kx0 − ωt).
The orbits are therefore circles (Figure 7.6a), of which the radius at the surface equals
a, the amplitude of the wave. The velocity components are
∂ξ
= aωekz cos(kx − ωt)
∂t
∂ζ
w=
= aωekz sin(kx − ωt),
∂t
u=
where we have omitted the subscripts on (x0 , z0 ). (For small amplitudes the difference
in velocity at the present and mean positions of a particle is negligible. The distinction
between mean particle positions and Eulerian coordinates is therefore not necessary,
unless finite amplitude effects are considered, as we will see in Section 14.) The
velocity vector therefore rotates clockwise (for a wave traveling in the positive x
direction) at frequency ω, while its magnitude remains constant at aωekz0 .
For deep-water waves, the perturbation pressure given in equation (7.44b) simplifies to
p ′ = ρgaekz cos(kx − ωt).
(7.59)
This shows that pressure change due to the presence of wave motion decays exponentially with depth, reaching 4% of its surface magnitude at a depth of λ/2. A sensor
placed at the bottom cannot therefore detect gravity waves whose wavelengths are
smaller than twice the water depth. Such a sensor acts like a “low-pass filter,” retaining
longer waves and rejecting shorter ones.
Shallow-Water Approximation
We know that tanh x ≃ x as x → 0 (Figure 7.9). For H /λ ≪ 1, we can therefore
write
2π H
2π H
tanh
≃
,
λ
λ
231
232
Gravity Waves
in which case the phase speed equation (7.41) simplifies to
c=
√
gH .
(7.60)
The approximation gives a better than 3% accuracy if H < 0.07λ. Surface waves are
therefore regarded as shallow-water waves if the water depth is <7% of the wavelength. (The water depth has to be really shallow for waves to behave as shallow-water
waves. This is consistent with the comments made in what follows (equation (7.58)),
that the water depth does not have to be really deep for water to behave as deep-water
waves.) For these waves equation (7.60) shows that the wave speed is independent of
wavelength and increases with water depth.
To determine the approximate form of particle orbits for shallow-water waves,
we substitute the following approximations into equation (7.46):
cosh k(z + H ) ≃ 1
sinh k(z + H ) ≃ k(z + H )
sinh kH ≃ kH.
The particle excursions given in equation (7.46) then become
a
sin(kx − ωt)
kH
z
cos(kx − ωt).
ζ =a 1+
H
ξ =−
These represent thin ellipses (Figure 7.6c), with a depth-independent semimajor axis
of a/kH and a semiminor axis of a(1 + z/H ), which linearly decreases to zero at
the bottom wall. From equation (7.39), the velocity field is found as
aω
cos(kx − ωt)
kH
z
sin(kx − ωt),
w = aω 1 +
H
u =
(7.61)
which shows that the vertical component is much smaller than the horizontal
component.
The pressure change from the undisturbed state is found from equation (7.44b)
to be
p ′ = ρga cos(kx − ωt) = ρgη,
(7.62)
where equation (7.33) has been used to express the pressure change in terms of η. This
shows that the pressure change at any point is independent of depth, and equals the
hydrostatic increase of pressure due to the surface elevation change η. The pressure
6. Approximations for Deep and Shallow Water
Figure 7.10 Refraction of a surface gravity wave approaching a sloping beach. Note that the crest lines
tend to become parallel to the coast.
field is therefore completely hydrostatic in shallow-water waves. Vertical accelerations are negligible because of the small w-field. For this reason, shallow water waves
are also called hydrostatic waves. It is apparent that a pressure sensor mounted at the
bottom can sense these waves.
Wave Refraction in Shallow Water
We shall now qualitatively describe the commonly observed phenomenon of refraction of shallow-water waves. Consider a sloping beach, with depth contours parallel
to the coastline (Figure 7.10). Assume that waves are propagating toward the coast
from the deep ocean, with their crests at an angle to the coastline. Sufficiently near the
coastline they begin to feel the effect of the bottom and finally become shallow-water
waves. Their frequency does not change along the
√ path (a fact that will be proved in
Section 10), but the speed of propagation c = gH and the wavelength λ become
smaller. Consequently, the crest lines, which are perpendicular to the local direction
of c, tend to become parallel to the coast. This is why we see that the waves coming
toward the beach always seem to have their crests parallel to the coastline.
An interesting example of wave refraction occurs when a deep-water wave with
straight crests approaches an island (Figure 7.11). Assume that the water depth
becomes shallower as the island is approached, and the constant depth contours are
circles concentric with the island. Figure 7.11 shows that the waves always come in
toward the island, even on the “shadow” side marked A!
The bending of wave paths in an inhomogeneous medium is called wave refraction. In this case the source of inhomogeneity is the spatial dependence of H . The
analogous phenomenon in optics is the bending of light due to density changes in
its path.
233
234
Gravity Waves
7. Influence of Surface Tension
It was explained in Section 1.5 that the interface between two immiscible fluids is in a
state of tension. The tension acts as a restoring force, enabling the interface to support
waves in a manner analogous to waves on a stretched membrane or string. Waves due
to the presence of surface tension are called capillary waves. Although gravity is not
needed to support these waves, the existence of surface tension alone without gravity
is uncommon. We shall therefore examine the modification of the preceding results
for pure gravity waves due to the inclusion of surface tension.
Let PQ = ds be an element of arc on the free surface, whose local radius of
curvature is r (Figure 7.12a). Suppose pa is the pressure on the “atmospheric” side,
and p is the pressure just inside the interface. The surface tension forces at P and Q,
Figure 7.11 Refraction of a surface gravity wave approaching an island with sloping beach. Crest lines,
perpendicular to the rays, are shown. Note that the crest lines come in toward the island, even on the
shadow side A. Reprinted with the permission of Mrs. Dorothy Kinsman Brown: B. Kinsman, Wind Waves,
Prentice-Hall Englewood Cliffs, NJ, 1965.
Figure 7.12 (a) Segment of a free surface under the action of surface tension; (b) net surface tension
force on an element.
235
7. Influence of Surface Tension
per unit length perpendicular to the plane of the paper, are each equal to σ and directed
along the tangents at P and Q. Equilibrium of forces on the arc PQ is considered in
Figure 7.12b. The force at P is represented by segment OA, and the force at Q is
represented by segment OB. The resultant of OA and OB in a direction perpendicular
to the arc PQ is represented by 2OC ≃ σ dθ. Therefore, the balance of forces in a
direction perpendicular to the arc PQ requires
−pa ds + p ds + σ dθ = 0.
It follows that the pressure difference is related to the curvature by
pa − p = σ
σ
dθ
= .
ds
r
The curvature 1/r of η(x) is given by
∂ 2 η/∂x 2
∂ 2η
1
=
≃ 2,
2
3/2
r
[1 + (∂η/∂x) ]
∂x
where the approximate expression is for small slopes. Therefore,
pa − p = σ
∂ 2η
.
∂x 2
Choosing the atmospheric pressure pa to be zero, we obtain the condition
p = −σ
∂ 2η
∂x 2
at
z = η.
(7.63)
Using the linearized Bernoulli equation
p
∂φ
+ + gz = 0,
∂t
ρ
condition (7.63) becomes
σ ∂ 2η
∂φ
− gη
=
∂t
ρ ∂x 2
at
z = 0.
(7.64)
As before, for small-amplitude waves it is allowable to apply the surface boundary
condition (7.64) at z = 0, instead at z = η.
Solution of the wave problem including surface tension is identical to the one for
pure gravity waves presented in Section 4, except that the pressure boundary condition
(7.32) is replaced by (7.64). This only changes the dispersion relation ω(k), which is
found by substitution of (7.33) and (7.38) into (7.64), to give
σ k2
tanh kH .
(7.65)
ω= k g+
ρ
236
Gravity Waves
Figure 7.13 Sketch of phase velocity vs wavelength in a surface gravity wave.
The phase velocity is therefore
g
gλ
σk
2π σ
c=
+
+
tanh kH =
k
ρ
2π
ρλ
tanh
2π H
.
λ
(7.66)
A plot of equation (7.66) is shown in Figure 7.13. It is apparent that the effect of surface
tension is to increase c above its value for pure gravity waves at all wavelengths. This
is because the free surface is now “tighter,” and hence capable of generating more
restoring forces. However, the effect of surface tension is only appreciable for very
small wavelengths. A measure of these wavelengths is obtained by noting that there
is a minimum phase speed at λ = λm , and surface tension dominates for λ < λm
(Figure 7.13). Setting dc/dλ = 0 in equation (7.66), and assuming the deep-water
approximation tanh(2π H /λ) ≃ 1 valid for H > 0.28λ, we obtain
cmin
4gσ
=
ρ
1/4
at
λm = 2π
σ
.
ρg
(7.67)
For an air–water interface at 20 ◦ C, the surface tension is σ = 0.074 N/m, giving
cmin = 23.2 cm/s
at
λm = 1.73 cm.
(7.68)
Only small waves (say, λ < 7 cm for an air–water interface), called ripples, are therefore affected by surface tension. Wavelengths <4 mm are dominated by surface tension
237
8. Standing Waves
and are rather unaffected by gravity. From equation (7.66), the phase speed of these
pure capillary waves is
2π σ
c=
,
(7.69)
ρλ
where we have again assumed tanh(2π H /λ) ≃ 1. The smallest of these, traveling
at a relatively large speed, can be found leading the waves generated by dropping a
stone into a pond.
8. Standing Waves
So far, we have been studying propagating waves. Nonpropagating waves can be generated by superposing two waves of the same amplitude and wavelength, but moving
in opposite directions. The resulting surface displacement is
η = a cos(kx − ωt) + a cos(kx + ωt) = 2a cos kx cos ωt.
It follows that η = 0 for kx = ±π/2, ±3π/2 . . . . Points of zero surface displacement
are called nodes. The free surface therefore does not propagate, but simply oscillates
up and down with frequency ω, keeping the nodal points fixed. Such waves are called
standing waves. The corresponding streamfunction, using equation (7.50), is both for
the cos(kx − ωt) and cos(kx + ωt) components, and for the sum. This gives
aω sinh k(z + H )
[cos(kx − ωt) − cos(kx + ωt)]
k
sinh kH
2aω sinh k(z + H )
=
sin kx sin ωt.
k
sinh kH
ψ=
(7.70)
The instantaneous streamline pattern shown in Figure 7.14 should be compared with
the streamline pattern for a propagating wave (Figure 7.7).
A limited body of water such as a lake forms standing waves by reflection from
the walls. A standing oscillation in a lake is called a seiche (pronounced “saysh”),
Figure 7.14 Instantaneous streamline pattern in a standing surface gravity wave. If this is mode n = 0,
then two successive vertical streamlines are a distance L apart. If this is mode n = 1, then the first and
third vertical streamlines are a distance L apart.
238
Gravity Waves
Figure 7.15 Normal modes in a lake, showing distributions of u for the first two modes. This is consistent
with the streamline pattern of Figure 7.14.
in which only certain wavelengths and frequencies ω (eigenvalues) are allowed by
the system. Let L be the length of the lake, and assume that the waves are invariant
along y. The possible wavelengths are found by setting u = 0 at the two walls.
Because u = ∂ψ/∂z, equation (7.70) gives
u = 2aω
cosh k(z + H )
sin kx sin ωt.
sinh kH
(7.71)
Taking the walls at x = 0 and L, the condition of no flow through the walls requires
sin(kL) = 0, that is,
kL = (n + 1)π
n = 0, 1, 2, . . . ,
which gives the allowable wavelengths as
λ=
2L
.
n+1
(7.72)
The largest wavelength is 2L and the next smaller is L (Figure 7.15). The allowed
frequencies can be found from the dispersion relation (7.40), giving
(n + 1)πH
πg(n + 1)
ω=
tanh
,
(7.73)
L
L
which are the natural frequencies of the lake.
9. Group Velocity and Energy Flux
An interesting set of phenomena takes place when the phase speed of a wave depends
on its wavelength. The most
√ common example is the deep water gravity wave, for
which c is proportional to λ. A wave phenomenon in which c depends on k is called
239
9. Group Velocity and Energy Flux
dispersive because, as we shall see in the next section, the different wave components
separate or “disperse” from each other.
In a dispersive system, the energy of a wave component does not propagate at
the phase velocity c = ω/k, but at the group velocity defined as cg = dω/dk. To see
this, consider the superposition of two sinusoidal components of equal amplitude but
slightly different wavenumber (and consequently slightly different frequency because
ω = ω(k)). Then the combination has a waveform
η = a cos(k1 x − ω1 t) + a cos(k2 x − ω2 t).
Applying the trigonometric identity for cos A + cos B, we obtain
η = 2a cos
1
2 (k2
− k1 )x − 21 (ω2 − ω1 )t cos
1
2 (k1
+ k2 )x − 21 (ω1 + ω2 )t .
Writing k = (k1 + k2 )/2, ω = (ω1 + ω2 )/2, dk = k2 − k1 , and dω = ω2 − ω1 ,
we obtain
η = 2a cos 21 dk x − 21 dω t cos(kx − ωt).
(7.74)
Here, cos(kx − ωt) is a progressive wave with a phase speed of c = ω/k. However,
its amplitude 2a is modulated by a slowly varying function cos[dk x/2 − dω t/2],
which has a large wavelength 4π/dk, a large period 4π/dω, and propagates at a speed
(=wavelength/period) of
cg =
dω
.
dk
(7.75)
Multiplication of a rapidly varying sinusoid and a slowly varying sinusoid, as in
equation (7.74), generates repeating wave groups (Figure 7.16). The individual wave
components propagate with the speed c = ω/k, but the envelope of the wave groups
travels with the speed cg , which is therefore called the group velocity. If cg < c,
then the wave crests seem to appear from nowhere at a nodal point, proceed forward
through the envelope, and disappear at the next nodal point. If, on the other hand,
cg > c, then the individual wave crests seem to emerge from a forward nodal point
and vanish at a backward nodal point.
2 a cos
1
(dk x 2tdv)
2
cg
c
node
Figure 7.16 Linear combination of two sinusoids, forming repeated wave groups.
240
Gravity Waves
Equation (7.75) shows that the group speed of waves of a certain wavenumber
k is given by the slope of the tangent to the dispersion curve ω(k). In contrast, the
phase velocity is given by the slope of the radius vector (Figure 7.17).
A particularly illuminating example of the idea of group velocity is provided
by the concept of a wave packet, formed by combining all wavenumbers in a certain narrow band δk around a central value k. In physical space, the wave appears
nearly sinusoidal with wavelength 2π/k, but the amplitude dies away in a length of
order 1/δk (Figure 7.18). If the spectral width δk is narrow, then decay of the wave
amplitude in physical space is slow. The concept of such a wave packet is more realistic than the one in Figure 7.16, which is rather unphysical because the wave groups
repeat themselves. Suppose that, at some initial time, the wave group is represented by
η = a(x) cos kx.
It can be shown (see, for example, Phillips (1977), p. 25) that for small times the
subsequent evolution of the wave profile is approximately described by
η = a(x − cg t) cos(kx − ωt),
(7.76)
where cg = dω/dk. This shows that the amplitude of a wave packet travels with the
group speed. It follows that cg must equal the speed of propagation of energy of a
certain wavelength. The fact that cg is the speed of energy propagation is also evident
in Figure 7.16 because the nodal points travel at cg and no energy can cross the nodal
points.
For surface gravity waves having the dispersion relation
(7.40)
ω = gk tanh kH ,
the group velocity is found to be
2kH
c
1+
cg =
.
2
sinh 2kH
Figure 7.17 Finding c and cg from dispersion relation ω(k).
(7.77)
241
9. Group Velocity and Energy Flux
Figure 7.18 A wave packet composed of a narrow band of wavenumbers δk.
The two limiting cases are
cg = 21 c
(deep water),
cg = c
(shallow water).
(7.78)
The group velocity of deep-water gravity waves is half the phase speed. Shallow-water
waves, on the other hand, are nondispersive, for which c = cg . For a linear nondispersive system, any waveform preserves its shape in time because all the wavelengths
that make up the waveform travel at the same speed. For a pure capillary wave, the
group velocity is cg = 3c/2 (Exercise 3).
The rate of transmission of energy for gravity waves is given by equation (7.57),
namely
c
2kH
F =E
1+
,
2
sinh kH
where E = ρga 2 /2 is the average energy in the water column per unit horizontal
area. Using equation (7.77), we conclude that
F = Ecg .
(7.79)
This signifies that the rate of transmission of energy of a sinusoidal wave component
is wave energy times the group velocity. This reinforces our previous interpretation
of the group velocity as the speed of propagation of energy.
We have discussed the concept of group velocity in one dimension only, taking
ω to be a function of the wavenumber k in the direction of propagation. In three
dimensions ω(k, l, m) is a function of the three components of the wavenumber
vector K = (k, l, m) and, using Cartesian tensor notation, the group velocity vector
is given by
cgi =
∂ω
,
∂Ki
where Ki stands for any of the components of K. The group velocity vector is then
the gradient of ω in the wavenumber space.
242
Gravity Waves
10. Group Velocity and Wave Dispersion
Physical Motivation
We continue our discussion of group velocity in this section, focussing on how the
different wavelength and frequency components are propagated. Consider waves in
deep water, for which
c
gλ
c=
cg = ,
2π
2
signifying that larger waves propagate faster. Suppose that a surface disturbance is
generated by dropping a stone into a pool. The initial disturbance can be thought of
as being composed of a great many wavelengths. A short time later, at t = t1 , the sea
surface may have the rather irregular profile shown in Figure 7.19. The appearance
of the surface at a later time t2 , however, is more regular, with the longer components
(which have been traveling faster) out in front. The waves in front are the longest
waves produced by the initial disturbance; we denote their length by λmax , typically
a few times larger than the stone. The leading edge of the wave system therefore
propagates at the group speed corresponding to these wavelengths, that is, at the
speed
1 gλmax
cg max =
.
2
2π
(Pure capillary waves can propagate faster than this speed, but they have small magnitude and get dissipated rather soon.) The region of initial disturbance becomes calm
because there is a minimum group velocity of gravity waves due to the influence of
surface tension, namely 17.8 cm/s (Exercise 4). The trailing edge of the wave system
therefore travels at speed
cg min = 17.8 cm/s.
With cg max > 17.8 cm/s for ordinary sizes of stones, the length of the disturbed region
gets larger, as shown in Figure 7.19. The wave heights are correspondingly smaller
because there is a fixed amount of energy in the wave system. (Wave dispersion,
therefore, makes the linearity assumption more accurate.) The smoothening of the
profile and the spreading of the region of disturbance continue until the amplitudes
become imperceptible or the waves are damped by viscous dissipation. It is clear
that the initial superposition of various wavelengths, running for some time, will sort
themselves out in the sense that the different sinusoidal components, differing widely
in their wavenumbers, become spatially separated, and are found in quite different
places. This is a basic feature of the behavior of a dispersive system.
The wave group as a whole travels slower than the individual crests. Therefore,
if we try to follow the last crest at the rear of the train, quite soon we find that it is the
second one from the rear; a new crest has been born behind it. In fact, new crests are
constantly “popping up from nowhere” at the rear of the train, propagating through
the train, and finally disappearing in front of the train. This is because, by following a
particular crest, we are traveling at twice the speed at which the energy of waves of a
243
10. Group Velocity and Wave Dispersion
Figure 7.19 Surface profiles at three values of time due to a disturbance caused by dropping a stone into
a pool.
particular length is traveling. Consequently, we do not see a wave of fixed wavelength
if we follow a particular crest. In fact, an individual wave constantly becomes longer
as it propagates through the train. When its length becomes equal to the longest wave
generated initially, it cannot evolve any more and dies out. Clearly, the waves in front
of the train are the longest Fourier components present in the initial disturbance.
Layer of Constant Depth
We shall now prove that an observer traveling at cg would see no change in k if the
layer depth H is uniform everywhere. Consider a wavetrain of “gradually varying
wavelength,” such as the one shown at later time values in Figure 7.19. By this we
mean that the distance between successive crests varies slowly in space and time.
Locally, we can describe the free surface displacement by
η = a(x, t) cos[θ (x, t)],
(7.80)
where a(x, t) is a slowly varying amplitude and θ(x, t) is the local phase. We know
that the phase angle for a wavenumber k and frequency ω is θ = kx − ωt. For a
gradually varying wavetrain, we can define a local wavenumber k(x, t) and a local
244
Gravity Waves
frequency ω(x, t) as the rate of change of phase in space and time, respectively.
That is,
∂θ
,
∂x
∂θ
ω = − .
∂t
(7.81)
∂k
∂ω
+
= 0.
∂t
∂x
(7.82)
k =
Cross differentiation gives
Now suppose we have a dispersion relation relating ω solely to k in the form
ω = ω(k). We can then write
∂ω
dω ∂k
=
,
∂x
dk ∂x
so that equation (7.82) becomes
∂k
∂k
+ cg
= 0,
∂t
∂x
(7.83)
where cg = dω/dk. The left-hand side of equation (7.83) is similar to the material
derivative and gives the rate of change of k as seen by an observer traveling at speed
cg . Such an observer will always see the same wavelength. Group velocity is therefore
the speed at which wavenumbers are advected. This is shown in the xt-diagram of
Figure 7.20, where wave crests are followed along lines dx/dt = c and wavelengths
are preserved along the lines dx/dt = cg . Note that the width of the disturbed region,
bounded by the first and last thick lines in Figure 7.20, increases with time, and that
the crests constantly appear at the back of the group and vanish at the front.
Layer of Variable Depth H (x)
The conclusion that an observer traveling at cg sees only waves of the same length is
true only for waves in a homogeneous medium, that is, a medium whose properties
are uniform everywhere. In contrast, a sea of nonuniform depth H (x) behaves like an
inhomogeneous medium, provided the waves are shallow enough to feel the bottom.
In such a case it is the frequency of the wave, and not its wavelength, that remains
constant along the path of propagation of energy. To demonstrate this, consider a case
where H (x) is gradually varying (on the scale of a wavelength) so that we can still
use the dispersion relation (7.40) with H replaced by H (x):
ω = gk tanh[kH (x)].
Such a dispersion relation has a form
ω = ω(k, x).
(7.84)
245
10. Group Velocity and Wave Dispersion
In such a case we can find the group velocity at a point as
cg (k, x) =
∂ω(k, x)
,
∂k
(7.85)
which on multiplication by ∂k/∂t gives
cg
∂ω ∂k
∂ω
∂k
=
=
.
∂t
∂k ∂t
∂t
(7.86)
Multiplying equation (7.82) by cg and using equation (7.86) we obtain
∂ω
∂ω
+ cg
= 0.
∂t
∂x
(7.87)
In three dimensions, this is written as
∂ω
+ cg • ∇ω = 0,
∂t
which shows that ω remains constant to an observer traveling with the group velocity
in an inhomogeneous medium.
Summarizing, an observer traveling at cg in a homogeneous medium sees constant
values of k, ω(k), c, and cg (k). Consequently, ray paths describing group velocity in
the xt-plane are straight lines (Figure 7.20). In an inhomogeneous medium only ω
Figure 7.20 Propagation of a wave group in a homogeneous medium, represented on an xt-plot. Thin
lines indicate paths taken by wave crests, and thick lines represent paths along which k and ω are constant. M. J. Lighthill, Waves in Fluids, 1978 and reprinted with the permission of Cambridge University
Press, London.
246
Gravity Waves
Figure 7.21 Propagation of a wave group in an inhomogeneous medium represented on an xt-plot. Only
ray paths along which ω is constant are shown. M. J. Lighthill, Waves in Fluids, 1978 and reprinted with
the permission of Cambridge University Press, London.
remains constant along the lines dx/dt = cg , but k, c, and cg can change. Consequently, ray paths are not straight in this case (Figure 7.21).
11. Nonlinear Steepening in a Nondispersive Medium
Until now we have assumed that the wave amplitude is small. This has enabled us to
neglect the higher-order terms in the Bernoulli equation and to apply the boundary
conditions at z = 0 instead of at the free surface z = η. One consequence of such
linear analysis has been that waves of arbitrary shape propagate unchanged in form
if the system is nondispersive, such as shallow water waves. The unchanging form is
a result of the fact that all wavelengths,
of which the initial waveform is composed,
√
propagate at the same speed c = gH , provided all the sinusoidal components satisfy
the shallow-water approximation H k ≪ 1. We shall now see that the unchanging
waveform result is no longer valid if finite amplitude effects are considered. Several
other nonlinear effects will also be discussed in the following sections.
Finite amplitude effects can be formally treated by the method of characteristics;
this is discussed, for example, in Liepmann and Roshko (1957) and Lighthill (1978).
Instead, we shall adopt only a qualitative approach here. Consider a finite amplitude
surface displacement consisting of an elevation and a depression, propagating in
shallow-water of undisturbed depth H (Figure 7.22). Let a little wavelet be superposed
on the elevation at point x, at which the water depth is H ′ (x) and the fluid velocity
due to the wave motion is u(x). Relative to an observer moving with
the fluid velocity
u, the wavelet propagates at the local shallow-water speed c′ = gH ′ . The speed of
the wavelet relative to a frame of reference fixed in the undisturbed fluid is therefore
c = c′ + u. It is apparent that the local wave speed c is no longer constant because
11. Nonlinear Steepening in a Nondispersive Medium
c′ (x) and u(x) are variables. This is in contrast to the linearized theory in which u is
negligible and c′ is constant because H ′ ≃ H .
Let us now examine the effect of such a variable c on the wave profile. The value
of c′ is larger for points on the elevation than for points on the depression. From
Figure 7.7 we also know that the fluid velocity u is positive (that is, in the direction
of wave propagation) under an elevation and negative under a depression. It follows
that wave speed c is larger for points on the hump than for points on the depression,
so that the waveform undergoes a “shearing deformation” as it propagates, the region
of elevation tending to overtake the region of depression (Figure 7.22).
We shall call the front face AB a “compression region” because the elevation here
is rising with time. Figure 7.22 shows that the net effect of nonlinearity is a steepening
of the compression region. For finite amplitude waves in a nondispersive medium
like shallow water, therefore, there is an important distinction between compression
and expansion regions. A compression region tends to steepen with time and form
a jump, while an expansion region tends to flatten out. This eventually would lead
to the shape shown at the top of Figure 7.22, implying the physically impossible
situation of three values of surface elevation at a point. However, before this happens
the wave slope becomes nearly infinite (profile at t2 in Figure 7.22), so that dissipative
Figure 7.22 Wave profiles at four values of time. At t2 the profile has formed a hydraulic jump. The
profile at t3 is impossible.
247
248
Gravity Waves
processes including wave breaking and foaming become important, and the previous
inviscid arguments become inapplicable. Such a waveform has the
a front
√ form of √
and propagates into still fluid at constant speed that lies between gH1 and gH2 ,
where H1 and H2 are the water depths on the two sides of the front. This is called
the hydraulic jump, which is similar to the shock wave in a compressible flow. This
is discussed further in the following section.
12. Hydraulic Jump
In the previous section we saw how steepening of the compression region of a surface
wave in shallow water leads to the formation of a jump, which subsequently propagates
into the undisturbed fluid at constant speed and without further change in form. In
this section we shall discuss certain characteristics of flow across such a jump. Before
we do so, we shall introduce certain definitions.
Consider the flow in a shallow canal of depth H . If the flow speed is u, we may
define a nondimensional speed by
Fr ≡ √
u
u
= .
c
gH
This is called the Froude number, which is the ratio of the speed of flow to the speed of
infinitesimal gravity waves. The flow is called supercritical if Fr > 1, and subcritical
if Fr < 1. The Froude number is analogous to the Mach number in compressible flow,
defined as the ratio of the speed of flow to the speed of sound in the medium.
It was seen in the preceding section that a hydraulic jump propagates into a
still fluid at a speed√(say, u1 ) that lies
√ between the long-wave speeds on the two
sides, namely, c1 = gH1 and c2 = gH2 (Figure 7.23c). Now suppose a leftward
propagating jump is made stationary by superposing a flow u1 directed to the right.
In this frame the fluid enters the jump at speed u1 and exits at speed u2 < u1
(Figure 7.23b). Because c1 < u1 < c2 , it follows that Fr 1 > 1 and Fr 2 < 1. Just as
a compressible flow suddenly changes from a supersonic to subsonic state by going
through a shock wave (Section 16.6), a supercritical flow in a shallow canal can change
into a subcritical state by going through a hydraulic jump. The depth of flow rises
downstream of a hydraulic jump, just as the pressure rises downstream of a shock
wave. To continue the analogy, mechanical energy is lost by dissipative processes
both within the hydraulic jump and within the shock wave. A common example of
a stationary hydraulic jump is found at the foot of a dam, where the flow almost
always reaches a supercritical state because of the free fall (Figure 7.23a). A tidal
bore propagating into a river mouth is an example of a propagating hydraulic jump.
Consider a control volume across a stationary hydraulic jump shown in Figure
7.23b. The depth rises from H1 to H2 and the velocity falls from u1 to u2 . If Q is
the volume rate of flow per unit width normal to the plane of the paper, then mass
conservation requires
Q = u1 H1 = u2 H2 .
12. Hydraulic Jump
Figure 7.23 Hydraulic jump.
249
250
Gravity Waves
Now use the momentum principle (Section 4.8), which says that the sum of the
forces on a control volume equals the momentum outflow rate at section 2 minus the
momentum inflow rate at section 1. The force at section 1 is the average pressure
ρgH1 /2 times the area H1 ; similarly, the force at section 2 is ρgH22 /2. If the distance
between sections 1 and 2 is small, then the force exerted by the bottom wall of the
canal is negligible. Then the momentum theorem gives
2
1
2 ρgH1
− 21 ρgH22 = ρQ(u2 − u1 ).
Substituting u1 = Q/H1 and u2 = Q/H2 on the right-hand side, we obtain
Q
Q
g 2
−
(H1 − H22 ) = Q
2
H2
H1
.
(7.88)
Canceling the factor (H1 − H2 ), we obtain
H2
H1
2
+
H2
− 2Fr 21 = 0,
H1
where Fr 21 = Q2 /gH13 = u21 /gH1 . The solution is
H2
= 21 (−1 + 1 + 8Fr 21 ).
H1
(7.89)
For supercritical flows Fr 1 > 1, for which equation (7.89) shows that H2 > H1 . Therefore, depth of water increases downstream of the hydraulic jump.
Although the solution H2 < H1 for Fr 1 < 1 is allowed by equation (7.89), such
a solution violates the second law of thermodynamics, because it implies an increase
of mechanical energy of the flow. To see this, consider the mechanical energy of a
fluid particle at the surface, E = u2 /2 + gH = Q2 /2H 2 + gH . Eliminating Q by
equation (7.88) we obtain, after some algebra,
E2 − E1 = −(H2 − H1 )
g(H2 − H1 )2
.
4H1 H2
This shows that H2 < H1 implies E2 > E1 , which violates the second law of thermodynamics. The mechanical energy, in fact, decreases in a hydraulic jump because
of the eddying motion within the jump.
A hydraulic jump not only appears at the free surface, but also at density interfaces
in a stratified fluid, in the laboratory as well as in the atmosphere and the ocean. (For
example, see Turner (1973), Figure 3.11, for his photograph of an internal hydraulic
jump on the lee side of a mountain.)
13. Finite Amplitude Waves of Unchanging Form in a
Dispersive Medium
In Section 11 we considered a nondispersive medium, and found that nonlinear
effects continually accumulate and add up until they become large changes. Such an
251
13. Finite Amplitude Waves of Unchanging Form in a Dispersive Medium
accumulation is prevented in a dispersive medium because the different Fourier components propagate at different speeds and become separated from each other. In a
dispersive system, then, nonlinear steepening could cancel out the dispersive spreading, resulting in finite amplitude waves of constant form. This is indeed the case. A
brief description of the phenomenon is given here; further discussion can be found in
Lighthill (1978), Whitham (1974), and LeBlond and Mysak (1978).
Note that if the amplitude is negligible, then in a dispersive system a wave of
unchanging form can only be perfectly sinusoidal because the presence of any other
Fourier component would cause the sinusoids to propagate at different speeds, resulting in a change in the wave shape.
Finite Amplitude Waves in Deep Water: The Stokes Wave
In 1847 Stokes showed that periodic waves of finite amplitude are possible in deep
water. In terms of a power series in the amplitude a, he showed that the surface
elevation of irrotational waves in deep water is given by
η = a cos k(x − ct) + 21 ka 2 cos 2k(x − ct)
+ 83 k 2 a 3 cos 3k(x − ct) + · · · ,
(7.90)
where the speed of propagation is
c=
g
(1 + k 2 a 2 ).
k
(7.91)
Equation (7.90) is the Fourier series for the waveform η. The addition of Fourier
components of different wavelengths in equation (7.90) shows that the wave profile
η is no longer exactly sinusoidal. The arguments in the cosine terms show that all the
Fourier components propagate at the same speed c, so that the wave profile propagates unchanged in time. It has now been established that the existence of periodic
wavetrains of unchanging form is a typical feature of nonlinear dispersive systems.
Another important result, generally valid for nonlinear systems, is that the wave speed
depends on the amplitude, as in equation (7.91).
Periodic finite-amplitude irrotational waves in deep water are frequently called
Stokes’ waves. They have a flattened trough and a peaked crest (Figure 7.24). The
maximum possible amplitude is amax = 0.07λ, at which point the crest becomes
a sharp 120◦ angle. Attempts at generating waves of larger amplitude result in the
appearance of foam (white caps) at these sharp crests. In finite amplitude waves, fluid
particles no longer trace closed orbits, but undergo a slow drift in the direction of
wave propagation; this is discussed in Section 14.
Figure 7.24 The Stokes wave. It is a finite amplitude periodic irrotational wave in deep water.
252
Gravity Waves
Finite Amplitude Waves in Fairly Shallow Water: Solitons
Next, consider nonlinear waves in a slightly dispersive system, such as “fairly long”
waves with λ/H in the range between 10 and 20. In 1895 Korteweg and deVries
showed that these waves approximately satisfy the nonlinear equation
∂η 3 η ∂η 1
∂ 3η
∂η
+ c0
+ c0
+ c0 H 2 3 = 0,
∂t
∂x
8 H ∂x
6
∂x
(7.92)
√
where c0 = gH . This is the Korteweg–deVries equation. The first two terms appear
in the linear nondispersive limit. The third term is due to finite amplitude effects and
the fourth term results from the weak dispersion due to the water depth being not
shallow enough. (Neglecting the nonlinear term in equation (7.92), and substituting
η = a exp(ikx − iωt), it is easy to show that the dispersion relation is c = c0 (1 −
(1/6)k 2 H 2 ). This agrees
√with the first two terms in the Taylor series expansion of the
dispersion relation c = (g/k) tanh kH for small kH , verifying that weak dispersive
effects are indeed properly accounted for by the last term in equation (7.92).)
The ratio of nonlinear and dispersion terms in equation (7.92) is
aλ2
η ∂η 2 ∂ 3 η
H
∼
.
H ∂x
∂x 3
H3
When aλ2 /H 3 is larger than ≈16, nonlinear effects sharpen the forward face of
the wave, leading to hydraulic jump, as discussed in Section 11. For lower values
of aλ2 /H 3 , a balance can be achieved between nonlinear steepening and dispersive spreading, and waves of unchanging form become possible. Analysis of the
Korteweg–deVries equation shows that two types of solutions are then possible, a
periodic solution and a solitary wave solution. The periodic solution is called cnoidal
Figure 7.25 Cnoidal and solitary waves. Waves of unchanging form result because nonlinear steepening
balances dispersive spreading.
253
14. Stokes’ Drift
wave, because it is expressed in terms of elliptic functions denoted by cn(x). Its waveform is shown in Figure 7.25. The other possible solution of the Korteweg–deVries
equation involves only a single hump and is called a solitary wave or soliton. Its
profile is given by
3a 1/2
2
η = a sech
(x − ct) ,
(7.93)
4H 3
where the speed of propagation is
a
,
c = c0 1 +
2H
showing that the propagation velocity increases with the amplitude of the hump. The
validity of equation (7.93) can be checked by substitution into equation (7.92). The
waveform of the solitary wave is shown in Figure 7.25.
An isolated hump propagating at constant speed with unchanging form and in
fairly shallow water was first observed experimentally by S. Russell in 1844. Solitons have been observed to exist not only as surface waves, but also as internal
waves in stratified fluid, in the laboratory as well as in the ocean; (See Figure 3.3,
Turner (1973)).
14. Stokes’ Drift
Anyone who has observed the motion of a floating particle on the sea surface knows
that the particle moves slowly in the direction of propagation of the waves. This is
called Stokes’ drift. It is a second-order or finite amplitude effect, due to which the
particle orbit is not closed but has the shape shown in Figure 7.26. The mean velocity
of a fluid particle (that is, the Lagrangian velocity) is therefore not zero, although
the mean velocity at a point (the Eulerian velocity) must be zero if the process is
periodic. The drift is essentially due to the fact that the particle moves forward faster
(when it is at the top of its trajectory) than backward (when it is at the bottom of its
orbit). Although it is a second-order effect, its magnitude is frequently significant.
To find an expression for Stokes’ drift, we use Lagrangian specification, proceeding as in Section 5 but keeping a higher order of accuracy in the analysis. Our analysis
is adapted from the presentation given in the work by Phillips (1977, p. 43). Let (x, z)
be the instantaneous coordinates of a fluid particle whose position at t = 0 is (x0 , z0 ).
The initial coordinates (x0 , z0 ) serve as a particle identification, and we can write
its subsequent position as x(x0 , z0 , t) and z(x0 , z0 , t), using the Lagrangian form of
specification. The velocity components of the “particle (x0 , z0 )” are uL (x0 , z0 , t) and
wL (x0 , z0 , t). (Note that the subscript “L” was not introduced in Section 5, since to
the lowest order we equated the velocity at time t of a particle with mean coordinates
(x0 , z0 ) to the Eulerian velocity at t at location (x0 , z0 ). Here we are taking the analysis to a higher order of accuracy, and the use of a subscript “L” to denote Lagrangian
velocity helps to avoid confusion.)
254
Gravity Waves
Figure 7.26 The Stokes drift.
The velocity components are
∂x
∂t
∂z
,
wL =
∂t
uL =
(7.94)
where the partial derivative signs mean that the initial position (serving as a particle
tag) is kept fixed in the time derivative. The position of a particle is found by integrating
equation (7.94):
t
x = x0 +
z = z0 +
uL (x0 , z0 , t ′ ) dt ′
0
(7.95)
t
wL (x0 , z0
, t ′ ) dt ′ .
0
At time t the Eulerian velocity at (x, z) equals the Lagrangian velocity of particle
(x0 , z0 ) at the same time, if (x, z) and (x0 , z0 ) are related by equation (7.95). (No
approximation is involved here! The equality is merely a reflection of the fact that
particle (x0 , z0 ) occupies the position (x, z) at time t.) Denoting the Eulerian velocity
components without subscript, we therefore have
uL (x0 , z0 , t) = u(x, z, t).
Expanding the Eulerian velocity u(x, z, t) in a Taylor series about (x0 , z0 ), we obtain
uL (x0 , z0 , t) = u(x0 , z0 , t) + (x − x0 )
∂u
∂x
0
+ (z − z0 )
∂u
∂z
0
+ · · · , (7.96)
and a similar expression for wL . The Stokes drift is the time mean value of equation (7.96). As the time mean of the first term on the right-hand side of equation (7.96)
255
15. Waves at a Density Interface between Infinitely Deep Fluids
is zero, the Stokes drift is given by the mean of the next two terms of equation (7.96).
This was neglected in Section 5, and the result was closed orbits.
We shall now estimate the Stokes drift for gravity waves, using the deep water
approximation for algebraic simplicity. The velocity components and particle displacements for this motion are given in Section 6 as
u(x0 , z0 , t) = aωekz0 cos(kx0 − ωt),
x − x0 = −aekz0 sin(kx0 − ωt),
z − z0 = aekz0 cos(kx0 − ωt).
Substitution into the right-hand side of equation (7.96), taking time average, and using
the fact that the time average of sin2 t over a time period is 1/2, we obtain
ūL = a 2 ωke2kz0 ,
(7.97)
which is the Stokes drift in deep water. Its surface value is a 2 ωk, and the vertical
decay rate is twice that for the Eulerian velocity components. It is therefore confined
very close to the sea surface. For arbitrary water depth, it is easy to show that
ūL = a 2 ωk
cosh 2k(z0 + H )
2 sinh2 kH
.
(7.98)
The Stokes drift causes mass transport in the fluid, due to which it is also called
the mass transport velocity. Vertical fluid lines marked, for example, by some dye
gradually bend over (Figure 7.26). In spite of this mass transport, the mean Eulerian
velocity anywhere below the trough is exactly zero (to any order of accuracy), if the
flow is irrotational. This follows from the condition of irrotationality ∂u/∂z = ∂w/∂x,
a vertical integral of which gives
z
∂w
u = u|z=−H +
dz,
−H ∂x
showing that the mean of u is proportional to the mean of ∂w/∂x over a wavelength,
which is zero for periodic flows.
15. Waves at a Density Interface between Infinitely Deep Fluids
To this point we have considered only waves at the free surface of a liquid. However,
waves can also exist at the interface between two immiscible liquids of different
densities. Such a sharp density gradient can, for example, be generated in the ocean
by solar heating of the upper layer, or in an estuary (that is, a river mouth) or a fjord into
which fresh (less saline) river water flows over oceanic water, which is more saline
and consequently heavier. The situation can be idealized by considering a lighter fluid
of density ρ1 lying over a heavier fluid of density ρ2 (Figure 7.27).
We assume that the fluids are infinitely deep, so that only those solutions that
decay exponentially from the interface are allowed. In this section and in the rest of
256
Gravity Waves
Figure 7.27 Internal wave at a density interface between two infinitely deep fluids.
the chapter, we shall make use of the convenience of complex notation. For example,
we shall represent the interface displacement ζ = a cos(kx − ωt) by
ζ = Re a ei(kx−ωt) ,
√
where Re stands for “the real part of,” and i = −1. It is customary to omit the Re
symbol and simply write
ζ = a ei(kx−ωt) ,
(7.99)
where it is implied that only the real part of the equation is meant. We are therefore
carrying an extra imaginary part (which can be thought of as having no physical
meaning) on the right-hand side of equation (7.99). The convenience of complex
notation is that the algebra is simplified, essentially because differentiating exponentials is easier than differentiating trigonometric functions. If desired, the constant a in equation (7.99) can be considered to be a complex number. For example,
the profile ζ = sin(kx − ωt) can be represented as the real part of ζ = −i exp
i(kx − ωt).
We have to solve the Laplace equation for the velocity potential in both layers, subject to the continuity of p and w at the interface. The equations are,
therefore,
∂ 2 φ1
∂ 2 φ1
+
=0
∂x 2
∂z2
(7.100)
∂ 2 φ2
∂ 2 φ2
+
= 0,
2
∂x
∂z2
subject to
φ1 →0
as
z→∞
(7.101)
φ2 →0
as
z → −∞
(7.102)
257
15. Waves at a Density Interface between Infinitely Deep Fluids
∂φ2
∂ζ
∂φ1
=
=
∂z
∂z
∂t
ρ1
∂φ1
∂φ2
+ ρ1 gζ =ρ2
+ ρ2 gζ
∂t
∂t
at
z=0
(7.103)
at
z = 0.
(7.104)
Equation (7.103) follows from equating the vertical velocity of the fluid on both
sides of the interface to the rate of rise of the interface. Equation (7.104) follows
from the continuity of pressure across the interface. As in the case of surface waves,
the boundary conditions are linearized and applied at z = 0 instead of at z = ζ .
Conditions (7.101) and (7.102) require that the solutions of equation (7.100) must be
of the form
φ1 = A e−kz ei(kx−ωt)
φ2 = B ekz ei(kx−ωt) ,
because a solution proportional to ekz is not allowed in the upper fluid, and a solution
proportional to e−kz is not allowed in the lower fluid. Here A and B can be complex.
As in Section 4, the constants are determined from the kinematic boundary conditions
(7.103), giving
A = −B = iωa/k.
The dynamic boundary condition (7.104) then gives the dispersion relation
ρ2 − ρ1
(7.105)
= ε gk,
ω = gk
ρ2 + ρ 1
where ε2 ≡ (ρ2 − ρ1 )/(ρ2 + ρ1 ) is a small number if the density difference between
the two liquids is small. The case of small density difference is relevant in geophysical
situations; for example, a 10 ◦ C temperature change causes the density of the upper
layer of the ocean to decrease by 0.3%. Equation (7.105) shows that waves at the
interface between two liquids √
of infinite thickness travel like deep water surface
waves, with ω proportional to gk, but at a much reduced frequency. In general,
therefore, internal waves have a smaller frequency, and consequently a smaller phase
speed, than surface waves. As expected, equation (7.105) reduces to the expression
for surface waves if ρ1 = 0.
The kinetic energy of the field can be found by integrating ρ(u2 + w2 )/2 over
the range z = ±∞. This gives the average kinetic energy per unit horizontal area of
(see Exercise 7):
Ek = 41 (ρ2 − ρ1 )ga 2 ,
The potential energy can be calculated by finding the rate of work done in deforming
a flat interface to the wave shape. In Figure 7.28, this involves a transfer of column
A of density ρ2 to location B, a simultaneous transfer of column B of density ρ1
to location A, and integrating the work over half the wavelength, since the resulting
258
Gravity Waves
Figure 7.28 Calculation of potential energy of a two-layer fluid. The work done in transferring element
A to B equals the weight of A times the vertical displacement of its center of gravity.
exchange forms a complete wavelength; see the previous discussion of Figure 7.8.
The potential energy per unit horizontal area is therefore
1 λ/2
ρ2 gζ dx −
ρ1 gζ 2 dx
λ 0
0
g(ρ2 − ρ1 ) λ/2 2
1
=
ζ dx = (ρ2 − ρ1 )ga 2 .
2λ
4
0
1
Ep =
λ
λ/2
2
The total wave energy per unit horizontal area is
E = Ek + Ep =
1
(ρ2 − ρ1 )ga 2 .
2
(7.106)
In a comparison with equation (7.55), it follows that the amplitude of internal waves
is usually much larger than those of surface waves if the same amount of energy is
used to set off the motion.
The horizontal velocity components in the two layers are
u1 =
∂φ1
= −ωae−kz ei(kx−ωt)
∂x
u2 =
∂φ2
= ωaekz ei(kx−ωt) ,
∂x
which show that the velocities in the two layers are oppositely directed (Figure 7.27).
The interface is therefore a vortex sheet, which is a surface across which the tangential
velocity is discontinuous. It can be expected that a continuously stratified medium, in
which the density varies continuously as a function of z, will support internal waves
whose vorticity is distributed throughout the flow. Consequently, internal waves in
a continuously stratified fluid are not irrotational and do not satisfy the Laplace
equation. This is discussed further in Section 16.
The existence of internal waves at a density discontinuity has explained an interesting phenomenon observed in Norwegian fjords (Gill, 1982). It was known for a
long time that ships experienced unusually high drags on entering these fjords. The
phenomenon was a mystery (and was attributed to “dead water”!) until Bjerknes, a
Norwegian oceanographer, explained it as due to the internal waves at the interface
generated by the motion of the ship (Figure 7.29). (Note that the product of the drag
259
16. Waves in a Finite Layer Overlying an Infinitely Deep Fluid
Figure 7.29 Phenomenon of “dead water” in Norwegian fjords.
times the speed of the ship gives the rate of generation of wave energy, with other
sources of resistance neglected.)
16. Waves in a Finite Layer Overlying an Infinitely Deep Fluid
As a second example of an internal wave at a density discontinuity, consider the case
in which the upper layer is not infinitely thick but has a finite thickness; the lower
layer is initially assumed to be infinitely thick. The case of two infinitely deep liquids,
treated in the preceding section, is then a special case of the present situation. Whereas
only waves at the interface were allowed in the preceding section, the presence of the
free surface now allows an extra mode of surface waves. It is clear that the present
configuration will allow two modes of oscillation, one in which the free surface and
the interface are in phase and a second mode in which they are oppositely directed.
Let H be the thickness of the upper layer, and let the origin be placed at the mean
position of the free surface (Figure 7.30). The equations are
∂ 2 φ1
∂ 2 φ1
+
=0
∂x 2
∂z2
∂ 2 φ2
∂ 2 φ2
+
= 0,
∂x 2
∂z2
subject to
at
z → −∞
(7.107)
∂φ1 ∂η
=
∂z
∂t
φ2 → 0
at
z=0
(7.108)
∂φ1
+ gη = 0
∂t
at
z=0
(7.109)
∂φ2
∂ζ
∂φ1
=
=
∂z
∂z
∂t
∂φ1
∂φ2
ρ1
+ ρ1 gζ =ρ2
+ ρ2 gζ
∂t
∂t
at
z = −H
(7.110)
at
z = −H.
(7.111)
260
Gravity Waves
Figure 7.30 Two modes of motion of a layer of fluid overlying an infinitely deep fluid.
Assume a free surface displacement of the form
η = aei(kx−ωt) ,
(7.112)
and an interface displacement of the form
ζ = bei(kx−ωt) .
(7.113)
As before, only the real part of the right-hand side is meant. Without losing generality,
we can regard a as real, which means that we are considering a wave of the form
η = a cos(kx − ωt). The constant b should be left complex, because ζ and η may
not be in phase. Solution of the problem determines such phase differences.
The velocity potentials in the layers must be of the form
φ1 = (A ekz + B e−kz ) ei(kx−ωt) ,
(7.114)
φ2 = C ekz ei(kx−ωt) .
(7.115)
The form (7.115) is chosen in order to satisfy equation (7.107). Conditions
(7.108)–(7.110) give the constants in terms of the given amplitude a:
g
ia ω
,
+
2 k
ω
g
ia ω
,
−
B=
2 k
ω
A=−
C=−
b=
(7.116)
(7.117)
ia ω
g ia ω
g 2kH
−
e
,
+
−
2 k
ω
2 k
ω
gk
a
1+ 2
2
ω
e−kH +
a
gk
1− 2
2
ω
ekH .
(7.118)
(7.119)
261
16. Waves in a Finite Layer Overlying an Infinitely Deep Fluid
Substitution into equation (7.111) gives the required dispersion relation ω(k). After
some algebraic manipulations, the result can be written as (Exercise 8)
2
ω
ω2
−1
[ρ1 sinh kH + ρ2 cosh kH ] − (ρ2 − ρ1 ) sinh kH = 0. (7.120)
gk
gk
The two possible roots of this equation are discussed in what follows.
Barotropic or Surface Mode
One possible root of equation (7.120) is
ω2 = gk,
(7.121)
which is the same as that for a deep water gravity wave. Equation (7.119) shows that
in this case
b = ae−kH ,
(7.122)
implying that the amplitude at the interface is reduced from that at the surface by the
factor e−kH . Equation (7.122) also shows that the motions of the interface and the
free surface are locked in phase; that is they go up or down simultaneously. This mode
is similar to a gravity wave propagating on the free surface of the upper liquid, in
which the motion decays as e−kz from the free surface. It is called the barotropic
mode, because the surfaces of constant pressure and density coincide in such
a flow.
Baroclinic or Internal Mode
The other possible root of equation (7.120) is
ω2 =
gk(ρ2 − ρ1 ) sinh kH
,
ρ2 cosh kH + ρ1 sinh kH
(7.123)
which reduces to equation (7.105) if kH → ∞. Substitution of equation (7.123) into
(7.119) shows that, after some straightforward algebra,
η = −ζ
ρ2 − ρ1
ρ1
e−kH ,
(7.124)
demonstrating that η and ζ have opposite signs and that the interface displacement
is much larger than the surface displacement if the density difference is small. This
mode of behavior is called the baroclinic or internal mode because the surfaces of
constant pressure and density do not coincide. It can be shown that the horizontal
velocity u changes sign across the interface. The existence of a density difference has
therefore generated a motion that is quite different from the barotropic behavior. The
case studied in the previous section, in which the fluids have infinite depth and no
free surface, has only a baroclinic mode and no barotropic mode.
262
Gravity Waves
17. Shallow Layer Overlying an Infinitely Deep Fluid
A very common simplification, frequently made in geophysical situations in which
large-scale motions are considered, involves assuming that the wavelengths are large
compared to the upper layer depth. For example, the depth of the oceanic upper layer,
below which there is a sharp density gradient, could be ≈50 m thick, and we may
be interested in interfacial waves that are much longer than this. The approximation
kH ≪ 1 is called the shallow-water or long-wave approximation. Using
sinh kH ≃ kH,
cosh kH ≃ 1,
the dispersion relation (7.123) corresponding to the baroclinic mode reduces to
ω2 =
k 2 gH (ρ2 − ρ1 )
.
ρ2
(7.125)
The phase velocity of waves at the interface is therefore
c=
g′H ,
(7.126)
where we have defined
g′ ≡ g
ρ2 − ρ1
,
ρ2
(7.127)
which is called the reduced gravity. Equation (7.126) is similar to the corresponding expression√for surface waves in a shallow homogeneous layer √
of thickness H ,
namely, c = gH , except that its speed is reduced by the factor (ρ2 − ρ1 )/ρ2 .
This agrees with our previous conclusion that internal waves generally propagate
slower than surface waves. Under the shallow-water approximation, equation (7.124)
reduces to
η = −ζ
ρ2 − ρ1
ρ1
.
(7.128)
In Section 6 we noted that, for surface waves, the shallow-water approximation
is equivalent to the hydrostatic approximation, and results in a depth-independent
horizontal velocity. Such a conclusion also holds for interfacial waves. The fact that
u1 is independent of z follows from equation (7.114) on noting that ekz ≃ e−kz ≃ 1. To
see that pressure is hydrostatic, the perturbation pressure in the upper layer determined
from equation (7.114) is
p′ = −ρ1
∂φ1
= iρ1 ω(A + B) ei(kx−ωt) = ρ1 gη,
∂t
(7.129)
18. Equations of Motion for a Continuously Stratified Fluid
where the constants given in equations (7.116) and (7.117) have been used. This
shows that p′ is independent of z and equals the hydrostatic pressure change due to
the free surface displacement.
So far, the lower fluid has been assumed to be infinitely deep, resulting in an
exponential decay of the flow field from the interface into the lower layer, with a
decay scale of the order of the wavelength. If the lower layer is now considered
thin compared to the wavelength, then the horizontal velocity will be depth independent, and the flow hydrostatic, in the lower layer. If both layers are considered thin
compared to the wavelength, then the flow is hydrostatic (and the horizontal velocity field depth-independent) in both layers. This is the shallow-water or long-wave
approximation for a two-layer fluid. In such a case the horizontal velocity field in the
barotropic mode has a discontinuity at the interface, which vanishes in the Boussinesq
limit (ρ2 − ρ1 )/ρ1 ≪ 1. Under these conditions the two modes of a two-layer system have a simple structure (Figure 7.31): a barotropic mode in which the horizontal
velocity is depth independent across the entire water column; and a baroclinic mode
in which the horizontal velocity is directed in opposite directions in the two layers
(but is depth independent in each layer).
We shall now summarize the results of interfacial waves presented in the preceding three sections. In the case of two infinitely√deep fluids, only the baroclinic
mode is possible, and it has a frequency of ω = ε gk. If the upper layer has finite
thickness, then both baroclinic and barotropic modes are possible. In the barotropic
mode, η and ζ are in phase, and the flow decreases exponentially away from the free
surface. In the baroclinic mode, η and ζ are out of phase, the horizontal velocity
changes direction across the interface, and the motion decreases exponentially away
from the interface. If we also make the long-wave approximation for the upper
layer,
then the phase speed of interfacial waves in the baroclinic mode is c = g ′ H , the
fluid velocity in the upper layer is almost horizontal and depth independent, and the
pressure in the upper layer is hydrostatic. If both layers are shallow, then the flow is
depth independent and hydrostatic in both layers; the two modes in such a system
have the simple structure shown in Figure 7.31.
18. Equations of Motion for a Continuously Stratified Fluid
We have considered surface gravity waves and internal gravity waves at a density
discontinuity between two fluids. Internal waves also exist if the fluid is continuously
Figure 7.31 Two modes of motion in a shallow-water, two-layer system in the Boussinesq limit.
263
264
Gravity Waves
stratified, in which the vertical density profile in a state of rest is a continuous function
ρ̄(z). The equations of motion for internal waves in such a medium will be derived
in this section, starting with the Boussinesq set (4.89) presented in Chapter 4. The
Boussinesq approximation treats density as constant, except in the vertical momentum
equation. We shall assume that the wave motion is inviscid. The amplitudes will be
assumed to be small, in which case the nonlinear terms can be neglected. We shall also
assume that the frequency of motion is much larger than the Coriolis frequency, which
therefore does not affect the motion. Effects of the earth’s rotation are considered in
Chapter 14. The set (4.89) then simplifies to
∂u
1 ∂p
=−
,
∂t
ρ0 ∂x
(7.130)
∂v
1 ∂p
=−
,
∂t
ρ0 ∂y
(7.131)
1 ∂p ρg
∂w
=−
−
,
∂t
ρ0 ∂z
ρ0
(7.132)
Dρ
= 0,
Dt
(7.133)
∂u ∂v
∂w
+
+
= 0,
∂x
∂y
∂z
(7.134)
where ρ0 is a constant reference density. As noted in Chapter 4, the equation
Dρ/Dt = 0 is not an expression of conservation of mass, which is expressed by
∇ • u = 0 in the Boussinesq approximation. Rather, it expresses incompressibility
of a fluid particle. If temperature is the only agency that changes the density, then
Dρ/Dt = 0 follows from the heat equation in the nondiffusive form DT /Dt = 0
and an incompressible (that is, ρ is not a function of p) equation of state in the
form δρ/ρ = −α δT , where α is the coefficient of thermal expansion. If the density changes are due to changes in the concentration S of a constituent, for example
salinity in the ocean or water vapor in the atmosphere, then Dρ/Dt = 0 follows
from DS/Dt = 0 (the nondiffusive form of conservation of the constituent) and an
incompressible equation of state in the form of δρ/ρ = β δS, where β is the coefficient describing how the density changes due to concentration of the constituent. In
both cases, the principle underlying the equation Dρ/Dt = 0 is an incompressible
equation of state. In terms of common usage, this equation is frequently called the
“density equation,” as opposed to the continuity equation ∇ • u = 0.
The equation set (7.130)–(7.134) consists of five equations in five unknowns
(u, v, w, p, ρ). We first express the equations in terms of changes from a state of rest.
That is, we assume that the flow is superimposed on a “background” state in which
the density ρ̄(z) and pressure p̄(z) are in hydrostatic balance:
0=−
1 d p̄ ρ̄g
−
.
ρ0 dz
ρ0
(7.135)
265
18. Equations of Motion for a Continuously Stratified Fluid
When the motion develops, the pressure and density change to
p = p̄(z) + p′ ,
ρ = ρ̄(z) + ρ ′ .
(7.136)
The density equation (7.133) then becomes
∂
∂
∂
∂
(ρ̄ + ρ ′ ) + u (ρ̄ + ρ ′ ) + v (ρ̄ + ρ ′ ) + w (ρ̄ + ρ ′ ) = 0.
∂t
∂x
∂y
∂z
(7.137)
Here, ∂ ρ̄/∂t = ∂ ρ̄/∂x = ∂ ρ̄/∂y = 0. The nonlinear terms in the second, third, and
fourth terms (namely, u ∂ρ ′ /∂x, v ∂ρ ′ /∂y, and w ∂ρ ′ /∂z) are also negligible for small
amplitude motions. The linear part of the fourth term, that is, w d ρ̄/dz, represents a
very important process and must be retained. Equation (7.137) then simplifies to
d ρ̄
∂ρ ′
+w
= 0,
∂t
dz
(7.138)
which states that the density perturbation at a point is generated only by the vertical
advection of the background density distribution. This is the linearized form of equation (7.133), with the vertical advection of density retained in a linearized form. We
now introduce the definition
g d ρ̄
N2 ≡ −
.
(7.139)
ρ0 dz
Here, N(z) has the units of frequency (rad/s) and is called the Brunt–Väisälä
frequency or buoyancy frequency. It plays a fundamental role in the study of stratified flows. We shall see in the next section that it has the significance of being the
frequency of oscillation if a fluid particle is vertically displaced.
After substitution of equation (7.136), the equations of motion (7.130)–(7.134)
become
1 ∂p ′
∂u
=−
,
∂t
ρ0 ∂x
(7.140)
1 ∂p ′
∂v
=−
,
∂t
ρ0 ∂y
(7.141)
1 ∂p ′
ρ′g
∂w
=−
−
,
∂t
ρ0 ∂z
ρ0
(7.142)
N 2 ρ0
∂ρ ′
−
w = 0,
∂t
g
(7.143)
∂u ∂v
∂w
+
+
= 0.
∂x
∂y
∂z
(7.144)
266
Gravity Waves
In deriving this set we have also used equation (7.135) and replaced the density
equation by its linearized form (7.138). Comparing the sets (7.130)–(7.134) and
(7.140)–(7.144), we see that the equations satisfied by the perturbation density and
pressure are identical to those satisfied by the total ρ and p.
In deriving the equations for a stratified fluid, we have assumed that ρ is a
function of temperature T and concentration S of a constituent, but not of pressure.
At first this does not seem to be a good assumption. The compressibility effects in the
atmosphere are certainly not negligible; even in the ocean the density changes due to
the huge changes in the background pressure are as much as 4%, which is ≈10 times
the density changes due to the variations of the salinity and temperature. The effects
of compressibility, however, can be handled within the Boussinesq approximation if
we regard ρ̄ in the definition of N as the background potential density, that is the
density distribution from which the adiabatic changes of density due to the changes
of pressure have been subtracted out. The concept of potential density is explained
in Chapter 1. Oceanographers account for compressibility effects by converting all
their density measurements to the standard atmospheric pressure; thus, when they
report variations in density (what they call “sigma tee”) they are generally reporting
variations due only to changes in temperature and salinity.
A useful equation for stratified flows is the one involving only w. The u and v
can be eliminated by taking the time derivative of the continuity equation (7.144) and
using the horizontal momentum equations (7.140) and (7.141). This gives
1 2 ′
∂ 2w
∇H p =
,
ρ0
∂z ∂t
(7.145)
where ∇H2 ≡ ∂ 2 /∂x 2 + ∂ 2 /∂y 2 is the horizontal Laplacian operator. Elimination of
ρ ′ from equations (7.142) and (7.143) gives
∂ 2w
1 ∂ 2 p′
= − 2 − N 2 w.
ρ0 ∂t ∂z
∂t
(7.146)
Finally, p′ can be eliminated by taking ∇H2 of equation (7.146), and using equation (7.145). This gives
∂2
∂t ∂z
which can be written as
∂ 2w
∂t ∂z
= −∇H2
∂ 2w
+ N 2w ,
∂t 2
∂2 2
∇ w + N 2 ∇H2 w = 0,
∂t 2
(7.147)
where ∇ 2 ≡ ∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z2 = ∇H2 + ∂ 2 /∂z2 is the three-dimensional
Laplacian operator. The w-equation will be used in the following section to derive
the dispersion relation for internal gravity waves.
267
19. Internal Waves in a Continuously Stratified Fluid
19. Internal Waves in a Continuously Stratified Fluid
In this chapter we have considered gravity waves at the surface or at a density
discontinuity; these waves propagate only in the horizontal direction. Because every
horizontal direction is alike, such waves are isotropic, in which only the magnitude
of the wavenumber vector matters. By taking the x-axis along the direction of wave
propagation, we obtained a dispersion relation ω(k) that depends only on the magnitude of the wavenumber. We found that phases and groups propagate in the same
direction, although at different speeds. If, on the other hand, the fluid is continuously
stratified, then the internal waves can propagate in any direction, at any angle to the
vertical. In such a case the direction of the wavenumber vector becomes important.
Consequently, we can no longer treat the wavenumber, phase velocity, and group
velocity as scalars.
Any flow variable q can now be written as
q = q0 ei(kx+ly+mz−ωt) = q0 ei(K
•
x−ωt)
,
where q0 is the amplitude and K = (k, l, m) is the wavenumber vector with components k, l, and m in the three Cartesian directions. We expect that in this case the
direction of wave propagation should matter because horizontal directions are basically different from the vertical direction, along which the all-important gravity acts.
Internal waves in a continuously stratified fluid are therefore anisotropic, for which
the frequency is a function of all three components of K. This can be written in the
following two ways:
ω = ω(k, l, m) = ω(K).
(7.148)
However, the waves are still horizontally isotropic because the dependence of the
wave field on k and l is similar, although the dependence on k and m is dissimilar.
The propagation of internal waves is a baroclinic process, in which the surfaces of
constant pressure do not coincide with the surfaces of constant density. It was shown
in Section 5.4, in connection with the demonstration of Kelvin’s circulation theorem,
that baroclinic processes generate vorticity. Internal waves in a continuously stratified
fluid are therefore not irrotational. Waves at a density interface constitute a limiting
case in which all the vorticity is concentrated in the form of a velocity discontinuity
at the interface. The Laplace equation can therefore be used to describe the flow field
within each layer. However, internal waves in a continuously stratified fluid cannot
be described by the Laplace equation.
The first task is to derive the dispersion relation. We shall simplify the analysis
by assuming that N is depth independent, an assumption that may seem unrealistic at
first. In the ocean, for example, N is large at a depth of ≈200 m and small elsewhere
(see Figure 14.2). Figure 14.2 shows that N < 0.01 everywhere but N is largest
between ≈200 m and 2 km. However, the results obtained by treating N as constant
are locally valid if N varies slowly over the vertical wavelength 2π/m of the motion.
The so-called WKB approximation of internal waves, in which such a slow variation
of N (z) is not neglected, is discussed in Chapter 14.
268
Gravity Waves
Consider a wave propagating in three dimensions, for which the vertical velocity is
w = w0 ei(kx+ly+mz−ωt) ,
(7.149)
where w0 is the amplitude of fluctuations. Substituting into the governing
equation
∂2 2
∇ w + N 2 ∇H2 w = 0,
(7.147)
∂t 2
gives the dispersion relation
k2 + l2
N 2.
k 2 + l 2 + m2
(7.150)
kN
kN
.
=
ω= √
K
k 2 + m2
(7.151)
ω = N cos θ,
(7.152)
ω2 =
For simplicity of discussion we shall orient the xz-plane so as to contain the wavenumber vector K. No generality is lost by doing this because the medium is horizontally isotropic. For this choice of reference axes we have l = 0; that is, the wave
motion is two dimensional and invariant in the y-direction, and k represents the entire
horizontal wavenumber. We can then write equation (7.150) as
This is the dispersion relation for internal gravity waves and can also be written as
where θ is the angle between the phase velocity vector c (and therefore K) and the
horizontal direction (Figure 7.32). It follows that the frequency of an internal wave in a
stratified fluid depends only on the direction of the wavenumber vector and not on the
magnitude of the wavenumber. This is in sharp contrast with surface and interfacial
gravity waves, for which frequency depends only on the magnitude. The frequency
lies in the range 0 < ω < N, revealing one important significance of the buoyancy
frequency: N is the maximum possible frequency of internal waves in a stratified
fluid.
Before discussing the dispersion relation further, let us explore particle motion
in an incompressible internal wave. The fluid motion can be written as
u = u0 ei(kx+ly+mz−ωt) ,
(7.153)
plus two similar expressions for v and w. This gives
∂u
= iku0 ei(kx+ly+mz−ωt) = iku.
∂x
The continuity equation then requires that ku + lv + mw = 0, that is,
K • u = 0,
(7.154)
19. Internal Waves in a Continuously Stratified Fluid
showing that particle motion is perpendicular to the wavenumber vector (Figure 7.32).
Note that only two conditions have been used to derive this result, namely the incompressible continuity equation and a trigonometric behavior in all spatial directions. As
such, the result is valid for many other wave systems that meet these two conditions.
These waves are called shear waves (or transverse waves) because the fluid moves
parallel to the constant phase lines. Surface or interfacial gravity waves do not have
this property because the field varies exponentially in the vertical.
We can now interpret θ in the dispersion relation (7.152) as the angle between
the particle motion and the vertical direction (Figure 7.32). The maximum frequency
ω = N occurs when θ = 0, that is, when the particles move up and down vertically.
This case corresponds to m = 0 (see equation (7.151)), showing that the motion is
independent of the z-coordinate. The resulting motion consists of a series of vertical
columns, all oscillating at the buoyancy frequency N , the flow field varying in the
horizontal direction only.
The w = 0 Limit
At the opposite extreme we have ω = 0 when θ = π/2, that is, when the particle
motion is completely horizontal. In this limit our internal wave solution (7.151) would
seem to require k = 0, that is, horizontal independence of the motion. However, such
a conclusion is not valid; pure horizontal motion is not a limiting case of internal
waves, and it is necessary to examine the basic equations to draw any conclusion for
Figure 7.32 Basic parameters of internal waves. Note that c and cg are at right angles and have opposite
vertical components.
269
270
Gravity Waves
Figure 7.33 Blocking in strongly stratified flow. The circular region represents a two-dimensional body
with its axis along the y direction.
this case. An examination of the governing set (7.140)–(7.144) shows that a possible
steady solution is w = p′ = ρ ′ = 0, with u and v any functions of x and y satisfying
∂u ∂v
+
= 0.
∂x
∂y
(7.155)
The z-dependence of u and v is arbitrary. The motion is therefore two-dimensional
in the horizontal plane, with the motion in the various horizontal planes decoupled
from each other. This is why clouds in the upper atmosphere seem to move in flat
horizontal sheets, as often observed in airplane flights (Gill, 1982). For a similar
reason a cloud pattern pierced by a mountain peak sometimes shows Karman vortex streets, a two-dimensional feature; see the striking photograph in Figure 10.20.
A restriction of strong stratification is necessary for such almost horizontal flows, for
equation (7.143) suggests that the vertical motion is small if N is large.
The foregoing discussion leads to the interesting phenomenon of blocking in a
strongly stratified fluid. Consider a two-dimensional body placed in such a fluid, with
its axis horizontal (Figure 7.33). The two dimensionality of the body requires ∂v/∂y =
0, so that the continuity equation (7.155) reduces to ∂u/∂x = 0. A horizontal layer
of fluid ahead of the body, bounded by tangents above and below it, is therefore
blocked. (For photographic evidence see Figure 3.18 in the book by Turner (1973).)
This happens because the strong stratification suppresses the w field and prevents the
fluid from going around and over the body.
20. Dispersion of Internal Waves in a Stratified Fluid
In the case of isotropic gravity waves at a free surface and at a density discontinuity,
we found that c and cg are in the same direction, although their magnitudes can be
different. This conclusion is no longer valid for the anisotropic internal waves in a
continuously stratified fluid. In fact, as we shall see shortly, they are perpendicular to
each other, violating all our intuitions acquired by observing surface gravity waves!
In three dimensions, the definition cg = dω/dk has to be generalized to
cg = ix
∂ω
∂ω
∂ω
+ iy
+ iz
,
∂k
∂l
∂m
(7.156)
where ix , iy , iz are the unit vectors in the three Cartesian directions. As in the preceding
section, we orient the xz-plane so that the wavenumber vector K lies in this plane
271
20. Dispersion of Internal Waves in a Stratified Fluid
and l = 0. Substituting equation (7.151), this gives
Nm
(ix m − iz k).
K3
(7.157)
ω
ωK
= 2 (ix k + iz m),
KK
K
(7.158)
cg =
The phase velocity is
c=
where K/K represents the unit vector in the direction of K. (Note that c = ix (ω/k) +
iz (ω/m), as explained in Section 3.) It follows from equations (7.157) and (7.158)
that
cg • c = 0,
(7.159)
showing that phase and group velocity vectors are perpendicular.
Equations (7.157) and (7.158) show that the horizontal components of c and cg
are in the same direction, while their vertical components are equal and opposite. In
fact, c and cg form two sides of a right-angled triangle whose hypotenuse is horizontal
(Figure 7.34). Consequently, the phase velocity has an upward component when the
group velocity has a downward component, and vice versa. Equations (7.154) and
(7.159) are consistent because c and K are parallel and cg and u are parallel. The fact
that c and cg are perpendicular, and have opposite vertical components, is illustrated in
Figure 7.35. It shows that the phase lines are propagating toward the left and upward,
whereas the wave groups are propagating to the left and downward. Wave crests are
constantly appearing at one edge of the group, propagating through the group, and
vanishing at the other edge.
The group velocity here has the usual significance of being the velocity of propagation of energy of a certain sinusoidal component. Suppose a source is oscillating
at frequency ω. Then its energy will only be found radially outward along four beams
oriented at an angle θ with the vertical, where cos θ = ω/N. This has been verified
in a laboratory experiment (Figure 7.36). The source in this case was a vertically
oscillating cylinder with its axis perpendicular to the plane of paper. The frequency
was ω < N. The light and dark lines in the photograph are lines of constant density,
made visible by an optical technique. The experiment showed that the energy radiated
Figure 7.34 Orientation of phase and group velocity in internal waves.
272
Gravity Waves
Figure 7.35 Illustration of phase and group propagation in internal waves. Positions of a wave group at
two times are shown. The phase line PP at time t1 propagates to P′ P′ at t2 .
along four beams that became more vertical as the frequency was increased, which
agrees with cos θ = ω/N.
21. Energy Considerations of Internal Waves in a
Stratified Fluid
In this section we shall derive the various commonly used expressions for potential
energy of a continuously stratified fluid, and show that they are equivalent. We then
show that the energy flux p ′ u is cg times the wave energy.
A mechanical energy equation for internal waves can be derived from equations (7.140)–(7.142) by multiplying the first equation by ρ0 u, the second by ρ0 v, the
third by ρ0 w, and summing the results. This gives
∂ 1
ρ0 (u2 + v 2 + w 2 ) + gρ ′ w + ∇ • (p ′ u) = 0.
(7.160)
∂t 2
Here the continuity equation has been used to write u ∂p′ /∂x+v ∂p′ /∂y+w ∂p′ /∂z =
∇ • (p ′ u), which represents the net work done by pressure forces. Another interpretation is that ∇ • (p ′ u) is the divergence of the energy flux p ′ u, which must change
the wave energy at a point. As the first term in equation (7.160) is the rate of change
of kinetic energy, we can anticipate that the second term gρ ′ w must be the rate of
change of potential energy. This is consistent with the energy principle derived in
21. Energy Considerations of Internal Waves in a Stratified Fluid
Figure 7.36 Waves generated in a stratified fluid of uniform buoyancy frequency N = 1 rad/s. The forcing
agency is a horizontal cylinder, with its axis perpendicular to the plane of the paper, oscillating vertically at
frequency ω = 0.71 rad/s. With ω/N = 0.71 = cos θ, this agrees with the observed angle of θ = 45◦ made
by the beams with the horizontal. The vertical dark line in the upper half of the photograph is the cylinder
support and should be ignored. The light and dark radial lines represent contours of constant ρ ′ and are
therefore constant phase lines. The schematic diagram below the photograph shows the directions of c and
cg for the four beams. Reprinted with the permission of Dr. T. Neil Stevenson, University of Manchester.
273
274
Gravity Waves
Chapter 4 (see equation (4.62)), except that ρ ′ and p′ replace ρ and p because we
have subtracted the mean state of rest here. Using the density equation (7.143), the
rate of change of potential energy can be written as
∂Ep
∂ g 2 ρ ′2
= gρ ′ w =
,
(7.161)
∂t
∂t 2ρ0 N 2
which shows that the potential energy per unit volume must be the positive quantity Ep = g 2 ρ ′2 /2ρ0 N 2 . The potential energy can also be expressed in terms of the
displacement ζ of a fluid particle, given by w = ∂ζ /∂t. Using the density equation
(7.143), we can write
N 2 ρ0 ∂ζ
∂ρ ′
=
,
∂t
g ∂t
which requires that
ρ′ =
N 2 ρ0 ζ
.
g
(7.162)
The potential energy per unit volume is therefore
Ep =
g 2 ρ ′2
1
= N 2 ρ0 ζ 2 .
2
2ρ0 N 2
(7.163)
This expression is consistent with our previous result from equation (7.106) for
two infinitely deep fluids, for which the average potential energy of the entire water
column per unit horizontal area was shown to be
1
4 (ρ2
− ρ1 )ga 2 ,
(7.164)
where the interface displacement is of the form ζ = a cos(kx − ωt) and (ρ2 − ρ1 ) is
the density discontinuity. To see the consistency, we shall symbolically represent the
buoyancy frequency of a density discontinuity at z = 0 as
N2 = −
g
g d ρ̄
= (ρ2 − ρ1 )δ(z),
ρ0 dz
ρ0
(7.165)
where δ(z) is the Dirac delta function. (As with other relations involving the delta
function, equation (7.165) is valid in the integral
sense, that is, the integral (across the
origin) of the last two terms is equal because δ(z) dz = 1.) Using equation (7.165),
a vertical integral of equation (7.163), coupled with horizontal averaging over a wavelength, gives equation (7.164). Note that for surface or interfacial waves Ek and Ep
represent kinetic and potential energies of the entire water column, per unit horizontal
area. In a continuously stratified fluid, they represent energies per unit volume.
We shall now demonstrate that the average kinetic and potential energies are
equal for internal wave motion. Substitute periodic solutions
[u, w, p′ , ρ ′ ] = [û, ŵ, p̂, ρ̂] ei(kx+mz−ωt) .
275
21. Energy Considerations of Internal Waves in a Stratified Fluid
Then all variables can be expressed in terms of w:
ωmρ0
p ′ = − 2 ŵ ei(kx+mz−ωt) ,
k
iN 2 ρ0
ŵ ei(kx+mz−ωt) ,
ωg
m
u = − ŵ ei(kx+mz−ωt) ,
k
ρ′ =
(7.166)
where p ′ is derived from equation (7.145), ρ ′ from equation (7.143), and u from
equation (7.140). The average kinetic energy per unit volume is therefore
Ek =
1
1
ρ0 (u2 + w 2 ) = ρ0
2
4
m2
+ 1 ŵ2 ,
k2
(7.167)
where we have used the fact that the average of cos2 x over a wavelength is 1/2. The
average potential energy per unit volume is
Ep =
N 2 ρ0 2
g 2 ρ ′2
=
ŵ ,
2ρ0 N 2
4ω2
(7.168)
where we have used ρ ′2 = ŵ2 N 4 ρ02 /2ω2 g 2 , found from equation (7.166) after taking
its real part. Use of the dispersion relation ω2 = k 2 N 2 /(k 2 + m2 ) shows that
Ek = Ep ,
(7.169)
which is a general result for small oscillations of a conservative system without
Coriolis forces. The total wave energy is
E = Ek + Ep = 21 ρ0
m2
+ 1 ŵ2 .
k2
(7.170)
Last, we shall show that cg times the wave energy equals the energy flux. The
average energy flux across a unit area can be found from equation (7.166):
ρ0 ωmŵ 2 m
(7.171)
ix − iz .
F = p ′ u = ix p ′ u + iz p ′ w =
2
k
2k
Using equations (7.157) and (7.170), group velocity times wave energy is
Nm
ρ0 m2
2
+
1
ŵ
,
cg E = 3 [ix m − iz k]
2 k2
K
which reduces to equation (7.171) on using the dispersion relation (7.151). It follows
that
F = cg E.
(7.172)
276
Gravity Waves
This result also holds for surface or interfacial gravity waves. However, in that case
F represents the flux per unit width perpendicular to the propagation direction (integrated over the entire depth), and E represents the energy per unit horizontal area. In
equation (7.172), on the other hand, F is the flux per unit area, and E is the energy
per unit volume.
Exercises
1. Consider stationary surface gravity waves in a rectangular container of length
L and breadth b, containing water of undisturbed depth H . Show that the velocity
potential
φ = A cos(mπ x/L) cos(nπy/b) cosh k(z + H ) e−iωt ,
satisfies ∇ 2 φ = 0 and the wall boundary conditions, if
(mπ/L)2 + (nπ/b)2 = k 2 .
Here m and n are integers. To satisfy the free surface boundary condition, show that
the allowable frequencies must be
ω2 = gk tanh kH.
[Hint: combine the two boundary conditions (7.27) and (7.32) into a single equation
∂ 2 φ/∂t 2 = −g ∂φ/∂z at z = 0.]
2. This is a continuation of Exercise 1. A lake has the following dimensions
L = 30 km
b = 2 km
H = 100 m.
Suppose the relaxation of wind sets up the mode m = 1 and n = 0. Show that the
period of the oscillation is 31.7 min.
3. Show that the group velocity of pure capillary waves in deep water, for which
the gravitational effects are negligible, is
cg = 23 c.
4. Plot the group velocity of surface gravity waves, including surface tension
σ , as a function of λ. Assuming deep water, show that the group velocity is
1 g 1 + 3σ k 2 /ρg
.
cg =
2 k 1 + σ k 2 /ρg
Show that this becomes minimum at a wavenumber given by
2
σ k2
= √ − 1.
ρg
3
For water at 20 ◦ C (ρ = 1000 kg/m3 and σ = 0.074 N/m), verify that cg min =
17.8 cm/s.
Literature Cited
5. A thermocline is a thin layer in the upper ocean across which temperature
and, consequently, density change rapidly. Suppose the thermocline in a very deep
ocean is at a depth of 100 m from the ocean surface, and that the temperature drops
across it from 30 to 20 ◦ C. Show that the reduced gravity is g ′ = 0.025 m/s2 .
Neglecting Coriolis effects, show that the speed of propagation of long gravity waves
on such a thermocline is 1.58 m/s.
6. Consider internal waves in a continuously stratified fluid of buoyancy frequency N = 0.02 s−1 and average density 800 kg/m3 . What is the direction of ray
paths if the frequency of oscillation is ω = 0.01 s−1 ? Find the energy flux per unit
area if the amplitude of vertical velocity is ŵ = 1 cm/s and the horizontal wavelength
is π meters.
7. Consider internal waves at a density interface between two infinitely deep
fluids. Using the expressions given in Section 15, show that the average kinetic energy
per unit horizontal area is Ek = (ρ2 −ρ1 )ga 2 /4. This result was quoted but not proved
in Section 15.
8. Consider waves in a finite layer overlying an infinitely deep fluid, discussed
in Section 16. Using the constants given in equations (7.116)–(7.119), prove the
dispersion relation (7.120).
9. Solve the equation governing spherical waves ∂ 2 p/∂t 2 = (c2 /r 2 )(∂/∂r)
(r 2 ∂p/∂r) subject to the initial conditions: p(r, 0) = e−r , (∂p/∂t)(r, 0) = 0.
Literature Cited
Gill, A. (1982). Atmosphere–Ocean Dynamics, New York: Academic Press.
Kinsman, B. (1965). Wind Waves, Englewood Cliffs, New Jersey: Prentice-Hall.
LeBlond, P. H. and L. A. Mysak (1978). Waves in the Ocean, Amsterdam: Elsevier Scientific Publishing.
Liepmann, H. W. and A. Roshko (1957). Elements of Gasdynamics, New York: Wiley.
Lighthill, M. J. (1978). Waves in Fluids, London: Cambridge University Press.
Phillips, O. M. (1977). The Dynamics of the Upper Ocean, London: Cambridge University Press.
Turner, J. S. (1973). Buoyancy Effects in Fluids, London: Cambridge University Press.
Whitham, G. B. (1974). Linear and Nonlinear Waves, New York: Wiley.
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Chapter 8
Dynamic Similarity
1. Introduction . . . . . . . . . . . . . . . . . . . . .
2. Nondimensional Parameters
Determined from Differential
Equations . . . . . . . . . . . . . . . . . . . . .
3. Dimensional Matrix . . . . . . . . . . . . .
4. Buckingham’s Pi Theorem . . . . . . .
5. Nondimensional Parameters and
Dynamic Similarity . . . . . . . . . . . . . .
Prediction of Flow Behavior from
Dimensional Considerations . . .
6. Comments on Model Testing. . . . . .
Example 8.1 . . . . . . . . . . . . . . . . . . . .
279
280
284
285
287
289
290
290
7. Significance of Common
Nondimensional Parameters . . . . .
Reynolds Number . . . . . . . . . . . . . . .
Froude Number . . . . . . . . . . . . . . . . .
Internal Froude Number . . . . . . . . .
Richardson Number . . . . . . . . . . . . .
Mach Number . . . . . . . . . . . . . . . . . . .
Prandtl Number . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . .
292
292
292
292
293
293
294
294
294
294
1. Introduction
Two flows having different values of length scales, flow speeds, or fluid properties
can apparently be different but still “dynamically similar”. Exactly what is meant
by dynamic similarity will be explained later in this chapter. At this point it is only
necessary to know that in a class of dynamically similar flows we can predict flow
properties if we have experimental data on one of them. In this chapter, we shall
determine circumstances under which two flows can be dynamically similar to one
another. We shall see that equality of certain relevant nondimensional parameters is
a requirement for dynamic similarity. What these nondimensional parameters should
be depends on the nature of the problem. For example, one nondimensional parameter
must involve the fluid viscosity if the viscous effects are important in the problem.
The principle of dynamic similarity is at the heart of experimental fluid
mechanics, in which the data should be unified and presented in terms of nondimensional parameters. The concept of similarity is also indispensable for designing
models in which tests can be conducted for predicting flow properties of full-scale
objects such as aircraft, submarines, and dams. An understanding of dynamic similarity is also important in theoretical fluid mechanics, especially when simplifications
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50008-3
279
280
Dynamic Similarity
are to be made. Under various limiting situations certain variables can be eliminated
from our consideration, resulting in very useful relationships in which only the constants need to be determined from experiments. Such a procedure is used extensively
in turbulence theory, and leads, for example, to the well-known K −5/3 spectral law
discussed in Chapter 13. Analogous arguments (applied to a different problem) are
presented in Section 5 of the present chapter.
Nondimensional parameters for a problem can be determined in two ways. They
can be deduced directly from the governing differential equations if these equations
are known; this method is illustrated in the next section. If, on the other hand, the
governing differential equations are unknown, then the nondimensional parameters
can be determined by performing a simple dimensional analysis on the variables
involved. This method is illustrated in Section 4.
The formulation of all problems in fluid mechanics is in terms of the conservation
laws (mass, momentum, and energy), constitutive equations and equations of state
to define the fluid, and boundary conditions to specify the problem. Most often, the
conservation laws are written as partial differential equations and the conservation
of momentum and energy may include the constitutive equations for stress and heat
flux, respectively. Each term in the various equations has certain dimensions in terms
of units of measurements. Of course, all of the terms in any given equation must have
the same dimensions. Now, dimensions or units of measurement are human constructs for our convenience. No system of units has any inherent superiority over any
other, despite the fact that in this text we exhibit a preference for the units ordained
by Napoleon Bonaparte (of France) over those ordained by King Henry VIII (of
England). The point here is that any physical problem must be expressible in completely dimensionless form. Moreover, the parameters used to render the dependent
and independent variables dimensionless must appear in the equations or boundary
conditions. One cannot define “reference” quantities that do not appear in the problem; spurious dimensionless parameters will be the result. If the procedure is done
properly, there will be a reduction in the parametric dependence of the formulation,
generally by the number of independent units. This is described in Sections 3 and 4
in this chapter. The parametric reduction is called a similitude. Similitudes greatly
facilitate correlation of experimental data. In Chapter 9 we will encounter a situation
in which there are no naturally occurring scales for length or time that can be used
to render the formulation of a particular problem dimensionless. As the axiom that
a dimensionless formulation is a physical necessity still holds, we must look for a
dimensionless combination of the independent variables. This results in a contraction
of the dimensionality of the space required for the solution, that is, a reduction by
one in the number of independent varibles. Such a reduction is called a similarity and
results in what is called a similarity solution.
2. Nondimensional Parameters Determined from
Differential Equations
To illustrate the method of determining nondimensional parameters from the governing differential equations, consider a flow in which both viscosity and gravity
are important. An example of such a flow is the motion of a ship, where the drag
281
2. Nondimensional Parameters Determined from Differential Equations
experienced is caused both by the generation of surface waves and by friction on the
surface of the hull. All other effects such as surface tension and compressibility are
neglected. The governing differential equation is the Navier–Stokes equation
∂w
∂w
∂w
∂w
1 ∂p
µ
+u
+v
+w
=−
−g+
∂t
∂x
∂y
∂z
ρ ∂z
ρ
∂ 2w ∂ 2w ∂ 2w
+
,
+
∂x 2
∂y 2
∂z2
(8.1)
and two other equations for u and v. The equation can be nondimensionalized by
defining a characteristic length scale l and a characteristic velocity scale U . In the
present problem we can take l to be the length of the ship at the waterline and U
to be the free-stream velocity at a large distance from the ship (Figure 8.1). The
choice of these scales is dictated by their appearance in the boundary conditions; U
is the boundary condition on the variable u and l occurs in the shape function of
the ship hull. Dynamic similarity requires that the flows have geometric similarity
of the boundaries, so that all characteristic lengths are proportional; for example,
in Figure 8.1 we must have d/ l = d1 / l1 . Dynamic similarity also requires that the
flows should be kinematically similar, that is, they should have geometrically similar
streamlines. The velocities at the same relative location are therefore proportional;
if the velocity at point P in Figure 8.1a is U/2, then the velocity at the corresponding point P1 in Figure 8.1b must be U1 /2. All length and velocity scales are then
proportional in a class of dynamically similar flows. (Alternatively, we could take
the characteristic length to be the depth d of the hull under water. Such a choice is,
however, unconventional.) Moreover, a choice of l as the length of the ship makes
the nondimensional distances of interest (that is, the magnitude of x/ l in the region
around the ship) of order one. Similarly, a choice of U as the free-stream velocity
makes the maximum value of the nondimensional velocity u/U of order one. For
reasons that will become more apparent in the later chapters, it is of value to have all
dimensionless variables of finite order. Approximations may then be based on any
extreme size of the dimensionless parameters that will preface some of the terms.
Accordingly, we introduce the following nondimensional variables, denoted by
primes:
y
z
tU
x
x′ =
y′ =
z′ =
t′ =
,
l
l
l
l
(8.2)
u
v
w
p − p∞
.
u′ =
v′ =
w′ =
p′ =
U
U
U
ρU 2
Figure 8.1
Two geometrically similar ships.
282
Dynamic Similarity
It is clear that the boundary conditions in terms of the nondimensional variables in
equation (8.2) are independent of l and U . For example, consider the viscous flow over
a circular cylinder of radius R. We choose the velocity scale U to be the free-stream
velocity and the length scale to be the radius R. In terms of nondimensional velocity
u′ = u/U and the nondimensional coordinate r ′ = r/R, the boundary condition at
infinity is u′ → 1 as r ′ → ∞, and the condition at the surface of the cylinder is
u′ = 0 at r ′ = 1. (Here, u is taken to be the r-component of velocity.)
There are instances where the shape function of a body may require two length
scales, such as a length l and a thickness d. An additional dimensionless parameter,
d/ l would result to describe the slenderness of the body.
Normalization, that is, dimensionless representation of the pressure, depends on
the dominant effect in the flow unless the flow is pressure-gradient driven. In the
latter case for flow in ducts or tubes, the pressure should be made dimensionless
by a characteristic pressure difference in the duct so that the dimensionless term
is finite. In other cases, when the flow is not pressure-gradient driven, the pressure
is a passive variable and should be normalized to balance the dominant effect in
the flow. Because pressure enters only as a gradient, the pressure itself is not of
consequence; only pressure differences are important. The conventional practice is
to render p − p∞ dimensionless. Depending on the nature of the flow, this could be
in terms of viscous stress µU/ l, a hydrostatic pressure ρgl, or as in the preceding, a
dynamic pressure ρU 2 .
Substitution of equation (8.2) into equation (8.1) gives
′
′
′
ν
∂p′
gl
∂w ′
′ ∂w
′ ∂w
′ ∂w
+
u
+
v
+
w
=
−
− 2+
′
′
′
′
′
∂t
∂x
∂y
∂z
∂z
Ul
U
∂ 2 w′
∂ 2 w′
∂ 2 w′
.
+
+
∂x ′2
∂y ′2
∂z′2
(8.3)
It is apparent that two flows (having different values of U , l, or ν), will obey the same
nondimensional differential equation if the values of nondimensional groups gl/U 2
and ν/U l are identical. Because the nondimensional boundary conditions are also
identical in the two flows, it follows that they will have the same nondimensional
solutions.
√
The nondimensional parameters U l/ν and U/ gl have been given special
names:
Ul
= Reynolds number,
Re ≡
ν
(8.4)
U
Fr ≡ √ = Froude number.
gl
Both Re and Fr have to be equal for dynamic similarity of two flows in which both
viscous and gravitational effects are important. Note that the mere presence of gravity
does not make the gravitational effects dynamically important. For flow around an
object in a homogeneous fluid, gravity is important only if surface waves are generated.
Otherwise, the effect of gravity is simply to add a hydrostatic pressure to the entire
system, which can be eliminated by absorbing gravity into the pressure term.
283
2. Nondimensional Parameters Determined from Differential Equations
Under dynamic similarity the nondimensional solutions are identical. Therefore,
the local pressure at point x = (x, y, z) must be of the form
p (x) − p∞
x
=
f
Fr,
Re;
,
l
ρU 2
(8.5)
where (p − p∞ )/ρU 2 is called the pressure coefficient. Similar relations also hold
for any other nondimensional flow variable such as velocity u/U and acceleration
al/U 2 . It follows that in dynamically similar flows the nondimensional local flow
variables are identical at corresponding points (that is, for identical values of x/ l).
In the foregoing analysis we have assumed that the imposed boundary conditions
are steady. However, we have retained the time derivative in equation (8.3) because
the resulting flow can still be unsteady; for example, unstable waves can arise spontaneously under steady boundary conditions. Such unsteadiness must have a time
scale proportional to l/U , as assumed in equation (8.2). Consider now a situation
in which the imposed boundary conditions are unsteady. To be specific, consider an
object having a characteristic length scale l oscillating with a frequency ω in a fluid
at rest at infinity. This is a problem having an imposed length scale and an imposed
time scale 1/ω. In such a case a velocity scale can be derived from ω and l to be
U = lω. The preceding analysis then
through, leading to the conclusion that
√ goes √
Re = U l/ν = ωl 2 /ν and Fr = U/ gl = ω l/g have to be duplicated for dynamic
similarity of two flows in which viscous and gravitational effects are important.
All nondimensional quantities are identical for dynamically similar flows. For
flow around an immersed body, we can define a nondimensional drag coefficient
CD ≡
D
,
ρU 2 l 2 /2
(8.6)
where D is the drag experienced by the body; use of the factor of 1/2 in equation (8.6)
is conventional but not necessary. Instead of writing CD in terms of a length scale l,
it is customary to define the drag coefficient more generally as
CD ≡
D
,
ρU 2 A/2
where A is a characteristic area. For blunt bodies such as spheres and cylinders,
A is taken to be a cross section perpendicular to the flow. Therefore, A = π d 2 /4 for
a sphere of diameter d, and A = bd for a cylinder of diameter d and length b, with
the axis of the cylinder perpendicular to the flow. For flow over a flat plate, on the
other hand, A is taken to be the “wetted area”, that is, A = bl; here, l is the length of
the plate in the direction of flow and b is the width perpendicular to the flow.
The values of the drag coefficient CD are identical for dynamically similar flows.
In the present example in which the drag is caused both by gravitational and viscous
effects, we must have a functional relation of the form
CD = f (Fr, Re) .
(8.7)
284
Dynamic Similarity
For many flows the gravitational effects are unimportant. An example is the flow
around the body, such as an airfoil, that does not generate gravity waves. In that case
Fr is irrelevant, and
CD = f (Re) .
(8.8)
We recall from the preceding discussion that speeds are low enough to ignore compressibility effects.
3. Dimensional Matrix
In many complicated flow problems the precise form of the differential equations may
not be known. In this case the conditions for dynamic similarity can be determined
by means of a dimensional analysis of the variables involved. A formal method of
dimensional analysis is presented in the following section. Here we introduce certain
ideas that are needed for performing a formal dimensional analysis.
The underlying principle in dimensional analysis is that of dimensional homogeneity, which states that all terms in an equation must have the same dimension. This
is a basic check that we constantly apply when we derive an equation; if the terms do
not have the same dimension, then the equation is not correct.
Fluid flow problems without electromagnetic forces and chemical reactions
involve only mechanical variables (such as velocity and density) and thermal variables (such as temperature and specific heat). The dimensions of all these variables can be expressed in terms of four basic dimensions—mass M, length L,
time T, and temperature θ. We shall denote the dimension of a variable q
by [q]. For example, the dimension of velocity is [u] = L/T, that of pressure is [p] = [force]/[area] = MLT−2 /L2 = M/LT2 , and that of specific heat
is [C] = [energy]/[mass][temperature] = MLT−2 L/Mθ = L2 /θT2 . When thermal
effects are not considered, all variables can be expressed in terms of three fundamental dimensions, namely, M, L, and T. If temperature is considered only in combination
with Boltzmann’s constant (kθ) or a gas constant (Rθ), then the units of the combination are simply L2 /T2 . Then only the three dimensions M, L, and T are required.
The method of dimensional analysis presented here uses the idea of a “dimensional matrix” and its rank. Consider the pressure drop p in a pipeline, which is
expected to depend on the inside diameter d of the pipe, its length l, the average size
e of the wall roughness elements, the average flow velocity U , the fluid density ρ,
and the fluid viscosity µ. We can write the functional dependence as
f (p, d, l, e, U, ρ, µ) = 0.
(8.9)
The dimensions of the variables can be arranged in the form of the following matrix:
p
M
1
L −1
T −2
d
l
e
U
ρ
µ
0
1
0
0
1
0
0 0 1 1
1 1 −3 −1
0 −1 0 −1
(8.10)
285
4. Buckingham’s Pi Theorem
Where we have written the variables p, d, . . . on the top and their dimensions in a
vertical column underneath. For example, [p] = ML−1 T−2 . An array of dimensions
such as equation (8.10) is called a dimensional matrix. The rank r of any matrix is
defined to be the size of the largest square submatrix that has a nonzero determinant.
Testing the determinant of the first three rows and columns, we obtain
1
−1
−2
0
1
0
0
1 = 0.
0
However, there does exist a nonzero third-order determinant, for example, the one
formed by the last three columns:
0 1 1
1 −3 −1 = −1.
−1 0 −1
Thus, the rank of the dimensional matrix (8.10) is r = 3. If all possible third-order
determinants were zero, we would have concluded that r < 3 and proceeded to test
the second-order determinants.
It is clear that the rank is less than the number of rows only when one of the rows
can be obtained by a linear combination of the other rows. For example, the matrix
(not from equation (8.10)):
0
1
0 1
−1
2
1 −2
−1
4
1 0
has r = 2, as the last row can be obtained by adding the second row to twice the first
row. A rank of less than 3 commonly occurs in problems of statics, in which the mass
is really not relevant in the problem, although the dimensions of the variables (such
as force) involve M. In most problems in fluid mechanics without thermal effects,
r = 3.
4. Buckingham’s Pi Theorem
Of the various formal methods of dimensional analysis, the one that we shall describe
was proposed by Buckingham in 1914. Let q1 , q2 , . . . , qn be n variables involved in
a particular problem, so that there must exist a functional relationship of the form
f (q1 , q2 , . . . , qn ) = 0.
(8.11)
Buckingham’s theorem states that the n variables can always be combined to form
exactly (n − r) independent nondimensional variables, where r is the rank of the
dimensional matrix. Each nondimensional parameter is called a “ number,” or more
commonly a nondimensional product. (The symbol is used because the nondimensional parameter can be written as a product of the variables q1 , . . . , qn , raised to
286
Dynamic Similarity
some power, as we shall see.) Thus, equation (8.11) can be written as a functional
relationship
(8.12)
φ (1 , 2 , . . . , n−r ) = 0.
It will be seen shortly that the nondimensional parameters are not unique. However,
(n − r) of them are independent and form a complete set.
The method of forming nondimensional parameters proposed by Buckingham is
best illustrated by an example. Consider again the pipe flow problem expressed by
f (p, d, l, e, U, ρ, µ) = 0,
(8.13)
whose dimensional matrix (8.10) has a rank of r = 3. Since there are n = 7 variables
in the problem, the number of nondimensional parameters must be n − r = 4. We
first select any 3 (= r) of the variables as “repeating variables”, which we want to be
repeated in all of our nondimensional parameters. These repeating variables must have
different dimensions, and among them must contain all the fundamental dimensions
M, L, and T. In many fluid flow problems we choose a characteristic velocity, a
characteristic length, and a fluid property as the repeating variables. For the pipe flow
problem, let us choose U , d, and ρ as the repeating variables. Although other choices
would result in a different set of nondimensional products, we can always obtain other
complete sets by combining the ones we have. Therefore, any choice of the repeating
variables is satisfactory.
Each nondimensional product is formed by combining the three repeating variables with one of the remaining variables. For example, let the first dimensional
product be taken as
1 = U a d b ρ c p.
The exponents a, b, and c are obtained from the requirement that 1 is dimensionless.
This requires
c
a
ML−1 T−2 = Mc+1 La+b−3c−1 T−a−2 .
M0 L0 T0 = LT−1 (L)b ML−3
Equating indices, we obtain a = −2, b = 0, c = −1, so that
1 = U −2 d 0 ρ −1 p =
p
.
ρU 2
A similar procedure gives
2 = U a d b ρ c l =
l
,
d
3 = U a d b ρ c e =
e
,
d
4 = U a d b ρ c µ =
µ
.
ρU d
287
5. Nondimensional Parameters and Dynamic Similarity
Therefore, the nondimensional representation of the problem has the form
l e µ
p
=φ
, ,
.
d d ρU d
ρU 2
(8.14)
Other dimensionless products can be obtained by combining the four in the preceding.
For example, a group pd 2 ρ/µ2 can be formed from 1 /24 . Also, different nondimensional groups would have been obtained had we taken variables other than U , d,
and ρ as the repeating variables. Whatever nondimensional groups we obtain, only
four of these are independent for the pipe flow problem described by equation (8.13).
However, the set in equation (8.14) contains the most commonly used nondimensional parameters, which have familiar physical interpretation and have been given
special names. Several of the common dimensionless parameters will be discussed in
Section 7.
The pi theorem is a formal method of forming dimensionless groups. With some
experience, it becomes quite easy to form the dimensionless numbers by simple
inspection. For example, since there are three length scales d, e, and l in equation (8.13), we can form two groups such as e/d and l/d. We can also form p/ρU 2
as our dependent nondimensional variable; the Bernoulli equation tells us that ρU 2
has the same units as p. The nondimensional number that describes viscous effects
is well known to be ρU d/µ. Therefore, with some experience, we can find all the
nondimensional variables by inspection alone, thus no formal analysis is needed.
5. Nondimensional Parameters and Dynamic Similarity
Arranging the variables in terms of dimensionless products is especially useful in
presenting experimental data. Consider the case of drag on a sphere of diameter d
moving at a speed U through a fluid of density ρ and viscosity µ. The drag force can
be written as
D = f (d, U, ρ, µ) .
(8.15)
If we do not form dimensionless groups, we would have to conduct an experiment
to determine D vs d, keeping U , ρ, and µ fixed. We would then have to conduct an
experiment to determine D as a function of U , keeping d, ρ, and µ fixed, and so on.
However, such a duplication of effort is unnecessary if we write equation (8.15) in
terms of dimensionless groups. A dimensional analysis of equation (8.15) gives
ρU d
D
=f
,
(8.16)
µ
ρU 2 d 2
reducing the number of variables from five to two, and consequently a single experimental curve (Figure 8.2). Not only is the presentation of data united and simplified,
the cost of experimentation is drastically reduced. It is clear that we need not vary
the fluid viscosity or density at all; we could obtain all the data of Figure 8.2 in one
wind tunnel experiment in which we determine D for various values of U . However,
if we want to find the drag force for a fluid of different density or viscosity, we can
288
Dynamic Similarity
CD =
D
( ½ ) ρU 2A
Re =
ρUd
µ
Figure 8.2 Drag coefficient for a sphere. The characteristic area is taken as A = π d 2 /4. The reason for
the sudden drop of CD at Re ∼ 5 × 105 is the transition of the laminar boundary layer to a turbulent one,
as explained in Chapter 10.
still use Figure 8.2. Note that the Reynolds number in equation (8.16) is written as
the independent variable because it can be externally controlled in an experiment. In
contrast, the drag coefficient is written as a dependent variable.
The idea of dimensionless products is intimately associated with the concept
of similarity. In fact, a collapse of all the data on a single graph such as the one in
Figure 8.2 is possible only because in this problem all flows having the same value
of Re = ρU d/µ are dynamically similar.
For flow around a sphere, the pressure at any point x = (x, y, z) can be written as
p (x) − p∞ = f (d, U, ρ, µ; x) .
A dimensional analysis gives the local pressure coefficient:
p (x) − p∞
=f
ρU 2
ρU d x
;
µ d
,
(8.17)
requiring that nondimensional local flow variables be identical at corresponding points
in dynamically similar flows. The difference between relations (8.16) and (8.17)
should be noted. equation (8.16) is a relation between overall quantities (scales of
motion), whereas (8.17) holds locally at a point.
289
5. Nondimensional Parameters and Dynamic Similarity
Prediction of Flow Behavior from Dimensional Considerations
An interesting observation in Figure 8.2 is that CD ∝ 1/Re at small Reynolds
numbers. This can be justified solely on dimensional grounds as follows. At small
values of Reynolds numbers we expect that the inertia forces in the equations of
motion must become negligible. Then ρ drops out of equation (8.15), requiring
D = f (d, U, µ) .
The only dimensionless product that can be formed from the preceding is D/µU d.
Because there is no other nondimensional parameter on which D/µU d can depend,
it can only be a constant:
D ∝ µU d
(Re ≪ 1) ,
(8.18)
which is equivalent to CD ∝ 1/Re. It is seen that the drag force in a low Reynolds
number flow is linearly proportional to the speed U; this is frequently called the Stokes
law of resistance.
At the opposite extreme, Figure 8.2 shows that CD becomes independent of Re
for values of Re > 103 . This is because the drag is now due mostly to the formation
of a turbulent wake, in which the viscosity only has an indirect influence on the flow.
(This will be clear in Chapter 13, where we shall see that the only effect of viscosity
as Re → ∞ is to dissipate the turbulent kinetic energy at increasingly smaller scales.
The overall flow is controlled by inertia forces alone.) In this limit µ drops out of
equation (8.15), giving
D = f (d, U, ρ) .
The only nondimensional product is then D/ρU 2 d 2 , requiring
D ∝ ρU 2 d 2
(Re ≫ 1) ,
(8.19)
which is equivalent to CD = const. It is seen that the drag force is proportional to U 2
for high Reynolds number flows. This rule is frequently applied to estimate various
kinds of wind forces such as those on industrial structures, houses, automobiles, and
the ocean surface. Consideration of surface tension effects may introduce additional
dimensionless parameters depending on the nature of the problem. For example, if
surface tension is to balance against a gravity body force, the Bond number Bo =
ρgl 2 /σ would be the appropriate dimensionless parameter to consider. If surface
tension is in competition with a viscous stress, then it would be the capillary number,
Ca = µU/σ . Similarly, the Weber number expresses the ratio of inertial forces to
surface tension forces.
It is clear that very useful relationships can be established based on sound physical
considerations coupled with a dimensional analysis. In the present case this procedure
leads to D ∝ µU d for low Reynolds numbers, and D ∝ ρU 2 d 2 for high Reynolds
numbers. Experiments can then be conducted to see if these relations do hold and to
determine the unknown constants in these relations. Such arguments are constantly
used in complicated fluid flow problems such as turbulence, where physical intuition
290
Dynamic Similarity
plays a key role in research. A well-known example of this is the Kolmogorov K −5/3
spectral law of isotropic turbulence presented in Chapter 13.
6. Comments on Model Testing
The concept of similarity is the basis of model testing, in which test data on one flow
can be applied to other flows. The cost of experimentation with full-scale objects
(which are frequently called prototypes) can be greatly reduced by experiments on
a smaller geometrically similar model. Alternatively, experiments with a relatively
inconvenient fluid such as air or helium can be substituted by an experiment with an
easily workable fluid such as water. A model study is invariably undertaken when a
new aircraft, ship, submarine, or harbor is designed.
In many flow situations both friction and gravity forces are important, which
requires that both the Reynolds number and the
√ Froude number be duplicated in a
model testing. Since Re = U l/ν and Fr =√U/ gl, simultaneous satisfaction of both
criteria would require U ∝ 1/ l and U ∝ l as the model length is varied. It follows
that both the Reynolds and the Froude numbers cannot be duplicated simultaneously
unless fluids of different viscosities are used in the model and the prototype flows.
This becomes impractical, or even impossible, as the requirement sometimes needs
viscosities that cannot be met by common fluids. It is then necessary to decide which
of the two forces is more important in the flow, and a model is designed on the
basis of the corresponding dimensionless number. Corrections can then be applied to
account for the inequality of the remaining dimensionless group. This is illustrated
in Example 8.1, which follows this section.
Although geometric similarity is a precondition to dynamic similarity, this is
not always possible to attain. In a model study of a river basin, a geometrically
similar model results in a stream so shallow that capillary and viscous effects become
dominant. In such a case it is necessary to use a vertical scale larger than the horizontal
scale. Such distorted models lack complete similitude, and their results are corrected
before making predictions on the prototype.
Models of completely submerged objects are usually tested in a wind tunnel or
in a towing tank where they are dragged through a pool of water. The towing tank
is also used for testing models that are not completely submerged, for example, ship
hulls; these are towed along the free surface of the liquid.
Example 8.1. A ship 100 m long is expected to sail at 10 m/s. It has a submerged
surface of 300 m2 . Find the model speed for a 1/25 scale model, neglecting frictional
effects. The drag is measured to be 60 N when the model is tested in a towing tank at
the model speed. Based on this information estimate the prototype drag after making
corrections for frictional effects.
Solution: We first estimate the model speed neglecting frictional effects. Then
the nondimensional drag force depends only on the Froude number:
D/ρU 2 l 2 = f U/ gl .
(8.20)
291
6. Comments on Model Testing
Equating Froude numbers for the model (denoted by subscript “m”) and prototype
(denoted by subscript “p”), we get
Um = Up gm lm /gp lp = 10 1/25 = 2 m/s.
The total drag on the model was measured to be 60 N at this model speed. Of
the total measured drag, a part was due to frictional effects. The frictional drag can
be estimated by treating the surface of the hull as a flat plate, for which the drag
coefficient CD is given in Figure 10.12 as a function of the Reynolds number. Using
a value of ν = 10−6 m2 /s for water, we get
U l/ν (model) = [2 (100/25)]/10−6 = 8 × 106 ,
U l/ν (prototype) = 10 (100) /10−6 = 109 .
For these values of Reynolds numbers, Figure 10.12 gives the frictional drag coefficients of
CD (model) = 0.003,
CD (prototype) = 0.0015.
Using a value of ρ = 1000 kg/m3 for water, we estimate
Frictional drag on model = 21 CD ρU 2 A
= 0.5 (0.003) (1000) (2)2 300/252 = 2.88 N
Out of the total model drag of 60 N, the wave drag is therefore 60 − 2.88 = 57.12 N.
Now the wave drag still obeys equation (8.20), which means that D/ρU 2 l 2 for
the two flows are identical, where D represents wave drag alone. Therefore
Wave drag on prototype
= (Wave drag on model) ρp /ρm
lp / lm
2
Up /Um
2
= 57.12 (1) (25)2 (10/2)2 = 8.92 × 105 N
Having estimated the wave drag on the prototype, we proceed to determine its
frictional drag. We obtain
Frictional drag on prototype = 21 CD ρU 2 A
= (0.5) (0.0015) (1000) (10)2 (300) = 0.225 × 105 N
Therefore, total drag on prototype = (8.92 + 0.225) × 105 = 9.14 × 105 N.
If we did not correct for the frictional effects, and assumed that the measured
model drag was all due to wave effects, then we would have found from equation (8.20)
a prototype drag of
Dp = Dm ρp /ρm
lp / lm
2
Up /Um
2
= 60 (1) (25)2 (10/2)2 = 9.37 × 105 N.
292
Dynamic Similarity
7. Significance of Common Nondimensional Parameters
So far, we have encountered several nondimensional groups such as the pressure
coefficient (p − p∞ )/ρU 2 , the drag coefficient√2D/ρU 2 l 2 , the Reynolds number Re = U l/ν, and the Froude number U/ gl. Several independent nondimensional parameters that commonly enter fluid flow problems are listed and discussed briefly in this section. Other parameters will arise throughout the rest of the
book.
Reynolds Number
The Reynolds number is the ratio of inertia force to viscous force:
Re ≡
ρU 2 / l
Ul
ρu∂u/∂x
Inertia force
∝
=
∝
.
2
2
Viscous force
ν
µ∂ u/∂x
µU/ l 2
Equality of Re is a requirement for the dynamic similarity of flows in which viscous
forces are important.
Froude Number
The Froude number is defined as
Inertia force
Fr ≡
Gravity force
1/2
ρU 2 / l
∝
ρg
1/2
U
=√ .
gl
Equality of Fr is a requirement for the dynamic similarity of flows with a free surface
in which gravity forces are dynamically significant. Some examples of flows in which
gravity plays a significant role are the motion of a ship, flow in an open channel, and
the flow of a liquid over the spillway of a dam (Figure 8.3).
Internal Froude Number
In a density-stratified fluid the gravity force can play a significant role without the
presence of a free surface. Then the effective gravity force in a two-layer situation is
Figure 8.3 Examples of flows in which gravity is important.
293
7. Significance of Common Nondimensional Parameters
the “buoyancy” force (ρ2 − ρ1 )g, as seen in the preceding chapter. In such a case we
can define an internal Froude number as
Fr ′ ≡
Inertia force
Buoyancy force
1/2
∝
ρ1 U 2 / l
(ρ2 − ρ1 )g
1/2
U
=
g′l
,
(8.21)
where g ′ ≡ g (ρ2 − ρ1 ) /ρ1 is the “reduced gravity.” For a continuously stratified
fluid having a maximum buoyancy frequency N, we similarly define
Fr ′ ≡
U
,
Nl
which is analogous to equation (8.21) since g ′ = g (ρ2 − ρ1 ) /ρ1 is similar to
−ρ0−1 g (dρ/dz) l = N 2 l.
Richardson Number
Instead of defining the internal Froude number, it is more common to define a nondimensional parameter that is equivalent to 1/Fr ′2 . This is called the Richardson
number, and in a two-layer situation it is defined as
g′l
.
U2
Ri ≡
(8.22)
In a continuously stratified flow, we can similarly define
Ri ≡
N 2l2
.
U2
(8.23)
It is clear that the Richardson number has to be equal for the dynamic similarity of
two density-stratified flows.
Equations (8.22) and (8.23) define overall or bulk Richardson numbers in terms
of the scales l, N , and U . In addition, we can define a Richardson number involving
the local values of velocity gradient and stratification at a certain depth z. This is
called the gradient Richardson number, and it is defined as
Ri (z) ≡
N 2 (z)
(dU/dz)2
.
Local Richardson numbers will be important in our studies of instability and turbulence in stratified fluids.
Mach Number
The Mach number is defined as
M≡
Inertia force
Compressibility force
1/2
∝
ρU 2 / l
ρc2 / l
1/2
=
U
,
c
294
Dynamic Similarity
where c is the speed of sound. Equality of Mach numbers is a requirement for the
dynamic similarity of compressible flows. For example, the drag experienced by a
body in a flow with compressibility effects has the form
CD = f (Re, M) .
Flows in which M < 1 are called subsonic, whereas flows in which M > 1 are called
supersonic. It will be shown in Chapter 16 that compressibility effects can be neglected
if M < 0.3.
Prandtl Number
The Prandtl number enters as a nondimensional parameter in flows involving heat
conduction. It is defined as
Pr ≡
Cp µ
ν
µ/ρ
Momentum diffusivity
= =
=
.
Heat diffusivity
κ
k/ρCp
k
It is therefore a fluid property and not a flow variable. For air at ordinary temperatures and pressures, Pr = 0.72, which is close to the value of 0.67 predicted from
a simplified kinetic theory model assuming hard spheres and monatomic molecules
(Hirschfelder, Curtiss, and Bird (1954), pp. 9–16). For water at 20 ◦ C, Pr = 7.1. The
dynamic similarity of flows involving thermal effects requires equality of Prandtl
numbers.
Exercises
1. Suppose that the power to drive a propeller of an airplane depends on d
(diameter of the propeller), U (free-stream velocity), ω (angular velocity of propeller), c (velocity of sound), ρ (density of fluid), and µ (viscosity). Find the dimensionless groups. In your opinion, which of these are the most important and should
be duplicated in a model testing?
2. A 1/25 scale model of a submarine is being tested in a wind tunnel in which
p = 200 kPa and T = 300 K. If the prototype speed is 30 km/hr, what should be the
free-stream velocity in the wind tunnel? What is the drag ratio? Assume that the
submarine would not operate near the free surface of the ocean.
Literature Cited
Hirschfelder, J. O., C. F. Curtiss, and R. B. Bird (1954). Molecular Theory of Gases and Liquids, New York:
John Wiley and Sons.
Supplemental Reading
Bridgeman, P. W. (1963). Dimensional Analysis, New Haven: Yale University Press.
Chapter 9
Laminar Flow
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 295
2. Analogy between Heat and Vorticity
Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
3. Pressure Change Due to Dynamic
Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
4. Steady Flow between Parallel
Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Plane Couette Flow. . . . . . . . . . . . . . . . . . . 300
Plane Poiseuille Flow . . . . . . . . . . . . . . . . . 301
5. Steady Flow in a Pipe . . . . . . . . . . . . . . . . 302
6. Steady Flow between Concentric
Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Flow Outside a Cylinder Rotating in an
Infinite Fluid . . . . . . . . . . . . . . . . . . . . . . 304
Flow Inside a Rotating Cylinder. . . . . . . 305
7. Impulsively Started Plate: Similarity
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Formulation of a Problem in Similarity
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 306
8.
9.
10.
11.
12.
13.
14.
15.
Similarity Solution . . . . . . . . . . . . . . . . . .
An Alternative Method of Deducing
the Form of η . . . . . . . . . . . . . . . . . . . . .
Method of Laplace Transform . . . . . . .
Diffusion of a Vortex Sheet . . . . . . . . . . .
Decay of a Line Vortex. . . . . . . . . . . . . . .
Flow Due to an Oscillating
Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
High and Low Reynolds Number
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creeping Flow around a
Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonuniformity of Stokes’ Solution
and Oseen’s Improvement . . . . . . . . . . . .
Hele-Shaw Flow. . . . . . . . . . . . . . . . . . . . .
Final Remarks . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . .
309
312
312
313
315
317
320
322
327
332
334
335
337
337
1. Introduction
In Chapters 6 and 7 we studied inviscid flows in which the viscous terms in the
Navier–Stokes equations were dropped. The underlying assumption was that the viscous forces were confined to thin boundary layers near solid surfaces, so that the
bulk of the flow could be regarded as inviscid (Figure 6.1). We shall see in the next
chapter that this is indeed valid if the Reynolds number is large. For low values of
the Reynolds number, however, the entire flow may be dominated by viscosity, and
the inviscid flow theory is of little use. The purpose of this chapter is to present certain solutions of the Navier–Stokes equations in some simple situations, retaining the
viscous term µ∇ 2 u everywhere in the flow. While the inviscid flow theory allows the
fluid to “slip” past a solid surface, real fluids will adhere to the surface because of
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50009-5
295
296
Laminar Flow
intermolecular interactions, that is, a real fluid satisfies the condition of zero relative
velocity at a solid surface. This is the so-called no-slip condition.
Before presenting the solutions, we shall first discuss certain basic ideas about
viscous flows. Flows in which the fluid viscosity is important can be of two types,
namely, laminar and turbulent. The basic difference between the two flows was dramatically demonstrated in 1883 by Reynolds, who injected a thin stream of dye into
the flow of water through a tube (Figure 9.1). At low rates of flow, the dye stream
was observed to follow a well-defined straight path, indicating that the fluid moved
in parallel layers (laminae) with no macroscopic mixing motion across the layers.
This is called a laminar flow. As the flow rate was increased beyond a certain critical
value, the dye streak broke up into an irregular motion and spread throughout the
cross section of the tube, indicating the presence of macroscopic mixing motions perpendicular to the direction of flow. Such a chaotic fluid motion is called a turbulent
flow. Reynolds demonstrated that the transition from laminar to turbulent flow always
occurred at a fixed value of the ratio Re = V d/ν ∼ 3000, where V is the velocity
averaged over the cross section, d is the tube diameter, and ν is the kinematic viscosity.
Laminar flows in which viscous effects are important throughout the flow are the
subject of the present chapter; laminar flows in which frictional effects are confined to
boundary layers near solid surfaces are discussed in the next chapter. Chapter 12 considers the stability of laminar flows and their transition to turbulence; fully turbulent
flows are discussed in Chapter 13. We shall assume here that the flow is incompressible, which is valid for Mach numbers less than 0.3. We shall also assume that the
Figure 9.1 Reynolds’s experiment to distinguish between laminar and turbulent flows.
297
3. Pressure Change Due to Dynamic Effects
flow is unstratified and observed in a nonrotating coordinate system. Some solutions
of viscous flows in rotating coordinates, such as the Ekman layers, are presented in
Chapter 14.
2. Analogy between Heat and Vorticity Diffusion
For two-dimensional flows that take place in the xy-plane, the vorticity equation is
(see equation (5.13))
Dω
= ν∇ 2 ω,
Dt
where ω = ∂v/∂x − ∂u/∂y. (For the sake of simplicity, we have avoided the vortex
stretching term ω • ∇u by assuming two dimensionality.) This shows that the rate of
change of vorticity ∂ω/∂t at a point is due to advection (−u • ∇ω) and diffusion
(ν∇ 2 ω) of vorticity. The equation is similar to the heat equation
DT
= κ∇ 2 T ,
Dt
where κ = k/ρCp is the thermal diffusivity. The similarity of the equations suggests
that vorticity diffuses in a manner analogous to the diffusion of heat. The similarity
also brings out the fact that the diffusive effects are controlled by ν and κ, and not by
µ and k. In fact, the momentum equation
1
Du
= ν∇ 2 u − ∇p,
Dt
ρ
(9.1)
also shows that the acceleration due to viscous diffusion is proportional to ν. Thus,
air (ν = 15 × 10−6 m2 /s) is more diffusive than water (ν = 10−6 m2 /s), although µ
for water is larger. Both ν and κ have the units of m2 /s; the kinematic viscosity ν is
therefore also called momentum diffusivity, in analogy with κ, which is called heat diffusivity. (However, velocity cannot be simply regarded as being diffused and advected
in a flow because of the presence of the pressure gradient term in equation (9.1). The
analogy between heat and vorticity is more appropriate.)
3. Pressure Change Due to Dynamic Effects
The equation of motion for the flow of a uniform density fluid is
ρ
Du
= ρg − ∇p + µ∇ 2 u.
Dt
If the body of fluid is at rest, the pressure is hydrostatic:
0 = ρg − ∇ps .
Subtracting, we obtain
ρ
Du
= −∇pd + µ∇ 2 u,
Dt
(9.2)
298
Laminar Flow
where pd ≡ p − ps is the pressure change due to dynamic effects. As there is no
accepted terminology for pd , we shall call it dynamic pressure, although the term is
also used for ρq 2 /2, where q is the speed. Other common terms for pd are “modified
pressure” (Batchelor, 1967) and “excess pressure” (Lighthill, 1986).
For a fluid of uniform density, introduction of pd eliminates gravity from the differential equation as in equation (9.2). However, the process may not eliminate gravity
from the problem. Gravity reappears in the problem if the boundary conditions are
given in terms of the total pressure p. An example is the case of surface gravity waves,
where the total pressure is fixed at the free surface, and the mere introduction of pd
does not eliminate gravity from the problem. Without a free surface, however, gravity
has no dynamic role. Its only effect is to add a hydrostatic contribution to the pressure
field. In the applications that follow, we shall use equation (9.2), but the subscript on
p will be omitted, as it is understood that p stands for the dynamic pressure.
4. Steady Flow between Parallel Plates
Because of the presence of the nonlinear advection term u • ∇u, very few exact
solutions of the Navier–Stokes equations are known in closed form. In general, exact
solutions are possible only when the nonlinear terms vanish identically. An example is
the fully developed flow between infinite parallel plates. The term “fully developed”
signifies that we are considering regions beyond the developing stage near the entrance
(Figure 9.2), where the velocity profile changes in the direction of flow because of the
development of boundary layers from the two walls. Within this “entrance length,”
which can be several times the distance between the walls, the velocity is uniform in
the core increasing downstream and decreasing with x within the boundary layers. The
derivative ∂u/∂x is therefore nonzero; the continuity equation ∂u/∂x + ∂v/∂y = 0
Figure 9.2 Developing and fully developed flows in a channel. The flow is fully developed after the
boundary layers merge.
299
4. Steady Flow between Parallel Plates
then requires that v = 0, so that the flow is not parallel to the walls within the entrance
length.
Consider the fully developed stage of the steady flow between two infinite parallel
plates. The flow is driven by a combination of an externally imposed pressure gradient
(for example, maintained by a pump) and the motion of the upper plate at uniform
speed U . Take the x-axis along the lower plate and in the direction of flow (Figure 9.3).
Two dimensionality of the flow requires that ∂/∂z = 0. Flow characteristics are also
invariant in the x direction, so that continuity requires ∂v/∂y = 0. Since v = 0 at
y = 0, it follows that v = 0 everywhere, which reflects the fact that the flow is parallel
to the walls. The x- and y-momentum equations are
d 2u
1 ∂p
+ν 2,
ρ ∂x
dy
1 ∂p
.
0=−
ρ ∂y
0=−
The y-momentum equation shows that p is not a function of y. In the x-momentum
equation, then, the first term can only be a function of x, while the second term can
only be a function of y. The only way this can be satisfied is for both terms to be
constant. The pressure gradient is therefore a constant, which implies that the pressure
varies linearly along the channel. Integrating the x-momentum equation twice, we
obtain
y 2 dp
+ µu + Ay + B,
(9.3)
0=−
2 dx
where we have written dp/dx because p is a function of x alone. The constants of
integration A and B are determined as follows. The lower boundary condition u = 0
at y = 0 requires B = 0. The upper boundary condition u = U at y = 2b requires
A = b(dp/dx) − µU/2b. The velocity profile equation (9.3) then becomes
u=
Figure 9.3
yU
y dp
y
b−
.
−
2b
µ dx
2
Flow between parallel plates.
(9.4)
300
Laminar Flow
The velocity profile is illustrated in Figure 9.4 for various cases.
The volume rate of flow per unit width of the channel is
Q=
2b
0
2b2 dp
u dy = U b 1 −
,
3µU dx
so that the average velocity is
V ≡
U
2b2 dp
Q
=
1−
.
2b
2
3µU dx
Two cases of special interest are discussed in what follows.
Plane Couette Flow
The flow driven by the motion of the upper plate alone, without any externally imposed
pressure gradient, is called a plane Couette flow. In this case equation (9.4) reduces
to the linear profile (Figure 9.4c)
u=
yU
.
2b
The magnitude of shear stress is
τ =µ
µU
du
=
,
dy
2b
which is uniform across the channel.
Figure 9.4 Various cases of parallel flow in a channel.
(9.5)
301
4. Steady Flow between Parallel Plates
Plane Poiseuille Flow
The flow driven by an externally imposed pressure gradient through two stationary
flat walls is called a plane Poiseuille flow. In this case equation (9.4) reduces to the
parabolic profile (Figure 9.4d)
u=−
y
y dp
b−
.
µ dx
2
(9.6)
The magnitude of shear stress is
τ =µ
dp
du
= (b − y) ,
dy
dx
which shows that the stress distribution is linear with a magnitude of b(dp/dx) at the
walls (Figure 9.4d).
It is important to note that the constancy of the pressure gradient and the linearity
of the shear stress distribution are general results for a fully developed channel flow
and hold even if the flow is turbulent. Consider a control volume ABCD shown in
Figure 9.3, and apply the momentum principle (see equation (4.20)), which states that
the net force on a control volume is equal to the net outflux of momentum through the
surfaces. Because the momentum fluxes across surfaces AD and BC cancel each other,
the forces on the control volume must be in balance; per unit width perpendicular to
the plane of paper, the force balance gives
dp
L 2y ′ = 2Lτ,
(9.7)
p− p−
dx
where y ′ is the distance measured from the center of the channel. In equation (9.7), 2y ′
is the area of surfaces AD and BC, and L is the area of surface AB or DC. Applying
equation (9.7) at the wall, we obtain
dp
b = τ0 ,
dx
(9.8)
which shows that the pressure gradient dp/dx is constant. Equations (9.7) and (9.8)
give
y′
τ = τ0 ,
(9.9)
b
which shows that the magnitude of the shear stress increases linearly from the center
of the channel (Figure 9.4d). Note that no assumption about the nature of the flow
(laminar or turbulent) has been made in deriving equations (9.8) and (9.9).
Instead of applying the momentum principle, we could have reached the foregoing conclusions from the equation of motion in the form
ρ
dp dτxy
Du
=−
+
,
Dt
dx
dy
302
Laminar Flow
Figure 9.5 Laminar flow through a tube.
where we have introduced subscripts on τ and noted that the other stress components
are zero. As the left-hand side of the equation is zero, it follows that dp/dx must be
a constant and τxy must be linear in y.
5. Steady Flow in a Pipe
Consider the fully developed laminar motion through a tube of radius a. Flow through
a tube is frequently called a circular Poiseuille flow. We employ cylindrical coordinates (r, θ, x), with the x-axis coinciding with the axis of the pipe (Figure 9.5).
The only nonzero component of velocity is the axial velocity u(r) (omitting the
subscript “x” on u), and none of the flow variables depend on θ. The equations of
motion in cylindrical coordinates are given in Appendix B. The radial equation of
motion gives
0=−
∂p
,
∂r
showing that p is a function of x alone. The x-momentum equation gives
0=−
dp µ d
+
dx
r dr
r
du
.
dr
As the first term can only be a function of x, and the second term can only be a
function of r, it follows that both terms must be constant. The pressure therefore falls
linearly along the length of pipe. Integrating twice, we obtain
u=
r 2 dp
+ A ln r + B.
4µ dx
Because u must be bounded at r = 0, we must have A = 0. The wall condition u = 0
at r = a gives B = −(a 2 /4µ)(dp/dx). The velocity distribution therefore takes the
parabolic shape
u=
r 2 − a 2 dp
.
4µ dx
(9.10)
303
6. Steady Flow between Concentric Cylinders
From Appendix B, the shear stress at any point is
∂u
∂ur
+
.
τxr = µ
∂x
∂r
In the present case the radial velocity ur is zero. Dropping the subscript on τ , we
obtain
r dp
du
=
,
(9.11)
τ =µ
dr
2 dx
which shows that the stress distribution is linear, having a maximum value at the
wall of
a dp
.
(9.12)
τ0 =
2 dx
As in the previous section, equation (9.12) is also valid for turbulent flows.
The volume rate of flow is
a
π a 4 dp
u2 π r dr = −
,
Q=
8µ dx
0
where the negative sign offsets the negative value of dp/dx. The average velocity
over the cross section is
Q
a 2 dp
V ≡
=−
.
2
8µ dx
πa
6. Steady Flow between Concentric Cylinders
Another example in which the nonlinear advection terms drop out of the equations of
motion is the steady flow between two concentric, rotating cylinders. This is usually
called the circular Couette flow to distinguish it from the plane Couette flow in which
the walls are flat surfaces. Let the radius and angular velocity of the inner cylinder be
R1 and 1 and those for the outer cylinder be R2 and 2 (Figure 9.6). Using cylindrical
coordinates, the equations of motion in the radial and tangential directions are
−
u2θ
1 dp
=−
,
r
ρ dr
d 1 d
(ruθ ) .
0=µ
dr r dr
The r-momentum equation shows that the pressure increases radially outward due
to the centrifugal force. The pressure distribution can therefore be determined once
uθ (r)has been found. Integrating the θ-momentum equation twice, we obtain
uθ = Ar +
B
.
r
(9.13)
304
Laminar Flow
Figure 9.6 Circular Couette flow.
Using the boundary conditions uθ =
we obtain
2
2 R2
R22
A=
B=
(
at r = R1 , and uθ =
1 R1
1
2 R2
at r = R2 ,
2
1 R1
,
− R12
−
2 2
2 )R1 R2
.
R22 − R12
−
Substitution into equation (9.13) gives the velocity distribution
uθ =
1
1 − (R1 /R2 )2
2
−
1
R1
R2
2
r+
R12
(
r
1
−
2)
.
(9.14)
Two limiting cases of the velocity distribution are considered in the following.
Flow Outside a Cylinder Rotating in an Infinite Fluid
Consider a long circular cylinder of radius R rotating with angular velocity in an
infinite body of viscous fluid (Figure 9.7). The velocity distribution for the present
problem can be derived from equation (9.14) if we substitute 2 = 0, R2 = ∞,
1 = , and R1 = R. This gives
uθ =
R2
,
r
(9.15)
which shows that the velocity distribution is that of an irrotational vortex for which the
tangential velocity is inversely proportional to r. As discussed in Chapter 5, Section 3,
this is the only example in which the viscous solution is completely irrotational. Shear
stresses do exist in this flow, but there is no net viscous force at a point. The shear
stress at any point is given by
305
6. Steady Flow between Concentric Cylinders
Figure 9.7 Rotation of a solid cylinder of radius R in an infinite body of viscous fluid. The shape of the
free surface is also indicated. The flow field is viscous but irrotational.
∂ uθ 1 ∂ur
τrθ = µ r
+
,
∂r r
r ∂θ
which, for the present case, reduces to
τrθ = −
2µ R 2
.
r2
The forcing agent performs work on the fluid at the rate
2π Ruθ τrθ .
It is easy to show that this rate of work equals the integral of the viscous dissipation
over the flow field (Exercise 4).
Flow Inside a Rotating Cylinder
Consider the steady rotation of a cylindrical tank containing a viscous fluid. The
radius of the cylinder is R, and the angular velocity of rotation is (Figure 9.8). The
flow would reach a steady state after the initial transients have decayed. The steady
velocity distribution for this case can be found from equation (9.14) by substituting
1 = 0, R1 = 0, 2 = , and R2 = R. We get
uθ =
r,
(9.16)
306
Laminar Flow
Figure 9.8
indicated.
Steady rotation of a tank containing viscous fluid. The shape of the free surface is also
which shows that the tangential velocity is directly proportional to the radius, so that
the fluid elements move as in a rigid solid. This flow was discussed in greater detail
in Chapter 5, Section 3.
7. Impulsively Started Plate: Similarity Solutions
So far, we have considered steady flows with parallel streamlines, both straight and
circular. The nonlinear terms dropped out and the velocity became a function of one
spatial coordinate only. In the transient counterparts of these problems in which the
flow is impulsively started from rest, the flow depends on a spatial coordinate and
time. For these problems, exact solutions still exist because the nonlinear advection
terms drop out again. One of these transient problems is given as Exercise 6. However,
instead of considering the transient phase of all the problems already treated in the
preceding sections, we shall consider several simpler and physically more revealing
unsteady flow problems in this and the next three sections. First, consider the flow
due to the impulsive motion of a flat plate parallel to itself, which is frequently called
Stokes’ first problem. (The problem is sometimes unfairly associated with the name of
Rayleigh, who used Stokes’ solution to predict the thickness of a developing boundary
layer on a semi-infinite plate.)
Formulation of a Problem in Similarity Variables
Consider an infinite flat plate along y = 0, surrounded by fluid (with constant ρ
and µ) for y > 0. The plate is impulsively given a velocity U at t = 0 (Figure 9.9).
Since the resulting flow is invariant in the x direction, the continuity equation
∂u/∂x + ∂v/∂y = 0 requires ∂v/∂y = 0. It follows that v = 0 everywhere because
307
7. Impulsively Started Plate: Similarity Solutions
Figure 9.9
Laminar flow due to an impulsively started flat plate.
it is zero at y = 0. If the pressures at x = ±∞ are maintained at the same level,
we can show that the pressure gradients are zero everywhere as follows. The x- and
y-momentum equations are
ρ
∂u
∂p
∂ 2u
=−
+µ 2,
∂t
∂x
∂y
∂p
0=− .
∂y
The y-momentum equation shows that p can only be a function of x and t. This can
be consistent with the x-momentum equation, in which the first and the last terms
can only be functions of y and t only if ∂p/∂x is independent of x. Maintenance
of identical pressures at x = ±∞ therefore requires that ∂p/∂x = 0. Alternatively,
this can be established by observing that for an infinite plate the problem must be
invariant under translation of coordinates by any finite constant in x.
The governing equation is therefore
∂u
∂ 2u
= ν 2,
∂t
∂y
(9.17)
subject to
u(y, 0) = 0
u(0, t) = U
u(∞, t) = 0
[initial condition],
[surface condition],
[far field condition].
(9.18)
(9.19)
(9.20)
308
Laminar Flow
The problem is well posed, because equations (9.19) and (9.20) are conditions at two
values of y, and equation (9.18) is a condition at one value of t; this is consistent with
equation (9.17), which involves a first derivative in t and a second derivative in y.
The partial differential equation (9.17) can be transformed into an ordinary
differential equation from dimensional considerations alone. Its real reason is the
absence of scales for y and t as discussed on page 287. Let us write the solution as a
functional relation
u = φ(U, y, t, ν).
(9.21)
An examination of the equation set (9.17)–(9.20) shows that the parameter U appears
only in the surface condition (9.19). This dependence on U can be eliminated from
the problem by regarding u/U as the dependent variable, for then the equation set
(9.17)–(9.20) can be written as
∂ 2 u′
∂u′
=ν 2,
∂t
∂y
′
u (y, 0) = 0,
u′ (0, t) = 1,
u′ (∞, t) = 0,
where u′ ≡ u/U . The preceding set is independent of U and must have a solution of
the form
u
= f (y, t, ν).
(9.22)
U
Because the left-hand side of equation (9.22) is dimensionless, the right-hand side
can only be a dimensionless function
of y, t, and ν. The only nondimensional variable
√
formed from y, t, and ν is y/ νt, so that equation (9.22) must be of the form
u
=F
U
y
√
νt
.
(9.23)
√
Any function of y/ vt would be dimensionless and could be used as the new independent variable. Why have we chosen to write it this way rather than νt/y 2 or some
other equivalent form? We have done so because we want to solve for a velocity profile
as a function of distance from the plate. By thinking of the solution to this problem in
this way, our new dimensionless similarity variable will feature y in the numerator to
the first power. We could have obtained equation (9.23) by applying Buckingham’s pi
theorem discussed in Chapter 8, Section 4. There are four variables in equation (9.22),
and two basic dimensions are involved, namely, length and time. Two dimensionless
variables can therefore be formed, and they are shown in equation (9.23).
We write equation (9.23) in the form
u
= F (η),
U
(9.24)
309
7. Impulsively Started Plate: Similarity Solutions
where η is the nondimensional distance given by
y
η≡ √ .
2 νt
(9.25)
We see that the absence of scales for length and time resulted in a reduction of the
dimensionality of the space required for the solution (from 2 to 1). The factor of
2 has been introduced in the definition of η for eventual algebraic simplification.
The equation set (9.17)–(9.20) can now be written in terms of η and F (η). From
equations (9.24) and (9.25), we obtain
∂u
∂η
∂F
y
UF ′ η
=U
= UF′
= −UF ′ √ 3/2 = −
,
∂t
∂t
∂t
2t
4 νt
∂u
∂F
∂η
1
=U
= UF ′
= UF ′ √ ,
∂y
∂y
∂y
2 νt
∂ 2u
U ′′
∂η
U
=
F .
= √ F ′′
2
∂y
4νt
∂y
2 νt
Here, a prime on F denotes derivative with respect to η. With these substitutions,
equation (9.17) reduces to the ordinary differential equation
−2ηF ′ = F ′′ .
(9.26)
The boundary conditions (9.18)–(9.20) reduce to
F (∞) = 0,
F (0) = 1.
(9.27)
(9.28)
Note that both (9.18) and (9.20) reduce to the same condition F (∞) = 0. This is
expected because the original equation (9.17) was a partial differential equation and
needed two conditions in y and one condition in t. In contrast, (9.26) is a second-order
ordinary differential equation and needs only two boundary conditions.
Similarity Solution
Equation (9.26) can be integrated as follows:
dF ′
= −2η dη.
F′
Integrating once, we obtain
ln F ′ = −η2 + const.
which can be written as
dF
2
= A e−η ,
dη
310
Laminar Flow
where A is a constant of integration. Integrating again,
η
2
e−η dη + B.
F (η) = A
(9.29)
0
Condition (9.28) gives
F (0) = 1 = A
0
0
from which B = 1. Condition (9.27) gives
F (∞) = 0 = A
∞
e
−η2
0
2
e−η dη + B,
√
A π
+ 1,
dη + 1 =
2
(where we
√ have used the result of a standard definite integral), from which
A = −2/ π . Solution (9.29) then becomes
η
2
2
F =1− √
e−η dη.
(9.30)
π 0
The function
2
erf(η) ≡ √
π
η
2
e−η dη,
0
is called the “error function” and is tabulated in mathematical handbooks. Solution
(9.30) can then be written as
y
u
.
(9.31)
= 1 − erf √
U
2 νt
It is apparent that the solutions at different times all collapse into a single curve of
u/U vs η, shown in Figure 9.10.
The nature of the variation of u/U with y for various values of t is sketched in
Figure 9.9. The solution clearly has a diffusive nature. At t = 0, a vortex sheet (that
is, a velocity discontinuity) is created at the plate surface. The initial vorticity is in the
form of a delta function, which is infinite at the plate surface and zero elsewhere. It can
∞
be shown that the integral 0 ω dy is independent of time (see the following section
for a demonstration), so that no new vorticity is generated after the initial time. The
initial vorticity is simply diffused outward, resulting in an increase in the width of
flow. The situation is analogous to a heat conduction problem in a semi-infinite solid
extending from y = 0 to y = ∞. Initially, the solid has a uniform temperature, and at
t = 0 the face y = 0 is suddenly brought to a different temperature. The temperature
distribution for this problem is given by an equation similar to equation (9.31).
We may arbitrarily define the thickness of the diffusive layer as the distance at
which u falls to 5% of U . From Figure 9.10, u/U = 0.05 corresponds to η = 1.38.
311
7. Impulsively Started Plate: Similarity Solutions
Figure 9.10 Similarity solution of laminar flow due to an impulsively started flat plate.
Therefore, in time t the diffusive effects propagate to a distance of order
√
δ ∼ 2.76 νt
(9.32)
√
which increases as t. Obviously, the factor of 2.76 in the preceding is somewhat
arbitrary and can be changed by choosing a different ratio of u/U as the definition
for the edge of the diffusive layer.
The present problem illustrates an important class of fluid mechanical problems
that have similarity solutions. Because of the absence of suitable scales to render
the independent variables dimensionless, the only possibility was a combination of
variables that resulted in a reduction of independent variables (dimensionality of the
space) required to describe the problem. In this case the reduction was from two (y, t)
to one (η) so that the formulation reduced from a partial differential equation in y, t
to an ordinary differential equation in η.
The solutions at different times are self-similar in the sense that they all collapse
into a single curve if the velocity
is scaled by U and y is scaled by the thickness of the
√
layer taken to be δ(t) = 2 νt. Similarity solutions exist in situations in which there
is no natural scale in the direction of similarity. In the present problem, solutions at
different t and y are similar because no length or time scales are imposed through
the boundary conditions. Similarity would be violated if, for example, the boundary
conditions are changed after a certain time t1 , which introduces a time scale into the
problem. Likewise, if the flow was bounded above by a parallel plate at y = b, there
could be no similarity solution.
312
Laminar Flow
An Alternative Method of Deducing the Form of η
Instead of arriving at the form of η from dimensional considerations, it could be
derived by a different method as illustrated in the following. Denoting the thickness
of the flow by δ(t), we assume similarity solutions in the form
u
= F (η),
U
y
η=
.
δ(t)
(9.33)
∂ 2F
∂η
= νU 2 .
∂t
∂y
(9.34)
Then equation (9.17) becomes
UF′
The derivatives in equation (9.34) are computed from equation (9.33):
∂η
∂t
∂η
∂y
∂F
∂y
∂ 2F
∂y 2
=−
=
y dδ
η dδ
=−
,
δ dt
δ 2 dt
1
,
δ
∂η
F′
=
,
∂y
δ
1 ∂F ′
F ′′
=
= 2.
δ ∂y
δ
= F′
Substitution into equation (9.34) and cancellation of factors give
δ dδ
−
ηF ′ = F ′′ .
ν dt
Since the right-hand side can only be an explicit function of η, the coefficient in
parentheses on the left-hand side must be independent of t. This requires
δ dδ
= const. = 2,
ν dt
for example.
√
Integration gives δ 2 √
= 4νt, so that the flow thickness is δ = 2 νt. Equation (9.33)
then gives η = y/(2 νt), which agrees with our previous finding.
Method of Laplace Transform
Finally, we shall illustrate the method of Laplace transform for solving the problem. Let û(y, s) be the Laplace transform of u(y, t). Taking the transform of equation (9.17), we obtain
d 2 û
(9.35)
s û = ν 2 ,
dy
313
8. Diffusion of a Vortex Sheet
where the initial condition (9.18) of zero velocity has been used. The transform of
the boundary conditions (9.19) and (9.20) are
U
,
s
û(∞, s) = 0.
û(0, s) =
(9.36)
(9.37)
Equation (9.35) has the general solution
û = A ey
√
s/ν
+ B e−y
√
s/ν
,
where the constants A(s) and B(s) are to be determined from the boundary conditions.
The condition (9.37) requires that A = 0, while equation (9.36) requires that B =
U/s. We then have
√
U
û = e−y s/ν .
s
The inverse transform of the preceding equation can be found in any mathematical
handbook and is given by equation (9.31).
We have discussed this problem in detail because it illustrates the basic diffusive
nature of viscous flows and also the mathematical techniques involved in finding
similarity solutions. Several other problems of this kind are discussed in the following
sections, but the discussions shall be somewhat more brief.
8. Diffusion of a Vortex Sheet
Consider the case in which the initial velocity field is in the form of a vortex sheet
with u = U for y > 0 and u = −U for y < 0. We want to investigate how the vortex
sheet decays by viscous diffusion. The governing equation is
∂ 2u
∂u
= ν 2,
∂t
∂y
subject to
u(y, 0) = U sgn(y),
u(∞, t) = U,
u(−∞, t) = −U,
where sgn(y) is the “sign function,” defined as 1 for positive y and −1 for negative
y. As in the previous section, the parameter U can be eliminated from the governing
set by regarding u/U as the dependent variable. Then u/U must be a function of
(y, t, ν), and a dimensional analysis reveals that there must exist a similarity solution
in the form
314
Laminar Flow
u
= F (η),
U
y
η= √ .
2 νt
The detailed arguments for the existence of a solution in this form are given
in the preceding section. Substitution of the similarity form into the governing set
transforms it into the ordinary differential equation
F ′′ = −2ηF ′ ,
F (+∞) = 1,
F (−∞) = −1,
whose solution is
F (η) = erf(η).
The velocity distribution is therefore
u = U erf
y
.
√
2 νt
(9.38)
A plot of the velocity distribution is shown in Figure 9.11. If we define the width of
the transition layer as the distance between the points where u = ±0.95U , then the
Figure 9.11 Viscous decay of a vortex sheet. The right panel shows the nondimensional solution and the
left panel indicates the vorticity distribution at two times.
315
9. Decay of a Line Vortex
corresponding
value of η is ± 1.38 and consequently the width of the transition layer
√
is 5.52 νt.
It is clear that the flow is essentially identical to that due to the impulsive start
of a flat plate discussed in the preceding section. In fact, each half of Figure 9.11
is identical to Figure 9.10 (within an additive constant of ±1). In both problems
the initial delta-function-like vorticity is diffused away. In the present problem the
magnitude of vorticity at any time is
ω=
∂u
U
2
e−y /4νt .
=√
∂y
π νt
(9.39)
This is a Gaussian distribution, √
whose width increases with time as
maximum value decreases as 1/ t. The total amount of vorticity is
∞
−∞
√
ω dy = 2 νt
∞
2U
ω dη = √
π
−∞
∞
−∞
√
t, while the
2
e−η dη = 2U,
which is independent of time, and equals the y-integral of the initial (delta-functionlike) vorticity.
9. Decay of a Line Vortex
In Section 6 it was shown that when a solid cylinder of radius R is rotated at angular speed
in a viscous fluid, the resulting motion is irrotational with a velocity
distribution uθ = R 2 /r. The velocity distribution can be written as
uθ =
Ŵ
,
2π r
where Ŵ = 2π R 2 is the circulation along any path surrounding the cylinder. Suppose the radius of the cylinder goes to zero while its angular velocity correspondingly
increases in such a way that the product Ŵ = 2π R 2 is unchanged. In the limit we
obtain a line vortex of circulation Ŵ, which has an infinite velocity discontinuity at
the origin.
Now suppose that the limiting (infinitely thin and fast) cylinder suddenly stops
rotating at t = 0, thereby reducing the velocity at the origin to zero impulsively. Then
the fluid would gradually slow down from the initial distribution because of viscous
diffusion from the region near the origin. The flow can therefore be regarded as that of
the viscous decay of a line vortex, for which all the vorticity is initially concentrated
at the origin. The problem is the circular analog of the decay of a plane vortex sheet
discussed in the preceding section.
Employing cylindrical coordinates, the governing equation is
∂uθ
∂
=ν
∂t
∂r
1 ∂
(ruθ ) ,
r ∂r
(9.40)
316
Laminar Flow
subject to
uθ (r, 0) = Ŵ/2π r,
uθ (0, t) = 0,
uθ (r → ∞, t) = Ŵ/2π r.
(9.41)
(9.42)
(9.43)
We expect similarity solutions here because there are no natural scales for r and t
introduced from the boundary conditions. Conditions (9.41) and (9.43) show that the
dependence of the solution on the parameter Ŵ/2π r can be eliminated by defining a
nondimensional velocity
uθ
,
(9.44)
u′ ≡
Ŵ/2π r
which must have a dependence of the form
u′ = f (r, t, ν).
As the left-hand side of the preceding equation is nondimensional, the right-hand side
must be a nondimensional function of r, t, and ν. A dimensional analysis
quickly
√
shows that the only nondimensional group formed from these is r/ νt. Therefore,
the problem must have a similarity solution of the form
u′ = F (η),
(9.45)
r2
.
4νt
√
(Note that we could have defined η = r/2 νt as in the previous problems, but the
algebra is slightly simpler if we define it as in equation (9.45).) Substitution of the
similarity solution (9.45) into the governing set (9.40)–(9.43) gives
η =
F ′′ + F ′ = 0,
subject to
F (∞) = 1,
F (0) = 0.
The solution is
F = 1 − e−η .
The dimensional velocity distribution is therefore
uθ =
Ŵ
2
[1 − e−r /4νt ].
2π r
(9.46)
A sketch of the velocity√distribution for various values of t is given in Figure 9.12.
Near the center (r ≪ 2 √νt) the flow has the form of a rigid-body rotation, while in
the outer region (r ≫ 2 νt) the motion has the form of an irrotational vortex.
317
10. Flow Due to an Oscillating Plate
Figure 9.12 Viscous decay of a line vortex showing the tangential velocity at different times.
The foregoing discussion applies to the decay of a line vortex. Consider now
the case where a line vortex is suddenly introduced into a fluid at rest. This can be
visualized as the impulsive start of an infinitely thin and fast cylinder. It is easy to
show that the velocity distribution is (Exercise 5)
uθ =
Ŵ −r 2 /4νt
e
,
2π r
(9.47)
which should be compared to equation (9.46). The analogous problem in heat conduction is the sudden introduction of an infinitely thin and hot cylinder (containing a
finite amount of heat) into a liquid having a different temperature.
10. Flow Due to an Oscillating Plate
The unsteady parallel flows discussed in the three preceding sections had similarity
solutions, because there were no natural scales in space and time. We now discuss
an unsteady parallel flow that does not have a similarity solution because of the
existence of a natural time scale. Consider an infinite flat plate that executes sinusoidal
oscillations parallel to itself. (This is sometimes called Stokes’ second problem.) Only
the steady periodic solution after the starting transients have died will be considered;
thus there are no initial conditions to satisfy. The governing equation is
∂u
∂ 2u
= ν 2,
∂t
∂y
(9.48)
318
Laminar Flow
subject to
u(0, t) = U cos ωt,
(9.49)
u(∞, t) = bounded.
(9.50)
In the steady state, the flow variables must have a periodicity equal to the periodicity
of the boundary motion. Consequently, we use a separable solution of the form
u = eiωt f (y),
(9.51)
where what is meant is the real part of the right-hand side. (Such a complex form
of representation is discussed in Chapter 7, Section 15.) Here, f (y) is complex,
thus u(y, t) is allowed to have a phase difference with the wall velocity U cos ωt.
Substitution of equation (9.51) into the governing equation (9.48) gives
iωf = ν
d 2f
.
dy 2
(9.52)
This is an equation with constant coefficients and must have exponential
√ soluiω/ν =
tions. Substitution
of
a
solution
of
the
form
f
=
exp(ky)
gives
k
=
√
±(i + 1) ω/2ν, where the two square roots of i have been used. Consequently,
the solution of equation (9.52) is
f (y) = A e−(1+i)y
√
ω/2ν
+ B e(1+i)y
√
ω/2ν
.
(9.53)
The condition (9.50), which requires that the solution must remain bounded at y = ∞,
needs B = 0. The solution (9.51) then becomes
u = A eiωt e−(1+i)y
√
ω/2ν
.
(9.54)
The surface boundary condition (9.49) now gives A = U . Taking the real part of
equation (9.54), we finally obtain the velocity distribution for the problem:
√
ω
u = U e−y ω/2ν cos ωt − y
.
(9.55)
2ν
The cosine term in equation (9.55) represents a signal propagating in the direction
of y, while the exponential term represents a decay in y. The flow therefore resembles a damped wave (Figure 9.13). However, this is a diffusion problem and not a
wave-propagation problem because there are no restoring forces involved here. The
apparent
√ propagation is merely a result of the√oscillating boundary condition. For
y = 4 ν/ω, the amplitude of u is U exp(−4/ 2) = 0.06U , which means that the
influence of the wall is confined within a distance of order
δ ∼ 4 ν/ω,
(9.56)
which decreases with frequency.
319
10. Flow Due to an Oscillating Plate
Figure 9.13 Velocity distribution in laminar flow near an oscillating plate. The distributions at ωt = 0,
√
π/2, π, and 3π/2 are shown. The diffusive distance is of order δ = 4 ν/ω.
Note that the solution (9.55) cannot be represented by a single curve in terms of
the nondimensional variables. This is expected because the frequency of the boundary motion introduces a natural time scale 1/ω into the problem, thereby violating
the requirements of self-similarity. There are two parameters in the governing set
(9.48)–(9.50), namely, U and ω. The parameter U can be eliminated by regarding
u/U as the dependent variable. Thus the solution must have a form
u
= f (y, t, ω, ν).
U
(9.57)
As there are five variables and two dimensions involved, it follows that there must
be three dimensionless
variables. A dimensional analysis of equation (9.57) gives
√
u/U , ωt, and y ω/ν as the three nondimensional variables as in equation (9.55).
Self-similar solutions exist only when there is an absence of such naturally occurring
scales requiring a reduction in the dimensionality of the space.
An interesting
point is that the oscillating plate has a constant diffusion dis√
tance δ = 4 ν/ω that is in contrast to the case of the impulsively started plate
in which the diffusion distance increases with time. This can be understood from
the governing equation (9.48). In the problem of sudden acceleration of a plate,
∂ 2 u/∂y 2 is positive for all y (see Figure 9.10), which results in a positive ∂u/∂t
everywhere. The monotonic acceleration signifies that momentum is constantly
diffused outward, which results in an ever-increasing width of flow. In contrast,
in the case of an oscillating plate, ∂ 2 u/∂y 2 (and therefore ∂u/∂t) constantly
320
Laminar Flow
changes sign in y and t. Therefore, momentum cannot diffuse outward monotonically,
which results in a constant width of flow.
The analogous problem in heat conduction is that of a semi-infinite solid, the
surface of which is subjected to a periodic fluctuation of temperature. The resulting
solution, analogous to equation (9.55), has been used to estimate the effective “eddy”
diffusivity in the upper layer of the ocean from measurements of the phase difference
(that is, the time lag between maxima) between the temperature fluctuations at two
depths, generated by the diurnal cycle of solar heating.
11. High and Low Reynolds Number Flows
Many physical problems can be described by the behavior of a system when a certain
parameter is either very small or very large. Consider the problem of steady flow
around an object described by
ρu • ∇u = −∇p + µ∇ 2 u.
(9.58)
First, assume that the viscosity is small. Then the dominant balance in the flow is
between the pressure and inertia forces, showing that pressure changes are of order
ρU 2 . Consequently, we nondimensionalize the governing equation (9.58) by scaling
u by the free-stream velocity U , pressure by ρU 2 , and distance by a representative
length L of the body. Substituting the nondimensional variables (denoted by primes)
x′ =
x
L
u′ =
u
U
p′ =
p − p∞
,
ρU 2
(9.59)
the equation of motion (9.58) becomes
u′ • ∇u′ = −∇p′ +
1 2 ′
∇ u,
Re
(9.60)
where Re = U l/ν is the Reynolds number. For high Reynolds number flows, equation (9.60) is solved by treating 1/Re as a small parameter. As a first approximation,
we may set 1/Re to zero everywhere in the flow, thus reducing equation (9.60) to
the inviscid Euler equation. However, this omission of viscous terms cannot be valid
near the body because the inviscid flow cannot satisfy the no-slip condition at the
body surface. Viscous forces do become important near the body because of the high
shear in a layer near the body surface. The scaling (9.59), which assumes that velocity gradients are proportional to U/L, is invalid in the boundary layer near the solid
surface. We say that there is a region of nonuniformity near the body at which point
a perturbation expansion in terms of the small parameter 1/Re becomes singular.
The proper scaling in the boundary layer and the procedure of solving high Reynolds
number flows will be discussed in Chapter 10.
Now consider flows in the opposite limit of very low Reynolds numbers, that is,
ℜ → 0. It is clear that low Reynolds number flows will have negligible inertia forces
and therefore the viscous and pressure forces should be in approximate balance.
321
11. High and Low Reynolds Number Flows
For the governing equations to display this fact, we should have a small parameter
multiplying the inertia forces in this case. This can be accomplished if the variables are
nondimensionalized properly to take into account the low Reynolds number nature of
the flow. Obviously, the scaling (9.59), which leads to equation (9.60), is inappropriate
in this case. For if equation (9.60) were multiplied by Re, then the small parameter
Re would appear in front of not only the inertia force term but also the pressure force
term, and the governing equation would reduce to 0 = µ∇ 2 u as Re → 0, which is
not the balance for low Reynolds number flows. The source of the inadequacy of the
nondimensionalization (9.59) for low Reynolds number flows is that the pressure is
not of order ρU 2 in this case. As we noted in Chapter 8, for these external flows,
pressure is a passive variable and it must be normalized by the dominant effect(s),
which here are viscous forces. The purpose of scaling is to obtain nondimensional
variables that are of order one, so that pressure should be scaled by ρU 2 only in high
Reynolds number flows in which the pressure forces are of the order of the inertia
forces. In contrast, in a low Reynolds number flow the pressure forces are of the
order of the viscous forces. For ∇p to balance µ∇ 2 u in equation (9.58), the pressure
changes must have a magnitude of the order
p ∼ Lµ∇ 2 u ∼ µU/L.
Thus the proper nondimensionalization for low Reynolds number flows is
x′ =
x
L
u′ =
u
U
p′ =
p − p∞
.
µU/L
(9.61)
The variations of the nondimensional variables u′ and p ′ in the flow field are now
of order one. The pressure scaling also shows that p is proportional to µ in a low
Reynolds number flow. A highly viscous oil is used in the bearing of a rotating shaft
because the high pressure developed in the oil film of the bearing “lifts” the shaft and
prevents metal-to-metal contact.
Substitution of equation (9.61) into (9.58) gives the nondimensional equation
Re u′ • ∇u′ = −∇p ′ + ∇ 2 u′ .
(9.62)
In the limit Re → 0, equation (9.62) becomes the linear equation
∇p = µ∇ 2 u,
(9.63)
where the variables have been converted back to their dimensional form.
Flows at Re ≪ 1 are called creeping motions. They can be due to small velocity,
large viscosity, or (most commonly) the small size of the body. Examples of such
flows are the motion of a thin film of oil in the bearing of a shaft, settling of sediment
particles near the ocean bottom, and the fall of moisture drops in the atmosphere. In
the next section, we shall examine the creeping flow around a sphere.
Summary: The purpose of scaling is to generate nondimensional variables that
are of order one in the flow field (except in singular regions or boundary layers).
322
Laminar Flow
The proper scales depend on the nature of the flow and are obtained by equating
the terms that are most important in the flow field. For a high Reynolds number
flow, the dominant terms are the inertia and pressure forces. This suggests the scaling
(9.59), resulting in the nondimensional equation (9.60) in which the small parameter
multiplies the subdominant term (except in boundary layers). In contrast, the dominant
terms for a low Reynolds number flow are the pressure and viscous forces. This
suggests the scaling (9.61), resulting in the nondimensional equation (9.62) in which
the small parameter multiplies the subdominant term.
12. Creeping Flow around a Sphere
A solution for the creeping flow around a sphere was first given by Stokes in 1851.
Consider the low Reynolds number flow around a sphere of radius a placed in a uniform stream U (Figure 9.14). The problem is axisymmetric, that is, the flow patterns
are identical in all planes parallel to U and passing through the center of the sphere.
Since Re → 0, as a first approximation we may neglect the inertia forces altogether
and solve the equation
∇p = µ∇ 2 u∗ .
Figure 9.14 Creeping flow over a sphere. The upper panel shows the viscous stress components at the
surface. The lower panel shows the pressure distribution in an axial (ϕ = const.) plane.
323
12. Creeping Flow around a Sphere
We can form a vorticity equation by taking the curl of the preceding equation,
obtaining
0 = ∇ 2 ω∗ .
Here, we have used the fact that the curl of a gradient is zero, and that the order of the
operators curl and ∇ 2 can be interchanged. (The reader may verify this using indicial
notation.) The only component of vorticity in this axisymmetric problem is ωϕ , the
component perpendicular to ϕ = const. planes in Figure 9.14, and is given by
ωϕ =
1 ∂(ruθ ) ∂ur
−
.
r
∂r
∂θ
In axisymmetric flows we can define a streamfunction ψ given in Section 6.18. In
spherical coordinates, it is defined as u = −∇ϕ × ∇ψ, (6.74) so
ur ≡
1
r 2 sin θ
∂ψ
∂θ
uθ ≡ −
1 ∂ψ
.
r sin θ ∂r
In terms of the streamfunction, the vorticity becomes
ωϕ = −
1
1 ∂
1 ∂ 2ψ
1 ∂ψ
+
.
r sin θ ∂r 2
r 2 ∂θ sin θ ∂θ
The governing equation is
∇ 2 ωϕ = 0 ∗ .
Combining the last two equations, we obtain
sin θ ∂
∂2
+ 2
2
∂r
r ∂θ
1 ∂
sin θ ∂θ
2
ψ = 0.
(9.64)
The boundary conditions on the preceding equation are
ψ(a, θ) = 0
∂ψ/∂r(a, θ) = 0
ψ(∞, θ) =
2
2
1
2 U r sin θ
[ur = 0 at surface],
[uθ = 0 at surface],
[uniform flow at ∞].
(9.65)
(9.66)
(9.67)
The last condition follows from the fact that the stream function for a uniform flow
is (1/2)U r 2 sin2 θ in spherical coordinates (see equation (6.76)).
* In spherical polar coordinates, the operator in the footnoted equations is actually −∇ × ∇ ×
(−curl curl ), which is different from the Laplace operator defined in Appendix B. Eq. (9.64) is the
square of the operator, and not the biharmonic.
324
Laminar Flow
The upstream condition (9.67) suggests a separable solution of the form
ψ = f (r) sin2 θ.
Substitution of this into the governing equation (9.64) gives
f iv −
4f ′′
8f ′
8f
+
− 4 = 0,
2
3
r
r
r
whose solution is
f = Ar 4 + Br 2 + Cr +
D
.
r
The upstream boundary condition (9.67) requires that A = 0 and B = U/2. The
surface boundary conditions then give C = −3 U a/4 and D = U a 3 /4. The solution
then reduces to
1 3a
a3
(9.68)
−
+ 3 .
ψ = U r 2 sin2 θ
2
4r
4r
The velocity components can then be found as
ur =
1 ∂ψ
3a
a3
,
=
U
cos
θ
1
−
+
2r
r 2 sin θ ∂θ
2r 3
(9.69)
1 ∂ψ
3a
a3
= −U sin θ 1 −
− 3 .
uθ = −
r sin θ ∂r
4r
4r
The pressure can be found by integrating the momentum equation ∇p = µ∇ 2 u. The
result is
3aµU cos θ
p=−
+ p∞
(9.70)
2r 2
The pressure distribution is sketched in Figure 9.14. The pressure is maximum at
the forward stagnation point where it equals 3µU/2a, and it is minimum at the rear
stagnation point where it equals −3µU/2a.
Let us determine the drag force D on the sphere. One way to do this is to apply
the principle of mechanical energy balance over the entire flow field given in equation (4.63). This requires
DU =
φ dV ,
which states that the work done by the sphere equals the viscous dissipation over the
entire flow; here, φ is the viscous dissipation per unit volume. A more direct way to
determine the drag is to integrate the stress over the surface of the sphere. The force
per unit area normal to a surface, whose outward unit normal is n is
Fi = τij nj = [−pδij + σij ]nj = −pni + σij nj ,
325
12. Creeping Flow around a Sphere
where τij is the total stress tensor, and σij is the viscous stress tensor. The component
of the drag force per unit area in the direction of the uniform stream is therefore
[−p cos θ + σrr cos θ − σrθ sin θ]r=a ,
(9.71)
which can be understood from Figure 9.14. The viscous stress components are
σrr = 2µ
σrθ
3a 3
3a
∂ur
−
,
= 2µU cos θ
∂r
2r 2
2r 4
(9.72)
∂ uθ 1 ∂ur
3µU a 3
+
=µ r
=−
sin θ,
∂r r
r ∂θ
2r 4
so that equation (9.71) becomes
3µU
3µU
3µU
cos2 θ + 0 +
sin2 θ =
.
2a
2a
2a
The drag force is obtained by multiplying this by the surface area 4π a 2 of the sphere,
which gives
D = 6π µaU,
(9.73)
of which one-third is pressure drag and two-thirds is skin friction drag. It follows
that the resistance in a creeping flow is proportional to the velocity; this is known as
Stokes’ law of resistance.
In a well-known experiment to measure the charge of an electron, Millikan used
equation (9.73) to estimate the radius of an oil droplet falling through air. Suppose
ρ ′ is the density of a spherical falling particle and ρ is the density of the surrounding
fluid. Then the effective weight of the sphere is 4π a 3 g(ρ ′ − ρ)/3, which is the weight
of the sphere minus the weight of the displaced fluid. The falling body is said to reach
the “terminal velocity” when it no longer accelerates, at which point the viscous drag
equals the effective weight. Then
3
′
4
3 πa g(ρ
− ρ) = 6π µaU,
from which the radius a can be estimated.
Millikan was able to deduce the charge on an electron making use of Stokes’ drag
formula by the following experiment. Two horizontal parallel plates can be charged
by a battery (see Fig. 9.15). Oil is sprayed through a very fine hole in the upper plate
and develops static charge (+) by losing a few (n) electrons in passing through the
small hole. If the plates are charged, then an electric force neE will act on each of
the drops. Now n is not known but E = −Vb /L, where Vb is the battery voltage
and L is the gap between the plates, provided that the charge density in the gap is
very low. With the plates uncharged, measurement of the downward terminal velocity
allowed the radius of a drop to be calculated assuming that the viscosity of the drop
is much larger than the viscosity of the air. The switch is thrown to charge the upper
326
Laminar Flow
Figure 9.15 Millikan oil drop experiment.
plate negatively. The same droplet then reverses direction and is forced upwards. It
quickly achieves its terminal velocity Uu by virtue of the balance of upward forces
(electric + buoyancy) and downward forces (weight + drag). This gives
6πµUu a + (4/3)πa 3 g(ρ ′ − ρ) = neE,
where Uu is measured by the observation telescope and the radius of the particle is
now known. The data then allow for the calculation of ne. As n must be an integer,
data from many droplets may be differenced to identify the minimum difference that
must be e, the charge of a single electron.
The drag coefficient, defined as the drag force nondimensionalized by ρU 2 /2
and the projected area πa 2 , is
CD ≡
D
1
2
2
2 ρU π a
=
24
,
Re
(9.74)
where Re = 2aU/ν is the Reynolds number based on the diameter of the sphere.
In Chapter 8, Section 5 it was shown that dimensional considerations alone require
that CD should be inversely proportional to Re for creeping motions. To repeat the
argument, the drag force in a “massless” fluid (that is, Re ≪ 1) can only have the
dependence
D = f (µ, U, a).
The preceding relation involves four variables and the three basic dimensions of mass,
length, and time. Therefore, only one nondimensional parameter, namely, D/µU a,
can be formed. As there is no second nondimensional parameter for it to depend on,
D/µU a must be a constant. This leads to CD ∝ 1/Re.
The flow pattern in a reference frame fixed to the fluid at infinity can be found
by superposing a uniform velocity U to the left. This cancels out the first term in
equation (9.68), giving
a3
3a
2
2
ψ = U r sin θ −
+ 3 ,
4r
4r
which gives the streamline pattern as seen by an observer if the sphere is dragged
in front of him from right to left (Figure 9.16). The pattern is symmetric between
the upstream and the downstream directions, which is a result of the linearity of the
327
13. Nonuniformity of Stokes’ Solution and Oseen’s Improvement
Figure 9.16 Streamlines and velocity distributions in Stokes’ solution of creeping flow due to a moving
sphere. Note the upstream and downstream symmetry, which is a result of complete neglect of nonlinearity.
governing equation (9.63); reversing the direction of the free-stream velocity merely
changes u to −u and p to −p. The flow therefore does not have a “wake” behind the
sphere.
13. Nonuniformity of Stokes’ Solution and
Oseen’s Improvement
The Stokes solution for a sphere is not valid at large distances from the body because
the advective terms are not negligible compared to the viscous terms at these distances.
From equation (9.72), the largest viscous term is of the order
viscous force/volume = stress gradient ∼
µU a
r3
as r → ∞,
while from equation (9.69) the largest inertia force is
inertia force/volume ∼ ρur
ρU 2 a
∂uθ
∼
∂r
r2
as r → ∞.
328
Laminar Flow
Therefore,
inertia force
ρU a r
r
∼
=ℜ
viscous force
µ a
a
as r → ∞.
This shows that the inertia forces are not negligible for distances larger than r/a ∼
1/Re. At sufficiently large distances, no matter how small Re may be, the neglected
terms become arbitrarily large.
Solutions of problems involving a small parameter can be developed in terms
of the perturbation series in which the higher-order terms act as corrections on the
lower-order terms. Perturbation expansions are discussed briefly in the following
chapter. If we regard the Stokes solution as the first term of a series expansion in the
small parameter Re, then the expansion is “nonuniform” because it breaks down at
infinity. If we tried to calculate the next term (to order Re) of the perturbation series,
we would find that the velocity corresponding to the higher-order term becomes
unbounded at infinity.
The situation becomes worse for two-dimensional objects such as the circular
cylinder. In this case, the Stokes balance ∇p = µ∇ 2 u has no solution at all that can
satisfy the uniform flow boundary condition at infinity. From this, Stokes concluded
that steady, slow flows around cylinders cannot exist in nature. It has now been realized
that the nonexistence of a first approximation of the Stokes flow around a cylinder is
due to the singular nature of low Reynolds number flows in which there is a region
of nonuniformity at infinity. The nonexistence of the second approximation for flow
around a sphere is due to the same reason. In a different (and more familiar) class
of singular perturbation problems, the region of nonuniformity is a thin layer (the
“boundary layer”) near the surface of an object. This is the class of flows with Re →
∞, that will be discussed in the next chapter. For these high Reynolds number flows
the small parameter 1/Re multiplies the highest-order derivative in the governing
equations, so that the solution with 1/Re identically set to zero cannot satisfy all
the boundary conditions. In low Reynolds number flows this classic symptom of
the loss of the highest derivative is absent, but it is a singular perturbation problem
nevertheless.
In 1910 Oseen provided an improvement to Stokes’ solution by partly accounting
for the inertia terms at large distances. He made the substitutions
u = U + u′
v = v′
w = w′ ,
where (u′ , v ′ , w′ ) are the Cartesian components of the perturbation velocity, and are
small at large distances. Substituting these, the advection term of the x-momentum
equation becomes
′
′
′
∂u
∂u
∂u′
∂u
′ ∂u
′ ∂u
′ ∂u
+v
+w
=U
+ u
+v
+w
.
u
∂x
∂y
∂z
∂x
∂x
∂y
∂z
Neglecting the quadratic terms, the equation of motion becomes
ρU
∂u′i
∂p
=−
+ µ∇ 2 u′i ,
∂x
∂xi
329
13. Nonuniformity of Stokes’ Solution and Oseen’s Improvement
where u′i represents u′ , v ′ , or w ′ . This is called Oseen’s equation, and the approximation involved is called Oseen’s approximation. In essence, the Oseen approximation
linearizes the advective term u • ∇u by U (∂u/∂x), whereas the Stokes approximation
drops advection altogether. Near the body both approximations have the same order
of accuracy. However, the Oseen approximation is better in the far field where the
velocity is only slightly different than U . The Oseen equations provide a lowest-order
solution that is uniformly valid everywhere in the flow field.
The boundary conditions for a moving sphere are
u′ = v ′ = w′ = 0 at infinity
u′ = −U, v ′ = w′ = 0 at surface.
The solution found by Oseen is
2
Re r
3
r
a
ψ
2
(1 + cos θ ) 1 − exp −
(1 − cos θ) ,
sin θ −
=
+
4r
Re
4 a
U a2
2a 2
(9.75)
where Re = 2aU/ν is the Reynolds number based on diameter. Near the surface
r/a ≈ 1, and a series expansion of the exponential term shows that Oseen’s solution
is identical to the Stokes solution (9.68) to the lowest order. The Oseen approximation
predicts that the drag coefficient is
24
3
1 + Re ,
CD =
Re
16
which should be compared with the Stokes formula (9.74). Experimental results (see
Figure 10.22 in the next chapter) show that the Oseen and the Stokes formulas for
CD are both fairly accurate for Re < 5.
The streamlines corresponding to the Oseen solution (9.75) are shown in
Figure 9.17, where a uniform flow of U is added to the left so as to generate the
pattern of flow due to a sphere moving in front of a stationary observer. It is seen
that the flow is no longer symmetric, but has a wake where the streamlines are closer
together than in the Stokes flow. The velocities in the wake are larger than in front of
the sphere. Relative to the sphere, the flow is slower in the wake than in front of the
sphere.
In 1957, Oseen’s correction to Stokes’ solution was rationalized independently
by Kaplun and Proudman and Pearson in terms of matched asymptotic expansions.
Here, we will obtain only the first-order correction. The full vorticity equation is
∇ × ∇ × ω = Re∇ × (u × ω).
(9.76)
In terms of the Stokes streamfunction ψ, equation (9.64) is generalized to
2 2
1 ∂(ψ, D 2 ψ)
+ 2 D ψLψ ,
D ψ = Re 2
∂(r, µ)
r
r
4
(9.77)
330
Laminar Flow
Figure 9.17 Streamlines and velocity distribution in Oseen’s solution of creeping flow due to a moving
sphere. Note the upstream and downstream asymmetry, which is a result of partial accounting for advection
in the far field.
where ∂(ψ, D 2 ψ)/∂(r, µ) is shorthand notation for the Jacobian determinant with
those four elements, µ = cos θ, and the operators are
L=
µ
1 ∂
∂
+
,
2
r ∂µ
1 − µ ∂r
D2 =
∂2
1 − µ2 ∂ 2
+
.
∂r 2
r 2 ∂µ2
We have seen that the right-hand side of equation (9.76) or (9.77) becomes of the
same order as the left-hand side when Re r/a ∼ 1 or r/a ∼ 1/Re. We will define
the “inner region” as r/a ≪ 1/Re so that Stokes’ solution holds approximately. To
obtain a better approximation in the inner region, we will write
ψ(r, µ; Re) = ψ0 (r, µ) + Re ψ1 (r, µ) + o(Re),
(9.78)
where the second correction “o(Re)” means that it tends to zero faster than Re in
the limit Re → 0. (See Chapter 10, Section 12. Here ψ is made dimensionless by
U a 2 and Re = U a/ν.) Substituting equation (9.78) into (9.77) and taking the limit
Re → 0, we obtain D 4 ψ0 = 0 and recover Stokes’ result
1
1 µ2 − 1
2
ψ0 = −
.
2r − 3r +
2
r
2
(9.79)
331
13. Nonuniformity of Stokes’ Solution and Oseen’s Improvement
Subtracting this, dividing by Re and taking the limit Re → 0, we obtain
D 4 ψ1 =
2
1 ∂(ψ0 , D 2 ψ0 )
+ 2 D 2 ψ0 Lψ0 ,
2
∂(r, µ)
r
r
which reduces to
9
D ψ1 =
4
4
2
3
1
− 3+ 5
r2
r
r
µ(µ2 − 1),
(9.80)
by using equation (9.79). This has the solution
3
1 µ(µ2 − 1)
1
1 µ2 − 1
2
2
+
,
2r − 3r + 1 − + 2
ψ1 = C1 2r − 3r +
r
2
16
r
2
r
(9.81)
where C1 is a constant of integration for the solution to the homogeneous equation
and is to be determined by matching with the outer region solution.
In the outer region rRe = ρ is finite. The lowest-order outer solution must be
uniform flow. Then we write the streamfuntion as
1
1
1 ρ2
2
(ρ,
θ)
+
o
.
sin
θ
+
(ρ, θ; Re) =
1
2 Re2
Re
Re
Substituting in equation (9.77) and taking the limit Re → 0 yields
sin θ ∂
∂
2
+
D2 1 = 0,
D − cos θ
∂ρ
ρ ∂θ
(9.82)
where the operator
D2 = ∂ 2 /∂ρ 2 +
sin θ
ρ2
∂ 1 ∂
∂θ sin θ ∂θ
.
The solution to equation (9.82) is found to be
1 (ρ, θ) = −2C2 (1 + cos θ )[1 − e−ρ(1−cos θ)/2 ],
where the constant of integration C2 is determined by matching in the overlap region
between the inner and outer regions: 1 ≪ r ≪ 1/Re, Re ≪ ρ ≪ 1.
The matching gives C2 = 3/4 and C1 = −3/16. Using this in equation (9.81)
for the inner region solution, the O(Re) correction to the stream function (equation (9.81)) has been obtained, from which the velocity components, shear stress, and
pressure may be derived. Integrating over the surface of the sphere of radius = a, we
obtain the final result for the drag force
D = 6πµU a[1 + 3U a/(8ν)],
which is consistent with Oseen’s result. Higher-order corrections were obtained by
Chester and Breach (1969).
332
Laminar Flow
14. Hele-Shaw Flow
Another low Reynolds number flow has seen wide application in flow visualization
apparatus because of its peculiar and surprising property of reproducing the streamlines of potential flows (that is, infinite Reynolds number flows).
The Hele-Shaw flow is flow about a thin object filling a narrow gap between
two parallel plates. Let the plates be located at x = ±b with Re = Uo b/ν ≪ 1.
Here, U0 is the velocity upstream in the central plane (see Figure 9.18). Now place a
circular cylinder of radius = a and width = 2b between the plates. We will require
b/a = ǫ ≪ 1. The Hele-Shaw limit is Re ≪ ǫ 2 ≪ 1. Imagine flow about a thin coin
with parallel plates bounding the ends of the coin. We are interested in the streamlines
of the flow around the cylinder. The origin of coordinates (R, θ, x) (Appendix B) is
taken at the center of the cylinder.
Consider steady flow with constant density and viscosity in the absence of body
forces. The dimensionless variables are, x ′ = x/b, R ′ = r/a, v ′ = v/Uo , p′ =
(p − p∞ )/(µUo /b), Re = Uo b/ν, ǫ = b/a. Conservation of mass and momentum
then take the following form (primes suppressed):
1 ∂
1 ∂uθ
∂ux
+ǫ
(RuR ) +
= 0.
∂x
R ∂R
R ∂θ
u2θ
∂uR
uθ ∂uR
∂uR
Re ux
+ ǫ uR
+
−
∂x
∂R
R ∂θ
R
2
2
∂ uR
1 ∂ 2 uR
1 ∂uR
uR
2 ∂uθ
∂p
2 ∂ uR
+
+
,
+
ǫ
+
−
−
=−
∂R
R ∂R
R ∂θ
∂x 2
∂R 2
R 2 ∂θ 2
R2
∂uθ
∂uθ
uθ ∂uθ
u R uθ
+ ǫ uR
+
−
Re ux
∂x
∂R
R ∂θ
R
2
1 ∂ 2 uθ
uθ
∂uθ
∂
u
1
2 ∂uR
1 ∂p ∂ 2 uθ
θ
2
+
+
−
+
ǫ
+
−
,
=−
R ∂θ
R ∂R
∂x 2
∂R 2
R 2 ∂θ 2
R 2 ∂θ
R2
x
R
a
Uo
x5b
a
x 5 2b
side view
Figure 9.18 Hele-Shaw flow.
top view
θ
333
14. Hele-Shaw Flow
∂ux
∂ux
uθ ∂ux
Re ux
+ ǫ uR
+
∂x
∂R
R ∂θ
2
1 ∂ 2 ux
1 ∂ux
∂p
2 ∂ ux
+ǫ
+ 2
+
.
=−
∂x
R ∂R
∂R 2
R ∂θ 2
Because Re ≪ ǫ 2 ≪ 1, we take the limit Re → 0 first and drop the convective
acceleration. Next, we take the limit ǫ → 0 to obtain the outer region flow:
∂ux
= O(ǫ) → 0, ux (x = ±1) = 0,
∂x
∂p
∂ 2 uR
+ O(ǫ 2 ),
=
2
∂R
∂x
1 ∂p
∂ 2 uθ
+ O(ǫ 2 ).
=
2
R ∂θ
∂x
so ux = 0 throughout,
With ux = O(ǫ) at most, ∂p/∂x = O(ǫ) at most so p = p(R, θ). Integrating the
momentum equations with respect to x,
1 ∂ 1
∂ 1
uR = −
p(1 − x 2 ) ,
uθ = −
p(1 − x 2 ) ,
∂R 2
R ∂θ 2
where no slip has been satisfied on x = ±1. Thus we can write u = ∇φ for the
two-dimensional field uR , uθ . Here, φ = − 21 p(1 − x 2 ). Now we require that ux =
o(ǫ) so that the first term in the continuity equation is small compared with the
others. Then
1 ∂
1 ∂uθ
(RuR ) +
= o(1) → 0
R ∂R
R ∂θ
as
ǫ→0
Substituting in terms of the velocity potential φ, we have ∇ 2 φ = 0 in R, θ subject to
the boundary conditions:
∂φ
= 0 (no mass flow normal to a solid boundary)
∂R
R → ∞, φ → R cos θ (1 − x 2 )/2 (uniform flow in each x =
constant plane)
R = 1,
The solution is just the potential flow over a circular cylinder (equation (6.35))
1 (1 − x 2 )
φ = R cos θ 1 + 2
,
2
R
where x is just a parameter. Therefore, the streamlines corresponding to this velocity
potential are identical to the potential flow streamlines of equation (6.35). This allows
for the construction of an apparatus to visualize such potential flows by dye injection
between two closely spaced glass plates. The velocity distribution of this flow is
334
Laminar Flow
1
1 − x2
,
uθ = − sin θ 1 + 2
2
R
1
uR = cos θ 1 − 2
R
1 − x2
.
2
As R → 1, uR → 0 but there is a slip velocity uθ → −2 sin θ (1 − x 2 )/2.
As this is a viscous flow, there must exist a thin region near R = 1 where
the slip velocity uθ decreases rapidly to zero to satisfy uθ = 0 on R = 1. This
thin boundary layer is very close to the body surface R = 1. Thus, uR ≈ 0 and
∂p/∂R ≈ 0 throughout the layer. Now p = −R cos θ(1 + 1/R 2 ) so for R ≈ 1,
(1/R)∂p/∂θ ≈ 2 sin θ . In the θ momentum equation, R derivatives become very
large so the dominant balance is
2
1 ∂p
∂ 2 uθ
2 ∂ uθ
= 2 sin θ.
+
ǫ
=
2
2
R ∂θ
∂x
∂R
It is clear from this balance that a stretching by 1/ǫ is appropriate in the boundary
layer: R̂ = (R − 1)/ǫ. In these terms
∂ 2 uθ
∂ 2 uθ
+
= 2 sin θ,
∂x 2
∂ R̂ 2
subject to uθ = 0 on R̂ = 0 and uθ → −2 sin θ(1 − x 2 )/2 as R̂ → ∞ (match with
outer region). The solution to this problem is
uθ (R̂, θ, x) = −(1 − x 2 ) sin θ +
1
∞
An cos(kn x)e−kn R̂ sin θ,
n=0
1
π,
kn = n +
2
1
An =
(1 − x 2 ) cos n +
π x dx.
2
−1
We conclude that Hele-Shaw flow indeed simulates potential flow (inviscid) streamlines except for a very thin boundary layer of the order of the plate separation adjacent
to the body surface.
15. Final Remarks
As in other fields, analytical methods in fluid flow problems are useful in understanding the physics and in making generalizations. However, it is probably fair to say
that most of the analytically tractable problems in ordinary laminar flow have already
been solved, and approximate methods are now necessary for further advancing our
knowledge. One of these approximate techniques is the perturbation method, where
the flow is assumed to deviate slightly from a basic linear state; perturbation methods
are discussed in the following chapter. Another method that is playing an increasingly important role is that of solving the Navier–Stokes equations numerically using
a computer. A proper application of such techniques requires considerable care and
familiarity with various iterative techniques and their limitations. It is hoped that the
reader will have the opportunity to learn numerical methods in a separate study. In
Chapter 11, we will introduce several basic methods of computational fluid dynamics.
335
Exercises
Exercises
1. Consider the laminar flow of a fluid layer falling down a plane inclined at
an angle θ with the horizontal. If h is the thickness of the layer in the fully developed
stage, show that the velocity distribution is
u=
g sin θ 2
(h − y 2 ),
2ν
where the x-axis points along the free surface, and the y-axis points toward the plane.
Show that the volume flow rate per unit width is
Q=
gh3 sin θ
,
3ν
and the frictional stress on the wall is
τ0 = ρgh sin θ.
2. Consider the steady laminar flow through the annular space formed by two
coaxial tubes. The flow is along the axis of the tubes and is maintained by a pressure
gradient dp/dx, where the x direction is taken along the axis of the tubes. Show that
the velocity at any radius r is
1 dp 2
r
b2 − a 2
2
u(r) =
ln
r −a −
,
4µ dx
ln (b/a) a
where a is the radius of the inner tube and b is the radius of the outer tube. Find the
radius at which the maximum velocity is reached, the volume rate of flow, and the
stress distribution.
3. A long vertical cylinder of radius b rotates with angular velocity concentrically outside a smaller stationary cylinder of radius a. The annular space is filled
with fluid of viscosity µ. Show that the steady velocity distribution is
uθ =
r 2 − a 2 b2
.
b2 − a 2 r
Show that the torque exerted on either cylinder, per unit length, equals
4π µ a 2 b2 /(b2 − a 2 ).
4. Consider a solid cylinder of radius R, steadily rotating at angular speed in
an infinite viscous fluid. As shown in Section 6, the steady solution is irrotational:
uθ =
R2
.
r
336
Laminar Flow
Show that the work done by the external agent in maintaining the flow (namely, the
value of 2πRuθ τrθ at r = R) equals the total viscous dissipation rate in the flow field.
5. Suppose a line vortex of circulation Ŵ is suddenly introduced into a fluid at
rest. Show that the solution is
uθ =
Ŵ −r 2 /4νt
.
e
2π r
Sketch the velocity distribution at different times. Calculate and plot the vorticity, and
observe how it diffuses outward.
6. Consider the development from rest of a plane Couette flow. The flow is
bounded by two rigid boundaries at y = 0 and y = h, and the motion is started
from rest by suddenly accelerating the lower plate to a steady velocity U . The upper
plate is held stationary. Notice that similarity solutions cannot exist because of the
appearance of the parameter h. Show that the velocity distribution is given by
∞
y 2U 1
nπy
νt
u(y, t) = U 1 −
−
exp −n2 π 2 2 sin
.
h
π
n
h
h
n=1
Sketch the flow pattern at various times, and observe how the velocity reaches the
linear distribution for large times.
7. Planar Couette flow is generated by placing a viscous fluid between two infinite
parallel plates and moving one plate (say, the upper one) at a velocity U with respect
to the other one. The plates are a distance h apart. Two immiscible viscous liquids are
placed between the plates as shown in the diagram. Solve for the velocity distributions
in the two fluids.
8. Calculate the drag on a spherical droplet of radius r = a, density ρ ′ and
viscosity µ′ moving with velocity U in an infinite fluid of density ρ and viscosity µ.
Assume Re = ρU a/µ ≪ 1. Neglect surface tension.
9. Consider a very low Reynolds number flow over a circular cyclinder of radius
r = a. For r/a = O(1) in the Re = U a/ν → 0 limit, find the equation governing the
streamfunction ψ(r, θ) and solve for ψ with the least singular behavior for large r.
There will be one remaining constant of integration to be determined by asymptotic
matching with the large r solution (which is not part of this problem). Find the domain
of validity of your solution.
Supplemental Reading
10. Consider a sphere of radius r = a rotating with angular velocity ω about a
diameter so that Re = ωa 2 /ν ≪ 1. Use the symmetries in the problem to solve the
mass and momentum equations directly for the azimuthal velocity vϕ (r, θ). Then find
the shear stress and torque on the sphere.
Literature Cited
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.
Lighthill, M. J. (1986). An Informal Introduction to Theoretical Fluid Mechanics, Oxford, England:
Clarendon Press.
Chester, W. and D. R. Breach (with I. Proudman) (1969). “On the flow past a sphere at low Reynolds
number.” J. Fluid Mech. 37: 751–760.
Hele-Shaw, H. S. (1898). “Investigations of the Nature of Surface Resistance of Water and of Stream Line
Motion Under Certain Experimental Conditions,” Trans. Roy. Inst. Naval Arch. 40: 21–46.
Kaplun, S. (1957). “Low Reynolds number flow past a circular cylinder.” J. Math. Mech. 6: 585–603.
Millikan, R. A. (1911). “The isolation of an ion, a precision measurement of its charge, and the correction
of Stokes’ law.” Phys. Rev. 32: 349–397.
Oseen, C. W. (1910). “Über die Stokes’sche Formel, und über eine verwandte Aufgabe in der Hydrodynamik.” Ark Math. Astrom. Fys. 6: No. 29.
Proudman, I. and J. R. A. Pearson (1957). “Expansions at small Reynolds numbers for the flow past a
sphere and a circular cylinder.” J. Fluid Mech. 2: 237–262.
Supplemental Reading
Schlichting, H. (1979). Boundary Layer Theory, New York: McGraw-Hill.
337
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Chapter 10
Boundary Layers and
Related Topics
1. Introduction . . . . . . . . . . . . . . . . . . . . . 340
2. Boundary Layer Approximation. . 340
3. Different Measures of Boundary
Layer Thickness . . . . . . . . . . . . . . . . . 346
The u = 0.99 U Thickness . . . . . . . 346
Displacement Thickness . . . . . . . . . 346
Momentum Thickness . . . . . . . . . . . 347
4. Boundary Layer on a Flat Plate with
a Sink at the Leading Edge:
Closed Form Solution . . . . . . . . . . 348
Axisymmetric Problem . . . . . . . . . . . 350
5. Boundary Layer on a Flat Plate:
Blasius Solution . . . . . . . . . . . . . . . . . 352
Similarity Solution–Alternative
Procedure . . . . . . . . . . . . . . . . . . . . . 353
Matching with External Stream . . 355
Transverse Velocity . . . . . . . . . . . . . . 356
Boundary Layer Thickness . . . . . . . 356
Skin Friction . . . . . . . . . . . . . . . . . . . . 357
Falkner–Skan Solution of the Laminar
Boundary Layer Equations . . . . 358
Breakdown of Laminar Solution . 360
6. von Karman Momentum Integral . 362
7. Effect of Pressure Gradient . . . . . . . 364
8. Separation . . . . . . . . . . . . . . . . . . . . . . 366
9. Description of Flow past a Circular
Cylinder. . . . . . . . . . . . . . . . . . . . . . . . . 368
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50010-1
10.
11.
12.
13.
14.
15.
16.
17.
Low Reynolds Numbers. . . . . . . . .
von Karman Vortex Street . . . . . .
High Reynolds Numbers . . . . . . . .
Description of Flow past
a Sphere . . . . . . . . . . . . . . . . . . . . . . .
Dynamics of Sports Balls . . . . . . .
Cricket Ball Dynamics . . . . . . . . . .
Tennis Ball Dynamics . . . . . . . . . . .
Baseball Dynamics . . . . . . . . . . . . .
Two-Dimensional Jets . . . . . . . . . . .
The Wall Jet . . . . . . . . . . . . . . . . . . . .
Secondary Flows . . . . . . . . . . . . . . .
Perturbation Techniques . . . . . . . .
Order Symbols and Gauge
Functions . . . . . . . . . . . . . . . . . . . .
Asymptotic Expansion . . . . . . . . . .
Nonuniform Expansion . . . . . . . . .
An Example of a Regular
Perturbation Problem . . . . . . . . . . .
An Example of a Singular
Perturbation Problem . . . . . . . . . . .
Comparison with Exact Solution
Why There Cannot Be a
Boundary Layer at y = 1. . . . . . . .
Decay of a Laminar Shear Layer
Exercises . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . .
369
369
373
375
376
377
379
380
381
385
388
389
390
391
393
394
396
400
401
401
407
409
410
339
340
Boundary Layers and Related Topics
1. Introduction
Until the beginning of the twentieth century, analytical solutions of steady fluid flows
were generally known for two typical situations. One of these was that of parallel
viscous flows and low Reynolds number flows, in which the nonlinear advective terms
were zero and the balance of forces was that between the pressure and the viscous
forces. The second type of solution was that of inviscid flows around bodies of various
shapes, in which the balance of forces was that between the inertia and pressure forces.
Although the equations of motion are nonlinear in this case, the velocity field can
be determined by solving the linear Laplace equation. These irrotational solutions
predicted pressure forces on a streamlined body that agreed surprisingly well with
experimental data for flow of fluids of small viscosity. However, these solutions also
predicted a zero drag force and a nonzero tangential velocity at the surface, features
that did not agree with the experiments.
In 1905 Ludwig Prandtl, an engineer by profession and therefore motivated to
find realistic fields near bodies of various shapes, first hypothesized that, for small
viscosity, the viscous forces are negligible everywhere except close to the solid boundaries where the no-slip condition had to be satisfied. The thickness of these boundary
layers approaches zero as the viscosity goes to zero. Prandtl’s hypothesis reconciled
two rather contradictory facts. On one hand he supported the intuitive idea that the
effects of viscosity are indeed negligible in most of the flow field if ν is small. At the
same time Prandtl was able to account for drag by insisting that the no-slip condition
must be satisfied at the wall, no matter how small the viscosity. This reconciliation
was Prandtl’s aim, which he achieved brilliantly, and in such a simple way that it
now seems strange that nobody before him thought of it. Prandtl also showed how
the equations of motion within the boundary layer can be simplified. Since the time
of Prandtl, the concept of the boundary layer has been generalized, and the mathematical techniques involved have been formalized, extended, and applied to various
other branches of physical science. The concept of the boundary layer is considered
one of the cornerstones in the history of fluid mechanics.
In this chapter we shall explore the boundary layer hypothesis and examine its
consequences. We shall see that the equations of motion within the boundary layer
can be simplified because of the layer’s thinness, and solutions can be obtained in
certain cases. We shall also explore approximate methods of solving the flow within a
boundary layer. Some experimental data on the drag experienced by bodies of various
shapes in high Reynolds number flows, including turbulent flows, will be examined.
For those interested in sports, the mechanics of curving sports balls will be explored.
Finally, the mathematical procedure of obtaining perturbation solutions in situations
where there is a small parameter (such as 1/Re in boundary layer flows) will be briefly
outlined.
2. Boundary Layer Approximation
In this section we shall see what simplifications of the equations of motion within the
boundary layer are possible because of the layer’s thinness. Across these layers, which
exist only in high Reynolds number flows, the velocity varies rapidly enough for the
2. Boundary Layer Approximation
Figure 10.1 The boundary layer. Its thickness is greatly exaggerated in the figure. Here, U∞ is the
oncoming velocity and U is the velocity at the edge of the boundary layer.
viscous forces to be important. This is shown in Figure 10.1, where the boundary
layer thickness is greatly exaggerated. (Around a typical airplane wing it is of order
of a centimeter.) Thin viscous layers exist not only next to solid walls but also in the
form of jets, wakes, and shear layers if the Reynolds number is sufficiently high. To
be specific, we shall consider the case of a boundary layer next to a wall, adopting a
curvilinear “boundary layer coordinate system” in which x is taken along the surface
and y is taken normal to it. We shall refer to the solution of the irrotational flow
outside the boundary layer as the “outer” problem and that of the boundary layer flow
as the “inner” problem.
The thickness of the boundary layer varies with x; let δ̄ be the average thickness
of the boundary layer over the length of the body. A measure of δ̄ can be obtained by
considering the order of magnitude of the various terms in the equations of motion.
The steady equation of motion for the longitudinal component of velocity is
2
∂u
1 ∂p
∂ u ∂ 2u
∂u
+v
=−
+ν
+
.
(10.1)
u
∂x
∂y
ρ ∂x
∂x 2
∂y 2
The Cartesian form of the conservation laws is valid only when δ̄/R ≪ 1, where
R is the local radius of curvature of the body shape function. The more general
curvilinear form for arbitrary R(x) is given in Goldstein (1938) and Schlichting
(1979). We generally expect δ̄/R to be small for large Reynolds number flows over
slender shapes. The first equation to be affected is the y-momentum equation where
centrifugal acceleration will enter the normal component of the pressure gradient.
In equation (10.1) we have also neglected body forces and any variations of ρ and
µ. The essential features of viscous boundary layers can be more clearly illustrated
without additional complications.
Let a characteristic magnitude of u in the flow field be U∞ , which can be identified
with the upstream velocity at large distances from the body. Let L be the streamwise
distance over which u changes appreciably. The longitudinal length of the body can
serve as L, because u within the boundary layer does change by a large fraction of
U∞ in a distance L (Figure 10.2). A measure of ∂u/∂x is therefore U∞ /L, so that a
341
342
Boundary Layers and Related Topics
Figure 10.2 Velocity profiles at two positions within the boundary layer. The velocity arrows are drawn
at the same distance y from the surface, showing that the variation of u with x is of the order of the free
stream velocity U∞ . The boundary layer thickness is greatly exaggerated.
measure of the first advective term in equation (10.1) is
u
U2
∂u
∼ ∞,
∂x
L
(10.2)
where ∼ is to be interpreted as “of order.” We shall see shortly that the other advective term in equation (10.1) is of the same order. A measure of the viscous term in
equation (10.1) is
ν
νU∞
∂ 2u
∼ 2 .
∂y 2
δ̄
(10.3)
The magnitude of δ̄ can now be estimated by noting that the advective and viscous
terms should be of the same order within the boundary layer, if viscous terms are to
be important. Equating equations (10.2) and (10.3), we obtain
δ̄
1
νL
or
∼√ .
δ̄ ∼
U∞
L
Re
This estimate of δ̄ can also be obtained by using results of unsteady parallel flows
discussed in the preceding
chapter, in which we saw that viscous effects diffuse to
√
a distance of order νt in time t. As the time to flow along a body of length L is
of
√ order L/U∞ , the width of the diffusive layer at the end of the body is of order
νL/U∞ .
A formal simplification of the equations of motion within the boundary layer
can now be performed. The basic idea is that variations across the boundary layer are
much faster than variations along the layer, that is
∂
∂
≪
,
∂x
∂y
∂2
∂2
≪ 2.
2
∂x
∂y
343
2. Boundary Layer Approximation
The distances in the x-direction over which the velocity varies appreciably are of
order L, but those in the y-direction are of order δ̄, which is much smaller than L.
Let us first determine a measure of the typical variation of v within the boundary
layer. This can be done from an examination of the continuity equation ∂u/∂x +
∂v/∂y = 0. Because u ≫ v and ∂/∂x ≪ ∂/∂y, we expect the two terms of the
continuity equation to be of the same order. This requires U∞ /L ∼ v/δ̄, or that the
variations of v are of order
√
v ∼ δ̄U∞ /L ∼ U∞ / Re.
Next we estimate the magnitude of variation of pressure within the boundary
layer. Experimental data on high Reynolds number flows show that the pressure distribution is nearly that in an irrotational flow around the body, implying that the pressure
forces are of the order of the inertia forces. The requirement ∂p/∂x ∼ ρu(∂u/∂x)
shows that the pressure variations within the flow field are of order
2
p − p∞ ∼ ρU∞
.
The proper nondimensional variables in the boundary layer are therefore
y√
y
Re,
=
L
δ̄
v √
p − p∞
v
=
Re,
p′ =
,
v′ =
U∞
ρU 2
δ̄U∞ /L
x′ =
x
,
L
y′ =
(10.4)
u
,
U∞
√
where δ̄ = νL/U∞ . The important point to notice is that the distances across
the boundary
layer have been magnified or “stretched” by defining y ′ = y/δ̄ =
√
(y/L) Re.
In terms of these nondimensional variables, the complete equations of motion
for the boundary layer are
u′ =
u′
′
∂p′
1 ∂ 2 u′
∂u′
∂ 2 u′
′ ∂u
+
v
=
−
+
+
,
∂x ′
∂y ′
∂x ′
Re ∂x ′ 2
∂y ′ 2
1 ∂ 2 v′
∂v ′
1 ∂ 2 v′
∂p′
∂v ′
1
,
u′ ′ + v ′ ′ = − ′ + 2 ′2 +
Re
∂x
∂y
∂y
Re ∂y ′2
Re ∂x
∂v ′
∂u′
+
= 0,
∂x ′
∂y ′
(10.5)
(10.6)
(10.7)
where we have defined Re ≡ U∞ L/ν as an overall Reynolds number. In these
equations, each of the nondimensional variables and their derivatives is of order one.
For example, ∂u′ /∂y ′ ∼ 1 in equation (10.5), essentially because the changes in
u′ and y ′ within the boundary layer are each of order one, a consequence of our
normalization (10.4). It follows that the size of each term in the set (10.5) and (10.6)
is determined by the presence of a multiplying factor involving the parameter Re. In
particular, each term in equation (10.5) is of order one except the second term on the
344
Boundary Layers and Related Topics
right-hand side, whose magnitude is of order 1/Re. As Re → ∞, these equations
asymptotically become
u′
′
∂p′
∂ 2 u′
∂u′
′ ∂u
,
+
v
=
−
+
∂x ′
∂y ′
∂x ′
∂y ′2
∂p′
0 = − ′,
∂y
∂v ′
∂u′
+ ′ = 0.
∂x ′
∂y
The exercise of going through the nondimensionalization has been valuable,
since it has shown what terms drop out under the boundary layer assumption. Transforming back to dimensional variables, the approximate equations of motion within
the boundary layer are
u
∂u
1 ∂p
∂ 2u
∂u
+v
=−
+ν 2,
∂x
∂y
ρ ∂x
∂y
0=−
∂p
,
∂y
∂u ∂v
+
= 0.
∂x
∂y
(10.8)
(10.9)
(10.10)
Equation (10.9) says that the pressure is approximately uniform across the boundary layer, an important result. The pressure at the surface is therefore equal to that at
the edge of the boundary layer, and so it can be found from a solution of the irrotational
flow around the body. We say that the pressure is “imposed” on the boundary layer
by the outer flow. This justifies the experimental fact, pointed out in the preceding
section, that the observed surface pressure is approximately equal to that calculated
from the irrotational flow theory. (A vanishing ∂p/∂y, however, is not valid if the
boundary layer separates from the wall or if the radius of curvature of the surface is
not large compared with the boundary layer thickness. This will be discussed later
in the chapter.) The pressure gradient at the edge of the boundary layer can be found
from the inviscid Euler equation
−
dUe
1 dp
= Ue
,
ρ dx
dx
(10.11)
or from its integral√p + ρUe2 /2 = constant, which is the Bernoulli equation. This
is because ve ∼ 1/ Re → 0. Here Ue (x) is the velocity at the edge of the boundary layer (Figure 10.1). This is the matching of the outer inviscid solution with the
boundary layer solution in the overlap domain of common validity. However, instead
of finding dp/dx at the edge of the boundary layer, as a first approximation we
can apply equation (10.11) along the surface of the body, neglecting the existence
of the boundary layer in the solution of the outer problem; the error goes to zero
as the boundary layer becomes increasingly thin. In any event, the dp/dx term in
345
2. Boundary Layer Approximation
equation (10.8) is to be regarded as known from an analysis of the outer problem,
which must be solved before the boundary layer flow can be solved.
Equations (10.8) and (10.10) are then used to determine u and v in the boundary
layer. The boundary conditions are
u(x, 0) = 0,
(10.12)
v(x, 0) = 0,
(10.13)
u(x, ∞) = U (x),
(10.14)
u(x0 , y) = uin (y).
(10.15)
Condition (10.14) merely means that the boundary layer must join smoothly with
the inviscid outer flow; points outside the boundary layer are represented by
y = ∞, although
√ we mean this strictly in terms of the nondimensional distance
y/δ̄ = (y/L) Re → ∞. Condition (10.15) implies that an initial velocity profile
uin (y) at some location x0 is required for solving the problem. This is because the
presence of the terms u ∂u/∂x and ν ∂ 2 u/∂y 2 gives the boundary layer equations a
parabolic character, with x playing the role of a timelike variable. Recall the Stokes
problem of a suddenly accelerated plate, discussed in the preceding chapter, where
the equation is ∂u/∂t = ν ∂ 2 u/∂y 2 . In such problems governed by parabolic equations, the field at a certain time (or x in the problem here) depends only on its past
history. Boundary layers therefore transfer effects only in the downstream direction.
In contrast, the complete Navier–Stokes equations are of elliptic nature. Elliptic equations require specification on the bounding surface of the domain of solution. The
Navier–Stokes equations are elliptic in velocity and thus require boundary conditions
on the velocity (or its derivative normal to the boundary) upstream, downstream, and
on the top and bottom boundaries, that is, all around. The upstream influence of the
downstream boundary condition is always of concern in computations.
In summary, the simplifications achieved because of the thinness of the boundary
layer are the following. First, diffusion in the x-direction is negligible compared to that
in the y-direction. Second, the pressure field can be found from the irrotational flow
theory, so that it is regarded as a known quantity in boundary layer analysis. Here, the
boundary layer is so thin that the pressure does not change across it. Further, a crude
estimate of the shear stress at the wall or skin friction is available from
√ knowledge
of the order of the boundary layer thickness τ0 ∼ µU/δ̄ ∼ (µU /L) Re. The skin
friction coefficient is
2
2µU √
τ0
Re ∼ √ .
∼
2
(1/2)ρU
ρLU 2
Re
As we shall see from the solutions to the problems in the following sections, this is
indeed the correct order of magnitude. Only the finite numerical factor differs from
problem to problem.
It is useful to compare equation (10.5) with equation (9.60), where we nondimensionalized both x- and y-directions by the same length scale. Notice that in
equation (9.60) the Reynolds number multiplies both diffusion terms, whereas in
346
Boundary Layers and Related Topics
equation (10.5) the diffusion term in the y-direction has been explicitly made order
one by a normalization appropriate within the boundary layer.
3. Different Measures of Boundary Layer Thickness
As the velocity in the boundary layer smoothly joins that of the outer flow, we have
to decide how to define the boundary layer thickness. The three common measures
are described here.
The u = 0.99U Thickness
One measure of the boundary thickness is the distance from the wall where the
longitudinal velocity reaches 99% of the local free stream velocity, that is where
u = 0.99 U . We shall denote this as δ99 . This definition of the boundary layer thickness
is however rather arbitrary, as we could very well have chosen the thickness as the
point where u = 0.95 U .
Displacement Thickness
A second measure of the boundary layer thickness, and one in which there is no
arbitrariness, is the displacement thickness δ ∗ . This is defined as the distance by
which the wall would have to be displaced outward in a hypothetical frictionless flow
so as to maintain the same mass flux as in the actual flow. Let h be the distance from
the wall to a point far outside the boundary layer (Figure 10.3). From the definition
of δ ∗ , we obtain
h
u dy = U (h − δ ∗ ),
0
where the left-hand side is the actual mass flux below h and the right-hand side is the
mass flux in the frictionless flow with the walls displaced by δ ∗ . Letting h → ∞, the
aforementioned gives
δ∗ =
Figure 10.3 Displacement thickness.
0
∞
1−
u
dy.
U
(10.16)
3. Different Measures of Boundary Layer Thickness
The upper limit in equation (10.16) may be allowed to extend to infinity because, as
we shall show in the following, u/U → 0 exponentially fast in y as y → ∞.
The concept of displacement thickness is used in the design of ducts, intakes of
air-breathing engines, wind tunnels, etc. by first assuming a frictionless flow and then
enlarging the passage walls by the displacement thickness so as to allow the same flow
rate. Another use of δ ∗ is in finding dp/dx at the edge of the boundary layer, needed for
solving the boundary layer equations. The first approximation is to neglect the existence of the boundary layer, and calculate the irrotational dp/dx over the body surface.
A solution of the boundary layer equations gives the displacement thickness, using
equation (10.16). The body surface is then displaced outward by this amount and a next
approximation of dp/dx is found from a solution of the irrotational flow, and so on.
The displacement thickness can also be interpreted in an alternate and possibly
more illuminating way. We shall now show that it is the distance by which the streamlines outside the boundary layer are displaced due to the presence of the boundary
layer. Figure 10.4 shows the displacement of streamlines over a flat plate. Equating
mass flux across two sections A and B, we obtain
h
h+δ ∗
u dy + U δ ∗ ,
u dy =
Uh =
0
0
which gives
U δ∗ =
0
h
(U − u) dy.
Here h is any distance far from the boundary and can be replaced by ∞ without
changing the integral, which then reduces to equation (10.16).
Momentum Thickness
A third measure of the boundary layer thickness is the momentum thickness θ, defined
such that ρU 2 θ is the momentum loss due to the presence of the boundary layer. Again
choose a streamline such that its distance h is outside the boundary layer, and consider
Figure 10.4 Displacement thickness and streamline displacement.
347
348
Boundary Layers and Related Topics
the momentum flux (=velocity times mass flow rate) below the streamline, per unit
width. At section A the momentum flux is ρU 2 h; that across section B is
h+δ ∗
0
ρu2 dy =
h
0
ρu2 dy + ρ δ ∗ U 2 .
The loss of momentum due to the presence of the boundary layer is therefore the
difference between the momentum fluxes across A and B, which is defined as ρU 2 θ:
ρU 2 h −
h
ρu2 dy − ρδ ∗ U 2 ≡ ρU 2 θ.
0
Substituting the expression for δ ∗ gives
0
h
2
2
(U − u ) dy − U
2
0
h
1−
u
dy = U 2 θ,
U
from which
θ=
0
∞
u
u
1−
dy,
U
U
(10.17)
where we have replaced h by ∞ because u = U for y > h.
4. Boundary Layer on a Flat Plate with a Sink at
the Leading Edge: Closed Form Solution
Although all other texts start their boundary layer discussion with the uniform flow
over a semi-infinite flat plate, there is an even simpler related problem that can be
solved in closed form in terms of elementary functions. We shall consider the large
Reynolds number flow generated by a sink at the leading edge of a flat plate. The outer
inviscid flow is represented by ψ = mθ/2π , m < 0 so that ur = m/2π r, uθ = 0
[Chapter 6, Section 5, equation (6.24) and Figure 6.6]. This represents radially inward
flow towards the origin. A flat plate is now aligned with the x-axis so that its boundary
is represented by θ = 0. For large Re, the boundary layer is thin so x = r cos θ ≈ r
because θ ≪ 1. For simplicity in what follows we shall absorb the 2π into the m
by defining m′ = m/2π and then suppressing the prime. The velocity at the edge
of the boundary layer is Ue (x) = m/x, m < 0 and the local Reynolds number is
Ue (x)x/ν = m/ν = Rex . Boundary layer coordinates are used, as in Figure 10.1,
with y normal to the plate and the origin at the leading edge.
The boundary layer equations (10.8)–(10.10) with equation (10.11) become
∂u ∂v
+
= 0,
∂x
∂y
u
∂u
∂u
m2
∂ 2u
+v
=− 3 +ν 2
∂x
∂y
x
∂y
with the boundary conditions (10.12)–(10.15). We consider the limiting case Rex =
|m/ν| → ∞. Because m < 0, the flow is from right (larger x) to left (smaller x),
4. Boundary Layer on a Flat Plate with a Sink at the Leading Edge: Closed Form Solution
and the initial condition at x = x0 is specified upstream, that is, at the largest x. The
solution is then determined for all x < x0 , that is, downstream of the initial location.
The natural way to make the variables
√ dimensionless and finite in
√ the boundary layer
is to normalize x by x0 , y by x0 / Rex , u by m/x0 , v by m/(x0 Rex ). The problem
is fully two-dimensional and well posed for any reasonable initial condition (10.15).
Now, suppress the initial condition. The length scale x0 , crucial to rendering the
problem properly dimensionless, has disappeared. How is one to construct a dimensionless formulation? We have seen before that this situation results in a reduction in
the dimensionality of the space required for the solution.
The variable y can be made
√
dimensionless only by x and must be stretched
by
Re
x
√
√ to be finite in the boundary
layer. The unique choice is then (y/x) Rex = (y/x) |m/ν|√= η. This is consistent
with the similarity variable for Stokes’ first problem η = y/ νt when t is taken to
be x/U and U = m/x. Finite numerical
factors are irrelevant here. Further,√
we note
|m/ν|
that we have found that δ ∼ x0 / Rex0 so with the x0 scale absent, δ ∼ x
and η = y/δ. Next we will reduce mass and momentum conservation to an ordinary
differential equation for the xsimilarity streamfunction. To reverse the flow we will
define the streamfunction ψ via u = −∂ψ/∂y, v = ∂ψ/∂x (note sign change). We
now have:
m2
∂ 3ψ
∂ψ ∂ 2 ψ
∂ψ ∂ 2 ψ
=
−
−
ν
,
−
∂y ∂y ∂x
∂x ∂y 2
x3
∂y 3
y = 0:
ψ=
∂ψ
= 0,
∂y
∂ψ
m
→ .
∂y
x
y → overlap with inviscid flow:
The streamfunction is made dimensionless by its order of magnitude and put in
similarity form via
x
ψ(x, y) = Ue δ(x)f (η) = Ue (x) · √
f (η)
Rex
=
νUe (x) · xf (η) =
in this problem. The problem for f reduces to
|νm| f (η),
f ′′′ (η) − f ′ 2 = −1,
f (0) = 0,
f ′ (0) = 0,
This may be solved in closed form with the result
√
1 − αe− 2η
u
√
= f ′ (η) = 3
Ue (x)
1 + αe− 2η
2
− 2,
f ′ (∞) = 1.
√
3− 2
α=√
√ = 0.101 . . . .
3+ 2
√
349
350
Boundary Layers and Related Topics
A result equivalent to this was first obtained by Pohlhausen (1921) in his solution
for flow in a convergent channel. From this simple solution we can establish several
properties characteristic of laminar boundary layers. First, as η → ∞, the matching
√
with the inviscid solution occurs exponentially fast, as f ′ (η) ∼ 1 − 12αe− 2η +
smaller terms as η → ∞.
Next v/Ue is of the correct small order,
1
y
1
v
ηf ′ (η) ∼ √
.
= f ′ (η) = √
Ue
x
Rex
Rex
The behavior of the displacement thickness is obtained from the definition
∗
δ =
0
∞
1
δ∗
=√
x
Rex
u
1−
dy =
Ue
∞
0
∞
0
[1 − f ′ (η)] dη · √
[1 − f ′ (η)] dη =
x
,
Rex
12α
1
0.7785
∼√
.
= √
√ √
Rex
Rex
[(1 + α) 2 Rex ]
The shear stress at the wall is
τ0 = µ
∂u
∂y
0
= −µ
m
x2
m ′′
f (0),
ν
2
f ′′ (0) = √ .
3
Then the skin friction coefficient is
√
τ0
−4/ 3
,
= √
Cf =
(1/2)ρUe2
Rex
Rex =
m
.
ν
Aside from numerical factors, which are obviously problem specific, the preceding results are universally valid for all similarity solutions of the laminar boundary layer equations. Ue (x) is the velocity at the edge of the boundary layer and
Rex = Ue (x)x/ν. In these terms
η=
y
Rex ,
x
ψ(x, y) =
νUe (x) · x f (η),
√
fast as η → ∞. We find v/Ue ∼ 1/ Rex ,
f (η) = u/U
e (x) → 1 exponentially
√
√
δ ∗ /x ∼ 1/ Rex , Cf ∼ 1/ Rex .
Axisymmetric Problem
Now let us consider the axially symmetric version of the problem we just solved.
This is the flow in the neighborhood of an infinite flat plate generated by a sink
in the center of the plate. The inviscid outer flow is ur = −Q/r 2 where r is the
spherical radial coordinate centered on the sink. The boundary layer adjacent to the
plate is best treated in cylindrical coordinates r, θ, z with ∂/∂θ = 0 (see Figure 10.5).
Mass conservation for a constant density flow with symmetry about the z-axis is
351
4. Boundary Layer on a Flat Plate with a Sink at the Leading Edge: Closed Form Solution
z
θ
r,x
Figure 10.5 Axisymmetric flow into a sink at the center of an infinite plate.
∂/∂r(rur ) + ∂/∂z(ruz ) = 0. In the following, the streamwise coordinate r is replaced
by x. Since Ue = −Q/x 2 , the local Reynolds number can be written as Rex =
Ue x/ν = Q/xν. Assuming this is sufficiently large, the full Navier–Stokes equations
reduce to the boundary layer equations with an error that is small in powers of inverse
Rex . Thus we seek to solve
u∂u/∂x + w∂u/∂z = Ue dUe /dx + ν∂ 2 u/∂z2
subject to u = w = 0 on z = 0 and u → Ue as z leaves the boundary layer.
A similarity solution can be obtained provided the requirement for an initial velocity
distribution is not imposed. First, the streamwise momentum equation is put in terms
of the axisymmetric streamfunction, u = −(1θ /x) × ∇ψ, so that xu = −∂ψ/∂z,
xw = ∂ψ/∂x. With the modification of the streamfunction due to axial symmetry,
the universal dimensionless similarity form becomes
ψ(x, z) = x[νxUe (x)]1/2 f (η) = (νQx)1/2 f (η)
where η = (z/x)(Rex )1/2 = (Q/ν)1/2 z/x 3/2 . The velocity components transform to u = −x −1 ∂ψ/∂z = Ue f ′ (η), w = [(νQ)1/2 /(2x 3/2 )](f − 3ηf ′ ) =
{Ue /[2(Rex )1/2 ]}(f − 3ηf ′ ).
The streamwise momentum equation transforms to
f ′′′ − (1/2)ff ′′ + 2(1 − f ′2 ) = 0
subject to
(10.18)
f (0) = 0,
f ′ (0) = 0,
f ′ (∞) = 1.
Rosenhead provides a tabulation of the solution to f ′′′ − ff ′′ + 4(1 − f ′2 ) = 0,
which is related to the equation above by the scaling η/21/2 , and f/21/2 . (We have
tried not to add extraneous numerical factors to our universal dimensionless similarity
scaling.) The solution to (10.18) is displayed in Figure 10.6.
352
Boundary Layers and Related Topics
Figure 10.6 Dimensionless velocity profile for flow illustrated in Figure 10.5.
5. Boundary Layer on a Flat Plate: Blasius Solution
We shall next discuss the classic problem of the boundary layer on a semi-infinite
flat plate. Equations (10.8)–(10.10) are a valid asymptotic representation of the full
Navier–Stokes equations in the limit Rex → ∞. Thus with x measured from the
leading edge, the initial station x0 (see equation (10.15)) must be sufficiently far
downstream that Ue x0 /ν ≫ 1. A major question in boundary layer theory is the
extent of downstream memory of the initial state. If the external stream Ue (x) admits
a similarity solution, is the initial condition forgotten and how soon? Serrin (1967)
and Peletier (1972) showed that for favorable pressure gradients (Ue dUe /dx) of
similarity form, the initial condition is forgotten and the larger the acceleration the
sooner similarity is achieved. A decelerating flow will accentuate details of the initial
state and similarity will never be found despite its mathematical admissability. This
is consistent with the experimental findings of Gallo et al. (1970). A flat plate for
which Ue (x) = U = const. is the borderline case; similarity is eventually achieved
here. In the previous problem, the sink creates a rapidly accelerating flow so that, if
we could ever realize such a flow, similarity would be achieved quickly.
As the inviscid solution gives u = U = const. everywhere, ∂p/∂x = 0 and the
equations become
∂u
∂ 2u
∂u
+v
= ν 2,
u
∂x
∂y
∂y
(10.19)
∂u ∂v
+
= 0,
∂x
∂y
353
5. Boundary Layer on a Flat Plate: Blasius Solution
subject to: y = 0: u = v = 0, x > 0
y → overlap at edge of boundary layer:
x = x0 : u(y) given, Rex0 ≫ 1.
u → U,
(10.20)
For x large compared with x0 , we can argue that the initial condition is forgotten.
With x0 no longer available for rendering the independent variables dimensionless, a
similarity solution will be obtained. Using our previous results,
√
y
Ux
,
ψ(x, y) = νU xf (η), η =
Rex , Rex =
x
ν
and u = ∂ψ/∂y, v = −∂ψ/∂x. Now u/U = f ′ (η) and
f ′′′ + 21 ff ′′ = 0,
f (0) = f ′ (0) = 0,
f (∞) = 1.
A different but equally correct method of obtaining the similarity form is described in
what follows. The plate length L (Figure 10.4) has been taken very large so a solution
independent of L has been sought. In addition, we limit our consideration to a domain
far downstream of x0 so the initial condition has been forgotten.
Similarity Solution—Alternative Procedure
We shall regard δ(x) as an unknown function in the following analysis; the form
of δ(x) will follow from a requirement that a similarity solution must exist for this
problem.
As there is no externally imposed length scale along x, the solutions at various
downstream locations must be self similar. Blasius, a student of Prandtl, showed
that a similarity solution can indeed be found for this problem. Clearly, the velocity
distributions at various downstream points can collapse into a single curve only if the
solution has the form
u
= g(η),
U
(10.21)
y
.
δ(x)
(10.22)
where
η=
At this point it is useful to pause a little and compare the situation with that of
a suddenly accelerated plate (see Chapter 9, Section 7), for which similarity solutions exist. In that case we argued that the parameter U drops out of the equations
and boundary conditions if we define u/U as the dependent variable, leading to
u/U = f (y, t, ν). A dimensional analysis then immediately
showed that the func√
tional form must be u/U = F [y/δ(t)], where δ(t) ∼ νt. In the current problem the
downstream distance is timelike, but we cannot analogously write u/U = f (y, x, ν),
because ν cannot be made nondimensional with the help of x or y. The dynamic
354
Boundary Layers and Related Topics
reason for this is that U cannot be eliminated from the problem simply by regarding
u/U as the dependent variable, because U still remains in the problem through the
dependence of δ on U . The correct dimensional argument in this case is that we must
have a solution of the form u/U = g[y/δ(x)],
√ where δ(x) is a function of (U, x, ν)
and therefore can only be of the form δ ∼ νx/U .
We now resume our search for a similarity solution for the flat plate boundary
layer. As the problem is two-dimensional, it is easier to work with the streamfunction
defined by
u≡
∂ψ
,
∂y
v≡−
∂ψ
.
∂x
Using the similarity form (10.21), we obtain
ψ=
0
y
u dy = δ
where we have defined
0
η
u dη = δ
η
0
Ug(η) dη = U δf (η),
(10.23)
df
.
(10.24)
dη
(Equation (10.23) shows that the similarity form for the stream function is ψ/U δ =
f (η), signifying that the scale for the streamfunction is proportional to the local flow
rate U δ.)
In terms of the streamfunction, the governing sets (10.19) and (10.20) become
g(η) ≡
∂ψ ∂ 2 ψ
∂ 3ψ
∂ψ ∂ 2 ψ
−
=
ν
,
∂y ∂x ∂y
∂x ∂y 2
∂y 3
(10.25)
subject to
∂ψ
=ψ =0
∂y
at y = 0, x > 0,
∂ψ
→U
∂y
as
y
→ ∞.
δ
(10.26)
To express sets (10.25) and (10.26) in terms of the similarity streamfunction
f (η), we find the following derivatives from equation (10.23):
∂ψ
∂f
dδ
=U f
+δ
∂x
dx
∂x
=U
dδ
[f − f ′ η],
dx
dδ ∂
U ηf ′′ dδ
∂ 2ψ
=U
[f − f ′ η] = −
,
∂x ∂y
dx ∂y
δ dx
∂ψ
= Uf ′ ,
∂y
(10.27)
(10.28)
(10.29)
355
5. Boundary Layer on a Flat Plate: Blasius Solution
∂ 2ψ
Uf ′′
=
,
δ
∂y 2
(10.30)
∂ 3ψ
Uf ′′′
= 2 ,
(10.31)
3
∂y
δ
where primes on f denote derivatives with respect to η. Substituting these derivatives
in equation (10.25) and canceling terms, we obtain
U δ dδ
(10.32)
ff ′′ = f ′′′ .
−
ν dx
In equation (10.32), f and its derivatives do not explicitly depend on x. The equation
can be valid only if
U δ dδ
= const.
ν dx
Choosing the constant to be
1
2
for eventual algebraic simplicity, an integration gives
δ=
νx
.
U
(10.33)
Equation (10.32) then becomes
1 ′′
ff + f ′′′ = 0.
2
(10.34)
In terms of f , the boundary conditions (10.26) become
f ′ (∞) = 1,
f (0) = f ′ (0) = 0.
(10.35)
A series solution of the nonlinear equation (10.34), subject to equation (10.35),
was given by Blasius. It is much easier to solve the problem with a computer, using for
example the Runge–Kutta technique. The resulting profile of u/U = f ′ (η) is shown
in Figure 10.7. The solution makes the√profiles at various downstream distances
collapse into a single curve of u/U vs y U/νx, and is in excellent agreement with
experimental data for laminar flows at high Reynolds numbers. The profile has a
point of inflection (that is, zero curvature) at the wall, where ∂ 2 u/∂y 2 = 0. This
is a result of the absence of pressure gradient in the flow and will be discussed in
Section 7.
Matching with External Stream
We find in this case that the difference between f ′ and 1 ∼ (1/η)e−η
exponentially fast as η → ∞.
2 /4
→ 0
356
Boundary Layers and Related Topics
Figure 10.7 The Blasius similarity solution of velocity distribution in a laminar boundary layer on a flat
plate. The momentum thickness θ and displacement δ ∗ are indicated by arrows on the horizontal axis.
Transverse Velocity
The lateral component of velocity is given by v = −∂ψ/∂x. From equation (10.27),
this becomes
0.86
v
1
1 νU
(ηf ′ − f ) ∼ √
as η → ∞,
(ηf ′ − f ),
= √
2 x
U
2 Rex
Rex
a plot of which is shown in Figure 10.8. The transverse velocity increases from zero
at the wall to a maximum value at the edge of the boundary layer, a pattern that is in
agreement with the streamline shapes sketched in Figure 10.4.
v=
Boundary Layer Thickness
From Figure 10.7, the distance where u = 0.99 U is η = 4.9. Therefore
δ99 = 4.9
νx
U
or
4.9
δ99
=√
,
x
Rex
(10.36)
where we have defined a local Reynolds number
Ux
Rex ≡
ν
√
The parabolic growth (δ ∝ x) of the boundary layer thickness is in good agreement with experiments. For air at ordinary temperatures flowing at U = 1 m/s, the
Reynolds number at a distance of 1 m from the leading edge is Rex = 6 × 104 , and
equation (10.36) gives δ99 = 2 cm, showing that the boundary layer is indeed thin.
357
5. Boundary Layer on a Flat Plate: Blasius Solution
Figure 10.8 Transverse velocity component in a laminar boundary layer on a flat plate.
The displacement and momentum thicknesses, defined in equations (10.16) and
(10.17), are found to be
δ ∗ = 1.72 νx/U ,
θ = 0.664 νx/U .
These thicknesses are indicated along the abscissa of Figure 10.7.
Skin Friction
The local wall shear stress is τ0 = µ(∂u/∂y)0 = µ(∂ 2 ψ/∂y 2 )0 , where the subscript
zero stands for y = 0. Using ∂ 2 ψ/∂y 2 = Uf ′′ /δ given in equation (10.30), we obtain
τ0 = µUf ′′ (0)/δ, and finally
τ0 =
0.332ρU 2
.
√
Rex
(10.37)
The wall shear stress therefore decreases as x −1/2 , a result of the thickening of the
boundary layer and the associated decrease of the velocity gradient. Note that the
wall shear stress at the leading edge is predicted to be infinite. Clearly the boundary
layer theory breaks down near the leading edge where the assumption ∂/∂x ≪ ∂/∂y
is invalid. The local Reynolds number Rex in the neighborhood of the leading edge
is of order 1, for which the boundary layer assumptions are not valid.
The wall shear stress is generally expressed in terms of the nondimensional skin
friction coefficient
Cf ≡
0.664
τ0
=√
.
2
(1/2)ρU
Rex
(10.38)
358
Boundary Layers and Related Topics
The drag force per unit width on one side of a plate of length L is
D=
0
L
τ0 dx =
0.664ρU 2 L
,
√
ReL
where we have defined ReL ≡ U L/ν as the Reynolds number based on the plate
length. This equation shows that the drag force is proportional to the 23 power of
velocity. This should be compared with small Reynolds number flows, where the
drag is proportional to the first power of velocity. We shall see later in the chapter
that the drag on a blunt body in a high Reynolds number flow is proportional to the
square of velocity.
The overall drag coefficient defined in the usual manner is
CD ≡
1.33
D
=√
.
2
(1/2)ρU L
ReL
(10.39)
It is clear from equations (10.38) and (10.39) that
CD =
1
L
L
Cf dx,
0
which says that the overall drag coefficient is the average of the local friction coefficient (Figure 10.9).
We must keep in mind that carrying out an integration from x = 0 is of
questionable validity because the equations and solutions are valid only for very
large Rex .
Falkner–Skan Solution of the Laminar Boundary Layer Equations
No discussion of laminar boundary layer similarity solutions would be complete
without mention of the work of V. W. Falkner and S. W. Skan (1931). They found
Figure 10.9 Friction coefficient and drag coefficient in a laminar boundary layer on a flat plate.
359
5. Boundary Layer on a Flat Plate: Blasius Solution
that Ue (x) = ax n admits a similarity solution, as follows. We assume that Rex =
ax (n+1) /ν is sufficiently large so that the boundary layer equations are valid and any
dependence on an initial condition has been forgotten. Then the initial station x0
disappears from the problem and we may write
ψ(x, y) =
η=
νUe (x) · x f (η) =
y
Rex =
x
√
νa x (n+1)/2 f (η),
a
yx (n−1)/2 .
ν
Then u/Ue = f ′ (η) and Ue (dUe /dx) = na 2 x 2n−1 .
The x-momentum equation reduces to the similarity form
f ′′′ +
n+1
ff ′′ − nf ′ 2 + n = 0,
2
f ′ (0) = 0,
f (0) = 0,
(10.40)
f ′ (∞) = 1.
(10.41)
The Blasius equation (10.34) and (10.35) is a special case for n = 0, that is,
Ue (x) = U . Although there are similarity solutions possible for n < 0, these are not
likely to be seen in practice. For n 0, all solutions of equations (10.40) and (10.41)
have the proper behavior as detailed in the preceding. The numerical coefficients
depend on n. Solutions to equations (10.40) and (10.41) are displayed in Figure 5.9.1
of Batchelor (1967) and reproduced here in Figure 10.10. They show a monotonically
increasing shear stress [f ′′ (0)] as n increases. For n = −0.0904, f ′′ (0) = 0 so τ0 = 0
and separation is imminent all along the surface. Solutions for n < −0.0904 do not
represent boundary layers. In most real flows, similarity solutions are not available
and the boundary layer equations with boundary and initial conditions as written in
1.0
n=4
0.8
1
0.6
0
0.4
– 0.0654
– 0.0904
0.2
0
1
2
3
4
1
Figure 10.10 Velocity distribution in the boundary layer for external stream Ue = ax n . G. K Batchelor,
An Introduction to Fluid Dynamics, 1st ed. (1967), reprinted with the permission of Cambridge University
Press.
360
Boundary Layers and Related Topics
equations (10.8)–(10.15) must be solved. A simple approximate procedure, the von
Karman momentum integral, is discussed in the next section. More often the equations will be integrated numerically by procedures that are discussed in more detail in
Chapter 11.
Breakdown of Laminar Solution
Agreement of the Blasius solution with experimental data breaks down at large downstream distances where the local Reynolds number Rex is larger than some critical
value, say Recr . At these Reynolds numbers the laminar flow becomes unstable and
a transition to turbulence takes place. The critical Reynolds number varies greatly
with the surface roughness, the intensity of existing fluctuations (that is, the degree
of steadiness) within the outer irrotational flow, and the shape of the leading edge.
For example, the critical Reynolds number becomes lower if either the roughness of
the wall surface or the intensity of fluctuations in the free stream is increased. Within
a factor of 5, the critical Reynolds number for a boundary layer over a flat plate is
found to be
Recr ∼ 106
(flat plate).
Figure 10.11 schematically depicts the flow regimes on a semi-infinite flat plate. For
finite Rex = U x/ν ∼ 1, the full Navier–Stokes equations are required to describe the
leading edge region properly. As Rex gets large at the downstream limit of the leading
edge region, we can locate x0 as the maximal upstream extent of the boundary layer
Figure 10.11 Schematic depiction of flow over a semiinfinite flat plate.
5. Boundary Layer on a Flat Plate: Blasius Solution
equations. For some distance x > x0 , the initial condition is remembered. Finally,
the influence of the initial condition may be neglected and the solution becomes of
similarity form. For somewhat larger Rex , a bit farther downstream, the first instability
appears. Then a band of waves becomes amplified and interacts nonlinearly through
the advective acceleration. As Rex increases, the flow becomes increasingly chaotic
and irregular in the downstream direction. For lack of a better word, this is called
transition. Eventually, the boundary layer becomes fully turbulent with a significant
increase in shear stress at the plate τ0 .
After undergoing transition, the boundary layer thickness grows faster than x 1/2
(Figure 10.11), and the wall shear stress increases faster with U than in a laminar
boundary layer; in contrast, the wall shear stress for a laminar boundary layer varies
as τ0 ∝ U 1.5 . The increase in resistance is due to the greater macroscopic mixing in
a turbulent flow.
Figure 10.12 sketches the nature of the observed variation of the drag coefficient in a flow over a flat plate, as a function of the Reynolds number. The
lower curve applies if the boundary layer is laminar over the entire length of
the plate, and the upper curve applies if the boundary layer is turbulent over the
entire length. The curve joining the two applies if the boundary layer is laminar
over the initial part and turbulent over the remaining part, as in Figure 10.11. The
exact point at which the observed drag deviates from the wholly laminar behavior
depends on experimental conditions and the transition shown in Figure 10.12 is at
Recr = 5 × 105 .
Figure 10.12 Measured drag coefficient for a boundary layer over a flat plate. The continuous line shows
the drag coefficient for a plate on which the flow is partly laminar and partly turbulent, with the transition
taking place at a position where the local Reynolds number is 5 × 105 . The dashed lines show the behavior
if the boundary layer was either completely laminar or completely turbulent over the entire length of the
plate.
361
362
Boundary Layers and Related Topics
6. von Karman Momentum Integral
Exact solutions of the boundary layer equations are possible only in simple cases,
such as that over a flat plate. In more complicated problems a frequently applied
approximate method satisfies only an integral of the boundary layer equations across
the layer thickness. The integral was derived by von Karman in 1921 and applied to
several situations by Pohlhausen.
The point of an integral formulation is to obtain the information that is really
required with minimum effort. The important results of boundary layer calculations
are the wall shear stress, displacement thickness, and separation point. With the help
of the von Karman momentum integral derived in what follows and additional correlations, these results can be obtained easily.
The equation is derived by integrating the boundary layer equation
u
∂u
dU
∂ 2u
∂u
+v
=U
+ν 2,
∂x
∂y
dx
∂y
from y = 0 to y = h, where h > δ is any distance outside the boundary layer. Here
the pressure gradient term has been expressed in terms of the velocity U (x) at the
edge of the boundary layer, where the inviscid Euler equation applies. Adding and
subtracting u(dU/dx), we obtain
(U − u)
∂(U − u)
∂(U − u)
∂ 2u
dU
+u
+v
= −ν 2 .
dx
∂x
∂y
∂y
(10.42)
Integrating from y = 0 to y = h, the various terms of this equation transform as
follows.
The first term gives
0
h
(U − u)
dU
dU
dy = U δ ∗
.
dx
dx
Integrating by parts, the third term gives,
h
v
0
h
h
∂v
∂(U − u)
dy = v(U − u) 0 −
(U − u) dy
∂y
0 ∂y
h
∂u
(U − u) dy,
=
∂x
0
where we have used the continuity equation and the conditions that v = 0 at y = 0
and u = U at y = h. The last term in equation (10.42) gives
−ν
0
h
∂ 2u
τ0
dy = ,
ρ
∂y 2
363
6. von Karman Momentum Integral
where τ0 is the wall shear stress.
The integral of equation (10.42) is therefore
Uδ
∗ dU
dx
+
h
0
∂(U − u)
∂u
u
+ (U − u)
∂x
∂x
dy =
τ0
.
ρ
(10.43)
The integral in equation (10.43) equals
0
h
d
∂
[u(U − u)] dy =
∂x
dx
0
h
u(U − u) dy =
d
(U 2 θ ),
dx
where θ is the momentum thickness defined by equation (10.17). Equation (10.43)
then gives
d
dU
τ0
(U 2 θ ) + δ ∗ U
= ,
dx
dx
ρ
(10.44)
which is called the Karman momentum integral equation. In equation (10.44), θ,
δ ∗ , and τ0 are all unknown. Additional assumptions must be made or correlations
provided to obtain a useful solution. It is valid for both laminar and turbulent boundary
layers. In the latter case τ0 cannot be equated to molecular viscosity times the velocity
gradient and should be empirically specified. The procedure of applying the integral
approach is to assume a reasonable velocity distribution, satisfying as many conditions
as possible. Equation (10.44) then predicts the boundary layer thickness and other
parameters.
The approximate method is only useful in situations where an exact solution
does not exist. For illustrative purposes, however, we shall apply it to the boundary
layer over a flat plate where U (dU/dx) = 0. Using definition (10.17) for θ , equation (10.44) reduces to
δ
d
τ0
(10.45)
(U − u)u dy = .
dx 0
ρ
Assume a cubic profile
y
y2
y3
u
=a+b +c 2 +d 3.
U
δ
δ
δ
The four conditions that we can satisfy with this profile are chosen to be
u = 0,
u = U,
∂ 2u
=0
∂y 2
at
y = 0,
∂u
=0
∂y
at
y = δ.
364
Boundary Layers and Related Topics
The condition that ∂ 2 u/∂y 2 = 0 at the wall is a requirement in a boundary layer
over a flat plate, for which an application of the equation of motion (10.8) gives
ν(∂ 2 u/∂y 2 )0 = U (dU/dx) = 0. Determination of the four constants reduces the
assumed profile to
u
3 y 1 y 3
−
.
=
U
2 δ
2 δ
The terms on the left- and right-hand sides of the momentum equation (10.45)
are then
δ
39 2
U δ,
(U − u)u dy =
280
0
∂u
3 Uν
τ0
=ν
.
=
ρ
∂y 0
2 δ
Substitution into the momentum integral equation gives
3 Uν
39U 2 dδ
=
.
280 dx
2 δ
Integrating in x and using the condition δ = 0 at x = 0, we obtain
δ = 4.64 νx/U ,
which is remarkably close to the exact solution (10.36). The friction factor is
Cf =
τ0
(3/2)U ν/δ
0.646
,
=
=√
2
2
(1/2)ρU
(1/2)U
Rex
which is also very close to the exact solution of equation (10.38).
Pohlhausen found that a fourth-degree polynomial was necessary to exhibit sensitivity of the velocity profile to the pressure gradient. Adding another term below
equation (10.45), e(y/δ)4 requires an additional boundary condition, ∂ 2 u/∂y 2 = 0
at y = δ. With the assumption of a form for the velocity profile, equation (10.44)
may be reduced to an equation with one unknown, δ(x) with U (x), or the pressure
gradient specified. This equation was solved approximately by Pohlhausen in 1921.
This is described in Yih (1977, pp. 357–360). Subsequent improvements by Holstein
and Bohlen (1940) are recounted in Schlichting (1979, pp. 206–217) and Rosenhead
(1988, pp. 293–297). Sherman (1990, pp. 322–329) related the approximate solution
due to Thwaites.
7. Effect of Pressure Gradient
So far we have considered the boundary layer on a flat plate, for which the pressure
gradient of the external stream is zero. Now suppose that the surface of the body is
curved (Figure 10.13). Upstream of the highest point the streamlines of the outer flow
converge, resulting in an increase of the free stream velocity U (x) and a consequent
365
7. Effect of Pressure Gradient
Figure 10.13 Velocity profiles across boundary layers with favorable and adverse pressure gradients.
fall of pressure with x. Downstream of the highest point the streamlines diverge,
resulting in a decrease of U (x) and a rise in pressure. In this section we shall investigate
the effect of such a pressure gradient on the shape of the boundary layer profile u(x, y).
The boundary layer equation is
u
∂u
1 ∂p
∂ 2u
∂u
+v
=−
+ν 2,
∂x
∂y
ρ ∂x
∂y
where the pressure gradient is found from the external velocity field as dp/dx
= −ρU (dU/dx), with x taken along the surface of the body. At the wall, the boundary
layer equation becomes
2
∂p
∂ u
.
=
µ
2
∂x
∂y wall
In an accelerating stream dp/dx < 0, and therefore
2
∂ u
<0
(accelerating).
∂y 2 wall
(10.46)
As the velocity profile has to blend in smoothly with the external profile, the slope
∂u/∂y slightly below the edge of the boundary layer decreases with y from a positive
value to zero; therefore, ∂ 2 u/∂y 2 slightly below the boundary layer edge is negative.
Equation (10.46) then shows that ∂ 2 u/∂y 2 has the same sign at both the wall and the
boundary layer edge, and presumably throughout the boundary layer. In contrast, for
a decelerating external stream, the curvature of the velocity profile at the wall is
2
∂ u
>0
(decelerating).
(10.47)
∂y 2 wall
366
Boundary Layers and Related Topics
so that the curvature changes sign somewhere within the boundary layer. In other
words, the boundary layer profile in a decelerating flow has a point of inflection
where ∂ 2 u/∂y 2 = 0. In the limiting case of a flat plate, the point of inflection is at
the wall.
The shape of the velocity profiles in Figure 10.13 suggests that a decelerating
pressure gradient tends to increase the thickness of the boundary layer. This can also
be seen from the continuity equation
y
∂u
v(y) = −
dy.
∂x
0
Compared to a flat plate, a decelerating external stream causes a larger −∂u/∂x within
the boundary layer because the deceleration of the outer flow adds to the viscous
deceleration within the boundary layer. It follows from the foregoing equation that
the v-field, directed away from the surface, is larger for a decelerating flow. The
boundary layer therefore thickens not only by viscous diffusion but also by advection
away from the surface, resulting in a rapid increase in the boundary layer thickness
with x.
If p falls along the direction of flow, dp/dx < 0 and we say that the pressure
gradient is “favorable.” If, on the other hand, the pressure rises along the direction
of flow, dp/dx > 0 and we say that the pressure gradient is “adverse” or “uphill.”
The rapid growth of the boundary layer thickness in a decelerating stream, and the
associated large v-field, causes the important phenomenon of separation, in which
the external stream ceases to flow nearly parallel to the boundary surface. This is
discussed in the next section.
8. Separation
We have seen in the last section that the boundary layer in a decelerating stream
has a point of inflection and grows rapidly. The existence of the point of inflection
implies a slowing down of the region next to the wall, a consequence of the uphill
pressure gradient. Under a strong enough adverse pressure gradient, the flow next
to the wall reverses direction, resulting in a region of backward flow (Figure 10.14).
The reversed flow meets the forward flow at some point S at which the fluid near the
surface is transported out into the mainstream. We say that the flow separates from
the wall. The separation point S is defined as the boundary between the forward flow
and backward flow of the fluid near the wall, where the stress vanishes:
∂u
=0
(separation).
∂y wall
It is apparent from the figure that one streamline intersects the wall at a definite angle
at the point of separation.
At lower Reynolds numbers the reversed flow downstream of the point of separation forms part of a large steady vortex behind the surface (see Figure 10.17 in
Section 9 for the range 4 < Re < 40). At higher Reynolds numbers, when the flow
8. Separation
Figure 10.14 Streamlines and velocity profiles near a separation point S. Point of inflection is indicated
by I. The dashed line represents u = 0.
has boundary layer characteristics, the flow downstream of separation is unsteady and
frequently chaotic.
How strong an adverse pressure gradient the boundary layer can withstand without undergoing separation depends on the geometry of the flow, and whether the
boundary layer is laminar or turbulent. A steep pressure gradient, such as that behind
a blunt body, invariably leads to a quick separation. In contrast, the boundary layer on
the trailing surface of a thin body can overcome the weak pressure gradients involved.
Therefore, to avoid separation and large drag, the trailing section of a submerged body
should be gradually reduced in size, giving it a so-called streamlined shape.
Evidence indicates that the point of separation is insensitive to the Reynolds
number as long as the boundary layer is laminar. However, a transition to turbulence
delays boundary layer separation; that is, a turbulent boundary layer is more capable
of withstanding an adverse pressure gradient. This is because the velocity profile
in a turbulent boundary layer is “fuller” (Figure 10.15) and has more energy. For
example, the laminar boundary layer over a circular cylinder separates at 82◦ from
the forward stagnation point, whereas a turbulent layer over the same body separates
at 125◦ (shown later in Figure 10.17). Experiments show that the pressure remains
fairly uniform downstream of separation and has a lower value than the pressures on
the forward face of the body. The resulting drag due to pressure forces is called form
drag, as it depends crucially on the shape of the body. For a blunt body the form drag
is larger than the skin friction drag because of the occurrence of separation. (For a
streamlined body, skin friction is generally larger than the form drag.) As long as
the separation point is located at the same place on the body, the drag coefficient
of a blunt body is nearly constant at high Reynolds numbers. However, the drag
coefficient drops suddenly when the boundary layer undergoes transition to turbulence
(see Figure 10.22 in Section 9). This is because the separation point then moves
downstream, and the wake becomes narrower.
Separation takes place not only in external flows, but also in internal flows such as
that in a highly divergent channel (Figure 10.16). Upstream of the throat the pressure
367
368
Boundary Layers and Related Topics
Figure 10.15 Comparison of laminar and turbulent velocity profiles in a boundary layer.
Figure 10.16 Separation of flow in a highly divergent channel.
gradient is favorable and the flow adheres to the wall. Downstream of the throat a
large enough adverse pressure gradient can cause separation.
The boundary layer equations are valid only as far downstream as the point of
separation. Beyond it the boundary layer becomes so thick that the basic underlying assumptions become invalid. Moreover, the parabolic character of the boundary
layer equations requires that a numerical integration is possible only in the direction of advection (along which information is propagated), which is upstream within
the reversed flow region. A forward (downstream) integration of the boundary layer
equations therefore breaks down after the separation point. Last, we can no longer
apply potential theory to find the pressure distribution in the separated region, as the
effective boundary of the irrotational flow is no longer the solid surface but some
unknown shape encompassing part of the body plus the separated region.
9. Description of Flow past a Circular Cylinder
In general, analytical solutions of viscous flows can be found (possibly in terms
of perturbation series) only in two limiting cases, namely Re ≪ 1 and Re ≫ 1.
9. Description of Flow past a Circular Cylinder
In the Re ≪ 1 limit the inertia forces are negligible over most of the flow field; the
Stokes–Oseen solutions discussed in the preceding chapter are of this type. In the
opposite limit of Re ≫ 1, the viscous forces are negligible everywhere except close
to the surface, and a solution may be attempted by matching an irrotational outer
flow with a boundary layer near the surface. In the intermediate range of Reynolds
numbers, finding analytical solutions becomes almost an impossible task, and one has
to depend on experimentation and numerical solutions. Some of these experimental
flow patterns will be described in this section, taking the flow over a circular cylinder
as an example. Instead of discussing only the intermediate Reynolds number range,
we shall describe the experimental data for the entire range of small to very high
Reynolds numbers.
Low Reynolds Numbers
Let us start with a consideration of the creeping flow around a circular cylinder,
characterized by Re < 1. (Here we shall define Re = U∞ d/ν, based on the upstream
velocity and the cylinder diameter.) Vorticity is generated close to the surface because
of the no-slip boundary condition. In the Stokes approximation this vorticity is simply
diffused, not advected, which results in a fore and aft symmetry. The Oseen approximation partially takes into account the advection of vorticity, and results in an asymmetric velocity distribution far from the body (which was shown in Figure 9.17). The
vorticity distribution is qualitatively analogous to the dye distribution caused by a
source of colored fluid at the position of the body. The color diffuses symmetrically
in very slow flows, but at higher flow speeds the dye source is confined behind a
parabolic boundary with the dye source at the focus.
As Re is increased beyond 1, the Oseen approximation breaks down, and the vorticity is increasingly confined behind the cylinder because of advection. For Re > 4,
two small attached or “standing” eddies appear behind the cylinder. The wake is completely laminar and the vortices act like “fluidynamic rollers” over which the main
stream flows (Figure 10.17). The eddies get longer as Re is increased.
von Karman Vortex Street
A very interesting sequence of events begins to develop when the Reynolds number is
increased beyond 40, at which point the wake behind the cylinder becomes unstable.
Photographs show that the wake develops a slow oscillation in which the velocity
is periodic in time and downstream distance, with the amplitude of the oscillation
increasing downstream. The oscillating wake rolls up into two staggered rows of
vortices with opposite sense of rotation (Figure 10.18). von Karman investigated the
phenomenon as a problem of superposition of irrotational vortices; he concluded that
a nonstaggered row of vortices is unstable, and a staggered row is stable only if the
ratio of lateral distance between the vortices to their longitudinal distance is 0.28.
Because of the similarity of the wake with footprints in a street, the staggered row
of vortices behind a blunt body is called a von Karman vortex street. The vortices
move downstream at a speed smaller than the upstream velocity U∞ . This means
that the vortex pattern slowly follows the cylinder if it is pulled through a stationary
fluid.
369
370
Boundary Layers and Related Topics
Figure 10.17 Some regimes of flow over a circular cylinder.
Figure 10.18 von Karman vortex street downstream of a circular cylinder at Re = 55. Flow visualized by
condensed milk. S. Taneda, Jour. Phys. Soc., Japan 20: 1714–1721, 1965, and reprinted with the permission
of The Physical Society of Japan and Dr. Sadatoshi Taneda.
In the range 40 < Re < 80, the vortex street does not interact with the pair
of attached vortices. As Re is increased beyond 80 the vortex street forms closer to
the cylinder, and the attached eddies (whose downstream length has now grown to be
about twice the diameter of the cylinder) themselves begin to oscillate. Finally the
attached eddies periodically break off alternately from the two sides of the cylinder.
While an eddy on one side is shed, that on the other side forms, resulting in an unsteady
flow near the cylinder. As vortices of opposite circulations are shed off alternately
9. Description of Flow past a Circular Cylinder
from the two sides, the circulation around the cylinder changes sign, resulting in
an oscillating “lift” or lateral force. If the frequency of vortex shedding is close
to the natural frequency of some mode of vibration of the cylinder body, then an
appreciable lateral vibration has been observed to result. Engineering structures such
as suspension bridges and oil drilling platforms are designed so as to break up a
coherent shedding of vortices from cylindrical structures. This is done by including
spiral blades protruding out of the cylinder surface, which break up the spanwise
coherence of vortex shedding, forcing the vortices to detach at different times along
the length of these structures (Figure 10.19).
The passage of regular vortices causes velocity measurements in the wake to have
a dominant periodicity. The frequency n is expressed as a nondimensional parameter
known as the Strouhal number, defined as
S≡
nd
.
U∞
Experiments show that for a circular cylinder the value of S remains close to 0.21 for a
large range of Reynolds numbers. For small values of cylinder diameter and moderate
values of U∞ , the resulting frequencies of the vortex shedding and oscillating lift lie
in the acoustic range. For example, at U∞ = 10 m/s and a wire diameter of 2 mm,
the frequency corresponding to a Strouhal number of 0.21 is n = 1050 cycles per
second. The “singing” of telephone and transmission lines has been attributed to this
phenomenon.
Wen and Lin (2001) conducted very careful experiments that purported to be
strictly two-dimensional by using both horizontal and vertical soap film water tunnels. They give a review of the recent literature on both the computational and experimental aspects of this problem. The asymptote cited here of S = 0.21 is for a flow
including three-dimensional instabilities. Their experiments are in agreement with
two-dimensional computations and the data are asymptotic to S = 0.2417.
Below Re = 200, the vortices in the wake are laminar and continue to be so for
very large distances downstream. Above 200, the vortex street becomes unstable and
Figure 10.19 Spiral blades used for breaking up the spanwise coherence of vortex shedding from a
cylindrical rod.
371
372
Boundary Layers and Related Topics
irregular, and the flow within the vortices themselves becomes chaotic. However, the
flow in the wake continues to have a strong frequency component corresponding to
a Strouhal number of S = 0.21. Above a very high Reynolds number, say 5000, the
periodicity in the wake becomes imperceptible, and the wake may be described as
completely turbulent.
Striking examples of vortex streets have also been observed in the atmosphere.
Figure 10.20 shows a satellite photograph of the wake behind several isolated mountain peaks, through which the wind is blowing toward the southeast. The mountains
pierce through the cloud level, and the flow pattern becomes visible by the cloud
Figure 10.20 A von Karman vortex street downstream of mountain peaks in a strongly stratified atmosphere. There are several mountain peaks along the linear, light-colored feature running diagonally in the
upper left-hand corner of the photograph. North is upward, and the wind is blowing toward the southeast.
R. E. Thomson and J. F. R. Gower, Monthly Weather Review 105: 873–884, 1977, and reprinted with the
permission of the American Meteorlogical Society.
373
9. Description of Flow past a Circular Cylinder
pattern. The wakes behind at least two mountain peaks display the characteristics of a
von Karman vortex street. The strong density stratification in this flow has prevented
a vertical motion, giving the flow the two-dimensional character necessary for the
formation of vortex streets.
High Reynolds Numbers
At high Reynolds numbers the frictional effects upstream of separation are confined
near the surface of the cylinder, and the boundary layer approximation becomes
valid as far downstream as the point of separation. For Re < 3 × 105 , the boundary
layer remains laminar, although the wake may be completely turbulent. The laminar
boundary layer separates at ≈ 82◦ from the forward stagnation point (Figure 10.17).
The pressure in the wake downstream of the point of separation is nearly constant and
lower than the upstream pressure (Figure 10.21). As the drag in this range is primarily
due to the asymmetry in the pressure distribution caused by separation, and as the
point of separation remains fairly stationary in this range, the drag coefficient also
stays constant at CD ≃ 1.2 (Figure 10.22).
Important changes take place beyond the critical Reynolds number of
Recr ∼ 3 × 105
(circular cylinder).
Figure 10.21 Surface pressure distribution around a circular cylinder at subcritical and supercritical
Reynolds numbers. Note that the pressure is nearly constant within the wake and that the wake is narrower
for flow at supercritical Re.
374
Boundary Layers and Related Topics
Figure 10.22 Measured drag coefficient of a circular cylinder. The sudden dip is due to the transition of
the boundary layer to turbulence and the consequent downstream movement of the point of separation.
In the range 3 × 105 < Re < 3 × 106 , the laminar boundary layer becomes unstable
and undergoes transition to turbulence. We have seen in the preceding section that
because of its greater energy, a turbulent boundary layer, is able to overcome a larger
adverse pressure gradient. In the case of a circular cylinder the turbulent boundary
layer separates at 125◦ from the forward stagnation point, resulting in a thinner wake
and a pressure distribution more similar to that of potential flow. Figure 10.21 compares the pressure distributions around the cylinder for two values of Re, one with a
laminar and the other with a turbulent boundary layer. It is apparent that the pressures
within the wake are higher when the boundary layer is turbulent, resulting in a sudden
drop in the drag coefficient from 1.2 to 0.33 at the point of transition. For values of
Re > 3 × 106 , the separation point slowly moves upstream as the Reynolds number
is increased, resulting in an increase of the drag coefficient (Figure 10.22).
It should be noted that the critical Reynolds number at which the boundary
layer undergoes transition is strongly affected by two factors, namely the intensity
of fluctuations existing in the approaching stream and the roughness of the surface,
an increase in either of which decreases Recr . The value of 3 × 105 is found to be
valid for a smooth circular cylinder at low levels of fluctuation of the oncoming
stream.
Before concluding this section we shall note an interesting anecdote about the
von Karman vortex street. The pattern was investigated experimentally by the French
physicist Henri Bénard, well-known for his observations of the instability of a layer
of fluid heated from below. In 1954 von Karman wrote that Bénard became “jealous
because the vortex street was connected with my name, and several times . . . claimed
priority for earlier observation of the phenomenon. In reply I once said ‘I agree that
what in Berlin and London is called Karman Street in Paris shall be called Avenue
de Henri Bénard.’ After this wisecrack we made peace and became good friends.”
von Karman also says that the phenomenon has been known for a long time and is
even found in old paintings.
10. Description of Flow past a Sphere
We close this section by noting that this flow illustrates three instances where the
solution is counterintuitive. First, small causes can have large effects. If we solve for
the flow of a fluid with zero viscosity around a circular cylinder, we obtain the results
of Chapter 6, Section 9. The inviscid flow has fore-aft symmetry and the cylinder
experiences zero drag. The bottom two panels of Figure 10.17 illustrate the flow for
small viscosity. For viscosity as small as you choose, in the limit viscosity tends
to zero, the flow must look like the last panel in which there is substantial fore-aft
asymmetry, a significant wake, and significant drag. This is because of the necessity
of a boundary layer and the satisfaction of the no-slip boundary condition on the
surface so long as viscosity is not exactly zero. When viscosity is exactly zero, there
is no boundary layer and there is slip at the surface. The resolution of d’Alembert’s
paradox is through the boundary layer, a singular perturbation of the Navier–Stokes
equations in the direction normal to the boundary.
The second instance of counterintuitivity is that symmetric problems can have
nonsymmetric solutions. This is evident in the intermediate Reynolds number middle
panel of Figure 10.17. Beyond a Reynolds number of ≈ 40 the symmetric wake
becomes unstable and a pattern of alternating vortices called a von Karman vortex
street is established. Yet the equations and boundary conditions are symmetric about a
central plane in the flow. If one were to solve only a half-problem, assuming symmetry,
a solution would be obtained, but it would be unstable to infinitesimal disturbances
and unlikely to be seen in the laboratory.
The third instance of counterintuitivity is that there is a range of Reynolds numbers where roughening the surface of the body can reduce its drag. This is true for
all blunt bodies, such as a sphere (to be discussed in the next section). In this range
of Reynolds numbers, the boundary layer on the surface of a blunt body is laminar,
but sensitive to disturbances such as surface roughness, which would cause earlier
transition of the boundary layer to turbulence than would occur on a smooth body.
Although, as we shall see, the skin friction of a turbulent boundary layer is much
larger than that of a laminar boundary layer, most of the drag is caused by incomplete
pressure recovery on the downstream side of a blunt body as shown in Figure 10.21,
rather than by skin friction. In fact, it is because the skin friction of a turbulent boundary layer is much larger, as a result of a larger velocity gradient at the surface, that
a turbulent boundary layer can remain attached farther on the downstream side of a
blunt body, leading to a narrower wake and more complete pressure recovery and thus
reduced drag. The drag reduction attributed to the turbulent boundary layer is shown
in Figure 10.22 for a circular cylinder and Figure 10.23 for a sphere.
10. Description of Flow past a Sphere
Several features of the description of flow over a circular cylinder qualitatively apply
to flows over other two-dimensional blunt bodies. For example, a vortex street is
observed in a flow perpendicular to a flat plate. The flow over a three-dimensional
body, however, has one fundamental difference in that a regular vortex street is absent.
For flow around a sphere at low Reynolds numbers, there is an attached eddy in the
form of a doughnut-shaped ring; in fact, an axial section of the flow looks similar to
375
376
Boundary Layers and Related Topics
Figure 10.23 Measured drag coefficient of a smooth sphere. The Stokes solution is CD = 24/Re, and the
Oseen solution is CD = (24/Re)(1 + 3Re/16); these two solutions are discussed in Chapter 9, Sections 12
and 13. The increase of drag coefficient in the range AB has relevance in explaining why the flight paths
of sports balls bend in the air.
that shown in Figure 10.17 for the range 4 < Re < 40. For Re > 130 the ring-eddy
oscillates, and some of it breaks off periodically in the form of distorted vortex
loops.
The behavior of the boundary layer around a sphere is similar to that around
a circular cylinder. In particular it undergoes transition to turbulence at a critical
Reynolds number of
Recr ∼ 5 × 105
(sphere),
which corresponds to a sudden dip of the drag coefficient (Figure 10.23). As in the
case of a circular cylinder, the separation point slowly moves upstream for postcritical
Reynolds numbers, accompanied by a rise in the drag coefficient. The behavior of the
separation point for flow around a sphere at subcritical and supercritical Reynolds
numbers is responsible for the bending in the flight paths of sports balls, as explained
in the following section.
11. Dynamics of Sports Balls
The discussion of the preceding section could be used to explain why the trajectories
of sports balls (such as those involved in tennis, cricket, and baseball games) bend in
the air. The bending is commonly known as swing, swerve, or curve. The problem has
been investigated by wind tunnel tests and by stroboscopic photographs of flight paths
11. Dynamics of Sports Balls
in field tests, a summary of which was given by Mehta (1985). Evidence indicates
that the mechanics of bending is different for spinning and nonspinning balls. The
following discussion gives a qualitative explanation of the mechanics of flight path
bending. (Readers not interested in sports may omit this section!)
Cricket Ball Dynamics
The cricket ball has a prominent (1-mm high) seam, and tests show that the orientation
of the seam is responsible for bending of the ball’s flight path. It is known to bend when
thrown at high speeds of around 30 m/s, which is equivalent to a Reynolds number of
Re = 105 . Here we shall define the Reynolds number as Re = U∞ d/ν, based on the
translational speed U∞ of the ball and its diameter d. The operating Reynolds number
is somewhat less than the critical value of Recr = 5 × 105 necessary for transition of
the boundary layer on a smooth sphere into turbulence. However, the presence of the
seam is able to trip the laminar boundary layer into turbulence on one side of the ball
(the lower side in Figure 10.24), while the boundary layer on the other side remains
laminar. We have seen in the preceding sections that because of greater energy a turbulent boundary layer separates later. Typically, the boundary layer on the laminar side
separates at ≈ 85◦ , whereas that on the turbulent side separates at 120◦ . Compared
to region B, the surface pressure near region A is therefore closer to that given by the
2 )=
potential flow theory (which predicts a suction pressure of (pmin − p∞ )/( 21 ρU∞
−1.25; see equation (6.81)). In other words, the pressures are lower on side A, resulting in a downward force on the ball. (Note that Figure 10.24 is a view of the flow
pattern looking downward on the ball, so that it corresponds to a ball that bends to
the left in its flight. The flight of a cricket ball oriented as in Figure 10.24 is called an
“outswinger” in cricket literature, in contrast to an “inswinger” for which the seam is
oriented in the opposite direction so as to generate an upward force in Figure 10.24.)
Figure 10.25, a photograph of a cricket ball in a wind tunnel experiment, clearly
shows the delayed separation on the seam side. Note that the wake has been deflected
upward by the presence of the ball, implying that an upward force has been exerted
Figure 10.24 The swing of a cricket ball. The seam is oriented in such a way that the lateral force on the
ball is downward in the figure.
377
378
Boundary Layers and Related Topics
Figure 10.25 Smoke photograph of flow over a cricket ball. Flow is from left to right. Seam angle is 40◦ ,
flow speed is 17 m/s, Re = 0.85 × 105 . R. Mehta, Ann. Rev Fluid Mech. 17: 151–189, 1985. Photograph
c 1985 Annual Reviews
reproduced with permission from the Annual Review of Fluid Mechanics, Vol. 17
www.AnnualReviews.org.
by the ball on the fluid. It follows that a downward force has been exerted by the fluid
on the ball.
In practice some spin is invariably imparted to the ball. The ball is held along the
seam and, because of the round arm action of the bowler, some backspin is always
imparted along the seam. This has the important effect of stabilizing the orientation
of the ball and preventing it from wobbling. A typical cricket ball can generate side
forces amounting to almost 40% of its weight. A constant lateral force oriented in
the same direction causes a deflection proportional to the square of time. The ball
therefore travels in a parabolic path that can bend as much as 0.8 m by the time it
reaches the batsman.
It is known that the trajectory of the cricket ball does not bend if the ball is thrown
too slow or too fast. In the former case even the presence of the seam is not enough
to trip the boundary layer into turbulence, and in the latter case the boundary layer
on both sides could be turbulent; in both cases an asymmetric flow is prevented. It is
also clear why only a new, shiny ball is able to swing, because the rough surface of an
old ball causes the boundary layer to become turbulent on both sides. Fast bowlers in
cricket maintain one hemisphere of the ball in a smooth state by constant polishing.
It therefore seems that most of the known facts about the swing of a cricket ball
have been adequately explained by scientific research. The feature that has not been
explained is the universally observed fact that a cricket ball swings more in humid
11. Dynamics of Sports Balls
conditions. The changes in density and viscosity due to changes in humidity can
change the Reynolds number by only 2%, which cannot explain this phenomenon.
Tennis Ball Dynamics
Unlike the cricket ball, the path of the tennis ball bends because of spin. A ball hit
with topspin curves downward, whereas a ball hit with underspin travels in a much
flatter trajectory. The direction of the lateral force is therefore in the same sense as
that of the Magnus effect experienced by a circular cylinder in potential flow with
circulation (see Chapter 6, Section 10). The mechanics, however, are different. The
potential flow argument (involving the Bernoulli equation) offered to account for the
lateral force around a circular cylinder cannot explain why a negative Magnus effect
is universally observed at lower Reynolds numbers. (By a negative Magnus effect we
mean a lateral force opposite to that experienced by a cylinder with a circulation of
the same sense as the rotation of the sphere.) The correct argument seems to be the
asymmetric boundary layer separation caused by the spin. In fact, the phenomenon
was not properly explained until the boundary layer concepts were understood in
the twentieth century. Some pioneering experimental work on the bending paths
of spinning spheres was conducted by Robins about two hundred years ago; the
deflection of rotating spheres is sometimes called the Robins effect.
Experimental data on nonrotating spheres (Figure 10.23) shows that the boundary
layer on a sphere undergoes transition at a Reynolds number of ≈ Re = 5 × 105 ,
indicated by a sudden drop in the drag coefficient. As discussed in the preceding
section, this drop is due to the transition of the laminar boundary layer to turbulence.
An important point for our discussion here is that for supercritical Reynolds numbers
the separation point slowly moves upstream, as evidenced by the increase of the drag
coefficient after the sudden drop shown in Figure 10.23.
With this background, we are now in a position to understand how a spinning
ball generates a negative Magnus effect at Re < Recr and a positive Magnus effect
at Re > Recr . For a clockwise rotation of the ball, the fluid velocity relative to the
surface is larger on the lower side (Figure 10.26). For the lower Reynolds number
case (Figure 10.26a), this causes a transition of the boundary layer on the lower side,
while the boundary layer on the upper side remains laminar. The result is a delayed
separation and lower pressure on the bottom surface, and a consequent downward
force on the ball. The force here is in a sense opposite to that of the Magnus effect.
The rough surface of a tennis ball lowers the critical Reynolds number, so that
for a well-hit tennis ball the boundary layers on both sides of the ball have already
undergone transition. Due to the higher relative velocity, the flow near the bottom has
a higher Reynolds number, and is therefore farther along the Re-axis of Figure 10.23,
in the range AB in which the separation point moves upstream with an increase of
the Reynolds number. The separation therefore occurs earlier on the bottom side,
resulting in a higher pressure there than on the top. This causes an upward lift force
and a positive Magnus effect. Figure 10.26b shows that a tennis ball hit with underspin generates an upward force; this overcomes a large fraction of the weight of the
ball, resulting in a much flatter trajectory than that of a tennis ball hit with topspin.
A “slice serve,” in which the ball is hit tangentially on the right-hand side, curves to
379
380
Boundary Layers and Related Topics
Figure 10.26 Bending of rotating spheres, in which F indicates the force exerted by the fluid: (a) negative
Magnus effect; and (b) positive Magnus effect. A well-hit tennis ball is likely to display the positive Magnus
effect.
Figure 10.27 Smoke photograph of flow around a spinning baseball. Flow is from left to right, flow
speed is 21 m/s, and the ball is spinning counterclockwise at 15 rev/s. [Photograph by F. N. M. Brown,
University of Notre Dame.] Photograph reproduced with permission, from the Annual Review of Fluid
c 1985 by Annual Reviews www.AnnualReviews.org.
Mechanics, Vol. 17
the left due to the same effect. (Presumably soccer balls curve in the air due to similar
dynamics.)
Baseball Dynamics
A baseball pitcher uses different kinds of deliveries, a typical Reynolds number being
1.5 × 105 . One type of delivery is called a “curveball,” caused by sidespin imparted
by the pitcher to bend away from the side of the throwing arm. A “screwball” has
the opposite spin and curved trajectory. The dynamics of this is similar to that of
a spinning tennis ball (Figure 10.26b). Figure 10.27 is a photograph of the flow
12. Two-Dimensional Jets
over a spinning baseball, showing an asymmetric separation, a crowding together of
streamlines at the bottom, and an upward deflection of the wake that corresponds to
a downward force on the ball.
The knuckleball, on the other hand, is released without any spin. In this case
the path of the ball bends due to an asymmetric separation caused by the orientation
of the seam, much like the cricket ball. However, the cricket ball is released with
spin along the seam, which stabilizes the orientation and results in a predictable
bending. The knuckleball, on the other hand, tumbles in its flight because of a lack
of stabilizing spin, resulting in an irregular orientation of the seam and a consequent
irregular trajectory.
12. Two-Dimensional Jets
So far we have considered boundary layers over a solid surface. The concept of
a boundary layer, however, is more general, and the approximations involved are
applicable if the vorticity is confined in thin layers without the presence of a solid
surface. Such a layer can be in the form of a jet of fluid ejected from an orifice, a wake
(where the velocity is lower than the upstream velocity) behind a solid object, or a
mixing layer (vortex sheet) between two streams of different speeds. As an illustration
of the method of analysis of these “free shear flows,” we shall consider the case of
a laminar two-dimensional jet, which is an efflux of fluid from a long and narrow
orifice. The surrounding is assumed to be made up of the same fluid as the jet itself,
and some of this ambient fluid is carried along with the jet by the viscous drag at the
outer edge of the jet (Figure 10.28). The process of drawing in the surrounding fluid
from the sides of the jet by frictional forces is called entrainment.
The velocity distribution near the opening of the jet depends on the details of
conditions upstream of the orifice exit. However, because of the absence of an externally imposed length scale in the downstream direction, the velocity profile in the
jet approaches a self-similar shape not far from the exit, regardless of the velocity
distribution at the orifice.
Figure 10.28 Laminar two-dimensional jet. A typical streamline showing entrainment of surrounding
fluid is indicated.
381
382
Boundary Layers and Related Topics
For large Reynolds numbers, the jet is narrow and the boundary layer
approximation can be applied. Consider a control volume with sides cutting across the
jet axis at two sections (Figure 10.28); the other two sides of the control volume are
taken at large distances from the jet axis. No external pressure gradient is maintained
in the surrounding fluid, in which dp/dx is zero. According to the boundary layer
approximation, the same zero pressure gradient is also impressed upon the jet. There
is, therefore, no net force acting on the surfaces of the control volume, which requires
that the rate of flow of x-momentum at the two sections across the jet are the same.
Let uo (x) be the streamwise velocity on the x-axis and assume Re = uo x/ν is
sufficiently large for the boundary layer equations to be valid. The flow is steady,
two-dimensional (x, y), without body forces, and with constant properties (ρ, µ).
Then ∂/∂y ≫ ∂/∂x, v ≪ u, ∂p/∂y = 0, so
∂u/∂x + ∂v/∂y = 0,
(10.48)
2
u∂u/∂x + v∂u/∂y = ν∂ u/∂y
2
(10.49)
subject to the boundary conditions: y → ±∞ : u = 0; y = 0 : v = 0; x = xo :
u = ũ(xo , y). Form u· [equation (10.48)] + equation (10.49) and integrate over all y:
∞
−∞
2u(∂u/∂x)dy +
d/dx
∞
−∞
∞
−∞
(u∂v/∂y + v∂u/∂y)dy = ν∂u/∂y|∞
−∞
∞
u2 dy + uv|∞
−∞ = ν∂u/∂y|−∞ .
Since u(y = ±∞) = 0, all derivatives of u with repect to y must also be zero at
y = ±∞. Then the streamwise momentum flux must be preserved,
d/dx
∞
−∞
ρu2 dy = 0
(10.50)
Far enough downstream that (a) the boundary layer equations are valid, and (b) the
initial distribution ũ(xo , y), specified at the upstream limit of validity of the boundary
layer equations, is forgotten, a similarity solution is obtained. This similarity solution is of the universal dimensionless similarity form for the laminar boundary layer
equations, that is,
ψ(x, y) = [xνuo (x)]1/2 f (η), η = (y/x)[xuo (x)/ν]1/2 ,
Rex = xuo (x)/ν
(10.51)
where ψ is the usual streamfunction, u = −k × ∇ψ, and f and η are dimensionless.
We obtain the behavior of uo (x) by substitution of the similarity transformation
(10.51) into the condition (10.50)
383
12. Two-Dimensional Jets
u = ∂ψ/∂y = uo (x)f ′ (η), dy = dη[νx/uo (x)]1/2
ρd/dx{u2o (x)[νx/uo (x)]1/2
∞
−∞
f ′2 (η)dη} = 0.
3/2
Since the integral is a pure constant, we must have uo (x) · x 1/2 = C 3/2 where C
is a dimensional constant. Then uo = Cx −1/3 . C is clearly related to the intensity or
momentum flux in the jet. Now, (10.51) becomes
ψ(x, y) = (νC)1/2 · x 1/3 f (η), η = (C/ν)1/2 · y/x 2/3
In terms of the streamfunction, (10.49) may be written
∂ψ/∂y · ∂ 2 ψ/∂y∂x − ∂ψ/∂x · ∂ 2 ψ/∂y 2 = ν∂ 3 ψ/∂y 3 .
Evaluating the derivatives of the streamfunction and substituting into the
x-momentum equation, we obtain
3f ′′′ + ff ′′ + f ′2 = 0
subject to the boundary conditions
η = ±∞ : f ′ = 0; η = 0 : f = 0.
Integrating once,
3f ′′ + ff ′ = C1 .
Evaluating at η = ±∞, C1 = 0. Integrating again,
3f ′ + f 2 /2 = 18C22 ,
where the constant of integration is chosen to be “18C22 ” for convenience in the next
integration, as will be seen. Now consider the transformation f/6 = g ′ /g, so that
f ′ /6 = g ′′ /g − g ′2 /g 2 . This results in g ′′ − C22 g = 0. The solution for g is
g = C3 exp(C2 η) + C4 exp(−C2 η).
Then
f = 6g ′ /g = 6C2 [C3 exp(C2 η) − C4 exp(−C2 η)]/[C3 exp(C2 η)
+ C4 exp(−C2 η)].
Now, f ′ = 6C22 − f 2 /6 = 6C22 {1 − [(C3 eC2 η − C4 e−C2 η )/(C3 eC2 η + C4 e−C2 η )]2 }
must be even in η. Or, use the boundary condition f (0) = 0. This requires C3 = C4 .
Then
f ′ (η) = 6C22 [1 − tanh2 (C2 η)] and f (η) = 6C2 tanh(C2 η). Thus
f ′ (η) = 6C22 sech2 (C2 η).
384
Boundary Layers and Related Topics
To obtain C2 , recall u(x, y = 0) = uo (x)f ′ (0) = Cx −1/3 · 6C22 = uo (x) by our
definition of uo (x).
√
√
Thus 6C22 = 1 and C2 = 1/ 6. Then f ′ (η) = sech2 (η/ 6) and u(x, y) =
√
uo (x) sech2 (η/ 6). The constant “C” in uo (x) = Cx −1/3 is related to the momen√
∞
∞
ρu2 dy = 2ρC 3/2 ν 1/2 sech4 (η/ 6)dη =
tum flux in the jet via F =
−∞
0√
force per unit√depth. Carrying out the integration, F = (4 6/3)ρC 3/2 ν 1/2 , so
C = [3F /(4 6ρν 1/2 )]2/3 , in terms of the jet force per unit depth or momentum
flux. The mass flux in the jet is
ṁ =
∞
−∞
ρudy = ρ
∞
−∞
uo (x)f ′ (η)dη · [νx/uo (x)]1/2 = (36ρ 2 νF )1/3 x 1/3 .
This grows downstream because of entrainment in the jet. The entrainment may be
seen as inward flow (y component of velocity) from afar.
v = −∂ψ/∂x = −(νC)1/2 x −2/3 (f − 2ηf ′ )/3, so
v/uo = −(f − 2ηf ′ )/(3 Rex ), Rex = xuo (x)/ν.
As
√
6/(3 Rex ),
√
η → −∞, v/uo → + 6/(3 Rex ),
η → ∞, v/uo → −
downwards toward jet
upwards toward jet.
Thus the entrainment is an inward flow of mass from above and below.√
′
2
The jet spreads as it travels
√ downstream. Now f (η) = sech (η/ 6). If η = 5
is taken as width of jet, 5/ 6 = 2.04 and f ′ (2.04) = .065. Calling
√the transverse
extent y of the jet, δ, we have 5 ≈ (δ/x)(Cx 2/3 /ν)1/2 so that δ ≈ 5 ν/Cx 2/3 . The
jet grows downstream x 2/3 . We can express the Reynolds numbers in terms of the
force or momentum flux in the jet, F
√
Rex = Cx 2/3 /ν = [3F x/(4 6ρν 2 )]2/3 ,
√
Reδ = uo δ/ν = 5 · [3F x/(4 6ρν 2 )]1/3 .
and
By drawing sketches of the profiles of u, u2 , and u3 , the reader can verify that,
under similarity, the constraint
∞
d
u2 dy = 0,
dx −∞
must lead to
d
dx
∞
−∞
u dy > 0,
385
12. Two-Dimensional Jets
and
d
dx
∞
u3 dy < 0.
−∞
The laminar jet solution given here is not readily observable because the flow
easily breaks up into turbulence. The low critical Reynolds number for instability of
a jet or wake is associated with the existence of a point of inflection in the velocity
profile, as discussed in Chapter 12. Nevertheless, the laminar solution has revealed
several significant ideas (namely constancy of momentum flux and increase of mass
flux) that also apply to a turbulent jet. However, the rate of spreading of a turbulent
jet is faster, being more like δ ∝ x rather than δ ∝ x 2/3 (see Chapter 13).
The Wall Jet
An example of a two-dimensional jet that also shares some boundary layer characteristics is the “wall jet.” The solution here is due to M. B. Glauert (1956). We consider a
fluid exiting a narrow slot with its lower boundary being a planar wall taken along the
x-axis (see Figure 10.29). Near the wall y = 0 and the flow behaves like a boundary
layer, but far from the wall it behaves like a free jet. The boundary layer analysis
shows that for large Rex the jet is thin (δ/x ≪ 1) so ∂p/∂y ≈ 0 across it. The
pressure is constant in the nearly stagnant outer fluid so p ≈ const. throughout the
flow. The boundary layer equations are
∂u ∂v
+
= 0,
∂x
∂y
∂u
∂ 2u
∂u
+v
= ν 2,
u
∂x
∂y
∂y
(10.52)
(10.53)
subject to the boundary conditions y = 0: u = v = 0; y → ∞: u → 0. With an
initial velocity distribution forgotten sufficiently far downstream that Rex → ∞, a
similarity solution is available. However, unlike the free jet, the momentum flux is
not constant; instead, it diminishes downstream because of the wall shear stress. To
obtain the conserved property in the wall jet, we start by integrating equation (10.53)
from y to ∞:
∞
∞
∂u
∂u
∂u
u
v
dy +
dy = −ν .
∂x
∂y
∂y
y
y
Figure 10.29 The planar wall jet.
386
Boundary Layers and Related Topics
Multiply this by u and integrate from 0 to ∞:
∞ 2
∞ ∞
∞
∂u
u
ν ∞ ∂ 2
∂
v
u
u
dy dy +
dy dy +
u dy = 0.
∂x y 2
∂y
2 0 ∂y
y
0
0
The last term integrates to 0 because of the boundary conditions at both ends. Integrating the second term by parts and using equation (10.52) yields a term equal to the
first term. Then we have
∞
∞
∞
∂
u
u2 v dy = 0.
(10.54)
u2 dy dy −
∂x y
0
0
Now consider
d
dx
0
∞
u
∞
y
u2 dy dy =
∞ ∂u
∞
u2 dy dy
∂x y
0
∞
∞
∂
u2 dy dy.
u
+
∂x y
0
Using equation (10.52) in the first term on the right-hand side, integrating by parts,
and using equation (10.54), we finally obtain
∞ ∞
d
2
u dy dy = 0.
(10.55)
u
dx 0
y
This says that the flux of exterior momentum flux is constant downstream and is used
as the second condition to obtain the similarity exponents. Rewriting equation (10.53)
in terms of the streamfunction u = ∂ψ/∂y, v = −∂ψ/∂x, we obtain
∂ψ ∂ 2 ψ
∂ 3ψ
∂ψ ∂ 2 ψ
−
=
ν
,
∂y ∂y ∂x
∂x ∂y 2
∂y 3
(10.56)
subject to:
y=0: ψ =
∂ψ
= 0;
∂y
y→∞:
∂ψ
→ 0.
∂y
(10.57)
Let ū(x) be some average or characteristic speed of the wall jet. We will be
able to relate this to the mass flow rate and width of the jet at the completion of
this discussion. We can write the universal dimensionless similarity scaling for the
laminar boundary layer equations in terms of ū(x), via
ψ(x, y) = [νx ū(x)]1/2 · f (η), η = (y/x) Rex = (y/x)[x ū(x)/ν]1/2 ,
and expect this similarity to hold when x ≫ xo , where xo is the location where the
initial condition is specified, which we take to be the upstream extent of the validity
of the boundary layer equations. Then u(x, y) = ∂ψ/∂y = ū(x)f ′ (η). Substituting
this into the conserved flux [(10.55)], we obtain
∞ ∞
3
d/dx{ū(x) (νx/ū) f ′ [
f ′2 dη]dη} = 0,
0
y
387
12. Two-Dimensional Jets
0.3
1.0
f
0.75
f
0.2
f9
0.5
f9
0.1
0.25
0
1
2
3
4
5
6
Figure 10.30 Variation of normalized mass flux (f ) and normalized velocity (f ′ ) with similarly variable
η. Reprinted with the permission of Cambridge University Press.
where we expect the integral to be independent of x. Then (ū)2 x = C 2 , or ū(x) =
Cx −1/2 . This gives us the similarity transformation
√
ψ(x, y) = νC · x 1/4 f (η), where η = (C/ν)1/2 · y/x 3/4 .
Differentiating and substituting into (10.56), we obtain (after multiplication by
4x 2 /C 2 ),
4f ′′′ + ff ′′ + 2f ′2 = 0
subject to the boundary conditions (10.57): f (0) = 0; f ′ (0) = 0; f ′ (∞) = 0.
This third order equation can be integrated once after multiplying by the integrating
factor f , to yield ff ′′ − f ′2 /2 + f 2 f ′ /4 = 0, where the constant of integration
has been evaluated at η = 0. Dividing by the integrating factor f 3/2 gives an equation
that can be integrated once more. The result is
3/2
f −1/2 f ′ + f 3/2 /6 = C1 ≡ f∞ /6,
where f∞ = f (∞).
Since f (0) = 0, f ′ (0) = 0, a Tayor series for f starts with f (η) = f ′′ (0)η2 /2.
3 /36. Since f and η are dimensionless, f is a
Then f ′2 (0)/f (0) = 2f ′′ (0) = f∞
∞
pure number. The final integration can be performed after one more transformation:
f/f∞ = g 2 (η̄), η̄ = f∞ η. This results in the equation dg/(1 − g 3 ) = d η̄/12. Now
1 − g 3 = (1 − g) · (1 + g + g 2 ), so integration may be effected by partial fractions,
with the result in implicit form,
√
√
√
√
− ln(1−g)+ 3 tan−1 [(2g +1)/ 3]+ln(1+g +g 2 )1/2 = η̄/4+ 3 tan−1 (1/ 3),
where the boundary condition g(0) = 0 was used to evaluate the constant of integration. We can verify easily that f ′ → 0 exponentially fast in η or η̄ from our
solution for g(η̄). √
As η̄ → √
∞, g → 1, so for large η̄ √
the solution √
for g reduces
−1 (1/ 3). The first
to − ln(1 − g) + 3 tan−1 3 + (1/2) ln 3 ∼
η̄/4
+
3
tan
=
term on each side of the equation dominates, leaving 1 − g ≈ e−(1/4)η̄ . Now
388
Boundary Layers and Related Topics
f ′ = g(1 − g)(1 + g + g 2 )/6 ≈ (1/2)e−(1/4)η̄ . The mass flow rate in the jet is
ṁ =
∞
0
or since
∞
ρudy = ρ ū(x) f ′ (η)dη ν/C · x 3/4 ,
0
√
ū = Cx −1/2 , ṁ = ρ νCf∞ x 1/4 ,
indicating that entrainment increases the flow rate in the jet with x 1/4 . If we define
the edge
√ of the−1jet3/4as δ(x) and say it corresponds to η̄ = 6, for example, then
δ = 6 ν/Cf∞
x . If we define ū by requiring ṁ = ρ ū(x)δ(x), the two forms
2 = 6. The entrainment is evident from the form of v =
for ṁ are coincident
if f∞
√
√
−∂ψ/∂x = − νC(f − 3ηf ′ )/(4x 3/4 ) → − νCf∞ /(4x 3/4 ) as η → ∞, so the
flow is downwards, toward the jet.
13. Secondary Flows
Large Reynolds number flows with curved streamlines tend to generate additional
velocity components because of properties of the boundary layer. These components are called secondary flows and will be seen later in our discussion of instabilities (p. 454). An example of such a flow is made dramatically visible by putting
finely crushed tea leaves, randomly dispersed, into a cup of water, and then stirring
vigorously in a circular motion. When the motion has ceased, all of the particles have
collected in a mound at the center of the bottom of the cup (see Figure 10.31). An
explanation of this phenomenon is given in terms of thin boundary layers. The stirring motion imparts a primary velocity uθ (R) (see Appendix B1 for coordinates) large
enough for the Reynolds number to be large enough for the boundary layers on the
sidewalls and bottom to be thin. The largest terms in the R-momentum equation are
ρu2θ
∂p
=
.
∂R
R
Away from the walls, the flow is inviscid. As the boundary layer on the bottom is
thin, boundary layer theory yields ∂p/∂x = 0 from the x-momentum equation. Thus
the pressure in the bottom boundary layer is the same as for the inviscid flow just
outside the boundary layer. However, within the boundary layer, uθ is less than the
inviscid value at the edge. Thus p(R) is everywhere larger in the boundary layer than
that required for circular streamlines inside the boundary layer, pushing the streamlines inwards. That is, the pressure gradient within the boundary layer generates an
inwardly directed uR . This motion is fed by a downwardly directed flow in the sidewall boundary layer and an outwardly directed flow on the top surface. This secondary
flow is closed by an upward flow along the center. The visualization is accomplished
by crushed tea leaves which are slightly denser than water. They descend by gravity
or are driven outwards by centrifugal acceleration. If they enter the sidewall boundary
layer, they are transported downwards and thence to the center by the secondary flow.
If the tea particles enter the bottom boundary layer from above, they are quickly swept
14. Perturbation Techniques
Figure 10.31 Secondary flow in a tea cup: (a) tea leaves randomly dispersed—initial state; (b) stirred
vigorously—transient motion; and (c) final state.
to the center and dropped as the flow turns upwards. All the particles collect at the
center of the bottom of the teacup. A practical application of this effect, illustrated in
Exercise 9, relates to sand and silt transport by the Mississippi River.
14. Perturbation Techniques
The preceding sections, based on Prandtl’s seminal idea, have revealed the physical
basis of the boundary layer concept in a high Reynolds number flow. In recent years,
the boundary layer method has become a powerful mathematical technique used to
solve a variety of other physical problems. Some elementary ideas involved in these
methods are discussed here. The interested reader should consult other specialized
texts on the subject, such as van Dyke (1975), Bender and Orszag (1978), and Nayfeh
(1981).
The essential idea is that the problem has a small parameter ε in either the
governing equation or in the boundary conditions. In a flow at high Reynolds number
the small parameter is ε = 1/Re, in a creeping flow ε = Re, and in flow around an
airfoil ε is the ratio of thickness to chord length. The solutions to these problems
389
390
Boundary Layers and Related Topics
can frequently be written in terms of a series involving the small parameter, the
higher-order terms acting as a perturbation on the lower-order terms. These methods
are called perturbation techniques. The perturbation expansions frequently break
down in certain regions, where the field develops boundary layers. The boundary
layers are treated differently than other regions by expressing the lateral coordinate y
in terms of the boundary layer thickness δ and defining η ≡ y/δ. The objective is to
rescale variables so that they are all finite in the thin singular region.
Order Symbols and Gauge Functions
Frequently we have a complicated function f (ε) and we want to determine the nature
of variation of f (ε) as ε → 0. The three possibilities are
f (ε) → 0 (vanishing)
f (ε) → A (bounded)
as ε → 0,
f (ε) → ∞ (unbounded)
where A is finite. However, this behavior is rather vague because it does not say
how fast f (ε) goes to zero or infinity as ε → 0. To describe this behavior, we
compare the rate at which f (ε) goes to zero or infinity with the rate at which certain
familiar functions go to zero or infinity. The familiar functions used for comparison
purposes are called gauge functions. The most common example of a sequence of
gauge functions is 1, ε, ε2 , ε3 , . . . . As an example, suppose we want to find how sin ε
goes to zero as ε → 0. Using the Taylor series
sin ε = ε −
we find that
ε5
ε3
+
− ··· ,
3!
5!
ε4
ε2
sin ε
= lim 1 −
+
− · · · = 1,
ε→0
ε→0 ε
3!
5!
lim
which shows that sin ε tends to zero at the same rate at which ε tends to zero.
Another way of expressing this is to say that sin ε is of order ε as ε → 0, which we
write as
sin ε = O(ε) as ε → 0.
Other examples are that
cos ε = O(1)
cos ε − 1 = O(ε 2 )
as ε → 0.
We can generalize the concept of “order” by the following statement. A function
f (ε) is considered to be of order of a gauge function g(ε), and written
f (ε) = O[g(ε)]
as ε → 0,
391
14. Perturbation Techniques
if
lim
ε→0
f (ε)
= A,
g(ε)
where A is nonzero and finite. Note that the size of the constant A is immaterial
as far as the mathematics is concerned. Thus, sin 7ε = O(ε) just as sin ε = O(ε),
and likewise 1000 = O(1). Thus, the mathematical order considered here is different
from the physical order of magnitude. However, if the physical problem has been
properly nondimensionalized, with the relevant scales judiciously chosen, then the
constant A will be of reasonable size. (Incidentally, we commonly regard a factor of
10 as a change of one physical order of magnitude, so when we say that the magnitude
of u is of order 10 cm/s, we mean that the magnitude of u is expected (or hoped!) to
be between 30 and 3 cm/s.)
Sometimes a comparison in terms of a familiar gauge function is unavailable or
inconvenient. We may say f (ε) = o[g(ε)] in the limit ε → 0 if
lim
ε→0
f (ε)
= 0,
g(ε)
so that f is small compared with g as ε → 0. For example, | ln ε| = o(1/ε) in the
limit ε → 0.
Asymptotic Expansion
An asymptotic expansion of a function, in terms of a given set of gauge functions, is
essentially a series representation with a finite number of terms. Suppose the sequence
of gauge functions is gn (ε), such that each one is smaller than the preceding one in
the sense that
gn+1
= 0.
lim
ε→0 gn
Then the asymptotic expansion of f (ε) is of the form
f (ε) = a0 + a1 g1 (ε) + a2 g2 (ε) + O[g3 (ε)],
(10.58)
where an are independent of ε. Note that the remainder, or the error, is of order of the
first neglected term. We also write
f (ε) ∼ a0 + a1 g1 (ε) + a2 g2 (ε),
where ∼ means “asymptotically equal to.” The asymptotic expansion of f (ε) as
ε → 0 is not unique, because a different choice of the gauge functions gn (ε) would
lead to a different expansion. A good choice leads to a good accuracy with only a few
terms in the expansion. The most frequently used sequence of gauge functions is the
power series ε n . However, in many cases the series in integral powers of ε does not
work, and other gauge functions must be used. There is a systematic way of arriving
at the sequence of gauge functions, explained in van Dyke (1975), Bender and Orszag
(1978), and Nayfeh (1981).
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Boundary Layers and Related Topics
An asymptotic expansion is a finite sequence of limit statements of the type
written in the preceding. For example, because limε→0 (sin ε)/ε = 1, sin ε = ε+o(ε).
Following up using the powers of ε as gauge functions,
lim (sin ε − ε)/ε 3 = −
ε→0
1
,
3!
sin ε = ε −
ε3
+ o(ǫ 3 ).
3!
By continuing this process we can establish that the term o(ε3 ) is better represented
by O(ε5 ) and is in fact ǫ 5 /5!. The series terminates with the order symbol.
The interesting property of an asymptotic expansion is that the series (10.58) may
not converge if extended indefinitely. Thus, for a fixed ε, the magnitude of a term may
eventually increase as shown in Figure 10.32. Therefore, there is an optimum number
of terms N (ε) at which the series should be truncated. The number N (ε) is difficult
to guess, but that is of little consequence, because only one or two terms in the
asymptotic expansion are calculated. The accuracy of the asymptotic representation
can be arbitrarily improved by keeping n fixed, and letting ε → 0.
We here emphasize the distinction between convergence and asymptoticity. In
convergence we are concerned with terms far out in an infinite series, an . We must
have limn→∞ an = 0 and, for example, limn→∞ |an+1 /an | < 1 for convergence.
Asymptoticity is a different limit: n is fixed at a finite number and the approximation
is improved as ε (say) tends to its limit.
Figure 10.32 Terms in a divergent asymptotic series, in which N (ε) indicates the optimum number of
terms at which the series should be truncated. M. Van Dyke, Perturbation Methods in Fluid Mechanics,
1975 and reprinted with the permission of Prof. Milton Van Dyke for The Parabolic Press.
393
14. Perturbation Techniques
The value of an asymptotic expansion becomes clear if we compare the
convergent series for a Bessel function J0 (x), given by
J0 (x) = 1 −
x4
x6
x2
+
−
+ ··· ,
22
2 2 42
22 42 62
(10.59)
with the first term of its asymptotic expansion
J0 (x) ∼
2
π
cos x −
πx
4
as x → ∞.
(10.60)
The convergent series (10.59) is useful when x is small, but more than eight terms
are needed for three-place accuracy when x exceeds 4. In contrast, the one-term
asymptotic representation (10.60) gives three-place accuracy for x > 4. Moreover,
the asymptotic expansion indicates the shape of the function, whereas the infinite
series does not.
Nonuniform Expansion
In many situations we develop an asymptotic expansion for a function of two
variables, say
an (x)gn (ε) as ε → 0.
(10.61)
f (x; ε) ∼
n
If the expansion holds for all values of x, it is called uniformly valid in x, and the problem is described as a regular perturbation problem. In this case any successive term
is smaller than the preceding term for all x. In some interesting situations, however,
the expansion may break down for certain values of x. For such values of x, am (x)
increases faster with m than gm (ε) decreases with m, so that the term am (x)gm (ε) is
not smaller than the preceding term. When the asymptotic expansion (10.61) breaks
down for certain values of x, it is called a nonuniform expansion, and the problem is
called a singular perturbation problem. For example, the series
1
= 1 − εx + ε 2 x 2 − ε 3 x 3 + · · · ,
1 + εx
(10.62)
is nonuniformly valid, because it breaks down when εx = O(1). No matter how small
we make ε, the second term is not a correction of the first term for x > 1/ε. We say
that the singularity of the perturbation expansion (10.62) is at large x or at infinity.
On the other hand, the expansion
√
√
ε2
ε 1/2 √
ε
= x 1+
− 2 + ··· ,
x+ε = x 1+
(10.63)
x
2x
8x
is nonuniform because it breaks down when ε/x = O(1). The singularity of this
expansion is at x = 0, because it is not valid for x < ε. The regions of nonuniformity
are called boundary layers; for equation (10.62) it is x > 1/ε, and for equation (10.63)
394
Boundary Layers and Related Topics
it is x < ε. To obtain expansions that are valid within these singular regions, we need
to write the solution in terms of a variable η which is of order 1 within the region
of nonuniformity. It is evident that η = εx for equation (10.62), and η = x/ε for
equation (10.63).
In many cases, singular perturbation problems are associated with the small
parameter ε multiplying the highest order derivative (as in viscous boundary layer
problems), so that the differential equation drops by one order as ε → 0, resulting in
an inability to satisfy all the boundary conditions. In several other singular perturbation
problems the small parameter does not multiply the highest-order derivative. An
example is low Reynolds number flows, for which the nondimensional governing
equation is
εu · ∇u = −∇p + ∇ 2 u,
where ε = Re ≪ 1. In this case the singularity or nonuniformity is at infinity. This is
discussed in Section 9.13.
15. An Example of a Regular Perturbation Problem
As a simple example of a perturbation expansion that is uniformly valid everywhere,
consider a plane Couette flow with a uniform suction across the flow (Figure 10.33).
The upper plate is moving parallel to itself at speed U and the lower plate is stationary. The distance between the plates is d and there is a uniform downward suction
velocity vs′ , with the fluid coming in through the upper plate and going out through the
bottom. For notational simplicity, we shall denote dimensional variables by a prime
and nondimensional variables without primes:
y=
y′
,
d
u=
u′
,
U
v=
v′
.
U
Figure 10.33 Uniform suction in a Couette flow, showing the velocity profile u(y) for ε = 0 and ε ≪ 1.
395
15. An Example of a Regular Perturbation Problem
As ∂/∂x = 0 for all variables, the nondimensional equations are
∂v
=0
(continuity),
∂y
1 d 2u
du
(x-momentum),
=
v
dy
Re dy 2
(10.64)
(10.65)
subject to
v(0) = v(1) = −vs ,
u(0) = 0,
u(1) = 1,
(10.66)
(10.67)
(10.68)
where Re = U d/ν, and vs = vs′ /U .
The continuity equation shows that the lateral flow is independent of y and
therefore must be
v(y) = −vs ,
to satisfy the boundary conditions on v. The x-momentum equation then becomes
du
d 2u
+ε
= 0,
2
dy
dy
(10.69)
where ε = vs Re = vs′ d/ν. We assume that the suction velocity is small, so that ε ≪ 1.
The problem is to solve equation (10.69), subject to equations (10.67) and (10.68).
An exact solution can easily be found for this problem, and will be presented at the
end of this section. However, an exact solution may not exist in more complicated
problems, and we shall illustrate the perturbation approach. We try a perturbation
solution in integral powers of ε, of the form,
u(y) = u0 (y) + εu1 (y) + ε 2 u2 (y) + O(ε 3 ).
(10.70)
(A power series in ε may not always be possible, as remarked upon in the preceding
section.) Our task is to determine u0 (y), u1 (y), etc.
Substituting equation (10.70) into equations (10.69), (10.67), and (10.68), we
obtain
d 2 u1
d 2 u2
du0
d 2 u0
2 du1
+
+
+
ε
+
ε
+ O(ε 3 ) = 0,
dy
dy
dy 2
dy 2
dy 2
(10.71)
subject to
u0 (0) + εu1 (0) + ε 2 u2 (0) + O(ε 3 ) = 0,
u0 (1) + εu1 (1) + ε 2 u2 (1) + O(ε 3 ) = 1.
(10.72)
(10.73)
396
Boundary Layers and Related Topics
Equations for the various orders are obtained by taking the limits of equations
(10.71)–(10.73) as ε → 0, then dividing by ε and taking the limit ε → 0 again,
and so on. This is equivalent to equating terms with like powers of ε. Up to order ε,
this gives the following sets:
Order ε 0 :
d 2 u0
= 0,
dy 2
u0 (0) = 0,
u0 (1) = 1.
(10.74)
du0
d 2 u1
=−
,
2
dy
dy
u1 (0) = 0,
u1 (1) = 0.
(10.75)
Order ε1 :
The solution of the zero-order problem (10.74) is
u0 = y.
(10.76)
Substituting this into the first-order problem (10.75), we obtain the solution
u1 =
y
(1 − y).
2
The complete solution up to order ε is then
ε
u(y) = y + [y(1 − y)] + O(ε 2 ).
2
(10.77)
In this expansion the second term is less than the first term for all values of y as ε → 0.
The expansion is therefore uniformly valid for all y and the perturbation problem is
regular. A sketch of the velocity profile (10.77) is shown in Figure 10.33.
It is of interest to compare the perturbation solution (10.77) with the exact solution. The exact solution of (10.69), subject to equations (10.67) and (10.68), is easily
found to be
u(y) =
1 − e−εy
.
1 − e−ε
(10.78)
For ε ≪ 1, Equation (10.78) can be expanded in a power series of ε, where the first
few terms are identical to those in equation (10.77).
16. An Example of a Singular Perturbation Problem
Consider again the problem of uniform suction across a plane Couette flow, discussed
in the preceding section. For the case of weak suction, namely ε = vs′ d/ν ≪ 1, we
saw that the perturbation problem is regular and the series is uniformly valid for all
397
16. An Example of a Singular Perturbation Problem
values of y. A more interesting case is that of strong suction, defined as ε ≫ 1, for
which we shall now see that the perturbation expansion breaks down near one of the
walls. As before, the v-field is uniform everywhere:
v(y) = −vs .
The governing equation is (10.69), which we shall now write as
δ
d 2 u du
+
= 0,
dy
dy 2
(10.79)
u(0) = 0,
u(1) = 1,
(10.80)
(10.81)
subject to
where we have defined
δ≡
ν
1
= ′ ≪ 1,
ε
vs d
as the small parameter. We try an expansion in powers of δ:
u(y) = u0 (y) + δu1 (y) + δ 2 u2 (y) + O(δ 3 ).
(10.82)
Substitution into equation (10.79) leads to
du0
= 0.
dy
(10.83)
The solution of this equation is u0 = const., which cannot satisfy conditions at both
y = 0 and y = 1. This is expected, because as δ → 0 the highest order derivative
drops out of the governing equation (10.79), and the approximate solution cannot
satisfy all the boundary conditions. This happens no matter how many terms are
included in the perturbation series. A boundary layer is therefore expected near one
of the walls, where the solution varies so rapidly that the two terms in equation (10.79)
are of the same order.
The expansion (10.82), valid outside the boundary layers, is the “outer” expansion, the first term of which is governed by equation (10.83). If the outer expansion
satisfies the boundary condition (10.80), then the first term in the expansion is u0 = 0;
if on the other hand the outer expansion satisfies the condition (10.81), then u0 = 1.
The outer expansion should be smoothly matched to an “inner” expansion valid within
the boundary layer. The two possibilities are sketched in Figure 10.34, where it is
evident that a boundary layer occurs at the top plate if u0 = 0, and it occurs at the
bottom plate if u0 = 1. Physical reasons suggest that a strong suction would tend
to keep the profile of the longitudinal velocity uniform near the wall through which
the fluid enters, so that a boundary layer at the lower wall seems more reasonable.
Moreover, the ε ≫ 1 case is then a continuation of the ε ≪ 1 behavior (Figure 10.33).
398
Boundary Layers and Related Topics
Figure 10.34 Couette flow with strong suction, showing two possible locations of the boundary layer.
The one shown in (a) is the correct one.
We shall therefore proceed with this assumption and verify later in the section that it
is not mathematically possible to have a boundary layer at y = 1.
The location of the boundary layer is determined by the sign of the ratio of the
dominant terms in the boundary layer. This is the case because the boundary layer
must always decay into the domain and the decay is generally exponential. The inward
decay is required so as to match with the outer region solution. Thus a ratio of signs
that is positive (when both terms are on the same side of the equation) requires the
boundary layer to be at the left or bottom, that is, the boundary with the smaller
coordinate.
The first task is to determine the natural distance within the boundary layer, where
both terms in equation (10.79) must be of the same order. If y is a typical distance
within the boundary layer, this requires that δ/y 2 = O(1/y), that is
y = O(δ),
showing that the natural scale for measuring distances within the boundary layer is δ.
We therefore define a boundary layer coordinate
η≡
y
,
δ
which transforms the governing equation (10.79) to
−
d 2u
du
=
.
dη
dη2
(10.84)
As in the Blasius solution, η = O(1) within the boundary layer and η → ∞ far
outside of it.
The solution of equation (10.84) as η → ∞ is to be matched to the solution of
equation (10.79) as y → 0. Another way to solve the problem is to write a composite
expansion consisting of both the outer and the inner solutions:
u(y) = [u0 (y) + δu1 (y) + · · · ] + {û0 (η) + δ û1 (η) + · · · },
(10.85)
399
16. An Example of a Singular Perturbation Problem
where the term within { } is regarded as the correction to the outer solution within
the boundary layer. All terms in the boundary layer correction { } go to zero as
η → ∞. Substituting equation (10.85) into equation (10.79), we obtain
du1
d 2 u0
du0
2
+δ
+
+δ
+ O(δ 3 )
dy
dy
dy 2
d û1
d û0
d 2 û0
d 2 û1
+
+ O(δ) = 0.
+
+
+ δ −1
dη
dη
dη2
dη2
(10.86)
A systematic procedure is to multiply equation (10.86) by powers of δ and take limits
as δ → 0, with first y held fixed and then η held fixed. When y is held fixed (which
we write as y = O(1)) and δ → 0, the boundary layer becomes progressively thinner
and we move outside and into the outer region. When η is held fixed (i.e, η = O(1))
and δ → 0, we obtain the behavior within the boundary layer.
Multiplying equation (10.86) by δ and taking the limit as δ → 0, with η = O(1),
we obtain
d 2 û0
d û0
+
= 0,
dη
dη2
(10.87)
which governs the first term of the boundary layer correction. Next, the limit of
equation (10.86) as δ → 0, with y = O(1), gives
du0
= 0,
dy
(10.88)
which governs the first term of the outer solution. (Note that in this limit η → ∞,
and consequently we move outside the boundary layer where all correction terms
go to zero, that is d û1 /dη → 0 and d 2 û1 /dη2 → 0.) The next largest term
in equation (10.86) is obtained by considering the limit δ → 0 with η = O(1),
giving
d 2 û1
d û1
= 0,
+
dη
dη2
and so on. It is clear that our formal limiting procedure is equivalent to setting the
coefficients of like powers of δ in equation (10.86) to zero, with the boundary layer
terms treated separately.
As the composite expansion holds everywhere, all boundary conditions can be
applied on it. With the assumed solution of equation (10.85), the boundary condition
equations (10.80) and (10.81) give
u0 (0) + û0 (0) + δ[u1 (0) + û1 (0)] + · · · = 0,
u0 (1) + 0 + δ[u1 (1) + 0] + · · · = 1.
(10.89)
(10.90)
400
Boundary Layers and Related Topics
Equating like powers of δ, we obtain the following conditions
u0 (0) + û0 (0) = 0,
u0 (1) = 1,
u1 (0) + û1 (0) = 0,
u1 (1) = 0.
(10.91)
(10.92)
We can now solve equation (10.88) along with the first condition in equation (10.92),
obtaining
u0 (y) = 1.
(10.93)
Next, we can solve equation (10.87), along with the first condition in equation (10.91),
namely
û0 (0) = −u0 (0) = −1,
and the condition û0 (∞) = 0. The solution is
û0 (η) = −e−η .
To the lowest order, the composite expansion is, therefore,
u(y) = 1 − e−η = 1 − e−y/δ ,
(10.94)
which we have written in terms of both the inner variable η and the outer variable y, because the composite expansion is valid everywhere. The first term is the
lowest-order outer solution, and the second term is the lowest-order correction in the
boundary layer.
Comparison with Exact Solution
The exact solution of the problem is (see equation (10.78)):
u(y) =
1 − e−y/δ
.
1 − e−1/δ
(10.95)
We want to write the exact solution in powers of δ and compare with our perturbation
solution. An important result to remember is that exp (−1/δ) decays faster than any
power of δ as δ → 0, which follows from the fact that
εn
e−1/δ
=
lim
= 0,
ε→∞ eε
δ→0 δ n
lim
e−1/δ = o(δ n ),
n > 0,
for any n, as can be verified by applying the l’Hôpital rule n times. Thus, the denominator in equation (10.95) exponentially approaches 1, with no contribution in powers
of δ. It follows that the expansion of the exact solution in terms of a power series in
δ is
u(y) ≃ 1 − e−y/δ ,
(10.96)
401
17. Decay of a Laminar Shear Layer
which agrees with our composite expansion (10.94). Note that no terms in powers of
δ enter in equation (10.96). Although in equation (10.94) we did not try to continue
our series to terms of order δ and higher, the form of equation (10.96) shows that these
extra terms would have turned out to be zero if we had calculated them. However, the
nonexistence of terms proportional to δ and higher is special to the current problem,
and not a frequent event.
Why There Cannot Be a Boundary Layer at y = 1
So far we have assumed that the boundary layer could occur only at y = 0. Let us
now investigate what would happen if we assumed that the boundary layer happened
to be at y = 1. In this case we define a boundary layer coordinate
ζ ≡
1−y
,
δ
(10.97)
which increases into the fluid from the upper wall (Figure 10.34b). Then the
lowest-order terms in the boundary conditions (10.91) and (10.92) are replaced by
u0 (0) = 0,
u0 (1) + û0 (0) = 1,
where û0 (0) represents the value of û0 at the upper wall where ζ = 0. The first
condition gives the lowest-order outer solution u0 (y) = 0. To find the lowest-order
boundary layer correction û0 (ζ ), note that the equation governing it (obtained by
substituting equation (10.97) into equation (10.87)) is
d 2 û0
d û0
= 0,
−
dζ
dζ 2
(10.98)
subject to
û0 (0) = 1 − u0 (1) = 1,
û0 (∞) = 0.
A substitution of the form û0 (ζ ) = exp(aζ ) into equation (10.103) shows that a =
+1, so that the solution to equation (10.98) is exponentially increasing in ζ and cannot
satisfy the condition at ζ = ∞.
17. Decay of a Laminar Shear Layer
It is shown in Chapter 12 (pp. 515–516) that flows exhibiting an inflection point
in the streamwise velocity profile are highly unstable. These results, based on the
Orr-Sommerfeld analysis for parallel flows, have been re-examined recently and it has
been found that the instability is not manifested until sufficiently far downstream for
similarity to develop. This is discussed in more detail in Chapter 12. Here we note that
402
Boundary Layers and Related Topics
a detailed treatment of the decay of a laminar shear layer illustrates some interesting
points. The problem of the downstream smoothing of an initial velocity discontinuity
has not been completely solved even now, although considerable literature might
suggest otherwise. Thus it is appropriate to close this chapter with a problem that
remains to be put to rest. See Figure 10.35 for a general sketch of the problem. The
basic parameter is Rex = U1 x/ν. In these terms the problem splits into distinct regions
as illustrated in Figure 10.11. This shown in the paper by Alston and Cohen (1992),
which also contains a brief historical summary. In the region for which Rex is finite,
the full Navier–Stokes equations are required for a solution. As Rex becomes large,
δ ≪ x, v ≪ u and the Navier–Stokes equations asymptotically decay to the boundary
layer equations. The boundary layer equations require an initial condition, which is
provided by the downstream limit of the solution in the finite Reynolds number region.
Here we see that, because they are of elliptic form, the full Navier–Stokes equations
require downstream boundary conditions on u and v (which would have to be provided
by an asymptotic matching). Paradoxically it seems, the downstream limit of the
Navier–Stokes equations, represented by the boundary layer equations, cannot accept
a downstream boundary condition because they are of parabolic form. The boundary
layer equations govern the downstream evolution from a specified initial station of
the streamwise velocity profile. In this problem there must be a matching between
the downstream limit of the initial finite Reynolds number region and the initial
condition for the boundary layer equations. Although the boundary layer equations
are a subset of the full Navier–Stokes equations and are generally appreciated to be the
resolution of d’Alembert’s paradox via a singular perturbation in the normal (say y)
direction, they are also a singular perturbation in the streamwise (say x) direction.
That is, the highest x derivative is dropped in the boundary layer approximation and
the boundary condition that must be dropped is the one downstream. This becomes an
issue in numerical solutions of the full Navier–Stokes equations. It arises downstream
in this problem as well.
Figure 10.35 Decay of a laminar shear layer.
403
17. Decay of a Laminar Shear Layer
If in Figure 10.35 the pressure in the top and bottom flow is the same, the boundary
layer formulation valid for x > x0 , Rex0 ≫ 1 is
∂u ∂v
+
= 0,
∂x
∂y
y → +∞ : u → U1 ,
∂u
∂u
∂ 2u
+v
= ν 2,
∂x
∂y
∂y
y → −∞ : u → U2 ,
u
x = x0 : U (x0 , y) specified (initial condition). One boundary condition on v is
required.
We can look for a solution sufficiently far downstream that the initial condition
has been forgotten so that the similarity form has been achieved. Then,
η=
y
x
U1 x
ν
and ψ(x, y) =
In these terms u/U1 = f ′ (η) and
f ′′′ +
1
2
νU1 xf (η).
ff ′′ = 0, f ′ (∞) = 1, f ′ (−∞) = U2 /U1 .
Of course a third boundary condition is required for a unique solution. This represents
the need to specify one boundary condition on v. Let us see how far we can go towards
a solution and what the missing boundary condition actually pins down. Consider the
transformation f ′ (η) = F (f ) = u/U1 . Then
dF
d 2f
=F
2
df
dη
and
d 3f
d 2F
= F
+
3
dη
df 2
dF
df
2
F.
The Blasius equation transforms to
d 2F
dF 2 1 dF
F
+
+ f
= 0,
df
2 df
df 2
F (f = ∞) = 1,
F (f = −∞) = U2 /U1 .
(10.99)
(10.100)
This has a unique solution for the streamwise velocity u/U1 = F in terms of the
similarity streamfunction f (η) with the expected properties, which are shown in
Figure 10.36(a) and (b). The exact solution varies more steeply than the linearized
solution for small velocity difference, with the greatest difference between solutions
at the region of maximum curvature at the low velocity end. This difference is shown
more clearly in the magnified insets of each frame. The difference increases as the normalized velocity difference, (U1 − U2 )/U1 , increases. We can see from the (Blasius)
404
Boundary Layers and Related Topics
(a)
1
0.99
0.98
0.97
0.96
F 0.95
0.91
0.94
0.93
0.90
–3
–2
numerical
0.92
analytical-linearized
0.91
0.9
–8
(b)
–6
–4
–2
0
f
2
4
6
8
1
0.98
1
0.96
0.94
0.98
0.92
0.96
F 0.9
0.84
0.88
0.94
0
2
4
0.86 0.82
0.84
numerical
0.8
–4
–3
–2
analytical-linearized
–1
0.82
0.8
–8
–6
–4
–2
0
f
2
4
6
8
Figure 10.36 Solution for F (f ) from equation (10.99) subject to boundary conditions (10.100) when
(a) U2 /U1 = 0.9, and (b) U2 /U1 = 0.8. The “analytical—linearized” approximation is the asymptotic
solution for (U1 − U2 )/U1 ≪ 1 : F = 1 − [(U1 − U2 )/(2U1 )]erfc(f/2). Magnified insets show the
difference between the two curves.
equation in η-space that the maximum of the shear stress occurs where f = 0. This
is the inflection point in the velocity profile in η or y. However, the inflection point
in the F (f ) curve is located where f = −2 dF /df < 0. This is below the dividing
405
17. Decay of a Laminar Shear Layer
streamline f= 0. To put this back in physical space (x, y), the transformation must
be inverted, dη = df/F (f ).
The integral on the right-hand side can be calculated exactly but the correspondence between any integration limit on the right-hand side and that on the left-hand
side is ambiguous. This solution admits a translation of η by any constant. The ambiguity in the location in y (or η) of the calculated profile was known to Prandtl. In the
literature, five different third boundary conditions have been used. They are as follows:
(a)
(b)
(c)
(d)
(e)
f (η = 0) = 0 (v = 0 on y or η = 0);
f ′ (η = 0) = (1 + U2 /U1 )/2 (average velocity on the axis);
ηf ′ − f → 0 as η → ∞ (v → 0 as η → ∞);
ηf ′ − f → 0 as η → −∞ (v → 0 as η → −∞); and
uv]∞ + uv]−∞ = 0 or f ′ (ηf ′ − f )]∞ + f ′ (ηf ′ − f )]−∞ = 0 (von Karman;
zero net transverse force).
Alston and Cohen (1992) considered the limit of small velocity difference (U2 −U1 )/
U1 ≪ 1 and showed that none of these third boundary conditions are correct.
As the normalized velocity difference increases, we expect the error in using any of
the incorrect boundary conditions to increase. Of all of them, the last (e) is closest
to the correct result. D.-C. Hwang, in his doctoral dissertation (2005) has shown,
that as the normalized velocity difference (U1 − U2 )/U1 increases, the trends seen
by Alston continue. Figure 10.37 shows that the streamwise velocity on the dividing
streamline (f = 0) is larger than the average velocity of the two streams, when the
upper stream is the faster one. What is not determined from the solution to (10.99)
1
From solution of
Eq. (10.99) with
(10.100)
0.95
0.9
F (f 5 0)
0.85
(U11U2)/(2U1)
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
0.2
0.4
0.6
0.8
1
U2/U1
Figure 10.37 Comparison of streamwise velocity at dividing streamline (f = 0) with average velocity
(dashed line).
406
Boundary Layers and Related Topics
subject to (10.100) is the location of the dividing streamline, f = 0, because that
depends on the inverse transformation, which requires one more boundary condition
for a unique specification. When U1 > U2 , the dividing streamline ψ = 0, which
starts at the origin, bends slowly downwards and its path can be tracked only by starting the solution at the origin and following the evolution of the equations downstream.
Thus, no simple statement of a third boundary condition is possible to complete the
similarity formulation. Following Klemp and Acrivos (1972), the deflection of the
dividing streamline from the x-axis was written by Hwang
√ (2005) as ϕ(x). Then
a modified similarity variable η = (U1 /ν)1/2 (y − ϕ)/ 2x was defined. In these
terms, the dividing streamline is η = 0 so that the boundary condition on the dividing
streamline is f (η = 0) = 0. However, the behavior of ϕ(x) had to be computed by
solving the Navier-Stokes equations from the origin. The downstream asymptote for
ϕ(x), which can be used as the third boundary condition for the similarity solution,
is shown in Figure 10.38 as a function of velocity ratio. The downstream distance to
achieve similarity is found to be given by Re = U1 x/ν ≈ 104 . Although the results
of Alston and Cohen are difficult to distinguish on the graph from the von Karman
condition (e), the numerical calculation shows a clear distinction. Deviations become
larger as U2 /U1 diminishes. Beyond the point shown, computations became excessively tedious.
0.1
0.0
20.1
current solution (Hwang)
von Kármán condition
f(0) 5 Um 5 (U11U2)/2
v(h 5 2`) 5 0
v(h 5 `) 5 0
f(0) 5 0, v(0) 5 0
,
Alston and Cohen s result
20.2
20.3
0.5
0.6
0.7
0.8
0.9
1.0
Velocity ratio, U 2 /U 1
Figure 10.38 Asymptotic downstream dividing streamline deflection as a function of velocity ratio.
407
Exercises
Exercises
1. Solve the Blasius sets (10.34) and (10.35) with a computer, using the
Runge–Kutta scheme of numerical integration.
2. A flat plate 4 m wide and 1 m long (in the direction of flow) is immersed in
kerosene at 20 ◦ C (ν = 2.29 × 10−6 m2 /s, ρ = 800 kg/m3 ) flowing with an undisturbed velocity of 0.5 m/s. Verify that the Reynolds number is less than critical everywhere, so that the flow is laminar. Show that the thickness of the boundary layer and
the shear stress at the center of the plate are δ = 0.74 cm and τ0 = 0.2 N/m2 , and
those at the trailing edge are δ = 1.05 cm and τ0 = 0.14 N/m2 . Show also that the
total frictional drag on one side of the plate is 1.14 N. Assume that the similarity
solution holds for the entire plate.
3. Air at 20 ◦ C and 100 kPa (ρ = 1.167 kg/m3 , ν = 1.5 × 10−5 m2 /s) flows
over a thin plate with a free-stream velocity of 6 m/s. At a point 15 cm from the
leading edge, determine the value of y at which u/U = 0.456. Also calculate v and
∂u/∂y at this point. [Answer: y = 0.857 mm, v = 0.39 cm/s, ∂u/∂y = 3020 s−1 .
You may not be able to get this much accuracy, because your answer will probably
use certain figures in the chapter.]
4. Assume that the velocity in the laminar boundary layer on a flat plate has the
profile
πy
u
= sin
.
U
2δ
Using the von Karman momentum integral equation, show that
4.795
δ
=√
,
x
Rex
0.655
Cf = √
.
Rex
Notice that these are very similar to the Blasius solution.
5. Water flows over a flat plate 30 m long and 17 m wide with a free-stream velocity of 1 m/s. Verify that the Reynolds number at the end of the plate is larger than the
critical value for transition to turbulence. Using the drag coefficient in Figure 10.12,
estimate the drag on the plate.
6. Find the diameter of a parachute required to provide a fall velocity no larger
than that caused by jumping from a 2.5 m height, if the total load is 80 kg. Assume
that the properties of air are ρ = 1.167 kg/m3 , ν = 1.5 × 10−5 m2 /s, and treat the
parachute as a hemispherical shell with CD = 2.3. [Answer: 3.9 m]
7. Consider the roots of the algebraic equation
x 2 − (3 + 2ε)x + 2 + ε = 0,
for ε ≪ 1. By a perturbation expansion, show that the roots are
1 − ε + 3ε 2 + · · · ,
x=
2 + 3ε − 3ε 2 + · · · .
408
Boundary Layers and Related Topics
(From Nayfeh, 1981, p. 28 and reprinted by permission of John Wiley & Sons, Inc.)
8. Consider the solution of the equation
ε
dy
d 2y
+ 2y = 0,
− (2x + 1)
2
dx
dx
ε ≪ 1,
with the boundary conditions
y(0) = α,
y(1) = β.
Convince yourself that a boundary layer at the left end does not generate “matchable”
expansions, and that a boundary layer at x = 1 is necessary. Show that the composite
expansion is
y = α(2x + 1) + (β − 3α)e−3(1−x)/ε + · · · .
For the two values ε = 0.1 and 0.01, sketch the solution if α = 1 and β = 0. (From
Nayfeh, 1981, p. 284 and reprinted by permission of John Wiley & Sons, Inc.)
9. Consider incompressible, slightly viscous flow over a semi-infinite flat plate
with constant suction. The suction velocity v(x, y = 0) = v0 < 0 is ordered by
O(Re−1/2 ) < v0 /U < O(1) where Re = U x/ν → ∞. The flow upstream is parallel
to the plate with speed U . Solve for u, v in the boundary layer.
10. Mississippi River boatmen know that when rounding a bend in the river,
they must stay close to the outer bank or else they will run aground. Explain in fluid
mechanical terms the reason for the cross-sectional shape of the river at the bend:
11. Solve to leading order in ε in the limit ε → 0
d 2f
df
+ sin xf = 0,
+ cos x
2
dx
dx
f (1) = 0, f (2) = cos 2.
ε[x −2 + cos (ln x)]
1 x 2,
12. A laminar shear layer develops immediately downstream of a velocity discontinuity. Imagine parallel flow upstream of the origin with a velocity discontinuity
Supplemental Reading
at x = 0 so that u = U1 for y > 0 and u = U2 for y < 0. The density may be
assumed constant and the appropriate Reynolds number is sufficiently large that the
shear layer is thin (in comparison to distance from the origin). Assume the static
pressures are the same in both halves of the flow at x = 0. Describe any ambiguities
or nonuniquenesses in a similarity formulation and how they may be resolved. In the
special case of small velocity difference, solve explicitly to first order in the smallness
parameter (velocity difference normalized by average velocity, say) and show where
the nonuniqueness enters.
13. Solve equation (10.99) subject to equation (10.100) asymptotically for small
velocity difference and obtain the result in the caption to Figure 10.36.
Literature Cited
Alston, T. M. and I. M. Cohen (1992). “Decay of a laminar shear layer.” Phys. Fluids A4: 2690–2699.
Bender, C. M. and S. A. Orszag (1978). Advanced Mathematical Methods for Scientists and Engineers.
New York: McGraw-Hill.
Falkner, V. W. and S. W. Skan (1931). “Solutions of the boundary layer equations.” Phil. Mag. (Ser. 7) 12:
865–896.
Gallo, W. F., J. G. Marvin, and A. V. Gnos (1970). “Nonsimilar nature of the laminar boundary layer.”
AIAA J. 8: 75–81.
Glauert, M. B. (1956). “The Wall Jet.” J. Fluid Mech. 1: 625–643.
Goldstein, S. (ed.). (1938). Modern Developments in Fluid Dynamics, London: Oxford University Press;
Reprinted by Dover, New York (1965).
Holstein, H. and T. Bohlen (1940). “Ein einfaches Verfahren zur Berechnung laminarer Reibungsschichten
die dem Näherungsverfahren von K. Pohlhausen genügen.” Lilienthal-Bericht. S. 10: 5–16.
Hwang, Din-Chih (2005). “Evolution of a laminar mixing layer.” Ph.D. dissertation, University of
Pennsylvania. Submitted for publication.
Klemp, J. B. and A. Acrivos (1972). “A note on the laminar mixing of two uniform parallel semi-infinite
streams.” Journal of Fluid Mechanics 55: 25–30.
Mehta, R. (1985). “Aerodynamics of sports balls.” Annual Review of Fluid Mechanics 17, 151–189.
Nayfeh, A. H. (1981). Introduction to Perturbation Techniques, New York: Wiley.
Peletier, L. A. (1972). “On the asymptotic behavior of velocity profiles in laminar boundary layers.” Arch.
for Rat. Mech. and Anal. 45: 110–119.
Pohlhausen, K. (1921). “Zur näherungsweisen Integration der Differentialgleichung der laminaren
Grenzschicht.” Z. Angew. Math. Mech. 1: 252–268.
Rosenhead, L. (ed.). (1988). Laminar Boundary Layers, New York: Dover.
Schlichting, H. (1979). Boundary Layer Theory, 7th ed., New York: McGraw-Hill.
Serrin, J. (1967). “Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory.” Proc.
Roy. Soc. A299: 491–507.
Sherman, F. S. (1990). Viscous Flow, New York: McGraw-Hill.
Taneda, S. (1965). “Experimental investigation of vortex streets.” J. Phys. Soc. Japan 20: 1714–1721.
Thomson, R. E. and J. F. R. Gower (1977). “Vortex streets in the wake of the Aleutian Islands.” Monthly
Weather Review 105: 873–884.
Thwaites, B. (1949). “Approximate calculation of the laminar boundary layer.” Aero. Quart. 1: 245–280.
van Dyke, M. (1975). Perturbation Methods in Fluid Mechanics, Stanford, CA: The Parabolic Press.
von Karman, T. (1921). “Über laminare und turbulente Reibung.” Z. Angew. Math. Mech. 1: 233–252.
Wen, C.-Y. and C.-Y. Lin (2001). “Two-dimensional vortex shedding of a circular cylinder.” Phys. Fluids
13: 557–560.
Yih, C. S. (1977). Fluid Mechanics: A Concise Introduction to the Theory, Ann Arbor, MI: West River
Press.
409
410
Boundary Layers and Related Topics
Supplemental Reading
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.
Friedrichs, K. O. (1955). “Asymptotic phenomena in mathematical physics.” Bull. Am. Math. Soc. 61:
485–504.
Lagerstrom, P. A. and R. G. Casten (1972). “Basic concepts underlying singular perturbation techniques.”
SIAM Review 14: 63–120.
Meksyn, D. (1961). New Methods in Laminar Boundary Layer Theory, New York: Pergamon Press.
Panton, R. L. (1984). Incompressible Flow, New York: Wiley.
Chapter 11
Computational Fluid Dynamics
by Howard H. Hu
University of Pennsylvania
Philadelphia, PA, USA
1. Introduction . . . . . . . . . . . . . . . . . . . . . 411
2. Finite Difference Method . . . . . . . . . 413
Approximation to Derivatives . . . . 413
Discretization and Its Accuracy. . . 414
Convergence, Consistency, and
Stability . . . . . . . . . . . . . . . . . . . . . . 415
3. Finite Element Method . . . . . . . . . . . 418
Weak or Variational Form of Partial
Differential Equations . . . . . . . . . 418
Galerkin’s Approximation and
Finite Element Interpolations . . 420
Matrix Equations, Comparison
with Finite Difference Method. . 421
Element Point of View of the
Finite Element Method . . . . . . . . 424
4. Incompressible Viscous Fluid Flow 426
Convection-Dominated
Problems. . . . . . . . . . . . . . . . . . . . . . 427
Incompressibility Condition . . . . . .
Explicit MacCormack Scheme . . . .
MAC Scheme . . . . . . . . . . . . . . . . . . . .
-Scheme . . . . . . . . . . . . . . . . . . . . . . .
Mixed Finite Element
Formulation. . . . . . . . . . . . . . . . . . .
5. Three Examples . . . . . . . . . . . . . . . . .
Explicit MacCormack Scheme for
Driven Cavity Flow Problem . . .
Explicit MacCormack Scheme for
Flow Over a Square Block . . . . .
Finite Element Formulation for
Flow Over a Cylinder Confined
in a Channel . . . . . . . . . . . . . . . . . .
6. Concluding Remarks . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . .
429
430
433
437
438
440
440
444
450
461
463
464
1. Introduction
Computational Fluid Dynamics (CFD) is a science that, with the help of digital computers, produces quantitative predictions of fluid-flow phenomena based on those
conservation laws (conservation of mass, momentum, and energy) governing fluid
motion. These predictions normally occur under those conditions defined in terms of
flow geometry, the physical properties of a fluid, and the boundary and initial conditions of a flow field. The prediction generally concerns sets of values of the flow
variables, for example, velocity, pressure, or temperature at selected locations in the
domain and for selected times. It may also evaluate the overall behavior of the flow,
such as the flow rate or the hydrodynamic force acting on an object in the flow.
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50011-3
411
412
Computational Fluid Dynamics
During the past four decades different types of numerical methods have been
developed to simulate fluid flows involving a wide range of applications. These
methods include finite difference, finite element, finite volume, and spectral methods.
Some of them will be discussed in this chapter.
The CFD predictions are never completely exact. Because many sources of error
are involved in the predictions, one has to be very careful in interpreting the results
produced by CFD techniques. The most common sources of error are:
• Discretization error. This is intrinsic to all numerical methods. This error is
incurred whenever a continuous system is approximated by a discrete one where
a finite number of locations in space (grids) or instants of time may have been
used to resolve the flow field. Different numerical schemes may have different
orders of magnitude of the discretization error. Even with the same method,
the discretization error will be different depending upon the distribution of the
grids used in a simulation. In most applications, one needs to properly select a
numerical method and choose a grid to control this error to an acceptable level.
• Input data error. This is due to the fact that both flow geometry and fluid
properties may be known only in an approximate way.
• Initial and boundary condition error. It is common that the initial and boundary
conditions of a flow field may represent the real situation too crudely. For
example, flow information is needed at locations where fluid enters and leaves
the flow geometry. Flow properties generally are not known exactly and are
thus only approximated.
• Modeling error. More complicated flows may involve physical phenomena that
are not perfectly described by current scientific theories. Models used to solve
these problems certainly contain errors, for example, turbulence modeling,
atmospheric modeling, problems in multiphase flows, and so on.
As a research and design tool, CFD normally complements experimental and
theoretical fluid dynamics. However, CFD has a number of distinct advantages:
• It can be produced inexpensively and quickly. Although the price of most items
is increasing, computing costs are falling. According to Moore’s law based on
the observation of the data for the last 40 years, the CPU power will double
every 18 months into the foreseeable future.
• It generates complete information. CFD produces detailed and comprehensive
information of all relevant variables throughout the domain of interest. This
information can also be easily accessed.
• It allows easy change of the parameters. CFD permits input parameters to be
varied easily over wide ranges, thereby facilitating design optimization.
• It has the ability to simulate realistic conditions. CFD can simulate flows directly
under practical conditions, unlike experiments, where a small- or a large-scale
model may be needed.
413
2. Finite Difference Method
• It has the ability to simulate ideal conditions. CFD provides the convenience
of switching off certain terms in the governing equations, which allows one
to focus attention on a few essential parameters and eliminate all irrelevant
features.
• It permits exploration of unnatural events. CFD allows events to be studied that
every attempt is made to prevent, for example, conflagrations, explosions, or
nuclear power plant failures.
2. Finite Difference Method
The key to various numerical methods is to convert the partial different equations
that govern a physical phenomenon into a system of algebraic equations. Different
techniques are available for this conversion. The finite difference method is one of
the most commonly used.
Approximation to Derivatives
Consider the one-dimensional transport equation,
∂T
∂ 2T
∂T
+u
=D 2
∂t
∂x
∂x
for
0 x L.
(11.1)
This is the classic convection-diffusion problem for T (x, t), where u is a convective
velocity and D is a diffusion coefficient. For simplicity, let us assume that u and D
are two constants. This equation is written in nondimensional form. The boundary
conditions for this problem are
T (0, t) = g
and
∂T
(L, t) = q,
∂x
(11.2)
where g and q are two constants. The initial condition is
T (x, 0) = T0 (x)
for 0 x L,
(11.3)
where T0 (x) is a given function that satisfies the boundary conditions (11.2).
Let us first discretize the transport equation (11.1) on a uniform grid with a grid
spacing x, as shown in Figure 11.1. Equation (11.1) is evaluated at spatial location
x = xi and time t = tn . Define T (xi , tn ) as the exact value of T at the location x = xi
and time t = tn , and let Tin be its approximation. Using the Taylor series expansion,
we have
n
n
n
∂T n x 2 ∂ 2 T
x 3 ∂ 3 T
x 4 ∂ 4 T
n
5
Ti+1
+
= Tin +x
,
+
+
+O
x
∂x i
2
6
∂x 2 i
∂x 3 i 24 ∂x 4 i
(11.4)
n
Ti−1
= Tin −x
∂T
∂x
n
n
n
n
x 2 ∂ 2 T
x 3 ∂ 3 T
x 4 ∂ 4 T
5
+
,
−
+
+O
x
2
6
∂x 2 i
∂x 3 i 24 ∂x 4 i
i
(11.5)
414
Computational Fluid Dynamics
tn+1
∆x
tn
xi21
x050
∆x
xi
xi+1
xn5L
tn21
Figure 11.1 Uniform grid in space and time.
where O x 5 means terms of the order of x 5 . Therefore, the first spatial derivative
may be approximated as
∂T
∂x
n
i
=
=
n −Tn
Ti+1
i
Tin
x
n
− Ti−1
+ O (x)
(forward difference)
+ O (x)
(backward difference)
x
n
T n − Ti−1
= i+1
+ O x 2 (centered difference)
2x
(11.6)
and the second order derivative may be approximated as
∂ 2T
∂x 2
n
i
=
n − 2T n + T n
Ti+1
i
i−1
x 2
+ O x 2 .
(11.7)
The orders of accuracy of the approximations (truncation errors) are also indicated in
the expressions of (11.6) and (11.7). More accurate approximations generally require
more values of the variable on the neighboring grid points. Similar expressions can
be derived for nonuniform grids.
In the same fashion, the time derivative can be discretized as
∂T
∂t
n
i
Tin+1 − Tin
+ O (t)
t
T n − Tin−1
+ O (t)
= i
t
n+1
n−1
− Ti
T
+ O t 2
= i
2t
=
(11.8)
where t = tn+1 − tn = tn − tn−1 is the constant time step.
Discretization and Its Accuracy
A discretization of the transport equation (11.1) is obtained by evaluating the equation at fixed spatial and temporal grid points and using the approximations for the
individual derivative terms listed in the preceding section. When the first expression
in (11.8) is used, together with (11.7) and the centered difference in (11.6), (11.1)
415
2. Finite Difference Method
may be discretized by
or
n
n
T n − Ti−1
T n − 2Tin + Ti−1
Tin+1 − Tin
2
+ u i+1
= D i+1
, (11.9)
+
O
t,
x
t
2x
x 2
Tin+1 ≈ Tin − ut
n −Tn
Ti+1
i−1
2x
+ Dt
n − 2T n + T n
Ti+1
i
i−1
x 2
(11.10)
n
n
n
n
,
+ β Ti+1
− 2Tin + Ti−1
− Ti−1
= Tin − α Ti+1
where
α=u
t
,
2x
β=D
t
.
x 2
(11.11)
Once the values of Tin are known, starting with the initial condition (11.3), the expression (11.10) simply updates the variable for the next time step t = tn+1 . This scheme
is known as an explicit algorithm. The discretization (11.10) is first order accurate in
time and second order accurate in space.
As another example, when the backward difference expression in (11.8) is used,
we will have
n
n
T n − Ti−1
T n − 2Tin + Ti−1
Tin − Tin−1
2
+ u i+1
+
O
t,
x
= D i+1
, (11.12)
t
2x
x 2
or
n
n
n
n
Tin + α Ti+1
≈ Tin−1 .
− β Ti+1
− 2Tin + Ti−1
− Ti−1
(11.13)
At each time step t = tn , here a system of algebraic equations needs to be solved to
advance the solution. This scheme is known as an implicit algorithm. Obviously, for
the same accuracy, the explicit scheme (11.10) is much simpler than the implicit one
(11.13). However, the explicit scheme has limitations.
Convergence, Consistency, and Stability
The result from the solution of the explicit scheme (11.10) or the implicit scheme
(11.13) represents an approximate numerical solution to the original partial differential equation (11.1). One certainly hopes that the approximate solution will be close
to the exact one. Thus we introduce the concepts of convergence, consistency, and
stability of the numerical solution.
The approximate solution is said to be convergent if it approaches the exact
solution as the grid spacings x and t tend to zero. We may define the solution
error as the difference between the approximate solution and the exact solution,
ein = Tin − T (xi , tn ) .
(11.14)
416
Computational Fluid Dynamics
Thus the approximate solution converges when ein → 0 as x, t → 0. For a
convergent solution, some measure of the solution error can be estimated as
n
e Kx a t b ,
(11.15)
i
where the measure may be the root mean square (rms) of the solution error on all the
grid points; K is a constant independent of the grid spacing x and the time step
t; the indices a and b represent the convergence rates at which the solution error
approaches zero.
One may reverse the discretization process, and examine the limit of the discretized equations (11.10) and (11.13), as the grid spacing tends to zero. The discretized equation is said to be consistent if it recovers the original partial differential
equation (11.1) in the limit of zero grid spacing.
Let us consider the explicit scheme (11.10). Substitution of the Taylor series
expansions (11.4) and (11.5) into this scheme (11.10) produces,
∂T
∂t
n
i
n
∂ 2T
−D
∂x 2
∂ 3T
∂x 3
n
∂T
+u
∂x
i
n
i
+ Ein = 0,
(11.16)
where
Ein
t
=
2
∂ 2T
∂t 2
n
i
x 2
+u
6
i
x 2
−D
12
∂ 4T
∂x 4
n
i
+O t 2 , x 4 , (11.17)
is the truncation error. Obviously,
as the grid spacing x, t → 0, this truncation
error is of the order of O t, x 2 and tends to zero. Therefore, the explicit scheme
(11.10) or expression (11.16) recovers the original partial differential equation (11.1),
or it is consistent. It is said to be first-order accurate in time and second-order accurate
in space, according to the order of magnitude of the truncation error.
In addition to the truncation error introduced in the discretization process, other
sources of error may be present in the approximate solution. Spontaneous disturbances
(such as the round-off error) may be introduced during either the evaluation or the
numerical solution process. A numerical approximation is said to be stable if these
disturbances decay and do not affect the solution.
The stability of the explicit scheme (11.10) may be examined using the von
Neumann method. Let us consider the error at a grid point,
n
ξin = Tin − T i ,
(11.18)
n
where Tin is the exact solution of the discretized system (11.10) and T i is the approximate numerical solution of the same system. This error could be introduced due to the
round-off error at each step of the computation. We need to monitor its decay/growth
with time. It can be shown that the evolution of this error satisfies the same homogeneous algebraic system (11.10) or
n
n
+ (1 − 2β) ξin + (β − α) ξi+1
.
ξin+1 = (α + β) ξi−1
(11.19)
417
2. Finite Difference Method
The error distributed along the grid line can always be decomposed in Fourier
space as
∞
ξin =
(11.20)
g n (k) eiπkxi
k=−∞
√
where i = −1, k is the wavenumber in Fourier space, and g n represents the function
g at time t = tn . As the system is linear, we can examine one component of (11.20)
at a time,
ξin = g n (k)eiπkxi .
(11.21)
The component at the next time level has a similar form
ξin+1 = g n+1 (k)eiπkxi .
(11.22)
Substituting the preceding two equations (11.21) and (11.22) into error equation
(11.19), we obtain,
or
g n+1 eiπkxi = g n [(α + β)eiπkxi−1 + (1 − 2β)eiπkxi + (β − α)eiπkxi+1 ]
(11.23)
g n+1
= [(α + β)e−iπkx + (1 − 2β) + (β − α)eiπkx ].
gn
(11.24)
This ratio g n+1/g n is called the amplification factor. The condition for stability is that
the magnitude of the error should decay with time, or
g n+1
1,
gn
(11.25)
for any value of the wavenumber k. For this explicit scheme, the condition for stability
(11.25) can be expressed as
2
θ
2
1 − 4β sin2
+ (2α sin θ)2 1,
(11.26)
where θ = kπx. The stability condition (11.26) also can be expressed as (Noye,
1983),
0 4α 2 2β 1.
(11.27)
For the pure diffusion problem (u = 0), the stability condition (11.27) for this
explicit scheme requires that
0β
1
2
or
t
1 x 2
,
2 D
(11.28)
which limits the size of the time step. For the pure convection problem (D = 0),
condition (11.27) will never be satisfied, which indicates that the scheme is always
418
Computational Fluid Dynamics
unstable and it means that any error introduced during the computation will explode
with time. Thus, this explicit scheme is useless for pure convection problems. To
improve the stability of the explicit scheme for the convection problem, one may use
an upwind scheme to approximate the convective term,
n
Tin+1 = Tin − 2α Tin − Ti−1
,
(11.29)
where the stability condition requires that
u
t
1.
x
(11.30)
The condition (11.30) is known as the Courant-Friedrichs-Lewy (CFL) condition.
This condition indicates that a fluid particle should not travel more than one spatial
grid in one time step.
It can easily be shown that the implicit scheme (11.13) is also consistent and
unconditionally stable.
It is normally difficult to show the convergence of an approximate solution theoretically. However, the Lax Equivalence Theorem (Richtmyer and Morton, 1967)
states that: for an approximation to a well-posed linear initial value problem, which
satisfies the consistency condition, stability is a necessary and sufficient condition for
the convergence of the solution.
For convection-diffusion problems, the exact solution may change significantly
in a narrow boundary layer. If the computational grid is not sufficiently fine to resolve
the rapid variation of the solution in the boundary layer, the numerical solution may
present unphysical oscillations adjacent to or in the boundary layer. To prevent the
oscillatory solution, a condition on the cell Peclét number (or Reynolds number) is
normally required (see Section 4),
Rcell = u
x
2.
D
(11.31)
3. Finite Element Method
The finite element method was developed initially as an engineering procedure for
stress and displacement calculations in structural analysis. The method was subsequently placed on a sound mathematical foundation with a variational interpretation
of the potential energy of the system. For most fluid dynamics problems, finite element applications have used the Galerkin finite element formulation on which we will
focus in this section.
Weak or Variational Form of Partial Differential Equations
Let us consider again the one-dimensional transport problem (11.1). The form of
(11.1) with the boundary condition (11.2) and the initial conditions (11.3) is called
the strong (or classical) form of the problem.
We first define a collection of trial solutions, which consists of all functions
L
that have square-integrable first derivatives (H 1 functions, i.e. 0 (T,x )2 dx < ∞ if
419
3. Finite Element Method
T ∈ H 1 ) and satisfy the Dirichlet type of boundary condition (where the value of the
variable is specified) at x = 0. This is expressed as the trial functional space,
S = T T ∈ H 1 , T (0) = g .
(11.32)
The variational space of the trial solution is defined as
V = w
w ∈ H 1 , w (0) = 0 ,
(11.33)
which requires a corresponding homogeneous boundary condition.
We next multiply the transport equation (11.1) by a function in the variational space
(w ∈ V ), and integrate the product over the domain where the problem is defined,
L
0
∂T
w dx + u
∂t
L
0
∂T
w dx = D
∂x
L
0
∂ 2T
w dx.
∂x 2
(11.34)
Integrating the right-hand-side of (11.34) by parts, we have
L
∂T
w dx + u
∂t
0
= Dqw (L) ,
L
0
∂T
w dx + D
∂x
L
0
L
∂T ∂w
∂T
w
dx = D
∂x ∂x
∂x
0
(11.35)
where the boundary conditions ∂T/∂x = q and w (0) = 0 are applied. The integral
equation (11.35) is called the weak form of this problem. Therefore, the weak form
can be stated as: Find T ∈ S such that for all w ∈ V ,
L
0
∂T
w dx + u
∂t
0
L
∂T
w dx + D
∂x
L
0
∂T ∂w
dx = Dqw (L) .
∂x ∂x
(11.36)
It can be formally shown that the solution of the weak problem is identical to that
of the strong problem, or that the strong and weak forms of the problem are equivalent.
Obviously, if T is a solution of the strong problem (11.1) and (11.2), it must also be a
solution of the weak problem (11.36) using the procedure for derivation of the weak
formulation. However, let us assume that T is a solution of the weak problem (11.36).
By reversing the order in deriving the weak formulation, we have
0
L
∂T
∂T
∂ 2T
+u
−D 2
∂t
∂x
∂x
∂T
wdx + D
(L) − q w (L) = 0.
∂x
(11.37)
Satisfying (11.37) for all possible functions of w ∈ V requires that
∂T
∂T
∂ 2T
+u
− D 2 = 0 for x ∈ (0, L) ,
∂t
∂x
∂x
and
∂T
(L) − q = 0,
∂x
(11.38)
420
Computational Fluid Dynamics
which means that this solution T will be also a solution of the strong problem. It
should be noted that the Dirichlet type of boundary condition (where the value of
the variable is specified) is built into the trial functional space S, and is thus called
an essential boundary condition. However, the Neumann type of boundary condition
(where the derivative of the variable is imposed) is implied by the weak formulation
as indicated in (11.38) and is referred to as a natural boundary condition.
Galerkin’s Approximation and Finite Element Interpolations
As we have shown, the strong and weak forms of the problem are equivalent, and there
is no approximation involved between these two formulations. Finite element methods
start with the weak formulation of the problem. Let us construct finite-dimensional
approximations of S and V , which are denoted by S h and V h , respectively. The superscript refers to a discretization with a characteristic grid size h. The weak formulation
(11.36) can be rewritten using these new spaces, as: Find T h ∈ S h such that for all
wh ∈ V h ,
0
L
∂T h h
w dx + u
∂t
L
∂T h h
w dx + D
∂x
0
L
∂T h ∂wh
∂x ∂x
0
dx = Dqwh (L) .
(11.39)
Normally, S h and V h will be subsets of S and V , respectively. This means that if a
function φ ∈ S h then φ ∈ S, and if another function ψ ∈ V h then ψ ∈ V . Therefore,
(11.39) defines an approximate solution T h to the exact weak form of the problem
(11.36).
It should be noted that, up to the boundary condition T (0) = g, the function
spaces S h and V h are composed of identical collections of functions. We may take
out this boundary condition by defining a new function
v h (x, t) = T h (x, t) − g h (x) ,
(11.40)
where g h is a specific function that satisfies the boundary condition g h (0) = g.
Thus, the functions v h and wh belong to the same space V h . Equation (11.39) can be
rewritten in terms of the new function v h : Find T h = v h + g h , where v h ∈ V h , such
that for all wh ∈ V h ,
L
∂v h h
(11.41)
w dx + a wh , v h = Dqwh (L) − a w h , g h .
∂t
0
The operator a (·, ·)is defined as
a (w, v) = u
0
L
∂v
w dx + D
∂x
0
L
∂v ∂w
dx.
∂x ∂x
(11.42)
The formulation (11.41) is called a Galerkin formulation, because the solution
and the variational functions are in the same space. Again, the Galerkin formulation
of the problem is an approximation to the weak formulation (11.36). Other classes
421
3. Finite Element Method
of approximation methods, called Petrov-Galerkin methods, are those in which the
solution function may be contained in a collection of functions other than V h .
Next we need to explicitly construct the finite-dimensional variational space
V h . Let us assume that the dimension of the space is n and that the basis (shape or
interpolation) functions for the space are
NA (x) , A = 1, 2, ..., n.
(11.43)
Each shape function has to satisfy the boundary condition at x = 0,
NA (0) = 0, A = 1, 2, ..., n,
(11.44)
which is required by the space V h . The form of the shape functions will be discussed
later. Any function wh ∈ V h can be expressed as a linear combination of these shape
functions,
n
(11.45)
wh =
cA NA (x),
A=1
where the coefficients cA are independent of x and uniquely define this function. We
may introduce one additional function N0 to specify the function g h in (11.40) related
to the essential boundary condition. This shape function has the property
N0 (0) = 1.
(11.46)
Therefore, the function g h can be expressed as
g h (x) = gN0 (x),
and g h (0) = g.
(11.47)
With these definitions, the approximate solution can be written as
v h (x, t) =
n
dA (t) NA (x),
(11.48)
dA (t) NA (x) + gN0 (x) ,
(11.49)
A=1
and
T h (x, t) =
n
A=1
where dA ’s are functions of time only for time dependent problems.
Matrix Equations, Comparison with Finite Difference Method
With the construction of the finite-dimensional space V h , the Galerkin formulation
of the problem (11.41) leads to a coupled system of ordinary differential equations.
422
Computational Fluid Dynamics
Substitution of the expressions for the variational function (11.45) and for the approximate solution (11.48) into the Galerkin formulation (11.41) yields
n
L
n
n
n
dB NB
c A NA ,
cA NA dx + a
ḋB NB
0
= Dq
n
A=1
B=1
A=1
A=1
B=1
cA NA (L) − a
n
cA NA , gN0
A=1
(11.50)
where ḋB = d(dB )/dt. Rearranging the terms, (11.50) reduces to
n
cA GA = 0,
n
dB a (NA , NB ) − DqNA (L) + ga (NA , N0 ) .
A=1
(11.51)
where
GA =
n
ḋB
L
(NA NB ) dx +
0
B=1
B=1
(11.52)
As the Galerkin formulation (11.41) should hold for all possible functions of w h ∈ V h ,
the coefficients, cA , should be arbitrary. The necessary requirement for (11.51) to hold
is that each GA must be zero, that is,
n
ḋB
B=1
L
(NB NA ) dx +
0
n
B=1
dB a (NA , NB ) = DqNA (L)−ga (NA , N0 ) (11.53)
for A = 1, 2, . . . , n. System of equations (11.53) constitutes a system of n first-order
ordinary differential equations (ODEs) for the dB s. It can be put into a more concise
matrix form. Let us define,
where
MAB =
M = [MAB ], K = [KAB ], F = {FA }, d = {dB } ,
(11.54)
(NA NB ) dx,
(11.55)
L
0
KAB = u
0
L
NB,x NA dx + D
FA = DqNA (L) − gu
0
L
0
L
NB,x NA,x dx,
N0,x NA dx − gD
Equation (11.53) can then be written as
Mḋ + Kd = F.
0
(11.56)
L
N0,x NA,x dx.
(11.57)
(11.58)
423
3. Finite Element Method
The system of equations (11.58) is also termed the matrix form of the problem.
Usually, M is called the mass matrix, K is the stiffness matrix, F is the force vector,
and d is the displacement vector. This system of ODE’s can be integrated by numerical
methods, for example, Runge-Kutta methods, or discretized (in time) by finite difference schemes as described in the previous section. The initial condition (11.3) will be
used for integration. An alternative approach is to use a finite difference approximation
to the time derivative term in the transport equation
(11.1)at the beginning of the
process, for example, by replacing ∂T / ∂t with T n+1 − T n /t, and then using the
finite element method to discretize the resulting equation.
Now let us consider the actual construction of the shape functions for the finite
dimensional variational space. The simplest example is to use piecewise-linear finite
element space. We first partition the domain
[0, L] into
n nonoverlapping subintervals
(elements). A typical one is denoted as xA , xA+1 . The shape functions associated
with the interior nodes, A = 1, 2, . . . , n − 1, are defined as
x−x
A−1
, xA−1 x < xA ,
x −x
A
A−1
xA+1 − x
(11.59)
NA (x) =
, xA x xA+1 ,
x
−
x
A+1
A
0,
elsewhere.
Further, for the boundary nodes, the shape functions are defined as
Nn (x) =
x − xn−1
, xn−1 x xn ,
xn − xn−1
(11.60)
x1 − x
, x0 x x1 .
x1 − x0
(11.61)
and
N0 (x) =
These shape functions are graphically plotted in Figure 11.2. It should be noted that
these shape functions have very compact (local) support and satisfy NA (xB ) = δAB ,
where δAB is the Kronecker delta (i.e. δAB = 1 if A = B, whereas δAB = 0 if
A = B).
With the construction of the shape functions, the coefficients, dA s, in the expression for the approximate solution (11.49) represent the values of T h at the nodes
x = xA (A = 1, 2, . . . , n), or
dA = T h (xA ) = TA .
1
N0
x0 = 0 x1
N A−1 N A
N A+1
x A −1
x A +1
xA
Figure 11.2 Piecewise linear finite element space.
(11.62)
Nn
xn −1
xn =L
424
Computational Fluid Dynamics
To compare the discretized equations generated from the finite element method
with those from finite difference methods, we substitute (11.59) into (11.53) and
evaluate the integrals. For an interior node xA (A = 1, 2, . . . , n − 1), we have
d
dt
D
2TA
TA+1
TA−1
u
+
+
+ (TA+1 − TA−1 )− 2 (TA−1 − 2TA + TA+1 ) = 0,
6
3
6
2h
h
(11.63)
where h is the uniform mesh size. The convective and diffusive terms in expression
(11.63) have the same forms as those discretized using the standard second-order finite
difference method (centered difference) in (11.12). However, in the finite element
scheme, the time derivative term is presented with a three-point spatial average of the
variable T , which differs from the finite difference method. In general, the Galerkin
finite element formulation is equivalent to a finite difference method. The advantage
of the finite element method lies in its flexibility to handle complex geometries.
Element Point of View of the Finite Element Method
So far we have been using a global view of the finite element method. The shape
functions are defined on the global domain, as shown in Figure 11.2. However, it is
also convenient to present the finite element method using a local (or element) point of
view. This viewpoint is useful for the evaluation of the integrals in (11.55) to (11.57)
and the actual computer implementation of the finite element method.
Figure 11.3 depicts the global and local descriptions of the eth element. The
global description of the element e is just the “local” view of the full domain shown in
Figure 11.2. Only two shape functions are nonzero within this element, NA−1 and NA .
Using the local coordinate in the standard element (parent domain) as shown on the
right of Figure 11.3, we can write the standard shape functions as
N1 (ξ ) =
1
(1 − ξ )
2
and N2 (ξ ) =
1
(1 + ξ ) .
2
(11.64)
Clearly, the standard shape function N1 (or N2 ) corresponds to the global shape
function NA−1 (or NA ). The mapping between the domains of the global and local
descriptions can easily be generated with the help of these shape functions,
x (ξ ) = N1 (ξ ) x1e + N2 (ξ ) x2e =
1
N A−1
xA − 1
NA
he
element e
1
(xA − xA−1 ) ξ + xA + xA−1 ,
2
1
xA x
N1
ξ1 = −1
N2
2
=1
standard element in parent domain
Figure 11.3 Global and local descriptions of an element.
(11.65)
425
3. Finite Element Method
with the notation that x1e = xA−1 and x2e = xA . One can also solve (11.65) for the
inverse map
2x − xA − xA−1
.
(11.66)
ξ (x) =
xA − xA−1
Within the element e, the derivative of the shape functions can be evaluated using
the mapping equation (11.66),
and
dN1
dNA dξ
2
−1
dNA
=
=
=
dx
dξ dx
xA − xA−1 dξ
xA − xA−1
(11.67)
dN2
dNA+1 dξ
2
1
dNA+1
=
=
=
.
dx
dξ dx
xA − xA−1 dξ
xA − xA−1
(11.68)
The global mass matrix (11.55), the global stiffness matrix (11.56), and the global
force vector (11.57) have been defined as the integrals over the global domain [0, L].
These integrals may be written as the summation of integrals over each element’s
domain. Thus
nel
nel
nel
Fe ,
(11.69)
Ke , F =
Me , K =
M=
e=1
e=1
e=1
e
Me = MAB
,
Fe = FAe
e
,
Ke = KAB
where nel is the total number of finite elements (in this case nel = n), and
e
= (NA NB ) dx,
MAB
(11.70)
(11.71)
e
e
KAB
=u
FAe
e
NB,x NA dx + D
= Dqδenel δAn − gu
e
e
NB,x NA,x dx,
N0,x NA dx − gD
e
N0,x NA,x dx
(11.72)
(11.73)
and e = x1e , x2e = xA−1 , xA is the domain of the eth element; and the first term
on right-hand-side of (11.73) is nonzero only for e = nel and A = n.
Given the construction of the shape functions, most of the element matrices
and force vectors in (11.71) to (11.73) will be zero. The non-zero ones require that
A = e or e + 1 and B = e or e + 1. We may collect these nonzero terms and arrange
them into the element mass matrix, stiffness matrix, and force vector as follows:
e
, f e = fae , a, b = 1, 2
(11.74)
me = meab , ke = kab
where
meab =
e
(Na Nb ) dx,
(11.75)
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Computational Fluid Dynamics
e
kab
=u
e
Nb,x Na dx + D
e
−gka1
e
fa = 0
Dqδa2
e
Nb,x Na,x dx,
e = 1,
e = 2, 3, . . . , nel − 1,
e = nel .
(11.76)
(11.77)
Here, me , ke and f e are defined with the local (element) ordering, and represent the
nonzero terms in the corresponding Me , Ke and Fe with the global ordering. The
terms in the local ordering need to be mapped back into the global ordering. For this
example, the mapping is defined as
e − 1 if a = 1
A=
(11.78)
e
if a = 2
for element e.
Therefore, in the element viewpoint, the global matrices and the global vector
can be constructed by summing the contributions of the element matrices and the
element vector, respectively. The evaluation of both the element matrices and the
element vector can be performed on a standard element using the mapping between
the global and local descriptions.
The finite element methods for two- or three-dimensional problems will follow
the same basic steps introduced in this section. However, the data structure and the
forms of the elements or the shape functions will be more complicated. Refer to
Hughes (1987) for a detailed discussion. In Section 5, we will present an example of
a two dimensional flow over a circular cylinder.
4. Incompressible Viscous Fluid Flow
In this section, we will discuss numerical schemes for solving incompressible viscous
fluid flows. We will focus on techniques using the primitive variables (velocity and
pressure). Other formulations using streamfunction and vorticity are available in the
literature (see Fletcher 1988, Vol. II) and will not be discussed here since their extensions to three-dimensional flows are not straightforward. The schemes to be discussed
normally apply to laminar flows. However, by incorporating additional appropriate
turbulence models, these schemes will also be effective for turbulent flows.
For an incompressible Newtonian fluid, the fluid motion satisfies the
Navier-Stokes equation,
ρ
∂u
+ (u · ∇)u = ρg − ∇p + µ∇ 2 u,
∂t
(11.79)
and the continuity equation,
∇ · u = 0,
(11.80)
where u is the velocity vector, g is the body force per unit mass, which could be
the gravitational acceleration, p is the pressure, and ρ, µ are the density and viscosity of the fluid, respectively. With the proper scaling, (11.79) can be written in the
427
4. Incompressible Viscous Fluid Flow
dimensionless form,
∂u
1 2
+ (u · ∇)u = g − ∇p +
∇ u
∂t
Re
(11.81)
where Re is the Reynolds number of the flow. In some approaches, the convective
term is rewritten in conservative form,
(u · ∇) u = ∇ · (uu) ,
(11.82)
because u is solenoidal.
In order to guarantee that a flow problem is well-posed, appropriate initial and
boundary conditions for the problem must be specified. For time-dependent flow
problems, the initial condition for the velocity,
u (x, t = 0) = u0 (x) ,
(11.83)
is required. The initial velocity field has to satisfy the continuity equation ∇ · u0 = 0.
At a solid surface, the fluid velocity should equal the surface velocity (no-slip condition). No boundary condition for the pressure is required at a solid surface. If the
computational domain contains a section where the fluid enters the domain, the fluid
velocity (and the pressure) at this inflow boundary should be specified. If the computational domain contains a section where the fluid leaves the domain (outflow section),
appropriate outflow boundary conditions include zero tangential velocity and zero
normal stress, or zero velocity derivatives, as further discussed in Gresho (1991).
Because the conditions at the outflow boundary are artificial, it should be checked
that the numerical results are not sensitive to the location of this boundary. In order
to solve the Navier-Stokes equations, it is also appropriate to specify the value of the
pressure at one reference point in the domain, because the pressure appears only as a
gradient and can be determined up to a constant.
There are two major difficulties in solving the Navier-Stokes equations numerically. One is related to the unphysical oscillatory solution often found in a
convection-dominated problem. The other is the treatment of the continuity equation
that is a constraint on the flow to determine the pressure.
Convection-Dominated Problems
As mentioned in Section 2, the exact solution may change significantly in a narrow
boundary layer for convection dominated transport problems. If the computational
grid is not sufficiently fine to resolve the rapid variation of the solution in the boundary layer, the numerical solution may present unphysical oscillations adjacent the
boundary. Let us examine the steady transport problem in one dimension,
u
∂ 2T
∂T
=D 2
∂x
∂x
for 0 x L,
(11.84)
T (L) = 1.
(11.85)
with two boundary conditions
T (0) = 0
and
428
Computational Fluid Dynamics
The exact solution for this problem is
T =
where
eRx/L − 1
eR − 1
R = uL/D
(11.86)
(11.87)
is the global Peclét number. For large values of R, the solution (11.86) behaves as
T = e−R(1−x / L) .
(11.88)
The essential feature of this solution is the existence of a boundary layer at
x = L , and its thickness δ is of the order of,
δ
=O
L
1
.
|R|
(11.89)
At 1 − x / L = 1/ R, T is about 37% of the boundary value; while at 1 − x /
L = 2 / R, T is about 13.5% of the boundary value.
If centered differences are used to discretize the steady transport equation (11.84)
using the grid shown in Figure 11.1, the resulting finite difference scheme is,
or
ux
Tj +1 − Tj −1 = Tj +1 − 2Tj + Tj −1 ,
2D
0.5Rcell Tj +1 − Tj −1 = Tj +1 − 2Tj + Tj −1 ,
(11.90)
(11.91)
where the grid spacing x = L / n and the cell Peclét number Rcell = ux /
D = R / n. From the scaling of the boundary thickness (11.89) we know that it
is of the order,
L
x
δ=O
=O
.
(11.92)
nRcell
Rcell
Physically, if T represents the temperature in the transport problem (11.84), the
convective term brings the heat toward the boundary x = L, whereas the diffusive
term conducts the heat away through the boundary. These two terms have to be
balanced. The discretized equation (11.91) has the same physical meaning. Let us
examine this balance for a node next to the boundary, j = n − 1. When the cell
Peclét number Rcell > 2, according to (11.92) the thickness of the boundary layer
is less than half the grid spacing, and the exact solution (11.86) indicates that the
temperatures Tj and Tj −1 are already outside the boundary layer and are essentially
zero. Thus, the two sides of the discretized equation (11.91) cannot balance, or the
conduction term is not strong enough to remove the heat convected to the boundary,
assuming the solution is smooth. In order to force the heat balance, an unphysical
oscillatory solution with Tj < 0 is generated to enhance the conduction term in
the discretized problem (11.91). To prevent the oscillatory solution, the cell Peclét
number is normally required to be less than two, which can be achieved by refining
429
4. Incompressible Viscous Fluid Flow
the grid to resolve the flow inside the boundary layer. In some respect, an oscillatory
solution may be a virtue since it provides a warning that a physically important feature
is not being properly resolved. To reduce the overall computational cost, non-uniform
grids with local fine grid spacing inside the boundary layer will frequently be used to
resolve the variables there.
Another common method to avoid the oscillatory solution is to use a first-order
upwind scheme,
Rcell Tj − Tj −1 = Tj +1 − 2Tj + Tj −1 ,
(11.93)
where a forward difference scheme is used to discretize the convective term. It is
easy to see that this scheme reduces the heat convected to the boundary and thus
prevents the oscillatory solution. However, the upwind scheme is not very accurate
(only first-order accurate). It can be easily shown that the upwind scheme (11.93)
does not recover the original transport equation (11.84). Instead it is consistent with a
slightly different transport equation (when the cell Peclét number is kept finite during
the process),
∂T
∂ 2T
u
(11.94)
= D (1 + 0.5Rcell ) 2 .
∂x
∂x
Thus, another way to view the effect of the first-order upwind scheme (11.93) is
that it introduces a numerical diffusivity of the value of 0.5Rcell D, which enhances
the conduction of heat through the boundary. For an accurate solution, one normally
requires that 0.5Rcell << 1, which is very restrictive and does not offer any advantage
over the centered difference scheme (11.91).
Higher-order upwind schemes may be introduced to obtain more accurate
non-oscillatory solutions without excessive grid refinement. However, those schemes
may be less robust. Refer to Fletcher (1988, vol.I, chapter 9) for discussions.
Similarly, there are upwind schemes for finite element methods to solve
convection-dominated problems. Most of those are based on Petrov-Galerkin approach that permit an effective upwind treatment of the convective term along local
streamlines (Brooks and Hughes, 1982). More recently, stabilized finite element methods have been developed where a least-square term is added to the momentum balance
equation to provide the necessary stability for convection-dominated flows (see Franca
et al., 1992).
Incompressibility Condition
In solving the Navier-Stokes equations using the primitive variables (velocity and
pressure), another numerical difficulty lies in the continuity equation: The continuity
equation can be regarded either as a constraint on the flow field to determine the pressure or the pressure plays the role of the Lagrange multiplier to satisfy the continuity
equation.
In a flow field, the information (or disturbance) travels with both the flow and the
speed of sound in the fluid. Since the speed of sound is infinite in an incompressible
fluid, part of the information (pressure disturbance) is propagated instantaneously
430
Computational Fluid Dynamics
throughout the domain. In many numerical schemes the pressure is often obtained
by solving a Poisson equation. The Poisson equation may occur in either continuous
form or discrete form. Some of these schemes will be described here. In some of
them, solving the pressure Poisson equation is the most costly step.
Another common technique to surmount the difficulty of the incompressible
limit is to introduce an artificial compressibility (Chorin, 1967). This formulation
is normally used for steady problems with a pseudo-transient formulation. In the
formulation, the continuity equation is replaced by,
∂p
+ c2 ∇ · u = 0,
∂t
(11.95)
where c is an arbitrary constant and could be the artificial speed of sound in a
corresponding compressible fluid with the equation of state p = c2 ρ. The formulation
is called pseudo-transient because (11.95) does not have any physical meaning before
the steady state is reached. However, when c is large, (11.95) can be considered as an
approximation to the unsteady solution of the incompressible Navier-Stokes problem.
Explicit MacCormack Scheme
Instead of using the artificial compressibility in (11.95), one may start with the exact
compressible Navier-Stokes equations. In Cartesian coordinates, the component form
of the continuity equation (4.8) and compressible Navier-Stokes equation (4.44) in
two dimensions can be explicitly written as
∂ (ρu) ∂ (ρv)
∂ρ
+
+
= 0,
∂t
∂x
∂y
(11.96)
µ ∂
∂
∂ 2
∂p
∂
+ µ∇ 2 u +
ρu +
(ρu) +
(ρvu) = ρgx −
∂t
∂x
∂y
∂x
3 ∂x
∂u ∂v
+
,
∂x
∂y
(11.97)
∂
∂ 2
∂p
µ ∂
∂
ρv = ρgy −
+ µ∇ 2 v +
(ρv) +
(ρuv) +
∂t
∂x
∂y
∂y
3 ∂y
∂u ∂v
+
,
∂x
∂y
(11.98)
with the equation of state, p = c2 ρ
(11.99)
where c is speed of sound in the medium. As long as the flows are limited to low
Mach numbers and the conditions are almost isothermal, the solution to this set of
equations should approximate the incompressible limit.
The explicit MacCormack scheme, after R.W. MacCormack (1969), is
essentially a predictor-corrector scheme, similar to a second-order Runge-Kutta
431
4. Incompressible Viscous Fluid Flow
method commonly used to solve ordinary differential equations. For a system of
equations of the form,
∂U ∂E (U) ∂F (U)
+
+
= 0,
∂t
∂x
∂y
(11.100)
the explicit MacCormack scheme consists of two steps,
t
t n
n
n
n
∗
n
predictor : Ui,j
−
(11.101)
−
F
= Ui,j
−
Ei+1,j − Ei,j
Fi,j
i,j ,
+1
x
y
t
1
t ∗
n+1
∗
∗
∗
n
∗
corrector : Ui,j =
Ei,j − Ei−1,j −
Fi,j − Fi,j −1
Ui,j + Ui,j −
2
x
y
(11.102)
Notice that the spatial derivatives in (11.100) are discretized with opposite one-sided
finite differences in the predictor and corrector stages. The star variables are all evaluated at time level tn+1 . This scheme is second-order accurate in both time and space.
Applying the MacCormack scheme to the compressible Navier-Stokes equations
(11.96) to (11.98) and replacing the pressure with (11.99), we have the predictor step,
∗
n
ρi,j
(11.103)
= ρi,j
− c1 (ρu)ni+1,j − (ρu)ni,j − c2 (ρv)ni,j +1 − (ρv)ni,j
(ρu)∗i,j
= (ρu)ni,j
− c1
2
2
ρu + c ρ
n
i+1,j
n
2
2
− ρu + c ρ
i,j
4
− c2 (ρuv)ni,j +1 − (ρuv)ni,j + c3 uni+1,j − 2uni,j + uni−1,j
3
n
n
n
+ c4 ui,j +1 − 2ui,j + ui,j −1
n
n
n
n
+
v
−
v
−
v
(11.104)
+ c5 vi+1,j
+1
i−1,j −1
i+1,j −1
i−1,j +1
(ρv)∗i,j = (ρv)ni,j − c1 (ρuv)ni+1,j − (ρuv)ni,j
n
n
− ρv 2 + c2 ρ
− c2 ρv 2 + c2 ρ
i,j +1
i,j
4 n
n
n
−
2v
+
v
+ c4 vi,j
i,j
+1
i,j −1
3
n
n
n
− 2vi,j
+ vi−1,j
+ c3 vi+1,j
+ c5 uni+1,j +1 + uni−1,j −1 − uni+1,j −1 − uni−1,j +1
(11.105)
Similarly, the corrector step is given by
n+1
n
∗
2ρi,j
= ρi,j
+ρi,j
−c1 (ρu)∗i,j − (ρu)∗i−1,j −c2 (ρv)∗i,j − (ρv)∗i,j −1 (11.106)
432
Computational Fluid Dynamics
2 (ρu)n+1
i,j
= (ρu)ni,j
+ (ρu)∗i,j
− c1
2
2
ρu + c ρ
∗
i,j
2
2
− ρu + c ρ
∗
i−1,j
4
− c2 (ρuv)∗i,j − (ρuv)∗i,j −1 + c3 u∗i+1,j − 2u∗i,j + u∗i−1,j
3
∗
∗
∗
+ c4 ui,j +1 − 2ui,j + ui,j −1
∗
∗
∗
∗
+ c5 vi+1,j
(11.107)
+1 + vi−1,j −1 − vi+1,j −1 − vi−1,j +1
∗
∗
∗
n
−
c
+
=
−
2 (ρv)n+1
(ρuv)
(ρuv)
(ρv)
(ρv)
1
i,j
i,j
i−1,j
i,j
i,j
∗
∗
− ρv 2 + c2 ρ
− c2 ρv 2 + c2 ρ
i,j −1
i,j
4 ∗
∗
∗
+ c4 vi,j
−
2v
+
v
i,j
+1
i,j −1
3
∗
∗
∗
+ c3 vi+1,j
− 2vi,j
+ vi−1,j
+ c5 u∗i+1,j +1 + u∗i−1,j −1 − u∗i+1,j −1 − u∗i−1,j +1
(11.108)
The coefficients are defined as,
c1 =
t
,
x
c2 =
t
,
y
c3 =
µt
2
(x)
,
c4 =
µt
(y)
2
,
c5 =
µt
.
12xy
(11.109)
In both the predictor and corrector steps the viscous terms (the second-order derivative
terms) are all discretized with centered-differences to maintain second-order accuracy.
For brevity, body force terms in the momentum equations are neglected here.
During the predictor and corrector stages of the explicit MacCormack scheme
(11.103) to (11.108), one-sided differences are arranged in the FF and BB fashion,
respectively. Here, in the notation FF, the first F denotes the forward difference in
the x-direction and the second F denotes the forward difference in the y-direction.
Similarly, BB stands for backward differences in both x and y directions. We denote
this arrangement as FF/BB. Similary, one may get BB/FF, FB/BF, BF/FB arrangements. It is noted that some balanced cyclings of these arrangements generate better
results than others.
Tannehill, Anderson and Pletcher (1997) give the following semi-empirical stability criterion for the explicit MacCormack scheme,
σ
t
1 + 2 Re
|u|
|v|
1
1
+
+
+c
2
x
y
x
y 2
−1
,
(11.110)
where σ is a safety factor (≈ 0.9), Re = min (ρ |u| x / µ, ρ |v| y / µ) is the
minimum mesh Reynolds number. This condition is quite conservative for flows with
small mesh Reynolds numbers.
433
4. Incompressible Viscous Fluid Flow
One key issue for the explicit MacCormack scheme to work properly is the
boundary conditions for density (thus pressure). We leave this issue to the next section
where its implementation in two sample problems will be demonstrated.
MAC Scheme
Most of numerical schemes developed for computational fluid dynamics problems can
be characterized as operator splitting algorithms. The operator splitting algorithms
divide each time step into several substeps. Each substep solves one part of the operator
and thus decouples the numerical difficulties associated with each part of the operator.
For example, consider a system,
dφ
+ A (φ) = f,
dt
(11.111)
with initial condition φ (0) = φ0 , where the operator A may be split into two operators
A (φ) = A1 (φ) + A2 (φ) .
(11.112)
Using a simple first-order accurate Marchuk-Yanenko fractional-step scheme
(Yanenko, 1971, and Marchuk, 1975), the solution of the system at each time step
φ n+1 = φ ((n + 1) t) (n = 0,1, . . .) is approximated by solving the following two
successive problems:
φ n+1/ 2 − φ n
+ A1 φ n+1/ 2 = f1n+1 ,
t
φ n+1 − φ n+1/ 2
+ A2 φ n+1 = f2n+1 ,
t
(11.113)
(11.114)
where φ 0 = φ0 , t = tn+1 − tn , and f1n+1 + f2n+1 = f n+1 = f ((n + 1) t).
The time discretizations in (11.113) and (11.114) are implicit. Some schemes to be
discussed in what follows actually use explicit discretizations. However, the stability
conditions for those explicit schemes must be satisfied.
The MAC (marker-and-cell) method was first proposed by Harlow and Welsh
(1965) to solve flow problems with free surfaces. There are many variations of this
method. It basically uses a finite difference discretization for the Navier-Stokes equations and splits the equations into two operators
1 2
∇p
∇ u
(u · ∇) u −
, and A2 (u, p) =
. (11.115)
A1 (u, p) =
Re
∇ ·u
0
Each time step is divided into two substeps as discussed in the Marchuk-Yanenko
fractional-step scheme (11.113) and (11.114). The first step solves a convection and
diffusion problem, which is discretized explicitly,
un+1/ 2 − un
1 2 n
+ (un · ∇)un −
∇ u = gn+1 .
t
Re
(11.116)
434
Computational Fluid Dynamics
In the second step, the pressure gradient operator is added (implicitly) and, at the
same time, the incompressible condition is enforced,
un+1 − un+1/ 2
+ ∇p n+1 = 0,
t
(11.117)
∇ · un+1 = 0.
and
(11.118)
This step is also called a projection step to satisfy the incompressibility condition.
Normally, the MAC scheme is presented in a discretized form. A preferred feature
of the MAC method is the use of the staggered grid. An example of a staggered grid
in two dimensions is shown in Figure 11.4. On this staggered grid, pressure variables
are defined at the centers of the cells and velocity components are defined at the cell
faces, as shown in Figure 11.4.
Using the staggered grid, two components of the transport equation (11.116) can
be written as,
∂u
1 2
∂u
n+1 2
+v
−
∇ u
ui+1//2,j = uni+1/ 2,j −t u
∂x
∂y
Re
n
∂v
∂v
1 2
n+1 2
n
vi,j +1/ / 2 = vi,j
+v
−
∇ v
+1/ 2 − t u
∂x
∂y
Re
n
where u = (u, v), g = (f, g), u
1
− ∇ 2v
Re
i+1/ 2,j
i,j +1/ 2
∂u
∂u
1 2
+v
−
∇ u
∂x
∂y
Re
n
n+1
+t fi+1
/ 2,j, (11.119)
n+1
+ t gi,j
+1/ 2 , (11.120)
n
and u
i+1/ 2,j
∂v
∂v
+v
∂x
∂y
are the functions interpolated at the grid locations for the
i,j +1/ 2
Γ
2,5/2
3,5/2
u1/2,2
1,5/2
p1,2u3/2,2
p2,2 u5/2,2
u1/2,1
1,3/2
p u3/2,1
2,3/2
p2,1 u5/2,1
p3,1
2,1/2
3,1/2 Γ
1,1
1,1/2
Figure 11.4 Staggered grid and a typical cell around p2,2 .
p3,2
3,3/2
435
4. Incompressible Viscous Fluid Flow
x-component of the velocity at (i + 1/ 2, j ) and for the y-component of the velocity
at (i, j + 1/ 2), respectively, and at the previous time t = tn . The discretized form of
(11.117) is
n+1/ 2
un+1
i+1/ 2,j = ui+1/ 2,j −
t n+1
n+1
,
pi+1,j − pi,j
x
(11.121)
n+1/ 2
n+1
vi,j
+1/ 2 = vi,j +1/ 2 −
t n+1
n+1
,
pi,j +1 − pi,j
y
(11.122)
where x = xi+1 − xi and y = yj +1 − yj are the uniform grid spacing in the
x and y directions, respectively. The discretized continuity equation (11.118) can be
written as,
n+1
un+1
i+1/ 2,j − ui−1/ 2,j
x
+
n+1
n+1
vi,j
+1/ 2 − vi,j −1/ 2
y
= 0.
(11.123)
Substitution of the two velocity components from (11.121) and (11.122) into the
discretized continuity equation (11.123) generates a discrete Poisson equation for the
pressure,
1 n+1
1 n+1
n+1
n+1
n+1
n+1
+
p
p
−
2p
+
p
−
2p
+
p
i,j
i,j
i+1,j
i,j
+1
i−1,j
i,j
−1
x 2
y 2
n+1/ 2
n+1/ 2
n+1 2
n+1/ 2
vi,j +1/ 2 − vi,j −1/ / 2
1 ui+1/ 2,j − ui−1/ 2,j
.
(11.124)
+
=
t
x
y
n+1
∇d2 pi,j
≡
The major advantage of the staggered grid is that it prevents the appearance
of oscillatory solutions. On a normal grid, the pressure gradient would have to be
approximated using two alternate grid points (not the adjacent ones) when a central
difference scheme is used, that is
∂p
∂x
i,j
=
pi+1,j − pi−1,j
2x
and
∂p
∂y
i,j
=
pi,j +1 − pi,j −1
.
2y
(11.125)
Thus a wavy pressure field (in a zigzag pattern) would be felt like a uniform one
by the momentum equation. However, on a staggered grid, the pressure gradient is
approximated by the difference of the pressures between two adjacent grid points.
Consequently, a pressure field with a zigzag pattern would no longer be felt as a
uniform pressure field and could not arise as a possible solution. It is also seen that
the discretized continuity equation (11.123) contains the differences of the adjacent
velocity components, which would prevent a wavy velocity field from satisfying the
continuity equation.
436
Computational Fluid Dynamics
Another advantage of the staggered
grid is its accuracy. For example, the
truncation error for (11.123) is O x 2 , y 2 even though only four grid points
are involved. The pressure gradient evaluated at the cell faces,
∂p
∂x
i+1/ 2,j
=
pi+1,j − pi,j
,
x
and
∂p
∂y
i,j +1/ 2
=
pi,j +1 − pi,j
, (11.126)
y
are all second-order accurate.
On the staggered grid, the MAC method does not require boundary conditions for
the pressure equation (11.124). Let us examine a pressure node next to the boundary,
for example p1,2 as shown in Figure 11.4. When the normal velocity is specified at the
boundary, un+1
1/ 2,2 is known. In evaluating the discrete continuity equation (11.123) at
n+1/ 2
the pressure node (1, 2), the velocity un+1
1/ 2,2 should not be expressed in terms of u1/2,2
using (11.121). Therefore p0,2 will not appear in equation (11.120), and no boundary
condition for the pressure is needed. It should also be noted that (11.119) and (11.120)
only update the velocity components for the interior grid points, and their values at
the boundary grid points are not needed in the MAC scheme. Peyret and Taylor (1983,
chapter 6) also noticed that the numerical solution in the MAC method is independent
of the boundary values of un+1/ 2 and v n+1/ 2 , and a zero normal pressure gradient on
the boundary would give satisfactory results. However, their explanation was more
cumbersome.
In summary, for each time step in the MAC scheme, the intermediate velocity
n+1/ 2
n+1/ 2
and vi,j
components, ui+1
+1/ 2 , in the interior of the domain are first evaluated
/ 2,j
using (11.119) and (11.120), respectively. Next, the discrete pressure Poisson equation
(11.124) is solved. Finally, the velocity components at the new time step are obtained
from (11.121) and (11.122). In the MAC scheme, the most costly step is the solution
of the Poisson equation for the pressure (11.124).
Chorin (1968) and Temam (1969) independently presented a numerical scheme
for the incompressible Navier-Stokes equations, termed the projection method. The
projection method was initially proposed using the standard grid. However, when it
is applied in an explicit fashion on the MAC staggered grid, it is identical to the MAC
method as long as the boundary conditions are not considered, as shown in Peyret
and Taylor (1983, chapter 6).
A physical interpretation of the MAC scheme or the projection method is that
the explicit update of the velocity field does not generate a divergence free velocity
field in the first step. Thus an irrotational correction field, in the form of a velocity
potential which is proportional to the pressure, is added to the nondivergence-free
velocity field in the second step in order to enforce the incompressibility condition.
As the MAC method uses an explicit scheme in the convection-diffusion step,
the stability conditions for this method are (Peyret and Taylor, 1983, chapter 6),
1 2
u + v 2 tRe 1,
2
(11.127)
437
4. Incompressible Viscous Fluid Flow
4t
1,
Rex 2
and
(11.128)
when x = y. The stability conditions (11.127) and (11.128) are quite restrictive
on the size of the time step. These restrictions can be removed by using implicit
schemes for the convection-diffusion step.
-Scheme
The MAC algorithm described in the preceding section is only first-order accurate
in time. In order to have a second-order accurate scheme for the Navier-Stokes
equations, the -scheme of Glowinski (1991) may be used. The -scheme
splits each time step symmetrically into three substeps, which are described
here.
• Step 1:
α 2 n+θ
β 2 n
un+θ − un
−
∇ u
+ ∇p n+θ = gn+θ +
∇ u − (un · ∇)un ,
θ t
Re
Re
(11.129)
∇ · un+θ = 0.
(11.130)
• Step 2:
β 2 n+1−θ
un+1−θ − un+θ
−
∇ u
+ (u∗ · ∇)un+1−θ
(1 − 2θ)t
Re
α 2 n+θ
∇ u
− ∇p n+θ .
= gn+1−θ +
Re
(11.131)
• Step 3:
α 2 n+1
β 2 n+1−θ
un+1 − un+1−θ
−
∇ u
+ ∇p n+1 = gn+1 +
∇ u
θ t
Re
Re
− (un+1−θ · ∇)un+1−θ ,
(11.132)
∇ · un+1 = 0.
(11.133)
√
It was shown that when θ = 1 − 1/ 2 = 0.29289 . . . , α + β = 1 and β =
θ/(1 − θ ), the scheme is second-order accurate. The first and third steps of the
-scheme are identical and are the Stokes flow problems. The second step, (11.131),
represents a nonlinear convection-diffusion problem if u∗ = un+1−θ . However, it
was concluded that there is practically no loss in accuracy and stability if u∗ = un+θ
is used. Numerical techniques for solving these substeps are discussed in Glowinski
(1991).
438
Computational Fluid Dynamics
Mixed Finite Element Formulation
The weak formulation described in Section 3 can be directly applied to the
Navier-Stokes equations (11.81) and (11.80), and it gives
∂u
2
p (∇ · ũ) d = 0,
D [u] : D ũ d −
+ u · ∇u − g · ũd +
∂t
Re
(11.134)
p̃∇ · ud = 0,
(11.135)
where ũ and p̃ are the variations of the velocity and pressure, respectively. The rate
of strain tensor is given by
D [u] =
1
∇u + (∇u)T .
2
(11.136)
The Galerkin finite element formulation for the problem is identical to (11.134)
and (11.135), except that all the functions are chosen from finite dimensional subspaces and represented in the form of basis or interpolation functions.
The main difficulty with this finite element formulation is the choice of the interpolation functions (or the type of the elements) for velocity and pressure. The finite
element approximations that use the same interpolation functions for velocity and
pressure suffer from a highly oscillatory pressure field. As described in the previous
section, a similar behavior in the finite difference scheme is prevented by introducing
the staggered grid. There are a number of options to overcome this problem with spurious pressure. One of them is the mixed finite element formulation that uses different
interpolation functions (or finite elements) for velocity and pressure. The requirement for the mixed finite element approach is related to the so-called Babuska-Brezzi
(or LBB) stability condition, or inf-sup condition. The detailed discussions for this
condition can be found in Oden and Carey (1984). A common practice in the mixed
finite element formulation is to use a pressure interpolation function that is one order
lower than a velocity interpolation function. As an example in two dimensions, a
triangular element is shown in Figure 11.5a. On this mixed element, quadratic interpolation functions are used for the velocity components and are defined on all six
nodes, while linear interpolation functions are used for the pressure and are defined
(a)
Figure 11.5 Mixed finite elements.
(b)
439
4. Incompressible Viscous Fluid Flow
on three vertices only. A slightly different approach is to use a pressure grid that is
twice coarser than the velocity one, and then use the same interpolation functions on
both grids (Glowinski, 1991). For example, a piecewise-linear pressure is defined on
the outside (coarser) triangle; while a piecewise-linear velocity is defined on all four
subtriangles, as shown in Figure 11.5b.
Another option to prevent a spurious pressure field is to use the stabilized finite
element formulation while keeping the equal order interpolations for velocity and
pressure. A general formulation in this approach is the Galerkin/least-squares (GLS)
stabilization (Tezduyar, 1992). In the GLS stabilization, the stabilizing terms are
obtained by minimizing the squared residual of the momentum equation integrated
over each element domain. The choice of the stabilization parameter is discussed in
Franca et al. (1992) and Franca and Frey (1992).
Comparing the mixed and the stabilized finite element formulations, the mixed
finite element method is parameter free, as pointed out in Glowinski (1991). There
is no need to adjust the stabilization parameters, which could be a delicate problem.
More importantly, for a given flow problem the desired finite element mesh size
is generally determined based on the velocity behavior (e.g., it is defined by the
boundary or shear layer thickness). Therefore, equal order interpolation will be more
costly from the pressure point of view but without further gains in accuracy. However,
the GLS-stabilized finite element formulation has the additional benefit of preventing
oscillatory solutions produced in the Galerkin finite element method due to the large
convective term in high Reynolds number flows.
Once the interpolation functions for the velocity and pressure in the mixed finite
element approximations are determined, the matrix form of equations (11.134) and
(11.135) can be written as
A B
Mu̇
+
0
BT 0
u
f
= u ,
fp
p
(11.137)
where u and p are the vectors containing all unknown values of the velocity components and pressure defined on the finite element mesh, respectively. u̇ is the first time
derivative of u. M is the mass matrix corresponding to the time derivative term in
equation (11.134). Matrix A depends on the value of u due to the nonlinear convective
term in the momentum equation. The symmetry in the pressure terms in (11.134) and
(11.135) results in the symmetric arrangement of B and BT in the algebraic system
(11.137). Vectors fu and fp come from the body force term in the momentum equation
and from the application of the boundary conditions.
The ordinary differential equation (11.137) can be further discretized in time with
finite difference methods. The resulting nonlinear system of equations is typically
solved iteratively using Newton’s method. At each stage of the nonlinear iteration,
the sparse linear algebraic equations are normally solved either by using a direct
solver such as the Gauss elimination procedure for small system sizes or by using an
iterative solver such as the generalized minimum residual method (GMRES) for large
systems. Other iterative solution methods for sparse nonsymmetric systems can be
found in Saad (1996). An application of the mixed finite element method is discussed
as one of the examples in the next section.
440
Computational Fluid Dynamics
5. Three Examples
In this section, we will solve three sample problems. The first one is the classic driven
cavity flow problem. The second is flow around a square block confined between two
parallel plates. These two problems will be solved by using the explicit MacCormack
scheme, with details in Perrin and Hu (2006). The contribution by Andrew Perrin in
preparing results for these two problems is greatly appreciated. The last problem is
flow around a circular cylinder confined between two parallel plates. It will be solved
by using a mixed finite element formulation.
Explicit MacCormack Scheme for Driven Cavity Flow Problem
The driven cavity flow problem, in which a fluid-filled square box (“cavity”) is swirled
by a uniformly translating lid as shown in Figure 11.6, is a classic problem in CFD.
This problem is unambiguous with easily applied boundary conditions and has a
wealth of documented analytical and computational results, for example Ghia et al.
(1982). We will solve this flow using the explicit MacCormack scheme discussed in
the previous section.
We may nondimensionalize the problem with the following scaling: lengths with
D, velocity with U , time with D / U , density with a reference density ρ0 , and pressure
with ρ0 U 2 . Using this scaling, the equation of state (11.99) becomes p = ρ / M 2 ,
where M = U / c is the Mach number. The Reynolds number is defined as Re =
ρ0 U D / µ.
The boundary conditions for this problem are relatively simple. The velocity
components on all four sides of the cavity are well defined. There are two singularities
of velocity gradient at the two top corners where velocity u drops from U to 0 directly
underneath the sliding lid. However, these singularities will be smoothed out on a given
grid, since the change of the velocity occurs linearly between two grid points. The
boundary conditions for the density (hence the pressure) are more involved. Since the
density is not specified on a solid surface, we need to generate an update scheme for
U
D
D
y
x
Figure 11.6 Driven cavity flow problem. The cavity is filled with a fluid with the top lid sliding at a
constant velocity U .
441
5. Three Examples
values of density on all boundary points. A natural option is to derive that using the
continuity equation.
Consider the boundary on the left (at x = 0). Since v = 0 along the surface, the
continuity equation (11.96) reduces to
∂ρ
∂ρu
+
= 0.
∂t
∂x
(11.138)
We may use a predictor-corrector scheme to update density on this surface with
a one-sided second-order accurate discretization for the spatial derivative,
or
∂f
∂x
∂f
∂x
i
i
1
(−fi+2 + 4fi+1 − 3fi ) + O x 2
2x
−1
=
(−fi−2 + 4fi−1 − 3fi ) + O x 2 .
2x
=
Therefore, on the surface of x = 0 (for i = 0 including two corner points on the left),
we have the following update scheme for density,
t
∗
n
predictor ρi,j
− (ρu)ni+2,j + 4 (ρu)ni+1,j − 3 (ρu)ni,j ,
= ρi,j
−
2x
(11.139)
t
n+1
n
∗
corrector 2ρi,j
= ρi,j
+ ρi,j
−
− (ρu)∗i+2,j + 4 (ρu)∗i+1,j − 3 (ρu)∗i,j .
2x
(11.140)
Similarly, on the right side of the cavity x = D (for i = nx − 1, where nx is the
number of grid points in the x-direction, including two corner points on the right),
we have
t
∗
n
predictor ρi,j
= ρi,j
+
− (ρu)ni−2,j + 4 (ρu)ni−1,j − 3 (ρu)ni,j ,
2x
(11.141)
t
n+1
n
∗
− (ρu)∗i−2,j + 4 (ρu)∗i−1,j − 3 (ρu)∗i,j .
corrector 2ρi,j
= ρi,j
+ ρi,j
+
2x
(11.142)
On the bottom of the cavity y = 0 (j = 0),
t
∗
n
predictor ρi,j
= ρi,j
−
− (ρv)ni,j +2 + 4 (ρv)ni,j +1 − 3 (ρv)ni,j , (11.143)
2y
t
n+1
n
∗
− (ρv)∗i,j +2 + 4 (ρv)∗i,j +1 − 3 (ρv)∗i,j .
corrector 2ρi,j
= ρi,j
+ ρi,j
−
2y
(11.144)
Finally, on the top of the cavityy = D (j = ny − 1 where ny is the number of
grid points in the y-direction), the density needs to be updated from slightly different
expressions since ∂ρu/ ∂x = U ∂ρ / ∂x is not zero there,
442
Computational Fluid Dynamics
predictor
∗
n
ρi,j
= ρi,j
−
t
tU n
n
ρi+1,j − ρi−1,j
+
− (ρv)ni,j −2 + 4 (ρv)ni,j −1 − 3 (ρv)ni,j ,
2x
2y
(11.145)
corrector
tU ∗
t
n+1
n
∗
∗
2ρi,j
= ρi,j
+ ρi,j
−
+
ρi+1,j − ρi−1,j
− (ρv)∗i,j −2 + 4 (ρv)∗i,j −1
2x
2y
(11.146)
− 3 (ρv)∗i,j .
In summary, we may organize the explicit MacCormack scheme at each time
step (11.103) to (11.108) into the following six substeps.
Step 1:
For 0 i < nx and 0 j < ny (all nodes):
%
n
,
ui,j = (ρu)ni,j ρi,j
Step 2:
%
n
.
vi,j = (ρv)ni,j ρi,j
For 1 i < nx − 1 and 1 j < ny − 1 (all interior nodes):
∗
n
= ρi,j
− a1 (ρu)ni+1,j − (ρu)ni,j − a2 (ρv)ni,j +1 − (ρv)ni,j ,
ρi,j
n
n
n
− ρi,j
− a1 ρu2
(ρu)∗i,j = (ρu)ni,j − a3 ρi+1,j
i+1,j
n
− ρu2
i,j
− a2 (ρuv)ni,j +1 − (ρuv)ni,j − a10 ui,j + a5 ui+1,j + ui−1,j
+ a6 ui,j +1 + ui,j −1 +a9 vi+1,j +1 + vi−1,j −1 − vi+1,j −1 − vi−1,j +1 ,
n
n
n
n
−
a
−
−
ρ
(ρuv)
(ρv)∗i,j = (ρv)ni,j − a4 ρi,j
(ρuv)
1
i,j
i,j
i+1,j
+1
− a2
ρv 2
n
i,j +1
n
− a11 vi,j + a7 vi+1,j + vi−1,j
− ρv 2
i,j
+ a8 vi,j +1 + vi,j −1 + a9 ui+1,j +1 + ui−1,j −1 − ui+1,j −1 − ui−1,j +1 .
∗ , (ρu)∗ and (ρv)∗ .
Step 3: Impose boundary conditions (at time tn+1 ) for ρi,j
i,j
i,j
Step 4: For 0 i < nx and 0 j < ny (all nodes):
%
∗
,
u∗i,j = (ρu)∗i,j ρi,j
%
∗
∗
.
vi,j
= (ρv)∗i,j ρi,j
443
5. Three Examples
Step 5: For 1 i < nx − 1 and 1 j < ny − 1 (all interior nodes):
n+1
n
∗
2ρi,j
− a1 (ρu)∗i,j − (ρu)∗i−1,j − a2 (ρv)∗i,j − (ρv)∗i,j −1 ,
= ρi,j
+ ρi,j
2 (ρu)n+1
i,j
= (ρu)ni,j
+ (ρu)∗i,j
∗
∗
∗
∗
− ρu2
− a3 ρi,j − ρi−1,j − a1 ρu2
i,j
− a2 (ρuv)∗i,j − (ρuv)∗i,j −1 − a10 u∗i,j + a5 u∗i+1,j + u∗i−1,j
i−1,j
∗
∗
∗
∗
−
v
+
v
−
v
+ a6 u∗i,j +1 +u∗i,j −1 + a9 vi+1,j
i−1,j +1 ,
+1
i−1,j −1
i+1,j −1
∗
∗
∗
∗
∗
n
2 (ρv)n+1
i,j = (ρv)i,j + (ρv)i,j − a4 ρi,j − ρi,j −1 − a1 (ρuv)i,j − (ρuv)i−1,j
∗
∗
∗
∗
∗
+ vi−1,j
− a11 vi,j
+ a7 vi+1,j
− ρv
i,j −1
i,j
∗
∗
+ a8 vi,j +1 + vi,j −1 + a9 u∗i+1,j +1 +u∗i−1,j −1 − u∗i+1,j −1 − u∗i−1,j +1 .
− a2
ρv
2
2
n+1
n+1
,(ρu)n+1
Step 6: Impose boundary conditions for ρi,j
i,j and (ρv)i,j .
The coefficients are defined as,
t
t
t
t
4t
, a2 =
, a3 =
, a4 =
, a5 =
,
x
y
xM 2
yM 2
3Re (x)2
t
4t
t
t
a6 =
, a7 =
, a8 =
, a9 =
,
12Rexy
Re (y)2
Re (x)2
3Re (y)2
a10 = 2 (a5 + a6 ) , a11 = 2 (a7 + a8 ) .
a1 =
∗ ) can take the same storage
For coding purposes, the variables ui,j (vi,j ) and u∗i,j (vi,j
n ,
space. At the end of each time step, the starting values of ρi,j
(ρu)ni,j and (ρv)ni,j
n+1
n+1
will be replaced with the corresponding new values of ρi,j
, (ρu)n+1
i,j and (ρv)i,j .
Next we present some of the results and compare them with those in the paper by
Hou et al. (1995) obtained by a lattice Boltzmann method. To keep the flow almost
incompresible, the Mach number is chosen as M = 0.1. Flows with two Reynolds
numbers, Re = ρ0 U D / µ = 100 and 400 are simulated. At these Reynolds numbers,
the flow will eventually be steady. Thus calculations need to be run long enough to
get to the steady state. A uniform grid of 256 by 256 was used for this example.
Figure 11.7 shows comparisons of the velocity field calculated by the explicit
MacCormack scheme with the streamlines from Hou (1995) at Re =100 and 400. The
agreement seems reasonable. It was also observed that the location of the center of
the primary eddy agrees even better. When Re =100, the center of the primary eddy
is found at (0.62 ± 0.02, 0.74 ± 0.02) from the MacCormack scheme in comparison
with (0.6196, 0.7373) from Hou. When Re =400, the center of the primary eddy is
found at (0.57 ± 0.02, 0.61 ± 0.02) from the MacCormack scheme in comparison
with (0.5608, 0.6078) from Hou.
444
Computational Fluid Dynamics
(a)
(b)
Figure 11.7 Comparisons of results from the explicit MacCormack scheme (light gray, velocity vector
field) and those from Hou, et al. (1995) (dark solid streamlines) calculated using a Lattice Boltzmann
Method. (a) Re = 100, (b) at Re = 400.
1
0.8
0.6
y/D
Explicit MacCormack (Re5100)
Explicit MacCormack (Re5400)
Hou et al. Re5100
Hou et al. Re5400
0.4
0.2
0
20.4
20.2
0
0.2
0.4
0.6
0.8
1
u
Figure 11.8 Comparison of velocity profiles along a line cut through the center of the cavity (x = 0.5 D)
at Re = 100 and 400.
For a more quantitative comparison, Figure 11.8 plots the velocity profile along
a vertical line cut through the center of the cavity (x = 0.5D). The velocity profiles
for two Reynolds numbers, Re = 100 and 400, are compared. The results from the
explicit MacCormack scheme are shown in solid and dashed lines. The data points in
symbols were directly converted from Hou’s paper. The agreement is excellent.
Explicit MacCormack Scheme for Flow Over a Square Block
For the second example, we consider flow around a square block confined between
two parallel plates. Fluid comes in from the left with a uniform velocity profile U , and
the plates are sliding with the same velocity, as indicated in Figure 11.9. This flow
corresponds to the block moving left with velocity U along channel’s center line. In
the calculation we set the channel width H = 3D, the channel length L = 35D with
445
5. Three Examples
U
U
H
D
y
D
x
U
Figure 11.9 Flow around a square block between two parallel plates.
15D ahead of the block and 19D behind. The Mach number is set at M = 0.05 to
approximate the incompressible limit.
The velocity boundary conditions in this problem are specified as shown in
Figure 11.9, except that at the outflow section, conditions ∂ρu/ ∂x = 0 and ∂ρv / ∂x =
0 are used. The density (or pressure) boundary conditions are much more complicated, especially on the block surface. On all four sides of the outer boundary (top
and bottom plates, inflow and outflow), the continuity equation is used to update
density as in the previous example. However, on the block surface, it was found
that the conditions derived from the momentum equations give better results. Let us
consider the front section of the block, and evaluate the x-component of the momentum equation (11.97) with u = v = 0,
∂ρ
1 4 ∂ 2u 1 ∂ 2v
∂ 2u
2
=M
+
+
∂x
Re 3 ∂x 2
3 ∂x∂y
∂y 2
∂ 2
∂
∂
ρu −
−
(ρvu) −
(ρu)
∂x
∂y
∂t
front suface
=
M2
Re
4 ∂ 2u 1 ∂ 2v
+
3 ∂x 2
3 ∂x∂y
(11.147)
.
In (11.147), the variables are non-dimensionalized with the same scaling as the
driven cavity flow problem except that the block size D is used for length. Furthermore, the density gradient may be approximated with a second order backward finite
difference scheme,
∂ρ
∂x
i,j
=
−1
−ρi−2,j + 4ρi−1,j − 3ρi,j + O(x 2 ).
2x
(11.148)
And the second order derivatives for the velocities are expressed as,
∂ 2u
∂x 2
i,j
=
1
2ui,j − 5ui−1,j + 4ui−2,j − ui−3,j + O(x 2 )
x 2
(11.149)
446
Computational Fluid Dynamics
and
∂ 2v
∂x∂y
i,j
=
−1
− vi−2,j +1 − vi−2,j −1 + 4 vi−1,j +1 − vi−1,j −1
4xy
− 3 vi,j +1 − vi,j −1 + O x 2 , xy, y 2 .
(11.150)
Substituting (11.148) to (11.150) into (11.147), we have an expression for density
at the front of the block,
ρi,j
f ront
=
8 M2
1
4ρi−1,j − ρi−2,j +
−5ui−1,j + 4ui−2,j − ui−3,j
3
9x Re
1 M2
− vi−2,j +1 − vi−2,j −1 + 4 vi−1,j +1 − vi−1,j −1
−
18y Re
− 3 vi,j +1 − vi,j −1 .
(11.151)
Similarly at the back of the block,
ρi,j |back =
1
8 M2
4ρi+1,j − ρi+2,j −
−5ui+1,j + 4ui+2,j − ui+3,j
3
9x Re
1 M2
− vi+2,j +1 − vi+2,j −1 + 4 vi+1,j +1 − vi+1,j −1
−
18y Re
− 3 vi,j +1 − vi,j −1 .
(11.152)
At the top of the block, the y-component of the momentum equation should be used,
and it is easy to find that
ρi,j
top
=
8 M2
1
4ρi,j +1 − ρi,j +2 −
−5vi,j +1 + 4vi,j +2 − vi,j +3
3
9y Re
2
1 M
− ui+1,j +2 − ui−1,j +2 + 4 ui+1,j +1 − ui−1,j +1
−
18x
Re
− 3 ui+1,j − ui−1,j ,
(11.153)
and finally at the bottom of the block,
ρi,j |bottom =
1
8 M2
4ρi,j −1 − ρi,j −2 +
−5vi,j −1 + 4vi,j −2 − vi,j −3
3
9y Re
1 M2
− ui+1,j −2 − ui−1,j −2 + 4 ui+1,j −1 − ui−1,j −1
−
18x
Re
− 3 ui+1,j − ui−1,j .
(11.154)
447
5. Three Examples
At the four corners of the block, the average values from the two corresponding
sides may be used.
In computation, double precision numbers should be used: otherwise cumulative
round-off error may corrupt the simulation, especially for long runs. It is also helpful
to introduce a new variable for density, ρ ′ = ρ − 1, such that only the density
variation is computed. For this example, we may extend the FF/BB form of the
explicit MacCormack scheme to have a FB/BF arrangement for one time step and
a BF/FB arrangement for the subsequent time step. This cycling seems to generate
better results.
We first plot the drag coefficient, CD = Drag / ( 21 ρ0 U 2 D), and the lift coefficient, CL = Lif t / ( 21 ρ0 U 2 D), as functions of time for flows at two Reynolds
numbers, Re = 20 and 100, in Figure 11.10. For Re = 20, after the initial messy
transient (corresponding to sound waves bouncing around the block and reflecting
at the outflow) the flow eventually settles into a steady state. The drag coefficient
stabilizes at a constant value around CD = 6.94 (obtained on a grid of 701x61).
Calculation on a finer grid (1401x121) yields CD = 7.003. This is in excellent
agreement with the value of CD = 7.005 obtained from an implicit finite element
calculation for incompressible flows (similar to the one used in the next example
in this section) on a similar mesh to 1401x121. There is a small lift (CL = 0.014)
due to asymmetries in the numerical scheme. The lift reduces to CL = 0.003 on
the finer grid of 1401x121. For Re = 100, periodic vortex shedding occurs. Drag
and lift coefficients are shown in Figure 11.10(b). The mean value of the drag coefficient and the amplitude of the lift coefficient are CD = 3.35 and CL = 0.77,
respectively. The finite element results are CD = 3.32 and CL = 0.72 under similar
conditions.
The flow field around the block at Re = 20 is shown in Figure 11.11. A steady
wake is attached behind the block, and the circulation within the wake is clearly
visible. Figure 11.12 displays a sequence of the flow field around the block during
one cycle of vortex shedding at Re = 100.
Figure 11.13 shows the convergence of the drag coefficient as the grid spacing
is reduced. Tests for two Reynolds numbers, Re = 20 and 100, are plotted. It seems
that the solution with 20 grid points across the block (x = y = 0.05) reasonably
resolves the drag coefficient and the singularity at the block corners does not affect
this convergence very much.
The explicit MacCormack scheme can be quite efficient to compute flows at
high Reynolds numbers where small time steps are naturally needed to resolve high
frequencies in the flow and the stability condition for the time step is no longer too
restrictive. Since with x = y and large (grid) Reynolds numbers, the stability
condition (11.110) becomes approximately,
σ
t √ Mx.
2
(11.155)
As a more complicated example, the flow around a circular cylinder confined between
two parallel plates (the same geometry as the fourth example later in this section) is calculated at Re = 1000 using the explicit MacCormack scheme. For flow visualization, a
448
Computational Fluid Dynamics
Drag Coefficient
12.5
0.09
10
0.08
7.5
0.07
5
0.06
2.5
0.05
Drag Coefficient
Lift Coefficient
0
0.04
22.5
0.03
25
0.02
27.5
0.01
210
Lift Coefficient
0.1
15
0
0
5
10
15
time
(a)
3.4
20
25
30
2
3.35
1.5
1
Lift Coefficient
3.25
0.5
3.2
0
20.5
3.15
3.1
54 .2
Lift Coefficient
Drag Coefficient
Drag Coefficient
3.3
55.97
57.74
59.51
61.28
time
(b)
63.05
64.82
66.59
21
68.36
Figure 11.10 Drag and lift coefficients as functions of time for flow over a block. (a) Re = 20, on a grid
of 701 × 61, (b) Re = 100, on a grid of 1401 × 121.
smoke line is introduced at the inlet. Numerically, an additional convection-diffusion
equation for smoke concentration is solved similarly, with an explicit scheme at each
time step coupled with the computed flow field. Two snap shots of the flow field
are displayed in Figure 11.14. In this calculation, the flow Mach number is set at
449
5. Three Examples
Figure 11.11 Streamlines for flow around a block at Re = 20.
(a)
(b)
(c)
(d)
(e)
Figure 11.12 A sequence of flow fields around a block at Re = 100 during one period of vortex shedding.
(a) t = 40.53, (b) t = 41.50, (c) t = 42.48, (d) t = 43.45, (e) t = 44.17.
450
Computational Fluid Dynamics
8
Drag Coefficient
7
6
CD (Re = 20)
Mean CD (Re = 100)
5
4
3
0.02
0.03
0.04
0.05
0.06
0.07
Grid Spacing
0.08
0.09
0.1
Figure 11.13 Convergence tests for the drag coefficient as the grid spacing decreases. The grid spacing
is equal in both directions x = y, and time step t is determined by the stability condition.
(a)
(b)
Figure 11.14 Smoke lines in flow around a circular cylinder between two parallel plates at Re = 1000.
The flow geometry is the same as in the fourth example later in this section.
M = 0.3, and a uniform fine grid with 100 grid points across the cylinder diameter
is used.
Finite Element Formulation for Flow Over a Cylinder Confined in a Channel
We next consider the flow over a circular cylinder moving along the center of a
channel. In the computation, we fix the cylinder, and use the flow geometry as shown
in Figure 11.15. The flow comes from the left with a uniform velocity U . Both plates
of the channel are sliding to the right with the same velocity U . The diameter of the
cylinder is d and the width of the channel is W = 4d. The boundary sections for the
451
5. Three Examples
Γ3
U
U
y
Γ1
d
x
W
Γ2
Γ5
U
Γ4
Figure 11.15 Flow geometry of flow around a cylinder in a channel.
Figure 11.16 A finite element mesh around a cylinder.
computational domain are indicated in the figure. The location of the inflow boundary
Ŵ1 is selected to be at xmin = −7.5d, and the location of the outflow boundary section
Ŵ2 is at xmax = 15d. They are both far away from the cylinder so as to minimize their
influence on the flow field near the cylinder. In order to compute the flow at higher
Reynolds numbers, we relax the assumptions that the flow is symmetric and steady.
We will compute unsteady flow (with vortex shedding) in the full geometry and using
the Cartesian coordinates shown in Figure 11.15.
The first step in the finite element method is to discretize (mesh) the computational
domain described in Figure 11.15. We cover the domain with triangular elements.
A typical mesh is presented in Figure 11.16. The mesh size is distributed in a way
that finer elements are used next to the cylinder surface to better resolve the local flow
field. For this example, the mixed finite element method will be used, such that each
triangular element will have six nodes as shown Figure 11.5a. This element allows
for curved sides that better capture the surface of the circular cylinder. The mesh in
Figure 11.16 has 3320 elements, 6868 velocity nodes, and 1774 pressure nodes.
The weak formulation of the Navier-Stokes equations is given in (11.134) and
(11.135). For this example the body force term is zero, g = 0. In Cartesian coordinates,
the weak form of the momentum equation (11.134) can be written explicitly as
∂u
∂u
2
∂u ∂ ũ 1
∂u
+u
+v
+
· ũd +
∂t
∂x
∂y
Re
∂x ∂x
2
∂v ∂ ṽ
∂ ũ ∂ ṽ
+
p
+
d −
d = 0,
∂y ∂y
∂x
∂y
∂u ∂v
+
∂y
∂x
∂ ũ ∂ ṽ
+
∂y
∂x
(11.156)
where is the computational domain and ũ = (ũ, ṽ). Since the variational functions
ũ and ṽ are independent, the weak formulation (11.156) can be separated into two
452
Computational Fluid Dynamics
equations,
∂u
∂ ũ
∂u
∂u
+u
+v
ũd − p d
∂t
∂x
∂y
∂x
∂u ∂ ũ
∂u ∂v ∂ ũ
1
+
2
+
+
d
Re
∂x ∂x
∂y ∂x ∂y
∂v
∂v
∂ ṽ
∂v
+u
+v
p d
ṽd −
∂t
∂x
∂y
∂y
∂u ∂v ∂ ṽ
1
∂v ∂ ṽ
+
+2
+
d
Re
∂y
∂x ∂x
∂y ∂y
= 0,
(11.157)
= 0.
(11.158)
The weak form of the continuity equation (11.135) is expressed as
∂u ∂v
+
p̃ d = 0.
−
∂x
∂y
(11.159)
Given a triangulation of the computational domain, for example, the mesh shown
in Figure 11.16, the weak formulation of (11.157) to (11.159) can be approximated by
the Galerkin finite element formulation based on the finite-dimensional discretization
of the flow variables. The Galerkin formulation can be written as,
∂uh
∂ ũh
∂uh
∂uh
+ uh
+ vh
d
ũh d −
ph
h
h
∂t
∂x
∂y
∂x
1
∂uh ∂ ũh
∂uh
∂v h ∂ ũh
+
2
+
+
d = 0,
(11.160)
Re h
∂x ∂x
∂y
∂x
∂y
and
h
h
h
∂v h
∂ ṽ h
h ∂v
h ∂v
h
ph
+u
+v
d
ṽ d −
h
∂t
∂x
∂y
∂y
1
∂uh
∂v h ∂ ṽ h
∂v h ∂ ṽ h
+
+
+2
d = 0,
Re h
∂y
∂x
∂x
∂y ∂y
−
h
∂v h
∂uh
+
∂x
∂y
p̃h d
= 0,
(11.161)
(11.162)
where h indicates a given triangulation of the computational domain.
The time derivatives in (11.160) and (11.161) can be discretized by finite difference methods. We first evaluate all the terms in (11.160) to (11.162) at a given time
instant t = tn+1 (fully implicit discretization). Then the time derivative in (11.160)
and (11.161) can be approximated as
u (x, tn+1 ) − u (x, tn )
∂u
∂u
−β
(x, tn+1 ) ≈ α
(x, tn ),
∂t
t
∂t
(11.163)
453
5. Three Examples
where t = tn+1 − tn is the time step. The approximation in (11.163) is first-order
accurate in time when α = 1 and β = 0. It can be improved to second-order accurate
by selecting α = 2 and β = 1 which is a variation of the well-known Crank-Nicolson
scheme.
As (11.160) and (11.161) are nonlinear, iterative methods are often used for the
solution. In Newton’s method, the flow variables at the current time t = tn+1 are
often expressed as
uh (x, tn+1 ) = u∗ (x, tn+1 ) + u′ (x, tn+1 ),
ph (x, tn+1 ) = p∗ (x, tn+1 ) + p ′ (x, tn+1 ),
(11.164)
where u∗ and p ∗ are the guesstimated values of velocity and pressure during the
iteration. u′ and p ′ are the corrections sought at each iteration.
Substituting (11.163) and (11.164) into Galerkin formulation (11.160) to (11.162),
and linearizing the equations with respect to the correction variables, we have
α ′
∂ ũh
∂u′
∂u′
∂u∗ ′ ∂u∗ ′ h
p′
u + u∗
+ v∗
+
u +
v ũ d −
d
h
h
t
∂x
∂y
∂x
∂y
∂x
1
∂u′ ∂ ũh
∂u′
∂v ′ ∂ ũh
+
2
+
+
d
Re h
∂x ∂x
∂y
∂x
∂y
α ∗
∂u∗
∂u∗ h
∂u
=−
u − u (tn ) − β
+ v∗
ũ d
(tn ) + u∗
h
t
∂t
∂x
∂y
h
∂u∗ ∂ ũh
1
∂u∗
∂v ∗ ∂ ũh
∗ ∂ ũ
+
2
d −
+
+
d ,
p
h
∂x
Re h
∂x ∂x
∂y
∂x
∂y
(11.165)
∂ ṽ h
∂v ′
∂v ′
∂v ∗ ′ ∂v ∗ ′ h
α ′
v + u∗
+ v∗
+
u +
v ṽ d −
d
p′
h
h
t
∂x
∂y
∂x
∂y
∂y
1
∂u′
∂v ′ ∂ ṽ h
∂v ′ ∂ ṽ h
+
+
+2
d
Re h
∂y
∂x ∂x
∂y ∂y
∗
∗
α ∗
∂v ∗
∗ ∂v
∗ ∂v
+v
v − v (tn ) − β
ṽ h d
=−
(tn ) + u
h
t
∂t
∂x
∂y
∂u∗
∂ ṽ h
1
∂v ∗ ∂ ṽ h
∂v ∗ ∂ ṽ h
+
p∗
d −
+
+2
d ,
h
∂y
Re h
∂y
∂x
∂x
∂y ∂y
(11.166)
and
−
h
∂u′
∂v ′
+
∂x
∂y
p̃h d
=
h
∂u∗
∂v ∗
+
∂x
∂y
p̃h d .
(11.167)
454
Computational Fluid Dynamics
As the functions in the integrals, unless specified otherwise, are all evaluated at
the current time instant tn+1 , the temporal discretization in (11.165) and (11.166) is
fully implicit and unconditionally stable. The terms on the right-hand-side of (11.165)
to (11.167) represent the residuals of the corresponding equations and can be used to
monitor the convergence of the nonlinear iteration.
Similar to the one-dimensional case in Section 3, the finite-dimensional discretization of the flow variables can be constructed using shape (or interpolation)
functions,
u′ =
uA NAu (x, y),
A
NAu (x, y)
v′ =
vA NAu (x, y),
A
p′ =
p
pB NB (x, y),
B
(11.168)
p
NB
where
and
(x, y) are the shape functions for the velocity and the pressure, respectively. They are not necessarily the same. In order to satisfy the LBB
stability condition, the shape function NAu (x, y) in the mixed finite element formup
lation should be one order higher than NB (x, y), as discussed in Section 4. The
summation over A is through all the velocity nodes, while the summation over B runs
through all the pressure nodes. The variational functions may be expressed in terms
of the same shape functions,
ũh =
ũA NAu (x, y),
A
ṽ h =
ṽA NAu (x, y),
A
p̃h =
p
p̃B NB (x, y).
B
(11.169)
Since the Galerkin formulation (11.165) to (11.167) is valid for all possible
choices of the variational functions, the coefficients in (11.169) should be arbitrary.
In this way, the Galerkin formulation (11.165) to (11.167) reduces to a system of
algebraic equations,
u
A′
A′
h
∂N u ′
∂N u ′
∂u∗ u
α u
NA′ + u∗ A + v ∗ A +
N ′ NAu
t
∂x
∂y
∂x A
∂NAu ′ ∂NAu
∂NAu ′ ∂NAu
+
2
d
∂x ∂x
∂y ∂y
∂u∗ u u
1 ∂NAu ′ ∂NAu
+
NA′ NA +
vA′
h
∂y
Re ∂x ∂y
′
1
+
Re
A
d
−
B′
pB ′
u
p′ ∂NA
h
NB ′
∂x
d
∗
∗
α ∗
∂u
∗ ∂u
∗ ∂u
+v
u − u (tn ) − β
NAu d
=−
(tn ) + u
h
t
∂t
∂x
∂y
u
∂u∗ ∂NAu
1
∂u∗
∂v ∗ ∂NAu
∗ ∂NA
+
p
2
d −
+
+
d ,
h
∂x
Re h
∂x ∂x
∂y
∂x
∂y
(11.170)
455
5. Three Examples
∂N u ′
∂N u ′
α u
∂u∗ u
NA′ + u∗ A + v ∗ A +
NA′ NAu
h
t
∂x
∂y
∂y
A′
∂NAu ′ ∂NAu
∂N u ′ ∂NAu
1
+
+2 A
d
Re
∂x ∂x
∂y ∂y
u
1 ∂NAu ′ ∂NAu
∂v ∗ u u
p ∂NA
′
+
N
p
N
+
N
d
−
′
uA ′
B
B ′ ∂y d
h
h
∂x A A Re ∂y ∂x
B′
A′
∂v ∗
∂v ∗
α ∗
∂v ∗
+ v∗
v − v (tn ) − β
NAu d
=−
(tn ) + u∗
h
t
∂t
∂x
∂y
∂N u
∂u∗
1
∂v ∗ ∂NAu
∂v ∗ ∂NAu
+
p∗ A d −
+
+2
d ,
h
∂y
Re h
∂y
∂x
∂x
∂y ∂y
(11.171)
v A′
and
−
A′
=
uA′
h
h
∂NAu ′ p
N
∂x B
∂u∗
∂v ∗
+
∂x
∂y
d
−
A′
vA′
h
∂NAu ′ p
N
∂y B
d
p
(11.172)
NB d ,
for all the velocity nodes A and pressure nodes B. Equations (11.170) to (11.172) can
be organized into a matrix form,
Auu Auv Bup
u
fu
Avu Avv Bvp v = fv ,
p
fp
BTup BTvp 0
(11.173)
where
and
up
Auv = Auv
Bup = BAB ′ ,
Auu = Auu
AA′ ,
AA′ ,
vp
Avu = Avu
Avv = Avv
Bvp = BAB ′ ,
AA′ ,
AA′ ,
u = {uA′ } , v = {vA′ } , p = {pB ′ } ,
p
fu = fAu , fv = fAv , fp = fB ,
Auu
AA′ =
+
h
(11.174)
∂N u ′
∂N u ′
∂u∗ u
α u
NA′ + u∗ A + v ∗ A +
N ′ NAu
t
∂x
∂y
∂x A
∂NAu ′ ∂NAu
∂N u ′ ∂NAu
1
+
2 A
Re
∂x ∂x
∂y ∂y
d ,
(11.175)
456
Computational Fluid Dynamics
Auv
AA′ =
Avu
AA′ =
Avv
AA′ =
d ,
(11.176)
h
∂u∗ u u
1 ∂NAu ′ ∂NAu
NA′ NA +
∂y
Re ∂x ∂y
d ,
(11.177)
h
∂v ∗ u u
1 ∂NAu ′ ∂NAu
NA′ NA +
∂x
Re ∂y ∂x
∂N u ′
∂N u ′
∂v ∗ u
α u
NA′ + u∗ A + v ∗ A +
N ′ NAu
h
t
∂x
∂y
∂y A
∂NAu ′ ∂NAu
1 ∂NAu ′ ∂NAu
+
+2
d ,
(11.178)
Re
∂x ∂x
∂y ∂y
BAB ′ = −
vp
BAB ′
up
=−
p
∂NAu
d ,
∂x
(11.179)
h
NB ′
p
∂NAu
d ,
∂y
(11.180)
h
NB ′
α ∗
∂u∗
∂u∗
∂u
u − u (tn ) − β
+ v∗
NAu d
(tn ) + u∗
h
t
∂t
∂x
∂y
∂N u
∂u∗
∂v ∗ ∂NAu
∂u∗ ∂NAu
1
+
p∗ A d −
2
+
+
d ,
h
∂x
Re h
∂x ∂x
∂y
∂x
∂y
(11.181)
fAu = −
fAv
=−
+
h
h
∗
∗
α ∗
∂v ∗
∗ ∂v
∗ ∂v
+v
v − v (tn ) − β
NAu d
(tn ) + u
t
∂t
∂x
∂y
∂N u
∂u∗
1
∂v ∗ ∂NAu
∂v ∗ ∂NAu
p∗ A d −
+
+2
d ,
∂y
Re h
∂y
∂x
∂x
∂y ∂y
(11.182)
p
fB =
h
∂v ∗
∂u∗
+
∂x
∂y
p
NB d .
(11.183)
The practical evaluation of the integrals in (11.175) to (11.183) is done
element-wise. We need to construct the shape functions locally and transform these
global integrals into local integrals over each element.
In the finite element method, the global shape functions have very compact support. They are zero everywhere except in the neighborhood of the corresponding grid
point in the mesh. It is convenient to cast the global formulation using the element
point of view (Section 3). In this element view, the local shape functions are defined
inside each element. The global shape functions are the assembly of the relevant local
457
5. Three Examples
ones. For example, the global shape function corresponding to the grid point A in
the finite element mesh consists of the local shape functions of all the elements that
share this grid point. An element in the physical space can be mapped into a standard
element, as shown in Figure 11.17 and the local shape functions can be defined on
this standard element. The mapping is given by
x(ξ, η) =
6
xae φa (ξ, η)
and
a=1
y(ξ, η) =
6
yae φa (ξ, η),
(11.184)
a=1
where xae , yae are the coordinates of the nodes in the element e. The local shape
functions are φa . For a quadratic triangular element they are defined as
φ1 = ζ (2ζ − 1), φ2 = ξ (2ξ − 1), φ3 = η (2η − 1), φ4 = 4ξ ζ, φ5 = 4ξ η,
φ6 = 4ηζ,
(11.185)
where ζ = 1 − ξ − η. As shown in Figure 11.17 the mapping (11.184) is able to
handle curved triangles. The variation of the flow variables within this element can
also be expressed in terms of their values at the nodes of the element and the local
shape functions,
u′ =
6
uea φa (ξ, η),
a=1
v′ =
6
p′ =
vae φa (ξ, η),
a=1
3
pbe ψb (ξ, η).
(11.186)
b=1
Here the shape functions for velocities are quadratic and the same as the coordinates. The shape functions for the pressure are chosen to be linear, thus one order
less than those for the velocities. They are given by,
ψ1 = ζ, ψ2 = ξ, ψ3 = η.
(11.187)
Furthermore, the integration over the global computational domain can be written
as the summation of the integrations over all the elements in the domain. As most of
3
(0,1,0)
3
=1 − −
5
6
Ω
e
(0,1/2,1/2)
6
(1/2,1/2,0)
5
2
4
1
y
x
1
(0,0,1)
4
2
(1/2,0,1/2)
(1,0,0)
Figure 11.17 A quadratic triangular finite element mapping into the standard element.
458
Computational Fluid Dynamics
these integrations will be zero, the non-zero ones are grouped as element matrices
and vectors,
eup
Aeuv = Aeuv
Beup = Bab′ ,
Aeuu = Aeuu
aa ′ ,
aa ′ ,
evp
Aevv = Aevv
Bevp = Bab′ ,
(11.188)
Aevu = Aevu
aa ′ ,
aa ′ ,
eu
ev
ep
e
e
e
fu = fa , fv = fa , fp = fb ,
where
Aeuu
aa ′ =
e
∂φa ′
∂φa ′
∂u∗
α
φa ′ + u ∗
+ v∗
+
φa ′ φa
t
∂x
∂y
∂x
∂φa ′ ∂φa
∂φa ′ ∂φa
1
+
2
+
Re
∂x ∂x
∂y ∂y
∂u∗
1 ∂φa ′
Aeuv
φ a ′ φa +
aa ′ =
e
∂y
Re ∂x
∂v ∗
1 ∂φa ′
evu
Aaa ′ =
φa ′ φa +
e
∂x
Re ∂y
Aevv
aa ′ =
(11.189)
d ,
∂φa
∂y
d ,
(11.190)
∂φa
∂x
d ,
(11.191)
∂φa ′
∂φa ′
∂v ∗
α
φa ′ + u ∗
+ v∗
+
φa ′ φa
e
t
∂x
∂y
∂y
1 ∂φa ′ ∂φa
∂φa ′ ∂φa
+
+2
d ,
Re ∂x ∂x
∂y ∂y
∂φa
eup
Bab′ = −
d ,
ψb′
e
∂x
∂φa
evp
Bab′ = −
ψb′
d ,
e
∂y
(11.192)
(11.193)
(11.194)
∗
∗
α ∗
∂u
∗ ∂u
∗ ∂u
u − u (tn ) − β
+v
φa d
=−
(tn ) + u
e
t
∂t
∂x
∂y
∂u∗ ∂φa
1
∂u∗
∂v ∗ ∂φa
∗ ∂φa
p
+
2
d −
+
+
d ,
e
∂x
Re e
∂x ∂x
∂y
∂x
∂y
(11.195)
faev = −
+
faeu
e
∂v ∗
∂v ∗
α ∗
∂v ∗
v − v (tn ) − β
+ v∗
φa d
(tn ) + u∗
t
∂t
∂x
∂y
p
e
∗ ∂φa
∂y
d
1
−
Re
e
∂u∗
∂v ∗
+
∂y
∂x
∂v ∗ ∂φa
∂φa
+2
∂x
∂y ∂y
d
,
(11.196)
459
5. Three Examples
ep
fb
=
e
∂u∗
∂v ∗
+
∂x
∂y
ψb d .
(11.197)
The indices a and a ′ run from 1 to 6, and b and b′ run from 1 to 3.
The integrals in the above expressions can be evaluated by numerical integration
rules,
1 1−η
Nint
1
f (ξl , ηl )J (ξl , ηl )Wl ,
f (ξ, η)J (ξ, η) dξ dη =
f (x, y) d =
e
2
0
0
l=1
(11.198)
where the Jacobian of the mapping (11.184) is given by J = xξ yη − xη yξ . Here
Nint is the number of numerical integration points and Wl is the weight of the lth
integration point. For this example, a seven-point integration formula with degree of
precision of 5 (see Hughes, 1987) was used.
The global matrices and vectors in (11.173) are the summations of the element
matrices and vectors in (11.188) over all the elements. In the process of summation
(assembly), a mapping of the local nodes in each element to the global node numbers
is needed. This information is commonly available for any finite element mesh.
Once the matrix equation (11.173) is generated, we may impose the essential
boundary conditions for the velocities. One simple method is to use the equation of
the boundary condition to replace the corresponding equation in the original matrix
or one can multiply a large constant by the equation of the boundary condition and
add this equation to the original system of equations in order to preserve the structure
of the matrix. The resulting matrix equation may be solved using common direct or
iterative solvers for a linear algebraic system of equations.
Figures 11.18 and 11.19 display the streamlines and vorticity lines around the
cylinder at three Reynolds numbers Re = 1, 10, and 40. For these Reynolds numbers,
Figure 11.18 Streamlines for flow around a cylinder at three different Reynolds numbers.
460
Computational Fluid Dynamics
Figure 11.19 Vorticity lines for flow around a cylinder at three different Reynolds numbers.
the flow is steady and should be symmetric above and below the cylinder. However,
due to the imperfection in the mesh used for the calculation and as shown in Figure
11.16, the calculated flow field is not perfectly symmetric. From Figure 11.18 we
observe the increase in the size of the wake behind the cylinder as the Reynolds
number increases. In Figure 11.19, we see the effects of the Reynolds number in the
vorticity build up in front of the cylinder, and in the convection of the vorticity by
the flow.
We next compute the case with Reynolds number Re = 100. In this case, the
flow is expected to be unsteady. Periodic vortex shedding occurs. In order to capture
the details of the flow, we used a finer mesh than the one shown in Figure 11.16. The
finer mesh has 9222 elements, 18816 velocity nodes and 4797 pressure nodes. In this
calculation, the flow starts from rest. Initially, the flow is symmetric, and the wake
behind the cylinder grows bigger and stronger. Then, the wake becomes unstable,
undergoes a supercritical Hopf bifurcation, and sheds periodically away from the
cylinder. The periodic vortex shedding forms the well-known von Karman vortex
street. The vorticity lines are presented in Figure 11.20 for a complete cycle of vortex
shedding.
For this case with Re = 100, we plot in Figure 11.21 the history of the forces and
torque acting on the cylinder. The oscillations shown in the lift and torque plots are
typical for the supercritical Hopf bifurcation. The nonzero mean value of the torque
shown in Figure 11.21c is due to the asymmetry in the finite element mesh. It is clear
that the flow becomes fully periodic at the times shown in Figure 11.20. The period of
the oscillation is measured as τ = 0.0475s or τ̄ = 4.75 in the non-dimensional units.
This period corresponds to a nondimensional Strouhal number S = nd / U = 0.21,
where n is the frequency of the shedding. In the literature, the value of the Strouhal
number for an unbounded uniform flow around a cylinder is found to be around 0.167
6. Concluding Remarks
Figure 11.20 Vorticity lines for flow around a cylinder at Reynolds number Re = 100. t¯ = tU/d is the
dimensionless time.
at Re = 100 (e.g., see Wen and Lin, 2001). The difference could be caused by the
geometry in which the cylinder is confined in a channel.
6. Concluding Remarks
It should be strongly emphasized that CFD is merely a tool for analyzing fluid-flow
problems. If it is used correctly, it would provide useful information cheaply and
quickly. However, it could easily be misused or even abused. In today’s computer
age, people have a tendency to trust the output from a computer, especially when they
do not understand what is behind the computer. One certainly should be aware of the
assumptions used in producing the results from a CFD model.
As we have previously discussed, CFD is never exact. There are uncertainties
involved in CFD predictions. However, one is able to gain more confidence in CFD
predictions by following a few steps. Tests on some benchmark problems with known
solutions are often encouraged. A mesh refinement test is normally a must in order
461
462
Computational Fluid Dynamics
2.02
2
Drag
1
2
ρU 2d
1.98
1.96
1.94
1.92
1.9
1.88
0
20
40
60
80
100
120
t=t⋅U d
tb
(a)
0.5
0 0475s
Fy
1
2
ρU 2d
0
20.5
0
20
40
60
80
100
120
t=t⋅U d
(b)
0.006
0.004
0.002
Torque
1
2
ρU 2d 2
0
20.002
20.004
20.006
0
20
40
60
80
100
120
t=t⋅U d
(c)
Figure 11.21 History of forces and torque acting on the cylinder at Re = 100: (a) drag coefficient; (b) lift
coefficient; and (c) coefficient for the torque.
463
6. Concluding Remarks
to be sure that the numerical solution converges to something meaningful. A similar
test with the time step for unsteady flow problems is often desired. If the boundary
locations and conditions are in doubt, their effects on the CFD predictions should be
minimized. Furthermore, the sensitivity of the CFD predictions to some key parameters in the problem should be investigated for practical design problems.
In this chapter, we have discussed the basics of the finite difference and finite
element methods and their applications in CFD. There are other kinds of numerical methods, for example, the spectral method and the spectral element method,
which are often used in CFD. They share the common approach that discretizes
the Navier-Stokes equations into a system of algebraic equations. However, a class
of new numerical techniques including lattice gas cellular automata, lattice Boltzmann method, and dissipative particle dynamics do not start from the continuum
Navier-Stokes equations. Unlike the conventional methods discussed in this chapter,
they are based on simplified kinetic models that incorporate the essential physics of
the microscopic or mesoscopic processes so that the macroscopic-averaged properties
obey the desired macroscopic Navier-Stokes equations.
Exercises
1. Show that the stability condition for the explicit scheme (11.10) is the condition
(11.26).
2. For the heat conduction equation ∂T / ∂t − D ∂ 2 T / ∂x 2 = 0, one of the
discretized forms is
n+1
n
− sTjn+1
−sTjn+1
+1 + (1 + 2s) Tj
−1 = Tj
where s = Dt / x 2 . Show that this implicit algorithm is always stable.
3. An insulated rod initially has a temperature of T (x, 0) = 0◦ C (0 x 1).
At t = 0 hot reservoirs (T = 100◦ C) are brought into contact with the two ends,
A (x = 0) and B (x = 1): T (0, t) = T (1, t) = 100◦ C. Numerically find the temperature T (x, t) of any point in the rod. The governing
equation of the problem is
the heat conduction equation ∂T / ∂t − D ∂ 2 T / ∂x 2 = 0. The exact solution to this
problem is
M
T ∗ xj , tn = 100 −
400
sin (2m − 1) π xj exp −D (2m − 1)2 π 2 tn
(2m − 1) π
m=1
(11.199)
where M is the number of terms used in the approximation.
(a). Try to solve the problem with the explicit forward time, centered space
(FTCS) scheme. Use the parameter s = Dt / x 2 =0.5 and 0.6 to test the stability of
the scheme.
464
Computational Fluid Dynamics
(b). Solve the problem with a stable explicit or implicit scheme. Test the rate of
convergence numerically using the error atx = 0.5.
4. Derive the weak form, Galerkin form, and the matrix form of the following
strong problem:
Given functions D(x), f (x), and constants g, h, find u(x) such that
[D(x)u,x ],x + f (x) = 0 on = (0, 1),
with u(0) = g and − u,x (1) = h.
Write a computer program solving this problem using piecewise-linear shape
functions. You may set D = 1, g = 1, h = 1 and h = 1. Check your numerical result
with the exact solution.
5. Solve numerically the steady convective transport equation,
u
∂ 2T
∂T
= D 2 , for 0 x 1,
∂x
∂x
with two boundary conditions T (0) = 0 and T (1) = 1, where u and D are two
constants,
(a) using the centered finite difference scheme in equation (11.91), and compare with
the exact solution; and
(b) using the upwind scheme (11.93), and compare with the exact solution.
6. Code the explicit MacCormack scheme with the FF/BB arrangement for the
driven cavity flow problem as described in Section 5. Compute the flow field at
Re = 100 and 400, and explore effects of Mach number and the stability condition
(11.110).
Literature Cited
Brooks, A. N. and T. J. R. Hughes (1982). “Streamline-upwinding/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on incompressible Navier-Stokes equation.” Comput.
Methods Appl. Mech. Engrg. 30: 199–259.
Chorin, A. J. (1967). “A numerical method for solving incompressible viscous flow problems.” J. Comput.
Phys. 2: 12–26.
Chorin, A. J. (1968). “Numerical solution of the Navier-Stokes equations.” Math. Comput. 22: 745–762.
Dennis, S. C. R. and G. Z. Chang (1970). “Numerical solutions for steady flow past a circular cylinder at
Reynolds numbers up to 100.” J. Fluid Mech. 42: 471–489.
Fletcher, C. A. J. (1988). Computational Techniques for Fluid Dynamics, I–Fundamental and General
Techniques, and II–Special Techniques for Different Flow Categories, New York: Springer-Verlag.
Franca, L. P., S. L. Frey and T. J. R. Hughes (1992). “Stabilized finite element methods: I. Application to
the advective-diffusive model.” Comput. Methods Appl. Mech. Engrg. 95: 253–276.
Franca, L. P. and S. L. Frey (1992). “Stabilized finite element methods: II. The incompressible
Navier-Stokes equations,”Comput. Methods Appl. Mech. Engrg. 99: 209–233.
Ghia, U., K. N. Ghia, C. T. Shin (1982) ı̀High-Re solutions for incompressible flow using the Navier-Stokes
equations and a multigrid method.ı̂ J. Comput. Phys. 48: 387–411.
Glowinski, R. (1991). “Finite element methods for the numerical simulation of incompressible viscous
flow, introduction to the control of the Navier-Stokes equations,” in Lectures in Applied Mathematics
Vol.28: 219–301. Providence, R.I.: American Mathematical Society.
Literature Cited
Gresho, P. M. (1991). “Incompressible fluid dynamics: Some fundamental formulation issues,” Annu. Rev.
Fluid Mech. 23: 413–453.
Harlow, F. H. and J. E. Welch (1965). “Numerical calculation of time-dependent viscous incompressible
flow of fluid with free surface.” Phys. Fluids 8: 2182–2189.
Hou, S., Q. Zou, S. Chen, G. D. Doolen and A. C. Cogley (1995). ı̀Simulation of cavity flow by the lattice
Boltzmann method.ı̂ J. Comp. Phys. 118: 329–347.
Hughes, T. J. R. (1987). The Finite Element Method, Linear Static and Dynamic Finite Element Analysis,
Englewood Cliffs, NJ: Prentice-Hall.
MacCormack, R. W. (1969). “The effect of viscosity in hypervelocity impact cratering.” AIAA Paper
69–354, Cincinnati, Ohio.
Marchuk, G. I. (1975). Methods of Numerical Mathematics, New York: Springer-Verlag.
Noye, J (1983). Chapter 2 in Numerical Solution of Differential Equations, J. Noye, ed., Amsterdam:
North-Holland.
Oden, J. T. and G. F. Carey (1984). Finite Elements: Mathematical Aspects, Vol. IV, Englewood Cliffs,
N.J.: Prentice-Hall.
Perrin, A. and H.H. Hu (2006). “An explicit finite-difference scheme for simulation of moving particles,”
J. Comput. Phys. 212: 166–187.
Peyret, R. and T. D. Taylor (1983). Computational Methods for Fluid Flow, New York: Springer-Verlag.
Richtmyer, R. D. and K. W. Morton (1967). Difference Methods for Initial-Value Problems, New York:
Interscience.
Saad, Y. (1996). Iterative Methods for Sparse Linear Systems, Boston: PWS Publishing Company.
Sucker, D. and H. Brauer (1975). “Fluiddynamik bei der angeströmten Zylindern.” Wärme-Stoffübertrag.
8: 149.
Takami, H. and H. B. Keller (1969). “Steady two-dimensional viscous flow of an incompressible fluid past
a circular cylinder.” Phys. Fluids 12: Suppl.II, II-51-II-56.
Tannehill, J. C., D. A. Anderson, R. H. Pletcher (1997), Computational Fluid Mechanics and Heat Transfer,
Washington, DC, Taylor & Francis.
Temam, R. (1969). “Sur l’approximation des équations de Navier-Stokes par la méthode de pas fractionaires.” Archiv. Ration. Mech. Anal. 33: 377–385.
Tezduyar, T. E. (1992). “Stabilized Finite Element Formulations for Incompressible Flow Computations,”
in Advances in Applied Mechanics , J.W. Hutchinson and T.Y. Wu eds., Vol. 28: 1–44. New York:
Academic Press.
Yanenko, N. N. (1971). The Method of Fractional Steps, New York: Springer-Verlag.
Wen, C. Y. and C. Y. Lin (2001). “Two dimensional vortex shedding of a circular cylinder,” Phys. Fluids
13: 557–560.
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Chapter 12
Instability
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2. Method of Normal Modes . . . . . . . . . . . .
3. Thermal Instability: The Bénard
Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Formulation of the Problem . . . . . . . . .
Proof That σ Is Real for Ra > 0 . . . . . .
Solution of the Eigenvalue Problem
with Two Rigid Plates . . . . . . . . . . . . .
Solution with Stress-Free Surfaces . . .
Cell Patterns . . . . . . . . . . . . . . . . . . . . . . . .
4. Double-Diffusive Instability . . . . . . . . . .
Finger Instability . . . . . . . . . . . . . . . . . . . .
Oscillating Instability . . . . . . . . . . . . . . . .
5. Centrifugal Instability:
Taylor Problem . . . . . . . . . . . . . . . . . . . . .
Rayleigh’s Inviscid Criterion . . . . . . . . .
Formulation of the Problem . . . . . . . . .
Discussion of Taylor’s Solution . . . . . . .
6. Kelvin–Helmholtz Instability . . . . . . . . .
7. Instability of Continuously
Stratified Parallel Flows . . . . . . . . . . . . .
Taylor–Goldstein Equation . . . . . . . . . .
Richardson Number Criterion. . . . . . . .
Howard’s Semicircle Theorem . . . . . . .
8. Squire’s Theorem and
Orr–Sommerfeld Equation . . . . . . . . . . .
Squire’s Theorem. . . . . . . . . . . . . . . . . . . .
Orr–Sommerfeld Equation . . . . . . . . . . .
467
469
470
471
475
477
480
481
482
482
485
486
486
488
490
493
500
500
503
504
507
509
509
9. Inviscid Stability of Parallel Flows . .
Rayleigh’s Inflection Point Criterion .
Fjortoft’s Therorm . . . . . . . . . . . . . . . . .
Critical Layers . . . . . . . . . . . . . . . . . . . . .
10. Some Results of Parallel Viscous
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mixing Layer . . . . . . . . . . . . . . . . . . . . . .
Plane Poiseuille Flow . . . . . . . . . . . . . . .
Plane Couette Flow. . . . . . . . . . . . . . . . .
Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary Layers with Pressure
Gradients . . . . . . . . . . . . . . . . . . . . . . . .
How can Viscosity Destabilize a
Flow? . . . . . . . . . . . . . . . . . . . . . . . . . . .
11. Experimental Verification of
Boundary Layer Instability . . . . . . . . .
12. Comments on Nonlinear Effects . . . . .
13. Transition . . . . . . . . . . . . . . . . . . . . . . . . . .
14. Deterministic Chaos . . . . . . . . . . . . . . . .
Phase Space . . . . . . . . . . . . . . . . . . . . . . .
Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Lorenz Model of Thermal
Convection . . . . . . . . . . . . . . . . . . . . . .
Strange Attractors . . . . . . . . . . . . . . . . . .
Scenarios for Transition to Chaos . . .
Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . .
510
511
511
512
514
515
516
516
516
517
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523
525
526
527
528
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531
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535
1. Introduction
A phenomenon that may satisfy all conservation laws of nature exactly, may still
be unobservable. For the phenomenon to occur in nature, it has to satisfy one more
condition, namely, it must be stable to small disturbances. In other words, infinitesimal disturbances, which are invariably present in any real system, must not amplify
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50012-5
467
468
Instability
Figure 12.1 Stable and unstable systems.
spontaneously. A perfectly vertical rod satisfies all equations of motion, but it does
not occur in nature. A smooth ball resting on the surface of a hemisphere is stable (and
therefore observable) if the surface is concave upwards, but unstable to small displacements if the surface is convex upwards (Figure 12.1). In fluid flows, smooth laminar
flows are stable to small disturbances only when certain conditions are satisfied. For
example, in flows of homogeneous viscous fluids in a channel, the Reynolds number
must be less than some critical value, and in a stratified shear flow, the Richardson
number must be larger than a critical value. When these conditions are not satisfied,
infinitesimal disturbances grow spontaneously. Sometimes the disturbances can grow
to a finite amplitude and reach equilibrium, resulting in a new steady state. The new
state may then become unstable to other types of disturbances, and may grow to yet
another steady state, and so on. Finally, the flow becomes a superposition of various
large disturbances of random phases, and reaches a chaotic condition that is commonly described as “turbulent.” Finite amplitude effects, including the development
of chaotic solutions, will be examined briefly later in the chapter.
The primary objective of this chapter, however, is the examination of stability
of certain fluid flows with respect to infinitesimal disturbances. We shall introduce
perturbations on a particular flow, and determine whether the equations of motion
demand that the perturbations should grow or decay with time. In this analysis the
problem is linearized by neglecting terms quadratic in the perturbation variables
and their derivatives. This linear method of analysis, therefore, only examines the
initial behavior of the disturbances. The loss of stability does not in itself constitute
a transition to turbulence, and the linear theory can at best describe only the very
beginning of the process of transition to turbulence. Moreover, a real flow may be
stable to infinitesimal disturbances (linearly stable), but still can be unstable to sufficiently large disturbances (nonlinearly unstable); this is schematically represented in
Figure 12.1. These limitations of the linear stability analysis should be kept in mind.
469
2. Method of Normal Modes
Nevertheless, the successes of the linear stability theory have been considerable.
For example, there is almost an exact agreement between experiments and theoretical
prediction of the onset of thermal convection in a layer of fluid, and of the onset of
the Tollmien–Schlichting waves in a viscous boundary layer. Taylor’s experimental
verification of his own theoretical prediction of the onset of secondary flow in a
rotating Couette flow is so striking that it has led people to suggest that Taylor’s
work is the first rigorous confirmation of Navier–Stokes equations, on which the
calculations are based.
For our discussion we shall choose problems that are of importance in geophysical
as well as engineering applications. None of the problems discussed in this chapter,
however, contains Coriolis forces; the problem of “baroclinic instability,” which does
contain the Coriolis frequency, is discussed in Chapter 14. Some examples will also
be chosen to illustrate the basic physics rather than any potential application. Further
details of these and other problems can be found in the books by Chandrasekhar
(1961, 1981) and Drazin and Reid (1981). The review article by Bayly, Orszag, and
Herbert (1988) is recommended for its insightful discussions after the reader has read
this chapter.
2. Method of Normal Modes
The method of linear stability analysis consists of introducing sinusoidal disturbances
on a basic state (also called background or initial state), which is the flow whose
stability is being investigated. For example, the velocity field of a basic state involving
a flow parallel to the x-axis, and varying along the y-axis, is U = [U (y), 0, 0]. On
this background flow we superpose a disturbance of the form
u(x, t) = û( y) eikx+imz+σ t ,
(12.1)
where û(y) is a complex amplitude; it is understood that the real part of the right-hand
side is taken to obtain physical quantities. (The complex form of notation is explained
in Chapter 7, Section 15.) The reason solutions exponential in (x, z, t) are allowed
in equation (12.1) is that, as we shall see, the coefficients of the differential equation
governing the perturbation in this flow are independent of (x, z, t). The flow field is
assumed to be unbounded in the x and z directions, hence the wavenumber components
k and m can only be real in order that the dependent variables remain bounded as x,
z → ∞; σ = σr + iσi is regarded as complex.
The behavior of the system for all possible K = [k, 0, m] is examined in the
analysis. If σr is positive for any value of the wavenumber, the system is unstable to
disturbances of this wavenumber. If no such unstable state can be found, the system
is stable. We say that
σr < 0: stable,
σr > 0: unstable,
σr = 0: neutrally stable.
470
Instability
The method of analysis involving the examination of Fourier components such as
equation (12.1) is called the normal mode method. An arbitrary disturbance can be
decomposed into a complete set of normal modes. In this method the stability of each
of the modes is examined separately, as the linearity of the problem implies that the various modes do not interact. The method leads to an eigenvalue problem, as we shall see.
The boundary between stability and instability is called the marginal state, for
which σr = 0. There can be two types of marginal states, depending on whether σi is
also zero or nonzero in this state. If σi = 0 in the marginal state, then equation (12.1)
shows that the marginal state is characterized by a stationary pattern of motion;
we shall see later that the instability here appears in the form of cellular convection or
secondary flow (see Figure 12.12 later). For such marginal states one commonly says
that the principle of exchange of stabilities is valid. (This expression was introduced
by Poincaré and Jeffreys, but its significance or usefulness is not entirely clear.)
If, on the other hand, σi = 0 in the marginal state, then the instability sets in as
oscillations of growing amplitude. Following Eddington, such a mode of instability
is frequently called “overstability” because the restoring forces are so strong that the
system overshoots its corresponding position on the other side of equilibrium. We
prefer to avoid this term and call it the oscillatory mode of instability.
The difference between the neutral state and the marginal state should be noted
as both have σr = 0. However, the marginal state has the additional constraint that it
lies at the borderline between stable and unstable solutions. That is, a slight change
of parameters (such as the Reynolds number) from the marginal state can take the
system into an unstable regime where σr > 0. In many cases we shall find the stability
criterion by simply setting σr = 0, without formally demonstrating that it is indeed
at the borderline of unstable and stable states.
3. Thermal Instability: The Bénard Problem
A layer of fluid heated from below is “top heavy,” but does not necessarily undergo
a convective motion. This is because the viscosity and thermal diffusivity of the
fluid try to prevent the appearance of convective motion, and only for large enough
temperature gradients is the layer unstable. In this section we shall determine the
condition necessary for the onset of thermal instability in a layer of fluid.
The first intensive experiments on instability caused by heating a layer of fluid
were conducted by Bénard in 1900. Bénard experimented on only very thin layers
(a millimeter or less) that had a free surface and observed beautiful hexagonal cells
when the convection developed. Stimulated by these experiments, Rayleigh in 1916
derived the theoretical requirement for the development of convective motion in a
layer of fluid with two free surfaces. He showed that the instability would occur when
the adverse temperature gradient was large enough to make the ratio
Ra =
gαŴd 4
,
κν
(12.2)
471
3. Thermal Instability: The Bénard Problem
exceed a certain critical value. Here, g is the acceleration due to gravity, α is the
coefficient of thermal expansion, Ŵ = −d T̄ /dz is the vertical temperature gradient
of the background state, d is the depth of the layer, κ is the thermal diffusivity, and ν is
the kinematic viscosity. The parameter Ra is called the Rayleigh number, and we shall
see shortly that it represents the ratio of the destabilizing effect of buoyancy force to
the stabilizing effect of viscous force. It has been recognized only recently that most
of the motions observed by Bénard were instabilities driven by the variation of surface
tension with temperature and not the thermal instability due to a top-heavy density
gradient (Drazin and Reid 1981, p. 34). The importance of instabilities driven by
surface tension decreases as the layer becomes thicker. Later experiments on thermal
convection in thicker layers (with or without a free surface) have obtained convective
cells of many forms, not just hexagonal. Nevertheless, the phenomenon of thermal
convection in a layer of fluid is still commonly called the Bénard convection.
Rayleigh’s solution of the thermal convection problem is considered a major
triumph of the linear stability theory. The concept of critical Rayleigh number finds
application in such geophysical problems as solar convection, cloud formation in the
atmosphere, and the motion of the earth’s core.
Formulation of the Problem
Consider a layer confined between two isothermal walls, in which the lower wall is
maintained at a higher temperature. We start with the Boussinesq set
∂ ũi
1 ∂ p̃
∂ ũi
+ ũj
=−
− g[1 − α(T̃ − T0 )]δi3 + ν∇ 2 ũi ,
∂t
∂xj
ρ0 ∂xi
(12.3)
∂ T̃
∂ T̃
= κ∇ 2 T̃ ,
+ ũj
∂t
∂xj
along with the continuity equation ∂ ũi /∂xi = 0. Here, the density is given by the
equation of state ρ̃ = ρ0 [1 − α(T̃ − T0 )], with ρ0 representing the reference density
at the reference temperature T0 . The total flow variables (background plus perturbation) are represented by a tilde ( ˜ ), a convention that will also be used in the following chapter. We decompose the motion into a background state of no motion, plus
perturbations:
ũi = 0 + ui (x, t),
T̃ = T̄ (z) + T ′ (x, t),
(12.4)
p̃ = P (z) + p(x, t),
where the z-axis is taken vertically upward. The variables in the basic state are
represented by uppercase letters except for the temperature, for which the symbol is T̄ .
The basic state satisfies
1 ∂P
− g[1 − α(T̄ − T0 )]δi3 ,
0=−
ρ0 ∂xi
(12.5)
d 2 T̄
0=κ 2.
dz
472
Instability
Figure 12.2 Definition sketch for the Bénard problem.
The preceding heat equation gives the linear vertical temperature distribution
T̄ = T0 − Ŵ(z + d/2),
(12.6)
where Ŵ ≡ T /d is the magnitude of the vertical temperature gradient, and T0 is
the temperature of the lower wall (Figure 12.2). Substituting equation (12.4) into
equation (12.3), we obtain
∂ui
1 ∂
∂ui
+ uj
=−
(P + p)
∂t
∂xj
ρ0 ∂xi
∂T ′
∂t
− g[1 − α(T̄ + T ′ − T0 )]δi3 + ν∇ 2 ui ,
+ uj
(12.7)
∂
(T̄ + T ′ ) = κ∇ 2 (T̄ + T ′ ).
∂xj
Subtracting the mean state equation (12.5) from the perturbed state equation (12.7),
and neglecting squares of perturbations, we have
1 ∂p
∂ui
=−
+ gαT ′ δi3 + ν∇ 2 ui ,
∂t
ρ0 ∂xi
∂T ′
− Ŵw = κ∇ 2 T ′ ,
∂t
(12.8)
(12.9)
where w is the vertical component of velocity. The advection term in equation (12.9)
results from uj (∂ T̄ /∂xj ) = w(d T̄ /dz) = −wŴ. Equations (12.8) and (12.9) govern
the behavior of perturbations on the system.
At this point it is useful to pause and show that the Rayleigh number defined by
equation (12.2) is the ratio of buoyancy force to viscous force. From equation (12.9),
the velocity scale is found by equating the advective and diffusion terms, giving
w∼
κŴ/d
κ
κT ′ /d 2
∼
= .
Ŵ
Ŵ
d
473
3. Thermal Instability: The Bénard Problem
An examination of the last two terms in equation (12.8) shows that
Buoyancy force
gαT ′
gαŴd
gαŴd 4
∼
∼
=
,
Viscous force
νw/d 2
νw/d 2
νκ
which is the Rayleigh number.
We now write the perturbation equations in terms of w and T ′ only. Taking the
Laplacian of the i = 3 component of equation (12.8), we obtain
∂
1
∂p
(∇ 2 w) = − ∇ 2
+ gα∇ 2 T ′ + ν∇ 4 w.
∂t
ρ0
∂z
(12.10)
The pressure term in equation (12.10) can be eliminated by taking the divergence of
equation (12.8) and using the continuity equation ∂ui /∂xi = 0. This gives
0=−
1 ∂ 2p
∂T ′
+ gα
δi3 + 0.
ρ0 ∂xi ∂xi
∂xi
Differentiating with respect to z, we obtain
0=−
1 2 ∂p
∂ 2T ′
∇
+ gα 2 ,
ρ0
∂z
∂z
so that equation (12.10) becomes
∂
(∇ 2 w) = gα∇H2 T ′ + ν∇ 4 w,
∂t
(12.11)
where ∇H2 ≡ ∂ 2 /∂x 2 + ∂ 2 /∂y 2 is the horizontal Laplacian operator.
Equations (12.9) and (12.11) govern the development of perturbations on the
system. The boundary conditions on the upper and lower rigid surfaces are that the
no-slip condition is satisfied and that the walls are maintained at constant temperatures. These conditions require u = v = w = T ′ = 0 at z = ±d/2. Because the
conditions on u and v hold for all x and y, it follows from the continuity equation
that ∂w/∂z = 0 at the walls. The boundary conditions therefore can be written as
w=
∂w
= T′ = 0
∂z
d
at z = ± .
2
(12.12)
We shall use dimensionless independent variables in the rest of the analysis. For
this, we make the transformation
d2
t,
κ
(x, y, z) → (xd, yd, zd),
t→
where the old variables are on the left-hand side and the new variables are on the
right-hand side; note that we are avoiding the introduction of new symbols for the
474
Instability
nondimensional variables. Equations (12.9), (12.11), and (12.12) then become
Ŵd 2
∂
2
−∇ T′ =
w,
(12.13)
∂t
κ
1 ∂
gαd 2 2 ′
− ∇ 2 ∇2w =
∇H T ,
(12.14)
Pr ∂t
ν
∂w
1
w=
= T ′ = 0 at z = ±
(12.15)
∂z
2
where Pr ≡ ν/κ is the Prandtl number.
The method of normal modes is now introduced. Because the coefficients of the
governing set (12.13) and (12.14) are independent of x, y, and t, solutions exponential
in these variables are allowed. We therefore assume normal modes of the form
w = ŵ(z) eikx+ily+σ t ,
T ′ = T̂ (z) eikx+ily+σ t .
The requirement that solutions remain bounded as x, y → ∞ implies that the
wavenumbers k and l must be real. In other words, the normal modes must be periodic in the directions of unboundedness. The growth rate σ = σr + iσi is allowed
to be complex. With this dependence, the operators in equations (12.13) and (12.14)
transform as follows:
∂
→ σ,
∂t
∇H2 → −K 2 ,
∇2 →
d2
− K 2,
dz2
√
where K = k 2 + l 2 is the magnitude of the (nondimensional) horizontal wavenumber. Equations (12.13) and (12.14) then become
Ŵd 2
ŵ,
κ
σ
gαd 2 K 2
− (D 2 − K 2 ) (D 2 − K 2 )ŵ = −
T̂ ,
Pr
ν
[σ − (D 2 − K 2 )]T̂ =
(12.16)
(12.17)
where D ≡ d/dz. Making the substitution
Ŵd 2
ŵ ≡ W.
κ
Equations (12.16) and (12.17) become
σ
Pr
[σ − (D 2 − K 2 )]T̂ = W,
− (D 2 − K 2 ) (D 2 − K 2 )W = −Ra K 2 T̂ ,
(12.18)
(12.19)
475
3. Thermal Instability: The Bénard Problem
where
Ra ≡
gαŴd 4
,
κν
is the Rayleigh number. The boundary conditions (12.15) become
W = DW = T̂ = 0
1
at z = ± .
2
(12.20)
Before we can proceed further, we need to show that σ in this problem can only
be real.
Proof That σ Is Real for Ra > 0
The sign of the real part of σ (= σr + iσi ) determines whether the flow is stable or
unstable. We shall now show that for the Bénard problem σ is real, and the marginal
state that separates stability from instability is governed by σ = 0. To show this,
multiply equation (12.18) by T̂ ∗ (the complex conjugate of T̂ ), and integrate between
± 21 , by parts if necessary, using the boundary conditions (12.20). The various terms
transform as follows:
σ
∗
T̂ T̂ dz = σ
|T̂ |2 dz,
1/2
T̂ ∗ D 2 T̂ dz = [T̂ ∗ D T̂ ]−1/2 −
D T̂ ∗ D T̂ dz = −
|D T̂ |2 dz,
where the limits on the integrals have not been explicitly written. Equation (12.18)
then becomes
σ
|T̂ |2 dz +
|D T̂ |2 dz + K 2
|T̂ |2 dz =
T̂ ∗ W dz,
which can be written as
σ I 1 + I2 =
T̂ ∗ W dz,
where
I1 ≡
|T̂ |2 dz,
I2 ≡
[|D T̂ |2 + K 2 |T̂ |2 ] dz.
(12.21)
476
Instability
Similarly, multiply equation (12.19) by W ∗ and integrate by parts. The first term in
equation (12.19) gives
σ
Pr
σ K2
W ∗ D 2 W dz −
W ∗ W dz
Pr
σ
=−
[|DW |2 + K 2 |W |2 ] dz.
(12.22)
Pr
W ∗ (D 2 − K 2 )W dz =
σ
Pr
The second term in (12.19) gives
W ∗ (D 2 − K 2 )(D 2 − K 2 )W dz
= W ∗ (D 4 + K 4 − 2K 2 D 2 )W dz
∗ 4
4
∗
2
= W D W dz + K
W W dz − 2K
W ∗ D 2 W dz
1/2
= [W ∗ D 3 W ]−1/2 − DW ∗ D 3 W dz + K 4 |W |2 dz
1/2
2
∗
2
− 2K [W DW ]−1/2 + 2K
DW ∗ DW dz
= [|D 2 W |2 + 2K 2 |DW |2 + K 4 |W |2 ] dz.
(12.23)
Using equations (12.22) and (12.23), the integral of equation (12.19) becomes
σ
J1 + J2 = Ra K 2 W ∗ T̂ dz,
(12.24)
Pr
where
J1 ≡
[|DW |2 + K 2 |W |2 ] dz,
J2 ≡
[|D 2 W |2 + 2K 2 |DW |2 + K 4 |W |2 ] dz.
Note that the four integrals I1 , I2 , J1 , and J2 are all positive. Also, the right-hand
side of equation (12.24) is Ra K 2 times the complex conjugate of the right-hand side
of equation (12.21). We can therefore eliminate the integral on the right-hand side of
these equations by taking the complex conjugate of equation (12.21) and substituting
into equation (12.24). This gives
σ
J1 + J2 = Ra K 2 (σ ∗ I1 + I2 ).
Pr
Equating imaginary parts
σi
J
1
Pr
+ Ra K 2 I1 = 0.
477
3. Thermal Instability: The Bénard Problem
We consider only the top-heavy case, for which Ra > 0. The quantity within [ ] is
then positive, and the preceding equation requires that σi = 0.
The Bénard problem is one of two well-known problems in which σ is real.
(The other one is the Taylor problem of Couette flow between rotating cylinders,
discussed in the following section.) In most other problems σ is complex, and the
marginal state (σr = 0) contains propagating waves. In the Bénard and Taylor problems, however, the marginal state corresponds to σ = 0, and is therefore stationary
and does not contain propagating waves. In these the onset of instability is marked
by a transition from the background state to another steady state. In such a case we
commonly say that the principle of exchange of stabilities is valid, and the instability
sets in as a cellular convection, which will be explained shortly.
Solution of the Eigenvalue Problem with Two Rigid Plates
First, we give the solution for the case that is easiest to realize in a laboratory experiment, namely, a layer of fluid confined between two rigid plates where no-slip conditions are satisfied. The solution to this problem was first given by Jeffreys in 1928.
A much simpler solution exists for a layer of fluid with two stress-free surfaces. This
will be discussed later.
For the marginal state σ = 0, and the set (12.18) and (12.19) becomes
(D 2 − K 2 )T̂ = −W,
(D 2 − K 2 )2 W = Ra K 2 T̂ .
(12.25)
Eliminating T̂ , we obtain
(D 2 − K 2 )3 W = −Ra K 2 W.
(12.26)
The boundary condition (12.20) becomes
W = DW = (D 2 − K 2 )2 W = 0
1
at z = ± .
2
(12.27)
We have a sixth-order homogeneous differential equation with six homogeneous
boundary conditions. Nonzero solutions for such a system can only exist for a particular value of Ra (for a given K). It is therefore an eigenvalue problem. Note that the
Prandtl number has dropped out of the marginal state.
The point to observe is that the problem is symmetric with respect to the two
boundaries, thus the eigenfunctions fall into two distinct classes—those with the
vertical velocity symmetric about the midplane z = 0, and those with the vertical
velocity antisymmetric about the midplane (Figure 12.3). The gravest even mode
therefore has one row of cells, and the gravest odd mode has two rows of cells. It can be
shown that the smallest critical Rayleigh number is obtained by assuming disturbances
in the form of the gravest even mode, which also agrees with experimental findings
of a single row of cells.
478
Instability
Figure 12.3 Flow pattern and eigenfunction structure of the gravest even mode and the gravest odd mode
in the Bénard problem.
Because the coefficients of the governing equations (12.26) are independent of z,
the general solution can be expressed as a superposition of solutions of the form
W = eqz ,
where the six roots of q are given by
(q 2 − K 2 )3 = −Ra K 2 .
The three roots of this equation are
Ra 1/3
−
1
,
K4
√
1 Ra 1/3
2
2
(1 ± i 3) .
q =K 1+
2 K4
q 2 = −K 2
(12.28)
Taking square roots, the six roots finally become
±iq0 ,
±q,
and
± q ∗,
where
q0 = K
Ra
K4
1/3
1/2
,
−1
and q and its conjugate q ∗ are given by the two roots of equation (12.28).
The even solution of equation (12.26) is therefore
W = A cos q0 z + B cosh qz + C cosh q ∗ z.
To apply the boundary conditions on this solution, we find the following
derivatives:
DW = −Aq0 sin q0 z + Bq sinh qz + Cq ∗ sinh q ∗ z,
(D 2 − K 2 )2 W = A(q02 + K 2 )2 cos q0 z + B(q 2 − K 2 )2 cosh qz
+ C(q ∗2 − K 2 )2 cosh q ∗ z.
3. Thermal Instability: The Bénard Problem
The boundary conditions (12.27) then require
q0
q
q∗
A
cos
cosh
cosh
2
2
2
q
q∗
q0
B = 0.
q sinh
q ∗ sinh
−q0 sin
2
2
2
∗
q
q
q
0
C
(q 2 − K 2 )2 cosh
(q ∗2 − K 2 )2 cosh
(q02 + K 2 )2 cos
2
2
2
Here, A, B, and C cannot all be zero if we want to have a nonzero solution, which
requires that the determinant of the matrix must vanish. This gives a relation between
Ra and the corresponding eigenvalue K (Figure 12.4). Points on the curve K(Ra)
represent marginally stable states, which separate regions of stability and instability.
The lowest value of Ra is found to be Ra cr = 1708, attained at Kcr = 3.12. As all
values of K are allowed by the system, the flow first becomes unstable when the
Rayleigh number reaches a value of
Racr = 1708.
The wavelength at the onset of instability is
λcr =
2π d
≃ 2d.
Kcr
Figure 12.4 Stable and unstable regions for Bénard convection.
479
480
Instability
Laboratory experiments agree remarkably well with these predictions, and the
solution of the Bénard problem is considered one of the major successes of the linear
stability theory.
Solution with Stress-Free Surfaces
We now give the solution for a layer of fluid with stress-free surfaces. This case can
be approximately realized in a laboratory experiment if a layer of liquid is floating on
top of a somewhat heavier liquid. The main interest in the problem, however, is that it
allows a simple solution, which was first given by Rayleigh. In this case the boundary
conditions are w = T ′ = µ(∂u/∂z + ∂w/∂x) = µ(∂v/∂z + ∂w/∂y) = 0 at the
surfaces, the latter two conditions resulting from zero stress. Because w vanishes (for
all x and y) on the boundaries, it follows that the vanishing stress conditions require
∂u/∂z = ∂v/∂z = 0 at the boundaries. On differentiating the continuity equation
with respect to z, it follows that ∂ 2 w/∂z2 = 0 on the free surfaces. In terms of the
complex amplitudes, the eigenvalue problem is therefore
(D 2 − K 2 )3 W = −Ra K 2 W,
(12.29)
with W = (D 2 − K 2 )2 W = D 2 W = 0 at the surfaces. By expanding (D 2 − K 2 )2 ,
the boundary conditions can be written as
W = D2 W = D4 W = 0
1
at z = ± ,
2
which should be compared with the conditions (12.27) for rigid boundaries.
Successive differentiation of equation (12.29) shows that all even derivatives of
W vanish on the boundaries. The eigenfunctions must therefore be
W = A sin nπ z,
where A is any constant and n is an integer. Substitution into equation (12.29) leads
to the eigenvalue relation
Ra = (n2 π 2 + K 2 )3 /K 2 ,
(12.30)
which gives the Rayleigh number in the marginal state. For a given K 2 , the lowest
value of Ra occurs when n = 1, which is the gravest mode. The critical Rayleigh
number is obtained by finding the minimum value of Ra as K 2 is varied, that is, by
setting d Ra/dK 2 = 0. This gives
d Ra
3(π 2 + K 2 )2
(π 2 + K 2 )3
=
−
= 0,
2
2
dK
K
K4
2 = π 2 /2. The corresponding value of Ra is
which requires Kcr
Racr =
27 4
4 π
= 657.
For a layer with a free upper surface (where the stress is zero) and a rigid bottom
wall, the solution of the eigenvalue problem gives Racr = 1101 and Kcr = 2.68.
3. Thermal Instability: The Bénard Problem
This case is of interest in laboratory experiments having the most visual effects, as
originally conducted by Bénard.
Cell Patterns
The linear theory specifies the horizontal wavelength at the onset of instability, but
not the horizontal pattern of the convective cells. This is because a given wavenumber
vector K can be decomposed into two orthogonal components in an infinite number of
ways. If we assume that the experimental conditions are horizontally isotropic, with
no preferred directions, then regular polygons in the form of equilateral triangles,
squares, and regular hexagons are all possible structures. Bénard’s original experiments showed only hexagonal patterns, but we now know that he was observing a
different phenomenon. The observations summarized in Drazin and Reid (1981) indicate that hexagons frequently predominate initially. As Ra is increased, the cells tend
to merge and form rolls, on the walls of which the fluid rises or sinks (Figure 12.5).
The cell structure becomes more chaotic as Ra is increased further, and the flow
becomes turbulent when Ra > 5 × 104 .
The magnitude or direction of flow in the cells cannot be predicted by linear
theory. After a short time of exponential growth, the flow becomes large enough for
the nonlinear terms to be important and reaches a nonlinear equilibrium stage. The
flow pattern for a hexagonal cell is sketched in Figure 12.6. Particles in the middle
of the cell usually rise in a liquid and fall in a gas. This has been attributed to the
Figure 12.5 Convection rolls in a Bénard problem.
Figure 12.6 Flow pattern in a hexagonal Bénard cell.
481
482
Instability
property that the viscosity of a liquid decreases with temperature, whereas that of
a gas increases with temperature. The rising fluid loses heat by thermal conduction
at the top wall, travels horizontally, and then sinks. For a steady cellular pattern,
the continuous generation of kinetic energy is balanced by viscous dissipation. The
generation of kinetic energy is maintained by continuous release of potential energy
due to heating at the bottom and cooling at the top.
4. Double-Diffusive Instability
An interesting instability results when the density of the fluid depends on two opposing gradients. The possibility of this phenomenon was first suggested by Stommel
et al. (1956), but the dynamics of the process was first explained by Stern (1960).
Turner (1973), and review articles by Huppert and Turner (1981), and Turner (1985)
discuss the dynamics of this phenomenon and its applications to various fields such
as astrophysics, engineering, and geology. Historically, the phenomenon was first
suggested with oceanic application in mind, and this is how we shall present it. For
sea water the density depends on the temperature T̃ and salt content s̃ (kilograms of
salt per kilograms of water), so that the density is given by
ρ̃ = ρ0 [1 − α(T̃ − T0 ) + β(s̃ − s0 )],
where the value of α determines how fast the density decreases with temperature, and
the value of β determines how fast the density increases with salinity. As defined here,
both α and β are positive. The key factor in this instability is that the diffusivity κs of
salt in water is only 1% of the thermal diffusivity κ. Such a system can be unstable even
when the density decreases upwards. By means of the instability, the flow releases
the potential energy of the component that is “heavy at the top.” Therefore, the effect
of diffusion in such a system can be to destabilize a stable density gradient. This is in
contrast to a medium containing a single diffusing component, for which the analysis
of the preceding section shows that the effect of diffusion is to stabilize the system
even when it is heavy at the top.
Finger Instability
Consider the two situations of Figure 12.7, both of which can be unstable although
each is stably stratified in density (d ρ̄/dz < 0). Consider first the case of hot and
salty water lying over cold and fresh water (Figure 12.7a), that is, when the system
is top heavy in salt. In this case both d T̄ /dz and dS/dz are positive, and we can
arrange the composition of water such that the density decreases upward. Because
κs ≪ κ, a displaced particle would be near thermal equilibrium with the surroundings,
but would exchange negligible salt. A rising particle therefore would be constantly
lighter than the surroundings because of the salinity deficit, and would continue to
rise. A parcel displaced downward would similarly continue to plunge downward.
The basic state shown in Figure 12.7a is therefore unstable. Laboratory observations
show that the instability in this case appears in the form of a forest of long narrow
convective cells, called salt fingers (Figure 12.8). Shadowgraph images in the deep
ocean have confirmed their existence in nature.
4. Double-Diffusive Instability
Figure 12.7 Two kinds of double-diffusive instabilities. (a) Finger instability, showing up- and downgoing
salt fingers and their temperature, salinity, and density. Arrows indicate direction of motion. (b) Oscillating
instability, finally resulting in a series of convecting layers separated by “diffusive” interfaces. Across these
interfaces T and S vary sharply, but heat is transported much faster than salt.
Figure 12.8 Salt fingers, produced by pouring salt solution on top of a stable temperature gradient. Flow
visualization by fluorescent dye and a horizontal beam of light. J. Turner, Naturwissenschaften 72: 70–75,
1985 and reprinted with the permission of Springer-Verlag GmbH & Co.
483
484
Instability
We can derive a criterion for instability by generalizing our analysis of the Bénard
convection so as to include salt diffusion. Assume a layer of depth d confined between
stress-free boundaries maintained at constant temperature and constant salinity. If we
repeat the derivation of the perturbation equations for the normal modes of the system,
the equations that replace equation (12.25) are found to be
(D 2 − K 2 )T̂ = −W,
κs 2
(D − K 2 )ŝ = −W,
κ
(D 2 − K 2 )2 W = −Ra K 2 T̂ + Rs′ K 2 ŝ,
(12.31)
where ŝ(z) is the complex amplitude of the salinity perturbation, and we have defined
Ra ≡
gαd 4 (d T̄ /dz)
,
νκ
Rs′ ≡
gβd 4 (dS/dz)
.
νκ
and
Note that κ (and not κs ) appears in the definition of Rs′ . In contrast to equation (12.31),
a positive sign appeared in equation (12.25) in front of Ra because in the preceding
section Ra was defined to be positive for a top-heavy situation.
It is seen from the first two of equations (12.31) that the equations for T̂ and
ŝκs /κ are the same. The boundary conditions are also the same for these variables:
T̂ =
κs ŝ
=0
κ
1
at z = ± .
2
It follows that we must have T̂ = ŝκs /κ everywhere. Equations (12.31) therefore
become
(D 2 − K 2 )T̂ = −W,
(D 2 − K 2 )2 W = (Rs − Ra)K 2 T̂ ,
where
Rs ≡
gβd 4 (dS/dz)
Rs′ κ
=
.
κs
νκs
The preceding set is now identical to the set (12.25) for the Bénard convection, with
(Rs − Ra) replacing Ra. For stress-free boundaries, solution of the preceding section
shows that the critical value is
Rs − Ra =
27 4
4 π
= 657,
485
4. Double-Diffusive Instability
which can be written as
gd 4
ν
α d T̄
β dS
−
κs dz
κ dz
= 657.
(12.32)
Even if α(d T̄ /dz) − β(dS/dz) > 0 (i.e., ρ̄ decreases upward), the condition (12.32)
can be quite easily satisfied because κs is much smaller than κ. The flow can therefore
be made unstable simply by ensuring that the factor within [ ] is positive and making
d large enough.
The analysis predicts that the lateral width of the cell is of the order of d, but such
wide cells are not observed at supercritical stages when (Rs − Ra) far exceeds 657.
Instead, long thin salt fingers are observed, as shown in Figure 12.8. If the salinity
gradient is large, then experiments as well as calculations show that a deep layer
of salt fingers becomes unstable and breaks down into a series of convective layers,
with fingers confined to the interfaces. Oceanographic observations frequently show
a series of staircase-shaped vertical distributions of salinity and temperature, with a
positive overall dS/dz and d T̄ /dz; this can indicate salt finger activity.
Oscillating Instability
Consider next the case of cold and fresh water lying over hot and salty water
(Figure 12.7b). In this case both d T̄ /dz and dS/dz are negative, and we can choose
their values such that the density decreases upwards. Again the system is unstable, but
the dynamics are different. A particle displaced upward loses heat but no salt. Thus it
becomes heavier than the surroundings and buoyancy forces it back toward its initial
position, resulting in an oscillation. However, a stability calculation shows that a less
than perfect heat conduction results in a growing oscillation, although some energy
is dissipated. In this case the growth rate σ is complex, in contrast to the situation of
Figure 12.7a where it is real.
Laboratory experiments show that the initial oscillatory instability does not last
long, and eventually results in the formation of a number of horizontal convecting
layers, as sketched in Figure 12.7b. Consider the situation when a stable salinity gradient in an isothermal fluid is heated from below (Figure 12.9). The initial instability
starts as a growing oscillation near the bottom. As the heating is continued beyond the
initial appearance of the instability, a well-mixed layer develops, capped by a salinity
step, a temperature step, and no density step. The heat flux through this step forms a
thermal boundary layer, as shown in Figure 12.9. As the well-mixed layer grows, the
temperature step across the thermal boundary layer becomes larger. Eventually, the
Rayleigh number across the thermal boundary layer becomes critical, and a second
convecting layer forms on top of the first. The second layer is maintained by heat flux
(and negligible salt flux) across a sharp laminar interface on top of the first layer. This
process continues until a stack of horizontal layers forms one upon another. From
comparison with the Bénard convection, it is clear that inclusion of a stable salinity
gradient has prevented a complete overturning from top to bottom.
The two examples in this section show that in a double-component system in
which the diffusivities for the two components are different, the effect of diffusion
486
Instability
Figure 12.9 Distributions of salinity, temperature, and density, generated by heating a linear salinity
gradient from below.
can be destabilizing, even if the system is judged hydrostatically stable. In contrast,
diffusion is stabilizing in a single-component system, such as the Bénard system. The
two requirements for the double-diffusive instability are that the diffusivities of the
components be different, and that the components make opposite contributions to
the vertical density gradient.
5. Centrifugal Instability: Taylor Problem
In this section we shall consider the instability of a Couette flow between concentric
rotating cylinders, a problem first solved by Taylor in 1923. In many ways the problem
is similar to the Bénard problem, in which there is a potentially unstable arrangement
of an “adverse” temperature gradient. In the Couette flow problem the source of the
instability is the adverse gradient of angular momentum. Whereas convection in a
heated layer is brought about by buoyant forces becoming large enough to overcome
the viscous resistance, the convection in a Couette flow is generated by the centrifugal
forces being able to overcome the viscous forces. We shall first present Rayleigh’s
discovery of an inviscid stability criterion for the problem and then outline Taylor’s
solution of the viscous case. Experiments indicate that the instability initially appears
in the form of axisymmetric disturbances, for which ∂/∂θ = 0. Accordingly, we shall
limit ourselves only to the axisymmetric case.
Rayleigh’s Inviscid Criterion
The problem was first considered by Rayleigh in 1888. Neglecting viscous effects,
he discovered the source of instability for this problem and demonstrated a necessary
and sufficient condition for instability. Let Uθ (r) be the velocity at any radial distance. For inviscid flows Uθ (r) can be any function, but only certain distributions can
be stable. Imagine that two fluid rings of equal masses at radial distances r1 and r2
(>r1 ) are interchanged. As the motion is inviscid, Kelvin’s theorem requires that the
487
5. Centrifugal Instability: Taylor Problem
circulation Ŵ = 2πrUθ (proportional to the angular momentum rUθ ) should remain
constant during the interchange. That is, after the interchange, the fluid at r2 will have
the circulation (namely, Ŵ1 ) that it had at r1 before the interchange. Similarly, the fluid
at r1 will have the circulation (namely, Ŵ2 ) that it had at r2 before the interchange.
The conservation of circulation requires that the kinetic energy E must change during
the interchange. Because E = Uθ2 /2 = Ŵ 2 /8π 2 r 2 , we have
Efinal
1
=
8π 2
Ŵ22
Einitial
1
=
8π 2
Ŵ12
r12
r12
+
+
Ŵ12
r22
Ŵ22
r22
,
,
so that the kinetic energy change per unit mass is
E = Efinal − Einitial
1
1
1
2
2
(Ŵ − Ŵ1 ) 2 − 2 .
=
8π 2 2
r1
r2
Because r2 > r1 , a velocity distribution for which Ŵ22 > Ŵ12 would make E positive, which implies that an external source of energy would be necessary to perform
the interchange of the fluid rings. Under this condition a spontaneous interchange of
the rings is not possible, and the flow is stable. On the other hand, if Ŵ 2 decreases
with r, then an interchange of rings will result in a release of energy; such a flow is
unstable. It can be shown that in this situation the centrifugal force in the new location
of an outwardly displaced ring is larger than the prevailing (radially inward) pressure
gradient force.
Rayleigh’s criterion can therefore be stated as follows: An inviscid Couette flow
is unstable if
dŴ 2
<0
dr
(unstable).
The criterion is analogous to the inviscid requirement for static instability in a density
stratified fluid:
d ρ̄
>0
dz
(unstable).
Therefore, the “stratification” of angular momentum in a Couette flow is unstable
if it decreases radially outwards. Consider a situation in which the outer cylinder is
held stationary and the inner cylinder is rotated. Then dŴ 2 /dr < 0, and Rayleigh’s
criterion implies that the flow is inviscidly unstable. As in the Bénard problem, however, merely having a potentially unstable arrangement does not cause instability in
a viscous medium. The inviscid Rayleigh criterion is modified by Taylor’s solution
of the viscous problem, outlined in what follows.
488
Instability
Formulation of the Problem
Using cylindrical polar coordinates (r, θ, z) and assuming axial symmetry, the
equations of motion are
ũ2
D ũr
− θ
Dt
r
ũr ũθ
D ũθ
+
Dt
r
D ũz
Dt
∂ ũr
∂r
1 ∂ p̃
ũr
2
=−
+ ν ∇ ũr − 2 ,
ρ ∂r
r
ũ
θ
= ν ∇ 2 ũθ − 2 ,
r
1 ∂ p̃
=−
+ ν∇ 2 ũz ,
ρ ∂z
ũr
∂ ũz
+
+
= 0,
r
∂z
(12.33)
where
∂
∂
∂
D
≡
+ ũr
+ ũz ,
Dt
∂t
∂r
∂z
and
∇2 ≡
1 ∂
∂2
∂2
+
+
.
∂r 2
r ∂r
∂z2
We decompose the motion into a background state plus perturbation:
ũ = U + u,
(12.34)
p̃ = P + p.
The background state is given by (see Chapter 9, Section 6)
Ur = Uz = 0,
1 dP
V2
=
,
ρ dr
r
Uθ = V (r),
(12.35)
where
V = Ar + B/r,
(12.36)
with constants defined as
A≡
2 R22 − 1 R12
R22 − R12
,
B≡
(1 − 2 )R12 R22
R22 − R12
.
Here, 1 and 2 are the angular speeds of the inner and outer cylinders, respectively,
and R1 and R2 are their radii (Figure 12.10).
5. Centrifugal Instability: Taylor Problem
Figure 12.10 Definition sketch of instability in rotating Couette flow.
Substituting equation (12.34) into the equations of motion (12.33), neglecting
nonlinear terms, and subtracting the background state (12.35), we obtain the perturbation equations
2V
1 ∂p
ur
∂ur
−
uθ = −
+ ν ∇ 2 ur − 2 ,
∂t
r
ρ ∂r
r
2
u
V
dV
∂uθ
+
+
ur = ν ∇ 2 uθ − θ ,
∂t
dr
r
r
(12.37)
∂uz
1 ∂p
2
=−
+ ν∇ uz ,
∂t
ρ ∂z
ur
∂uz
∂ur
+
+
= 0.
∂r
r
∂z
As the coefficients in these equations depend only on r, the equations admit solutions
that depend on z and t exponentially. We therefore consider normal mode solutions
of the form
(ur , uθ , uz , p) = (ûr , ûθ , ûz , p̂) eσ t+ikz .
The requirement that the solutions remain bounded as z → ±∞ implies that the
axial wavenumber k must be real. After substituting the normal modes into (12.37)
and eliminating ûz and p̂, we get a coupled system of equations in ûr and ûθ . Under the
489
490
Instability
narrow-gap approximation, for which d = R2 −R1 is much smaller than (R1 +R2 )/2,
these equations finally become (see Chandrasekhar (1961) for details)
(D 2 − k 2 − σ )(D 2 − k 2 )ûr = (1 + αx)ûθ ,
(D 2 − k 2 − σ )ûθ = −Ta k 2 ûr ,
(12.38)
where
2
− 1,
1
r − R1
x≡
,
d
d ≡ R2 − R1 ,
d
D≡
.
dr
α≡
We have also defined the Taylor number
1 R12 − 2 R22 1 d 4
.
Ta ≡ 4
ν2
R22 − R12
(12.39)
It is the ratio of the centrifugal force to viscous force, and equals 2(V1 d/ν)2 (d/R1 )
when only the inner cylinder is rotating and the gap is narrow.
The boundary conditions are
ûr = D ûr = ûθ = 0
at x = 0, 1.
(12.40)
The eigenvalues k at the marginal state are found by setting the real part of σ to zero.
On the basis of experimental evidence, Taylor assumed that the principle of exchange
of stabilities must be valid for this problem, and the marginal states are given by
σ = 0. This was later proven to be true for cylinders rotating in the same directions,
but a general demonstration for all conditions is still lacking.
Discussion of Taylor’s Solution
A solution of the eigenvalue problem (12.38), subject to equation (12.40), was
obtained by Taylor. Figure 12.11 shows the results of his calculations and his own
experimental verification of the analysis. The vertical axis represents the angular
velocity of the inner cylinder (taken positive), and the horizontal axis represents the
angular velocity of the outer cylinder. Cylinders rotating in opposite directions are
represented by a negative 2 . Taylor’s solution of the marginal state is indicated, with
the region above the curve corresponding to instability. Rayleigh’s inviscid criterion is
also indicated by the straight dashed line. It is apparent that the presence of viscosity
can stabilize a flow. Taylor’s viscous solution indicates that the flow remains stable
until a critical Taylor number of
Tacr =
1708
,
(1/2) (1 + 2 /1 )
(12.41)
5. Centrifugal Instability: Taylor Problem
Figure 12.11 Taylor’s observation and narrow-gap calculation of marginal stability in rotating Couette
flow of water. The ratio of radii is R2 /R1 = 1.14. The region above the curve is unstable. The dashed line
represents Rayleigh’s inviscid criterion, with the region to the left of the line representing instability.
is attained. The nondimensional axial wavenumber at the onset of instability is found
to be kcr = 3.12, which implies that the wavelength at onset is λcr = 2π d/kcr ≃ 2d.
The height of one cell is therefore nearly equal to d, so that the cross-section of a cell
is nearly a square. In the limit 2 / 1 → 1, the critical Taylor number is identical
to the critical Rayleigh number for thermal convection discussed in the preceding
section, for which the solution was given by Jeffreys five years later. The agreement
is expected, because in this limit α = 0, and the eigenvalue problem (12.38) reduces
to that of the Bénard problem (12.25). For cylinders rotating in opposite directions
the Rayleigh criterion predicts instability, but the viscous solution can be stable.
Taylor’s analysis of the problem was enormously satisfying, both experimentally
and theoretically. He measured the wavelength at the onset of instability by injecting
dye and obtained an almost exact agreement with his calculations. The observed onset
of instability in the 1 2 -plane (Figure 12.11) was also in remarkable agreement.
This has prompted remarks such as “the closeness of the agreement between his
theoretical and experimental results was without precedent in the history of fluid
mechanics” (Drazin and Reid 1981, p. 105). It even led some people to suggest happily
that the agreement can be regarded as a verification of the underlying Navier–Stokes
equations, which make a host of assumptions including a linearity between stress and
strain rate.
The instability appears in the form of counter-rotating toroidal (or doughnutshaped) vortices (Figure 12.12a) called Taylor vortices. The streamlines are in the
form of helixes, with axes wrapping around the annulus, somewhat like the stripes
on a barber’s pole. These vortices themselves become unstable at higher values of
Ta, when they give rise to wavy vortices for which ∂/∂θ = 0 (Figure 12.12b). In
effect, the flow has now attained the next higher mode. The number of waves around
the annulus depends on the Taylor number, and the wave pattern travels around the
491
492
Instability
Figure 12.12 Instability of rotating Couette flow. Panels a, b, c, and d correspond to increasing Taylor
number. D. Coles, Journal of Fluid Mechanics 21: 385–425, 1965 and reprinted with the permission of
Cambridge University Press.
annulus. More complicated patterns of vortices result at a higher rates of rotation,
finally resulting in the occasional appearance of turbulent patches (Figure 12.12d),
and then a fully turbulent flow.
Phenomena analogous to the Taylor vortices are called secondary flows because
they are superposed on a primary flow (such as the Couette flow in the present case).
493
6. Kelvin–Helmholtz Instability
Figure 12.13 Görtler vortices in a boundary layer along a concave wall.
There are two other situations where a combination of curved streamlines (which
give rise to centrifugal forces) and viscosity result in instability and steady secondary
flows in the form of vortices. One is the flow through a curved channel, driven by
a pressure gradient. The other is the appearance of Görtler vortices in a boundary layer flow along a concave wall (Figure 12.13). The possibility of secondary
flows signifies that the solutions of the Navier–Stokes equations are nonunique in
the sense that more than one steady solution is allowed under the same boundary
conditions. We can derive the form of the primary flow only if we exclude the secondary flow by appropriate assumptions. For example, we can derive the expression
(12.36) for Couette flow by assuming that Ur = 0 and Uz = 0, which rule out the
secondary flow.
6. Kelvin–Helmholtz Instability
Instability at the interface between two horizontal parallel streams of different
velocities and densities, with the heavier fluid at the bottom, is called the
Kelvin–Helmholtz instability. The name is also commonly used to describe the instability of the more general case where the variations of velocity and density are continuous and occur over a finite thickness. The more general case is discussed in the
following section.
Assume that the layers have infinite depth and that the interface has zero thickness.
Let U1 and ρ1 be the velocity and density of the basic state in the upper layer and U2
and ρ2 be those in the bottom layer (Figure 12.14). By Kelvin’s circulation theorem,
the perturbed flow must be irrotational in each layer because the motion develops from
an irrotational basic flow of uniform velocity in each layer. The flow can therefore be
described by a velocity potential that satisfies the Laplace equation. Let the variables
in the perturbed state be denoted by a tilde ( ˜ ). Then
∇ 2 φ̃ 1 = 0,
∇ 2 φ̃ 2 = 0.
(12.42)
494
Instability
Figure 12.14 Discontinuous shear across a density interface.
The flow is decomposed into a basic state plus perturbations:
φ̃1 = U1 x + φ1 ,
(12.43)
φ̃2 = U2 x + φ2 ,
where the first terms on the right-hand side represent the basic flow of uniform streams.
Substitution into equation (12.42) gives the perturbation equations
∇ 2 φ1 = 0,
∇ 2 φ2 = 0,
(12.44)
subject to
φ1 → 0
φ2 → 0
as z → ∞,
(12.45)
as z → −∞.
As discussed in Chapter 7, there are kinematic and dynamic conditions to be
satisfied at the interface. The kinematic boundary condition is that the fluid particles
at the interface must move with the interface. Considering particles just above the
interface, this requires
∂φ1
∂ζ
Dζ
∂ζ
∂ζ
=
=
+ (U1 + u1 )
+ v1
∂z
Dt
∂t
∂x
∂y
at z = ζ.
This condition can be linearized by applying it at z = 0 instead of at z = ζ and
by neglecting quadratic terms. Writing a similar equation for the lower layer, the
kinematic boundary conditions are
∂φ1
∂ζ
∂ζ
=
+ U1
∂z
∂t
∂x
∂φ2
∂ζ
∂ζ
=
+ U2
∂z
∂t
∂x
at z = 0,
(12.46)
at z = 0.
(12.47)
The dynamic boundary condition at the interface is that the pressure must be
continuous across the interface (if surface tension is neglected), requiring p1 = p2
495
6. Kelvin–Helmholtz Instability
at z = ζ . The unsteady Bernoulli equations are
1
p̃1
∂ φ̃ 1
+ (∇ φ̃ 1 )2 +
+ gz = C1 ,
∂t
2
ρ1
∂ φ̃ 2
1
p̃2
+ (∇ φ̃ 2 )2 +
+ gz = C2 .
∂t
2
ρ2
(12.48)
In order that the pressure be continuous in the undisturbed state (P1 = P2 at z = 0),
the Bernoulli equation requires
ρ1
1 2
U − C1
2 1
= ρ2
1 2
U2 − C2 .
2
(12.49)
Introducing the decomposition (12.43) into the Bernoulli equations (12.48), and
requiring p̃1 = p̃2 at z = ζ , we obtain the following condition at the interface:
∂φ1
ρ1
− [(U1 + u1 )2 + v12 + w12 ] − ρ1 gζ
∂t
2
∂φ2
ρ2
= ρ2 C2 − ρ2
− [(U2 + u2 )2 + v22 + w22 ] − ρ2 gζ.
∂t
2
ρ1 C1 − ρ1
Subtracting the basic state condition (12.49) and neglecting nonlinear terms, we obtain
ρ1
∂φ
1
∂t
+ U1
∂φ
∂φ1
∂φ2
2
+ gζ
= ρ2
+ U2
+ gζ
.
z=0
z=0
∂x
∂t
∂x
(12.50)
The perturbations therefore satisfy equation (12.44), and conditions (12.45),
(12.46), (12.47), and (12.50). Assume normal modes of the form
(ζ, φ1 , φ2 ) = (ζ̂ , φ̂ 1 , φ̂ 2 ) eik(x−ct) ,
where k is real (and can be taken positive without loss of generality), but c = cr + ici
is complex. The flow is unstable if there exists a positive ci . (Note that in the preceding
sections we assumed a time dependence of the form exp(σ t), which is more convenient
when the instability appears in the form of convective cells.) Substitution of the normal
modes into the Laplace equations (12.44) requires solutions of the form
φ̂ 1 = A e−kz ,
φ̂ 2 = B ekz ,
where solutions exponentially increasing from the interface are ignored because of
equation (12.45).
Now equations (12.46), (12.47), and (12.50) give three homogeneous linear algebraic equations for determining the three unknowns ζ̂ , A, and B; solutions can
496
Instability
therefore exist only for certain values of c(k). The kinematic conditions (12.46)
and (12.47) give
A = −i(U1 − c)ζ̂ ,
B = i(U2 − c)ζ̂ .
The Bernoulli equation (12.50) gives
ρ1 [ik(U1 − c)A + g ζ̂ ] = ρ2 [ik(U2 − c)B + g ζ̂ ].
Substituting for A and B, this gives the eigenvalue relation for c(k):
kρ2 (U2 − c)2 + kρ1 (U1 − c)2 = g(ρ2 − ρ1 ),
for which the solutions are
U1 − U2 2 1/2
g ρ2 − ρ 1
ρ2 U2 + ρ1 U1
.
±
− ρ 1 ρ2
c=
ρ2 + ρ 1
k ρ2 + ρ 1
ρ2 + ρ 1
(12.51)
It is seen that both solutions are neutrally stable (c real) as long as the second term
within the square root is smaller than the first; this gives the stable waves of the
system. However, there is a growing solution (ci > 0) if
g(ρ22 − ρ12 ) < kρ1 ρ2 (U1 − U2 )2 .
Equation (12.51) shows that for each growing solution there is a corresponding decaying solution. As explained more fully in the following section, this happens because
the coefficients of the differential equation and the boundary conditions are all real.
Note also that the dispersion relation of free waves in an initial static medium, given
by Equation (7.105), is obtained from equation (12.51) by setting U1 = U2 = 0.
If U1 = U2 , then one can always find a large enough k that satisfies the requirement for instability. Because all wavelengths must be allowed in an instability analysis,
we can say that the flow is always unstable (to short waves) if U1 = U2 .
Consider now the flow of a homogeneous fluid (ρ1 = ρ2 ) with a velocity discontinuity, which we can call a vortex sheet. Equation (12.51) gives
c=
i
1
(U1 + U2 ) ± (U1 − U2 ).
2
2
The vortex sheet is therefore always unstable to all wavelengths. It is also seen that
the unstable wave moves with a phase velocity equal to the average velocity of the
basic flow. This must be true from symmetry considerations. In a frame of reference moving with the average velocity, the basic flow is symmetric and the wave
therefore should have no preference between the positive and negative x directions
(Figure 12.15).
The Kelvin–Helmholtz instability is caused by the destabilizing effect of shear,
which overcomes the stabilizing effect of stratification. This kind of instability is easy
6. Kelvin–Helmholtz Instability
Figure 12.15 Background velocity field as seen by an observer moving with the average velocity
(U1 + U2 )/2 of two layers.
Figure 12.16 Kelvin–Helmholtz instability generated by tilting a horizontal channel containing two
liquids of different densities. The lower layer is dyed. Mean flow in the lower layer is down the plane and
that in the upper layer is up the plane. S. A. Thorpe, Journal of Fluid Mechanics 46: 299–319, 1971 and
reprinted with the permission of Cambridge University Press.
to generate in the laboratory by filling a horizontal glass tube (of rectangular cross
section) containing two liquids of slightly different densities (one colored) and gently
tilting it. This starts a current in the lower layer down the plane and a current in the
upper layer up the plane. An example of instability generated in this manner is shown
in Figure 12.16.
Shear instability of stratified fluids is ubiquitous in the atmosphere and the ocean
and believed to be a major source of internal waves in them. Figure 12.17 is a striking
photograph of a cloud pattern, which is clearly due to the existence of high shear across
a sharp density gradient. Similar photographs of injected dye have been recorded in
oceanic thermoclines (Woods, 1969).
497
498
Instability
Figure 12.17 Billow cloud near Denver, Colorado. P. G. Drazin and W. H. Reid, Hydrodynamic Stability,
1981 and reprinted with the permission of Cambridge University Press.
Figures 12.16 and 12.17 show the advanced nonlinear stage of the instability in
which the interface is a rolled-up layer of vorticity. Such an observed evolution of the
interface is in agreement with results of numerical calculations in which the nonlinear
terms are retained (Figure 12.18).
The source of energy for generating the Kelvin–Helmholtz instability is derived
from the kinetic energy of the shear flow. The disturbances essentially smear out the
gradients until they cannot grow any longer. Figure 12.19 shows a typical behavior, in
which the unstable waves at the interface have transformed the sharp density profile
ACDF to ABEF and the sharp velocity profile MOPR to MNQR. The high-density
fluid in the depth range DE has been raised upward (and mixed with the lower-density
fluid in the depth range BC), which means that the potential energy of the system has
increased after the instability. The required energy has been drawn from the kinetic
energy of the basic field. It is easy to show that the kinetic energy of the initial profile
MOPR is larger than that of the final profile MNQR. To see this, assume that the
initial velocity of the lower layer is zero and that of the upper layer is U1 . Then the
linear velocity profile after mixing is given by
1
z
U (z) = U1
+
− h z h.
2 2h
6. Kelvin–Helmholtz Instability
Figure 12.18 Nonlinear numerical calculation of the evolution of a vortex sheet that has been given a
small sinusoidal displacement of wavelength λ. The density difference across the interface is zero, and U0
is the velocity difference across the sheet. J. S. Turner, Buoyancy Effects in Fluids, 1973 and reprinted with
the permission of Cambridge University Press.
Figure 12.19 Smearing out of sharp density and velocity profiles, resulting in an increase of potential
energy and a decrease of kinetic energy.
499
500
Instability
Consider the change in kinetic energy only in the depth range −h < z < h, as the
energy outside this range does not change. Then the initial and final kinetic energies
per unit width are
ρ
Einitial = U12 h,
2
ρ h 2
ρ
U (z) dz = U12 h.
Efinal =
2 −h
3
The kinetic energy of the flow has therefore decreased, although the total momentum
(= U dz) is unchanged. This is a general result: If the integral of U (z) does not
change, then the integral of U 2 (z) decreases if the gradients decrease.
In this section we have considered the case of a discontinuous variation across
an infinitely thin interface and shown that the flow is always unstable. The case of
continuous variation is considered in the following section. We shall see that a certain
condition must be satisfied in order for the flow to be unstable.
7. Instability of Continuously Stratified Parallel Flows
An instability of great geophysical importance is that of an inviscid stratified fluid
in horizontal parallel flow. If the density and velocity vary discontinuously across
an interface, the analysis in the preceding section shows that the flow is unconditionally unstable. Although only the discontinuous case was studied by Kelvin and
Helmholtz, the more general case of continuous distribution is also commonly called
the Kelvin–Helmholtz instability.
The problem has a long history. In 1915, Taylor, on the basis of his calculations with assumed distributions of velocity and density, conjectured that a gradient
Richardson number (to be defined shortly) must be less than 41 for instability. Other values of the critical Richardson number (ranging from 2 to 41 ) were suggested by Prandtl,
Goldstein, Richardson, Synge, and Chandrasekhar. Finally, Miles (1961) was able to
prove Taylor’s conjecture, and Howard (1961) immediately and elegantly generalized
Miles’ proof. A short record of the history is given in Miles (1986). In this section we
shall prove the Richardson number criterion in the manner given by Howard.
Taylor–Goldstein Equation
Consider a horizontal parallel flow U (z) directed along the x-axis. The z-axis is taken
vertically upwards. The basic flow is in equilibrium with the undisturbed density field
ρ̄(z) and the basic pressure field P (z). We shall only consider two-dimensional disturbances on this basic state, assuming that they are more unstable than three-dimensional
disturbances; this is called Squires’ theorem and is demonstrated in Section 8 in
another context. The disturbed state has velocity, pressure, and density fields of
[U + u, 0, w],
P + p,
The continuity equation reduces to
∂u ∂w
+
= 0.
∂x
∂z
ρ̄ + ρ.
501
7. Instability of Continuously Stratified Parallel Flows
The disturbed velocity field is assumed to satisfy the Boussinesq equation
∂
g
1 ∂
∂
(Ui + ui ) + (Uj + uj )
(Ui + ui ) = − (ρ̄ + ρ)δi3 −
(P + p),
∂t
∂xj
ρ0
ρ0 ∂xi
where the density variations are neglected except in the vertical equation of motion.
Here, ρ0 is a reference density. The basic flow satisfies
0=−
1 ∂P
g ρ̄
δi3 −
.
ρ0
ρ0 ∂xi
Subtracting the last two equations and dropping nonlinear terms, we obtain the perturbation equation of motion
∂Ui
∂ui
gρ
1 ∂p
∂ui
+ uj
+ Uj
= − δi3 −
.
∂t
∂xj
∂xj
ρ0
ρ0 ∂xi
The i = 1 and i = 3 components of the preceding equation are
∂U
∂u
1 ∂p
∂u
+w
+U
=−
,
∂t
∂z
∂x
ρ0 ∂x
∂w
∂w
gρ
1 ∂p
+U
=−
−
.
∂t
∂x
ρ0
ρ0 ∂z
(12.52)
In the absence of diffusion the density is conserved along the motion, which
requires that D(density)/Dt = 0, or that
∂
∂
∂
(ρ̄ + ρ) + (U + u) (ρ̄ + ρ) + w (ρ̄ + ρ) = 0.
∂t
∂x
∂z
Keeping only the linear terms, and using the fact that ρ̄ is a function of z only, we
obtain
∂ρ
d ρ̄
∂ρ
+U
+w
= 0,
∂t
∂x
dz
which can be written as
ρ0 N 2 w
∂ρ
∂ρ
+U
−
= 0,
∂t
∂x
g
(12.53)
where we have defined
N2 ≡ −
g d ρ̄
.
ρ0 dz
N is the buoyancy frequency. The last term in equation (12.53) represents the density
change at a point due to the vertical advection of the basic density field across the
point.
502
Instability
The continuity equation can be satisfied by defining a streamfunction through
u=
∂ψ
,
∂z
w=−
∂ψ
.
∂x
Equations (12.52) and (12.53) then become
1
px ,
ρ0
gρ
1
−ψxt − ψxx U = −
− pz ,
ρ0
ρ0
ρ0 N 2
ρt + Uρx +
ψx = 0,
g
ψzt − ψx Uz + ψxz U = −
(12.54)
where subscripts denote partial derivatives.
As the coefficients of equation (12.54) are independent of x and t, exponential
variations in these variables are allowed. Consequently, we assume normal mode
solutions of the form
[ρ, p, ψ] = [ρ̂(z), p̂(z), ψ̂(z)] eik(x−ct) ,
where quantities denoted by ( ˆ ) are complex amplitudes. Because the flow is
unbounded in x, the wavenumber k must be real. The eigenvalue c = cr + ici can be
complex, and the solution is unstable if there exists a ci > 0. Substituting the normal
modes, equation (12.54) becomes
1
p̂,
ρ0
1
g ρ̂
− p̂z ,
k 2 (U − c)ψ̂ = −
ρ0
ρ0
ρ0 N 2
(U − c)ρ̂ +
ψ̂ = 0.
g
(U − c)ψ̂z − Uz ψ̂ = −
(12.55)
(12.56)
(12.57)
We want to obtain a single equation in ψ̂. The pressure can be eliminated by
taking the z-derivative of equation (12.55) and subtracting equation (12.56). The
density can be eliminated by equation (12.57). This gives
2
d
N2
2
(U − c)
ψ̂
−
U
ψ̂
+
−
k
ψ̂ = 0.
(12.58)
zz
dz2
U −c
This is the Taylor–Goldstein equation, which governs the behavior of perturbations
in a stratified parallel flow. Note that the complex conjugate of the equation is also
a valid equation because we can take the imaginary part of the equation, change the
sign, and add to the real part of the equation. Now because the Taylor–Goldstein
equation does not involve any i, a complex conjugate of the equation shows that if ψ̂
is an eigenfunction with eigenvalue c for some k, then ψ̂ ∗ is a possible eigenfunction
with eigenvalue c∗ for the same k. Therefore, to each eigenvalue with a positive ci there
503
7. Instability of Continuously Stratified Parallel Flows
is a corresponding eigenvalue with a negative ci . In other words, to each growing mode
there is a corresponding decaying mode. A nonzero ci therefore ensures instability.
The boundary conditions are that w = 0 on rigid boundaries at z = 0, d. This
requires ψx = ik ψ̂ exp(ikx − ikct) = 0 at the walls, which is possible only if
ψ̂(0) = ψ̂(d) = 0.
(12.59)
Richardson Number Criterion
A necessary condition for linear instability of inviscid stratified parallel flows can be
derived by defining a new variable φ by
φ≡√
ψ̂
U −c
or
ψ̂ = (U − c)1/2 φ.
Then we obtain the derivatives
φUz
,
2(U − c)1/2
φUz2
Uz φz + (1/2)φUzz
1
= (U − c)1/2 φzz +
−
.
(U − c)1/2
4 (U − c)3/2
ψ̂z = (U − c)1/2 φz +
ψ̂zz
The Taylor–Goldstein equation then becomes, after some rearrangement,
(1/4)Uz2 − N 2
d
1
2
{(U − c)φz } − k (U − c) + Uzz +
φ = 0.
dz
2
U −c
(12.60)
Now multiply equation (12.60) by φ ∗ (the complex conjugate of φ), integrate from
z = 0 to z = d, and use the boundary conditions φ(0) = φ(d) = 0. The first term
gives
d
d
∗
{(U − c)φz }φ dz =
{(U − c)φz φ ∗ } − (U − c)φz φz∗ dz
dz
dz
= − (U − c)|φz |2 dz,
where we have used φ = 0 at the boundaries. Integrals of the other terms in equation (12.60) are also simple to manipulate. We finally obtain
N 2 − (1/4)Uz2 2
|φ| dz = (U − c){|φz |2 + k 2 |φ|2 } dz
U −c
1
Uzz |φ|2 dz.
(12.61)
+
2
The last term in the preceding is real. The imaginary part of the first term can be found
by noting that
1
U − c∗
U − cr + ici
=
=
.
2
U −c
|U − c|
|U − c|2
504
Instability
Then the imaginary part of equation (12.61) gives
N 2 − (1/4)Uz2 2
|φ| dz = −ci {|φz |2 + k 2 |φ|2 } dz.
ci
|U − c|2
The integral on the right-hand side is positive. If the flow is such that N 2 > Uz2 /4
everywhere, then the preceding equation states that ci times a positive quantity equals
ci times a negative quantity; this is impossible and requires that ci = 0 for such a
case. Defining the gradient Richardson number
Ri(z) ≡
N2
,
Uz2
(12.62)
we can say that linear stability is guaranteed if the inequality
Ri >
1
4
(stable),
(12.63)
is satisfied everywhere in the flow.
Note that the criterion does not state that the flow is necessarily unstable if
Ri < 41 somewhere, or even everywhere, in the flow. Thus Ri < 41 is a necessary
but not sufficient condition for instability. For example, in a jetlike velocity profile
u ∝ sech2 z and an exponential density profile, the flow does not become unstable until
the Richardson number falls below 0.214. A critical Richardson number lower than 41
is also found in the presence of boundaries, which stabilize the flow. In fact, there is no
unique critical Richardson number that applies to all distributions of U (z) and N (z).
However, several calculations show that in many shear layers (having linear, tanh,
or error function profiles for velocity and density) the flow does become unstable to
disturbances of certain wavelengths if the minimum value of Ri in the flow (which is
generally at the center of the shear layer) is less than 41 . The “most unstable” wave,
defined as the first to become unstable as Ri is reduced below 41 , is found to have a
wavelength λ ≃ 7h, where h is the thickness of the shear layer. Laboratory (Scotti
and Corcos, 1972) as well as geophysical observations (Eriksen, 1978) show that the
requirement
Rimin < 41 ,
is a useful guide for the prediction of instability of a stratified shear layer.
Howard’s Semicircle Theorem
A useful result concerning the behavior of the complex phase speed c in an inviscid
parallel shear flow, valid both with and without stratification, was derived by Howard
(1961). To derive this, first substitute
F ≡
ψ̂
,
U −c
505
7. Instability of Continuously Stratified Parallel Flows
in the Taylor–Goldstein equation (12.58). With the derivatives
ψ̂z = (U − c)Fz + Uz F,
ψ̂zz = (U − c)Fzz + 2Uz Fz + Uzz F,
Equation (12.58) gives
(U − c)[(U − c)Fzz + 2Uz Fz − k 2 (U − c)F ] + N 2 F = 0,
where terms involving Uzz have canceled out. This can be rearranged in the form
d
[(U − c)2 Fz ] − k 2 (U − c)2 F + N 2 F = 0.
dz
Multiplying by F ∗ , integrating (by parts if necessary) over the depth of flow, and
using the boundary conditions, we obtain
2
∗
2
2
2
(U − c) |F | dz + N 2 |F |2 dz = 0,
− (U − c) Fz Fz dz − k
which can be written as
(U − c)2 Q dz =
N 2 |F |2 dz,
where
Q ≡ |Fz |2 + k 2 |F |2 ,
is positive. Equating real and imaginary parts, we obtain
2
2
[(U − cr ) − ci ]Q dz = N 2 |F |2 dz,
ci (U − cr )Q dz = 0.
(12.64)
(12.65)
For instability ci = 0, for which equation (12.65) shows that (U − cr ) must change
sign somewhere in the flow, that is,
Umin < cr < Umax ,
(12.66)
which states that cr lies in the range of U . Recall that we have assumed solutions of
the form
eik(x−ct) = eik(x−cr t) ekci t ,
which means that cr is the phase velocity in the positive x direction, and kci is the
growth rate. Equation (12.66) shows that cr is positive if U is everywhere positive,
506
Instability
and is negative if U is everywhere negative. In these cases we can say that unstable
waves propagate in the direction of the background flow.
Limits on the maximum growth rate can also be predicted. Equation (12.64) gives
[U 2 + cr2 − 2U cr − ci2 ]Q dz > 0,
which, on using equation (12.65), becomes
(U 2 − cr2 − ci2 )Q dz > 0.
(12.67)
Now because (Umin − U ) < 0 and (Umax − U ) > 0, it is always true that
(Umin − U )(Umax − U )Q dz 0,
which can be recast as
[Umax Umin + U 2 − U (Umax + Umin )]Q dz 0.
Using equation (12.67), this gives
[Umax Umin + cr2 + ci2 − U (Umax + Umin )]Q dz 0.
On using equation (12.65), this becomes
[Umax Umin + cr2 + ci2 − cr (Umax + Umin )]Q dz 0.
Because the quantity within [ ] is independent of z, and Q dz > 0, we must have
[ ] 0. With some rearrangement, this condition can be written as
2
1
2
1
cr − (Umax + Umin ) + ci2 (Umax − Umin ) .
2
2
This shows that the complex wave velocity c of any unstable mode of a disturbance
in parallel flows of an inviscid fluid must lie inside the semicircle in the upper half of
the c-plane, which has the range of U as the diameter (Figure 12.20). This is called
the Howard semicircle theorem. It states that the maximum growth rate is limited by
kci <
k
(Umax − Umin ).
2
The theorem is very useful in searching for eigenvalues c(k) in numerical solution of
instability problems.
507
8. Squire’s Theorem and Orr–Sommerfeld Equation
Figure 12.20 The Howard semicircle theorem. In several inviscid parallel flows the complex eigenvalue
c must lie within the semicircle shown.
8. Squire’s Theorem and Orr–Sommerfeld Equation
In our studies of the Bénard and Taylor problems, we encountered two flows in which
viscosity has a stabilizing effect. Curiously, viscous effects can also be destabilizing,
as indicated by several calculations of wall-bounded parallel flows. In this section we
shall derive the equation governing the stability of parallel flows of a homogeneous
viscous fluid. Let the primary flow be directed along the x direction and vary in the
y direction so that U = [U (y), 0, 0]. We decompose the total flow as the sum of the
basic flow plus the perturbation:
ũ = [U + u, v, w],
p̃ = P + p.
Both the background and the perturbed flows satisfy the Navier–Stokes equations.
The perturbed flow satisfies the x-momentum equation
∂
∂
∂u
+ (U + u) (U + u) + v (U + u)
∂t
∂x
∂y
∂
1 2
= − (P + p) +
∇ (U + u),
∂x
Re
(12.68)
where the variables have been nondimensionalized by a characteristic length scale L
(say, the width of flow), and a characteristic velocity U0 (say, the maximum velocity
of the basic flow); time is scaled by L/U0 and the pressure is scaled by ρU02 . The
Reynolds number is defined as Re = U0 L/ν.
508
Instability
The background flow satisfies
0=−
∂P
1 2
+
∇ U.
∂x
Re
Subtracting from equation (12.68) and neglecting terms nonlinear in the perturbations,
we obtain the x-momentum equation for the perturbations:
∂u
∂U
∂p
1 2
∂u
+U
+v
=−
+
∇ u.
∂t
∂x
∂y
∂x
Re
(12.69)
Similarly the y-momentum, z-momentum, and continuity equations for the
perturbations are
∂v
∂v
∂p
1 2
+U
=−
+
∇ v,
∂t
∂x
∂y
Re
∂w
∂w
∂p
1 2
+U
=−
+
∇ w,
∂t
∂x
∂z
Re
∂w
∂u ∂v
+
+
= 0.
∂x
∂y
∂z
(12.70)
The coefficients in the perturbation equations (12.69) and (12.70) depend only on y,
so that the equations admit solutions exponential in x, z, and t. Accordingly, we
assume normal modes of the form
[u, p] = [û(y), p̂(y)] ei(kx+mz−kct) .
(12.71)
As the flow is unbounded in x and z, the wavenumber components k and m must be
real. The wave speed c = cr + ici may be complex. Without loss of generality, we
can consider only positive values for k and m; the sense of propagation is then left
open by keeping the sign of cr unspecified. The normal modes represent
√ waves that
travel obliquely to the basic flow with a wavenumber of magnitude k 2 + m2 and
have an amplitude that varies in time as exp(kci t). Solutions are therefore stable if
ci < 0 and unstable if ci > 0.
On substitution of the normal modes, the perturbation equations (12.69) and
(12.70) become
1
[ûyy − (k 2 + m2 )û],
Re
1
[v̂yy − (k 2 + m2 )v̂],
ik(U − c)v̂ = −p̂y +
Re
1
ik(U − c)ŵ = −imp̂ +
[ŵyy − (k 2 + m2 )ŵ],
Re
ik û + v̂y + imŵ = 0,
ik(U − c)û + v̂Uy = −ik p̂ +
(12.72)
where subscripts denote derivatives with respect to y. These are the normal mode
equations for three-dimensional disturbances. Before proceeding further, we shall
first show that only two-dimensional disturbances need to be considered.
509
8. Squire’s Theorem and Orr–Sommerfeld Equation
Squire’s Theorem
A very useful simplification of the normal mode equations was achieved by Squire in
1933, showing that to each unstable three-dimensional disturbance there corresponds
a more unstable two-dimensional one. To prove this theorem, consider the Squire
transformation
k̄ = (k 2 + m2 )1/2 ,
k̄ ū = k û + mŵ,
p̄
p̂
= ,
k
k̄
c̄ = c,
v̄ = v̂,
(12.73)
k̄ Re = k Re.
In substituting these transformations into equation (12.72), the first and third of equation (12.72) are added; the rest are simply transformed. The result is
1
[ūyy − k̄ 2 ū],
Re
1
[v̄yy − k̄ 2 v̄],
i k̄(U − c)v̄ = −p̄y +
Re
i k̄ ū + v̄y = 0.
i k̄(U − c)ū + v̄Uy = −i k̄ p̄ +
These equations are exactly the same as equation (12.72), but with m = ŵ = 0. Thus,
to each three-dimensional problem corresponds an equivalent two-dimensional one.
Moreover, Squire’s transformation (12.73) shows that the equivalent two-dimensional
problem is associated with a lower Reynolds number as k̄ > k. It follows that the
critical Reynolds number at which the instability starts is lower for two-dimensional
disturbances. Therefore, we only need to consider a two-dimensional disturbance if
we want to determine the minimum Reynolds number for the onset of instability.
The three-dimensional disturbance (12.71) is a wave propagating obliquely to the
basic flow. If we orient the coordinate system with the new x-axis in this direction, the
equations of motion are such that only the component of basic flow in this direction
affects the disturbance. Thus, the effective Reynolds number is reduced.
An argument without using the Reynolds number is now given because Squire’s
theorem also holds for several other problems that do not involve the Reynolds number.
Equation (12.73) shows that the growth rate for a two-dimensional disturbance is
exp(k̄ c̄i t), whereas equation (12.71) shows that the growth rate of a three-dimensional
disturbance is exp(kci t). The two-dimensional growth rate is therefore larger because
Squire’s transformation requires k̄ > k and c̄ = c. We can therefore say that the
two-dimensional disturbances are more unstable.
Orr–Sommerfeld Equation
Because of Squire’s theorem, we only need to consider the set (12.72) with
m = ŵ = 0. The two-dimensionality allows the definition of a streamfunction
ψ(x, y, t) for the perturbation field by
u=
∂ψ
,
∂y
v=−
∂ψ
.
∂x
510
Instability
We assume normal modes of the form
[u, v, ψ] = [û, v̂, φ] eik(x−ct) .
(To be consistent, we should denote the complex amplitude of ψ by ψ̂; we are using
φ instead to follow the standard notation for this variable in the literature.) Then we
must have
û = φy ,
v̂ = −ikφ.
A single equation in terms of φ can now be found by eliminating the pressure
from the set (12.72). This gives
(U − c)(φyy − k 2 φ) − Uyy φ =
1
[φyyyy − 2k 2 φyy + k 4 φ],
ik Re
(12.74)
where subscripts denote derivatives with respect to y. It is a fourth-order ordinary
differential equation. The boundary conditions at the walls are the no-slip conditions
u = v = 0, which require
φ = φy = 0
at y = y1 and y2 .
(12.75)
Equation (12.74) is the well-known Orr–Sommerfeld equation, which governs
the stability of nearly parallel viscous flows such as those in a straight channel or in
a boundary layer. It is essentially a vorticity equation because the pressure has been
eliminated. Solutions of the Orr–Sommerfeld equations are difficult to obtain, and
only the results of some simple flows will be discussed in the later sections. However,
we shall first discuss certain results obtained by ignoring the viscous term in this
equation.
9. Inviscid Stability of Parallel Flows
Useful insights into the viscous stability of parallel flows can be obtained by first
assuming that the disturbances obey inviscid dynamics. The governing equation can
be found by letting Re → ∞ in the Orr–Sommerfeld equation, giving
(U − c)[φyy − k 2 φ] − Uyy φ = 0,
(12.76)
which is called the Rayleigh equation. If the flow is bounded by walls at y1 and y2
where v = 0, then the boundary conditions are
φ=0
at y = y1 and y2 .
(12.77)
The set (12.76) and (12.77) defines an eigenvalue problem, with c(k) as the eigenvalue
and φ as the eigenfunction. As the equations do not involve i, taking the complex
conjugate shows that if φ is an eigenfunction with eigenvalue c for some k, then
φ ∗ is also an eigenfunction with eigenvalue c∗ for the same k. Therefore, to each
eigenvalue with a positive ci there is a corresponding eigenvalue with a negative ci .
511
9. Inviscid Stability of Parallel Flows
In other words, to each growing mode there is a corresponding decaying mode. Stable
solutions therefore can have only a real c. Note that this is true of inviscid flows only.
The viscous term in the full Orr–Sommerfeld equation (12.74) involves an i, and the
foregoing conclusion is no longer valid.
We shall now show that certain velocity distributions U ( y) are potentially unstable according to the inviscid Rayleigh equation (12.76). In this discussion it should
be noted that we are only assuming that the disturbances obey inviscid dynamics; the
background flow U ( y) may be chosen to be chosen to be any profile, for example,
that of viscous flows such as Poiseuille flow or Blasius flow.
Rayleigh’s Inflection Point Criterion
Rayleigh proved that a necessary (but not sufficient) criterion for instability of an
inviscid parallel flow is that the basic velocity profile U (y) has a point of inflection.
To prove the theorem, rewrite the Rayleigh equation (12.76) in the form
φyy − k 2 φ −
Uyy
φ = 0,
U −c
and consider the unstable mode for which ci > 0, and therefore U − c = 0. Multiply
this equation by φ ∗ , integrate from y1 to y2 , by parts if necessary, and apply the
boundary condition φ = 0 at the boundaries. The first term transforms as follows:
y2
∗
∗
∗
φ φyy dy = [φ φy ]y1 − φy φy dy = − |φy |2 dy,
where the limits on the integrals have not been explicitly written. The Rayleigh
equation then gives
Uyy
2
2
2
(12.78)
|φ|2 dy = 0.
[|φy | + k |φ| ] dy +
U −c
The first term is real. The imaginary part of the second term can be found by multiplying the numerator and denominator by (U − c∗ ). The imaginary part of equation
(12.78) then gives
Uyy |φ|2
dy = 0.
(12.79)
ci
|U − c|2
For the unstable case, for which ci = 0, equation (12.79) can be satisfied only if
Uyy changes sign at least once in the open interval y1 < y < y2 . In other words, for
instability the background velocity distribution must have at least one point of inflection (where Uyy = 0) within the flow. Clearly, the existence of a point of inflection
does not guarantee a nonzero ci . The inflection point is therefore a necessary but not
sufficient condition for inviscid instability.
Fjortoft’s Theorem
Some seventy years after Rayleigh’s discovery, the Swedish meteorologist Fjortoft in
1950 discovered a stronger necessary condition for the instability of inviscid parallel
512
Instability
flows. He showed that a necessary condition for instability of inviscid parallel flows
is that Uyy (U − UI ) < 0 somewhere in the flow, where UI is the value of U at the
point of inflection. To prove the theorem, take the real part of equation (12.78):
Uyy (U − cr ) 2
|φ| dy = − [|φy |2 + k 2 |φ|2 ] dy < 0.
(12.80)
|U − c|2
Suppose that the flow is unstable, so that ci = 0, and a point of inflection does exist
according to the Rayleigh criterion. Then it follows from equation (12.79) that
(cr − UI )
Uyy |φ|2
dy = 0.
|U − c|2
(12.81)
Adding equations (12.80) and (12.81), we obtain
Uyy (U − UI ) 2
|φ| dy < 0,
|U − c|2
so that Uyy (U − UI ) must be negative somewhere in the flow.
Some common velocity profiles are shown in Figure 12.21. Only the two flows
shown in the bottom row can possibly be unstable, for only they satisfy Fjortoft’s
theorem. Flows (a), (b), and (c) do not have any inflection point: flow (d) does satisfy
Rayleigh’s condition but not Fjortoft’s because Uyy (U − UI ) is positive. Note that
an alternate way of stating Fjortoft’s theorem is that the magnitude of vorticity of the
basic flow must have a maximum within the region of flow, not at the boundary. In
flow (d), the maximum magnitude of vorticity occurs at the walls.
The criteria of Rayleigh and Fjortoft essentially point to the importance of having
a point of inflection in the velocity profile. They show that flows in jets, wakes, shear
layers, and boundary layers with adverse pressure gradients, all of which have a point
of inflection and satisfy Fjortoft’s theorem, are potentially unstable. On the other
hand, plane Couette flow, Poiseuille flow, and a boundary layer flow with zero or
favorable pressure gradient have no point of inflection in the velocity profile, and are
stable in the inviscid limit.
However, neither of the two conditions is sufficient for instability. An example
is the sinusoidal profile U = sin y, with boundaries at y = ±b. It has been shown
that the flow is stable if the width is restricted to 2b < π , although it has an inflection
point at y = 0.
Critical Layers
Inviscid parallel flows satisfy Howard’s semicircle theorem, which was proved in
Section 7 for the more general case of a stratified shear flow. The theorem states that
the phase speed cr has a value that lies between the minimum and the maximum
values of U (y) in the flow field. Now growing and decaying modes are characterized
by a nonzero ci , whereas neutral modes can have only a real c = cr . It follows that
neutral modes must have U = c somewhere in the flow field. The neighborhood y
around yc at which U = c = cr is called a critical layer. The point yc is a critical
9. Inviscid Stability of Parallel Flows
Figure 12.21 Examples of parallel flows. Points of inflection are denoted by I. Only (e) and (f) satisfy
Fjortoft’s criterion of inviscid instability.
point of the inviscid governing equation (12.76), because the highest derivative drops
out at this value of y. The solution of the eigenfunction is discontinuous across this
layer. The full Orr–Sommerfeld equation (12.74) has no such critical layer because
the highest-order derivative does not drop out when U = c. It is apparent that in a
real flow a viscous boundary layer must form at the location where U = c, and the
layer becomes thinner as Re → ∞.
The streamline pattern in the neighborhood of the critical layer where U = c was
given by Kelvin in 1888; our discussion here is adapted from Drazin and Reid (1981).
513
514
Instability
Figure 12.22 The Kelvin cat’s eye pattern near a critical layer, showing streamlines as seen by an observer
moving with the wave.
Consider a flow viewed by an observer moving with the phase velocity c = cr . Then
the basic velocity field seen by this observer is (U − c), so that the streamfunction
due to the basic flow is
= (U − c) dy.
The total streamfunction is obtained by adding the perturbation:
ψ̃ = (U − c) dy + Aφ( y) eikx ,
(12.82)
where A is an arbitrary constant, and we have omitted the time factor on the second
term because we are considering only neutral disturbances. Near the critical layer
y = yc , a Taylor series expansion shows that equation (12.82) is approximately
ψ̃ =
1
U yc ( y − yc )2 + Aφ(yc ) cos kx,
2
where Uyc is the value of Uy at yc ; we have taken the real part of the right-hand side,
and taken φ(yc ) to be real. The streamline pattern corresponding to the preceding
equation is sketched in Figure 12.22, showing the so-called Kelvin cat’s eye pattern.
10. Some Results of Parallel Viscous Flows
Our intuitive expectation is that viscous effects are stabilizing. The thermal and centrifugal convections discussed earlier in this chapter have confirmed this intuitive
expectation. However, the conclusion that the effect of viscosity is stabilizing is not
always true. Consider the Poiseuille flow and the Blasius boundary layer profiles in
Figure 12.21, which do not have any inflection point and are therefore inviscidly
stable. These flows are known to undergo transition to turbulence at some Reynolds
number, which suggests that inclusion of viscous effects may in fact be destabilizing in these flows. Fluid viscosity may thus have a dual effect in the sense that it
can be stabilizing as well as destabilizing. This is indeed true as shown by stability
calculations of parallel viscous flows.
515
10. Some Results of Parallel Viscous Flows
The analytical solution of the Orr–Sommerfeld equation is notoriously
complicated and will not be presented here. The viscous term in (12.74) contains
the highest-order derivative, and therefore the eigenfunction may contain regions of
rapid variation in which the viscous effects become important. Sophisticated asymptotic techniques are therefore needed to treat these boundary layers. Alternatively,
solutions can be obtained numerically. For our purposes, we shall discuss only certain features of these calculations. Additional information can be found in Drazin and
Reid (1981), and in the review article by Bayly, Orszag, and Herbert (1988).
Mixing Layer
Consider a mixing layer with the velocity profile
U = U0 tanh
y
.
L
A stability diagram for solution of the Orr–Sommerfeld equation for this velocity
distribution is sketched in Figure 12.23. It is seen that at all Reynolds numbers the
flow is unstable to waves having low wavenumbers in the range 0 < k < ku , where
the upper limit ku depends on the Reynolds number Re = U0 L/ν. For high values of
Re, the range of unstable wavenumbers increases to 0 < k < 1/L, which corresponds
to a wavelength range of ∞ > λ > 2π L. It is therefore essentially a long wavelength
instability.
Figure 12.23 implies that the critical Reynolds number in a mixing layer is zero. In
fact, viscous calculations for all flows with “inflectional profiles” show a small critical
Reynolds number; for example, for a jet of the form u = U sech2 ( y/L), it is Recr = 4.
These wall-free shear flows therefore become unstable very quickly, and the inviscid
criterion that these flows are always unstable is a fairly good description. The reason
the inviscid analysis works well in describing the stability characteristics of free shear
Figure 12.23 Marginal stability curve for a shear layer u = U0 tanh( y/L).
516
Instability
flows can be explained as follows. For flows with inflection points the eigenfunction
of the inviscid solution is smooth. On this zero-order approximation, the viscous
term acts as a regular perturbation, and the resulting correction to the eigenfunction
and eigenvalues can be computed as a perturbation expansion in powers of the small
parameter 1/Re. This is true even though the viscous term in the Orr–Sommerfeld
equation contains the highest-order derivative.
The instability in flows with inflection points is observed to form rolled-up blobs
of vorticity, much like in the calculations of Figure 12.18 or in the photograph of
Figure 12.16. This behavior is robust and insensitive to the detailed experimental
conditions. They are therefore easily observed. In contrast, the unstable waves in a
wall-bounded shear flow are extremely difficult to observe, as discussed in the next
section.
Plane Poiseuille Flow
The flow in a channel with parabolic velocity distribution has no point of inflection and
is inviscidly stable. However, linear viscous calculations show that the flow becomes
unstable at a critical Reynolds number of 5780. Nonlinear calculations, which consider the distortion of the basic profile by the finite amplitude of the perturbations,
give a critical number of 2510, which agrees better with the observed transition.
In any case, the interesting point is that viscosity is destabilizing for this flow. The
solution of the Orr–Sommerfeld equation for the Poiseuille flow and other parallel
flows with rigid boundaries, which do not have an inflection point, is complicated.
In contrast to flows with inflection points, the viscosity here acts as a singular perturbation, and the eigenfunction has viscous boundary layers on the channel walls
and around critical layers where U = cr . The waves that cause instability in these
flows are called Tollmien–Schlichting waves, and their experimental detection is discussed in the next section. In his text, C. S. Yih gives a thorough discussion of the
solution of the Orr-Sommerfeld equation using asymptotic expansions in the limit
sequence Re → ∞, then k → 0 (but kRe ≫ 1). He follows closely the analysis
of W. Heisenberg (1924). Yih presents C. C. Lin’s improvements on Heisenberg’s
analysis with S. F. Shen’s calculations of the stability curves.
Plane Couette Flow
This is the flow confined between two parallel plates; it is driven by the motion of
one of the plates parallel to itself. The basic velocity profile is linear, with U = Ŵy.
Contrary to the experimentally observed fact that the flow does become turbulent
at high values of Re, all linear analyses have shown that the flow is stable to small
disturbances. It is now believed that the instability is caused by disturbances of finite
magnitude.
Pipe Flow
The absence of an inflection point in the velocity profile signifies that the flow is
inviscidly stable. All linear stability calculations of the viscous problem have also
shown that the flow is stable to small disturbances. In contrast, most experiments
10. Some Results of Parallel Viscous Flows
show that the transition to turbulence takes place at a Reynolds number of about
Re = Umax d/ν ∼ 3000. However, careful experiments, some of them performed
by Reynolds in his classic investigation of the onset of turbulence, have been able to
maintain laminar flow until Re = 50,000. Beyond this the observed flow is invariably turbulent. The observed transition has been attributed to one of the following
effects: (1) It could be a finite amplitude effect; (2) the turbulence may be initiated at
the entrance of the tube by boundary layer instability (Figure 9.2); and (3) the instability could be caused by a slow rotation of the inlet flow which, when added to the
Poiseuille distribution, has been shown to result in instability. This is still under investigation. New insights into the instability and transition of pipe flow were described by
Eckhardt et al. (2007) by analysis via dynamical systems theory and comparison with
recent very carefully crafted experiments by them and others. They characterized the
turbulent state as a “chaotic saddle in state space.” The boundary between laminar and
turbulent flow was found to be exquisitely sensitive to initial conditions. Because pipe
flow is linearly stable, finite amplitude disturbances are necessary to cause transition,
but as Reynolds number increases, the amplitude of the critical disturbance diminishes. The boundary between laminar and turbulent states appears to be characterized
by a pair of vortices closer to the walls which give the strongest amplification of the
initial disturbance.
Boundary Layers with Pressure Gradients
Recall from Chapter 10, Section 7 that a pressure falling in the direction of flow is said
to have a “favorable” gradient, and a pressure rising in the direction of flow is said to
have an “adverse” gradient. It was shown there that boundary layers with an adverse
pressure gradient have a point of inflection in the velocity profile. This has a dramatic
effect on the stability characteristics. A schematic plot of the marginal stability curve
for a boundary layer with favorable and adverse gradients of pressure is shown in
Figure 12.24. The ordinate in the plot represents the longitudinal wavenumber, and
the abscissa represents the Reynolds number based on the free-stream velocity and
the displacement thickness δ ∗ of the boundary layer. The marginal stability curve
divides stable and unstable regions, with the region within the “loop” representing
instability. Because the boundary layer thickness grows along the direction of flow,
Reδ increases with x, and points at various downstream distances are represented by
larger values of Reδ .
The following features can be noted in the figure. The flow is stable for low
Reynolds numbers, although it is unstable at higher Reynolds numbers. The effect of
increasing viscosity is therefore stabilizing in this range. For boundary layers with a
zero pressure gradient (Blasius flow) or a favorable pressure gradient, the instability
loop shrinks to zero as Reδ → ∞. This is consistent with the fact that these flows do
not have a point of inflection in the velocity profile and are therefore inviscidly stable.
In contrast, for boundary layers with an adverse pressure gradient, the instability
loop does not shrink to zero; the upper branch of the marginal stability curve now
becomes flat with a limiting value of k∞ as Reδ → ∞. The flow is then unstable
to disturbances of wavelengths in the range 0 < k < k∞ . This is consistent with the
existence of a point of inflection in the velocity profile, and the results of the mixing
517
518
Instability
Figure 12.24 Sketch of marginal stability curves for a boundary layer with favorable and adverse pressure
gradients.
TABLE 12.1
Flow
Jet
Shear layer
Blasius
Plane Poiseuille
Pipe flow
Plane Couette
Linear Stability Results of Common Viscous Parallel Flows
U ( y)/U0
Recr
Remarks
sech2 ( y/L)
4
0
520
5780
∞
∞
Always unstable
Re based on δ ∗
L = half-width
Always stable
Always stable
tanh ( y/L)
1 − ( y/L)2
1 − (r/R)2
y/L
layer calculation (Figure 12.23). Note also that the critical Reynolds number is lower
for flows with adverse pressure gradients.
Table 12.1 summarizes the results of the linear stability analyses of some common
parallel viscous flows.
The first two flows in the table have points of inflection in the velocity profile
and are inviscidly unstable; the viscous solution shows either a zero or a small critical
Reynolds number. The remaining flows are stable in the inviscid limit. Of these, the
Blasius boundary layer and the plane Poiseuille flow are unstable in the presence of
viscosity, but have high critical Reynolds numbers.
In Section 10.17 we discussed the decay of a laminar shear layer. Mass conservation requires that a transverse velocity be generated so the flow cannot be parallel.
Although the idealized tanh profile for a shear layer, assuming straight and parallel
streamlines is immediately unstable, recent work by Bhattacharya et al. (2006), which
allowed for the basic flow to be two-dimensional, has yielded a finite critical Reynolds
number, modifying somewhat Table 12.1 (above).
519
10. Some Results of Parallel Viscous Flows
How can Viscosity Destabilize a Flow?
Let us examine how viscous effects can be destabilizing. For this we derive an integral
form of the kinetic energy equation in a viscous flow. The Navier–Stokes equation
for the disturbed flow is
∂
∂
(Ui + ui ) + (Uj + uj )
(Ui + ui )
∂t
∂xj
=−
∂2
1 ∂
(P + p) + ν
(Ui + ui ).
ρ ∂xi
∂xj ∂xj
Subtracting the equation of motion for the basic state, we obtain
∂ui
∂ui
∂ui
∂Ui
1 ∂p
∂ 2 ui
+ uj
+ Uj
+ uj
=−
+ν
,
∂t
∂xj
∂xj
∂xj
ρ ∂xi
∂xj2
which is the equation of motion of the disturbance. The integrated mechanical energy
equation for the disturbance motion is obtained by multiplying this equation by ui
and integrating over the region of flow. The control volume is chosen to coincide with
the walls where no-slip conditions are satisfied, and the length of the control volume
in the direction of periodicity is chosen to be an integral number of wavelengths
(Figure 12.25). The various terms of the energy equation then simplify as follows:
∂ui
dV
∂t
∂ui
ui uj
dV
∂xj
∂ui
ui U j
dV
∂xj
∂p
ui
dV
∂xi
∂ 2 ui
ui 2 dV
∂xj
ui
d u2i
dV ,
dt
2
1
∂
1
2
=
(u uj ) dV =
u2i uj dAj = 0,
2 ∂xj i
2
1
∂
1
=
(u2i Uj ) dV =
u2i Uj dAj = 0,
2 ∂xj
2
∂
=
(pui ) dV = pui dAi = 0,
∂xi
∂ui ∂ui
∂ui
∂
ui
dV −
dV
=
∂xj
∂xj
∂xj ∂xj
∂ui ∂ui
dV .
=−
∂xj ∂xj
=
Here, dA is an element of surface area of the control volume, and d V is an
element of volume. In these the continuity equation ∂ui /∂xi = 0, Gauss’ theorem,
and the no-slip and periodic boundary conditions have been used to show that the
divergence terms drop out in an integrated energy balance. We finally obtain
d
dt
1 2
u dV = −
2 i
ui uj
∂Ui
dV − φ,
∂xj
520
Instability
Figure 12.25 A control volume with zero net flux across boundaries.
where φ = ν (∂ui /∂xi )2 dV is the viscous dissipation. For two-dimensional disturbances in a shear flow defined by U = [U ( y), 0, 0], the energy equation
becomes
d
dt
1 2
(u + v 2 ) dV = −
2
uv
∂U
dV − φ.
∂y
(12.83)
This equation has a simple interpretation. The first term is the rate of change of kinetic
energy of the disturbance, and the second term is the rate of production of disturbance
energy by the interaction of the “Reynolds stress” uv and the mean shear ∂U/∂y. The
concept of Reynolds stress will be explained in the following chapter. The point to
note here is that the value of the product uv averaged over a period is zero if the
velocity components u and v are out of phase of 90◦ ; for example, the mean value of
uv is zero if u = sin t and v = cos t.
In inviscid parallel flows without a point of inflection in the velocity profile, the
u and v components are such that the disturbance field cannot extract energy from
the basic shear flow, thus resulting in stability. The presence of viscosity, however,
changes the phase relationship between u and v, which causes Reynolds stresses such
that the mean value of −uv(∂U/∂y) over the flow field is positive and larger than the
viscous dissipation. This is how viscous effects can cause instability.
11. Experimental Verification of Boundary Layer Instability
In this section we shall present the results of stability calculations of the Blasius
boundary layer profile and compare them with experiments. Because of the nearly
parallel nature of the Blasius flow, most stability calculations are based on an analysis
of the Orr–Sommerfeld equation, which assumes a parallel flow. The first calculations
were performed by Tollmien in 1929 and Schlichting in 1933. Instead of assuming
11. Experimental Verification of Boundary Layer Instability
exactly the Blasius profile (which can be specified only numerically), they used the
profile
0 y/δ 0.1724,
1.7( y/δ)
U
2
= 1 − 1.03 [1 − ( y/δ) ]
0.1724 y/δ 1,
U∞
1
y/δ 1,
which, like the Blasius profile, has a zero curvature at the wall. The calculations
of Tollmien and Schlichting showed that unstable waves appear when the Reynolds
number is high enough; the unstable waves in a viscous boundary layer are called
Tollmien–Schlichting waves. Until 1947 these waves remained undetected, and the
experimentalists of the period believed that the transition in a real boundary layer was
probably a finite amplitude effect. The speculation was that large disturbances cause
locally adverse pressure gradients, which resulted in a local separation and consequent
transition. The theoretical view, in contrast, was that small disturbances of the right
frequency or wavelength can amplify if the Reynolds number is large enough.
Verification of the theory was finally provided by some clever experiments conducted by Schubauer and Skramstad in 1947. The experiments were conducted in
a “low turbulence” wind tunnel, specially designed such that the intensity of fluctuations of the free stream was small. The experimental technique used was novel.
Instead of depending on natural disturbances, they introduced periodic disturbances
of known frequency by means of a vibrating metallic ribbon stretched across the flow
close to the wall. The ribbon was vibrated by passing an alternating current through it
in the field of a magnet. The subsequent development of the disturbance was followed
downstream by hot wire anemometers. Such techniques have now become standard.
The experimental data are shown in Figure 12.26, which also shows the
calculations of Schlichting and the more accurate calculations of Shen. Instead of
the wavenumber, the ordinate represents the frequency of the wave, which is easier
to measure. It is apparent that the agreement between Shen’s calculations and the
experimental data is very good.
The detection of the Tollmien–Schlichting waves is regarded as a major accomplishment of the linear stability theory. The ideal conditions for their existence require
two dimensionality and consequently a negligible intensity of fluctuations of the free
stream. These waves have been found to be very sensitive to small deviations from
the ideal conditions. That is why they can be observed only under very carefully
controlled experimental conditions and require artificial excitation. People who care
about historical fairness have suggested that the waves should only be referred to as
TS waves, to honor Tollmien, Schlichting, Schubauer, and Skramstad. The TS waves
have also been observed in natural flow (Bayly et al., 1988).
Nayfeh and Saric (1975) treated Falkner-Skan flows in a study of nonparallel
stability and found that generally there is a decrease in the critical Reynolds number.
The decrease is least for favorable pressure gradients, about 10% for zero pressure gradient, and grows rapidly as the pressure gradient becomes more adverse. Grabowski
(1980) applied linear stability theory to the boundary layer near a stagnation point
on a body of revolution. His stability predictions were found to be close to those of
parallel flow stability theory obtained from solutions of the Orr–Sommerfeld equation.
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522
Instability
Figure 12.26 Marginal stability curve for a Blasius boundary layer. Theoretical solutions of Shen and
Schlichting are compared with experimental data of Schubauer and Skramstad.
Reshotko (2001) provides a review of temporally and spatially transient growth as
a path from subcritical (Tollmien–Schlichting) disturbances to transition. Growth or
decay is studied from the Orr–Sommerfeld and Squire equations. Growth may occur
because eigenfunctions of these equations are not orthogonal as the operators are not
self-adjoint. Results for Poiseuille pipe flow and compressible blunt body flows are
given.
Fransson and Alfredsson (2003) have shown that the asymptotic suction profile
[solved in Exercise 9, Chapter 10] significantly delays transition stimulated by free
stream turbulence or by Tollmien-Schlichting waves. Specifically, the value of Recr =
520 based on δ∗ in Table 12.1 is increased for suction velocity ratio to vo /U∞ =
−.00288 over 54000. The very large stabilizing effect is a result of the change in
the shape of the streamwise velocity from the Blasius profile to an exponential. The
normal suction velocity has a very small effect on stability.
12. Comments on Nonlinear Effects
To this point we have discussed only linear stability theory, which considers infinitesimal perturbations and predicts exponential growth when the relevant parameter
13. Transition
exceeds a critical value. The effect of the perturbations on the basic field is neglected in
the linear theory. An examination of equation (12.83) shows that the perturbation field
must be such that the mean Reynolds stress uv (the “mean” being over a wavelength)
be nonzero for the perturbations to extract energy from the basic shear; similarly, the
heat flux uT ′ must be nonzero in a thermal convection problem. These rectified fluxes
of momentum and heat change the basic velocity and temperature fields. The linear
instability theory neglects these changes of the basic state. A consequence of the constancy of the basic state is that the growth rate of the perturbations is also constant,
leading to an exponential growth. Within a short time of such initial growth the perturbations become so large that the rectified fluxes of momentum and heat significantly
change the basic state, which in turn alters the growth of the perturbations.
A frequent effect of nonlinearity is to change the basic state in such a way as
to stop the growth of the disturbances after they have reached significant amplitude
through the initial exponential growth. (Note, however, that the effect of nonlinearity
can sometimes be destabilizing; for example, the instability in a pipe flow may be
a finite amplitude effect because the flow is stable to infinitesimal disturbances.)
Consider the thermal convection in the annular space between two vertical cylinders
rotating at the same speed. The outer wall of the annulus is heated and the inner wall
is cooled. For small heating rates the flow is steady. For large heating rates a system of
regularly spaced waves develop and progress azimuthally at a uniform speed without
changing their shape. (This is the equilibrated form of baroclinic instability, discussed
in Chapter 14, Section 17.) At still larger heating rates an irregular, aperiodic, or
chaotic flow develops. The chaotic response to constant forcing (in this case the
heating rate) is an interesting nonlinear effect and is discussed further in Section 14.
Meanwhile, a brief description of the transition from laminar to turbulent flow is given
in the next section.
13. Transition
The process by which a laminar flow changes to a turbulent one is called transition.
Instability of a laminar flow does not immediately lead to turbulence, which is a
severely nonlinear and chaotic stage characterized by macroscopic “mixing” of fluid
particles. After the initial breakdown of laminar flow because of amplification of small
disturbances, the flow goes through a complex sequence of changes, finally resulting
in the chaotic state we call turbulence. The process of transition is greatly affected by
such experimental conditions as intensity of fluctuations of the free stream, roughness
of the walls, and shape of the inlet. The sequence of events that lead to turbulence is
also greatly dependent on boundary geometry. For example, the scenario of transition
in a wall-bounded shear flow is different from that in free shear flows such as jets
and wakes.
Early stages of the transition consist of a succession of instabilities on increasingly complex basic flows, an idea first suggested by Landau in 1944. The basic
state of wall-bounded parallel shear flows becomes unstable to two-dimensional TS
waves, which grow and eventually reach equilibrium at some finite amplitude. This
steady state can be considered a new background state, and calculations show that
523
524
Instability
it is generally unstable to three-dimensional waves of short wavelength, which vary
in the “spanwise” direction. (If x is the direction of flow and y is the directed normal to the boundary, then the z-axis is spanwise.) We shall call this the secondary
instability. Interestingly, the secondary instability does not reach equilibrium at finite
amplitude but directly evolves to a fully turbulent flow. Recent calculations of the
secondary instability have been quite successful in reproducing critical Reynolds
numbers for various wall-bounded flows, as well as predicting three-dimensional
structures observed in experiments.
A key experiment on the three-dimensional nature of the transition process in a
boundary layer was performed by Klebanoff, Tidstrom, and Sargent (1962). They conducted a series of controlled experiments by which they introduced three-dimensional
disturbances on a field of TS waves in a boundary layer. The TS waves were as usual
artificially generated by an electromagnetically vibrated ribbon, and the three dimensionality of a particular spanwise wavelength was introduced by placing spacers
(small pieces of transparent tape) at equal intervals underneath the vibrating ribbon
(Figure 12.27). When the amplitude of the TS waves became roughly 1% of the
free-stream velocity, the three-dimensional perturbations grew rapidly and resulted
Figure 12.27 Three-dimensional unstable waves initiated by vibrating ribbon. Measured distributions of
intensity of the u-fluctuation at two distances from the ribbon are shown. P. S. Klebanoff et al., Journal of
Fluid Mechanics 12: 1–34, 1962 and reprinted with the permission of Cambridge University Press.
14. Deterministic Chaos
in a spanwise irregularity of the streamwise velocity displaying peaks and valleys
in the amplitude of u. The three-dimensional disturbances continued to grow until
the boundary layer became fully turbulent. The chaotic flow seems to result from the
nonlinear evolution of the secondary instability, and recent numerical calculations
have accurately reproduced several characteristic features of real flows (see Figures 7
and 8 in Bayly et al., 1988).
It is interesting to compare the chaos observed in turbulent shear flows with
that in controlled low-order dynamical systems such as the Bérnard convection or
Taylor vortex flow. In these low-order flows only a very small number of modes
participate in the dynamics because of the strong constraint of the boundary conditions. All but a few low modes are identically zero, and the chaos develops in
an orderly way. As the constraints are relaxed (we can think of this as increasing the number of allowed Fourier modes), the evolution of chaos becomes less
orderly.
Transition in a free shear layer, such as a jet or a wake, occurs in a different manner.
Because of the inflectional velocity profiles involved, these flows are unstable at a very
low Reynolds numbers, that is, of order 10 compared to about 103 for a wall-bounded
flow. The breakdown of the laminar flow therefore occurs quite readily and close
to the origin of such a flow. Transition in a free shear layer is characterized by the
appearance of a rolled-up row of vortices, whose wavelength corresponds to the one
with the largest growth rate. Frequently, these vortices group themselves in the form
of pairs and result in a dominant wavelength twice that of the original wavelength.
Small-scale turbulence develops within these larger scale vortices, finally leading to
turbulence.
14. Deterministic Chaos
The discussion in the previous section has shown that dissipative nonlinear systems
such as fluid flows reach a random or chaotic state when the parameter measuring
nonlinearity (say, the Reynolds number or the Rayleigh number) is large. The change
to the chaotic stage generally takes place through a sequence of transitions, with the
exact route depending on the system. It has been realized that chaotic behavior not only
occurs in continuous systems having an infinite number of degrees of freedom, but
also in discrete nonlinear systems having only a small number of degrees of freedom,
governed by ordinary nonlinear differential equations. In this context, a chaotic system
is defined as one in which the solution is extremely sensitive to initial conditions. That
is, solutions with arbitrarily close initial conditions evolve into quite different states.
Other symptoms of a chaotic system are that the solutions are aperiodic, and that the
spectrum is broadband instead of being composed of a few discrete lines.
Numerical integrations (to be shown later in this section) have recently demonstrated that nonlinear systems governed by a finite set of deterministic ordinary differential equations allow chaotic solutions in response to a steady forcing. This fact is
interesting because in a dissipative linear system a constant forcing ultimately (after
the decay of the transients) leads to constant response, a periodic forcing leads to
periodic response, and a random forcing leads to random response. In the presence
525
526
Instability
of nonlinearity, however, a constant forcing can lead to a variable response, both
periodic and aperiodic. Consider again the experiment mentioned in Section 12,
namely, the thermal convection in the annular space between two vertical cylinders
rotating at the same speed. The outer wall of the annulus is heated and the inner wall
is cooled. For small heating rates the flow is steady. For large heating rates a system
of regularly spaced waves develops and progresses azimuthally at a uniform speed,
without the waves changing shape. At still larger heating rates an irregular, aperiodic,
or chaotic flow develops. This experiment shows that both periodic and aperiodic flow
can result in a nonlinear system even when the forcing (in this case the heating rate)
is constant. Another example is the periodic oscillation in the flow behind a blunt
body at Re ∼ 40 (associated with the initial appearance of the von Karman vortex
street) and the breakdown of the oscillation into turbulent flow at larger values of the
Reynolds number.
It has been found that transition to chaos in the solution of ordinary nonlinear
differential equations displays a certain universal behavior and proceeds in one of a
few different ways. At the moment it is unclear whether the transition in fluid flows is
closely related to the development of chaos in the solutions of these simple systems;
this is under intense study. In this section we shall discuss some of the elementary
ideas involved, starting with certain definitions. An introduction to the subject of
chaos is given by Bergé, Pomeau, and Vidal (1984); a useful review is given in
Lanford (1982). The subject has far-reaching cosmic consequences in physics and
evolutionary biology, as discussed by Davies (1988).
Phase Space
Very few nonlinear equations have analytical solutions. For nonlinear systems, a
typical procedure is to find a numerical solution and display its properties in a space
whose axes are the dependent variables. Consider the equation governing the motion
of a simple pendulum of length l:
Ẍ +
g
sin X = 0,
l
where X is the angular displacement and Ẍ (= d 2 X/dt 2 ) is the angular acceleration.
(The component of gravity parallel to the trajectory is −g sin X, which is balanced by
the linear acceleration l Ẍ.) The equation is nonlinear because of the sin X term. The
second-order equation can be split into two coupled first-order equations
Ẋ = Y,
g
Ẏ = − sin X.
l
(12.84)
Starting with some initial conditions on X and Y , one can integrate set (12.84). The
behavior of the system can be studied by describing how the variables Y (=Ẋ) and X
vary as a function of time. For the pendulum problem, the space whose axes are Ẋ and
X is called a phase space, and the evolution of the system is described by a trajectory
14. Deterministic Chaos
in this space. The dimension of the phase space is called the degree of freedom of the
system; it equals the number of independent initial conditions necessary to specify
the system. For example, the degree of freedom for the set (12.84) is two.
Attractor
Dissipative systems are characterized by the existence of attractors, which are structures in the phase space toward which neighboring trajectories approach as t → ∞.
An attractor can be a fixed point representing a stable steady flow or a closed
curve (called a limit cycle) representing a stable oscillation (Figure 12.28a, b). The
nature of the attractor depends on the value of the nonlinearity parameter, which
Figure 12.28 Attractors in a phase plane. In (a), point P is an attractor. For a larger value of R, panel
(b) shows that P becomes an unstable fixed point (a “repeller”), and the trajectories are attracted to a limit
cycle. Panel (c) is the bifurcation diagram.
527
528
Instability
will be denoted by R in this section. As R is increased, the fixed point representing a steady solution may change from being an attractor to a repeller with spirally outgoing trajectories, signifying that the steady flow has become unstable to
infinitesimal perturbations. Frequently, the trajectories are then attracted by a limit
cycle, which means that the unstable steady solution gives way to a steady oscillation (Figure 12.28b). For example, the steady flow behind a blunt body becomes
oscillatory as Re is increased, resulting in the periodic von Karman vortex street
(Figure 10.18).
The branching of a solution at a critical value Rcr of the nonlinearity parameter
is called a bifurcation. Thus, we say that the stable steady solution of Figure 12.28a
bifurcates to a stable limit cycle as R increases through Rcr . This can be represented
on the graph of a dependent variable (say, X) vs R (Figure 12.28c). At R = Rcr , the
solution curve branches into two paths; the two values of X on these branches (say,
X1 and X2 ) correspond to the maximum and minimum values of X in Figure 12.28b.
It is seen that the size of the limit cycle grows larger as (R − Rcr ) becomes larger.
Limit cycles, representing oscillatory response with amplitude independent of initial
conditions, are characteristic features of nonlinear systems. Linear stability theory
predicts an exponential growth of the perturbations if R > Rcr , but a nonlinear theory
frequently shows that the perturbations eventually equilibrate to a steady oscillation
whose amplitude increases with (R − Rcr ).
The Lorenz Model of Thermal Convection
Taking the example of thermal convection in a layer heated from below (the Bénard
problem), Lorenz (1963) demonstrated that the development of chaos is associated
with the attractor acquiring certain strange properties. He considered a layer with
stress-free boundaries. Assuming nonlinear disturbances in the form of rolls invariant
in the y direction, and defining a streamfunction in the xz-plane by u = −∂ψ/∂z and
w = ∂ψ/∂x, he substituted solutions of the form
ψ ∝ X(t) cos π z sin kx,
T ′ ∝ Y (t) cos π z cos kx + Z(t) sin 2π z,
(12.85)
into the equations of motion (12.7). Here, T ′ is the departure of temperature from the
state of no convection, k is the wavenumber of the perturbation, and the boundaries
are at z = ± 21 . It is clear that X is proportional to the intensity of convective motion, Y
is proportional to the temperature difference between the ascending and descending
currents, and Z is proportional to the distortion of the average vertical profile of
temperature from linearity. (Note in equation (12.85) that the x-average of the term
multiplied by Y (t) is zero, so that this term does not cause distortion of the basic
temperature profile.) As discussed in Section 3, Rayleigh’s linear analysis showed that
solutions of the form (12.85), with X and Y constants and Z = 0, would develop if Ra
slightly exceeds the critical value Racr = 27 π 4 /4. Equations (12.85) are expected to
give realistic results when Ra is slightly supercritical but not when strong convection
occurs because only the lowest terms in a “Galerkin expansion” are retained.
529
14. Deterministic Chaos
On substitution of equation (12.85) into the equations of motion, Lorenz finally
obtained
Ẋ = Pr(Y − X),
Ẏ = −XZ + rX − Y,
(12.86)
Ż = XY − bZ,
where Pr is the Prandtl number, r = Ra/Racr , and b = 4π 2 /(π 2 + k 2 ). Equations
(12.86) represent a set of nonlinear equations with three degrees of freedom, which
means that the phase space is three-dimensional.
Equations (12.86) allow the steady solution X = Y = Z = 0, representing the
state of no convection. For r > 1 the system possesses
two additional steady-state
√
solutions, which we shall denote by X̄ = Ȳ = ± b(r − 1), Z̄ = r − 1; the two signs
correspond to the two possible senses of rotation of the rolls. (The fact that these steady
solutions satisfy equation (12.86) can easily be checked by substitution and setting
Ẋ = Ẏ = Ż = 0.) Lorenz showed that the steady-state convection becomes unstable
if r is large. Choosing Pr = 10, b = 8/3, and r = 28, he numerically integrated the
set and found that the solution never repeats itself; it is aperiodic and wanders about
in a chaotic manner. Figure 12.29 shows the variation of X(t), starting with some
initial conditions. (The variables Y (t) and Z(t) also behave in a similar way.) It is
seen that the amplitude
√of the convecting motion initially oscillates around one of the
steady values X̄ = ± b(r − 1), with the oscillations growing in magnitude. When
it is large enough, the amplitude suddenly goes through zero to start oscillations of
opposite sign about the other value of X̄. That is, the motion switches in a chaotic
Figure 12.29 Variation of X(t) in the Lorenz model. Note that the solution oscillates erratically around
the two steady values X̄ and X̄′ . P. Bergé,Y. Pomeau, and C. Vidal, Order Within Chaus, 1984 and reprinting
permitted by Heinemann Educational, a division of Reed Educational & Professional Publishing Ltd.
530
Instability
manner between two oscillatory limit cycles, with the number of oscillations between
transitions seemingly random. Calculations show that the variables X, Y , and Z have
continuous spectra and that the solution is extremely sensitive to initial conditions.
Strange Attractors
The trajectories in the phase plane in the Lorenz model of thermal convection are
shown in Figure 12.30.
√ The centers of the two loops represent the two steady convections X̄ = Ȳ = ± b(r − 1), Z̄ = r − 1. The structure resembles two rather flat
loops of ribbon, one lying slightly in front of the other along a central band with the
two joined together at the bottom of that band. The trajectories go clockwise around
the left loop and counterclockwise around the right loop; two trajectories never intersect. The structure shown in Figure 12.30 is an attractor because orbits starting with
initial conditions outside of the attractor merge on it and then follow it. The attraction
is a result of dissipation in the system. The aperiodic attractor, however, is unlike the
normal attractor in the form of a fixed point (representing steady motion) or a closed
curve (representing a limit cycle). This is because two trajectories on the aperiodic
attractor, with infinitesimally different initial conditions, follow each other closely
only for a while, eventually diverging to very different final states. This is the basic
reason for sensitivity to initial conditions.
For these reasons the aperiodic attractor is called a strange attractor. The idea of
a strange attractor is quite nonintuitive because it has the dual property of attraction
and divergence. Trajectories are attracted from the neighboring region of phase space,
but once on the attractor the trajectories eventually diverge and result in chaos. An
ordinary attractor “forgets” slightly different initial conditions, whereas the strange
Figure 12.30 The Lorenz attractor. Centers of the two loops represent the two steady solutions (X̄, Ȳ , Z̄).
531
14. Deterministic Chaos
attractor ultimately accentuates them. The idea of the strange attractor was first
conceived by Lorenz, and since then attractors of other chaotic systems have also
been studied. They all have the common property of aperiodicity, continuous spectra,
and sensitivity to initial conditions.
Scenarios for Transition to Chaos
Thus far we have studied discrete dynamical systems having only a small number
of degrees of freedom and seen that aperiodic or chaotic solutions result when the
nonlinearity parameter is large. Several routes or scenarios of transition to chaos in
such systems have been identified. Two of these are described briefly here.
(1) Transition through subharmonic cascade: As R is increased, a typical nonlinear system develops a limit cycle of a certain frequency ω. With further
increase of R, several systems are found to generate additional frequencies
ω/2, ω/4, ω/8, . . . . The addition of frequencies in the form of subharmonics does not change the periodic nature of the solution, but the period
doubles each time a lower harmonic is added. The period doubling takes
place more and more rapidly as R is increased, until an accumulation point
(Figure 12.31) is reached, beyond which the solution wanders about in a chaotic
manner. At this point the peaks disappear from the spectrum, which becomes
continuous. Many systems approach chaotic behavior through period doubling.
Feigenbaum (1980) proved the important result that this kind of transition
develops in a universal way, independent of the particular nonlinear systems
studied. If Rn represents the value for development of a new subharmonic,
then Rn converges in a geometric series with
Rn − Rn−1
→ 4.6692
Rn+1 − Rn
as n → ∞.
That is, the horizontal gap between two bifurcation points is about a fifth of the
previous gap. The vertical gap between the branches of the bifurcation diagram
also decreases, with each gap about two-fifths of the previous gap. In other
words, the bifurcation diagram (Figure 12.31) becomes “self similar” as the
accumulation point is approached. (Note that Figure 12.31 has not been drawn
to scale, for illustrative purposes.) Experiments in low Prandtl number fluids
(such as liquid metals) indicate that Bénard convection in the form of rolls
develops oscillatory motion of a certain frequency ω at Ra = 2Racr . As Ra is
further increased, additional frequencies ω/2, ω/4, ω/8, ω/16, and ω/32 have
been observed. The convergence ratio has been measured to be 4.4, close to the
value of 4.669 predicted by Feigenbaum’s theory. The experimental evidence
is discussed further in Bergé, Pomeau, and Vidal (1984).
(2) Transition through quasi-periodic regime: Ruelle and Takens (1971) have
mathematically proved that certain systems need only a small number of
bifurcations to produce chaotic solutions. As the nonlinearity parameter is
increased, the steady solution loses stability and bifurcates to an oscillatory
532
Instability
Figure 12.31 Bifurcation diagram during period doubling. The period doubles at each value Rn of the
nonlinearity parameter. For large n the “bifurcation tree” becomes self similar. Chaos sets in beyond the
accumulation point R∞ .
limit cycle with frequency ω1 . As R is increased, two more frequencies
(ω2 and ω3 ) appear through additional bifurcations. In this scenario the ratios
of the three frequencies (such as ω1 /ω2 ) are irrational numbers, so that the
motion consisting of the three frequencies is not exactly periodic. (When the
ratios are rational numbers, the motion is exactly periodic. To see this, think
of the Fourier series of a periodic function in which the various terms represent sinusoids of the fundamental frequency ω and its harmonics 2ω, 3ω, . . . .
Some of the Fourier coefficients could be zero.) The spectrum for these systems suddenly develops broadband characteristics of chaotic motion as soon
as the third frequency ω3 appears. The exact point at which chaos sets in is
not easy to detect in a measurement; in fact the third frequency may not be
identifiable in the spectrum before it becomes broadband. The Ruelle–Takens
theory is fundamentally different from that of Landau, who conjectured that
turbulence develops due to an infinite number of bifurcations, each generating a new higher frequency, so that the spectrum becomes saturated with
peaks and resembles a continuous one. According to Bergé, Pomeau, and
Vidal (1984), the Bénard convection experiments in water seem to suggest
that turbulence in this case probably sets in according to the Ruelle–Takens
scenario.
The development of chaos in the Lorenz attractor is more complicated and does
not follow either of the two routes mentioned in the preceding.
533
Exercises
Closure
Perhaps the most intriguing characteristic of a chaotic system is the extreme sensitivity
to initial conditions. That is, solutions with arbitrarily close initial conditions evolve
into two quite different states. Most nonlinear systems are susceptible to chaotic
behavior. The extreme sensitivity to initial conditions implies that nonlinear phenomena (including the weather, in which Lorenz was primarily interested when he
studied the convection problem) are essentially unpredictable, no matter how well we
know the governing equations or the initial conditions. Although the subject of chaos
has become a scientific revolution recently, the central idea was conceived by Henri
Poincaré in 1908. He did not, of course, have the computing facilities to demonstrate
it through numerical integration.
It is important to realize that the behavior of chaotic systems is not intrinsically
indeterministic; as such the implication of deterministic chaos is different from that of
the uncertainty principle of quantum mechanics. In any case, the extreme sensitivity
to initial conditions implies that the future is essentially unknowable because it is
never possible to know the initial conditions exactly. As discussed by Davies (1988),
this fact has interesting philosophical implications regarding the evolution of the
universe, including that of living species.
We have examined certain elementary ideas about how chaotic behavior may
result in simple nonlinear systems having only a small number of degrees of freedom.
Turbulence in a continuous fluid medium is capable of displaying an infinite number
of degrees of freedom, and it is unclear whether the study of chaos can throw a great
deal of light on more complicated transitions such as those in pipe or boundary layer
flow. However, the fact that nonlinear systems can have chaotic solutions for a large
value of the nonlinearity parameter (see Figure 12.29 again) is an important result by
itself.
Exercises
1. Consider the thermal instability of a fluid confined between two rigid plates,
as discussed in Section 3. It was stated there without proof that the minimum critical
Rayleigh number of Racr = 1708 is obtained for the gravest even mode. To verify
this, consider the gravest odd mode for which
W = A sin q0 z + B sinh q z + C sinh q ∗ z.
(Compare this with the gravest even mode structure: W = A cos q0 z + B cosh q z+
C cosh q ∗ z.) Following Chandrasekhar (1961, p. 39), show that the minimum
Rayleigh number is now 17,610, reached at the wavenumber Kcr = 5.365.
2. Consider the centrifugal instability problem of Section 5. Making the
narrow-gap approximation, work out the algebra of going from equation (12.37)
to equation (12.38).
3. Consider the centrifugal instability problem of Section 5. From equations (12.38) and (12.40), the eigenvalue problem for determining the marginal
534
Instability
state (σ = 0) is
(D 2 − k 2 )2 ûr = (1 + αx)ûθ ,
2
2 2
2
(D − k ) ûθ = −Ta k ûr ,
(12.87)
(12.88)
with ûr = D ûr = ûθ = 0 at x = 0 and 1. Conditions on ûθ are satisfied by assuming
solutions of the form
ûθ =
∞
Cm sin mπ x.
(12.89)
m=1
Inserting this in equation (12.87), obtain an equation for ûr , and arrange so that the
solution satisfies the four remaining conditions on ûr . With ûr determined in this
manner and ûθ given by equation (12.89), equation (12.88) leads to an eigenvalue
problem for Ta(k). Following Chandrasekhar (1961, p. 300), show that the minimum
Taylor number is given by equation (12.41) and is reached at kcr = 3.12.
4. Consider an infinitely deep fluid of density ρ1 lying over an infinitely deep
fluid of density ρ2 > ρ1 . By setting U1 = U2 = 0, equation (12.51) shows that
g ρ2 − ρ 1
.
(12.90)
c=
k ρ2 + ρ 1
Argue that if the whole system is given an upward vertical acceleration a, then g in
equation (12.90) is replaced by g ′ = g +a. It follows that there is instability if g ′ < 0,
that is, the system is given a downward acceleration of magnitude larger than g. This
is called the Rayleigh–Taylor instability, which can be observed simply by rapidly
accelerating a beaker of water downward.
5. Consider the inviscid instability of parallel flows given by the Rayleigh
equation
(U − c)(v̂yy − k 2 v̂) − Uyy v̂ = 0,
(12.91)
where the y-component of the perturbation velocity is v = v̂ exp(i kx − i kct).
(i) Note that this equation is identical to the Rayleigh equation (12.76) written in
terms of the stream function amplitude φ, as it must because v̂ = −ikφ. For
a flow bounded by walls at y1 and y2 , note that the boundary conditions are
identical in terms of φ and v̂.
(ii) Show that if c is an eigenvalue of equation (12.91), then so is its conjugate
c∗ = cr − i ci . What aspect of equation (12.91) allows the result to be valid?
(iii) Let U ( y) be an antisymmetric jet, so that U ( y) = −U (−y). Demonstrate that
if c(k) is an eigenvalue, then −c(k) is also an eigenvalue. Explain the result
physically in terms of the possible directions of propagation of perturbations
in an antisymmetric flow.
(iv) Let U ( y) be a symmetric jet. Show that in this case v̂ is either symmetric or
antisymmetric about y = 0.
Literature Cited
[Hint: Letting y → −y, show that the solution v̂(−y) satisfies equation (12.91)
with the same eigenvalue c. Form a symmetric solution S( y) = v̂( y) + v̂(−y) =
S(−y), and an antisymmetric solution A( y) = v̂( y)−v̂(−y) = −A(−y). Then write
A[S-eqn] − S[A-eqn] = 0, where S-eqn indicates the differential equation (12.91)
in terms of S. Canceling terms this reduces to (SA′ − AS ′ )′ = 0, where the prime
(′ ) indicates y-derivative. Integration gives SA′ − AS ′ = 0, where the constant of
integration is zero because of the boundary condition. Another integration gives S =
bA, where b is a constant of integration. Because the symmetric and antisymmetric
functions cannot be proportional, it follows that one of them must be zero.]
Comments: If v is symmetric, then the cross-stream velocity has the same sign
across the entire jet, although the sign alternates every half of a wavelength along the
jet. This mode is consequently called sinuous. On the other hand, if v is antisymmetric,
then the shape of the jet expands and contracts along the length. This mode is now
generally called the sausage instability because it resembles a line of linked sausages.
6. For a Kelvin–Helmholtz instability in a continuously stratified ocean, obtain
a globally integrated energy equation in the form
1 d
(u2 + w 2 + g 2 ρ 2 /ρ02 N 2 ) dV = − uwUz dV .
2 dt
(As in Figure 12.25, the integration in x takes place over an integral number
of wavelengths.) Discuss the physical meaning of each term and the mechanism of
instability.
Literature Cited
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Review of Fluid Mechanics 20: 359–391.
Bergé, P., Y. Pomeau, and C. Vidal (1984). Order Within Chaos, New York: Wiley.
Bhattacharya, P., M. P. Manoharan, R. Govindarajan, and R. Narasimha (2006). “The critical Reynolds
number of a laminar incompressible mixing layer from minimal composite theory.” Journal of Fluid
Mechanics 565: 105–114.
Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability, London: Oxford University Press;
New York: Dover reprint, 1981.
Coles, D. (1965). “Transition in circular Couette flow.” Journal of Fluid Mechanics 21: 385–425.
Davies, P. (1988). Cosmic Blueprint, New York: Simon and Schuster.
Drazin, P. G. and W. H. Reid (1981). Hydrodynamic Stability, London: Cambridge University Press.
Eckhardt, B., T. M. Schneider, B. Hof, and J. Westerweel (2007). “Turbulence transition in pipe flow.”
Annual Review of Fluid Mechanics 39: 447–468.
Eriksen, C. C. (1978). “Measurements and models of fine structure, internal gravity waves, and wave
breaking in the deep ocean.” Journal of Geophysical Research 83: 2989–3009.
Feigenbaum, M. J. (1978). “Quantitative universality for a class of nonlinear transformations.” Journal of
Statistical Physics 19: 25–52.
Fransson, J. H. M. and P. H. Alfredsson (2003). “On the disturbance growth in an asymptotic suction
boundary layer.” Journal of Fluid Mechanics 482: 51–90.
Grabowski, W. J. (1980). “Nonparallel stability analysis of axisymmetric stagnation point flow.” Physics
of Fluids 23: 1954–1960.
Heisenberg, W. (1924). “Über Stabilität und Turbulenz von Flüssigkeitsströmen.” Annalen der Physik
(Leipzig) (4) 74: 577–627.
535
536
Instability
Howard, L. N. (1961). “Note on a paper of John W. Miles.” Journal of Fluid Mechanics 13: 158–160.
Huppert, H. E. and J. S. Turner (1981). “Double-diffusive convection.” Journal of Fluid Mechanics 106:
299–329.
Klebanoff, P. S., K. D. Tidstrom, and L. H. Sargent (1962). “The three-dimensional nature of boundary
layer instability”. Journal of Fluid Mechanics 12: 1–34.
Lanford, O. E. (1982). “The strange attractor theory of turbulence.” Annual Review of Fluid Mechanics
14: 347–364.
Lin, C. C. (1955). The Theory of Hydrodynamic Stability, London: Cambridge University Press, Chapter 8.
Lorenz, E. (1963). “Deterministic nonperiodic flows.” Journal of Atmospheric Sciences 20:
130–141.
Miles, J. W. (1961). “On the stability of heterogeneous shear flows.” Journal of Fluid Mechanics 10:
496–508.
Miles, J. W. (1986). “Richardson’s criterion for the stability of stratified flow.” Physics of Fluids 29:
3470–3471.
Nayfeh, A. H. and W. S. Saric (1975). “Nonparallel stability of boundary layer flows.” Physics of Fluids
18: 945–950.
Reshotko, E. (2001). “Transient growth: A factor in bypass transition.” Physics of Fluids 13: 1067–1075.
Ruelle, D. and F. Takens (1971). “On the nature of turbulence.” Communications in Mathematical Physics
20: 167–192.
Scotti, R. S. and G. M. Corcos (1972). “An experiment on the stability of small disturbances in a stratified
free shear layer.” Journal of Fluid Mechanics 52: 499–528.
Shen, S. F. (1954). “Calculated amplified oscillations in plane Poiseuille and Blasius Flows.” Journal of
the Aeronautical Sciences 21: 62–64.
Stern, M. E. (1960). “The salt fountain and thermohaline convection.” Tellus 12: 172–175.
Stommel, H., A. B. Arons, and D. Blanchard (1956). “An oceanographic curiosity: The perpetual salt
fountain.” Deep-Sea Research 3: 152–153.
Thorpe, S. A. (1971). “Experiments on the instability of stratified shear flows: Miscible fluids.” Journal of
Fluid Mechanics 46: 299–319.
Turner, J. S. (1973). Buoyancy Effects in Fluids, London: Cambridge University Press.
Turner, J. S. (1985). “Convection in multicomponent systems.” Naturwissenschaften 72: 70–75.
Woods, J. D. (1969). “On Richardson’s number as a criterion for turbulent–laminar transition in the atmosphere and ocean.” Radio Science 4: 1289–1298.
Yih, C. S. (1979). Fluid Mechanics: A Concise Introduction to the Theory, Ann Arbor, MI: West River
Press, pp. 469–496.
Chapter 13
Turbulence
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
Historical Notes . . . . . . . . . . . . . . . . . . . . .
Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correlations and Spectra . . . . . . . . . . . .
Averaged Equations of Motion . . . . . . .
Mean Continuity Equation . . . . . . . . . . .
Mean Momentum Equation . . . . . . . . . .
Reynolds Stress . . . . . . . . . . . . . . . . . . . . .
Mean Heat Equation . . . . . . . . . . . . . . . .
Kinetic Energy Budget of
Mean Flow . . . . . . . . . . . . . . . . . . . . . . . . . .
Kinetic Energy Budget of
Turbulent Flow. . . . . . . . . . . . . . . . . . . . . .
Turbulence Production and
Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectrum of Turbulence in Inertial
Subrange . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wall-Free Shear Flow . . . . . . . . . . . . . . . .
Intermittency . . . . . . . . . . . . . . . . . . . . . . .
Entrainment . . . . . . . . . . . . . . . . . . . . . . . .
Self-Preservation . . . . . . . . . . . . . . . . . . . .
Consequence of Self-Preservation in
a Plane Jet . . . . . . . . . . . . . . . . . . . . . . .
Turbulent Kinetic Energy Budget in
a Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wall-Bounded Shear Flow . . . . . . . . . . .
537
539
541
543
547
548
549
550
553
12.
13.
14.
554
556
559
15.
562
564
565
566
566
567
16.
568
570
Inner Layer: Law of the Wall . . . . . . . .
Outer Layer: Velocity Defect Law . . . .
Overlap Layer: Logarithmic Law . . . .
Rough Surface . . . . . . . . . . . . . . . . . . . . . .
Variation of Turbulent Intensity . . . . . .
Eddy Viscosity and Mixing
Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coherent Structures in
a Wall Layer . . . . . . . . . . . . . . . . . . . . . . . .
Turbulence in a Stratified
Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Richardson Numbers . . . . . . . . . . . .
Monin–Obukhov Length . . . . . . . . . . . . .
Spectrum of Temperature
Fluctuations. . . . . . . . . . . . . . . . . . . . . .
Taylor’s Theory of Turbulent
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . .
Rate of Dispersion of a Single
Particle . . . . . . . . . . . . . . . . . . . . . . . . . .
Random Walk . . . . . . . . . . . . . . . . . . . . . . .
Behavior of a Smoke Plume in
the Wind . . . . . . . . . . . . . . . . . . . . . . . . .
Effective Diffusivity . . . . . . . . . . . . . . . . .
Concluding Remarks. . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . .
571
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573
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580
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586
587
588
589
591
591
595
596
597
598
598
600
601
1. Introduction
Most flows encountered in engineering practice and in nature are turbulent. The
boundary layer on an aircraft wing is likely to be turbulent, the atmospheric boundary
layer over the earth’s surface is turbulent, and the major oceanic currents are turbulent. In this chapter we shall discuss certain elementary ideas about the dynamics of
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50013-7
537
538
Turbulence
turbulent flows. We shall see that such flows do not allow a strict analytical study, and
one depends heavily on physical intuition and dimensional arguments. In spite of our
everyday experience with it, turbulence is not easy to define precisely. In fact, there
is a tendency to confuse turbulent flows with “random flows.” With some humor,
Lesieur (1987) said that “turbulence is a dangerous topic which is at the origin of
serious fights in scientific meetings since it represents extremely different points of
view, all of which have in common their complexity, as well as an inability to solve
the problem. It is even difficult to agree on what exactly is the problem to be solved.”
Some characteristics of turbulent flows are the following:
(1) Randomness: Turbulent flows seem irregular, chaotic, and unpredictable.
(2) Nonlinearity: Turbulent flows are highly nonlinear. The nonlinearity serves
two purposes. First, it causes the relevant nonlinearity parameter, say the
Reynolds number Re, the Rayleigh number Ra, or the inverse Richardson number Ri−1 , to exceed a critical value. In unstable flows small perturbations grow
spontaneously and frequently equilibrate as finite amplitude disturbances. On
further exceeding the stability criteria, the new state can become unstable to
more complicated disturbances, and the flow eventually reaches a chaotic state.
Second, the nonlinearity of a turbulent flow results in vortex stretching, a key
process by which three-dimensional turbulent flows maintain their vorticity.
(3) Diffusivity: Due to the macroscopic mixing of fluid particles, turbulent flows
are characterized by a rapid rate of diffusion of momentum and heat.
(4) Vorticity: Turbulence is characterized by high levels of fluctuating vorticity.
The identifiable structures in a turbulent flow are vaguely called eddies. Flow
visualization of turbulent flows shows various structures—coalescing, dividing, stretching, and above all spinning. A characteristic feature of turbulence is
the existence of an enormous range of eddy sizes. The large eddies have a size
of order of the width of the region of turbulent flow; in a boundary layer this is
the thickness of the layer (Figure 13.1). The large eddies contain most of the
Figure 13.1 Turbulent flow in a boundary layer, showing that a large eddy has a size of the order of
boundary layer thickness.
2. Historical Notes
energy. The energy is handed down from large to small eddies by nonlinear
interactions, until it is dissipated by viscous diffusion in the smallest eddies,
whose size is of the order of millimeters.
(5) Dissipation: The vortex stretching mechanism transfers energy and vorticity
to increasingly smaller scales, until the gradients become so large that they are
smeared out (i.e., dissipated) by viscosity. Turbulent flows therefore require a
continuous supply of energy to make up for the viscous losses.
These features of turbulence suggest that many flows that seem “random,” such
as gravity waves in the ocean or the atmosphere, are not turbulent because they are
not dissipative, vortical, and nonlinear.
Incompressible turbulent flows in systems not large enough to be influenced by
the Coriolis force will be studied in this chapter. These flows are three-dimensional. In
large-scale geophysical systems, on the other hand, the existence of stratification and
Coriolis force severely restricts vertical motion and leads to a chaotic flow that is nearly
“geostropic” and two-dimensional. Geostrophic turbulence is briefly commented on
in Chapter 14.
2. Historical Notes
Turbulence research is currently at the forefront of modern fluid dynamics, and some
of the well-known physicists of this century have worked in this area. Among them
are G. I. Taylor, Kolmogorov, Reynolds, Prandtl, von Karman, Heisenberg, Landau,
Millikan, and Onsagar. A brief historical outline is given in what follows; further
interesting details can be found in Monin and Yaglom (1971). The reader is expected
to fully appreciate these historical remarks only after reading the chapter.
The first systematic work on turbulence was carried out by Osborne Reynolds
in 1883. His experiments in pipe flows, discussed in Section 9.1, showed that the
flow becomes turbulent or irregular when the nondimensional ratio Re = UL/ν,
later named the Reynolds number by Sommerfeld, exceeds a certain critical value.
(Here ν is the kinematic viscosity, U is the velocity scale, and L is the length scale.)
This nondimensional number subsequently proved to be the parameter that determines the dynamic similarity of viscous flows. Reynolds also separated turbulent
variables as the sum of a mean and a fluctuation and arrived at the concept of turbulent stress. The discovery of the significance of Reynolds number and turbulent
stress has proved to be of fundamental importance in our present knowledge of
turbulence.
In 1921 the British physicist G. I. Taylor, in a simple and elegant study of
turbulent diffusion, introduced the idea of a correlation function. He showed that the
rms distance of a particle
√ from its source point initially increases with time as ∝ t,
and subsequently as ∝ t, as in a random walk. Taylor continued his outstanding
work in a series of papers during 1935–1936 in which he laid down the foundation
of the statistical theory of turbulence. Among the concepts he introduced were those
of homogeneous and isotropic turbulence and of turbulence spectrum. Although real
turbulent flows are not isotropic (the turbulent stresses, in fact, vanish for isotropic
flows), the mathematical techniques involved have proved valuable for describing the
539
540
Turbulence
small scales of turbulence, which are isotropic. In 1915 Taylor also introduced the
idea of mixing length, although it is generally credited to Prandtl for making full use of
the idea.
During the 1920s Prandtl and his student von Karman, working in Göttingen,
Germany, developed the semiempirical theories of turbulence. The most successful
of these was the mixing length theory, which is based on an analogy with the concept
of mean free path in the kinetic theory of gases. By guessing at the correct form for the
mixing length, Prandtl was able to deduce that the velocity profile near a solid wall is
logarithmic, one of the most reliable results of turbulent flows. It is for this reason that
subsequent textbooks on fluid mechanics have for a long time glorified the mixing
length theory. Recently, however, it has become clear that the mixing length theory is
not helpful since there is really no rational way of predicting the form of the mixing
length. In fact, the logarithmic law can be justified from dimensional considerations
alone.
Some very important work was done by the British meteorologist Lewis
Richardson. In 1922 he wrote the very first book on numerical weather prediction.
In this book he proposed that the turbulent kinetic energy is transferred from large
to small eddies, until it is destroyed by viscous dissipation. This idea of a spectral
energy cascade is at the heart of our present understanding of turbulent flows. However, Richardson’s work was largely ignored at the time, and it was not until some
20 years later that the idea of a spectral cascade took a quantitative shape in the hands
of Kolmogorov and Obukhov in Russia. Richardson also did another important piece
of work that displayed his amazing physical intuition. On the basis of experimental
data on the movement of balloons in the atmosphere, he proposed that the effective
diffusion coefficient of a patch of turbulence is proportional to l 4/3 , where l is the scale
of the patch. This is called Richardson’s four-third law, which has been subsequently
found to be in agreement with Kolmogorov’s five-third law of spectrum.
The Russian mathematician Kolmogorov, generally regarded as the greatest
probabilist of the twentieth century, followed up on Richardson’s idea of a spectral
energy cascade. He hypothesized that the statistics of small scales are isotropic and
depend on only two parameters, namely viscosity ν and the rate of dissipation ε. On
dimensional grounds, he derived that the smallest scales must be of size η = (ν 3 /ε)1/4 .
His second hypothesis was that, at scales much smaller than l and much larger than
η, there must exist an inertial subrange in which ν plays no role; in this range the
statistics depend only on a single parameter ε. Using this idea, in 1941 Kolmogorov
and Obukhov independently derived that the spectrum in the inertial subrange must
be proportional to ε 2/3 K −5/3 , where K is the wavenumber. The five-third law is
one of the most important results of turbulence theory and is in agreement with
observations.
There has been much progress in recent years in both theory and observations.
Among these may be mentioned the experimental work on the coherent structures
near a solid wall. Observations in the ocean and the atmosphere (which von
Karman called “a giant laboratory for turbulence research”), in which the Reynolds
numbers are very large, are shedding new light on the structure of stratified
turbulence.
541
3. Averages
3. Averages
The variables in a turbulent flow are not deterministic in details and have to be treated
as stochastic or random variables. In this section we shall introduce certain definitions
and nomenclature used in the theory of random variables.
Let u(t) be any measured variable in a turbulent flow. Consider first the case
when the “average characteristics” of u(t) do not vary with time (Figure 13.2a). In
such a case we can define the average variable as the time mean
1
ū ≡ lim
t0 →∞ t0
t0
u(t) dt.
(13.1)
0
Now consider a situation in which the average characteristics do vary with time. An
example is the decaying series shown in Figure 13.2b, which could represent the
velocity of a jet as the pressure in the supply tank falls. In this case the average is a
function of time and cannot be formally defined by using equation (13.1), because we
cannot specify how large the averaging interval t0 should be made in evaluating the
integral (13.1). If we take t0 to be very large then we may not get a “local” average,
and if we take t0 to be very small then we may not get a reliable average. In such
a case, however, one can still define an average by performing a large number of
experiments, conducted under identical conditions. To define this average precisely,
we first need to introduce certain terminology.
A collection of experiments, performed under an identical set of experimental
conditions, is called an ensemble, and an average over the collection is called an
ensemble average, or expected value. Figure 13.3 shows an example of several records
of a random variable, for example, the velocity in the atmospheric boundary layer
measured from 8 am to 10 am in the morning. Each record is measured at the same
place, supposedly under identical conditions, on different days. The ith record is
denoted by ui (t). (Here the superscript does not stand for power.) All records in the
figure show that for some dynamic reason the velocity is decaying with time. In other
words, the expected velocity at 8 am is larger than that at 10 am. It is clear that the
average velocity at 9 am can be found by adding together the velocity at 9 am for
Figure 13.2 Stationary and nonstationary time series.
542
Turbulence
Figure 13.3 An ensemble of functions u(t).
each record and dividing the sum by the number of records. We therefore define the
ensemble average of u at time t to be
ū(t) ≡
N
1 i
u (t),
N
(13.2)
i=1
where N is a large number. From this it follows that the average derivative at a certain
time is
1 ∂u1 (t) ∂u2 (t) ∂u3 (t)
∂u
=
+
+
+ ···
∂t
N
∂t
∂t
∂t
∂ ū
∂ 1 1
=
{u (t) + u2 (t) + · · · } =
.
∂t N
∂t
This shows that the operation of differentiation commutes with the operation of ensemble averaging, so that their orders can be interchanged. In a similar manner we can
show that the operation of integration also commutes with ensemble averaging. We
therefore have the rules
∂u ∂ ū
=
,
∂t
∂t
b
b
ū dt.
u dt =
a
a
(13.3)
(13.4)
543
4. Correlations and Spectra
Similar rules also hold when the variable is a function of space:
∂ ū
∂u
=
,
∂xi
∂xi
u dx = ū dx.
(13.5)
(13.6)
The rules of commutation (13.3)–(13.6) will be constantly used in the algebraic
manipulations throughout the chapter.
As there is no way of controlling natural phenomena in the atmosphere and the
ocean, it is very difficult to obtain observations under identical conditions. Consequently, in a nonstationary process such as the one shown in Figure 13.2b, the average
value of u at a certain time is sometimes determined by using equation (13.1) and
choosing an appropriate averaging time t0 , small compared to the time during which
the average properties change appreciably. In any case, for theoretical discussions, all
averages defined by overbars in this chapter are to be regarded as ensemble averages.
If the process also happens to be stationary, then the overbar can be taken to mean
the time average.
The various averages of a random variable, such as its mean and rms value,
are collectively called the statistics of the variable. When the statistics of a random
variable are independent of time, we say that the underlying process is stationary.
Examples of stationary and nonstationary processes are shown in Figure 13.2. For a
stationary process the time average (i.e., the average over a single record, defined by
equation (13.1)) can be shown to equal the ensemble average, resulting in considerable
simplification. Similarly, we define a homogeneous process as one whose statistics
are independent of space, for which the ensemble average equals the spatial average.
The mean square value of a variable is called the variance. The square root of
variance is called the root-mean-square (rms) value:
variance ≡ u2 ,
urms ≡ (u2 )1/2 .
The time series [u(t) − ū], obtained after subtracting the mean ū of the series, represents the fluctuation of the variable about its mean. The rms value of the fluctuation
is called the standard deviation, defined as
uSD ≡ [(u − ū)2 ]1/2 .
4. Correlations and Spectra
The autocorrelation of a single variable u(t) at two times t1 and t2 is defined as
R(t1 , t2 ) ≡ u(t1 )u(t2 ).
(13.7)
544
Turbulence
In the general case when the series is not stationary, the overbar is to be regarded
as an ensemble average. Then the correlation can be computed as follows: Obtain a
number of records of u(t), and on each record read off the values of u at t1 and t2 .
Then multiply the two values of u in each record and calculate the average value of
the product over the ensemble.
The magnitude of this average product is small when a positive value of u(t1 ) is
associated with both positive and negative values of u(t2 ). In such a case the magnitude
of R(t1 , t2 ) is small, and we say that the values of u at t1 and t2 are “weakly correlated.”
If, on the other hand, a positive value of u(t1 ) is mostly associated with a positive
value of u(t2 ), and a negative value of u(t1 ) is mostly associated with a negative value
of u(t2 ), then the magnitude of R(t1 , t2 ) is large and positive; in such a case we say
that the values of u(t1 ) and u(t2 ) are “strongly correlated.” We may also have a case
with R(t1 , t2 ) large and negative, in which one sign of u(t1 ) is mostly associated with
the opposite sign of u(t2 ).
For a stationary process the statistics (i.e., the various kinds of averages) are
independent of the origin of time, so that we can shift the origin of time by any
amount. Shifting the origin by t1 , the autocorrelation (13.7) becomes u(0)u(t2 − t1 )
= u(0)u(τ ), where τ = t2 − t1 is the time lag. It is clear that we can also write this
correlation as u(t)u(t + τ ), which is a function of τ only, t being an arbitrary origin
of measurement. We can therefore define an autocorrelation function of a stationary
process by
R(τ ) = u(t)u(t + τ ).
As we have assumed stationarity, the overbar in the aforementioned expression can
also be regarded as a time average. In such a case the method of estimating the
correlation is to align the series u(t) with u(t + τ ) and multiply them vertically
(Figure 13.4).
Figure 13.4 Method of calculating autocorrelation u(t)u(t + τ ).
545
4. Correlations and Spectra
We can also define a normalized autocorrelation function
r(τ ) ≡
u(t)u(t + τ )
u2
,
(13.8)
where u2 is the mean square value. For any function u(t), it can be proved that
u(t1 )u(t2 ) [u2 (t1 )]1/2 [u2 (t2 )]1/2 ,
(13.9)
which is called the Schwartz inequality. It is analogous to the rule that the inner product
of two vectors cannot be larger than the product of their magnitudes. For a stationary
process the mean square value is independent of time, so that the right-hand side of
equation (13.9) equals u2 . Using equation (13.9), it follows from equation (13.8) that
r 1.
Obviously, r(0) = 1. For a stationary process the autocorrelation is a symmetric
function, because then
R(τ ) = u(t)u(t + τ ) = u(t − τ )u(t) = u(t)u(t − τ ) = R(−τ ).
A typical autocorrelation plot is shown in Figure 13.5. Under normal conditions r goes
to 0 as τ → ∞, because a process becomes uncorrelated with itself after a long time.
A measure of the width of the correlation function can be obtained by replacing
the measured autocorrelation distribution by a rectangle of height 1 and width ᐀
(Figure 13.5), which is therefore given by
᐀≡
∞
r(τ ) dτ.
0
Figure 13.5 Autocorrelation function and the integral time scale.
(13.10)
546
Turbulence
This is called the integral time scale, which is a measure of the time over which u(t)
is highly correlated with itself. In other words, ᐀ is a measure of the “memory” of
the process.
Let S(ω) denote the Fourier transform of the autocorrelation function R(τ ). By
definition, this means that
S(ω) ≡
1
2π
∞
e−iωτ R(τ ) dτ.
(13.11)
−∞
It can be shown that, for equation (13.11) to be true, R(τ ) must be given in terms of
S(ω) by
R(τ ) ≡
∞
eiωτ S(ω) dω.
(13.12)
−∞
We say that equations (13.11) and (13.12) define a “Fourier transform pair.” The
relationships (13.11) and (13.12) are not special for the autocorrelation function, but
hold for any function for which a Fourier transform can be defined. Roughly speaking,
a Fourier transform can be defined if the function decays to zero fast enough at infinity.
It is easy to show that S(ω) is real and symmetric if R(τ ) is real and symmetric
(Exercise 1). Substitution of τ = 0 in equation (13.12) gives
u2
=
∞
(13.13)
S(ω) dω.
−∞
This shows that S(ω) dω is the energy (more precisely, variance) in a frequency
band dω centered at ω. Therefore, the function S(ω) represents the way energy is
distributed as a function of frequency ω. We say that S(ω) is the energy spectrum, and
equation (13.11) shows that it is simply the Fourier transform of the autocorrelation
function. From equation (13.11) it also follows that
S(0) =
1
2π
∞
−∞
R(τ ) dτ =
u2
π
0
∞
r(τ ) dτ =
u2 ᐀
,
π
which shows that the value of the spectrum at zero frequency is proportional to the
integral time scale.
So far we have considered u as a function of time and have defined its autocorrelation R(τ ). In a similar manner we can define an autocorrelation as a function of
the spatial separation between measurements of the same variable at two points. Let
u(x0 , t) and u(x0 + x, t) denote the measurements of u at points x0 and x0 + x. Then
the spatial correlation is defined as u(x0 , t)u(x0 + x, t). If the field is spatially homogeneous, then the statistics are independent of the location x0 , so that the correlation
depends only on the separation x:
R(x) = u(x0 , t)u(x0 + x, t).
547
5. Averaged Equations of Motion
We can now define an energy spectrum S(K) as a function of the wavenumber vector K
by the Fourier transform
∞
1
S(K) =
e−iK · x R(x) dx,
(13.14)
(2π)1/3 −∞
where
R(x) =
∞
eiK · x S(K) dK.
(13.15)
−∞
The pair (13.14) and (13.15) is analogous to equations (13.11) and (13.12). In the
integral (13.14), dx is the shorthand notation for the volume element dx dy dz. Similarly, in the integral (13.15), dK = dk dl dm is the volume element in the wavenumber
space (k, l, m).
It is clear that we need an instantaneous measurement u(x) as a function of
position to calculate the spatial correlations R(x). This is a difficult task and so we
frequently determine this value approximately by rapidly moving a probe in a desired
direction. If the speed U0 of traversing of the probe is rapid enough, we can assume
that the turbulence field is “frozen” and does not change during the measurement.
Although the probe actually records a time series u(t), we may then transform it
to a spatial series u(x) by replacing t by x/U0 . The assumption that the turbulent
fluctuations at a point are caused by the advection of a frozen field past the point is
called Taylor’s hypothesis.
So far we have defined autocorrelations involving measurements of the same
variable u. We can also define a cross-correlation function between two stationary
variables u(t) and v(t) as
C(τ ) ≡ u(t)v(t + τ ).
Unlike the autocorrelation function, the cross-correlation function is not a symmetric
function of the time lag τ , because C(−τ ) = u(t)v(t − τ ) = C(τ ). The value of the
cross-correlation function at zero lag, that is u(t)v(t), is simply written as uv and
called the “correlation” of u and v.
5. Averaged Equations of Motion
A turbulent flow instantaneously satisfies the Navier–Stokes equations. However, it
is virtually impossible to predict the flow in detail, as there is an enormous range
of scales to be resolved, the smallest spatial scales being less than millimeters and
the smallest time scales being milliseconds. Even the most powerful of today’s computers would take an enormous amount of computing time to predict the details of
an ordinary turbulent flow, resolving all the fine scales involved. Fortunately, we are
generally interested in finding only the gross characteristics in such a flow, such as
the distributions of mean velocity and temperature. In this section we shall derive the
equations of motion for the mean state in a turbulent flow and examine what effect
the turbulent fluctuations may have on the mean flow.
548
Turbulence
We assume that the density variations are caused by temperature fluctuations
alone. The density variations due to other sources such as the concentration of a solute
can be handled within the present framework by defining an equivalent temperature.
Under the Boussinesq approximation, the equations of motion for the instantaneous
variables are
∂ ũi
∂ ũi
1 ∂ p̃
∂ 2 ũi
+ ũj
=−
− g[1 − α(T̃ − T0 )]δi3 + ν
,
∂t
∂xj
ρ0 ∂xi
∂xj ∂xj
∂ ũi
= 0,
∂xi
∂ T̃
∂ T̃
∂ 2 T̃
+ ũj
=κ
.
∂t
∂xj
∂xj ∂xj
(13.16)
(13.17)
(13.18)
As in the preceding chapter, we are denoting the instantaneous quantities by a
tilde ( ˜ ). Let the variables be decomposed into their mean part and a deviation from
the mean:
ũi = Ui + ui ,
p̃ = P + p,
(13.19)
′
T̃ = T̄ + T .
(The corresponding density is ρ̃ = ρ̄ +ρ ′ .) This is called the Reynolds decomposition.
As in the preceding chapter, the mean velocity and the mean pressure are denoted by
uppercase letters, and their turbulent fluctuations are denoted by lowercase letters.
This convention is impossible to use for temperature and density, for which we use an
overbar for the mean state and a prime for the turbulent part. The mean quantities
(U, P , T̄ ) are to be regarded as ensemble averages; for stationary flows they can also
be regarded as time averages. Taking the average of both sides of equation (13.19),
we obtain
ūi = p̄ = T ′ = 0,
showing that the fluctuations have zero mean.
The equations satisfied by the mean flow are obtained by substituting the
Reynolds decomposition (13.19) into the instantaneous Navier–Stokes equations
(13.16)–(13.18) and taking the average of the equations. The three equations transform
as follows.
Mean Continuity Equation
Averaging the continuity equation (13.17), we obtain
∂Ui
∂Ui
∂ ūi
∂
∂ui
(Ui + ui ) =
+
=
+
= 0,
∂xi
∂xi
∂xi
∂xi
∂xi
549
5. Averaged Equations of Motion
where we have used the commutation rule (13.5). Using ūi = 0, we obtain
∂Ui
= 0,
∂xi
(13.20)
which is the continuity equation for the mean flow. Subtracting this from the continuity
equation (13.17) for the total flow, we obtain
∂ui
= 0,
∂xi
(13.21)
which is the continuity equation for the turbulent fluctuation field. It is therefore seen
that the instantaneous, the mean, and the turbulent parts of the velocity field are all
nondivergent.
Mean Momentum Equation
The momentum equation (13.16) gives
∂
∂
(Ui + ui ) + (Uj + uj )
(Ui + ui )
∂t
∂xj
=−
∂2
1 ∂
(P + p) − g[1 − α(T̄ + T ′ − T0 )] δi3 + ν 2 (Ui + ui ). (13.22)
ρ0 ∂xi
∂xj
We shall take the average of each term of this equation. The average of the time
derivative term is
∂Ui
∂Ui
∂ ūi
∂Ui
∂
∂ui
(Ui + ui ) =
+
=
+
=
,
∂t
∂t
∂t
∂t
∂t
∂t
where we have used the commutation rule (13.3), and ūi = 0. The average of the
advective term is
(Uj + uj )
∂ui
∂
∂Ui
∂ ūi
∂Ui
(Ui + ui ) = Uj
+ Uj
+ ūj
+ uj
∂xj
∂xj
∂xj
∂xj
∂xj
∂Ui
∂
= Uj
+
(ui uj ),
∂xj
∂xj
where we have used the commutation rule (13.5) and ūi = 0; the continuity equation
∂uj /∂xj = 0 has also been used in obtaining the last term.
The average of the pressure gradient term is
∂P
∂ p̄
∂P
∂
(P + p) =
+
=
.
∂xi
∂xi
∂xi
∂xi
550
Turbulence
The average of the gravity term is
g[1 − α(T̄ + T ′ − T0 )] = g[1 − α(T̄ − T0 )],
where we have used T̄ ′ = 0. The average of the viscous term is
ν
∂ 2 Ui
∂2
(Ui + ui ) = ν
.
∂xj ∂xj
∂xj ∂xj
Collecting terms, the mean of the momentum equation (13.22) takes the form
∂Ui
∂
∂ 2 Ui
1 ∂P
∂Ui
+ Uj
+
(ui uj ) = −
− g[1 − α(T̄ − T0 )] δi3 + ν
.
∂t
∂xj
∂xj
ρ0 ∂xi
∂xj ∂xj
(13.23)
The correlation ui uj in equation (13.23) is generally nonzero, although ūi = 0. This
is discussed further in what follows.
Reynolds Stress
Writing the term ui uj on the right-hand side, the mean momentum equation (13.23)
becomes
∂Ui
∂
1 ∂P
DUi
(13.24)
ν
− g[1 − α(T̄ − T0 )] δi3 +
− ui uj ,
=−
Dt
ρ0 ∂xi
∂xj
∂xj
which can be written as
1 ∂ τ̄ij
DUi
=
− g[1 − α(T̄ − T0 )] δi3 ,
Dt
ρ0 ∂xj
(13.25)
where
τ̄ij = −P δij + µ
∂Uj
∂Ui
+
∂xj
∂xi
− ρ 0 ui uj .
(13.26)
Compare equations (13.25) and (13.26) with the corresponding equations for the
instantaneous flow, given by (see equations (4.13) and (4.36))
D ũi
1 ∂ τ̃ij
=
− g[1 − α(T̃ − T0 )] δi3 ,
Dt
ρ0 ∂xj
∂ ũj
∂ ũi
.
+
τ̃ij = −p̃δij + µ
∂xj
∂xi
It is seen from equation (13.25) that there is an additional stress −ρ0 ui uj acting in a
mean turbulent flow. In fact, these extra stresses on the mean field of a turbulent flow
5. Averaged Equations of Motion
are much larger than the viscous contribution µ(∂Ui /∂xj + ∂Ui /∂xj ), except very
close to a solid surface where the fluctuations are small and mean flow gradients are
large.
The tensor −ρ0 ui uj is called the Reynolds stress tensor and has the nine Cartesian
components
−ρ0 u2 −ρ0 uv −ρ0 uw
−ρ0 uv −ρ0 v 2 −ρ0 vw .
−ρ0 uw −ρ0 vw −ρ0 w 2
This is a symmetric tensor; its diagonal components are normal stresses, and the
off-diagonal components are shear stresses. If the turbulent fluctuations are completely isotropic, that is, if they do not have any directional preference, then the
off-diagonal components of ui uj vanish, and u2 = v 2 = w 2 . This is shown in
Figure 13.6, which shows a cloud of data points (sometimes called a “scatter plot”)
on a uv-plane. The dots represent the instantaneous values of the uv-pair at different
times. In the isotropic case there is no directional preference, and the dots form a
spherically symmetric pattern. In this case a positive u is equally likely to be associated with both a positive and a negative v. Consequently, the average value of the
product uv is zero if the turbulence is isotropic. In contrast, the scatter plot in an
anisotropic turbulent field has a polarity. The figure shows a case where a positive u
is mostly associated with a negative v, giving uv < 0.
It is easy to see why the average product of the velocity fluctuations in a turbulent
flow is not expected to be zero. Consider a shear flow where the mean shear dU/dy
is positive (Figure 13.7). Assume that a particle at level y is instantaneously traveling
upward (v > 0). On the average the particle retains its original velocity during the
migration, and when it arrives at level y + dy it finds itself in a region where a larger
velocity prevails. Thus the particle tends to slow down the neighboring fluid particles
Figure 13.6 Isotropic and anisotropic turbulent fields. Each dot represents a uv-pair at a certain time.
551
552
Turbulence
Figure 13.7 Movement of a particle in a turbulent shear flow.
after it has reached the level y + dy, and causes a negative u. Conversely, the particles
that travel downward (v < 0) tend to cause a positive u in the new level y − dy. On the
average, therefore, a positive v is mostly associated with a negative u, and a negative
v is mostly associated with a positive u. The correlation uv is therefore negative for
the velocity field shown in Figure 13.7, where dU/dy > 0. This makes sense, since
in this case the x-momentum should tend to flow in the negative y-direction as the
turbulence tends to diffuse the gradients and decrease dU/dy.
The procedure of deriving equation (13.26) shows that the Reynolds stress arises
out of the nonlinear term ũj (∂ ũi /∂xj ) of the equation of motion. It is a stress exerted
by the turbulent fluctuations on the mean flow. Another way to interpret the Reynolds
stress is that it is the rate of mean momentum transfer by turbulent fluctuations. Consider again the shear flow U (y) shown in Figure 13.7, where the instantaneous velocity
is (U + u, v, w). The fluctuating velocity components constantly transport fluid particles, and associated momentum, across a plane AA normal to the y-direction. The
instantaneous rate of mass transfer across a unit area is ρ0 v, and consequently the
instantaneous rate of x-momentum transfer is ρ0 (U + u)v. Per unit area, the average
rate of flow of x-momentum in the y-direction is therefore
ρ0 (U + u)v = ρ0 U v̄ + ρ0 uv = ρ0 uv.
Generalizing, ρ0 ui uj is the average flux of j-momentum along the i-direction, which
also equals the average flux of i-momentum along the j-direction.
The sign convention for the Reynolds stress is the same as that explained in
Chapter 2, Section 4: On a surface whose outward normal points in the positive
x-direction, a positive τxy points along the y-direction. According to this convention, the Reynolds stresses −ρ0 uv on a rectangular element are directed as in Figure 13.8, if they are positive. The discussion in the preceding paragraph shows
that such a Reynolds stress causes a mean flow of x-momentum along the negative
y-direction.
553
5. Averaged Equations of Motion
Figure 13.8 Positive directions of Reynolds stresses on a rectangular element.
Mean Heat Equation
The heat equation (13.18) is
∂
∂
∂2
(T̄ + T ′ ) + (Uj + uj )
(T̄ + T ′ ) = κ 2 (T̄ + T ′ ).
∂t
∂xj
∂xj
The average of the time derivative term is
∂ T̄
∂ T̄ ′
∂
∂ T̄
(T̄ + T ′ ) =
+
=
.
∂t
∂t
∂t
∂t
The average of the advective term is
(Uj + uj )
∂T ′
∂
∂ T̄
∂T ′
∂ T̄
(T̄ + T ′ ) = Uj
+ Uj
+ ūj
+ uj
∂xj
∂xj
∂xj
∂xj
∂xj
= Uj
∂
∂ T̄
+
(uj T ′ ).
∂xj
∂xj
The average of the diffusion term is
∂2
∂ 2 T̄
∂ 2T ′
∂ 2 T̄
(T̄ + T ′ ) =
+
=
.
2
2
2
∂xj
∂xj
∂xj
∂xj2
Collecting terms, the mean heat equation takes the form
∂ T̄
∂ T̄
∂
∂ 2 T̄
+ Uj
+
(uj T ′ ) = κ 2 ,
∂t
∂xj
∂xj
∂xj
554
Turbulence
which can be written as
D T̄
∂
=
Dt
∂xj
κ
∂ T̄
− uj T ′ .
∂xj
(13.27)
Multiplying by ρ0 Cp , we obtain
ρ 0 Cp
∂Qj
D T̄
=−
,
Dt
∂xj
(13.28)
∂ T̄
+ ρ 0 Cp u j T ′ ,
∂xj
(13.29)
where the heat flux is given by
Qj = −k
and k = ρ0 Cp κ is the thermal conductivity. Equation (13.29) shows that the fluctuations cause an additional mean turbulent heat flux of ρ0 Cp uT ′ , in addition to the
molecular heat flux of −k∇ T̄ . For example, the surface of the earth becomes hot
during the day, resulting in a decrease of the mean temperature with height, and an
associated turbulent convective motion. An upward fluctuating motion is then mostly
associated with a positive temperature fluctuation, giving rise to an upward heat flux
ρ0 Cp wT ′ > 0.
6. Kinetic Energy Budget of Mean Flow
In this section we shall examine the sources and sinks of mean kinetic energy of a
turbulent flow. As shown in Chapter 4, Section 13, a kinetic energy equation can be
obtained by multiplying the equation for DU/Dt by U. The equation of motion for
the mean flow is, from equations (13.25) and (13.26),
∂Ui
1 ∂ τ̄ij
g
∂Ui
+ Uj
=
− ρ̄δi3 ,
∂t
∂xj
ρ0 ∂xj
ρ0
(13.30)
where the stress is given by
τ̄ij = −P δij + 2µEij − ρ0 ui uj .
(13.31)
Here we have introduced the mean strain rate
∂Uj
1 ∂Ui
Eij ≡
.
+
2 ∂xj
∂xi
Multiplying equation (13.30) by U i (and, of course, summing over i), we obtain
∂ 1 2
1 2
∂Ui
1
g
∂
1 ∂
Ui + Uj
Ui =
(Ui τ̄ij ) − τ̄ij
− ρ̄Ui δi3 .
∂t 2
∂xj 2
ρ0 ∂xj
ρ0 ∂xj
ρ0
555
6. Kinetic Energy Budget of Mean Flow
On introducing expression (13.31) for τ̄ij , we obtain
1
∂
D 1 2
− Ui P δij + 2νUi Eij − ui uj Ui
U =
Dt 2 i
∂xj
ρ0
∂Ui
∂Ui
g
∂Ui
1
− 2νEij
+ ui uj
− ρ̄U3 .
+ P δij
ρ0
∂xj
∂xj
∂xj
ρ0
The fourth term on the right-hand side is proportional to δij (∂Ui /∂xj ) = ∂Ui /∂xi =
0 by continuity. The mean kinetic energy balance then becomes
P Uj
∂
D 1 2
−
U =
+ 2νUi Eij − ui uj Ui
Dt 2 i
∂xj
ρ0
transport
− 2νEij Eij + ui uj
viscous
dissipation
g
∂Ui
− ρ̄U3 .
∂xj
ρ0
loss to
turbulence
(13.32)
loss to
potential
energy
The left-hand side represents the rate of change of mean kinetic energy, and the
right-hand side represents the various mechanisms that bring about this change. The
first three terms are in the “flux divergence” form. If equation (13.32) is integrated
over all space to obtain the rate of change of the total (or global) kinetic energy, then
the divergence terms can be transformed into a surface integral by Gauss’ theorem.
These terms then would not contribute if the flow is confined to a limited region in
space, with U = 0 at sufficient distance. It follows that the first three terms can only
transport or redistribute energy from one region to another, but cannot generate or
dissipate it. The first term represents the transport of mean kinetic energy by the mean
pressure, the second by the mean viscous stresses 2νEij , and the third by Reynolds
stresses.
The fourth term is the product of the mean strain rate Eij and the mean viscous
stress 2νEij . It is a loss at every point in the flow and represents the direct viscous
dissipation of mean kinetic energy. The energy is lost to the agency that generates the
viscous stress, and so reappears as the kinetic energy of molecular motion (heat).
The fifth term is analogous to the fourth term. It can be written as
ui uj (∂Ui /∂xj ) = ui uj Eij , so that it is a product of the turbulent stress and the
mean strain rate field. (Note that the doubly contracted product of a symmetric tensor
ui uj and any tensor ∂Ui /∂xj is equal to the product of ui uj and symmetric part of
∂Ui /∂xj , namely, Eij ; this is proved in Chapter 2, Section 11.) If the mean flow is
given by U (y), then ui uj (∂Ui /∂xj ) = uv(dU/dy). We saw in the preceding section
that uv is likely to be negative if dU/dy is positive. The fifth term ui uj (∂Ui /∂xj )
is therefore likely to be negative in shear flows. By analogy with the fourth term, it
must represent an energy loss to the agency that generates turbulent stress, namely the
fluctuating field. Indeed, we shall see in the following section that this term appears
on the right-hand side of an equation for the rate of change of turbulent kinetic energy,
556
Turbulence
but with the sign reversed. Therefore, this term generally results in a loss of mean
kinetic energy and a gain of turbulent kinetic energy. We shall call this term the shear
production of turbulence by the interaction of Reynolds stresses and the mean shear.
The sixth term represents the work done by gravity on the mean vertical motion.
For example, an upward mean motion results in a loss of mean kinetic energy, which
is accompanied by an increase in the potential energy of the mean field.
The two viscous terms in equation (13.32), namely, the viscous transport
2ν∂(Ui Eij )/∂xj and the viscous dissipation −2νEij Eij , are small in a fully turbulent
flow at high Reynolds numbers. Compare, for example, the viscous dissipation and
the shear production terms:
2νEij2
ui uj (∂Ui /∂xj )
∼
ν
ν(U/L)2
∼
≪ 1,
2
urms U/L
UL
where U is the scale for mean velocity, L is a length scale (for example, the width of
the boundary layer), and urms is the rms value of the turbulent fluctuation; we have
also assumed that urms and U are of the same order, since experiments show that
urms is a substantial fraction of U . The direct influence of viscous terms is therefore
negligible on the mean kinetic energy budget. We shall see in the following section
that this is not true for the turbulent kinetic energy budget, in which the viscous terms
play a major role. What happens is the following: The mean flow loses energy to
the turbulent field by means of the shear production; the turbulent kinetic energy so
generated is then dissipated by viscosity.
7. Kinetic Energy Budget of Turbulent Flow
An equation for the turbulent kinetic energy is obtained by first finding an equation
for ∂u/∂t and taking the scalar product with u. The algebra becomes compact if we
use the “comma notation,” introduced in Chapter 2, Section 15, namely, that a comma
denotes a spatial derivative:
∂A
,
A,i ≡
∂xi
where A is any variable. (This notation is very simple and handy, but it may take a
little practice to get used to it. It is used in this book only if the algebra would become
cumbersome otherwise. There is only one other place in the book where this notation
has been applied, namely Section 5.7. With a little initial patience, the reader will
quickly see the convenience of this notation.)
Equations of motion for the total and mean flows are, respectively,
∂
(Ui + ui ) + (Uj + uj )(Ui + ui ),j
∂t
1
= − (P + p),i − g[1 − α(T̄ + T ′ − T0 )]δi3 + ν(Ui + ui ),jj ,
ρ0
∂Ui
1
+ Uj Ui,j = − P,i − g[1 − α(T̄ − T0 )] δi3 + νUi,jj − (ui uj ),j .
∂t
ρ0
557
7. Kinetic Energy Budget of Turbulent Flow
Subtracting, we obtain the equation of motion for the turbulent velocity ui :
∂ui
1
+ Uj ui,j + uj Ui,j + uj ui,j − (ui uj ),j = − p,i + gαT ′ δi3 + νui,jj .
∂t
ρ0
(13.33)
The equation for the turbulent kinetic energy is obtained by multiplying this equation
by ui and averaging.
The first two terms on the left-hand side of equation (13.33) give
1 2
u ,
2 i
1 2
= Uj
u
.
2 i ,j
∂ui
∂
ui
=
∂t
∂t
ui Uj ui,j
The third, fourth and fifth terms on the left-hand side of equation (13.33) give
ui uj Ui,j = ui uj Ui,j ,
ui uj ui,j = ( 21 u2i uj ),j − 21 u2i uj,j = 21 (u2i uj ),j ,
−ui (ui uj ),j = −ūi (ui uj ),j = 0,
where we have used the continuity equation ui,i = 0 and ūi = 0.
The first and second terms on the right-hand side of equation (13.33) give
−ui
1
1
p,i = − (ui p),i ,
ρ0
ρ0
ui gαT ′ δi3 = gαwT ′ .
The last term on the right-hand side of equation (13.33) gives
νui ui,jj = ν{ui ui,jj + 21 (ui,j + uj,i )(ui,j − uj,i )},
where we have added the doubly contracted product of a symmetric tensor (ui,j +uj,i )
and an antisymmetric tensor (ui,j − uj,i ), such a product being zero. In the first term
on the right-hand side, we can write ui,jj = (ui,j + uj,i ),j because of the continuity
equation. Then we can write
νui ui,jj = ν{ui (ui,j + uj,i ),j + (ui,j + uj,i )(ui,j − 21 ui,j − 21 uj,i )}
= ν{[ui (ui,j + uj,i )],j − 21 (ui,j + uj,i )2 }.
Defining the fluctuating strain rate by
eij ≡ 21 (ui,j + uj,i ),
558
Turbulence
we finally obtain
νui ui,jj = 2ν[ui eij ],j − 2νeij eij .
Collecting terms, the turbulent energy equation becomes
D
Dt
1 2
u
2 i
=−
∂
∂xj
1
1
puj + u2i uj − 2νui eij
ρ0
2
transport
− ui uj Ui,j + gαwT ′ − 2νeij eij .
shear prod
buoyant prod
(13.34)
viscous diss
The first three terms on the right-hand side are in the flux divergence form and consequently represent the spatial transport of turbulent kinetic energy. The first two
terms represent the transport by turbulence itself, whereas the third term is viscous
transport.
The fourth term ui uj Ui,j also appears in the kinetic energy budget of the
mean flow with its sign reversed, as seen by comparing equation (13.32) and equation (13.34). As argued in the preceding section, −ui uj Ui,j is usually positive, so
that this term represents a loss of mean kinetic energy and a gain of turbulent kinetic
energy. It must then represent the rate of generation of turbulent kinetic energy by the
interaction of the Reynolds stress with the mean shear Ui,j . Therefore,
Shear production = −ui uj
∂Ui
.
∂xj
(13.35)
The fifth term gαwT ′ can have either sign, depending on the nature of the background temperature distribution T̄ (z). In a stable situation in which the background
temperature increases upward (as found, e.g., in the atmospheric boundary layer at
night), rising fluid elements are likely to be associated with a negative temperature
fluctuation, resulting in wT ′ < 0, which means a downward turbulent heat flux. In
such a stable situation gαwT ′ represents the rate of turbulent energy loss by working against the stable background density gradient. In the opposite case, when the
background density profile is unstable, the turbulent heat flux wT ′ is upward, and
convective motions cause an increase of turbulent kinetic energy (Figure 13.9). We
shall call gαwT ′ the buoyant production of turbulent kinetic energy, keeping in mind
that it can also be a buoyant “destruction” if the turbulent heat flux is downward.
Therefore,
Buoyant production = gαwT ′ .
(13.36)
The buoyant generation of turbulent kinetic energy lowers the potential energy
of the mean field. This can be understood from Figure 13.9, where it is seen that the
heavier fluid has moved downward in the final state as a result of the heat flux. This
559
8. Turbulence Production and Cascade
Figure 13.9 Heat flux in an unstable environment, generating turbulent kinetic energy and lowering the
mean potential energy.
can also be demonstrated by deriving an equation for the mean potential energy, in
which the term gαwT ′ appears with a negative sign on the right-hand side. Therefore,
the buoyant generation of turbulent kinetic energy by the upward heat flux occurs at
the expense of the mean potential energy. This is in contrast to the shear production
of turbulent kinetic energy, which occurs at the expense of the mean kinetic energy.
The sixth term 2νeij eij is the viscous dissipation of turbulent kinetic energy, and
is usually denoted by ε:
ε = Viscous dissipation = 2νeij eij .
(13.37)
This term is not negligible in the turbulent kinetic energy equation, although an
analogous term (namely 2νEij2 ) is negligible in the mean kinetic energy equation, as
discussed in the preceding section. In fact, the viscous dissipation ε is of the order of
the turbulence production terms (ui uj Ui,j or gαwT ′ ) in most locations.
8. Turbulence Production and Cascade
Evidence suggests that the large eddies in a turbulent flow are anisotropic, in the
sense that they are “aware” of the direction of mean shear or of background density
gradient. In a completely isotropic field the off-diagonal components of the Reynolds
stress ui uj are zero (see Section 5 here), as is the upward heat flux wT ′ because there
is no preference between the upward and downward directions. In such an isotropic
case no turbulent energy can be extracted from the mean field. Therefore, turbulence
must develop anisotropy if it has to sustain itself against viscous dissipation.
A possible mechanism of generating anisotropy in a turbulent shear flow is discussed by Tennekes and Lumley (1972, p. 41). Consider a parallel shear flow U (y)
560
Turbulence
Figure 13.10 Large eddies oriented along the principal directions of a parallel shear flow. Note that the
vortex aligned with the α-axis has a positive v when u is negative and a negative v when u is positive,
resulting in uv < 0.
shown in Figure 13.10, in which the fluid elements translate, rotate, and undergo
shearing deformation. The nature of deformation of an element depends on the orientation of the element. An element oriented parallel to the xy-axes undergoes only
a shear strain rate Exy = 21 dU/dy, but no linear strain rate (Exx = Eyy = 0). The
strain rate tensor in the xy-coordinate system is therefore
1
0
2 dU/dy
.
E= 1
0
2 dU/dy
As shown in Chapter 3, Section 10, such a symmetric tensor can be diagonalized by
rotating the coordinate system by 45◦ . Along these principal axes (denoted by α and
β in Figure 13.10), the strain rate tensor is
1
0
2 dU/dy
,
E=
0
− 21 dU/dy
so that there is a linear extension rate of Eαα = 21 dU/dy, a linear compression
rate of Eββ = − 21 dU/dy, and no shear (Eαβ = 0). The kinematics of stretching and
compression along the principal directions in a parallel shear flow is discussed further
in Chapter 3, Section 10.
The large eddies with vorticity oriented along the α-axis intensify in strength due
to the vortex stretching, and the ones with vorticity oriented along the β-axis decay
in strength. The net effect of the mean shear on the turbulent field is therefore to
cause a predominance of eddies with vorticity oriented along the α-axis. As is evident
in Figure 13.10, these eddies are associated with a positive u when v is negative,
8. Turbulence Production and Cascade
and with a negative u when v is positive, resulting in a positive value for the shear
production −uv(dU/dy).
The largest eddies are of order of the width of the shear flow, for example the
diameter of a pipe or the width of a boundary layer along a wall or along the upper
surface of the ocean. These eddies extract kinetic energy from the mean field. The
eddies that are somewhat smaller than these are strained by the velocity field of the
largest eddies, and extract energy from the larger eddies by the same mechanism of
vortex stretching. The much smaller eddies are essentially advected in the velocity
field of the large eddies, as the scales of the strain rate field of the large eddies are much
larger than the size of a small eddy. Therefore, the small eddies do not interact with
either the large eddies or the mean field. The kinetic energy is therefore cascaded
down from large to small eddies in a series of small steps. This process of energy
cascade is essentially inviscid, as the vortex stretching mechanism arises from the
nonlinear terms of the equations of motion.
In a fully turbulent shear flow (i.e., for large Reynolds numbers), therefore, the
viscosity of the fluid does not affect the shear production, if all other variables are
held constant. The viscosity does, however, determine the scales at which turbulent
energy is dissipated into heat. From the expression ε = 2νeij eij , it is clear that the
energy dissipation is effective only at very small scales, which have high fluctuating
strain rates. The continuous stretching and cascade generate long and thin filaments,
somewhat like “spaghetti.” When these filaments become thin enough, molecular
diffusive effects are able to smear out their velocity gradients. These are the smallest scales in a turbulent flow and are responsible for the dissipation of the turbulent
kinetic energy. Figure 13.11 illustrates the deformation of a fluid particle in a turbulent motion, suggesting that molecular effects can act on thin filaments generated
by continuous stretching. The large mixing rates in a turbulent flow, therefore, are
essentially a result of the turbulent fluctuations generating the large surfaces on which
the molecular diffusion finally acts.
It is clear that ε does not depend on ν, but is determined by the inviscid properties
of the large eddies, which supply the energy to the dissipating scales. Suppose l is
a typical length scale of the large eddies (which may be taken equal to the integral
length scale defined from a spatial correlation function, analogous to the integral time
scale defined by equation (13.10)), and u′ is a typical scale of the fluctuating velocity
Figure 13.11 Successive deformations of a marked fluid element. Diffusive effects cause smearing when
the scale becomes of the order of the Kolmogorov microscale.
561
562
Turbulence
(which may be taken equal to the rms fluctuating speed). Then the time scale of large
eddies is of order l/u′ . Observations show that the large eddies lose much of their
energy during the time they turn over one or two times, so that the rate of energy
transferred from large eddies is proportional to u′2 times their frequency u′ / l. The
dissipation rate must then be of order
ε∼
u′3
,
l
(13.38)
signifying that the viscous dissipation is determined by the inviscid large-scale
dynamics of the turbulent field.
Kolmogorov suggested in 1941 that the size of the dissipating eddies depends
on those parameters that are relevant to the smallest eddies. These parameters are the
rate ε at which energy has to be dissipated by the eddies and the diffusivity ν that
does the smearing out of the velocity gradients. As the unit of ε is m2 /s3 , dimensional
reasoning shows that the length scale formed from ε and ν is
η=
ν3
ε
1/4
,
(13.39)
which is called the Kolmogorov microscale. A decrease of ν merely decreases the scale
at which viscous dissipation takes place, and not the rate of dissipation ε. Estimates
show that η is of the order of millimeters in the ocean and the atmosphere. In laboratory
flows the Kolmogorov microscale is much smaller because of the larger rate of viscous
dissipation. Landahl and Mollo-Christensen (1986) give a nice illustration of this.
Suppose we are using a 100-W household mixer in 1 kg of water. As all the power is
used to generate the turbulence, the rate of dissipation is ε = 100 W/kg = 100 m2 /s3 .
Using ν = 10−6 m2 /s for water, we obtain η = 10−2 mm.
9. Spectrum of Turbulence in Inertial Subrange
In Section 4 we defined the wavenumber spectrum S(K), representing turbulent
kinetic energy as a function of the wavenumber vector K. If the turbulence is isotropic,
then the spectrum becomes independent of the orientation of the wavenumber vector
and depends on its magnitude K only. In that case we can write
∞
2
S(K) dK.
u =
0
In this section we shall derive the form of S(K) in a certain range of wavenumbers
in which the turbulence is nearly isotropic.
Somewhat vaguely, we shall associate a wavenumber K with an eddy of size K −1 .
Small eddies are therefore represented by large wavenumbers. Suppose l is the scale
of the large eddies, which may be the width of the boundary layer. At the relatively
563
9. Spectrum of Turbulence in Inertial Subrange
small scales represented by wavenumbers K ≫ l −1 , there is no direct interaction
between the turbulence and the motion of the large, energy-containing eddies. This is
because the small scales have been generated by a long series of small steps, losing
information at each step. The spectrum in this range of large wavenumbers is nearly
isotropic, as only the large eddies are aware of the directions of mean gradients. The
spectrum here does not depend on how much energy is present at large scales (where
most of the energy is contained), or the scales at which most of the energy is present.
The spectrum in this range depends only on the parameters that determine the nature
of the small-scale flow, so that we can write
S = S(K, ε, ν)
K ≫ l −1 .
The range of wavenumbers K ≫ l −1 is usually called the equilibrium range. The
dissipating wavenumbers with K ∼ η−1 , beyond which the spectrum falls off very
rapidly, form the high end of the equilibrium range (Figure 13.12). The lower end
of this range, for which l −1 ≪ K ≪ η−1 , is called the inertial subrange, as only
the transfer of energy by inertial forces (vortex stretching) takes place in this range.
Both production and dissipation are small in the inertial subrange. The production of
energy by large eddies causes a peak of S at a certain K ≃ l −1 , and the dissipation
of energy causes a sharp drop of S for K > η−1 . The question is, how does S vary
with K between the two limits in the inertial subrange?
Figure 13.12 A typical wavenumber spectrum observed in the ocean, plotted on a log–log scale. The
unit of S is arbitrary, and the dots represent hypothetical data.
564
Turbulence
Kolmogorov argued that, in the inertial subrange part of the equilibrium range,
S is independent of ν also, so that
S = S(K, ε)
l −1 ≪ K ≪ η−1 .
Although little dissipation takes place in the inertial subrange, the spectrum here does
depend on ε. This is because the energy that is dissipated must be transferred across
the inertial subrange, from low to high wavenumbers. As the unit of S is m3 /s2 and
that of ε is m2 /s3 , dimensional reasoning gives
S = Aε2/3 K −5/3
l −1 ≪ K ≪ η−1 ,
(13.40)
where A ≃ 1.5 has been found to be a universal constant, valid for all turbulent
flows. Equation (13.40) is usually called Kolmogorov’s K −5/3 law. If the Reynolds
number of the flow is large, then the dissipating eddies are much smaller than the
energy-containing eddies, and the inertial subrange is quite broad.
Because very large Reynolds numbers are difficult to generate in the laboratory,
the Kolmogorov spectral law was not verified for many years. In fact, doubts were
being raised about its theoretical validity. The first confirmation of the Kolmogorov
law came from the oceanic observations of Grant et al. (1962), who obtained a velocity
spectrum in a tidal flow through a narrow passage between two islands near the west
coast of Canada. The velocity fluctuations were measured by hanging a hot film
anemometer from the bottom of a ship. Based on the depth of water and the average
flow velocity, the Reynolds number was of order 108 . Such large Reynolds numbers
are typical of geophysical flows, since the length scales are very large. The K −5/3
law has since been verified in the ocean over a wide range of wavenumbers, a typical
behavior being sketched in Figure 13.12. Note that the spectrum drops sharply at
Kη ∼ 1, where viscosity begins to affect the spectral shape. The figure also shows
that the spectrum departs from the K −5/3 law for small values of the wavenumber,
where the turbulence production by large eddies begins to affect the spectral shape.
Laboratory experiments are also in agreement with the Kolmogorov spectral law,
although in a narrower range of wavenumbers because the Reynolds number is not as
large as in geophysical flows. The K −5/3 law has become one of the most important
results of turbulence theory.
10. Wall-Free Shear Flow
Nearly parallel shear flows are divided into two classes—wall-free shear flows and
wall-bounded shear flows. In this section we shall examine some aspects of turbulent
flows that are free of solid boundaries. Common examples of such flows are jets,
wakes, and shear layers (Figure 13.13). For simplicity we shall consider only plane
two-dimensional flows. Axisymmetric flows are discussed in Townsend (1976) and
Tennekes and Lumley (1972).
10. Wall-Free Shear Flow
Intermittency
Consider a turbulent flow confined to a limited region. To be specific we shall consider
the example of a wake (Figure 13.13b), but our discussion also applies to a jet, a shear
layer, or the outer part of a boundary layer on a wall. The fluid outside the turbulent
Figure 13.13 Three types of wall-free turbulent flows: (a) jet; (b) wake; and (c) shear layer.
565
566
Turbulence
region is either in irrotational motion (as in the case of a wake or a boundary layer), or
nearly static (as in the case of a jet). Observations show that the instantaneous interface
between the turbulent and nonturbulent fluid is very sharp. In fact, the thickness of the
interface must equal the size of the smallest scales in the flow, namely the Kolmogorov
microscale. The interface is highly contorted due to the presence of eddies of various
sizes. However, a photograph exposed for a long time does not show such an irregular
and sharp interface but rather a gradual and smooth transition region.
Measurements at a fixed point in the outer part of the turbulent region (say at
point P in Figure 13.13b) show periods of high-frequency fluctuations as the point P
moves into the turbulent flow and quiet periods as the point moves out of the turbulent
region. Intermittency γ is defined as the fraction of time the flow at a point is turbulent.
The variation of γ across a wake is sketched in Figure 13.13b, showing that γ = 1
near the center where the flow is always turbulent, and γ = 0 at the outer edge of
the flow.
Entrainment
A flow can slowly pull the surrounding irrotational fluid inward by “frictional” effects;
the process is called entrainment. The source of this “friction” is viscous in laminar
flow and inertial in turbulent flow. The entrainment of a laminar jet was discussed in
Chapter 10, Section 12. The entrainment in a turbulent flow is similar, but the rate is
much larger. After the irrotational fluid is drawn inside a turbulent region, the new
fluid must be made turbulent. This is initiated by small eddies (which are dominated
by viscosity) acting at the sharp interface between the turbulent and the nonturbulent
fluid (Figure 13.14).
The foregoing discussion of intermittency and entrainment applies not only to
wall-free shear flows but also to the outer edge of boundary layers.
Self-Preservation
Far downstream, experiments show that the mean field in a wall-free shear flow
becomes approximately self-similar at various downstream distances. As the mean
field is affected by the Reynolds stress through the equations of motion, this means that
the various turbulent quantities (such as Reynolds stress) also must reach self-similar
Figure 13.14 Entrainment of a nonturbulent fluid and its assimilation into turbulent fluid by viscous
action at the interface.
567
10. Wall-Free Shear Flow
states. This is indeed found to be approximately true (Townsend, 1976). The flow is
then in a state of “moving equilibrium,” in which both the mean and the turbulent
fields are determined solely by the local scales of length and velocity. This is called
self-preservation. In the self-similar state, the mean velocity at various downstream
distances is given by
y
U
=f
Uc
δ
y
U∞ − U
=f
U∞ − U c
δ
y
U − U1
=f
U2 − U 1
δ
(jet),
(wake),
(13.41)
(shear layer).
Here δ(x) is the width of flow, Uc (x) is the centerline velocity for the jet and the wake,
and U1 and U2 are the velocities of the two streams in a shear layer (Figure 13.13).
Consequence of Self-Preservation in a Plane Jet
We shall now derive how the centerline velocity and width in a plane jet must vary if
we assume that the mean velocity profiles at various downstream distances are self
similar. This can be done by examining the equations of motion in differential form.
An alternate way is to examine an integral form of the equation of motion, derived
2 in
Chapter 10, Section 12. It was shown there that the momentum
flux
M
=
ρ
U dy
across the jet is independent of x, while the mass flux ρ U dy increases downstream
due to entrainment. Exactly the same constraint applies to a turbulent jet. For the
sake of readers who find cross references annoying, the integral constraint for a
two-dimensional jet is rederived here.
Consider a control volume shown by the dotted line in Figure 13.13a, in which the
horizontal surfaces of the control volume are assumed to be at a large distance from
the jet axis. At these large distances, there is a mean V field toward the jet axis due to
entrainment, but no U field. Therefore, the flow of x-momentum through the horizontal surfaces of the control volume is zero. The pressure is uniform throughout the flow,
and the viscous forces are negligible. The net force on the surface of the control volume is therefore zero. The momentum principle for a control volume (see Chapter 4,
Section 8) states that the net x-directed force on the boundary equals the net rate of
outflow of x-momentum through the control surfaces. As the net force here is zero,
the influx of x-momentum must equal the outflow of x-momentum. That is
M=ρ
∞
−∞
U 2 dy = independent of x,
(13.42)
where M is the momentum flux of the jet (= integral of mass flux ρU dy times velocity U ). The momentum flux is the basic externally controlled parameter for a jet and
isknown from an evaluation of equation (13.42) at the orifice opening. The mass flux
ρ U dy across the jet must increase because of entrainment of the surrounding fluid.
568
Turbulence
The assumption of self similarity can now be used to predict how δ and Uc in a
jet should vary with x. Substitution of the self-similarity assumption (13.41) into the
integral constraint (13.42) gives
∞
y
.
f2d
M = ρUc2 δ
δ
−∞
The preceding integral is a constant because it is completely expressed in terms of
the nondimensional function f (y/δ). As M is also a constant, we must have
Uc2 δ = const.
(13.43)
At this point we make another important assumption. We assume that the
Reynolds number is large, so that the gross characteristics of the flow are independent
of the Reynolds number. This is called Reynolds number similarity. The assumption
is expected to be valid in a wall-free shear flow, as viscosity does not directly affect
the motion; a decrease of ν, for example, merely decreases the scale of the dissipating eddies, as discussed in Section 8. (The principle is not valid near a smooth wall,
and as a consequence the drag coefficient for a smooth flat plate does not become
independent of the Reynolds number as Re → ∞; see Figure 10.12.) For large Re,
then, Uc is independent of viscosity and can only depend on x, ρ, and M:
Uc = Uc (x, ρ, M).
A dimensional analysis shows that
Uc ∝
M
ρx
( jet),
(13.44)
so that equation (13.43) requires
δ∝x
( jet).
(13.45)
This should be compared with the δ ∝ x 2/3 behavior of a laminar jet, derived in
Chapter 10, Section 12. Experiments show that the width of a turbulent jet does grow
linearly, with a spreading angle of 4◦ .
For two-dimensional wakes and shear layers, it can be shown (Townsend, 1976;
Tennekes and Lumley, 1972) that the assumption of self similarity requires
√
U∞ − Uc ∝ x −1/2 , δ ∝ x
(wake),
U1 − U2 = const., δ ∝ x
(shear layer).
Turbulent Kinetic Energy Budget in a Jet
The turbulent kinetic energy equation derived in Section 7 will now be applied to
a two-dimensional jet. The energy budget calculation uses the experimentally measured distributions of turbulence intensity and Reynolds stress across the jet. Therefore, we present the distributions of these variables first. Measurements show that
569
10. Wall-Free Shear Flow
Figure 13.15 Sketch of observed variation of turbulent intensity and Reynolds stress across a jet.
the turbulent intensities and Reynolds stress are distributed as in Figure 13.15. Here
u2 is the intensity of fluctuation in the downstream direction x, v 2 is the intensity along the cross-stream direction y, and w 2 is the intensity in the z-direction;
q 2 ≡ (u2 + v 2 + w 2 )/2 is the turbulent kinetic energy per unit mass. The Reynolds
stress is zero at the center of the jet by symmetry, since there is no reason for v at the
center to be mostly of one sign if u is either positive or negative. The Reynolds stress
reaches a maximum magnitude roughly where ∂U/∂y is maximum. This is also close
to the region where the turbulent kinetic energy reaches a maximum.
Consider now the kinetic energy budget. For a two-dimensional jet under the
boundary layer assumption ∂/∂x ≪ ∂/∂y, equation (13.34) becomes
0 = −U
∂q 2
∂ 2
∂U
∂q 2
−V
− uv
−
q v + pv/ρ − ε,
∂x
∂y
∂y
∂y
(13.46)
where the left-hand side represents ∂q 2 /∂t = 0. Here the viscous transport and
a term (v 2 − u2 )(∂U/∂x) arising out of the shear production have been neglected
on the right-hand side because they are small. The balance of terms is analyzed in
Townsend (1976), and the results are shown in Figure 13.16, where T denotes turbulent
transport represented by the fourth term on the right-hand side of (13.46). The shear
production is zero at the center where both ∂U/∂y and uv are zero, and reaches a
maximum close to the position of the maximum Reynolds stress. Near the center, the
dissipation is primarily balanced by the downstream advection −U (∂q 2 /∂x), which is
positive because the turbulent intensity q 2 decays downstream. Away from the center,
but not too close to the outer edge of the jet, the production and dissipation terms
balance. In the outer parts of the jet, the transport term balances the cross-stream
570
Turbulence
Figure 13.16 Sketch of observed kinetic energy budget in a turbulent jet. Turbulent transport is indicated by T .
advection. In this region V is negative (i.e., toward the center) due to entrainment
of the surrounding fluid, and also q 2 decreases with y. Therefore the cross-stream
advection −V (∂q 2 /∂y) is negative, signifying that the entrainment velocity V tends
to decrease the turbulent kinetic energy at the outer edge of the jet. The stationary
state is therefore maintained by the transport term T carrying turbulent kinetic energy
away from the center (where T < 0) into the outer parts of the jet (where T > 0).
11. Wall-Bounded Shear Flow
The gross characteristics of free shear flows, discussed in the preceding section, are
independent of viscosity. This is not true of a turbulent flow bounded by a solid wall,
in which the presence of viscosity affects the motion near the wall. The effect of
viscosity is reflected in the fact that the drag coefficient of a smooth flat plate depends
on the Reynolds number even for Re → ∞, as seen in Figure 10.12. Therefore,
the concept of Reynolds number similarity, which says that the gross characteristics
are independent of Re when Re → ∞, no longer applies. In this section we shall
examine how the properties of a turbulent flow near a wall are affected by viscosity.
Before doing this, we shall examine how the Reynolds stress should vary with distance
from the wall.
Consider first a fully developed turbulent flow in a channel. By “fully developed”
we mean that the flow is no longer changing in x (see Figure 9.2). Then the mean
equation of motion is
0=−
∂ τ̄
∂P
+
,
∂x
∂y
571
11. Wall-Bounded Shear Flow
Figure 13.17 Variation of shear stress across a channel and a boundary layer: (a) channel; and (b) boundary
layer.
where τ̄ = µ(dU/dy) − ρuv is the total stress. Because ∂P /∂x is a function of x
alone and ∂ τ̄ /∂y is a function of y alone, both of them must be constants. The stress
distribution is then linear (Figure 13.17a). Away from the wall τ̄ is due mostly to the
Reynolds stress, but close to the wall the viscous contribution dominates. In fact, at
the wall the velocity fluctuations and consequently the Reynolds stresses vanish, so
that the stress is entirely viscous.
In a boundary layer on a flat plate there is no pressure gradient and the mean flow
equation is
ρU
∂U
∂ τ̄
∂U
+ ρV
=
,
∂x
∂y
∂y
where τ̄ is a function of x and y. The variation of the stress across a boundary layer
is sketched in Figure 13.17b.
Inner Layer: Law of the Wall
Consider the flow near the wall of a channel, pipe, or boundary layer. Let U∞ be the
free-stream velocity in a boundary layer or the centerline velocity in a channel and
pipe. Let δ be the width of flow, which may be the width of the boundary layer, the
channel half width, or the radius of the pipe. Assume that the wall is smooth, so that
the height of the surface roughness elements is too small to affect the flow. Physical
considerations suggest that the velocity profile near the wall depends only on the
parameters that are relevant near the wall and does not depend on the free-stream
velocity U∞ or the thickness of the flow δ. Very near a smooth surface, then, we
expect that
U = U (ρ, τ0 , ν, y),
(13.47)
572
Turbulence
where τ0 is the shear stress at the wall. To express equation (13.47) in terms of
dimensionless variables, note that only ρ and τ0 involve the dimension of mass, so
that these two variables must always occur together in any nondimensional group.
The important ratio
u∗ ≡
τ0
,
ρ
(13.48)
has the dimension of velocity and is called the friction velocity. Equation (13.47) can
then be written as
U = U (u∗ , ν, y).
(13.49)
This relates four variables involving only the two dimensions of length and time.
According to the pi theorem (Chapter 8, Section 4) there must be only 4 − 2 = 2
nondimensional groups U/u∗ and yu∗ /ν, which should be related by some universal
functional form
yu
U
∗
= f (y+ )
(law of the wall),
(13.50)
=f
u∗
ν
where y+ ≡ yu∗ /ν is the distance nondimensionalized by the viscous scale ν/u∗ .
Equation (13.50) is called the law of the wall, and states that U/u∗ must be a universal
function of yu∗ /ν near a smooth wall.
The inner part of the wall layer, right next to the wall, is dominated by viscous
effects (Figure 13.18) and is called the viscous sublayer. It used to be called the
“laminar sublayer,” until experiments revealed the presence of considerable fluctuations within the layer. In spite of the fluctuations, the Reynolds stresses are still
small here because of the dominance of viscous effects. Because of the thinness of
the viscous sublayer, the stress can be taken as uniform within the layer and equal
to the wall shear stress τ0 . Therefore the velocity gradient in the viscous sublayer is
given by
µ
dU
= τ0 ,
dy
which shows that the velocity distribution is linear. Integrating, and using the no-slip
boundary condition, we obtain
U=
yτ0
.
µ
In terms of nondimensional variables appropriate for a wall layer, this can be written as
U
= y+
u∗
(viscous sublayer).
(13.51)
Experiments show that the linear distribution holds up to yu∗ /ν ∼ 5, which may be
taken to be the limit of the viscous sublayer.
573
11. Wall-Bounded Shear Flow
Figure 13.18 Law of the wall. A typical data cloud is shaded.
Outer Layer: Velocity Defect Law
We now explore the form of the velocity distribution in the outer part of a turbulent
layer. The gross characteristics of the turbulence in the outer region are inviscid and
resemble those of a wall-free turbulent flow. The existence of Reynolds stresses in the
outer region results in a drag on the flow and generates a velocity defect (U∞ − U ),
which is expected to be proportional to the wall friction characterized by u∗ . It follows
that the velocity distribution in the outer region must have the form
y
U − U∞
=F
= F (ξ )
u∗
δ
(velocity defect law),
(13.52)
where ξ ≡ y/δ. This is called the velocity defect law.
Overlap Layer: Logarithmic Law
The velocity profiles in the inner and outer parts of the boundary layer are governed
by different laws (13.50) and (13.52), in which the independent variable y is scaled
differently. Distances in the outer part are scaled by δ, whereas those in the inner part
are measured by the much smaller viscous scale ν/u∗ . In other words, the small distances in the inner layer are magnified by expressing them as yu∗ /ν. This is the typical
behavior in singular perturbation problems (see Chapter 10, Sections 14 and 16). In
these problems the inner and outer solutions are matched together in a region of overlap by taking the limits y+ → ∞ and ξ → 0 simultaneously. Instead of matching
velocity, in this case it is more convenient to match their gradients. (The derivation
574
Turbulence
given here closely follows Tennekes and Lumley (1972).) From equations (13.50)
and (13.52), the velocity gradients in the inner and outer regions are given by
u2 df
dU
,
= ∗
dy
ν dy+
dU
u∗ dF
=
.
dy
δ dξ
(13.53)
(13.54)
Equating (13.53) and (13.54) and multiplying by y/u∗ , we obtain
ξ
df
1
dF
= y+
= ,
dξ
dy+
k
(13.55)
valid for large y+ and small ξ . As the left-hand side can only be a function of ξ and
the right-hand side can only be a function of y+ , both sides must be equal to the same
universal constant, say 1/k, where k is called the von Karman constant. Experiments
show that k ≃ 0.41. Integration of equation (13.55) gives
1
ln y+ + A,
k
1
F (ξ ) = ln ξ + B.
k
f (y+ ) =
(13.56)
Experiments show that A = 5.0 and B = −1.0 for a smooth flat plate, for which
equations (13.56) become
U
1 yu∗
= ln
+ 5.0,
u∗
k
ν
U − U∞ 1 y
= ln − 1.0.
u∗
k δ
(13.57)
(13.58)
These are the velocity distributions in the overlap layer, also called the inertial sublayer or simply the logarithmic layer. As the derivation shows, these laws are only
valid for large y+ and small y/δ.
The foregoing method of justifying the logarithmic velocity distribution near a
wall was first given by Clark B. Millikan in 1938, before the formal theory of singular
perturbation problems was fully developed. The logarithmic law, however, was known
from experiments conducted by the German researchers, and several derivations based
on semiempirical theories were proposed by Prandtl and von Karman. One such
derivation by the so-called mixing length theory is presented in the following section.
The logarithmic velocity distribution near a surface can be derived solely on
dimensional grounds. In this layer the velocity gradient dU/dy can only depend on
the local distance y and on the only relevant velocity scale near the surface, namely u∗ .
(The layer is far enough from the wall so that the direct effect of ν is not relevant
and far enough from the outer part of the turbulent layer so that the effect of δ is not
575
11. Wall-Bounded Shear Flow
relevant.) A dimensional analysis gives
u∗
dU
=
,
dy
ky
where the von Karman constant k is introduced for consistency with the preceding
formulas. Integration gives
U=
u∗
ln y + const.
k
(13.59)
It is therefore apparent that dimensional considerations alone lead to the logarithmic
velocity distribution near a wall. In fact, the constant of integration can be adjusted
to reduce equation (13.59) to equation (13.57) or (13.58). For example, matching the
profile to the edge of the viscous sublayer at y = 10.7ν/u∗ reduces equation (13.59)
to equation (13.57) (Exercise 8). The logarithmic velocity distribution also applies to
rough walls, as discussed later in the section.
The experimental data on the velocity distribution near a wall is sketched in
Figure 13.18. It is a semilogarithmic plot in terms of the inner variables. It shows that
the linear velocity distribution (13.51) is valid for y+ < 5, so that we can take the
viscous sublayer thickness to be
δν ≃
5ν
u∗
(viscous sublayer thickness).
The logarithmic velocity distribution (13.57) is seen to be valid for 30 < y+ < 300.
The upper limit on y+ , however, depends on the Reynolds number and becomes
larger as Re increases. There is therefore a large logarithmic overlap region in flows
at large Reynolds numbers. The close analogy between the overlap region in physical
space and inertial subrange in spectral space is evident. In both regions, there is little
production or dissipation; there is simply an “inertial” transfer across the region by
inviscid nonlinear processes. It is for this reason that the logarithmic layer is called
the inertial sublayer.
As equation (13.58) suggests, a logarithmic velocity distribution in the overlap
region can also be plotted in terms of the outer variables of (U − U∞ )/u∗ vs y/δ.
Such plots show that the logarithmic distribution is valid for y/δ < 0.2. The logarithmic law, therefore, holds accurately in a rather small percentage (∼20%) of the
total boundary layer thickness. The general defect law (13.52), where F (ξ ) is not
necessarily logarithmic, holds almost everywhere except in the inner part of the wall
layer.
The region 5 < y+ < 30, where the velocity distribution is neither linear nor
logarithmic, is called the buffer layer. Neither the viscous stress nor the Reynolds
stress is negligible here. This layer is dynamically very important, as the turbulence
production −uv(dU/dy) reaches a maximum here due to the large velocity gradients.
Wosnik et al. (2000) very carefully reexamined turbulent pipe and channel flows
and compared their results with superpipe data and scalings developed by Zagarola
and Smits (1998), and others. Very briefly, Figure 13.18 is split into more regions
576
Turbulence
in that a “mesolayer” is required between the buffer layer and the inertial sublayer.
Proper description of the velocity in this mesolayer requires an offset parameter in the
logarithm of equations (13.56). This is obtained by generalizing equation (13.55) to
(ξ + ā)
df
1
dF
= (y+ + a+ )
= ,
d(ξ + ā)
d(y+ + a+ )
k
where ā = a/δ, a+ = au∗ /ν.
Equations (13.56) become
f (y+ ) = k −1 ln(y+ + a+ ) + A,
F (ξ ) = k −1 ln(ξ + ā) + B.
The value for a+ suggested by Wosnik et al. that best fits the superpipe data is
a+ = −8.
A more rational asymptotic treatment was given by Buschmann
and Gad-el-Hak
(2003a) in terms of an expansion for large Karman number δ + = (Cf /2)·(δ/θ)Reθ
in the case of a zero pressure gradient turbulent boundary layer. Here Cf is the
skin friction coefficient defined in (10.38) and θ is the momentum thickness defined
in (10.17). Reθ is the Reynolds number based on the local momentum thickness of
the boundary layer. The second author had previously found δ + = 1.168(Reθ ).875
empirically over a wide range of Re. U/u∗ is expanded in both the inner layer (y + )
and the outer layer (η = y/δ) in negative powers of δ + . To lowest order we recover the
simple log velocity profile [(13.59)]. Higher-order terms include powers of the inner
and outer variables. After matching in an overlap region, the remaining coefficients
are ultimately determined by comparison with experiments. Comparing with alternative forms for the turbulent velocity profiles, Buschmann and Gad-el-Hak (2003b)
conclude that the generalized log law gives a better fit over an extended range of y +
than any alternative velocity profile. Also, as Reθ increases, the higher-order terms in
the Karman number expansion become asymptotically small.
The outer region of turbulent boundary layers (y+ > 100) is the subject of a
similarity analysis by Castillo and George (2001). They found that 90% of a turbulent flow under all pressure gradients is characterized by a single pressure gradient
parameter,
=
dp∞
δ
.
2 dδ/dx dx
ρU∞
A requirement for “equilibrium” turbulent boundary layer flows, to which their analysis is restricted, is that = const., and this leads to similarity. Examination of
data from many sources led them to conclude that “. . . there appear to be almost
no flows that are not in equilibrium . . . .” Their most remarkable result is that only
three values of correlate the data for all pressure gradients: = 0.22 (adverse
pressure gradients); = −1.92 (favorable pressure gradients); and = 0 (zero
−1/
pressure gradient). A direct consequence of = const. is that δ(x) ∼ U∞ . Data
was well correlated by this result for both favorable and adverse pressure gradients.
Walker and Castillo (2002) then correlated velocity defect profiles for favorable, zero,
11. Wall-Bounded Shear Flow
and adverse pressure gradients by plotting [(U ∞ − U)/U ∞ ] (δ ∗ /δ 99 ) vs. y/δ99 . (See
Section 10.3 for the definitions of δ99 and δ ∗ ). Remarkably, only three distinct turbulent velocity profiles resulted. This correlation of data with only three values of
was contested by Maciel, Rossignol, and Lemay (2006) in their examination of
data bases for adverse pressure gradient turbulent boundary layers. They found that
scalings developed by Zagarola and Smits (1998) worked best but that varied by a
factor of 2 (from 0.16 to 0.33), while in each of the flows was held constant and
the flow was observed to be self-similar. The value of = 0.22 held for only two
of the nine adverse pressure gradient data sets listed by Maciel et al. Moreover, the
velocity defect profiles for adverse pressure gradient flows did not collapse onto a
single limiting profile, as asserted by Walker and Castillo.
Very close to separation, the boundary layer assumptions that ∂/∂x ≪ ∂/∂y and
v ≪ u break down and new scalings become necessary as discussed in Indinger,
Buschmann, and Gad-el-Hak (2006).
A review paper by W. K. George (2006) puts much of the analysis and discussion
into a wide-view perspective. Three scalings for the outer 90% of the zero pressure
gradient turbulent velocity deficit profiles are written, the first a generalization of
Eq. (13.52) to include behavior with δu∗ /ν. The second is of the same form but the
normalization is by U ∞ instead of u∗ . The third scaling is of the form of the second
with the functional behavior prefaced by δ ∗ /δ, where the displacement thickness
δ ∗ is defined in Eq. (10.16). The first scaling gives the logarithmic behavior in the
overlap between the inner and outer regions of the turbulent boundary layer whereas
the second gives a power law.
Professor George pointed out that although the momentum integral equation for
constant pressure turbulent boundary layers [Eq. (10.44)] holds, dθ/dx = (u∗ /U ∞ )2 ,
and the main contribution to θ [momentum thickness, Eq. (10.17)] comes from near
wall regions, almost the entire value of dθ/dx comes from distances far from the
wall.
The third scaling is due to Zagarola and Smits (1998) and can reduce to either
of the first two depending on the Re → ∞ asymptotic behavior. If δ ∗ /δ → u∗ /U ∞ ,
then the first scaling leading to a logarithmic overlap is obtained. If, on the other
hand, δ ∗ /δ → const., then the second scaling leading to a power law is found. Here
the limit Re → ∞ must be taken as x → ∞ downstream with fixed upstream and
external conditions. It is also shown here that the logarithmic overlap is the lowest
order in an expansion of the power law. The two results are very close largely because
du/dy ∼ 1/y1 yields a logarithm (upon integration) but du/dy ∼ 1/y γ where γ is
anything other than = 1 but may be very close to 1, yields a power upon integration.
Since the latter is more general and results from a difference in turbulence scales in
the two regions involved in the overlap, it is likely to be correct, but more detailed
and careful data is required to distinguish the two forms.
It is easier to discuss the details of the balance among regions of turbulence for
a planar fully developed pressure driven turbulent shear flow. Consider a flow in the
x-direction between two parallel plates at y = 0 and y = 2δ that is no longer evolving
in the x-direction. The convective acceleration terms are then identically zero and the
x-momentum equation reduces to −∂P/∂x + d/dy(µ∂U/∂y − ρ 0 uv) = 0. Since the
577
578
Turbulence
pressure gradient is constant in x, it must be balanced by the shear stress on the two
walls over any distance L, which yields the equality τ 0 = −δ · ∂P/∂x. Then with the
dimensionless variables, U/u∗ = u+ , yu∗ /ν = y+ , δu∗ /v = δ + , (uv)/u2∗ = (uv)+ ,
2
2
the x-momentum equation becomes δ −1
+ + d u+ /dy+ − d/dy(uv)+ = 0, representing a balance among the pressure gradient, viscous stress gradient, and Reynolds
stress gradient. This is the starting point of the analysis by Fife, Wei, Klewicki, and
McMurtry (2005). The authors find distinct scalings representing different balances
of terms. These are described by the following in order of increasing distance from
the wall. For y+ < 3, there is the sublayer where the viscous stress gradient balances
the mean pressure gradient, and the Reynolds stress gradient is small by comparison.
As we go outwards from the wall, the viscous stress gradient balances the Reynolds
stress gradient. For large enough Re this layer extends into the logarithmic region of
the earlier models. Near the location of maximum Reynolds stress, since the Reynolds
stress gradient is small, the viscous stress gradient and pressure gradient are again in
balance. This is true despite the Reynolds stress being much larger than the viscous
stress. This region is called the viscous/pressure gradient mesolayer. Still further from
the wall the Reynolds stress gradient and pressure gradient are in balance.
Wei, Fife, and Klewicki (2007) codified this analysis for any combination of
Couette and Poiseuille flows. We will concentrate on the latter. Explicitly, the inner
normalized equations are written in terms of u+ and y+ . The innermost viscous layer
is then a simple generalization of Eq. (13.51) to include the pressure gradient. The
balance of the inner normalized equations is between the viscous and Reynolds stress
gradients. The outer normalization retains u+ but uses ξ = y/δ as the outer length
scale. In the outermost region, the Reynolds stress and pressure gradients balance
giving: 1 − d/dξ (uv)+ = O(δ −1
+ ) → 0. Between these, in the neighborhood of
maximum Reynolds stress, there is a mesolayer in which all terms are in balance.
This is scaled by:
−1/2
= δ 1/2
ŷ = (y+ − ym+ )δ + , (uv)
+ (uv)+ − (uv)m+ , û(ŷ)
= u+ − um+ − (du+ /dy+ )m · (y+ − ym+ ).
Here, all ˆ variables are finite in the limit δ+ ≡ Re∗ → ∞. Although the location ym+
and the value of (uv)m+ are not known a priori, good approximations are obtained if
−1/2
ym+ δ+ = 1 and (uv)m+ = 1.
Fife et al. (2005) and Wei et al. (2007) introduced the notion of scaling patches
such that a patch is “. . . defined to be a region in the flow field specified by an interval
of distance from the wall, together with a scaling or non-dimensionalization of the
variables which is natural for that region.” Here “natural” means that in appropriately
normalized variables, the data within that region are independent of Reynolds number
in the turbulent limit, Re → ∞. The authors assert that knowledge of the local
scaling properties of the mean momentum and Reynolds stress profiles is essential to
understanding wall bounded turbulent flows. The scaling to render all three terms in
the mean momentum balance of the same order (in Re or δ + ) in the neighborhood
of the Reynolds stress maximum is indeterminate, leaving one free parameter. All
coefficients are = 1 in the Re → ∞ limit in this patch centered about the Reynolds
579
11. Wall-Bounded Shear Flow
stress maximum. A very special choice of this free parameter leads to a logarithm
for u+ ; any other choice gives power law growth. A different scaling patch may be
found in the neighborhood of the channel centerline, again with one free parameter.
A very special choice of the free parameter results in the classical defect law for the
outermost layer.
Rough Surface
In deriving the logarithmic law (13.57), we assumed that the flow in the inner layer
is determined by viscosity. This is true only in hydrodynamically smooth surfaces,
for which the average height of the surface roughness elements is smaller than the
thickness of the viscous sublayer. For a hydrodynamically rough surface, on the other
hand, the roughness elements protrude out of the viscous sublayer. An example is
the flow near the surface of the earth, where the trees and buildings act as roughness elements. This causes a wake behind each roughness element, and the stress is
transmitted to the wall by the “pressure drag” on the roughness elements. Viscosity
becomes irrelevant for determining either the velocity distribution or the overall drag
on the surface. This is why the drag coefficients for a rough pipe and a rough flat
surface become independent of the Reynolds number as Re → ∞.
The velocity distribution near a rough surface is again logarithmic, although it
cannot be represented by equation (13.57). To find its form, we start with the general
logarithmic law (13.59). The constant of integration can be determined by noting that
the mean velocity U is expected to be negligible somewhere within the roughness
elements (Figure 13.19b). We can therefore assume that (13.59) applies for y > y0 ,
where y0 is a measure of the roughness heights and is defined as the value of y at
which the logarithmic distribution gives U = 0. Equation (13.59) then gives
1
y
U
= ln .
u∗
k y0
(13.60)
Figure 13.19 Logarithmic velocity distributions near smooth and rough surfaces: (a) smooth wall; and
(b) rough wall.
580
Turbulence
Figure 13.20 Sketch of observed variation of turbulent intensity and Reynolds stress across a channel
of half-width δ. The left panels are plots as functions of the inner variable y+ , while the right panels are
plots as functions of the outer variable y/δ.
Variation of Turbulent Intensity
The experimental data of turbulent intensity and Reynolds stress in a channel flow are
given in Townsend (1976). Figure 13.20 shows a schematic representation of these
data, plotted both in terms of the outer and the inner variables. It is seen that the
turbulent velocity fluctuations are of order u∗ . The longitudinal fluctuations are the
largest because the shear production initially feeds the energy into the u-component;
the energy is subsequently distributed into the lateral components v and w. (Incidentally, in a convectively generated turbulence the turbulent energy is initially fed to the
vertical component.) The turbulent intensity initially rises as the wall is approached,
but goes to zero right at the wall in a very thin wall layer. As expected from physical considerations, the normal component vrms starts to feel the wall effect earlier.
Figure 13.20 also shows that the distribution of each variable very close to the wall
becomes clear only when the distances are magnified by the viscous scaling ν/u∗ .
The Reynolds stress profile in terms of the inner variable shows that the stresses are
negligible within the viscous sublayer (y+ < 5), beyond which the Reynolds stress
is nearly constant throughout the wall layer. This is why the logarithmic layer is also
called the constant stress layer.
12. Eddy Viscosity and Mixing Length
The equations for mean motion in a turbulent flow, given by equation (13.24), cannot
be solved for Ui (x) unless we have an expression relating the Reynolds stresses
581
12. Eddy Viscosity and Mixing Length
ui uj in terms of the mean velocity field. Prandtl and von Karman developed certain
semiempirical theories that attempted to provide this relationship.
These theories are based on an analogy between the momentum exchanges both
in turbulent and in laminar flows. Consider first a unidirectional laminar flow U (y),
in which the shear stress is
τlam
dU
=ν
,
ρ
dy
(13.61)
where ν is a property of the fluid. According to the kinetic theory of gases, the diffusive
properties of a gas are due to the molecular motions, which tend to mix momentum
and heat throughout the flow. It can be shown that the viscosity of a gas is of order
ν ∼ aλ,
(13.62)
where a is the rms speed of molecular motion, and λ is the mean free path defined as
the average distance traveled by a molecule between collisions. The proportionality
constant in equation (13.62) is of order 1.
One is tempted to speculate that the diffusive behavior of a turbulent flow may
be qualitatively similar to that of a laminar flow and may simply be represented by a
much larger diffusivity. By analogy with (13.61), Boussinesq proposed to represent
the turbulent stress as
−uv = νe
dU
,
dy
(13.63)
where νe is the eddy viscosity. Note that, whereas ν is a known property of the fluid, νe
in (13.63) depends on the conditions of the flow. We can always divide the turbulent
stress by the mean velocity gradient and call it νe , but this is not progress unless
we can formulate a rational method for finding the eddy viscosity from other known
parameters of a turbulent flow.
The eddy viscosity relation (13.63) implies that the local gradient determines
the flux. However, this cannot be valid if the eddies happen to be larger than the
scale of curvature of the profile. Following Panofsky and Dutton (1984), consider the
atmospheric concentration profile of carbon monoxide (CO) shown in Figure 13.21.
An eddy viscosity relation would have the form
−wc = κe
dC
,
dz
(13.64)
where C is the mean concentration (kilograms of CO per kilogram of air), c is its
fluctuation, and κe is the eddy diffusivity. A positive κe requires that the flux of CO at
P be downward. However, if the thermal convection is strong enough, the large eddies
so generated can carry large amounts of CO from the ground to point P, and result in
an upward flux there. The direction of flux at P in this case is not determined by the
local gradient at P, but by the concentration difference between the surface and point
P. In this case, the eddy diffusivity found from equation (13.64) would be negative
and, therefore, not very meaningful.
582
Turbulence
Figure 13.21 An illustration of breakdown of an eddy diffusivity type relation. The eddies are larger than
the scale of curvature of the concentration profile C(z) of carbon monoxide.
In cases where the concept of eddy viscosity may work, we may use the analogy
with equation (13.62), and write
νe ∼ u′ lm ,
(13.65)
where u′ is a typical scale of the fluctuating velocity, and lm is the mixing length,
defined as the cross-stream distance traveled by a fluid particle before it gives up its
momentum and loses identity. The concept of mixing length was first introduced by
Taylor (1915), but the approach was fully developed by Prandtl and his coworkers.
As with the eddy viscosity approach, little progress has been made by introducing the
mixing length, because u′ and lm are just as unknown as νe is. Experience shows that
in many situations u′ is of the order of either the local mean speed U or the friction
velocity u∗ . However, there does not seem to be a rational approach for relating lm to
the mean flow field.
Prandtl derived the logarithmic velocity distribution near a solid surface by using
the mixing length theory in the following manner. The scale of velocity fluctuations
in a wall-bounded flow can be taken as u′ ∼ u∗ . Prandtl also argued that the mixing
length must be proportional to the distance y. Then equation (13.65) gives
νe = ku∗ y.
For points outside the viscous sublayer but still near the wall, the Reynolds stress can
be taken equal to the wall stress ρu2∗ . This gives
ρu2∗ = ρku∗ y
dU
,
dy
which can be written as
dU
u∗
=
.
dy
ky
(13.66)
583
12. Eddy Viscosity and Mixing Length
This integrates to
1
U
= ln y + const.
u∗
k
In recent years the mixing length theory has fallen into disfavor, as it is incorrect
in principle (Tennekes and Lumley, 1972). It only works when there is a single length
scale and a single time scale; for example in the overlap layer in a wall-bounded
flow the only relevant length scale is y and the only time scale is y/u∗ . However, its
validity is then solely a consequence of dimensional necessity and not of any other
fundamental physics. Indeed it was shown in the preceding section that the logarithmic velocity distribution near a solid surface can be derived from dimensional
considerations alone. (Since u∗ is the only characteristic velocity in the problem, the
local velocity gradient dU/dy can only be a function of u∗ and y. This leads to equation (13.66) merely on dimensional grounds.) Prandtl’s derivation of the empirically
known logarithmic velocity distribution has only historical value.
However, the relationship (13.65) is useful for estimating the order of magnitude
of the eddy diffusivity in a turbulent flow, if we interpret the right-hand side as
simply the product of typical velocity and length scales of large eddies. Consider the
thermal convection between two horizontal plates in air. The walls are separated by
a distance L = 3 m, and the lower layer is warmer by T = 1 ◦ C. The equation of
motion (13.33) for the fluctuating field gives the vertical acceleration as
gT
Dw
∼ gαT ′ ∼
,
Dt
T
(13.67)
where we have used the fact that the temperature fluctuations are expected to be of
order T and that α = 1/T for a perfect gas. The time to rise through a height L is
t ∼ L/w, so that equation (13.67) gives a characteristic velocity fluctuation of
√
w ∼ gLT /T ≈ 0.1 m/s ≈ 0.316 m/s.
It is fair to assume that the largest eddies are as large as the separation between the
plates. The eddy diffusivity is therefore
κe ∼ wL ∼ 0.95 m2 /s,
which is much larger than the molecular value of 2 × 10−5 m2 /s.
As noted in the preceding, the Reynolds averaged Navier–Stokes equations do
not form a closed system. In order for them to be predictive and useful in solving
problems of scientific and engineering interest, closures must be developed. Reynolds
stresses or higher correlations must be expressed in terms of themselves or lower correlations with empirically determined constants. An excellent review of an important
class of closures is provided by Speziale (1991). Critical discussions of various closures together with comparisons with each other, with experiments, or with numerical
simulations are given for several idealized problems. Very tragically, Charles Speziale
died while at the peak of his intellectual productivity. He contributed numerous papers
584
Turbulence
on turbulence modeling and other subjects on the foundations of fluid mechanics. A
memorial tribute and a number of papers on turbulence in his honor by some of the
prominent authorities may be found in the May 2006 issue of Journal of Applied
Mechanics (v. 73, no. 3).
A different approach to turbulence modeling is represented by renormalization
group (RNG) theories. Rather than use the Reynolds averaged equations, turbulence
is simulated by a solenoidal isotropic random (body) force field f (force/mass). Here
f is chosen to generate the velocity field described by the Kolmogorov spectrum in
the limit of large wavenumber K. For very small eddies (larger wavenumbers beyond
the inertial subrange), the energy decays exponentially by viscous dissipation. The
spectrum in Fourier space (K) is truncated at a cutoff wavenumber and the effect
of these very small scales is represented by a modified viscosity. Then an iteration
is performed successively moving back the cutoff into the inertial range. Smith and
Reynolds (1992) provide a tutorial on the RNG method developed several years
earlier by Yakhot and Orszag. Lam (1992) develops results in a different way and
offers insights and plausible explanations for the various artifices in the theory.
13. Coherent Structures in a Wall Layer
The large-scale identifiable structures of turbulent events, called coherent structures,
depend on the type of flow. A possible structure of large eddies found in the outer parts
of a boundary layer, and in a wall-free shear flow, was illustrated in Figure 13.10. In
this section we shall discuss the coherent structures observed within the inner layer
of a wall-bounded shear flow. This is one of the most active areas of current turbulent
research, and reviews of the subject can be found in Cantwell (1981) and Landahl
and Mollo-Christensen (1986).
These structures are deduced from spatial correlation measurements, a certain
amount of imagination, and plenty of flow visualization. The flow visualization
involves the introduction of a marker, one example of which is dye. Another involves
the “hydrogen bubble technique,” in which the marker is generated electrically. A thin
wire is stretched across the flow, and a voltage is applied across it, generating a line
of hydrogen bubbles that travel with the flow. The bubbles produce white spots in the
photographs, and the shapes of the white regions indicate where the fluid is traveling
faster or slower than the average.
Flow visualization experiments by Kline et al. (1967) led to one of the most
important advances in turbulence research. They showed that the inner part of the
wall layer in the range 5 < y+ < 70 is not at all passive, as one might think. In fact,
it is perhaps dynamically the most active, in spite of the fact that it occupies only
about 1% of the total thickness of the boundary layer. Figure 13.22 is a photograph
from Kline et al. (1967), showing the top view of the flow within the viscous sublayer
at a distance y+ = 2.7 from the wall. (Here x is the direction of flow, and z is the
“spanwise” direction.) The wire producing the hydrogen bubbles in the figure was
parallel to the z-axis. The streaky structures seen in the figure are generated by regions
of fluid moving downstream faster or slower than the average. The figure reveals that
the streaks of low-speed fluid are quasi-periodic in the spanwise direction. From
13. Coherent Structures in a Wall Layer
Figure 13.22 Top view of near-wall structure (at y+ = 2.7) in a turbulent boundary layer on a horizontal
flat plate. The flow is visualized by hydrogen bubbles. S. J. Kline et al., Journal of Fluid Mechanics 30:
741–773, 1967 and reprinted with the permission of Cambridge University Press.
time to time these slowly moving streaks lift up into the buffer region, where they
undergo a characteristic oscillation. The oscillations end violently and abruptly as
the lifted fluid breaks up into small-scale eddies. The whole cycle is called bursting,
or eruption, and is essentially an ejection of slower fluid into the flow above. The
flow into which the ejection occurs decelerates, causing a point of inflection in the
profile u(y) (Figure 13.23). The secondary flow associated with the eruption motion
causes a stretching of the spanwise vortex lines, as sketched in the figure. These vortex
lines amplify due to the inherent instability of an inflectional profile, and readily break
up, producing a source of small-scale turbulence. The strengths of the eruptions vary,
and the stronger ones can go right through to the edge of the boundary layer.
It is clear that the bursting of the slow fluid associates a positive v with a negative u, generating a positive Reynolds stress −uv. In fact, measurements show that
most of the Reynolds stress is generated by either the bursting or its counterpart,
called the sweep (or inrush) during which high-speed fluid moves toward the wall.
The Reynolds stress generation is therefore an intermittent process, occurring perhaps
25% of the time.
Largely due to numerical simulations of turbulent flows, it is now understood
that the very large turbulent wall shear stress (as compared with that in laminar
flow) is due to streamwise vorticity in the buffer or inner wall layer (y+ = 10–50).
Kim (2003) reports on the history of discovery by computation and experimental
verification of insight into the details of turbulent flows. This insight led to strategies to reduce the wall shear stress by active or passive controls. The availability
of microsensors and MEMS actuators creates the possibility of actively modifying
585
586
Turbulence
Figure 13.23 Mechanics of streak break up. S. J. Kline et al., Journal of Fluid Mechanics 30: 741–773,
1967 and reprinted with the permission of Cambridge University Press.
the flow near the wall to significantly reduce the shear stress. Passive modification
is exemplified by adding riblets to the surface. These are fine streamwise corrugations that interfere with the interaction between the streamwise vortices and the wall.
Much smaller drag reduction is achieved this way. An example of active modification
of the near-wall flow is blowing and suctioning alternately on the surface to counter
the streamwise vorticity. A surprising result of these studies is that linear control
theory (for the Navier–Stokes equation linearized about a mean flow) provides excellent results for a strategy for reducing wall shear stress, provided that function to be
extremized (which cannot be drag) is carefully chosen. All of these results apply only
for small turbulence Reynolds number (Re∗ = u∗ δ/ν). However, there has been a
history of success in applying insights gained for small Re∗ to larger, more realistic
values.
14. Turbulence in a Stratified Medium
Effects of stratification become important in such laboratory flows as heat transfer
from a heated plate and in geophysical flows such as those in the atmosphere and in
the ocean. Some effects of stratification on turbulent flows will be considered in this
section. Further discussion can be found in Tennekes and Lumley (1972), Phillips
(1977), and Panofsky and Dutton (1984).
As is customary in geophysical literature, we shall take the z-direction as upward,
and the shear flow will be denoted by U (z). For simplicity the flow will be assumed
homogeneous in the horizontal plane, that is independent of x and y. The turbulence
in a stratified medium depends critically on the static stability. In the neutrally stable
state of a compressible environment the density decreases upward, because of the
decrease of pressure, at a rate dρa /dz called the adiabatic density gradient. This
587
14. Turbulence in a Stratified Medium
is discussed further in Chapter 1, Section 10. A medium is statically stable if the
density decreases faster than the adiabatic decrease. The effective density gradient
that determines the stability of the environment is then determined by the sign of
d(ρ −ρa )/dz, where ρ −ρa is called the potential density. In the following discussion,
we shall assume that the adiabatic variations of density have been subtracted out, so
that when we talk about density or temperature, we shall really mean potential density
or potential temperature.
The Richardson Numbers
Let us first examine the equation for turbulent kinetic energy (13.34). Omitting the
viscous transport and assuming that the flow is independent of x and y, it reduces to
∂
dU
1
D 2
(q ) = −
pw + q 2 w − uw
+ gαwT ′ − ε,
Dt
∂z ρ0
dz
where q 2 = (u2 + v 2 + w 2 )/2. The first term on the right-hand side is the transport
of turbulent kinetic energy by fluctuating w. The second term −uw(dU/dz) is the
production of turbulent energy by the interaction of Reynolds stress and the mean
shear; this term is almost always positive. The third term gαwT ′ is the production of
turbulent kinetic energy by the vertical heat flux; it is called the buoyant production,
and was discussed in more detail in Section 7. In an unstable environment, in which
the mean temperature T̄ decreases upward, the heat flux wT ′ is positive (upward),
signifying that the turbulence is generated convectively by upward heat fluxes. In the
opposite case of a stable environment, the turbulence is suppressed by stratification.
The ratio of the buoyant destruction of turbulent kinetic energy to the shear production
is called the flux Richardson number:
Rf =
buoyant destruction
−gαwT ′
=
,
−uw(dU/dz)
shear production
(13.68)
where we have oriented the x-axis in the direction of flow. As the shear production
is positive, the sign of Rf depends on the sign of wT ′ . For an unstable environment
in which the heat flux is upward Rf is negative and for a stable environment it is
positive. For Rf > 1, the buoyant destruction removes turbulence at a rate larger than
the rate at which it is produced by shear production. However, the critical value of Rf
at which the turbulence ceases to be self-supporting is less than unity, as dissipation
is necessarily a large fraction of the shear production. Observations indicate that the
critical value is Rf cr ≃ 0.25 (Panofsky and Dutton, 1984, p. 94). If measurements
indicate the presence of turbulent fluctuations, but at the same time a value of Rf
much larger than 0.25, then a fair conclusion is that the turbulence is decaying.
When Rf is negative, a large −Rf means strong convection and weak mechanical
turbulence.
Instead of Rf, it is easier to measure the gradient Richardson number, defined as
Ri ≡
αg(d T̄ /dz)
N2
=
,
2
(dU/dz)
(dU/dz)2
(13.69)
588
Turbulence
where N is the buoyancy frequency. If we make the eddy coefficient assumptions
d T̄
,
dz
dU
−uw = νe
,
dz
−wT ′ = κe
then the two Richardson numbers are related by
Ri =
νe
Rf.
κe
(13.70)
The ratio νe /κe is the turbulent Prandtl number, which determines the relative efficiency of the vertical turbulent exchanges of momentum and heat. The presence
of a stable stratification damps the vertical transports of both heat and momentum;
however, the momentum flux is reduced less because the internal waves in a stable environment can transfer momentum (by moving vertically from one region to
another) but not heat. Therefore, νe /κe > 1 for a stable environment. Equation (13.70)
then shows that turbulence can persist even when Ri > 1, if the critical value of 0.25
applies on the flux Richardson number (Turner, 1981; Bradshaw and Woods, 1978).
In an unstable environment, on the other hand, νe /κe becomes small. In a neutral environment it is usually found that νe ≃ κe ; the idea of equating the eddy coefficients of
heat and momentum is called the Reynolds analogy.
Monin–Obukhov Length
The Richardson numbers are ratios that compare the relative importance of mechanical
and convective turbulence. Another parameter used for the same purpose is not a ratio,
but has the unit of length. It is the Monin–Obukhov length, defined as
LM ≡ −
u3∗
kαgwT ′
,
(13.71)
where u∗ is the friction velocity, wT ′ is the heat flux, α is the coefficient of thermal
expansion, and k is the von Karman constant introduced for convenience. Although
wT ′ is a function of z, the parameter LM is effectively a constant for the flow, as
it is used only in the logarithmic surface layer in which both the stress and the
heat flux wT ′ are nearly constant. The Monin–Obukhov length then becomes a
parameter determined from the boundary conditions of drag and the heat flux at
the surface. Like Rf, it is positive for stable conditions and negative for unstable
conditions.
The significance of LM within the surface layer becomes clearer if we write
Rf in terms of LM , using the logarithmic velocity distribution (13.60), from which
dU/dz = u∗ /kz. (Note that we are now using z for distances perpendicular to the
surface.) Using uw = u2∗ because of the near uniformity of stress in the logarithmic
589
14. Turbulence in a Stratified Medium
layer, equation (13.68) becomes
Rf =
z
.
LM
(13.72)
As Rf is the ratio of buoyant destruction to shear production of turbulence, (13.72)
shows that LM is the height at which these two effects are of the same order.
For both stable and unstable conditions, the effects of stratification are slight if
z ≪ |LM |. At these small heights, then, the velocity profile is logarithmic, as in a
neutral environment. This is called a forced convection region, because the turbulence is mechanically forced. For z ≫ |LM |, the effects of stratification dominate.
In an unstable environment, it follows that the turbulence is generated mainly by
buoyancy at heights z ≫ −LM , and the shear production is negligible. The region
beyond the forced convecting layer is therefore called a zone of free convection
(Figure 13.24), containing thermal plumes (columns of hot rising gases) characteristic
of free convection from heated plates in the absence of shear flow.
Observations as well as analysis show that the effect of stratification on the velocity distribution in the surface layer is given by the log-linear profile (Turner, 1973)
u∗
z
z
U=
.
+5
ln
k
z0
LM
The form of this profile is sketched in Figure 13.25 for stable and unstable conditions.
It shows that the velocity is more uniform than ln z in the unstable case because of
the enhanced vertical mixing due to buoyant convection.
Spectrum of Temperature Fluctuations
An equation for the intensity of temperature fluctuations T ′2 can be obtained in a
manner identical to that used for obtaining the turbulent kinetic energy. The procedure
is therefore to obtain an equation for DT ′ /Dt by subtracting those for D T̃ /Dt and
Figure 13.24 Forced and free convection zones in an unstable atmosphere.
590
Turbulence
Figure 13.25 Effect of stability on velocity profiles in the surface layer.
D T̄ /Dt, and then to multiply the resulting equation for DT ′ /Dt by T ′ and take the
average. The result is
1 DT ′2
d T̄
∂ 1 ′2
dT ′2
T w−κ
= −wT ′
−
− εT ,
2 Dt
dz
∂z 2
dz
where εT ≡ κ(∂T ′ /∂xj )2 is the dissipation of temperature fluctuation, analogous
to the dissipation of turbulent kinetic energy ε = 2νeij eij . The first term on the
right-hand side is the generation of T ′2 by the mean temperature gradient, wT ′ being
positive if d T̄ /dz is negative. The second term on the right-hand side is the turbulent
transport of T ′2 .
A wavenumber spectrum of temperature fluctuations can be defined such that
T ′2
≡
∞
Ŵ(K) dK.
0
As in the case of the kinetic energy spectrum, an inertial range of wavenumbers
exists in which neither the production by large-scale eddies nor the dissipation by
conductive and viscous effects are important. As the temperature fluctuations are
intimately associated with velocity fluctuations, Ŵ(K) in this range must depend not
only on εT but also on the variables that determine the velocity spectrum, namely ε
and K. Therefore
Ŵ(K) = Ŵ(εT , ε, K)
l −1 ≪ K ≪ η−1 .
591
15. Taylor’s Theory of Turbulent Dispersion
The unit of Ŵ is ◦ C2 m, and the unit of εT is ◦ C2 /s. A dimensional analysis gives
Ŵ(K) ∝ εT ε−1/3 K −5/3
l −1 ≪ K ≪ η−1 ,
(13.73)
which was first derived by Obukhov in 1949. Comparing with equation (13.40), it is
apparent that the spectra of both velocity and temperature fluctuations in the inertial
subrange have the same K −5/3 form.
The spectrum beyond the inertial subrange depends on whether the Prandtl number ν/κ of the fluid is smaller or larger than one. We shall only consider the case of
ν/κ ≫ 1, which applies to water for which the Prandtl number is 7.1. Let ηT be the
scale responsible for smearing out the temperature gradients and η be the Kolmogorov
microscale at which the velocity gradients are smeared out. For ν/κ ≫ 1 we expect
that ηT ≪ η, because then the conductive effects are important at scales smaller than
the viscous scales. In fact, Batchelor (1959) showed that ηT ≃ η(κ/ν)1/2 ≪ η. In
such a case there exists a range of wavenumbers η−1 ≪ K ≪ ηT−1 , in which the
scales are not small enough for the thermal diffusivity to smear out the temperature
fluctuation. Therefore, Ŵ(K) continues farther up to ηT−1 , although S(K) drops off
sharply. This is called the viscous convective subrange, because the spectrum is dominated by viscosity but is still actively convective. Batchelor (1959) showed that the
spectrum in the viscous convective subrange is
Ŵ(K) ∝ K −1
η−1 ≪ K ≪ ηT−1 .
(13.74)
Figure 13.26 shows a comparison of velocity and temperature spectra, observed in a
tidal flow through a narrow channel. The temperature spectrum shows that the spectral
slope increases from − 35 in the inertial subrange to −1 in the viscous convective
subrange.
15. Taylor’s Theory of Turbulent Dispersion
The large mixing rate in a turbulent flow is due to the fact that the fluid particles
gradually wander away from their initial location. Taylor (1921) studied this problem
and calculated the rate at which a particle disperses (i.e., moves away) from its initial
location. The presentation here is directly adapted from his classic paper. He considered a point source emitting particles, say a chimney emitting smoke. The particles
are emitted into a stationary and homogeneous turbulent medium in which the mean
velocity is zero. Taylor used Lagrangian coordinates X(a, t), which is the present
location at time t of a particle that was at location a at time t = 0. We shall take the
point source to be the origin of coordinates and consider an ensemble of experiments
in which we measure the location X(0, t) at time t of all the particles that started
from the origin (Figure 13.27). For simplicity we shall suppress the first argument in
X(0, t) and write X(t) to mean the same thing.
Rate of Dispersion of a Single Particle
Consider the behavior of a single component of X, say Xα (α = 1, 2, or 3). (We are
using a Greek subscript α because we shall not imply the summation convention.)
592
Turbulence
105
1
103
1022
10
1024
1021
1026
1023
1028
1025
1023
1022
1021
1
10
102
Figure 13.26 Temperature and velocity spectra measured by Grant et al. (1968). The measurements were
made at a depth of 23 m in a tidal passage through islands near the coast of British Columbia, Canada.
Wavenumber K is in cm−1 . Solid points represent Ŵ in (◦ C)2 /cm−1 , and open points represent S in
(cm/s)2 /cm−1 . Powers of K that fit the observation are indicated by straight lines. O. M. Phillips, The
Dynamics of the Upper Ocean, 1977 and reprinted with the permission of Cambridge University Press.
The average rate at which the magnitude of Xα increases with time can be found by
finding d(Xα2 )/dt, where the overbar denotes ensemble average and not time average.
We can write
dXα
d
(X 2 ) = 2Xα
,
dt α
dt
(13.75)
where we have used the commutation rule (13.3) of averaging and differentiation.
Defining
uα =
dXα
,
dt
as the Lagrangian velocity component of a fluid particle at time t, equation (13.75)
becomes
593
15. Taylor’s Theory of Turbulent Dispersion
Figure 13.27 Three experimental outcomes of X(t), the current positions of particles from the origin at
time t = 0.
d
(X 2 ) = 2Xα uα = 2
dt α
=2
0
t
0
t
uα
(t ′ ) dt ′
uα
uα (t ′ )uα (t) dt ′ ,
(13.76)
where we have used the commutation rule (13.4) of averaging and integration. We
have also written
t
uα (t ′ ) dt ′ ,
Xα =
0
which is valid as Xα and uα are associated with the same particle. Because the flow is
assumed to be stationary, u2α is independent of time, and the autocorrelation of uα (t)
and uα (t ′ ) is only a function of the time difference t − t ′ . Defining
rα (τ ) ≡
uα (t)uα (t + τ )
u2α
,
to be the autocorrelation of Lagrangian velocity components of a particle, equation (13.76) becomes
t
d
(Xα2 ) = 2u2α
rα (t ′ − t) dt ′
dt
0
t
= 2u2α
rα (τ ) dτ,
(13.77)
0
594
Turbulence
where we have changed the integration variable from t ′ to τ = t − t ′ . Integrating, we
obtain
Xα2 (t)
=
t
2u2α
dt
′
t′
(13.78)
rα (τ ) dτ,
0
0
which shows how the variance of the particle position changes with time.
Another useful form of equation (13.78) is obtained by integrating it by parts.
We have
0
t
dt
′
0
t′
rα (τ ) dτ = t
=t
=t
′
t′
rα (τ ) dτ
0
t
t ′ =0
t
t
0
rα (τ ) dτ −
t
0
1−
−
t
t ′ rα (t ′ ) dt ′
0
t ′ rα (t ′ ) dt ′
0
τ
rα (τ ) dτ.
t
Equation (13.78) then becomes
Xα2 (t)
=
2u2α t
t
0
1−
τ
rα (τ ) dτ.
t
(13.79)
Two limiting cases are examined in what follows.
Behavior for small t: If t is small compared to the correlation scale of rα (τ ), then
rα (τ ) ≃ 1 throughout the integral in equation (13.78) (Figure 13.28). This gives
Xα2 (t) ≃ u2α t 2 .
(13.80)
Taking the square root of both sides, we obtain
Xαrms = urms
α t
t ≪ ᐀,
(13.81)
which shows that the rms displacement increases linearly with time and is proportional
to the intensity of turbulent fluctuations in the medium.
Behavior for large t: If t is large compared with the correlation scale of rα (τ ), then
τ/t in equation (13.79) is negligible, giving
Xα2 (t) ≃ 2u2α ᐀t,
where
(13.82)
595
15. Taylor’s Theory of Turbulent Dispersion
Figure 13.28 Small and large values of time on a plot of the correlation function.
᐀≡
∞
rα (τ ) dτ,
0
is the integral time scale determined from the Lagrangian correlation rα (τ ). Taking
the square root, equation (13.82) gives
√
Xαrms = urms
2᐀ t
α
t ≫ ᐀.
(13.83)
The t 1/2 behavior of equation (13.83) at large times is similar to the behavior in a
random walk, in which the distance traveled in a series of random (i.e., uncorrelated)
steps increases as t 1/2 . This similarity is due to the fact that for large t the fluid
particles have “forgotten” their initial behavior at t = 0. In contrast, the small time
behavior Xαrms = urms
α t is due to complete correlation, with each experiment giving
Xα ≃ uα t. The concept of random walk is discussed in what follows.
Random Walk
The following discussion is adapted from Feynman et al. (1963, pp. 6–5 and 41–8).
Imagine a person walking in a random manner, by which we mean that there is
no correlation between the directions of two consecutive steps. Let the vector Rn
represent the distance from the origin after n steps, and the vector L represent the nth
step (Figure 13.29). We assume that each step has the same magnitude L. Then
Rn = Rn−1 + L,
596
Turbulence
Figure 13.29 Random walk.
which gives
Rn2 = Rn · Rn = (Rn−1 + L) · (Rn−1 + L)
2
= Rn−1
+ L2 + 2Rn−1 · L.
Taking the average, we get
2
+ L2 + 2Rn−1 · L.
Rn2 = Rn−1
(13.84)
The last term is zero because there is no correlation between the direction of the
nth step and the location reached after n − 1 steps. Using rule (13.84) successively,
we get
2
2
+ L2 = Rn−2
+ 2L2
Rn2 = Rn−1
= R12 + (n − 1)L2 = nL2 .
The rms distance traveled after n uncorrelated steps, each of length L, is therefore
√
Rnrms = L n,
(13.85)
which is called a random walk.
Behavior of a Smoke Plume in the Wind
Taylor’s analysis can be adapted to account for the presence of mean velocity. Consider
the dispersion of smoke into a wind blowing in the x-direction (Figure 13.30). Then a
photograph of the smoke plume, in which the film is exposed for a long time, would
outline the average width Y rms . As the x-direction in this problem is similar to time in
597
15. Taylor’s Theory of Turbulent Dispersion
Figure 13.30 Average shape of a smoke plume in a wind blowing uniformly along the x-axis. G. I. Taylor,
Proc. London Mathematical Society 20: 196–211, 1921.
Taylor’s problem, the limiting behavior in equations (13.81) and (13.83) shows that
the smoke plume is parabolic with a pointed vertex.
Effective Diffusivity
An equivalent eddy diffusivity can be estimated from Taylor’s analysis. The equivalence is based on the following idea: Consider the spreading of a concentrated source,
say of heat or vorticity, in a fluid of constant diffusivity. What should the diffusivity be in order that the spreading rate equals that predicted by equation (13.77)?
The problem of the sudden introduction of a line vortex of strength Ŵ, considered in
Chapter 9, Section 9, is such a problem of diffusion of a concentrated source. It was
shown there that the tangential velocity in this flow is given by
uθ =
Ŵ −r 2 /4νt
e
.
2π r
The solution is therefore proportional to exp(−r 2 /4νt), which has a Gaussian shape
√ in
the radial direction r, with a characteristic width (“standard deviation”) of σ = 2νt.
It follows that the momentum diffusivity ν in this problem is related to the variance
σ 2 as
ν=
1 dσ 2
,
2 dt
(13.86)
which can be calculated if σ 2 (t) is known. Generalizing equation (13.86), we can
say that the effective diffusivity in a problem of turbulent dispersion of a patch of
598
Turbulence
particles issuing from a point is given by
1 d
κe ≡
(X2 ) = u2α
2 dt α
t
rα (τ ) dτ,
(13.87)
0
where we have used equation (13.77). From equations (13.80) and (13.82), the two
limiting cases of equation (13.87) are
κe ≃ u2α t
t ≪ ᐀,
(13.88)
κe ≃ u2α ᐀
t ≫ ᐀.
(13.89)
Equation (13.88) shows the interesting fact that the eddy diffusivity initially
increases with time, a behavior different from that in molecular diffusion with constant diffusivity. This can be understood as follows. The dispersion (or separation)
of particles in a patch is caused by eddies with scales less than or equal to the scale
of the patch, since the larger eddies simply advect the patch and do not cause any
separation of the particles. As the patch size becomes larger, an increasing range of
eddy sizes is able to cause dispersion, giving κα ∝ t. This behavior shows that it is
frequently impossible to represent turbulent diffusion by means of a large but constant eddy diffusivity. Turbulent diffusion does not behave like molecular diffusion.
For large times, on the other hand, the patch size becomes larger than the largest eddies
present, in which case the diffusive behavior becomes similar to that of molecular
diffusion with a constant diffusivity given by equation (13.89).
16. Concluding Remarks
Turbulence is an area of classical fluid mechanics that is the subject of continued
intensive research. Frequent symposia are held to summarize and bring the research
community up-to-date on new results and focus on promising approaches. Some
noteworthy proceedings are listed in the Supplementary Reading section at the end
of this chapter.
Exercises
1. Let R(τ ) and S(ω) be a Fourier transform pair. Show that S(ω) is real and
symmetric if R(τ ) is real and symmetric.
2. Calculate the mean, standard deviation, and rms value of the periodic time
series
u(t) = U0 cos ωt + Ū .
3. Show that the autocorrelation function u(t)u(t + τ ) of a periodic series
u = U cos ωt is itself periodic.
4. Calculate the zero-lag cross-correlation u(t)v(t) between two periodic series
u(t) = cos ωt and v(t) = cos (ωt + φ). For values of φ = 0, π/4, and π/2, plot the
599
Exercises
scatter diagrams of u vs v at different times, as in Figure 13.6. Note that the plot is
a straight line if φ = 0, an ellipse if φ = π/4, and a circle if φ = π/2; the straight
line, as well as the axes of the ellipse, are inclined at 45◦ to the uv-axes. Argue that
the straight line signifies a perfect correlation, the ellipse a partial correlation, and the
circle a zero correlation.
5. Measurements in an atmosphere at 20 ◦ C show an rms vertical velocity of
wrms = 1 m/s and an rms temperature fluctuation of Trms = 0.1 ◦ C. If the correlation
coefficient is 0.5, calculate the heat flux ρCp wT ′ .
6. A mass of 10 kg of water is stirred by a mixer. After one hour of stirring, the
temperature of the water rises by 1.0 ◦ C. What is the power output of the mixer in
watts? What is the size η of the dissipating eddies?
7. A horizontal smooth pipe 20 cm in diameter carries water at a temperature
of 20 ◦ C. The drop of pressure is dp/dx = 8 N/m2 per meter. Assuming turbulent flow, verify that the thickness of the viscous sublayer is ≈0.25 mm. [Hint: Use
dp/dx = 2τ0 /R, as given in equation (9.12). This gives τ0 = 0.4 N/m2 , and therefore
u∗ = 0.02 m/s.]
8. Derive the logarithmic velocity profile for a smooth wall
U
1 yu∗
= ln
+ 5.0,
u∗
k
ν
by starting from
U=
u∗
ln y + const.
k
and matching the profile to the edge of the viscous sublayer at y = 10.7 ν/u∗ .
9. Estimate the Monin–Obukhov length in the atmospheric boundary layer if the
surface stress is 0.1 N/m2 and the upward heat flux is 200 W/m2 .
10. Consider a one-dimensional turbulent diffusion of particles issuing from a
point source. Assume a Gaussian Lagrangian correlation function of particle velocity
r(τ ) = e−τ
2 /t 2
c
,
where tc is a constant. By integrating the correlation function from τ = 0 to ∞, find
the integral time scale ᐀ in terms of tc . Using the Taylor theory, estimate the eddy
diffusivity at large times t/᐀ ≫ 1, given that the rms fluctuating velocity is 1 m/s
and tc = 1 s.
11. Show by dimensional reasoning as outlined in Section
√ 10 that for
self-preserving flows far downstream, U∞ − Ue ∼ x −1/2 , δ ∼ x, for a wake,
and U1 − U2 = const., δ ∼ x, for a shear layer.
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Turbulence
Literature Cited
Batchelor, G. K. (1959). “Small scale variation of convected quantities like temperature in turbulent fluid.
Part I: General discussion and the case of small conductivity.” Journal of Fluid Mechanics 5: 113–133.
Bradshaw, P. and J. D. Woods (1978). “Geophysical turbulence and buoyant flows,” in: Turbulence,
P. Bradshaw, ed., New York: Springer-Verlag.
Buschmann, M. H. and M. Gad-el-Hak (2003a). “Generalized logarithmic law and its consequences.”
AIAA Journal 41: 40–48.
Buschmann, M. H. and M. Gad-el-Hak (2003b). “Debate concerning the mean velocity profiles of a
turbulent boundary layer.” AIAA Journal 41: 565–572.
Cantwell, B. J. (1981). “Organized motion in turbulent flow.” Annual Review of Fluid Mechanics 13:
457–515.
Castillo, L. and W. K. George (2001). “Similarity analysis for turbulent boundary layer with pressure
gradient: Outer flow.” AIAA Journal 39: 41–47.
Feynman, R. P., R. B. Leighton, and M. Sands (1963). The Feynman Lectures on Physics, New York:
Addison-Wesley.
Fife, P., T. Wei, J. Klewicki, and P. McMurtry (2005). “Stress gradient balance layers and scale hierarchies
in wall-bounded turbulent flows.” Journal of Fluid Mechanics 532: 165–189.
George, W. K. (2006). “Recent advancements toward the understanding of turbulent boundary layers.”
AIAA Journal 44: 2435–2449.
Grant, H. L., R. W. Stewart, and A. Moilliet (1962). “The spectrum of a cross-stream component of
turbulence in a tidal stream.” Journal of Fluid Mechanics 13: 237–240.
Grant, H. L., B. A. Hughes, W. M. Vogel, and A. Moilliet (1968). “The spectrum of temperature fluctuation
in turbulent flow.” Journal of Fluid Mechanics 34: 423–442.
Indinger, T., M. H. Buschmann, and M. Gad-el-Hak (2006). “Mean velocity profile of turbulent boundary
layers approaching separation.” AIAA Journal 44: 2465–2474.
Kim, John (2003). “Control of turbulent boundary layers.” Physics of Fluids 15: 1093–1105.
Kline, S. J., W. C. Reynolds, F. A. Schraub, and P. W. Runstadler (1967). “The structure of turbulent
boundary layers.” Journal of Fluid Mechanics 30: 741–773.
Lam, S. H. (1992). “On the RNG theory of turbulence.” The Physics of Fluids A 4: 1007–1017.
Landahl, M. T. and E. Mollo-Christensen (1986). Turbulence and Random Processes in Fluid Mechanics,
London: Cambridge University Press.
Lesieur, M. (1987). Turbulence in Fluids, Dordrecht, Netherlands: Martinus Nijhoff Publishers.
Maciel, Y., K. S. Rossignol, and J. Lemay (2006). “Self-similarity in the outer region of adversepressure-gradient turbulent boundary layers.” AIAA Journal 44: 2450–2464.
Monin, A. S. and A. M. Yaglom (1971). Statistical Fluid Mechanics, Cambridge, MA: MIT Press.
Panofsky, H. A. and J. A. Dutton (1984). Atmospheric Turbulence, New York: Wiley.
Phillips, O. M. (1977). The Dynamics of the Upper Ocean, London: Cambridge University Press.
Smith, L. M. and W. C. Reynolds (1992). “On theYakhot-Orszag renormalization group method for deriving
turbulence statistics and models.” The Physics of Fluids A 4: 364–390.
Speziale, C. G. (1991). “Analytical methods for the development of Reynolds-stress closures in turbulence.”
Annual Review of Fluid Mechanics 23: 107–157.
Taylor, G. I. (1915). “Eddy motion in the atmosphere.” Philosophical Transactions of the Royal Society of
London A215: 1–26.
Taylor, G. I. (1921). “Diffusion by continuous movements.” Proceedings of the London Mathematical
Society 20: 196–211.
Tennekes, H. and J. L. Lumley (1972). A First Course in Turbulence, Cambridge, MA: MIT Press.
Townsend, A. A. (1976). The Structure of Turbulent Shear Flow, London: Cambridge University Press.
Turner, J. S. (1973). Buoyancy Effects in Fluids, London: Cambridge University Press.
Turner, J. S. (1981). “Small-scale mixing processes,” in: Evolution of Physical Oceanography, B. A. Warren
and C. Wunch, eds, Cambridge, MA: MIT Press.
Walker, D. J. and L. Castillo (2002). “Effect of initial conditions on turbulent boundary layers.” AIAA
Journal 40: 2540–2542.
Supplemental Reading
Wei, T., P. Fife, and J. Klewicki (2007). “On scaling the mean momentum balance and its solutions in
turbulent Couette-Poiseuille flow.” Journal of Fluid Mechanics 573: 371–398.
Wosnik, M., L. Castillo, and W. K. George (2000). “A theory for turbulent pipe and channel flows.” Journal
of Fluid Mechanics 421: 115–145.
Zagarola, M. V. and A. J. Smits (1998). “Mean-flow scaling of turbulent pipe flow.” Journal of Fluid
Mechanics 373: 33–79.
Supplemental Reading
Hinze, J. O. (1975). Turbulence, 2nd ed., New York: McGraw-Hill.
Hunt, J. C. R., N. D. Sandham, J. C. Vassilicos, B. E. Launder, P. A. Monkewitz, and G. F. Hewitt(2001).
Developments in turbulence research: are view based on the 1999 Programme of the Isaac Newton
Institute, Cambridge. Published in Journal of Fluid Mechanics, 436: 353–391.
Yakhot, V. and S. A. Orszag (1986). “Renormalization group analysis of turbulence. I. Basic theory.”
Journal of Scientific Computing 1: 3–51.
Proceedings of the Boeing Symposium on Turbulence. Published in Journal of Fluid Mechanics (1970),
41: Parts 1 (March) and 2 (April).
Symposium on Fluid Mechanics of Stirring and Mixing, IUTAM. Published in Physics of Fluids A (1991),
3 (5), May, Part 2.
The Turbulent Years: John Lumley at 70, A Symposium in Honor of John L. Lumley on his 70th Birthday.
Published in Physics of Fluids (2002), 14: 2424–2557.
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Chapter 14
Geophysical Fluid Dynamics
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . .
2. Vertical Variation of Density in
Atmosphere and Ocean . . . . . . . . . . . . . .
3. Equations of Motion . . . . . . . . . . . . . . . . .
Formulation of the Frictional
Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Approximate Equations for a Thin
Layer on a Rotating Sphere . . . . . . . . . .
f -Plane Model . . . . . . . . . . . . . . . . . . . . . .
β-Plane Model . . . . . . . . . . . . . . . . . . . . . .
5. Geostrophic Flow . . . . . . . . . . . . . . . . . . . .
Thermal Wind . . . . . . . . . . . . . . . . . . . . . . .
Taylor–Proudman Theorem . . . . . . . . . .
6. Ekman Layer at a Free Surface . . . . . .
Explanation in Terms of
Vortex Tilting . . . . . . . . . . . . . . . . . . . .
7. Ekman Layer on a Rigid Surface . . . . .
8. Shallow-Water Equations . . . . . . . . . . . .
9. Normal Modes in a Continuously
Stratified Layer . . . . . . . . . . . . . . . . . . . . .
Boundary Conditions on ψn . . . . . . . . . .
Solution of Vertical Modes for
Uniform N . . . . . . . . . . . . . . . . . . . . . . .
10. High- and Low-Frequency Regimes in
Shallow-Water Equations . . . . . . . . . . . .
603
605
607
608
610
612
612
613
614
615
617
622
622
625
628
630
631
634
11. Gravity Waves with Rotation . . . . . . . . .
Particle Orbit . . . . . . . . . . . . . . . . . . . . . . .
Inertial Motion . . . . . . . . . . . . . . . . . . . . . .
12. Kelvin Wave . . . . . . . . . . . . . . . . . . . . . . . . .
13. Potential Vorticity Conservation in
Shallow-Water Theory. . . . . . . . . . . . . . .
14. Internal Waves . . . . . . . . . . . . . . . . . . . . . .
WKB Solution . . . . . . . . . . . . . . . . . . . . . . .
Particle Orbit . . . . . . . . . . . . . . . . . . . . . . .
Discussion of the Dispersion
Relation. . . . . . . . . . . . . . . . . . . . . . . . . .
Lee Wave . . . . . . . . . . . . . . . . . . . . . . . . . . .
15. Rossby Wave . . . . . . . . . . . . . . . . . . . . . . . .
Quasi-geostrophic Vorticity
Equation. . . . . . . . . . . . . . . . . . . . . . . . .
Dispersion Relation . . . . . . . . . . . . . . . . . .
16. Barotropic Instability. . . . . . . . . . . . . . . .
17. Baroclinic Instability . . . . . . . . . . . . . . . .
Perturbation Vorticity Equation . . . . . .
Wave Solution . . . . . . . . . . . . . . . . . . . . . . .
Boundary Conditions . . . . . . . . . . . . . . . .
Instability Criterion. . . . . . . . . . . . . . . . . .
Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . .
18. Geostrophic Turbulence . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . .
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638
639
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647
650
652
654
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657
658
660
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668
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669
671
673
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677
1. Introduction
The subject of geophysical fluid dynamics deals with the dynamics of the atmosphere
and the ocean. It has recently become an important branch of fluid dynamics due to
our increasing interest in the environment. The field has been largely developed by
meteorologists and oceanographers, but non-specialists have also been interested in
the subject. Taylor was not a geophysical fluid dynamicist, but he held the position of
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50014-9
603
604
Geophysical Fluid Dynamics
a meteorologist for some time, and through this involvement he developed a special
interest in the problems of turbulence and instability. Although Prandtl was mainly
interested in the engineering aspects of fluid mechanics, his well-known textbook
(Prandtl, 1952) contains several sections dealing with meteorological aspects of fluid
mechanics. Notwithstanding the pressure for specialization that we all experience
these days, it is worthwhile to learn something of this fascinating field even if one’s
primary interest is in another area of fluid mechanics.
The importance of the study of atmospheric dynamics can hardly be overemphasized. We live within the atmosphere and are almost helplessly affected by the
weather and its rather chaotic behavior. The motion of the atmosphere is intimately
connected with that of the ocean, with which it exchanges fluxes of momentum, heat
and moisture, and this makes the dynamics of the ocean as important as that of the
atmosphere. The study of ocean currents is also important in its own right because of
its relevance to navigation, fisheries, and pollution disposal.
The two features that distinguish geophysical fluid dynamics from other areas
of fluid dynamics are the rotation of the earth and the vertical density stratification
of the medium. We shall see that these two effects dominate the dynamics to such an
extent that entirely new classes of phenomena arise, which have no counterpart in the
laboratory scale flows we have studied in the preceding chapters. (For example, we
shall see that the dominant mode of flow in the atmosphere and the ocean is along
the lines of constant pressure, not from high to low pressures.) The motion of the
atmosphere and the ocean is naturally studied in a coordinate frame rotating with
the earth. This gives rise to the Coriolis force, which is discussed in Chapter 4. The
density stratification gives rise to buoyancy force, which is introduced in Chapter 4
(Conservation Laws) and discussed in further detail in Chapter 7 (Gravity Waves). In
addition, important relevant material is discussed in Chapter 5 (Vorticity), Chapter 10
(Boundary Layer), Chapter 12 (Instability), and Chapter 13 (Turbulence). The reader
should be familiar with these before proceeding further with the present chapter.
Because Coriolis forces and stratification effects play dominating roles in both
the atmosphere and the ocean, there is a great deal of similarity between the dynamics of these two media; this makes it possible to study them together. There are also
significant differences, however. For example the effects of lateral boundaries, due to
the presence of continents, are important in the ocean but not in the atmosphere. The
intense currents (like the Gulf Stream and the Kuroshio) along the western boundaries
of the ocean have no atmospheric analog. On the other hand phenomena like cloud
formation and latent heat release due to moisture condensation are typically atmospheric phenomena. Processes are generally slower in the ocean, in which a typical
horizontal velocity is 0.1 m/s, although velocities of the order of 1–2 m/s are found
within the intense western boundary currents. In contrast, typical velocities in the
atmosphere are 10–20 m/s. The nomenclature can also be different in the two fields.
Meteorologists refer to a flow directed to the west as an “easterly wind” (i.e., from the
east), while oceanographers refer to such a flow as a “westward current.” Atmospheric
scientists refer to vertical positions by “heights” measured upward from the earth’s
surface, while oceanographers refer to “depths” measured downward from the sea
surface. However, we shall always take the vertical coordinate z to be upward, so no
confusion should arise.
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2. Vertical Variation of Density in Atmosphere and Ocean
We shall see that rotational effects caused by the presence of the Coriolis force
have opposite signs in the two hemispheres. Note that all figures and descriptions
given here are valid for the northern hemisphere. In some cases the sense of the
rotational effect for the southern hemisphere has been explicitly mentioned. When
the sense of the rotational effect is left unspecified for the southern hemisphere, it has
to be assumed as opposite to that in the northern hemisphere.
2. Vertical Variation of Density in Atmosphere and Ocean
An important variable in the study of geophysical fluid dynamics is the density stratification. In equation (1.38) we saw that the static stability of a fluid medium is
determined by the sign of the potential density gradient
dρpot
dρ
gρ
=
+ 2,
dz
dz
c
(14.1)
where c is the speed of sound. A medium is statically stable if the potential density
decreases with height. The first term on the right-hand side corresponds to the in situ
density change due to all sources such as pressure, temperature, and concentration of
a constituent such as the salinity in the sea or the water vapor in the atmosphere. The
second term on the right-hand side is the density gradient due to the pressure decrease
with height in an adiabatic environment and is called the adiabatic density gradient.
The corresponding temperature gradient is called the adiabatic temperature gradient.
For incompressible fluids c = ∞ and the adiabatic density gradient is zero.
As shown in Chapter 1, Section 10, the temperature of a dry adiabatic atmosphere
decreases upward at the rate of ≈10 ◦ C/km; that of a moist atmosphere decreases
at the rate of ≈5–6 ◦ C/km. In the ocean, the adiabatic density gradient is gρ/c2
∼4×10−3 kg/m4 , taking a typical sonic speed of c = 1520 m/s. The potential density
in the ocean increases with depth at a much smaller rate of 0.6 × 10−3 kg/m4 , so
that the two terms on the right-hand side of equation (14.1) are nearly in balance.
It follows that most of the in situ density increase with depth in the ocean is due to
the compressibility effects and not to changes in temperature or salinity. As potential
density is the variable that determines the static stability, oceanographers take into
account the compressibility effects by referring all their density measurements to the
sea level pressure. Unless specified otherwise, throughout the present chapter potential
density will simply be referred to as “density,” omitting the qualifier “potential.”
The mean vertical distribution of the in situ temperature in the lower 50 km of
the atmosphere is shown in Figure 14.1. The lowest 10 km is called the troposphere,
in which the temperature decreases with height at the rate of 6.5 ◦ C/km. This is
close to the moist adiabatic lapse rate, which means that the troposphere is close to
being neutrally stable. The neutral stability is expected because turbulent mixing due
to frictional and convective effects in the lower atmosphere keeps it well-stirred and
therefore close to the neutral stratification. Practically all the clouds, weather changes,
and water vapor of the atmosphere are found in the troposphere. The layer is capped by
the tropopause, at an average height of 10 km, above which the temperature increases.
This higher layer is called the stratosphere, because it is very stably stratified. The
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Geophysical Fluid Dynamics
Figure 14.1 Vertical distribution of temperature in the lower 50 km of the atmosphere.
increase of temperature with height in this layer is caused by the absorption of the sun’s
ultraviolet rays by ozone. The stability of the layer inhibits mixing and consequently
acts as a lid on the turbulence and convective motion of the troposphere. The increase
of temperature stops at the stratopause at a height of nearly 50 km.
The vertical structure of density in the ocean is sketched in Figure 14.2, showing
typical profiles of potential density and temperature. Most of the temperature increase
with height is due to the absorption of solar radiation within the upper layer of the
ocean. The density distribution in the ocean is also affected by the salinity. However,
there is no characteristic variation of salinity with depth, and a decrease with depth
is found to be as common as an increase with depth. In most cases, however, the
vertical structure of density in the ocean is determined mainly by that of temperature,
the salinity effects being secondary. The upper 50–200 m of ocean is well-mixed,
due to the turbulence generated by the wind, waves, current shear, and the convective
overturning caused by surface cooling. The temperature gradients decrease with depth,
becoming quite small below a depth of 1500 m. There is usually a large temperature
gradient in the depth range of 100–500 m. This layer of high stability is called the
thermocline. Figure 14.2 also shows the profile of buoyancy frequency N, defined by
N2 ≡ −
g dρ
,
ρ0 dz
where ρ of course stands for the potential density and ρ0 is a constant reference density.
The buoyancy frequency reaches a typical maximum value of Nmax ∼ 0.01 s−1
(period ∼ 10 min) in the thermocline and decreases both upward and downward.
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3. Equations of Motion
Figure 14.2 Typical vertical distributions of: (a) temperature and density; and (b) buoyancy frequency
in the ocean.
3. Equations of Motion
In this section we shall review the relevant equations of motion, which are derived and
discussed in Chapter 4. The equations of motion for a stratified medium, observed in
a system of coordinates rotating at an angular velocity with respect to the “fixed
stars,” are
∇ • u = 0,
Du
1
gρ
+ 2 × u = − ∇p −
k + F,
Dt
ρ0
ρ0
Dρ
= 0,
Dt
(14.2)
where F is the friction force per unit mass. The diffusive effects in the density equation
are omitted in set (14.2) because they will not be considered here.
Set (14.2) makes the so-called Boussinesq approximation, discussed in Chapter 4,
Section 18, in which the density variations are neglected everywhere except in the
gravity term. Along with other restrictions, it assumes that the vertical scale of the
motion is less than the “scale height” of the medium c2 /g, where c is the speed
of sound. This assumption is very good in the ocean, in which c2 /g ∼ 200 km. In
the atmosphere it is less applicable, because c2 /g ∼ 10 km. Under the Boussinesq
approximation, the principle of mass conservation is expressed by ∇ • u = 0. In
contrast, the density equation Dρ/Dt = 0 follows from the nondiffusive heat equation
DT /Dt = 0 and an incompressible equation of state of the form δρ/ρ0 = −αδT .
(If the density is determined by the concentration S of a constituent, say the water
vapor in the atmosphere or the salinity in the ocean, then Dρ/Dt = 0 follows from
608
Geophysical Fluid Dynamics
the nondiffusive conservation equation for the constituent in the form DS/Dt = 0,
plus the incompressible equation of state δρ/ρ0 = βδS.)
The equations can be written in terms of the pressure and density perturbations
from a state of rest. In the absence of any motion, suppose the density and pressure
have the vertical distributions ρ̄(z) and p̄(z), where the z-axis is taken vertically
upward. As this state is hydrostatic, we must have
d p̄
= −ρ̄g.
dz
(14.3)
In the presence of a flow field u(x, t), we can write the density and pressure as
ρ(x, t) = ρ̄(z) + ρ ′ (x, t),
p(x, t) = p̄(z) + p ′ (x, t),
(14.4)
where ρ ′ and p ′ are the changes from the state of rest. With this substitution, the first
two terms on the right-hand side of the momentum equation in (14.2) give
−
gρ
1
g(ρ̄ + ρ ′ )
1
∇p −
k = − ∇(p̄ + p′ ) −
k
ρ0
ρ0
ρ0
ρ0
g(ρ̄ + ρ ′ )
1 d p̄
′
k + ∇p −
k.
=−
ρ0 dz
ρ0
Subtracting the hydrostatic state (14.3), this becomes
−
gρ
1
gρ ′
1
∇p −
k = − ∇p ′ −
k,
ρ0
ρ0
ρ0
ρ0
which shows that we can replace p and ρ in equation (14.2) by the perturbation
quantities p′ and ρ ′ .
Formulation of the Frictional Term
The friction force per unit mass F in equation (14.2) needs to be related to the velocity
field. From Chapter 4, Section 7, the friction force is given by
Fi =
∂τij
,
∂xj
where τij is the viscous stress tensor. The stress in a laminar flow is caused by the
molecular exchanges of momentum. From equation (4.41), the viscous stress tensor
in an isotropic incompressible medium in laminar flow is given by
∂uj
∂ui
τij = ρν
.
+
∂xj
∂xi
In large-scale geophysical flows, however, the frictional forces are provided by turbulent mixing, and the molecular exchanges are negligible. The complexity of turbulent
609
3. Equations of Motion
behavior makes it impossible to relate the stress to the velocity field in a simple way.
To proceed, then, we adopt the eddy viscosity hypothesis, assuming that the turbulent
stress is proportional to the velocity gradient field.
Geophysical media are in the form of shallow stratified layers, in which the
vertical velocities are much smaller than horizontal velocities. This means that the
exchange of momentum across a horizontal surface is much weaker than that across a
vertical surface. We expect then that the vertical eddy viscosity νv is much smaller than
the horizontal eddy viscosity νH , and we assume that the turbulent stress components
have the form
∂w
∂u
+ ρνH
,
∂z
∂x
∂w
∂v
+ ρνH
,
τyz = τzy = ρνv
∂z
∂y
∂u ∂v
τxy = τyx = ρνH
+
,
∂y
∂x
∂u
∂v
∂w
= 2ρνH , τyy = 2ρνH , τzz = 2ρνv
.
∂x
∂y
∂z
τxz = τzx = ρνv
τxx
(14.5)
The difficulty with set (14.5) is that the expressions for τxz and τyz depend on the fluid
rotation in the vertical plane and not just the deformation. In Chapter 4, Section 10,
we saw that a requirement for a constitutive equation is that the stresses should be
independent of fluid rotation and should depend only on the deformation. Therefore, τxz should depend only on the combination (∂u/∂z + ∂w/∂x), whereas the
expression in equation (14.5) depends on both deformation and rotation. A tensorially correct geophysical treatment of the frictional terms is discussed, for example,
in Kamenkovich (1967). However, the assumed form (14.5) leads to a simple formulation for viscous effects, as we shall see shortly. As the eddy viscosity assumption is
of questionable validity (which Pedlosky (1971) describes as a “rather disreputable
and desperate attempt”), there does not seem to be any purpose in formulating the
stress–strain relation in more complicated ways merely to obey the requirement of
invariance with respect to rotation.
With the assumed form for the turbulent stress, the components of the frictional
force Fi = ∂τij /∂xj become
Fx =
∂τxy
∂τxz
∂τxx
+
+
= νH
∂x
∂y
∂z
Fy =
∂τyy
∂τyz
∂τyx
+
+
= νH
∂x
∂y
∂z
∂τzy
∂τzx
∂τzz
Fz =
+
+
= νH
∂x
∂y
∂z
∂ 2u ∂ 2u
+ 2
∂x 2
∂y
∂ 2v
∂ 2v
+
∂x 2
∂y 2
+ νv
∂ 2u
,
∂z2
+ νv
∂ 2v
,
∂z2
∂ 2w ∂ 2w
+
∂x 2
∂y 2
+ νv
(14.6)
∂ 2w
.
∂z2
Estimates of the eddy coefficients vary greatly. Typical suggested values are
νv ∼ 10 m2 /s and νH ∼ 105 m2 /s for the lower atmosphere, and νv ∼ 0.01 m2 /s
610
Geophysical Fluid Dynamics
and νH ∼ 100 m2 /s for the upper ocean. In comparison, the molecular values are
ν = 1.5 × 10−5 m2 /s for air and ν = 10−6 m2 /s for water.
4. Approximate Equations for a Thin Layer on
a Rotating Sphere
The atmosphere and the ocean are very thin layers in which the depth scale of flow
is a few kilometers, whereas the horizontal scale is of the order of hundreds, or even
thousands, of kilometers. The trajectories of fluid elements are very shallow and
the vertical velocities are much smaller than the horizontal velocities. In fact, the
continuity equation suggests that the scale of the vertical velocity W is related to that
of the horizontal velocity U by
W
H
∼ ,
U
L
where H is the depth scale and L is the horizontal length scale. Stratification and
Coriolis effects usually constrain the vertical velocity to be even smaller than U H /L.
Large-scale geophysical flow problems should be solved using spherical polar
coordinates. If, however, the horizontal length scales are much smaller than the radius
of the earth (= 6371 km), then the curvature of the earth can be ignored, and the
motion can be studied by adopting a local Cartesian system on a tangent plane
(Figure 14.3). On this plane we take an xyz coordinate system, with x increasing
eastward, y northward, and z upward. The corresponding velocity components are u
(eastward), v (northward), and w (upward).
Figure 14.3 Local Cartesian coordinates. The x-axis is into the plane of the paper.
611
4. Approximate Equations for a Thin Layer on a Rotating Sphere
The earth rotates at a rate
= 2π rad/day = 0.73 × 10−4 s−1 ,
around the polar axis, in a counterclockwise sense looking from above the north
pole. From Figure 14.3, the components of angular velocity of the earth in the local
Cartesian system are
x
= 0,
y
=
cos θ,
z
=
sin θ,
where θ is the latitude. The Coriolis force is therefore
i
j
k
2 × u = 0 2 cos θ 2 sin θ
u
v
w
= 2 [i(w cos θ − v sin θ) + ju sin θ − ku cos θ].
In the term multiplied by i we can use the condition w cos θ ≪ v sin θ, because the
thin sheet approximation requires that w ≪ v. The three components of the Coriolis
force are therefore
(2 × u)x = −(2 sin θ )v = −f v,
(2 × u)y = (2 sin θ)u = f u,
(14.7)
(2 × u)z = −(2 cos θ)u,
where we have defined
f = 2 sin θ ,
(14.8)
to be twice the vertical component of . As vorticity is twice the angular velocity, f is called the planetary vorticity. More commonly, f is referred to as the
Coriolis parameter, or the Coriolis frequency. It is positive in the northern hemisphere and negative in the southern hemisphere, varying from ±1.45 × 10−4 s−1 at
the poles to zero at the equator. This makes sense, since a person standing at the
north pole spins around himself in an counterclockwise sense at a rate , whereas
a person standing at the equator does not spin around himself but simply translates.
The quantity
Ti = 2π/f,
is called the inertial period, for reasons that will be clear in Section 11.
612
Geophysical Fluid Dynamics
The vertical component of the Coriolis force, namely −2 u cos θ, is generally
negligible compared to the dominant terms in the vertical equation of motion, namely
gρ ′ /ρ0 and ρ0−1 (∂p ′ /∂z). Using equations (14.6) and (14.7), the equations of motion
(14.2) reduce to
Du
1 ∂p
− fv =−
+ νH
Dt
ρ0 ∂x
∂ 2u ∂ 2u
+ 2
∂x 2
∂y
+ νv
∂ 2u
,
∂z2
∂ 2v
∂ 2v
∂ 2v
+ νv 2 ,
+
2
2
∂x
∂y
∂z
2
2
Dw
1 ∂p gρ
∂ 2w
∂ w ∂ w
+
ν
=−
−
+ νH
+
.
v
Dt
ρ0 ∂z
ρ0
∂x 2
∂y 2
∂z2
Dv
1 ∂p
+ fu=−
+ νH
Dt
ρ0 ∂y
(14.9)
These are the equations of motion for a thin shell on a rotating earth. Note that only
the vertical component of the earth’s angular velocity appears as a consequence of the
flatness of the fluid trajectories.
f -Plane Model
The Coriolis parameter f = 2 sin θ varies with latitude θ. However, we shall see
later that this variation is important only for phenomena having very long time scales
(several weeks) or very long length scales (thousands of kilometers). For many purposes we can assume f to be a constant, say f0 = 2 sin θ0 , where θ0 is the central
latitude of the region under study. A model using a constant Coriolis parameter is
called an f-plane model.
β-Plane Model
The variation of f with latitude can be approximately represented by expanding f in
a Taylor series about the central latitude θ0 :
f = f0 + βy,
(14.10)
where we defined
β≡
df
dy
θ0
=
df dθ
dθ dy
θ0
=
2 cos θ0
.
R
Here, we have used f = 2 sin θ and dθ/dy = 1/R, where the radius of the earth is
nearly
R = 6371 km.
A model that takes into account the variation of the Coriolis parameter in the simplified
form f = f0 + βy, with β as constant, is called a β-plane model.
613
5. Geostrophic Flow
5. Geostrophic Flow
Consider quasi-steady large-scale motions in the atmosphere or the ocean, away from
boundaries. For these flows an excellent approximation for the horizontal equilibrium
is a balance between the Coriolis force and the pressure gradient:
−f v = −
1 ∂p
,
ρ0 ∂x
1 ∂p
.
fu=−
ρ0 ∂y
(14.11)
Here we have neglected the nonlinear acceleration terms, which are of order U 2 /L,
in comparison to the Coriolis force ∼f U (U is the horizontal velocity scale, and L
is the horizontal length scale.) The ratio of the nonlinear term to the Coriolis term is
called the Rossby number :
Rossby number =
Nonlinear acceleration
U
U 2 /L
∼
=
= Ro.
Coriolis force
fU
fL
For a typical atmospheric value of U ∼ 10 m/s, f ∼ 10−4 s−1 , and L ∼ 1000 km,
the Rossby number turns out to be 0.1. The Rossby number is even smaller for many
flows in the ocean, so that the neglect of nonlinear terms is justified for many flows.
The balance of forces represented by equation (14.11), in which the horizontal
pressure gradients are balanced by Coriolis forces, is called a geostrophic balance. In
such a system the velocity distribution can be determined from a measured distribution of the pressure field. The geostrophic equilibrium breaks down near the equator
(within a latitude belt of ±3◦ ), where f becomes small. It also breaks down if the
frictional effects or unsteadiness become important.
Velocities in a geostrophic flow are perpendicular to the horizontal pressure gradient. This is because equation (14.11) implies that v • ∇p = 0, i.e., . . .
∂p
1
∂p
∂p
∂p
•
+j
+j
−i
i
= 0.
(iu + jv) • ∇p =
ρ0 f
∂y
∂x
∂x
∂y
Thus, the horizontal velocity is along, and not across, the lines of constant pressure.
If f is regarded as constant, then the geostrophic balance (14.11) shows that p/fρ0
can be regarded as a streamfunction. The isobars on a weather map are therefore
nearly the streamlines of the flow.
Figure 14.4 shows the geostrophic flow around low and high pressure centers
in the northern hemisphere. Here the Coriolis force acts to the right of the velocity
vector. This requires the flow to be counterclockwise (viewed from above) around
a low pressure region and clockwise around a high pressure region. The sense of
circulation is opposite in the southern hemisphere, where the Coriolis force acts to
the left of the velocity vector. (Frictional forces become important at lower levels in
the atmosphere and result in a flow partially across the isobars. This will be discussed
614
Geophysical Fluid Dynamics
Figure 14.4 Geostrophic flow around low and high pressure centers. The pressure force (−∇p) is
indicated by a thin arrow, and the Coriolis force is indicated by a thick arrow.
in Section 7, where we will see that the flow around a low pressure center spirals
inward due to frictional effects.)
The flow along isobars at first surprises a reader unfamiliar with the effects
of the Coriolis force. A question commonly asked is: How is such a motion set up?
A typical manner of establishment of such a flow is as follows. Consider a horizontally
converging flow in the surface layer of the ocean. The convergent flow sets up the
sea surface in the form of a gentle “hill,” with the sea surface dropping away from
the center of the hill. A fluid particle starting to move down the “hill” is deflected to
the right in the northern hemisphere, and a steady state is reached when the particle
finally moves along the isobars.
Thermal Wind
In the presence of a horizontal gradient of density, the geostrophic velocity develops
a vertical shear. This is easy to demonstrate from an analysis of the geostrophic and
hydrostatic balance
−f v = −
1 ∂p
,
ρ0 ∂x
(14.12)
fu = −
1 ∂p
,
ρ0 ∂y
(14.13)
∂p
− gρ.
∂z
(14.14)
0=−
615
5. Geostrophic Flow
Eliminating p between equations (14.12) and (14.14), and also between equations
(14.13) and (14.14), we obtain, respectively,
g ∂ρ
∂v
=−
,
∂z
ρ0 f ∂x
∂u
g ∂ρ
=
.
∂z
ρ0 f ∂y
(14.15)
Meteorologists call these the thermal wind equations because they give the vertical variation of wind from measurements of horizontal temperature gradients. The
thermal wind is a baroclinic phenomenon, because the surfaces of constant p and ρ
do not coincide.
Taylor–Proudman Theorem
A striking phenomenon occurs in the geostrophic flow of a homogeneous fluid. It can
only be observed in a laboratory experiment because stratification effects cannot be
avoided in natural flows. Consider then a laboratory experiment in which a tank of
fluid is steadily rotated at a high angular speed and a solid body is moved slowly
along the bottom of the tank. The purpose of making large and the movement of
the solid body slow is to make the Coriolis force much larger than the acceleration
terms, which must be made negligible for geostrophic equilibrium. Away from the
frictional effects of boundaries, the balance is therefore geostrophic in the horizontal
and hydrostatic in the vertical:
−2 v = −
1 ∂p
,
ρ ∂x
(14.16)
2 u=−
1 ∂p
,
ρ ∂y
(14.17)
1 ∂p
− g.
ρ ∂z
(14.18)
0=−
It is useful to define an Ekman number as the ratio of viscous to Coriolis forces
(per unit volume):
Ekman number =
viscous force
ρνU/L2
ν
=
=
= E.
Coriolis force
ρf U
f L2
Under the circumstances already described here, both Ro and E are small.
Elimination of p by cross differentiation between the horizontal momentum
equations gives
∂v
∂u
2
+
= 0.
∂y
∂x
616
Geophysical Fluid Dynamics
Using the continuity equation, this gives
∂w
= 0.
∂z
(14.19)
Also, differentiating equations (14.16) and (14.17) with respect to z, and using
equation (14.18), we obtain
∂u
∂v
=
= 0.
∂z
∂z
(14.20)
Equations (14.19) and (14.20) show that
∂u
= 0,
∂z
(14.21)
showing that the velocity vector cannot vary in the direction of . In other words,
steady slow motions in a rotating, homogeneous, inviscid fluid are two dimensional.
This is the Taylor–Proudman theorem, first derived by Proudman in 1916 and demonstrated experimentally by Taylor soon afterwards.
In Taylor’s experiment, a tank was made to rotate as a solid body, and a small
cylinder was slowly dragged along the bottom of the tank (Figure 14.5). Dye was
introduced from point A above the cylinder and directly ahead of it. In a nonrotating fluid the water would pass over the top of the moving cylinder. In the rotating
experiment, however, the dye divides at a point S, as if it had been blocked by an
upward extension of the cylinder, and flows around this imaginary cylinder, called
the Taylor column. Dye released from a point B within the Taylor column remained
there and moved with the cylinder. The conclusion was that the flow outside the
upward extension of the cylinder is the same as if the cylinder extended across the
entire water depth and that a column of water directly above the cylinder moves with
it. The motion is two dimensional, although the solid body does not extend across
the entire water depth. Taylor did a second experiment, in which he dragged a solid
body parallel to the axis of rotation. In accordance with ∂w/∂z = 0, he observed
that a column of fluid is pushed ahead. The lateral velocity components u and v
were zero. In both of these experiments, there are shear layers at the edge of the
Taylor column.
In summary, Taylor’s experiment established the following striking fact for steady
inviscid motion of homogeneous fluid in a strongly rotating system: Bodies moving
either parallel or perpendicular to the axis of rotation carry along with their motion
a so-called Taylor column of fluid, oriented parallel to the axis. The phenomenon is
analogous to the horizontal blocking caused by a solid body (say a mountain) in a
strongly stratified system, shown in Figure 7.33.
6. Ekman Layer at a Free Surface
Figure 14.5 Taylor’s experiment in a strongly rotating flow of a homogeneous fluid.
6. Ekman Layer at a Free Surface
In the preceding section, we discussed a steady linear inviscid motion expected to be
valid away from frictional boundary layers. We shall now examine the motion within
frictional layers over horizontal surfaces. In viscous flows unaffected by Coriolis
forces and pressure gradients, the only term which can balance the viscous force is
either the time derivative ∂u/∂t or the advection u • ∇u. The balance of ∂u/∂t and
the viscous force gives rise to a viscous layer whose thickness increases with time,
as in the suddenly accelerated plate discussed in Chapter 9, Section 7. The balance
of u • ∇u and the viscous force give rise to a viscous layer whose thickness increases
in the direction of flow, as in the boundary layer over a semi-infinite plate discussed
in Chapter 10, Sections 5 and 6. In a rotating flow, however, we can have a balance
between the Coriolis and the viscous forces, and the thickness of the viscous layer
can be invariant in time and space. Two examples of such layers are given in this and
the following sections.
617
618
Geophysical Fluid Dynamics
Consider first the case of a frictional layer near the free surface of the ocean,
which is acted on by a wind stress τ in the x-direction. We shall not consider how
the flow adjusts to the steady state but examine only the steady solution. We shall
assume that the horizontal pressure gradients are zero and that the field is horizontally
homogeneous. From equation (14.9), the horizontal equations of motion are
−f v = νv
d 2u
,
dz2
(14.22)
d 2v
.
dz2
(14.23)
f u = νv
Taking the z-axis vertically upward from the surface of the ocean, the boundary
conditions are
ρνv
du
= τ
dz
at z = 0,
(14.24)
dv
= 0
dz
at z = 0,
(14.25)
as z → −∞.
(14.26)
u, v → 0
Multiplying equation (14.23) by i =
√
−1 and adding equation (14.22), we obtain
d 2V
if
= V,
dz2
νv
(14.27)
where we have defined the “complex velocity”
V ≡ u + iv.
The solution of equation (14.27) is
V = A e(1+i)z/δ + B e−(1+i)z/δ ,
(14.28)
(14.29)
where we have defined
δ≡
2 νv
.
f
We shall see shortly that δ is the thickness of the Ekman layer. The constant B
is zero because the field must remain finite as z → −∞. The surface boundary
conditions (14.24) and (14.25) can be combined as ρνv (dV /dz) = τ at z = 0, from
which equation (14.28) gives
619
6. Ekman Layer at a Free Surface
A=
τ δ(1 − i)
.
2ρνv
Substitution of this into equation (14.28) gives the velocity components
z π
τ/ρ z/δ
,
e cos − +
u= √
δ
4
f νv
z π
τ/ρ z/δ
v = −√
.
e sin − +
δ
4
f νv
The Swedish oceanographer Ekman worked out this solution in 1905. The solution is shown in Figure 14.6 for the case of the northern hemisphere, in which f
is positive. The velocities at various depths are plotted in Figure 14.6a, where each
arrow represents the velocity vector at a certain depth. Such a plot of v vs u is sometimes called a “hodograph” plot. The vertical distributions of u and v are shown
in Figure 14.6b. The hodograph shows that the surface velocity is deflected 45◦ to
the right of the applied wind stress. (In the southern hemisphere the deflection is to
the left of the surface stress.) The velocity vector rotates clockwise (looking down)
with depth, and the magnitude exponentially decays with an e-folding scale of δ,
which is called the Ekman layer thickness. The tips of the velocity vector at various
depths form a spiral, called the Ekman spiral.
Figure 14.6 Ekman layer at a free surface. The left panel shows velocity at various depths; values of
−z/δ are indicated along the curve traced out by the tip of the velocity vectors. The right panel shows
vertical distributions of u and v.
620
Geophysical Fluid Dynamics
The components of the volume transport in the Ekman layer are
0
−∞
u dz = 0,
0
−∞
v dz = −
τ
.
ρf
(14.30)
This shows that the net transport is to the right of the applied stress and is independent
of νv . In fact, the result v dz = −τ/fρ follows directly from a vertical integration of
the equation of motion in the form −ρf v = d(stress)/dz, so that the result does not
depend on the eddy viscosity assumption. The fact that the transport is to the right of
the applied stress makes sense, because then the net (depth-integrated) Coriolis force,
directed to the right of the depth-integrated transport, can balance the wind stress.
The horizontal uniformity assumed in the solution is not a serious limitation.
Since Ekman layers near the ocean surface have a thickness (∼50 m) much smaller
than the scale of horizontal variation (L > 100 km), the solution is still locally applicable. The absence of horizontal pressure gradient assumed here can also be relaxed
easily. Because of the thinness of the layer, any imposed horizontal pressure gradient
remains constant across the layer. The presence of a horizontal pressure gradient
merely adds a depth-independent geostrophic velocity to the Ekman solution. Suppose
the sea surface slopes down to the north, so that there is a pressure force acting northward throughout the Ekman layer and below (Figure 14.7). This means that at the
bottom of the Ekman layer (z/δ → −∞) there is a geostrophic velocity U to the
right of the pressure force. The surface Ekman spiral forced by the wind stress joins
smoothly to this geostrophic velocity as z/δ → −∞.
Figure 14.7 Ekman layer at a free surface in the presence of a pressure gradient. The geostrophic velocity
forced by the pressure gradient is U .
6. Ekman Layer at a Free Surface
Pure Ekman spirals are not observed in the surface layer of the ocean, mainly
because the assumptions of constant eddy viscosity and steadiness are particularly
restrictive. When the flow is averaged over a few days, however, several instances
have been found in which the current does look like a spiral. One such example is
shown in Figure 14.8.
Figure 14.8 An observed velocity distribution near the coast of Oregon. Velocity is averaged over 7 days.
Wind stress had a magnitude of 1.1 dyn/cm2 and was directed nearly southward, as indicated at the top of
the figure. The upper panel shows vertical distributions of u and v, and the lower panel shows the hodograph
in which depths are indicated in meters. The hodograph is similar to that of a surface Ekman layer (of
depth 16 m) lying over the bottom Ekman layer (extending from a depth of 16 m to the ocean bottom).
P. Kundu, in Bottom Tubulence, J. C. J. Nihoul, ed., Elsevier, 1977 and reprinted with the permission of
Jacques C. J. Nihoul.
621
622
Geophysical Fluid Dynamics
Explanation in Terms of Vortex Tilting
We have seen in previous chapters that the thickness of a viscous layer usually grows
in a nonrotating flow, either in time or in the direction of flow. The Ekman solution,
in contrast, results in a viscous layer that does not grow either in time or space. This
can be explained by examining the vorticity equation (Pedlosky, 1987). The vorticity
components in the x- and y-directions are
ωx =
∂w ∂v
dv
−
=− ,
∂y
∂z
dz
ωy =
du
∂u ∂w
−
=
,
∂z
∂x
dz
where we have used w = 0. Using these, the z-derivative of the equations of motion
(14.22) and (14.23) gives
−f
d 2 ωy
dv
= νv
,
dz
dz2
(14.31)
du
d 2 ωx
−f
= νv
.
dz
dz2
The right-hand side of these equations represent diffusion of vorticity. Without
Coriolis forces this diffusion would cause a thickening of the viscous layer. The
presence of planetary rotation, however, means that vertical fluid lines coincide with
the planetary vortex lines. The tilting of vertical fluid lines, represented by terms on
the left-hand sides of equations (14.31), then causes a rate of change of horizontal
component of vorticity that just cancels the diffusion term.
7. Ekman Layer on a Rigid Surface
Consider now a horizontally independent and steady viscous layer on a solid surface
in a rotating flow. This can be the atmospheric boundary layer over the solid earth or
the boundary layer over the ocean bottom. We assume that at large distances from the
surface the velocity is toward the x-direction and has a magnitude U . Viscous forces
are negligible far from the wall, so that the Coriolis force can be balanced only by a
pressure gradient:
fU = −
1 dp
.
ρ dy
(14.32)
This simply states that the flow outside the viscous layer is in geostrophic balance,
U being the geostrophic velocity. For our assumed case of positive U and f , we
must have dp/dy < 0, so that the pressure falls with y—that is, the pressure force is
directed along the positive y direction, resulting in a geostrophic flow U to the right
623
7. Ekman Layer on a Rigid Surface
of the pressure force in the northern hemisphere. The horizontal pressure gradient
remains constant within the thin boundary layer.
Near the solid surface the viscous forces are important, so that the balance within
the boundary layer is
−f v = νv
f u = νv
d 2u
,
dz2
(14.33)
d 2v
+ f U,
dz2
(14.34)
where we have replaced −ρ −1 (dp/dy) by f U in accordance with equation (14.32).
The boundary conditions are
u = U,
v=0
as z → ∞,
(14.35)
u = 0,
v=0
at z = 0,
(14.36)
where z is taken vertically upward from the solid surface. Multiplying equation (14.34)
by i and adding equation (14.33), the equations of motion become
d 2V
if
= (V − U ),
2
dz
νv
(14.37)
where we have defined the complex velocity V ≡ u + iv. The boundary
conditions (14.35) and (14.36) in terms of the complex velocity are
V =U
as z → ∞,
(14.38)
V =0
at z = 0.
(14.39)
The particular solution of equation (14.37) is V = U . The total solution is, therefore,
V = A e−(1+i)z/δ + B e(1+i)z/δ + U,
(14.40)
√
where δ ≡ 2νv /f . To satisfy equation (14.38), we must have B = 0. Condition
(14.39) gives A = −U . The velocity components then become
u = U [1 − e−z/δ cos (z/δ)],
v = U e−z/δ sin (z/δ).
(14.41)
According to equation (14.41), the tip of the velocity vector describes a spiral for
various values of z (Figure 14.9a). As with the Ekman layer
√ at a free surface, the
frictional effects are confined within a layer of thickness δ = 2νv /f , which increases
with νv and decreases with the rotation rate f . Interestingly, the layer thickness is
independent of the magnitude of the free-stream velocity U ; this behavior is quite
different from that of a steady nonrotating boundary layer on a semi-infinite plate
√ (the
Blasius solution of Section 10.5) in which the thickness is proportional to 1/ U .
624
Geophysical Fluid Dynamics
Figure 14.9 Ekman layer at a rigid surface. The left panel shows velocity vectors at various heights;
values of z/δ are indicated along the curve traced out by the tip of the velocity vectors. The right panel
shows vertical distributions of u and v.
Figure 14.9b shows the vertical distribution of the velocity components. Far from
the wall the velocity is entirely in the x-direction, and the Coriolis force balances the
pressure gradient. As the wall is approached, retarding effects decrease u and the
associated Coriolis force, so that the pressure gradient (which is independent of z)
forces a component v in the direction of the pressure force. Using equation (14.41),
the net transport in the Ekman layer normal to the uniform stream outside the layer is
∞
0
νv
v dz = U
2f
1/2
=
1
U δ,
2
which is directed to the left of the free-stream velocity, in the direction of the pressure
force.
If the atmosphere were in laminar motion, νv would be equal to its molecular
value for air, and the Ekman layer thickness at a latitude of 45◦ (where f ≃ 10−4 s−1 )
would be ≈ δ ∼ 0.4 m. The observed thickness of the atmospheric boundary layer
is of order 1 km, which implies an eddy viscosity of order νv ∼ 50 m2 /s. In fact,
Taylor (1915) tried to estimate the eddy viscosity by matching the predicted velocity
distributions (14.41) with the observed wind at various heights.
The Ekman layer solution on a solid surface demonstrates that the three-way
balance among the Coriolis force, the pressure force, and the frictional force within
the boundary layer results in a component of flow directed toward the lower pressure.
8. Shallow-Water Equations
Figure 14.10 Balance of forces within an Ekman layer, showing that velocity u has a component toward
low pressure.
The balance of forces within the boundary layer is illustrated in Figure 14.10. The
net frictional force on an element is oriented approximately opposite to the velocity
vector u. It is clear that a balance of forces is possible only if the velocity vector has a
component from high to low pressure, as shown. Frictional forces therefore cause the
flow around a low-pressure center to spiral inward. Mass conservation requires that
the inward converging flow should rise over a low-pressure system, resulting in cloud
formation and rainfall. This is what happens in a cyclone, which is a low-pressure
system. In contrast, over a high-pressure system the air sinks as it spirals outward
due to frictional effects. The arrival of high-pressure systems therefore brings in clear
skies and fair weather, because the sinking air does not result in cloud formation.
Frictional effects, in particular the Ekman transport by surface winds, play a
fundamental role in the theory of wind-driven ocean circulation. Possibly the most
important result of such theories was given by Henry Stommel in 1948. He showed
that the northward increase of the Coriolis parameter f is responsible for making the
currents along the western boundary of the ocean (e.g., the Gulf Stream in the Atlantic
and the Kuroshio in the Pacific) much stronger than the currents on the eastern side.
These are discussed in books on physical oceanography and will not be presented
here. Instead, we shall now turn our attention to the influence of Coriolis forces on
inviscid wave motions.
8. Shallow-Water Equations
Both surface and internal gravity waves were discussed in Chapter 7. The effect
of planetary rotation was assumed to be small, which is valid if the frequency ω
of the wave is much larger than the Coriolis parameter f . In this chapter we are
considering phenomena slow enough for ω to be comparable to f . Consider surface
625
626
Geophysical Fluid Dynamics
gravity waves on a shallow layer of homogeneous fluid whose mean depth is H . If we
restrict ourselves to wavelengths λ much larger than H , then the vertical velocities
are much smaller than the horizontal velocities. In Chapter 7, Section 6 we saw that
the acceleration ∂w/∂t is then negligible in the vertical momentum equation, so that
the pressure distribution is hydrostatic. We also demonstrated that the fluid particles
execute a horizontal rectilinear motion that is independent of z. When the effects
of planetary rotation are included, the horizontal velocity is still depth-independent,
although the particle orbits are no longer rectilinear but elliptic on a horizontal plane,
as we shall see in the following section.
Consider a layer of fluid over a flat horizontal bottom (Figure 14.11). Let z be
measured upward from the bottom surface, and η be the displacement of the free
surface. The pressure at height z from the bottom, which is hydrostatic, is given by
p = ρg(H + η − z).
The horizontal pressure gradients are therefore
∂p
∂η
= ρg ,
∂x
∂x
∂p
∂η
= ρg .
∂y
∂y
(14.42)
As these are independent of z, the resulting horizontal motion is also depth
independent.
Now consider the continuity equation
∂w
∂u ∂v
+
+
= 0.
∂x
∂y
∂z
As ∂u/∂x and ∂v/∂y are independent of z, the continuity equation requires that w
vary linearly with z, from zero at the bottom to the maximum value at the free surface.
Integrating vertically across the water column from z = 0 to z = H + η, and noting
that u and v are depth independent, we obtain
(H + η)
∂u
∂v
+ (H + η)
+ w(η) − w(0) = 0,
∂x
∂y
Figure 14.11 Layer of fluid on a flat bottom.
(14.43)
627
8. Shallow-Water Equations
where w(η) is the vertical velocity at the surface and w(0) = 0 is the vertical velocity
at the bottom. The surface velocity is given by
w(η) =
Dη
∂η
∂η
∂η
=
+u
+v .
Dt
∂t
∂x
∂y
The continuity equation (14.43) then becomes
(H + η)
∂u
∂v
∂η
∂η
∂η
+ (H + η)
+
+u
+v
= 0,
∂x
∂y
∂t
∂x
∂y
which can be written as
∂
∂
∂η
+
[u(H + η)] +
[v(H + η)] = 0.
∂t
∂x
∂y
(14.44)
This says simply that the divergence of the horizontal transport depresses the free
surface. For small amplitude waves, the quadratic nonlinear terms can be neglected
in comparison to the linear terms, so that the divergence term in equation (14.44)
simplifies to H ∇ • u.
The linearized continuity and momentum equations are then
∂η
+H
∂t
∂u ∂v
+
∂x
∂y
= 0,
∂u
∂η
− f v = −g ,
∂t
∂x
(14.45)
∂v
∂η
+ f u = −g .
∂t
∂y
In the momentum equations of (14.45), the pressure gradient terms are written in the
form (14.42) and the nonlinear advective terms have been neglected under the small
amplitude assumption. Equations (14.45), called the shallow water equations, govern
the motion of a layer of fluid in which the horizontal scale is much larger than the
depth of the layer. These equations will be used in the following sections for studying
various types of gravity waves.
Although the preceding analysis has been formulated for a layer of homogeneous
fluid, equations (14.45) are applicable to internal waves in a stratified medium, if we
replaced H by the equivalent depth He , defined by
c2 = gHe ,
(14.46)
where c is the speed of long nonrotating internal gravity waves. This will be demonstrated in the following section.
628
Geophysical Fluid Dynamics
9. Normal Modes in a Continuously Stratified Layer
In the preceding section we considered a homogeneous medium and derived the
governing equations for waves of wavelength larger than the depth of the fluid layer.
Now consider a continuously stratified medium and assume that the horizontal scale
of motion is much larger than the vertical scale. The pressure distribution is therefore
hydrostatic, and the equations of motion are
∂u ∂v
∂w
+
+
= 0,
∂x
∂y
∂z
(14.47)
1 ∂p
∂u
− fv = −
,
∂t
ρ0 ∂x
(14.48)
∂v
1 ∂p
+ fu = −
,
∂t
ρ0 ∂y
(14.49)
0=−
∂p
− gρ,
∂z
∂ρ
ρ0 N 2
−
w = 0,
∂t
g
(14.50)
(14.51)
where p and ρ represent perturbations of pressure and density from the state of
rest. The advective term in the density equation is written in the linearized form
w(d ρ̄/dz) = −ρ0 N 2 w/g, where N (z) is the buoyancy frequency. In this form the
rate of change of density at a point is assumed to be due only to the vertical advection
of the background density distribution ρ̄(z), as discussed in Chapter 7, Section 18.
In a continuously stratified medium, it is convenient to use the method of separation of variables and write q = qn (x, y, t)ψn (z) for some variable q. The solution
is thus written as the sum of various vertical “modes,” which are called normal modes
because they turn out to be orthogonal to each other. The vertical structure of a mode is
described by ψn and qn describes the horizontal propagation of the mode. Although
each mode propagates only horizontally, the sum of a number of modes can also
propagate vertically if the various qn are out of phase.
We assume separable solutions of the form
[u, v, p/ρ0 ] =
w=
ρ=
∞
n=0
∞
[un , vn , pn ]ψn (z),
z
wn
n=0
ψn (z) dz,
(14.53)
−H
n=0
∞
(14.52)
ρn
dψn
,
dz
(14.54)
629
9. Normal Modes in a Continuously Stratified Layer
where the amplitudes un , vn , pn , wn , and ρn are functions of (x, y, t). The z-axis
is measured from the upper free surface of the fluid layer, and z = −H represents
the bottom wall. The reasons for assuming the various forms of z-dependence in
equations (14.52)–(14.54) are the following: Variables u, v, and p have the same
vertical structure in order to be consistent with equations (14.48) and (14.49). Continuity equation (14.47) requires that the vertical structure of w should be the integral of ψn (z). Equation (14.50) requires that the vertical structure of ρ must be the
z-derivative of the vertical structure of p.
Subsititution of equations (14.53) and (14.54) into equation (14.51) gives
∞
n=0
∂ρn dψn
ρ0 N 2
−
wn
∂t dz
g
z
−H
ψn dz = 0.
This is valid for all values of z, and the modes are linearly independent, so the quantity
within [ ] must vanish for each mode. This gives
dψn /dz
ρ0 wn
1
=
≡ − 2.
z
g ∂ρn /∂t
cn
N 2 −H ψn dz
(14.55)
As the first term is a function of z alone and the second term is a function of (x, y, t)
alone, for consistency both terms must be equal to a constant; we take the “separation
constant” to be −1/cn2 . The vertical structure is then given by
1
1 dψn
=− 2
N 2 dz
cn
z
ψn dz.
−H
Taking the z-derivative,
d
dz
1 dψn
N 2 dz
+
1
ψn = 0,
cn2
(14.56)
which is the differential equation governing the vertical structure of the normal modes.
Equation (14.56) has the so-called Sturm–Liouville form, for which the various solutions are orthogonal.
Equation (14.55) also gives
wn = −
g ∂ρn
.
ρ0 cn2 ∂t
Substitution of equations (14.52)–(14.54) into equations (14.47)–(14.51) finally gives
the normal mode equations
∂un
∂vn
1 ∂pn
+
+ 2
= 0,
∂x
∂y
cn ∂t
(14.57)
630
Geophysical Fluid Dynamics
∂pn
∂un
− f vn = −
,
∂t
∂x
(14.58)
∂vn
∂pn
+ f un = −
,
∂t
∂y
(14.59)
g
ρn ,
ρ0
(14.60)
1 ∂pn
.
cn2 ∂t
(14.61)
pn = −
wn =
Once equations (14.57)–(14.59) have been solved for un , vn and pn , the amplitudes ρn
and wn can be obtained from equations (14.60) and (14.61). The set (14.57)–(14.59)
is identical to the set (14.45) governing the motion of a homogeneous layer, provided
pn is identified with gη and cn2 is identified with gH . In a stratified flow each mode
(having a fixed vertical structure) behaves, in the horizontal dimensions and in time,
just like a homogeneous layer, with an equivalent depth He defined by
cn2 ≡ gHe .
(14.62)
Boundary Conditions on ψn
At the layer bottom, the boundary condition is
w=0
at z = −H.
To write this condition in terms of ψn , we first combine the hydrostatic equation
(14.50) and the density equation (14.51) to give w in terms of p:
g(∂ρ/∂t)
1 ∂ 2p
1
w=
=
−
=− 2
ρ0 N 2
ρ0 N 2 ∂z ∂t
N
∞
n=0
∂pn dψn
.
∂t dz
(14.63)
The requirement w = 0 then yields the bottom boundary condition
dψn
=0
dz
at z = −H.
(14.64)
We now formulate the surface boundary condition. The linearized surface
boundary conditions are
w=
∂η
,
∂t
p = ρ0 gη
at z = 0,
(14.65′ )
631
9. Normal Modes in a Continuously Stratified Layer
where η is the free surface displacement. These conditions can be combined into
∂p
= ρ0 gw
∂t
at z = 0.
Using equation (14.63) this becomes
g ∂ 2p
∂p
+
=0
2
N ∂z ∂t
∂t
at z = 0.
Substitution of the normal mode decomposition (14.52) gives
dψn
N2
+
ψn = 0
dz
g
at z = 0.
(14.65)
The boundary conditions on ψn are therefore equations (14.64) and (14.65).
Solution of Vertical Modes for Uniform N
For a medium of uniform N , a simple solution can be found for ψn . From equations (14.56), (14.64), and (14.65), the vertical structure of the normal modes is
given by
d 2 ψn
N2
+
ψn = 0,
dz2
cn2
(14.66)
with the boundary conditions
dψn
N2
+
ψn = 0
dz
g
dψn
=0
dz
at z = 0,
(14.67)
at z = −H.
(14.68)
The set (14.66)–(14.68) defines an eigenvalue problem, with ψn as the eigenfunction
and cn as the eigenvalue. The solution of equation (14.66) is
ψn = An cos
Nz
Nz
+ Bn sin
.
cn
cn
(14.69)
Application of the surface boundary condition (14.67) gives
Bn = −
cn N
An .
g
The bottom boundary condition (14.68) then gives
tan
NH
cn N
=
,
cn
g
whose roots define the eigenvalues of the problem.
(14.70)
632
Geophysical Fluid Dynamics
Figure 14.12 Calculation of eigenvalues cn of vertical normal modes in a fluid layer of depth H and
uniform stratification N.
The solution of equation (14.70) is indicated graphically in Figure 14.12. The
first root occurs for N H /cn ≪ 1, for which we can write tan(NH /cn ) ≃ N H /cn ,
so that equation (14.70) gives (indicating this root by n = 0)
c0 =
gH .
The vertical modal structure is found from equation (14.69). Because the magnitude
of an eigenfunction is arbitrary, we can set A0 = 1, obtaining
ψ0 = cos
N z c0 N
Nz
N 2z
−
sin
≃1−
≃ 1,
c0
g
c0
g
where we have used N|z|/c0 ≪ 1 (with N H /c0 ≪ 1), and N 2 z/g ≪ 1 (with
N 2 H /g = (NH /c0 )(c0 N/g) ≪ 1, both sides of equation (14.70) being much less
than 1). For this mode the vertical structure of u, v, and p is therefore nearly
depth-independent. The corresponding structure for w (given by ψ0 dz, as indicated in equation (14.53)) is linear in z, with zero at the bottom and a maximum at the
upper free surface. A stratified medium therefore has a mode of motion that behaves
like that in an unstratified medium; this mode does not feel the stratification. The
n = 0 mode is called the barotropic mode.
The remaining modes n 1 are baroclinic. For these modes cn N/g ≪ 1 but
N H /cn is not small, as can be seen in Figure 14.12, so that the baroclinic roots of
equation (14.70) are nearly given by
tan
NH
= 0,
cn
633
9. Normal Modes in a Continuously Stratified Layer
which gives
cn =
NH
,
nπ
n = 1, 2, 3, . . . .
(14.71)
Taking a typical depth-average oceanic value of N ∼ 10−3 s−1 and H ∼ 5 km, the
eigenvalue for the first baroclinic mode is c1 ∼ 2 m/s. The corresponding equivalent
depth is He = c12 /g ∼ 0.4 m.
An examination of the algebraic steps leading to equation (14.70) shows that
neglecting the right-hand side is equivalent to replacing the upper boundary condition (14.65′ ) by w = 0 at z = 0. This is called the rigid lid approximation. The
baroclinic modes are negligibly distorted by the rigid lid approximation. In contrast,
the rigid lid approximation applied to the barotropic mode would yield c0 = ∞, as
equation (14.71) shows for n = 0. Note that the rigid lid approximation does not
imply that the free surface displacement corresponding to the baroclinic modes is
negligible in the ocean. In fact, excluding the wind waves and tides, much of the
free surface displacements in the ocean are due to baroclinic motions. The rigid lid
approximation merely implies that, for baroclinic motions, the vertical displacements
at the surface are much smaller than those within the fluid column. A valid baroclinic
solution can therefore be obtained by setting w = 0 at z = 0. Further, the rigid lid
approximation does not imply that the pressure is constant at the level surface z = 0;
if a rigid lid were actually imposed at z = 0, then the pressure on the lid would vary
due to the baroclinic motions.
The vertical mode shape under the rigid lid approximation is given by the cosine
distribution
ψn = cos
nπ z
,
H
n = 0, 1, 2, . . . ,
because it satisfies dψn /dz = 0 at z = 0, −H . The nth mode ψn has n zero crossings
within the layer (Figure 14.13).
A decomposition into normal modes is only possible in the absence of
topographic variations and mean currents with shear. It is valid with or without Coriolis forces and with or without the β-effect. However, the hydrostatic approximation
here means that the frequencies are much smaller than N. Under this condition the
eigenfunctions are independent of the frequency, as equation (14.56) shows. Without the hydrostatic approximation the eigenfunctions ψn become dependent on the
frequency ω. This is discussed, for example, in LeBlond and Mysak (1978).
Summary: Small amplitude motion in a frictionless continuously stratified ocean
can be decomposed in terms of noninteracting vertical normal modes. The vertical
structure of each mode is defined by an eigenfunction ψn (z). If the horizontal scale
of the waves is much larger than the vertical scale, then the equations governing
the horizontal propagation of each mode are identical to those of a shallow homogeneous layer, with the layer depth H replaced by an equivalent depth He defined
by cn2 = gHe . For a medium of constant N , the baroclinic (n 1) eigenvalues are
634
Geophysical Fluid Dynamics
Figure 14.13 Vertical distribution of a few normal modes in a stratified medium of uniform buoyancy
frequency.
given by cn = NH /π n, while the barotropic eigenvalue is c0 =
approximation is quite good for the baroclinic modes.
√
gH . The rigid lid
10. High- and Low-Frequency Regimes in
Shallow-Water Equations
We shall now examine what terms are negligible in the shallow-water equations for
the various frequency ranges. Our analysis is valid for a single homogeneous layer
or for a stratified medium. In the latter case H has to be interpreted as the equivalent
depth, and c has to be interpreted as the speed of long nonrotating internal gravity
waves. The β-effect will be considered in this section. As f varies only northward,
horizontal isotropy is lost whenever the β-effect is included, and it becomes necessary
to distinguish between the different horizontal directions. We shall follow the usual
geophysical convention that the x-axis is directed eastward and the y-axis is directed
northward, with u and v the corresponding velocity components.
The simplest way to perform the analysis is to examine the v-equation. A single
equation for v can be derived by first taking the time derivatives of the momentum
equations in (14.45) and using the continuity equation to eliminate ∂η/∂t. This gives
∂ 2u
∂v
∂ ∂u ∂v
−f
= gH
+
,
(14.72)
∂t 2
∂t
∂x ∂x
∂y
∂ 2v
∂u
∂ ∂u ∂v
+f
= gH
+
.
(14.73)
∂t 2
∂t
∂y ∂x
∂y
635
10. High- and Low-Frequency Regimes in Shallow-Water Equations
Now take ∂/∂t of equation (14.73) and use equation (14.72), to obtain
∂ 3v
∂v
∂ ∂u ∂v
∂2
∂u ∂v
+f f
+ gH
+
+
= gH
.
∂t 3
∂t
∂x ∂x
∂y
∂y ∂t ∂x
∂y
(14.74)
To eliminate u, we first obtain a vorticity equation by cross differentiating and subtracting the momentum equations in equation (14.45):
∂
∂t
∂u ∂v
−
∂y
∂x
− f0
∂u ∂v
+
∂x
∂y
− βv = 0.
Here, we have made the customary β-plane approximation, valid if the y-scale is small
enough so that f/f ≪ 1. Accordingly, we have treated f as constant (and replaced
it by an average value f0 ) except when df/dy appears; this is why we have written
f0 in the second term of the preceding equation. Taking the x-derivative, multiplying
by gH , and adding to equation (14.74), we finally obtain a vorticity equation in terms
of v only:
∂ 3v
∂v
∂
∂v
− gH ∇H2 v + f02
− gHβ
= 0,
3
∂t
∂t
∂t
∂x
(14.75)
where ∇H2 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 is the horizontal Laplacian operator.
Equation (14.75) is Boussinesq, linear and hydrostatic, but otherwise quite general in the sense that it is applicable to both high and low frequencies. Consider wave
solutions of the form
v = v̂ ei(kx+ly−ωt) ,
where k is the eastward wavenumber and l is the northward wavenumber. Then equation (14.75) gives
ω3 − c2 ωK 2 − f02 ω − c2 βk = 0,
(14.76)
√
where K 2 = k 2 + l 2 and c = gH . It can be shown that all roots of equation (14.76)
are real, two of the roots being superinertial (ω > f ) and the third being subinertial
(ω ≪ f ). Equation (14.76) is the complete dispersion relation for linear shallow-water
equations. In various parametric ranges it takes simpler forms, representing simpler
waves.
First, consider high-frequency waves ω ≫ f . Then the third term of equation (14.76) is negligible compared to the first term. Moreover, the fourth term is
also negligible in this range. Compare, for example, the fourth and second terms:
c2 βk
β
∼
∼ 10−3 ,
c2 ωK 2
ωK
636
Geophysical Fluid Dynamics
where we have assumed typical values of β = 2 × 10−11 m−1 s−1 , ω = 3f
∼ 3 × 10−4 s−1 , and 2π/K ∼ 100 km. For ω ≫ f , therefore, the balance
√ is between
gH , which
the first and second terms in equation (14.76), and
the
roots
are
ω
=
±K
√
correspond to a propagation speed of ω/K = gH . The effects of both f and β are
therefore negligible for high-frequency waves, as is expected as they are too fast to
be affected by the Coriolis effects.
Next consider ω > f , but ω ∼ f . Then the third term in equation (14.76) is not
negligible, but the β-effect is. These are gravity waves influenced by Coriolis forces;
gravity waves are discussed in the next section. However, the time scales are still too
short for the motion to be affected by the β-effect.
Last, consider very slow waves for which ω ≪ f . Then the β-effect becomes
important, and the first term in equation (14.76) becomes negligible. Compare, for
example, the first and the last terms:
ω3
≪ 1.
c2 βk
Typical values for the ocean are c ∼ 200 m/s for the barotropic mode, c ∼ 2 m/s for
the baroclinic mode, β = 2 × 10−11 m−1 s−1 , 2π/k ∼ 100 km, and ω ∼ 10−5 s−1 .
This makes the forementioned ratio about 0.2 × 10−4 for the barotropic mode and
0.2 for the baroclinic mode. The first term in equation (14.76) is therefore negligible
for ω ≪ f .
Equation (14.75) governs the dynamics of a variety of wave motions in the
ocean and the atmosphere, and the discussion in this section shows what terms can
be dropped under various limiting conditions. An understanding of these limiting
conditions will be useful in the following sections.
11. Gravity Waves with Rotation
In this chapter we shall examine several free-wave solutions of the shallow-water
equations. In this section we shall study gravity waves with frequencies in the
range ω > f , for which the β-effect is negligible, as demonstrated in the preceding section. Consequently, the Coriolis frequency f is regarded as constant here.
Consider progressive waves of the form
(u, v, η) = (û, v̂, η̂)ei(kx+ly−ωt) ,
where û, v̂, and η̂ are the complex amplitudes, and the real part of the right-hand side
is meant. Then equation (14.45) gives
−iωû − f v̂ = −ikg η̂,
(14.77)
−iωv̂ + f û = −ilg η̂,
(14.78)
−iωη̂ + iH (k û + l v̂) = 0.
(14.79)
637
11. Gravity Waves with Rotation
Solving for û and v̂ between equations (14.77) and (14.78), we obtain
g η̂
(ωk + if l),
− f2
g η̂
v̂ = 2
(−if k + ωl).
ω − f2
û =
ω2
(14.80)
Substituting these in equation (14.79), we obtain
ω2 − f 2 = gH (k 2 + l 2 ).
(14.81)
This is the dispersion relation of gravity waves in the presence of Coriolis forces.
(The relation can be most simply derived by setting the determinant of the set of linear
homogeneous equations (14.77)–(14.79) to zero.) It can be written as
ω2 = f 2 + gH K 2 ,
(14.82)
√
where K = k 2 + l 2 is the magnitude of the horizontal wavenumber. The dispersion relation shows that the waves can propagate in any horizontal direction and have
ω > f . Gravity waves affected by Coriolis forces are called Poincaré waves, Sverdrup
waves, or simply rotational gravity waves. (Sometimes the name “Poincaré wave” is
used to describe those rotational gravity waves that satisfy the boundary conditions
in a channel.) In spite of their name, the solution was first worked out by Kelvin (Gill,
1982, p. 197). A plot of equation (14.82) is shown in Figure 14.14. It is seen that the
2
2
waves are dispersive except for ω ≫ f when
√ equation (14.82) gives ω ≃ gH K ,
so that the propagation speed is ω/K = gH . The high-frequency limit agrees
with our previous discussion of surface gravity waves unaffected by Coriolis
forces.
Figure 14.14 Dispersion relations for Poincaré and Kelvin waves.
638
Geophysical Fluid Dynamics
Particle Orbit
The symmetry of the dispersion relation (14.81) with respect to k and l means that the
x- and y-directions are not felt differently by the wavefield. The horizontal isotropy
is a result of treating f as constant. (We shall see later that Rossby waves, which
depend on the β-effect, are not horizontally isotropic.) We can therefore orient the
x-axis along the wavenumber vector and set l = 0, so that the wavefield is invariant
along the y-axis. To find the particle orbits, it is convenient to work with real quantities.
Let the displacement be
η = η̂ cos(kx − ωt),
where η̂ is real. The corresponding velocity components can be found by multiplying
equation (14.80) by exp(ikx − iωt) and taking the real part of both sides. This gives
ωη̂
cos(kx − ωt),
kH
f η̂
v=
sin(kx − ωt).
kH
u=
(14.83)
To find the particle paths, take x = 0 and consider three values of time corresponding
to ωt = 0, π/2, and π . The corresponding values of u and v from equation (14.83)
show that the velocity vector rotates clockwise (in the northern hemisphere) in elliptic
paths (Figure 14.15). The ellipticity is expected, since the presence of Coriolis forces
means that f u must generate ∂v/∂t according to the equation of motion (14.45).
(In equation (14.45), ∂η/∂y = 0 due to our orienting the x-axis along the direction
of propagation of the wave.) Particles are therefore constantly deflected to the right
by the Coriolis force, resulting in elliptic orbits. The ellipses have an axis ratio of
ω/f, and the major axis is oriented in the direction of wave propagation. The ellipses
become narrower as ω/f increases, approaching the rectilinear orbit of gravity waves
Figure 14.15 Particle orbit in a rotational gravity wave. Velocity components corresponding to ωt = 0,
π/2, and π are indicated.
639
12. Kelvin Wave
unaffected by planetary rotation. However, the sea surface in a rotational gravity wave
is no different than that for ordinary gravity waves, namely oscillatory in the direction
of propagation and invariant in the perpendicular direction.
Inertial Motion
Consider the limit ω → f , that is when the particle paths are circular. The dispersion
relation (14.82) then shows that K → 0, implying a horizontal uniformity of the flow
field. Equation (14.79) shows that η̂ must tend to zero in this limit, so that there
are no horizontal pressure gradients in this limit. Because ∂u/∂x = ∂v/∂y = 0, the
continuity equation shows that w = 0. The particles therefore move on horizontal
sheets, each layer decoupled from the one above and below it. The balance of forces is
∂u
− f v = 0,
∂t
∂v
+ f u = 0.
∂t
The solution of this set is of the form
u = q cos f t,
v = −q sin f t,
√
where the speed q = u2 + v 2 is constant along the path. The radius r of the orbit
can be found by adopting a Lagrangian point of view, and noting that the equilibrium
of forces is between the Coriolis force f q and the centrifugal force rω2 = rf 2 ,
giving r = q/f . The limiting case of motion in circular orbits at a frequency f is
called inertial motion, because in the absence of pressure gradients a particle moves
by virtue of its inertia alone. The corresponding period 2π/f is called the inertial
period. In the absence of planetary rotation such motion would be along straight
lines; in the presence of Coriolis forces the motion is along circular paths, called
inertial circles. Near-inertial motion is frequently generated in the surface layer of
the ocean by sudden changes of the wind field, essentially because the equations of
motion (14.45) have a natural frequency f . Taking a typical current magnitude of
q ∼ 0.1 m/s, the radius of the orbit is r ∼ 1 km.
12. Kelvin Wave
In the preceding section we considered a shallow-water gravity wave propagating in
a horizontally unbounded ocean. We saw that the crests are horizontal and oriented in
a direction perpendicular to the direction of propagation. The absence of a transverse
pressure gradient ∂η/∂y resulted in a transverse flow and elliptic orbits. This is clear
from the third equation in (14.45), which shows that the presence of f u must result in
∂v/∂t if ∂η/∂y = 0. In this section we consider a gravity wave propagating parallel
to a wall, whose presence allows a pressure gradient ∂η/∂y that can decay away from
the wall. We shall see that this allows a gravity wave in which f u is geostrophically
640
Geophysical Fluid Dynamics
balanced by −g(∂η/∂y), and v = 0. Consequently the particle orbits are not elliptic
but rectilinear.
Consider first a gravity wave propagating in a channel. From Figure 7.7 we know
that the fluid velocity under a crest is “forward” (i.e., in the direction of propagation),
and that under a trough it is backward. Figure 14.16 shows two transverse sections of
the wave, one through a crest (left panel) and the other through a trough (right panel).
The wave is propagating into the plane of the paper, along the x-direction. Then the
fluid velocity under the crest is into the plane of the paper and that under the trough is
out of the plane of the paper. The constraints of the side walls require that v = 0 at the
walls, and we are exploring the possibility of a wave motion in which v is zero everywhere. Then the equation of motion along the y-direction requires that f u can only be
geostrophically balanced by a transverse slope of the sea surface across the channel:
f u = −g
∂η
.
∂y
In the northern hemisphere, the surface must slope as indicated in the figure, that is
downward to the left under the crest and upward to the left under the trough, so that
the pressure force has the current directed to its right. The result is that the amplitude
of the wave is larger on the right-hand side of the channel, looking into the direction
of propagation, as indicated in Figure 14.16. The current amplitude, like the surface
displacement, also decays to the left.
If the left wall in Figure 14.16 is moved away to infinity, we get a gravity wave
trapped to the coast (Figure 14.17). A coastally trapped long gravity wave, in which
the transverse velocity v = 0 everywhere, is called a Kelvin wave. It is clear that it can
propagate only in a direction such that the coast is to the right (looking in the direction
of propagation) in the northern hemisphere and to the left in the southern hemisphere.
The opposite direction of propagation would result in a sea surface displacement
increasing exponentially away from the coast, which is not possible.
An examination of the transverse momentum equation
∂v
∂η
+ f u = −g ,
∂t
∂y
Figure 14.16 Free surface distribution in a gravity wave propagating through a channel into the plane of
the paper.
641
12. Kelvin Wave
Figure 14.17 Coastal Kelvin wave propagating along the x-axis. Sea surface across a section through a
crest is indicated by the continuous line, and that along a trough is indicated by the dashed line.
reveals fundamental differences between Poincaré waves and Kelvin waves. For a
Poincaré wave the crests are horizontal, and the absence of a transverse pressure
gradient requires a ∂v/∂t to balance the Coriolis force, resulting in elliptic orbits. In a
Kelvin wave a transverse velocity is prevented by a geostrophic balance of f u and
−g(∂η/∂y).
From the shallow-water set (14.45), the equations of motion for a Kelvin wave
propagating along a coast aligned with the x-axis (Figure 14.17) are
∂η
∂u
+H
= 0,
∂t
∂x
∂u
∂η
= −g ,
∂t
∂x
f u = −g
(14.84)
∂η
.
∂y
Assume a solution of the form
[u, η] = [û(y), η̂(y)]ei(kx−ωt) .
Then equation (14.84) gives
−iωη̂ + iH k û = 0,
−iωû = −igk η̂,
f û = −g
d η̂
.
dy
(14.85)
642
Geophysical Fluid Dynamics
The dispersion relation can be found solely from the first two of these equations; the
third equation then determines the transverse structure. Eliminating û between the
first two, we obtain
η̂[ω2 − gH k 2 ] = 0.
√
A nontrivial solution is therefore possible only if ω = ±k gH , so that the wave
propagates with a nondispersive speed
c=
√
gH .
(14.86)
The propagation speed of a Kelvin wave is therefore identical to that of nonrotating
gravity waves. Its dispersion equation is a straight line and is shown in Figure 14.14.
All frequencies are possible.
To determine the transverse structure, eliminate û between the first and third of
equation (14.85), giving
d η̂ f
± η̂ = 0.
dy
c
The solution that decays away from the coast is
η̂ = η0 e−fy/c ,
where η0 is the amplitude at the coast. Therefore, the sea surface slope and the velocity
field for a Kelvin wave have the form
η = η0 e−fy/c cos k(x − ct),
g −fy/c
e
cos k(x − ct),
u = η0
H
(14.87)
where we have taken the real parts, and have used equation (14.85) in obtaining the
u field.
Equations (14.87) show that the transverse decay scale of the Kelvin wave is
≡
c
,
f
which is called the Rossby radius of deformation. For a deep sea
√ of depth H = 5 km,
and a midlatitude value of f = 10−4 s−1 , we obtain c = gH = 220 m/s and
= c/f = 2200 km. Tides are frequently in the form of coastal Kelvin waves of
semidiurnal frequency. The tides are forced by the periodic changes in the gravitational
attraction of the moon and the sun. These waves propagate along the boundaries of
an ocean basin and cause sea level fluctuations at coastal stations.
643
12. Kelvin Wave
Analogous to the surface or “external” Kelvin waves discussed in the preceding,
we can have internal Kelvin waves at the interface between two fluids of different
densities (Figure 14.18). If the lower layer is very deep, then the speed of propagation
is given by (see equation (7.126))
c=
g′H ,
where H is the thickness of the upper layer and g ′ = g(ρ2 − ρ1 )/ρ2 is the reduced
gravity. For a continuously stratified medium of depth H and buoyancy frequency N,
internal Kelvin waves can propagate at any of the normal mode speeds
c = N H /nπ,
n = 1, 2, . . . .
The decay scale for internal Kelvin waves, = c/f, is called the internal Rossby
radius of deformation, whose value is much smaller than that for the external Rossby
radius of deformation. For n = 1, a typical value in the ocean is = NH /πf
∼ 50 km; a typical atmospheric value is much larger, being of order ∼ 1000 km.
Internal Kelvin waves in the ocean are frequently forced by wind changes near
coastal areas. For example, a southward wind along the west coast of a continent
in the northern hemisphere (say, California) generates an Ekman layer at the ocean
surface, in which the mass flow is away from the coast (to the right of the applied wind
stress). The mass flux in the near-surface layer is compensated by the movement of
deeper water toward the coast, which raises the thermocline. An upward movement of
the thermocline, as indicated by the dashed line in Figure 14.18, is called upwelling.
The vertical movement of the thermocline in the wind-forced region then propagates
poleward along the coast as an internal Kelvin wave.
Figure 14.18 Internal Kelvin wave at an interface. Dashed line indicates position of the interface when
it is at its maximum height. Displacement of the free surface is much smaller than that of the interface and
is oppositely directed.
644
Geophysical Fluid Dynamics
13. Potential Vorticity Conservation in
Shallow-Water Theory
In this section we shall derive a useful conservation law for the vorticity of a shallow layer of fluid. From Section 8, the equations of motion for a shallow layer of
homogeneous fluid are
∂u
∂u
∂u
∂η
+u
+v
− f v = −g ,
∂t
∂x
∂y
∂x
(14.88)
∂v
∂v
∂v
∂η
+u
+v
+ f u = −g ,
∂t
∂x
∂y
∂y
(14.89)
∂
∂
∂h
+
(uh) +
(vh) = 0,
∂t
∂x
∂y
(14.90)
where h(x, y, t) is the depth of flow and η is the height of the sea surface measured
from an arbitrary horizontal plane (Figure 14.19). The x-axis is taken eastward and the
y-axis is taken northward, with u and v the corresponding velocity components. The
Coriolis frequency f = f0 + βy is regarded as dependent on latitude. The nonlinear
terms have been retained, including those in the continuity equation, which has been
written in the form (14.44); note that h = H + η. We saw in Section 8 that the constant
density of the layer and the hydrostatic pressure distribution make the horizontal
pressure gradient depth-independent, so that only a depth-independent current can be
generated. The vertical velocity is linear in z.
A vorticity equation can be derived by differentiating equation (14.88) with
respect to y, equation (14.89) with respect to x, and subtracting. The pressure is
eliminated, and we obtain
∂
∂t
∂u
∂v
∂u
∂v
∂
∂v
∂
∂u
−
+v
+v
+
u
−
u
∂x
∂y
∂x
∂x
∂y
∂y
∂x
∂y
∂u ∂v
+
+ βv = 0.
+ f0
∂x
∂y
Figure 14.19 Shallow layer of instantaneous depth h(x, y, t).
(14.91)
645
13. Potential Vorticity Conservation in Shallow-Water Theory
Following the customary β-plane approximation, we have treated f as constant
(and replaced it by an average value f0 ) except when df/dy appears. We now introduce
ζ ≡
∂u
∂v
−
,
∂x
∂y
as the vertical component of relative vorticity, that is, the vorticity measured relative
to the rotating earth. Then the nonlinear terms in equation (14.91) can easily be
rearranged in the form
∂ζ
∂u ∂v
∂ζ
+v
+
+
ζ.
u
∂x
∂y
∂x
∂y
Equation (14.91) then becomes
∂ζ
∂ζ
∂ζ
+u
+v
+
∂t
∂x
∂y
∂u ∂v
+
∂x
∂y
(ζ + f0 ) + βv = 0,
which can be written as
Dζ
∂u ∂v
+ (ζ + f0 )
+
+ βv = 0,
Dt
∂x
∂y
(14.92)
where D/Dt is the derivative following the horizontal motion of the layer:
D
∂
∂
∂
≡
+u
+v .
Dt
∂t
∂x
∂y
The horizontal divergence (∂u/∂x+∂v/∂y) in equation (14.92) can be eliminated
by using the continuity equation (14.90), which can be written as
Dh
∂u ∂v
+h
+
= 0.
Dt
∂x
∂y
Equation (14.92) then becomes
Dζ
ζ + f0 Dh
=
− βv.
Dt
h Dt
This can be written as
ζ + f0 Dh
D(ζ + f )
=
,
Dt
h Dt
where we have used
Df
∂f
∂f
∂f
=
+u
+v
= vβ.
Dt
∂t
∂x
∂y
(14.93)
646
Geophysical Fluid Dynamics
Because of the absence of vertical shear, the vorticity in a shallow-water model
is purely vertical and independent of depth. The relative vorticity measured with
respect to the rotating earth is ζ , while f is the planetary vorticity, so that the absolute
vorticity is (ζ +f ). Equation (14.93) shows that the rate of change of absolute vorticity
is proportional to the absolute vorticity times the vertical stretching Dh/Dt of the
water column. It is apparent that Dζ /Dt can be nonzero even if ζ = 0 initially. This is
different from a nonrotating flow in which stretching a fluid line changes its vorticity
only if the line has an initial vorticity. (This is why the process was called the vortex
stretching; see Chapter 5, Section 7.) The difference arises because vertical lines in a
rotating earth contain the planetary vorticity even when ζ = 0. Note that the vortex
tilting term, discussed in Chapter 5, Section 7, is absent in the shallow-water theory
because the water moves in the form of vertical columns without ever tilting.
Equation (14.93) can be written in the compact form
D
Dt
ζ +f
h
= 0,
(14.94)
where f = f0 +βy, and we have assumed βy ≪ f0 . The ratio (ζ +f )/ h is called the
potential vorticity in shallow-water theory. Equation (14.94) shows that the potential
vorticity is conserved along the motion, an important principle in geophysical fluid
dynamics. In the ocean, outside regions of strong current vorticity such as coastal
boundaries, the magnitude of ζ is much smaller than that of f . In such a case (ζ + f )
has the sign of f . The principle of conservation of potential vorticity means that an
increase in h must make (ζ + f ) more positive in the northern hemisphere and more
negative in the southern hemisphere.
As an example of application of the potential vorticity equation, consider an
eastward flow over a step (at x = 0) running north–south, across which the layer
thickness changes discontinuously from h0 to h1 (Figure 14.20). The flow upstream
of the step has a uniform speed U , so that the oncoming stream has no relative vorticity.
To conserve the ratio (ζ + f )/ h, the flow must suddenly acquire negative (clockwise)
relative vorticity due to the sudden decrease in layer thickness. The relative vorticity
of a fluid element just after passing the step can be found from
f
ζ +f
=
,
h0
h1
giving ζ = f (h1 − h0 )/ h0 < 0, where f is evaluated at the upstream latitude of the
streamline. Because of the clockwise vorticity, the fluid starts to move south at x = 0.
The southward movement decreases f , so that ζ must correspondingly increase so
as to keep (f + ζ ) constant. This means that the clockwise curvature of the stream
reduces, and eventually becomes a counterclockwise curvature. In this manner an
eastward flow over a step generates stationary undulatory flow on the downstream
side. In Section 15 we shall see that the stationary oscillation is due to a Rossby wave
generated at the step whose westward phase velocity
is canceled by the eastward
√
current. We shall see that the wavelength is 2π U/β.
14. Internal Waves
√
Figure 14.20 Eastward flow over a step, resulting in stationary oscillations of wavelength 2π U/β.
Suppose we try the same argument for a westward flow over a step. Then a
particle should suddenly acquire clockwise vorticity as the depth of flow decreases
at x = 0, which would require the particle to move north. It would then come into a
region of larger f , which would require ζ to decrease further. Clearly, an exponential
behavior is predicted, suggesting that the argument is not correct. Unlike an eastward
flow, a westward current feels the upstream influence of the step so that it acquires a
counterclockwise curvature before it encounters the step (Figure 14.21). The positive
vorticity is balanced by a reduction in f , which is consistent with conservation of
potential vorticity. At the location of the step the vorticity decreases suddenly. Finally,
far downstream of the step a fluid particle is again moving westward at its original
latitude. The westward flow over a topography is not oscillatory.
14. Internal Waves
In Chapter 7, Section 19 we studied internal gravity waves unaffected by Coriolis forces. We saw that they are not isotropic; in fact the direction of propagation with respect to the vertical determines their frequency. We also saw that
their frequency satisfies the inequality ω N, where N is the buoyancy frequency.
Their phase-velocity vector c and the group-velocity vector cg are perpendicular and
have oppositely directed vertical components (Figure 7.32 and Figure 7.34). That is,
phases propagate upward if the groups propagate downward, and vice versa. In this
section we shall study the effect of Coriolis forces on internal waves, assuming that
f is independent of latitude.
Internal waves are ubiquitous in the atmosphere and the ocean. In the lower atmosphere turbulent motions dominate, so that internal wave activity represents a minor
component of the motion. In contrast, the stratosphere contains very little convective
motion because of its stable density distribution, and consequently a great deal of
internal wave activity. They generally propagate upward from the lower atmosphere,
where they are generated. In the ocean they may be as common as the waves on
647
648
Geophysical Fluid Dynamics
Figure 14.21 Westward flow over a step. Unlike the eastward flow, the westward flow is not oscillatory
and feels the upstream influence of the step.
the surface, and measurements show that they can cause the isotherms to go up and
down by as much as 50–100 m. Sometimes the internal waves break and generate
small-scale turbulence, similar to the “foam” generated by breaking waves.
We shall now examine the nature of the fluid motion in internal waves. The
equations of motion are
∂w
∂u ∂v
+
+
= 0,
∂x
∂y
∂z
∂u
1 ∂p
− fv = −
,
∂t
ρ0 ∂x
1 ∂p
∂v
+ fu = −
,
∂t
ρ0 ∂y
(14.95)
∂w
1 ∂p ρg
,
=−
−
∂t
ρ0 ∂z
ρ0
∂ρ
ρ0 N 2
−
w = 0.
∂t
g
We have not made the hydrostatic assumption because we are not assuming that the
horizontal wavelength is long compared to the vertical wavelength. The advective
term in the density equation is written in a linearized form w(d ρ̄/dz) = −ρ0 N 2 w/g.
Thus the rate of change of density at a point is assumed to be due only to the vertical advection of the background density distribution ρ̄(z). Because internal wave
activity is more intense in the thermocline where N varies appreciably (Figure 14.2),
we shall be somewhat more general than in Chapter 7 and not assume that N is
depth-independent.
An equation for w can be formed from the set (14.95) by eliminating all other
variables. The algebraic steps of such a procedure are shown in Chapter 7, Section 18
without the Coriolis forces. This gives
649
14. Internal Waves
2
∂2 2
2 2
2∂ w
∇
w
+
N
∇
w
+
f
= 0,
H
∂t 2
∂z2
(14.96)
where
∇2 ≡
∂2
∂2
∂2
+
+
∂x 2
∂y 2
∂z2
and
∇H2 ≡
∂2
∂2
+
.
∂x 2
∂y 2
Because the coefficients of equation (14.96) are independent of the horizontal directions, equation (14.96) can have solutions that are trigonometric in x and y. We
therefore assume a solution of the form
[u, v, w] = [û(z), v̂(z), ŵ(z)] ei(kx+ly−ωt) .
(14.97)
Substitution into equation (14.96) gives
d2
d 2 ŵ
2
2
2
(−iω) (ik) + (il) + 2 ŵ + N 2 [(ik)2 + (il)2 ]ŵ + f 2 2 = 0,
dz
dz
from which we obtain
d 2 ŵ (N 2 − ω2 )(k 2 + l 2 )
+
ŵ = 0.
dz2
ω2 − f 2
(14.98)
Defining
m2 (z) ≡
(k 2 + l 2 )[N 2 (z) − ω2 ]
,
ω2 − f 2
(14.99)
Equation (14.98) becomes
d 2 ŵ
+ m2 ŵ = 0.
dz2
(14.100)
For m2 < 0, the solutions of equation (14.100) are exponential in z signifying that
the resulting motion is surface-trapped. It represents a surface wave propagating horizontally. For a positive m2 , on the other hand, solutions are trigonometric in z, giving
internal waves propagating vertically as well as horizontally. From equation (14.99),
therefore, internal waves are possible only in the frequency range:
f < ω < N,
where we have assumed N > f , as is true for much of the atmosphere and the ocean.
650
Geophysical Fluid Dynamics
WKB Solution
To proceed further, we assume that N (z) is a slowly varying function in that its
fractional change over a vertical wavelength is much less than unity. We are therefore
considering only those internal waves whose vertical wavelength is short compared
to the scale of variation of N. If H is a characteristic vertical distance over which N
varies appreciably, then we are assuming that
H m ≫ 1.
For such slowly varying N (z), we expect that m(z) given by equation (14.99) is also
a slowly varying function, that is, m(z) changes by a small fraction in a distance 1/m.
Under this assumption the waves locally behave like plane waves, as if m is constant.
This is the so-called WKB approximation (after Wentzel–Kramers–Brillouin), which
applies when the properties of the medium (in this case N) are slowly varying.
To derive the approximate WKB solution of equation (14.100), we look for a
solution in the form
ŵ = A(z)eiφ(z) ,
where the phase φ and the (slowly varying) amplitude A are real. (No generality is
lost by assuming A to be real. Suppose it is complex and of the form A = Ā exp(iα),
where Ā and α are real. Then ŵ = Ā exp [i(φ + α)], a form in which (φ + α) is the
phase.) Substitution into equation (14.100) gives
2
dφ
d 2φ
dA dφ
d 2A
2
+
A
m
−
+
iA
= 0.
+
i2
dz2
dz
dz dz
dz2
Equating the real and imaginary parts, we obtain
2
dφ
d 2A
2
+A m −
= 0,
2
dz
dz
(14.101)
dA dφ
d 2φ
+ A 2 = 0.
dz dz
dz
(14.102)
2
In equation (14.101) the term d 2 A/dz2 is negligible because its ratio with the second
term is
d 2 A/dz2
1
∼ 2 2 ≪ 1.
Am2
H m
Equation (14.101) then becomes approximately
dφ
= ±m,
dz
(14.103)
651
14. Internal Waves
whose solution is
z
φ=±
m dz,
the lower limit of the integral being arbitrary.
The amplitude is determined by writing equation (14.102) in the form
dA
(d 2 φ/dz2 ) dz
(dm/dz) dz
1 dm
=−
=−
=−
,
A
2(dφ/dz)
2m
2 m
where equation (14.103) has been used. Integrating, we obtain ln A = − 21 ln m +
const., that is,
A0
A= √ ,
m
where A0 is a constant. The WKB solution of equation (14.100) is therefore
A0
ŵ = √ e±i
m
z
m dz
(14.104)
.
Because of neglect of the β-effect, the waves must behave similarly in x and y,
as indicated by the symmetry of the dispersion relation (14.99) in k and l. Therefore,
we lose no generality by orienting the x-axis in the direction of propagation, and
taking
k>0
l=0
ω > 0.
To find u and v in terms of w, use the continuity equation ∂u/∂x + ∂w/∂z = 0,
noting that the y-derivatives are zero because of our setting l = 0. Substituting the
wave solution (14.97) into the continuity equation gives
ik û +
d ŵ
= 0.
dz
(14.105)
The
√ z-derivative of ŵ in equation (14.104) can be obtained by treating the denominator
m as approximately constant because the variation of ŵ is dominated by the wiggly
behavior of the local plane wave solution. This gives
A0
d ŵ
= √ (±im)e±i
dz
m
z
m dz
√
= ±iA0 me±i
z
m dz
,
so that equation (14.105) becomes
√
A0 m ±i
û = ∓
e
k
z
m dz
.
(14.106)
652
Geophysical Fluid Dynamics
An expression for v̂ can now be obtained from the horizontal equations of motion
in equation (14.95). Cross differentiating, we obtain the vorticity equation
∂
∂t
∂u ∂v
−
∂y
∂x
=f
∂u ∂v
+
∂x
∂y
.
Using the wave solution equation (14.97), this gives
û
iω
=
.
v̂
f
Equation (14.106) then gives
√
if A0 m ±i
v̂ = ±
e
ω
k
z
m dz
.
(14.107)
Taking real parts of equations (14.104), (14.106), and (14.107), we obtain the velocity
field
√
z
A0 m
u=∓
m dz − ωt ,
cos kx ±
k
√
z
A0 f m
v=∓
m dz − ωt ,
sin kx ±
(14.108)
ωk
z
A0
w = √ cos kx ±
m dz − ωt ,
m
where the dispersion relation is
m2 =
k 2 (N 2 − ω2 )
.
ω2 − f 2
(14.109)
The meaning of m(z) is clear from equation (14.108). If we call the argument of the
trigonometric terms the “phase,” then it is apparent that ∂(phase)/∂z = m(z), so that
m(z) is the local vertical wavenumber. Because we are treating k, m, ω > 0, it is also
apparent that the upper signs represent waves with upward phase propagation, and
the lower signs represent downward phase propagation.
Particle Orbit
To find the shape of the hodograph in the horizontal plane, consider the point
x = z = 0. Then equation (14.108) gives
u = ∓ cos ωt,
f
v = ± sin ωt,
ω
(14.110)
653
14. Internal Waves
Figure 14.22 Particle orbit in an internal wave. The upper panel (a) shows projection on a horizontal plane;
points corresponding to ωt = 0, π/2, and π are indicated. The lower panel (b) shows a three-dimensional
view. Sense of rotation shown is valid for the northern hemisphere.
where the amplitude of u has been arbitrarily set to one. Taking the upper signs in
equation (14.110), the values of u and v are indicated in Figure 14.22a for three values
of time corresponding to ωt = 0, π/2, and π . It is clear that the horizontal hodographs
are clockwise ellipses, with the major axis in the direction of propagation x, and the
axis ratio is f/ω. The same conclusion applies for the lower signs in equation (14.110).
The particle orbits in the horizontal plane are therefore identical to those of Poincaré
waves (Figure 14.15).
However, the plane of the motion is no longer horizontal. From the velocity
components equation (14.108), we note that
u
m
= ∓ = ∓ tan θ,
w
k
(14.111)
where θ = tan−1 (m/k) is the angle made by the wavenumber vector K with the
horizontal (Figure 14.23). For upward phase propagation, equation (14.111) gives
u/w = − tan θ, so that w is negative if u is positive, as indicated in Figure 14.23.
A three-dimensional sketch of the particle orbit is shown in Figure 14.22b. It is easy
to show (Exercise 6) that the phase velocity vector c is in the direction of K, that c
and cg are perpendicular, and that the fluid motion u is parallel to cg ; these facts are
demonstrated in Chapter 7 for internal waves unaffected by Coriolis forces.
654
Geophysical Fluid Dynamics
Figure 14.23 Vertical section of an internal wave. The three parallel lines are constant phase lines, with
the arrows indicating fluid motion along the lines.
The velocity vector at any location rotates clockwise with time. Because of the
vertical propagation of phase, the tips of the instantaneous vectors also turn with depth.
Consider the turning of the velocity vectors with depth when the phase velocity is
upward, so that the deeper currents have a phase lead over the shallower currents
(Figure 14.24). Because the currents at all depths rotate clockwise in time (whether
the vertical component of c is upward or downward), it follows that the tips of the
instantaneous velocity vectors should fall on a helical spiral that turns clockwise with
depth. Only such a turning in depth, coupled with a clockwise rotation of the velocity
vectors with time, can result in a phase lead of the deeper currents. In the opposite case
of a downward phase propagation, the helix turns counterclockwise with depth. The
direction of turning of the velocity vectors can also be found from equation (14.108),
by considering x = t = 0 and finding u and v at various values of z.
Discussion of the Dispersion Relation
The dispersion relation (14.109) can be written as
ω2 − f 2 =
k2
(N 2 − ω2 ).
m2
(14.112)
Introducing tan θ = m/k, equation (14.112) becomes
ω2 = f 2 sin2 θ + N 2 cos2 θ,
which shows that ω is a function of the angle made by the wavenumber with the
horizontal and is not a function of the magnitude of K. For f = 0 the forementioned
expression reduces to ω = N cos θ, derived in Chapter 7, Section 19 without Coriolis
forces.
14. Internal Waves
Figure 14.24 Helical spiral traced out by the tips of instantaneous velocity vectors in an internal wave
with upward phase speed. Heavy arrows show the velocity vectors at two depths, and light arrows indicate
that they are rotating clockwise with time. Note that the instantaneous vectors turn clockwise with depth.
Figure 14.25 Dispersion relation for internal waves. The different regimes are indicated on the left-hand
side of the figure.
A plot of the dispersion relation (14.112) is presented in Figure 14.25, showing
ω as a function of k for various values of m. All curves pass through the point ω = f ,
which represents inertial oscillations. Typically, N ≫ f in most of the atmosphere
and the ocean. Because of the wide separation of the upper and lower limits of the
internal wave range f ω N , various limiting cases are possible, as indicated in
Figure 14.25. They are
(1) High-frequency regime (ω ∼ N, but ω N): In this range f 2 is negligible
in comparison with ω2 in the denominator of the dispersion relation (14.109),
which reduces to
655
656
Geophysical Fluid Dynamics
m2 ≃
k 2 (N 2 − ω2 )
,
ω2
that is,
ω2 ≃
N 2 k2
.
m2 + k 2
Using tan θ = m/k, this gives ω = N cos θ . Thus, the high-frequency internal waves are the same as the nonrotating internal waves discussed in
Chapter 7.
(2) Low-frequency regime (ω ∼ f, but ω f ): In this range ω2 can be neglected
in comparison to N 2 in the dispersion relation (14.109), which becomes
m2 ≃
k2 N 2
,
ω2 − f 2
that is,
ω2 ≃ f 2 +
k2 N 2
.
m2
The low-frequency limit is obtained by making the hydrostatic assumption,
that is, neglecting ∂w/∂t in the vertical equation of motion.
(3) Midfrequency regime (f ≪ ω ≪ N): In this range the dispersion relation
(14.109) simplifies to
m2 ≃
k2 N 2
,
ω2
so that both the hydrostatic and the nonrotating assumptions are applicable.
Lee Wave
Internal waves are frequently found in the “lee” (that is, the downstream side) of
mountains. In stably stratified conditions, the flow of air over a mountain causes
a vertical displacement of fluid particles, which sets up internal waves as it moves
downstream of the mountain. If the amplitude is large and the air is moist, the upward
motion causes condensation and cloud formation.
Due to the effect of a mean flow, the lee waves are stationary with respect to the
ground. This is shown in Figure 14.26, where the westward phase speed is canceled
by the eastward mean flow. We shall determine what wave parameters make this
cancellation possible. The frequency of lee waves is much larger than f , so that
rotational effects are negligible. The dispersion relation is therefore
ω2 =
N 2 k2
.
m2 + k 2
(14.113)
However, we now have to introduce the effects of the mean flow. The dispersion
relation (14.113) is still valid if ω is interpreted as the intrinsic frequency, that is, the
frequency measured in a frame of reference moving with the mean flow. In a medium
moving with a velocity U, the observed frequency of waves at a fixed point is Doppler
shifted to
ω0 = ω + K • U,
657
15. Rossby Wave
Figure 14.26 Streamlines in a lee wave. The thin line drawn through crests shows that the phase propagates downward and westward.
where ω is the intrinsic frequency; this is discussed further in Chapter 7, Section 3.
For a stationary wave ω0 = 0, which requires that the intrinsic frequency is
ω = −K • U = kU . (Here −K • U is positive because K is westward and U is
eastward.) The dispersion relation (14.113) then gives
N
.
U=√
2
k + m2
If the flow speed U is given, and the mountain introduces a typical horizontal
wavenumber k, then the preceding equation determines the vertical wavenumber
m that generates stationary waves. Waves that do not satisfy this condition would
radiate away.
The energy source of lee waves is at the surface. The energy therefore must propagate upward, and consequently the phases propagate downward. The intrinsic phase
speed is therefore westward and downward in Figure 14.26. With this information,
we can determine which way the constant phase lines should tilt in a stationary lee
wave. Note that the wave pattern in Figure 14.26 would propagate to the left in the
absence of a mean velocity, and only with the constant phase lines tilting backwards
with height would the flow at larger height lead the flow at a lower height.
Further discussion of internal waves can be found in Phillips (1977) and Munk
(1981); lee waves are discussed in Holton (1979).
15. Rossby Wave
To this point we have discussed wave motions that are possible with a constant Coriolis
frequency f and found that these waves have frequencies larger than f . We shall now
consider wave motions that owe their existence to the variation of f with latitude.
With such a variable f , the equations of motion allow a very important type of wave
motion called the Rossby wave. Their spatial scales are so large in the atmosphere that
they usually have only a few wavelengths around the entire globe (Figure 14.27). This
658
Geophysical Fluid Dynamics
Figure 14.27 Observed height (in decameters) of the 50 kPa pressure surface in the northern hemisphere. The center of the picture represents the north pole. The undulations are due to Rossby waves
(dm = km/100). J. T. Houghton, The Physics of the Atmosphere, 1986 and reprinted with the permission
of Cambridge University Press.
is why Rossby waves are also called planetary waves. In the ocean, however, their
wavelengths are only about 100 km. Rossby-wave frequencies obey the inequality
ω ≪ f . Because of this slowness the time derivative terms are an order of magnitude
smaller than the Coriolis forces and the pressure gradients in the horizontal equations
of motion. Such nearly geostrophic flows are called quasi-geostrophic motions.
Quasi-Geostrophic Vorticity Equation
We shall first derive the governing equation for quasi-geostrophic motions. For simplicity, we shall make the customary β-plane approximation valid for βy ≪ f0 , keeping in mind that the approximation is not a good one for atmospheric Rossby waves,
which have planetary scales. Although Rossby waves are frequently superposed on
a mean flow, we shall derive the equations without a mean flow, and superpose a
uniform mean flow at the end, assuming that the perturbations are small and that a
linear superposition is valid. The first step is to simplify the vorticity equation for
quasi-geostrophic motions, assuming that the velocity is geostrophic to the lowest
order. The small departures from geostrophy, however, are important because they
determine the evolution of the flow with time.
659
15. Rossby Wave
We start with the shallow-water potential vorticity equation
D ζ +f
= 0,
Dt
h
which can be written as
h
Dh
D
(ζ + f ) − (ζ + f )
= 0.
Dt
Dt
We now expand the material derivative and substitute h = H + η, where H is the
uniform undisturbed depth of the layer, and η is the surface displacement. This gives
∂ζ
∂ζ
∂ζ
∂η
∂η
∂η
(H + η)
+u
+v
+ βv − (ζ + f0 )
+u
+v
= 0.
∂t
∂x
∂y
∂t
∂x
∂y
(14.114)
Here, we have used Df/Dt = v(df/dy) = βv. We have also replaced f by f0
in the second term because the β-plane approximation neglects the variation of f
except when it involves df/dy. For small perturbations we can neglect the quadratic
nonlinear terms in equation (14.114), obtaining
H
∂ζ
∂η
+ Hβv − f0
= 0.
∂t
∂t
(14.115)
This is the linearized form of the potential vorticity equation. Its quasi-geostrophic version is obtained if we substitute the approximate geostrophic expressions for velocity:
g ∂η
,
f0 ∂y
g ∂η
v≃
.
f0 ∂x
u≃−
(14.116)
From this the vorticity is found as
ζ =
g
f0
∂ 2η ∂ 2η
+
,
∂x 2
∂y 2
so that the vorticity equation (14.115) becomes
gH ∂ ∂ 2 η ∂ 2 η
∂η
gHβ ∂η
+
− f0
= 0.
+
f0 ∂t ∂x 2
∂y 2
f0 ∂x
∂t
√
Denoting c = gH , this becomes
∂η
∂ ∂ 2 η ∂ 2 η f02
+ 2 − 2 η +β
= 0.
2
∂t ∂x
∂y
c
∂x
(14.117)
660
Geophysical Fluid Dynamics
This is the quasi-geostrophic form of the linearized vorticity equation, which governs
the flow of large-scale motions. The ratio c/f0 is recognized as the Rossby radius.
Note that we have not set ∂η/∂t = 0, in equation (14.115) during the derivation
of equation (14.117), although a strict validity of the geostrophic relations (14.116)
would require that the horizontal divergence, and hence ∂η/∂t, be zero. This is because
the departure from strict geostrophy determines the evolution of the flow described
by equation (14.117). We can therefore use the geostrophic relations for velocity
everywhere except in the horizontal divergence term in the vorticity equation.
Dispersion Relation
Assume solutions of the form
η = η̂ ei(kx+ly−ωt) .
We shall regard ω as positive; the signs of k and l then determine the direction of
phase propagation. A substitution into the vorticity equation (14.117) gives
ω=−
k2
βk
.
+ f02 /c2
+ l2
(14.118)
This is the dispersion relation for Rossby waves. The asymmetry of the dispersion
relation with respect to k and l signifies that the wave motion is not isotropic in
the horizontal, which is expected because of the β-effect. Although we have derived
it for a single homogeneous layer, it is equally applicable to stratified flows if c is
replaced by the corresponding internal value, which is c = g ′ H for the reduced
gravity model (see Chapter 7, Section 17) and c = N H /nπ for the nth mode of a
continuously stratified model. For the barotropic mode c is very large, and f02 /c2 is
usually negligible in the denominator of equation (14.118).
The dispersion relation ω(k, l) in equation (14.118) can be displayed as a surface,
taking k and l along the horizontal axes and ω along the vertical axis. The section of
this surface along l = 0 is indicated in the upper panel of Figure 14.28, and sections
of the surface for three values of ω are indicated in the bottom panel. The contours
of constant ω are circles because the dispersion relation (14.118) can be written as
2
f2
β
β 2
+ l2 =
− 02 .
k+
2ω
2ω
c
The definition of group velocity
cg = i
∂ω
∂ω
+j ,
∂k
∂l
shows that the group velocity vector is the gradient of ωin the wavenumber space. The
direction of cg is therefore perpendicular to the ω contours, as indicated in the lower
15. Rossby Wave
Figure 14.28 Dispersion relation ω(k, l) for a Rossby wave. The upper panel shows ω vs k for l = 0.
Regions of positive and negative group velocity cgx are indicated. The lower panel shows a plan view of the
surface ω(k, l), showing contours of constant ω on a kl-plane. The values of ωf0 /βc for the three circles
are 0.2, 0.3, and 0.4. Arrows perpendicular to ω contours indicate directions of group velocity vector cg .
A. E. Gill, Atmosphere–Ocean Dynamics, 1982 and reprinted with the permission of Academic Press and
Mrs. Helen Saunders-Gill.
panel of Figure 14.28. For l = 0, the maximum frequency and zero group speed
are attained at kc/f0 = −1, corresponding to ωmax f0 /βc = 0.5. The maximum
frequency is much smaller than the Coriolis frequency. For example, in the ocean
the ratio ωmax /f0 = 0.5βc/f02 is of order 0.1 for the barotropic mode, and of order
0.001 for a baroclinic mode, taking a typical midlatitude value of f0 ∼ 10−4 s−1 ,
661
662
Geophysical Fluid Dynamics
a barotropic gravity wave speed of c ∼ 200 m/s, and a baroclinic gravity wave speed
of c ∼ 2 m/s. The shortest period of midlatitude baroclinic Rossby waves in the ocean
can therefore be more than a year.
The eastward phase speed is
cx =
β
ω
.
=− 2
k
k + l 2 + f02 /c2
(14.119)
The negative sign shows that the phase propagation is always westward. The phase
speed reaches a maximum when k 2 +l 2 → 0, corresponding to very large wavelengths
represented by the region near the origin of Figure 14.28. In this region the waves are
nearly nondispersive and have an eastward phase speed
cx ≃ −
βc2
.
f02
With β = 2 × 10−11 m−1 s−1 , a typical baroclinic value of c ∼ 2 m/s, and a midlatitude value of f0 ∼ 10−4 s−1 , this gives cx ∼ 10−2 m/s. At these slow speeds the
Rossby waves would take years to cross the width of the ocean at midlatitudes. The
Rossby waves in the ocean are therefore more important at lower latitudes, where
they propagate faster. (The dispersion relation (14.118), however, is not valid within
a latitude band of 3◦ from the equator, for then the assumption of a near geostrophic
balance breaks down. A different analysis is needed in the tropics. A discussion of
the wave dynamics of the tropics is given in Gill (1982) and in the review paper by
McCreary (1985).) In the atmosphere c is much larger, and consequently the Rossby
waves propagate faster. A typical large atmospheric disturbance can propagate as a
Rossby wave at a speed of several meters per second.
Frequently, the Rossby waves are superposed on a strong eastward mean current,
such as the atmospheric jet stream. If U is the speed of this eastward current, then the
observed eastward phase speed is
cx = U −
β
.
k 2 + l 2 + f02 /c2
(14.120)
Stationary Rossby waves can therefore form when the eastward current cancels the
westward phase speed, giving cx = 0. This is how stationary waves are formed downstream of the topographic step in Figure 14.20. A simple expression for the wavelength
results if we assume l = 0 and the flow is barotropic, so that f02 /c2 is negligible in
equation (14.120).
This gives U = β/k 2 for stationary solutions, so that the wave√
length is 2π U/β.
Finally, note that we have been rather cavalier in deriving the quasi-geostrophic
vorticity equation in this section, in the sense that we have substituted the approximate
geostrophic expressions for velocity without a formal ordering of the scales. Gill
(1982) has given a more precise derivation, expanding in terms of a small parameter.
Another way to justify the dispersion relation (14.118) is to obtain it from the general
dispersion relation (14.76) derived in Section 10:
663
16. Barotropic Instability
ω3 − c2 ω(k 2 + l 2 ) − f02 ω − c2 βk = 0.
(14.121)
For ω ≪ f , the first term is negligible compared to the third, reducing
equation (14.121) to equation (14.118).
16. Barotropic Instability
In Chapter 12, Section 9 we discussed the inviscid stability of a shear flow U (y) in a
nonrotating system, and demonstrated that a necessary condition for its instability is
that d 2 U/dy 2 must change sign somewhere in the flow. This was called Rayleigh’s
point of inflection criterion. In terms of vorticity ζ̄ = −dU/dy, the criterion states
that d ζ̄ /dy must change sign somewhere in the flow. We shall now show that, on a
rotating earth, the criterion requires that d(ζ̄ + f )/dy must change sign somewhere
within the flow.
Consider a horizontal current U (y) in a medium of uniform density. In the absence
of horizontal density gradients only the barotropic mode is allowed, and U (y) does
not vary with depth. The vorticity equation is
∂
•
+ u ∇ (ζ + f ) = 0.
∂t
(14.122)
This is identical to the potential vorticity equation D/Dt[(ζ + f )/ h] = 0, with the
added simplification that the layer depth is constant because w = 0. Let the total flow
be decomposed into background flow plus a disturbance:
u = U (y) + u′ ,
v = v′.
The total vorticity is then
ζ = ζ̄ + ζ ′ = −
dU
+
dy
∂v ′
∂u′
−
∂x
∂y
=−
dU
+ ∇ 2 ψ,
dy
where we have defined the perturbation streamfunction
u′ = −
∂ψ
,
∂y
v′ =
∂ψ
.
∂x
Substituting into equation (14.122) and linearizing, we obtain the perturbation vorticity equation
∂
∂
d 2 U ∂ψ
(∇ 2 ψ) + U (∇ 2 ψ) + β −
= 0.
∂t
∂x
dy 2 ∂x
(14.123)
664
Geophysical Fluid Dynamics
Because the coefficients of equation (14.123) are independent of x and t, there can
be solutions of the form
ψ = ψ̂(y) eik(x−ct) .
The phase speed c is complex and solutions are unstable if its imaginary part ci > 0.
The perturbation vorticity equation (14.123) then becomes
2
d
d 2U
2
(U − c)
− k ψ̂ + β −
ψ̂ = 0.
dy 2
dy 2
Comparing this with equation (12.76) derived without Coriolis forces, it is seen that
the effect of planetary rotation is the replacement of −d 2 U/dy 2 by (β − d 2 U/dy 2 ).
The analysis of the section therefore carries over to the present case, resulting in the
following criterion: A necessary condition for the inviscid instability of a barotropic
current U (y) is that the gradient of the absolute vorticity
d
d 2U
(ζ̄ + f ) = β −
,
dy
dy 2
(14.124)
must change sign somewhere in the flow. This result was first derived by Kuo (1949).
Barotropic instability quite possibly plays an important role in the instability of
currents in the atmosphere and in the ocean. The instability has no preference for any
latitude, because the criterion involves β and not f . However, the mechanism presumably dominates in the tropics because midlatitude disturbances prefer the baroclinic
instability mechanism discussed in the following section. An unstable distribution of
westward tropical wind is shown in Figure 14.29.
Figure 14.29 Profiles of velocity and vorticity of a westward tropical wind. The velocity distribution is
barotropically unstable as d(ζ̄ + f )/dy changes sign within the flow. J. T. Houghton, The Physics of the
Atmosphere, 1986 and reprinted with the permission of Cambridge University Press.
17. Baroclinic Instability
17. Baroclinic Instability
The weather maps at midlatitudes invariably show the presence of wavelike horizontal
excursions of temperature and pressure contours, superposed on eastward mean flows
such as the jet stream. Similar undulations are also found in the ocean on eastward
currents such as the Gulf Stream in the north Atlantic. A typical wavelength of these
disturbances is observed to be of the order of the internal Rossby radius, that is, about
4000 km in the atmosphere and 100 km in the ocean. They seem to be propagating as
Rossby waves, but their erratic and unexpected appearance suggests that they are not
forced by any external agency, but are due to an inherent instability of midlatitude
eastward flows. In other words, the eastward flows have a spontaneous tendency
to develop wavelike disturbances. In this section we shall investigate the instability
mechanism that is responsible for the spontaneous relaxation of eastward jets into a
meandering state.
The poleward decrease of the solar irradiation results in a poleward decrease of
the temperature and a consequent increase of the density. An idealized distribution of
the atmospheric density in the northern hemisphere is shown in Figure 14.30. The density increases northward due to the lower temperatures near the poles and decreases
upward because of static stability. According to the thermal wind relation (14.15),
an eastward flow (such as the jet stream in the atmosphere or the Gulf Stream in
the Atlantic) in equilibrium with such a density structure must have a velocity that
increases with height. A system with inclined density surfaces, such as the one in
Figure 14.30, has more potential energy than a system with horizontal density surfaces, just as a system with an inclined free surface has more potential energy than a
system with a horizontal free surface. It is therefore potentially unstable because
it can release the stored potential energy by means of an instability that would
cause the density surfaces to flatten out. In the process, vertical shear of the mean
flow U (z) would decrease, and perturbations would gain kinetic energy.
Instability of baroclinic jets that release potential energy by flattening out the
density surfaces is called the baroclinic instability. Our analysis would show that the
preferred scale of the unstable waves is indeed of the order of the Rossby radius, as
observed for the midlatitude weather disturbances. The theory of baroclinic instability
Figure 14.30 Lines of constant density in the northern hemispheric atmosphere. The lines are nearly
horizontal and the slopes are greatly exaggerated in the figure. The velocity U (z) is into the plane of
paper.
665
666
Geophysical Fluid Dynamics
was developed in the 1940s by Bjerknes et al. and is considered one of the major
triumphs of geophysical fluid mechanics. Our presentation is essentially based on the
review article by Pedlosky (1971).
Consider a basic state in which the density is stably stratified in the vertical
with a uniform buoyancy frequency N, and increases northward at a constant rate
∂ ρ̄/∂y. According to the thermal wind relation, the constancy of ∂ ρ̄/∂y requires that
the vertical shear of the basic eastward flow U (z) also be constant. The β-effect is
neglected as it is not an essential requirement of the instability. (The β-effect does
modify the instability, however.) This is borne out by the spontaneous appearance of
undulations in laboratory experiments in a rotating annulus, in which the inner wall
is maintained at a higher temperature than the outer wall. The β-effect is absent in
such an experiment.
Perturbation Vorticity Equation
The equations for total flow are
∂u
∂u
∂u
1 ∂p
+u
+v
− fv = −
,
∂t
∂x
∂y
ρ0 ∂x
∂v
∂v
1 ∂p
∂v
+u
+v
+ fu = −
,
∂t
∂x
∂y
ρ0 ∂y
0=−
∂p
− ρg,
∂z
(14.125)
∂u ∂v
∂w
+
+
= 0,
∂x
∂y
∂z
∂ρ
∂ρ
∂ρ
∂ρ
+u
+v
+w
= 0,
∂t
∂x
∂y
∂z
where ρ0 is a constant reference density. We assume that the total flow is composed of
a basic eastward jet U (z) in geostrophic equilibrium with the basic density structure
ρ̄(y, z) shown in Figure 14.30, plus perturbations. That is,
u = U (z) + u′ (x, y, z),
v = v ′ (x, y, z),
w = w′ (x, y, z),
(14.126)
ρ = ρ̄(y, z) + ρ ′ (x, y, z),
p = p̄(y, z) + p′ (x, y, z).
The basic flow is in geostrophic and hydrostatic balance:
fU = −
1 ∂ p̄
,
ρ0 ∂y
∂ p̄
0=−
− ρ̄g.
∂z
(14.127)
667
17. Baroclinic Instability
Eliminating the pressure, we obtain the thermal wind relation
dU
g ∂ ρ̄
=
,
dz
fρ0 ∂y
(14.128)
which states that the eastward flow must increase with height because ∂ ρ̄/∂y > 0.
For simplicity, we assume that ∂ ρ̄/∂y is constant, and that U = 0 at the surface z = 0.
Thus the background flow is
U=
U0 z
,
H
where U0 is the velocity at the top of the layer at z = H .
We first form a vorticity equation by cross differentiating the horizontal equations
of motion in equation (14.125), obtaining
∂ζ
∂ζ
∂ζ
∂w
+u
+v
− (ζ + f )
= 0.
∂t
∂x
∂y
∂z
(14.129)
This is identical to equation (14.92), except for the exclusion of the β-effect here; the
algebraic steps are therefore not repeated. Substituting the decomposition (14.126),
and noting that ζ = ζ ′ because the basic flow U = U0 z/H has no vertical component
of vorticity, (14.129) becomes
∂ζ ′
∂w ′
∂ζ ′
+U
−f
= 0,
∂t
∂x
∂z
(14.130)
where the nonlinear terms have been neglected. This is the perturbation vorticity
equation, which we shall now write in terms of p ′ .
Assume that the perturbations are large-scale and slow, so that the velocity is
nearly geostrophic:
u′ ≃ −
1 ∂p ′
,
ρ0 f ∂y
v′ ≃
1 ∂p ′
,
ρ0 f ∂x
(14.131)
from which the perturbation vorticity is found as
ζ′ =
1 2 ′
∇ p.
ρ0 f H
(14.132)
We now express w′ in equation (14.130) in terms of p ′ . The density equation gives
∂
∂
∂
∂
(ρ̄ + ρ ′ ) + (U + u′ ) (ρ̄ + ρ ′ ) + v ′ (ρ̄ + ρ ′ ) + w ′ (ρ̄ + ρ ′ ) = 0.
∂t
∂x
∂y
∂z
Linearizing, we obtain
∂ρ ′
∂ρ ′
∂ ρ̄
ρ0 N 2 w ′
+U
+ v′
−
= 0,
∂t
∂x
∂y
g
(14.133)
668
Geophysical Fluid Dynamics
where N 2 = −gρ0−1 (∂ ρ̄/∂z). The perturbation density ρ ′ can be written in terms of
p′ by using the hydrostatic balance in equation (14.125), and subtracting the basic
state (14.127). This gives
0=−
∂p′
− ρ ′ g,
∂z
(14.134)
which states that the perturbations are hydrostatic. Equation (14.133) then gives
w′ = −
1
ρ0 N 2
∂
∂
+U
∂t
∂x
dU ∂p ′
∂p′
−
,
∂z
dz ∂x
(14.135)
where we have written ∂ ρ̄/∂y in terms of the thermal wind dU/dz. Using equations (14.132) and (14.135), the perturbation vorticity equation (14.130) becomes
∂
∂
+U
∂t
∂x
∇H2 p ′
f 2 ∂ 2 p′
+ 2
= 0.
N ∂z2
(14.136)
This is the equation that governs the quasi-geostrophic perturbations on an eastward
current U (z).
Wave Solution
We assume that the flow is confined between two horizontal planes at z = 0 and
z = H and that it is unbounded in x and y. Real flows are likely to be bounded in the
y direction, especially in a laboratory situation of flow in an annular region, where the
walls set boundary conditions parallel to the flow. The boundedness in y, however,
simply sets up normal modes in the form sin(nπy/L), where L is the width of the
channel. Each of these modes can be replaced by a periodicity in y. Accordingly, we
assume wavelike solutions
p′ = p̂(z) ei(kx+ly−ωt) .
(14.137)
The perturbation vorticity equation (14.136) then gives
d 2 p̂
− α 2 p̂ = 0,
dz2
(14.138)
where
α2 ≡
N2 2
(k + l 2 ).
f2
(14.139)
The solution of equation (14.138) can be written as
H
H
p̂ = A cosh α z −
+ B sinh α z −
.
2
2
(14.140)
669
17. Baroclinic Instability
Boundary conditions have to be imposed on solution (14.140) in order to derive an
instability criterion.
Boundary Conditions
The conditions are
w′ = 0
at z = 0, H.
The corresponding conditions on p′ can be found from equation (14.135) and
U = U0 z/H . We obtain
−
∂ 2 p′
U0 z ∂ 2 p ′
U0 ∂p′
−
+
=0
∂t ∂z
H ∂x ∂z
H ∂x
at z = 0, H,
where we have also used U = U0 z/H . The two boundary conditions are therefore
∂ 2 p′
U0 ∂p′
−
=0
∂t ∂z
H ∂x
∂ 2 p′
∂ 2 p′
U0 ∂p′
−
+ U0
=0
∂t ∂z
H ∂x
∂x ∂z
at z = 0,
at z = H.
Instability Criterion
Using equations (14.137) and (14.140), the foregoing boundary conditions require
U0
αH
αH
−
cosh
A αc sinh
2
H
2
U0
αH
αH
+
sinh
= 0,
+ B −αc cosh
2
H
2
αH
U0
αH
A α(U0 − c) sinh
−
cosh
2
H
2
αH
U0
αH
+ B α(U0 − c) cosh
−
sinh
= 0,
2
H
2
where c = ω/k is the eastward phase velocity.
This is a pair of homogeneous equations for the constants A and B. For nontrivial
solutions to exist, the determinant of the coefficients must vanish. This gives, after
some straightforward algebra, the phase velocity
U0
αH
U0
αH
αH
αH
c=
±
− tanh
− coth
.
(14.141)
2
αH
2
2
2
2
Whether the solution grows with time depends on the sign of the radicand. The
behavior of the functions under the radical sign is sketched in Figure 14.31. It is
apparent that the first factor in the radicand is positive because αH /2 > tanh(αH /2)
670
Geophysical Fluid Dynamics
Figure 14.31 Baroclinic instability. The upper panel shows behavior of the functions in equation (14.141),
and the lower panel shows growth rates of unstable waves.
for all values of αH . However, the second factor is negative for small values of αH
for which αH /2 < coth(αH /2). In this range the roots of c are complex conjugates,
with c = U0 /2 ± ici . Because we have assumed that the perturbations are of the form
exp(−ikct), the existence of a nonzero ci implies the possibility of a perturbation
that grows as exp(kci t), and the solution is unstable. The marginal stability is given
by the critical value of α satisfying
αc H
αc H
= coth
,
2
2
whose solution is
αc H = 2.4,
and the flow is unstable if αH < 2.4. Using the definition of α in equation (14.139),
it follows that the flow is unstable if
2.4
HN
<√
.
f
k2 + l2
671
17. Baroclinic Instability
As all values of k and l are allowed, we can always find a value of k 2 + l 2 low enough
to satisfy the forementioned inequality. The flow is therefore always unstable (to low
wavenumbers). For a north–south wavenumber l = 0, instability is ensured if the
east–west wavenumber k is small enough such that
HN
2.4
<
.
f
k
(14.142)
In a continuously stratified ocean, the speed of a long internal wave for the n = 1
baroclinic mode is c = N H /π , so that the corresponding internal Rossby radius is
c/f = NH /πf . It is usual to omit the factor π and define the Rossby radius in a
continuously stratified fluid as
≡
HN
.
f
The condition (14.142) for baroclinic instability is therefore that the east–west wavelength be large enough so that
λ > 2.6.
However, the wavelength λ = 2.6 does not grow at the fastest rate. It can be
shown from equation (14.141) that the wavelength with the largest growth rate is
λmax = 3.9.
This is therefore the wavelength that is observed when the instability develops. Typical
values for f , N , and H suggest that λmax ∼ 4000 km in the atmosphere and 200 km
in the ocean, which agree with observations. Waves much smaller than the Rossby
radius do not grow, and the ones much larger than the Rossby radius grow very slowly.
Energetics
The foregoing analysis suggests that the existence of “weather waves” is due to the
fact that small perturbations can grow spontaneously when superposed on an eastward
current maintained by the sloping density surfaces (Figure 14.30). Although the basic
current does have a vertical shear, the perturbations do not grow by extracting energy
from the vertical shear field. Instead, they extract their energy from the potential
energy stored in the system of sloping density surfaces. The energetics of the baroclinic
instability is therefore quite different than that of the Kelvin–Helmholtz instability
(which also has a vertical shear of the mean flow), where the perturbation Reynolds
stress u′ w ′ interacts with the vertical shear and extracts energy from the mean shear
flow. The baroclinic instability is not a shear flow instability; the Reynolds stresses
are too small because of the small w in quasi-geostrophic large-scale flows.
The energetics of the baroclinic instability can be understood by examining the
equation for the perturbation kinetic energy. Such an equation can be derived by
672
Geophysical Fluid Dynamics
multiplying the equations for ∂u′ /∂t and ∂v ′ /∂t by u′ and v ′ , respectively, adding
the two, and integrating over the region of flow. Because of the assumed periodicity
in x and y, the extent of the region of integration is chosen to be one wavelength in
either direction. During this integration, the boundary conditions of zero normal flow
on the walls and periodicity in x and y are used repeatedly. The procedure is similar
to that for the derivation of equation (12.83) and is not repeated here. The result is
dK
= −g
dt
w′ ρ ′ dx dy dz,
where K is the global perturbation kinetic energy
K≡
ρ0
2
(u′ 2 + v ′ 2 ) dx dy dz.
In unstable flows we must have dK/dt > 0, which requires that the volume integral of w′ ρ ′ must be negative. Let us denote the volume average of w ′ ρ ′ by w ′ ρ ′ .
A negative w ′ ρ ′ means that on the average the lighter fluid rises and the heavier fluid
sinks. By such an interchange the center of gravity of the system, and therefore its
potential energy, is lowered. The interesting point is that this cannot happen in a stably
stratified system with horizontal density surfaces; in that case an exchange of fluid
particles raises the potential energy. Moreover, a basic state with inclined density
surfaces (Figure 14.30) cannot have w ′ ρ ′ < 0 if the particle excursions are vertical.
If, however, the particle excursions fall within the wedge formed by the constant density lines and the horizontal (Figure 14.32), then an exchange of fluid particles takes
lighter particles upward (and northward) and denser particles downward (and southward). Such an interchange would tend to make the density surfaces more horizontal,
releasing potential energy from the mean density field with a consequent growth of the
Figure 14.32 Wedge of instability (shaded) in a baroclinic instability. The wedge is bounded by constant
density lines and the horizontal. Unstable waves have a particle trajectory that falls within the wedge.
673
18. Geostrophic Turbulence
perturbation energy. This type of convection is called sloping convection. According
to Figure 14.32 the exchange of fluid particles within the wedge of instability results
in a net poleward transport of heat from the tropics, which serves to redistribute the
larger solar heat received by the tropics.
In summary, baroclinic instability draws energy from the potential energy of
the mean density field. The resulting eddy motion has particle trajectories that are
oriented at a small angle with the horizontal, so that the resulting heat transfer has a
poleward component. The preferred scale of the disturbance is the Rossby radius.
18. Geostrophic Turbulence
Two common modes of instability of a large-scale current system were presented in the
preceding sections. When the flow is strong enough, such instabilities can make a flow
chaotic or turbulent. A peculiarity of large-scale turbulence in the atmosphere or the
ocean is that it is essentially two dimensional in nature. The existence of the Coriolis
force, stratification, and small thickness of geophysical media severely restricts the
vertical velocity in large-scale flows, which tend to be quasi-geostrophic, with the
Coriolis force balancing the horizontal pressure gradient to the lowest order. Because
vortex stretching, a key mechanism by which ordinary three-dimensional turbulent
flows transfer energy from large to small scales, is absent in two-dimensional flow,
one expects that the dynamics of geostrophic turbulence are likely to be fundamentally different from that of three-dimensional laboratory-scale turbulence discussed
in Chapter 13. However, we can still call the motion “turbulent” because it is unpredictable and diffusive.
A key result on the subject was discovered by the meteorologist Fjortoft (1953),
and since then Kraichnan, Leith, Batchelor, and others have contributed to various
aspects of the problem. A good discussion is given in Pedlosky (1987), to which the
reader is referred for a fuller treatment. Here, we shall only point out a few important
results.
An important variable in the discussion of two-dimensional turbulence is enstrophy, which is the mean square vorticity ζ 2 . In an isotropic turbulent field we can
define an energy spectrum S(K), a function of the magnitude of the wavenumber
K, as
∞
u2 =
S(K) dK.
0
It can be shown that the enstrophy spectrum is K 2 S(K), that is,
ζ2 =
∞
K 2 S(K) dK,
0
which makes sense because vorticity involves the spatial gradient of velocity.
We consider a freely evolving turbulent field in which the shape of the velocity
spectrum changes with time. The large scales are essentially inviscid, so that both
674
Geophysical Fluid Dynamics
energy and enstrophy are nearly conserved:
∞
d
dt
d
dt
∞
0
S(K) dK = 0,
(14.143)
K 2 S(K) dK = 0,
(14.144)
0
where terms proportional to the molecular viscosity ν have been neglected on
the right-hand sides of the equations. The enstrophy conservation is unique to
two-dimensional turbulence because of the absence of vortex stretching.
Suppose that the energy spectrum initially contains all its energy at wavenumber
K0 . Nonlinear interactions transfer this energy to other wavenumbers, so that the
sharp spectral peak smears out. For the sake of argument, suppose that all of the
initial energy goes to two neighboring wavenumbers K1 and K2 , with K1 < K0 < K2 .
Conservation of energy and enstrophy requires that
S0 = S1 + S2 ,
K02 S0 = K12 S1 + K22 S2 ,
where Sn is the spectral energy at Kn . From this we can find the ratios of energy and
enstrophy spectra before and after the transfer:
K2 − K0 K2 + K0
S1
=
,
S2
K0 − K 1 K1 + K 0
K12 S1
K 2 K 2 − K02
= 12 22
.
2
K 2 S2
K2 K0 − K12
(14.145)
As an example, suppose that nonlinear smearing transfers energy to wavenumbers
K1 = K0 /2 and K2 = 2K0 . Then equations (14.145) show that S1 /S2 = 4 and
K12 S1 /K22 S2 = 41 , so that more energy goes to lower wavenumbers (large scales),
whereas more enstrophy goes to higher wavenumbers (smaller scales). This important result on two-dimensional turbulence was derived by Fjortoft (1953). Clearly, the
constraint of enstrophy conservation in two-dimensional turbulence has prevented a
symmetric spreading of the initial energy peak at K0 .
The unique character of two-dimensional turbulence is evident here. In smallscale three-dimensional turbulence studied in Chapter 13, the energy goes to smaller
and smaller scales until it is dissipated by viscosity. In geostrophic turbulence, on the
other hand, the energy goes to larger scales, where it is less susceptible to viscous
dissipation. Numerical calculations are indeed in agreement with this behavior, which
shows that the energy-containing eddies grow in size by coalescing. On the other hand,
the vorticity becomes increasingly confined to thin shear layers on the eddy boundaries; these shear layers contain very little energy. The backward (or inverse) energy
cascade and forward enstrophy cascade are represented schematically in Figure 14.33.
It is clear that there are two “inertial” regions in the spectrum of a two-dimensional
turbulent flow, namely, the energy cascade region and the enstrophy cascade region.
18. Geostrophic Turbulence
Figure 14.33 Energy and enstrophy cascade in two-dimensional turbulence.
If energy is injected into the system at a rate ε, then the energy spectrum in the
energy cascade region has the form S(K) ∝ ε2/3 K −5/3 ; the argument is essentially
the same as in the case of the Kolmogorov spectrum in three-dimensional turbulence
(Chapter 13, Section 9), except that the transfer is backwards.A dimensional argument
also shows that the energy spectrum in the enstrophy cascade region is of the form
S(K) ∝ α 2/3 K −3 , where α is the forward enstrophy flux to higher wavenumbers.
There is negligible energy flux in the enstrophy cascade region.
As the eddies grow in size, they become increasingly immune to viscous dissipation, and the inviscid assumption implied in equation (14.143) becomes increasingly
applicable. (This would not be the case in three-dimensional turbulence in which
the eddies continue to decrease in size until viscous effects drain energy out of the
system.) In contrast, the corresponding assumption in the enstrophy conservation
equation (14.144) becomes less and less valid as enstrophy goes to smaller scales,
where viscous dissipation drains enstrophy out of the system. At later stages in the
evolution, then, equation (14.144) may not be a good assumption. However, it can be
shown (see Pedlosky, 1987) that the dissipation of enstrophy actually intensifies the
process of energy transfer to larger scales, so that the red cascade (that is, transfer to
larger scales) of energy is a general result of two-dimensional turbulence.
The eddies, however, do not grow in size indefinitely. They become increasingly
slower as their length scale l increases, while their velocity scale u remains constant.
The slower dynamics makes them increasingly wavelike, and the eddies transform
into Rossby-wave packets as their length scale becomes of order (Rhines, 1975)
u
l∼
(Rhines length),
β
675
676
Geophysical Fluid Dynamics
where β = df/dy and u is the rms fluctuating speed. The Rossby-wave propagation
results in an anisotropic elongation of the eddies in the east–west (“zonal”)
direction,
√
while the eddy size in the north–south direction stops growing at u/β. Finally, the
velocity
field consists of zonally directed jets whose north–south extent is of order
√
u/β. This has been suggested as an explanation for the existence of zonal jets in
the atmosphere of the planet Jupiter (Williams, 1979). The inverse energy cascade
regime may not occur in the earth’s atmosphere and the ocean at midlatitudes because
the Rhines length (about 1000 km in the atmosphere and 100 km in the ocean) is of
the order of the internal Rossby radius, where
√ the energy is injected by baroclinic
instability. (For the inverse cascade to occur, u/β needs to be larger than the scale
at which energy is injected.)
Eventually, however, the kinetic energy has to be dissipated by molecular effects
at the Kolmogorov microscale η, which is of the order of a few millimeters in the
ocean and the atmosphere. A fair hypothesis is that processes such as internal waves
drain energy out of the mesoscale eddies, and breaking internal waves generate
three-dimensional turbulence that finally cascades energy to molecular scales.
A recent review of intense storm motion (lower atmosphere dynamics and thermodynamics) was published by Chan (2005), whereas upper atmospheric motion was
discussed by Haynes (2005). Oceanic flow transport was treated by Wiggins (2005).
Exercises
1. The Gulf Stream flows northward along the east coast of the United States
with a surface current of average magnitude 2 m/s. If the flow is assumed to be in
geostrophic balance, find the average slope of the sea surface across the current at a
latitude of 45◦ N. [Answer: 2.1 cm per km]
2. A plate containing water (ν = 10−6 m2 /s) above it rotates at a rate of 10
revolutions per minute. Find the depth of the Ekman layer, assuming that the flow is
laminar.
3. Assume that the atmospheric Ekman layer over the earth’s surface at a latitude
of 45◦ N can be approximated by an eddy viscosity of νv = 10 m2 /s. If the geostrophic
velocity above the Ekman layer is 10 m/s, what is the Ekman transport across isobars?
[Answer: 2203 m2 /s]
4. Find the axis ratio of a hodograph plot for a semidiurnal tide in the middle
of the ocean at a latitude of 45◦ N. Assume that the midocean tides are rotational
surface gravity waves of long wavelength and are unaffected by the proximity of
coastal boundaries. If the depth of the ocean is 4 km, find the wavelength, the phase
velocity, and the group velocity. Note, however, that the wavelength is comparable to
the width of the ocean, so that the neglect of coastal boundaries is not very realistic.
5. An internal Kelvin wave on the thermocline of the ocean propagates along
the west coast of Australia. The thermocline has a depth of 50 m and has a nearly
discontinuous density change of 2 kg/m3 across it. The layer below the thermocline
677
Literature Cited
is deep. At a latitude of 30◦ S, find the direction and magnitude of the propagation
speed and the decay scale perpendicular to the coast.
6. Using the dispersion relation m2 = k 2 (N 2 − ω2 )/(ω2 − f 2 ) for internal
waves, show that the group velocity vector is given by
[cgx , cgz ] =
(N 2 − f 2 ) km
[m, −k]
(m2 + k 2 )3/2 (m2 f 2 + k 2 N 2 )1/2
[Hint: Differentiate the dispersion relation partially with respect to k and m.] Show
that cg and c are perpendicular and have oppositely directed vertical components.
Verify that cg is parallel to u.
7. Suppose the atmosphere at a latitude of 45◦ N is idealized by a uniformly
stratified layer of height 10 km, across which the potential temperature increases by
50 ◦ C.
(i) What is the value of the buoyancy frequency N?
(ii) Find the speed of a long gravity wave corresponding to the n = 1 baroclinic
mode.
(iii) For the n = 1 mode, find the westward speed of nondispersive (i. e., very large
wavelength) Rossby waves. [Answer: N = 0.01279 s−1 ; c1 = 40.71 m/s;
cx = −3.12 m/s]
8. Consider a steady flow rotating between plane parallel boundaries a distance
L apart. The angular velocity is and a small rectilinear velocity U is superposed.
There is a protuberance of height h ≪ L in the flow. The Ekman and Rossby numbers
are both small: Ro ≪ l, E ≪ l. Obtain an integral of the relevant equations of motion
that relates the modified pressure and the streamfunction for the motion, and show
that the modified pressure is constant on streamlines.
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Geophysical Fluid Dynamics
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Phillips, O. M. (1977). The Dynamics of the Upper Ocean, London: Cambridge University Press.
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Rhines, P. B. (1975). “Waves and turbulence on a β-plane.” Journal of Fluid Mechanics 69: 417–443.
Taylor, G. I. (1915). “Eddy motion in the atmosphere.” Philosophical Transactions of the Royal Society of
London A215: 1–26.
Wiggins, S. (2005). “The dynamical systems approach to Lagrangian transport in oceanic flows.” Annual
Review of Fluid Mechanics 37: 295–328.
Williams, G. P. (1979). “Planetary circulations: 2. The Jovian quasi-geostrophic regime.” Journal of Atmospheric Sciences 36: 932–968.
Chapter 15
Aerodynamics
1. Introduction . . . . . . . . . . . . . . . . . . . . .
2. The Aircraft and Its Controls . . . . .
Control Surfaces . . . . . . . . . . . . . . . . .
3. Airfoil Geometry . . . . . . . . . . . . . . . . .
4. Forces on an Airfoil . . . . . . . . . . . . . .
5. Kutta Condition . . . . . . . . . . . . . . . . .
Historical Notes . . . . . . . . . . . . . . . . .
6. Generation of Circulation . . . . . . . .
7. Conformal Transformation for
Generating Airfoil Shape . . . . . . . . .
Transformation of a Circle into
a Straight Line . . . . . . . . . . . . . . .
Transformation of a Circle into
a Circular Arc . . . . . . . . . . . . . . . .
Transformation of a Circle into
a Symmetric Airfoil. . . . . . . . . . .
Transformation of a Circle into
a Cambered Airfoil . . . . . . . . . . .
8. Lift of Zhukhovsky Airfoil . . . . . . . .
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9. Wing of Finite Span. . . . . . . . . . . . .
10. Lifting Line Theory of Prandtl
and Lanchester . . . . . . . . . . . . . . . . .
Bound and Trailing Vortices. . . . .
Downwash . . . . . . . . . . . . . . . . . . . . .
Induced Drag. . . . . . . . . . . . . . . . . . .
Lanchester versus Prandtl . . . . . .
11. Results for Elliptic Circulation
Distribution . . . . . . . . . . . . . . . . . . . .
12. Lift and Drag Characteristics of
Airfoils . . . . . . . . . . . . . . . . . . . . . . . . .
13. Propulsive Mechanisms of Fish
and Birds . . . . . . . . . . . . . . . . . . . . . .
Locomotion of Fish . . . . . . . . . . . . .
Flight of Birds and insects . . . . . .
14. Sailing against the Wind . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . .
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1. Introduction
Aerodynamics is the branch of fluid mechanics that deals with the determination
of the flow past bodies of aeronautical interest. Gravity forces are neglected, and
viscosity is regarded as small so that the viscous forces are confined to thin boundary
layers (Figure 10.1). The subject is called incompressible aerodynamics if the flow
speeds are low enough (Mach number < 0.3) for the compressibility effects to be
negligible. At larger Mach numbers the subject is normally called gas dynamics,
which deals with flows in which compressibility effects are important. In this chapter
we shall study some elementary aspects of incompressible flow around aircraft wing
shapes. The blades of turbomachines (such as turbines and compressors) have the
same cross section as that of an aircraft wing, so that much of our discussion will also
apply to the flow around the blades of a turbomachine.
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50015-0
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Aerodynamics
Because the viscous effects are confined to thin boundary layers, the bulk of the
flow is still irrotational. Consequently, a large part of our discussion of irrotational
flows presented in Chapter 6 is relevant here. It is assumed that the reader is familiar
with that chapter.
2. The Aircraft and Its Controls
Although a book on fluid mechanics is not the proper place for describing an aircraft
and its controls, we shall do this here in the hope that the reader will find it interesting.
Figure 15.1 shows three views of an aircraft. The body of the aircraft, which houses the
passengers and other payload, is called the fuselage. The engines (jets or propellers)
are often attached to the wings; sometimes they may be mounted on the fuselage.
Figure 15.2 shows the plan view of a wing. The outer end of each wing is called the
wing tip, and the distance between the wing tips is called the wing span s. The distance
between the leading and trailing edges of the wing is called the chord length c, which
Figure 15.1 Three views of a transport aircraft and its control surfaces (NASA).
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2. The Aircraft and Its Controls
varies along the spanwise direction. The plan area of the wing is called the wing
area A. The narrowness of the wing planform is measured by its aspect ratio
≡
s2
s
= ,
A
c̄
where c̄ is the average chord length.
The various possible rotational motions of an aircraft can be referred to three
axes, called the pitch axis, the roll axis, and the yaw axis (Figure 15.3).
Figure 15.2 Wing planform geometry.
Figure 15.3 Aircraft axes.
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Aerodynamics
Control Surfaces
The aircraft is controlled by the pilot by moving certain control surfaces described in
the following paragraphs.
Aileron: These are portions of each wing near the wing tip (Figure 15.1), joined
to the main wing by a hinged connection, as shown in Figure 15.4. They move
differentially in the sense that one moves up while the other moves down.
A depressed aileron increases the lift, and a raised aileron decreases the lift,
so that a rolling moment results. The object of situating the ailerons near the
wing tip is to generate a large rolling moment. The pilot generally controls the
ailerons by moving a control stick, whose movement to the left or right causes
a roll to the left or right. In larger aircraft the aileron motion is controlled by
rotating a small wheel that resembles one half of an automobile steering wheel.
Elevator: The elevators are hinged to the trailing edge of the tail plane. Unlike
ailerons they move together, and their movement generates a pitching motion of
the aircraft. The elevator movements are imparted by the forward and backward
movement of a control stick, so that a backward pull lifts the nose of the aircraft.
Rudder: The yawing motion of the aircraft is governed by the hinged rear
portion of the tail fin, called the rudder. The pilot controls the rudder by pressing
his feet against two rudder pedals so arranged that moving the left pedal forward
moves the aircraft’s nose to the left.
Flap: During take off, the speed of the aircraft is too small to generate enough
lift to support the weight of the aircraft. To overcome this, a section of the rear
of the wing is “split,” so that it can be rotated downward to increase the lift
(Figure 15.5). A further function of the flap is to increase both lift and drag
during landing.
Figure 15.4 The aileron.
Figure 15.5 The flap.
3. Airfoil Geometry
Modern jet transports also have “spoilers” on the top surface of each wing. When
raised slightly, they separate the boundary layer early on part of the top of the wing
and this decreases its lift. They can be deployed together or individually. Reducing
the lift on one wing will bank the aircraft so that it would turn in the direction of
the lowered wing. Deployed together, lift would be decreased and the aircraft would
descend to a new equilibrium altitude. Spoilers have another function as well. Upon
touchdown during landing they are deployed fully as flat plates nearly perpendicular
to the wing surface. As such they add greatly to the drag to slow the aircraft and
shorten its roll down the runway.
An aircraft is said to be in trimmed flight when there are no moments about
its center of gravity. Trim tabs are small adjustable surfaces within or adjacent to
the major control surfaces described in the preceding: ailerons, elevators, and rudder.
Deflections of these surfaces may be set and held to adjust for a change in the aircraft’s
center of gravity in flight due to consumption of fuel or a change in the direction of
the prevailing wind with respect to the flight path. These are set for steady level flight
on a straight path with minimum deflection of the major control surfaces.
3. Airfoil Geometry
Figure 15.6 shows the shape of the cross section of a wing, called an airfoil section
(spelled aerofoil in the British literature). The leading edge of the profile is generally
rounded, whereas the trailing edge is sharp. The straight line joining the centers of
curvature of the leading and trailing edges is called the chord. The meridian line of the
section passing midway between the upper and lower surfaces is called the camber
line. The maximum height of the camber line above the chord line is called the camber
of the section. Normally the camber varies from nearly zero for high-speed supersonic
wings, to ≈5% of chord length for low-speed wings. The angle α between the chord
line and the direction of flight (i.e., the direction of the undisturbed stream) is called
the angle of attack or angle of incidence.
Figure 15.6 Airfoil geometry.
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Aerodynamics
4. Forces on an Airfoil
The resultant aerodynamic force F on an airfoil can be resolved into a lift force L
perpendicular to the direction of undisturbed flight and a drag force D in the direction
of flight (Figure 15.7). In steady level flight the drag is balanced by the thrust of
the engine, and the lift equals the weight of the aircraft. These forces are expressed
nondimensionally by defining the coefficients of lift and drag:
CL ≡
L
,
(1/2)ρU 2 A
CD ≡
D
.
(1/2)ρU 2 A
(15.1)
The drag results from the tangential stress and normal pressure distributions on the
surface. These are called the friction drag and the pressure drag, respectively. The lift
is almost entirely due to the pressure distribution. Figure 15.8 shows the distribution
of the pressure coefficient Cp = (p − p∞ )/ 21 ρU 2 at a moderate angle of attack. The
outward arrows correspond to a negative Cp , while a positive Cp is represented by
inward arrows. It is seen that the pressure coefficient is negative over most of the
surface, except over small regions near the nose and the tail. However, the pressures
over most of the upper surface are smaller than those over the bottom surface, which
results in a lift force. The top and bottom surfaces of an airfoil are popularly referred
to as the suction side and the compression side, respectively.
5. Kutta Condition
In Chapter 6, Section 11 we showed that the lift per unit span in an irrotational flow
over a two-dimensional body of arbitrary cross section is
L = ρU Ŵ,
(15.2)
where U is the free-stream velocity and Ŵ is the circulation around the body. Relation
(15.2) is called the Kutta–Zhukhovsky lift theorem. The question is, how does a flow
develop such a circulation? Obviously, a circular or elliptic cylinder does not develop
Figure 15.7 Forces on an airfoil.
5. Kutta Condition
Figure 15.8 Distribution of the pressure coefficient over an airfoil. The upper panel shows Cp plotted
normal to the surface and the lower panel shows Cp plotted normal to the chord line.
any circulation around it, unless it is rotated. It has been experimentally observed that
only bodies having a sharp trailing edge, such as an airfoil, can generate circulation
and lift.
Figure 15.9 shows the irrotational flow pattern around an airfoil for increasing
values of clockwise circulation. For Ŵ = 0, there is a stagnation point A located just
below the leading edge and a stagnation point B on the top surface near the trailing
edge. When some clockwise circulation is superimposed, both stagnation points move
slightly down. For a particular value of Ŵ, the stagnation point B coincides with the
trailing edge. (If the circulation is further increased, the rear stagnation point moves
to the lower surface.) As far as irrotational flow of an ideal fluid is concerned, all
these flow patterns are possible solutions. A real flow, however, develops a specific
amount of circulation, depending on the airfoil shape and the angle of attack.
Consider the irrotational flow around the trailing edge of an airfoil. It is shown in
Chapter 6, Section 4 that, for flow in a corner of included angle γ , the velocity at the
corner point is zero if γ < 180◦ and infinite if γ > 180◦ (see Figure 6.4). In the upper
two panels of Figure 15.9 the fluid goes from the lower to the upper side by turning
around the trailing edge, so that γ is slightly less than 360◦ . The resulting velocity
at the trailing edge is therefore infinite in the upper two panels of Figure 15.9. In the
bottom panel, on the other hand, the trailing edge is a stagnation point because γ is
slightly less than 180◦ .
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Aerodynamics
Figure 15.9 Irrotational flow pattern over an airfoil for various values of clockwise circulation.
Photographs of flow around airfoils reveal that the pattern sketched in the
bottom panel of Figure 15.9 is the one developed in practice. The German aerodynamist Wilhelm Kutta proposed the following rule in 1902: In flow over a
two-dimensional body with a sharp trailing edge, there develops a circulation of
magnitude just sufficient to move the rear stagnation point to the trailing edge. This
is called the Kutta condition, sometimes also called the Zhukhovsky hypothesis. At
the beginning of the twentieth century it was merely an experimentally observed fact.
Justification for this empirical rule became clear after the boundary layer concepts
were understood. In the following section we shall see why a real flow should satisfy
the Kutta condition.
Historical Notes
According to von Karman (1954, p. 34), the connection between the lift of airplane
wings and the circulation around them was recognized and developed by three persons. One of them was the Englishman Frederick Lanchester (1887–1946). He was a
multisided and imaginative person, a practical engineer as well as an amateur mathematician. His trade was automobile building; in fact, he was the chief engineer and
general manager of the Lanchester Motor Company. He once took von Karman for a
ride around Cambridge in an automobile that he built himself, but von Karman “felt a
little uneasy discussing aerodynamics at such rather frightening speed.” The second
person is the German mathematician Wilhelm Kutta (1867–1944), well-known for
the Runge–Kutta scheme used in the numerical integration of ordinary differential
equations. He started out as a pure mathematician, but later became interested in
aerodynamics. The third person is the Russian physicist Nikolai Zhukhovsky, who
developed the mathematical foundations of the theory of lift for wings of infinite span,
independently of Lanchester and Kutta. An excellent book on the history of flight and
the science of aerodynamics was recently authored by Anderson (1998).
6. Generation of Circulation
6. Generation of Circulation
We shall now discuss why a real flow around an airfoil should satisfy the Kutta
condition. The explanation lies in the frictional and boundary layer nature of a real
flow. Consider an airfoil starting from rest in a real fluid. The flow immediately after
starting is irrotational everywhere, because the vorticity adjacent to the surface has
not yet diffused outward. The velocity at this stage has a near discontinuity adjacent
to the surface. The flow has no circulation, and resembles the pattern in the upper
panel of Figure 15.9. The fluid goes around the trailing edge with a very high velocity
and overcomes a steep deceleration and pressure rise from the trailing edge to the
stagnation point.
Within a fraction of a second (in a time of the order of that taken by the flow
to move one chord length), however, boundary layers develop on the airfoil, and the
retarded fluid does not have sufficient kinetic energy to negotiate the steep pressure
rise from the trailing edge toward the rear stagnation point. This generates a back-flow
in the boundary layer and a separation of the boundary layer at the trailing edge. The
consequence of all this is the generation of a shear layer, which rolls up into a spiral
form under the action of its own induced vorticity (Figure 15.10). The rolled-up shear
layer is carried downstream by the flow and is left at the location where the airfoil
started its motion. This is called the starting vortex.
The sense of circulation of the starting vortex is counterclockwise in Figure 15.10,
which means that it must leave behind a clockwise circulation around the airfoil. To
see this, imagine that the fluid is stationary and the airfoil is moving to the left.
Consider a material circuit ABCD, made up of the same fluid particles and large
enough to enclose both the initial and final locations of the airfoil (Figure 15.11).
Initially the trailing edge was within the region BCD, which now contains the starting
vortex only. According to the Kelvin circulation theorem, the circulation around any
material circuit remains constant, if the circuit remains in a region of inviscid flow
(although viscous processes may go on inside the region enclosed by the circuit).
The circulation around the large curve ABCD therefore remains zero, since it was
zero initially. Consequently the counterclockwise circulation of the starting vortex
around DBC is balanced by an equal clockwise circulation around ADB. The wing is
therefore left with a circulation Ŵ equal and opposite to the circulation of the starting
vortex.
It is clear from Figure 15.9 that a value of circulation other than the one that
moves the rear stagnation point exactly to the trailing edge would result in a sequence
of events as just described and would lead to a readjustment of the flow. The only value
Figure 15.10 Formation of a spiral vortex sheet soon after an airfoil begins to move.
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Aerodynamics
Figure 15.11 A material circuit ABCD in a stationary fluid and an airfoil moving to the left.
of the circulation that would not result in further readjustment is the one required by
the Kutta condition. With every change in the speed of the airflow or in the angle of
attack, a new starting vortex is cast off and left behind. A new value of circulation
around the airfoil is established so as to place the rear stagnation point at the trailing
edge in each case.
It is apparent that the viscosity of the fluid is not only responsible for the drag,
but also for the development of circulation and lift. In developing the circulation, the
flow leads to a steady state where a further boundary layer separation is prevented.
The establishment of circulation around an airfoil-shaped body in a real fluid is a
remarkable result.
7. Conformal Transformation for Generating Airfoil Shape
In the study of airfoils, one is interested in finding the flow pattern and pressure
distribution. The direct solution of the Laplace equation for the prescribed boundary
shape of the airfoil is quite straightforward using a computer, but analytically difficult.
In general the analytical solutions are possible only when the airfoil is assumed
thin. This is called thin airfoil theory, in which the airfoil is replaced by a vortex
sheet coinciding with the camber line. An integral equation is developed for the local
vorticity distribution from the condition that the camber line be a streamline (velocity
tangent to the camber line). The velocity at each point on the camber line is the
superposition (i.e., integral) of velocities induced at that point due to the vorticity
distribution at all other points on the camber line plus that from the oncoming stream
(at infinity). Since the maximum camber is small, this is usually evaluated on the
x–y-plane. The Kutta condition is represented by the requirement that the strength of
the vortex sheet at the trailing edge is zero. This is treated in detail in Kuethe and
Chow (1998, chapter 5) and Anderson (2007, chapter 4). An indirect way of solving
the problem involves the method of conformal transformation, in which a mapping
function is determined such that the arbitrary airfoil shape is transformed into a circle.
Then a study of the flow around the circle would determine the flow pattern around
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7. Conformal Transformation for Generating Airfoil Shape
the airfoil. This is called Theodorsen’s method, which is complicated and will not be
discussed here.
Instead, we shall deal with a case in which a given transformation maps a circle
into an airfoil-like shape and determines the properties of the airfoil generated thereby.
This is the Zhukhovsky transformation
z=ζ+
b2
,
ζ
(15.3)
where b is a constant. It maps regions of the ζ -plane into the z-plane, some examples
of which are discussed in Chapter 6, Section 14. Here, we shall assume circles of
different configurations in the ζ -plane and examine their transformed shapes in the
z-plane. It will be seen that one of them will result in an airfoil shape.
Transformation of a Circle into a Straight Line
Consider a circle, centered at the origin in the ζ -plane, whose radius b is the same as
the constant in the Zhukhovsky transformation (Figure 15.12). For a point ζ = b eiθ
on the circle, the corresponding point in the z-plane is
z = b eiθ + b e−iθ = 2b cos θ.
As θ varies from 0 to π , z goes along the x-axis from 2b to −2b. As θ varies from π
to 2π , z goes from −2b to 2b. The circle of radius b in the ζ -plane is thus transformed
into a straight line of length 4b in the z-plane. It is clear that the region outside the
circle in the ζ -plane is mapped into the entire z-plane. (It can be shown that the region
inside the circle is also transformed into the entire z-plane. This, however, is of no
concern to us, since we shall not consider the interior of the circle in the ζ -plane.)
Transformation of a Circle into a Circular Arc
Let us consider a circle of radius a (>b) in the ζ -plane, the center of which is displaced
along the η-axis and which cuts the ξ -axis at (±b, 0), as shown in Figure 15.13. If a
Figure 15.12 Transformation of a circle into a straight line.
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Aerodynamics
Figure 15.13 Transformation of a circle into a circular arc.
point on the circle in the ζ -plane is represented by ζ = Reiθ , then the corresponding
point in the z-plane is
z = Reiθ +
b2 −iθ
e ,
R
whose real and imaginary parts are
x = (R + b2 /R) cos θ,
y = (R − b2 /R) sin θ.
(15.4)
Eliminating R, we obtain
x 2 sin2 θ − y 2 cos2 θ = 4b2 sin2 θ cos2 θ.
(15.5)
To understand the shape of the curve represented by equation (15.5) we must express
θ in terms of x, y, and the known constants. From triangle OQP, we obtain
QP2 = OP2 + OQ2 − 2(OQ)(OP) cos (QÔP).
Using QP = a = b/ cos β and OQ = b tan β, this becomes
b2
= R 2 + b2 tan2 β − 2Rb tan β cos(90◦ − θ ),
cos2 β
which simplifies to
2b tan β sin θ = R − b2 /R = y/ sin θ,
(15.6)
where equation (15.4) has been used. We now eliminate θ between equations (15.5)
and (15.6). First note from equation (15.6) that cos2 θ = (2b tan β − y)/2b tan β, and
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7. Conformal Transformation for Generating Airfoil Shape
cot 2 θ = (2b tan β − y)/y. Then divide equation (15.5) by sin2 θ, and substitute these
expressions of cos2 θ and cot 2 θ. This gives
x 2 + (y + 2b cot 2β)2 = (2b csc 2β)2 ,
where β is known from cos β = b/a. This is the equation of a circle in the z-plane,
having the center at (0, −2b cot 2β) and a radius of 2b csc 2β. The Zhukhovsky
transformation has thus mapped a complete circle into a circular arc.
Transformation of a Circle into a Symmetric Airfoil
Instead of displacing the center of the circle along the imaginary axis of the ζ -plane,
suppose that it is displaced to a point Q on the real axis (Figure 15.14). The radius of
the circle is a (>b), and we assume that a is slightly larger than b:
a ≡ b(1 + e)
e ≪ 1.
(15.7)
A numerical evaluation of the Zhukhovsky transformation (15.3), with assumed values for a and b, shows that the corresponding shape in the z-plane is a streamlined body
that is symmetrical about the x-axis. Note that the airfoil in Figure 15.14 has a rounded
nose and thickness, while the one in Figure 15.13 has a camber but no thickness.
Transformation of a Circle into a Cambered Airfoil
As can be expected from Figures 15.13 and 15.14, the transformed figure in the z-plane
will be a general airfoil with both camber and thickness if the circle in the ζ -plane is
displaced in both η and ξ directions (Figure 15.15). The following relations can be
proved for e ≪ 1:
c ≃ 4b,
camber ≃ 21 βc,
tmax /c ≃ 1.3 e.
Figure 15.14 Transformation of a circle into a symmetric airfoil.
(15.8)
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Aerodynamics
Figure 15.15 Transformation of a circle into a cambered airfoil.
Figure 15.16 Shapes of the trailing edge: (a) trailing edge with finite angle; and (b) cusped trailing edge.
Here tmax is the maximum thickness, which is reached nearly at the quarter chord
position x = −b. The “camber,” defined in Figure 15.6, is indicated in Figure 15.15.
Such airfoils generated from the Zhukhovsky transformation are called
Zhukhovsky airfoils. They have the property that the trailing edge is a cusp, which
means that the upper and lower surfaces are tangent to each other at the trailing
edge. Without the Kutta condition, the trailing edge is a point of infinite velocity,
as discussed in Section 5. If the trailing edge angle is nonzero (Figure 15.16a), the
coincidence of the stagnation point with the point of infinite velocity still makes the
trailing edge a stagnation point, because of the following argument: The fluid velocity
on the upper and lower surfaces is parallel to its respective surface. At the trailing
edge this leads to normal velocities in different directions, which cannot be possible.
The velocities on both sides of the airfoil must therefore be zero at the trailing edge.
This is not true for the cusped trailing edge of a Zhukhovsky airfoil (Figure 15.16b).
In that case the tangents to the upper and lower surfaces coincide at the trailing edge,
and the fluid leaves the trailing edge smoothly. The trailing edge for the Zhukhovsky
airfoil is simply an ordinary point where the velocity is neither zero nor infinite.
8. Lift of Zhukhovsky Airfoil
The preceding section has shown how a circle is transformed into an airfoil with
the help of the Zhukhovsky transformation. We are now going to determine certain
flow properties of such an airfoil. Consider flow around the circle with clockwise
circulation Ŵ in the ζ -plane, in which the approach velocity is inclined at an angle α
with the ξ -axis (Figure 15.17). The corresponding pattern in the z-plane is the flow
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8. Lift of Zhukhovsky Airfoil
Figure 15.17 Transformation of flow around a circle into flow around an airfoil.
around an airfoil with circulation Ŵ and angle of attack α. It can be shown that the
circulation does not change during a conformal transformation. If w = φ + iψ is the
complex potential, then the velocities in the two planes are related by
dw
dw dζ
=
.
dz
dζ dz
Using the Zhukhovsky transformation (15.3), this becomes
dw
dw ζ 2
.
=
dz
dζ ζ 2 − b2
(15.9)
Here dw/dz = u − iv is the complex velocity in the z-plane, and dw/dζ is the
complex velocity in the ζ -plane. Equation (15.9) shows that the velocities in the two
planes become equal as ζ → ∞, which means that the free-stream velocities are
inclined at the same angle α in the two planes.
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Aerodynamics
Point B with coordinates (b, 0) in the ζ -plane is transformed into the trailing
edge B′ of the airfoil. Because ζ 2 − b2 vanishes there, it follows from equation (15.9)
that the velocity at the trailing edge will in general be infinite. If, however, we arrange
that B is a stagnation point in the ζ -plane at which dw/dζ = 0, then dw/dz at the
trailing edge will have the 0/0 form. Our discussion of Figure 15.16b has shown that
this will in fact result in a finite velocity at B′ .
From equation (6.39), the tangential velocity at the surface of the cylinder is
given by
uθ = −2U sin θ −
Ŵ
,
2π a
(15.10)
where θ is measured from the diameter CQE. At point B, we have uθ = 0 and
θ = −(α + β). Therefore equation (15.10) gives
Ŵ = 4π U a sin(α + β),
(15.11)
which is the clockwise circulation required by the Kutta condition. It shows that the
circulation around an airfoil depends on the speed U , the chord length c (≃ 4a), the
angle of attack α, and the camber/chord ratio β/2. The coefficient of lift is
CL =
L
≃ 2π(α + β),
(1/2)ρU 2 c
(15.12)
where we have used 4a ≃ c, L = ρU Ŵ, and sin(α + β) ≃ (α + β) for small angles
of attack. Equation (15.12) shows that the lift can be increased by adding a certain
amount of camber. The lift is zero at a negative angle of attack α = −β, so that the
angle (α + β) can be called the “absolute” angle of attack. The fact that the lift of an
airfoil is proportional to the angle of attack is important, as it suggests that the pilot
can control the lift simply by adjusting the attitude of the airfoil.
A comparison of the theoretical lift equation (15.12) with typical experimental
results on a Zhukhovsky airfoil is shown in Figure 15.18. The small disagreement
can be attributed to the finite thickness of the boundary layer changing the effective
shape of the airfoil. The sudden drop of the lift at (α + β) ≃ 20◦ is due to a severe
boundary layer separation, at which point the airfoil is said to stall. This is discussed
in Section 12.
Zhukhovsky airfoils are not practical for two basic reasons. First, they demand a
cusped trailing edge, which cannot be practically constructed or maintained. Second,
the camber line in a Zhukhovsky airfoil is nearly a circular arc, and therefore the
maximum camber lies close to the center of the chord. However, a maximum camber
within the forward portion of the chord is usually preferred so as to obtain a desirable
pressure distribution. To get around these difficulties, other families of airfoils have
been generated from circles by means of more complicated transformations. Nevertheless, the results for a Zhukhovsky airfoil given here have considerable application
as reference values.
9. Wing of Finite Span
Figure 15.18 Comparison of theoretical and experimental lift coefficients for a cambered Zhukhovsky
airfoil.
9. Wing of Finite Span
So far we have considered only two-dimensional flows around wings of infinite span.
We shall now consider wings of finite span and examine how the lift and drag are
modified. Figure 15.19 shows a schematic view of a wing, looking downstream from
the aircraft. As the pressure on the lower surface of the wing is greater than that on
the upper surface, air flows around the wing tips from the lower into the upper side.
Therefore, there is a spanwise component of velocity toward the wing tip on the underside of the wing and toward the center on the upper side, as shown by the streamlines
in Figure 15.20a. The spanwise momentum continues as the fluid goes over the wing
and into the wake downstream of the trailing edge. On the stream surface extending
downstream from the wing, therefore, the lateral component of the flow is outward
(toward the wing tips) on the underside and inward on the upper side. On this surface,
then, there is vorticity with axes oriented in the streamwise direction. The vortices
have opposite signs on the two sides of the central axis OQ. The streamwise vortex
filaments downstream of the wing are called trailing vortices, which form a vortex
sheet (Figure 15.20b). As discussed in Chapter 5, Section 9, a vortex sheet is composed of closely spaced vortex filaments and generates a discontinuity in tangential
velocity.
Downstream of the wing the vortex sheet rolls up into two distinct vortices, which
are called tip or trailing vortices. The circulation around each of the tip vortices is
equal to Ŵ0 , the circulation at the center of the wing (Figure 15.21). The existence
of the tip vortices becomes visually evident when an aircraft flies in humid air. The
decreased pressure (due to the high velocity) and temperature in the core of the tip
vortices often cause atmospheric moisture to condense into droplets, which are seen
in the form of vapor trails extending for kilometers across the sky.
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Aerodynamics
Figure 15.19 Flow around wind tips.
Figure 15.20 Flow over a wing of finite span: (a) top view of streamline patterns on the upper and lower
surfaces of the wing; and (b) cross section of trailing vortices behind the wing.
Figure 15.21 Rolling up of trailing vortices to form tip vortices.
One of Helmholtz’s vortex theorems states that a vortex filament cannot end in
the fluid, but must either end at a solid surface or form a closed loop or “vortex ring.”
In the case of the finite wing, the tip vortices start at the wing and are joined together
at the other end by the starting vortices. The starting vortices are left behind at the
point where the aircraft took off, and some of them may be left where the angle of
attack was last changed. In any case, they are usually so far behind the wing that
10. Lifting Line Theory of Prandtl and Lanchester
their effect on the wing may be neglected, and the tip vortices may be regarded as
extending to an infinite distance behind the wing.
As the aircraft proceeds the tip vortices get longer, which means that kinetic
energy is being constantly supplied to generate the vortices. It follows that an additional drag force is experienced by a wing of finite span. This is called the induced
drag, which is explored in the following section.
10. Lifting Line Theory of Prandtl and Lanchester
In this section we shall formalize the concepts presented in the preceding section and
derive an expression for the lift and induced drag of a wing of finite span. The basic
assumption of the theory is that the value of the aspect ratio span/chord is large,
so that the flow around a section is approximately two dimensional. Although a
formal mathematical account of the theory was first published by Prandtl, many of
the important underlying ideas were first conceived by Lanchester. The historical
controversy regarding the credit for the theory is noted at the end of the section.
Bound and Trailing Vortices
It is known that a vortex, like an airfoil, experiences a lift force when placed in a
uniform stream. In fact, the disturbance created by an airfoil in a uniform stream is in
many ways similar to that created by a vortex filament. It therefore follows that a wing
can be replaced by a vortex, with its axis parallel to the wing span. This hypothetical
vortex filament replacing the wing is called the bound vortex, “bound” signifying that
it moves with the wing. We say that the bound vortex is located on a lifting line, which
is the core of the wing. Recall the discussion in Section 7 where the camber line was
replaced by a vortex sheet in thin airfoil theory. This sheet may be regarded as the
bound vorticity. According to one of the Helmholtz theorems (Chapter 5, Section 4),
a vortex cannot begin or end in the fluid; it must end at a wall or form a closed loop.
The bound vortex therefore bends downstream and forms the trailing vortices.
The strength of the circulation around the wing varies along the span, being
maximum at the center and zero at the wing tips. A relation can be derived between
the distribution of circulation along the wing span and the strength of the trailing
vortex filaments. Suppose that the clockwise circulation of the bound vortex changes
from Ŵ to Ŵ − dŴ at a certain point (Figure 15.22a). Then another vortex AC of
strength dŴ must emerge from the location of the change. In fact, the strength and
sign of the circulation around AC is such that, when AC is folded back onto AB, the
circulation is uniform along the composite vortex tube. (Recall the vortex theorem of
Helmholtz, which says that the strength of a vortex tube is constant along its length.)
Now consider the circulation distribution Ŵ(y) over a wing (Figure 15.22b). The
change in circulation in length dy is dŴ, which is a decrease if dy > 0. It follows
that the magnitude of the trailing vortex filament of width dy is
−
dŴ
dy,
dy
The trailing vortices will be stronger near the wing tips where dŴ/dy is the largest.
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Aerodynamics
Figure 15.22 Lifting line theory: (a) change of vortex strength; and (b) nomenclature.
Downwash
Let us determine the velocity induced at a point y1 on the lifting line by the trailing
vortex sheet. Consider a semi-infinite trailing vortex filament, whose one end is at the
lifting line. Such a vortex of width dy, having a strength −(dŴ/dy) dy, will induce
a downward velocity of magnitude
−(dŴ/dy) dy
.
4π(y − y1 )
dw(y1 ) =
Note that this is half the velocity induced by an infinitely long vortex, which equals
(circulation)/(2πr) where r is the distance from the axis of the vortex. The bound
vortex makes no contribution to the velocity induced at the lifting line itself.
The total downward velocity at y1 due to the entire vortex sheet is therefore
w(y1 ) =
1
4π
s/2
−s/2
dŴ dy
,
dy (y1 − y)
(15.13)
which is called the downwash at y1 on the lifting line of the wing. The vortex sheet
also induces a smaller downward velocity in front of the airfoil and a larger one behind
the airfoil (Figure 15.23).
The effective incident flow on any element of the wing is the resultant of U and w
(Figure 15.24). The downwash therefore changes the attitude of the airfoil, decreasing
the “geometrical angle of attack” α by the angle
ε = tan
w
w
≃ ,
U
U
so that the effective angle of attack is
αe = α − ε = α −
w
.
U
(15.14)
Because the aspect ratio is assumed large, ε is small. Each element dy of the finite
wing may then be assumed to act as though it is an isolated two-dimensional section
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10. Lifting Line Theory of Prandtl and Lanchester
Figure 15.23 Variation of downwash ahead of and behind an airfoil.
Figure 15.24 Lift and induced drag on a wing element dy.
set in a stream of uniform velocity Ue , at an angle of attack αe . According to the
Kutta–Zhukhovsky lift theorem, a circulation Ŵ superimposed on the actual resultant
velocity Ue generates an elementary aerodynamic force dLe = ρUe Ŵ dy, which acts
normal to Ue . This force may be resolved into two components, the conventional
lift force dL normal to the direction of flight and a component dDi parallel to the
direction of flight (Figure 15.24). Therefore
dL = dLe cos ε = ρUe Ŵ dy cos ε ≃ ρU Ŵ dy,
dDi = dLe sin ε = ρUe Ŵ dy sin ε ≃ ρwŴ dy.
In general w, Ŵ, Ue , ε, and αe are all functions of y, so that for the entire wing
s/2
L=
ρU Ŵ dy,
−s/2
s/2
Di =
(15.15)
ρwŴ dy.
−s/2
These expressions have a simple interpretation: Whereas the interaction of U and Ŵ
generates L, which acts normal to U , the interaction of w and Ŵ generates Di , which
acts normal to w.
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Aerodynamics
Induced Drag
The drag force Di induced by the trailing vortices is called the induced drag, which is
zero for an airfoil of infinite span. It arises because a wing of finite span continuously
creates trailing vortices and the rate of generation of the kinetic energy of the vortices
must equal the rate of work done against the induced drag, namely Di U . For this reason
the induced drag is also known as the vortex drag. It is analogous to the wave drag
experienced by a ship, which continuously radiates gravity waves during its motion.
As we shall see, the induced drag is the largest part of the total drag experienced by
an airfoil.
A basic reason why there must be a downward velocity behind the wing is the
following: The fluid exerts an upward lift force on the wing, and therefore the wing
exerts a downward force on the fluid. The fluid must therefore constantly gain downward momentum as it goes past the wing. (See the photograph of the spinning baseball
(Figure 10.27), which exerts an upward force on the fluid.)
For a given Ŵ(y), it is apparent that w(y) can be determined from equation (15.13)
and Di can then be determined from equation (15.15). However, Ŵ(y) itself depends
on the distribution of w(y), essentially because the effective angle of attack is changed
due to w(y). To see how Ŵ(y) may be estimated, first note that the lift coefficient for
a two-dimensional Zhukhovsky airfoil is nearly CL = 2π(α + β). For a finite wing
we may assume
w(y)
+ β(y) ,
(15.16)
CL = K α −
U
where (α −w/U ) is the effective angle of attack, −β(y) is the angle of attack for zero
lift (found from experimental data such as Figure 15.18), and K is a constant whose
value is nearly 6 for most airfoils. (K = 2π for a Zhukhovsky airfoil.) An expression
for the circulation can be obtained by noting that the lift coefficient is related to the
circulation as CL ≡ L/( 21 ρU 2 c) = Ŵ/( 21 U c), so that Ŵ = 21 U cCL . The assumption
equation (15.16) is then equivalent to the assumption that the circulation for a wing
of finite span is
K
w(y)
Ŵ(y) = U c(y) α −
+ β(y) .
(15.17)
2
U
For a given U , α, c(y), and β(y), equations (15.13) and (15.17) define an integral
equation for determining Ŵ(y). (An integral equation is one in which the unknown
function appears under an integral sign.) The problem can be solved numerically by
iterative techniques. Instead of pursuing this approach, in the next section we shall
assume that Ŵ(y) is given.
Lanchester versus Prandtl
There is some controversy in the literature about who should get more credit for
developing modern wing theory. Since Prandtl in 1918 first published the theory in
a mathematical form, textbooks for a long time have called it the “Prandtl Lifting
Line Theory.” Lanchester was bitter about this, because he felt that his contributions
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11. Results for Elliptic Circulation Distribution
were not adequately recognized. The controversy has been discussed by von Karman
(1954, p. 50), who witnessed the development of the theory. He gives a lot of credit to
Lanchester, but falls short of accusing his teacher Prandtl of being deliberately unfair.
Here we shall note a few facts that von Karman brings up.
Lanchester was the first person to study a wing of finite span. He was also the
first person to conceive that a wing can be replaced by a bound vortex, which bends
backward to form the tip vortices. Last, Lanchester was the first to recognize that
the minimum power necessary to fly is that required to generate the kinetic energy
field of the downwash field. It seems, then, that Lanchester had conceived all of the
basic ideas of the wing theory, which he published in 1907 in the form of a book
called “Aerodynamics.” In fact, a figure from his book looks very similar to our
Figure 15.21.
Many of these ideas were explained by Lanchester in his talk at Göttingen, long
before Prandtl published his theory. Prandtl, his graduate student von Karman, and
Carl Runge were all present. Runge, well-known for his numerical integration scheme
of ordinary differential equations, served as an interpreter, because neither Lanchester
nor Prandtl could speak the other’s language. As von Karman said, “both Prandtl and
Runge learned very much from these discussions.”
However, Prandtl did not want to recognize Lanchester for priority of ideas,
saying that he conceived of them before he saw Lanchester’s book. Such controversies
cannot be settled. And great men have been involved in controversies before. For
example, astrophysicist Stephen Hawking (1988), who occupied Newton’s chair at
Cambridge (after Lighthill), described Newton to be a rather mean man who spent
much of his later years in unfair attempts at discrediting Leibniz, in trying to force
the Royal astronomer to release some unpublished data that he needed to verify his
predictions, and in heated disputes with his lifelong nemesis Robert Hooke.
In view of the fact that Lanchester’s book was already in print when Prandtl published his theory, and the fact that Lanchester had all the ideas but not a formal mathematical theory, we have called it the “Lifting Line Theory of Prandtl and Lanchester.”
11. Results for Elliptic Circulation Distribution
The induced drag and other properties of a finite wing depend on the distribution of
Ŵ(y). The circulation distribution, however, depends in a complicated way on the
wing planform, angle of attack, and so on. It can be shown that, for a given total lift
and wing area, the induced drag is a minimum when the circulation distribution is
elliptic. (See, for e.g., Ashley and Landahl, 1965, for a proof.) Here we shall simply
assume an elliptic distribution of the form (see Figure 15.22b)
Ŵ = Ŵ0 1 −
2y
s
2 1/2
,
and determine the resulting expressions for downwash and induced drag.
(15.18)
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Aerodynamics
The total lift force on a wing is then
L=
s/2
−s/2
π
ρU Ŵ0 s.
4
ρU Ŵ dy =
(15.19)
To determine the downwash, we first find the derivative of equation (15.18):
4Ŵ0 y
dŴ
=−
.
dy
s s 2 − 4y 2
From equation (15.13), the downwash at y1 is
w(y1 ) =
1
4π
s/2
−s/2
dŴ dy
Ŵ0
=
dy y1 − y
πs
s/2
−s/2
y dy
.
(y − y1 ) s 2 − 4y 2
Writing y = (y − y1 ) + y1 in the numerator, we obtain
Ŵ0
w(y1 ) =
πs
s/2
−s/2
dy
s 2 − 4y 2
+ y1
s/2
−s/2
dy
.
(y − y1 ) s 2 − 4y 2
The first integral has the value π/2. The second integral can be reduced to a standard
form (listed in any mathematical handbook) by substituting x = y − y1 . On setting
limits the second integral turns out to be zero, although the integrand is not an odd
function. The downwash at y1 is therefore
w(y1 ) =
Ŵ0
,
2s
(15.20)
which shows that, for an elliptic circulation distribution, the induced velocity at the
wing is constant along the span.
Using equations (15.18) and (15.20), the induced drag is found as
Di =
s/2
−s/2
ρwŴ dy =
π
ρŴ 2 .
8 0
In terms of the lift equation (15.19), this becomes
Di =
2L2
,
ρU 2 π s 2
which can be written as
CD i =
CL2
,
π
(15.21)
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11. Results for Elliptic Circulation Distribution
where we have defined the coefficients (here A is the wing planform area)
≡
CDi ≡
s2
= aspect ratio
A
Di
,
(1/2)ρU 2 A
CL ≡
L
.
(1/2)ρU 2 A
Equation (15.21) shows that CDi → 0 in the two-dimensional limit → ∞. More
important, it shows that the induced drag coefficient increases as the square of the
lift coefficient. We shall see in the following section that the induced drag generally
makes the largest contribution to the total drag of an airfoil.
Since an elliptic circulation distribution minimizes the induced drag, it is of interest to determine the circumstances under which such a circulation can be established.
Consider an element dy of the wing (Figure 15.25). The lift on the element is
dL = ρU Ŵ dy = CL 21 ρU 2 c dy,
(15.22)
where c dy is an elementary wing area. Now if the circulation distribution is elliptic,
then the downwash is independent of y. In addition, if the wing profile is geometrically similar at every point along the span and has the same geometrical angle of
attack α, then the effective angle of attack and hence the lift coefficient CL will be
independent of y. Equation (15.22) shows that the chord length c is then simply proportional to Ŵ, and so c(y) is also elliptically distributed. Thus, an untwisted wing
with elliptic planform, or composed of two semiellipses (Figure 15.25), will generate
an elliptic circulation distribution. However, the same effect can also be achieved with
nonelliptic planforms if the angle of attack varies along the span, that is, if the wing
is given a “twist.”
Figure 15.25 Wing of elliptic planform.
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Aerodynamics
12. Lift and Drag Characteristics of Airfoils
Before an aircraft is built its wings are tested in a wind tunnel, and the results are
generally given as plots of CL and CD vs the angle of attack. A typical plot is shown in
Figure 15.26. It is seen that, in a range of incidence angle from α = −4◦ to α = 12◦ ,
the variation of CL with α is approximately linear, a typical value of dCL /dα being
≈0.1 per degree. The lift reaches a maximum value at an incidence of ≈15◦ . If the
angle of attack is increased further, the steep adverse pressure gradient on the upper
surface of the airfoil causes the flow to separate nearly at the leading edge, and a very
large wake is formed (Figure 15.27). The lift coefficient drops suddenly, and the wing
is said to stall. Beyond the stalling incidence the lift coefficient levels off again and
remains at ≈0.7–0.8 for fairly large angles of incidence.
The maximum lift coefficient depends largely on the Reynolds number Re. At
lower values of Re ∼ 105 –106 , the flow separates before the boundary layer undergoes
Figure 15.26 Lift and drag coefficients vs angle of attack.
Figure 15.27 Stalling of an airfoil.
12. Lift and Drag Characteristics of Airfoils
transition, and a very large wake is formed. This gives maximum lift coefficients <0.9.
At larger Reynolds numbers, say Re > 107 , the boundary layer undergoes transition
to turbulent flow before it separates. This produces a somewhat smaller wake, and
maximum lift coefficients of ≈1.4 are obtained.
The angle of attack at zero lift, denoted by −β here, is a function of the section
camber. (For a Zhukhovsky airfoil, β = 2(camber)/chord.) The effect of increasing
the airfoil camber is to raise the entire graph of CL vs α, thus increasing the maximum
values of CL without stalling. A cambered profile delays stalling essentially because
its leading edge points into the airstream while the rest of the airfoil is inclined to the
stream. Rounding the airfoil nose is very helpful, for an airfoil of zero thickness would
undergo separation at the leading edge. Trailing edge flaps act to increase the camber
when they are deployed. Then the maximum lift coefficient is increased, allowing for
lower landing speeds.
Various terms are in common usage to describe the different components of the
drag. The total drag of a body can be divided into a friction drag due to the tangential
stresses on the surface and pressure drag due to the normal stresses. The pressure
drag can be further subdivided into an induced drag and a form drag. The induced
drag is the “drag due to lift” and results from the work done by the body to supply
the kinetic energy of the downwash field as the trailing vortices increase in length.
The form drag is defined as the part of the total pressure drag that remains after
the induced drag is subtracted out. (Sometimes the skin friction and form drags are
grouped together and called the profile drag, which represents the drag due to the
“profile” alone and not due to the finiteness of the wing.) The form drag depends
strongly on the shape and orientation of the airfoil and can be minimized by good
design. In contrast, relatively little can be done about the induced drag if the aspect
ratio is fixed.
Normally the induced drag constitutes the major part of the total drag of a wing.
As CDi is nearly proportional to CL2 , and CL is nearly proportional to α, it follows
that CDi ∝ α 2 . This is why the drag coefficient in Figure 15.26 seems to increase
quadratically with incidence.
For high-speed aircraft, the appearance of shock waves can adversely affect the
behavior of the lift and drag characteristics. In such cases the maximum flow speeds
can be close to or higher than the speed of sound even when the aircraft is flying at subsonic speeds. Shock waves can form when the local flow speed exceeds
the local speed of sound. To reduce their effect, the wings are given a sweepback
angle, as shown in Figure 15.2. The maximum flow speeds depend primarily on the
component of the oncoming stream perpendicular to the leading edge; this component is reduced as a result of the sweepback. As a result, increased flight speeds are
achievable with highly swept wings. This is particularly true when the aircraft flies
at supersonic speeds, in which there is invariably a shock wave in front of the nose
of the fuselage, extending downstream in the form of a cone. Highly swept wings
are then used in order that the wing does not penetrate this shock wave. For flight
speeds exceeding Mach numbers of order 2, the wings have such large sweepback
angles that they resemble the Greek letter ; these wings are sometimes called delta
wings.
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Aerodynamics
13. Propulsive Mechanisms of Fish and Birds
The propulsive mechanisms of many animals utilize the aerodynamic principle of lift
generation on winglike surfaces. We shall now describe some of the basic ideas of
this interesting subject, which is discussed in more detail by Lighthill (1986).
Locomotion of Fish
First consider the case of a fish. It develops a forward thrust by horizontally oscillating
its tail from side to side. The tail has a cross section resembling that of a symmetric
airfoil (Figure 15.28a). One-half of the oscillation is represented in Figure 15.28b,
which shows the top view of the tail. The sequence 1 to 5 represents the positions of
the tail during the tail’s motion to the left. A quick change of orientation occurs at one
extreme position of the oscillation during 1 to 2; the tail then moves to the left during 2
to 4, and another quick change of orientation occurs at the other extreme during 4 to 5.
Suppose the tail is moving to the left at speed V , and the fish is moving forward
at speed U . The fish controls these magnitudes so that the resultant fluid velocity Ur
(relative to the tail) is inclined to the tail surface at a positive “angle of attack.” The
resulting lift L is perpendicular to Ur and has a forward component L sin θ. (It is easy
to verify that there is a similar forward propulsive force when the tail moves from left
to right.) This thrust, working at the rate U L sin θ , propels the fish. To achieve this
propulsion, the tail of the fish pushes sideways on the water against a force of L cos θ,
which requires work at the rate V L cos θ. As V /U = tan θ , ideally the conversion
of energy is perfect—all of the oscillatory work done by the fish tail goes into the
translational mode. In practice, however, this is not the case because of the presence
of induced drag and other effects that generate a wake.
Most fish stay afloat by controlling the buoyancy of a swim bladder inside their
stomach. In contrast, some large marine mammals such as whales and dolphins
Figure 15.28 Propulsion of fish. (a) Cross section of the tail along AA is a symmetric airfoil. Five
positions of the tail during its motion to the left are shown in (b). The lift force L is normal to the resultant
speed Ur of water with respect to the tail.
13. Propulsive Mechanisms of Fish and Birds
develop both a forward thrust and a vertical lift by moving their tails vertically.
They are able to do this because their tail surface is horizontal, in contrast to the vertical tail shown in Figure 15.28. A recent review by Fish and Lauder (2006) provided
evidence that leading edge tubercles as seen on humpback whale flippers increase lift
and reduce drag at high angles of attack. This is because separation is delayed due
to the creation of streamwise vortices on the suction side. Cetacean flukes or flippers
and fish tail fins as well as dorsal and pectoral fins are flexible and can vary their
camber during a stroke. As a result they are very efficient propulsive devices.
Flight of Birds and Insects
Now consider the flight of birds, who flap their wings to generate both the lift to
support their body weight and the forward thrust to overcome the drag. Figure 15.29
shows a vertical section of the wing positions during the upstroke and downstroke
of the wing. (Birds have cambered wings, but this is not shown in the figure.) The
angle of inclination of the wing with the airstream changes suddenly at the end of each
stroke, as shown. The important point is that the upstroke is inclined at a greater angle
to the airstream than the downstroke. As the figure shows, the downstroke develops a
lift force L perpendicular to the resultant velocity of the air relative to the wing. Both
Figure 15.29 Propulsion of a bird. A cross section of the wing is shown during upstroke and downstroke.
During the downstroke, a lift force L acts normal to the resultant speed Ur of air with respect to the wing.
During the upstroke, Ur is nearly parallel to the wing and very little aerodynamic force is generated.
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Aerodynamics
a forward thrust and an upward force result from the downstroke. In contrast, very
little aerodynamic force is developed during the upstroke, as the resultant velocity
is then nearly parallel to the wing. Birds therefore do most of the work during the
downstroke, and the upstroke is “easy.”
A recent study by Liu et al. (2006) provides the most complete description to date
of wing planform, camber, airfoil section, and spanwise twist distribution of seagulls,
mergansers, teals, and owls. Moreover, flapping as viewed by video images from free
flight was digitized and modeled by a two-jointed wing at the quarter chord point.
The data from this paper can be used to model the aerodynamics of bird flight.
Using previously measured kinematics and experiments on an approximately
100X upscaled model, Ramamurti and Sandberg (2001) calculated the flow about a
Drosophila (fruit fly) in flight. They matched Reynolds number (based on wing-tip
speed and average chord) and found that viscosity had negligible effect on thrust and
drag at a flight Reynolds number of 120. The wings were near elliptical plates with
axis ratio 3:1.2 and thickness about 1/80 of the span. Averaged over a cycle, the mean
thrust coefficient (thrust/[dynamic pressure × wing surface]) was 1.3 and the mean
drag coefficient close to 1.5.
14. Sailing against the Wind
People have sailed without the aid of an engine for thousands of years and have
known how to arrive at a destination against the wind. Actually, it is not possible
to sail exactly against the wind, but it is possible to sail at ≈40–45◦ to the wind.
Figure 15.30 shows how this is made possible by the aerodynamic lift on the sail,
which is a piece of large stretched cloth. The wind speed is U , and the sailing speed
is V , so that the apparent wind speed relative to the boat is Ur . If the sail is properly
oriented, this gives rise to a lift force perpendicular to Ur and a drag force parallel to
Ur . The resultant force F can be resolved into a driving component (thrust) along the
motion of the boat and a lateral component. The driving component performs work
in moving the boat; most of this work goes into overcoming the frictional drag and
in generating the gravity waves that radiate outward. The lateral component does not
cause much sideways drift because of the shape of the hull. It is clear that the thrust
decreases as the angle θ decreases and normally vanishes when θ is ≈40–45◦ . The
energy for sailing comes from the wind field, which loses kinetic energy after passing
through the sail.
In the foregoing discussion we have not considered the hydrodynamic forces
exerted by the water on the hull. At constant sailing speed the net hydrodynamic force
must be equal and opposite to the net aerodynamic force on the sail. The hydrodynamic
force can be decomposed into a drag (parallel to the direction of motion) and a
lift. The lift is provided by the “keel,” which is a thin vertical surface extending
downward from the bottom of the hull. For the keel to act as a lifting surface, the
longitudinal axis of the boat points at a small angle to the direction of motion of the
boat, as indicated near the bottom right part of Figure 15.30. This “angle of attack”
is generally <3◦ and is not noticeable. The hydrodynamic lift developed by the keel
opposes the aerodynamic lateral force on the sail. It is clear that without the keel the
lateral aerodynamic force on the sail would topple the boat around its longitudinal axis.
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Exercises
Figure 15.30 Principle of a sailboat.
To arrive at a destination directly against the wind, one has to sail in a zig-zag
path, always maintaining an angle of ≈45◦ to the wind. For example, if the wind is
coming from the east, we can first proceed northeastward as shown, then change the
orientation of the sail to proceed southeastward, and so on. In practice, a combination
of a number of sails is used for effective maneuvering. The mechanics of sailing
yachts is discussed in Herreshoff and Newman (1966).
Exercises
1. Consider an airfoil section in the xy-plane, the x-axis being aligned with the
chordline. Examine the pressure forces on an element ds = (dx, dy) on the surface,
and show that the net force (per unit span) in the y-direction is
c
c
pl dx,
pu dx +
Fy = −
0
0
where pu and pl are the pressures on the upper and the lower surfaces and c is the
chord length. Show that this relation can be rearranged in the form
Cy ≡
Fy
=
(1/2)ρU 2 c
Cp d
x
,
c
where Cp ≡ (p − p∞ )/( 21 ρU 2 ), and the integral represents the area enclosed in a
Cp vs x/c diagram, such as Figure 15.8. Neglect shear stresses. [Note that Cy is not
exactly the lift coefficient, since the airstream is inclined at a small angle α with the
x-axis.]
710
Aerodynamics
2. The measured pressure distribution over a section of a two-dimensional airfoil
at 4◦ incidence has the following form:
Upper Surface: Cp is constant at −0.8 from the leading edge to a distance
equal to 60% of chord and then increases linearly to 0.1 at the trailing edge.
Lower Surface: Cp is constant at −0.4 from the leading edge to a distance
equal to 60% of chord and then increases linearly to 0.1 at the trailing edge.
Using the results of Exercise 1, show that the lift coefficient is nearly 0.32.
3. The Zhukhovsky transformation z = ζ + b2 /ζ transforms a circle of radius
b, centered at the origin of the ζ -plane, into a flat plate of length 4b in the z-plane.
The circulation around the cylinder is such that the Kutta condition is satisfied at the
trailing edge of the flat plate. If the plate is inclined at an angle α to a uniform stream
U , show that
(i) The complex velocity in the ζ -plane is
1
iŴ
ln (ζ e−iα ),
w = U ζ e−iα + b2 eiα +
ζ
2π
where Ŵ = 4πU b sin α. Note that this represents flow over a circular cylinder
with circulation, in which the oncoming velocity is oriented at an angle α.
(ii) The velocity components at point P (−2b, 0) in the ζ -plane are [ 43 U cos α,
9
4 U sin α].
(iii) The coordinates of the transformed point P′ in the xy-plane are [−5b/2, 0].
(iv) The velocity components at [−5b/2, 0] in the xy-plane are [U cos α, 3U sin α].
4. In Figure 15.13, the angle at A′ has been marked 2β. Prove this. [Hint : Locate
the center of the circular arc in the z-plane.]
5. Consider a cambered Zhukhovsky airfoil determined by the following
parameters:
a = 1.1,
b = 1.0,
β = 0.1.
Using a computer, plot its contour by evaluating the Zhukhovsky transformation. Also
plot a few streamlines, assuming an angle of attack of 5◦ .
6. A thin Zhukhovsky airfoil has a lift coefficient of 0.3 at zero incidence. What
is the lift coefficient at 5◦ incidence?
7. An untwisted elliptic wing of 20-m span supports a weight of 80,000 N in a
level flight at 300 km/hr. Assuming sea level conditions, find (i) the induced drag and
(ii) the circulation around sections halfway along each wing.
Literature Cited
8. The circulation across the span of a wing follows the parabolic law
4y 2
Ŵ = Ŵ0 1 − 2
s
Calculate the induced velocity w at midspan, and compare the value with that obtained
when the distribution is elliptic.
Literature Cited
Anderson, John D., Jr. (1998). A History of Aerodynamics, London: Cambridge University Press.
Anderson, John D., Jr. (2007). Fundamentals of Aerodynamics, New York: McGraw-Hill.
Ashley, H. and M. Landahl (1965). Aerodynamics of Wings and Bodies, Reading, MA: Addison-Wesley.
Fish, F. E. and G. V. Lauder (2006). “Passive and Active Control by Swimming Fishes and Mammals.”
Annual Rev. Fluid Mech. 38: 193–224.
Hawking, S. W. (1988). A Brief History of Time, New York: Bantam Books.
Herreshoff, H. C. and J. N. Newman (1986). “The study of sailing yachts.” Scientific American 215 (August
issue): 61–68.
von Karman, T. (1954). Aerodynamics, New York: McGraw-Hill. (A delightful little book, written for the
nonspecialist, full of historical anecdotes and at the same time explaining aerodynamics in the easiest
way.)
Kuethe, A. M. and C. Y. Chow (1998). Foundations of Aerodynamics: Basis of Aerodynamic Design,
New York: Wiley.
Lighthill, M. J. (1986). An Informal Introduction to Theoretical Fluid Mechanics, Oxford, England:
Clarendon Press.
Liu, T., K. Kuykendoll, R. Rhew, and S. Jones (2006). “Avian Wing Geometry and Kinematics.” AIAA J.
44: 954–963.
Ramamurti, R. and W. C. Sandberg (2001). “Computational Study of 3-D Flapping Foil Flows.” AIAA
Paper 2001–0605.
Supplemental Reading
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.
Karamcheti, K. (1980). Principles of Ideal-Fluid Aerodynamics, Melbourne, FL: Krieger Publishing Co.
Prandtl, L. (1952). Essentials of Fluid Dynamics, London: Blackie & Sons Ltd. (This is the English edition
of the original German edition. It is very easy to understand, and much of it is still relevant today.)
Printed in New York by Hafner Publishing Co. If this is unavailable, see the following reprints in
paperback that contain much if not all of this material:
Prandtl, L. and O. G. Tietjens (1934) [original publication date]. Fundamentals of Hydro and Aeromechanics, New York: Dover Publ. Co.; and
Prandtl, L. and O. G. Tietjens (1934) [original publication date]. Applied Hydro and Aeromechanics,
New York: Dover Publ. Co. This contains many original flow photographs from Prandtl’s laboratory.
711
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Chapter 16
Compressible Flow
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 713
Criterion for Neglect of Compressibility
Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
Classification of Compressible
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
Useful Thermodynamic Relations . . . . 716
2. Speed of Sound . . . . . . . . . . . . . . . . . . . . . . 717
3. Basic Equations for
One-Dimensional Flow . . . . . . . . . . . . . . 721
Continuity Equation. . . . . . . . . . . . . . . . . 721
Energy Equation . . . . . . . . . . . . . . . . . . . . 721
Bernoulli and Euler
Equations . . . . . . . . . . . . . . . . . . . . . . . . 723
Momentum Principle for a Control
Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 723
4. Stagnation and Sonic Properties . . . . . 724
Table 16.1: Isentropic Flow of
a Perfect Gas (γ = 1.4) . . . . . . . . . . . . 727
5. Area–Velocity Relations in
One-Dimensional Isentropic Flow . . . . 729
Example 16.1 . . . . . . . . . . . . . . . . . . . . . . . 732
6. Normal Shock Wave . . . . . . . . . . . . . . . . . 733
Normal Shock Propagating in a Still
Medium . . . . . . . . . . . . . . . . . . . . . . . . . . 738
Shock Structure . . . . . . . . . . . . . . . . . . . . 738
7. Operation of Nozzles at Different
Back Pressures . . . . . . . . . . . . . . . . . . . . . 741
Convergent Nozzle . . . . . . . . . . . . . . . . . . 741
Convergent–Divergent Nozzle . . . . . . . 741
Example 16.2 . . . . . . . . . . . . . . . . . . . . . . 742
Table 16.2: One-Dimensional Normalshock Relations (γ = 1.4) . . . . . . . . 745
8. Effects of Friction and Heating in
Constant-Area Ducts . . . . . . . . . . . . . . . 747
Effect of Friction . . . . . . . . . . . . . . . . . . . 748
Effect of Heat Transfer . . . . . . . . . . . . . 749
Choking by Friction or Heat
Addition . . . . . . . . . . . . . . . . . . . . . . . . 750
9. Mach Cone . . . . . . . . . . . . . . . . . . . . . . . . . 750
10. Oblique Shock Wave . . . . . . . . . . . . . . . . 752
Generation of Oblique Shock
Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
The Weak Shock Limit . . . . . . . . . . . . . 756
11. Expansion and Compression in
Supersonic Flow. . . . . . . . . . . . . . . . . . . . 756
12. Thin Airfoil Theory in Supersonic
Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
Literature Cited . . . . . . . . . . . . . . . . . . . . 763
Supplemental Reading . . . . . . . . . . . . . . 763
1. Introduction
To this point we have neglected the effects of density variations due to pressure
changes. In this chapter we shall examine some elementary aspects of flows in which
the compressibility effects are important. The subject of compressible flows is also
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50016-2
713
714
Compressible Flow
called gas dynamics, which has wide applications in high-speed flows around objects
of engineering interest. These include external flows such as those around airplanes,
and internal flows in ducts and passages such as nozzles and diffusers used in jet
engines and rocket motors. Compressibility effects are also important in astrophysics.
Two popular books dealing with compressibility effects in engineering applications
are those by Liepmann and Roshko (1957) and Shapiro (1953), which discuss in
further detail most of the material presented here.
Our study in this chapter will be rather superficial and elementary because this
book is essentially about incompressible flows. However, this small chapter on compressible flows is added because a complete ignorance about compressibility effects
is rather unsatisfying. Several startling and fascinating phenomena arise in compressible flows (especially in the supersonic range) that go against our intuition developed
from a knowledge of incompressible flows. Discontinuities (shock waves) appear
within the flow, and a rather strange circumstance arises in which an increase of flow
area accelerates a (supersonic) stream. Friction can also make the flow go faster and
adding heat can lower the temperature in subsonic duct flows. We will see this later in
this chapter. Some understanding of these phenomena, which have no counterpart in
low-speed flows, is desirable even if the reader may not make much immediate use of
this knowledge. Except for our treatment of friction in constant area ducts, we shall
limit our study to that of frictionless flows outside boundary layers. Our study will,
however, have a great deal of practical value because the boundary layers are especially thin in high-speed flows. Gravitational effects, which are minor in high-speed
flows, will be neglected.
Criterion for Neglect of Compressibility Effects
Compressibility effects are determined by the magnitude of the Mach number
defined as
u
M≡ ,
c
where u is the speed of flow, and c is the speed of sound given by
∂p
,
c2 =
∂ρ s
where the subscript “s” signifies that the partial derivative is taken at constant entropy.
To see how large the Mach number has to be for the compressibility effects to be
appreciable in a steady flow, consider the one-dimensional steady flow version of the
continuity equation ∇ · (ρu) = 0, that is,
u
∂u
∂ρ
+ρ
= 0.
∂x
∂x
The incompressibility assumption requires that
u
∂u
∂ρ
≪ρ
∂x
∂x
715
1. Introduction
or that
δu
δρ
≪
.
ρ
u
(16.1)
Pressure changes can be estimated from the definition of c, giving
δp ≃ c2 δρ.
(16.2)
δp
.
ρ
(16.3)
The Euler equation requires
u δu ≃
By combining equations (16.2) and (16.3), we obtain
u2 δu
δρ
≃ 2 .
ρ
c u
From comparison with equation (16.1) we see that the density changes are negligible if
u2
= M 2 ≪ 1.
c2
The constant density assumption is therefore valid if M < 0.3, but not at higher Mach
numbers.
Although the significance of the ratio u/c was known for a long time, the Swiss
aerodynamist Jacob Ackeret introduced the term “Mach number,” just as the term
Reynolds number was introduced by Sommerfeld many years after Reynolds’ experiments. The name of the Austrian physicist Ernst Mach (1836–1916) was chosen
because of his pioneering studies on supersonic motion and his invention of the
so-called Schlieren method for optical studies of flows involving density changes;
see von Karman (1954, p. 106). (Mach distinguished himself equally well in philosophy. Einstein acknowledged that his own thoughts on relativity were influenced by
“Mach’s principle,” which states that properties of space had no independent existence but are determined by the mass distribution within it. Strangely, Mach never
accepted either the theory of relativity or the atomic structure of matter.)
Classification of Compressible Flows
Compressible flows can be classified in various ways, one of which is based on the
Mach number M. A common way of classifying flows is as follows:
(i) Incompressible flow: M < 0.3 everywhere in the flow. Density variations due
to pressure changes can be neglected. The gas medium is compressible but the
density may be regarded as constant.
(ii) Subsonic flow: M exceeds 0.3 somewhere in the flow, but does not exceed 1
anywhere. Shock waves do not appear in the flow.
716
Compressible Flow
(iii) Transonic flow: The Mach number in the flow lies in the range 0.8–1.2. Shock
waves appear and lead to a rapid increase of the drag. Analysis of transonic
flows is difficult because the governing equations are inherently nonlinear,
and also because a separation of the inviscid and viscous aspects of the flow
is often impossible. (The word “transonic” was invented by von Karman and
Hugh Dryden, although the latter argued in favor of having two s’s in the word.
von Karman (1954, p. 116) stated that “I first introduced the term in a report to
the U.S. Air Force. I am not sure whether the general who read the word knew
what it meant, but his answer contained the word, so it seemed to be officially
accepted.”)
(iv) Supersonic flow: M lies in the range 1–3. Shock waves are generally present.
In many ways analysis of a flow that is supersonic everywhere is easier than an
analysis of a subsonic or incompressible flow as we shall see. This is because
information propagates along certain directions, called characteristics, and a
determination of these directions greatly facilitates the computation of the flow
field.
(v) Hypersonic flow: M > 3. The very high flow speeds cause severe heating in
boundary layers, resulting in dissociation of molecules and other chemical
effects.
Useful Thermodynamic Relations
As density changes are accompanied by temperature changes, thermodynamic principles will be constantly used here. Most of the necessary concepts and relations have
been summarized in Sections 8 and 9 of Chapter 1, which may be reviewed before
proceeding further. Some of the most frequently used relations, valid for a perfect gas
with constant specific heats, are listed here for quick reference:
Equation of state
p = ρRT ,
Internal energy
e = Cv T ,
Enthalpy
h = Cp T ,
Specific heats
Speed of sound
Entropy change
γR
,
γ −1
R
Cv =
,
γ −1
Cp =
Cp − Cv = R,
c = γ RT ,
p2
T2
− R ln ,
T1
p1
ρ2
T2
− R ln .
= Cv ln
T1
ρ1
S2 − S1 = Cp ln
(16.4)
(16.5)
717
2. Speed of Sound
An isentropic process of a perfect gas between states 1 and 2 obeys the following
relations:
γ
p2
ρ2
,
=
p1
ρ1
γ −1 (γ −1)/γ
p2
ρ2
T2
=
.
=
T1
ρ1
p1
Some important properties of air at ordinary temperatures and pressures are
R = 287 m2 /(s2 K),
Cp = 1005 m2 /(s2 K),
Cv = 718 m2 /(s2 K),
γ = 1.4.
These values will be useful for solution of the exercises.
2. Speed of Sound
We know that a pressure pulse in an incompressible flow behaves in the same way
as that in a rigid body, where a displaced particle simultaneously displaces all the
particles in the medium. The effects of pressure or other changes are therefore instantly
felt throughout the medium. A compressible fluid, in contrast, behaves similarly to
an elastic solid, in which a displaced particle compresses and increases the density of
adjacent particles that move and increase the density of the neighboring particles, and
so on. In this way a disturbance in the form of an elastic wave, or a pressure wave,
travels through the medium. The speed of propagation is faster when the medium is
more rigid. If the amplitude of the elastic wave is infinitesimal, it is called an acoustic
wave, or a sound wave.
We shall now find an expression for the speed of propagation of sound.
Figure 16.1a shows an infinitesimal pressure pulse propagating to the left with speed c
into a still fluid. The fluid properties ahead of the wave are p, T , and ρ, while the flow
speed is u = 0. The properties behind the wave are p + dp, T + dT , and ρ + dρ,
whereas the flow speed is du directed to the left. We shall see that a “compression
wave” (for which the fluid pressure rises after the passage of the wave) must move
the fluid in the direction of propagation, as shown in Figure 16.1a. In contrast, an
“expansion wave” moves the fluid “backwards.”
To make the analysis steady, we superimpose a velocity c, directed to the right,
on the entire system (Figure 16.1b). The wave is now stationary, and the fluid enters
the wave with velocity c and leaves with a velocity c − du. Consider an area A on
the wavefront. A mass balance gives
Aρc = A(ρ + dρ)(c − du).
718
Compressible Flow
Figure 16.1 Propagation of a sound wave: (a) wave propagating into still fluid; and (b) stationary wave.
Because the amplitude is assumed small, we can neglect the second-order terms,
obtaining
du = c(dρ/ρ).
(16.6)
This shows that du > 0 if dρ is positive, thus passage of a compression wave leaves
behind a fluid moving in the direction of the wave, as shown in Figure 16.1a.
Now apply the momentum equation, which states that the net force in the
x-direction on the control volume equals the rate of outflow of x-momentum minus
the rate of inflow of x-momentum. This gives
pA − (p + dp)A = (Aρc)(c − du) − (Aρc)c,
where viscous stresses have been neglected. Here, Aρc is the mass flow rate. The first
term on the right-hand side represents the rate of outflow of x-momentum, and the
second term represents the rate of inflow of x-momentum. Simplifying the momentum
equation, we obtain
dp = ρc du.
(16.7)
Eliminating du between equations (16.6) and (16.7), we obtain
c2 =
dp
.
dρ
(16.8)
719
2. Speed of Sound
If the amplitude of the wave is infinitesimal, then each fluid particle undergoes a
nearly isentropic process as the wave passes by. The basic reason for this is that
the irreversible entropy production is proportional to the squares of the velocity and
temperature gradients (see Chapter 4, Section 15) and is therefore negligible for
weak waves. The particles do undergo small temperature changes, but the changes
are due to adiabatic expansion or compression and are not due to heat transfer from
the neighboring particles. The entropy of a fluid particle then remains constant as a
weak wave passes by. This will also be demonstrated in Section 6, where it will be
shown that the entropy change across the wave is dS ∝ (dp)3 , implying that dS goes
to zero much faster than the rate at which the amplitude dp tends to zero.
It follows that the derivative dp/dρ in equation (16.8) should be replaced by the
partial derivative at constant entropy, giving
c2
=
∂p
∂ρ
.
(16.9)
s
For a perfect gas, the use of p/ρ γ = const. and p = ρRT reduces the speed of sound
(16.9) to
c=
γp
= γ RT .
ρ
(16.10)
For air at 15 ◦ C, this gives c = 340 m/s. We note that the nonlinear terms that we
have neglected do change the shape of a propagating wave depending on whether it
is a compression or expansion, as follows. Because γ > 1, the isentropic relations
show that if dp > 0 (compression), then dT > 0, and from equation (16.10) the
sound speed c is increased. Therefore, the sound speed behind the front is greater
than that at the front and the back of the wave catches up with the front of the wave.
Thus the wave steepens as it travels. The opposite is true for an expansion wave, for
which dp < 0 and dT < 0 so c decreases. The back of the wave falls farther behind
the front so an expansion wave flattens as it travels.
Finite amplitude waves, across which there is a discontinuous change of pressure,
will be considered in Section 6. These are called shock waves. It will be shown that
the finite waves are not isentropic and that they propagate through a still fluid faster
than the sonic speed.
The first approximate expression for c was found by Newton, who assumed that
dp was proportional to dρ, as would be true if the process undergone by a√fluid
particle was isothermal. In this manner Newton arrived at the expression c = RT .
He attributed the discrepancy of this formula with experimental measurements as
due to “unclean air.” The science of thermodynamics was virtually nonexistent at the
time, so that the idea of an isentropic process was unknown to Newton. The correct
expression for the sound speed was first given by Laplace.
720
Compressible Flow
To show explicitly that small disturbances in a compressible fluid obey a wave
equation, we consider a slightly perturbed uniform flow in the x-direction so that
u = U∞ (ix + u′ ),
p = p∞ (1 + p ′ ),
ρ = ρ∞ (1 + ρ ′ ),
and so on
where the perturbations ()′ are all << 1. We substitute this assumed flow into the
equations for conservation of mass, momentum, and energy. We shall neglect the
effects of viscous stresses and heat conduction here but we will include them at
the end of Section 6, where they are determinative of shock structure. We may write
the conservation laws in the form
Dρ/Dt + ρ∇ · u = 0
ρDu/Dt + ∇p = 0
ρDh/Dt − Dp/Dt = 0
where body forces have also been neglected and D/Dt denotes the derivative following the fluid particle, D/Dt = ∂/∂t + u · ∇. Substituting the assumed flow into mass
conservation first,
ρ∞ ∂ρ ′ /∂t + ρ∞ U∞ ∂ρ ′ /∂x + ρ∞ U∞ u′ · ∇ρ ′ + ρ∞ U∞ ∇ · u′ + ρ∞ U∞ ρ ′ ∇ · u′ = 0.
We neglect the squares and products of the perturbations, leaving
∂ρ ′ /∂t + U∞ ∂ρ ′ /∂x + U∞ ∇ · u′ = 0.
Similarly, the momentum equation yields
∂u′ /∂t + U∞ ∂u′ /∂x + [p∞ /(ρ∞ U∞ )]∇p ′ = 0.
We may eliminate u′ by taking the divergence of the momentum equation and substituting into mass conservation, giving,
(∂/∂t +U∞ ∂/∂x)∇ ·u′ = −(1/U∞ )(∂/∂t +U∞ ∂/∂x)2 ρ ′ = −[p∞ /(ρ∞ U∞ )]∇ 2 p ′ .
The energy equation is put in terms of p ′ , ρ ′ for a perfect gas with constant specific
heats, h = (Cp /R)(p/ρ) and Cp /R = γ /(γ − 1). This results in D/Dt (p/ρ γ ) = 0
γ
but p/ρ γ = (p∞ /ρ∞ )(1 + p′ − γρ ′ ), with squares and products of the perturbations
neglected. Then (∂/∂t + U∞ ∂/∂x)p ′ − γ (∂/∂t + U∞ ∂/∂x)ρ ′ = 0. Using this to
eliminate ρ ′ , (∂/∂t + U∞ ∂/∂x)2 p ′ = (γp∞ /ρ∞ )∇ 2 p ′ = c2 ∇ 2 p ′ . This is a classical
linear wave equation for p′ . We can translate this back to a frame at rest by a Galilean
transformation, (x, y, z, t) → (x ′ , y ′ , z′ , t ′ ) with t ′ = t + x/U∞ , x ′ = x, y ′ =
y, z′ = z. Thus ∂/∂t ′ = ∂/∂t + U∞ ∂/∂x and we are left with
∂ 2 p/∂t 2 = c2 ∇ 2 p
(primes suppressed), as seen in Section 7.2, p.200. The solution in one dimension is
given there and it is seen that c is the wave speed.
721
3. Basic Equations for One-Dimensional Flow
3. Basic Equations for One-Dimensional Flow
In this section we begin our study of certain compressible flows that can be analyzed
by a one-dimensional approximation. Such a simplification is valid in flow through a
duct whose centerline does not have a large curvature and whose cross section does
not vary abruptly. The overall behavior in such flows can be studied by ignoring the
variation of velocity and other properties across the duct and replacing the property
distributions by their average values over the cross section (Figure 16.2). The area of
the duct is taken as A(x), and the flow properties are taken as p(x), ρ(x), u(x), and
so on. Unsteadiness can be introduced by including t as an additional independent
variable. The forms of the basic equations in a one-dimensional compressible flow
are discussed in what follows.
Continuity Equation
For steady flows, conservation of mass requires that
ρuA = independent of x.
Differentiating, we obtain
dρ
du dA
+
+
= 0.
ρ
u
A
(16.11)
Energy Equation
Consider a control volume within the duct, shown by the dashed line in Figure 16.2.
The first law of thermodynamics for a control volume fixed in space is
d
u2
u2
e+
ρ e+
dV +
ρuj d Aj = ui τij dAj − q · dA,
dt
2
2
(16.12)
where u2 /2 is the kinetic energy per unit mass. The first term on the left-hand side
represents the rate of change of “stored energy” (the sum of internal and kinetic
Figure 16.2 A one-dimensional flow.
722
Compressible Flow
energies) within the control volume, and the second term represents the flux of energy
out of the control surface. The first term on the right-hand side represents the rate
of work done on the control surface, and the second term on the right-hand side
represents the heat input through the control surface. Body forces have been neglected
in equation (16.12). (Here, q is the heat flux per unit area per unit time, and dA is
directed along the outward normal, so that q · dA is the rate of outflow of heat.)
Equation (16.12) can easily be derived by integrating the differential form given by
equation (4.65) over the control volume.
Assume steady state, so that the first term on the left-hand side of equation (16.12)
is zero. Writing ṁ = ρ1 u1 A1 = ρ2 u2 A2 (where the subscripts denote sections 1
and 2), the second term on the left-hand side in equation (16.12) gives
1 2
1 2
1 2
e + u ρuj dAj = ṁ e2 + u2 − e1 − u1 .
2
2
2
The work done on the control surfaces is
ui τij dAj = u1 p1 A1 − u2 p2 A2 .
Here, we have assumed no-slip on the sidewalls and frictional stresses on the endfaces
1 and 2 are negligible. The rate of heat addition to the control volume is
− q · dA = Qṁ,
where Q is the heat added per unit mass. (Checking units, Q is in J/kg, and ṁ is in
kg/s, so that Qṁ is in J/s.) Then equation (16.12) becomes, after dividing by ṁ,
1
1
1
e2 + u22 − e1 − u21 = [u1 p1 A1 − u2 p2 A2 ] + Q.
2
2
ṁ
(16.13)
The first term on the right-hand side can be written in a simple manner by noting that
uA
= v,
ṁ
where v is the specific volume. This must be true because uA = ṁv is the volumetric
flow rate through the duct. (Checking units, ṁ is the mass flow rate in kg/s, and
v is the specific volume in m3 /kg, so that ṁv is the volume flow rate in m3 /s.)
Equation (16.13) then becomes
e2 + 21 u22 − e1 − 21 u21 = p1 v1 − p2 v2 + Q.
(16.14)
It is apparent that p1 v1 is the work done (per unit mass) by the surroundings in
pushing fluid into the control volume. Similarly, p2 v2 is the work done by the fluid
inside the control volume on the surroundings in pushing fluid out of the control
723
3. Basic Equations for One-Dimensional Flow
volume. Equation (16.14) therefore has a simple meaning. Introducing the enthalpy
h ≡ e + pv, we obtain
h2 + 21 u22 = h1 + 21 u21 + Q.
(16.15)
This is the energy equation, which is valid even if there are frictional or nonequilibrium
conditions (e.g., shock waves) between sections 1 and 2. It is apparent that the sum
of enthalpy and kinetic energy remains constant in an adiabatic flow. Therefore,
enthalpy plays the same role in a flowing system that internal energy plays in a
nonflowing system. The difference between the two types of systems is the flow work
pv required to push matter across a section.
Bernoulli and Euler Equations
For inviscid flows, the steady form of the momentum equation is the Euler equation
u du +
dp
= 0.
ρ
(16.16)
Integrating along a streamline, we obtain the Bernoulli equation for a compressible flow:
dp
1 2
u +
= const.,
(16.17)
2
ρ
which agrees with equation (4.78).
For adiabatic frictionless flows the Bernoulli equation is identical to the energy
equation. To see this, note that this is an isentropic flow, so that the T dS equation
T dS = dh − v dp,
gives
dh = dp/ρ.
Then the Euler equation (16.16) becomes
u du + dh = 0,
which is identical to the adiabatic form of the energy equation (16.15). The collapse
of the momentum and energy equations is expected because the constancy of entropy
has eliminated one of the flow variables.
Momentum Principle for a Control Volume
If the centerline of the duct is straight, then the steady form of the momentum principle
for a finite control volume, which cuts across the duct at sections 1 and 2, gives
p1 A1 − p2 A2 + F ≡ ρ2 u22 A2 − ρ1 u21 A1 ,
(16.18)
724
Compressible Flow
Figure 16.3 Application of the momentum principle to an infinitesimal control volume in a duct.
where F is the x-component of the resultant force exerted on the fluid by the walls.
The momentum principle (16.18) is applicable even when there are frictional and
dissipative processes (such as shock waves) within the control volume:
x2
F =
(−pδij + σij ) dAj =
p dA(x) − (fσ )x ,
sides
x1
x
fσ,x = −
σij dAj .
sides
x
If frictional processes are absent, then equation (16.18) reduces to the Euler
equation (16.16). To see this, consider an infinitesimal area change between sections 1
and 2 (Figure 16.3). Then the average pressure exerted by the walls on the control
surface is (p + 21 dp), so that F = dA(p + 21 dp). Then equation (16.18) becomes
pA − (p + dp)(A + dA) + p + 21 dp dA = ρuA(u + du) − ρu2 A,
where by canceling terms and neglecting second-order terms, this reduces to the Euler
equation (16.16).
4. Stagnation and Sonic Properties
A very useful reference state for computing compressible flows is the stagnation state
in which the velocity is zero. Suppose the properties of the flow (such as h, ρ, u)
are known at a certain point. The stagnation properties at a point are defined as those
that would be obtained if the local flow were imagined to slow down to zero velocity
isentropically. The stagnation properties are denoted by a subscript zero. Thus the
stagnation enthalpy is defined as
h0 ≡ h + 21 u2 .
725
4. Stagnation and Sonic Properties
For a perfect gas, this gives
Cp T0 = Cp T + 21 u2 ,
(16.19)
which defines the stagnation temperature.
It is useful to express the ratios such as T0 /T in terms of the local Mach number.
From equation (16.19), we obtain
u2
γ − 1 u2
T0
=1+
=1+
,
T
2Cp T
2 γ RT
where we have used Cp = γ R/(γ − 1). Therefore
γ −1 2
T0
=1+
M ,
T
2
(16.20)
from which the stagnation temperature T0 can be found for a given T and M.
The isentropic relations can then be used to obtain the stagnation pressure and
stagnation density:
p0
=
p
T0
T
ρ0
=
ρ
T0
T
γ /(γ −1)
1/(γ −1)
γ − 1 2 γ /(γ −1)
,
= 1+
M
2
γ − 1 2 1/(γ −1)
.
= 1+
M
2
(16.21)
(16.22)
In a general flow the stagnation properties can vary throughout the flow field. If,
however, the flow is adiabatic (but not necessarily isentropic), then h + u2 /2 is constant√throughout the flow as shown in equation (16.15). It follows that h0 , T0 , and c0
(= γ RT0 ) are constant throughout an adiabatic flow, even in the presence of friction. In contrast, the stagnation pressure p0 and density ρ0 decrease if there is friction.
To see this, consider the entropy change in an adiabatic flow between sections 1 and 2,
with 2 being the downstream section. Let the flow at both sections hypothetically be
brought to rest by isentropic processes, giving the local stagnation conditions p01 , p02 ,
T01 , and T02 . Then the entropy change between the two sections can be expressed as
S2 − S1 = S02 − S01 = −R ln
T02
p02
+ Cp ln
,
p01
T01
where we have used equation (16.4) for computing entropy changes. The last term is
zero for an adiabatic flow in which T02 = T01 . As the second law of thermodynamics
requires that S2 > S1 , it follows that
p02 < p01 ,
which shows that the stagnation pressure falls due to friction.
726
Compressible Flow
Figure 16.4 An isentropic process starting from a reservoir. Stagnation properties are uniform everywhere
and are equal to the properties in the reservoir.
It is apparent that all stagnation properties are constant along an isentropic flow.
If such a flow happens to start from a large reservoir where the fluid is practically at
rest, then the properties in the reservoir equal the stagnation properties everywhere
in the flow (Figure 16.4).
In addition to the stagnation properties, there is another useful set of reference
quantities. These are called sonic or critical conditions and are denoted by an asterisk.
Thus, p∗ , ρ ∗ , c∗ , and T ∗ are properties attained if the local fluid is imagined to
expand or compress isentropically until it reaches M = 1. It is easy to show (Exercise
1) that the area of the passage A∗ , at which the sonic conditions are attained, is
given by
1
2
γ − 1 2 (1/2)(γ +1)/(γ −1)
A
.
=
M
1+
A∗
M γ +1
2
(16.23)
We shall see in the following section that sonic conditions can only be reached at the
throat of a duct, where the area is minimum. Equation (16.23) shows that we can find
the throat area A∗ of an isentropic duct flow if we know the Mach number M and
the area A at some point of the duct. Note that it is not necessary that a throat actually should exist in the flow; the sonic variables are simply reference values that are
reached if the flow were brought to the sonic state isentropically. From its definition
it is clear that the value of A∗ in a flow remains constant along an isentropic flow. The
presence of shock waves, friction, or heat transfer changes the value of A∗ along the
flow.
The values of T0 /T , p0 /p, ρ0 /ρ, and A/A∗ at a point can be determined
from equations (16.20)–(16.23) if the local Mach number is known. For γ = 1.4,
these ratios are tabulated in Table 16.1. The reader should examine this table
at this point. Examples 16.1 and 16.2 given later will illustrate the use of this
table.
727
4. Stagnation and Sonic Properties
TABLE 16.1
Isentropic Flow of a Perfect Gas (γ = 1.4)
M
p/p0
ρ/ρ0
T /T0
A/A∗
0.0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0.5
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.26
1.28
1.3
1.32
1.34
1.36
1.38
1.4
1.42
1.44
1.0
0.9997
0.9989
0.9975
0.9955
0.9930
0.9900
0.9864
0.9823
0.9776
0.9725
0.9668
0.9607
0.9541
0.9470
0.9395
0.9315
0.9231
0.9143
0.9052
0.8956
0.8857
0.8755
0.8650
0.8541
0.8430
0.5039
0.4919
0.4800
0.4684
0.4568
0.4455
0.4343
0.4232
0.4124
0.4017
0.3912
0.3809
0.3708
0.3609
0.3512
0.3417
0.3323
0.3232
0.3142
0.3055
0.2969
1.0
0.9998
0.9992
0.9982
0.9968
0.9950
0.9928
0.9903
0.9873
0.9840
0.9803
0.9762
0.9718
0.9670
0.9619
0.9564
0.9506
0.9445
0.9380
0.9313
0.9243
0.9170
0.9094
0.9016
0.8935
0.8852
0.6129
0.6024
0.5920
0.5817
0.5714
0.5612
0.5511
0.5411
0.5311
0.5213
0.5115
0.5019
0.4923
0.4829
0.4736
0.4644
0.4553
0.4463
0.4374
0.4287
0.4201
1.0
0.9999
0.9997
0.9993
0.9987
0.9980
0.9971
0.9961
0.9949
0.9936
0.9921
0.9904
0.9886
0.9867
0.9846
0.9823
0.9799
0.9774
0.9747
0.9719
0.9690
0.9659
0.9627
0.9594
0.9559
0.9524
0.8222
0.8165
0.8108
0.8052
0.7994
0.7937
0.7879
0.7822
0.7764
0.7706
0.7648
0.7590
0.7532
0.7474
0.7416
0.7358
0.7300
0.7242
0.7184
0.7126
0.7069
∞
28.9421
14.4815
9.6659
7.2616
5.8218
4.8643
4.1824
3.6727
3.2779
2.9635
2.7076
2.4956
2.3173
2.1656
2.0351
1.9219
1.8229
1.7358
1.6587
1.5901
1.5289
1.4740
1.4246
1.3801
1.3398
1.0013
1.0029
1.0051
1.0079
1.0113
1.0153
1.0198
1.0248
1.0304
1.0366
1.0432
1.0504
1.0581
1.0663
1.0750
1.0842
1.0940
1.1042
1.1149
1.1262
1.1379
M
p/p0
ρ/ρ0
T /T0
A/A∗
0.52
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1.0
1.02
2.04
2.06
2.08
2.1
2.12
2.14
2.16
2.18
2.2
2.22
2.24
2.26
2.28
2.3
2.32
2.34
2.36
2.38
2.4
2.42
2.44
0.8317
0.8201
0.8082
0.7962
0.7840
0.7716
0.7591
0.7465
0.7338
0.7209
0.7080
0.6951
0.6821
0.6690
0.6560
0.6430
0.6300
0.6170
0.6041
0.5913
0.5785
0.5658
0.5532
0.5407
0.5283
0.5160
0.1201
0.1164
0.1128
0.1094
0.1060
0.1027
0.0996
0.0965
0.0935
0.0906
0.0878
0.0851
0.0825
0.0800
0.0775
0.0751
0.0728
0.0706
0.0684
0.0663
0.0643
0.8766
0.8679
0.8589
0.8498
0.8405
0.8310
0.8213
0.8115
0.8016
0.7916
0.7814
0.7712
0.7609
0.7505
0.7400
0.7295
0.7189
0.7083
0.6977
0.6870
0.6764
0.6658
0.6551
0.6445
0.6339
0.6234
0.2200
0.2152
0.2104
0.2058
0.2013
0.1968
0.1925
0.1882
0.1841
0.1800
0.1760
0.1721
0.1683
0.1646
1.1609
0.1574
0.1539
0.1505
0.1472
0.1439
0.1408
0.9487
0.9449
0.9410
0.9370
0.9328
0.9286
0.9243
0.9199
0.9153
0.9107
0.9061
0.9013
0.8964
0.8915
0.8865
0.8815
0.8763
0.8711
0.8659
0.8606
0.8552
0.8498
0.8444
0.8389
0.8333
0.8278
0.5458
0.5409
0.5361
0.5313
0.5266
0.5219
0.5173
0.5127
0.5081
0.5036
0.4991
0.4947
0.4903
0.4859
0.4816
0.4773
0.4731
0.4688
0.4647
0.4606
0.4565
1.3034
1.2703
1.2403
1.2130
1.1882
1.1656
1.1451
1.1265
1.1097
1.0944
1.0806
1.0681
1.0570
1.0471
1.0382
1.0305
1.0237
1.0179
1.0129
1.0089
1.0056
1.0031
1.0014
1.0003
1.0000
1.0003
1.7451
1.7750
1.8056
1.8369
1.8690
1.9018
1.9354
1.9698
2.0050
2.0409
2.0777
2.1153
2.1538
2.1931
2.2333
2.2744
2.3164
2.3593
2.4031
2.4479
2.4936
728
Compressible Flow
TABLE 16.1
(Continued)
M
p/p0
ρ/ρ0
T /T0
A/A∗
M
p/p0
ρ/ρ0
T /T0
A/A∗
1.46
1.48
1.5
1.52
1.54
1.56
1.58
1.6
1.62
1.64
1.66
1.68
1.7
1.72
1.74
1.76
1.78
1.8
1.82
1.84
1.86
1.88
1.9
1.92
1.94
1.96
1.98
2.0
2.02
3.04
3.06
3.08
3.1
3.12
3.14
3.16
3.18
3.2
3.22
3.24
3.26
3.28
3.3
3.32
3.34
3.36
3.38
0.2886
0.2804
0.2724
0.2646
0.2570
0.2496
0.2423
0.2353
0.2284
0.2217
0.2151
0.2088
0.2026
0.1966
0.1907
0.1850
0.1794
0.1740
0.1688
0.1637
0.1587
0.1539
0.1492
0.1447
0.1403
0.1360
0.1318
0.1278
0.1239
0.0256
0.0249
0.0242
0.0234
0.0228
0.0221
0.0215
0.0208
0.0202
0.0196
0.0191
0.0185
0.0180
0.0175
0.0170
0.0165
0.0160
0.0156
0.4116
0.4032
0.3950
0.3869
0.3789
0.3710
0.3633
0.3557
0.3483
0.3409
0.3337
0.3266
0.3197
0.3129
0.3062
0.2996
0.2931
0.2868
0.2806
0.2745
0.2686
0.2627
0.2570
0.2514
0.2459
0.2405
0.2352
0.2300
0.2250
0.0730
0.0715
0.0700
0.0685
0.0671
0.0657
0.0643
0.0630
0.0617
0.0604
0.0591
0.0579
0.0567
0.0555
0.0544
0.0533
0.0522
0.0511
0.7011
0.6954
0.6897
0.6840
0.6783
0.6726
0.6670
0.6614
0.6558
0.6502
0.6447
0.6392
0.6337
0.6283
0.6229
0.6175
0.6121
0.6068
0.6015
0.5963
0.5910
0.5859
0.5807
0.5756
0.5705
0.5655
0.5605
0.5556
0.5506
0.3511
0.3481
0.3452
0.3422
0.3393
0.3365
0.3337
0.3309
0.3281
0.3253
0.3226
0.3199
0.3173
0.3147
0.3121
0.3095
0.3069
0.3044
1.1501
1.1629
1.1762
1.1899
1.2042
1.2190
1.2344
1.2502
1.2666
1.2836
1.3010
1.3190
1.3376
1.3567
1.3764
1.3967
1.4175
1.4390
1.4610
1.4836
1.5069
1.5308
1.5553
1.5804
1.6062
1.6326
1.6597
1.6875
1.7160
4.3990
4.4835
4.5696
4.6573
4.7467
4.8377
4.9304
5.0248
5.1210
5.2189
5.3186
5.4201
5.5234
5.6286
5.7358
5.8448
5.9558
6.0687
2.46
2.48
2.5
2.52
2.54
2.56
2.58
2.6
2.62
2.64
2.66
2.68
2.7
2.72
2.74
2.76
2.78
2.8
2.82
2.84
2.86
2.88
2.9
2.92
2.94
2.96
2.98
3.0
3.02
4.04
4.06
4.08
4.1
4.12
4.14
4.16
4.18
4.2
4.22
4.24
4.26
4.28
4.3
4.32
4.34
4.36
4.38
0.0623
0.0604
0.0585
0.0567
0.0550
0.0533
0.0517
0.0501
0.0486
0.0471
0.0457
0.0443
0.0430
0.0417
0.0404
0.0392
0.0380
0.0368
0.0357
0.0347
0.0336
0.0326
0.0317
0.0307
0.0298
0.0289
0.0281
0.0272
0.0264
0.0062
0.0061
0.0059
0.0058
0.0056
0.0055
0.0053
0.0052
0.0051
0.0049
0.0048
0.0047
0.0046
0.0044
0.0043
0.0042
0.0041
0.0040
0.1377
0.1346
0.1317
0.1288
0.1260
0.1232
0.1205
0.1179
0.1153
0.1128
0.1103
0.1079
0.1056
0.1033
0.1010
0.0989
0.0967
0.0946
0.0926
0.0906
0.0886
0.0867
0.0849
0.0831
0.0813
0.0796
0.0779
0.0762
0.0746
0.0266
0.0261
0.0256
0.0252
0.0247
0.0242
0.0238
0.0234
0.0229
0.0225
0.0221
0.0217
0.0213
0.0209
0.0205
0.0202
0.0198
0.0194
0.4524
0.4484
0.4444
0.4405
0.4366
0.4328
0.4289
0.4252
0.4214
0.4177
0.4141
0.4104
0.4068
0.4033
0.3998
0.3963
0.3928
0.3894
0.3860
0.3827
0.3794
0.3761
0.3729
0.3696
0.3665
0.3633
0.3602
0.3571
0.3541
0.2345
0.2327
0.2310
0.2293
0.2275
0.2258
0.2242
0.2225
0.2208
0.2192
0.2176
0.2160
0.2144
0.2129
0.2113
0.2098
0.2083
0.2067
2.5403
2.5880
2.6367
2.6865
2.7372
2.7891
2.8420
2.8960
2.9511
3.0073
3.0647
3.1233
3.1830
3.2440
3.3061
3.3695
3.4342
3.5001
3.5674
3.6359
3.7058
3.7771
3.8498
3.9238
3.9993
4.0763
4.1547
4.2346
4.3160
11.1077
11.3068
11.5091
11.7147
11.9234
12.1354
12.3508
12.5695
12.7916
13.0172
13.2463
13.4789
13.7151
13.9549
14.1984
14.4456
14.6965
14.9513
3.4
3.42
0.0151
0.0147
0.0501
0.0491
0.3019
0.2995
6.1837
6.3007
4.4
4.42
0.0039
0.0038
0.0191
0.0187
0.2053
0.2038
15.2099
15.4724
729
5. Area–Velocity Relations in One-Dimensional Isentropic Flow
TABLE 16.1
M
p/p0
ρ/ρ0
T /T0
A/A∗
3.44
3.46
3.48
3.5
3.52
3.54
3.56
3.58
3.6
3.62
3.64
3.66
3.68
3.7
3.72
3.74
3.76
3.78
3.8
3.82
3.84
3.86
3.88
3.9
3.92
3.94
3.96
3.98
4.0
4.02
0.0143
0.0139
0.0135
0.0131
0.0127
0.0124
0.0120
0.0117
0.0114
0.0111
0.0108
0.0105
0.0102
0.0099
0.0096
0.0094
0.0091
0.0089
0.0086
0.0084
0.0082
0.0080
0.0077
0.0075
0.0073
0.0071
0.0069
0.0068
0.0066
0.0064
0.0481
0.0471
0.0462
0.0452
0.0443
0.0434
0.0426
0.0417
0.0409
0.0401
0.0393
0.0385
0.0378
0.0370
0.0363
0.0356
0.0349
0.0342
0.0335
0.0329
0.0323
0.0316
0.0310
0.0304
0.0299
0.0293
0.0287
0.0282
0.0277
0.0271
0.2970
0.2946
0.2922
0.2899
0.2875
0.2852
0.2829
0.2806
0.2784
0.2762
0.2740
0.2718
0.2697
0.2675
0.2654
0.2633
0.2613
0.2592
0.2572
0.2552
0.2532
0.2513
0.2493
0.2474
0.2455
0.2436
0.2418
0.2399
0.2381
0.2363
6.4198
6.5409
6.6642
6.7896
6.9172
7.0471
7.1791
7.3135
7.4501
7.5891
7.7305
7.8742
8.0204
8.1691
8.3202
8.4739
8.6302
8.7891
8.9506
9.1148
0.2817
9.4513
9.6237
9.7990
9.9771
10.1581
10.3420
10.5289
10.7188
10.9117
(Continued)
M
p/p0
ρ/ρ0
T /T0
A/A∗
4.44
4.46
4.48
4.5
4.52
4.54
4.56
4.58
4.6
4.62
4.64
4.66
4.68
4.7
4.72
4.74
4.76
4.78
4.8
4.82
4.84
4.86
4.88
4.9
4.92
4.94
4.96
4.98
5.0
0.0037
0.0036
0.0035
0.0035
0.0034
0.0033
0.0032
0.0031
0.0031
0.0030
0.0029
0.0028
0.0028
0.0027
0.0026
0.0026
0.0025
0.0025
0.0024
0.0023
0.0023
0.0022
0.0022
0.0021
0.0021
0.0020
0.0020
0.0019
0.0019
0.0184
0.0181
0.0178
0.0174
0.0171
0.0168
0.0165
0.0163
0.0160
0.0157
0.0154
0.0152
0.0149
0.0146
0.0144
0.0141
0.0139
0.0137
0.0134
0.0132
0.0130
0.0128
0.0125
0.0123
0.0121
0.0119
0.0117
0.0115
0.0113
0.2023
0.2009
0.1994
0.1980
0.1966
0.1952
0.1938
0.1925
0.1911
0.1898
0.1885
0.1872
0.1859
0.1846
0.1833
0.1820
0.1808
0.1795
0.1783
0.1771
0.1759
0.1747
0.1735
0.1724
0.1712
0.1700
0.1689
0.1678
0.1667
15.7388
16.0092
16.2837
16.5622
16.8449
17.1317
17.4228
17.7181
18.0178
18.3218
18.6303
18.9433
19.2608
19.5828
19.9095
20.2409
20.5770
20.9179
21.2637
21.6144
21.9700
22.3306
22.6963
23.0671
23.4431
23.8243
24.2109
24.6027
25.0000
5. Area–Velocity Relations in One-Dimensional
Isentropic Flow
Some surprising consequences of compressibility are dramatically demonstrated by
considering an isentropic flow in a duct of varying area. Before we demonstrate this
effect, we shall make some brief comments on two common devices of varying area
in which the flow can be approximately isentropic. One of them is the nozzle through
which the flow expands from high to low pressure to generate a high-speed jet. An
example of a nozzle is the exit duct of a rocket motor. The second device is called
the diffuser, whose function is opposite to that of a nozzle. (Note that the diffuser has
nothing to do with heat diffusion.) In a diffuser a high-speed jet is decelerated and
compressed. For example, air enters the jet engine of an aircraft after passing through
a diffuser, which raises the pressure and temperature of the air. In incompressible
flow, a nozzle profile converges in the direction of flow to increase the velocity, while
730
Compressible Flow
a diffuser profile diverges. We shall see that this conclusion is true for subsonic flows,
but not for supersonic flows.
Consider two sections of a duct (Figure 16.3). The continuity equation gives
du dA
dρ
+
+
= 0.
ρ
u
A
(16.24)
In a constant density flow dρ = 0, for which the continuity equation requires that a
decreasing area leads to an increase of velocity.
As the flow is assumed to be frictionless, we can use the Euler equation
u du = −
dp dρ
dρ
dp
=−
= −c2 ,
ρ
dρ ρ
ρ
(16.25)
where we have used the fact that c2 = dp/dρ in an isentropic flow. The Euler
equation requires that an increasing speed (du > 0) in the direction of flow must
be accompanied by a fall of pressure (dp < 0). In terms of the Mach number,
equation (16.25) becomes
du
dρ
= −M 2 .
ρ
u
(16.26)
This shows that for M ≪ 1, the percentage change of density is much smaller than the
percentage change of velocity. The density changes in the continuity equation (16.24)
can therefore be neglected in low Mach number flows, a fact also demonstrated in
Section 1.
Substituting equation (16.26) into equation (16.24), we obtain
−dA/A
du
=
.
u
1 − M2
(16.27)
This relation leads to the following important conclusions about compressible flows:
(i) At subsonic speeds (M < 1) a decrease of area increases the speed of flow.
A subsonic nozzle therefore must have a convergent profile, and a subsonic
diffuser must have a divergent profile (upper row of Figure 16.5). The behavior
is therefore qualitatively the same as in incompressible flows.
(ii) At supersonic speeds (M > 1) the denominator in equation (16.27) is negative,
and we arrive at the surprising conclusion that an increase in area leads to an
increase of speed. The reason for such a behavior can be understood from
equation (16.26), which shows that for M > 1 the density decreases faster
than the velocity increases, thus the area must increase in an accelerating flow
in order that the product Aρu is constant.
The supersonic portion of a nozzle therefore must have a divergent profile, while
the supersonic part of a diffuser must have a convergent profile (bottom row of
Figure 16.5).
Suppose a nozzle is used to generate a supersonic stream, starting from low
speeds at the inlet (Figure 16.6). Then the Mach number must increase continuously
from M = 0 near the inlet to M > 1 at the exit. The foregoing discussion shows
5. Area–Velocity Relations in One-Dimensional Isentropic Flow
Figure 16.5 Shapes of nozzles and diffusers in subsonic and supersonic regimes. Nozzles are shown in
the left column and diffusers are shown in the right column.
Figure 16.6 A convergent–divergent nozzle. The flow is continuously accelerated from low speed to
supersonic Mach number.
731
732
Compressible Flow
Figure 16.7 Convergent–divergent passages in which the condition at the throat is not sonic.
that the nozzle must converge in the subsonic portion and diverge in the supersonic
portion. Such a nozzle is called a convergent–divergent nozzle. From Figure 16.6 it
is clear that the Mach number must be unity at the throat, where the area is neither
increasing nor decreasing. This is consistent with equation (16.27), which shows that
du can be nonzero at the throat only if M = 1. It follows that the sonic velocity can
be achieved only at the throat of a nozzle or a diffuser and nowhere else.
It does not, however, follow that M must necessarily be unity at the throat.
According to equation (16.27), we may have a case where M = 1 at the throat if
du = 0 there. As an example, note that the flow in a convergent–divergent tube may
be subsonic everywhere, with M increasing in the convergent portion and decreasing in the divergent portion, with M = 1 at the throat (Figure 16.7a). The first half
of the tube here is acting as a nozzle, whereas the second half is acting as a diffuser. Alternatively, we may have a convergent–divergent tube in which the flow is
supersonic everywhere, with M decreasing in the convergent portion and increasing
in the divergent portion, and again M = 1 at the throat (Figure 16.7b).
Example 16.1
The nozzle of a rocket motor is designed to generate a thrust of 30,000 N when
operating at an altitude of 20 km. The pressure inside the combustion chamber is
1000 kPa while the temperature is 2500 K. The gas constant of the fluid in the jet is
R = 280 m2 /(s2 K), and γ = 1.4. Assuming that the flow in the nozzle is isentropic,
calculate the throat and exit areas. Use the isentropic table (Table 16.1).
Solution: At an altitude of 20 km, the pressure of the standard atmosphere
(Section A4 in Appendix A) is 5467 Pa. If subscripts “0” and “e” refer to the stagnation
and exit conditions, then a summary of the information given is as follows:
pe = 5467 Pa,
p0 = 1000 kPa,
733
6. Normal Shock Wave
T0 = 2500 K,
Thrust = ρe Ae u2e = 30,000 N.
Here, we have used the facts that the thrust equals mass flow rate times the exit velocity,
and the pressure inside the combustion chamber is nearly equal to the stagnation
pressure. The pressure ratio at the exit is
5467
pe
=
= 5.467 × 10−3 .
p0
(1000)(1000)
For this ratio of pe /p0 , the isentropic table (Table 16.1) gives
Me = 4.15,
Ae
= 12.2,
A∗
Te
= 0.225.
T0
The exit temperature and density are therefore
Te = (0.225)(2500) = 562 K,
ρe = pe /RTe = 5467/(280)(562) = 0.0347 kg/ m3 .
The exit velocity is
ue = Me γ RTe = 4.15 (1.4)(280)(562) = 1948 m/s.
The exit area is found from the expression for thrust:
Ae =
Thrust
30,000
=
= 0.228 m2 .
2
ρe ue
(0.0347)(1948)2
Because Ae /A∗ = 12.2, the throat area is
A∗ =
0.228
= 0.0187 m2 .
12.2
6. Normal Shock Wave
A shock wave is similar to a sound wave except that it has finite strength. The thickness
of such a wavefront is of the order of micrometers, so that the properties vary almost
discontinuously across a shock wave. The high gradients of velocity and temperature
result in entropy production within the wave, due to which the isentropic relations
734
Compressible Flow
cannot be used across the shock. In this section we shall derive the relations between
properties of the flow on the two sides of a normal shock, where the wavefront is
perpendicular to the direction of flow. We shall treat the shock wave as a discontinuity;
a treatment of Navier-Strokes shock structure is given at the end of this section.
To derive the relationships between the properties on the two sides of the shock,
consider a control volume shown in Figure 16.8, where the sections 1 and 2 can
be taken arbitrarily close to each other because of the discontinuous nature of the
wave. The area change between the upstream and the downstream sides can then be
neglected. The basic equations are
Continuity:
x-momentum:
Energy:
ρ1 u1 = ρ2 u2 ,
p1 − p2 =
h1 +
1 2
2 u1
ρ2 u22
=
− ρ1 u21 ,
h2 + 21 u22 .
(16.28)
(16.29)
In the application of the momentum theorem, we have neglected any frictional drag
from the walls because such forces go to zero as the wave thickness goes to zero.
Note that we cannot use the Bernoulli equation because the process inside the wave is
dissipative. We have written down four unknowns (h2 , u2 , p2 , ρ2 ) and three equations.
The additional relation comes from the perfect gas relationship
h = Cp T =
γp
γR p
=
,
γ − 1 ρR
(γ − 1)ρ
so that the energy equation becomes
1
γ p2
1
γ p1
+ u21 =
+ u22 .
γ − 1 ρ1
2
γ − 1 ρ2
2
Figure 16.8 Normal shock wave.
(16.30)
735
6. Normal Shock Wave
We now have three unknowns (u2 , p2 , ρ2 ) and three equations (16.28)–(16.30).
Elimination of ρ2 and u2 from these gives, after some algebra,
ρ1 u21
2γ
p2
=1+
−1 .
p1
γ + 1 γp1
This can be expressed in terms of the upstream Mach number M1 by noting that
ρu2 /γp = u2 /γ RT = M 2 . The pressure ratio then becomes
p2
2γ
=1+
(M 2 − 1).
p1
γ +1 1
(16.31)
Let us now derive a relation between M1 and M2 . Because ρu2 = ρc2 M 2 =
ρ(γp/ρ)M 2 = γpM 2 , the momentum equation (16.29) gives
p1 + γp1 M12 = p2 + γp2 M22 .
Using equation (16.31), this gives
M22 =
(γ − 1)M12 + 2
2γ M12 + 1 − γ
,
(16.32)
which is plotted in Figure 16.9. Because M2 = M1 (state 2 = state 1) is a solution
of equations (16.28)–(16.30), that is shown as well indicating two possible solutions
for M2 for all M1 > [(γ − 1)/2γ ]1/2 . We show in what follows that M1 1 to avoid
violation of the second law of thermodynamics. The two possible solutions are: (a) no
change of state; and (b) a sudden transition from supersonic to subsonic flow with
consequent increases in pressure, density, and temperature. The density, velocity, and
temperature ratios can be similarly obtained. They are
(γ + 1)M12
u1
ρ2
=
=
,
ρ1
u2
(γ − 1)M12 + 2
(16.33)
T2
2(γ − 1) γ M12 + 1 2
=1+
(M1 − 1).
T1
(γ + 1)2
M12
(16.34)
The normal shock relations (16.31)–(16.34) were worked out independently by
the British engineer W. J. M. Rankine (1820–1872) and the French ballistician
Pierre Henry Hugoniot (1851–1887). These equations are sometimes known as the
Rankine–Hugoniot relations.
736
Compressible Flow
Figure 16.9 Normal shock-wave solution M2 (M1 ) for γ = 1.4. Trivial (no change) solution is also
shown. Asymptotes are [(γ − 1)/2γ ]1/2 = 0.378.
An important quantity is the change of entropy across the shock. Using equation (16.4), the entropy change is
γ
S2 − S1
p2 ρ1
= ln
Cv
p 1 ρ2
(γ − 1)M12 + 2 γ
2γ
= ln 1 +
,
(M12 − 1)
γ +1
(γ + 1)M12
(16.35)
which is plotted in Figure 16.10. This shows that the entropy across an expansion
shock would decrease, which is impermissible for the perfect gas equation of state.
However, for a heavy fluorocarbon gas (FC-70), Fergason et al. (2001), using different
equations of state, have numerically simulated a rarefaction (expansion) shock in a
shock tube type flow. Equation (16.36) calculates the entropy change for a perfect gas
with constant specific heats explicitly in the neighborhood of M1 = 1. Now assume
that the upstream Mach number M1 is only slightly larger than 1, so that M12 − 1 is
737
6. Normal Shock Wave
Figure 16.10 Entropy change (S2 − S1 )/Cv as a function of M1 for γ = 1.4. Note higher-order contact
at M = 1.
a small quantity. It is straightforward to show that equation (16.35) then reduces to
(Exercise 2)
S 2 − S1
2γ (γ − 1) 2
≃
(M1 − 1)3 .
Cv
3(γ + 1)2
(16.36)
This shows that we must have M1 > 1 because the entropy of an adiabatic
process cannot decrease. Equation (16.32) then shows that M2 < 1. Thus, the
Mach number changes from supersonic to subsonic values across a normal shock;
a discontinuous change from subsonic to supersonic conditions would lead to a
violation of the second law of thermodynamics. (A shock wave is therefore analogous to a hydraulic jump (Chapter 7, Section 12) in a gravity current, in which
the Froude number jumps from supercritical to subcritical values; see Figure 7.23.)
Equations (16.31), (16.33), and (16.34) then show that the jumps in p, ρ, and T
are also from low to high values, so that a shock wave compresses and heats
a fluid.
Note that the terms involving the first two powers of (M12 − 1) do not appear in
equation (16.36). Using the pressure ratio (16.31), equation (16.36) can be written as
γ 2 − 1 p 3
S2 − S1
.
≃
Cv
12γ 2 p1
This shows that as the wave amplitude decreases, the entropy jump goes to zero
much faster than the rate at which the pressure jump (or the jumps in velocity
or temperature) goes to zero. Weak shock waves are therefore nearly isentropic.
738
Compressible Flow
Figure 16.11 Stationary and moving shocks.
This is why we argued that the propagation of sound waves is an isentropic
process.
Because of the adiabatic nature of the process, the stagnation properties T0 and
h0 are constant across the shock. In contrast, the stagnation properties p0 and ρ0
decrease across the shock due to the dissipative processes inside the wavefront.
Normal Shock Propagating in a Still Medium
Frequently, one needs to calculate the properties of flow due to the propagation
of a shock wave through a still medium, for example, due to an explosion. The
transformation necessary to analyze this problem is indicated in Figure 16.11. The
left panel shows a stationary shock, with incoming and outgoing velocities u1 and u2 ,
respectively. On this flow we add a velocity u1 directed to the left, so that the fluid
entering the shock is stationary, and the fluid downstream of the shock is moving to
the left at a speed u1 − u2 , as shown in the right panel of the figure. This is consistent
with our remark in Section 2 that the passage of a compression wave “pushes” the
fluid forward in the direction of propagation of the wave. The shock speed is therefore
u1 , with a supersonic Mach number M1 = u1 /c1 > 1. It follows that a finite pressure disturbance propagates through a still fluid at supersonic speed, in contrast to
infinitesimal waves that propagate at the sonic speed. The expressions for all the thermodynamic properties of the flow, such as those given in equations (16.31)–(16.36),
are still applicable.
Shock Structure
We shall now note a few points about the structure of a shock wave. The viscous and
heat conductive processes within the shock wave result in an entropy increase across
the front. However, the magnitude of the viscosity µ and thermal conductivity k only
determines the thickness of the front and not the magnitude of the entropy increase.
The entropy increase is determined solely by the upstream Mach number as shown
by equation (16.36). We shall also see later that the wave drag experienced by a body
due to the appearance of a shock wave is independent of viscosity or thermal conductivity. (The situation here is analogous to the viscous dissipation in fully turbulent
739
6. Normal Shock Wave
flows (Chapter 13, Section 8), in which the dissipation rate ε is determined by the
velocity and length scales of a large-scale turbulence field (ε ∼ u3 / l) and not by the
magnitude of the viscosity; a change in viscosity merely changes the scale at which
the dissipation takes place (namely, the Kolmogorov microscale).)
The shock wave is in fact a very thin boundary layer. However, the velocity
gradient du/dx is entirely longitudinal, in contrast to the lateral velocity gradient
involved in a viscous boundary layer near a solid surface. Analysis shows that the
thickness δ of a shock wave is given by
δu
∼ 1,
ν
where the left-hand side is a Reynolds number based on the velocity change across
the shock, its thickness, and the average value of viscosity. Taking a typical value for
air of ν ∼ 10−5 m2 /s, and a velocity jump of u ∼ 100 m/s, we obtain a shock
thickness of
δ ∼ 10−7 m.
This is not much larger than the mean free path (average distance traveled by a
molecule between collisions), which suggests that the continuum hypothesis becomes
of questionable validity in analyzing shock structure.
To gain some insight into the structure of shock waves, we shall consider the
one-dimensional steady Navier–Stokes equations, including heat conduction and
Newtonian viscous stresses. Despite the fact that the significant length scale for the
structure pushes the limits of validity of the continuum formulation, the solution we
obtain provides a smooth transition between upstream and downstream states, looks
reasonable, and agrees with experiments and kinetic theory models for upstream Mach
numbers less than about 2. The equations for conservation of mass, momentum, and
energy are, respectively,
d(ρu)/dx = 0
ρudu/dx + dp/dx = d(µ′′ du/dx)/dx, µ′′ = 2µ + λ
ρudh/dx − udp/dx = µ′′ (du/dx)2 + d(kdT /dx)/dx.
By adding to the energy equation the product of u with the momentum equation, these
can be integrated once to yield,
ρu = m
mu + p − µ′′ du/dx = mV
m(h + u2 /2) − µ′′ udu/dx − kdT /dx = mI,
where m, V , I are the constants of integration. These are evaluated upstream (state 1)
and downstream (state 2) where gradients vanish and yield the Rankine-Hugoniot
relations derived above. We also need the equations of state for a perfect gas with
constant specific heats to solve for the structure: h = Cp T , p = ρRT . Multiplying
the energy equation by Cp /k we obtain the form
(mCp /k)(Cp T + u2 /2) − (µ′′ Cp /k)d(u2 /2)/dx − d(Cp T )/dx = mCp I /k.
740
Compressible Flow
This has an exact integral in the special case Pr ′′ ≡ µ′′ Cp /k = 1. This was found by
Becker in 1922. If Stokes relation is assumed [(4.42)], 3λ + 2µ = 0 then µ′′ = 4µ/3
and Pr = µCp /k = 3/4, which is quite close to the actual value for air. The Becker
integral is Cp T + u2 /2 = I . Eliminating all variables but u from the momentum
equation, using the equations of state, mass conservation, and the energy integral,
mu + (m/u)(R/Cp )(I − u2 /2) − µ′′ du/dx = mV .
With Cp /R = γ /(γ − 1), multiplying by u/m, we obtain
−[2γ /(γ + 1)](µ′′ /m)udu/dx = −u2 + [2γ /(γ + 1)]uV − 2I (γ − 1)/(γ + 1)
≡ (U1 − U )(U − U2 )
Divide by V 2 and let u/V = U . The equation for the structure becomes
−U (U1 − U )−1 (U − U2 )−1 dU = [(γ + 1)/2γ ](m/µ′′ )dx,
where the roots of the quadratic are
U1,2 = [γ /(γ + 1)]{1 ± [1 − 2(γ 2 − 1)I /(γ 2 V 2 )]1/2 },
the dimensionless speeds far up- and downstream of the shock. The left-hand side
of the equation for the structure is rewritten in terms of partial fractions and then
integrated to obtain
[U1 ln(U1 − U ) − U2 ln(U − U2 )]/(U1 − U2 )
= [(γ + 1)/(2γ )]m dx/µ′′ ≡ [(γ + 1)/(2γ )]η
The structure is shown in Figure 16.12 in terms of the stretched coordinate
η = (m/µ′′ )dx where µ′′ is often a strong function of temperature and thus of x.
A similar structure is obtained for all except quite small values of Pr ′′ . In the limit
Pr ′′ → 0, Hayes (1958) points out that there must be a “shock within a shock” because
U
0.9
U1 = .848…
0.8
0.7
0.6
05
0.4
U2 = .318…
0.3
0.2
0.1
0
–10
–8
–6
–4
–2
η
0
2
4
6
Figure 16.12 Shock structure velocity profile for the case U1 = 0.848485, U2 = 0.31818, corresponding
to M1 = 2.187.
7. Operation of Nozzles at Different Back Pressures
heat conduction alone cannot provide the entire structure. In fact, Becker (1922)
(footnote, p. 341) credits Prandtl for originating this idea. Cohen and Moraff (1971)
provided the structure of both the outer (heat conducting) and inner (isothermal viscous) shocks. The variable η is a dimensionless length scale measured very roughly
in units of mean free paths. We see that a measure of shock thickness is of the order
of 5 mean free paths.
7. Operation of Nozzles at Different Back Pressures
Nozzles are used to accelerate a fluid stream and are employed in such systems as
wind tunnels, rocket motors, and steam turbines. A pressure drop is maintained across
it. In this section we shall examine the behavior of a nozzle as the exit pressure is
varied. It will be assumed that the fluid is supplied from a large reservoir where the
pressure is maintained at a constant value p0 (the stagnation pressure), while the
“back pressure” pB in the exit chamber is varied. In the following discussion, we
need to note that the pressure pexit at the exit plane of the nozzle must equal the back
pressure pB if the flow at the exit plane is subsonic, but not if it is supersonic. This
must be true because sharp pressure changes are only allowed in a supersonic flow.
Convergent Nozzle
Consider first the case of a convergent nozzle shown in Figure 16.13, which examines
a sequence of states a through c during which the back pressure is gradually lowered.
For curve a, the flow throughout the nozzle is subsonic. As pB is lowered, the Mach
number increases everywhere and the mass flux through the nozzle also increases.
This continues until sonic conditions are reached at the exit, as represented by curve b.
Further lowering of the back pressure has no effect on the flow inside the nozzle. This
is because the fluid at the exit is now moving downstream at the velocity at which no
pressure changes can propagate upstream. Changes in pB therefore cannot propagate
upstream after sonic conditions are reached at the exit. We say that the nozzle at this
stage is choked because the mass flux cannot be increased by further lowering of
back pressure. If pB is lowered further (curve c in Figure 16.13), supersonic flow is
generated outside the nozzle, and the jet pressure adjusts to pB by means of a series
of “oblique expansion waves,” as schematically indicated by the oscillating pressure
distribution for curve c. (The concepts of oblique expansion waves and oblique shock
waves will be explained in Sections 10 and 11. It is only necessary to note here that
they are oriented at an angle to the direction of flow, and that the pressure decreases
through an oblique expansion wave and increases through an oblique shock wave.)
Convergent–Divergent Nozzle
Now consider the case of a convergent–divergent passage (Figure 16.14). Completely
subsonic flow applies to curve a. As pB is lowered to pb , sonic condition is reached
at the throat. On further reduction of the back pressure, the flow upstream of the
throat does not respond, and the nozzle has “choked” in the sense that it is allowing
the maximum mass flow rate for the given values of p0 and throat area. There is a
741
742
Compressible Flow
Figure 16.13 Pressure distribution along a convergent nozzle for different values of back pressure pB :
(a) diagram of nozzle; and (b) pressure distributions.
range of back pressures, shown by curves c and d, in which the flow initially becomes
supersonic in the divergent portion, but then adjusts to the back pressure by means of
a normal shock standing inside the nozzle. The flow downstream of the shock is, of
course, subsonic. In this range the position of the shock moves downstream as pB is
decreased, and for curve d the normal shock stands right at the exit plane. The flow
in the entire divergent portion up to the exit plane is now supersonic and remains so
on further reduction of pB . When the back pressure is further reduced to pe , there
is no normal shock anywhere within the nozzle, and the jet pressure adjusts to pB
by means of oblique shock waves outside the exit plane. These oblique shock waves
vanish when pB = pf . On further reduction of the back pressure, the adjustment to
pB takes place outside the exit plane by means of oblique expansion waves.
Example 16.2
A convergent–divergent nozzle is operating under off-design conditions, resulting in
the presence of a shock wave in the diverging portion. A reservoir containing air at
7. Operation of Nozzles at Different Back Pressures
Figure 16.14 Pressure distribution along a convergent–divergent nozzle for different values of back
pressure pB . Flow patterns for cases c, d, e, and g are indicated schematically on the right. H. W. Liepmann
and A. Roshko, Elements of Gas Dynamics, Wiley, New York 1957 and reprinted with the permission of
Dr. Anatol Roshko.
400 kPa and 800 K supplies the nozzle, whose throat area is 0.2 m2 . The upstream
Mach number of the shock is M1 = 2.44. The area at the exit is 0.7 m2 . Find the area
at the location of the shock and the exit temperature.
Solution: Figure 16.15 shows the profile of the nozzle, where sections 1 and 2
represent conditions across the shock. As a shock wave can exist only in a supersonic
stream, we know that sonic conditions are reached at the throat, and the throat area
equals the critical area A∗ . The values given are therefore
p0 = 400 kPa,
T0 = 800 K,
743
744
Compressible Flow
Figure 16.15 Example 16.2.
Athroat = A∗1 = 0.2 m2 ,
M1 = 2.44,
A3 = 0.7 m2 .
Note that A∗ is constant upstream of the shock, up to which the process is isentropic;
this is why we have set Athroat = A∗1 .
The technique of solving this problem is to proceed downstream from the given
stagnation conditions. Corresponding to the Mach number M1 = 2.44, the isentropic
table Table 16.1 gives
A1
= 2.5,
A∗1
so that
A1 = A2 = (2.5)(0.2) = 0.5 m2 .
This is the area at the location of the shock. Corresponding to M1 = 2.44, the normal
shock Table 16.2 gives
M2 = 0.519,
p02
= 0.523.
p01
There is no loss of stagnation pressure up to section 1, so that
p01 = p0 ,
which gives
p02 = 0.523p0 = 0.523(400) = 209.2 kPa.
745
7. Operation of Nozzles at Different Back Pressures
The value of A∗ changes across a shock wave. The ratio A2 /A∗2 can be found from
the isentropic table (Table 16.1) corresponding to a Mach number of M2 = 0.519.
(Note that A∗2 simply denotes the area that would be reached if the flow from state 2
were accelerated isentropically to sonic conditions.) Corresponding to M2 = 0.519,
Table 16.1 gives
A2
= 1.3,
A∗2
TABLE 16.2
One-Dimensional Normal-Shock Relations (γ = 1.4)
M1
M2
p2 /p1
T2 /T1
(p0 )2 /(p0 )1
M1
M2
p2 /p1
T2 /T1
(p0 )2 /(p0 )1
1.00
1.02
1.04
1.06
1.000
0.980
0.962
0.944
1.000
1.047
1.095
1.144
1.000
1.013
1.026
1.039
1.000
1.000
1.000
1.000
1.96
1.98
2.00
2.02
0.584
0.581
0.577
0.574
4.315
4.407
4.500
4.594
1.655
1.671
1.688
1.704
0.740
0.730
0.721
0.711
1.08
1.10
1.12
1.14
0.928
0.912
0.896
0.882
1.194
1.245
1.297
1.350
1.052
1.065
1.078
1.090
0.999
0.999
0.998
0.997
2.04
2.06
2.08
2.10
0.571
0.567
0.564
0.561
4.689
4.784
4.881
4.978
1.720
1.737
1.754
1.770
0.702
0.693
0.683
0.674
1.16
1.18
1.20
1.22
0.868
0.855
0.842
0.830
1.403
1.458
1.513
1.570
1.103
1.115
1.128
1.140
0.996
0.995
0.993
0.991
2.12
2.14
2.16
2.18
0.558
0.555
0.553
0.550
5.077
5.176
5.277
5.378
1.787
1.805
1.822
1.837
0.665
0.656
0.646
0.637
1.24
1.26
1.28
1.30
0.818
0.807
0.796
0.786
1.627
1.686
1.745
1.805
1.153
1.166
1.178
1.191
0.988
0.986
0.983
0.979
2.20
2.22
2.24
2.26
0.547
0.544
0.542
0.539
5.480
5.583
5.687
5.792
1.857
1.875
1.892
1.910
0.628
0.619
0.610
0.601
1.32
1.34
1.36
1.38
0.776
0.766
0.757
0.748
1.866
1.928
1.991
2.055
1.204
1.216
1.229
1.242
0.976
0.972
0.968
0.963
2.28
2.30
2.32
2.34
0.537
0.534
0.532
0.530
5.898
6.005
6.113
6.222
1.929
1.947
1.965
1.984
0.592
0.583
0.575
0.566
1.40
1.42
1.44
1.46
0.740
0.731
0.723
0.716
2.120
2.186
2.253
2.320
1.255
1.268
1.281
1.294
0.958
0.953
0.948
0.942
2.36
2.38
2.40
2.42
0.527
0.525
0.523
0.521
6.331
6.442
6.553
6.666
2.003
2.021
2.040
2.060
0.557
0.549
0.540
0.532
1.48
1.50
1.52
1.54
0.708
0.701
0.694
0.687
2.389
2.458
2.529
2.600
1.307
1.320
1.334
1.347
0.936
0.930
0.923
0.917
2.44
2.46
2.48
2.50
0.519
0.517
0.515
0.513
6.779
6.894
7.009
7.125
2.079
2.098
2.118
2.138
0.523
0.515
0.507
0.499
1.56
1.58
1.60
1.62
0.681
0.675
0.668
0.663
2.673
2.746
2.820
2.895
1.361
1.374
1.388
1.402
0.910
0.903
0.895
0.888
2.52
2.54
2.56
2.58
0.511
0.509
0.507
0.506
7.242
7.360
7.479
7.599
2.157
2.177
2.198
2.218
0.491
0.483
0.475
0.468
1.64
1.66
0.657
0.651
2.971
3.048
1.416
1.430
0.880
0.872
2.60
2.62
0.504
0.502
7.720
7.842
2.238
2.260
0.460
0.453
746
Compressible Flow
TABLE 16.2
(Continued)
M1
M2
p2 /p1
T2 /T1
(p0 )2 /(p0 )1
M1
M2
p2 /p1
T2 /T1
(p0 )2 /(p0 )1
1.68
1.70
0.646
0.641
3.126
3.205
1.444
1.458
0.864
0.856
2.64
2.66
0.500
0.499
7.965
8.088
2.280
2.301
0.445
0.438
1.72
1.74
1.76
1.78
0.635
0.631
0.626
0.621
3.285
3.366
3.447
3.530
1.473
1.487
1.502
1.517
0.847
0.839
0.830
0.821
2.68
2.70
2.72
2.74
0.497
0.496
0.494
0.493
8.213
8.338
8.465
8.592
2.322
2.343
2.364
2.386
0.431
0.424
0.417
0.410
1.80
1.82
1.84
1.86
0.617
0.612
0.608
0.604
3.613
3.698
3.783
3.869
1.532
1.547
1.562
1.577
0.813
0.804
0.795
0.786
2.76
2.78
2.80
2.82
0.491
0.490
0.488
0.487
8.721
8.850
8.980
9.111
2.407
2.429
2.451
2.473
0.403
0.396
0.389
0.383
1.88
1.90
1.92
1.94
2.92
2.94
2.96
0.600
0.596
0.592
0.588
0.480
0.479
0.478
3.957
4.045
4.134
4.224
9.781
9.918
10.055
1.592
1.608
1.624
1.639
2.586
2.609
2.632
0.777
0.767
0.758
0.749
0.352
0.346
0.340
2.84
2.86
2.88
2.90
2.98
3.00
0.485
0.484
0.483
0.481
0.476
0.475
9.243
9.376
9.510
9.645
10.194
10.333
2.496
2.518
2.541
2.563
2.656
2.679
0.376
0.370
0.364
0.358
0.334
0.328
which gives
A∗2 =
0.5
A2
=
= 0.3846 m2 .
1.3
1.3
The flow from section 2 to section 3 is isentropic, during which A∗ remains
constant. Thus
A3
A3
0.7
= ∗ =
= 1.82.
A∗3
A2
0.3846
We should now find the conditions at the exit from the isentropic table (Table 16.1).
However, we could locate the value of A/A∗ = 1.82 either in the supersonic or the
subsonic branch of the table. As the flow downstream of a normal shock can only
be subsonic, we should use the subsonic branch. Corresponding to A/A∗ = 1.82,
Table 16.1 gives
T3
= 0.977.
T03
The stagnation temperature remains constant in an adiabatic process, so that
T03 = T0 . Thus
T3 = 0.977(800) = 782 K.
747
8. Effects of Friction and Heating in Constant-Area Ducts
8. Effects of Friction and Heating in Constant-Area Ducts
In a duct of constant area, the equations of mass, momentum, and energy reduced to
one-dimensional steady form become
ρ1 u1 = ρ2 u2 ,
p1 + ρ1 u21 = p2 + ρ2 u22 + p1 f,
h1 + 21 u21 + h1 q = h2 + 21 u22 .
Here, f = (fσ )x /(p1 A) is a dimensionless friction parameter and q = Q/ h1 is a
dimensionless heating parameter. In terms of Mach number, for a perfect gas with
constant specific heats, the momentum and energy equations become, respectively,
p1 1 + γ M12 − f = p2 1 + γ M22 ,
γ −1 2
γ −1 2
M1 + q = h2 1 +
M2 .
h1 1 +
2
2
Using mass conservation, the equation of state p = ρRT , and the definition of Mach
number, all thermodynamic variables can be eliminated resulting in
1 + γ M22
1 + ((γ − 1)/2)M12 + q 1/2
M2
=
.
M1
1 + γ M12 − f
1 + ((γ − 1)/2)M22
Bringing the unknown M2 to the left-hand side and assuming q and f are specified
along with M1 ,
M22 1 + ((γ − 1)/2)M22
1 + γ M22
2
=
M12 1 + ((γ − 1)/2)M12 + q
1 + γ M12 − f
2
≡ A.
This is a biquadratic equation for M2 with the solution
M22 =
−(1 − 2Aγ ) ± [1 − 2A(γ + 1)]1/2
.
(γ − 1) − 2Aγ 2
(16.37)
Figures 16.16 and 16.17 are plots of equation (16.37), M2 = F (M1 ) first with f
as a parameter (16.16) and q = 0 and then with q as a parameter and f = 0
(16.17). Generally, we specify the properties of the flow at the inlet station (station 1)
and wish to calculate the properties at the outlet (station 2). Here, we will regard
the dimensionless friction and heat transfer f and q as specified. Then we see that
once M2 is calculated from (16.37), all of the other properties may be obtained from
the dimensionless formulation of the conservation laws. When q and f = 0, two
solutions are possible: the trivial solution M1 = M2 and the normal shock solution
that we obtained in Section 6 in the preceding. We also showed that the upper left
branch of the solution M2 > 1 when M1 < 1 is inaccessible because it violates the
second law of thermodynamics, that is, it results in a spontaneous decrease of entropy.
748
Compressible Flow
Figure 16.16 Flow in a constant-area duct with friction f as parameter; q = 0. Upper left quadrant is
inaccessible because S < 0. γ = 1.4.
Effect of Friction
Referring to the left branch of Figure 16.16, the solution indicates that for M2 > M1 so
that friction accelerates a subsonic flow. Then the pressure, density, and temperature
are all diminished with respect to the entrance values. How can friction make the
flow go faster? Friction is manifested by boundary layers at the walls. The boundary
layer displacement thickness grows downstream so that the flow behaves as if it is in
a convergent duct, which, as we have seen, is a subsonic nozzle. We will discuss in
what follows what actually happens when there is no apparent solution for M2 . When
M1 is supersonic, two solutions are generally possible—one for which 1 < M2 < M1
and the other where M2 < 1. They are connected by a normal shock. Whether or not
a shock occurs depends on the downstream pressure. There is also the possibility of
M1 insufficiently large or f too large so that no solution is indicated. We will discuss
that in the following but note that the two solutions coalesce when M2 = 1 and the
8. Effects of Friction and Heating in Constant-Area Ducts
Figure 16.17 Flow in a constant-area duct with heating/cooling q as parameter; f = 0. Upper left
quadrant is inaccessible because S < 0. γ = 1.4.
flow is said to be choked. At this condition the maximum mass flow is passed by the
duct. In the case 1 < M2 < M1 , the flow is decelerated and the pressure, density,
and temperature all increase in the downstream direction. The stagnation pressure is
always decreased by friction as the entropy is increased.
Effect of Heat Transfer
The range of solutions is twice as rich in this case as q may take both signs.
Figure 16.17 shows that for q > 0 solutions are similar in most respects to those
with friction (f > 0). Heating accelerates a subsonic flow and lowers the pressure
and density. However, heating generally increases the fluid temperature except in
√
the limited range 1/ γ < M1 < 1 in which the tendency to accelerate the fluid
is greater than the ability of the heat flux to raise the temperature. The energy from
heat addition goes preferentially into increasing the kinetic energy of the fluid. The
fluid temperature is decreased by heating in this limited range of Mach number. The
supersonic branch M2 > 1 when M1 < 1 is inaccessible because those solutions
violate the second law of thermodynamics. Again, as with f too large or M1 too close
to 1, there is a possibility with q too large of no solution indicated; this is discussed
in what follows. When M1 > 1, two solutions for M2 are generally possible and they
are connected by a normal shock. The shock is absent if the downstream pressure is
low and present if the downstream pressure is high. Although q > 0 (and f > 0)
does not always indicate a solution (if the flow has been choked), there will always
749
750
Compressible Flow
be a solution for q < 0. Cooling a supersonic flow accelerates it, thus decreasing
its pressure, temperature, and density. If no shock occurs, M2 > M1 . Conversely,
cooling a subsonic flow decelerates it so that the pressure and density increase. The
temperature decreases when heat is removed from the flow except in the limited range
√
1/ γ < M1 < 1 in which the heat removal decelerates the flow so rapidly that the
temperature increases.
For high molecular weight gases, near critical conditions (high pressure, low
temperature), gasdynamic relationships as developed here for perfect gases may be
completely different. Cramer and Fry (1993) found that such gases may support
expansion shocks, accelerated flow through “antithroats,” and generally behave in
unfamiliar ways.
Choking by Friction or Heat Addition
We can see from Figures 16.16 and 16.17 that heating a flow or accounting for
friction in a constant-area duct will make that flow tend towards sonic conditions.
For any given M1 , the maximum f or q > 0 that is permissible is the one for which
M = 1 at the exit station. The flow is then said to be choked, and no more mass/time
can flow through that duct. This is analogous to flow in a convergent duct. Imagine
pouring liquid through a funnel from one container into another. There is a maximum
volumetric flow rate that can be passed by the funnel, and beyond that flow rate, the
funnel overflows. The same thing happens here. If f or q is too large, such that no
(steady-state) solution is possible, there is an external adjustment that reduces the
mass flow rate to that for which the exit speed is just sonic. Both for M1 < 1 and
M1 > 1 the limiting curves for f and q indicating choked flow intersect M2 = 1 at
right angles. Qualitatively, the effect is the same as choking by area contraction.
9. Mach Cone
So far in this chapter we have considered one-dimensional flows in which the flow
properties varied only in the direction of flow. In this section we begin our study of
wave motions in more than one dimension. Consider a point source emitting infinitesimal pressure disturbances in a still fluid in which the speed of sound is c. If the point
disturbance is stationary, then the wavefronts are concentric spheres. This is shown
in Figure 16.18a, where the wavefronts at intervals of t are shown.
Now suppose that the source propagates to the left at speed U < c. Figure 16.18b
shows four locations of the source, that is, 1 through 4, at equal intervals of time t,
with point 4 being the present location of the source. At point 1, the source emitted
a wave that has spherically expanded to a radius of 3c t in an interval of time
3 t. During this time the source has moved to location 4, at a distance of 3U t
from point 1. The figure also shows the locations of the wavefronts emitted while the
source was at points 2 and 3. It is clear that the wavefronts do not intersect because
U < c. As in the case of the stationary source, the wavefronts propagate everywhere
in the flow field, upstream and downstream. It therefore follows that a body moving
at a subsonic speed influences the entire flow field; information propagates upstream
as well as downstream of the body.
751
9. Mach Cone
Figure 16.18 Wavefronts emitted by a point source in a still fluid when the source speed U is: (a) U = 0;
(b) U < c; and (c) U > c.
Now consider a case where the disturbance moves supersonically at U > c
(Figure 16.18c). In this case the spherically expanding wavefronts cannot catch up
with the faster moving disturbance and form a conical tangent surface called the Mach
cone. In plane two-dimensional flow, the tangent surface is in the form of a wedge,
and the tangent lines are called Mach lines. An examination of the figure shows that
the half-angle of the Mach cone (or wedge), called the Mach angle µ, is given by
sin µ = (c t)/(U t), so that
sin µ =
1
.
M
(16.38)
752
Compressible Flow
The Mach cone becomes wider as M decreases and becomes a plane front (that is,
µ = 90◦ ) when M = 1.
The point source considered here could be part of a solid body, which sends out
pressure waves as it moves through the fluid. Moreover, Figure 16.18c applies equally
if the point source is stationary and the fluid is approaching at a supersonic speed U .
It is clear that in a supersonic flow an observer outside the Mach cone would not
“hear” a signal emitted by a point disturbance, hence this region is called the zone
of silence. In contrast, the region inside the Mach cone is called the zone of action,
within which the effects of the disturbance are felt. This explains why the sound of a
supersonic airplane does not reach an observer until the Mach cone arrives, after the
plane has passed overhead.
At every point in a planar supersonic flow there are two Mach lines, oriented at
±µ to the local direction of flow. Information propagates along these lines, which
are the characteristics of the governing differential equation. It can be shown that the
nature of the governing differential equation is hyperbolic in a supersonic flow and
elliptic in a subsonic flow.
10. Oblique Shock Wave
In Section 6 we examined the case of a normal shock wave, oriented perpendicular to
the direction of flow, in which the velocity changes from supersonic to subsonic values.
However, a shock wave can also be oriented obliquely to the flow (Figure 16.19a),
the velocity changing from V1 to V2 . The flow can be analyzed by considering a
normal shock across which the normal velocity varies from u1 to u2 and superposing
a velocity v parallel to it (Figure 16.19b). By considering conservation of momentum
in a direction tangential to the shock, we may show that v is unchanged across a shock
(Exercise 12). The magnitude and direction of the velocities on the two sides of the
shock are
u21 + v 2
V 2 = u2 + v 2
V1 =
oriented at σ = tan−1 (u1 /v),
oriented at σ − δ = tan−1 (u2 /v).
Figure 16.19 (a) Oblique shock wave in which δ = deflection angle and σ = shock angle; and
(b) analysis by considering a normal shock and superposing a velocity v parallel to the shock.
753
10. Oblique Shock Wave
The normal Mach numbers are
Mn1 = u1 /c1 = M1 sin σ > 1,
Mn2 = u2 /c2 = M2 sin(σ − δ) < 1.
Because u2 < u1 , there is a sudden change of direction of flow across the shock; in
fact the flow turns toward the shock by an amount δ. The angle σ is called the shock
angle or wave angle and δ is called the deflection angle.
Superposition of the tangential velocity v does not affect the static properties,
which are therefore the same as those for a normal shock. The expressions for the ratios
p2 /p1 , ρ2 /ρ1 , T2 /T1 , and (S2 −S1 )/Cv are therefore those given by equations (16.31),
(16.33)–(16.35), if M1 is replaced by the normal component of the upstream Mach
number M1 sin σ . For example,
p2
2γ
=1+
(M 2 sin2 σ − 1),
p1
γ +1 1
(16.39)
(γ + 1)M12 sin2 σ
u1
tan σ
ρ2
=
=
=
.
ρ1
u2
tan (σ − δ)
(γ − 1)M12 sin2 σ + 2
(16.40)
The normal shock table, Table 16.2, is therefore also applicable to oblique shock
waves if we use M1 sin σ in place of M1 .
The relation between the upstream and downstream Mach numbers can be found
from equation (16.32) by replacing M1 by M1 sin σ and M2 by M2 sin (σ − δ). This
gives
M22 sin2 (σ − δ) =
(γ − 1)M12 sin2 σ + 2
2γ M12 sin2 σ + 1 − γ
.
(16.41)
An important relation is that between the deflection angle δ and the shock angle σ
for a given M1 , given in equation (16.40). Using the trigonometric identity for tan (σ −
δ), this becomes
tan δ = 2 cot σ
M12 sin2 σ − 1
M12 (γ + cos 2σ ) + 2
.
(16.42)
A plot of this relation is given in Figure 16.20. The curves represent δ vs σ for constant
M1 . The value of M2 varies along the curves, and the locus of points corresponding
to M2 = 1 is indicated. It is apparent that there is a maximum deflection angle δmax
for oblique shock solutions to be possible; for example, δmax = 23 ◦ for M1 = 2.
For a given M1 , δ becomes zero at σ = π/2 corresponding to a normal shock, and
at σ = µ = sin−1 (1/M1 ) corresponding to the Mach angle. For a fixed M1 and
δ < δmax , there are two possible solutions: a weak shock corresponding to a smaller
σ , and a strong shock corresponding to a larger σ . It is clear that the flow downstream
of a strong shock is always subsonic; in contrast, the flow downstream of a weak
shock is generally supersonic, except in a small range in which δ is slightly smaller
than δmax .
754
Compressible Flow
Figure 16.20 Plot of oblique shock solution. The strong shock branch is indicated by dashed lines, and
the heavy dotted line indicates the maximum deflection angle δmax . (From NACA Report 1135.)
Generation of Oblique Shock Waves
Consider the supersonic flow past a wedge of half-angle δ, or the flow over a wall
that turns inward by an angle δ (Figure 16.21). If M1 and δ are given, then σ can be
obtained from Figure 16.20, and Mn2 (and therefore M2 = Mn2 /sin(σ − δ)) can be
obtained from the shock table, Table 16.2. An attached shock wave, corresponding
to the weak solution, forms at the nose of the wedge, such that the flow is parallel
to the wedge after turning through an angle δ. The shock angle σ decreases to the
Mach angle µ1 = sin−1 (1/M1 ) as the deflection δ tends to zero. It is interesting that
the corner velocity in a supersonic flow is finite. In contrast, the corner velocity in
a subsonic (or incompressible) flow is either zero or infinite, depending on whether
the wall shape is concave or convex. Moreover, the streamlines in Figure 16.21 are
straight, and computation of the field is easy. By contrast, the streamlines in a subsonic
flow are curved, and the computation of the flow field is not easy. The basic reason
for this is that, in a supersonic flow, the disturbances do not propagate upstream of
Mach lines or shock waves emanating from the disturbances, hence the flow field can
be constructed step by step, proceeding downstream. In contrast, the disturbances
10. Oblique Shock Wave
Figure 16.21 Oblique shocks in supersonic flow.
Figure 16.22 Detached shock.
propagate both upstream and downstream in a subsonic flow, so that all features in
the entire flow field are related to each other.
As δ is increased beyond δmax , attached oblique shocks are not possible, and
a detached curved shock stands in front of the body (Figure 16.22). The central
streamline goes through a normal shock and generates a subsonic flow in front of the
wedge. The strong shock solution of Figure 16.20 therefore holds near the nose of the
body. Farther out, the shock angle decreases, and the weak shock solution applies.
If the wedge angle is not too large, then the curved detached shock in Figure 16.22
becomes an oblique attached shock as the Mach number is increased. In the case
of a blunt-nosed body, however, the shock at the leading edge is always detached,
although it moves closer to the body as the Mach number is increased.
We see that shock waves may exist in supersonic flows and their location and
orientation adjust to satisfy boundary conditions. In external flows, such as those just
755
756
Compressible Flow
described, the boundary condition is that streamlines at a solid surface must be tangent
to that surface. In duct flows the boundary condition locating the shock is usually the
downstream pressure.
The Weak Shock Limit
A simple and useful expression can be derived for the pressure change across a weak
shock by considering the limiting case of a small deflection angle δ. We first need to
simplify equation (16.42) by noting that as δ → 0, the shock angle σ tends to the
Mach angle µ1 = sin−1 (1/M1 ).
Also from equation (16.39) we note that (p2 −p1 )/p1 → 0 as M12 sin2 σ −1 → 0,
(as σ → µ and δ → 0). Then from equations (16.39) and (16.42)
1
γ + 1 p2 − p1
.
(16.43)
tan δ = 2 cot σ
2
2γ
p1
M1 (γ + 1 − 2 sin2 σ ) + 2
As δ → 0, tan δ ≈ δ, cot µ =
M12 − 1, sin σ ≈ 1/M1 , and
p2 − p 1
≃
p1
γ M12
δ.
(16.44)
M12 − 1
The interesting point is that relation (16.44) is also applicable to a weak expansion
wave and not just a weak compression wave. By this we mean that the pressure
increase due to a small deflection of the wall toward the flow is the same as the
pressure decrease due to a small deflection of the wall away from the flow. This is
because the entropy change across a shock goes to zero much faster than the rate at
which the pressure difference across the wave decreases as our study of normal shock
waves has shown. Very weak “shock waves” are therefore approximately isentropic
or reversible. Relationships for a weak shock wave can therefore be applied to a weak
expansion wave, except for some sign changes. In Section 12, equation (16.44) will
be applied in estimating the lift and drag of a thin airfoil in supersonic flow.
11. Expansion and Compression in Supersonic Flow
Consider the supersonic flow over a gradually curved wall (Figure 16.23). The
wavefronts are now Mach lines, inclined at an angle of µ = sin−1 (1/M) to the local
direction of flow. The flow orientation and Mach number are constant on each Mach
line. In the case of compression, the Mach number decreases along the flow, so that
the Mach angle increases. The Mach lines therefore coalesce and form an oblique
shock. In the case of gradual expansion, the Mach number increases along the flow
and the Mach lines diverge.
If the wall has a sharp deflection away from the approaching stream, then the
pattern of Figure 16.23b takes the form of Figure 16.24. The flow expands through
a “fan” of Mach lines centered at the corner, called the Prandtl–Meyer expansion
11. Expansion and Compression in Supersonic Flow
Figure 16.23 Gradual compression and expansion in supersonic flow: (a) gradual compression, resulting
in shock formation; and (b) gradual expansion.
Figure 16.24 The Prandtl–Meyer expansion fan.
fan. The Mach number increases through the fan, with M2 > M1 . The first Mach line
is inclined at an angle of µ1 to the local flow direction, while the last Mach line is
inclined at an angle of µ2 to the local flow direction. The pressure falls gradually along
a streamline through the fan. (Along the wall, however, the pressure remains constant
along the upstream wall, falls discontinuously at the corner, and then remains constant
along the downstream wall.) Figure 16.24 should be compared with Figure 16.21, in
which the wall turns inward and generates a shock wave. By contrast, the expansion
in Figure 16.24 is gradual and isentropic.
The flow through a Prandtl–Meyer fan is calculated as follows. From
Figure 16.19b, conservation of momentum tangential to the shock shows that the
tangential velocity is unchanged, or
V1 cos σ = V2 cos(σ − δ) = V2 (cos σ cos δ + sin σ sin δ).
We are concerned here with very small deflections, δ → 0 so σ → µ. Here,
√ cos δ ≈ 1,
sin δ ≈ δ, V1 ≈ V2 (1 + δ tan σ ), so (V2 − V1 )/V1 ≈ δ tan σ ≈ −δ/ M12 − 1.
Regarding this as appropriate for infinitesimal
change in V for an infinitesimal
√
deflection, we can write this as dδ = −dV M 2 − 1/V (first
√ quadrant deflection).
Because V = Mc, dV /V = dM/M + dc/c. With c = γ RT for a perfect gas,
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758
Compressible Flow
dc/c = dT /2T . Using equation (16.20) for adiabatic flow of a perfect gas, dT /T
= −(γ − 1)M dM/[1 + ((γ − 1)/2)M 2 ].
Then
√
dM
M2 − 1
dδ = −
.
M
1 + ((γ − 1)/2)M 2
Integrating δ from 0 (radians) and M from 1 gives
δ + ν(M) = const.,
where
M
√
dM
M2 − 1
2 M
1
+
((γ
−
1)/2)M
1
γ +1
γ −1 2
−1
tan
(M − 1) − tan−1 M 2 − 1,
=
γ −1
γ +1
ν(M) =
(16.45)
√
M 2 − 1 originates
is called the Prandtl–Meyer function. The √sign of
2
from the identification of tan σ = tan µ = 1/ M − 1 for a first quadrant
deflection (upper
√ half-plane). For a fourth quadrant deflection (lower half-plane),
tan µ = −1/ M 2 − 1. For example, in Figure 16.23 we would write
δ1 + ν1 (M1 ) = δ2 + ν2 (M2 ),
where, for example, δ1 , δ2 , and M1 are given. Then
ν2 (M2 ) = δ1 − δ2 + ν1 (M1 ).
In panel (a), δ1 − δ2 < 0, so ν2 < ν1 and M2 < M1 . In panel (b), δ1 − δ2 > 0, so
ν2 > ν1 and M2 > M1 .
12. Thin Airfoil Theory in Supersonic Flow
Simple expressions can be derived for the lift and drag coefficients of an airfoil in
supersonic flow if the thickness and angle of attack are small. The disturbances caused
by a thin airfoil are small, and the total flow can be built up by superposition of small
disturbances emanating from points on the body. Such a linearized theory of lift and
drag was developed by Ackeret. Because all flow inclinations are small, we can use
the relation (16.44) to calculate the pressure changes due to a change in flow direction.
We can write this relation as
p − p∞
γ M2 δ
= ∞ ,
2 −1
p∞
M∞
(16.46)
12. Thin Airfoil Theory in Supersonic Flow
where p∞ and M∞ refer to the properties of the free stream, and p is the pressure at
a point where the flow is inclined at an angle δ to the free-stream direction. The sign
of δ determines the sign of (p − p∞ ).
To see how the lift and drag of a thin body in a supersonic stream can be estimated,
consider a flat plate inclined at a small angle α to a stream (Figure 16.25). At the
leading edge there is a weak expansion fan on the top surface and a weak oblique
shock on the bottom surface. The streamlines ahead of these waves are straight. The
streamlines above the plate turn through an angle α by expanding through a centered
fan, downstream of which they become parallel to the plate with a pressure p2 < p∞ .
The upper streamlines then turn sharply across a shock emanating from the trailing
edge, becoming parallel to the free stream once again. Opposite features occur for
the streamlines below the plate. The flow first undergoes compression across a shock
coming from the leading edge, which results in a pressure p3 > p∞ . It is, however,
not important to distinguish between shocks and expansion waves in Figure 16.25,
because the linearized theory treats them the same way, except for the sign of the
pressure changes they produce.
Figure 16.25 Inclined flat plate in a supersonic stream. The upper panel shows the flow pattern and the
lower panel shows the pressure distribution.
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Compressible Flow
The pressures above and below the plate can be found from equation (16.46),
giving
γ M2 α
p2 − p∞
= − ∞ ,
2 −1
p∞
M∞
γ M2 α
p3 − p∞
= ∞ .
2 −1
p∞
M∞
The pressure difference across the plate is therefore
2
2αγ M∞
p3 − p 2
=
.
2 −1
p∞
M∞
If b is the chord length, then the lift and drag forces per unit span are
2αγ M 2 p∞ b
,
L = (p3 − p2 )b cos α ≃ ∞
2 −1
M∞
2α 2 γ M 2 p∞ b
D = (p3 − p2 )b sin α ≃ ∞
.
2 −1
M∞
(16.47)
The lift coefficient is defined as
CL ≡
L
L
=
,
2 b
2 b
(1/2)ρ∞ U∞
(1/2)γp∞ M∞
where we have used the relation ρU 2 = γpM 2 . Using equation (16.47), the lift and
drag coefficients for a flat lifting surface are
4α
CL ≃
,
2 −1
M∞
4α 2
CD ≃
.
2 −1
M∞
(16.48)
These expressions do not hold at transonic speeds M∞ → 1, when the process of
linearization used here breaks down. The expression for the lift coefficient should be
compared to the incompressible expression CL ≃ 2π α derived in the preceding chapter. Note that the flow in Figure 16.25 does have a circulation because the velocities
at the upper and lower surfaces are parallel but have different magnitudes. However,
in a supersonic flow it is not necessary to invoke the Kutta condition (discussed in
the preceding chapter) to determine the magnitude of the circulation. The flow in
Figure 16.25 does leave the trailing edge smoothly.
The drag in equation (16.48) is the wave drag experienced by a body in a supersonic stream, and exists even in an inviscid flow. The d’Alembert paradox therefore
does not apply in a supersonic flow. The supersonic wave drag is analogous to the
gravity wave drag experienced by a ship moving at a speed greater than the velocity
761
Exercises
of surface gravity waves, in which a system of bow waves is carried with the ship.
The magnitude of the supersonic wave drag is independent of the value of the viscosity, although the energy spent in overcoming this drag is finally dissipated through
viscous effects within the shock waves. In addition to the wave drag, additional drags
due to viscous and finite-span effects, considered in the preceding chapter, act on a
real wing.
In this connection, it is worth noting the difference between the airfoil shapes
used in subsonic and supersonic airplanes. Low-speed airfoils have a streamlined
shape, with a rounded nose and a sharp trailing edge. These features are not helpful
in supersonic airfoils. The most effective way of reducing the drag of a supersonic
airfoil is to reduce its thickness. Supersonic wings are characteristically thin and have
a sharp leading edge.
Exercises
1. The critical area A∗ of a duct flow was defined in Section 4. Show that
the relation between A∗ and the actual area A at a section, where the Mach number
equals M, is that given by equation (16.23). This relation was not proved in the text.
[Hint: Write
ρ ∗ c∗
ρ ∗ ρ0 c ∗ c
ρ ∗ ρ0 T ∗ T0 1
A
=
=
=
.
A∗
ρu
ρ0 ρ c u
ρ0 ρ T0 T M
Then use the relations given in Section 4.]
2. The entropy change across a normal shock is given by equation (16.35). Show
that this reduces to expression (16.36) for weak shocks. [Hint: Let M12 − 1 ≪ 1.
Write the terms within the two brackets [ ] [ ] in equation (16.35) in the form
[1 + ε1 ][1 + ε2 ]γ , where ε1 and ε2 are small quantities. Then use series expansion
ln (1 + ε) = ε − ε2 /2 + ε 3 /3 + · · · . This gives equation (16.36) times a function of
M1 in which we can set M1 = 1.]
3. Show that the maximum velocity generated from a reservoir in which the
stagnation temperature equals T0 is
umax = 2Cp T0 .
What are the corresponding values of T and M?
4. In an adiabatic flow of air through a duct, the conditions at two points are
u1 = 250 m/s,
T1 = 300 K,
p1 = 200 kPa,
u2 = 300 m/s,
p2 = 150 kPa.
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Compressible Flow
Show that the loss of stagnation pressure is nearly 34.2 kPa. What is the entropy
increase?
5. A shock wave generated by an explosion propagates through a still
atmosphere. If the pressure downstream of the shock wave is 700 kPa, estimate the
shock speed and the flow velocity downstream of the shock.
6. A wedge has a half-angle of 50◦ . Moving through air, can it ever have an
attached shock? What if the half-angle were 40◦ ? [Hint: The argument is based entirely
on Figure 16.20.]
7. Air at standard atmospheric conditions is flowing over a surface at a Mach
number of M1 = 2. At a downstream location, the surface takes a sharp inward turn
by an angle of 20◦ . Find the wave angle σ and the downstream Mach number. Repeat
the calculation by using the weak shock assumption and determine its accuracy by
comparison with the first method.
8. A flat plate is inclined at 10◦ to an airstream moving at M∞ = 2. If the chord
length is b = 3 m, find the lift and wave drag per unit span.
9. A perfect gas is stored in a large tank at the conditions specified by po ,
To . Calculate the maximum mass flow rate that can exhaust through a duct of
cross-sectional area A. Assume that A is small enough that during the time of interest
po and To do not change significantly and that the flow is adiabatic.
10. For flow of a perfect gas entering a constant area duct at Mach number M1 ,
calculate the maximum admissable values of f, q for the same mass flow rate. Case (a)
f = 0; Case (b) q = 0.
11. Using thin airfoil theory calculate the lift and drag on the airfoil shape given
by yu = t sin(πx/c) for the upper surface and yl = 0 for the lower surface. Assume
a supersonic stream parallel to the x-axis. The thickness ratio t/c ≪ 1.
12. Write momentum conservation for the volume of the small “pill box” shown
in Figure 4.22 (p. 121) where the interface is a shock with flow from side 1 to side 2.
Let the two end faces approach each other as the shock thickness → 0 and assume
viscous stresses may be neglected on these end faces (outside the structure). Show
that the n component of momentum conservation yields (16.29) and the t component
gives u · t is conserved or v is continuous across the shock.
Supplemental Reading
Literature Cited
Ames Research Staff (1953). NACA Report 1135: “Equations, Tables, and Charts for Compressible Flow.”
Becker, R. (1922). “Stosswelle und Detonation.” Z. Physik 8: 321–362.
Cohen, I. M. and C. A. Moraff (1971). “Viscous inner structure of zero Prandtl number shocks.” Phys.
Fluids 14: 1279–1280.
Cramer, M. S. and R. N. Fry (1993). “Nozzle flows of dense gases.” The Physics of Fluids A 5: 1246–1259.
Fergason, S. H., T. L. Ho, B. M. Argrow, and G. Emanuel (2001). “Theory for producing a single-phase
rarefaction shock wave in a shock tube.” Journal of Fluid Mechanics 445: 37–54.
Hayes, W. D. (1958). “The basic theory of gasdynamic discontinuities,” Sect. D of Fundamentals of
Gasdynamics, Edited by H. W. Emmons, Vol. III of High Speed Aerodynamics and Jet Propulsion,
Princeton, NJ: Princeton University Press.
Liepmann, H. W. and A. Roshko (1957). Elements of Gas Dynamics, New York: Wiley.
Shapiro, A. H. (1953). The Dynamics and Thermodynamics of Compressible Fluid Flow, 2 volumes.
New York: Ronald.
von Karman, T. (1954). Aerodynamics, New York: McGraw-Hill.
Supplemental Reading
Courant, R. and K. O. Friedrichs (1977). Supersonic Flow and Shock Waves, New York: Springer-Verlag.
Yahya, S. M. (1982). Fundamentals of Compressible Flow, New Delhi: Wiley Eastern.
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Chapter 17
Introduction to Biofluid
Mechanics
Portonovo S. Ayyaswamy
University of Pennsylvania
Philadelphia, PA
1. Introduction . . . . . . . . . . . . . . . . . . . . . 765
2. The Circulatory System in the
Human Body . . . . . . . . . . . . . . . . . . . . 766
The Heart as a Pump . . . . . . . . . . . . 769
Nature of Blood . . . . . . . . . . . . . . . . . . 773
Nature of the Blood Vessels . . . . . . . 779
3. Modelling of Flow in Blood Vessels 782
General Introduction . . . . . . . . . . . . . 782
Hagen-Poiseuille Flow . . . . . . . . . . . 783
Pulsatile Flow Theory . . . . . . . . . . . . 791
Blood Vessel Bifurcation: An
Application of Poiseuille’s Formula
and Murray’s Law . . . . . . . . . . . . . 807
Flow in a Rigid Walled Curved
Tube . . . . . . . . . . . . . . . . . . . . . . . . . . 812
Flow in Collapsible Tubes . . . . . . . . 818
Laminar Flow of a Casson Fluid
in a Rigid Walled Tube . . . . . . . . 826
Pulmonary Circulation . . . . . . . . . . . 829
The Pressure Pulse Curve in the Right
Ventricle . . . . . . . . . . . . . . . . . . . . . . 830
Effect of Pulmonary Arterial Pressure
on Pulmonary Resistance . . . . . . 830
4. Introduction to the Fluid Mechanics
of Plants . . . . . . . . . . . . . . . . . . . . . . . . 831
Xylem . . . . . . . . . . . . . . . . . . . . . . . . . . . 833
Xylem Flow . . . . . . . . . . . . . . . . . . . . . 835
Phloem . . . . . . . . . . . . . . . . . . . . . . . . . . 835
Phloem Flow . . . . . . . . . . . . . . . . . . . . 836
Exercises . . . . . . . . . . . . . . . . . . . . . . . . 837
Acknowledgment . . . . . . . . . . . . . . . . 838
Literature Cited . . . . . . . . . . . . . . . . . 838
1. Introduction
This chapter is intended to be of an introductory nature to the vast field of biofluid
mechanics. Here, we shall consider the ideas and principles of the preceding chapters
in the context of fluid motion in biological systems. First we will learn about some
aspects of the fluid motion in the human body, and later we will learn about some
aspects of fluid mechanics of plants.
The human body is a complex system that requires materials such as air, water,
minerals and nutrients for survival and function. Upon intake, these materials have to
be transported and distributed around the body as required. The associated biotransport and distribution processes involve interactions with membranes, cells, tissues, and
organs comprising the body. Subsequent to cellular metabolism in the tissues, waste
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50017-4
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Introduction to Biofluid Mechanics
by products have to be transported to the excretory organs for synthesis and removal.
In addition to these functions, biotransport systems and processes are required for
homeostasis (physiological regulation–for example, maintenance of pH and of body
temperature), and for enabling the movement of immune substances to aid in body’s
defense and recovery from infection and injury. Furthermore, in certain other specialized systems such as the cochlea in the ear, fluid transport enables hearing and motion
sensing. Evidently, in the human body, there are multiple types of fluid dynamic systems that operate at multiple and widely disparate scales. These scales are at various
levels such as macro, micro, nano, pico and so on. Systems at the micro and macro
levels, for example, include cells (micro), tissue (micro–macro), and organs (macro).
Transport at the micro, nano and pico levels would include ion channeling, binding,
signaling, endocytosis, and so on. Tissues constitute organs, and organs as systems
perform various functions. For example, the cardiovascular system consists of the
heart, blood vessels (arteries, arterioles, venules, veins, capillaries), lymphatic vessels, and the lungs. Its function is to provide adequate blood flow and regulate the
flow as required by the various organs of the body. In this chapter, as related to the
human body, we shall restrict attention to some aspects of the cardiovascular system
for blood circulation.
2. The Circulatory System in the Human Body
The primary functions of the cardiovascular system are: (i) to pick up oxygen and
nutrients from the lungs and the intestine, respectively, and deliver them to tissues
(cells) of various parts, (ii) to remove waste and carbon dioxide from the body for
excretion through the kidneys and the lungs, respectively, and (iii) to regulate body
temperature by convecting the heat generated and dissipating it through transport
across the skin. The circulatory system in the normal human body (as in all vertebrates
and some other select group of species) can be considered as a closed system, meaning
that the blood never leaves the system of blood vessels. The driving potential for blood
flow is the prevailing pressure gradient.
The circulations associated with the cardiovascular system may be considered
under three subsystems. These are the (i) systemic circulation, (ii) pulmonary circulation, and (iii) coronory circulation. (See Fig. 17.1.) In the systemic circulation,
blood flows to all of the tissues in the body except the lungs. Contraction of the left
ventricle of the heart pumps oxygen-rich blood to a relatively high pressure and ejects
it through the aortic valve into the aorta. Branches from the aorta supply blood to the
various organs via systemic arteries and arterioles. These, in turn, carry blood to the
capillaries in the tissues of various organs. Oxygen and nutrients are transported by
diffusion across the walls of the capillaries to the tissues. Cellular metabolism in the
tissues generates carbon dioxide and byproducts (waste). Carbon dioxide dissolves
in the blood and waste is carried by the blood stream. Blood drains into venules and
veins. These vessels ultimately empty into two large veins called the superior vena
cava (SVC) and and inferior vena cava (IVC) that return carbon dioxide rich blood to
the right atrium. The mean blood pressure of the systemic circulation ranges from a
high of 93 mmHg in the arteries to a low of few mmHg in the venae cavae. Fig. 17.2
2. The Circulatory System in the Human Body
Figure 17.1 Schematic of blood flow in systemic and pulmonary circulation. (Reproduced with permission from Silverthorn, D.U. (2001) Human Physiology: An Integrated Approach, 2nd ed., Prentice Hall,
Upper Saddle River, NJ.).
shows that pressure falls continuously as blood moves farther from the heart. The
highest pressure in the vessels of the circulatory system is in the aorta and in the
systemic arteries while the lowest pressure is in the venae cavae.
In pulmonary circulation, contraction of the right atrium ejects carbon dioxide
rich blood through the tricuspid valve into the right ventricle. Contraction of the right
ventricle pumps the blood through the pulmonic valve (also called semilunar valve)
into the pulmonary arteries. These arteries bifurcate and transport blood into the
complex network of pulmonary capillaries in the lungs. These capillaries lie between
and around the alveoli walls. During respiratory inhalation, the concentration of oxygen in the air is greater in the air sacs of the alveolar region than in the capillary blood.
Oxygen diffuses across capillary walls into blood. Simultaneously, the concentration
of carbon dioxide in the blood is higher than in the air and carbon dioxide diffuses
from the blood into the alveoli. Carbon dioxide exits through the mouth and nostrils.
Oxygenated blood leaves the lungs through the pulmonary veins and enters the left
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Introduction to Biofluid Mechanics
Figure 17.2 Pressure gradient in the blood vessels. (Reproduced with permission from Silverthorn,
D.U. (2001) Human Physiology: An Integrated Approach, 2nd ed., Prentice Hall, Upper Saddle
River, NJ.).
atrium. When the left atrium contracts, it pumps blood through the bicuspid (mitral)
valve into the left ventricle. Figs. 17.3 and 17.4 provide an overview of external and
cellular respiration and the branching of the airways, respectively.
Blood is pumped through the systemic and pulmonary circulations at a rate of
about 5.2 liters per minute under normal conditions. The systemic and pulmonary
circulations described above constitute one cardiac cycle. The cardiac cycle denotes
any one or all of such events related to the flow of blood that occur from the beginning
of one heartbeat to the beginning of the next. Throughout the cardiac cycle, the blood
pressure increases and decreases. The frequency of the cardiac cycle is the heart rate.
The cardiac cycle is controlled by a portion of the autonomic nervous system (that
part of the nervous system which does not require the brain’s involvement in order to
function).
In coronary circulation, blood is supplied to and from the heart muscle itself.
The muscle tissue of the heart, or myocardium, is thick and it requires coronary blood
vessels to deliver blood deep into the myocardium. The vessels that supply blood with
a high concentration of oxygen to the myocardium are known as coronary arteries.
The main coronary artery arises from the root of the aorta and branches into the left
and right coronary arteries. Up to about seventy five percent of the coronary blood
supply goes to the left coronary artery, the remainder going to the right coronary artery.
Blood flows through the capillaries of the heart and returns through the cardiac veins
which remove the deoxygenated blood from the heart muscle. The coronary arteries
that run on the surface of the heart are relatively narrow vessels and are commonly
affected by atherosclerosis and can become blocked, causing angina or a heart attack.
The coronary arteries are classified as “end circulation,” since they represent the only
source of blood supply to the myocardium.
2. The Circulatory System in the Human Body
Figure 17.3 Overview of external and cellular respiration. (Reproduced with permission from Silverthorn, D.U. (2001) Human Physiology: An Integrated Approach, 2nd ed., Prentice Hall, Upper Saddle
River, NJ.).
The Heart as a Pump
The heart has four pumping chambers–two atria (upper) and two ventricles (lower).
The left and right parts of the heart are separated by a muscle called the septum which
keeps the blood volumes in each part separate. The upper chambers interact with
the lower chambers via the heart valves. The heart has four valves which ensure that
blood flows only in the desired direction. The atrio-ventricular valves (AV) consist
of the tricuspid (three flaps) valve between the right atrium and the right ventricle,
and the bicuspid (two flaps, also called the mitral) valve between the left atrium
and the left ventricle. The pulmonary valve is between the right ventricle and the
pulmonary artery, and the aortic valve is between the left ventricle and the aorta. Both
the pulmonary and aortic valves have three symmetrical half moon shaped valve flaps
(cusps), and are called the semilunar valves. The function of the four chambers in
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Figure 17.4 Branching of the airways. Areas have units of cm2 . (Reproduced with permission from
Silverthorn, D.U. (2001) Human Physiology: An Integrated Approach, 2nd ed., Prentice Hall, Upper
Saddle River, NJ.).
the heart is to pump blood through pulmonary and systemic circulations. The atria
receive blood from the veins–right atrium receives carbon dioxide rich blood from
the SVC and IVC, and the left atrium receives oxygen rich blood from the pulmonary
veins. The heart is controlled by a single electrical impulse and both sides of the
heart act synchronously. Electrical activity stimulates the heart muscle (myocardium)
of the chambers of the heart to make them contract. This is immediately followed
by mechanical contraction of the heart. Both atria contract at the same time. The
contraction of the atria moves the blood from the upper chambers through the valves
into the ventricles. The atrial muscles are electrically separated from the ventricular
muscles except for one pathway through which an electrical impulse is conducted
from the atria to the ventricles. The impulse reaching the ventricles is delayed by
about 110 ms while the conduction occurs through the pathway. This delay allows the
ventricles to be filled before they contract. The left ventricle is a high pressure pump
and its contraction supplies systemic circulation while the right ventricle is a low
pressure pump supplying pulmonary circulation (Lungs offer much less resistance to
flow than systemic organs).
From the above discussions, we see that the pumping action of the heart can be
regarded as a two phase process–a contraction phase (systole) and a filling (relaxation)
phase (diastole). Systole describes that portion of the heartbeat during which contraction of the heart muscle and hence ejection of blood takes place. A single “beat” of
the heart involves three operations: atrial systole, ventricular systole and complete
cardiac diastole. Atrial systole is the contraction of the heart muscle of the left and
right atria, and occurs over a period of 0.1s. As the atria contract, the blood pressure
in each atrium increases, which forces the mitral and tricuspid valves to open forcing
blood into the ventricles. The AV valves remain open during atrial systole. Following
atrial systole, ventricular systole which is the contraction of the muscles of the left
and right ventricles occurs over a period of 0.3s. The ventricular systole generates
enough pressure to force the AV valves to close, and the aortic and pulmonic valves
open. (The aortic and pulmonic valves are always closed except for the short period
of ventricular systole when the pressure in the ventricle rises above the pressure in
2. The Circulatory System in the Human Body
the aorta for the left ventricle and above the pressure in the pulmonary artery for the
right ventricle.) During systole, the typical pressures in the aorta and the pulmonary
artery rise to 120 mmHg and 24 mmHg, respectively, (note conversion, 1 mmHg =
133 Pa). In normal adults, blood flow through the aortic valve begins at the start of
ventricular systole, and rapidly accelerates to a peak value of approximately 1.35
m/s during the first one-third of systole. Thereafter, the blood flow begins to decelerate. Pulmonic valve peak velocities are lower and in normal adults, they are about
0.75 m/s. Contraction of the ventricles in systole ejects about two thirds of the blood
from these chambers. As the left ventricle empties, its pressure falls below the pressure
in the aorta, and the aortic valve closes. Similarly, as the pressure in the right ventricle
falls below the pressure in the pulmonary artery, the pulmonic valve closes. Thus, at
the end of the the ventricular systole, the aortic and pulmonic valves close, with the
aortic valve closing a little earlier than the pulmonic valve. Diastole describes that
portion of the heart beat during which the chamber refilling takes place. The cardiac
diastole is the period of time when the heart relaxes after contraction in preparation
for refilling with circulating blood. The ventricles refill or ventricular diastole occurs
during atrial systole. When the ventricle is filled and ventricular systole begins, then
the AV valves are closed and the atria begin refilling with blood or atrial diastole
occurs. About a period of 0.4s following ventricular systole, both the atria and the
ventricles begin refilling and both chambers are in diastole. During this period, both
AV valves are open and aortic and pulmonic valves are closed. The typical diastolic
pressure in the aorta is 80 mmHg and, in the pulmonary artery, it is 8 mmHg. Thus, the
typical systolic and diastolic pressure ratios are 120/80 mmHg for the aorta and 24/8
mmHg for the pulmonary artery. The systolic pressure minus the diastolic pressure is
called the pressure pulse, and for the aorta (left ventricle) it is 40 mmHg. The pulse
pressure is a measure of the strength of the pressure wave. It increases with increased
stroke volume (say, due to activity or exercise). Pressure waves created by the ventricular contraction diminish in amplitude with the distance and are not perceptible
in the capillaries. Fig. 17.5 shows the pressure throughout the systemic circulation.
Figure 17.5 Pressure throughout the systemic circulation. (Reproduced with permission from Silverthorn,
D.U. (2001) Human Physiology: An Integrated Approach, 2nd ed., Prentice Hall, Upper Saddle River, NJ.).
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Net Work Done by the Ventricle on Blood During One Cardiac Cycle
The work done by the ventricle on blood may be calculated from the area enclosed by
the pressure–volume curve for the ventricle. Consider, for example, the left ventricle
(LV). Fig. 17.6 shows the pressure–volume curve for the LV.
Blood pressure is measured in mm of Hg, and the volume in ml. At A, the ventricular pressure and volume are at their lowest values. With the increase of atrial
pressure, the bicuspid valve will open and let blood flow into the ventricle. AB
represents diastolic ventricular filling. During AB work is being done by the blood in
the LV to increase the volume. At B, the ventricular volume is filled to its maximum
and this volume is called the end diastolic volume (EDV ). The ventricular muscles
begin to contract, pressure increases, and the bicuspid valve closes. BC is the constant
volume contraction of the ventricle. No work is done during BC but energy is stored
as elastic energy in the muscles. At C, ventricular pressure is greater than that in the
aorta, the aortic valve opens and blood is ejected into the aorta. Ventricular volume
decreases, but the ventricle continues to contract and the pressure increases. However,
at D, pressure in the aorta exceeds that of the ventricular pressure and the aortic valve
closes. During CD, work is done by the heart muscles on blood. The volume in the
Figure 17.6 Left ventricular pressure–volume curve. (Reproduced with permission from Silverthorn,
D.U. (2001) Human Physiology: An Integrated Approach, 2nd ed., Prentice Hall, Upper Saddle
River, NJ.).
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2. The Circulatory System in the Human Body
LV at D is at its lowest value, and this is called the end systolic volume (ESV ). DA
is the constant volume pressure decrease in the ventricle due to muscle relaxation
and no work is done during this process. Ventricular pressure falls below that in the
aorta causing the aortic valve to close. ABCD constitutes one cardiac cycle, and the
area within the pressure-volume diagram represents the net work done by the LV on
blood. The energy required to perform this work is derived from the oxygen in the
blood. A similar development applies for the right ventricle.
Typically, work done by the heart is only about 10–15% of the total input energy,
and the remainder is dissipated as heat.
The volume of blood pumped by the LV into the systemic circulation in a cardiac
cycle is called the stroke volume (SV ), and it is expressed in ml/beat. The normal
stroke volume is 70 ml/beat.
SV = EDV − ESV
(17.1)
A parameter that is related to stroke volume is ejection fraction (EF). EF is the fraction
of blood ejected by the LV during systole. At the start of systole, the LV is filled with
blood to the EDV. During systole, the LV contracts and ejects blood until it reaches
ESV. EF is given by
EF = (SV/EDV) × 100%
(17.2)
Cardiac output (CO) is the volume of blood being pumped by the heart (in particular,
by a ventricle) in a minute. It is the time averaged flow rate. It is equal to the heart
rate multiplied by the stroke volume. Thus,
CO = SV × HR,
(17.3)
where HR is the heart rate in beats/min. For a normal adult, the typical HR is between
70 and 75 beats per minute. With 70 beats per minute, and 70 ml blood ejection
with each beat of the heart, the CO is 4900 ml/m. This value is typical for a normal
adult at rest, although CO may reach up to 30 l/m during extreme activity (say,
exercise). Heart rate can vary by a factor of approximately 3, between 60 and 180
beats per minute, while the stroke volume can vary between 70 and 120 ml, a factor
of only 1.7. The cardiac index (CI) relates CO with the body surface area, BSA as
given by,
CI = CO/BSA = SV × HR/BSA,
(17.4)
where, BSA is in square meters.
Nature of Blood
Composition of Blood
Blood is about 7% of the human body weight. Its density is approximately
1054 kg/m3 . The pH of normal blood is in the range 7.35 < pH < 7.45. The normal adult has a blood volume of about 5 liters. At any given time, about 13% of
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the total blood volume resides in the arteries and about 7% resides in the capillaries.
Blood is a complex circulating liquid tissue consisting of several types of formed
elements (corpuscles or cells) (about 45% by volume) suspended in a fluid medium
known as plasma (about 55% by volume; 2.7−3.0 liters in a normal human). The
plasma is a dilute electrolyte solution (almost 92% water) containing, about 8% by
weight, three major types of blood proteins—fibrinogen (5%), globulin (45%), and
albumin (50%) in water. Beta lipoprotein and lipalbumin are also present in trace
amounts. Plasma proteins are large molecules with high molecular weight and do
not pass through the capillary wall. The formed elements (cells) consist of red blood
cells (erythrocytes; about 45% of blood volume), white blood cells (leukocytes; about
1% of blood volume), and platelets (thrombocytes; <1% of blood volume). Thus, the
formed elements in blood consist of 95% red blood cells, 0.13% white blood cells, and
about 4.9% platelets. The specific gravity of red blood cells is about 1.06. The white
blood cells further consist of monocytes, lymphocytes, neutrophils, eosinophils, and
basophils.
In humans, mature red blood cells lack a nucleus and organelles. They are produced in the bone marrow, and the cell life span is about 125 days. The red blood cell is
biconcave in shape. It consists of a concentrated solution of hemoglobin, an oxygen
carrying protein, surrounded by a flexible membrane. The hemoglobin transports
oxygen from the lungs to capillaries in various tissues, and some carbon dioxide. The
cell is about 8.5µm in diameter with transverse dimensions of 2.5µm at the thickest
portion and about 1µm at the thinnest portion. However, its flexibility is such that it
can bend and pass through capillaries as small as 5µm in diameter. The surface area
of the cell is about 163(µm)2 , and the intracellular fluid volume is about 87(µm)3 .
There are approximately 5 × 106 red blood cells in each mm3 of blood. The biconcave shape of the cell provides it with a very large ratio of surface area to volume.
This enables efficient gas exchange in the capillaries. The percentage of blood volume made up by red blood cells is referred to as the hematocrit. Hematocrit ranges
from 42 to 45 in normal blood, and plays a major role in determining the rheological
properties of blood. White blood cells or leukocytes are cells of the immune system
which defend the body against both infectious disease and foreign materials. Several different and diverse types of leukocytes exist and they are all produced in the
bone marrow. There are normally about 104 white blood cells in each mm3 of blood.
Platelets or thrombocytes are cell fragments circulating in blood that are involved in
the cellular mechanisms of hemostasis leading to the formation of blood clots. They
are smaller in size than red or white blood cells. Low levels of platelets predisposes
to bleeding, while high levels increase the risk of thrombosis (coagulation of blood
in the heart or a blood vessel).
Viscosity of Blood
An important property of a flowing fluid is its viscosity. The viscosity of blood depends
on the viscosity of the plasma and its protein content, the hematocrit, the temperature,
the shear rate (also called the rate of shearing strain), and the narrowness of the vessel
in which it is flowing (for example, a narrow diameter capillary). The dependence
2. The Circulatory System in the Human Body
on the narrowness of the vessel diameter is called the Fahraeus-Lindqvist effect.
The dependence on the prevailing shear rate and the Fahraeus-Lindqvist effect, each
classify blood as a non-Newtonian fluid.
The presence of white cells and platelets do not significantly affect the viscosity
since they are in such small proportions. We will briefly discuss these various features
of blood viscosity.
The viscosity of plasma and blood are often given in terms of relative viscosity
as compared to that of water (viscosity of water is about 0.8 centipoise at room
temperature; 1 centipoise (1 cP) = 0.01 Poise, conversion: 1 Poise = 1dyne s/cm2 =
0.1 N s/m2 ). The viscosity of plasma depends on the protein composition of the
plasma and ranges between 1.1 and 1.6 centipoise. The viscosity of whole blood at a
physiological hematocrit of 45% is about 3.2 cP. Higher hematocrit results in higher
viscosity. At a hematocrit of 60%, the relative viscosity of blood is about 8. Viscosity
of blood increases with decreasing temperature, and the increase is approximately
2% for each ◦ C. The dependence of viscosity on flow rate in vessels is complicated.
As noted in earlier chapters, viscous flow rates in vessels are significantly influenced
by the shear stress, τ , and the associated rate of shearing strain (or shear rate), γ̇ . For
Newtonian fluids, τ is linearly related to γ̇ , and the slope of this characteristic is the
viscosity, µ. For whole blood, this relationship is complicated due to the following
reasons. In a blood volume at rest, above a minimum hematocrit of about 5−8%,
blood cells form a continuous structure. A finite stress (called the yield stress), τy ,
is required to break this continuous structure into a suspension of aggregates in the
plasma. This yield stress depends also on the concentration of plasma proteins, in
particular, fibrinogen. An empirical correlation for the yield stress is given by the
expression:
√
τy = (H − 0.1)(CF + 0.5),
(17.5)
where H is the hematocrit expressed as a fraction and it is >0.1, and CF is the fibrinogen content in grams per 100 ml and 0.21 < CF < 0.46. For 45% hematocrit
blood, the yield stress is in the range 0.01 < τy < 0.06 dynes/cm2 , (conversion:
1 dyne/cm2 = 0.1 N/m2 ). Beyond the yield, when sheared in the bulk, up to about
γ̇ < 50 sec−1 , the aggregates in blood break into smaller units called rouleaux formations. For shear rates up to about 200 sec−1 , the rouleaux progressively break into
individual cells. Beyond this, no further reduction in structure is noted to occur with
an increase in the shearing rate.
For whole blood, at low shear rates, γ̇ < 200 sec−1 , the variation of τ with γ̇ is
noted to be nonlinear. This behavior at low γ̇ is non-Newtonian. Low γ̇ values are
associated with flows in small arteries and capillaries (microcirculation). At higher
shear rates, γ̇ > 200 sec−1 , the relationship between τ and γ̇ is linear, and the viscosity
approaches an asymptotic value of about 3.5 cP. Blood flows in large arteries have
such high shear rates, and the viscosity in such cases may be assumed as constant and
equal to 3.5 cP. Since whole blood behaves like a non-Newtonian yield stress fluid,
the slope of the shear stress—rate of strain characteristic at any given point on the
curve is defined as the apparent viscosity of blood at that point, µapp . Clearly, µapp
is not a constant but depends on the prevailing γ̇ at that point. (See Fig. 17.7.) There
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Figure 17.7 Shear stress vs. Shear rate for blood flow. (Reproduced with permission from
Whitmore, R. L. (1968) Rheology of Circulation, Pergamon Press, New York).
are a number of constitutive equations available in the literature that attempt to model
the relationship between shear stress and shear rate of flowing blood. A commonly
used one is called the Casson model and it is expressed as follows:
τy
τ
= kc γ̇ +
,
(17.6)
µp
µp
where µp is plasma viscosity and kc is the Casson viscosity coefficient (a
dimensionless number). An expression based on a least square fit of the experimental
data and expressed in Casson form is that of Whitmore (1968),
τ
(17.7)
= 1.53 γ̇ + 2.0.
µp
This expression is plotted in Fig. 17.8. Apparent viscosity significantly increases at
low rates of shear. It must be noted that although the Casson model is suitable at low
shear rates, it still assumes that blood can be modelled as a homogeneous fluid.
In blood vessels of less than about 500 µm in diameter, the inhomogeneous
nature of blood starts to have an effect on the apparent viscosity. This feature will be
discussed next.
Fahraeus-Lindqvist Effect
When blood flows through narrow tubes of decreasing radii, approximately in
the range (15 µm < d < 500 µm), the apparent viscosity, µapp , decreases with
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2. The Circulatory System in the Human Body
Figure 17.8 A least square fit of apparent viscosity as a function of shear rate in Casson form. (Reproduced
with permission from Whitmore, R. L. (1968) Rheology of Circulation, Pergamon Press, New York).
HT < HF, H
D
HT
HF
H
D
Figure 17.9 The Fahraeus effect.
decreasing radius of the vessel. This is a second non-Newtonian characteristic of
blood and is called the Fahraeus-Lindqvist (FL) effect. The reduced viscosity in
narrow tubes is beneficial to the pumping action of the heart.
The basis for the FL effect is the Fahraeus effect.
When blood of constant hematocrit (feed hematocrit or bulk hematocrit, HF )
flows from a large vessel into a small vessel (vessel sizes in the ranges cited above),
the hematocrit in the small vessel (dynamic or tube hematocrit, HT ) decreases as the
tube diameter decreases. (See Fig. 17.9.) This phenomenon is called Fahraeus effect
and must not be confused with a diminution of particle concentration in the smaller
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vessel because of an entrance effect whereby particle entry into the smaller vessel
is hindered (see, Goldsmith et al. (1989) for detailed discussions). To separate this
possible “screening effect” and confirm the Fahraeus effect, HT may be compared
with the hematocrit in the blood flowing out (discharge hematocrit, HD ) from the
smaller tube into a discharge vessel of comparable size to the feed vessel. In the
steady state, HF = HD . In vivo and in vitro experiments show that, HT < HD
in tubes up to about 15 µm in diameter. The HT /HD ratio decreases from about
1 to about 0.46 as the capillary diameter decreases from about 600 µm to about
15 µm. While the discharge hematocrit value may be 45%, the corresponding
dynamic hematocrit in a narrow sized vessel such as an arteriole may just be
20%. As a consequence, the apparent viscosity decreases in the diameter range
15 µm < d < 500 µm. However, for tubes less than about 15 µm in diameter, the
ratio HT /HD starts to increase.
Why does the hematocrit decrease in small blood vessels? The reason for this
effect is not fully understood at this time. In blood vessel flow, there seems to be
a tendency for the red cells to move toward the axis of the tube, leaving a layer of
plasma, whose width, usually designated by δ, increases with increase in the shear
rate. This tendency to move away from the wall is not observed with rigid particles;
thus, the deformability of the red cell appears to be the reason for lateral migration.
Deformable particles are noted to experience a net radial hydrodynamic force even at
low Reynolds numbers and tend to migrate towards the tube axis. (see, Fung (1993)
for detailed discussions). Chandran et al. (2007) state that as the blood flows through a
tube, the blood cells (with their deformable biconcave shape) rotate (spin)in the shear
field. Due to this spinning, they tend to move away from the wall and toward the
center of the tube. The cell free plasma layer reduces the tube hematocrit. As the size
of the vessel gets smaller, the fraction of the volume occupied by the cell-free layer
increases, and the tube hematocrit is further lowered. A numerical validation of this
reasoning is available in a recent paper by Liu and Liu (2006). There is yet another
reason. Blood vessels have many smaller sized branches. If a branching daughter
vessel is so located that it draws blood from the larger parent vessel mainly from the
cell free layer, the hematocrit in the branch will end up being lower. This is called
plasma skimming. In all these circumstances, the tube hematocrit is lowered. The
viscosity of blood at the core may be higher due to a higher core hematocrit, Hc ,
there, but the overall apparent viscosity in the tube flow is lower.
As the tube diameter becomes less than about 6 µm, the apparent viscosity
increases dramatically. The erythrocyte is about 8 µm in diameter and can enter
tubes somewhat smaller in size, and a tube of about 2.7 microns is about the smallest
size that an RBC can enter, Fournier (2007), Fung (1993). When the tube diameter
becomes very small, the pressure drop associated with the flow increases greatly and
there is increase in apparent viscosity.
If we consider laminar blood flow in straight, horizontal, circular, feed and capillary tubes, a number of straightforward relationships between QF , Qc , Qp , HF , HT ,
Hc , δ, and a may be established based on the law of conservation of blood cells. Here,
Q denotes flow rate, subscripts c and p denote core and plasma regions, respectively,
and a is the radius of the capillary tube. Thus,
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2. The Circulatory System in the Human Body
QF HF = Qc Hc ,
Qc + Qp = QF ,
and,
HT a 2 = Hc (a − δ)2 ,
(17.8)
where a is the radius of the capillary tube. Equation (17.8) will be useful in modelling
the FL phenomenon. A simple mathematical model for the FL effect is included in a
subsequent section.
Nature of the Blood Vessels
All blood vessels other than capillaries are usually composed of three layers: the
tunica intima, tunica media, and tunica adventitia. The tunica intima consists of a
layer of endothelial cells lining the lumen of the vessel (the hollow internal cavity in
which the blood flows), as well as a subendothelial layer made up of mostly loose
connective tissue. The endothelial cells are in direct contact with the blood flow. An
internal elastic lamina often separates the tunica intima from the tunica media. The
tunica media is composed chiefly of circumferentially arranged smooth muscle cells.
Again, an external elastic lamina often separates the tunica media from the tunica
adventitia. The tunica adventitia is primarily composed of loose connective tissue
made up of fibroblasts and associated collagen fibers. In the largest arteries, such
as the aorta, the amount of elastic tissue is very considerable. Veins have the same
three layers as arteries, but boundaries are indistinct, walls are thinner, and elastic
components are not as well developed.
Blood flows under high pressure in the aorta (about 120 mmHg systolic,
80 mmHg diastolic, pressure pulse of 40 mm Hg at the root) and the major arteries.
These vessels have strong walls. The aorta is an elastic artery, about 25 mm in
diameter with a wall thickness of about 2 mm, and is quite distensible. During
left ventricular systole (about 1/3 of the cardiac cycle), the aorta expands. This
stretching gives the potential energy that will help maintain blood pressure during
diastole. During the diastole (about 2/3 of the cardiac cycle), the pressure pulse
decays exponentially and the aorta contracts passively. Medium arteries are about
4 mm in diameter with a wall thickness of about 1 mm. Arterioles are about 50 µm
in diameter and have thin muscular walls (usually only one to two layers of smooth
muscle) of about 20 µm thickness. Their vascular tone is controlled by regulatory mechanisms, and they constrict or relax as needed to maintain blood pressure.
Arterioles are the primary site of vascular resistance and blood flow distribution to
various regions is controlled by changes in resistance offered by various arterioles.
True capillaries average from 9 to 12 µm in diameter, just large enough to permit passage of cellular components of blood. The thin wall consists of extremely
attenuated endothelial cells. In cross section, the lumen of small capillaries may
be encircled by a single endothelial cell, while larger capillaries may be made
up of portions of 2 or 3 cells. No smooth muscle is present. Venules are about
20 µm in diameter and allow deoxygenated blood to return from the capillary beds
to the larger veins. They have three layers. An inner endothelium layer which acts a
membrane, a middle layer of muscle and elastic tissue, and an outer layer of fibrous
connective tissue. The middle layer is poorly developed. The walls of venules are
about 2 µm in thickness, and thus are very much thinner than those of arterioles.
Veins are thin walled, distensible, and collapsible tubes. Some of them may be
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collapsed in normal function. They transport blood at a lower pressure than the
arteries. They are about 5 mm in diameter and the wall thickness is about 500 µm.
They are surrounded by helical bands of smooth muscles which help maintain blood
flow to the right atrium. Most veins have one-way flaps called venous valves. These
valves prevent gravity from causing blood to flow back and collect in the lower
extremities. Veins more distal to the heart have more valves. Pulmonary veins and
the smallest venules have no valves. Veins also have a thick collagen outer layer,
which helps maintain blood pressure. In the venous system, a large increase in the
blood volume results in a relatively small increase in pressure compared to the
arterial system (see, Chandran et al. (2007)). The veins act as the main reservoir
for blood in the circulatory system and the total capacity of the veins is more than
sufficient to hold the entire blood volume of the body. This capacity is reduced
through the constriction of smooth muscles, minimizing the cross-sectional area
(and hence volume) of the individual veins and therefore the total venous system.
The superior vena cava is a large, yet short vein that carries de-oxygenated blood
from the upper half of the body to the heart’s right atrium. The inferior vena cava is
the large vein that carries de-oxygenated blood from the lower half of the body into
the heart. The vena cava is about 30 mm in diameter with a wall thickness of about
1.5 mm. The venae cavae have no valves. Fig. 17.10 shows the cross-sectional areas
of different parts of the systemic circulation with velocity of blood flow in each
part. The fastest flow is in the arterial system. The slowest flow is in the capillaries
and venules.
Total Peripheral Resistance Concept
As stated earlier, arterioles are the primary site of vascular resistance, and blood
flow distribution to various regions is controlled by changes in resistance offered by
various arterioles. To quantify the resistance of the arterioles in an averaged sense,
the concept of total peripheral resistance is introduced. Total peripheral resistance
essentially refers to the cumulative resistance of the thousands of arterioles involved
in the systemic or pulmonary circulation, respectively. For systemic circulation, with
time averaging of quantities over a cardiac cycle,
Total Peripheral Resistance = R =
p̄
,
Q
(17.9)
where R denotes resistance, p̄ is the difference between the time averaged pressure
at the aortic valve and the time averaged venous pressure at the right atrium, and
Q is the time averaged flow rate (cardiac output). The units of peripheral resistance
would therefore be in mmHg per cm3 /s. This unit of measuring resistance is called
the Peripheral Resistance Unit (PRU). Letting p̄A and p̄V to denote the time averaged
pressures at the aortic valve and at the right atrium, respectively,
p = p̄A − p̄V ,
(17.10)
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2. The Circulatory System in the Human Body
Figure 17.10 Vessel diameter, total cross-sectional area, and velocity of flow. (Reproduced with permission from Silverthorn, D. U. (2001) Human Physiology: An Integrated Approach, 2nd ed., Prentice Hall,
Upper Saddle River, NJ.).
and, with p̄V = 0, p̄ = p̄A , the time averaged arterial pressure. Then, p̄A = QR.
The average pressure, p̄A , may be estimated as:
p̄A =
2
1
1
pS + pD = pD + (pS − pD ),
3
3
3
(17.11)
where, pS is the systolic pressure, pD is the diastolic pressure, and (pS − pD )
is the pressure pulse (see, Kleinstreuer (2006)). For a normal person at rest, with
p̄A = 100 mmHg, Q = 86.6 cm3 /s, R = 1.2 PRU . An expression similar to that in
equation (17.9) would apply for pulmonary circulation and would involve the difference between time averaged pressures at the pulmonary artery and at the left atrium,
and the flow rate in pulmonary circulation (same as that in systemic circulation). Since
the difference between time averaged pressures in pulmonary circulation is about an
order of magnitude smaller than in the systemic circulation, the corresponding PRU
would be an order of magnitude smaller.
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3. Modelling of Flow in Blood Vessels
There are approximately 100 000 km of blood vessels in the adult human body (Brown
et al. (1999)). In this section, we will examine several models for describing blood
flow in some important vessels.
General Introduction
Blood flow in the circulatory system is in general unsteady. In most regions it is
pulsatile due to the systolic and diastolic pumping. In pulsatile flow, the flow has a
periodic behavior and a net directional motion over the cycle. Pressure and velocity
profiles vary periodically with time, over the duration of a cardiac cycle. A dimensionless parameter called the Womersley number, α, is used to characterize the pulsatile
nature of blood flow, and it is defined by:
ω
α=a
,
(17.12)
ν
where, a is the radius of the tube, ω is the frequency of the pulse wave (heart rate
expressed in radians/sec), and ν is the kinematic viscosity. This definition shows that
Womersley number is a composite parameter of the Reynolds number, Re = u 2a/ν,
and the Strouhal number, St = ω 2a/u. The square of the Womersley number is
called the Stokes number. The Womersley number denotes the ratio of unsteady
inertial forces to viscous forces in the flow. It ranges from as large as about 20 in the
aorta, significantly greater than 1 in all large arteries, to as small as about 10−3 in the
capillaries. Let us estimate the Womersley number for an illustration. With a normal
heart rate of 72 beats per minute, ω = (2 π 72/60) ≈ 8 rad/s. Take ρ = 1.05 g cm−3 ,
µ = 0.04 g cm−1 s−1 and an artery of radius a = 0.5 cm. Then α ≈ 7. Decreasing
α values correspond to increasing role of viscous forces and, for α < 1, viscous
effects are dominant. In that highly viscous regime, the flow may be regarded as
quasi-steady. With increasing α, inertial forces become important. In pulsatile flows,
flow separation may occur both by a geometric adverse pressure gradient and/or by
time varying changes in the driving pressure. Geometric adverse pressure gradients
may arise due to varying cross sectional areas through which the flow occurs. On
the other hand, time varying changes in a cardiac cycle result in acceleration and
deceleration during the cycle. An adverse pressure gradient during the deceleration
phase may result in flow separation.
Blood vessel walls are viscoelastic in their behavior. The ability of a blood vessel wall to expand and contract passively with changes in pressure is an important
function of large arteries and veins. This ability of a vessel to distend and increase volume with increasing transmural pressure difference (inside minus outside pressure)
is quantified as vessel compliance. During systole, pressure from the left ventricle is
transmitted as a wave due to the elasticity of the arteries. Due to the compliant nature
of the arteries and their finite thickness, the pressure travels like a wave at a speed
much faster than the flow velocity. Since blood vessels may have many branches, the
reflection and transmission of waves in such branching vessels significantly complicate the understanding of such flows. In this chapter, a reasonably simplified picture
3. Modelling of Flow in Blood Vessels
of these various complex features will be presented. Further reading in advanced
treatments such as the book by Fung (1997) will be necessary to obtain a comprehensive understanding.
First we start with the study of laminar, steady flow of blood in circular tubes,
and in subsequent sections, we shall consider more realistic models.
Hagen-Poiseuille Flow
In the simplest model, blood flow in a vessel is modelled as a laminar, steady, incompressible, fully developed flow of a Newtonian fluid through a straight, rigid, cylindrical, horizontal tube of constant circular cross section (see Fig. 17.11). Such a flow
is called the circular Poiseuille or more commonly as the Hagen-Poiseuille flow. This
flow has been treated in Chapter 9, Section 5.
How valid is the Hagen-Poiseuille model?
In the normal body, blood flow in vessels is generally laminar. However, at high
flow rates, particularly in the ascending aorta, the flow may become turbulent at or
near to peak systole. Disturbed flow may occur during the deceleration phase of the
cardiac cycle (Chandran et al. (2007)). Turbulent flow may also occur in large arteries
at branch points. However, under normal conditions, the critical Reynolds number,
Rec , for blood flow in long, straight, smooth blood vessels is relatively high, and the
flow remains laminar. Let us consider some estimates. The aorta is about 40 cm long
and the average velocity of flow in it is about 40 cm/s. The lumen diameter at the
root of the aorta is d = 25 mm, and the corresponding Re = ρ u d/µ is 3000. The
maximum Reynolds number may be as high as 9000. The average value for Re in the
vena cava is about 3000. Arteries have varying sizes and the maximum Re is about
1000. For Newtonian fluid flow in a straight cylindrical rigid tube, the critical Rec is
about 2300. However, aorta and arteries are distensible tubes, and the Rec of 2300
criterion does not apply. In the case of blood flow, laminar flow conditions generally
prevail even at a high Reynolds number of 10,000 (Mazumdar (2004)). In summary,
the laminar flow assumption is reasonable in many cases.
Blood flow in the circulatory system is in general unsteady and pulsatile. The large
arteries have elastic walls and are subject to substantially pulsatile flow. The steady
flow assumption is inapplicable until the flow has reached smaller muscular arteries
and arterioles in the circulatory system. Blood flow in arteries has been described
by several authors (see, McDonald (1974), Pedley (1980), Ku (1997)). In the heart
chambers and blood vessels, blood may be considered incompressible. In the walls
Figure 17.11 Poiseuille flow.
783
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Introduction to Biofluid Mechanics
of the heart and in the blood vessel walls, it may not be considered as incompressible
(Fung (1997)). Since blood flow remains laminar at very high Reynolds numbers, the
entry length is very large in many cases. Branches and curved vessels hinder flow
development. The fully developed flow assumption is very restrictive in describing
blood flow in vessels.
Flow in large blood vessels may be generally regarded as Newtonian. The Newtonian fluid assumption is inapplicable at low shear rates such as those that would
occur in arterioles and capillaries.
Many blood vessels are not straight but are curved and have branches. However,
flow may be regarded to occur in straight sections in many cases of interest.
Arterial walls are not rigid but are viscoelastic and distensible. The pressure pulse
generated during left ventricular contraction travels through the arterial wall. The
speed of wave propagation depends upon the elastic properties of the wall and the
fluid—structure interaction. Arterial branches and curves may cause reflections of
the wave.
Gravitational and hydrostatic effects become very important for orientations of
the body other than the supine position.
Systemic arteries are generally circular tubes but may have tapering cross sections, while the veins and pulmonary arteries tend to be elliptical.
Since there are many situations where the Hagen-Poiseuille model is reasonably
applicable, we will now start with the recapitulation of the flow results provided in
Chapter 9. The pertinent results are:
Axial flow velocity, u = u(r), in a pipe of radius, a (see, equation (9.10)):
r 2 − a2
u=
4µ
dp
.
dx
(17.13)
Pressure drop: In a fully developed flow, the pressure gradient, (dp/dx), is a constant,
and, it may be expressed in terms of the pressure gradient along the entire tube:
dp
dx
=−
(p1 − p2 )
p
=−
,
L
L
(17.14)
where, p is the imposed pressure difference, subscripts 1 and 2 denote inlet and
exit ends, respectively, and L is the length of the entire tube. With equation (17.14),
equation (17.13) becomes,
r 2 − a2
u=
4µ
dp
dx
r 2
p a 2
p1 − p2 2
2
.
(a − r ) =
=
1−
4µL
4µL
a
(17.15)
The maximum velocity occurs at the center of the tube, r = 0, and is given by
umax
p a 2
=
4µL
(17.16)
785
3. Modelling of Flow in Blood Vessels
The volume flow rate is:
a
Q=
0
u2πrdr = −
πa 4
8µ
dp
dx
=
π a 4 p
umax 2
π a 4 (p1 − p2 )
=
=
πa
8µ
L
8µ L
2
(17.17)
This equation (17.17) is called the Poiseuille formula. The average velocity over the
cross section is :
Q
Q
umax
V =
=
,
(17.18)
=
A
2
π a2
where A is the cross section of the tube. The shear stress at tube wall is:
τxr |r=a = τ = −µ
du
dr
r=a
=−
a
2
dp
dx
=−
a p
,
2 L
(17.19)
where the negative sign has been included to give τ > 0 with du
dr < 0 (the velocity
decreases from the tube centerline to the tube wall). The maximum shear stress occurs
at the walls, and the stress decreases towards the center of the vessel.
The Hagen-Poiseuille equation and its derivatives are most applicable to flow in
the muscular arteries, but modifications are likely to be required outside this range
(see Brown et al. (1999)). For an application of Poiseuille flow relationships in the
context of perfused tissue heat transfer and thermally significant blood vessels, see
Baish et al. (1986a, 1986b).
With the results for the Hagen-Poiseuille flow, we have from Eq. (17.9),
Total Peripheral Resistance = R =
p̄
8µl
.
=
Q
π a4
(17.20)
Equation (17.20) shows that peripheral resistance to the flow of blood is inversely
proportional to the fourth power of vessel diameter.
Hagen-Poiseuille Flow and the Fahraeus-Lindqvist Effect
Consider laminar, steady flow of blood through a straight, rigid, cylindrical, horizontal
tube of constant circular cross section and radius a, as shown in Fig. 17.12.
Let the flow be divided into two regions: a central core containing RBCs and a cell
free plasma layer of thickness δ surrounding the core. Let the viscosities of the core
and the plasma layer be µc and µp , respectively. Let the shear rates be such that each
region can be considered Newtonian, and that we could employ Hagen-Poiseuille
theory.
The shear stress distribution in the core region is governed by,
τxr = −µc
duc
r p
=−
,
dr
2 L
(17.21)
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Introduction to Biofluid Mechanics
Plasma
Core
Figure 17.12 Fahraeus-Lindqvist effect.
subject to conditions,
duc
= 0, at r = 0,
dr
τxr |c = τxr |p , at r = (a − δ).
(17.22)
(17.23)
The shear stress distribution in the plasma region is governed by,
dup
r p
=−
,
dr
2 L
(17.24)
uc = up , at r = (a − δ),
(17.25)
τxr = −µp
subject to conditions,
p
u = 0, at r = a.
(17.26)
Integration of equations (17.21) and (17.24) subject to the indicated conditions
yield the following expressions for the axial velocities in the plasma and core regions:
r 2
a 2 p
p
u =
, for a − δ ≤ r ≤ a,
(17.27)
1−
4µp L
a
and,
µ p r 2 µ p a − δ 2
+
−
, for 0 ≤ r ≤ a−δ.
µc a
µc
a
(17.28)
The volume flow rates in the plasma, Qp , and core region, Qc , are:
a 2 p
u =
1−
4µp L
c
a−δ
a
2
a
Qp = 2π
a−δ
up rdr =
2
π p 2
a − (a − δ)2 ,
8µp L
(17.29)
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3. Modelling of Flow in Blood Vessels
and,
a−δ
uc rdr
µp (a − δ)4
πa 2 p 2
=
.
a − 1−
4µp L
2µc
a2
Qc = 2π
0
(17.30)
The total flow rate of blood within the tube, Q, is the sum of the flow rates in the
plasma and core regions. Therefore,
µp
πa 4 p
δ 4
.
(17.31)
1− 1−
1−
Q = Qp + Qc =
8µp L
a
µc
From the equation (17.31), we could calculate the apparent viscosity of the two region
fluid by measuring Q, and p/L. Define µapp , by analogy with Hagen-Poiseuille
flow, as given by,
π a 4 p
.
(17.32)
Q=
8µapp L
From equations (17.31) and (17.32), the apparent viscosity, µapp , may be expressed
in terms of µp as,
−1
µp
δ 4
.
1−
1− 1−
a
µc
µapp = µp
In the limit (δ/a) ≪ 1, 1 −
δ 4
a
(17.33)
≈ (1 − 4δ/a). Then, equation (17.33) reduces to:
−1
δ µc
µapp = µc 1 + 4
→ µc → µ.
−1
a µp
(17.34)
In equations (17.31) and (17.33), δ and µc are unknown. From equation (17.8), we
have Hc /HF = 1 + (Qp /Qc ). We still need input from experimental data to set up
a modelling procedure for FL. Fournier (2007) recommends the use of Charm and
Kurland’s equation for this purpose, (see Charm and Kurland (1974) for details),
µc = µp
1
,
1 − α c Hc
(17.35)
where,
1107
exp(−1.69Hc ) ,
αc = 0.070 exp 2.49Hc +
T
(17.36)
where T is temperature in K. Equation (17.36) may be used to a hematocrit of 0.60.
With this input, a modelling procedure can be developed for various flow and
tube parameters.
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Introduction to Biofluid Mechanics
Effect of Developing Flow
When we discussed Poiseuille flow, we noted that the fully developed flow assumption
that is often invoked in the study of blood flow in vessels is very restrictive. We will
now learn about some of the limitations of this assumption.
When a fluid under the action of a pressure gradient enters a cylindrical tube,
it takes a certain distance called the inlet or entrance length, ℓ, before the flow in
the tube becomes steady and fully developed. When the flow is fully developed and
laminar, the velocity profile is parabolic that is characteristic of Poiseuille flow. Within
the inlet length, the velocity profile changes in the direction of the flow and the fluid
accelerates or decelerates as it flows. There is a balance among pressure, viscous, and
inertia (acceleration) forces. Compared to fully developed flow, the entrance region
is subject to large velocity gradients near the wall and these result in high wall shear
stresses. The entry of blood from the ventricular reservoir into the aortic tube or
from a large artery into a smaller branch will involve an entrance length. It must be
understood, however, that the inlet length with pulsating flow (say, in the proximal
aorta) is different from that for a steady flow.
If we assumed that the fluid enters the tube from a reservoir, the profile at the inlet
is virtually flat. The transition from a flat velocity distribution, at the entrance of a
tube, to the fully developed parabolic velocity profile is illustrated in Fig. 17.13. Once
inside the tube, the layer of fluid immediately in contact with the wall will become
stationary (no-slip condition) and the laminae adjacent to it slide on it subject to
viscous forces and a boundary layer is formed. The presence of the endothelial lining
on the inside of a blood vessel wall does not negate the no-slip condition. The motion
of the bulk of fluid in the central region of the tube will not be affected by the viscous
forces and will have a flat velocity profile. As flow progresses down the tube, the
boundary layer will grow in thickness as the viscous drag involves more and more of
the fluid.
Eventually, the boundary layer fills the whole of the tube and the steady viscous
flow is established or the flow is fully developed. In the literature (see, for example, Mohanty and Asthana (1979)), there are discussions which divide the entrance
region into two parts, the inlet region and the filled region. At the end of the inlet
region, the boundary layers meet at the tube axis but the velocity profiles are not yet
similar. In the filled region, adjustment of the completely viscous profile takes place
until the Poiseuille similar profile is attained at the end of it. In our discussion here,
we will treat the entrance region as a region with a potential core and a developing
Figure 17.13 Developing velocity profile in a tube flow.
789
3. Modelling of Flow in Blood Vessels
boundary layer at the wall. The shape of the velocity profile in the tube depends on
whether the flow is laminar or turbulent, as does the length of the entrance region,
ℓ. This is a direct consequence of the differences in the nature of the shear stress in
laminar and turbulent flows. The magnitude of the pressure gradient, ∂p/∂x, is larger
in the entrance region than in the fully developed region. There is also an expenditure
of kinetic energy involved in transition from a flat to a parabolic profile. For steady
flow of a Newtonian fluid in a rigid walled horizontal circular tube, the entrance length
may be estimated from,
ℓ
= 0.06 Re for laminar flow and Re > 50,
d
ℓ
= 0.693 Re1/4 for turbulent flow
d
(17.37)
For steady flow at low Reynolds number, the entrance region is approximately
one tube radius long (for Re ≤ 0.01, say in capillaries, ℓ/d = 0.65). In large arteries,
the entrance length is relatively long and over a significant length of the artery the
velocity gradients are high near the wall. This affects the mass exchange of gas and
nutrient molecules between the blood and artery wall.
Unsteady flow through the entrance region with a pulsating flow depends on the
Womersley and Reynolds numbers. For a medium sized artery, the Reynolds number
is typically on the order of 100 to 1000, and the Womersley number ranges from
1 to 10. Pedley (1980) has estimated the wall shear stress in the entrance region
for pulsatile flow using asymptotic boundary layer theory while He and Ku (1994)
have employed a spectral element simulation to investigate unsteady entrance flow
in a straight tube. For a mean Re of 200 and α varying from 1.8 to 12.5 and an
inlet wave form 1 + sin ωt, He and Ku have computed variations in entrance length
during the pulsatile cycle. The amplitude of the entrance length variation decreases
with an increase in α. The phase lag between the entrance length and the inlet flow
waveform increases for α up to 5.0 and decreases for larger values of α. For low α,
the maximum entrance length during pulsatile flow is approximately the same as the
steady entrance length for the peak flow and is primarily dependent on the Reynolds
number. For high α, the Stokes boundary layer growth is faster and the entrance
length is more uniform during the cycle. For α ≥ 12.5, the pulsatile entrance length
is approximately the same length as the entrance length of the mean flow. At all α,
the wall shear rate converges to its fully developed value at about half the length at
which the centerline velocity converges to its fully developed value. This leads to the
conclusion that the upstream flow conditions leading to a specific artery may or may
not be fully developed and can be predicted only by the magnitudes of the Reynolds
number and Womersley number.
Effect of Tube Wall Elasticity on Poiseuille Flow
Here, we will include the elastic behavior of the vessel wall and examine the effect
on the Hagen-Poiseuille model.
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Introduction to Biofluid Mechanics
Consider a pressure gradient driven, laminar, steady flow of a Newtonian fluid
in a long, circular, cylindrical, thin walled, elastic tube. Let the initial radius of the
tube be a0 , and h be the wall thickness and it is small compared to a0 . Because the
tube is elastic, it will distend more at the high pressure end (inlet) than at the outlet
end. The tube radius, a, will now be a function of x.
The variation in tube radius due to wall elasticity has to be ascertained. The
difference between the pressure external to the tube (on the outside of the tube), pe ,
and the pressure inside the tube, p(x), at any cross section of the tube (the negative of
transmural pressure difference), is (pe − p(x)). This pressure difference acts across
h at every cross section, and will induce a circumferential stress. There will be a
corresponding circumferential strain. This strain is the ratio of the change in radius
to the original radius of the tube. In this way, we can ascertain the cross section at x.
Consider the static force equilibrium on a cylindrical segment of the blood vessel
consisting of the top half cross section and of unit length. Let σθθ denote the average
circumferential (hoop) stress in the tube wall. The net downward force due to the
pressure difference will be balanced by the net upward force, and this balance is,
π
2σθθ h =
0
which results in,
σθθ =
(p(x) − pe ) a(x) sin θdθ,
(17.38)
(p(x) − pe ) a(x)
.
h
(17.39)
From Hooke’s law, the circumferential strain, eθθ is given by,
σθθ
a(x)
(a(x) − a0 )
eθθ =
− 1,
=
=
E
a0
a0
(17.40)
where, E is the Young’s modulus of the tube wall material, and we have neglected the
radial stress σrr as compared to σθθ in the thin walled tube. The wall is considered
thin if (h/a) ≪ 1. From equations (17.39) and (17.40), we get, the pressure—radius
relationship,
−1
a0
(17.41)
a(x) = a0 1 −
(p(x) − pe )
Eh
Now since the flow is laminar and steady, we can still apply Hagen-Poiseuille formula,
equation (17.17), to the flow. Thus,
π dp
Q=−
(a(x))4
(17.42)
8µ dx
Therefore,
dp
8µQ
=−
dx
π(a(x))4
With equation (17.41),
−4
a0
8µ
1−
dp = −
Q dx
(p(x) − pe )
Eh
π a0 4
(17.43)
(17.44)
791
3. Modelling of Flow in Blood Vessels
This is subject to the conditions, p = p1 at x = 0, and p = p2 at x = L. By integration of equation (17.44) and from the boundary conditions,
−3
−3
a0
8µ
a0
Eh
L Q.
− 1−
=−
1−
(p2 − pe )
(p1 − pe )
3a0
Eh
Eh
π a0 4
(17.45)
Solving for Q,
Q=
−3
−3
a0
a0
πa0 3 Eh
− 1−
1−
(p1 − pe )
(p2 − pe )
24µL
Eh
Eh
(17.46)
From equation (17.46), we see that the flow is a nonlinear function of pressure
drop if wall elasticity is taken into account. In the above development, we have
assumed Hookean behavior for the stress-strain relationship. However, blood vessels
do not necessarily obey Hooke’s law, their zero-stress states are open sectors, and
their constitutive equations may be non-linear (see, Zhou and Fung (1997)).
Pulsatile Flow Theory
As stated earlier, blood flow in the arteries is pulsatile in nature. One of the earliest
attempts to model pulsatile flow was carried out by Otto Frank in 1899 (see Fung
(1997)).
Elasticity of the Aorta and the Windkessel Theory
Recall that when the left ventricle contracts during systole, pressure within the chamber increases until it is greater than the pressure in the aorta, leading to the opening
of the aortic valve. The ventricular muscles continue to contract increasing the chamber pressure while ejecting blood into the aorta. As a result, the ventricular volume
decreases. The pressure in the aorta starts to build up and the aorta begins to distend
due to wall elasticity. At the end of the systole, ventricular muscles start to relax,
the ventricular pressure rapidly falls below that of the aorta and the aortic valve
closes. Not all of the blood pumped into the aorta, however, immediately goes into
systemic circulation. A part of the blood is used to distend the aorta and a part of the
blood is sent to peripheral vessels. The distended aorta acts as an elastic reservoir
or a Windkessel (the name in German for an elastic reservoir), the rate of outflow
from which is determined by the total peripheral resistance of the system (systemic).
As the distended aorta contracts, the pressure diminishes in the aorta. The rate of
pressure decrease in the aorta is much slower compared to that in the heart chamber.
In other words, during the systole part of the heart pumping cycle, the large fluctuation of blood pressure in the left ventricle is converted to a pressure wave with a
high mean value and a smaller fluctuation in the distended aorta (Fung (1997)). This
behavior of the distended aorta was thought to be analogous to the high-pressure
air chamber (Windkessel) of 19th century fire engines in Germany, and hence the
name Windkessel theory was used by Otto Frank to describe this phenomenon.
In the Windkessel theory, blood flow at a rate Q(t) from the left ventricle enters
an elastic chamber (the aorta) and a part of this flows out into a single rigid tube
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Introduction to Biofluid Mechanics
representative of all of the peripheral vessels. The rigid tube offers constant resistance,
R, equal to the total peripheral resistance that was evaluated in the Hagen-Poiseuille
model, equation (17.9). From the law of conservation of mass, assuming blood is
incompressible,
Rate of Inflow into Aorta = Rate of change of volume of elastic chamber
+ Rate of outflow into rigid tube.
(17.47)
Let the instantaneous blood pressure in the elastic chamber be p(t), and its volume be
v(t). The pressure on the outside of the aorta is taken to be zero. The rate of change
of volume of an elastic chamber is given by,
dv
dp
dv
=
.
(17.48)
dt
dp
dt
dv
is the compliance, K, of the vessel and is a
In equation (17.48), the quantity dp
measure of the distensibility. Compliance at a given pressure is the rate of change in
volume with respect to a change in pressure. Here pressures are always understood
to be transmural pressure differences. Compliance essentially represents the distensibility of the vascular walls in response to a certain pressure. Also, from equation
(17.9), rate of flow into peripherals is given by (p(t)/R), where we have assumed
p̄V = 0. Therefore, equation (17.47) becomes,
p(t)
dp
+
.
(17.49)
Q(t) = K
dt
R
The equation (17.49) is a linear equation of the form,
Q=
dy
+ P y,
dx
(17.50)
whose solution is,
ye
P dx
=A+
Qe
P dx
dx.
(17.51)
From equations (17.49) and (17.51), with p0 denoting p at t = 0, the instantaneous
pressure p in the aorta as a function of the left ventricular ejection rate Q(t) is
given by,
t
1
Q(τ )eτ/R K dτ + p0 e−t/R K .
(17.52)
p(t) = e−t/R K
K
0
In equation (17.52), p0 would be the aortic pressure at the end of diastolic phase.
A fundamental assumption in the Windkessel theory is that the pressure pulse
wave generated by the heart is transmitted instantaneously throughout the arterial
system and disappears before the next cardiac cycle. In reality, pressure waves require
finite but small transmission times, and are modified by reflection at bifurcations,
bends, tapers, and at the end of short tubes of finite length, and so on. We will now
account for some of these features.
793
3. Modelling of Flow in Blood Vessels
Pulse Wave Propagation in an Elastic Tube: Inviscid Theory
Consider a homogeneous, incompressible, and inviscid fluid in an infinitely long,
horizontal, cylindrical, thin walled, elastic tube. Let the fluid be initially at rest. The
propagation of a disturbance wave of small amplitude and long wave length compared
to the tube radius is of interest to us. In particular, we wish to calculate the wave speed.
Since the disturbance wave length is much greater than the tube diameter, the time
dependent internal pressure can be taken to be a function only of (x, t).
Before we embark on developing the solution, we need to understand the inviscid
approximation. For flow in large arteries, the Reynolds and Womersley numbers are
large, the wall boundary layers are very thin compared to the radius of the vessel.
The inviscid approximation may be useful in giving us insights in understanding
such flows. Clearly, this will not be the case with arterioles, venules and capillaries.
However, the inviscid analysis is strictly of limited use since it is the viscous stress
that is dominant in determining flow stability in large arteries.
Under the various conditions prescribed, the resulting flow may be treated as one
dimensional.
Let A(x, t) and u(x, t) denote the the cross sectional area of the tube and the
longitudinal velocity component, respectively. The continuity equation is:
∂A ∂(Au)
+
= 0,
∂t
∂x
and, the equation for the conservation of momentum is:
∂u
∂ ((p − pe ) A)
∂u
+u
,
=−
ρA
∂t
∂x
∂x
(17.53)
(17.54)
where (p − pe ) is the transmural pressure difference. Since the tube wall is assumed
to be elastic (not viscoelastic), under the further assumption that A depends on the
transmural pressure difference (p − pe ) alone, and the material obeys Hooke’s law,
we have from equation (17.41), the pressure—radius relationship (referred to as
“tube law”),
1
a0 Eh
Eh
A0 2
p − pe =
1−
,
(17.55)
1−
=
a0
a
a0
A
where A = πa 2 , and A0 = πa02 . The equations (17.53), (17.54), and (17.55) govern
the wave propagation. We may simplify this equation system further by linearizing
it. This is possible if the pressure amplitude (p − pe ) compared to p0 , the induced
fluid speed u, and (A − A0 ) compared to A0 , and their derivatives are all small.
If the pulse is moving slowly relative to the speed of sound in the fluid, the wave
amplitude is much smaller than the wave length, and the distension at one cross
section has no effect on the distension elsewhere, the assumptions are reasonable.
As discussed by Pedley (2000), in normal human beings, the mean blood pressure,
relative to atmospheric, at the level of the heart is about 100 mmHg, and there is a
cyclical variation between 80 and 120 mmHg, so the amplitude-to-mean ratio is 0.2,
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Introduction to Biofluid Mechanics
which is reasonably small. Also, in the ascending aorta, the pulse wave speed, c, is
about 5 m/s, and the maximum value of u is about 1 m/s, and (u/c) is also around
0.2. In that case, the system of equations reduce to
∂u
∂A
+ A0
= 0,
∂t
∂x
(17.56)
and,
ρ
∂p
∂u
=− ,
∂t
∂x
(17.57)
and,
p − pe =
∂p
Eh
Eh
=
(A − A0 ) , and
2a0 A0
∂A
2a0 A0
(17.58)
Differentiating equation (17.56) with respect to t and equation (17.57) with respect
to x, and subtracting the resulting equations, we get,
A0 ∂ 2 p
∂ 2A
=
,
2
ρ ∂x 2
∂t
(17.59)
and with equation (17.58), we obtain,
Eh ∂ 2 A
∂p A0 ∂ 2 p
∂ 2p
=
=
.
2
2
2a0 A0 ∂t
∂A ρ ∂x 2
∂t
(17.60)
Combining equations (17.59) and (17.60), we produce,
1 ∂ 2p
∂ 2p
∂ 2p
∂ 2p
2
=
,
or,
=
c
(A
)
,
0
∂x 2
c2 ∂t 2
∂t 2
∂x 2
where, c2 =
Eh
2ρa0
=
A dp
ρ dA .
(17.61)
Equation (17.61) is the wave equation, and the quantity,
c=
Eh
=
2ρa0
A dp
,
ρ dA
(17.62)
is the speed of propagation of the pressure pulse. This is known as the
Moens-Korteweg wave speed. If the thin wall assumption is not made, following Fung
(1997), by evaluating the strain on the midwall of the tube,
Eh
c=
,
(17.63)
2ρ (a0 + h/2)
Next, similar to equation (17.61), we can develop,
1 ∂ 2u
∂ 2u
=
,
∂x 2
c2 ∂t 2
(17.64)
795
3. Modelling of Flow in Blood Vessels
for the velocity component u. The wave equation (17.61) has the general solution,
x
x
+ f2 t +
,
(17.65)
p = f1 t −
c
c
where f1 and f2 are arbitrary functions; f2 is zero if the wave propagates only in
the +x direction. This result states that the small amplitude disturbance can propagate
along the tube, in either direction, without change of shape of the wave form, at speed
c(a0 ). Also, the velocity wave form is predicted to be of the same shape as the pressure
wave form.
In principle, the Moens-Korteweg wave speed given in equation (17.63) must
enable the determination of the arterial modulus E as a function of a by noninvasive
measurement of the values of arterial dimensions (a, h), the wave forms of the arterial
inner radius at two sites, the transit time (as the time interval between the wave form
peaks), and hence the pulse wave velocity. More details in this regard are available in
the book by Mazumdar (1999).
Next, consider the solutions of wave equations (17.61) and (17.64),
p = p̂1 f (x − ct) + p̂2 g(x + ct),
and,
u = û1 f (x − ct) + û2 g(x + ct),
(17.66)
(17.67)
where p̂1 , û1 , p̂2 , and, û2 , are the pressure and velocity amplitudes for waves
travelling in the positive x-direction and negative x-direction, respectively. From
equation (17.57),
(17.68)
p̂1 = ρcû1 , and, p̂2 = −ρcû2 .
This equation (17.68) relates the amplitudes of the pressure and velocity waves.
The above analysis would equally apply if the inviscid fluid in the tube was
initially in steady motion, say from left to right. In that case, u would have to be
regarded as a small perturbation superposed on the steady flow, and c would be the
speed of the perturbation wave relative to the undisturbed flow.
Let us now examine the limitations of the above model. For typical flow in the
aorta, the speed of propagation of the pulse is about 4 m/s (Brown et al. (1999)),
about 5 m/s in the ascending aorta, rising to about 8 m/s in more peripheral arteries.
These predictions are very close to measured values in normal subjects, either dogs
or humans (Pedley (2000)). The peak flow speed is about 1 m/s. The speed of propagation in a collapsible vein might be as low as 1 m/s, and this may lead to phenomena
analogous to sonic flow ( Brown et al. (1999)). From equation (17.62), for given E,
h, ρ, and size of vessel, the wave speed is a constant. Experimental studies indicate,
however, that the wave speed is a function of frequency. The shape of the wave form
does not remain the same. The theory must be modified to account for peaking of
the pressure pulse due to wave reflection from arterial junctions, wave front steepening due to nonlinear dispersion effects ( Lighthill (1978)), and observed velocity
wave form by including dissipative effects due to viscosity (Lighthill (1975), Pedley
(2000)). The neglect of the inertial terms and the effects of viscosity have therefore
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Introduction to Biofluid Mechanics
to be examined to address these concerns and to develop a systematic understanding.
These issues will be addressed in later sections in the following order. First, we will
learn about pulsatile viscous flow in a rigid walled single straight tube. This would
imply the assumption of an infinite wave speed. Subsequent to that, we will examine
the effects of wall elasticity on pulsatile viscous flow in a single tube to gain a more
realistic understanding. This would allow us to understand wave transmission at finite
speed. Following this, we will study blood vessel bifurcation. This will be extended
to understand the effects of wave reflection from arterial junctions under the inviscid
flow approximation.
Pulsatile Flow in a Rigid Cylindrical Tube: Viscous Effects Included, Infinite
Wave Speed Assumption
Consider the axi-symmetric flow of a Newtonian incompressible fluid in a long, thin,
circular, cylindrical, horizontal, rigid walled tube. Clearly, the assumption of a rigid
wall implies that the speed of wave propagation is infinite and unrealistic. However,
the development presented here will provide us with useful insights and these will be
helpful in formulating a much improved theory in the next section.
We shall employ the cylindrical coordinates (r, θ, x) with velocity components
(ur , uθ , and, ux ), respectively. Let λ be the wave length of the pulse. This is long, and
a ≪ λ. Since the wave speed is infinite, all the velocity components are very much
smaller than the wave speed. These assumptions would enable us to drop the inertial
terms in the momentum equations. With the additional assumptions of axi-symmetry
∂
(uθ = 0, and ∂θ
= 0), and rigid tube wall, (ur = 0), and omitting the subscript x in
ux for convenience, the continuity equation may be written:
∂u
= 0,
∂x
(17.69)
∂p
∂r
(17.70)
and the r-momentum equation is:
0=−
and the x momentum equation is:
2
∂u
∂p
∂ u 1 ∂u
ρ
=−
+µ
+
∂t
∂x
r ∂r
∂r 2
(17.71)
We see that u = u(r, t) and p = p(x, t). Therefore, we are left with just one equation:
∂ 2 u 1 ∂u
∂p
∂u
µ
=
.
−ρ
+
2
r ∂r
∂t
∂x
∂r
(17.72)
∂p
In equation (17.72), since p = p(x, t), ∂x
will be a function only of t. Since the
pressure wave form is periodic, it is convenient to express the partial derivative of
pressure using a Fourier series. Such a periodic function depends on the fundamental
797
3. Modelling of Flow in Blood Vessels
frequency of the signal, ω, heart rate (unit, rad/s), and the time t. Recall that ω is also
called the circular frequency, ω/2π is the frequency (unit, Hz), and λ is the wave
length, (unit, m). Also, λ = c/(ω/2π ), where c is wave speed. The wave length is
the wave speed divided by frequency, or the distance travelled per cycle.
We set
∂p
= −Geiωt ,
(17.73)
∂x
where G is a constant denoting the amplitude of the pressure gradient pulse and
eiωt = cos ωt + i sin ωt. With this representation for p(t), equation (17.72) becomes:
µ
∂ 2 u 1 ∂u
∂u
∂p
= −Geiωt
−
ρ
=
+
r ∂r
∂t
∂x
∂r 2
(17.74)
This is a linear, second order, partial differential equation with a forcing function.
For ω = 0, the flow is described by the Hagen-Poiseuille model. Womersley (1955a,
1955b), has solved this problem, and we will provide essential details.
For ω = 0, we may try solutions of the form,
u(r, t) = U (r)eiωt ,
(17.75)
where, U (r) is the velocity profile in any cross section of the tube. The real part
in equation (17.75) gives the velocity for the pressure gradient G cos ωt and the
imaginary part gives the velocity for the pressure gradient G sin ωt. Assume that the
flow is identical at each cross section along the tube. From equations (17.74) and
(17.75), we get:
d 2U
iωρ
G
1 dU
−
U= .
(17.76)
+
r dr
µ
µ
dr 2
This is a Bessel’s differential equation, and the solution would involve Bessel functions of zeroth order and complex arguments. Thus,
G
U (r) = C1 J0 i (iωρ/µ) r + C2 Y0 i (iωρ/µ) r +
,
ωρi
(17.77)
where C1 and C2 are constants. In equation (17.77), from the requirement that U is
finite at r = 0, C2 = 0. For a rigid walled tube, U = 0 at r = a. Therefore,
G
C1 J0 i 3/2 (ωρ/µ) a +
= 0.
ωρi
(17.78)
√
From equation (17.12), the Womersley number is defined by α = a ω/ν. Therefore,
from equation (17.78), we may write,
C1 =
iG
1
.
ωρ J0 i 3/2 α
(17.79)
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Introduction to Biofluid Mechanics
Therefore, from equation (17.77),
J0 i 3/2 α r/a
iG
U (r) = −
1−
.
ωρ
J0 i 3/2 α
(17.80)
Introduce, for convenience,
F1 (α) =
J0 i 3/2 α r/a
J0 i 3/2 α
.
(17.81)
Now, from equation (17.75),
u(r, t) = U (r)eiωt = −
iG
Ga 2
(1 − F1 (α)) eiωt.
(1 − F1 (α)) eiωt =
ωρ
iµα 2
(17.82)
In the above development, we have found the velocity as a function of radius r and
∂p
and
time t for the entire driving pressure gradient. Since we have represented both ∂x
u(r, t) in terms of Fourier modes, we could also express the solution for both these
quantities in terms of individual Fourier modes or harmonics explicitly as,
N
∂p
Gn einωt,
=−
∂x
(17.83)
n=0
where N is the number of modes (harmonics), and the n = 0 term represents the
mean pressure gradient. Similarly, for velocity,
u(r, t) = u0 (r) +
N
un (r)einωt
(17.84)
1
In equation (17.84),
u0 (r) =
G0 a 2
4µ
r2
1− 2 ,
a
(17.85)
is the mean flow and is recognized as the steady Hagen-Poiseuille flow with G0 as
the mean pressure gradient, and, for each harmonic,
un (r) =
Gn a 2
(1 − F1 (αn ))
iµαn 2
(17.86)
We can now write down the expressions for un (r) in the limits of αn small and large.
These are, for αn small,
Gn a 2
un (r) ≈
4µ
r2
1− 2 ,
a
(17.87)
799
3. Modelling of Flow in Blood Vessels
which represents a quasi-steady flow, and for αn large,
ω
Gn a 2
(1 + i)(a − r) ,
1 − exp −
un (r) ≈
2ν
iµαn 2
(17.88)
which is the velocity boundary layer on a plane wall in an oscillating flow. This flow
was discussed in Chapter 9 (Stokes second problem).
The volume flow rate, Q(t), may be obtained by integrating the velocity profile
across the cross section. Thus, from equations (17.85) and (17.86)
∞
a
2
a 2 Gn
inωt
2 G0 a
Q(t) =
,
u 2 πrdr = πa
+
[1 − F2 (αn )] e
8µ
iµ
αn 2
0
1
(17.89)
or equivalently, with equation (17.82),
a
Q(t) =
0
2πeiωt
π a4
Ga 2
−
F
(α))
rdr
=
G (1 − F2 (α)) eiωt , (17.90)
(1
1
iµα 2
iµα 2
where,
F2 (α) =
2J1 i 3/2 α
i 3/2 αJ0 i 3/2 α
.
(17.91)
The real part of equation (17.90) gives the volume flow rate when the pressure gradient
is G cos ωt and the imaginary part gives the rate when the pressure gradient is G sin ωt.
Next, the wall shear rate, τ (t)|r=a is given by,
τ (t)|r=a =
∂u
∂r
r=a
=
N
a
G0 a
Gn F (αn )einωt
+
2
2
(17.92)
1
We may now examine the flow rates in the limit cases of α → 0 and α → ∞. As
α → 0, by Taylor’s expansion,
F2 (α) ≈ 1 −
iα 2
− O(α 4 ),
8
(17.93)
and, from equation (17.90), in the limit as α → 0,
Q=
π Ga 4 inωt
e
,
8µ
(17.94)
and the magnitude of the volumetric flow rate, Q0 , in the limit as α → 0 is,
|Q0 | =
π Ga 4
,
8µ
as would be expected (Hagen-Poiseuille result). As α → ∞,
1
2
F2 (α) ≈ 1/2
1+
,
2α
i α
(17.95)
(17.96)
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Introduction to Biofluid Mechanics
Next, in Hagen-Poiseuille flow, the steady flow rate is the maximum attainable
and there is no phase lag between the applied pressure gradient and the flow. To
understand the phase difference between the applied pressure gradient pulse and the
flow rate in the present flow model, we set,
(1 − F2 (α)) = Z(α), Z(α) = X(α) + iY (α)
(17.97)
Then from equation (17.90),
Q=
πGa 4
{[Y cos(ωt) + X sin(ωt)] − i [X cos(ωt) − Y sin(ωt)]}
µα 2
(17.98)
The magnitude of Q is,
|Q| =
π Ga 4 2
X + Y 2.
µα 2
(17.99)
The phase angle between the applied pressure gradient Geiωt and the flow rate (equation (17.90)) is now noted to be,
tan φ =
X
.
Y
(17.100)
With increasing ω, the phase lag between the pressure gradient and the flow rate
increases, and the flow rate decreases. Thus, the magnitude of the volumetric flow
rate, |Q|, given by equation (17.99) will be considerably less than the magnitude |Q0 |
given by equation (17.95) as would be expected. For an arterial flow, with α = 8,
X ≈ 0.85, Y ≈ 0.16, the pulsed volumetric flow rate, |Q| would be just about one
tenth of the steady value, |Q0 |. For more detailed discussions and comparisons with
measured values of pressure gradients and flow rates in blood vessels, see Nichols
and O’Rourke (1998).
The above analysis assumed an infinite wave speed of propagation. In order to
accommodate the requirement of wave transmission at a finite wave speed, we need
to account for vessel wall elasticity. This will be discussed in the next section.
Wave Propagation in a Viscous Liquid Contained in an Elastic Cylindrical Tube
Blood vessel walls are viscoelastic. But in large arteries the effect of nonlinear
viscoelasticity on wave propagation is not so severe (Fung, 1997). Even where viscoelastic effects are important, an understanding based on elastic walls will be useful.
In this section, we will first study the effects of elastic walls. Then, we will briefly
discuss the effects of wall viscoelasticity.
Consider a long, thin, circular, cylindrical, horizontal elastic tube containing
a Newtonian, incompressible fluid. Let this system be set in motion solely due to a
pressure wave, and the amplitude of the disturbance be small enough so that quadratic
terms in the amplitude are negligible compared with linear ones.
In the formulation, we have to consider the fluid flow equations together with the
equations of motion governing tube wall displacements. Assume that the tube wall
801
3. Modelling of Flow in Blood Vessels
material obeys Hooke’s law. Since the tube is thin, membrane theory for modelling
the wall displacements is adequately accurate, and we will neglect bending stresses.
The primary question is, how does viscosity attenuate velocity and pressure in
this flow?
We shall employ the cylindrical coordinates (r, θ, x) with velocity components (ur , uθ , and, ux ), respectively. With the assumption of axi-symmetry, uθ =
∂
= 0. For convenience, we write the ur component as v, and we omit the
0 and ∂θ
subscript x in ux .
Restricting the analysis to small disturbances, the governing equations for the
fluid are:
∂u 1 ∂ (rv)
+
= 0,
∂x
r ∂r
2
∂p
∂ u 1 ∂u ∂ 2 u
∂u
+
=−
+µ
+ 2 ,
ρ
∂t
∂x
r ∂r
∂r 2
∂x
2
1 ∂v
v
∂p
∂ v
∂ 2v
∂v
+
=−
+µ
+ 2− 2 ,
ρ
∂t
∂r
r ∂r
∂r 2
∂x
r
(17.101)
(17.102)
(17.103)
where u and v are the velocity components in the axial and radial directions, respectively.
These have to be supplemented with the tube wall displacement equations. Let
the tube wall displacements in the (r, θ, x) directions be (η, ζ, and ξ ), respectively,
and the tube material density be ρw . The initial radius of the tube is a0 , and the wall
thickness is h.
For this thin elastic tube, the circumferential (hoop) tension and the tension in
the axial direction are related by Hooke’s law as follows:
Eh
∂ξ
η
+
ν̂
,
(17.104)
Tθ =
∂x
1 − ν̂ 2 a0
and
Tx =
Eh
1 − ν̂ 2
η
∂ξ
+ ν̂
∂x
a0
,
(17.105)
where ν̂ is Poisson’s ratio.
By a force balance on a wall element of volume (h rdθ dx), the equations
governing wall displacements may be written as:
• r-direction
∂ 2η
Tθ
= σrr |r=a − ,
a0
∂t 2
(17.106)
∂ 2ξ
∂Tx
− σrx |r=a .
=+
∂x
∂t 2
(17.107)
ρw h
and
• x-direction
ρw h
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Introduction to Biofluid Mechanics
There is no displacement equation for the θ direction. In equations (17.106) and
(17.107), σrr |r=a and σrx |r=a refer to radial and shear stresses, respectively, which
the fluid exerts on the tube wall. These equations are based on the assumptions that
shear and bending stresses in the tube wall material are negligible and the slope of
the disturbed tube wall (∂a/∂x) is small. These also imply that the ratios (a/λ) and
(h/λ), where λ is the wavelength of disturbance, are small.
From equations (17.104), (17.105), (17.106), (17.107), we obtain,
ν̂ ∂ξ
Eh
η
∂ 2η
ρw h 2 = σrr |r=a −
+
,
(17.108)
a0 ∂x
∂t
1 − ν̂ 2 a02
and
∂u ∂v
∂ 2ξ
+
ρw h 2 = −µ
∂r
∂x
∂t
r=a
Eh
+
1 − ν̂ 2
ν̂ ∂η
∂ 2ξ
+
. (17.109)
a0 ∂x
∂x 2
In the above equations, from the theory of fluid flow, the normal compressive stress
due to fluid flow on element of area perpendicular to the radius is given by,
σrr = +p − 2µ
∂v
,
∂r
(17.110)
and the shear stress due to fluid flow acting in a direction parallel to the axis of the
tube on an element of area perpendicular to a radius is,
∂u ∂v
σrx = µ
+
.
(17.111)
∂r
∂x
These are the radial and shear stresses exerted by the fluid on the wall of the vessel.
With equations (17.110) and (17.111), equations (17.108), and (17.109) become,
ν̂ ∂ξ
∂v
η
Eh
∂ 2η
ρw h 2 = +p|r=a − 2µ
+
−
,
(17.112)
∂r r=a 1 − ν̂ 2 a02
a0 ∂x
∂t
and
ρw h
∂ 2ξ
= −µ
∂t 2
∂u ∂v
+
∂r
∂x
r=a
+
Eh
1 − ν̂ 2
ν̂ ∂η
∂ 2ξ
.
+
a0 ∂x
∂x 2
(17.113)
We have to solve equations (17.101), (17.102), (17.103), together with (17.112),
and (17.113) subject to prescribed conditions. The boundary conditions at the wall
are that the velocity components of the fluid be equal to those of the wall. Thus,
u|r=a0 =
∂ξ
∂t
r=a0
v|r=a0 =
∂η
∂t
r=a0
,
(17.114)
.
(17.115)
and
803
3. Modelling of Flow in Blood Vessels
We note that the boundary conditions given in equations (17.114) and (17.115) are
linearized conditions, since we are evaluating u and v at the undisturbed radius a0 .
We now represent the various quantities in terms of Fourier modes. Thus,
u(x, r, t) = û(r)ei(kx−ωt) , v(x, r, t) = v̂(r)ei(kx−ωt) ,
p(x, t) = p̂ei(kx−ωt) , ξ(x, t) = ξ̂ ei(kx−ωt) ,
η(x, t) = η̂ei(kx−ωt) ,
(17.116)
where û(r), v̂(r), p̂, ξ̂ , and η̂ are the amplitudes, ω = 2π/T is a real constant, the
frequency of the forced disturbance, T is the period of the heart cycle, and k = k1 +ik2
is a complex constant, k1 being the wave number and k2 is a measure of
the decay of
the disturbance as it travels along the vessel (damping constant), |k| = k1 2 + k2 2 =
2π/λ where λ is the wave length of disturbance, and c = ω/k1 is the wave speed.
The above formulation has been solved by Morgan and Kiely (1954) and
by Womersley (1957a, 1957b), and we will provide the essential details here. The
analysis will be restricted to disturbances of long wavelength, that is, a/λ ≪ 1, and
large Womersley number, α ≫ 1.
From equation (17.101),
v
v̂(r)
=
= O(|ak|).
u
û(r)
(17.117)
For small damping, we note that |k| ≈ k1 = 2π/λ, and c = ω/k1 is the wave speed.
From equations (17.102) and (17.103), we may make the following observations.
∂2u
In equation (17.102), ∂x
2 may be neglected in comparison with the other terms since
a/λ ≪ 1 and λα ≫ 1. In equation (17.103), ∂p
∂r is of a higher order of magnitude in
∂p
a/λ than is ∂x
. In fact, we may neglect all terms that are of order a/λ. In effect, we
are neglecting radial acceleration and damping terms and taking the pressure to be
uniform over each cross section. The fluid equations become,
∂u 1 ∂ (rv)
+
= 0,
∂x
r ∂r
2
∂p
∂ u 1 ∂u
∂u
+
=−
+µ
,
ρ
∂t
∂x
r ∂r
∂r 2
∂p
= 0,
∂r
p = p̂ei(kx−ωt) .
(17.118)
(17.119)
(17.120)
(17.121)
Now substitute the assumed forms given in equation (17.116) into equations (17.118)
and (17.119) to produce,
d(r v̂)
= −ikr û,
dr
ik p̂
d 2 û 1 d û iωρ
+
+
û =
,
2
r ∂r
µ
µ
dr
(17.122)
(17.123)
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Introduction to Biofluid Mechanics
The boundary conditions given by equations (17.114) and (17.115) become:
û(a0 )ei(kx−ωt) = −iωξ̂ ei(kx−ωt) ,
v̂(a0 )ei(kx−ωt) = −iωη̂ei(kx−ωt) .
(17.124)
(17.125)
We may now note that the linearization of the boundary conditions will involve an error
of the same order as that caused by neglecting the nonlinear terms in the equations.
The error would be small if ξ̂ and η̂ are very small compared to a.
Next, introduce the assumed form given in equation (17.116), and use equation
(17.120) in the displacement equations (17.112) and (17.113) to develop,
ˆ
Eh
η̂
iνk
d v̂
2
−
+
ξ̂ , (17.126)
−ρw hω η̂ = p̂ − 2µ
dr r=a0 1 − ν̂ 2 a0 2
a0
ˆ
iνk
d û
Eh
2
2
−k ξ̂ +
−ρw hω ξ̂ = −µ
+ ik v̂
η̂ . (17.127)
+
dr
a0
1 − ν̂ 2
r=a0
Now invoke the assumptions that h/a ≪ 1, ρ is of the same order of magnitude as
ρw , and a 2 /λ2 ≪ 1 in equations (17.126) and (17.127). This amounts to neglecting
the terms which represent tube inertia, and approximating σrx in equation (17.111)
∂v
by µ ∂x
and σrr in (17.110) by p. After considerable algebra, equations (17.126)
and (17.127) reduce to:
p̂ =
d û
i ν̂
Eh
µ
η̂ −
2
a0 k dr
a0
ξ̂ =
i ν̂
1 − ν̂ 2 d û
η̂ −
µ
ka0
dr
Ehk 2
(17.128)
,
r=a0
(17.129)
r=a0
We are now left with equations (17.122), (17.123), (17.128), and (17.129), subject
to boundary conditions given by (17.124) and (17.125) and the pseudo boundary
condition that u(r) be nonsingular at r = 0.
Equations (17.123) and (17.128) can be combined to give:
ik Eh
d 2 û 1 d û iωρ
ν̂ d û
+
+
û =
η̂ +
2
2
r ∂r
µ
µ a0
a0 dr
dr
.
(17.130)
r=a0
Satisfying the pseudo boundary condition, the solution to this Bessel’s differential
equation is given by:
û(r) = AJ0 (βr) +
where, β =
k Eh
ν̂
A J1 (βa0 ),
η̂ −
2
ω ρa0
βa0
(17.131)
√
iω/ν, and A is an arbitrary constant. Next, from equation (17.122),
v̂ = −
ik
r
r
r û(r)dr.
0
(17.132)
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3. Modelling of Flow in Blood Vessels
From equation (17.131) and equation (17.132),
v̂(r) = −
ikA
ik ν̂
ik 2 Ehη̂ r
r
J1 (βr) −
+
A J1 (βa0 ).
2
β
ω ρa0 2 βa0
2
(17.133)
Equations (17.131) and (17.133) give the expressions for û(r) and v̂(r), respectively.
Subjecting them to the boundary conditions given in equations (17.124) and (17.125),
introducing β̂ = βa0 , and eliminating ξ̂ by the use of equation (17.129), the following
two linear homogeneous equations for η̂ are developed:
ω ν̂
kEh
ν̂
iβωµ(1 − ν̂ 2 )
η̂
−
= A J0 (β̂) + J1 (β̂)
, (17.134)
−
k a0
Ehk 2
ωρa02
β̂
k 2 Eh
k
k ν̂
η̂ 1 − 2
= AJ1 (β̂)
−
.
ωβ
2ωβ
ω 2ρa0
(17.135)
For non zero solutions, the determinant of the above set of linear algebraic equations in
η̂ and A must be zero. As a result, the following characteristic equation is developed:
2
2
k Eh
k 2 Eh
J0 (β̂)
J0 (β̂)
−4 +
2β̂
4ν̂ − 1 − 2β̂
ω2 2ρa0
ω2 2ρa0
J1 (β̂)
J1 (β̂)
(17.136)
+ 1 − ν̂ 2 = 0.
The solution to this quadratic equation will give k 2 /ω2 in terms of known quantities.
Then we can find, k/ω = (k1 +ik2 )/ω. The wave speed, ω/k1 , and the damping factor
may be evaluated by determining the real and imaginary parts of k/ω.
Morgan and Kiely (1954) have provided explicit results for the wave speed, c,
and the damping constant, k2 , in the limits of small and large α. Mazumdar (1999)
has indicated that by an in vivo study, the wave speed, ω/k1 , can be evaluated
non-invasively by monitoring the transit time as the time interval between the peaks
of ultrasonically measured waveforms of the arterial diameter at two arterial sites at a
known distance apart. Then from equation (17.136), E can be calculated. From either
of the equations (17.134) or (17.135), A can be expressed in terms of η̂, and with that
û(r) can be related to p̂. Mazumdar gives details as to how the cardiac output may
be calculated with the information so developed in conjunction with pulsed Doppler
flowmetry.
Figure 17.14 shows velocity profiles at intervals of ωt = 15◦ , of the flow
resulting from a pressure gradient varying as cos(ωt) in a tube. As this is harmonic
motion, only half cycle is illustrated and for ωt > 180◦ , the velocity profiles are of
the same form but opposite in sign. α is the Womersley number. The reversal of flow
starts in the laminae near the wall. As the Womersley number increases, the profiles
become flatter in the central region, there is a reduction in the amplitudes of the flow,
and the rate of reversal of flow increases close to the wall. At α = 6.67, the central
mass of the fluid is seen to reciprocate like a solid core.
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Introduction to Biofluid Mechanics
Figure 17.14 Velocity profiles of a sinusoidally oscillating flow in a pipe. (Reproduced from McDonald,
D. A. (1974) Blood Flow in Arteries, The Williams & Wilkins Company, Baltimore).
Effect of Viscoelasticity of Tube Material
In general, the wall of a blood vessel must be treated as viscoelastic. This means
that the relations given in equations (17.104) and (17.105) must be replaced by
corresponding relations for a tube of viscoelastic material. In this problem, all the
stresses and strains in the problem are assumed to vary as ei(kx−ωt) , and we will
further assume that the effect of the strain rates on the stresses is small compared to
the effect of the strains. For the purely elastic case, only two real elastic constants
were needed. Morgan and Kiely (1954) have shown that by substituting suitable
complex quantities for the elastic modulus and the Poisson’s ratio the viscoelastic
behavior of the tube wall may be accommodated. They introduce,
′
′
E ∗ = E − iωE , and, ν̂ ∗ = ν̂ − iων̂ ,
′
′
(17.137)
where, E and ν̂ are new constants. In equations (17.104) and (17.105), E ∗ and ν̂ ∗
will replace E and ν̂, respectively. The formulation will otherwise remain the same.
An equation for k/ω will arise as before. The fact that E ∗ and ν̂ ∗ are complex has
to be taken into account while evaluating the wave velocity and the damping factor.
Morgan and Kiely provide results appropriate for small and large α.
Morgan and Ferrante (1955) have extended the study by Morgan and Kiely
(1954) discussed above to the situation for small α values where there is Poiseuille
like flow in the thin, elastic walled tube. The flow oscillations are small and they are
superimposed on a large steady stream velocity. The steady flow modifies the wave
velocity. The wave velocity in the presence of a steady flow is the algebraic sum
3. Modelling of Flow in Blood Vessels
of the normal wave velocity and the steady flow velocity. Morgan and Ferrante
predict a decrease in the damping of a wave propagated in the direction of the
stream and an increase in the damping when propagated upstream. However, the
steady flow component in arteries is so small in comparison with the pulse wave
velocity that its role in damping is of little importance (see McDonald (1974)).
Womersley (1957a) has considered the situation where the flow oscillations are
large in amplitude compared to the mean stream velocity. This is similar to the
situation in an artery. He predicts that the presence of a steady stream velocity
would produce a small increase in the damping.
Next, we will study blood flow in branching tubes.
Blood Vessel Bifurcation: An Application of Poiseuille’s Formula and
Murray’s Law
Blood vessels bifurcate into smaller daughter vessels which in turn bifurcate to even
smaller ones. On the basis that the flow satisfies Poiseuille’s formula in the parent
and all the daughter vessels, and by invoking the principle of minimization of energy
dissipation in the flow, we can determine the optimal size of the vessels and the
geometry of bifurcation. We recall that Hagen-Poiseuille flow involves established
(fully developed) flow in a long tube. Here, for simplicity, we will assume that established Poiseuille flow exists in all the vessels. This is obviously a drastic assumption
but the analysis will provide us with some useful insights.
Let the parent and daughter vessels be straight, circular in cross section, and lie
in a plane.
Consider a parent vessel AB of length L0 of radius a0 in which the flow rate is Q
which bifurcates into two daughter vessels BC and BD with lengths L1 , and L2 , radii
a1 and a2 , and flow rates Q1 and Q2 , respectively. The axes of vessels BC and BD
are inclined at angles θ and φ with respect to the axis of AB, as shown in Fig. (17.15).
Points A,C,D are fixed. The optimal sizes of the vessels and the optimal location of
B have to determined from the principle of minimization of energy dissipation.
Figure 17.15 Schematic of an arterial bifurcation.
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Introduction to Biofluid Mechanics
The total rate of energy dissipation by flow rate Q in a blood vessel of length
of L and radius a is equal to sum of the rate at which work is done on the blood,
Qp, and the rate at which energy is used up by the blood vessel by metabolism,
Kπa 2 L, where K is a constant. For Hagen-Poiseuille flow, from equation (17.17),
4
Q = π8µa p
L . Therefore,
Total energy dissipation =
8µL 2
Q + Kπa 2 L = Ê1 , (say).
π a4
(17.138)
To obtain the optimal size of a vessel for transport, for a given length of vessel, we
need to minimize this quantity with respect to radius of the vessel. Thus,
32µL 2 −5
∂ Ê1
=−
Q a + 2KπLa = 0.
∂a
π
(17.139)
Solving for a,
16µ
K
a=
π2
1/6
Q1/3 .
(17.140)
The equation (17.140) gives the optimal radius for the blood vessel indicating that
minimum energy dissipation occurs under this condition. The optimal relationship,
Q ∼ a 3 , is called Murray’s Law.
With equation (17.140), the minimum value for energy dissipation is
Ê1,min =
3π
KLa 2 .
2
(17.141)
Next, consider the flow with the branches. The minimum value for energy dissipation
with branches is
Ê2,min =
3π
K L0 a02 + L1 a12 + L2 a22 .
2
(17.142)
Also,
16µ
a0 =
K
π2
1/6
1/3
Q0 ,
16µ
a1 =
K
π2
1/6
1/3
Q1 ,
16µ
and, a2 =
K
π2
1/6
1/3
Q2 ,
(17.143)
and, from mass conservation,
Q = Q1 + Q2 → a03 = a13 + a23 .
(17.144)
The lengths L0 , L1 , L2 depend on the location of point B. The optimum location of
the point B is determined by examining associated variational problems (see Fung
(1997)).
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3. Modelling of Flow in Blood Vessels
Any small movement of B changes Ê2,min by δ Ê2,min , and,
δ Ê2,min =
3π
K δL0 a02 + δL1 a12 + δL2 a22
2
(17.145)
The optimal location of B would be such as to make δ Ê2,min = 0 for arbitrary small
movement δL of point B. By making such displacements of B, one at a time, in the
direction of AB, in the direction of BC, and finally in the direction of DB, and setting
the value of the corresponding δ Ê2,min to zero, we develop a set of three conditions
governing optimization. These are:
cos θ =
a04 + a14 − a24
2a02 a12
, cos φ =
a04 − a14 + a24
2a02 a22
, cos(θ + φ) =
a04 − a14 − a24
2a12 a22
.
(17.146)
Together with equation (17.144), equation set (17.146) may be solved for the optimum
angle θ as,
4/3
a04 + a14 − a03 − a13
,
(17.147)
cos θ =
2a02 a12
and a similar equation for φ. Comparison of these optimization results with experimental data are noted to be excellent (see Fung (1997)).
Reflection of Waves at Arterial Junctions: Inviscid Flow and Long Wave
Length Approximation
Arteries have branches. When a pressure or a velocity wave reaches a junction where
the parent artery 1 bifurcates into daughter tubes 2 and 3 as shown in the Figure 17.16,
the incident wave is partially reflected at the junction into the parent tube and partially
transmitted down the daughters. In the long wave length approximation, we may
neglect the flow at the junction. Let the longitudinal coordinate in each tube be x,
Figure 17.16 Schematic of an arterial bifurcation: Reflection.
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Introduction to Biofluid Mechanics
with x = 0 at the bifurcation. the incident wave in the parent tube comes from
x = −∞.
Let pI be the oscillatory pressure associated with the incident wave, pR that associated with the reflected wave, and pT 1 and pT 2 , those associated with the transmitted
waves. Let the pressure be a single valued and continuous function at the junction for
all time t. The continuity requirement ensures that there are no local accelerations.
Under these conditions, at the junction,
pI + pR = pT 1 = pT 2 .
(17.148)
Next, let QI be the flow rate associated with the incident wave, QR that associated with
the reflected wave, and QT 1 and QT 2 , those associated with the transmitted waves.
The flow rate is also taken to be single valued and continuous at the junction for all
time t. The continuity requirement ensures conservation of mass. At the junction,
QI − QR = QT 1 + QT 2 .
(17.149)
Let the undisturbed cross sectional areas of the tubes be A1 , A2 , and A3 , and the
intrinsic wave speeds be c1 , c2 , c3 , respectively. In general, for a fluid of density ρ
flowing under the influence of a wave with intrinsic wave speed c, through a tube of
cross sectional area A, the flow rate Q is related to the mean velocity u by,
Q = Au = ±
A
p,
ρc
(17.150)
where we have employed the relationship given in equation (17.68). The plus or the
minus sign applies depending on whether the wave is going in the positive x direction
or in the negative x direction. The quantity A/ρc is called the characteristic admittance
of tube and is denoted by Y , while, ρc/A, is called the characteristic impedance of
the tube and is denoted by Z. Admittance is seen to be the ratio of the oscillatory
flow to the oscillatory pressure when the wave goes in the direction of +x axis. With
these definitions,
p
(17.151)
Q = Au = ±Yp = ± .
Z
The equation (17.149) may be written in terms of admittances or impedances as:
Y1 (pI − pR ) =
3
j =2
3
Yj pTj , or
(pI − pR ) pTj
=
.
Z1
Zj
(17.152)
j =2
We can simultaneously solve equations (17.148) and (17.152) to produce,
Y1 − Yj
pTj
pR
2Y1
=
= R, and
=
=T,
(17.153)
pI
Y1 + Yj
pI
Y1 + Yj
or,
Z1−1 − Zj−1
2Z1−1
pTj
pR
= −1 −1 , and
= −1
.
pI
pI
Z1 + Zj
Z1 + Zj−1
(17.154)
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3. Modelling of Flow in Blood Vessels
In equation (17.153), R and T are called the reflection and transmission coefficients,
respectively. From equation (17.153), the amplitudes of the reflected and transmitted
pressure waves are R and T times the amplitude of the incident pressure wave,
respectively. These relations can be written in more explicit manner as follows (see
Lighthill (1978)):
The contribution of the incident wave to the pressure in the parent tube is given
by,
x
,
(17.155)
pI = PI f t −
c1
where, PI is an amplitude parameter, and f is a continuous, periodic function whose
maximum value is 1. The corresponding contribution to the flow rate is,
QI = A1 u = Y1 PI f
x
.
t−
c1
(17.156)
The contributions to pressure from the reflected and transmitted waves to the parent
and daughter tubes, respectively, are:
x
x
, and, pTj = PTj hj t −
, (j = 2, 3).
pR = PR g t +
c1
cj
(17.157)
where PR and PT are amplitude parameters, and g and h are are continuous, periodic
functions. The corresponding contributions to the flow rates are:
x
x
, and, QTj = Yj PTj hj t −
, (j = 2, 3).
QR = −Y1 PR g t +
c1
cj
(17.158)
Therefore, the pressure perturbation in the parent tube is given by equation (17.155)
and (17.157) to be:
p
=f
PI
x
t−
c1
PR
+
f
PI
x
,
t+
c1
(17.159)
and the flow rate, from equations (17.156) and (17.158), is:
PR
x
x
−
.
f t+
Q = Y1 PI f t −
c1
PI
c1
(17.160)
The transmission of energy by the pressure waves is of interest. The rate of work
done by the wave motion through the cross section of the tube or equivalently, the
rate of transmission of energy by the wave is clearly, p A u or pQ, which is the same
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Introduction to Biofluid Mechanics
as p2 /Z from equation (17.151). Now we could calculate the incident, reflected and
transmitted quantities at the junction. Thus,
pI2
,
Z1
(RpI )2
Rate of energy transmission by reflected wave =
Z1
p2
= R2 I
Z1
Rate of energy transmission by incident wave =
(17.161)
(17.162)
The quantity R2 is called the energy reflection coefficient. Similarly, the energy
transmission coefficient which is the rate of energy transfer in the two transmitted
waves compared with that in the incident wave may be defined by,
pT2 2
Z2
+
pI2
Z1
pT2 3
Z3
=
Z2−1 + Z3−1
Z1−1
pT 2
pI
2
=
Z2−1 + Z3−1
Z1−1
T 2,
(17.163)
where we have noted that in our case, pT 2 = pT 3 .
A comparison of equations (17.159) and (17.160) shows that, if we include
reflection at bifurcations, the pressure and flow wave forms are no longer of the
same shape. Pedley (1980) has offered interesting discussions about the behavior
of
the waves at the junction. From equation (17.153), for real values of cj and Yj ,
if
Yj < Y1 , then the reflected wave has the same sign as the incident wave, and
the pressures in the two waves are in phase at x = 0. They combine additively to
form a large-amplitude fluctuation at the junction,
and the effect of the junction is
similar to that of a closed end (PR = PI ). If
Yj > Y1 , there is a phase change
at x = 0, the smallest-amplitude pressure fluctuation
occurs there, and the junction
resembles an open end (PR = −PI ). If
Yj = Y1 , there is no reflected wave,
and the junction is said to be perfectly matched. Pedley (2000) has noted that the
increase in the pressure wave amplitude in the aorta with distance down the vessel
may indicate that there is a closed end type of reflection at (or beyond) the iliac
bifurcation. Peaking of the pressure pulse is a consequence of closed end type of
reflection in a blood vessel.
Waves in more complex systems consisting of many branches may be analyzed
by repeated application of the results presented above.
Next, we will study blood flow in curved tubes. Almost all blood vessels have
curvature and the curvature affects both the nature (stability) and volume rate of flow.
Flow in a Rigid Walled Curved Tube
Blood vessels are typically curved and the curvature effects have to be accounted
for in modelling in order to get a realistic understanding. The aortic arch is a 3D
bend twisting through more than 180◦ , Ku (1997). In a curved tube, fluid motion is
not everywhere parallel to the curved axis of the tube (see Fig. 17.17), secondary
motions are generated, the velocity profile is distorted, and there is increased energy
3. Modelling of Flow in Blood Vessels
Figure 17.17 Schematic of flow in a curved tube.
dissipation. However, curving of a tube increases the stability of flow, and the critical
Reynolds number increases significantly, and a critical Reynolds number of 5,000 is
easily obtained (see, McDonald (1974)). Flows in curved tubes are discussed in detail
by McConalogue and Srivastava (1968), Singh (1974), Pedley (1980) and by Berger
et al. (1983). In this Section, we will concentrate on some of the most important
aspects and will focus on the flow in a uniformly curved vessel of small curvature.
The wall is considered to be rigid. Pulsatile flow through a curved tube can induce
complicated secondary flows with flow reversals and is very difficult to analyze. It
may be noted that steady viscous flow in a symmetrical bifurcation resembles that in
two curved tubes stuck together. Thus, an understanding gained in studying curved
flows will be beneficial in that regard as well.
Consider fully developed, steady, laminar, viscous flow in a curved tube of radius
a and a uniform radius of curvature R. Let us employ the toroidal coordinate system
(r ′ , α, θ), where r ′ denotes the distance from the center of the circular cross section
of the pipe, α is the angle between the radius vector and the plane of symmetry, and θ
is the angular distance of the cross section from the entry of the pipe (see Fig. 17.18).
Let the corresponding dimensional velocity components be (u′ , v ′ , w′ ). As a fluid
particle traverses a curved path of radius R (radius of curvature) with a (longitudinal)
speed w ′ along the θ direction, it will experience a lateral (centrifugal) acceleration
of w′2 /R, and a lateral force equal to mp w ′2 /R, where mp is the mass of the particle.
The radii of curvature of the particle paths near the inner bend, the central axis, and
the outer bend will be of increasing magnitude as we move away from the inner bend.
Also, due to the no-slip condition, the velocities, w ′ , of particles near the inner and
outer bends will be lower, while that of the particle at the central axis will be the
highest. The particle at the central axis will experience the highest centrifugal force
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Introduction to Biofluid Mechanics
Figure 17.18 Toroidal coordinate system.
while that near the outer bend will experience the least. A lateral pressure gradient
will cause the faster flowing fluid near the center to be swept towards the outside of
the bend and be replaced at the inside by the slower moving fluid near the wall. In
effect, a secondary circulation will be set up resulting in two vortices, called Dean
vortices because Dean (1928) was the first to systematically study these secondary
motions in curved tubes (see Fig. 17.17). Dean vortices significantly influence the
axial flow. The wall shear near the outside of the bend is relatively higher than the
(much reduced) wall shear on the inside of the bend. Fully developed flow upstream
of or through curved tubes exhibits velocity that skews toward the outer wall of the
bend. For most arterial flows, skewing will be toward the outer wall. If the flow into
the entrance region of a curved tube is not developed, then the inviscid core of the
fluid in the curve can act like a potential vortex with velocity skewing toward the
inner wall.
Secondary flow in curved tubes is utilized in heart-lung machines to promote
oxygenation of blood ( Fung (1997)). In the machine, blood flows inside the curved
tube and oxygen flows on the outside. The tube is permeable to oxygen. The secondary
flow in the tube stirs up the blood and results in faster oxygenation.
Let us now analyze the flow in a curved tube so as to understand the salient
features. Introduce non-dimensional variables, r = r ′ /a, s = Rθ/a, u = u′ /W̄0 ,
and p = p ′ /ρ W̄02 , where u = (u, v, w) is the velocity vector, p is the pressure, ρ is
the density, and W̄0 is the mean axial velocity in the pipe. Restrict consideration to
815
3. Modelling of Flow in Blood Vessels
the case where the flow is fully developed (∂u/∂s = 0). Introduce the dimensionless
ratio,
δ=
radius of tube cross section
a
= ,
radius of curvature of the centerline
R
(17.164)
We restrict consideration to a uniformly curved tube, δ = constant, and with a
slight curvature (weakly curved), δ ≪ 1. Since δ is a constant, the velocity field is
independent of s, the components are functions only of r and θ, and the pressure
gradient ∂p/∂s is independent of s. With δ constant, the only way that the transverse
velocities are affected by the axial velocity is through the centrifugal force, and
it is the centrifugal force that drives the secondary motion. This means that the
centrifugal force terms must be of the same order of magnitude as the viscous and
inertial terms in the momentum equation, and this requires√rescaling
√ the velocities.
The transformation that√accomplishes this is (u, v, w) → ( δ û, δ v̂, ŵ ). We will
also let s = Rθ/a = 1/δ s̃ for convenience.
In the following, we shall omit writing the “ ˆ ” on u, v, w, and the “ ˜ ” on s for
convenience. When δ ≪ 1, the major contribution to the axial pressure gradient may
be separated from the transverse component, and we may write,
p = p0 (s) + δp1 (r, α, s) + . . . ,
(17.165)
Under all these restrictions, the governing equations become,
u
∂u v ∂u
+
∂r
r ∂α
u
v ∂v
∂v
+
∂r
r ∂α
∂u u 1 ∂v
+ +
= 0,
(17.166)
∂r
r
r ∂α
∂p1
2 1 ∂ ∂v
v
1 ∂u
v2
2
−
− w cos α = −
−
+ −
, (17.167)
r
∂r
κ r ∂α ∂r
r
r ∂α
uv
1 ∂p1
2 ∂ ∂v
v
1 ∂u
2
+
+ w sin α = −
+
+ −
, (17.168)
r
r ∂α
κ ∂r ∂r
r
r ∂α
∂p0
2
∂w v ∂w
+
=−
+
u
∂r
r ∂α
∂s
κ
1 ∂ 2w
∂ 2 w 1 ∂w
+
.
+ 2
r ∂r
∂r 2
r ∂α 2
(17.169)
The boundary conditions are:
u = v = w = 0 at r = 1, no singularity at r = 0.
(17.170)
The flow is governed by just one parameter κ in the equations, and it is called
the Dean number. It is given by,
κ=
√ 2 a W̄0
√
= δ 2 Re,
δ
ν
(17.171)
where W̄0 is the mean axial velocity in the pipe. The Dean number is the Reynolds
number modified by the pipe curvature. The appearance of the numerical constant 2
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Introduction to Biofluid Mechanics
in the definition of the Dean number is by convention. At higher Dean numbers, the
flow can separate along the inner boundary curve.
There are many different definitions of Dean number in the literature and the
reader must be careful to see which particular form is being used in any given
discussion.
From equation (17.169), ∂p0 /∂s is independent of s, and p0 can be written as
p0 (s) = −G s, where G is a constant. The equation (17.166) admits the existence of
a stream function for the secondary flow, ψ, defined by
u=
1 ∂ψ
,
r ∂α
v=−
∂ψ
.
∂r
Substitution of equation (17.172) into equation (17.169) yields,
κ ∂p0
κ ∂ψ ∂w ∂ψ ∂w
∇12 w −
=
−
,
2 ∂s
2r ∂α ∂r
∂r ∂α
(17.172)
(17.173)
while elimination of pressure from equations (17.167) and (17.168) yields,
2 4
1 ∂ψ ∂
∂ψ ∂
∂w cos α ∂w
2
∇ ψ−
−
+
∇1 ψ = −2w sin α
, (17.174)
κ 1
r ∂r ∂α
∂α ∂r
∂r
r ∂α
where,
∇12 ψ =
1 ∂
∂2
1 ∂2
+
.
+
r ∂r
∂r 2
r 2 ∂α 2
(17.175)
The boundary conditions are:
ψ=
∂ψ
= w = 0,
∂r
at r = 1.
(17.176)
The equations (17.173) and (17.174) subject to conditions (17.176) have to be solved.
For small values of Dean number, following Dean (1928), we expand w and ψ
in terms of a series in powers of the Dean number as follows:
w=
∞
n=0
κ 2n wn (r, α), and, ψ = κ
∞
κ 2n ψn (r, α).
(17.177)
n=0
The w0 term corresponds to Poiseuille flow in a straight tube with rigid walls. The ψ0
term is O(κ). The series expansion in κ is equivalent to the successive approximation
of inertia terms in lubrication theory. The leading term in the secondary flow takes the
form of a pair of counter rotating helical vortices, placed symmetrically with respect
to the plane of symmetry. This flow pattern arises because of a centrifugally induced
pressure gradient, approximately uniform over the cross section. The dimensionless
volume flux is,
K 4
K 2
Q
(17.178)
+
0.0120
+
O
K6 ,
=
1
−
0.0306
576
576
πa 2 W̄
817
3. Modelling of Flow in Blood Vessels
where K = (2a/R)(Wmax a/ν)2 = 2(κ)2 , is another frequently used definition of
Dean’s number. Here, Wmax = 2W̄ ; Wmax and W̄ are the maximum and mean
velocities, respectively, in a straight pipe of the same radius under the same axial
pressure gradient and under fully developed flow conditions. The first term corresponds to the Poiseuille straight pipe solution. The effect of curvature is seen to
reduce the flux.
Many other authors define Dean’s number by,
D=
√ Ĝa 2 a
2δ
,
µ ν
(17.179)
where, −Ĝ is the dimensional pressure gradient,
Ĝ = −
8µW̄
.
a2
In terms of D, equation (17.178) becomes,
8
4
D
D
Q
12
,
+
0.0120
+
O
D
=
1
−
0.0306
96
96
πa 2 W̄
(17.180)
(17.181)
Next, consider the friction factor for flow in a curved tube. Let λc and λs denote
the flow resistance in a curved and a straight pipe, respectively, while the flows in
them are subject to pressure gradients equal in magnitude. The ratio λ is,
−1
K 2
K 4
Qc
λc
= 1 + 0.0306
− 0.0110
+ . . . , (17.182)
=
λ=
λs
Qs
576
576
where, Qc and Qs are the fluxes in straight and curved pipes, respectively. The flow
resistance in a curved tube is not affected by the first order terms and is increased
only by higher order terms. With regard to shear stress, the curvature increases axial
wall shear on the outside wall and decreases it on the inside, and it also generates a
positive secondary shear in the α direction.
The size of the coefficients suggests that the small D expansion is valid for values
of D up to about 100 or K ≈ 600, and the results here are useful only for smaller
blood vessels. Pedley points out that in the canine aorta, where δ ≈ 0.2, the mean D
is greater than 2000. As mentioned earlier, flow in a curved tube is much more stable
than that in a straight tube and the critical Reynolds number could be as high as 5000
which corresponds to K ≈ 1.6 × 106 .
For intermediate values of D, only numerical solutions are possible due to the
importance of non-linear terms. Numerical results of Collins and Dennis (1975) for
developed flow up to a D of 5000 are stated to compare very well with experimental
results. At intermediate values of D, a boundary layer develops on the outside wall of
the bend where the axial shear is high. The secondary flow in the core is approximately
uniform and continues to manifest a two vortex structure. At higher values of D, there
is greater distortion of the secondary streamlines. The wall shear at r = 1, α = 0, is
proportional to D (≈ 0.85D); see Pedley (2000).
818
Introduction to Biofluid Mechanics
At large Dean numbers, the centers of the two vortices move toward the outer
bend, α = 0, and the flow is very much reduced compared with a straight pipe
for equal magnitude pressure gradients. Detailed studies using advanced computational methods are required to resolve the flow structure at large D. They are as yet
unavailable in the published literature.
Pedley (2000) discusses nonuniqueness of curved tube flow results. When D
is sufficiently small, the steady flow equations have just one solution and there is a
single secondary flow vortex in each half of the tube. However, there is a critical value
of D, above which more than one steady solution exists and these may correspond to
four vortices, two in each half. Again, detailed computational studies are necessary
to resolve these features.
We will next study the flow of blood in collapsible tubes. The role of pressure
difference, (pe − p(x)), on the vessel wall will be significant in such flows.
Flow in Collapsible Tubes
At large negative values of the transmural pressure difference (the difference between
the pressure inside and the pressure outside), the cross sectional area of a blood vessel
is either very small, the lumen being reduced to two narrow channels separated by a
flat region of contact between the opposite walls or it may even fall to zero. There is an
intermediate range of values of transmural pressure difference in which the cross section is very compliant and even the small viscous or inertial pressure drop of the flow
may be enough to cause a large reduction in area, that is, collapse. Collapse occurs in
a number of situations and a listing is given by Kamm and Pedley (1989). Collapse
occurs, for example, in systemic veins above the heart (and outside the skull), as a
result of the gravitational decrease in internal pressure with height; intramyocardial
coronary blood vessels during systole; systemic arteries compressed by a sphygmomanometer cuff, or within the chest during cardiopulmonary resuscitation; pulmonary
blood vessels in the upper levels of the lung; large intrathoracic airways during forced
expiration or coughing; the urethra during micturition and in the ureter during peristaltic pumping. Collapse, therefore occurs both in small and large blood vessels, and
as a result both at low and high Reynolds numbers. In certain cases, at high Reynolds
number, collapse is accompanied by self-excited, flow-induced oscillations. There is
audible sound. For example, Korotkoff sounds heard during sphygmomanometry are
associated with this.
A Note on Korotkoff Sounds
Korotkoff sounds, named after Dr. Nikolai Korotkoff, a physician who described them
in 1905, are sounds that physicians listen for when they are taking blood pressure.
When the cuff of a sphygmomanometer is placed around the upper arm and inflated
to a pressure above the systolic pressure, there will be no sound audible because the
pressure in the cuff would be high enough to completely occlude the blood flow. If
the pressure is now dropped, the first Korotkoff sound will be heard. As the pressure
in the cuff is the same as the pressure produced by the heart, some blood will be able
to pass through the upper arm when the pressure in the artery rises during systole.
3. Modelling of Flow in Blood Vessels
This blood flows in spurts as the pressure in the artery rises above the pressure in the
cuff and then drops back down, resulting in turbulence that results in audible sound.
As the pressure in the cuff is allowed to fall further, thumping sounds continue to be
heard as long as the pressure in the cuff is between the systolic and diastolic pressures,
as the arterial pressure keeps on rising above and dropping back below the pressure
in the cuff. Eventually, as the pressure in the cuff drops further, the sounds change in
quality, then become muted, then disappear altogether when the pressure in the cuff
drops below the diastolic pressure. Korotkoff described 5 types of Korotkoff sounds.
The first Korotkoff sound is the snapping sound first heard at the systolic pressure.
The second sounds are the murmurs heard for most of the area between the systolic
and diastolic pressures. The third and the fourth sounds appear at pressures within
10 mmHg above the diastolic blood pressure, and are described as “thumping” and
“muting”. The fifth Korotkoff sound is silence as the cuff pressure drops below the
diastolic pressure. Traditionally, the systolic blood pressure is taken to be the pressure
at which the first Korotkoff sound is first heard and the diastolic blood pressure is the
pressure at which the fourth Korotkoff sound is just barely audible. There has recently
been a move towards the use of the 5th Korotkoff sound (i.e. silence) as the diastolic
pressure, as this has been felt to be more reproducible.
Starling Resistor: A Motivating Experiment for Flow in Collapsible Tubes
The study of flows in collapsible tubes is facilitated by a well known experiment
carried out under varying conditions by different researchers. In the experiment, a
length of uniform collapsible tube is mounted at each end to a shorter length of rigid
tube and is enclosed in a chamber whose pressure pe can be adjusted. Fluid, say water,
flows through the tube. The inlet and outlet pressures at the ends of the collapsible
tube are p1 and p2 . The volume rate of flow is Q. The pressures and the flow rate are
next varied in a systematic way and the results are noted. The set up described is called
a Starling resistor after physiologist Starling (see Fung (1997)). This experiment will
enable us understand some aspects of actual flows in physiological systems. There are
many different versions of the description of the Starling resistor experiment in the
literature. The experiments have been carried out under both steady flow and unsteady
flow conditions. We will describe the experiments as reported by Kamm and Pedley
(1989).
Case (1): (p1 − p2 ) is increased while (p1 − pe ) is held constant.
This is accomplished either by reducing p2 with p1 and pe fixed, or by simultaneously increasing p1 and pe while p2 is held constant. With either procedure, Q at
first increases, but above a critical value it levels off and the condition of flow limitation is reached. In this condition, however much the driving pressure is increased
the flow rate remains constant, or may even fall as a result of increasingly severe tube
collapse. This experiment is relevant to forced expiration from the lung, to venous
return, and to micturition.
Case (2): (p1 − p2 ) or Q is increased while (p2 − pe ) is held constant at some
negative value.
819
820
Introduction to Biofluid Mechanics
In this case, the tube is collapsed at low flow rates, but starts to open up from the
upstream end as Q increases above a critical value, so that the resistance falls and
(p1 − p2 ) ceases to rise. This is termed pressure-drop limitation. This experiment
does not seem to apply to any particular physiological condition.
Case (3): (p1 − p2 ) is held constant while (p2 − pe ) is decreased from a large
positive value.
In this case, the tube first behaves as though it were rigid and the flow rate is
nearly constant. Then as (p2 − pe ) becomes sufficiently negative to produce partial
collapse, the resistance rises and Q begins to fall. This experiment is relevant to
pulmonary capillary flows.
Case (4): pe fixed. The outlet end is connected to a flow resistor. The pressure
downstream of the flow resistor is fixed (flow is exposed to atmosphere). Thus p2 is
equal to atmospheric pressure plus Q times the fixed resistance. p1 is varied.
In this case, p2 varies with Q due to the presence of a fixed downstream resistance.
The degree of tube collapse (progressive collapse) also varies with Q for the same
reason. At high flow rates, the tube is distended and its resistance is low. As the flow
rate is reduced below a critical value the tube starts to collapse. Its resistance and
(p1 − p2 ) both increase as Q is decreased. Only when the tube is severely collapsed
along most of its length does (p1 − p2 ) start to decrease again as Q approaches zero.
When p1 is approximately equal to pe , virtually the entire tube is collapsed (Fung
(1997)). The tube often flutters in Case 4 (see discussions in Fung).
Case (5): Unsteady flow experiments
Excepting at small Reynolds numbers, there is always some parameter range
where flow oscillations occur. The oscillations have a wide variety of modes.
The experiments reveal the importance of a tube law relating transmural pressure
difference with the area of cross section of the collapsible tube and the flow and
pressure drop limitations when analyzing collapsible tubes. Shapiro (1977a, 1977b)
has developed a comprehensive one dimensional theory for steady flow based on a
suitable tube law. Kamm and Shapiro (1979) have extended it to unsteady flow in a
collapsible tube. In the following, we shall discuss the steady flow theory.
One-Dimensional Flow Treatment
The equations describing one-dimensional flow in a collapsible tube are similar to
those in gas dynamics or channel flow of a liquid with a free surface (see,Shapiro
(1977a)). Here, we will study the one dimensional steady flow formulation for the
collapsible tube. However, before that let us recapitulate the traditional basic equations
for one dimensional flow in a smoothly varying elastic tube. (see Section 3.3.2 on
Pulse wave propagation in an elastic tube: Inviscid theory).
We had studied flow in an elastic tube with cross section A(x, t) and longitudinal
velocity u(x, t). The constant external pressure on the tube was set at pe . The primary
mechanism of unsteady flow in the tube was wave propagation. The transmural pressure difference, (p − pe ) was related to the local cross sectional area by a “tube law”
which involved hoop tension, and it may be expressed as,
(p − pe ) = P̂ (A)
(17.183)
821
3. Modelling of Flow in Blood Vessels
where the functional form P̂ depends on data. For disturbances of small amplitude
and long wave length compared to the tube diameter,
A = A0 + A′ , p − pe = P̂ (A0 ) + p ′ , |A′ | ≪ A0 , |p ′ | ≪ P̂ (A0 )
(17.184)
and the wave speed is given by,
c2 =
A d P̂
A d(p − pe )
=
.
ρ dA
ρ
dA
(17.185)
Tube collapse is associated with negative transmural pressure difference and the
pressure difference is supported by bending stiffness of the tube wall (see Fig. 17.19).
Contrast this with positive transmural pressure difference discussed earlier which was
supported by hoop tension. Following Shapiro (1977a), introduce,
P=
(p − pe )
A
, and, α =
,
Kp
A0
(17.186)
Figure 17.19 Behavior of a collapsible tube. (Reproduced with permission from the American Society
of Mechanical Engineers, NY.).
822
Introduction to Biofluid Mechanics
where Kp is a parameter proportional to the bending stiffness of the wall material,
and A0 is the reference area of the tube for zero transmural pressure difference. The
pressure difference is supported primarily by the bending stiffness of the tube wall.
For a linear elastic tube wall material, Kp is proportional to the modulus of elasticity
E, and the bending moment I , as in,
Kp ∝ EI,
I = (h/a0 )3 /(1 − ν̂ 2 ),
(17.187)
where h is wall thickness and ν̂ is Poisson’s ratio.
From a fit of experimental data (see, Shapiro (1977a)), the tube law for flow in a
collapsible tube is taken to be,
−P ≈ α −n − 1, and n =
3
.
2
(17.188)
For P < 0, the tube is partially collapsed. If the tube is in longitudinal tension, say, TL ,
then there will be a local curvature RL in the longitudinal plane. The effect of TL is to
change pe by TL /RL , and the tube law equation (17.188) will not hold (see, Cancelli
and Pedley (1985)) . We will here assume that TL /RL ≪ (p − pe ). Now, if the tube
law equation (17.188) and transmural pressure difference are assumed to be uniform
along the length of the tube, then with equation (17.185), at any location x, the phase
velocity of long area waves is given by
nKp α −n
A ∂(p − pe )
=
,
(17.189)
c2 =
ρ
∂A
ρ
for the square of the wave speed.
The assumptions of uniformity of tube law and transmural pressure difference
are not valid under most physiological circumstances and these have to be relaxed.
The physical causes that negate uniformity include: friction, gravity, variations of
external pressure or of muscular tone, longitudinal variations in A0 , and longitudinal
changes in the mechanical properties of the tube. To address some of these issues, we
consider a more general formulation given by Shapiro.
The flow will still be considered steady, one dimensional, and incompressible.
The governing equations now are:
dA du
+
= 0,
A
u
and,
−Adp − τw sdx − ρgAdz = ρAudu = ρAu2
(17.190)
du
,
u
(17.191)
where, τw is the wall shear stress, s is the perimeter of the tube, z is the elevation in
the gravity field g. For the shear stress, Shapiro (1977a) considers the cases of fully
developed turbulent flow and fully developed laminar Poiseuille flow in the tube. For
turbulent flow,
1
4 fT dx
τw s dx
= ρu2
,
(17.192)
A
2
de
823
3. Modelling of Flow in Blood Vessels
where de = 4A/s is the equivalent hydraulic diameter and fT is skin friction coefficient for turbulent flow, while for laminar flow,
µ u 1 4 fL′ dx
τw s dx
=
, where, fL′ (α) =
A
d0 α
d0
A
Ae
fL ,
(17.193)
and d0 is the diameter for A0 , and fL is laminar skin friction coefficient.
With equation (17.190), the equation (17.191) may be written,
d(p + ρgz) +
dA
τw sdx
− ρu2
= 0,
A
A
(17.194)
where the appropriate expression for the shear stress must be introduced depending
on the nature of the flow.
Shapiro (1977a) introduces a dimensionless speed-index, S,
u
S = , so that
c
dS 2
S2
=2
du
dc
−2
.
u
c
(17.195)
This index facilitates in the development of the theory and in the interpretation of
results. Its role is comparable in significance to that of Mach number and Froude
number in gas dynamics and in free-surface channel flow, respectively (Shapiro
(1977a)). By analogy with results of gas dynamics, in steady flow, when S < 1
(subcritical), friction causes the area and pressure to decrease in the downstream
direction, and the velocity to increase. When S > 1 (supercritical), the area and
pressure increase along the tube, while the velocity decreases. In general, whatever
the effect of changes of A0 , pe , z, etc. in a subcritical flow, the effect is of opposite
sign in supercritical flow. For example, let pe be increased while all other independent
variables such as A0 , elasticity, etc. are held constant. Then A and p will decrease
for S < 1, but they will increase for S > 1. When S = 1, choking of flow and flow
limitation as at the throat of a Laval nozzle will occur. Again, as in gas dynamics,
there is the possibility of continuous transitions from supercritical to subcritical flow,
and also rapid transitions from supercritical to subcritical as in shock waves.
In the steady flow problem, the known quantities are dA0 , dpe , gdz, f dx, dKp ,
∂P/∂x, ∂P/∂α, while the unknowns are du, dA, dp, dα, dS and so on.
In order to develop the final set of equations relating the dependent and independent variables, a number of useful relationships may be established between the
differential quantities.
The external pressure is pe (x), dpe = (dpe /dx) dx, the area A0 = A0 (x), and
dA0 = (dA0 /dx) dx. Since α = A/A0 ,
dα
=
α
dA dA0
−
.
A
A0
(17.196)
The bending stiffness parameter is Kp = Kp (x), dKp = (dKp /dx) dx, and the tube
law is,
824
Introduction to Biofluid Mechanics
P=
p − pe
= P(α, x), → dp = dpe + Kp dP + PdKp .
Kp (x)
(17.197)
The appropriate form of equation (17.185) is,
c2 (A, x) =
∂P
A ∂(p − pe )
α
→ c2 (α, x) = Kp
ρ
∂A
ρ
∂α
x
.
(17.198)
x=constant
In equation (17.197),
dP =
∂P
∂P
dα +
dx.
∂α
∂x
(17.199)
With equations (17.198) and (17.199), equation (17.197) becomes,
dp = dpe + ρ c2
∂P
dα
+ Kp
dx + PdKp .
α
∂x
(17.200)
With equations (17.198) and (17.197), we get,
2
dKp
αKp ∂ ∂P
dc
α ∂ 2 P/∂α 2 dα
= 1+
+
+
dx,
c
∂P/∂α
α
Kp
ρc2 ∂x ∂x
(17.201)
and, with equation (17.195), equation (17.196) becomes,
dS 2
S2
= −2
dA0
dc
dα
−2
−2
.
α
A0
c
(17.202)
We now have equations (17.194), (17.196), (17.200), (17.201), and (17.202).
With these, Shapiro (1977a) has developed a series of equations that relate each
dependent variable as a linear sum of terms each containing an independent variable multiplied by appropriate coefficients (influence coefficients by analogy with
one-dimensional gas dynamics). A comprehensive listing of equations is provided in
the paper by Shapiro. From the listing, the most important dependent variables turn
out to be dα/dx and dS 2 /dx. Once these are known, other dependent quantities such
as P, u, c etc. may be calculated easily. We now list these equations.
Let us consider cases where P is just a function of α alone, that is, P(α). For the
tube law,
p − pe (x) = Kp (x)P(α),
(17.203)
the equation governing the variation in α is,
1 dα
dKp
S 2 dA0
1 dpe
dz
1−S
=
− 2
+ ρg
+ RQ + P
,
α dx
A0 dx
dx
dx
dx
ρc
2
(17.204)
825
3. Modelling of Flow in Blood Vessels
where R is viscous resistance per unit length (laminar or turbulent) and Q is flow
rate, and the equation governing the speed index is,
1 dS 2
1 dA0
2
1 − S2
−2
+
(2
−
M)
S
=
A0 dx
S 2 dx
dz
M dpe
+ ρg
+ RQ
+ 2
dx
dx
ρc
dP
1 dKp
2
+ 2
MP − 1 − S α
, (17.205)
dα
ρc dx
where
M=3+
α∂ 2 P/∂α 2
.
∂P/∂α
(17.206)
The equations for dα/dx and dS 2 /dx are coupled and must be solved simultaneously by using numerical procedures. Shapiro (1977a) has included results for
several limit cases. These include several examples in which a smooth transition
through the critical condition S = 1 is possible, that is, continuous passage of flow
from regime S < 1 through S = 1 into S > 1 might occur. Fig. 17.20 shows the
transition from subcritical to supercritical flow by means of a minimum in the neutral area A0 . The pressure decreases continuously in the axial direction, and the
area A of the deformed cross section would also decrease continuously in the axial
direction. Fig. 17.20 shows the transition through S = 1 caused by a weight or
clamp, a sphincter or pressurized cuff, due to changing pe . The fluid pressure and
the area, both decrease continuously in the axial direction. S = 1 occurs in the
region where a sharp constriction exists.
Pedley (2000) points out that when S = 1, the right hand side of equation
(17.205) is −M times that of equation (17.204). Therefore, at S = 1, it is possible
for dα/dx or dS 2 /dx to be non-zero as long as the right hand sides are zero. Of the
terms on the right hand side, RQ is associated with friction and is always positive. This
means that at least one of d(pe + ρgz)/dx, dKp /dx or −dA0 /dx should be negative,
that is, the external pressure, the height or the stiffness should decrease with x or the
undisturbed cross-sectional area should increase. An example where dz/dx in a vertical collapsible tube is negative (= −1) is the jugular vein of an upright giraffe and
this problem has been discussed in detail by Pedley. Apparently, the giraffe jugular
vein is normally partially collapsed!
In the next section, we shall learn about the modelling of a Casson fluid flow in a
tube. We recall that blood behaves as a non-Newtonian fluid at low shear rates below
about 200/s, and the apparent viscosity increases to relatively large magnitudes at
low rates of shear. The modelling of such a fluid flow is important and will enable us
understand blood flow at various shear rates.
826
Introduction to Biofluid Mechanics
Figure 17.20 Smooth transition through the critical condition. (Reproduced with permission from the
American Society of Mechanical Engineers, NY.).
Laminar Flow of a Casson Fluid in a Rigid Walled Tube
As shear rates decrease below about 200/s, the apparent viscosity of blood rapidly
increases. (See Fig. 17.7). As mentioned earlier, the variation of shear stress in blood
flow with shear rate is accurately expressed by the equation (17.6):
1/2
τ 1/2 = τy
+ Kc γ̇ 1/2 , for τ ≥ τy , and γ̇ = 0, for τ < τy ,
(17.207)
where τy and Kc are determined from viscometer data. The yield stress τy for normal
blood at 37 ◦ C is about 0.04 dynes/cm2 . In modelling the flow, this behavior must
be included.
Consider the steady laminar axi-symmetric flow of a Casson fluid in a rigid
walled, horizontal, cylindrical tube under the action of an imposed pressure gradient,
827
3. Modelling of Flow in Blood Vessels
(p1 − p2 ) /L. We shall employ cylindrical coordinates (r, θ, x) with velocity components (ur , uθ , and ux ), respectively. With the assumption of axi-symmetry,
∂
= 0 . For convenience, we write ur component as v, and we omit
uθ = 0, and ∂θ
the subscript x in ux .
The maximum shear stress in the flow, τw , would be at the vessel wall. If the
magnitude of τw is equal to or greater than the yield stress, τy , then there will be
flow. We may estimate the minimum pressure gradient required to cause flow of
a yield stress fluid in a cylindrical tube by a straightforward force balance on a
cylindrical volume of fluid of radius r and length x. For steady flow, the viscous
force opposing motion must be balanced by the force due to the applied pressure
gradient. Thus,
τrx 2πrx = −π r 2 ( p|x+x − p|x ),
(17.208)
and, as x → 0,
τrx (r) =
r dp
(p1 − p2 )r
=
.
2 dx
2L
(17.209)
The shear stress at the wall, τw = −(a/2)(dp/dx) = (p1 − p2 )a/2L. When τy is
equal to or less than τw , there will be fluid motion. The minimum pressure differential
to cause flow is given by (p1 − p2 )|min = 2Lτy /a. With τy = 0.04 dynes/cm2 ,
for a blood vessel of L/a = 500, the minimum pressure drop required for flow is
40 dynes/cm2 or 0.03 mmHg. Recall that during systole, the typical pressures in the
aorta and the pulmonary artery rise to 120 mm Hg and 24 mm Hg, respectively.
For axi-symmetric blood flow in a cylindrical tube, at low shear rates, the fully
developed flow is noted to consist of a central core region where the shear rate is zero
and the velocity profile is flat, surrounded by a region where the flow has a varying
velocity profile (see Fig. 17.21). In the core, the fluid moves as if it were a solid body
(also called, plug flow).
Figure 17.21 Velocity profile for axi-symmetric blood flow in a circular tube.
828
Introduction to Biofluid Mechanics
Let the radius of this core region be ac . Then,
τ = τy at r = ac , and γ̇ = 0 for 0 ≤ r < ac ,
τy
,
ac = 2Lτy /(p1 − p2 ) = a
τw
1/2
τ 1/2 = τy
+ Kc γ̇ 1/2 for ac < r ≤ a.
(17.210)
In the core region, γ̇ = 0 ⇒ (du/dr) = 0 ⇒ u = constant = uc (say).
Outside the core region, the velocity is a function of r only, and,
√
τ + τy − 2 τ τy
du
γ̇ = −
=
dr
Kc2
(17.211)
Let (p1 − p2 ) = p, τ = p r/2L, and τy = p ac /2L. From equation (17.211),
−
√
du
1 p
=
r + ac − 2 rac
2
dr
2Kc L
(17.212)
By integration,
1 p
u=
2Kc2 L
r2
4
3
ac r −
− ac r + C ,
3
2
(17.213)
where C is the integration constant. With the no-slip boundary condition at the wall
of the vessel, u = 0 at r = a,
a2
4
3
− ac a .
ac a −
C=−
3
2
(17.214)
Therefore,
8 √ 3 3
1 p
2
2
(a − r ) −
ac
a − r + 2ac (a − r) ,
u=
3
4Kc2 L
(17.215)
in (ac ≤ r ≤ a). With u = uc at r = rc , in terms of τw and τy , the equation
(17.215) becomes,
u=
aτw
2Kc2
r 2 8 τ
r 3/2
τy
r
y
1−
1−
−
1−
+2
,
a
3 τw
a
τw
a
(17.216)
in (ac ≤ r ≤ a). We get the velocity in the core, uc , by setting,
r
a
=
a
c
a
=
τy
,
τw
(17.217)
829
3. Modelling of Flow in Blood Vessels
in equation (17.216). In terms of pressure gradient, a, and ac , uc becomes,
√ 3 √
1 p √
1√
uc =
a − ac
a+
ac .
(17.218)
3
4Kc2 L
The volume rate of flow is given by,
Q = πac2 uc +
a
2π rudr.
(17.219)
ac
After considerable algebra,
π 1 p 4
1 ac 4
16 ac 1/2 4 ac
Q=
−
+
.
a
1
−
8 Kc2 L
7 a
3 a
21 a
(17.220)
The Casson model predicts results that are in very good agreement with experimental results for blood flow over a large range of shear rates (see, Charm and Kurland
(1974)).
Pulmonary Circulation
Pulmonary circulation is the movement of blood from the heart, to the lungs, and back
to the heart again. The veins bring oxygen depleted blood back to the right atrium.
The contraction of the right ventricle ejects blood into the pulmonary artery. In the
human heart, the main pulmonary artery begins at the base of the right ventricle. It
is short and wide—approximately 5 cm in length and 3 cm in diameter, and extends
about 4 cm before it branches into the right and left pulmonary arteries that feed the
two lungs. The pulmonary arteries are larger in size and more distensible than the
systemic arteries and the resistance in pulmonary circulation is lower. In the lungs,
red blood cells release carbon dioxide and pick up oxygen during respiration. The
oxygenated blood then leaves the lungs through the pulmonary veins, which return
it to the left heart, completing the pulmonary cycle. The pulmonary veins, like the
pulmonary arteries, are also short, but their distensibility characteristics are similar
to those of the systemic circulation (Guyton (1968)). The blood is then distributed to
the body through the systemic circulation before returning again to the pulmonary
circulation. The pulmonary circulation loop is virtually bypassed in fetal circulation.
The fetal lungs are collapsed, and blood passes from the right atrium directly into
the left atrium through the foramen ovale, an open passage between the two atria.
When the lungs expand at birth, the pulmonary pressure drops and blood is drawn
from the right atrium into the right ventricle and through the pulmonary circuit.
The rate of blood flow through the lungs is equal to the cardiac output except
for the 1 to 2% that goes through the bronchial circulation ( Guyton (1968)). Since
almost the entire cardiac output flows through the lungs, the flow rate is very high.
However, the low pulmonic pressures generated by the right ventricle are still sufficient to maintain this flow rate because pulmonary circulation involves a much shorter
flow path than systemic circulation, and the pulmonary arteries are, as noted earlier,
are larger and more distensible.
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Introduction to Biofluid Mechanics
Figure 17.22 Pressure pulse contours in the right ventricle, and pulmonary artery. (Reproduced with
permission from Guyton, A. C. and Hall, J. E. (2000) Textbook of Medical Physiology, W. B. Saunders
Company, Philadelphia).
The nutrition to lungs themselves are supplied by bronchial arteries which are a
part of systemic circulation. The bronchial circulation empties into pulmonary veins
and returns to the left atrium by passing alveoli.
The Pressure Pulse Purve in the Right Ventricle
The pressure pulse curves of the right ventricle and pulmonary artery are illustrated
in the Fig. (17.22). As described by Guyton (1968), approximately 0.16 second prior
to ventricular systole, the atrium contracts, pumping a small quantity of blood into
the right ventricle, and thereby causing about 4 mmHg initial rise in the right ventricular diastolic pressure even before the ventricle contracts. Following this, the right
ventricle contracts, and the right ventricular pressure rises rapidly until it equals the
pressure in the pulmonary artery. The pulmonary valve opens, and for approximately
0.3 second blood flows from the right ventricle into the pulmonary artery. When the
right ventricle relaxes, the pulmonary valve closes, and the right ventricular pressure
falls to its diastolic level of about zero. The systolic pressure in the right ventricle of
the normal human being averages approximately 22 mmHg, and the diastolic pressure
averages about 0 to 1 mmHg.
Effect of Pulmonary Arterial Pressure on Pulmonary Resistance
At the end of systole, the ventricular pressure falls while the pulmonary arterial
pressure remains elevated, then falls gradually as blood flows through the capillaries
of the lungs. The pulse pressure in the pulmonary arteries averages 14 mmHg which
is almost two thirds as much as the systolic pressure. Fig. (17.23) shows the variation
in pulmonary resistance with pulmonary arterial pressure. At low arterial pressures,
pulmonary resistance is very high and at high pressures the resistance falls to low
values. The rapid fall is due to the high distensibility of the pulmonary vessels.
4. Introduction to the Fluid Mechanics of Plants
Figure 17.23 Effect of pulmonary arterial pressure on pulmonary resistance. (Reproduced with permission from Guyton, A. C. and Hall, J. E. (2000) Textbook of Medical Physiology, W. B. Saunders Company,
Philadelphia).
The ability of lungs to accommodate greatly increased blood flow with little
increase in pulmonary arterial pressure helps to conserve the energy of the heart.
As described by Guyton, the only reason for flow of blood through the lungs is to
pick up oxygen and to release carbon dioxide. The ability of pulmonary vessels to
accommodate greatly increased blood flow without an increase in pulmonary arterial
pressure accomplishes the required gaseous exchange without overworking the right
ventricle.
In the earlier sections, we have discussed several modelling procedures in relation
to systemic blood circulation. The modelling of the blood flow in pulmonary vessels
are similar to what we studied in those sections.
A discussion of gas and material exchange in the capillary beds is beyond the
scope of this introductory chapter and a reference may be made to the article by
Grotberg (1994) for details.
4. Introduction to the Fluid Mechanics of Plants
Plant life comprises 99% of the earth’s biomass (Bidwell (1974), Rand (1983)).
The basic unit of a plant is a plant cell. Plant cells are formed at meristems, and
then develop into cell types which are grouped into tissues. Plants have three tissue
types: 1) Dermal; 2) Ground; and 3) Vascular. Dermal tissue covers the outer surface
and is composed of closely packed epidermal cells that secrete a waxy material that
aids in the prevention of water loss. The ground tissue comprises the bulk of the
primary plant body. Parenchyma, collenchyma, and sclerenchyma cells are common
in the ground tissue. Vascular tissue transports food, water, hormones and minerals
within the plant.
831
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Introduction to Biofluid Mechanics
Basically, a plant has two organ systems: 1) the shoot system, and 2) the root
system. The shoot system is above ground and includes the organs such as leaves,
buds, stems, flowers and fruits. The root system includes those parts of the plant below
ground, such as the roots, tubers, and rhizomes. There is transport between the roots
and the shoots (see Fig. (17.24)).
Transport in plants occurs on three levels: (1). The uptake and loss of water and
solutes by individual cells, (2). Short-distance transport of substances from cell to
cell at the level of tissues or organs, and, (3). Long-distance transport of sap within
xylem and phloem at the level of the whole plant.
The transport occurs as a result of gradients in chemical concentration (Fickian
diffusion), hydrostatic pressure, and gravitational potential. These three driving potentials are grouped under one single quantity, the water potential. The water potential
is designated ψ, and
ψ = p − RTc + ρgz,
(17.221)
where p is hydrostatic pressure (bar), R is gas constant (= 83.141cm3 − bar/
mole K), T is temperature (K), c is the concentration of all solutes in assumed
Figure 17.24 Overview of plant fluid mechanics. (Reproduced with permission from Annual Review of
c
Fluid Mechanics, Vol. 15 1983
Annual Reviews www.AnnualReviews.org.).
4. Introduction to the Fluid Mechanics of Plants
dilute solution (mole/cm3 ), ρ is density of water (g/cm3 ), g is acceleration due
to gravity (= 980 cm/sec2 ), and z is height (cm). ψ is in bars (Conversion:
1 bar = 106 dyne/cm2 ).
Transport at the cellular level in a plant depends on the selective permeability
of plasma membranes which controls the movement of solutes between the cell and
the extracellular solution. Molecules move down their concentration gradient across
a membrane without the direct expenditure of metabolic energy (Fickian diffusion).
Transport proteins embedded in the membrane speed up the movement across the
membrane. Differences in water potential, ψ, drive water transport in plant cells.
Uptake or loss of water by a cell occurs by osmosis across a membrane. Water moves
across a membrane from a higher water potential to a lower water potential. If a plant
cell is introduced into a solution with a higher water potential than that of the cell,
osmotic uptake of water will cause the cell to swell. As the cell swells, it will push
against the elastic wall, creating a “turgor” pressure inside the cell. Loss of water
causes loss of turgor pressure and may result in wilting.
In contrast to the human circulatory system, the vascular system of plants is open.
Unlike the blood vessels of human physiology, the vessels (conduits) of plants are
formed of individual plant cells placed adjacent to one another. During cell differentiation the common walls of two adjacent cells develop pores which permit fluid to
pass between them. Vascular tissue includes xylem, phloem, parenchyma, and cambium cells. Xylem and phloem make up the big transportation system of vascular
plants. Long distance transport of materials (such as nutrients) in plants is driven by
the prevailing pressure gradient.
In this section we will restrict attention to the vascular system that includes xylem
and phloem cells.
Xylem
The term Xylem applies to woody walls of certain cells of plants. Xylem cells tend
to conduct water and minerals from roots to leaves. Generally speaking, the xylem of
a plant is the system of tubes and transport cells that circulates water and dissolved
minerals. Xylem is made of vessels that are connected end to end to enable efficient
transport. The xylem contains tracheids and vessel elements (see Fig. (17.25) from
Rand (1983)). Xylem tissue dies after one year and then develops a new (rings in the
tree trunk).
Water and mineral salts from soil enter the plant through the epidermis of roots,
cross the root cortex, pass into the stele, and then flow up xylem vessels to the shoot
system. The xylem flow is also called transpirational flow. Perforated end walls of
xylem vessel elements enhance the bulk flow.
The movement of water and solutes through xylem vessels occurs due to a
pressure gradient. In xylem, it is actually tension (negative pressure) that drives
long-distance transport. Transpiration (evaporation of water from a leaf) reduces pressure in the leaf xylem and creates a tension that pulls xylem sap upward from the
roots. While transpiration enables the pull, the cohesion of water due to hydrogen
bonding transmits the upward pull along the entire length of the xylem from the
leaves to the root tips. The pull extends down only through an unbroken chain of
833
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Introduction to Biofluid Mechanics
Figure 17.25 Fluid-conducting cells in the vascular tissue of plants. (Reproduced with permission from
c 1983 Annual Reviews www.AnnualReviews.org.).
Annual Review of Fluid Mechanics, Vol. 15
water molecules. Cavitation, formation of water vapor pockets in the xylem vessel,
may break the chain. Cavitation will occur when xylem sap freezes in water and as
a result the vessel function will be compromised. Absorption of solar energy drives
transpiration by causing water to evaporate from the moist walls of mesophyll cells
of a leaf and by maintaining a high humidity in the air spaces within the leaf. To
facilitate gas exchange between the inner parts of leaves, stems, and fruits, plants
have a series of openings known as stomata. These enable exchange of water vapor,
oxygen and carbon dioxide.
The pressure gradient for transpiration flow is essentially created by solar power,
and in principle, a plant expends no energy in transporting xylem sap up to the leaves
by bulk flow. The detailed mechanism of transpiration from a leaf is very complicated
and depends on the interplay of adhesive and cohesive forces of water molecules
at mesophyll cell—air space interfaces resulting in surface tension gradients and
capillary forces. This will not be discussed in this section.
Xylem sap flows upward to veins that branch throughout each leaf, providing each
with water. Plants lose a huge amount of water by transpiration—an average-sized
maple tree loses more than about 200 liters of water per hour during the summer. Flow
of water up the xylem replaces water lost by transpiration and carries minerals to the
shoots. At night, when transpiration is very low, root cells are still expending energy
to pump mineral ions into the xylem, accumulation of minerals in the stele lowers
water potential, generating a positive pressure, called root pressure, that forces fluid
up the xylem. It is the root pressure that is responsible for guttation, the exudation of
4. Introduction to the Fluid Mechanics of Plants
water droplets that can be seen in the morning on tips of grass blades or leaf margins
of some plants. Root pressure is not the main mechanism driving the ascent of xylem
sap. It can force water upward by only a few meters, and many plants generate no root
pressure at all. Small plants may use root pressure to refill xylem vessels in spring.
Thus, for the most part, xylem sap is not pushed from below but pulled upward by
the leaves.
Xylem Flow
Water and minerals absorbed in the roots are brought up to the leaves through the
xylem. The upward flow in the xylem (also called the transpiration flow) is driven
by evaporation at the leaves. In the xylem, the flow may be treated as quasi-steady.
The rigid tube model for flow description is appropriate because plant cells have
stiff walls. The xylem is about 0.02 mm in radius and the typical values for flow are,
velocity 0.1 cm/s, the kinematic viscosity of the fluid 0.01 cm2 /s and the Reynolds
number, Re = ud/ν is 0.04. In view of the low Reynolds number, the Stokes flow in
a rigid tube approximation is appropriate.
Phloem
Phloem cells are usually located outside the xylem and conduct food from leaves to
rest of the plant. The two most common cells in the phloem are the companion cells
and sieve cells. Phloem cells are laid out end-to-end throughout the plant to form long
tubes with porous cross walls between cells. These tubes enable translocation of the
sugars and other molecules created by the plant during photosynthesis. Phloem flow
is also called translocation flow. Phloem sap is an aqueous solution with sucrose as the
most prevalent solute. It also contains minerals, amino acids, and hormones. Dissolved
food, such as sucrose, flows through the sieve cells. In general, sieve tubes carry food
from a sugar source (for example, mature leaves) to a sugar sink (roots, shoots or
fruits). A tuber or a bulb, may be either a source or a sink, depending on the season.
Sugar must be loaded into sieve-tube members before it can be exported to sugar
sinks. Companion cells pass sugar they accumulate into the sieve-tube members via
plasmodesmata. Translocation through the phloem is dependent on metabolic activity
of the phloem cells (in contrast to transport in the xylem).
Unlike the xylem, phloem is always alive. In contrast to xylem sap, the direction
that phloem sap travels is variable depending on locations of source and sink.
The pressure-flow hypothesis is employed to explain the movement of nutrients
through the phloem. It proposes that water containing nutrient molecules flows under
pressure through the phloem. The pressure is created by the difference in water concentration of the solution in the phloem and the relatively pure water in the nearby
xylem ducts.
At their “source”—the leaves—sugars are pumped by active transport into the
companion cells and sieve elements of the phloem. The exact mechanism of sugar
transport in the phloem is not known, but it cannot be simple diffusion. As sugars and
other products of photosynthesis accumulate in the phloem, the water potential in the
leaf phloem is decreased and water diffuses from the neighboring xylem vessels by
835
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Introduction to Biofluid Mechanics
osmosis. This increases the hydrostatic pressure in the phloem. Turgor pressure builds
up in the sieve tubes (similar to the creation of root pressure). Water and dissolved
solutes are forced downwards to relieve the pressure. As the fluid is pushed down
(and up) the phloem, sugars are removed by the cortex cells of both stem and root
(the “sinks”) and consumed or converted into starch. Starch is insoluble and exerts
no osmotic effect. Therefore, the osmotic pressure of the contents of the phloem
decreases. Finally, relatively pure water is left in the phloem. At the same time, ions
are being pumped into the xylem from the soil by active transport, reducing the water
potential in the xylem. The xylem now has a lower water potential than the phloem, so
water diffuses by osmosis from the phloem to the xylem. Water and its dissolved ions
are pulled up the xylem by tension from the leaves. Thus it is the pressure gradient
between “source” (leaves) and “sink” (shoot and roots) that drives the contents of the
phloem up and down through the sieve tubes.
Phloem Flow
Phloem flow occurs mainly through cells called sieve tubes which are arranged end
to end and are joined by perforated cell walls called sieve plates (see Fig. (17.26)
from Rand and Cooke (1978)). As a model of Phloem flow, Rand et al. (1980) have
derived an approximate formula for the pressure drop for flow through a series of
sieve tubes with periodically placed sieve sieve plates with pores (see Fig. (17.27)
from Rand et al. (1980)). The approximation arises from treating the transport through
the pore as creeping conical flow (see (Happel and Brenner, 1983)).
The approximate formula given by Rand et al. (1980) is:
ℓ a 4
8µQ
L+
+ 2 p ′ , where,
p =
N r
πa 4
8µQ ae
ae 3
r
′
p =
− 1 − 1.5 1 −
. (17.222)
0.57N
r
ae
πa 3 r
Figure 17.26 Sieve tube with sieve plate. (Reproduced with permission from the American Society of
Agricultural Engineers, MI.).
Exercises
Figure 17.27 Sieve tube with pores and stream lines for conical flow through one pore. (Reproduced
with permission from the American Society of Agricultural Engineers, MI.).
In equation (17.222), p is the pressure drop due to one sieve tube and one sieve
plate, µ is the viscosity of fluid in (g/cm − s), Q is the flow rate in (cm3 /s), N is
the number of pores in sieve plate, a is sieve tube radius in cm, r is average radius
of sieve pore in cm, L is the sieve tube
√length in cm, ℓ is sieve plate thickness in cm,
and the effective tube radius ae = a/ N.
Rand et al. (1980) note that the approximate formula has not been tested for
N = 1.
Exercises
1. Consider steady laminar flow of a Newtonian fluid in a long, cylindrical,
elastic tube of length L. The radius of the tube at any cross section is a = a(x).
Poiseuille’s formula for the flow rate is a good approximation in this case.
(a) Develop an expression for the outlet pressure p(L) in terms of the higher
inlet pressure, the flow rate Q̇, fluid viscosity µ, and a(x).
(b) For a pulmonary blood vessel, we may assume that the pressure-radius relaαp
tionship is linear: a(x) = a0 +
, where a0 is the tube radius when the transmural
2
pressure is zero and α is the compliance of the tube. For a tube of length L, show that
π
Q̇ =
[a(0)]5 − [a(L)]5 ,
20µαL
where a(0) and a(L) are the values of a(x) at x = 0 and x = L, respectively.
2. For pulsatile flow in a rigid cylindrical tube of length L, the pressure drop
p may be expressed as: p = f (L, a, ρ, µ, ω, U ), where a is tube radius, ρ is
837
838
Literature Cited
density, µ is viscosity, ω is frequency, and U is the average velocity of flow. Using
dimensional analysis, show that
C2
p
L
= C1
(Re)C3 (St)C4 ,
2
a
ρU
where C1 , C2 , C3 , and C4 are constants, Re is Reynolds number, and St is Strouhal
number defined as aω/U .
3. Localized narrowing of an artery may be caused by the formation of arthero
sclerotic plaque in that region. Such localized narrowing is called stenosis. It is
important to understand the flow characteristics in the vicinity of a stenosis. Flow
in a tube with mild stenosis may be approximated by axi-symmetric flow through a
converging-diverging tube. In this context, follow the details described in the paper
by B.E. Morgan and D.F. Young, Bull. Math. Biol. 36, (1974), pp. 39–53, and obtain
expressions for the velocity profile and wall shear stress.
4. Shapiro in his analysis of the steady flow in collapsible tubes (Trans. ASME,
August 1977, pp. 126–147) has developed a series of equations that relate the dependent variables du, dA,dp, dα,dS etc., with the independent variables such as dA0 , dpe ,
g dz, fT dx etc. In Section IV of that study, explicit calculations of certain simple
flows are presented. In particular, consider pure pressure-gravity flows. Discuss the
flow behavior patterns in this case.
5. Consider the Power-law model to describe the non-Newtonian behavior of
blood. In this model, τ = µγ̇ n , where τ is the shear stress and the γ̇ is the rate of
shearing strain. Determine the flux for the flow of such a fluid in a rigid cylindrical
tube of radius R. Show that when n = 1, the results correspond to the Poiseuille flow.
6. Consider the Herschel-Bulkely model to describe the non-Newtonian behavior
of blood. In this model,
τ = µγ̇ n + τ0 ,
γ̇ = 0,
τ ≥ τ0
τ < τ0
Determine the flux for the flow of such a fluid in a rigid cylindrical tube of radius R.
Show that in the limit τ0 = 0, the results for the Herschel-Bulkley model coincide
with those for the Power-law model.
Acknowledgment
The help received from Dr. K. Mukundakrishnan and Mrs. Olivia Brubaker during
the development of this chapter is gratefully acknowledged.
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Appendix A
Some Properties of
Common Fluids
A1. Useful Conversion Factors . . . . . . 841
A2. Properties of Pure Water at
Atmospheric Pressure . . . . . . . . . . . 842
A3. Properties of Dry Air at
Atmospheric Pressure . . . . . . . . . . . 842
A4. Properties of Standard
Atmosphere . . . . . . . . . . . . . . . . . . . . 843
A1. Useful Conversion Factors
Length:
1 m = 3.2808 ft
1 in. = 2.540 cm
1 mile = 1.609 km
1 nautical mile = 1.852 km
Mass:
1 kg = 2.2046 lb
1 metric ton = 1000 kg
Time:
1 day = 86,400 s
Density:
1 kg/m3 = 0.062428 lb/ft3
Velocity:
1 knot = 0.5144 m/s
Force:
1 N = 105 dyn
Pressure:
1 dyn/cm2 = 0.1 N/m2 = 0.1 Pa
1 bar = 105 Pa
Energy:
1 J = 107 erg = 0.2389 cal
1 cal = 4.186 J
Energy flux:
1 W/m2 = 2.39 × 10−5 cal cm−2 s−1
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50018-6
841
842
Appendix A: Some Properties of Common Fluids
A2. Properties of Pure Water at Atmospheric Pressure
Here, ρ = density, α = coefficient of thermal expansion, µ = shear viscosity,
ν = kinematic viscosity = µ/ρ, κ = thermal diffusivity = k/(ρCp ), [k is first
defined on p.6] Pr = Prandtl number, and 1.0 × 10−n is written as 1.0E − n
◦C
ρ
kg/m3
α
K −1
µ
kg m−1 s−1
ν
m2 /s
κ
m2 /s
Cp
J kg−1 K−1
Pr
ν/κ
0
10
20
30
40
50
1000
1000
997
995
992
988
−0.6E − 4
+0.9E − 4
2.1E − 4
3.0E − 4
3.8E − 4
4.5E − 4
1.787E − 3
1.307E − 3
1.002E − 3
0.799E − 3
0.653E − 3
0.548E − 3
1.787E − 6
1.307E − 6
1.005E − 6
0.802E − 6
0.658E − 6
0.555E − 6
1.33E − 7
1.38E − 7
1.42E − 7
1.46E − 7
1.52E − 7
1.58E − 7
4217
4192
4182
4178
4178
4180
13.4
9.5
7.1
5.5
4.3
3.5
T
Latent heat of vaporization at 100 ◦ C = 2.257 × 106 J/kg.
Latent heat of melting of ice at 0 ◦ C = 0.334 × 106 J/kg.
Density of ice = 920 kg/m3 .
Surface tension between water and air at 20 ◦ C = 0.0728 N/m.
Sound speed at 25 ◦ C ≃ 1500 m/s.
A3. Properties of Dry Air at Atmospheric Pressure
◦C
ρ
kg/m3
µ
kg m−1 s−1
ν
m2 /s
κ
m2 /s
Pr
ν/κ
0
10
20
30
40
60
80
100
1.293
1.247
1.200
1.165
1.127
1.060
1.000
0.946
1.71E − 5
1.76E − 5
1.81E − 5
1.86E − 5
1.87E − 5
1.97E − 5
2.07E − 5
2.17E − 5
1.33E − 5
1.41E − 5
1.50E − 5
1.60E − 5
1.66E − 5
1.86E − 5
2.07E − 5
2.29E − 5
1.84E − 5
1.96E − 5
2.08E − 5
2.25E − 5
2.38E − 5
2.65E − 5
2.99E − 5
3.28E − 5
0.72
0.72
0.72
0.71
0.71
0.71
0.70
0.70
T
At 20 ◦ C and 1 atm,
Cp = 1012 J kg−1 K−1
Cv = 718 J kg−1 K−1
γ = 1.4
α = 3.38 × 10−3 K −1
c = 340.6 m/s (velocity of sound)
Constants for dry air :
Gas constant R = 287.04 J kg−1 K−1
Molecular mass m = 28.966 kg/kmol
843
A4. Properties of Standard Atmosphere
A4. Properties of Standard Atmosphere
The following average values are accepted by international agreement. Here, z is the
height above sea level.
z
km
◦C
T
p
kPa
ρ
kg/m3
0
0.5
1
2
3
4
5
6
8
10
12
14
16
18
20
15.0
11.5
8.5
2.0
−4.5
−11.0
−17.5
−24.0
−37.0
−50.0
−56.5
−56.5
−56.5
−56.5
−56.5
101.3
95.5
89.9
79.5
70.1
61.6
54.0
47.2
35.6
26.4
19.3
14.1
10.3
7.5
5.5
1.225
1.168
1.112
1.007
0.909
0.819
0.736
0.660
0.525
0.413
0.311
0.226
0.165
0.120
0.088
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Appendix B
Curvilinear Coordinates
B1. Cylindrical Polar Coordinates . . 845
B2. Plane Polar Coordinates . . . . . . . . 847
B3. Spherical Polar Coordinates . . . . 847
B1. Cylindrical Polar Coordinates
The coordinates are (R, θ, x), where θ is the azimuthal angle (see Figure 3.1b, where
ϕ is used instead of θ). The equations are presented assuming ψ is a scalar, and
u = i R uR + i θ uθ + i x ux ,
where iR , iθ , and ix are the local unit vectors at a point.
Gradient of a scalar
∇ψ = iR
∂ψ
iθ ∂ψ
∂ψ
+
+ ix
.
∂R
R ∂θ
∂x
Laplacian of a scalar
1 ∂
∇ ψ=
R ∂R
2
∂ 2ψ
∂ψ
1 ∂ 2ψ
.
R
+ 2 2 +
∂R
R ∂θ
∂x 2
Divergence of a vector
∇·u =
1 ∂(RuR )
1 ∂uθ
∂ux
+
+
.
R ∂R
R ∂θ
∂x
Curl of a vector
∂uR
1 ∂(Ruθ )
1 ∂ux
∂uθ
∂ux
1 ∂uR
∇ × u = iR
−
−
−
+ iθ
+ ix
.
R ∂θ
∂x
∂x
∂R
R ∂R
R ∂θ
Laplacian of a vector
uθ
2 ∂uR
2 ∂uθ
uR
2
2
2
∇ u = iR ∇ uR − 2 − 2
− 2 + i x ∇ 2 ux .
+ i θ ∇ uθ + 2
R
R ∂θ
R ∂θ
R
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50019-8
845
846
Appendix B: Curvilinear Coordinates
Strain rate and viscous stress (for incompressible form σij = 2µeij )
eRR =
eθθ =
exx =
eRθ =
eθx =
exR =
1
∂uR
=
σRR ,
∂R
2µ
uR
1
1 ∂uθ
+
=
σθθ ,
R ∂θ
R
2µ
1
∂ux
=
σxx ,
∂x
2µ
R ∂ uθ
1 ∂uR
1
+
=
σRθ ,
2 ∂R R
2R ∂θ
2µ
1 ∂uθ
1
1 ∂ux
+
=
σθx ,
2R ∂θ
2 ∂x
2µ
1 ∂uR
1 ∂ux
1
+
=
σxR .
2 ∂x
2 ∂R
2µ
Vorticity (ω = ∇ × u)
1 ∂ux
∂uθ
−
,
R ∂θ
∂x
∂ux
∂uR
−
,
ωθ =
∂x
∂R
1 ∂
1 ∂uR
ωx =
(Ruθ ) −
.
R ∂R
R ∂θ
ωR =
Equation of continuity
∂ρ
1 ∂
∂
1 ∂
+
(ρRuR ) +
(ρuθ ) +
(ρux ) = 0.
∂t
R ∂R
R ∂θ
∂x
Navier–Stokes equations with constant ρ and ν, and no body force
u2θ
∂uR
1 ∂p
uR
2 ∂uθ
2
+ (u · ∇)uR −
=−
+ ν ∇ uR − 2 − 2
,
∂t
R
ρ ∂R
R
R ∂θ
∂uθ
u R uθ
1 ∂p
2 ∂uR
uθ
+ (u · ∇)uθ +
=−
+ ν ∇ 2 uθ + 2
− 2 ,
∂t
R
ρR ∂θ
R ∂θ
R
∂ux
1 ∂p
+ (u · ∇)ux = −
+ ν∇ 2 ux ,
∂t
ρ ∂x
where
∂
uθ ∂
∂
+
+ ux ,
∂R
R ∂θ
∂x
∂2
1 ∂
∂
1 ∂2
R
+ 2 2 + 2.
∇2 =
R ∂R
∂R
R ∂θ
∂x
u · ∇ = uR
847
B3. Spherical Polar Coordinates
B2. Plane Polar Coordinates
The plane polar coordinates are (r, θ), where r is the distance from the origin
(Figure 3.1a). The equations for plane polar coordinates can be obtained from
those of the cylindrical coordinates presented in Section B1, replacing R by r and
suppressing all components and derivatives in the axial direction x. Some of the
expressions are repeated here because of their frequent occurrence.
Strain rate and viscous stress (for incompressible form σij = 2µeij )
1
∂ur
=
σrr ,
∂r
2µ
1 ∂uθ
ur
1
=
+
=
σθθ ,
r ∂θ
r
2µ
1 ∂ur
1
r ∂ uθ
=
+
=
σrθ .
2 ∂r r
2r ∂θ
2µ
err =
eθθ
erθ
Vorticity
ωz =
1 ∂
1 ∂ur
(ruθ ) −
.
r ∂r
r ∂θ
Equation of continuity
1 ∂
1 ∂
∂ρ
+
(ρrur ) +
(ρuθ ) = 0.
∂t
r ∂r
r ∂θ
Navier–Stokes equations with constant ρ and ν, and no body force
u2
2 ∂uθ
∂ur
ur
uθ ∂ur
1 ∂p
∂ur
+ ur
+
− θ =−
+ ν ∇ 2 ur − 2 − 2
,
∂t
∂r
r ∂θ
r
ρ ∂r
r
r ∂θ
∂uθ
uθ ∂uθ
u r uθ
1 ∂p
2 ∂ur
uθ
∂uθ
+ ur
+
+
=−
+ ν ∇ 2 uθ + 2
− 2 ,
∂t
∂r
r ∂θ
r
ρr ∂θ
r ∂θ
r
where
∇2 =
1 ∂
r ∂r
∂
1 ∂2
r
+ 2 2.
∂r
r ∂θ
B3. Spherical Polar Coordinates
The spherical polar coordinates used are (r, θ, ϕ), where ϕ is the azimuthal angle
(Figure 3.1c). Equations are presented assuming ψ is a scalar, and
u = i r ur + i θ uθ + i ϕ uϕ ,
where ir , iθ , and iϕ are the local unit vectors at a point.
848
Appendix B: Curvilinear Coordinates
Gradient of a scalar
∇ψ = ir
1 ∂ψ
1 ∂ψ
∂ψ
+ iθ
+ iϕ
.
∂r
r ∂θ
r sin θ ∂ϕ
Laplacian of a scalar
∇2ψ =
1 ∂
r 2 ∂r
∂
∂ψ
1
∂ψ
1
∂ 2ψ
.
r2
+ 2
sin θ
+
∂r
∂θ
r sin θ ∂θ
r 2 sin2 θ ∂ϕ 2
Divergence of a vector
∇·u =
1 ∂(uθ sin θ)
1 ∂uθ
1 ∂(r 2 ur )
+
+
.
2
∂r
r sin θ
∂θ
r sin θ ∂ϕ
r
Curl of a vector
∇×u =
∂(ruϕ )
∂(uϕ sin θ ) ∂uθ
ir
iθ
1 ∂ur
−
−
+
r sin θ
∂θ
∂ϕ
r sin θ ∂ϕ
∂r
iϕ ∂(ruθ ) ∂ur
+
−
.
r
∂r
∂θ
Laplacian of a vector
2 ∂(uθ sin θ )
2ur
2 ∂uϕ
∇ 2 u = ir ∇ 2 ur − 2 − 2
− 2
∂θ
r
r sin θ
r sin θ ∂ϕ
2 ∂ur
uθ
2 cos θ ∂uϕ
2
+ i θ ∇ uθ + 2
−
−
r ∂θ
r 2 sin2 θ
r 2 sin2 θ ∂ϕ
uϕ
2 cos θ ∂uθ
2 ∂ur
+
−
+ i ϕ ∇ 2 uϕ + 2
.
r sin θ ∂ϕ
r 2 sin2 θ ∂ϕ
r 2 sin2 θ
Strain rate and viscous stress (for incompressible form σij = 2µeij )
err =
eθθ =
eϕϕ =
eθϕ =
eϕr =
erθ =
1
∂ur
=
σrr ,
∂r
2µ
1 ∂uθ
ur
1
+
=
σθθ ,
r ∂θ
r
2µ
ur
uθ cot θ
1
1 ∂uϕ
+
+
=
σϕϕ ,
r sin θ ∂ϕ
r
r
2µ
sin θ ∂ uϕ
1
1
∂uθ
+
=
σθϕ ,
2r ∂θ sin θ
2r sin θ ∂ϕ
2µ
1
1
r ∂ uϕ
∂ur
=
+
σϕr ,
2r sin θ ∂ϕ
2 ∂r r
2µ
1 ∂ur
1
r ∂ uθ
+
=
σrθ .
2 ∂r r
2r ∂θ
2µ
B3. Spherical Polar Coordinates
Vorticity
∂uθ
∂
1
(uϕ sin θ ) −
,
ωr =
r sin θ ∂θ
∂ϕ
∂(ruϕ )
1
1 ∂ur
ωθ =
−
,
r sin θ ∂ϕ
∂r
∂ur
1 ∂
(ruθ ) −
ωϕ =
.
r ∂r
∂θ
Equation of continuity
1 ∂
1
1
∂
∂
∂ρ
+ 2 (ρr 2 ur ) +
(ρuθ sin θ ) +
(ρuϕ ) = 0.
∂t
r sin θ ∂θ
r sin θ ∂ϕ
r ∂r
Navier–Stokes equations with constant ρ and ν, and no body force
u2θ + u2ϕ
∂ur
+ (u · ∇)ur −
∂t
r
2 ∂(uθ sin θ)
1 ∂p
2ur
2 ∂uϕ
=−
+ ν ∇ 2 ur − 2 − 2
− 2
,
ρ ∂r
∂θ
r
r sin θ
r sin θ ∂ϕ
u2ϕ cot θ
∂uθ
u r uθ
+ (u · ∇)uθ +
−
∂t
r
r
2
uθ
1 ∂p
∂ur
2 cos θ ∂uϕ
2
+ ν ∇ uθ + 2
−
,
=−
−
ρr ∂θ
r ∂θ
r 2 sin2 θ
r 2 sin2 θ ∂ϕ
u ϕ ur
uθ uϕ cot θ
∂uϕ
+ (u · ∇)uϕ +
+
∂t
r
r
uϕ
1
2 ∂ur
2 cos θ ∂uθ
∂p
2
=−
+ ν ∇ uϕ + 2
+
−
,
ρr sin θ ∂ϕ
r sin θ ∂ϕ
r 2 sin2 θ ∂ϕ
r 2 sin2 θ
where
uϕ ∂
uθ ∂
∂
+
+
,
∂r
r ∂θ
r sin θ ∂ϕ
∂
1 ∂
1
∂
1
∂2
2
2 ∂
.
r
+ 2
sin θ
+
∇ = 2
∂r
∂θ
r ∂r
r sin θ ∂θ
r 2 sin2 θ ∂ϕ 2
u · ∇ = ur
849
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Appendix C
Founders of
Modern Fluid Dynamics
Ludwig Prandtl (1875–1953) . . . . . . . . . 851
Geoffrey Ingram Taylor (1886–1975) . . 852
Supplemental Reading . . . . . . . . . . . . 853
Ludwig Prandtl (1875−1953)
Ludwig Prandtl was born in Freising, Germany, in 1875. He studied mechanical
engineering in Munich. For his doctoral thesis he worked on a problem on elasticity
under August Föppl, who himself did pioneering work in bringing together applied
and theoretical mechanics. Later, Prandtl became Föppl’s son-in-law, following the
good German academic tradition in those days. In 1901, he became professor of
mechanics at the University of Hanover, where he continued his earlier efforts to
provide a sound theoretical basis for fluid mechanics. The famous mathematician
Felix Klein, who stressed the use of mathematics in engineering education, became
interested in Prandtl and enticed him to come to the University of Göttingen. Prandtl
was a great admirer of Klein and kept a large portrait of him in his office. He served as
professor of applied mechanics at Göttingen from 1904 to 1953; the quiet university
town of Göttingen became an international center of aerodynamic research.
In 1904, Prandtl conceived the idea of a boundary layer, which adjoins the surface
of a body moving through a fluid, and is perhaps the greatest single discovery in the
history of fluid mechanics. He showed that frictional effects in a slightly viscous fluid
are confined to a thin layer near the surface of the body; the rest of the flow can
be considered inviscid. The idea led to a rational way of simplifying the equations
of motion in the different regions of the flow field. Since then the boundary layer
technique has been generalized and has become a most useful tool in many branches
of science.
His work on wings of finite span (the Prandtl–Lanchester wing theory) elucidated the generation of induced drag. In compressible fluid motions he contributed the
Prandtl–Glauert rule of subsonic flow, the Prandtl–Meyer expansion fan in supersonic
flow around a corner, and published the first estimate of the thickness of a shock wave.
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50020-4
851
852
Appendix C: Founders of Modern Fluid Dynamics
He made notable innovations in the design of wind tunnels and other aerodynamic
equipment. His advocacy of monoplanes greatly advanced heavier-than-air aviation.
In experimental fluid mechanics he designed the Pitot-static tube for measuring velocity. In turbulence theory he contributed the mixing length theory.
Prandtl liked to describe himself as a plain mechanical engineer. So naturally he
was also interested in solid mechanics; for example, he devised a soap-film analogy
for analyzing the torsion stresses of structures with noncircular cross sections. In
this respect he was like G. I. Taylor, and his famous student von Karman; all three
of them did a considerable amount of work on solid mechanics. Toward the end of
his career Prandtl became interested in dynamic meteorology and published a paper
generalizing the Ekman spiral for turbulent flows.
Prandtl was endowed with rare vision for understanding physical phenomena. His
mastery of mathematical tricks was limited; indeed many of his collaborators were
better mathematicians. However, Prandtl had an unusual ability of putting ideas in
simple mathematical forms. In 1948, Prandtl published a simple and popular textbook
on fluid mechanics, which has been referred to in several places here. His varied
interest and simplicity of analysis is evident throughout this book. Prandtl died in
Göttingen 1953.
Geoffrey Ingram Taylor (1886−1975)
Geoffrey Ingram Taylor’s name almost always includes his initials G. I. in references,
and his associates and friends simply refer to him as “G. I.” He was born in 1886 in
London. He apparently inherited a bent toward mathematics from his mother, who was
the daughter of George Boole, the originator of “Boolean algebra.” After graduation
from the University of Cambridge, Taylor started to work with J. J. Thomson in pure
physics.
He soon gave up pure physics and changed his interest to mechanics of fluids
and solids. At this time a research position in dynamic meteorology was created at
Cambridge and it was awarded to Taylor, although he had no knowledge of meteorology! At the age of 27 he was invited to serve as meteorologist on a British ship
that sailed to Newfoundland to investigate the sinking of the Titanic. He took the
opportunity to make measurements of velocity, temperature, and humidity profiles
up to 2000 m by flying kites and releasing balloons from the ship. These were the very
first measurements on the turbulent transfers of momentum and heat in the frictional
layer of the atmosphere. This activity started his lifelong interest in turbulent flows.
During World War I he was commissioned as a meteorologist by the British
Air Force. He learned to fly and became interested in aeronautics. He made the first
measurements of the pressure distribution over a wing in full-scale flight. Involvement
in aeronautics led him to an analysis of the stress distribution in propeller shafts.
This work finally resulted in a fundamental advance in solid mechanics, the “Taylor
dislocation theory.”
Taylor had a extraordinarily long and productive research career (1909–1972).
The amount and versatility of his work can be illustrated by the size and range of
his Collected Works published in 1954: Volume I contains “Mechanics of Solids”
(41 papers, 593 pages); Volume II contains “Meteorology, Oceanography, and
Supplemental Reading
Turbulent Flow” (45 papers, 515 pages); Volume III contains “Aerodynamics and
the Mechanics of Projectiles and Explosions” (58 papers, 559 pages); and Volume IV
contains “Miscellaneous Papers on Mechanics of Fluids” (49 papers, 579 pages).
Perhaps G. I. Taylor is best known for his work on turbulence. When asked, however,
what gave him maximum satisfaction, Taylor singled out his work on the stability of
Couette flow.
Professor George Batchelor, who has encountered many great physicists at
Cambridge, described G. I. Taylor as one of the greatest physicists of the century. He
combined a remarkable capacity for analytical thought with physical insight by which
he knew “how things worked.” He loved to conduct simple experiments, not to gather
data to understand a phenomenon, but to demonstrate his theoretical calculations; in
most cases he already knew what the experiment would show. Professor Batchelor
has stated that Taylor was a thoroughly lovable man who did not suffer from the
maladjustment and self-concern that many of today’s institutional scientists seem to
suffer (because of pressure!), and this allowed his creative energy to be used to the
fullest extent.
He thought of himself as an amateur, and worked for pleasure alone. He did not
take up a regular faculty position at Cambridge, had no teaching responsibilities, and
did not visit another institution to pursue his research. He never had a secretary or
applied for a research grant; the only facility he needed was a one-room laboratory
and one technical assistant. He did not “keep up with the literature,” tended to take up
problems that were entirely new, and chose to work alone. Instead of mastering tensor
notation, electronics, or numerical computations, G. I. Taylor chose to do things his
own way, and did them better than anybody else.
Supplemental Reading
Batchelor, G. K. (1976). “Geoffrey Ingram Taylor, 1886 – 1975.” Biographical Memoirs of Fellows of the
Royal Society 22: 565–633.
Batchelor, G. K. (1986). “Geoffrey Ingram Taylor, 7 March 1886–27 June 1975.” Journal of Fluid
Mechanics 173: 1–14.
Oswatitsch, K. and K. Wieghardt (1987). “Ludwig Prandtl and his Kaiser-Wilhelm-Institute.” Annual
Review of Fluid Mechanics 19: 1–25.
Von Karman, T. (1954). Aerodynamics, New York: McGraw-Hill.
853
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Appendix D
Visual Resources
Following is a list of films, all but the first by the National Committee for Fluid
Mechanics Films (NCFMF), founded in 1961 by the late Ascher H. Shapiro, then
Professor of Mechanical Engineering at M.I.T. Descriptive text for the films was
published separately as described below.
The Fluid Dynamics of Drag, Parts I, II, III, IV (1960)
Text: Ascher H. Shapiro, Shape and Flow:The Fluid Dynamics of Drag,
Doubleday and Co., New York (1961).
Vorticity, Parts I, II (1961)
The text for this and all following films is: NCFMF, Illustrated Experiments in
Fluid Mechanics, MIT Press, Cambridge, MA (1972).
Deformation of Continuous Media (1963)
Flow Visualization (1963)
Pressure Fields and Fluid Acceleration (1963)
Surface Tension in Fluid Mechanics (1964)
Waves in Fluids (1964)
*Boundary Layer Control (1965)
Rheological Behavior of Fluids (1965)
Secondary Flow (1965)
Channel Flow of a Compressible Fluid (1967)
Low-Reynolds-Number Flows (1967)
Magnetohydrodynamics (1967)
Cavitation (1968)
Eulerian and Lagrangian Descriptions in Fluid Mechanics (1968)
Flow Instabilities (1968)
Fundamentals of Boundary Layers (1968)
Rarefied Gas Dynamics (1968)
Stratified Flow (1968)
Aerodynamic Generation of Sound (1969)
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.50021-6
855
856
Appendix D: Visual Resources
Rotating Flows (1969)
Turbulence (1969)
Although these films are decades old, they remain excellent visualizations of the
principles of fluid mechanics. All but the one marked “*” are available for viewing
on the MIT web site as follows: http://web.mit.edu/fluids/www/Shapiro/ncfmf.html
It would be a very good idea to view the film appropriate to the corresponding section
of the text.
Index
Ackeret, Jacob, 715, 758
Acoustic waves, 717
Adiabatic density gradient, 586, 605
Adiabatic process, 24, 737, 746
Adiabatic temperature gradient, 19, 605
Advection, 56
Advective derivative, 56
Aerodynamics
aircraft parts and controls, 680–683
airfoil forces, 684
airfoil geometry, 683
conformal transformation, 688–692
defined, 653
finite wing span, 695–697
gas, 679
generation of circulation, 687–688
incompressible, 679
Kutta condition, 684–686
lift and drag characteristics, 704–705
Prandtl and Lanchester lifting line
theory, 697–701
propulsive mechanisms of fish and birds,
706–708
sailing, 708–709
Zhukhovsky airfoil lift, 692–694
Air, physical properties of, 735
Aircraft, parts and controls, 680–683
Airfoil(s)
angle of attack/incidence, 683
camber line, 683
chord, 683
compression side, 684
conformal transformation, 688–692
drag, induced/vortex, 673–674, 697, 700
finite span, 695–697
forces, 684
geometry, 683
lift and drag characteristics, 704–705
stall, 694, 704
suction side, 684
supersonic flow, 758–761
thin airfoil theory, 688
Zhukhovsky airfoil lift, 692–695
Alston, T. M., 402, 405, 409
Alternating tensors, 36–37
Analytic function, 170
Anderson, John, D., Jr., 432, 465, 686,
688, 711
Angle of attack/incidence, 683, 698
Angular momentum principle/theorem, for
fixed volume, 98–100
Antisymmetric tensors, 40–41
Aorta
elasticity, 791–792
Arterioles, 779
resistance, 780–781
Aris, R., 51, 79, 101, 136
Ashley, H., 701, 711
Aspect ratio of wing, 681
Asymptotic expansion, 391–392
Atmosphere
properties of standard, 843
scale height of, 22
Attractors
aperiodic, 530
dissipative systems and, 527–528
fixed point, 527
limit cycle, 527
strange, 530–531
Autocorrelation function, 544
normalized, 545
of a stationary process, 544
Averages, 541–543
Axisymmetric irrotational flow, 201–202
Babuska–Brezzi stability condition, 438
Baroclinic flow, 147–148
Baroclinic instability, 665–673
Baroclinic/internal mode, 261
Barotropic flow, 118, 146, 147
Barotropic instability, 663–664
Barotropic/surface mode, 261–262, 633
Baseball dynamics, 380
Batchelor, G. K., 24, 105, 131, 133, 136, 163,
212, 298, 337, 410, 591, 600, 673,
711, 853
Bayly, B. J., 469, 515, 521, 528, 535
Becker, R., 740, 741, 763
Bergé, P., 529, 531, 535
Bénard, H., 374
convection, 471
thermal instability, 470–479
Bender, C. M., 409
Bernoulli equation, 118–120
applications of, 122–123
energy, 121
one-dimensional, 723–724
steady flow, 119–120
857
858
Index
Bernoulli equation (continued)
unsteady irrotational flow, 121
β-plane model, 612
Bifurcation, 527
Biofluid mechanics
flow in blood vessels, 782–831
human circulatory system, 766–781
plants, 831–838
Biot and Savart, law of, 151
Bird, R. B., 294
Birds, flight of, 707–708
Blasius solution, boundary layer, 352–361
Blasius theorem, 184–185
Blocking, in stratified flow, 270
Blood
composition, 773–775
coronary circulation, 768
Fahraeus-Lindqvist effect, 775–776,
777–779, 785–787
flow, 782
flow in vessels, modelling of, 782–831
plasma, 774
pulmonary circulation, 767–768,
829–831
systemic circulation, 780–781
total peripheral resistance, 780–785
viscosity, 774–776
Blood vessels
bifurcation, 807–812
Casson fluid flow in rigid tube, 826–829
composition of, 775
flow in, 782–783
flow in collapsible tube, 818–819
flow in rigid walled curved tube,
812–818
Hagen-Poiseuille flow, 783–791
nature, 779–782
pulsatile flow, 791–796
Body forces, 88
Body of revolution
flow around arbitrary, 208–209
flow around streamlined, 206–208
Bohlen T., 364, 409
Bond number, 289
Boundary conditions, 129–133, 669
geophysical fluids, 630
at infinity, 168
kinematic, 220
on solid surface, 168
Boundary layer
approximation, 340–346
Blasius solution, 352–361
breakdown of laminar solution, 360–361
closed form solution, 348–351
concept, 340
decay of laminar shear layer, 401–406
displacement thickness, 346–347
drag coefficient, 358
dynamics of sports balls, 376–381
effect of pressure gradient, 364–366,
517–518
Falkner–Skan solution, 358–360
flat plate and, 348–352
flow past a circular cylinder, 368–375
flow past a sphere, 375–376
instability, 520–522
Karman momentum integral, 362–364
momentum thickness, 347–348
perturbation techniques, 389–394
secondary flows, 388–389
separation, 366–368
simplification of equations, 319–324
skin friction coefficient, 357–358
technique, 2, 154
transition to turbulence, 367
two-dimensional jets, 381–388
u = 0.99U thickness, 346
Bound vortices, 697
Boussinesq approximation, 73, 86, 108–109
continuity equation and, 125–126
geophysical fluid and, 583–585
heat equation and, 127–128
momentum equation and, 126–127
Bradshaw, P., 588, 600
Brauer, H., 465
Breach, D. R., 337
Bridgeman, P. W., 294
Brooks, A. N., 429, 464
Brunt–Väisälä frequency, 265
Buckingham’s pi theorem, 285–287
Buffer layer, 575
Bulk strain rate, 60
Bulk viscosity, coefficient of, 102
Buoyancy frequency, 265, 606
Buoyant production, 558, 587
Bursting in turbulent flow, 585
Buschmann, M. H., 576, 600
Camber line, airfoil, 683
Cantwell, B. J., 584, 600
Capillarity, 9
Capillary number, 289
Capillary waves, 234, 237
Cardiac cycle, 768
net work done by ventricle on blood in
one, 772–773
Cardiac output, 773
Cardiovascular system (human), functions, 766
Carey G. F., 438, 465
Cascade, enstrophy, 674
Casson fluid, laminar flow in a rigid walled
tube, 826–829
Casson model, 776, 777
Casten R. G., 415
859
Index
Castillo, L., 576, 600–601
Cauchy–Riemann conditions, 167, 170
Cauchy’s equation of motion, 93
Cavitation, 834
Centrifugal force, effect of, 108–109
Centrifugal instability (Taylor), 486–493
Chandrasekhar, S., 137, 469, 490, 500,
533, 535
Chang G. Z., 464
Chaos, deterministic, 525–533
Characteristics, method of, 246
Chester, W., 331, 337
Chord, airfoil, 683
Chorin, A. J., 430, 436, 464
Chow C. Y., 688, 711
Circular Couette flow, 303
Circular cylinder
flow at various Re, 368–375
flow past, boundary layer, 368–375
flow past, with circulation, 180–184
flow past, without circulation, 178–179
Circular Poiseuille flow, 302–303
Circulation, 62–64
Kelvin’s theorem, 144–149
Clausius-Duhem inequality, 103
Cnoidal waves, 252, 253
Coefficient of bulk viscosity, 102
Cohen I. M., iii, xviii, 402, 405, 409, 741, 763
Coherent structures, wall layer, 584–586
Coles, D., 492, 535
Collapsible tubes
flow in, 818
one-dimensional steady flow in,
820–826
Starling resistor experiment, 819–820
Comma notation, 49, 152
Complex potential, 170
Complex variables, 169–171
Complex velocity, 171
Compressible flow
classification of, 715–716
friction and heating effects, 747–749
internal versus external, 714
Mach cone, 750–752
Mach number, 714–715
one-dimensional, 721–724, 730–733
shock waves, normal, 734–742
shock waves, oblique, 752–757
speed of sound, 717–720
stagnation and sonic properties, 724–730
supersonic, 756–758
Compressible medium, static equilibrium of,
18–19
potential temperature and density, 20–22
scale height of atmosphere, 22
Compression waves, 214
Computational fluid dynamics (CFD)
advantages of, 412–413
conclusions, 462
defined, 411
examples of, 440–463
finite difference method, 413–418
finite element method, 418–426
incompressible viscous fluid flow,
426–439
sources of error, 412
Concentric cylinders, laminar flow between,
303–306
Conformal mapping, 190–192
application to airfoil, 688–692
Conservation laws
Bernoulli equation, 118–124
boundary conditions, 129–134
Boussinesq approximation, 124–128
differential form, 82
integral form, 82
of mass, 84–86
mechanical energy equation, 111–115
of momentum, 92–93
Navier-Stokes equation, 104–105
rotating frame, 105–111
thermal energy equation, 115–116
time derivatives of volume integrals,
82–84
Conservative body forces, 88, 147
Consistency, 415–418
Constitutive equation, for Newtonian fluid,
100–104
Continuity equation, 73–75, 84, 86
Boussinesq approximation and, 125–126
one-dimensional, 722
Continuum hypothesis, 4–5
Control surfaces, 82
Control volume, 82
Convection, 57
-dominated problems, 427–429
forced, 589
free, 589
sloping, 673
Convergence, 415–418
Conversion factors, 841
Corcos G. M., 504, 536
Coriolis force, effect of, 109–111
Coriolis frequency, 611
Coriolis parameter, 611
Coronary arteries, 768
Coronary circulation, 766, 768
Correlations and spectra, 543–547
Couette flow
circular, 303
plane, 300, 516
Courant, R., 764
Cramer, M. S., 751, 763
Creeping flow, around a sphere, 322–327
860
Index
Creeping motions, 321
Cricket ball dynamics, 377–379
Critical layers, 512–514
Critical Re
blood flow, 783
Critical Re for transition
over circular cylinder, 373–374
over flat plate, 360
over sphere, 375–376
Cross-correlation function, 547
Cross product, vector, 38
Curl, vector, 40
Curtiss, C. F., 294
Curvilinear coordinates, 845–850
D’Alembert’s paradox, 179, 188
D’Alembert’s solution, 215
Davies, P., 526, 533, 535
Dead water phenomenon, 258
Dean number, 815–818
Decay of laminar shear layer, 401–406
Defect law, velocity, 573
Deflection angle, 754
Deformation
of fluid elements, 112–113
Rossby radius of, 643
Degree of freedom, 527
Delta wings, 705
Dennis, S. C. R., 464
Density
adiabatic density gradient, 586, 605
potential, 20–22
stagnation, 725
Derivatives
advective, 56
material, 56–57
particle, 56
substantial, 56
time derivatives of volume integrals,
82–84
Deviatoric stress tensor, 100
Diastole, 770, 771
Differential equations, nondimensional
parameters determined from,
280–284
Diffuser flow, 729–733
Diffusion of vorticity
from impulsively started plate, 306–312
from line vortex, 315–317
from vortex sheet, 313–315
Diffusivity
eddy, 580–584
effective, 597–598
heat, 297
momentum, 297
thermal, 116, 128
vorticity, 147, 313–315
Dimensional homogeneity, 284
Dimensional matrix, 284–285
Dipole. See Doublet
Dirichlet problem, 195
Discretization error, 412
of transport equation, 414–415
Dispersion
of particles, 591–595
relation, 223, 654–656, 660–664
Taylor’s theory, 591–598
Dispersive wave, 203, 221–225, 248–250
Displacement thickness, 340–341
Dissipation
of mean kinetic energy, 513
of temperature fluctuation, 545
of turbulent kinetic energy, 517
viscous, 112–113
Divergence
flux, 111–112
tensor, 39–40
theorem, 45, 85
vector, 39
Doormaal, J. P., 411, 413, 450
Doppler shift of frequency, 219
Dot product, vector, 37
Double-diffusive instability, 482–486
Doublet
in axisymmetric flow, 205
in plane flow, 174–175
Downwash, 698–699
Drag
characteristics for airfoils, 704–705
on circular cylinder, 374
coefficient, 288, 357–358
on flat plate, 357–358
force, 684
form, 367, 705
induced/vortex, 697, 700
pressure, 684, 705
profile, 705
skin friction, 357–358, 684, 705
on sphere, 375–376
wave, 290–291, 700, 760
Drazin, P. G., 469, 471, 481, 491, 498, 513,
515, 535
Dussan, V., E. B., 137
Dutton J. A., 581, 586–587, 600
Dynamic pressure, 123, 297–298
Dynamic similarity
nondimensional parameters and,
287–290
role of, 279–280
Dynamic viscosity, 7
Eddy diffusivity, 580–584
Eddy viscosity, 580–584
861
Index
Effective gravity force, 109
Eigenvalues and eigenvectors of symmetric
tensors, 41–44
Einstein summation convention, 28
Ekman layer
at free surface, 617–622
on rigid surface, 622–625
thickness, 619
Ekman number, 615
Ekman spiral, 619–620
Ekman transport at a free surface, 620
Elastic waves, 214, 717
Element point of view, 424–426
Elliptic circulation, 701–703
Elliptic cylinder, ideal flow, 192–194
Elliptic equation, 168
End diastolic volume (EDV), 772
End systolic volume (ESV), 773
Energy
baroclinic instability, 671–673
Bernoulli equation, 121–122
spectrum, 546
Energy equation
integral form, 82
mechanical, 111–115
one-dimensional, 721–723
thermal, 115–116
Energy flux
group velocity and, 238–241
in internal gravity wave, 272–276
in surface gravity wave, 229
Ensemble average, 541–542
Enstrophy, 673
Enstrophy cascade, 674
Enthalpy
defined, 14
stagnation, 724
Entrainment
in laminar jet, 381
turbulent, 566
Entropy
defined, 15
production, 116–118
Epsilon delta relation, 37, 105
Equations of motion
averaged, 547–554
Boussinesq, 126, 607–608
Cauchy’s, 93
for Newtonian fluid, 100–104
in rotating frame, 105–111
for stratified medium, 607–610
for thin layer on rotating sphere,
610–612
Equations of state, 13–14
for perfect gas, 16–18
Equilibrium range, 563
Equipartition of energy, 228
Equivalent depth, 627
Eriksen, C. C., 504, 535
Euler equation, 104, 118, 317
one-dimensional, 723–724
Euler momentum integral, 96–97, 188
Eulerian description, 55
Eulerian specifications, 54–55
Exchange of stabilities, principle of, 470
Expansion coefficient, thermal, 16–18
Fahraeus effect, 777–778
Fahraeus-Lindqvist (FL) effect, 775, 776–779
mathematical model, 785–787
Falkner, V. W., 358, 409
Falkner–Skan solution, 358–360
Feigenbaum, M. J., 531, 535
Fermi, E., 131, 137
Feynman, R. P., 595, 600
Fick’s law of mass diffusion, 6
Finite difference method, 413–418, 421–424
Finite element method
element point of view, 424–426
Galerkin’s approximation, 420–421
matrix equations, 421–424
weak or variational form, 418–420
First law of thermodynamics, 13
thermal energy equation and, 115–116
Fish, locomotion of, 706–708
Fixed point, 527
Fixed region, mechanical energy equation and,
114
Fixed volume, 83
angular momentum principle for,
98–100
momentum principle for, 93–97
Fjortoft, R., 673, 674, 677
Fjortoft’s theorem, 511–512
Flat plate, boundary layer and
Blasius solution, 352–361
closed form solution, 348–351
drag coefficient, 358
Fletcher, C. A. J., 426, 429, 464
Flow limitation, 819
Fluid mechanics, applications, 1–2
Fluid mechanics, visual resources, 855–856
Fluid statics, 9–12
Flux divergence, 111–112
Flux of vorticity, 64
Force field, 88
Force potential, 88
Forces
conservative body, 88, 147
Coriolis, 109–111
on a surface, 33–36
Forces in fluid
body, 88
line, 89
862
Index
Forces in fluid (continued)
origin of, 88–89
surface, 88
Form drag, 367, 705
Fourier’s law of heat conduction, 6
f -plane model, 612
Franca, L. P., 429, 439, 464
Frank, Otto, 791
Frequency, wave
circular or radian, 217
Doppler shifted, 219
intrinsic, 218
observed, 218
Frey S. L., 439, 464
Friction drag, 357–358, 684, 705
Friction, effects in constant-area ducts,
747–750
Friedrichs K. O., 410, 763
Froude number, 248, 282, 292
internal, 292–293
Fry R. N., 750, 763
Fully developed flow, 298
Fuselage, 680
Gad-el-Hak, M., 576, 577, 600
Galerkin least squares (GLS), 439
Galerkin’s approximation, 420–421
Gallo, W. F., 352, 409
Gas constant
defined, 16–18
universal, 16
Gas dynamics, 679
See also Compressible flow
Gases, 3–4
Gauge functions, 390–391
Gauge pressure, defined, 9
Gauss’ theorem, 44–47, 82
Geophysical fluid dynamics
approximate equations for thin layer on
rotating sphere, 610–612
background information, 603–605
baroclinic instability, 665–673
barotropic instability, 663–664
Ekman layer at free surface, 617–622
Ekman layer on rigid surface, 622–625
equations of motion, 607–610
geostrophic flow, 613–617
gravity waves with rotation, 636–639
Kelvin waves, 639–643
normal modes in continuous stratified
layer, 628–634
Rossby waves, 657–663
shallow-water equations, 625–627,
634–636
vertical variations of density, 605–607
vorticity conservation in shallow-water
theory, 644–647
George W. K., 576, 600–601
Geostrophic balance, 613
Geostrophic flow, 613–617
Geostrophic turbulence, 673–676
Ghia, U., 440, 464
Ghia, K. N., 464
Gill, A. E., 258, 270, 277, 637, 661, 662, 677
Glauert, M. B., 385, 409
Glowinski scheme, 437
Glowinski, R., 437, 439, 464
Gnos, A. V., 409
Goldstein, S., 409, 500
Görtler vortices, 493
Gower J. F. R., 372, 409
Grabowski, W. J., 521, 535
Gradient operator, 38–39
Gravity force, effective, 109
Gravity waves
deep water, 230–231
at density interface, 255–259
dispersion, 233, 242–246, 270–272
energy issues, 272–276
equation, 214–216
finite amplitude, 250–253
in finite layer, 259–261
group velocity and energy flux, 238–241
hydraulic jump, 248–250
internal, 267–270
motion equations, 263–266
nonlinear steepening, 246–248
parameters, 216–219
refraction, 233
with rotation, 636–639
shallow water, 231–233, 262–263,
246–248
standing, 237–238
Stokes’ drift, 253–255
in stratified fluid, 267–270
surface, 205–209, 209–215, 219–223,
223–229
surface tension, 234–237
Gresho, P. M., 427, 465
Group velocity
concept, 215, 224–231, 240, 238–246
of deep water wave, 230–231
energy flux and, 238–241
Rossby waves, 660–662
wave dispersion and, 242–246
Hagen-Poiseuille flow, 783–785
application, 807–809
effect of developing flow, 788–789
effect of vessel wall elasticity, 789–791
Fahraeus-Lindqvist effect and, 785–787
Half-body, flow past a, 175–178
Hardy, G. H., 2
Harlow, F. H., 433, 465
Harmonic function, 168
863
Index
Hatsopoulos, G. N., 24
Hawking, S. W., 701, 711
Hayes, W. D., xvii, 740, 763
Heart, pumping action, 769–771
Heat diffusion, 297
Heat equation, 115–116
Boussinesq equation and, 127–128
Heat flux, turbulent, 554
Heating, effects in constant-area ducts,
747–750
Heisenberg, W., 535, 539
Hele-Shaw, H. S., 337
Hele–Shaw flow, 332–334
Helmholtz vortex theorems, 149
Hematocrit, 774
plasma skimming, 778
Herbert, T., 469, 515, 535
Herreshoff, H. C., 709, 711
Hinze, J. O., 601
Hodograph plot, 619
Holstein, H., 364, 409
Holton, J. R., 105, 137, 657, 678
Homogeneous turbulent flow, 525
Hooke’s law, blood vessels and, 790–791
Hou, S., 465
Houghton, J. T., 658, 664, 678
Howard, L. N., 500, 504, 506, 507, 512, 536
Howard’s semicircle theorem, 504–506
Hughes T. J. R., 426, 429, 459, 464, 465, 600
Hugoniot, Pierre Henry, 735
Human body, biotransport and distribution
processes, 765–766
Huppert, H. E., 482, 536
Hydraulic jump, 248–250
Hydrostatics, 11
Hydrostatic waves, 233
Hypersonic flow, 716
Images, method of, 158, 189–190
Incompressible aerodynamics. See
Aerodynamics
Incompressible fluids, 102
Incompressible viscous fluid flow, 426
convection-dominated problems,
427–429
Glowinski scheme, 437
incompressibility condition, 429
MAC scheme, 433–437
mixed finite element, 438–439
SIMPLE-type formulations, 410–413
Induced/vortex drag, 697, 700
coefficient, 703
Inertia forces, 321
Inertial circles, 639
Inertial motion, 639
Inertial period, 611, 639
Inertial sublayer, 574–576
Inertial subrange, 562–564
Inflection point criterion, Rayleigh, 511, 663
inf-sup condition, 438
Initial and boundary condition error, 412
Inlet (entrance) length, 788–789
Inner layer, law of the wall, 571–573
Input data error, 412
Instability
background information, 469
baroclinic, 665–673
barotropic, 663–664
boundary layer, 517–518, 520–522
centrifugal (Taylor), 486–493
of continuously stratified parallel flows,
500–507
destabilizing effect of viscosity, 519–520
double-diffusive, 482–486
inviscid stability of parallel flows,
510–514
Kelvin–Helmholtz instability, 493–500
marginal versus neutral state, 470
method of normal modes, 469–470
mixing layer, 515–516
nonlinear effects, 522–523
Orr–Sommerfeld equation, 509–510
oscillatory mode, 470
pipe flow, 516–517
plane Couette flow, 516
plane Poiseuille flow, 516
principle of exchange of stabilities, 470
results of parallel viscous flows,
514–520
salt finger, 482, 483, 485
sausage instability, 535
secondary, 524
sinuous mode, 535
Squire’s theorem, 492–493, 507, 509
thermal (Bénard), 470–482
Integral time scale, 546
Interface, conditions at, 130, 131
Intermittency, 565–566
Internal energy, 12, 115–117
Internal Froude number, 292–293
Internal gravity waves, 214
See also Gravity waves
energy flux, 272, 275
at interface, 255, 259
in stratified fluid, 267–270
in stratified fluid with rotation, 646–657
WKB solution, 650–652
Internal Rossby radius of deformation, 643
Intrinsic frequency, 218, 656
Inversion, atmospheric, 19
Inviscid stability of parallel flows, 510–514
Inviscid theory, blood flow, 793–796
long wave length approximation,
809–812
864
Index
Irrotational flow, 63
application of complex variables,
169–171
around body of revolution, 206–208
axisymmetric, 201–205
conformal mapping, 190–192
doublet/dipole, 174–175
forces on two-dimensional body,
184–189
images, method of, 158, 189–190
numerical solution of plane, 195–201
over elliptic cylinder, 192–194
past circular cylinder with circulation,
180–184
past circular cylinder without
circulation, 178–180
past half-body, 175–178
relevance of, 165–167
sources and sinks, 173
uniqueness of, 194–195
unsteady, 121–122
velocity potential and Laplace equation,
167–169
at wall angle, 171–173
Irrotational vector, 40
Irrotational vortex, 70, 142–143, 174
Isentropic flow, one-dimensional, 729–732
Isentropic process, 17
Isotropic tensors, 37, 100
Isotropic turbulence, 551
Iteration method, 195–201
path lines, 57–59
polar coordinates, 75–77
reference frames and streamline pattern,
59–60
relative motion near a point, 64–67
shear strain rate, 61–62
streak lines, 59
stream function, 73–75
streamlines, 57–59
viscosity, 7
vortex flows and, 67–71
vorticity and circulation, 62–64
Kinetic energy
of mean flow, 554–556
of turbulent flow, 556–559
Kinsman, B., 234, 277
Klebanoff, P. S., 524, 536
Kline, S. J., 584, 585, 586, 600
Kolmogorov, A. N., 521
microscale, 522
spectral law, 290, 562–564
Korotkoff sounds, 818–819
Korteweg–deVries equation, 252
Kronecker delta, 36–37
Krylov V. S., 137
Kuethe, A. M., 688, 711
Kundu, P. K., 621, 678
Kuo, H. L., 664, 678
Kutta condition, 684–686
Kutta, Wilhelm, 183
Kutta–Zhukhovsky lift theorem, 183, 185–189,
684
Jets, two-dimensional laminar, 381–385
Kaplun, S., 329, 337
Karamcheti, K., 711
Karman. See under von Karman
Karman number, 576
Karman, T., 24, 409, 539–540, 701, 711, 763,
853
Keenan, J. H., 24
Keller, H. B., 465
Kelvin–Helmholtz instability, 493–500
Kelvin’s circulation theorem, 144–149
Kelvin waves
external, 639–643
internal, 618–619, 643
Kim, John, 585, 600
Kinematics
defined, 53
Lagrangian and Eulerian specifications,
54–55
linear strain rate, 60–61
material derivative, 55–57
one-, two-, and three-dimensional flows,
71–73
parallel shear flows and, 67–68
Lagerstrom, P. A., 410
Lagrangian description, 54
Lagrangian specifications, 54–55
Lam, S. H., 584, 600
Lamb, H., 120, 131, 137
Lamb, Horace, 120
Lamb surfaces, 120
Laminar boundary layer equations,
Falkner–Skan solution, 358–360
Laminar flow
creeping flow, around a sphere, 322–327
defined, 296
diffusion of vortex sheet, 313–315
Hele–Shaw, 332–334
high and low Reynolds number flows,
320–322
oscillating plate, 317–320
pressure change, 297–298
similarity solutions, 306–313
steady flow between concentric
cylinders, 303–306
steady flow between parallel plates,
298–302
steady flow in a pipe, 302–303
865
Index
Laminar flow, of a Casson fluid in a rigid
walled tube, 826–829
Laminar jet, 381–388
Laminar shear layer, decay of, 401–407
Laminar solution, breakdown of, 360–361
Lanchester, Frederick, 686
lifting line theory, 697–701
Landahl, M., 562, 584, 600, 701, 711
Lanford, O. E., 526, 536
Laplace equation, 167
numerical solution, 195–201
Laplace transform, 312
Law of the wall, 571–573
LeBlond, P. H., 251, 271, 633, 673
Lee wave, 656–657
Leibniz theorem, 82, 83
Leighton, R. B., 600
Lesieur, M., 538, 600
Levich, V. G., 131, 137
Liepmann, H. W., 246, 277, 686, 713, 732
Lift force, airfoil, 683–684
characteristics for airfoils, 704–705
Zhukhovsky, 692–694
Lifting line theory
Prandtl and Lanchester, 697–701
results for elliptic circulation, 701–703
Lift theorem, Kutta–Zhukhovsky, 183,
185–188, 684
Lighthill, M. J., 159, 163, 245–246, 246, 251,
277, 298, 337, 701, 706, 711
Limit cycle, 527
Lin C. Y., 371, 409, 465, 536
Lin, C. C., 448, 500, 518
Linear strain rate, 60–61
Line forces, 89
Line vortex, 140, 315–317
Liquids, 3–4
Logarithmic law, 573–578
Long-wave approximation. See Shallow-water
approximation
Lorenz, E., 467, 528, 529, 530, 532, 536
Lorenz, E.
model of thermal convection, 528–529
strange attractor, 530–531
Lumley J. L., 559, 564, 568, 574, 583, 586, 600
MacCormack, R. W., 411, 430, 432, 440, 444,
449, 450
McCreary, J. P., 662, 678
Mach, Ernst, 715
angle, 751
cone, 750–752
line, 751
number, 248–293, 686–687
MAC (marker-and-cell) scheme, 433
Magnus effect, 183
Marchuk, G. I., 433, 465
Marginal state, 470
Marvin, J. G., 409
Mass, conservation of, 84–86
Mass transport velocity, 255
Material derivative, 53–54
Material volume, 84
Mathematical order, physical order of
magnitude versus, 391
Matrices
dimensional, 284–285
multiplication of, 29–30
rank of, 285–286
transpose of, 26
Matrix equations, 421–424
Mean continuity equation, 548–549
Mean heat equation, 553–554
Mean momentum equation, 549–550
Measurement, units of SI, 2–3
conversion factors, 841
Mechanical energy equation, 111–115
Mehta, R., 377, 378, 409
Miles, J. W., 500, 536
Millikan, R. A., 325, 326, 337, 539
Milne-Thompson, L. M., 212
Mixed finite element, 438–439
Mixing layer, 515–516
Mixing length, 580–584
Modeling error, 412
Model testing, 290–291
Moens-Kortewag wave speed, 794–795
Moilliet, A., 600
Mollo-Christensen, E., 600
Momentum
conservation of, 92–93
diffusivity, 297
thickness, 347–348
Momentum equation, Boussinesq equation
and, 126–127
Momentum integral, von Karman, 362–364
Momentum principle, for control volume, 723
Momentum principle, for fixed volume, 93–97
angular, 98–100
Monin, A. S., 539, 600
Monin–Obukhov length, 588–589
Moore, D. W., 110, 137
Moraff, C. A., 741, 763
Morton K. W., 418, 465
Munk, W., 657, 678
Murray’s Law, 808
application, 808–809
Mysak L. A., 251, 277, 633, 679
Narrow-gap approximation, 490
National Committee for Fluid Mechanics
Films (NCFMF), 855
866
Index
Navier–Stokes equation, 104–105, 281
convection-dominated problems,
427–429
incompressibility condition, 429–430
Nayfeh, A. H., 389, 391, 408–409, 521, 536
Neumann problem, 195
Neutral state, 470
Newman J. N., 709, 711
Newtonian fluid, 100–103
non-, 103–104
Newton’s law
of friction, 7
of motion, 92
Nondimensional parameters
determined from differential equations,
280–284
dynamic similarity and, 287–290
significance of, 292–294
Non-Newtonian fluid, 103–104
Nonrotating frame, vorticity equation in,
146–150
Nonuniform expansion, 393–394
at low Reynolds number, 394
Nonuniformity
See also Boundary layers
high and low Reynolds number flows,
320–322
Oseen’s equation, 329–331
region of, 393–394
of Stokes’ solution, 327–331
Normal modes
in continuous stratified layer, 628–634
instability, 469–470
for uniform N, 631–634
Normal shock waves, 733–741
Normal strain rate, 60–61
Normalized autocorrelation function, 545
No-slip condition, 296
Noye, J., 419, 464
Nozzle flow, compressible, 729–733
Numerical solution
Laplace equation, 195–201
of plane flow, 195–201
Oblique shock waves, 752–756
Observed frequency, 656
Oden, J. T., 438, 464
One-dimensional approximation, 71
One-dimensional flow
area/velocity relations, 729–733
equations for, 721–724
One-dimensional flow, in a collapsible tube,
820–826
Order, mathematical versus physical order of
magnitude, 391
Ordinary differential equations (ODEs),
422–423
Orifice flow, 123–124
Orr–Sommerfeld equation, 509–510
Orszag S. A., 389, 391, 409, 469, 515, 535,
584, 600
Oscillating plate, flow due to, 317–320
Oscillatory mode, 470, 485–486
Oseen, C. W., 327, 328, 329, 331, 337, 369
Oseen’s approximation, 329–331
Oseen’s equation, 329
Oswatitsch, K., 853
Outer layer, velocity defect law, 573
Overlap layer, logarithmic law, 573–579
Panofsky, H. A., 581, 586, 587, 600
Panton, R. L., 410
Parallel flows
instability of continuously stratified,
500–506
inviscid stability of, 510–514
results of viscous, 514–520
Parallel plates, steady flow between, 298–302
Parallel shear flows, 67–68
Particle derivative, 55
Particle orbit, 638–639, 652–654
Pascal’s law, 10
Patankar, S. V., 410, 411, 412, 432, 450
Path functions, 13
Path lines, 57–59
Pearson J. R. A., 329, 337
Pedlosky, J., 105, 136, 163, 609, 622, 666,
673, 675, 678
Peletier, L. A., 352, 409
Perfect differential, 195
Perfect gas, 16–17
Peripheral resistance unit (PRU), 780–781
Permutation symbol, 37
Perturbation pressure, 224
Perturbation techniques, 389
asymptotic expansion, 391–393
nonuniform expansion, 393–394
order symbols/gauge functions, 390–391
regular, 394–396
singular, 396–401
Perturbation vorticity equation, 666–668
Petrov–Galerkin methods, 421
Peyret, R., 436, 465
Phase propagation, 662
Phase space, 526–527
Phenomenological laws, 6
Phillips, O. M., 240, 253, 277, 586, 592, 600,
658, 678
Phloem, 835–836
flow, 836–837
Physical order of magnitude, mathematical
versus, 391
Pipe flow, instability and, 516
Pipe, steady laminar flow in a, 302–303
867
Index
Pitch axis of aircraft, 681
Pi theorem, Buckingham’s, 285–287
Pitot tube, 122–123
Plane Couette flow, 300, 516
Plane irrotational flow, 195–201
Plane jet
self-preservation, 567–568
turbulent kinetic energy, 568–570
Plane Poiseuille flow, 301–302
instability of, 516
Planetary vorticity, 155, 156–157, 611
Planetary waves. See Rossby waves
Plants
fluid mechanics, 831–837
physiology, 831–832
Plasma, blood, 774–776
skimming, 778
viscosity, 775, 776
Plastic state, 4
Platelets (thrombocytes), 774
Pohlhausen, K., 350, 362, 364, 409
Poincaré, Pitot, Henri, 533
Poincaré waves, 637
Point of inflection criterion, 366
Poiseuille flow
circular, 302–303
instability of, 516
plane laminar, 300–301
Polar coordinates, 75–77
cylindrical, 845–846
plane, 847
spherical, 847–849
Pomeau, Y., 526, 531, 532, 535
Potential, complex, 170
Potential density gradient, 22, 587
Potential energy
baroclinic instability, 671–673
mechanical energy equation and,
113–115
of surface gravity wave, 228
Potential flow. See Irrotational flow
Potential temperature and density, 20–22
Potential vorticity, 646
Prager, W., 51
Prandtl, L., 2, 24, 79, 163, 183, 209, 212, 294,
340, 353, 389, 405, 409, 474, 477,
500, 529, 531, 539, 540, 574, 581,
582, 588, 591, 604, 678, 697, 700,
701, 711, 741, 756, 757, 763, 851,
852, 853
Prandtl, Ludwig, 2, 340, 851, 853
biographical information, 851–852
mixing length, 580–584
Prandtl and Lanchester lifting line
theory, 697–701
Prandtl–Meyer expansion fan, 756–758
Prandtl number, 294
turbulent, 588
Pressure
absolute, 9
coefficient, 177, 283
defined, 5, 9
drag, 684, 705
dynamic, 123, 297–298
gauge, 9
stagnation, 123
waves, 214, 717
Pressure-drop limitation, 820
Pressure gradient
boundary layer and effect of, 364–366,
517–518
constant, 299
Pressure pulse, 771
Principal axes, 42, 64–67
Principle of exchange of stabilities, 470
Probstein, R. F., 131, 137
Profile drag, 705
Proudman theorem, Taylor-, 615–616
Proudman, I., 329, 337
Pulmonary circulation, 766, 767–768, 829–831
Pulsatile flow, 791
aorta elasticity and Windkessel theory,
791–792
inviscid theory, 793–796
in rigid cylindrical tube, 796–800
tube material viscoelasticity, 806–807
wall viscoelasticity, 800–806
Quasi-geostrophic motion, 658–660
Quasi-periodic regime, 531
Raithby, G. D., 411, 413, 450
Random walk, 595–596
Rankine, W.J.M., 735
vortex, 71
Rankine–Hugoniot relations, 735
Rayleigh
equation, 510
inflection point criterion, 511, 663
inviscid criterion, 486–487
number, 471
Rayleigh, Lord (J. W. Strutt), 133, 137
Red blood cells, 774
Reduced gravity, 262
Reducible circuit, 194
Refraction, shallow-water wave, 233
Regular perturbation, 394–396
Reid W. H., 469, 471, 481, 491, 513, 498, 513,
515, 535, 678
Relative vorticity, 645
Relaxation time, molecular, 12
Renormalization group theories, 558
Reshotko, E., 522, 536
Reversible processes, 13
868
Index
Reynolds analogy, 588
decomposition, 547–548
experiment on flows, 278
similarity, 568
stress, 550–553
transport theorem, 84
Reynolds W. C., 296, 539, 584, 600
Reynolds, O., 514
Reynolds number, 166, 282, 292, 369
high and low flows, 320–322, 369,
373–375
Rhines, P. B., 675, 678
Rhines length, 675, 679
Richardson, L. F., 540
Richardson number, 293, 587, 588
criterion, 503, 504
flux, 587
gradient, 293, 504
Richtmyer, R. D., 465
Rigid lid approximation, 633–634
Ripples, 236
Roll axis of aircraft, 680
Root-mean-square (rms), 543
Rosenhead, L., 352, 364, 409
Roshko A., 246, 277, 714, 741, 762
Rossby number, 613
Rossby radius of deformation, 642
Rossby waves, 658–663
Rotating cylinder
flow inside, 305–306
flow outside, 304–305
Rotating frame, 105–111
vorticity equation in, 121–125
Rotation, gravity waves with, 637–639
Rotation tensor, 65
Rough surface turbulence, 576–578
Ruelle, D., 533, 536
Runge–Kutta technique, 355, 423
Saad, Y., 440, 469
Sailing, 707–709
Salinity, 21
Salt finger instability, 482–485
Sands, M., 599
Sargent, L. H., 525, 535
Saric W. S., 521, 535
Scalars, defined, 25
Scale height, atmosphere, 22
Schlichting, H., 337, 341, 364, 409, 469, 516,
520
Schlieren method, 715
Schraub, F. A., 600
Schwartz inequality, 545
Scotti, R. S., 536
Secondary flows, 388–389, 492
Secondary instability, 524
Second law of thermodynamics, 15
entropy production and, 116–118
Second-order tensors, 30–32
Seiche, 237
Self-preservation, turbulence and, 566–568
Separation, 366–368
Serrin, J., 352, 409
Shallow-water approximation, 262–263
Shallow-water equations, 625–627
high and low frequencies, 634–636
Shallow-water theory, vorticity conservation
in, 644–647
Shames, I. H., 212
Shapiro, A. H., 714, 763
Shapiro, Ascher H., 855
Shaw, H. S., 295, 332, 334, 337
Shear flow
wall-bounded, 570–580
wall-free, 564–570
Shear production of turbulence, 556, 559–562
Shear strain rate, 61
Shen, S. F., 516, 521–522, 536
Sherman, F. S., 364, 409
Shin, C. T., 464
Shock angle, 752
Shock structure, 720, 734
Shock waves
normal, 733–741
oblique, 752–756
structure of, 736–741
SI (système international d’unités), units of
measurement, 2–3
conversion factors, 841
Similarity
See also Dynamic similarity
geometric, 281
kinematic, 281
Similarity solution, 280
for boundary layer, 352–361
decay of line vortex, 315–317
diffusion of vortex sheet, 313–315
for impulsively started plate, 306–313
for laminar jet, 381–388
Singly connected region, 194
Singularities, 170
Singular perturbation, 396–401, 516
Sink, boundary layer, 348–352
Skan, S. W., 358, 409
Skin friction coefficient, 357–358
Sloping convection, 674
Smith, L. M., 584, 600
Smits A. J., 575, 601
Solenoidal vector, 40
Solid-body rotation, 68–69, 141
Solids, 3–4
Solitons, 252–253
Sommerfeld, A., 32, 51, 149, 159, 163, 539,
715
Sonic conditions, 726
Sonic properties, compressible flow, 724–729
869
Index
Sound
speed of, 16, 17, 717–720
waves, 717–720
Source-sink
axisymmetric, 206
near a wall, 189–190
plane, 173
Spalding D. B., 410, 450
Spatial distribution, 10
Specific heats, 14
Spectrum
energy, 546
as function of frequency, 546
as function of wavenumber, 547
in inertial subrange, 562–564
temperature fluctuations, 589–591
Speziale, C. G., 583, 600
Sphere
creeping flow around, 322–327
flow around, 206
flow at various Re, 375–376
Oseen’s approximation, 329
Stokes’ creeping flow around, 322–327
Spiegel, E. A., 124, 137
Sports balls, dynamics of, 376–381
Squire’s theorem, 500, 507, 509
Stability, 415–418
See also Instability
Stagnation density, 725
Stagnation flow, 172
Stagnation points, 167
Stagnation pressure, 123, 725
Stagnation properties, compressible flow,
724–729
Stagnation temperature, 725
Standard deviation, 543
Standing waves, 237–238
Starling resistor, flow in a collapsible tube,
819–820
Starting vortex, 661–662, 687
State functions 13, 15
surface tension, 8–9
Stationary turbulent flow, 543
Statistics of a variable, 543
Steady flow
Bernoulli equation and, 119–120
between concentric cylinders, 303–306
between parallel plates, 298–302
in a pipe, 302–303
Steady flow, in a collapsible tube, 820–826
Stern, M. E., 482, 536
Stewart, R. W., 600
Stokes’ assumption, 102
Stokes’ creeping flow around spheres, 315–322
Stokes’ drift, 253–255
Stokes’ first problem, 306
Stokes’ law of resistance, 288, 325
Stokes’ second problem, 317
Stokes’ stream function, 203
Stokes’ theorem, 47–49, 63
Stokes’ waves, 251
Stommel, H. M., 110, 137, 482, 536, 625
Strain rate
linear/normal, 60–61
shear, 61–62
tensor, 61
Strange attractors, 530–531
Stratified layer, normal modes in continuous,
628–634
Stratified turbulence, 540
Stratopause, 506
Stratosphere, 605
Streak lines, 59
Streamfunction
generalized, 87–88
in axisymmetric flow, 203–205
in plane flow, 73–75
Stokes, 203
Streamlines, 57–59
Stress, at a point, 90–91
Stress tensor
deviatoric, 100
normal or shear, 89
Reynolds, 551
symmetric, 90–91
Strouhal number, 371
Sturm–Liouville form, 629
Subcritical gravity flow, 248
Subharmonic cascade, 531–532
Sublayer
inertial, 547–576
streaks, 584
viscous, 572
Subrange
inertial, 562–564
viscous convective, 591
Subsonic flow, 294, 715
Substantial derivative, 56
Sucker, D., 436, 465
Supercritical gravity flow, 248
Supersonic flow, 294, 716
airfoil theory, 758–761
expansion and compression, 756–758
Surface forces, 89, 93
Surface gravity waves, 214, 219–223
See also Gravity waves
in deep water, 230–231
features of, 223–229
in shallow water, 231–233
Surface tension, 8–9
Surface tension, generalized, 130
Sverdrup waves, 637
Sweepback angle, 681, 705
Symmetric tensors, 40–41
eigenvalues and eigenvectors of, 41–44
870
Index
Systemic circulation, 766-767
pressure throughout, 771
Systole, 770–771
systolic blood pressure, measurement,
818–819
Takami, H., 465
Takens F., 531, 536
Taneda, S., 370, 409
Tannehill, J. C., 432, 465
Taylor T. D., 2, 24, 436, 465, 486, 539, 582,
591, 596–597, 600, 624, 678,
852–853
Taylor, G. I., 2, 24, 600, 678, 852–853
biographical information, 852–853
centrifugal instability, 486–493
column, 616
hypothesis, 547
number, 490, 491
theory of turbulent dispersion, 591–598
vortices, 491
Taylor–Goldstein equation, 500–503
Taylor–Proudman theorem, 615–617
TdS relations, 15
Temam, R., 436, 465
Temperature
adiabatic temperature gradient, 19, 605
fluctuations, spectrum, 589–591
potential, 20–22
stagnation, 725
Tennekes, H., 559, 564, 568, 574, 600, 656
Tennis ball dynamics, 379–380
Tensors, Cartesian
boldface versus indicial notation, 49–50
comma notation, 49
contraction and multiplication, 32–33
cross product, 38
dot product, 37
eigenvalues and eigenvectors of
symmetric, 41–44
force on a surface, 33–36
Gauss’ theorem, 44–47
invariants of, 33
isotropic, 37, 100
Kronecker delta and alternating, 36–37
multiplication of matrices, 29–30
operator del, 38–40
rotation of axes, 26–29
scalars and vectors, 25–26
second-order, 30–32
Stokes’ theorem, 47–49
strain rate, 60–61
symmetric and antisymmetric, 40–41
vector or dyadic notation, 49–51
Tezduyar, T. E., 439, 465
Theodorsen’s method, 689
Thermal conductivity, 6
Thermal convection, Lorenz model of, 528–530
Thermal diffusivity, 116, 128
Thermal energy equation, 115–116
Boussinesq equation and, 124–128
Thermal energy, 13
Thermal expansion coefficient, 16
Thermal instability (Bénard), 470–482
Thermal wind, 614–615
Thermocline, 606
Thermodynamic pressure, 100
Thermodynamics
entropy relations, 15
equations of state, 13, 16
first law of, 12–14, 115–116
review of, 688–689
second law of, 15–16, 116–118
specific heats, 14
speed of sound, 16
thermal expansion coefficient, 16
Thin airfoil theory, 689, 759–761
Thompson, L. M., 212
Thomson, R. E., 409
Thorpe, S. A., 497, 536
Three-dimensional flows, 71–73
Thwaites, B., 364, 409
Tidstrom, K. D., 524, 536
Tietjens, O. C., 24, 79, 163, 712
Time derivatives of volume integrals
general case, 82–83
fixed volume, 83
material volume, 84
Time lag, 544
Tip vortices, 695
Tollmien–Schlichting wave, 469, 516
Total peripheral resistance, 780–781, 785
Townsend, A. A., 564, 567, 568, 569, 600
Trailing vortices, 696, 698–699
Transition to turbulence, 366–367, 523–525
Translocation, 835
Transonic flow, 716
Transpiration, 833–834
Transport phenomena, 5–8
Transport terms, 112
Transpose, 26
Tropopause, 606
Troposphere, 606
Truesdell, C. A., 102, 137
Tube collapse, 818
Turbulent flow/turbulence
averaged equations of motion, 547–554
averages, 541–543
buoyant production, 558–559, 565
cascade of energy, 587
characteristics of, 538–539
coherent structures, 584–586
commutation rules, 542–543
correlations and spectra, 543–547
871
Index
Turbulent flow/turbulence (continued)
defined, 272
dispersion of particles, 591–595
dissipating scales, 561
dissipation of mean kinetic energy, 555
dissipation of turbulent kinetic energy,
559
eddy diffusivity, 581–583
eddy viscosity, 580–584
entrainment, 566
geostrophic, 673–676
heat flux, 554
homogeneous, 543
inertial sublayer, 574–576
inertial subrange, 562–564
integral time scale, 546
intensity variations, 580
intermittency, 565–566
isotropic, 551
in a jet, 567–570
kinetic energy of, 556–559
kinetic energy of mean flow, 554–556
law of the wall, 571–573
logarithmic law, 573–579
mean continuity equation, 548–549
mean heat equation, 553–554
mean momentum equation, 549–550
mixing length, 580–584
Monin–Obukhov length, 588–589
research on, 539–540
Reynolds analogy, 588
Reynolds stress, 550–553
rough surface, 579
self-preservation, 566–567
shear production, 556, 558–561
stationary, 543
stratified, 586–591
Taylor theory of, 591–598
temperature fluctuations, 589–591
transition to, 367, 523–525
velocity defect law, 573
viscous convective subrange, 591
viscous sublayer, 572
wall-bounded, 570–580
wall-free, 564–570
Turner J. S., 250, 253, 270, 277, 482–483, 499,
536, 588–589, 600
Two-dimensional flows, 71–73, 184–189
Two-dimensional jets. See Jets,
two-dimensional, 381–388
Unbounded ocean, 639
Uniform flow, axisymmetric flow, 205
Uniformity, 116
Unsteady irrotational flow, 121
Upwelling, 643
Vallentine, H. R., 212
Van Dyke, M., 389, 391, 392, 409
Vapor trails, 695
Variables, random, 541–543
Variance, 543
Vascular system, plant, 833
phloem, 835–837
xylem, 833–835
Vector(s)
cross product, 38
curl of, 40
defined, 25, 26, 29
divergence of, 39
dot product, 37
operator del, 38
Velocity defect law, 573
Velocity gradient tensor, 65
Velocity potential, 121, 161–169
Ventricles, work done on blood, 772–773
Veronis G., 124, 137
Vertical shear, 614
Vidal, C., 526, 529, 531, 532, 535
Viscoelastic, 4
Viscosity
coefficient of bulk, 102
destabilizing, 507
dynamic, 7
eddy, 580–584
irrotational vortices and, 141–144
kinematic, 7
net force, 142, 143
rotational vortices and, 139–140
viscosity, blood, 774–776
Viscous convective subrange, 591
Viscous dissipation, 112–113
Viscous fluid flow, incompressible, 426–439
Viscous sublayer, 572
Vogel, W. M., 600
Volumetric strain rate, 60
von Karman, 686
constant, 574
momentum integral, 362–364
vortex streets, 270, 369–373
Vortex
bound, 701
decay, 315–317
drag, 697–700
Görtler, 493
Helmholtz theorems, 149
interactions, 157–160
irrotational, 174
lines, 140, 315–317
sheet, 149–150, 295–296, 480, 670
starting, 661–662
stretching, 145, 621
Taylor, 491
tilting, 156, 622, 646
tip, 695
872
Index
Vortex (continued)
trailing, 695, 697–698
tubes, 140
von Karman vortex streets, 270,
369–373
Vortex flows
irrotational, 70–71
Rankine, 71–72
solid-body rotation, 68–69
Vorticity, 62–64
absolute, 155, 646
baroclinic flow and, 147–148
diffusion, 147, 297, 313–315
equation in nonrotating frame, 149–151
equation in rotating frame, 152–157
flux of, 64
Helmholtz vortex theorems, 149
Kelvin’s circulation theorem, 144–149
perturbation vorticity equation, 666–669
planetary, 155, 156, 611
potential, 646
quasi-geostrophic, 658–660
relative, 645
shallow-water theory, 644–647
Wall angle, flow at, 171–173
Wall-bounded shear flow, 570–580
Wall-free shear flow, 564–570
Wall jet, 385–388
Wall, law of the, 571–573
Wall layer, coherent structures in, 584–586
Water, physical properties of, 842
Wavelength, 216
Wavenumber, 216–217
Waves
See also Internal gravity waves; Surface
gravity waves
acoustic, 717
amplitude of, 216
angle, 753
capillary, 234
cnoidal, 252
compression, 214
deep-water, 230–231
at density interface, 255–259
dispersive, 223, 242–246, 270–272
drag, 291, 700, 760–761
elastic, 214, 717
energy flux, 229, 242–246
equation, 214–216
group speed, 229, 242–246
hydrostatic, 233
Kelvin, 639–643
lee, 656–657
packet, 240–241
parameters, 216–219
particle path and streamline, 224–227
phase of, 214
phase speed of, 217
Poincaré, 637
potential energy, 228
pressure, 214, 717
pressure change, 224
refraction, 233
Rossby, 657–663
shallow-water, 231–233
shock, 733–741
solitons, 252–253
solution, 668–669
sound, 717–720
standing, 237–238
Stokes’, 251
surface tension effects, 234–237
Sverdrup, 637
Wedge instability, 672–673
Welch J. E., 464
Wen, C. Y., 371, 409, 461, 464
White blood cells (leukocytes), 774
Whitham, G. B., 251, 277
Wieghardt K., 853
Williams, G. P., 676, 678
Windkessel theory, 791-792
Wing(s)
aspect ratio, 681
bound vortices, 697–698
drag, induced/vortex, 697, 700
delta, 705
finite span, 695–697
lift and drag characteristics, 704–705
Prandtl and Lanchester lifting line
theory, 697–701
span, 680
tip, 680
tip vortices, 695
trailing vortices, 695, 697–698
WKB approximation, 650–652
Womersley number, 782
Woods J. D., 497, 536, 588, 600
Wosnik, M., 575–576, 601
Xylem, 833-835
flow, 835
Yaglom A. M., 539, 600
Yahya, S. M., 763
Yakhot, V., 584, 601
Yanenko, N. N., 433, 465
Yaw axis of aircraft, 681
Yih, C. S., 364, 409, 516, 536
Zagarola, M. V., 575, 577, 601
Zhukhovsky, N.,
airfoil lift, 692–695
hypothesis, 686
lift theorem, 183, 185–188, 684
transformation, 689–692
Zone of action, 752
Zone of silence, 752