PHYSICS OF FLUIDS 19, 076101 共2007兲
Unsteady flow organization of compressible planar base flows
R. A. Humble,a兲 F. Scarano, and B. W. van Oudheusden
Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS, Delft,
The Netherlands
共Received 15 January 2007; accepted 5 April 2007; published online 12 July 2007兲
The unsteady flow features of a series of two-dimensional, planar base flows are examined, within
a range of low-supersonic Mach numbers in order to gain a better understanding of the effects of
compressibility on the organized global dynamics. Particle image velocimetry is used as the primary
diagnostic tool in order to characterize the instantaneous near wake behavior, in combination with
data processing using proper orthogonal decomposition. The results show that the mean flowfields
are simplified representations of the instantaneous flow organizations. Generally, each test case can
be characterized by a predominant global mode, which undergoes an evolution with compressibility,
within the Mach number range considered. 共The term “global mode” is defined herein as an
organized global dynamical behavior of the near wake region, recognizing that the near wake
dynamics may be describable in terms of several global modes.兲 At Mach 1.46, the predominant
global mode can be characterized by a sinuous or flapping motion. With increasing compressibility,
this flapping mode decreases, and the predominant global mode evolves into a pulsating motion
aligned with the wake axis at Mach 2.27. These global modes play an important role in the
distributed nature of the turbulence properties. The turbulent mixing processes become increasingly
confined to a narrower redeveloping wake with increasing compressibility. Global maximum levels
of the streamwise turbulence intensity and the kinematic Reynolds shear stress occur within the
vicinity of the mean reattachment location, and show no systematic trend with compressibility. In
contrast, the global maximum level of the vertical turbulence intensity moves upstream from the
redeveloping wake toward the mean reattachment location. The vertical turbulence intensity decays
thereafter more slowly than the other turbulence quantities. Overall, the local maximum levels of the
turbulence properties decrease appreciably with increasing compressibility. © 2007 American
Institute of Physics. 关DOI: 10.1063/1.2739411兴
INTRODUCTION
The compressible near wake region behind a blunt-based
body contains a series of complicated flow phenomena that
constitute a fundamental fluid dynamics research problem.
The near wake contains a myriad of flow structures, characterized by a large separated flow region, bounded by separated free shear layers and the base wall. Because of its fundamental, as well as practical relevance in a variety of fluid
dynamic applications, such as high-speed projectiles and
powered missiles, numerous experimental efforts over the
decades have sought to improve the physical understanding
of its complex behavior. Schlieren and shadowgraph methods have historically been used to characterize shear layer
growth rates, determine the location of shock and expansion
waves, as well as provide a visualization of the large-scale
structures found to be present. Yet these techniques are
somewhat unsuited for the accurate determination of turbulent features at high convective Mach numbers 共i.e., high
compressibility兲, since they have been shown to become
highly three-dimensional in nature.1 Planar imaging techniques, such as Mie scattering, have therefore been used, to
provide instantaneous structural information on the flow organization of the shear layer,1,2 as well as in the near wake
Author to whom correspondence should be addressed. Tel: ⫹31共0兲15
2785169; Fax: ⫹31共0兲15 2787077. Electronic mail: r.a.humble@tudelft.nl
a兲
1070-6631/2007/19共7兲/076101/17/$23.00
region.3,4 Such studies have provided direct evidence of
large-scale structures along the shear layers, and at the wake
interface, using spatial covariance analyses to describe their
size, eccentricity, and orientation. Yet planar imaging techniques are incapable of providing quantitative turbulence information, which is critically important for a better understanding of the fluid dynamic mechanisms present.
Numerous studies have therefore focused attention on
characterizing in detail the mean velocity distribution, turbulence intensities, turbulent kinetic energy, and kinematic
Reynolds shear stress distribution on both planar5–7 and
axisymmetric8–10 base flow configurations. The majority of
such studies have primarily made use of laser Doppler velocimetry 共LDV兲, with the statistical data being used by the
computational community to help validate the most promising computational codes, which have typically made use of
both Reynolds-averaged Navier-Stokes 共RANS兲 and large
eddy simulation 共LES兲 techniques.11–13 Of particular interest
in recent years has been the study of the instantaneous flow
organization of the near wake region. This has been motivated by the recognized importance of the unsteady motion
of the reattachment point and the associated recompression
shock wave system, which can lead to fluctuating base
pressures,14 as well as fluctuating loads on the body itself.
Furthermore, it has become increasingly evident that data on
the unsteady flow features of compressible base flows may
19, 076101-1
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076101-2
Humble, Scarano, and van Oudheusden
be useful for the validation of computational models, since it
has been shown that the consideration of the near wake unsteadiness significantly improves the numerical prediction of
the average flowfield properties.15 Although the use of pointby-point velocity measurements provides valuable mean and
turbulence statistics, the inability to make large-scale instantaneous velocity flowfield measurements represents a significant obstacle in obtaining a more thorough description of the
near wake global dynamics.
Advances in laser and digital imaging technology have
led to rapid developments in nonintrusive, planar flow diagnostic tools, such as particle image velocimetry 共PIV兲 in particular. This technique is capable of making instantaneous
whole-field velocity measurements, rendering it suitable to
investigate large-scale unsteady flow features. Although PIV
has historically found widespread application as a standard
diagnostic tool in low-speed incompressible flows,16 extending the technique into the high-speed compressible flow regime has become possible with the introduction of highenergy short-pulsed lasers, as well as short interframe
transfer CCD cameras. PIV has been successfully applied to
a variety of high-speed flow problems of practical interest,
such as jets,17 mixing layers,18 and boundary layers,19 making the study of the compressible planar base flow problem
well posed.
Although large-scale organized motions are obvious in
visualization studies of compressible base flows, it has been
difficult to incorporate them into an underlying description
of the global dynamics, and to describe the role of these
dynamics in the overall distributed nature of the turbulence
properties. While there is a general consensus as to the effects of compressibility in turbulent shear flows, such as
leading to a reduction in shear layer growth rates,20,21 turbulent fluctuating velocities, and shear stresses,22,23 as well as
large-scale turbulent structures appearing more threedimensional and disorganized than under incompressible
conditions,1,24 less agreement is found when attempting to
formulate a more unified picture of the role of compressibility in the underlying global dynamics. It is desirable then to
describe the flowfield in terms of its most dominant features,
or processes, and aim to provide a simpler conceptual picture
that can be more readily understood. Much attention has
therefore been focussed upon the development of low-order
or simplified representations of complicated fluid dynamical
systems. This has led to interest in statistical techniques such
as Proper Orthogonal Decomposition 共POD兲 in particular.
This technique has been introduced into fluid mechanics25 as
a tool to highlight the most prominent coherent motions
within turbulent flows, as well as to facilitate the development of a low-order description of the system’s overall dynamics. The technique has been successfully applied to a
variety of practical flow problems, including boundary
layers,26 turbulent jets,27 as well as compressible flows.28
PIV data are particularly suited for POD analyses since the
entire spatial velocity field is available, leading to the construction of global eigenmodes.
Previous studies that have attempted to address the organized global dynamics of compressible planar base flows29,30
have visualized the large-scale near wake motion, leading to
Phys. Fluids 19, 076101 共2007兲
a characterization of the instantaneous near wake behavior.
The application of POD to the problem31 has led to hypotheses being made regarding the underlying flow organization.
Generally, under transonic conditions, the most dynamically
significant feature of the flow could be characterized by a
sinuous or flapping motion. Under fully supersonic conditions, the near wake could be characterized by a streamwiseoriented pulsating motion. Predominant modes have also
been independently documented for flat plate wakes using
Schlieren32 and holographic interferometry,33 as well as axisymmetric cylinders and cones under both subsonic and supersonic conditions using RANS/LES computations.11 Yet no
systematic study documenting the behavior of these global
modes with compressibility has been reported for the planar
base flow case, nor have their role in the distributed nature of
the turbulence properties been properly addressed.
The present paper aims to study the unsteady flow features of two-dimensional, planar base flows, in order to gain
a better understanding of the effects of compressibility on the
organized global dynamics of the near wake region, as well
as their role in the distributed nature of the turbulence properties. Experiments are carried out at several Mach numbers
in the low-supersonic range. PIV is used as the primary diagnostic tool in combination with data processing using
POD. Mean and instantaneous velocity measurements are
obtained, enabling a visualization of the near wake behavior.
Turbulence statistics are then presented, and the role of organized global dynamics in the distributed nature of the turbulence properties is addressed. The velocity fields are then
analyzed by POD to make further statements regarding the
unsteady flow organization. Anticipating our discussion later
on, we first define a key compressibility parameter; the convective Mach number M c = ⌬U / 共a1 + a2兲, where ⌬U is the
velocity change across the initial constant-pressure mixing
shear layer, and a1 and a2 are the corresponding speeds of
sound. The convective Mach number is considered as an
appropriate parameter to scale the effects of compressibility
in supersonic shear flows, and even though a single parameter may not be universally applicable,34 it is still considered
a useful parameter to delineate the effects of compressibility
and will be used for this purpose in the present work.
APPARATUS AND EXPERIMENTAL TECHNIQUE
Experiments were performed in the blow-down
transonic-supersonic wind tunnel 共TST-27兲 of the HighSpeed Aerodynamics Laboratories at Delft University of
Technology. The facility generates flows within the Mach
number range 0.5–4.2 in a test section of dimensions
280 mm⫻ 270 mm. The Mach number was set by means of
a continuous variation of the throat section and flexible
nozzle walls. The tunnel operates at unit Reynolds numbers
ranging from 30⫻ 106 to 130⫻ 106 m−1, enabling a blowdown operating use of the tunnel of approximately 300 s.
The model consisted of a symmetric double-wedge with a
sharp leading edge, giving a deflection angle of 7.13°. This
was followed by a 30 mm long thick plate with constant
thickness of h = 20 mm. The total chord length was 110 mm.
The model terminated with a vertical base and spanned the
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076101-3
TABLE I. Experimental conditions.
Case
Flow quantity
M⬁
Mc
U⬁ 共m/s兲
P0 共kPa兲
T0 共K兲
Reh, 1 ⫻ 104
Phys. Fluids 19, 076101 共2007兲
Unsteady flow organization of compressible planar
1
2
3
Uncertainties
共%兲
1.46
0.65
398
192
273
68
1.78
0.84
461
194
273
60
2.27
1.01
528
255
263
62
2
⬍10
1
1
1
3
entire width of the test section. In the present study, experiments at three different Mach numbers were conducted that
are summarized in Table I.
Wall static pressure, plenum stagnation pressure, and
stagnation temperature measurements were used to determine the flow conditions. In Table I, the freestream Mach
number M ⬁ corresponds to that along the thick flat plate
immediately upstream of the terminating base. A schematic
representation of the experimental configuration is shown in
Fig. 1. The inset illustrates the reference coordinate system.
A characterization of the incoming flow conditions for the
range of Mach numbers considered in the present study determined that the undisturbed freestream flow was uniform,
with a turbulence level ⬃1%. The boundary layer thickness
␦99 immediately upstream of the base edge was within the
range 1 – 2 mm, within the Mach number range 1.4–3.0, respectively. The maximum streamwise turbulence intensity
within the boundary layers was typically of the order
具u⬘典 / U⬁ ⬃ 0.15 共where 具·典 denotes the root-mean-square
quantity兲. In the present study, the average statistical uncertainties 共for all three test cases兲 associated with the limited
number of realizations for the mean velocity ū, turbulence
intensity 具u⬘典, and kinematic Reynolds shear stress u⬘v⬘,
were estimated to be approximately 2%, 3%, and 25%, respectively, based upon the maximum standard deviation and
correlation coefficient measured in each experiment, using a
95% confidence interval.35
Two-component PIV is employed in the present study.
Flow seeding constitutes one of the most critical aspects of
PIV in high-speed flows. A 2D distributor rake was used to
seed a fraction of the flow in the settling chamber with titanium dioxide 共TiO2兲 particles with a nominal median diameter of 50 nm and estimated bulk density of 200 kg/ m3. Hot-
wire anemometry measurements performed in the freestream
of the facility revealed no noticeable difference in the mean
velocity field, and only a 0.2% increase in turbulence intensity as a result of the seeding device. A previous particle
response assessment study has been carried out across an
oblique shock wave in the facility under similar experimental
conditions to the present study, and returned a particle relaxation time of p = 2.4 s, corresponding to a frequency response of f p = 417 kHz.6 From this relaxation time, it could
be inferred that the effective particle size, due to the phenomenon of particle agglomeration, is approximately
400 nm. The seeded streamtube area was approximately
10 cm2 as it entered the center of the test section. The seeded
flow was illuminated by a Spectra-Physics Quanta Ray
double-pulsed Nd:YAG laser with 400 mJ pulsed energy and
a 6 ns pulse duration at wavelength 532 nm. Laser light tunnel access was provided by a probe inserted into the flow
downstream of the model. The laser pulse separation applied
in the measurements was 1 s, which produced a maximum
freestream particle displacement within the range
0.4– 0.5 mm. The light sheet was approximately 1 mm thick.
Images were recorded by a PCO Sensicam QE, a 12-bit
Peltier-cooled CCD digital camera with frame-straddling architecture, and a 1376⫻ 1040 pixel sensor. The camera was
equipped with a Nikon 60 mm focal objective with an
f-number f # = 8, which yielded a diffraction limited particle
image spacing of approximately 2 pixels. A narrow-bandpass 532 nm filter was used to suppress background illumination from daylight. All test cases were imaged over a fieldof-view of 70 mm⫻ 54 mm, resulting in a digital resolution
of approximately 19.6 pixels/ mm. A dataset size of 400 image pairs was acquired in each case at a framing rate of 5 Hz.
The recorded images were analyzed using the twodimensional cross-correlation technique WIDIM.36 This
method is based upon the deformation of correlation windows with an iterative multigrid scheme, which is particularly suited for highly sheared flows. All images were interrogated using square windows of size 31⫻ 31 pixels and an
overlap factor of 75%.
POD TECHNIQUE
The POD is a statistical technique that decomposes a
signal into a basis of nonspecified functions chosen to represent the energy of the signal in the fewest number of
modes.37 In the present study, a decomposition of the time
variation of each flowfield into a limited number of modes is
FIG. 1. Schematic representation of the experimental
configuration. Reference coordinate system is also
shown 共inset兲.
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076101-4
Phys. Fluids 19, 076101 共2007兲
Humble, Scarano, and van Oudheusden
made. Consider a system in which N data measurements are
simultaneously taken at M time instants tn, such that the
samples are uncorrelated and linearly independent. The data
in the present study represent velocity obtained from the PIV
study. The mean velocity at a point ū共x , y兲 is subtracted from
the instantaneous velocity u共x , y , tn兲, leaving only data containing fluctuations from the mean u⬘共x , y , tn兲. The POD extracts k time-independent orthonormal basis functions, empirical eigenfunctions, or eigenmodes, k共x , y兲, and timedependent orthonormal amplitude coefficients, ak共tn兲, such
that the reconstruction
C⌽ = ⌽.
共7兲
In this way, the eigenvectors of the N ⫻ N matrix R can be
found, by computing the M ⫻ M matrix C—an attractive
method since in the present study M Ⰶ N 关where M = 400 for
each test case, with each snapshot containing vector fields of
size N = 168共Nx兲 ⫻ 126共Ny兲 ⯝ 20000兴. The reconstruction of
any of the original snapshots using an arbitrary number of
modes K can be performed with
K
u共x,y,tn兲 = ū共x,y兲 + 兺 ak共tn兲k共x,y兲.
共8兲
k=1
M
u⬘共x,y,tn兲 = 兺 ak共tn兲k共x,y兲
共1兲
k=1
is optimal, in the sense that the functions maximize the
normalized averaged projection of onto u⬘, viz.,
具兩共u⬘, 兲兩2典
max
.
储储2
共2兲
Here, 储·储 denotes the L2 norm 储f储2 = 共· , · 兲, where 共·,·兲 is the
standard Euclidean inner product. 兩·兩 is the modulus. It is
usual to invoke the ergodic hypothesis, so that ensemble averages 具·典 are considered as representing time averages. The
problem can be recast as the solution to the following EulerLagrange integral equation:
冕冕
⍀
具u⬘共x,y兲 丢 u⬘*共x⬘,y ⬘兲典共x⬘,y ⬘兲dx⬘dy ⬘ = 共x,y兲,
共3兲
where 丢 is the tensor product, ⴱ denotes complex conjugation, and 关x , y兴 苸 ⍀. The kernel of Eq. 共3兲 is the averaged
autocorrelation tensor
R共x,x⬘ ;y,y ⬘兲 = 具u⬘共x,y,tn兲 丢 u⬘ 共x⬘,y ⬘,tn兲典.
*
共4兲
The non-negative and self-adjoint properties of R共x , x⬘ ; y , y ⬘兲
ensure that all eigenvalues are real and non-negative, and can
therefore be ordered such that n 艌 n+1 艌 ¯ 艌 0. In practice, results from the experiments give snapshots of data at a
finite number of discrete points, and these data can be placed
into matrix form. R共x , x⬘ ; y , y ⬘兲 is thus given by
M
R共x,x⬘ ;y,y ⬘兲 =
1
兺 u⬘共x,y,tn兲u⬘共x⬘,y⬘,tn兲.
M n=1
共5兲
The snapshot POD method, as first proposed by Sirovich,38
is implemented in the present study. The snapshot method
makes use of the fact that u⬘共x , y , tn兲 and k共x , y兲 span the
same linear space. Thus, the POD eigenmodes can be written
as a linear sum of the data vectors
M
k共x,y兲 = 兺 ⌽knu⬘共x,y,tn兲,
k = 1, . . . ,M ,
共6兲
n=1
where ⌽kn is the nth component of the kth eigenvector. The
eigenmodes can then be found by solving the following eigenvalue problem:
Important properties are the following orthogonality conditions:
„k共x,y兲, *l 共x,y兲… = ␦kl ,
具ak共tn兲a*l 共tn兲典 = k␦kl ,
共9兲
共10兲
where ␦kl is the Kronecker delta. The total energy of the flow
is defined as the mean square fluctuating value of velocity,
and is given by the sum of the eigenvalues k, each eigenmode being assigned an energy percentage based upon the
eigenmode’s specific eigenvalue, such that
冒兺
M
Ek = k
i .
共11兲
i=1
The ensemble size M now becomes important because it is
directly linked with the size of the matrix C. A previous
study on the application of POD to PIV data of compressible
planar base flows31 has indicated that an acceptable degree of
statistical convergence is reached if the ensemble typically
contains more than 100 realizations.
RESULTS AND DISCUSSION
Mean flow organization
Mean velocity streamlines are displayed with velocity
vectors and vertical velocity contours in Figs. 2共a兲–2共c兲. Vectors are undersampled in the streamwise direction 共showing
1 in 20 in order to better visualize the important flow features兲. The outer flow can be seen to pass through a series of
expansion waves, which emanate from the shoulder of the
base. Here, the flow accelerates isentropically to a maximum
velocity magnitude at Mach 1.46, 1.78, and 2.27 of approximately 1.19, 1.18, and 1.17U⬁, respectively. The system of
expansion waves does not focus exactly at the shoulder of
the base due to the finite thickness of the boundary layer.
Since the local expansion strength decreases with distance
from the shoulder, fluid near to the shoulder experiences a
more sudden expansion than fluid farther away. The measurement data are therefore compromised in the near shoulder region due to the large flow gradients present, which
cause the flow to rapidly accelerate. This introduces a difference between the measured and theoretical velocity pattern
due to particle lag. A previous planar base flow study using
PIV6 has determined that the associated error falls to below
1% for x / h ⬎ 0.25 from the shoulder of the base. Another
common concern in the measurement of these types of sepa-
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076101-5
Unsteady flow organization of compressible planar
Phys. Fluids 19, 076101 共2007兲
FIG. 3. 共Color兲 Spanwise survey of the near wake region. Isosurfaces of
mean streamwise velocity are shown flooded with mean vertical velocity.
FIG. 2. 共Color兲 Mean flow topology: 共a兲 Mach 1.46, 共b兲 Mach 1.78, 共c兲
Mach 2.27. Mean velocity streamlines are shown with vertical velocity contours. Velocity vectors show 1 in 20.
rated flows is the downstream interference from the initial
bow shock wave, as it reflects off the tunnel walls and back
toward the redeveloping wake. The streamwise location of
the impingement of the reflected bow shock wave back onto
the redeveloping wake xreflect was estimated using isentropic
flow theory. It was determined to be xreflect / h ⬎ 7 at Mach
1.78, as measured from the base edge. This distance increases with increasing Mach number. Note that the flow
approaching the model in the Mach 1.46 case was subsonic,
and thus there is no bow shock wave.
Mean flow streamlines verify the symmetry of the mean
flow topology. The shear layers undergo a recompression and
realignment process, eventually reattaching at a distance xr
along the centerline of the wake. For Mach 1.46, 1.78, and
2.27, reattachment occurs at xr / h = 1.76, 1.25, and 1.1, respectively, consistent with other results.6 Fluid that has insufficient momentum to overcome the pressure gradient at
reattachment is directed back toward the base, leading to the
formation of two well-defined recirculating flow regions. The
recirculating fluid accelerates from the reattachment point to
a maximum reversed-flow velocity of 0.19, 0.27, and
0.28U⬁, at Mach 1.46, 1.78, and 2.27, respectively. It appears
that a similarity relationship exists for the centerline velocity
distribution, since the present distributions were found to
change little over the Mach number range considered. Spu-
rious streamlines close to the base wall could be observed in
all test cases 共typically for x / h ⬍ 0.1兲. These were due to
laser light reflections from the base.
The separated free shear layers appear as densely spaced
velocity contours, which approach each other in the reattachment region, spreading slightly due to the turbulent mixing.
Downstream of reattachment, a mutual interaction takes
place, and the spreading of the velocity profiles becomes
more pronounced. Flow recompression is marked by the appearance of compression waves, which emanate from the
reattachment region. These coalesce to form the oblique
shock waves. A recovery process of the wake deficit occurs
farther downstream, although the complete recovery process
cannot be observed within the present measurement domain.
Overall, it is clear that the spatial extent of the near wake
decreases with increasing Mach number, with the expansion
and oblique shock waves becoming more inclined toward the
wake axis.
Two-dimensionality of the mean flowfield
As noted by Amatucci et al.,5 nominally twodimensional flowfields that are characterized by a large-scale
separated flow region may exhibit spanwise cellular nonuniformity. To examine this effect in the present study, a multiplanar assessment of the near wake was carried out at Mach
1.78, within the range −2.0艋 z / ␦ 艋 2.0, in increments of 0.5␦
共i.e., nine planes兲. A total of 60 image pairs were obtained at
each spanwise location. Figure 3 shows isosurfaces of the
mean streamwise velocity, flooded with contours of vertical
velocity. From these results, it can be seen that the mean
velocity remains quite uniform over the spanwise region
considered. There appears to be no significant influence of
the sidewall boundary layers, nor evidence of wind tunnel
irregularities. However, the measured flow properties show
an appreciable deviation from the centerline values at distances from the centerline greater than 15% of the test sec-
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076101-6
Humble, Scarano, and van Oudheusden
Phys. Fluids 19, 076101 共2007兲
FIG. 4. 共Color兲 Instantaneous streamwise velocity distribution u / U⬁: 共left兲 Mach 1.46, 共center兲 Mach 1.78, 共right兲 Mach 2.27.
tion width. This behavior is ascribed to the lower measurement confidence level, due to the increasingly intermittent
nature of the incoming seeded streamtube.
Instantaneous flow organization
In order to first give a relatively general description of
the organized global dynamics, several fields of the instantaneous streamwise velocity component are illustrated in Figs.
4共a兲–4共i兲. These fields typify the dynamical events that are
observed to take place. The time that elapses between consecutive recordings 共5 Hz framing rate兲 is significantly
greater than any characteristic flow time scale, which leads to
uncorrelated velocity snapshots. The instantaneous results reveal that the structure of the near wake region varies considerably in time. In general, it can be seen that the more asymmetric the near wake region becomes, the greater the
movement of the reattachment region. This behavior is similar to what has been observed in POD flow analyses of an
annular jet.39 Interestingly, while a unique reattachment point
can be easily determined from the mean velocity field, the
flow often stagnates at several locations instantaneously.
The structure of the near wake region consists of a variety of organized dynamics, or dynamical features, which extend over streamwise length scales that are considered large
in comparison to the length scales associated with the instabilities that cause them. The underlying global dynamics that
arise from these instabilities can therefore be quite complicated. Before discussing them in more detail, let us first establish the concept of a global mode, as referring to an organized global dynamical behavior of the near wake region.
Thus far, we have been rather casual in our description of the
organized global dynamics, and we must now return to this
notion. A global mode will be describable in terms of a global eigenmode, which can essentially be viewed as being a
perturbation of the mean flow. Each eigenmode is constructed out of the spatial organization of the fluctuating part
of the flowfield in the most efficient manner. That is, an
optimal set of basis functions are generated that represent the
“energy” of the data, defined by our user-selected norm,
which is the mean square fluctuating value of the velocity.
This basis is optimal in the sense that a finite number of
these modes represent more of the energy than any other set
of orthogonal modes. In reality, the complex behavior of the
near wake region can be shown to consist of many global
eigenmodes, which can in fact be used together in an appropriate combination to reconstruct an individual dynamical
state, or velocity field obtained from the experiments. In
what follows, we restrict our attention to discussing only the
most predominant global modes observed in the experiments, which correspond to the most dynamically significant
organized motions. 共It will be shown that the most predominant global modes are related to the most energetic eigen-
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076101-7
Phys. Fluids 19, 076101 共2007兲
Unsteady flow organization of compressible planar
modes.兲 With this terminology at hand, we can see by inspection that each of the considered flows exhibit a different
instantaneous flow organization, whose overall general behavior can be characterized by a predominant global mode.
A significant vertical motion of the near wake is observed at Mach 1.46, and we qualitatively characterize the
global mode in this case as being a sinuous or flapping motion, similar to the well-known von Kármán vortex street
commonly observed in many incompressible wakes. This
flapping motion has an amplitude that is comparable to the
mean wake width. The unsteady motion is particularly evident within the redeveloping wake region, where the vortex
shedding process takes place. Bourdon and Dutton40 have
examined shear layer flapping in a supersonic axisymmetric
base flow. They found that the flapping generally increases
with downstream development, reaching root-mean-square
displacements of up to 40% of the local thickness. They
further noted that within the wake redevelopment region, the
flapping becomes particularly significant when compared to
the base radius. These trends are consistent with the present
observations made at Mach 1.46. 共The reader can confirm
this by looking ahead to the turbulence statistics that will be
presented, bearing in mind that organized motions have a
significant effect on the overall turbulence structure.兲 Interestingly, it was also shown in the work by Bourdon and
Dutton that planar shear layer results were typically larger
than in their axisymmetric base flow. This is also substantiated by the present results, which show a flapping motion
that is of the order of the redeveloping wake width. It has
been suggested that the geometrical constraints imposed by
the axisymmetric shear layer dampens the large-scale motions.
The present results show that as the Mach number increases, the amplitude of the flapping motion decreases,
along with the large vertical velocity fluctuations typically
associated with the vortex shedding process. Indeed, it is
well known that vortex shedding is weak in supersonic
flows.32,33 Upon closer inspection, it can be seen that not
only does this flapping motion decrease with increasing
Mach number, but the motion of the reattachment region
develops a more streamwise orientation. 共This observation
will be further evidenced in the POD analysis to be presented
later, and is substantiated by the turbulence statistics that
follow.兲 We may therefore conceptualize the predominant
global mode at Mach 2.27 as being characterized by a pulsating motion aligned with the wake axis. Observe how the
reattachment location moves a streamwise distance that is
typically of the order of the redeveloping wake width.
Scarano and van Oudheusden,6 using a low-pass filtered version of the velocity field, have determined the spatial occurrence of the instantaneous reattachment location for the same
planar base flow at Mach 2. They found that the spatial fluctuations were indeed larger in the streamwise direction than
in the vertical direction, suggesting that the overall global
dynamics were dominated by a pulsating motion. Interestingly, there is evidence suggested by Clemens and Mungal41
that the organization of the large-scale turbulent structures
develop a more streamwise orientation with increasing compressibility.
One particularly interesting fluid dynamic feature of the
present results is that the predominant global modes appear
to influence the motion of the recompression shock waves,
which emanate from the reattachment region. Inspection of
the individual instantaneous velocity vector fields reveals
that the recompression shock waves generally move upstream when wake streamlines in the reattachment region
become temporarily concave, and move downstream when
wake streamlines become temporarily convex. This observation is more apparent at Mach 1.46, as would be expected.
关Figure 4共d兲 is a typical example.兴 This behavior has in fact
also been noted by Kastengren et al.42 in their recent reattachment shock wave visualization studies using Mie scattering. Furthermore, the recompression shock waves can be
seen to generally move downstream when the near wake region becomes temporarily larger, and move upstream when
the near wake region becomes temporarily smaller. This observation is more apparent at Mach 2.27, and has also been
documented in earlier work by the present authors.31 Another
interesting feature is the occurrence of an expansion fan
where the redeveloping wake becomes locally convex 关Figs.
4共d兲 and 4共g兲 are typical examples illustrating this兴. Here, the
expansion fan can be seen to influence the overall compression that takes place in the outer flow. Although particularly
evident at Mach 1.46, this observation can also be made at
Mach 1.78 and 2.27. The foregoing results highlight some
important observations regarding the instantaneous near
wake behavior, which will be used in our subsequent discussion. It is important to emphasize that these observations are
supported by experience with hundreds of velocity fields, as
well as those obtained in other PIV compressible planar base
flow studies,6,31 which have considered additional intermediate Mach numbers not considered here.
Turbulence statistics
Turbulence statistics are now presented because we wish
to examine the effect of the organized global dynamics on
the distributed nature of the turbulence properties. Distributions of the streamwise turbulence intensity 具u⬘典 for the test
cases considered are shown in Figs. 5共a兲–5共c兲. Relatively
moderate levels can be observed within the separated shear
layers. The location of the peak 具u⬘典 in this region appears to
coincide with the inflection point of the mean velocity profile, which is centered between the edges of the shear layer,
consistent with observations that have been made in planar
compressible base flows.6 Detailed LDV studies of axisymmetric separating shear layers8 have shown that the effect of
the separation process through the rapid expansion at the
shoulder is in fact to magnify 具u⬘典 along the inner edge of the
shear layer. Further work is required to clarify the similarities and differences between planar and axisymmetric shear
layers.
The present values of 具u⬘典 throughout the near wake region compare favorably with the results of other supersonic
planar base flows,5,6 as well as backward facing steps43 and
axisymmetric configurations,8 which reach levels of up to
25% of U⬁ 共see also Fig. 8兲. As the flow enters the reattachment region, a significant broadening of 具u⬘典 occurs due to
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076101-8
Humble, Scarano, and van Oudheusden
Phys. Fluids 19, 076101 共2007兲
FIG. 5. Streamwise turbulence intensity distribution 具u⬘典 / U⬁: 共a兲 Mach
1.46, 共b兲 Mach 1.78, 共c兲 Mach 2.27.
FIG. 6. Vertical turbulence intensity distribution 具v⬘典 / U⬁: 共a兲 Mach 1.46, 共b兲
Mach 1.78, 共c兲 Mach 2.27.
the large-scale interaction, as well as the increase in turbulence activity along the wake axis, as a result of the mutual
shear layer interaction. Here, global maximum levels of the
具u⬘典 component are obtained. The two shear layers of opposite spanwise vorticity begin to merge, and the velocity deficit thereafter begins to recover. Although two distinct streamwise turbulence intensity peaks indicating the approaching
shear layers are initially present, their merging appears to be
rather different at Mach 1.46 than at Mach 1.78 and 2.27.
While in the latter two cases there is a strong merging, resulting in a single turbulence peak within the reattachment
region, in the former case there is a somewhat weaker merging farther downstream. Clearly, there is some question of
how the global mode dynamics affect the structure of the
turbulence properties, but the present evidence suggests that
there is an important role being played by the predominant
global mode in determining the distributed nature of the turbulence properties within the near wake region.
The corresponding vertical turbulence intensity 具v⬘典 distributions are shown in Figs. 6共a兲–6共c兲. The scale is now
twice as sensitive as 具u⬘典. The results show initially moderate
levels within the separated shear layers, with significantly
elevated levels observed within the reattachment and wake
redevelopment regions. At Mach 1.46, 具v⬘典 appears to spread
rather broadly over the vertical height of the interaction
downstream within the redeveloping wake. In contrast, at
Mach 1.78 and 2.27, there is an appreciable decay of 具v⬘典
downstream. This is consistent with the idea that the flapping
mode decreases with increasing Mach number, being replaced with a more streamwise-oriented unsteady wake mo-
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076101-9
Unsteady flow organization of compressible planar
Phys. Fluids 19, 076101 共2007兲
FIG. 8. Evolution of the maximum streamwise turbulence intensity
具u⬘典max / U⬁.
FIG. 7. Kinematic Reynolds shear stress distribution −u⬘v⬘ / U⬁2 ⫻ 102: 共a兲
Mach 1.46, 共b兲 Mach 1.78, 共c兲 Mach 2.27.
tion. The turbulent mixing processes therefore remain confined to a much narrower redeveloping wake width with
increasing compressibility, as a result of the diminished vortex street. This again highlights the important impact of the
predominant global modes on the turbulence properties. At
some point downstream of the reattachment region, we can
anticipate that the change in flow regime from a shear layer
into a wake will eventually lead to significantly reduced turbulence levels, since the wake does not contain the relatively
large mean shear rates as those present in the approaching
shear layers. Note the higher level of fluctuations associated
with the oblique shock waves, which is typically encountered
in these experimental conditions due their unsteady motion.
The kinematic Reynolds shear stress distributions
u⬘v⬘ / U⬁2 are shown in Figs. 7共a兲–7共c兲. Such measurements
are principally carried out to aid the modeling of turbulent
effects by computational methods. Initially moderate levels
are present within the separated shear layers, similar to what
has been observed in the turbulence intensity distributions,
except along the centerline, where the kinematic Reynolds
shear stress must vanish by symmetry. These levels increase
significantly as the reattachment region is approached, and
two well-defined, broad regions of kinematic Reynolds shear
stress persist downstream into the redeveloping wake. Note
that within the shear layers, the magnitude of the turbulence
properties increases with streamwise development, indicating that a mixing and entrainment process takes place along
the shear layer boundaries.
The maximum contributions from each component are
important to the improved understanding of the evolving nature of the near wake turbulence structure. The evolution of
the local maximum levels of 具u⬘典 and 具v⬘典 are shown in Figs.
8 and 9, respectively, along the normalized coordinate 共x
− xr兲 / h. Both components increase from their relatively moderate levels within the shear layers, to reach global maximum
levels within the reattachment region. To be precise, 具u⬘典
reaches its global maximum in the vicinity of the mean reattachment point, and does not appear to exhibit any systematic trend with compressibility, within the Mach number
range considered. Farther downstream, a rapid decrease of
具u⬘典 occurs within the redeveloping wake region. This can be
contrasted with the behavior of the 具v⬘典 component. Here, we
see that a systematic movement of the global maximum of
具v⬘典 occurs with increasing compressibility; namely, a movement from the redeveloping wake, upstream toward the mean
reattachment point. This behavior is attributed to the decreasing action of the alternately shed vortices, which lead to a
reduced alternating flow entrainment pattern of vertical velocity. It is interesting to note that the 具v⬘典 component typically undergoes a slower recovery downstream of its global
maximum in comparison to 具u⬘典.
To further explain these trends, it is necessary to better
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076101-10
Humble, Scarano, and van Oudheusden
Phys. Fluids 19, 076101 共2007兲
FIG. 9. Evolution of the maximum vertical turbulence intensity 具v⬘典max / U⬁.
FIG. 10. Evolution of the maximum kinematic Reynolds shear stress.
understand the physical processes taking place. The transfer
of kinetic energy from the mean flow is essentially to 具u⬘典 by
normal advection across the mean velocity gradient. Since
具u⬘典 is generally larger than both 具v⬘典 and the spanwise component 具w⬘典, energy is redistributed to these latter two terms.
Therefore, similar to what occurs in axisymmetric base
flows, the transfer of energy from the mean flow to the turbulence field takes place via the classical production mechanisms to the streamwise component 具u⬘典, whereas the vertical and spanwise components acquire energy through the
more passive pressure-strain correlation terms and turbulent
diffusion.44 This results in significant turbulence anisotropy
within the near wake region and explains the very different
turbulence amplifications of the turbulence intensities.
Direct numerical simulations 共DNS兲 performed by
Friedrich et al.45 have demonstrated that the absolute
pressure-strain correlation terms are typically smaller in
compressible flows than in corresponding incompressible
flows. One might therefore intuitively expect that the energy
transfer process due to the pressure-strain correlation terms
will diminish with increasing compressibility, thereby suppressing the redistribution of streamwise turbulence energy
to the other components. This may explain why the maximum streamwise intensity remains somewhat unaffected, if
not decreased with increasing compressibility, whereas the
vertical turbulence intensity decreases significantly, and recovers more slowly than the streamwise component. Interestingly, the streamwise component remains systematically
higher than the vertical component, despite its rapid recovery
downstream of the reattachment region. It would therefore
appear that the relatively large 具u⬘典 production does not necessarily balance the tendency toward isotropy, an observation
that has been attributed to the insufficient streamwise extent
of the interaction process.46
Maximum levels of the kinematic Reynolds shear stress
throughout the near wake region are shown in Fig. 10. The
kinematic Reynolds shear stress increases rapidly upon approaching the mean reattachment point, reaching global
maximum levels of approximately 1.8%, 1.6%, and 0.8% at
Mach 1.46, 1.78, and 2.27, respectively. Clearly, there is a
systematic decrease in the maximum kinematic shear stress
with increasing compressibility. These levels are consistent
with the study made by Samimy et al.47 on a twodimensional, reattaching shear layer at Mach 2.46, which
shows a maximum shear stress 共without the density term兲
leveling off at approximately 0.5% in the reattachment region. Also, in a study of the interaction between two compressible, turbulent shear layers at Mach 1.50 and 2.07,
Samimy and Addy7 report a maximum kinematic Reynolds
shear stress of approximately 1.5% at Mach 1.50, and 1.1%
at Mach 2.07, again consistent with the present results. The
increase in kinematic Reynolds shear stress is the result of
the overall destabilizing effects of streamline curvature and
bulk compression.8 The kinematic Reynolds shear stress profiles tend to peak sharply in the vicinity of the mean reattachment point, followed by a rapid decay farther downstream, very similar to what has been observed by Amatucci
et al.5 in their two-dimensional supersonic base flow investigation. This is the result of the shear layer alignment process and the progressive decrease in mean shear rates as the
shear layers develop into a wake flow. This behavior can be
contrasted with what is known to occur in compressible axisymmetric base flows. Here, the turbulent Reynolds stresses
have been shown to actually decrease throughout the recompression and reattachment regions.8 This difference in behavior between planar and axisymmetric base flows has been
hypothesized to be the result of the overwhelming effects of
lateral streamline curvature, which provides a stabilizing influence that is not present in the planar case. A similar qualitative trend also occurs in subsonic, solid wall cases.48 Although the precise cause共s兲 for this behavior are not known,
it seems that compressibility effects are more predominant in
supersonic flow reattachment,47 at least in the twodimensional, supersonic compliant case.8
It is now clear that differences in the underlying unsteady flow organization 共as well as the type of configura-
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076101-11
Unsteady flow organization of compressible planar
Phys. Fluids 19, 076101 共2007兲
tion兲 play an important role in determining the distributed
nature of the turbulence properties within the near wake region. As well as the differences observed in their distributed
nature, there is also a disparity between the overall turbulence levels with increasing compressibility. Generally, the
local maximum turbulence intensities and kinematic Reynolds shear stress can be seen to decrease with increasing
compressibility, within the Mach number range considered.
This behavior may be the result of an increasingly weaker
separation and expansion process with increasing Mach
number, as suggested by Samimy and Addy7 共and somewhat
substantiated by the decrease in the maximum velocity ratio
attained within the expansion region reported in the present
study兲, or it may be the consequence of a higher convective
Mach number, as suggested by Papamoschou and Roshko.49
The correlation coefficient Ruv = u⬘v⬘ / 具u⬘典具v⬘典, however, appears to remain unchanged from its incompressible value,
being typically within the range −0.5 to −0.6 for the test
cases considered.
POD analysis
Having conceptually outlined some important observations regarding the underlying unsteady flow organizations
of the flows considered, we now wish to construct a loworder representation of their complex dynamical behavior. A
statistical evaluation of the velocity fluctuations based upon
the PIV data is carried out using the POD snapshot method
described above. Physically, each eigenmode can be considered as capturing an independent predominant dynamical
characteristic of the flow, which may not be intuitively revealed by the instantaneous realizations. Each eigenmode
therefore represents a global mode. Although the interpretation of eigenmodes as representing physical flow phenomena
has long been a source of debate, a general consensus is that
it relies chiefly upon the energy convergence. As a motivating prelude, the eigenmode energy and cumulative eigenmode energy distributions for the test cases considered are
presented in Figs. 11 and 12, respectively.
The present eigenmode energy distributions generally reflect a poor energy convergence when compared to other
POD analyses, including wake flows,50–52 where a larger
amount of energy is typically captured in a relatively smaller
number of eigenmodes. This discrepancy is attributed to the
comparably high Reynolds numbers in the present experiments, which lead to energy being distributed among a larger
number of modes, since many flow scales are captured in an
instantaneous flowfield. Furthermore, the presence of random noise and occasional poor data quality, naturally present
in experimental data, also contribute to flattening the
eigenspectra distribution, by transferring energy toward the
higher-order eigenmodes.
The results show that a systematically lower energy convergence occurs with increasing Mach number. Almost 50%
of the total energy is captured by the first 10 eigenmodes at
Mach 1.46, compared with 39% and 33% at Mach 1.78 and
2.27, respectively. This trend is ascribed to the fact that the
motion of the large-scale organized dynamics becomes increasingly deterministic with decreasing Mach number, in
FIG. 11. Eigenmode energy distribution.
the sense that a well-organized, large-scale von Kármán type
wake becomes predominant, such that the energy associated
with this motion is captured increasingly more effectively by
the lower-order eigenmodes. The present eigenmodes are
considered capable of providing a good basis for a general
discussion about the nature of the flows, since it will be
shown that conclusions can be drawn that are consistent with
the observations made in the PIV instantaneous realizations.
A selection of eigenmodes depicting the normalized
streamwise velocity component are shown in Figs.
13共a兲–13共l兲. These eigenmodes are considered to be those
that best represent the unsteady nature of the flows. Interme-
FIG. 12. Cumulative eigenmode energy distribution.
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076101-12
Humble, Scarano, and van Oudheusden
Phys. Fluids 19, 076101 共2007兲
FIG. 13. 共Color兲 POD eigenmodes: streamwise velocity component u⬘ / U⬁: 共left兲 Mach 1.46, 共center兲 Mach 1.78, 共right兲 Mach 2.27. Mode number and energy
content shown 共inset兲. Note that 2u⬘ / U⬁ is shown for modes 艌3.
diate eigenmodes typically show the same fundamental features and have therefore been omitted. The eigenmodes are
sorted by decreasing fractional energy and also display velocity vectors, undersampled in order to improve the clarity
of the results. Also, note the offset in the abscissa, since data
for x / h ⬍ 0.5 were not considered in the POD analysis. This
was due to the spurious velocity fluctuations that occurred,
as a result of the strong laser light reflections in the proximity of the base. These spurious velocity fluctuations were
found to interfere with the eigenmode constructions.
The low-order eigenmodes represent the most energetic
modes, and therefore best represent the fluctuating part of the
flowfield. The first eigenmode of the Mach 1.46, 1.78, and
2.27 cases contain approximately 12%, 11%, and 8% of the
total kinetic energy, respectively. Generally, within these first
eigenmodes, the largest streamwise velocity fluctuations
typically occur within the reattachment region, which is
known to be the region of highest turbulent activity in compressible base flows.53 It is therefore expected that fluid motions in this region will likely contribute the most toward the
total energy of the flow, since it is defined as the mean square
fluctuating value of the velocity. More specifically, the eigenmodes at Mach 1.46 display distinct regions of relatively
large velocity fluctuations of alternating sign, indicative of
the train of quasistreamwise vortices associated with a convected vortex street.54 Here, pairing occurs between several
of these eigenmodes 共e.g., eigenmodes 5 and 6兲. This pairing
can be explained by the fact that the mechanism of the sinuous mode can be represented by a traveling wave composed
of eigenmodes in phase quadrature. When one eigenmode in
the pair is at a maximum, the other is at a minimum stage.
This relation reverses itself at the phase shift between them,
with an energy exchange taking place that thereby propagates the flow pattern. This can be contrasted with what is
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076101-13
Phys. Fluids 19, 076101 共2007兲
Unsteady flow organization of compressible planar
observed to occur in the Mach 2.27 case. Here, streamwiseelongated regions of velocity fluctuations can be seen, which
are typical of a pulsating motion aligned with the wake
axis.31 No significant pairing can be observed between the
eigenmodes in this case. This is consistent with the idea that
the predominant global mode can be characterized by a pulsating motion, represented by a standing wave. Previous planar base flow experiments performed by Kemmoun55 at
Mach 1.95, using hot-wire anemometry in the same facility,
have also found evidence that supports the existence of a
predominant pulsating motion. A peak frequency of 7.5 kHz
in the region between the expansion fan and free shear layer
has been reported. This corresponds to a Strouhal number of
0.3 when based upon the base height h and freestream velocity U⬁, which is not dissimilar to the Strouhal numbers obtained in studies of flat plate supersonic wakes.32,33 Interestingly, the first two eigenmodes of the Mach 1.78 case appear
to form a pair, symmetrically arranged with the wake axis.
These eigenmodes can be seen to exhibit both the flapping
and pulsating motion observed in the Mach 1.46 and 2.27
cases, respectively, tentatively suggesting that the overall
global dynamics may be a combination of these two predominant global modes. This is not immediately obvious
from the instantaneous velocity realizations.
Higher-order eigenmodes in all the test cases typically
show coherent flow features becoming progressively smaller
in spatial extent, as they contain an increasingly smaller fraction of the total kinetic energy. Regions of velocity fluctuations that alternate in sign often appear, symmetrically arranged with the wake axis. This behavior is particularly
evident at Mach 1.46, and occurs in the separated shear layers, reattachment, and redeveloping wake regions. Maximum
streamwise velocity fluctuations can still, however, be observed in the reattachment region. It is now undisputable
that, with increasing compressibility, the streamwise velocity
fluctuations become increasingly elongated and aligned with
the wake axis. Interestingly, in all eigenmodes considered,
very little flow activity is observed in the recirculating flow
region. This is consistent with the idea that while turbulence
within this region is regarded as being typically high, there is
relatively little correlative behavior induced by the smallscale velocity fluctuations.
The corresponding eigenmodes of the normalized vertical velocity component are shown in Figs. 14共a兲–14共l兲. Like
the streamwise component, the largest velocity fluctuations
in the lower-order eigenmodes typically occur in the reattachment region, however the magnitudes are significantly
less than those observed in the streamwise component. Nevertheless, it is clear, and important to emphasize, that all near
wake regions undergo a significant motion in both the
streamwise and vertical directions. The alternating distribution of vertical velocity fluctuations at Mach 1.46 substantiates the presence of a predominant sinuous mode. This flapping motion can be clearly seen to decrease with increasing
Mach number, as evidenced by the vortex shedding process
downstream of reattachment becoming significantly weaker.
While at Mach 1.46 many of the higher-order eigenmodes
typically show the same fundamental features as their lowerorder counterparts, this is not generally the case at the higher
Mach numbers. The higher-order eigenmodes become increasingly difficult to interpret with increasing compressibility. This has also been reported in earlier work, which considered intermediate Mach numbers not considered here.31
As noted by Clemens and Mungal,21 numerical studies based
upon linear stability analyses have shown that at low compressibility 共M c ⬍ 0.6 say兲, the flow structure is dominated by
Kelvin-Helmholtz instabilities. At higher convective Mach
numbers, the dominant instabilities become oblique, and result in a more three-dimensional flow structure organization.
It has therefore been proposed that structural coherence degeneration occurs with increasing compressibility, and can be
attributed to the role of these three-dimensional instability
modes.21,56 This may provide a possible explanation for the
difficulty in interpreting the present higher-order eigenmodes
with increasing compressibility, since the results are based
upon a two-dimensional representation of an essentially
three-dimensional flowfield. It should be remarked, however,
that in the present experiments, there were increased difficulties associated with optical diagnostics with increasing Mach
number; namely, flow seeding of an increasingly smaller
wake of lower density ratio.
In order to further clarify the observations made above,
projections of several of the low-order eigenmodes onto their
mean flow are made. Recall from Eq. 共8兲 that one can arbitrarily choose a finite number K of the most energetic modes
to form a subspace spanned by the first K eigenmodes. Similarly, subspaces can also be formulated based on a single
eigenmode, by first ordering the temporal coefficients of all
M observations, such that ak共tn兲 艋 ak共tn+1兲 艋 ¯ 艋 ak共t M 兲. An
eigenmode can then be projected onto the mean velocity field
to yield M subspaces, the nth subspace of the kth eigenmode
ukn共x , y兲 given by
ukn共x,y兲 = ū共x,y兲 + ak共tn兲k共x,y兲,
n = 1, . . . ,M .
共12兲
These subspaces provide a convenient method to analyze the
dynamical behavior given by the kth eigenmode. The motion
of the reattachment point contained within these subspaces
共determined from the streamline topology兲 was chosen for
comparison. The subspace reattachment point is defined by
the streamwise and vertical coordinates xr and y r, respectively, and we arbitrarily define a reattachment point angle
from the wake axis,
= tan−1
冉冊
yr
.
xr
共13兲
Note that M data points are available for a given eigenmode,
but only a sample of these results showing 1 in 30 data
points is displayed for clarity. Irregularities in the data are
due to the uncertainty in accurately defining the reattachment
point within a subspace.
A polar plot of the subspace reattachment point trajectory for the first eigenmode at Mach 1.46 is shown in Fig. 15.
Generally, it can be seen that both an appreciable streamwise
and vertical deviation of the reattachment point from its
mean position 共located at the origin兲 occurs. The first eigenmode of the Mach 1.46 case reveals that a significant vertical
motion of the reattachment point occurs, consistent with the
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076101-14
Humble, Scarano, and van Oudheusden
Phys. Fluids 19, 076101 共2007兲
FIG. 14. 共Color兲 POD eigenmodes: vertical velocity component v⬘ / U⬁: 共left兲 Mach 1.46, 共center兲 Mach 1.78, 共right兲 Mach 2.27. Mode number and energy
content shown 共inset兲.
observations made in the PIV results, and supports the claim
that the predominant global mode is a sinuous flapping motion. The first two eigenmodes of the Mach 1.78 case are
shown in Fig. 16. These two reattachment trajectories are
symmetrically arranged about the wake axis, and reflect the
symmetry observed between the two eigenmodes. These results support the claim that the predominant global mode
consists of both a significant sinuous and pulsating motion—
the predominant global modes of the Mach 1.46 and 2.27
cases, respectively. In contrast, the Mach 2.27 case, shown in
Fig. 17, appears to lack eigenmode pairing. A significant
streamwise motion of the reattachment point can be observed
in the second eigenmode of the Mach 2.27 case. It is somewhat surprising to see that the corresponding first eigenmode
does not exhibit the same behavior. It should be remarked,
however, that the motion of the reattachment point itself does
not necessarily give a clear indication of the underlying flow
organization. A visual animation of the subspaces of this
eigenmode reveals that it is in fact not dissimilar 共in a global
sense兲 to its higher-order counterpart. Such motion of the
reattachment point and associated recompression shock wave
system may have important consequences for the fluctuating
base pressure, as well as fluctuating loads on the body itself.
It is worth noting that attempts were made to represent a
coherent, phase-resolved component, by considering the correlation of the temporal coefficients of the eigenmodes that
appeared to be in phase-quadrature. This approach has
proven successful in characterizing the vortex shedding process in low-speed near wakes.54 No significant correlation
could be found, however, between any of the temporal coefficients considered in the present study. This is thought to be
the result of variations in the cyclic vortex shedding process
induced by the small-scale velocity fluctuations, or turbulence within the flow, as well as the presence of random
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076101-15
Phys. Fluids 19, 076101 共2007兲
Unsteady flow organization of compressible planar
FIG. 15. Polar plot of the eigenmode subspace reattachment point trajectory
at Mach 1.46.
FIG. 17. Polar plot of the eigenmode subspace reattachment point trajectory
at Mach 2.27.
noise and occasional poor data quality associated with the
experimental data. The eigenmode energy distributions support this statement, since small-scale velocity fluctuations are
present in higher-order eigenmodes, which are still generally
associated with an appreciable fraction of the total kinetic
energy.
Overall, it would appear that the eigenmodes presented
above, along with the instantaneous velocity fields, describe
an evolution of the predominant global mode with compressibility, within the Mach number range considered. The predominant flow mechanisms responsible for the fluctuating
characteristics of transonic and supersonic base flows have
been explored by Kawai11 in his systematic LES/RANS studies of subsonic, transonic, and supersonic cylindrical base
flows. Under transonic conditions, they are thought to be
associated with the close relation between unsteady vortex
shedding induced by the instability of the shear layers and
the local shock waves, which oscillate along the axial direction of the free shear layers. Under fully supersonic conditions, the predominant flow mechanisms are thought to be
based upon the oscillation of the free shear layers and the
recompression shock waves. The evolution of these two
modes has been further hypothesized to be associated with
the changes that occur in the characteristics of free shear
layers with increasing compressibility; namely, from a
Kelvin-Helmholtz type of instability at subsonic speeds, to a
coupling between multiple waves at supersonic speeds, as a
result of a possible feedback loop from the oscillation of the
recompression shock waves. Clearly, further research needs
to be conducted in this area to help unify experimental and
computational efforts concerning both planar and axisymmetric base flows. It is interesting to note that the flapping
motion reported in the present study has in fact been proposed to help explain the substantial time variation of base
pressure fluctuations commonly observed in supersonic base
flow experiments.14
CONCLUSIONS
FIG. 16. Polar plot of the eigenmode subspace reattachment point trajectory
at Mach 1.78.
The unsteady flow features of compressible, twodimensional, planar base flows have been studied. Experiments were performed under a range of low-supersonic con-
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076101-16
Phys. Fluids 19, 076101 共2007兲
Humble, Scarano, and van Oudheusden
ditions in order to gain a better understanding of the effects
of compressibility on the organized global dynamics, and
their role in the distributed nature of the turbulence properties. PIV was used as the primary diagnostic tool in combination with data processing using POD. Mean and instantaneous velocity measurements enabled a visualization of the
near wake behavior. Turbulence statistics were then presented, and the role of organized global dynamics in the
distributed nature of the turbulence properties was addressed.
The data were then used in combination with POD to make
further statements regarding the unsteady flow organization.
From these data, the following conclusions can be drawn:
A predominant global mode is present in each test case
considered, determined both by visual inspection of the PIV
results and confirmed by the POD eigenmodes. These predominant global modes undergo an evolution with compressibility, within the Mach number range considered. At Mach
1.46, both the PIV results and the eigenmodes show a fluctuating velocity pattern, indicative of the train of quasistreamwise vortices associated with a convected vortex street.
The predominant global mode can therefore be characterized
by a sinuous flapping motion. With increasing Mach number,
this sinuous mode decreases in amplitude. The eigenmodes
show streamwise velocity fluctuations becoming increasingly
elongated, and the predominant global mode evolves into a
pulsating motion aligned with the wake axis at Mach 2.27.
The predominant global mode of the intermediate case of
Mach 1.78 consists of both the global modes of the Mach
1.46 and 2.27 cases. All eigenmodes show very little flow
activity in the recirculating flow region.
The predominant global modes have a significant effect
on the motion of the recompression shock wave system. The
sinuous mode is typically associated with an alternating
translating motion of the recompression shock waves,
whereas the pulsating mode is typically associated with the
translation of both recompression shock waves together. A
region of expansion where the redeveloping wake becomes
locally convex often occurs, and influences the overall compression that takes place in the outer flow.
The predominant global modes play an important role in
the distributed nature of the turbulence properties. The turbulent mixing processes become increasingly confined to a
narrower wake with increasing compressibility. Global maximum levels of the streamwise turbulence intensity and the
kinematic Reynolds shear stress occur within the vicinity of
the mean reattachment location, and show no systematic
trend with compressibility. In contrast, the maximum vertical
turbulence intensity moves systematically upstream, from the
redeveloping wake, toward the mean reattachment location.
The vertical turbulence intensity decays thereafter more
slowly than the other turbulence quantities. Overall, the local
maximum levels of the turbulence properties decrease appreciably with increasing compressibility.
ACKNOWLEDGMENT
This work is supported by the Dutch Technology Foundation STW under the VIDI—Innovation Impulse program,
Grant No. DLR.6198.
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