An Overview of the Lagrangian Dispersion Modeling of Heavy Particles in Homogeneous Isotropic Turbulence and Considerations on Related LES Simulations
<p>Sketch of the eddy interaction model.</p> "> Figure 2
<p>Construction principle of an inertial particle trajectory in the random walk model (RWM).</p> "> Figure 3
<p>Matrices A & B used to calculate the turbulent fluid velocity seen by the particle along its path.</p> "> Figure 4
<p>Particle Lagrangian time scale (g = 0).</p> "> Figure 5
<p>Normalized variance of particle velocity (g = 0).</p> "> Figure 6
<p>Particle dispersion coefficient (g = 0).</p> "> Figure 7
<p>Normalized variance of particle velocity (g = 0).</p> "> Figure 8
<p>Particle dispersion coefficient (g = 0).</p> "> Figure 9
<p>Particle Lagrangian time scale (g ≠ 0).</p> "> Figure 10
<p>Particle dispersion coefficient (g ≠ 0).</p> "> Figure 11
<p>Normalized variance of particle velocity (g ≠ 0).</p> "> Figure 12
<p>Normalized variance of particle velocity (g = 0).</p> "> Figure 13
<p>Particle Lagrangian time scale (g = 0).</p> "> Figure 14
<p>Particle Dispersion coefficient (g = 0).</p> "> Figure 15
<p>Normalized variance of particle velocity (g ≠ 0).</p> "> Figure 16
<p>Particle dispersion coefficients (g ≠ 0).</p> "> Figure 17
<p>Particle Lagrangian time scale (g ≠ 0).</p> "> Figure 18
<p>Normalized variance of particle velocity (g ≠ 0).</p> "> Figure 19
<p>Particle dispersion coefficient (g ≠ 0).</p> "> Figure 20
<p>Particle Lagrangian time scale (g = 0).</p> "> Figure 21
<p>Normalized variance of particle velocity (g = 0).</p> "> Figure 22
<p>Particle dispersion coefficient (g = 0).</p> "> Figure 23
<p>Particle Lagrangian time scale (g ≠ 0).</p> "> Figure 24
<p>Normalized variance of particle velocity (g ≠ 0).</p> "> Figure 25
<p>Particle dispersion coefficients (g ≠ 0).</p> "> Figure A1
<p>Autocorrelation coefficients as proposed by Equation (A2).</p> ">
Abstract
:1. Introduction
2. Theoretical Background
2.1. Taylor’s Turbulent Diffusion Theory and Batchelor’s Generalization
2.2. Toward Turbulent Dispersion
2.2.1. Scales of Turbulent Motion
- the Kolmogorov micro-length scale ,
- the Kolmogorov time scale ,
2.2.2. Equation of Motion
- the Stokes number , based on the Eulerian moving scale is a measure of the relative importance of the particle inertia; it characterizes the particle’s response to the turbulent fluid velocity fluctuations;
- the drift parameter , which is a dimensionless drift velocity related to the turbulence level, given by
- the drag correction factor , a function that increases with Reynolds number (fluid–particle drift velocity or particle size).
Note on Stokes Numbers
- based on the Kolmogorov microscale,
- based on the Eulerian moving macroscale,
- based on the classical Eulerian macroscale,
- based on the Lagrangian time scale,
- based on a time scale of the mean flow indicating if the particles follow the mean fluid flow.
2.2.3. Qualitative Analysis of Turbulent Dispersion
2.2.4. Short Review of Analytical Approaches
- –
- –
- since the terminal velocity is a measure of the inertia, the particle does not completely follow the high-frequency fluctuations of the turbulent fluid velocity; thus, Yudine did not separate inertia and gravity effects;
- –
- if it has an appreciable settling velocity, a particle will fall from one eddy to another, whereas a fluid point will remain in the same eddy throughout the lifetime of the eddy; this is one of the first papers mentioning “overshooting”.
- for long time diffusion .
- the ratio of the fluctuating velocity variances is
3. Numerical Methods
3.1. Eddy Interaction Models (EIM–DRW)
3.1.1. General Description
- the particle (P) leaves the fluid eddy (F) for entering into a new one when the eddy (F) lifetime is elapsed (condition 1),
- or when the distance between the particle and the eddy center does exceed the eddy length a radial dimension (condition 2).
3.1.2. Historical Background
- a first estimate of the fluid velocity with the criteria ,
- a spatial step that calculates the fluid–particle velocity at time at with a correlation function:
- the integration of the equation of motion with and to obtain the new discrete particle velocity.
3.2. Random Walk Models (RWM-CWM)
3.2.1. Classical Approach
3.2.2. Two-Step Space–Time Approach
3.3. CRW–Matrix Methods
3.4. Numerical Context
4. Numerical Results
4.1. EIM–DRW Results
- (1)
- the distance between the eddy center and the heavy particle position does not exceed the random eddy length (no overshooting condition),
- (2)
- the interaction time of the particle does not exceed the random eddy lifetime.
4.1.1. Classical Dispersion Without Gravity (g = 0)
4.1.2. Modified EIM with Wang and Stock Correction (No Gravity)
4.1.3. Dispersion under Gravity Effect
4.1.4. Conclusion on EIM–DRW Monte-Carlo Methods
4.2. Markovian Methods
- A more classical Markovian model with a Langevin equation based on the correlation functions proposed by Wang and Stock (Equation (A2)):
4.2.1. CRW–Lu Model
- Dispersion without gravity (g = 0)
- RWM–Lu with gravity effects (g ≠ 0)
4.2.2. Langevin-WS Model
4.2.3. Conclusions on Markovian Models
4.3. Matrix Method and Related Considerations
4.3.1. Matrix Method without Gravity (g = 0)
4.3.2. Matrix Method with Gravity (g ≠ 0)
5. Discussion and Present Trends of Modeling Approaches
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
- variances of the particle turbulent velocity in the i-th direction:
- long time coefficient of the heavy particle in the i-th direction:
- particle Lagrangian integral time scale in the i-th direction:
- variances of the particle turbulent velocity in the i-th direction (zero drift velocity):
- particle Lagrangian integral time scale in the i-th direction (zero drift velocity):
- long time coefficient of the heavy particle in the i-th direction (zero drift velocity):
Nomenclature
Particle diameter | |
Fluid turbulent diffusion coefficient | |
Particle turbulent diffusion coefficient | |
Lagrangian turbulent fluid energy spectrum | |
Non-Stokesian drag correction factor | |
Longitudinal Eulerian space correlation function | |
Lateral Eulerian space correlation function | |
Turbulent kinetic energy | |
Typical eddy length scale | |
Eulerian integral length scale (i and j directions) | |
Size of the largest eddies | |
m | Structure parameter of turbulence |
Particle Reynolds number | |
Eulerian fluid correlation function related to i,j directions | |
Lagrangian fluid correlation function related to i,j directions | |
Lagrangian particle correlation function related to i,j directions | |
Stokes number | |
Lagrangian integral fluid time scale related to i,j directions | |
Lagrangian integral particle time scale related to i,j directions | |
Eddy turnover time | |
Eulerian integral time scale | |
Moving Eulerian time | |
Characteristic time for the crossing-trajectories effect | |
Interaction time as defined by Graham [ ] | |
Turbulent fluid velocity | |
Turbulent fluid velocity at the particle position | |
Root mean-square fluid velocity | |
Turbulent fluid velocity | |
Fluid velocity variance | |
Particle velocity variance | |
Particle velocity variance | |
Mean particle-fluid drift velocity | |
Particle position | |
Mean lateral square fluid point displacement | |
Greek symbols | |
Kolmogorov length scale | |
Rate of dissipation of turbulence kinetic energy | |
Lagrangian integral length scale | |
Fluid dynamic viscosity | |
, | kinematic viscosity of the fluid |
Fluid density | |
Particle density | |
Relative density | |
Eulerian microscale (Taylor) | |
Lagrangian microscale (Taylor) | |
Kolmogorov time scale | |
Non-Stokesian relaxation time | |
Stokesian relaxation time | |
Superscript/subscript | |
f | For fluid properties |
p | For particle properties |
Abbreviations
EIM | Eddy Interaction Model |
IE | Inertial effect |
RWM | Random Walk Model |
CE | Continuity effect |
CTE | Crossing Interaction effect |
HIST | Homogeneous Isotropic Stationary Turbulence |
References
- Crowe, C.T.; Schwarzkopf, J.D.; Sommerfeld, M.; Tsuji, Y. Multiphase Flows with Droplets and Bubbles, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2011; ISBN 9780429106392. [Google Scholar] [CrossRef]
- Podgorska, W. Multiphase Particulate Systems in Turbulent Flows; CRC Press, Taylor & Francis Group: New York, NY, USA, 2019. [Google Scholar] [CrossRef]
- Varaskin, A.Y. Turbulent Particle-Laden Gas Flows; Springer-Verlag: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Zaichnik, L.I.; Alipchenkov, V.M.; Sinaiski, E.G. Particles in Turbulent Flows; Wiley-VCH Verlag: Hoboken, NJ, USA, 2008. [Google Scholar]
- Crowe, C.T. Review—Numerical Models for Dilute Gas-Particle Flows. J. Fluids Eng. 1982, 104, 297–303. [Google Scholar] [CrossRef]
- Crowe, C.T.; Troutt, T.R.; Chung, J.N. Numerical Models for Two-Phase Turbulent Flows. Annu. Rev. Fluid Mech. 1996, 28, 11–43. [Google Scholar] [CrossRef]
- Thomson, D.J. Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J. Fluid Mech. 1987, 180, 529–556. [Google Scholar] [CrossRef]
- Taylor, G.I. Diffusion by Continuous Movements. Proc. Lond. Math. Soc. 1921, 20, 196–211. [Google Scholar] [CrossRef]
- Degrazia, G.; Anfossi, D. Estimation of the Kolmogorov constant C0 from classical statistical diffusion theory. Atmos. Environ. 1998, 32, 3611–3614. [Google Scholar] [CrossRef]
- Huilier, D. Relationships between Lagrangian and Eulerian Scales: A Review. In Proceedings of the 2002 Joint US ASME-European Fluids Summer Conference, Forum on Environmental Flows, Le Centre Sheraton Hotel, Montreal, QC, Canada, 14–18 July 2002. [Google Scholar]
- Graham, D.I.; James, P.W. Turbulent dispersion of particles using eddy interaction models. Int. J. Multiph. Flow 1996, 22, 157–175. [Google Scholar] [CrossRef]
- Thomson, D.J. Random Walk Models of Turbulent Dispersion. Ph.D. Thesis, Brunel University, Brunel, UK, November 1988. [Google Scholar]
- Walklate, P.J. A random-walk model for dispersion of heavy particles in turbulent air flow. Bound. Layer Meteorol. 1987, 39, 175–190. [Google Scholar] [CrossRef]
- Wilson, J.D.; Sawford, B.L. Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere. Bound. Layer Meteorol. 1996, 78, 191–210. [Google Scholar] [CrossRef]
- Snyder, W.; Lumley, J. Some measurements of particle velocity autocorrelation functions in a turbulent flow. J. Fluid Mech. 1971, 48, 41–71. [Google Scholar] [CrossRef]
- Wells, M.; Stock, D. The effects of crossing trajectories on the dispersion of particles in a turbulent flow. J. Fluid Mech. 1983, 136, 31–62. [Google Scholar] [CrossRef]
- Calabrese, R.V.; Middleman, S. The dispersion of discrete particles in a turbulent fluid field. Aiche J. 1979, 25, 1025–1035. [Google Scholar] [CrossRef]
- Wang, L.P.; Stock, D.E. A theoretical method for obtaining Lagrangian statistics from measurable Eulerian statistics for homogeneous turbulence. In Proceedings of the 11th Symposium on Turbulence, Rolla, MO, USA, 17–19 October 1988; pp. B14-1–B14-12. [Google Scholar]
- Wang, L.P.; Stock, D.E. Numerical simulation of heavy particle dispersion: Time-step and nonlinear drag considerations. J. Fluids Engng. 1992, 114, 100–106. [Google Scholar] [CrossRef]
- Wang, L.-P.; Stock, D.E. Dispersion of Heavy Particles by Turbulent Motion. J. Atmos. Sci. 1993, 50, 1897–1913. [Google Scholar] [CrossRef] [Green Version]
- Kampe de Feriet, M.-J. Les fonctions aléatoires stationnaires et la théorie statistique de la turbulence homogène. J. Ann. Soc. Sci. Bruxelles Ser. 1939, 59, 145–194. [Google Scholar]
- Kampe de Feriet, M.J.; Pai, S. Introduction to the Statistical Theory of Turbulence. II. J. Soc. Ind. Appl. Math. 1954, 2, 143–174. [Google Scholar] [CrossRef]
- Frenkiel, F.N. Etude Statistique de la Turbulence. Fonctions Spectrales et Coefficients de Corrélation; Technique Rapport n°34; ONERA: Lille, France, 1948. [Google Scholar]
- Frenkiel, F.N. Statistical Study of Turbulence: Spectral Functions and Correlation Coefficients, NACA-TM-1436 Technical Report. 1958. Available online: https://ntrs.nasa.gov/search.jsp?R=20030067904 (accessed on 10 July 2020).
- Batchelor, G.K. Diffusion in a Field of Homogeneous Turbulence. I. Eulerian Analysis. Aust. J. Sci. Res. 1949, A2, 437–450. [Google Scholar] [CrossRef]
- Batchelor, G.K. Diffusion in a Field of Homogeneous Turbulence, II. The Relative Motion of Particles. Proc. Camb. Phil. Soc. 1952, 48, 345–362. [Google Scholar] [CrossRef]
- Tennekes, H.; Lumley, J.L. A First Course in Turbulence; MIT Press: Cambridge, MA, USA, 1972. [Google Scholar]
- Hinze, J.O. Turbulence, 2nd ed.; McGraw-Hill Inc.: New York, NY, USA, 1975. [Google Scholar]
- Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Richardson, S.F. Weather Prediction by Numerical Process; Cambridge University Press: Cambridge, UK, 1922. [Google Scholar]
- Taylor, G.I. Statistical Theory of Turbulence. Proc. Roy. Soc. Lond. 1935, A151, 421–478. [Google Scholar] [CrossRef] [Green Version]
- Michaelides, E.E. Review—The Transient Equation of Motion for Particles, Bubbles, and Droplets. J. Fluids Eng. 1997, 119, 233. [Google Scholar] [CrossRef]
- Michaelides, E.E. Particles, Bubbles & Drops: Their Motion, Heat and Mass Transfer; World Scientific Publishing Company: Singapore, 2006; ISBN 9789812566478. [Google Scholar]
- Clift, R.; Grace, J.R.; Weber, M.E. Bubbles, Drops, and Particles; Academic Press: London, UK, 1978; ISBN 012176950X9780121769505. [Google Scholar]
- Maxey, M.R. The Equation of Motion for a Small Rigid Sphere in a Nonuniform or Unsteady Flow ASME/FED. Gas-Solid Flows 1993, 166, 57–62. [Google Scholar]
- Maxey, M.R.; Riley, J.J. Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 1983, 26, 883–889. [Google Scholar] [CrossRef]
- Schiller, L.; Naumann, A.Z. Ueber die grundlegende Berechnung bei der Schwerkraftaufbereitung. Z. Ver. Dtsch. Ing. 1933, 77, 318–320. [Google Scholar]
- Turton, R.; Levenspiel, O. A short note on the drag correlation for spheres. Powder Technol. 1986, 47, 83–86. [Google Scholar] [CrossRef]
- Clift, R.; Gauvin, W.H. Motion of entrained particles in gas streams. Can. J. Chem. Eng. 1971, 49, 439–448. [Google Scholar] [CrossRef]
- Reeks, M.W. On the dispersion of small particles suspended in an isotropic turbulent fluid. J. Fluid Mech. 1977, 83, 529–546. [Google Scholar] [CrossRef]
- Yudine, M.I. Physical considerations on heavy particle diffusion. In Advances in Geophysics; Academic Press: New York, NY, USA, 1959; Volume 6, pp. 185–191. [Google Scholar] [CrossRef]
- Csanady, G.T. Turbulent Diffusion of Heavy Particles in the Atmosphere. J. Atmos. Sci. 1963, 20, 201–208. [Google Scholar] [CrossRef]
- Tchen, C.M. Mean and Correlation Problems Connected with the Motion of Small Particles Suspended in a Turbulent Fluid. Ph.D. Thesis, Delft University, The Hague, The Netherlands, 1947. [Google Scholar]
- Soo, S.L. Statistical properties of momentum transfer in two-phase flow Chem. Eng. Sci. 1956, 5, 57–67. [Google Scholar] [CrossRef]
- Chao, B.T. Turbulent transport behavior of small particles in dilute suspensions. Oest. Ing. Arch. 1964, 18, 7–21. [Google Scholar]
- Hinze, J.O. Turbulence; McGraw Hill: New York, NY, USA, 1959. [Google Scholar]
- Liu, V.-C. Turbulent dispersion of dynamic particles. J. Meteorol. 1956, 13, 399–405. [Google Scholar] [CrossRef] [2.0.CO;2" target='_blank'>Green Version]
- Friedlander, S.K. Behavior of suspended particles in a turbulent fluid. AICHE J. 1957, 3, 381–385. [Google Scholar] [CrossRef]
- Hay, J.S.; Pasquill, F. Diffusion from a continuous source in relation to the spectrum and scale of turbulence. Adv. Geophys. 1959, 6, 345–365. [Google Scholar]
- Pasquill, F. Atmospheric Diffusion; D. van Nostrand Co.: London, UK, 1962. [Google Scholar]
- Townsend, A.A. The Structure of Turbulent Shear Flow, 2nd ed.; Cambridge University Press: Cambridge, UK, 1976. [Google Scholar]
- Lumley, J.L. Two-Phase and Non-Newtonian Flows, in Topics in Applied Physics; Bradshaw, P., Ed.; Springer-Verlag: Berlin/Heidelberg, Germany, 1976; Volume 12. [Google Scholar]
- Hjelmfelt, A.T.; Mockros, L.F. Motion of discrete particles in a turbulent fluid. Appl. Sci. Res. 1966, 16, 149–161. [Google Scholar] [CrossRef]
- Meek, C.C.; Jones, B.G. Studies of the Behavior of heavy Particles in a Turbulent Fluid Flow. J. Atmos. Sci. 1973, 30, 239–244. [Google Scholar] [CrossRef] [2.0.CO;2" target='_blank'>Green Version]
- Peskin, R.L. Comments on “Studies of the Behavior of Heavy Particles in a Turbulent Fluid Flow”. J. Atmos. Sci. 1974; 31, 1167–1168. [Google Scholar] [CrossRef] [Green Version]
- Phythian, R. Dispersion by random velocity fields. J. Fluid Mech. 1975, 67, 145–153. [Google Scholar] [CrossRef]
- Pismen, L.M.; Nir, A. On the motion of suspended particles in stationary homogeneous turbulence. J. Fluid Mech. 1978, 84, 193. [Google Scholar] [CrossRef]
- Kraichnan, R.H. Diffusion by a Random Velocity Field. Phys. Fluids 1970, 13, 22. [Google Scholar] [CrossRef]
- Nir, A.; Pismen, L.M. The effect of a steady drift on the dispersion of a particle in turbulent fluid. J. Fluid Mech. 1979, 94, 369. [Google Scholar] [CrossRef]
- Wang, L.-P.; Stock, D.E. Stochastic trajectory models for turbulent diffusion: Monte Carlo process versus Markov chains. Atmospheric Environment. Part A. Gen. Top. 1992, 26, 1599–1607. [Google Scholar] [CrossRef]
- Hutchinson, P.; Hewitt, G.F.; Dukler, A.E. Deposition of liquid or solid dispersions from turbulent gas streams: A stochastic model. Chem. Engr. Sci. 1971, 26, 419–439. [Google Scholar] [CrossRef]
- Brown, D.J.; Hutchinson, P. The Interaction of Solid or Liquid Particles and Turbulent Fluid Flow Fields—A Numerical Simulation. J. Fluids Eng. 1979, 101, 265. [Google Scholar] [CrossRef]
- Hotchkiss, R.S.; Hirt, C.W. Particulate transport in highly distorted three-dimensional flow fields. In Proceedings of the 1972 Summer Simulation Conference, SHARE, San Diego, CA, USA, 14–16 June 1972; pp. 1037–1046. [Google Scholar]
- Yuu, S.; Yasukouchi, N.; Hirosawa, Y.; Jotaki, T. Particle turbulent diffusion in a dust laden round jet. Aiche J. 1978, 24, 509–519. [Google Scholar] [CrossRef]
- Dukowicz, J.K. A particle-fluid numerical model for liquid sprays. J. Comput. Phys. 1980, 35, 229–253. [Google Scholar] [CrossRef]
- Hirt, C.W.; Nichols, B.D.; Romero, N.C. SOLA: A Numerical Solution Algorithm for Transient Fluid Flows; Report LA-5852; Los Alamos Scientific Laboratory: Los Alamos, NM, USA, 1975; p. 50. [Google Scholar]
- Gosman, A.; Ioannides, E. Aspects of computer simulation of liquid-fuelled combustors. In Proceedings of the19th Aerospace Sciences Meeting, St Louis, MO, USA, 12–15 January 1981. [Google Scholar] [CrossRef]
- Gosman, A.D.; Loannides, E. Aspects of Computer Simulation of Liquid-Fueled Combustors. J. Energy 1983, 7, 482–490. [Google Scholar] [CrossRef]
- Shuen, J.-S.; Chen, L.-D.; Faeth, G.M. Evaluation of a stochastic model of particle dispersion in a turbulent round jet. AIChE J. 1983, 29, 167–170. [Google Scholar] [CrossRef]
- Shuen, J.-S.; Solomon, A.S.P.; Faeth, G.M.; Zhang, Q.-F. Structure of particle-laden jets-Measurements and predictions. AIAA J. 1985, 23, 396–404. [Google Scholar] [CrossRef]
- Solomon, A.S.P.; Shuen, J.-S.; Zhang, Q.-F.; Faeth, G.M. Measurements and Predictions of the Structure of Evaporating Sprays. J. Heat Transf. 1985, 107, 679. [Google Scholar] [CrossRef]
- Solomon, A.S.P.; Shuen, J.-S.; Zhang, Q.-F.; Faeth, G.M. Structure of nonevaporating sprays. I-Initial conditions and mean properties. AIAA J. 1985, 23, 1548–1555. [Google Scholar] [CrossRef]
- Solomon, A.S.P.; Shuen, J.-S.; Zhang, Q.-F.; Faeth, G.M. Structure of nonevaporating sprays. II-Drop and turbulence properties. AIAA J. 1985, 23, 1724–1730. [Google Scholar] [CrossRef]
- Chen, P.P.; Crowe, C.T. On the Monte-Carlo method for modeling particle dispersion in turbulence. In Proceedings of the ASME FED, Gas-Solid Flows, Energy Sources Technology Conference, New Orleans, LA, USA, 9–14 December 1984; Volume 10, pp. 37–41. [Google Scholar]
- Arnason, G. Measurement of Particle Dispersion in Turbulent Pipe Flow. Ph.D. Thesis, Washington State University, Pullman, WA, USA, 1982. [Google Scholar]
- Arnason, G.; Stock, D.E. A new method to measure particle turbulent dispersion using laser Doppler anemometer. Exp. Fluids 1984, 2, 89–93. [Google Scholar] [CrossRef]
- Durst, F.; Milojevic, D.; Schönung, B. Eulerian and Lagrangian predictions of particulate two-phase flows: A numerical study. Appl. Math. Model. 1984, 8, 101–115. [Google Scholar] [CrossRef]
- Milojevic, D. Lagrangian Stochastic-Deterministic (LSD) Predictions of Particle Dispersion in Turbulence. Part. Part. Syst. Charact. 1990, 7, 181–190. [Google Scholar] [CrossRef]
- Sommerfeld, M. Particle Dispersion in Turbulent Flow: The effect of particle size distribution. Part. Part. Syst. Charact. 1990, 7, 209–220. [Google Scholar] [CrossRef]
- Sommerfeld, M.; Ando, A.; Wennerberg, D. Swirling, Particle-Laden Flows Through a Pipe Expansion. J. Fluids Eng. 1992, 114, 648. [Google Scholar] [CrossRef]
- Mostafa, A.; Mongia, H.; MCDonnell, V.; Samuelsen, G. On the evolution of particle-laden jet flows-A theoretical and experimental study. In Proceedings of the 23rd Joint Propulsion Conference, San Diego, CA, USA, 29 June–2 July 1987. [Google Scholar] [CrossRef] [Green Version]
- Mostafa, A.A.; Mongia, H.C. On the modeling of turbulent evaporating sprays: Eulerian versus Lagrangian approach. Int. J. Heat Mass Transf. 1987, 30, 2583–2593. [Google Scholar] [CrossRef]
- Mostafa, A.A.; Mongia, H.C. On the interaction of particles and turbulent fluid flow. Int. J. Heat Mass Transf. 1988, 31, 2063–2075. [Google Scholar] [CrossRef]
- Mostafa, A.; Mongia, H.; MCDonnell, V.; Samuelsen, G.S. Evolution of particle-laden jet flows-A theoretical and experimental study. AIAA J. 1989, 27, 167–183. [Google Scholar] [CrossRef]
- Govan, A.H.; Hewitt, G.F.; Ngan, C.F. Particle motion in a turbulent pipe flow. Int. J. Multiph. Flow 1989, 22, 177–184. [Google Scholar] [CrossRef]
- Ormancey, A.; Martinon, J. Simulation numérique du comportement de particules dans un écoulement turbulent. Rech. Aérospatiale 1984, 5, 353–362. [Google Scholar]
- Ormancey, A.; Martinon, J. Prediction of particle dispersion in turbulent flows. PCH Phys. Chem. Hydrodyn. 1984, 15, 229–244. [Google Scholar]
- Frenkiel, F.N. Application of the statistical theory of turbulent diffusion to micrometeorology. J. Meteorol. 1952, 9, 252–259. [Google Scholar] [CrossRef] [2.0.CO;2" target='_blank'>Green Version]
- Hajji, L.; Pascal, P.; Oesterlé, B. A simple description of some inertia effects in the behaviour of heavy particles in a turbulent gas flow. Int. J. Non-Linear Mech. 1996, 31, 387–403. [Google Scholar] [CrossRef]
- Kallio, G.A.; Reeks, M.W. A numerical simulation of particle deposition in turbulent boundary layers. Int. J. Multiph. Flow 1989, 15, 433–446. [Google Scholar] [CrossRef]
- Burnage, S.H. Moon Prédétermination de la dispersion de particules matérielles dans un écoulement turbulent C. R. Acad. Sci. Paris 1990, 310, 1595–1600. [Google Scholar]
- Karl, J.-J.; Huilier, D.; Burnage, H. Mean behavior of a coaxial air-blast atomized spray in a co-flowing air stream. At. Spray 1996, 6, 409–433. [Google Scholar] [CrossRef]
- Huilier, D.; Burnage, H. Numerical simulation of particle dispersion in a grid turbulent flow: Influence of the lift forces. Mech. Res. Commun. 1996, 23, 433–439. [Google Scholar] [CrossRef]
- Domgin, J.-F.; Huilier, D.; Burnage, H.; Gardin, P. Coupling of a Lagrangian Model with a CFD Code: Application to the Numerical Modelling of the Turbulent Dispersion of Droplets in a Turbulent Pipe Flow. J. Hydraul. Res. 1997, 35, 473–490. [Google Scholar] [CrossRef]
- Graham, D.I. On the inertia effect in eddy interaction models. Int. J. Multiph. Flow 1996, 22, 177–184. [Google Scholar] [CrossRef]
- Graham, D.I. An Improved Eddy Interaction Model for Numerical Simulation of Turbulent Particle Dispersion. J. Fluids Eng. 1996, 118, 819–823. [Google Scholar] [CrossRef]
- Graham, D.I. Improved Eddy Interaction Models with Random Length and Time Scales. Int. J. Multiph. Flow 1998, 24, 335–345. [Google Scholar] [CrossRef]
- Squires, K.D.; Eaton, J.K. Measurements of particle dispersion obtained from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 1991, 226, 1. [Google Scholar] [CrossRef]
- Deutsch, E.; Simonin, O. Large eddy simulation applied to the motion of particles in stationary homogeneous fluid turbulence,” Turbulence Modification in Multiphase Flows. Proc. ASME-FED 1991, 110, 35–42. [Google Scholar]
- Chen, X.-Q.F.; Pereira, J.C. Computation of Particle-Laden Turbulent Gas Flows Using Two Dispersion Models. AIAA J. 1998, 36, 539–546. [Google Scholar] [CrossRef]
- Chen, X.-Q. Heavy particle dispersion in inhomogeneous, anisotropic, turbulent flows. Int. J. Multiph. Flow 2000, 26, 635–661. [Google Scholar] [CrossRef]
- MacInnes, J.M.; Bracco, F.V. Stochastic particle dispersion modeling and the tracer-particle limit. Phys. Fluids 1992, 4, 2809–2824. [Google Scholar] [CrossRef]
- Thomson, D.J. Random walk modelling of diffusion in inhomogeneous turbulence. Quart. J. R. Meteorol. Soc. 1984, 110, 1107–1120. [Google Scholar] [CrossRef]
- Walklate, P.J. A Markov-chain particle dispersion model based on air flow data: Extension to large water droplets. Bound. Layer Meteorol. 1986, 37, 313–318. [Google Scholar] [CrossRef]
- Walklate, P.J. Reply to comments on a relationship between fluid and immersed-particle velocity fluctuations. Bound. Layer Meteorol. 1988, 43, 99–100. [Google Scholar] [CrossRef]
- Sawford, B.L. Lagrangian statistical simulation of concentration mean and fluctuation fields. J. Clim. Appl. Met. 1985, 24, 1152–1166. [Google Scholar] [CrossRef]
- Legg, B.J.; Wall, C. Movement of plant pathogens in the crop canopy. Phil. Trans. R. Soc. Lond. 1983, B302, 559–574. [Google Scholar] [CrossRef]
- Legg, B.J.; Raupach, M.R. Markov-chain simulation of particles dispersion in inhomogeneous flows: The mean drift velocity induced by a gradient in Eulerian velocity variance. Bound. Layer Meteorol. 1982, 24, 3–13. [Google Scholar] [CrossRef]
- Rodean, H.C. Stochastic Lagrangian Models in Turbulent Diffusion, Meteorological Monographs; American Meteorological Society: Boston, MA, USA, 1996. [Google Scholar]
- Thomson, D.J.; Wilson, J.D. History of Lagrangian Stochastic Models for Turbulent Dispersion. Chap. 3 in Lagrangian modeling of the atmosphere. Geophys. Monogr. Ser. 2013, 200, 19–36. [Google Scholar] [CrossRef] [Green Version]
- Wilson, J.D. Trajectory Models for Heavy Particles in Atmospheric Turbulence: Comparison with Observations. J. Appl. Meteorol. 2000, 39, 1894–1912. Available online: www.jstor.org/stable/26184381 (accessed on 6 April 2021).
- Sawford, B.L.; Guest, F.M. Lagrangian statistical simulation of the turbulent motion of heavy particles. Bound. Layer Meteorol. 1991, 54, 147–166. [Google Scholar] [CrossRef]
- Reynolds, A.M. A Lagrangian Stochastic Model for Heavy Particle Deposition. J. Colloid Interface Sci. 1999, 215, 85–91. [Google Scholar] [CrossRef] [PubMed]
- Reynolds, A.M. On the Formulation of Lagrangian Stochastic Models for Heavy-Particle Trajectories. J. Colloid Interface Sci. 2000, 232, 260–268. [Google Scholar] [CrossRef]
- Reynolds, A.M.; Cohen, J.E. Stochastic simulation of heavy-particle trajectories in turbulent flows. Phys. Fluids 2002, 14, 342–351. [Google Scholar] [CrossRef]
- Reynolds, A.M. Incorporating terminal velocities into Lagrangian stochastic models of particle dispersal in the atmospheric boundary layer. Sci. Rep. 2018, 8, 1–7. [Google Scholar] [CrossRef]
- Zhuang, Y.; Wilson, J.D.; Lozowski, E.P. A trajectory simulation model of heavy particle motion in turbulent flows. J. Fluids Eng. 1989, 111, 492–494. [Google Scholar] [CrossRef]
- Sommerfeld, M.; Kohnen, G.; Rueger, M. Some open questions and inconsistencies of Lagrangian particle dispersion models. In Proceedings of the 8th Symposium on Turbulent Shear Flow, Kyoto, Japan, 16–18 August 1993. [Google Scholar]
- Hunt, J.C.R.; Nalpanis, P. Saltating and Suspended Particles over Flat and Sloping Surfaces’, in 0. E. Bamdorff-Nielsen (ed.). In Proceedings of the International Workshop on the Physics of Blown Sand, Aarhus, Denmark, 28–31 May 1985. [Google Scholar]
- Huilier, D. On the necessity of including the turbulence experienced by an inertial particle in Lagrangian random-walk models. Mech. Res. Commun. 2004, 31, 237–242. [Google Scholar] [CrossRef]
- Desjonqueres, P.; Berlemont, A.; Gouesbet, G. A lagrangian approach for the prediction of particle dispersion in turbulent flows. J. Aerosol. Sci. 1988, 19, 99–103. [Google Scholar] [CrossRef]
- Berlemont, A.; Desjonqueres, P.; Gouesbet, G. Particle lagrangian simulation in turbulent flows. Int. J. Multiph. Flow 1990, 16, 19–34. [Google Scholar] [CrossRef]
- Gouesbet, G.; Berlemont, A. Eulerian and Lagrangian approaches for predicting the behaviour of discrete particles in turbulent flows. Prog. Energy Combust. Sci. 1999, 25, 133–159. [Google Scholar] [CrossRef]
- Zhou, Q.; Leschziner, M.A. A time-correlated stochastic model for particle dispersion in anisotropic turbulence. In Proceedings of the 8th Symposium on Turbulent Shear Flow, Munich, Germany, 9–11 September 1991; pp. 10-3-1–10-3-6. [Google Scholar]
- Burry, D.; Bergeles, G. Dispersion of particles in anisotropic turbulent flows. Int. J. Multiph. Flow 1993, 19, 651–664. [Google Scholar] [CrossRef]
- Lu, Q.Q.; Fontaine, J.R.; Aubertin, G. Particle Motion in Two-Dimensional Confined Turbulent Flows. Aerosol. Sci. Technol. 1992, 17, 169–185. [Google Scholar] [CrossRef] [Green Version]
- Lu, Q.Q.; Fontaine, J.R.; Aubertin, G. Numerical study of the solid particle motion in grid-generated turbulent flows. Int. J. Heat Mass Transf. 1993, 36, 79–87. [Google Scholar] [CrossRef]
- Lu, Q.Q.; Fontaine, J.R.; Aubertin, G. Particle Dispersion in Shear Turbulent Flows. Aerosol Sci. Technol. 1993, 18, 85–99. [Google Scholar] [CrossRef]
- Lu, Q.Q.; Fontaine, J.R.; Aubertin, G. A lagrangian model for solid particles in turbulent flows. Int. J. Multiph. Flow 1993, 19, 347–367. [Google Scholar] [CrossRef]
- Lu, Q.Q. An approach to modeling particle motion in turbulent flows—I. Homogeneous, isotropic turbulence. Atmos. Environ. 1995, 29, 423–436. [Google Scholar] [CrossRef]
- Mashayek, F. Stochastic simulations of particle-laden isotropic turbulent flow. Int. J. Multiph. Flow 1999, 25, 1575–1599. [Google Scholar] [CrossRef]
- Mei, R.; Adrian, R.J.; Hanratty, T.J. Particle dispersion in isotropic turbulence under Stokes drag and Basset force with gravitational settling. J. Fluid Mech. 1991, 225, 481. [Google Scholar] [CrossRef]
- Huilier, D.; Saintlos, S.; Karl, J.J.; Burnage, H. Numerical modelling of the turbulent dispersion of heavy particles: Inertia and gravity effects on turbulent diffusivity. In Proceedings of the Third International Symposium on Engineering Turbulence Modelling and Measurements, Heraklion, Greece, 27–29 May 1996; Rodi, W., Bergeles, G., Eds.; pp. 861–870. [Google Scholar] [CrossRef]
- Laín, S.; Grillo, C.A. Comparison of turbulent particle dispersion models in turbulent shear flows. Braz. J. Chem. Eng. 2007, 24, 351–363. [Google Scholar] [CrossRef]
- Hishida, K.; Maeda, M. Turbulent Characteristics of Gas-Solids Two-phase Confined Jet: Effect of Particle Density, Japanese. J. Multiph. Flow 1987, 1, 56–69. [Google Scholar] [CrossRef]
- Zhou, Q.; Leschziner, M.A. Modelling Particle Dispersion in Anisotropic Turbulence. In Proceedings of the ECCOMAS Computational Fluid Dynamics Conference, Paris, France, 9–13 September 1996; Wiley & Sons Ltd.: Paris, France, 1996; pp. 577–583. [Google Scholar]
- Shirolkar, J.S.; Coimbra, C.F.M.; McQuay, M.Q. Fundamental aspects of modeling turbulent particle dispersion in dilute flows. Prog. Energy Combust. Sci. 1996, 22, 363–399. [Google Scholar] [CrossRef]
- Shirolkar, J.S.; McQuay, M.Q. Probability density function propagation model for turbulent particle dispersion. Int. J. Multiph. Flow 1998, 24, 663–678. [Google Scholar] [CrossRef]
- Pozorski, J.; Minier, J.-P. On the Lagrangian turbulent dispersion models based on the Langevin equation. Int. J. Multiph. Flow 1998, 24, 913–945. [Google Scholar] [CrossRef]
- Iliopoulos, I.; Hanratty, T.J. Turbulent dispersion in a non-homogeneous field. J. Fluid Mech. 1999, 392, 45–71. [Google Scholar] [CrossRef]
- Pascal, P.; Oesterlé, B. On the dispersion of discrete particles moving in a turbulent shear flow. Int. J. Multiph. Flow 2000, 26, 293–325. [Google Scholar] [CrossRef]
- Bocksell, T.L.; Loth, E. Random Walk Models for Particle Diffusion in Free-Shear Flows. AIAA J. 2000, 39, 1086–1096. [Google Scholar] [CrossRef]
- Mito, Y.; Hanratty, T.J. Use of a modified Langevin equation to describe turbulent dispersion of fluid particles in a channel flow Flow. Turbul. Combust. 2002, 68, 1–26. [Google Scholar] [CrossRef]
- Iliopoulos, I.; Hanratty, T.J. A non-Gaussian stochastic model to describe passive tracer dispersion and its comparison to a direct numerical simulation. Phys. Fluids 2004, 16, 3006–3030. [Google Scholar] [CrossRef]
- Iliopoulos, I.; Mito, Y.; Hanratty, T.J. A stochastic model for solid particle dispersion in a nonhomogeneous turbulent field. Int. J. Multiph. Flow 2003, 29, 375–394. [Google Scholar] [CrossRef]
- Oesterlé, B.; Zaichik, L.I. On Lagrangian time scales and particle dispersion modeling in equilibrium turbulent shear flows. Phys. Fluids 2004, 16, 3374–3384. [Google Scholar] [CrossRef]
- Zaichik, L.I.; Oesterlé, B.; Alipchenkov, V.M. On the probability density function model for the transport of particles in anisotropic turbulent flow. Phys. Fluids 2004, 16, 1956–1964. [Google Scholar] [CrossRef]
- Carlier, J.P.; Khalij, M.; Oesterlé, B. An Improved Model for Anisotropic Dispersion of Small Particles in Turbulent Shear Flows. Aerosol Sci. Technol. 2005, 39, 196–205. [Google Scholar] [CrossRef] [Green Version]
- Bocksell, T.L.; Loth, E. Stochastic modeling of particle diffusion in a turbulent boundary layer. Int. J. Multiph. Flow 2006, 32, 1234–1253. [Google Scholar] [CrossRef]
- Dehbi, A. Turbulent particle dispersion in arbitrary wall-bounded geometries: A coupled CFD-Langevin-equation based approach. Int. J. Multiph. Flow 2008, 34, 819–828. [Google Scholar] [CrossRef]
- Arcen, B.; Tanière, A.; Zaichik, L.I. Assessment of a statistical model for the transport of discrete particles in a turbulent channel flow. Int. J. Multiph. Flow 2008, 34, 419–426. [Google Scholar] [CrossRef]
- Tanière, A.; Arcen, B.; Oesterlé, B.; Pozorski, J. Study on Langevin model parameters of velocity in turbulent shear flows. Phys. Fluids 2010, 22, 115101. [Google Scholar] [CrossRef]
- Arcen, B.; Tanière, A. Simulation of a particle-laden turbulent channel flow using an improved stochastic Lagrangian model. Phys. Fluids 2009, 21, 043303. [Google Scholar] [CrossRef] [Green Version]
- Van Aartrijk, M.; Clercx, H.J.H. Dispersion of heavy particles in stably stratified turbulence. Phys. Fluids 2009, 21, 033304. [Google Scholar] [CrossRef] [Green Version]
- Dehbi, A. Validation against DNS statistics of the normalized Langevin model for particle transport in turbulent channel flows. Powder Technol. 2010, 200, 60–68. [Google Scholar] [CrossRef]
- Jin, C.; Potts, I.; Reeks, M.W. A simple stochastic quadrant model for the transport and deposition of particles in turbulent boundary layers. Phys. Fluids 2015, 27, 053305. [Google Scholar] [CrossRef] [Green Version]
- Minier, J.-P.; Chibbaro, S.; Pope, S.B. Guidelines for the formulation of Lagrangian stochastic models for particle simulations of single-phase and dispersed two-phase turbulent flows. Phys. Fluids 2014, 26, 113303. [Google Scholar] [CrossRef] [Green Version]
- Tanière, A.; Arcen, B. Overview of existing Langevin models formalism for heavy particle dispersion in a turbulent channel flow. Int. J. Multiph. Flow 2016, 82, 106–118. [Google Scholar] [CrossRef]
- Launay, K. Analysis of Lagrangian Models for Predicting the Turbulent Particle Dispersion. Ph.D. Thesis, University of Strasbourg, Strasbourg, France, 4 June 1998. [Google Scholar]
- Launay, K.; Huilier, D.; Burnage, H. An improved Lagrangian method for predicting the long-time turbulent dispersion in gas-particle flows. In Proceedings of the ASME Summer Fluids Engineering Meeting, FEDSM98-5012, Washington, DC, USA, 21–25 June 1998. [Google Scholar]
- Launay, K.; Huilier, D.; Burnage, H. Numerical Predictions of the Heavy particles dispersion in a turbulent flow. In Proceedings of the Fourth ECCOMAS Computational Fluid Dynamics Conference, Athens, Greece, 7–11 September 1998; pp. 158–161, ISBN 0-471-98579-1. [Google Scholar]
- Pétrissans, A.; Tanière, A.B.; Oesterlé, B. Effects of Nonlinear Drag and Negative Loop Correlations on Heavy Particle Motion in Isotropic Stationary TurbulenceUsing a New Lagrangian Stochastic Model. Aerosol Sci. Technol. 2002, 36, 963–971. [Google Scholar] [CrossRef] [Green Version]
- Vames, J.S.; Hanratty, T.J. Turbulent dispersion of droplets for air flow in a pipe. Exp. Fluids 1988, 6, 94–104. [Google Scholar] [CrossRef]
- Lee, M.M.; Hanratty, T.J.; Adrian, R.J. An axial viewing photographic technique to study turbulence characteristics of particles. Int. J. Multiph. Flow 1989, 15, 787–802. [Google Scholar] [CrossRef]
- Young, J.B.; Hanratty, T.J. Optical studies on the turbulent motion of solid particles in a pipe flow. J. Fluid Mech. 1991, 231, 665. [Google Scholar] [CrossRef]
- Tsuji, Y.; Morikawa, Y.; Shiomi, H. LDV measurements of an air-solid two-phase flow in a vertical pipe. J. Fluid Mech. 1984, 139, 417. [Google Scholar] [CrossRef]
- Call, C.J.; Kennedy, I.M. Measurements of droplet dispersion in heated and unheated turbulent jets. AIAA J. 1994, 32, 874–875. [Google Scholar] [CrossRef]
- Call, C.J.; Kennedy, I.M. Droplet dispersion in a round turbulent jet. In Proceedings of the 28th Aerospace Sciences Meeting, Reno, NV, USA, 8–11 January 1990; American Institute of Aeronautics and Astronautics Inc., AIAA: Reno, NV, USA, 1990. [Google Scholar] [CrossRef]
- Call, C.J.; Kennedy, I.M. A technique for measuring Lagrangian and Eulerian particle statistics in a turbulent flow. Exp. Fluids 1991, 12, 125–130. [Google Scholar] [CrossRef]
- Call, C.J.; Kennedy, I.M. Measurements and simulations of particle dispersion in a turbulent flow. Int. J. Multiph. Flow 1992, 18, 891–903. [Google Scholar] [CrossRef]
- Hishida, K.; Maeda, M. Application of Laser/Phase Doppler Anemometer to Dispersed Two-Phase Jet Flow. Part. Part. Syst. Charact. 1990, 7, 152–159. [Google Scholar] [CrossRef]
- Hishida, K.; Takemoto, M. Maeda: Turbulence characteristics of gas- solid two-phase confined jet (in Japanese). Jpn. J. Multiph. Flow I 1987, 56–69. [Google Scholar] [CrossRef]
- Hishida, K.; Ando, A.; Maeda, M. Experiments on particle dispersion in a turbulent mixing layer. Int. J. Multiph. Flow 1992, 18, 181–194. [Google Scholar] [CrossRef]
- Sato, Y.; Yamamoto, K. Lagrangian measurement of fluid-particle motion in an isotropic turbulent field. J. Fluid Mech. 1987, 175, 183–199. [Google Scholar] [CrossRef]
- Burnage, H.; Huilier, D. Diffusion of a submicronic Spray in an Homogeneous Turbulent Flow. Aerosol Sci. Technol. 1990, 12, 637–649. [Google Scholar] [CrossRef]
- Launay, K.; Huilier, D.; Burnage, H. Lagrangian simulation of the turbulent dispersion of heavy particles using a Wang and Stock correction. In Proceedings of the 1997 ASME Fluids Engineering Division Summer Meeting: FEDSM ‘97, Vancouver, BC, Canada, 22–26 June 1997. [Google Scholar]
- Launay, K.; Huilier, D.; Burnage, H. Lagrangian predictions of the dispersion of heavy particles in a dilute two-phase flow: On the inertia effect. Mech. Res. Commun. 1998, 25, 251–256. [Google Scholar] [CrossRef]
- Deutsch, E. Dispersion de Particules dans une Turbulence Homogéne, Isotrope, Stationnaire Calculée par Simulation Numérique des Grandes Echelles. Ph.D. Thesis, Ecole Centrale de Lyon, Lyon, France, 1992. [Google Scholar]
- Huang, X.; Stock, D.E.; Wang, L.-P. Using the Monte-Carlo process to simulate two-dimensional heavy particle dispersion. ASME/FED, Gas-Solid Flows 1993, 166, 153–167. [Google Scholar]
- Matt, K.; Huilier, D. Monte-Carlo simulations of turbulent gas-particle dispersion without gravity: Effect of intertia and nonlinear drag. Int. Commun. Heat Mass Transf. 2001, 28, 631–640. [Google Scholar] [CrossRef]
- Obukhov, A.M. Description of Turbulence in Terms of Lagrangian Variables in Advances in Geophysics. Adv. Geophys. 1959, 113–116. [Google Scholar] [CrossRef]
- Nijkamp, E.L. A Dance with the Langevin Equation. Master’s Thesis, University of California, Los Angeles, CA, USA, 2018. Available online: https://escholarship.org/uc/item/1j04685x (accessed on 6 July 2020).
- Tabar, M.R.R. Equivalence of Langevin and Fokker–Planck Equations. In Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems. Understanding Complex Systems; Springer: Cham, Switzerland, 2019; Available online: https://doi.org/10.1007/978-3-030-18472-8_7 (accessed on 6 July 2020).
- Wilson, J.D.; Thurtell, G.W.; Kidd, G.E. Numerical simulation of particle trajectories in inhomogeneous turbulence, I: Systems with constant turbulent velocity scale. Bound. Layer Meteorol. 1981, 21, 295–313. [Google Scholar] [CrossRef]
- Wilson, J.D.; Thurtell, G.W.; Kidd, G.E. Numerical simulation of particle trajectories in inhomogeneous turbulence, II: Systems with variable turbulent velocity scale. Bound. Layer Meteorol. 1981, 21, 423–441. [Google Scholar] [CrossRef]
- Sawford, B.L. The basis for, and some limitations of, the Langevin equation in atmospheric relative dispersion modelling. Atmos. Environ. 1984, 18, 2405–2411. [Google Scholar] [CrossRef]
- Van Dop, H.; Nieuwstadt, F.T.M.; Hunt, J.C.R. Random walk models for particle displacements in inhomogeneous unsteady turbulent flows. Phys. Fluids 1985, 28, 1639–1653. [Google Scholar] [CrossRef]
- Haworth, D.C.; Pope, S.B. A generalized Langevin model for turbulent flows. Phys. Fluids 1986, 29, 387. [Google Scholar] [CrossRef]
- Pope, S.B. PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 1985, 11, 119–192. [Google Scholar] [CrossRef]
- Pope, S.B. Consistency conditions for random-walk models of turbulent dispersion. Phys. Fluids 1987, 30, 2374. [Google Scholar] [CrossRef]
- Haworth, D.C.; Pope, S.B. A pdf modeling study of self-similar turbulent free shear flows. Phys. Fluids 1987, 30, 1026. [Google Scholar] [CrossRef]
- Simonin, O.; Deutsch, E.; Minier, J.P. Eulerian prediction of the fluid/particle correlated motion in turbulent two-phase flows. Appl. Sci. Res. 1993, 51, 275–283. [Google Scholar] [CrossRef]
- Minier, J.-P.; Peirano, E. The pdf approach to turbulent polydispersed two-phase flow. Phys. Rep. 2001, 352, 1–214. [Google Scholar] [CrossRef]
- Peirano, E.S.; Chibbaro, J. Pozorski, J.; Minier, J.P. Mean-field/PDF numerical approach for polydispersed turbulent two-phase flows, Prog. Energy Combust. Sci. 2006, 32, 315–371. [Google Scholar] [CrossRef] [Green Version]
- Minier, J.P.; Peirano, E.; Chibbaro, S. PDF model based on Langevin equation for polydispersed two-phase flows applied to a bluff-body gas-solid flow. Phys. Fluids 2004, 16, 2419–2431. [Google Scholar] [CrossRef] [Green Version]
- Chibbaro, S.; Minier, J.P. Stochastic modelling of polydisperse turbulent two-phase flows. In Stochastic Methods in Fluid Mechanics. CISM International Centre for Mechanical Sciences; Chibbaro, S., Minier, J., Eds.; Springer: Vienna, Austria, 2014; p. 548. [Google Scholar] [CrossRef]
- Minier, J.-P. Statistical descriptions of polydisperse turbulent two-phase flows. Phys. Rep. 2016, 665, 1–122. [Google Scholar] [CrossRef]
- Reeks, M.W. On a kinetic equation for the transport of particles in turbulent flows. Phys. Fluids A Fluid Dyn. 1991, 3, 446–456. [Google Scholar] [CrossRef]
- Reeks, M.W. On the constitutive relations for dispersed particles in nonuniform flows. I: Dispersion in a simple shear flow. Phys. Fluids A Fluid Dyn. 1993, 5, 750–761. [Google Scholar] [CrossRef]
- Reeks, M.W. On the continuum equations for dispersed particles in nonuniform flows. Phys. Fluids A Fluid Dyn. 1992, 4, 1290–1303. [Google Scholar] [CrossRef]
- Reeks, M.W. Transport, Mixing and Agglomeration of Particles in Turbulent Flows. Flow Turbul. Combust. 2014, 92, 3–25. [Google Scholar] [CrossRef]
- Reeks, M.; Simonin, O.; Fede, P. Models for Particle Transport, Mixing and Collisions in Turbulent Gas in Multiphase Flow Handbook, 2nd ed.; Michaelides, E., Crowe, C.T., Schwarzkopf, J.D., Eds.; CRC Press: Boca Raton, FL, USA, 2017; pp. 144–202. [Google Scholar]
- Minier, J.-P.; Profeta, C. Kinetic and dynamic probability-density-function descriptions of disperse turbulent two-phase flows. Phys. Rev. E 2015, 92. [Google Scholar] [CrossRef]
- Reeks, M.; Swailes, D.C.; Bragg, A.D. Is the kinetic equation for turbulent gas-particle flows ill posed? Phys. Rev. E 2018, 97. [Google Scholar] [CrossRef] [Green Version]
- Zhong, D.-Y.; Wang, G.-Q.; Zhang, M.-X.; Li, T.-J. Kinetic equation for particle transport in turbulent flows. Phys. Fluids 2020, 32, 073301. [Google Scholar] [CrossRef]
- Wang, Q.; Squires, K.D. Large eddy simulation of particle-laden turbulent channel flow. Phys. Fluids 1996, 8, 1207–1223. [Google Scholar] [CrossRef]
- Fukagata, K.; Zahrai, S.; Bark, F.H. Dynamics of Brownian particles in a turbulent channel flow. Heat Mass Transf. 2004, 40, 715–726. [Google Scholar] [CrossRef]
- Shotorban, B.; Mashayek, F. A stochastic model for particle motion in large-eddy simulation. J. Turbul. 2006, 7, N18. [Google Scholar] [CrossRef]
- Berrouk, A.S.; Douce, A.; Laurence, D.; Riley, J.J.; Stock, D.E. RANS and LES of Particle Dispersion in Turbulent Pipe Flow: Simulations Versus Experimental Results. In Proceedings of the ASME 2006 2nd Joint U.S.-European Fluids Engineering Summer Meeting Collocated with the 14th International Conference on Nuclear Engineering, Miami, FL, USA, 17–20 July 2006; Volume 1: Symposia, Parts A and B. [Google Scholar] [CrossRef]
- Berrouk, A.; Laurence, D.; Riley, J.; Stock, D. Stochastic modelling of inertial particle dispersion by subgrid motion for LES of high Reynolds number pipe flow. J. Turbul. 2007, 8, N50. [Google Scholar] [CrossRef]
- Berrouk, A.S.; Stock, D.E.; Laurence, D.; Riley, J.J. Heavy particle dispersion from a point source in turbulent pipe flow. Int. J. Multiph. Flow 2008, 34, 916–923. [Google Scholar] [CrossRef]
- Weil, J.C.; Sullivan, P.P.; Moeng, C.-H. The Use of Large-Eddy Simulations in Lagrangian Particle Dispersion Models. J. Atmos. Sci. 2004, 61, 2877–2887. [Google Scholar] [CrossRef]
- Amiri, A.E.; Hannani, S.K.; Mashayek, F. Large-Eddy Simulation of Heavy-Particle Transport in Turbulent Channel Flow. Numer. Heat Transf. Part B Fundam. 2006, 50, 285–313. [Google Scholar] [CrossRef]
- Kuerten, J.G.M.; Vreman, A.W. Can turbophoresis be predicted by large-eddy simulation? Phys. Fluids 2005, 17, 011701. [Google Scholar] [CrossRef] [Green Version]
- Kuerten, J.G.M. Subgrid modeling in particle-laden channel flow. Phys. Fluids 2006, 18, 025108. [Google Scholar] [CrossRef] [Green Version]
- Pozorski, J.; Apte, S. Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion. Int. J. Multiph. Flow 2009, 35, 118–128. [Google Scholar] [CrossRef]
- Fede, P.; Simonin, O.; Villedieu, P.; Squires, K. Stochastic Modeling of the Turbulent Subgrid Fluid Velocity along Inertial Particle Trajectories. In Proceedings of the Summer Program, Center for Turbulence Research, Stanford, CA, USA; 2006; pp. 247–258. Available online: https://web.stanford.edu/group/ctr/ctrsp06/fede.pdf (accessed on 7 April 2021).
- Bini, M.; Jones, W.P. Large-eddy simulation of particle-laden turbulent flows. J. Fluid Mech. 2008, 614, 207–252. [Google Scholar] [CrossRef] [Green Version]
- Gobert, C. Analytical assessment of models for large eddy simulation of particle laden flow. J. Turbul. 2010, 11, 1–24. [Google Scholar] [CrossRef] [Green Version]
- Vinkovic, I.; Aguirre, C.; Ayrault, M.; Simoëns, S. Large-eddy Simulation of the Dispersion of Solid Particles in a Turbulent Boundary Layer. Bound. Layer Meteorol. 2006, 121, 283–311. [Google Scholar] [CrossRef]
- Khan, M.; Luo, X.; Nicolleau, F.; Tucker, P.; Iacono, G. Effects of LES sub-grid flow structure on particle deposition in a plane channel with a ribbed wall. Int. J. Numer. Methods Biomed. Eng. 2010, 26, 999–1015. [Google Scholar] [CrossRef]
- Gobert, C.; Manhart, M. Subgrid modelling for particle-LES by spectrally optimised interpolation (SOI). J. Comput. Phys. 2011, 230, 7796–7820. [Google Scholar] [CrossRef]
- Chibbaro, S.; Minier, J. The FDF or LES/PDF method for turbulent two-phase flows. J. Phys. Conf. Ser. 2011, 318, 042049. [Google Scholar] [CrossRef]
- Michalek, W.; Kuerten, J.; Zeegers, J.; Liew, R.; Pozorski, J.; Geurts, B. A hybrid stochastic deconvolution model for large-eddy simulation of particle-laden flow. Phys. Fluids 2013, 25, 123302. [Google Scholar] [CrossRef] [Green Version]
- Gobert, C.; Manhart, M. A priori and a posteriori analysis of models for large-eddy simulation of particle-laden flow. Phys. Fluid Dyn. 2010, 1004, 1–18. [Google Scholar]
- Cernick, M.J.; Tullis, S.W.; Lightstone, M.F. Particle subgrid scale modelling in large-eddy simulations of particle-laden turbulence. J. Turbul. 2015, 16, 101–135. [Google Scholar] [CrossRef]
- Marchioli, C.; Soldati, A.; Salvetti, M.V.; Kuerten, J.G.M.; Konan, A.; Fede, P.; Portela, L.M. Benchmark Test on Particle-Laden Channel Flow with Point-Particle LES. In Direct and Large-Eddy Simulation VIII. 2011, 177. Available online: https://doi.org/10.1007/978-94-007-2482-2_28 (accessed on 6 April 2021).
- Marchioli, C. Large-eddy simulation of turbulent dispersed flows: A review of modelling approaches. Acta Mech. 2017, 228, 741–771. [Google Scholar] [CrossRef] [Green Version]
- Caporaloni, M.; Tampieri, F.; Trombetti, F.; Vittori, O. Transfer of Particles in Nonisotropic Air Turbulence. J. Atmos. Sci. 1975, 32, 565–568. [Google Scholar] [CrossRef] [2.0.CO;2" target='_blank'>Green Version]
- Reeks, M.W. The transport of discrete particles in inhomogeneous turbulence. J. Aerosol Sci. 1983, 14, 729–739. [Google Scholar] [CrossRef]
- Crowe, C.T.; Gore, R.A.; Troutt, T.R. Particle dispersion by coherent structures in free shear flows. Part. Sci. Technol. 1985, 3, 149–158. [Google Scholar] [CrossRef]
- Crowe, C.T.; Chung, J.N.; Troutt, T.R. Particle mixing in free shear flows. Prog. Energy Combust. Sci. 1988, 14, 171–194. [Google Scholar] [CrossRef]
- Crowe, C.T.; Chung, J.N. Troutt, Particle Dispersion Models and Drag Coefficients for Particles in Turbulent Flows. Available online: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890001787.pdf (accessed on 6 April 2021).
- Chein, R.; Chung, J.N. Simulation of particle dispersion in a two-dimensional mixing layer. Aiche J. 1988, 34, 946–954. [Google Scholar] [CrossRef]
- Squires, K.D.; Eaton, J.K. Preferential concentration of particles by turbulence. Phys. Fluids A Fluid Dyn. 1991, 3, 1169–1178. [Google Scholar] [CrossRef]
- Eaton, J.K.; Fessler, J.R. Preferential concentration of particles by turbulence. Int. J. Multiph. Flow 1994, 20, 169–209. [Google Scholar] [CrossRef]
- Fessler, J.R.; Kulick, J.D.; Eaton, J.K. Preferential concentration of heavy particles in a turbulent channel flow. Phys. Fluids 1994, 6, 3742–3749. [Google Scholar] [CrossRef]
- Wood, A.M.; Hwang, W.; Eaton, J.K. Preferential concentration of particles in homogeneous and isotropic turbulence. Int. J. Multiph. Flow 2005, 31, 1220–1230. [Google Scholar] [CrossRef]
- Rouson, D.W.I.; Eaton, J.K. On the preferential concentration of solid particles in turbulent channel flow. J. Fluid Mech. 2001, 428, 149–169. [Google Scholar] [CrossRef]
- Aliseda, A.; Cartellier, A.; Hainaux, A.; Lasheras, J.C. Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 2002, 468, 77–105. [Google Scholar] [CrossRef] [Green Version]
- Obligado, M.; Missaoui, M.; Monchaux, R.; Cartellier, A.; Bourgoin, M. Reynolds number influence on preferential concentration of heavy particles in turbulent flows. J. Phys. Conf. Ser. 2011, 318, 052015. [Google Scholar] [CrossRef]
- Monchaux, R.; Bourgoin, M.; Cartellier, A. Analyzing preferential concentration and clustering of inertial particles in turbulence. Int. J. Multiph. Flow 2012, 40, 1–18. [Google Scholar] [CrossRef]
- Obligado, M.; Teitelbaum, T.; Cartellier, A.; Mininni, P.; Bourgoin, M. Preferential concentration of heavy particles in turbulence. J. Turbul. 2014, 15, 293–310. [Google Scholar] [CrossRef] [Green Version]
- Falkinhoff, F.; Obligado, M.; Bourgoin, M.; Mininni, P.D. Preferential Concentration of Free-Falling Heavy Particles in Turbulence. Phys. Rev. Lett. 2020, 125. [Google Scholar] [CrossRef]
- Sumbekova, S.; Aliseda, A.; Cartellier, A.; Bourgoin, M. Clustering and Settling of Inertial Particles in Turbulence. In Proceedings of the 5th International Conference on Jets, Wakes and Separated Flows (ICJWSF2015), Stockholm, Sweden, 15–18 June 2015; pp. 475–482. [Google Scholar] [CrossRef]
- Sumbekova, S.; Cartellier, A.; Aliseda, A.; Bourgoin, M. Preferential concentration of inertial sub-kolmogorov particles. The roles of mass loading of particles, Stokes and Reynolds numbers. Phys. Rev. Fluids 2016, 2. [Google Scholar] [CrossRef] [Green Version]
- Obligado Cartellier, A.; Aliseda, A.; Calmant, T.; De Palma, N. Study on preferential concentration of inertial particles in homogeneous isotropic turbulence via Big-Data techniques. Phys. Rev. Fluids 2020, 5, 024303. [Google Scholar] [CrossRef] [Green Version]
- Goto, S.; Vassilicos, J.C. Sweep-Stick Mechanism of Heavy Particle Clustering in Fluid Turbulence. Phys. Rev. Lett. 2008, 100. [Google Scholar] [CrossRef] [Green Version]
- Coleman, S.W.; Vassilicos, J.C. A unified sweep-stick mechanism to explain particle clustering in two- and three-dimensional homogeneous, isotropic turbulence. Phys. Fluids 2009, 21, 113301. [Google Scholar] [CrossRef] [Green Version]
- Kaimal, J.C.; Finnigan, J.J. Atmospheric Boundary Layer Flows: Their Structure and Measurement; Oxford University Press: Oxford, UK, 1994. [Google Scholar]
- Maxey, M.R. Simulation Methods for Particulate Flows and Concentrated Suspensions. Annu. Rev. Fluid Mech. 2017, 49, 171–193. [Google Scholar] [CrossRef]
- Geurts, B.J.; Clercx, H.; Uijttewaal, W. ; Particle-Laden Flow, from Geophysical to Kolmogorov Scales, Ercoftac Series; Springer: Berlin/Heidelberg, Germany, 2007; Volume 11. [Google Scholar]
- Loth, E. Numerical approaches for motion of dispersed particles, droplets and bubbles. Prog. Energy Combust. Sci. 2000, 26, 161–223. [Google Scholar] [CrossRef]
- Soo, S.L. Multiphase Fluid Dynamics; Science Press: London, UK; New York, NY, USA, 1990. [Google Scholar]
- Soo, S.L. Fluid Dynamics of Multiphase Systems; Blaisdell Pub. Co.: Waltham, MA, USA, 1967. [Google Scholar]
- Yeo, G.H.; Tun, J. Computational Techniques for Multiphase Flows; Elsevier: Amsterdam, The Netherlands, 2009. [Google Scholar]
- Stock, D.E. Particle Dispersion in Flowing Gases—1994 Freeman Scholar Lecture. ASME J. Fluids Eng. 1996, 118, 4. [Google Scholar] [CrossRef]
- Balachandar, S.; Eaton, J.K. Turbulent Dispersed Multiphase Flow. Annu. Rev. Fluid Mech. 2010, 42, 111–133. [Google Scholar] [CrossRef]
- Toschi, F.; Bodenschatz, E. Lagrangian Properties of Particles in Turbulence. Annu. Rev. Fluid Mech. 2009, 41, 375–404. [Google Scholar] [CrossRef]
- Biferale, L.; Bodenschatz, E.; Cencini, M.; Lanotte, A.S.; Ouellette, N.T.; Toschi, F.; Xu, H. Lagrangian structure functions in turbulence: A quantitative comparison between experiment and direct numerical simulation. Phys. Fluids 2008, 20, 065103. [Google Scholar] [CrossRef] [Green Version]
- Richardson, L.F. Atmospheric Diffusion Shown on a Distance-Neighbour Graph. Proceedings of the Royal Society A: Mathematical. Phys. Eng. Sci. 1926, 110, 709–737. [Google Scholar] [CrossRef] [Green Version]
- Batchelor, G.K. The application of the similarity theory of turbulence to atmospheric diffusion. Q. J. R. Meteorol. Soc. 1950, 76, 133–146. [Google Scholar] [CrossRef]
- Biferale, L.; Boffetta, G.; Celani, A.; Devenish, B.J.; Lanotte, A.; Toschi, F. Lagrangian statistics of particle pairs in homogeneous isotropic turbulence. Phys. Fluids 2005, 17, 115101. [Google Scholar] [CrossRef]
- Ouellette, N.T.; Xu, H.; Bourgoin, M.; Bodenschatz, E. An experimental study of turbulent relative dispersion models. New J. Phys. 2006, 8, 109. [Google Scholar] [CrossRef]
- Bourgoin, M.; Ouellette, N.T.; Xu, H.; Berg, J.; Bodenschatz, E. The role of pair dispersion in turbulent flow. Science 2006, 311, 835–838. [Google Scholar] [CrossRef] [Green Version]
- Sawford, B. Turbulent relative dispersion. Annu. Rev. Fluid Mech. 2001, 33, 289–317. [Google Scholar] [CrossRef]
- Salazar, J.P.L.C.; Collins, L.R. Two-Particle Dispersion in Isotropic Turbulent Flows. Annu. Rev. Fluid Mech. 2009, 41, 405–432. [Google Scholar] [CrossRef] [Green Version]
- Rani, S.; Gupta, V.; Koch, D. Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 1. Drift and diffusion flux closures. J. Fluid Mech. 2019, 871, 450–476. [Google Scholar] [CrossRef] [Green Version]
- Rani, S.L.; Dhariwal, R.; Koch, D.L. A stochastic model for the relative motion of high Stokes number particles in isotropic turbulence. J. Fluid Mech. 2014, 756, 870–902. [Google Scholar] [CrossRef]
- Dhariwal, R.; Rani, S.L.; Koch, D.L. Stochastic theory and direct numerical simulations of the relative motion of high-inertia particle pairs in isotropic turbulence. J. Fluid Mech. 2017, 813, 205–249. [Google Scholar] [CrossRef]
- Zaichik, L.I.; Alipchenkov, V.M. Pair dispersion and preferential concentration of particles in isotropic turbulence. Phys. Fluids 2003, 15, 1776. [Google Scholar] [CrossRef]
Case | Dispersion Regime | ||
---|---|---|---|
1 | yes | yes | Large structure dispersion with high-frequency cut-off and turbulence modification, inertia and CTE-CE if drift |
2 | no | yes | Large structure dispersion without high-frequency cut-off, turbulence modification, CTE-CE |
3 | yes | no | Small and large structure influence with damped particle response to high-frequency fluctuationsInertia effect (IE) and CTE-CE if drift |
4 | no | no | Turbulent diffusion |
10 | 0.003 | 0.31 | 0.31 | 0.0017 | 0.0017 | 0.023 | 0.002 |
20 | 0.012 | 1.23 | 1.24 | 0.0068 | 0.0068 | 0.093 | 0.017 |
50 | 0.07 | 7.35 | 7.75 | 0.042 | 0.0404 | 0.55 | 0.25 |
70 | 0.135 | 13.7 | 15.2 | 0.084 | 0.0753 | 1.3 | 0.65 |
100 | 0.25 | 25 | 31 | 0.170 | 0.137 | 1.95 | 1.7 |
120 | 0.34 | 34 | 44.64 | 0.245 | 0.187 | 2.57 | 2.8 |
150 | 0.47 | 48 | 69.75 | 0.38 | 0.26 | 3.62 | 4.9 |
200 | 0.710 | 72 | 124 | 0.68 | 0.39 | 5.42 | 9.8 |
250 | 0.944 | 96 | 194 | 1.06 | 0.52 | 7.21 | 16.3 |
400 | 1.61 | 164 | 496 | 2.72 | 0.90 | 12.3 | 47 |
500 | 2.02 | 205 | 775 | 4.26 | 1.13 | 15.4 | 74 |
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Huilier, D.G.F. An Overview of the Lagrangian Dispersion Modeling of Heavy Particles in Homogeneous Isotropic Turbulence and Considerations on Related LES Simulations. Fluids 2021, 6, 145. https://doi.org/10.3390/fluids6040145
Huilier DGF. An Overview of the Lagrangian Dispersion Modeling of Heavy Particles in Homogeneous Isotropic Turbulence and Considerations on Related LES Simulations. Fluids. 2021; 6(4):145. https://doi.org/10.3390/fluids6040145
Chicago/Turabian StyleHuilier, Daniel G. F. 2021. "An Overview of the Lagrangian Dispersion Modeling of Heavy Particles in Homogeneous Isotropic Turbulence and Considerations on Related LES Simulations" Fluids 6, no. 4: 145. https://doi.org/10.3390/fluids6040145