Abstract
Salient features of single-phase turbulent flow modelling are recalled first, including the closure problem, the statistical RANS models, the Lagrangian stochastic approach (one-point PDF method) together with its extension for near-wall turbulence, and the basics of Large-Eddy simulation (LES). In the second part of the chapter, two-phase dispersed turbulent flows in the Eulerian-Lagrangian approach are addressed. The issue of turbulent dispersion in RANS is succintly presented. Then, the subfilter dispersion in LES is discussed at length; functional and structural models are described, and some recent ideas about closures in terms of stochastic diffusion processes are discussed. Examples of computational results are presented for homogeneous isotropic and wall-bounded turbulence. At last, a specific modelling study of particle-laden channel flow is recalled where a low-order dynamical system with a reduced number of fluid velocity modes is constructed.
Keywords
- Probability Density Function
- Proper Orthogonal Decomposition
- Smooth Particle Hydrodynamic
- Proper Orthogonal Decomposition Mode
- Reynolds Stress Model
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aidun, C. K., & Clausen, J. R. (2010). Lattice-Boltzmann method for complex flows. Annual Review of Fluid Mechanics, 42, 439–472.
Allery, C., Béghein, C., Wacławczyk, M., & Pozorski, J. (2014). Application of POD-based dynamical systems to dispersion and deposition of particles in turbulent channel flow. International Journal of Multiphase Flow, 58, 97–113.
Apte, S. V., Mahesh, K., Moin, P., & Oefelein, J. C. (2003). Large-eddy simulation of swirling particle-laden flows in a coaxial-jet combustor. International Journal of Multiphase Flow, 29, 1311–1331.
Armenio, V., Piomelli, U., & Fiorotto, V. (1999). Effect of the subgrid scales on particle motion. Physics of Fluids, 11, 3030–3042.
Aubry, N., Holmes, P., Lumley, J. L., & Stone, E. (1988). The dynamics of coherent structures in the wall region of turbulent boundary layer. Journal of Fluid Mechanics, 192, 115–173.
Babler, M. U., Biferale, L., Brandt, L., Feudel, U., Guseva, K., Lanotte, A.S., et al. (2015). Numerical simulations of aggregate breakup in bounded and unbounded turbulent flows. Journal of Fluid Mechanics, 766, 104–128.
Balachandar, S., & Eaton, J. (2010). Turbulent dispersed multiphase flow. Annual Review of Fluid Mechanics, 42, 111–133.
Bianco, F., Chibbaro, S., Marchioli, C., Salvetti, M. V., & Soldati, A. (2012). Intrinsic filtering errors of Lagrangian particle tracking in LES flow fields. Physics of Fluids, 24, art. 045103.
Brennen, C. E. (2005). Fundamentals of multiphase flow. Cambridge: Cambridge University Press.
Burton, G. C., & Dahm, W. J. A. (2005). Multifractal subgrid-scale modeling for large-eddy simulation. I. Model development and a priori testing. Physics of Fluids, 17, art. 075111.
Casey, M., & Wintergerste, T. (Eds.). (2000). Best practice guidelines: quality and trust in industrial CFD, ERCOFTAC.
Colucci, P. J., Jaberi, F. A., Givi, P., & Pope, S. B. (1998). The filtered density function for large-eddy simulation of turbulent reactive flows. Physics of Fluids, 10, 499–515.
Crowe, C., Sommerfeld, M., & Tsuji, T. (1998). Multiphase flows with droplets and particles. New York: CRC Press.
Dreeben, T. D., & Pope, S. B. (1997). Wall-function treatment in PDF methods for turbulent flows. Physics of Fluids, 9, 2692–2703.
Dreeben, T. D., & Pope, S. B. (1998). PDF/Monte Carlo simulation of near-wall turbulent flows. Journal of Fluid Mechanics, 357, 141–166.
Duan, G., & Chen, B. (2015). Large Eddy Simulation by particle method coupled with Sub-Particle-Scale model and application to mixing layer flow. Applied Mathematical Modelling, 39, 3135–3149.
Eaton, J., & Fessler, J.R. (1994). Preferential concentration of particles by turbulence. International Journal of Multiphase Flow, 20, Suppl., 169–209.
Ernst, M., Dietzel, M., & Sommerfeld, M. (2013). LBM for simulating transport and agglomeration of resolved particles. Acta Mechanica, 224, 2425.
Fede, P., & Simonin, O. (2006). Numerical study of the subgrid turbulence effects on the statistics of heavy colliding particles. Physics of Fluids, 17, art. 045103.
Fede, P., Simonin, O., Villedieu, P., & Squires, K. D. (2006). Stochastic modelling of the turbulent subgrid fluid velocity along inertial particle trajectories. In Proceedings of the Summer Program: Center for Turbulence Research, Stanford University, (pp. 247–258).
Gardiner, C. W. (1990). Handbook of stochastic methods for physics, chemistry and the natural sciences (2nd ed.). Berlin: Springer.
Gatski, T. B., Hussaini, M. Y., & Lumley, J. L. (Eds.). (1996). Simulation and modeling of turbulent flows. Oxford University Press.
Geurts, B. J., & Kuerten, J. G. M. (2012). Ideal stochastic forcing for the motion of particles in large-eddy simulation extracted from direct numerical simulation of turbulent channel flow. Physics of Fluids, 24, art. 081702.
Gicquel, L. Y. M., Givi, P., Jaberi, F. A., & Pope, S. B. (2002). Velocity filtered density function for large eddy simulation of turbulent flows. Physics of Fluids, 14, 1196–1213.
Grabowski, W. W., & Wang, L.-P. (2013). Growth of cloud droplets in a turbulent environment. Annual Review of Fluid Mechanics, 45, 293–324.
Guha, A. (2008). Transport and deposition of particles in turbulent and laminar flow. Annual Review of Fluid Mechanics, 40, 311–341.
Gustavsson, K., & Mehlig, B. (2016). Statistical models for spatial patterns of heavy particles in turbulence. Advances in Physics, 65, 1–57.
Haworth, D. C. (2010). Progress in probability density function methods for turbulent reacting flows. Progress in Energy and Combustion Science, 36, 168–259.
Henry, C., Minier, J.-P., Mohaupt, M., Profeta, C., Pozorski, J., & Tanière, A. (2014). A stochastic approach for the simulation of collisions between colloidal particles at large time steps. International Journal of Multiphase Flow, 61, 94–107.
Hoyas, S., & Jimenez, J. (2006). Scaling of the velocity fluctuations in turbulent channels up to \(Re_\tau =2003\). Physics of Fluids, 18, art. 011702.
Jenny, P., Roekaerts, D., & Beishuizen, N. (2012). Modeling of turbulent dilute spray combustion. Progress in Energy and Combustion Science, 38, 846–887.
Jin, B., Potts, I., & Reeks, M. W. (2015). A simple stochastic quadrant model for the transport and deposition of particles in turbulent boundary layers. Physics of Fluids, 27, art. 053305.
Johansson, A. V. (2002). Engineering turbulence models and their development. In Oberlack, M., & Busse, F. H. (Eds.) Theories of Turbulence. CISM Courses and Lectures (Vol. 442). Springer.
Kajzer, A., Pozorski, J., & Szewc, K. (2014). Large-eddy simulations of 3D Taylor-Green vortex: Comparison of smoothed particle hydrodynamics, lattice Boltzmann and finite volume methods. Journal of Physics: Conference Series, 530, art. 012019.
Karlin, S. (1966). A first course in stochastic processes. New York: Academic Press.
Khan, M. A. I., Luo, X. Y., Nicolleau, F. C. G. A., Tucker, P. G., & Lo, Iacono G. (2010). Effects of LES sub-grid flow structure on particle deposition in a plane channel with a ribbed wall. International Journal for Numerical Methods in Biomedical Engineering, 26, 999–1015.
Kim, J., Moin, P., & Moser, R. (1987). Turbulence statistics in fully developed channel flow at low Reynolds number. Journal of Fluid Mechanics, 177, 133–166.
Kloeden, P. E., & Platen, E. (1992). Numerical solution of stochastic differential equations. Springer.
Knorps, M., & Pozorski, J. (2015). An inhomogeneous stochastic subgrid scale model for particle dispersion in Large-Eddy Simulation. In Fröhlich, J. et al. (Eds.) Direct and Large-Eddy simulation (Vol IX, pp. 671–678). Springer.
Kuerten, J. G. M. (2006). Subgrid modeling in particle-laden channel flows. Physics of Fluids, 18, art. 025108.
Launder, B. E., & Sandham, N. D. (Eds.). (2002). Closure strategies for turbulent and transitional flows. Cambridge University Press.
Lovecchio, S., Zonta, F., & Soldati, A. (2014). Influence of thermal stratification on the surfacing and clustering of floaters in free surface turbulence. Advances in Water Resources, 72, 22–31.
Lozano-Duran, A., & Jimenez, J. (2014). Effect of the computational domain on direct simulations of turbulent channels up to \(Re_\tau =4200\). Physics of Fluids, 26, art. 011702.
Lundgren, T. S. (1967). Distribution functions in the statistical theory of turbulence. Physics of Fluids, 10, 969–975.
Łuniewski, M., Kotula, P., & Pozorski, J. (2012). Large-eddy simulations of particle-laden turbulent jets. TASK Quarterly, 16, 33–51.
Manceau, R. (2015). Recent progress in the development of the Elliptic Blending Reynolds-stress model. International Journal of Heat and Fluid Flow, 51, 195–220.
Manceau, R., & Hanjalić, K. (2002). Elliptic blending model: a new near-wall Reynolds-stress turbulence closure. Physics of Fluids, 14, 744–754.
Marchioli, C., Armenio, V., & Soldati, A. (2007). Simple and accurate scheme for fluid velocity interpolation for Eulerian-Lagrangian computation of dispersed flows in 3D curvilinear grids. Computers & Fluids, 36, 1187–1198.
Marchioli, C., Salvetti, M. V., & Soldati, A. (2008). Appraisal of energy recovering sub-grid scale models for large-eddy simulation of turbulent dispersed flows. Acta Mechanica, 201, 277–296.
Marchioli, C., & Soldati, A. (2002). Mechanisms for particle transfer and segregation in turbulent boundary layer. Journal of Fluid Mechanics, 468, 283–315.
Marchioli, C., Soldati, A., Kuerten, J. G. M., Arcen, B., Tanière, A., Goldensoph, G., et al. (2008). Statistics of particle dispersion in direct numerical simulations of wall-bounded turbulence: Results of an international collaborative benchmark test. International Journal of Multiphase Flow, 34, 879–893.
Maxey, M. R. (1987). The motion of small spherical particles in a cellular flow field. Physics of Fluids, 30, 1915–1928.
Maxey, M. R., & Riley, J. J. (1983). Equation of motion for a small rigid sphere in a nonuniform flow. Physics of Fluids, 26, 883–889.
Mayrhofer, A., Laurence, D., Rogers, B. D., & Violeau, D. (2015). DNS and LES of 3-D wall-bounded turbulence using Smoothed Particle Hydrodynamics. International Journal of Heat and Fluid Flow, 51, 195–220.
McComb, W. D. (1990). The physics of fluid turbulence. Oxford: Clarendon Press.
Michałek, W. R., Kuerten, J. G. M., Liew, R., Zeegers, C. H., Pozorski, J., & Geurts, B. J. (2013). A hybrid deconvolution stochastic model for LES of particle-laden flow. Physics of Fluids, 25, art. 123202.
Minier, J.-P. (2015). On Lagrangian stochastic methods for turbulent polydisperse two-phase reactive flows. Progress in Energy and Combustion Science, 50, 1–62.
Minier, J.-P., & Chibbaro, S., (Eds.). (2014). Stochastic methods in fluid mechanics. CISM Courses and Lectures (Vol. 548). Springer.
Minier, J.-P., Chibbaro, S., & Pope, S.B. (2014). Guidelines for the formulation of Lagrangian stochastic models for particle simulations of single-phase and dispersed two-phase turbulent flows. Physics of Fluids, 26, art. 113303.
Minier, J.-P., & Peirano, E. (2001). The PDF approach to turbulent polydispersed two-phase flows. Physics Reports, 352, 1–214.
Minier, J.-P., & Pozorski, J. (1997). Propositions for a PDF model based on fluid particle acceleration. In Hanjalić, K., & Peeters, T. W. J. (Eds.) Turbulence, Heat and Mass Transfer (Vol. 2, pp. 771–778). Delft University Press.
Minier, J.-P., & Pozorski, J. (1999). Wall boundary conditions in PDF methods and application to a turbulent channel flow. Physics of Fluids, 11, 2632–2644.
Minier, J.-P., & Profeta, C. (2015). Kinetic and dynamic probability-density-function descriptions of disperse two-phase turbulent flows. Physical Review E, 92, art. 53020.
Monchaux, R., Bourgoin, M., & Cartellier, A. (2012). Analyzing preferential concentration and clustering of inertial particles in turbulence. International Journal of Multiphase Flow, 40, 1–18.
Moser, R. D., Kim, J., & Mansour, N. N. (1999). Direct numerical simulation of turbulent channel flow up to \(Re_\tau =590\). Physics of Fluids, 11, 943–945.
Peirano, E., Chibbaro, S., Pozorski, J., & Minier, J.-P. (2006). Mean-field/PDF numerical approach for polydispersed turbulent two-phase flows. Progress in Energy and Combustion Science, 32, 315–371.
Piomelli, U., & Balaras, E. (2002). Wall-layer models for Large-Eddy Simulations. Annual Review of Fluid Mechanics, 34, 349–374.
Pope, S. B. (2000). Turbulent flows. Cambridge University Press.
Pope, S. B. (2002). A stochastic Lagrangian model for acceleration in turbulent flows. Physics of Fluids, 14, 2360–2375.
Pozorski, J. (2004). Stochastic modelling of turbulent flows. Zeszyty Naukowe IMP PAN 536/1495, Gdańsk.
Pozorski, J., & Apte, S. V. (2009). Filtered particle tracking in isotropic turbulence and stochastic modelling of subgrid-scale dispersion. International Journal of Multiphase Flow, 35, 118–128.
Pozorski, J., Knorps, M., & Łuniewski, M. (2011). Effects of subfilter velocity modelling on dispersed phase in LES of heated channel flow. Journal of Physics: Conference Series, 333, art. 012014.
Pozorski, J., Knorps, M., Minier, J.-P., & Kuerten, J. G. M. (2012). Anisotropic stochastic dispersion model for LES of particle-laden turbulent flows. Engineering Turbulence Modelling and Measurements, 9. Thessaloniki, Greece, June 6–8.
Pozorski, J., & Łuniewski, M. (2008). Analysis of SGS particle dispersion model in LES of channel flow. In Meyers, J., Geurts, B., & Sagaut, P. (Eds.), Quality and Reliability of Large-Eddy Simulations (pp. 331–342). Springer.
Pozorski, J., & Minier, J.-P. (1998). On the Lagrangian turbulent dispersion models based on the Langevin equation. International Journal of Multiphase Flow, 24, 913–945.
Pozorski, J., & Minier, J.-P. (1999). PDF modeling of dispersed two-phase turbulent flows. Physical Review E, 59, 855–863.
Pozorski, J., & Minier, J.-P. (2006). Stochastic modelling of conjugate heat transfer in near-wall turbulence. International Journal of Heat and Fluid Flow, 27, 867–877.
Pozorski, J., Sazhin, S., Wacławczyk, M., Crua, C., Kennaird, D., & Heikal, M. (2002). Spray penetration in a turbulent flow. Flow Turbulence and Combustion, 68, 153–165.
Reeks, M. W. (1991). On a kinetic equation for the transport of particles in turbulent flows. Physics of Fluids A, 3, 446–456.
Reeks, M. W. (1992). On the continuum equations for dispersed particles in nonuniform flows. Physics of Fluids A, 4, 1290–1303.
Rosa, B., Parishani, H., Ayala, O., Wang, L.-P., & Grabowski, W. W. (2013). Kinematic and dynamic collision statistics of cloud droplets from high-resolution simulations. New Journal of Physics, 15, art. 045032.
Rosa, B., & Pozorski, J. (2016). Analysis of subfilter effects on inertial particles in forced isotropic turbulence. 9th International Conference on Multiphase Flow. Firenze, Italy, May 22–27.
Scotti, A., & Meneveau, C. (1999). A fractal interpolation model for large eddy simulation of turbulent flows. Physica D, 127, 198–232.
Sobczyk, K. (1991). Stochastic differential equations. Kluwer Academic Publishers.
Soldati, A. (2005). Particles turbulence interactions in boundary layers. ZAMM, 85, 683–699.
Soldati, A., & Marchioli, C. (2009). Physics and modelling of turbulent particle deposition and entrainment: Review of a systematic study. International Journal of Multiphase Flow, 35, 827–839.
Squires, K. D. (2007). Point-particle methods for disperse flows. In Prosperetti, A., & Tryggvason, G. (Eds.) Computational Methods for Multiphase Flow. Cambridge: Cambridge University Press.
Squires, K. D., & Eaton, J. K. (1991). Preferential concentration of particles by turbulence. Physics of Fluids A, 3, 1169–1178.
Subramanian, S. (2013). Lagrangian-Eulerian methods for multiphase flows. Progress in Energy and Combustion Science, 39, 215–245.
Tanière, A., Arcen, B., Oesterlé, B., & Pozorski, J. (2010). Study on Langevin model parameters of velocity in turbulent shear flows. Physics of Fluids, 22, art. 115101.
Tenneti, S., & Subramanian, S. (2014). Particle-resolved direct numerical simulation for gas-solid flow model development. Annual Review of Fluid Mechanics, 46, 199–230.
Traczyk, M., & Knorps, M. (2012). Private communication.
Violeau, D. (2012). Fluid mechanics and the SPH method. Oxford University Press.
Voßkuhle, M., Pumir, A., Lévêque, E., & Wilkinson, M. (2014). Collision rate for suspensions at large Stokes numbers—comparing Navier-Stokes and synthetic turbulence. Journal of Turbulence, 16, 15–25.
Wacławczyk, M., & Pozorski, J. (2002). Two-point velocity statistics and the POD analysis of the near-wall region in a turbulent channel flow. Journal of Theoretical and Applied Mechanics, 40, 895–916.
Wacławczyk, M., & Pozorski, J. (2007). Modelling of near-wall turbulence with large-eddy velocity modes. Journal of Theoretical and Applied Mechanics, 45, 705–724.
Wacławczyk, M., Pozorski, J., & Minier, J.-P. (2004). PDF computation of turbulent flows with a new near-wall model. Physics of Fluids, 16, 1410–1422.
Wacławczyk, M., Pozorski, J., & Minier, J.-P. (2008). New molecular transport model for FDF/LES of turbulence with passive scalar. Flow Turbulence and Combustion, 81, 235–260.
Wang, L.-P., & Maxey, M. R. (1993). Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. Journal of Fluid Mechanics, 256, 27–68.
Yu, W., Vinkovic, I., & Buffat, M. (2016). Acceleration statistics of finite-size particles in turbulent channel flow in the absence of gravity. Flow Turbulence and Combustion, 96, 183–205.
Zamansky, R., Vinkovic, I., & Gorokhovski, M. (2013). Acceleration in turbulent channel flow: Universalities in statistics, subgrid stochasticmodels and application. Journal of Fluid Mechanics, 721, 627–668.
Acknowledgments
I am grateful to my colleagues and Ph.D. students for a common interest in this fascinating subject: Marta Wacławczyk, Mirosław Łuniewski, Maria Knorps and Christophe Henry at IMP Gdańsk, Claudine Béghein and Cyrille Allery at University of La Rochelle, Sourabh Apte at Oregon, Bogdan Rosa at IMGW Warsaw. I am most grateful to Jean-Pierre Minier (Electricité de France R & D, Chatou) for many stimulating discussions and common research on stochastic turbulence modelling over the years. I wish to express my sincere thanks to Professor Hans Kuerten (TU Eindhoven) for common insights and the kind permission to use his DNS code. The research presented here has partly been supported by the National Science Centre (NCN, Poland) through the project 2011/03/B/ST8/05677.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 CISM International Centre for Mechanical Sciences
About this chapter
Cite this chapter
Pozorski, J. (2017). Models of Turbulent Flows and Particle Dynamics. In: Minier, JP., Pozorski, J. (eds) Particles in Wall-Bounded Turbulent Flows: Deposition, Re-Suspension and Agglomeration. CISM International Centre for Mechanical Sciences, vol 571. Springer, Cham. https://doi.org/10.1007/978-3-319-41567-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-41567-3_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-41566-6
Online ISBN: 978-3-319-41567-3
eBook Packages: EngineeringEngineering (R0)