Models of Turbulent Flows and Particle Dynamics

Chapter
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 571)

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.

References

  1. Aidun, C. K., & Clausen, J. R. (2010). Lattice-Boltzmann method for complex flows. Annual Review of Fluid Mechanics, 42, 439–472.MathSciNetCrossRefMATHGoogle Scholar
  2. 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.CrossRefGoogle Scholar
  3. 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.CrossRefMATHGoogle Scholar
  4. Armenio, V., Piomelli, U., & Fiorotto, V. (1999). Effect of the subgrid scales on particle motion. Physics of Fluids, 11, 3030–3042.CrossRefMATHGoogle Scholar
  5. 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.MathSciNetCrossRefMATHGoogle Scholar
  6. 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.Google Scholar
  7. Balachandar, S., & Eaton, J. (2010). Turbulent dispersed multiphase flow. Annual Review of Fluid Mechanics, 42, 111–133.CrossRefMATHGoogle Scholar
  8. 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.Google Scholar
  9. Brennen, C. E. (2005). Fundamentals of multiphase flow. Cambridge: Cambridge University Press.CrossRefMATHGoogle Scholar
  10. 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.Google Scholar
  11. Casey, M., & Wintergerste, T. (Eds.). (2000). Best practice guidelines: quality and trust in industrial CFD, ERCOFTAC.Google Scholar
  12. 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.MathSciNetCrossRefMATHGoogle Scholar
  13. Crowe, C., Sommerfeld, M., & Tsuji, T. (1998). Multiphase flows with droplets and particles. New York: CRC Press.Google Scholar
  14. Dreeben, T. D., & Pope, S. B. (1997). Wall-function treatment in PDF methods for turbulent flows. Physics of Fluids, 9, 2692–2703.MathSciNetCrossRefMATHGoogle Scholar
  15. Dreeben, T. D., & Pope, S. B. (1998). PDF/Monte Carlo simulation of near-wall turbulent flows. Journal of Fluid Mechanics, 357, 141–166.MathSciNetCrossRefMATHGoogle Scholar
  16. 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.MathSciNetCrossRefGoogle Scholar
  17. Eaton, J., & Fessler, J.R. (1994). Preferential concentration of particles by turbulence. International Journal of Multiphase Flow, 20, Suppl., 169–209.Google Scholar
  18. Ernst, M., Dietzel, M., & Sommerfeld, M. (2013). LBM for simulating transport and agglomeration of resolved particles. Acta Mechanica, 224, 2425.MathSciNetCrossRefMATHGoogle Scholar
  19. 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.Google Scholar
  20. 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).Google Scholar
  21. Gardiner, C. W. (1990). Handbook of stochastic methods for physics, chemistry and the natural sciences (2nd ed.). Berlin: Springer.MATHGoogle Scholar
  22. Gatski, T. B., Hussaini, M. Y., & Lumley, J. L. (Eds.). (1996). Simulation and modeling of turbulent flows. Oxford University Press.Google Scholar
  23. 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.Google Scholar
  24. 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.MathSciNetCrossRefMATHGoogle Scholar
  25. Grabowski, W. W., & Wang, L.-P. (2013). Growth of cloud droplets in a turbulent environment. Annual Review of Fluid Mechanics, 45, 293–324.MathSciNetCrossRefMATHGoogle Scholar
  26. Guha, A. (2008). Transport and deposition of particles in turbulent and laminar flow. Annual Review of Fluid Mechanics, 40, 311–341.MathSciNetCrossRefMATHGoogle Scholar
  27. Gustavsson, K., & Mehlig, B. (2016). Statistical models for spatial patterns of heavy particles in turbulence. Advances in Physics, 65, 1–57.Google Scholar
  28. Haworth, D. C. (2010). Progress in probability density function methods for turbulent reacting flows. Progress in Energy and Combustion Science, 36, 168–259.CrossRefGoogle Scholar
  29. 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.CrossRefGoogle Scholar
  30. Hoyas, S., & Jimenez, J. (2006). Scaling of the velocity fluctuations in turbulent channels up to \(Re_\tau =2003\). Physics of Fluids, 18, art. 011702.Google Scholar
  31. Jenny, P., Roekaerts, D., & Beishuizen, N. (2012). Modeling of turbulent dilute spray combustion. Progress in Energy and Combustion Science, 38, 846–887.CrossRefGoogle Scholar
  32. 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.Google Scholar
  33. 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.Google Scholar
  34. 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.Google Scholar
  35. Karlin, S. (1966). A first course in stochastic processes. New York: Academic Press.Google Scholar
  36. 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.Google Scholar
  37. 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.Google Scholar
  38. Kloeden, P. E., & Platen, E. (1992). Numerical solution of stochastic differential equations. Springer.Google Scholar
  39. 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.Google Scholar
  40. Kuerten, J. G. M. (2006). Subgrid modeling in particle-laden channel flows. Physics of Fluids, 18, art. 025108.Google Scholar
  41. Launder, B. E., & Sandham, N. D. (Eds.). (2002). Closure strategies for turbulent and transitional flows. Cambridge University Press.Google Scholar
  42. 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.CrossRefGoogle Scholar
  43. 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.Google Scholar
  44. Lundgren, T. S. (1967). Distribution functions in the statistical theory of turbulence. Physics of Fluids, 10, 969–975.CrossRefGoogle Scholar
  45. Łuniewski, M., Kotula, P., & Pozorski, J. (2012). Large-eddy simulations of particle-laden turbulent jets. TASK Quarterly, 16, 33–51.Google Scholar
  46. 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.CrossRefGoogle Scholar
  47. Manceau, R., & Hanjalić, K. (2002). Elliptic blending model: a new near-wall Reynolds-stress turbulence closure. Physics of Fluids, 14, 744–754.CrossRefMATHGoogle Scholar
  48. 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.CrossRefMATHGoogle Scholar
  49. 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.CrossRefMATHGoogle Scholar
  50. Marchioli, C., & Soldati, A. (2002). Mechanisms for particle transfer and segregation in turbulent boundary layer. Journal of Fluid Mechanics, 468, 283–315.CrossRefMATHGoogle Scholar
  51. 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.CrossRefGoogle Scholar
  52. Maxey, M. R. (1987). The motion of small spherical particles in a cellular flow field. Physics of Fluids, 30, 1915–1928.CrossRefGoogle Scholar
  53. 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.CrossRefMATHGoogle Scholar
  54. 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.MathSciNetCrossRefGoogle Scholar
  55. McComb, W. D. (1990). The physics of fluid turbulence. Oxford: Clarendon Press.MATHGoogle Scholar
  56. 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.Google Scholar
  57. Minier, J.-P. (2015). On Lagrangian stochastic methods for turbulent polydisperse two-phase reactive flows. Progress in Energy and Combustion Science, 50, 1–62.CrossRefGoogle Scholar
  58. Minier, J.-P., & Chibbaro, S., (Eds.). (2014). Stochastic methods in fluid mechanics. CISM Courses and Lectures (Vol. 548). Springer.Google Scholar
  59. 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.Google Scholar
  60. Minier, J.-P., & Peirano, E. (2001). The PDF approach to turbulent polydispersed two-phase flows. Physics Reports, 352, 1–214.MathSciNetCrossRefMATHGoogle Scholar
  61. 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.Google Scholar
  62. 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.CrossRefMATHGoogle Scholar
  63. 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.Google Scholar
  64. 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.CrossRefGoogle Scholar
  65. 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.CrossRefMATHGoogle Scholar
  66. 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.CrossRefGoogle Scholar
  67. Piomelli, U., & Balaras, E. (2002). Wall-layer models for Large-Eddy Simulations. Annual Review of Fluid Mechanics, 34, 349–374.MathSciNetCrossRefMATHGoogle Scholar
  68. Pope, S. B. (2000). Turbulent flows. Cambridge University Press.Google Scholar
  69. Pope, S. B. (2002). A stochastic Lagrangian model for acceleration in turbulent flows. Physics of Fluids, 14, 2360–2375.MathSciNetCrossRefMATHGoogle Scholar
  70. Pozorski, J. (2004). Stochastic modelling of turbulent flows. Zeszyty Naukowe IMP PAN 536/1495, Gdańsk.Google Scholar
  71. 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.CrossRefGoogle Scholar
  72. 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.Google Scholar
  73. 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.Google Scholar
  74. 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.Google Scholar
  75. 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.CrossRefMATHGoogle Scholar
  76. Pozorski, J., & Minier, J.-P. (1999). PDF modeling of dispersed two-phase turbulent flows. Physical Review E, 59, 855–863.CrossRefGoogle Scholar
  77. 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.CrossRefGoogle Scholar
  78. 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.CrossRefMATHGoogle Scholar
  79. Reeks, M. W. (1991). On a kinetic equation for the transport of particles in turbulent flows. Physics of Fluids A, 3, 446–456.CrossRefMATHGoogle Scholar
  80. Reeks, M. W. (1992). On the continuum equations for dispersed particles in nonuniform flows. Physics of Fluids A, 4, 1290–1303.CrossRefMATHGoogle Scholar
  81. 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.Google Scholar
  82. 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.Google Scholar
  83. Scotti, A., & Meneveau, C. (1999). A fractal interpolation model for large eddy simulation of turbulent flows. Physica D, 127, 198–232.MathSciNetCrossRefMATHGoogle Scholar
  84. Sobczyk, K. (1991). Stochastic differential equations. Kluwer Academic Publishers.Google Scholar
  85. Soldati, A. (2005). Particles turbulence interactions in boundary layers. ZAMM, 85, 683–699.MathSciNetCrossRefMATHGoogle Scholar
  86. 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.CrossRefGoogle Scholar
  87. 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.Google Scholar
  88. Squires, K. D., & Eaton, J. K. (1991). Preferential concentration of particles by turbulence. Physics of Fluids A, 3, 1169–1178.CrossRefGoogle Scholar
  89. Subramanian, S. (2013). Lagrangian-Eulerian methods for multiphase flows. Progress in Energy and Combustion Science, 39, 215–245.CrossRefGoogle Scholar
  90. 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.Google Scholar
  91. Tenneti, S., & Subramanian, S. (2014). Particle-resolved direct numerical simulation for gas-solid flow model development. Annual Review of Fluid Mechanics, 46, 199–230.MathSciNetCrossRefMATHGoogle Scholar
  92. Traczyk, M., & Knorps, M. (2012). Private communication.Google Scholar
  93. Violeau, D. (2012). Fluid mechanics and the SPH method. Oxford University Press.Google Scholar
  94. 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.CrossRefGoogle Scholar
  95. 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.Google Scholar
  96. Wacławczyk, M., & Pozorski, J. (2007). Modelling of near-wall turbulence with large-eddy velocity modes. Journal of Theoretical and Applied Mechanics, 45, 705–724.Google Scholar
  97. 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.CrossRefMATHGoogle Scholar
  98. 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.CrossRefMATHGoogle Scholar
  99. 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.CrossRefGoogle Scholar
  100. 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.CrossRefGoogle Scholar
  101. 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.Google Scholar

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© CISM International Centre for Mechanical Sciences 2017

Authors and Affiliations

  1. 1.Institute of Fluid-Flow Machinery, Polish Academy of SciencesGdańskPoland

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