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Consistent subgrid scale modelling for lattice Boltzmann methods

  • Orestis Malaspinas (a1) and Pierre Sagaut (a1)
Abstract

The lattice Boltzmann method has become a widely used tool for the numerical simulation of fluid flows and in particular of turbulent flows. In this frame the inclusion of subgrid scale closures is of crucial importance and is not completely understood from the theoretical point of view. Here, we propose a consistent way of introducing subgrid closures in the BGK Boltzmann equation for large eddy simulations of turbulent flows. Based on the Hermite expansion of the velocity distribution function, we construct a hierarchy of subgrid scale terms, which are similar to those obtained for the Navier–Stokes equations, and discuss their inclusion in the lattice Boltzmann method scheme. A link between our approach and the standard way on including eddy viscosity models in the lattice Boltzmann method is established. It is shown that the use of a single modified scalar relaxation time to account for subgrid viscosity effects is not consistent in the compressible case. Finally, we validate the approach in the weakly compressible case by simulating the time developing mixing layer and comparing with experimental results and direct numerical simulations.

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Corresponding author
Email address for correspondence: malaspinas@lmm.jussieu.fr
References
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1. Aidun, C. K. & Clausen, R. J. 2010 Lattice-Boltzmann method for complex flows. Annu. Rev. Fluid Mech. 42 (1), 439472.
2. Ansumali, S., Karlin, I. V., Iliya, V. & Succi, S. 2004 Kinetic theory of turbulence modeling: smallness parameter, scaling and microscopic derivation of Smagorinsky model. Physica A: Statist. Mech. Appl. 338 (3–4), 379394.
3. Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94 (3), 511525.
4. Chapman, S. & Cowling, T. G. 1960 The Mathematical Theory of Non-uniform Gases. Cambridge University Press.
5. Chen, S. 2009 A large-eddy-based lattice Boltzmann model for turbulent flow simulation. Appl. Maths Comput. 215 (2), 591598.
6. Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30 (1), 329364.
7. Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S. & Yakhot, V. 2003 Extended Boltzmann kinetic equation for turbulent flows. Science 301 (5633), 633636.
8. Chen, H., Orszag, S. A., Staroselsky, I. & Succi, S. 2004 Expanded analogy between Boltzmann kinetic theory of fluids and turbulence. J. Fluid Mech. 519, 301314.
9. Chen, H., Succi, S. & Orszag, S. 1999 Analysis of subgrid scale turbulence using the Boltzmann Bhatnagar–Gross–Krook kinetic equation. Phys. Rev. E 59 (3), R2527R2530.
10. Chikatamarla, S. S., Frouzakis, C. E., Karlin, I. V., Tomboulides, A. G. & Boulouchos, K. B. 2010 Lattice Boltzmann method for direct numerical simulation of turbulent flows. J. Fluid Mech. 656, 298308.
11. Dellar, P. J. 2001 Bulk and shear viscosities in lattice Boltzmann equations. Phys. Rev. E 64 (3), 031203.
12. Dong, Y.-H., Sagaut, P. & Marié, S. 2008 Inertial consistent subgrid model for large-eddy simulation based on the lattice Boltzmann method. Phys. Fluids 20 (3), 035104.
13. Eggels, J. G. M. 1996 Direct and large-eddy simulation of turbulent fluid flow using the lattice-Boltzmann scheme. Intl J. Heat Fluid Flow 17 (3), 307323.
14. Filippova, O., Succi, S., Mazzocco, F., Arrighetti, C., Bella, G. & Hänel, D. 2001 Multiscale lattice Boltzmann schemes with turbulence modelling. J. Comput. Phys. 170 (2), 812829.
15. Garnier, E., Adams, N. & Sagaut, P. 2009 Large-eddy Simulation for Compressible Flows. Springer.
16. Girimaji, S. S. 2007 Boltzmann kinetic equation for filtered fluid turbulence. Phys. Rev. Lett. 99 (3), 034501.
17. Grad, H. 1949a Note on the -dimensional Hermite polynomials. Commun. Pure Appl. Maths 9, 325.
18. Grad, H. 1949b On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 9, 331.
19. Guo, Z., Shi, B. & Zheng, C. 2002 A coupled lattice BGK model for the Boussinesq equations. Intl J. Numer. Meth. Fluids 39, 325342.
20. Hou, S., Sterling, J., Chen, S. & Doolen, G. D. 1996 A lattice Boltzmann subgrid model for high Reynolds number flows. Fields Inst. Comm. 6, 151-66.
21. Huang, K. 1987 Statistical Mechanics. John Wiley & Sons.
22. Kerimo, J. & Girimaji, S. 2007 Boltzmann-BGK approach to simulating weakly compressible 3D turbulence: comparison between lattice Boltzmann and gas kinetic methods. J. Turbul. 8.
23. Krafczyk, M., Tölke, J. & Luo, L.-S. 2003 Large-eddy simulations with a multiple-relaxation-time LBE model. Intl J. Mod. Phys. B 17 (1–2), 3339.
24. Labbé, O., Montreuil, E. & Sagaut, P. 2002 Large-eddy simulation of heat transfer over a backward facing step. Intl J. Numer. Heat Transfer, Part A 42 (1–2), 7390.
25. Leonard, A. 1974 Energy cascade in large-eddy simulations of turbulent fluid flows. Adv. Geophys. A 18, 237248.
26. Malaspinas, O. 2009 Lattice Boltzmann method for the simulation of viscoelastic fluid flows. PhD dissertation, EPFL, Lausanne, Switzerland.
27. Malaspinas, O. & Sagaut, P. 2011 Advanced large-eddy simulation for lattice Boltzmann methods: the approximate deconvolution model. Phys. Fluids 23.
28. Meyers, J. & Sagaut, P. 2006 On the model coefficients for the standard and the variational multi-scale Smagorinsky model. J. Fluid Mech. 569, 287319.
29. Meyers, J., Sagaut, P. & Geurts, B. J. 2006 Optimal model parameters for multi-obective large-eddy simulations. Phys. Fluids 18 (9), 095103.
30. Meyers, J., Sagaut, P. & Geurts, B. J. 2007 A computational error assessment of central finite-volume discretizations in large-eddy simulation using a Smagorinsky model. J. Comput. Phys. 227 (1), 156173.
31. Nie, X. B., Shan, X. & Chen, H. 2008 Galilean invariance of lattice Boltzmann models. Europhys. Lett. 81 (3), 34005.
32. Premnath, K. N., Pattison, M. J. & Banerjee, S. 2009 Dynamic subgrid scale modeling of turbulent flows using lattice-Boltzmann method. Physica A: Statist. Mech. Appl. 388 (13), 26402658.
33. Quéméré, P., Sagaut, P. & Couaillier, V. 2001 A new multidomain/multiresolution method for large-eddy simulation. Intl J. Numer. Meth. Fluids 36 (4), 391416.
34. Quéméré, P., Sagaut, P. & Couaillier, V. 2002 Zonal multi-domain RANS/LES simulations of turbulent flows. Intl J. Numer. Meth. Fluids 40 (7), 903925.
35. Rogers, M. M. & Moser, R. D. 1994 Direct simulation of a self-similar turbulent mixing layer. Phys. Fluids 6 (2), 903923.
36. Sagaut, P. 2005 Large Eddy Simulation for Incompressible Flows: An Introduction. Springer.
37. Sagaut, P. 2010 Toward advanced subgrid models for lattice-Boltzmann-based large-eddy simulation: theoretical formulations. In Mesoscopic Methods in Engineering and Science, International Conferences on Mesoscopic Methods in Engineering and Science, Comput. Maths Applics. 59 (7), 21942199.
38. Sagaut, P., Deck, S. & Terracol, M. 2006 Multiscale and Multiresolution Approaches in Turbulence. Imperial College Press.
39. Scagliarini, A., Biferale, L., Sbragaglia, M., Sugiyama, K. & Toschi, F. 2010a Lattice Boltzmann methods for thermal flows: Continuum limit and applications to compressible Rayleigh–Taylor systems. Phys. Fluids 22 (5), 055101.
40. Scagliarini, A., Biferale, L., Sbragaglia, M., Sugiyama, K. & Toschi, F. 2010b Numerical simulations of compressible Rayleigh–Taylor turbulence in stratified fluids. Phys. Scr. 2010 (T142), 014017.
41. Seror, C., Sagaut, P., Bailly, C. & Juvé, D. 2001 On the radiated noise computed by large-eddy simulation. Phys. Fluids 13 (2), 476487.
42. Shan, X. & Chen, H. 2007 A general multiple-relaxation-time Boltzmann collision model. Intl J. Mod. Phys. C 18, 635.
43. Shan, X., Yuan, X.-F. & Chen, H. 2006 Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation. J. Fluid Mech. 550, 413441.
44. Smagorinsky, J. 1963 General circulation experiments with the primitive equations: I. The basic equations. Mon. Weath. Rev. 91, 99164.
45. Succi, S. 2001 The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press.
46. Terracol, M., Sagaut, P. & Basdevant, C. 2003 A time self-adaptive multilevel algorithm for large-eddy simulation. J. Comput. Phys. 184 (2), 339365.
47. Weickert, M., Teike, G., Schmidt, O. & Sommerfeld, M. 2010 Investigation of the LES WALE turbulence model within the lattice Boltzmann framework. Comput. Maths Applic. 59 (7), 22002214.
48. Wolf-Gladrow, D. A. 2000 Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction. Springer.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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