Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-23T08:32:47.239Z Has data issue: false hasContentIssue false
  • English
  • Français

Derivation of Hydrodynamics for Multi-Relaxation Time Lattice Boltzmann using the Moment Approach

Published online by Cambridge University Press:  03 June 2015

Goetz Kaehler  and
Alexander J. Wagner
Get access

Abstract

A general analysis of the hydrodynamic limit of multi-relaxation time lattice Boltzmann models is presented. We examine multi-relaxation time BGK collision operators that are constructed similarly to those for the MRT case, however, without explicitly moving into a moment space representation. The corresponding ‘moments’ are derived as left eigenvectors of said collision operator in velocity space. Consequently we can, in a representation independent of the chosen base velocity set, generate the conservation equations. We find a significant degree of freedom in the choice of the collision matrix and the associated basis which leaves the collision operator invariant. We explain why MRT implementations in the literature reproduce identical hydrodynamics despite being based on different orthogonalization relations. More importantly, however, we outline a minimal set of requirements on the moment base necessary to maintain the validity of the hydrodynamic equations. This is particularly useful in the context of position and time-dependent moments such as those used in the context of peculiar velocities and some implementations of fluctuations in a lattice-Boltzmann simulation.

Keywords

76P0582B40Lattice Boltzmannkinetic theoryNavier-Stokeshydrodynamics
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 3 , March 2013 , pp. 614 - 628
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Eggels, J. G. M.Direct and large-eddy simulation of turbulent fluid flow using the lattice-Boltzmann scheme. Int. J. Heat Fluid Flow, 17(3): 307–323, 1996.Google Scholar
[2]Shan, X. and Chen, H.Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E, 47(3): 1815–1819, Mar 1993.Google Scholar
[3]Frisch, U., Hasslacher, B., and Pomeau, Y.Lattice-Gas Automata for the Navier-Stokes Equation. Phys. Rev. Lett., 56(14): 1505–1508, 1986.Google Scholar
[4]Succi, S.The lattice Boltzmann equation for fluid dynamics and beyond. Oxford University Press, USA, first edition, 2001.Google Scholar
[5]Huang, K.Statistical Mechanics. Wiley, second edition, 1987.Google Scholar
[6]Dellar, P. J.Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations. Phys. Rev. E, 65: 036309, February 2002.Google Scholar
[7]Grad, H.On the Kinetic Theory of Rarefied Gases. Commun. Pure Appl. Math., 2(4): 331–407, 1949.CrossRefGoogle Scholar
[8]Swift, M. R., Orlandini, E., Osborn, W. R., and M, J.Yeomans. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Phys. Rev. E, 54(5): 5041–5052, 1996.Google Scholar
[9]Giraud, L., dHumieres, D., and Lallemand, P.A lattice-Boltzmann model for visco-elasticity. Int. J. Mod. Phys. C, 8(4): 805–815, 1997.Google Scholar
[10]Levermore, C.Moment closure hierarchies for kinetic theories. Journal of Statistical Physics, 83: 1021–1065, 1996. 10.1007/BF02179552.Google Scholar
[11]Koplik, J. and Banavar, J. R.Continuum Deductions from Molecular Hydrodynamics. Annu. Rev. Fluid Mech., 27: 257–292, 1995.Google Scholar
[12]Yudistiawan, W. P., Kwak, S. K., Patil, D. V., and Ansumali, SantoshHigher-order galilean-invariant lattice boltzmann model for microflows: Single-component gas. Phys. Rev. E, 82: 046701, Oct 2010.Google Scholar
[13]Wagner, A. J.Thermodynamic consistency of liquid-gas lattice boltzmann simulations. Phys. Rev. E, 74(5): 056703, 2006.Google Scholar
[14]McNamara, G. R. and Zanetti, G.Use of the boltzmann equation to simulate lattice-gas automata. Phys. Rev. Lett., 61(20): 2332–2335, Nov 1988.CrossRefGoogle ScholarPubMed
[15]Higuera, F. J. and Jimenez, J.Boltzmann approach to lattice gas simulations. EPL (Euro-physics Letters), 9(7): 663, 1989.Google Scholar
[16]Qian, Y. H., d’Humieres, D., and Lallemand, P.Lattice BGK Models for Navier-Stokes Equation. Europhys. Lett., 17(6): 479, 1992.Google Scholar
[17]d’Humieres, D.Generalized lattice-Boltzmann equations. Rarefied Gas Dynamics: Theory and Simulations, Prog. Astronaut. Aeronaut., 159: 450–458, 1992.Google Scholar
[18]Dunweg, B. and Ladd, A.Lattice boltzmann simulations of soft matter systems. In Holm, Christian and Kremer, Kurt, editors, Advanced Computer Simulation Approaches for Soft Matter Sciences III, volume 221 of Advances in Polymer Science, pages 89–166. Springer Berlin/Heidelberg, 2009.Google Scholar
[19]Junk, M., Klar, A., and Luo, L. S.Asymptotic analysis of the lattice Boltzmann equation. J. Comput. Phys., 210: 676–704, July 2005.Google Scholar
[20]Ansumali, S., Arcidiacono, S., Chikatamarla, S. S., Prasianakis, N. I., Gorban, A. N., and Karlin, I. V.Quasi-equilibrium lattice boltzmann method. The European Physical Journal B – Condensed Matter and Complex Systems, 56: 135–139, 2007. 10.1140/epjb/e2007-00100-1.Google Scholar
[21]Kaehler, G. and Wagner, A. J.Galilean Invariance in fluctuating Lattice Boltzmann. in preparation, 2011.Google Scholar
[22]Higuera, F. J., Succi, S., and Benzi, R.Lattice Gas-Dynamics with enhanced Collisions. Europhys. Lett., 9(4): 345–349, June 1989.Google Scholar
[23]Bhatnagar, P. L., Gross, E. P., and Krook, M.A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems. Phys. Rev., 94(3): 511–525, 1954.Google Scholar
[24]Wagner, A. J. and Li, Q.Investigation of galilean invariance of multiphase lattice boltzmann methods. Physica A, 362(1): 105–110, March 2006.Google Scholar
[25]He, Xiaoyi and Luo, Li-ShiA priori derivation of the lattice boltzmann equation. Phys. Rev. E, 55(6): R6333–R6336, Jun 1997.Google Scholar
[26]Wagner, A. J. and Yeomans, J. M.Phase separation under shear in two-dimensional binary fluids. Phys. Rev. E, 59(4): 4366–4373, October 1999.Google Scholar
[27]Lallemand, P. and Luo, L. S.Theory of the lattice Boltzmann method: Acoustic and thermal properties in two and three dimensions. Phys. Rev. E, 68(3, Part 2): 036706, September 2003.Google Scholar
[28]Dellar, P. J.Multiple-Relaxation-Time Collision Operators in the Lattice Boltzmann Method. Lecture notes for a tutorial presented at the 17th Discrete Simulation of Fluid Dynamics conference, Florianopolis, Brazil, 4-8 August, August 2008.Google Scholar
[29]Benzi, R., Succi, S., and Vergassola, M.The Lattice Boltzmann-Equation-Theory and Applications. Phys. Rep.-Rev. Sec. Phys. Lett., 222(3): 145–197, December 1992.Google Scholar
[30]Lallemand, P. and Luo, L. S.Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E, 61(6, Part A): 6546–6562,June 2000.Google Scholar
[31]Adhikari, R., Stratford, K., Cates, M. E., and Wagner, A. J.Fluctuating lattice Boltzmann. Europhys. Lett., 71(3): 473–479, August 2005.Google Scholar
[32]Dunweg, B., Schiller, U. D., and Ladd, A. J. C.Statistical mechanics of the fluctuating lattice boltzmann equation. Phys. Rev. E, 76(3): 036704, Sep 2007.Google Scholar
[33]Adhikari, R. and Succi, S.Duality in matrix lattice Boltzmann models. Phys. Rev. E, 78(6): 066701, December 2008.Google Scholar