Incompressible limits of lattice Boltzmann equations using multiple relaxation times

P. J. Dellar  (2003) Incompressible limits of lattice Boltzmann equations using multiple relaxation times   J. Comput. Phys. 190 351--370

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Abstract

Lattice Boltzmann equations using multiple relaxation times are intended to be more stable than those using a single relaxation time. The additional relaxation times may be adjusted to suppress non-hydrodynamic modes that do not appear directly in the continuum equations, but may contribute to instabilities on the grid scale. If these relaxation times are fixed in lattice units, as in previous work, solutions computed on a given lattice are found to diverge in the incompressible (small Mach number) limit. This non-existence of an incompressible limit is analysed for an inclined one dimensional jet. An incompressible limit does exist if the non-hydrodynamic relaxation times are not fixed, but scaled by the Mach number in the same way as the hydrodynamic relaxation time that determines the viscosity.

This paper extends the decomposition of the 9 velocity lattice Boltzmann equation introduced in my earlier paper

P. J. Dellar (2002) Non-hydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations  Phys. Rev. E 65 036309 (12 pages)
 

For recent work on multiple relaxation time collision operators see

P. Lallemand and L.-S. Luo (2000) Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability Phys. Rev. E 61 6546--6562  DOI: 10.1103/PhysRevE.61.6546    Also ICASE Report 2000-17 from NASA Langley Research Center

D. d'Humieres, I. Ginzburg, M. Krafczyk, P. Lallemand, and L.-S. Luo (2002) Multiple-relaxation-time lattice Boltzmann models in three dimensionsPhil. Trans. R. Soc. Lond. A 360 437-451  DOI: 10.1098/rsta.2001.0955   Also ICASE Report 2002-20 from NASA Langley Research Center
 

For a general overview of collision operators in lattice Boltzmann equations see

S. Succi, I. V. Karlin, and H. Chen (2002) Role of the H theorem in lattice Boltzmann hydrodynamic simulations  Rev. Mod. Phys. 74 1203-1220  ( cond-mat/0205639 ) DOI: 10.1103/RevModPhys.74.1203

BibTeX citation information:

@article{Dellar03incompLB,
author = "P. J. Dellar",
title = "Incompressible limits of lattice Boltzmann equations using multiple relaxation times",
year = "2003",
journal = "J. Comput. Phys.",
volume = "190",
pages = "351--370"
}

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