Discrete Rotational Symmetry, Moment Isotropy, and Higher Order Lattice Boltzmann Models

Abstract Conventional lattice Boltzmann models only satisfy moment isotropy up to fourth order. In order to accurately describe important physical effects beyond the isothermal Navier-Stokes fluid regime, higher-order isotropy is required. In this paper, we present some basic results on moment isotropy and its relationship to the rotational symmetry of a generating discrete vector set. The analysis provides a geometric understanding for popular lattice Boltzmann models, while offering a systematic procedure to construct higher-order models.

[1]  C. Bunn,et al.  Modern Crystallography , 1973, Nature.

[2]  X. Yuan,et al.  Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation , 2006, Journal of Fluid Mechanics.

[3]  Akiyama,et al.  Thermal lattice Bhatnagar-Gross-Krook model without nonlinear deviations in macrodynamic equations. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases , 1939 .

[5]  Raphael Aronson,et al.  Theory and application of the Boltzmann equation , 1976 .

[6]  R. Benzi,et al.  Lattice Gas Dynamics with Enhanced Collisions , 1989 .

[7]  S. Orszag,et al.  Expanded analogy between Boltzmann kinetic theory of fluids and turbulence , 2004, Journal of Fluid Mechanics.

[8]  S. Wolfram.  Cellular automaton fluids 1: Basic theory , 1986 .

[9]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[10]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[11]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[12]  M. Gad-el-Hak.  The Fluid Mechanics of Microdevices—The Freeman Scholar Lecture , 1999 .

[13]  Hudong Chen.  VOLUMETRIC FORMULATION OF THE LATTICE BOLTZMANN METHOD FOR FLUID DYNAMICS : BASIC CONCEPT , 1998 .

[14]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[15]  Antony Jameson,et al.  Gas-kinetic finite volume methods, flux-vector splitting, and artificial diffusion , 1995 .

[16]  Ramesh K. Agarwal,et al.  Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime , 2001 .

[17]  R. Schwarzenberger,et al.  N-dimensional crystallography , 1980 .

[18]  X. He,et al.  Discretization of the Velocity Space in the Solution of the Boltzmann Equation , 1997, comp-gas/9712001.

[19]  Hudong Chen,et al.  Digital Physics Approach to Computational Fluid Dynamics: Some Basic Theoretical Features , 1997 .

[20]  Hudong Chen,et al.  H-theorem and generalized semi-detailed balance condition for lattice gas systems , 1995 .

[21]  Michihisa Tsutahara,et al.  Possibility of constructing a multispeed Bhatnagar-Gross-Krook thermal model of the lattice Boltzmann method. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Hudong Chen,et al.  H-theorem and origins of instability in thermal lattice Boltzmann models , 2000 .

[23]  R. Bishop,et al.  Tensor Analysis on Manifolds , 1980 .

[24]  David Freed,et al.  Multi-speed thermal lattice Boltzmann method stabilization via equilibrium under-relaxation , 2000 .

[25]  Frisch,et al.  Lattice gas automata for the Navier-Stokes equations. a new approach to hydrodynamics and turbulence , 1989 .

[26]  R Zhang,et al.  Effective volumetric lattice Boltzmann scheme. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Kun Xu,et al.  A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method , 2001 .

[28]  Christopher M. Teixeira.  Continuum limit of lattice gas fluid dynamics , 1992 .

[29]  Sauro Succi,et al.  Colloquium: Role of the H theorem in lattice Boltzmann hydrodynamic simulations , 2002 .

[30]  Matthaeus,et al.  Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.