Efficient kinetic method for fluid simulation beyond the Navier-Stokes equation

Abstract

We present a further theoretical extension to the kinetic theory based formulation of the lattice Boltzmann method of Shan et al (2006). In addition to the higher order projection of the equilibrium distribution function and a sufficiently accurate Gauss-Hermite quadrature in the original formulation, a new regularization procedure is introduced in this paper. This procedure ensures a consistent order of accuracy control over the nonequilibrium contributions in the Galerkin sense. Using this formulation, we construct a specific lattice Boltzmann model that accurately incorporates up to the third order hydrodynamic moments. Numerical evidences demonstrate that the extended model overcomes some major defects existed in the conventionally known lattice Boltzmann models, so that fluid flows at finite Knudsen number (Kn) can be more quantitatively simulated. Results from force-driven Poiseuille flow simulations predict the Knudsen's minimum and the asymptotic behavior of flow flux at large Kn.

High-order regularization in lattice-Boltzmann equations

“…This choice was investigated by Ladd 15 , Chen et al 16 , Zhang et al 17 , and by Latt and Chopard 18 for the D2Q9 model; very recently Mattila et al 19 compared this regularization with the entropic stabilizer. Here we propose a method for dealing with general flow problems which require utilization of high-order LBEs: the method is based on a systematic improvement of the standard regularization.…”

“…Refs. 17,[20][21][22][23]. Moreover, Dubois et al 24,25 recently showed that the choice of moments does influence the stability properties of a LB scheme: a poor choice of moments can lead to suboptimal stability ranges.…”

Moment isotropy and discrete rotational symmetry of two-dimensional lattice vectors

“…With proper choices of a set of discrete velocity vectors, continuum hydrodynamics can be recovered at the Navier-Stokes level [3,4] and beyond (cf. [5,6]). The main theme of LBM studies is how to recover continuum fluid behaviour out of only a finite set of discrete vector values.…”

A lattice Boltzmann approach for solving scalar transport equations

“…Since the 19-speed LB model has a higher order lattice symmetry than the six-speed LB model, the same 19-speed lattice model for scalar is used here. Unlike the regular BGK, the so-called BGK regularized/filtered collision operator form is used here [13,14]. Such a form ensures that only the first-order non-equilibrium moment contributes to scalar diffusion in the hydrodynamic range.…”