A Comparative Study of the Lattice Boltzmann and Pseudo-Spectral Methods for Decaying Homogeneous Isotropic Turbulence

Abstract

We compare the lattice Boltzmann method (LBM) and Pseudo-Spectral (PS) method for direct numerical simulation of decaying homogeneous isotropic turbulence. In this study we use the generalized lattice Boltzmann equation (GLBE) with multiple-relaxationtime (MRT) collision model, which overcomes all the apparent defects in the popular lattice BGK equation. We first compare the instantaneous flow field and they agree well with each other. We then compare the statistical quantities including the energy spectra, compensated spectra, skewness, flatness and so on. The LBM and SP results agree well with each other although the LBM results have oscillations due to the acoustics. I. Introduction The lattice Boltzmann method (LBM) 1–7 has emerged as an alternative method for computational fluid dynamics (CFD). The LBM is a kinetic method derived from the Boltzmann equation, as opposed to conventional computational fluid dynamics (CFD) methods based on direct discretizations of the Navier-Stokes equations. Two distinctive features of kinetic methods immediately appear. First, kinetic methods include extended hydrodynamics beyond the validity regime of the Navier-Stokes equations, because they are based on kinetic theory. It is known that the Boltzmann equation provides the theoretical connection between hydrodynamics and the underlying microscopic physics. Kinetic methods are often called mesoscopic methods for they bridge between the macroscopic conservation laws and the underlying microscopic dynamics. And second, the Boltzmann equation is a first-order integro-partial-differential equation with a linear advection term, while the Navier-Stokes equation is a second-order partial differential equation with a nonlinear advection term. The nonlinearity in the Boltzmann equation resides in its collision term, which is local. This feature may lead to a number of computational advantages. 8 For these two reasons, kinetic methods have attracted some interest recently. Although LBM is a relatively new method, its efficiency and effectiveness for direct numerical simulation (DNS) of turbulence has not yet been thoroughly investigated. In an effort to assess the ability of LBM in turbulence, we perform DNS of decaying homogeneous isotropic turbulence (DHIT). Decaying turbulence is a standard problem in study of turbulence. 9–21 In fact, the first attempt at DNS with incompressible NS equation involved this problem. 9 Since then several numerical investigations of decaying HIT have been carried out. Some preliminary studies of three-dimensional (3D) decaying HIT using LBM have also been performed, 22,23 but these investigations stop well short of a rigorous and thorough comparison with a well