A Comparative Study of the LBE and GKS Methods for DNS of Decaying Turbulence

Abstract

We compare the lattice Boltzmann equation (LBE) and the gas-kinetic scheme (GKS) for direct numerical simulation of decaying homogeneous isotropic turbulence. Although both methods are derived from the Boltzmann equation thus share a common kinetic origin, numerically they are rather different. The LBE is a finite difference method, while the GKS is a finite volume one. In addition, the LBE is limited to incompressible flows with the Mach number Ma < 0.3, while the GKS is valid for fully compressible fluids without the low-Mach-number restriction. In this study we use the generalized lattice Boltzmann equation (GLBE) with multiple-relaxation-time (MRT) collision model, which overcomes all the apparent defects in the popular lattice BGK equation. We use both the LBE and GKS methods to simulate the decaying homogeneous isotropic turbulence. We compared the time evolution of kinetic energy, dissipation rate and other higher order statistical parameters. Our results show that both the LBE and GKS yield quantitatively similar results for low order statistical quantities and there are differences in higher order statistical quantities. In general, kinetic methods for computational fluid dynamics (CFD) are derived from the Boltzmann equation, as opposed to conventional CFD methods based on direct discretizations of the Navier-Stokes equations. Two distinctive features of kinetic methods immediately appear. First, kinetic methods can 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 can relate the macroscopic conservation laws to 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 some computational advantages. 1 For these two reasons, kinetic methods have attracted some interest recently. Due to their mesoscopic nature, kinetic methods are particularly appealing in modeling and simulations of complex 2 and non-equilibrium 3–6 flows. There are a number of kinetic or mesoscopic methods, such as the lattice gas cellular automata (LGCA), the lattice Boltzmann equation (LBE), the gas-kinetic schemes (GKS), the smoothed particle hydrodynamics (SPH), and the dissipative particle dynamics (DPD). Among these methods, the LBE and GKS methods are specifically designed as numerical methods for CFD: the former is for low-Mach-number flows while the latter is for fully compressible flows. Both methods have been applied to simulate viscous flows, 7,8 heat transfer problems, 9,10 shallow water equations, 11 multiphase 12–15 and multicomponent 16–20 flows, magnetohydrodynamics, 21–23 and microflows. 3,24–26 Although both the LBE and GKS methods share a common kinetic origin, they are quite different otherwise. First of all, the LBE is a finite difference (FD) method while the GKS is a finite volume (FV) one. Second, the LBE is a system evolving on the discrete phase space ,