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November 24, 1997 15:23 Annual Reviews AR049-12
Annu. Rev. Fluid Mech. 1998. 30:329–64
Copyright c© 1998 by Annual Reviews Inc. All rights reserved
LATTICE BOLTZMANN METHOD
FOR FLUID FLOWS
Shiyi Chen1,2 and Gary D. Doolen2
1IBM Research Division, T. J. Watson Research Center, P.O. Box 218, Yorktown
Heights, NY 10598; 2Theoretical Division and Center for Nonlinear Studies, Los
Alamos National Laboratory, Los Alamos, NM 87545; e-mail: syc@cnls.lanl.gov
KEY WORDS: lattice Boltzmann method, mesoscopic approach, fluid flow simulation
ABSTRACT
We present an overview of the lattice Boltzmann method (LBM), a parallel and
efficient algorithm for simulating single-phase and multiphase fluid flows and
for incorporating additional physical complexities. The LBM is especially useful
for modeling complicated boundary conditions and multiphase interfaces. Recent
extensions of this method are described, including simulations of fluid turbulence,
suspension flows, and reaction diffusion systems.
INTRODUCTION
In recent years, the lattice Boltzmann method (LBM) has developed into an
alternative and promising numerical scheme for simulating fluid flows and
modeling physics in fluids. The scheme is particularly successful in fluid flow
applications involving interfacial dynamics and complex boundaries. Unlike
conventional numerical schemes based on discretizations of macroscopic con-
tinuum equations, the lattice Boltzmann method is based on microscopic mod-
els and mesoscopic kinetic equations. The fundamental idea of the LBM is
to construct simplified kinetic models that incorporate the essential physics of
microscopic or mesoscopic processes so that the macroscopic averaged prop-
erties obey the desired macroscopic equations. The basic premise for using
these simplified kinetic-type methods for macroscopic fluid flows is that the
macroscopic dynamics of a fluid is the result of the collective behavior of many
microscopic particles in the system and that the macroscopic dynamics is not
sensitive to the underlying details in microscopic physics (Kadanoff 1986).
By developing a simplified version of the kinetic equation, one avoids solving
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330 CHEN & DOOLEN
complicated kinetic equations such as the full Boltzmann equation, and one
avoids following each particle as in molecular dynamics simulations.
Even though the LBM is based on a particle picture, its principal focus is
the averaged macroscopic behavior. The kinetic equation provides many of
the advantages of molecular dynamics, including clear physical pictures, easy
implementation of boundary conditions, and fully parallel algorithms. Because
of the availability of very fast and massively parallel machines, there is a current
trend to use codes that can exploit the intrinsic features of parallelism. The LBM
fulfills these requirements in a straightforward manner.
The kinetic nature of the LBM introduces three important features that dis-
tinguish it from other numerical methods. First, the convection operator (or
streaming process) of the LBM in phase space (or velocity space) is linear.
This feature is borrowed from kinetic theory and contrasts with the nonlinear
convection terms in other approaches that use a macroscopic representation.
Simple convection combined with a relaxation process (or collision operator)
allows the recovery of the nonlinear macroscopic advection through multi-scale
expansions. Second, the incompressible Navier-Stokes (NS) equations can be
obtained in the nearly incompressible limit of the LBM. The pressure of the
LBM is calculated using an equation of state. In contrast, in the direct nu-
merical simulation of the incompressible NS equations, the pressure satisfies a
Poisson equation with velocity strains acting as sources. Solving this equation
for the pressure often produces numerical difficulties requiring special treat-
ment, such as iteration or relaxation. Third, the LBM utilizes a minimal set of
velocities in phase space. In the traditional kinetic theory with the Maxwell-
Boltzmann equilibrium distribution, the phase space is a complete functional
space. The averaging process involves information from the whole velocity
phase space. Because only one or two speeds and a few moving directions are
used in LBM, the transformation relating the microscopic distribution func-
tion and macroscopic quantities is greatly simplified and consists of simple
arithmetic calculations.
The LBM originated from lattice gas (LG) automata, a discrete particle ki-
netics utilizing a discrete lattice and discrete time. The LBM can also be
viewed as a special finite difference scheme for the kinetic equation of the
discrete-velocity distribution function. The idea of using the simplified kinetic
equation with a single-particle speed to simulate fluid flows was employed by
Broadwell (Broadwell 1964) for studying shock structures. In fact, one can view
the Broadwell model as a simple one-dimensional lattice Boltzmann equation.
Multispeed discrete particle velocities models have also been used for studying
shock-wave structures (Inamuro & Sturtevant 1990). In all these models, al-
though the particle velocity in the distribution function was discretized, space
and time were continuous. The full discrete particle velocity model, where