B202 格子ボルツマン法に基づく流体運動のモデル化と数値シミュレーション(LBMによる流体解析)

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This paper describes a review of the lattice Boltzmann method (LBM) to simulate fluid flows, in which a macroscopic fluid consists of mesoscopic particles repeating collisions and translations. The main advantages are, high efficiency of parallel computing, easy boundary condition for solid body with complicated geometry, and that an interface in multiphase fluid can be reproduced without its geometric boundary condition in a self-organizing way by repulsive interaction between particles. LBM uses distribution function as variable of real number density of particle distinguished by their discrete velocities, to define macroscopic variable, fluid density, velocity, pressure, temperature and so on. Set of particle velocity is selected under the constraint that a velocity space is spherical-symmetric enough to reproduce isotropic fluid motion. The time- and spatial development of particle velocity distribution functions is governed by so-called lattice Boltzmann equation (LBE) that is discretized with finite difference schemes and includes convection, collision, gravity force and other force terms. The lattice BGK model is a well-known collision operator, based on an assumption that the particle distributions relax to a state of local equilibrium at constant rate during collision steps. The relaxation time is concerned with transport coefficients, viscosity and heat conductivity. An available equilibrium distribution form to derive the Navier-Stokes equations from LBE corresponds to Taylor-expansion polynomial of the Maxwell-Boltzmann distribution function with flow velocity on low Mach number. LBM models for fluid motions are developed flexibly with one or combination of various techniques, that is, addition of external term in LBE, multi-speed particle velocity set, time- or/and space-depending relaxation time period, full-matrix collision operator, modified equilibrium distribution functions, multiple distribution function system for scalar variables, etc. As a result, any macroscopic fluid dynamics can be recovered from ensemble average behavior of mesoscopic particles, according to the Chapman-Enskog expansion procedure. In addition, LBM also possesses potential to describe other dynamic systems, where the equilibrium state of particle velocity takes non-Maxwell formula and particle is regarded just as a carrier for information of variable in the field. In this paper, two kinds of numerical result are presented, three-dimensional 2-bubble motion in a circular tube filled with stagnant liquid under gravity, and large eddy simulation of incompressible fluid flow in a two-dimensional square cavity for high Reynolds number. The former simulation was carried out with the binary fluid model proposed by Swift et al., while, in the latter one, the standard Smagorinsky eddy viscosity model was incorporated into the relaxation time depending on local strain rate, which can be calculated from local non-equilibrium components of particle distribution functions without differential operation of flow velocity.
2001-11-14

B202 格子ボルツマン法に基づく流体運動のモデル化と数値シミュレーション(LBMによる流体解析)

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