Theory of Lattice Boltzmann Equation

Li-Shi Luo

Motivation and Objective

To develope a rigorous theory of the lattice Boltzmann equation in connection with the continuous Boltzmann equation.

Approach and Accomplishments

The lattice Boltzmann equation (LBE) is directly derived from the Boltzmann equation by discretization in both time and phase space. A procedure to systematically derive discrete velocity models is presented. A new LBE algorithm with arbitrary mesh grids is proposed and a numerical simulation of the backward-facing step is conducted (Figure 1). The numerical result agrees well with experimental and previous numerical results. Various improvements on the LBE models are discussed, and an explanation of the instability of the existing LBE thermal models is also provided.

Figure 1. (a) The nonuniform mesh for the backward-facing step flow simulation. The mesh size is Nx x N_y = 61 x 48. (b) The stream lines of the backward-facing step flow. The solid lines and dashed lines are the results from the simulations by using the nonuniform mesh and the uniform mesh of size $N_x \times N_y = 385 \times 48$, respectively. The boundary conditions in both simulations are the maximum velocity at entrance $U = 0.1$, and the pressure at the exit $P_1 = 1.0$. The mass density $\rho = 1.0 $, and $\tau = 0.596$ in the simulations. The convergence criterion is the relative global difference of the velocity fields (with $L^2$ norm) between two successive time iterations less than $10^{-7}$. In both simulations, the convergence is attained after $50,000$ time iterations.

Future Plans

To extend the theory to multi-phase, multi-component fluids.

References

X. He and Li-Shi Luo, A priori derivation of the lattice Boltzmann equation, Phys. Rev. E, Rap. Comm. 55:R6333 (1997).

X. He and Li-Shi Luo, Theory of the lattice Boltzmann equation: From the Boltzmann equation to the attice Boltzmann equation, Phys. Rev. E 56 (1997).

Point of Contact: Li-Shi Luo, luo@icase.edu