Lattice Boltzmann Method And Some Application To Hydrodynamics

In recent years, the lattice Boltzmann method (LBM) has been developped into an alternative and promising numerical scheme for simulating fluid flows and modelling physics in fluids. The LBM was first proposed to simulate fluid dynamics by McNamara and Zanetti in 1988. Historically it is originated and evolved from lattice gas automata (LGA). It can be either regarded as an extension of the LGA or as a special discrete form of the continous Boltzmann equation from the kinetic theory. Unlike conventional numerical schemes based on discretizations of macroscopic equations, the LBM is based on the statistical physics and describes the microscopic picture of particles movement in an extremely simplified way, but at the macroscopic level it gives a correct average description of the motion. Because of many advantages, including simple calculation, intrinsic parallelism, and easy implementation of boundary conditions, the LBM has achieved great success in solving numerous problems in many areas in the last decade or so. It has been particularly successful in simulations of complicated boundaries or/and complex fluid, such as turbulent flow, multi-phase flow, multi-component fluids through porous media, chemical reactive flows, combustions. That shows a wide application prospect to us and opens a revolutionary avenue in the computational fluid dynamics.The present dissertation is discussed focusing on the LBM. First we introduce briefly the history, the fundamental idea and the characters, and the applications and the development of the LBM. And a viewpoint that the LBM, a particular and novel numerical method, is very different from other conventional ones is indicated. Then the statistical mechanics and the lattice Boltzmann models are presented. The connections between the LBM and the LGA, and the continous Boltzmann equation are expatiated. In paticular, we emphasize the simplest, the most common and the most classical model-BGK lattice Boltzmann model. The characteristics of the fluid, which the physical quantities of the macroscopic equation describe, are obtained by solving the microscopic kinetics model in the LBM. The main analysic means—the Chapman-Enskog expansion or multiscale analysis, is the bridge between the microscopic equation and macroscopic one.Because the analysis applied to other methods has not been systematically used to study the LBM, a unconventional one. The theory about the LBM is not per- feet. Some recent achievements on the consistency, the stability and the asymptotic behavior of the LBM are proposed in the paper. The implementation of boundary conditions for the LBM is very important, and has great effect on the accuracy and the stability of the method. The movement of the fluid is described with the kinetic equations of the distribution function of the particles, and the distribution functions are unknown on the boundary. So we must construct schemes to give the boundary condtions on the microscopic level. Some usual implementations of boundary conditions are outlined, such as bounce-back scheme, hydrodynamic boundary schemes, extrapolation schemes and the treatment of complex boundary.Our work is outlined in the following several aspects. A comprehensive survey and impersonal review on the LBM are proposed. The corresponding lattice Boltzmann models, 2-bit model and D2Q4 model, are developped to slove a sort of important model equation-the nonlinear evolution one such as the modified Burgers' equation, the two-dimensional unsteady Burgers' equation and the red-and-green light model for traffic flow. The computational efficiency of the LBM for the problems with severe gradients or multi-discontinuities is shown. The multiscale analysis is used and the maximum principle and the stability of the models are proved. The reaction-diffusion systems(1D and 2D), especially for the formation of Turing patterns in CIMA, are modeled with lattice Boltzmann 3-bit and D2Q5 models respectively. The numerical results indicate that the LBM is highly stable and efficient for the nonlinear complex system. The LBM is also investigated to solve incompressible viscous flow and free-surface problems in hydraulic dynamics. Some cases of the cavity flows (square cavity and triangular cavity), dam breaking problem (square dam and circle dam), the propagation and diffraction of dam-break wave around rectangular and circular cylinder, the backward step flow and the jet-forced flow are simulated successfully. The results further show that the lattice Boltzmann method is a robust and competitive tool for the computational fluid dynamics.