The Lattice Boltzmann Methods and Their Applications to Fluid Flows

The history of the Lattice Boltzmann Method and its application to fluid mechanics are investigated here. Detailed formulations are provided to form a basis for the Lattice Boltzmann Method and its many variations. These variations are designed to overcome shortcomings in the standard single relaxation time Lattice Boltzmann model. Presented here are: a model that utilizes the non-equilibrium parts of the stress tensor, the Regularized Lattice Boltzmann model; a model that converts over to momentum space, the Multi-Relaxation Time Lattice Boltzmann model; and a model that corrects itself using the entropy equation, the entropic Lattice Boltzmann model. Extensions for the Lattice Boltzmann method are derived that include: external forces, multiphase flows, and thermal flows. Various types of boundary conditions are modeled using different approaches. A detailed explanation on extracting common macroscopic flow properties in physical units is provided. These extracted properties can be used to check temporal and spatial convergence. A two dimensional, nine velocity model and a three dimensional, fifteen velocity model are used to provide examples of a number of the approaches mentioned. A two dimensional and three dimensional lid-driven cavity flow is used to illustrate these methods.