| 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. |
