Lattice Boltzmann Models (LBMs)

Abstract

Lattice Boltzmann models vastly simplify Boltzmann’s original conceptual view by reducing the number of possible particle spatial positions and microscopic momenta from a continuum to just a handful and similarly discretizing time into distinct steps. Particle positions are confined to the nodes of the lattice. Variations in momenta that could have been due to a continuum of velocity directions and magnitudes and varying particle mass are reduced (in the simple 2-D model we focus on here) to 8 directions, 3 magnitudes, and a single particle mass. Figure 21 shows the Cartesian lattice and the velocities ea where a = 0, 1, ..., 8 is a direction index and e₀ = 0 denotes particles at rest. This model is known as D2Q9 as it is 2 dimensional and contains 9 velocities. This LBM classification scheme was proposed by Qian et al. (1992) and is in widespread use. Because particle mass is uniform (1 mass unit or mu in the simplest approach), these microscopic velocities and momenta are always effectively equivalent. The lattice unit (lu) is the fundamental measure of length in the LBM models and time steps (ts) are the time unit.