Lattice Boltzmann methods with anisotropic equilibrium distributions

Lattice Boltzmann methods are usually derived under the assumption of isotropy. In this work, we present a derivation of a Lattice Boltzmann method for anisotropic fluid flow. Starting from an anisotropic equilibrium distribution, we show a full derivation of the resulting lattice Boltzmann method. We ensure that our method correctly reproduces macroscopic behavior via Chapman-Enskog analysis for a single-relaxation time collision operator. As a result, we are able to show that a properly discretized anisotropic Maxwell-Boltzmann equilibrium does macroscopically in fact lead to an anisotropic variation of the Navier-Stokes equations. All desired properties of lattice Boltzmann methods, such as locality of the collision operator, isotropic discrete position and velocity space, or mass and momentum conservation are retained. While it is explicitly shown in the context of fluid flow, the presented scheme is straightforward to adopt to advection-diffusion problems.