A novel velocity discretization for lattice Boltzmann method: Application to compressible flow

The lattice Boltzmann method (LBM) has emerged as a powerful tool in computational fluid dynamics and materials science. However, due to the failure to accurately reproduce the third moment of the equilibrium distribution function, the standard LBM formulation does not recover the correct compressible Navier-Stokes equations at macroscale. It only recovers Navier-Stokes in the incompressible limit. The errors include terms that destroy Galilean invariance in the presence of density variations. In this paper, we introduce a new velocity discretization method to overcome some of these challenges. In this new formulation, the particle populations are discretized using a bump function that has a mean and a variance. This introduces enough independent degrees of freedom to set the equilibrium moments to the moments of Maxwell-Boltzmann distribution up to and including the third moments. Consequently, the correct macroscopic fluid dynamics equations for compressible fluids are recovered. This new method does neither require introducing ad hoc correction terms or coupling to an extra potential nor significantly alter the implementation. As a result, it is not considerably more complicated or computationally heavy compared to the original LBM but provides a significantly improved hydrodynamics. We validate our method using several benchmark simulations of isothermal compressible flows, including Poiseuille flow, sound wave decay, Couette flow in the presence of a density gradient, and flow over a cylinder. We show that the new formulation restores Galilean Invariance, by comparison to analytical solutions (less than 0.1% mean error), and is capable of capturing flows with large density variations, high Reynolds numbers ( similar to 1000), and Mach numbers near 1.