Grad's 13 moments to the rescue of compressible lattice Boltzmann methods

A double-distribution-function based lattice Boltzmann method (DDF-LBM) is proposed for the simulation of polyatomic gases in the supersonic regime. The model relies on an extended equilibrium state that is constructed to mimic Grad's 13 moments of the Maxwell-Boltzmann distribution. This allows the correct simulation of thermal, compressible flows with only 39 discrete velocities. The stability of this BGK-LBM is reinforced by relying on Knudsen-number-dependent relaxation times that are computed analytically. Hence, high-Reynolds number, supersonic flows can be simulated in an efficient and elegant manner. While the 1D Riemann problem shows the ability of the proposed approach to handle discontinuities in the zero-viscosity limit, the simulation of the flow past a NACA0012 airfoil (Mach number $\mathrm{Ma}=1.5$, Reynolds number $\mathrm{Re=10^4}$) confirms the excellent behavior of this model in a turbulent and supersonic regime. The proposed model is substantially more efficient than previous models and opens up a whole new world of compressible flow applications that can be realistically tackled with a purely LB approach.