Finite difference lattice boltzmann method for compressible thermal fluids

A finite difference lattice Boltzmann method based on the Bhatnagar-Gross-Krook-type modeled Boltzmann equation is proposed. The method relies on a different lattice equilibrium particle distribution function and the use of a splitting method to solve the modeled lattice Boltzmann equation. The splitting technique permits the boundary conditions for the lattice Boltzmann equation to be set as conveniently as those required for the finite difference solution of the Navier-Stokes equations. It is shown that the compressible Navier-Stokes equation can be recovered fully from this approach; however, the formulation requires the solution of a Poisson equation governing a secondorder tensor. Thus constructed, the method has no arbitrary constants. The proposed method is used to simulate thermal Couette flow, aeroacoustics, and shock structures with an extended thermodynamics model. The simulations are carried out using a high-order finite difference scheme with a two-dimensional, nine-velocity lattice. All simulations are performed using a single relaxation time and a set of constants deduced from the derivation. It is found that the finite difference lattice Boltzmann method is able to correctly replicate viscous effects in thermal Couette flows, aeroacoustics, and shock structures. The solutions obtained are identical either to analytical results, or obtained by solving the compressible Navier-Stokes equations using a direct numerical simulation technique.