Gas kinetic principles in Navier-Stokes finite-volume solvers

Abstract

We present a new approach for introducing more physics into the flux function of a finite-volume solver for compressible Navier-Stokes equations. This approach is conceived as an alternative solution to classical approximate Riemann solvers.The peculiarity of the solution we designed comes from the use of the gas distribution function, which is employed in the kinetic theory of gases; once computed from the Boltzmann equation with a simplified collision source term (BGK model), this distribution function is used, then, for the computation of the cell interface flux.Two solutions are investigated for computing the distribution function. In the first solution, this function is approximated as a first-order approximation of the BGK equation; in that case the resulting numerical flux smoothly evolves between a Godunov-like flux and a flux vector splitting: this first choice generates the simple gas kinetic scheme (SCG scheme). In the second solution, we use the kinetic meaning of the local sound speed at cell interfaces, to introduce macroscopic wave informations into the gas distribution function; this way, the resulting numerical flux is a convex combination between a HLLE-type flux and a modified flux-vector-splitting: this is what we call the acoustic gas kinetic scheme (ASCGE scheme). Both solutions are inserted into a space-time separation algorithm that is solved by a positive Runge-Kutta (SSPRK) method; in addition, a MUSCL procedure for generating a high-order positivity-preserving scheme, is selected.Viscous fluxes are computed by using kinetic informations coming from both sides of the cell face of the finite-volume method. Lastly, both schemes (SCG and ASCGE schemes) are investigated and analyzed over a variety of test-cases including inviscid/viscous and steady/unsteady problems.