Gas kinetic principles in Navier–Stokes finite-volume solvers

Abstract

We present two simplified gas kinetic models for the computation of the Navier–Stokes equations for compressible flows. Upon starting from the non-linear BGK kinetic model, we derive a simplified procedure for computing the normalized particle velocity distribution function, ψ.From a first-order approximation in time and a directional splitting of the BGK equation, the function ψ is computed over each time-step and at each cell interface of a finite-volume algorithm, as a convex combination between its value at the beginning of each time-step and an equilibrium state of the gas.Hence, depending on the definition of this equilibrium state, we can generate two kinetic theory inspired schemes.The first scheme (SCG scheme), is designed by computing the initial equilibrium state from two Maxwellian distributions of the gas, at each cell interface. The second scheme (ASCGE scheme) is designed by introducing acoustic principles into the definition of this initial equilibrium state. Then, the resulting flux function for convective terms is computed from the gas distribution function, ψ, at each time-step.Viscous fluxes are weakly coupled with convective fluxes by using kinematic features coming from both sides of the generic cell face.Finally, these convective and diffusive fluxes are introduced into a fourth-order MUSCL-like finite-volume procedure for modeling compressible flows.These two resulting simplified gas kinetic models are tested and compared with an HLLC Riemann solver scheme and recent simplified gas kinetic schemes over a variety of physical problems including inviscid/viscous and steady/unsteady problems for a compressible flow.