Explicit formulations of gas-kinetic flux solver for simulation of incompressible and compressible viscous flows

In this paper, a gas-kinetic flux solver (GKFS) is presented for the simulation of incompressible and compressible viscous flows. In this solver, the finite volume method is applied to discretize the Navier–Stokes equations. The inviscid and viscous fluxes at the interface are obtained simultaneously via the gas-kinetic scheme, which locally reconstruct the solution for the continuous Boltzmann equation. Different from the conventional gas-kinetic BGK scheme [1], a simple way is presented in this work to evaluate the non-equilibrium distribution function, which is calculated by the difference of equilibrium distribution functions at the cell interface and its surrounding points. As a consequence, explicit formulations for computing the conservative flow variables and fluxes are simply derived. In particular, three specific schemes are proposed and validated via several incompressible and compressible test examples. Numerical results show that all three schemes can provide accurate numerical results for incompressible flows. On the other hand, Scheme III is much more stable and consistent in simulation of compressible flows.