Unified gas-kinetic scheme with simplified multi-scale numerical flux for thermodynamic non-equilibrium flow in all flow regimes

In this paper, a BGK-type kinetic model for diatomic gases is proposed to describe the high-temperature thermodynamic non-equilibrium effect, which is a phenomenological relaxation model with continuous distribution modes of rotational and vibrational energies. In order to obtain the correct Prandtl number and reasonable relaxation rate of heat fluxes, the equilibrium distribution function is constructed by using a multi-dimensional Hermitian expansion around Maxwellian distribution. Based on this kinetic model equation, a unified gas-kinetic scheme (UGKS) with simplified multi-scale numerical flux is proposed for thermodynamic non-equilibrium flows involving the excitation of molecular vibrational degrees of freedom in all flow regimes. The present UGKS keeps the basic conservation laws of the macroscopic flow variables and microscopic gas distribution function in a discretized space. In order to improve the efficiency of UGKS, a simplified multi-scale numerical flux is directly constructed from the characteristic difference solution of complex kinetic model equation. Furthermore, the applications of unstructured discrete velocity space (DVS) and a simple integration error correction reduce the number of velocity mesh significantly and make the present method be rather efficient for flow simulation in all flow regimes. The new scheme is examined in a series of cases, such as Sod's shock tube, high non-equilibrium shock structure, hypersonic flow around a circular cylinder with Knudsen (Kn) number Kn=0.01, supersonic rarefied flow over a flat plate with a sharp leading edge, and hypersonic rarefied flow past a blunt wedge. The present UGKS results agree well with the benchmark data of DSMC and other validated methods.