A Kinetic BGK Scheme for Non-equilibrium Gaseous Flow Computation

Abstract

Accurately computing the inner structure of normal shock waves or oblique shock waves is crucial for many hypersonic applications. As such, it will improve the prediction accuracy of aerodynamics properties and aerothermal effects on hypersonic vehicles and spacecraft during atmospheric entries. Because a shock wave usually has a thickness of a few mean free paths, it is quite difficult to accurately compute the detailed non-equilibrium inner structure across a shock wave with a continuum method. This paper reports a Gas-kinetic Bhatnagar-Gross-Krook (BGK) scheme for computations of one-dimensional, vibrationally non-equilibrium nitrogen flows through a planar shock wave. The present Gaskinetic-BGK scheme is a generalization of the work of Xu,in that it solves for the shock structure with multiple temperatures, including two translational temperatures, one rotational temperature and one vibrational temperature. The salient features of the present Gaskinetic-BGK method are multi-fold. Its applicability covers a wide simulation regime extending that of continuum flows to the transition flows; it is more computationally efficient than the traditional direct simulation Monte Carlo (DSMC) method in time for shock wave simulation; it does not require additional or special techniques to stabilize the shock wave. To provide proper downstream subsonic boundary conditions for very strong shock waves requires the determination of a proper post-shock equilibrium state where all temperatures have accomplished relaxation processes to a common equilibrium temperature. Analytical expressions of a complete set of generalized Rankine-Hugoniot Relations across a planar shock wave are obtained to account for the variant specific heat ratio due to inner energy excitations. Numerical simulation results by the present Gaskinetic-BGK scheme and the DSMC method are found to be in good agreement.