One-dimensional multiple-temperature gas-kinetic Bhatnagar-Gross-Krook scheme for shock wave computation

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 very difficult to accurately compute the detailed nonequilibrium inner structure across a shock wave with a continuum method. This paper reports a gas-kinetic Bhatnagar-Gross-Krook scheme for computations of one-dimensional vibrationally nonequilibrium nitrogen flows through a planar shock wave. The present gas-kinetic Bhatnagar-Gross-Krook scheme 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 gas-kinetic Bhatnagar-Gross-Krook method are multifold. Its applicability covers a wide simulation regime, extending that of continuum flows to the transition flows; it is more computationally efficient in time than the traditional direct simulation Monte Carlo method for shock wave simulation. To provide proper downstream subsonic boundary conditions for very strong shock waves, it is required to determine a proper postshock equilibrium state in which 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 gas-kinetic Bhatnagar-Gross-Krook scheme and the direct simulation Monte Carlo method are found to be in good agreement.