Computations of rarefied hypersonic blunt body flow in binary inert gas mixtures using the generalized Boltzmann equation

Abstract

The results of 2-D numerical simulations of hypersonic flow of a single diatomic gas, e.g., Nitrogen and a binary inert mixture of two gases (which are constituents of air namely N2, O2, Ar) past a 2-D blunt body in rotational non-equilibrium from low to high Knudsen Numbers are obtained using the Wang-Chang Uhlenbeck equation [1] or the Generalized Boltzmann Equation (GBE) [2]. The computational framework available for the classical Boltzmann equation for a monoatomic gas with translational degrees of freedom [3] is extended by including the rotational degrees of freedom in the GBE. The general computational methodology for the solution of the GBE for a diatomic gas is similar to that for the classical Boltzmann equation except that the evaluation of the collision integral becomes significantly more complex due to the quantization of rotational energy levels. There are two main difficulties encountered in computation of high Mach number flows of diatomic gases with rotational degrees of freedom using the GBE: (1) a large velocity domain is needed for accurate numerical description of molecular velocity distribution function resulting in enormous computational effort in calculation of the collision integral and (2) about 50 to 70 energy levels are needed for accurate representation of the rotational spectrum of the gas. These two problems result in very large CPU and memory requirements for shock wave computations at high Mach numbers (> 6). We employ a two level Rotational-Translational (RT) relaxation model to address this problem [4]; as a result the efficiency of calculations increases by several orders of magnitude. For numerical solution of GBE for an inert binary gas mixture, the GBE is formulated in the impulse space. The gas mixtures may consist of both monatomic and diatomic gases with arbitrary constituents, concentrations, and mass ratios. The method is exercised for various concentration ratios, mass ratios, and density ratios to evaluate its ability to simulate a wide range of binary gas mixtures of monoatomic and diatomic gases. In particular, the method is applied to simulate two of the three primary constituents of air (N2, O2, Ar) in a binary mixture at 1:1 density ratio and air concentration ratio with gases in translational and rotational non-equilibrium. The results of GBE are compared with DSMC calculations; a reasonably good agreement is obtained. The solutions presented in this paper can also serve as validation test cases for other methods as well as an important building block in developing complex 3D simulations for shock waves in a mixture of multiple gases.