Computation of Hypersonic Shock Wave Flows of Diatomic Gases and Gas Mixtures Using the Generalized Boltzmann Equation

Abstract

Hypersonic flows about space vehicles produce flow fields in thermodynamic nonequilibrium with local Knudsen numbers L K n /   (where is the mean free path of gas molecules and L is a characteristic length) which may lie in all the three regimes – continuum, transition and rarefied. Flows in continuum regime can be modeled accurately by the Navier-Stokes (NS) equations; however the flows in transition and rarefied regimes require a kinetic approach such as the Direct Simulation Monte Carlo (DSMC) method or the solution of the Boltzmann equation. This paper describes the development of a computational methodology and a code for computing hypersonic non-equilibrium shock wave flows of diatomic gases using the Generalized Boltzmann Equation (GBE) at Knudsen numbers in transitional and rarefied flow regimes. The GBE solver has been validated by computing the 1D shock structure in nitrogen for Rotational-Translational (R-T) relaxations and comparing the numerical results with the experimental data for Mach numbers up to 15. The solver has been exercised successfully for computing the 2D blunt body flows in nitrogen and 3D flow from a rectangular jet of nitrogen in vacuum for R-T relaxations. The issues of stability of the algorithm and the possibility of reducing the number of rotational levels in the computations without compromising the accuracy of the solutions have been rigorously addressed. A new two-level kinetic model has been developed for computing the RT relaxations in a diatomic gas and has been validated by comparing the results with the solutions of complete GBE. The model is about twenty times more efficient than the GBE in computing the shock structure. It should be noted that the model is different than the BGK model; it accounts for both elastic and inelastic collisions. The computational methodology has been extended to compute the hypersonic shock structure in diatomic gases including both the RT and Vibrational-Translational (V-T) relaxations. 1-D shock structure in nitrogen has been computed including both R-T and V-T relaxations and has been validated by comparing the results with the experimental data. A computational methodology has also been developed to compute the hypersonic shock structure in a non-reactive mixture of two diatomic gases. 1-D shock structure has been computed in an inert mixture of nitrogen and oxygen for R-T relaxations. To accomplish this, the GBE is formulated and solved in “impulse space” instead of velocity space.