Computation of hypersonic shock wave flows of diatomic gases using the generalized boltzmann equation

This paper describes the computational methodology for computing hypersonic non-equilibrium shock wave (SW) flows of diatomic gases such as Nitrogen and Oxygen using the Generalized Boltzmann Equation (GBE). The GBE can be considered a more general form of Wang-Chang Uhlenbeck Equation (WC-UE) and is also applicable when the energy levels are degenerated. In the GBE (similar to WC-UE), the internal and translational degrees of freedom are considered in the framework of quantum and classical mechanics respectively. The computational framework available for the classical Boltzmann equation (for a monoatomic gas with translational degrees of freedom only) is extended by including both the rotational and vibrational 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 BE except that the evaluation of the collision integral becomes significantly more complex due to the quantization of rotational and vibrational energy levels. The solution of GBE requires modeling of transition probabilities, elastic and inelastic cross-sections etc. of a diatomic gas molecule, needed for the solution of the collision integral. An efficient computational methodology has been developed for the solution of GBE for computing the shock structure in diatomic gases at high Mach numbers. 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). Our computational methodology has addressed these problems, and as a result efficiency of calculations has increased by several orders of magnitude. The code has been validated by computing the shock structure in Nitrogen for Mach numbers up to 15 including the translational and rotational degrees of freedom.