Computation of hypersonic shock structure in diatomic gases with rotational and vibrational relaxation using the Generalized Boltzmann Equation

The 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) at Knudsen numbers in transitional and rarefied flow regimes. In the GBE (similar to Wang-Chang Uhlenbeck equation (WC-UE) [1]), the internal and translational degrees of freedom are considered in the framework of quantum and classical mechanics respectively. The GBE [2] can be considered as a more general form of WC-UE and is also applicable when the energy levels are degenerated. The computational framework available for the standard Boltzmann equation (for a monoatomic gas with translational degrees of freedom) [3] is extended by including both the rotational and vibrational degrees of freedom in the GBE. 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. The whole problem that includes both the vibrational - translational (VT) and rotational - translational (RT) transfers is solved by applying a three-stage splitting procedure to the GBE. The three stages consist of free molecular transport, VT relaxation, and RT relaxation. For the VT relaxation, GBE is always solved. For the RT relaxation, two approaches are employed. In the first approach, for the RT relaxation GBE is solved. This approach is computationally very intensive since it requires solving the complete GBE for both vibrational and rotational degrees of freedom. In the second approach, a two-level BGK type model of RT relaxation that equilibrates rotational and translational energies is employed. The second approach is computationally less intensive than the first and therefore is much more efficient (about 20 times faster than the first approach). The paper describes the two-level RT relaxation model with a single relaxation time. The model is validated by computing the shock structure at high Mach numbers by comparing the results with the complete GBE solution for RT relaxations. Computations are then performed for the shock structure at high Mach numbers accounting for both the vibrational and rotational excitations; the second approach is employed for including the RT relaxations.