Internal energy relaxation in shock wave structure

Abstract

The Wang Chang-Uhlenbeck (WCU) equation is numerically integrated to characterize the internal structure of Mach 3 and Mach 5 shock waves in a gas with excitation in the internal energy states for the treatment of inelastic collisions. Elastic collisions are modeled with the hard sphere collision model and the transition rates for the inelastic collisions modified appropriately using probabilities based on relative velocities of the colliding particles. The collision integral is evaluated by the conservative discrete ordinate method [F. Tcheremissine, “Solution of the Boltzmann kinetic equation for high-speed flows,” Comput. Math. Math. Phys. 46, 315–329 (2006); F. Cheremisin, “Solution of the Wang Chang-Uhlenbeck equation,” Dokl. Phys. 47, 487–490 (2002)] developed for the Boltzmann equation. For the treatment of the diatomic molecules, the internal energy modes in the Boltzmann equation are described quantum mechanically given by the WCU equation. As a first step in the treatment of the inelastic collisions by the WCU equation, a two- and three-quantum system is considered to study the effect of the varying of (1) the inelastic cross section and (2) the energy gap between the quantum energy states. An alternative method, the direct simulation Monte Carlo method, is used for the Mach 3 shock wave to ensure the consistency of implementation in the two methods and there is an excellent agreement between the two methods. The results from the WCU implementation showed consistent trends for the Mach 3 and Mach5 standing shock waves simulations. Inelastic contributions change the downstream equilibrium state and allow the flow to transition to the equilibrium state further upstream.