Kinetic View of the Influence of Internal Energy States on Shock 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 were 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 of Tcheremissine1, 2 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 twoand three-quantum system is considered to study the effect of the varying (1) the inelastic cross section and (2) the energy gap between the quantum energy states. Consistent results for the Mach 3 and 5 standing shock waves simulations were achieved in the study. The boundary conditions for the standing shock are obtained from modified Rankine-Hugoniot jump conditions accounting for the contribution of the internal energy states. For the two level system there exists a range of temperatures, close to the maximum in the specific heat, within which an intensive population of particles from the ground level to the excited level takes place. In this range the ratio of the specific heats decreases from 5/3 approaching 3/2 resulting in substantial changes in the downstream properties.