Extension of the △-ε Method to Diatomic Gases

Abstract

The delta-epsilon method for solving the Boltzmann equation is extended to allow for internal energy modes. The delta-epsilon method utilizes a discrete approximation of the infinite velocity phase space. The velocity distribution function is only defined over regions where its values are deemed large enough to be of importance to the physics of the problem. This allows distribution points to exist anywhere in the phase space. A fourth order finite-difference scheme is used to model the convection terms and a Monte Carlo-like method is applied to the discrete velocity space to model the elastic collision integral. Internal energy mode effects on the distribution function are modeled by an inelastic contribution to the collision integral. The exchange of energy between translational and internal modes is modeled empirically by a relaxation rate equation that contains a characteristic relaxation time and the mean internal mode energies. Using this approach the internal mode energy distributions do not have to be computed as only mean values are used. A relation governing the variation of the velocity distribution function with temperature is developed from the Maxwellian distribution. Numerical examples of homogeneous relaxation are given and compared to solutions obtained via Direct Simulation Monte Carlo (DSMC).