Continuum Onset Parameter Based on Entropy Gradients Using Boltzmann's H-Theorem

Abstract

In an eort to provide a means to quantify the validity and breakdown of the continuum equations of uid o w, we examine the concept of entropy generation. It is reasoned that since this quantity is fundamentally related to the physics of nonequilibrium it should provide a good tool for the examination of such phenomena. Furthermore, since one may formulate this parameter utilizing statistical mechanics and kinetic theory, there are no inherent mathematical limitations necessary in its calculation. We present an analysis based on statistical mechanics and kinetic theory which leads to a form of entropy generation rate in terms of energy distribution functions. This analysis is then applied to monatomic and diatomic molecules using the rigid-rotator and harmonic oscillator assumptions. A numerical procedure is presented which allows for computation of such values via DSMC. Results are presented for normal shock waves in argon at Mach numbers of 1.55, 5, and 10. These results are then compared to the entropy generation rate predicted via the one-dimensional Navier-Stokes equations for the same shocks. It is shown that the increase in entropy production entering the shock is predicted later by the Navier-Stokes equations than by DSMC, showing evidence that breakdown parameters based upon continuum data may fail to capture the initial nonequilibrium eects. The results also show at lower Mach numbers that the entropy production spike is on the order of the scatter in the DSMC data, which somewhat speaks to the ability of this parameter to characterize equilibrium onset.