An evaluation of the second order constitutive relations for rarefied gas dynamics based on the second law of thermodynamics.

摘要

In recent years there has been a resurgence of interest in using the Burnett constitutive equations to describe rarefied flows in which the Navier-Stokes equations no longer provide an accurate approximation. Applications of the Burnett equations have met with some success when used for calculating one-dimensional shock waves and two-dimensional hypersonic blunt body flows. Accurate solutions for more complex flows, however, have been elusive. The original objective of this research was to apply the Burnett equations to the cowl lip shock interaction problem associated with the engine inlet of hypersonic air-breathing vehicles. However, during the course of the investigation, it was discovered that the Burnett equations may fail to satisfy the second law of thermodynamics.; The entropy balance relation for the Burnett equations is constructed from two points of origin. It is first derived from classical thermodynamic theory using the Gibbs equation and the continuum conservation relations for mass, momentum and energy. A similar relation is also derived from kinetic theory using Boltzmann's H-theorem in conjunction with the Chapman-Enskog expansion. The entropy source appearing in the balance relations is found to be negative for certain situations, in violation of the second law of thermodynamics. Numerical simulations of one- dimensional shock waves are used to support this conclusion.; Moreover, the two derivations are shown to be completely equivalent to second order in the Knudsen number, indicating that the Gibbs equation is indeed compatible with the Burnett equations, contrary to common understanding. The implications are such that neither the classical Gibbs equation, nor the Burnett equations, is adequate to describe the effects of translational non-equilibrium. A better continuum approach to modeling a non-equilibrium gas might include an expanded set of thermodynamic variables and the corresponding relaxation equations. The Burnett equations do not accomplish this. However, the 13-moment approximation of Grad and the relatively new discipline of extended irreversible thermodynamics may offer such an alterative.