SYNOPTIC: Breakdown of Translational and Rotational Equilibrium in Gaseous Expansions

Abstract

An application of the direct simulation Monte Carlo method, which is claimed to give a solution of the full Boltzmann equation. The breakdown of translational equilibrium in steady cylindrical and spherical expansions was studied for both hard sphere and Maxwell molecules. The study of spherical expansions was extended to the combined translational and rotational breakdown in a gas of rough sphere molecules. The breakdown of translational equilibrium in a complete one-dimensional rarefaction wave in a hard sphere gas was also studied. The application of the method generally followed standard procedures, except for the unsteady flow, in which a Lagrangian system of cells moving with the fluid was used in place of the usual Eulerian cells fixed in physical space. The onset of nonequilibrium was marked by the divergence of the separate kinetic temperatures based on the molecular velocity components parallel and normal to the flow direction. The region of simulation of the steady expansion generally covered only a restricted range of Mach number in the vicinity of the point of breakdown. However, several runs were made over a wide range of Mach number and the temperatures were plotted against radius in order to obtain an over-all picture of the process. The results gave qualitative support to one of the major predictions of the BGK theory— that the freezing of the parallel temperature Tx occurs gradually over a wide range of Mach number and is much less rapid for Maxwell molecules than for hard sphere molecules. In the case of the normal temperature Tn, there is a qualitative difference between the results for the two molecular models. The Monte Carlo calculation for the Maxwell molecules gave a Tn curve that remains above the r~ / 3 continuum curve and could well be consistent with the r prediction of the BGK theory. However, for the simulation with hard sphere molecules, the Tn curve falls below the continuum curve. A rarefaction parameter P was defined by the ratio of the logarithmic time derivative of density p, following the motion of the fluid, to the collision frequency v in the gas. That is, The departure of the temperature ratio Tx/Tn from unity gives the best indication of translational breakdown, and it was plotted against P for all the steady expansion runs. These involved two geometries, three molecular models and Mach numbers ranging from 2 to 20. In all cases, the breakdown of equilibrium occurred at a value of P of approximately 0.04. If the hard sphere formula for collision frequencyin terms of viscosity coefficient is substituted into Eq. (1), this leads to the following empirical breakdown criterion for steady expansions