Plane Waves in a Multispeed Discrete-Velocity Gas

Abstract

A kinetic flux-splitting procedure used in conjunction with local thermodynamic equilibrium in a finite volume allows us to investigate numerically discrete-velocity gas flows. The procedure, outlined for a general discrete-velocity gas, is used to simulate flows of the nine-velocity gas, a simple two dimensional multiple-speed discrete-velocity gas, wherein a multiplicity of speeds ensures nontrivial thermodynamics. After verifying the linear wave limit and the non-linear steepening of wavefronts, the stability and propagation of planar discontinuities in that model gas is studied. The supersonic-subsonic requirement for the stable propagation of a discontinuity, being kinematic in nature is the same in the model gas, as e.g., in a perfect gas. However, the finiteness of the velocity space in the model gas does not allow a translation of the above kinematic condition to the thermodynamic requirement of increasing entropy across a compressive shock: a case of an entropy decreasing compressive shock in the model gas is presented. Finally, the interaction of various types of waves--shock waves, rarefactions and contact surfaces--in the model gas are shown in a simulation of the shock tube problem.