Kinetics of thermalization in shock waves

The recently developed interlaced system which can be employed for the numerical calculation of flow problems with arbitrary cell-Knudsen numbers, Kn(c)=lambda/Deltax (lambda=mean free path length, Deltax=typical computational cell dimension), has been extended by introducing a Lennard-Jones (LJ) interaction potential. The scattering behavior of the LJ-model is quite different from that of the previously used hard-sphere (HS) model: The HS-model has a symmetric dipole-type scattering lobe, whereas the scattering lobe of the LJ-model shows a strong forward scattering, due to the large cross sections for low energy collisions. The influence of this scattering behavior on the collisional loss frequency nu and the gain function f(g)=G/nu (with G being the gain rate) in the kinetic equation is considerable and may be roughly stated as follows: For the HS-model the loss rate is small but the collisional redistribution is very efficient, for the LJ-model the situation is reversed. The kinetics of thermalization controlled by the loss and gain terms is studied, in a first step, using a pseudo-shock (with space homogeneity). In a second step, the structure of normal shock waves is calculated by using both models (HS; LJ for helium and argon). Comparison of density profiles with existing shock tube experiments show fair agreement for the LJ-model but too small shock thickness for the HS-model. Results for the stress term sigma(xx) and the heat flux q(x) are in good agreement with simple estimates for large Mach numbers, M-1>1, where we find, for the extrema of these moments, that \sigma(xx)\similar toM(1)(2) and \q(x)\similar toM(1)(3). The complex interaction of loss and gain terms and their dependence on the energy of the interacting particles has some influence on the shock thickness. We find that, with decreasing upstream temperature T-1, also the shock wave thickness decreases, while the minima of sigma(xx) and q(x) practically remain unchanged. When testing the collisional terms used in the BGK-model, with a constant loss term nu and a gain function f(g)=f(M) (M=Maxwellian), we find the typical upstream tail in the density and temperature profiles, as it was calculated before. (C) 2002 American Institute of Physics.