The Structure of an Infinitely Strong Shock Wave for Hard Sphere Molecules

The structure of an infinitely strong shock wave (i.e., a shock wave with infinitely large upstream Mach number) is investigated on the basis of the Boltzmann equation. The velocity distribution function is expressed as a sum of a multiple of the Dirac delta function, centered at the upstream bulk velocity, and a remainder. Strong evidence that the remainder has a singularity in the molecular velocity space was provided by a previous Monte Carlo simulation for a hard-sphere gas [Cercignani et a/., Phys. Fluids 11, 2757 (1999)]. Then, the singularity was confirmed and clarified with sufficient accuracy by a precise numerical analysis by means of a finite-difference method. More specifically, the equation for the remainder, which contains the linear collision term linearized around the delta function and the nonlinear collision term, is solved numerically for a hard-sphere gas after the nonlinear collision term is replaced by the BGK collision model. The present paper reports on the main result of this analysis. INTRODUCTION H. Grad [1] suggested that the limit of the shock profile for the upstream Mach number going to infinity exists (at least for collision operators with a finite collision frequency) and is given by a multiple of the delta function, centered at the upstream bulk velocity, plus a comparatively smooth function, i.e., the velocity distribution function / can be decomposed as / = psS(£ — u_) + /, where 6 is the Dirac delta function, £ is the molecular velocity, and u_ is the flow velocity at upstream infinity. The equation for the remainder /, which is not hard to derive, seems to be more complicated than the Boltzmann equation itself, but the presumed smoothness of its solution should allow a simple approximate solution to be obtained. Grad investigated the simplest choice for the smooth remainder, a Maxwellian distribution, the parameters of which are determined by the conservation equations. Recently the problem was revisited by Cercignani et al. [2], who provided a survey on the shock wave problem with particular attention to an infinitely strong shock and showed that a Monte Carlo simulation for a hard sphere gas provides strong evidence for a singularity of the remainder / in velocity space. The singularity appears to be given by |£ — u_ ~. This singularity was the object of a further investigation by a deterministic, more accurate method presented by Takata et al. [3]. They obtained a deterministic (rather than Monte Carlo) numerical solution for a hardsphere gas, which confirms the previous results with considerable accuracy. The method used in their paper is based on the idea that the singularity is essentially determined by the Boltzmann collision operator L$ linearized about the delta function, first introduced by Grad [1] and studied by Caflisch [4]. That is, Takata et al. replaced the nonlinear collision term for the remainder / by a BGK collision model [5,6], keeping the hardsphere interaction in L$. This replacement, which simplifies the numerical procedure dramatically, enabled them to perform an accurate numerical analysis. The interest of the problem lies in the fact that, as conjectured by Cercignani et al. [2], the aforementioned singularity depends on the molecular model. If the remainder / multiplied by the molecular velocity component £_i_ perpendicular to the direction of gas flow is integrated over the full range of the component, and if the result is plotted as a function of the component £y parallel to the flow, the singularity manifests itself as a CP585, Rarefied Gas Dynamics: 22 International Symposium, edited by T. J. Bartel and M. A. Gallis © 2001 American Institute of Physics 0-7354-0025-3/01/$18.00 313 ta,* l i anr tics, i , , i i t ti , lit i i l , l