Numerical Hydrodynamics from Gas-Kinetic Theory

Abstract

We present a new hydrodynamics code, based on the solution of the Bhatnagar-Gross-Krook model of the Boltzmann equation. The basic idea is to construct an approximate, locally valid solution of a set of non-linear integral equations for the equilibrium Maxwell-Boltzmann distribution, and use this solution to solve the BGK equation for the velocity distribution function at cell walls. Once this is known it is possible to compute time-dependent fluxes of mass, momentum, and energy between cells. All steps are carried out explicitly for the case of a perfect gas; the details of the distribution functions disappear in the final code. In common with other "Boltzmann-type" codes employing distribution functions, the Riemann problem does not arise. The novelty of the present code consists in solving for the distribution function, taking account of collisions, rather than assuming a convenient form for it. Results are presented for a number of standard test cases, with features ranging from cavitation (Sjögreen test) to the collision of strong shocks (Woodward-Colella test). All cases are run with precisely the same code, and all cells are treated in the same way. Our code appears to behave as well as current high-order difference schemes at shocks and to give better results for rarefaction waves. It is competitive with current codes that do not employ special means of diagnosing and treating contact discontinuities—the inclusion of such devices in our scheme remains an option.