A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows

Abstract

In this paper, an alternative approach to the traditional continuum analysis of flow problems is presented. The traditional methods, that have been popular with the CFD community in recent times, include potential flow, Euler and Navier–Stokes solvers. The method presented here involves solving the governing equation of the molecular gas dynamics that underlies the macroscopic behaviour described by the macroscopic governing equations. The equation solved is the Boltzmann kinetic equation in its simplified collisionless and BGK forms. The algorithm used is a discontinuous Taylor–Galerkin type and it is applied to the 2D problems of a highly rarefied gas expanding into a vacuum, flow over a vertical plate, rarefied hypersonic flow over a double ellipse, and subsonic and transonic flow over an aerofoil. The benefit of this type of solver is that it is not restricted to continuum regime (low Knudsen number) problems. However, it is a computationally expensive technique.