Affordable robust moment closures for CFD based on the maximum-entropy hierarchy

Abstract

The use of moment closures for the prediction of continuum and moderately non-equilibrium flows offers modelling and numerical advantages over other methods. The maximum-entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations, however their are two issues that limit their practical implementation. Firstly, for closures that have a treatment for heat transfer, fluxes cannot be written in closed form and a very expensive iterative procedure is required at every flux evaluation. Secondly, for these same closures, there are physically possible moment states for which the entropy-maximization problem has no solution and the entire framework breaks down. This paper demonstrates that affordable closed-form moment closures that are inspired by the maximum-entropy framework can be proposed. It is known that closing fluxes in the maximum-entropy hierarchy approach a singularity as the region of non-solvability is approached. This paper shows that, far from a disadvantage, this singularity allows for smooth and accurate prediction of shock-wave structure, even for high Mach numbers. The presence of unphysical “sub-shocks” within shock-profile predictions of traditional closures has long been regarded as an unfortunate limitation of the entire moment-closure technique. The realization that smooth shock profiles are, in fact, possible for moment methods with a moderate number of moments greatly increases the method’s applicability to high-speed flows. In this paper, a 5-moment system for a simple one-dimensional gas and a 14-moment system for realistic gases are developed and examined. Numerical solution for shock-waves at a variety of incoming flow Mach numbers demonstrate both the robustness and the accuracy of the closures.