The influence of stabilization parameters in the SUPG finite element method for hypersonic flows

This paper examines the influence of particular forms of the stabilization parameters which appear in the streamline-upwind Petrov-Galerkin (SUPG) finite element discretization of the calorically perfect Euler and Navier-Stokes equations at hypersonic conditions. The particular emphasis of the investigation is (i) the construction of the upwind-bias weighting function t and (ii) a comparison of discontinuity capturing operators. Various representations of these terms have appeared in the literature over the years, and a number of comparison studies have been performed, primarily in the inviscid 2D setting. The length scale used in the SUPG weighting function τ is shown to have a critical effect on the stagnation-point heat transfer predicted to blunt bodies in three dimensions. The structure of the discontinuity capturing operator (required to achieve stable solutions in the presence of strong shock waves) is also considered in detail. Two classical approaches are compared, which differ primarily in the coordinate system in which they operate. Of the two approaches, it is found that the operator formed in reference element computational space (the so-called ν formulation) has superior convergence properties and produced a crisper shock than a physical space formulation (the δ scheme), however both schemes produce essentially the same jump conditions. To achieve the proper jump conditions we show that it is critical that the interplay of the SUPG and shock capturing operators be adequately addressed to avoid "over-stabilizing" the problem.