A class of structurally complete approximate Riemann solvers for trans- and supercritical flows with large gradients

Abstract

The wave structure of approximate Riemann solvers has a significant impact on the accuracy and computational requirements of finite volume codes. We propose a class of structurally complete approximate Riemann solvers (StARS) and provide an efficient means for analytically restoring the expansion wave to pre-existing three-wave solvers. The method analytically restores the expansion, is valid for arbitrary thermodynamics, and has comparable complexity to the popular Harten-Hyman entropy fix. The StARS modification is applied to a Roe scheme, resulting in Roe-StARS with noticeable improvements in unsteady transcritical and supercritical conditions with large flow gradients. A novel scaling analysis is performed on the flow conditions that cause rarefaction fluxes and the magnitude of errors if the rarefaction is omitted. Four test cases are examined: a transcritical shock tube, a shock tube with periodic bounds resulting in interfering shocks and rarefactions, a two-dimensional Riemann problem, and a “gradient” Riemann problem—a variant on the traditional Riemann problem featuring an initial gradient of varying slope rather than an initial step function. The results highlight the complex causes and effects of entropy violations, and encourage further study of StARS-type solvers for modern flow problems in which high flow speeds, large gradients, and non-ideal thermodynamics are increasingly common.