A robust and contact preserving flux splitting scheme for compressible flows

Abstract

Low-dissipation shock-capturing methods usually suffer from various forms of shock instabilities, such as the notorious carbuncle phenomenon, in simulations of hypersonic flows. Many strategies for tackling the shock instability of flux difference splitting schemes, such as HLLC and Roe schemes, have been proposed while little work has been undertaken on the shock instability of flux splitting schemes. In this paper, we are concerned with the shock instability of a convection-pressure flux splitting scheme. By means of a linearized perturbation analysis, it shows that all of the perturbations in the streamwise direction are damped, while the perturbations of density and shear velocity in the transverse direction are undamped. Due to the special symmetry, the two-dimensional Sedov blast wave problem is studied to demonstrate the relation between the transverse flux and shock instability. Based on the theoretical analyses, a shock stable flux splitting scheme is presented through introducing additional viscosities of entropy wave and shear wave to damp out all perturbations in the transverse direction. For restoring the contact surface and shear layer, a pressure-based function depending on the shock-wave strength is defined to control the amount and location of additional viscosities. Numerical results of several challenging test problems demonstrate that the new scheme has greatly improved the robustness without undermining the accuracy.