Towards an Accurate and Robust Rotated Riemann Solver for Hypersonic Flow Computations

Abstract

Although shock-capturing methods have been well developed in the past few decades, there are still some challenging issues that need to be addressed with caution, especially in hypersonic flows. It is well known that Godunov-type schemes will suffer from shock anomalies such as carbuncle phenomenon and post shock oscillations for hypersonic flow computations. What is worse is that numerical schemes which have minimal dissipation on linear waves are more prone to the shock instability. To circumvent the instability problem, a rotated methodology is proposed to combine two interlinked Godunov-type schemes, i.e. a low dissipative HLLEM scheme and a shear dissipative HLLEC scheme. It is demonstrated that the new scheme can be implemented in a simple and economic manner in the form of the Roe solver, which provides comparative resolution to Roe-type and Osher-type schemes while being free from the shock instability. A stability analysis is conducted to verify its robustness against strong shocks. Numerical results of several carefully chosen strong shock wave problems are investigated to demonstrate the robustness of the proposed method.