An hybrid numerical flux for supersonic flows with application to rocket nozzles

The numerical simulation of shock waves in supersonic flows is challenging because of several instabilities which can affect the solution. Among them, the carbuncle phenomenon can introduce unphysical perturbations in captured shock waves. In the present work, an hybrid numerical flux is proposed for the evaluation of the convective fluxes that avoids carbuncle and keeps high-accuracy on shocks and boundary layers. In particular, the proposed flux is a combination between an upwind approximate Riemann solver and the Local Lax-Friedrichs scheme. A simple strategy to mix the two fluxes is proposed and tested in the framework of a discontinuous Galerkin discretisation. The approach is investigated on the supersonic flow around a cylinder, on the supersonic flow on a flat plate and on the flow in a rocket nozzle.