Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers

In this paper, we propose new Euler flux functions for use in a finite-volume Euler/Navier–Stokes code, which are very simple, carbuncle-free, yet have an excellent boundary-layer-resolving capability, by combining two different Riemann solvers into one based on a rotated Riemann solver approach. We show that very economical Euler flux functions can be devised by combining the Roe solver (a full-wave solver) and the Rusanov/HLL solver (a fewer-wave solver), based on a rotated Riemann solver approach: a fewer-wave solver automatically applied in the direction normal to shocks to suppress carbuncles and a full-wave solver applied, again automatically, across shear layers to avoid an excessive amount of dissipation. The resulting flux functions can be implemented in a very simple and economical manner, in the form of the Roe solver with modified wave speeds, so that converting an existing Roe flux code into the new fluxes is an extremely simple task. They require only 7–14% extra CPU time and no problem-dependent tuning parameters. These new rotated fluxes are not only robust for shock-capturing, but also accurate for resolving shear layers. This is demonstrated by an extensive series of numerical experiments with standard finite-volume Euler and Navier–Stokes codes, including various shock instability problems and also an unstructured grid case.