Assessment of spurious numerical oscillations in high-order spectral difference solvers for supercritical flows

We have investigated single-component supercritical flows using the spectral difference (SD) method. We compare the use of a fully conservative approach with the solution of the pressure evolution equation: the former, violating mechanical equilibrium at contact discontinuities, can result in unstable computations, while the latter allows for fast, accurate, and robust computations in absence of shocks. We provide h-and p-convergence study for 1D entropy wave advection, showing grid refinement difficulties for the fully conservative method if conditions are too close to the critical point. We then perform 2D computations of a buoyant supercritical configuration showing the visibly altered numerical solution (up until blow-up) if a conservative approach is employed, while the non-conservative method allows to obtain a stable computation. Finally, we apply the SD discretization with both conservative and non-conservative methods to the solution of supercritical isothermall-wall channel flow, pointing out differences in turbulent statistics and flow visualizations.