Assessment of shock capturing methods for numerical simulations of compressible turbulence with shock waves

The direct numerical simulation (DNS) and large eddy simulation (LES) of turbulent flows raise challenges to the numerical schemes, especially when they are combined with shock waves. The two demands of the schemes seem to be contrary: the simulation of small turbulence scales requires non-dissipative schemes while the capture of shock waves requires dissipative schemes. To have a systematic comparison of some popular schemes' properties in solving the problems above, a suite of test cases were displayed, Taylor-Green vortex, Shu-Osher problem, and compressible isotropic turbulence. Through systematic analysis and comparisons, results show that the widely used high order schemes are capable of satisfying the two demands required above in some extent. However, both of them have their advantages and disadvantages compared to each other and no one can be called perfect. Thus, they are still far from the theoretic object, and their improvements are in high demands.