On the use of the discontinuous Galerkin method for numerical simulation of two-dimensional compressible turbulence with shocks

In this paper, the discontinuous Galerkin (DG) method combined with localized artificial diffusivity is investigated in the context of numerical simulation of broadband compressible turbulent flows with shocks for under-resolved cases. Firstly, the spectral property of the DG method is analyzed using the approximate dispersion relation (ADR) method and compared with typical finite difference methods, which reveals quantitatively that significantly less grid points can be used with DG for comparable numerical error. Then several typical test cases relevant to problems of compressible turbulence are simulated, including one-dimensional shock/entropy wave interaction, two-dimensional decaying isotropic turbulence, and two-dimensional temporal mixing layers. Numerical results indicate that higher numerical accuracy can be achieved on the same number of degrees of freedom with DG than high order finite difference schemes. Furthermore, shocks are also well captured using the localized artificial diffusivity method. The results in this work can provide useful guidance for further applications of DG to direct and large eddy simulation of compressible turbulent flows.