Comparison of entropy stable collocation high-order DG methods for compressible turbulent flows

Abstract

High-order methods are well-suited for the numerical simulation of complex compressible turbulent flows, but require additional stabilization techniques to capture instabilities arising from the underlying non-linear hyperbolic equations. This paper provides a detailed comparison of the effectiveness of entropy stable discontinuous Galerkin methods for the stabilization of compressible (wall-bounded) turbulent flows. For this investigation, an entropy stable discontinuous Galerkin spectral element method is applied on Gauss–Legendre and Gauss–Lobatto nodes. In the compressible regime, an additional stabilization technique for shock capturing based on a convex blending of a low-order finite volume with the high-order discontinuous Galerkin operator is utilized. The present investigation provides a systematic study from convergence tests, to the Taylor–Green vortex and finally to a more intricate turbulent wall-bounded 3D diffuser flow, encompassing both weakly compressible and compressible regimes. The comparison demonstrates that the DGSEM on Gauss–Lobatto nodes is generally less accurate for an equal amount of degrees of freedom. Conversely, it is faster than the DGSEM on Gauss–Legendre nodes due to a less severe time step restriction and simpler numerical operator. A performance comparison reveals that the DGSEM on Gauss–Lobatto nodes generally outperforms the DGSEM on Gauss nodes for under-resolved turbulence in the subsonic regime on a periodic domain. Conversely, the opposite effect can be observed for wall-bounded flows as well as the supersonic regime, the latter depending of course on the chosen shock-capturing scheme. To the author’s knowledge, this is the first time for which a comparison of entropy stable DGSEM on Gauss–Lobatto and Gauss–Legendre has been performed for compressible, wall-bounded turbulent flows with separation.