Assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number

This paper focuses on the assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number. The Taylor-Green vortex at Re=1600 is considered. The results are compared with those obtained using a pseudo-spectral solver, converged on a 512(3) grid and taken as the reference. The temporal evolution of the dissipation rate, visualisations of the vortical structures and the kinetic energy spectrum at the instant of maximal dissipation are compared to assess the results. At an effective resolution of 288(3), the fourth-order accurate discontinuous Galerkin method (DGM) solution (p=3) is already very close to the pseudo-spectral reference; the error on the dissipation rate is then essentially less than a percent, and the vorticity contours at times around the dissipation peak overlap everywhere. At a resolution of 384(3), the solutions are indistinguishable. Then, an order convergence study is performed on the slightly under-resolved grid (resolution of 192(3)). From the fourth order, the decrease of the error is no longer significant when going to a higher order. The fourth-order DGM is also compared with an energy conserving fourth-order finite difference method (FD4). The results show that, for the same number of DOF and the same order of accuracy, the errors of the DGM computation are significantly smaller. In particular, it takes 768(3) DOF to converge the FD4 solution. Finally, the method is also successfully applied on unstructured high quality meshes. It is found that the dissipation rate captured is not significantly impacted by the element type. However, the element type impacts the energy spectrum in the large wavenumber range and thus the small vortical structures. In particular, at the same resolution, the results obtained using a tetrahedral mesh are much noisier than those obtained using a hexahedral mesh. Those obtained using a prismatic mesh are already much better, yet still slightly noisier. Copyright (c) 2013 John Wiley & Sons, Ltd.