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Numerical Simulation of the Taylor-Green Vortex at Re = 1600 with the Discontinuous Galerkin Spectral Element Method

In the following work, we present the results of selected simulations of the classical Taylor-Green vortex problem with a variant of the Discontinuous Galerkin method (DG) labeled the “Discontinuous Galerkin Spectral Element Method” (DGSEM). In the classical DGSEM formulation, the non-linear fluxes are colocated on the solution grid, leading to a highly efficient scheme but possible aliasing errors. Polynomial de-aliasing techniques proposed by Kirby and Karniadakis [5] avoid these errors, but incur a higher computational cost. We show results for the co-location and fully de-aliased versions of DGSEM, along with results for a locally adaptive de-aliasing approach. Taylor-Green Vortex flow The Taylor-Green vortex flow problem constitutes the simplest flow for which a turbulent energy cascade can be observed numerically. Starting from an initial analytical solution containing only a single length scale, the flow field undergoes a rapid build-up of a fully turbulent dissipative spectrum because of non-linear interactions of the developing eddies (Fig. 1). The resulting flow field exhibits the features of an isotropic, homogeneous turbulence and is often used in code validation or evaluation of numerical approaches to subgrid scale modeling [2], [3], [4]. All our computations were run on a structured Cartesian grid of hexahedral elements, covering a tripleperiodic box of size [−π, π]. The physical time frame from 0s to 20s was covered according to the problem description, starting from the initial analytical solution with given velocity and pressure fields, a constant temperature and an essentially incompressible flow field with a Mach number of Ma = 0.1. Figure 1: Taylor-Green Vortex (Re = 5000). Isocontours of vorticity magnitude, colored by helicity at t = 0.5s, 1.9s and 9.0s