Simulation of the Taylor-Green Vortex Using High-Order Flux Reconstruction Schemes

In this paper, the ability of high-order flux reconstruction numerical schemes to perform accurate and stable computations of compressible turbulent flows on coarse meshes is investigated. Two new flux reconstruction schemes, which are optimized for wave dissipation and dispersion properties, are compared to the nodal discontinuous Galerkin and spectral difference methods recovered via the energy-stable flux reconstruction method. The Taylor-Green vortex benchmark problem at Re=1600 is used as a simple a priori test of the numerics. Dissipation rates computed from kinetic energy, vorticity, and pressure dilatation are plotted against reference solutions. Results show that, at low mesh resolution, the flux reconstruction schemes are highly accurate across a range of orders of accuracy, although oscillations can appear in the solution at orders of six and above. Although the flux reconstruction method has a built-in stabilization mechanism, an additional means of damping these instabilities is required. The schemes vary in the amount of numerical dissipation and resolution of the turbulent spectrum. One of the optimized flux reconstruction schemes (the optimized flux reconstruction scheme) is shown to have greater spectral accuracy than any of the others tested, motivating its future usage for high-order high-fidelity computational fluid dynamics.