NASA/TM-2013-217850 1 I. IntroductionThe understanding and prediction of turbulent fluid flow continues to be one of the pacing items in the physical sciences. The bulk of turbulent flow predictions are done using Reynolds-Averaged NavierStokes (RANS) methods. RANS solves a time-averaged form of the Navier-Stokes equations and the effect of the unsteady turbulent motion on the time-mean flowfield is approximated using a turbulence model. RANS methods perform well for "well-behaved" flows where the models have been calibrated, e.g.. attached boundary layers and mildly separated flows. They fail in more complex flows beyond the bounds of the modeling assumptions, e.g.. large separations, highly anisotropic flows, shock boundary layer interactions, etc. RANS methods rely completely on the model for the prediction of the turbulent flowfield. To reduce or remove the reliance on modeling, methods that resolve some or all of the turbulent motion are gaining usage. Direct numerical simulation (DNS) solves the unsteady Navier-Stokes equations and computes all scales of the turbulent motion and eliminates modeling entirely. Large-eddy simulation (LES) directly computes the large-scale turbulent structures, those that contain most of the turbulent kinetic energy, and uses a model to capture the effects of the smaller unresolved scales, minimizing the model's influence. The success of both DNS and LES rely on the ability of the numerical scheme to accurately compute the unsteady turbulent motion.The Taylor-Green vortex (TGV) is a canonical problem in fluid dynamics developed to study vortex dynamics, turbulent transition, turbulent decay and the energy dissipation process.1 The TGV problem contains several key physical processes in turbulence in a simple construct and therefore is an excellent case for the evaluation of turbulent flow simulation methodologies. The problem consists of a cubic volume of fluid that contains a smooth initial distribution of vorticity. Periodic boundary conditions are applied to all boundary surfaces. As time advances the vortices roll-up, stretch and interact, eventually breaking down into turbulence. Because there is no external forcing the small-scale turbulent motion will eventually dissipate all the energy in the fluid and it will come to rest.In this work we explore the TGV problem using the Wave Resolving Large-Eddy Simulation (WRLES) code 2, 3 to gain a better understanding of the interplay between the numerical methods, grid resolution and turbulence, and to validate the code. An initial set of solutions was generated for the First International Workshop on High-Order Methods in Computational Fluid Dynamics, which was held at the American Institute of Aeronautics and Astronautics's Aerospace Sciences Meeting in Nashville Tennessee on January 7-8, 2012. The TGV problem was one of several cases that participants could solve and submit their results for comparison. Additional solutions have since been generated for this paper.NASA/TM-2013-217850 2 II. Code DescriptionThe code used...

The flux reconstruction (FR) method offers a simple, efficient, and easy to implement method, and it has been shown to equate to a differential approach to discontinuous Galerkin (DG) methods. The FR method is also accurate to an arbitrary order and the isentropic Euler vortex problem is used here to empirically verify this claim. This problem is widely used in computational fluid dynamics (CFD) to verify the accuracy of a given numerical method due to its simplicity and known exact solution at any given time. While verifying our FR solver, multiple obstacles emerged that prevented us from achieving the expected order of accuracy over short and long amounts of simulation time. It was found that these complications stemmed from a few overlooked details in the original problem definition combined with the FR and DG methods achieving high-accuracy with minimal dissipation. This paper is intended to consolidate the many versions of the vortex problem found in the literature and to highlight some of the consequences if these overlooked details remain neglected.

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