On the Stability and Accuracy of Flux Reconstruction ... - WCCM 2016

On the Stability and Accuracy of Flux Reconstruction Schemes for Implicit Large Eddy simulation of Turbulent Flows Brian C. Vermeire Department of Aeronautics, Imperial College London Peter E. Vincent Department of Aeronautics, Imperial College London Recently, Huynh [1] introduced the high-order unstructured flux reconstruction (FR) scheme for advectiondiffusion problems. FR can recover several other schemes including the nodal DG, SD, and SV methods. Single-parameter families of linearly stable FR schemes, referred to as the original energy stable flux reconstruction (O-ESFR) schemes, were later introduced. Recently, a range of multi-parameter stable-symmetricconservative FR schemes were discovered, referred to here as E-ESFR. It has been demonstrated that the FRDG scheme can be used for implicit large eddy simulation (ILES) of turbulent flows. However, the relative performance of other ESFR schemes have not been investigated in detail for ILES. In the current work we investigate the performance of a range of linearly stable FR schemes for ILES considering accuracy, efficiency, and stability. The objective of this work is to determine whether any linearly stable schemes, or ranges of schemes, can outperform the commonly used FRDG method for ILES of turbulent flows. We show that the dissipation and dispersion properties of linearly stable FR schemes can vary greatly. To identify useful schemes, we perform marginally-resolved simulations of the Taylor-Green vortex, which is prone to aliasing instabilities. We use the amount of time a simulation can run before succumbing to nonlinear instabilities as a measurement of its relative stability for ILES. For each scheme we systematically vary q0 and q1 and run a Taylor-Green vortex simulation to failure, or a non-dimensional time of tc = 20, whichever occurs first. Our first observation is that the FRDG method is not the most stable for this test case. For P3 there is a wide range of schemes that outperform the FRDG scheme in terms of stability. For P4 there is a group of schemes with small positive q0 and q1 that are significantly more stable than FRDG. Finally, we perform ILES of turbulent flow over an SD7003 aerofoil using an unstructured hexahedral mesh with a P4 scheme with q0 = 0.14 and q1 = 0.06, one of two schemes to complete the Taylor-Green vortex simulation. Results demonstrate that this scheme is stable and accurate for ILES using unstructured grids. 0.6

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Figure 1: Maximum simulation time contours for ILES of the TGV test case for P3 (left) and P4 (right).

Keywords: flux, reconstruction, implicit, large, eddy, simulation, extended, range, energy, stable References [1] H. T. Huynh. A flux reconstruction approach to high-order schemes including discontinuous galerkin methods. In 18th AIAA Computational Fluid Dynamics Conference, Miami, FL, June 2007. AIAA.