A Recovery-Assisted Discontinuous Galerkin Method for Direct Numerical Simulation of Compressible Turbulence

Abstract

Computational Fluid Dynamics (CFD) serves as a valuable complement to analytical and experimental methods in the study of fluid mechanics. However, the engineering and fundamental research communities are continually plagued by the conflict between solution accuracy and computational resource availability. Increasingly accurate simulations demand more computational memory and more wall time, so many flows of interest, particularly turbulent flows at moderate to high Reynolds numbers, cannot be simulated with high fidelity. This issue motivates further development of high-order CFD methods, which promise to deliver better return on investment with respect to accuracy versus computational resource consumption compared to second-order methods (the standard in applied CFD). In the high-order community, the discontinuous Galerkin (DG) method is a popular approach due to its capability for arbitrarily high orders of accuracy and near-trivial extension to unstructured meshes and distributed-memory architectures. In this work, the state-of-the-art DG method is paired with a relatively new tool known as the recovery operator. The objective is to improve the resolution properties of the DG method in the context of advection-diffusion systems (such as the compressible Navier-Stokes equations), thereby facilitating more accurate simulations of turbulent flows for a given gridpoint count and solution order p.

To discretize the advective fluxes of the governing PDE(s), a biased version of the recovery operator is combined with the traditional upwind DG formulation. This simple modification improves the DG method's order of accuracy while retaining its well-regarded tendency to damp away spurious nonphysical oscillations. Where the traditional upwind DG formulation approximates the exact translation operator with order 2p+1 accuracy, the new scheme achieves order 2p+2 accuracy. Optimal convergence is achieved in the global L_2 norm. For the diffusive flux terms, the recovery operator is combined with the traditional mixed formulation to leverage the extraordinary accuracy of the recovery operator while maintaining stability on a compact computational stencil. The result of our efforts is a new advection-diffusion discretization, now known as the Recovery-assisted DG method. Detailed analysis and a comprehensive suite of test problems are presented to demonstrate that for flows containing a broad range of length scales, the new approach is consistently more accurate than the state-of-the-art DG method for the compressible Navier-Stokes equations. As a bonus topic, we show how the familiar Fourier analysis technique can be extended to predict the performance of Flux Reconstruction methods (including a novel Recovery-assisted formulation) on unstructured meshes.

Overall, this study indicates that with regard to error minimization at fixed computational cost, manipulation of interface flux terms in the DG weak form is a reliable path to scheme improvement; in our case, the performance gain is achieved via the recovery operator. Fourier analysis proves the Recovery-assisted method superior to the conventional approach for linear problems. The exceptional performance of the new method extends to 3D turbulence simulations, validating the general scheme development philosophy of performing detailed analysis with simple model PDEs. By design, the Recovery-assisted method is relatively easy to implement in an existing DG code and avoids a prohibitive increase in computational cost; these traits, combined with the method's superior resolution properties, make it a valuable new tool for CFD analysis.