A Three-Dimensional Recovery-Based Discontinuous Galerkin Method for Turbulence Simulations

Abstract

Discontinuous Galerkin (DG) methods have recently received much attention because of their portability to complex geometries, scalability in parallel architectures and relatively simple extension to high order. However, their implementation for compressible turbulence problems is not straightforward, e.g., due to parameter-free limiting for orders greater than first and the lack of a consistent high-order diffusion scheme for DG. To address this last point, Van Leer proposed the idea of recovery-based discontinuous Galerkin (RDG) approaches. In the present work, an explicit recovery-based diffusion scheme is developed in three dimensions on a Cartesian grid to solve incompressible and slightly compressible turbulence problems. It is shown that the selection of an optimal recovery and enhancement basis preserves the super-convergence property of this scheme (eighth order for P = 2, where P is the order of the polynomial basis) for the three-dimensional Navier-Stokes equations. Test problems confirm the efficient and accurate application of the present recovery-based discontinuous Galerkin method to incompressible and slightly compressible turbulence problems.

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

“…The TGV was one of the problems in the 1st, 2nd and 3rd International Workshops on High-Order CFD Methods [38][39][40]) and is considered to be a challenging test for high-order methods. Various authors have used high-order discontinuous methods on the TGV test case with excellent results, including classical DG [5,36,37], DGSEM [9,29] and a preliminary study by Bull and Jameson using the FR-SD scheme [41]. Wang gives a review of the current status of high-order methods for several problems including the TGV [1].…”

“…Gassner [9], Diosady and Murman [29], Chapelier et al [5], Don et al [35], Johnsen et al [36] and Carton de Wiart et al [37]. The TGV was identified as a challenging problem for high-order methods in the 1st, 2nd and 3rd International Workshops on High-Order CFD Methods [38][39][40].…”

A Simple Method to Improve the Accuracy of Advection in Discontinuous Galerkin Methods for Navier-Stokes Simulations

“…The reader is referred to past work on this topic for additional information. 12,13 A. Combining DG and MUSCL Using Recovery for the diffusive terms, accuracy of the order 3p + 2 in the cell-average of the conservative quantities are readily achieved by using information from the neighboring cells.…”

“…In past work,, 12 it could be observed that, while kinetic energy and enstrophy were well resolved already on grids of 64 3 with the standard approach for advection, dilatation was not. It is interesting to note that such a feature has been observed with other approaches for compressible turbulence.…”

“…12,13 While the temporal evolution of the kinetic energy and enstrophy were well-resolved on grids of 64 3 , that dilatation was not, particuarly in the more compressible simulations. It is suspected that lower accuracy of the advection scheme is responsible for this phenomenon.…”