Reconstructed discontinuous Galerkin methods for high Reynolds number flows

Abstract

Reconstructed Discontinuous Galerkin (rDG) methods aim to provide a unified framework between Discontinuous Galerkin (DG) and finite volume (FV) methods. This unification leads to a new family of spatial discretization schemes from order three upwards. The first of these new schemes is the rDG(P1P2) method, which represents the solution on each element as linear functions while reconstructing quadratic contributions to compute the fluxes inside the element and over the faces. For the rDG(P1P2) method, two different reconstruction methods were implemented. The first of these reconstruction methods is a least-squares based reconstruction. For this reconstruction, an inverse distance weighting was introduced to improve the discretization error in anisotropic mesh regions, as well as an extended reconstruction stencil variant, which aims to stabilise the reconstruction on simplicial meshes. The inclusion of an inverse distance weighting was found to be beneficial for high Reynolds number flows on the example of the two-dimensional zero pressure gradient flat plate. As a second method, a variational reconstruction method was implemented. For the variational reconstruction rDG methods it was shown that they can offer significantly reduced discretization errors compared to DG methods for smooth flows. It was shown on the example of a method of manufactured solutions, that all implemented methods reach their designed order of accuracy and can provide lower spatial discretization errors than a DG method of a comparable order on regular and randomly perturbed hexahedral meshes as well as on tetrahedral meshes. The rDG methods was applied to several two and three-dimensional RANS test cases. For these test cases, a stronger influence of the Reynolds number on the discretization error of rDG methods was found compared to the weaker influence observed for DG methods. For all test-cases, it was shown that rDG methods converge faster on the same mesh, however, yield a higher absolute error, due to the lower number of degrees of freedom compared to native DG methods.