Split form ALE discontinuous Galerkin methods with applications to under-resolved turbulent low-Mach number flows

Abstract

e construction of discontinuous Galerkin (DG) methods for the compressible Euler or Navier-Stokes equations (NSE) includes the approximation of non-linear ux terms in the volume integrals. e terms can lead to aliasing and stability issues in turbulence simulations with moderate Mach numbers (Ma 0.3), e.g. due to under-resolution of vortical dominated structures typical in large eddy simulations (LES). e kinetic energy or entropy are elevated in smooth, but under-resolved parts of the solution which are a ected by aliasing. It is known that the kinetic energy is not a conserved quantity for compressible ows, but for small Mach numbers minor deviations from a conserved evolution can be expected. While it is formally possible to construct kinetic energy preserving (KEP) and entropy conserving (EC) DG methods for the Euler equations, due to the viscous terms in case of the NSE, we aim to construct kinetic energy dissipative (KED) or entropy stable (ES) DG methods on moving curved hexahedral meshes.e Arbitrary Lagrangian-Eulerian (ALE) approach is used to include the e ect of mesh motion in the split form DG methods. First, we use the three dimensional Taylor-Green vortex to investigate and analyze our theoretical ndings and the behavior of the novel split form ALE DG schemes for a turbulent vortical dominated ow. Second, we apply the framework to a complex aerodynamics application. An implicit LES split form ALE DG approach is used to simulate the transitional ow around a plunging SD7003 airfoil at Reynolds number Re = 40, 000 and Mach number Ma = 0.1. We compare the standard nodal ALE DG scheme, the ALE DG variant with consistent overintegration of the non-linear terms and the novel KED and ES split form ALE DG methods in terms of robustness, accuracy and computational e ciency.

Numerical evidence of anomalous energy dissipation in incompressible Euler flows: Towards grid-converged results for the inviscid Taylor-Green problem

“…In [41,42,43], it is argued that a fundamentally different behavior in terms of energy dissipation and time reversibility is expected between viscous flows in the limit Re → ∞ and inviscid flows at Re = ∞. Especially in numerical studies, it is often expected that energy conservation holds for an exact solution of the Euler equations (since ν = 0), see for example [21,23,27,39,40,44,45] and the recent review article [46] to mention just a few. It is therefore often considered a desirable quality criterion if a numerical method preserves the kinetic energy exactly in the inviscid limit ν = 0.…”