High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations

Abstract

This paper presents robust discontinuous Galerkin methods for the incompressible Navier-Stokes equations on moving meshes. High-order accurate arbitrary Lagrangian-Eulerian formulations are proposed in a unified framework for both monolithic as well as projection or splitting-type Navier-Stokes solvers. The framework is flexible, allows implicit and explicit formulations of the convective term, and adaptive time-stepping. The Navier-Stokes equations with ALE transport term are solved on the deformed geometry storing one instance of the mesh that is updated from one time step to the next. Discretization in space is applied to the time discrete equations so that all weak forms and mass matrices are evaluated at the end of the current time step. This design ensures that the proposed formulations fulfill the geometric conservation law automatically, as is shown theoretically and demonstrated numerically by the example of the free-stream preservation test. We discuss the peculiarities related to the imposition of boundary conditions in intermediate steps of projectiontype methods and the ingredients needed to preserve high-order accuracy. We show numerically that the formulations proposed in this work maintain the formal order of accuracy of the Navier-Stokes solvers. Moreover, we demonstrate robustness and accuracy for under-resolved turbulent flows.

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

“…while this step is skipped, ûh = ûh , in the inviscid limit when solving the incompressible Euler equations. In a final postprocessing step, consistent divergence and continuity penalty terms are applied to weakly enforce the incompressibility constraint and normal continuity of the velocity field [54,61]…”