Numerical comparisons of finite element stabilized methods for high Reynolds numbers vortex dynamics simulations

TL;DR

This paper compares various stabilized finite element methods within the VMS framework for high Reynolds number vortex simulations, finding SUPG most effective without turbulence modeling or eddy viscosity.

Contribution

It provides a comprehensive numerical comparison of stabilized FE-VMS methods for vortex dynamics at high Reynolds numbers, highlighting the superior performance of SUPG.

Findings

SUPG method performs best among tested methods.

No need for eddy viscosity modeling in these simulations.

Higher order RB-VMS does not improve results over SUPG.

Abstract

In this paper, we consider up-to-date and classical Finite Element (FE) stabilized methods for time-dependent incompressible flows. All studied methods belong to the Variational MultiScale (VMS) framework. So, different realizations of stabilized FE-VMS methods are compared in high Reynolds numbers vortex dynamics simulations. In particular, a fully Residual-Based (RB)-VMS method is compared with the classical Streamline-Upwind Petrov--Galerkin (SUPG) method together with grad-div stabilization, a standard one-level Local Projection Stabilization (LPS) method, and a recently proposed LPS method by interpolation. These procedures do not make use of the statistical theory of equilibrium turbulence, and no ad-hoc eddy viscosity modeling is required for all methods. Applications to the simulations of high Reynolds numbers flows with vortical structures on relatively coarse grids are showcased, by focusing on two-dimensional plane mixing-layer flows. Both Inf-Sup Stable (ISS) and Equal Order (EO) FE pairs are explored, using a second-order semi-implicit Backward Differentiation Formula (BDF2) in time. Based on the numerical studies, it is concluded that the SUPG method using both ISS and EO FE pairs performs best among all methods. Furthermore, there seems to be no reason to extend SUPG method by the higher order terms of the RB-VMS method.