Benchmark stress tests for flow past a cylinder at higher Reynolds numbers using EMAC

TL;DR

This paper evaluates the performance of the EMAC formulation of Navier-Stokes equations in flow past a cylinder at high Reynolds numbers, highlighting its advantages and limitations for different finite element discretizations.

Contribution

It introduces a benchmark for high Reynolds number flow simulations and assesses EMAC's effectiveness compared to standard methods across various finite element orders.

Findings

EMAC outperforms standard formulations with lowest-order elements.

Higher-order EMAC can become unstable on under-resolved meshes.

Flow becomes periodic at high Reynolds numbers when sufficiently resolved.

Abstract

We consider a test problem for Navier-Stokes solvers based on the flow around a cylinder at Reynolds numbers 500 and 1000, where the solution is observed to be periodic when the problem is sufficiently resolved. Computing the resulting flow is a challenge, even for exactly divergence-free discretization methods, when the scheme does not include sufficient numerical dissipation. We examine the performance of the energy, momentum and angular momentum conserving (EMAC) formulation of the Navier-Stokes equations. This incorporates more physical conservation into the finite element method even when the numerical solution is not exactly divergence-free. Consequently, it has a chance to outperform standard methods, especially for long-time simulations. We find that for lowest-order Taylor-Hood elements, EMAC outperforms the standard convective formulations. However, for higher-order elements, EMAC can become unstable on under-resolved meshes.