Reliable chaotic transition in incompressible fluid simulations

TL;DR

This paper evaluates various numerical methods for simulating chaotic flow around a cylinder in incompressible fluids, emphasizing the importance of pressure-robustness, high resolution, and suitable dissipation to accurately capture chaotic dynamics.

Contribution

It identifies key numerical requirements for reliable simulation of chaotic fluid flows and assesses the performance of existing finite element methods in this context.

Findings

Some established methods fail to capture correct chaotic behavior.

Pressure-robustness is crucial for accurate long-term simulations.

High resolution and proper dissipation are necessary for resolving small scales.

Abstract

We consider a test problem for Navier-Stokes solvers based on the flow around a cylinder that exhibits chaotic behavior, to examine the behavior of various numerical methods. We choose a range of Reynolds numbers for which the flow is time-dependent but can be characterized as essentially two-dimensional. The problem requires accurate resolution of chaotic dynamics over a long time interval. It also requires the use of a relatively large computational domain, part of which is curved. We review the performance of different finite element methods for the proposed range of Reynolds numbers. These tests indicate that some of the most established methods do not capture the correct behavior. The key requirements identified are pressure-robustness of the method, high resolution, and appropriate numerical dissipation when the smallest scales are under-resolved.