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Reliable chaotic transition in incompressible fluid simulations

Henry von Wahl, L. Ridgway Scott

TL;DR

The paper tackles the reliability of finite element discretizations for the chaotic, two-dimensional wake behind a cylinder across a broad Reynolds range. By comparing H(div)-conforming MSC, Taylor–Hood (with/without grad-div), and Scott–Vogelius methods, it shows that high-order, pressure-robust, and divergence-free schemes with controlled numerical dissipation best capture long-time chaotic dynamics, while low-order methods fail or become unreliable. The authors provide detailed benchmarks for drag, lift, and Strouhal period, revealing how numerical dissipation and exact incompressibility influence the transition to chaos around $\text{Re}\gtrsim 1000$. The work offers a concrete, high-quality benchmark and methodology to guide reliable CFD in chaotic wake regimes and informs practical choices for simulations of cylinder-related technologies.

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.

Reliable chaotic transition in incompressible fluid simulations

TL;DR

The paper tackles the reliability of finite element discretizations for the chaotic, two-dimensional wake behind a cylinder across a broad Reynolds range. By comparing H(div)-conforming MSC, Taylor–Hood (with/without grad-div), and Scott–Vogelius methods, it shows that high-order, pressure-robust, and divergence-free schemes with controlled numerical dissipation best capture long-time chaotic dynamics, while low-order methods fail or become unreliable. The authors provide detailed benchmarks for drag, lift, and Strouhal period, revealing how numerical dissipation and exact incompressibility influence the transition to chaos around . The work offers a concrete, high-quality benchmark and methodology to guide reliable CFD in chaotic wake regimes and informs practical choices for simulations of cylinder-related technologies.

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.
Paper Structure (21 sections, 20 equations, 6 figures, 8 tables)

This paper contains 21 sections, 20 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Setting the problem and model equations
  3. Weak formulation of the Navier--Stokes equations
  4. Drag and lift evaluation
  5. Alternate drag evaluation
  6. Problem Set-up
  7. Challenge computation
  8. Discretization methods
  9. H(div)-conforming FEM
  10. Taylor--Hood
  11. Scott--Vogelius
  12. Meshes
  13. Computing the period
  14. Computational experiments
  15. Reynolds 120--500
  16. ...and 6 more sections

Figures (6)

  • Figure 1: Relf experimental data ref:errelflowRex compared to the data computed in lrsBIBki with the Scot--Vogelius finite elements scheme and the new data presented here.
  • Figure 1: Left: Left section of mesh for the $\texttt{TH}{}$, $\texttt{SV}$, and $\texttt{MCS}{}$ methods with $h=8$. Right: Zoom in to the cylinder of the same mesh.
  • Figure 1: Vorticity of the flow with Reynolds number 500 at $t=300$ computed using the $\texttt{MCS}_{4}$ method on the mesh with $h_\text{max}=2$.
  • Figure 2: Vorticity of the flow with Reynolds number 1000 at $t=300$ computed using the $\texttt{MCS}_{4}$ method on the mesh with $h_\text{max}=2$.
  • Figure 3: Vorticity of the flow with Reynolds number 2000 computed using the $\texttt{MCS}_{4}$ method on the mesh with $h_\text{max}=2$. Top: $t=310$, Bottom: $t=350.4$.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Remark 4.1: Other $H(\text{div})$-conforming methods