3D Euler about a 2D Symmetry Plane

TL;DR

This paper presents new high-resolution simulations of anti-parallel Euler vortices to investigate potential finite-time singularities, proposing improved analysis methods and challenging previous conclusions about vortex behavior.

Contribution

It introduces more robust criteria for simulation termination, compares classical and spectral convergence tests, and provides new insights into vortex singularity scaling.

Findings

Support for enstrophy growth rate $eta_ ext{Omega} o 1/2$

Vorticity growth rate $eta > 1$

Analysis on higher resolution meshes confirms trends

Abstract

Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr (1993) and the lack thereof by Hou and Li (2006). Core profiles designed to reproduce the two results are presented, new more robust analysis is proposed, and new criteria for when calculations should be terminated are introduced and compared with classical resolution studies and spectral convergence tests. Most of the analysis is on a $512 \times 128 \times 2048$ mesh, with new analysis on a just completed $1024 \times 256 \times 2048$ used to confirm trends. One might hypothesize that there is a finite-time singularity with enstrophy growth like $\Omega \sim (T_c-t)^{-\gamma_\Omega}$ and vorticity growth like $||\omega||_\infty \sim (T_c-t)^{-\gamma}$. The new analysis would then support $\gamma_\Omega \approx 1/2$ and $\gamma > 1$. These represent modifications of the conclusions of Kerr (1993). Issues that might arise at higher resolution are discussed.