Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations

TL;DR

This study investigates the behavior of vortex stretching in 3D Euler equations, showing that vortex lines' geometry prevents finite-time blowup and leads to bounded vorticity growth.

Contribution

It provides numerical evidence that geometric regularity of vortex lines causes dynamic depletion of vortex stretching, preventing singularity formation in 3D Euler flows.

Findings

Maximum vorticity grows no faster than double exponential.

Velocity, enstrophy, and enstrophy production remain bounded.

Vortex tubes flatten into sheets and roll up, with vortex lines remaining relatively straight.

Abstract

We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition \cite{Kerr93}[Phys. Fluids {\bf 5} (1993), 1725]. We use a pseudo-spectral method with resolution up to $1536\times 1024 \times 3072$ to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than double exponential in time, up to $t=19$, beyond the singularity time $t=18.7$ predicted by Kerr's computations \cite{Kerr93,Kerr04}. The velocity, the enstrophy and enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.