Local near-Beltrami structure and depletion of the nonlinearity in the 3D Navier-Stokes flows

TL;DR

This paper investigates the geometric structure of 3D Navier-Stokes flows, showing that near-Beltrami configurations with small velocity-vorticity angle are incompatible with finite-time singularity formation, thus providing insights into turbulence behavior.

Contribution

It demonstrates that the persistence of near-Beltrami structures cannot lead to singularities if the velocity-vorticity alignment is sufficiently strong.

Findings

Near-Beltrami regions are incompatible with finite-time singularities under certain alignment conditions.

Flow structures with small velocity-vorticity angle are geometrically constrained.

Results suggest geometric conditions prevent blow-up in turbulent flows.

Abstract

Computational simulations of turbulent flows indicate that the regions of low dissipation/enstrophy production feature high degree of local alignment between the velocity and the vorticty, i.e., the flow is locally near-Beltrami. Hence one could envision a geometric scenario in which the persistence of the local near-Beltramy property might be consistent with a (possible) finite-time singularity formation. The goal of this note is to show that this scenario is in fact prohibited if the sine of the angle between the velocity and the vorticty is small enough with respect to the local enstrophy.