Cascades and Reconnection in Interacting Vortex Filaments

TL;DR

This paper investigates how the interaction between two vortex tubes at different angles and Reynolds numbers leads to various instability mechanisms and vortex structures, revealing a transition from Crow to elliptical instability and the formation of vortex sheets and filaments.

Contribution

It demonstrates that the transition between Crow and elliptical instability mechanisms depends on the initial angle between vortex tubes, highlighting a universal feature of vortex interactions at high Reynolds numbers.

Findings

Transition from Crow to elliptical instability with changing initial angle.

Formation of vortex sheets and their breakdown into filaments.

Robustness of fine-scale vortex structure formation at high Reynolds numbers.

Abstract

At high Reynolds number, the interaction between two vortex tubes leads to intense velocity gradients, which are at the heart of fluid turbulence. This vorticity amplification comes about through two different instability mechanisms of the initial vortex tubes, assumed anti-parallel and with a mirror plane of symmetry. At moderate Reynolds number, the tubes destabilize via a Crow instability, with the nonlinear development leading to strong flattening of the cores into thin sheets. These sheets then break down into filaments which can repeat the process. At higher Reynolds number, the instability proceeds via the elliptical instability, producing vortex tubes that are perpendicular to the original tube directions. In this work, we demonstrate that these same transition between Crow and Elliptical instability occurs at moderate Reynolds number when we vary the initial angle $\beta$ between two straight vortex tubes. We demonstrate that when the angle between the two tubes is close to $\pi/2$, the interaction between tubes leads to the formation of thin vortex sheets. The subsequent breakdown of these sheets involves a twisting of the paired sheets, followed by the appearance of a localized cloud of small scale vortex structures. At smaller values of the angle $\beta$ between the two tubes, the breakdown mechanism changes to an elliptic cascade-like mechanism. Whereas the interaction of two vortices depends on the initial condition, the rapid formation of fine-scales vortex structures appears to be a robust feature, possibly universal at very high Reynolds numbers.