Twisting vortex lines regularize Navier-Stokes turbulence

TL;DR

This paper reveals a self-regularizing mechanism in vortex line dynamics within Navier-Stokes turbulence, showing how twisting and anti-twisting interactions prevent singularities without viscosity.

Contribution

It uncovers an inviscid regularization process driven by vortex line twisting and anti-twisting, providing new insights into turbulence regularity.

Findings

Self-stretching induces vortex line twisting leading to amplification.

A spontaneous anti-twist emerges to prevent unbounded vortex growth.

Vortex regularization occurs even in the absence of viscosity.

Abstract

Fluid flows are intrinsically characterized via the topology and dynamics of underlying vortex lines. Turbulence in common fluids like water and air, mathematically described by the incompressible Navier-Stokes equations (INSE), engenders spontaneous self-stretching and twisting of vortex lines, generating a complex hierarchy of structures. While the INSE are routinely used to describe turbulence, their regularity remains unproven; the implicit assumption being that the self-stretching is ultimately regularized by viscosity, preventing any singularities. Here, we uncover an inviscid regularizing mechanism stemming from self-stretching itself, by analyzing the flow topology as perceived by an observer aligned with the vorticity vector undergoing amplification. While, initially, vorticity amplification occurs via increasing twisting of vortex lines, a regularizing anti-twist spontaneously emerges to prevent unbounded growth. By isolating a vortex, we additionally demonstrate the genericity of this self-regularizing anti-twist. Our work, directly linking dynamics of vortices to turbulence statistics, reveals how the Navier-Stokes dynamics avoids the development of singularities even without the aid of viscosity.