Geometry of turbulent dissipation and the Navier-Stokes regularity problem

TL;DR

This paper introduces a new geometric framework based on vorticity sparseness to analyze the Navier-Stokes regularity problem, providing numerical evidence that it can help bridge the gap in understanding singularity formation in fluid flows.

Contribution

It applies a novel geometric scale of sparseness to numerical simulations, offering the first computational validation of this approach for the Navier-Stokes regularity problem.

Findings

The scale effectively detects the onset of dissipation.

Numerical results support the potential of the framework to address the regularity problem.

Evidence suggests mathematical efforts may close the scaling gap.

Abstract

The question of whether a singularity can form in an initially regular flow, described by the 3D incompressible Navier-Stokes (NS) equations, is a fundamental problem in mathematical physics. The NS regularity problem is super-critical, i.e., there is a 'scaling gap' between what can be established by mathematical analysis and what is needed to rule out a singularity. A recently introduced mathematical framework--based on a suitably defined `scale of sparseness' of the regions of intense vorticity--brought the first scaling reduction of the NS super-criticality since the 1960s. Here, we put this framework to the first numerical test using a spatially highly resolved computational simulation performed near a 'burst' of the vorticity magnitude. The results confirm that the scale is well suited to detect the onset of dissipation and provide strong numerical evidence that ongoing mathematical efforts may succeed in closing the scaling gap.