Mathematical reformulation of the Kolmogorov-Richardson energy cascade in terms of vortex stretching

TL;DR

This paper reformulates the Kolmogorov-Richardson energy cascade using vortex stretching, proves a new regularity criterion for Navier-Stokes flows, and derives the -5/3 energy spectrum law through vortex hierarchy and self-similarity.

Contribution

It introduces a vortex-based mathematical reformulation of the energy cascade and derives the energy spectrum law without Kolmogorov hypotheses, supported by DNS results.

Findings

Supports a new regularity criterion for Navier-Stokes flows.

Derives the -5/3 energy spectrum law from vortex hierarchy.

DNS results support the regularity criterion.

Abstract

In this paper, with the aid of direct numerical simulations (DNS) of forced turbulence in a periodic domain, we mathematically reformulate the Kolmogorov-Richardson energy cascade in terms of vortex stretching. By using the description, we prove that if the Navier-Stokes flow satisfies a new regularity criterion in terms of the enstrophy production rate, then the flow does not blow up. Our DNS results seem to support this regularity criterion. Next, we mathematically construct the hierarchy of tubular vortices, which is statistically self-similar in the inertial range. Under the assumptions of the scale-locally of the vortex stretching/compressing (i.e. energy cascade) process and the statistical independence between vortices that are not directly stretched or compressed, we can derive the $-5/3$ power law of the energy spectrum of statistically stationary turbulence without directly using the Kolmogorov hypotheses.