A hierarchy of length scales for weak solutions of the three-dimensional Navier-Stokes equations

TL;DR

This paper introduces a hierarchy of inverse length scales for weak solutions of 3D Navier-Stokes equations, linking moments of vorticity to small-scale structures and their computational resolution.

Contribution

It defines and estimates a hierarchy of length scales based on vorticity moments, extending understanding of small-scale structures in fluid turbulence.

Findings

The smallest inverse length scale matches the inverse Kolmogorov length.

Higher moments lead to rapidly increasing Reynolds number exponents.

Implications for resolving small-scale vortical structures computationally.

Abstract

Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales for weak solutions of the three-dimensional, incompressible Navier-Stokes equations on a periodic box. The estimate for the smallest of these inverse scales coincides with the inverse Kolmogorov length but thereafter the exponents of the Reynolds number rise rapidly for correspondingly higher moments. The implications of these results for the computational resolution of small scale vortical structures are discussed.