Identifying the multifractal set on which energy dissipates in a turbulent Navier-Stokes fluid

TL;DR

This paper links the multifractal properties of turbulence to Leray's solutions of Navier-Stokes equations, identifying the set where energy dissipates and its associated dimensions and scales.

Contribution

It explicitly relates multifractal turbulence properties to mathematical solutions of Navier-Stokes, deriving the dimensions and scales of energy dissipation sets.

Findings

Energy dissipation set $_{m}$ has dimension $ ext{Dim}=3/m$.

Range of dissipation scales spans from $Re^{3/4}$ to $Re^{3}$.

Multifractal model parameter $h$ obeys $h ext{ } ext{min} ext{ } ext{to} ext{ } -2/3 ext{ } ext{to} ext{ } 1/3$.

Abstract

The rich multifractal properties of fluid turbulence illustrated by the work of Parisi and Frisch are related explicitly to Leray's weak solutions of the three-dimensional Navier-Stokes equations. Directly from this correspondence it is found that the set on which energy dissipates, $\mathbb{F}_{m}$, has a range of dimensions $\Dim=3/m$ ($1 \leq m \leq \infty$), and a corresponding range of sub-Kolmogorov dissipation inverse length scales $L\eta_{m}^{-1} \leq Re^{3/(1+\Dim)}$ spanning $Re^{3/4}$ to $Re^{3}$. Correspondingly, the multifractal model scaling parameter $h$, must obey $h \geq h_{min}$ with $-\twothirds \leq h_{min} \leq \third$.