Universal Area Law in Turbulence

TL;DR

This paper revisits and simplifies the derivation of the Area Law in turbulence, explores its applicability limits, and relates velocity circulation statistics to turbulence properties, suggesting the scaling index is less than 2/3.

Contribution

The paper provides a simpler derivation of the Area Law in turbulence and analyzes the limits of its applicability, connecting circulation statistics to turbulence scaling functions.

Findings

The Area Law applies to circulation moments as area approaches infinity.

Numerical experiments indicate the scaling index is less than 2/3.

Velocity circulation observables are calculable using the PDF and minimal surface shape.

Abstract

We re-visit the Area Law in Turbulence discovered many years ago \cite{M93} and verified recently in numerical experiments\cite{S19}. We derive this law in a simpler way, at the same time outlining the limits of its applicability. Using the PDF for velocity circulation as a functional of the loop in coordinate space, we obtain explicit formulas for vorticity correlations in presence of velocity circulation. These functions are related to the shape of the scaling function of the PDF as well as the shape of the minimal surface inside the loop. The background of velocity circulation does not eliminate turbulence but makes observable quantities in inertial range \textbf{calculable}. The scaling dimension of velocity circulation as a function of large area remains unknown. Numerical experiments \cite{S19} suggest transition for log-log derivative of circulation moments $\left<\Gamma^p\right>$ by the loop area from Kolmogorov index $\frac{2p}{3}$ at $p <4$ down to approximately $0.58 p$ for $4 \leq p \leq 10$ within available Reynolds numbers. We argue that Area Law applies to these moments only in the limit $p\rightarrow \infty$ when they are dominated by the tails of the PDF. So, these numerical experiments suggest that the scaling index in Area law is less then $\frac{2}{3}$.