Does Fully-Developed Turbulence Exist? Reynolds Number Independence versus Asymptotic Covariance

TL;DR

This paper explores whether fully-developed turbulence exists at high Reynolds numbers, proposing that corrections to Kolmogorov's 2/3 law diminish logarithmically with Re, supported by recent experimental data.

Contribution

It introduces a theoretical framework suggesting Re-invariant statistical averages with logarithmic corrections, challenging the traditional notion of turbulence universality at high Re.

Findings

Observed structure functions are bounded by an envelope consistent with Kolmogorov's form.

Experimental data indicates the Kolmogorov constant varies with Re, supporting the proposed corrections.

Two generic scenarios for the behavior of structure functions near the envelope are discussed.

Abstract

By analogy with recent arguments concerning the mean velocity profile of wall-bounded turbulent shear flows, we suggest that there may exist corrections to the 2/3 law of Kolmogorov, which are proportional to $(\ln\,\Re)^{-1}$ at large Re. Such corrections to K41 are the only ones permitted if one insists that the functional form of statistical averages at large Re be invariant under a natural redefinition of Re. The family of curves of the observed longitudinal structure function $D_{LL}(r, \Re)$ for different values of Re is bounded by an envelope. In one generic scenario, close to the envelope, $D_{LL}(r, \Re)$ is of the form assumed by Kolmogorov, with corrections of $O((\lnRe)^{-2})$. In an alternative generic scenario, both the Kolmogorov constant $C_K$ and corrections to Kolmogorov's linear relation for the third order structure function $D_{LLL} (r)$ are proportional to $(\ln\,\Re)^{-1}$. Recent experimental data of Praskovsky and Oncley appear to show a definite dependence of $C_K$ on Re, which if confirmed, would be consistent with the arguments given here.