Reynolds number required to accurately discriminate between proposed trends of skin friction and normal stress in wall turbulence

TL;DR

This paper reviews the Reynolds number thresholds needed to distinguish between different trends in skin friction and normal stress in wall turbulence, emphasizing the importance of high-Reynolds-number data for accurate analysis.

Contribution

It consolidates recent analyses on turbulence metrics, clarifies the Reynolds number requirements for discrimination, and challenges the attached eddy model's predictions with new data insights.

Findings

Inner peak of <uu>^+ remains finite at infinite Re

Reynolds number > 10^6 needed for trend discrimination

Agreement among recent analyses on turbulence peak behavior

Abstract

In Nagib, Chauhan and Monkewitz~\cite{NCM07} we concluded that nearly all available $C_f$ relations for zero-pressure-gradient boundary layers are in remarkable agreement over the entire range $Re_\theta$ $<$ O($10^8$), provided one coefficient is adjusted in each relation by anchoring it to accurate measurements. Regarding the peak of the streamwise turbulence intensity $<uu>^+_P$, we conclude here that accurate measurements in flows with $Re_\tau$ $>$ O($10^6$) are required, especially when looking only at the peak $<uu>^+_P$ to discriminate between recently proposed trends. We also find remarkable agreement between the three analyses of Monkewitz \cite{M22}, Chen and Sreenivasan \cite{CS22} and Monkewitz and Nagib \cite{MN15}, with some coefficients slightly modified, by underpinning them with the same accurate measurements of $<uu>^+_P$ from reliable channel and boundary layer data. All the three analyses conclude that the inner peak of $<uu>^+$ remains finite in the limit of infinite Reynolds number, which is at variance with the unlimited growth of $<uu>^+_P$ as $\ln{Re}$ $_\tau$ predicted by the attached eddy model \cite{MM19}. Accurate measurements of high-order moments and the guidance of consistent asymptotic expansions may help clarify the issue at lower $Re_\tau$ values.