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

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.