Reynolds number asymptotics of wall-turbulence fluctuations

In continuation of our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, R3; Chen & Sreenivasan, J. Fluid Mech., vol. 933, 2022a, A20 - together referred to as CS hereafter), we present a self-consistent Reynolds number asymptotics for wall-normal profiles of variances of streamwise and spanwise velocity fluctuations as well as root-mean-square pressure, across the entire flow region of channel and pipe flows and flat-plate boundary layers. It is first shown that, when normalized by peak values, the Reynolds number dependence and wall-normal variation of all three profiles can be decoupled, in excellent agreement with available data, sharing the common inner expansion of the type phi(y(+)) = f(0)(y(+)) + f(1)(y(+))/Re-tau(1/4), where phi is one of the quantities just mentioned, the functions f(0) and f(1) depend only on y(+), and Re-tau is the friction Reynolds number. Here, the superscript + indicates normalization by wall variables. We show that this result is completely consistent with CS. Secondly, by matching the above inner expansion and the outer flow similarity form, a bounded variation phi(y*) = alpha(phi) - beta(phi)y*(1/4) is derived for the outer region where, for each phi, the constants alpha(phi) and beta(phi) are independent of Re-tau and y* equivalent to y(+)/Re-tau - also in excellent agreement with simulations and experimental data. One of the predictions of the analysis is that, for asymptotically high Reynolds numbers, a finite plateau phi approximate to alpha(phi) appears in the outer region. This result sheds light on the intriguing issue of the outer shoulder of the variance of the streamwise velocity fluctuation, which should be bounded by the asymptotic plateau of approximately 10.

Automated summary

In this article, a self-consistent Reynolds number asymptotics is presented to describe the variations of velocity fluctuation variances and root-mean-square pressure with respect to the wall-normal position in channel flows, pipe flows, and flat-plate boundary layers. The study shows that when normalized by peak values, the Reynolds number dependence and wall-normal variation of these profiles can be decoupled and are in good agreement with experimental and simulation data. Additionally, it is predicted that a finite plateau appears in the outer region for asymptotically high Reynolds numbers.