Reynolds number asymptotics of wall-turbulence fluctuations

Abstract

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 $\phi (y^+)=f_0(y^+)+f_1(y^+)/Re^{1/4}_\tau$ , 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 $\phi (y^\ast )=\alpha _\phi -\beta _{\phi }y^{{\ast {1}/{4}}}$ is derived for the outer region where, for each $\phi$ , the constants $\alpha _\phi$ and $\beta _{\phi }$ are independent of $Re_\tau$ and $y^\ast$ $\equiv y^+/Re_\tau$ – 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 $\phi \approx \alpha _\phi$ 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.

Turbulent Channel Flow: Direct Numerical Simulation-Data-Driven Modeling

“…These values are in general agreement with some of the most recent von Kármán constant estimates reported in the literature. For further details the reader is referred to references [28][29][30]. 1.…”

The Asymptotic Structure of Canonical Wall-Bounded Turbulent Flows

“…The understanding of the structure of wall-bounded turbulent flows has been a vibrant topic of classical fluid mechanics for almost a century [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The problem that the Reynolds number (Re) usually has a strong influence on the flow structure, and our ability to reliably study turbulent flows at very high Re using direct numerical simulation (DNS) or experiments is rather limited [15].…”

The Law of the Wall and von Kármán Constant: An Ongoing Controversial Debate

“…The Princeton superpipe (PSP) measurements (involving data up to Re τ = 530,023) were available at this time [46], but concerns about the data accuracy still exist [47]. Supported by increasing access to high Re data, the controversial debate of the validity and universality of the law of the wall vibrantly continuous over the last 15 years: traditional views in favor of universality are challenged by opposite views [48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]. Recent analyses in favor of universality were presented, e.g., in Refs.…”