Reynolds number asymptotics of wall-turbulence fluctuations

In furtherance of our earlier work (Chen \& Sreenivasan, {\it J. Fluid Mech.} {\bf 908}, 2021, p. R3; {\bf 933}, 2022, p. 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, and the functions $f_0$ and $f_1$ depend only on $y^+$. Here, the superscript $+$ indicates normalization by wall variables. Secondly, by matching the above inner expansion and the inviscid outer flow similarity form, a bounded variation $\phi^+(y^\ast)=\alpha_\phi-\beta_{\phi}y^{\ast{{1}/{4}}}$ is derived beyond the peak where, for each $\phi^+$, the constants $\alpha_\phi$ and $\beta_{\phi}$ are independent of $Re_\tau$ and $y^\ast$ ($=y^+/Re_\tau$, $Re_\tau$ being the Reynolds number based on the friction velocity) -- 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 about 10.