  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.
