  The scaling of different features of stream-wise normal stress profiles
$\langle uu\rangle^+(y^+)$ in turbulent wall-bounded flows, in particular in
truly parallel flows, such as channel and pipe flows, is the subject of a long
running debate. Particular points of contention are the scaling of the "inner"
and "outer" peaks of $\langle uu\rangle^+$ at $y^+\approxeq 15$ and $y^+
=\mathcal{O}(10^3)$, respectively, their infinite Reynolds number limit, and
the rate of logarithmic decay in the outer part of the flow. Inspired by the
landmark paper of Chen and Sreenivasan (2021), two terms of the inner
asymptotic expansion of $\langle uu\rangle^+$ in the small parameter
$Re_\tau^{-1/4}$ are extracted for the first time from a set of direct
numerical simulations (DNS) of channel flow. This inner expansion is completed
by a matching outer expansion, which not only fits the same set of channel DNS
within 1.5\% of the peak stress, but also provides a good match of laboratory
data in pipes and the near-wall part of boundary layers, up to the highest
$Re_\tau$'s of order $10^5$. The salient features of the new composite
expansion are first, an inner $\langle uu\rangle^+$ peak, which saturates at
11.3 and decreases as $Re_\tau^{-1/4}$, followed by a short "wall loglaw" with
a slope that becomes positive for $Re_\tau \gtrapprox 20'000$, leading up to an
outer peak, and an outer logarithmic overlap with a negative slope continuously
going to zero for $Re_\tau \to\infty$.
