  The original turbulent pipe flow experiments in the Princeton "Superpipe" by
Zagarola & Smits (1997, 1998) at unprecedented laboratory Reynolds numbers have
started an ongoing vigorous debate on the logarithmic law in the mean velocity
profile $U^+(y^+)$ and the intimately related question of Pitot probe
corrections for mean shear, viscous effects and turbulence level. Considering
that the Pitot probe diameter $d^+$ exceeded 7000 wall units at the highest
Reynolds number, the various \textcolor{black}{traditional Pitot} corrections
had to be extended into uncharted territory \textcolor{black}{where they may no
longer be additive}. In this note, the inverse approach is adopted, where the
net result of all the corrections is assumed to be compatible with the model
for $U^+$ developed by \cite{Monk17}. The latter has an inner part which is, up
to higher order corrections, identical to the zero pressure gradient turbulent
boundary layer profile and switches around $y^+_{\mathrm{break}} \approx 400$
to a logarithmic overlap layer with a K\'arm\'an "parameter" $\kappa$
\textcolor{black}{that depends on pressure gradient and possibly on other flow
parameters}. The simplicity of the resulting global Pitot correction
proportional to $(d^+)^{0.9}(R^+)^{-0.4}$, with only two fitting parameters,
indirectly supports this model. \textcolor{black}{Based on the required
equality of the overlap and centerline $\kappa$'s, it is furthermore shown that
$y^+_{\mathrm{break}}$ must be a constant.} \textcolor{black}{Finally,} the
outer "wake" part of the profile is argued to be asymptotically linear between
the wall and about half the pipe radius. This gives rise to a linear higher
order tail $\propto y^+/R^+$ in the logarithmic overlap layer, which has been
\textcolor{black}{the subject} of asymptotic analysis over the last decades.

Fluid Dynamics