Derivation of Pitot corrections for the Zagarola & Smits Superpipe data and their composite fit

Abstract

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.