Inertial-layer Mean Velocity Profiles for extreme Reynolds number flows

For fully developed statistically stationary channel and pipe flows with smooth walls, any averaged quantity of the flow at a distance y from the wall can be specified by the control parameters $\rho$, $\mu$, $u_{\tau}$ and $y$ itself. By the Pi theorem, there exist only three non-dimensional groups that can be formed from the combination of $\langle u\rangle(y)$ or $\frac{d\langle u\rangle}{dy}(y)$, and those quantities. In an earlier work, by observing that center of the log-law scales as the geometric mean $\sqrt{\delta \delta_{\nu}}$, we have proposed an attached eddy framework, which results on a new friction factor formula for extreme Reynolds number, namely $f\sim \frac{1}{Re^{2/13}}$. In this work, by assuming incomplete self-similarity, we show that the new friction factor is compatible with a new MVP formula, namely $\langle u\rangle(y)=u_{\tau}\Phi(\frac{y}{\delta},\frac{y}{\sqrt{\delta \delta_{\nu}}}),$ which in wall units result in the new expression $ u^+=A (y^+)^{1/12}+\frac{B}{Re_{\tau}^{1/12}} (y^+)^{1/6} .$ This formula, with only two free parameters, results in a very good fit for the MVP data obtained from Princeton superpipe experiments, for a large range of extreme Reynolds number, $Re>10^7$, and for a large radial extension above the log-law range.