Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region

We study the global, i.e. radially averaged, high Reynolds number (asymptotic) scaling of streamwise turbulence intensity squared defined as ${I^2=\overline{u^2}/U^2}$, where $u$ and $U$ are fluctuating mean velocities, respectively (overbar is time averaging). The investigation based on mathematical abstraction that logarithmic region in wall extends across entire inner outer layers. Results matched to spatially integrated Princeton Superpipe measurements [Hultmark M, Vallikivi Bailey SCC Smits AJ. Logarithmic smooth- rough-wall pipe flow. J. Fluid Mech. Vol. 728, 376-395 (2013)]. Scaling expressions derived both for log-law power-law functions radius. A transition asymptotic found at a friction $Re_{\tau} \sim 11000$.