Complete and incomplete similarity for the mean velocity profile of turbulent pipe and channel flows at extreme Reynolds number

The recent availability of  experimental data on turbulent pipe flows in the extreme Reynolds number range has shown strong evidence that when Re >> O(10⁵), the flow exhibits a transition to new scaling laws for  important physical quantities, such as the friction factor and the Reynolds stress. In this work, we discuss a transition concerning the scaling of the mean velocity profile (MVP) of turbulent pipe and channel flows. We will first show that for sufficiently large Re number, the MVP exhibits complete similarity asymptotics in bulk coordinates. Then, using a recent friction power-law formulation for extreme-Re flows, we show that this is equivalent to an incomplete similarity asymptotics. We then use the incomplete similarity asymptotics to relate the representation of the flow in bulk coordinates to the representation of the flow in inner coordinates, so that u⁺ can be approximated by a Re_τ-dependent functional. We  also propose a multiscale polynomial model of the MVP, which yields excellent approximations of available data in both bulk and inner coordinates.