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

• Published in: Physics of fluids (1994) Vol. 33; no. 8; pp. 85118 - 85129
• Main Authors: Sanfins, G., Anbarlooei, H. R., Cruz, D. O. A., Ramos, F.
• Format: Journal Article
• Language: English
• Published: Melville American Institute of Physics 01.08.2021
• Subjects: Asymptotic properties; Channel flow; Computational fluid dynamics; Extreme values; Fluid dynamics; Fluid flow; Friction factor; Physics; Pipe flow; Polynomials; Representations; Reynolds number; Reynolds stress; Scaling laws; Similarity; Velocity distribution
• ISSN: 1070-6631; 1089-7666
• DOI: 10.1063/5.0060258

The recent availability of experimental data on turbulent pipe flows in the extreme Reynolds number range has shown strong evidence that when R e ≫ O ( 10 5 ), 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 show that for sufficiently large Re number, the MVP exhibits complete similarity asymptotics in bulk coordinates, as U / U ¯ → Ψ E ( 1 ) ( η ). Then, using a recent friction power-law formulation for extreme-Re flows, we show that this is equivalent to an incomplete similarity asymptotics, as u + → R e τ 12 Φ E ( 1 ) ( y + / R e τ ). We then use the incomplete similarity asymptotics to relate the representation of the flow in bulk coordinates, ( η , U / U ¯ ), to the representation of the flow in inner coordinates, ( y + , u + ), so that u+ can be approximated by a R e τ-dependent functional as u + ≈ Φ E ( y + , R e τ ). We also propose a multiscale polynomial model of the MVP, which yields excellent approximations of available data in both bulk and inner coordinates.