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Reynolds number scaling of the peak turbulence intensity in wall flows

Published online by Cambridge University Press:  15 December 2020

Xi Chen Open the ORCID record for Xi Chen [Opens in a new window]  and
Katepalli R. Sreenivasan
Xi Chen*
Affiliation:
Key Laboratory of Fluid Mechanics of Ministry of Education, Beihang University (Beijing University of Aeronautics and Astronautics), 100191Beijing, PR China
Katepalli R. Sreenivasan
Affiliation:
Tandon School of Engineering, Courant Institute of Mathematical Sciences, Department of Physics, New York University, New York, NY 10012, USA
*
†Email address for correspondence: [email protected]
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Abstract

The celebrated wall-scaling works for many statistical averages in turbulent flows near smooth walls, but the streamwise velocity fluctuation, $u^{\prime }$, is thought to be among the few exceptions. In particular, the near-wall mean-square peak, $\overline {u'u'}^+_p$ – where the superscript $+$ indicates normalization by the friction velocity $u_\tau$, the subscript $p$ indicates the peak value and the overbar indicates time averaging – is known to increase with increasing Reynolds number. The existing explanations suggest a logarithmic growth with respect to $Re$, where $Re$ is the Reynolds number based on $u_\tau$ and the thickness of the wall flow. We show that this boundless growth calls into question the veracity of wall-scaling and so cannot be sustained, and we establish an alternative formula for the peak magnitude that approaches a finite limit $\overline {u'u'}^+_\infty$ owing to the natural constraint of boundedness on the dissipation rate at the wall. This new formula agrees well with the existing data and, in contrast to the logarithmic growth, supports the classical wall-scaling for turbulent intensity at asymptotically high Reynolds numbers.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers Boundary Layers: Pipe flow boundary layer
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , R3
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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