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Law of bounded dissipation and its consequences in turbulent wall flows

Published online by Cambridge University Press:  23 December 2021

Xi Chen Open the ORCID record for Xi Chen [Opens in a new window]  and
Katepalli R. Sreenivasan Open the ORCID record for Katepalli R. Sreenivasan [Opens in a new window]
Xi Chen*
Affiliation:
Key Laboratory of Fluid Mechanics of Ministry of Education, Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, PR China
Katepalli R. Sreenivasan
Affiliation:
Tandon School of Engineering, Courant Institute of Mathematical Sciences, Department of Physics, New York University, New York, USA
*
†Email address for correspondence: chenxi97@outlook.com
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Abstract

The dominant paradigm in turbulent wall flows is that the mean velocity near the wall, when scaled on wall variables, is independent of the friction Reynolds number $Re_\tau$. This paradigm faces challenges when applied to fluctuations but has received serious attention only recently. Here, by extending our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, p. R3) we present a promising perspective, and support it with data, that fluctuations displaying non-zero wall values, or near-wall peaks, are bounded for large values of $Re_\tau$, owing to the natural constraint that the dissipation rate is bounded. Specifically, $\varPhi _\infty - \varPhi = C_\varPhi \,Re_\tau ^{-1/4},$ where $\varPhi$ represents the maximum value of any of the following quantities: energy dissipation rate, turbulent diffusion, fluctuations of pressure, streamwise and spanwise velocities, squares of vorticity components, and the wall values of pressure and shear stresses; the subscript $\infty$ denotes the bounded asymptotic value of $\varPhi$, and the coefficient $C_\varPhi$ depends on $\varPhi$ but not on $Re_\tau$. Moreover, there exists a scaling law for the maximum value in the wall-normal direction of high-order moments, of the form $\langle \varphi ^{2q}\rangle ^{{1}/{q}}_{max}= \alpha _q-\beta _q\,Re^{-1/4}_\tau$, where $\varphi$ represents the streamwise or spanwise velocity fluctuation, and $\alpha _q$ and $\beta _q$ are independent of $Re_\tau$. Excellent agreement with available data is observed. A stochastic process for which the random variable has the form just mentioned, referred to here as the ‘linear $q$-norm Gaussian’, is proposed to explain the observed linear dependence of $\alpha _q$ on $q$.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers Boundary Layers: Pipe flow boundary layer
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A20
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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