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Law of bounded dissipation and its consequences in turbulent wall flows
Published online by Cambridge University Press: 23 December 2021
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
$q$.
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- © The Author(s), 2021. Published by Cambridge University Press
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REFERENCES
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