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Properties of turbulent channel flow similarity solutions

Published online by Cambridge University Press:  11 March 2021

J.C. Klewicki Open the ORCID record for J.C. Klewicki [Opens in a new window]
J.C. Klewicki*
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Victoria3010, Australia
*
†Email address for correspondence: klewicki@unimelb.edu.au
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Abstract

High resolution direct numerical simulation (DNS) data are used to investigate properties of similarity solutions for the mean velocity ($U$) and Reynolds shear stress ($T$) in turbulent channel flow. The analysis uses the method of Fife et al. (Multiscale Model. Simul., vol. 4, 2005, p. 936) for determining the asymptotic scaling properties of indeterminate equations. Subject to its primary assumption, the analysis yields an invariant form of the mean momentum equation that is valid over a significant portion of the flow domain. Because the requisite coordinate transformations yield a self-similar relationship between scaled wall-normal derivatives of $U$ and $T$, this procedure inherently closes the mean equation. These transformations incorporate use of the so-called Fife similarity parameter, $\phi$. The asymptotic constancy of $\phi$ ($\to \phi _c$) on the inertial sublayer is analytically related to the von Kármán constant by $\kappa = \phi _c^{-2}$. Integrations of the transformed mean equation are shown to independently yield similarity solutions for both $U$ and $T$, and the veracity of these solutions is examined in detail. DNS data at $\delta ^+ \simeq 8000$ reveal that the primary assumption of the analysis is satisfied to within a deviation of less than $0.1\,\%$. The present solutions and their associated similarity structure are further used to generate a number of new results. These include a cogent specification for the both the inner and outer boundaries of the inertial sublayer and a variety of well-founded ways to estimate $\phi _c$ at finite Reynolds number. Extensions of the analytical arguments by Klewicki et al. (Phys. Rev. E, vol. 90, 2014, 063015) support the conjecture that $\phi _c \to \varPhi = (1 + \sqrt {5})/2$ (or equivalently, $\kappa = 2/(3 + \sqrt {5})$) at large Reynolds number. Connections between the channel solutions and those in the other canonical wall flows are briefly discussed, as are potential implications of the present results relative to wall-flow model development.

JFM classification

Turbulent Flows: Shear layer turbulence Turbulent Flows: Turbulent boundary layers Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A39
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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