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Velocity gradient statistics in turbulent shear flow: an extension of Kolmogorov's local equilibrium theory

Published online by Cambridge University Press:  21 October 2021

Yukio Kaneda Open the ORCID record for Yukio Kaneda [Opens in a new window]  and
Yoshinobu Yamamoto Open the ORCID record for Yoshinobu Yamamoto [Opens in a new window]
Yukio Kaneda*
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
Yoshinobu Yamamoto
Affiliation:
Department of Mechanical Engineering, University of Yamanashi, Kofu 400-8511, Japan
*
†Email address for correspondence: kaneda@math.nagoya-u.ac.jp
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Abstract

This paper presents an extension of Kolmogorov's local similarity hypotheses of turbulence to include the influence of mean shear on the statistics of the fluctuating velocity in the dissipation range of turbulent shear flow. According to the extension, the moments of the fluctuating velocity gradients are determined by the local mean rate of the turbulent energy dissipation $\left \langle \epsilon \right \rangle$ per unit mass, kinematic viscosity $\nu$ and parameter $\gamma \equiv S (\nu /\left \langle \epsilon \right \rangle )^{1/2}$, provided that $\gamma$ is small in an appropriate sense, where $S$ is an appropriate norm of the local gradients of the mean flow. The statistics of the moments are nearly isotropic for sufficiently small $\gamma$, and the anisotropy of moments decreases approximately in proportion to $\gamma$. This paper also presents a report on the second-order moments of the fluctuating velocity gradients in direct numerical simulations (DNSs) of turbulent channel flow (TCF) with the friction Reynolds number $Re_\tau$ up to $\approx 8000$. In the TCF, there is a range $y$ where $\gamma$ scales approximately $\propto y^ {-1/2}$, and the anisotropy of the moments of the gradients decreases with $y$ nearly in proportion to $y^ {-1/2}$, where $y$ is the distance from the wall. The theoretical conjectures proposed in the first part are in good agreement with the DNS results.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 929 , 25 December 2021 , A13
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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