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Intermediate scaling and logarithmic invariance in turbulent pipe flow

Published online by Cambridge University Press:  23 February 2021

Sourabh S. Diwan Open the ORCID record for Sourabh S. Diwan [Opens in a new window]  and
Jonathan F. Morrison Open the ORCID record for Jonathan F. Morrison [Opens in a new window]
Sourabh S. Diwan*
Affiliation:
Department of Aeronautics, Imperial College London, SW7 2AZ, UK
Jonathan F. Morrison
Affiliation:
Department of Aeronautics, Imperial College London, SW7 2AZ, UK
*
Email address for correspondence: [email protected]
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Abstract

A three-layer asymptotic structure for turbulent pipe flow is proposed revealing, in terms of intermediate variables, the existence of a Reynolds-number-invariant logarithmic region for the streamwise mean velocity and variance. The formulation proposes a local velocity scale (which is not the friction velocity) for the intermediate layer and results in two overlap layers. We find that the near-wall overlap layer is governed by a power law for the pipe for all Reynolds numbers, whereas the log law emerges in the second overlap layer (the inertial sublayer) for sufficiently high Reynolds numbers ($Re_{\tau }$). This provides a theoretical basis for explaining the presence of a power law for the mean velocity at low $Re_{\tau }$ and the coexistence of power and log laws at higher $Re_{\tau }$. The classical von Kármán ($\kappa$) and Townsend–Perry ($A_1$) constants are determined from the intermediate-scaled log-law constants; $\kappa$ shows a weak trend at sufficiently high $Re_{\tau }$ but falls within the commonly accepted uncertainty band, whereas $A_1$ exhibits a systematic Reynolds-number dependence until the largest available $Re_{\tau }$. The key insight emerging from the analysis is that the scale separation between two adjacent layers in the pipe is proportional to $\sqrt {Re_{\tau }}$ (rather than $Re_{\tau }$) and therefore the approach to an asymptotically invariant state can be expected to be slow.

JFM classification

Boundary Layers: Boundary layer structure Turbulent Flows: Turbulent boundary layers Boundary Layers: Pipe flow boundary layer
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , R1
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

Present address: Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012.

References

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