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Approach to the 4/3 law for turbulent pipe and channel flows examined through a reformulated scale-by-scale energy budget

Published online by Cambridge University Press:  26 November 2021

Spencer J. Zimmerman Open the ORCID record for Spencer J. Zimmerman [Opens in a new window] ,
R.A. Antonia ,
L. Djenidi Open the ORCID record for L. Djenidi [Opens in a new window] ,
J. Philip Open the ORCID record for J. Philip [Opens in a new window]  and
J.C. Klewicki Open the ORCID record for J.C. Klewicki [Opens in a new window]
Spencer J. Zimmerman*
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia
R.A. Antonia
Affiliation:
School of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
L. Djenidi
Affiliation:
School of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
J. Philip
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia
J.C. Klewicki
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia
*
†Email address for correspondence: [email protected]
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Abstract

In this study, we propose a scale-by-scale (SBS) energy budget equation for flows with homogeneity in at least one direction. This SBS budget represents a modified form of the equation first proposed by Danaila et al. (J. Fluid Mech., vol. 430, 2001, pp. 87–109) for the channel centreline – the primary difference is that, here, we consider the role of pressure along with the errors associated with the isotropic approximations of the interscale divergence and Laplacian of the squared velocity increment. The term encompassing the effects of mean shear is also characterised such that the present analysis can be extended straightforwardly to locations away from the centreline. We show, based on a detailed analysis of previously published channel flow direct numerical simulations and pipe flow experiments near the centreline, how several terms in the present SBS budget equation (including the third-order velocity structure function) behave with increasing Reynolds number. The behaviour of these terms is shown to imply a rate of emergence and subsequent growth of the 4/3 law scale subrange at the channel centreline and pipe axis. The analysis also suggests that the peak magnitude of the third-order velocity structure function occurs at a scale that is fixed in proportion to the Taylor microscale at sufficiently high Reynolds number.

JFM classification

Turbulent Flows: Pipe flow Turbulent Flows: Isotropic turbulence Turbulent Flows: Homogeneous turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 931 , 25 January 2022 , A28
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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