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The logarithmic variance of streamwise velocity and $k^{-1}$ conundrum in wall turbulence

Published online by Cambridge University Press:  20 December 2021

Yongyun Hwang Open the ORCID record for Yongyun Hwang [Opens in a new window] ,
Nicholas Hutchins  and
Ivan Marusic Open the ORCID record for Ivan Marusic [Opens in a new window]
Yongyun Hwang*
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington SW7 2AZ, UK
Nicholas Hutchins
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Victoria 3010, Australia
Ivan Marusic
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Victoria 3010, Australia
*
†Email address for correspondence: y.hwang@imperial.ac.uk
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Abstract

The logarithmic dependence of streamwise turbulence intensity has been observed repeatedly in recent experimental and direct numerical simulation data. However, its spectral counterpart, a well-developed $k^{-1}$ spectrum ($k$ is the spatial wavenumber in a wall-parallel direction), has not been convincingly observed from the same data. In the present study, we revisit the spectrum-based attached eddy model of Perry and co-workers, who proposed the emergence of a $k^{-1}$ spectrum in the inviscid limit, for small but finite $z/\delta$ and for finite Reynolds numbers ($z$ is the wall-normal coordinate, and $\delta$ is the outer length scale). In the upper logarithmic layer (or inertial sublayer), a reexamination reveals that the intensity of the spectrum must vary with the wall-normal location at order of $z/\delta$, consistent with the early observation argued with ‘incomplete similarity’. The streamwise turbulence intensity is subsequently calculated, demonstrating that the existence of a well-developed $k^{-1}$ spectrum is not a necessary condition for the approximate logarithmic wall-normal dependence of turbulence intensity – a more general condition is the existence of a premultiplied power-spectral intensity of $O(1)$ for $O(1/\delta ) < k < O(1/z)$. Furthermore, it is shown that the Townsend–Perry constant must be weakly dependent on the Reynolds number. Finally, the analysis is semi-empirically extended to the lower logarithmic layer (or mesolayer), and a near-wall correction for the turbulence intensity is subsequently proposed. All the predictions of the proposed model and the related analyses/assumptions are validated with high-fidelity experimental data (Samie et al., J. Fluid Mech., vol. 851, 2018, pp. 391–415).

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers

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JFM Papers
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Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A8
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© The Author(s), 2021. Published by Cambridge University Press

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