The logarithmic variance of streamwise velocity and k−1 conundrum in wall turbulence
File(s)
Accepted version
Author(s)
Hwang, Yongyun
Hutchins, Nicholas
Marusic, Ivan
Type
Journal Article
Abstract
The logarithmic dependence of streamwise turbulence intensity has repeatedly been
observed 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 k
−1
spectrum in the inviscid limit, for
small but finite z/δ and for finite Reynolds numbers (z is the wall-normal co-ordinate
and δ 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 wallnormal location at order of z/δ, 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 pre-multiplied power-spectral intensity of
O(1) for O(1/δ) < k < O(1/z). Furthermore, it is shown that the Townsend-Perry
constant must weakly be 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., 2018, vol. 851, pp. 391–415).
observed 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 k
−1
spectrum in the inviscid limit, for
small but finite z/δ and for finite Reynolds numbers (z is the wall-normal co-ordinate
and δ 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 wallnormal location at order of z/δ, 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 pre-multiplied power-spectral intensity of
O(1) for O(1/δ) < k < O(1/z). Furthermore, it is shown that the Townsend-Perry
constant must weakly be 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., 2018, vol. 851, pp. 391–415).
Date Issued
2021-12-20
Date Acceptance
2021-11-22
Citation
Journal of Fluid Mechanics, 2021, 933, pp.1-24
ISSN
0022-1120
Publisher
Cambridge University Press
Start Page
1
End Page
24
Journal / Book Title
Journal of Fluid Mechanics
Volume
933
Copyright Statement
© The Author(s), 2021. Published by Cambridge University Press. This article has been published in a revised form in
Journal of Fluid Mechanics https://doi.org/10.1017/jfm.2021.1057. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works.
Journal of Fluid Mechanics https://doi.org/10.1017/jfm.2021.1057. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works.
Sponsor
The Leverhulme Trust
Engineering & Physical Science Research Council (E
Engineering & Physical Science Research Council (E
Identifier
https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/logarithmic-variance-of-streamwise-velocity-and-k1-conundrum-in-wall-turbulence/1EF3B223C23015595222D8454FC483CA
Grant Number
RPG-2019-123
EP/T009365/1
EP/V502354/1
Subjects
01 Mathematical Sciences
09 Engineering
Fluids & Plasmas
Publication Status
Published
Date Publish Online
2021-12-20