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Two-dimensional cross-spectrum of the streamwise velocity in turbulent boundary layers

Published online by Cambridge University Press:  02 March 2020

Rahul Deshpande Open the ORCID record for Rahul Deshpande [Opens in a new window] ,
Dileep Chandran Open the ORCID record for Dileep Chandran [Opens in a new window] ,
Jason P. Monty Open the ORCID record for Jason P. Monty [Opens in a new window]  and
Ivan Marusic Open the ORCID record for Ivan Marusic [Opens in a new window]
Rahul Deshpande*
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Parkville, VIC3010, Australia
Dileep Chandran
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Parkville, VIC3010, Australia
Jason P. Monty
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Parkville, VIC3010, Australia
Ivan Marusic
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Parkville, VIC3010, Australia
*
†Email address for correspondence: [email protected]
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Abstract

In this paper, we present the two-dimensional (2-D) energy cross-spectrum of the streamwise velocity ($u$) component and use it to test the notion of self-similarity in turbulent boundary layers. The primary focus is on the cross-spectrum ($\unicode[STIX]{x1D6F7}_{cross}^{w}$) measured across the logarithmic ($z_{o}$) and near-wall ($z_{r}$) wall-normal locations, providing the energy distribution across the range of streamwise ($\unicode[STIX]{x1D706}_{x}$) and spanwise ($\unicode[STIX]{x1D706}_{y}$) wavelengths (or length scales) that are coherent across the wall-normal distance. $\unicode[STIX]{x1D6F7}_{cross}^{w}$ may thus be interpreted as a wall-filtered subset of the full 2-D $u$-spectrum ($\unicode[STIX]{x1D6F7}$), the latter providing information on all coexisting eddies at $z_{o}$. To this end, datasets comprising synchronized two-point $u$-signals at $z_{o}$ and $z_{r}$, across the friction Reynolds number range $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3}){-}O(10^{4})$, are analysed. The published direct numerical simulation (DNS) dataset of Sillero et al. (Phys. Fluids, vol. 26 (10), 2014, 105109) is considered for low-$Re_{\unicode[STIX]{x1D70F}}$ analysis, while the high-$Re_{\unicode[STIX]{x1D70F}}$ dataset is obtained by conducting synchronous multipoint hot-wire measurements. High-$Re_{\unicode[STIX]{x1D70F}}$ cross-spectra reveal that the wall-attached large scales follow a $\unicode[STIX]{x1D706}_{y}/z_{o}\sim \unicode[STIX]{x1D706}_{x}/z_{o}$ relationship more closely than seen for $\unicode[STIX]{x1D6F7}$, where this self-similar trend is obscured by coexisting scales. The present analysis reaffirms that a self-similar structure, conforming to Townsend’s attached eddy hypothesis, is ingrained in the flow.

JFM classification

Boundary Layers: Boundary layer structure Turbulent Flows: Turbulent boundary layers
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 890 , 10 May 2020 , R2
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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