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Quadrupolar flows around spots in internal shear flows

Published online by Cambridge University Press:  06 April 2020

Zhe Wang Open the ORCID record for Zhe Wang [Opens in a new window] ,
Claude Guet Open the ORCID record for Claude Guet [Opens in a new window] ,
Romain Monchaux Open the ORCID record for Romain Monchaux [Opens in a new window] ,
Yohann Duguet  and
Bruno Eckhardt Open the ORCID record for Bruno Eckhardt [Opens in a new window]
Zhe Wang*
Affiliation:
Energy Research Institute at Nanyang Technological University (ERI@N), Nanyang Technological University, 639798Singapore Interdisciplinary Graduate School, Nanyang Technological University, 639798Singapore School of Materials Science and Engineering, Nanyang Technological University, 639798Singapore
Claude Guet
Affiliation:
Energy Research Institute at Nanyang Technological University (ERI@N), Nanyang Technological University, 639798Singapore School of Materials Science and Engineering, Nanyang Technological University, 639798Singapore
Romain Monchaux
Affiliation:
IMSIA, ENSTA-ParisTech/CNRS/CEA/EDF, Institut Polytechnique de Paris, 91762Palaiseau, France
Yohann Duguet
Affiliation:
LIMSI-CNRS, Université Paris Saclay, 91403Orsay, France
Bruno Eckhardt
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, 35032Marburg, Germany
*
† Email address for correspondence: zhe.wang@ntu.edu.sg
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Abstract

Turbulent spots occur in shear flows confined between two walls and are surrounded by robust quadrupolar flows. Although the far-field decay of such large-scale flows has been reported to be exponential, we predict a different algebraic decay for the case of plane Couette flow. We address this problem theoretically, by modelling an isolated spot as an obstacle in a linear plane shear flow with free-slip boundary conditions at the walls. By seeking invariant solutions in a co-moving Lagrangian frame and using geometric scale separation, a set of differential equations governing large-scale flows is derived from the Navier–Stokes equations and solved analytically. The wall-normal velocity turns out to be exponentially localised in the plane, while the quadrupolar in-plane velocity field, after wall-normal averaging, features a superposition of algebraic and exponential decays. The algebraic decay exponent is $-3$. The quadrupolar angular dependence stems from (i) the shearing of the streamwise velocity and (ii) the breaking of the spanwise homogeneity. Near the spot, exponentially decaying solutions can generate reversed quadrupolar flows. Eventually, by noting that the algebraically decaying in-plane flow is two-dimensional and harmonic, we suggest a topological origin to the quadrupolar large-scale flow.

JFM classification

Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 892 , 10 June 2020 , A27
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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