Nonlinear neutral modes in zero-mass-flux plane Poiseuille-Couette flow
File(s)
Published version
Author(s)
Walton, AG
Barnes, CC
Type
Journal Article
Abstract
Plane Poiseuille–Couette flow with zero mean advection velocity, a flow well known to be linearly
stable, is analysed at asymptotically large Reynolds number R. A nonlinear neutral mode structure
is uncovered which is self-sustained due to the interaction of the viscous wall layers with a
predominantly inviscid nonlinear internal critical layer. The O(1) phase speed and wavenumber of
the disturbance are computed as functions of the O(R−1/3) wave amplitude, with the formula for
the amplitude determined analytically. Two distinct coherent structures are found: one involving
a single critical layer and a second with two critical layers. The single critical layer case has
a long wavelength/small amplitude limit for which an analytical solution exists, while the dual
critical layer structure possesses two solution branches and exists above a threshold amplitude.
Both states distort the mean flow by an O(R−1/6) amount, and in the long wavelength limit the
distortion becomes comparable in size with the original basic flow. The relevance of this work to
the experimental observation of self-sustained states in this flow is discussed.
stable, is analysed at asymptotically large Reynolds number R. A nonlinear neutral mode structure
is uncovered which is self-sustained due to the interaction of the viscous wall layers with a
predominantly inviscid nonlinear internal critical layer. The O(1) phase speed and wavenumber of
the disturbance are computed as functions of the O(R−1/3) wave amplitude, with the formula for
the amplitude determined analytically. Two distinct coherent structures are found: one involving
a single critical layer and a second with two critical layers. The single critical layer case has
a long wavelength/small amplitude limit for which an analytical solution exists, while the dual
critical layer structure possesses two solution branches and exists above a threshold amplitude.
Both states distort the mean flow by an O(R−1/6) amount, and in the long wavelength limit the
distortion becomes comparable in size with the original basic flow. The relevance of this work to
the experimental observation of self-sustained states in this flow is discussed.
Date Issued
2023-11
Date Acceptance
2024-02-26
Citation
Quarterly Journal of Mechanics and Applied Mathematics, 2023, 76 (4), pp.471-497
ISSN
0033-5614
Publisher
Oxford University Press
Start Page
471
End Page
497
Journal / Book Title
Quarterly Journal of Mechanics and Applied Mathematics
Volume
76
Issue
4
Copyright Statement
© The Author(s) 2024. Published by Oxford University Press.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
License URL
Identifier
http://dx.doi.org/10.1093/qjmam/hbae002
Publication Status
Published
Date Publish Online
2024-04-15