A self-sustaining mechanism for plane Couette flow with a flexible boundary
File(s)
Published version
Author(s)
Chotai, Avni
Walton, Andrew
Type
Journal Article
Abstract
The stability of plane Couette flow to travelling-wave disturbances is studied asymptotically at high Reynolds numbers Re when the lower boundary possesses a degree of flexibility modelled by a spring-backed plate. First it is shown that a three-dimensional linear instability exists, with streamwise and spanwise wavelengths comparable with the channel width. Building on this, nonlinear effects from the self-interaction of the wave are introduced, leading to a self-sustaining interaction between a roll/streak flow and the three-dimensional wave. Governing nonlinear vortex-wave
interaction (VWI) equations are derived and a perturbation analysis is carried out to guide a numerical investigation of the equations. The co-existence of two families of finite-amplitude solutions, each with different flow structures, is found. Numerical solutions of the VWI equations in each case show that a small wave amplitude of O(Re−1(log Re)−1/2) is all that is necessary to provoke an O(1) change to the basic Couette flow.
interaction (VWI) equations are derived and a perturbation analysis is carried out to guide a numerical investigation of the equations. The co-existence of two families of finite-amplitude solutions, each with different flow structures, is found. Numerical solutions of the VWI equations in each case show that a small wave amplitude of O(Re−1(log Re)−1/2) is all that is necessary to provoke an O(1) change to the basic Couette flow.
Date Issued
2023-08
Date Acceptance
2023-11-10
Citation
Quarterly Journal of Mechanics and Applied Mathematics, 2023, 76 (3), pp.371-403
ISSN
0033-5614
Publisher
Oxford University Press
Start Page
371
End Page
403
Journal / Book Title
Quarterly Journal of Mechanics and Applied Mathematics
Volume
76
Issue
3
Copyright Statement
© The Author, 2023. Published by Oxford University Press.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License
(http://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
(http://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
https://doi.org/10.1093/qjmam/hbad009
Publication Status
Published
Date Publish Online
2023-12-12