The cascade structure of linear instability in collapsible channel flows

Luo, X.Y. , Cai, Z.X., Li, W.G. and Pedley, T.J. (2008) The cascade structure of linear instability in collapsible channel flows. Journal of Fluid Mechanics, 600, pp. 45-76. (doi:10.1017/S0022112008000293)

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Abstract

This paper studies the unsteady behaviour and linear stability of the flow in a collapsible channel using a fluid–beam model. The solid mechanics is analysed in a plane strain configuration, in which the principal stretch is defined with a zero initial strain. Two approaches are employed: unsteady numerical simulations solving the nonlinear fully coupled fluid–structure interaction problem; and the corresponding linearized eigenvalue approach solving the Orr–Sommerfeld equations modified by the beam. The two approaches give good agreement with each other in predicting the frequencies and growth rates of the perturbation modes, close to the neutral curves. For a given Reynolds number in the range of 200–600, a cascade of instabilities is discovered as the wall stiffness (or effective tension) is reduced. Under small perturbation to steady solutions for the same Reynolds number, the system loses stability by passing through a succession of unstable zones, with mode number increasing as the wall stiffness is decreased. It is found that this cascade structure can, in principle, be extended to many modes, depending on the parameters. A puzzling ‘tongue’ shaped stable zone in the wall stiffness–Re space turns out to be the zone sandwiched by the mode-2 and mode-3 instabilities. Self-excited oscillations dominated by modes 2–4 are found near their corresponding neutral curves. These modes can also interact and form period-doubling oscillations. Extensive comparisons of the results with existing analytical models are made, and a physical explanation for the cascade structure is proposed.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Luo, Professor Xiaoyu
Authors: Luo, X.Y., Cai, Z.X., Li, W.G., and Pedley, T.J.
Subjects:R Medicine > R Medicine (General)
Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Fluid Mechanics
Journal Abbr.:J. Fluid. Mech.
Publisher:Cambridge University Press
ISSN:0022-1120
ISSN (Online):1469-7645
Published Online:26 March 2008
Copyright Holders:Copyright © 2008 Cambridge University Press
First Published:First published in Journal of Fluid Mechanics 600:45-76
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher.

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