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The cascade structure of linear instability in collapsible channel flows

Published online by Cambridge University Press:  26 March 2008

X. Y. LUO ,
Z. X. CAI ,
W. G. LI  and
T. J. PEDLEY
X. Y. LUO
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow Q12 8QW, UKx.y.Luo@maths.gla.ac.uk
Z. X. CAI
Affiliation:
Department of Mechanics, Tianjin University, China
W. G. LI
Affiliation:
Department of Mechanical Engineering, University of Sheffield, Sheffield S1 3JD, UK
T. J. PEDLEY
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WAUK
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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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 600 , 10 April 2008 , pp. 45 - 76
Copyright
Copyright © Cambridge University Press 2008

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