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Resonance-driven oscillations in a flexible-channel flow with fixed upstream flux and a long downstream rigid segment

Published online by Cambridge University Press:  03 April 2014

Feng Xu
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
John Billingham
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Oliver E. Jensen*
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
Email address for correspondence: oliver.jensen@manchester.ac.uk

Abstract

Flow driven through a planar channel having a finite-length membrane inserted in one wall can be unstable to self-excited oscillations. In a recent study (Xu, Billingham & Jensen J. Fluid Mech., vol. 723, 2013, pp. 706–733), we identified a mechanism of instability arising when the inlet flux and outlet pressure are held constant, and the rigid segment of the channel downstream of the membrane is sufficiently short to have negligible influence on the resulting oscillations. Here we identify an independent mechanism of instability that is intrinsically coupled to flow in the downstream rigid segment, which becomes prominent when the downstream segment is much longer than the membrane. Using a spatially one-dimensional model of the system, we perform a three-parameter unfolding of a degenerate bifurcation point having four zero eigenvalues. Our analysis reveals how instability is promoted by a 1:1 resonant interaction between two modes, with the resulting oscillations described by a fourth-order amplitude equation. This predicts the existence of saturated sawtooth oscillations, which we reproduce in full Navier–Stokes simulations of the same system.

Type
Papers
Copyright
© 2014 Cambridge University Press 

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Xu et al. supplementary movie

A saturated oscillation corresponding to figure 11(a), showing the axial flow field, computed using oomph-lib. The inlet profile is parabolic. The vertical axis is rescaled by a factor of 10. The constriction at the downstream end of the flexible membrane opens more quickly than it closes.

Download Xu et al. supplementary movie(Video)
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