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Sloshing and slamming oscillations in a collapsible channel flow

Published online by Cambridge University Press:  25 August 2010

PETER S. STEWART
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
MATTHIAS HEIL
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
SARAH L. WATERS
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK
OLIVER E. JENSEN*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
*
Email address for correspondence: oliver.jensen@nottingham.ac.uk

Abstract

We consider laminar high-Reynolds-number flow through a finite-length planar channel, where a portion of one wall is replaced by a thin massless elastic membrane that is held under longitudinal tension T and subject to a linear external pressure distribution. The flow is driven by a fixed pressure drop along the full length of the channel. We investigate the global stability of two-dimensional Poiseuille flow using a method of matched local eigenfunction expansions, which is compared to direct numerical simulations. We trace the neutral stability curve of the primary oscillatory instability of the system, illustrating a transition from high-frequency ‘sloshing’ oscillations at high T to vigorous ‘slamming’ motion at low T. Small-amplitude sloshing at high T can be captured using a low-order eigenmode truncation involving four surface-based modes in the compliant segment of the channel coupled to Womersley flow in the rigid segments. At lower tensions, we show that hydrodynamic modes increasingly contribute to the global instability, and we demonstrate a change in the mechanism of energy transfer from the mean flow, with viscous effects being destabilizing. Simulations of finite-amplitude oscillations at low T reveal a generic slamming motion, in which the flexible membrane is drawn close to the opposite rigid wall before recovering rapidly. A simple model is used to demonstrate how fluid inertia in the downstream rigid channel segment, coupled to membrane curvature downstream of the moving constriction, together control slamming dynamics.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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References

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Stewart et al. supplementary movie

Animation of the large-amplitude "slamming" oscillations at Re=250, T=21, corresponding to the time-traces shown in Figs. 7 and 8 in the paper. The movie identifies several key features of the flow: The rapid deceleration of the flow in the downstream rigid section during "slamming"; the disturbances downstream of the collapsible section; and the presence of higher-frequency waves in the collapsible section.

Download Stewart et al. supplementary movie(Video)
Video 13.7 MB

Stewart et al. supplementary movie

Animation of the large-amplitude "slamming" oscillations at Re=250, T=21, corresponding to the time-traces shown in Figs. 7 and 8 in the paper. The movie identifies several key features of the flow: The rapid deceleration of the flow in the downstream rigid section during "slamming"; the disturbances downstream of the collapsible section; and the presence of higher-frequency waves in the collapsible section.

Download Stewart et al. supplementary movie(Video)
Video 5.5 MB