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Sloshing and slamming oscillations in a collapsible channel flow


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.

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M. Heil & A. L. Hazel 2006 Oomph–lib: an object-oriented multi-physics finite-element library. In Fluid–Structure Interaction (ed. M. Schäfer & H.-J. Bungartz ), pp. 1949. Springer.

R. Peyret 2002 Spectral Methods for Incompressible Viscous Flow. Springer.

P. J. Schmid & D. S. Henningson 2001 Stability and Transition in Shear Flows. Springer.

P. S. Stewart , S. L. Waters , J. Billingham & O. E. Jensen 2010 a Spatially localised growth within global instabilities of flexible channel flows. In Seventh IUTAM Symposium on Laminar–Turbulent Transition (ed. P. Schlatter & D. S. Henningson ), vol. 18, pp. 397402. Springer.

R. J. Whittaker , M. Heil , O. E. Jensen & S. L. Waters 2010 bThe onset of high-frequency self-excited oscillations in elastic-walled tubes. Proc. R. Soc. Lond. A, doi:10.1098/rspa.2009.0641.

R. J. Whittaker , M. Heil , O. E. Jensen & S. L. Waters 2010 cA rational derivation of a tube law from shell theory. Q. J. Mech. Appl. Maths, doi:10.1093/qjmam/hbq020.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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