Journal of Fluid Mechanics

Papers

Sloshing and slamming oscillations in a collapsible channel flow

PETER S. STEWARTa1, MATTHIAS HEILa2, SARAH L. WATERSa3 and OLIVER E. JENSENa4 c1

a1 Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA

a2 School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

a3 Oxford Centre for Industrial and Applied Mathematics, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK

a4 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, 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.

(Received February 12 2010)

(Revised June 08 2010)

(Accepted June 12 2010)

(Online publication August 25 2010)

Key words:

  • flow–vessel interactions

Correspondence:

c1 Email address for correspondence: oliver.jensen@nottingham.ac.uk

Metrics
Related Content