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Sloshing and slamming oscillations in a collapsible channel flow

  • PETER S. STEWART (a1), MATTHIAS HEIL (a2), SARAH L. WATERS (a3) and OLIVER E. JENSEN (a4)
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.

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Corresponding author
Email address for correspondence: oliver.jensen@nottingham.ac.uk
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J. P. Armitstead , C. D. Bertram & O. E. Jensen 1996 A study of the bifurcation behaviour of a model of flow through a collapsible tube. Bull. Math. Biol. 58, 611641.

C. D. Bertram 2008 Flow-induced oscillation of collapsed tubes and airway structures. Respir. Physiol. Neurobiol. 163, 256265.

C. D. Bertram & T. J. Pedley 1982 A mathematical model of unsteady collapsible tube behaviour. J. Biomech. 15, 3950.

C. D. Bertram , C. D. Raymond & T. J. Pedley 1990 Mapping of instabilities for flow through collapsed tubes of different length. J. Fluids Struct. 4, 125154.

C. D. Bertram , M. D. Sheppeard & O. E. Jensen 1994 Prediction and measurement of area–distance profile of collapsed tubes during self-excited oscillation. J. Fluids Struct. 8, 637660.

E. V. Bogdanova & O. S. Ryzhov 1983 Free and induced oscillations in Poiseuille flow. Q. J. Mech. Appl. Math. 36, 271287.

F. P. Bretherton 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166188.

T. J. Bridges & P. J. Morris 1984 Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55, 437460.

P. W. Carpenter & A. D. Garrad 1985 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 1. Tollmien–Schlichting instabilities. J. Fluid Mech. 155, 465510.

P. W. Carpenter & A. D. Garrad 1986 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities. J. Fluid Mech. 170, 199232.

C. Davies & P. W. Carpenter 1997 Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205243.

J. A. Dempsey , S. C. Veasey , B. J. Morgan & C. P. O'Donnell 2010 Pathophysiology of sleep apnea. Physiol. Rev. 90, 47112.

J. A. Domaradzki & R. W. Metcalfe 1987 Stabilization of laminar boundary layers by compliant membranes. Phys. Fluids 30, 695705.

C. P. H. Elemans , M. Muller , O. Næsbye Larsen & J. L. van Leeuwen 2009 Amplitude and frequency modulation control of sound production in a mechanical model of the avian syrinx. J. Exp. Biol., 212, 12121224.

J. B. Grotberg & O. E. Jensen 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121147.

A. Guaus & A. Bottaro 2007 Instabilities of the flow in a curved channel with compliant walls. Proc. R. Soc. Lond. A 463, 22012222.

J. C. Guneratne & T. J. Pedley 2006 High-Reynolds-number steady flow in a collapsible channel. J. Fluid Mech. 569, 151184.

S. Hayashi , T. Hayase & H. Kwamura 1998 Numerical analysis for stability and self-excited oscillation in collapsible tube flow. Trans. ASME J. Biomech. Engng 120, 468475.

M. Heil & J. Boyle 2010 Self-excited oscillations in three-dimensional collapsible tubes: onset and large amplitude oscillations. J. Fluid Mech. 652, 405426.

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.

M. Heil & O. E. Jensen 2003 Flows in deformable tubes and channels. In Flow in Collapsible Tubes and Past Other Highly Compliant Boundaries (ed. P. W. Carpenter & T. J. Pedley ). Kluwer.

M. Heil & S. L. Waters 2008 How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. J. Fluid Mech. 601, 199227.

O. E. Jensen & M. Heil 2003 High-frequency self-excited oscillations in a collapsible-channel flow. J. Fluid Mech. 481, 235268.

F. P. Knowlton & E. H. Starling 1912 The influence of variations in temperature and blood pressure on the performance of the isolated mammalian heart. J. Physiol. Lond. 44, 206219.

H. F. Liu , X. Y. Luo , Z. X. Cai & T. J. Pedley 2009 Sensitivity of unsteady collapsible channel flows to modelling assumptions. Commun. Numer. Meth. Engng 25, 483504.

X. Y. Luo , Z. X. Cai , W. G. Li & T. J. Pedley 2008 The cascade structure of linear instability in collapsible channel flows. J. Fluid Mech. 600, 4576.

X. Y. Luo & T. J. Pedley 1996 A numerical simulation of unsteady flow in a two-dimensional collapsible channel. J. Fluid Mech. 314, 191225.

S. Mandre & L. Mahadevan 2010 A generalized theory of viscous and inviscid flutter. Proc. R. Soc. Lond. A 466, 141156.

S. V. Manuilovich 2004 Propagation of a Tollmien–Schlichting wave over the junction between rigid and compliant surfaces. Fluid Dyn. 39, 702717.

J. W. Miles 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185199.

T. J. Pedley & X. Y. Luo 1998 Modelling flow and oscillations in collapsible tubes. Theor. Comput. Fluid Dyn. 10, 277294.

T. J. Pedley & K. D. Stephanoff 1985 Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves. J. Fluid Mech. 160, 337367.

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. K. Sen , P. W. Carpenter , S. Hegde & C. Davies 2009 A wave driver theory for vortical waves propagating across junctions with application to those between rigid and compliant walls. J. Fluid Mech. 625, 146.

K. D. Stephanoff , T. J. Pedley , C. J. Lawrence & T. W. Secomb 1983 Fluid flow along a channel with an asymmetric oscillating constriction. Nature 305, 692695.

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.

P. S. Stewart , S. L. Waters & O. E. Jensen 2009 Local and global instabilities of flow in a flexible-walled channel. Eur. J. Mech. B/Fluids 28, 541557.

P. S. Stewart , S. L. Waters & O. E. Jensen 2010 bLocal instabilities of flow in a flexible channel: asymmetric flutter driven by a weak critical layer. Phys. Fluids 22, 031902.

S. L. Thomson , L. Mongeau & S. H. Frankel 2005 Aerodynamic transfer of energy to the vocal folds. J. Acoust. Soc. Am. 118, 16891700.

J. W. Wang , Y. T. Chew & H. T. Low 2009 Effects of downstream system on self-excited oscillations in collapsible tubes. Commun. Numer. Meth. Engng 25, 429445.

R. J. Whittaker , M. Heil , J. B. Boyle , O. E. Jensen & S. L. Waters 2010 a The energetics of flow through a rapidly oscillating tube. Part 2. Application to an elliptical tube. J. Fluid Mech. 648, 123153.

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.

R. J. Whittaker , S. L. Waters , O. E. Jensen , J. B. Boyle & M. Heil 2010 d The energetics of flow through a rapidly oscillating tube. Part 1. General theory. J. Fluid Mech. 648, 83121.

J. R. Womersley 1955 Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127, 553563.

Y. Xia , T. Hayase , S. Hayashi & T. Hamaya 2000 Effect of initial axial strain of collapsible tube on self-excited oscillation. JSME Intl J. Ser. C Mech. Syst. Mach. Elem. Manuf. 43, 882888.

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Journal of Fluid Mechanics
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  • EISSN: 1469-7645
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