Skip to main content Accessibility help
×
Home

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.

Copyright

Corresponding author

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

References

Hide All
Armitstead, J. P., Bertram, C. D. & Jensen, O. E. 1996 A study of the bifurcation behaviour of a model of flow through a collapsible tube. Bull. Math. Biol. 58, 611641.
Bertram, C. D. 2008 Flow-induced oscillation of collapsed tubes and airway structures. Respir. Physiol. Neurobiol. 163, 256265.
Bertram, C. D. & Pedley, T. J. 1982 A mathematical model of unsteady collapsible tube behaviour. J. Biomech. 15, 3950.
Bertram, C. D., Raymond, C. D. & Pedley, T. J. 1990 Mapping of instabilities for flow through collapsed tubes of different length. J. Fluids Struct. 4, 125154.
Bertram, C. D., Sheppeard, M. D. & Jensen, O. E. 1994 Prediction and measurement of area–distance profile of collapsed tubes during self-excited oscillation. J. Fluids Struct. 8, 637660.
Bogdanova, E. V. & Ryzhov, O. S. 1983 Free and induced oscillations in Poiseuille flow. Q. J. Mech. Appl. Math. 36, 271287.
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166188.
Bridges, T. J. & Morris, P. J. 1984 Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55, 437460.
Carpenter, P. W. & Garrad, A. D. 1985 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 1. Tollmien–Schlichting instabilities. J. Fluid Mech. 155, 465510.
Carpenter, P. W. & Garrad, A. D. 1986 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities. J. Fluid Mech. 170, 199232.
Davies, C. & Carpenter, P. W. 1997 Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205243.
Dempsey, J. A., Veasey, S. C., Morgan, B. J. & O'Donnell, C. P. 2010 Pathophysiology of sleep apnea. Physiol. Rev. 90, 47112.
Domaradzki, J. A. & Metcalfe, R. W. 1987 Stabilization of laminar boundary layers by compliant membranes. Phys. Fluids 30, 695705.
Elemans, C. P. H., Muller, M., Næsbye Larsen, O. & van Leeuwen, J. L. 2009 Amplitude and frequency modulation control of sound production in a mechanical model of the avian syrinx. J. Exp. Biol., 212, 12121224.
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121147.
Guaus, A. & Bottaro, A. 2007 Instabilities of the flow in a curved channel with compliant walls. Proc. R. Soc. Lond. A 463, 22012222.
Guneratne, J. C. & Pedley, T. J. 2006 High-Reynolds-number steady flow in a collapsible channel. J. Fluid Mech. 569, 151184.
Hayashi, S., Hayase, T. & Kwamura, H. 1998 Numerical analysis for stability and self-excited oscillation in collapsible tube flow. Trans. ASME J. Biomech. Engng 120, 468475.
Heil, M. & Boyle, J. 2010 Self-excited oscillations in three-dimensional collapsible tubes: onset and large amplitude oscillations. J. Fluid Mech. 652, 405426.
Heil, M. & Hazel, A. L. 2006 Oomph–lib: an object-oriented multi-physics finite-element library. In Fluid–Structure Interaction (ed. Schäfer, M. & Bungartz, H.-J.), pp. 1949. Springer.
Heil, M. & Jensen, O. E. 2003 Flows in deformable tubes and channels. In Flow in Collapsible Tubes and Past Other Highly Compliant Boundaries (ed. Carpenter, P. W. & Pedley, T. J.). Kluwer.
Heil, M. & Waters, S. L. 2008 How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. J. Fluid Mech. 601, 199227.
Jensen, O. E. & Heil, M. 2003 High-frequency self-excited oscillations in a collapsible-channel flow. J. Fluid Mech. 481, 235268.
Knowlton, F. P. & Starling, E. H. 1912 The influence of variations in temperature and blood pressure on the performance of the isolated mammalian heart. J. Physiol. Lond. 44, 206219.
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. URSS 17, 4254.
Liu, H. F., Luo, X. Y., Cai, Z. X. & Pedley, T. J. 2009 Sensitivity of unsteady collapsible channel flows to modelling assumptions. Commun. Numer. Meth. Engng 25, 483504.
Luo, X. Y., Cai, Z. X., Li, W. G. & Pedley, T. J. 2008 The cascade structure of linear instability in collapsible channel flows. J. Fluid Mech. 600, 4576.
Luo, X. Y. & Pedley, T. J. 1996 A numerical simulation of unsteady flow in a two-dimensional collapsible channel. J. Fluid Mech. 314, 191225.
Mandre, S. & Mahadevan, L. 2010 A generalized theory of viscous and inviscid flutter. Proc. R. Soc. Lond. A 466, 141156.
Manuilovich, S. V. 2004 Propagation of a Tollmien–Schlichting wave over the junction between rigid and compliant surfaces. Fluid Dyn. 39, 702717.
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185199.
Pedley, T. J. & Luo, X. Y. 1998 Modelling flow and oscillations in collapsible tubes. Theor. Comput. Fluid Dyn. 10, 277294.
Pedley, T. J. & Stephanoff, K. D. 1985 Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves. J. Fluid Mech. 160, 337367.
Peyret, R. 2002 Spectral Methods for Incompressible Viscous Flow. Springer.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.
Sen, P. K., Carpenter, P. W., Hegde, S. & Davies, C. 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.
Stephanoff, K. D., Pedley, T. J., Lawrence, C. J. & Secomb, T. W. 1983 Fluid flow along a channel with an asymmetric oscillating constriction. Nature 305, 692695.
Stewart, P. S., Waters, S. L., Billingham, J. & Jensen, O. E. 2010 a Spatially localised growth within global instabilities of flexible channel flows. In Seventh IUTAM Symposium on Laminar–Turbulent Transition (ed. Schlatter, P. & Henningson, D. S.), vol. 18, pp. 397402. Springer.
Stewart, P. S., Waters, S. L. & Jensen, O. E. 2009 Local and global instabilities of flow in a flexible-walled channel. Eur. J. Mech. B/Fluids 28, 541557.
Stewart, P. S., Waters, S. L. & Jensen, O. E. 2010 b Local instabilities of flow in a flexible channel: asymmetric flutter driven by a weak critical layer. Phys. Fluids 22, 031902.
Thomson, S. L., Mongeau, L. & Frankel, S. H. 2005 Aerodynamic transfer of energy to the vocal folds. J. Acoust. Soc. Am. 118, 16891700.
Wang, J. W., Chew, Y. T. & Low, H. T. 2009 Effects of downstream system on self-excited oscillations in collapsible tubes. Commun. Numer. Meth. Engng 25, 429445.
Whittaker, R. J., Heil, M., Boyle, J. B., Jensen, O. E. & Waters, S. L. 2010 a The energetics of flow through a rapidly oscillating tube. Part 2. Application to an elliptical tube. J. Fluid Mech. 648, 123153.
Whittaker, R. J., Heil, M., Jensen, O. E. & Waters, S. L. 2010 b The onset of high-frequency self-excited oscillations in elastic-walled tubes. Proc. R. Soc. Lond. A, doi:10.1098/rspa.2009.0641.
Whittaker, R. J., Heil, M., Jensen, O. E. & Waters, S. L. 2010 c A rational derivation of a tube law from shell theory. Q. J. Mech. Appl. Maths, doi:10.1093/qjmam/hbq020.
Whittaker, R. J., Waters, S. L., Jensen, O. E., Boyle, J. B. & Heil, M. 2010 d The energetics of flow through a rapidly oscillating tube. Part 1. General theory. J. Fluid Mech. 648, 83121.
Womersley, J. R. 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.
Xia, Y., Hayase, T., Hayashi, S. & Hamaya, T. 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.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

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)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed