Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-12-01T02:36:10.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-excited oscillations in three-dimensional collapsible tubes: simulating their onset and large-amplitude oscillations

Published online by Cambridge University Press:  13 April 2010

MATTHIAS HEIL  and
JONATHAN BOYLE
MATTHIAS HEIL*
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
JONATHAN BOYLE
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We employ numerical simulations to explore the development of flow-induced self-excited oscillations in three-dimensional collapsible tubes which are subject to boundary conditions (flow rate prescribed at the outflow boundary) that encourage the development of high-frequency oscillations via an instability mechanism originally proposed by Jensen & Heil (J. Fluid Mech., vol. 481, 2003, p. 235). The simulations show that self-excited oscillations tend to arise preferentially from steady equilibrium configurations in which the tube is buckled non-axisymmetrically. We follow the growing oscillations into the large-amplitude regime and show that short tubes tend to approach an approximately axisymmetric equilibrium configuration in which the oscillations decay, whereas sufficiently long tubes develop sustained large-amplitude limit-cycle oscillations. The period of the oscillations and the critical Reynolds number beyond which their amplitude grows are found to be in good agreement with theoretical scaling estimates.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 652 , 10 June 2010 , pp. 405 - 426
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Present address: Research Computing Services, University of Manchester, Oxford Road, Manchester M13 9PL, UK

References

REFERENCES

Benjamin, T. 1963 The threefold classification for unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16, 436–450.CrossRefGoogle Scholar
Bertram, C. D. 2003 Experimental studies of collapsible tubes. In Flow in Collapsible Tubes and Past Other Highly Compliant Boundaries (ed. Pedley, T. J. & Carpenter, P. W.), pp. 51–65. Kluwer.CrossRefGoogle Scholar
Bertram, C. D. & Elliot, N. S. J. 2003 Flow-rate limitation in a uniform thin-walled collapsible tube, with comparison to a uniform thick-walled tube and a tube of tapering thickness. J. Fluids Struct. 7, 541–559.CrossRefGoogle Scholar
Bertram, C. D. & Pedley, T. J. 1982 A mathematical model of unsteady collapsible tube behaviour. J. Biomech. 15, 39–50.CrossRefGoogle ScholarPubMed
Gresho, P. & Sani, R. 2000 Incompressible Flow and the Finite Element Method. Volume Two: Isothermal Laminar Flow. John Wiley and Sons.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.CrossRefGoogle Scholar
Hazel, A. L. & Heil, M. 2003 Steady finite Reynolds number flow in three-dimensional collapsible tubes. J. Fluid Mech. 486, 79–103.CrossRefGoogle Scholar
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. 19–49. Springer. oomph-lib is available as open-source software at http://www.oomph-lib.org.CrossRefGoogle Scholar
Heil, M., Hazel, A. L. & Boyle, J. 2008 Solvers for large-displacement fluid-structure interaction problems: segregated vs. monolithic approaches. Comput. Mech. 43, 91–101.CrossRefGoogle Scholar
Heil, M. & Jensen, O. E. 2003 Flows in deformable tubes and channels – theoretical models and biological applications. In Flow in Collapsible Tubes and Past Other Highly Compliant Boundaries (ed. Pedley, T. J. & Carpenter, P. W.), pp. 15–50. Kluwer.CrossRefGoogle Scholar
Heil, M. & Waters, S. 2006 Transverse flows in rapidly oscillating, elastic cylindrical shells. J. Fluid Mech. 547, 185–214.CrossRefGoogle Scholar
Heil, M. & Waters, S. 2008 How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. J. Fluid Mech. 601, 199–227.CrossRefGoogle Scholar
Jensen, O. E. & Heil, M. 2003 High-frequency self-excited oscillations in a collapsible-channel flow. J. Fluid Mech. 481, 235–268.CrossRefGoogle Scholar
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. 44, 206–219.CrossRefGoogle ScholarPubMed
Kouanis, K. & Mathioulakis, D. S. 1999 Experimental flow study within a self oscillating collapsible tube. J. Fluids Struct. 13, 61–73.CrossRefGoogle Scholar
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, 483–504.CrossRefGoogle Scholar
Luo, X. Y., Cai, Z. X., Li, W. G. & Pedley, T. J. 2008 The cascade structure of linear stabilities of flow in collapsible channels. J. Fluid Mech. 600, 45–76.CrossRefGoogle Scholar
Luo, X. Y. & Pedley, T. J. 1996 A numerical simulation of unsteady flow in a two-dimensional collapsible channel. J. Fluid Mech. 314, 191–225.CrossRefGoogle Scholar
Luo, X. Y. & Pedley, T. J. 1998 The effects of wall inertia on flow in a two-dimensional collapsible channel. J. Fluid Mech. 363, 253–280.CrossRefGoogle Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flow. J. Fluid Mech. 3, 185–199.CrossRefGoogle Scholar
Truong, N. K. & Bertram, C. D. 2009 The flow-field downstream of a collapsible tube during oscillation onset. Commun. Numer. Meth. Engng 25, 405–428.CrossRefGoogle Scholar
Whittaker, R. J., Heil, M., Boyle, J., 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, 83–121.CrossRefGoogle Scholar
Whittaker, R. J., Waters, S. L., Jensen, O. E., Boyle, J. & Heil, M. 2010 b The energetics of flow through a rapidly oscillating tube. Part 1. General theory. J. Fluid Mech. 648, 123–153.CrossRefGoogle Scholar