Hostname: page-component-f554764f5-nqxm9 Total loading time: 0 Render date: 2025-04-10T14:36:51.331Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability and energy budget of pressure-driven collapsible channel flows

Published online by Cambridge University Press:  07 September 2011

H. F. Liu ,
X. Y. Luo  and
Z. X. Cai
H. F. Liu
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
X. Y. Luo*
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
Z. X. Cai
Affiliation:
Department of Mechanics, Tianjin University, 300072, People’s Republic of China
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Although self-excited oscillations in collapsible channel flows have been extensively studied, our understanding of their origins and mechanisms is still far from complete. In the present paper, we focus on the stability and energy budget of collapsible channel flows using a fluid–beam model with the pressure-driven (inlet pressure specified) condition, and highlight its differences to the flow-driven (i.e. inlet flow specified) system. The numerical finite element scheme used is a spine-based arbitrary Lagrangian–Eulerian method, which is shown to satisfy the geometric conservation law exactly. We find that the stability structure for the pressure-driven system is not a cascade as in the flow-driven case, and the mode-2 instability is no longer the primary onset of the self-excited oscillations. Instead, mode-1 instability becomes the dominating unstable mode. The mode-2 neutral curve is found to be completely enclosed by the mode-1 neutral curve in the pressure drop and wall stiffness space; hence no purely mode-2 unstable solutions exist in the parameter space investigated. By analysing the energy budgets at the neutrally stable points, we can confirm that in the high-wall-tension region (on the upper branch of the mode-1 neutral curve), the stability mechanism is the same as proposed by Jensen & Heil. Namely, self-excited oscillations can grow by extracting kinetic energy from the mean flow, with exactly two-thirds of the net kinetic energy flux dissipated by the oscillations and the remainder balanced by increased dissipation in the mean flow. However, this mechanism cannot explain the energy budget for solutions along the lower branch of the mode-1 neutral curve where greater wall deformation occurs. Nor can it explain the energy budget for the mode-2 neutral oscillations, where the unsteady pressure drop is strongly influenced by the severely collapsed wall, with stronger Bernoulli effects and flow separations. It is clear that more work is required to understand the physical mechanisms operating in different regions of the parameter space, and for different boundary conditions.

JFM classification

Instability: Parametric instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 705: Special issue dedicated to Professor T. J. Pedley on his 70th birthday , 25 August 2012 , pp. 348 - 370
Copyright
Copyright © Cambridge University Press 2011

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.)

Article purchase

Temporarily unavailable

References

1. Bertram, C. D. 1982 2 modes of instability in a thick-walled collapsible tube conveying a flow. J. Biomech. 15 (3), 223–224.CrossRefGoogle Scholar
2. Bertram, C. D. & Elliott, 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. 17 (4), 541–559.CrossRefGoogle Scholar
3. Bertram, C. D. & Pedley, T. J. 1982 A mathematical-model of unsteady collapsible tube behavior. J. Biomech. 15 (1), 39–50.CrossRefGoogle Scholar
4. Bertram, C. D., Raymond, C. J. & Pedley, T. J. 1990 Mapping of instabilities for flow through collapsed tubes of differing length. J. Fluids Struct. 4 (2), 125–153.CrossRefGoogle Scholar
5. Cai, Z. X. & Luo, X. Y. 2003 A fluid–beam model for flow in collapsible channels. J. Fluids Struct. 17, 123–144.CrossRefGoogle Scholar
6. Cancelli, C. & Pedley, T. J. 1985 A separated-flow model for collapsible-tube oscillations. J. Fluid Mech. 157, 375–404.CrossRefGoogle Scholar
7. Conrad, W. A. 1969 Pressure-flow relationships in collapsible tubes. IEEE Trans. Biomed. Engng 16 (4), 284–295.CrossRefGoogle ScholarPubMed
8. Étienne, S., Garon, A. & Pelletier, D. 2009 Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow. J. Comput. Phys. 228 (7), 2313–2333.CrossRefGoogle Scholar
9. Garbow, B. S. 1978 The QZ algorithm to solve the generalized eigenvalue problem for complex matrices. ACM Trans. Math. Softw. 4 (4), 404–410.CrossRefGoogle Scholar
10. Hazel, A. L. & Heil, M. 2003 Steady finite-Reynolds-number flows in three-dimensional collapsible tubes. J. Fluid Mech. 486, 79–103.CrossRefGoogle Scholar
11. Heil, M. & Waters, S. L. 2008 How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. J. Fluid Mech. 601, 199–227.CrossRefGoogle Scholar
12. Ishizaka, K. 1972 Synthesis of voiced sounds from a two-mass model of the vocal cords. Bell Syst. Tech. J. 51, 1233–1268.CrossRefGoogle Scholar
13. Jensen, O. E. 1990 Instabilities of flow in a collapsed tube. J. Fluid Mech. 220, 623–659.CrossRefGoogle Scholar
14. Jensen, O. E. & Heil, M. 2003 High-frequency self-excited oscillations in a collapsible-channel flow. J. Fluid Mech. 481, 235–268.CrossRefGoogle Scholar
15. Kamm, R. D. & Shapiro, A. H. 1979 Unsteady flow in a collapsible tube subjected to external pressure or body forces. J. Fluid Mech. 95 (1), 1–78.CrossRefGoogle Scholar
16. Katz, A. I., Chen, Y. & Moreno, A. H. 1969 Flow through a collapsible tube: experimental analysis and mathematical model. Biophys. J. 9 (10), 1261–1279.CrossRefGoogle ScholarPubMed
17. 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 (5), 483–504.CrossRefGoogle Scholar
18. 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, 45–76.CrossRefGoogle Scholar
19. Luo, X. Y. & Pedley, T. J. 1995 A numerical simulation of steady flow in a 2-D collapsible channel. J. Fluids Struct. 9 (2), 149–174.CrossRefGoogle Scholar
20. 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
21. 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
22. Luo, X. Y. & Pedley, T. J. 2000 Multiple solutions and flow limitation in collapsible channel flows. J. Fluid Mech. 420, 301–324.CrossRefGoogle Scholar
23. Marzo, A., Luo, X. Y. & Bertram, C. D. 2005 Three-dimensional collapse and steady flow in thick-walled flexible tubes. J. Fluids Struct. 20 (6), 817–835.CrossRefGoogle Scholar
24. Pedley, T. J. & Luo, X. Y. 1998 Modelling flow and oscillations in collapsible tubes. Theor. Comput. Fluid Dyn. 10 (1), 277–294.CrossRefGoogle Scholar
25. Shapiro, A. H. 1977 Steady flow in collapsible tubes. J. Biomech. Engng 99, 126.CrossRefGoogle Scholar
26. Stewart, P. S., Heil, M., Waters, S. L. & Jensen, O. E. 2010 Sloshing and slamming oscillations in collapsible channel flow. J. Fluid Mech. 662, 288–319.CrossRefGoogle Scholar
27. 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 (4), 541–557.CrossRefGoogle Scholar
28. Whittaker, R. J., Heil, M., Boyle, J., Jensen, O. E. & Waters, S. L. 2010a The energetics of flow through a rapidly oscillating tube. Part 2. Application to an elliptical tube. J. Fluid Mech. 648, 123–153.CrossRefGoogle Scholar
29. Whittaker, R. J., Heil, M., Jensen, O. E. & Waters, S. L. 2010b A rational derivation of a tube law from shell theory. Q. J. Mech. Appl. Maths 63 (4), 465.CrossRefGoogle Scholar
30. Whittaker, R. J., Heil, M., Jensen, O. E. & Waters, S. L. 2010c Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes. Proc. R. Soc. Lond. A 466, 3635.Google Scholar
31. Whittaker, R. J., Waters, S. L., Jensen, O. E., Boyle, J. & Heil, M. 2010d The energetics of flow through a rapidly oscillating tube. Part 1. General theory. J. Fluid Mech. 648, 83–121.CrossRefGoogle Scholar
32. Zhu, Y., Luo, X. Y. & Ogden, R. W. 2008 Asymmetric bifurcations of thick-walled circular cylindrical elastic tubes under axial loading and external pressure. Intl J. Solids Struct. 45 (11–12), 3410–3429.CrossRefGoogle Scholar
33. Zhu, Y., Luo, X. Y. & Ogden, R. W. 2010 Nonlinear axisymmetric deformations of an elastic tube under external pressure. Eur. J. Mech. A/Solids 29, 216–229.CrossRefGoogle Scholar