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The cascade structure of linear instability in collapsible channel flows

Published online by Cambridge University Press:  26 March 2008

X. Y. LUO ,
Z. X. CAI ,
W. G. LI  and
T. J. PEDLEY
X. Y. LUO
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow Q12 8QW, UKx.y.Luo@maths.gla.ac.uk
Z. X. CAI
Affiliation:
Department of Mechanics, Tianjin University, China
W. G. LI
Affiliation:
Department of Mechanical Engineering, University of Sheffield, Sheffield S1 3JD, UK
T. J. PEDLEY
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WAUK
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Abstract

This paper studies the unsteady behaviour and linear stability of the flow in a collapsible channel using a fluid–beam model. The solid mechanics is analysed in a plane strain configuration, in which the principal stretch is defined with a zero initial strain. Two approaches are employed: unsteady numerical simulations solving the nonlinear fully coupled fluid–structure interaction problem; and the corresponding linearized eigenvalue approach solving the Orr–Sommerfeld equations modified by the beam. The two approaches give good agreement with each other in predicting the frequencies and growth rates of the perturbation modes, close to the neutral curves. For a given Reynolds number in the range of 200–600, a cascade of instabilities is discovered as the wall stiffness (or effective tension) is reduced. Under small perturbation to steady solutions for the same Reynolds number, the system loses stability by passing through a succession of unstable zones, with mode number increasing as the wall stiffness is decreased. It is found that this cascade structure can, in principle, be extended to many modes, depending on the parameters. A puzzling ‘tongue’ shaped stable zone in the wall stiffness–Re space turns out to be the zone sandwiched by the mode-2 and mode-3 instabilities. Self-excited oscillations dominated by modes 2–4 are found near their corresponding neutral curves. These modes can also interact and form period-doubling oscillations. Extensive comparisons of the results with existing analytical models are made, and a physical explanation for the cascade structure is proposed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 600 , 10 April 2008 , pp. 45 - 76
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Barrodale, I. & Wilson, K. B. 1978 A Fortran program for solving a nonlinear equation by Muller's method. J. Comput. Appl. Maths 4 (2), 159–166.CrossRefGoogle Scholar
Bearman, P. W. & Huarte, F. H. 2006 The hydrodynamic forces acting on long flexible circular cylinder responding to VIV. Proc. 6th FSI, AE & FIV and N Symp. Vancouver.CrossRefGoogle Scholar
Bertram, C. D. 1982 Two modes of instability in a thick-walled collapsible tube conveying a flow. J. Biomech. 15, 223–224.CrossRefGoogle Scholar
Bertram, C. D. & Elliott, N. S. J. 2001 Aqueous flow limitation in uniform collapsible tubes: multiple flow-limited flow-rates at the same pressure drop and upstream transmural pressure. Proc. ASME Bioengng Conf. 27 June–1 July, Snowbird, vol. 50 (ed. Kamm, R. D., Schmid-Shonbein, G. W., Ateshian, G. A. & Hefzy, M. S.), pp. 383–384.Google Scholar
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, 125–153.CrossRefGoogle Scholar
Cai, Z. X. & Luo, X. Y. 2003 A fluid-beam model for flow in a collapsible channel. J. Fluids Struct. 17, 125–146.CrossRefGoogle Scholar
Cancelli, C. & Pedley, T. J. 1985 A separated-flow model for collapsible-tube oscillations. J. Fluid Mech. 157, 375–404.CrossRefGoogle Scholar
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, 188–232.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 1997 a Numerical simulation of the evolution of Tollmien–Schlichting waves over finite compliant panels. J. Fluid Mech. 335, 361–392.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 1997 b Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205–243.CrossRefGoogle Scholar
Davies, C., Carpenter, P. W., Ali, R. & Lockerby, D. A. 2006 Disturbance development in boundary layers over compliant surfaces. Laminar–Turbulent Transition, IUTAM Symp. Bangalore, India (ed. R. Govindarajan). pp. 225–230. Springer.CrossRefGoogle Scholar
Ehrenstein, U. & Rossi, M. 1993 Nonlinear Tollmien–Schlichting waves for plane Poiseuille flow with compliant walls. Eur. J. Mech. B/Fluids 12, 789–810.Google Scholar
Garbow, B. S. 1978 The QZ algorithm to solve the generalized eigenvalue problem for complex matrices. ACM Trans. Math. Software 4, 404–410.CrossRefGoogle Scholar
Green, C. H. & Ellen, C. H. 1972 The stability of plane Poiseuille flow between flexible walls. J. Fluid Mech. 51, 403–416.CrossRefGoogle Scholar
Grimshaw, R. 1990 Nonlinear Ordinary Differential Equations. Blackwell Scientific/ Wolfe.Google Scholar
Grotberg, J. B. & Gavriely, N. 1989 Flutter in collapsible tubes: a theoretical model of wheezes. J. Appl. Physiol. 66, 2262–2273.CrossRefGoogle ScholarPubMed
Guneratne, J. C. & Pedley, T. J. 2006 High-Reynolds number steady flow in a collapsible channel. J. Fluid Mech. 569, 151–184.CrossRefGoogle Scholar
Haughton, D. 1980 Post-bifurcation of perfect and imperfect spherical elastic membranes. Intl J. Solid Struct. 16, 1123–1133.CrossRefGoogle Scholar
Hayashi, S., Hayase, T. & Kawamura, H. 1998 Numerical analysis for stability and self-excited oscillation in collapsible tube flow. Trans. ASME K: J. Biomech. Engng 120, 468–475.Google ScholarPubMed
Hazel, A. L. & Heil, M. 2003 Steady finite-Reynolds-number flows in three-dimensional collapsible tubes. J. Fluid Mech. 486, 79–103.CrossRefGoogle Scholar
Heil, M. 1997 Stokes flow in collapsible tubes: computation and experiment. J. Fluid Mech. 353, 285–312.CrossRefGoogle Scholar
Heil, M. & Waters, S. L. 2006 Transverse flows in a rapidly oscillating, elastic cylindrical shell. J. Fluid Mech. 547, 185–214.CrossRefGoogle Scholar
Hemon, P., de Langre, E. & Schmid, P. 2006 Experimental evidence of transient growth of energy before airfoil flutter. J. Fluids Struct. 22, 391–400.CrossRefGoogle Scholar
Jensen, O. E. 1990 Instabilities of flow in a collapsed tube. J. Fluid Mech. 220, 623–659.CrossRefGoogle Scholar
Jensen, O. E. 1992 Chaotic oscillations in a simple collapsible tube model. Trans. ASME K: J. Biomech. Engng 114, 55–59.Google Scholar
Jensen, O. E. & Heil, M. H. 2003 High-frequency self-excited oscillations in a collapsible-channel flow. J. Fluid Mech. 481, 235–268.CrossRefGoogle Scholar
Jensen, O. E. & Pedley, T. J. 1989 The existence of steady flow in a collapsed tube. J. Fluid Mech. 206, 339–374.CrossRefGoogle Scholar
Kamm, R. D. & Pedley, T. J. 1989 Flow in collapsible tubes: a brief review. Trans. ASME K: J. Biomech. Engng 111, 117–179.Google ScholarPubMed
de Langre, E., Paidoussis, M. P., Doare, O. & Modarres-Sadeghi, Y. 2007 Flutter of long flexible cylinders in axial flow. J. Fluid Mech., 571, 371–389.CrossRefGoogle Scholar
Liu, H. F., Luo, X. Y. & Cai, Z. X. 2007 Sensitivity of unsteady collapsible channel flows to modelling assumptions. Comput. Struct. (submitted).Google Scholar
Lopes, J. L., Paidoussis, M. P. & Semler, C. 2002 Linear and nonlinear dynamics of cantilevered cylinders in axial flow. Part 2. The equations of motion. J. Fluids Struct. 16, 715–737.CrossRefGoogle Scholar
Lucey, A. D. & Pitman, D. M. 2006 A new method for determining the eigenmodes of finite flow-structure systems. Proc. 6th FSI, AE & FIV and N Symp. Vancouver.CrossRefGoogle Scholar
Luo, X. Y. & Cai, Z. X. 2004 Effects of wall stiffness on the linear stability of flow in an elastic channel. Proc. 8th Intl Conf. on Fluid Induced Vibrations, Paris, France (ed. Langre, E. de & Axisa, F.), vol. 2, pp. 167–170.Google Scholar
Luo, X. Y. & Pedley, T. J. 1995 A numerical simulation of steady flow in a two-dimensional collapsible channel. J. Fluids Struct. 9, 149–174.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
Luo, X. Y. & Pedley, T. J. 2000 Multiple solutions and flow limitation in two-dimensional collapsible channel flows. J. Fluid Mech. 420, 301–324.CrossRefGoogle Scholar
Luo, X. Y., Calderhead, B., Liu, H. F. & Li, W. G. 2007 On the initial configurations of collapsible channel flow. Comput. Struct. 85, 977–987.CrossRefGoogle Scholar
Marzo, A. X. Y., Luo, X. Y. & Bertram, C. D. 2005 Three-dimensional flow through a thick walled collapsible tube. J. Fluids Struct. 20, 817–835.CrossRefGoogle Scholar
Matsuzaki, Y. & Fujimura, K. 1995 Reexamination of steady solutions of a collapsible channel conveying fluid, a technical brief. Trans. ASME K: J. Biomech. Engng 117, 492–494.Google Scholar
Nayfeh, A. H. 1973 Perturbation Methods. Wiley-Interscience.Google Scholar
Paidoussis, M. P. 1998 Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 1. Elsevier.Google Scholar
Paidoussis, M. P. 2004 Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 2. Elsevier.Google Scholar
Paidoussis, M. P., Grinevich, E., Adamovic, D., & Semler, C. 2002 Linear and nonlinear dynamics of cantilevered cylinders in axial flow. Part 1. Physical dynamics. J. Fluids Struct. 16, 691–713.CrossRefGoogle Scholar
Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.CrossRefGoogle Scholar
Pedley, T. J. & Luo, X. Y. 1998 Modelling flow and oscillations in collapsible tubes. Theor. Comput. Fluid Dyn. 10, 277–294.CrossRefGoogle Scholar
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, 337–367.CrossRefGoogle Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 2002 Numerical Recipes in Fortran, the Art of Scientific Computing, 2nd edn. Cambridge University Press.Google Scholar
Semler, C., Lopes, J. L., Augu, N. & Paidoussis, M. P. 2002 Linear and nonlinear dynamics of cantilevered cylinders in axial flow. Part 3. Nonlinear dynamics. J. Fluids Struct. 16, 739–759.CrossRefGoogle Scholar
Shapiro, A. H. 1977 Steady flow in collapsible tubes. Trans. ASME K: J. Biomech. Engng 99, 126–147.Google Scholar
Tetlow, A., Lucey, A. D. & Balint, T. S. 2006 Instability of a cantilevered flexible plate in viscous channel flow driven by constant pressure drop. Proc. 6th FSI, AE & FIV and N Symp. Vancouver.CrossRefGoogle Scholar
Timoshenko, A. P. & Gere, J. M. 1963 Theory of Elastic Stability, 2nd edn. McGraw–Hill.Google Scholar
Webster, P. M., Sawatzky, R. P., Hoffstein, V., Leblang, R., Hinchey, J. M. & Sullivan, P. A. 1985 Wall motion in expiratory flow limitation: choke and flutter. J. Appl. Physiol. 59, 1304–1312.CrossRefGoogle ScholarPubMed
Wu, X. S. & Luo, J. S. 2006 Influence of small imperfections on the stability of plane Poiseuille flow and the limitation of Squire's theorem. Phys. Fluids 18, 044104.CrossRefGoogle Scholar
Yamaki, N. 1984 Elastic Stability of Circular Cylindrical Shells. North-Holland.Google Scholar
Zhu, Y. F., Luo, X. Y. & Ogden, R. W. 2007 Asymmetric bifurcations of thick-walled circular cylindrical elastic tubes under axial loading and external pressure. Intl J. Solids Struct. (accepted).CrossRefGoogle Scholar