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A numerical simulation of unsteady flow in a two-dimensional collapsible channel

Published online by Cambridge University Press:  26 April 2006

X. Y. Luo  and
T. J. Pedley
X. Y. Luo
Affiliation:
Department of Applied Mathematical Studies, The University of Leeds, Leeds LS2 9JT, UK
T. J. Pedley
Affiliation:
Department of Applied Mathematical Studies, The University of Leeds, Leeds LS2 9JT, UK Present address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK.
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Abstract

The collapse of a compressed elastic tube conveying a flow occurs in several physiological applications and has become a problem of considerable interest. Laboratory experiments on a finite length of collapsible tube reveal a rich variety of self-excited oscillations, indicating that the system is a complex, nonlinear dynamical system. Following our previous study on steady flow in a two-dimensional model of the collapsible tube problem (Luo & Pedley 1995), we here investigate the instability of the steady solution, and details of the resulting oscillations when it is unstable, by studying the time-dependent problem. For this purpose, we have developed a time-dependent simulation of the coupled flow – membrane problem, using the Spine method to treat the moving boundary and a second-order time integration scheme with variable time increments.

It is found that the steady solutions become unstable as tension falls below a certain value, say Tu, which decreases as the Reynolds number increases. As a consequence, steady flow gives way to self-excited oscillations, which become increasingly complicated as tension is decreased from Tu. A sequence of bifurcations going through regular oscillations to irregular oscillations is found, showing some interesting dynamic features similar to those observed in experiments. In addition, vorticity waves are found downstream of the elastic section, with associated recirculating eddies which sometimes split into two. These are similar to the vorticity waves found previously for flow past prescribed, time-dependent indentations. It is speculated that the mechanism of the oscillation is crucially dependent on the details of energy dissipation and flow separation at the indentation.

As tension is reduced even further, the membrane is sucked underneath the downstream rigid wall and, although this causes the numerical scheme to break down, it in fact agrees with another experimental observation for flow in thin tubes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 314 , 10 May 1996 , pp. 191 - 225
Copyright
© 1996 Cambridge University Press

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References

Bertram, C. D. 1982 Two modes of instability in a thick-walled collapsible tube conveying a flow. J. Biomech. 15, 223–224.Google Scholar
Bertram, C. D. 1986 Unstable equilibrium behaviour in collapsible tubes. J. Biomech. 19, 61–69.Google Scholar
Bertram, C. D. & Pedley, T. J. 1982 A mathematical model of unsteady collapsible tube behaviour. J. Biomech. 15, 39–50.Google Scholar
Bertram, C. D. & Pedley, T. J. 1983 Steady and unsteady separation in an approximately two-dimensional indented channel. J. Fluid Mech. 130, 315–345.Google Scholar
Bertram, C. D., Raymond, C. J. & Pedley, T. J. 1990 Mapping of instabilities for flow through collapsed tubes of different length. J. Fluids Struct. 4, 125–154.Google Scholar
Bertram, C. D., Raymond, C. J. & Pedley, T. J. 1991 Application of nonlinear dynamics concepts to the analysis of self-excited oscillations of a collapsible tube conveying a fluid. J. Fluids Struct. 5, 391–426.Google Scholar
Bonis, M. & Ribreau, C. 1978 Etude de quelques propriétés de l’écoulement dans une conduite collabable. La Houille Blanche 3/4, 165–173.Google Scholar
Borgas, M. S. 1986 Waves, singularities and non-uniqueness in channel and pipe flows. PhD thesis, Cambridge University.
Brower, R. W. & Scholten, C. 1975 Experimental evidence on the mechanism for the instability of flow in collapsible vessels. Med. Biol. Engng 13, 839–845.Google Scholar
Cancelli, C. & Pedley, T. J. 1985 A separated-flow model for collapsible-tube oscillations. J. Fluid Mech. 157, 375–404.Google 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, 199–232.Google Scholar
Carpenter, P. W. & Morris, P. J. 1990 The effect of anisotropic wall compliance on boundary-layer stability and transition. J. Fluid Mech. 218, 171–223.Google Scholar
Davies, C. 1995 Evolution of Tollmien-Schlichting waves over a compliant panel. PhD thesis, University of Warwick.
Davies, C. & Carpenter, P. W. 1995 Numerical simulation of the evolution of Tollmien-Schlichting waves over finite compliant panels. J. Fluid Mech. (submitted).Google Scholar
Davies, C. & Carpenter, P. W. 1996 Instabilities in a panel channel flow between compliant walls. to be submitted.
Ehrenstein, U. & Koch, W. 1989 Three-dimensional wavelike equilibrium states in plane Poiseuille flow. J. Fluid Mech. 228, 111–148.Google Scholar
Engelman, M. S., Sani, R. L. & Gresho, P. M. 1982 The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow. Intl J. Numer. Meth. Fluids 2, 225–238.Google Scholar
Gresho, P. M., Lee, R. L. & Sani, R. L. 1979 On the time-dependent solutions of the Navier-Stokes equations in two and three dimensions. In Recent Advances in Numerical Methods in Fluids (ed. C. Taylor & K. Morgan), pp. 27–79.
Grotberg, J. B. & Reiss, E. L. 1984 Subsonic flapping flutter. J. Sound Vib. 92, 349–361.Google Scholar
Grotberg, J. B. & Shee, T. R. 1985 Compressible-flow channel flutter. J. Fluid Mech. 159, 175–193.Google Scholar
Hood, P 1976 Frontal solution program for unsymmetric matrices. Intl J. Numer. Meth. Engng 10, 379–399.Google Scholar
Irons, B. M. 1970 A frontal solution program for finite element analysis. Intl J. Numer. Meth. Engng 2, 5–32.Google Scholar
Jensen, O. E. 1990 Instabilities of flow in a collapsed tube. J. Fluid Mech. 220, 623–659.Google Scholar
Jensen, O. E. 1992 Chaotic oscillations in a simple collapsible tube model. J. Biomech. Engng 114, 55–59.Google Scholar
Katz, A. I., Chen, Y. & Moreno, A. H. 1969 Flow through a collapsible tube. Biophys. J. 9, 1261–1279.Google Scholar
Kheshgi, H. S. & Scriven, L. E. 1984 Penalty finite element analysis of unsteady free surface flows. Finite Elements in Fluids 5 393–434.Google Scholar
Kramer, M. O. 1962 Boundary layer stabilization by distributed damping. J. Am. Soc. Naval Engrs 74, 341–385.Google Scholar
Lowe, T. W., Luo, X. Y. & Rast, M. P. 1995 A comparison of three solution methods for flow in a collapsible channel. J. Fluids Struct. (submitted).Google Scholar
Lowe, T. W. & Pedley, T. J. 1995 Computation of Stokes flow in a channel with a collapsible segment. J. Fluids Struct. 9, 885–905.Google Scholar
Lucey, A. D. & Carpenter, P. W. 1992 A numerical simulation of the interaction of a compliant wall and inviscid flow. J. Fluid Mech. 234, 147–170.Google Scholar
Luo, X. Y. & Pedley, T. J. 1995 A numerical simulation of steady flow in a 2-D collapsible channel. J. Fluids Struct. 9, 149–174.Google Scholar
Matsuzaki, Y. & Fung, Y. C. 1979 Nonlinear stability analysis of a two-dimensional model of an elastic tube conveying a compressible flow. J. Appl. Mech. 46, 31–36.Google Scholar
Matsuzaki, Y. & Matsumoto, T. 1989 Flow in a two-dimensional collapsible channel with rigid inlet and outlet. J. Biomech. Engng 111, 180–184.Google Scholar
Ohba, K., Yoneyama, N., Shimanaka, Y. & Maeda, H. 1984 Self-excited oscillation of flow in collapsible tube. Tech. Rep. Kansai Univ. 25, 1–13.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Pedley, T. J. 1992 Longitudinal tension variation in collapsible channels: a new mechanism for the breakdown of steady flow. J. Biomech. Engng 114, 60–67.Google 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.Google Scholar
Ralph, M. E. & Pedley, T. J. 1988 Flow in a channel with a moving indentation. J. Fluid Mech. 190, 87–112.Google Scholar
Ralph, M. E. & Pedley, T. J. 1989 Viscous and inviscid flows in a channel with a moving indentation. J. Fluid Mech. 109, 543–566.Google Scholar
Ralph, M. E. & Pedley, T. J. 1990 Flow in a channel with a time-dependent indentation in one wall. J. Fluids Engng 112, 468–475.Google Scholar
Rast, M. P. 1994 Simultaneous solution of the Navier-Stokes and elastic membrane equations by a finite-element method. Intl J. Numer. Meth. Fluids 19, 1115–1135.Google Scholar
Reyn, J. W. 1974 On the mechanism of self-excited oscillations in the flow through collapsible tubes. Delft Prog. Rep. 1, 51–67.Google Scholar
Riley, J. J., Gad-el-Hak, M. & Metcalfe, R. W. 1988 Compliant coatings. Ann. Rev. Fluid Mech. 20, 393–420.Google Scholar
Ruschak, K. J. 1980 A method for incorporating free boundaries with surface tension in finite element fluid-flow simulators. Intl J. Numer. Meth. Engng 15, 639–648.Google Scholar
Saito, H. & Scriven, L. E. 1981 Studying of coating flow by the finite element method. J. Comput. Phys. 42, 53–76.Google Scholar
Schoendorfer, D. W. & Shapiro, A. H. 1977 The collapsible tube as a prosthetic vocal source. Proc. San Diego Biomed. Symp. 16, 349–356.
Shapiro, A. H. 1977 Steady flow in collapsible tubes. J. Biomech. Engng 99, 126–147.Google Scholar
Silliman, W. J. 1979 Viscous film flows with contact lines. PhD thesis, University of Minnesota.
Weaver, D. S. & Paidoussis, M. P. 1977 On collapse and flutter phenomena in thin tubes conveying fluid. J. Sound Vib. 50, 117–132.Google Scholar