Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T14:05:42.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition to chaos in a two-sided collapsible channel flow

Published online by Cambridge University Press:  07 September 2021

Qiuxiang Huang Open the ORCID record for Qiuxiang Huang [Opens in a new window] ,
Fang-Bao Tian Open the ORCID record for Fang-Bao Tian [Opens in a new window] ,
John Young Open the ORCID record for John Young [Opens in a new window]  and
Joseph C.S. Lai Open the ORCID record for Joseph C.S. Lai [Opens in a new window]
Qiuxiang Huang
Affiliation:
School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia
Fang-Bao Tian*
Affiliation:
School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia
John Young
Affiliation:
School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia
Joseph C.S. Lai
Affiliation:
School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia
*
†Email addresses for correspondence: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The nonlinear dynamics of a two-sided collapsible channel flow is investigated by using an immersed boundary-lattice Boltzmann method. The stability of the hydrodynamic flow and collapsible channel walls is examined over a wide range of Reynolds numbers $Re$, structure-to-fluid mass ratios $M$ and external pressures $P_e$. Based on extensive simulations, we first characterise the chaotic behaviours of the collapsible channel flow and explore possible routes to chaos. We then explore the physical mechanisms responsible for the onset of self-excited oscillations. Nonlinear and rich dynamic behaviours of the collapsible system are discovered. Specifically, the system experiences a supercritical Hopf bifurcation leading to a period-1 limit cycle oscillation. The existence of chaotic behaviours of the collapsible channel walls is confirmed by a positive dominant Lyapunov exponent and a chaotic attractor in the velocity-displacement phase portrait of the mid-point of the collapsible channel wall. Chaos in the system can be reached via period-doubling and quasi-periodic bifurcations. It is also found that symmetry breaking is not a prerequisite for the onset of self-excited oscillations. However, symmetry breaking induced by mass ratio and external pressure may lead to a chaotic state. Unbalanced transmural pressure, wall inertia and shear layer instabilities in the vorticity waves contribute to the onset of self-excited oscillations of the collapsible system. The period-doubling, quasi-periodic and chaotic oscillations are closely associated with vortex pairing and merging of adjacent vortices, and interactions between the vortices on the upper and lower walls downstream of the throat.

JFM classification

Biological Fluid Dynamics: Blood flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 926 , 10 November 2021 , A15
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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.)

References

REFERENCES

Aidun, C.K. & Clausen, J.R. 2010 Lattice-Boltzmann method for complex flows. Annu. Rev. Fluid Mech. 42, 439–472.CrossRefGoogle Scholar
Battaglia, F., Tavener, S.J., Kulkarni, A.K. & Merkle, C.L. 1997 Bifurcation of low Reynolds number flows in symmetric channels. AIAA J. 35, 99–105.CrossRefGoogle Scholar
Bertram, C.D. 1986 Unstable equilibrium behaviour in collapsible tubes. J. Biomech. 19, 61–69.CrossRefGoogle ScholarPubMed
Bertram, C.D., Raymond, C.J. & Butcher, K.S.A. 1989 Oscillations in a collapsed-tube analog of the brachial artery under a sphygmomanometer cuff. Trans. ASME J. Biomech. Engng 111, 185–191.CrossRefGoogle 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
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.CrossRefGoogle Scholar
Bertram, C.D. & Tscherry, J. 2006 The onset of flow-rate limitation and flow-induced oscillations in collapsible tubes. J. Fluids Struct. 22, 1029–1045.CrossRefGoogle Scholar
Bluestein, D., Gutierrez, C., Londono, M. & Schoephoerster, R.T 1999 Vortex shedding in steady flow through a model of an arterial stenosis and its relevance to mural platelet deposition. Ann. Biomed. Engng 27, 763–773.CrossRefGoogle Scholar
Borazjani, I., Ge, L. & Sotiropoulos, F. 2008 Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies. J. Comput. Phys. 227, 7587–7620.CrossRefGoogle ScholarPubMed
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
Chen, S. & Doolen, G.D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–64.CrossRefGoogle Scholar
Connell, B.S.H. & Yue, D.K.P. 2007 Flapping dynamics of a flag in a uniform stream. J. Fluid Mech. 581, 33–67.CrossRefGoogle Scholar
Dai, G., Gertler, J.P. & Kamm, R.D. 1999 The effects of external compression on venous blood flow and tissue deformation in the lower leg. Trans. ASME J. Biomech. Engng 121, 557–564.CrossRefGoogle ScholarPubMed
Davies, C. & Carpenter, P.W. 1997 a Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205–243.CrossRefGoogle Scholar
Davies, C. & Carpenter, P.W. 1997 b Numerical simulation of the evolution of Tollmien-Schlichting waves over finite compliant panels. J. Fluid Mech. 335, 361–392.CrossRefGoogle Scholar
Favier, J., Revell, A. & Pinelli, A. 2014 A lattice Boltzmann-immersed boundary method to simulate the fluid interaction with moving and slender flexible objects. J. Comput. Phys. 261, 145–161.CrossRefGoogle Scholar
Feigenbaum, M.J. 1978 Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52.CrossRefGoogle Scholar
Feigenbaum, M.J. 1979 The universal metric properties of nonlinear transformations. J. Stat. Phys. 21, 669–706.CrossRefGoogle Scholar
Feng, L.Y., Gao, H., Griffith, B., Niederer, S. & Luo, X.Y. 2019 Analysis of a coupled fluid–structure interaction model of the left atrium and mitral valve. Intl J. Numer. Meth. Biomed. Engng 35, e3254.CrossRefGoogle ScholarPubMed
Feng, Z.G. & Michaelides, E.E. 2004 The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems. J. Comput. Phys. 195, 602–628.CrossRefGoogle Scholar
Gilmanov, A., Le, T. & Sotiropoulos, F. 2015 A numerical approach for simulating fluid structure interaction of flexible thin shells undergoing arbitrarily large deformations in complex domains. J. Comput. Phys. 300, 814–843.CrossRefGoogle Scholar
Goza, A., Colonius, T. & Sader, J.E. 2018 Global modes and nonlinear analysis of inverted-flag flapping. J. Fluid Mech. 857, 312–344.CrossRefGoogle Scholar
Griffith, B.E. & Patankar, N.A. 2020 Immersed methods for fluid–structure interaction. Annu. Rev. Fluid Mech. 52, 421–448.CrossRefGoogle ScholarPubMed
Grotberg, J.B. & Jensen, O.E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.CrossRefGoogle Scholar
Guiot, G., Pianta, P.G., Cancelli, C. & Pedley, T.J. 1990 Prediction of coronary blood flow with a numerical based on collapsible tube dynamics model. Am. J. Physiol. Heart Circ. Physiol. 258, H1606–H1614.CrossRefGoogle Scholar
Guo, Z.-L., Zheng, C.-G. & Shi, B.-C. 2002 Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method. Chin. Phys. 11, 366–374.Google Scholar
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. & Boyle, J. 2010 Self-excited oscillations in three-dimensional collapsible tubes: simulating their onset and large-amplitude oscillations. J. Fluid Mech. 652, 405–426.CrossRefGoogle Scholar
Heil, M. & Hazel, A.L. 2011 Fluid–structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162.CrossRefGoogle Scholar
Hua, R., Zhu, L. & Lu, X.-Y. 2014 Dynamics of fluid flow over a circular flexible plate. J. Fluid Mech. 759, 56–72.CrossRefGoogle Scholar
Huang, W.-X., Shin, S.J. & Sung, H.J. 2007 Simulation of flexible filaments in a uniform flow by the immersed boundary method. J. Comput. Phys. 226, 2206–2228.CrossRefGoogle Scholar
Huang, W.-X. & Tian, F.-B. 2019 Recent trends and progress in the immersed boundary method. Proc. Inst. Mech. Engng C 233, 7617–7636.CrossRefGoogle Scholar
Hussain, A.K.M.F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303–356.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 J. Biomech. Engng 114, 55–59.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
Kang, S.K. & Hassan, Y.A. 2011 A comparative study of direct-forcing immersed boundary-lattice Boltzmann methods for stationary complex boundaries. Intl J. Numer. Meth. Fluids 66, 1132–1158.CrossRefGoogle Scholar
Kheiri, M. 2020 Nonlinear dynamics of imperfectly-supported pipes conveying fluid. J. Fluids Struct. 93, 102850.CrossRefGoogle Scholar
Kounanis, K. & Mathioulakis, D.S. 1999 Experimental flow study within a self oscillating collapsible tube. J. Fluids Struct. 13, 61–73.CrossRefGoogle Scholar
Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G. & Viggen, E.M. 2017 The Lattice Boltzmann Method. Springer International Publishing.CrossRefGoogle Scholar
Krüger, T., Varnik, F. & Raabe, D. 2011 Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method. Comput. Maths Applics. 61, 3485–3505.CrossRefGoogle Scholar
Ku, D.N. 1997 Blood flow in arteries. Annu. Rev. Fluid Mech. 29, 399–434.CrossRefGoogle Scholar
Lallemand, P. & Luo, L.-S. 2000 Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 61, 6546–6562.CrossRefGoogle ScholarPubMed
Liu, H.F., Luo, X.Y. & Cai, Z.X. 2012 Stability and energy budget of pressure-driven collapsible channel flows. J. Fluid Mech. 705, 348–370.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
Lucey, A.D. & Carpenter, P.W. 1992 A numerical simulation of the interaction of a compliant wall and inviscid flow. J. Fluid Mech. 234, 121–146.CrossRefGoogle Scholar
Lucey, A.D. & Carpenter, P.W. 1995 Boundary layer instability over compliant walls: comparison between theory and experiment. Phys. Fluids 7, 2355–2363.CrossRefGoogle Scholar
Luo, L.-S., Liao, W., Chen, X., Peng, Y. & Zhang, W. 2011 Numerics of the lattice Boltzmann method: effects of collision models on the lattice Boltzmann simulations. Phys. Rev. E 83, 1–24.CrossRefGoogle ScholarPubMed
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
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
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.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, Y., Xiao, Q., Zhu, Q. & Pan, G. 2020 Pulsed-jet propulsion of a squid-inspired swimmer at high Reynolds number. Phys. Fluids 32, 111901.CrossRefGoogle Scholar
Ma, J.T., Wang, Z., Young, J., Lai, J.C.S., Sui, Y. & Tian, F.-B. 2020 An immersed boundary-lattice Boltzmann method for fluid–structure interaction problems involving viscoelastic fluids and complex geometries. J. Comput. Phys. 415, 109487.CrossRefGoogle Scholar
Manneville, P. & Pomeau, Y. 1980 Different ways to turbulence in dissipative dynamical systems. Physica D 1, 219–226.CrossRefGoogle Scholar
Marzo, A., Luo, X.Y. & Bertram, C.D. 2005 Three-dimensional collapse and steady flow in thick-walled flexible tubes. J. Fluids Struct. 20, 817–835.CrossRefGoogle Scholar
Miozzi, M., Querzoli, G. & Romano, G.P. 1998 The investigation of an unstable convective flow using optical methods. Phys. Fluids 10, 2995–3008.CrossRefGoogle Scholar
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261.CrossRefGoogle Scholar
Newhouse, S., Ruelle, D. & Takens, F. 1978 Occurrence of strange axioma attractors near quasi periodic flows on $T^{m}$, $m\ge 3$. Commun. Math. Phys. 64, 35–40.CrossRefGoogle Scholar
Ohba, K, Skurai, A. & Oka, J. 1997 Laser Doppler measurement of local flow field in collapsible tube during self-excited oscillation. JSME Intl J. 40, 665–670.CrossRefGoogle Scholar
Oliveira, M.S.N., Rodd, L.E., McKinley, G.H. & Alves, M.A. 2008 Simulations of extensional flow in microrheometric devices. Microfluid Nanofluidics 5, 809.CrossRefGoogle Scholar
Pedley, T.J. 1992 Longitudinal tension variation in collapsible channels: a new mechanism for the breakdown of steady flow. Trans. ASME J. Biomech. Engng 114, 60–67.CrossRefGoogle ScholarPubMed
Peskin, C.S. 1972 Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10, 252–271.CrossRefGoogle Scholar
Peskin, C.S. 2002 The immersed boundary method. Acta Numer. 11, 479–517.CrossRefGoogle Scholar
Rocha, G.N., Poole, R.J. & Oliveira, P.J. 2007 Bifurcation phenomena in viscoelastic flows through a symmetric 1:4 expansion. J. Non-Newtonian Fluid 141, 1–17.CrossRefGoogle Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167–192.CrossRefGoogle Scholar
Shapiro, A.H. 1977 Steady flow in collapsible tubes. Trans. ASME J. Biomech. Engng 99, 126–147.CrossRefGoogle Scholar
Sobey, I.J. & Drazin, P.G. 1986 Bifurcations of two-dimensional channel flows. J. Fluid Mech. 171, 263–287.CrossRefGoogle Scholar
Sotiropoulos, F. & Yang, X. 2014 Immersed boundary methods for simulating fluid–structure interaction. Prog. Aerosp. Sci. 65, 1–21.CrossRefGoogle Scholar
Stewart, P.S., Heil, M., Waters, S.L. & Jensen, O.E. 2010 Sloshing and slamming oscillations in a collapsible channel flow. J. Fluid Mech. 662, 288–319.CrossRefGoogle Scholar
Sui, Y., Chew, Y.T., Roy, P. & Low, H.T. 2008 A hybrid method to study flow-induced deformation of three-dimensional capsules. J. Comput. Phys. 227, 6351–6371.CrossRefGoogle Scholar
Tang, C., Zhu, L., Akingba, G. & Lu, X.-Y. 2015 Viscous flow past a collapsible channel as a model for self-excited oscillation of blood vessels. J. Biomech. 48, 1922–1929.CrossRefGoogle Scholar
Tian, F.-B. 2014 FSI modeling with the DSD/SST method for the fluid and finite difference method for the structure. Comput. Mech. 54, 581–589.CrossRefGoogle Scholar
Tian, F.-B., Dai, H., Luo, H.X., Doyle, J.F. & Rousseau, B. 2014 Fluid–structure interaction involving large deformations: 3D simulations and applications to biological systems. J. Comput. Phys. 258, 451–469.CrossRefGoogle Scholar
Tian, F.-B., Luo, H., Zhu, L., Liao, J.C. & Lu, X.-Y. 2011 a An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments. J. Comput. Phys. 230, 7266–7283.CrossRefGoogle ScholarPubMed
Tian, F.-B., Luo, H., Zhu, L. & Lu, X.-Y. 2010 Interaction between a flexible filament and a downstream rigid body. Phys. Rev. E 82, 026301.CrossRefGoogle Scholar
Tian, F.-B., Luo, H., Zhu, L. & Lu, X.-Y. 2011 b Coupling modes of three filaments in side-by-side arrangement. Phys. Fluids 23, 111903.CrossRefGoogle Scholar
Tian, F.-B., Wang, Y., Young, J. & Lai, J.C.S. 2015 An FSI solution technique based on the DSD/SST method and its applications. Math. Models Meth. Appl. Sci. 25, 2257–2285.CrossRefGoogle Scholar
Tsigklifis, K. & Lucey, A.D. 2017 Asymptotic stability and transient growth in pulsatile Poiseuille flow through a compliant channel. J. Fluid Mech. 820, 370–399.CrossRefGoogle Scholar
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, 429–445.CrossRefGoogle Scholar
Wang, L., Currao, G.M.D., Han, F., Neely, A.J., Young, J. & Tian, F.-B. 2017 An immersed boundary method for fluid–structure interaction with compressible multiphase flows. J. Comput. Phys. 346, 131–151.CrossRefGoogle Scholar
Wang, L. & Tian, F.-B. 2018 Heat transfer in non-Newtonian flows by a hybrid immersed boundary–lattice Boltzmann and finite difference method. Appl. Sci. 8, 559.CrossRefGoogle Scholar
Williams, H.A.R., Fauci, L.J. & Gaver III, D.P. 2009 Evaluation of interfacial fluid dynamical stresses. Discrete Continuous Dyn. Syst. 11, 519–540.CrossRefGoogle ScholarPubMed
Wolf, A. 2021 Wolf lyapunov exponent estimation from a time series. MATLAB Central File Exchange. Retrieved June 6, 2021, https://www.mathworks.com/matlabcentral/fileexchange/48084-wolf-lyapunov-exponent-estimation-from-a-time-series.Google Scholar
Wolf, A., Swift, J.B., Swinney, H.L. & Vastano, J.A. 1985 Determining Lyapunov exponents from a time series. Physica D 16, 285–317.CrossRefGoogle Scholar
Xu, L.C., Tian, F.-B., Young, J. & Lai, J.C.S. 2018 A novel geometry-adaptive Cartesian grid based immersed boundary–lattice Boltzmann method for fluid–structure interactions at moderate and high Reynolds numbers. J. Comput. Phys. 375, 22–56.CrossRefGoogle Scholar
Zhang, J., Liu, N.-S. & Lu, X.-Y. 2009 Route to a chaotic state in fluid flow past an inclined flat plate. Phys. Rev. E 79, 045306.CrossRefGoogle Scholar
Zhang, S., Luo, X.Y. & Cai, Z.X. 2018 Three-dimensional flows in a hyperelastic vessel under external pressure. Biomech. Model. Mechanobiol. 17, 1187–1207.CrossRefGoogle Scholar
Zhu, L., He, G., Wang, S., Miller, L., Zhang, X., You, Q. & Fang, S. 2011 An immersed boundary method based on the lattice Boltzmann approach in three dimensions, with application. Comput. Maths Applics. 61, 3506–3518.CrossRefGoogle Scholar

Huang et al. Supplementary Movie 1

Re = 200, M=1, Ks=2400 and Pe=1.95

Download Huang et al. Supplementary Movie 1(Video)
Video 6.8 MB

Huang et al. Supplementary Movie 2

Re = 550, M=1, Ks=2400 and Pe=1.95

Download Huang et al. Supplementary Movie 2(Video)
Video 5.6 MB

Huang et al. Supplementary Movie 3

Re = 600, M=1, Ks=2400 and Pe=1.95

Download Huang et al. Supplementary Movie 3(Video)
Video 6.2 MB

Huang et al. Supplementary Movie 4

Re = 650, M=1, Ks=2400 and Pe=1.95

Download Huang et al. Supplementary Movie 4(Video)
Video 9.7 MB

Huang et al. Supplementary Movie 5

Re = 800, M=1, Ks=2400 and Pe=1.95

Download Huang et al. Supplementary Movie 5(Video)
Video 7.8 MB

Huang et al. Supplementary Movie 6

Re = 250, M=5, Ks=2400 and Pe=1.95

Download Huang et al. Supplementary Movie 6(Video)
Video 7.1 MB

Huang et al. Supplementary Movie 7

Re = 250, M=6, Ks=2400 and Pe=1.95

Download Huang et al. Supplementary Movie 7(Video)
Video 7 MB

Huang et al. Supplementary Movie 8

Re = 250, M=7, Ks=2400 and Pe=1.95

Download Huang et al. Supplementary Movie 8(Video)
Video 7.1 MB

Huang et al. Supplementary Movie 9

Re = 250, M=8, Ks=2400 and Pe=1.95

Download Huang et al. Supplementary Movie 9(Video)
Video 7.1 MB