Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-19T06:02:02.409Z Has data issue: false hasContentIssue false
  • English
  • Français

Coupling modes between two flapping filaments

Published online by Cambridge University Press:  22 May 2007

LAI-BING JIA ,
FANG LI ,
XIE-ZHEN YIN  and
XIE-YUAN YIN
LAI-BING JIA
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China
FANG LI
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China
XIE-ZHEN YIN*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China
XIE-YUAN YIN
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China
*
†Author to whom correspondence should be addressed: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The flapping coupling between two filaments is studied theoretically and experimentally in this paper. A temporal linear instability analysis is carried out based on a simplified hydrodynamic model. The dispersion relationship between the eigen-frequency ω and wavenumber k is expressed by a quartic equation. Two special cases of flapping coupling, i.e. two identical filaments having the same length and two filaments having different lengths, are studied in detail. In the case of two identical filaments, the theoretical analysis predicts four coupling modes, i.e. the stretched-straight mode, the antisymmetrical in-phase mode, the symmetrical out-of-phase mode and the indefinite mode. The theory also predicts the existence of an eigenfrequency jump during transition between the in-phase and out-of-phase modes, which has been observed in previous experiments and numerical simulations. In the case of two filaments having different lengths, four modes similar to those in the former case are identified theoretically. The distribution of coupling modes for both the cases is shown in two planes. One is a dimensionless plane of S vs. U, where S is the density ratio of solid filament to fluid and U2 is the ratio of fluid kinetic energy to solid elastic potential energy. The other is a dimensional plane of the half-distance (h) between two filaments vs. the filament length (L). Relevant experiments are carried out in a soap-film tunnel and the stable and unstable modes are observed. Theory and experiment are compared in detail. It should be noted that the model used in our analysis is a very simplified one that can provide intuitional analytical results of the coupling modes as well as their qualitative distributions. The factors neglected in our model, such as vortex shedding, viscous and nonlinear effects, do not allow the model to predict results precisely consistent with the experiments. Moreover, the Strouhal numbers of the flapping filaments are found to be generally around a fixed value in the experiments for both cases, implying that the filaments try to maintain a lower potential energy state.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 581 , 25 June 2007 , pp. 199 - 220
Copyright
Copyright © Cambridge University Press 2007

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

Argentina, M. & Mahadevan, L. 2005 Fluid-flow-induced flutter of a flag. Proc. Natl Acad. Sci. 102 (6), 1829–1834.CrossRefGoogle ScholarPubMed
Burmann, W. 1975 Water hammer in coaxial pipe systems. J. Hydrauli. Div. ASCE. 101, 699–715.CrossRefGoogle Scholar
Chen, J. N. & Lin, S. P. 2002 Instability of an annular jet surrounded by a viscous gas in a pipe. J. Fluid Mech. 450, 235–258.CrossRefGoogle Scholar
Chen, X. P., Jia, L. B., Yin, X. Z., Cheng, J. S. & Lu, J. 2005 Spraying modes in coaxial jet electrospray with outer driving liquid. Phys. Fluids 17, Art. No. 032101.CrossRefGoogle Scholar
Chomaz, J. M. 2001 The dynamics of a viscous soap film with soluble surfactant. J. Fluid Mech. 442, 387–409.CrossRefGoogle Scholar
Chomaz, J. M. & Cathalau, B. 1990 Soap films as 2-dimensional classical fluids. Phys. Rev. A 41 (4), 2243–2245.CrossRefGoogle Scholar
Couder, Y. 1981 The observation of a shear flow instability in a rotating system with a soap membrane. J. Phys. Lett. 42, 429–431.CrossRefGoogle Scholar
Couder, Y., Chomaz, J. M. & Rabaud, M. 1989 On the hydrodynamics of soap films. Physica D: 37 (1–3), 384–405.Google Scholar
Crighton, D. G. & Oswell, J. E. 1991 Fluid loading with mean flow. I. Response of an elastic plate to localized excitation. Phil. Trans. R. Soc. Lond. A 335, 557–592.Google Scholar
Du, Q. & Li, X. 2005 Effect of gas stream swirls on the instability of viscous annular liquid jets. Acta Mechanica 176 (1–2), 61–81.CrossRefGoogle Scholar
Evangelinos, C. & Karniadakis, G. E. M. 1999 Dynamics and flow structures in the turbulent wake of rigid and flexible cylinders subject to vortex-induced vibrations. J. Fluid Mech. 400, 91–124.CrossRefGoogle Scholar
Farnell, D. J. J., David, T. & Barton, D. C. 2004 a Coupled states of flapping flags. J. Fluids Struct. 19 (1), 29–36.CrossRefGoogle Scholar
Farnell, D. J. J., David, T. & Barton, D. C. 2004 b Numerical simulations of a filament in a flowing soap film. Intl J. Numer. Meth. Fluids 44 (3), 313–330.CrossRefGoogle Scholar
Gharib, M. & Derango, P. 1989 A liquid film (soap film) tunnel to study two-dimensional laminar and turbulent shear. Physica D: 37 (1–3), 406–416.Google Scholar
Ginzburg, V. L. 1989 Applications of Electrodynamics in Theoretical Physics and Astrophysics. Taylor & Francis..Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.CrossRefGoogle Scholar
Hagerty, W. W. & Shea, J. F. 1955 A study of the stability of plane fluid sheets. Trans. ASME: J. Appl. Mech. 22, 509–514.CrossRefGoogle Scholar
Han, C. D. & Shetty, R. 1977 Flow instability in tubular film blowing: 1. Experimental-study. Indust. Engng Chem. Fundam. 16 (1), 49–56.CrossRefGoogle Scholar
Hu, H. H. & Patankar, N. 1995 Non-axisymmetrical instability of core-annular flow. J. Fluid Mech. 290, 213–224.CrossRefGoogle Scholar
Huang, L. 1995 Flutter of cantilevered plates in axial-flow. J. Fluids Struct. 9 (2), 127–147.CrossRefGoogle Scholar
Ivanov, I. B. 1988 Thin Liquid Films. Marcel Dekker.Google Scholar
Lee, S. W., Fischer, P. F., Loth, F., Royston, T. J., Grogan, J. K. & Bassiouny, H. S. 2005 Flow-induced vein-wall vibration in an arteriovenous graft. J. Fluids Struct. 20 (6), 837–852.CrossRefGoogle Scholar
Li, F., Yin, X. Y. & Yin, X. Z. 2006 Instability analysis of a coaxial jet under a radial electric field in the nonequipotential case. Phys. Fluids 18 (3), 037101.CrossRefGoogle Scholar
Lopez-Herrera, J. M., Barrero, A., Lopez, A., Loscertales, I. G. & Marquez, M. 2003 Coaxial jets generated from electrified Taylor cones. Scaling laws. J. Aerosol Sci. 34 (5), 535–552.CrossRefGoogle Scholar
Loscertales, I. G., Barrero, A., Guerrero, I., Cortijo, R., Marquez, M. & Ganan-Calvo, A. M. 2002 Micro/nano encapsutation via electrified coaxial liquid jets. Science 295 (5560), 1695–1698.CrossRefGoogle ScholarPubMed
Lucassen, J., Tempel, M. Van Den, Vrij, A. & Hesselink, F. 1970 Waves in thin liquid films i. the different modes of vibration. Proc. Koninklijke Nederlandse Akad. Van Wetenschappen B 73, 109.Google Scholar
Lynch, D. G., Waters, S. L. & Pedley, T. J. 1996 Flow in a tube with non-uniform, time-dependent curvature: Governing equations and simple examples. J. Fluid Mech. 323, 237–265.CrossRefGoogle Scholar
Maa, Y. F., Nguyen, P. A. T. & Hsu, S. W. 1998 Spray-drying of air-liquid interface sensitive recombinant human growth hormone. J. Pharm. Sci. 87 (2), 152–159.CrossRefGoogle ScholarPubMed
Martin, B. & Wu, X. I. 1995 Shear flow in a two-dimensional couette cell: A technique for measuring the viscosity of free-standing liquid films. Rev. Sci. Instrum. 66 (12), 5603–5608.CrossRefGoogle Scholar
Rayleigh, Lord 1878 On the instability of jets. Proc. Lond. Math. Soc. 10, 4–13.CrossRefGoogle Scholar
Rutgers, M. A., Wu, X. L. & Daniel, W. B. 2001 Conducting fluid dynamics experiments with vertically falling soap films. Rev. Sci. Instrum. 72 (7), 3025–3037.CrossRefGoogle Scholar
Shelley, M., Vandenberghe, N. & Zhang, J. 2005 Heavy flags undergo spontaneous oscillations in flowing water. Phys. Rev. Lett. 94 (9), 094302.CrossRefGoogle ScholarPubMed
Shen, J. & Li, X. 1996 Instability of an annular viscous liquid jet. Acta Mechanica 114 (1–4), 167–183.CrossRefGoogle Scholar
Smith, B. W., Chase, J. G., Nokes, R. I., Shaw, G. M. & Wake, G. 2004 Minimal haemodynamic system model including ventricular interaction and valve dynamics. Med. Engng Phys. 26 (2), 131–139.CrossRefGoogle ScholarPubMed
Squire, H. B. 1953 Investigation of the instability of a moving liquid film. Bit. J. Appl. Phys. 4, 167–169.Google Scholar
Stanley, S. & Sarkar, S. 1997 Simulations of spatially developing two-dimensional shear layers and jets. Theor. Comput. Fluid Dyn. 9 (2), 121–147.CrossRefGoogle Scholar
Taylor, G. K., Nudds, R. L. & Thomas, A. L. R. 2003 Flying and swimming animals cruise at a strouhal number tuned for high power efficiency. Nature. 425, 707–711.CrossRefGoogle Scholar
Trapeznikov, A. A. 1957 Application of the method of two-dimensional viscosity and shear strength to the investigation of the structure and composition of two-sided films and surface layers in solutions of soaps and saponins. Proc. 2nd Intl Congress on Surface Activity, pp. 242–258.Google Scholar
Triantafyllou, M. S., Triantafyllou, G. S. & Gopalkrishnan, R. 1991 Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A 3 (12), 2835–2837.CrossRefGoogle Scholar
Veligorskiy, V. I. 1991 The effect of external axisymmetrical pressure on the characteristics of a 2-layer optical fiber. Telecom. Radio Engng 46 (10), 5–12.Google Scholar
Villermaux, E. & Clanet, C. 2002 Life of a flapping liquid sheet. J. Fluid Mech. 462, 341–363.CrossRefGoogle Scholar
Vorobieff, P. & Ecke, R. E. 1999 Cylinder wakes in flowing soap films. Phys. Rev. E 60 (3), 2953–2956.Google ScholarPubMed
Watanabe, Y., Isogai, K., Suzuki, S. & Sugihara, M. 2002 a A theoretical study of paper flutter. J. Fluids Struct. 16 (4), 543–560.CrossRefGoogle Scholar
Watanabe, Y., Suzuki, S., Sugihara, M. & Sueoka, Y. 2002 b An experimental study of paper flutter. J. Fluids Struct. 16 (4), 529–542.CrossRefGoogle Scholar
Wen, C. Y. & Lin, C. Y. 2001 Two-dimensional vortex shedding of a circular cylinder. Phys. Fluids 13 (3), 557–560.CrossRefGoogle Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.CrossRefGoogle Scholar
York, J. L., Stubbs, H. E. & Tek, M. R. 1953 The mechanism of disintegration of liquid sheets. Trans. ASME pp. 1279–1286.CrossRefGoogle Scholar
Zhang, J., Childress, S., Libchaber, A. & Shelley, M. 2000 Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature. 408, 835–839.CrossRefGoogle Scholar
Zhu, L. D. & Peskin, C. S. 2002 Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method. J. Comput. Phys. 179 (2), 452–468.CrossRefGoogle Scholar
Zhu, L. D. & Peskin, C. S. 2003 Interaction of two flapping filaments in a flowing soap film. Phys. Fluids 15 (7), 1954–1960.CrossRefGoogle Scholar