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Nonlinear waves on the surface of a fluid covered by an elastic sheet

Published online by Cambridge University Press:  23 September 2013

Luc Deike ,
Jean-Claude Bacri  and
Eric Falcon
Luc Deike*
Affiliation:
Université Paris Diderot, Sorbonne Paris Cité, MSC, UMR 7057 CNRS, F-75 013 Paris, France
Jean-Claude Bacri
Affiliation:
Université Paris Diderot, Sorbonne Paris Cité, MSC, UMR 7057 CNRS, F-75 013 Paris, France
Eric Falcon
Affiliation:
Université Paris Diderot, Sorbonne Paris Cité, MSC, UMR 7057 CNRS, F-75 013 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

We experimentally study linear and nonlinear waves on the surface of a fluid covered by an elastic sheet where both tension and flexural waves occur. An optical method is used to obtain the full space–time wave field, and the dispersion relation of the waves. When the forcing is increased, a significant nonlinear shift of the dispersion relation is observed. We show that this shift is due to an additional tension of the sheet induced by the transverse motion of a fundamental mode of the sheet. When the system is subjected to a random-noise forcing at large scales, a regime of hydroelastic wave turbulence is observed with a power-law spectrum of the scale, in disagreement with the wave turbulence prediction. We show that the separation between relevant time scales is well satisfied at each scale of the turbulent cascade as expected theoretically. The wave field anisotropy, and finite size effects are also quantified and are not at the origin of the discrepancy. Finally, the dissipation is found to occur at all scales of the cascade, contrary to the theoretical hypothesis, and could thus explain this disagreement.

JFM classification

Waves/Free-surface Flows: Elastic waves Turbulent Flows: Turbulent Flows Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 733 , 25 October 2013 , pp. 394 - 413
Copyright
©2013 Cambridge University Press 

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References

Boudaoud, A., Cadot, O., Odille, B. & Touzé, C. 2008 Observation of wave turbulence in vibrating plates. Phys. Rev. Lett. 100, 234504.Google Scholar
Cobelli, P., Maurel, A., Pagneux, V. & Petitjeans, P. 2009 Global measurement of water waves by Fourier transform profilometry. Exp. Fluids 46, 1037–1047.Google Scholar
Cobelli, P., Przadka, A., Petitjeans, P., Lagubeau, G., Pagneux, V. & Maurel, A. 2011 Different regimes for water wave turbulence. Phys. Rev. Lett. 107, 214503.Google Scholar
Connaughton, C., Nazarenko, S. & Newell, A. C. 2003 Dimensional analysis and weak turbulence. Physica D 184 (1–4), 86–97.CrossRefGoogle Scholar
Davys, J. W., Hosking, R. J. & Sneyd, A. D. 1985 Waves due to a steadily moving source on a floating ice plate. J. Fluid Mech. 158, 269–287.Google Scholar
Deike, L., Berhanu, M. & Falcon, E. 2012 Decay of capillary wave turbulence. Phys. Rev. E 85, 066311.Google Scholar
Denissenko, P., Lukaschuk, S. & Nazarenko, S. 2007 Gravity wave turbulence in a laboratory flume. Phys. Rev. Lett. 99, 014501.Google Scholar
Düring, G., Josserand, C. & Rica, S. 2006 Weak turbulence for a vibrating plate: can one hear a Kolmogorov spectrum? Phys. Rev. Lett. 97, 025503.Google Scholar
Falcon, E. 2010 Laboratory experiment on wave turbulence. Discrete Contin. Dyn. Syst. B 13, 819–840.Google Scholar
Falcon, E., Laroche, C. & Fauve, S. 2007 Observation of gravity-capillary wave turbulence. Phys. Rev. Lett. 98 (9), 094503.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.Google Scholar
Herbert, E., Mordant, N. & Falcon, E. 2010 Observation of the nonlinear dispersion relation and spatial statistics of wave turbulence on the surface of a fluid. Phys. Rev. Lett. 105, 144502.Google Scholar
Humbert, T., Cadot, O., During, G., Josserand, C., Rica, S. & Touze, C. 2013 Wave turbulence in vibrating plates: the effect of damping. Eur. Phys. Lett. 102, 30002.Google Scholar
Kolmakov, G. V., Brazhnikov, M. Y., Levchenko, A. A., Abdurakhimov, L. V., McClintock, P. V. E. & Mezhov-Deglin, L. P. 2009 Capillary turbulence on the surfaces of quantum fluids. In Progress in Low Temperature Physics, 16 edn. chap. 6, Elsevier.Google Scholar
Lamb, H. 1932 Hydrodynamics. Dover.Google Scholar
Landau, L. & Lifchitz, F. 1951 Theory of Elasticity. Editions Mir.Google Scholar
Laurie, J., Bortolozzo, U., Nazarenko, S. & Residori, S. 2012 One-dimensional optical wave turbulence: experiment and theory. Phys. Rep. 514 (4), 121–175.Google Scholar
Legge, K. A. & Fletcher, N. H. 1984 Nonlinear generation of missing modes on a vibrating string. J. Acoust. Soc. Am. 76 (1), 5–12.Google Scholar
Marchenko, A. V. & Shrira, V. I. 1991 Theory of two-dimensional nonlinear waves in liquid covered by ice. Fluid Dyn. 26, 580–587.Google Scholar
McGoldrick, L. F., Phillips, O. M., Huang, N. E. & Hodgson, T. H. 1966 Measurements of third-order resonant wave interactions. J. Fluid Mech. 25, 437–456.Google Scholar
Miquel, B., Alexakis, A. & Mordant, N. 2013 The role of dissipation in flexural wave turbulence: from experimental spectrum to Kolmogorov–Zakharov spectrum. Phys. Rev. E (submitted).CrossRefGoogle Scholar
Miquel, B. & Mordant, N. 2011a Nonlinear dynamics of flexural wave turbulence. Phys. Rev. E 84, 066607.Google Scholar
Miquel, B. & Mordant, N. 2011b Nonstationary wave turbulence in an elastic plate. Phys. Rev. Lett. 107, 034501.Google Scholar
Mordant, N. 2008 Are there waves in elastic wave turbulence? Phys. Rev. Lett. 100, 234505.Google Scholar
Mordant, N. 2010 Fourier analysis of wave turbulence in a thin elastic plate. Eur. Phys. J. B 76, 537–545.Google Scholar
Morse, P. M. & Ingard, K. U. 1968 Theoretical Acoustics. Princeton University Press.Google Scholar
Nazarenko, S. 2011 Wave Turbulence. Springer.Google Scholar
Newell, A. C., Nazarenko, S. & Biven, L. 2001 Wave turbulence and intermittency. Physica D 152–153, 520–550.Google Scholar
Newell, A. C. & Rumpf, B. 2011 Wave Turbulence. Annu. Rev. Fluid Mech. 43 (1), 59–78.Google Scholar
Parau, E. & Dias, F. 2002 Nonlinear effects in the response of a floating ice plate to a moving load. J. Fluid Mech. 460, 281–305.Google Scholar
Peake, N. 2001 Nonlinear stability of a fluid-loaded elastic plate with mean flow. J. Fluid Mech. 434, 101–118.Google Scholar
Peake, N. & Sorokin, S. V. 2006 A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading. Proc. R. Soc.Lond. A 462 (2072), 2205–2224.Google Scholar
Schulkes, R. M. S. M., Hosking, R. J. & Sneyd, A. D. 1987 Waves due to a steadily moving source on a floating ice plate. Part 2. J. Fluid Mech. 180, 297–318.Google Scholar
Shelley, M. J. & Zhang, J. 2011 Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43 (1), 449–465.Google Scholar
Sommeria, J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–168.Google Scholar
Squire, V. A. 2007 Of ocean waves and sea-ice revisited. Cold Reg. Sci. Technol. 49, 110–133.Google Scholar
Squire, V. A., Hosking, R. J. & Kerr, A. D. 1996 Moving Loads on Ice Plates. Kluwer.Google Scholar
Squire, V. A., Robinson, W. H., Langhorne, P. J. & Haskell, T. G. 1988 Vehicles and aircraft on floating ice. Nature 33, 159–161.Google Scholar
Takizawa, T. 1985 Response of a floating sea ice sheet to a steadily moving load. J. Geophys. Res. 93, 5100–5112.Google Scholar
Watanabe, E., Utsunomiya, T. & Wang, C. M. 2004 Hydroelastic analysis of pontoon-type vlfs: a literature survey. Engng Struct. 26 (2), 245–256.Google Scholar
Wright, W. B., Budakian, R. & Putterman, S. J. 1996 Diffusing light photography of fully developed isotropic ripple turbulence. Phys. Rev. Lett. 76, 4528–4531.Google Scholar
Xia, H., Shats, M. & Punzmann, H. 2010 Modulation instability and capillary wave turbulence. Eur. Phys. Lett. 91 (1), 14002.Google Scholar
Zakharov, V. E., Falkovitch, G. C. & Lv´ov, V. S. 1992 Kolmogorov Spectra of Turbulence. I. Wave Turbulence. Springer.Google Scholar