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Measurement of the velocity field in parametrically excited solitary waves

Published online by Cambridge University Press:  14 August 2014

Leonardo Gordillo  and
Nicolás Mujica
Leonardo Gordillo*
Affiliation:
Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile Laboratoire ‘Matière et Systèmes Complexes’ (MSC), UMR 7057 CNRS, Université Paris 7 Diderot, 75205 Paris CEDEX 13, France
Nicolás Mujica
Affiliation:
Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
*
†Email address for correspondence: leonardo.gordillo@univ-paris-diderot.fr
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Abstract

Parametrically excited solitary waves emerge as localized structures in high-aspect-ratio free surfaces subject to vertical vibrations. Herein, we provide the first experimental characterization of the hydrodynamics of these waves using particle image velocimetry. We show that the underlying velocity field of parametrically excited solitary waves is primarily composed of a subharmonic oscillatory component. Our results confirm the accuracy of Hamiltonian models with added dissipation in describing this field. Remarkably, our measurements also uncover the onset of a streaming velocity field which we show to be as important as other crucial nonlinear terms in the current theory. Using vorticity equations, we show that the streaming pattern arises from the coupling of the potential bulk flow with the oscillating boundary layers on the vertical walls. Numerical simulations provide good agreement between this model and experiments.

JFM classification

Instability: Parametric instability Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 754 , 10 September 2014 , pp. 590 - 604
Copyright
© 2014 Cambridge University Press 

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References

Arbell, H. & Fineberg, J. 2000 Temporally harmonic oscillons in Newtonian fluids. Phys. Rev. Lett. 85 (4), 756–759.CrossRefGoogle ScholarPubMed
Barashenkov, I. V., Bogdan, M. M. & Korobov, V. I. 1991 Stability diagram of the phase-locked solitons in the parametrically driven, damped nonlinear Schrödinger equation. Europhys. Lett. 15, 113–118.CrossRefGoogle Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Chen, W., Tu, J. & Wei, R. 1999 Onset instability to non-propagating hydrodynamic solitons. Phys. Lett. A 255 (4), 272–276.CrossRefGoogle Scholar
Clerc, M. G., Coulibaly, S., Gordillo, L., Mujica, N. & Navarro, R. 2011 Coalescence cascade of dissipative solitons in parametrically driven systems. Phys. Rev. E 84 (3), 036205.CrossRefGoogle ScholarPubMed
Clerc, M. G., Coulibaly, S., Mujica, N., Navarro, R. & Sauma, T. 2009 Soliton pair interaction law in parametrically driven Newtonian fluid. Phil. Trans. R. Soc. Lond. A 367, 3213–3226.Google Scholar
Denardo, B., Galvin, B., Greenfield, A., Larraza, A., Putterman, S. J. & Wright, W. B. 1992 Observations of localized structures in nonlinear lattices: domain walls and kinks. Phys. Rev. Lett. 68 (1), 1730–1733.CrossRefGoogle ScholarPubMed
Denardo, B., Wright, W. B., Putterman, S. J. & Larraza, A. 1990 Observation of a kink soliton on the surface of a liquid. Phys. Rev. Lett. 64 (1), 1518–1521.CrossRefGoogle ScholarPubMed
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409.CrossRefGoogle Scholar
Gordillo, L.2012 Non-propagating hydrodynamic solitons in a quasi-one-dimensional free surface subject to vertical vibrations. PhD thesis, Universidad de Chile, Santiago.Google Scholar
Gordillo, L., Sauma, T., Zárate, Y., Espinoza, I., Clerc, M. G. & Mujica, N. 2011 Can non-propagating hydrodynamic solitons be forced to move? Eur. Phys. J. D 62 (1), 39–49.CrossRefGoogle Scholar
Laedke, E. W. & Spatschek, K. H. 1991 On localized solutions in nonlinear Faraday resonance. J. Fluid Mech. 223, 589–601.CrossRefGoogle Scholar
Lide, D. R. 2004 Handbook of Chemistry and Physics, 85th edn. Chemical Rubber Company Press.Google Scholar
Longuet-Higgins, M. S. 1953 Mass Transport in Water Waves. Phil. Trans. R. Soc. Lond. A 245, 535–581.Google Scholar
Martín, E., Martel, C. & Vega, J. M. 2002 Drift instability of standing Faraday waves. J. Fluid Mech. 467, 57–79.CrossRefGoogle Scholar
Martín, E. & Vega, J. M. 2005 The effect of surface contamination on the drift instability of standing Faraday waves. J. Fluid Mech. 546, 203–225.CrossRefGoogle Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Miles, J. W. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75, 419–448.CrossRefGoogle Scholar
Miles, J. W. 1977 On Hamilton’s principle for surface waves. J. Fluid Mech. 83, 153–158.CrossRefGoogle Scholar
Miles, J. W. 1984a Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.CrossRefGoogle Scholar
Miles, J. W. 1984b Parametrically excited solitary waves. J. Fluid Mech. 148, 451–460.CrossRefGoogle Scholar
Rajchenbach, J., Leroux, A. & Clamond, D. 2011 New standing solitary waves in water. Phys. Rev. Lett. 107 (2), 024502.CrossRefGoogle ScholarPubMed
Sanchis, A. & Jensen, A. 2011 Dynamic masking of PIV images using the Radon transform in free surface flows. Exp. Fluids 51 (4), 871–880.CrossRefGoogle Scholar
Tannehill, J., Anderson, D. & Pletcher, R. 1997 Computational Fluid Mechanics and Heat Transfer, 2nd edn. Taylor & Francis.Google Scholar
Umeki, M. 1991 Parametric dissipative nonlinear Schrödinger equation. J. Phys. Soc. Japan 60 (1), 146–167.CrossRefGoogle Scholar
Wang, W., Wang, X., Wang, J. & Wei, R. 1996 Dynamical behavior of parametrically excited solitary waves in Faraday’s water trough experiment. Phys. Lett. A 219, 74–78.CrossRefGoogle Scholar
Wang, X. & Wei, R. 1994 Observations of collision behavior of parametrically excited standing solitons. Phys. Lett. A 192 (1), 1–4.CrossRefGoogle Scholar
Wu, J., Keolian, R. & Rudnick, I. 1984 Observation of a non-propagating hydrodynamic soliton. Phys. Rev. Lett. 52 (1), 1421–1424.CrossRefGoogle Scholar
Zhang, L., Wang, X. & Tao, Z. 2007 Spatiotemporal bifurcations of a parametrically excited solitary wave. Phys. Rev. E 75 (3), 036602.CrossRefGoogle ScholarPubMed

Gordillo supplementary movie

The movie shows the evolution through a whole cycle of the oscillatory velocity field of a parametrically excited solitary wave. The sequence displays the sloshing motion of the localized structure at the center of the trough viewed from the front (x-z plane). The laser sheet was placed at y s=1.07 cm (see axis orientation in figure 1). The white background indicates regions occupied by the fluid. The free-surface and bottom position as well as the velocity field (displayed in a coarse resolution) were obtained experimentally from images.

Download Gordillo supplementary movie(Video)
Video 844.9 KB

Gordillo supplementary movie

This sequence of images shows the emergence of a streaming pattern that becomes noticeable after the camera and laser pulses are synchronized with the oscillating structure. The phase of the parametrically excited solitary wave is here fixed at θs=π. The other parameters are the same than in Movie 1. The video also shows the streaming velocity field obtained by means of PIV. For other physical or processing parameters, please refer to the full article.

Download Gordillo supplementary movie(Video)
Video 6.9 MB

Gordillo supplementary movie

In this video, we provide a complementary view from the side (y-z plane) of the oscillating velocity field of a parametrically excited solitary wave. In this case, the structure is pinned to a lateral wall and the laser sheet was fixed at xs=9.23 cm. As in Movie 1, the white background displays the regions occupied by the fluid. The higher resolution allows to solve the advancing and receding meniscii at front and lateral walls of the trough.

Download Gordillo supplementary movie(Video)
Video 2.1 MB

Gordillo supplementary movie

As in Movie 2, we show a sequence of images where the streaming field becomes perceptible but now from a side view (y-z plane). Notice that near both meniscii, particle clusters are hard to follow due to the high shear within the region. We also depict the velocity field obtained by applying the PIV on a sequence of images. For more details, please refer to the full article.

Download Gordillo supplementary movie(Video)
Video 8.8 MB