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Parametrically forced gravity waves in a circular cylinder and finite-time singularity

Published online by Cambridge University Press:  06 March 2008

S. P. DAS  and
E. J. HOPFINGER
S. P. DAS
Affiliation:
LEGI/CNRS/UJF, BP 53, 38041 Grenoble Cedex, [email protected]
E. J. HOPFINGER
Affiliation:
LEGI/CNRS/UJF, BP 53, 38041 Grenoble Cedex, [email protected]
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Abstract

In this paper we present results on parametrically forced gravity waves in a circular cylinder in the limit of large fluid-depth approximation. The phase diagram that shows the stability-forcing-amplitude threshold and the wave-breaking threshold has been determined in the frequency range of existence of the lowest axisymmetric wave mode. The instability is shown to be supercritical for forcing frequencies at and above the natural frequency and subcritical below in a frequency range where the instability and breaking thresholds do not coincide. Above the instability threshold, the growth in wave amplitude is exponential, but with an initial time delay. The wave-amplitude response curve of stationary wave motions exhibits steady-state wave motion, amplitude modulations and bifurcations to other wave modes at frequencies where the parametric instability boundary of the axisymmetric mode overlaps with the neighbouring modes. The amplitude modulations are either on a slow time scale or exhibit period tripling and intermittent period tripling, without wave breaking. In the wave-breaking regime, a finite-time singularity may occur with intense jet formation, a phenomenon demonstrated by others in fluids of high viscosity and large surface tension. Here, this singular behaviour with jet formation is demonstrated for a low viscosity and low kinematic surface tension liquid. The results indicate that the jet is driven by inertial collapse of the cavity created at the wave trough. Therefore, the jet velocity is determined by the wave fluid velocity but depends, in addition, on kinematic surface tension and viscosity as these affect the last stable wave crest shape and the cavity size.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 599 , 25 March 2008 , pp. 205 - 228
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Abramson, H. N. (ed.) 1966 The dynamic behavior of liquids in moving containers. NASA TR SP-106.Google Scholar
Benielli, D. & Sommeria, J. L. 1998 Excitation and breaking of internal gravity waves by parametric instability. J. Fluid Mech. 374, 117–144.CrossRefGoogle Scholar
Benjamin, T. B. & Scott, J. C. 1979 Gravity–capillary waves with edge constraints. J. Fluid Mech. 92, 241–267.CrossRefGoogle Scholar
Benjamin, T. B. & Ursell, F. 1954 The stability of plane free surface of a liquid in a vertical periodic motion. Proc. R. Soc. Lond. A 225, 505–515.Google Scholar
Bredmose, H., Brocchini, M., Peregrine, D. H. & Thais, L. 2003 Experimental investigation and numerical modelling of steep forced water waves. J. Fluid Mech. 490, 217–249.CrossRefGoogle Scholar
Dodge, F. T. 1966 Vertical excitation of a propellant tank. In ‘I The dynamic behavior of liquids in moving containers’, chap. 8 (ed. H. N. Abramson). NASA TR SP-106.Google Scholar
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148.CrossRefGoogle Scholar
Faltinsen, O. M. Rognebakke, O. F. Lukovsky, I. A. & Timokha, A. N. 2000 Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. J. Fluid Mech. 407, 201–234.CrossRefGoogle Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2003 Resonant three-dimensional nonlinear sloshing in a square-base basin. J. Fluid Mech. 487, 1–42.CrossRefGoogle Scholar
Feng, J. C. & Sethna, P. R. 1989 Symmetry-breaking bifurcations in resonant surface waves. J. Fluid Mech. 199, 495–518.CrossRefGoogle Scholar
Henderson, D. M. & Miles, J. W. 1990 Single-mode Faraday waves in small cylinders. J. Fluid Mech. 213, 95–109.CrossRefGoogle Scholar
Henderson, D. M. & Miles, J. W. 1994 Surface-wave damping in a circular cylinder with a fixed contact line. J. Fluid Mech. 275, 285–299.CrossRefGoogle Scholar
Hopfinger, E. J. & Baumbach, V. 2007 Liquid sloshing in cylindrical fuel tanks. 2nd EUCASS Proc. Brussels.Google Scholar
Ibrahim, R. A. 2005 Liquid Sloshing Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Jiang, L., Ting, C.-L. Perlin, M. & Schultz, W. W. 1996 Moderate and steep Faraday waves: instabilities, modulation and temporal asymmetries. J. Fluid Mech. 329, 275–307.CrossRefGoogle Scholar
Jiang, L., Perlin, M. & Schultz, W. W. 1998 Period tripling and energy dissipation of breaking standing waves. J. Fluid Mech. 369, 273–299.CrossRefGoogle Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.CrossRefGoogle Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Longuet-Higgins, M. S. 1983 On the joint distribution of wave period and amplitudes in a random wave field. Proc. R. Soc. Lond. A 389, 241–258.Google Scholar
Martel, C., Nicolas, J. A. & Vega, J. M. 1998 Surface-wave damping in a brimful circular cylinder. J. Fluid Mech. 360, 213–228.CrossRefGoogle Scholar
Miles, J. W. 1984 Resonantly forced surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31.CrossRefGoogle Scholar
Miles, J. W. & Henderson, D. M. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143–165.CrossRefGoogle Scholar
Miles, J. W. & Henderson, D. M. 1998 A note on interior vs. boundary-layer damping of surface waves in a circular cylinder. J. Fluid Mech. 364, 319–323.CrossRefGoogle Scholar
Milgram, J. H. 1969 The motion of a fluid in a cylindrical container with a free surface following vertical impact. J. Fluid Mech. 37, 453–448.CrossRefGoogle Scholar
Nayfeh, A. H. 1987 Surface waves in closed basins under parametric and internal resonances. Phys. Fluids 30, 2976–2983.CrossRefGoogle Scholar
Peregrine, D. H., Bredmose, H. McCobe, A. Bullock, G. Obhrai, C. Müller, G. & Wolters, G. 2004 Violent water wave impact on walls and the role of air. Proc. 29th Intl Conf. Coastal Engng Lisbon (ed. J. McK. Smith), World Sci. 4, 4005–4017.Google Scholar
Perlin, M. & Schultz, W. 2000 Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32, 241–274.CrossRefGoogle Scholar
Royon-Lebeaud, A. Hopfinger, E. J. & Cartellier, A. 2007 Liquid sloshing and wave breaking in circular and square-base cylindrical containers. J. Fluid Mech. 577, 467–494.CrossRefGoogle Scholar
Simonelli, F. & Gollub, J. P. 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471–494.CrossRefGoogle Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in the direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201, 192.Google Scholar
Taylor, G. I. 1953 An experimental study of standing waves. Proc. R. Soc. Lond. A 218, 44–59.Google Scholar
Zeff, B. W., Kleber, B., Fineberg, J. & Lathrop, D. P. 2000 Singularity dynamics in curvature collapse and jet eruption on a fluid surface. Nature 403, 401–404.CrossRefGoogle ScholarPubMed