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Resonant interactions between Kelvin ship waves and ambient waves

Published online by Cambridge University Press:  01 February 2008

QIANG ZHU ,
YUMING LIU  and
DICK K. P. YUE
QIANG ZHU
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
YUMING LIU
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
DICK K. P. YUE*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
‡Author to whom correspondence should be addressed: yue@mit.edu
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Abstract

We consider the nonlinear interactions between the steady Kelvin waves behind an advancing ship and an (unsteady) ambient wave. It is shown that, for moderately steep ship waves and/or ambient waves, third-order (quartet) resonant interaction among the two wave systems could occur, leading to the generation of a new propagating wave along a specific ray in the Kelvin wake. The wave vector of the generated wave as well as the angle of the resonance ray are determined by the resonance condition and are functions of the ship forward speed and the wave vector of the ambient wave. To understand the resonance mechanism and the characteristics of the generated wave, we perform theoretical analyses of this problem using two related approaches. To obtain a relatively simple model in the form of a nonlinear Schrödinger (NLS) equation for the evolution of the resonant wave, we first consider a multiple-scale approach assuming locally discrete Kelvin wave components, with constant wave vectors but varying amplitudes along the resonance ray. This NLS model captures the key resonance mechanism but does not account for the detuning effect associated with the wave vector variation of Kevin waves in the neighbourhood of the resonance ray. To obtain the full quantitative features and evolution characteristics, we also consider a more complete model based on Zakharov's integral equation applied in the context of a continuous wave vector spectrum. The resulting evolution equation can be reduced to an NLS form with, however, cross-ray variable coefficients, on imposing a narrow-band assumption valid in the neighbourhood of the resonance ray. As expected, the two models compare well when wave vector detuning is small, in the near wake close to the ray. To verify the analyses, direct high-resolution simulations of the nonlinear wave interaction problem are obtained using a high-order spectral method. The simulations capture the salient features of the resonance in the near wake of the ship, with good agreements with theory for the location of the resonance and the growth rate of the generated wave.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 597 , 25 February 2008 , pp. 171 - 197
Copyright
Copyright © Cambridge University Press 2008

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Footnotes

†

Present address: Department of Structure Engineering, UCSD, USA

References

REFERENCES

Akylas, T. R. 1987 Unsteady and nonlinear effects near the cusp lines of the Kelvin ship-wave pattern. J. Fluid Mech. 175, 333–342.CrossRefGoogle Scholar
Benny, D. J. 1962 Nonlinear gravity wave interactions. J. Fluid Mech. 14, 577–589.CrossRefGoogle Scholar
Benny, D. J. & Roskes, G. 1969 Wave instabilities. Stud. Appl. Maths 48, 377–385.CrossRefGoogle Scholar
Booij, N., Ris, R. C. & Holthuijsen, L. H. 1999 A third-generation wave model for coastal regions. Part 1. Model description and validation. J. Geophys. Res. 104, 7649–7666.CrossRefGoogle Scholar
Brown, E. D., Buchsbaum, S. B., Hall, R. E., Penhum, J. P., Schmitt, K. F., Watson, K. M. & Wyatt, D. C. 1989 Observations of nonlinear solitary wave packet in the Kelvin wake of a ship. J. Fluid Mech. 204, 263–293.CrossRefGoogle Scholar
Cao, Y 1991 Computation of nonlinear gravity waves by a desingularized boundary integral method. PhD thesis, University of Michigan.Google Scholar
Crawford, D. R., Lake, B. M., Saffman, P. G. & Yuen, H. C. 1981 Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 105, 177–191.CrossRefGoogle Scholar
Dommermuth, D. G. & Yue, D. K. P. 1987 A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.CrossRefGoogle Scholar
Dommermuth, D. G. & Yue, D. K. P. 1988 The nonlinear three-dimensional waves generated by a moving surface disturbance. In Proc. 17th Symp. on Naval Hydro., Hague, Netherlands.Google Scholar
Eggers, K. & Schultz, W. W. 1992 Investigation on time-harmonic disturbances for inner-Kelvin-angle wave packets. In 7th International Workshop on Water Waves and Floating Bodies. Val de Reuil, France.Google Scholar
Fu, L. L. & Holt, B. 1982 SESAT view of oceans and sea ice with synthetic aperture radar. Jet Propulsion Lab. Publ. 81–120.Google Scholar
Hall, H. & Buchsbaum, S. 1990 A model for the generation and evolution of an inner-angle soliton in a Kelvin wake. In Proc. 18th ONR Symp. Naval Hydrodynamics, pp. 453–463.Google Scholar
Hasselmann, S., Hasselmann, K., Bauer, E., Janssen, P. A. E. M., Komen, G. J., Bertotti, L., Lionello, P., Guillaume, A., Cardone, V. C., Greenwood, J. A., Reistad, M., Zambresky, L. & Ewing, J. A. 1988 The WAM model: A third generation ocean wave prediction model. J. Phys. Oceanogr. 18, 1775–1810.Google Scholar
Krasitskii, V. P. 1994 On reduced equation in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
Li, J. J. & Tulin, M. P. 1999 On the stability and nonlinear dynamics of ocean-like wave systems with energy continuously distributed in direction. J. Engng Maths 35, 59–70.CrossRefGoogle Scholar
Liu, Y., Dommermuth, D. G. & Yue, D. K. P. 1992 A high-order spectral method for nonlinear wave-body interactions. J. Fluid Mech. 245, 115–136.CrossRefGoogle Scholar
Liu, Y. & Yue, D.K.P. 1998 On generalized Bragg scattering of surface waves by bottom ripples. J. Fluid Mech. 356, 297–356.CrossRefGoogle Scholar
Mei, C. C., Stiassnie, M. & Yue, D. K. P. 2005 Theory and Applications of Ocean Surface Waves. World Scientific.Google Scholar
Mei, C. C. & Naciri, M. 1991 Note on ship oscillations and wake solitons. Proc. R. Soc. Lond. A 432, 535–546.Google Scholar
Munk, W. H., Scully-Power, P. & Zachariasen, F. 1987 Ships from space. Proc. R. Soc. Lond. A 412, 231–254.Google Scholar
Newman, J. N. 1971 Third order interactions in the Kelvin ship wave systems. J. Ship Res. 15, 1–10.CrossRefGoogle Scholar
Newman, J. N. 1977 Marine Hydrodynamics. The MIT Press.CrossRefGoogle Scholar
Newman, J. N. 1987 Evaluation of the wave-resistance Green function. Part 1. The double integral. J. Ship Res. 31, 79–90.CrossRefGoogle Scholar
Newman, J. N. 1992 The Green function for potential flow in a rectangular channel. J. Engng Math. 26, 51–59.CrossRefGoogle Scholar
Peregrine, D. H. 1971 A ship's waves and its wake. J. Fluid Mech. 49, 353–360.CrossRefGoogle Scholar
Phillips, O. M. 1960 Unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.CrossRefGoogle Scholar
Reed, A. M. & Milgram, J. H. 2002 Ship wakes and their radar images. Annu. Rev. Fluid Mech. 34, 469–502.CrossRefGoogle Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes' expansion for gravity waves. J. Fluid Mech. 62, 553–578.CrossRefGoogle Scholar
Shemdin, O. 1987 SAR imaging of ship wakes in the gulf of Alaska. Report, Jet Propulsion Lab.Google Scholar
Tulin, M. & Miloh, T. 1990 Ship internal waves in a shallow thermocline: the supersonic case. Proc. 18th Symp. Naval Hydrodynamics.Google Scholar
Yue, D. K. P. & Mei, C. C. 1980 Forward diffraction of Stokes waves by a thin wedge. J. Fluid Mech. 99, 33–52.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (English translation.)CrossRefGoogle Scholar
Zhu, Q. 2000 Features of nonlinear wave-wave and wave-body interactions. PhD Thesis, Massachusetts Institute of Technology.Google Scholar
Zhu, Q., Liu, Y., Tjavaras, A. A., Yue, D. K. P. & Triantafyllou, M. S. 1999 Mechanics of nonlinear short-wave generation by a moored near-surface buoy. J. Fluid Mech. 381, 305–335.CrossRefGoogle Scholar