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On the weakly nonlinear seakeeping solution near the critical frequency

Published online by Cambridge University Press:  11 May 2018

Chengxi Li  and
Yuming Liu Open the ORCID record for Yuming Liu [Opens in a new window]
Chengxi Li
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
*
†Email address for correspondence: yuming@mit.edu
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Abstract

We study theoretically and numerically the nonlinear seakeeping problem of a submerged or floating body translating with constant forward speed $U$ parallel to the undisturbed free surface while at the same time undergoing a small oscillatory motion and/or encountering small-amplitude waves at frequency $\unicode[STIX]{x1D714}$ . It is known that at the critical frequency corresponding to $\unicode[STIX]{x1D70F}\equiv \unicode[STIX]{x1D714}U/g=1/4$ , where $g$ is the gravitational acceleration, the classical linear solution is unbounded for a single point source, and the inclusion of third-order free-surface nonlinearity due to cubic self-interactions of waves is necessary to remove the associated singularity. Although it has been shown that the linear solution is in fact bounded for a body with full geometry rather than a point source, the solution still varies sharply near the critical frequency. In this work, we show theoretically that for a submerged body, the nonlinear correction to the linear solution due to cubic self-interactions of resonant waves in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$ is of first order in the wave steepness (or body motion amplitude), which is the same order as the linear solution. With the inclusion of nonlinear effects in the dispersion relation, the wavenumbers of resonant waves become complex-valued and the resonant waves become evanescent, with their amplitudes vanishing with the distance away from the body. To assist in understanding the theory, we derive the analytic nonlinear solution for the case of a submerged two-dimensional circular cylinder in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$ . Independent numerical simulations confirm the analytic solution for the submerged circular cylinder. Finally, we also demonstrate by numerical simulations that similar significant nonlinear effects for a surface-piercing body exist in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$ .

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , pp. 999 - 1022
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Copyright
© 2018 Cambridge University Press 

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References

Akylas, T. R. 1984 On the excitation of nonlinear water waves by a moving pressure distribution oscillating at resonant frequency. Phys. Fluids 27, 2803–2807.Google Scholar
Alam, M.-R, Liu, Y. & Yue, D. K. P. 2009 Waves due to an oscillating and translating disturbance in a two-layer density-stratified fluid. J. Engng Maths 65, 179–200.Google Scholar
Alam, M.-R, Liu, Y. & Yue, D. K. P. 2011 Resonant-wave signature of an oscillating and translating disturbance in a two-layer density stratified fluid. J. Fluid Mech. 675, 477–494.Google Scholar
Bingham, H. B.1994 Simulating ship motions in the time domain. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
Bracewell, N. 1986 The Fourier Transform and its Applications, vol. 31999. McGraw-Hill.Google Scholar
Brard, R. 1972 The representation of a given ship form by singularity distributions when the boundary condition on the free surface is linearized. J. Ship Res. 16, 79–92.Google Scholar
Dagan, D. & Miloh, T. 1980 Flow past oscillating bodies at resonant frequency. In Proc. 13th Symp. of Naval Hydrodynamics (ed. Inui, T.), pp. 355–373. The Shipbuilding Research Association of Japan.Google Scholar
Dagan, D. & Miloh, T. 1982 Free-surface flow past oscillating singularities at resonant frequency. J. Fluid Mech. 120, 139–154.Google Scholar
Grue, J. & Palm, E. 1985 Wave radiation and wave diffraction from a submerged body in a uniform current. J. Fluid Mech. 151, 257–278.Google Scholar
Haskind, M. D. 1954 On wave motion of a heavy fluid. Prikl. Mat. Mekh. 18, 15–26.Google Scholar
Havelock, T. H. 1958 The effect of speed of advance upon the damping of heave and pitch. Quart. Trans. Inst. Nav. Archit. 100, 131–135.Google Scholar
John, F. 1950 On the motion of floating bodies. Part II. Commun. Pure Appl. Maths 3, 45–101.Google Scholar
Kring, D. C. 1998 Ship seakeeping through the 𝜏 = 1/4 critical frequency. J. Ship Res. 42, 113–119.Google Scholar
Li, Y. & Ellingsen, S. Å. 2016 Multiple resonances of a moving oscillating surface disturbance on a shear current. J. Fluid Mech. 808, 668–689.Google 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.Google Scholar
Liu, Y., Xue, M. & Yue, D. K. P. 2001 Computations of fully nonlinear three-dimensional wave–wave and wave–body interactions. Part 2. Nonlinear waves and forces on a body. J. Fluid Mech. 438, 41–66.Google Scholar
Liu, Y. & Yue, D. K. P. 1993 On the solution near the critical frequency for an oscillating and translating body in or near free surface. J. Fluid Mech. 254, 251–266.Google Scholar
Maruo, H. 1967 Application of the slender body theory to the longitudinal motion of ships among waves. Bull. Fac. Engng Yokohama Natl Univ. 16, 29–61.Google Scholar
Mo, A. & Palm, E. 1987 On radiated and scattered waves from a submerged elliptic cylinder in a uniform current. J. Ship Res. 31, 23–33.Google Scholar
Nakos, D. E. & Sclavounos, P. D. 1990 Ship motions by a three-dimensional Rankine panel method. In Proc. 18th Symp. of Naval Hydrodynamics, pp. 21–44. National Academy Press.Google Scholar
Newman, J. N. 1978 The theory of ship motions. Adv. Appl. Mech. 18, 221–283.CrossRefGoogle Scholar
Palm, E. & Grue, J. 1999 On the wave field due to a moving body performing oscillations in the vicinity of the critical frequency. J. Engng Maths 35, 219–232.CrossRefGoogle Scholar
Sclavounos, P. D., Nakos, D. E. & Huang, Y. 1993 Seakeeping and wave induced loads on ship with flare by a Rankine panel method. In Proc. 6th Int. Conf. on Numerical Ship Hydrodynamics, p. 62. National Academy Press.Google Scholar
Shen, M. & Liu, Y. 2017 Current effects on global motions of a floating platform in waves. Ocean Syst. Engng 7, 121–141.Google Scholar
Tyvand, P. A. & Torheim, T. 2012 Surface waves from bottom vibrations in uniform open-channel flow. Eur. J. Mech. (B/Fluids) 36, 39–47.Google Scholar
Wehausen, J. W. & Laitone, E. V. 1960 Surface waves. In Encyclopedia of Physics (ed. Flugge, S.), pp. 446–778. Springer.Google Scholar
Yan, H. & Liu, Y. 2011 An efficient high-order boundary element method for nonlinear wave–wave and wave–body interactions. J. Comput. Phys. 230, 402–424.Google Scholar
Zhang, Y. & Zhu, S. 1995 Resonant interaction between a uniform current and an oscillating object. Appl. Ocean. Res. 17, 259–264.CrossRefGoogle Scholar
Zhu, Q., Liu, Y. & Yue, D. K. P. 2006 Dynamics of a three-dimensional oscillating foil near the free surface. AIAA J. 12, 2997–3009.Google Scholar