Abstract
The small-time expansion of the 2-D problem of a heaving semi-submerged circular cylinder, starting from rest, in analyzed. The first three terms of the small-time expansion of the outer solution (away from the cylinder) are developed. A logarithmic singularity is found for the second derivative in time of the surface elevation at the intersection point. This singularity is of second order in the heaving velocity of the cylinder, and the inner expansion developed for the linear wavemaker problem therefore does not cover this case. The problem of the horizontal motion of the semi-submerged cylinder is analyzed as well. For the case with the initial condition for the velocity potential chosen as zero on the free surface, the logarithmic singularity found for the wavemaker is recovered. Selecting the initial condition as a vanishing normal velocity on the free surface leads to a solution where the singularity appears in the fifth derivative in time of the free-surface elevation. The solution of this problem shows the characteristic behaviour of formation of asymmetry, which finally would lead to formation of lee waves. The inner solution of the problem of the heaving cylinder is discussed, showing that the oscillatory behaviour reported for the wavemaker problem appears in the present solution as well.
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References
M.S. Longuet-Higgins and E.D. Cokelet, The deformation of steep surface waves on water, I. A numerical method for computation. Proc. Roy. Soc. London A350 (1976) 1–28.
D.H. Peregrine, Flow due to vertical plate moving in a channel. Unpublished notes (1972).
A.T. Chwang, Nonlinear hydrodynamic pressure on accelerating plate. Phys. Fluids 26 (1983) 383–387.
W.-M. Lin, Nonlinear Motion of the Free Surface near a Moving Body. Ph. D. Thesis Dept. Ocean Eng., Mass. Inst. Techn., Cambridge, Mass., U.S.A. (1984).
A.L. Roberts, Transient free-surface flows generated by a moving vertical plate. Q. J. Mech. Appl. Math. 40 (1987) 129–158.
S.W. Joo, W.W. Schultz and A.F. Messiter, Uniformly valid solutions to the initial-value wavemaker problem. Rept. No. 89–01,, Univ. Mich. Prog. in Ship Hydrodynamics, Coll. Eng., Univ. Mich., Ann Arbor, Mich., U.S.A. (1989).
L.M. Milne-Thomson, Theoritical Hydrodynamics, MacMillan, London (1968).
T. Miloh, On the initial stage slamming of a rigid spherein a vertical water entry. Appl. Ocean Res. 13 (1991) 43–48.
M.J. Greenhow, Water-entry and exit of a horizontal cylinder. Appl. Ocean Res. 10 (1988) 191–198.
T. Miloh, Irregularities in solutions of nonlinear wave diffraction problem by vertical cylinder. J. Waterway, Port, Coastal and Ocean Div. 106 (1980) 279–284.
P.D. Sclavounos, Radiation and diffraction of second-order surface waves by floating bodies. J. Fluid Mech. 196 (1988) 65–91.
O.M. Faltinsen, Numerical solutions of transient nonlinear free-surface motion outside and inside moving bodies. In: 2nd Int. Conf. Num. Ship Hydrodyn., Berkeley, Cal., U.S.A. (1985).
T. Vinje and P. Brevig, Nonlinear ship motion. In: 3rd Int. Conf. Num. Ship Hydrodyn., Paris, France (1981).
M.A. Grosenbaugh and R.W. Yeung, Nonlinear bow flow — An experimental and theoretical investigation. In: 17th ONR Symp. Naval Hydrodyn., The Hague, The Netherlands (1988).
T. Vinje and P. Brevig, Nonlinear, two-dimensional ship motion. SIS Rep., Norwegian Hydrodyn. Lab., Trondheim, Norway (1980).
F. Ursell, The decay of the free motion of a floating body. J. Fluid Mech. 19 (1964) 305–319.
S.J. Maskell and F. Ursell, The transient motion of a floating body. J. Fluid Mech. 44 (1970) 303–313.
F. Ursell, On the heaving motion of a circular cylinder on the surface of a fluid. Q. J. Mech. Appl. Math. 2 (1949) 218–231.
M.J. Greenhow and M. Simon, A note on the equivalent wavemaker method. Appl. Ocean Res. 7 (1985) 107–112.
E.H. Kennard, Generation of surface waves by a moving partition. Q. J. Appl. Math. 7 (1949) 303–312.
J.V. Wehausen and E.V. Laitone, Surface waves. In: Handbuch der Physik, Vol. 9, Springer Verlag, Berlin, Germany (1960).
M. Abramowitz and I.E. Stegun, Handbook of mathematical functions, 5th Edition Dover, New York, U.S.A. (1968).
H. Lamb, Hydrodynamics, Cambridge Univ. Press, Cambridge, U.K. (1932).
A. Erdelyi, Asymptotic expansions, Dover, New York, U.S.A. (1956).
K.-H. Wang and A.T. Chwang, Nonlinear free-surface flow around an impulsively moving cylinder. J. Ship Res. 33 (1989) 194–202.
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Vinje, T. On small-time expansion of nonlinear free surface problems. J Eng Math 28, 173–190 (1994). https://doi.org/10.1007/BF00127584
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DOI: https://doi.org/10.1007/BF00127584