Journal of Engineering Mathematics

, Volume 28, Issue 2, pp 173–190 | Cite as

On small-time expansion of nonlinear free surface problems

  • T. Vinje
Article
Received:
Accepted:

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.

Keywords

Mathematical Modeling Free Surface Industrial Mathematic Intersection Point Circular Cylinder 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. 1.
    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.Google Scholar
  2. 2.
    D.H. Peregrine, Flow due to vertical plate moving in a channel. Unpublished notes (1972).Google Scholar
  3. 3.
    A.T. Chwang, Nonlinear hydrodynamic pressure on accelerating plate. Phys. Fluids 26 (1983) 383–387.Google Scholar
  4. 4.
    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).Google Scholar
  5. 5.
    A.L. Roberts, Transient free-surface flows generated by a moving vertical plate. Q. J. Mech. Appl. Math. 40 (1987) 129–158.Google Scholar
  6. 6.
    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).Google Scholar
  7. 7.
    L.M. Milne-Thomson, Theoritical Hydrodynamics, MacMillan, London (1968).Google Scholar
  8. 8.
    T. Miloh, On the initial stage slamming of a rigid spherein a vertical water entry. Appl. Ocean Res. 13 (1991) 43–48.Google Scholar
  9. 9.
    M.J. Greenhow, Water-entry and exit of a horizontal cylinder. Appl. Ocean Res. 10 (1988) 191–198.Google Scholar
  10. 10.
    T. Miloh, Irregularities in solutions of nonlinear wave diffraction problem by vertical cylinder. J. Waterway, Port, Coastal and Ocean Div. 106 (1980) 279–284.Google Scholar
  11. 11.
    P.D. Sclavounos, Radiation and diffraction of second-order surface waves by floating bodies. J. Fluid Mech. 196 (1988) 65–91.Google Scholar
  12. 12.
    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).Google Scholar
  13. 13.
    T. Vinje and P. Brevig, Nonlinear ship motion. In: 3rd Int. Conf. Num. Ship Hydrodyn., Paris, France (1981).Google Scholar
  14. 14.
    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).Google Scholar
  15. 15.
    T. Vinje and P. Brevig, Nonlinear, two-dimensional ship motion. SIS Rep., Norwegian Hydrodyn. Lab., Trondheim, Norway (1980).Google Scholar
  16. 16.
    F. Ursell, The decay of the free motion of a floating body. J. Fluid Mech. 19 (1964) 305–319.Google Scholar
  17. 17.
    S.J. Maskell and F. Ursell, The transient motion of a floating body. J. Fluid Mech. 44 (1970) 303–313.Google Scholar
  18. 18.
    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.Google Scholar
  19. 19.
    M.J. Greenhow and M. Simon, A note on the equivalent wavemaker method. Appl. Ocean Res. 7 (1985) 107–112.Google Scholar
  20. 20.
    E.H. Kennard, Generation of surface waves by a moving partition. Q. J. Appl. Math. 7 (1949) 303–312.Google Scholar
  21. 21.
    J.V. Wehausen and E.V. Laitone, Surface waves. In: Handbuch der Physik, Vol. 9, Springer Verlag, Berlin, Germany (1960).Google Scholar
  22. 22.
    M. Abramowitz and I.E. Stegun, Handbook of mathematical functions, 5th Edition Dover, New York, U.S.A. (1968).Google Scholar
  23. 23.
    H. Lamb, Hydrodynamics, Cambridge Univ. Press, Cambridge, U.K. (1932).Google Scholar
  24. 24.
    A. Erdelyi, Asymptotic expansions, Dover, New York, U.S.A. (1956).Google Scholar
  25. 25.
    K.-H. Wang and A.T. Chwang, Nonlinear free-surface flow around an impulsively moving cylinder. J. Ship Res. 33 (1989) 194–202.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • T. Vinje
    • 1
  1. 1.Norwegian Contractors a.sNorway

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