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Slender oscillating ships at zero forward speed

Published online by Cambridge University Press:  28 March 2006

F. Ursell
F. Ursell
Affiliation:
Department of Mathematics, University of Manchester
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Abstract

The ship is assumed to be a slender body of revolution with its axis in the mean free surface and making periodic oscillations of small amplitude. The theory presented here is a generalization of the well-known slender-body theory of incompressible aerodynamics in which the fluid is externally unbounded. One version of that theory goes as follows: approximate the body by axial line-distributions of known point singularities (sources and multipoles), whose strength is to be determined; by means of the Fourier convolution theorem express the velocity potentials of these line distributions in terms of the Fourier transforms (in the axial direction) of the point-singularity potentials; expand these Fourier transforms in powers of the radius and retain only the leading terms (it is here that the slender-body assumption is introduced); by means of the Fourier convolution theorem interpret the resulting expressions. By this procedure it is not only shown that near the body the potential is two-dimensional harmonic in every plane normal to the axis; but also the interaction between sections is shown to be involved in the ‘constant’ term and to depend in an explicit manner on the coefficient functions, which can be found without difficulty by applying the prescribed boundary conditions. This foregoing procedure can be justified when the body is slender and sharply pointed.

In the present paper the same procedure is adapted to an oscillating surface ship a t zero speed. The fluid is now bounded by the ship, and also by a horizontal plane (the mean free surface) on which a wave boundary condition must be applied. The point singularities are now wave sources and wave-free potentials, each satisfying the free-surface condition. The Fourier transforms of these singularities are found and are expanded near the axis; the expansions near the axis are the only parts of the argument that present any serious difficulty. When only the leading terms are retained and the results are interpreted by the convolution theorem, explicit two-dimensional potentials are again obtained.

It is assumed that the ratios (ship-radius/ship-length) and (ship-radius/wavelength) are small whereas the ratio (ship-length/wave-length) may have any value. Expressions are given which are valid according as this last ratio is not large or not small. The potential on the body is found, and forces, moments and wave damping are calculated. It is believed that the expansions can be extended with little trouble to certain other ranges of those ratios, to other cross-sections and to the boundary condition for ships moving a t non-zero speeds.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 14 , Issue 4 , December 1962 , pp. 496 - 516
Copyright
© 1962 Cambridge University Press

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References

Erdélyi, A. (ed.) 1953 Higher Transcendental Functions, vol. 2, Bateman Manuscript Project. New York: McGraw-Hill.
Grim, O. 1957 Durch Wellen an einem Schiffskörper erregte Kráfte. Proc. Symp. on the Behaviour of Ships in a Seaway, pp. 232–65. Wageningen: H. Veenman and Zonen.
Grim, O. 1960 A method for a more precise computation of heaving and pitching motions both in smooth water and in waves. Third Sym. on Naval Hydrodynamics, Scheveningen, pp. 271–326. Wageningen: H. Veenman and Zonen.
Havelock, Sir Thomas 1955 Waves due to a floating sphere making periodic heaving oscillations. Proc. Roy. Soc. A, 231, 1–7.Google Scholar
Havelock, Sir Thomas 1956 The damping of heave and pitch: a comparison of two-dimensional and three-dimensional calculations. Trans. Inst. Nav. Archit., Lond., 98, 464–8.Google Scholar
Lighthill, M. J. 1958 Fourier Analysis and Generalised Functions. Cambridge University Press.
Michell, J. H. 1898 The wave resistance of a ship. Phil. Mag. (5) 45, 106–23.Google Scholar
Peters, A. S. & Stoker, J. J. 1957 The motion of a ship, as a floating rigid body, in a seaway. Comm. Pure Appl. Math. 10, 399–490.Google Scholar
Tasai, F. 1959 On the damping force and added mass of ships heaving and pitching. J. Zosen Kiokai (Soc. Naval Architects of Japan), 105, 47–56. Translation published by Inst. Engng Res., Univ. of California, Berkeley, Series 82, Issue 15, 1960.Google Scholar
Thorne, R. C. 1953 Multipole expansions in the theory of surface waves. Proc. Camb. Phil. Soc. 49, 707–16.Google Scholar
Thwaites, B. (ed.) 1960 Incompressible Aerodynamics. Oxford University Press.
Tuck, E. O. 1962 Unpublished.
Ursell, F. 1949a On the heaving motion of a circular cylinder on the surface of a fluid. Quart. J. Mech. Appl. Math., 2, 218–31.Google Scholar
Ursell, F. 1949b On the rolling motion of cylindrs in the surface of a fluid. Quart. J. Mech. Appl. Math., 2, 335–53.Google Scholar
Ursell, F. 1953 Short surface waves due to an oscillating immersed body. Proc. Roy. Soc. A, 220, 90–103.Google Scholar
Ursell, F. 1957 On the virtual mass and damping of floating bodies at zero speed ahead. Proc. Symp. on the Behaviour of Ships in a Seaway, pp. 374–87. Wageningen: H. Veenman and Zonen.
Vossers, G. 1962 Some applications of the slender body theory in ship hydrodynamics. Publication no. 214, Netherlands Ship Model Basin, Wageningen.
Ward, G. N. 1955 Linearized Theory of Steady High-Speed Flow, Ch. 9. Cambridge University Press.
Watson, G. N. 1944 Bessel Functions, 2nd ed. Cambridge University Press.
Wehausen, J. 1957 Wave resistance of thin ships. Symp. on Naval Hydrodynamics, Washington, D.C., 1956, pp. 109–33. Washington: National Academy of Sciences—National Research Council.