Hydrodynamics of non-axisymmetric oblate spheroids below a free surface

We develop an analytical methodology for solving the hydrodynamic diffraction and the radiation problems by immersed oblate spheroids in a liquid field of infinite extend bounded from above by a free-surface. In turn, free-surface effects are included in the formulation of the hydrodynamic problem allowing the investigation of the hydrodynamic characteristics of the multi-purpose oblate spheroidal geometry that runs from a sphere to a circular disk depending on its slenderness parameter. The case considered here is that of a “non-axisymmetric” placement of a fully wetted spheroid, namely when its semi-minor axis is parallel to the free-surface, extending the existing efforts which have considered only the “symmetric” case (i.e. semi-minor axis perpendicular to the free-surface). The analytic solution is sought by constructing and using the multipole potentials of the problem that express the fundamental Green’s function in terms of oblate spheroidal harmonics. The adopted methodology is based on a Fredholm integral equation (of the second kind) approach that provides the source distribution over the surface of the spheroid. The source distribution is accordingly obtained by considering the potentials in the interior and the exterior of the spheroid and expressing them in spheroidal harmonics. The derivation of the multipole potentials, enable us to obtain analytic expressions for the diffraction and the radiation velocity potentials in terms of the involved harmonics. Several numerical tests have been performed to calculate the exciting forces and moments as well as the hydrodynamic parameters of the spheroid. Special attention has been paid to the limiting case of a circular disk subjected to head seas, for which the hydrodynamic effects are most pronounced. The other limiting case, i.e. that of a sphere, is well documented and was used for validation purposes.