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fluid density


source strength

disturbance velocity potential


disturbance velocity potential in double-body flow


disturbance velocity potential due to free-surface effects


For theoretical prediction of the hydrodynamic forces acting on a maneuvering ship, the ship's hull may be considered to be a lifting body of very small aspect ratio. Although a free surface is present, it may be replaced by a rigid horizontal plane in the range of low speeds. In such a case, the effective aspect ratio of the hull is twice its draft-to-length ratio, and the flow is equivalent to the so-called double-body flow to which various theoretical methods developed in aerodynamics may be applied directly. On the other hand, for ships at higher speeds, the free-surface displacement may have a significant influence on the hydrodynamic forces, thus the free-surface effects must be included in the theoretical analysis.

Since the maneuvering behavior of a ship is mainly determined by its lateral underwater profile, a vertical flat plate with the same draft and length may be a useful model for calculating the hydrodynamic forces acting on the ship. For a surface-piercing plate in maneuvering motion, a successful numerical method was proposed by Chapman (1), (2) under the slender-body assumption. Since the flow separation along the lower edge is neglected in this method, its applicability is confined to small lateral motions, and although the aspect ratio is assumed to be small, better agreements between numerical and experimental results were obtained for plates of larger aspect ratios.

Another important fact is that the wake influence is fully neglected in the slender-body theory. As we known, the aspect ratio is an important measure of the effects of three dimensionality and the trailing wake is an inevitable consequence of these effects. In the case of very large aspect ratio, the flow around a vertical plate may be viewed as two dimensional in the horizontal plane, whereas in the limiting case of zero aspect ratio the flow is essentially two dimensional cross flow in the lateral plane. On the other hand, for a plate with small but finite aspect ratio, three dimensional flow effects may be important, and the influence of trailing wake must be taken into account.

For a yawed surface-piercing flat plate, Maniar et al. (3) proposed a three-dimensional panel method using Kelvin singularities. In correspondence with the conventional linear lifting-surface theory, the normal dipole distribution on the plate and in the wake downstream of the trailing edge is put on the plane of the plate. Thus this method, just as Chapman's method, is only applicable to a plate of moderate or larger aspect ratio at small drift angle.

Recently, for a surface-piercing plate in steady turning motion, Landrini and Campana (4) presented a three dimensional numerical method using Rankine singularities. The Neumann-Kelvin boundary-value problem was solved by coupling a standard panel method on the free surface with a vortex lattice procedure on the body and wake surface, where corresponding to the nonlinear lifting-surface theory a nonlinear wake modeling was used.

In the present paper, a three-dimensional Rankine panel method for free-surface flow around a vertical plate of small aspect ratio in steady oblique motion and/or turning motion is presented. The flow is assumed to separate from the trailing edge and the lower edge. The wake is modeled by a nonlinear vortex sheet. The boundary condition on the free surface is linearized with respect to the double-body flow. To solve the linearized boundary-value problem, a source distribution on the free surface and a vortex distribution on the plate and in the wake are used, whereas the singularity strengths are determined by satisfying the corresponding boundary conditions.

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