A finite pressure element approach to the planing problem of high speed craft

Abstract

A finite element method is presented for the steady motion of a craft planing over the surface of calm water. The fluid is assumed to be infinitely deep, inviscid, incompressible and without surface tension and the free surface is assumed to be of infinite extent. In addition, the angle of attack is assumed to be small and linearized potential flow theory is used. The method applies to the case of arbitrary Froude number and aspect ratio. The presence of the craft is modelled by an unknown pressure distribution on its wetted bottom projected on the plane of the undisturbed free surface. This is represented by a finite element mesh consisting of a number of pressure elements, each of constant but different strength. The shape of the element can be arbitrary and therefore the theory can be applied to wetted planforms of any shape or configuration. The shape and extent of the wetted bottom is assumed to be known and the immersions along the transom are determined together with the pressures by satisfying the kinematic hull boundary condition at the centre of each element and the Kutta condition at a discrete number of points along the transom. The finite element method has been applied to planing flat plate and prismatic hulls of constant deadrise angle. The derived lifts centre of pressure locations and pressures have compared reasonably well with other experimental and theoretical results. An interpolating scheme for determining the operating trim angle and wetted length for a craft of specified loading condition and speed has also been developed. The work also comprises a study of the hydrodynamics of planing craft under two conditions: firstly, when it is heeled at a small angle, and secondly, when it is yawed at a small angle. For the heel case, the theory predicts a decrease in roll stability when the craft is planing at high speed. The computed hydrodynamic force and moment derivatives have shown reasonably good agreement with the experiment data obtained by other authors. The behaviour of these hydrodynamic derivatives at high speed has also been investigated. For the yaw case, the theory predicts an interesting feature of the development of suction under the outboard side of the hull at high planing speed. The theory also predicts a change in the direction of the induced roll moment which could well be directly related to the phenomena that some high speed crafts bank inwards during turning while others bank outwards.

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