The differences between a wing and a planing plate in two-dimensional flow

Abstract

The normal force coefficient on a flat planing surface having arbitrary heave and pitch motion in two-dimensional flow is compared with the lift coefficient of a thin wing in an infinite fluid. Despite the totally different derivations, they are found to be identical (at large Froude numbers and low trim angles and allowing for the wing's interaction with twice as much fluid) at low reduced frequencies. For higher frequency motions, the wing's angle of attack induced lift and its pitch and heave damping are less than those of a planing surface, but the acceleration terms remain identical. The differences at the higher reduced frequencies are due to the fact that, in invisad irrotational flow, the planning plate cannot leave a vortex wake, whereas a wing does. It seems to follow that the “virtual mass” planing hull analysis can be applied to “quasi-static” problems involving wings and bodies in an infinite fluid without the slenderness restriction originally imposed by Jones (1946). Certainly, it is remarkable that the so called “quasi-steady” forces on a two-dimensional wing can be obtained in a few lines of elementary analysis. On the other hand, the method fails entirely when used to compute the pitching moment on a two-dimensional plate, even though it has been found to give good results for the three-dimensional case (Payne, 1981c). This work is offered as a very incomplete study of an intriguing relationship between two very different bodies of analysis. Much more work will need to be done before the relationship between the two approaches will be fully understood.