Contributions to the virtual mass theory of hydrodynamic planning

Abstract

Virtual mass solutions (Payne, 1974, 1980a) for the normal force on a flat planing plate gave good agreement with experiments for wetted length to beam ratios ( l ̂ = l/b) in excess of unity. In an article by Payne (1981a), the region below l ̂ = 1 was faired into the known two-dimensional solution (infinite aspect ratio) by means of a bridging function. A subsequent study (Payne, 1980c) of the virtual mass of a rectangular plate led to another bridging function which seemed to be more fundamental, and might give better agreement between theory and experiment below l ̂ = 1 than was the case in the work by Payne (1981a). In the present note this is shown to be so. And although the lift curve slope of a wing and a planing plate (when the cavity lift contribution is neglected) are identical for slender planforms ( l ̂ > 1) and in the two dimensional flow ( l ̂ = 0) the plate is found to have a higher lift curve slope in between. There is no relationship between the pitching moment of a planing plate and an aerofoil. This subsequent work has also resulted in a more elegant version of the basic equations, and the new version is presented for both normal force and moment. For slender ( l ̂ = 1) flat plates the non-dimensional center of pressure ( X ̂ G) of the “virtual mass” force is close to the stagnation line and moves aft with increasing τ and l ̂ . It is rarely further aft than 10% of the wetted length. The theoretical prediction is compared with the measurements of Sottorf and found to be within the scatter of the data. In contrast, X ̂ G = 2 3 for a chines dry prismatic hull, and 1 4 for a wing or hydrofoil. The theory enables the effect of camber and rocker to be studied. Using simplified approximations to the general theory it is shown that camber reduces pressure drag (by as much as half) and the bow-up pitching moment; the latter by so much that a cambered planing surface can become unstable. (This was already known theoretically for two-dimensional flow, from the work of Wagner (1932a,b), Cumberbatch (1958), Wu and Whitney (1972), Doctors (1974), Wellicome and Jahangeer (1978). It was known for slender planing surfaces from the work of Maruo (1967) and Tulin (1957). It was also known experimentally for finite aspect ratios from the work of Sottorf (1932, 1934), Clements (1964) and Moore (1967). Rocker, on the other hand, gives a large bow-up couple, and results in increased pressure drag. Presumably this large bow-up moment is why rocker is almost invariably employed with planing sailing dinghies and surfboards, despite the resistance penalty.