The normal force on a flat planing plate, including low length to beam ratios

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Abstract A previous paper (Payne, 1980) developed the theory of a slender planing surface from first principles, and found it to agree with experiment without the need for “factors determined from experiments”. The theory broke down, however, for low wetted length to beam ratios because the flow is then no longer two-dimensional in character in respect to a section normal to the flow. It is in a transition to the condition where, at very small length to beam ratios, it is two-dimensional in a vertical plane parallel to the flow, rather than normal to it. In the present paper we determine, albeit somewhat heuristically, the virtual mass coefficient of a flat plate in the transition region. This is then employed to compute the normal force on a planing plate in both steady state and transient motion. For both conditions the agreement with the available experimental data is within the accuracy of the data. Both theory and experiment lead to the conclusion that certain traditional concepts in the of trim angle and Pierson's suggestion that ‘wetted length’ is more fundamental than the length of intersection with the still water surface. The widespread acceptance of this latter suggestion has meant that a key parameter in the present theory—the nominal or undisturbed water intersection length—is almost never measured in modern experiments. Thus, since we could not find any modern data, we had to appeal to the classical experimental papers which originated the scientific study of planing in the thirties and forties. Oddly, the earliest papers gave data which had much less scatter than the later papers.