Planing and impacting plate forces at large trim angles

Abstract

Added mass theory has been shown to give excellent agreement with experimental measurements on planing surfaces at normal planing angles [e.g. Payne, P.R. (1982, Ocean Engng 9, 515–545; 1988, Design of High-Speed Boats, Volume 1: Planning. Fishergate. Inc., Annapolis, Maryland)] and to agree exactly with more complex conformal transformations where such a comparison is possible. But at large trim angles, it predicts non-transient pressures that are greater than the free-stream dynamic pressure and so cannot be correct. In this paper, I suggest that the reason is because, unlike a body or a wing in an infinite fluid, a planing plate only has fluid on one side—the “high pressure” side. So the fluid in contact with the plate travels more slowly as the plate trim angle (and therefore static pressure) increases. This results in lower added mass forces than Munk, M. (1924) The Aerodynamic Forces on Airship Hulls (NACA TR-184) and Jones, R. T. (1946) Properties of Low-Aspect-Ratio Pointed Wings at Speeds Below and Above the Speed of Sound (NACA TR-835) originally calculated for wings and other bodies in an infinite fluid. For simplicity of presentation, I have initially considered the example of a triangular (vertex forward) planning plate. This makes the integration of elemental force very simple and so the various points are made without much trouble. But the penalty is that there seem to be no experimental data for such a configuration; at least none that I have been able to discover. But at least the equations obtained in the limits of zero and infinite aspect ratio, small trim angles (τ) and τ = 90° all agree with established concepts and the variation of normal force with trim angle looks like what we would expect from our knowledge of how delta wings behave in air. I then employed the new equation to calculate the force on a rectangular planing surface at a trim angle τ, having a constant horizontal velocity u o and a vertical impact velocity of ż. This happens to have been explored experimentally by Smiley, R. F. [(1951) An Experimental Study of Water Pressure-Distributions During Landings and Planing of a Heavily Loaded Rectangular Flat Plate Model (NACA TM 2453)] up to trim angles of τ = 45°, and so a comparison between theory and experiment is possible. The results of this comparison are encouraging, as is also a comparison with the large trim angle planing plate measurements of Shuford, C. L. [(1958) A Theoretical and Experimental Study of Planing Surfaces Including Effects of Cross Section and Plan Form (NACA Report)]. As two practical applications, I first employed the new equations to calculate the “design pressures” needed to size the plating of a transom bow on a high-speed “Wavestrider” hull. The resulting pressures were significantly different to those obtained using semi-empirical design rules in the literature. Then I used the theory to critically review data obtained from tank tests of a SES bow section during water impact to identify how the “real world” of resilient deck plating diverged from the “model world” of extreme structural rigidity.