Low-Aspect-Ratio Flat-Ship Theory

Abstract

The integral equation, first obtained by H. Maruo, which determines the pressure distribution generating flow past a slender ip of vanishing is studied further. New results obtained include predictions of singular centerplane effects of gravity for pointed bodies, a solution for ships with cusped parabolic waterplanes, and some preliminary numerical solutions of the integral equation in the general case. I. Introduction T HIS paper is an extension of a study by MaruoJ on ''ships of small draft, ships/' or surfaces, all of which are equivalent descriptions. The small-draft assumption allows linearization of the free-surface boundary condition, as in the comparable case of thin ships, or ships of small beam.2 However the present linearized problem is much harder to solve, since the generating singularity distribution (effectively a distribution of pressure points on the limiting waterplane) is not explicitly given in terms of hull shape, but requires solution of an integral equation. This problem is analogous to the lifting-surface problem of aerodynamics, whereas the thin-ship problem corresponds to the simpler thickness problem of aerodynamics. Although this paper contains a brief reconsideration of the general flat-ship problem, to emphasize some aspects not discussed by Maruo, the paper is devoted mainly to the lowaspect-ratio limit. Thus the wetted length of the ship is supposed much greater than its beam, the latter having already been assumed much greater than the draft by the flat-ship requirement. The ship is therefore not only flat, but also slender. An alternative derivation is given here of an integral equation equivalent to one obtained by Maruo, having as its unknown function a pressure distribution representing the ship. This integral equation is also obtainable from the highFroude-number slender-body theory of Ogilvie, 3 by assuming that the ship is not only slender, but also flat. Maruo's low-aspect-ratio flat-ship integral equation is formally valid only at moderately-high Froude numbers, specifically such that U2B/gL2 is a quantity of order unity, where U is ship speed, B its beam and L its length, and g is gravity. The equation reduces to that of low-aspect-ratio wing theory in aerodynamics as g->0. One approach to practical solution of any planing problem, whether or not the aspect ratio is low, is to expand in an asymptotic series for very high Froude number, commencing with the aerodynamic g = 0 limit as the leading term.4 Maruol obtains the first two terms in this series for the lift on a flat delta wing, and an alternative treatment of this class of expansion, for general hull shapes, is presented here. In particular, very strong effects of gravity near the center plane of pointed bodies are demonstrated. At all Froude numbers, the low-aspect-ratio flat-ship integral equation possesses a similarity solution, such that the pressure distribution has the same shape at all stations. This linearized but gravity-dependent result should not be confused with the well-known conical solution for nonlinear planing or water entry in the absence of gravity (Gilbarg,5 p.360). In fact the present geometrical requirement is for a cusped parabolic waterplane shape but an arbitrary section shape, whereas the nonlinear zero-gravity solution requires a triangular plan form and section shape. The low-aspect-ratio flat-ship integral equation is amendable to direct computation, and we present here some preliminary examples of its numerical solution. Much more work needs to be done to derive efficient procedures, and the present computer program can only be considered as a crude first attempt. However, the results are of considerable interest, indicating rather dramatic gravity effects especially near the center plane, as predicted analytically, and confirming Maruo's1 estimate of the lift coefficient of a delta wing at sufficiently high Froude number.