A note on the optimum profile of a sprayless planing surface

Abstract

The paper presents an exact analytical solution to the problem of finding the optimum profile of a two-dimensional plate which planes on a water surface without spray formation and maximizes the lift force. The lift is maximized under the only isoperimetric constraint of fixed total arclength of the plate. The exact solution is compared with approximate analytical and numerical results by Wu & Whitney (1972). The shape of the optimum plate turns out to be technically unrealizable because of small, tightly wound spirals near the end points. It was shown numerically that cutting off small segments near the end points leads on the one hand to insignificant change in the lift force and on the other hand to a non-separating boundary layer along the remaining part of the optimum plate.

On the lift and drag of cavitating profiles and the maximum lift and drag

“…For instance, if we take the initial r = 1 m, after a half-revolution its length will be 0.05 mm, and the remaining part of the spiral will be like a point. The spirals of this kind appeared in all previous papers, where isoperimetric optimization problems with free streamlines were solved (see Maklakov 1988Maklakov , 1997Maklakov , 1999Maklakov , 2005Maklakov & Uglov 1995;Maklakov et al 2007;Elizarov et al 2008). As was demonstrated by Maklakov (1997Maklakov ( , 1999 and Elizarov et al (2008), the contribution of these tiny spirals to the hydrodynamic forces is negligible.…”

“…Maklakov (1999) has proved that for this flow the lift coefficient that, for the infinite cavity flow and for the planing surface, the upper estimates of the lift force and the optimal velocity distributions coincide. The explanation for this curious coincidence is rather simple.…”

“…The papers by Wu & Whitney (1972) and Maklakov (1999) have studied the problem of finding the optimum profile of a two-dimensional plate that planes on a water surface without spray formation and maximizes the lift force. The scheme of the flow region and the corresponding parametric lower half-plane are demonstrated in figure 7(a,b).…”

“…But the shapes of the optimum profiles will certainly be different. For comparison we reproduce in figure 8 the optimum planing surface found in figure 3 of Maklakov (1999).…”

On parachutes of optimum shape in a subsonic gas flow

“…In Maklakov & Uglov [7] the problem of the ideal optimum parachute for the wake model by Joukovsky, Roshko and Eppler was solved. In Maklakov [9] the optimum shape of a planing plate was found. In Maklakov [10] the problem of the optimum parachute in Helmholtz flow was discussed.…”