For this hullform, the block coefficient is Cb = 0.417 and the prismatic coefficient is Cp = 4 Cb/pi = 0.533. Clearly this is a much finer type of hull than that of a typical merchant ship, but is relevant to sporting canoes and hulls of special high-speed vessels. It is particularly appropriate for slender vessels with the high length/beam ratios that we shall find optimal.
Michell's integral depends for its validity on the ship being thin, and is sometimes considered (perhaps unfairly) to be insufficiently accurate for use with ships of conventional proportions. However, the hulls produced by the optimisation process in this study are significantly thinner than conventional ships, and there is good evidence that for such slender vessels Michell's integral is satisfactory. For example, Hanhirova et al 1995 (see also Tuck 1989 and Chapman 1972) report that for length-based Froude numbers above 0.35, accuracies relative to measured residuary resistance of better than 10% are achieved by Michell's integral for hulls with length/beam ratios of the order of 10.0. The optimised hulls in the present study are even more slender.
In any case, the hulls resulting from the optimisation process also have the property that their wave resistance is generally only about 10% of the total, so that the absolute accuracy of the wave resistance measure is not critical. This proportion of wave resistance to total drag is lower than what is usually encountered with conventional ships, since the present optimum is in part achieved by increasing the length beyond the conventional, so as to reduce the influence of wave resistance. Even though the wave resistance is then only a small component of the final total drag, it remains a critically important component nevertheless in controlling the optimisation process; after all, if there was no wave resistance at all, short ships of minimum surface area would be preferred.
Rv=1/2*rho*U2*S*Cv
where rho is the water density and S the wetted surface area of the hull. When skin friction dominates, the drag coefficient Cv approximately equals Cf, where Cf is a skin friction coefficient which can be estimated using the ITTC 1957 ship correlation line (Proc. 8th ITTC).
Cf = 0.075/(log10R-2)2
where R = UL/nu is the Reynolds number; nu approx equals 1.054 x 10-6m2s-1 is the kinematic viscosity.
We have used the full length of the waterline for L in the definition of the Reynolds number; however there are other possibilities. Gerritsma et al. (1981) use 0.7L in their study of the resistance of a systematic yacht hull series, reasoning that this defines a kind of average length.
In their examination of eight-oared rowing shells, which have a hullform not unlike the canoe body examined here, Scragg and Nelson (1993) found a simple empirical formula for the form factor of these hulls. The viscous resistance coefficient is written as
Cv = (1+k)Cf
where
k = 0.0097(thetaentry + thetaexit)
Here, thetaentry and thetaexit are the half-angles (in degrees) of the bow and stern, respectively, at the waterplane.
We assume that there is no effect of dynamic vertical forces, which at low speeds account for sinkage and trim. At high speeds, dynamic forces are upward and yield a lift rather than sinkage; hence planing, and we neglect that. The present results are for displacement rather than planing conditions, although for completeness we exhibit them even in speed ranges where planing would be expected.
Shallow water effects can be important in some applications, e.g. see Millward (1992) for catamarans, and Scragg and Nelson (1993) for eight-oared rowing shells. However, we retain the infinite-depth assumption here. We also neglect any lateral flow domain restrictions; see Doctors and Day (1995) and Day and Doctors (1996) for the case of a ship moving in a channel.
For rowing vessels, we do not make any allowance for changes in the centre of gravity or the consequent change in trim due to the continually changing position of the rowers. The present analysis is done on a steady-flow basis, and hence relates to the average conditions during a race, neglecting speed variations due to racing conditions, as well as unsteady variations during an individual rowing stroke.