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.
There is really no actual Michell integral for multihulls. What we use here is the assumption that each separate hull can be represented by the same singularity distribution (namely sources distributed over its own centreplane) as if that hull were alone. This neglects one type of interaction between the hulls, namely the influence of one hull on another in creating a cross-flow which modifies this singularity distribution, in particular inducing vortices as well as sources. On the other hand, it does not prevent interference between the wave systems generated by the centreplane sources. Little is known of the relative importance of these two types of interactions, but the present assumption seems to yield quite good results for the wave resistance (Tuck 1987, Salvesen et al 1985). It is notable that the assumption that there are no induced vortices due to other hulls can be exactly satisfied by allowing the hull centrelines to possess a suitable small camber (Lin 1974). This camber has no effect on the wave resistance, and may be desirable in eliminating induced drag. It is also important (Tuck 1987, Newman and Tuck 19xx) that the size of this induced-crossflow effect relative to that due to the hull's own thickness is proportional to the draft/length ratio, and hence negligible for conventional (and a forteriori the present optimal) slender ships of small draft, even if formally of the same order as self-thickness effects for the thin ships of finite draft for which Michell derived his integral.
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 (1991) 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.
Care must be taken in applying this form factor. In an optimisation problem where shape is not constrained, there could be many undesirable side-effects. For example, since there is a tendency of the optimisation to reduce the above half-angles, bows could tend to be overly cuspy. In addition, since more of the displacement can be placed deeper without penalty, there could be a tendency away from wall-sided hulls and towards some tumblehome below the waterline. This is not an issue here, since our parabolic waterlines and elliptic sections do not allow so much freedom. We shall also find that use of this form factor leads to an improved optimisation process in the high-speed range.
We assume that there is no effect of dynamic vertical forces, which at low speeds account for sinkage and trim. These are small effects, but notably for multihulls can be substantially different than for monohulls. At high speeds, dynamic forces are upward and yield a lift rather than a 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.
Asymmetric flow around each demihull of a catamaran has been observed. This manifests itself in differences in draft and wetted surface area between both sides of the demihulls. Asymmetrical flow can cause lift and inevitably, induced drag; see Insel and Molland (1991).
Viscous interference between the demihulls of catamarans also seems to be an as yet incompletely understood effect, which can complicate the estimation and application of simple form factors. Insel and Molland (1991) state that "catamarans show substantially higher resistance than twice that of the monohulls, even at ... low speeds where wave interactions are negligible, therefore indicating viscous interactions. Additionally, flow visualisation experiments ... on a catamaran model indicated a change of flow lines and pressure field, hence some form of viscous interaction."
Shallow water effects can be important in some applications, e.g. see Millward (1992) for catamarans, and Scragg and Nelson (1993) for 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.