dynamic behaviour has something in common with that of the bicycle, on which much has been published [4-7]. But the water does not behave quite like a road, so this paper offers an analysis of the motion of such a vessel, and discussion of the control problem.
It also considers
some more practical matters, such as how to avoid capsize in the absence of forward motion.
2. HULL SHAPE OF LEAST RESISTANCE FOR A GIVEN DISPLACEMENT AND SPEED
To get an exact answer to the question “what hull form will permit a vessel of given displacement to be driven at a given speed with the least power?” is a major and perhaps insoluble problem. But a rough idea can be got using the Froude hypothesis and published experimental results for
residuary resistance, e.g. considering low-powered hydrofoil-borne craft are not in question.
As an example, we consider a small human-powered vessel of displacement 100 kg, to be propelled at 2.2 m/s (4.27 knots).
This speed approximates to 8 km/hour,
which is the speed limit for mechanically-powered boats on the River Thames in England, the author’s local waterway. A new type of human-powered boat that could maintain this speed over several hours might have some attractions. Figure 1a shows how the total hull resistance is estimated to vary with waterline length for a single-hulled vessel having beam/draught ratios of 2.25, 3 and 3.75 (for which the residuary resistance is quoted in [9]. For the first case the frictional resistance is plotted too. As the hull length is increased the frictional resistance grows gradually, because of the increased wetted area, while the residuary resistance falls rapidly, ultimately becoming almost negligible. Appendix One gives some details of how the hull shape was defined.
It
differs from that of the “Taylor Series” to which the residuary resistance data applies, but it is believed that the resulting errors are fairly small. A point showing the measured drag of a rowing scull at 100 kg displacement is included [10], and appears roughly to confirm the validity of that part of the graph at least. The scull has of course been optimised for a rather
higher speed
(sustained usually only for a matter of minutes), so is non-optimum for
2.2 m/s (too large a prismatic coefficient as well as too long).
For this speed it can be seen that the total resistance is minimised with the lowest beam/draught ratio (almost a semicircular cross-section) and at a waterline length of about 5m.
It is then about 12.6N. vessels, so planing or
Figure 1: Hull [8,9]. We are
resistance against length for a 100kg
vessel at 2.2m/s; (A) single hull at three beam/draught ratios; (B) single hull v. catamaran, beam/draught = 2.25
But for this 5m hull the maximum beam and draught are 0.323m and 0.143m respectively, and the metacentre is 0.014m above the waterline. It is difficult to conceive how such a vessel, together with the occupant who is propelling it and navigating it, could be designed to have its mass centre below its metacentre, and thus be statically stable, unless by the use of a ballast keel, which would both increase the resistance and reduce the payload. Even at a beam/draught ratio of 3.75, the metacentre is still only 0.098m above the waterline.
To get around this problem, human-powered boats with propeller drive are frequently built as catamarans, which solves the stability problem. But the extra wetted area then increases the resistance. Figure 1b shows the total hull resistance (and frictional resistance), estimated as above for a pair of hulls each of displacement 50 kg, and beam/draught ratio 2.25, on the same plot as a single 100kg hull. Any hydrodynamic interaction between the two hulls has been ignored.
At very low waterline
lengths the catamaran has the advantage, due to its lower residual resistance, but the frictional resistance is larger than that of the single hull of the same length by a factor of about √2, as one would expect, so at longer lengths, where the residuary resistance is small, it loses out to the single hull. The minimum resistance is now about 17N at a waterline length of about 4.7m, or 35% more than for a single hull of the same beam/draught ratio.
So ignoring
appendage and air resistance, the effective power to propel it would be 12.6N x 2.2m/s = 28W. Even after including appendage and air resistances and allowing a propulsive efficiency of say 70%, the propulsive power needed should be sustainable by a human user for a considerable time.
A similar difference appears at somewhat larger sizes, in spite of the higher Reynolds Numbers. A vessel of 100 tonnes displacement cruising at 7m/s (13.6 knots) would have
about the same Froude Number based on
displacement as one of 100kg at 2.2m/s (F = Speed/√(g⅓) = about 1.03 in each case). In this case the catamaran has the lower resistance below a length of about 36m, but whereas the catamaran’s least resistance comes out as 10760N at 50m, the single hull achieves 8065N at 55m. This is about a 25% reduction, and an effective
power saving of 19kW. For the same
B-2
©2007: Royal Institution of Naval Architects
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