which would be a function of Fnv alone were it not for the fact that the skin friction coefficient depends on Reynolds number.
In order to exhibit this scale effect of displacement, we carry out the optimisation at fourteen fixed (dimensional) displacements, ranging from 0.075 tonnes up to one tonne. In fact we have also computed results for even larger vessels, up to one million tonnes.
For definiteness, we give most results for the fixed displacement of one tonne. Some such results have already been given in Figures 1.1 and 1.2. It is notable that for this displacement, L*=1 metre, so that the non-dimensional length can also be interpreted as the actual length in metres. The volumetric Froude number is also uniquely proportional to the actual speed in metres/second or knots, and Fnv=1 occurs at 6.1 knots for a one-tonne vessel.
It is important to bear in mind that none of the Figures 2 and 3 to follow, where the total drag coefficient Ct is plotted against the volumetric Froude number Fnv, can be interpreted in the usual naval architectural manner as a graph of drag versus speed for a given ship. As Fnv varies, the ship itself changes its shape, and in particular its length, so as to keep the drag as small as possible.
To produce the results in Figures 2(a)-2(e), we performed the optimisations at 83 different speeds. For the case with no form drag, the volumetric Froude numbers corresponding to these speeds are 0.100, 0.125,...,0.675; 0.680, 0.685,...,0.700; 0.725, 0.750...,0.900; 0.905, 0.910...,0.925; 0.950, 0.975; 0.980, 0.985,...,1.0; 1.025, 1.050,...,1.2; and 1.3, 1,4,...,4.0. The very small speed increments at the low end of the range were needed to capture the discontinuities described earlier. Similar ranges and increments were used to produce the plots for the case where form drag has been included.
We used a population of 64 during the optimisation. Each of the design problems was run with at least five different initial populations. A minimum of 5,000 resistance evaluations were performed during each run. To investigate further the previously discussed discontinuities, additional runs were performed with the search domain constrained in such a manner as to disallow one of the alternative solutions. This is an extremely tedious process because we don't know in advance exactly where the discontinuities might occur. Whether we like it or not, human input and understanding is still essential to complex engineering design systems.
|Fig. 2(a): The effect of form factor on the optimum total resistance
of a one
The dashed curve of Fig. 2(a) shows Ct as a function of volumetric Froude number Fnv, for a one-tonne boat. This is the residual value of the total drag, after the boat's dimensions have been optimised to minimise Ct without any allowance being made for form drag. The hull parameters that produce these optimal Ct's are shown as the dashed curves in Figures 2(c)-2(e).
|Fig. 2(b): The effect of form factor on the optimal
proportion of wave
resistance of a one tonne monohull.
Fig. 2(b) shows wave resistance as a fraction of the total drag. There is considerable scatter at low Fnv. This could be due to long shallow regions in the "fitness" landscape, where for example, one length is as good as another. Although Ct remains the same, Cw/Ct may vary. GODZILLA searches for the lowest total resistance and if it encounters two or more combinations of parameters with almost the same Ct, it cannot prefer one to the other. In these regions, it could be important to perform the integrations more accurately. In any case, wave resistance is less than 12% of the total for all Fnv<3.25.
The most obvious feature of the dashed curve in Figure 2(b), however, is the sudden increase in the proportion of wave resistance for Fnv>3.25, a rather high speed (of the order of 20 knots for a one-tonne vessel) near the upper end of the range being considered in this study. Figure 2(c) shows that the optimum (non-dimensional) length also drops sharply to a very low level at this speed. This discontinuity is essentially an interchange in the roles of two local minima, as in Figure 1.1. For Fnv<3.25, the longer boat is best; for Fnv>3.25 the shorter boat is best, and in the present case, the shorter boat is so short as to be quite unrealistic. Indeed, this boat almost eliminates its wave resistance by going to a very high rather than a very low conventional Froude number. Minimum viscous drag dictates minimum surface area, and that inevitably pushes the optimum toward a hemispherical geometry. In the present case, other constraints prevent this hemisphere being achieved exactly, but this class of "optimum" boat does tend to have length comparable to the beam and draft. Clearly this is not a realistic conclusion, and in particular would lead us to question the validity of neglecting form drag.
|Fig. 2(c): The effect of form factor on the optimal length of a one tonne
Returning to the "realistic" boats produced for lower speeds, with Fnv<3.25, as the "design speed" increases from zero in that range, the optimum length L shown in Figure 2(c) increases to a maximum of about 22 at a Fnv value of about 1.8 before decreasing slowly as the speed increases further. This volumetric Froude number of 1.8 corresponds to a conventional (actual length-based) Froude number of 0.38, or a speed of 11 knots for a one-tonne vessel. At speeds below this value, the usual very dramatic large rise in wave resistance occurs as the length-based Froude number increases. Not surprisingly, longer boats are then preferred as the speed rises.
This trend cannot continue for ever. Eventually, the optimal boatlength reaches a maximum, and further increases in speed can no longer be met by increasing length to keep operating well below the wave resistance main peak. Instead, the length-based Froude number passes (quite rapidly) through the value where wave resistance is maximal, but the proportion of wave resistance is nevertheless kept sufficiently low to achieve an optimal design because of the large boatlength. Eventually as the speed increases further, the optimal boatlength starts to decrease again, since we are now operating at a length-based Froude number above the main wave resistance peak. Then the wave resistance decreases with Froude number, and hence shorter boats have less rather than more wave resistance at any given speed, and are preferred in the optimisation.
|Fig. 2(d): The effect of form factor on the optimal beam-to-draft ratio
one tonne monohull.
When the length is so great, the surface area strongly controls the optimisation, and to minimise the increase in frictional resistance, semi-circular sections tend to be preferred. This is clear in Figure 2(d), where it can be seen that the beam-to-draft ratio B/T stays at a value of roughly 2 for Fnv between 1.0 and 2.5.
|Fig. 2(e): The effect of form factor on the optimal length-to-beam ratio
one tonne monohull.
The optimum boats are very slender. Figure 2(e) shows the length-to-beam ratio, which is very high indeed (reaching a maximum of about 42 at Fnv=1.8) by conventional ship standards, though not entirely unreasonable for rowing shells.
Figure 2(a) indicates that there is only a quite small increase in the residual total drag Ct for all speeds, consistent with the fact that the form drag is small, especially for the present very fine hulls. The greatest impact of form effects on the optimisation process occurs at very low and very high Fnv. The solid-line Ct curves of Figure 2(a) are smoother at low Fnv than the dashed curves, and the ultimate decrease in Ct at high Fnv is no longer as rapid.
Figure 2(b) shows that with form drag included, the proportion of wave resistance now remains below 10% for all speeds and all displacements. The scatter at low Fnv is not so pronounced as in the optimisations without form effects. Most important of all, however, is that there is no longer a sudden discontinuous increase in the proportion of wave resistance for Fnv>3.25. We have already anticipated this, since the very short boats that were suggested at high speeds by the optimisation without form drag are now heavily penalised by their large entrance and exit angles, and fail in total drag competition with a local minimum corresponding to a longer boat.
Figure 2(c) confirms this point, indicating that the optimum boat stays "long" for all speeds, with no discontinuity at any high-end speed. Indeed, with the inclusion of form effects, there is a tendency towards slightly longer optimum boats. The beam-to-draft ratios shown in Figure 2(d) are generally about 10% smaller with form drag included. For our canoe body, small entrance and exit angles can only be achieved by reducing the beam, so there is a slight tendency toward non-circular cross-sections, with B/T<2.
At the intermediate speeds which are of the greatest practical interest, there is only a small effect of the form factor on all outputs, and the qualitative discussion in the previous section about transition through the speeds where the wave resistance and hence the optimum length is maximal applies equally with or without form factor. Nevertheless, because as we have seen, inclusion of a form factor makes for a smoother and more realistic optimisation process at all speeds, such a factor is included in all of the remaining computations presented here.
One commonly-used method to rate rowing performance is the U.S. gold medal standard time over 2000m. For example, standard time over 2,000m. for the light women's single sculls (D=0.075 tonnes) is 7 min. 42 sec., for the open men's coxless fours (D=0.4 tonnes) standard time is 5 min. 55 sec., and for the men's open coxed eights (D=0.9 tonnes) it is 5 min. 29 sec., (Gwadz, M., pers. comm.). Using these standard times to calculate average boatspeeds, the corresponding volumetric Froude numbers of the three classes are approximately equal to 2.0, 2.1 and 2.0, respectively. Thus the range of volumetric Froude numbers we are considering here extends well above and well below the standard times for each class of rowing, and the central value Fnv=2 seems representative of good speeds for a wide range of displacements.
Since we are not constraining length or draft, the following results also apply to canoes and kayaks. Of course, the narrow hulls that result from the optimisation process have low static stability and are probably less suitable for paddling styles. It is important, however, to know at what cost in total resistance the necessary extra stability can be achieved.
To reduce computation time, 33 waterlines, 33 stations, and 240 intervals of theta were used in calculations. Length was constrained to lie in the range 0.5m <=L<=30.0m; draft was limited to the range 0.01m <=T<=2.0m. For the optimisations, we used a small population size of 32. Each of the 112 design problems (14 displacements, 8 speeds) was run with three different initial populations, and a minimum of 5,000 resistance evaluations were performed during each run. Optimisations were performed on 14 Sun workstations (some IPX models, some Sparc4s some Sparc 10s) over four nights.
In all cases convergence was quite fast due to the small number of design parameters, and because the wave resistance varies smoothly with length-based Froude number greater than about 0.35. In hindsight, the number of evaluations performed seems quite excessive for this set of problems. In nearly all cases, the eventual optimum was found (to at least the 4th decimal place in Ct) during the first run. Subsequent runs made little difference to either the optimum total resistance, or the parameters producing that optimum resistance. Of course, if we had used smaller search domains, for example small regions surrounding the dimensions of existing shells, we could have used hill-climbing operators alone. However, we are looking for unusual hullforms, one's that we as humans might not normally conceive of. If the search domain is too small, we could miss, for example, the low-speed and high-speed discontinuities discussed previously.
Figures 3(a) and 3(b) show the variation with displacement of the total resistance coefficient and length respectively. Results for the one-tonne boats are the same as in figures 2(a) and 2(c). Note that the proportionality constant relating actual speed to volumetric Froude number varies as the one-sixth power of displacement. Specifically, the actual speed at the central Froude number Fnv=2.0 is 7.9 knots for a 0.075-tonne vessel, 10.8 knots for a 0.5-tonne vessel, and 12.2 knots for a 1.0-tonne vessel. Actual speeds at other Froude numbers are obtained by scaling these central speeds.
We do not present graphs of Cw/Ct, beam-to-draft ratio, or length-to-beam ratio to save space, and also because they are very similar to those of the one tonne results in figures 2(b), 2(d) and 2(e).
|Fig. 3(a): The effect of displacement on the optimal total
resistance of a monohull.
In figure 3(a) the curves at the top of the graph are for 0.075-tonne boats; the curves at the bottom are for the one-tonne shells.
|Fig. 3(b): The effect of displacement on the optimal length of a
Smaller boats have larger Ct because they are shorter, their Reynolds numbers are smaller, and consequently the skin friction coefficient is larger. Of course the actual total drag Rt is much larger for larger shells, once we multiply Ct by L*2=D2/3.
In this range of speeds, the dependence of the results on displacement is quite smooth and predictable by interpolation within the curves presented here.
To provide a more realistic comparison, we repeated the optimisation of a 0.871-tonne boat at a speed of 11.7 knots, but now with the beam constrained to B=0.57m., roughly the same as that of the existing shells considered by Scragg and Nelson. GODZILLA found that this sub-optimum hull had Ct=0.0243, which still represents a 2% improvement over the Ct estimated by Scragg and Nelson for the extant hulls. Our 0.57m-beam shell has L=16.9m, which is about 10% shorter than the above optimum length of 18.5m. On the other hand, this length compares very well with the three existing shells: the Janocek hull has L=16.3m; the Vespoli A has L=17.0m; the Vespoli B has L=16.6m. However, the draft for our optimum shell, T=0.216m., is 15.0% greater than the extant hulls which have drafts of about T=0.187m.
Scragg and Nelson's experiments with the Vespoli B and other hulls showed
considerable scatter around 11-12 knots and Ct's between 0.0234
and 0.0247 were reported. The scatter was so considerable in fact, that
they concluded (page 98)
"...the empirical approach is less reliable in discerning small differences in performance than the systematic results obtained from numerical hydrodynamics."
The present authors take great heart from this!