3. RESULTS AND DISCUSSION

3.1 Variation in Displacement

In this section, we present the dimensions and other characteristics of the optimal hulls found by the optimisation process. Results are presented as a function of the volumetric Froude number, F_nv, defined previously. In fact, were it not for scale (Reynolds number) effects, all results would be universal functions of this Froude number. For example, the final minimum total drag R_t=R_v+R_w is expressed in terms of the coefficient

which would be a function of F_nv alone were it not for the fact that the skin friction coefficient depends on Reynolds number.

3.1.1 Total Resistance

Figure 3.1(a) shows the optimum total resistance coefficient C_t as a function of F_nv for the nine displacements considered in this study. It is important to note that we cannot interpret this graph in the usual naval architectural manner as a graph of drag versus speed for a given boat. In this graph, as F_nv varies, the kayak hull itself changes its shape so as to keep the drag as small as possible. The hull dimensions and other parameters that lead to the optimum total drag are given in figures 3.1(b)-3.1(f).

Fig. 3.1(a): Comparison of total resistance coefficients.

Figure 3.1(a) shows that smaller boats have larger drag coefficients. This is because they are shorter, their Reynolds numbers are smaller, and consequently the skin friction coefficient is larger. Of course the actual total drag, R_t, is much larger for larger boats, once we multiply C_t by 1/2*rho*U²D^2/3. In this range of speeds, the dependence of the results on displacement is quite smooth and predictable by interpolation within the curves for each class presented here.

3.1.2 Wave Resistance

Fig. 3.1(b): Comparison of R_w/R_t.

Figure 3.1(b) shows the wave resistance of the optimal designs as a fraction of the total resistance. For the K1 class, wave resistance comprises between 14% and 20% of the total resistance. Smaller displacement boats (0.080 and 0.090 tonnes) have proportionally less wave resistance than larger boats (0.100 tonnes). Evidently, the length constraint does not affect the smaller displacement craft quite so severely.

For the K2 class R_w/R_t varies between about 16% and 22%. As withthe K1 hulls, the proportion of wave resistance decreases with displacement.

For the K4 class hulls, which are somewhat finer than the K1 and K2 hulls, the proportion of wave resistance varies between about 8% and 13% of the total. Although the variation with displacement is small, note that the trend is opposite to the K1 and K2 classes - here R_w/R_t decreases with increasing displacement.

3.1.3 Beam

For all design displacements and all design speeds considered here, the beam found by GODZILLA was equal to the minimum allowed by ICF rules. That is, this minimum-beam constraint exerted a controlling effect on the design, and if it were not present, the optimal designs would have smaller beam. Thus for the K1 class, the design beam is 0.51m, for the K2 class it is 0.55m, and for the K4 class it is 0.6m.

3.1.4 Length

Fig. 3.1(c): Comparison of optimum lengths (metres).

For the K1 class, figure 3.1(c) shows that the optimum length for all design speeds is equal to, or very close to, the ICF limit. Only at the two highest design speeds does the optimum length drop below that limit, and then only slightly. The length found for all K2 class kayaks is 6.5m, the maximum allowable, so that relaxing this constraint would have led to longer boats.

In contrast to the other two classes, the optimum length for the K4 class kayaks shows a stronger dependence on displacement and design speed. Optimum length decreases with displacement and as the design speed increases. The ICF maximum length constraint in general exerts no control on the optimum kayak, which is shorter than this constraint.

Figures 3(b) and 3(c) suggest that the ICF constraints affect the K4 class kayaks in a significantly different manner to the K1 and K2 kayaks.

3.1.5 Draft

Fig. 3.1(d): Comparison of length-to-draft ratios.

The optimum length-to-draft ratios are shown in Figure 3(d). For the K1 and K2 kayaks, this ratio decreases sharply with increasing displacement, but only slowly with increasing F_nv. The curve for the 0.155 tonne kayak shows the length-to-draft ratio decreasing in a step-wise manner with increasing F_nv. We have found similar discontinuities in our examination of rowing shells (Tuck and Lazauskas 1996). In that study we held the hull shape constant, so it is not surprising that in the present study, where we are allowing shape variations, that the optimum surface should also contain discontinuities as a function of design speed.

For the K4 kayaks, the length-to-draft ratio decreases more quickly with increasing speed, but there are only small differences with respect to displacement.

3.1.6 Shape Parameters

3.1.6.2 Waterlines

In all cases, the optimum waterline shape factor, a₁, was 1.0. That is, parabolic waterlines were found to be optimal (at least for the limited class of shapes we have considered here). This is not unexpected given the dependence of the form factor on the entrance and exit angles. Although we could have reduced the search space significantly by considering only parabolic waterlines, we might have missed out on some non-standard hull. After all, one reason for using genetic algorithms is to examine configurations that we as humans might not normally consider. In the present investigation, and with the benefit of hindsight, however, it seems our hopes were misplaced as far as waterlines are concerned.

3.1.6.2 Cross-sections

Fig. 3.1(e): Comparison of cross-section shape parameters (a₂).

For all classes, the optimum cross-section parameter, a₂, lies between 0.6 and 0.75. This means that the cross-section shapes are "between" elliptical (a₂=0.5) and parabolic (a₂=1.0).

The small "wobbles" in the curves of figure 3.1(e) suggests that small changes in the cross-section shape are not critically important. Large wobbles can be indicative of regions where the optimum parameter surface may be discontinuous. For example, the curve for the 0.300 tonne K4 kayak shows that the optimal cross-section shape parameter at F_nv=1.7 is about 0.27. For F_nv=1.6 and F_nv=1.8 the optimum value of this parameter is about 0.36. There is in fact another hull, (with a total resistance differing only in the 4th decimal place) at F_nv=1.7, with a₂=0.36. This hull has a different length, draft and keel-line shape parameter to the hull with optimum cross-section shape parameter a₂=0.27. Thus it is quite possible that the off-design performance of these two "equally optimal" hulls will be significantly different, a factor that could be important when it comes to ultimately choosing one or other of the two hulls. Stability considerations could also be used to break the apparent "deadlock", but we leave these more detailed design problems to a later time.

3.1.6.3 Keel-lines

Fig. 3.1(f): Comparison of buttock shape parameters (a₃).

Figure 3.1(f) shows the variation of the optimal value of the keel-line shape parameter, a₃. For the K1 and K2 classes this parameter is small, indicating that constant draft hulls (i.e. rectangular sideviews) are near optimal. For these two classes, a₃ increases in a step-like manner with increasing speed, again indicating possible discontinuities in the optimum parameter space.

The optimal keel-line shape parameter varies from 0.359309 to 0.359311 for the K4 class. This is somewhere "between" a constant draft hull (a₃=0.0) and an elliptical keel-line (a₃=0.5). It is surprising that for all ten design speeds, this small range of values should be optimal.

3.2 Expected Improvement

We have found no publications concerning the resistance of Olympic class kayaks in the recent literature or on the electronic Internet. In order to assess the effects of shape, we will compare our optimal hulls with a set of "PER" hulls, which have parabolic waterlines, elliptical cross-sections and rectangular sideviews. As our "optimal" hulls we use those found at F_nv=2.3 for all classes and displacements. Furthermore, the lengths of the PER hulls have been chosen to be close to those of the corresponding optimal hulls.

Tables 4(a)-4(c) show the hull dimensions and shape parameters of the hulls used in the comparisons in this section. In all cases, waterlines are parabolic (i.e. parameter a₁=1.0); the beam for each class and displacement is the ICF minimum for that class.

Fig. 3.2(a): Total resistance coefficients for PER hulls.

Fig. 3.2(b): Total resistance coefficients for Optimal hulls.

Figure 3.2(a) shows the total resistance coefficient of the PER hulls as a function of boat speed in metres per second. Figure 3.2(b) is similar to figure 3.2(a) but for the "optimal" hulls. It is difficult to compare differences between the two sets of curves of figures 3.2(a) and 3.2(b) so we have included figure 3.2(c) which shows the improvement that could be expected if our optimal hulls were used in place of kayaks with PER hulls. This "improvement" is defined as
1 - R_opt/R_PER where R_opt is the total resistance of the optimal hull and R_PER is the total resistance of the corresponding PER hull.

Fig. 3.2(c): Expected relative improvement(%) of Optimal hulls over PER hulls.

It can be seen that for the K1 and K2 classes, and at speeds in excess of 3.5 ms^-1, there is almost nothing to be gained by using the optimal hullforms relative to PER hulls. For the K4 class kayaks, the advantage of using optimised hulls is much greater, and improvements of about 1% over the PER hulls can be expected for speeds in excess of about 5.0 ms^-1. The larger expected improvements at low speeds would, under normal racing conditions, only be evident at the start of the event.

The curve for the 0.100 tonne K1 kayak is interesting. For this hull there is a very small, almost constant improvement over the range of speeds considered. This hull has a rectangular sideview, the same as the PER hull, only the draft and the cross-section shape parameter are different.

For K4 kayaks at racing speeds, greater than 5.5 ms^-1 say, the expected improvement is about 1.0%. The low speed performance is up to 6% better than for the unoptimised PER hulls.