4. RESULTS

4.1 Method of presentation

C_t=R_t/(1/2*rho*U²L^*2)

which would be a function of F_nv 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 three fixed (dimensional) displacements of one, one hundred, and ten thousand tonnes. This large range of displacements means that in some cases the speeds are not realistic, but results are nevertheless provided for completeness in such cases. 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 F_nv=1 occurs at 6.1 knots for a one-tonne vessel.

It is important to bear in mind that none of the Figures 2-4 to follow, where the total drag coefficient C_t is plotted against the volumetric Froude number F_nv, can be interpreted in the usual naval architectural manner as a graph of drag versus speed for a given ship. As F_nv varies, the ship itself changes its shape, and in particular its length, so as to keep the drag as small as possible.

4.2 Monohull without form drag

Fig. 2(a): The effect of form factor on the optimum total resistance of a one tonne monohull.

The dashed curve of Fig. 2(a) shows C_t as a function of volumetric Froude number F_nv, for a one-tonne "ship". This is the residual value of the total drag, after the ship's dimensions have been optimised to minimise C_t without any allowance being made for form drag. The hull parameters that produce these optimal C_t'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 F_nv. This could be due to long shallow regions in the "fitness" landscape, where for example, one length is as good as another. Although C_t remains the same, C_w/C_t may vary. GODZILLA searches for the lowest total resistance and if it encounters two or more combinations of parameters with almost the same C_t, 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 F_nv<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 F_nv>3.25, a rather high speed (of the order of 20 knots for a 1-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 F_nv<3.25, the longer ship is best; for F_nv>3.25 the shorter ship is best, and in the present case, the shorter ship is so short as to be quite unrealistic. Indeed, this "ship" 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" ship 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 monohull.

Returning to the "realistic" ships produced for lower speeds, with F_nv<3.25, as the "design speed" increases from zero in that range, the optimum length L/L^* (same as length in metres for a one-tonne ship) shown in Figure 2(c) increases to a maximum of about 22 at a F_nv 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 ships are then preferred as the speed rises.

This trend cannot continue for ever. Eventually, the optimal shiplength 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 shiplength. Eventually as the speed increases further, the optimal shiplength 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 ships 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 of a 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 F_nv between 1.0 and 2.5.

Fig. 2(e): The effect of form factor on the optimal length-to-beam ratio of a one tonne monohull.

The optimum ships are very slender. Figure 2(e) shows the length/beam ratio, which is very high indeed (reaching a maximum of about 42 at F_nv=1.8) by conventional ship standards, though not entirely unreasonable for rowing shells.

4.3 Monohull with form drag

Figure 2(a) indicates that there is only a quite small increase in the residual total drag C_t 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 F_nv. The solid-line C_t curves of Figure 2(a) are smoother at low F_nv than the dashed curves, and the ultimate decrease in C_t at high F_nv 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 F_nv 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 F_nv>3.25. We have already anticipated this, since the very short ships 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 ship.

Figure 2(c) confirms this point, indicating that the optimum ship 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 ships. 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.

4.4 Variation in Displacement

Fig. 3(a): The effect of displacement on the optimal total resistance of a monohull.

Fig. 3(b): The effect of displacement on the optimal proportion of wave resistance of a monohull.

Fig. 3(c): The effect of displacement on the optimal length of a monohull.

Fig. 3(d): The effect of displacement on the optimal beam-to-draft ratio of a monohull.

Fig. 3(e): The effect of displacement on the optimal length-to-beam ratio of a monohull.

In Figures 3(a)-(e) and 4(a)-(c), the blue, green and red curves correspond to displacements of 1, 100, and 10,000 tonnes respectively.

Figures 3(a)-(e) indicate variation with displacement of the same quantities that were discussed earlier for the one-tonne ship. 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 F_nv=1 is 6.1 knots for a 1-tonne vessel, 13.1 knots for a 100-tonne vessel, and 28.3 knots for a 10,000-tonne vessel.

Smaller ships have larger C_t 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 ships, once we multiply C_t by 1/2*rho*U*D^2/3.

In the most important middle range of speeds, the dependence of the results on displacement is quite smooth and predictable by interpolation within the curves presented here.

4.5 Catamarans

From a survey of modern high speed catamaran dimensions, Insel and Molland (1991) concluded that the general range of parameters was: L/B=6 to 12, L/L^*=6 to 9, B/T=1.0 to 3.0 and C_b=0.33 to 0.45. Our optimum hulls are very much longer; however C_b and B/T are within the above range.

Fig. 4(a): The effect of displacement on the optimal total resistance of a catamaran.

The C_t curves in Figure 4(a) are similar in general character to those for the monohull in Figure 3(a). It is obvious that there is no speed or displacement at which a catamaran has lower total resistance than an optimum monohull of the same displacement. The proportion of wave resistance for optimum catamarans is generally similar to that for optimum monohulls as given in Figure 3(b), and is always less than 10%.

Fig. 4(b): The effect of displacement on the optimal length of a catamaran.

A comparison of the optimum length results in Figures 4(b) and 3(c) shows that at low speeds, catamarans have an optimal length roughly the same as the equivalent monohulls. At higher speeds around F_nv=1.6 (namely 10 knots for a one-tonne vessel), where the optimal length of a catamaran reaches its maximum a little earlier than a monohull, optimal catamarans tend to be roughly 25% shorter than the optimal monohull of the same displacement. Length-to-beam ratios for the demihulls of optimal catamarans are similar to that for monohulls (Figure 2(e)) at all speeds. For example, each demihull beam is also about 25% less than that of the full equivalent monohull at about F_nv=1.6 when L/B takes its maximum value of about 41. That is, each demihull of an optimum catamaran is approximately as slender as the optimum monohull, and is much more slender than conventional catamaran hulls. Beam-to-draft ratios are also similar to those of monohulls, and nearly semi-circular sections are preferred.

Fig. 4(c): The effect of displacement on the optimal width-to-length ratio of a catamaran.

Figure 4(c) shows the optimum hull separation W/L. It can be seen that for F_nv between 0.2 and 1.1 the optimum separation is roughly 20%-30% of the length of the catamaran. For F_nv between 1.1 and 2.2 there is no optimum finite spacing. To all intents and purposes, if one only wishes to minimise total drag, there is no reason why the two hulls need to be close to each other, because they cannot favourably interfere with each other to reduce wave resistance in that speed range. Indeed, they must interfere with each other unfavourably, and our conclusion is that the further they are apart the better from the drag point of view. This speed range is one of considerable practical interest, and it is one in which the decision on choice of hull separation distance must be made on grounds other than drag minimisation. The existence of a speed range where there is no best separation distance is in rough accord with the results of Turner and Taplin (1968), wherein it was pointed out that this conclusion tends toward catamarans that are impractically wide for all but sailing boats. At still higher speeds, there again seems to be a band of F_nv (say between 2.3 and 3.4) where there is an optimum finite hull separation, again of about 20% of the length. For F_nv>3.4, again there is no best separation distance.

Insel and Molland (1991) comment on some aspects of this phenomenon, stating that "The wave interference can effectively be neglected above a particular speed which is both separation and L/B dependent. This is an interesting and important result since it suggests that, for higher speed designs, the choice of hull spacing may be based on other requirements such as seakeeping performance without incurring significant penalties in calm water resistance." However, they do not seem to have observed the second range of speeds where destructive interference again becomes useful.

4.6 Trimarans

In order to reduce the amount of output data that has to be presented for trimarans, we select a relatively small set of speeds (displacement-based Froude numbers of 1.3,1.7 and 2.1), and plot results at each fixed speed versus the above-defined displacement ratio sigma. The speeds chosen are in the "interesting" range, namely speeds above those where discontinuities occur, but below those where planing and other presently-neglected flow phenomena might be important. This range (say 8-10-13 knots for a 1-tonne vessel) is also the competition range for some sporting applications.

Plotting trimaran results versus the displacement ratio sigma has the feature that the monohull results are reproduced when sigma=0 and the catamaran results when sigma=1, which is a useful check. The trimaran thus interpolates between monohulls and catamarans, for 0<sigma<1.

Fig. 6(a): Optimum total resistance of one tonne generalised trimarans.

Figure 6(a) gives the residual total drag as a function of sigma and shows that trimarans are never competitive with the best monohull. As sigma increases from the monohull value of 0, the drag rises to a maximum at about sigma=0.8 before falling again toward the catamaran limit at sigma=1, which (as already discussed) is inferior to the monohull, and also to any trimaran with sigma less than about 0.2.

Fig. 6(b): Optimum lengths of one tonne generalised trimarans.

Figure 6(b) gives the optimum lengths of the main and wing hulls. The former decreases steadily and the latter increases steadily with increasing sigma. The three hulls are all of the same length at about sigma=0.6.

Fig. 6(c): Optimum lateral separation of one tonne generalised trimarans.

Fig. 6(d): Optimum longitudinal separation of one tonne generalised trimarans.

Figures 6(c) and 6(d) give the optimum lateral and longitudinal offsets of the wing hulls. The lateral offset b would approach W/2 in the catamaran limit sigma=1, where W is as before the lateral separation of the catamaran demihulls. However, in the speed range being examined, the optimum value of W for a catamaran is actually infinite. Thus as sigma increases from the monohull value of zero, the lateral offset remains relatively small at less than 4 metres until sigma exceeds about 0.8, then rises rapidly as the trimaran turns into a catamaran.

The optimum longitudinal offset a given in Figure 6(d) displays a somewhat complicated variation with sigma. The limiting value at the monohull end sigma=0 seems to be about one-half of the main hull length, and since the wing hull lengths are tending to zero in this limit with zero lateral separation, the wing hulls simply "tuck in" at the stern of the main hull. At the other extreme, in the catamaran limit sigma=1, there is again a tendency for a to increase rapidly but of course the "main" hull then becomes a hydrodynamically insignificant "dagger board" far ahead of the dominating wing hulls.

The isosceles triangle formed by the centres of the three hulls of the trimaran has a half-angle that is quite small for near-monohull cases with sigma<0.3, but tends to range between 10 and 15 degrees for larger sigma. This is consistent with estimates of the optimum half angle for minimum wave resistance (references?), noting that it implies that the wing hulls lie just inside the Kelvin angle of the wave pattern of the main hull.

Fig. 6(e): Optimum overall length of one tonne generalised trimarans.

Figure 6(e) gives the overall length of the trimaran. This is the sum of the main and wing hull half-lengths, plus the longitudinal separation a. For near-monohulls, because the wing hulls are short and the longitudinal separation a is less than half the main hull length, the overall length is close to that of the monohull, decreasing slightly as sigma increases from zero. For the lower speeds, the overall length reaches a minimum of about 10% less than the monohull length at about sigma=0.5-0.6; there is a more complicated variation at the higher speed. The overall length becomes large as the trimaran approaches a catamaran, but loses meaning as the "main" hull becomes of vanishing size and significance relative to the wing hulls.