represents a zero-order approximation for the wave resistance. This is not good enough for our purposes, however.

An exact theory was given by Havelock in 1932. A simplified, but still quite general form of Havelock's equation has been found by Castles for hulls with lateral and longitudinal symmetry. A discussion of the physical basis of Havelock's theory and Castle's equations for single or multiple hulls is given in Appendix A. Castles' equation for a single hull is

where

" .00 x2e-aX2dx

The quantity 5 is the depth of the centroid of the maximum cross section for which a is the area. The prismatic coefficient CP is defined by

pHgaL

and therefore describes the distribution of cross sectional area along the length of the boat.

The wave resistance is more or less insensitive to the shape of the hull cross section. This is not true for the frictional resistance which depends directly upon the wetted surface area Aw. This quantity is given quite accurately ( + 1 percent) for a wide variety of forms by

The minimum girth for a given enclosed area is featured by the semicircular cross section for which Gm = \nB and Aw = 1.16BL. Since the wave resistance is shape independent, we are free to concentrate our interest on semi-circular sections. For this case a = 7iB2/8, 3 = B/n, and the total resistance can be written in terms of the Froude number F, the length-to-beam ratio L/B, and the prismatic coefficient CP as r - 1 48 f2il\c +1 lb w~cP U; ' 4<vU

-2 BF2X2

lgh speed One of the first numbers to be generated when planning a boat is sailing the displacement-length ratio DLR = A/(.01L)3 where A is the displacement in long tons (A = jy/2240). Using Eq. (2-19), we find for semicircular cross sections that

In formulating a design, one is always limited by material strength-to-weight ratios and other factors to some minimum DLR. From Eq. (2-22) we see that an infinite variety of values of CP and L/B correspond to a given DLR. The problem is to find the best set of values from the point of view of minimizing the total resistance. Both components of the resistance, friction and wave, increase with F2, however the wave term also contains an additional function of F2 that is

16 I*ig 2-3. Specific running rcsistancc as a function of L/B for V^JyjL - 2,

16 I*ig 2-3. Specific running rcsistancc as a function of L/B for V^JyjL - 2, zero for very small or very large value of F2 and reaches a maximum hulls and value for Fi * 0.281(7*^/1 » 1.78). It therefore makes good sense to outriggers compare the resistance of hulls for this value of Froude number or one only slightly higher. We have calculated R/W using Eq. (2-21) for a range of prismatic coefficients from 0.50 to 0.80 and length-to-beam ratio from 8 to 32 at a fixed value of F2 = 0.355 {VjjL = 2.0).

The results are shown in Fig. 2-3. We see that higher prismatic coefficients are better than lower ones and that hulls in the L/B range from 12-16 are maximally efficient. The waterline length L has been taken as 25 feet in calculating the friction, however, the overall result is a very weak function of size, hence this choice does not limit the generality of these results.

!gh speed Interesting though it is, Fig. 2-3 still does not answer the basic sailing question which is: for a given weight and length (DLR), what is the optimum value of prismatic coefficient in order that R (not R/W) be a minimum? This is answered by replotting the data as shown in Fig. 2-4. (The calculated values from which Figs. 2-3 and 2-4 were drawn are presented in tabular form in Appendix B.) The figure shows that high prismatics are best and, indeed implies that coefficients even higher than 0.80 are desirable. The choice of CP is seen to be much less critical for the lower DLR's than for higher ones. In order to properly evaluate the data of Fig. 2-4, we must remember that eddy and flow separation effects owing to projections or small local radius of hull curvature are ignored. For high CP, the hull tends to a scow shape (for which CP = 1.0) and flow discontinuities at the bow can be expected to arise. These effects will tend to increase the resistance on the high CP end of the curves. Thus it seems likely that the ideal value of CP for the low-DLR hulls in which we are interested will lie in the range 0.65-0.75.

The effect of some variations from our standard hull (see Appendix A) for which the above results are derived should be noted. The effect of moving the cross section of maximum area forward of the midship position is an increase in the total resistance at all speeds. By moving the maximum section somewhat aft of amidships an average reduction of RjW by about 5 percent for L/B = 12 between the SLR values 0.8 to 3.0 is possible. Outside this speed range the logitudinally symmetrical hull is superior. The sensitivity of R/W to the position of the maximum section decreases with increasing L/B and CP. For the high L/B, high CP hulls of interest to us, placement of the maximum section can be dictated to a large extent by design factors other than resistance minimization.

Another possible modification is to broaden and flatten the sections of the after half of the hull. The effect of this on R/W is similar in magnitude and dependence on DLR to that of moving the maximum section aft. As in that case, stern flattening is disadvantageous for SLR's less than 0.8 and greater than 3.0. This modification also has the effect of damping pitching motion. The mechanism of this damping effect is discussed in some detail in Chapter 6. The pitching motion of low-DLR hulls is already highly damped even without stern flattening, thus proas pay only a marginal penalty for their longitudinal symmetry.

In order to make specific recommendations concerning the various multihull configurations, we must establish a working definition. We shall always refer to the weight-carrying hull as the hull and the float (or ama) as the outrigger. As a general rule, outriggers should be as light as possible and should not be used for stores and certainly not for accommodation.

A catamaran is a configuration of two identical hulls, or, in the case ofassymetrical hulls, mirror images. In the daysailing sizes, catamarans are faster than trimarans owing to their ability to fly the windward hull and sail on one hull at a modest angle of heel. We shall discuss this question of hull flying further in Chapter 4. For catamarans to be sailed in relatively smooth waters, the hull sections should be serni-circular for most of the length with some flattening in the after third and a fairly rapid transition to elliptical, parabolic, and vee sections hulls and at the bow. For larger craft intended for offshore sailing, the drag outriggers that arises owing to the presence of ocean waves must be taken into account. This drag roughly doubles the effective resistance of a typical monohull racer sailing to windward in seas having a wavelength greater than the length of the boat. Rough water drag decreases with increasing

L/B and is more or less insensitive to the beam-to-draught ratio B/H.

This suggests that a practical optimum hull section for offshore use will correspond to B/H somewhat less than the semicircular value of 2.

In rough water, large portions of the windward hull will be exiting and entering the water at high speed. In order to avoid pounding, the semi-circular section should be distorted into a rounded vee or parabolic section.

The question of whether to make catamaran hulls symmetrical or asymmetrical can be argued both ways. The intended purpose of an asymmetrical hull is, usually, to create horizontal lift on the more highly curved side. In the case of a catamaran with more or less flat outer sides and curved inner sides, the horizontal lift of the two hulls only serves to compress the cross beams unless heeling occurs. This is shown in Fig. 2-5. Only daysailers are sailed at such angles of heel and then only for short times. Even more discouraging to the notion of using hull asymmetry to enhance the lifting action of a long shallow hull is the fact that only for large angles of attack (>10°) does asymmetry make a significant contribution. Such a high angle of attack between the centreline of the boat and the course line would create an intolerable drag and is therefore out of the question. Asymmetric hulls have, however, been found to be highly resistant to broaching

when running in heavy seas. This can be understood as shown in Fig. 2-6. When the boat, beginning to broach, reaches a yaw angle of 10 or so, the lift effect of the asymmetry in the leeward hull is strongly excited; the windward hull is at a negative angle of attack ^

GH SPEED and is not producing a significant lift. The lift of the leeward hull is SAILING accompanied by a large induced drag. The excess in drag of the leeward hull over the windward hull times the overall beam of the boat constitutes a torque to counter the broach. The superiority of asymmetrical hulls under these conditions is a matter of practical experience as well as theory. In making the decision of whether or not to use asymmetrical hulls, bear in mind that for a hull section having B/H =1.5, a reasonable amount of asymmetry will cost a 5 percent increase in the wetted surface area and thus in frictional resistance. Friction is the dominant component of hull resistance when sailing in light airs (where multihulls are at a natural disadvantage anyway) and at very high speeds (SLR > 2.8).

Fig. 2-6. Hull asymmetry as an anti-broaching feature.

The question of full, transom-type sterns or fine canoe-type sterns for catamarans where load carrying is not a major consideration can be settled in favour of the fine stern. At low speeds (SLR < 1.8) form drag acts against a full-sterned hull. The pressure of the water against the hull forward of the maximum section resulting in a resistive force is cancelled in a fine-sterned hull by the vector sum of the pressures aft of the maximum section except for a small amount that we lumped into the friction calculation [see Eq. (2-6)]. If the hull is terminated suddenly as is the case with full sterns, then this cancellation is not achieved. This is not the case in air where, for example, racing sports car bodies are found to give less resistance if the rear ends are chopped abruptly. At high speeds (SLR > 1.8) the difference between the resistance of full- and fine-sterned hulls is small with a slight advantage to the full stern. If we are considering an ocean racer, then we must take into account the fact that the sterns will often be buried in the seas that accompany high winds and fast sailing. Under these conditions, fine stcrncd hulls experience significantly less rough water drag.

20 Trimarans pose a different set of problems. Since all of the weight

is effectively carried by the central hull, this hull should have a semicircular section over most of its length. This section may be somewhat flattened toward the stern and should be sharpened toward the bow. Since the DLR of the trimaran hull will be roughly twice that of either hull of a catamaran of similar overall specifications, it makes sense to use a transom stern. The transom should be narrow, however, and should not extend below the load water line.

The design of trimaran outriggers and their positioning with respect to the hull require special discussion. There are two schools of thought on the question of whether to fit full-buoyancy outriggers, either one of which can support the full weight of the craft without being driven under, or submersible outriggers that heel easily within a larger range of stable angles and give a better indication of when the boat is being over-driven. This question was settled (for me, at least) by a rash of capsizes in 1976-77 involving tris with low-buoyancy outriggers. It seems that when lying ahull in bad conditions, a wave may heel the trimaran in such a way as to drive the lee outrigger under. This outrigger having a high resistance to lateral motion then acts as a fixed pivot axis about which the boat can be capsized. The choice of low or full buoyancy outriggers is therefore the choice between the increased possibility of a wave capsize and the increased possibility of sailing the boat over. I personally feel that the latter is more acceptable.

For high performance, the outriggers should have semicircular sections over the after 70 percent of their length going over into a sharpening spade section toward the bow. In designing the outrigger and hull bows we want a configuration that will pierce small waves with minimum retardation and rise to large waves in order to avoid burying the bows with the possible consequence of a diagonal or stern-over-bow capsize. These requirements call for reasonably fine bows with moderate overhang and sheer, but little flare except in the main hull. The outrigger bows can be fitted with lifting plates as shown in Fig. 2-7.

lig. 2-7. Lift plates and sheer as dive preventors for outriggers.

w speed sailing

In driving hard to windward, the deep running lee outrigger will generate a large resistance acting along a line to leeward of the driving force. The result is a torque that tends to yaw the boat to leeward (lee helm). This can be countered by designing the outrigger so that its centre of lateral resistance is 8-15 percent (depending on overall beam) ahead of the centre of lateral resistance of the hull. The keel action of the outrigger then acts along a line forward of the line of action of the centreboard and cancels the above-described lee helm. This is shown schematically in Fig. 2-8. The outriggers should be

Fig. 2-8. Balance of yawing torques in a trimaran sailing to windward.

mounted in such a way that both are clear of the water with the boat at rest under average load conditions. In this way the trimaran can sail on its central hull alone when running and thereby gain a distinct advantage in resistance over a similar catamaran. For windward work, the high positioning will allow a somewhat greater heel angle. This has the effect of putting the windward outrigger several feet out of the water where its round bottom will not often encounter a wave. When going to windward, the centreline of the hull lies at an angle X, the leeway angle, to the course line if the keel (centreboard, dagger board, leeboard, etc.) is laterally symmetrical. In this case drag can be reduced by toeing the outriggers out by an angle equal to the leeway angle experienced on a beam reach (Edwin Doran, Jr., AYRS 83 B, 18 (1976).) It is also advantageous to incline the vertical centreline of the outriggers outward at the bottom by an angle of not more than 15°. This has the effect of making the outrigger a smooth extension of the curved cross beams, thus reducing the stresses at that junction. As the boat heels the outrigger is brought into an upright position corresponding to minimum drag.

Fig. 2-8. Balance of yawing torques in a trimaran sailing to windward.

In order to prevent a rapid rise in outrigger drag with increasing hulls and immersion, the DLR of the fully pressed outrigger must be quite low. outriggers This means that the reserve volume must be contained in length rather than freeboard. The limitation of such a long needle-like outrigger is the strength-to-weight ratio of its construction. It is likely that the current (1977) practice of making outriggers about 80% as long as the hull is too conservative and that longer outriggers should be contemplated.

Proas are the least understood multihull type. The original Micro-nesian proa consisted of a lean asymmetric hull to leeward and a heavy log outrigger (counterbalance weight, really) to windward. This craft was sailed by a large and agile crew who arranged themselves to windward as needed to keep the log flying just clear of the water. The few modern adaptations of the proa that have been built in a size suitable for offshore sailing have been 'Atlantic' proas with the hull to windward and a submersible or low-buoyancy outrigger to leeward. The exception to this is Newick's Cheers, a schooner-rigged proa that featured equal hulls. Cheers was the only one of the lot to have enjoyed any racing success.

The best way to think of a proa in modern terms is to visualise a trimaran with the windward outrigger and cross beams sawn off. The Micronesian outrigger or counterweight becomes our hull and the Micronesian hull becomes our full-buoyancy outrigger. So far as the hull and outrigger shapes are concerned, they should resemble the forward half of the trimaran repeated on both ends.

Comparing the proa with a catamaran in terms of performance, we see that in the daysailing sizes, the concentration of crew weight to windward gives the catamaran all the advantage of the proa. In the larger size where mobile crew weight is not a factor, the proa retains (he advantage of permanent weight bias to windward. In comparison with the trimaran, the fact of not having to carry a windward outrigger and cross beams constitutes a big advantage in weight and windage. The weight saved can go into huskier and longer cross beams to put the centre of gravity further to windward. Clearly, in the oceangoing sizes, proas will be faster than either catamarans or trimarans on all courses. Catamarans may be faster than trimarans going to windward owing to a possible windage advantage. Trimarans will usually be faster than catamarans on a run or in light airs to the extent that outrigger drag can be minimized or eliminated. The difference in performance between these two types is much less than the performance advantage of the proa.

On the basis of Eq. (2-21) and the fact that rough water drag is proportional to WF2, we might suppose that multihulls having a sufficiently low DLR might obey a simplified drag equation such as

where a is approximately constant. This turns out to be true. In a paper MTHcntcd at the 1977 Royal Yachting Association Speed Sailing Symposium, Derek Kelsall reported that tank testing of a five-foot inmumn model and resistance calculations for several multihulls using 23

high speed the International Offshore Multihull Rule (IOMR) equations both sailing resulted in smooth parabolic curves of the form of Eq. (2-23) where the constant a varied from boat to boat over a range of 0.025-0.032. Notably, Kelsall sees no hump in the curves owing to wave drag as is seen in monohull data. This is apparently obscured by the rough water drag.

Now let us discuss the question of accommodation arrangement. The minimum requirement is a bunk for each crew member, a galley, head, a few shelves and storage lockers, and a place to sit in comfort for eating, navigating, or what-have-you. The facilities and arrangements required by individuals vary too much for detailed recommendations containing my own biases to be useful. Some general observations on accommodation where performance is the overriding consideration are in order, however.

The waterline beam of the hull must be kept small as we have seen; however the hull can be flared or stepped above the waterline. This allows bunks, lockers, shelves, and so on to be fitted in the narrow hull and still give room for movement without too much elbow friction.

In a catamaran, there will be a strong temptation to build accommodation space on the deck between the hulls, because of the narrowness of the individual hulls. This has the effect of raising the centre of gravity higher off the water and adding windage. As we shall see when we discuss structural problems, there is good reason to have some sort of thick connecting structure which can comprise a cockpit and enclosed space with seated headroom.

In the trimaran and proa, accommodation is restricted to the hull. The proa, needing longitudinal symmetry, will have a centre cockpit. In the trimaran, cockpit location is optional. Other than that, the accommodation space and layout of proa and trimaran may be similar.

Weight must be kept out of the ends of the hull or hulls in order to keep the moment of inertia about the pitching axis low. This will have the effect of limiting the amplitude of pitching motions and ensure that they are rapidly damped. This is vital in reducing rough water drag. Only the central half of the hull should be regarded as habitable. Human nature being what it is, any small spaces that you as a designer do not wish to have heavy stores put into can be filled with plastic foam. This will serve to absorb shock in case of damage, though foam is heavy in large volumes and should not be overdone. Do not regard standing headroom as a necessity in small yachts. I would not build a coach house structure at all but would continue a fair line from the beams straight across the hull. In the fore-and-aft direction, the sheer line of the hull should curve smoothly into this raised deck. Flat areas should be avoided everywhere. They are structurally, aero-dynamically, and aesthetically unsound. The trimaran Three Cheers designed by Dick Newick and shown in Fig. 2-9 is a good example of the type of continuous deck and outrigger beam structure recommended.

To close this chapter on hulls and outriggers, I would like to pass along some thoughts on drawing hull lines. This method is used 24 by a number of naval architects but docs not seem to have entered

the text books; I learned it from Newick who revealed it at the World Multihull Symposium in Toronto, Canada, 14-17 June, 1976.

Hull fairness is all important. Hollows or abrupt changes of hull curvature through deviations from fairness constitute sources of eddies and turbulence that can ruin a boat's performance.

One first draws the profile and load waterline onto the station lines (Fig. 2-10a). Next draw the plan view showing the sheer line and keel (Fig. 2-10b). Finally, draw the maximum cross section (Fig. 2-10c). The centreline and sheer intersect points for the cross sections can now be transferred from the profile and plan views to a body plan. The problem is now to draw the remaining cross sections such that the hull will everywhere be fair, without curvature changes or reversals over short distances. For a hull of fairly simple shape such as the proa hull (by Newick) shown here, a template can be constructed that includes the curve of the maximum section with a fair extension on either end. The master template for Newick's proa is shown in Fig. 2-10d). By keeping the x mark on the template along the reference line AA' and either set of intersect points on the template curve, all cross sections will change proportionally and the lines will be fair. The problem is therefore reduced from one of constructing the sections by experienced eyeball to one of finding one suitable reference line. It will usually be a straight line as is the case here, although it can also be a smooth curve. 25

igh speed sailing

Fig. 2-10. Line drawing technique illustrated by Newick's PROa.

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