The tendency to view hydrofoils only as a means to lift the hull or hulls clear of the sea is conditioned by the fact that hydrofoils have been applied mainly to engine-powered craft which are subject to a less complex set of forces than sailing craft. In Fig. 6-1, to obtain a quantitative feeling for the problem, we have plotted the vertical lift-to-drag ratio (weight-to-resistance) of a typical multihull craft as a function of VJ^/L using Eq. (1-8). Also shown is a curve labelled hydrofoils, which is characteristic of a set of deeply immersed hydrofoils together with the necessary struts. From this figure we see that the buoyancy/resistance figure for a good hull is greater than the lift/drag value of hydrofoils up to a value of VB/y/L * 2.5. The attainment of such high speeds requires that the boat be able to counter a large heeling moment and still keep the sail as near vertical as possible. It is in this regard that the primary application of hydrofoils to sailing craft arises. Although foil stabilization was used in the outrigger craft of Madagascar and Dar es Salaam, Edmond Bruce was the first to formulate the physics correctly.
In Fig. 6-2 we show a multihull craft equipped with a canted hydrofoil to leeward. In the (a) part of this figure the craft is at rest and the only forces in effect are the weight W which is opposed by the buoyancy B operating along the same vertical line. In the (b) part of the figure, the boat is in motion at a constant speed and a state of dynamic equilibrium exists. The basis for leaving the buoyancy in the same vertical line as the weight in this case is the assumption that we will be successful in eliminating the heeling moment. Were this assumption not justified, then the centre of buoyancy would move to leeward as the boat heels.
In order to enjoy a state of equilibrium, an extended body must experience no net forces in the vertical or horizontal directions. Also, the moment of the forces (torque) about any point must vanish. For the present purpose we can neglect the forces normal to the page, that is, the driving force and the drag. The vanishing of the forces implies
Fig. 6-1. A comparison of the lift/drag characteristics of fine hulls and hydrofoils.
Multiplying Eq. (6-1) by sin 9 and Eq. (6-2) by cos 0 (where 0 is the dihedral angle of the foil) and adding, we find
By virtue of Eq. (6-2), we see that the vertical component of the hydrofoil force is f cos 0 - Fy ctn 0.
Fig. 6-2. A Bruce foil-equipped sailing craft in static and dynamic equilibrium.
Fig. 6-2. A Bruce foil-equipped sailing craft in static and dynamic equilibrium.
Using Eq. (6-3), we can now express the buoyancy as
Thus our hydrofoil sailing craft in dynamic equilibrium can be reduced to the force diagram shown in Fig. 6-3. By taking moments about any point on this figure and setting them equal to zero, we are led to the following key relation:
Since Fy is never zero except in the trivial static case, the quantity in parentheses must vanish in order to ensure the vanishing of the heeling torque. Thus y = h tan 0. (6-7)
Having found the dimensional relationship that causes the heeling torque to vanish, we must now ask ourselves the following question. How large can Fy be (or, equivalently, how much sail can be carried in a given wind) in order that Eq. (6-6) still be satisfied? The answer to this question is contained in Eq. (6-5). This equation describes the decrease of B, the buoyancy, as Fy increases, thus increasing the vertical component of the foil force. As F approaches a value W tan 0, the 62 buoyancy approaches zero as the nulls lift out. At the point of liftoff, y
Fw ctn 6
Fig. 6-3. Equivalent force diagram for sub-lift-off speeds.
W tan 0
Fig. 6-4. Equivalent force diagram at lift-off.
Fig. 6-5. A Bruce foil craft with windward stabilization.
high speed the force diagram tends to that shown in Fig. 6-4. The force components sailing Fy ctn 6 of the righting couple have reached a maximum value W; the corresponding righting moment is therefore
and any further increase in Fy over the value W tan 9 will lead to capsize. If we were to replace the foil unit by a light nonsubmersible hull, the same value of maximum righting moment obtains. The virtue of the leeward canted foil arrangement, known as a Bruce foil, is that 1) no heeling at all is experienced up to the point of liftoff, thus keeping the driving effort of the sails at a maximum, and 2) the heeling force Fh has been converted into vertical lift thereby reducing the drag of the hulls.
It is also possible to cancel the heeling torque by using a canted foil to windward that depresses rather than lifts the boat as shown in Fig. 6-5. One may also derive Eq. (6-6) for this case. There are two outstanding disadvantages to this arrangement, however. First, the hulls are being depressed and so the hull drag is a rapidly increasing function of Fy. Second, and even more important, the windward foil arrangement is unstable. If a leeward foil pops out of the water owing to wave action, then the heeling torque acts to quickly reimmerse the foil. If a windward foil comes unstuck, the heeling torque is then uncountered by a windward depression and the craft will probably capsize very quickly. For this reason, a leeward Bruce foil or foils having a dihedral angle 9 such that y = h tan 0, is the best choice for vertical lift and lateral roll stabilization. In choosing the dihedral angle 0, we find that angles in the range 40-45 degs are probably best. If the angle is very much greater than 45 degs, then the beam necessary to satisfy Eq. (6-7) becomes excessive. If 9 is too small, the foil area or leeway angle must be large in order to generate a horizontal force component equal to Fy.
Now let us see what type of hydrofoils are suitable for use on sailing craft. Hydrofoils can be separated into two classes depending on whether the blades are wholly immersed by struts or whether the foils themselves pierce the surface. In order for the boat to have vertical stability, the hydrofoils must somehow manage to "see" the air-water interface and thus be able to respond to a vertical displacement perturbation in such a way as to rapidly restore the original flight altitude. Fully immersed foils can only do this by operating very near the surface where lift is a sensitive function of depth, or by having a surface sensor that transmits orders to the hydrodoil for required changes in angle of attack. The first method, employed extensively by the Russians in their large powered river craft, is useless in any sort of sea. The second method has been explored by Hook and others. At the present stage of development, this feedback method is, in the author's estimation, too heavy and complicated to be appealing.
The outlook for surface-piercing hydrofoils is much better. To a first approximation, the lift exerted by a hydrofoil at a given speed varies linearly with its depth of immersion and with the angle of attack of the water llow onto the foil a. In Fig. 6-6a we see a hydrofoil 64 experiencing a llow with angle of attack a. If a downward displacement
occurs, the hydrofoil experiences an additional component of fluid velocity from below. This is interpreted as an increase in the angle of attack as shown in Fig. 6-6b. If the displacement is upward, the effect is a reduction of a and the foil tends to loose lift in opposition to the perturbation (Fig. 6-6c.) The effect is entirely analogous to that described in Chapter 5 for yawing sails and keels. Thus we see that the hydrofoil resists vertical displacements with a force proportional both to the displacement and the rate of displacement. This tendency to resist changes in the vertical direction is powerfully augmented in the case of surface-piercing foils by the automatic variation of the foil area; that is, if a perturbation depresses the foil below its equilibrium water-line, the action of the increased area tends to restore the equilibrium while the change of apparent angle of attack tends to damp the motion and prevent overshoot.
Vee foil Ladder foil
Fig. 6-7. Monoplanar and multiplanar surface-piercing hydrofoils.
Surface-piercing hydrofoils may be mono- or multi-planar as shown in Fig. 6-7. For smooth-water sailing the monoplanar foil works quite well, however for offshore sailing the ladder foil should be preferred. The large reserve of unimmersed foil in the ladder arrangement can exert high lift as the foil enters a wave. The monoplanar foil is generally used in powered vessels designed to operate foil-borne only over a narrow range of speeds. In sailing, our source of power and 65
consequent speed varies over a wide range. In a multiplanar ladder, the foil section can be varied from a high lift, low section near the static waterline to a section more appropriate to high speeds at the bottom of the ladder.
Now let us examine the effect of wave action upon hydrofoils. In Fig. 6-8 we have indicated the motions of individual elements of water
Fig. 6-8. Water particle motion under wave action.
as a wave passes through from left to right. These orbits are circular with a diameter equal to the wave height at the surface and tend to a shuffling back and forth at greater depth. Now consider a hydrofoil-borne craft sailing to weather, that is, moving against the wave motion. Since the water on the front of the wave is rising, the foil sees this as an increase in angle of attack and the lift is increased. The hydrofoil therefore tends to climb the wave rather than hold a constant altitude. On the back side of the wave, the water is falling. This is seen as a decrease in angle of attack, lift decreases, and the boat tends to contour the back side of the wave as well In a following sea, the situation is more serious. If a foil-borne craft moves to leeward fast enough to overtake the waves, then it will tend to plow into the back side as the apparent angle of attack of the foil decreases. It is for just this situation that the ladder foil arrangement realizes its maximum advantage.
66 Tig. 6-9. An antiventilation fcncc.
The pressure on the curved surface of a hydrofoil is lower than that hydrofoil on the flat side. In surface-piercing operation, a portion of the curved applica-surface of the hydrofoil near the air-water surface may experience a tions pressure below atmospheric. This leads to the formation of a cavity and the entry of air from the surface, thus resulting in a loss of lift. This phenomenon is known as ventilation. It is controlled by the use of fences as shown in Fig. 6-9. It is also helpful if the foil or strut can be angled forward in order that the surface water encountering the member is given a velocity up the strut. This technique was used by Keiper on the trans-Pacific foil trimaran Williwaw with considerable success.
At higher speeds for a given chord length, the pressure at some point on the hydrofoil finally falls below the vapour pressure of the surrounding water. Bubbles then form as local boiling commences. These bubbles move aft along the foil surface and as the pressure rises, the bubbles collapse. The impulsive pressures on the foil surface as these cavitation bubbles collapse are several thousand pounds per square inch and pitting of the foil surface usually results. The speed at which cavitation begins lies in the 40 knots plus range for most hydrofoils and so does not concern us other than for a large sailing speed record machine. Such an ultimate sailing craft might well be equipped with a ladder foil arrangement having a supercavitating foil as its bottom rung. Such a section is shown in Fig. 6-10.
We would now like to address ourselves to the question of the configuration into which the hydrofoils should be arranged on a sailing boat. It is immediately evident that the hydrofoil array must have considerable extent in both the transverse and longitudinal directions, hence the buoyancy for sub-foiling conditions must be provided by an arrangement of multiple hulls, that is, a catamaran, trimaran, or proa.
The hydrofoil configuration will be symmetrical about the longitudinal centreline when applied to a symmetrical hull layout such as the catamaran or trimaran. The two simplest configurations in such a case are the aeroplane and canard. These two configurations are shown in Figs. 6-11 and 6-12. The aeroplane configuration features the Bruce foils in a forward location, carrying most of the weight, and the steerable stern foil serves as a pitch stabilizer. The canard configuration has its lightly loaded pitch stabilizer in the bow and the Bruce foils near the stern. The question of which hull arrangement is best with which foil arrangement may have to be decided by the question of accommodation or crew placement, however, for offshore work a trimaran-cunurd is likely to be best. 67
As we have previously noted, a hydrofoil unit is analogous to a damped spring by virtue of the dependence of its lift on the depth of immersion and rate of immersion. The stiffness of the spring is proportional to the rate of change of lift with depth of immersion (small chord = stiff foil) and the damping rate is proportional to the rate of change of lift with angle of attack since vertical velocity of the unit results in a proportional change in angle of attack. A foil of large chord operated at a low angle of attack has such a high damping rate. If the bow and stern foils have identical characteristics or if the stern foil is stiffer, a pitching perturbation can induce a porpoising type of instability. In conditions where one is running into a following 68 sea, a highly damped stern foil and a stiff bow foil provide excellent
pitch control. In practical terms, this calls for a lightly loaded bow foil operated at a high angle of attack and a stern foil carrying about £5 percent of the weight operated at the angle of attack corresponding to maximum In a hull-borne craft, these characteristics are obtained by using a fine bow with generous flare above the waterline and a broad, flat run off at the stern. Thus the canard configuration should be expected to be far superior to the aeroplane configuration in pitch control. In lateral roll control there is not much to choose between the two. If the two main foils are both canted lifters (the leeward Bruce foil configuration) then the angle of leeway will tend to increase the angle of attack of the lee foil and decrease the angle of attack of the weather foil. In order to operate on either tack in satisfaction of Eq. (6-7), the beam would have to be approximately twice the height of the centre of effort which is unlikely. Thus a laterally symmetrical hydrofoil craft will experience some heeling to leeward which also serves to nullify the lift of the windward foil by lifting it partially or wholly out of the water.
Now let us examine the laterally asymmetric proa foil configuration. This arrangement has a decided advantage in heeling control since, using a sail plan of modest overall aspect ratio such as the pyramid rig, the condition for heeling cancelation can be met. In order to gain pitch control, it is necessary to split the Bruce foil into two units located at either end of a long, slim leeward hull as shown in Fig. 6-13. The bow unit can then be operated at a slightly higher angle of attack than the stern unit in order to provide the necessary longitudinal distribution of stiffness and damping.
At this point I would like to address the question of extending the righting moment of an ideal Bruce foil proa beyond the value given by Eq. (6-8). It is obvious that this can only be done by adding weight to the windward hull or by using a windward hydrofoil unit that exerts a downward force. Both of these suggestions have advantages and disadvantages.
high speed Harry Stover has suggested that a water scoop might be used to sailing increase the weight of the windward hull as boat speed and heeling moment increase. The main problems with this suggestion would seem to be the large drag induced by such a scoop and the inability to regain buoyancy in the windward hull on short notice. The windward depressing foil also must pay the penalty of increased drag owing to the fact that the Bruce foils have not only to lift the weight of the boat, but also the windward foil force K. For this case, assuming the foil units to operate at a lift/drag ratio of 10, it can be shown that the effective horizontal lift/drag ratio in terms of which the drag angle 3h is defined is
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