Section 5 The steering and balance of a boat
There is no doubt that a boat can follow a circular path. When it does there must be an inward force that is equal to the product of the mass of the boat and its centripetal acceleration. The centripetal acceleration is the consequence not of a change in the speed of the boat through the water but of having a continual change in direction at constant speed. Figure 55 sorts this out. The boat is following a circular course at constant tangential speed. During the time that the boat takes to move from A to B it changes its direction and therefore its velocity. A velocity triangle can be drawn for the two positions of the boat A and B. The velocity OB at B can be thought of as being equal to OA, the velocity at A, plus the velocity AB added vectorially. If AB is divided by the time taken to go from A to B the result is the centripetal acceleration that is known to be either or where is the angular velocity and is the tangential velocity. Then the inward force is or where is the mass of the boat.
So what makes the boat follow a circular path? Obviously the rudder is the agent but it produces an outward force when we need an inward force so how does it work? Let me start by letting the boat be driven by an airscrew so that the rudder action is separated from the drive.
Suppose that the boat is following a straight course under its airscrew drive. Now let the rudder be turned. The obvious result is that the leading face of the rudder throws up water and there can be no doubt that a force is exerted on the rudder by the water flowing over it. That force acts in the opposite direction to the intended direction of turning, ie it acts outwards. As a result the boat turns about some vertical axis near to amidships.
So the boat continues to make way and turns at the same time. This gives us the mechanism of turning, the airscrew continues to drive the boat and the rudder makes the boat turn continuously about some axis. This means that the boat will follow a curved path and there must then be an acceleration towards the centre of the turn just as is shown in figure 55. We have seen in the first paragraph that a force acting towards the centre is required to maintain the acceleration. This force can only come about if the boat makes leeway ie it moves sideways as well as forwards. This alters the flow pattern round the hull and the boat follows a curve that depends on the magnitude of the force that is generated.
It follows from the above that the inward force on the hull can be thought of as acting at some point on the hull. For most hulls this point is about 40% of the waterline length from the stem.
I have explained this in detail in an article on this website called "How to locate the fin and mast of a model yacht." It is the same as for all yachts and I will condense it for this text.
In figure 53 it is clear that the force on the single sail varies from directly across the hull and well behind the mast to in line with the hull and way out to the right. In these cases and for every other point of sailing there will be a moment tending to turn the boat into wind and this moment must be resisted somehow. It must be resisted by another moment in the contrary direction. One might use the rudder but, if this is used to balance the moment, the drag caused by the rudder will slow the boat and when the rudder is called on to steer as well it may just give up and stall. Somehow the sailing rig must be designed and fitted to the hull so that the use of the rudder is minimised. This is called balancing the boat.
The moment that is exerted on the boat is caused by two forces acting in opposite directions and not in line. The forces are the net force on the rig and the transverse force produced by the hull as it moves with leeway. What would seem to be desirable is for the rig to be located on the hull so that, wherever the sails of the rig may be set, the force on the rig just equals the force exerted by the water on the hull and that the two forces are in line when seen from above. In practice the best that we can do is to arrange to have a small moment tending to turn the boat to windward when it is beating.
Let me set out the problem. At the design stage the layout of the rig can be chosen to suit the application and the need to balance the boat has to be included in the process. Once the layout has been settled the rig has to be located, fore and aft, on the deck and getting this wrong could be expensive. So we need a way to locate the rig.
We start by noting that, if a sail is properly set, its centre of effort is not far away from its centre of area and that we can locate a centre of area if we try. At the design stage it is normal to draw a sail plan of the rig with the sails drawn flat. It is a simple calculation to find the centre of area of the whole sail plan. It is the only point that we can find in a simple way. The side elevation of the hull and its water line will also be available at the design stage and it is known that hulls behave as if the transverse force created by leeway acts at about 40% of the water line length back from the stem. If the appendages to limit leeway are fitted so that their centres of area are also at the 40% point we can decide our strategy. We now have a fair idea of the position of the centre of area of the rig and a position for the transverse force on the hull. The point of sailing for which the sails are nearest to the arrangement in the sail plan is when it is beating when we would like the boat to tend to turn to windward. In fact, if one recognises that the centre of area is likely to be aft of the centre of effort, a good starting point is to have the centre of area of the rig a short distance aft of the 40% point. This strategy seems to work quite well and despite the fact that the sails are never used as depicted in the sail plan the balance is much better than eyeballing can achieve.
Leeway and the fin and rudder
There seems to have been no technology transfer from aerodynamics to sailing to solve the problem of finding a way of resisting the transverse force generated by a sailing rig with a minimum of penalty in drag. I found an author who claimed that figure 56 showed the evolutionary sequence from the deep-hulled Brixham trawler with integral rudder through blended keels with integral rudder to a separate fin and rudder. When I was thinking about this some time ago I altered the diagram to replace the blended keel with a fin with inset rudder to give figure 36. It is then obvious that the keel in the middle was well placed to balance the boat but not for steering it and that the movement aft improved the steering but upset the balance of the boat. It was a dead end and led to the final change which was to separate these two requirements and fit a rigid keel to balance the boat and a rudder to control the keel and to steer.
That keel is the same thing as an aerofoil and we can get a good idea of the performance of underwater keels from aerofoil data. Quite clearly the keel must have a symmetrical section but it cannot be any old shape because we want to generate the necessary transverse force with a minimum drag. The only shape that will do is a symmetrical aerofoil shape. I think that the best all round section is NACA 0012-64 because it stalls so gently. If a keel works between the hull at the top and a streamlined bulb at the bottom the keel effectively works between endplates and the lift and drag data for two-dimensional flow is valid for investigating the performance. The first thing we need is a graph of transverse force per square foot of area of a practical keel for different angles of attack ie of leeway.
In figure 58 there are five graphs of the transverse force per square foot of keel for angles of attack of 2°, 4°, 6°, 8° and 10° for a range of boat speeds that would be possible in a 50 foot boat. In 15 knots of wind a sail area of 300 square feet will generate a maximum transverse force of about 250 pounds. So a keel of say 7.5 square feet would reduce the leeway to an undetectable 3° without any contribution from the hull.
This has consequences. At this small angle of leeway a keel of modest area produces a much greater transverse force than the hull. This means that the hull can be designed solely for forward motion and that the balance of the boat can be struck between the rig and the keel and the position of the rig/keel combination on the hull is not then so critical.
Combine this with a lift/drag ratio of 20 at the least and the hydrodynamic keel looks to be an excellent device to limit leeway.
However hydrodynamic keels do not work by themselves, a stabilising surface is needed. We need to look towards light aeroplanes.
The wing/stabiliser system
The hydrodynamic keel is clearly a small wing working under water. Aeroplane wings do not work unaided but as one element in a wing/stabiliser combination. This is the arrangement used almost universally on aeroplanes. In the same way the keel works with the rudder. This means that we can look at the information available in aeronautics for an explanation of the mode of operation of the wing/stabiliser system and then use as it a guide to the operation of a fin and rudder. However we must consider the case of a wing and stabiliser with symmetrical sections because the fin and the rudder have to produce a force in both directions and so must be symmetrical.
Figure 59 shows a model of a wing and stabiliser in the working section of a wind tunnel. The wing is of parallel chord and is pivoted at its tips in the sides of the tunnel. A rod, acting as a fuselage, is fixed to the wing to support the stabiliser at one end and a balance weight at the other. Provision is made for the adjustment of the angle that the stabiliser makes with the axis of the rod.
Let us start with the stabiliser aligned with the rod. When the tunnel is running neither of the two surfaces experiences a vertical force although both are subject to a skin drag. Our interest starts when the stabiliser is set at a small angle of just a few degrees to the rod.
The immediate effect is to produce a force on the stabiliser that deflects the stabiliser downward and tilts the wing to give it an angle of attack. This movement brings two forces and two moments into existence. The aerodynamic force on the wing is exerted directly on the pivots. As we have seen this force is usually regarded as the combination of a lift and a drag and it is inclined towards the trailing edge. As we have also seen, a moment about the pivot will also be exerted on the wing. These will both change with the angle of attack. A similar force and a moment of smaller magnitude will be exerted on the stabiliser. The final position adopted by the model is shown in Figure 60. In this position the wing and stabiliser are in equilibrium with the stabiliser providing an upward force to balance out the combined moments on the wing and on itself and to do this it must make an angle of attack (probably smaller) in the same direction as that of the wing. The net force exerted by the two aerodynamic surfaces will be exerted on the pivots. It must be evident that this system permits the control of the angle of attack of the wing, and therefore the force exerted on it, by adjusting the angle that the stabiliser makes with the fuselage.
Before we can carry this over to a yacht we must see what happens to the wing/stabiliser combination when another force is applied to this model at some point other than the pivot. Two suitable points A and B are indicated in the wind tunnel diagram. Let us start with a force applied at point A. The final position is shown in Figure 61. The wing is shown in the same position as before but now the stabiliser has to provide a much greater upward force to balance the combined moments on the wing, its own moment, and the moment about the pivot caused by the added force at A. The angle of attack of the stabiliser must be increased. In Figure 61 it is shown with just about the maximum angle it can have without stalling.
If now we consider the case of a force exerted downwards at B the outcome is shown in Figure 62 where the wing is again in the same position. Here the force required at the stabiliser acts downwards mainly to balance the force at B. The stabiliser now has to have a “negative” angle of attack and, in the equilibrium position, the force and moment it generates balances the force and moment on the wing and the moment exerted about the pivots by the force at B.
We need to know what happens when the point of application of the added force is too far from the pivots. Suppose that the position of A were to be moved backwards progressively. The angle of attack of the stabiliser would have to be increased and eventually the angle would exceed the stalling angle. Then the front of the wing would rise and the system would enter a state from which recovery is impossible. In the case of B, the progressive movement forwards would require an increasingly negative angle of attack until the stabiliser stalls and again there would be no recovery.
Clearly, if we choose to use aerofoil sections for the keel and rudder, they will operate in just the same way as a wing and stabiliser even if they have different names. Then the behaviour of the wing and stabiliser derived from the model in a wind tunnel is directly transferable to the same system working in water. The first thing to recognise is that the rudder will work best if it does not have to cope with forces like those at A and B for the wing and stabiliser, that is, with forces that arise because the boat is not balanced. We want the forces on our yacht to be as well balanced as we can get them so that the rudder can be used for simultaneously controlling the angle of leeway, and hence the transverse force produced, and the course. This means that if the rig and the keel are set in the best position relative to each other the rudder is also a very effective device.
Of course one must design a keel and a rudder and the study of aerodynamics points to the use of high aspect ratios for both. It would seem that high aspect ratios present no special problem when draught is not limited by some extraneous factor but the requirements for stability do add design constraints and I must look at stability now.
 People are surprised that a hydroplane or a hovercraft cannot be steered like a boat using either a water rudder or an air rudder
 The use of pivots ensures that the model can move only in two dimensions but it also introduces a simplification when compared with an aeroplane. For an aerofoil moving freely in the air the pressure forces on the aerofoil reduce to a single force and a couple. The point of application of the force and its magnitude and that of the couple vary with the angle of attack. The use of pivots permits us to locate the force at the expense of changing the couple to a moment about the pivot.