The stability of sailing vessels
Figure 63 is a drawing of a Viking ship that may not be accurate for the want of reliable data about Viking ships. Nevertheless I shall take it to be sufficiently accurate for my purpose. The hull is long and narrow and the sail is low and wide. Such a sail has an aspect ratio of less than one and cannot have been efficient. I am sure that the Vikings knew that a taller sail would have been better but the hull was not capable of carrying a taller sail without excessive heel. This was a war vessel to be rowed when in action and had to be kept light yet the only way to make the boat more stable was to add lots of ballast in the form of stones in the bottom of the hull. Here is the basic contradiction of sailing vessels, they need to be very stable to carry a rig yet the obvious methods of increasing stability compromise other desirable characteristics. We need to understand this.
The starting point must be the idea of a metacentric height and the starting point for that is to note that when a sailing vessel heels under the force on its rig only the shape of the hole made by the hull in the water changes. Somehow this change produces forces that stop the vessel from capsizing.
In figure 64(a) I have shown a cross-section of a pontoon floating at rest on an even keel. I have shown the centre of gravity G of the pontoon and also the centre of volume B of the hole that the pontoon makes in the water. The upthrust exerted by the water on the pontoon acts upwards through B and point B is called the centre of buoyancy. As the pontoon is at rest the upthrust must equal the weight of the pontoon.
If the pontoon is somehow made to heel about its longitudinal axis the shape of the hole in the water will change and the centre of buoyancy will move to the new point B¢. If the upthrust acting vertically through B¢ is to the right of G a couple will form between the weight and the upthrust to try to right the pontoon. This is how vessels can be stable even though the centre of gravity is above the water line.
We need a way of quantifying the stability.
In figure 65(a) I have shown, for a heeled vessel, the relative positions of and and the vertical axis of symmetry of the vessel. The righting couple is equal to the weight multiplied by the horizontal distance between and . If the vertical through is extended to cut the axis of symmetry at p the righting couple is equal to the weight multiplied by where is the angle of heel. If we could find we could calculate the righting couple.
Others have looked at this. They reasoned that is not necessarily a fixed point relative to the vessel and asked what happens to point when the angle of heel is reduced? In practice moves but they saw that a diagram could be drawn for a very tiny angle of heel and that the upthrust would still intersect with the centre line and that would still have a position even when the angle is zero. This point is now called the metacentre and given the symbol and is a measure of stability and is called the metacentric height. I have redrawn the centre line when the vessel is on an even keel in 65b to show the positions of .
Now we need to be able to calculate the metacentric height and we have to do this in two stages. The first is to find a value for the distance to . The mechanics leads to the statement that can be calculated from where I is the second moment of area of the water plane section of the hull about the longitudinal axis and is the displaced volume of the hull. The second moment of area can be defined from figure 66. It is the sum of the product for all small areas like the one shown in figure 66 where is the distance of the small area from the centre line. It crops up quite often in engineering.
Then all we need to know is the position of the centre of buoyancy and we can locate .
All this is entirely a matter of mensuration and the position of relative to say, the outer bottom, is a dimension of the boat.
The position of is in part determined by the structure of the vessel, but within limits, can be moved by the distribution of payload within the hull. Given that statement we need to have some idea of the desirable value for the metacentric height. Even for large ships it is small, only 4 or 5 feet and possibly much less, because, at this sort of value, the movement of the ship in response to waves is “easy” and imposes least strain on the structure of the ship. Cruise liners look like blocks of flats and containers are loaded on the deck of container ships to make them have the same shape as a block of flats. This is done to bring the centre of gravity up high enough for the metacentric height to be correct. Square-rigged sailing ships were loaded carefully and dunnage was used to raise dense cargoes within the hull to give an “easy” response to wind and wave and having a “stiff” boat was not really the best solution nor was having a high centre of gravity because the ship would have rolled excessively and have been very wet and slow to clear its decks of water. Getting the metacentric height right before sailing has always been important. However we are interested in sailing vessels.
The earliest sailing vessels look like open boats fitted with a square sail or a triangular sail. No doubt the early ones were used on rivers or in coastal waters and did not face the conditions on the ocean. In figure 67 I have outlined a Viking war ship and added , , and . We can look at this ship. must be below the free surface and, were it not for the sailing rig, the centre of gravity of the hull and its contents would be at or slightly above the water line. depends on the dimensions of the hull and the open boat is typically wide enough to be quite stable and to have somewhere near where it is shown. If this open boat is fitted with a square sail as shown the centre of gravity will be raised and reduced. It would be all too easy to have the position of too high and the hull become too “tender” which is not an option for an open boat.
Attempts to solve this problem depended on carrying ballast in the bottom of the hull. It made long voyages possible but only for sails with low aspect ratios. The ballast made the hull sink more deeply and this immediately put a practical limit to the amount of ballast that could be carried because it reduced the freeboard, occupied space that could be employed more usefully, and increased the moment of inertia about both axes. This sinking into the water more deeply increased both the value of the second moment of the water plane section and the displaced volume. For war vessels did not change much. The centre of gravity was lowered but the configuration of the hull precluded a really useful increase in . The early years of the 20th century saw the emergence of hulls with built-in keels containing ballast but, as we have seen, this was a dead end. The best solution was to attach the ballast to a hydrodynamic keel. Unbelievably designers were still wondering how to do this in the 1990’s 50 years after the design of the Mustang fighter of wartime fame. Its wing was symmetrical and it was the most efficient of the fighters and it carried fuel in tip tanks and that was just what is needed for a hydrodynamic keel and a ballast weight.
How should this appendage of keel and weight be designed? It is not by any means as simple as it seems. Figure 68 shows a yacht that is fitted with a keel and ballast weight (bulb) heeling under the wind force on the sails. I have shown the centre of gravity well below the outer bottom of the hull and somewhere in the keel. Then the upthrust acting through the centre of buoyancy and the weight acting through the centre of gravity produce a righting couple. As there is now evidence for rigs being more efficient when they work without heel it seems that, for racing yachts, we should aim for the maximum righting couple. Then given the need to have the least volume of hull in the water the goal is to have a yacht that is light overall. If a maximum weight is set then absolutely everything, other than the bulb, should be essential to the function of the boat and have the minimum weight and the bulb should be as deep as possible. There is no other way. How simple it sounds but there are other factors to consider.
The first is the area of the keel. It is not difficult to calculate the maximum transverse force that might be exerted on a sailing rig. Then, using, a graph like that in figure 58 it is possible to choose an area for the keel. It will be a compromise because we want the keel to produce the minimum drag and that comes from having the best combination of angle of attack and area and we want a large enough area for a keel of high aspect ratio to be strong enough to support a heavy bulb. This latter is difficult to meet because we really want a thin keel (7% perhaps) yet be capable of having strength in both bending and torsion. This leads to the use of materials like carbon fibre but these are not like mild steel that can yield before breaking, their high strength is associated with sudden brittle failure although the consequences of this inherent behaviour does not seem not to be properly appreciated.
So let me suppose that an area can be chosen and a fin constructed. In figure 68 I have also shown the force on the rig. If the hydrodynamic keel is properly designed it will be quite capable of resisting this force without a contribution from the hull. As the force on the rig acts to leeward, the force on the keel must act to windward and the two forces produce a large upsetting couple. The deeper the bulb is carried the greater this upsetting couple becomes. This is reflected in the design of the shape of the keel. As the force on the keel acts at what aerodynamicists call the centre of pressure and the sailing world calls the centre of effort there is an incentive to choose profiles that let the bulb be deep but the centre of effort be high. The centre of effort is not far away from the centre of area of the keel and it follows that the extreme design is a triangle with its base at the hull. This is not practical for two reasons. The first is that the keel at its bottom end has to be of sufficient size to permit the secure attachment of the bulb and the second is that hydrofoils of small chord are not very efficient. It is likely that, if the keel is tapered as much as is consistent with the mechanical requirements for attachment of the bulb to the keel and the keel to the hull, it will be satisfactory hydrodynamically.
The whole process is changed if there is some restriction on draught. Then a shallow, wide keel becomes necessary and because of the limitation that this imposes on the lowering of the centre of gravity the weight of the bulb must be increased. Mechanically it is much easier to use.
Whatever the ultimate design the surface finish must be as good as can be maintained. Happily a polished, well-designed keel and bulb will have a very low drag indeed.
However everything in engineering carries some penalty and the hydrodynamic keel is no exception. We have seen that the rudder controls the transverse lift that is generated by the keel. Suppose that the yacht is running before the wind with the sails out to one side. The rudder will have to generate a significant force just to produce a moment to balance the moment from the asymmetrical rig. Conditions could arise that cause the rudder to be required to produce too big a force and to make it stall. Then things become interesting. The stalling of the rudder removes the control from the hydrodynamic keel and also lets the hull swing in response to the asymmetrical force on the sails. The angle of attack of the keel suddenly increases and the keel lifts to windward and the force is almost certain to be large enough to lift the bulb and for the kinetic energy of the whole yacht to go into lifting the bulb until the yacht stops. Then the bulb sinks again to right the yacht. I suppose that broaching for a deep-keeled yacht is even more exciting than it was on a square rigged ship.
 Viking merchant ships carried much taller sails with much higher aspect ratios because they had much different hulls with greater freeboard and, of course, cargo that acted as ballast.
 I saw a yacht ready to launch at Ramsgate. The hull was gleaming in the sunshine. The keel and bulb were as cast. I wondered how the owner could polish the hull right up to the junction of the keel and hull and not see the contradictory philosophies at work.