Chapter 14 The sailing diagrams for a conventional Bermuda rig
The polar performance diagram for a conventional rig will only differ in detail from that for a swing rig. (See Diagram 10-1) so there is little point to plotting it. However, for the conventional rig, the combined force on the sails is no longer in a fixed position but has a line of action that changes as the sails are sheeted out. We need sailing diagrams to see what happens.
It seems to me that we are not likely to improve on the outcome of Chapters 12 and 13 and that the angles for the sails that result from the geometry of single-winch sheeting can be used to draw sailing diagrams for the ordinary Bermuda rig. The forces on the sails can be shown on those diagrams and we can see how the sails work to propel the boat. When drawing the first set of sailing diagrams for the swing rig I used angular increments of 10° to find out in detail how things change. It is probably adequate to draw only five diagrams for the conventional rig and to let those match the angles used in the Hele-Shaw pictures.
In order to draw sailing diagrams we have to find relative magnitudes for the forces on the two sails. For the swing rig we let the forces be proportional to the sail areas and allowed both to vary with the square of the speed of the apparent wind. For our first polar diagram we used an area ratio between the two sails that was appropriate to a swing rig. Now we need an area ratio for a conventional rig. It might be best if we used the area ratio for a metre yacht which, for the top suit, is very nearly 3 to 2.
We have seen that it is easy to draw these diagrams if they are set relative to the apparent wind as in Figure 13-1 but we need diagrams relative to the true wind. This requires a knowledge of the angle between the true wind and the apparent wind which is not yet available. This also means that we need to know the speed of the yacht which is also not yet available. No doubt a computer program could be written to sort this out by converging on to the correct diagrams. However it can be done by drawing if one is prepared to draw three sets of diagrams with each set giving figures to improve the next set. The starting point is to use the polar diagram in Chapter 10 to give a first guess to the angle between the true and apparent winds. As the polar diagram was drawn for a maximum speed of 3 mph in a wind of 10 mph and the critical speed for a metre boat is 2.8 mph it is still relevant.
In order to draw these diagrams it is necessary to draw the arrows representing the forces on the two sails. As before I have drawn these arrows at right angles to the booms but there remains the problem of where to put them on the booms. This requires us to address the problem of where the resultant force acts on a sail. If the pressure were to be the same all over the windward side of a sail and if the pressure on the leeward side were to be the same all over but different and if the sail were to be flat without a twist the resultant force would act at the centre of area. As the pressure is not uniform on either side of the sail and the sail is not flat it is unlikely that the force will, just by chance, act through the centre of area. We have little to help us but, if data from aerofoils is relevant, the force on an aerofoil acts at between 35% and 40% back from the leading edge. I have taken this as better than using the centre of area and estimated a position of the resultant force on each of the two sails of a top rig for a metre boat.
I have given the final set of diagrams. All the diagrams are drawn in the same way as that in the labelled first one, Diagram 14-1a. The course is shown more or less in line with the centre line of
the hull, the forces on the sails are shown and the combined force on the rig is shown with a dotted extension. The component of this force along the course (the driving force) is shown in line with the course. As the speed of the apparent wind falls from that in 14-1a to that in 14-1b the forces on the sails fall and, of course, the net force on the rig also falls. However even though the rig force falls the change in position of the sails relative to the yacht means that the driving force changes little for 14-1b, c and d. For these three angles the speed of the yacht is much the same. For 14-1a and 14-1e the drive is smaller and the speed of the yacht can be derived from the shape of Diagram 10-1. The leeway depends on the heeling force which, whilst not shown, can easily be judged. The leeway is shown falling from 3° to zero.
The diagrams show how the line of action of the combined force on the rig moves as the sails are sheeted out. The dotted line moves from in front of the mast in 14-1a to about one quarter of the length of the foot of the main sail to one side of the mast in 14-1e. (There is nothing special about the mast as a reference point other than its convenience.) The action of the combined force at this large offset means that a moment is applied to the yacht tending to make it luff into wind. This must be resisted in some way by the hull, the fin and the rudder working together.
Sailing diagrams between reaching and running
When the main sail boom is sheeted out to 90° and the angle of attack of the main sail is 32.5° the course of the yacht makes an angle of about 42° to the true wind. For all courses between this and running before the wind the sail positions will be unchanged relative to the hull but the angle of attack will increase to 90° for both sails. As the angles of attack increase the flow over the leeward side will break away, the flow on the windward side will progressively split further away from the luff as in Picture 9-4c until it splits more or less in the middle. This general flow pattern will be affected by the uncertainties of the disturbed approach flow and the two sails will interact to further complicate the flow pattern. In these circumstances the best we can do is draw the relative positions of the winds (true and apparent), the hull and the sails and see what we can deduce from them.
I have chosen to draw sailing diagrams for running and for an angle of 22.5° between the true wind and the course of the yacht. As the fore sail can be on either side of the yacht this leads to four diagrams 14-2a, b, c and d.
These diagrams are adequate for us to look at these two points of sailing. In Diagrams 14-2a and 14-2b I have attempted to locate the line of action of the combined force on the rig. In order to do so I have assumed that the forces on the two sails are proportional to their areas. This cannot be true because the main sail blankets the fore sail but the effect of this is to reduce the drive from the fore sail and push the line of action of the combined force still further out. In Diagram 14-2b the yacht is goose-winged and then both sails work and the line of action has a much smaller offset.
Diagrams 14-2c and 14-2d are for the yacht sailing a course that makes 22.5° to the true wind. We stand no chance of estimating where the forces might act but the goose-winged position is certain to give less offset to the combined force. A glance at Diagram 14-2d suggested that with the fore sail goose-winged the flow is at least in part from leech to luff. This means that the unsupported leech can flutter and lead to a switch back to Diagram 14-2c. This means that somewhere in the last 22.5°the fore sail can become stable but, given the state of the natural wind and the fact that the sails are square to it, that stability may be upset very easily.
 I am taking the efficiency of the sails to be low. If the sails are better than I think then the drive will be increased which is, of course, desirable.
 I have taken this to be the best angle of attack.
 It would be smaller still if a radial jib fitting were to be used.
 This switching is easily detectable.