Chapter 9. Flow visualisation for the beating yacht
It became clear in the previous chapter that we need a practical way of looking at flow patterns.
There are plenty of decorative looking pictures of models of yachts, complete with Bermuda rigs, being tested in large wind tunnels but not much in the way of results from such tests. I am always sceptical of such tests because too many variables are unknown when whole rigs are tested. I prefer the methods adopted by the NACA where the most simple, two-dimensional model is tested to give a firm starting point for the other complexities.
This suggests that all we have to do is make a rectangular sail from, say, mylar and set it up between the walls of the working section of a wind tunnel and carry out the tests. One tends to think of the leading and trailing edges being supported and letting the film find its own shape. The problem is that the mylar will buckle and crease on the sides when a force is applied to it. Suddenly it all looks to be more complicated. One might decide to test “sails” made of sheet metal but this has a fixed shape. Perhaps if time and money were to be available the results from testing plate sails would be instructive and interesting. We need something that can be done at home by the competent model maker for little outlay. The only equipment I know is the Hele-Shaw rig. I built one and used it.
This equipment is really a special water tunnel. Water tunnels are just like wind tunnels in overall design but with adaptations to cope with different fluids. In the Hele-Shaw rig the working section is made from two sheets of plate glass fixed at their sides using servo tape. This makes a duct that is very wide compared with its height which is about 0.035². The lower glass plate extends to form the bottom of an open top tank. Water is fed through a sprinkler into the feed box and flows between the plates and leaves freely. The junction between the top plate and the front of the feed box forms a smooth radius to aid the flow into the working section and it incorporates a set of equally spaced nipples through which dye can be fed to give the flow lines. I used ordinary dyes mixed with water and left the rig to run without dye to clean it after use. If two-dimensional shapes, corresponding to the models used in wind tunnels, are cut from 0.035² sheet and introduced between the plates the dye lines will show the flow pattern.
This rig is not the equivalent of a wind tunnel and, if we are not to be deceived, we have to decide what can be deduced from any flow pattern that we may see. In order to do so we must decide what is going on between the plates. The crucial point is that the flow is laminar or non-mixing. That is evident from the fact that the lines of dye retain their definition and do not spread into the water surrounding them. There are serious differences between laminar flow as we see it in a Hele-Shaw rig and the turbulent or mixing flow that takes place round a real sail. The most obvious difference is that as it flows round, say, a stalled aerofoil, the laminar flow never breaks down into an eddying wake as it would in turbulent flow. However flow patterns do not break down for turbulent flow approaching an obstruction and it is a matter of observation that the approach flow in laminar flow compares well with the approach flow in turbulent flow. The principal difference comes in the leaving flow and, if we treat the leaving flow cautiously, we can learn a great deal about the two-dimensional flow round a single sail or about the interacting flow round two sails. At the very least we shall be looking at a real flow and not at what someone imagines that a real flow might look like.
Before we can use the Hele-Shaw rig we have to make some shapes to correspond to the sails and so we must decide what shape a sail takes when it is working. We really do not know the precise shape that the sail might assume but an arc of a circle would be a reasonable approximation. This means that we need a way of defining an arc in just a few useful dimensions. Ever since the very early aeroplanes when wings were made of a frame of ribs and spars and covered with fabric the shape has been described as a camber. This established method requires only the chord and the offset as shown in Diagram 9-2. The offset is usually given as a percentage of the chord and most people refer to the camber as a percentage.
Most sailors of model yachts set up their rigs by eye and talk of increasing the camber or of flattening the sail. However the flow pattern is sensitive to the angle of the luff not to the camber. The angle can easily be calculated in terms of the offset. The figures are
Camber as a percentage 2 4 6 8 10 12
Angle at luff in degrees 4.6 9.15 13.6 18.17 22.5 27
It is instructive to look and see how camber and the angle at the luff are related. In Diagram 9-3 I have drawn the arcs for a 12% camber and for a 6% camber for comparison with sail shapes. Observation of successful sails suggests that 12% would not be uncommon and that some reduction is made for strong winds. The angles at the luff are surprisingly large. The models to go in the Hele-Shaw rig have about 10% camber.
The Single Sail in the Hele-Shaw Rig
We can now see whether the Hele-Shaw rig supports the ideas put forward in chapter 7. The obvious starting place is to look at the flow pattern round a sail, which is complete with its mast, set at 32.5° to the undisturbed flow. I made a suitably sized model of a sail and set it up in the Hele-Shaw rig. The resulting flow pattern is shown in Picture 9-4a and a close up of the same flow is shown in Picture 9-4b. It is worth repeating the flow pattern over a stalled aerofoil from chapter 7. It is included here as picture 9-4c.
Now we have to interpret what we see. Clearly the approach flow is diverted upwards just as it was for the aerofoil. This means that there is, ahead of the sail, the diverted flow in which a fore sail can be made to work. The water flows smoothly up into the concave curve of the sail diverging as it enters and converging as it leaves. If anything the flow seems to be better for our purpose than that over the underside of the aerofoil. The flow over the top of the aerofoil curved downwards to “dive” into the mixing region and the flow lines in the Hele-Shaw rig dive in the same way. Of course there is no mixing region. However the lines of dye do become very wide and there is information in this for us.
We have already said that the flow of water in the rig is laminar and I called it non-mixing yet here we have lines of dye spreading as if they are mixing. They are not and we need an explanation. It is evident from all sorts of experiment that when laminar flow occurs between two flat plates the fluid in contact with the plates does not move. Nevertheless, as there is flow, the rest of the fluid must move. In fact the flow becomes organised with its maximum velocity at the mid-point between the plates and everywhere else has velocities between the maximum velocity and zero at the surfaces. The velocity distribution is shown in Diagram 9-5 and is parabolic. (A parabola is the graph of y = a.x2.)
This has significance where the flow has to curve if it is to follow the surface of the model. In order for the water to follow a curved path there must be a force acting across the flow which, in this case, will be produced by a transverse pressure difference. The force acts on all the water in the same way and bends the slow moving water near to the plates much more than it bends the fast moving water in the middle. As a result the cross-section of the dye lines takes a shape very much like the shape of the velocity distribution above and appears to widen. Careful inspection of the flow lines shows that one side is well defined and the other shades away as one might expect.
The Hele-Shaw rig tells us where large transverse pressure gradients occur and these regions are where, in ordinary turbulent flow, the flow is most likely to break down into eddies. We should also expect that the break down will start at points where there is a sudden change in direction such as that over the leading edge just as it did for the aerofoil at a high angle of attack
The Bermuda Rig in the Hele-Shaw Equipment
It seems that the Hele-Shaw rig offers us a cheap way of looking at the flow pattern round two sails working together as in a Bermuda rig. The rig will tell us where to expect a break down in flow but not what that breakdown would look like. Perhaps its value should be taken to lie in the understanding of the flow so that the process of learning to set up a sailing rig is shortened. With the warning that we must not expect too much from this equipment we can go on to simulate the flow over the sails of a Bermuda rig when it is beating close to the wind.
We need an experimental plan. I suggested in chapter 8 that one might consider the fore sail as working in a different apparent wind to the main sail. In truth the two sails share a flow pattern and the idea is too simple. We are trying to decide how the fore sail can best be set relative to the main sail so that it produces a useful force to drive the yacht. We do know that the boom of the fore sail on a model yacht must be able to swing past the mast so that sets one limit. We also know that the swivel point for the fore sail boom will be on the centre line of the deck and that, most frequently, the swivel is attached to the boom at about a quarter of its length from the tack. So it makes sense to let the centre line of the yacht make 32.5° to the undisturbed flow, put the model of the main sail on the centre line, and then set the fore sail model at angles of, say, 10°, 15°, 20°, 25°, 30°, and 35° which should cover the likely range of angles.
The following photographs were taken with a digital camera to facilitate their inclusion in the text. I could not avoid the shadowy outline of the photographer as the rig is an outdoor exercise and the plates must be horizontal. Perhaps I should have used non- reflecting glass.
Now we have to assess these pictures to find out what we can from them. As it happens one flow line passes through the gap between the sails in every picture. In 9-8b there is a sharp kink in this line near to the mast and the flow over the main sail is greatly altered from when it is operating as a single sail. The flow over the luff of the fore sail is going to lead to a serious break down of flow.
In Picture 9-6l this flow line has lost its kink and two other flow lines also pass through the gap. One of these lines has come from under the main sail and the other has come from over the fore sail. The sharp deflection of the one going round the mast is not at all desirable. The shape of the one going under the fore sail suggests that it could easily break away near to the luff.
This leads us to look in the middle of our chosen range for a better pattern. Before doing so we should recall that we are not just looking for a large force on the fore sail, it must also have a large component in the forward direction. Furthermore we would like to have this force without a vigorously eddying wake to add to the drag.
Study of the pictures suggests that the two best flow patterns are those for 20° and 25°. In both cases the presence of the main sail is diverting the flow smoothly upwards into the concave side
of the fore sail and the flow over the convex side of the fore sail may be such that the flow does not break away. The flow over the main sail is not much different from that of the sail acting alone. There is still a kink in the common flow line but that is to be expected. In the 30° and the 35° pictures the flow into the fore sail is changing to require an unlikely bend upwards and the flow round the main sail also requires a new rapid change in direction.
It seems that it is best if the fore sail makes an angle of between 20° and 25° to the centre line of the yacht.
Having come this far with the Hele-Shaw rig it is evident that we could form an opinion on the possibility of producing an even better flow pattern by letting the main sail make a small angle with the centre line. I have taken pictures with fore sail at 25° and the main sail at 5° and 10°.
The change does not improve the flow over the fore sail and the only way to re-establish the flow is to reduce the angle of the fore sail. That is not what is wanted.
The reader will have his own view of the success of the Hele-Shaw equipment as a cheap way of seeing how fluid flows over the sails of a Bermuda rig. I think that most readers will have a better idea of the characteristics of the flow and may be better placed to experiment.
The Hele-Shaw rig has given us only a two-dimensional flow and now we must think about the sails in three dimensions.
 Hele Shaw was a mathematician and he was interested in predicting flow patterns using mathematics. He could only deal with a very few, very simple cases, which would still be true if it were not for the existence of very powerful computers. Hele-Shaw required some practical way of confirming his mathematics and he designed his rig to check the laminar flow of a fluid round a cylinder. It worked but few people have used it to advantage since. Hele-Shaw worked about 1898.
 This does not match the angle at which the fore sail boom is normally set but it does match the angle of the sail at a point about half way up the luff. We will return to this.