# Section 2 The single soft sail

The single soft sail.

The obvious starting point is to look at the behaviour of the single sail and then to consider combinations of sails.

In almost every case the single sail will have a cross-section made up of a mast with a piece of fabric attached to it in such a way that the fabric is curved as shown in figure 1. The fabric takes up its shape as a result of pressure differences between the concave and the convex surfaces. If the flow past the sail is not in the right direction the sail will not inflate and, if the pressures are not suitably distributed the sail may have an S shape just behind the mast.

We need to be able to describe the shape in some way so that we can refer to it. In the lower diagram I have shown the leading edge of the sail that is called the luff and the trailing edge that is called the leech. A line joining the centre of the mast and the leech is called the chord or the chord line and the maximum distance between the inflated sail and the chord line is called the camber. It gives a first order way to describe the curvature of the sail.

Single sails come in several shapes and with various degrees of twist. Figure 2 shows a Topper with its single triangular sail. Luff to leech the sail is fairly flat but it is curved at the luff as a result of the mast bending. This gives the curve at the leech. The sail will have an uphaul, a downhaul, an outhaul and a kicking strap all of which can be adjusted to alter the shape of the sail within the limits of the physical constraints.

Figures 3a and 3b shows my model of a Norfolk wherry to show, quite faithfully, the plan-form of the sail and the camber, the belly and the twist that is used on the full size to get the boat to follow small rivers under sail as far as is possible without poling. (Poling is called quanting because the poles have flat ends like those used for jumping dykes.) The flow over such a sail is complex. No doubt it flows past the mast and the gaff but the presence of the sail will affect the air as it approaches the mast and the gaff and divert it from windward to leeward and simultaneously, upwards or downwards. Once past the mast and the gaff the air on the windward side follows the curve of the sail to continue upwards to leave the leech and downwards to leave from the foot. The air on the leeward side separates from the convex shape to form an eddy pattern that creates a thick wake. Outside this wake the air flows smoothly. In all this the air speeds up and slows down in an orderly fashion and, as a consequence, the pressure in the flow increases or decreases relative to atmospheric pressure to produce forces that combine into a single force that drives the yacht and makes it heel and move sideways.

In the main, sailors set their sails to try to get the maximum force to drive the boat for any point of sailing and to cope with the resulting heeling and leeway by whatever means is possible.

Generally sailors learn how to get the best performance out of such a sail by a combination of trial and instruction and sails are fitted with tell-tales and streamers that are used as visual cues to what is happening to the flow so that a sail may be set to give the best performance. A guide to how the whole yacht is performing comes from the behaviour of the rudder and the sound of the boat that can be interpreted by an experienced helmsman.

However some sailors still desire to have a rational explanation of how the sail actually works and perhaps some way of quantifying the forces on the sail in all of its many attitudes to the wind. For rigid shapes like propellers for aeroplanes, or wings, complex mathematical models can be devised to give values for the various forces, powers and efficiencies but for sails the models would be very complex and spending time and money is not justified by the function of a yacht as a device used for pleasure. We have to decide what can actually be done to get an insight to the behaviour of sails and whether it is worth doing.

What we would really like is to have a sail with two-dimensional flow over it and to see how to use it for the best. That is not possible. Given that the wind is never really steady and the yacht pitches and heels the flow over the sail changes continually but one might hope to find a representative wind to use in analysis and computation. We cannot transfer a yacht to a wind tunnel because the flow pattern over the sails is created by the movement of the yacht that cannot be replicated in a tunnel. So we have to aim for the possible and do what we can to gain an insight to sail action. In my experience any serious attempt to analyse a mechanical device leads to a better understanding of how it works and perhaps that is all we can hope for.

This all requires some clear idea of how to approach this analysis. I have explained that the flow is very complex and effectively too complex to analyse without simplification. I think that one should consider looking at the most simple version of a sail to find out how it works and expect to have to take the complexities of the flow and the constraints of it application into account qualitatively.

In figure 4 I have shown a yacht sailing at an angle to the apparent wind. The wind flowing over the sail produces a force that is at an angle to the apparent wind. That force could be thought of as the vector sum of two forces at right angles, one parallel to the course that drives the yacht and the other across the yacht. I have shown these in the diagram.

The sailor will attempt to set the sail so that the force in line with the course drives the yacht as fast as possible along the course. The transverse force on the sail produces two effects, it makes the yacht heel and it produces leeway. It all looks very simple but this diagram comes into existence when all the variables like the speed of the yacht combines with the true wind to give the right angle of attack to produce the right component of force to drive the yacht at the right speed. I have just shown the sail as a single line even though it will have belly and twist. In doing so I have really said that I am going to deal with sails by treating the flow as two-dimensional across the sail. I think that is all that is justified by the outcome and the use to which it will be put.

We shall get nowhere without some data on how the force exerted by the wind on a sail varies with speed and sail setting and this must be experimental data. I have explained why such data cannot be obtained for sails but we still need something to use in analysis. Any reliable data, however obtained, would be useful to add to our knowledge of how sails work. We must look towards data gathered for aeronautical use and therefore for rigid test models.

If we are to make use of experimental data we really need to know in what form it is gathered. In aeronautics the angle of attack is measured relative to the direction of the free stream which, in practice, usually equates to the direction of motion of the aerofoil. For sails the angle of attack is measured relative to the apparent wind and the apparent wind can be in any direction relative to the yacht.

If we replace the force on the sail by two new components, one at right angles to the apparent wind and the other parallel to the apparent wind as shown in figure 5 we could call them the lift and drag just as we would for aerofoils. But, if we are to do that, we need to understand what is involved.

## Lift and drag

Right from the start the early experimenters had ideas of lift for a wing, which was needed to get into the air, and drag that would have to be overcome continuously by an engine if flight were to become a reality. We still measure the same forces using equipment that has improved over the years and express them in a form that facilitates the storage of the data that has been obtained by experiment. We write:-

Lift =

Drag =

where are the coefficient of lift is  and the coefficient of drag is ,  is the density of the flowing fluid and  is the speed of the undisturbed fluid. In both cases the area  is the same, that is, the plan area of the aerofoil. These expressions are not the outcome of analysis and they have been developed solely to give us a way to experiment to find values for  and . They have this structure because  is the pressure that is actually produced when a fluid flowing at  is brought to rest and, when it is imagined that this pressure is exerted on some easily defined and relevant area A, the result is a comprehensible force on which to base a coefficient. The two coefficients vary primarily with the angle of attack and the size of the sail.

For sails our reference vector is the apparent wind. This may not be as convenient as that for aerofoils but it is the best reference vector that we can find so we make use of it. Then, in the rational expression that defines our two coefficients, we must change the velocity  to the speed of the apparent wind  and then the expressions can be used for sails.

The expressions become :-Lift =  and Drag=  where  is the area of the sail-plan.

In the case of aerofoils the net force is only ever replaced by its two components lift and drag, but the net force on a sail is replaced to considerable advantage by components in line with the course and across the course. The link between these two pairs of components is the force on the sail. We could use a coefficient of force to advantage and it would be defined as Force = . If we had values of  and .they could be replaced by .

We would like to experiment to find these components and use them to find coefficients of lift and drag and force and record the outcome.

If we wanted such data for an aerofoil it would present no insuperable problems because we can make test models of short lengths of wings and measure the forces exerted on these models when they are tested in a wind tunnel. There is a profusion of experimental data on aerofoils of all shapes. When it comes to sails it will be apparent that trying to measure the forces exerted on a piece of fabric attached to a dummy mast is not going to be as easy as testing rigid models of rigid wings especially when the range of angle of attack is from 0º to 90º compared with 0º to 20º for a model wing. It is probably impossible but the arrangement in figure 6 would permit some measurements.

The suspension arms and the wire would be attached to the lift-and-drag balance of a wind tunnel. There are snags. The model would be a thin plate fitted with a dummy mast and it would fit snugly between the side walls of the tunnel. The suspension arms would have to be offset to permit the high angles of attack and the plate would need to be off the centre line so that it ends up some where near the middle of the tunnel at an angle of attack of 90º.

I have never come across any definitive measurements on such a model sail that give values for these coefficients at various angles of attack. This means that we need to think of a way of deducing them from other things we know.

The first thing that we can do is to find out how the air actually flows over the sail.

The sail may look like an aerofoil but it is not. An aerofoil depends crucially on the shape of its nose for its performance and it simply could not work with a cylinder attached to its leading edge as shown in figure 7. The cylinder would ensure that the flow over the upper surface became detached from the surface to form a wide wake. If the aerofoil were to be set at an angle to a steady wind a force that was upward and backwards would act on it but the backwards component would be much greater than it would be without the cylinder.

So we want to know how the force exerted on a sail varies with the angle of attack in a steady wind. It would be very convenient if we had reliable flow patterns for such a sail at the angles between 0º and 90º but I doubt if they exist. This means that I have to draw flow patterns and I shall have to turn to aerofoils. Figure 8 shows the flow pattern round a real aerofoil in a wind tunnel. The lines showing the way in which the air divides to flow over and under the aerofoil are of smoke fed into the flow by a smoke rake. They are called flow lines and they do not divide or unite, nor do they cross. The horizontal lines are parallel to the axis of the tunnel. The aerofoil makes an angle of 30° to these lines. The question that needs an answer is “how does this aerofoil produce lift?”

There is an obvious statement to be made first and that is that the only way that a force can be exerted by the air on the aerofoil is by pressure. People are quick to quote the Bernoulli theorem and attempt to apply it in the most simplistic way and produce the most unlikely assertions. For our purpose it is sufficient to observe that, in a stream of air flowing steadily past some object the sum of the kinetic energy and the pressure energy, in suitable units, does not change. This means that where the speed is high the pressure is low and vice versa. The two-dimensional flow pattern lets us see where pressures are higher than average and where they are lower than average because, where the flow lines are wider apart, the pressure is high and the speed is low and vice versa. Then, where the lines diverge the speed is decreasing and the pressure is rising and where they converge the pressure is falling. We can then see that the region at the trailing edge and beyond it is of low pressure, the region under the leading edge is at high pressure and the region above the leading edge is a low pressure. The flow diagram tells us nothing about the region over the upper surface of the aerofoil where we must expect the pressure to be low.

We need to know what is happening in the large white area of the diagram in order to see why the pressure is low.

In the flow pattern shown in figure 8 the absence of flow lines in the white area means that the smoke trails have lost their definition because they entered a region of mixing, eddying and turbulence. Mixing is the coalescence of two or more streams of fluid; eddying is a motion in which the fluid flows round closed loops and turbulence is the small scale eddying and mixing that goes on inside most fluids when they move. If we are to extract more from this diagram we must look at the heavy black line aft of the trailing edge and the wide black line over the leading edge. What goes on in these two regions?

At the top it is clear that the air is flowing down into our white space. That air is carrying momentum that is not lost just because it is mixing and the air in the white space must continue to flow from left to right and slightly downwards and it will mix with the air that is in the top of the white space and affect the air moving below it. At the bottom we have fast moving air that is marking the boundary of the white space. By viscous drag and by mixing this will also try to drag air in the white space along with it. It will draw air out of the white region and into the wake. So we must expect to find some region of the flow pattern where air is taken in to be sucked out at the trailing edge.

Figure 9 shows the flow pattern round an asymmetrical aerofoil due to Ludwig Prandtl. The pattern was made by wet paint on glass. It is complex and we must ask how it comes about. In order to do so we must extract more from our first diagram. Prandtl’s diagram shows what actually happens but we must know what we are looking at. This is obviously eddying flow like that which can be seen at any time around the supports of a bridge over a river. Observation of such a flow shows that, whilst the eddies remain in position all the time, they continuously change the detail of their shapes. The eddies that are so clear in Prandtl’s picture also behave in this way. What we are looking at is an “average” position and shape for the eddies built up over time in wet paint, that moves very slowly, on glass. It is a magnificent picture that is at least 70 years old. Now we have to extract information from it.

In figure 10 I have changed Prandtl’s picture to its negative and picked out the primary features of the flow. I think that the most important line that has been added starts on the left, loops upwards, and then back down to leave from the right hand side. It encloses a bubble that contains at least four eddies. Now it follows from the laws of motion that in order for a flowing fluid to follow a sharply curved path as the air does in the top right hand corner, where I have added an asterisk, there must be a continuously-operating force acting towards the centre of curvature. There is only one way for this force to be created and that is for a pressure difference to act towards the centre of curvature. As the pressure in the region above the right hand corner is atmospheric the pressure inside the bubble and therefore on the upper surface of the aerofoil must be lower than atmospheric. The aerofoil is subjected to high, though non-uniform pressure, over it’s under side and low, and again non-uniform pressure, over the upper surface. The net effect will be a single force acting upwards and to the right and a moment.

It will be evident that there is agreement between the flow lines and the paint. However there is still the bubble to examine. Seemingly the main feature of the bubble is the large eddy with the three smaller eddies ahead of it but I think that, given the stable existence of this family of eddies, the more interesting region is close to the upper side of the trailing edge where I have added one sharply hooked line with an arrowhead. The flow between the boundary of the bubble and the main eddy splits just above the trailing edge and this hooked line marks the division of the flow. Air to the left continues as part of the main eddy and that to the right goes round sharply to be entrained into the flow from the lower face of the aerofoil. This is where the air that was mixed at the top leaves the bubble after following a path through the various eddies. It is most unlikely that the flow will be steady in the region of this separation. Energy will be lost continuously in the eddying but it is replenished from the air that flows into the top of the bubble and out at the trailing edge. This makes the bubble stable so that it is not shed continuously and replaced like the eddies that form alternately on the sides of a spinnaker and make it unstable.

I think that this single picture tells us that the wake is not just a mass of squiggles as is so often implied by technical diagrams. There is a pattern within the wake. I have no doubt that the pattern changes for an aerofoil as its angle of attack increases up to the maximum of 90º but I doubt if it ever just becomes random. There is no reason to suppose that the flow round a sail will behave in a significantly different way. I think that it is possible to draw tentative flow patterns for a sail at various angles to the wind and show what happens but not what happens in the wake.

In figure 11 I have drawn flow patterns for a sail with a camber of 12% and I have drawn them for angles of attack of 10º, 30º, 50º, 70º, and 90º. It is clearly a progression of change from what looks like an aerofoil with detached flow on the upper surface to obstructing the flow just like a parachute does.

In each pattern I have drawn the stagnation line and that starts at the mast for α = 10º and moves progressively to nearly the middle of the sail when it is at 90º to the wind. The stagnation line is the dividing line between the flow that will go over the sail and that which will go under the sail and its position is dictated by the fact that the momentum flow away from the stagnation line and upwards equals the momentum flow downwards. This follows from the fact that the forces due to pressure acting across the stagnation line must be equal.

The asymmetry in the 90º position is due to the presence of the mast.

The wake from the sail increases in width as the angle increases but, as one might expect from the geometry, it increases more rapidly at the lower angles than as it approaches 90º. This suggests that the lift will decrease and the drag increase as the angle of attack increases until there is no lift just drag.

I have argued that the pressure over the top of the sail will be non-uniform and lower than atmospheric pressure. Clearly the flow lines crowd together more at the luff and leech as the angle increases and we might expect a progressive drop in the pressure over the convex surface.

The curvature of the flow lines on the concave side will lead to an increase in pressure just to force the flow lines to make the large change in direction to produce a centripetal force. We must expect to find a steady increase in the net force on the sail and of course a progressive change in its direction.

It seems to me that these diagrams are consistent with the actual behaviour of sails.

We need to relate the force on the sail to the angle of attack. The first observation must be that the chord line at the foot for a real sail will be coincident with the boom and that if the boom is at zero angle of attack the sail will not fill. If the angle of attack is increased steadily at some angle the sail will stiffen under the pressure forces exerted on it and fill. It can then produce a force. The angle at which the sail fills is quite small, just a few degrees. Once the sail has filled the angle of attack can be increased and the force on the sail will increase. However we do not know either the magnitude or direction of the force and the five flow patterns show that the course must turn to become in line with the wind at 90º. We have a problem to decide how best to think about this.

If one wants to measure a force in this way presumably the result will be recorded for future use so we need to fit in with existing methods that have been tried and not found wanting. So, for our sails we might contemplate finding out how  and  vary with the angle of attack measured to the apparent wind.

What we have to do is complete the graph in figure 12. It may seem perverse to start to do what looks like guessing but there are physical limitations on what is possible for this sail and this narrows down the possibilities. We know that the graphs for  and  lie inside this rectangle because for the best aerofoils  does not exceed 1.6 and that  could only be as high as 2 if the pressure on the windward were to be  and on the leeward side . We also know initially that the coefficient of lift at  = 90º is zero.

Now I must consider what the maximum value of  might be. First it cannot be greater than 2 when the pressure over the whole of the concave face would be  and over the convex face would be . That is not feasible but aerofoils can reach 1.6. We might expect large sails to reach 1.4 but for small sails 1.2 is more likely. So let me settle for 1.3.

This means that we can add another line to the graph to give figure 13. Then the  curve must fit in the bottom rectangle whilst the  graph still may go up to 2.

I now want to consider the position and shape of the maximum especially whether it is a peak like that for an aerofoil or whether it is curved like the top of a sine curve. I think that this is important because, if it curved, the value of  will not change much over a useful range of angles of attack and make the sail less sensitive to fluctuation in the direction of the wind. The peak exhibited by an aerofoil is the consequence of a sudden change in the flow pattern when the flow over the top becomes detached. There is no such change in the flow pattern for sails because the flow is always detached so the curve will be continuous. In order to make a first stab at its shape I think that the slope of the graph of  for an aerofoil over the first 15º at 0.1 per degree is some guide. The graph for a sail will be lower than this but not by much I will add that slope as in figure 14.

In the figure 14 I have added the start of the aerofoil graph and the start of the sail graph. The actual curves will break away from these two lines. Keeping in mind that the value of  will be zero at  90º, if the graph is a smooth curve that starts at the origin it will either have a shape that is asymmetrical with a maximum value for  between about 20º to 45º or it will be more or less symmetrical and take a maximum at about 45º. In figure 15 I have drawn the symmetrical curve as a dotted line.

As I was working through this it seemed to me that this dotted line might be the line for all sails and aerofoils operating with detached flow. It might be a sort of default line to which aerofoils switch when they stall. I needed some genuine experimental data for a real aerofoil over the range of angles of attack from 0º to 90º.

I found experimental data via aerospaceweb from Sandia National Labs report SAND 80-2114 giving lift and drag coefficients for NACA 0015 for angles of attack from 0º to 180º. (They were interested in vertical axis wind turbines.) The NACA 0015 is shown in figure 16. They gave their results in two graphs of  in figure 17 and of  in figure 18

The graph of  looked to be exactly what I thought might be the case. The data needed to be plotted for the range 0º to 90º and I have done for both graphs in figure 19.

Then it is a short step to interpolating the default graph as I have done in figure 20 and it is hard to escape the fact that the effect of the shaping of an aerofoil is to add the "ear" to the default graph for devices with detached flow, and let us have the special case of aviation where the flow is nearly always attached because of the special shape of the aerofoil. There is a link here between sails and aerofoils. To go into that "ear" the aerofoil must have a correctly shaped nose.

These two graphs of  looked to me like a sine wave and a cosine wave. I tried curve fitting to find models for these curves and the result is in figure 21 where the curves are just sine waves with that for the drag displaced by 5º.

The experimental points from the Sandia National Labs report are superimposed on these graphs as shown in figure 22 as black dots. The coincidence is remarkably good.

I think that it is necessary to assess its reliability. I think that we must accept the source of the experimental data as reliable. But we have data for a symmetrical aerofoil section, the NACA 0015, when what we want is similar data for a soft sail. I have described the points on the  graph as the default line to which aerofoils go when they stall, that is, when the flow over the upper surface becomes detached. This sinusoidal shape is so clear that one might expect there to be some fundamental mechanism at work here but I have nothing to offer.

So we must now consider whether the graph can be used for a single soft sail. There is a clear difference in that the sail has one convex surface and one concave surface whereas the aerofoil is symmetrical. If I were to draw flow patterns for the NACA 0015 at high angles of attack the flow over the lower face would be very different to that over the concave face of a sail as shown in figures 11a to 11e. The flow would pass the aerofoil without the sharp bends in the belly of the sail that are so apparent for the concave sail and overall the pressures would be lowered. This is supported by the fact that redrawing of the aerofoil section about a curved chord line gives higher peak values of the coefficient of lift and for a sail this peak value seems to be around 1.3. For engineers the graph in figure 22 is as much as one might hope to find and the magnitudes of the coefficients are consistent with other data for sails. I do not think that I shall find a better graph for either of these coefficients. If it is used, then mathematical modelling using these sinusoidal expressions makes any modelling of a sail relatively easy.

If we change the maximum value of  to 1.3 we appear to have an excellent model for the values of  for a single soft sail.

Figure 24 shows the models of  for a single soft sail where the maximum value of  is 1.3 and the maximum value of  is 1.8.

The first thing that we can use the models for is to find out how the lift/drag ratio changes with angle of attack. It is shown in figure 24. One should not interpret this graph too literally and think that this graph bears some direct relationship to the force that drives the yacht. Yachts are driven by drag as much as lift, indeed when running the yacht is driven wholly by drag. When compared with aerofoils this figure seems to be abysmal but the wind is free and the sails in common use are large but not by any means unmanageable as single-handed sailors have shown us, and if they were smaller, sailing may not be so much fun. If we wanted to improve this value of the lift/drag ratio it could not come from an increase in  because it is already better than a symmetrical aerofoil would manage at the same angle of attack and as there is no way to reduce the values of  we have a fact of physics that we must work with as it stands.

However we are looking at graphs where the angle of attack and the coefficients of lift and drag are measured to the apparent wind when what we want is to know the force that is generated to drive the yacht along its course. I bring down figure 5 as figure 25 to show that it would be more useful to know the force on the sail than to know the lift and drag because it is the component of this force in line with the course that drives the yacht. We do not know the lift and drag but we do know  and, as I have already suggested the coefficients of lift and drag can be combined into a third coefficient  to give:-

Force .

If we plot a graph of the value of  and its angle relative to the apparent wind it is a curve as shown in figure 26. It is interesting that this force takes its maximum at about 67º and not at 90º.

We can also plot the direction of the force to the apparent wind from the trigonometry of figure 31 as shown in figure 27.

In fact it is probably better to plot this graph in polar co-ordinates. I have reworked the Mathcad plot to give figure 28 where, of course, the radial length gives the magnitude of  and the angle of the force to the apparent wind.

If we chose a course for the yacht we could set the sail at any angle of attack that we like but only one angle of attack will give the maximum value of the component of  along the course to drive the yacht that I shall denote it as . That will be when the component of  across the course  is tangential to the graph of  relative to the apparent wind. We can draw these two components on figure 28 without difficulty to give figure 29.

We see immediately that the component of  to drive the yacht  starts at zero when the course is directly upwind which is to be expected. The component does not increase significantly until the course is changed to about 40º. We need to think about this from a practical point of view. If the sail were to be allowed to be free to flap so that there is no aerodynamic force on it the yacht would just drift downwind because of the wind drag on all the parts above the water including the sailor. This wind drag is always there and no sail-driven boat will make way until the force acting along the course is greater than the wind drag. It follows that the smallest angle to the true wind that any given boat may sail is greater for boats with high wind drag. We shall need to look at this in more detail.

This angle between the true wind and the course of the yacht is made up of smaller angles. Figure 30 shows a hull when the yacht is beating. There is an angle between the true wind and the course of the yacht that will usually be greater than about 40º. There is an angle between the true wind and the apparent wind of between 7º and about 11º depending on the length of the hull. The hull will have leeway that sets the hull at about 3º to windward of the course. This leaves an angle of about 26º to 30º say 30º that must be attributable to the need to overcome wind drag. So the angle between the course and the apparent wind of about 30º is probably near to the minimum possible angle and, if the yacht is to make any sensible speed against the wind drag it will be greater by perhaps 15º.

Figure 29 also shows us that the maximum value of  occurs for a course of about 150º to the apparent wind when one might expect it to be at 180º and, at this angle, there is no heeling component. The driving force depends on the product of  and  and, as  at 150º is greater than that at 180º it might pay to lay off the wind by 10º or 20º to get the best speed and gybe. Experienced sailors will know whether this is the case.

Generally when viewing the diagram in figure 29 it is important to recognise that in order to convert these coefficients and components of coefficients into forces they must be multiplied by . For a given sail this means multiplying by  which varies considerably with the course. So despite the fact that the value of  seems to be high for courses between say 140º and 180º the value of  is very low because the course of the boat is effectively downwind We get the best combination of  and  from about 70º to 120º.

This model makes it possible to use the diagram in figure 29 to find out how the best angle of attack to the apparent wind varies with the course.

In figure 31 I have used figure 29 again to show the method. I have added the course that makes 70º to the apparent wind and made the heeling component  tangential to the graph of  v angle to the apparent wind.  will then be the vector from the origin to the point of tangency. This vector will be the vector sum  of the coefficients of lift and drag as I have shown. The ratio of lift/drag is equal to .

We have a graph of lift/drag against angle of attack to the apparent wind and it can be used to find the angle of attack. A graph can then be drawn of angle of attack versus angle of course both measured to the apparent wind. It is figure 33. The details are in the table, figure 32.

Figure 32

 Angle of course to apparent wind degrees Angle Lift/Drag Angle of attack to apparent wind degrees 20 75 3.73 10 30 70 2.74 18.5 40 67 2.35 23 50 63 1.96 28 60 59.5 1.7 33 70 57 1.54 35 80 54 1.38 40 90 52 1.28 43 100 49 1.15 47 110 45 1 52 120 42 0.9 55 130 38 0.78 58 140 35 0.7 63 150 30 0.58 67 160 24 0.45 72 170 10 0.29 78 180 7 0.12 88

The graph in figure 33 shows the outcome. The graph is effectively linear and the angle of attack to the apparent wind is about one half of the angle of the course to the apparent wind. Certainly it gives a first order idea of where to set the sail.

Given this graph it is possible to draw a diagram relating the hull, the course and the sail to the apparent wind. I think that it is self-explanatory.

I think that this is as much as we can extract from the models of lift and drag that I have used. It would be attractive if the models could lead to a way to evaluate speed for a given yacht when following a given course in a known wind. Some progress could be made but there are two intractable problems in finding the speed and direction of the apparent wind and having a relationship between driving force and speed for the yacht. We must see what can be done.

The problem of finding the relationship between the force produced by a sail and the course relative to either the true wind or the apparent wind

The true wind is the natural wind when viewed from a fixed position. This wind is clearly not always steady yet, even when it is unsteady, it soon becomes obvious that there is a mean direction to the wind as if there is a flow in one direction with disturbances moving in this flow, for example, a rotating system that produces wind shifts and gusts, veering and backing. The apparent wind is real enough to a sailor on a moving boat but notionally it is the vectorial combination of the true wind and the speed of the boat. The sailor has no visual cues that might allow him to uncouple these two velocities and gauge the true wind so he must learn to sail using the apparent wind and by using some mental image or perhaps electronic image of the relationship between the true and apparent winds.

In figure 35 I have drawn three vector triangles for a boat moving at the same speed in three different directions relative to the true wind. They are in red. The apparent winds are in green. It is very obvious that the apparent wind is greatest when the course is into wind and least when the course is directly down wind.

It would be very useful if we could calculate the magnitude and direction of the apparent wind. In order to do so we must first calculate the speed of the boat along a course at some angle to the apparent wind and know the speed of the true wind. Let us look at the essential steps in what is inevitably an iterative process. We start by putting a realistic value to the speed of the boat. Then, from this speed we can find a value for the speed and direction of the apparent wind. If we had a real boat in mind we could put a value to the sail area and to . This would enable us to quantify the force that drives the boat. In order to calculate the speed of the boat a relationship between the force acting on the boat and the resulting speed is needed. If these calculations prove to be possible a second iteration can be performed using this better value for the speed. There is nothing general in this; it is for one boat and very speculative.

We can make some progress if we find the driving component  of the coefficient of force  for any course relative to the apparent wind and multiply it by the square of the apparent wind. For any given sail this will be proportional to the actual driving force exerted on the sail. However this is not straightforward because we would only know the speed of the true wind. It is possible if the boat speed is taken to be constant.

In figure 36 I have drawn the vector triangle for a yacht following a course that makes angle  to the apparent wind. The yacht moves at a constant speed  and the vector sum of this speed and that of the true wind gives the speed of the apparent wind. The angle between the true and apparent winds is . From the trigonometry of the diagram:-

where  is the speed of the true wind. I solved this equation by trial to find values for  in terms of the other quantities and . In addition we have .

The values of  can be found from figure 29 and this leads to the table in figure 37.

True wind speed = 15 knots; speed of yacht = 7 knots.

Figure 37

 degrees degrees 30 426 13.5 0.14 64 40 387 127.5 0.31 119 50 344 20.8 0.56 172 60 305 23.8 0.68 208 70 252 26 1.05 231 80 211 27.4 1.12 235 90 176 27.8 1.3 229 100 146 27.4 1.5 217 110 122 26 1.62 198 120 104 23.8 1.75 182 130 90 21 1.83 166 140 80 17.4 1.86 149 150 72 13.5 1.86 134 160 67 9.2 1.87 126 170 65 4.6 1.86 120 180 64 0 1.8 117

Figure 38 is really two graphs and needs some care in its interpretation. The red one is for the product of  plotted against the angle of the course to the apparent wind. The blue one is for  plotted against the angle of the course to the true wind.

In order to interpret this graph we need to have some idea of the relationship between the force produced by the sail to drive the yacht and the resulting speed. Figure 39 shows a resistance/speed graph for a typical yacht hull. It is the shape of this graph that interests us here. However it is more useful if it is re-plotted as a graph of speed versus force to drive the yacht to give figure 40. Then we can see that, in the region where the speed is greatest, a large change in force produces only a small change in speed.

I have drawn attention to two points on the graph. The first point A shows that for only 7% of the driving force that is required to give maximum speed the yacht will travel at 70% of the maximum speed. Between A and B the resistance of the hull due to wave-making is starting to build up and, from point B, where the speed is about 83% of the maximum, the force required to drive the hull increases from 20% to 100% of the maximum force. No doubt this is the range in which people want to race and so most sailing takes place when the driving force is greater than about 20% of the maximum.

If we use this information in conjunction with figure 38 we see that, for the red graph, where angles are measured from the apparent wind, the product  is in the range of 50 to 250 for courses from 25º to 180º. For the blue graph, where angles are measured from the true wind, the range will be from about 40º to 180º

It should be remembered that before the yacht can make way in a controlled manner the driving force must exceed the wind drag on the yacht. It follows that the angles of 25º and 40º do not represent courses at which the yacht can be sailed.

So, for every course with the exception of those at very small angles to the apparent wind, the yacht will sail at speeds above the speed at point B in figure 40. This means that the decision to let the speed of the yacht be constant does not invalidate the implications of figure 38.

It seems that the single sail is satisfactory over the whole range of points of sailing except those points close to the wind and, of course, running. We need to look to ways of improving the performance when beating and the most commonly used way is to split the total area of the sail into two smaller sails and use them to form a Bermuda rig. The only way to increase the speed when running is to increase the sail area and this means using a spinnaker or a gennaker but that sail is most unlikely to be the subject of any realistic analysis.

### The single sail in three dimensions

So far I have been working in two dimensions having said at the outset that the flow is very complex. The outcome seems to me to have the potential to be useful but only if some means can be found to allow for the three-dimensional character of the flow. Given the extraordinary range of different shapes for sails there is no hope that any process can be found that will permit the numerical analysis of a given sail so that its performance can be predicted accurately. Probably the most that we can expect is to get an idea of the magnitudes of forces by using coefficients that have been altered to take into account the effects of three-dimensional flow and any other relevant factor.

We have little to go on in connection with sails but we have some information from aerofoils that can be carried over to sails. The first of these is the creation of rotating flow in the wake from the sail.

Figure 41 is a photo of a crop dusting aeroplane. Its wings create two counter-rotating swirls in the air. The mixture being sprayed on to the crops serves to show a part of the flow pattern and it indicates that the shape of each swirl is essentially helical like the handrail of a multi-turn spiral (actually helical) staircase. It is part of a much larger flow pattern that is generally called a vortex.

We have seen that, in general terms, the pressure on the upper side of an aerofoil is low and on the underside is high relative to atmospheric pressure. During the approach to the wing, and as it flows over the wing, the air is given an impetus to make it flow in circles that, when they are superimposed on the stream of air, produce the helical flow. This rotating flow looks rather like a free vortex but others who have looked at this have produced a mathematical expression for the paths that are superimposed on the flow and drawn them as shown in figure 42. The impression one gets of the vortices produced by a wing is that they are small and at the tips. In fact they are large with high velocities in the rotating flow at the tips and much slower, but still rotating, flow starting from every point on the lower surface right back to the wing roots. The crop duster is too close to the ground to show this larger flow. Most of the energy imparted to the air in these “vortices” is in the large radius flow simply because so much air is affected compared with that in the high-speed flow near to the tips. In effect, because the wings are of finite length air can escape round the tips and this reduces the pressure on the underside of the wing and increases it on the upper side. In aeroplanes and in birds this effect is materially reduced by the use of long wings of small chord, the so-called high aspect ratio wing. They are very effective. High aspect ratio sails also have a lower drag for a given lift, that is, they are more efficient than sails of low aspect ratio. However lots of aeroplanes and most birds do not have high aspect ratio wings because there are other constraints. Aeroplanes have to be matched to their function, birds have to fold their wings and sailing boats cannot all have tall, slender, stayed masts on which to rig high aspect ratio sails. Birds that have wings of low aspect ratio and must soar get round this problem by using five or so large feathers at their wing tips to control the flow pattern.

There are not many soft sails shaped like aeroplane wings. They are generally more or less triangular. They are cut so that they curve to form a concave surface with a “belly”. This means that they have a greater camber in the middle of the sail than towards the head or foot. We have seen that the wind can flow smoothly over the windward face of the sail and that it will form a bubble containing eddies on the leeward face. The air can obviously flow through the space between the foot of the sail and the deck and “leak” off the top of the sail although the curve from head to foot will tend to limit these flows. In practice, at some point somewhere lower than half way up, the flow splits with one stream bending upwards to form a broad, diffuse, rotating wake off the head of the sail as it combines with the flow from the leeward side and the other bending downwards to form a fairly intense rotating wake off the foot as it combines with the flow from the leeward side. This is a very complex flow pattern and it is almost beyond my draughting skill to illustrate it. I have tried in figures 43 and 44 but regard these diagrams as very tentative.[1] The flow is further complicated by the fact that the sail is normally used on a mast that is heeling. Broadly, the flow on the windward face will spread and the flow on the leeward face will converge. Neither flow will be disturbed by this, indeed the flow on the leeward side will be “held together” to make it even more stable although the eddy pattern must change from the head to the foot of the sail and with angle of attack. We might reasonably expect the profile of the bubble containing the eddies to change according to its position on the sail but not to change in character. Sails also twist to add to the difficulty of describing the flow whether mathematically or in words or in pictures.

Yachts also heel by a relatively small angle that would usually be less than 10º. It is difficult to know the effect of this but it must be evident that any heel will make it easier for the air to flow over the sail and so reduce the pressures. During the opening years of the 21st century the maximum distance covered in a day by large mono-hulled yachts has increased dramatically by 200 miles to the current 600 miles. The principal reason for this is the use of swinging keels that keep the rig and the hull upright. The upright hulls are designed for planing to get round the wave making problem and the upright sailing rig seems to work better. This points to the flow pattern over the sails being vulnerable to heeling. This is not surprising, as a change in the whole flow pattern must occur when the dominant flow starts to change to foot to head and not from luff to leech. The trend in yachts is toward greater stability using keels and weights and I shall proceed as if sails are effectively upright when the heel is less than say 10º and the flow is dominantly across the sail and alter coefficients accordingly.

Then the presence of the two superimposed vortices and the complex wake that they produce does not seem to cause the sail to be unreliable. But the process of generating the vortices does reduce the coefficient of lift and increase the coefficient of drag within the constraint that the coefficient of drag cannot exceed 2.

I am told that the performance of sails either improves with use or gets worse with use. Sails are usually made from a composite of man-made threads arranged to carry load between the points of attachment to the rig and sealed between two films. After assembly into a sail such materials can undergo a gradual change in shape as the threads and the films move and stretch under the load of sailing. There can be only one conclusion from this observation; there is a best shape for a sail. It seems that this was well known to the Vikings who made their sails from only one sort of wool and regarded a good sail as more difficult to replace than a good hull.

I think that this gives a picture of a complex flow pattern that is far beyond analysis by mathematics and, if it is amenable to any mathematics, only a very few will benefit. As an engineer I know that I have to proceed by creating a mental picture of the flow and refining that picture as and when the opportunity arises and making the best use I can of it. Then I have to either devise, or look out for, rules of good practice in the design and use of sails. There is nothing new in this; it is what engineers do when faced with an intractable problem. Fortunately the sail, provided that it is more or less upright, appears to behave in same way whenever it is used because the flow pattern is robust and reliable. Given this, gathering data by trial is also reliable and rules of good practice are possible. I am sure that they already exist but, by the look of the rigs used on yachts, they are not fully developed.

This all points to the fact that sails and sailing rigs will have to improve by trial and no doubt the process of trial can be facilitated by the use of computers especially in creating shapes so that we know what is tested.

I think that I can proceed as if the three-dimensional sail behaves like a two-dimensional sail but with a lower lift coefficient and a proportionally higher drag.

[1] I have yet to see a really persuasive diagram of this flow.