Section 1 The single soft sail

 

The wind

Sails exist is to utilise the wind to drive boats. If we are to think about sails we must first have some ideas about the natural wind and keep these in mind when attempting to analyse the performance of sails.

 

There is no question that the average speed of a natural wind is lowest near to the ground or the surface of the sea and that the average speed increases with height. Glider pilots make their approach to landings at a speed in excess of the stalling speed of the glider so that, as the glider flies down through the diminishing wind speed, the airspeed does not fall below the stalling speed. It would be nice to think that the wind speed changed with height in an orderly way but this is not the case. The wind in the first 10 feet appears to eddy vigorously and then settle down with further increase in height. Some how the sails must cope with this motion.

 

There is the further problem of the way in which the wind appears to swing from side to side in some sort of cycle. The photos in figures 1 and 2 are obviously of a windsock. It was at the down wind end of the very long runway at Lasham Airfield in Hampshire, England. The wind was blowing from end to end of the runway. I observed that the windsock was changing its direction repeatedly and chose to photograph the two extremes of the same cycle of swinging. The time between these photos was short, to be measured in tens of seconds, as is evident from the clouds. At the time I did not have any idea how this motion was generated but I think now that it is the result of the passing of vortices that are in the wind and moving with it.

 

This motion is of large scale when compared with the yacht and action by the crew is needed to respond to it.

 

Thinking about sails

It seems to me that sailing as a sport involves working with a natural wind that swings around and increases and decreases in speed on a water surface that has waves with many natural variations. Sailing would be no fun if this was not to be the case and what sorts out a good sailor is the ability to make the boat go quickly despite these problems. A good skipper on a big boat will anticipate the arrival of a wind shift that is in fact the passing of a vortex and will have his crew ready to make all the trimming changes that will be needed to take advantage of the wind shift. Dinghy sailors will not just sit in their boats but will be ever moving their body weight and adjusting the rudder and trimming the rig to help the boat up a wave and down the other side.

 

These are not activities that lend themselves to lots of measured data. What is wanted is a working knowledge of the mechanics of sails and sailing rigs so that the behaviour of a boat can be understood and taken into account when developing a skill through practice. This involves having an approach to the problem of sails and it seems to me that the approach used in aerodynamics for wings can be carried over to sails provided that it is recognised that sails are like deeply-stalled wings with all that this implies.

 

In order to do so I need to explain the concepts of a coefficient of lift and a coefficient of drag.

 

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  and  are the coefficient of lift and the coefficient of drag,  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 have this structure because  is the pressure produced when a fluid flowing at  is brought to rest and, when this is exerted on an easily defined area[1][1], , gives a comprehensible force on which to base a coefficient. The two coefficients vary with Reynolds’ number, surface finish and angle of attack.

 

The ratio of lift to drag is an obvious way of assessing aerofoil efficiency. It is the same as the ratio of  to . For early aerofoils a ratio of 12 was all one might hope for. By the time of the Second World War this ratio had risen to 30  which is still a practical value.

 

In figure 3 I have shown an aerofoil set up to make an angle to the undisturbed flow. This angle is called the angle of attack. I have shown, to scale, the three forces lift, drag and total force for a lift over drag ratio of 25. It would be difficult to draw the diagram for higher ratios and this emphasises just how small the drag is when compared with the lift. The aerofoil is a very efficient device. Modern gliders may have a glide ratio of 60 to 1 and can travel about 60 miles from a starting height of 5,000 feet in the absence of vertical movements of the air. More revealing is that the drag on a glider spanning perhaps 100 feet is only 10lb for each 600lb of all up weight. The lift over drag ratio is 60, which is the ratio for the lift of the wings to the drag of the whole glider! This is quite astonishing but this performance is very fragile as just a few insect strikes can reduce it to 40 to 1 instantly.

 

I want to draw your attention to figure 4. It is a graph of coefficient of lift,  as used in aerodynamics, versus angle of attack, a. It starts off with graph ABC that is for an aerofoil of symmetrical section, typically NACA 0012-64. All practical aerofoils share a graph from 0° to about 10° (  = 0.8 at 8°) but the graph between B and C varies with the shape of the aerofoil. For all sections, the graph takes a peak that tends to be rounded for asymmetrical sections and quite pointed for symmetrical sections. This peak is the point of stalling where the aerofoil loses its lift. The asymmetrical section might stall gently but symmetrical sections tend to stall suddenly.

 

Aerodynamicists are not generally interested in the performance of stalled aerofoils but those interested in soft sails most certainly are.

 

It is a matter of observation that the soft sail does not fill until it has an angle of attack of more than 30°. This can be demonstrated. Sheet in the rig on a yacht until the yacht is sailing as close to the wind as it can. Use the rudder to come up into wind and go into irons. Centre the rudder and leave the yacht to fall off the wind until the sails fill. Note the heading. Repeat this for the yacht to come out of irons on the opposite tack and note the heading again. The angle between these two headings is twice the angle of attack at which the sails first fill. It is anything from 70° to 90° so the angle of attack is more than 35°. I am not sure whether the sail would collapse at the same angle as it fills, my impression is that, once filled, a sail can remain filled to a smaller angle of attack but it still collapses at an angle of about 30°. Given that the very, very best wing on a high performance glider stalls at about 16° there can be no hope that a soft sail can operate other than in a condition where it is aerodynamically stalled. I do not think that anything is to be gained by pretending that this is an aerofoil. It is a soft sail with its own characteristics and I propose to explain these characteristics.

 

However it is useful to look first at the rest of the graph for a symmetrical aerofoil. In the absence of experimental data I have to infer a graph. It obviously starts at C and, in some way, goes to D where the coefficient of lift is zero at 90°. There will be no peaks or bumps so the graph is likely to be continuous and to occur within the area that I have shown hatched in red. There is no reason to think that a soft sail, once filled, has other than a similar relationship between coefficient of lift and angle of attack. Therefore the soft sail operates from say 35° to 90°. I have drawn the relationship as a band hatched in red because I have no experimental data. This looks to be poor compared with the aerofoil but it is not. The soft sail is eminently practical. Let me explain.

 

We shall need to have some idea of the coefficient of drag, . Once more I have no data but the value of  when a = 90° cannot be greater than 2 which is very large compared with the value at the stall. It is likely that the graph of  versus  is contained in the area hatched in blue.

 

If we ignore the practical problem of what to do with a rigid wing when it is not driving the boat, the wing sail looks to have the potential to be far more efficient than the soft sail with a coefficient of lift twice that of the soft sail and, just as importantly, a very much lower coefficient of drag. There is a snag. In order to take advantage of these attractive characteristics the wing sail must be set at the most practical angle of about 10° to the apparent wind. This is a tall order because it requires a control system that can sense the direction of the apparent wind very accurately and we do not have such a device. It might even be impossible because of swirls and eddies in the natural wind. But it pales into insignificance compared with coping with wind shifts. The photographs of the windsock show veering of  10° or more to the true wind and, if a wing sail operates in this wind without adequately swift response to wind shifts, the angle of attack could change from about  ° to + 18° in a very short time. Then the force on the wing could change quickly from being in the wrong direction to being beyond the stall. Such a process is very violent and puts great strain on the yacht.[2][2] [3][3]

 

A look at the graph shows that there is not much difference between the coefficient of lift of the wing at 10° and the coefficient of lift of a soft sail when it first fills. There will be a greater drag but, given the advantage of being able to furl and change soft sails and to stow them away, the soft sail is much more practical than a wing sail. In my view the soft sail is a very good solution to a difficult engineering problem.

 

The single soft sail

I shall confine my interest to sails used in fore-and-aft rigs because the analysis of the three-masted, square-rigger is outside my competence[4] although I think that it s possible to see how they work. I think that the analysis of the Bermuda rig opens the way to understanding most of the other rigs. Certainly it was possible to predict correctly the relative positions of the sails on the ill-fated Team Phillips for reaching. The Bermuda rig has two sails, the main and the jib, and is the most-used rig. I want to start with a single sail that is operating in a steady wind.

 

In my view the use of mathematics is most unlikely to yield any understanding of this rig especially if you start by treating it as an aerofoil. The way forward is through the mass of data on the behaviour of fluids flowing round bodies of various shapes all of which has been found by experiment. I know that this data is stored using formulae in the form of mathematical equations, like those for  and , but the equations do not in themselves contribute to our knowledge and do not make the data theoretical. Can there be any experimental data that has been more carefully gathered than that due to the NACA? Yet its storage system looks to be very theoretical.

 

It seems to me that the first thing that is needed is to decide how a single sail can interact with the wind to generate a force to drive a boat. We need a flow pattern. Figure 5 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 come from smoke. 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. And the aerofoil makes an angle of 30° to these lines. It is, therefore, at a slightly smaller angle than the smallest angle for which a soft sail fills. 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, in suitable units, does not change. This means that where the speed is high the pressure is low and vice versa. The flow diagram lets us see where pressures are higher than average and where they are lower because where the flow lines diverge the speed is decreasing and the pressure is rising and where they converge the pressure is falling. Where they are parallel and close together the speed is high and the pressure low. 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 5 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 stream 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.[5][4] It will draw air out of the white region and into the wake. So we must expect to find some flow pattern that takes in air at the top and sucks it out at the trailing edge.

 

Figure 6 shows the flow pattern round an asymmetrical aerofoil due to Ludwig Prandtl[6][5. 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 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”[7] position and shape for the eddies built up over time in wet paint on glass. It is a magnificent picture that is at least 70 years old. Now we have to extract information from it.

 

In figure 7 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.

 

This flow pattern is stable and so it should be transferable to the soft sail. As I have never seen anything equivalent to the paint picture for a single surface like an inflated soft sail I shall have to construct a diagram with all these features in a way that is consistent with science.

 

This is shown in figure 8. Before I could draw it there was a decision to take about the mast. One might try to decide what the best shape should be for the luff of a sail. Some might think that a sharp edge would be best but this seems to me to be unlikely.[8][6] This matter is academic because the sail must be practical. It will lead with a wire, a mast or a bolt-rope. Jib sails do not seem to be materially affected by having a wire in a pocket or a wire set ahead of the luff of the sail, indeed the jib is used to indicate how close the rig is to pointing too high and losing drive. For sails rigged fore and aft a mast is essential. The mast can be of any cross-section although the obvious section is circular. Claims have been made for the superiority of one shape over another, but, it seems to me, that the circular section is as correct for a sail as the bulbous bow is for the stem of a boat and for the same reason. It causes the air to part ahead of the luff and not at the luff. However the sail has to be attached to the mast and this usually leads to a gap between the mast and the luff through which air can flow. This gives a second point of entry to the bubble behind the sail. It would be best to avoid this because it increases the drag of the sail/mast combination. I have chosen to draw a circular mast with the sail coming from the “middle” of it and sealed. If I were to plan a development programme for the best shape for a mast this is where I would start but I would not automatically discount the possibility that a gap might have some advantage.

 

Prandtl did not show much of the upstream flow nor any of the downstream flow. This is the normal thing to do but I have extended the diagram because both are important to us. Let me list its features. The overall effect of the sail on the air flowing over the sail in my diagram is that the air affected by the sail is deflected downwards. This must occur if energy is to be extracted by the sail to drive the boat. A most important feature for us to notice is that the air is not just deflected downwards from its original path but is first deflected upwards ahead of the sail. The flow over the underside of the sail goes up into the curve of the sail and over most of the sail the flow is convergent. There is no reason to suppose that the bubble containing the four eddies is any different to the bubble over a stalled aerofoil so I reproduce it here especially as I am convinced that the point of separation near to the leech occurs on real sails. The air leaves the sail as a wake of highly disturbed air that contains eddies that break up into smaller eddies.

 

This is what really goes on round a soft sail when it is working. The flow pattern is robust in that it does not alter dramatically with angle of attack or wind-speed. The important breakdown occurs when the angle of attack becomes too small and the sail collapses. I do not know the sequence of events that leads to this collapse but, fortunately for sailors, it is initiated at the leech as is evident from fluttering of the sail. I would speculate that the breakdown starts when the point of division of flow moves behind the leech and opens the bubble to let the system of four eddies be swept away into the wake in preparation for what would be a re-attachment if the sail were to be rigid.

 

It is a matter of observation that disturbances in convergent flow quickly die away and that disturbances in divergent flow normally grow down-stream. Our sail will not be troubled by discontinuities in the concave face nor will it be affected by discontinuities on the convex face because the flow is already disrupted. However these discontinuities do add to the drag exerted on the sail by the wind. Generally a sail is insensitive to changes in angle of attack.

It follows that it should be possible to draw a graph of  and  against a for a sail. I have done so in figure 9 by cropping the original graph for the aerofoil at a = 35° because the operation of the sail is not certain between 0° and 35°. Given the poor aerodynamic shape of sails it is likely that the lower values of  and the higher values of  will be appropriate. Clearly, on this graph, the highest value of  and the lowest value of  occur at a = 35°  but only fully-crewed ocean racers can continually adjust the rig during gusting and veering to make it possible to work at this angle continuously. Normally the sail must operate at some greater angle. I have added a line for 40° and hopefully the sail might operate in this 5° band at, say, 37.5°.[9][7]

 

Then the sail will operate with a value of  of about 0.5 and a value of  about 0.25 giving a ratio of  /  of 2 and perhaps higher.

 

With this information we can represent the force on a sail as in figure 10. The sail is, of course, set to the apparent wind. I have set it at 37.5°. I have drawn the lift at right angles to the apparent wind and the drag in line with the apparent wind and combined them to give the force on the sail.

 

From the start we recognised that the force on the sail caused by wind flowing over it would be exerted as pressure that would be above and below atmospheric pressure and be distributed in some consistent way over the sail. The net effect of this pressure pattern can be represented by this single force and a moment. The moment comes into existence because the pressure pattern does not balance out between the two sides of the sail; the high-pressure area on the underside is near the luff and the main force on the upper side is near to the leech.

 

Even though this diagram is for a two-dimensional flow the pressure pattern in three dimensions over a real sail or even a whole rig would also reduce to a single force (acting at the centre of effort) and a moment. The moment can be balanced out and, if the force acts forward relative to the course of the boat it will drive the boat.

This opens the way to an explanation of the way that a sail can drive a boat. I have reduced figure 11  to the sail, the apparent wind, and the force on the sail because this is the basic driving unit of all sail driven boats with fore and aft sailing rigs. We need to understand the characteristic behaviour of this unit when it is attached to a hull and used to drive the hull in any of the possible directions.

 

Now I want to find out what happens when this simple device is added to a suitable hull to form a sailing yacht but first I must look at the sail in three dimensions.

 

The single sail in three dimensions

Figure 12 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 vortex is essentially helical like the handrail of a multi-turn spiral (actually helical) staircase. It is part of a much bigger 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 the diagram. 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. This energy is lost and is paid for as drag. In aeroplanes and in birds this drag 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 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 14 and 15 but regard these diagrams as very tentative.[10] 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. We might reasonably expect the profile of the bubble to change according to its position on the sail but not to change in character.

 

During the opening years of the 21^st century the maximum distance covered in a day by large mono-hulled yachts has increased dramatically 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 hulls are designed for planing to get round Froude’s critical speed and the upright 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 is from 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 my sails are effectively upright and the flow dominantly across the sail.

 

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 vortices do reduce the coefficient of lift and increase the coefficient of drag.

 

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 experiment 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.

 

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

 

The performance of the single soft sail.

I want to draw diagrams to show how a sail behaves when it is employed to drive a boat. It is very complicated and some complications will have to be taken into account by preliminary decisions. The first decision is to use the same values for the coefficients of lift and drag as I used previously and to look at the outcome before changing them. I shall have to make some decision about whether to draw diagrams relative to the true wind or the apparent wind. Analytical purism says the true wind but I am not so sure. I think usefulness is at least a factor. I need to have a clearer idea of the other factors.

 

 Let me start with the relationship between true wind and apparent wind. For different reasons both of these are notional winds. 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. The apparent wind is real enough to a sailor on a moving boat, its direction can be observed with a masthead burgee. 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.

 

Apparent wind is not something special to sailing. Try walking along a seaside promenade that runs east-west in a wind from the north-east in winter. The terms “walking into the wind” and “walking with the wind” come to mind. The speed of walking is enough to make a difference in all but the most fierce wind.

 

For sailing there is some value in plotting the strength of the apparent wind for various courses and the angle between the true and apparent winds. It is easily done using any computing package for mathematics. In figure 16 I have taken a fixed boat speed and a fixed speed for the true wind and I have taken the true wind to have a fixed direction. Using this basic diagram I have drawn a family of graphs in figure 17 for a true wind of 15 knots and boat speeds of 4, 6, 8, and 10 knots and this will bracket all the likely boat speeds

 

The angle between the apparent wind and the true wind can come in the same way. The first 40° of the graph have been omitted because the boat cannot sail in this angle.

 

The lower family of graphs is for the apparent wind and the other for the angle between the true and apparent winds. Clearly the angle is greatest where the boat is reaching and is going to be at least 15° rising to perhaps 30°. The apparent wind is not much different in strength from that of the true wind. For reaching the helmsman will set the course using marks or compass or land features and the crew set the rig using the burgee and any other cues they can contrive. It is the apparent wind that is important.

 

For beating the strength of the apparent wind is quite high but it is at a much smaller angle to the true wind. The helmsman will be steering relative to the apparent wind to lay a course that is giving the greatest speed made good and the crew getting the best from the rig using the same cues.

 

For running down wind the true and apparent winds are in the same direction but of different magnitudes.

 

It seems to me that a sailing diagram for a sail using the apparent wind would be useful and easy to draw.

 

In drawing figure 18 I have supposed that a sailor can set the single sail at the angle of 37.5° to the apparent wind for any of the possible courses that can be sailed. I have drawn eight representative courses each with a hull set along it. (I have ignored leeway  which will, of course, skew the boat to windward of the course by a few degrees.) To each hull I have added a sail, in black, in the correct relative position.

 

The force on the sail is not the same for every point of sailing but, for this initial diagram, I have ignored the variation. Then the force available to drive the sail is the component of the force in line with the course. In effect I have to draw right-angled triangles and these will all lie in a circle of a diameter equal to the force on the sail. I have drawn this circle and inserted both the component along the course in red and the component acting across the boat in blue.

 

The first observation from this diagram is that, in principle, the boat could sail any course between 90° forward of the direction of the force to 90° behind the direction of the force. The diagram also shows that, as the course approaches 90° forward of the force the drive disappears. However the component driving the boat increases rapidly as the boat bears away and only a small change in course to leeward is needed to generate a useful drive and to let the sail be set to leeward. Clearly the sail operates best in the angle round to the fifth of the courses I have shown. For this course the sail is square across the boat and at the limit of the sheeting equipment. I have drawn the sail in the “theoretical” positions but they are obviously not practical and I have added the practical positions in green.

 

Between the fifth and eighth positions the angle of attack of the sail is increasing and, as a result the coefficient of lift decreases and the coefficient of drag increases. The sail changes progressively from being deeply-stalled to being square to the wind. It would be quite possible to use the graph of  and  against a above to redraw the changing force on the sail and its components. However I have extracted enough from this diagram to show that a single sail works best over a range of course between beating and reaching but is not so good when close to the wind and when changing from broad reaching to running. These deficiencies lead to the use of rigs comprising a main sail and a fore sail and the use of spinnakers and other special sails.

 

In the previous diagram I took the force on the sail to be fixed. This cannot be the case because the force is proportional to the square of the apparent wind speed. Now I want to draw a diagram to show the way that a sail behaves when it is harnessed to a boat to drive it through the water. If the result is to be realistic I need to know the strength of the apparent wind for practical values of the true wind speed and for the boat speed. I will simplify the process by using a true wind of 15 knots and then I need a value for a typical boat speed.

 

Boat speeds, for displacement boats in a given wind, vary with the length. William Froude gave us a way of sorting this out. Froude was interested in finding a way of predicting the power requirement to drive a merchant ship at an economical speed. This involved a study of the surface waves made by such ships and he noted that there would be a speed at which the distance between crests of the wave created at the bow would be equal to the length of the ship. It would be impossible for a ship to achieve this speed but quite possible for a yacht. However any increase in speed beyond this critical speed would cause the stern to drop and the boat to start “climbing its own bow wave” with a prohibitive requirement for propulsive power. Even yachts cannot go beyond this speed to any advantage and the next step is planing.

 

I want to draw graphs using polar coordinates of boat speed versus course to show how the single sail loses its drive when it sails close to the wind and when it runs. This is not easy to draw because the course must be measured either to the true wind or to the apparent wind and the force on the sails varies with the apparent wind and that varies with the course, the boat speed and the speed of the true wind. This might be possible if there was a simple mathematical expression relating the force driving a boat and the resultant speed but there is not. Progress would be possible if, in fact, the speed of the boat did not vary too much with the speed of the apparent wind. In order to find out what is possible I need information.

 

 There is an expression for this critical speed, it is:-

             critical speed =  knots where  is in feet.

It is usually changed to  and called the hull speed. Most yachts, including cruising yachts, can reach this hull speed in winds of 15 to 20 knots and racing yachts can easily achieve this speed. The range of speed for any displacement hull is from 0 to its critical speed and in figure 19 I have given the graph of critical speed against length for lengths up to 100 feet. The most commonly used boats have lengths between 25 feet and 40 feet and have hull speeds between about 7 knots and 8.8 knots.

 

Judging by the sailing boats that I see on the River Thames at Gravesend only people who are racing attempt to get the best out of their sails. Those who are just in transit take advantage of the forgiving characteristics of sails and simply set the rig to give a drive and steer. So I need to think about rigs when they are used for racing.

 

I need a relationship between the boat speed and the resistance to motion for a given hull speed. I know that this all varies with angle of heel but I want to see the general shape of such graphs not put numbers to it.

 

The graph in figure 20 is typical of the shape of the graph of resistance versus speed. The right hand edge corresponds to the critical speed. It is drawn mathematically to have the same shape as a published curve derived from measurement. As expected the graph shows a pronounced “knee” as the bow wave builds up. But it also shows that at low speed the resistance is very low and this makes sailing in light winds possible. This is really a quite important feature of the behaviour of a hull.

 

When the boat is under way the force from the sails to drive the boat is equal to the resistance of the hull. Then it is more instructive to plot speed against driving force  which is the next graph.

 

In figure 21 the top edge of the graph corresponds to the hull speed or critical speed. It now becomes obvious how easily the boat can be driven at low speed. For 10% of the force required to produce the hull speed the boat will move at 55% of the hull speed. However racers will want their hulls to move in the range of speeds from 80% of the hull speed and upwards with an effective maximum of the hull speed. The range of speed is very small for the range of force required to produce it. This range of speed can be produced by true winds of 15 to 20 knots.

 

I think that I can usefully take the boat speed to be constant at 90% of the hull speed and attempt to draw my graph.

 

Suppose that the length of the boat is 25 feet. Its hull speed will be 7 knots and 90% of that is 6.3 knots. If the true wind speed is 15 knots simple trigonometrical relationships will give the apparent wind speed and the angle between the apparent wind and the true wind. I have used a polar plot so that the speed and angle is spatially related to the course of the boat relative to the true wind.

 

Of course the boat cannot sail upwind closer than say 40° but I will cope with this later.

 

Given the relationships above it is possible to draw the polar graph in figure 22 of the force produced by the sail for any course relative to the true wind. In figure 23 the black triangle is the vector diagram combining the true wind and the speed of the boat to give the apparent wind. The lift on the sail is at right angles to the apparent wind and the drag in line with this wind. I have let these two forces be in the ratio of 2/1 as before but one should keep in mind the fact that a three-dimensional sail will have a lower lift and a larger drag. They are combined to give the net force on the sail. I have added in green the two components of the net force on the sail, one along the course and driving the boat and the other at right angles to the course and more or less transverse to the hull.

 

A development of the calculation for the apparent wind will permit the calculation of the force driving the boat. I have treated the force as being proportional to the square of the apparent wind. The outcome is the polar plot in figure 24.

 

There is a region that I have labelled “no sail” where no course is possible because the sail will not fill. There is a second region in which the sail is not set at 37.5° to the apparent wind but sheeted out only to about 90° to the hull. This region starts at a course of about 130° to the true wind and from this course to 180° the angle of attack changes to 90° when it will be square to the wind and not lifting. So the plot is valid for about 90° where the boat is reaching.

 

It would be interesting to know how the drive changes during the change to being square to the wind but this is another exercise that must be taken separately. It ends in a whole new application of the sail when it is being used for a job that is better served by the use of s spinnaker. It might just be worthwhile to look at the mechanics of running directly down wind before I look at the Bermuda rig.

 

We have for a sail set at right angles to the wind that Drag = . This drag is now the force available to drive the boat,  is the speed of the apparent wind which is the difference between the true wind speed and the boat speed,  is the area of the sail and  takes a value of about 1.5.

 

It follows that we can write force to drive the boat =  where T is the speed of the true wind and B is the speed of the boat and k is a constant. Graphs to a suitable scale can be superimposed on the graph relating hull resistance to speed. In figure 25 I have shown the resistance in red and three graphs of drive for values of k of 1, 2, and 4 in green, blue and magenta. The three drive graphs all intersect at large angles and this means that, for a real boat where the hull and the rig are both effectively unalterable, the equilibrium speed will be reached quickly and can only change in response to a change in the speed of the true wind. Rig adjustment will not alter the speed.

 

If higher speeds are desirable then the only thing that can be changed is the area of the sail and that will require an extra sail eg a spinnaker.