This section on aerodynamics is needed for the understanding of the function of the fin and rudder of a model yacht. We shall see that little of aerodynamics is of direct value in the study of sails but the science of aerodynamics has a well-established vocabulary of descriptive words, which can be “poached”, and used for sails. Sails clearly lack some of the important features of aerofoils and have more in common with kites and hang gliders. They are best regarded as a study in their own right
Picture 6-1 shown here will help us to make a start. It is a picture of a model of an aeroplane wing in a wind tunnel. The model is set up so that we can see its cross section, which is obviously an aerofoil. The lines of smoke indicate the pattern of the flow. It is most likely that the aerofoil was tested in a tunnel with a square working section and that the model fitted quite closely to the side-walls so that there was little leakage between the aerofoil and the walls. The velocity of the air was probably quite low in order to avoid diffusing the smoke too quickly. The smoke would have been introduced to the tunnel by a smoke rake, which is a piece of pipe work looking like a rake with hollow tines. The tines are set up at equal spacing and the rake is jogged at regular intervals of time to produce lines of smoke each having blips which were inserted at the same instant of time. These are the blips on the flow lines. There are important observations to make immediately.
There is a simplistic explanation of the way in which lift is generated that supposes that the air flowing across the upper surface must flow more quickly because the upper surface is longer than the lower one. By implication the suggestion is that the approaching air parts in front of the aerofoil and rejoins in the same respective positions behind the aerofoil. This is just not true; the air flowing over the upper surface is some distance ahead of the air flowing over the lower surface when they rejoin. A closer look will show us that a set of blips is just arriving on the left side at the top but has yet to appear at the bottom. Clearly the air to the top of the diagram is already moving more quickly than that at the bottom. We need more information before we can make more progress.
For this we can go back to our physics chapter in which we looked at the Bernoulli equation and the Venturi. We found that when fluid flows through a converging tube its speed rises and its pressure falls and vice versa for a diverging tube. With this reminder let us return to our aerofoil.
We have a single shot of the air flowing over an aerofoil in a wind tunnel. There is no reason to suppose that the shot would change if we moved the rake across the wind tunnel, ie along the leading edge of the aerofoil. It follows that what we see as a smoke line is just one of many all across the tunnel which together form a curved surface. Air flowing between two such adjacent surfaces is trapped. It must continue to flow between the surfaces just like the air in a Venturi tube. Just as in the Venturi tube when the surfaces part the pressure rises, when they come together the pressure falls.
So, in region A, the pressure is lower than the average, in region B it is higher than average, in C and D it is near to the average. Clearly there is a net force on the aerofoil acting upwards and for separate reasons backwards. (There is also a moment tending to turn the aerofoil nose over tail but this cannot be deduced from the picture.)
Let us see what else can be extracted from our smoke lines if we link them to the regions of high and low pressure. We can discern a stream of air between two flowlines that flows past the underside of the aerofoil and in contact with it. Keeping in mind that flow lines diverge as the pressure rises, we can see that the pressure at the leading edge and over the front part of the underside is higher than the free stream pressure as we might expect. The existence of this region of high pressure is inevitable because the aerofoil is solid and exerts an “active” force on the air. The very existence of this region of high pressure affects all the air moving past the aerofoil. We can see that, in the air moving below the aerofoil, it produces a slowing down upstream (with a consequent rise in pressure ahead of the aerofoil), and a speeding up downstream (with a drop in pressure). The presence of the other air and ultimately the side-walls of the tunnel prevents a sideways spread. The rise in pressure upstream of the high-pressure region is also exerted on the air which will eventually pass over the upper surface to divert it upwards and to make it go faster as it approaches the sharply curved part of this surface. So the presence of the aerofoil affects the flow well upstream to divert it upwards and to accelerate the air that will flow over the top and slow down the air that will flow underneath. This will be seen to be crucial for our understanding of a sailing rig. Now we must look at the air flowing over the top.
Here we meet the region that is totally different to anything we have described before. The air stream is trying to flow away from the surface and there is no obvious reason why it should not do so. Yet we want it to flow smoothly over the upper surface and not separate from it so that the pressure does not rise and reduce the lift.
Now, for anything with substance to follow a curved path, a force pointing towards the centre of the arc of travel is essential. This is the force exerted in a rope when a weight on the end of it is swung in a circle. It is called the centripetal force. It is common knowledge that, if the rope were to be released so that the centripetal force no longer exists, the weight would carry on along a path which is tangential to the circular path of the weight. Air has substance and, if it is to follow the curve of the upper surface of the aerofoil, a centripetal force is needed, that is a force acting on the air towards the surface. It can only come from a drop in pressure below the free stream pressure. Our picture shows that the drop in pressure occurs over the sharply curving first part of the aerofoil. But it is evident that even though the curvature becomes much more gentle at the after parts of the surface the flow lines diverge and the pressure rises although it is still below the free stream pressure. Obviously conditions can and do occur when the pressure available to keep the air in contact with the surface is too small and the flow separates from the upper surface. It can be shown by analysis that the magnitude of the pressure gradient needed to make fluid flow in a circular path increases with decreasing radius. In this context it must be remembered that the radii of an aerofoil of 3 inch chord are everywhere smaller than corresponding radii for an aerofoil of similar shape but of 6 inch chord.
The pressure distribution round an aerofoil has been measured but there is no really good way of showing the result on a diagram. The best we can do is to draw lines at right angles to the surface of lengths which are proportional to the pressure and show which way the pressure acts. Lines may be drawn joining the heads (or tails) of the arrows as shown in the Diagram 6-2. Clearly both sides of the aerofoil have net forces on them and both act upwards in this diagram.
There are a few more points to be made. The change in sense of the pressure at the nose of the aerofoil really is very sudden. On some light aeroplanes the stall-warning device is a whistle which is connected to a hole in the leading edge of one wing. When the wing is flying normally the hole is in the high pressure region but, if the angle of attack increases to a dangerous value, the pressure pattern moves round the wing and the hole “goes” into the low pressure region, starts to suck, and blows the whistle. As we might expect our fore sails to have similar pressure distributions and a sharp leading edge the behaviour of the air at the luff may be of special interest.
The failure of the flow to follow the upper surface is called separation. Separation is not solely dependent on the curvature of the aerofoil. It also depends on the behaviour of the boundary layer. We must explain this idea of a boundary layer. It seems that the fluid, in this case air, in contact with a surface behaves as if the surface is a stationary layer of fluid. Then the effect of viscosity (internal friction) is to prevent the layers of air in the immediate vicinity of the surface from moving as quickly as the fluid more remote from the surface. The resulting variation of velocity in a stream of air flowing over a solid surface is shown in Diagram 6-3. At each distance from the surface the length of the arrow indicates the velocity. The diagram is drawn to a much-exaggerated vertical scale. Clearly the velocity changes from zero at the surface to almost the maximum velocity in a small distance and the air in this small distance is called the boundary layer. The existence of this layer severely limits the performance of aerofoils. Let us see how this comes about.
Let us go back to our aerofoil in the wind tunnel and its smoke lines. These lines give us a good guide to what happens to the airflow but they are too coarse to show what happens near to the surface. However we can tell that, over the aft part of the upper surface, the pressure is increasing as the air slows down. Recalling that a rise in pressure can only come at the expense of kinetic energy we see that the rise in pressure requires a matching drop in velocity. This may be possible for the air in the main flow but the air in the boundary layer does not have enough velocity to start with to produce a large enough rise in pressure. So, at some point on the upper surface, air in the boundary layer and close to the surface comes to rest and air starts to flow forwards from the trailing edge towards this point. This effectively separates the main stream from the surface and causes an eddy to form as shown in Diagram 6-4. Now, where we had low pressure over the aft part of the aerofoil before, we have the pressure at the trailing edge which is more or less the free stream pressure which means that the upward force on the aerofoil is reduced. In addition the energy in the eddy is lost and appears as an increase in drag.
The same result can be produced by a discontinuity in the surface. A fault in the surface such as a joint in the wing skin or surface damage or dead insects and so on are obvious discontinuities. These all produce an arrest in the boundary layer which produces a wake like that in Picture 5-6 and this is all that is required to initiate separation. However there is another way of upsetting the flow and that is by a change in curvature. This is illustrated in Diagram 6-5. Two circular arcs AB and BC of radii R1 and R2 are joined at a common tangent. If air flows over ABC in contact with the surface the inward pressure will be that required to suit the radius R1. At the junction the reduction in radius means that suddenly a greater inward pressure is required if the flow is not to separate. The boundary layer may have been moving before the junction but the sudden need to convert more kinetic energy could well stop the boundary layer at the junction or at best soon after it.
All this gives us a guide to good practice in the design of aerofoils because it means that aerofoils (and streamlined lead weights for keels) should be designed to have gradual changes in curvature unless the point of separation needs to be located for some special reason.
Aerofoils have evolved for over a century now by a gradual process of refinement. Such a process is impossible unless some means of recording aerofoil performance can be devised so that comparisons between aerofoils can be made. Some method of testing is essential and our well-tried methods are the wind tunnel and direct application. From the time of Lilienthal and the Wright brothers the performance data has been recorded in substantially the same way and their method has its roots in James Smeaton’s work on lighthouses and sea defences. The Wright brothers turned to wind tunnel testing fairly quickly after attempts to make measurements on an aerofoil mounted on a bicycle. So, what was this method?
Smeaton was interested in wind forces and wave forces. He recognised that the wind force on a square surface held at right angles to the wind varied mainly as the square of the wind speed and that, whilst the same held true for water impinging on the same surface, the water forces were much greater. He went on to express the force on other surfaces in the following way :-
Force = C. k. . S
where v was the velocity of the air or water
k was the coefficient of air (or water ) pressure such that the product k. equalled the force on one square foot of surface held at right angles to a flow at velocity v.
S was the area of the surface
and C was a coefficient which depended on the shape of the surface and its attitude to the flow.
Lilienthal and the Wright brothers wanted to predict wing areas for the man-lifting gliders that they were flying. They followed Smeaton’s approach initially for man-carrying gliders and the Wright brothers were ultimately successful in flying a controllable powered aircraft. During their work they found Smeaton’s coefficient of pressure to be too great but, whilst over-estimation of wind and water forces was on the safe side for Smeaton, the over-estimation led the Wrights to underestimate their wing areas.
We use the same method today with a refinement. We write:-
Force = C. ½ r .A
where A is the area of the surface (this was S) and k. has been replaced by ½ r where r is the density of the flowing fluid.
Now we have to decide what to record. Right from the start the early experimenters had ideas of lift for a wing, which was needed to get into the air, and drag which would have to be overcome continuously by an engine if flight were to become a reality. We still measure the same forces and express them :-
Lift = . ½ r .A
Drag = . ½ r .A
where and are the coefficient of lift and the coefficient of drag. In both cases the area is the same, that is, the plan area of the aerofoil.
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.
It is worth looking at Figure 6-6 where the three forces are drawn to scale on an aerofoil for a lift over drag ratio of 30. It is clear 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 that is the ratio for the lift of the wings to the drag of the whole glider.
The successful use of aerofoils is to a large extent attributable to the work of the NACA. This was a body sponsored by the US government and charged with the production of reliable data on the performance of aerofoils and for the evolution of new ones. Its work was carried out between the wars and then, after the war, the NACA became the NASA disbanded. The data is still the most commonly used because, except for the evolution of special purpose aerofoils, they are as fully developed as is justified for ordinary manufacture. The aerofoils are in families, each family starting from a symmetrical section that is then altered systematically by thickening and/or giving it a camber and testing each member of the family. Test models of these aerofoils were of 24 inches chord spanning a tunnel 36 inches wide by 7.5 feet high. The air flow was almost free from turbulence and data from the tunnel closely matched data from flight tests. The data was published as graphs and is readily available.
The sections that interest us are the symmetrical ones because these are used for our fins and rudders. Figure 6-7 gives four NACA symmetrical aerofoil sections. The sections clearly reflect the basic design concept of using smooth curves for top and bottom and a smoothly blended nose radius to join them together. NACA 0006 is the basic symmetrical section and has a maximum thickness of 6% of the chord at 30% of the chord from the nose. NACA 0009 is the same section scaled up to have 9% maximum thickness. NACA 0010-34 is 10% thick but has a different profile indicated by the 3. Its maximum thickness is at 40% of the chord. NACA 0012-64 has yet another profile with a maximum thickness of 12% at 50% of the chord.
All these sections are practical shapes for use in aeroplanes and they are readily transferable for use for fins. Experimentally obtained data is available for these and many other sections.
Data on aerofoils is given primarily as a graph of the coefficient of lift, , plotted against the angle of attack a. In order to understand this graph we need an explanation of the term “angle of attack”. Let us consider the testing of an aerofoil section in a wind tunnel. The tunnel will have been designed and developed so that, with the tunnel empty, the air flows through smoothly in a direction that is parallel with the walls with more or less the same speed everywhere except very close to the walls. The aerofoil to be tested is like a short rectangular wing fitting between the side-walls of the tunnel which prevent span-wise flow. The primary measurement to be made is the lift that may be upwards, downwards or zero even for an asymmetrical section. The lift varies with the way in which the aerofoil is set to the stream of air that means that we have to measure its attitude in some way. Before it can be measured we have to decide what exactly we shall measure and we have a choice. The most common decision, and the one used by NACA, is to measure the angle between a designated line on the aerofoil (the axis of symmetry for symmetrical sections and some other specified line for asymmetrical sections) and the axis of the tunnel. The angle is called the angle of attack and given the symbol a. This has been done in Graph 6-8
For all the symmetrical sections shown above there is only one line on the graph until a reaches about 8°. It is a straight line passing through = 0.8 at a = 8°. The rest of the graph depends on the shape of the aerofoil and the effect of this shape on the flow over the upper side of the aerofoil. There are four dotted lines for test aerofoils with typical surfaces and one for a smooth 0012-64. All the dotted parts of the lines represent the change in lift during the stall. Sections 0006, 0009, and 0010-34 display gradual loss of lift. However 0012-64 shows that the section can be taken to a higher maximum lift if a sudden stall is acceptable. The line for the smooth 0012-64 shows what can be done with a good finish or perhaps by ingesting the boundary layer.
The similarity of performance should not surprise us. A look at Picture 6-1 shows that the underside of the aerofoil is not much different from the underside of any of the symmetrical sections and it is this surface that determines the flow pattern upstream and, to a great extent, the lift. It is the upper surface with its curvature that determines whether the separation and breakdown of flow takes place suddenly at some angle of attack or progressively over a range of angles. In general the sections with their maximum thickness near to the leading edge go to high values of and a (eg1.2 at 12°) and then stall suddenly and those with the maximum thickness at 40% or 50% back tend to stall more gently at lower angles of attack.
We now have information on the variation of with a but have said nothing about . We need to think more deeply about drag and its origins. Ordinary day to day experience leads us to think that a rough surface would offer a greater resistance than a smooth one to a fluid flowing across it and this proves to be the case. We need a more detailed explanation. Suppose that we had an aerofoil section such as might be tested in a wind tunnel with a very high surface finish and a surface profile which had been made to very high standard so that the aerofoil had truly smooth curves without sudden changes in radius of curvature. Now suppose it to be set up in a tunnel so that air flows over it at an angle of attack of zero. The fluid still “sees” the surface as a stationary film of air and in the layer close to the surface the velocity increases with distance from the surface from nothing to nearly the main-stream velocity in a very short distance. In this boundary layer the flow is orderly and is said to be laminar. The thickness of the layer increases with distance from the leading edge and, at some point, as we have seen, may separate from the surface. For the high quality surface we are considering the air appears to be unaffected by the inevitable small imperfections and the point of separation will be much nearer to the trailing edge than it would be if the surface were to be even slightly rougher or have a small discontinuity. Separation may not occur at all. The air leaving the aerofoil will form a thin eddying wake. The value of for this aerofoil may be as low as 0.004 and this may not vary much up to perhaps a = 6° when / = 150.
This high figure is not achieved in practical application and so we must know what goes wrong. The first and obvious problem is that the quality of the test aerofoil. A poorer surface finish will cause the boundary layer to thicken much more and make separation almost inevitable. The errors in the profile will simply add to the effect of roughness to give rise to a much thicker eddying wake and a rise in drag because more kinetic energy is imparted to the wake. It follows that is not predictable in the same way as
The effect of Aspect Ratio
So far we have been concerned with data from wind tunnels. We must now consider what happens when we have an aerofoil where there are no side walls to prevent the flow having a component along the aerofoil and indeed to explain the origin of this type of flow.
Suppose that we had a wind tunnel that was very wide and we attached an aerofoil to one side so that it protruded, cantilever fashion, across one half of the tunnel. Suppose that this aerofoil had a length that was 3 times its chord. It would have the same proportions as one wing of a light aeroplane. We now have an aerofoil with no wall at one end. We can call the end at the wall the root and the free end the tip.
If the tunnel is run with the aerofoil set at a practical angle of attack there is no reason to suppose that the flow over the aerofoil very close to the wall will be affected. This means that the region of high pressure under the aerofoil will be created as before and the pressure will drop over the upper surface exactly as before. But now the high pressure is no longer constrained to act only backwards, forwards and up and down, but can now act sideways. At the other wall there is no reason to suppose that the flow will differ from that which would prevail if the tunnel were to be empty. This gives rise to a further complication in that the low-pressure region which comes into existence over the upper surface of the aerofoil is low relative to the pressure on this wall. The result is that the air approaching the aerofoil follows broadly the same flow pattern as before but now the flow which will go under the aerofoil moves towards the tip and the flow which will go over the aerofoil tends to move towards the root. This is now a complex three-dimensional flow that cannot easily be investigated by use of smoke and perspective representations in two dimensions become excessively difficult to draw.
If we separated the two effects and looked just at the flow from the underside of the wing to the upper side we could see it as a two-dimensional flow. Others who have looked at this have produced a mathematical expression for these paths and drawn them as shown in Diagram 6-9. With it is Picture 6-10 which is of a crop duster. Accepting that the aeroplane is flying fairly close to the ground and that the cloud of fine particles does not extend into the air beyond the wing tips, it is hard to avoid the conclusion that the combination of the flow over the wings and the rotation produces a pair of large spiral vortices. Normally the main observable feature would appear in humid air as a small vortex leaving the tip but this is but a small part of the flow and probably not the most important. These vortices are unwanted. They reduce the lift and, as a result of imparting kinetic energy of rotation, increase the work to be done by the engine to overcome the extra drag.
Now suppose that the wings of an aeroplane could be replaced by new wings of the same area but of twice the span. Clearly the mean value of the chord will be one half of the original value. The new wings have to provide the same lift by deflecting air downwards but now they affect twice as much air. The result is that the greater mass of air moves at about half the speed and absorbs perhaps a quarter of the kinetic energy. The result is a reduction in drag.
Let us now consider a real wing. Suppose that the wing is to generate a given lift to fly an aeroplane and that we are free to choose a wing plan. We see now that unless there are good reasons to do otherwise we should choose a wing that has a large span and a small chord. In other words to give the wing a high value of the ratio of the wing-
span to the chord that is a high aspect ratio. We might further consider tapering the wing at the tips to reduce the chord and perhaps twist it at the tips to reduce the angle of attack at the tips. All these things would give an efficient wing with very small tip vortices and low drag. Unfortunately it also gives a thin wing that is not much good for storing fuel, undercarriages, etcetera and is likely to be weak in torsion and a nuisance on the ground. Such a wing is also difficult to control near to the ground during landing and these considerations lead us to look more closely at wings with lower aspect ratios. We want to know how the aspect ratio affects the lift and drag to help us choose.
Graphs 6-11 and 6-12 are for rectangular wings of aspect ratios from 1:1 to 7:1. The first shows the plot of against a. This graph shows clearly that the gains to be made from any further increase above 7:1 are going to be small. It also shows that practical values of aspect ratio start at about 5:1. The second graph of against for the same aspect ratios show how the drag increases for low aspect ratios.
Aerofoils at high angles of attack
The sails of model yachts work at high angles of attack by comparison with those used in aeroplanes. Picture 8-6 shows flow lines round a symmetrical section at an angle of attack of 30°. The approach flow is curved upwards quite sharply by the aerofoil and the main features of the flow pattern are determined by the underside of the aerofoil. The air flowing over the upper side has separated completely. Aeroplanes do not operate at such high angles of attack but the sails of model yachts do. The data available for aerofoils will not help us much with sails but will be invaluable for fins.
This is what often happens to the air flowing over a sail and coupling this with the fact that the sail is essentially a single surface it becomes evident that the sail is not simply another aerofoil.
 Some let it extend to 99% of the maximum, some to 90%. It is not significant for us.
 Over many years aeronautical engineers have sought to avoid this reversal of flow by sucking the boundary layer through tiny holes in the upper surface of aeroplane wings. Practical methods have been very elusive.
 One must be careful not to confuse this with angle of incidence. This is the term used for the rigging angles of early aeroplanes. For example it might refer to the angle which the wing should make to the centre line of the fuselage.
 Some aerobatic aeroplanes have the maximum thickness as little as 15% behind the leading edge to deliberately make the stall extremely sudden.