# Chapter 19     The aerofoil

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# Chapter 19 The aerofoil

## Introduction

I want to start this section on engineering practice with the aerofoil simply because the aerofoil is ubiquitous and not very well understood. Aerofoils work singly as in aeroplane wings and other flying surfaces and they work in sets of identical blades as in turbines and compressors. In this chapter, I want to explain the way isolated aerofoils work and how experimental data is stored and may be used by engineers.

In some ways we are fortunate because aerofoils only go back to about 1900 and have no long history in which to generate and consolidate spurious explanations in the minds of those who use aerofoils. Nevertheless aerofoils have attracted their fair share of pseudo-science not to mention lots of mathematical analysis that never seems to go anywhere and just muddies the water for engineers.

In this chapter I shall work wholly from recognised science and include only such explanations as might be useful to an engineer and then deal with some important applications in subsequent chapters.

Preamble

For many years the aerofoil and its application seemed to me to be in the hands of mathematicians. When I consulted text-books I was confronted with mathematics and accounts of experiments based on mathematics on every page. I had very little faith in any of it because I had spent many years making model aeroplanes and could easily see that almost without exception the profiles of aerofoils drawn in the diagrams in these books showed that the authors had never actually looked at a real wing nor yet had any idea how the aerofoil actually worked. I do not know how we came to be in this situation but one must remember that there is a greater cachet associated with anything mathematical than with anything practical. In its way this mathematical treatment is just as troublesome as pseudo-science.

# I was sucked into this idea that mathematics was necessary to understand aerofoil action and gave up on the mathematics as beyond my ability. But gradually I began to build up a sensible explanation for the mode of operation of aerofoils and to understand their limitations in practice. I realised that an approach from the practical end of things would yield dividends.

Where is started

When one talks of aerofoils one is really talking about wings. Wings have always been with us because we can see that birds can fly and they use wings. However the wings of birds are living things that can be adjusted continually to give the best performance. Just watch the wings of crows as they go about the business of being a crow. Watch their wings during a landing approach to see what happens to the feathers. Watch them fold their wings. Birds have evolved to use wings that can be folded more or less in the middle so that they are manageable when the bird is not flying. When this requirement is acknowledged in conjunction with the way that the bird seeks its food, wings of different designs have evolved. Raptors live on land and roost at night and so must stow their wings. They often use rising currents of air such as thermals or ridge waves to scour the land for food. They usually have short, wide (front to back) wings with an array of large feathers like fingers on the tips. By comparison birds that live continuously at sea (except for breeding) soar close to the surface and use very long wings with no obviously special feathers at their tips. They cannot fly in calm conditions because their muscles are not strong enough to drive the large wings. There are many other designs between these two extremes.

What is clear is that engineers cannot emulate bird flight, they must use rigid[1] wings. They can design long wings for gliders and short wings for light aeroplanes and everything in between. As soon as you decide to use a rigid wing the shape of the cross-section of the wing becomes of great interest and generally we call this shape an aerofoil section.

There can be no question that the aerofoil section has developed by trial and that it has its origins in fabric sails. Previously Lilienthal 1848 – 1896, who was an engineer and applied his knowledge to flying, had made many flights in gliders that were forerunners of current hang gliders. He flew from a man-made hill[2] using the up currents from the natural wind just as hang glider pilots do today. He also controlled his gliders by shifting the centre of gravity of the whole combination of pilot and machine just as is done now but, unlike a modern hang glider where the pilot is suspended from the wing, Lilienthal had his wing strapped to his shoulders and this limited the possible range of movement of his body to alter the position of the centre of gravity. Lilienthal died in an accident. He had shown that manned flight was possible and that control of a flying machine was not easy. He based his designs on the wings of birds and he made them from wood and fabric. In doing so he inevitably lost the great advantage enjoyed by birds in that a bird’s wing being a live structure can continually adjust its shape to the needs of flight. He found that, where birds used an adjustable feathered tail that is essentially horizontal but can be twisted, he needed a vertical surface to control his glider as well as a horizontal surface.

The Wright Brothers also made their wings from wood and fabric but unlike Lilienthal saw the need for some system of three-axis control in order firstly to keep the flying machine at the correct attitude for flight and secondly to control the direction of flight in three-dimensional space. The Wright Brothers proved that controlled flight using machines was possible and showed that this could be done with the pilot sitting in a seat and therefore without the movement of the centre of gravity. This was an extraordinary step forward and was not achieved just by trial but by their own form of analysis and experiment combined with trial by two brothers who financed their own programme.[3] Theirs was a work of genius.

Their successful aeroplane shown in figure 19-1 used two pairs of wings, a double stabiliser mounted forward of the wing, two rudders mounted behind the wings and the pilot sat in a cradle above the lower wing. We need to understand the system of control. If one attempted to fly using just the two main wings it would become evident that those wings may lift but they also tend to turn over backwards. In order to prevent this the two forward control surfaces are moveable to enable the pilot to apply a downwards force if need be although, in fact, it is likely the centre of gravity will be far enough forward for there to be a downwards force on these control surfaces and that they actually lift. They can then be used both to stabilise the main wings and to control ascent and descent. This arrangement with a forward stabiliser is more forgiving than the normal rear-mounted
stabilisers in use today.

This takes care of the fore and aft control but leaves the problem of making a horizontal turn. This actually requires some means of making the aeroplane roll and the Wrights used a system of twisting the wings so that, in flight, one wing lifted as the other dropped. (We still use the same system using ailerons instead.) Then a turn is produced by a combination of this wing twisting,[4] and the use of the stabiliser combined with the use of the rudder. The Wrights were the first to use this system of three axis control.

If the Wrights could fly successfully they must have had practical wings with a practical shape to their aerofoil. However the Wrights were flying at very low speeds of say 40 mph or 65 kph and at the minimum all up weight that they could contrive. So we must not expect to find them using modern wing sections that are suited to much higher speeds. They used a section like that shown in figure 19-2.[5] The ribs were cut from ash and steamed to give them the required curvature and the main spar was in spruce. There was a leading edge probably also in spruce and this and the rearward main spars were joined to inter-plane struts and bracing wires to form a very light framed structure. The wing had a wire instead of a trailing edge and the whole wing was covered with sateen (A cloth that looks like silk.).

The Wrights' aerofoil is very thin. Any claim that an aerofoil works because the air has to go farther over the top than the bottom does not apply here. What is interesting is that, as a result of extensive testing the Wrights abandoned Lilienthal's circular arc in favour of a curve with its highest point at about one third back from the leading edge. Modern aerofoils do the same. The Wrights used a minimum offset of 1/30 of the length but could increase it by tightening the bracing wires.

Once the Wrights had shown how to achieve controlled flight the development of the aeroplane advanced very quickly and the idea of twisting the wing was soon abandoned in favour of rigid wings and inset ailerons. The aerofoils became thicker but the idea that they must have camber was not seriously challenged for another couple of decades.

Wings and aerofoils

We would have no interest in wings for flight if the properties of atmospheric air were such that powered flight was impossible.

On some other planet the density may be so high that very small wings would be adequate or, the density might to be so low that flight would be very difficult to achieve. The point is that getting a weight into the air by using wings is not so easy that any old wing will do. It would have been very convenient had we lived on a planet with an atmosphere of greater density but the same viscosity. We have but one planet so we have to design our wings to work in the atmosphere that we have. What we want is to have wings that will lift useful weights in the atmosphere that we have and to do it with a low resistance to motion to be overcome by the engine. What we need to know is how our wings work so that we know what we are up against when we try to use them. It has turned out that the density of atmospheric air at ground level has been great enough for wings of a practical size to lift a useful weight off the ground and the viscosity of atmospheric air is low enough for an engine of relatively low weight and power to maintain the forward speed and the lift. Once airborne the aeroplane can climb into the upper atmosphere where conditions are much more suitable for long distance flight.

In order to discuss how wings work I have to talk about wings in general first and then take simplifying decisions to get rid of the complications. Aeroplane wings must lift. They do so by having high pressure under the wings and low pressure on top of the wings. Wings are just cantilevers sticking out sideways and, at the tips, there is nothing to separate the high pressure on the lower face from the low pressure on the upper face. Inevitably the air flowing over the wings flows round the tips. This motion cannot be local and leads to oblique flow over the whole wing being towards the fuselage on the top and away from the fuselage on the bottom. This leads to two vortices as shown in figure 19-3 and I will explain this in detail later in this text but, for my immediate purpose, I want to get rid of this complication and consider the flow to be two dimensional as it would be in a wind tunnel where the test model fits snugly between the sides of the working section and the vortices are suppressed. Then we are only concerned with two dimensional flow over a cross-section of the wing and this cross-section we have called an aerofoil section.

Terminology and measurements for aerofoils

I want to tackle my explanation of the way aerofoils work in an essentially practical way using diagrams to explain what is going on without unnecessary recourse to mathematical analysis. This does not mean that I am guessing. When I draw a flow pattern from scratch I am accepting the fact that the rules that must be applied to draw these flow patterns preclude nonsensical patterns where the flow lines will not fit together and the rules have been broken. I find the act of drawing these patterns both informative and mentally rewarding.

Figure 19-4 shows a typical aerofoil section. It just looks as though air or water would flow smoothly over it as indeed is the case. It is one of the NACA sections for which there is some[6] absolutely impeccable test data. It is widely used as a section for stabilisers, as shown in figure 19-5, and fins on aeroplanes and for wings. It is used in marine applications like the control surfaces on submarines and on stabilisers for passenger ships. We can see the shape and now we have to explain how it works and why it is so useful.

Clearly, if we are to talk about cross-sections of wings we enter the realm of the geometry of the section and its attitude to the fluid flowing over it. This leads to the commonly-used terminology. First we should note that the length of the aerofoil is called the chord and it is regarded as the basic length and is used in calculating Reynolds number. It is likely that its name originates with Lilienthal who used ribs in the shape of an arc of a circle to make his wings. The behaviour of aerofoils is to significant extent determined by the maximum thickness. The aerofoil in figure 19-4 has a maximum thickness of 12% of the chord. Then there are the other features like the fact that the aerofoil is symmetrical and that it is constructed from four parabolas that share a common tangent at the position of maximum thickness. The two parabolas that form the front part are joined with a circular arc and the radius of that arc is an important dimension. Indeed, by the NACA system of coding the 00 signifies that this aerofoil is symmetrical and the 12 says its maximum thickness is 12% , the 6 gives the radius of the nose as 6% of the chord and the 4 gives the position of the point of maximum thickness as 40% of the chord. The number of possible shapes for an aerofoil is infinite and NACA used families of aerofoils where each family is based of some common geometrical construction. Each family can then be coded. There are lots of other ways in use as well.

Now we must consider the attitude of the aerofoil. We have something of a problem here. For a wing in flight we can know the direction of motion of the aeroplane and we can see that the wing makes an angle to that direction and moreover that the angle changes for different phases of flight. We can define this angle without difficulty for a symmetrical section if we know the direction of flight as is evident from figure 19-6a. However it goes without saying that when we are trying to gather test data for an aerofoil that is stationary in a wind tunnel we still need to be able to quantify the angle that the aerofoil makes to the direction of the flow of air in the tunnel. We have to suppose that the flow of air through the empty tunnel is along the line of the axis of the tunnel and, even though the presence of the aerofoil alters the flow we still measure the angle between the axis of the tunnel and the centre line of the aerofoil section, as shown in figure 19-6b, and call it the angle of attack.[7] In truth we have no alternative but at least there can be no doubt about what has been measured nor about whether it is relevant.

I have shown the aerofoil of figure 19-6 at an angle of attack of 5º. I always think that this looks like a large angle in this context but this aerofoil can work properly up to about 10º or perhaps 12º although 5º is more likely.

We need to have a clear idea about how an aerofoil can produce lift and also of the way that aerofoils behave. I think that the aeroplane is the only[8] major machine that moves through a fluid to support its own weight and be controllable without some form of mechanical contact with the ground. The wings are the all important element of the aeroplane but the stabilising and control surfaces make it possible for these wings to function in a predictable and stable manner. We need to see how it is all achieved and I must start with the wing.

I have said that the aim for an aerofoil is to produce a large force, that is normally called the lift and denoted , at right angles to the direction of motion for a small resistance to motion that is normally called the drag denoted . A good wing will have a high ratio of lift to drag and this high ratio will occur when the wing is lifting as much as it is safe to lift at its speed through the air.[9] It is relatively easy to get a ratio of 12 for a wood and fabric wing, a well-built wing in metal or fibreglass will have a ratio of 30+ and a superlative wing for a glider will exceed 60. These figures mean that wings are quite practical as is, of course, quite evident from the incessant aviation activity.

We have to find out how lift and drag are created and how lift and drag can be affected by the shape of the aerofoil used in the wing. In order to make a start I shall follow my golden rule and look for the forces and for this I have to be able to draw some two-dimensional flow patterns round an aerofoil when it is in use. I devoted Chapter 4 to flow patterns and now I want to use them. The rules for drawing flow patterns are that these patterns are made up of paths that are followed by fluid as it flows through space and they cannot cross nor can a path divide into two or three new paths. So I shall represent an orderly two dimensional flow by a pattern of lines that can only get closer or further apart and mostly, for an aerofoil, it will be drawn for just one plane. Such flow patterns can be "seen" by injecting smoke into the air flowing over a test aerofoil in a wind tunnel.

Figure 19-7a shows the perspex working section of a small wind tunnel. Figure 19-7b shows a close up of the model wing (It had a NACA 0012-64 section.) in the working section of the tunnel. The model is suspended by two streamlined supports from a lift and drag balance mounted over the working section and they also act as pivots for the model to rotate under the control of a wire to change the angle of attack. The model appears to be upside down but, in practice, it is better to let the vertical force on the model, i.e. the lift, act downwards as if it were to be a weight to be weighed on the balance and to get the suspensions in the "underside" of the model where they cause the least effect on the flow.

Now this tunnel has been designed carefully with an intake incorporating a double contraction with honeycomb and gauze between the two contractions in order to produce an air stream in the working section that is closely uniform when the tunnel is empty. When a prismatic model like the one shown is mounted in the tunnel the flow pattern is very nearly the same for every vertical plane through the tunnel and provided that the pattern does not include a region where the flow separates from the model it will have a low level of turbulence. It is possible to set up a smoke rake to inject smoke upstream of the model in order to see how the flow takes place over the model[10].

I need some genuine flow patterns against which to compare any patterns that I may draw. So it is reasonable for me to start with a real flow pattern to see what actually happens.

Figure 19-8 is a picture of a model of a symmetrical aerofoil section in a wind tunnel. The lines of smoke indicate the pattern of the flow. It is most likely that the aerofoil was tested in a tunnel like the one in figure 19-7a 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 average 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 through which smoke is injected to the airstream. 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.

The first is that the air must have approached the model in parallel horizontal paths but it leaves with the air that is directly affected by the aerofoil having been deflected downwards. This is inevitable because, if there is lift, there is an equal and opposite force exerted on the flow of air over the aerofoil. The force can only exist if there is a steady imparting of momentum to the air as it passes the aerofoil and that must appear as a downward velocity.

The second is that the air that is affected by the aerofoil is diverted upwards ahead of the aerofoil as if it was preparing to part as indeed it is. In addition we can see from the blips that the air that will flow over the top of the aerofoil is already moving more quickly than the distant approach flow.

Finally the blips show that the two flows of air that leave the aerofoil are a long way out of step and we must expect that the "gap" will go on widening.[11] But that leads us into the wake produced by the aerofoil and that is another story that puzzles me.

It seems to me that the most important flow line in a flow pattern is the stagnation line that I have shown here in red. It is the line that hits the aerofoil and stops to give the stagnation pressure equal to  where  is the velocity of the undisturbed flow. This line shown in red is an end view of a surface of separation between the air that flows over the aerofoil and the air that flows under the aerofoil. If the flow is steady this stagnation line must be in equilibrium between opposing distributed forces caused by the non-uniform pressures acting on its two sides and, for this to be true, the stagnation line must terminate at the aerofoil normally to the surface at the point of contact. (It will be square to the tangent to the nose at the point of contact.) In figure 19-8 the stagnation line in red divides the flow pattern into areas that are either above or below whatever pressure is used as a datum pressure. I have also added, in white, lines that join the points of highest pressure for the flows between adjacent flow lines. The pressure rises and the velocity falls as the flow approaches these two lines and the pressure falls again and the velocity rises as the flow leaves.

Now we can look at the flow pattern to pick out regions of high pressure and low pressure. In region A the flow-lines are closer together that they are upstream so the pressure is lower than the datum. In region B where the lines are wider apart the pressure is higher than datum, in C it is near to but below datum and D it is near to but above the datum.

The pressure distribution round aerofoils has been measured many times 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 tangent to the surface of lengths that are proportional to the pressure measured relative to the datum pressure and show which way the net pressure acts. Lines may be drawn joining the heads (or tails) of the arrows as shown in the figure 19-9.

It is not easy to decide what pressure to use as a datum pressure. It is not a matter to trouble engineers who want to use data not gather it but they do need to know what was done.

Figure 19-10 shows a multi-tube manometer that has been designed for use with wind tunnels to measure pressures relative to some pressure picked up by a tapping on the tunnel. It is easy to suppose that this tapping picks up the pressure of the free stream when the tunnel is empty but there is no guarantee that it will measure the same pressure when a model is being tested. The test data should make clear the method of measuring the datum pressure. I shall call it the free stream pressure.[12]

This manometer can be inclined to change its sensitivity.

When we look at the top surface in figure 19-9 everywhere the pressure is lower than the free stream pressure. However the pressures on the lower surface is everywhere greater than the free stream pressure.

There is clearly a point on the nose where the pressure changes from being above to being below the free stream pressure. It is not coincident with the stagnation point except for the case of a symmetrical section set at zero angle of attack.

These pressures act on areas and as a result both sides of the aerofoil have net forces on them and these add to generate the total force with its two components, the lift and the drag. The point of application of this total force is usually called the centre of pressure. (Or centre of effort in sailing.) The position of the centre of pressure is obviously of interest as is any movement of the centre of pressure with angle of attack or speed.

The pressure distribution is such that there is also a couple acting on the aerofoil and that couple acts clockwise on the aerofoil in figure 19-11 and tends to tip the aerofoil over backwards. A couple is a combination of two equal and opposite forces that are parallel but not in line. We cannot locate the couple as one might when taking moments about a polar point in computation but it is usual to mount the test model on pivots at some specified point such as one quarter back from the leading edge and then quote the moment and denote it as . I do not think that anyone would attempt to deduce the resultant force on the aerofoil or the moment of the couple from measurements of the pressure distribution over the aerofoil because it is far too tedious. However, if the profile is capable of being described by mathematical functions, values for the lift, the drag and the moment can be computed for a limited range of angle of attack.

The flow over aerofoils

As I have come to understand how the aerofoil works I have come to wish that those who have photographed flow patterns had paid more attention to the flow upstream of the aerofoil because this is where the flow pattern starts to take shape and it is important. As you look at figure 19-8 it is hard to gauge how far upstream you would have to look to see flow lines that are parallel. It could be as much as one length of the aerofoil. If it is, the aerofoil is affecting the flow a long way ahead. That region is part of the flow pattern. The same thing is true of the downstream flow particularly for wind turbines.

In figure 19-8 the implication of the blips is that the air arrives at the vicinity of the aerofoil with a velocity gradient already established in the approach flow and the flow lines have all been diverted upwards. Of course the gradient is not uniform nor is the diversion the same for all the flow lines. The approaching air "sees" an obstruction ahead and prepares to flow round it. I have never seen photographs of the upstream flow and so I am not wholly certain what it looks like. What is certain that it does nothing untoward or it would have been photographed and recorded.

We can make some progress with finding an understanding how aerofoils generate lift. I think that the place to start is with the stagnation line, that is, the one that comes to rest at the stagnation point on the nose of the aerofoil. The line that we see is a cross-section of a surface and that surface is subjected to a pressure that rises the nearer the point on the surface is to the stagnation point. This surface separates the flow into two streams and the forces impressed on the two streams are the same because the stagnation line does not move in steady flow. They each impart momentum to the flow, one force to the fluid that flows over the aerofoil and the other to the fluid that flows under the aerofoil. These two flows exist in very different circumstances.

I have said that I wish that there were more photographs of real flow patterns that extend upstream. Figure 19-12 is taken from Prandtl. It is of paint on glass that shows the mean flow over this asymmetrical aerofoil. In figure 19-12. I have extended the flow lines of this paint picture from Prandtl by extrapolating. My impression is that the added flow lines, that I have spaced equally, diverge up and down in preparation for flowing round the aerofoil with the divergence upwards being greater than that downwards. Once again we can see that the net effect of the aerofoil is to deflect the whole flow downwards. I want to look at the region that is in the vicinity of the nose. I have extracted a portion of figure 19-12 to become figure 19-13. It shows what goes on at the nose. The stagnation line is not evident on this photo but it must lie between the two white lines that go one over and one under the aerofoil. The pressure is greatest at the stagnation point. The distributed force acting downwards on the flow under the aerofoil is resisted by a region of elevated pressure under the nose of the aerofoil and the flow curves under the nose to be the start of converging flow that continues right to the trailing edge of the aerofoil. The distance between the flow lines shows that the pressure is high. The solid boundary of the aerofoil acts passively to produce any force that is needed. By contrast the upper surface cannot do this because the flow is curving over that surface and necessarily a centripetal force is required to keep the fluid in contact with the surface and flowing steadily. This centripetal force can only come from a pressure gradient extending from the distant flow to the upper surface. This means that the pressure at the upper surface is low by an amount that depends on the speed of the flow and the radius of curvature. There is a much greater difference between the pressure at the stagnation point and the fluid flowing round the nose and we can see that it accelerates upwards in a short distance to form the swiftly flowing layer over the nose. This tells us why the profile of the nose and say the first 15% of the upper surface is important. This region just above the nose appears to be the place where the velocity is greatest anywhere on the aerofoil and the rest of the flow over the upper surface is divergent as the velocity falls.

So now we know that we have converging flow under the aerofoil and divergent flow over it. We already know that disturbances in convergent flow tend to decay and we also know that disturbances in divergent flow tend to grow and spread causing separation and other unwanted changes. This means that the aerofoil is vulnerable to anything that might interfere with the flow over the upper surface, even something as small as dead insects or minute pits caused by rain. I will deal with that shortly but first we must look at two basic problems that affect the behaviour of aerofoils.

The stall

Wings are something of a problem to use. The lift produced at a given forward speed increases fairly uniformly with the angle of attack but, typically, at some angle exceeding about 10º, the lift will suddenly drop in what is generally called a stall. For some real wings the stall may occur at an angle of attack as low as 8º, but for some others it can occur at angles as great as 16º depending on the aerofoil section that is in use and the quality of the wing. For normal flight unintentional stalling is to be avoided and this means that the range of the useful angles of attack is relatively small. As I have said if the efficiency of a wing is measured by the lift/drag ratio we need to know whether the highest values of lift/drag are associated with the practical range of angle of attack. But first we need to understand the stall.

In order to explain the stall I want to draw some flow patterns by eye. I have drawn flow patterns in this way elsewhere in this text. In my experience if the rules are followed some elements of the resulting approximate flow pattern are seen to be inescapable. This is the case here; the flow patterns must not be examined closely as if they were drawn from smoke lines in a tunnel but they are useful and often establish what is possible and what is impossible. So let me draw some flow patterns for my symmetrical section. I have drawn the patterns in figures 19-14 a, b and c for the NACA 0012-64 symmetrical section at three angles of attack, 0º, 5º and for 10º.

It is immediately clear that the flow over the nose makes a curve that takes a smaller radius as the angle of attack increases and the velocity of the flow increases just over the nose. This causes trouble.

When a mass of fluid follows a curved path it is subject to an acceleration towards the centre of curvature. That acceleration is called the centripetal acceleration and it can be evaluated from .

However, if a mass of fluid flowing close to a solid surface is subjected to a centripetal acceleration, the force needed to produce the acceleration must either come from the solid boundary or it must come from a pressure difference acting towards a solid boundary. In this case it is a pressure difference.

In figures 19-14b and 14c I have added a small mark at the 10% point of the chord. It is inevitable that the flow over the upper part in this first 10% of the chord is making a sharp change in direction just where it is also accelerating along the line of flow. The result will be a drop in pressure in this region. There will not be the same drop in pressure between flow lines that are farther removed from the aerofoil so there must be a pressure gradient acting to produce an inward force as is required to get the flow round the nose.

Inspection of figures 19-13a, b and c show that the radius of the flow lines in this first 10% decrease with increasing angle of attack. This means that a greater centripetal force is required as the angle of attack increases and this requires an increased pressure difference. As the angle of attack increases there must come an angle when the pressure difference required to keep the flow in contact with the surface cannot be generated and at this angle of attack the flow must detach from the upper surface. This is the stall. I have shown the flow lines for one aerofoil but there are other symmetrical aerofoils that are either much thinner or of the same thickness but with a much smaller nose radius. In both cases the radius of curvature at the nose is reduced and the stall occurs at a lower angle of attack.

I have been using a symmetrical aerofoil to date because it is really our fundamental section. It has the very obvious advantage that it can produce a transverse force in either direction and I have shown it in use on a glider. However, there are lots of applications where the transverse force is required to act in one direction, e.g. the light aeroplane. Then we can use an asymmetrical section to considerable advantage.

We can produce an asymmetrical section by modifying the NACA 0012-64 in figure 19-15a by re-plotting it on a curved line, for example, an arc of a circle. I have done this in figure 19-15b where the original symmetrical section changes to the more familiar asymmetrical section. The arc touches the chord line at its end and the central offset is now called the camber so the radius of the circle is determined by the length of the chord and the camber.

In figure 19-5c I have doubled the camber and the underside now becomes concave or under-cambered and the upper surface now has a pronounced hump. All this is just geometrical construction but we must see how it affects the flow pattern.

Figure 19-16 shows a slightly under cambered aerofoil derived from NACA 0012-64 set at an angle of attack of 10º. There is a marked improvement in the flow pattern over the first 10% of the aerofoil in that the radius of curvature of the flow lines is increased over that for the symmetrical section at the same angle of attack.

It is the way of engineering that such improvements do not come without some other complication and I must now consider the problem of separation on the upper surface of the aerofoil.

Separation

Figure 19-17 shows a flow of air that starts on the left and curves round to follow the solid surface. I have drawn flow lines that all curve and we can see that the flow diverges. At section AA the flow lines are more or less evenly spaced and close together. At section BB the lines have diverged and are still more or less evenly spaced. If this were to be one-dimensional flow the energy equation would tell us that kinetic energy is being given up in exchange for pressure energy and that the pressure would rise between AA and BB. In fact we also have the curving flow to consider and its affect on the pressure. I think that we can safely say that the pressure will rise between AA and BB and, of course, if this were to be the rear part of an aerofoil the pressure at BB will not be much different from the free stream pressure.

However there is a complication. When a fluid flows over a smooth, solid surface it behaves as if the surface is a stationary layer of fluid. (This was Newton’s contention when he defined his concept of viscosity.) Then the effect of viscosity 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 with distance from the boundary in a stream of fluid is given in figure 19-18. 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 very small distance and the fluid in this very small distance is called the boundary layer. The existence of this layer limits the performance of aerofoils because the fluid in the boundary layer has too little kinetic energy to give up to give an increase in pressure energy of the magnitude required. Always there is uncertainty about what is happening in the boundary layer.

In figure 19-15 a, b and c it is evident that whilst the re-plotting of the symmetrical section about an arc of a circle improves the flow over the first 10% or so of the upper surface it also decreases the average radius of curvature of the upper surface towards the trailing edge. This means that in order to give the inevitable centripetal acceleration there is a need to further reduce the pressure over the rear of the upper surface. Conditions at the upper surface near to the trailing edge are very complex and it seems that, at some combination of speed and angle of attack, the flow in the boundary layer reverses and, at the point of zero velocity, breaks away and gets swept down stream as shown in figure 19-19 where the scale is exaggerated to show the reversal.

At some angle of attack eddies will form as shown in figure 19-20. This reversal of flow and the eddies that start in the boundary layer can grow in size and advance towards the leading edge as the combination of speed and angle of attack changes and lead to total separation of the flow and to a major loss of lift. It is hard to know whether the stall is the result of separation that starts at the trailing edge or a breakaway at the nose. Whichever it is it must not be allowed to occur in normal flight[13].

Disturbance of the boundary layer

We know from previous chapters that convergent flow is quite robust and will quickly attenuate any disturbance. We also know that divergent flow is much more fragile and any disturbance in the flow tends to grow. It follows that the flow over the upper surface is vulnerable to any imperfection in the surface. So it is the upper surface that we must treat very carefully and I will list the several types of imperfections.

We have Newton's concept of a surface that behaves like a stationary layer of fluid. It seems to me that there is an implication in this that the surface is smooth because it must be apparent that, if the surface is rough and by that I mean anything rougher that very fine sandpaper or a diamond hone, the flow will not behave as if the surface is a stationary layer of fluid. We are back with Nikuradse who suggested that the surface imperfections protrude through the laminar boundary layer. Then the boundary layer has a different character that some call "turbulent" although this seems to me to be a dramatic word for what may just be a disturbance in the orderliness of the flow. So the flow in the boundary layer is affected by the surface finish and, for example, the application of paint by brushing, as distinct from spraying, to an otherwise production wing can increase the drag by as much as a half. Roughness is especially troublesome when it occurs in the first 10% of the chord above the nose.

Even if the surface is smooth discontinuities can initiate separation. Real wings have discontinuities at the ailerons and the flaps and any other interruptions of the smooth character of the surface. Even additions like insect debris can severely affect the boundary layer. The trouble with insect debris is that the debris creates a widening wake like that in figure 19-21. It is a picture of the surface of a small river at a point where a small twig protrudes through the surface. It shows the surface waves that are created downstream. When insect debris or a deep scratch occurs on the surface of a wing a wake like this is created in the boundary layer. If there are lots of dead insects the wakes interact to upset the whole of the boundary layer.

However a mathematical discontinuity that can also affect the flow in the boundary layer. The NACA 0012-64 is made up of five curves, four parabolas and one circular arc. There are four junctions and at each the joining curves share a common tangent. Even so at every junction there is a sudden change in radius of curvature of the combined profile. That change in radius is a discontinuity and can initiate separation.

So the wing is potentially a rather troublesome device to use and it pays to keep these problems in mind. Figure 19-22 shows a cross section of the wing of a Shorts 360 feeder liner. The upper surface is continuous as far as manufacturing will allow but the lower surface is interrupted by the retracted flap. Clearly the design concept accepts the large discontinuity in the lower surface in order to produce a suitably shaped gap for the air to flow through when the flap is down. The aerofoil has a pronounced downward angle that is close to the angle of zero lift. It is a rigging angle for the aeroplane.

Shapes of aerofoils

# In his paper of 1961 Riegels[14] gives 327 silhouettes of aerofoils and this was not exhaustive. It makes one wonder what the designers of aerofoils were actually hoping to find. No doubt they would have looked for a high value of lift at a useful angle of attack when compared with the inevitable drag and for this high value to occur well away from the stall. Perhaps they were looking for small movement of the centre of pressure or a small moment or possibly a high angle of attack at the stall and a gentle loss on lift during the stall. The sheer number of these aerofoil sections suggest that the best combination of desirable properties was quite elusive and, keeping in mind the multiple uses for aerofoils, it may be that even deciding on the most desirable combination of properties was elusive.

It is clear that the physical testing of this large number of aerofoil sections would be prohibitively expensive and no doubt there is a desire to try to use mathematics and computers to winnow out some promising sections to test. This process has been going on steadily but, in fact, whilst waiting for the new "super" section that might be thrown up by this method, there is always the problem of constructing a wing or other device to the standard needed to give this new "super" performance, and the problems of adaptation to the intended design for say roll control as well as the affects of the environment in which it will work.

There is a feeling of déjà vu when reading about the testing of aerofoils. Whole programmes that seem to have been self-perpetuating have been mounted to gather data about families of sections where each family is based on some central concept. For example all the Joukowski sections were devised to have a mathematically continuous profile. It reminds me of the investigation of the friction loss in pipes up to the point when Moody brought it all to an end.

There are whole families of aerofoils that have the general principle that their shapes are all mathematically continuous. Kutta, Joukowsky, von Karman and others appear to have spent a long time in finding shapes that might prove to be practical by mathematically manipulating a circle. Whilst they could produce symmetrical sections that were practical it seems that as soon as the section became asymmetrical it also became under-cambered. The sections did not find application on aeroplanes but some sections were used in turbo machinery where aerofoils work together in sets. It did seem that the effects of the changes in radius of curvature that followed from constructing shapes like that of NACA 0012-64 are more that offset by the desirable characteristics of such aerofoils. This is hardly surprising as wings have to be manufactured and manufacturing methods are likely to produce plenty of discontinuities that are greater than those attributable to changes in radius of curvature. In a similar way the Gottingen sections did not include many shapes that came into common use. A large proportion of the sections tested by the NACA do not even look to be practical and, of the rest, only a few have found favour with engineers.

# Aerofoils seem to be very simple devices, after all there is only the aerofoil and the air. But the air can have innumerable combinations of pressure, temperature and turbulence and the aerofoil can have innumerable shapes and surfaces. When the Wrights started their tests on wing sections[15] they had a clear goal and their data was for one application. If the goal is to gather data for use by anyone for any purpose the whole magnitude of the undertaking changes. A programme of testing has to be started and before that the immense quantity of data that will be generated makes it essential to consider how this data is to be compressed for storage into something that is manageable. In addition the means of storing should be comprehensible to most people who might want to use the data; otherwise it is wasted.

This is not as easy as it sounds. We have Lord Rayleigh to thank for devising his method of dimensions just in time for aeronautical investigations.

The problem of presenting experimental data in a concise form must have exercised the minds of the scientists who were exploring the natural world in the late 18 hundreds. Lord Rayleigh, in a seminal but very short paper to the Philosophical Magazine in October 1899, reported experiments linking the frequency of drop formation at the ends of vertical glass capillary tubes with thick walls and the ends ground square through which liquids flowed slowly and dripped from the end. This seems to be a trivial experiment but Rayleigh used it to explore ideas of dynamical similarity between systems and to devise a way of presenting the results in a plot between non-dimensional quantities. In this paper he showed how to form non-dimensional groups almost in passing and this is now called Rayleigh’s method. As a result of the non-dimensional plot a very considerable amount of data ended up as one line on a graph. Rayleigh gave us a way of compressing data into a manageable form but in doing so effectively encoded it so that a would-be user had to decode it for use. This is not often stated so explicitly and experimenters do not always pay sufficient attention to its ramifications. Put bluntly you have to learn how to use this compressed data if you want to use it.

Rayleigh’s method could be used to find other non-dimensional parameters for other physical systems and it is likely that all the useful ones have been derived. They are now known by name, for example, Reynolds number, Froude number, Prandtl number, Mach number and so on. They have become very useful for the understanding of fluid mechanics and of heat transfer and for the storage, retrieval and application of experimental data. The ones that interests us and permit the compression of aerofoil data are the Reynolds’ number and the Mach number. Reynolds introduced the non-dimensional group that is named after him in connection with the flow of water in pipes in 1883 when he demonstrated that there were two modes of flow for water in a glass pipe and speculated that the change from one to the other took place at a particular value of the non–dimensional group where  is the density of the fluid flowing in the pipe,  is the viscosity of the fluid and  is the diameter of the pipe and the mean velocity of flow. Subsequently, in several of the well established expressions that involved a non-dimensional coefficient it turned out that the coefficient was some function of a form of the same group that became known as Reynolds’ number and denoted Re. For aerofoils Re involves  is the density of the fluid,  is the viscosity of the fluid,  is the velocity of the aerofoil having a specified profile and surface set at the same attitude and  is the chord that is the single dimension that determines the size of the aerofoil. Then any aerofoil of the specified profile and finish at the same attitude in any fluid will share a single function of Re. We shall see just how important this is to the compression of aerofoil data.

This meant that Rayleigh had introduced the idea of non-dimensional plots being used to compress data and Stanton and Pannell in 1914 with their smooth pipe curve showed just how powerful non-dimensional plots can be.

The compression of this data depends first on the use of a rational expression. The rational expression is to be required for defining coefficients for forces and moments and must relate the force to the size of the aerofoil, the fluid in which it flows and the speed of the fluid relative to the aerofoil. The obvious thing to use is the product of stagnation pressure and a relevant area to give a notional but quite understandable force. We use:-

where  is the density of the flowing fluid,  is the free stream velocity and  is a relevant area that, in this case, is the plan area of the test model aerofoil.

Then coefficients can be devised to relate this notional force to a real measured force.

We can define a coefficient of lift  using:-

.

A coefficient of drag can be defined in the same way:-

[16]

A moment coefficient  can be defined using the chord as a typical dimension to give:-

where the plan area has been multiplied by the chord to extend the idea of a notional force to a notional moment.

Reynolds number can be evaluated for any measurement.

Now we must think about how data may be collected. Once again we need a strategy. What designers want is to be able to design wings with a reasonable level of confidence in the predicted performance. Real wings might look like the one in figure 19-23 with all sorts of things sticking out of them. A designer is most likely to want to start with test data on the aerofoils that have been used successfully in the past. The data will have been obtained in a wind tunnel using some specified test conditions and related to actual performance.

He will need to understand these test conditions that must include the details of what was actually tested. It may be that the test model was made as accurately as is possible with contemporary manufacturing facilities with a high quality finish but what he really wanted to know was how a model of a real wing performed. Somehow he has to make a correction and then further corrections for all the departures from a solid wing that must be made to permit take-off and landing. He will know about the behaviour of aerofoils as I have explained them above to help him in this task.

Knowing about this problem, those who decide what to test have some quite important decisions to take. One might argue that they should test a range of aerofoil sections that are likely to be used for aeroplanes. Then one might argue that the sections should be made to the highest standards of accuracy of profile and of surface finish to give the best possible figures for performance as a basis for the design procedure. Then one might argue for a test model to be made by normal manufacturing methods to give realistic data for a real wing to give a starting point for allowances to be made for the fitting of control surfaces etc.. Who knows how the decisions will be taken and by whom? I have said enough in Chapter 18 to show that there is a level of unpredictability about such a process. Whatever is decided, designers have to make the best use they can of the outcome.

Wherever I look for anything to do with aerofoils I come across the work of the National Advisory Committee for Aeronautics (1915 – 1958) which was an American Federal agency. They gathered and published data on the low speed (100 mph) performance of a set of families of aerofoil sections. The work was done to such exacting standards that there would be no point to repeating it and it will stand for all time. Unhappily it is not very much use to engineers who will have to get what little they can from it and augment it from other sources.

The book “Theory of Wing Sections” by Abbott and von Doenhoff published in 1949 is in two sections. The first is a section on theory with a heavy mathematical treatment and some explanation of the test programme and its outcome. In effect the introduction is written in a sort of mathematical code that makes little sense to most people who might wish to use the book. This is the sort of presentation that Kuhn says that scientists use to talk to others in the same field. It is wasted on me. The second is a section on the data published by the NACA and that runs to about 100 aerofoil sections all of which were tested exhaustively. The presentation is, in modern parlance, not user-friendly. There is no indication that the authors actively thought that they were creating a storage and retrieval system for use by people with widely differing levels of familiarity with either mathematics or fluid mechanics. The graphs are all lost behind clouds of little circles, triangles and squares against a uniform background of squared paper that is so difficult to use and looks to have been chosen so that it is easy to make mistakes. They use scales in twos instead of the universal fives. However one must remember that this started about 80 years ago as I write in 2010 and, no doubt, as the work proceeded it must have become ever more difficult to revisit the original decisions and do something better. In modern parlance it is not user-friendly.

The test procedure was impeccable The aerofoils were tested in a wind tunnel with a working section of 3 feet in width and 7.5 feet in height and capable of operating at any pressure up to 150 psi. This latter provision permits changes of density and presumably temperature of the air in the tunnel. The turbulence in the unobstructed working section was so low that it was very nearly the same as that in the free atmosphere. The use of compressed air permits change in density and viscosity and this permits testing at high values of Reynolds’ number to make the data even more general.

NACA chose to test their models at three different values of Re;  and  presumably reasoning that data for other values of Re could be found by interpolation and perhaps limited extrapolation. The high values of Re were obtained by increasing the pressure and not the speed. NACA thought that this would let their data be used for aeroplanes operating at higher speeds than that produced in the wind tunnel.

Their test models were of 2 feet chord, that is, length from leading edge to trailing edge. They spanned the whole width of the tunnel to give effectively two-dimensional flow. The models were made to very high standards and with relatively few exceptions[18] the only “rough” sections tested had carborundum powder spread thinly over the first 8% of the surface. The powder had a mean size of about 0.011². This area is, of course, the place where the flow is most vulnerable to disturbance of the boundary layer.

Now there must be a choice of section. In the early days there were two schools of thought. The practical men were making thin wings using wood and fabric and, at the same time, others were trying to find a theory of wing sections and mathematical ways of creating sections eg Joukowski, Kutta. My impression is that the mathematical methods did not produce a practical aerofoil because they were all too fat at the front and too sharply tapered at the back and probably under cambered but they may well have found an application in rotodynamic machines like gas turbine engines but I doubt it. The need for depth in wings to permit internal structure led to sections like Clark Y and RAF 30 and 32 that were quite different in shape and no doubt these were tested. The NACA operated before digital computers and it was not easy to explore possible profiles by computation and the NACA chose to test families of sections where each family was derived by simple mathematical processes from a single symmetrical section. Abbott and von Doenhoff gives details of the processes but most users would look at the profiles to assess suitability for some purpose and then the performance data and only need to know how to draw the sections later.

Now we have to find out what was measured and how.

Lift and drag

The ratio of lift to drag is an obvious way of assessing aerofoil efficiency. 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 and this is still a practical value although the cost of fuel is great incentive to improve wings and all the rest of the aeroplane for that matter.

It is worth looking at figure 19-11 that I have brought down for convenience where I have attempted to draw the three forces to scale on an aerofoil for a lift over drag ratio of 25. 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 10 lb for each 600 lb of all up weight. The lift over drag ratio is 60 and this is the ratio for the lift of the wings to the drag of the whole glider!

It must be obvious that the NACA experimenters had to decide how to test their models. If the ratio of lift to drag is seen to be important, the decision to use the plan area for defining the coefficient of drag ensures that this ratio is the same as the ratio of  to . Then the accurate determination of  and  become obvious goals and the primary mechanical variable must then be the angle that the aerofoil makes to the flow. This angle is called the angle of attack and denoted .

The first decision to be settled in such a programme is the definition of the angle of attack. The angle of attack is an angle between a direction of the flow in the tunnel and the test aerofoil. Angles are measured between two intersecting straight lines and we can settle one if we suppose that the air, in the absence of a model, flows along the centreline of the tunnel. But the other line is somewhere on the test aerofoil. Presumably it will be a line that passes through the trailing edge but where will it be? For all sections it could pass through the centre of the nose radius and its location be clear; for flat-bottomed or under-cambered sections it could be a tangent to the underside and again the location be clear; or an engineer might see some advantage in choosing the line that is parallel to the centre line of the tunnel when the aerofoil is set at the angle for zero lift. This last reference line is not so easy to locate but is much more useful for computation. A decision must be taken and the user of the data then collected must be aware of the decision. The NACA did not test any sections that were flat-bottomed or under-cambered but they constructed all their sections by plotting a symmetrical section made up of four parabolas and an arc of a circle about some cambered mean line. Abbott and von Doenhoff say that it is in "the plane of the aerofoil". Presumably they used the chord line as their datum.[19][20]

The chord line as shown in figure 19-4 was regarded as a reference line and test sections were set up in the tunnel with the chord at some known angle to the centre line of the tunnel which was the direction of the undisturbed flow in the absence of a test model. This angle they called the angle of attack and denoted it by.

The NACA did not use a lift and drag balance, which eliminated the two arms shown in figure 19-7b, instead they measured the pressure distributions along the top and bottom of the tunnel. As the lift on the model must be equal to the net force on the air flowing in the tunnel the lift can be deduced from these pressure traverses. The drag was deduced in a similar way from traverses across the tunnel. The moment was measured directly as a force about the suspension points. This calls for a decision on the location of the suspension pivots in the walls of the tunnel. As has been noted above the pressures acting all over the aerofoil produce a distributed force and such an arrangement can be resolved into a single force and a couple. It is inconceivable that the force and the couple will have values that are wholly independent of angle of attack so we must expect the force to change in magnitude and in position relative to the aerofoil and the couple to change as well. The force will be the vector sum of the lift and the drag both of which are measurable but the couple is another problem altogether. Measuring couples is difficult and normally we let the aerofoil move freely on the pivots and put the position of the pivots at the quarter point of the aerofoil. This is done because it is known that the force on the aerofoil acts very nearly at the quarter point. Then the moment on the aerofoil is nearly equal to the moment of the couple and more or less independent of angle of attack. Purists might say that this ducks the issue but one must look at how the resultant information is to be used and measure it in a practical way. The NACA chose to pivot their aerofoils at the quarter point of the chord and they measured the moment by a support wire. It is comprehensible to everyone.

No doubt considerable experimental skill went into the exact way that these measurements were made and it is claimed that the results were very accurate and agreed with direct measurements made elsewhere. In short they were and are totally reliable for the sections that were tested.

The NACA chose to give their results in graphical form. They plotted, on one pair of axes, graphs of lift coefficient and moment coefficient against angle of attack. They chose to use scales for the axes in increments of two instead of 10 or five like everyone else and their plots are on squared paper without the bold pattern of squares that is normally provided to facilitate plotting. It is hard to think of any presentation that would be more troublesome to use except having no squares. It turns out that the values of  are many times greater than the values of  and this precludes the plotting of  and on the same pair of axes. Two graphs or at least two vertical scales are required. No doubt NACA thought that there was quite enough clutter on their graph of  versus  and chose to use two graphs. Having used the angle of attack as the independent variable and understandable by everyone for the first graph, NACA change to  as the abscissa when they plot . It is incomprehensible to me as an engineer. (On page 3 Abbott and von Doenhoff show a "typical" plot against .) Drag coefficient and moment coefficient were plotted against lift coefficient. This seems to be perverse but examination of all the graphs of  against  shows that over a range of  of about 10° the slope of the graph is the much the same for all sections and is very nearly 0.1 per degree. This means that a mental change in the scale of the drag and moment graphs can convert the abscissa to angle of attack over the practical range of angles of attack by multiplying by 10. In addition if you prefer to measure angle of attack from the attitude for zero lift a new scale for angle of attack can start at the intersection of the  graph and the horizontal axis.

Aerofoil data

I am not going to copy out the methods used to construct the profiles of the sections that were tested. The information is given in Abbott and von Doenhoff. I want to look at the test data to see whether a it is possible to get a better mental picture of the behaviour of aerofoils for the purposes of engineering.

In my experience almost all engineering devices that either consume or generate power have important parameters like power and efficiency that, when they are plotted against, say, speed, take maximum values. Whilst we would like to have maximum power at maximum efficiency it is not often possible. We seek the best compromise. It is the same for aerofoils and, in turn, wings.

In the first instance engineers will need to know how the lift, the drag and the ratio of lift/drag vary with angle of attack (Just as Abbott and von Doenhoff did on page 3.)and they really need to know this for real wings at the predicted value of Reynolds number. From such a plot they might be able to decide whether there is a useful range of  that coincides with a useful range of .

It is a fact of life for designers of aeroplanes that symmetrical sections are needed for stabilisers and asymmetrical sections for the wings if only to accommodate fuel tanks and to make high lift devices for landing mechanically practical for aerofoils. They do not need to know how impractical sections behave even if they do come, by development, out of the geometry of a family of aerofoils. (In fact only a few of the NACA sections are in common use.) So it makes sense for me to concentrate on a typical symmetrical section and an asymmetrical section that I know is in current use.

I have chosen the symmetrical section that I have been using throughout this text and the asymmetrical section NACA 632-415 that is used on the wing of the Shorts 360. I have to decide what an engineer might extract from the NACA data that can be of use to him.

In order to start I have copied the graphs for NACA 0012-64 and NACA 632-415 from Abbott and von Doenhoff. They are figure 19-24.

Scroll

The problems of presentation of experimental data in the light of the methods of reproduction of the times become evident. The graphs are very cluttered. It is hard to see why NACA chose to plot their experimentally determined points. They just make it difficult to see the actual graphs and plotting just one graph with the points to show scatter would have been sufficient. NACA give us data[21] for their high quality test sections and for these same sections with the artificial roughening. They test their standard sections at Re =  and ; they test the roughened sections at Re. So what can be deduced from this data without further effort?

If you have the book you can riffle through these plots and see that the graphs of  against  all look basically the same. This is the limiting slope for the  graph for near-perfect aerofoils in two dimensional flow. For all real aerofoils in three dimensional flow will have a smaller slope.

I want to look at the lift graphs first to see what can be deduced from them for engineers. I need to re-plot them in a normal way, that is, with decimal scales. I have done that in figures 19-25 and 19-26 in black for the smooth, accurate, test model at three values of Re =  and . In figure 19-25 the three graphs share a trace until = about 12º and then the three traces part and then they are very different. We are clearly looking at the sudden loss of lift during stalling, one very sudden and complete, one a more gentle process and one in between. The asymmetrical section does much the same thing but the stall is quite different for all three values of Re and is quite gentle.

There are two points of interest here. For each Reynolds number and for both sections the line up to the maximum is an "envelope" graph in that, given that these test models were made to such high standard, no other model having the same section could have a graph above this one; it could only lie on or below this line. Then there is the unlikely fact that the straight line parts of the graphs have a slope of approximately one unit of for 10ºof  which is a great help for computation.

This first point raises the question of where in fact the lines would go for models having imperfections of some sort. NACA roughened their sections over the first 8% of the chord with "0.011 inch grain carborundum spread thinly to cover 5 to 10% of the area from the leading edge to 0.08c along both surfaces of a section with a chord of 24 inches". An engineer needs to have some idea of what this might have looked like. First the grain size corresponds to that used for P60 abrasive paper that would be described as coarse grade. The spread of 5 to 10% is not much different from the area covered with ink on this page if it were to be printed. So there is a lot of space between the grains the size of which is quite large compared with the thickness of the boundary NACA called it "standard roughness" and tested their roughened sections at Re =. I know what has been done but, as an engineer, I do not find it very helpful. It is too artificial. I have a mental picture of wakes like that of figure 19-21 spreading out from each particle of carborundum.

NACA give data for this section with standard roughness. The graph for this model is shown in red in figures 19-25 and 19-26. One might expect to see significant changes in slope but this does not happen. For the symmetrical section the roughening has little effect until = about 10º and then the section stalls very abruptly. It seems that the effect of the roughening is incipient and that it might cause the reversal in the boundary layer on the upper surface to creep from the trailing edge towards the leading edge and suddenly cause the total separation that we call the stall. The small divergence from the envelope curve is consistent with this view[22]. For the asymmetrical section there is an observable change is slope as a result of roughening but it is not much and takes the slope more nearly equal to that of the symmetrical section. The stall is very gentle.

Now I want to look at the graph for both sections on the right hand sides of figure 19-24. I do not think that it is as useful as it might be. In figure 19-27 I have re-plotted  against  for the standard NACA 0012-64 section instead of against  omitting the negative side because the graph is symmetrical. The lower three curves are in red labelled 3, 6 and 9 are for . The single upper curve is for a roughened section at . I cannot read the data for the range -2º and +2º so I have omitted it. This is the range of  in which the section operates most often so this is unfortunate but it is clear the  is about 0.006 in this range. We know what happens to the  versus  graph.

For the asymmetrical section the three graphs for the standard section are most unlikely in shape. The only graphs that I know that have such discontinuities are all those for drag when fluids flow round bodies of various shaped like cylinders, spheres and cuboids and are the result of a sudden change in flow pattern. For aerofoils these discontinuities are also real and they appear in most of the drag graphs. They will have an effect on the lift/drag ratio producing peaks etc. For the roughened section there are no discontinuities.

What we really want to know where the graph of  versus  will lie for a real wing. For Re =  it may lie between the two lines marked 6 but we do not know from this data whether this roughening has given us a second envelope curve. However Abbott and von Doenhoff say that effect of the artificial roughening at the leading edge is severe when compared with that of ordinary manufacturing irregularities but is less severe than that caused by accumulations of ice or mud. This is not surprising as one affects the boundary layer and the other the profile but they also tell us that roughness strips located further back than 20% of the chord has little effect on the maximum lift coefficient or the slope of the graph of versus . So, for these two graphs the upper red line will be the boundary of what is possible and the lower red line marked 6 is more or less the worst practical case so that real sections lie between these two lines. With this information we can go on to plot graphs of ,for these two sections at the single value of  but for standard sections and roughened sections.

I have done for NACA 0012-64 in figure 19-26 a, and b and for NACA 632-415 in figures 19-27 a and b.

If now I plot Figures 19-26a and 26b on a single plot I can cross hatch areas that will contain practical values of . It is figure 19-28. Similarly figure 19-29 is the plot of figures 19-27a and 27b for the asymmetrical section.

One piece of information emerged immediately. The worst case using NACA data is when, for any value of , the lowest value of  is divided by the highest value of . For the symmetrical section this is the lower line in green and this takes a peak value of = 37. This appears to be a usefully high value but it must be remembered that symmetrical sections are normally used for stabilisers and fins where they work in a range of just a few degrees either side of zero. Then the range of is much more modest. At least these surfaces are not cluttered like wings and, give or take a little, we make a similar plot for the asymmetrical section and the peak value is now about 65.

Aspect Ratio

This chapter has been about two-dimensional flow over prismatic aerofoils. In practice we may need to design devices where the flow is not two-dimensional. Such applications include the aeroplane wing, the aerodynamic keel for yachts, and the blades of, say, a propeller or of a wind turbine. In all these we have cantilever wings or blades that inevitably have tips where there is a pressure difference between the two sides that produces a flow round the tip and, as is evident in figure 19-3, the total flow over a pair of wings in flight is three dimensional. We need to understand what happens to the flow and the effect on the performance of the wing as compared with our two-dimensional aerofoils.

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 figure 19-30 where the red bar represents a prismatic wing. (It is the pattern of iron filings in the magnetic field round a bar magnet shaped like the wing but magnetised with opposite poles on the long faces.)

The combination of this flow and the cross flow on the wings produces a wake that is like that shown in figure 10-31. It  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 a greater mass of air is involved in each vortex and moves at about half the speed and absorbs perhaps a quarter of the energy of the shorter wing. What this means is that long wings are more efficient that short wings of the same area.

We need some simple means of quantifying the shape of the wings. The obvious thing would be to use the ratio of the length of a wing divided by its mean chord but for historical reasons we use the wing span divided by the mean chord. We call it the aspect ratio. It is not as definite as one might hope. If you applied it to the aeroplane in figure 19-32 it would include the fuselage and the two engines in the span. Using the definition above this aeroplane has an aspect ratio of 12.

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 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 low energy vortices and low drag. Unfortunately it also gives a thin wing and the wing in figure 19-32 is in fact braced with struts as shown in figure 19-33.

High aspect ratio wings do not lend themselves to storing fuel, accommodating undercarriages, etcetera and is likely to be weak in torsion and a nuisance on the ground and these considerations leads us to look more closely at wings with lower aspect ratios. As a guide a typical the wings of a light aeroplane will have an aspect ratio of about 6. So in order to make a choice we need to know how the aspect ratio affects the lift and drag.

I quite glibly suggested that a cantilever wing could be tested in a wind tunnel but I must look at the reality. NACA used models with a chord of 2 feet and, if the object of the testing is to determine the affect of aspect ratio on the aerofoil performance, then the wing should also have a chord of 2 feet. The Shorts 360 uses an aspect ratio of 12 and a top class standard glider an aspect ratio in excess of 20. Then the test models would need to be up to 20 feet in length and be tested on a tunnel of about twice this width and a height to match. This is possible now but was not in the between war period. NACA did not test for aspect ratio.

Abbott and von Doenhoff give information of the effect of aspect ratio but they took it from Prandtl who could not possibly have been able to test in the required way. Figures 19-34 and 19-35 are derived from Abbott and von Doenhoff. They give no information about what was tested to give these graphs and even though Prandtl is normally reliable it is possible that his work is taken out of context. So we need to look at the graphs very carefully because the first impression is that once you get to an aspect ratio of 7 there is little need to go to higher figures. This must be wrong. For a start the graphs look as though they were drawn by a commercial artist and not a scientist especially 19-35. There is cause for concern here.

The fact that the zero of the coefficient of lift is at a negative angle of attack tells us that the section tested was asymmetrical and the angle of attack is plotted as if the angle of attack was measured from the chord line as the NACA did or perhaps Abbott and von Doenhoff had the graphs re-worked. Whatever has been done these two graphs have not been plotted as NACA chose to plot their graphs of  versus so I must re-plot them if I am to relate them to other NACA data.

Figures 19-28 and 19-29 show for NACA 0012-64 and NACA 632-415 the data for smooth sections and for sections that have been roughened and the data from Prandtl via Abbott and von Doenhoff could be plotted on the graph for NACA 632-415 to permit comparison.

In the event this is not practical and three graphs are needed for ,  and  against the angle of attack . They were difficult to draw because of the poor quality of the original data but I tried to treat it impartially. Figure 19-36 is figure 19-29 repeated for convenience.

Figure 19-37 gives the plots of  against  for NACA 632-415 and for the "Prandtl" family of curves. I could not separate the curves for aspect ratios of 6 and 7. The slope of the  v  lines increase with increase in aspect ratio but come nowhere near to the slope of the NACA 632-415 whether rough or smooth. The slope of the NACA 632-415 graph is about 0.1 per degree and of the graph for highest aspect ratio of 7 is about 0.08 per degree. The gap is so large that it seems unlikely that further tests at higher aspect ratios would have filled it in an acceptable way. The most that can be deduced is probably that aspect ratios of less than 6 or 7 are not very practical.

In figure 19-38 of  against  it was impossible to draw a consistent set of curves for the Prandtl data. All that we can say is that the values of  for the Prandtl data is many times greater than even the roughened test section. It is hard to imagine how the test sections were made.

When you look at figure 19-39 the Prandtl curves take a maximum for  of about 25 when compared with 125 for NACA 632-415.

It seems to me that the data in figures 19-34 and 35 should be regarded as misinformation and discarded. The only thing that must just be useful is that Prandtl, who must be respected as a very competent man, must have thought that the slope of the  v increased with aspect ratio even if we are not sure how. The more worrying thing is that the sections tested appear to have been of low quality or very small or both and so we do not know whether to attribute these changes in slope to the change in aspect ratio alone or to a combination of the two.

Conclusion

I have spent a long time on the aerofoil to see how the data available can be used by engineers. It seems to me that re-plotting the lift and drag coefficients and the ratio of the lift over the drag against the angle of attack reveals more about aerofoils than is immediately obvious from the presentation used by NACA[23].. What is revealed is that the lift over drag ratio will always have a maximum in the useful range of angle of attack and that the lift over drag curve can vary in its position relative to the graph of for different sections. The sections that were developed last have a pronounced bucket shape to their curves of  against  and that bucket moves to higher values of  as the section becomes thicker. This moves the maximum value of  as well.

This bucket shape is important because it means that the drag is very low over a very useful range of angle of attack of 5° where the coefficient is rising to about 0.5 which is a very useful value. This gives us an efficient wing without which economical flight would not be possible.

For flight, the demands on the wings are different for take-off, landing and for cruise or maximum speed and this means that matching an aerofoil section to a design of an aeroplane is not easy. Unfortunately the re-plotting is time consuming to the extent that one would only do it for selected sections and NACA has narrowed the possible choice.

I said that the data for the smooth sections gives us envelope curves for maximum values but I think that there might be room for doubt about whether the artificial roughening gives a reliable guide to the performance of sections as they are normally manufactured and used. This leaves the engineer with a difficult problem to solve.

I looked into books on the design of light aeroplanes and this situation was sidestepped by recommending one or two NACA sections as having been found to be satisfactory. No doubt aeroplane companies have accumulated data of their own to serve the same function.

I think that the way in which the behaviour of a section is related to the flow pattern around it is very useful in applications other than flight.

This chapter has highlighted the problems associated with aspect ratio and its effect on lift and drag.. Wings come in a wide variety of plan forms and to just evaluate aspect ratio as the span, including the fuselage, divided by the mean chord is most unlikely to be sufficient for our purposes. Yet it is hard to see any other simple parameter that might be used. It seems to me that we shall never get much further that saying that, for a given area that is required to produce some desired lift, long wings are better than short ones and let the choice of length be settled by other constraints.[24]

So we do know a great deal more about aerofoils than we did at the outset of this chapter and this will be needed later in applications. I think that the first need is to consolidate this chapter by looking at the way that an aeroplane flies and that is the subject of Chapter 20.

[1] Wings are never rigid in that they flex up and down but, as far as is possible, they are torsionally stiff.

[3] Shades of Froude for just the same reasons. They wanted no interference from know-alls.

[4] This is usually called wing warping.

[5] This diagram is taken from drawings of the Wright Flyer.

[6] NACA gave us the data for models made to very high standards tested in good tunnels. Their data tells us nothing about the effect on the data of using these sections on real wings of finite length.

[7] One should not confuse angle of attack with angle of incidence. The angle of attack is, as I have said, between the centreline and the direction of the undisturbed flow. It can change in flight. The angle of incidence is the angle on an aeroplane between the centreline of the symmetrical aerofoil and the horizontal reference line of the aeroplane. It is fixed during construction. In days past these two names were used as if they were interchangeable.

[8] Helicopters are relatively small.

[9] It is a fact of engineering that often machines turn out to produce high efficiencies but not when they are producing high powers. Engineers must be on the watch for marketing ploys of quoting maximum efficiency and maximum power as if they occurred simultaneously.

[10] It is not a straightforward technique. In picture 4-2 in chapter 4 I pointed out how the dyelines seem to just fade as the water approaches the orifice in the fast-flowing water and, of course, smoke fades in the same way in fast-flowing air. As there is a limit to how dense the smoke can be to start with it is all too easy for the smoke lines to disappear.

[11] 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 to flow over and under 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 at the top of the diagram is already moving more quickly than that at the bottom where the air will have slowed down.

[12] This is a problem for aircraft where a measurement in flight of the atmospheric pressure is needed when flying. It is usual to find a point on the fuselage where, by chance, the pressure is closely equal to the atmospheric pressure.

[13] Aerobatic aeroplanes are built to stall as and when required.

[14] Dr Friedrich Wilhelm Riegels, " Aerofoil Sections" Results from wind tunnel investigations: Theoretical foundations                    Butterworths 1961

[15] I am not sure that the method that the Wrights used was really sound. We really need an unbiased assessment of their method.

[16] Stowed away in that simple statement is another decision. In the absence of a desire to have an aerofoil efficiency that is defined in a simple way one might have chosen to use the frontal area of the aerofoil and not the plan view to construct the rational expression to define . This has consequences.

[17] Experience tells me that the NACA would have been seen as a scientific establishment and to produce science. It seem to me to be unlikely that there were any engineers involved in setting up the programme. My experience tells me that had there been the full weight of the ordinary hierarchical structure of science would have over ridden their views just as it did before the Challenger  disaster.

[18] It is easy to say that typical surfaces should be tested by NACA but it is extremely difficult if not impossible to decide how to quantify a typical surface. If you ignore discontinuities there is are still the problems of waviness and roughness to quantify.

[19] There is a very slight difference between the chord length and the distance between perpendiculars.

[20] I do not like this decision because it destroys the normal mental model that expects the lift to be zero when the angle of attack is zero. Furthermore NACA have chosen to develop sections by geometry from symmetrical sections where the lift is zero when the angle of attack is zero. This makes plotting data for symmetrical and asymmetrical sections for comparison very confusing. Fortunately it is possible to re-plot the data even if it is time consuming.

[21]The presentation used for NACA 0012-64 is used for only five sections. The rest are like that used for NACA 632-415.

[22] Read Abbott and von Doenhoff on turbulent boundary layers.

[23] I suppose that if you ask scientists to collect data for engineers you should expect this outcome.

[24] This problem of being unable to gather data occurs elsewhere notably in gathering test data for soft sails.