# Preamble

Uses of aerofoils

How aerofoils work

# Experimental determination of aerofoil data

Lift and drag

Aerofoil data

The effect of Aspect Ratio

## Units

Typical calculations

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 almost without exception (As always Prantl got it right.)  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.

# Just take a look at the websites that come up when the keywords aerofoil theory are offered to a search engine. What appears is an array of sites all devoted to mathematics at doctoral level. Who uses these? It is certain that the number of people who might understand these sites must be very low indeed and the chances of these very few being engaged in design almost non-existent. At the same time lots of people try to use aerofoils in the total absence of any understanding of the way they work and have no hope of consulting and using aerofoil data because it is hidden (probably unavoidably) behind what looks like mathematically based storage systems.

When I was teaching mechanical engineering to undergraduates our building was just across the road from the publishers of Kempe’s Engineering Handbook. It seemed to me to be ironic that none of our lecturers used Kempe’s, none would have thought of reading it and most would have regarded it as very passé. Yet Kempe’s was the book engineers actually used and not the abstruse material that we taught. The gulf between academic engineering and practical engineering was then very wide and it is still growing. When I took the plunge and attempted to use both Kempe’s and our academic work I found that, with a small change in mindset, they went together quite well and that I could get much farther using both than I could using either alone. I think that this marriage of practice and applied physics is much more productive than progress by trial. The trouble has always been that, as in, say, the case of aeroplanes, the aircraft makers wanted planes to sell now not in five years time when the slow-moving, science-based research produced guidance on how to design them. The design of wartime aeroplanes clearly depended on inspired design and shrewd feedback from pilots and test pilots and was an evolutionary process.

When I was working, the college bought a small wind tunnel. It was rated to give a wind speed of 25 metres/second in the working section. One day I looked at the fan and noticed that the blades were as-cast, all different, and set to no common angle on the hub. The hub had a flat front face. The technicians reshaped the thinnest blade to my instruction to give it a proper aerofoil and then reshaped the other thicker ones to match it. They made a wooden spinner and I calculated the angles for the blades and when it was all re-assembled the tunnel ran at 35 metres/sec. Then we had to rebuild all the instrumentation! Whoever designed the tunnel had no idea how aerofoils or fans work but lots of these tunnels were sold. All that was needed to get this enormous improvement was knowledge of aerofoil action to no great depth. It is a common problem.

Now I do have some idea about how to use aerofoils and, as there is space on my website, this section is a record of that knowledge for anyone to read.

Uses of aerofoils

I think that most would see aerofoils as being used on aeroplanes, gliders and wind-powered generators. They might see an application in helicopters and perhaps, in another world completely, on submarines. They might think of sport kites as well.

Aeroplanes appeared at about 1900, had become a weapon of war by 1917, evolved into a passenger carrying device in the inter war years and then into potent weapons during the 39-45 war and finally into the modern airliners and fighting machines of today. This is an extraordinarily short time for such a level of sophistication to be reached. However what seems to me to be the most significant observation is that Cessna light aeroplanes and the DC 3’s are still flying after 70 years. Designs that are good enough to last so long evolved after only about 40 years of powered flight.

Quite obviously wings lift aeroplanes. In order to do so they have to be driven through the air because the upward force is accompanied by a force that opposes the forward motion. The upward force, the lift, is what is wanted, The resistance to motion, the drag, is the price that has to be paid to get it and the ratio of the lift to the drag is a measure of the efficiency of the wing. No light aeroplane that followed the Cessna was so much better that the Cessna became obsolete, indeed they are being made again. Wings that were made by very ordinary manufacturing methods stood the test of time for light aeroplanes and not much has changed.

The question that this raises is “Why not?” The answer to this lies in the way that the aeroplane is used. Light aeroplanes can been seen lined up on airfields. They may have covers over their cockpits to protect the perspex from the elements but the wings are not generally protected. Dirt and mould must accumulate in the crevices just as they do on cars yet the owners expect the aeroplane, that is totally dependent on its wings and tail, to be usable at any time with no cleaning and only regular maintenance and inspection. Aerofoils that depend for their performance on accuracy of profile and the quality of their surface would not be suitable for light aeroplanes. There is simply no incentive to do anything other than build wings, tails and fins to adequate standards for their intended use. Exhaustive programmes of mathematics or testing to find a new and better section for such aeroplanes is pointless as the Cessna has shown.

We must recognise another aspect of light aeroplanes vis-a-vis airliners. They both fly in natural winds but at very different heights and speeds. The wind at low altitudes contains all sorts of disturbances as is evidenced by clouds and windsocks. Light aeroplanes have to fly fast enough for these disturbances not to affect the flying surfaces unduly and aeroplanes with pilots of ordinary proficiency in aeroplanes with engines of modest power expect to be able to fly safely. Aerofoils with forgiving characteristics are needed and the very fact that an aeroplane designed and built 70 years ago is still flying shows that such sections are available.

Airliners also have long lives and they are normally scrapped because of new regulations about engine noise or the like not in order to introduce new flying surfaces. They fly at much higher altitudes where the air is much more free from disturbance. There is great incentive to improve wing performance to reduce fuel consumption during long flights. A small percentage of a lot is worth having. This is achieved by refinement of the manufacturing methods, detail changes to the profiles of the flying surface, and by an organised programme of maintenance of the flying surfaces.

Gliders are another matter altogether. A good one costs in excess of £150,000. Such machines are flown in competition when a closed course of 300 miles or so is specified and the one that gets round in the shortest time wins. The first requirement is to remain airborne for all that distance using thermals to gain height as required. The second is to fly quickly between thermals without losing height unnecessarily. In order to satisfy these needs the wings are made to extraordinary standards of both profile and surface finish and every aspect of the fuselage, the stabiliser and the fin is designed and built to the most exacting standards. The glider is then looked after very carefully and anyone who can buy such a glider can employ professional help. Even so, if a glider of this calibre runs into a cloud of flying insects, its performance is instantly reduced to that of fairly standard gliders. Some are fitted with wipers to clean off insect debris! Gliders have not quite reached the limits of what is possible with wings and no engine. Inevitably one aerofoil section will ultimately be seen to be the best and as that section is approached the law of diminishing returns will apply. It is hard to know whether computational fluid dynamics will shorten this process but it is likely to refine the aerofoil and the wing plan rather than design it. There are inevitably constraints. These gliders have very long wings and these are difficult to control during the landing phase to give a limit to increasing span.

These high performance wings are made of composites in moulds that have been cut on computer controlled machine tools and polished to high finish. The cured wing is then polished by hand to a very high standard and, as the wings may each be 75 feet long, this is labour intensive.

Aerofoils are used in wind-powered generators. Such generators have diameters up to about 300 feet and rotate at about 5 rpm. This gives tip speeds of up to 50 mph. At the centre the blades have very low velocities. The turbines are extracting energy from the natural wind and this is anything but orderly. The average horizontal speed of the wind varies with height being least near to the ground. The wind will also contain randomly orientated swirls. Large swirls often make the wind appear to swing from side to side and conditions are by no means ideal for having a steady interaction between aerofoils and the wind. If we imagine the wind to be steady at say, 10 mph, the velocity of the wind relative to the blades makes angles between 90° and 11° so the blades are twisted. A blade is rigid and can have only one shape, its designed shape, and all that can be done to match the turbine to the prevailing wind speed is to rotate the blade as a whole about a radial axis. This small angle for the relative velocity at the tip imposes constraints on how much adjustment is possible. Only a politician would think that a wind-powered turbine is a workable device with the sort of versatility of the steam turbine or the gas turbine or the aeroplane wing.

How aerofoils work

We would have no interest in the aerofoil if the properties of atmospheric air were such that powered flight was impossible. 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 power to make the lift.

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 be careful how we design our wings. What we want is to have wings that will lift useful weights on our planet and do it with a low resistance to motion to be overcome by the engine.

I look at airliners coming in to land at Kent International Airport during landing practice and it seems to me, as an engineer, that it is impossible that so heavy a machine can fly so slowly. It seems to be so reliable that the landing approach is directly over Ramsgate and no one looks up. They might have a different attitude if they could see the air flowing over the aeroplane. The aeroplane bristles with little flight surfaces that are extended in front of and behind the wings just to give sufficient lift at a low enough speed for landing to be practical. The aeroplane comes in with significant power being developed by the engines and the nose tilted upwards noticeably. Air is being swept upwards and over and under the wings and then downwards in a great stream and the net effect is to provide a very large lifting force with a great expenditure of power. It would be an interesting sight.

When that same aeroplane has its undercarriage retracted and the high lift devices drawn back into the wing it can fly at altitude at high speed using fuel very economically.

Wings can be complicated things and they have to be a compromise between at least three requirements, cruising and landing and climbing. This is all too complicated as a starting point but it does tell us that we are not dealing with a simple reliable device that can be abused like a bus in India and that we need to understand how it works. We need to start with the most simple. I will start with an aerofoil with a symmetrical section but first we need some physics.

## Essential physics

Clearly aerofoils are made up of curved surfaces over which air flows. Sometimes the curved surface curves in a direction to actively divert the flow of air (it gets in the way) and sometimes the surface curves away from the flow. In order to deal with these two cases we need some information that comes from applied physics.

It is common to draw flow patterns to show how air flows round aerofoils and round many other devices. Usually flow patterns are for two-dimensional flow. In a wind tunnel it is usual to mount an aerofoil horizontally between the two side-walls of the working section so that there can be no flow between the aerofoil and the side-walls. The stream of air in the tunnel starts off with as uniform flow as possible and with as little fine grain turbulence as possible. Then it is reasonable to expect the flow pattern over the aerofoil to be much the same for any vertical section through the working section. Then if one looks at any single flow pattern it is representative of the entire flow pattern. It is not dependent on its position across the tunnel ie the third dimension. So two-dimensional flow patterns are really made up of air (or any other fluid) flowing between surfaces that are viewed from the side. I have drawn two such surfaces between the sides of the tunnel in figure 1. In effect we have a curved, divergent duct. Clearly air cannot escape from this imaginary duct and as it diverges the velocity falls and the pressure rises in accordance with the energy equation. Generally all this is taken as read and a flow pattern is just drawn as lines that represent the end view of the curved surfaces. They are called flow lines or perhaps, for analyses using the concept of an ideal fluid, streamlines. If follows that, despite the wild notions entertained by several august bodies, whilst flow lines can converge or diverge or simply disappear into a region of eddying, they cannot cross or split or coalesce. This helps us to interpret a flow pattern.

When a mass follows a curved path it is subject to an acceleration towards the centre of curvature. That acceleration can be evaluated from . But, if a mass is subjected to an acceleration, there must be a force to cause it and that force acts through the centre of curvature. If the mass is a quantity of gas such as air, the force must either come from some solid boundary or it must come from a pressure difference acting inwards.

Figure 2 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 are spaced out so 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. Now, as I said above, it is accepted in fluid flow that when the velocity of a flow falls its pressure will rise. Here the lines have diverged and therefore the velocity fallen so it follows that the pressure must rise between AA and BB.

However there is a snag. It seems that when a fluid flows over a 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 (internal friction) is to prevent the layers of air in the immediate vicinity of the surface from moving as quickly as the fluid more remote from the surface. The resulting variation of velocity in a stream of fluid is given in figure 3. At each distance from the surface the length of the arrow indicates the velocity. The diagram is drawn to a much-exaggerated vertical scale. Clearly the velocity changes from zero at the surface to almost the maximum velocity in a small distance and the fluid in this small distance is called the boundary layer. The existence of this layer severely limits the performance of aerofoils. Figure 4 shows why.  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. If a tiny obstruction such as 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 to make it very fragile. A joint in the skin or a control surface has the same effect. However such imperfections do not affect the lower surface because the wakes quickly die away. The flow over the lower surface is much more robust.

Returning to the flow over the curved surface and recalling that a rise in pressure can only come at the expense of kinetic energy we see that the rise in pressure requires a matching drop in velocity. This may be possible for the air in the main flow but the air in the boundary layer does not have enough velocity to start with to produce a large enough rise in pressure. So, at some point on the upper surface, air in the boundary layer and close to the surface comes to rest and air starts to flow forwards from the trailing edge towards this point. This effectively separates the main stream from the surface and causes an eddy to form as shown in figure 5.

Despite the fact that aerofoils are used in the natural wind engineers do not, at the outset, attempt to take this into account. Instead they consider something much more simple to start with and then try to take other complexities into consideration. Here I shall have to imagine a steady wind that approaches in a straight line at uniform velocity. This has the great advantage of being comprehensible even if it is not attainable. Figure 6 shows a symmetrical aerofoil (NACA 0012-64) that is aligned with the steady uniform wind. We know things about this system. First the flow will be symmetrical and there will be no net vertical force on the aerofoil. We know that the aerofoil will be subject to a force acting from front to back. This force is usually called the drag. One must ask how this drag is actually exerted on the aerofoil. I think that we can say with certainty that, in the process of displacing air upwards and downwards, the pressure over the whole of the leading part of the aerofoil back to the point of maximum thickness will rise above the free stream pressure. Of course the rise in pressure will not be uniform and one would expect it to be greatest at the nose and fall away to nothing. We can be confident because the rigid aerofoil can produce any passive force that is required to satisfy the laws of motion. This distributed pressure will produce a distributed force that has a net component in the direction of the free stream but no net vertical force.

This deals with the forward part but what of the rest? Over the “tail” of the aerofoil the air closes in again and does so over a pair of curved surfaces. The air must follow curved paths and necessarily a centripetal acceleration is needed and a centripetal force to go with it. That force can only come into existence if the pressure over the tail drops below the free stream pressure. The net result is another force acting in the direction of the free stream on the tail to add to the one on the nose. The two together give the drag. But this force is not of the same character as that at the nose because, as we have seen, the flow can become detached from the solid surface whereas, on the nose, separation is not at all likely.

So this flow is not as simple as it looks.

Figure 7 will help us to make a start. It is a picture of a model of an aeroplane wing having a symmetrical 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 with a square working section and that the model fitted quite closely to the side-walls so that there was little leakage between the aerofoil and the walls. The velocity of the air was probably quite low in order to avoid diffusing the smoke too quickly. The smoke would have been introduced to the tunnel by a smoke rake, which is a piece of pipe work looking like a rake with hollow tines. The tines are set up at equal spacing and the rake is jogged at regular intervals of time to produce lines of smoke each having blips which were inserted at the same instant of time. These are the blips on the flow lines. There are important observations to make immediately.

There is a simplistic explanation of the way in which lift is generated that supposes that the air flowing across the upper surface must flow more quickly because the upper surface is longer than the lower one. By implication the suggestion is that the approaching air parts in front of the aerofoil and rejoins in the same respective positions behind the aerofoil. This is just not true; the air flowing over the upper surface is some distance ahead of the air flowing over the lower surface when they rejoin. A closer look will show us that a set of blips is just arriving on the left side at the top but has yet to appear at the bottom. Clearly the air at the top of the diagram is already moving more quickly than that at the bottom where the air will have slowed down.

So, in region A, the pressure is lower than the average, in region B it is higher than average, in C and D it is near to the average. Inevitably there is a net force on the aerofoil acting upwards and for separate reasons backwards. (There is also a moment tending to turn the aerofoil nose over tail but this cannot be deduced from the picture.)

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 and show which way the pressure acts. Lines may be drawn joining the heads (or tails) of the arrows as shown in the figure 8. The trouble is that, whilst the lower surface seems to be subject to pressure acting on the surface, in fact it is subjected to pressures inside and outside and the arrows show the net difference in pressure. When we look at the top surface the outside pressure is lower than the pressure inside the aerofoil and once more we are looking at the net difference that is commonly called “suction”. Clearly pressures act on areas and as a result both sides of the aerofoil have net forces on them and these add to generate the lift and of course the drag. But that would involve a computation.

Let us see what else can be extracted from our smoke lines if we link them to the regions of high and low pressure. We can discern a stream of air between two flow lines that flows past the underside of the aerofoil and in contact with it. Keeping in mind that flow lines diverge as the pressure rises, we can see that the pressure at the leading edge and over the front part of the underside is higher than the free stream pressure as we might expect. The existence of this region of high pressure is inevitable because the aerofoil is solid and exerts a passive force (it exerts whatever force is required) on the air. The very existence of this region of high pressure affects all the air moving past the aerofoil. We can see that, in the air moving below the aerofoil, it produces a slowing down upstream (with a consequent rise in pressure ahead of the aerofoil), and a speeding up downstream (with a drop in pressure). The presence of the other air and ultimately the side-walls of the tunnel prevents a sideways spread. The rise in pressure upstream of the high-pressure region is also exerted on the air that will eventually pass over the upper surface to divert it upwards and to make it go faster as it approaches the sharply curved part of this surface. So the presence of the aerofoil affects the flow well upstream to divert it upwards and to accelerate the air that will flow over the top and slow down the air that will flow underneath.

Now I need to look at the flow pattern round an aerofoil in greater detail.

The most important part of an aerofoil is the nose and in figures 9 and 10 I have drawn flow patterns round the nose of a symmetrical section at various angles to the undisturbed flow. Experience tells me that people pay little attention to these patterns but it is a great advantage to understand them. Even though these figures are not from photographs taken in wind tunnels, they cannot be far from reality and I think that they are reliable. I want to draw attention to the change in the shape of the flow lines over the nose as the angle increases. When the angle is zero the radius of curvature is large, but as the angle increases the radius becomes very small. I pointed out above that the centripetal acceleration is given by . Now we have increasing velocity and decreasing radius. The centripetal acceleration must grow if the flow is not to break away. Inevitably at some angle to the undisturbed flow the air will be unable to make the turn and will break away. When it does the contribution to the total lift produced by the upper surface will disappear. In the normal parlance the aerofoil will stall.

This leading edge is also the place most likely to collect insect debris and ice and dirt and it becomes obvious that there is an incentive to look for a new shape that is not so vulnerable.

Intuitively one might think of making the nose sharp but too many would-be flyers have died in accidents with sharp-edged wings for that to be a serious option. Our only option is to go away from the symmetrical section, accept that any new section will really only lift in one direction, and see what is possible.

In figure 11 I have drawn an asymmetrical aerofoil. It is derived from the NACA 0012-64 drawn above. The symmetrical aerofoil is drawn about an arc of a circle to give this shape. I am not advocating that aerofoils should be drawn this way merely drawing one to show what happens when a symmetrical section is plotted on a curved line and the arc is the most simple. Now, of course, we want to see what happens to the flow pattern when this new section is at 10° to the undisturbed flow.

I have drawn it in figure 12. When compared with the symmetrical section at the same attitude in figure 10 it is evident that the flow-lines before and over the nose have larger radii of curvature and that was the object of the change. It would be reasonable to expect this asymmetrical section to go to a larger angle before it stalls and this is in fact the case. However the radius of curvature at the trailing edge is considerably reduced and this will have an effect on separation initiated from the trailing edge.

Summing up, I think that this gives a practical explanation of the way that aerofoils interact with a stream of air and now we need to see what happens when aerofoils are tested diligently in a wind tunnel.

Before we do there are a few more points to be made. On every diagram I have shown one flow line that goes up to the nose and stops. This line is called the stagnation line and, in practice, it is sharply defined as the line between air that flows over and the air that flows under the aerofoil. The change from pressure to suction at the nose of the aerofoil really is very sudden. On some light aeroplanes the stall-warning device is a whistle that is connected to a hole in the leading edge of one wing. When the wing is flying normally the hole is in the high pressure region but, if the angle of attack increases to a dangerous value, the pressure pattern moves round the wing and the hole “goes” into the low pressure region, starts to suck, and blows the whistle.

This all means that it is quite possible to predict the behaviour of an aerofoil. Both the lift and the drag are made up of two forces added together. The lift is the sum of the force on the under side and the force on the upper side. The force on the underside will always have a component that acts upwards except perhaps as its angle of attack approaches 90°. One might expect it to grow with angle of attack. The force on the upper surface will grow with angle of attack and then either suddenly or perhaps over a small change in angle of attack go to nothing as the flow breaks away and the negative pressure disappear. So the sum of these forces will grow with angle of attack and then drop substantially  when the flow on the upper surface breaks away.

The drag is made up of the friction force exerted by the air on the aerofoil and a force that is really the horizontal component of the net dynamic force on the aerofoil. The friction force will always be there but the dynamic force will be small when the angle of attack is small and then, when the flow over the upper surface breaks down, the horizontal component of the dynamic force will grow and ultimately, at 90°, be the dominant force.

# 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 and temperature and the aerofoil can have innumerable shapes and surfaces. When the Wrights started their tests on wing sections 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. Some programme of testing must be devised and some means of storing the data must be chosen and that means of storing must be comprehensible to most people who might want to use the data.

The only thing that I can do is to explain what the National Advisory Committee for Aeronautics (1915 – 1958), an American Federal agency, did in the 1940’s and hope that this will enable me to use other data. 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.

In the book “Theory of Wing Sections” by Abbott and von Doenhoff published in 1949 there 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 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.

One might reasonably ask why the tunnels could be pressurised in this way when aerofoils in ordinary use fly in air of 14.7 psi or lower. The use of compressed air permits a change in density and viscosity and this permits testing at high values of Reynolds’ number to make the data even more general. This calls for an explanation of Reynolds’ number.

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.

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 eg Reynolds number, Froude number, Prandtl number, Mach number and so on. They have become very useful for the understanding of fluid mechanics and in heat transfer and for the storage, retrieval and application of experimental data. The one that interests us and permits the compression of aerofoil data is the Reynolds’ number. Reynolds introduced this 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 a single dimension namely the chord 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.

This meant that Rayleigh had shown us non-dimensional plots and probably Stanton and Pannell in 1914 with their smooth pipe curve showed just how powerful non-dimensional plots can be.

NACA chose to test their models at constant values of Re using three values of  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, ie 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 a few exceptions the only “rough” sections tested had carborundum powder spread thinly over the first 8% of the top 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.

They are in several families so that interpolation is possible.

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 Joukowshy, 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.

Lift and drag

From the time of Lilienthal and the Wright brothers the performance data has been recorded in substantially the same way and their method has its roots in James Smeaton’s work on lighthouses and sea defences. The Wright brothers turned to wind tunnel testing fairly quickly after attempts to make measurements on an aerofoil mounted on a bicycle. So, what was this method?

Smeaton was interested in wind forces and wave forces. He recognised that the wind force on a square surface held at right angles to the wind varied mainly as the square of the wind speed and that, whilst the same held true for water impinging on the same surface, the water forces were much greater. He went on to express the force on other surfaces in the following way :-

Force = . where  was the velocity of the air or water

was the coefficient of air (or water ) pressure such that the product  equals the force on one square foot of surface held at right angles to a flow at velocity v.

was the area of the surface

and  was a coefficient which depended on the shape of the surface and its attitude to the flow.

We use the same method today with a refinement. We write:-

Force =

where S is the area of the surface and  has been replaced by  where r is the density of the flowing fluid.

There is more to this expression than a simple change. What it amounts to is that the force is now expressed as a coefficient times a defined quantity. If the terms in that quantity are inserted in consistent units the quantity as a whole has the units of force. Then the coefficient does not change with a change in the system of units. It is non-dimensional. I call this quantity a rational expression because each term has a meaning but as a whole the quantity is not theoretical in the sense that it is the outcome of theorising about a particular physical system. The group  is the rise in pressure that occurs when a fluid flowing at velocity  is brought to rest. This is the pressure where the stagnation line meets the nose of an aerofoil. The area  is chosen as being either significant or typical for the system in hand. For aerofoils it is the plan area of the aerofoil. The product of these two is a comprehensible force and, as a consequence, a value for the coefficient has meaning. For instance it would be unthinkable that a coefficient of lift could be 2 because that would mean that the lift could be twice the stagnation pressure multiplied by the area of the aerofoil and that outcome is absurd.

Now we have to decide what to record.  The distributed forces acting on an aerofoil can be reduced to one force and a moment. Right from the start the early experimenters had ideas of lift for a wing, which was needed to get into the air, and drag that would have to be overcome continuously by an engine if sustained flight were to become a reality. These two forces are the components of the total force and we call them the lift and the drag. We still measure these same forces and express them :-

Lift =

Drag =

where  and  are the coefficient of lift and the coefficient of drag. In both cases the area is the same, that is, the plan area of the aerofoil. The moment is measured and expressed as  where  is the moment coefficient,  is the chord and the other symbols have the same meanings as above with a change to  for velocity.

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

It is worth looking at Figure 11 where I have attempted to draw the three forces to scale on an aerofoil for a lift over drag ratio of 30. It is clear just how small the drag is when compared with the lift. The aerofoil is a very efficient device. Modern gliders may have a glide ratio of 60 to 1 and can travel about 60 miles from a starting height of 5,000 feet in the absence of vertical movements of the air. More revealing is that the drag on a glider spanning perhaps 100 feet is only 10lb for each 600lb of all up weight. The lift over drag ratio is 60 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. The chord line as shown in figure 9 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 is called the angle of attack and denoted by . As it is the angle between the chord line of the wing of an aeroplane and the direction of flight so it has direct application.

The NACA did not use a lift and drag balance, 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. 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 reliable.

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 independent of angle of attack so we must expect the force to change in value 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 pivots, let the pivots resist a force equal to the resultant of the lift and drag and measure the moment of the resultant about the pivots. 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 satisfactory way.  The NACA chose to pivot their aerofoils at the quarter point of the chord because this is not far away from where the resultant force appears to act and it is comprehensible to everyone. Then they measured the moment.

An aerofoil when flying as a part of an aeroplane must be stabilised in some way. Normally a second, usually smaller, aerofoil is attached to the main aerofoil and adjusted to achieve stable flight. I devoted chapter 18 of my book on the model yacht to the problems of such a system and I refer the reader to that. The moment on the aerofoil is a just one element in the whole system and it is evident that extreme accuracy is not called for.

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. One might expect that the drag coefficient would be plotted in the same way but this was not so. 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 same for all sections and is very nearly 0.1 per degree. This simplifies cross relating of data.

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 relate the shapes to the data.

Figure 12 is a copy of two pages of Abbott and von Doenhoff. They show the data for section NACA 632 –415 that would be appropriate for a feeder airliner. The section is relatively thick to facilitate a construction to suit wing-mounted engines and high lift devices. It is obviously asymmetrical.

The left hand page is complicated by the data for the deflected flap and the main data lies in the two “middle” graphs. Ignore the lower graph of the four plotted of  versus . It is for a roughened section. The remaining three give us three points on non-dimensional plots for Re =  and . They are much the same and no one would want to use this aerofoil in the region where they divide. The maximum value of  is about 1.4.

The shape of this graph is what we might expect. The sum of the forces on the lower and upper surfaces increase steadily with angle of attack and the relatively suddenly the coefficient of lift starts to fall. The force on the upper surface is decaying rapidly. The drag coefficient has a significant lowest value, that of the friction drag, and then increases with increase of the angle of attack.  NACA did not test beyond the stall but the coefficient of drag must go on increasing after the stall.

The shape of the section shows that the forward part to the upper surface has a relatively large radius of curvature just as the flow lines suggest is desirable to avoid breakaway. The after part of the upper surface is a very good compromise between thickness and curvature. Between them these features give the gentle stall.

The right hand page gives  versus  and it is interesting to find a few values of . At   = 8°,  =1.1 and  = about 0.011 giving a ratio of 100 for the superior sections that were tested. When I first saw this I was astonished but I have got used to it now. If the ratio is evaluated for the same angle but for the roughened section it becomes70.

In viewing these figures one must recognise a basic characteristic of all aerofoils when actually used. Most often they have to provide a constant force. This can be evaluated from Lift = . ½ r .S . Then  is constant and it follows that, as speed increases, the angle of attack must fall and vice versa. Aerofoils operate mostly at low angles of attack perhaps up to 4°.  Then the lift over drag ratio is 145 for the smooth section and about 70 as before for the roughened section.

The graphs show that the aerofoil does best at the high values of Reynolds number and that should the section ever be taken into the region where it stalls the stall is gentle and not sudden as it is for some other sections.

Figure 13 is for symmetrical section NACA 0012-64

Clearly the symmetry shifts graphs that are very similar to those for NACA 632 –415 to be symmetrical relative to the axes. As we have seen the symmetrical section is likely to stall at a lower angle of attack and so it proves. In addition the smaller radius of curvature just behind the nose makes the stall much more sudden. Otherwise the symmetrical section is just one of a family of sections and is not in any way special. Its great attraction is in applications like control surfaces for aeroplanes, submarines, keels and rudders for yachts  and in aerobatic aeroplanes where inverted flight is expected. Personally I like this section. It behaves very reliably.

The effect of Aspect Ratio

So far we have been concerned with data from wind tunnels. We must now consider what happens when we have an aerofoil where there are no side walls to prevent the flow having a component along the aerofoil and indeed to explain the origin of this type of flow.

Suppose that we had a wind tunnel that was very wide and we attached an aerofoil to one side so that it protruded, cantilever fashion, across one half of the tunnel. Suppose that this aerofoil had a length that was 3 times its chord. It would have the same proportions as one wing of a light aeroplane. We now have an aerofoil with no wall at one end. We can call the end at the wall the root and the free end the tip.

If the tunnel is run with the aerofoil set at a practical angle of attack there is no reason to suppose that the flow over the aerofoil very close to the wall will be affected. This means that the region of high pressure under the aerofoil will be created as before and the pressure will drop over the upper surface exactly as before. But now the high pressure is no longer constrained to act only backwards, forwards and up and down, but can now act sideways. At the other wall there is no reason to suppose that the flow will differ from that which would prevail if the tunnel were to be empty. This gives rise to a further complication in that the low-pressure region which comes into existence over the upper surface of the aerofoil is low relative to the pressure on this wall. The result is that the air approaching the aerofoil follows broadly the same flow pattern as before but now the flow which will go under the aerofoil moves towards the tip and the flow which will go over the aerofoil tends to move towards the root. This is now a complex three-dimensional flow that cannot easily be investigated by use of smoke and perspective representations in two dimensions become excessively difficult to draw.

If we separated the two effects and looked just at the flow from the underside of the wing to the upper side we could see it as a two-dimensional flow. Others who have looked at this have produced a mathematical expression for these paths and drawn them as shown in figure 13 With it is figure 14 which is of a crop duster. Accepting that the aeroplane is flying fairly close to the ground and that the cloud of fine particles does not extend into the air beyond the wing tips, it is hard to avoid the conclusion that the combination of the flow over the wings and the rotation produces a pair of large spiral vortices. Normally the main observable feature would appear in humid air as a small vortex leaving the tip but this is but a small part of the flow and probably not the most important. These vortices are unwanted. They reduce the lift and, as a result of imparting kinetic energy of rotation, increase the work to be done by the engine to overcome the extra drag.

Now suppose that the wings of an aeroplane could be replaced by new wings of the same area but of twice the span. Clearly the mean value of the chord will be one half of the original value. The new wings have to provide the same lift by deflecting air downwards but now they affect twice as much air. The result is that a greater mass of air is involved in each vortex and moves at about half the speed and absorbs perhaps a quarter of the kinetic energy. The result is a reduction in drag.

Let us now consider a real wing. Suppose that the wing is to generate a given lift to fly an aeroplane and that we are free to choose a wing plan. We see now that unless there are good reasons to do otherwise we should choose a wing that has a large span and a small chord. In other words to give the wing a high value of the ratio of the wing-span to the chord that is a high aspect ratio. We might further consider tapering the wing at the tips to reduce the chord and perhaps twist it at the tips to reduce the angle of attack at the tips. All these things would give an efficient wing with low energy vortices and low drag. Unfortunately it also gives a thin wing and that is not much good for storing fuel, undercarriages, etcetera and is likely to be weak in torsion and a nuisance on the ground. Such a wing is also difficult to control near to the ground during landing and these considerations lead us to look more closely at wings with lower aspect ratios. We want to know how the aspect ratio affects the lift and drag to help us choose.

Figures 15 and 16 are derived from Abbott and von Doenhoff where the originals can be consulted together with their explanatory text. They are for rectangular wings of aspect ratios from 1:1 to 7:1. The first shows the plot of  against a. This graph shows clearly that the gains to be made from any further increase above 7:1 are going to be small. It also shows that practical values of aspect ratio start at about 5:1. The second graph of  against  for the same aspect ratios show how the drag increases for low aspect ratios. The two graphs show a pattern of diminishing returns for increases in aspect ratio.

## Units

If calculations are to be made from aerofoil data some system of units is essential. We have three systems of consistent units based on the basic quantities mass, length and time. They are the SI system where one Newton acting on one kilogram gives it an acceleration of 1 metre per second per second the Imperial based systems where either one poundal acting on one pound mass gives it an acceleration of 1 foot per second per second or one pound force gives one slug an acceleration of 1 foot per second per second.

It is important to use one of these three but whichever is chosen the values of density and viscosity must be found from published data in the correct units. These are kilogram/metre second, pound/foot second or slug/foot second. These get manipulated in all sorts of ways and looking up viscosity is fraught with possibilities. Be careful.

As a guide to data. The viscosity of air at 20°C and one atmosphere is about 1.76  kg/ms or about 3.8  slug/ft sec. The viscosity of water is about 1  kg/ms or about 1.06  slug/ft sec at 20°C.

Speeds must be in metres/second or feet/second. Lengths must be in metres or feet.

Typical calculations

The two sections that I have used can be plotted from the following data. The plot uses stations along the chord as percentages of the chord and ordinates that are also percentages of the chord.

NACA 0012-64

 Station 0 1.25 2.5 5 7.5 10 15 20 30 40 Ordinate 0 1.813 2.453 3.267 3.813 4.24 4.867 5.293 5.827 6

 Station 50 60 70 80 90 95 100 Ordinate 5.827 5.32 4.48 3.32 1.867 1.027 0.12

NACA 632 –415

 Upper surface Lower Surface Station Ordinate Station Ordinate 0.300 1.287 0.700 -1.087 0.525 1.585 0.975 -1.305 0.991 2.074 1.509 -1.646 2.198 2.964 2.802 -2.220 4.660 4.264 5.340 -3.000 7.147 5.261 7.853 -3.565 9.647 6.077 10.353 -4.009 14.669 7.348 15.331 -4.656 19.705 8.279 20.295 -5.095 24.750 8.941 25.250 -5.361 29.800 9.362 30.200 -5.474 34.852 9.559 35.148 -5.439 39.905 9.527 40.095 -5.243 44.955 9.289 45.045 -4.909 50.000 8.871 50.000 -4.459 55.039 8.298 54.961 -3.918 60.070 7.595 59.930 -3.311 65.093 6.780 64.907 -2.660 70.106 5.877 69.894 -1.989 75.109 4.907 74.891 -1.327 80.102 3.900 79.898 -0.716 85.085 2.885 84.915 -0.193 90.059 1.884 89.941 0.184 95.028 0.931 94.972 0.333 100 0 100 0

Let me give two examples.

Keel of yacht. Typically touring yachts have keels of about 5 feet in length and about 2 feet wide and they carry a streamlined weight at the lower end. The keel is an aerofoil or hydrofoil that makes an angle to the course to generate a sideways force to resist the transverse force produced by the sailing rig.

Suppose that the yacht makes 6 knots.

Re is calculated from .

Re=

Where the first bracket is the density of seawater in slug per cubic foot, 2 is the chord length, the next bracket is the speed converted to feet per second and the last bracket is the inverse of the viscosity in slug/foot sec.

Clearly this puts this keel on to our plot of  against a.

The force can be calculated from

Where the area is 10 square feet. Now, as  = 0.1a from the data, this becomes :-

Lift =102a lbs and at, say 8°, this gives a force of 816 lbs.

The keel has a low aspect ratio of 2.5 but the hull at one end and its weight at the other act to counter the generation of vortices by the keel. If the calculated force were to be reduced by 30% (See the graph in figure 15) this would give a conservative figure.

Reference to figure 13 shows that the value of Re takes us to the upper of the three curves for  and so  will be in excess of say 0.012 even if the keel is polished. Then the drag at a = 8° is about 102´0.012 = 1.2 lb. But there is a widely held view that the keel should be as cast and the drag will be much higher perhaps as much as 2 lb from the upper graph. This tells us why the rough keel does not produce a discernible disadvantage. The drag is just very low.

The Shorts 360 aeroplane flies at about 195 knots with a wing area of 42 square metres. The aspect ratio is about 12 and the effective chord is about 1.48 metre. It has very nearly the NACA 632 –415 section. For this wing :-

where the first bracket is the density of air at sea level, the second is the chord, the third the speed converted to metre/sec and the last the inverse of the density. This value of Re means that we can use the data for the section.

Then the lift by similar methods becomes:-

Then for an angle of attack of, say 8°, the lift is 22,303 kg or 49,000 lb.

Figure 15 suggests that, for this aspect ratio of 12 there is little loss of lift caused by vortex generation so this figure is a probably reliable.

The maximum take off weight of the aeroplane is given as 26,000lb so there is a large margin of safety at take off and for the reduction in lift with height..

The drag can be calculated from drag = 2,800 ´ a kgf or drag = 6,200 ´ a lb. Looking at figure 12 the value of  at a = 8° is about 0.005 for any value of Re so the drag would be about 31 lb if the wing were to be as good as the NACA models. NACA suggest that even with the flap down and a rough model the vale of  would be about 0.012 and the drag about 74 lb. This wing can then be explored to find its envelope. When you look at all the rest of the aeroplane it is hard to imagine that the drag of the wings is the major component of the total drag.

Ivor Bittle October 2007