Aerodynamics

To generate lift, the wings of an aircraft must move through the air. But people who study aerodynamics usually picture the scene the other way around, with the aircraft held stationary and the air moving over the wings in a steady stream. The wing profile is curved, and the fluid parcels speed up as they pass over the top. Here, the pressure falls, so the wing applies a force to the air, pulling it downwards, and in turn, the air pulls the wing upwards, providing the ‘lift’ needed to keep the aircraft flying. Our problem is to fill in the gaps: to explain in simple language what is going on, and how designers have been able to re-shape the profile to handle fluid speeds up to and beyond the speed of sound.

The boundary layer

The atmosphere can apply a force to the wing in either of two ways. The first is through pressure acting at right-angles to the surface, which is largely responsible for the lift. Where does it come from? The answer is that the source lies some distance away. The air parcels ahead of the wing divide into two streams as they approach the leading edge, some passing over the top, and some below. Their paths are curved as they deviate from a straight line, and across the whole of the flow field, other air parcels adjust their paths to accommodate the deviation. It may be small, but each parcel has inertia, and when accelerating and decelerating along a curved path, it applies pressure to its neighbours, which is transmitted onwards through a chain of actions and reactions among adjacent parcels to emerge finally at the wing surface.

By contrast, the origins of the second force lie within the fluid layer closest to the wing surface. It is responsible for much of the drag force that resists the aircraft’s forward movement. It arises because like any fluid, air is ‘viscous’. As explained in Section F1917, the viscosity is small, but when it flows over an aircraft wing, the air doesn’t slip past in the same way that you or I might slip on an icy path. Some of the molecules are attracted to the wing surface and remain there for a moment as if glued in position. When they depart, they are replaced by others from further away, so in aggregate, the parcels of molecules closest to the surface move quite slowly, while those further away move a little faster, as shown in figure 1.

Figure 1

These sluggishly-moving parcels make up a layer that may be only a few millimetres in depth, but it has a marked effect on the wing’s behaviour. Named the boundary layer, its properties were investigated during the early years of the last century by Ludwig Prandtl and his team of researchers at Göttingen in Germany. Their discoveries brought a new scientific dimension to aeronautical engineering: they found that the thickness of the boundary layer rises and falls with the speed of the aircraft, and it varies from place to place along the wing surface, starting at zero near the leading edge, and growing aft towards the rear.

Inside the boundary layer the air parcels begin their journey by moving smoothly together along the surface, a kind of motion known as laminar flow. But it soon degenerates into turbulent flow, with eddies and fluctuations that disturb the underlying pattern. We still picture the motion in terms of smooth, parallel streamlines, but this is a convenient fiction. The fluctuations are random, and they take place simultaneously on several different scales of measurement, so the streamlines represent the general or ‘average’ direction of motion.

Figure 2

figure 2 shows how the boundary layer evolves over the wing of a large aircraft. For clarity, the vertical scale is exaggerated. In this example, the free stream velocity is 250 m s\({}^{-1}\) and the kinematic viscosity of the air is taken as \(1.4607 \times 10{}^{-5}\; \text{m}{}^{2}\; \text{s}{}^{-1}\). To begin with, the flow is laminar, and there is relatively little friction. But it doesn’t take much to trigger turbulent flow, and for an aircraft wing, the point of transition lies close to the leading edge. After a short distance, the boundary layer becomes turbulent and thickens thereafter; for a wing whose chord measures 5 m, it will reach a depth of around 50 mm at the trailing edge. Aft of the transition point, the erratic motion of the air parcels transfers energy from the outer boundary layer to parcels that lie next to the surface. So they move faster than they would if the flow were laminar, and transmit a greater shear stress to the wing – in other words, the drag rises.

You’d imagine that as well as causing drag, the boundary layer interferes in some way with the normal pressure that fluid parcels exert on the wing. The normal pressure, which accounts for most of the lift force, is usually estimated on the basis of potential flow theory, which does not take the boundary layer into account. But as Prandtl realised, this doesn’t matter: the lift force acts in a direction perpendicular to the wing surface, and is the outcome of fluid motion in the wider flow field, whose behaviour is largely unaffected by what happens in the layer immediately next to wing surface. The lift force is transmitted through it, and its magnitude can be predicted (at least, approximately) using potential flow theory. However, skin friction affects the aircraft in a different way, by slowing it down. In fact, it accounts for roughly half the total drag, and before the Second World War, designers tried to reduce it by developing ‘laminar flow aerofoils’, the aim being to delay the transition so that the area of skin affected by turbulence was as small as possible. It was done by re-shaping the profile and by polishing the wing surface. The experiment was not altogether successful, and we’ll see why later.

Separation

It would be convenient for aircraft designers if the boundary layer remained in place at all times, as if stuck to the aircraft like a coat of paint, albeit paint that is creeping slowly along the surface. Unfortunately, it doesn’t. From time to time, it is liable to separate and peel away from the skin. Separation is not confined to aircraft wings, but can occur in the boundary layer around any moving object. The places where it happens fall into two categories: the first consists of projections or bulges that compel the fluid parcels to make a detour around them on a curved path. A fluid parcel moving at a given speed along a curved path needs a centripetal force to hold it on track, and the smaller the radius, the greater the force. It can only come from the surrounding fluid, so there must be a well of low pressure at the centre of the curve, rising to a higher pressure in the outer flow field.

Figure 3

Then, as shown in figure 3, an air parcel travelling round the nose of a wing will experience a transverse pressure gradient that deflects it sideways, as if attached to the nose by a piece of string. But the pressure at the centre cannot be less than zero, so there’s a lower limit to the radius of the curve, and particles travelling around a corner of very small radius will fly off at a tangent.

It follows that the sharper the nose, the more likely the flow is to separate, and in this sense, a thin wing is more vulnerable than a thick one.

Figure 4

figure 4 shows the flow pattern round the leading edge of a flat plate. For the purpose of illustration, the angle of attack is exaggerated. The radius of the leading edge is small, and as it approaches zero, in theory the velocity increases without limit. At this point, an air parcel must accelerate to a high speed, and lose contact en route, as if propelled by centrifugal force away from the nose. The fluid detaches itself from the wing surface and re-attaches downstream, leaving a ‘separation bubble’ of turbulent flow underneath [5]. Inside the bubble, the velocity is reduced, the pressure rises, and the lift falls. As the angle of attack increases, the bubble stretches further aft until it reaches the trailing edge and leaves a turbulent wake as shown in figure 5.

Figure 5

The second kind of separation arises in a different way. As described in the previous Section, with many aerofoils, the air pressure on the upper surface of the wing increases towards the rear. So when the wing is aligned at a high angle of attack, the flow can come to a halt before it reaches the trailing edge because the rising pressure acts as a barrier, slowing the air parcels down. Crucially, the parcels in the boundary layer are travelling more slowly than their neighbours, and having lost much of their kinetic energy to friction, they come to a halt first. Those further away take longer. As a result, the boundary layer detaches at an angle as shown diagrammatically in figure 6. Beyond this point, the flow reverses, and further downstream, the motion degenerates into a sluggish wake whose effect is to reduce the lift and increase the drag.

Figure 6

In principle, therefore, separation can take place anywhere along the upper wing surface. It’s important for the designer to know whereabouts on the wing surface this is likely to happen – the further forward, the greater the loss of lift. It’s not an easy problem to solve. One can use inviscid flow theory to model the outer flow field, and a viscous model for the boundary layer, but at the point of separation, the two interact, and neither model explains the interaction. You can find a more on this topic in [1], together with details of a mathematical model that bridges the gap in [8].

Lift, speed, and angle of attack

At all times during a flight, the lift created by the aerofoil together with the speed and the angle of attack are closely connected. So let’s look at the relationship between them. In this web site, we have encountered two different formulae for the lift \(L\): the first appeared earlier in the previous Section A1816. It relates \(L\) to the circulation \(\Gamma\) around the wing cross-section when the aerofoil is moving at speed \(V\) in a fluid of density \(\rho\):

The second formula is set out in Section F1817, and it’s the one we need here. When applied to an aircraft, it views the lift as a function of \(\rho\), \(V\), the plan area \(A\) of the wings, and the coefficient of lift \(C_{L}\):

The equation tells us that the lift rises and falls with the density of the atmosphere, and with the square of the aircraft speed. The shape of the aircraft is embodied in the lift coefficient. However, at this stage we are dealing with a small part of the aircraft, a two-dimensional cross-section of the wing. As such, it doesn’t behave like the entire wing, and to reflect this it is usual to change the notation from \(C_{L}\) to \(c_{l}\), so equation 2 becomes

Notice that the angle of attack doesn’t appear explicitly on the right-hand side – its influence is hidden in the section lift coefficient \(c_{l}\). If the angle of attack changes, as far as the oncoming air flow is concerned, the shape of the body has changed, and \(c_{l}\) changes too. However, its value can be measured using models in a wind tunnel at different angles of attack, and the results scaled up to aerofoils of full size.

Let’s pursue the angle of attack for a moment. It refers to the angle at which the wings of an aircraft bite into the approaching air, and it changes when the pilot pitches the nose up or down. In normal flight, it rarely exceeds 15\({^\circ}\). To measure its value, we need a reference line that defines the fore-and-aft axis of the wing cross-section.

Figure 7

As shown in figure 7, it is the chord line, usually taken as the straight line joining the leading edge and the trailing edge (its length is known simply as the ‘chord’). Then \(\alpha\) is the angle between the chord line and the undisturbed air stream. At any given speed, \(c_{l}\) (together with the resulting value of lift) rises with \(\alpha\), and the curve that describes the relationship between the two is an important indicator of the wing’s handling behaviour – a notional curve is sketched in the upper part figure 8.

Figure 8

It is known as the lift curve. For many aerofoil sections, the main body of the curve follows approximately a straight line, whose gradient \(a_{0}\) is known as the lift curve slope, or just the lift slope:

In some ways, the relationship expressed by the lift curve resembles the relationship between the lateral force and the ‘slip angle’ of a car tyre, or the lateral thrust and the ‘drift angle’ of a ship’s hull. They are all relationships that link a transverse force to the alignment of the body in question relative to the direction of motion. In the case of the flat plate aerofoil, the relationship has a rigorous mathematical basis. In the last Section, we showed that for a flat plate moving through an ideal fluid, the slope was approximately \(2\pi\), and in fact, this applies to a range of thin aerofoils [4]. Also, note that a cambered profile generates lift even when the angle of attack is zero, in which case the lift curve shifts bodily to the left, as indicated by the dashed line in the lower part of the Figure. At a particular angle of attack, the lift coefficient reaches a maximum value that is often in the region of 1.5. But what happens if the angle of attack continues to rise?

Stall

At this point there is a marked change in the aerodynamic behaviour of the wing. Essentially, the air flow breaks down, either because (a) the nose of the aircraft is pitched too high, or (b) the wing isn’t moving fast enough, and the result is a stall. The process begins with the separation of the boundary layer, and since potential flow theory cannot account for separation, the maximum lift coefficient that occurs just before the stall can only be determined by experiment. The breakdown can take either of two distinct forms, depending on the shape of the aerofoil. Both the thickness and camber are important. Consider the thickness first: for a thin aerofoil such as a flat plate or a conventional aerofoil whose thickness-to-chord ratio lies in the range 10 – 16%, the stall occurs suddenly and with little warning. At the point of maximum lift, the air flow over the upper surface separates completely as shown earlier in figure 5, starting near the leading edge and extending across the full chord. A thicker aerofoil with a rounded nose is more resilient. The stall takes place gradually, starting at the trailing edge (figure 6). The lift coefficient doesn’t reach such a high peak, but the peak itself is more rounded and the stall is less sudden [5]. Now for the camber: at the peak of the lift curve, a cambered aerofoil produces more lift, and at the same time, the curve shifts bodily to the left so the critical angle of attack is reduced.

When a stall occurs, the aircraft sinks so the oncoming air stream meets the wing from below, with the angle of attack continuing to increase of its own accord. As the downward component of speed increases, the lift will rise again, but it won’t return to its original value. To restore lift, the pilot must either pitch the nose down or increase the aircraft’s forward speed, and unless the port and starboard wings are producing exactly the same lift, in practice it will tend to roll to one side, and under these circumstances it’s difficult to keep control. We’ll be looking at the consequences later, in Section A0418.

The wing cross-section

Earlier, you might have spotted from equation 2 that other things being equal, the lift provided by an aircraft wing is proportional to its plan area \(A\). This doesn’t necessarily mean that a heavy aircraft needs large wings. Lift is also proportional to \(V^{2}\), so with relatively small wings, an aircraft can extract the lift it needs by moving fast. The problem, therefore, is not just to obtain more lift, but to obtain it economically, and to tailor the wing profile so the aircraft can fly slowly when needed.

Requirements

Flying an aircraft is more complicated than driving a car. The lift needed to keep it in the air varies over time during each flight, and in principle, it can be adjusted by (a) opening and closing the throttle to change the speed, (b) by moving the control column to change the pitch, or (c) a combination of the two. Compared with a car, an aircraft responds slowly to throttle control, and to keep the process manageable, the pilot will hold the engine power at a constant level for long periods during the flight, use trim tabs on the wings to optimise the configuration, and make minor corrections to the aircraft speed and lift by raising or lowering the pitch. In this way, account can be taken of the diminishing weight of the aircraft as it consumes fuel.

When taking off and landing, the wings must deliver maximum lift when travelling at a relatively low speed, which for large aircraft, can be achieved by varying the camber – indeed the whole wing cross-section – by means of mechanical flaps and slats. Note that in general, flying slowly is more dangerous than flying fast because at low speed, an aircraft is more likely to stall. To maintain an appropriate safety margin, the pilot may increase the engine power and simultaneously raise the pitch – if needed, the speed can be restored by reducing the pitch again.

Apart from the lift, the principal force acting on the wing is the drag force. It opposes the thrust supplied by the engines and slows the aircraft down. Aircraft engines burn a great deal of fuel, and the amount consumed on a long journey may account for half the total weight on take-off, so minimising the drag not only reduces fuel consumption and pollution, but it also releases carrying capacity for passengers and cargo. It follows that when designing a wing, reducing drag is almost as important as creating lift, and ultimately, it’s the ratio of lift to drag that determines the efficiency of the aircraft as a means of carrying a payload from A to B.

Is there a ‘best’ shape for the wing profile? In the early days of flying machines, inventors knew that some profiles performed better than others, but could not forecast their behaviour in advance - they had to work out the best configuration by trial-and-error. It’s a complicated issue because a shape that works well at low speeds might not work well at high speeds, and vice versa. A commercial aircraft spends much of its time travelling at its design cruising speed at a particular altitude, and ideally will generate the least possible drag while doing so. Yet, the same aerofoil should sustain maximum lift at the relatively low speeds that arise during take-off and landing, and it should give the pilot a margin of safety in which to recover if things go wrong.

The leading edge

Let us look, therefore, at different parts of the aerofoil in turn, and see how they are adapted to the task. Figure 9 shows an idealised profile.

Figure 9

Two features stand out: the rounded nose and the pointed tail, and immediately, the rounded nose presents a puzzle. I once used to imagine that a sharp edge would cut through the air more easily than a blunt one, but this is not what happens. If judiciously proportioned, at subsonic speeds a rounded nose produces less drag than a sharp one. This might seem surprising. You’d imagine that the atmosphere would resist, putting pressure on the nose as it pushes the air aside; after all, this is what happens to a bluff body such as an automobile when travelling at speed. But an aircraft wing is slender and streamlined, so most of the resistance arises in a different way, through friction with the air parcels as they pass along the upper and lower surfaces.

Figure 10

And as figure 10 shows, other things being equal, a wing with a rounded leading edge has a smaller perimeter than one with a sharp leading edge of the same cross-sectional area. A round nose means less surface area, and less friction.

More important, however, is the question of lift. Here, the radius of the nose plays an important part. To see why, let’s track the progress of a fluid parcel as it curves over the leading edge as shown in figure 11.

Figure 11

Assume we are dealing with an ideal fluid in irrotational flow. We know that when it moves along a curved path, the fluid parcels nearest the centre of the curve travel faster than those further away. For example, in a free vortex, each fluid parcel travels in a circular orbit at a velocity that is inversely proportional to the radius \(r\) of the orbit. Here, we’ll assume that the nose of our aerofoil is circular in shape, and the flow pattern around the nose is approximately the same as the flow pattern in a free vortex. The fluid parcel we are about to track starts with a square boundary of side \(\delta r\), and since potential flow theory tells us that it cannot rotate, the shape of the parcel must deform in a particular manner as it progresses round the curve. Inside the boundary, the constituent parts move at different speeds and in different directions.

Figure 12

The small arrows in figure 12 indicate the velocities at different locations within the parcel, measured relative to the bottom left-hand corner of the parcel at P. The arrows down the right-hand side indicate rotation around P in an anticlockwise direction, while those along the top indicate rotation in the opposite direction, and since the pattern is symmetrical about the diagonal PQ, the net rotation is zero. As a result, the square parcel deforms into a diamond shape, in which the points closest to the wing surface are catapulted forward, travelling round the nose faster than those further away.

So what happens to the flow at the wing surface? Ignoring the boundary layer, very roughly, the vortex analogy tells us that the flow velocity around the leading edge is inversely proportional to the radius \(r\) of the nose. Here, the air pressure is low, lower in fact than anywhere else on the wing surface, because of the small radius and locally high velocity. This explains why lift force is concentrated at the front of the wing rather than the rear. However, there is a limit to the velocity increase that can be generated round the nose in this way. Too large a velocity will cause the flow to detach, in which case the lift-making process will break down. This is essentially what happens with a flat plate. In contrast, a nose with a larger radius will enable the pilot to maintain a higher angle of attack before the air flow separates, and the result is a higher maximum section lift coefficient \(c_{l} {}_{max}\) [13]. The aircraft can travel at a lower speed during take-off and landing, and operate safely from a shorter runway.

The trailing edge

The pointed tail raises a different question. It seems that the lift produced by a thin aerofoil such as an inclined plate is concentrated near the front, and in fact this is true of many conventional aerofoils. Aft of the midpoint, the pressure on the wing surface contributes relatively little to the total lift, which leads one to ask what the rest of the wing is for. Section F1616 explains how for any streamlined body, the aft part is a ‘fairing’ that enables the air to converge smoothly without leaving a turbulent wake and creating unnecessary drag. But in the case of a wing, it is possible in addition to extract lift from the wing surface aft of the midpoint by adjusting the profile, and as we’ll see shortly, this has become a major preoccupation of aerofoil designers in recent years.

Thickness and camber

So let’s consider the remainder of the cross-section. Most wings are asymmetrical, with an arched profile. In the past, it was customary to break down the shape into two components, (a) thickness, and (b) curvature. They could be quantified separately, which enabled manufacturers to catalogue the profiles in a systematic way and enabled customers to make informed comparisons between alternative designs. Both thickness and curvature are measured relative to the chord line. In principle, the thickness profile can be represented by the aerofoil depth measured perpendicular to the chord line at regular intervals from the nose to the tail. The most important of these measurements is the maximum thickness. For subsonic aircraft today, the ratio of the maximum thickness to the chord is typically in the range 8 – 16% [10], with the thickest part located just ahead of mid-chord. Compared with that of a thin wing, such a profile allows for a more rounded nose so there is less risk of separation, and other things being equal, it is also stronger because it allows room for a thicker wing spar to resist bending loads. These benefits were first realised by Ludwig Prandtl in 1917, and as a result, thick aerofoils were taken up by the aircraft manufacturer Fokker for its famous Triplane. Later, metal skins enabled designers to switch to a cantilevered monoplane layout, with no struts or wires to create extra drag.

At high speeds the situation is different. Here, a thin wing is essential to pierce through the supersonic shock waves that build up in the flow field ahead. A good example is the 1954 Lockheed Starfighter, which cruised at twice the speed of sound. It had short, stubby wings with a sharp leading edge, whose thickness measured just 4% of their chord (Anderson 99 ‘Performance’ p37).

Finally, the curvature is most easily defined in terms of the camber line, which runs through the middle of the cross-section. Some aerofoils are symmetrical so the mean camber line is straight - and it coincides with the chord line as shown in the upper part of figure 13.

Figure 13

More often though, the camber line arches upwards as shown in the lower part of the Figure. The maximum distance between the mean camber line and the chord line is called simply the camber. It is measured perpendicular to the chord line, and usually expressed as the percentage ratio of camber to chord. The camber has a considerable influence on the behaviour of a wing: adding a small amount (3 – 4%) can double the ratio \(L/D\) of lift to drag by comparison with a flat plate [7]. The reason seems to be that effectively, it lowers the leading edge in relation to the rest of the aerofoil. This has two advantages.

Figure 14

The first is that as shown in figure 14, it enables the oncoming air flow to meet the nose in a direction more-or-less tangential to the camber line at the leading edge, so the wing can sustain a high angle of attack without separation, and develop a greater maximum lift. This is particlarly important for small aircraft, but less so for commercial jets, which are equipped with high-lift devices such as a flaps and slats that effectively change the profile of the wing cross-section during take-off and landing. The second is that it acts as a barrier to the air flow under the wing, creating a pocket of slow-moving air below the crest. Here, the pressure rises, and with it, the total lift on the aerofoil. In addition, the arch raises the speed of travel over the upper surface, so the pressure falls, further increasing the total lift. The effect is most clearly apparent in the flow pattern around a curved plate. Figure 15 shows the plate pitched at a zero angle of attack: it is generating lift, and within limits, will continue to do so even when the angle is negative.

Figure 15

Aerofoils

The pioneers of aviation faced many difficulties. One of them was to build wings that could lift an aircraft off the ground. Having little information to go on, they experimented with profiles inspired by the cross-section of a bird’s wing. The profiles were wide, thin, and cambered, but they weren’t very efficient, and since a small change in the shape of an aerofoil can lead to a large change in the lift and drag, it wasn’t obvious what to do next. At the same time, combustion engines were improving quickly, and new wing profiles were needed for the faster and more powerful machines that began to emerge during the 1920s and 1930s. Rapid growth in the aircraft industry enabled a more systematic approach to research and development, in which the wind tunnel played an important part.

The subsonic regime

During the inter-war period, research programmes were launched at the national level in Germany and the USA. Existing aerofoils had evolved by trial-and-error, and the aim was to measure their behaviour more accurately and make the results available across the rapidly growing industry. In the USA, the geometric shape of each aerofoil was represented by a series of numbers within a standard classification system. This resulted in a series of profiles whose details were published by the US National Advisory Committee for Aeronautics (NACA). The key parameters were (a) the camber, (b) the shape of the camber line, and (c) to a lesser extent, the thickness distribution. The first series of standard aerofoils appeared in 1931, each profile identified with a 4-digit code number indicating the camber and thickness [2], which was followed in 1935 by a 5-digit series with higher lift coefficients. All these profiles were useful for aircraft constructors, particularly smaller companies that lacked testing facilities such as wind tunnels to experiment with their own designs.

Meanwhile, speeds for larger commercial aircraft continued to rise, which led to a new problem. Conventional aerodynamic theory assumed that the air density remains constant throughout the flow field. But at higher subsonic speeds, the air parcels at some locations around the wings of any aircraft become a little compressed, and at others, they expand. As a result, during the 1930s, the predicted pattern of streamlines differed significantly from the actual ones. However, the estimates could be corrected by using a formula that was developed by Prandtl and first published in 1927. It was derived initially from the velocity potential for incompressible inviscid flow, in the form of an equation analogous to the Laplace equation that included two additional variables: density and heat (we’ll have more to say about this later in Section A1615). Unlike the Laplace equation it was non-linear, but it could be linearised through an approximation. The result was a simple ‘compressibility correction’ for adjusting the results for thin aerofoil sections at low angles of attack [6]. It takes the form of an equation that adjusts the lift coefficient \(C_{L} {}_{,0}\) for incompressible flow (which we assume is already known from previous analysis and wind tunnel tests) to predict the lift at higher subsonic speeds. If we measure the speed \(V\) of the aircraft in terms of its Mach number \({\rm Ma}\) (the ratio \(V/c\) where \(c\) is the speed of sound at the current altitude), it turns out that the accuracy of the lift coefficient is acceptable for free stream velocities up to \(0.7{\rm \; Ma}\):

The equation tells us that the lift coefficient rises with increasing velocity.

Laminar flow aerofoils

As well as affecting the air density, higher speeds led to greater drag. Aircraft designers knew that the largest single contribution came from friction between the boundary layer and the wing surface. The friction was greatest in regions of turbulent flow, which tended to occur over the latter half of the aerofoil where the pressure gradient was negative, acting against the motion of the air particles along the surface and leading to separation. During the 1930s, NACA researchers set about developing a modified aerofoil that delayed the transition. This was done (a) by polishing the wing surface to make it smooth, and (b) by raising the aft part of the upper surface as shown in figure 16.

Figure 16

This shifted the point of maximum thickness to a location around mid-chord, which increased the speed of the boundary layer and produced a favourable pressure gradient along a greater proportion of the upper surface. By delaying separation, the drag acting on a smoothly polished model ‘flown’ in a wind tunnel could be reduced by more than a half. This led to a new series of aerofoil profiles [3] that were tested with full-sized aircraft in 1938-9, and adopted for the P-51 Mustang propeller-driven fighter that went into operation during the Second World War. After 1945, the laminar flow aerofoil was used on many of the post-war generation of jet turbine aircraft [12].

Supercritical aerofoils

During the 1930s, manufacturers produced military aircraft that were much faster than their predecessors. Although propeller-driven like the Mustang, they were capable of speeds approaching 500 km/h. Typically, for a modern aircraft, shock waves will begin to form over the wings at a relatively low Mach number of around 0.65 [11]. This is because, in order to generate lift, the wings are designedto make the air parcels speed up as they pass over the upper surface, and relative to the wing, they can travel at supersonic speed even though the aircraft itself is not. The speed at which the aircraft is travelling when the air flow locally becomes supersonic is denoted by the critical Mach number \(Ma_{Crit}\).

Figure 17

figure 17 shows a patch of supersonic air flow of the kind that typically occurs on the upper surface of a conventional aerofoil of an aircraft travelling faster than \(Ma_{Crit}\). When an air parcel crosses the dotted line, the pressure falls and it accelerates to supersonic speed. On exit, the pressure must rise again but the parcel is moving too fast to transmit the pressure change aft to the parcels following on behind, and as a result, they ram into their predecessors and the flow collapses into a shock wave. The pressure rise is abrupt, and where the shock wave intersects the boundary layer, it causes the boundary layer to separate. This creates a turbulent wake, that in turn reduces the lift and increase the drag.

The problem is that with conventional aerofoils, when once aircraft speed \(V\) rises above the critical Mach number and a supersonic patch forms, if \(V\) continues to rise, at some point there is a very large increase in drag, by a factor of ten or more. The curve is sketched diagrammatically in figure 18.

Figure 18

The phenomenon is known as drag divergence, and the speed at which it happens is denoted by \(Ma_{Div}\). Effectively \(Ma_{Div}\) constitutes an upper limit to the aircraft’s cruising speed, and the question is whether and how the interval between \(Ma_{Crit}\) and \(Ma_{Div}\) can be increased. It’s known that swept wings and a thin aerofoil section help, but here we’re interested in the aerofoil geometry. Here are two key facts:

(i) If the supersonic patch lies on the forward part of the aerofoil, it does not greatly affect the wave drag, but becomes more significant when located at or behind the crest [11].

(ii) The height of the supersonic patch measured perpendicular to the wing is important. Around the nose, the radius of the wing surface is small and the velocity falls away sharply with distance, so the height of the patch is small and only a small amount of fluid actually passes through the shock wave. By comparison, further aft, the radius is larger, and the patch extends further out into the flow field where it affects more fluid. Moreover, air parcels within the boundary layer are moving more slowly and are more prone to separate [11].

So what does this tell us? Aerodynamicists reasoned that if there must be a patch of supersonic flow, it should lie ahead of the thickest part of the aerofoil, where the boundary layer was still accelerating under the influence of a favourable pressure gradient [12]. It was achieved by reducing the curvature of the upper surface of the aerofoil abruptly behind the leading edge. This changed the pressure distribution along the upper surface, with the suction pressure peaking close to the front, and falling gradually thereafter to generate a relatively small supersonic patch with a weak shock wave ahead of the crest. The resulting aerofoils were called peaky aerofoils, and a profile of this kind was adopted for British VC-10, and for the Boeing 747.

The final stage in this progression of aerofoil design resembled that of a peaky aerofoil but with subtle modifications. They included a rounded nose, an extended flat top, and pronounced camber in the aft-most section to restore the lost lift, together with a cusped trailing edge that helped to minimise the wake and associated drag (figure 19).

Figure 19

The supercriticalaerofoil was developed by Richard Whitcomb in 1965 at the NASA Langley Research Center, and used on the Boeing 747-400, 757 and 767, together with the Airbus 310. The resulting aerofoil was thicker, which gave the designers a further advantage. With more room for the wing spar, the structure could be stiffer and lighter structure with more room for fuel.

To summarise, the development of aerofoils during this period took place through subtle changes in the section profile. Visually, the geometries don’t look very different, but we can see the outcome in terms of their aerodynamic performance. For each of three main categories (conventional, laminar flow, and supercritical), the pressure distribution over the upper surface when flying at cruising speed is shown diagrammatically in figure 20 for comparison.

Figure 20

Postscript

Today, although commercial aircraft fly quite fast, they’re not much faster than they were during the 1960s. Their cruising speed is usually a little under the speed of sound rather than above it, because in subsonic flight, aircraft use less fuel, make less noise, and costs less to run per passenger-kilometre. Of course, military aircraft such as jet fighters and bombers are not limited in this way, their wings being designed for sustained supersonic flight. The pattern of air flow around the wing is different, and we’ll return to it later in Section A1615. Throughout this Section, we’ve assumed that the wing cross-section and the pattern of air flow is the same at all points across the wingspan, which in practice, isn’t true. To understand how an aircraft wing works, we shall need to broaden our scope recognising that the cross-section is only one factor within a three-dimensional whole. In the next Section, we’ll see that the movement of air over the wing surface involves a transverse or sideways component, and that the circulation around the aerofoil is part of a larger vortex system that trails behind any moving aircraft.