Observing the birds always made it obvious what was needed: a surface and movement of the air. Through the ages, experience with kites and similar man-made devices substantiated the basic understanding and, in more recent times, motivated attempts by men to fly.

In the early days of aviation, late in the nineteenth and early twentieth centuries, knowledge of what a wing can do to obtain lift was limited. The physics we understand today was not available. Early on, the lift per square foot of wing area that could be achieved later was limited by the design of wing airfoil sections, the shape of the hardware designed to deflect air nicely. The first generation of aerodynamic surfaces were thin and delicate because they had to be light in weight, made with flexible fabric, and not able to produce a lot of lift. In order to get a sufficient total lift from the wing, the wing had to be large in area. This presented another challenge in that large span wings were hard, if not impossible, to build with the materials that were available while staying within weight limits. A brilliant solution was to employ, as the Wright brothers did, a box-beam approach to building the wings where the top and bottom surfaces form two lifting surfaces of the wings and the other two (front and rear) sides were reduced to vertical struts and wires to carry the shear and bending loads and thus let the air flow around the lifting surfaces. This was the classical design of a biplane (Fig. 2.1). Knowledge of airfoil and wing design and construction technology improved and with the desire for higher speeds, the biplane eventually became a historical artifact.

Fig. 2.1
figure 1

Top: 1902 Wright Brothers glider undergoing testing (Wikimedia Commons GPN-2002-00125). Bottom: DeHavilland DH.82 Tiger Moth biplane introduced in 1932 (Photo: NIMH)

A look at the current state of the art in wing design reveals that a number of features on the biplane will be replaced, if not eliminated, by improvements in technology: the struts and wires, one of the wings, the rather sharp leading edge, and the materials with which the wing is built. While the internal combustion engine that powered early airplanes will survive, it was supplanted for large and fast airplanes by the jet engine.

How a modern wing provides lift is well understood by professional engineers and the public should rest comfortably with the thought that magic is not involved. Flight is a risky undertaking only when pilots and ground personnel, who are, after all, human, make decisions that have the very small potential of being bad.

A first question is central to flight: How does the wing function? There are curious stories about what causes pressure differences to be experienced by the top and bottom surfaces of the wing and the speeds of air over these surfaces. Some arguments are thought to require that two molecules of air separated at the leading edge have to be reunited at the trailing edge. This is nonsense even if they include vestiges of reality. The pressure differences exerted by the moving air on the upper and lower surfaces of the wing are real and the question may be better stated as: How do these pressure differences arise?

Engineers and scientists work with models. These models may be in their minds, they maybe physical (like a wind tunnel model) or, most commonly, they are mathematical. The mathematical model is very useful because it is quick to be used and can be changed to incorporate a new or better description of the physical phenomena involved. Nowadays such work is almost always done on a computer. We will leave mathematical modelling to the engineers and consider just the physical modelling of air flow, that invisible medium in which flight takes place. We will be unable to avoid description of mathematical relationships entirely. They will have to be invoked and we will do so in hopefully simple easy-to-understand terms.

Our modelling will rely on sketches to represent the basic phenomena because visual illustration can be very helpful to augment a verbal narrative. Or, as somebody once said, a picture is worth a thousand words. Our first question—how does a wing generate lift—is often raised by visitors to flight museums, by students of aviation, and the lay public generally. The answer is very simple if one looks at the big picture but, the details are more complicated.

Let’s look at the big picture and take the wing to be some mysterious device that causes air flow onto the wing to be deflected downward. That simple fact means that air that had no velocity (or better momentum,Footnote 1 our first abstract idea) in the downward direction has been acted upon by the wing to have downward velocity (called downwash). The high-lift system shown in the Fig. 2.2 should make that function clear if this wing succeeds in directing the air as suggested by the hardware. According to Isaac Newton, a force is required to affect a change in momentum and the wing does just that; it creates such a momentum where none was present before the wing came moving along. The reaction force is the lift we seek to understand. That is the easy explanation. Harder to understand are the mechanics of fluid motion and its ability to impart pressures to the surfaces of the wing.

Fig. 2.2
figure 2

A wing of a Boeing 727 with all high lift devices deployed (Photo by author, Museum of Flight, Seattle)

The image in Fig. 2.3 shows the external consequences of an airplane flying in air made visible by clouds. What is apparent is that two rotating masses of air are involved; rotating in opposite senses. Each such flow feature of rotating air mass is called a vortex. It has a center and its influence decreases further from that center. In isolation, a vortex flow has streamlinesFootnote 2 that are circular. Because such motion trails from the passing wing, i.e., shed by the wing, they are referred to as trailing vortices.

Fig. 2.3
figure 3

An airplane flying through a cloud layer displaying the downwash and the associated and necessary wing tip vortices (Credit: istock.com: image 1125702712/alextov)

The details of vortex flow will become clearer further on but for the time being, we can view the vortex as similar to a tornado or hurricane. The high (circular) velocities near the center can be violent and destructive as indeed they are in storm systems, and they also have to be reckoned with in flight situations. For example, a small airplane following a large airplane that created a strong vortex, say in a landing pattern, can experience an unpleasant, if not deadly, encounter because the small airplane may be rotated by the air in which it tries fly.

2.1 The Bound Vortex

In addition to the trailing vortices created by the passing airplane, there is also vortex-like motion around the wing itself as we suggested when we mentioned wings shedding two vortices. Our first task is to show that this is true and specifically that it is intimately associated with the wing’s lift. In short, we will show that the bound and trailing vortices form a system. The first step in understanding the pressures on the wing is to understand the velocity field. The relation between pressure and velocity is set aside for now.

Consider a uniform flow around an object whose function is to create a downward flow. We adopt a viewing platform at rest relative to the device and let the flow arrive uniformly from the left, far away. The picture and our perspective will be in two dimensions in the plane of this sheet of paper. A student might suggest that an airfoilFootnote 3 shaped as in Fig. 2.4 would do the trick: turn the flow downward in a smooth fashion. In practice, that turns out to be easy to do, provided we do not ask that the turning angle be too severe.

Fig. 2.4
figure 4

A flow deflecting airfoil and a 1909 application by Alberto Santos-Dumont flying his Demoiselle no. 20. The image is provided by Stuart Wier. A similar image may be found at the National Air and Space Museum online archive, NASM 1B35740. This image is flipped left to right from the original for illustrative purposes

The totality of the oncoming flow will be divided into two parts, one above the airfoil and the other below. The lower flow between the airfoil and any boundary near or far away will be slowed and, to conserve mass flow of the oncoming freestream, the upper one will be accelerated, at least locally. The idea of mass conservation is that any fluid coming in from the left must exit on the far right. This picture implies that the insertion of the lifting airfoil into the uniform flow field results in the addition of rotation onto the uniform field. An observer would also add that whatever rotational phenomena are involved should be vanishingly small far away from the airfoil.

We are going to argue that this rotation is associated with a vortex connected with the presence of the lifting airfoil and what the airfoil demands the oncoming flow to do. This conclusion is not trivial but let us give it a try.

2.2 Circulation

We can describe the rotational aspects of the flow by drawing a closed path around the far field that includes the airfoil and ask: “what is the circulation along that path?” The circulation is the total sum of velocity along the closed path.Footnote 4 It is a measure of the rotation of the fluid enclosed by the path. There is zero circulation in a uniform flow because there are portions along the path where the velocity and the path are aligned leading to a positive contribution to the circulation and there will be an equal path length where the flow is in the opposition direction to the path contributing negative circulation. These two elements will cancel out and the conclusion is that uniform flow has no circulation.

If one draws a path around the airfoil, the fact that the upper surface flow velocity is faster than that along the lower surface flow implies that the lifting airfoil must have circulation (rotational flow) associated with it (Fig. 2.5).

Fig. 2.5
figure 5

Circulation is the sum of velocities measured along a path such as the circles shown. In a uniform flow (at upper left) circulation is zero. Around the airfoil it is finite. The boundaries shown could be far away or not

Whatever circulation or rotational flow is associated with the airfoil is a feature of its geometry and cannot be associated with the way we measure it. In other words, the circulation must be independent of the diameter of any large circular path we draw around the airfoil. That path length is proportional to the radius (or diameter) of our chosen path. This is possible only if the rotational velocity varies inversely with distance from the center of rotation and includes the center. Such a flow field is called a vortex. The circulation is the strength of the vortex. Mathematically, the rotational velocity around the vortex at any point is the circulation divided by the radius to this point, or more precisely, divided by the circumference of the circle. Students of vector calculus would say that the velocity field is curl-free, except where vorticity is located.

2.3 Friction to the Rescue

There is a fundamental problem with this model of a vortex: the implied rotational velocity at the center is infinite because the radius is zero! Nature abhors infinite quantities so that a mechanism will have to be invoked to prevent that. Real vortices in nature are thereby distinguishable from mathematical ones by the flow behavior near the center. The mechanism invoked by nature is friction. Near the center of the vortex, friction becomes so effective that the fluid there rotates as a solid body. This happens because shearing forces dominate, and the entire central portion of the vortex rotates free of internal relative motion. In such a body the rotational velocity is zero at the center of rotation and it grows larger with increasing radius. Instead of an infinite velocity at the center, we end up with a zero velocity. Notice that we have, so far, not talked about turbulence. In the model vortex with its infinite speeds, it plays no role. In a vortex involving real air, turbulence is the friction mechanism that converts heavily sheared fluid motion to the solid body rotation in the core by means of three-dimensional eddies that decay to ever-smaller size and ultimately to thermal motion at a molecular level, that we perceive as heat.

In a real vortex like a hurricane, the center or eye, of the storm is relatively calm as it passes over a location. At the center, the circular velocity is low and increases as one looks at speeds further from the center. The nature of a real vortex is of two very different characteristics: circular velocity proportional to radius near the center and varying inversely with radius far away from the center. Somewhere between these regions this azimuthal (to use the right term) velocity will exhibit a maximum as the character of the flow blends between the two types of velocity variation described above. In order to describe the strength of a storm vortex, the weatherman would stress the wind speeds in the transition zone because they are highest there.

The real vortex is therefore not a singularity (with an infinite velocity) but operates with the rotating part of the flow distributed over space. That space could be an area on our page (the rotational core) rather than a point. A way of visualizing this is to think of a vortex as a line, like a wooden broom handle, and when one looks at the details, the handle appears more like a bundle of smaller sticks. These can be spread out over an area and their density as dots on the page may vary. To accommodate that possibility, one may speak of vorticityFootnote 5 as a local description of the rotation of the flow. Vorticity over a finite area will look to the observer as a vortex with a singularity (a non-existent infinite velocity point) when viewed from far way. This breakdown of a vortex into distributed vorticity will be very handy to describe the function of the airfoil. Figures 2.6 and 2.7 show two types of vortex structures in nature.

Fig. 2.6
figure 6

Satellite view of hurricane Katrina (2005) from space. The image shows the dominant circumferential flow around a core with a modest addition of radial flow and vertical flow between the earth’s surface and the upper atmosphere (US National Oceanographic and Atmospheric Administration)

Fig. 2.7
figure 7

Tornados in the American prairie in various stages of development. The opaque cloudiness is not a direct display of the rotational core flow boundary but rather a measure of the local air speed within the tornado, the temperature, and the humidity of the air (NOAA)

Figure 2.8 shows the schematic variation of the rotational velocity component around a real vortex (lower part of the figure). The upper portion shows the spatial distribution of the vorticity that is collected and relatively uniform near the center and there is none outside the transition zone. In a vortex like a storm system, there are additional velocities, radial and along the rotation axis, that complicate the physics.

Fig. 2.8
figure 8

Velocity and vorticity distributions in a vortex in nature. The lower black line shows the azimuthal velocity variation around the axis of the vortex. The upper part of the plot shows the vorticity distribution, uniformly distributed in the solid body rotation part of the real vortex. Going outward, the transition zone is decreasingly rotational and the outer region (past the arrows) is irrotational

The entire flow field, with the exception of the vortex core, is circulation-free and is called irrotational. The physical argument is that in order to have local rotation, a torque must have been applied to the material in question. There were no torques applied to fluid that just came uniformly from the upstream direction. On a real airfoil with flowing fluid, there will be torques applied to parts of the fluid that rub along the surfaces. Friction, as we will see, is the origin of the vorticity involved.

The natural tornado’s vorticity is created when two weather systems flow past each other, one above the other. The friction experienced by the atmospheric layer close to the ground slows that air mass and so the stage is set for the air involved to become a tornado. While there are similarities between the natural weather-related vortices and those associated with flight, these similarities do not include the size scale of the vortices, their internal secondary flows and the physics of their creation. The natural vortices are sources of awe and damage. Our concern is focused on the more modest ones associated with flying airplanes.

Setting the friction aspects aside for now, we note that a single vortex associated with a lifting airfoil makes no sense. The reason is that a single vortex on the airfoil will have flow velocities involved on some part of the airfoil that are perpendicular to its surface which is, of course, impossible because the airfoil is presumably rigid and does not allow flow through it. Imagine inserting a stationary and rigid, vertical sheet of metal into a water whirlpool and holding on to it. The flow would be severely disrupted, and the vortex would cease to be a single vortex. In all likelihood, two vortices would be created and they would have to move away from the metal sheet. The conclusion must be that a single vortex cannot be located on a rigid surface. Figure 2.9 illustrates this point. Shown there is a single vortex located halfway between points A and B. At the points A and B, the vortex induced velocities would be as noted by the arrows. That is clearly an impossible situation. We need a better model.

Fig. 2.9
figure 9

Superposition of a single vortex on the airfoil and the resulting velocity field. The vortex is centered on the airfoil half-way between points A and B on the heavy line representing the airfoil. The impossibility of a single vortex satisfying the lifting requirement is noted in that at points like A and B, the flow velocity would be through the airfoil

So, what to do about modelling the vortex that seems to be necessary to generate lift? We invoke the idea that a vortex can be viewed as a bundle of smaller vortices whose strengths add up to the total. Picture this vortex in three dimensions as a collection of wooden sticks, or better, a bundle of hard spaghetti noodles. If we take this bundle of noodles and spread them out onto the table or some other geometry, the circulation associated with the bundle in its new shape will be the same as if concentrated into round bundle or any other shape. The flow field very, very far from the bundle must be independent of the bundle configuration. Thus, from very far away, the distributed spaghetti vortex is not distinguishable from a tight vortex bundle.

We could postulate that the vortex (or better, the vorticity) on our airfoil is distributed along a line co-located with the airfoil in such a manner that the flow resulting from the superposition of the distributed vorticity and the oncoming flow will result in a flow that exactly conforms to the shape of the airfoil. That such a postulate makes sense is suggested by the following thought experiment. Consider that the single vortex in Fig. 2.9 is halved and the two resulting vortices are located at points A and B. The net velocity associated with these two vortices at some location between A and B will be zero because the two induced velocities associated with the vortices cancel out. At this point on the airfoil, we have met the requirement that the flow be parallel to the airfoil surface. This thought experiment could be expanded to a very large number of very small vortices of varying strengths, all adding up to the bound vortex strength. This is the distributed strength we seek.

Finding that correct distribution is a formidable mathematical task. Fortunately, theoretical aerodynamicists succeeded in solving this challenge in the 1920s and 30s and since the solution can be found (and agrees with observations), we may conclude that the vortex is indeed distributed in some way along the shape of the airfoil. That allows the ability to investigate airfoil shape analytically. A real step forward.