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# 3  Airfoils and Airflow

— Have you heard how to make a small fortune in the aviation business?

## 3.1  The Airplane and the Air

In ordinary steady flight, the airplane must develop enough upward force to support its weight, i.e. to counteract the downward force of gravity. It is a defining property of an aircraft (as opposed to a ballistic missile, spacecraft, or watercraft) that virtually all of this upward force comes from the air.

In any case, there is one thing we know for sure:

1.
Whenever the airplane applies a downward force to the air, the air applies an equal amount of upward force to the airplane.

Idea 1 is the cornerstone of any understanding of how the airplane is able to fly. It is 100% true. It is a direct consequence of a fundamental principle of physics, namely conservation of momentum; for details on this, see section 19.2. Everything else in this chapter is just a clarification or an elaboration of this simple idea. For example, we shall see in section 3.3 that it is better to think of the wing as pulling down on the air, rather than pushing.

As always, whenever you come across a new idea, you should mull it over, checking to see how it connects – or conflicts – with other things you know. See section 21.10 for more on this. In this case:

2.
Everybody knows that if you try to push on the air with your hand, the air moves out of the way before you can develop much force.

To reconcile idea 1 with idea 2, observe that the airplane is moving sideways, so that at each moment, it is pulling down on a new parcel of air. It transfers some momentum before the parcel has time to move out of the way. Combining these two ideas allows us to make a prediction:

3.
If you try to fly too slowly, you will have problems.

At this point we have three ideas, all of which are true, and all of which are connected in a logical way.

## 3.2  Flow Patterns Near a Wing

In this chapter I will explain a few things about how air behaves as it flows past a wing. Many of the illustrations – such as figure 3.1 – were produced by a wind-tunnel simulation1 program that I wrote for my computer. The wing is stationary in the middle of the wind tunnel; air flows past it from left to right. A little ways upstream of the wing (near the left edge of the figure) I have arranged a number of smoke injectors. Seven of them are on all the time, injecting thin streams of purple smoke. The smoke is carried past the wing by the airflow, making visible stream lines.

Figure 3.1: Flow Past a Wing

In addition, on a five-times closer vertical spacing, I inject pulsed streamers. The smoke is turned on for 10 milliseconds out of every 20. In the figure, the blue smoke was injected starting 70 milliseconds ago, the green smoke was injected starting 50 milliseconds ago, the orange smoke was injected starting 30 milliseconds ago, and the red smoke was injected starting 10 milliseconds ago. The injection of the red smoke was ending just as the snapshot was taken.

Figure 3.2: Upwash and Downwash

Figure 3.2 points out some important properties of the airflow pattern. For one thing, we notice that the air just ahead of the wing is moving not just left to right but also upward; this is called upwash. Similarly, the air just aft of the wing is moving not just left to right but also downward; this is called downwash. Downwash behind the wing is relatively easy to understand; the whole purpose of the wing is to impart some downward motion to the air.

The upwash in front of the wing is a bit more interesting. As discussed in section 3.8, air is a fluid, which means it can exert pressure on itself as well as other things. The air pressure strongly affects the air, even the air well in front of the wing. Upwash and downwash are discussed in more detail in section 3.4.

Along the leading edge of the wing there is something called a stagnation line, which is the dividing line between air that flows over the top of the wing and air that flows under the bottom of the wing. On an airplane, the stagnation line runs the length of the wingspan, but since figure 3.2 shows only a cross section of the wing, all we see of the stagnation line is a single point.

Another stagnation line runs spanwise along the trailing edge. It marks the place where air that passed above the wing rejoins air that passed below the wing.

We see that at moderate or high angles of attack, the forward stagnation line is found well below and aft of the leading edge of the wing. The air that meets the wing just above the stagnation line will backtrack toward the nose of the airplane, flow up over the leading edge, and then flow aft along the top of the wing.

Figure 3.3: Velocity Field of a Wing

The set of all points that passed the injector array at a given time defines a timeline. The right-hand edge of the orange smoke is the “30 millisecond” timeline. Several of the timelines are labeled according to their age in figure 3.3.

Since the air near the wing is flowing at all sorts of different speeds and directions, the question arises of what is the “true” airspeed in the wind tunnel. The logical thing to do is to measure the velocity of the free stream; that is, at a point well upstream, before it has been disturbed by the wing.

The pulsed streamers give us a lot of information. Regions where the pulsed streamers have been stretched out are high velocity regions. This is pretty easy to see; each pulsed streamer lasts exactly 10 milliseconds, so if it covers a long distance in that time it must be moving quickly. The maximum velocity produced by this wing at this angle of attack is approximately twice the free-stream velocity. Airfoils can be very effective at speeding up the air.

Conversely, regions where the pulsed streamers cover a small distance in those 10 milliseconds must be low-velocity regions. The minimum velocity is zero. That occurs near the front and rear stagnation lines.

The relative wind vanishes on the stagnation lines. A small bug walking on the wing of an airplane in flight could walk along the stagnation line without feeling any wind.2

Stream lines have a remarkable property: the air can never cross a stream line. That is because of the way the stream lines were defined: by the smoke. If any air tried to flow past a point where the smoke was, it would carry the smoke with it. Therefore a particular parcel of air bounded by a pair of stream lines (above and below) and a pair of timelines (front and rear) never loses its identity. It can change shape, but it cannot mix with another such parcel.3

Another thing we should notice is that in low velocity regions, the stream lines are farther apart from each other. This is no accident. At reasonable airspeeds, the wing doesn’t push or pull on the air hard enough to change its density significantly (see section 3.6.3 for more on this). Therefore the air parcels mentioned in the previous paragraph do not change in area when they change their shape. In one region, we have a long, skinny parcel of air flowing past a particular point at a high velocity. (If the same amount of fluid flows through a smaller region, it must be flowing faster.) In another region, we have a short fat parcel flowing by at a low velocity.

The most remarkable thing about this figure is that the blue smoke that passed slightly above the wing got to the trailing edge 10 or 15 milliseconds earlier than the corresponding smoke that passed slightly below the wing.

This is not a mistake. Indeed, we shall see in section 3.12.3 that if this were not true, it would be impossible for the wing to produce lift.

This may come as a shock to many readers, because all sorts of standard references claim that the air is somehow required to pass above and below the wing in the same amount of time. I have seen this erroneous statement in elementary-school textbooks, advanced physics textbooks, encyclopedias, and well-regarded pilot training handbooks. Bear with me for a moment, and I’ll convince you that figure 3.3 tells the true story.

First, I must convince you that there is no law of physics that prevents one bit of fluid from being delayed relative to another.

Figure 3.4: Delay is Not Forbidden

Consider the scenario depicted in figure 3.4. A river of water is flowing left to right. Using a piece of garden hose, I siphon some water out of the river, let it waste some time going through several feet of coiled-up hose, and then return it to the river. The water that went through the hose will be delayed. The delayed parcel of water will never catch up with its former neighbors; it will not even try to catch up.

Note that delaying the water did not require compressing the water, nor did it require friction.

Let’s now discuss the behavior of air near a wing. We will see that there are two parts to the story: The obstacle effect, and the circulation effect.

The first part of the story is that the wing is an obstacle to the air. Air that passes near such an obstacle will be delayed. In fact, air that comes arbitrarily close to a stagnation line will be delayed an arbitrarily long time. The air molecules just hang around in the vicinity of the stagnation line, like the proverbial donkey midway between two bales of hay, unable to decide which alternative to choose.

Air near the wing is delayed relative to an undisturbed parcel of air. The obstacle effect is about the same for a parcel passing above the wing as it is for the parcel passing a corresponding distance below the wing. This effect falls off very quickly as a function of distance from the wing. You can see that the air that hits the stagnation line dead-on (the middle blue streamer) never makes it to the trailing edge, as you can see in all three panels of figure 3.5. When the wing is producing zero lift, this obstacle effect is pretty much the whole story, as shown in the top panel of figure 3.5.

Figure 3.5: Airflow at Various Angles of Attack

Now we turn to the second part of the story, the circulation effect. In figure 3.5 the three panels are labelled as to angle of attack. Lift is proportional to angle of attack whenever the angle is not too large. In particular, the zero-lift case is what we are calling zero angle of attack, even for cambered wings, as discussed in section 2.2.

For the rest of this section, we assume the wing is producing a positive amount of lift. This makes the airflow patterns much more interesting, as you can see from the second and third panels of figure 3.5. An air parcel that passes above the wing arrives at the trailing edge early. It arrives early compared to the parcel a corresponding distance below the wing, with no exceptions. This is because of something called circulation, as will be discussed in section 3.12.

We can also see that most of the air passing above the wing arrives early in absolute terms, early compared to an undisturbed parcel of air. The exception occurs very close to the wing, where the obstacle effect (as previously discussed) overwhelms the circulation effect.

Unlike the obstacle effect, the circulation effect drops off quite slowly. It extends for quite a distance above and below the wing – a distance comparable to the wingspan.

A wing is amazingly effective at producing circulation, which speeds up the air above it. Even though the air that passes above the wing has a longer path, it gets to the back earlier than the corresponding air that passes below the wing.

Note the contrast:

 The change in speed is temporary. As the air reaches the trailing edge and thereafter, it quickly returns to its original, free-stream velocity (plus a slight downward component). This can been seen in the figures, such as figure 3.3 — the spacing between successive smoke pulses returns to its original value. The change in relative position is permanent. If we follow the air far downstream of the wing, we find that the air that passed below the wing will never catch up with the corresponding air that passed above the wing. It will not even try to catch up.

## 3.3  Pressure Patterns Near a Wing

Figure 3.6 is a contour plot that shows what the pressure is doing in the vicinity of the wing. All pressures will be measured relative to the ambient atmospheric pressure in the free stream. The blue-shaded regions indicate suction, i.e. negative pressure relative to ambient, while the red-shaded regions indicate positive pressure relative to ambient. The dividing line between pressure and suction is also indicated in the figure.

Figure 3.6: Pressure Near a Wing
Note on units: The pressure and suction near the wing are conveniently measured in multiples of the dynamic pressure,4 Q. In figures such as figure 3.6, each contour represents exactly 0.2 Q. We choose units of Q, rather than more prosaic units such as PSI, because it allows the figure to remain quantitatively accurate over a rather wide range of airspeed and density conditions. If you know the dynamic pressure, you can figure out what the wing is doing; you don’t need to know the airspeed or density separately.

As a numerical example: If you are doing 100 knots under standard sea level conditions, we have:

 Q := ½ ρ V2 = ½ × 1.2250 kg/m3 × (51.44 m/s)2 = 1621 pascals = 0.235 PSI = 0.016 Atm
(3.1)

Whenever we are talking about pressure in connection with lift and drag, it is safe to assume we mean gauge pressure, i.e. pressure relative to the ambient free-stream pressure – not absolute pressure – unless the context clearly demands otherwise. Ordinary light-aircraft speeds are small compared to the speed of sound, which guarantees that the dynamic pressure Q is always small compared to 1 Atm. Therefore if you hear somebody talking about a pressure on the order of 1Q, you know it must be gauge pressure, not absolute pressure. Furthermore it should go without saying that any mention of suction refers to gauge pressure, since there is no such thing as negative absolute pressure.

The maximum positive pressure on any airfoil is exactly equal to Q. This occurs right at the stagnation lines. This stands to reason, since by Bernoulli’s principle, the slowest air has the highest pressure. At the stagnation lines, the air is stopped — which is slow as it can get. See section 3.6, especially figure 3.10.

The maximum suction near an airfoil depends on the angle of attack, and on the detailed shape of the airfoil. Similarly-shaped airfoils tend to exhibit broadly similar behavior. By way of example, the angle of attack in figure 3.6 is 3 degrees, a reasonable “cruise” value. For this airfoil under these conditions, the max suction is just over 0.8 Q.

There is a lot we can learn from studying this figure. For one thing, we see that the front quarter or so of the wing does half of the lifting, which is typical of general-aviation airfoils. That means the wing produces relatively little pitch-wise torque around the so-called “quarter chord” point. This is why engineers typically put the main wing spar at or near the quarter chord point. Another thing to notice is that suction acting on the top of the wing is vastly more important than pressure acting on the bottom of the wing.

For the airfoil in figure 3.6, under cruise conditions, there is almost no high pressure on the bottom of the wing; indeed there is mostly suction there.5 The only reason the wing can support the weight of the airplane is that there is more suction on the top of the wing. (There is a tiny amount of positive pressure on the rear portion of the bottom surface, but the fact remains that suction above the wing does more than 100% of the job of lifting the airplane.)6

This pressure pattern would be really hard to explain in terms of bullets bouncing off the wing. Remember, the air is a fluid, as discussed in section 3.8. It has a well-defined pressure everywhere in space. When this pressure field meets the wing, it exerts a force: pressure times area equals force.

At higher angles of attack, above-atmospheric pressure does develop below the wing, but it is always less pronounced than the below-atmospheric pressure above the wing.

## 3.4  Upwash and Downwash

Both upwash and downwash make a positive contribution to lift. This can be understood as follows. In figure 3.7, the plank plays the role of the airplane. We imagine you are riding on the plank. Now suppose somebody on the ground throws a baseball upwards. This is upwash. When the baseball hits the plank, it exerts an upward force on the plank. If you caught the baseball at this point, that would be the end of the story. You would be getting lift from upwash alone.

Figure 3.7: Upwash and Downwash

Next, suppose that after catching the ball, you throw it downwards. This is downwash. This provides additional upward force, i.e. additional lift.

In fact, you don’t need to catch or throw the ball; all you need to do is let it bounce off the bottom of the plank. Each bounce provides two units of upward impulse, one from the incoming upwash and one from the outgoing downwash.

Both the upwash and the downwash are related to the circulation in the air, as discussed in section 3.12. Lift requires circulation and vorticity, as discussed in section 3.14.

Figure 3.8: Upwash Flowing Uphill

Beware: It is a common mistake to think that upwash exists because the wing is sucking upward on the air. Such suction would cause a downward force on the wing, making a negative contribution to lift. In fact the upwash flows uphill, into a region of high and increasing pressure, slowing down as it goes. You can see this in figure 3.8, in the region framed by the purple rectangle, just below and ahead of the wing. Bottom line: Both the upwash and the downwash make a positive contribution to the lift.

## 3.5  Stream Line Curvature

Let’s consider the relationship between pressure and velocity. It turns out that given the velocity field, it is rather straightforward to calculate the pressure field. Indeed there are two ways to do this; we discuss one of them here, and the other in section 3.6.

Figure 3.9 shows the pressure and velocity at various angles of attack. The velocity distribution depends on angle of attack, and so does the pressure distribution.

Figure 3.9: Airflow and Pressure Near Wings

We know that air has mass. Moving air has momentum. If the air parcel follows a curved path, there must be a net force on it, as required by Newton’s laws.7

Pressure alone does not make a net force; you need a pressure difference so that one side of the air parcel is being pressed harder than the other. Therefore the rule is this: If at any place the stream lines are curved, the pressure at nearby places is different.

You can see in the figures that tightly-curved streamlines correspond to big pressure gradients and vice versa.

If you want to know the pressure everywhere, you can start somewhere and just add up all the changes as you move from place to place to place. This is mathematically tedious, but it works. It works even in situations where Bernoulli’s principle isn’t immediately applicable.

## 3.6  Bernoulli’s Principle

We now discuss a second way in which pressure is related to velocity, namely Bernoulli’s principle, aka Bernoulli’s formula. In situations where this formula can be applied (which includes most situations – but not all), this is by far the slickest way of doing things.

Bernoulli’s principle is very easy to understand provided the principle is correctly stated. However, we must be careful, because seemingly-small changes in the wording can lead to completely wrong conclusions.

For simplicity, let’s consider a scenario where you are sitting in the airplane, in flight. We restrict attention to situations where the effects of friction can be neglected. We will analyze the same situation in two different ways.

First analysis: We pick a particular location in your reference frame, located at some fixed distance relative to you. As a premise of the scenario, we assume the air pressure, velocity, density, etc. at this location are constant. If you measure things at this location now, and come back and measure them again later, everything is the same. We call this a steady flow situation.

Second analysis: Rather than considering a particular location in space, we ask what happens to a particular parcel of air as it flows along a streamline. Even though the properties of pressure, velocity, density, etc. that pertain to a particular location are not changing, the properties that pertain to a particular parcel of air will change as the parcel flows from location to location.

We will now state the general idea of Bernoulli’s principle. In this scenario, for any particular parcel of fluid:

 higher pressure ⇔ lower airspeed lower pressure ⇔ higher airspeed
(3.2)

The explanation for this principle is completely logical and straightforward: The idea is that as the parcel moves along, following a streamline, as it moves into an area of higher pressure there will be higher pressure ahead (higher than the pressure behind) and this will exert a force on the parcel, slowing it down. Conversely if the parcel is moving into a region of lower pressure, there will be an higher pressure behind it (higher than the pressure ahead), speeding it up. As always, any unbalanced force will cause a change in momentum (and velocity), as required by Newton’s laws of motion.

There are various ways of quantifying this idea, depending on what sort of simplifications and approximations you want to make. Suppose we have two points B and A (denoting “before” and “after”) not too far apart. We continue to neglect viscosity and to assume steady flow. Then we can describe the flow of a single parcel of air as follows:

 PA − PB = −½ (ρ vA2 − ρ vB2)              (3.3)

where P denotes pressure, v denotes airspeed, and ρ denotes the density, i.e. mass per unit volume. In general ρA will be different from ρB but we are not going to worry about it for the moment, because the whole equation is only valid to first order, and worrying about ρA − ρB would be a second-order correction.

As a fancier way of writing this formula, we have

 Δ(P) = − ½ ρ Δ(v2)              (3.4)

which means exactly the same thing, since Δ(⋯) is just a fancy way of writing “small difference in ⋯” (namely the difference between point B and point A).

If we are careful, we can simplify this expression as follows:

 P + ½ ρ v2 = stagnation pressure (in this scenario) = const (to first order)
(3.5)

There are several important provisos to keep in mind:

1. We must be careful, because the constant in equation 3.5 is only constant to first order. Indeed, the whole equation is only valid to first order. That is, the equation is only valid when the pressure P is close to the ambient pressure and the actual density ρ is close to the ambient density. (If you want to understand how close, see equation 3.7 and/or or equation 3.8.)
2. This means that if we are going to invoke the simplifications and approximations that lead to equation 3.5, the velocity vA must be close to the velocity vB, or that both must be small compared to the speed of sound. Note that there are plenty of general-aviation aircraft (including some piston-driven aircraft and practically all jets) that go fast enough to make the first-order approximation unacceptable. This explains why we distinguish CAS versus EAS – that is, calibrated airspeed versus equivalent airspeed. (This is sometimes called the “compressibility correction” but that is a terrible misnomer, as discussed in the next item.)
3. Beware that people sometimes claim that Bernoulli’s principle only applies to “incompressible” fluids, but this claim is nonsense. There are no incompressible fluids. Air is highly compressible. The density ρ changes when the pressure P changes. The temperature changes too. All these changes are quite significant. Equations such as equation 3.5 already account for this correctly to first order; if they did not, the equations would give spectacularly wrong answers. The underlying idea is that since all these contributions are proportional to one another (to first order), you don’t need a temperature-dependent term and a density-dependent term and a pressure-dependent term; you can lump all the dependencies into a single term, which shows up as the pressure-dependent term in equation 3.5. For details on this, see section 3.6.3.

 There are no incompressible fluids.

Besides, we are free to analyze things in a way that is correct to second order (as in equation 3.8) or even correct to all orders (as in equation 3.7); we are not limited to the first-order approximations that are embodied in equation 3.5.

4. These equations apply only to steady flow. That is, the pressure, velocity, density, etc. can change from place to place, but they must not change as a function of time at any particular place (in your frame of reference).

For example, these equations do not apply inside a cylinder with a piston, such as you find in a bicycle pump, or in the cylinders of a piston engine. In such a cylinder, at any particular location, the pressure changes as a function of time. (It is possible to analyze such devices, but it requires formulas that are more complicated than equation 3.5.)

Here’s another important example: Consider a vortex (such as a tornado or hurricane) spinning counterclockwise and embedded in an airmass that is moving northward relative to the ground. At a point ahead of the vortex and to the left of the centerline, the pressure will drop and the windspeed will drop as the vortex approaches. This is just the opposite of what you would expect if you rashly tried to apply equation 3.5 to this situation.

In this scenario, it would be valid to apply Bernoulli’s principle in a frame moving along with the overall air mass, but not in a frame attached to the ground.

So, we see that even when we do have steady flow, it will only be steady in one frame of reference; it will be time-varying in almost any other frame. This is one of the reasons why we usually choose to analyze things using a frame moving along with the airplane.

5. The equations given here apply only to a particular parcel of fluid moving along a particular streamline. The same logic applies to every other parcel of fluid, but different parcels will in general have different stagnation pressures.
• Since we are restricting attention to steady flow, any parcel anywhere on the same streamline will have the same stagnation pressure. That’s because they are all essentially earlier or later versions of the same parcel.
• In some cases,8 we may have reason to believe that every parcel in some vicinity started out with the same stagnation pressure. This is not something that comes out of the Bernoulli equation, but rather something that can be put into the Bernoulli equation if/when we know it to be true.
• Otherwise you should not use equation 3.5 to compare one parcel with another.
6. None of the equations given here be trusted in any situation where frictional forces are playing a significant role. In particular, in the “boundary layer” very near the surface of a wing, there is a tremendous amount of friction, due to the large difference in velocity between nearby points.

This proviso is not as much trouble as you might think, for the following reasons. (1) In normal flight (not near the stall) the boundary layer is usually very thin, and (2) we can apply Bernoulli’s principle outside the boundary layer and then infer by other means what the pressure must be inside the boundary layer.

In many cases (but not all) the pressure inside the boundary layer is very nearly the same as the pressure just outside the boundary layer ... but you cannot use Bernoulli’s principle to establish this fact. If you completely ignored the existence of a boundary layer and tried to use Bernoulli’s formula right near the surface of the wing, you would be making two mistakes: using the wrong formula and using the wrong airspeed. Oddly enough, in many cases these two mistakes cancel each other out, but you should not make a habit of doing things this way.

Bernoulli’s principle is intimately related to the idea of streamline curvature discussed in section 3.5. If the parcel experiences a side-to-side pressure gradient, the direction of motion will change. If the parcel experiences a front-to-back pressure gradient, the speed of motion will change. This is exactly what we would expect from Newton’s laws of motion.

It must be emphasized that you do not get to choose Bernoulli’s principle “instead of” Newton’s laws or vice versa. Bernoulli’s principle is a consequence of Newton’s laws. See section 3.16 for more on this.

Equation 3.5 can be interpreted as saying the enthalpy of the parcel remains constant as it flows along a streamline. For more about what this means, see section 3.6.3 and reference 24.

Sometimes people who use equation 3.5 are tempted to interpret it as an application of the principle of conservation of energy. That is, they try to interpret Bernoulli’s equation as equivalent to the law of the roller coaster (figure 1.9) in the sense that the parcel loses speed when it climbs up a pressure gradient and gains speed when it slides down a pressure gradient. This is plausible at the level of dimensional analysis, since ½ ρ v2 is in fact the kinetic energy per unit volume, and pressure has the same dimensions as energy per unit volume. Alas, this interpretation is not correct. There is more to physics than dimensional analysis. The parcel of air is unlike a roller coaster in that it changes size and shape as it flows up and down the pressure gradient. Furthermore, the pressure is not numerically equal to the potential energy per unit volume. Actually, for nonmoving air, the pressure is numerically equal to about 40% of the energy per unit volume.

To summarize:

• It is correct to interpret equation 3.4 as a force-balance equation.
• It is correct to interpret equation 3.5 as saying the enthalpy of the parcel remains constant.
• It is not correct to say that the energy of the parcel remains constant. There is typically a qualitative correlation between the enthalpy and the energy, in the sense that they both go up together or both go down together … but they are not numerically equal.

### 3.6.1  Magnitude

It makes sense to measure the local velocity (lower-case v) at each point as a multiple of the free-stream velocity (capital V) since they vary in proportion to each other. Similarly it makes sense to measure relative pressures in terms of the free-stream dynamic pressure:

 Q = ½ρV2              (3.6)

which is always small compared to atmospheric pressure (assuming V is small compared to the speed of sound). Remember, this Q (with a capital Q) is a property of the free stream, as measured far from the wing.

Turning now to the local velocity v (with a small v) and other details of the local flow pattern, the pressure versus velocity relationship is shown graphically in figure 3.10. The highest possible pressure (corresponding to completely stopped air) is one Q above atmospheric, while fast-moving air can have pressure several Q below atmospheric.

It doesn’t matter whether we measure P as an absolute pressure or as a relative pressure (relative to atmospheric). If you change from absolute to relative pressure it just shifts both sides of Bernoulli’s equation by a constant, and the new value (just as before) remains constant as the air parcel flows past the wing. Similarly, if we use relative pressure in figure 3.10, we can drop the word “Atm” from the pressure axis and just speak of “positive one Q” and “negative two Q” — keeping in mind that all the pressures are only slightly above or below one atmosphere.

Figure 3.10: Pressure versus Velocity

Bernoulli’s principle allows us to understand why there is a positive pressure bubble right at the trailing edge of the wing (which is the last place you would expect if you thought of the air as a bunch of bullets). The air at the stagnation line is the slowest-moving air in the whole system; it is not moving at all. It has the highest possible pressure, namely 1 Q.

As we saw in the bottom panel of figure 3.9, at high angles of attack a wing is extremely effective at speeding up the air above the wing and retarding the air below the wing. The maximum local velocity above the wing can be more than twice the free-stream velocity. This creates a negative pressure (suction) of more than 3 Q.

### 3.6.2  Altimeters; Static versus Stagnation Pressure

Consider the following line of reasoning:

1. The airplane’s altimeter operates by measuring the pressure at the static port. See section 20.2.2 for more on this.
2. The static port is oriented sideways to the airflow, at a point where the air flows past with a local velocity just equal to the free-stream velocity.
3. In accordance with Bernoulli’s principle, this velocity must be associated with a “lower” pressure there.
4. You might think this lower pressure would cause huge errors in the altimeter, depending on airspeed. In fact, though, there are no such errors. The question is, why not?

The answer has to do with the notion of “lower” pressure. You have to ask, lower than what? Indeed the pressure there is 1 Q lower than the stagnation pressure of the air. However, in your reference frame, the stagnation pressure is 1 Atm + 1 Q. When we subtract 1 Q from that, we see that the pressure in the static port is just equal to atmospheric. Therefore the altimeter gets the right answer, independent of airspeed.

Another way of saying it is that the air near the static port has 1 Atm of static pressure and 1 Q of dynamic pressure. The altimeter is sensitive only to static pressure, so it reads 1 Atm — as it should.

In contrast, the air in the Pitot tube has the same stagnation pressure, 1 Atm + 1 Q, but it is all in the form of pressure since (in your reference frame) it is not moving.

We can now see why the constant on the right hand side of equation 3.5 is officially called the stagnation pressure, since it is the pressure that you observe in the Pitot tube or any other place where the air is stagnant, i.e. where the local velocity v is zero (relative to the airplane).

In ordinary language “static” and “stagnant” mean almost the same thing, but in aerodynamics they designate two very different concepts. The static pressure is the pressure you would measure in the reference frame of the air, for instance if you were in a balloon comoving with the free stream. As you increase your airspeed, the stagnation pressure goes up, but the static pressure does not.

Also: we can contrast this with what happens in a carburetor. There is no change of reference frames, so the stagnation pressure remains 1 Atm. The high-speed air in the throat of the Venturi has a pressure below the ambient atmospheric pressure.

### 3.6.3  Compressibility

First, a bit of terminology:

• Pressure denotes a force per unit area.
• Compressibility denotes a change in density in response to pressure.

Non-experts may not make much distinction between a “pressurized” fluid and a “compressed” fluid, but in the engineering literature there is a world of difference between the two concepts.

Every substance on earth is compressible — be it air, water, cast iron, or anything else. It must increase its density when you apply pressure; otherwise there would be no way to balance the energy equations.

However, changes in density are not very important to understanding the basic features of how wings work, as long as the airspeed is not near or above the speed of sound. Typical general aviation airspeeds correspond to Mach 0.2 or 0.3 or thereabouts (even when we account for the fact that the wing speeds up the air locally), and at those speeds the density never changes more than a few percent.

For an ideal gas such as air, density is proportional to pressure, so you may be wondering why pressure-changes are important but density-changes are not. Here’s why:

 We are directly interested in differences in pressure. We are only rarely interested in differences in density.

 We are only rarely interested in the total pressure. We are directly interested in the total density.

That is, lift depends on a pressure difference between the top and bottom of the wing. Similarly pressure drag depends on pressure differences. Therefore the relevant differential pressures are zero plus important terms proportional to ½ρV2. Meanwhile, the relevant pressures are proportional to the total density, which is some big number plus or minus unimportant terms proportional to ½ρV2.

To say it again: Flight depends directly on total density but not directly on total atmospheric pressure, just differences in pressure.

Many books say the air is “incompressible” in the subsonic regime. That’s bizarrely misleading. In fact, when those books use the words “incompressible flow” it generally means that the density undergoes only small-percentage changes. This has got nothing to do with whether the fluid has a high or low compressibility. The real explanation is that the density-changes are small because the pressure-changes are small compared to the total atmospheric pressure.

As previously mentioned, many books claim that equation 3.4 only applies to an “incompressible” fluid, but this claim is nonsense. Here’s the real story:

1. Compressibility specifies to first order how density depends on pressure. Equation 3.4 specifies to first order how the kinetic energy depends on pressure. It already accounts for the effects of compressibility and all other first-order quantities. Therefore equation 3.4 is valid whenever the pressure-changes are a small percentage of the total pressure, regardless of compressibility.
2. At high airspeeds, the pressure changes are bigger, and you need a more sophisticated form of Bernoulli’s equation. Using the full equation of state, you can derive Bernoulli’s equation in a form that is valid over the whole range of pressures and speeds.

Here is Bernoulli’s equation in a much less restricted form:

 H m
=
 γ γ−1

 P0 ρ0
[
 P P0
](γ−1)/γ g h + ½ v2
= constant along a streamline
(3.7)

Here H/m is the specific enthalpy, i.e. the enthalpy per unit mass, as explained in reference 24. Also, P0 is some “reference” pressure (usually taken to be the ambient atmospheric pressure), ΔP is the difference between the actual pressure and the reference pressure, ρ0 is the density the air would have at the reference pressure, and γ (gamma) is a constant that appears in the equation of state for the fluid. It is sometimes called the adiabatic exponent, and sometimes called the ratio of specific heats, for reasons that need not concern us at the moment. The γ value for a few fluids are given in the table below.

If we expand equation 3.7 to first order, we recover equation 3.5.

It is more interesting to expand equation 3.7 to second order. That gives us:

 ΔP ρ0
[1 −
 ΔP 2 γ P0
]   + ½ v2 = constant              (3.8)

The validity of the approximations involved in equation 3.4 do not depend on any notion of “incompressible” fluid, as we can see from the fact that the correction term depends on γ, which is not correlated with the actual compressibility.

 dimensionless adiabatic γ compressibility helium 1.666 0.6 nitrogen 1.4 0.714 oxygen 1.4 0.714 air 1.4 0.714 methane 1.31 0.763

The meaning of the numbers in the rightmost column in the table is this: If you start with a sample of air and increase the pressure by 1%, the volume goes down by 0.7%.

In equation 3.8, when the pressure P is near atmospheric, the term in square brackets approaches unity, and the expression becomes equivalent to the elementary version, equation 3.4, as it should.

Don’t let anybody tell you that Bernoulli’s principle can’t cope with compressibility. Even the elementary version (equation 3.4) accounts for compressibility to first order.

## 3.7  Stall Warning Devices

We are now in a position to understand how stall warning devices work. There are two types of stall-warning devices commonly used on light aircraft. The first type (used on most Pipers, Mooneys, and Beechcraft) uses a small vane mounted slightly below and aft of the leading edge of the wing as shown in the left panel of figure 3.11. The warning is actuated when the vane is blown up and forward. At low angles of attack (e.g. cruise) the stagnation line is forward of the vane, so the vane gets blown backward and everybody is happy. As the angle of attack increases, the stagnation line moves farther and farther aft underneath the wing. When it has moved farther aft than the vane, the air will blow the vane forward and upward and the stall warning will be activated.

The second type of stall-warning device (used on the Cessna 152, 172, and some others, not including the 182) operates on a different principle. It is sensitive to suction at the surface rather than flow along the surface. It is positioned just below the leading edge of the wing, as indicated in the right panel of figure 3.11. At low angles of attack, the leading edge is a low-velocity, high-pressure region; at high angles of attack it becomes a high-velocity, low-pressure region. When the low-pressure region extends far enough down around the leading edge, it will suck air out of the opening. The air flows through a harmonica reed, producing an audible warning.

Figure 3.11: Stall Warning Devices

Note that neither device actually detects the stall. Each one really just measures angle of attack. It is designed to give you a warning a few degrees before the wing reaches the angle of attack where the stall is expected. Of course if there is something wrong, such as frost on the wings (see section 3.15), the stall will occur at a lower-than-expected angle of attack, and you will get no warning from the so-called stall warning device.

## 3.8  Air Is A Fluid, Not A Bunch of Bullets

We all know that at the submicroscopic level, air consists of particles, namely molecules of nitrogen, oxygen, and various other substances. Starting from the properties of these molecules and their interactions, it is possible to calculate macroscopic properties such as pressure, velocity, viscosity, speed of sound, et cetera.

However, for ordinary purposes such as understanding how wings work, you can pretty much forget about the individual particles, since the relevant information is well summarized by the macroscopic properties of the fluid. This is called the hydrodynamic approximation.

In fact, when people try to think about the individual particles, it is a common mistake to overestimate the size of the particles and to underestimate the importance of the interactions between particles.

Figure 3.12: The Bullet Fallacy

If you erroneously imagine that air particles are large and non-interacting, perhaps like the bullets shown in figure 3.12, you will never understand how wings work. Consider the following comparisons. There is only one important thing bullets and air molecules have in common:

 Bullets hit the bottom of the wing, transferring upward momentum to it. Similarly, air molecules hit the bottom of the wing, transferring upward momentum to it.

Otherwise, all the important parts of the story are different:

 No bullets hit the top of the wing. Air pressure on top of the wing is only a few percent lower than the pressure on the bottom.

 The shape of the top of the wing doesn’t matter to the bullets. The shape of the top of the wing is crucial. A spoiler at location “X” in figure 3.12 could easily double the drag of the entire airplane.

 The bullets don’t hit each other, and even if they did, it wouldn’t affect lift production. Each air molecule collides with one or another of its neighbors 10,000,000,000 times per second. This is crucial.

 Each bullet weighs a few grams. Each nitrogen molecule weighs 0.00000000000000000000005 grams.

 Bullets that miss the wing are undeflected. The wing creates a pressure field that strongly deflects even far-away bits of fluid, out to a distance of a wingspan or so in every direction.

 Bullets could not possibly knock a stall-warning vane forward. Fluid flow nicely explains how such a vane gets blown forward and upward. See section 3.7.

The list goes on and on, but you get the idea. Interactions between air molecules are a big part of the story. It is a much better approximation to think of the air as a continuous fluid than as a bunch of bullet-like particles.

## 3.9  Other Fallacies

You may have heard stories that try to use the Coandǎ effect or the teaspoon effect to explain how wings produce lift. These stories are completely fallacious, as discussed in section 18.4.4 and section 18.4.3.

There are dozens of other fallacies besides. It is beyond the scope of this book to discuss them, or even to catalog them all.

## 3.10  Inverted Flight, Cambered vs. Symmetric Airfoils

You’ve probably been told that an airfoil produces lift because it is curved on top and flat on the bottom. This is “common knowledge” ... but alas it’s not true. You shouldn’t believe it, not even for an instant.

Presumably you are aware that airshow pilots routinely fly for extended periods of time upside down. Doesn’t that make you suspicious that there might be something wrong with the story about curved on top and flat on the bottom?

Here are some of things you expect to see in an airplane optimized for upside-down flight:

• Four-point or five-point seatbelts, to keep the pilot from flopping around.
• Flop tubes in the fuel tanks, so that the engine continues to run during extended inverted flight.
• Possibly a stronger structure, to handle stresses in funny directions.
• Possibly additional windows, for looking out in funny directions.
• Possibly a less-cambered wing, to optimize the inverted stall, but not necessarily. A C-150 Aerobat has exactly the same wing as a regular C-150.
• Possibly less washout, again to optimize the inverted stall, but again not necessarily.
• Fuel injection (as opposed to carburetion), although this is hardly worth mentioning, because nowadays practically everything bigger than a walk-behind lawn mower is fuel injected, and even some lawn mowers.
• et cetera.

It must be emphasized that changing the cross-sectional shape of the wing is not a necessity. Any ordinary wing flies just fine inverted. Even a wing that is flat on one side and curved on the other flies just fine inverted, as shown in figure 3.13. It may look a bit peculiar, and the inverted stall is not optimized, but basically it just works. The basic lift-producing process is the same.

Figure 3.13: Inverted Flight

The misconception that wings must be curved on top and flat on the bottom is commonly associated with the previously-discussed misconception that the air is required to pass above and below the wing in equal amounts of time. In fact, an upside-down wing produces lift by exactly the same principle as a rightside-up wing.

Figure 3.14: Airfoil Terminology

To help us discuss airfoil shapes, figure 3.14 illustrates some useful terminology.

1. The chord line is the straight line drawn from the leading edge to the trailing edge.
2. The term camber in general means “bend”. If you want to quantify the amount of camber, draw a curved line from the leading edge to the trailing edge, staying always halfway between the upper surface and the lower surface; this is called the mean camber line. The maximum difference between this and the chord line is the amount of camber. It can be expressed as a distance or (more commonly) as a percentage of the chord length.

A symmetric airfoil, where the top surface is a mirror image of the bottom surface, has zero camber. The airflow and pressure patterns for such an airfoil are shown in figure 3.15.

Figure 3.15: Symmetric Airfoil

This figure could be considered the side view of a symmetric wing, or the top view of a rudder. Rudders are airfoils, too, and work by the same principles.

At small angles of attack, a symmetric airfoil works better than a highly cambered airfoil. Conversely, at high angles of attack, a cambered airfoil works better than the corresponding symmetric airfoil. An example of this is shown in figure 3.16. The airfoil designated “631-012” is symmetric, while the airfoil designated “631-412” airfoil is cambered; otherwise the two are pretty much the same.9 At any normal angle of attack (up to about 12 degrees), the two airfoils produce virtually identical amounts of lift. Beyond that point the cambered airfoil has a big advantage because it does not stall until a much higher relative angle of attack. As a consequence, its maximum coefficient of lift is much greater.

Figure 3.16: Camber Fends Off The Stall

At high angles of attack, the leading edge of a cambered wing will slice into the wind at less of an angle compared to the corresponding symmetric wing. This doesn’t prove anything, but it provides an intuitive feeling for why the cambered wing has more resistance to stalling.

On some airplanes, the airfoils have no camber at all, and on most of the rest the camber is barely perceptible (maybe 1 or 2 percent). One reason wings are not more cambered is that any increase would require the bottom surface to be concave — which would be a pain to manufacture. A more profound reason is that large camber is only really beneficial near the stall, and it suffices to create lots of camber by extending the flaps when needed, i.e. for takeoff and landing.

Reverse camber is clearly a bad idea (since it causes earlier stall) so aircraft that are expected to perform well upside down (e.g. Pitts or Decathlon) have symmetric (zero-camber) airfoils.

We have seen that under ordinary conditions, the amount of lift produced by a wing depends on the angle of attack, but hardly depends at all on the amount of camber. This makes sense. In fact, the airplane would be unflyable if the coefficient of lift were determined solely by the shape of the wing. Since the amount of camber doesn’t often change in flight, there would be no way to change the coefficient of lift. The airplane could only support its weight at one special airspeed, and would be unstable and uncontrollable. In reality, the pilot (and the trim system) continually regulate the amount of lift by regulating the all-important angle of attack; see chapter 2 and chapter 6.

## 3.11  Thin Wings

The wing used on the Wright brothers’ first airplane is shown in figure 3.17.

Figure 3.17: The Wrights’ 1903 Airfoil

It is thin, highly cambered, and quite concave on the bottom. There is no significant difference between the top surface and the bottom surface — same length, same curvature. Still, the wing produces lift, using the same lift-producing principle as any other airfoil. This should further dispel the notion that wings produce lift because of a difference in length between the upper and lower surfaces.

Similar remarks apply to the sail of a sailboat. It is a very thin wing, oriented more-or-less vertically, producing sideways lift.

Even a thin flat object such as a barn door will produce lift, if the wind strikes it at an appropriate angle of attack. The airflow pattern (somewhat idealized) for a barn door (or the wing on a dime-store balsa glider) is shown in figure 3.18. Once again, the lift-producing mechanism is the same.

## 3.12  Circulation

### 3.12.1  Visualizing the circulation

You may be wondering whether the flow patterns shown in figure 3.18 or the earlier figures are the only ones allowed by the laws of hydrodynamics. The answer is: almost, but not quite. Figure 3.19 shows the barn door operating with the same angle of attack (and the same airspeed) as in figure 3.18, but the airflow pattern is different.

Figure 3.18: Barn Door — Natural Airflow

Figure 3.19: Barn Door — Unnatural Stream Lines

Figure 3.20: Barn Door — Pure Circulation

Figure 3.21: Barn Door — Natural Stream Lines

The new airflow pattern (figure 3.19) is highly symmetric. I have deleted the timing information, to make it clear that the stream lines are unchanged if you flip the figure right/left and top/bottom. The front stagnation line is a certain distance behind the leading edge; the rear stagnation line is the same distance ahead of the trailing edge. This airflow pattern produces no lift. (There will be a lot of torque — the so-called Rayleigh torque — but no lift.)

The key idea here is circulationfigure 3.18 has circulation while figure 3.19 does not. (Figure 3.21 is the same as figure 3.18 without the timing information.)

To understand circulation and its effects, first imagine an airplane with barn-door wings, parked on the ramp on a day with no wind. Then imagine stirring the air with a paddle, setting up a circulatory flow pattern, flowing nose-to-tail over the top of the wing and tail-to-nose under the bottom (clockwise in this figure). This is the flow pattern for pure circulation, as shown in figure 3.20. The magnitude of this circulatory flow is greatest near the wing, and is negligible far from the wing. It does not affect the airmass as a whole.

Then imagine that a headwind springs up, a steady overall wind blowing in the nose-to-tail direction (left to right in the figure), giving the parked airplane some true airspeed relative to the airmass as a whole. At each point in space, the velocity fields will add. The circulatory flow and the airmass flow will add above the wing, producing high velocity and low pressure there. The circulatory flow will partially cancel the airmass flow below the wing, producing low velocity and high pressure there.

If we take the noncirculatory nose-to-tail flow in figure 3.19 and add various amounts of circulation, we can generate all the flow patterns consistent with the laws of hydrodynamics — including the actual natural airflow shown in figure 3.18 and figure 3.21.10

There is nothing special about barn doors; real airfoils have analogous airflow patterns, as shown in figure 3.22, figure 3.23, and figure 3.24.

Figure 3.22: Unnatural Airflow — Angle of Attack but No Circulation

Figure 3.23: Pure Circulation

Figure 3.24: Normal, Natural Airflow

If you suddenly accelerate a wing from a standing start, the initial airflow pattern will be noncirculatory, as shown in figure 3.22. Fortunately for us, the air absolutely hates this airflow pattern, and by the time the wing has traveled a short distance (a couple of chord-lengths or so) it develops enough circulation to produce the normal airflow pattern shown in figure 3.24.

### 3.12.2  How Much Circulation? The Kutta Condition

In real flight situations, precisely enough circulation will be established so that the rear stagnation line is right at the trailing edge, so no air needs to turn the corner there. The counterclockwise flow at the trailing edge in figure 3.19 is cancelled by the clockwise flow in figure 3.20. Meanwhile, at the leading edge, both figure 3.19 and figure 3.20 contribute clockwise flow, so the real flow pattern (figure 3.21) has lots and lots of flow around the leading edge.

The general rule — called the Kutta condition — is that the air hates to turn the corner at a sharp trailing edge. To a first approxmation, the air hates to turn the corner at any sharp edge, because the high velocity there creates a lot of friction. For ordinary wings, that’s all we need to know, because the trailing edge is the only sharp edge.

The funny thing is that if the trailing edge is sharp, an airfoil will work even if the leading edge is sharp, too. This explains why dime-store balsa-wood gliders work, even with sharp leading edges.

It is a bit of a mystery why the air hates turning a corner at the trailing edge, and doesn’t mind so much turning a sharp corner at the leading edge — but that’s the way it is.11 This is related to the well-known fact that blowing is different from sucking. (Even though you can blow out a candle from more than a foot away, you cannot suck out a candle from more than an inch or two away.) In any case, the rule is:

 The air wants to flow cleanly off the trailing edge.

As the angle of attack increases, the amount of circulation needed to meet the Kutta condition increases.

Here is a nice, direct way of demonstrating the Kutta condition:

• Choose an airplane where the stall warning indicator is on the flapped section of the wing. This includes the Cessna C-152 and C-172, but not the C-182. It includes most Mooneys and the Grumman Tiger, but excludes Piper Cherokees and the Beech Bonanza.
• At a safe altitude, start with the airplane in the clean configuration in level flight, a couple of knots above the speed where the stall warning horn comes on.
• Maintaining constant pitch attitude and maintaining level flight, extend the flaps. The stall warning horn will come on.
The following items are not what we are trying to emphasize here, but for completeness they should perhaps be mentioned: (a) since extending the flaps increases the coefficient of lift the wing can produce, you can expect to need a lower airspeed, in order to maintain lift equal to weight; (b) you may need to fiddle with the throttle in order to maintain level flight; and (c) you may need to fiddle with the yoke to keep the fuselage at a constant pitch angle.

The goal is to create a situation where increasing the incidence of the wing section – by extending the flaps – increases the section’s angle of attack and increases its circulation. The increased circulation trips the stall-warning detector, as described in section 3.7.

We need to maintain the fuselage at a constant angle relative to the direction of flight, so that changing the incidence directly changes the wing’s angle of attack, in accordance with the formula pitch + incidence = angle of climb + angle of attack, as discussed in section 2.4.

There is no need to stall the airplane; the warning horn itself makes the point.

This demonstration makes it clear that the flap (which is at the back of the wing) is having a big effect on the airflow around the entire wing, including the stall-warning detector (which is near the front).

### 3.12.3  How Much Lift? The Kutta-Zhukovsky Theorem

Here is a beautifully simple and powerful result: The lift is equal to the airspeed, times the circulation, times the density of the air, times the span of the wing. This is called the Kutta-Zhukovsky theorem.12

 Lift = airspeed × circulation × density × span              (3.9)

Since circulation is proportional to the coefficient of lift and to the airspeed, this new notion is consistent with our previous knowledge that the lift should be proportional to the coefficient of lift times airspeed squared.

You can look at a velocity field and visualize the circulation. In figure 3.25, the right-hand edge of the blue streamers shows where the air is 70 milliseconds after passing the reference point. For comparison, the vertical black line shows where the 70 millisecond timeline would have been if the wing had been completely absent. However, this comparison is not important; you should be comparing each air parcel above the wing with the corresponding parcel below the wing.

Figure 3.25: Circulation Advances Upper & Retards Lower Streamers

Because of the circulatory contribution to the velocity, the streamers above the wing are at a relatively advanced position, while the streamers below the wing are at a relatively retarded position.

If you refer back to figure 3.9, you can see that circulation is proportional to angle of attack. In particular, note that when the airfoil is not producing lift there is no circulation — the upper streamers are not advanced relative to the lower streamers.

The same thing can be seen by comparing figure 3.22 to figure 3.24 — when there is no circulation the upper streamers are not advanced relative to the lower streamers.

### 3.12.4  Quantifying the Circulation

Circulation can be measured, according to the following procedure. Set up an imaginary loop around the wing. Go around the loop clockwise, dividing it into a large number of small segments. For each segment, multiply the length of that segment times the speed of the air along the direction of the loop at that point. (If the airflow direction is opposite to the direction of the loop, the product will be negative.) Add up all the products. The total velocity-times-length will be the circulation. This is the official definition.

Interestingly, the answer is essentially independent of the size and shape of the loop.13 For instance, if you go farther away, the velocity will be lower but the loop will be longer, so the velocity-times-length will be unchanged.

## 3.13  Mechanically-Induced Circulation

There is a widely-held misconception that it is the velocity relative to the skin of the wing that produces lift. This causes no end of confusion.

Remember that the air has a well defined velocity and pressure everywhere, not just at the surface of the wing. Using a windmill and a pressure gauge, you can measure the velocity and pressure anywhere in the air, near the wing or elsewhere. The circulatory flow set up by the wing creates low pressure in a huge region extending far above the wing. The velocity at each point determines the pressure at that point.

The circulation near a wing is normally set up by the interaction of the wind with the shape of the wing. However, there are other ways of setting up circulatory flow. In figure 3.26, the wings are not airfoil-shaped but paddle-shaped. By rotating the paddle-wings, we can set up a circulatory airflow pattern by brute force.

Bernoulli’s principle applies point-by-point in the air near the wing, creating low pressure that pulls up on the wings, even though the air near the wing has no velocity relative to the wing – it is “stuck” between the vanes of the paddle. The Kutta-Zhukovsky theorem remains the same as stated above: lift is equal to the airspeed, times the circulation, times the density of the air, times the span of the wing.

This phenomenon — creating the circulation needed for lift by mechanically stirring the air — is called the Magnus effect.

The airplane in figure 3.26 would have definite controllability problems, since the notion of angle of attack would not exist (see chapter 2 and chapter 6). The concept, though, is not as ridiculous as might seem. The famous aerodynamicist Flettner once built a ship that “sailed” all the way across the Atlantic using huge rotating cylinders as “sails” to catch the wind.

Figure 3.27: Tumbling Card — Lift Created by Circulation

Also, it is easier than you might think to demonstrate this important concept. You don’t need four vanes on the rotating paddle; a single flat surface will do. A business card works fairly well. Drop the card from shoulder height, with its long axis horizontal. As you release it, give it a little bit of backspin around the long axis. It will fly surprisingly well; the lift-to-drag ratio is not enormous, but it is not zero either. The motion is depicted in figure 3.27.

You can improve the performance by giving the wing a finer aspect ratio (more span and/or less chord). I once took a manila folder and cut out several pieces an inch wide and 11 inches long; they work great.

As an experiment, try giving the wing the wrong direction of circulation (i.e. topspin) as you release it. What do you think will happen?

I strongly urge you to try this demonstration yourself. It will improve your intuition about the relationship of circulation and lift.

Figure 3.28: Curve Balls

We can use these ideas to understand some (but not all) of the aerodynamics of tennis balls and similar objects. As portrayed in figure 3.28, if a ball is hit with a lot of backspin, the surface of the spinning ball will create the circulatory flow pattern necessary to produce lift, and it will be a “floater”. Conversely, the classic “smash” involves topspin, which produces negative lift, causing the ball to “fly” into the ground faster than it would under the influence of gravity alone. Similar words apply to leftward and rightward curve balls.

To get even close to the right answer, we must ask where the relative wind is fast or slow, relative to undisturbed parcels of air — not relative to the rotating surface of the ball. Remember that the fluid has a velocity and a pressure everywhere, not just at the surface of the ball. Air moving past a surface creates drag, not lift. Bernoulli says that high velocity is associated with low pressure and vice versa. For the floater, the circulatory flow created by the backspin combines with the free-stream flow created by the ball’s forward motion to create high-velocity, low-pressure air above the ball — that is, lift.

 The air has velocity and pressure everywhere ... not just at surfaces.

This simple picture of mechanically-induced circulation applies best to balls that have evenly-distributed roughness. Cricket balls are in a different category, since they have a prominent equatorial seam. If you spin-stabilize the orientation of the seam, and fly the seam at an “angle of attack”, airflow over the seam causes extra turbulence which promotes attached flow on one side of the ball. See section 18.3 for some discussion of attached versus separated flow. Such effects can overwhelm the mechanically-induced circulation.

To really understand flying balls or cylinders, you would need to account for the direct effect of spin on circulation, the effect of spin on separation, the effect of seams on separation, et cetera. That would go beyond the scope of this book. A wing is actually easier to understand.

## 3.14  Lift Requires Circulation & Vortices

### 3.14.1  Vortices

A vortex is a bunch of air circulating around itself. The axis around which the air is rotating is called a vortex line. It is mathematically impossible for a vortex line to have loose ends. A smoke ring is an example of a vortex. It closes on itself so it has no loose ends.

The circulation necessary to produce lift can be attributed to a bound vortex line. It binds to the wing and travels with the airplane. The question arises, what happens to this vortex line at the wingtips?

In the simplest case, the answer is that the vortex spills off each wingtip. Each wing forms a trailing vortex (also called wake vortex) that extends for miles behind the airplane. These trailing vortices constitute the continuation of the bound vortex. See figure 3.29. Far behind the airplane, possibly all the way back at the place where the plane left ground effect, the two trailing vortices join up to form an unbroken14 vortex line.

Figure 3.29: Bound Vortex, Trailing Vortices

The air rotates around the vortex line in the direction indicated in the figure. We know that the airplane, in order to support its weight, has to yank down on the air. The air that has been visited by the airplane will have a descending motion relative to the rest of the air. The trailing vortices mark the boundary of this region of descending air.

It doesn’t matter whether you consider the vorticity to be the cause or the effect of the descending air — you can’t have one without the other.

Lift must equal weight times load factor, and we can’t easily change the weight, or the air density, or the wingspan. Therefore, when the airplane flies at a low airspeed, it must generate lots of circulation.

#### *   Winglets, Fences, “Lateral” Flow, etc.

It is a common misconception that the wingtip vortices are somehow associated with unnecessary spanwise flow (sometimes called “lateral” flow), and that they can be eliminated using fences, winglets, et cetera. The reality is that the vortices are completely necessary; you cannot produce lift without producing vortices.

 Lift and trailing vortices are intimately and necessarily associated with air flowing around the span. Neither lift nor trailing vortices are in any important way associated with “lateral” flow along the span.

Also keep in mind that “circulation” and “vorticity” are two quite different ways of expressing the same idea: When we draw a vortex line, it represents the core of the vortex, which is the axis of the circulatory motion. The air circulates around the vortex line. Circulation refers to flow around the vortex line, not along the vortex line.

If you look closely, you find that the overall flow pattern is more accurately described by a large number of weak vortex lines, rather than by the one strong vortex line shown in figure 3.29. By fiddling with the shape of the wing the designers can control (to some extent) where along the span the vorticity is shed.

It turns out that behind each wing, the weak vortex lines get twisted around each other. (This is the natural consequence of the fact that each vortex line gets carried along in the circulatory flow of each of the other vortex lines.) If you look at a point a few span-lengths behind the aircraft, all the weak vortex lines have rolled up into what is effectively one strong vortex. That means that visualizing the wake in terms of one strong vortex (per wingtip), as shown in figure 3.29, is good enough for most pilot purposes. However, you might care about the details of the roll-up process if you are flying in close formation behind another aircraft, such as a glider being towed.

Winglets encourage the vorticity to be shed nearer the wingtips, rather than somewhere else along the span. This produces more lift, since each part of the span contributes lift in proportion to the amount of circulation carried by that part of the span, in accordance with the Kutta-Zhukovsky theorem. In any case, as a general rule, adding a pair of six-foot-tall winglets has no aerodynamic advantage compared to adding six feet of regular, horizontal wing on each side.15

The important point remains that there is no way to produce lift without producing wake vortices. Remember: The trailing vortices mark the boundary between the descending air behind the wing and the undisturbed air outboard of the descending region.

The bound vortex that produces the circulation that supports the weight of the airplane should not be confused with the little vortices produced by vortex generators (to re-energize the boundary layer) as discussed in section 18.3.

### 3.14.2  Wake Turbulence

When air traffic control (ATC) tells you “caution — wake turbulence” they are really telling you that some previous airplane has left a wake vortex in your path. The wake vortex from a large, heavy aircraft can easily flip a small aircraft upside down.

A heavy airplane like a C5-A flying slowly is the biggest threat, because it needs lots of circulation to support all that weight at a low airspeed. So the most important rule is to beware of an aircraft that is heavy and slow.

Conventional pilot lore says that an aircraft with flaps extended should be less dangerous than one with the flaps retracted, on the grounds that there is more circulation around the flapped section of wing, and less circulation around the remaining (outboard) section of each wing. That means that a goodly amount of circulation will be shed at the boundary between the flapped and unflapped section, so you get two half-strength vortices per wing, rather than one full-strength one.

That’s undoubtedly relevant if you are flying in close formation behind a heavy, slow aircraft … but in the other 99.999% of general-aviation flying, you won’t be close enough for the other plane’s flaps to give you any protection. At any reasonable distance behind the other aircraft, all the trailing vorticity will have rolled up into what is effectively one strong vortex. When you couple that with the fact that the aircraft with flaps extended might be flying slower than the one without, you should not imagine that flaps reduce the threat of wake turbulence. Besides, I don’t plan on getting close enough to the other aircraft to even see whether it’s got flaps extended or not.

To summarize: Although conventional pilot lore says to beware of heavy, slow, and clean, it is simpler and better to beware of heavy and slow (whether clean or not).

 Beware of vortices behind heavy and slow aircraft.

Like a common smoke ring, the wake vortex does not just sit there, it moves. In this case it moves downward. A common rule of thumb says they normally descend at about 500 feet per minute, but the actual rate will depend on the wingspan and coefficient of lift of the airplane that produced the vortex.

Vortices are part of the air. A vortex in a moving airmass will be carried along with the air. In fact, the reason wake vortices descend is that the right vortex is carried downward by the flow field associated with the left vortex, and the left vortex is carried downward by the flow field associated with the right vortex. Superimposed on this vertical motion, the ordinary wind blows the vortices downwind, usually more-or-less horizontally.

When a vortex line gets close to the ground, it “sees its reflection”as discussed in section 3.14.5. That is, a vortex at height H above ground moves as if it were being acted on by a mirror-image vortex a distance H below ground. This causes the wake vortices to spread out. Specifically, the left vortex moves to the left, carried along by the flow field associated with its mirror image. Similarly the right vortex moves the right.

#### *   Avoiding Wake Turbulence Problems

If you are flying a light aircraft, avoid the airspace below and behind a large aircraft. Avoiding the area for a minute or two suffices, because a vortex that is older than that will have lost enough intensity that it is probably not a serious problem.

If you are landing on the same runway as a preceding large aircraft, you can avoid its wake vortices by flying a high, steep approach, and landing at a point well beyond the point where it landed. Remember, it doesn’t produce vortices unless it is producing lift. Assuming you are landing into the wind, the wind can only help clear out the vortices for you.

If you are departing from the same runway as a preceding large aircraft, you can avoid its vortices — in theory — if you leave the runway at a point well before the point where it did, and if you make sure that your climb-out profile stays above and/or behind its. In practice, this might be hard to do, since the other aircraft might be able to climb more steeply than you can. Also, since you are presumably taking off into the wind, you need to worry that the wind might blow the other plane’s vortices toward you.

A light crosswind might keep a vortex on the runway longer, by opposing its spreading motion. A less common problem is that a crosswind might blow vortices from a parallel runway onto your runway.

The technique that requires the least sophistication is to delay your takeoff a few minutes, so the vortices can spread out and be weakened by friction.

### 3.14.3  Induced Drag

Here are some more benefits of understanding circulation and vortices: it explains induced drag, and explains why gliders have long skinny wings. Induced drag is commonly said to be the “cost” of producing lift. However, there is no law of physics that requires a definite cost. If you could take a very large amount of air and pull it downward very gently, you could support your weight at very little cost. The cost you absolutely must pay is the cost of making that trailing vortex. For every mile that the airplane flies, each wingtip makes another mile of vortex. The circulatory motion in that vortex involves nontrivial amounts of kinetic energy, and that’s why you have induced drag. A long skinny wing will need less circulation than a short fat wing producing the same lift. Gliders (which need to fly slowly with minimum drag) therefore have very long skinny wings (limited only by strength; it’s hard to build something long, skinny, and strong).

### 3.14.4  Soft-Field Takeoff

We can use what we know about vortices to help understand soft-field takeoff procedure.

By way of background, consider the following parable: Suppose you throw a basketball downwards. By the law of action and reaction, this gives you some upward momentum. Now the ball bounces, and you catch it as it comes back up. The catch gives you some additional upward momentum. The ball winds up with the same momentum before and after, and the same energy before and after, but still you were able to use it as a means to transfer some momentum.

The parable works like this:

 Not in Ground Effect In Ground Effect

 Normal flight is like flinging a series of baskeballs downward, using each one only once. Lift requires imparting some momentum to each basketball, and this in turn requires some energy. In ground effect, you get to catch the basketball as it comes back up after bouncing. Ideally, this doubles the amount of useful momentum transfer (a 100% increase) and approximately cancels the required energy (a 100% decrease).

Let’s go through the analysis again:

 In normal flight, producing lift requires transferring momentum to the air. This momentum “eventually” gets transferred to the ground, but that is a messy process that dissipates energy. In ground effect, the momentum gets rather quickly transferred through the air into the ground. Ideally the air winds up with the same momentum before and after, and the same energy before and after.

 You wind up with less momentum stored in the air, but that’s not a problem, because lift depends on transferring momentum through the air.

### 3.14.5  Ground Effect

Here’s a good way of thinking about how ground effect works: The aircraft “sees its reflection” in the ground. That is, suppose you are flying 10 feet above the ground. Then you have a choice: (a) You can analyze the situation directly by accounting for the way air bounces off the ground, or (b) you can achieve the same result by imagining the ground isn’t there and instead imagining a mirror-image aircraft flying upside down, 10 feet below where the ground level used to be. Instead of air bouncing upward from the ground, you see air flowing upward from the mirror-image aircraft, as it produces upside-down lift. This is called the method of images.

Because you are flying in the upwash created by the mirror-image airplane, you need a lower angle of attack to produce a given amount of lift, at any given airspeed. You can think of this as a pseudo-updraft. It is a relatively minor effect, comparable to the pseudo-tailwind mentioned in section 3.14.6. It is most noticeable in situations where it affects the tail differently from the main wing, for instance when there is a pitch-change associated with takeoff or with the landing flare. In such situations, the airplane might exhibit nonlinear “squirrely” behavior.

It remains true that lift requires circulation and vortices, as discussed in section 3.14.1. The circulatory flow-pattern associated with your wing is still grabbing air and throwing it downwards. This accounts for the force of lift supporting your airplane.

Meanwhile, the mirror-image airplane is grabbing air and throwing it upwards at you. Now, it turns out that in terms of force, a downward flow of downward momentum is the same as an upward flow of upward momentum. The forces don’t cancel. If you are very very near the ground, you wind up with nearly 100% less momentum stored in the air, and nearly 100% more momentum being transferred through the air.

The mirror-image aircraft’s trailing vortices spin in the opposite direction and largely cancel your trailing vortices — greatly reducing induced drag. The bound vortex attached to your wing is not canceled. It is still there, as it must be to produce lift in accordance with the Kutta-Zhukovsky theorem. However, after the vortex spills off the wing it can mix with and cancel the mirror-image trailing vortex. This is by far the most significant part of what we call ground effect.

Here are a few ways this relates to pilot procedure:

• As discussed in section 13.4, in a soft-field takeoff, you leave the ground at a very low airspeed, and then fly in ground effect for a while. There will be no wheel friction (or damage) because the wheels are not touching the ground. There will be very little induced drag because of the ground effect, and there will be very little parasite drag because you are going slowly. The airplane will accelerate like crazy. When you reach normal flying speed, you raise the nose and fly away.
• Sometimes in an emergency, you can save energy by flying in ground effect, as discussed in section 15.2.7.
• Conversely: if you are approaching a not-very-long runway and have a bit of excess energy on short final, get rid of it before entering ground effect. Fly horizontally for a while if you have to, well above ground level. If it’s hard to get rid of the energy up there, it will only be harder in ground effect. A timely go-around might be wise.

If you want to practice skimming in ground effect, find a long, long, long runway to practice on. Enter ground effect at a speed well above normal landing speed, and observe what happens. Be careful not to run off the far end.

### 3.14.6  Bound Vortex

Let’s not forget about the bound vortex, which runs spanwise from wingtip to wingtip, as shown in figure 3.29.

When you are flying in ground effect, you are influenced by the mirror image of your bound vortex. Specifically, the flow circulating around the mirror-image bound vortex will reduce the airflow over your wing. I call this a pseudo-tailwind.16 It is a relatively minor effect, comparable to the pseudo-updraft mentioned in section 3.14.4.

Operationally, this means that for any given angle of attack, you need a higher true airspeed to support the weight of the airplane. This in turn means that a low-wing airplane will need a longer runway than the corresponding high-wing airplane, other things being equal. It also means – in theory – that there are tradeoffs involved during a soft-field takeoff: you want to be sufficiently deep in ground effect to reduce induced drag, but not so deep that your speeds are unduly increased. In practice, though, feel free to fly as low as you want during a soft-field takeoff, since in an ordinary-shaped airplane the bad effect of the reflected bound vortex (greater speed) never outweighs the good effect of the reflected trailing vortices (lesser drag).

As a less-precise way of saying things, you could say that to compensate for ground effect, at any given true airspeed, you need more coefficient of lift. This explains why all airplanes – some more so than others – exhibit “squirrely” behavior when flying near the ground, including:

• Immediately after liftoff, the airplane may seem to leap up a few feet, as you climb out of the pseudo-tailwind. This is generally a good thing, because when you become airborne you generally want to stay airborne.
• Conversely, on landing, the airplane may seem to drop suddenly, as the pseudo-tailwind takes effect. In some slight, very theoretical way this could be helpful, insofar as when you land you generally want to stay landed, but in reality, you usually prefer the landing to be more gentle rather than more firm, so it would be better to have negative feedback, not positive feedback. In any case, it’s not really a big problem once you learn to anticipate it. It does mean that practicing flaring at altitude (as discussed in section 12.11.3) will never entirely prepare you for real landings.
• The wing and the tail will be influenced by ground effect to different degrees. (This is particularly pronounced if your airplane has a low wing and a high T-tail, but no airplane is entirely immune.) That means that when you enter or exit ground effect, there will be squirrely pitch-trim changes ... in addition to the effects mentioned in the previous items. Just to rub salt in the wound, the behavior will be different from flight to flight, depending on how the aircraft is loaded, i.e. depending on whether the center of mass is near the forward limit or the aft limit.

During landing, ground effect is a lose/lose/lose proposition. You regret greater speed, you regret lesser drag, and you regret squirrely handling.

## 3.15  Frost on the Wings

The Federal Aviation Regulations prohibit takeoff when there is frost adhering in critical places including wings and control surfaces. The primary reason for concern is that the frost causes roughness on the surface of the airfoil. (In contrast, the weight of the frost is usually negligible.)

Wind-tunnel data indicates that roughness can cause a surprisingly large amount of trouble.

• The most obvious effect of roughness on the wings is to create a lot more drag, as seen in the right panel in figure 3.30, which shows wind-tunnel data for a real airfoil (the NACA 631-412 airfoil; see reference 26). At cruise angle of attack, the drag is approximately doubled; at higher angles of attack (corresponding to lower airspeeds) it is even worse.
• The less obvious (yet more critical) problem is that roughness causes the wing to stall at a considerably lower angle of attack, lower coefficient of lift, and higher airspeed. This can be seen in the left panel of figure 3.30. The pilot of the frosty airplane could get a very nasty surprise.

Figure 3.30: Roughness Degrades Wing Performance

As mentioned in section 3.6, Bernoulli’s principle cannot be trusted when frictional forces are at work. Frost, by sticking up into the breeze, is very effective in causing friction. This tends to de-energize the boundary layer, leading to separation which produces the stall.17

It is interesting that at moderate and low angles of attack (cruise airspeed and above) the frost has hardly any effect on the coefficient of lift. This reinforces the point made in section 3.13 that the velocity of the air right at the surface, relative to the surface, is not what produces the lift.

An interesting situation arises when the airplane has been sitting long enough to pick up a big load of frost, but the present air temperature is slightly above freezing. By far the easiest way to get rid of the frost is by dousing the plane with five-gallon jugs of warm water. That will melt the frost and heat the wings to an above-freezing temperature (so that frost will not re-form).

## 3.16  Consistent (Not Cumulative) Laws of Physics

We have seen that several physical principles are involved in producing lift. Each of the following statements is correct as far as it goes:

• The wing produces lift “because” it is flying at an angle of attack.
• The wing produces lift “because” of circulation.
• The wing produces lift “because” of Bernoulli’s principle.
• The wing produces lift “because” of Newton’s law of action and reaction.

We now examine the relationship between these physical principles. Do we get a little bit of lift because of Bernoulli, and a little bit more because of Newton? No, the laws of physics are not cumulative in this way.

There is only one lift-producing process. Each of the explanations itemized above concentrates on a different aspect of this one process. The wing produces circulation in proportion to its angle of attack (and its airspeed). This circulation means the air above the wing is moving faster. This in turn produces low pressure in accordance with Bernoulli’s principle. The low pressure pulls up on the wing and pulls down on the air in accordance with all of Newton’s laws.

See section 19.2 for additional discussion of how Newton’s laws apply to the airplane and to the air.

## 3.17  Summary: How a Wing Produces Lift

• The flow pattern created by a wing is the sum of the obstacle effect (which is significant only very near the wing, and is the same whether or not the wing is producing lift) plus the circulation effect (which extends for huge distances above and below the wing, and is proportional to the amount of lift, other things being equal).
• A wing is very effective at changing the speed of the air. The air above is speeded up relative to the corresponding air below. Each air parcel gets a temporary change in speed and a permanent offset in position.
• Bernoulli’s principle asserts that a given parcel of air has high velocity when it has low pressure, and vice versa. This is an excellent approximation under a wide range of conditions. This can be seen as a consequence of Newton’s laws.
• Below-atmospheric pressure above the wing is much more pronounced than above-atmospheric pressure below the wing.
• There is significant upwash ahead of the wing and even more downwash behind the wing.
• The front stagnation line is well below and behind the leading edge.
• The rear stagnation line is at or very near the trailing edge. The Kutta condition says the air wants to flow cleanly off the sharp trailing edge. This determines the amount of circulation.
• An airfoil does not have to be curved on top and/or flat on the bottom in order to work. A rounded leading edge is a good idea, but even a barn door will fly.
• Air passing above and below the wing does not do so in equal time. When lift is being produced, every air parcel passing above the wing arrives substantially early (compared to corresponding parcel below the wing) even though it has a longer path.
• Most of the air above the wing arrives early in absolute terms (compared to undisturbed air), but this is not important, and the exceptions are doubly unimportant.
• Lift is equal to circulation, times airspeed, times density, times wingspan.
• Well below the stalling angle of attack, the coefficient of lift is proportional to the angle of attack; the circulation is proportional to the coefficient of lift times the airspeed.
• Air is a fluid, not a bunch of bullets. The fluid has pressure and velocity everywhere, not just where it meets the surface of the wing.
• There is downward momentum in any air column behind the wing. There is zero momentum in any air column ahead of the wing, outboard of the trailing vortices, or aft of the starting vortex.
• Vortex lines cannot have loose ends; therefore you cannot produce lift without producing wake vortices.
• Induced drag arises when you have low speed and/or short span, because you are visiting a small amount of air and yanking it down violently, producing strong wake vortices. In contrast there is very little induced drag when you have high speed and/or long span, because you are visiting a large amount of air, pulling it down gently, producing weak wake vortices.

1
These simulations are based on a number of assumptions, including that the viscosity is small (but not zero), the airspeed is small compared to the speed of sound, the airflow is not significantly turbulent, no fluid can flow through the surface of the wing, and the points of interest are close to the wing and not too close to either wingtip.
2
To be more precise: there is no wind in either of the two dimensions that show up in figure 3.3. There might be some flow in the third dimension (i.e. spanwise along the stagnation line) but that isn’t relevant to the present discussion.
3
... although for turbulent flow, the stream lines can get so tangled that they lose any useful meaning.
4
5
This low pressure is associated with fast-moving air in this region. You may be wondering why some of this fast-moving air arrives at the trailing edge late. The answer is that it spent a lot of time hanging around near the leading-edge stagnation line, moving much slower than the ambient air. Then as it passes the wing, it moves faster than ambient, but not faster enough to make up for the lost time.
6
Of course, if there were no atmospheric pressure below the wing, there would be no way to have reduced pressure above the wing. Fundamentally, atmospheric pressure below the wing is responsible for supporting the weight of the airplane. The point is that pressure changes above the wing are more pronounced than the pressure changes below the wing.
7
Newton’s laws are discussed in section 19.1.
8
... but not always. See section 18.4 for a counterexample.
9
The airfoil designations aren’t just serial numbers; the digits actually contain information about the shape of the airfoil. For details see reference 26.
10
We are still assuming negligible viscosity, small percentage pressure changes, no turbulence in the fluid, no fluid flowing through the surface of the wing, and a few other reasonable assumptions.
11
Actually, you never get 100% of the circulation predicted by the Kutta condition, especially for crummy airfoils like barn doors. For nice airfoils with a rounded leading edge, you get something like 99% of the Kutta circulation.
12
The second author’s name is properly spelled Жуковский. When Russian scientists write this name in English, they almost always spell it Zhukovsky ... which is the spelling used in this book. Not coincidentally, that conforms to standard transliteration rules and is a reasonable guide to the pronunciation. Beware: you may encounter the same name spelled other ways. In particular, “Joukowski” was popular once upon a time, for no good reason.
13
This assumes that the loop is big enough to include the places where circulation is being produced (i.e. the wing and the boundary layer).
14
There is a rule that says vortex lines can never have loose ends. They form closed loops, like magnetic field lines. This is not a mere law of physics; it is a mathematical identity.
15
This assumes the goal is to produce wings, as opposed to (say) rudders. Also note that the winglet solution may provide a practical advantage when taxiing and parking. This is why Boeing put winglets (instead of additional span) on the 747-400 — they wanted to be able to park in a standard slot at the airport.
16
It’s only a pseudo-tailwind, not a real tailwind, because wind is officially supposed to be measured in the ambient air, someplace where the air is not disturbed by the airplane — or by its mirror image. Similarly airspeed is measured relative to the ambient air.
17
Boundary layers, separation, etc. are discussed in more detail in section 18.3.

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