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Copyright 1996-2001 jsd

2   Angle of Attack Awareness and Angle of Attack Management


  • If you want to go up, pull back on the yoke.
  • If you want to go down, pull back a little more.
  • If you want to go down real fast and spin around and around and around, just keep pulling back.
— Aviation proverb.

2.1   The Importance of Angle of Attack

Angle of attack is a very important and useful concept. Most of the airplane's critical performance numbers are more closely related to angle of attack than to anything else. Let's explore what this means.

You've probably heard that it is good to fly the airplane ``by the numbers''. The question is, what numbers?

Suppose we wish to achieve the best rate of climb:

A)
You could try to control the airplane by reference to the ``rate of climb'' number shown on the vertical speed indicator. This is not recommended!
B)
It would be better to maintain VY, the nominal best-rate-of-climb speed, as shown on the airspeed indicator, and accept whatever rate of climb results. This is almost exactly the right idea.
C)
It would be even better to realize that the best rate of climb is achieved at a particular angle of attack. In particular, if the airplane is lightly loaded compared to what was anticipated in the handbook, the best rate of climb will be achieved at a lower speed than is reflected in the handbook's VY value.
This is not an isolated example. Many of the airplane's critical performance numbers are really angle of attack numbers:
    
The stall occurs at a particular angle of attack.
    
The smallest power-off descent rate occurs at a particular angle of attack.
    
The best power-off glide ratio occurs at a particular angle of attack.
    
The recommended ``approach speed'' is really an angle of attack recommendation.
    
The best rate of climb occurs at a particular angle of attack.
    
The best angle of climb occurs at a particular angle of attack.12
Here is a summary of the main ideas that will be explained in this chapter:

2.2   Definition of Angle of Attack

I will now explain what angle of attack is, why it is important, and how it is related to things a pilot can actually observe and control.

The basic idea is simple: the angle of attack is the angle at which the air hits the wing. The Wright brothers had only one flight instrument on their first airplane — an angle of attack instrument. It was all they needed.

Their angle of attack indicator consisted of a stick attached to the wing, with a piece of yarn dangling from the front end, as indicated in figure 2.1. The yarn aligns itself with the relative wind.3 The stick serves as a reference line, and also serves to locate the yarn in a region of air that has not been too badly disturbed by the wing.

aoa-def2
Figure 2.1: Simple Angle of Attack Instrument

The angle between the stick and the yarn indicates angle of attack.

The exact alignment of the indicator stick relative to the airplane is not critical. The most elegant scheme is to orient the stick in the zero-lift direction so that zero angle of attack corresponds to zero coefficient of lift. That choice will be used throughout this book; see section 2.14 for a discussion of other possibilities.

Most aircraft do not have any instruments that give the you a direct indication of angle of attack. Surprisingly, many airliners and other aircraft that do have fancy angle-of-attack sensors don't make the information available to the flight crew — only to the autopilot. The bottom line is that most pilots have to use a few tricks in order to perceive angle of attack. We now discuss how this is done.

It turns out to be easier to maintain some constant angle of attack than to know precisely what angle of attack you've got. The strategy is summarized in the following outline.

        1
— There are several ways to maintain a constant angle of attack.
   1.1
– The airplane is trimmed for a definite angle of attack (see section 2.3).
   1.2
– You can perceive the angle of attack and regulate it by hand. To perceive the angle of attack, you need to compare the pitch attitude to the relative wind.
1.2(a)
– There are at least four ways to perceive the pitch attitude (see section 2.5).
1.2(b)
– There are a couple of ways to estimate the direction of the relative wind (see section 2.11).
        2
— You can use the airspeed and other considerations to decide if you are maintaining the right angle of attack (see section 2.12).
Now let's investigate each of the items in this outline.

2.3   Trim for Angle of Attack!

The simplest and best way to get the airplane to fly at a constant angle of attack is to leave it alone! An airplane, by its very structure, is trimmed for a definite angle of attack. The reason for this is discussed in chapter 6. Even a dime-store balsa-wood glider wants to fly at a definite angle of attack.

This concept is so important that it is the focal point of the first lesson I give student pilots, who sometimes arrive with the misconception that pilots must use great skill and continual intervention to keep the airplane under control. I trim the airplane for straight and level flight and then take my hands off the controls, demonstrating that the airplane will fly just fine for quite a while with no intervention at all. I emphasize a professional pilot does not grab the controls firmly and move them quickly; a real pro grabs them lightly and moves them smoothly .

The second lesson is this: I trim the airplane for a speed near VY, straight and level. I then roll the trim wheel back a little, which results in a decrease in the trim speed. It does not result in a steady climb. I explain that the trim wheel controls angle of attack, and that airspeed is related to angle of attack. Trim for angle of attack!

To make changes in the angle of attack, you should adjust the pitch attitude using pressure the yoke, then trim to remove the pressure, as discussed in section 2.6.

Configuration changes can affect the airplane's preferred angle of attack. In a Cessna 152, 172, or 182, if you extend the flaps while the engine is at a high power setting or if you increase the power while the flaps are extended it will cause a nasty decrease in the trim speed. This is highly undesirable and dangerous behavior. This means that when you perform a go-around, the airplane tends to pitch up drastically and lose airspeed; to maintain control you need to push on the yoke while you retract the flaps and retrim. This pitch-up behavior is particularly treacherous because it is not familiar. The trim speed changes very little if you extend the flaps at low power settings, and/or change the power with the flaps retracted, so if you haven't recently performed many go-arounds or similar maneuvers you might be in for a nasty surprise.

For a typical Cherokee, extending two notches of flaps lowers the trim speed ten or fifteen knots. This is discussed further in section 5.5 and section 12.10. Increasing or decreasing engine power affects the trimmed angle of attack only slightly. As discussed in section 1.3.2, if you just reduce power the airplane should just descend. It should not slow down appreciably; in fact it will probably speed up a little.

An advanced lesson serves to demonstrate that constant angle of attack is not quite the same as constant airspeed. When the airplane is subjected to a high G-loading, as in a steep turn, the trim mechanism causes it to speed up, so that it can support the increased load at the same angle of attack. This is important, since (as discussed in section 6.2) it helps explain graveyard spirals, and why it is a bit tricky to recover from them safely.

Conclusions:
Do not trim for pitch attitude.
Do not trim for rate of climb.
Trim for airspeed, approximately.
Trim for angle of attack!

2.4   Three Contributions to Angle of Attack

As mentioned earlier, it is difficult to directly perceive angle of attack. Fortunately, there are three other quantities that can be perceived, and together they determine the angle of attack. They are:These quantities are related to the angle of attack by a very simple formula:
Pitch Attitude + Incidence = Angle of Climb + Angle of Attack

This relationship is illustrated in figure 2.2. Perhaps the simplest case is straight and level flight at cruise airspeed. In this case, the pitch attitude is zero, the angle of climb is zero, and the angle of attack is equal to the angle of incidence. Some more examples, with specific numbers for a typical airplane, are included in table 2.1.

incidence
Figure 2.2: Pitch + Incidence = Climb + Attack

Extending the flaps has the effect of increasing the incidence5 by several degrees. You need to be always aware of what flap setting you are using, and to recognize the distinction between ``pitch attitude'' and ``pitch attitude plus incidence''. For any given flap setting, you can take the incidence to be constant, whereupon angle of attack depends only on pitch attitude and direction of flight.

The table mentions VX and VY, which denote the airspeeds for best angle of climb and best rate of climb, respectively, as discussed in section 7.5. The relationship of airspeed to angle of attack will be discussed in section 2.12.

Airspeed (KCAS)
Pitch Attitude
Incidence
Angle of Climb
Angle of Attack
stall  59 14.0 4.5 0 18.5
level at VX  64  8.5 4.5 0 13.0
level at VY  76  4.0 4.5 0  8.5
climbing at VY  76  7.0 4.5 3  8.5
cruise 115  0.0 4.5 0  4.5
Table 2.1: Angles in various situations

2.5   Perceiving Pitch Angle

In straight and level flight you can control angle of attack by controlling pitch attitude. You won't be able to pick a particular angle of attack such as 6.37 degrees, but whatever angle of attack you've got can be maintained.6

There are at least four ways of perceiving pitch attitude. Perhaps the best way is to use a mark on the windshield, as shown in figure 2.3. The line of sight from your eye through the mark makes a good pointer. (Try not to move your head up and down too much!) If you can't find a scratch or bug-corpse in exactly the right place, you can make a mark, or a pair of marks, as discussed in section 11.5.2. It is even simpler to rest your hand atop the instrument panel, holding the tip of your finger in the right place, as shown in figure 11.2.

Suppose you identify (or make) the mark when the airplane is flying at the angle of attack that corresponds to VY. Then if you re-trim for a higher angle of attack7 the sight line through that mark will point two or three degrees above the horizon. Similarly, if you re-trim for high-speed cruise, the sight mark will appear three or four degrees below the horizon.

perceive-pitch-front
Figure 2.3: Perceiving Pitch Using The Forward Horizon

The second way of perceiving pitch attitude also involves looking out the front, but uses a sight line through a point on the cowling. This is also indicated in figure 2.3. Be sure you chose a point on the cowling directly ahead of your dominant8 eye; if your seat is way over on one side of the airplane and you choose a sight mark on the middle of the cowling, your sight line will be angled sideways, which will mess up your pitch attitude perception as soon as you try to bank the airplane. A Cessna 152 or 172 has a bolt on the cowling, directly ahead of the pilot, that makes a good sight mark.

A sight mark on the cowling has the advantage that it is farther away from your eye, so it is easier to keep both it and the horizon in focus at the same time. The disadvantage is that the sight line constructed this way sometimes points quite a ways below the horizon. This means the angle you are trying to perceive — the angle between this reference line and the relative wind — is larger. It is always harder to perceive a small change in something large than a small change in something that was small to begin with.

Using the cowling has one big advantage over using marks on the windshield: the cowling is a permanent part of the airplane and is in the same place on all airplanes of that make and model.

The third way to perceive pitch attitude is to observe the angle between the wing and the lateral horizon, as shown in figure 2.4. On a high-wing airplane, the bottom surface of the wing makes a good reference. In particular, on a Cessna 152 / 172 / 182, the bottom surface has a rather large flat section, which makes an ideal reference — and this reference is very nearly aligned with the horizon at cruise angle of attack (in level flight).

perceive-pitch-lateral
Figure 2.4: Perceiving Pitch Using The Lateral Horizon

On a low-wing airplane, you typically have to use a little more imagination to use the wing as a reference pointer — but it is definitely possible and definitely worth the effort. Sometimes it helps to envision the chord line with your mind's eye. If you control the angle between the chord line and the lateral horizon, you are controlling pitch attitude.

The idea that you can control pitch attitude while looking out the side window is very important. Aerobatics pilots often attach crosshair-like pointers to their wings, just so they can be sure to have an easy-to-use pitch attitude reference when they're looking out the side. Conversely, it is common to find students who (although they can fly OK while looking out the front) lose control of pitch as soon as they try to look out the side; this makes it tough to check landmarks or scan for traffic.

The fourth way of perceiving pitch attitude is to use the attitude indicator instrument — the artificial horizon. This has the drawback that it is much too close to your eye; you can't look at the attitude indicator and look for traffic at the same time. You should use outside pitch references whenever possible.

2.6   Making Changes in Angle of Attack

The push/pull motion of the yoke and the trim wheel are part of the same system, jointly controlling the angle of attack. They also jointly control airspeed, as discussed in section 2.12.

If you want to make a temporary increase in angle of attack, just raise the nose by applying a little back pressure on the yoke. When you reach the new pitch attitude, you can release most of the pressure, and for the first few moments the airplane will maintain the new pitch attitude. Then, as it decelerates, you will need to maintain progressively more back pressure in order to maintain the new pitch attitude (and new angle of attack). After a few seconds things will stabilize at a new pitch attitude, a new angle of attack, and a new airspeed. At this point, if you release the back pressure, the airplane will want to drop its nose so it can return to its trimmed angle of attack.

If you push or pull the airplane off its trim speed and then suddenly let go of the yoke, the airplane will not return smoothly and immediately to its trim speed; there will be some phugoid oscillation (as discussed in section 6.1.12). To undo a temporary change in angle of attack, the proper technique requires observing and controlling the pitch attitude. Let the nose drop to the correct pitch attitude, then apply enough back pressure to keep it from dropping farther. Then, as the airplane gradually returns to its trim speed, you will need progressively less pressure.

Similar logic applies to making long-term changes in angle of attack. Use the yoke to change the pitch attitude. At first very little pressure will be required to maintain the new pitch attitude. Then, as the airspeed changes, use pressure on the yoke to keep the attitude where you want it. Make the change permanent by using the trim wheel to trim off the applied pressure.

Let's see how these ideas apply to a typical maneuver: levelling off from a climb. Initially let's suppose you start out nicely trimmed, climbing at 475 feet per minute at 90 knots true airspeed.9 As discussed in section 2.11, that means your direction of flight is 3 degrees above the horizon. As shown in figure 2.5, the first step in the level-off is to change your direction of flight so it becomes horizontal. During the brief time that the direction of flight is changing, the aircraft will be out of equilibrium; lift will be less than weight. The load on the aircraft and its occupants will be slightly less than one G.

level-off
Figure 2.5: Level-Off Maneuver

As the direction of flight changes, you will need to lower the nose the same amount (three degrees). At this point, since the direction of flight and the pitch attitude have changed together, the angle of attack is (for the moment) the same as it was during the climb. This can be seen by comparing the top two parts of figure 2.6. The airspeed is still 90 knots, which is the trim speed, so no yoke force will be needed to maintain the new attitude (for the moment). So far so good.
aoa-change
Figure 2.6: Angle of Attack during Level-Off

Now we need to think about the energy situation (as discussed in section 1.3.1). Since the airplane is no longer climbing, the engine power that had previously been devoted to increasing the altitude is now being devoted to increasing the airspeed. As the airplane gradually accelerates from climb speed to cruise speed, the direction of flight remains horizontal, so the pitch attitude will gradually decrease as the angle of attack decreases. This is shown in the bottom part of figure 2.6. You will need to apply progressively more forward pressure. You can trim off this pressure all at once, or trim it off in stages during the acceleration. 10
Eventually, the airplane will reach cruise speed. At this point, the airplane has all the altitude (potential energy) and airspeed (kinetic energy) that it needs, so you should throttle back to cruise power. Now11 you make your final trim adjustment and the level-off maneuver is complete.

Here is a useful trick: make a note of how much trim change is required in your favorite airplane to make the transition from climb to cruise. It is some definite amount, and remembering this amount obviates a lot of guessing and fiddling. I remember the amounts in terms of ``sectors'' and ``bumps''. That is, on most airplanes only a certain sector of the trim wheel is exposed, and this defines how much trim change can be achieved with a single hand motion; I call this one sector. Similarly, the trim wheel typically features a series of bumps, to make it easier to grasp. Each bump represents 1/4th or 1/5th of a sector.

Suppose after cruising in level flight for a while, you decide to climb to a higher altitude. If you roll in three sectors of nose-up trim as you start the climb, you can bet that you will need roll those three sectors back out to return to cruise airspeed afterwards.12

Similarly, suppose you are cruising along and encounter an updraft. If you roll in half a bump of nose-down trim to help you maintain altitude, you can bet that you will need to roll that half-bump back out when you exit the updraft and return to normal airspeed. Keep track of the amount! Say to yourself, ``I'm carrying a half-bump of nose-down trim which I'll have to get rid of sooner or later''.

2.7   Fly with a Light Touch

Here's some really important advice: Fly with a light touch. You need to feel if you are pulling the airplane off its trim speed.

Some airplanes have such heavy control forces that it's hard to imagine anyone accidentally pulling the plane off its trim speed. You need to trim it properly lest you wear yourself out trying to hold the yoke. For some maneuvers (such as the landing flare) you need to apply a significant amount of force — but please be aware of what you are doing.

Some other planes have such light control forces that if you have the proverbial death grip on the yoke, you could pull the airplane ten knots away from its trim speed without feeling it.

I once flew with a pilot who held the yoke so tightly that his knuckles turned white, literally. Every time he looked to the right, the airplane pitched down 10 or 15 degrees. Every time he looked to the left, the airplane pitched up 10 or 15 degrees. It's a good thing he didn't look to the left very long; otherwise we might have stalled.

For any plane, C-152 or Airbus, if you trim it properly you will be able to fly most maneuvers using just your thumb and one or two fingertips.

The yoke is not just a control carrying commands from you to the airplane — it is also a valuable sensor carrying information from the airplane to you. This is discussed in more detail in section 12.12.

You should make sure the airplane is at all times trimmed for the right airspeed (or, rather, angle of attack). You should be aware of (and wary of) any force you apply to the yoke, forcing the airplane off its trim speed.

Fly with a light touch!

2.8   Trim Won't Solve All The World's Problems

Although the airplane's tendency to return to its trimmed angle of attack is very powerful, very important, and usually very helpful, there is more to the story.

If the airplane is disturbed from its trimmed angle of attack, it will not just return; it will overshoot. It will oscillate a few times before settling down. These phugoid oscillations are slow enough that you can easily extinguish them by timely pressure on the yoke, as discussed in section 6.1.12.

In smooth air, you can trim the airplane and let it fly itself. However, turbulent air will frequently provoke new phugoid oscillations so you will frequently need to apply small nudges to the yoke.

For similar reasons, it is not normal procedure to use the trim wheel to initiate a change in pitch attitude, airspeed, or angle of attack. That would just provoke an oscillation. Initiate the change with the yoke as described above. Put the pitch attitude where it belongs, keep it there with the yoke, and then trim off the pressure.

Finally, in some airplanes the trim speed is perturbed when you add power, when you extend flaps, and especially when you have power and flaps at the same time. See section 5.5 and section 12.10.

2.9   Pitch Attitude versus Angle of Attack

The previous sections pointed out that while pitch attitude and angle of attack are related, they are not quite the same. Pitch attitude is measured relative to the horizon, but angle of attack involves the direction of the relative wind. In any situation where the relative wind is not horizontal, we have to be careful.

I forgot the distinction once; let me tell you the story. One summer I spent several weeks at the Aspen Center for Physics. This was my first opportunity to do any mountain flying, so I arranged for a lesson from the flight school at Aspen. The lesson included flying over the continental divide and landing at Leadville. Leadville is famous for being the highest airport in the United States — 9900 feet above sea level. On the day in question, it was about 90F in the shade, so the density altitude at Leadville was around 13,000 feet, and I knew takeoff performance would be critical.
I used my best short-field procedure, even though the runway was 5000 feet long. I accelerated on the runway to the proper climb-out speed (75 knots indicated, 90 knots true) and then rotated to what I assumed was the correct climb-out attitude. Based on my experience at lowland airports, I knew that 11 degrees of nose-up attitude was usually just right for climb out. Following my usual habit, I scanned the airspeed indicator after we had climbed a few feet. To my horror, I observed that the airspeed was decreasing rapidly. I immediately lowered the nose, and flew the airplane in ground effect while it regained speed. (What had been intended as a short-field procedure ended with a peculiar imitation of soft-field procedure.) I used up almost the entire runway getting back to 75 knots. At 75 knots I rotated again, choosing a much lower pitch attitude this time. We climbed out at 75 KIAS and the rest of the lesson was relatively uneventful.

This situation is depicted in the following figures and table. Figure 2.7 shows the normal takeoff procedure at a low-altitude airport. Figure 2.8 shows that using the normal pitch attitude does not produce the normal angle of attack at a high-altitude airport, because the angle of climb is an indispensable part of the equation. Figure 2.9 shows how to do it right. Table 2.2 summarizes the arithmetic.

takeoff-low
Figure 2.7: Climbing Out From Lowland International

takeoff-wrong
Figure 2.8: Climbing Out From Leadville (Wrong)

takeoff-right
Figure 2.9: Climbing Out From Leadville (Right)

Calib. Airspeed
Pitch Attitude
Incidence
Climb Rate @ True Airspeed
Angle of Climb
Angle of Attack
sea level
76 KCAS 11.0  4.5
900 fpm @ 76 KTAS
  7  8.5
Leadville (wrong)
dropping rapidly
11.0  4.5
200 fpm @ 90 KTAS
  1 14.5
Leadville (right)
76 KCAS  5.0  4.5
200 fpm @ 90 KTAS
  1  8.5
Table 2.2: Right versus wrong climb attitude
Understanding what went wrong in this scenario is very instructive. The main difference between a sea-level takeoff and a mountain takeoff is that the airplane does not climb nearly so steeply. The direction of flight is much more nearly horizontal. As can be seen by comparing figure 2.8 with figure 2.9, this means a much lower pitch attitude is needed to achieve the same angle of attack.

The really embarrassing part of my story is that I had actually calculated the climb gradient as part of my preflight preparation, to make sure I could clear obstructions. I just didn't make the connection between the climb gradient (which I calculated) the best-climb angle of attack (which I knew) and the pitch attitude (which I used for controlling the airplane). Fortunately I did know the connection between airspeed and angle of attack, and I scanned the airspeed indicator before the situation got too far out of hand.

2.10   Power plus Attitude does not equal Performance

You may have heard the assertion that ``Power plus Attitude equals Performance''. Well, that assertion is not quite right, and has caused all sorts of unnecessary confusion.

Consider the following scenario: You are cruising along in a typical 180 horsepower, one-ton aircraft. You have constant power and constant attitude, so you expect constant performance. You do indeed get constant performance, and everything seems just fine.

Now, just raise the nose to a 15 degree nose-up attitude, and hold that attitude as accurately as you can. You will once again have constant power and constant attitude, so you might expect constant performance — but that is definitely not what you will get. Instead, you will get decreasing airspeed and increasing angle of attack. The initial climb that looked so promising will peter out and you will wind up on the edge of a stall.

If you think about this situation in terms of energy and angle of attack, the airplane's behavior is completely predictable.

First of all, we need to remember that not all climbs are steady climbs. As portrayed in figure 2.10, it is possible for a roller-coaster with no engine at all to zoom up a little ways by cashing in its initial kinetic energy. Just because it starts out on a certain climb trajectory doesn't mean it can continue.

aoa-roller
Figure 2.10: Climb Powered by Speed

Airplanes, too, can be placed on climb trajectories that cannot be sustained by the available engine power. The initial climb succeeds only because airspeed is being cashed in to purchase altitude.

Unlike a roller-coaster, the airplane will not stay on its initial trajectory until it runs out of speed altogether. As the airspeed decays, the airplane will have to fly at a higher angle of attack in order to support its weight. Since, as discussed above, the angle of attack depends on the angle between the pitch attitude and the direction of flight, a constant attitude implies a non-constant direction of flight, as indicated in figure 2.11.

chandelle
Figure 2.11: Constant Power & Attitude but Changing Performance

If you are lucky, the changing flight path will result in a trajectory where the rate of climb and the drag budget can be sustained by engine power, with no further decrease of airspeed; otherwise the maneuver will end in a stall.

One of the maneuvers you have to perform in order to get a commercial pilot certificate is called a chandelle. As discussed in section 16.11, it involves turning as well as climbing, but if you disregard the turning part, the maneuver is exactly what is portrayed in figure 2.11. This maneuver is an important part of the syllabus because it forces people to learn that constant power and constant attitude do not imply constant performance.

As discussed in section 2.6, a pitch attitude excursion is not necessarily the same as an angle of attack excursion. Suppose due to turbulence or whatever, the pitch attitude and direction of flight both increase by 15 degrees. If you correct the situation promptly, the airspeed and altitude will not have time to change much. If on the other hand you allow the pitch excursion to persist, the airplane will begin to follow the chandelle trajectory shown in figure 2.11. The altitude will increase (at least at first), the airspeed will decrease, and the angle of attack will increase. It is good pilot technique to correct pitch attitude excursions before they turn into altitude / airspeed / angle of attack excursions.

To summarize: the Leadville scenario and the chandelle scenario prove that angle of attack is far more important than pitch attitude in determining performance. But this does not mean you disregard pitch attitude — far from it. I recommend that you use pitch attitude as a means of controlling angle of attack — just don't use pitch attitude as a substitute for controlling angle of attack.

2.11   Estimating the Relative Wind

As discussed above, to control the angle of attack you need to know both the pitch attitude and the direction of flight.13 I have given several methods for estimating the pitch attitude. Now it is time to explain how to estimate the direction of the relative wind. This is almost the same thing as estimating direction of flight.

In level flight, the task is easy: The relative wind is coming at you horizontally. (Again, I am assuming there are no major updrafts or downdrafts.)

If the airplane is climbing or descending, the origin of the relative wind will be above or below the horizon, respectively. The amount above or below depends on the ratio of your vertical speed to your airspeed. I have committed some of the numbers to memory; for instance, I know that flying a standard 3 degree glideslope at 90 knots involves a 480 fpm descent. Using the same little fact in reverse tells me that if I am climbing out at 90 knots and the vertical speed indicator (VSI) is reporting 480 fpm, I must be flying toward a point 3 degrees above the horizon; to say it the other way, the relative wind must be coming toward me from that point, three degrees above the horizon. That means that I can relabel the VSI as a ``direction of flight'' indicator, as shown in figure 2.12. Any particular relabeling is only valid for one airspeed.14

perceive-direction-vsi
Figure 2.12: Vertical, Horizontal Speed Gauges Determine Angle

If you maintain 90 knots and transition from level flight to a 480 fpm climb, you will have to raise the pitch attitude 3 degrees in order to maintain the same angle of attack.15

If you want to know the vertical speed that corresponds to some other angle and/or some other horizontal speed, you can refer to table 2.3; a similar table appears in every ``instrument approach procedures'' booklet published by the US government. The inverse table (finding the angle, given horizontal and vertical speeds) is shown in table 2.4.

Angle
  Horizontal speed / knots
60 75 90 105 120
3 320 400 480 555 635
4 425 530 635 745 850
5 530 665 795 930 1065
6 640 800 960 1120 1275
7 745 935 1120 1305 1490
8 855 1065 1280 1495 1710
Table 2.3: Vertical Speed vs. Angle and Horizontal Speed

Vertical Speed
fpm
Horizontal speed / knots
60  75  90  105  120 
250 2.4 1.9 1.6 1.3 1.2
500 4.7 3.8 3.1 2.7 2.4
750 7.0 5.6 4.7 4.0 3.5
1000 9.3 7.5 6.3 5.4 4.7
Table 2.4: Angle vs. Vertical Speed and Horizontal Speed
The VSI is not the only way of determining the direction of flight. If you are established on an ILS approach, as long as the glideslope needle stays centered you are descending at a known angle (usually three degrees). Similarly, there might be a VASI or other approach slope indicator that you could follow. As always, it is better to use outside references instead of instruments.

Perhaps the best way to judge the angle of descent is to use the ``rule of thumb'' as discussed in section 12.3. That frees you from relying on any fancy equipment.

If you control the direction of flight using any of these techniques, and control the pitch attitude using the techniques discussed elsewhere in this chapter, then you are also controlling the angle of attack.

Actually, there is one more ingredient in this recipe: the wind. Three of the methods just mentioned (VASI, electronic glideslope, and rule of thumb) give you information about your direction of flight relative to the ground, but the angle of attack depends on your direction of flight through the air. In the presence of wind, the two are not quite the same. This is discussed in section 12.4.3.

The scheme of estimating the direction of flight using the VSI gives the correct answer even when nature's wind is blowing (provided, again, there are no major updrafts or downdrafts).

Outside references should be your primary means of controlling angle of attack. Every so often you should look at the airspeed indicator to make sure you have got the right angle of attack (as discussed in section 2.12), but you should maintain that angle of attack by outside references.

Suggestion: one look out of ten, look at the instruments; nine looks out of ten, look at the outside references.

2.12   Airspeed Is Related to Angle of Attack

2.12.1   Airspeed versus Coefficient of Lift

So far in this chapter I have mentioned that the critical performance numbers usually specified by airspeeds such as VY are really angle of attack recommendations.

Therefore you are probably beginning to suspect that there might be a relationship between angle of attack and airspeed. That's right! The purpose of this section is to tell you why you can use the airspeed indicator to control angle of attack, when you have to compensate for its imperfections, and when you can't trust it at all.

The basic line of reasoning is this: the amount of lift produced by the wing depends on angle of attack and calibrated airspeed. We can turn this around to get a simple relationship between airspeed and angle of attack (assuming lift is known, as it usually is). The key formula is
lift = ½ rV2 coefficient of lift area              (2.1)
The coefficient of lift will be discussed below, and (in more detail) in section 4.4. The quantity ½ rV2 is called the dynamic pressure, also called Q for short, but more often than not people just call it one-half rho vee squared.

You don't need to calculate ½ rV2 because your airspeed indicator does it for you. You may have thought that an airspeed indicator would ideally measure the true airspeed (which is simply the genuine speed of the air relative to the aircraft, denoted V in all the formulas). However, the airspeed indicator doesn't even try to measure V (i.e. the square root of V2); instead it tries to measure something called calibrated airspeed (CAS), which is proportional to the square root of ½ rV2. Note the factor of r in the CAS formula.16 While we're on the subject, indicated airspeed (IAS) refers to whatever is indicated on your airspeed indicator. It is the same as calibrated airspeed, plus whatever errors there are in the mechanism. This discussion assumes that your instrument is not too wildly inaccurate, so that formulas that apply to CAS exactly also apply to IAS accurately enough for present purposes.17

In flight, the lift is nearly always equal to the weight times load factor.18 The weight is presumably not changing much from moment to moment. This leads us to rearrange the lift equation as follows:
coefficient of lift = (weight load factor) / ( ½ rV2 area)              (2.2)
If the airspeed goes down, the coefficient of lift must go up. This relationship is illustrated in figure 2.13.

ias-cl
Figure 2.13: Airspeed versus Coefficient of Lift

Three of the critical V-numbers are marked in figure 2.13; each corresponds to a particular coefficient of lift.

2.12.2   Coefficient of Lift versus Angle of Attack

Now we bring in a new fact: The coefficient of lift is a simple function of the angle of attack. This dependence is shown in figure 2.14. Note that for small angles of attack, the coefficient of lift is essentially proportional to the angle of attack. The angle of attack that gives the maximum coefficient of lift is called the ``critical angle of attack'' and is marked in the figure.

cl-aoa
Figure 2.14: Coefficient of Lift versus Angle of Attack

ias-cl-aoa
Figure 2.15: Airspeed Is Related to Angle of Attack

By combining this fact with what we already know, we can establish the relationship between angle of attack and indicated airspeed. We combine figure 2.13 with figure 2.14, as is done in figure 2.15. We see that a particular V-number, such as VNE, corresponds to a particular coefficient of lift, which in turn corresponds to a certain angle of attack. The same goes for most of the other V-numbers, such as VY. The argument works in reverse, too: any particular angle of attack corresponds to a particular airspeed (assuming we know how much lift is being produced).

We conclude that the airspeed indicator is really a pretty good angle of attack indicator — with one major exception.

Here's the exception: There is a whole range of angles of attack near the critical angle of attack that all produce about the same coefficient of lift (because the coefficient of lift versus angle of attack curve is quite flat on top). All the coefficient-of-lift values in this small range correspond to nearly the same airspeed — namely VS, the stalling airspeed.

The stall is a very critical flight regime. This is a regime where you would very much like to have an accurate instrument to indicate angle of attack, and alas it is the one regime where the airspeed indicator doesn't tell you anything you need to know.

You want to land the airplane at a very high angle of attack. You will have to perceive the angle of attack using outside visual cues, as discussed in the previous sections. During the flare, the airspeed indicator doesn't tell you anything you need to know. I once asked an airline captain to tell me at what airspeed his airliner touched down. He said ``I don't know; I never looked. I've always had more important things to look at''. That was a good pilot's honest answer.

2.12.3   Correcting for Reduced Density

In all non-stalling regimes of flight, including (especially) final approach, the airspeed indicator provides your most quantitative information about angle of attack. We now discuss some corrections that may be needed.

The airspeed indicator is basically a pressure gauge; the pressure that moves the airspeed needle is the same dynamic pressure that holds up the wings in accordance with the lift formula (equation 2.1). Knowing the pressure that holds up the wing is more important than knowing your true airspeed.

The airspeed indicator is doing you a favor by not measuring speed per se. For instance, on final approach you should fly the proper indicated airspeed. At high density altitudes this will be a higher-than-normal true airspeed. Remember that calibrated airspeed is what holds up your wings.

In other words: do not correct VX, VY, VS, or approach speed (1.3 VS0) for density altitude. Trust the calibrated airspeed.

The true airspeed that corresponds to this calibrated airspeed will be higher, by about 2% per thousand feet of density altitude. Your groundspeed will also be greater.

When landing at a high-altitude airport, the greater groundspeed means you will need more runway length, by about 4% per thousand feet of density altitude. Check the charts in your POH.

A high-altitude takeoff is even worse than the landing, because the engine (unless turbocharged) will be producing less power. Check the charts in your POH. Apply a generous safety factor, since many of the handbooks are disgracefully overoptimistic. Do the takeoff planning (not just the landing planning) before you land, lest you go into an airport you can't get out of.

2.12.4   Correcting for Reduced Lift Requirements

So far we have been assuming the weight was equal to some standard value. Let's relax that assumption and see what happens.

As mentioned in section 7.5.7, it is easy to imagine flying a Cherokee Six at half of its maximum legal weight.

The problem is that the Pilot's Operating Handbook for the airplane specifies all the critical angle of attack information in terms of speeds — speeds that only apply at max weight. We know that the airplane stalls at a definite angle of attack, not at a definite airspeed or anything else.

In general, if you keep the angle of attack constant and lower the weight of the airplane by 10%, the airspeed needed to support that weight goes down by 5%. This is because the lift depends on the square of the airspeed in equation 2.1; the square root of 0.90 is 0.95 and the square root of 1.10 is 1.05. For really large changes in weight, the speed change is even somewhat greater; the square root of 0.50 is not 0.75 but rather 0.707.

At reduced weights, your approach speed and all the other critical performance speeds must be reduced below their standard-weight handbook values.

The percentage change in speed
is half of the percentage change in weight.

Since the cruise speed depends on power and hardly depends on angle of attack, it does not decrease as the weight is decreased; the situation is depicted in figure 7.13 in section 7.5.7. The maneuvering speed does decrease as weight is decreased, as discussed in section 2.13.2.

2.12.5   Correcting for Increased Lift Requirements

There is one fairly common situation where maintaining a given angle of attack requires flying at airspeeds above the V-numbers given in the Pilot's Operating Handbook.

In a steep turn, the wings are required to produce enough lift not only to support the airplane's weight, but also to shove it around the corner. In a 60 degree bank, the lift requirement is doubled. We say there is a load factor of 2.0. The airspeed necessary to produce this lift at a given angle of attack is increased by a factor of 2, which is 1.41.

If you are going to use the airspeed indicator as a source of angle of attack information, you have to take this into account. If you fly at a speed near the bottom of the green arc in a steep turn, the airplane will stall. For example, if the airplane stalls at 60 knots in unaccelerated flight, it will stall at 85 knots in a 60 degree banked turn (since 60 1.41 = 85).

Also remember that the airplane is trimmed for a definite angle of attack, and it really wants to maintain that angle of attack. If you are cruising along, trimmed for 120 knots in straight and level flight, and the airplane gets into a 60 degree bank, it will accelerate to 169 knots (120 times the square root of 2) in order to meet the increased lift requirement at the same angle of attack. This situation is described in more detail in section 6.2.

2.12.6   Compute with Calibrated not Indicated Airspeed

In a wide range of airplanes, it turns out that a good airspeed for final approach is 1.3 times the stalling speed.19

When applying this rule, a little sophistication is necessary, or you might get into trouble. In particular, you must not just look at the indicated stalling speed on the airspeed indicator, multiply by 1.3, and then try to use the result as your indicated airspeed on final.

The only safe way to calculate the approach speed is to multiply the calibrated stalling speed by 1.3, and then convert the result an indicated airspeed. That is, if you know the indicated stalling speed, the correct procedure is:

a)
convert the indicated speed to a calibrated speed, using the conversion information in the Pilot's Operating Handbook;
b)
multiply the calibrated speed by 1.3; and
c)
convert this calibrated approach speed back to an indicated airspeed you can use in the cockpit.
Table 2.5 shows an example which contrasts the right and wrong calculations.

CAS IAS
safe
approach
speed?
stall 50 43  
1.3 indicated stall 58 56 no
1.3 calibrated stall 65 65 yes
Table 2.5: Calibrated versus Indicated Approach Speed
The origin of the problem is this: It is possible to position the Pitot tube and static port so that the IAS is a few knots higher than the CAS in cruise conditions, yet a few knots lower than the CAS near the stall. Manufacturers commonly do this, presumably in hopes of making pilots think the airplane performs better than it really does.

These errors would not be much of a problem if the the IAS were simply proportional to the CAS. The constant of proportionality would drop out of the calculation, and you could skip steps (a) and (c). Alas, in many airplanes the errors are highly nonlinear. The indicated airspeeds are much too low at the low end of the scale. If you multiply such a low number by 1.3, you get a number that is still much too low, but falls at a place where the gauge is more accurate, so you wind up with a real airspeed that is dangerously low.

You may we wondering about other calculations, such as the corrections for nonstandard weight. Should calculations also be done using calibrated airspeed? The answer, alas, is not 100% obvious. It depends on whether you think the errors in the system depend on airspeed itself, or depend on angle of attack. You might want to measure your airplane, as follows: Fly it at its maximum weight, at a safe altitude, and observe the indicated airspeed at which the stall warning horn comes on. Do this in the clean configuration and in the landing configuration. Then repeat the measurements at the lowest convenient weight. Then you will know for sure how the indicated airspeed varies with weight, at particular angles of attack.

2.12.7   Correcting for Slip

It is easy to get into situations where the indicated airspeed is wildly inaccurate. In some airplanes the opening that is supposed to measure the static pressure is located on the side of the fuselage, and during a slip that point is subject to some dynamic pressure in addition to the static pressure.20

In such a case you must remember that it is angle of attack that really matters. You can use the airspeed indicator if you wish before the slip to help figure out what angle of attack you want, but during the slip you must maintain that angle of attack by looking at the angles themselves (pitch angle and direction of flight). See section 11.2 for more on this.

2.12.8   Drag and Lift-to-Drag Ratio

Let's return to the scenario of the airplane flying at half of its standard weight, and ask (a) what is the best glide speed, and (b) how well will it glide at that speed.

To answer these questions we need to think about drag as well as lift. (Section 2.12.4 concentrated on topics like VS and VA which depend on total lift, not lift-to-drag ratio.) Fortunately, the answer comes out the same. This is because the formula for drag,
drag = ½ rV2 coefficient of drag area              (2.3)
has the same form as the famous formula for lift:
lift = ½ rV2 coefficient of lift area              (2.4)
The key idea is that the coefficient of drag depends on angle of attack; at any particular angle of attack the coefficient does not perceptibly depend on weight or airspeed. The same is true of the coefficient of lift and the lift-to-drag ratio.

If you want to glide from point A to point B in no-wind conditions,21 the main thing you care about is lift-to-drag ratio. For example, if your airplane is capable of a 10-to-1 lift-to-drag ratio, then you can glide to a point that is 1/10th of a radian (i.e. six degrees) below the horizon.

The optimal lift-to-drag ratio is achieved at a definite angle of attack. To support the weight of the airplane at that angle of attack, you will need to fly at a speed proportional to the square root of the weight, for the reasons given in section 2.12.4.

The lightly-loaded gliding airplane will have the same angle of descent, the same direction of flight, and the same total gliding distance, as indicated in figure 2.16. The only difference is that it will have a slower descent rate and a slower forward speed; this is indicated in the figure by stopwatches that show how long it takes the plane to reach a particular point.

weighted-glider
Figure 2.16: Angle of Glide Independent of Weight

The moral of the story is if you are flying a lightly-loaded airplane, you should fly it ``by the numbers'', namely the angle of attack numbers. The critical airspeed numbers (climb speed, approach speed, stalling speed, etc.) are all reduced according to half the weight-change percentage. That is, if you are 10% light, reduce the handbook speeds by 5%.

There is one well-known exception to the rule of thumb that says important performance speeds decrease as the weight decreases. That is, the cruising speed actually increases at reduced weights. This is not an exception to the real rule that speeds should vary with weight at a given angle of attack, because cruising speed is not tied to a particular angle of attack. If the airplane is lightly loaded, you can cruise at a lower angle of attack and a higher airspeed, since the wings need to do less work to support the weight of airplane.

2.13   Not Everything Depends on Angle of Attack

Some of the airplane's critical performance numbers depend directly on angle of attack, while others don't. It's somewhat useful to know which are which, so you can know which ones change with the weight of the airplane and which ones don't.

2.13.1   Explicit Airspeed Limits

There is a normal-operations airspeed, VNO. This is indicated by the top of the green arc on the airspeed indicator. You should not exceed this speed except in smooth air, and then only with caution. The idea here is that you don't want to break the wing. There is a maximum coefficient of lift, and the lift force depends on this coefficient times calibrated airspeed squared. By limiting the airspeed, you limit the maximum force that the wing can produce. This is typically what determines VNO.

There is also a never-exceed airspeed, VNE. This is indicated by the top of the yellow arc, and by a red radial line on the airspeed indicator. As the name suggests, you should never exceed this speed under any circumstances. This limit depends on many things, including drag force on the primary structure (wings, tail, landing gear etc.); drag force on secondary items (antennas, fairings, etc.); instability of the structure and control systems due to flutter; and other nasty complications.

2.13.2   Maneuvering Speed

If you are flying in moderate or severe turbulence, you should keep your airspeed below the maneuvering airspeed, VA. By the same token, you should avoid large, sudden deflections of the controls unless your airspeed is below VA. The idea behind VA is that you want the wing to stall before anything breaks. You may think that a stall is bad, but remember that you can recover from a stall much more easily than you can recover from a broken airplane.

Maneuvering speed means the wing is supposed to stall
before it produces enough Gs to break any part of the airplane.

We say it is supposed to stall, not guaranteed to stall, because the formal definition of VA takes into account only certain types of rough control usage, and only certain types of turbulence (namely purely vertical updrafts and downdrafts). In real life, other possibilities must be considered. For instance, if you start out at VA and encounter a large horizontal wind shear, arbitrarily large forces can be developed. For this and several other reasons, the exact value of VA should not be taken too literally.

Still, the general idea of VA makes sense: If you observe or anticipate a situation that imposes large G loads on the airplane, you should slow down and/or confine yourself to gentler maneuvers.

Unlike VNO, the maneuvering speed varies in proportion to the square root of the mass of the airplane. The reason for this is a bit tricky. The trick is that VA is not a force limit but rather an acceleration limit. When the manufacturers determine a value for VA, they are not worried about breaking the wing, but are worried about breaking other important parts of the airplane, such as the engine mounts. These items don't directly care how much force the wing is producing; they just care about the acceleration they are undergoing.

By increasing the mass of the airplane, you decrease the overall acceleration that results from any overall force. (Of course, if you increase the mass of cargo, it increases the stress on the cargo-compartment floor — but it decreases the stress on unrelated components such as engine mounts, because the acceleration is less.)

This means you should put VA along with VS and VY etc. on your list of critical airspeeds that vary in proportion to the square root of the mass of the airplane. However, VA depends on real mass not on weight, so unlike the others it does not increase with load factor.

To illustrate this point, consider what happens when the airplane is in a steep turn. Compared to unaccelerated flight:

  1. The stalling speed increases (because the stalling angle of attack stays the same), and
  2. the airspeed for best rate of climb increases (because the optimum angle of attack stays the same), but
  3. the maneuvering speed remains the same (since it doesn't directly depend on angle of attack).
Finally, we should note that there are two different concepts that, loosely speaking, are called maneuvering speeds.

This is a book for pilots, not designers, so when we use VA it always means VA(L). But you should be careful when reading the FARs and other books, because they sometimes use the same symbol to mean two different things, which makes it very hard to think clearly.

2.13.3   Overview of Limits and Performance Numbers

We see that there are four main classes of numbers:

2.14   Relative versus Absolute Angle of Attack

You can skip this section unless you are trying to compare this book with another book that uses a different definition of ``the'' angle of attack.

As mentioned in connection with figure 2.1, we are free to choose how the angle-of-attack reference stick is aligned relative to the rest of the wing. Throughout this book, we choose to align it so that zero angle of attack corresponds to zero coefficient of lift. According to the standard terminology, the angle measured relative to the zero-lift direction is called the relative angle of attack.

Some other books try to align the reference with the chord line22 of the wing. According to the standard terminology, the angle measured relative to chord line is called the absolute angle of attack.

If you try to compare books, there is potential for confusion, because this book uses ``angle of attack'' as shorthand for relative angle of attack, while some other books use the same term as shorthand for absolute angle of attack. To make sense when comparing books, you must do away with all the shorthand and use the fully explicit terms. In particular:
relative angle of attack = absolute angle of attack + k
absolute angle of attack = relative angle of attack - k
             (2.5)
where -k is the X-intercept of the graph of coefficient of lift versus absolute angle of attack, when graphed using whatever reference the other book is using to define absolute angle of attack. The X-intercept is always zero in this book.

Note that the names ``absolute'' versus ``relative'' don't mean anything. Neither quantity is more absolute or more relative than the other.

Also note that are not just two possibilities; the choice of reference is really quite arbitrary. It is perfectly valid to measure angles relative to the bottom surface of the wing, or the long axis of the fuselage, or any other reference you choose, provided you are consistent about it.

Using the chord as a reference works OK if you are only talking about one section of a plain wing. On the other hand: The simple rule ``pitch plus incidence equals angle of attack plus angle of incidence'' (figure 2.2) is always technically valid (because the arbitrariness in the angle of incidence cancels the arbitrariness in the angle of attack). But if you want the rule to be useful in situations where flap settings are changing, you need to choose relative angle of attack or something very similar; otherwise you don't know what angle of attack produces a given amount of lift.

2.15   Summary


1
Wind may affect what angle of attack gives the best angle of climb or best angle of glide; see section 7.5.6.
2
Changes in engine performance may slightly affect what angle of attack gives the best climb rate or best climb angle.
3
The relative wind is defined to be the speed and direction that the air is moving relative to the airplane. (It is very, very different from the velocity of the wind relative to the ground.) Unless otherwise specified, the relative wind is measured at a place where the airmass has not been greatly disturbed by the passage of the airplane. See section 2.11 for more details.
4
See section 19.6.2 for a more formal definition.
5
The effect of flaps is discussed in more detail in section 5.5. See also the table in section 12.11.
6
I am imagining a day without appreciable updrafts and downdrafts, so that the relative wind is horizontal; otherwise the story gets a little more complicated.
7
adjusting power if necessary to maintain level flight
8
Most people are right-eye dominant. If neither eye is strongly dominant, you can choose one arbitrarily and close the other when checking the sight line.
9
The meaning of true (versus calibrated) airspeed will be discussed in section 2.12.
10
A ``clean'' airplane (compared to a draggy one) requires more stages of trimming, since it accelerates to a higher speed, and this acceleration takes longer.
11
Not sooner, since on most airplanes, power changes slightly affect trim speed.
12
Maybe not exactly, because your indicated airspeed at the new cruise altitude may be slightly different. Apply the expected amount of trim, see how it works, and then trim off any remaining yoke force.
13
Incidence is important too, but you rarely need to worry about it. It only changes when you are changing the flap setting.
14
I am glossing over the distinction between horizontal speed and total speed. The tables tabulate the tangent and arctangent, based on true horizontal and vertical motion. In a really steep dive, the airspeed indicator would indicate the total motion, which is the resultant of the horizontal and vertical components. If you really wanted to calibrate the angles against the VSI and the (total) airspeed, you should use the sine and arcsine. However, at any halfway reasonable angle, the difference is less than a percent or two. Don't worry about it.
15
See section 12.4.3, including figure 12.13, for an analogous discussion of angles relative to the ground.
16
The constant of proportionality is arranged so that at sea level, in standard conditions, the calibrated airspeed is identical with the true airspeed. Since you are almost always flying at altitudes above sea level, your true airspeed will almost always be larger than your calibrated airspeed.
17
... but beware: as discussed in section 2.12.6, there are cases where it is quite important to keep track of the distinction between IAS and CAS.
18
Load factor is defined in section 6.2.3. You don't need to worry about it except during aggressive maneuvers.
19
See section 12.11.3 for more on this and related topics.
20
In other airplanes the static pressure is measured on a mast, far from the side of the fuselage, so this problem does not occur.
21
If you want to maximize gliding time instead of distance, or if you want to account for tailwinds and/or updrafts, see section 7.5.
22
The chord line is the straight line drawn from the leading edge to the trailing edge, as defined in figure 3.12.

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