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:
This is not an isolated example. Many of the airplane’s critical performance numbers are really angle of attack numbers:
Here is a summary of the main ideas that will be explained in this chapter:
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 an angle of attack indicator on their first airplane. It 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.4 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.
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 2.15 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.
Now let’s investigate each of the items in this outline.
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 on 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.
Conclusion, valid when load factor = 1:
You don’t need to worry about load factor except during steep turns and suchlike, so usually you just trim for airspeed. More generally, you trim for angle of attack. Section 2.6 discusses making changes in angle of attack.
Conclusion, valid always:
Do not trim for pitch attitude. Do not trim for rate of climb. Trim for airspeed at 1 G. Trim for 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:
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.
Extending the flaps has the effect of increasing the incidence6 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 2.13.
Airspeed (KCAS) 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
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.7
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 11.7.2. 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 11.6.
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 attack8 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.
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 dominant9 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. Given some definite amount of change, it will manifest itself as a small-percentage change in something that was large, or as a large-percentage change in something that was small to begin with. The latter is usually easier to perceive.
One advantage of using the cowling as a reference is permanence. That is, the cowling is in the same place on all airplanes of that make and model, whereas marks on the windshield will not. A second advantage is that it is less awkward than using your finger. The big disavantage of using the cowling crops up during turns, as discussed in section 11.7.2.
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).
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. Also note that the view out the front depends on various factors, such as whether your seat is adjusted high or adjusted low ... whereas the view out the side gives a less variable, more reliable perception of pitch angle.
This is worth practicing. Since you will be emphasizing the side view, take along an instructor or at least a trusted safety pilot who can check for traffic etc. by looking out the other side and looking out the front. Trim the airplane for level flight, and practice maintaining straight and level flight solely (or at least mostly) by reference to the side view. Once every minute or so, peek at the altimeter and the heading indicator to see how you are doing.Once you get the hang of straight and level flight, practice some vertical-S maneuvers. That is, transition from level flight to a 500 fpm climb. Climb for one minute, then level off ... all by reference to the side view. Similarly practice making 30 degree turns, left and right, all by reference to the side view.
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.
Note: Most of the time, you are primarily concerned with changes in the pitch angle. That is, you don’t usually need to know that the pitch angle is 1.234 degrees, or any other specific value. If you wanted to really quantify the pitch angle, you would have to decide whether to measure it relative to the wing, relative to the cowling, or relative to a mark on the windshield, et cetera ... but for practical piloting purposes you don’t need to quantify it. Usually the more important thing is to perceive changes, and any or all of the aforementioned references will work fine for that.
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 2.13.
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 slows down, 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 6.1.14).
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. Decide what pitch attitude you want. Use the yoke to obtain and maintain that 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. Don’t lead with the trim.
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.10 As discussed in section 2.12, 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.
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.
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. (See section 1.3.1.)
As the airplane gradually accelerates from climb speed to cruise speed, the direction of flight remains horizontal, so the pitch attitude gradually decreases as the angle of attack decreases. This is shown in the bottom part of figure 2.6. You need to apply progressively more forward pressure. In a trainer you might trim off this pressure all at once, but in a high-powered airplane you need to re-trim repeatedly, in stages, as the airplane keeps accelerating and accelerating.
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. Maybe the right answer won’t be exactly three sectors, because your indicated airspeed at the new cruise altitude may be slightly different. However, having some idea is better than having no idea! Apply the expected amount of trim, see how it works, and then trim off any slight yoke force that remains.
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”.
Here’s some really important advice: You should at all times be aware of how much force you are putting on the yoke. You don’t want to accidentally pull the airplane off its trim speed.
Usually this is summarized by saying “keep it trimmed, and fly with a light touch”, but “light touch” is a relative concept, and somewhat hard to quantify:
| Some airplanes have such heavy control forces that it’s difficult to imagine anyone accidentally pulling the airplane off its trim speed. You need to trim it properly lest you wear yourself out trying to hold the yoke. | Many planes have such light control forces that if you keep a tight grip on the yoke, you could easily pull the airplane ten knots away from its trim speed without feeling it. |
In all cases, the important thing is to be aware of how much force you’re putting on the yoke.
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 almost any plane, from C-152 to Airbus, if you trim it
properly you will be able to fly most maneuvers using just your thumb
and one or two fingertips. There are some exceptions; for instance the B-24 was notorious for its heavy control forces. However, that just makes trimming even more important. |
There are of course some maneuvers, notably the landing flare, where everything is changing so quickly that it’s not worth re-trimming, and goodly amounts of force may be needed. |
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.
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 6.1.14.
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.
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 90∘F 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.
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.
Understanding what went wrong in this scenario is very instructive. Let’s be clear: Figure 2.7 and figure 2.8 have the same pitch attitude, which is the wrong procedure. Figure 2.7 and figure 2.9 have the same angle of attack, which is the right procedure. 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.
Angle of Attack 76 KCAS 11.0 4.5 7 8.5 11.0 4.5 1 14.5 76 KCAS 5.0 4.5 1 8.5 Table 2.2: Right versus wrong climb attitude
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.
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.
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 16.15, 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. However, 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.
As mentioned in section 1.2.5, the power curve is so important that there is no good way to explain the details all at once. Now that we know about angle of attack, we can use that to better understand the power curve:
In particular, it is best to think of the power curve as two functions: airspeed as a function of angle of attack, and rate of climb as a function of angle of attack.
To say the same thing the other way: Looking at the axes of the conventional power-curve diagram (as in figure 2.12) may tempt you to think of the rate of climb as a function of airspeed, but this is not the smart way to think about things. In particular, you cannot tell the difference between the mushing regime and the stalled regime based on airspeed. Instead, the smart approach is to tell the difference based on angle of attack, as shown by the blue numbers in figure 2.13. Remember that airspeed can be used as an approximate indication of angle of attack, provided the angle of attack is not near or above the critical angle of attack. For more about this approximation, see section 2.13.
Let’s be clear: In normal flight, on the front side of the power curve, it is OK to think of the rate of climb as a function of airspeed. Indeed in most of the mushing regime (not too close to the stall) you can think of the rate of climb as a function of airspeed. However, if you want to understand the power curve as a whole, including the stalled regime, you need to understand the whole thing as a function of angle of attack. Both the airspeed and the rate of climb are determined by the angle of attack.
The point at the boundary between the mushing regime and the stalled regime has a number of special properties. The corresponding angle of attack is called the critical angle of attack. The corresponding airspeed is called the stall speed.
For a discussion of why the stalled regime of the power curve lies below and to the right of this special point, see section 4.5 and especially section 4.5.
Warning about terminology: When pilots talk about “the stall”, sometimes they are talking about this one point (i.e. the critical angle of attack) ... but sometimes they are talking about the stalled regime as a whole (i.e. any angle of attack greater than or equal to critical). For a more-detailed explanation of what a stall is, see section 5.3.
It is not possible to maintain flight at an airspeed below the stalling airspeed, but it is possible to maintain flight at an angle of attack greater than the critical angle of attack. This is discussed in section 5.3.2.
As discussed above, to control the angle of attack you need to know both the pitch attitude and the direction of flight.12 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 2.14. Any particular relabeling is only valid for one airspeed.13
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.14
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 2.13), but you should maintain that angle of attack by outside references.
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.
Specifically:
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 = ½ρV2 × coefficient of lift × area (2.1) |
The coefficient of lift will be discussed below, and (in more detail) in section 4.5. The quantity ½ρV2 is called the dynamic pressure, also called Q for short, but more often than not people just call it one-half rho vee squared.
The quantity ½ρV2 is tremendously important, as discussed in section 2.13.3.
You don’t need to calculate ½ρV2 because your airspeed indicator does it for you. You may have thought that an airspeed indicator would ideally measure the true airspeed (TAS), 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 ½ρV2. Note the factor of ρ in the CAS formula. Numerical values for the TAS/CAS conversion factor, as a function of altitude, can be found in table 7.1.15 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.16
Note: In what follows, we will make use of the weight as observed in the laboratory reference frame, denoted weightlab. This is what would be observed by an engineer standing on the ground, or in a chase-plane that is maintaining unaccelerated flight. This stands in contrast to the weight as observed in a reference frame attached to your aircraft, denoted weightac. This is a departure from the usual practice in this book of analyzing things from the pilot’s point of view, but in this case it is easier to use the unaccelerated engineer’s point of view.
In flight, the lift is nearly always equal to the weightlab times load factor. The weightlab is presumably not changing much from moment to moment. This leads us to rearrange the lift equation as follows:
| coefficient of lift = (weightlab × load factor) / (½ρV2 × area) (2.2) |
If the airspeed goes down, the coefficient of lift must go up. This relationship is illustrated in figure 2.15.
Three of the critical V-numbers are marked in figure 2.15; each corresponds to a particular coefficient of lift.
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 2.16. 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.
By combining this fact with what we already know, we can establish the relationship between angle of attack and indicated airspeed. We combine figure 2.15 with figure 2.16, as is done in figure 2.17. 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: Near the stall, there is a largish range of angles of attack that all produce nearly the same coefficient of lift. (That’s because the coefficient of lift versus angle of attack curve is quite flat on top, as shown in figure 2.17.) This narrow range of coefficient-of-lift values corresponds to a narrow range of airspeeds, all near 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, but alas it is the one regime where the airspeed indicator doesn’t tell you what you need to know. A given airspeed near the stall could correspond to one degree below the critical angle of attack, or one degree above the critical angle of attack, and looking at the airspeed indicator won’t tell you which is which.
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.
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. It is telling you what you most need to know. Remember that calibrated airspeed is what holds up your wings. In principle, the calibrated airspeed depends on true airspeed and on density, and the density depends on altitude, temperature, and humidity ... but the wing doesn’t care about any of those details; it only cares about the calibrated airspeed in accordance with equation 2.1. 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.
In other words: do not correct Vy, Vs, glide speed, or approach speed (roughly 1.3 Vs0) for altitude or temperature. Trust the calibrated airspeed. These speeds need to be corrected for weight (section 2.13.4) but not for density altitude.
The true airspeed that corresponds to any given 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.
So far we have been assuming the weight was equal to some standard value. Let’s relax that assumption and see what happens.
It is easy to imagine flying a Cherokee Six at half of its maximum legal weight. (See section 7.5.8 for more on this.)
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 calibrated 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 approach speed and best-glide speed must be reduced below their standard-weight handbook values, according to the square root of the weight. The Vx and Vy values should be reduced by approximately the same factor. The maneuvering speed must also be reduced, although for different reasons, as discussed in section 2.14.2.
Since the cruise speed depends mainly on power and parasite drag, it hardly depends on angle of attack. That means it does not decrease as the weight is decreased; the situation is depicted in figure 7.13 in section 7.5.8. Also, in a multi-engine airplane, Vmc may or may not depend on lift requirements, so the safest thing is to not reduce it.
All the speeds in this section are calibrated airspeeds, for reasons discussed in section 2.13.7.
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 laboratory-frame 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.
On final approach, there is a definite angle of attack that works best. You should make a point of learning what this angle looks like, so you can maintain it without looking at the airspeed indicator very much (if at all).
The corresponding airspeed is called the “reference” speed, Vref. You might reasonably choose a speed higher or lower than Vref, depending on conditions (notably wind gusts).
As a rule of thumb, Vref is is about 1.3 times the stall speed.17 Naturally, this is based on the stall speed in the landing configuration, with whatever flap setting you have chosen, at the actual weight. This is just a rule of thumb, definitely not a hard-and-fast rule, for reasons we are about to discuss, here and in section 2.13.7.
The calculation is complicated by the fact that the airspeed-indicator system is never ideal. Never. The FAA allows it to be slightly imperfect, and manufacturers take advantage of this. They fudge the system so that it reads low at the low end of the scale and high at the high end. This is so people think that their airplane has a lower landing speed and a higher cruise speed. This is pure hype, but market forces demand it.
The nonidealities are big enough to cause serious problems if you’re not careful. In simple cases, you can calculate Vref using the following procedure:
The handbook should tell you how to convert IAS to CAS and vice versa. So you should do all your calculations in terms of calibrated airspeed, and then convert the final result to an indicated airspeed. That is, if you know the indicated stalling speed at “max gross” weight, the procedure goes like this, in simple cases:
Table 2.5 shows an example which contrasts the right and wrong calculations. The wrong calculation is typeset in red, as a warning.
CAS IAS stall: 50 ← 43 1.3 × indicated stall: ↓ wrong! unsafe approach speed: 58 ← 56 CAS IAS stall: 50 ← 43 (a) 1.3 × calibrated stall: ↓ (b) normal approach speed: 65 → 65 (c) Table 2.5: Calibrated versus Indicated Approach Speed
As mentioned in section 2.13.6, manufacturers make sure the airspeed indicator system underestimates the stall speed and overestimates the cruise speed. (In contrast, you will probably never see an aircraft that mis-estimates things the other way around.)
There are various ways of fudging things:
Scenario 1 is relatively straightforward. You have to do some work, but the work is guaranteed to be doable. That’s because if you know the IAS you can determine the CAS and vice versa. This is the simple situation discussed in section 2.13.6.
Scenario 2 is more common. It is in one way simpler, but in other ways much more complicated.
Here’s the simple part: Suppose you just want to calculate the stall speed as a function of weight (as in section 2.13.4). The airplane always stalls at the same angle of attack, so Pitot-related nonidealities drop out of the calculation. You can calculate the stall speed directly in terms of IAS, using the usual square-root-of-load factor.
However, alas, it becomes essentially impossible to calculate Vref in terms of the stall speed. The problem is, in this scenario you cannot determine the CAS just by looking at the airspeed indicator. Any given IAS could support a large weight at a small angle of attack, or a smaller weight at a larger angle of attack. The IAS/CAS discrepancy depends on angle of attack, but you don’t know the angle of attack, and you can’t figure it out by looking at the airspeed indicator.
In this situation, the handbook “should” provide explicit Vs and Vref numbers as a function of weight. If you’ve ever wondered why the handbook approach speed is not equal to 1.3 times the stall speed, now you know.
Ideally, the handbook would provide a whole set of IAS/CAS calibration tables, depending on weight.
There are various ways of figuring out the nonidealities of your airspeed system. For starters, while the airplane is parked, you could apply a carefully-calibrated pressure to the Pitot probe and observe whether the airspeed indicator reading is accurately proprtional to the square root of pressure. It is also possible to quantify the nonidealities that depend on angle of attack, but that requires some difficult test-pilot work. The details are beyond the scope of this discussion.
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 one side of the fuselage. During a slip, if that side is facing into the relative wind, it is subject to some dynamic pressure in addition to the static pressure.18 This is not a small effect; I have seen the indicated airspeed go to zero during intentional slips.
In a slip (or any other maneuver) where the airspeed indicator cannot be trusted, you must remember that it is angle of attack that really matters. You can use the airspeed indicator if you wish before the maneuver to help figure out what angle of attack you want, but during the maneuver you must maintain that angle of attack by looking at the angles themselves (pitch angle and direction of flight). See section 11.5 for more on this.
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 2.13.4 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 = ½ρV2 × coefficient of drag × area (2.3) |
has the same form as the famous formula for lift:
| lift = ½ρV2 × 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,19 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 2.13.4.
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 2.18. 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.
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, stall 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.
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.
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.
If you are flying in moderate or severe turbulence, you should keep your airspeed below the maneuvering speed, 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.
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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 fly through an arbitrarily intense 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:
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). However, 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.
We see that there are four main classes of numbers:
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 the reference with the zero-lift direction. That means that zero angle of attack corresponds to zero coefficient of lift. According to the standard terminology, the angle measured in this way is called the absolute angle of attack.
Some other books try to align the reference with the chord line20 of the wing. The angle measured in this way is called the geometric angle of attack.
If you try to compare books, there is potential for confusion, because this book uses “angle of attack” as shorthand for absolute angle of attack, while some other books use the same words as shorthand for other things, commonly geometric angle of attack. To make sense when comparing books, you must avoid shorthand and use the fully explicit terms. The relationship between the two ideas is shown in figure 2.19.
Quantitatively, to convert from one system to another:
| (2.5) |
where −k is the X-intercept of the graph of the coefficient of lift according to the “geometric” scheme.
In this book, we always use the absolute scheme, so the X-intercept is always zero.
Also note that are many possibilities, not just absolute versus geometric; the choice of reference is really quite arbitrary. It is perfectly valid to measure angles relative any reference you choose, provided you are consistent about it. (Aligning the reference stick with the fuselage is useful in some situations; see section 5.5.3.)
Using the chord as a reference works OK if you are only talking about one section of a plain wing. On the other hand:
Thinking about geometric angle of attack would be advantageous if you were building an airplane, or conducting wind-tunnel research on wing sections. Engineers can look at a wing section and determine the geometric angle of attack.
In contrast, if you are piloting the airplane, geometric angle of attack has no advantages and several big disadvantages: it’s hard to define, it’s hard to perceive, and it doesn’t tell you what you need to know anyway! We care about coefficient of lift, which is proportional to absolute angle of attack over a wide range (i.e. not too close to the stall). Each degree of angle of attack is worth about 0.1 units of coefficient of lift.
The simple rule “pitch plus incidence equals angle of climb plus angle of attack” (figure 2.2) is always mathematically valid, no matter what reference you’re using to measure angle of attack. (That’s because the arbitrariness in the angle of incidence cancels the arbitrariness in the angle of attack.) However, if you want the rule to be useful in the cockpit, especially in situations where flap settings are changing (as discussed in section 5.5), you need to focus on absolute angle of attack.
As a point of terminology: In the aeronautical engineering literature, the angle of attack is almost universally represented by α (Greek letter alpha). We mention this so that if you ever hear people talking about “alpha this” and “alpha that” you will know they are talking about angle of attack.
