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8  Yaw-Wise Torque Budget

In aircraft (unlike cars, bikes, or small boats) you have separate control over which way it is pointing relative to which way it is going.

8.1  Overview

This chapter discusses the yaw-wise motion of the airplane, which has to do with which way the airplane is pointing.1

Normally you want the aircraft to be pointing the same direction as it is going through the air. That is, you want the slip angle to be small. There are several reasons for this:

Maintaining zero slip angle while maneuvering requires coordinated use of the ailerons and rudder, so pilots speak of “zero slip angle” and “good coordination” almost interchangeably. (Situations that call for an intentional slip are discussed in section 11.5.)

This chapter considers, one by one, the various phenomena that affect the airplane’s yaw-wise motion. There are surprisingly many such phenomena, including the helical propwash, yaw-wise inertia, adverse yaw, P-factor, and gyroscopic precession — plus the stability and damping created by the vertical fin and rudder.

8.2  Yaw Stability

For ordinary objects such as cars, bicycles, and small boats, there is only one “steering” control, and it serves to control both the direction you are pointing and the direction you are going. However, the two ideas are not necessarily linked.

As an extreme example of non-linkage, take a Frisbee and draw on it the picture of an airplane. When you throw the Frisbee, the picture of the airplane will be yawing like crazy, turning around and around and around. You have no control over which way it is pointing. You do, however, have some modest control over which way the Frisbee as a whole is going: if there is a nonzero bank angle, the Frisbee will follow a curved flight-path.

Far and away the most powerful technique2 for changing the direction an airplane is going is to put it into a bank, so that the horizontal component of lift forces a change in flight-path, as mentioned in section 4.3 and especially section 6.2. This is not yaw; bank by itself will not change the direction you are pointing.

The vertical fin and rudder are responsible for controlling the yaw angle, which is the main topic of this chapter.

An airplane has partial linkage between the direction you are going and the direction you are pointing. That is:

Figure 8.1 shows a situation where the airplane’s heading has been disturbed out of its usual alignment with the airflow. There are lots of ways this could happen, including a gust of wind, a momentary uncoordinated deflection of the controls, or whatever.

crab
Figure 8.1: Response to a Yaw Angle

In this situation, the relative wind is striking the vertical fin and rudder at an angle. Like any other airfoil, the fin/rudder produces lift in proportion to its angle of attack, so it will produce a force (and therefore a torque) that tends to re-align the airplane with the wind. We say that the airplane has lots of yaw-wise stability.

The colloquial name for yaw-wise stability is “weathervaning tendency”. That is, the airplane tends to align itself with the relative wind, just as a weathervane does. Section 8.12 discusses weathervaning during taxi.

8.3  Yaw Damping

In most airplanes, pure yawing motions are reasonably well damped. The process is analogous to the process that produces damping of pure vertical motions and pure rolling motions (see chapter 5). When the tail is swinging to the right with an appreciable velocity, it sees a relative wind coming from ahead and to the right. The resulting angle of attack produces a leftward force that damps the rightward motion.

A leftward force in proportion to a rightward velocity is exactly what constitutes damping.

In some airplanes, there is a lightly-damped Dutch roll mode, involving the yaw axis along with others, as discussed in section 10.6.1.

8.4  Helical Propwash

One of the very first things that people find out about when they start learning to fly is that it takes right3 rudder (sometimes a lot of right rudder) to keep the airplane going straight at the beginning of the takeoff roll. The physics of the situation is portrayed4 in figure 8.2.

cork-rudder
Figure 8.2: Helical Propwash

It would be nice if the propeller would just take the air and throw it straight backwards, but it doesn’t. The propeller airfoil necessarily has some drag, so it drags the air in the direction of rotation to some extent. Therefore the slipstream follows a helical (corkscrew-like) trajectory, rotating as it flows back over the aircraft.

The next thing to notice is that on practically all aircraft, the vertical fin and rudder stick up, not down, projecting well above the centerline of the slipstream. That means the helical propwash will strike the left side of the tail, knocking it to the right, which makes the nose go to the left, which means you need right rudder to compensate.

You don’t notice the effect of the helical propwash in cruise, because the aircraft designers have anticipated the situation. The vertical fin and rudder have been installed at a slight angle, so they are aligned with the actual airflow, not with the axis of the aircraft.

In a high-airspeed, low-power situation (such as a power-off descent) the built-in compensation is more than you need, so you need to apply explicit left rudder (or dial in left-rudder trim) to undo the compensation and get the tail lined up with the actual airflow.

Conversely, in a high-power, low-airspeed situation (such as initial takeoff roll, or slow flight) the helix is extra-tightly wound, so you have to apply explicit right rudder.

Helical propwash sometimes contributes to left/right asymmetry in multi-engine aircraft, as discussed in section 17.1.11.

8.5  P-Factor

The term P-factor is defined to mean “asymmetric disk loading”. It is an extremely significant effect for helicopters. When the helicopter is in forward flight, the blade on one side has a much higher airspeed than the other. If you tried to fly the blades at constant angle of attack, the advancing blade would produce quite a bit more lift than the retreating blade.

8.5.1  Blade Speed

For airplanes, the same effect can occur, although it is usually small. For the effect to occur at all, you need to have an angle between the propeller axis and the relative wind. To be specific, imagine that the aircraft is in a nose-high attitude, but its direction of motion is horizontal (i.e. the relative wind is horizontal). Then the downgoing blade will be going down and a little bit forward, while the upgoing blade will be going up and a little bit backward. The downgoing blade will effectively have a slightly higher airspeed. Since this blade is on the right-hand side of the airplane (assuming a typical American engine) it will tend to torque the airplane around to the left and you’ll need right rudder to compensate.

p-factor-intro
Figure 8.3: P-Factor

The situation is depicted in figure 8.3. The airplane is in level flight, with a 10 degree nose-up attitude. The motion of the blade through the air is shown in magenta. It consists of the rotational motion (shown in green) plus the forward motion of the whole airplane (shown in red). The motion of the downgoing blade is shown with solid lines, while the motion of the upgoing blade is shown with dotted lines. You can see that the speed of the downgoing blade is larger than the speed of the upgoing blade.

This is the main contribution to P-factor: the advancing blade sees more relative wind, while the retreating blade sees less relative wind.

8.5.2  Blade Angle

There is a widespread misconception that P-factor arises because the angle of the right (downgoing) propeller blade is larger than the angle of the left (upgoing) propeller blade. Many books erroneously call attention to the angle of the blade relative to the ground. The blade doesn’t care about the ground; the only thing that matters is the angle of attack, i.e. the angle between the blade and its own motion through the air.

The correct analysis is shown in figure 8.4. As a point of reference, the left panel shows level pitch attitude in normal level flight, where no P-factor occurs. Meanwhile, the right panel shows the airplane in a 10 degree nose-up attitude (still in level flight). Since we want to discuss angle of attack, I have attached a “reference line” pointer to each of the blades, just like the reference line used in section 2.2. The angle of attack of the propeller blade is just the angle between the reference line and the blade’s motion through the air.

You can also think of the blade’s angle of attack as the angle between the reference and the blade’s relative wind. Relative wind and direction of motion are the same concept, just reversed 180 degrees. Be careful though, because there are various different relative winds, including the instantaneous wind relative to the moving blade and the average wind relative to the overall airplane.

p-factor-aoa
Figure 8.4: P-Factor and Angle of Attack

When the propeller disk is inclined to the direction of flight (so that P-factor really is occurring) the upgoing blade has slightly less angle of attack (compared to the downgoing blade). That is to say, the angle Xup shown in figure 8.4 is slightly less than the angle Xdown. We could figure this out by adding the airplane’s forward speed to the blade’s rotational motion to form the resultant, and then comparing this to the blade’s reference line, but it is perhaps easier to work backward, starting from the resultant and subtracting the airplane’s forward motion.

– Downgoing Blade –   – Upgoing Blade –

The resultant is longer and more inclined to the airplane’s motion, so the motion vector has “worse leverage” for creating an angle between the rotational motion and the resultant.   The resultant is shorter and more nearly perpendicular to the airplane’s motion, so the motion vector has “better leverage” for creating an angle between the rotational motion and the resultant.

This angle-of-attack effect is of course zero when propeller axis is aligned with the direction of flight. Also, at the opposite extreme, it goes to zero again when the axis is perpendicular to the direction of flight, as in a helicopter. (The airspeed effect discussed in section 8.5.1 is very important for helicopters.) The angle-of-attack effect is never very large, because

  1. At low speeds, the airplane’s forward velocity (as represented by the horizontal red arrow in figure 8.4 is so small that it can’t have much effect on anything.
  2. At high speeds, the airplane has a low angle of attack, so the angle between the propeller disk and relative wind is necessarily small (except for helicopters, tilt-rotors, and such).
  3. At very high speeds, when you are going fast enough to over-run the geometric pitch of the propeller (so that the resultant coincides with the reference line in figure 8.4), you might think that a small difference in blade angle of attack would be a 100% effect. I suppose that’s true, but in this case the total thrust is practically zero, and 100% of nothing is nothing.

This angle-of-attack effect is in addition to (and usually smaller than) the airspeed effect discussed previously. Both are small compared to the helical propwash effect.

Remember, we don’t care whether the downgoing blade makes a bigger angle to the vertical than does the upgoing blade. The blade doesn’t care which way is up — all it cares about is where the relative wind is coming from. Imagine a tailwheel-type airplane stationary in the run-up area on a windless day. You can incline the propeller disk as much as you want relative to vertical, but there will be no P-factor unless there is wind blowing through the propeller disk at an angle.

8.5.3  Initial Takeoff Roll

There are quite a lot of myths surrounding P-factor. For some reason, P-factor gets blamed for the fact that typical aircraft require right rudder on initial takeoff roll. This is impossible for several reasons.

The real reason that you need right rudder on initial takeoff roll is because of the helical propwash, as discussed in section 8.4. P-factor exists in some circumstances, but it cannot possibly explain the behavior we observe during initial takeoff roll.

8.5.4  Observing P-Factor

It is not easy to observe P-factor. It is usually swamped by other effects such as helical propwash (section 8.4) and twisted lift (section 8.9.5).

An important preliminary experiment is to observe what happens during the takeoff roll in a multi-engine aircraft. (To be specific, let’s consider the case where both engines rotate clockwise as seen from the rear.) In some airplanes, the propwash hits the tail, and you must apply right rudder to compensate, just like in single-engine planes.

In other airplanes, most of the helical propwash misses the vertical tail in normal flight. This causes no problems and no compensation is required. See section 17.1.11 for details on this. This is the perfect way to illustrate that there is no P-factor when the propeller disk is not inclined.

If you really want to observe nonzero P-factor, you can proceed as follows: Take a twin-engine (or four-engine) aircraft with non-counter-rotating propellers. Attach a slip string, as discussed in section 11.3. Establish coordinated cruising flight, with the same amount of power on both sides. Confirm that the ball and string are centered. Pull up into a nonturning climb at very low airspeed (i.e. very high angle of attack), maintaining cruise power. Maintain coordinated flight as indicated by the slip string. Observe the rolling tendency due to propeller drag. Shift weight (e.g. fuel) from left to right to get rid of the rolling tendency, so you can fly straight without deflecting the ailerons, i.e. without incurring any twisted lift.

You will observe the inclinometer ball will be slightly off-center. This can be attributed to P-factor. To be explicit:

The effect of P-factor is not very large. You can easily compensate using a little bit of right rudder and right bank. Indeed, in typical situations you can just ignore it entirely.

You can never use rudder deflection as an indication of P-factor, because any situation that exhibits P-factor will also exhibit a large amount of helical propwash.

The single-engine version of the previous experiment goes like this: Put a slip string on each wing, far enough out on the wing that it is not unduly disturbed by the propwash, yet close enough in that you can see it. In a high-wing aircraft, you’ll have to put the string on the bottom of the wing. Put strings on both sides in symmetric locations, so you can tell for sure what string position corresponds to symmetric airflow. Then confirm that in normal nonturning cruising flight, you have symmetric airflow (as indicated by the strings) and zero inclination (as indicated by the inclinometer ball). Finally, set up a situation in which the largest possible P-factor occurs: flaps retracted, minimum airspeed, and full power.

Once again, the indication of P-factor in this situation would be to have the ball be off-center when the strings were centered. I have tried this experiment, but the P-factor was too small to observe.

Here’s another possible experiment. Take your favorite aerobatic airplane and paint the starboard rudder pedal green and the port rudder pedal red, just so we can keep straight which is which. Now go to a safe altitude and set up for inverted slow flight. In this high-power, low-speed situation, do you need to push the port (red) pedal or the starboard (green) pedal? If P-factor is more important, the answer will be port, because that is now the downgoing, advancing blade. If helical propwash is more important, the answer is starboard, because the relationship between the propeller, rudder, and rudder pedals is unchanged by the inversion.

8.6  Gyroscopic Precession

A spinning object will respond to a torque in one direction with a motion in another direction. This remarkable and counterintuitive phenomenon — gyroscopic precession — is discussed in more detail in section section 19.10.

Gyroscopic precession is often quite noticeable at the point where a taildragger raises the tail, early in the takeoff roll.5 If the airplane were an ordinary non-spinning object, you could raise the tail using the flippers alone. The flippers do not actually dictate the motion of the fuselage; they just produce a force and a pitch-wise torque. For a gyroscope, this pitch-wise torque produces a yaw-wise motion. If you try to raise the tail of a real airplane using flippers alone, it will yaw to the left because of precession.

To get a gyroscope to actually start moving in the pitch-wise direction, you need to apply a torque in the yaw-wise direction. This is what the vertical fin and rudder are for. See section 19.10.

Of course, an airplane has some plain old mass in addition to its gyroscopic properties. In order to lift this ordinary mass you need to use the flippers. Therefore, the tail-raising maneuver requires both flippers and rudder — flippers to change the pitch of the ordinary mass, and rudder to change the pitch of the gyroscope.

8.7  Canted Engine

Often the engine is mounted in such a way that direction of the thrust vector is a little to one side of the axis of the airplane. This is done in order to compensate for various nonidealities such as helical propwash. It contributes to the yaw-wise torque budget in the obvious way.

8.8  Yaw-Wise Rotation of Propwash

As discussed in section 9.5, propeller drag imparts a certain amount of rotational motion to the propwash (in addition to the desired straight-line motion). In a normal airplane, to a good approximation, we think of this rotation as being entirely in a roll-wise direction, so it contributes nothing to the yaw budget.

However, if we look more closely, things are not quite so simple. If the straight-line motion of the propwash has some downward component, the rotational motion will have some yaw-wise component.

In a wide range of aircraft, this effect is not noticeable when the flaps are retracted, but becomes noticeable when the flaps are extended. This makes sense, because the flaps deflect the propwash, giving a downward component to the straight-line motion and therefore a yaw-wise component to the rotational motion.

To demonstrate this, start in the clean configuration at a speed within the white arc. Observe how much rudder deflection is required to maintain coordinated flight. Then extend the flaps. Maintain the same power setting and the same airspeed; the vertical speed will change (due to the added drag) but that is not important. If you observe that incrementally more rudder deflection is required to maintain coordination, the increment can be attributed to the vertical component of the propwash.

8.9  Rudder Usage During Rolls

Turning the airplane properly requires coordinated use of ailerons and rudder. Getting it exactly right is a bit tricky.

Remember that in an airplane, the direction you are moving is not necessarily the same as the direction you are pointing. There are several crucial things that happen during a turn:

1)
You use the wings to change the direction you are going, i.e. to re-orient your momentum vector. I call this the MV-turn.
2a)
You use the rudder to change your heading (i.e. to overcome yaw-wise inertia, i.e. to provide yaw-wise acceleration).
2b)
You use the rudder to overcome steady adverse yaw due to twisted lift, as discussed below.
2c)
You use the rudder to overcome transitory adverse yaw due to differential drag.

Item 1 is relatively straightforward: you put the airplane into a bank. The horizontal component of lift will change the direction of motion. Note that MV is a bit of a pun; it might stand for “momentum vector” or “mass times velocity” ... or both.

Item 2a is important because if the airplane didn’t have any vertical tail, banking would cause it to just slip off in the new direction without changing its heading. It is much nicer to yaw the plane to align its axis with the new direction of motion, so you apply the rudder, thereby creating a yaw rate that matches the MV-turn rate.6

Now we come to item 2b. We must consider adverse yaw. As discussed in section 8.9.5, during a steady roll, the aerodynamic forces produced by the two wings are equal in magnitude, but one force vector is twisted slightly forward while the other one is twisted slightly rearward. This causes a yawing moment in exactly the wrong direction: if you are rolling to the right it tries to make the airplane yaw to the left. To compensate you must deflect the rudder whenever the ailerons are deflected.

Finally, we come to item 2c. Suppose you are flying an airplane where there is a lot of mass out on the wings. Whenever you are starting or ending a roll maneuver, you need to accelerate one wing upward and the other wing downward. As discussed in section 8.9.4, this briefly requires extra lift on one wing and reduced lift on the other wing. This unequal lift produces unequal induced drag. This drag causes additional adverse yaw.

For any given rate of roll, you need to use lots more rudder at low airspeeds, for reasons discussed in section 8.9.7.

Procedures for maintaining coordination during turns are summarized in section 8.9.8; the intervening sections describe in a little more detail what is the problem we are trying to solve.

8.9.1  Analysis of a Roll

To make the discussion more concrete, let’s consider a roll starting from straight-and-level flight and rolling to the right. As we can see from figure 8.5, there are multiple timescales in the problem.

yaw-timeline
Figure 8.5: Timeline for Roll Maneuver

Let’s analyze what happens if you move the ailerons fairly abruptly. Although generally I recommend flying with a smooth, gentle touch, (1) there will be times when you want to roll the airplane on short notice, so let’s learn how to do it; and (2) the abrupt case makes it easier to understand what is going on.7

In some airplanes, such as a Piper Cub, the roll rate will reach its final very quickly (within a small fraction of a second), because the airplane has very little roll-wise inertia. Practically all the mass (pilot, passenger, fuel, and engine) is arranged in a straight line right on top of the roll axis, so they don’t contribute much moment of inertia. In other airplanes, such as a Cessna 310, the roll rate responds much more slowly, because lots of mass (engines and tip tanks) is situated far from the roll axis.

Before the roll rate is established (i.e. during the time [t1, t2]) the plane will experience transitory adverse yaw due to differential induced drag. The nose will swing a little toward the outside of the turn. The effect is usually rather small, since

  1. these differential drag forces are typically small (during slow flight) or very, very small (during cruise), compared to the differential lift forces that cause the roll, and
  2. these forces must act against the yaw-wise inertia, which is at least as large as the roll-wise inertia.

The rest of the discussion applies no matter how slowly or abruptly you moved the ailerons.

After the time t2, a steady roll rate exists. Even though the ailerons are deflected, there is no difference in lift from one wing to the other, for reasons discussed in section 8.9.5. Since there is no difference in lift, there will be no difference in induced drag, hence no transitory adverse yaw.

However, one wingtip is diving, so its force vector is twisted slightly forward. The other wingtip is rising, so its force vector is twisted slightly rearward. Even though each force has practically the same magnitude as it would in non-rolling flight, the twist means there is a slight component of force in just the right direction to produce a steady adverse yawing moment.

In addition, because the airplane is rolling, a bank is developing. This bank causes an MV-turn; that is, the airplane is changing its direction of motion. In order to keep it pointing in the same direction it is moving, you need to deflect the rudder during the roll, as discussed in section 8.9.6.

At time t6, the ailerons are neutralized, but the rolling motion has not yet stopped. (Again, there is a delay due to roll-wise inertia.) At this point there are several things going on:

  1. There is a difference in lift between the two wings, as needed to damp out the roll. This creates a negative amount of transitory adverse yaw. This requires a left-rudder contribution to compensate.
  2. However, the airplane is still rolling, and a still-increasing rate of yaw is needed to coordinate with the still-increasing rate of MV-turn. This requires a right-rudder contribution.
  3. Similarly, because the airplane is still rolling, the twisted lift requires a right rudder contribution.

In practical situations, the first item (transitory adverse yaw) is usually smaller than the other two. During the interval [t5, t7] the roll rate is decreasing, so you need less and less rudder deflection.

Analogous statements would apply if you started from a left turn and used right aileron and right rudder to roll out of the turn. Similarly, it is easy to do a similar analysis for rolling into a left turn and/or rolling out of a right turn.

8.9.2  Designers’ Tricks

Imagine an airplane without a vertical fin. It would behave be more like a Frisbee than a boat — if you gave it a yaw rate, inertia would make it just keep on yawing until some torque acted to stop it. Even if it were not yawing, there would be no reason to expect the yaw angle (i.e. heading) to be anywhere close to the desired value.

In a real airplane, of course, the vertical fin and rudder supply the forces required to keep the yaw angle and yaw rate under control. An overview of how you use the rudder during turns can be found in section 8.9.

Aircraft manufacturers know about how turns are affected by twisted lift and yaw-wise inertia. They generally try to provide the needed yaw-wise torque automatically, using various tricks. One trick is to interconnect the rudder and ailerons with a spring. That means you automatically get a certain amount of rudder deflection in proportion to the aileron deflection. They choose the proportionality factor so that you can more or less fly “with your feet on the floor” at cruise airspeeds. Of course, vastly more rudder is needed at lower airspeeds; fortunately you can easily overpower the interconnect spring by pushing on the controls in the obvious way.

Here’s another trick, which you may have noticed on many airplanes: when one aileron goes down a little, the other one goes up a lot. (This is called differential aileron deflection.) The designers were trying to arrange for the upward-deflected aileron to generate a lot of parasite drag. If they do it just right, the drag force is just enough to overcome twisted lift and yaw-wise inertia during a steady roll. The so-called Frise aileron uses a similar trick. It has lip on the bottom, well ahead of the hinge. The lip sticks down into the airstream when the main part of the aileron is deflected up. Again, the purpose of the lip is to generate drag on the wing with the upward-deflected aileron.

In addition to overcoming yaw-wise inertia (during a steady roll), the designers also want to overcome transitory adverse yaw (when ailerons have been deflected but the roll hasn’t yet started). Fortunately, transitory adverse yaw is rather small, and by adjusting the amount of differential deflection, and the amount of the Frise effect, pretty good cancellation can be achieved.

The bad news is that this compensation only works at one airspeed. The designers arrange it so you can fly with your feet on the floor during cruise. This is a mixed blessing, because it can lull you into complacency. At lower airspeeds, where it is most important, you still need to use lots of rudder to keep things coordinated. Don’t forget!

8.9.3  Two Modes and Four Zones

Any action that affects the left wing and the right wing equally we will call a common-mode action. Changing the pitch attitude is an example in this category. Extending the flaps is another example. Actions in this category can affect the overall lift, but (to a good approximation) do not affect the rolling moment or yawing moment.

In contrast, any action that affects the left wing and the right wing oppositely we will call a differential-mode action. A rolling motion is an example in this category, as discussed in section 8.9.5. Deflecting the ailerons is another example, as discussed in section 8.9.4. Actions in this category can affect the rolling and yawing moments, but (to a good approxmation) do not affect the total lift.

There is a fairly deep principle of physics here: For a rigid body, if you know the total force vector and the total moment bivector, you can write down the equation of motion. In any case, from the pilot’s point of view, you have only a limited number of controls (elevator, ailerons, flaps) so you couldn’t control much of anything beyond the common mode and the differential mode, even if you wanted to.

We now discuss how the various parts of the airplane contribute to the common mode and the differential mode. (If you’re not interested in details, you can skip this and go on to the next section.)

Rather than thinking about the four zones separately, let’s combine them in various ways, using addition and subtraction (and a little bit of multiplication). Addition of zones gives us the common mode, and subtraction of zones gives us the differential mode, as you can see in the following table:

Inboard zones,        Outboard zones,
left/right same.        left/right same.
Ex: flaps.        Ex: pitch.
Common mode.        Common mode.
Affects lift.        Affects lift.
             
Inboard zones,        Outboard zones,
left/right opposite,        left/right opposite,
         Ex: rolling motion.
         Ex: aileron deflection.
no effect        Differential mode.
         Affects moments.

Anything you do that affects both sides equally contributes to the common mode. This is true for the inboard zones as well as the outboard zones.

Anything you do that affects the outboard zones, affecting left and right oppositely, contributes to the differential mode.

We imagine that the inboard zones of the wing are not much affected by the rolling or yawing motion of the aircraft (because they are close to the centerline) and conversely nothing that happens to the inboard zones has much effect on the roll-wise or yaw-wise torque budget (again because they are close to the centerline, so there is very little lever arm).

Within the four-zone model, if the lift of the left wing is equal to the lift of the right wing, then there is no net rolling moment. Pilots prefer the simple idea and simple terminology: “same lift on both sides”. That sounds simpler than “no net rolling moment” and means the same thing, for practical purposes.

The four-zone model works quite well. Even when it is not exactly right, the mistakes are hard to notice. If the model makes a mistake that affects the two wings equally, we just lump it in with the inboard zones and change the pitch attitude a tiny bit to compensate. If the model makes a mistake that affects the two wings oppositely, we just lump it in with the outboard zones and deflect the ailerons a tiny bit to compensate.

The two-mode model works even better than the four-zone model. That’s because you don’t care exactly how the lift is distributed along the wing. Ordinarily all you care about is (a) having enough total lift to support the airplane’s weight and load factor, and (b) having the desired rolling moment.

8.9.4  Transitory Adverse Yaw

Suppose you wish to roll into a right turn. You will deflect the ailerons to the right, as shown in figure 8.6. During the brief time after the ailerons are deflected and before the steady roll is established, this will increase the lift created by the left wing, and decrease the lift created by the right wing. Unfortunately, there is no way to produce lift without producing drag, so the left wing will be dragged backwards while the right wing lunges forward. This is the exact opposite of what we wanted; the airplane yaws to the left even though we wanted it to turn to the right. Being a good pilot, you have anticipated this, so you apply right rudder as well as right aileron, to make sure the nose swings the right way.

adverse-yaw-1
Figure 8.6: Transitory Adverse Yaw

Even if you don’t get the footwork exactly right, the nose will eventually swing around and point more-or-less the right way, because of the airplane’s inherent yaw stability (as discussed in section 8.2).

Now suppose a steady roll rate is established; that is, there is no roll-wise acceleration. Using the four-zone model, we can say both wings producing the same amount of lift. That means the type of adverse yaw we are discussing here – transitory adverse yaw – is no longer a factor. Instead, there is another kind of adverse yaw, as discussed in section 8.9.5.

Now let’s consider what happens when you roll out of the turn. The airplane is banked to the right and already turning to the right. You will deflect the ailerons to the left. This will cause extra drag on the right wing, and reduced drag on the left wing. The airplane will yaw to the right, continuing and exaggerating the turn that you were trying to stop. Anticipating this, you apply left rudder along with the left aileron.

8.9.5  Steady Adverse Yaw – Twisted Lift

Now let’s consider what happens during a steady roll. As illustrated in figure 8.7, the airplane as a whole is moving forward, but the left wingtip is moving forward and up while the right wingtip is moving forward and down (because of the rolling motion).

adverse-yaw-steady
Figure 8.7: Steady Adverse Yaw – Twisted Lift

Let’s see what the local angle of attack is at the wingtip. We use the trusty formula

angle of attack + angle of climb  = pitch + incidence              (8.1)

As you can see in the figure:

The left wingtip has a positive angle of climb, since it is going forward and up.   The right wingtip has a negative angle of climb, since it is going forward and down.

To say the same thing the other way:

The relative wind near the left wingtip is coming from ahead and above.   The relative wind near the right wingtip is coming from ahead and below.

By geometry, that tells us about the orientation of the lift vector, since it is always perpendicular to the local relative wind:

On the upgoing wing, the lift vector vector points up and a little bit aft.   On the downgoing wing, the lift vector points up and a little bit forward.

These fore-an-aft components of lift produce a yawing moment, in proportion to the roll rate. You need to deflect the rudder to compensate, if you want to maintain coordinated flight.

Tangential remark: Some people try to argue that the fore-and-aft components of the lift vector should be renamed “drag” forces since they point in the same direction as the overall (non-local) relative wind. However, it is much better to analyze such things in local terms. The lift vector is always (by definition!) perpendicular to the local relative wind. As a related point: a drag force would dissipate energy in proportion to force times airspeed, but the twisted lift forces do not dissipate energy.

There is of course such a thing as twisted drag, for all the same reasons. Twisted drag contributes to the roll-wise torque budget, just as twisted lift contributes to the yaw-wise torque budget (except to a lesser degree, because drag is small compared to lift).

At each point along the wing, the amount of twisted lift force depends on the rate of roll, multiplied by the distance of that point from the roll axis. The amount of yaw-wise torque depends on this force and on the lever-arm, which gives us another factor of distance. That is, the amount of yaw-wise torque depends on roll rate and on distance squared. Using the four-zone model (section 8.9.3), we say that all of the effect comes from the outboard zones of the wing, and none from the inboard zones.

It must be emphasized that twisted lift depends directly on the roll rate; it does not have any direct or necessary dependence on aileron deflection. In particular, consider a multi-engine airplane with tip tanks, meaning there is a lot of roll-wise inertia. If the airplane is rolling, it will continue to roll for a while just due to inertia, even if the ailerons are not deflected. There will be a yawing moment due to twisted lift, so long as the rolling motion continues.

Figure 8.7 shows the ailerons deflected, which is typical during a simple, steady roll. To maintain a steady roll, the ailerons must be deflected, for reasons having to do with the roll-wise torque budget: Within the four-zone model, we can say that the aileron deflection changes the incidence of the outboard zone of the wing, by an amount that compensates for the twist in the relative wind, so that in accordance with equation 8.1 the two wingtips have the same angle of attack. This means the two wings have the same amount of lift, or (more precisely) there is no net rolling moment, which is what we need for a steady roll. (For more on the roll-wise torque budget, see chapter 9.)

For present purposes – to understand twisted lift – we concentrate on the direction of the lift vector, which depends on the roll rate, whether or not the ailerons are deflected, and whether or not the roll is steady.

Twisted lift is only one of about ten contributions to the yaw-wise torque budget, as discussed in other sections of this chapter. In practice, it is often hard to notice the difference between twisted lift, transitory adverse yaw (section 8.9.4), and yaw-wise inertia (section 8.9.6). This is particularly hard to notice if you begin and end the rolling maneuver smoothly, by deflecting the ailerons gradually, as you should. In this case the roll rate, aileron deflection, and rudder deflection are all nicely proportional, so you don’t need to worry about the theory too much.

8.9.6  Yaw-Wise Inertia

(In this section, we will assume that you are flying at such a low airspeed that the designers’ tricks discussed in section 8.9.2 are not sufficient to produce automatically coordinated turns.)

Whenever the airplane is in a bank, it will make a MV-turn. A pure MV-turn, however, is not what you want. A pure MV-turn means that even though the airplane is moving in a new direction, the heading hasn’t changed. The airplane has a nonzero slip angle. The uncoordinated airflow acting on the tail will eventually set up a yawing motion that matches the MV-turn rate, converting it from a pure MV-turn to a more-or-less8 coordinated turn. If the yaw-wise damping is weak, as it usually is, the nose will slosh back and forth several times as it tries to catch up with the MV-turn.

At any particular MV-turn rate, once the yaw rate is established, no further yaw-wise torque is required. Like a toy top, once the airplane starts rotating in the yaw-wise directon it will be happy to continue rotating.9 The only time you need a yaw-wise torque is when the yaw rate is changing.

So, we see that during a steady roll,

Conclusion: the rudder should be deflected when the ailerons are deflected.

8.9.7  Amount of Rudder Required

As we have seen, there are actually three different reasons why you need to apply the rudder during roll maneuvers: twisted lift, differential induced drag, and yaw-wise inertia. The amount of rudder deflection you need depends on the shape of your airplane, and also depends on airspeed.

*   Twisted Lift

Example 1: Consider an airplane with long wings and with most of the mass concentrated near the middle of the airplane. A typical glider is an excellent example, but almost any ordinary-shaped airplane will do. In this case there will be very little roll-wise inertia, and accordingly very little transitory adverse yaw. There will also be rather little yaw-wise inertia. Therefore in such a plane, the dominant effect will be steady adverse yaw due to twisted lift.

Example 2: Suppose you are flying along in any airplane on a sunny summer day. You encounter a situation where your right wing is in an updraft, while your left wing is in a downdraft. You deflect the ailerons in order to maintain zero bank, zero roll rate, and constant heading. This combination of non-horizontal relative wind and deflected ailerons creates twisted lift, the same as shown in figure 8.7 (except that the roll rate is zero in this case). Therefore this is a perfect example of steady adverse yaw, and you must deflect the rudder to compensate. (This could not be explained by differential drag or yaw-wise inertia. This is pure twisted lift.)

The yawing moment due to twisted lift is essentially independent of airspeed. It just depends on the deflection-angle of the ailerons. Meanwhile, though, the force produced by the rudder is proportional to airspeed squared. Therefore you need lots more rudder deflection (per unit aileron deflection) when the airspeed is low.

*   Differential Induced Drag

Example 3: Consider an aircraft where there is a lot of mass located far away from the roll axis. A twin with heavy engines mounted way out on the wings, plus tip-tanks full of fuel, is a good example. Such a plane will have lots of roll-wise inertia, and therefore lots of transitory adverse yaw. You will still have to worry about yaw-wise inertia and twisted lift, but in addition to those effects you will need to apply extra rudder deflection when ailerons are first deflected, before the steady roll develops.

The amount of rudder required depends dramatically on airspeed. In addition to the rudder-force issue discussed above, the amount of transitory yawing moment itself increases when the airspeed decreases. The key to understanding this is to realize that whereas the coefficient of lift is more or less proportional to the angle of attack (for moderate angles of attack), the coefficient of induced drag is more or less proportional to the square of the angle of attack.

adverse-yaw-slow
Figure 8.8: Slow Flight Means More Transitory Adverse Yaw

The left side of figure 8.8 shows the same situation as in figure 8.6, along with the coefficient of drag curve. On this curve I have indicated the different angles of attack for the two wingtips, and the correspondingly different amounts of drag. We see that the coefficient of drag curve is relatively flat on the bottom, so at relatively small angles of attack (high airspeeds), a difference in angle of attack doesn’t cause too much difference in drag.

In contrast, the right side of figure 8.8 shows the same aircraft in slow flight. Both wings are operating at a higher angle of attack. Because the coefficient of drag curve is steeper in this regime, the same difference in angle of attack (i.e. the same aileron deflection) creates more difference in drag (i.e. more transitory adverse yaw).

*   Yaw-Wise Inertia

Example 4: Consider a long, thin, single-engine biplane carrying lots of cargo. Since it has a rather short wingspan, there will be rather little twisted lift, i.e. rather little steady adverse yaw. Similarly, since all the mass is close to the roll axis, there will be very little roll-wise inertia, i.e. very little transitory adverse yaw. There will, however, be lots of yaw-wise inertia.

Example 5: Let’s return to the case where your right wing is in an updraft, while your left wing is in a downdraft. This time, however, you don’t deflect the ailerons; you just accept the resulting roll rate. During the steady roll, you will need to deflect the rudder to supply the yaw-wise angular momentum to match the ever-increasing MV-turn rate. (This rudder requirement could not be explained by twisted lift or differential drag. This is pure yaw-wise inertia. Also note that no designers’ tricks could maintain coordination in this situation, since the ailerons are not deflected.)

Once again, the amount of rudder required increases markedly at low airspeeds. There are three main contributions; the first two essentially cancel each other:

8.9.8  Summary: Coordinated Turning Procedures

A proper turn consists of two ingredients: a MV-turn and a heading change. In an idealized “basic” airplane, you would use the ailerons to bank the airplane and lift the MV around the corner, and you would use the rudder to change the heading and combat adverse yaw. In a typical modern airplane at cruise airspeeds, deflecting the ailerons alone creates a fair approximation of the proper torques in both directions (roll-wise and yaw-wise). In all airplanes at low airspeeds, proper rudder usage is vitally important.

The basic rule is simple:

The amount of rudder will depend inversely on the airspeed.

Another version of the rule substitutes the word “aileron” for “roll”:

In a steady roll, the two versions are more or less equivalent; at the beginning and end of a roll (when the roll rate does not match the aileron deflection) the truth lies somewhere in between. Split the difference.

These rules allow you to anticipate the need for rudder deflection. As discussed in section 11.7, you have many ways of knowing when you’ve got it right:

  1. The acid test involves looking out the window. You should perceive that the rate of heading change is proportional to the amount of bank.
  2. You can also look to the side and perceive that the wings flap straight up and down, not slicing fore and aft as you roll.
  3. You can see that the inclinometer ball remains almost centered.
  4. Yet more information comes from the seat of your pants.

By the way: If you think about it for a moment, you can see that in inverted flight (negative angle of attack) you will have a negative amount of adverse yaw — if you deflect the yoke to the left you will need to push on the right rudder pedal, and vice versa — just the opposite of what you would do in noninverted flight. When you are actually in the plane, hanging upside down, this is not as confusing as it seems on paper. A little thought and a little practice will make it fairly self-evident which wing you should lower to make a MV-turn and which rudder pedal you should push to change the heading.

As mentioned at the beginning of this chapter, there are lots of reasons why you should use the rudder properly during turns. Alas, the learning process is complicated by the fact that in many cases the airplane will “cover up” small mistakes for you. In particular, whenever the airplane is in a slip, the vertical fin will automatically try to return the plane to zero slip angle. This is the yaw-wise stability discussed in section 8.2. The plane will (under most conditions) eventually establish an approximately correct rate of heading change anyway. The goal of correct rudder usage is to establish the correct yaw-wise motion without a slip developing even temporarily.

The dependence of adverse yaw on airspeed can lead to trouble. Pilots spend almost all of their time buzzing around at cruise airspeeds, where ignoring the rudder is OK or nearly so. Sometimes this leads to complacency. The problem arises on approach and/or departure, where airspeeds are much lower. Proper coordination becomes more challenging, exactly at the place where it is most important (since the margins for error are also smaller). If you mishandle the ailerons at low speed and low altitude, you could well cause a spin or a snap roll, with no chance for recovery.

Section 11.7 describes a few useful tricks for perceiving exactly how much rudder is needed to achieve perfect coordination.

8.10  Long-Tail Slip

Now let’s see what happens while the airplane is in an established turn. In particular, let’s consider an airplane with a fairly long fuselage, flying in a fairly tight turn. As shown in figure 8.9, there is no way that the airflow can be lined up with the front part of the fuselage and the back part of the fuselage at the same time. The fuselage is straight, and the path through the air is curved. You can’t have a straight line be tangent to a circle at two different points. You have to choose.

long-tail-slip
Figure 8.9: Airplane in a Tight Turn — Rudder Neutral

If left to its own devices, the airplane will choose to have the vertical fin and rudder lined up with the airflow. The fin/rudder combination is, after all, an airfoil. Airfoils are good at producing tremendous forces if the wind hits them at an angle of attack. Besides, the tail is way back there where it has a lot of leverage.

Because of the air hitting the sides of the fuselage, and other effects, the fin/rudder might not completely determine the slip angle, but it will be the main determining factor. For sure, the airflow at the front of the fuselage — and over the wing — will have a significant slip component.

This will occur whenever the airplane is in a turn (unless you explicitly deflect the rudder to compensate). I call this the long-tail slip effect. This slip sounds like a bad thing, but in fact it can be put to good use; without it there would be no roll-wise stability, for reasons discussed in section 9.3. Remember: an inadvertent turn will be a slipping turn.

You can see from the geometry of the situation that the amount of long-tail slip is proportional to the length of the airplane and inversely proportional to the turning radius. The latter depends on the square of the airspeed, as well as the bank angle.

In a stubby, fast aircraft like a V-tailed Bonanza in a 15 degree bank at 165 knots, the long-tail slip effect will be small fraction of a degree — hardly noticeable. On the other hand, in a long, slow glider, maneuvering to stay in a thermal using a 45 degree bank at 50 knots, the effect will be fifty or a hundred times greater! You will need several degrees of rudder deflection. You may need to push the rudder pedal all the way to the floor just to keep the air flowing straight over the wings. (Even if you decide to accept a little slip over the wings in order to reduce the crossflow over the fuselage and stabilizer, you will still want inside rudder, and lots of it.)

long-tail-noslip
Figure 8.10: Airplane in a Tight Turn — Rudder Deflected

I emphasize that even though you are holding inside rudder (bottom rudder) during the turn, this is definitely not a skidding turn (unless you get carried away and use too much inside rudder). This rudder usage is completely unrelated to the uncoordinated “boat turn” discussed in section 8.11.

We would like the airflow to be aligned perpendicular to the wings and parallel to the fuselage everywhere, but in a tight turn this is not possible. We have to compromise and “split the difference”. The lowest-drag arrangement is to have at the nose a slight crossflow from inside the turn, and at the tail a slight crossflow from outside the turn.

The best way to check the alignment is with a slip string — a piece of yarn exposed to the airflow where the pilot can see it. Non-experts commonly call this a yaw string, but this is a misnomer. In fact the string measures slip angle, not yaw angle. This is discussed in more detail in section 11.3.

If (as is usually the case) you don’t have a slip string, you can try to infer the alignment by looking at the inclinometer ball. Remember, however, that the inclinometer ball and the slip string actually measure quite different things. The distinction is noticeable whenever the rudder is deflected, particularly in a twin with an inoperative engine, as discussed in section 17.1.3.

8.11  Boat Turn

My friend Larry has a sailboat. It doesn’t have ailerons. You steer it with the rudder.10 This changes the direction the boat is pointing. As shown in figure 8.11, this causes the water to flow crosswise past the hull, creating a sideways force that eventually changes the direction the boat is going.

boat-boat-turn
Figure 8.11: Boat Making a Boat Turn
plane-boat-turn
Figure 8.12: Airplane Making a Boat Turn

All the same words can be applied to an airplane. Keeping the wings level, you press the right rudder pedal. This causes the airplane to yaw to starboard. As shown in figure 8.12, air will then hit the fuselage on the port side, creating a sideways force11 that will gradually shove the airplane around in a right-hand turn. (There will also be a lot of drag, but that is not our concern at the moment.) The force of the wind on the rudder (needed to yaw the plane) is smaller than, and in the opposite direction to, the resulting force of the wind on the fuselage.

In powered flight, the horizontal component of thrust will make an additional contribution to the boat turn. Remember, a turn results from a net force that is not aligned with the way you are going. This includes engine thrust, whenever there is a nonzero slip angle.

To reiterate: the airplane will turn to the right if you hold the right rudder pedal down — even if the wings are not banked. Of course, turning the airplane properly (using the wings) is ten times more effective and more efficient than a boat turn

8.12  Weathervaning During Taxi

When the airplane is on the ground, it feels the force of the ground and the force of the wind.

Since the tail is far, far behind the wheels, a crosswind will create a yaw-wise torque. It will tend to blow the tail downwind, forcing the nose to turn upwind, just like a weathervane.

Now, suppose you are moving (as opposed to parked). The weathervaning tendency causes the nose to turn into the wind. The wheels are still on the ground, making lots of friction, so the airplane will roll in the direction determined by the wheels, i.e. the direction it is heading. Therefore the airplane will travel toward the upwind side of the runway. This may seem ironic or even paradoxical, but it’s true — the crosswind causes the airplane to move upwind.12 You have to deflect the rudder to downwind to compensate.

8.13  Asymmetric Thrust

In a multi-engine airplane, if the engine on one side has failed, or for any reason is developing less thrust than its counterpart on the other side, this will produce a torque (possibly a very large torque) in the yaw-wise direction. This is discussed in section 17.1.3.

8.14  Yaw-Wise Torque Budget — Summary

We have finally come to the end of this section, having covered the most important causes and effects of yaw-wise torques and motions. There are quite a number of such processes:

Some of these ideas will be revisited when we discuss “Dutch roll” in section 10.6.1.

Perceiving coordination and maintaining coordinated flight is important. Further discussion of this topic appears in chapter 11, along with a discussion of how and why to perform intentional slips.


1
For a more-precise definition of what we mean by yaw, see figure 19.10 in section 19.7.1. For a definition of terms such as yaw angle, heading, and slip angle, please refer to section 19.7.3. The terminology and general principles of forces and moments are discussed in section 19.8. Roll-wise and pitch-wise motion are discussed in chapter 9 and chapter 6.
2
... but not the only technique, as discussed in section 8.11.
3
All the examples in this section assume a typical American engine that rotates clockwise as seen from behind.
4
The figure exaggerates the curvature of the stream lines.
5
but if you pay attention you can notice it in many other situations
6
Yaw-wise acceleration (which may be a somewhat unfamiliar subject) is discussed in more detail in section 8.9.1.
7
This analysis ignores the overbanking tendency and various other small effects.
8
It won’t be exactly coordinated because of the long-tail slip effect, as discussed in section 8.10.
9
You will generally need some rudder deflection to compensate for the long-tail slip effect.
10
Boat lovers’ note: there are some ocean liners that do use roll-control devices rather like ailerons, although they are primarily for passengers’ comfort, not for steering. Also, to be sure, there are some boats that can be steered by banking them. On my sailboard, for instance, you have to bank it the wrong way (i.e. to the outside of the turn) by shifting your weight. On some light racing yachts you can steer them pretty well just by shifting the weight of the crew around. Many speedboats bank into the turns. However, we’re getting off the subject. The point is that Larry’s boat (like lots of others) leans to leeward whether you are turning left, turning right, or going straight. The reason it doesn’t tilt any more than it does is because it has tons of lead in the keel. You can’t bank it by shifting your weight, and it wouldn’t turn much if you did. You steer it with the rudder.
11
This force is classified as a lift force, since it is perpendicular to the relative wind — even though it is produced by the fuselage (not the wings), and even though it is horizontal. See the official definitions in chapter 4.
12
In those rare cases where there is inadequate friction on the wheels (such as a seaplane, or an airplane taxiing on a slick icy surface) it is quite possible for the wind to blow the airplane downwind. This of course has nothing to do with torque; it’s just a plain force.

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