Let us say I am flying a constant airspeed turn while maintaining my altitude, when I enter into the turn, lift has to be increased and since airspeed remains constant, the AOA has to be increased to increase lift; so I pulled up the column and increased the elevator deflections to nose up the aircraft to maintain altitude. How is the pitch attitude changed during this maneuver? Does the pitch angle remain the same as the pitch before entering the turn? Is the equation Pitch = AOA + FPA (FPA is zeroed in this case, I think) still applicable during this turn maneuver?
Does the pitch angle remain the same as the pitch before entering the turn? Is the equation Pitch = AOA + FPA (FPA is zeroed in this case, I think) still applicable during this turn maneuver?
No, that equation is not applicable during banked flight. Consider the following thought experiment: assume zero wing incidence. Assume we are flying horizontally with wings level (no bank) at 10 degrees angle-of-attack. The pitch attitude will be 10 degrees. Now we roll (around the aircraft's longitudinal axis) instantly to a bank angle of 90 degrees. We've converted 100% of our angle-of-attack into sideslip, which we might also call "yaw angle".1 But the pitch attitude, as shown on the attitude indicator (see this related answer for depiction of attitude indicator) is still the same. Now, we might note that the aircraft will almost certainly not be in equilibrium in this new attitude unless pilot makes major changes to control inputs (applies top rudder, relaxes all elevator back pressure) and power setting, and possibly also allows the airspeed to change. But the key point that's relevant to the current question is that once the aircraft is banked 90 degrees, anything that the pilot does with the elevator at this point to change the aircraft's angle-of-attack will have zero immediate2 effect on the pitch attitude, shown on the attitude indicator. Likewise, if the aircraft were banked 45 degrees, a given change in angle-of-attack would have less effect on the aircraft's pitch attitude, as shown on the attitude indicator, than if the aircraft were wings-level.
So no, the formula "pitch angle = flight path angle + angle-of-attack" does not apply to banked flight. The correct formula is:
pitch angle = flight path angle + (angle-of-attack * cosine bank angle)
It follows therefore that if the flight path remains horizontal (i.e. if altitude remains constant), any given increase in pitch angle represents a greater increase in angle-of-attack if the aircraft is banked than if the aircraft is wings-level. And the difference is given by the cosine of the bank angle.
So what happens to the aircraft's pitch attitude as the aircraft enters a banked turn, while holding altitude and airspeed constant? Let's start by simplifying the problem by assuming that the wing has no twist (washout), has a fully symmetrical airfoil, and is mounted to the fuselage at zero degrees incidence. Since the required lift force (for horizontal flight) is equal to the weight / cosine bank angle, it logically follows that as the pilot increases the bank angle, while also increasing the angle-of-attack as needed to hold airspeed constant, and also increasing power as needed to hold altitude constant, the aircraft's pitch attitude should remain constant, if the slope of the curve of lift coefficient versus angle-of-attack is linear.3,4
Why might this seem counterintuitive? I'd suggest that one reason is that in actual practice, pilots rarely attempt to hold airspeed constant while entering a turn. If cruising at a high airspeed, well on the "right side" of the power-required curve, many light airplanes will be unable to maintain altitude in a steep turn unless the pilot increases the angle-of-attack enough to actually cause some decrease in airspeed, despite the increased wing loading. For shallower turns, if short in duration, many pilots may just leave the throttle alone and accept a decrease in airspeed as they apply back pressure as needed to maintain altitude while entering the turn, even if more power would actually be available if the throttle position were changed.
But read on for other reasons why a pilot typically will observe an increase in pitch attitude as he or she enters a turn while maintaining constant altitude and airspeed.
Consider the effect of adjustments to the height of the "index" dot on the attitude indicator. If the pilot has, for convenience, set the attitude indicator to show a pitch angle of zero in unbanked horizontal flight, even though the aircraft is actually slightly nose-up to create the required angle-of-attack, will it still be true that no change in indicated pitch attitude will be seen if the pilot enters a banked turn while holding airspeed and altitude constant, given the assumptions noted above of a lift curve with a linear slope, etc? One way to think about this is to imagine an attitude indicator with two reference "dots" rather than one, one vertically (in the aircraft's reference frame) below the other. It's clear that as the aircraft rolls from wings-level into a steep bank, only one of these "dots" can remain on a constant pitch attitude reference line on the attitude indicator. If the upper "dot" remains on a constant pitch attitude reference line, the lower dot will end up on a higher pitch attitude reference line than it started at, indicating that the aircraft's pitch attitude has increased. Similarly, if the pilot has selected a visual reference point on the cowling that is below the "ideal" or "correct" reference point, then as the aircraft enters a banked turn while holding altitude and airspeed constant, that visual reference point will appear to be rise, in terms of its apparent distance above or below the horizon -- and here we're using "up" and "down" relative to the reference frame of the earth, i.e. perpendicular to the line of the horizon.
So if the "height" of the reference dot on the attitude indicator (or the "height" of the visual reference point on the aircraft's cowling, windscreen, etc) makes a difference, what is the "correct" way to set the height of the reference dot on the attitude indicator, or to choose a visual reference point? If the observed pitch attitude is to remain the same regardless of bank angle, assuming a linear slope to the curve of lift coefficient versus angle-of-attack, it appears that the pitch attitude must be measured in relation to a line that is parallel to the zero-lift line of the wing5. If the wing has positive incidence relative to the fuselage, or if the wing has a non-symmetrical airfoil so that it still produces some lift at zero degrees angle-of-attack, then the appropriate reference line will be inclined in the nose up, tail-down direction in relation to the actual longitudinal axis of the fuselage. In this case, setting the reference "dot" on the attitude indicator to be right on the horizon in actual wings-level cruising flight will cause the indicated pitch attitude to increase as the aircraft enters a banked turn while maintaining altitude and airspeed. And similarly, a visual reference point that was right on the visual horizon6 in actual wings-level cruising flight would rise above the horizon as the aircraft enters a banked turn while maintaining altitude and airspeed.7 But a visual reference point on the canopy chosen such that a line from the pilot's eyes to that point was parallel to the zero-lift line of the wing should stay a fixed distance above or below the visual horizon-- measuring perpendicular to the horizon, rather than in the aircraft's own local plane of "up" and "down"-- as we enter a banked turn while holding altitude and airspeed constant.
In summary, here are some reasons why we typically observe an increase in pitch attitude as we enter a banked turn, while holding altitude and airspeed constant, even given the assumption of a wing with a linear curve of lift coefficient versus angle-of-attack:
If we are making observations simply by picking a point on the cowling or nose that lies directly ahead of the pilot (in the left-right sense) in wings-level flight, a line from the pilot's eyes to that point will always be inclined significantly downwards in relation to the longitudinal axis of the fuselage, which means that that point would always tend to rise toward the horizon (measuring perpendicular to the horizon, not in the aircraft's own local up-and-down plane) if the aircraft were to increase bank angle while maintaining a constant pitch attitude relative to the horizon. This would give the false impression of a nose-up change in pitch attitude.
Due to wing incidence and due to having a non-symmetrical (cambered) airfoil, the zero-lift line of the airfoil is not aligned with the actual longitudinal axis of the aircraft. This causes an actual nose-up change in (fuselage) pitch attitude as the aircraft enters a banked turn while holding altitude and airspeed constant, even in the ideal case where the slope of lift coefficient versus angle-of-attack is linear.
Even in aircraft with a fully symmetrical airfoil, mounted to the fuselage at zero incidence, if we "zero" out the reference dot on the attitude indicator in wings-level flight, or if we pick a visual reference point on the windscreen that is right on the horizon in wings-level flight, we're still using a reference line that is not parallel to the zero-lift line of the wing (and the longitudinal axis of the fuselage.) This will cause an increase in the indicated or apparent pitch attitude, even if the actual pitch attitude remains exactly constant as the aircraft enters a banked turn while holding altitude and airspeed constant. To see an indication of a constant pitch attitude as we increase the bank angle, we'd have to set the reference dot on the attitude indicator to correspond to the aircraft's actual pitch attitude in wings-level flight, or pick a visual reference point on the windscreen, etc that was an appropriate distance above the horizon to correspond to the aircraft's actual pitch attitude in wings-level flight.
Disclaimer-- the geometry involved here is complex. My intuitive hunch would have been that the aircraft's pitch attitude would need to increase as bank angle is increased in a constant-altitude constant-airspeed turn, even in the idealized case of a fully symmetrical wing mounted to the fuselage with zero incidence. I haven't confirmed any of the results given in this answer by reference to any outside sources, nor have I actually made any in-flight observations to confirm or contradict these results.
It would be most interesting to see answers to this question based on careful observations in actual flight (particularly, recent observations made with a fresh and open mind, for the specific purpose of answering this question.) Links to video including a clear view of the attitude indicator would provide added value. The most interesting results might be obtained from an aircraft with a fully symmetrical airfoil, mounted to the fuselage at zero degrees of incidence. Answers based on observations from high-quality flight simulators might also be of some interest.
Using "yaw angle" to describe the angle, in the yaw plane of rotation, between the aircraft's longitudinal axis and the actual direction of the flight path, would be an informal or non-standard usage. In this answer to a related question, in the discussion of "Euler angles", the concept of "yaw angle" would clearly correspond to the aircraft's heading, not the sideslip angle. The basic point remains that rolling toward a steeper bank angle tends to convert angle-of-attack into sideslip. However, unless the roll rate is very high and the aircraft's intrinsic pitch stability and directional/yaw/"weathervane" stability are quite low, this dynamic will tend to be obscured by the aircraft's tendency to return to the "trimmed" angle-of-attack and slip angle, i.e. the angle-of-attack and slip angle corresponding to the current position of the flight controls. So this "conversion" of angle-of-attack into sideslip is surely not the primary reason that we need to apply rudder along with ailerons to keep the ball centered while rolling in most aircraft. Normally, aerodynamic yaw torque due to displaced control surfaces, and also due to the "twisted" nature of the relative wind while rolling and the resulting "twist" in the lift and drag vectors, are the key drivers of slip while rolling. "Your mileage may vary", particularly if you are flying an aircraft with minimal aerodynamic adverse yaw, relaxed directional/weathervane/yaw stability, and a lightning-fast roll rate.
Re "zero immediate effect" -- we're referring to the immediate relationship between a rotation about the aircraft's pitch axis, and the aircraft's pitch attitude. At 90 degrees bank, they are completely de-coupled. Of course, in the longer term, changing the angle-of-attack will change the drag coefficient, which will influence the airspeed, which will influence the amount of skyward aerodynamic force produced as the aircraft flies along in a slipping attitude (knife-edge flight), which will affect the trajectory of the flight path, which will affect the aircraft's pitch attitude.
Re "if the slope of the curve of lift coefficient versus angle-of-attack is linear in that part of the flight envelope"-- this is indeed a good approximation for much of the flight envelope for most aircraft, especially with unswept wings. See for example this section of the excellent "See How It Flies" website by John S. Denker. See also this image. (Source of image is unknown, but similar content appears in the book "Aerodynamics for Naval Aviators".
This observation is inspired by similar content in this related answer to the same question-- yet the alert reader will notice some significant differences in the explanation as to "why" this is so, including the mathematical formulae.
Re "If the observed pitch attitude is to remain the same regardless of bank angle, assuming a linear slope to the curve of lift coefficient versus angle-of-attack, it appears that the pitch attitude must be measured in relation to a line that is parallel to the zero-lift line of the wing."-- the point is that if the curve of lift coefficient versus angle-of-attack is linear but the y-intercept is not at 0 degrees angle-of-attack, we essentially must shift our angle-of-attack measurement so that it's zeroed when the wing is creating zero lift. For a given airspeed, lift will be directly proportional to this "shifted" angle-of-attack, not the actual angle-of-attack, and for a given bank angle and airspeed, lift will be directly proportional to (1 / (cosine bank) multiplied by this "shifted" angle-of-attack, not the actual angle-of-attack. Similarly, a "shifted" pitch attitude that is measured relative to the wing's zero-lift line, not the aircraft's actual pitch attitude measured relative to the longitudinal axis of the fuselage, is what we should expect to remain constant as we vary the bank angle while holding airspeed and altitude constant. If the pitch attitude measured relative to one particular reference line (e.g. the zero-lift line of wing) remains constant as we vary the bank angle (under any given set of constraints, such as while holding altitude and airspeed constant), then the pitch attitude measured relative to any other reference line (e.g. the longitudinal axis of fuselage) cannot also remain constant. See footnote 7 for more.
Here we're assuming that the visual horizon line is not depressed below the true horizonal plane due to the earth's curvature. If the visual horizon line is significantly depressed below the true "horizontal" plane, this will cause a shift in the aircraft's perceived pitch attitude, but this shift appears to be independent of bank angle, bearing in mind that pitch attitude is measured perpendicular to the horizon, not in the aircraft's own local plane of "up" and "down". So any depression of the visual horizon line due to the earth's curvature ought not complicate the relationship between changes in bank angle, and changes in resulting (apparent) pitch attitude, while airspeed and altitude are held constant.
One way to think about this is to recognize that if the wing has positive incidence, then for horizontal flight at any given angle-of-attack, the aircraft's nose is depressed earthwards less when the aircraft is banked than when the aircraft is wings-level, again measuring perpendicular to the horizon, not in the aircraft's local up-down plane. The same logic applies to a sightline defined by the line between the pilot's eyes and a spot on the cowling out in front of the pilot-- this line always slopes downward (earthward) in wings-level flight, and the earthward slope of this line always becomes less when aircraft is banked, so the selected spot on the cowling will always move up toward the horizon as the aircraft is banked while holding the pitch attitude (as defined either by the longitudinal axis of the fuselage or the zero-lift line of the wing) constant.
Some other related content on ASE:
Q: How does the angle of attack vary in turns? -- (see various answers)
Pitch attitude will be increased slightly as one rolls into a bank to increase the AoA in order to maintain a vertical component of lift equal to the weight of the aircraft. In addition power or thrust will be increased slightly when entering the turn to compensate for an increase in induced drag, as a consequence of increased AoA, and maintain a constant airspeed. Rudder will be used, as required, to counteract adverse yaw, both during the roll into the bank and once the turn is established.
For very shallow banked turns an increase and pitch will only be slight, if anything at all. For moderate or steeply banks turns, the increase and pitch will be much more noticeable, as the required lift in the turn increases.
Let us say I am flying a constant airspeed turn while maintaining my altitude, when I enter into the turn, lift has to be increased and since airspeed remains constant, the AOA has to be increased to increase lift; so I pulled up the column and increased the elevator deflections to nose up the aircraft to maintain altitude. How is the pitch attitude changed during this maneuver? Does the pitch angle remain the same as the pitch before entering the turn?
Observations from actual in-flight experiments--
Aircraft: Glassflugel Libelle sailplane
The aircraft had no attitude indicator (artifical horizon), but the pitch attitude was observed in two different ways, described below.
Results: When the aircraft transitioned from wings-level flight at 50 mph to a stabilized 45-degree banked turn at the same airspeed, the pitch attitude shifted slightly in the nose-up direction.
Conclusion: If the same aircraft had been flying under power at the same airspeed, with the power varied as needed to keep the vertical speed at zero so that the altitude remained constant, the variation in pitch attitude would have been in the same direction as described above, but slightly larger in magnitude. So in this case as well, the pitch attitude would have been more nose-high in a constant-banked turn than in wings-level flight at the same airspeed.
Methods of observing pitch attitude:
While pilot's head was held firmly in position against the top of the canopy (using a folded cloth as a cushion), a mark was made on the front of the canopy to represent the position of the horizon in wings-level flight. Then it was carefully noted whether the mark ended up above or below the horizon when the glider was established in a turn.1,2
An array of bubble levels (to cover a wider range of angles than could be measured with a single bubble level) was attached to each side of the cockpit wall. Each bubble level was oriented in the fore-and-aft direction, i.e. as viewed from above, parallel to the aircraft's longitudinal axis. The bubbles were observed to note whether they were displaced forward (indicating a pitch change in the nose-up direction) or aft (indicating a pitch change in the nose-down direction) in circling flight, compared to in wings-level flight.3
The observed change in pitch attitude was small enough that it was only consistently detectable when the great care was taken to keep the pilot's head exactly fixed in position. The relatively close proximity of the front of the canopy to the pilot's eyes in this particular aircraft added to the challenge of getting reliable measurements. This method would be easier to use if the windscreen or canopy were further from the pilot's eyes. If the windscreen or canopy is out of the pilot's reach, a series of closely spaced reference marks, oriented vertically, should be applied before flight.
Note that there's a presumption that this sight line is nearly parallel to the longitudinal axis of the aircraft. Looking over the geometry of this aircraft, this appears to be a reasonable presumption. The related answer Aircraft pitch and AOA change during a coordinated turn maneuver goes into more detail in the geometry involved here, but in short, in any given transition between wings-level flight and a steady-state turn, regardless of what parameters we specify are being held constant (e.g. airspeed, angle-of-attack, pitch attitude), there is always some fore-and-aft reference line on the aircraft that stays in a constant "pitch attitude" relative to the horizon. (If this reference line happens to coincide with the aircraft's longitudinal axis, then we say that the pitch attitude is remaining constant.) In the same transition between wings-level flight and turning flight, any reference line oriented more tail-up/nose-down than the line described above will experience a nose-up shift in pitch attitude, and any reference line oriented more tail-down/nose-up than the line described above will experience a nose-down shift in pitch attitude. This is a simple function of geometry, not aerodynamics. Another way to explain this is to note that, while transitioning to wings-level flight to turning flight, if the reference "dot" on an attitude indicator is seen to stay at a fixed pitch indication, then adjusting the height of the dot upwards would cause the indicated pitch attitude to shift in the nose-down direction during the same transition, and adjusting the height of the dot downwards would cause the indicated pitch attitude to shift in the nose-up direction during the same transition. So if an aircraft were flying in an extreme nose high pitch attitude, then a reference line drawn from the pilot's eyes to the horizon would be aimed well below the actual longitudinal axis of the aircraft, and would tend to give an overestimate of any increase in aircraft pitch attitude during the transition from wings-level flight to turning flight, and might actually give an appearance that the pitch attitude was increasing, when it was in fact staying constant or decreasing. Note also that by the same logic, since in most aircraft (including the glider described here) the zero lift line of the wing is inclined significantly upwards (nose-up, tail-down) in relation to the aircraft's longitudinal axis in horizontal flight (or in a sailplane that is gliding efficiently), the observed increase in pitch attitude described here is actually not inconsistent with the idea that the zero-lift line of the wing is keeping a constant angle with the horizon, as viewed from the side. This would be consistent with the idea expressed in other answers that as the bank angle is increased, the increase in angle-of-attack might be exactly geometrically compensated for by the increase in bank angle, so that the actual pitch angle of some given reference line does not actually end up increasing. This statement was clearly not true in these experiments if we pick the aircraft's longitudinal axis for the reference line, but it might be true of the wing's zero-lift line. It would be most interesting to repeat these experiments in an aerobatic aircraft with a symmetrical airfoil and zero wing incidence. It seems possible that in such a case, the aircraft's pitch attitude might be the same in a constant-altitude banked turn, as in wings-level flight at the same airspeed.
Note that in a steady-state turn (constant airspeed, vertical speed, bank angle, and pitch attitude), as viewed from the side, the bubble in a level mounted in this fashion will seek to position itself in curved tube of the level so that it is oriented "straight up" relative to the earth. As long as the level is not mounted so that it is "toed in" or "toed out" relative to the aircraft's longitudinal axis as viewed from above, it will not be affected by the forces that displace the slip-skid ball during a slip or a skid. A slip-skid ball instrument, mounted parallel to the aircraft's longitudinal axis as viewed from the side, rather than in the usual orientation on the instrument panel, could also have been used for this experiment. Naturally, it would tend to orient itself "straight" down relative to the earth, not "straight up". And it goes without saying that none of these instruments will orient themselves "straight up" or "straight down" relative to the earth, if the flight path is curving earthward or skyward-- e.g. during a loop-- that's why we can't use bubbles or balls to fly in clouds.
Let's say you are in level constant speed linear flight where AoA = Pitch to the horizon.
Roll to 45 degrees and vertical lift is now cosine bank angle × lift = .707 × Lift. Pitch (to the horizon) is now also .707 of (level flight) AoA, or 70.7%, because the aircraft has rolled 45 degrees from horizontal into a new plane of flight. As a result, it's AoA points into the circle, as does the lift vector.
Lift is approximately linear to AoA so:
AoA must be increased by around 41% to keep flight path in a circle level to the horizon.
1.41 × .707 = 1! Your pitch to the horizon is the same in the turn as in level flight because the plane is banked, but AoA is higher.
It is easy to confuse "pitch to the horizon" and "pitching the aircraft". Validation of the term "pitch angle" as the longitudinal axis of the aircraft relative to the horizon can be found here.
Aircraft pitch motion is defined relative to the aircraft: when it rolls, the pitch axis rolls with it. And seen from the plane, pitch angle is still Angle of Attack plus Flight Path Angle - zero deg. in a coordinated turn at constant altitude.
On the instrument, indicated pitch is relative to the banked horizontal, all quantity reference lines bank with it. The horizon on the instrument is a memory of earth gravitational field, and is therefore relative to earth axes. Indicated pitch angle on the instrument is relative to where the horizon was in horizontal flight = pure pitch.
If you want to maintain 2Gs, your bank angle must be 60 degrees. And your a/c will be needed more lift to maintain altitude. So the only way to produce more lift without changing config’ is pulling up your stick to make your nose up. This pulling up motion will increase the gap between flight path and nose(AOA). So for maintaining 60 bank angle and altitude, your nose have to be upped more than level.