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As you make a turn, an airplane does not naturally straighten itself out (like a car does to some extent). Are ailerons engineered to stay in whatever position they are placed, or is this simply a byproduct of aerodynamics?

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I think it's a little bit of both -- for most aircraft, how they behave in a turn is a byproduct of their designed aerodynamic properties. This question is actually quite complex to answer. Let's assume you're talking about a simple light aircraft, where control surfaces are mechanically linked to the controls that move them -- this ensures that anything you feel is related to what's happening to the plane, and not necessarily to anything more complex (like a fly-by-wire computer, or a hydraulic linkage with some degree of hysteresis).

Aircraft turns are divided, by the CAA & FAA at least (see the FAA's Airplane Flying Handbook, Chapter 3) into three main categories:

  1. Shallow turns (roughly speaking, <20º angle of bank). These tend to become shallower if left alone.
  2. Medium turns (roughly speaking, 20º < $\theta$ < 45º angle of bank). These turns tend to be stable without further pilot input.
  3. Steep turns ($\theta > 45$º). These tend to become steeper if left alone, sometimes dramatically so.

Let's look at these in a bit more detail, with some explanation as to why these effects occur.

Shallow turns. These tends to become less steep as the turn progresses, and most aircraft have a natural tendency to decrease the angle of bank throughout the turn. This tendency to lessen the angle of bank arises, I believe, entirely as a designed aerodynamic feature of the aircraft to have lateral stability. As shallow angles of bank are essentially a situation that's is but a small perturbation about flying straight and level, it's desirable for the angle of bank to decrease back to zero if left alone. If that isn't the case, the aircraft stability has to be actively maintained by constant manipulation of the control surfaces (which is tiring or impossible for a human). This stability can be achieved through design features, such as (i) the dihedral of the aircraft's wings, (ii) their sweep, and (iii) their height position -- if high, there is an increased degree of lateral stability due to the centre of mass of the aircraft being below the tilted plane of the wings, and, if disturbed slightly, it experiences a force $F\approx mg\sin\theta$ to "pull" the wings back upright again. The sole purpose of any degree of wing dihedral -- the angle they make with the horizontal -- is to improve lateral stability. If a perturbation causes one wing to drop, the resultant (unbalanced) force will produce a sideslip in the direction of the downward wing. In effect this is analogous to "creating" a flow of air in the opposite direction to the slip. This flow of air will strike the lower wing at a greater angle of attack than it strikes the upper wing. The lower wing will thus receive more lift and the airplane will roll back into its neutral position. Many small aircraft have a slight degree of dihedral on the wings. A similar effect occurs with (ii) swept wings -- when a disturbance causes an airplane with swept back wings to slip or drop a wing, the low wing presents its leading edge at an angle that is almost perpendicular to the relative airflow. As a result, the low wing acquires more lift, rises and the airplane is restored to its original attitude. As an aside, you don't really notice you're in a shallow turn, aside from the fact that you're continually applying a little bit of stick and rudder. The $g$-loading is minimal.

Medium turns. In this case, the aircraft is in a metastable state -- it will carry on at a constant angle of bank, but a `shove' to either side will result in the bank increasing or decreasing as appropriate. By this point, $\cos\theta\neq1$, and the outer wing is really generating a lot more lift than the other one -- and thus you'd effect the angle of bank to increase, as it pulls the aircraft into the circle. The effects discussed above, however, still apply, and provide an effective restoring force back to the normal attitude. In a medium turn, aircraft are designed to have these forces balanced, and, in practice, you often need little-to-no aileron and almost no rudder to do a coordinated medium turn in a light aircraft. If it's properly trimmed, you can take your hands off the controls and sit there merrily going around in a circle. Depending on your speed, the $g$-loading is apparent, but not great.

Steep turns. By this point, one wing is generating almost no lift, and the other is generating quite a lot. We've gone beyond the designed stability point of a medium turn, and the angle of bank tends to increase as the outward wing pulls the plane of the aircraft towards its axis of turn. This type of turn actively requires pilot input to maintain a constant angle of bank, and, depending upon the speed, it can physically be quite hard work. The $g$-loading of the wings is significant, and, as the stall speed of an aircraft approximately increases as the square-root of the $g$ factor experienced, it's entirely possible to enter an accelerated stall or even a spin here. You notice the angle of bank, and feel the odd attitude of the aircraft. This phenomenon is called the overbanking tendency.

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All control surfaces are designed to return to a neutral position if no control force is applied. This is part of the certification requirements for every aircraft. JAR 23.177(d) reads:

In straight, steady sideslips at 1·2 VS1 for any landing gear and flap positions and for any symmetrical power conditions up to 50% of maximum continuous power, the aileron and rudder control movements and forces must increase steadily (but not necessarily in constant proportion) as the angle of sideslip is increased up to the maximum appropriate to the type of aeroplane. At larger sideslip angles up to the angle at which full rudder or aileron control is used or a control force limit contained in JAR 23.143 is reached, the aileron and rudder control movements and forces must not reverse as the angle of sideslip is increased. (…)

Note that if you keep ailerons in a deflected position, the aircraft will pick up roll speed until an equilibrium with roll damping is reached. In order to maintain a desired bank angle, the pilot has to actively stop the rolling motion by deflecting the ailerons briefly against the roll. This is different from steering a car where the steering angle is proportional to the turn rate.

Since the ailerons are hinged close to their forward edge, the aerodynamic forces will "weathervane" them into the direction of the local flow. By selecting a hinge line and sizing the part of the control surface ahead of the hinge line, the aircraft designer can tailor the control forces so they are positive at all speeds, but low enough to not tire the pilot. Many ingenious solutions have been tried especially for ailerons, and only the application of hydraulics has let this art form within the field of engineering wither in the last decades.

Don't get me started on how exactly you can tailor roll stick forces - this will be a post which will be much longer than anything I have written here before.

How and why an aircraft can or cannot right itself up when flying a turn is discussed at length in this question.

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Aircraft can be designed to be aerodynamically stable around each of its axis: pitch, roll and yaw.

Most aircraft (except modern fighters) are designed to be aerodynamically stable in pitch (in angle of attack, actually) and yaw. However making the aircraft aerodynamically stable in both yaw and roll causes an uncomfortable dutch roll oscillation. Therefore they are designed to be only slightly stable in roll with the result that without pilot input shallow banks tend to be, slowly, corrected, but larger bank angles increase in time.

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It is a byproduct of aerodynamics.

A constant aileron deflection gives a constant roll moment. A constant, non null, roll moment gives an increasing roll rate.

During a turn you want a constant bank angle, not a rate and even less a rate acceleration.

To achieve a constant bank angle, once you are in a turn, you only need minimal aileror deflection (the amount depends on the direction of turn if you have an asymmetrical aircraft such a single propeller or a twin with co-rotating engines).

The amount of deflection needed will also be affected by the wing anhedral/dihedral: a dihedral, for example, will give you a roll moment that will tend to level off the aircraft, the ailerons will have to be deflected just enough to balance this moment with an equal and opposite one.

To enter and leave a turn you need a larger aileron deflection because you want to change your bank attitude, you want then to have a net non-zero roll moment (for a relatively brief period of time).

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