Why do the wings on a banked plane have different angles of attack?

Factual Questions
1

On a mentour pilot video the host said that when a plane is banking (i.e. rolling) the wing upwards wing has a different angle of attack than the lower wing and this causes the plane to want to roll even further. He explained this has something to do with the airflow across the fuselage to the wing. I’m only mildly curious about planes, aviation and physics, so this makes no sense to me.

As I understand it, angle of attack is the distance from level (i.e. not tilted) along the front to back axis (i.e. higher at the front than the back, or vice-versa). I can not picture how this would change just by the plane banking. I can absolutely see the wings flexing, and a yawing action, but no way can I make sense of the wings somehow tilting (front-to-back) differently (or effectively differently) without the control surfaces (slats, flaps, ailerons, etc.) moving.

2

It requires sideslip–that is, the plane not moving exactly in the direction that the nose points in.

Look at figure 2 here. You can see that the dihedral angle on the wings plus some degree of sideslip results in a higher angle of attack on the lower wing than the upper.

Contrary to popular belief, the dihedral angle does not directly contribute to a wings-level situation (it’s sometimes said that the lower wing experiences more lift due to there being a greater projection on the horizontal axis, but this isn’t really right). But it does contribute to stability of the “spiral mode” due to its combined effect with sideslip.

3

Consider the fact that the outboard wing (the higher wing) is traveling slightly faster than than the inboard wing so therefore the outboard wing would have slightly more lift than the inboard wing. Could this be what is causing one wing to have more lift than the other. However, the effect is so slight that I never remember noticing it when I flew 30+ years ago.

I see no reason why the angles of attack would be different.

4

Thank you. Unfortunately, that doesn’t really help understanding very much. It adds the step I kind of missed from the Mentour video about relative airflow being slightly off-center, but doesn’t explain how sideslip + bank = different angles of attack. Which is what I’ve come to expect from wikipedia.

5

That’s pretty cool.

I think I always heard the incorrect description. Unfortunately that Wikipedia page has poor references, so it isn’t clear where the information came from.
Do you have a good reference to this piece of intriguing information?

6

All I can say is to examine the figures closely. The “camera” is pointing downwind. In figure 1, it’s easy to see that it doesn’t matter how the plane is banked; that can’t change the angle of attack. And the wings are symmetrical so there is never a torque on the plane.

However, there is a lateral force. And this causes a sideslip–the airstream is no longer directly down the long axis of the craft, but slightly angled. This leads to figure 2. You can see, just based on the geometry, that the lower wing has a higher angle of attack than the upper one.

I’ll see what I can dig up, but really you can demonstrate this with a crude model. Fold a strip of paper into a shallow V shape. Look down the axis as if they were the wings on a plane with its nose pointed to you.

You can see, first, that banking the “plane” has no effect on the angle of attack on the wings. Applying nose up or nose down affects the angle of attack, but only symmetrically, so there is no net torque on it.

Now try turning the model on the horizontal plane, as if you had applied rudder. Doing this, one wing now has a positive AoA and the other a negative AoA. And that does clearly produce a torque on the plane. Flat wings don’t produce the effect.

7

The new information is not what I was questioning; I remember seeing images in textbooks of dihedral wing airplanes next to high-wing airplanes along with an explanation of the high wing seeking equilibrium via a pendulum effect and the dihedral wing seeking equilibrium via the different projected horizontal areas of the wings. If that is all hogwash (at least the dihedral part) then it would be nice to see a proper cite that refutes those old textbooks.

And I have no problem with them being wrong; it’s probably something like the “tongue map” we all saw in our science books as children.

8

It tends to be hard to refute things that are nonsense in the first place :).

Let me try to give a taste of why it’s wrong, though. Imagine a plane with a 22.5 degree dihedral. And imagine that it’s banking at 22.5 degrees, so that one wing is horizontal with respect to the ground and the other is at 45 degrees.

Let’s consider segments that are at equal distances from the base. Say, a 1-cm segment that’s 1000 cm from the base. If each of these segments balance, then there must be no net torque on the plane.

It’s true that if the horizontal segment has a vertical lift of 1 N (just to pick an easy number), then the vertical component of the angled segment if 0.707 N. But that is not everything that’s going on: we need to compute the torque along the axis of flight.

The torque from the horizontal wing segment is 1000 cm * 1 N = 1000 N-cm. Easy. The torque from the vertical component of lift on the elevated wing is the projected force times the projected distance: so 0.707 N * 707 cm = 500 N-cm. But there’s the horizontal component as well: that’s the same thing (since it’s 45 degrees), 0.707 N * 707 cm = 500 N-cm. And 500+500 N-cm = 1000 N-cm, so the torque completely balances out. The same goes for paired segments at every distance from the base, so there’s no “pendulum effect”.

It’s not hard to generalize to different angles, of course, but hopefully that give you some idea of what’s wrong with the conventional wisdom.

9

Figure 2 is the airplane seen from the direction it’s moving, relative to the air. It’s moving forward and slightly sideways (sideslip). And as you can see, when the airplane is moving in that direction, the left wing has a higher angle of attack (you can see the underside of the left wing but not the right).

10

Unfortunately, it is not inherently obvious and you haven’t convinced me, even though I tried to follow along with a drawing on my whiteboard. No worries, I’ll just put this on the long list of stuff I have to unlearn for whatever reason. Just like the traditional way that we were taught about how the wing shape provides lift, and nowadays they say it doesn’t work that way.

ETA: I really do appreciate that even if you thought it nonsense you put in the effort to explain it. We need more of that at the 'Dope!

11

I’m happy to try to explain further–though I need a hint as to where you think you’re getting stuck. Nothing beats a whiteboard, though!

I guess I’ll just emphasize that the traditional explanation doesn’t work because it simply ignores some forces. If the plane is going to roll, it must have a torque along the flight axis. And to compute torque you must look at all the forces, both vertical and lateral.

12

First, the effect in sideslip is to stabilize. Look at the diagram - he wind resistance on the exposed side of the aircraft will tend to straighten it out. I’ve experienced this, to sideslip you have to actively hold the rudder and ailerons to force it.

When banked, yes, the torque (sort of “lift”, but not vertical) is balanced. But the fuselage, and presumably most of the weight, is pulling downward (vertically) on both wings relatively equally. Because of dihedral, the level wing provides more lift to counter this than the wing pointing more upward, thus levelling the aircraft. (Same concept as trying to drag a tilted catamaran down under water, for example - the deeper side has more buoyancy, so harder to flip) Simple torque is balanced; torque against gravity is not balanced.

The danger in a bank is that the wings, dihedral or not, do not present the same horizontal component, and hence the same amount of vertical component lift, as they do in horizontal flight. This is easy to understand - bank to 90 degrees and nothing is keeping the aircraft up, it will fall sideways downward. The “lift” is pulling the aircraft toward the top of the fuselage direction, so the plane will be pulled to the side facing fuselage-up. Bank 45 degrees, and the same speed that kept you level will no longer work. You will start to fall as you turn. As anyone who takes flight training knows, when you go into a serious turn, you must push the nose upward, and increase power, to ensure that you maintain level flight in the turn.

Usually a bank is accompanied by a rudder movement to start the aircraft turning, too.

The biggest risk is a spin. At an extreme banked angle, the risk is the inside wing, going slower from being inside the turn, will “stall” or stop generating lift, before the outside wing. Then, the inevitable - the outside wing pulls the aircraft over onto its back, the tail causes the nose to point downward, and the aircraft goes into a deadly spiral downward, eventually straight down. If the propeller is powered, the torque of the engine aggravates this. If it happens too close to the ground, uncontrolled flight into terrain is the byproduct.

13

It’s really simple. If there is no sideslip, then the direction of lift produced by each wing is fixed relative to the airplane. If the airplane tilts to the right, there is no corrective force to roll the plane to the left. Draw a front-end picture of the plane on a piece of paper, and draw arrows representing the lift from each wing. Then tilt the whole piece of paper. Is there now an arrow that is trying to tilt the plane back to the original orientation? No. The arrows are fixed relative to the plane. There is no new torque around the roll axis.

The reason high-wing aircraft are stable is because, when the plane tilts to one side, the plane starts sliding towards that direction (sideslip), and as a result, there is a high pressure region under the wing on that side. Which tries to tilt the plane back to the original orientation. With a low-wing airplane, the same thing happens under the wing, but since there is no fuselage there to trap air, it takes a much greater dihedral to have the same effect.

14

This isn’t exactly right. The comparison to water buoyancy isn’t valid: catamarans are stable because there’s a tremendous difference in buoyancy between water and air. Pushing a floating object just a little further into water takes a great deal of effort. The same is not true of airplanes: the difference in buoyancy from top to bottom at the scale of an airplane is minuscule (just the weight of that column of air).

That said, there is a kind of pendulum effect with regards to the difference between center of lift and center of mass. But this is not really because of the dihedral. A positive dihedral is one means of raising the center of lift, but so is just mounting the wings on the top. Some planes, like a C-5 Galaxy, have a negative dihedral (an anhedral), but because their wings are mounted high (for military reasons: reduction in foreign object damage on poor runways, etc.) the center of lift is still above the center of mass (otherwise it would be extremely unstable).

15

Clearly everything I ever heard about this topic is wrong.

Isn’t the reason high-wing aircraft are stable because the center of gravity is below the center of lift?
Indeed, I thought those downward-angled wings on heavy freight aircraft were intended to counter the severe effect of the heavy freight + high wing combo to make the aircraft more controllable.

ETA: Didn’t see the post just above mine mentioning these heavy planes.
I think I’ll quietly back out of the room and start machining tiny parts for the baby steam engine I’m working on. At least that’s a process I can get my head around.

16

No. That’s like saying a rocket is unstable if the engine is behind the center of mass. Admittedly it’s a common misconception - Goddard’s first rocket was built with a nozzle at the top based on this misconception.

If you had a balloon instead of a wing (or rocket engine), the lift is always upwards (away from the ground), so having the center of lift above the center of mass makes it stable. But wings and rocket engines don’t work like that. Their thrust/lift is in a specific direction relative to the vehicle. If the vehicle tilts, the thrust tilts with it. If the vehicle tilts 90 degrees to the right, the lift/thrust is also 90 degrees to the right, not up. There’s no component of this force that tries to tilt the vehicle back to the left.

17

No, it doesn’t matter where the center of mass vs. center of lift is.

A high wing aircraft is inherently stable because when it sideslips, the fuselage blocks the air flow from the side and creates a high-pressure area under the wing on that side. So even without dihederal, this effect pushes the aircraft back to level. And on some aircraft, this effect by itself makes the aircraft too stable (i.e. not maneuverable enough). A negative dihedral partially cancels out this effect and makes the airplane more maneuverable (less stable).

18

There’s another effect, though–when the airplane banks, the total lift is no longer enough to support the weight of the plane (unless the pilot corrects by altering the airspeed or AoA). The plane loses altitude and experiences an upward airflow. This will tend to make the wings level.

I probably should not have said center of lift, though–this is more of a weathervaning effect based on the total shape of the fuselage. I guess it’s more related to the aerodynamic center of the craft. And it doesn’t appear if the craft stays at a stable altitude–that is, the pilot has corrected for the loss of lift in the banked turn.

19

Possibly a clearer way of looking at it:
As said, the lift vector on a plane always points down (from the plane’s perspective). High or low mounted wings don’t change this, nor does dihedral or anhedral. The wings are symmetrical and any lateral lift is completely canceled.

However, lift is not the complete story. Wings don’t have a zero cross-section, and the fuselage has to be shaped to support the wing. And this is (one place) where other effects come into play–the fuselage shadowing effect you mentioned, and the weathervaning effect I mentioned (and really these two effects overlap). Probably there are other effects as well.

This only arises if the airflow is not directly down the main axis–either because of sideslip or loss of altitude. So the usual explanation is still wrong.

20

OK thanks, yes I see that. If the craft is moving downwards, then it does matter if the center of aerodynamic drag (which I suppose is the center of lift) is above or below the center of mass. That’s why a parachute is stable - because it’s moving down, and the center of mass is below the center of drag.

Similarly, on a rocket, the position of the engine doesn’t matter, but the center of aerodynamic drag does matter. You want it to be behind the center of mass. That’s why rockets often have tail fins - to move the center of drag to the rear. Same with the fletchings (feathers) on arrows.