Is the amount of dihedral on low wing aircraft based more on landing considerations than on aerodynamic benefits?

Question

During a recent discussion on glider design, a quandary arose over what the aerodynamic effects of moving an otherwise unchanged wing from the top of an aircraft to the bottom would be.

One claim was that it would reduce "self righting" characteristics due to a change in the interaction of the relative wind to the upwind wing/fuselage root and reduced effect on the lift of the downwind wing. This would be the equivalent of increased "anhedral effect".

Another point of view was that lowering the wing lowers CG, increasing area above the CG, which (as an upright tail fin or fuselage component) would increase tendency to roll away from a side wind, or increased "dihedral effect".

In the mean time, high wings and low wings continue to fly on in abundance, with passionate arguments supporting the virtues of each.

With so much ambiguity, even after more than 120 years of aviation to sort things out, is it possible that both designs still exist because one really does not have a significant aerodynamic advantage over the other?

Landing considerations of long winged aircraft are obvious considering the dangers of catching a wing tip and "cartwheeling". Increased dihedral and a high wing position help prevent this. Others state low wings are easier maintenance.

In the air, are they really closer than we think?

Answer 1

Dihedral cannot be chosen in isolation. For good flying characteristics the parameter it controls, namely $\text{c}_{l\beta}$, must have a certain relation to weathervane stability $\text{c}_{n\beta}$. While $\text{c}_{n\beta}$ is positive, $\text{c}_{l\beta}$ is negative, so if you plot both as in the figure below (taken shamelessly from Ray Whitford's Fundamentals of Fighter Design), the ratio can be found in the second quadrant of a coordinate system.

If weathervane stability is high, dihedral must also be higher than usual so the ratio of both coefficients falls into the "satisfactory" area. If the engine-out case drives tail size up, the wing dihedral has to follow so that flying characteristics do not suffer.

The hasty development of the Heinkel 162 resulted in too much dihedral and adverse flying qualities. The quick fix was to add Lippischohren (Lippisch ears), named after the person who used them first, Alexander Lippisch.

He-162 front view (picture source). No, these are not inverted winglets but Lippischohren to correct excessive dihedral.

Answer 2

Some points to consider:

1. Dihedral effect requires sideslip to work. When an airplane banks due to a disturbance, you must have some sideslip develop to create the self-righting differential lift in the first place. The vertical tail volume must be low enough to allow some sideslip to develop. If the fin/rudder is too large, sideslip instantly generates a weathervaning yaw and you get a spiral diving tendency instead of an automatic return to wings level tendency. So there is a whole balancing act on fin sizing you have to deal with first.
2. Pendulum effect to the extent that it exists, isn't a pendulum so much, rather than a form of static stability in a vertical plane (you're floating in a gas - a pendulum requires a fixed attachment). If the aerodynamic center of the body in the vertical axis, the vertical Neutral Point, is offset from the roll rotation axis, the vertical C of G, then when a lateral movement is present the Neutral Point will seek to trail, or weathervane, behind the center of mass so to speak, just as in the yaw and pitch axis. If the Neutral Point is above the vertical C of G, a restorative moment is created similar to dihedral effect because the body with the wind acting on it wants to "trail" the vertical C of G. Take a Cessna 172, and add a heavy weight on the end of a long rigid pole below the fuselage, and skid the plane, and it will roll away from the skid from static stability forces acting in the vertical axis (it's not the "pendulum effect" of the weight down below, it's the body of the plane trying to weathervane behind the lowered center of mass, about which the airplane is rolling, and in fact the same effect will work in weightlessness). I would say that although this effect is present to some degree in most airplanes, slightly positive in high wing airplanes and close to neutral in low wing ones, it's too subtle to notice because the vertical Neutral Point and center of mass are too close together in most cases. However this effect can be observed on float planes due to the mass of the floats moving the vertical C of G down more than they move the aerodynamic center down, and in my experience airplanes have somewhat noticeably better roll stability when on floats than when on wheels.
3. Then there's dihedral effect from actual geometric dihedral of the wings, and dihedral effect that high wing airplanes enjoy from the T configuration, where sideslip generates a righting force just from the lack of side surface area above the wing. A low wing airplane will need more wing physical dihedral than a high wing one to have the same righting effect during sideslip. Lots of high wing airplanes do fine with no wing dihedral at all, just relying on the T junction of wing-to-fuselage. The Wittman Tailwind homebuilt and the CL215/415 water bomber come to mind, designs where mild dihedral effect was all that was wanted, so the wings have zero geometric dihedral. In fact, when the CL415 was developed from the 215, one of the nasty surprises was that the flat PW120 engine nacelles sticking up above the wing killed off some of the T junction effect, as if it had become a mid-wing airplane, and roll stability was insufficient. The odd turned up wingtips of the CL415 are there to create some geometric dihedral effect to compensate, without having to redesign the entire wing (one of the ultimate aerodynamic bandaids).

So in the end, a low wing airplane with a dihedral angle of, say 3 degrees, might have the same roll stability, that is, righting force from sideslip, as a high wing airplane with a T configuration plus 1.5 degrees of geometric dihedral. In the end it's what do you have to do to achieve a given righting tendency from sideslip with whatever configuration you're using, and if you moved a wing from the top to the bottom, you would expect to have to increase (or introduce) dihedral angle to maintain the same righting tendency due to sideslip.

As far as landing conditions go, not really. The best example is the Corsair, where anhedral is used to shorten the gear legs, but must be offset by major dihedral outboard to achieve the desired roll stability. If the Corsair's wings were made straight with longer gear legs, the overall dihedral angle would be way less than the outboard gull wing's has to be.