I think I understand how a high wing differentiates from a mid wing or a low wing; what I am asking is how, for instance, a very high high-wing makes itself unique in flight from a very low high-wing.
So let's say you have a high-wing aircraft. Raise the wing vertically to make the high-wing even higher. What does this do to the airplane's flight characteristics?
First off, it's important to realize that a wing does not constitute a fulcrum in a pendulum. There's no magical hinge that a plane rotates about, hidden in the wing. This is known as the pendulum fallacy.
Instead, think of an airplane purely in terms of incoming airflow. If the air is coming from straight ahead, there's not much difference between a low wing and a high wing. The main difference is that interference effects at the wing root are most prominent at the suction side (top of the wing) in a low wing aircraft, which may affect performance. Mid fuselage would be ideal, like on most gliders.
A more prominent effect is when the wind is not from straight ahead but from the side, like in a gust or sideslip. This will increase the pressure at the windward side of the fuselage. For a low wing, this high pressure pushes the windward wing down, whereas a high wing will be pushed up. Ideally, you want the latter, since the induced roll will make the plane slip the other way, stabilizing towards a coordinated turn. So a high wing is superior in this regard, although dihedral can also provide this stabilization. A high wing far away from the fuselage would lose this benefit.
In reality, wing placement is governed mostly by practical issues like obstacle or water clearance, landing gear and engine placement, and not having a wing spar in the middle of the passenger cabin. Dihedral or anhedral is then used to achieve the desired stability.
The most important advantage that a higher wing gives you, is the ability to land on 'airports' that have obstacles directly adjacent to the runway, such as tall grass or snow banks, or surfaces that may provoke roll, such as water. It also allows you to have the CG determine your airplanes tendency to keep speed constant, rather than only the wing and tail shape. With a very high wing, pitching up or down will change the projection of the lengthwise distance between neutral point and CG on the horizontal plane much more than on mid- or low wing aircraft. This means that at high pitch angle your static stability is larger than at low pitch angle.
A good example of such an aircraft is the Consolidated PBY Catalina (picture source).
So let's say you have a high-wing aircraft
Allright, I choose the Dornier Do-24 ATT, a very well behaved high wing airplane with excellent seaworthiness. Below I show it in frontal and side view with lift and weight forces symbolized by arrows. I decomposed the lift into its zero-lift moment and put the lift vector at the neutral point, where all angle-of-attack-dependent lift acts, so the lengthwise distance between weight and lift vector is proportional to the static longitudinal stability and is delineated by a pair of arrows. Of course, when you shift the lift vector into the center of pressure, it will be aligned with the weight vector and the zero-lift moment will disappear.
This is my baseline high wing design, and next I raise wing and tail by a grotesque amount. No less will do to make the differences obvious.
Raise the wing vertically to make the high-wing even higher. What does this do to the airplane's flight characteristics?
If we neglect the additional vertical surface at the back, what has changed are inertias in all axes, which have grown by the increased distance between the horizontal surfaces and the fuselage. Also, the center of gravity has moved up a bit.
What are the consequences for the flight characteristics?
Now what about lateral stability? And what about pitch changes away from level flight at cruise speed? For this we need another sketch. The baseline first:
On the left, the airplane flies a coordinated turn with a bank angle of 45° and on the right it is in level flight close to stall at 15° angle of attack. Turning adds a centrifugal force which acts at the center of gravity and needs a bank angle and lift increase so lift is sufficient to balance the resulting mass forces (denoted as R here). Since both forces act along the centerline, no imbalance or instability comes with the high wing arrangement. However, the side sketch now reveals a larger distance between the mass and lift forces, which means that the airplane becomes longitudinally more stable at low speed. The zero-lift moment has to become larger to trim this high pitch angle by incasing the negative elevator deflection. Note that the green circle has grown in size and weight to reflect this. The increased elevator travel helps to keep stick forces at low dynamic pressure up and requires more elevator travel for stalling than in a low wing configuration.
And now we do the same for the high wing version:
Again, not much has changed, only that the stabilizing effect of a high pitch attitude is now even more pronounced and the zero-lift moment needs to become even larger than before. This will require a bigger tail, or only a small range of angles of attack can be trimmed. Turning feels the same, apart from the need to overcome the higher inertia with more forceful commands, and again no instability can be seen.
But a low wing ... a low wing will still show the same results. Angle of attack changes mean less change in static longitudinal stability and lateral stability is unaffected. Only the effect of the fuselage on the yaw-induced rolling moment will require a low wing to have more dihedral which makes it ever so slightly less efficient. But mayhem and carnage fail to manifest themselves.
And what does an uncoordinated turn do? Now the airplane will sideslip and the weight vector will not be aligned with the centerline in the frontal view. But the lift vector, still orthogonal to the wing, will, so no rolling moment from weight or wing lift develops. Only the side force of the vertical tail and the fuselage might cause a very small roll contribution which can easily be balanced with a bit of aileron. Again, mayhem and carnage fail to manifest themselves.
The central fallacy here is the comparison with the broom, balanced on a fingertip. Airplanes (and drones or rockets, for that measure) are different. Conservation of momentum dictates that all rotations take place around the center of gravity, and lift, being the result of surface pressures, is always orthogonal to the surface of the wing. In consequence, regardless of wing position, a bank angle will not cause a destabilizing rolling moment.
Placing the wing above the center of mass of the airframe means that perturbations in the three axes will generate a righting moment which tends to oppose the perturbation. The plane can then be positively stable in the hands-off condition and will to some extent "fly itself".
Placing the wing (the center of support) below the center of mass is then like balancing a pencil (with its center of mass at its midpoint) by its tip on your finger (the center of support). Once the pencil begins to rotate and fall, the rolling moment becomes stronger (as the center of mass gets offset relative to the center of support) which means those perturbations will grow instead, and the plane will be divergent (dynamically unstable, i.e. it will display "negative stability") in all axes- and if flown hands-off, it will try very hard to invert itself and fly upside-down. Mayhem then ensues, and carnage will result.
You can imagine that any change to the outside shape of an aircraft or anything that affects its centre of gravity or angular inertia will affect its flight characteristics. On the face of it, a high wing will impart more stability but it may compromise other things such as placement of engines and undercarriage, airflow over the empennage and control surfaces, characteristics while in ground effect and so on. In aircraft design there are very many factors to take into account and it’s difficult to consider one in isolation from the others. Trivially, high wing has more inherent stability than low wing but there are many ther factors.
Weight distribution is critical in determining the effects of making a high wing even higher.
While trial and error with balsa models will keep one busy, some educational background is very helpful in eliminating the more embarrassing moments.
As a designer, try to plot any changes in center of gravity and center of drag.
Generally with aircraft, raising the wing moves the center of drag higher. This can make the plane more difficult to control in a crosswind. As seen with the PBY 3 Catalina, the designers raised the center of gravity by placing the engines on the wings to reduce this effect$^1$.
Having the Cdrag above the Cg means that ailerons must be held in a turn or the plane will tend to roll away. This is the "self righting tendency".
Again depending on the new CG, raising the wing may also give the plane a greater tendency to pitch up. As this will vary with forward airspeed and AoA, one might wonder if too much of this may actually be undesirable.
$^1$ keeping the props away from the water as well
