Copyright © 1999 jsd
In a number of mailing-list articles and in his web page http://www.amasci.com/wing/rotbal.txt,
Bill Beaty advocates using a collection of disk-shaped balloons
as a model for how wings work.
This model has some interesting features. For one thing, the size
of the balloons is chosen to be proportional to the wingspan of
the airplane. This is a good choice. This portrays the airplane,
as it flies along, as affecting a swath of air not just right
near the wing, but extending horizontally and vertically
for a distance comparable to a wingspan. This is a good match
to what a wing really does. (This is in pleasant contrast to other
guesses about how wings work, many of which assume that the swath
of air has a height comparable only to the chord or some other
small distance.)
The most noticeable weakness of the disk-balloon model is its arbitrariness. Why should the size of the disk balloons be chosen to match the wingspan? Why not to match the chord? In particular, suppose two airplanes fly through the same airspace a few moments apart. They are identical in all ways (weight, wing area, airspeed et cetera) except for one: the first airplane has a short, fat wing (small aspect ratio) whereas the second airplane has a long, skinny wing (large aspect ratio). Why should the air in one case be modeled by small balloons, and in the other case by large balloons? It's the same air. Why should not the long-winged airplane support itself on a larger number of same-sized balloons?
Now it turns out that a real airplane can be and should be thought of as affecting the air in all directions for a distance that depends on the wingspan. There are good physical reasons for this, having to do with the "trailing vortices" that are shed by the wingtips. At distances large compared to a wingspan, the velocity field of one vortex largely cancels the velocity field of the other vortex.
The point is that the disk-balloon model arbitrarily assumes the right answer. It is rather like carving a block of ebony into the shape of a crow and using it to explain why crows are black. Sure, it is a good portrayal of a crow, but it doesn't really explain anything. You could just as easily have carved a canary out of ebony, or carved a bluebird out of white pine.
In this sense the disk-balloon model is a decent portrayal
of some features of the airflow near a wing, but it cannot be
considered an explanation of how a wing works. As discussed below,
it does not even portray all features correctly.
Let us look more closely at the disk-balloon model.
A real vortex (such as is produced by a real wing) has a flow pattern where the velocity decreases as we move away from the center. The velocity is inversely proportional to the distance from the vortex core. The disk-balloon model gets this exactly backwards: A disk has the property that the rotational velocity increases as we move away from the center, in direct proportion to distance.
Because of this inaccuracy, the disk-balloon model makes wrong predictions about what happens when airplanes fly in formation. Let's start by considering five airplanes, in rather loose formation, as shown in this figure:
===== ===== ===== ===== =====
A B C D E
Each airplane has a wingspan of 5 meters, and there is a space of about 6 meters between each of them. Each one is supported by its own set of disk balloons, where the balloons have a diameter of roughly 5 meters. Neighboring planes do not interfere with each other in this situation.
Now suppose the aircraft converge into a tight formation, with only a 1-meter gap between neighbors:
===== ===== ===== ===== =====
A B C D E
12345678901234567890123456789
At this point intuition suggests (and experiment or careful analysis confirms) that it is better to consider this as one long wing (29 meters long) rather than five shorter wings. According to the disk-balloon theory, it should be supported by one set of 29-meter disks at the left wingtip (location A) and another set of 29-meter disks at the right wingtip (location E) and nothing in between. But this really begs the question of how, why, and when does the wingtip at location A switch from 5-meter disks to 29-meter disks. And what happens if additional airplanes (F, G, H, ... Z) join up in tight formation extending to the right of E? Do the disk balloons at A get larger and larger? How do they know to do that in response to such far-away changes in the formation?
If we rely on the real physics of real vortices, these questions
do not arise. The vortex strength at location A is unaffected
by the presence or absence of the other airplanes. The vortices
in the interior of the formation cancel in pairs, to a good approximation.
The result is exactly the flow pattern you would expect for one
long wing (plus small corrections for the leakage in the gaps
between the neighboring wingtips).
These questions about the size of the disk-balloons become crucially important if you try to calculate the induced drag produced by the wing, that is, the energy (per unit distance) that the wing imparts to the air. Since the proponents of the disk-balloon theory recently did such a calculation, and pointed to it as evidence of the success of the theory, this line of inquiry seems like fair game.
Recall that in a real vortex, there is high velocity near the center and much less velocity farther out. Most of the energy is near the core. There is no outer edge; it just peters out. In contrast, a disk has high velocity near the edge. Most of the energy is near the edge. You've got to know where the edge is, or you'll never be able to figure out the magnitude or the location of the energy.
To illustrate the trouble this causes for the disk-balloon model, suppose that our airplanes re-arrange themselves into a formation that is only moderately tight:
===== ===== ===== ===== =====
A B C D E
12345678
In this situation, there is a 3-meter gap between neighboring planes. In the gap, there is not enough room for the radius of one disk plus the radius of the neighbor's disk. If we imagine real balloons, there will be a tragic collision. If we imagine that the disk balloons are just metaphors for rotating disks of air, then the air velocity at mid-gap will be very high, because the velocity induced by the plane on the left will be added to the velocity induced by the plane on the right. The velocity will be doubled and the kinetic energy will be increased fourfold. Therefore the model predicts a significant increase in induced drag for this formation. Alas, just the opposite occurs in real life: When airplanes fly in formation like this they experience a reduction in induced drag.
Describing a real airplane in terms of real vortices is trouble-free. Two counter-rotating vortices have less energy when they are near each other. Yes, there is a region between the two vortices where the velocity is higher, but there are larger regions to the left of the pair, to the right of the pair, above the pair, and below the pair, where the velocity induced by one vortex cancels part of the velocity induced by the other vortex, reducing the energy. Disks lack these cancellation areas (whenever their centers are separated by more than a radius but less than a diameter).
Similar considerations apply when considering ground effect.
The airplane can be thought of as flying "in formation"
with the ground. The disk-balloon theory makes predictions which
do not correspond with reality.
A vortex is a well-known solution to the laws of motion. It involves a vortex line running down the core of the vortex, with no vortex lines elsewhere. There is a law of physics (indeed a mathematical theorem) that says that vortex lines can never have loose ends; they can only form closed loops. The trailing vortices of a real wing are joined to the bound vortex that runs along the wing, satisfying this theorem.
In contrast, it is very hard to describe a rotating disk of air in a way that is consistent with the laws of motion (unless the air is contained within the superstructure of a real, non-metaphorical, rotating balloon). To the extent it can be described at all, it requires vortex lines penetrating the disk everywhere, including places far from the wing. When you consider the boundary between the face of the last disk already visited (which is spinning), and the face of the next disk about to be visited (which is not yet spinning), you find a huge violation of the closed-loops-only theorem.
If that violation were not bad enough, we must consider what happens
at the outer edges of the spinning disks, where they meet the
non-spinning ambient air. Along this boundary there is another
gross violation of the laws of motion.
Some people say that disks are preferred because students have
seen rotating disks but have never seen vortices. Well, that
problem is fixable. Get a vat of water. Chuck up a paddle in
a variable-speed electric drill and make a vortex for them.
We have two hypotheses. We can model the wake of the airplane
in terms of spinning vortices or in terms of spinning disks.
| disk | vortex | |
| The wing imparts energy and downward momentum to the air. | yes | yes |
| The wing affects a swath of air comparable in width to the wingspan. | yes | yes |
| The wing imparts some rotational motion to the air. | yes | yes |
| The wing affects a swath of air up to a height that depends on the wingspan. | yes | yes |
| The theory requires an arbitrary assumption about the size of the region of rotating air. | required | not required |
| The theory correctly and naturally describes how the wing interacts with the ground and with other wings. | no | yes |
| The theory is consistent with the known laws of motion. | no | yes |
The first two points are so obvious that we hardly need a model at all in order to portray them correctly. The third point is almost as obvious: there will always be a boundary where the air that was pulled down by the wing meets the relatively undisturbed air, and along this boundary there will obviously be some vorticity, some circulatory flow.
On the fourth point, the disk theory portrays it correctly, but it seems contrived: the ebony crow is the right color. On the remaining points the disk theory doesn't do too well.
So the disk theory gets credit for three obvious points plus one non-obvious point, but misses several other points. One has to wonder whether introducing the theory is worth the trouble.
By way of constructive counter-offer, a pilot-oriented explanation of how wings really work can be found at
http://www.av8n.com/how/htm/airfoils.html
Comments can be sent to jsd@av8n.com