AIRPLANE FLIGHT ANALOGY            1997 William Beaty

(See also J. Denker's critique and my response, 8/99)

The controversy about wings and the lifting force has a definite origin. It arises because we are taught about the flow patterns surrounding two-dimensional airfoil crossections... and then we apply those concepts to 3D wings.

This is a major mistake. The behavior of 3D wings is fundamentally different than the behavior of 2D airfoils.

In a 3D world, an airplane produces a downstream wake with net downwash, but in a 2D wind-tunnel there is no such wake, and the 2D upwash must always be equal to the 2D downwash. Even more important, 3D wings have finite area, while 2D wings act as if they are infinitely long. An infinite wing gives some strange results; results never produced by finite 3D wings. For example, if a 2D infinite wing should ever deflect even a tiny portion of the oncoming air downwards, it would deflect an infinite amount of air and produce an infinite lifting force. As a result, a 2D infinite airfoil does an odd thing: it applies a FINITE force to an infinite mass of air. In response a net amount of air does NOT move downwards (since it has infinite mass!) The wing acts like a reaction engine, but an engine where the "exhaust gasses" have zero velocity and infinite mass. This strange effect only applies to 2D airfoils and infinitely long wings, and is never seen with 3D airplanes flying through 3D air.

The controversy about "Bernoulli versus Newton" is really a controversy about two-D versus three-D. It's a controversy over the physics of airfoils in two-dimensional "flatland" worlds, versus the more ordinary physics of short 3D wings in a 3D world.

I could attempt to explain the problem in words, but words are easily misunderstood (especially on hot-button issues where emotions run high.) A visual analogy works much better. Below is my explanation for how a three-dimensional airplane flys through 3D space. It is very different than the typical 2D explanations found in most textboooks. My "circulation" is flipped ninety degrees!

Imagine a huge, disk-shaped helium balloon floating in the air. The disk stands on edge. It is weighted for neutral buoyancy so it neither rises nor sinks. A small platform sticks out of its rim. (If you feel the need, you can imagine a counterweight on the opposite rim to the platform, so the balloon hovers without rotating.) See fig. 1 below

            _____
         _--     --_
        /           \    fig. 1  DISK-BALLOON WITH
     __|      .      |           A SMALL PLATFORM
       |             |
        \_         _/
          --_____--

Next, suppose we have TWO giant disk-shaped balloons stacked adjacent to each other like pancakes standing on edge.

           ____
        _-- _____
      /  _--     --_
   __|  /           \    fig. 2 TWO DISK-BALLOONS,
     __|      .      |          STACKED ADJACENTLY
       |             |
        \_         _/
          --_____--

Next, suppose we have a row of these disk-balloons one KM long. It looks like fig. 2 above, but with hundreds of hovering balloons. Now I can run from platform to platform, and I will stay aloft until I run out of balloons. Behind me I leave a trail of rotating, downward-moving balloons. I can remain suspended against gravity because I am flinging mass downwards. The mass takes the form of helium mass trapped inside the balloons. I am also doing much more work than necessary, since the energy I expend in rotating the balloons does not contribute to my fight against gravity. (In truth, all my work is really not necessary, I could simply walk along the Earth's surface with no need to move any massive gasbags!)

To make the situation more symmetrical, let me add a second row of platform-bearing balloons in parallel to the first row:

            _____                 _____
         _--     --_           _--     --_     fig. 3
        /           \         /           \    END VIEW OF TWO LONG
       |      .      |__   __|      .      |   ROWS OF DISK-BALLOONS
       |             |       |             |
        \_         _/         \_         _/
          --_____--             --_____--

Fig. 4 Forcing the balloons downwards

Also see: Smoke Ring animation

"DISK BALLOONS" BEHIND AIRPLANES

So, how do airplanes fly? Real aircraft use "invisible disk-balloons" to stay aloft. The two rows of "invisible balloons" form a single, very long, downwards-moving cylinder of air. This single cylinder has significant mass and carries a large momentum downwards.

         _____                 _____
      _--     --_           _--     --_     fig. 5
     /           \    |    /           \    FRONT VIEW OF AIRCRAFT, W/
    |      ___    |   |   |    ___      |   AIR MASSES ROTATED BY THE
    |         ---____/ \____---         |   WINGS' PRESSURE DIFFERENCE
     \_         _/   \_/   \_         _/ 
       --_____--             --_____--



               \      |      /
                 \    |    /
      ______      |   |   |     ______      fig. 6
     /  ___  \    |   |   |    /  ___  \    THE CROSS-SECTION OF AN
   /   /   \   \   |  |  |   /   /   \   \  ACTUAL WAKE HAS STREAMLINES
   |  |  o  |  |   |  |  |   |  |  o  |  |  WHICH DO NOT PERFECTLY
   \   \___/   /  |   |   |  \   \___/   /  RESEMBLE A PAIR OF ROTATING
     \_______/    |   |   |    \_______/    BALLOONS.  THE BALLOONS ARE
                 /    |    \                A CRUDE ANALOGY.
               /      |      \

The "exhaust" below a wing made visible
(See Hyperphysics and other photos.)

FLY FASTER FOR LESS DRAG

Similarly, if a real aircraft flys slowly, it must fling the vortex-pairs violently downward. It performs extra work and experiences a very large "induced drag." If it flys fast, it spreads out the necessary momentum-changes, and therefore it needs only to barely touch each parcel of mass (each "balloon.") Hence, faster flight is desirable because it requires far less work to be performed in moving the air downwards. And if a slow-flying, heavily-loaded aircraft should fly very low over you, its powerful wake vortices will blow you over and put dust in your eyes.

All of my reasoning implies that modern aircraft actually remain aloft by launching "smoke rings" downwards. Imagine one of the flying cars in the old 'Jetsons' cartoon, the ones with those little white rings shooting down out of the underside. But rather than launching a great number of individual rings, modern aircraft throw just one very long ring downwards, and they are lifted by the upward reaction force.

A CRUDE PREDICTION

Suppose the "disk-balloons" contain air which rotates as a solid object, (or imagine radial membranes in the balloons.) If I add together the work done in creating the circulatory flow, plus the work done in projecting the constrained air downwards, I arrive at a predicted aircraft power expenditure of:

Power = 8 * (M * g)^2 / [ pi * span^2 * V * density ]

Induced Drag = 8 * (M * g)^2 / [ pi * span^2 * V^2 * density ]

How does this match reality? I'm looking for information on this at the moment. I'm told that these two equations are identical to the equations of real aircraft, except that the number "8" is replaced by a factor which is dependent upon the particular geometry of the wing. Pretty good for an "amateur aerodynamicist", eh?