Two rows of stacked disks approach from the left, then at one spot they suddenly move downards together while counterrotating

1997 William Beaty

The decades-old controversy about wings and the lifting force has a definite origin. It arises because two-dimensional airfoils are used to teach us about pressure and flow patterns... 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. Going from 2D to 3D is not a trivial change.

In our 3D world, airplanes behave like hovering helicopters, producing a downwards-moving plume of air. When flying horizontally, the plume of air becomes a downstream wake which carries away downwards momentum, and constitutes a net downwash. The wing injects downwards momentum into local air parcels, and this air keeps moving downward long after the aircraft has departed.

We encounter this as a wide, descending air mass commonly labeled "tip-vorticies."

But in the 2D world of the wind-tunnel, there is no such downward-moving wake behind the wing. Instead, the 2D upwash must always be exactly equal to the 2D downwash, and the wing does not push upon the air. Instead, all the momentum is injected directly and instantly into the distant floor/ceiling. Even more important, 3D wings have finite dimensions, while 2D wings act as if they have infinite span. An infinite wingspan gives some exceeding strange results; odd results never produced by finite 3D wings in the real world.

For example, if a 2D infinitely-wide wing should ever deflect even a tiny portion of the oncoming air downwards (deflecting a streamline,) 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 incoming and outgoing streamlines remain horizontal. The wing doesn't deflect any air at all, on average. The wing acts like a weird reaction engine, a strange engine where the "exhaust gasses" have zero velocity and infinite mass. This strange effect only applies to 2D airfoils and to infinitely-wide wings, and obviously is never encountered with real 3D airplanes flying through 3D air.

The controversy about "Bernoulli versus Newton" is probably a controversy about two-D wings versus three-D. It's a fight between the reaction- motor-like flight of real wings, versus the instant-forces of ground-effect flight seen in airfoil diagrams and in 2D wind tunnels. It's a controversy over fluid propulsion versus venturi-effect, between the the physics of short 3D wings in a 3D world, versus flight in strange a two-dimensional "flatland" or "Dewdney Planiverse" world.

In other words, flight of airplanes in the real world is an example of reaction engines and propulsion, same as with hovering helicopters and hovering rocket engines. Real wings are reaction engines. But the flight of a 2D airfoil is not propulsion, and doesn't inherently require any fuel. Real 3D airplanes require fuel, even when flying in an idealized zero-friction environment. (Real helicopters and rocket engines are the same. But a rocket engine with an infinitely-wide engine-bell does not need fuel and is not a reaction engine. The same is true of a helicopter with infinitely-wide rotor blades ...and true for a two-dimensional airfoil in a world of 2D streamlines.)

I 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 flies through 3D air. It is very different than the typical 2D explanations found in most textboooks. My "circulation" is flipped ninety degrees!

Take a look at my animated crude GIF diagram, and also see the green smoke laser-scanned video segment of a model 747 producing a downwash pattern.

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
                   _--     --_
                  /           \
               __|      .      |
                 |             |
                  \_         _/

Now suppose I were to leap from the top of a ladder and onto the balloon's small platform. The balloon would move downwards. It would also rotate rapidly counterclockwise, and I would be dumped off.

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



fig. 2
                  _-- _____
                /  _--     --_
             __|  /           \
               __|      .      |
                 |             |
                  \_         _/

They do not touch each other. Both have platforms. If I jump onto the first platform, but then I immediately leap onto the next platform, I can stay up there for a tiny bit longer.

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
              _____                 _____
           _--     --_           _--     --_
          /           \         /           \
         |      .      |__   __|      .      |
         |             |       |             |  
          \_         _/         \_         _/
            --_____--             --_____--

There's one platform for each of my feet. I can run forwards, leaving a trail of "wake turbulence" behind me. The "wake" is composed of rotating, descending balloons. Fig. 4 below shows an animated GIF of this process.

[GIF animation: descending balloons]
Fig. 4 Forcing the balloons downwards

Also see: Smoke Ring animation



A 3D aircraft does much the same thing as me and my balloons: it remains aloft by shedding vortices: by throwing down a spinning region of mass. This mass consists of two long, thin, vortex-threads and the tubular regions of air which are constrained to circulate around them. The balloons crudely represent the separatrix of a vortex-pair: the cylindrical parcels of air which must move with closed streamlines.

So, how do airplanes fly? Real aircraft shed vortices. They inject momentum into air which as a result moves downwards. They employ "invisible disk-balloons" to stay aloft. The two rows of invisible balloons together form a single, very long, downwards-moving cylinder of air. This single cylinder has significant mass and carries a large momentum downwards. Airplane flight is vortex-shedding flight.

      _____                 _____
   _--     --_           _--     --_
  /           \    |    /           \
 |      ___    |   |   |    ___      |
 |         ---____/ \____---         |  
  \_         _/   \_/   \_         _/ 
    --_____--             --_____--

fig. 5

              \      |      /
                \    |    /
     ______      |   |   |     ______      
    /  ___  \    |   |   |    /  ___  \    
  /   /   \   \   |  |  |   /   /   \   \  
  |  |  o  |  |   |  |  |   |  |  o  |  |  
  \   \___/   /  |   |   |  \   \___/   /  
    \_______/    |   |   |    \_______/    
                /    |    \                
              /      |      \
fig. 6

view behind Learjet flying just above fogbank: miles long downwash vortex-pair punches a clear slot
The downward-moving "reaction exhaust" below a wing, made visible.
The aircraft flys horizontally above the fog bank,
while the vortex-pattern descends into the fog.
(See Hyperphysics and other photos.)

Scanned laser smoke downwash



My forward speed makes a difference in how much work I perform. If I walk slowly along my rows of balloons, each platform sinks downwards significantly. I must always leap upwards to the next platform, and each balloon is thrown violently downward as I leap. I tire quickly. On the other hand, if I run very fast, my feet touch each platform briefly, the balloons barely move, and the situation resembles my running along the solid ground.

Similarly, if a real aircraft flies slowly, it must fling the vortex-pairs violently downward. It performs extra work and experiences a very large "induced drag." If it flies fast, it spreads out the necessary momentum-shedding across a much larger volume of decending air, and therefore it needs only barely touch each mass-parcel (each "balloon.") Hence, faster flight is desirable because it requires far less work to be performed in moving the air downwards. And if a slowly-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 vortex-shedding; by launching a long chain of combined "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.


In wondering why the entire aero community seems ignorant of these simple concepts, I stumbled upon something interesting. Klaus Weltner found that L. Prandtl was the initial source of the equal-transit-time fallacy. Prandtl's 1924 paper teaches that, parcels split by the leading edge, must rejoin at the trailing edge.

Looking at more Prandtl work, I find that he was using two simplifications, changes which make the math easier by erasing any moving vortexes (no vorticity in translation.) This eliminates any propulsion effects, any fuel use or induced-drag. He did two things: analyzed 2D airfoils, and also analyzed 3D wings in flight, but where the tip-vortices extended back perfectly horizontally. (So, a "Prandtl Horse-shoe" diagram, but without any tilt to the vortex-lines.) These two changes have profound effects: direct violations of Newton's laws.

First, if we analyze a 2D airfoil, we should find that the lift-force is equal and opposite to a downward force on a nearby solid surface. The system must include both the wing and the ground. If not, if we do like Prandtl and refuse to sketch in the ground surface, then we're creating a single-ended force which violates Newton's 3rd. Well, can't we move the ground far away? No, that doesn't work, since the force upon the ground is constant, and doesn't decrease with flight-altitude. (It's only permissible to remove something to infinite distance, if this reduces the pertinant variables to zero, and the force on the ground doesn't reduce.) The reason behind this phenomenon is interesting. A 2D airfoil is an infinitely-wide wing. Infinite span. It's forever trapped in ground-effect flight-mode, even if it flys at miles of altitude. (The span is inifinte, so in comparison, the wing is right next to the ground, even when miles high.) Whenever we explain a 2D airfoil, really we're explaining a "W.I.G." or wing-in-ground aircraft. A Caspian Sea Monster, those weird Russian planes with the ICBM nuke warheads. If a real airplane is like a hovering rocket, then a 2D airfoil is like a rocket with an infinitely-wide engine-bell. With a rocket motor, if we make the motor infinitely wide, then the exhaust-velocity decreases to zero, and no fuel is needed to create the thrust. It stops being a reaction engine. No propulsion, just a static thrust appearing from nowhere. In other words, a 2D airfoil, which is supposed to be like any airplane, was supposed to be a reaction-engine, but the infinite-wingspan effect screws up everything. (It causes an instant-force against the ground; a force which doesn't exist with real airplanes, except when those airplanes are flying at altitudes below about twenty feet!)

Prandtl does it again too. When working with 3D airfoils and their closed circle of vorticity (the tip-vortices, and the distant start-vortex,) Prandtl makes things simpler by analyzing horizontal tip-vortices. Big mistake. HUUUUGE! Having horizontal tip-vortices is the same as flying your aircraft faster and faster, and finally infinitely fast ...while attempting to remove the down-momentum being injected into the air. Unfortunately it doesn't work. The weight of the airplane is constant, so the rate of down-momentum injected into the air is constant. Speed of flight has no effect. If we try to remove this complicated momentum effect, by flying infinitely fast, by making the tip-vorticies horizontal, then again, we directly violate Newton, because flight-speed doesn't affect the net down-momentum! Again, a rocket-analogy. It's like having a rocket-engine where the exhaust-velocity is infinite. With infinite gas-velocity, then we dont' need any gas at all, and our rocket uses zero energy, zero fuel when producing thrust. (Wings as reaction engines are harder to analyze. So Prandtl just violates Newton, and declares that wings aren't based on propulsion, or creating any "exhaust plume," any momentum-bearing tip-vortices moving downwards.)

So, if you don't understand how wings work ...perhaps it's because real wings are reaction-motors, they're propulsion devices, same as hovering rocket engines (or hovering helicopters.) They work by pulling in air from all directions, then launching it downwards. But this involves changing vorticity, which brings in the horrible mathematics of turbulence. To avoid this, to make the equations have actual solutions, Prandtl cheated. He accepted the fundamental Newton-violations, in pursuit of simple math models. But then he crossed the line, because he never explicitly stated that this is what he was doing, or admitted that his models were completely un-physical (flagrantly violating simple Newtonian physics.)

OK, enough Prandtl-bashing. I'll let others take up the big club and have a go at him. (I'm thinking, he deserves it.)


How well does the "disk balloons" model correspond to the real world? Well, we can pull an equation out of the motions of the balloons, and use it to predict both aircraft energy-use and induced drag. If the equation is at all similar to the actual aerodynamics of a real-world airplane, then the "disk balloons" are a useful model. If my equation turns out to be faulty, then my model only has weak ties with reality.

Suppose the "disk-balloons" contain air which rotates as a solid object, (or imagine radial membranes in the balloons. Or stuff them with aerogel.) If I then 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 ]
M * g being aircraft weight, V is velocity of horizontal flight, and "density" is the density of air. Induced drag should then be power/V:
Induced Drag = 8 * (M * g)^2 / [ pi * span^2 * V^2 * density ]
What happens if I assume that the air within the disk-balloons is not "solid", but instead it's made to whirl faster near the center of the balloon, such that the tangential velocity of the air is constant, regardless of its distance from the center of the balloon? (Imagine a wing which produces a downward velocity of net downwash which is constant at each point along the whole span of the wing.) If the "downwash" is constant across the wingspan, then the modified "balloon equation" predicts a power expenditure of 2x that above.

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. So, calculated from first principles, without prior instruction in fluid dynamics. Pretty good for an "amateur aerodynamicist", eh?

One final note. The downwash vortices of real airplanes contains rapidly rotating air. This represents wasted energy, since only the "shell" of each "balloon" needs to rotate as the region of air moves downwards. Is there a wing which can produce a downwash vortex-pair without any spinning cores? Maybe it would use less fuel than modern wings.

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


Misc. thought experiments

If a balloon inside a sealed chamber experiences an upwards force, well, Newton'd 3rd must be obeyed, so where does the down-force appear? Obviously it appears as a slight increase in static pressure on the floor of the chamber. The balloon accelerates upwards, and the chamber accelerates downward, conserving momentum. Now if our balloon carries a load, and load KG is adjusted so fhs Lloon hovers in level flight, what then? The down-force of the load is exactly countered by the buoyancy up-force of the balloon ...but the physics isn't local-only. Therefore, the floor of the chamber must experience a small pressure increase whenever a balloon lifts a small mass, a down-force exactly equal to the weight of the suspended mass. (And no, we cannot eliminate the weight of a truck's load upon the Earth by suspending the truck via balloons! If the chamber is sealed, the balloons cannot reduce the weight of the truck. If the balloons are outside the chamber, then the "footprint" of down-force wide across the ground, but it still nets to exactly the weight of the suspended mass.

OK, how about this one. Suppose I'm high above the ground, and I have a huge piston and cylinder. I pull the piston out; forming an evacuated volume inside the cylinder. I pull it so far out that the buoyant force of the evacuated cylinder now lifts the weight of the cylinder, piston, and myself (I'm that strong.) I'm now suspended, hovering. So, what's the footprint of my weight upon the ground? :) And, what are the dynamic changes, if any, since the ground cannot experience force-changes except after speed-of-sound propagation delays.
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