William J. Beaty

First let me say that, in my opinion, John Denker's critique of my "Disk
Balloons" article has exposed no fatal flaws.  True, there are flaws, but
"flaws" exist upon a spectrum ranging from tiny quibbles which only a
nitpicker would notice, all the way up to glaring errors which stab into
the very heart of a concept and kill it dead.

I see nothing in John's critique which "punctures" the disk-balloons.  :) 
No fatal flaws have been discovered as yet. 

Mr. Denker writes that my "Disk Balloons" model is arbitrary.  I'm sorry,
I don't understand: in what sense "arbitrary?"  They are based upon
reality.  This "balloon model"  matches certain features of the real air
movements around a real aircraft.  I have read various descriptions of
these air movements, so I took it upon myself to construct a simplified
visual model in order to clarify certain elements of the physics of
flight.  I look at the air-flows around an aircraft in flight, I mentally
slice them into thin segments, then I replace each segment with a pair of
counterrotating disk-balloons where the center of each balloon is aligned 
with the wingtip (just as the thread of vorticity connects near the 
wingtip in a realworld aircraft.)

If such a process must be denegrated as being arbitrary, when why aren't
most other descriptive physical models arbitrary as well?  I do try to
take criticism seriously, but in this case I'll need more.  I'll need an
explanation on how to make my model *NOT* be "arbitrary."  Please tell me
how to alter the disk-balloons model so it is not arbitrary, and yet it
still approximately describes the actual air motions surrounding a
real-world aircraft.

Perhaps the problem is that my model is DESCRIPTIVE.  

It is not a full-blown physical theory which predicts the airflows from
first principles.  The "disk balloons" are obviously a simplified
mental model of the vortex-wake which is produced behind a real-world
aircraft.  I chose this model in order to accentuate certain features of
the physics of flight, while covering up others.  My goal was to create a
highly simplified and "Newton-compatible" description of what happens when
a wing applies a downwards force to the air passing near it.

Why don't my "balloons" rotate chordwise?  Because they are a model for
the aircraft wake; for the trailing circulation patterns *behind* a wing,
and the air trailing for hundreds of yards behind a wing does not rotate
chordwise about the wing.  Surely this is obvious.  If we slice up the
vortex-wake found behind an aircraft, we obtain something that resembles
the magnetic field around a DC transmission line consisting of two
conductors with a space between them:

                  \      |      /
                    \    |    /
         ______      |   |   |
        /  ___  \    |   |   |    /  ___  \
      /   /   \   \   |  |  |   /   /   \   \
      |  |  o  |  |   |  |  |   |  |  o  |  |
      \   \___/   /  |   |   |  \   \___/   /
        \_______/    |   |   |    \_______/
                    /    |    \ 
                  /      |      \

The cores of the two vortices are centered on the wingtips of the
aircraft.  The pattern also vaguely resembles a pair of counter-rotating
disks of air, therefore I chose these pairs of disk-shaped bouyant gasbags
as an appropriate model.

Are the radii of the disks a random choice?  No, the radii are fixed by
the shape of the vortex-wake behind the aircraft, which in turn are fixed
by the geometry of the aircraft. The center of each disk is aligned with a
wingtip of the aircraft, and the diameter of the huge disks is adjusted so
that they nearly touch each other in the center of the span where the
fuselage would normally lie.  Hardly an "arbitrary" shape.  I thought this
was obvious to anyone, but apparently I was wrong.  Perhaps I need
something better than ASCII art to show how disk-balloons work.

Why balloons?  I used balloons because I wanted to create a mass-bearing
"conceptual object" to which I could apply Newton's laws without much
grief, (and hopefully without introducing any bizarre artifacts into the
concepts.)  A balloon entrains mass and carries momentum.  So does a
rotating vortex-pair.  Throw either a balloon or a vortex-pair downwards,
and you will feel a reaction force which pushes you in the upwards

OK, if we must call the balloons a "visual model" of real-world air flows,
rather than an "explanation" of the lifting force, that's fine, because
that's what they are.  If the model has faults, I'll listen to the details
of any complaints.  Perhaps the reasons for various features of the model
seem obscure.  If so, please ask for clarification and I'll try to

*WHY* does the air behind a wing organize itself into a pair counter-
rotating cylindrical structures?  Similarly, WHY does a ring-vortex appear
whenever a parcel of moving viscous fluid is injected into a non-moving
volume of fluid?  Drip a droplet of dye into a glass of water, and WHY
does it form a ring-vortex?  These are very interesting questions, but
they're a distraction.  The goal here is to achieve a gut-level "feel" for
the mechanism behind lift, not to explore the depths of every detailed
facet of viscous fluid dynamics.  If we concentrate on details, it
distracts us into ignoring the overall picture.

(If for some strange reason we wish to *intentionally* ignore the overall
picture, then the tactic of concentrating on the details is one method to
fulfill this goal, but it is not honest, and it is not science.)  Only
after we understand the overall picture of the production of
lifting-force should we take a look at the interesting details.

I make this assertion: "When the wing throws air downwards all along its
trailing edge, a long series of rotating 'disk balloons' are created, but
no downward-moving blocks of air are created."  If we wish, we can drip a
droplet of blue food-coloring into a cup of water and note that the
downward-moving dye organizes itself into a ring of moving fluid.  This
simple phenomena shows how vorticity can arise.  I do not understand the
details behind this event (perhaps somebody can explain how viscosity and
turbulence creates vortex-rings?)  For some obscure reason, viscous
interactions usually cause these particular patterns of flow to appear
behind aircraft as well as surrounding their wings.  Knowing this, we can
note the general shape of the downward-moving regions of air, figure out
the mass that happens to be entrained within them, and end up with a
simple way to apply Newton's laws to the lift-generating process:
disk-shaped balloons.

About "Qualitative Inconsistencies".  Mr. Denker claims that a "real" 
vortex is a flow pattern where the (tangential) velocity is greater near
the center of the vortex in a 1/r profile of velocity.  NOT TRUE.  While
John's is a clear description of a mathematically-simple textbook vortex,
(where the "vorticity" is entirely concentrated as an infinitely thin
thread at the center of the the flow pattern,) this type of vortex is not
"real," if by "real" we mean that it's the only pattern allowed in any
real-world situation.  In reality, many different sorts of vorticity
distributions are perfectly possible, each with a different profile of
tangential velocity (and each with a resulting different profile of
distributed vorticity which is not necessarily concentrated at the center
of the rotating mass of fluid.) 

If the vorticity is concentrated as an infinitely thin thread running down
the center of the rotating fluid, then the tangential velocities are
distributed as Mr. Denker describes.  In that case, for a lone vortex ring
or thread, each parcel of fluid will have a tangential velocity which is
inversely related to its distance from the vortex-thread.  However,
whenever a quantity of rotating fluid instead happens to resemble a
rotating solid cylinder, then obviously the vorticity is not concentrated
in a central "vortex thread."  Instead it must be distributed throughout
the volume of the rotating air.

Why is this a problem?  It's a problem because aircraft vorticies don't
have thin vortex-threads down their centers!  It's well known that the
vorticity behind an aircraft IS NOT concentrated totally at the wingtips.
(I hope that Mr. Denker knows this.  After all, he's the airplane expert
here, not me.)  If we use our hand to stir a bucket filled with water, we
*don't* tend to produce one of these textbook-ideal vortex-thread patterns
(although it's not impossible to do so if that is our goal.)

OK, let us see who is right:  Where is the vorticity concentrated in the
wake of a *real* aircraft?  Is it exclusively concentrated in two thin
threads extending from the wingtips as John seems to imagine?  Or does
vorticity appear elsewhere as well?   What do textbooks say?

I'm not too experienced in such things, but I do recall several
descriptions in books I read long ago.  They mentioned an "elliptical" 
distribution of tangential (downwards) velocity, not a 1/r distribution,
and they described a "sheet of vorticity" which pours off the trailing
edge of a wing.  If this is accurate, then the air behind a wing does not
move like Mr. Denker imagines.  It does not have a 1/r tangential
velocity profile relative to the wingtip.  Instead, the vorticity behind a
wing exists as a sheet-like structure, and it's NOT totally contained in
two narrow threadlike structures which extend from the wingtips. 
Therefore the motions of the air-parcels within the rotating cylinders of
air behind a real wing do *not* have huge downwards velocity near the
wingtip and much lower downwards velocity in the center of the span, Their
their downwards velocity profile does not at all resemble the 1/r rule for
ideal vorticies. 

So, these 'disk balloons' are supposedly faulty because the velocity of
air within them is not inversely related to the distance from the center
of rotation?  Apparantly air doesn't behave that way in a real-world
aircraft either.  Therefore any description which only uses vortex-threads
is faulty as well.  Mr. Denker is totally wrong when he states that this
aspect of the disk-balloons "VIOLATES THE LAWS OF MOTION."  The laws of
motion do not require that vortex-motions have threads of vorticity
running down their centers.

With the disk-balloons, do we find a discontinuity at the edge of the
balloon where the balloon meets the outside air?  Yes, but not in the way
Mr. Denker imagines.  Look at the region found near the center of the
wingspan.  The two disk balloons are almost in contact, and they shield
each other from the effects of the outside atmosphere.  On the other hand,
at the two regions on the far edges of the disk-balloons, far to the left
and right of the wing, the rotating disk-balloons must meet the unmoving
atmosphere with no "shielding effect" whatsoever.  However, the vertical
relative velocity of the surface of the disk balloon is zero there, since
the pair of disk-balloons is moving downwards as a system, while at the
same time they rotate so as to move their outer edges upwards.  When the
rotating motion is added to the net downwards motion, the result is a
near-zero velocity between the air and the balloon surfaces in the region
where the balloons are in direct "unsheilded"  contact with the outside
atmosphere.  This effect is obvious to anyone who can see these balloons
rotating in their minds.  I did not realize that I had to specifically
explain it.  Are most people unable to visualize the "rotating disk
balloons."  Maybe my article is biased towards visual thinkers.  Maybe I
need to create an animated image of the disk-balloons moving downwards
while the "little man" runs forwards along them.  [NOTE:  I ADDED AN

The balloon-pair behaves in much the same way that a real-world wake-
vortex pair does.  Because it moves downwards as a whole, the rotation
of each vortex eliminates any gross discontinuity at the far edges of the
rotating mass of air. 

Actually, this conversation gives me quite a bit of insight into one
particular phenomenon in fluid dynamics which I had lacked before.  It has
been clear to me that the wake vorticies do not obey a 1/R velocity
rule.  But if this is so, and if the tangential velocity of each parcel of
air which is far from the vortex-core is actually very large, then
shouldn't there be a discontinuity where the rotating air meets the
environment?  Don't we REQUIRE a 1/R velocity distribution in order
to avoid a huge shear at the "surface" of these cylindrical vortices?  Not

Think of it this way.  When an automobile tire meets the road, the tire
does not "skid."  Instead the velocity of the tire matches the velocity of
the road.  It's clear that the wake-vortices of an aircraft must do
something very similar: they "roll against" each other as they descend,
and their outer regions "roll against" the unmoving outside atmosphere and
push downwards through it.  The entrained air is separated from the
outside air by a circular-shaped "separatrix", and the air at this
"surface" can match the velocity of outside air which it "touches."  This
well-organized motion must lead to very low friction, and as a result, the
wake-vortex pair probably travels for quite a large distance downwards
before it is halted by viscosity.  Aha!  THAT'S why ring-vorticies in
general can travel with such low friction. They are like a row of wheels
threaded upon a metal ring.  A row of solid wheels.  Rotating solid
cylinders.  Like "disk balloons." 

One thing I've noticed in many explanations of flight: they almost
universally concentrate on inviscid flow.  However, vortex patterns cannot
be created in an inviscid fluid.  If viscosity is zero, then parcels of
fluid cannot drag against each other, and therefore we cannot create
smoke-rings, nor tornados, nor can we build wings which leak vorticity
from their trailing edges at the same time that they create the necessary
new vorticity associated with chordwise circulation. The entire topic of
trailing wake-vorticies requires a VISCOUS model of air.  If the usual
explanations of flight exclusively focus on inviscid aerodynamics and
simplified computer simulations, then they are missing something
important.  They are ignoring an entire section of aerodynamics where
Newton connects with Bernoulli.

The other points raised in Mr. Denker's article are totally accurate. 
Real-world physical balloons cannot superpose upon each other, yet
patterns of air-motion can and do superpose, as when aircraft fly with
wingtips nearly touching.  As with any other physical model, this
balloon-based model has areas of proper application, as well as particular
limits beyond which it becomes foolish to employ them.

However, if "disk balloons" are to be disparaged because they are not
perfect, or because they are only a model, then we should also
disparage all other models for the same reason, since a "model" is not
reality.  The same sort of attack could easily be launched against any
models which use inviscid flow or 2D flow:  these models are
unrealistic because the real world is three-dimensional and
viscous.  They are unrealistic because Circulation must be added
artificially by way of the Kutta condition.  Also, they do not explain
induced drag, and they suggest that upwash equals downwash even in a 3D
aircraft.  As a result, if 2D inviscid models are applied beyond their
limits, they can lead us to profound misconceptions regarding how aircraft
really work.  They can even lead to blatant violations of Newton's laws. 
Yet these models are incredibly useful when they are applied correctly. 
Models are tools.  A screwdriver is a very useful tool, but we should not
attack it because it makes a lousy wrench.  And if it shatters when we try
to use it as a chisel, we should blame ourselves and not the tool.

The "disk balloons" model totally ignores one important facet of flight:  
the mechanism which forms the vortex-pair.  I provide a pair of human feet
stepping onto a platform and spinning the balloons up to speed.  In any
realworld aircraft the balloons (the vortex pattern) envelop the entire
wing, and the "disk balloons" connect smoothly to the pattern of chordwise
circulation associated with the wing.  My goal was not to explain how that
part works.  My goal was to focus everyone's attention on the DECENDING
MASS that is launched by the wing, and to show why there is no need for a
an aircraft to reach downwards for miles and create a direct force-pair
between itself and the earth.  This whole article is a response to a
common incorrect statement: "airplanes fly because they push upon the
Earth."  No, an airplane does not have to push upon the earth.  This is
because it can give the nearby air a net acceleration and leave it
descending afterwards.  The decending air eventually applies its force to
the earth, but the airplane does not participate in that exchange of

As a model aimed at K-6 textbooks, 'disk balloons' are a vast improvement
compared to the usual faulty models presented in those books.  It is
impossible to explain flight to little children if we limit ourselves to
circulation-based models.  Even high-school physics teachers have trouble
making sense of explanations based upon circulation, so we should think
twice before insisting that children learn these models.  At the same
time, if "disk balloons" were used to replace full-blown aerodynamics
theory, explanations would be ridiculously limited and distorted.  They
are a child's version of vortex theory.  The right tool for the right job?

What misconceptions do 'disk balloons' breed?  Mr. Denker has found a few.  
Undoubtedly there are more.  Hopefully any new misconceptions are as minor
as the current ones being spread by textbooks, and hopefully there are
none which cause a fundamental misunderstanding of how aircraft work.  
They are not without misconceptions, but we must compare them to the
damage being done by the current crop of K-6 textbook explanations, and
even the damage being done by 2D inviscid-flow models used in more
advanced textbooks.  2D wings and 3D wings do not use the same mechanism
to generate lift.

The disk balloons are not "real" in that they don't match reality
perfectly.  Is a mathematically sophisticated, inviscid-fluid-based
description of flight "real"?  Of course not.  No single model is.  If we
wish to understand the world, then we must employ a variety of separate
models which "overlap" in order to cover all possibilities.  We also need
to be very aware of their individual flaws and their ranges of

One issue might have been missed here.  Perhaps the 'disk balloons' are a
disgusting and unsophisticated oversimplification which are appropriate
only for little children?  Correct.  Everyone please read the title of my
website.  "K-6" means *children* (or specifically, science teachers who
work at those grade levels.)  Some of the material does occasionally
extend up to the K-12 grades (e.g. trying to use disk-balloons to
calculate lift, instead of only using them as a visual intuitive picture
of the lift-generating process.) 

If we desire a simplified, basic explanation of flight which gives us a
feel of how Newton's laws apply to aircraft, then "disk balloons" work
fairly well.  If we want to explain why migrating geese should fly in "V"  
formations for efficiency, or what happens when two aircraft approach each
other wingtip-to-wingtip, then disk balloons are worse than useless
because they make crazy predictions.  However, the inviscid-fluid computer
models which can correctly treat the problem of multiple-aircraft
interactions will make crazy predictions when used to calculate fuel usage
at various low airspeeds.  In my opinion, they're also useless when it
comes to explaining airplanes in a clear, easily-understood manner which
even a child can understand.  The right tool for the right job.

As with newspapers and science museums, my target audience is the general
public.  Therefore I do the same thing as the New York Times:  I try to
aim everything at the 6th-grade level.  Yes, there are many other models
which could be used to explain how aircraft fly.  But at the 6th grade
level, most of them are hideously distorted (e.g. that stuff about parcels
of air rushing to meet each other at the trailing edge of an airfoil.)  
At higher grade levels, sophisticated models employ abstract concepts and
mathematics, and they tend to obscure the simple, basic ideas behind
flight.  Such models are totally unsuitable for explaining flight to


Let's review John Denker's table of predictions made by "Disk Balloons" 
compared against the predictions regarding the behavior real vortices: 

                                                DISK-BALLOON   REAL VORTEX

The wing imparts energy and downward momentum   yes            yes
to the air.

The wing affects a swath of air comparable      yes            yes
in width to the wingspan. 

The wing imparts some rotational motion         yes            yes
to the air.              

The wing affects a swath of air up to a         yes            yes
height that depends on the wingspan.

The theory requires an arbitrary assumption     determined     emergent
about the size of the region of rotating air.   empirically    phenomenon

The theory correctly and naturally describes    NO             yes
interactions with the ground and with other wings.

The theory is consistent with the known laws    yes            yes
of motion.

It appears that the disk-balloons theory stands up in every case but one:
it cannot be easily used to explain the ground-effect phenomena, nor the
changes in lifting force which occur when airplanes fly in formation. And
,as I said earlier, I give no explanation for why the wing can make those
"balloons" start moving.

In my opinion, this is pretty impressive for a "theory" which was
developed in order to explain flight to children!

If we are required to make things visible, basic, and to eliminate from
our explanations as many abstract concepts and mathematics as possible,
then "disk balloons" is one result.  It's a "toy explanation," and it
might cause any important and sophisticated aerodynamicist to laugh
scornfully.  Exactly. That's what I *intended* it to be from the start. 
Too bad about any scornful laughter though.  Perhaps the important
aerodynamicists should develop a small bit of humility and think of
themselves not as experts who are unable to make embarassing errors, but
as students who are always learning and who are expected to make huge
errors as part of the trial-and-error process. 

I still find it amazing that the "disk balloon" description contains
features of which some aerodynamicists are apparantly unaware: it points
out that the gravity-fighting flow of momentum must end up in the
descending wake-vortex behind the aircraft.  It shows how the lifting
force upon a wing is the result of a 3rd-law "reaction motor" effect.  It
illustrates how a pair of non-1/R vorticies can interface with the
unmoving atmosphere without any high-shear regions.  It allows an
ambitious highschool kid to derive the equation for induced aerodynamic
drag.  And it clearly illustrates the fact that a two-dimensional
description cannot hope to explain the flight of a real aircraft: the
patterns of flow before and behind a 3D airplane are dissimilar, and also
the lifting mechanism originates in the creation (and the downward
acceleration) of a vortex-wake in three dimensions.  None of this fits
into a 2D simulation, and it cannot exist at all in an inviscid simulation
unless we artificially place it there by hand.  As a result of our 2-D
thinking, we often do something which is unfortunately equivalent to this: 
ignoring the exhaust from a rocket motor, but then attempting to explain
rockets by applying Bernoulli's equation to the tangential velocity of
gases across their solid surfaces.  Yet rockets cannot be explained if we
close our eyes to the existence of their exhaust stream!!!  The
downwards-moving wake vortices behind an aircraft are the "exhaust" from
the reaction motors, where the wings are the reaction motors.  In a 2D
simulation, all of these simple 3D concepts are impossible and unphysical.

The right tools for the right job, and sometimes the incorrect mental
tools can make the job impossible. 

Created and maintained by Bill Beaty. Mail me at: .