planes flying abreast: the Newtonian solution

[William Beaty <billb@ESKIMO.COM>]

n Mon, 23 Aug 1999, John Denker wrote:


> What about theory 1b?  In Fermi's immortal words, it is "not even wrong".
> Indeed, theory 1b is *not* wrong.  It's *true* that each person has a
> weight-to-height ratio.  But this is not physics.  It's not science.  It
> has no predictive power.


I agree with everything that John Denker says above.  Who could disagree?


John, correct me if I'm wrong, but isn't this quite obviously related to
your low opinion of the "Newton/Downwash" theory of flight?  Isn't the
above stuff actually referring to your earlier messages about
Nonlinearity, where you point out that the Induced Drag of two airplanes
is greater when they are far apart, and smaller they are flying in tight
formation abreast?  Also, might you be be trying to say that the
"Newton/Downwash" theory is "not even wrong," because you believe that it
cannot predict what happens to induced drag when planes fly
wingtip-to-wingtip?

If so, then please note that I rebutted your assertion regarding this
"adjacent flight" issue a few days ago. (see attached message).  At least
I had imagined that I did.  Different planes which have their
lifting-force distributed differently across their wingspans will each
have a different value "vortex-creation drag" as part of their total
Induced Drag.  I assumed that as an aerodynamics expert you understood
this, and I didn't need to go into details and preach to the choir.

The "Newton/downwash" model predicts both the "lifting" drag and the
"vortex" drag.  If two planes fly in close formation, wingtip to wingtip,
they create a single larger wing with a very strange distribution of
lifting force indeed.  Was this concept buried too deeply in the message
below?  I can see it in my mind and it appears very obvious, but I guess I
wasn't clear enough in describing it.  Time for more detail.

The "Newton/downwash" explanation of flight says that a certain aircraft,
if it has a certain wingspan/velocity/weight, must divert a certain amount
of air downwards per second in order to create an F=ma lifting force, and
this requires a certain amount of energy per second.  This energy/second
represents a power loss which must be made up by the engine.  It is one
component of induced drag.  Another component of induced drag exists as
well, because the downwash isn't moving downwards as if it was a solid
"cube" of air.  Instead it is swirling in a vortex-pair pattern.  The
swirling motion of the air contains a significant amount of kinetic
energy.  Stirring that "vortex pair" into existence represents a
significant power loss for the airplane, and the engine must do
significant work to generate those long, growing cylinders of rotating
air.

Is this fairly clear?  The induced drag has two parts: one part is a
downwards motion of mass which creates lift by conservation of momentum.
The other part is an unwanted (but unavoidable) rotating motion of the air
that is present within each vortex of the plane's trailing wake-vortex
pair.

Note that this topic is not broached in the Anderson/Eberhardt paper.
Their paper is just a simplified version of the full-blown, Newton-based
vortex theory of aerodynamics.  Their paper is a stripped-down version
intended for public consumption.  If people are interested in the detailed
version of the theory, I suspect that the authors would be happy to supply
some references.

OK, the details:

Suppose two identical airplanes fly far apart, flying at the same altitude
and velocity.  Each one will suffer the same amount of induced drag.

Now suppose they fly wingtip-to-wingtip.  If we look only at the part of
the induced drag (and the net downwash) which supplies the lifting force,
we will see that the "Newton/downwash" model insists that this portion of
the drag must have the *same* total value regardless of whether the planes
fly far apart, or whether they fly with wingtips touching.  Mr. Denker is
correct in noticing this fact.  The weight of the planes has not changed,
so the lifting force has not changed, therefor that part of the induced
drag cannot change.  "Newton said so."  :)

What about the Kinetic-Energy/sec placed into that "swirling motion"
imparted to the wake-vortices by the downwash-pattern distributed across
the wingspan?  What about the corresponding portion of energy-loss and
induced drag which this creates?  Well, I've heard in the past that the
value of this drag component changes depending on the distribution of lift
across the wing.  In the Newton-based induced-drag calculation, it appears
as a "geometrical factor".  The presence of this
lift-distribution-dependant correction factor suggests that when two
separate airplanes come together to form a single "longer wing" of twice
the mass of a single airplane, this "correction factor" changes
significantly, so we should *not* be suprised if the energy flow needed to
spin the wake-vortices of this "longer wing" is very different than the
sum of the energy flow of two separate airplanes.  After all, the shape of
the trailing wake-vortex pattern has significantly changed (it is related
to the square of the wingspan.)  This means that the shape of the
wake-vortex pair is very different for two planes flying abreast than for
two planes flying far apart, and so the drag must be different.  Since the
Newton/downwash model predicts this, Mr. Denker's argument against the
Newton/downwash model is invalid.  That's what I was *trying* to say in
the attached message.

There's a far simpler way to look at this.  When the two planes are flying
separately, they generate FOUR vortex-cores in the wakes behind them, and
this needlessly expends energy and fuel.  When they fly
wingtip-to-wingtip, they only generate TWO significant vortex-cores (in
other words, together the planes behave as one long wing with only two
wingtips.)  If there is a significant amount of kinetic energy expended in
producing the fast-spinning air in regions near the vortex cores, then we
would expect that the *NET* induced drag would fall by a large value when
two planes come together and fly in tight formation.

The above is a sensible prediction of the "Newton/Downwash" theory if we
make sure to include the "downwash" at every location along the span of
the wings, and also to take account of the fact that the "downwash" curves
into a flowing shape which resembles the magnetic field of a pair of
parallel wires containing electric currents of opposite polarity.  (This
as opposed to just pretending that each airplane generates a single large,
un-swirling chunk of downwashing air.)  The "downwash" looks something
like this ASCII figure below, with the wing of the airplane running
horizontally across the diagram, and the wingtips touching the two central
dots.  This is a view looking towards the rear of the plane, and the
vertical lines show the net downwards motion which the wing has given to
the air.  Note that this is also the shape of the "scoop" which encounters
oncoming air and forces it downwards.  The "invisible scoop" is far, far
larger than the visible airplane wing.


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


BTW, the reduced number of wake-vortex cores is supposedly the reason why
Canada geese fly in formation.  They use less "fuel" if they aren't doing
the equivalent of creating an entire goose-flock full of wingtip
vorticies.  Perhaps this makes it hard on the "wingtip" geese, and they
might trade positions frequently?  Ah, this might explain why they don't
seem to fly in symmetrical "V" shapes.  Only one "wing", so only one major
"wingtip vortex."


John, please do me the favor of responding to my above "rebuttal" of your
contention that "Newton/downwash" cannot explain the drop in drag present
when planes fly abreast.

In several past messages you have remained silent about my counters to
your arguments.  For this reason I don't know if you accept them, if you
reject them, or if I was simply too unclear with my writing and you didn't
realize I was making them.  If you remain silent, it effectively ends the
argument, but ANYONE can "win" an argument if silence is their defensive
strategy.  Rather than silence, I would prefer that you say something
similar to this:

  "I understand your arguments, but my intuition still tells me that you
   are wrong, so I will continue to search for more powerful
   counterarguments."


((((((((((((((((((((( ( (  (   (    (O)    )   )  ) ) )))))))))))))))))))))
William J. Beaty                                  SCIENCE HOBBYIST website
billb@eskimo.com                                  http://www.amasci.com
EE/programmer/sci-exhibits          science projects, tesla, weird science
Seattle, WA   206-781-3320          freenrg-L taoshum-L vortex-L webhead-L




Date: Fri, 20 Aug 1999 21:33:47 -0700
From: William Beaty <billb@eskimo.com>
To: PHYS-L@lists.nau.edu
Subject: Re: physics can be so nonlinear

On Fri, 20 Aug 1999, John Denker wrote:

> Hi Folks --
>
> Here's another attempt to help people understand why certain theories which
> are intuitively appealing are doomed to failure.
>
> People like nice, simple, linear theories.  The problem is, fluid dynamics
> is nonlinear -- highly nonlinear.
>
> Consider two airplanes with reasonably long wings, initially flying in
> formation, wingtip to wingtip.  Then they move apart, maintaining constant
> airspeed at all times.  This produces an increase in the induced drag
> force.  This is fundamentally a nonlinear effect.  It is not a small
> nonlinearity;  there is (to a good approximation) a 100% increase in
> induced drag.
>
> Any theory that involves only a simple "throw something down" process is
> too simple to provide a passable description of induced drag.

We agree totally on this.  The "throwing something down" theory might
supply the fundamental process, but the fundamental process cannot explain
all of the details of a three-dimensional "lifting force" event.

It's obvious to me that the wake-vortex behind any aircraft in high-
altitude flight will display two main motions:  A downwards velocity, and
a spinning motion.  Amazingly enough, the two are not related.  Depending
on the distribution of vorticity within the wake, we can have a wake which
spins fiercely but which does not decend, or we can have a wake of exactly
the same size and shape which moves downwards very fast but barely spins
at all.

I've long been aware of this effect.  Go and play for hours with
smoke-rings made by a cardboard oatmeal container and a stick of incense
and you too will see why.  (Hey, why not launch smoke-rings with a
loudspeaker and a computer-based waveform generator?  Tailor the waveform
as needed.  Make smoke-rings which spin without advancing, or smoke rings
which advance without spinning.  Or smoke rings which crawl slowly along
for a time, but which then undergo a turbulent transition and "explode"
into a disorganised cloud.  I've never tried doing any of this, you be the
first. See:  http://www.amasci.com/amateur/vortgen.html )

Ahem.

If an aircraft develops lift by accelerating the mass of its vortex-wake
downwards, then the rate of doing work does not depend only on the
downwards momentum-change of the air.  After all, it also takes work to
cause the vortices to spin, and this spinning motion is not at all a part
of the downwards momentum change being given to the vortex wake as a
whole.

For example, suppose that aircraft #1 throws a pair of vortices downwards,
If the core region of those vorticies spin at a high rate, then that
aircraft is doing lots of unnecessary work, and it will have a needlessly
high induced drag.

If aircraft #2 throws a pair of vorticies downwards, and if the general
size of the vorticies is identical to those of aircraft #1, and if the
weights of the two craft are the same, then we can see that the downwards
momentum of the wake-vorticies must be the same... but the spinning motion
need not be the same.  If aircraft #2 can cause its wake-vorticies to
barely spin at all, then that aircraft must have a far lower induced drag
than aircraft #1.  Because its vorticies spin slowly, aircraft #2 does far
less work upon the air, and it experiences less drag as a consequence,
even though it produces the same change in downwards-directed momentum as
aircraft #1.

Why are the changes in induced drag apparantly unrelated to the volume of
the air which must be thrown downwards?  There's a simple answer.  The
answer can be found in the spin of the air.  The spinning motion of the
central regions of the wake-vortex pair is unwanted component in the
lift-generation process, and if it varies, then induced drag will vary as
well.   If it can be eliminated, then fuel costs will be saved.

> I wish I could offer you a nice linear theory of induced drag.  But I
> can't.  Mother nature won't let me.


"Disk balloons" and smoke-ring boxes and "messing around."  Staying up far
too late at night and suffering massive "physics AHA!" experiences as a
consequence of lowered inhibitions to creativity.  Conversing directly
with Mother Nature in the nonverbal language of the scientist (it's the
same nonverbal language used by little kids.)  That's where my above ideas
came from. Are they right?  I don't know!!!!  They sound sensible, but we
obviously require experimental evidence before we can put any trust in
them.

Or, more likely, this is well-trodden ground in aerodynamics, so perhaps
I can find a textbook which says the same thing as I do above, but uses
the proper terminology and contains the math that backs it up.  But make
no mistake, if my stuff above is true, then the math is nothing exotic.
It's probably as linear as... as the math of the magnetic field around
electromagnet coils wound pancake-style!  Yeaaah, that's the ticket.  Vary
the spacing of the turns of wire in the pancake, and you "program" the
vorticity distribution in the wake behind a similarly-shaped wing.  Hey,
maybe the energy contained in the resulting magnetic field corresponds to
the energy contained in the spinning wake-vortex? )


((((((((((((((((((((( ( (  (   (    (O)    )   )  ) ) )))))))))))))))))))))
William J. Beaty                                  SCIENCE HOBBYIST website
billb@eskimo.com                                  http://www.amasci.com
EE/programmer/sci-exhibits          science projects, tesla, weird science
Seattle, WA   206-781-3320          freenrg-L taoshum-L vortex-L webhead-L