Re: physics can be so nonlinear

[William Beaty <billb@ESKIMO.COM>]

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? )


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