Re: vortices can be so nonlinear

[John Denker <jsd@MONMOUTH.COM>]

At 09:33 PM 8/20/99 -0700, William Beaty wrote:
> 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,

OK, good question.

> then we can see that the downwards
> momentum of the wake-vorticies must be the same...

I don't want to sound nit-picky, but it's not good to refer to the momentum
of the vortices.  The momentum is in the *air* (as you say correctly below)
not in the vortices per se.  So let's assume you meant "air" and go on.

> 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 ....

The phenomenon you cite has been known to aircraft designers since the
earliest days.  They were so worried about it that some early aircraft had
elliptical wings, which can be shown to shed vortex lines equally spaced
along the span, thereby minimizing the energy in the vortex core(s).

Nowadays wings tend to have straight edges.  It turns out that the
resulting shape is "close enough" to an ellipse that the vorticity is not
shed all at one place, so the energy in the vortex core is not a problem.
At cruise speed induced drag is nearly negligible anyway.

For a nice picture of how all these little vortex lines wrap around each
other to eventually form "the" trailing vortex behind each wing, refer to
figure 7.3 in E. L. Houghton and N. B. Carruthers, _Aerodynamics for
Engineering Students_.  See figure 7.5 ibid for a slightly more realistic
picture of the distribution of bound and trailing vortices on a rectangular
wing.

========

Related bit of physics:

The circulatory velocity in a vortex goes like 1/r.  The kinetic energy
density goes like 1/r^2.  The total energy goes like
  integral(from core to X) of 1/r^2  r d(theta)  d(r)
That's logarithmic, so it diverges only *slowly* as the lower limit of
integration gets small.  Physically this lower limit is the size of the
"core" of the vortex -- the eye of the hurricane, where the velocity is a
more like a disk (proportional to r) than like a vortex (1/r).  Because of
this *slow* divergence, people don't worry too much about the exact core size.

The outer limit X doesn't matter either, because at large distances we
don't have the 1/r velocity field of a single vortex.  Instead we have the
field of a *pair* of counter-rotating vortices, in which case the integrand
goes to zero quickly enough that the X->infinity limit converges nicely.