Re: misconceptions: 2D model of 3D flight

[William Beaty <billb@ESKIMO.COM>]

On Sat, 14 Aug 1999, John Denker wrote:

> > In other words, the air within the wake-vortex pair behind an aircraft has
> > been given a net downwards motion and this acts like a "rocket exhaust."
>
> No.

OK, this is an argument which can be settled by presenting evidence.
First let me clear up this point: do I understand that the wake-vortex
pair behind an aircraft is *not* supposed have any net downwards motion?
I am almost certain that it does move downwards, and so I should be able
to find some evidence which supports this idea.  My reasoning at
http://www.amasci.com/wing/rotbal.html certainly supports this idea.

> > The wings of an airplane do not simply create a spinning pair of
> > wake-vorticies, they also project those vorticies downwards.
>
> It is not the downward motion of the vortices that lifts the airplane.  It
> is the downward motion of the *air*.

I totally agree.  In order to create lift, an airfoil must cause the air
to experience a downwards-directed change in momentum.  If I am right,
then this occurs when the wake-vortex pair entrains a certain amount of
air within the circular streamlines, and then the wake-vortices move
downwards as a whole.  The wake-vortex pair entrains a certain mass per
unit length, and they carry this mass along with them as they move.  They
behave much like an object which has been thrown downwards.  If the
vorticies move downwards, the constrained mass moves downwards.

If the air was not moving downwards before the aircraft arrived, and if it
does move down afterwards, then that air experienced a downwards momentum
change, and there must be an equal momentum change somewhere else.  If the
wings created the wake-vortex pair, then I'd predict that the upwards
momentum change appears at the wings.  Since the wake-vortices are being
extended in length as the aircraft flys forwards, then the aircraft is
accelerating a certain number of kilograms of air downwards per second.
It looks to me like this is the source of the lifting force by f=ma.


> Two airplanes with equal vortices can
> produce greatly different amounts of lift.  For homework, explain how.

This is interesting.  I assume that "equal vorticies" means "equal energy
expenditure in creating those vortices?"

There is a typical flaw in the "bernoulli-ist" argument regarding energy
and flight.  Here it is below.  In the past I've had difficulty
communicating my points, so this time I'm going to try using excessive
detail so there can be no confusion.  (Or does that make things *more*
confused?)  :)


  A wing need not do any work on the atmosphere since we can make a wing
  arbitrarily long, and therefor we can reduce the rate of work to an
  arbitrarily small level.  Agreed?

OK, let's follow the implications of this sort of reasoning.  If a wing
deflects air, then a *longer* wing deflects more air, yet the weight of
the aircraft need not change as the wings are made longer.

Gravitational attraction injects a certain downwards momentum into the
aircraft per second, and to remain aloft, the wings must apply an equal
downwards momentum-change to the air per second.  The aircraft flys level,
and the air moves down instead.  If the wings intercept m2 mass of air per
second and give it a downwards velocity v2, then:

   m(craft) * G = m2 * v2 / sec

Because it accelerates the air by v2 velocity per second, the aircraft
expends U = 1/2*m2*v2^2 energy per second to fly.  This is the "induced
drag."

OK, lets make the wings twice as long.  Now the wings intercept twice as
much air per second, yet the weight of the aircraft has not changed.  The
wings must be adjusted to produce half the downwards deflection of air as
before, therefore:

  m(craft) * G = 2 * m2 * (v2/2) / sec

The weight is the same to the momentum-change must be the same.  What has
happened to the work done on the air per second? It has decreased to:

  U2 = 1/2*2*m2*(v2/2)^2

...or half the rate of work compared to the shorter wings.  In this
simplified model, doubling the length of the wings allows us to use half
as much energy to remain in level flight.  Does this prove that wings are
not a reaction engine, or that flight does not require fuel?  No.  Let's
compare the above situation to a simple reaction engine.

Suppose I hover above the ground by firing a machine gun downwards.  If my
machine gun and I together weigh m(craft)*G, then I must eject m2
kilograms of bullets per second in order to stay aloft, and I must fire
them at v2 meters per second so that:

  m(craft)*G = m2 * v2 / sec

My gun expends U = 1/2*m2*v2^2 in energy per second to accelerate those
bullets and to stay aloft.  Fuel in the form of gunpowder is being burned.

I want to use less energy per second.  What if I fire TWO machine guns
downwards simultaneously?  The mass/second of the bullets is doubled, yet
my weight is the same, so I must reduce their velocity by a factor of two
so that:

  m(craft)*G = 2*m2 * (1/2*v2) / sec

My energy expenditure per second is now U2 = 1/2*2*m2*(v2/2)^2, or half
as much as with one gun.  I can add more guns and use even less energy.
Here is the key argument:

   Because I can add an arbitrarily large number of machine guns to my
   craft, I can reduce the work done per second to an arbitrarily small
   value.  Therefor my craft does not use a reaction engine.

Huh?  Of course it's a reaction engine, since IT REACTS AGAINST BULLETS, a
continuous change in momentum produces a continuous upwards force.  True,
the power required for flight can be reduced by adding more and more
machine guns, but this applies to ALL reaction engines.  If I increase the
area of a rocket-exhaust, then I must reduce the exhaust velocity to
remain hovering, and so I use less energy. A rocket with an arbitrarily
large exhaust aperature can use arbitrarily small energy for flight, but
this does not prove that rockets aren't reaction engines.

Therefor, I argue that wings *ARE* reaction engines, and that the
downwards-moving pair of wake-vortices is as critical to their operation
as the exhaust is to a rocket.  The wake-vortex is as necessary as the
bullets fired by my machine-gun craft.  The momentum-change is placed in
the wake vorticies, and this value does not increase in proportion to
wing-length.  When attempting to explain aircraft, ignoring the
wake-vortex makes as much sense as trying to explain how a rocket can
hover but at the same time excluding the rocket's exhaust from all
explanations.


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