Re: Airplane Drag

[John Denker <jsd@MONMOUTH.COM>]

At 02:04 AM 9/13/99 -0600, Jim Green wrote:
> I have been told most assuredly that extending flaps at high speed produces
> less drag than extending them at lower speed. (I don't know what "high" and
> "lower" are.)
> I find the physics of this a little hard to fathom -- if true (:-)

I say again: it's not true.

Let me formalize the statement of the question, to remove possible sources
of misunderstanding.

  Delta_F(v1) = force_of_drag(v1, flapped) - force_of_drag(v1, clean)

  Delta_F(v2) = force_of_drag(v2, flapped) - force_of_drag(v2, clean)

where without loss of generality we assume v1 < v2

and the claim is that Delta_F(v2) is less than Delta_F(v1) for at least
some combination of v1 and v2 within the reasonable range.

We can even define
  delta_Delta_F := Delta_F(v2)  -  Delta_F(v1)
so the assertion is that delta_Delta_F is less than zero.

Let's write it out:

        force_of_drag = ki v^(-2) + kp(v^2)
where ki has to do with induced drag and kp has to do with parasite drag,
i.e. friction and everything else except induced drag.

Plugging in and collecting terms we have

 delta_Delta_F =  (ki_flapped - ki_clean) {v2^(-2) - v1^(-2)}
                + (kp_flapped - kp_clean) [v2^2    - v1^2   ]

where the factor in [square brackets] is manifestly positive.  The factor
(kp_flapped - kp_clean) is *large* and positive.  The factor in {curly
braces} is negative, but the remaining factor, namely (ki_flapped -
ki_clean) is zero to a first approximation, because the amount of energy
the airplane must leave in the wake vortices at any particular airspeed
does not strongly depend on details of the configuration.  To a second
approximation, for typical modern planforms, this factor (ki_flapped -
ki_clean) is almost certainly slightly negative, because it is more
efficient for each wing to make two weak vortices than one strong vortex.
So this term, too, makes a small positive contribution.  Even if it made a
negative contribution, I don't see how it could overcome the large positive
contribution of the parasite-drag term.

Therefore we conclude:

* The bold assertion that delta_Delta_F is "most assuredly" always negative
is most assuredly false.

* To prove the falsity of the bold assertion, it suffices to show a single
reasonable counterexample.  But I will go farther than that.  I believe
that there are very few, if any, reasonable circumstances where
delta_Delta_F is negative.