I was thumbing through Martin Simons book model Aircraft Aerodynamics book 4th ed & came across the induced drag coefficient equation presented as: Cdi = k * (Cl^2 / 3.142 * A) where
Cdi is the vortex induced drag coefficient
K = “a correcting figure to allow for wing planform, for a well designed wing it is only a little over 1.0”
Cl = lift coefficient
A = aspect ratio = span^2/area

It goes on to describe the nasty stall characteristics based on Cl distributions of different planforms like these bad boys:
http://www.faatest.com/books/FLT/Cha...ngPlanform.htm

I know the elliptical wing thing was beat to death in past RCU posts (which I’ve read). Simons says pretty much the same thing: “mathematical analysis & experiment show that the only type of wing that will produce at all speeds constant downwash & load distribution matching the area is one with elliptical planform area” ....“To aim for an elliptical area distribution is not quite the same as saying the wing should be a perfect elipse. It may be so, but any other form which gives a chord at each point the same as the pure elipse will have the same effect.”

That’s all fine & good, but I see an equation where ‘k’ & ‘A’ have identical impact; increasing A by 10% reduces Cdi by the same amount as increasing k by 10%.

So lets say for arguments sake I humbly accept the elliptically based wing, that freezes A in the equation. Now I have created 3 wing planforms, A, B, C. All have the same span, chord, area & aspect ratio. A is a perfect elipse, B is a straight trailing edge derivative, C is an arbitrary wonky shape. Both B&C were derived from elipse A by meeting the criteria “gives a chord at each point the same as a pure elipse”.

So am I to assume that the k value shape factor is the same, therefore A,B,C have the identical Cdi values based on the equation? Or does k encompass some other geometry parameters that makes these planforms different? Where does one find a table of ‘k’ values? Who generated them & how? Simons says "k in this equation is a correcting figure to allow for wing planform. For a well designed wing it is only a little over 1.0". Unfortunately he doesnt go on to show corresponding values or variations thereof.

Using the same methods, I could actually generate very bizarre looking shapes. I could pull the tips back to something approaching the Sabre jet in the example which has stall approaching the tips from the rear. Yet these are all based on the same elipse. Something tells me they can’t all be ‘equal’, but where in the equations does that come out or am I misinterpreting something?

Hopefully one of the other more "seasoned" AE's will weigh in on this and add to my post somewhat, but I can tell you that k is an efficiency factor. It is also expressed as "1/e" sometimes as in: Cdi=Cl^2/pi*e*AR. I like e better so I'm going to use it from here on, just remember it is the reciprocal of k.

"e" is called the span efficiency factor and is 1 for the "ideal" elliptical wing. For typical subsonic wings it generally ranges from .85 to .95. "e" accounts for differences between a theoretical wing and the "real world" wing and is determined empirically.

I doubt there is a table of k values for you to reference. If you're trying to figure something for your model then you can use 1 since it won't make much of a difference based on the typical models size and performance envelope.

for models -such as we fly - most of that is --- not important.
The ellipticals I have done do fly well but for aerobatics - no real benifit
Having a swept LE makes thing a bit more stable at high AOA but aside from that
Zip
It did look good on paper tho.

Nope, not talking about an aerobatic model. Im interested specifically in the high performance sector where they are trying to eek every last advantage out of planforms of constrained dimesnions. Mostly, I want to understand the underlying equations & math in this post, thanks.

I read where an elliptical wing planform, the theoretical optimum, only produced, in wind tunnel tests, about two percent less drag than a wing with straight taper of about 60 percent or so, thus there is little reason to use the structurally complicated elliptical planform, unless you just like the way it looks.

There is such a table on Page 99 of "Low Power Laminar Aircraft Design" by Alex Strojnik (1984). The table is multi dimensional because the correction factor depends again on aspect ratio in addition to span wise twist and planform. And, the correction factors vary with angle of attack. The best planform shown in the table is constant C for 1/3 of the half span and then the TE tapers forward to .4C at the tip. The wing twist is -3 degrees. The resulting K is 1.009 for Cl of .2 through 1.0.

This is book has the best discussion of practical design that I have found.

We made em -flew em and from a purely pracical standpoint -on a model - not worth the effort.
a good alternative is a double sweep on the LE only .
I do love the looks of the early Spit wing tho------------------

Thanks fo rthat. In the absense of havng the text, can you tell what the maximum variation in k is among similar aspect ration profiles? I find it odd that it's such a specifc parameter, yet very little (that I can find) information details its history or use. Im wondering if values were determined through early testing & this method of correction is now kind of abandoned with VLM type methods that are a more rigourous analysis.

I'll email a copy to you - give me a little time, I want to get in some flying this weekend.

Just sent 2 PMs with the chart and text - enjoy.

You can get Strojnik's books here:

http://www.reactionresearch.com/airc...jnikbooks.html