Reducing Induced Drag 1

[StrikeEagle]

I just finished reading an old thread (thread2-4996)-- "Winglet effect on wing lift and drag?" I'm curious--I've heard that you can approximate an elliptical lift distribution by having the inner portion of the wing rectangular and the outer portion tapered along with three degrees of washout. Are there rules of thumb for determining how much of the wing should be rectangular and how much should be tapered (half and half?) or what the taper ratio of the outer portion should be? Thank you!

 

[InteroperabilityFix]

if I remember correctly, the tip chord should be around 0.9 of the root chord on a straight wing to best approximate an elliptical shape.

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[StrikeEagle]

Thanks--I'll experiment with that and see how it performs!

 

[FastMouse]

The tapered wing that you describe is like an irregular octagon. You could write a short program that finds the "best fit" octagon for an ideal elliptical planform, maybe keeping the total area and the span constant for both planforms. This would give you the "best fit" root chord, tip chord, and spanwise break point, where you go from the retangular portion to the tapered portion.

Exactly what "best fit" means in this case might be up for debate, but you could loop through all sensible octagons, and pick the one that has a minimum of area where it overlaps the ellipse, or where the ellipse overlaps the octagon, i.e. where the common area of the octagon and the ellipse is maximised. Hope I've managed to state that clearly, a sketch would be better!

Assuming spanwise and chordwise axes of symmetry would mean that you would have to analyse only one quadrant.

I'm sure that the "best fit" octagon could also be found analytically, too.

Besides washout/twist that you also mention, you could additionally consider changing the airfoil section with spanwise position to get an acceptable lift distribution.

Then of course, there is the complication of fuselage and nacelle (if present) effects...

FastMouse

 

[StrikeEagle]

I think I understand what you've said Fastmouse-- It makes sense and I created an excel file to implement it. If I understood you correctly, I found (using excel's solver) that the taper ratio should be about .4 and that the rectangular span is 41% of the whole span. I changed the aspect ratio of the ellipse (is there an ideal AR for an ellipse?) and found the same solution. That seems like sort of a strong taper, but maybe it's right. Is there a way for me to attach the excel file for others to review and see if I did understand you and if my methodology made sense?

Thanks!
StrikeEagle

 

[FastMouse]

I regret my tardy response! Too many things going on at once.

I do not know if there is a general rule about what octagonal planform dimensions fit an ellipse the best, but if there is, it could probably be found analytically. What I was thinking about was a program where you enter the parameters of the ellipse, and it returns a best-fit octagon. It sounds like you got there first with the spreadsheet that you wrote! I don't know how you would upload it here for others to see, but maybe it's possible to put it on a server somewhere and post the link.

Anyway, I wrote a small program to do the best-fit mentioned above. You plug in the ellipse (root chord and wingspan), and it calculates a best-fit octagon. It keeps the wingspan the same as the ellipse when it's finding the octagon, but not necessarily the wing area. It searches for the octagon for which ?(y.ellipse - y.octagon)² is minimised. To me also, the taper of the octagon appears quite marked, but the fit looks okay in the plot. There's a typical screenshot below.

As for an ideal AR for an ellipse, basically the more you increase AR for a given wing area (for any given platform), the lower the induced drag will be. I guess that at some value of AR, the chord and thus the Reynolds number becomes so low that the profile is no longer working efficiently and the total drag of the wing starts to increase with AR, but that's probably a limiting case.