Why do the elliptical and the rectangular wing show different aerodynamic efficiency?

I understand the elliptical planform is aerodynamically the most efficient type of wing planform and generates the greatest L/Dmax, but it's difficult for me to understand why.

What makes the elliptical shape of a wing aerodynamically more efficient, say, compared to the rectangular type of a wing?

• Does this answer your question? if not, could you mention what exactly is not clear from those explanations? aviation.stackexchange.com/q/39566/1467 – Federico Sep 27 '17 at 8:26
• @Federico Thank you for the link. I've read all the answers from it, but I think I'll need more time to study them before I ask you any questions. Thank you! – lemonincider Sep 27 '17 at 8:54

2 Answers

(This may be an anti-answer.) Why is difficult. It's just how air flows past one obstacle or another: something that a wind tunnel can put numbers on, but not give an eloquent explanation for. Amuse yourself with debates about the Coanda effect, which devolve into anthropomorphism.

For a given definition of efficiency, one combination of planform and washout and sweep and taper and camber and vortex generators and aerodynamic elasticity and who knows what else will outperform another.

For the famous Spitfire case, everything else being equal, an elliptical planform yields, per unit wingspan, more lift and less induced drag, over a wide range of angle of attack, than any other planform, rectangular (cheap to manufacture) or otherwise.

Theorists writing "consider a spherical cow" papers attribute this to the downwash being constant spanwise, so each spanwise sliver of the wing is working as hard as it could be. (See what I mean about anthropomorphism.)

It turns out, elliptical lift distribution is not the most efficient planform. It is, however, the most efficient planform in terms of L/D for any given span length, and the reason has to do with its constant span induced downwash (see For the elliptical wing, what is elliptical, and why is drag regularly distributed?).

If you relax the span length requirement and instead optimize with regards to root bending moments, the Bell-Shaped Lift Distribution is actually superior to elliptical (see derivation and NASA paper). For the same root bending moment and the same lift, the bell-shaped distribution has less induced drag.