On Sunday, December 27, 2015 at 1:56:26 AM UTC-8, J. Nieuwenhuize wrote:

If we go back to theory, the perfect wing is a double super-ellipse (Lamé curve):
http://en.wikipedia.org/wiki/Superellipse

I know we've had this discussion before and maybe something is lost in translation - but I cannot find a reference in an aerodynamics or aircraft design text anywhere to confirm use of super-elipse as a wing planform plus dihedral choice - optimal or otherwise. The only reference to super-elipse anywhere in any aircraft design text I could come up with was he

http://www.dtic.mil/dtic/tr/fulltext/u2/a461449.pdf

It refers to the cross-section shape for a fuselage as a super-elipse is generally a rounded rectangle shape. I'm not clear how this shapes maps to an elliptical shaped wing planform with continuous polyhedral - which is what I think the post is describing. Even the included wikipedia page doesn't show that. Double-elipse refers to a wing where both the leading and trailing edges are eliptical - as in the Supermarine Spitfire of WW II. Do you have a reference to any technical papers on double super-elipse (planform AND polyhedral) as an optimal design for wings. I can't find anything and am curious about where this comes from.

http://www.militaryairshows.co.uk/must.jpg


Both the top view and the dihedral of the wing should have the shape of a super-ellipse for lowest induced drag for a given bending moment (structural weight) and wetted area (profile drag at high speeds). Both the A350 and the Dreamliner are very close to this ideal:
wallpaperswide.com/download/boeing_787_dreamliner-wallpaper-1920x1200.jpg

Here is a nice planform view of the 787.

http://i.stack.imgur.com/1OUtR.jpg

I'd be hard-pressed to describe the planform as a perfect eclipse - it seems to be the typical swept double-taper design with a higher aspect ratio outer panel than typical in earlier designs. It's a transonic wing design so you'd expect some different considerations than for a glider. The shape of the dihedral is more a function of wing flex than anything else, somewhat enhanced by the raked winglet. I've seen no claims by Boeing that the dihedral is for anything other than handling and ride qualities - that they do claim. They do make aerodynamic claims about the raked winglet, but it is a more complex rationale than simple elliptical planform and they have since introduced a double-raked winglet on the 737 Max. Flow fields at wing tips are...complicated.

https://thedesignair.files.wordpress...30-620x413.jpg


Another plus compared to a wing without dihedral and winglets is that the interference drag between the winglet and the wing is much reduced since it's proportional to the radius of the fillet and how sharp the break is.

That seems to be true from published studies but for a glider Reynolds number the radius of curvature above which there is no additional benefit is less than a meter. The V3 radius of curvature is more than 10 meters for the polyhedral section and considerably tighter for the winglet transition, so I suspect more complex considerations are involved - possibly handling and ground clearance of the tips as has been previously discussed. I am skeptical of the arguments that you can sneak in more wing area or aspect ratio onto a polyhedral wing as only the wing planform component in the direction perpendicular to gravity (in level flight) matters. Any component of lifting force inward on one wing is cancelled by the opposite wing so you get net zero lift and some additional drag, which is why most aircraft don't have 45-degrees of dihedral and spans along the wing of 21 meters instead of 15.


Such a gradually curved wing is impossible to build because all the control surfaces would have a bend in them. The wing with sections progressively canted more and more (polyhedral) is a good compromise.

True that.


It also helps with flutter apparantly. The Vortex shedding frequency of the various sections make the critical flutter speed for such a wing higher, allowing either a higher VNE, or a less stiff and thus lighter wing structure.

Flutter is an aeroelastic effect so it's a function of stiffness, sweep, twist and bending - among other things. It's hard to say very much about polyhedral all by itself. Maybe.