# For the elliptical wing, which property is actually elliptically distributed?

Elliptical wings are sometimes used:

One of the 50 Spitfire still flying in the World. Source

I read it's because this shape allows a perfect distribution of aerodynamic forces. But actually:

• What is actually elliptical in the elliptical wing?
• What is regularly distributed: Is it lift? or drag? or something else?

It is both the planform and the circulation distribution. Note that circulation is not lift coefficient but bound vortex intensity. You can interpret it as local lift coefficient times local chord.

On the untwisted elliptical wing the local lift coefficient is constant over span, and changes in angle of attack over the linear range will change the lift coefficient equally everywhere. This, when combined with the elliptical chord distribution over span, means that the circulation distribution will stay elliptical over the linear angle of attack range. This is the special characteristic of an elliptical wing: While any wing can have an elliptical circulation distribution at one angle of attack (given the right twist distribution), the elliptical wing will keep that elliptical circulation distribution over the whole operating range.

With an elliptic circulation distribution comes also a constant induced angle of attack and downwash angle over span. I guess this is expressed by some authors with the term "regular".

However, only the aerodynamicists will see that as an advantage. Both weight and stall characteristics of elliptical wings are less than optimum; the low induced drag coefficient is bought with higher structural mass and, consequently, lift. A more triangular circulation distribution will yield the lowest wing weight and overall drag for a given non-lifting mass (that is, all mass that is not involved in lift generation, especially the payload). Note that for such a triangular distribution drag will be highest near the center.

When people talk about elliptical lift distribution, they mean lift per span. I prefer to use the more correct term circulation, since lift is a force as in pressure times area and can only be produced by a whole wing or at least wing section, not one spanwise station.

(Edited after comments by mods. Thanks for guiding me through the process!)

The only thing that is elliptical is the planform shape of an elliptical wing.

Many people unfortunately confuse an elliptical pressure distribution with the pressure distribution over an elliptical wing. They are not equivalent.

On an untwisted elliptical wing the local lift coefficient is not constant over span, as some here and in many other places on the internet maintain.

Arguments from lifting line theory are not appropriate for discussing the behaviour of the flow near wing tips. It is unreasonable to expect lifting line theory to be valid near the wing tips because is not a consistent large aspect ratio asymptotic expansion. See: Van Dyke, "Perturbation methods in fluid mechanics", 1964.

The span loading includes a logarithmic term, hence it is not elliptical. Furthermore, it cannot induce a constant downwash, and so at the trailing edge the vortex wake does not start as a flat sheet.

The process whereby the vortex sheet rolls up is far more vigorous than if it started out as a flat sheet because the relatively weak effect of viscosity would be the main mechanism driving that rolling-up process. There is a strong upwash in the flow-field near to and around the wing tips which initiates the rolling-up process far more vigorously.

See, for example, among many other papers:

Peter F. Jordan, "Exact Solutions for Lifting Surfaces", AIAA Journal, Vol. 11, No. 8, 1973., pp. 1123-1129.

Peter F. Jordan, "On Lifting Wings with Parabolic Tips", ZAMM 54, pp. 463-477, 1974.

• Hello Lysistrata, welcome to Aviation.stackexchange.com. I think your answer has great potential, but currently it is only a comment to Peter Kämpf's answer. Can you edit your post and answer the original question, thereby addressing where you think Peter's answer falls short? Otherwise it may be flagged as "comment to other post" and subsequently deleted. Commented Oct 11, 2017 at 12:12
• My comment is a criticism of Peter's answer, specifically regarding constant spanwise lift coefficient. (I have no problem with the other parts of his thoughtful answer.) I gave reasons for why I believe that part of his answer is incorrect, and I have cited appropriate references. I'm not sure that I can do much more than that. Commented Oct 11, 2017 at 12:43
• Welcome too, I have to agree with DeltaLima. It's not that your comment is bad, it's great, but the site rules don't allow this format. Please answer the original question in your post as well, to prevent your post from being deleted. It's just to make sure your post fits the community rules, because the community on stack-exchange is strict. Commented Oct 11, 2017 at 12:52
• For a moment I thought you deleted your answer and all the good references were gone. Thank you for editing your answer and putting it back in! By the way, we (@NoahKrasser and me) are not mods, we're just ordinary users like you. It's the combined effort of the community that makes this site great. Commented Oct 11, 2017 at 14:21

The beautiful shape of the Spitfire planform also made it difficult and expensive to produce. The target was an elliptical lift distribution, at the time of design regarded as resulting in the highest lift/drag ratio.

This lift distribution is obtained by shaping the wing geometry, in the case of the Spitfire by making the chord an elliptical function of wing span. The other way of achieving this is to use a combination of wing taper and wing twist: the chord reduces linearly with span, plus the wing is twisted with the wing tip having a lower angle of attack than the wing root.

This shape creates elliptical lift distribution as well, but at only one AoA of the aircraft. The elliptical wing has an elliptical planform and elliptical lift distribution throughout all aircraft AoA.

The eliptical planform wing has a constant AoA, and drag changes as a function of wing chord only. The twisted wing has a wing root with a higher AoA, and a wing tip with lower AoA than aircraft AoA: the fat bit of the wing has the higher drag, since wing drag is a function of chord and AoA. So total wing drag of the tapered wing is higher than that of the elliptical wing.

The tapered twisted wing rules though, because:

• It is cheaper and faster to produce.
• The wing tip stalls last, and that is where the ailerons are.
• Actually, the reason for the elliptic shape was to house the guns in the thinnest possible wing. Mitchell was cited by Shemstone as follows: "I remember once discussing the wing shape with him and he commented: “! don’t give a b..... whether it’s elliptical or not, so long as it covers the guns!” Commented Jul 4, 2017 at 21:19
• This answer is quite wrong, it wholly mis-characterises the Spitfire's wing. It was twisted, with (I think) around 3 deg of washout. The aerofoil changed along the span, being thinner and less cambered at the tip. In consequence its lift distribution was closer to the structurally optimal bell-shape than the elliptical. This was of course deliberate, in order to combine superb handling qualities with high efficiency. As @PeterKämpf says, the elliptical shape was an accident of the specification change from four guns to eight. Commented May 11, 2021 at 19:06
• @GuyInchbald I think you are too strict. The archetype for the Spitfire wing was the Heinkel 70 wing, and here the elliptic shape was chosen to have enough internal space for the retractable landing gear. Both camber and thickness were reduced towards the tip as well, so Mitchell simply stuck to what the Günther brothers had chosen. But since the Günther brothers were obsessed by optimum aerodynamic shaping, the elliptic result wasn't entirely unwelcome, either. Commented May 12, 2021 at 1:22
• @PeterKämpf That He 70 ripoff meme is another myth peddled by too many mainstream writers who fail to check their sources; the designs were quite independent. Any resemblances are convergent evolution not copying. The He 70 was designed with fixed u/c and the retraction mechanism introduced later; the Spit wing was evolved from a straight taper to fit more guns within its chord. It was not Mitchell who designed the Spitfire wing, it was Shenstone and another colleague whose name slips my mind at the moment. Even one error from you is unusual, three in one comment is remarkable. Take care. Commented May 12, 2021 at 9:05
• @GuyInchbald He who sits in a glass house should not throw stones. The fixed gear design was the Heinkel 65. The He-70 had a retractable gear from the start. Commented May 12, 2021 at 10:23