I know that it is associated with Prandtl lifting-line theory and that ideal elliptical wing actually meets the condition of my question. Unfortunately I am not good at fluid dynamics and I would like a relatively simple explanation why downwash (and lift) differs along wingspan (of non-elliptical wings). I would like to understand at least the basics and satisfy my curiosity :)
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Why should non-elliptical wings not produce constant downwash over span? They do at one angle of attack, given a proper twist distribution so that the circulation distribution over span is elliptical.
However, once the angle of attack is different from that of the elliptical distribution, the change in local circulation varies in proportion to local chord. If the chord distribution is not elliptical, the new circulation distribution due to that angle of attack change is also not elliptical, and so is the change in local downwash angle.
Sometimes it is desirable to not produce an elliptical circulation distribution rsp. have an elliptical planform. If the objective is to minimise drag, it helps to reduce weight and pure lift/drag optimization is too narrow. What should count is the best ratio of lift minus wing weight relative to drag. R. T. Jones wrote a NACA Technical Note back in 1950 in which he looked at this problem analytically. Wing weight goes up when much lift is created near the tips, because this lift will cause a disproportional root bending moment, and the wing spar, which has to carry this bending moment, is a significant part of the wing structure. Therefore, reducing lift at the tips and adding more lift at the root will create a lighter wing for a modest drag increase, resulting in an overall optimum with an almost triangular lift distribution. When compared to an elliptical wing planform, the total wing span of such an optimized wing is bigger for the same overall drag, but this wing will weigh less.
Another example is wing stall. An elliptical wing without washout will have a constant lift coefficient over span, and Reynolds number effects ensure that the wing will stall at the tips first. An unsymmetrical stall will result in an accelerated roll departure: The stalled side of the wing loses lift, drops and sees an increase in angle of attack which worsens the stall condition. This effect gives an elliptical wing poor handling characteristics and is the reason why many GA aircraft have rectangular or trapezoidal planforms of little taper.
answered Nov 12, 2016 at 19:43
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$\begingroup$ Thank you, Peter! Your answer really helps to understand the issues of elliptical circulation. However I do not understand the principle why elliptical circulation produces the least induced drag. $\endgroup$ Nov 12, 2016 at 20:27
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$\begingroup$ @RokasZilinskas: Did you read this answer already? Generally, since induced drag depends on the square of the lift coefficient, there exists an optimum and not all distributions are equally efficient. $\endgroup$ Nov 13, 2016 at 6:38
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$\begingroup$ Well, I understand that there should be the most optimum circulation because of the quadratic dependency and how induced drag is mathematically derived. But, I do not understand why each part of span of, for example, a rectangular untwisted wing, does not produce the same lift? $\endgroup$ Nov 13, 2016 at 7:45
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$\begingroup$ @RokasZilinskas: That is because span is finite. If the wing would not end, lift would indeed be equal at every span station. The lift at the wingtip has to be zero, however, because pressure will equalise around the wingtip. From there inwards lift will gradually increase until it reaches the maximum at the center. Even the center of high aspect ratio wings will never quite reach the lift coefficient of the infinite wing due to the influence of the tip. $\endgroup$ Nov 13, 2016 at 8:08
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$\begingroup$ Is it the same as saying that air below the wing spills to the top of the wing and therefore pressure equalises? $\endgroup$ Nov 14, 2016 at 14:39