Ludwig Prandtl’s 1933 Paper Concerning Wings for Minimum Induced Drag, Translation and Commentary

Minimum Induced Drag for Tapered Wings Including Structural Constraints

“…, which corresponds to Prandtl's 1933 lift distribution. This supports Prandtl's claim that "tapered wings have an advantage over those with a nearly rectangular profile" [14,15], Fig. 1 Ratio of the induced drag and corresponding wingspan produced by an elliptic planform to those produced by a rectangular planform, plotted as a function of B3.…”

“…In this paper, the work of Phillips et al [12,38] will be extended analytically to identify expressions for the induced drag of non-rectangular wings, including the effects of the chord distribution on the wing structure. The new expressions will then be used to evaluate Prandtl's claim that tapered wings have an advantage over rectangular wings [14,15]. As will be shown, the results in this paper demonstrate that Prandtl's claim is indeed correct for elliptically-tapered and linearly-tapered planforms when the effects of planform shape are considered.…”

“…corresponds more closely to that produced by what he called "spitzendigen Flügel" [14] (which can be translated as "pointed-end wings" or "tapered wings") than to the elliptic lift distribution. Within the framework of his solution [14], the elliptic lift distribution performs "noticeably worse" [14,15] than the lift distribution given by 3 1…”

“…. Therefore, near the end of his 1933 paper, Prandtl concluded that "tapered wings have an advantage over those with a nearly rectangular profile" [14,15]. Prandtl's statement is somewhat vague because any wing having a chord distribution that decreases with span could be considered a tapered wing.…”

“…Equations (54) and (55) show that when compared to the rectangular planform, the triangular planform can reduce induced drag by up to 24.44%, with a wingspan increase of 15.04%. Thus, Prandtl's argument that "tapered wings have an advantage over those with a nearly rectangular profile" [14,15] holds when the effects of taper on wing-structure weight are taken into account for all of the lift distributions considered here, with the greatest advantage at 3 1…”