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# Rectangular Wings

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Thorp devotes some space to the structural aspect of taper, which Bergey ignores. A certain amount of material is unavoidably needed to give a wing its aerodynamic shape, Thorp says, and that structure is naturally able to carry a certain portion of the total wing load. "The additional material needed to make the wing strong enough to sustain all the bending," he writes, "is a small percentage of the total weight." In other words, the weight of the spars is a small fraction of the wing weight, and the reduction in spar weight achieved by tapering the wing is a much smaller fraction still. But Thorp goes further, anticipating his aerodynamic argument: "Aerodynamic scale effect makes it possible to use a smaller rectangular wing for a given stalling speed," and therefore the tapered wing, though inherently slightly lighter, must be larger, erasing the weight saving.

"Aerodynamic scale effect" is crucial to both Thorp's and Bergey's reasonings, although Bergey does not mention it explicitly. He merely states that tapered wings tend to stall first at the tips. Thorp explains the reason in terms of Reynolds Number, a rather obscure but wonderfully useful index of the properties of fluid flows. The Reynolds Number, or RN, identified by the 19th century British scientist Sir Osborne Reynolds, combines the size of an object, the viscosity of the fluid in which it is moving and its speed into a single number, and states that fluid dynamics are the same at a given RN, however speed, size or viscosity may vary. Thus, air feels like motor oil to a gnat, and sea water feels like air to a whale?or something like that?and the flow around a small wing going very fast looks similar to that around a large wing going slowly. The thing about RN that matters to Thorp is that the lower the RN, the lower the maximum lift coefficient that a given airfoil can attain. In other words, a shorter chord means less lifting capability per square foot. It happens that this deleterious effect of scale is quite pronounced at the chord lengths and landing speeds typical of light airplanes, whereas it is less influential for airplanes with larger wings and higher landing speeds.

Keep scale effect in mind for a minute while we backtrack with Bergey to re-examine the assumption that wing shape coincides with spanwise lift distribution. "Whatever the shape of the wing, tapered or straight, the nature of the airflow around the wing tends to force the spanwise lift distribution toward the ideal elliptical shape," Bergey writes. The real lift distribution lies about halfway between the area distribution and a true ellipse. The lift distribution for a moderately tapered wing indeed closely approaches an ellipse, a fact, Bergey somewhat sarcastically adds, "that is noted in nearly all textbooks and repeated in many articles on the subject." But there's a hitch?and this is where Thorp, having made a detour to visit Sir Osborne, rejoins Bergey: "Tapered wings tend to stall outboard, reducing aileron effectiveness and increasing the likelihood of a rolloff into a spin."

As Thorp puts it, "Wing stall starts where some section along the span first reaches its maximum section lift coefficient." Because of scale effects, the tip has a lower maximum lift coefficient than the root, and so it's the first to stall. The diminished lifting ability of the tip is also the reason for Thorp's assertion that for a given landing speed, the tapered wing needs to have more area.