Comparison of DIFFERENT WING SHAPES

Following the Lanchester-Prandtl prediction of downwash angle and induced Drag for elliptical loadings, other investigators, starting with Glauert, in 1926, and going on through a chain, started finding ways to modify the basic equations to account for straight sided plans, wing twists, and rake (sweepback.) By 1941, when Robert T. Jones published "Correction of the Lifting Line Theory for the effect of the chord," - NACA Technical Note 617,- the theory was essentially complete. It allowed precise design of foils of arbitrary straight-sided plan shape, twist, taper, aspect ratio and sweep-back in non-compressible flow.

In 1949, Abbott and von Doenhoff (who were closely identified with NACA - which was about to become NASA) consolidated a vast quantity of theory and test data and published it as THEORY OF WING SECTIONS. [This famous book is now published in reprint form by Dover Publications, New York, under a 1959 date at a very reasonable price.] It is among the most powerful and enduring packets of information that I have ever seen. It gives much "direct" section data, as well as mean-line performance data, thickness distribution performance data, wing theory, compressible flow theory, a discussion of viscosity and drag, and a review of high-lift devices, their performance and limitations. Much of this data is provided in graphs and charts. It may then be put into (sometimes long) but straightforward equations. Sometimes a first result goes into a second equation. As I have indicated in earlier parts of this series, in its theoretical parts, Abbott and von Doenhoff might be daunting for non-mathematical readers. But around pages 16 to 22 there is enough data to allow anyone who is not too readily "turned off" by the sight of ordinary algebra to understand and design arbitrary wings, based on section performance.

The following few paragraphs are intended to try to offer to the sailor who is not interested in details of design, some intuitive sense of the relative merits and effects of wings of various shapes and twists. Following that, I offer more detailed notes for the benefit of designers and others with a more technical interest.

In general, all wing half-planforms between triangle and rectangle (without twist or change of section, in a constant stream) tend to develop Lift Distributions between their own physical shape and the ideal ellipse. Of the rectilinear forms, a taper ratio of about 0.4 gives an LDC very close to the ellipse. A rectangle with a little washhout does very well too.

Of common shapes, the triangle is significantly poorer (for any given Apect Ratio) that the others. (Did you ever wonder why you don't see it on airplanes, despite the fact that it does provides a large Aspect Ratio for its half span? In fact its "shape-performance deficiency" if we can call it that, increases with Aspect Ratio to a greater extent than other less-extreme shapes.

Without twist, at Aspect Ratios up to 4, straight taper wings with taper ratios from 0.15 to 1.0 (from narrow tip to full rectangular shape) have an induced Drag no more than 1 percent in excess of the induced drag of the ideal ellipse of the same Aspect Ratio. But at A = 4 the triangular shape has induced drag about 5 percent higher than the elliptical form.

At Aspect Ratio 10, the triangular shape has an induced Drag 20 percent more than the ellipse. But at the the same Aspect Ratio, taper ratios ranging from about 0.28 to 0.65 still have no more than about 2 percent more induced Drag than the ideal ellipse.

At Aspect ratio 20, the drag of an untwisted wing of triangular shape, is about 28 percent more than that of the ellipse, while wings with taper ratios from .28 to 0.55 are no worse than 3.5 percent more than the ellipse, and the rectangle stands at 14 percent more.

Again, these conclusions are drawn from Abbott and von Doenhoff, based on the most complete and workable analysis of Wings that has ever been made readily available (to my knowledge.)

In general terms, this suggests that, for sails, as Aspect Ratio gets lower, the head of the sail should get broader. And a "sagging gaff" is nowhere near the inefficient anachronism that modern pseudo-scientific mythology has made it out to be. A triangular sail going to windward should try to minimize twist in order to increase head loading.

Notice that all of these comments are based on "the same Aspect Ratio." Assuming a perfect "closing plate" at the bottom, a triangular sail has twice the Aspect Ratio of a rectangle with the same height and foot. On that ground, it does indeed have less Induced Drag than the rectangle despite its "shape deficiency."

Now, some readers have been asking for technical detail. Others whose eyes glaze over at the prospect may skip the following passage between the two lines of stars without missing much applicable to an understanding of improved sailing performance.

The wing factors of greatest interest to us as sailors are induced Drag Coefficient and slope of the Lift curve. (How much change of Lift Coefficient per degree increment of angle of attack in the linear (straight) part of the curve. Remember part 3, where we saw that for decent sections the slope was about 0.1 per degree.) We might also be interested in the angle of attack on the root section coresponding to a given Lift Coefficient.

The first big break in the analysis of straight taper wings was made by Glauert. In "Elements of Aerofoil and Airscrew Theory," Cambridge University Press, 1926, he solved the equations for rectilinear foils without twist. He presented his result as a factor U, to be placed in the denominator of equation 5-2, so that (C-D-i) is given as:

U is a function of both aspect ratio and taper ratio, presented by Glauert in graphical form in his boook, and reproduced by Abott and von Doehnoff as figure 10, page 17. For elliptical wings of any aspect ratio, U has the value 1. All other planforms have a smaller U value, which being in the denominator makes the induced Drag higher. Figure 5.7 is an attempt to show the general intent of A&vD's Figure 10.

Showing the General form of the results of Glauert's correction factor U to be applied to the Lanchester-Prandtl result for rectilinear untwisted wings with taper ratios from 0 to 1. The original graph is a series of curves, convex up, more or less flat on top, rising sharply from low values on the left, and sloping down more gradually on the right. U gets lower with increasing Aspect Ratio. U is above 0.98 for all values of taper ratio from 0.2 to 0.95 for all Aspect Ratios below 6. But for Aspect ratio 6, the U factor has dropped to below 0.88 for the triangular form.

There are three additive terms in the full equation for induced Drag Coefficient. It takes the form

We have just seen the form of term A in equation 5-3. Terms B and C contain a twist factor (among other things) so that without twist they are zero, So equation 5-3 as first presented by Glauert is the full expression of induced Drag for an untwisted wing. This applies directly to keels, centreboards (etc), rudders, rudders on skegs and flaps on keels (after allowing for the fact that the latter two are high-lift devices.) Twisted sails in a twisted (wind-sheared) flow are a complication requiring artful thought.

Terms B and C contain expressions for wing twist, Lift Coefficient, and the *effective* Lift Curve slope of the section (among other things). Each term contains a factor read from graphs in Abbott and von Doenhoff, which we'll look at a little later. Each term also contains a factor representing the product of wing twist and the effective Lift curve slope of the section (this is a refinement of the idea of section lift curve that we developed in part 4.)

To take terms B and C one at a time:

The idea of the section *effective* Lift curve slope (a-e), is a reflection of the fact that the foil is not really a line but a surface. (If you had a slightly queasy feeling about the transformation of flow around a wing surface to a redirected flow about a lifting line, your intuition was working well.) The redirection of flow about the theoretical line must be corrected for the fact that the chord of the wing has a real influence in that rotation. This was one of the last great additions to the lifting line theory, made by Jones. His correction, called the "edge velocity correction factor," is given the symbol E.* Its value is the ratio of the semi perimeter of the wing to the span. (See Figure 5.8) The effective lift curve slope of a section at any chord point in the wing is its two dimensional value, divided by E. You will note that for the lifting line itself, E has the value one (its semi perimeter out and back equals 2s, equals the span.) Any real wing having a finite width must have E greater than unity.

A demonstration of the meaning of the Jones' Edge Velocity Correction Factor, E, and its influence on the section effective lift curve slope (a-e). The long horizontal line represents the centre line of the wing or a closing plate. All root chords have the value 1. Values are shown for half-span-to-chord ratios of 3:1 and 1:1 for tip chord ratios of 1.0, 0 and 0.4. SP equals Semi Perimeter. For convenience, the lift curve slope of all sections in two-dimensional flow is taken as 0.1 (Remember or refer to part 4 of this series.)

There are two points to be noted here. First is that the effective lift curve slope of a section (a-e) in the wing is NOT the Lift curve slope of the wing. That is a separate and later calculation. But (a-e) is a factor in both the wing lift curve slope, and in induced Drag, which we'll go on with now.

What I have nicknamed Term B in the induced Drag equation contains the product of twist in degrees (positive for wash in) times the section effective lift curve slope. The square of this product appears also in term C. The factor v in term B is zero for elliptical wings, so that even if an elliptical wing is twisted, there is no contribution to induced Drag from this term, no matter what its lift Coefficient. Figure 11 in A&vD is difficult to reproduce sensibly in ASCII (and I'm reluctant to go too far anyway. This part of my work would be better considered as a students' guide to A&vD rather than a reproduction of it. But I'll try to describe it. Triangular half plans have a v = -0.008 from Aspect Ratio 2 to 7, whence it rises in a straight line to a value of 0.006 at Aspect Ratio 17. Rectangular foils have v approximately +0.005 from high Aspect Ratios (20) down to Aspect Ratio 8, when they start to decline through a value of 0.003 at Aspect Ratio 4, towards zero at Aspect Ratio 2. Because twist is negative for washout, this means that triangular half plans are worsened by washout (a negative times a negative is a positive, causing an additive drag factor). But rectangular half plans gather a subtractive Drag contribution from washout and high Lift coefficient in this term B.

The third term C in the overall induced Drag Coefficient depends on the factor w. This is always positive and never larger than 0.004 for the rectangle, and not larger than .003 for the triangle. These maximums appear at aspect ratio 7. All other taper ratios and the elliptical form are between the triangular and rectangular curves, which fall respectively to about v = 0.003 and v = 0.0025 at aspect ratios of 2 and 20 (i.e. towards both ends of the spectrum.)

As downwash increases (as a result of high Lift Coefficient or low Aspect Ratio) section lift (C-l) is rotated far back, (Figure 5.9a). Then its projection on the normal line which is Wing Lift (C-L), is no longer even approximately equal to it in magnitude. In these conditions, for a given geometrical angle of attack, the Wing develops less Lift than the section. (a slightly clumsy way of putting it, because the Wing Lift if the summation of all section lifts.)

  a = f {(a-e) / [ 1 + 57.3 (a-e) / (Pi)A]}             Equation 5-4