An Explanation of Aspect Ratio and how it affects sailing


Ivor G. Slater, P. Eng.

Part V (AR-5B)

WING THEORY (part 2)

This document is called AR-5B. It is the completion of AR-5. Both together constitute Part 5 of the Aero-hydrodynamics of Sailing.

(This is really a short discussion of the aero-hydrodynamics of sailing. Let's save space by calling it AHdynamics.)


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.

A Non-Technical Discussion of the Effects of Wing Shape

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.

Twist in the form of washout (think of the head of the sail falling off more than the rotation of the apparent wind) worsens the induced Drag of the triangle and improves the rectangle. (In both cases by unloading the tip.) This increases the imbalance of the triangle (more root loading, less tip loading, causing greater variation of downwash.) It improves the rectangle's performance by decreasing tip loading, so making downwash more uniform. A taper ratio of about 0.45 is relatively immune to the effect of twist, although as the Aspect Ratio gets lower, the taper ratios move towards improvement in the range of about 0.6 to 0.7 at A = 2.

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."

The high-aspect-ratio triangle needs a way to REDUCE its Aspect Ratio to overcome its markedly poorer performance off the wind. That explains the need for, and the appearance of spinnakers made wide compared to height, set high above the water (to minimize the "closing plate" effect) hence keeping Aspect Ratio as low as possible.

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.

A More TECHNICAL DISCUSSION Of the Effects of Foil Shape and Twist and Wing Lift Coefficient

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.

*Glauert's Analysis, 1926*

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:

     (C-D-i) = (C-L)^2 / A(Pi)U                          (Eq'n 5-3)

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.

Figure 5.7

U ^ |  All Elliptical Foils (loadings) have U = 1
1.0 |-------.-----.-----.-------------------. A=4
    |        \A=4  \A=6 \ A=8                 A=6
 .96|.A=2           ^A=20                     A=8
 .92|                                     A=10
 .88|A=6                                     ^A=20
    |   |   |   |   |   |   |   |   |   |   |
    0  .1  .2  .3  .4  .5  .6  .7  .8  .9  1.0
                 Taper Ratio
  Triangle                                Rectangle
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.
Figure 5.7

Other Terms in the Equation for Induced Drag Coefficient

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

(C-D-i) = A + B + C

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:

    B = (C-L) (epsilon) (a-e) v     C =[(epsilon)(a-e)]^2 w

In these expressions(C-L)is Wing Lift Coefficient as before.
 (epsilon) shown as the Greek letter looking somewhat like (-is the wing twist in degrees from the root (central) chord to the tip, taken between the angles of zero lift of the sections, positive for increasing angle of attack towards the tips (wash in).
 (a-e)is the section *effective* lift curve slope (See following text.).
 vis a factor given in Figure 11 of Abbott and von Doenhoff, page 18.
 wis a factor given in Abbott and von Doenhoff in Figure 12, page 18.

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.

For some reason there is another symbol E in A&vD, shown in a graph in Figure 12, for calculating pitching moment. (Such duplication is a rarity.) Do not mistake it for Jones' edge velocity correction factor, which you calculate yourself.

Figure 5.8
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.)
   ______1_______                                     __0.4_
3 |              | 3                   /|3               /      |3
  |              |                      |              3.06     |
  | SP   = 7     |                   /  |span = 6               |
  | Span = 6     |                      |              /        |
  | E = 7/6      |         root 10      |              E=6.46/6 |
  |   = 1.17     |         = 3.16  /    |               =1.077  |
  |(a-e) = 0.1/E |        SP= 6.16      |            /          |
  |      = 0.0855|        E= 6.16/6     |          (a-e)=0.0929 |
  |              |         =1.03        |                       |
  |       1      |      (a-e)=0.097     |                   0.4 |
  |--------------|                    / |          /      ------|
  |  Semi        |            /     /   |                /      |
1 |  perimeter 3 | 1       root 2 /     | 1            /        |
  |  Span 2      |        =1.41 /E=2.41/2            1.17       |1
  | E = 3/2 =1.5 |            /   =1.2  |           /           |
  | (a-e)= 0.067 |          /(a-e)=0.083|         /(a-e)=0.078  |
          \ Wing Centre line or Closing Plate

Rectangular               Triangular            Taper Ratio 0.4
3:1 and 1:1               3:1 and 1:1           3:1 and 1:1
     (latter ratios are half-span to root-chord ratios of the upper
      and lower figures )

Figure 5.8

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.)

The Effective Lift Curve Slope of the Wing

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.)

Figure 5.9a

                          .     |normal
                     .          |     Wing Lift Coeffient (C-L)
                 _ _ _ _ _ _ _ _|     equals summation of project-
                \               |     ions of section lift
                  \             |     Coefficients on the normal
                    \ C-l       |     line.
                      \         |
                        \       |
                          \     |
                            \   |
             V                \ |
                                     Figure 5.9b
                                     section lift curve
                                     \     .  .
                     1.0 |                  \             wing 1   .
                         |               .               .
                         |             .            .
               Scale of  |           .          .            wing 2
               Lift      |         .       .  \              .    .
            Coefficients |       .     .    wing Lift curves /
                         |     .   .w1     . based of same sections
                         |   . .   .w2    depend on wing geometries
                         | .
                       0 |-----------------------|----------
                         0   Angle of attack    10 degrees

Figures 5.9a and 5.9b

You may remember from part 4 of this series that almost all decent SECTIONS had a 0.1 change of lift Coefficient per degree of angle of attack (the slope of the section Lift curve.). A little earlier, in discussing section *effective* lift curve slope we have seen that the slope of the section lift curve within a wing is reduced in a way depending on wing shape. Now figure 5.9 shows that even this effective section lift curve slope is not delivered to the wing when Lift Coefficient is large and Aspect Ratio is small.

The slope of the Wing Lift curve (symbol a) has been calculated, to include a factor f, which is displayed graphically in Figure 8, page 16 of A&vD. The equation takes the form

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

This says that the Wing Lift curve slope is factor f times the section effective slope (average for the wing if section shape changes) divided by a denominator. The denominator is one plus the ratio (a-e)/A times 57.3/(Pi). The very last term removes "degrees" from the expression in square brackets, making the denominator a non-dimensional pure number. Although the denominator is "tempered" by the additive factor 1, clearly, the overall value of a is raised by higher values of A and (a-e).

The value of f for elliptical wings of all aspect ratios is 1.00. A straight taper wing of tip chord ratio 0.4 falls very close (looks better than 0.998). Tip-chord ratios of 0.2 and 0.6 are nearly as good. Tip-chord ratio 0.8 runs along f = 0.99 from A = 20 down to A=10, when it starts to climb towards f=1.0 at A=2. The rectangle (tip chord ratio 1.0) runs along at level f = 0.983 from A=20 down to A=10, when it too begins a climb to unity at A=2, through f=0.99 at Aspect Ratio 5. Tip chord ratio 0 (triangular half plan) is odd man out in this picture. At A = 2, the triangle's f is only about 0.94. rising on a curve through f=0.96 at Aspect Ratio 5 and catching up to the rectangle at f=0.983 at Aspect Ratio 20.

The relationship between wing Lift Coefficient and angle of attack is useful to know. Wing angle of attack (alpha-s) is measured in degrees at the root section between the chord and the free stream. It is give by another expression with three additive terms, as

(alpha-s) = (C-L)/a + (alpha-l-0-s) + J(epsilon)

where(C-L)is the Wing Lift Coefficient
 ais the Wing Lift curve slope as determined above
 (alpha-l-0-s)is the angle of zero lift of the root section
 Jis a factor dependent on Aspect Ratio and taper ratio as shown in Figure 9, p 17 of A&vD
 (epsilon), as beforeis the twist in the wing from root to tip, in degrees, measured between angles of zero lift, positive for wash in.

Figure 9 in A&vD shows that J for an elliptical wing has a constant value of -0.4 for all values of Aspect ratio. A straight taper wing of tip ratio 0.4 is a close parallel. A triangular wing has J of about -0.35 from Apect Ratio 20 down to A = 7, when it rises gently towards J =-0.38 at Aspect Ratio 2. A rectangular wing has J approximately -0.45 from A=20 down to A=10, where its starts a gentle decline towards -0.4 at Aspect Ratio 2.

Abbott and von Doenhoff also give formulae and graphs for determining the aerodynamic centre and pitching moment of wings, including those with rake to the aerodynamic axis (positive or negative sweepback.) I don't propose to touch those subjects here.

I had thought of creating a sample wing, and working through these calculations as an example for you. But then I decided not to be selfish, and to leave the pleasure to you.


So you see that for at least 45 years, a body of work has been available that tackled AHdynamics with great rigour. It applies quite well to underwater surfaces of "fin-like" form (but not so well to forms like clipper ships and catamaran hulls without appendages.) Only one more bit of well organized data was needed to make it fully applicable to sails. But, so far as I am aware, the work to fill the gap has never been done.

The problem with sailing rigs is that they do not generally fit down tightly onto smooth flat decks, or the water, to make a clear "closing plate." To some degree or other, that gap under the rig is another "end" to the "wing." This leaves us with the problem of guessing the flow-constraining effect of the gap, so hoping that we can thereby approximate the Aspect Ratio of the rig.

Please be warned that in the next couple of paragraphs I am expressing my own opinions. Twenty five or so years ago, when I was spending some time in this field (out of my own personal interest) it was a great disappointment to find that people (well publicised researchers) were spending much time, money, and resources in sailing rig research without tackling what I saw, (and continue to see) as the principal outstanding problem. One book by a now-noted authority effectively duplicated the work of Manfred Curry in wind tunnel tests on sails done in the 1920s. But without a consistent frame of reference the work is inapplicable to other rigs. It is as though the author had not read the decades of NACA work reported and consolidated by Abbott and von Doenhoff, or if he had read, he hadn't understood the great power of the work already done many years before. Much of this sort of "sailing research" has been sadly too much like reinventing the wheel without realizing that the wheel is lacking a tire. It seems to me that there is little to be learned about the actions of real sails from wind tunnel tests of model sails. The wind tunnel cannot reproduce the wind gradient (increasing speed with height.) Because the model cannot move across the tunnel wind, it could not generate the twist of the apparent wind even if a gradient existed. So tunnel testing of model sails does not deal with two major elements of sailing rality any more than aerodynamic wing theory does.

As a result, even today, design data for underwater foils of any sort, and for sails, is less likely to come out of sailing research labs, wind tunnels blowing across model boats, and test tanks, than out of the pages of Abbott and von Doenhoff. If our design or performance-prediction work is of sufficient importance to be worth the effort, we can make estimates of the effect of the detailed complications, and hope thereby to make adequate approximations.

For the sailor, wing theory as we have followed it, even to the Lanchester-Prandtl level, gives very bright insights into sailing performance.

Editor: Eppo R. Kooi; email: E.R.Kooi@XS4all.NL
Last updated: 010716
Back to the top of this page or the homepage of Yacht-L

Counted since Sept99