Following on from the last technical post on induced drag, this week i’ll give an overview of e, that quantity i left provisionally defined as 1 in the equation for induced drag;

.                                                $C_{D_i} = \dfrac{{C_L}^2}{\pi{e\Lambda}}$

$\Lambda$ is the aspect ratio.

Planform is the word that describes the shape of a foil when looking at it’s broad side, for instance profile view of a boat’s keel.

As you cannot have discontinuities in the fluid medium , the lift produced does not stop at the end of the wing abruptly, rather it tapers off gradually. It follows that if the planform is rectangular (leading and trailing edges parallel to each other) the pressure will be less at the tip than over the root of the foil. In other words the tip will be underloaded due to the fact that the lift must taper off but the chord retains it full width all the way to the end.

Lift distribution along span of foils with varying taper ratios.

Similarly, for each planform with its varying rate of taper, there will be a corresponding lift distribution curve. For a triangular wing, as expected, the lift distribution tapers down more quickly than the lift distribution for a rectangular wing. However, it does not taper down as quickly as the surface does, meaning the tips are relatively overloaded.

One way to even out the lift coefficient along the entire span is to twist the wing along its length so that each section of wing is at a different angle of attack. In the case of a rectangular wing, one would have to twist it such that the underloaded tips are twisted to greater angle of attack than the root, in such a way that each section of wing is loaded the same.

For triangular wings, the tips are relatively overloaded, so the wing would have to be twisted the other way, with the angle of attack reducing towards the ends.

The problem with this is that the amount of twist needed depends on the overall coefficient of lift, which varies. Thus there is no way to create a twist that will correctly compensate for the variations of loading at all global lift coefficients.

Furthermore, Max Munk  determined that for the least possible induced drag for a given span, the downwash angle has to be constant across the whole span, so that the air stream immediately behind the wing is deflected in a perfectly uniform way.

Without getting too far into it, for uniform planar flows, an elliptical lift distribution curve will result in a constant downwash angle across the whole span. Also it will give the best possible value of $e$

It turns out that there is a planform somewhere between a rectangle (big tips) and a triangle (vanishing tips) that has a lift curve that matches the area distribution curve, thus making each piece of the wing work at the same coefficient of lift, or loading if you will, and that, at all global coefficients of  lift. In this case the local and global lift coefficient would in fact be equal at every position along the span.

An untwisted elliptical planform will produce the required elliptical lift distribution.

Another factor of planform is sweep; the foil can be swept back, or forwards, rather than being at right angles to the flow. This will also affect the value of e. Note that what is important in determining the sweep back angle of a foil is neither the chord midline, nor the leading edge, nor the trailing edge. It is the quarter chord line, about which one can consider as being the aerodynamic “center of lift” of any foil. Explaining the rather complex effects of sweepback and sweepforward will have to wait for another post though.

For now suffice to say, that in general, any sweep is a deviation from the optimum.

An early example of all this being put into practice is the wing of the british Spitfire. Observe not only the eliptical planform but also the practically straight quarter chord line.

The famous elliptical wings of the spitfire

There is however another issue to take into consideration and one which may not evident at first. A pure elliptical wing, despite having the optimal area distribution, has a trailing edge that blends smoothly to the tip and on to the leading edge. Where one starts and the other ends is quite ambiguous, and this has the effect of pulling the tip vortex in towards the wing root. The tip vortex tends to follow the radius of the tip around towards the trailing edge until the angle becomes too great, forcing the vortex to break away. This makes the vortex separation unnescessarily messy and means the tip vortices are closer together than the actual extremity of the foil, thus reducing the effective span of the wing.

This would force a substitution of $\Lambda_e$ for $\Lambda$ with $\Lambda_e \leq \Lambda$

In order to get the tip vortex to peel away as far outboard as possible and neatly, requires a sharp corner between the tip and the trailing edge.  This then requires a little manipulation of the quarter chord line near the tip.

Elliptical area distribution with straight quarter chord line modified near tip so trailing edge is straight and tip has vortex shedding corner.

In the above image i have manipulated the quarter chord line in such a way that the trailing edge straightens out at 0.8 of the span, this is close to being an optimum planform for planar (non twisted) foils. It loses a tiny bit of $e$  but maintains $\Lambda_e = \Lambda$, ie , as high as possible.

The 1988 US catamaran Stars and Stripes with a vortex shedding tip planform designed by Burt Rutan.
Image from François Chevalier

Notice too that in all this, that the triangular planform is about the worst possible area distribution. There exist empirical tables for the value of $e$ and the further one deviates from an elliptical area distribution the lower $e$ becomes, increasing induced drag. Yet it has been generally considered over the last fifty years or so that the bermudian (triangular) is naturally the best shape for going to windward etc. Marchaj did a number of wind tunnel tests on this and confirmed that the triangular planform is actually quite poor. This belief mainly stems from whatever is the current trend in raceboats, setting general ‘idealizations’ about what makes a boat ‘fast’, when in fact raceboats have to optimize to arbitrary rules just as much as actually go fast. This conflict inevitably produces design distortions that are completely innapropriate for sailboats that do not have to conform to any race rule.

To be fair, sailboat rig span loading is actually considerably more complex than that due to the fact that a sailboat rig is not span (height) constrained but rather heeling moment constrained, which imposes optimization along somewhat different lines than as described above. But we’ll come back to the finer points of optimum rig lift distribution in due course.

The foot of the jib is in close contact with foredeck, effectively eliminating one of the airfoil tips.

The value for b or span is taken as the distance separating the separation points of the tip vortices. But what if the lower tip is closed off completely as pictured on the jib of the boat above? This would effectively eliminate the lower tip vortex. How to measure b? In that case, mirror theory states that one can model the three dimensional flow as being one half of the aerodynamic geometry and its mirror image as reflected through the fluid boundary, which in this case is the surface of the water. So b becomes from the upper tip vortex down to its mirror image (underwater).

In simple terms this means that closing off the gap doubles the effective span and thus also doubles the effective aspect ratio, halving induced drag.

This is a vast increase in aspect ratio and one that comes at no cost in heeling moment, so eliminating this gap is the single most effective way of improving a sailboat’s rig performance. This applies to not just jibs, but every sail. Of course, there are plenty of other reasons that may make it impractical to close off the gap completely, but if performance is high on the list of priorities, every effort should me made to reduce the gap as much as possible, for even if the gap is not reduced to the point of having a significant effect on induced drag, it still increases aspect ratio and sail effectiveness at no cost.