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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.

Stationary smoke wall shows the tip vortices as well as the downwash created by the passage of a lifting foil.

Induced drag , or vortex induced drag is a topic one often hears mentioned at the technical end of sailing enthusiast circles.

Induced drag is the energy cost of lift.

In this post I’ll elaborate a bit on the post of ‘How is lift made‘ of two weeks back as well as to lay some groundwork in place that will be useful in future posts that will attempt to correct some commonly held fallacies as to some of the implications of induced drag.

As i explained in the penultimate post, lift is produced by creating a pair of counter rotating vortices; one that is shed at the starting point, and one that is bound by the wing and travelling along with it. These vortices are created by the act of moving an inclined plane or streamlined plane through a fluid.

What i neglected to mention is that a free flying wing must have ends; i.e. it cannot be infinitely long. So what happens at the ends of the wing?

Since there is by definition a pressure difference across the thickness of any lifting wing (otherwise it would not be lifting!) , when one gets close to the ends, the higher pressure underneath tends to push the streamlines out towards the free area beyond the wing tip, and similarly, the lower pressure above the wing tends to pull in air from beyond the tip. Both of these together make the air curl around the end of the lifting foil, at the same time that the fluid particles are flowing aft. When looking at the streamlines it looks like they get twisted up around the tip of the wing. They carry on twisting behind the wing under the newly acquired momentum, forming a long contiuous vortex going back along the path of where the wing came.

Tip vortex streamlines

Kutta and Joukowski both discovered and defined the concept of circulation;

$\Gamma = -\oint_C V\cdot{ds}$

Now, Helmholtz’s theorems, state that  a vortex can never start or end within the fluid, apart from instantaneously when being formed; instead any vortex must either join up with itself forming a loop without ends, or terminate at a fluid boundary.

Putting this together, we realize that the circulation vortex that bounds the wing and the vortices being shed off the wings are one and the same. Then, remembering that a vortex cannot have an end, we follow these tip vortices back all the way to where the wing began its journey,… and where it left behind its starting vortex, and there, the tip vortices join up with the ends of the starting vortex.

So we see that there is no ‘end’, it is all actually one continuous rectangular vortex. One short side is where the trailing edge of the wing was at the beginning, the two longer, and continually lengthening, sides are the lines the wingtips traced out in space, and finally the other short side goes through the wing from tip to tip and moves along with it. This last is the “bound” vortex part of the total rectangular vortex ring, and the part that is actually creating the lift which is the always manifested by the moving of a bound vortex through the fluid.

Around the outside of this rectangle air is moving up, and within it is moving down. This downwards moving air is the air that the passage of the foil forced downwards, called the downwash.

Not only do the tip vortices not contribute any useful lift they took energy to create so represent lost energy for the plane. Also, the ‘spillage’ of air from the tip makes the foil less effective at accelerating air downwards, so represents a loss of lift too.

Lift distribution and trailing vortices
source; Olivier Cleynen

The anatomy of a wing rectangular vortex wake .

So how does this tip vortex create the extra resistance?
It actually does not, at least not directly, what it does instead is induce the resistance by affecting the overall airflow over the wing. Think about this; the extra downwash attributable to the tip vortices does not contribute to lift, yet is causing air to sink more than if there were no tips ‘leaking’. This in turn means that the foil is continually forced to climb out of the hole it is creating for itself. Since lift is produced at right angles to the airflow and this airflow is pivoted downwards it means the lift force is rotated aft as compared to the direction normal to the axis of travel. This creates a component of the lift force to be adding itself to the rest of the drag.

Downwash induced by the tip vortices negatively affect the orientation of the lift vector.

Now to quantize some of these concepts…

I will spare readers the full derivation of these equations, and just jump straight to the conclusions. For those interested in knowing more, wikipedia has a surprisingly thorough amount of information on this.

The Coefficient of lift of a foil is defined in the following dimensionless manner;

.                                                 $C_L = \dfrac{F}{\frac{1}{2}\rho{SV^2}}$

Where F is the lift force produced, $\rho$ is fluid density, S is foil surface area, V is flow velocity, all in SI (standard international) units. The  $C_L$ is a measure of how effective the foil is at producing lift.

And the coefficient of induced drag works out to be;

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

Where $\Lambda$   is the aspect ratio, the measure of how long and skinny the foil is (when looking down from and above on an airplane) and is;

.                                                $\Lambda = \dfrac{b^2}{S}$

where b is the span of the foil from tip to tip. $e$ is the planform efficiency factor, which i will explain later but for now can be taken as 1 for the perfect elliptical surface area distribution.

So we see that the longer and skinnier (higher aspect ratio) the lower the induced drag. Also the lower the $C_L$ the lower the aspect ratio, but squared.