The span is the whole story

A wing with ends pays a drag that an infinite one does not, in a fluid with no viscosity at all. The circulation cannot be carried off the tip, so it is shed as a sheet of trailing vorticity, that sheet induces a downwash over the wing, the wing meets the air at a smaller angle than it meets the stream, and the lift it makes is tilted backwards.

The question this rung asks is what to do about it, and the answer is unusually blunt: make the wing longer. Nothing else in the whole of aerodynamics comes close.

The comparison has to be at fixed lift #

Before any of the numbers mean anything, the experiment has to be set up correctly, and the temptation is to set it up wrongly.

Comparing wings at fixed incidence compares wings carrying different loads. A short wing at five degrees makes less lift than a long one at five degrees, because the downwash over the short wing is larger, so it meets the air at a smaller effective angle. Its induced drag is smaller too, and crediting the span with that difference is double counting.

The comparison that answers the question people actually ask — what does a longer wing buy at the same job — holds the lift coefficient fixed and lets the incidence be whatever it needs to be. That is what the figure above does: at aspect ratio 4 the wing needs 8.21° to reach $C_L = 0.6$, and at 28 it needs 5.86°.

The incidence differences are worth reading on their own. A short wing needs three degrees more attitude to carry the same load, which is a large part of why short-winged aircraft land nose-high and why a glider does not.

What the solver computed, and how it was checked #

Prandtl’s lifting-line theory is solved by Glauert’s method: the circulation along the span is written as a Fourier sine series, the monoplane equation is collocated at a set of stations, and the coefficients come out of a linear system. Eight harmonics, odd only, since the loadings here are symmetric.

For each aspect ratio the incidence is found by bisection, so every row of the table below is a wing carrying exactly the same coefficient.

The check is that the solve reproduces $C_{D_i} = C_L^2 / \pi A\!R\, e$ at every row without having been told to, to within half a per cent — and separately that the residual of the monoplane equation off the collocation points is below $5 \times 10^{-3}$, which is the test that the Fourier series solved the equation rather than merely matching it at the points it was fitted to.

The rejection test is the other half: a table of drags scaled by 1.2 is refused, as is any sweep in which the drag fails to fall with span.

Why the planform hardly matters #

The other lever a designer has is the shape of the wing in plan, and the folklore around it is disproportionate to what the numbers say.

At aspect ratio 8 the solve gives a span efficiency of 1.0000 for elliptic loading, 0.9869 for a tapered wing and 0.9368 for a rectangular one. That is the whole range: a rectangular wing pays about 6% more induced drag than the theoretical optimum.

Set that beside the factor of seven from the span sweep and the priority is obvious. The elliptical planform is the optimum, the optimum is worth a few per cent, and a wing one aspect ratio longer is worth more than getting the planform exactly right.

The build asserts that the elliptic loading really is optimal — that no other planform in the set achieves a lower induced drag at the same lift — and refuses a table in which one does, or in which any span efficiency exceeds one.

Why the effect is so strong #

The scaling is worth deriving rather than quoting, because it explains why span is uniquely powerful.

Induced drag comes from the downwash, the downwash comes from the trailing vortex sheet, and the strength of that sheet is set by how much circulation has to be got rid of at the tips. For a given total lift, a longer wing carries the same load spread over more span, so the circulation at any station is smaller, so the shed vorticity is weaker, so the downwash is smaller — and the drag is the product of two of those effects rather than one.

Written out, $C_{D_i} = C_L^2/\pi A\!R\, e$ at fixed $C_L$ is simply inverse in the aspect ratio, and the aspect ratio is $b^2/S$. At fixed area the drag goes as $1/b^2$; at fixed chord it goes as $1/b$. Either way the exponent is on the span, and nothing else in the expression is available to change.

The downwash, which is the thing being reduced #

Everything above is bookkeeping on one physical quantity, and it is worth looking at directly.

The uniformity is the reason elliptic loading wins, and the argument is a general one about minimising a quadratic form. The induced drag is the integral of the local lift times the local downwash angle. Redistributing load from a station with a small downwash to one with a large downwash always increases that integral, so the minimum is at the distribution for which the downwash is the same everywhere — and the loading that produces uniform downwash is the elliptic one.

That also explains why the penalty for getting it wrong is so mild. The optimum is a minimum, so the first derivative there is zero: a distribution some way from elliptic costs only second order, which is why 6% is the whole of what a rectangular wing gives away.

What a longer wing does to the whole aircraft #

A factor of seven in induced drag does not become a factor of seven in anything a passenger would notice, because induced drag is only part of the total.

At high speed and low lift coefficient — cruise — the total is mostly friction, and span buys little. At low speed and high lift coefficient — climb, loiter, thermalling, take-off — the total is mostly induced, and span buys nearly everything. Aircraft designed for the first look nothing like aircraft designed for the second, and the whole of that difference is which side of this crossing they spend their time on.

The crossing itself is where the two curves are equal, which is the best-glide point, and it moves to a higher lift coefficient as the aspect ratio rises.

What stops anybody building a wing of aspect ratio 40 #

If span is the whole story, the obvious question is why aircraft are not built with much more of it, and the answers are entirely outside aerodynamics.

Bending. A wing is a cantilever carrying the aircraft’s weight, and the root bending moment goes up with span at fixed load. A longer wing needs a deeper or heavier spar, and past some point the structure weighs more than the drag saving is worth.

Aeroelasticity. Long slender wings twist and flutter. The dynamics get worse faster than the aerodynamics get better.

Ground handling. Airport gates, taxiways and hangars have widths. The folding wingtips on recent airliners exist precisely because the aerodynamically preferred span exceeded the gate.

Friction. More span at fixed chord is more wetted area, and a wing pays for its surface as well as for its ends. At high speed that bill dominates, which is why fast aircraft have short wings and slow ones have long ones.

Sailplanes, which are optimised for exactly one thing and are not required to fit anywhere, run aspect ratios of 25 to 50 and glide at better than 50:1. They are the experiment being described here, conducted without the other constraints.

The same result, said as a length #

There is a way of stating the induced-drag result that strips out the coefficients and makes the role of span unmistakable.

For a wing carrying weight $W$ at speed $V$, the induced drag can be written

$$D_i = \frac{W^2}{\tfrac{1}{2}\rho V^2 \pi b^2 e}$$

with the area gone. Not hidden in a coefficient, not cancelled — genuinely absent. The induced drag of a wing depends on its weight, its speed, and its span, and on nothing else about its shape except the few per cent that $e$ carries.

That is a remarkable form to arrive at, and it says several things at once. Two wings of the same span and different areas have the same induced drag at the same weight and speed. Making a wing bigger in chord does nothing for it. And the quantity a designer is really choosing when they choose “aspect ratio” is span at a given weight, with the area following from whatever wing loading the landing speed permits.

It also gives the cleanest statement of what a bird or a glider is optimising. Minimum sink at a given weight means minimum power, and the induced part of the power goes as $W^2/(\rho V b^2)$ — so a soaring bird’s problem is to be light and to have long wings, in that order, and every soaring bird looks like the solution to that problem.

Optimal subject to what #

Every optimality claim above carries a constraint that has not been written down, and writing it down changes the answer. Elliptic loading is optimal at a fixed span — that is the quantity held while the load is redistributed, and it is what the solver holds. It is a perfectly sensible thing to hold if the span is set by a hangar door.

It is not what a structure cares about. A wing is a cantilever, and what limits it is the bending moment at the root, which is the load on each half-wing multiplied by how far outboard that load sits. So a designer choosing between wings is not usually choosing at fixed span; the choice is at fixed strength, and a loading that carries less of its lift near the tips buys span for nothing.

Prandtl worked that version out in 1933 and it gives a different answer. Minimising induced drag at fixed lift and fixed root bending moment produces a bell-shaped span loading rather than an elliptic one — unloaded at the tips, more heavily loaded inboard — flown on a span about 22 per cent greater for the same structure, and yielding about 11 per cent less induced drag than the elliptic wing it replaces.

Both results are correct and they are answers to different questions. Nothing in the sweep above is wrong; what is wrong is the habit of quoting “elliptic is optimal” without the clause, since the clause is the part a designer is actually negotiating.

And the bell loading has a consequence that is nothing to do with drag. Its downwash is not uniform: towards the tips the induced flow reverses, so the outer wing meets a slight upwash and its local lift is tilted forwards. That region produces induced thrust rather than induced drag, and it sits exactly where the ailerons are — so a down-going aileron, adding load to a region already generating thrust, yaws the aircraft into the turn instead of out of it.

That is proverse yaw, and it is the opposite of the adverse yaw every conventional wing produces and every rudder exists to cancel. The Horten brothers built tailless sailplanes on the principle in the 1930s and 1940s and flew them without vertical surfaces; NASA re-derived and flight-tested it seventy years later.

Which is the honest qualification of this essay’s title. Span is the whole story when span is what is being chosen. When strength is what is being chosen, span is the output — and the loading that maximises it is one this essay’s own figures rank as sixth-best, because they are ranking it against a constraint the aeroplane does not have.

What the picture cannot show #

Lifting-line theory assumes the wing is a line — that the chord is small compared with the span — so everything on this page degrades as the aspect ratio falls. At aspect ratio 4 the theory is already approximate, and below about 3 it stops being usable; delta wings and missile fins need something else entirely.

It also assumes the wake leaves straight downstream and stays flat. A real trailing sheet rolls up into two concentrated vortices within a few chords, and while that turns out to make little difference to the induced drag, it makes all the difference to the hazard behind a large aircraft — and to how much circulation is in the sheet at all, which is a question about an area integral rather than about a pair of lines.

Nothing here is viscous. The wing has no friction drag, no separation and no stall, so the tables above will happily report a lift coefficient any wing would have stalled at — and the incidence column reaches values at which the flow would have left the section long before the drag was collected.

Where the model stops #

The one physically important thing the analysis omits is what happens near the tip. The theory predicts an infinite downwash at the end of a non-elliptic wing, which is why the figures here do not draw the outermost two per cent of the span — the model is singular there, not the flow.

Real tips are where designers spend disproportionate effort, and winglets are the visible result. The honest way to describe a winglet is not as a device that “blocks the vortex” but as a way of increasing the effective span without increasing the geometric one, by putting lifting surface where the flow is already turning. Whether that is worth its weight and its own friction drag is an arithmetic problem, and the answer varies by aircraft.

An honest accounting of the tip vortices #

The trailing sheet is usually drawn as two tidy vortices trailing from the tips, and the picture is right in one sense and misleading in another.

What is right: the sheet does roll up, within a few spans, into two concentrated cores, and those cores persist for a very long way. The kinetic energy in them, per unit length of flight path, is exactly the induced drag — so the drag is not lost to friction, it is deposited, as a rotating structure the aircraft leaves behind.

What is misleading: the vortices are not the cause. They are what the shed vorticity has organised itself into, and the induced drag was determined at the wing by the downwash there, before the roll-up happened. An argument that begins “the tip vortices cause induced drag” has the sequence backwards, and it leads to the belief that a device which disrupts the vortex would recover the drag. Nothing at the tip can recover energy that was already spent at the wing.

This is also the practical reason wake turbulence separation exists at airports. A heavy aircraft at low speed and high lift coefficient sheds the most energetic wake it ever will, precisely when the following aircraft is slow and close to the ground.

Who found it, and when #

Prandtl’s lifting-line theory was developed at Göttingen between 1911 and 1918, with the key results published after the war. Glauert’s Fourier method, which is what the solver here uses, dates from the early 1920s.

The elliptical planform’s fame owes more to the Spitfire than to the arithmetic. Its designers chose the ellipse partly for the induced drag and substantially because it allowed a thin wing with room for the undercarriage and eight guns — and the Hurricane, with a much less elegant wing, was not six per cent worse at anything that mattered.

Where the ladder goes next #

Below this rung, the price of having ends establishes that induced drag exists at all, in a fluid with no viscosity. This rung is the design consequence: the one number that moves it.

Beside it, the two drags a wing pays puts induced drag against friction and finds the lift at which they are equal. Above that lies the question of what to fly at — the speed for least drag being a different speed from the speed for least power.