The price of having ends

The ideal theory predicts that nothing has any drag. That is d’Alembert’s paradox, it is the most instructive failure in the subject, and it is not quite universal.

There is one kind of drag the ideal theory does predict, in a fluid with no viscosity whatever, and it exists for a reason that has nothing to do with friction. It exists because a wing has ends.

Circulation cannot stop #

A wing lifts because it has circulation round it. At the tip there is no wing, so there is no circulation. Somewhere between the middle and the tip the circulation must therefore fall from its full value to nothing.

Helmholtz’s theorems forbid a vortex line from simply ending in the fluid. So the circulation that is lost between one station and the next has to go somewhere, and the only direction left is downstream: the wing sheds a sheet of vorticity from its trailing edge, of strength equal to the rate at which the circulation falls along the span.

That sheet is not an incidental feature. It is the direct consequence of the wing being finite, and everything in this essay follows from it.

Far downstream the sheet rolls up into two concentrated cores, one behind each tip, turning in opposite senses. These are the trailing vortices — the things that make condensation trails curl, that mark the wake of a crop-duster, and that force air traffic control to space aircraft several miles apart on approach.

The sheet’s strength is worth reading off the first figure directly rather than taking on trust. The vorticity shed per unit span is the slope of the circulation curve, so the sheet is strongest where the curve falls fastest — which is at the tips, where every one of the three planforms drops steeply — and vanishes at the centre line, where all three are flat. That is the whole reason the roll-up produces two cores at the tips rather than a smear across the span.

It also explains a shape choice that otherwise looks like styling. A rectangular wing’s circulation curve has a corner at the tip and falls almost vertically; an elliptic one meets zero smoothly. The rectangular wing therefore concentrates its shed vorticity into a smaller region, which is a more violent tip vortex for the same lift — and a more violent tip vortex is exactly what a wing does not want.

The sheet acts back on the wing #

The vorticity trailing behind the wing does not just sit there. Like any vorticity, it induces a velocity field, and part of that field is at the wing itself.

The direction is the one that matters. Two trailing vortices of opposite sign, one at each tip, induce a downward velocity between them — over the whole span, at the wing. That is the downwash, and it is not a small effect.

The consequence is a rotation of the whole local picture. Each section of the wing meets air that is already moving downwards, so the direction it thinks is “oncoming” is tilted down by an angle αᵢ. The lift that section makes is perpendicular to that direction rather than to the free stream — and a vector perpendicular to a tilted direction has a component pointing backwards.

That backwards component is induced drag. No friction was involved at any point in this argument.

What the solver computed, and how it was checked #

The circulation distribution is found by Glauert’s method: write it as a Fourier sine series in a transformed spanwise coordinate, impose the condition that each section’s lift matches what its local angle of attack demands, and collocate at as many stations as there are terms. Eight odd harmonics, eight stations, one small linear system.

That is a construction with plenty of room to be silently wrong, so four things are checked and none of them was told to the solver.

The equation holds between the collocation points. It holds at them by construction, so testing there proves nothing. Tested at sixty stations deliberately off the grid, the residual for the elliptic wing is 1.4 × 10⁻¹⁷, and for the rectangular and tapered wings 1.5 × 10⁻³ and 3.6 × 10⁻³ — small, and non-zero because eight harmonics cannot represent those loadings exactly.

The two routes to induced drag agree. One sums the Fourier coefficients; the other integrates the local circulation against the local downwash across the span numerically. They agree to 10⁻¹⁵.

No wing beats elliptic. Scaled to the same lift coefficient, the elliptic wing’s induced drag must be the lowest of any planform, and the assertion refuses a set in which it is not.

Elliptic loading produces uniform downwash. Nothing in the code arranges this. It comes out: for the elliptic wing the induced angle varies across the span by less than 10⁻⁴, and the other two vary by a factor of four.

The numbers that fall out are the textbook ones. Span efficiency 1.0000 for elliptic, 0.9890 for a taper ratio of 0.4, 0.9453 for rectangular. And the induced drag of the elliptic wing comes out as CL² / πAR to nine decimal places, without that formula appearing anywhere in the solver.

Why elliptic is best #

The optimality result has a clean argument behind it, and it is worth having rather than accepting.

Induced drag is the work done against the downwash. Total lift is the integral of circulation across the span. The question is which distribution of circulation gives a required total lift for the least drag, and it is a constrained minimisation with a standard answer: the cost is least when the marginal cost is the same everywhere.

The marginal cost of adding a little circulation at a station is proportional to the downwash there. So the optimum is the distribution whose downwash is uniform across the span — because if it were not, circulation could be moved from a high-downwash station to a low-downwash one and the drag would fall.

Uniform downwash, worked back through the relation between circulation and induced angle, requires an elliptic distribution. That is the whole theorem, and the figure above is its verification: the elliptic curve is flat and the others are not.

The Spitfire’s elliptical wing is the famous consequence and it is slightly misleading, because elliptic loading is what is wanted and an elliptic planform is only one way to get it. Twist will do it. Taper will nearly do it — the figure’s tapered wing reaches 98.9% of the benefit with a shape far easier to build. The elliptical planform is an expensive solution to a problem that a straight taper mostly solves.

The margins involved are worth stating, because they explain why almost nobody builds elliptic wings and yet everybody knows about them. The gap between the best possible planform and an ordinary tapered one is about one per cent of the induced drag, and induced drag is perhaps a third of the total in cruise — so the elliptical planform is worth a few parts in a thousand of the aircraft’s drag, in exchange for compound curvature in every rib. It is bought when there is a reason other than drag, and on the Spitfire the reason was that an elliptical planform gave the deepest possible spar near the root while leaving room for the undercarriage and the guns.

Aspect ratio is the whole story #

For elliptic loading the induced drag coefficient is

$$C_{D_i} = \frac{C_L^2}{\pi A\!R}$$

and almost every visible feature of aircraft design is in that expression.

It goes as the square of the lift coefficient. Induced drag is negligible at high speed and dominant at low speed, which is the opposite of friction drag and is why the two trade off against each other.

It goes as one over aspect ratio. Doubling the span at constant area halves the induced drag. That is why a glider has a wing like a plank, why an albatross has a wingspan of three metres and a pigeon’s is thirty centimetres, and why the airliners of the 2020s have winglets that the airliners of the 1970s did not.

There is a second effect of aspect ratio, and it is the one that surprises. Because the downwash reduces the angle each section sees, a finite wing needs more geometric incidence to produce the same lift than an infinite one — so the lift-curve slope is reduced as well. The slope becomes 2π/(1 + 2/AR), and the solver reproduces that to four significant figures at every aspect ratio it was run at. A short-span wing is not just draggier; it is less responsive.

The drag is in the wake, not on the wing #

There is a second place the same number can be computed, and it turns out to be the more revealing one.

Take a plane fixed in space, far downstream, and let the aircraft fly through it. What it leaves behind is a cross-flow — the rolled-up sheet, circulating, with the free stream removed — and that cross-flow has a kinetic energy per unit length of wake. Multiply by the speed and it is a power, and that power is exactly the induced drag times the speed. The induced drag is the rate at which the aircraft leaves kinetic energy in the air behind it, and it is measurable in a plane the wing never reaches.

Two things follow that the near-field account makes hard to see.

The chordwise arrangement drops out entirely. The far plane knows only the span load and the shape the sheet has in the cross-flow — how much circulation at each spanwise station, and where that station sits in the vertical. It knows nothing about where along the aircraft the load was applied. That is Munk’s stagger theorem: sliding the lifting surfaces fore and aft, keeping their loads, does not change the total induced drag at all. A biplane’s induced drag depends on the gap between its wings and not on their stagger; a canard and a tailplane carrying the same load cost the same; and an aircraft flying in a neighbour’s wake is being priced by where it sits across the wake, not behind it.

And the vertical dimension is a free variable. The cross-flow’s energy depends on the sheet’s shape in that plane, so a wake that is not flat can hold the same circulation at lower energy. That is what a winglet buys, and it is why the theory prices it as effective span rather than as a device: the tip device is a change to what the wake looks like in the only plane the drag is computed in.

What the picture cannot show #

The lifting line is a wing collapsed onto a single line, and the collapse throws away everything about chordwise distribution. The figures show what each station carries and nothing about how it is carried, so nothing in this essay can say where on the chord the load sits — which is a separate question with its own answer.

The downwash figure stops short of the tips, and the note under it says so. Lifting-line theory is genuinely singular at the tip for any loading that is not elliptic: the induced angle goes to infinity there, and it does so in the theory rather than in the numerics. The last two per cent of span is not drawn because there is nothing sensible to draw. A real tip has a finite chord, a rolled-up core of finite size, and a flow that is thoroughly three-dimensional, none of which this model contains.

The trailing sheet is also treated as flat and straight, extending downstream forever in the plane of the wing. In reality it rolls up within a few spans into two cores and then descends slowly as the pair induces a downward velocity on itself. That roll-up changes the induced drag by only a few per cent, which is why the approximation survives, but it changes the wake — and the wake is what matters for aircraft separation.

Where the model stops #

Lifting line assumes a wing of high aspect ratio, straight, unswept, and lightly loaded, in incompressible flow.

Low aspect ratio breaks it first. Below about four the assumption that each section behaves like a two-dimensional section in a locally tilted stream is no longer defensible, and lifting surface methods — a sheet of vorticity rather than a line — are needed. A delta wing is outside this theory entirely.

Sweep breaks it in a different way, because the trailing sheet from an inboard station passes close to an outboard one and the simple line integral no longer represents the induced velocity properly.

Stall breaks it because the local sections stop obeying a linear lift curve, and the whole method rests on that linearity. Which is a pity, since spanwise stall progression is exactly the thing a designer most wants to predict.

And nothing here is compressible. Above about Mach 0.3 the corrections start, and near Mach 1 the whole framework is replaced.

There is a further limitation that is not a limitation of the theory but of what the theory is for. Lifting line gives the induced drag of a given loading and says which loading is best; it does not give the loading of a given wing shape without the section data being supplied from outside. Every number in this essay depends on an assumed section lift-curve slope of 2π, which is the thin-aerofoil value and which a real section only approximates. The three-dimensional theory is exact about the three-dimensional part and inherits whatever error the two-dimensional input carried.

Who found it, and when #

Lanchester had the picture by 1897 and published it in 1907, in a book written in his own private terminology that few people read. Prandtl and his students at Göttingen — Betz, Munk, and later Glauert at Farnborough — turned it into the theory between 1913 and 1921.

The timing is worth noticing. The theory arrived during and immediately after a war fought with aeroplanes designed by trial, and it arrived on the losing side. Prandtl’s group had the theory of induced drag while the aircraft of every combatant were being designed without it. The results were classified and only fully published in the early 1920s, at which point they reorganised the subject worldwide in about five years.

The method it displaced was not another theory but a catalogue. Before it, the effect of aspect ratio was known from wind-tunnel data and tabulated; after it, the effect was a formula with a derivation, and the tables became a check on the formula rather than the source of the answer.

That change of status is worth dwelling on, because it is what a theory is for and it is easy to lose sight of. A table of measured aspect-ratio corrections is not useless — it will give the right answer for any wing resembling one that was tested. What it cannot do is tell anyone which wing to test next, or why the correction has the shape it has, or that there is a best possible loading and what it is. The formula does all three, and it does them because it came from an argument about vorticity rather than from a fit.

The clearest evidence that it worked is negative. After 1921 nobody discovered a wing planform that beat the theory’s prediction, and nobody has since. Winglets, which look like an exception, are not: they are a way of achieving a larger effective span within a limited physical one, and the theory prices them correctly.

Where the ladder goes next #

Next rungs on this anchor: the horseshoe vortex, which is the simplest model of the whole system and the one every subsequent method generalises; lifting-surface and vortex-lattice methods, which restore the chordwise dimension the line threw away; winglets and end plates, which reduce induced drag by changing what the sheet does at the tip rather than by lengthening the span; and formation flight, where a following aircraft sits in a neighbour’s upwash and genuinely saves fuel, for exactly the reason a wing near the ground does.

Then across to the drag budget, where the induced drag computed here is added to the friction drag that has nothing to do with it, and to d’Alembert’s paradox, which this essay has just found the one exception to.