The optimum that does not matter

The span is the whole story makes the case that induced drag is decided by how far apart the wing tips are and that everything else is a correction of a few per cent. This essay is about the few per cent, which turn out to be a sum of squares — and about what that does to the design decision.

Delta is a sum of squares, and that is the entire proof #

Write the spanwise loading of a wing as a Fourier sine series in the substitution angle $\theta$, with $y = -(b/2)\cos\theta$. Prandtl’s lifting line then gives the lift and the induced drag as

$$C_L = \pi A\!R\,A_1, \qquad C_{D_i} = \frac{C_L^2}{\pi A\!R}\,(1 + \delta), \qquad \delta = \sum_{n\ \text{odd},\ n\ge 3} n\left(\frac{A_n}{A_1}\right)^2.$$

Look at $\delta$. Every term in it is a positive integer times a square. It cannot be negative. It is zero when every harmonic above the first vanishes — which is elliptic loading — and positive otherwise.

That is the whole proof that elliptic loading is optimal. Not a variational argument, not a Lagrange multiplier: an observation about a sum of squares. It is one of the shortest optimality proofs in engineering and it is exact.

Solved numerically at twenty-four harmonics without being told the answer, the elliptic planform returns $A_1 = 0.017453$ and every subsequent coefficient at $10^{-19}$. Its downwash is the same at every station across the span to four parts in $10^{16}$ — which is the reason it is optimal, and is nowhere in the code.

Why the harmonics above the first cost anything at all #

The sum of squares is the proof and it is not the mechanism, so it is worth having the mechanism.

The lift a wing makes is the first harmonic and nothing else: $C_L = \pi A\!R\,A_1$, because every other $\sin n\theta$ integrates to zero against the spanwise station. So the higher harmonics contribute no lift. What they do contribute is downwash — the induced angle at a station is $\sum n A_n \sin n\theta / \sin\theta$, where each harmonic enters with a factor of $n$ — and downwash tilts the local lift vector backwards, which is drag.

A wing carrying a non-elliptic loading is therefore doing something specific and wasteful: it is generating a velocity field that costs drag and delivers no lift, on top of the one that does both. That field is the wake’s, and in the surface in the wake sense the whole calculation can be done in a plane far downstream, where the higher harmonics are visibly extra kinetic energy left in the air.

The factor of $n$ is where the penalty’s weighting comes from, and it says something useful: a high-order wiggle in the loading is worse than a low-order one of the same amplitude, in exact proportion to its order. A loading error concentrated near the tips — which is what an over-tapered wing has — is made of high harmonics and is punished accordingly.

No straight taper reaches it #

A real wing is not elliptic in planform, because an elliptic wing is expensive to build and stalls along its whole span at once. It is straight-tapered, and a straight taper cannot produce elliptic loading: a linear chord and an elliptic circulation are different functions, so $\delta$ is never zero.

The best a straight taper does at aspect ratio eight is $e = 0.98749$ at a taper ratio of 0.364. That is $\delta = 0.0127$ — one and a quarter per cent of induced drag given away, permanently, for the convenience of a straight leading edge and a straight trailing one.

Drawing the loadings shows what the shortfall is made of. A wing with too little taper carries too much lift near the tip; one with too much carries too little. Neither error is dramatic, and neither can be removed by a chord distribution that is a straight line.

And the penalty is quadratic, so the optimum is flat #

Here is where the sum of squares comes back with the other half of its meaning.

Add a third harmonic of relative amplitude $a$ to an elliptic loading and $\delta = 3a^2$ exactly. Not approximately, not to leading order: it is the definition of $\delta$ evaluated for one term. So a loading ten per cent away from elliptic costs three per cent in induced drag, one two per cent away costs a tenth of one per cent, and one one per cent away costs three parts in ten thousand.

Applied to the taper sweep, that quadratic law makes the maximum extremely flat. Every taper ratio from 0.31 to 0.42 is within a tenth of a per cent of the best. Every ratio from 0.25 to 0.51 is within half a per cent. The most-used taper ratios in aviation are in that band and none of them is at 0.364, and the loss for being elsewhere is smaller than the uncertainty in any of the other terms in the drag budget.

This is not special pleading about lifting-line theory being approximate. The theory is exact within its own assumptions and the optimum is exactly where it says. The point is that an exact optimum of a stationary quantity is worth what its curvature says it is worth, and this one’s curvature is low.

The number the optimum cannot see #

If the efficiency does not choose the taper ratio, something else does.

Root bending moment per unit lift rises monotonically with taper ratio while the efficiency has a maximum in the middle, so two taper ratios either side of the optimum share a span efficiency and put measurably different loads into the wing root. Since a heavier spar costs weight and weight costs induced drag through the lift it has to make, the real optimisation is a coupled one — and it lands below the aerodynamic optimum, at taper ratios nearer 0.25 to 0.3, which is where most transport wings are.

That is the practical shape of the whole essay: the aerodynamic optimum is exact and flat, so the structural constraint decides. The same reasoning taken further is the loading nobody used, where constraining the root bending moment rather than the lift moves the optimum away from elliptic altogether — and moves it a long way, because the objective it is stationary in is a different one.

What the flatness is worth in the wind tunnel #

There is a measurement consequence that is worth separating from the design one.

Span efficiency is measured by fitting a straight line to $C_D$ against $C_L^2$ and reading the slope as $1/\pi A\!R e$. That fit inherits every other quadratic-in-$C_L$ term in the drag — the lift-dependent part of the profile drag, chiefly — so the number it returns is not $e$ but an effective value that lumps them together.

At the accuracies involved that matters. Two taper ratios differing by half a per cent in true $e$ differ by 0.0005 in the fitted slope’s implied efficiency, which is inside the scatter of an ordinary force-balance polar. So the flatness is not merely a design irrelevance; it is below the resolution of the standard measurement, and a tunnel campaign comparing taper ratios is measuring the profile drag’s lift dependence rather than the induced drag’s.

The site’s own warning about this is the instrument in the answer: a quantity extracted by a fit carries whatever else the fit could not separate.

What aspect ratio does to it #

The optimum drifts from 0.371 at an aspect ratio of four to 0.340 at twenty — a move of eight per cent over a fivefold change in the wing — while the efficiency it buys falls from 0.9914 to 0.9669.

The second half of that is worth stating plainly because it inverts a common intuition. A long wing is further from elliptic than a short one, at each one’s own best taper. The reason is in the monoplane equation: the term that couples the chord distribution to the downwash carries a factor of $4b/(a_0 c)$, so a slender wing is more strongly governed by its own chord distribution and departs more from the ideal loading. The gain from more span is still overwhelming — it is the whole of the price of having ends — but the efficiency multiplying it degrades slowly as the span grows.

What the number is worth to how many figures #

One limit is worth recording before anybody quotes $\delta$ to four figures. A straight-tapered chord has a kink at the centreline, so its Fourier series converges slowly: $\delta$ at a taper ratio of 0.4 falls from 0.013249 at six harmonics to 0.012980 at eighty, a drift of two per cent, and it is still moving at the top of that range.

So the honest statement is $\delta \approx 0.0130$ and $e \approx 0.987$, three figures, with the fourth belonging to the truncation. That does not affect anything above — the flatness of the optimum is a hundred times larger than the truncation — but it does mean a comparison of two published $\delta$ values is a comparison of two harmonic counts unless both are stated.

Two numbers in that table are of different kinds and it is worth saying which. The elliptic wing’s $\delta$ at $3\times10^{-33}$ is round-off: it is a computation returning an exact zero, and the digits are the arithmetic’s. The best taper’s $\delta$ at 0.0127 is a physical quantity converged to three figures, and the fourth is the harmonic truncation discussed above. Quoting them to the same precision would be a mistake of the kind this collection tries to avoid.

Where the flat optimum stops being flat #

It would be wrong to leave the impression that every choice near an optimum is free. Two things sharpen the curve, and both are visible in the figures above.

Going far enough away. The band is flat because the penalty is quadratic, and quadratic penalties grow. A taper ratio of 0.05 — a wing that comes to a point — gives $e = 0.917$, eight per cent down; an untapered rectangular wing gives 0.944. Those are not free, and the second of them is the reason a rectangular wing is a compromise rather than a simplification.

And constraining something else. The flatness is a property of the objective, so changing the objective changes it. Optimise the induced drag at a fixed root bending moment rather than at a fixed lift and the stationary point moves off elliptic by a long way, to the bell-shaped loading that is the loading nobody used — and there the difference between the two optima is eleven per cent in induced drag, which is not flat and is not negotiable.

So the correct summary is narrower than “optima are flat”: this optimum, in this objective, over this range of taper, is flat. The general statement is the one below, which is about how the flatness is related to the precision of the objective rather than about how flat any particular curve happens to be.

The general shape of this #

The pattern here is not about wings. A quantity that is stationary at its optimum is known well there and locates its optimum badly: if the objective is known to $\varepsilon$ and has curvature $c$, the position is known only to $\sqrt{2\varepsilon/c}$. Half the significant figures are gone, and no amount of care in the objective recovers them.

That is why the elliptic wing’s efficiency is exactly one and the planform that approaches it is a band rather than a shape; it is why the flow with the least energy is identified by a quantity that cannot tell a good picture from a mediocre one; and it is why a bearing designer’s optimum tilt is quoted to five figures in handbooks that disagree in the fourth. An exact optimum is a real thing and a flat one is a cheap thing, and almost every optimum in this subject is both.

One more reading of delta #

There is a way of writing $\delta$ that makes both halves of this essay visible at once.

Because the harmonics are orthogonal, the mean-square departure of the loading from the elliptic one carrying the same lift is $\sum_{n\ge3} (A_n/A_1)^2$ — a sum of the same squares, without the factor of $n$. Call that $\eta^2$. Then

$$3\eta^2 \le \delta \le N\eta^2,$$

with $N$ the highest harmonic present, so the drag penalty and the square of the loading error are the same quantity to within the order of the harmonics involved. For a straight taper, whose error is dominated by the third harmonic, the two are within a few per cent of each other and $\delta \approx 3\eta^2$.

That inequality is the whole essay compressed. The left-hand side says the penalty is at least three times the square of the error, so a wing cannot be badly non-elliptic and cheap. The right-hand side says the penalty is at most $N$ times it, so a wing that is nearly elliptic is cheap in exact proportion to how nearly — and “nearly” is being squared.

It also explains why the loading picture is more useful than the efficiency number. A designer looking at four loadings can see at a glance which is closest to the ellipse; the efficiency numbers those loadings produce differ in the third decimal place.

The second knob, and what it costs to turn #

Taper is not the only way to shape a loading, and the other way behaves differently enough to be worth separating.

An untwisted wing with a linear section lift curve has a loading whose shape does not depend on incidence at all: raise the angle and every harmonic scales by the same factor, so the ratios $A_n/A_1$ are fixed and $\delta$ is a constant of the planform. That is why the whole essay above could quote a span efficiency without saying at what lift coefficient.

Add twist and it stops being true. Twist contributes its own loading, which does not scale with incidence, so the total is a sum of two shapes in a proportion that changes as the wing is flown faster or slower. The harmonic ratios move, and with them $\delta$.

Two consequences follow. A twisted straight taper can be made exactly elliptic — there are enough free parameters — but only at one lift coefficient, and it is non-elliptic at every other. So a twisted wing’s span efficiency is a curve against $C_L$ with a minimum at its design point rather than a single number, and quoting one without the condition is quoting a point on a curve.

And the two reasons for installing washout turn out not to fight each other. It is fitted for stall behaviour — unloading the tips so the root gives way first — which is a benefit wanted at high lift coefficient, while the elliptic point can be placed at cruise. The drag it costs is paid at the angles where the stall margin is being bought.

What is not claimed #

Lifting-line theory, and nothing beyond it. The whole calculation is Prandtl’s, which assumes a planar wake at the free-stream speed, a high aspect ratio and a linear section lift curve. At aspect ratio four the theory is already forty per cent out on the induced angle, and the taper optima quoted for low aspect ratios inherit that.

One aerofoil section, and a linear one. The section lift-curve slope is $2\pi$ everywhere in the solve, so every wing here has the same section at every station. A real taper is usually accompanied by a section change, and the loading that results is not the one computed here.

Untwisted wings only. Every number here is for an untwisted wing; the section above says what twist does to them, and none of it is computed.

The bending moment is the lifting line’s, not a structure’s. It is the spanwise integral of the circulation times the station, which is the aerodynamic load and not the load a spar carries once inertia relief and fuel weight are counted.

The efficiency is the induced one and nothing else. Everything in this essay is $C_{D_i} = C_L^2/\pi A\!R e$ with $e$ from the harmonics. The Oswald efficiency an aircraft performance calculation uses is a different number that absorbs the lift-dependent profile drag as well, and the two are routinely quoted under the same symbol — which is the length in the number problem in another field.

And the flatness is measured at one aspect ratio. The band quoted — 0.31 to 0.42 within a tenth of a per cent — is for aspect ratio eight at five degrees. It narrows slowly as the aspect ratio rises, and nothing here reports how much.