The loading nobody used

Every argument on this site so far has ended at the same answer. Elliptic loading is optimal: it gives uniform downwash, it minimises induced drag, and every other distribution the solver is handed is required not to beat it.

That statement has a constraint hidden in it, and the constraint is one no aeroplane has.

Minimum induced drag at a given lift, on a given span. A wing is not built to a given span. It is built to a structure, and what the structure has to survive is the bending moment the lift makes about the root — the integral of the lift against distance out along the wing. Two wings can carry the same lift on the same span and load the spar completely differently, and the one that loads it less can be made longer for the same weight.

Change the constraint and the answer changes. It was worked out in 1933, published, and not used for eighty years.

Setting the problem up so it can be answered #

The apparatus is the Trefftz plane: the cross-section of the wake far downstream, where by Munk’s theorem the whole induced drag is decided. Cut the wake trace into segments, and the drag is a quadratic form in the circulations on them,

$$D = \tfrac12\,\Gamma^{\mathsf T} A\,\Gamma,$$

with A built from the velocity each shed vortex induces at each control point. The lift is a linear functional of Γ, and so is the root bending moment. Minimising a quadratic subject to linear constraints is one symmetric system, solved directly.

The first thing to do with such a machine is point it at the answer that is already known.

That check earns its keep in a way worth recording. The lattice originally placed its nodes evenly and its control points at the segment midpoints, and got the elliptic drag wrong by 2.1 per cent at forty segments, converging at only first order — 0.13 per cent still wrong at six hundred and forty. Cosine-spaced nodes with geometric midpoints were worse. Cosine-spaced nodes with control points at the half-indices of the same parametric formula give the closed form to machine precision at forty segments, because that placement is Glauert’s collocation written out as a lattice.

The check that the optimum is elliptic is what found that, because nothing else in the calculation had an exactly known answer to be measured against.

The other constraint #

Now add the bending moment. Minimise the drag subject to a fixed lift and a fixed moment about the root, and a different curve comes out.

The curve is close to $\Gamma \propto (1-\eta^2)^{3/2}$ — the bell. Its arithmetic is unusually clean, and every number in it is a small rational.

Write the loading as a Fourier series in the Glauert variable. The bell is $\sin^3\theta$, and $\sin^3\theta = (3\sin\theta - \sin 3\theta)/4$, so it has exactly two harmonics with $A_3/A_1 = -1/3$. The lift comes from $A_1$; the drag weights the nth harmonic by n. So

$$e = \frac{A_1^2}{A_1^2 + 3A_3^2} = \frac{9/16}{9/16 + 3/16} = \frac34.$$

Span efficiency exactly three quarters, and the solver requires it to five parts in a hundred thousand. On a fixed span the bell is a third worse than the ellipse.

The root bending moment goes the other way, and by another rational: exactly four fifths of the elliptic wing’s, at the same lift and the same span. Required to a part in ten thousand.

The trade, which is where the rationals collide #

Those two numbers are on opposite sides of a ledger and the span is the thing that settles it.

Drag falls as the square of the span. Bending moment, at fixed lift, rises in proportion to it. So a loading that is worse on drag and better on moment can be made longer until the moment is back where it started, and by then the drag may be ahead.

The bending moment at 4/5 rises as the span, so it is back to one at a span of 5/4. The drag, which is 4/3 at equal span, falls as the square, so at 5/4 span it is

$$\frac{4/3}{(5/4)^2} = \frac{4}{3}\cdot\frac{16}{25} = \frac{64}{75} = 0.8533\ldots$$

Twenty-five per cent more span, the same lift, the same root bending moment, and 14.7 per cent less induced drag. Both fractions are exact, both are checked against the rationals by a solver that found them by quadrature, and neither was put in by hand.

Prandtl’s own published comparison uses a different structural constraint and gets 22.5 per cent more span for 11.1 per cent less drag — which is the same trade at a span of $\sqrt{3/2}$ rather than 5/4. The difference between the two answers is the difference between holding the root moment and holding the moment of inertia of the structure, and it is worth saying aloud that the celebrated numbers depend on which one is held.

What the two constraints are actually about #

It is worth stepping back from the arithmetic, because the structure of the argument recurs and is more useful than the numbers.

An optimum is a pair: an objective and a constraint set. Change either and the answer changes, and an answer quoted without its constraint set is a slogan. This subject is unusually full of them, and the elliptic result is the most-quoted one on the whole site.

The two constraints here differ in what they suppose is scarce. Span is scarce if the wing has to fit somewhere — a hangar, a runway, a carrier deck, a gate. Bending moment is scarce if the wing has to be light, and it is the right proxy for that because a spar’s weight scales with the moment it carries and the depth it has to carry it in.

Which one binds is a fact about the aeroplane and not about aerodynamics. An airliner is gate-limited and therefore span-limited, which is why the industry spent thirty years putting winglets and raked tips on wings whose span was fixed by an airport rather than by a structure. A sailplane is weight-limited and can be as long as it likes, and sailplane wings have got steadily longer for seventy years. The two aircraft are optimising the same quantity under different scarcities and arriving at different shapes, and neither is making a mistake.

The bell is not quite the optimum #

This is worth stating plainly because the bell is usually presented as the answer.

Solve the constrained problem properly and what comes back is not $(1-\eta^2)^{3/2}$. It is close — within five per cent in shape, and one per cent in drag — and it carries slightly differently across the middle of the span. The reason is structural rather than numerical: the root bending moment’s weight function has a kink at mid-span, so its Fourier coefficients do not vanish past the third harmonic, and the optimum therefore has fifth and seventh harmonics in it that the bell does not.

The exact bell is the answer to Prandtl’s constraint, which was the moment of inertia of the wing’s structure, and the near-bell is the answer to a root-moment constraint. A per cent apart, and they are answers to different questions, which is exactly the caution this essay opened with applied one level further down.

The upwash, and the yaw it produces #

There is a second consequence of bell loading that has nothing to do with drag and is the reason it has been flown recently.

For the bell, the induced angle is proportional to

$$\sum n A_n \sin n\theta\big/\sin\theta = \tfrac34\,(1 - \sin 3\theta/\sin\theta) = -\tfrac32\cos 2\theta,$$

which vanishes at $\theta = \pi/4$ — that is, at $\eta = 1/\sqrt2 = 0.7071$. The solver locates the crossing by interpolation and requires it within three thousandths of that value.

Outboard of it the induced velocity is an upwash. The local lift vector is tilted forward and the section produces induced thrust — in a calculation with no viscosity, no propulsion and nothing but a wake.

That sign change is what produces proverse yaw. Roll a conventional wing and the down-going wing — which is being asked for more lift — gains induced drag and is dragged backwards, so the aircraft yaws away from the turn. Every pilot learns to correct that with the rudder, and every conventional aeroplane needs a vertical tail partly to deal with it.

Roll a bell-loaded wing by twisting its outer panels and the outer panels are in the upwash region, so asking one for more lift makes it produce more thrust. The aircraft yaws into the turn. It is a wing that coordinates its own turns, and the aircraft does not need a vertical tail for that reason.

The Horten brothers built tailless wings on this principle in the 1930s and 40s and said so; the argument was dismissed for decades; NASA’s Prandtl-D flew it and measured it in 2015 and found the proverse yaw where the arithmetic puts it.

Why nobody used it for eighty years #

Four reasons, and none of them is that it was wrong.

The comparison is at the wrong constraint. Elliptic is optimal is true, is memorable, and is what a first course teaches. Almost nobody who repeats it states the constraint, and once the constraint is unstated the sentence sounds like a fact about wings.

A bell-loaded wing is 25 per cent longer, and a wing 25 per cent longer does not fit in the hangar, the gate, the ground-handling envelope or the flutter clearance. The drag saving is real and the span is a hard constraint of a different kind.

And a bell-loaded wing is badly loaded for stalling. It carries its lift inboard, which is where the section coefficients are highest, so it stalls at the root — but it is also carrying very little outboard, and the outboard sections are at low local coefficients doing very little useful work. That is a lot of wetted area for the lift it makes, which is profile drag the induced-drag saving has to pay for.

The structural constraint is not really the root moment either. A real spar is sized by a distribution of moments, a shear, a torsion and a flutter speed, and reducing one of them by twenty per cent does not reduce the weight by twenty per cent. The trade is genuine and the exchange rate is softer than the arithmetic makes it look.

So the honest summary is that the 1933 result is right, the 14.7 per cent is real under its own constraint, and turning it into an aeroplane requires the constraint to be the binding one — which on a sailplane or a flying wing it can be, and on an airliner it is not.

The proverse-yaw claim, stated carefully #

There is a version of the yaw argument that overstates itself and it is worth heading off, because this site’s whole business is the difference between a mechanism and a slogan.

What the figure shows is that the local induced drag on a bell-loaded wing changes sign at η = 1/√2, so that the outboard sections have a forward-tilted lift vector.

What follows is that a control input which increases the loading outboard of that station produces a forward force there, and therefore a yawing moment towards the wing that is being asked for more lift — that is, into the turn.

What does not follow is that a bell-loaded aeroplane needs no vertical surface. A vertical tail does several jobs: it resists sideslip, it damps yaw, it provides directional stability, and it counters adverse yaw. Only the last of those is addressed here, and a tailless aircraft still needs the other three from somewhere — which on a Horten is sweep, on a modern flying wing is drag rudders, and on the Prandtl-D is both.

And what is not shown at all is any relation between the aileron deflection and the loading change. Getting the yawing moment out of this requires knowing how much circulation an aileron adds and where, which is a control-surface calculation and not a spanload one.

So the honest claim is narrow and it is still remarkable: the induced drag distribution of a bell-loaded wing has a sign change in it, and a conventional wing’s does not. Everything about turn coordination follows from that and is a separate piece of engineering.

The loading is only that loading once #

Choosing a spanload rather than a planform has a consequence the whole essay has assumed away: a wing delivers its chosen loading at one lift coefficient and something else everywhere else.

A twisted wing’s loading is the sum of two parts. One is set by the twist and does not change with incidence at all; the other is set by the planform and scales with the lift coefficient. Getting a bell needs a great deal of washout — of the order of ten degrees from root to tip — so the fixed part is large, and the two parts are in the intended proportion at exactly one flight condition.

Fly slower, and the incidence-proportional part grows until it dominates: the loading drifts back towards whatever the planform alone would have given. Fly faster, and the twist part dominates instead, until the outer panels are at negative lift and the wing is carrying more than its own weight inboard to make up for tips that are pushing down.

So the two prizes are not collected together. The proverse yaw needs the sign change to be where it was designed to be, and the benign root-first stall needs a high lift coefficient — which is the condition at which the loading has drifted furthest from the bell that was supposed to produce them.

What the model does not contain #

No profile drag. Every comparison here is induced drag alone. A 25 per cent longer wing has 25 per cent more wetted area, and on a real aircraft the induced saving and the profile penalty are the same order of magnitude at cruise.

No structural model. The root bending moment is a proxy for weight and it is a crude one. Nothing here computes a spar.

The wake is flat and rigid. It is a straight line in the Trefftz plane that does not roll up, does not move and does not deform. Roll-up happens within a few spans, and the classical result is that it does not change the induced drag — which is a theorem worth knowing and is not proved here.

No viscosity, so no stall. The remarks about stalling behaviour above are read off the loading and the chord distribution and are not computed.

And proverse yaw is a sign, not a number. The figure shows where the local induced drag changes sign. Turning that into a yawing moment needs an aileron deflection, a change in loading and a moment arm, none of which is here.

Who found it, and when #

Prandtl published Über Tragflügel kleinsten induzierten Widerstandes in 1933, fifteen years after the lifting line and the elliptic result that made his name. It is a short paper and it says plainly that the earlier answer was the answer to a constraint nobody has.

It was ignored. The elliptic result was already in every textbook, the Spitfire was two years away, and an argument that the famous answer was answering the wrong question had no constituency. Reinhold Platz and then the Horten brothers used bell loadings on tailless designs and were understood to be doing something eccentric; Ludwig Prandtl’s own students did not pursue it.

Robert T. Jones re-derived the fixed-bending-moment problem in 1950 and reached the same conclusion from the American side, and it was ignored again. It reappeared through the Hortens’ advocates in the 1980s, was treated as a curiosity, and was finally flown as a research aeroplane — the Prandtl-D — by NASA in 2015, which measured the proverse yaw and the span efficiency and found both where the 1933 paper puts them.

Eighty-two years between the publication and the flight test, on a result that needs no new mathematics and can be checked on a laptop. The reason is not that anybody doubted the algebra. It is that the elliptic answer was so completely established that the question it answers stopped being visible as a question.

Where the ladder goes next #

Every optimum so far has been on a wake that lies in a plane. Munk’s theorem says the induced drag depends on the wake’s shape in that cross-section and on nothing else — which invites the obvious question of what happens if the wake is allowed to leave the plane, and the answer is what a winglet is and is not.