A wake that closes on itself

Every wing on this site so far has had ends. That is what makes induced drag: the circulation has to fall to nothing at the tip, the drop is shed as trailing vorticity, and the wake that vorticity forms costs energy for as long as the aeroplane keeps flying.

The obvious response is to build a wing without ends. Join the tips of an upper wing and a lower one with vertical members, and the wake becomes a closed curve — no free ends, nowhere for the circulation to be shed at, and, as Prandtl showed in 1924, the least induced drag available for a stated span and height.

The calculation of what that buys is short, because Munk’s theorem reduces every lifting system to a curve in one plane. It also contains a trap that no open wing has, and the trap is the interesting part.

The circulation that costs nothing #

Put a constant circulation round the closed curve. Every corner then has as much circulation arriving as leaving, so nothing is shed anywhere. No trailing vorticity, no induced velocity, no drag. And no lift either: the lift is $\rho U\oint \Gamma\,dy$, and $\oint dy$ round any closed curve is zero.

So the closed wake carries a mode that does nothing whatever. The site’s influence matrix says so to $8\times10^{-16}$ — every row sums to that — and the drag of that mode comes to $4\times10^{-16}$.

An open wing has no such mode, because a constant circulation on it would have to be shed at the tips. The moment a wake is closed, the drag becomes a function with a flat direction in it, the matrix that represents it becomes singular, and a straightforward least-drag solve returns a division by zero rather than an answer.

The fix is a gauge condition: minimise the drag subject to the lift and to a stated mean circulation. Which mean is arbitrary, and every choice must give the same drag.

Here the honesty is worth more than the tidiness. It does not, quite. Moving along the null direction changes this site’s computed drag by 0.62 per cent, and refining the wake from forty-eight segments to three hundred and eighty-four does not improve it: 0.60 per cent at the coarsest, 0.71 at the finest. The invariance is exact in the continuum — a constant added to $\Gamma$ sheds nothing, so the wake’s energy cannot change — and it survives discretely only as far as $\oint w_n\,ds = 0$ survives, which is an integral taken along the very line the shed vortices sit on. It is the same defect that leaves the influence matrix visibly asymmetric on any wake with a corner in it. Every number below is quoted at the mean-zero gauge, and the 0.62 per cent is smaller than the quantity this essay argues about — which is the only reason it is tolerable.

Why the drag is the wake, and nothing else #

The whole calculation rests on one identification, and it is worth restating because it is what makes a closed wake a legitimate object at all. Induced drag is the kinetic energy left in the crossflow of the wake, per unit distance flown. It is not a pressure force on a surface, it is not friction, and it does not depend on where the surfaces are along the flight direction — which is Munk’s theorem, and which is why a picture of a box wing’s wake is a complete description of its induced drag.

That is also why the vertical members can matter without lifting. They move vorticity, and the energy in the crossflow depends on where the vorticity is.

What the gap buys #

At a gap of a fifth of the span — which is a large aeroplane — the box costs 0.669 of what a monoplane of the same span and lift would, a saving of a third. Prandtl’s fit says 0.680. At a tenth of a span it is 0.776 against 0.791, and at four fifths it is 0.403 against 0.414.

Two features of that curve matter more than the numbers.

Nothing flattens out. The drag keeps falling as the gap grows, with no floor, because the two wakes keep getting further from each other and the mutual induction keeps weakening. There is no optimum height; there is only a structure that becomes impossible.

And the saving is entirely a matter of where the vorticity is. Both surfaces carry lift, the total lift is fixed, and the span, which is usually the whole story, is fixed. What changes is the distance between the sheets of trailing vorticity, which is the only geometric quantity in the Trefftz plane.

What the vertical members actually do #

They lift nothing. It is worth saying twice because a picture of a box wing invites the opposite reading: the vertical members are big, they carry circulation, and they contribute exactly zero to the lift, because lift in the Trefftz plane is $\Gamma$ weighted by the spanwise extent of each element and a vertical element has none.

What they do is carry the circulation from the lower tip to the upper one without shedding it. On an unjoined biplane the circulation on each surface has to reach zero at its own tip, and the drop is shed there as a vortex; on a box it does not have to, and the trailing vorticity that would have been concentrated at four tips is distributed round the whole loop instead. The optimum trades a concentrated wake for a spread one, which is the same trade a winglet makes with only one end of it available.

The two limits, and only one of them is a wing #

The trade curve has an end at each side and both are checks rather than results.

As the gap goes to zero the two surfaces merge, the vertical members shrink to nothing, and the box must cost what a single wing costs. The computation gives 0.932 of the monoplane at a gap of two per cent of span, which is close and is not one — and the discrepancy is the nearly-degenerate geometry rather than physics: at that gap the upper and lower sheets are a hundredth of a span apart and the lattice is resolving a difference between two large and nearly equal influences.

As the gap grows the drag keeps falling, and there is no limit in the mathematics. At a gap equal to the span the box costs a third of the monoplane — 0.359 against Prandtl’s 0.377 — and at two spans it would cost less. What stops this is not aerodynamics.

The optimum is a minimum, and that had to be checked #

A stationary point of a quadratic form is a minimum only if the form is positive on the constraint surface, and the form used here has been symmetrised — the discrete reciprocity that would make that harmless is exact only for a flat wake. So the answer is checked by brute force: two hundred perturbations of the optimum loading, each rescaled to carry the same lift, and every one of them must cost more drag.

Every one of them does. That check exists because this site has been caught by the alternative: the Trefftz drag matrix once went in with the wrong overall sign, every induced drag came out negative, and the optimiser cheerfully returned the worst loading available while looking entirely convincing.

There is no lower bound, and that is the real result #

The trade curve falling with no floor under it looks at first like a defect of the model — a calculation that has left something out and is therefore promising too much. It is not. It is the central fact about induced drag, and the box wing is one way of demonstrating it rather than a special case that happens to.

The cleanest demonstration is Prandtl’s own second answer to the same question. Take $N$ wings of the same span, stacked far enough apart that each is effectively alone, and share the lift equally between them. Induced drag goes as the square of the lift, so each wing costs $1/N^2$ of the monoplane’s, and there are $N$ of them: the total is $D_1/N$. Bring them closer and mutual induction spoils that, so a real stack does worse than $1/N$ — but the sequence still falls, and there is no $N$ at which it stops.

So the induced drag of a lifting system at a fixed span and a fixed lift has no positive lower bound. It can be made as small as anyone likes, by any of several arrangements, and the only thing preventing it is that each arrangement is made of aluminium.

That is worth restating in the form it changes an intuition. Induced drag is habitually described as the price of making lift, in the same breath as skin friction is described as the price of having a surface — as though both were unavoidable consequences of doing the job. The second is: a surface has area and the area is wetted. The first is not. Induced drag is the price of making lift on one curve of limited extent, and it is a property of the arrangement of the lifting elements rather than of the lift. Nothing in the physics requires it to be paid; what requires it is that an aeroplane must be an object, and an object has a size.

Which puts the two drags in their proper relation, and explains why almost all aerodynamic ingenuity since 1920 has been pointed at one of them. Profile drag can be reduced by making a surface smoother or smaller, and every reduction runs against a floor set by the area the aircraft needs. Induced drag can be reduced without limit by rearrangement — a longer span, a winglet, a box, a stack, a formation of birds — and every such rearrangement is a structural argument rather than an aerodynamic one. The aerodynamicist’s answer to induced drag is always available and always someone else’s problem.

It also explains the shape of the design history. The biplane was abandoned for reasons that had nothing to do with the argument above: it is a better induced-drag arrangement than a monoplane of the same span and it lost anyway, because bracing wires and struts and two sets of wetted area cost more in profile drag than the induced saving was worth once speeds rose. The saving is a fraction of a term that shrinks as the square of the speed; the penalty is a fraction of a term that grows as the square of the speed. Every closed or stacked wing system is the same bet, and which way it pays depends entirely on where the aircraft sits on that crossover — which is why the arrangement keeps being reinvented for slow, heavy, span-limited aircraft, and keeps losing for fast ones.

What the picture cannot show #

Four limits, and the first two are why box wings are rare.

There is no structure in this calculation. The whole argument is that the induced drag falls as the gap grows, and the mass of the vertical members grows with it — as does their profile drag, their cost and their interference with everything else. The optimum height is set by a trade this calculation contains no term for.

There is no profile drag anywhere. The vertical members are large surfaces with skin friction on them, and the induced saving has to exceed that. At the gaps a real aircraft could carry — a tenth of a span or so — the saving here is 22 per cent of the induced drag, which for an airliner in cruise is perhaps 8 per cent of the total, and the added wetted area eats a good part of that.

Every loading in the trade is the constrained optimum on its own wake, which no fixed piece of aluminium achieves. A real box wing is built to some approximation of these curves, and the approximation costs.

And the closed wake’s gauge freedom is a numerical fact as well as a physical one, quantified above at 0.62 per cent and not improving with refinement. That is the honest bar on every ratio in this essay.

What it would take to build one #

The aerodynamic case is settled and the aircraft is rare, which is worth an honest paragraph.

A box wing at a gap of a fifth of the span saves a third of the induced drag. For an airliner in cruise, induced drag is roughly a third of the total, so the saving is around ten per cent of drag — a very large number in an industry that fights for one. Nothing in the calculation above is in doubt, and Prandtl’s own numbers agree with it.

What the calculation does not contain is the aeroplane. The vertical members are structure with air flowing over it: they add wetted area, and their profile drag comes off the saving directly. They must carry load, so they are spars rather than fairings, and a closed structural loop is stiffer than an open one — which is an argument in the box wing’s favour that nothing here can evaluate. They change the aircraft’s ground handling, its hangar footprint, its emergency egress and its certification basis, none of which appear in a Trefftz plane.

And the two surfaces interfere in ways the ideal theory has no term for. The rear surface flies in the wake of the front one, at a different effective incidence, with a boundary layer that has already been through somebody else’s pressure gradient. The induced-drag calculation is exact and it is one line of the budget. Being right about a third of the drag is not the same as being right about the aeroplane, and the honest form of this essay’s result is a ceiling rather than a promise.

Who worked it out, and the surprise underneath #

Prandtl published Der induzierte Widerstand von Mehrdeckern in 1924, and the closed system in it is sometimes called the best wing system. The name is precise: it is the least induced drag for a stated span and a stated height, which is a different optimisation from the one that gives the ellipse and answers a different question about aeroplanes. The two live together in this collection as an example of how much the choice of constraint decides — the bell-shaped loading is the same lesson with a structural constraint instead of a geometric one.

The surprising part is the null mode, and it is worth stating in general because it is not really about wings. Closing a boundary introduces a gauge freedom. A closed wake has a circulation that does nothing, exactly as a closed body has a potential that can be shifted by any constant, and exactly as a body’s Neumann problem has a solution only up to one. In each case the freedom is harmless physically and fatal numerically, and in each case the fix is to say which member of the family is wanted before asking for one.

Where the ladder goes next #

The obvious rung is the same optimisation with the structure in it: minimise induced drag subject to lift, span, height and a root bending moment, which is Prandtl’s 1933 problem with a third constraint and produces a family of box loadings rather than one.

The nearer one is what happens when the closed curve is not closed by choice but by a wall. A wing near the ground is a wing and its own mirror image, which is a box wing with an unusual lower half — the loading below is fixed rather than free, it lifts the wrong way, and the drag saving that results has the same origin as this one.