A wing that leaves the plane

A wing’s induced drag is decided in a place the wing is not.

Take a plane at right angles to the flight direction, far enough downstream that the wing’s own near field has died away. Everything the wing did to the air is now expressed as a distribution of circulation in that cross-section — a curve, and a strength along it. Munk’s theorem says the induced drag is a functional of that alone.

Not of the chord. Not of the section. Not of where along the flight direction the surfaces sat. One curve in one plane.

That is a strong enough statement to be worth using as a tool rather than as a fact. If the drag is a functional of a curve, then the question what is the best loading on this curve is a minimisation, and the question what is the best curve is a design problem with the answer computed rather than argued about.

The machine #

Discretise the trace into segments, each carrying a bound circulation, shedding at the nodes. The induced drag is a quadratic form,

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

built from the two-dimensional velocity each shed vortex induces at each control point. Lift and side force are linear functionals of Γ. Minimising the quadratic under a lift constraint is one symmetric linear system.

Two things about the matrix A are worth stating because both are checks rather than constructions.

It comes out symmetric, and the code does not make it so. The influence coefficients are built by a one-sided sum — the velocity segment k’s shed vorticity induces at segment i’s control point — and the raw matrix is symmetric to 2 × 10⁻¹⁴ before anything is symmetrised. That is Munk’s reciprocal theorem: the drag surface one suffers from surface two’s wake equals the drag surface two suffers from surface one’s. Nothing in the arithmetic arranges it and it is the reason the stagger theorem is true.

And the placement decides the accuracy. Nodes cosine-spaced with control points at the half-indices of the same parametric formula give the elliptic wing’s induced drag to machine precision at forty segments; evenly spaced nodes with midpoint control points get 2.1 per cent wrong at forty and converge at first order. The essay that follows would be untrustworthy on the second placement, because everything below is a comparison of numbers a few per cent apart.

Bending the trace #

Now let the trace leave the plane.

The optimum loading runs right round the corner. It does not stop at the junction and start again on the winglet; it is one continuous distribution on one curve, because the arithmetic sees one curve.

That is the whole of what a winglet is, and it is why the blocking story is wrong in a way worth being precise about. A winglet is a wing. It carries circulation, it makes a force, and the force points sideways because the surface is vertical. The sideways force does not lift the aeroplane and does not have to: what it contributes is a wake that extends in z, and the induced drag is a functional of the wake’s whole shape.

Read the other way, the same statement says why an end plate — a flat vertical surface with no section, no camber and no incidence — buys much less. It carries little circulation, so it puts little into the wake, so it does little. Whitcomb’s contribution in 1976 was precisely to point out that the device should be designed as a lifting surface, at its own incidence to the local flow, and not as a barrier.

Why the tip-leak story survives #

The blocking account is worth taking seriously for a moment rather than simply contradicting, because it is describing something real and the reason it fails is instructive.

Near a real wingtip there is a flow round the end from the high-pressure side to the low, and it does roll into a concentrated vortex, and photographs of it are among the most familiar images in the subject. Anyone who has watched an aircraft land in damp air has seen the two cores come off the flaps. So the story has a picture attached to it, and a picture is very hard to argue with.

What the picture does not contain is a cause. The flow round the tip is a consequence of the pressure difference, which is a consequence of the lift, and the vortex is where the vorticity the wing shed has ended up. Blocking the local flow at the tip does not remove the pressure difference — the wing is still lifting — and the vorticity still has to leave. It leaves along the whole span, in proportion to the rate at which the circulation changes, and the tip is merely where the circulation changes fastest.

An end plate that genuinely stopped the flow at the tip would change the loading — it would let the wing carry circulation further out — and that is where its small benefit comes from. It is a loading effect and it appears in the Trefftz plane like everything else, which is why the theorem can account for it and the barrier story cannot account for the theorem.

The tip vortex is an effect of lift, not a cause of drag. Induced drag and the vortex are two descriptions of the same shed vorticity, and a device that reduces one reduces the other because it changed the loading, not because it interfered with a leak.

What it is worth, and what it replaces #

Taller is better, monotonically, and the solver requires it.

Those numbers are ceilings and should be read as such. A real winglet buys a few per cent of total drag, not twenty per cent of anything, because a real winglet has wetted area of its own, has profile drag, is not optimally loaded, has to be structurally attached, and is a compromise with cruise Mach number and with the flutter clearance. The figure says what the induced-drag mechanism can offer; it does not say what a design gets.

The comparison that matters is not against nothing. It is against the obvious alternative — making the wing longer.

A winglet and a span extension can buy the same drag, and they charge different amounts. The winglet’s load is a side force at the tip; the spar bends about a horizontal axis and does not much care. The extension’s load is lift, further out, and that is exactly the moment the spar is sized by.

That is the design case, it is structural rather than aerodynamic, and it is the reason a retrofit winglet exists as a product at all. An airline with an existing wing, an existing spar and an existing gate cannot make the wing longer — the span is fixed by the airport and the structure is fixed by the certificate — and a winglet buys some of the same benefit inside both constraints.

Where the taller-is-better argument stops #

Three places, and none of them is in the arithmetic above.

Profile drag. A winglet’s own wetted area costs drag that scales with its size, while the induced saving has diminishing returns — the curve above is flattening. There is a height at which the two derivatives cross, and it is where real winglets are.

The bending moment does not stay small. The figure holds the root moment and it rises with height too, just more slowly than the extension’s. A tall winglet is a large side load on a long lever arm and the wing box has to carry it.

And the loading here is optimal. Every point on those curves is the best possible loading on that wake, achieved by whatever twist and chord the surfaces would need. A real winglet is a fixed piece of aluminium at one incidence and is optimal at one lift coefficient at most.

So the honest reading of the gain figure is: this is the ceiling, it is real, and the distance between it and a product is where the engineering is.

What the optimum satisfies, written down #

The solver above finds the best loading by minimising a quadratic form, which is the right way to compute it and says nothing about what the answer is. The condition it satisfies has a clean form and it generalises the one result everybody knows.

On a flat trace the answer is famous: the least-drag loading is the one whose induced downwash is uniform across the span, and the loading that produces it is elliptic. That is the statement the planar essay rests on, and it follows from a variational argument — move a scrap of circulation from one station to another, and the drag is stationary only if every station is equally expensive.

On a bent trace the same argument runs, and the constraint changes it. What is being held fixed is the lift, which is the vertical component of the force, so a segment of wake inclined at dihedral angle $\varphi$ to the horizontal contributes only $\cos\varphi$ of its force to the quantity being constrained. The stationarity condition therefore becomes

$$w_n(s) = w_0\cos\varphi(s),$$

with $w_n$ the induced velocity normal to the trace: the downwash is uniform only where the wake is horizontal, and it falls off as the cosine of the local inclination.

Two readings follow, and both are visible in the figures above.

On a vertical winglet the condition says $w_n = 0$. A perfectly vertical segment contributes nothing to lift, so at the optimum it should be doing nothing that costs drag either — the normal induced velocity there vanishes. That is why the optimum loading tapers over the winglet rather than being carried out to the tip at strength.

And it explains the continuum. Cant angle enters the answer through one cosine, so a raked tip, a canted winglet and a vertical one are the same optimisation with a different $\varphi(s)$, exactly as the section below says.

The most striking consequence of the theorem is a configuration rather than a device. Ask for the least induced drag of any system of given span and given height, and the answer is a closed box — two horizontal wings joined by vertical end members, which Prandtl in 1924 called the best wing system. Its induced drag falls with the height-to-span ratio and, at a ratio of a fifth, is around two thirds of the planar wing’s at the same span and lift. Nobody flies one, for reasons of structure, weight and where the control surfaces go — but it is the bound every other device in this essay is working towards.

The continuum this puts everything on #

The most useful consequence of doing it this way is that a set of devices that look different turn out to be the same calculation at different cant angles.

At ninety degrees of cant the trace is flat: that is a span extension. At zero cant it is vertical: a winglet. In between are raked tips, which are a span extension with a swept, tapered outboard panel, and blended winglets, which are a smooth transition rather than a corner. A wingtip fence is a small winglet above and below. A spiroid is a closed loop.

None of those is a different mechanism. Each is a different curve in the same plane, and each has an optimum loading and an induced drag that the same twenty lines of arithmetic produce. Which one an aircraft gets is decided by profile drag, structure, ground clearance and what will fit at the gate, and not by any aerodynamic distinction between them.

That is what a theorem buys. Without Munk, each device is its own subject with its own folklore. With it, they are points in a design space and the comparison between them is a computation.

Reading the trade the way an airline does #

The arithmetic above is a mechanism. The decision it feeds is a commercial one and the shape of it is worth stating, because it explains why winglets appeared on existing aeroplanes rather than on new ones.

A wing already in service has three things fixed: its span, by the gate; its structure, by the certificate; and its section, by the tooling. The only variables left are at the tip. So the question is not what is the best wing but what is the best change to this wing, and the answer is a device that adds drag benefit without adding root bending moment — which is exactly the property the equivalent-span figure isolates.

The bill it does not show is weight and torsion. A winglet adds mass at the extreme end of the wing, which lowers the flutter speed and can require the wing box to be reinforced; the reinforcement is weight, and weight is fuel. The retrofits that succeeded are the ones where that reinforcement was small, and the ones that did not happen are the ones where it was not.

And there is a cruise-condition catch worth naming. The induced drag saving is largest at high lift coefficient — a heavy aircraft, early in a long flight, or a short-field climb — and smallest at low lift coefficient. Profile drag is the other way round. So a winglet is worth more on a long-haul aircraft at high weight than on a short-haul one, and the marketing number and the fleet-average number are not the same number.

What the model does not contain #

No profile drag anywhere. Every number is induced drag. On a real aircraft at cruise the two are comparable and no conclusion about total drag can be drawn from these figures.

The wake does not roll up. It is a fixed curve. Real wakes roll up within a few spans, and the classical result that the roll-up does not change the induced drag is a theorem this essay uses and does not prove.

The wake is not deformed by the winglet either. In reality the trace behind a bent wing moves and distorts, and a fully nonlinear Trefftz calculation lets it. The linear one does not, and near the corner that is the least accurate part of the picture.

No compressibility. A winglet on a transonic wing has to be designed so that its own suction peak does not go supersonic in the corner where two surfaces meet, and that constraint frequently sizes the device. Nothing here has a Mach number.

And the structural argument is a proxy. The root bending moment stands in for weight, which it does adequately at fixed geometry and badly across configurations. A winglet also loads the wing in torsion, which is not computed here at all and which is often what actually limits it.

Who found it, and when #

Max Munk’s 1919 dissertation is the source, and the stagger theorem it contains is one of the most quietly powerful results in the subject: the induced drag of a multiplane system does not depend on the longitudinal positions of its surfaces. His reciprocal theorem is the underlying statement, and it is what makes the whole Trefftz-plane apparatus work.

Frederick Lanchester patented an end plate in 1897 and it did very little, for the reason above. The device only became worth having when Richard Whitcomb at NASA Langley reasoned, in 1976, that the surface should be a wing — cambered, at an incidence to the local flow, and designed to carry a side force. That is the intellectual step, and it is a small one that had eighty years of dismissal behind it because the earlier devices had been fences.

There is a moral here that this site keeps meeting. The theorem was available in 1919 and the device arrived in 1976, and the gap is not a computational one — the arithmetic in this essay would have been within reach of a slide rule. What was missing was the reading of the theorem: that a wake which leaves the plane is a wake with more of it, and that a vertical surface at the tip is a way of putting some there.

Where the ladder goes next #

If the induced drag depends only on the wake’s cross-section, then two wings one above the other are the same kind of problem — and the theorem then says something a designer would not guess about where they are allowed to sit relative to each other, which is: anywhere.