Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Span efficiency — where it appears

The factor by which a wing's induced drag exceeds that of the elliptically loaded wing of the same span and lift. It is one for the optimum and a few per cent worse for anything else, which is why the span matters so much more than the planform.

Named by 15 essays across 2 fields — each of them below, with the objects they name alongside it.

Circulation across the span, for three planforms. How much circulation each part of the wing carries, plotted across the span. It has to reach zero at both tips, because a wing cannot carry circulation off its end, and the rate at which it falls is what determines the vorticity shed into the wake.

The price of having ends

An infinitely long wing in a fluid with no viscosity has no drag at all. Cut two ends into it and the drag appears — not from friction, which is still absent, but from the fact that circulation cannot be carried off the end of anything.

circulation · Finite wing Induced drag against aspect ratio, all at C_L = 0.6. Induced drag for wings of different aspect ratio, each flown at whatever incidence makes it carry the same lift coefficient. The drag falls as one over the aspect ratio, so the longest wing here pays a fraction of what the shortest does for exactly the same load.

The span is the whole story

Induced drag is the price of having ends, and the only thing that lowers it at a given lift is putting those ends further apart. Not more area, not a better section, not a cleverer planform — span, and everything else is a correction of a few per cent to it.

circulation · Finite wing The rectangular wing stalls at the root; the tapered one stalls at the tip. Section lift coefficient across the half span for three planforms, each drawn at the wing incidence where its own worst section first reaches 1.5. A rectangular wing's peak is at the root, which is where a designer wants it: the stall starts inboard, ahead of the ailerons, and the pilot feels it. A tapered wing's peak has moved out to 0.62 of the semi-span — over the ailerons — because taper takes chord away from the tip faster than it takes circulation. The elliptic wing is the degenerate case: every section reaches the limit at once, which is elegant and is the worst possible stall behaviour.

Which part of a wing stalls first

A wing has one lift coefficient and its sections have a hundred, and no section is at the wing's. Which of them runs out first is decided by the planform, it decides whether the pilot keeps the ailerons, and on any wing that was close to elliptic the standard fix costs span efficiency for every degree of it applied.

circulation · Taper Five quarters of the span, the same bending moment, 64/75 of the drag. The bell-loaded wing's induced drag and root bending moment against its span, both as fractions of an elliptic wing of unit span carrying the same lift. At equal span the bell is worse: its span efficiency is exactly three quarters. But its bending moment is exactly four fifths, and bending moment grows in proportion to span while drag falls as its square — so there is a span at which the bell has bought back the structure and is ahead on drag. It is at exactly five quarters, where the two curves are at 1 and 0.8533. Both numbers are rational and neither was put in by hand.

The loading nobody used

Elliptic loading is the least-drag answer to a question no aeroplane asks. Constrain the moment the lift makes about the wing root instead of the span, and a different curve comes out — five quarters of the span for sixty-four seventy-fifths of the drag, both exact — with an upwash over the outer wing and a yaw that turns the right way.

circulation · Bell shape A closed wake, and the loading of least drag on it. The wake of a box wing in the plane that decides its drag, with the circulation of least induced drag drawn as a thickness along it. The horizontal members carry a loading close to elliptic and the vertical ones carry a share that lifts nothing — they contribute no lift, since lift is Γ dy and dy is zero on a vertical, and they change the drag by changing where the wake's vorticity is. At a gap of 20 per cent of span this system costs 67.1 per cent of what a single wing of the same span and lift would.

A wake that closes on itself

Prandtl's best wing system is a rectangle, not a wing. Solving for the loading on a closed wake turns up a circulation that costs nothing and does nothing — a gauge freedom in the middle of an optimisation — and a drag that keeps falling with no floor under it.

circulation · Box wing How the weight is shared between the two surfaces, against the centre of gravity. The wing's load and the tail's, for an aeroplane in level flight, against the static margin. The two must sum to the weight and their moments must cancel, and those two equations decide the split. At a forward centre of gravity the tail carries down and the wing carries more than the weight; the crossing is where the tail carries nothing.

More lift than weight

An aeroplane in level flight is drawn with one arrow up and one arrow down, equal and opposite. That equation collapses the whole configuration onto one number, and it is not true of any aeroplane with a tail behind it: there are two surfaces, two equations, and the second decides the split.

circulation · Trim How flat the optimum is. The same curve near its maximum, with the bands within a tenth and a half of one per cent of the best shaded. Every taper ratio from 0.31 to 0.42 is within a tenth of a per cent of the optimum, and the whole band from 0.25 to 0.51 is within half a per cent. The optimum is exact, and choosing it rather than its neighbour buys nothing a wing can measure.

The optimum that does not matter

Elliptic loading gives the least induced drag, exactly, and the proof is one line: the penalty is a sum of squares. Which is also why the optimum is flat enough that a quarter of the design space sits inside a tenth of a per cent of it.

circulation · Taper A washed-out rectangular wing is at its best at one lift coefficient. The span efficiency of a rectangular wing of aspect ratio 8 against lift coefficient, untwisted and with 2°, 4° and 6° of washout. Untwisted, it is 0.9367 at every lift coefficient. Each washout has one lift coefficient where it is best — 0.254, 0.509 and 0.763 — proportional to the twist, and each reaches the same peak, 0.9911, there. Below the peak the efficiency falls fast.

Washout is right at one lift coefficient

A twisted wing's loading is the sum of two shapes in a proportion that changes with speed, so its induced drag is a parabola in lift coefficient with its least at one design point, and that point moves in proportion to the twist. On a rectangular wing four degrees of washout puts it at a lift coefficient of 0.51 and recovers almost all the span efficiency the planform lost. On a well-tapered wing the same four degrees puts it at nineteen, and the washout is a drag paid at every speed for the stall alone.

circulation · Taper A trainer's sortie wants the washout of its cruise, not of its average. The span efficiency of a rectangular wing of aspect ratio 8 averaged over a training sortie — eight minutes' climb, thirty of cruise, ten of slow flight and twelve of circuits — against its linear washout. Untwisted it is 0.9367. The best washout for the sortie is 3.43°, reaching 0.9774: the optimum for a lift coefficient of 0.436, the distance-weighted harmonic mean of the sortie's. The cruise alone would want 2.6°; the time-averaged lift coefficient, 0.671, would want 5.28° and lose 1.2% of the efficiency.

One twist is chosen for the speed flown per metre

A twisted wing is best at one lift coefficient and an aeroplane flies at many. Average its induced drag over a flight the way fuel and height are actually spent, per metre, and the twist that wins is the one for the root-mean-square speed of the flight — so the cruise outvotes the slow flight, a glider's circling barely counts, and a trainer's usual three or four degrees of washout turn out to be right for reasons nobody fitted them for.

circulation · Taper The face behind a finite wing carries the share its own shape gives it. The share of the lift leaving through the face of the box 500 spans behind the wing as downward momentum, and the share carried as pressure on the faces above and below, against the box's height over its width, for boxes of thirty-six square spans, with the far-wake share (2/π)·arctan(H/W). A box sixteen times as tall as it is wide sends 0.96 of the weight through the face behind, a square one 0.501, and one sixteen times as wide 0.04.

Behind a finite wing the wake face keeps its share

Draw a box round a two-dimensional section and the face behind it carries almost none of the lift, so "the air is thrown down" keeps the wrong term. A finite wing is different. Its trailing vortices carry downward momentum through the face behind it however far back the face is. The share is set by the face's own shape — all of the weight through a tall face, half through a square one, none through a wide one — and the only things a box cannot change are the total and the induced drag.

misconceptions · Momentum lift A bending-moment wing has no best span until its tips are forbidden to push down. The least induced drag for the elliptic wing's lift and bending moment, against span, for a spar sized by its root moment and one sized by the moment along its whole span. Free, both fall without limit: at 1.7 times the span the root-sized optimum is 0.815 of the elliptic drag and the other 0.79, with the tips carrying downloads. With no section allowed to push down both stop where the free loading's tip load first reaches zero: at 1.225 and 0.8889 for the whole-span measure, and near 1.33 and 0.84375 for the root moment. Longer spans then leave their tips unloaded and gain nothing.

A bending moment never chooses a span

Hold a wing's lift and its bending moment and ask for the least induced drag, and the answer does not stop: the span grows for ever and the tips push down. Both celebrated answers — Prandtl's bell at √(3/2) of the span for 8/9 of the drag, and the root-moment wing at 4/3 for 27/32 — are the span at which the least-drag loading's tips stop lifting. The structure sets how much is saved; a condition nobody states sets where the saving ends.

circulation · Bell shape The washout a mission wants rises with the lift coefficient; a twist driven by dynamic pressure rises the other way. Washout against lift coefficient for the trainer's sortie: each lift coefficient's own best washout, 7.87° per unit of it; the best fixed washout, 3.43°; the best twist that varies with dynamic pressure, which has to wash in as the aeroplane speeds up; and a twist of the natural sign, one degree of extra washout at cruise growing with dynamic pressure, from a nose-down pitching moment. The four segments sit at Cₗ 0.82, 0.33, 1.3, 0.92.

A wing that twists with its speed washes out the wrong way

A fixed washout is right at one lift coefficient, and a flight that spends its time at four of them pays for the spread. A wing whose twist changed in flight could follow — but the twist a real wing gets for free is driven by dynamic pressure, and it adds washout as the aeroplane speeds up, exactly where less is wanted. On a trainer's sortie a degree of it at cruise raises the induced energy by a third of a point. The best twist driven by dynamic pressure washes in, and only one that follows the incidence itself removes the spread entirely.

circulation · Taper A symmetric panel with no spring stiffness washes out exactly as each lift coefficient wants. The outer panel's washout against the wing's lift coefficient over the trainer's sortie, with a spring stiffness of 0 cruise dynamic pressures: each lift coefficient's own best, 5.31° per unit of it; the best fixed washout, 2.31°; a symmetric panel at its best hinge and preload, 1.34% above an elliptic wing; and a panel with a cambered trainer section's moment of −0.05, 9.31%. The fixed twist spends 2.65% and the floor is 1.34%.

A pivoted wingtip follows the lift only with no spring stiffness and no camber

The washout a trainer's sortie wants grows in proportion to the lift coefficient, and the twist a wing gets for free follows the dynamic pressure instead. A tip panel free to pivot on a hinge ahead of its own aerodynamic centre can do better, and on one condition it does exactly what is wanted: held by a constant torque and carrying no camber moment, its own lift coefficient stays in proportion to the wing's, and every segment of the sortie flies at its own best twist. Every unit of spring stiffness gives part of that back, a cambered section's moment turns the panel into the wrong law altogether, and the free panel sheds a quarter of a gust's lift for nothing.

circulation · Taper Losing the leading-edge suction doubles a slender wing's drag due to lift and multiplies a long wing's by about half its aspect ratio. The drag due to lift with no leading-edge suction, Cₗ²/Cₗα, over the induced drag a full suction leaves, Cₗ²/(πAe), against aspect ratio, from a vortex lattice for rectangular, delta and elliptic flat wings; with the closed form 1 + √(1 + A²/4) that Helmbold's lift slope gives an elliptic wing. Rectangular: 2.15 at A = 1, 3.46 at 4, 5.33 at 8, 7.11 at 12; delta: 2.22 at 0.5 and 2.83 at 2. The slender limit is 2.

A wing that cannot hold its suction gains little from its span

The leading-edge suction is a small force concentrated at one edge, and on a finite wing it decides what the aspect ratio is worth. With it, the drag due to lift is the induced drag, which falls as the span grows. Without it, the drag due to lift is the lift tilted back by the incidence, and that falls only as fast as the lift slope rises — which stops at the section's 2π. A slender delta that loses its suction pays exactly twice its induced drag; a wing of aspect ratio eight pays more than five times it. Without the suction no span can lift a wing's best lift-to-drag ratio past a ceiling set by the section alone.

circulation · Leading edge suction Counting the spar's weight stops the span, and the best wing then sheds a little load from its tips to grow five per cent. Total drag at a fixed payload, induced drag of the lift that carries the spar too, against span, both in units of the best elliptic wing's, for a wing whose area is held so that its spar thins as it lengthens. Elliptic, the best span carries a spar of a quarter of the lift; with the third harmonic free the best ratio is ε = -0.1356 (closed form (√33 − 9)/24 = -0.1356), the span 1.05 times and the drag 0.9575 times the ellipse's, with the spar still a quarter.

A spar that must lift itself stops the span at a quarter of the weight

Held only to a lift and a bending moment, the least-drag wing never stops growing. An aeroplane's wing stops because its spar has weight, and the weight has to be lifted. Count it, and the best span is the one at which the spar is exactly a quarter of the lift when the wing's area is fixed, and a third when its chord is — whatever the loading. The best loading then sheds some of its tip load, but not all the way to Prandtl's bell; with the chord held and nothing else charged, it is the bell exactly, and the more of the drag the wing's own area makes, the nearer it comes back to the ellipse.

circulation · Bell shape

Named alongside it

The objects these essays reach for when they reach for this one.

Induced dragLifting lineOptimisationSpan loadingModel limitDownwashAspect ratioThe Trefftz planeCirculationBending momentConstraintLift coefficient

All concepts