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

Span loading — where it appears

The distribution of lift along a wing's span, as distinct from its total. The drag depends on the distribution and the lift does not, which is why an elliptic loading is optimal and why a planform is a way of arranging one.

Named by 23 essays across 2 fields — each of them below, with the objects they name alongside it.

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 The wake, and the velocity it gives itself. The cross-section of the wake far behind a wing with turned-up tips, with the velocity the wake induces on itself drawn as arrows normal to the trace. Induced drag is the integral of the circulation against that velocity and nothing else — this picture contains the entire quantity. It also contains no information whatever about where the surfaces were: two wings a chord apart and two wings ten chords apart produce the same picture and therefore the same drag, which is Munk's stagger theorem stated as a fact about what the arithmetic can see.

A wing that leaves the plane

A winglet is not a fence and it does not block anything escaping round the tip. The induced drag of any system of lifting surfaces depends on one cross-section of its wake and on nothing else whatever, and a wake that reaches upwards is cheaper for the same reason a wake that reaches sideways is.

circulation · Winglet Four aeroplanes, one number. Four biplanes with the same gap and the same loadings, staggered by nothing, by four tenths, by nine tenths and by one and six tenths of a chord. Their induced drags agree to the last bit of the arithmetic — the calculation cannot even express the stagger, because the Trefftz plane is a cross-section and everything drawn here projects onto the same one. That is Munk's stagger theorem, and stating it as the drag is unchanged understates it: there is no place in the computation where the stagger could be entered.

Two wings and it does not matter where

Move one wing of a biplane a chord forward and the induced drag does not change. Not approximately, not to a good approximation — the calculation that gives the induced drag has nowhere to put the stagger, because everything projects onto the same cross-section of the wake.

circulation · Biplane Tip to tip, two wings cost exactly half of what they cost apart. The induced-drag saving of a pair of wings against the gap between their tips, as a fraction of what the two pay flying alone. With the tips touching the pair is one wing of twice the span carrying twice the lift, and the arithmetic of that is exact: the drag halves, and the computation returns 0.499257 of the separate figure. Pull them apart and the saving falls away with the square of the distance. Birds fly in a V because the tips are the part worth overlapping, and the spacing that pays is a small fraction of a span rather than any distance a formation could hold by eye.

The lift beside a wing

Fly two aeroplanes with their wingtips touching and the pair costs exactly half what the two cost apart. Not approximately half — the arithmetic is a closed form, because two wings tip to tip are one wing of twice the span, and induced drag goes as the square of it.

circulation · Formation Elliptic loading rolls up to πb/4, and it is π that puts it there. Three loadings on the same span carrying the same lift, with the rolled-up core positions marked below each. The core sits at the centroid of the shed vorticity, which is a quadrature over the loading and needs nothing about the roll-up itself. Elliptic loading gives πb/8 from the centreline, so the pair ends up 0.7854 of the span apart — the number every wake-separation rule is written against, and one of the few places in this subject where π turns up in an answer an engineer uses directly. The bell rolls up to 0.586 and a nearly rectangular loading to 0.978, because it sheds at the tips.

Where the wake ends up

The sheet a wing sheds rolls up into two cores within a few spans, and nobody can compute the roll-up cheaply. Nobody has to: what the cores conserve is fixed before they form, and for an elliptically loaded wing the answer contains π and comes out at 78.5 per cent of the span.

circulation · Tip vortex 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 The local sweep of the isobars, across the span. The sweep of the half-load line at each spanwise station, for four geometric sweeps. Over the middle of the span it is the wing's own sweep, which is the simple theory being right. At the root it collapses — by twenty-one degrees at a geometric thirty-five — and at the tip it falls again. That root region is where the shock forms first on every swept wing ever built, and it is why they have waisted fuselages.

The sweep a root does not have

Simple sweep theory is one of the cleanest arguments in aerodynamics: an infinite yawed wing cannot know about the velocity along its own span, so only the normal component matters. A real wing has a root and two tips, and at the root of a thirty-five-degree wing the isobars are swept fourteen.

circulation · Sweep 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 What every wing pays to bend its root less. Least induced drag against the root bending moment of the lift, both as fractions of the elliptic monoplane of the same span and lift, for the monoplane and for box wings with gaps of a tenth, a fifth and two-fifths of the span. Each curve is a parabola with its minimum at that wing's unconstrained optimum. The box wings' minima sit to the right of the monoplane's — they load their lift further out — and their parabolas are shallower: the gap-of-a-fifth box can bend its root 21 per cent less than the elliptic monoplane and still match its drag.

A lighter spar turns a box wing into a biplane

Prandtl's best wing system has two-thirds of a monoplane's induced drag at a gap of a fifth of the span, and part of that saving is carried by circulation turning the corner into its fins. Ask the box to bend its root less and it pays about half what a monoplane pays — but a biplane with no fins pays nearly as little, and by the time the spar is a fifth lighter the fins carry almost nothing and the box has become the biplane it was built from.

circulation · Box wing A V seen from above, and what each member pays. Nine wings in a V swept 45 degrees, tips one span apart, each carrying the same lift, seen from above with the flight direction up the page. Beside each is its induced drag as a fraction of what it would pay flying alone. The leader at the point pays 0.89; every other member pays between 0.35 and 0.39. The saving has flowed backwards through the formation.

The bird at the point pays for the V

A flock's saving in a V is fixed by where its members sit across the stream, and the angle of the V cannot change it by a single per cent. What the angle changes is who gets the saving. It flows backwards through the formation, so that in a V swept forty-five degrees the leader pays nine-tenths of what it would pay alone while every bird behind it pays under four-tenths.

circulation · Formation The V in which every bird pays the same is curved. Nine birds one span apart, flying up the page, in plan: each short line is a wing, its height its distance behind the leader in spans. The straight V at 7.5° — the fairest straight V — and the V whose members' positions are solved so that every one pays exactly the same share of the induced drag. The equal-share arms leave the leader almost abreast and bend back ever more steeply: 1.6°, 7.6°, 17°, 32° from the apex to the tips.

A fair V is a curved V

In a straight V of birds somebody always pays more than somebody else: at the fairest angle the leader and the two birds at the tips pay half again what the birds between them pay. The V in which every bird pays exactly the same can be solved for, and it is not straight. Its arms leave the leader almost abreast and bend back ever more steeply, to thirty-two degrees at the tips of a flock of nine. The share every bird then pays is the flock's average, fixed by Munk's theorem before any position is chosen — so fairness costs nothing, and the only thing that can make the flock cheaper is flying closer together sideways.

circulation · Formation A phantom biplane partner doubles the induced drag. Induced drag at a given lift, as a multiple of its deep-water value, against the foil's depth in spans. The surface's image of the foil is an identical wing, equally loaded, twice the depth above it, so the real foil pays its own induced drag plus the mutual drag of a biplane with a gap of twice its depth: 1 + σ, with σ Prandtl's biplane factor. The elliptic biplane's 1 + σ and the aspect-ratio-9 foil's lattice both run to two as the depth closes.

A foil under the surface flies with a phantom

At foiling speed the water's surface cannot hold a pressure, and the image that condition requires is not the reversed one a wall makes. For a horizontal foil it is an identical wing, equally loaded, above the surface — a biplane partner that takes lift and gives nothing back, doubling the induced drag as the foil rises. For the board that pierces the surface it is a reversed copy, which makes the board pay two and a half times what a keel under a hull pays. It is the board, not the foil, that decides how deep a foiling boat rides.

applied · Sailing An elliptic wing filters the gust like a round aperture. How much of a gust component with spanwise wavenumber k₂ survives the span average, against κ = k₂b/2, for three weights. A uniform strip is a slit and its filter is sinc², falling as κ⁻² between its zeros. An elliptic loading is a circular aperture and its filter is the Airy pattern (2J₁(κ)/κ)², whose first zero is at 3.83 and whose envelope falls as κ⁻³, but which stays near one to larger κ: the ellipse's weight is narrower, so its filter is wider. A rectangular wing of aspect ratio nine weights the span by its own loading and lies between, with the ellipse's κ⁻³ fall because its loading too goes to zero at the tips as a square root.

A wing averages a gust through its own loading

A finite span averages a turbulent gust and so tames the load it causes, but it does not average uniformly. The reverse-flow theorem says exactly how it weights the span: by the loading the wing makes at a uniform incidence. An elliptic wing therefore filters the gust the way a round aperture diffracts light, passes eight per cent more of its short scales than a uniform strip, and crosses its mean load up to four per cent more often.

circulation · Gust 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 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 A root's bending moment weighs the gust by the loading of a wing twisted like its moment arm. Across the span of an elliptic wing of aspect ratio 9, each normalised to unit area: the lift's weight, the loading at uniform incidence; the two roots' bending moment's, the loading of a wing washed in as |y|, with a hole at the root; and the right root's alone, the loading of a wing washed in as y on the right half and not at all on the left. The one-root weight reaches across to the left wing through the induced flow, and it is not symmetric.

One wing root feels the rolling gust its lift cannot

The reverse-flow theorem says the gust weight of any load is the loading the wing would carry at an incidence shaped like that load's arm. For the lift that is the familiar uniform-incidence loading. For the bending moment at one root it is the loading of a wing washed in towards one tip only — a weight that is not symmetric, and so passes the rolling half of a gust that the lift and the two roots together cancel. That half is a sliver of the variance and adds up to a fifth to how often the root's moment crosses its mean.

circulation · Gust 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 Sweep slows every load's crossings by about a tenth at 35°, and leaves one root's excess over the lift standing. Against the sweep, for an elliptic wing of aspect ratio nine whose span is eight hundredths of the turbulence's integral scale: left, the lift's and one root's crossing rates as fractions of the straight wing's; right, one root's rate over the lift's. At 35° the lift crosses its mean 0.899 times as often as on the straight wing and the root 0.887; the root's excess over the lift goes from 1.19 to 1.17, and at 45° to 1.16.

Sweep slows a wing root's gust cycles and leaves their excess over the lift

On a straight wing every station meets a gust at the same instant, and one wing root's bending moment crosses its mean nearly a fifth more often than the lift, because it reads the rolling half of the gust. Sweep the wing back and the outer stations meet the gust later, by their distance aft. That delay averages the gust along the flight path as a second, softer chord would: at 35° every load crosses its mean about a tenth less often, the root slightly more than the lift since its weight sits further out, and the variance hardly changes at all. The rolling excess survives almost whole.

circulation · Gust 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 Inboard the spar is sized by its load, outboard by a stiffness floor its load cannot lower. The spar's weight per unit span along the semi-span, as a share of the payload's weight per unit of the reference span, at the best wing when the stiffness floor is 0.4: the weight the bending moment needs, and the floor. The floor governs the outer 53.1 per cent of the semi-span; the best loading has a third-harmonic ratio of -0.0993 and the spar is 32.1 per cent of the lift.

A stiffness floor keeps the span and gives the tips their load back

A spar sized by strength alone thins to nothing at the tips, and charged as lift it stops the best span where it weighs a quarter of everything the wing carries, with the tips a little unloaded. A real outer wing is sized by stiffness, and stiffness sets a floor under the spar's weight that no reduction of the tip load can lower. Put that floor in and the tips get their load back — the best loading slides from a sixth of the way to the bell towards the ellipse — the quarter does not survive, rising to two fifths as the floor spreads over the outer two thirds of the span, and the span itself barely moves. A weight per metre cannot stop a span.

circulation · Bell shape

Named alongside it

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

Model limitInduced dragOptimisationLifting lineSpan efficiencyThe Trefftz planeBending momentCirculationMunk's stagger theoremConstraintDownwashGust

All concepts