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

Pitching moment — where it appears

The moment of the aerodynamic forces about a stated point, which changes with the point chosen. About the aerodynamic centre it does not change with angle of attack, which is why that point is the useful one to quote it at.

Named by 13 essays across 2 fields — each of them below, with the objects they name alongside it.

Finding the aerodynamic centre by sweeping the chord. How fast the pitching moment changes with incidence, plotted against where along the chord the moment is taken. The curve crosses zero once, and that crossing is the aerodynamic centre — the single point about which pitching the wing does not change the moment.

Where the lift acts

Lift gets drawn as an arrow, and an arrow has to start somewhere. The pressure is spread over the whole surface, so the choice of where to put the arrow is free — except that exactly one point on the chord has the property of not changing its answer when the wing is pitched.

circulation · Moment Four camber lines, and the angle at which each stops lifting. Four mean lines on the same chord: symmetric, a circular arc, a four-digit line with its crest at forty per cent, and a reflexed line whose tail turns up. The zero-lift angle beside each is computed from that line's own slope by quadrature and is a property of the shape alone — no incidence, no speed, no thickness enters it. The symmetric line's is exactly zero, the arc's is −2m to ten decimal places, and the reflexed line's is positive: it needs to be pointed up before it stops lifting.

Where lift starts

A wing at zero incidence is not a wing making no lift. The angle at which a section stops lifting is a property of its camber line and of nothing else — not of its thickness, not of its speed, not of the air — and it is an integral anybody can take.

circulation · Thin-aerofoil A loading that integrates to nothing and turns the aeroplane over. The loading a slender body carries at 6 degrees, drawn along it. It is positive over the front half and negative over the back, in equal measure, so the lift integrates to -3.6e-18 — nothing, which is d'Alembert's paradox for a body of revolution. The moment does not: the two halves act at different stations, and the couple that survives is nose-up, grows with the square of the speed, and is what a tail is sized against.

A body with no lift, and a moment anyway

A fuselage in ideal flow carries no lift at any incidence and still tries to turn the aeroplane over. The couple is computable in one line, it is why tails are the size they are, and the line comes from applying the wall condition to a place where there is no wall.

inviscid · Slender body The downwash approaches twice the value at the wing — from above. The downwash behind an elliptically loaded wing of aspect ratio 8, as a multiple of the induced angle at the wing itself, against distance in spans. Every account of tail sizing quotes a factor of two here. Two is the value at infinity: the trailing legs of the horseshoe system contribute a factor (1 + x/√(x² + a²)) which is one at the lifting line and two far downstream. Close behind, the bound vortex dominates and the field is much larger, and the curve comes down to its limit. A tailplane sits two or three chords behind, which on this wing is 0.31 of a span — where the factor is 2.46, a quarter above the number in the formula.

The surface in the wake

A wing has no opinion about its own incidence, which is why it needs a second surface behind it. How much that surface is worth depends on how much of the wing's downwash it is sitting in, and the factor of two everybody quotes for that is the value at infinity — where no tailplane has ever been put.

circulation · Moment 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 Rolling effectiveness against dynamic pressure. The rolling moment an aileron produces, as a fraction of what it would produce on a rigid wing. It falls from one, passes through zero at the reversal pressure, and goes negative: beyond that point deflecting the aileron down rolls the aeroplane the other way. There is no oscillation anywhere in this figure and no frequency — it is a static failure.

The control that works backwards

Divergence is the static aeroelastic failure everybody names, and a wing with its elastic axis at its aerodynamic centre cannot diverge at any speed. It can still reverse — deflect the aileron down above a certain dynamic pressure and the aeroplane rolls the other way — because the aileron's own nose-down moment is there whatever the elastic axis is doing.

circulation · Flutter The six bodies, drawn at the same scale. Two circles, two ellipses and two Joukowski sections, each at the incidence that gives it a circulation of exactly two. There is no family resemblance and no common parameter; what they share is one number, and the theorem needs nothing else.

One formula, and it does not ask what the shape is

Kutta–Joukowski gives the lift of any two-dimensional body from one number. Six bodies with nothing else in common are put at that number here and come out with identical lifts — and with pitching moments, load distributions and suction peaks that are not even close.

circulation · Lift Two camber lines, one lift and one moment. A NACA 2412 mean line and the same line with a fourth harmonic added to its slope. They are eight tenths of a per cent of the chord apart, which is forty per cent of the section's own camber, and they have the same lift and the same pitching moment at every incidence — to the last bit of double precision.

Three numbers out of a camber line

Thin-aerofoil theory takes a whole function and returns a lift and a moment. Only three coefficients of that function survive: two camber lines matched in the first three, and eight tenths of a per cent of chord apart, have the same lift and the same moment at every incidence and load distributions thirty-seven per cent apart.

circulation · Thin-aerofoil One bowl, and a line across it for every way of flying. The least induced drag of a wing, a canard and a tail together, as a function of the canard's and the tail's shares of the lift, drawn as rings of equal drag about the bowl's foot, where each carries 1.9 per cent. A static margin of a tenth of a chord and a wing pitching moment draw a straight line of trimmed splits across the bowl. The tail aircraft trims where its line crosses the axis of zero canard load; the three-surface aircraft slides along the same line to the point nearest the foot.

A third surface is worth a square

A canard and a tail each leave an aircraft one free share of its lift, and the static margin spends it. Give an aircraft both and one share stays free after trim, so the split can slide along a line to the point of least induced drag. What that slide is worth turns out to be a square: half the bowl's curvature times the square of how far the tail aircraft already trims from twice the bowl's foot. For a cruising wing the saving is under one per cent and less than the canard's own skin friction; it pays only for a wing whose pitching moment is as large as a flapped section's.

circulation · Moment No elevon acts through more than a quarter of the chord. The distance behind the quarter chord at which the load an elevon adds acts — its arm — against the elevon's share of the chord. A vanishing tab at the trailing edge has an arm of exactly a quarter chord; a tenth-chord elevon 0.217, a fifth 0.185, a third 0.145, half 0.0972, and a flap of the whole chord, which is simply a change of incidence, none. However far aft the hinge, the deflection loads the whole chord and most of the load sits near the leading edge.

No elevon reaches past a quarter chord

A wing with no tail trims itself with the back of its own section, and the back of a section is a short lever. Thin-aerofoil theory prices it exactly: the load an elevon adds acts at most a quarter of a chord behind the quarter-chord point, however small and far aft the elevon is. So every unit of trim moment costs at least four units of lift, where a tail three chords back costs a third, and a camber change shaped as a pure couple would cost nothing.

circulation · Trim The camber lines that carry them. The camber lines that carry the four loads at their ideal angles, heights in chords at a design lift coefficient of one. The uniform load's line is symmetric about mid-chord, 5.52% high; tapering the load over the last fifth moves the peak to 0.515 of the chord and raises it to 6.79%; tapering it over the whole chord, a = 0, puts the peak at 0.323.

A load carried to the trailing edge needs a hook

Thin-aerofoil theory runs backwards: ask for a chordwise load and it returns the camber line that carries it. Ask for the simplest load of all, the same everywhere, and the line comes back with a vertical tangent at both ends. The one at the nose is the price of a clean entry; the one at the tail is the Kutta condition being broken, and the hook it makes is what a Gurney flap is. Every NACA a-series line is a way of not paying it.

circulation · Thin-aerofoil 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

Named alongside it

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

Model limitAerodynamic centreThin-aerofoil theoryStatic marginCamberCirculationInduced dragOptimisationTrimAeroelasticityLiftLift coefficient

All concepts