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

Neutral point — where it appears

The point along an aircraft's length about which its total pitching moment does not change with angle of attack, the aerodynamic centre of the whole configuration. An aircraft is statically stable in pitch when its centre of gravity lies ahead of it, and the distance between the two is the static margin.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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 The drag is set by the split, and stability sets the split. The least induced drag of a wing and a smaller surface together, as a multiple of the elliptic wing's alone, against the share of the lift the smaller surface carries. By Munk's stagger theorem the curve is the same whichever surface is in front. Its minimum, 0.9984, is at a share of 1.9 per cent. At a static margin of a tenth of a chord, a tail trims with 6.7 per cent of the weight on it, upwards, and pays 1.009; a canard trims with 19 per cent and pays 1.128.

A canard pays for its stability in induced drag

The argument for a canard is that both of its surfaces lift upwards, while a tail pushes down and makes the wing carry the difference. Munk's stagger theorem turns the question into arithmetic: two surfaces' least induced drag depends only on how the lift is split between them, not on which is in front. Static margin sets the split. At a margin of a tenth of a chord a tail carries a small upload and costs under one per cent; a canard must carry a fifth of the weight on a third of the span and costs twelve, and the more stable it is made, the more it pays.

circulation · Moment 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 Holding the neutral point with a canard ahead takes a tail two and a half times the size. The tail area, as a fraction of the wing's, that puts the neutral point where the tail aircraft with a tail of 0.2 has it, against the tail's height above the wing's plane in spans: with no canard, with the canard, and with the canard but its wake at the tail removed. At the reference height the tail aircraft needs 0.2; with the canard it needs 0.496, of which 0.22 is for the canard's own lift ahead of the centre of gravity and 0.077 for its wake at the tail. A tail 0.25 span up needs 0.354 with the canard and 0.154 without.

The tail a canard needs costs more than the canard saves

A canard added to an aircraft as a second trimming surface saves induced drag only if the neutral point is held where it was, and holding it is not free. The canard's own lift ahead of the centre of gravity pulls the neutral point forward, its wake reaches the tail and weakens it, and the tail has to grow to put the neutral point back. For a canard of a tenth of the wing's area the tail grows two and a half times, and its skin friction is twenty to thirty counts against a saving of one to twenty. A T-tail escapes most of the wake and a third of the bill, and still only a flapped wing comes near to paying.

circulation · Moment

Named alongside it

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

Model limitCanardDownwashInduced dragThe Trefftz planeTrimHorseshoe vortexMunk staggerPitching momentSkin frictionStatic stabilityTail volume

All concepts