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

Trim — where it appears

The condition in which an aircraft's forces and its moments both balance, so that it holds its attitude as well as its flight path. It is a second equation beside lift equals weight, and it decides how the load is shared between the wing and the tail — usually with the tail carrying down and the wing carrying more than the weight.

Named by 5 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 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 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 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

Named alongside it

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

Model limitPitching momentAerodynamic centreDownwashInduced dragNeutral pointStatic marginCanardMunk staggerOptimisationTail volumeThe Trefftz plane

All concepts