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

Tail volume — where it appears

The tail's area times its arm, divided by the wing's area times its chord — the one group that decides how much stabilising moment a tail supplies. It is why a small surface far back does the work of a large one close in.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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

Named alongside it

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

Pitching momentStatic marginAerodynamic centreDownwashLiftModel limitTrimAdded massApproximation errorAxisymmetricThe Biot–Savart lawBoundary condition

All concepts