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

Slender body theory — where it appears

An approximation for a body long compared with its width, in which the disturbance it makes is replaced by a line of sources along its axis. Only the distribution of cross-sectional area survives the approximation, so two bodies with the same area distribution disturb the flow identically.

Named by 3 essays across one field — 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 Three aircraft, and the only thing the drag can see. The cross-sectional area against station, for a clean body, the same body with a wing added, and the body cut away where the wing is. The wave drag depends on this curve and on nothing else about the three shapes — so the middle one pays for its lump and the third does not, although the third has the same wing.

The drag that can only see one curve

A slender body's far field is a line of sources of strength A′(x), and nothing else about its shape reaches it. So its supersonic wave drag depends on the cross-sectional area distribution alone — and a fuselage cut away where the wing joins it can carry the wing for nothing.

inviscid · Slender body The free waist is flat-bottomed, and follows the sweep aft. The cross-section each waist removes along the body, as a fraction of the fuselage's greatest, for the root strip unswept, at 30° and at 45°. The linear programme's waists have flat bottoms and steep sides: depth is spent only where the strip needs it. With sweep they lengthen aft rather than deepen — 24.8, 24.3 and 26.9 per cent. The best single dents, dashed, must widen and deepen instead: 39.8, 45.7 and 52.6 per cent.

A free waist needs half the depth, and the sweep costs it nothing

A cosine-squared dent in a fuselage cancels its overspeed at an unswept wing's root for about a third of the fuselage's cross-section, and nothing can remove the overspeed, only move it. Let a linear programme choose the waist's shape instead, and half of that third turns out to have been the dent's fault. The free waist is flat-bottomed and steep-sided, and for a swept root it simply runs aft with the chord, where a single dent has to widen and deepen.

inviscid · Axisymmetric

Named alongside it

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

Area ruleAxisymmetricInviscidModel limitOptimisationSuperpositionAdded massApproximation errorBoundary conditionCritical machd'Alembert's paradoxDimensionless

All concepts