Concept

Slender body — where it appears

A body long compared with its width, whose disturbance to a flow can be represented by a line of sources along its axis. That makes its supersonic wave drag a functional of the cross-sectional area distribution alone, with nothing else about its shape entering.

Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.

One signature, aged four times. The pressure signature of a slender body at four distances, computed by the exact Lax formula for the nonlinear propagation. Each point of the waveform moves forward in proportion to its own overpressure, so the compression at the front catches the undisturbed air and a shock forms there, while the expansion at the rear falls behind and forms a second one. What is left is an N-wave: two discontinuities and a straight line between them, spreading and weakening.

The signature that forgets the shape

The pressure field near a supersonic aeroplane depends on every part of it. What reaches the ground has two parameters. Two bodies whose near-field signatures differ by fifty-five per cent in peak and by their whole shape age into the same N-wave, to two and a half per cent.

compressible · Sonic boom

Four bodies of identical length and volume, and their wave drags. Each body has the same length and the same volume; only the distribution of area along it differs. The Sears–Haack body — the spindle whose area goes as the three-halves power of x(L−x) — has the least wave drag of the four, and every other shape pays between thirty-seven per cent and a hundred and seventy-five per cent more for carrying the same volume the same distance. Nothing about the cross-sections' shape enters: only the area distribution does.

The least drag a volume can have

A body's supersonic wave drag depends on nothing about it except how its cross-sectional area is distributed along its length. Minimising that for a given volume gives one shape — and the answer goes as the volume squared over the fourth power of the length.

compressible · Wave drag

Three exponents for one dimensionless group. The local slope of each error, measured over one decade at a time. The duct's is exactly 1, the long wave's is exactly 2, and the slender body's drifts from 1.900 to 1.733 across the range and never reaches either. The same geometric ratio, in three problems that look alike, and the third one has no exponent at all.

One group, three exponents

Lubrication theory, shallow water and slender-body theory are taught in three places and are one expansion in one group — a ratio of two lengths, with no speed, no viscosity and no fluid in it at all. The error is supposed to be second order. In three problems that look alike it is first order, second order, and an exponent that does not exist.

regimes · Slenderness

Two theories, one composite, and the aspect ratio between them. The lift-curve slope against aspect ratio. Prandtl's lifting line is exact as the aspect ratio goes to infinity and Jones's slender-wing theory is exact as it goes to zero, and each is generous outside its own limit. Helmbold's formula reduces to both with no free constant, which is what a composite expansion is, and runs under them where they disagree.

Where the line stops being a line

Prandtl's lifting line replaces a wing with a single bound vortex and its trailing sheet, and the formula that comes out is the most quoted in low-speed aerodynamics. Solved numerically at aspect ratio one it returns its own closed form to sixteen decimals — and the answer is forty-one per cent too high.

circulation · Finite wing

The only term that turns spin into thrust, and what it is made of. The coupling term of the propulsion matrix against the drag anisotropy, with the geometry held fixed. It is exactly proportional to the difference of the two drag coefficients, so it is zero when they are equal — not small, zero — and no helix of any pitch turned at any rate would move. A real filament sits at 1.66, which is a third of the way from useless to the unreachable limit.

Two drags, or nothing swims

A bacterium turns a corkscrew and goes forward, and the reason is not the corkscrew. It is that a thin filament dragged broadside resists more than the same filament dragged end-on. Make the two resistances equal and the thrust is not small but exactly zero, for every pitch and every rate.

viscous · Swimming

The same thickness, drawn as a body of revolution and as a wing section. The surface speed along three prolate spheroids of thickness ratio a half, a quarter and a tenth, and along the plane ellipses of the same ratios, in units of the stream speed. Each plane section speeds the flow over its middle by exactly its own thickness ratio. The body of revolution of the same ratio does so by much less — at a tenth, by two per cent where the ellipse manages ten.

Thickness costs a fuselage far less than a wing

A wing section a tenth as thick as it is long speeds the air over its middle by ten per cent. A body of revolution with the same proportions speeds it by two. The difference is not a detail of shape: a plane section pays for its thickness linearly and a body of revolution pays quadratically, and that one exponent is why a fuselage reaches the speed of sound on its surface long after its wing.

inviscid · Axisymmetric $A waist moves the fuselage's overspeed off the root chord. The fuselage's axial perturbation velocity at its own surface, as a fraction of flight speed, along its length at Mach 0.8, with the wing's root chord between the two rules: the plain fuselage, which adds 0.0251 everywhere along the chord; a narrow waist 6.2 per cent deep that cancels it at mid-chord and raises it at the chord's ends; and a wide one 31 per cent deep that leaves none of it positive along the chord and lifts it on the body ahead of and behind the wing.$

A waist moves the overspeed and cannot remove it

A fuselage adds two per cent of overspeed at the wing root, and a waist — a local narrowing of the body where the wing joins it — is the obvious cure. Slender-body theory says exactly what a waist can do. Its sources and sinks sum to nothing, so the velocity it adds along the body integrates to zero: it cannot remove the overspeed, only move it. Moved off the whole root chord it needs a third of the fuselage's cross-section at the wing, a sixth of what the transonic area rule would take, and it lands on the fuselage just ahead of and behind the wing, where there is room for it.

inviscid · Axisymmetric

Named alongside it

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

Model limit Area rule Added mass Asymptotics Compressibility Critical mach Measurement Prandtl–Glauert correction Sears haack Sonic boom Superposition Aeroacoustics

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