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

Axisymmetric — where it appears

A flow whose properties are the same in every plane through one axis. The symmetry reduces three velocity components to two and admits a stream function, at the cost of saying nothing about swirl, which that scalar cannot represent.

Named by 5 essays across 3 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 Ideal flow past a sphere, drawn in a meridional plane. The Stokes stream function of ideal flow past a sphere, contoured at equal intervals. Its relations to the velocity carry factors of r sin θ that the plane stream function does not have, and differencing it reproduces the closed-form velocity to 10⁻¹¹. The surface speed at the equator is exactly one and a half times the free stream, against twice for a circular cylinder: a three-dimensional body lets the flow past in two directions rather than one.

The one number that runs out at three dimensions

A stream function is one function where the velocity is two, and four essays here are built on it. It exists because the divergence vanishes, it is single-valued only if nothing inside is making fluid, and in three dimensions it is not one function at all.

kinematics · Stream function 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 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 A round jet held further from the plate arrives slower in proportion and leaves a radial wall jet stronger in proportion. For the round water jet from a 2 mm nozzle, against the nozzle's distance from the plate, each over its value at 10 cm: the arriving centreline speed, the radial invariant F = Kνh/(2π) it hands on, and the radial wall jet's peak speed 5 cm from the stagnation point; dashed, a plane jet's speed and invariant for comparison. At 40 cm the round jet arrives at 0.25 of its speed at 10 cm and hands on 4 times the invariant, and its wall jet runs 2 times as fast; a plane jet's invariant would have grown 1.59 times.

A round jet hands its wall jet a strength proportional to its throw

A round jet striking a plate spreads into a radial wall jet, and like the plane jet before it, the wall jet keeps not the jet's momentum but Glauert's flux of momentum flux — the momentum the jet brought times the fluid it gathered on the way. A round jet gathers fluid at a rate set by the viscosity alone, eight π ν per metre of travel, whatever its strength. So the invariant it hands on grows in exact proportion to the distance it was thrown, where a plane jet's grew as the cube root: hold the nozzle four times further away and the wall jet is four times as strong and runs twice as fast.

viscous · Free shear

Named alongside it

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

Added massBoundary conditionCritical machModel limitPotential flowPrandtl–Glauert correctionSlender body theorySourceApproximation errorArea ruleAsymptoticsCompressibility

All concepts