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 2 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 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

Named alongside it

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

Boundary conditionAdded massApproximation errord'Alembert's paradoxDivergenceDoubletFlow visualisationHelmholtz decompositionInviscidLiftMass conservationModel limit

All concepts