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

Poisson equation — where it appears

The equation relating a field's Laplacian to a stated source, which appears whenever a quantity is set by a distribution rather than by a boundary alone. Pressure in an incompressible flow satisfies one, and so does the velocity in fully developed duct flow.

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

The pressure, relaxed rather than quoted. The pressure field round a cylinder, obtained by relaxing ∇²p = −ρ∇·(u·∇u) on a body-fitted polar lattice with the exact pressure on the surface and on a far circle. The interior was told nothing except the velocity gradients. It agrees with Bernoulli's closed form everywhere to under two parts in ten thousand of the dynamic pressure, which is the strongest statement this site can make that the elliptic equation is the pressure's own.

Pressure has no speed

Take the divergence of the momentum equation for an incompressible flow and the time derivative disappears, the viscosity disappears, and what is left is Poisson's equation. Pressure is not carried anywhere: it is whatever satisfies an elliptic equation everywhere at once, and that is a statement about a fluid nobody has.

inviscid · Pressure A factor of 1.8 between the extremes. The same numbers as a bar. The spread is exactly 96 over 160/3, which is 1.8 — and it is a spread in a quantity that a single correlation using the hydraulic diameter treats as a constant. A rectangle twenty times as wide as it is deep is 78 per cent more resistant than a circle of the same hydraulic diameter.

A number that is only the shape of the hole

The friction factor times the Reynolds number for laminar flow in a duct is a pure number that depends on the cross-section's shape and on nothing else — not the fluid, not the size, not the flow rate. It is exactly 64 for a circle and exactly 96 for a slot, and the correlation that treats them as the same is wrong by a third.

applied · Internal flow

Named alongside it

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

Bernoulli's equationBoundary conditionConvergenceDimensionlessDiscretisationEllipticFriction factorHydraulic diameterIncompressibleLaminarMach numberMeasurement

All concepts