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

Linearisation — where it appears

The replacement of a nonlinear problem by its first-order approximation about a known state. It is exact in the limit of small disturbances and it is the source of most of what this subject can compute in closed form, including every stability result and every wave speed.

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

The correction belongs to the wing, not to the air. Lift-curve slope against Mach number for three aspect ratios, each computed by solving the lifting line for the wing the transformation actually implies — one of aspect ratio βAR — and dividing by β. The dashed curve is the two-dimensional rule, which is what an aerofoil section gets. At Mach 0.7 the aspect ratio of 20 has gained 35 per cent of slope and the aspect ratio of 4 has gained 24, against the 40 per cent the section rule promises both. The β in the finite-wing term cancels the β in front of it, so the shorter the wing the less compressibility does to it.

The wing the equation is really solving

The Prandtl–Glauert rule is usually quoted as a factor on the answer. It is a change of shape — and in three dimensions the shape it changes is the aspect ratio, so a short wing is far less affected by compressibility than a long one.

regimes · Similarity The free surface a submerged body leaves behind it, and the flat water in front. The linearised free-surface problem solved as a Fourier integral with a radiation condition. Behind the body a wave train of the wavelength that stands still relative to it, 2 pi U²/g; ahead of it, an amplitude a hundred and twenty times smaller. The asymmetry is the drag: an ideal fluid with a free surface can carry energy away.

The drag that is made of waves

D'Alembert's paradox says a body in a steady, irrotational, incompressible, inviscid flow feels no drag. Put a free surface above it and every one of those words still holds — and the drag is not zero. It is the energy walking away in the wave train behind.

inviscid · Dalembert

Named alongside it

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

Aspect ratioBoundary conditionCompressibilityCritical machd'Alembert's paradoxDispersionDoubletDragEnergy fluxFree surfaceFroude numberGroup velocity

All concepts