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

Radiation condition — where it appears

The requirement that waves generated by a body travel away from it rather than towards it, imposed because the steady problem alone does not distinguish the two. It is what puts a ship's waves behind it and it is the reason a wave-resistance calculation has an answer.

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

Each cancellation costs two powers of the Mach number. Radiated power against compactness for three source clusters: a single monopole, two of opposite sign, and four on a square with alternating signs. The fitted slopes are 0.00, 2.00, 4.00 — zero, two and four in (kd), measured by integrating the far field over a sphere rather than assumed. A turbulent eddy turns over in about the time sound crosses it, so kd is of order the Mach number, and those exponents become the fourth, sixth and eighth powers of speed. A flow with no moving surfaces has no monopole and no dipole available to it, which is Lighthill's whole argument, and the eighth power is what is left.

The sound that only leaves

A flow is a catastrophically bad radiator, and the reason is that it has no monopole and no dipole available to it. What is left is the eighth power of speed — and the equation is equally happy with sound converging on a jet, which is ruled out by a condition imposed at infinity.

compressible · Aeroacoustics 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.

Acoustic analogyAeroacousticsBoundary conditionCorrelationd'Alembert's paradoxDispersionDissipationDopplerDoubletDragEfficiencyEnergy flux

All concepts