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

Phase speed — where it appears

The speed at which the crests of a wave travel, its frequency divided by its wavenumber. In a stability problem it is the real part of a complex eigenvalue whose imaginary part is the growth, and where it equals the local flow speed it marks a critical layer.

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

Also named here as rayleigh equation — the same set of essays touches all of them, so they are one junction rather than several.

Every unstable mode of every profile, inside one circle. The complex phase speeds of the unstable modes, scaled so that each profile's own semicircle is the unit one. Howard's theorem says every one of them must lie inside — the centre and the radius are the mean and half-range of the velocity profile and nothing else — and every one of them does, with the closest approach at 0.915 of the radius.

Every unstable wave is inside one circle

Before solving anything, you know where the answer is. Howard's theorem says the complex phase speed of any growing disturbance in a shear flow lies inside a circle fixed by the fastest and slowest parts of the profile — and by nothing else about it at all.

turbulence · Instability A band of growing waves that opens at 5772 and narrows as the viscosity goes. The wavenumbers at which a two-dimensional wave on plane Poiseuille flow neither grows nor decays, against the Reynolds number on a logarithmic axis. Inside the tongue waves grow; outside they decay. The tongue's tip is the critical point. Both edges slope downward and towards each other in wavenumber as the Reynolds number rises, so the band of unstable waves shrinks towards long waves — the direction in which the inviscid problem, which has no growing wave at all, is reached.

The profile Rayleigh cleared and viscosity did not

Flow between two plates has no inflection point, so without viscosity no wave on it can grow. With viscosity one does, above a Reynolds number of 5772. Taking the viscosity away again slows that wave and narrows the band it grows in, because the stress that feeds it is made by viscosity in the first place.

turbulence · Instability

Named alongside it

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

EigenvalueInflection pointLinear stabilityRayleigh equationBoundConvergenceCritical layerDissipationGalilean invarianceMeasurementMixingReynolds number

All concepts