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

Instability — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A shear layer, and the point of inflection in it. The velocity profile U = tanh y across a layer of finite thickness, with the inflection point located by searching for a sign change in the second derivative rather than by reading it off the algebra. Rayleigh's theorem says an inviscid parallel flow can only be unstable if such a point exists — a necessary condition, not a sufficient one.

    A layer with a kink in it

    Rayleigh proved in 1880 that an inviscid shear flow cannot be unstable unless its velocity profile bends the other way somewhere. It is one line of algebra, it is necessary and not sufficient, and it ties instability to separation through the sign of a single derivative at the wall.

    part 1 · turbulence
  2. Every wavelength grows, and the shortest grows fastest. Growth rate against wavenumber for a vortex sheet of zero thickness and for a layer of thickness δ. The sheet's curve rises without limit, which is the model announcing that it has no shortest scale in it; the layer's turns over and dies at kδ = 1, the neutral wavenumber computed from Rayleigh's equation rather than fitted.

    Every wavelength at once

    A vortex sheet of zero thickness is unstable at every wavelength, and the shorter the wavelength the faster it grows. The answer has no smallest scale in it, which is not a fact about fluids — it is the model reporting that it left something out.

    part 2 · turbulence
  3. 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.

    part 3 · turbulence
  4. 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.

    part 4 · turbulence

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