Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Instability — the series

6 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
  5. The drip spacing is set by the depth of the layer. The spacing of the fastest-growing wave, in units of 2π capillary lengths, against the depth of the hanging layer in capillary lengths, for water, glycerol and honey. A thin film of any of them drips at √2. Deep water drips at √3. Deep glycerol and honey keep growing past √3, their fastest wave as long as the layer is deep, because viscosity slows short waves more than long ones.

    How far apart a ceiling drips

    A layer of liquid hanging from a ceiling is heavy fluid over light, and every ripple on it longer than about seventeen millimetres grows. Which ripple grows fastest, and so how far apart the drips form, is usually given as one number. It is at least three, and what chooses between them is not the liquid's surface tension but the depth of the layer.

    part 5 · turbulence
  6. Whether a sloping ceiling drips in place is one ray's growth. How fast a disturbance grows as seen by an observer moving along the ceiling at speed v, for four speeds at which the film carries its disturbances downhill, in the film's own units. On a flat ceiling (V = 0) the growth peaks at a quarter for the observer standing still and falls to zero for one running at ±1.622. Tilting the ceiling slides the whole curve downhill. While the observer at the point disturbed, v = 0, still sees growth, the ceiling drips where the disturbance began; at V = 1.622 that observer sees none, and above it the disturbance grows only while it is carried away.

    A sloping ceiling drips downhill, or not at all

    A film hanging from a level ceiling drips where its ripples form. Tilt the ceiling and the film flows downhill, carrying its ripples with it, and at some slope they are carried away faster than they spread and the ceiling stops dripping in place. That slope is set by one number, 1.622 — the speed at which a level ceiling's disturbance spreads — and for water it is tiny: a degree for a tenth of a millimetre of film, six for half a millimetre. Past it the ripples still grow, and still drip, but downhill, at a distance that grows with the slope: a ceiling shorter than that delivers its water to the edge.

    part 6 · turbulence

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