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

Stratification — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A cross at 30.0° for every wavelength it makes. A body oscillating at ω = 0.5N in a stratified fluid, and the four beams along which its energy leaves. The angle is arccos(ω/N) from the vertical — 30.0 degrees from the horizontal here — and it is the same for every wavelength the body excites, because the dispersion relation has no length in it. The short strokes are the crests, which lie along the beams rather than across them: the phase advances perpendicular to the energy, and the two are exactly at right angles. Raise the frequency and the cross closes towards the vertical; reach ω = N and it shuts entirely, because nothing above the buoyancy frequency propagates.

    The number that stops the mixing

    A fluid whose density falls with height resists being stirred, and the resistance has a threshold at exactly one quarter, from an energy balance with no fluid mechanics in it. The waves such a fluid carries are stranger still — their frequency decides the direction they travel in and says nothing about their wavelength.

    part 1 · turbulence
  2. Five numbers, one flow, one name. A hyperbolic-tangent shear layer with a matching density profile, measured five ways, each of which appears in the literature as "the Richardson number". The minimum gradient value is what Miles' and Howard's theorem is about and is the only one the quarter belongs to. The bulk numbers depend on which thickness and which velocity difference were used; the depth-averaged one depends on how far from the layer the measurement extended, and grows without limit as it extends further.

    Five numbers, one name

    The Richardson number has a threshold at a quarter, and the quarter belongs to one of the five quantities that go by the name. On a single tanh shear layer they run from J to seventeen J, and the largest of them grows without limit as the measurement is extended further from the layer.

    part 2 · turbulence
  3. Instability up to a quarter, and none past it. The fastest growth rate of a stratified shear layer against its Richardson number, on a profile whose gradient Richardson number is the same at every height. It falls smoothly towards zero and reaches it at a quarter: at Ri = 0.2499 the fastest mode still grows at 0.00106, and at 0.26 the solver finds no unstable mode at all.

    Sufficient, and not necessary

    A stratified shear layer whose Richardson number exceeds a quarter everywhere cannot go unstable. That is a theorem with an exact number in it. What it does not say — and what it is constantly read as saying — is that a layer below a quarter will.

    part 3 · turbulence
  4. A thin interface stays unstable far past a quarter. The fastest growth rate of any disturbance against the bulk Richardson number J, on a logarithmic scale, for a density interface as thick as the shear and two, two and a half and three times thinner. The matched layer's billow dies at a quarter, as Miles' theorem requires. Two times thinner, the billow dies sooner, its last growth at J = 0.12, and nothing replaces it. Three times thinner, the billow's last is at 0.08 and a travelling wave takes over: 0.0335 at a quarter, 0.0109 at one and 0.00629 at 1.3, falling steadily with no threshold in the range solved. At 2.5 the waves are weaker and reach 0.001 at 1.3.

    A thin interface keeps its waves past a quarter

    Miles' quarter rules a shear layer whose density changes over the same depth as its velocity. Make the density interface three times thinner and the stationary billow dies early, but a pair of travelling waves takes its place and is still growing at five times the quarter. No theorem is broken: at the edges of the shear, where the waves draw their energy, the local Richardson number has fallen to nothing.

    part 4 · turbulence
  5. A free interface's thickness ratio heads to the root of its Prandtl number. The ratio of the shear layer's thickness to the density interface's against time, for an interface starting four times thinner than its shear, heat and salt in water. A free shear layer thickens by viscosity faster than the interface by conduction, so the ratio heads to √Pr: 2.64 for heat, 25.8 for salt. Held at its thickness, the shear lets the ratio fall through 2 — at about 50 s for heat and 4725 s for salt.

    What holds the shear decides whether the waves last

    Holmboe's travelling waves need a density interface about half as thick as the shear around it, and diffusion thickens both. If the shear layer is held at its thickness, the interface catches up — in under a minute for heat in a laboratory tank, over an hour for salt — and the waves get two e-folds or a hundred and seventy. If the shear layer is free it thickens faster than the interface, the ratio heads to the root of the Prandtl number, and the waves last until the Richardson number ends them instead.

    part 5 · turbulence

All series