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

Coanda — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Every curved streamline has a pressure gradient across it. Twelve points in the flow past a cylinder, with the arrow at each showing the pressure gradient across the streamline. It points away from the centre of curvature everywhere, and its size is ρq²κ: the fluid is being pushed round a bend, and something has to do the pushing. Both sides are computed here and they share no arithmetic — one is Bernoulli's pressure differenced across the flow, the other is the turning rate of the velocity direction along it — and they agree to 6.7e-5. This is the whole content of the effect usually named after Coandă, and it is happening on every curved streamline of every flow.

    The effect that explains nothing

    A jet follows a curved wall and the wall feels a suction. Both are real, both are famous, and naming them after Coandă explains neither — what is happening is the normal component of Euler's equation, and it holds on every curved streamline in every flow.

    part 1 · misconceptions
  2. Where the suction on a cylinder is actually made. The share of the surface's pressure deficit generated within a given distance of it. Twelve per cent comes from the first twentieth of a radius, half from within a third, and a tenth of it from beyond one and a half radii. There is no thin layer doing the work: the suction is made by the curvature of the whole outer field.

    The effect that is real, and where it stops

    The Coandă effect has a genuine mechanism, and it is the same equation that makes the suction on a wing. What separates the two is where the integral comes from: a wall jet generates all of its pressure deficit inside a layer two per cent of the radius thick, and a cylinder in a stream needs a third of a radius for half of it.

    part 2 · misconceptions
  3. A water sheet two millimetres thick leaves the lip above 0.270 m/s. The effective tension of a sheet of water, 2σ − ρU²h per metre of its width, against the speed it is poured at, for sheets one, two and four millimetres thick. At rest every sheet carries the tension of its two surfaces, 145.4 mN/m, and the momentum it carries along itself subtracts from that. A 1 mm sheet reaches zero at 0.382 m/s, a 2 mm sheet reaches zero at 0.270 m/s and a 4 mm sheet reaches zero at 0.191 m/s. Below its own crossing a sheet bent round a lip is pulled onto it; above, it is flung off. The quantity that decides is a tension, not a pressure, and the speed at which it vanishes is the speed of waves along the sheet.

    The teapot effect is a tension, not a pressure

    A slow pour runs back under a spout and down its outside, and the name it is usually given is the Coandă effect. The Coandă effect borrows the ambient pressure, and a liquid in air has none to borrow. A liquid sheet is held to a lip by its own surface tension, and it lets go at the one speed that tension cannot carry — the speed of waves along the sheet — whatever the lip's radius.

    part 3 · misconceptions
  4. One rim, one drop, every angle in the band. Water on a disc the size of a US cent, 19.1 mm across, pinned at its rim. The faint outlines are the drop at apparent angles of 60°, 90°, 120° and 150°; the dark one is the drop at the angle set, 120°, holding 1.13 mL. The rim stays put while the angle climbs, and every shape is an equilibrium: 0.52 mL at 60°, 1.44 mL at 150°. They are flattened by gravity into puddles with rounded edges, not spherical caps.

    An edge holds any angle it is given

    A liquid's contact angle is taught as a property of the liquid and the solid. On a smooth face it is. On a sharp edge the line stops, the angle is free to take any value in a band as wide as the edge is sharp, and that freedom is how a coin carries a dome of water and a glass stands full above its brim.

    part 4 · misconceptions
  5. An edge is worth its sharpness against the capillary length. The extra height a rounded edge holds over a flat plate, as a fraction of what a sharp edge holds, against the rounding radius in capillary lengths, for Young angles of 30°, 60° and 90°. An edge rounded to a thirtieth of a capillary length, about a tenth of a millimetre for water, keeps about ninety-five per cent. One rounded to a third of a millimetre keeps between eight and nine tenths; one rounded to a whole capillary length, about a third; and past that the worth falls as the inverse of the radius.

    A rounded edge spills before its line goes round

    Gibbs's band says a sharp edge lets a liquid stand at any angle across a range as wide as the edge's turn. No real edge is sharp, and on a rounded one the line does not stop; it slides round the curve, meeting it at the Young angle everywhere. The band survives the sliding. What the rounding costs is height, because the line drops as it goes round, and past a point the drop outweighs the steeper angle and the liquid spills before the line has reached the far face. An edge is worth its sharpness measured against the capillary length, and nothing smaller.

    part 5 · misconceptions

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