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

Vorticity — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The velocity field, arrows to scale. The same flow drawn as arrows. Scaled to the local speed the picture is honest and crowded; drawn all the same length it is legible and hides the very variation the figure is about.

    Spin is not the same as going round

    A whirlpool whose streamlines are perfect circles can have no rotation in it anywhere. A flow whose streamlines are dead straight and parallel can be rotating everywhere. Both statements are true, and getting them the wrong way round is the most expensive confusion in the subject.

    part 1 · kinematics
  2. Stokes' theorem on a solved wake at Re 40. A rectangle drawn in a viscous flow that was solved on a grid. The circulation round its boundary is computed by walking the four sides and adding up the velocity along them; the vorticity inside it is computed by adding up the stored vorticity cell by cell. The two computations share no sample point and the theorem says they must agree.

    Circulation is vorticity, added up

    One of these two quantities is measured by walking round a loop and one by summing over the area inside it, and a theorem says they are the same number. That reconciles the site's most confusable pair — and explains how a flow with circulation can have no spin in it anywhere.

    part 2 · kinematics
  3. The one place the stretching argument closes. Burgers' vortex: an axisymmetric strain carrying vorticity inwards at exactly the rate viscosity spreads it outwards. The vorticity profile is a Gaussian of radius √(4ν/α), the swirl velocity peaks at 1.12 core radii rather than at the core radius itself, and the circulation reaches its full value by about two. The steady vorticity equation is evaluated on this profile by differencing it, not by re-deriving it.

    The spin that feeds itself

    Stretch a vortex tube and its spin rises in exact proportion, because the circulation round it cannot change and its area has fallen. Nothing in that argument sets a limit — and the one flow where the limit can be written down exactly puts it at a length of √(4ν/α).

    part 3 · kinematics
  4. The wall makes vorticity at a rate with no viscosity in it. At a stationary wall the momentum equation collapses to ν ∂²u/∂y² = (1/ρ) ∂p/∂x, and the left-hand side is the diffusive flux of vorticity out of the surface. So the pressure gradient along the wall is the vorticity source, and the viscosity that made the no-slip condition necessary has cancelled out of what the condition produces. The curve is that flux across the Falkner–Skan family, computed from profiles solved by shooting and differenced at the wall; the straight line is the pressure gradient each of those flows has. They agree to 2.3e-14. At zero pressure gradient the flux is exactly zero: a flat plate creates no vorticity at all after its leading edge, and everything in its layer arrived from there.

    Where vorticity comes from

    Every scrap of vorticity in a flow past a body entered through its surface, and the rate at which it enters contains no viscosity at all — it is the pressure gradient along the wall. A flat plate makes none, and a closed body makes exactly as much of each sign.

    part 4 · kinematics
  5. A spun cylinder carries its whole circulation at once, and hides it until the vorticity has left. The circulation round circles of radius r about a cylinder of radius a started spinning at once, as a share of the circulation of its own surface, 2πa²Ω, against r/a on a logarithmic axis, at νt/a² = 0.01, 0.1, 1, 10 and 100. At the surface it is the whole of it from the first instant, because no slip makes the fluid there turn with the cylinder. Just outside, the spin-up has laid down an equal and opposite ring of vorticity, so the circulation round a larger circle is only what has diffused past it: at two radii 0.000, 0.035, 0.611, 0.936 and 0.993 of the surface's at the same five times. The circulation a Magnus rotor needs is in the fluid the moment it spins; the far field learns of it only as fast as the counter-vorticity moves out.

    A wall puts in exactly its own speed

    A wall sliding in its own plane makes vorticity at a rate equal to its acceleration, with no viscosity in the rate. So however a wall is started, the vorticity it has put into the fluid is its speed, to the last digit; a wall that stops takes all of it back and leaves the fluid moving; and a spinning cylinder carries its whole circulation from the first instant, hidden behind an equal and opposite ring until viscosity carries the ring away.

    part 5 · kinematics

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