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

Images — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Flow net — a free vortex. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

    A wall made by reflection

    Imposing a boundary condition on a plane is work. Putting a mirrored copy of everything on the far side of where the plane would be is not, and the plane then appears on its own — as a consequence of the symmetry rather than as a condition anybody enforced.

    part 1 · inviscid
  2. A wing at h/c = 0.4. A wing close to the ground, with the mirrored vortex row that makes the ground a streamline drawn faintly beneath it. The air between wing and ground is squeezed through a narrowing gap, and the wing carries more lift than it would in free air.

    The wall that pushes back

    A wing near the ground carries more lift than the same wing in free air, and the usual explanation — a cushion of compressed air underneath — is not what the equations say. The ground is a mirror, and what changes is not the pressure under the wing but the circulation the wing is forced to carry.

    part 2 · circulation
  3. Flow net — a stream past a cylinder. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

    The mirror that is a circle

    A flat wall is made by reflecting everything in it. A round one is made the same way, except that the mirror is an inversion — the image of a point at distance d sits at a²/d, and a vortex acquires a second image at the centre that nothing about the wall requires.

    part 3 · inviscid
  4. The image system of a wedge of pi/3. The vortex and its images, with the two walls. Reflection in one wall and then the other generates a dihedral group, and the group is finite exactly when the angle is pi over a whole number — here 2n vortices, alternating in sign round the circle, with both walls streamlines to a part in 10¹⁶.

    The corners that can be done with mirrors

    The method of images works for a wall and for a circle, and for a corner it works only when the angle is pi over a whole number. At every other angle the reflections never come back, the image set is infinite and dense, and the flow exists anyway — which says the method is a statement about symmetry rather than about fluids.

    part 4 · inviscid

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