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

Coherent structures — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Q and the vorticity along a radius of one vortex. Two candidate measures of where the vortex is, along a radius of a Lamb–Oseen vortex. The vorticity is a Gaussian: positive at every radius, so a threshold on it puts the edge wherever the threshold is put. Q — the excess of rotation over strain — changes sign exactly once, at 1.121 core radii, and that radius is a property of the flow rather than of the person drawing it. Inside it, 71.5 per cent of the circulation.

    Where a vortex stops

    Four criteria decide where a vortex ends, and in two dimensions three of them are the same criterion. The fourth is a knob. And the one that is not a knob is not objective: a co-rotating pair of vortices occupies four per cent of a window to one observer and more than half of it to another.

    part 1 · kinematics
  2. The stretching a window of history did, drawn as a field. The largest finite-time Lyapunov exponent over eight units of time in the double gyre, darkest where two neighbouring parcels were pulled furthest apart. The bright crest is a curve across the domain, and it is a property of the eight units rather than of any instant inside them.

    A boundary that only exists over a window

    The curve that separates fluid going one way from fluid going another is not in any snapshot of the flow. It is the crest of a field built from a stretch of history, it moves when the stretch is changed, and reversing the direction of time gives a different curve entirely — both of them real.

    part 2 · kinematics
  3. The two observers disagree twelvefold about where Q says the vortices are, and not at all about the LAVD edge. The Q = 0 contours and the outermost convex LAVD level over one orbit, in the laboratory and in the frame turning with the pair. Q > 0 covers 2.00 of the window in the laboratory and 24.94 in the turning frame; the LAVD edges enclose 6.29 in the laboratory and 6.29 in the turning frame, and the two LAVD fields differ nowhere by more than 8.0e-8 of their peak.

    An edge no observer can move still depends on what it is averaged against

    The Q-criterion puts a co-rotating vortex pair in four per cent of a window for one observer and in more than half of it for another. The cure is to average, along each fluid path, how far the vorticity departs from its mean — a number no observer's rotation can change — and take the outermost convex level of it round each core. Computed, it works: both observers find two discs of radius 1.00, twice the core radius, and no fluid inside them strays from its core in an orbit. But the edge has a setting of its own. The mean must be taken over some region, and as that region grows from a radius of 4 to 20 the edge moves from 0.90 to 1.13.

    part 3 · kinematics

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