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

Kelvin — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Wound up, and worth exactly what it started with. A material loop in a steady cellular flow — an exact solution of Euler's equations — drawn at four times. Each streamline in the cell has its own period, so the loop is stretched steadily into a spiral: by the last frame its perimeter is 5.7 times what it started as. The circulation round it is 0.903741 at the start and 0.903666 at the end. Nothing about the curve survives except the number.

    What survives being wound up

    Draw a loop of marked fluid particles and let the flow carry it. It will be stretched, folded and wound into a spiral until nothing about its shape is recognisable, and the circulation round it will not have moved at all — provided three conditions hold, each of which can be broken on purpose.

    part 1 · inviscid
  2. One material loop, at five stages of being drawn out. A circle of fluid particles carried by four point vortices, drawn at equal intervals over fourteen time units. Its length grows by a factor of six and its shape becomes unrecognisable; the circulation round it does not move at all.

    The drift was the instrument

    Kelvin's theorem was checked on this site by carrying a loop and watching its circulation move by eight parts in a hundred thousand. That drift is not the flow forgetting. The exact number is a count of what is inside the loop, it does not move by anything at all, and the drift belongs entirely to the two instruments used to measure it.

    part 2 · inviscid
  3. Over a day the breeze turns right round. The tip of the surface wind vector over one day, onshore to the right and along the coast upwards, at 15°, 30° and 45° north, with a spin-down time of twelve hours, in units of the push over the daily frequency. Without rotation the wind would swing on and off shore along one line. The Coriolis force turns it clockwise through the day into an ellipse; at 30°, where the Earth's inertial period is exactly a day, the turning keeps pace with the push and the ellipse is nearly a circle, of radius 1.8 against the ellipses' 1.2 at 15° and 1.2 at 45°.

    At thirty degrees the Earth keeps time with the sea breeze

    A sea breeze is Kelvin's circulation theorem failing: where warm air over land meets cool air over the sea, pressure and density surfaces cross and a circulation grows out of still air. Bjerknes' theorem prices the push, and the textbook's numbers give twenty-six metres a second in an hour — far more than any sea breeze blows. What stops it first is not friction but the evening: the push reverses before the wind can grow, and the day, not the drag, sets the breeze's size. Friction sets its hour, the Earth's rotation turns it through the day, and at thirty degrees, where the inertial period is a day, the turning keeps time with the push.

    part 3 · inviscid
  4. At the equator the front runs as long as the land is warmer; at forty-five degrees it stops hours before. The sea-breeze front's distance inland against the hours since the land first became warmer than the sea, at the equator and at 30°, 45° and 60°. At 0° it reaches 185 km. At 30° it reaches 133 km, 99 per cent of that by hour 11. At 45° it reaches 99.2 km, 99 per cent of that by hour 9.1. At 60° it reaches 80.3 km, 99 per cent of that by hour 8. The quasi-steady estimate, the head speed of a current fed from a deep reservoir at each hour's contrast, integrated, is 189 km.

    The sea-breeze front runs till evening only near the equator

    A real sea breeze arrives: the cool sea air runs inland under the warm air as a gravity current, its head moving at a speed set by the density contrast and the depth of the cool layer. How far it gets depends on the latitude. Near the equator the head keeps running as long as the land stays warmer, and the length of the afternoon decides the reach. In mid-latitudes the Earth's rotation turns the current along the coast and stops the head hours before the contrast ends, a third further inland than the Rossby radius, and a longer afternoon adds nothing.

    part 4 · inviscid

All series