Acceleration — the series
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Steady does not mean nothing is happening
Photograph the flow past a cylinder twice and the two pictures are identical. Every parcel of air in them is being thrown about — braked to a dead stop, hauled round the shoulder at nearly twice the free-stream speed, braked again. Both statements are exactly true.
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How long the fluid has been in there
Age is the simplest thing a flow can remember. It obeys the shortest transport equation in the subject — its material derivative is one — and no instrument pointed at a steady flow can read it, because a steady flow's every field is constant and its fluid is getting older all the time.
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A parcel goes straight while the streamlines curve
In a steady flow every parcel accelerates while the picture never changes. The opposite also happens: a flow whose picture changes all the time while no parcel accelerates at all, because the local and convective halves of the acceleration cancel exactly. Such a flow has no pressure gradient anywhere, and that is a severe demand. An incompressible flow can meet it only as a pure shear at every point; a cloud that can compress meets it freely and pays later, when its straight paths cross in a caustic.
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Drag decides whether a cloud folds, and the air decides at what
A cloud of free particles keeps its velocities, and wherever it converges its paths cross in finite time: a caustic, the density infinite along a sheet. Give each particle a drag on the air and the answer depends on what the air is doing. In still air the drag only brakes, and a cloud folds only if its Stokes number is above one. In air that is itself converging the drag keeps pushing, and a quarter is enough — the same quarter that decides whether a droplet hits a wing.
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A lattice of vortices folds a cloud at the saddles' quarter
A cloud of heavy particles folds in still air above a Stokes number of one, and in converging air above a quarter. Air that turns as well as converges might have settled between them. In a lattice of vortices it settles exactly on the quarter: the vortices fling the particles to the cell walls, but only the saddles where the walls meet can make them cross. Below the quarter nothing folds, and the cloud is gathered onto the walls without limit instead.
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Wobbling saddles keep the quarter, and raise it
Heavy particles in a steady lattice of vortices fold onto the cell walls only above a Stokes number of a quarter, the converging saddles' own threshold. Random flows fold at every Stokes number, so it was natural to suspect the quarter of belonging to steady saddles. Set the lattice wobbling and the particles cross from cell to cell, seventy per cent of them in forty time units, and still none folds below a quarter. The threshold belongs to the strongest strain the flow has; moving the saddles only moves particles out of reach of it, and raises the threshold.
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A stronger strain folds a cloud only if it lasts
Heavy particles in a lattice of vortices fold onto the walls above a Stokes number of a quarter of the saddles' strain. Let the strain pulse and the threshold moves — down to the quarter of the peak strain when the pulse is slow, back up to the quarter of the mean when it is fast — because a particle responds to the strain it has been in for about four strain times, not to the strain at the instant. In a random strain the same rule makes the folds arrive in bursts, at the excursions that are strong and long at once.
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A cloud folds where a saddle is strong, not when
When every saddle of a lattice strengthens and weakens together, a cloud of heavy particles folds in one step, at the strain's first strong excursion. Give each saddle its own random strength and the folding spreads out into a steady rate — and falls, because a particle now has to be at a saddle while that saddle is strong. The rate follows the Gaussian tail of the strain, at about a tenth of the time the strain spends past its fold condition, and a longer-lived strain helps only up to a few strain times.