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The thread: What is conserved — page 8

Page 8 of 33, continuing through the 292 essays this motif runs through.

292 essays carry this thread — page 8 of 33.

The street, as two rows of point vortices. The exact velocity field of a staggered double row of point vortices at the stable spacing ratio, with the row spacing and the circulation of one core printed from a line integral of the field rather than from the number that built it. This is a model of a wake. It contains no body, no viscosity and no mechanism that would shed anything, and the viscous stepper used here does not produce a street at any Reynolds number.

The street this site cannot draw

The alternating wake behind a cylinder is the most photographed structure in fluid mechanics, and this site's solver does not produce one. What can honestly be drawn instead is a model of it — and the model settles one thing exactly, which is the spacing.

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.

Flows and fields

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.

Three waves out of one discontinuity. The x–t diagram of the burst diaphragm. A shock runs right at a speed of its own, a contact surface follows it more slowly, and an expansion fan spreads left as a family of rays that opens with time — the three wave families the Euler equations possess, produced at once by an initial condition with no waves in it at all. Every straight line here is a speed the solution computed, and the fan is drawn as the rays it actually consists of.

Compressible flow

One diaphragm, every wave

Two states of the same gas at rest, separated by nothing, is the simplest initial condition compressible flow admits — and its answer contains all three waves the equations possess at once: a shock one way, an expansion fan the other, and between them a surface across which the density jumps and the pressure does not.

$Ninety microseconds, and most of it spent barely moving. The radius of a collapsing cavity against time, both as fractions of their own totals. The bubble spends most of the collapse near its original size and the last few per cent of the radius in the last fraction of a per cent of the time. The total is 91.47 microseconds for a millimetre cavity at one bar, computed by quadrature and agreeing with the closed form in gamma functions to a part in 10⁹.$ Fluids at work

The bubble that hammers

A vapour cavity swept into higher pressure does not deflate. It collapses, in ninety microseconds for a millimetre bubble, and the model that describes the collapse predicts a wall speed that reaches the speed of sound in water at three per cent of the original radius — which is to say it predicts its own failure, and locates it.

A steady pressure field, from a flow with no steady part. The time-averaged pressure round a cylinder in a stream that oscillates as U₀cos ωt. The mean velocity is exactly zero at every point — the flow spends as long going one way as the other — and the mean pressure is not, because pressure depends on the square of the speed and a square has no sign. The mean coefficient reaches -2.00 at the shoulders and averages -1.00 over the surface, and its resultant is 6.6e-16: a real field with no force in it. The pale lines are the instantaneous streamlines, which reverse every half cycle.

Flows and fields

The mean is not the flow

Average an unsteady flow and the result is a new object with its own properties, and it is not a solution of anything. An inviscid stream oscillating about zero has a mean velocity of exactly nothing everywhere, a mean pressure that reaches minus two dynamic pressures at the shoulders, and a missing term in its own momentum equation that can be written down in closed form.

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.

Ideal flow

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.

The section the pressure asked for. The designed section over the one it started from, both drawn to their own chords. Asking for 28 per cent more speed over the forward 62 per cent of the upper surface produces a section 15.0 per cent thick against the original's 10, with the extra thickness forward and the camber changed — none of which was asked for, and all of which is what that pressure distribution is. The pale outline is the baseline. The one thing the method cannot be told is where along the chord any of it happens: the speed is prescribed against the circle's parameter, and where a given station ends up is an output of the same solve that produces the shape.

Ideal flow

Ask for the pressure, and see what shape that is

A designer knows what the pressure distribution has to do long before knowing what shape does it. Running the problem that way round is possible, it is exact, and it refuses more asks than it grants.

80 per cent of the energy goes to the larger scale. One unit of energy is taken out of the middle wavenumber and shared between its two neighbours. Two conservation laws decide the split completely: the energy must add up, and so must the enstrophy, which weights each wavenumber by k². For (1, 2, 4) the answer is that 80.0 per cent of the energy goes up in scale and 80.0 per cent of the enstrophy goes down. There is no model of turbulence anywhere in that: it is two linear equations in two unknowns, and it is why a two-dimensional flow organises itself into large vortices while its gradients get finer.

Transition and turbulence

The cascade that runs backwards

Three-dimensional turbulence carries energy from large scales to small ones and dissipates it. Take away one dimension and the term that does it vanishes identically, a second quantity becomes conserved, and two conservation laws between them force the energy to go the other way — up in scale, into ever larger vortices.

Four turns of a wave, and the parcel is not back. Two parcels traced through four periods of a linear deep-water wave of steepness 0.1, by integrating the exact velocity field. Each orbit is very nearly a closed circle and misses closing by a little, every time, in the same direction — that miss is the whole of the Stokes drift. The near-surface parcel advances 0.0357 of a wavelength over the four cycles and the one a tenth of a wavelength down advances 0.0122, a third as far — because the drift falls off twice as fast with depth as the orbit's own size does.

Flows and fields

The drift in a wave that has none

The velocity at any fixed point under a passing wave averages to exactly zero, and every parcel of water in it moves steadily forward anyway. The orbits do not close, they miss by the same amount every time, and the missing amount is the square of the steepness times the wave speed.

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