The collection

Every essay — page 30

Page 30 of 61, continuing through the fields in the same order.

Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search

Flows and fields

Streamlines, particle paths and the field that carries them. What is conserved, what a picture of a flow can show, and what it cannot.

A double integral that comes out an integer. The Gauss linking integral evaluated on six pairs of closed curves. It is not constrained to be a whole number by anything in its own definition — it is a double integral of a smooth kernel — and it returns one to within two parts in ten thousand on two hundred points per curve, because what it is computing is a topological count.

The knot a flow cannot untie

Ideal flow conserves energy, circulation and impulse, and all three are what they look like. It conserves a fourth quantity that is not: a volume integral of the velocity dotted into the vorticity, whose value counts how many times the vortex lines are linked through one another.

7 figures $One rate per moment, and none of them the same. lambda_p = ln<l^p>/(p t) against p. As p goes to zero it is the Lyapunov exponent, the rate of the typical element; at p = 1 it is the rate of the average length, which is nearly twice as large. If ln l were exactly Gaussian this would be a straight line with the Lyapunov exponent as its intercept, and the departure from that line is the same multifractality the velocity increments have.$

The stretching rate that is not one number

A material line in a flow gets longer, and there is a theorem saying its length grows at a definite exponential rate. There is also a rate at which the average length grows, and it is nearly twice as large — and a different rate for every moment of the distribution.

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A patch of dye, folded. A circle of marker particles carried by the flow, at four times. It is stretched into a filament and folded through itself, and its area does not change at any point of that — which is not visible in the picture, and is the whole difficulty. A scheme that lost eight per cent an orbit would produce a picture indistinguishable from this one.

The area that must not move

A patch of dye in an incompressible two-dimensional flow keeps exactly the area it started with, for ever. Two respectable integrators are put on the same flow: one respects that identically at any step size, the other does not, and the pictures they draw are the same picture.

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And the net drift, which is where they disagree. The Stokes drift plus each return flow. All three drift forward at the surface, because the Stokes drift there swamps any return current of the right size. Below that they part company completely: the uniform current has most of the column moving upstream, and the two that satisfy no slip have almost none of it. The reversal depths span 77 per cent of the water column.

The drift a closed box will not allow

A wave in a wave tank carries mass forward, and the tank has nowhere to put it. So a return current appears carrying exactly the opposite transport — exactly, from mass conservation and nothing else. Which fixes a total and leaves the answer anybody wants entirely open.

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A uniform scalar in a fluid at rest, under two face rules. Nothing is flowing and the scalar starts at one everywhere. The swept-volume rule leaves it at one to the last bit, at every step of the two time units. The midpoint rule moves it by two parts in ten thousand, on a mesh motion that begins and ends in the same place, and the excursion looks exactly like a physical transient.

The mesh that makes its own mass

The transport theorem holds for a region moving at any velocity, which is what makes a moving-mesh calculation possible. Discretised carelessly it is not an identity but an approximation, and a fluid at rest with a uniform density then gains density from the motion of a grid — smoothly, plausibly, and looking exactly like a physical transient.

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What a probe in a flame sees. Half the time hot light fluid at a hundred metres a second, half the time cold heavy fluid at twenty. That is what intermittency in a jet flame looks like at a point, and it is the simplest field in which the two averages of the velocity are different numbers.

Two averages of one flow

In a flow whose density varies there are two mean velocities, they are both correct, and across a flame they differ by a factor of two. One of them is what a hot wire returns; the other is what every compressible turbulence model is written in; and the mass flux is the single product they agree on.

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One pair of strainings, two orders, two lengths. The stretch of the most-stretched material direction against time, for a simple shear followed by a pure strain and for the same two in the other order. The two curves are identical until the swap and separate afterwards, ending a factor of 2.16 apart.

Two strainings, and the order they came in

A material line is stretched by a shear and then by a pure strain, and then by the same two in the other order. Every instantaneous measure of how hard the fluid was being worked is identical in the two cases. The lengths at the end differ by a factor of 2.16.

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Five points, and where each one's fluid came from. Back-trajectories through six units of time in an unsteady double gyre. Each curve ends at the place the fluid now at the marked point started; nothing about that place can be read off the velocity at the marker.

A scalar is a record of where its fluid was

A conserved scalar has no value of its own. Its value at a point is whatever it was at the place that point's fluid started from, which makes a dye field a photograph of the past — and makes the map from now to then the only thing in the flow that carries the past at all.

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A drift made of two things that average to zero

Stokes drift is usually explained as a parcel spending longer in the forward half of its orbit. That is true and it is not a formula. The formula is a correlation between a displacement and a gradient, each of which averages to exactly nothing, and it splits into two halves that are equal to twelve figures.

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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.

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What a mean profile cannot tell anybody

Two flows are built here with mean velocity profiles that agree to four parts in 10¹⁷. One of them carries momentum across the shear and dissipates forty per cent more energy; the other carries nothing. Everything that distinguishes them is second order in a disturbance the mean cannot see at all.

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A duct that does not change, and a parcel that does. A contraction of area ratio four, with the parcel marked at five equal intervals of time. The walls do not move, the field at every point is the same at every instant, and the spacing of the markers grows because the parcel is carrying its own history through the duct.

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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