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The thread: What is conserved — page 11

Page 11 of 33, continuing through the 292 essays this motif runs through.

292 essays carry this thread — page 11 of 33.

An oscillation with no mean, and the steady flow it drives. The steady second-order velocity through a Stokes layer, in units of U U′/ω. The first-order flow averages to zero at every height; the average of its own nonlinear term does not, and the pale curve is that forcing. Integrating it twice across the layer, with no slip at the wall and no stress at the top, gives a steady velocity that rises through the layer and settles at -0.749998 — Rayleigh's −3/4, which was not put in anywhere. Beyond about five layer thicknesses nothing more happens, which is why the number is a boundary condition for the flow outside.

An oscillation with somewhere to go

Shake a fluid back and forth over a body and it develops a steady circulation that never reverses. The driving flow has no mean at all; the mean of its own nonlinear term does, and integrating that twice across the oscillatory layer gives a slip velocity of exactly three-quarters of U dU/dx over the frequency.

One spectrum, two fields. Two one-dimensional fields built from the same amplitudes and different phases. Their energy spectra are identical to the last bit, their two-point correlations agree to 2e-15, their second-order structure functions are the same function — and no measurement of either kind can tell them apart. Everything that distinguishes them is in the phases, which is where the cascade lives.

Transition and turbulence

The moment a spectrum cannot hold

A spectrum and a two-point correlation are the same object, transformed, so everything one of them records the other records too. Give a field the exact Kolmogorov amplitudes and independent phases and every second-order measurement comes out right — while the cascade, which lives in the phases, is not there at all.

The same reading, and only one of them gives it back. A Venturi and an orifice plate at the same diameter ratio, with the pressure along the axis drawn beneath each. Both narrow the flow by the same amount, both read the same difference between the pipe and the narrowest section, and both infer the same flow rate from it. Downstream they part company: the Venturi's diffuser turns the throat's speed back into pressure, and the orifice's jet expands into the pipe and destroys 73% of the reading. The picture is a section rather than a solved field: nothing here computes the jet, and the recirculating corner is not drawn.

Fluids at work

The price of knowing the flow rate

Two flowmeters can narrow a pipe by the same amount, read the same pressure difference and infer the same flow rate, and cost pressures that differ by an order of magnitude. What separates them is not viscosity, and not workmanship — it is whether the flow is decelerated or abandoned.

How big before gravity shows. A drop's height over its width against the Bond number, which is the ratio of its weight to the force its own skin can supply. The number is one where those two are equal, and by then the drop is a bun: it is one per cent from a ball at Bo = 0.0079, five per cent at 0.054 and ten at 0.13. Every one of those is below one, and the first is below it by a factor of a hundred and twenty-six.

Regimes and numbers

The size a drop is allowed

The Bond number sets a drop's weight against the force its own skin can supply, and it is one when they are equal. By then the drop is a bun — it is a per cent from being a ball at Bond number 0.0079, which is a water drop half a millimetre across.

The sound is what fails to cancel. A compact quadrupole of kd = 0.01, followed outwards. The upper curve is what one of its four sources produces on its own at each radius — every one of them is as loud as a monopole — and the lower curve is what all four produce together. Inside the source they barely cancel at all; by one radian of wavelength the sum is 9.3e-5 of what a single source is doing, and it is falling as 1/R from there on because that is what radiation does. Between the two the field falls as the cube of the distance, which is a near field rather than a sound.

Compressible flow

The sound is what does not cancel

A quadrupole is not a weak source. Every one of the four monopoles in it is as loud as a monopole of the same strength, and what makes the assembly quiet is that they very nearly cancel — one part in ten thousand survives at a hundredth of a wavelength. The eighth-power law is a statement about how nearly, not about how little.

Two divergence-free fields with the same boundary conditions. On the left, the exact potential flow round a cylinder moving through fluid at rest. On the right, the same flow with a divergence-free eddy added — one that has no normal velocity on the body or on the outer circle, so it changes nothing about what crosses a boundary. Both fields conserve mass, both satisfy the wall condition, and only one is the flow. Nothing in the drawing says which.

Ideal flow

The flow with the least energy in it

Draw a flow that conserves mass and does not go through the walls, and it will look exactly like a solution. There are infinitely many of them and one is the flow. What separates it from the others is not visible anywhere in the picture — it is a number, and the number is an energy.

A material region, and the dye that stays inside it. The same fluid at four times, carried by an unsteady straining flow whose strain rate oscillates. The outline is a circle of the fluid at the first instant, tracked by integrating the velocity field; the shading is a blob of passive dye. The region is stretched to nearly seven to one and its area is unchanged to fifteen decimal places, because the flow is incompressible. The amount of dye inside it is unchanged to thirteen, because the dye is carried by the same fluid.

Flows and fields

A rate of change that will not hold still

Three boxes drawn in one flow at one instant give three different answers to how fast the dye inside them is changing — one falling, one falling twice as fast, one rising. All three reconcile with a single material rate, and that rate is zero.

$Five quarters of the span, the same bending moment, 64/75 of the drag. The bell-loaded wing's induced drag and root bending moment against its span, both as fractions of an elliptic wing of unit span carrying the same lift. At equal span the bell is worse: its span efficiency is exactly three quarters. But its bending moment is exactly four fifths, and bending moment grows in proportion to span while drag falls as its square — so there is a span at which the bell has bought back the structure and is ahead on drag. It is at exactly five quarters, where the two curves are at 1 and 0.8533. Both numbers are rational and neither was put in by hand.$ Circulation and lift

The loading nobody used

Elliptic loading is the least-drag answer to a question no aeroplane asks. Constrain the moment the lift makes about the wing root instead of the span, and a different curve comes out — five quarters of the span for sixty-four seventy-fifths of the drag, both exact — with an upwash over the outer wing and a yaw that turns the right way.

The one exact law, and where viscosity takes it back. The two terms of the Kármán–Howarth relation against separation, measured in Kolmogorov lengths. The four-fifths term rises linearly with r and is the whole of the law at large separation; the viscous term, 6ν dS₂/dr, falls as r^(−1/3) and takes over below 5.64η. Nothing in the calculation was told what η is: the crossing is at the same multiple of it at every viscosity tried, which is what makes the dissipation scale the lower end of the inertial range rather than a separate assumption.

Transition and turbulence

The one exact result

Almost nothing in turbulence follows from the equations without a model in it. One thing does — Kolmogorov's four-fifths law, which fixes the third moment of the velocity differences at −(4/5)εr with no adjustable constant anywhere in it. The companion two-thirds law is not exact, and the 4.02 everybody quotes in it turns out to be a gamma function.

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