Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: A smooth picture proves nothing — page 12

Page 12 of 20, continuing through the 172 essays this motif runs through.

172 essays carry this thread — page 12 of 20.

Seven shock structures, and the two states they all connect. The velocity through the shock for seven dissipation models — Prandtl numbers from a quarter to two, viscosities from constant to linear in temperature. Each curve is shifted so its midpoint sits at the origin. They start at the same speed, end at the same speed, and are nothing alike in between. Compressible flow

The jump does not ask what made it

Seven different dissipation mechanisms are made to smear the same shock. Their interiors are a factor of two and a third apart in thickness, their entropies overshoot the final value by between a quarter and a doubling, and the state they all arrive at agrees to seven parts in ten billion — because the end states are conservation and the interior is transport.

 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. Flows and fields

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.

 How much the peak cares about the valve at each instant. A small bump is added to the valve's opening at one moment and the peak is recomputed; the difference, divided by the bump, is plotted against the moment. It is flat and zero for the first 0.45 seconds — that part of the closure is invisible to the peak — rises through the reflection arrivals, and falls to nothing after the peak has happened, which is the one part of the shape that needs no fluid mechanics. Fluids at work

The part of the closure a pipe cannot see

A surge specification says how long the valve takes to shut. The line does not integrate the duration, it integrates the shape — and it is measurably blind to the first half-second of an eight-second closure, while three closures of identical length give peaks 97 per cent apart.

 The sound speed of a mixture, against how much of it is gas. Wood's formula for air in water. Both ends are the pure phases at 343 and 1,481 metres a second; in between the mixture takes the water's inertia and the air's springiness and the speed collapses to twenty-four metres a second — a fourteenth of the slower constituent. Compressible flow

Slower than either of them

Sound travels at 343 metres a second in air and 1,481 in water. In a mixture of the two it travels at twenty-four, because the mixture takes the water's inertia and the air's springiness — and one per cent of air by volume is enough to take water down to a twelfth of its own speed.

 Six orbits that do not close. One parcel's path under a linear deep-water wave of steepness 0.05, at a fifth of a wavelength down, released at the phase that centres the orbit on its release depth. Each loop returns almost to where it began and not quite. Flows and fields

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.

 When the tracer leaves, given when it went in. The residence-time distribution of three beds with the same mean residence time and different Peclet numbers. Every one of them has its mean at exactly one, and they are nothing alike: the loosest lets a tenth of the tracer out before a third of the mean time has passed, and the tightest is nearly the spike a plug-flow calculation assumes. Fluids at work

The outlet is the inlet, a while ago

A bed has a mean residence time and everybody quotes it. Six beds with the same mean let their first hundredth through at 0.20 and at 0.85 of it, mix a window of inlet history between 1.53 and 0.18 wide, and convert a first-order reaction by amounts the mean cannot distinguish.

 Three instruments, three derivatives, one field. A density field — a shock, smoothed to its own thickness — and what each of the three optical techniques records across it, each scaled to its own peak so the shapes can be compared. Interferometry follows the density itself; schlieren follows its first derivative and peaks where the density is changing fastest; shadowgraph follows the second and is a light-and-dark pair straddling the same place. None of them is looking at the flow: the refractive index of a gas is linear in its density, so every one of them is a densitometer and the differences between them are differences of calculus rather than of apparatus. What is taught wrongly

An instrument that takes a derivative

Dye, smoke and seeded particles mark the fluid and read the marks. The optical ones mark nothing — light passes through and is bent — and what a plate records is not the flow but a derivative of its density. Two of the three are therefore exactly blind to a uniform stream, at any speed it happens to have.

 The singularity a thin aerofoil drives onto its own nose. The Joukowski map has a critical point that maps to a place inside the body, a distance 4 mu²/(1 + 2 mu) from the leading edge. Against thickness that distance is a clean square: a twelve per cent section is analytic only within eight thousandths of a chord of its own nose, and the thin-aerofoil limit is the limit in which the singularity arrives on the surface. Ideal flow

The part of the flow inside the body

A potential flow outside a body is an analytic function, and an analytic function does not stop at the boundary it was defined on. It continues inward until it meets a singularity — and every body in this collection has at least one inside it, in a place that decides how the flow behaves outside.

 The two dissipations, side by side. The strain form on the left and the enstrophy form on the right, for the same field, on the same scale. They have their maxima in different places — 0.46 apart on a box of side 2 pi — and neither is a smoothed version of the other. One says the dissipation is in the strained regions and the other says it is in the rotating ones, which is nearly a complete disagreement about what a turbulent flow is doing. Transition and turbulence

Equal on average, and nothing else

The rate at which a fluid turns motion into heat can be written two ways, and every textbook says the two are equivalent. Their averages are equal to fourteen decimal places. Point by point they are uncorrelated, and their maxima are in different places.

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