Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
The collection

Every essay — page 32

Page 32 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.

The kept transport spirals into nothing as the sea deepens. The net Lagrangian transport as a vector, scaled on the Stokes transport, traced as the water depth increases from a quarter of an Ekman depth to eight, for an 8-second swell with an eddy viscosity of 0.01 m²/s. Shallow water keeps the whole transport pointing with the waves, at the right-hand end. As the sea deepens the vector shortens and swings to the right, crosses the across-wave axis near two Ekman depths, and winds into the origin, which is the open ocean's exact cancellation.

The floor that gives the drift back

In the open ocean the Coriolis force drives a current that cancels a swell's Stokes transport exactly. Over a continental shelf the sea floor holds a stress, and whatever it holds is transport the rotation does not take back. How much survives depends almost only on the depth in Ekman depths; which way it points depends on the wave.

7 figures The best efficiency runs from nine-eighths of φ² to one. The best efficiency a peristaltic pump can reach, against the fraction of the channel its wave closes, with its two limits. For a shallow wave it is 9φ²/8, which is small — a wave closing a fifth of the channel is at best 4.5 per cent efficient. As the wave closes the channel the best efficiency tends to one, and its shortfall shrinks in proportion to the remaining gap: about 1.9(1 − φ). Nothing in between is independent of the amplitude.

The pump that is better the more it squeezes

A waving sheet swims at a cost per metre with no amplitude in it. A waving wall pumping fluid is the same mechanism turned round, and its efficiency is nothing like that: it starts at nine-eighths of the amplitude ratio squared, is exactly 2 − √3 at half closure, and rises towards one as the wave closes the tube — where the pump stops being a wave and becomes a piston.

8 figures The line that decides how much a thickener can pass. The batch settling flux against concentration, the same curve the closed column was solved on, with the operating line of a continuous thickener drawn across it. The line runs from the underflow concentration on the axis to the flux the plant can take, and it is placed by one condition: it must touch the flux curve rather than cut it. Where it touches is the concentration at which the thickener is working hardest.

The tangent that sizes a plant

A closed cylinder of settling suspension is a kinematic wave problem with a flux curve. Open the cylinder — feed it at the top and draw thickened slurry from the floor — and the capacity of the whole plant is a straight line laid against that same curve, touching it rather than cutting it.

8 figures Three speeds, and the Froude number at which two of them cross. The kinematic wave speed and the two dynamic wave speeds, all divided by the speed of a shallow-water wave on still water, against the Froude number. The kinematic wave is three halves of the water speed and the downstream dynamic wave is the water speed plus one, so they cross at Froude two — and beyond that crossing the news of a change in flux arrives before the news of a change in depth, which uniform flow cannot survive.

When the flux outruns the pressure

A kinematic wave is derived by throwing the momentum equation away, and the derivation never says when that is allowed. It is allowed until the kinematic wave, which travels faster than the water, overtakes the downstream dynamic wave as well — at which point uniform flow ceases to exist and a concrete chute carries a train of surges instead of a sheet.

7 figures Every gradient in one plane, and where each one goes. The two invariants of a trace-free velocity gradient, R across and Q up. Under the restricted Euler equation the combination 27R²/4 + Q³ never changes, so every trajectory is one of these curves, and R never decreases, so every curve is travelled from left to right. Vieillefosse's line, where the combination is zero, has two branches: the left one runs into the origin and is the only way to reach it, and the right one is where every other trajectory ends, at infinity, in finite time.

A gradient left to itself

Follow a parcel's velocity gradient with nothing acting on it but its own square and the part of the pressure a single parcel can know about. Every starting gradient but a set of measure zero reaches infinity in a finite time, and it gets there as a sheet with its spin lying in the sheet.

7 figures The gas flows in, against the heat. The radial velocity of the gas, in units of α/a, at the same four times — from the motion of the gas shells themselves — with the heat flux, reversed in sign and in the same units, drawn as dots. The two coincide: at constant pressure the gas moves at (γ − 1)/(γp) times the heat flux, against it. Heat leaves the spot outward and the gas comes in, everywhere.

The air flows in against the heat

A gas moving at a ten-thousandth of the speed of sound is as incompressible as anything gets, by the usual arithmetic. Heat one spot of it and let the spot cool: the gas changes its volume threefold, and it flows towards the hot spot while the heat flows out, at a speed fixed by the heat flux alone. At constant pressure, a joule is a volume.

6 figures The unburnt gas bends round a wrinkle, and the bending makes it grow. Streamlines of the unburnt gas flowing up towards a flame with a small sinusoidal wrinkle, one wavelength across, for a density ratio of seven, with the wrinkle exaggerated. The expansion behind the flame pushes back on the gas ahead of it where the flame bulges forward, so the streamlines spread there and the gas arrives slower, and they crowd together where the flame lags, so the gas arrives faster. A flame that burns into the gas at a fixed speed therefore advances further where it was already ahead.

A flat flame is unstable at every size

A flame expands the gas it burns, and the expansion pushes back on the fresh gas ahead. Where the flame bulges forward the fresh gas slows, so the bulge burns further forward; where it lags, the gas speeds up and it falls further back. Every wrinkle grows, the shorter ones faster, and what finally gives a real flame a size is how its burning speed responds to its own curvature.

6 figures Every particle leaves a flow that is a vortex at every instant. Eight fluid particles starting on a small circle in Haller's rotating-saddle flow, followed for 2.4 time units. At every instant the velocity gradient has complex eigenvalues and a positive Q — the flow is a vortex by every criterion that reads a snapshot — and every particle spirals outward, its distance growing as e to the time. The fluid is not held; it is flung.

Every snapshot says vortex, and every particle leaves

There is a flow in which the velocity gradient has complex eigenvalues at every point and every instant — a vortex by every criterion that reads a snapshot, in the frame the flow is measured in — and in which every fluid particle is flung away exponentially. The snapshot is not wrong about the gradient. It is wrong about the fluid, and it is wrong exactly when the strain's axes turn.

6 figures In the pair's own frame, two eddies with no vorticity. The flow round two equal co-rotating vortices (orange dots), seen from the frame that turns with them, where it is steady; shading is the stream function. Round each vortex a lobe of fluid circulates; a band round both is bounded by a figure-eight through the saddle at the centre; and above and below, centred on the two points that make equilateral triangles with the vortices (blue dots), are two large eddies of fluid that circulate in this frame and carry no vorticity at all, bounded by the streamline through the two outer saddles at √5 half-separations.

A vortex pair carries eddies no snapshot can see

Two equal vortices circling each other are, by every snapshot of the velocity gradient, two small vortices in a straining flow: outside their cores the flow is irrotational, and the gradient there is pure strain. Seen from the frame that turns with them, the same flow contains two large eddies, one on each side, centred where a third point would complete an equilateral triangle with the pair. Their fluid goes round with the pair for ever, and it carries no vorticity at all. They hold seventy times the area of the cores, and the only way to see them is to follow the fluid or to turn with it.

6 figures A wrinkled flame settles into arcs meeting at a cusp. The steady front of a flame in a periodic domain 5, 10 and 20 neutral wavelengths wide, from the exact pole solution of the Michelson–Sivashinsky equation, with the burnt gas below and the flame advancing upwards; each is drawn across one period, scaled to the same width. Every one is a single smooth arc bulging into the fresh gas, meeting its neighbour in a sharp cusp pointing back into the burnt gas, and in these units the three arcs nearly coincide: only the cusp sharpens as the domain widens. In physical units the arc's depth grows in proportion to the domain's width, so the three flames are the same shape at three sizes.

A wrinkled flame has one cusp and a speed limit

The linear theory of a flame says every long wrinkle grows and none is favoured. The weakly nonlinear theory — the Michelson–Sivashinsky equation — says where the growth goes: small wrinkles merge, the front settles into smooth arcs bulging into the fresh gas and meeting in sharp cusps, and in a domain of any width it ends with a single arc and a single cusp. That front is an exact solution made of poles in the complex plane, and its speed is a closed form that rises in steps as the domain admits more poles and then stops: beyond about five neutral wavelengths a wider flame is no faster, because it is the same shape at a larger size.

6 figures Remembered pressure turns the runaway into a teardrop. The velocity gradient's two invariants, Q against R, both scaled by the mean enstrophy, for an ensemble of parcels whose pressure remembers one Kolmogorov time of deformation, at a memory of a tenth of a large-eddy time. The restricted Euler equation sends every parcel off to infinity along the right-hand branch of Vieillefosse's curve, dashed. With the remembered pressure the parcels stay: they crowd along that branch in the strain-dominated quadrant and above the axis on the left where vortices are being stretched — the teardrop measured in turbulence.

The pressure a parcel remembers keeps it finite

The restricted Euler equation follows a parcel's velocity gradient with the pressure's shape thrown away, and every gradient it follows blows up. Give the parcel back a pressure that remembers how its neighbourhood was deformed over the last Kolmogorov time — nothing more — and no gradient blows up at all. The ensemble settles into the teardrop measured in turbulence, its vorticity lines up with the middle strain axis, and its intermittency grows as the memory shortens. What the memory cannot do is keep the one identity homogeneity demands, and that miss says where the rest of the pressure lives.

6 figures A quiet flame has one cusp; a noisy one keeps making more. Two flame fronts twenty neutral wavelengths wide, burning upwards, drawn apart for clarity: above, the quiet front, the exact pole solution with one cusp per period and the most poles it can hold; below, the same flame with a noise of a millionth kicking its growing wavelengths. The noise keeps seeding small wrinkles on the smooth arcs; each grows as it is swept along the arc into the cusp, so the noisy front carries a train of sub-cusps the quiet one never has, and advances at 0.862 against the quiet 0.5.

A flame's speed limit holds only in silence

A quiet wrinkled flame settles into a single cusp and a speed it never exceeds, however wide it grows. Add noise and the limit is gone. A disturbance of one part in a million million already speeds the front up; one in a million makes it a third faster; and a noisy flame, unlike a quiet one, keeps getting faster as it gets wider, because every added width is room for more wrinkles to be born on its arcs and swept into its cusp. The quiet flame's speed is a property of a silence no burner has.

5 figures