The collection

Every essay — page 47

Page 47 of 53, 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

Transition and turbulence

Where the laminar solutions stop being the ones the flow takes, and what can honestly be said about what follows — which is less than the textbooks imply and more than nothing.

Forced at the integral scale, the third moment nearly reaches four-fifths. The Kármán–Howarth balance at a Taylor-scale Reynolds number of 200, on the same model spectrum, for a flow forced in a band at the spectrum's peak and for one decaying. Each term is divided by (4/5)εr and plotted against separation in Kolmogorov lengths. The viscous term is the same for both. The forcing term is negligible until the separation approaches the integral scale, and −Dₗₗₗ/((4/5)εr) for the forced flow peaks at 0.965 at 90 Kolmogorov lengths, where the decaying flow's peaks at 0.747 at 32.

Forcing at the integral scale leaves the cascade alone

A decaying flow's third moment falls a quarter short of the four-fifths law at the Reynolds numbers a grid reaches. A flow forced at its largest scales, on the same spectrum at the same Reynolds number, falls short by three and a half per cent. The shortfall is not what a finite Reynolds number does to every flow; it is what the source does, and how fast it closes is set by how far the source reaches into the inertial range.

7 figures

Bursts in the velocity put the tails back. The kurtosis of the pairs' separation against how long each pair remembers its velocity's direction, β, in turnover times. The lowest curve is a Gaussian velocity, as in the earlier calculation. The others give the velocity a flatness of 4, as measured in the inertial range, with its amplitude remembered for α turnover times. At the memory real pairs are estimated to have, β = 0.7, the Gaussian cloud's kurtosis is 1.89; with bursts remembered for three turnover times it is 3.41, and with the amplitude frozen 4.4 — either side of Richardson's 3.76, dashed.

Bursts and memory pull a cloud both ways

A pair of fluid particles that remembers its relative velocity spreads into a cloud with shorter tails than Richardson's, and the earlier calculation proposed reading the memory off the cloud's shape. Real relative velocities come in bursts, their amplitude set by a local dissipation that varies, and a burst that lasts pushes the tails back out — hard, because separation grows as the cube of diffusivity. At the memory real pairs are estimated to have, the two effects nearly cancel, and a cloud can have Richardson's exact shape for entirely the wrong reason.

6 figures $A flow's window mean converges a power faster. The variance of the mean over an L × L window, as a fraction of the point variance, against the window's side in integral lengths, on logarithmic axes. The scalar field's falls as the inverse area. The plane cut through a three-dimensional flow falls the same way at half the level. The velocity of a two-dimensional incompressible flow falls as the inverse cube of the side, because its mean over any window is a streamfunction difference around the window's edge. Lines are closed forms; dots are Monte Carlo means over fifty generated fields.$

A snapshot counts areas, and a flow counts its edge

A record at one point holds one independent value for every two integral scales it lasts. A snapshot of a field holds one for every integral area it covers, so at the same number of samples it holds far fewer, and sampling it more finely adds nothing. Except for a velocity in a two-dimensional incompressible flow, whose mean over a window is fixed by the window's edge alone: it converges a whole power faster than its area allows, and a large enough snapshot of it beats the record.

6 figures

A cloud's shape falls as it grows into calmer eddies. The kurtosis of a cloud of pairs released at a ten-thousandth of the integral scale, against its rms size, when the velocity's flatness follows the 1962 law, and for clouds whose velocity has one flatness, 3, 4 or 5, at every separation. The fixed-flatness clouds settle at a shape and keep it. The growing cloud rises to a kurtosis of 6.85 at 0.0022 L and then falls without settling, to 2.9 by 0.57 L, crossing all three.

A cloud grows out of its bursts and keeps their shape

Measured velocity differences are burstiest across the smallest separations and nearly Gaussian across the largest, so a cloud of particle pairs released close together starts in the fiercest intermittency and grows out of it. Its shape follows, but late: the kurtosis falls steadily as the cloud grows, always above what its present statistics would give, and the cube law's constant falls with it, so the growth exponent climbs towards three and never arrives.

7 figures

A frozen run counts the area it sweeps. Independent values in a run of frames while a frozen pattern is swept 100 integral lengths past a window 12.8 integral lengths square, against the distance the pattern moves between frames. For a scalar the run holds 353 values with a frame every half integral length, 362 with one every four and 388 with one every window width: the swept area in integral areas, whatever the frame rate. Only past a window width, when gaps open between frames, does the count fall. Treating frames an integral scale or two apart as independent, as a record would allow, counts 2075 at two integral lengths — 5.8 times too many.

Frames that share their eddies count as one

A particle-image run is a sequence of snapshots, and snapshots closer together than an integral time photograph the same eddies. When a mean flow carries a frozen pattern past the window, a run holds exactly the area it sweeps, counted in integral areas, and a faster camera adds nothing until frames stop overlapping — a threshold set by the window, not by the turbulence. In a flat flow one component escapes the rule: its run mean is fixed by the pattern at the two ends.

6 figures

A thin interface stays unstable far past a quarter. The fastest growth rate of any disturbance against the bulk Richardson number J, on a logarithmic scale, for a density interface as thick as the shear and two, two and a half and three times thinner. The matched layer's billow dies at a quarter, as Miles' theorem requires. Two times thinner, the billow dies sooner, its last growth at J = 0.12, and nothing replaces it. Three times thinner, the billow's last is at 0.08 and a travelling wave takes over: 0.0335 at a quarter, 0.0109 at one and 0.00629 at 1.3, falling steadily with no threshold in the range solved. At 2.5 the waves are weaker and reach 0.001 at 1.3.

A thin interface keeps its waves past a quarter

Miles' quarter rules a shear layer whose density changes over the same depth as its velocity. Make the density interface three times thinner and the stationary billow dies early, but a pair of travelling waves takes its place and is still growing at five times the quarter. No theorem is broken: at the edges of the shear, where the waves draw their energy, the local Richardson number has fallen to nothing.

7 figures $Half the forced summit is gone in a fifth of an eddy time. The third moment of the velocity differences as a fraction of four-fifths εr, against separation in Kolmogorov lengths of the forced flow, at the instant the forcing is switched off and 0.1, 0.2, 0.5, 1 and 4 large-eddy times later. Forced, the curve peaks at 0.94 near 67 η. A tenth of an eddy time later the summit is 0.899, at a fifth 0.838, at a half 0.759, and after that it hardly moves: 0.742 at one and 0.72 at four, where the flow is simply decaying. The large separations fall first and farthest.$

The summit forgets the forcing before the dissipation does

A flow forced at its largest scales carries the four-fifths law closer to exact than a decaying one. Switch the forcing off and the third moment has to pass from one value to the other. It does not wait for the cascade to drain: half the forced summit is gone in a fifth of an eddy turnover, while the dissipation has hardly moved, and the largest separations overshoot the decaying value before they settle on it.

7 figures

Four of six decayed states sit on the sinh branch; the other two lie above it. The streamfunction's flatness against the enstrophy-to-energy ratio Z/E: the sinh and tanh relaxed dipoles as curves, from the linear dipole at Z/E = 1 outwards, and the six decayed states at t = 600 as points. The tanh branch falls below 9/4 and stays near Z/E of one; the sinh branch rises. The two runs begun as two-level patches lie on the sinh branch at their own Z/E, 0.0049 to 0.055 from it, and so do the two begun on a k⁻³ spectrum, 0.04 to 0.072. The two random-phase runs lie above it, by 0.3 to 0.31: on the sinh side of 9/4, but not sinh dipoles.

Decaying flows end on the sinh side

Three theories predict the state a decaying two-dimensional flow relaxes to, and the streamfunction's flatness tells them apart: above 9/4 for the point-vortex theory's sinh, below for the two-level theory's tanh. Run the decay from three kinds of start and every run ends above 9/4 — including the one begun as two-level patches, the case the tanh theory was built for. Four of six land on the sinh branch itself; the other two are sinh-like without being sinh.

7 figures

The wake is a tenth of the velocity and a third of the displacement

The logarithmic law describes a band in the middle of a turbulent boundary layer, and above it the profile lifts away by an amount Coles called the wake. It is a tenth of the edge velocity, so it looks like a correction. It is not: it carries a third of the layer's displacement, it is what turns the log law into a friction law for a boundary layer, and with it the friction comes out within a few per cent of a measured correlation that contains no logarithm at all.

6 figures $An adverse gradient hollows the layer from the outside in. The velocity profile of an equilibrium layer, as a fraction of the edge velocity against y/δ, at Clauser's β = 0, 2, 10 and 50, with the wake strength from Das's fit. The inner part keeps the wall law; the outer part's wake grows — Π = 0.476, 1.59, 4.67 and 15.2 — and the profile sags until most of the layer is moving slowly over a thin fast wall region.$

An adverse gradient grows the wake, not the log law

Push a turbulent boundary layer up a rising pressure and its profile changes from the outside in. The wall region keeps its law; the wake grows, the layer hollows, the shape factor climbs and the friction falls. Clauser found the layers that stay similar while this happens, and their arithmetic has a surprise in it: there is a steepest pressure rise any such layer can climb, close to a free stream falling as the inverse fourth root of the distance, and the friction approaches zero only as the layer becomes all wake.

5 figures

Viscosity

The thin layer next to a surface that ideal flow ignores, and which supplies drag, separation and the wake.

Flow past a cylinder at Re 40. A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.

Everything happens in a layer you cannot see

Air has so little viscosity that ignoring it works almost everywhere. Almost everywhere leaves out a film next to the surface, perhaps a millimetre thick, and that film decides drag, stall and whether an aircraft flies at all.

6 figures

Flow past a cylinder at Re 100. A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.

When the flow lets go

Every body asks the air behind it to slow down and climb back up to the pressure it started at. Sometimes the air cannot, and the moment it refuses is separation — the source of most drag, the cause of stall, and the reason a golf ball has dimples.

6 figures