Theme

The thread: The exact theory is wrong — page 29

Page 29 of 31, continuing through the 276 essays this motif runs through.

276 essays carry this thread — page 29 of 31.

$Posts slip as one over the root of their fraction. The slip length of a surface of posts, over the period of their square array, against the fraction of the surface that is solid, beside Philip's stripes along and across and the dilute limit in which each post is a lone disc dragged edgewise, (3/16)√(π/φ). At a hundredth solid the posts slip 2.87 periods against the stripes' 1.32; at a tenth 0.597 against 0.591, nearly equal; at a half 0.0663 against 0.11 along the stripes and 0.0552 across them.$ Flows and fields

A post holds liquid back by its radius, a stripe by its length

A surface of no-slip posts standing in a gas-filled texture lets liquid slip over it, and the slip length grows as one over the square root of the solid fraction, where a surface of stripes grows only as its logarithm. The reason is the oldest contrast in slow flow: a disc dragged through a viscous liquid has a drag proportional to its radius, which is Stokes' law, and a line has no finite drag at all, which is Stokes' paradox. But sparse posts beat stripes only while they are sparse.

Local transparency is the best tree only when the ends absorb. The root's reflection at the heart rate against the reflection at the tree's leaves, for the map tree, for the single exponent that is best at that leaf reflection, and for Murray's. With leaves that reflect nothing the map tree is the best of all, 0.035 against 0.0419: making every junction transparent is then the whole job. As the leaves start to reflect, the best single exponent pulls ahead, because it leaves its junctions slightly mismatched in the way that cancels the echo coming back from the ends; at a leaf reflection of 0.5 it reflects 0.0735 to the map tree's 0.13.

Regimes and numbers

A tree transparent at every junction is not the quietest

The branching rule that lets a heart's pulse through one junction without reflection is set by that junction's Womersley number: area-preserving in the aorta, Murray's in the small arteries. Build a whole arterial tree that way, every junction at its own transparent rule, and it reflects more of the pulse than the best tree built to a single exponent. The single exponent wins by leaving its junctions slightly mismatched, in the way that cancels the echo from the tree's own ends — the trick an antireflection coating plays on light.

Mark's criterion lets the wall gas through above Mach 6.5; the gas inside the layer never gets through. The stagnation pressure of the incident shock's boundary-layer gas in the reflected shock's frame, over the reservoir pressure p₅, against the incident shock's Mach number in air: for the gas at the wall, which is Mark's criterion, and for the lowest anywhere in the layer. Below one the gas cannot pass into the reservoir and the shock bifurcates. The wall gas is refused from Mach 1.32 to 6.5; the layer's lowest from 1.32 on, levelling near 0.69. At the tailored helium condition, Mach 3.41, both give 0.51; at hydrogen's, Mach 6, the wall gas gives 0.89 and the layer 0.64.

Compressible flow

The gas a reflected shock refuses is inside the layer

A reflected shock in a shock tube bifurcates when the gas in the wall's boundary layer cannot be pushed into the reservoir behind it: its stagnation pressure, in the shock's frame, is below the reservoir's. Mark's criterion asks that of the gas at the wall, and in air it says the bifurcation stops above an incident Mach number of 6.5. But the gas that fails at high Mach numbers is not at the wall. It is inside the layer, heated by friction until it meets the shock too slowly for its sound speed, and on that measure a reflected shock in air bifurcates at every Mach number above 1.3.

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

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.

With a small plenum, stall feedback closes the hysteresis loop. The stall amplitude against the throttle as it is closed slowly past the peak and opened again, at B = 0.2, with no stall feedback and with feedback at twice the critical gain; closing solid, opening dashed. Without feedback the cell appears suddenly near γ₀ = 0.6094 and vanishes only at 0.6428, and the loop between the two legs has an area of 0.078. With feedback the cell grows gradually as the throttle closes and shrinks along the same curve as it opens: the area is 0.0014.

Fluids at work

A throttle can hold the peak only in a small plenum

A compressor makes its most pressure at the peak of its characteristic, and at the peak it either surges or stalls. A throttle moved by feedback can fight both. Driven by the plenum's pressure it damps surge, with a gain that grows as the square of Greitzer's B; driven by the flow, it cannot. Driven by the stall cell's amplitude it turns the sudden jump into deep stall into a gradual one and closes the hysteresis loop — but only while the plenum is small. Above B of about a third the same feedback turns the jump into a cycle.

The entrance grows with the Péclet number, peaks, and then shrinks. The distance from a wall-temperature step at which the local Nusselt number has come within 5 per cent of its developed value, in diameters, against the Péclet number, for turbulent pipes at Reτ = 500, 1000, 2000 and 5000, with the slug's laminar line L/D = 0.029·Pe. The liquid metals sit on the line; the entrance peaks at 11.6 diameters at Pe ≈ 4700 for Reτ = 1000 and falls to about three and a half diameters for water.

Regimes and numbers

The longest thermal entrance belongs to neither metal nor gas

A liquid metal carries its heat across a turbulent pipe by conduction, as a laminar flow would, and so its thermal entrance should be long. It is long only when its Péclet number is large. Below a few hundred the entrance grows in proportion to the Péclet number with the constant of a fluid moving as a solid block, and it leaves that scaling close to the threshold at which the eddies first match conduction. The longest entrance in a turbulent pipe belongs to the fluids between the metals and the gases, and how long it is depends on a number the measurements have never pinned down.

A wake throws a following cylinder out to the side, and the pair still meets. One cylinder's centre relative to the other's, the stream from left to right, for pairs released six radii apart at 2°, 5°, 10° and 20° from tandem at Re = 200. Without wakes (faint) the near-tandem pairs drift apart, out to 13 radii, before the drag lets the turn bring them in. With wakes the follower first drafts in, is thrown sideways out of the leader's wake, and never gets beyond 6.26 radii. Every pair touches, near side by side.

Ideal flow

A trailing wake hurries a pair together

Two cylinders free in a stream and damped by their drag always end up meeting near side by side, and the obvious candidate for what holds real pairs apart is their wakes: a body beside another's wake is pushed towards the faster fluid, away from it. Give each cylinder a wake that carries its drag and the push is there — but it points the wrong way. It throws a following cylinder out of the leader's wake and round towards side by side, where neither is in the other's wake and nothing opposes the stream's pull. Near tandem the wakes make the pair meet seven to sixty times sooner, and the only balance they create is a saddle.

Compressible flow

A cold wall holds its layer on longer

A compressible boundary layer's pressure gradient pushes on its density, and its density is its temperature. Solve the velocity and the enthalpy together and the wall's temperature reaches the velocity: a cold wall holds its layer on to an adverse gradient two-thirds steeper, a hot one lets it go a third sooner, the stagnation point's heat-to-friction ratio more than doubles between them, and the cold band an uncoupled layer put above a cooled nose disappears.

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.

Transition and turbulence

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.

All themes