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The thread: Everything happens in a thin layer — page 11

Page 11 of 11, continuing through the 99 essays this motif runs through.

99 essays carry this thread — page 11 of 11.

$The wall lets go long before the textbook's shock arrives. Where the flow leaves the nozzle's wall, as a fraction of the divergent section's length, against the ratio of chamber to ambient pressure. The inviscid picture runs full to the exit until the pressure ratio falls to 24.5, where a normal shock stands at the exit, and then moves the shock inside. The wall flow separates instead, when its pressure falls to Summerfield's 0.4 of ambient — at a pressure ratio of 192 — or to Schmucker's Mach-dependent value, at 138: six to eight times higher. Across that whole band the textbook's nozzle is full and the real one is not.$ Compressible flow

An overexpanded nozzle lets go before its shock arrives

The textbook's overexpanded nozzle runs full until its back pressure is high enough to hold a normal shock at the exit, and then draws the shock walking inside. A real nozzle never shows that sequence. Its wall's boundary layer cannot climb the pressure rise a normal shock imposes; it separates at a wall pressure of about four-tenths of ambient, which for a rocket nozzle happens at six to eight times the pressure ratio the textbook's shock needs. The separation is not a failure: it is what keeps the nozzle's thrust, and it is why a sea-level engine can be built twice the size it expands to.

Five gas films never meet their thermal crossing. Six air films placed by their squeeze number, which decides whether the gas leaves the gap or is trapped in it, and their thermal number, which decides whether the heat of compression leaves. The dashed diagonal is the estimate that the two differ by the Prandtl number. The devices lie between one and eleven decades below it, because heat leaves across the gap and the gas along the disc. Only the levitator at a twenty-micron gap comes within a decade of the thermal crossing at ωh²/κ ≈ 18.

Viscosity

The heat of a squeeze leaves across the gap

Squeeze a film of air and it warms, and whether that heat reaches the walls in time decides whether the gas is compressed isothermally or adiabatically. The natural estimate has the heat leaving the way the gas does, along the disc, and puts the thermal crossing beside the viscous one. It leaves across the gap instead, a distance hundreds of times shorter, so the crossing sits decades away — and when a film does reach it, what it gains is not stiffness but a second band of loss.

$Smooth below one Mach number, a jump and a tail above it. The velocity change through steady shocks in carbon dioxide, as a fraction of the whole change, against distance in relaxation lengths — the equilibrium sound speed times the relaxation time, 0.737 mm at one atmosphere. Shocks slower than the frozen sound speed, M below 1.041 referred to the equilibrium speed, are smooth throughout: the whole rise is the vibration catching up. Faster ones jump first, at the frozen speed's own shock, and relax afterwards: at M = 1.3 the jump takes 83.9 per cent of the change at once.$ Viscosity

A bulk viscosity holds a shock together until it splits

Carbon dioxide's bulk viscosity, fifteen hundred times its shear viscosity, is a vibrational relaxation seen from below its frequency, and a shock is where that description is tested hardest. Carried through a steady shock, the relaxation reproduces the coefficient exactly for the weakest shocks. At a pressure rise of nine and a half per cent the shock outruns the frozen sound speed and splits into a jump and a tail, and the coefficient then draws a shock that does not exist.

Helium wastes least for most stiffnesses; xenon reaches furthest. The film's loss tangent — damping over stiffness — against its stiffness, each gas traced from a tenth of an atmosphere up to the pressure of its greatest stiffness. Raising the pressure buys stiffness and costs loss for every gas. Up to about a hundred newtons per unit relative amplitude helium's film loses least at any stiffness — 0.29 at 50 N against air's 0.4 and xenon's 0.46. Beyond, helium is near its peak of 127 N and the heavy monatomic gases take over: xenon reaches 149 N, and at 120 N loses 0.73 against helium's 0.78.

Viscosity

The gas sets a squeeze film's loss, not its stiffness

A squeeze-film levitator wastes part of every cycle as heat that crosses the gap, and the gas and its pressure decide how much. Fill one levitator with six gases at one atmosphere and its stiffness hardly changes while its loss nearly doubles from helium to xenon. Raise the pressure and every gas stiffens towards a peak near ten atmospheres. Helium wastes least at any stiffness up to a hundred newtons; above that, only the heavy monatomic gases get there.

Elasticity adds a spring at first order and a slip only at second. The damping's deficit, one minus the damping ratio, and the in-phase force against Λ on logarithmic axes. The in-phase force rises in proportion to Λ — a fitted slope of 1.0000 — and the damping's deficit as its square, 2.0000. A slip length changes the damping at first order and adds no spring at all, so a soft wall and a slipping one are distinct in kind: the soft wall announces itself first in the phase of the force.

What is taught wrongly

A soft wall is read as slip only at second order

A drainage force smaller than Taylor's has been read as the liquid slipping at the wall. A wall that gives under the drainage pressure lowers the force too, and it can be mistaken for slip. Solved together, the flow and the elastic wall say how: the softness first adds a spring to the force, in phase with the motion, and only at second order takes anything from the damping a slip length is read from. When it does, the slip it imitates grows as the square of the frequency and falls as the square of the gap.

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.

Transition and turbulence

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.

Flows and fields

A swept wall makes vorticity along its isobars, not across its flow

A still wall in a pressure gradient puts vorticity into the fluid at a rate set by the gradient, and the source is a vector: it lies in the wall, along the isobars. On an unswept body the isobars run across the flow and so does the vorticity, which is the ordinary boundary layer. On a swept one the isobars run along the span and the flow does not, so part of every new vortex line points along the flow — half of it, over the accelerating front of a cylinder swept 35°. That part is where a swept wing's crossflow comes from.

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

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.

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