Theme

The thread: One number decides the regime — page 16

Page 16 of 34, continuing through the 305 essays this motif runs through.

305 essays carry this thread — page 16 of 34.

The wall is a boundary condition, solved for rather than reflected. A model's trailing vortices in a closed working section, with the sheet of sources that makes the wall a wall drawn as a displacement of the wall itself. The interference is an upwash — the model looks better than it is, and the correction factor comes to 0.1242 against the exact value 0.1250 that the image at the inverse point gives for a circle. The arrows are the interference velocity alone, with the model's own downwash removed, drawn to a common scale set by the longest of them.

The walls are in the answer

A wind tunnel measures a wing in a box the aeroplane will never fly in, and the box is worth a fifteenth of the induced drag. The correction is exactly an eighth for a closed circular section and exactly minus an eighth for an open jet, and the sign is the whole argument.

$Where Einstein's line stops being the measurement. The viscosity of a suspension of rigid spheres relative to the liquid's, against the volume fraction. Einstein's 1 + 5φ/2 is exact for one sphere and holds while the spheres cannot feel one another, which is up to about five per cent by volume. Batchelor and Green's two-sphere term takes it a little further; beyond about a fifth nothing derived works and the curve drawn is a fit, which diverges at a maximum packing that is itself a measurement.$ Viscosity

A viscosity made of particles

Stir rigid spheres into a liquid and the mixture is thicker, by five halves of the volume fraction. Einstein's coefficient is not an empirical constant — it is a dissipation calculation on one sphere — and doing it as an energy rather than as a stress shows that four-fifths of it comes from somewhere nobody mentions.

Ideal flow

The body the outer flow actually sees

A boundary layer lets less fluid past than an inviscid one would. The outer flow can be given exactly the same reduced flow rate by leaving the fluid inviscid and moving the wall out — so the potential flow that matters is not the flow past the body, but the flow past the body plus a thickness the boundary layer computes.

Four bodies of identical length and volume, and their wave drags. Each body has the same length and the same volume; only the distribution of area along it differs. The Sears–Haack body — the spindle whose area goes as the three-halves power of x(L−x) — has the least wave drag of the four, and every other shape pays between thirty-seven per cent and a hundred and seventy-five per cent more for carrying the same volume the same distance. Nothing about the cross-sections' shape enters: only the area distribution does.

Compressible flow

The least drag a volume can have

A body's supersonic wave drag depends on nothing about it except how its cross-sectional area is distributed along its length. Minimising that for a given volume gives one shape — and the answer goes as the volume squared over the fourth power of the length.

Storage and loss moduli, for one relaxation time and for a spectrum. The two moduli of a Maxwell fluid and of a Rouse chain, against frequency in units of the longest relaxation time. One relaxation time makes the storage modulus overtake the loss modulus at λω = 1 and then leave it behind without limit. A spectrum of 1000 modes makes them rise together as the square root of frequency and stay a fixed ratio apart, so the material never becomes the solid the single time predicts.

Regimes and numbers

A solid, if it is not given time

The Deborah number is the only group on this site with no fluid in it — two times and nothing else — and it says a material is a solid or a liquid depending on how long anybody watches. What it throws away is that no real material has one time, and the spectrum it replaces changes the answer in kind rather than in degree.

Model spectra at four Reynolds numbers, compensated. The spectrum multiplied by k^(5/3) and divided by eps^(2/3), so that a true inertial range is a horizontal line at the Kolmogorov constant. What a finite Reynolds number has instead is a single maximum: it reaches 1.4996 at the highest and 1.49 at the lowest, and the band over which it is flat to one per cent goes from a third of a decade to two.

Transition and turbulence

The range a real Reynolds number does not have

Kolmogorov's minus five thirds is a statement about a band of scales that has forgotten the forcing and does not feel the viscosity. Both conditions are about separation, and separation is exactly what a finite Reynolds number does not have much of.

A patch of dye, folded. A circle of marker particles carried by the flow, at four times. It is stretched into a filament and folded through itself, and its area does not change at any point of that — which is not visible in the picture, and is the whole difficulty. A scheme that lost eight per cent an orbit would produce a picture indistinguishable from this one.

Flows and fields

The area that must not move

A patch of dye in an incompressible two-dimensional flow keeps exactly the area it started with, for ever. Two respectable integrators are put on the same flow: one respects that identically at any step size, the other does not, and the pictures they draw are the same picture.

A factor of 1.8 between the extremes. The same numbers as a bar. The spread is exactly 96 over 160/3, which is 1.8 — and it is a spread in a quantity that a single correlation using the hydraulic diameter treats as a constant. A rectangle twenty times as wide as it is deep is 78 per cent more resistant than a circle of the same hydraulic diameter.

Fluids at work

A number that is only the shape of the hole

The friction factor times the Reynolds number for laminar flow in a duct is a pure number that depends on the cross-section's shape and on nothing else — not the fluid, not the size, not the flow rate. It is exactly 64 for a circle and exactly 96 for a slot, and the correlation that treats them as the same is wrong by a third.

One sign change, and both of a stall's surprises follow from it. A lift curve with a peak, and the rolling moment a wing makes against its own roll at the same incidence. Below the peak the slope of the lift curve is positive, the down-going wing makes more lift, and the roll is opposed. Past the peak the slope is negative, the down-going wing makes less, and the roll is reinforced. The stalling angle here is 16.46° and the damping changes sign at 17.07°. Nothing in this picture is a spin yet — a spin needs yaw as well — but the engine that drives one is the crossing of that line.

What is taught wrongly

A roll that feeds itself

A spin is routinely described as a stall that got worse, and it is not a stall at all in the sense of an angle rather than a speed. It is autorotation — a roll that sustains itself because past the peak of the lift curve the down-going wing makes less lift rather than more — and the arithmetic says it begins a little past the stalling angle rather than at it.

All themes