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The thread: One number decides the regime — page 7

Page 7 of 34, continuing through the 305 essays this motif runs through.

305 essays carry this thread — page 7 of 34.

The flap moves the curve and leaves its slope alone. Lift coefficient against incidence for several flap deflections. The lines are parallel: deflecting the flap gives the section lift at an incidence where it had none, and does not change how much lift each further degree of incidence buys.

What a flap does, and what it does not

Lowering a flap gives a wing lift at an incidence where it had none. It does not make the wing more responsive to being pitched — the lift curve moves sideways and its slope does not change, and those are two quite different things to buy.

The Ohnesorge diagram, with the boundaries where they belong. The classical map of jet break-up: the Ohnesorge number against the jet Reynolds number, with the five nozzles placed on it. The three sloping lines are Reitz's transitions in the gas Weber number, and their geometry is computed rather than sketched — a fixed We_g means Oh·Re is fixed, which is a straight line of slope exactly −1 in these coordinates, and the assertion checks that a decade in Reynolds number moves each line by exactly one decade. Where they sit is borrowed; that they are straight and parallel is not. A nozzle below and to the right of the last line atomises.

Regimes and numbers

Where a jet stops being a jet

A tap makes drops a few centimetres down, a garden hose makes a stream that carries, a sprayer makes a mist and a diesel injector makes fog. Same liquid, same mechanism, four regimes — and the number that separates them is not the jet's inertia but the surrounding air's.

Fluids at work

The fastest way is not the straight one

A boat racing to a mark dead upwind sails seventy per cent further than the distance to it, and arrives first. The best angle is forty-five degrees plus half the apparent wind angle, it comes out of one line of trigonometry, and the same argument says to gybe downwind rather than run.

$Induced drag against aspect ratio, all at C_L = 0.6. Induced drag for wings of different aspect ratio, each flown at whatever incidence makes it carry the same lift coefficient. The drag falls as one over the aspect ratio, so the longest wing here pays a fraction of what the shortest does for exactly the same load.$ Circulation and lift

The span is the whole story

Induced drag is the price of having ends, and the only thing that lowers it at a given lift is putting those ends further apart. Not more area, not a better section, not a cleverer planform — span, and everything else is a correction of a few per cent to it.

Three equations, and the set they never leave. The Lorenz trajectory at r = 28, projected on x and z, after the transient has been discarded. It never repeats, never leaves, and never crosses itself in three dimensions. The Lyapunov exponent printed beside it is measured on this system by separating a nearby pair, so the claim of sensitive dependence is a computation.

Transition and turbulence

Three numbers left of a fluid

Saltzman truncated convection to three Fourier modes and Lorenz studied what was left. The result changed science, and it stopped being a description of a fluid at about a fifth of the way to the parameter everybody quotes it at.

A million times faster, and the constant is 48.0. The effective diffusivity along a pipe, divided by the molecular one, against the Péclet number — both logarithmic. Below Pe ≈ 7 the tracer simply diffuses and the curve is flat at one. Above it the dispersion is all Taylor's, rising as the square of the Péclet number, so a thousandfold Péclet number is a millionfold enhancement. The constant in D(1 + Pe²/48) is not quoted here: it is recovered from a numerical solution of the cell problem across the section, giving 48.0000 for a tube and 52.5 for a plane channel, which is Aris' 2/105.

Regimes and numbers

Two slow things make a fast one

Shear stretches a slug of dye and mixes nothing, because it is reversible. Molecular diffusion is hopeless at any scale bigger than a hair. Put the two together in a pipe and the dye spreads along it with an effective diffusivity two million times the molecular one — which gets larger as the molecular one gets smaller.

Past 23.0° at Mach 2.00 there is no attached shock. The same wedge at two half-angles. On the left the θ–β–M relation has a root and the shock sits on the nose. On the right it has none, and the solver throws rather than returning the nearest thing — which matters, because a solver that quietly clamped to the maximum would draw a neat attached shock on a body that cannot carry one. The bow shock on the right is indicative: its shape is not solved here.

Compressible flow

When the wedge is too blunt

Every curve of shock angle against deflection has a maximum. Past it there is no attached shock at any angle — the solver has no root to return and must say so, rather than quietly handing back the nearest thing and drawing a picture that cannot exist.

A millionth, doubling every three-quarters of a second. The separation of two trajectories started a millionth apart, on log axes against time. It grows as a straight line until it saturates at the size of the attractor, and the slope of that line is the largest Lyapunov exponent — measured here on the system by renormalising a nearby pair, not quoted. Sensitive dependence is what the straightness of the line means.

Transition and turbulence

A millionth is enough

Two trajectories a millionth apart separate by a factor of e every three-quarters of a second, so a millionfold improvement in the measurement buys about ten seconds of extra prediction. That exchange rate, and not the size of the error, is what limits forecasting.

The left wall is not a speed, it is an angle drawn in speed coordinates. The V–n diagram: every combination of speed and load factor the aircraft can reach. The curved left boundary is the wing at its stalling angle — n = ½ρV²S C_Lmax /W, a parabola — and it is the same limit at every point along it. The flat top and bottom are the structure. Where the two meet is the corner speed, 56.6 metres per second here: the slowest speed at which the aircraft can reach its limit load factor, and therefore the speed at which it turns hardest. Above it the wing can pull more than the structure allows — at the never-exceed speed it could reach 9.6g before stalling — and the pilot's limit stops being the air.

What is taught wrongly

An angle, not a speed

The number is printed in the handbook, marked in white on the airspeed indicator and used in every briefing, and the wing has no way of knowing it. A wing stalls at an angle. The speed at which an aeroplane reaches that angle is an answer with four other variables in it, and every one of them moves.

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