The thread: A smooth picture proves nothing — page 2

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.

Viscosity

The two theories, side by side

The exact solution and the real flow, for the same body in the same stream. One is beautiful and predicts nothing has drag; the other is approximate and has a wake in it. Where they agree and where they part is the whole map of the subject.

The velocity field, arrows to scale. The same flow drawn as arrows. Scaled to the local speed the picture is honest and crowded; drawn all the same length it is legible and hides the very variation the figure is about.

Flows and fields

What a flow is

A fluid is made of molecules and nobody models it that way. Treating it as a continuous field with a velocity at every point is an approximation, an extremely good one, and knowing why it works is knowing where it stops.

Every wavelength grows, and the shortest grows fastest. Growth rate against wavenumber for a vortex sheet of zero thickness and for a layer of thickness δ. The sheet's curve rises without limit, which is the model announcing that it has no shortest scale in it; the layer's turns over and dies at kδ = 1, the neutral wavenumber computed from Rayleigh's equation rather than fitted.

Transition and turbulence

Every wavelength at once

A vortex sheet of zero thickness is unstable at every wavelength, and the shorter the wavelength the faster it grows. The answer has no smallest scale in it, which is not a fact about fluids — it is the model reporting that it left something out.

The Kelvin wake, and the angle it comes out at. The wake behind something moving over deep water. Each curve joins everything radiated at one moment, wherever it has since travelled to. All of it stays inside a wedge whose half-angle is the inverse sine of one third, about nineteen and a half degrees, and that number does not depend on the speed, on gravity, or on the size of the vessel.

Regimes and numbers

The angle that does not care

Every ship on deep water leaves a wake inside a wedge of the same angle. Not roughly the same — the same, for a rowing boat and a supertanker, at any speed either of them can manage, on any planet with any gravity.

The acceleration field of a steady flow. How hard the fluid is being accelerated at each point of a steady flow past a cylinder. The flow does not change with time anywhere in this picture, and yet almost nowhere in it is a parcel travelling at constant velocity — the pattern stands still while the fluid running through it is thrown about.

Flows and fields

Steady does not mean nothing is happening

Photograph the flow past a cylinder twice and the two pictures are identical. Every parcel of air in them is being thrown about — braked to a dead stop, hauled round the shoulder at nearly twice the free-stream speed, braked again. Both statements are exactly true.

Four ways of photographing one flow, and what each of them records. The same solved flow, rendered as four different laboratory techniques would record it. Smoke from a port gives a streakline; tufts give direction with no speed in it at all; an oil film gives the direction of the friction on the surface rather than the flow above it; pressure taps give a scalar with no direction in it. None of the four is the velocity field, and only the first happens to coincide with a streamline, because this flow is steady.

What is taught wrongly

What a photograph of a flow shows

Wind-tunnel pictures are the evidence this whole subject is argued from, and hardly anybody says which quantity a given technique records. They are not interchangeable — smoke, tufts, oil and pressure taps measure four different things, and none of them is the velocity field.

Flows and fields

Spin is not the same as going round

A whirlpool whose streamlines are perfect circles can have no rotation in it anywhere. A flow whose streamlines are dead straight and parallel can be rotating everywhere. Both statements are true, and getting them the wrong way round is the most expensive confusion in the subject.

Reynolds number: one number, four different flows. Reynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.

Regimes and numbers

The model that cannot be matched

A scale model behaves like the real thing when its dimensionless numbers agree. With one number that is a matter of choosing the tunnel speed. With two it is usually impossible, and every wind-tunnel result ever published has been obtained in spite of that.

The throat stops listening: mass flow against back pressure. Mass flow through a convergent nozzle, normalised on its choked value, as the back pressure is lowered. It rises until the throat reaches Mach one and then stops, exactly. Below that pressure the throat is sonic and nothing downstream can send a signal upstream to ask for more, so the flow does not respond however far the back pressure falls.

Compressible flow

The throat that stops listening

Lower the pressure downstream of a nozzle and more gas flows through it. Keep lowering it and, at a definite point, the flow stops responding — not gradually, but completely, because the news that the pressure has fallen can no longer travel upstream.