Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: One number decides the regime — page 2

Page 2 of 34, continuing through the 305 essays this motif runs through.

305 essays carry this thread — page 2 of 34.

A source at Mach 2.00, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all. Compressible flow

When the warning cannot arrive

Supersonic flow is not fast flow. It is flow in which the fluid ahead has been told nothing, because the body is outrunning its own pressure signals — and that single change turns an equation of one type into an equation of another.

 Two drag laws, one derived and one fitted. Flat-plate drag coefficient against Reynolds number, laminar and turbulent, on log axes. The laminar curve is Blasius' similarity solution, solved by shooting; the turbulent one is the 1/7-power correlation, which was fitted to experiment and is drawn dashed-in-kind to keep the difference visible. Their slopes differ — −1/2 against −1/5 — so the gap between them widens rather than staying put. Transition and turbulence

The number that is not a number

Transition Reynolds numbers are quoted to three figures and vary by two decades. That is not sloppiness in the measurement — it is the honest report of a quantity that depends on the laboratory as much as on the fluid, and knowing which part is which decides what may be designed on it.

 A shear layer, and the point of inflection in it. The velocity profile U = tanh y across a layer of finite thickness, with the inflection point located by searching for a sign change in the second derivative rather than by reading it off the algebra. Rayleigh's theorem says an inviscid parallel flow can only be unstable if such a point exists — a necessary condition, not a sufficient one. Transition and turbulence

A layer with a kink in it

Rayleigh proved in 1880 that an inviscid shear flow cannot be unstable unless its velocity profile bends the other way somewhere. It is one line of algebra, it is necessary and not sufficient, and it ties instability to separation through the sign of a single derivative at the wall.

 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.

 A Joukowski aerofoil at 6°. A cambered aerofoil in a uniform stream. The circulation is not chosen: it is whatever value makes the flow leave the sharp trailing edge smoothly, and that single condition fixes the lift. Circulation and lift

The lift curve, and why it is a straight line

Lift against angle of attack is a straight line, it does not pass through the origin, and its slope is very close to a number that has no business being there. All three facts fall out of the theory.

 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.

 Mach number: one number, four different flows. Mach number is speed ÷ speed of sound. 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

When air stops being incompressible

Air is a gas and can obviously be squeezed, yet most of aerodynamics treats its density as fixed. The assumption holds until the flow approaches the speed at which pressure information travels — and then everything changes at once.

 The laminar boundary-layer profile. Speed against height through a laminar boundary layer on a flat plate, in the similarity variable that collapses every station along the plate onto one curve. The straight line is the slope at the wall, which is what the skin friction is proportional to. Viscosity

How thick is thin

The boundary layer has no edge. It approaches the free stream and never arrives, so any thickness quoted for it is a convention — and the three conventions in use measure three different things, one of which is not a height at all.

 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.

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