Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Mach cone — where it appears

The cone of influence trailing a supersonic source, whose half-angle is the inverse sine of one over the Mach number. Outside it nothing has been heard, which is what it means for a flow to arrive without warning.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

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.

compressible · Speed of sound A 10° wedge at Mach 2.00 has two shocks that solve it. The wedge turns the flow through a fixed angle, and the θ–β–M relation offers two shock angles that achieve it. The weak solution, drawn steeply forward, leaves the flow supersonic and is what a wedge in a free stream produces. The strong solution leaves it subsonic and appears where downstream pressure forces it. Nothing local to the wedge chooses between them.

A shock that leans

Tilt a shock and only the velocity component across it is changed — the component along it passes through untouched. That single observation turns every oblique shock into a normal shock in disguise, and it is why a wedge at Mach 2 leaves the flow supersonic while a blunt nose does not.

compressible · Oblique shock A 20° cone at Mach 2: the shock at 37.80°, and a flow still compressing behind it. The conical flow round a cone of 20° half-angle at Mach 2, from the Taylor–Maccoll equation. The shock sits at 37.80° from the axis and turns the flow crossing it through only 8.57°, leaving it at Mach 1.693. Between the shock and the surface every ray from the apex carries its own state: the Mach number falls from 1.693 just behind the shock to 1.568 at the surface and the pressure rises from 1.586 to 1.912 times the free stream's — the rest of the turn, made without a shock. Nothing depends on the distance from the apex, so the rays are lines of constant state.

A cone finishes its turn after the shock

A wedge turns a supersonic stream all at once, at its shock. A cone of the same angle does not: its shock turns the flow only part of the way and leaves the rest to a smooth compression between the shock and the surface. Solved from Taylor and Maccoll's equation, the cone's shock is weaker, keeps more of the total pressure, carries less than half the wedge's surface pressure, and stays attached to 40.7° at Mach 2 where the wedge gives up at 23°.

compressible · Oblique shock

Named alongside it

The objects these essays reach for when they reach for this one.

Shock waveMach numberOblique shockThe θ–β–M relationTotal pressureWedgeCharacteristicsDetachmentDomain of dependenceEntropyHyperbolicIsentropic

All concepts