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

Expansion fan — where it appears

The continuous turning of a supersonic flow round a convex corner, spread over a fan of Mach waves rather than concentrated in a discontinuity. It is isentropic, which is why turning away from the flow is free and turning into it is not.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The same 10° turn, taken both ways. A supersonic stream turned away from itself expands through a fan of Mach waves and keeps every bit of its total pressure. Turned into itself through the same angle it shocks, and pays. Nothing in the equations distinguishes the two cases except the sign of the angle: compression waves converge and steepen into a front, expansion waves diverge and spread.

Turning the other way is free

Compression through ten degrees at Mach 2 costs 1.54 per cent of the total pressure. Expansion through the same ten degrees costs exactly nothing — not approximately nothing, nothing — and the two are the same equations with the sign of one angle changed.

compressible · Expansion fan Three waves out of one discontinuity. The x–t diagram of the burst diaphragm. A shock runs right at a speed of its own, a contact surface follows it more slowly, and an expansion fan spreads left as a family of rays that opens with time — the three wave families the Euler equations possess, produced at once by an initial condition with no waves in it at all. Every straight line here is a speed the solution computed, and the fan is drawn as the rays it actually consists of.

One diaphragm, every wave

Two states of the same gas at rest, separated by nothing, is the simplest initial condition compressible flow admits — and its answer contains all three waves the equations possess at once: a shock one way, an expansion fan the other, and between them a surface across which the density jumps and the pressure does not.

compressible · Shock tube The wall is what cancels the waves it made. The characteristic net of a minimum-length nozzle designed for Mach 2.4, with 18 waves. The pale lines run from the sharp throat down to the axis, reflect there by symmetry, and run back up to the wall; the wall turns through exactly the angle needed to cancel each one as it arrives, so nothing reflects back into the flow and the exit is uniform at Mach 2.400 and parallel to 0.0 degrees. The area ratio decides the Mach number and this net decides the shape, and the two agree on the exit height to 0.51 per cent at this resolution.

The wall that cancels its own waves

The area ratio of a supersonic nozzle fixes its exit Mach number and says nothing whatever about its shape. What fixes the shape is a wave-by-wave construction in which the wall turns through exactly the angle needed to absorb each expansion as it arrives — and getting it wrong leaves a stream full of oblique shocks at precisely the right Mach number.

compressible · Characteristics A compression piston, and where its characteristics first cross. Sixty C+ characteristics from an accelerating piston, drawn in the distance-time plane. Each is a straight line, because the invariant makes the state along it constant; later ones are faster, because the gas ahead of them has been compressed; so they converge, and the first crossing is the shock. The envelope formula gives 2.7529 and the first actual crossing is at 2.7510.

Two numbers that do not change

One-dimensional unsteady gas flow carries two quantities that are exactly constant along two families of curves. That single fact turns a pair of coupled partial differential equations into a family of straight lines, and gives an exact speed at which a gas outruns its own expansion.

compressible · Characteristics The four waves, and the one that never goes away. The shock tube in space and time: a shock running right, an expansion fan running left, and the contact surface between them. The shock and the fan are travelling disturbances that leave; the contact is made of fluid, so it is carried along and is there for ever.

A surface that remembers the diaphragm

Between the shock and the expansion in a shock tube there is a surface across which the pressure and the velocity are identical and the temperature differs by a factor of two. It is made of fluid, so it never goes away, and nothing in the pressure field says it is there.

compressible · Shock tube

Named alongside it

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

CharacteristicsEntropyIsentropicRiemann invariantsContact surfaceMeasurementPrandtl–Meyer expansionRiemann problemShock tubeShock waveArea machConservation

All concepts