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

Contact surface — where it appears

A surface separating two bodies of gas that move together at the same pressure but are otherwise different. Temperature, density and composition all jump across it, and it carries a permanent record of where each side came from.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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 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 Where the reflected shock meets the driver gas. Distance against time in a helium-driven air tube, the incident shock in red running from the diaphragm to the end wall, the contact surface behind it in green, and the reflected shock coming back. Where it meets the contact, an under-tailored tube sends an expansion back to the wall — in ochre — which lowers the reservoir's pressure, and the contact drifts back towards the diaphragm; an over-tailored one sends a shock, which raises it. At the tailored Mach number, 3.41, nothing is sent back, the contact stops dead, and the reservoir at the end wall holds until something slower arrives.

The shock that passes without an echo

A shock tunnel's reservoir lasts until a wave comes back from the driver gas to spoil it. For each pair of gases there is one incident shock strength at which nothing comes back at all, and it is found by asking the reflected shock to stop two different gases at the same pressure.

compressible · Shock tube Tailored, the reservoir waits for the driver's own expansion. Distance against time in a helium-driven air tube at the tailored Mach number, 3.41, with a driver half as long as the driven tube: the incident shock, the contact surface, the reflected shock and the shock it transmits into the driver gas, and the driver's expansion — its head running back to the driver's closed end, and the head reflected from there. Nothing returns from the contact, and the reservoir at the end wall holds from 0.293 until the reflected head arrives at 0.555 L/a₁: a test time of 0.261.

A tailored tube buys its test time with its driver

Tailoring a shock tube removes the wave the contact surface would send back to the reservoir. What ends the reservoir then is slower: the driver's own expansion, which runs back to the driver's closed end, reflects, and has to cross the whole tube to reach the end wall. Its arrival is exact in one dimension, because the reflected head crosses the incident fan as a simple wave, and the answer is that test time is bought with driver length — about seven-tenths of a driven-tube crossing time per driver length for helium — and that tailoring is worth nothing with a driver shorter than a quarter of the tube.

compressible · Shock tube

Named alongside it

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

Expansion fanRiemann problemShock tubeEntropyRiemann invariantsCharacteristicsModel validityNormal shockRankine–Hugoniot conditionsShockConserved quantityImpedance

All concepts