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

Riemann invariants — where it appears

A combination of flow variables that is constant along a characteristic. It reduces a hyperbolic problem to book-keeping along curves, and it is how a shock tube's whole solution is written down without a grid.

Named by 7 essays across 3 fields — 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 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 121 m after the cavity closes, against 69 m from the closure. The head at a valve shut instantly on water flowing at 0.36 m/s through 600 m of 100 mm pipe, a = 1200 m/s, with a steady head of 25 m. The closure raises it to 69.1 m, the Joukowsky head; the reflection returns at one round trip, 1.00 s, and takes the head down to the vapour head, −10.1 m, where a cavity opens (shaded). It closes 2.146 round trips after the closure, and the first pulse after it reaches 121.3 m — 52.3 m above the Joukowsky head — for 146 ms. The step line is the exact solution between events; the thin line is a 240-reach grid solver that was told nothing about it and agrees with its first pulse to better than a millimetre.

Twice the margin, on top of the hammer

Shut a valve on a line whose pressure is low and the returning wave boils the water beside it. When that cavity closes, the head at the valve can pass the Joukowsky rise — by up to twice the margin that let the water boil, in a sawtooth that jumps each time one more round trip fits into the cavity's life, and for a time that is shortest exactly when the pulse is tallest.

applied · Water hammer 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 At the equator the front runs as long as the land is warmer; at forty-five degrees it stops hours before. The sea-breeze front's distance inland against the hours since the land first became warmer than the sea, at the equator and at 30°, 45° and 60°. At 0° it reaches 185 km. At 30° it reaches 133 km, 99 per cent of that by hour 11. At 45° it reaches 99.2 km, 99 per cent of that by hour 9.1. At 60° it reaches 80.3 km, 99 per cent of that by hour 8. The quasi-steady estimate, the head speed of a current fed from a deep reservoir at each hour's contrast, integrated, is 189 km.

The sea-breeze front runs till evening only near the equator

A real sea breeze arrives: the cool sea air runs inland under the warm air as a gravity current, its head moving at a speed set by the density contrast and the depth of the cool layer. How far it gets depends on the latitude. Near the equator the head keeps running as long as the land stays warmer, and the length of the afternoon decides the reach. In mid-latitudes the Earth's rotation turns the current along the coast and stops the head hours before the contrast ends, a third further inland than the Rossby radius, and a longer afternoon adds nothing.

inviscid · Kelvin

Named alongside it

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

Expansion fanCharacteristicsModel limitContact surfaceMethod of characteristicsRiemann problemShock tubeEntropyExact solutionIsentropicNormal shockRankine–Hugoniot conditions

All concepts