Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Fanno rayleigh — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two different physics, one destination. The Fanno and Rayleigh lines, each with temperature up the page and entropy across, each referred to its own sonic state. Neither is a curve anybody drew: the entropy was swept over Mach number and its maximum located, and it sits at Mach 1 on both to within the resolution of the sweep. The subsonic branch runs up to the nose from the left and the supersonic branch runs up from below, so whatever is being done to the flow, it can only be moved towards that point.

    Two ways to choke

    Friction and heat are different physics acting on different conservation laws, and they drive a duct's flow to exactly the same place. Both have their entropy maximum at Mach 1, so neither can push a subsonic flow past it and both drag a supersonic one down to it — and where their two curves cross is a shock.

    part 1 · compressible
  2. The Fanno line, which has only one direction on it. The entropy relative to the sonic state, against Mach number, for both branches. Friction raises the entropy, so a duct flow moves to the right along this curve whichever branch it is on — up in Mach number from below and down from above — and it stops at the sonic point.

    A duct that cannot be run backwards

    Friction drives a compressible duct flow towards the speed of sound from either side, and the entropy rises the whole way. So the state of the gas at a station is an odometer: it records how much duct is behind it, and no amount of further duct can take it back.

    part 2 · compressible
  3. A pipe at the gas's temperature does not keep the gas at it. Left, the static temperature of the gas along a pipe, as a fraction of the wall's, against the local Mach number, entered at Mach 0.1: with no heat transfer, with the heat transfer Reynolds' analogy gives, and with ten thousand times that. Right, the stagnation temperature. The isothermal model holds the static temperature at the wall's and needs the stagnation temperature to climb by 14 per cent; every real case keeps it within two per cent, cools, and runs on to Mach one.

    A pipe cannot hold its gas at the wall's temperature

    The textbook model of a long gas pipe in contact with the ground holds the gas at the ground's temperature and has it choke at 0.845 of the speed of sound. No pipe does either. A wall at the gas's temperature draws heat out of it rather than putting heat in, and with any strength of heat transfer at all the flow runs on to Mach one, within a tenth of a per cent of the length a perfectly insulated pipe would need.

    part 3 · compressible
  4. A long line empties from the break backwards. The pressure along a gas line three hundred friction lengths long — fL/D = 300 — at five times after it breaks at its far end into still air at a twentieth of its pressure, the break on the right. Half a transit time in, the rarefaction has crossed half the line and the gas beyond it has not moved. Friction then holds the pressure in a slope that lengthens back towards the closed end; at six transit times the closed end is at 0.55 of its starting pressure, at fifteen 0.21, and the break has long since stopped being choked.

    A broken line empties at the pace of its friction

    When a gas line breaks, its open end chokes and the textbook stops there: a choked outlet passes gas at its own speed of sound and nothing downstream can change that. On a long line the choke is the least of it. The break stays choked only while friction lets enough gas reach it, which on a line a thousand friction lengths long is the first eighth of the time to half pressure. The pressure falls at a pace friction sets, the half-time grows as the square root of the line's length in friction lengths, and the wall's heat — negligible in the steady line — makes the blowdown a third slower.

    part 4 · compressible

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