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

Choking — where it appears

The condition in which a duct's mass flow can no longer be increased by lowering the downstream pressure, because the throat has reached the speed of sound. The information that the pressure has fallen cannot travel upstream past that point.

Named by 16 essays across 3 fields — each of them below, with the objects they name alongside it.

The throat stops listening: mass flow against back pressure. Mass flow through a convergent nozzle, normalised on its choked value, as the back pressure is lowered. It rises until the throat reaches Mach one and then stops, exactly. Below that pressure the throat is sonic and nothing downstream can send a signal upstream to ask for more, so the flow does not respond however far the back pressure falls.

The throat that stops listening

Lower the pressure downstream of a nozzle and more gas flows through it. Keep lowering it and, at a definite point, the flow stops responding — not gradually, but completely, because the news that the pressure has fallen can no longer travel upstream.

compressible · Area mach One nozzle, five back pressures, five different flows. The duct above, and the static pressure along it below, computed station by station from the local area. Where a shock stands inside the divergent section its position was solved for rather than placed: the shock spends total pressure, which sets the subsonic Mach number at the exit, which has to match the imposed back pressure.

One area, two answers

The area of a duct at a station fixes the Mach number there — twice over. A ratio of 2.5 is satisfied at Mach 0.24 and again at Mach 2.44, nothing local chooses between them, and where the choice cannot be made consistently a shock appears inside the duct to join the two.

compressible · Area mach Two depths for the same energy, and one for the least. Specific energy against depth for a discharge of 0.5 square metres per second per metre of width. Every energy above the minimum is carried by two different depths — one fast and shallow, one slow and deep — and the minimum is carried by exactly one. That depth is the critical depth, the Froude number there is one, and the least energy is three halves of it; all three are found here by search and checked against their closed forms.

The depth that costs least

For a given flow there are two depths that carry it at any energy above a floor, and exactly one at the floor. That one depth is where the Froude number is one, the least energy is exactly three halves of it, and a bump in the bed that asks for more than the flow has does not thin the water — it backs it up.

applied · Open-channel 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.

compressible · Fanno rayleigh 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.

compressible · Fanno rayleigh With friction the flow goes sonic after the throat, and leaves slower. The Mach number along a convergent–divergent nozzle of exit area ratio 2.5, for friction lengths 4fL/Dₜ of 0, 0.2 and 1, each solution passing smoothly through Mach 1 at the point where the sonic condition holds. Without friction that point is the throat, at x = 0.42. With friction it moves downstream — to 0.4357 and 0.4996 — and the throat itself is subsonic, at Mach 0.965 and 0.852. The exit Mach number falls from 2.443 to 2.247 and 1.747. The throat is where the area is least; the sonic point is where the widening has caught up with the friction, and the two coincide only when there is none.

Friction moves the sonic point past the throat

A choked nozzle is sonic at its throat — in a nozzle with frictionless walls. With friction the flow reaches Mach one where the section's widening rate has caught up with the friction, which is downstream of the throat, and the throat itself is subsonic. The solution through that point is a saddle that can only be found from the inside, and the mass flow it passes is less than the throat's area allows.

compressible · Area mach At 70 per cent speed the first stage runs at 0.64 of its flow coefficient and the last at 1.17. The flow coefficient of each stage of a compressor of 8 stages at 300 m/s mean blade speed, drawn for a flow coefficient of 0.5 and a work coefficient of 0.35, as a share of the value its blades were cut for, along the operating line a choked exit nozzle sets, at 110 per cent, 100 per cent, 90 per cent, 80 per cent, 70 per cent of design speed. At design speed every stage is at exactly one. Below it the front stages fall towards the stall limit (shaded below 0.82) and the rear ones rise towards the choke limit (shaded above 1.3): at 70 per cent the first stage is at 0.642 and the eighth at 1.170. Above design speed the pattern reverses, the front stages rising and the rear falling.

Matched at one speed and at no other

Every stage of a compressor passes the same mass flow, and the annulus behind each one is cut for the density the air will have reached there at design speed. Slow the shaft and the air is less dense than the metal expects, so the rear stages carry more volume than they were drawn for while the front ones starve — and below a definite speed no throttle setting keeps all of them working at once.

applied · Turbomachine The throat holds the hammer back only while its cavity lasts. Left, pressure at the closing valve (red) and at the upstream face of the venturi (gold); right, the throat's cavity volume; after the valve shuts with a 5 mL cavity in the throat. The valve sees the full Joukowsky rise of 14.8 bar at once. The wave reaches the venturi 33.3 ms later, and for the next 7.9 ms the upstream pipe hears nothing: its pressure stays at 5 bar while the cavity is squeezed. When the cavity closes at 41.3 ms the surge passes into the upstream pipe at 13.8 bar above its steady pressure.

A choked throat buys time, not silence

A venturi whose throat has reached vapour pressure passes a flow the downstream pressure cannot change, and it is tempting to read that as isolation: whatever happens downstream, the upstream pipe will not hear it. Slam a valve downstream and it hears it. The cavity at the throat holds the surge back only for as long as it takes to fill, and then lets 93 per cent of it through.

misconceptions · Venturi The characteristic gets a wall, and the wall does not care about the discharge. One jet pump's characteristic, at an area ratio of 0.275, with the flow ratio at which its throat entry reaches vapour pressure drawn for three values of the cavitation parameter σ = (Pₛ − pᵥ)/(Pₘ − Pₛ). Left of a wall the machine runs on its curve. At the wall no lower discharge pressure raises the flow: the head ratio can fall to zero along the vertical and the flow ratio stays where it is. The wall's position contains the nozzle and suction losses and nothing downstream of the throat entry.

The wall the suction puts in the curve

A liquid jet pump's lowest pressure is where the entrained stream enters the throat, and when that reaches vapour pressure the pump curve stops being a curve. The flow ratio freezes at a value no lower discharge pressure can move — and raising the motive pressure, the obvious cure, brings the wall closer.

applied · Ejector Flat until the back pressure reaches a value, then gone. Entrainment ratio against back pressure, as a multiple of the suction pressure, for three mixing-section sizes of one steam ejector driven from a motive supply a hundred times its suction pressure. Each is exactly flat while the entrained stream is choked beside the jet, up to its critical back pressure. The dashed lines join that point to the back pressure at which the entrainment has fallen to nothing, 1.4 per cent higher for the middle machine. The model fixes those two ends and not the path between them. A larger mixing section entrains more and breaks at a lower back pressure.

A choke that belongs to two streams

A steam ejector entrains a fixed amount of gas whatever its back pressure, up to a pressure where it stops. The flat part is a choke, and the entrained gas is not at Mach one when it happens: it is at 0.886, because the supersonic jet beside it is part of the same throat. One area ratio then trades that entrainment for compression, and the trade decides how a vacuum train is built.

applied · Ejector 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.

compressible · Fanno rayleigh 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.

compressible · Fanno rayleigh The exit area is a condition, not a second choice. Critical entrainment and critical compression ratio for a steam ejector with a mixing tube of 60 nozzle throats, against the nozzle's exit area as a multiple of the exit that delivers the jet at exactly the pressure it shares with the entrained gas. Both peak at the matched exit, at every suction drawn, so no exit area buys entrainment at the cost of compression or the other way. Halving the exit costs 2.4 per cent of the entrainment at a hundredth of the motive pressure, doubling it 3.2 per cent.

An ejector's nozzle exit is a condition, not a choice

A steam ejector has two areas a designer can pick, the mixing tube's and the nozzle exit's, and one of them looked like a second way round the trade between entrainment and compression. It is not. Both are largest with the exit matched to the pressure the jet meets, as a rocket's thrust is, and every other exit draws a curve inside the matched one. What moves the machine is loss, and the losses sort themselves: only the nozzle's reaches the entrainment, and the sharp edge of the characteristic belongs to the ideal machine alone.

applied · Ejector Zero wave drag at one Mach number, inside a loop it cannot enter from below. The wave drag of the pair, per unit of one element's chord, against Mach number, beside a single diamond of the pair's combined thickness. The biplane's drag is zero at its design Mach number, 1.944, and rises either side. Shaded: below 1.6 the channel cannot pass the flow at all; between 1.6 and 2.15 it can, but only if it was started above the band. Accelerating from subsonic, the biplane stays choked until Mach 2.15, past its design point.

Busemann's biplane has to be flown past its design point

Two half-diamonds facing across a channel cancel each other's waves at one Mach number and carry their thickness at no wave drag. The cancellation needs the channel narrowed at its middle, and a narrowed supersonic channel will not swallow the shock that forms in it on the way up. A biplane 3 per cent thick per element, designed for Mach 1.94, stays choked until Mach 2.15, and only one that is thinner than about 2.6 per cent can start by itself at any speed.

compressible · Wave drag Behind the lip the ceiling is set by starting, not by the cowl. The best total-pressure recovery with every compression isentropic but the terminal normal shock, against free-stream Mach number: an external ramp turning to the free stream's detachment angle; the same ramp followed by the largest internal contraction a fixed duct can start through, Kantrowitz's limit; and the same ramp followed by a contraction to a throat Mach number of 1.2, which only a duct that opened its throat to start can hold. At Mach 3 the three are 88.5%, 95.1% and 99.3%; at Mach 4, 71%, 82.5% and 99.3%.

Compression behind the lip is bought by opening the throat

The cowl caps how far an intake's ramps can turn the flow, and moving some of the compression inside the duct escapes the cap. It does not escape the duct's other limit: a duct that contracts a supersonic stream has to swallow the shock that stands at its lip while it starts. With fixed walls that allows a contraction of about a tenth, and at Mach 3 six shocks with three inside recover exactly what six outside do. A duct that opens its throat to start and closes it after recovers 94 per cent — the internal compression is paid for by the moving wall, not by the walls being inside.

compressible · Oblique shock A flap a fifth of the chord long, turned in three and a half degrees, starts the biplane at Mach 1.72 instead of 2.15. The Mach number at which the channel between the elements swallows the normal shock at its mouth, against the leading-edge flap's deflection, for flaps of 0.1, 0.2, 0.3 and 0.4 of the chord; the deflection runs from shut to the ramp's own angle, 3.43°, where the flap lies flat. Shut, the channel starts at 2.15; flat, at 1.93 (0.1), 1.72 (0.2), 1.53 (0.3), 1.32 (0.4). The design Mach number is 1.944.

A flap that closes the mouth starts Busemann's biplane below its design point

Busemann's biplane cancels its own waves at one Mach number and cannot get there: its channel narrows to a throat that will not swallow the shock standing in its mouth until the aeroplane is flying faster than the design point. Hinge the front of each element's inner ramp and turn it into the channel, and the mouth closes towards the throat while the throat stays put. A flap a fifth of the chord long, turned in three and a half degrees, starts the channel at Mach 1.72 instead of 2.15; once started, the flaps can shut at any speed above 1.60 and the zero at the design point returns exactly.

compressible · Wave drag

Named alongside it

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

Model limitArea machNormal shockTotal pressureMach numberSonic throatCompressibilityde Laval nozzleEfficiencyEjectorEntropyFriction factor

All concepts