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

Signal speed — where it appears

The speed at which a disturbance propagates through a fluid, which is the speed of sound in a gas and the shallow-water wave speed in a channel. Whether the flow is faster than it decides what the fluid upstream can know.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

A source at Mach 0.55, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all.

What a signal travels at

Air finds out that an aeroplane is coming, and it finds out at a definite speed. That speed does not depend on how hard the air is squeezed or on how much of it there is — only on how hot it is — and every result in compressible flow is downstream of that one fact.

compressible · Speed of sound A source at Mach 2.00, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all.

When the warning cannot arrive

Supersonic flow is not fast flow. It is flow in which the fluid ahead has been told nothing, because the body is outrunning its own pressure signals — and that single change turns an equation of one type into an equation of another.

compressible · Speed of sound 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 The Hugoniot of a gas that can burn, and the gap in the middle of it. Pressure against specific volume, both scaled on the unburnt gas. The lower curve is the ordinary shock Hugoniot, which passes through the initial state because a jump of zero strength satisfies mass, momentum and energy. Adding a heat release lifts it away, and the initial state now sits in a region no wave can reach: between the two branches a Rayleigh line would need a positive slope, and its slope is minus the square of the mass flux. A burning gas has no weak waves available to it at all.

The other branch of the same curve

Put heat into the jump conditions and the Hugoniot lifts away from the initial state, leaving a gap no wave can occupy. A burning gas has no weak waves: it must run supersonically or subsonically, and the conservation laws pick the first speed exactly and say nothing at all about the second.

compressible · Detonation 12 bar from stopping one metre per second. The head at the valve after it shuts, computed by the method of characteristics on a 600 m pipe. The rise is 122.4 m of water, which is ρaΔV/ρg to 1.4e-14 m — and the scheme was told neither ρaΔV nor anything else about the answer. The wave then runs to the reservoir and back every 2.000 s, and with no friction in the model it never decays: a real pipe damps this out in a few tens of cycles.

Stopping water costs more than moving it

Shut a valve on water running at one metre per second and the pressure that appears is twelve bar — not because the water was pushing hard, but because the only way to stop a column of fluid is to send a message back along it, and the message travels at the speed of sound in the pipe.

applied · Water hammer The chord and the tangent, which are the two speeds. The flux of a conserved quantity against its own density, for a wide river and for traffic. At any point the slope of the chord from the origin is the speed the material moves at, and the slope of the tangent is the speed a disturbance moves at. They are the same number only if the curve is a straight line through the origin. For the river the tangent is five-thirds of the chord at every depth; for traffic the tangent turns negative above half the jam density while the chord never does.

A wave nothing in it travels with

A flood crest moves at five-thirds the speed of the water it is made of, at every depth, whatever the roughness and whatever the slope. A traffic wave moves backwards through cars that are all going forwards. Neither result contains a momentum equation.

kinematics · Kinematic waves One signature, aged four times. The pressure signature of a slender body at four distances, computed by the exact Lax formula for the nonlinear propagation. Each point of the waveform moves forward in proportion to its own overpressure, so the compression at the front catches the undisturbed air and a shock forms there, while the expansion at the rear falls behind and forms a second one. What is left is an N-wave: two discontinuities and a straight line between them, spreading and weakening.

The signature that forgets the shape

The pressure field near a supersonic aeroplane depends on every part of it. What reaches the ground has two parameters. Two bodies whose near-field signatures differ by fifty-five per cent in peak and by their whole shape age into the same N-wave, to two and a half per cent.

compressible · Sonic boom The backwater curve behind a weir. The depth at the control is 1.6 times the normal depth; integrating upstream the profile relaxes onto normal depth over about seven hundred metres and stays there. That relaxation is the reach forgetting the weir, and nothing about the weir survives past it.

The section that decides the river

A reach of open channel has a normal depth and a critical depth, and the water has neither. What it has is a profile obeying a first-order equation, which needs exactly one condition — and whether that condition belongs at the upstream end or the downstream end is not a choice, because the equation is stable in one direction and unstable in the other.

applied · Open-channel A sinusoid, distorting on its way to a shock. A finite-amplitude sound wave at four fractions of the distance to shock formation, computed by inverting the implicit simple-wave solution. Each point of the waveform travels at its own speed, so the compressions catch up with the rarefactions ahead of them and the profile leans forward. At σ = 1 the front is vertical. The linear theory says the first panel is the answer at every distance, for ever.

Every compression becomes a shock in the end

Linear acoustics has no time scale in it, which is the sign that something has been thrown away. A 120-decibel tone shocks after three hundred metres and a jet engine after twenty; the distance goes exactly as the reciprocal of the amplitude, and nothing is exempt.

compressible · Characteristics The column in height and time: a falling interface, a rising shock, a fan. A batch settling test from a uniform φ₀ = 0.1, height above the bottom against time, both scaled on the column height and the single-particle settling time. The interface with clear water (thick) falls in a straight line at 0.4538; the sediment shock rises from the bottom at 0.1484 until the two meet at t = 1.661, height 0.2464; the thin lines are characteristics of the fan, each carrying one concentration between 0.317 and packing, and the interface bends as it crosses them. Dots are the finite-volume solve on 400 cells: the interface and the sediment front.

The column the chord rule cannot settle

A suspension settling in a closed column is a kinematic wave, and its flux curve bends both ways. At the top the chord rule works: clear water meets the suspension at a single falling front. At the bottom it does not, and the bed grows behind a shock that stops short of packing and a graded layer beneath it — so the interface, instead of arriving, slows for ever.

kinematics · Kinematic waves Below Mach 1.153 the boom turns back before it reaches the ground. The ray leaving the Mach cone straight down from an aeroplane at 11 km, at Mach 1.1, 1.15, 1.2, 1.5, 2, traced through a standard atmosphere whose sound speed rises from 295.1 m/s at the aeroplane to 340.3 m/s at the ground. A ray bends back upward where the local sound speed equals the aeroplane's speed, so Mach 1.1: turns at 4.00 km; Mach 1.15: turns at 0.25 km; Mach 1.2: lands 24.0 km on; Mach 1.5: lands 11.4 km on; Mach 2: lands 7.1 km on. The dividing speed, Mach 1.1533, is the ratio of the two sound speeds.

The boom that turns back before the ground

Sound is faster in the warm air near the ground, so a sonic boom's rays bend back upward on the way down. Whether any of them arrive is one comparison — the aeroplane's speed against the fastest sound beneath it — and the ray that just grazes the ground sets the edge of the carpet, which the uniform air of the ageing calculation cannot give it.

compressible · Sonic boom The arrival map, and the place where it goes backwards. Where each boom ray lands on the ground, against the Mach number the aeroplane was doing when it launched it, for four accelerations from 15 km. In level flight this would be a straight line rising at the aeroplane's own speed. Here it falls before it rises: a ray launched at a higher Mach number is shorter and steeper, and near the cut-off it shortens faster than the aeroplane advances. Every minimum in these curves is a fold — two emission times delivering to one place — and on the fold itself neighbouring rays converge onto a single line.

The carpet an accelerating aeroplane folds

Level flight launches every boom ray with the same invariant, so they run parallel and each place hears one boom. Accelerate, and each successive ray is shorter than the last — shorter, near the cut-off, than the aeroplane's own advance — so later rays overtake earlier ones and the arrival map folds onto a line.

compressible · Sonic boom

Named alongside it

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

Model limitShock waveCharacteristicsDiscontinuityMach numberSpeed of soundNonlinear steepeningSonic boomConservationConstitutive lawDomain of dependenceFar field

All concepts