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

Irreversibility — where it appears

The property of a process that cannot be run backwards without violating the second law. Shocks, hydraulic jumps, sudden expansions and turbulent dissipation are all of this kind, and each converts an amount of mechanical energy that can be computed exactly.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

The box, and the one thing assumed about it. The control volume across a sudden enlargement. Mass and momentum crossing the two ends are known exactly. The only modelling statement in the whole derivation is written on the annular step: the pressure there is taken to be the upstream pressure, because the fluid in the corner is nearly stationary. Measurement supports it well. Nothing else is assumed, and in particular nothing at all is assumed about the eddy that lives in that corner — which this figure therefore does not draw.

A loss with no viscosity in it

Where a pipe suddenly widens, energy is destroyed. The amount is exact, it has been known since 1766, and the derivation never mentions viscosity, Reynolds number or roughness — because momentum does not care where the energy went, only that it left.

applied · Internal flow Two depths, and a gap the model will not describe. The surface either side of a hydraulic jump at an arriving Froude number of 5.05. Both depths are exact consequences of the momentum balance. The distance between them is not: the shallow-water model has no length scale in it and cannot say how far the transition takes, so the region between the two levels is left blank and the six-depth rule of thumb beside it is somebody's measurement rather than this site's result.

The shock in a river

Shallow water is a gas whose ratio of specific heats is two. The white water below a weir is a shock wave, momentum is conserved across it exactly, energy is not, and one direction is forbidden for the same reason an expansion shock is forbidden — which makes the analogy exact to first order and wrong at the second.

applied · Open-channel The second law is the only thing that forbids the other half of this curve. Entropy change across a normal shock, against the Mach number in front of it. The solid branch is the compression shock that exists. The dashed branch below Mach one is the expansion shock, and it satisfies mass, momentum and energy exactly — the residuals are zero to machine precision. It is refused by the second law alone, the one statement in the problem that no conservation residual can show.

The only law that forbids it

The jump conditions permit a discontinuity in either direction. An expansion shock conserves mass, momentum and energy exactly — the residuals are zero to machine precision — and it does not exist. Nothing that can be drawn rules it out.

compressible · Shock The same 10° turn, taken both ways. A supersonic stream turned away from itself expands through a fan of Mach waves and keeps every bit of its total pressure. Turned into itself through the same angle it shocks, and pays. Nothing in the equations distinguishes the two cases except the sign of the angle: compression waves converge and steepen into a front, expansion waves diverge and spread.

Turning the other way is free

Compression through ten degrees at Mach 2 costs 1.54 per cent of the total pressure. Expansion through the same ten degrees costs exactly nothing — not approximately nothing, nothing — and the two are the same equations with the sign of one angle changed.

compressible · Expansion fan The same number, by two integrals that share no arithmetic. Three flows whose dissipation is in closed form both ways. The volume route integrates the dissipation function over the fluid; the boundary route multiplies a force or a torque by the speed of whatever is applying it. Neither calculation contains the other, and the residual column is what is left when they are subtracted.

The price of a gradient

Viscosity does not charge for motion. It charges for the rate at which a parcel is being deformed, and a fluid in solid-body rotation at any speed whatever destroys nothing at all. What is charged for is a sum of squares, which is why the bill can be computed twice.

viscous · Dissipation The parabola is the cheapest shape the walls allow. The dissipation of the profile u = A(1 − |y/h|ⁿ) carrying a fixed flux between fixed walls, against the exponent n, divided by the parabola's. Every member of the family satisfies the same boundary conditions and carries the same fluid; they differ only in shape. The minimum is at n = 2 exactly, which is not a coincidence — it is Helmholtz's theorem, and the parabola is a solution of the equations because it is the least dissipative shape rather than the other way round.

The cheapest shape the walls allow

The parabola in a pipe is usually presented as what the equations give. It is better understood the other way round — of every profile that could carry that flow between those walls, it is the one that destroys the least energy, and the equations give it for that reason.

viscous · Minimum-dissipation The velocity through a shock at Mach two, as a function of position. The Becker profile, integrated outwards from its own inflection point. It runs from the upstream velocity to the downstream one — the two states the jump conditions give, which appear here as the equilibria of a first-order differential equation — and its steepest gradient matches the closed form to one part in ten thousand. The horizontal axis is in units of the thickness, which for this shock is 139 nanometres.

The discontinuity that has a thickness

The jump conditions do not contain the viscosity, which is why they are exact. The thickness is entirely viscosity — 289 nanometres at Mach 1.5, 35 at Mach 5, against a mean free path of 64. At Mach five the continuum equations have produced a structure thinner than the distance between collisions.

compressible · Shock So splitting a turn into N ramps costs one over N squared. A twelve-degree compression at Mach 3, done in one ramp and in up to sixty-four. The entropy is N times a cube of one Nth, so it falls as exactly the inverse square of the number of ramps — the measured exponent is −2.00 — and sixty-four ramps cost a two-hundred-and-fifty-sixth of what one costs.

A compression that costs nothing in the end

Turning a supersonic stream away from itself is free and turning it into itself is not. But the price of a compression is the cube of its strength, so splitting one turn into N turns costs one over N squared — and in the limit the compression is free too.

compressible · Expansion fan Seven shock structures, and the two states they all connect. The velocity through the shock for seven dissipation models — Prandtl numbers from a quarter to two, viscosities from constant to linear in temperature. Each curve is shifted so its midpoint sits at the origin. They start at the same speed, end at the same speed, and are nothing alike in between.

The jump does not ask what made it

Seven different dissipation mechanisms are made to smear the same shock. Their interiors are a factor of two and a third apart in thickness, their entropies overshoot the final value by between a quarter and a doubling, and the state they all arrive at agrees to seven parts in ten billion — because the end states are conservation and the interior is transport.

compressible · Shock 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 Two totals across a shock. The ratio of the total temperature and the ratio of the total pressure across a normal shock, against the shock's Mach number. One of them is one at every Mach number, to the last bit of double precision; the other falls to under a hundredth by Mach eight.

Two totals, one of which a shock cannot touch

Across a normal shock the total temperature ratio is 1.000000000000000 at every Mach number, and the total pressure ratio falls to 0.0085 by Mach 8. One of the two records the energy that has been added to the gas and nothing else; the other records every irreversibility on the way.

compressible · Bernoulli's equation

Named alongside it

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

EntropyDissipationTotal pressureConservationDiscontinuityShock waveViscosityControl volumeMeasurementRegimeThe second law of thermodynamicsCharacteristics

All concepts