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

Conserved quantity — where it appears

A quantity whose total does not change with time, usually because the terms that would change it cancel identically. Energy, circulation, impulse and helicity are the ideal flow's four, and each holds only under stated hypotheses that can be broken on purpose.

Named by 20 essays across 8 fields — each of them below, with the objects they name alongside it.

Elliptic loading rolls up to πb/4, and it is π that puts it there. Three loadings on the same span carrying the same lift, with the rolled-up core positions marked below each. The core sits at the centroid of the shed vorticity, which is a quadrature over the loading and needs nothing about the roll-up itself. Elliptic loading gives πb/8 from the centreline, so the pair ends up 0.7854 of the span apart — the number every wake-separation rule is written against, and one of the few places in this subject where π turns up in an answer an engineer uses directly. The bell rolls up to 0.586 and a nearly rectangular loading to 0.978, because it sheds at the tips.

Where the wake ends up

The sheet a wing sheds rolls up into two cores within a few spans, and nobody can compute the roll-up cheaply. Nobody has to: what the cores conserve is fixed before they form, and for an elliptically loaded wing the answer contains π and comes out at 78.5 per cent of the span.

circulation · Tip vortex A double integral that comes out an integer. The Gauss linking integral evaluated on six pairs of closed curves. It is not constrained to be a whole number by anything in its own definition — it is a double integral of a smooth kernel — and it returns one to within two parts in ten thousand on two hundred points per curve, because what it is computing is a topological count.

The knot a flow cannot untie

Ideal flow conserves energy, circulation and impulse, and all three are what they look like. It conserves a fourth quantity that is not: a volume integral of the velocity dotted into the vorticity, whose value counts how many times the vortex lines are linked through one another.

kinematics · Helicity The mean flux, flat across the inertial shells and equal to the dissipation. The time-averaged transfer out of the first n shells, computed as the rate at which the nonlinearity changes their energy rather than from a remembered formula. It is constant to a tenth across the middle of the ladder and equal to the dissipation, which is the cascade — and it is an average.

A flux that runs both ways

The cascade is a statement about a mean. Kolmogorov's four-fifths law fixes an average and the constant flux through the inertial range is an average, and neither says anything about what the transfer is doing at any instant — which turns out to be running backwards a substantial part of the time.

turbulence · Cascade The four waves, and the one that never goes away. The shock tube in space and time: a shock running right, an expansion fan running left, and the contact surface between them. The shock and the fan are travelling disturbances that leave; the contact is made of fluid, so it is carried along and is there for ever.

A surface that remembers the diaphragm

Between the shock and the expansion in a shock tube there is a surface across which the pressure and the velocity are identical and the temperature differs by a factor of two. It is made of fluid, so it never goes away, and nothing in the pressure field says it is there.

compressible · Shock tube 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 The patches' centroids, against the point-vortex circle. Two circular patches of uniform vorticity, advected by nothing but the velocity their own boundaries induce, over one full co-rotation. Their centroids stay within one per cent of a separation of the exact point-vortex orbit — which they must, because the exterior field of a circular patch is the point vortex's and a harmonic function's area average over a disc is its value at the centre.

What a point vortex is not

Two circular patches of vorticity move exactly as two point vortices do — the centroid velocity is the point model's with no correction of any order, and that is a theorem rather than an approximation. The trouble starts the moment they stop being circular, which is immediately.

inviscid · Vortex dynamics Where a rotor's pressure rise comes from, as the radius moves. The static pressure rise across a rotor, split into the two terms rothalpy gives it. The diffusion term is held at the de Haller limit throughout — the blade is being asked to slow the relative flow as hard as a boundary layer will allow — so it is a flat 11558.4 Pa at every radius ratio. Everything above that line is the centrifugal term, which costs no diffusion and has no limit of its own. At a radius ratio of 2 it supplies 49.92 per cent of the rise and at 3, 72.66 per cent. An axial machine, at a ratio of exactly one, gets none of it.

What a turning frame keeps

Euler's equation prices the work and says nothing about where the pressure comes from. In the frame turning with the blades — which is accelerating, and carries two fictitious forces — a Bernoulli-like quantity survives both of them, and it splits the pressure rise into a term a boundary layer limits and a term that is free if the radius moves.

applied · Turbomachine 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 Two wakes with one drag. The two initial velocity deficits: a slab, which is roughly what a bluff body leaves, and a pair of separated lobes, which is roughly what a body with a jet through the middle of it leaves. Their integrals are identical, so the two bodies have exactly the same drag.

A wake that keeps the drag and forgets the body

Two very different wakes with the same momentum deficit converge to the same profile, because the deficit is conserved and everything else diffuses away. The convergence is a power law rather than an exponential, so it takes two hundred widths for twenty per cent agreement and nine hundred for five.

turbulence · Wake One material loop, at five stages of being drawn out. A circle of fluid particles carried by four point vortices, drawn at equal intervals over fourteen time units. Its length grows by a factor of six and its shape becomes unrecognisable; the circulation round it does not move at all.

The drift was the instrument

Kelvin's theorem was checked on this site by carrying a loop and watching its circulation move by eight parts in a hundred thousand. That drift is not the flow forgetting. The exact number is a count of what is inside the loop, it does not move by anything at all, and the drift belongs entirely to the two instruments used to measure it.

inviscid · Kelvin What is left behind, and what it is made of. The two rolled-up vortices behind a large aircraft, drawn to scale against its span. They sit at 78.5 per cent of the span apart, each carries 508 square metres a second of circulation, and the pair descends at 1.72 metres a second under its own induction.

A wake that says what made it

Two vortices sitting behind an aeroplane carry its weight and its span in a form that can be read back out: circulation times separation times density times speed is the lift, exactly. The reading is exact for about a minute and worth nothing after four.

circulation · Tip vortex There and back again. A blob of a hundred and twenty tracer particles at the start, after four time units of stirring, and after the same four run backwards. The third set is drawn over the first and the worst particle is 1.4·10⁻⁹ from where it began.

Reversible, and unusable

Ideal flow has no arrow of time in it. Run a stirring backwards and the dye comes back — here to 1.4 parts in a thousand million. Nudge the state by a hundred-millionth first and the same reversal returns a blob almost five hundred times further from home than the nudge was large.

inviscid · Vortex dynamics Six ways of reaching one speed. Six velocity histories, all starting from rest and all reaching exactly one at the same moment. Two are ramps, two are eased, one overshoots and comes back, and one goes backwards before it goes forwards.

Everything about the start, except one vector

Six ways of accelerating a body from rest to the same speed produce six force histories with nothing in common — peaks spanning a factor of thirty-nine, two of them negative for part of the journey. The impulse left in the fluid is the same ten-figure number in every case, and so is the energy.

inviscid · Impulse Inside a strong blast: a shell of gas around a hot, nearly empty core. Left, the density, velocity and pressure inside a spherical strong blast wave at γ = 1.4, each as a fraction of its value just behind the shock, against the distance from the centre as a fraction of the shock radius. The density falls to nothing well inside the shock while the pressure levels off at 0.37 of its post-shock value. Right, the temperature, which is the pressure over the density and so rises without limit towards the centre.

The fireball is hollow

Inside a strong blast wave there is almost nothing. Half the air the shock has swept up lies in the outer four per cent of its radius; at half the radius the density is a hundredth of the air's; the centre is empty and, formally, infinitely hot. Integrating Sedov's equations to see this also shows that a published table of his constants was wrong at three of its four entries.

compressible · Blast wave Time per litre rises in a straight line with the litres already filtered. The elapsed time divided by the filtrate collected, against the filtrate, for a constant-pressure filtration of a silica slurry through a cloth: integrated step by step, and Ruth's closed form. The line's slope is the cake's resistance, which grows with every litre, and its intercept is the cloth's, which does not. After a quarter of a cubic metre the cake is already the larger.

A filter is slowed by what it has caught

Push a slurry through a cloth and within minutes the cloth no longer matters: the particles it has caught form a cake, the cake is a packed bed, and every litre filtered makes the bed thicker for the next. The filtrate grows only as the square root of time. And if the cake is soft, it packs tighter against the cloth the harder it is pushed, until more pressure buys almost nothing.

applied · Porous A free jet keeps its momentum; a wall jet keeps the flux of its momentum flux. Each quantity against distance, relative to its value at 2 cm, on logarithmic axes. The free jet's momentum flux is constant and its volume flux grows as x^(1/3). The wall jet's momentum flux falls as x^(−1/4) — the wall's shear takes it — while its volume flux grows as x^(1/4), and the one thing it conserves is Glauert's ∫u(∫u²)dy, the flux, through each height, of the momentum flux carried above it.

A jet along a wall keeps the flux of its momentum flux

A jet in open fluid keeps its momentum and gathers fluid as it goes. Blow the same jet along a wall and the wall takes its momentum away as drag, so it keeps neither. What it keeps is stranger: the flux, through each height, of the momentum flux carried above that height. That one invariant fixes everything else — the jet thickens as the three-quarter power of distance, slows as the inverse square root, and gathers fluid as the fourth root, more slowly than the free jet's cube root.

viscous · Free shear Hold the nozzle further from the plate and the jet arrives slower but leaves a stronger wall jet. For a laminar jet from a 2 mm slot at 5 cm/s in water, against the nozzle's distance from the plate, each over its value at 10 cm: the arriving centreline speed, which falls as h^−⅓; the inherited invariant F = 3KQ/32, which grows as h^⅓ because the jet has gathered fluid; and the wall jet's peak speed a metre along the plate, which grows as h^⅙. At 2 cm the arriving speed is 1.71 of its value at 10 cm and the far wall jet 0.765; at 2 m, 0.368 and 1.65.

A jet striking a plate leaves a stronger wall jet the further it travelled

A plane jet that strikes a wall splits into two wall jets, and each keeps not the jet's momentum but Glauert's flux of momentum flux. Turned without loss at the stagnation point, each half arrives laid along the wall with its axis on the wall, and its invariant is the momentum it brought times the fluid it had gathered on the way: F = 3KQ/32. Hold the nozzle further away and the jet arrives slower but leaves a stronger wall jet, by the cube root of the distance — and the wall jet takes a distance of about the Reynolds number in jet widths to forget it was ever free.

viscous · Free shear Of the four common plumes, only the laminar one above a line speeds up as it rises. Centreline speed against height for four plumes, each scaled to its own speed at one reference height: laminar above a line source, rising as the fifth root of height; laminar above a point source and turbulent above a line, both constant; and turbulent above a point, falling as height to the minus one third. Each exponent follows from the heat flux being fixed and the width growing by diffusion, √(νx/u), when laminar, or in proportion to height when turbulent.

Above a warm wire the plume speeds up as it rises

A plume above a line of heat has no wall, no imposed speed and one conserved quantity, the heat it carries. That alone fixes a centreline speed growing as the fifth root of height — the only one of the four common plumes that accelerates — a temperature excess falling as the three-fifths power, and a warm core that is narrower than the moving sheet around it in any fluid more viscous than five ninths of its own diffusivity. At that Prandtl number the equations have a closed form, and the solver lands on it to three parts in a million.

regimes · Grashof Freed by its inertia, a settling particle falls through a lattice of vortices several times faster than through still air. The mean settling speed of a cloud released over one cell, over its speed in still air, against the Stokes number, for still-air speeds of 0.1, 0.3 and 0.6 of the vortices' speed. 0.1: 4.82 at St = 0.02, 4.39 at 1 and 1.86 at 10; 0.3: 2.36 at St = 0.02, 1.22 at 1 and 0.605 at 10; 0.6: 1.43 at St = 0.02, 0.984 at 1 and 0.959 at 10. Without inertia most of the cloud never falls at all.

Inertia empties the vortices that would hold a settling particle, and then speeds its fall

A particle that settles slower than a lattice of vortices turns can be carried round a cell for ever: without inertia, its path is a closed curve, and Stommel showed in 1949 that half a cell can be such a trap. Give the particle the slightest inertia and every trap leaks, in a time that grows as one over its Stokes number; the freed particle is flung to the cells' edges, rides their downdraughts, and falls on average several times faster than it would through still air. More inertia undoes some of that, and a narrow band of it can hold a slow particle up indefinitely on an orbit of its own.

regimes · Particle A round jet held further from the plate arrives slower in proportion and leaves a radial wall jet stronger in proportion. For the round water jet from a 2 mm nozzle, against the nozzle's distance from the plate, each over its value at 10 cm: the arriving centreline speed, the radial invariant F = Kνh/(2π) it hands on, and the radial wall jet's peak speed 5 cm from the stagnation point; dashed, a plane jet's speed and invariant for comparison. At 40 cm the round jet arrives at 0.25 of its speed at 10 cm and hands on 4 times the invariant, and its wall jet runs 2 times as fast; a plane jet's invariant would have grown 1.59 times.

A round jet hands its wall jet a strength proportional to its throw

A round jet striking a plate spreads into a radial wall jet, and like the plane jet before it, the wall jet keeps not the jet's momentum but Glauert's flux of momentum flux — the momentum the jet brought times the fluid it gathered on the way. A round jet gathers fluid at a rate set by the viscosity alone, eight π ν per metre of travel, whatever its strength. So the invariant it hands on grows in exact proportion to the distance it was thrown, where a plane jet's grew as the cube root: hold the nozzle four times further away and the wall jet is four times as strong and runs twice as fast.

viscous · Free shear

Named alongside it

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

Model limitMeasurementModel validityMemory kernelRegimeCirculationSimilarity solutionEntrainmentBoundary layerEntropyJetMomentum flux

All concepts