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

Diffusion — where it appears

The spreading of a quantity by molecular motion, at a rate set by a diffusivity and the local gradient. It is hopeless over any distance larger than a hair, which is why every practical mixing process uses a flow to make the distances short.

Named by 20 essays across 5 fields — each of them below, with the objects they name alongside it.

A million times faster, and the constant is 48.0. The effective diffusivity along a pipe, divided by the molecular one, against the Péclet number — both logarithmic. Below Pe ≈ 7 the tracer simply diffuses and the curve is flat at one. Above it the dispersion is all Taylor's, rising as the square of the Péclet number, so a thousandfold Péclet number is a millionfold enhancement. The constant in D(1 + Pe²/48) is not quoted here: it is recovered from a numerical solution of the cell problem across the section, giving 48.0000 for a tube and 52.5 for a plane channel, which is Aris' 2/105.

Two slow things make a fast one

Shear stretches a slug of dye and mixes nothing, because it is reversible. Molecular diffusion is hopeless at any scale bigger than a hair. Put the two together in a pipe and the dye spreads along it with an effective diffusivity two million times the molecular one — which gets larger as the molecular one gets smaller.

regimes · Dispersion One layer, and the four others inside and outside it. The velocity profile over a flat plate, and the temperature profile in the same layer at four Prandtl numbers: a liquid metal at 0.01, air at 0.71, water at 7 and a heavy oil at 100. The equations differ by one number and the profiles differ by a factor of twenty in thickness. At Pr = 1 the two are the same function — not similar, identical, to eight decimal places — because the equations and the conditions are then the same, which is what every statement called a Reynolds analogy rests on.

The other layer, and the one number that separates them

A wall in a stream carries two boundary conditions and grows two layers. Their thicknesses differ by a factor of twenty across ordinary fluids, and at exactly one Prandtl number the two profiles are not similar but identical.

regimes · Peclet Ballistic first, diffusive later. The mean square displacement of a parcel carried by a fluctuating velocity, from Taylor's 1921 integral with an exponential correlation of time scale 1. Below the correlation time it follows the straight-line law u²t² exactly — the parcel has not yet changed its mind — and above it the curve joins the diffusion law 2u²T·t, offset by the head start the ballistic phase gave it. The eddy diffusivity is u²T = 1.000, and it is a property of the correlation rather than of any equation of motion.

How far a parcel gets

Turbulent transport is one integral. A parcel carried by a fluctuating velocity goes in a straight line while it still remembers its own motion and performs a random walk once it has forgotten, and the crossover is the correlation time — so an eddy diffusivity is not a property a flow has, it is the limit of a measurement that has run for long enough.

turbulence · Mixing The number that settles the argument. How many turns the earth's contribution gets through in the time the vessel takes to drain: the draining time divided by the rotation period at the drain. A bathtub manages a tenth of one and there is nothing to see. The apparatus that settled it manages nearly nineteen hundred, and that is the whole of what its enormous area-to-drain ratio was for.

The bath that was only ever a wait

The Coriolis force is far too weak to steer a draining bath, and it does steer a large enough tank that has been left alone long enough. Both are true, and the number that separates them is not a force ratio — it is how many turns the earth's contribution gets through before the vessel is empty.

misconceptions · Coriolis scale The scalar spectrum, with its two ranges. A model scalar spectrum at a Schmidt number of two thousand — dye in water. Below the Kolmogorov wavenumber it is Obukhov and Corrsin's five-thirds, inherited from the velocity; above it there is no turbulence left and the spectrum is Batchelor's minus one, which contains no velocity spectrum at all.

The scalar has its own cascade

Below the Kolmogorov scale there is no turbulence left, and a dye stirred into the flow goes on cascading anyway — on a spectrum whose exponent is minus one and whose amplitude contains no velocity spectrum at all. Resolving it costs the three-halves power of the Schmidt number, which for dye in water is a factor of ninety thousand.

turbulence · Mixing A particle released in still fluid, with the history and without. The velocity of a hundred-micron particle released at one metre a second in air, computed with the unsteady Stokes force and with the quasi-steady drag alone. After five particle time constants the one that remembers is still moving nearly four times as fast.

The drag that integrates a whole history

A particle released in still air does not stop exponentially. The unsteady drag on it carries a term that is an integral over everything the particle has already done, weighted by the inverse square root of how long ago — so there is no time constant, and five particle time constants later it is still moving four times faster than the quasi-steady answer allows.

viscous · Diffusion What the fluid at one height is listening to. The weight the fluid two millimetres above a moving wall gives to the wall's velocity a given delay earlier, in water. It peaks at two thirds of a second and has a tail that falls as the delay to the power minus three halves — so the fluid is responding to a broad stretch of the wall's past rather than to a moment of it.

The wall the fluid is listening to

Water two millimetres above a moving wall is responding to what the wall did two thirds of a second ago — most likely. Half of its response is older than four and a half seconds, a tenth is older than two minutes, and the average age of what it is responding to does not exist at all.

viscous · Exact layer 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 How long a fluid takes to forget it was not rotating. The fraction of solid-body rotation a container's interior has reached, against time, by the two available routes. The Ekman layers on the end walls pump fluid radially and carry angular momentum inwards in a hundred seconds; diffusion alone would need ten thousand.

How long a fluid takes to forget it was not rotating

Spin a container of water and the fluid inside reaches solid-body rotation in a hundred seconds rather than the three hours diffusion would need. The shortcut is the thin layers on the end walls, and the advantage they give is exactly the reciprocal of the square root of the Ekman number.

viscous · Rotating How fast a duct forgets each part of its inlet. The decay rate along the duct for each mode of a disturbance to the developed profile. It goes as the square of the mode number, so the third mode is forgotten nine times faster than the first and the tenth a hundred times faster.

A duct that forgets everything but one number

Whatever is fed into a pipe, what survives a little way down it is one shape. The disturbance's higher modes decay as the square of their mode number, so the sixth is gone in thirteen centimetres where the first survives four and a half metres — and the entrance length is that one mode's decay rate and an arbitrary threshold.

viscous · Entrance A wave that travels and a wave that spreads. A harmonic pressure wave's amplitude and its instantaneous value along a tube, over two wavelengths of the inviscid wave, at four Womersley numbers. At α = 15 the wave marches on, a little weaker each wavelength. At α = 5 it is visibly damped. At α = 2 it is nearly gone within a wavelength. At α = 0.5 there is no wave to speak of: the disturbance falls away within a small fraction of the inviscid wavelength, as heat does into a wall.

The pulse that has to travel

In a rigid tube the Womersley number decides the shape of an oscillating flow. Make the wall elastic and the pressure pulse has to travel, a second number appears — the tube's length in wavelengths — and the first number turns out to decide something more basic than the profile: whether the tube carries a wave at all, or only a disturbance that spreads like heat.

regimes · Womersley The spreading per metre has a best speed; the spreading per second has none. The plate height — how much variance a slug picks up per unit length travelled — against the velocity, for four retention factors. Each curve falls as one over the velocity, because molecular diffusion along the tube matters when the flow is slow, and rises with it, because that is Taylor's dispersion. The minimum is what a chromatograph is run at, and Taylor's effective diffusivity has no such point anywhere.

The best a tube can do is its own radius

Taylor's dispersion coefficient rises without limit with velocity, so there is no best speed to run a tube at. Ask instead how much a slug spreads per metre travelled and an optimum appears at once — and the spreading there is the tube's radius over the square root of three, with no diffusivity and no velocity left in it.

regimes · Dispersion Three speeds, and the Froude number at which two of them cross. The kinematic wave speed and the two dynamic wave speeds, all divided by the speed of a shallow-water wave on still water, against the Froude number. The kinematic wave is three halves of the water speed and the downstream dynamic wave is the water speed plus one, so they cross at Froude two — and beyond that crossing the news of a change in flux arrives before the news of a change in depth, which uniform flow cannot survive.

When the flux outruns the pressure

A kinematic wave is derived by throwing the momentum equation away, and the derivation never says when that is allowed. It is allowed until the kinematic wave, which travels faster than the water, overtakes the downstream dynamic wave as well — at which point uniform flow ceases to exist and a concrete chute carries a train of surges instead of a sheet.

kinematics · Kinematic waves What an oscillation carries, against how fast it is asked. The extra axial diffusivity of a zero-mean oscillation, at fixed velocity amplitude, against the Womersley number. At low frequency it is exactly Taylor's dispersion evaluated at the mean square velocity — the profile has time to be Poiseuille's at every instant. At high frequency the shear is confined to a Stokes layer, the tracer in the core is never sheared, and the transport falls as the cube of the Womersley number.

Transport with nothing transported

Taylor's mechanism needs shear and diffusion and nothing else — and in particular it does not need a mean flow. Oscillate a tube about a fixed position and the tracer still spreads along it, by hundreds of times the molecular rate, which is how a patient is ventilated with a tidal volume smaller than the airway it goes down.

regimes · Dispersion Three pairs released at different distances end on one line. The mean square separation of two parcels against time, both in Kolmogorov units, for pairs released one, thirty and a thousand Kolmogorov lengths apart in turbulence whose integral scale is a million of them. Each starts flat — the pair has not yet moved — and each joins the same line, g ε t³ with g = 0.5, after which nothing about where it started survives. Beyond the integral scale all three become ordinary diffusion, growing as t.

The further apart, the faster they part

Release two specks of smoke close together in turbulence and the distance between them does not diffuse the way a single speck wanders. It grows faster the larger it already is, because larger separations are pulled apart by larger eddies — as the cube of the time, forgetting where it started. The law is ninety years old, its constant is still argued about, and it needs a very large flow to be seen at all.

turbulence · Mixing The orbits a rod's axis can be on, seen along the vorticity. The tip of the unit vector along a rod of aspect ratio five, tumbling in a simple shear, seen from along the vorticity axis, for five values of the orbit constant. Every orbit is closed and every one takes the same time. A rod near the centre is spinning about the vorticity axis; a rod on the outer circle tumbles end over end in the plane of shear. The flow never moves a rod from one orbit to another.

A viscosity the flow cannot decide

Spheres stirred into a liquid thicken it by a definite amount. Rods do not. A rod in a shear flow tumbles round a closed orbit, the flow never moves it to another, and the extra viscosity depends on which orbit it is on. The equations of slow flow permit a whole range of values and choose none of them. The smallest amount of noise chooses one, and it does not matter how small.

viscous · Suspension The longer a pair remembers, the less its cloud has tails. The kurtosis of the pairs' separation, ⟨r⁴⟩/⟨r²⟩², against how long each pair keeps its relative velocity, in units of the turnover time of eddies of its own size. Richardson's memoryless diffusion gives 3.76, dashed, with long tails of pairs that separate fast by chance; a Gaussian cloud gives 5/3. A memory of a tenth of a turnover time already takes the kurtosis to 2.55; one turnover time, to 1.82. The cloud's shape is a measurement of the memory.

A pair's memory shapes the cloud it spreads into

Richardson's diffusion of pair separations has no memory: each moment's push is independent of the last, and the cloud of separations grows as t³ with a peaked shape and long tails. Real pairs keep their relative velocity for about the turnover time of eddies their own size. Give them that memory and the cube law survives, because it is dimensional, but everything else about the cloud changes: its constant falls, its tails shrink, and its shape becomes a measure of how long a pair remembers. A memory of a tenth of a turnover time already takes the kurtosis from 3.76 to 2.5.

turbulence · Mixing Dissolved gas takes height from the largest nuclei first. The tallest crown a siphon runs over, against the radius of its largest nucleus at the reservoir, when a nucleus at the crown must neither break the water at once nor grow to break it by diffusion, for water at 0, 10, 30, 60 and 100 per cent of saturation. Fully degassed water gives the static limit. For a one-micron nucleus, gas below about a fifth of saturation changes nothing — 14.41 m against 14.41 — and saturated water brings it to 10.7 m. A five-micron nucleus in saturated water breaks the siphon at 2.4 m.

A crown dissolves its nuclei or breaks on them, and fast

A siphon of degassed water runs until its crown's tension reaches the breaking threshold of its largest nucleus. A nucleus lodged at the crown does not stay the size it arrived: it gains or loses gas by diffusion, and across a micron diffusion takes milliseconds, so the crown gives its verdict almost at once. Every stable nucleus loses gas unless the water holds more than half the crown's tension, and past a knee in gas content, dissolved air takes height from a siphon's crown that no amount of further degassing gives back.

misconceptions · Siphon The pocket escapes at the least pressure its states can stand. The liquid pressure each equilibrium of the pocket needs, against the pocket's volume, for a crevice with a one-micrometre mouth in a wall the water wets at 40° and one it does not, at 110°. As the crown's pressure falls the pocket follows its curve to the right: the rim slides out of the cone, pins at the mouth, and the meniscus bulges into a cap. The curve's minimum is the escape threshold — -106 kPa for the wetting wall and -138 kPa for the non-wetting one — and below it no state exists: the pocket becomes a cavity.

A crevice keeps the nucleus a free bubble loses

A free bubble at a siphon's crown dissolves in milliseconds, so it cannot be what breaks a siphon that has run for an hour. A pocket of gas in a crack of the hose wall can be, because its meniscus is curved by the wall and need not dissolve. Followed through its equilibria, the pocket escapes at a tension set almost entirely by the crack's mouth. What the wall's wettability decides is slower and more consequential: whether the crack fills, or keeps drawing gas in until the siphon breaks.

misconceptions · Siphon A free interface's thickness ratio heads to the root of its Prandtl number. The ratio of the shear layer's thickness to the density interface's against time, for an interface starting four times thinner than its shear, heat and salt in water. A free shear layer thickens by viscosity faster than the interface by conduction, so the ratio heads to √Pr: 2.64 for heat, 25.8 for salt. Held at its thickness, the shear lets the ratio fall through 2 — at about 50 s for heat and 4725 s for salt.

What holds the shear decides whether the waves last

Holmboe's travelling waves need a density interface about half as thick as the shear around it, and diffusion thickens both. If the shear layer is held at its thickness, the interface catches up — in under a minute for heat in a laboratory tank, over an hour for salt — and the waves get two e-folds or a hundred and seventy. If the shear layer is free it thickens faster than the interface, the ratio heads to the root of the Prandtl number, and the waves last until the Richardson number ends them instead.

turbulence · Stratification

Named alongside it

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

Model limitMeasurementTransportDispersionMemory kernelModel validityPeclet numberRegimeDimensionlessMixingRelaxation timeBoundary condition

All concepts