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 11 essays across 4 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

Named alongside it

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

MeasurementMemory kernelModel validityRegimeTransportPeclet numberRelaxation timeBoundary conditionConvolutionDispersionMixingAngular momentum

All concepts