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

Averaging — where it appears

The operation that replaces a fluctuating quantity by its mean over time, over realisations, or over a cycle. It commutes with every linear operation and with no nonlinear one, which is why an averaged equation is short of terms the mean field cannot supply.

Named by 23 essays across 8 fields — each of them below, with the objects they name alongside it.

Ten unknowns, four equations. What is left after the Navier–Stokes equations are averaged. The mean velocities and mean pressure were there before; the six Reynolds stresses are new, and they arrived from the one term that does not average away. Nothing in the count is an approximation — the averaged equations are exact — and that is what makes the gap uncomfortable.

What averaging costs

Split the velocity into a mean and a fluctuation, average the equations, and the result is exact. It is also short of six equations, because the one nonlinear term does not average away and leaves six new unknowns behind that nothing determines.

turbulence · Closure A steady pressure field, from a flow with no steady part. The time-averaged pressure round a cylinder in a stream that oscillates as U₀cos ωt. The mean velocity is exactly zero at every point — the flow spends as long going one way as the other — and the mean pressure is not, because pressure depends on the square of the speed and a square has no sign. The mean coefficient reaches -2.00 at the shoulders and averages -1.00 over the surface, and its resultant is 6.6e-16: a real field with no force in it. The pale lines are the instantaneous streamlines, which reverse every half cycle.

The mean is not the flow

Average an unsteady flow and the result is a new object with its own properties, and it is not a solution of anything. An inviscid stream oscillating about zero has a mean velocity of exactly nothing everywhere, a mean pressure that reaches minus two dynamic pressures at the shoulders, and a missing term in its own momentum equation that can be written down in closed form.

kinematics · Averaging Four turns of a wave, and the parcel is not back. Two parcels traced through four periods of a linear deep-water wave of steepness 0.1, by integrating the exact velocity field. Each orbit is very nearly a closed circle and misses closing by a little, every time, in the same direction — that miss is the whole of the Stokes drift. The near-surface parcel advances 0.0357 of a wavelength over the four cycles and the one a tenth of a wavelength down advances 0.0122, a third as far — because the drift falls off twice as fast with depth as the orbit's own size does.

The drift in a wave that has none

The velocity at any fixed point under a passing wave averages to exactly zero, and every parcel of water in it moves steadily forward anyway. The orbits do not close, they miss by the same amount every time, and the missing amount is the square of the steepness times the wave speed.

kinematics · Stokes drift One number instead of a porous medium. The velocity through the bottom of a channel whose lower wall is a porous block of permeability 1e-4. Inside the block the flow decays over the pore scale √K = 1.0e-2 to Darcy's seepage velocity; above it the channel profile arrives at the interface with a slip velocity rather than at rest. A channel told nothing but u = √K du/dy at a flat wall reproduces that profile to 0.058 per cent of the flow rate, against 3.03 per cent for a wall told to hold the fluid still. The grid solve of the coupled problem agrees with the closed form to 0.0077 per cent.

A wall that is not quite there

A porous surface has structure on every scale below the pore, and no calculation resolves it. The whole of it can be replaced by one length — the square root of the permeability — and the replacement is exact to first order, with what it leaves out identifiable as the flow the wall itself carries.

kinematics · Boundary conditions A millionth, doubling every three-quarters of a second. The separation of two trajectories started a millionth apart, on log axes against time. It grows as a straight line until it saturates at the size of the attractor, and the slope of that line is the largest Lyapunov exponent — measured here on the system by renormalising a nearby pair, not quoted. Sensitive dependence is what the straightness of the line means.

The randomness that is not in the equations

Turbulence is described in the language of statistics — means, variances, spectra, probability distributions — and none of that language appears in the equations it is a description of. The Navier–Stokes equations have no random term anywhere in them. What is random is the observer's ignorance of the initial data, and the flow's habit of amplifying it.

misconceptions · Randomness The two bodies the far field cannot tell apart. A circular cylinder and the Rankine oval that has the same doublet strength: 1.17 radii long against the circle's one, and 0.94 tall against its one, with a source and a sink 1.2 apart inside it. On the pale ring, one and a half radii out, the two flows differ by 19 per cent of the disturbance; at six radii by one per cent; at infinity not at all. What a far field records of a body is three numbers — its circulation, its net outflow and its doublet — and nothing else survives the journey.

What the far field remembers

Three numbers survive the journey to infinity — a circulation, a net outflow and a dipole — and nothing else about a body does. Two shapes with nothing in common can therefore make the same flow a few radii away, and the difference between them dies two orders faster than the disturbance either one makes.

inviscid · Far field An oscillation with no mean, and the steady flow it drives. The steady second-order velocity through a Stokes layer, in units of U U′/ω. The first-order flow averages to zero at every height; the average of its own nonlinear term does not, and the pale curve is that forcing. Integrating it twice across the layer, with no slip at the wall and no stress at the top, gives a steady velocity that rises through the layer and settles at -0.749998 — Rayleigh's −3/4, which was not put in anywhere. Beyond about five layer thicknesses nothing more happens, which is why the number is a boundary condition for the flow outside.

An oscillation with somewhere to go

Shake a fluid back and forth over a body and it develops a steady circulation that never reverses. The driving flow has no mean at all; the mean of its own nonlinear term does, and integrating that twice across the oscillatory layer gives a slip velocity of exactly three-quarters of U dU/dx over the frequency.

viscous · Streaming The sound is what fails to cancel. A compact quadrupole of kd = 0.01, followed outwards. The upper curve is what one of its four sources produces on its own at each radius — every one of them is as loud as a monopole — and the lower curve is what all four produce together. Inside the source they barely cancel at all; by one radian of wavelength the sum is 9.3e-5 of what a single source is doing, and it is falling as 1/R from there on because that is what radiation does. Between the two the field falls as the cube of the distance, which is a near field rather than a sound.

The sound is what does not cancel

A quadrupole is not a weak source. Every one of the four monopoles in it is as loud as a monopole of the same strength, and what makes the assembly quiet is that they very nearly cancel — one part in ten thousand survives at a hundredth of a wavelength. The eighth-power law is a statement about how nearly, not about how little.

compressible · Aeroacoustics The exposure lengths that give the true mean, and the ones that do not. How far a finite exposure of a shedding wake lands from the long-exposure mean, against how many shedding periods the shutter was open for. It is exactly zero at every whole number of periods — an average over a complete cycle is the mean, with no error at all — and between the zeros it falls as one over the exposure. Nothing about the picture tells the reader which of these they are looking at.

The shutter is part of the answer

Flow photographs are compared as though the exposure were a detail of the camera. It is a term in the measurement: an average over exactly one shedding period returns the true mean to fourteen figures, any other length carries a residue that falls only as one over the exposure, and a two-pulse velocity reading is short by exactly the sinc of the swept angle.

misconceptions · Visualisation Two velocities, 2.50 apart, and only one of them is anybody's. The velocity in Darcy's law is the flow rate divided by the whole cross-section, solid included: a speed no fluid particle ever has, since the fluid occupies only the fraction ε of that area. The speed the fluid actually averages is larger by exactly 1/ε — 2.50 times here — and it is the one that belongs in a residence time, in a pore Reynolds number and in any statement about when a tracer arrives. The grains are drawn to say that the pore-scale flow is not computed anywhere: no figure on this site claims to resolve it.

A velocity nobody has

The velocity in Darcy's law is the flow rate divided by the whole cross-section, solid included — a speed no fluid particle in the bed ever has. Averaging buys a linear law and charges for it in exactly this coin, and the constant that comes with it is an area of about a square micron.

applied · Porous One rate per moment, and none of them the same. lambda_p = ln<l^p>/(p t) against p. As p goes to zero it is the Lyapunov exponent, the rate of the typical element; at p = 1 it is the rate of the average length, which is nearly twice as large. If ln l were exactly Gaussian this would be a straight line with the Lyapunov exponent as its intercept, and the departure from that line is the same multifractality the velocity increments have.

The stretching rate that is not one number

A material line in a flow gets longer, and there is a theorem saying its length grows at a definite exponential rate. There is also a rate at which the average length grows, and it is nearly twice as large — and a different rate for every moment of the distribution.

kinematics · Material lines Where Einstein's line stops being the measurement. The viscosity of a suspension of rigid spheres relative to the liquid's, against the volume fraction. Einstein's 1 + 5φ/2 is exact for one sphere and holds while the spheres cannot feel one another, which is up to about five per cent by volume. Batchelor and Green's two-sphere term takes it a little further; beyond about a fifth nothing derived works and the curve drawn is a fit, which diverges at a maximum packing that is itself a measurement.

A viscosity made of particles

Stir rigid spheres into a liquid and the mixture is thicker, by five halves of the volume fraction. Einstein's coefficient is not an empirical constant — it is a dissipation calculation on one sphere — and doing it as an energy rather than as a stress shows that four-fifths of it comes from somewhere nobody mentions.

viscous · Suspension The mean lift is not the lift at the mean angle. A finite wing's lift curve, with a gust distribution of standard deviation 3° about a mean angle of 10° drawn along the foot. The wing spends its time spread across that distribution, so what it averages is the average of the curve: 0.7951 against the 0.8368 the mean angle promises, a deficit of 5.0 per cent. Nothing has stalled, and no gust has taken the wing past the stall angle: the deficit comes entirely from the curve bending over, and it is there at every angle where the curve is not straight.

The lift at the mean angle

A wing in rough air flies at every angle in turn, so what it averages is the average of its lift curve rather than the lift at its average angle. Where the curve bends over near the stall the two differ by five per cent at three degrees of gust and by fourteen at five, and the drag goes the other way.

circulation · Gust 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 What a probe in a flame sees. Half the time hot light fluid at a hundred metres a second, half the time cold heavy fluid at twenty. That is what intermittency in a jet flame looks like at a point, and it is the simplest field in which the two averages of the velocity are different numbers.

Two averages of one flow

In a flow whose density varies there are two mean velocities, they are both correct, and across a flame they differ by a factor of two. One of them is what a hot wire returns; the other is what every compressible turbulence model is written in; and the mass flux is the single product they agree on.

kinematics · Averaging Five approach profiles a meter might be looking at. The velocity across the pipe upstream of a contraction, for a uniform flow, fully developed laminar flow, two turbulent power laws and an annular jet of the kind a bend or a partly open valve leaves. All five carry the same volume flow. The meter reads a pressure difference and cannot see any of this.

The profile a meter cannot see

A differential-pressure flowmeter measures a force balance and reports a flow rate. The step between them needs two integrals of a velocity profile the instrument has no access to — and two profiles differing by half the mean velocity across the pipe give identical readings, which is why the standards specify straight pipe rather than a correction.

applied · Metering Six orbits that do not close. One parcel's path under a linear deep-water wave of steepness 0.05, at a fifth of a wavelength down, released at the phase that centres the orbit on its release depth. Each loop returns almost to where it began and not quite.

A drift made of two things that average to zero

Stokes drift is usually explained as a parcel spending longer in the forward half of its orbit. That is true and it is not a formula. The formula is a correlation between a displacement and a gradient, each of which averages to exactly nothing, and it splits into two halves that are equal to twelve figures.

kinematics · Stokes drift When the tracer leaves, given when it went in. The residence-time distribution of three beds with the same mean residence time and different Peclet numbers. Every one of them has its mean at exactly one, and they are nothing alike: the loosest lets a tenth of the tracer out before a third of the mean time has passed, and the tightest is nearly the spike a plug-flow calculation assumes.

The outlet is the inlet, a while ago

A bed has a mean residence time and everybody quotes it. Six beds with the same mean let their first hundredth through at 0.20 and at 0.85 of it, mix a window of inlet history between 1.53 and 0.18 wide, and convert a first-order reaction by amounts the mean cannot distinguish.

applied · Porous The two dissipations, side by side. The strain form on the left and the enstrophy form on the right, for the same field, on the same scale. They have their maxima in different places — 0.46 apart on a box of side 2 pi — and neither is a smoothed version of the other. One says the dissipation is in the strained regions and the other says it is in the rotating ones, which is nearly a complete disagreement about what a turbulent flow is doing.

Equal on average, and nothing else

The rate at which a fluid turns motion into heat can be written two ways, and every textbook says the two are equivalent. Their averages are equal to fourteen decimal places. Point by point they are uncorrelated, and their maxima are in different places.

turbulence · Dissipation Two flows with one mean profile. The time-averaged velocity of a plain shear and of the same shear carrying a zero-mean disturbance. There is one line on this plot: the largest difference anywhere across the channel is four parts in 10¹⁷.

What a mean profile cannot tell anybody

Two flows are built here with mean velocity profiles that agree to four parts in 10¹⁷. One of them carries momentum across the shear and dissipates forty per cent more energy; the other carries nothing. Everything that distinguishes them is second order in a disturbance the mean cannot see at all.

kinematics · Averaging A window one core wide reads the vortex 7.5 per cent slow. The tangential velocity across a Lamb–Oseen vortex, in units of its core radius and of Γ/2π divided by it, as it is and as particle image velocimetry reports it with square interrogation windows of three widths — the average of the velocity over each window, which is what a correlation over the window returns to first order. The true peak is 0.6382 at 1.1209 core radii. A window 0.5 core radii wide reports 98.0 per cent of it, 1.021 times as far out, a window 1 core radii wide reports 92.5 per cent of it, 1.084 times as far out and a window 2 core radii wide reports 77.1 per cent of it, 1.334 times as far out. The instrument that measures velocity directly still reports a slower, fatter vortex than the one there, by an amount set entirely by the window against the core.

The window every vector is averaged over

Particle image velocimetry is the one flow-visualisation technique that reports the velocity itself, and it still applies an operator: every vector is an average over an interrogation window. A window is a filter with a transfer function, and it makes a vortex slower and fatter, a thin shear layer exactly as thick as the window, and some features smaller than the window point the wrong way.

misconceptions · Visualisation A float under swell on a rotating planet goes round instead of away. The track of a float at the surface over 1 inertial periods (16.9 hours) after a 8 s swell of amplitude 1 m arrives at latitude 45°, in kilometres, the waves travelling to the right. Without friction the float runs round a circle of radius Uₛ/f = 0.479 km and comes back to where it started every 16.92 hours. With a drag on the Eulerian current it spirals out into a steady drift veered to the right: 24.3 per cent of the drift at 76.0° for a drag of 0.25 f; 70.7 per cent of the drift at 45.0° for a drag of 1 f; 94.9 per cent of the drift at 18.4° for a drag of 3 f. In a non-rotating ocean the same float would have gone 3.0 km straight on.

The drift a rotating planet takes back

In a wave tank the Stokes drift is cancelled by a return current because the tank has walls. The open ocean has none, and the drift is cancelled anyway: the Coriolis force acts on the water's real motion, drives an Eulerian current that answers it, and leaves the depth-integrated transport exactly zero at every viscosity. A float under steady swell with nothing to stop it goes round a circle instead of away.

kinematics · Stokes drift Rolls turning side by side, with the fastest downwind water where they sink. The fastest-growing mode at a Langmuir number of 0.13, looking downwind, over two roll spacings of 2.89 decay depths and 4 decay depths down. The closed curves are streamlines of the overturning; the dashed curves are contours of the downwind velocity the rolls carry, positive under the lines where the water sinks. At the surface the cross-wind flow converges onto those lines, which is where floating foam and weed collect as windrows. The amplitude is arbitrary, as in any linear mode.

The drift that turns a current into rolls

A current carrying a Stokes drift feels a force the drift makes out of the current's own vorticity, and under a wind that force is unstable. It turns the surface layer into rolls lined up downwind, with windrows where they sink. The rolls need both the current's shear and the drift's; their growth rate sees only the product; and the split between the two decides which motion gets the energy.

kinematics · Stokes drift

Named alongside it

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

MeasurementNonlinearityClosureModel validityReynolds stressCorrelationDissipationMemory kernelMixingStokes driftTurbulenceVorticity

All concepts