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

Particle — where it appears

A small discrete body carried by a flow, either a real one being transported or a tracer added to make the flow visible. It follows the fluid only up to a frequency set by its own inertia, so it is a low-pass filter with a corner rather than a perfect marker.

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

A sphere does not go at the speed of the flow it is in. How far a force-free sphere on the axis of a round tube lags the fluid at its own centre, against its size. The lag is two-thirds of the square of the size ratio and comes from the reciprocal theorem in one line: the sphere moves at the average of the ambient flow over its own surface, and a parabolic profile is slower everywhere on that surface than at the middle. At a fifth of the tube's radius the lag is 2.7 per cent, which is the difference between a tracer and a thing being measured.

A force without the flow that makes it

A sphere carried along by a flow does not travel at the speed of the fluid at its centre. It travels at the average of the flow over its own surface — and getting that result needs no solution of the flow around the sphere at all, only the answer to a completely different problem that everybody already knows.

viscous · Reciprocity The one curve of the four that measures a gradient. A row of particles released along a line at one instant, seen at 4 later times, in a layer profile. Each line has been displaced by the local velocity times the elapsed time and by nothing else, so dividing the displacement back out returns the profile exactly — checked here to machine precision at every one of two hundred heights. A streamline, a pathline and a streakline each report where fluid went; this one reports how fast neighbouring fluid was going relative to its neighbour, which is the quantity a boundary layer is made of and the one the other three never show.

The curve that measures a gradient

Three of the four curves drawn through a flow answer the same question — where did the fluid go. The fourth answers a different one. A line of particles released together is displaced by the local velocity and by nothing else, so its shape is the velocity profile, and dividing the elapsed time back out returns that profile exactly rather than approximately.

kinematics · Flow curves 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 A particle is a low-pass filter. The fraction of a fluctuating flow's velocity that a particle follows, and the phase by which it lags, against the Stokes number. At one the particle follows 71 per cent of the motion and lags by 45 degrees — which is the corner frequency of a first-order filter, arrived at from mechanics rather than from electronics.

A particle is a low-pass filter

A seeding particle does not report the flow; it reports the flow through its own transfer function. At a Stokes number of one it follows 71 per cent of the motion and lags it by 45 degrees, and a fifty-micron droplet at a kilohertz is following two per cent of what it is supposed to be measuring.

regimes · Particle Moving air needs a quarter of the inertia still air does. The first crossing against Stokes number for a cloud thrown into still air — time in units of the cloud's fastest convergence rate — and for a cloud carried by air converging on a line, in units of the air's strain rate. Still air only brakes: its threshold is one. Converging air keeps pushing: its threshold is a quarter, and heavy particles in it cross in about one strain time. Dots: particles integrated in the converging sine flow; the curve is the linearised oscillator.

Drag decides whether a cloud folds, and the air decides at what

A cloud of free particles keeps its velocities, and wherever it converges its paths cross in finite time: a caustic, the density infinite along a sheet. Give each particle a drag on the air and the answer depends on what the air is doing. In still air the drag only brakes, and a cloud folds only if its Stokes number is above one. In air that is itself converging the drag keeps pushing, and a quarter is enough — the same quarter that decides whether a droplet hits a wing.

kinematics · Acceleration Below a quarter the cloud settles on the cell walls; above it, it crosses them. Paths of particles released with the air's velocity inside one cell of the vortex lattice, at Stokes numbers of 0.15 and 0.4, over 25 time units; the cell's walls, the separatrices joining the saddles, are drawn. At 0.15 the particles are flung outwards by the rotation and approach the walls ever more closely without crossing, so the cloud is compressed onto lines. At 0.4 they reach a wall with enough speed to overshoot it into the next cell, where they meet particles coming the other way: the cloud folds.

A lattice of vortices folds a cloud at the saddles' quarter

A cloud of heavy particles folds in still air above a Stokes number of one, and in converging air above a quarter. Air that turns as well as converges might have settled between them. In a lattice of vortices it settles exactly on the quarter: the vortices fling the particles to the cell walls, but only the saddles where the walls meet can make them cross. Below the quarter nothing folds, and the cloud is gathered onto the walls without limit instead.

kinematics · Acceleration Moving the saddles never lowers the quarter; fast large wobbles raise it. The smallest Stokes number at which a cloud of particles in the oscillating lattice folds within sixty time units, against the oscillation's amplitude ε, at frequencies of 1, 3 and 10 times the saddles' strain rate. With ε = 0 it is the steady lattice's 0.251. Moving the lattice never takes it below a quarter; at ε = 1 it is 0.283, 0.311 and 0.324 at the three frequencies.

Wobbling saddles keep the quarter, and raise it

Heavy particles in a steady lattice of vortices fold onto the cell walls only above a Stokes number of a quarter, the converging saddles' own threshold. Random flows fold at every Stokes number, so it was natural to suspect the quarter of belonging to steady saddles. Set the lattice wobbling and the particles cross from cell to cell, seventy per cent of them in forty time units, and still none folds below a quarter. The threshold belongs to the strongest strain the flow has; moving the saddles only moves particles out of reach of it, and raises the threshold.

kinematics · Acceleration A stronger strain folds particles below a quarter only if it lasts. The Stokes number above which particles in a lattice with pulsing strain 1 + a sin ωt fold onto its walls, against the pulse's frequency, for amplitudes of 0.15, 0.3 and 0.5, with the steady lattice's quarter dashed. A slow pulse folds them at the peak strain's quarter, 1/(4(1 + a)): 0.194 at a = 0.3 and ω = 0.05, against 0.192. A fast one folds them only at the mean's: 0.249 at ω = 5.

A stronger strain folds a cloud only if it lasts

Heavy particles in a lattice of vortices fold onto the walls above a Stokes number of a quarter of the saddles' strain. Let the strain pulse and the threshold moves — down to the quarter of the peak strain when the pulse is slow, back up to the quarter of the mean when it is fast — because a particle responds to the strain it has been in for about four strain times, not to the strain at the instant. In a random strain the same rule makes the folds arrive in bursts, at the excursions that are strong and long at once.

kinematics · Acceleration Saddles that share one signal fold a cloud in a single step; independent saddles fold it steadily, and less. The share of a cloud seeded over the whole patch that has folded, against time, at a Stokes number of 0.16, in four seeded fields: with every saddle's strain the same signal, and with each saddle its own. After 400 time units the shared signal has folded 0.89, 0.89, 0.89, 0.89; the independent saddles 0.77, 0.85, 0.66, 0.69. A ninth of each cloud starts at a vortex centre, where the shared lattice holds it still.

A cloud folds where a saddle is strong, not when

When every saddle of a lattice strengthens and weakens together, a cloud of heavy particles folds in one step, at the strain's first strong excursion. Give each saddle its own random strength and the folding spreads out into a steady rate — and falls, because a particle now has to be at a saddle while that saddle is strong. The rate follows the Gaussian tail of the strain, at about a tenth of the time the strain spends past its fold condition, and a longer-lived strain helps only up to a few strain times.

kinematics · Acceleration 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 flicker of a few per cent empties the suspension band and drains Stommel's traps more slowly. The share of a cloud released over one cell that is still within four cells of its start after the run, against the flicker's standard deviation as a fraction of the cells' mean strength, with a memory of three turnovers: inertialess particles falling at 0.3 of the cells' speed in still air, held in Stommel's traps, and particles of Stokes number 3 falling at 0.1, held in the suspension band. Stommel's traps hold 0.52, 0.41, 0.33, 0.16, 0.03, 0; the suspension band 0.22, 0.055, 0.015, 0.015, 0.005, 0, at flickers of 0, 0.05, 0.1, 0.2, 0.35, 0.5.

Flickering vortices cannot hold a settling particle up, and still make it fall faster

A steady lattice of vortices can do two things to a heavy particle settling through it: hold it up, on closed paths or on orbits of its own, and speed its fall, by flinging it into the down-draughts. Let the vortices strengthen and fade at random, as real eddies do, and the two come apart. A flicker of five per cent of the cells' strength empties three quarters of the band of particles held up by their inertia; a flicker of a third drains Stommel's traps almost entirely. The speed-up hardly notices a flicker of a tenth, and is halved only when the cells swing by half their strength and remember it for a few turnovers. Trapping needs a steady cell; the speed-up does not.

regimes · Particle

Named alongside it

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

Stokes numberModel limitRelaxation timeInertiaThresholdMeasurementStrain rateVelocity gradientSaddleVortexConvolutionFlow map

All concepts