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

Stokes number — where it appears

The time a particle takes to relax towards the local fluid velocity, divided by the time the flow has to turn. Below a critical value a body in a stream collects nothing at all, and above it the particle increasingly ignores the flow and travels straight.

Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.

Particles at St = 1, against the flow that carries them. Particle paths and the streamlines they were released on, in this site's exact cylinder solution. At small Stokes number the two are indistinguishable and the body catches nothing; as the particles get heavier their paths straighten, cross the streamlines, and begin to strike. The paths are integrated with Stokes drag and nothing else — no gravity, no lift, no effect of the particles on the flow.

Whether the droplet turns

The air goes round the wing. Whether what is carried in it goes round too is decided by one number — and below a critical value of that number the body collects nothing at all, however many droplets are thrown at it, because the flow turns every one of them in time.

regimes · Particle The relaxation time a particle actually has. Two quantities against the particle-to-fluid density ratio. β = 3ρ_f/(2ρ_p + ρ_f) is three for a bubble, one for a neutrally buoyant particle and nearly zero for anything heavy; it is the factor by which the fluid's own acceleration is felt. The other curve is the true relaxation time over the usual formula's, which is one for a heavy droplet, exactly three halves for a neutrally buoyant tracer, and unbounded for a bubble — the usual formula gives a bubble a relaxation time of zero, and therefore no dynamics at all.

The tracer that is not one

Every Stokes number is built on a relaxation time that counts the particle's own inertia and nothing else. Adding the two terms it leaves out gives a bubble a relaxation time where the usual formula gives zero, makes a neutrally buoyant tracer half as slow again as advertised, and sends bubbles into vortex cores that droplets are flung out of.

regimes · Particle 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 Eight numbers, one construction. The dimensionless groups this collection has produced, placed on a logarithmic axis at a representative value. Each is a memory time divided by a process time, each was named separately in a different field, and each decides the same question: whether the past is still present.

Every memory number is one time over another

These essays produced eight dimensionless groups in eight different fields, named after eight different people, spanning a factor of four hundred in value. Written out, all eight are a memory time divided by a process time, and all eight govern the same curve.

regimes · Dimensional Below a Stokes number of ten a ball does not come back. The restitution of an impact through a liquid film, as a fraction of the same impact's dry restitution, against Stokes number. Nothing rebounds below the threshold and the recovery above it is a hyperbola: the restitution, as a fraction of its dry value, is one minus the critical Stokes number over the Stokes number, with that critical value 10.18 computed from the film and the dry restitution alone. The measured curve, drawn beside it, is the same expression with ten in place of that number — and ten is what four decades of viscosity and four materials all give.

A ball that bounces in water and not in oil

A squeeze film cannot be closed with any finite energy, so nothing should ever touch anything. A sphere dropped into a tank nevertheless rebounds, and whether it does is decided by a number near ten that four materials and four decades of viscosity all agree on.

viscous · Squeeze film A film pulled apart asks for more tension than a liquid has. The pressure below ambient in the film under a sphere moving away from a wall, scaled by the tension the liquid can bear, against the distance from the axis in sphere radii, at the contact gap. The lubrication solution (dashed) asks for four times that tension on the axis. A liquid that cannot give it cavitates: a disc of vapour opens where the demand exceeds the floor, and outside it the pressure is exactly the solution it would have had. Here the disc reaches 0.12 sphere radii.

A torn film still pulls

A sphere bouncing off a wall under liquid has to climb back out through the film it squeezed, and the film pulls it back with a suction no real liquid can supply. Let the liquid cavitate and the obvious guess is that the sphere escapes the torn part of the film for free. It does not. The liquid round the vapour disc goes on pulling, and the disc itself holds the full tension over its area, so a film that tears at five times the contact gap saves a third of what the guess says — and the rebound threshold moves by five per cent where the guess said a third. At an atmosphere, in the liquids the threshold was measured in, it barely moves at all.

viscous · Squeeze film 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.

Model limitThresholdParticleRelaxation timeInertiaMeasurementSaddleStagnation pointStrain rateVelocity gradientVortexCavitation

All concepts