Inertia empties the vortices that would hold a settling particle, and then speeds its fall

A particle is a low-pass filter wrote a small heavy particle’s response to the flow carrying it as a transfer function: at a Stokes number of one it follows 71 per cent of an oscillation and lags it by 45 degrees, and a fifty-micron droplet at a kilohertz reports two per cent of what it is meant to measure. Whether the droplet turns found the same number deciding whether a droplet carried towards a wing goes round it or strikes it. In both the particle was a passenger with a lag, and gravity was absent.

Every rain drop, dust grain and seed of tracer settles. In a turbulent flow the settling particle crosses eddies that push it up on one side and down on the other, and the question that matters for rain, for sediment and for every measurement made with heavy tracers is what those eddies do to its fall on average. The simplest flow that asks the question cleanly is a lattice of steady vortices, the cellular flow that a lattice of vortices folds a cloud at the saddles’ quarter used for clouds without gravity, and that wobbling saddles keep the quarter left without it on purpose.

The answer turns on the Stokes number, and it turns twice. Without inertia a slow particle can be held aloft for ever. With any inertia it cannot, and once free it falls faster than through still air. Only at larger Stokes numbers does the lattice begin to slow the fall, and a narrow band of them can suspend a slow particle again, on an orbit its inertia makes for it.

The lattice and the particle #

The flow is $\psi = (U/k)\sin kx \sin ky$: square cells of width $\pi/k$, alternating in sense, each turning at speed $U$, with upflow on one side and downflow on the other. Lengths are measured in $1/k$ and times in $1/(Uk)$, the cells’ turnover time. The particle is small and heavy, so the only force the fluid exerts on it is a linear drag towards the local velocity, with a relaxation time $\tau$, and gravity gives it a still-air settling speed $V_t$. Two numbers then describe everything: the Stokes number $\mathrm{St} = \tau U k$, the particle’s lag in turnover times, and $W = V_t/U$, its settling speed against the vortices’.

Particles are released on a grid over one cell, moving with the local air less their settling speed, and followed by fourth-order Runge–Kutta for hundreds of turnover times. A particle that has fallen four cells is counted free, and the free particles’ mean settling speed is measured over the run’s second half.

Without inertia, half a cell is a trap #

A particle with no inertia moves at the air’s velocity plus its settling, and its path has an invariant: $H = \psi + W x$ is constant along it, so its paths are the contours of $H$. Where $W$ is less than one, the cells’ vortices are strong enough to close some of those contours, and a particle on a closed one goes round it for ever, never falling. Stommel found these retention zones in 1949, studying plankton held in Langmuir circulations.

The first figure measures their size two ways. One follows inertialess paths from a grid over the cell and counts those that never fall a cell’s height. The other uses only the invariant: a path that leaves the cell must cross one of its side walls, so a point whose $H$ lies outside the range $H$ takes on those walls can never leave. The two agree — 0.515 of the cell at $W = 0.3$ both ways — and the held share is 0.773 at $W = 0.1$, 0.254 at 0.6 and zero once the particle settles as fast as the vortices turn. A slow particle released at random in such a lattice has an even chance of never reaching the ground.

Any inertia empties the trap #

The retention zones are a property of the inertialess limit, and they do not survive leaving it. A heavy particle going round a vortex lags the air, and a lag on a curved path is a centrifugal drift: the particle moves outwards across the closed contours, spiralling slowly out of the zone. The second figure releases five particles inside the zone at $W = 0.3$ and times their escape. At $\mathrm{St} = 0.005$ they take about 450 turnover times to fall a cell; at 0.01, 230; at 0.02, 130; at 0.1, 40. The time is close to $2.3/\mathrm{St}$ at small Stokes numbers: the outward drift is proportional to the Stokes number, so the time to cross the zone goes as its inverse.

So the inertialess particle is a singular limit. However small its Stokes number, a real particle is only delayed by a retention zone, and a delay of a few hundred turnover times in a flow whose eddies themselves live a few turnovers is no delay at all. The trap that Stommel’s analysis found is a trap only for a particle with exactly no mass.

Freed, it rides the downdraughts #

The third figure follows one particle released inside a zone. With no inertia, at the dial’s first setting, it circles for ever. With a Stokes number of 0.05 it circles, spirals outwards, reaches the cell’s edge, and then falls — not straight down, but along the edges of the cells where the air is descending, crossing from one cell’s downflow to the next diagonally below. Over 200 turnover times it falls about a hundred units, thirty cells. At 0.5 it leaves the zone almost at once; at 3 it barely notices the cells, its path a slow, wavering descent.

The path along the downflowing edges is the reason for the next figure. A particle at a cell’s edge where the air goes down is carried down by the air as well as by gravity; one at an edge where the air goes up is pushed back towards the cell’s interior, spirals out again, and arrives at the next downflow. Inertia sorts particles onto the descending side.

Several times faster than still air #

The fourth figure is the result. At small Stokes numbers the freed cloud settles 4.8 times faster than in still air at $W = 0.1$, 2.4 times at 0.3 and 1.4 times at 0.6. In absolute terms those are about $0.48U$, $0.71U$ and $0.86U$: the slowest-settling particles gain the most, because the air’s descent along the edges, of order half the vortices’ speed, is a large addition to a small settling speed and a small one to a large.

As the Stokes number grows the gain shrinks. At $\mathrm{St} = 1$ the cloud settles 4.4, 1.2 and 1.0 times faster than in still air; at 10, 1.9, 0.6 and 1.0. A very inertial particle barely responds to any one cell; it samples upflow and downflow alike as it falls through them, and with a linear drag its mean fall then returns towards its still-air speed. On the way, at $W = 0.3$ and a Stokes number of ten, the mean fall is slower than in still air — 0.6 of it — as the lattice’s upflows hold the particle for longer than its downflows hurry it.

Gravity and inertia together therefore make the lattice a sorting machine whose sign depends on the Stokes number. Small inertia sorts particles into the downdraughts and speeds them; large inertia lets them linger in the updraughts and slows them; zero inertia traps half of them. Which a rain drop or a dust grain experiences depends on its Stokes number in the eddies that matter, not on its settling speed alone.

Why the trap must leak #

The escape has a one-line explanation. A particle on a closed path round a vortex is turning, and to turn it needs a force towards the centre; the drag supplies it only if the particle slips outwards relative to the air, by about its relaxation time times its centripetal acceleration — the Stokes number times the square of the local speed over the radius of curvature, in the cells’ units. That slip is small, but it always points out of the vortex, and on a closed contour it never reverses. Each circuit therefore moves the particle a little outwards across the contours of $H$, by an amount proportional to the Stokes number, until it crosses the zone’s edge. The escape time is the zone’s width over that drift, which is why it goes as one over the Stokes number, and why no Stokes number above zero holds a particle for ever.

It is the same lag the tracer that is not one gives a heavy particle in any curved flow, and it has a mirror: a particle lighter than the air around it — a bubble in water — slips inwards and is drawn to the vortices’ centres instead of flung from them. A bubble rising slowly through the same lattice would be held in the cores, not freed from them, which is the reason bubbles gather in vortices and why they make poor tracers of exactly the structures they collect in.

The sorting onto the downflowing edges is the same lag seen at the cells’ corners. Particles flung out of every vortex arrive at the cell boundaries, and there the flow is a set of stagnation points joined by straight streamlines, alternately up and down. A heavy particle on an upflowing boundary is decelerated against gravity and turned back into a cell; one on a downflowing boundary is accelerated along it. Over many cells the cloud ends up concentrated on the downflows — the preferential concentration that drag decides whether a cloud folds found without gravity, here given a direction.

Rain, dust and seeding #

The numbers translate directly. A cloud droplet of twenty micrometres has a relaxation time of about a millisecond and a still-air settling speed of about a centimetre a second; in eddies a centimetre across turning at ten centimetres a second its Stokes number is about 0.04 and its settling speed 0.12 of theirs — the left-hand end of the fourth figure, where the lattice multiplies its fall several times over. That is the mechanism proposed for part of the faster-than-expected growth of rain in turbulent clouds: droplets that would fall slowly and evenly instead crowd into the downdraughts, fall faster, and collide more. Dust in a dust devil, sediment over a rippled bed and seed particles in a water tunnel’s vortices sit at other places on the same map, and the map’s lesson is that the settling speed alone is not the number to read: with $W$ fixed, moving the Stokes number from 0.02 to 10 changes the mean fall by a factor of four.

For a measurement, the warning is the same one a particle is a low-pass filter gave for frequency response, extended to position. A seeding particle heavy enough to settle visibly is not uniformly distributed across the eddies it is meant to reveal; it is concentrated in their descending sides and absent from their cores, and a velocity field measured from it is weighted towards the flow there. The bias does not need a large Stokes number: the fourth figure’s largest effects are at the smallest.

A band of inertia that holds a particle up #

The fifth figure finds the exception. At $W = 0.1$ and Stokes numbers between 2 and 5, part of the cloud never falls: 8 per cent at $\mathrm{St} = 2$, 23 at 3, 9 at 5, and none at 10. These particles have found periodic orbits of their own — paths that are not the inertialess particle’s closed contours but loops their inertia sustains, going down on one side of a cell and being carried back up on the other — and they hold them for the whole run. At $W = 0.3$ no such orbit holds anything. They are a second trap, narrower than Stommel’s and made by inertia rather than destroyed by it.

The convention: Stokes number and settling, in the cells’ units #

The Stokes number is the particle’s relaxation time over the cells’ turnover time, $\tau U k$; the settling speed is the still-air terminal speed over the vortices’ speed. “Held” means not yet fallen four cells by the end of a run; “free” means fallen further. Mean settling speeds are the free particles’ fall over the run’s second half divided by that time, over the still-air speed. The flow is steady and two-dimensional, and the particles do not affect it or each other.

What the lattice leaves out #

A steady flow. Real eddies live a few turnover times and move. A particle in a retention zone of a short-lived eddy is released when the eddy dies, so the zones’ importance in turbulence is bounded by the eddies’ lifetime; the inertial speed-up, which works within a few turnovers, is the effect that survives — and is the one measured in turbulence, where heavy particles are found to settle faster than in still air.

Only drag and gravity. The particle is taken as much denser than its fluid, which justifies leaving out the pressure gradient, the added mass and the history force; a sediment grain in water, only two and a half times denser, feels all three, and they reduce its effective Stokes number.

Linear drag. A drop whose Reynolds number is not small feels a drag growing faster than its slip, which penalises fast slips more and can reverse the sign of the large-Stokes-number effect; the slowing found here depends on the linear law. That law is Stokes’s drag on a sphere, the result of the world with no inertia, and it holds only while the particle’s own Reynolds number, built on its diameter and its slip, stays below about one — a condition a raindrop of a millimetre breaks by two orders of magnitude and a cloud droplet of twenty micrometres comfortably meets.

Two dimensions. In three dimensions the vortices are tubes, and a particle can leave a cell along its axis.

The particle’s own wake and its neighbours. Collisions, collective drag and the effect of a cloud on its air are all absent; the cloud is a set of independent particles.

Who found it #

Stommel’s retention zones are from 1949; Maxey and Corrsin’s study of inertial particles settling through cellular flows, with the escape from Stommel’s zones and the preferential sweeping along the downflow, is from 1986, periodic orbits that suspend inertial particles were analysed in the studies that followed it, and Wang and Maxey measured the same speed-up in simulated turbulence in 1993. The figures here recompute those results for the stated lattice: the zones’ size two ways, the escape time’s inverse dependence on the Stokes number, the settling speed across four decades of it and the window of suspension.

How it was checked #

A particle settling fifty times faster than the vortices turn falls at its still-air speed to five parts in a million, since the cells average out across its fall. Along an inertialess path the invariant $\psi + Wx$ drifts by $2\cdot10^{-11}$ over thirty turnover times. And the retention zone’s size from the invariant’s levels, which follows no path, matches the share of paths that never fall, to a part in a thousand at $W = 0.3$.

Still open: cells that live and die #

The lattice is steady, and the only time in it is the turnover. A flow whose cells are born and decay over a few turnovers — a lattice whose amplitude flickers, or one that drifts — would release any trapped particle when its cell dies and recapture it in another. The next calculation would let the lattice’s amplitude fluctuate with a stated memory time and ask how the speed-up and the suspension band depend on that memory: whether the speed-up, which needs only a few turnovers, survives cells that live two, and whether the suspended orbits, which need a steady cell, disappear as soon as the cells’ lives are shorter than the orbits’ own period.