Steady, three-dimensional, and mixing anyway

This field’s first essay on stirring proved something clean: a steady two-dimensional flow cannot mix. Its trajectories are its streamlines, a streamline is a curve, and a particle released on one is confined to it for ever — however fast the flow is and however complicated the curve. Nothing about the vigour of the stirring changes that, and the way round it there was to make the flow unsteady.

That proof used the plane twice, and both uses are worth naming, because only one of them is usually the one people remember.

Which step needs the plane #

Step one: the streamlines are the level sets of a single scalar. That is the stream function, and it exists because the flow is two-dimensional and divergence-free. In three dimensions there is no such scalar, and a streamline is the intersection of two surfaces rather than a level set of one.

Step two: a curve in a plane separates it. That is a topological fact about two dimensions and it is what makes the confinement matter: a particle on a streamline is not merely on a curve, it is on a curve that walls off the region inside it from the region outside.

Step three, which does not need the plane at all: a particle released on a streamline of a steady flow stays on it. That is true in any number of dimensions, and it is the half everybody remembers.

So the conclusion collapses in three dimensions while its most memorable premise does not. A streamline is still a curve and a particle is still confined to it — and a curve in a volume can come arbitrarily close to every point of that volume.

The counterexample #

Something is needed that is unarguably steady, unarguably exact, and unarguably a flow rather than a model. The ABC flow is all three:

$$\mathbf u = (A\sin z + C\cos y,\ B\sin x + A\cos z,\ C\sin y + B\cos x)$$

on the $2\pi$-periodic box. Two things are true of it and both are measured here rather than quoted:

$$\nabla\cdot\mathbf u = 0\ \text{exactly},\qquad \nabla\times\mathbf u = \mathbf u\ \text{to}\ 6\times10^{-10}.$$

The second makes it a Beltrami flow, and that is what makes it an exact steady solution of the Euler equations with no approximation anywhere: $\mathbf u\times\boldsymbol\omega = 0$, so the entire nonlinear term is a gradient and the pressure absorbs it.

It is not a model of a flow, a linearisation of one, or a numerical approximation to one. It is a flow.

What one streamline does #

The way to see what a single trajectory is doing in three dimensions is to plot a point every time it crosses a plane going upwards — a Poincaré section. Two runs, from the same starting point, differing only in one coefficient of the same exact solution:

With $C = 0$ the flow is integrable: it has an invariant, its trajectories lie on surfaces, and the section of one is a curve. With $C = 1$ the section of the same trajectory is a scatter of points covering sixteen per cent of the plane.

Telling a curve from a region #

A scatter plot is not an argument: a curve drawn with too few points looks like a scatter, and a scatter drawn on a coarse enough grid looks like a curve. The measurement that separates them is what happens when the trajectory is run longer.

Count how many cells of a forty-by-forty grid the section points have landed in, against the number of crossings:

The integrable trajectory visits exactly the same forty boxes at five hundred crossings and at four thousand. A curve has a fixed set of boxes and running longer only revisits them. The other keeps finding new ones.

Neighbours #

The other measurement is what happens to two particles released a hair apart. Start them a hundred-millionth of a box from each other and measure the separation with the box’s own periodicity, so that particles on opposite faces count as neighbours rather than as a box apart.

They reach a tenth of a box after $114$ time units and a whole box after $118$, and then saturate at the size of the region they are confined to.

The behaviour class that describes is another site’s subject and this essay does not fit an exponent to it. What is owned here is the kinematic fact: a steady velocity field separates neighbouring particles, and the field doing it solves the Euler equations exactly.

Regular and irregular in the same flow #

One more picture, because it kills the idea that the flow is either one thing or the other.

Four starting points, one section plane, one steady flow. Three of the trajectories lie on closed curves and stay on them for ever; the fourth wanders through the space between them.

There is no parameter to turn. This is a single flow, drawn once, and it contains both behaviours at once. A parcel of dye released in one of the islands will still be in that island after any length of time; a parcel released between them will be spread through the region between them. The two parcels are in the same fluid, obeying the same steady field.

What a Poincaré section is measuring #

A word about the instrument, because the whole essay rests on it and it is easy to over-read.

A section replaces a three-dimensional trajectory by the sequence of points at which it crosses a chosen plane in a chosen direction. That throws away everything about the trajectory between crossings and keeps the only thing that matters for the question being asked: whether the trajectory returns to the neighbourhood of where it has been.

If the trajectory lies on a surface, the section is the intersection of that surface with the plane — a curve, traced over and over. If it does not lie on any surface, the points fill an area. So the section turns is there an invariant surface into is the picture a curve or a patch, which is a question a grid can answer.

What it cannot do is prove a negative. A trajectory that appears to fill an area might lie on a surface so convoluted that its intersection with the plane looks two-dimensional at the resolution used, and no finite computation rules that out. What the coverage measurement does instead is much weaker and much more useful: it says that the two trajectories in the figure behave differently, on the same grid, from the same start, in flows differing by one coefficient.

The two kinds of mixing this collection now has #

It is worth putting the three stirring mechanisms side by side, because they are usually run together.

Unsteadiness in two dimensions. The blinking vortex: each half-period is an exact rotation, the map from one period to the next is analytic, and the mixing comes from the two rotations not commuting. Nothing in it is three-dimensional and nothing in it is random.

Three-dimensionality with steadiness. This essay: no time dependence at all, and the mixing comes from the streamlines not lying on surfaces. Nothing in it is unsteady.

And straining, which is neither. Material lines are pulled apart wherever the strain beats the rotation, and that happens in steady two-dimensional flows too — but the stretching there is confined to a streamline’s own neighbourhood, so it thins filaments without ever bringing distant fluid together.

The last is the mechanism; the first two are the two ways a flow arranges for it to keep happening in new directions. A flow that only strains produces long thin filaments that stay where they were. A flow that strains and folds produces the same filaments wrapped through each other, and that is mixing.

Why this was surprising when it was found #

Some history, because the result is more recent than its simplicity suggests.

The two-dimensional theorem is old and correct, and it hardened into a general intuition that laminar steady flow does not mix. Chemical engineering practice reflected that: mixers were built to be turbulent or to be unsteady, and a steady laminar mixer was thought to be a contradiction.

The recognition that streamlines in three dimensions need not lie on surfaces is due to Arnold in the 1960s, and the ABC flow carries his initials along with Beltrami’s and Childress’s. The demonstration that it matters practically — that a static mixer with no moving parts and a Reynolds number of ten can mix — followed in the 1980s under the name chaotic advection, and it is now how a great deal of microfluidic mixing is done, where turbulence is not available at any price.

The reason the practical use came so late is instructive. The mechanism is invisible in a streamline plot: the field looks smooth, the pictures look tidy, and the pathological behaviour is in a property of the trajectories that no snapshot of the field displays. That is the same warning flow-curves gives about photographs of flows, in its most expensive form.

How it is arranged on purpose, and what it buys #

The mechanism is used deliberately, and the way it is arranged is worth having, because it turns out to be the blinking vortex realised in a flow that never changes with time.

The problem it solves is acute. A microchannel runs at a Reynolds number of order one or less, so turbulence is not available at any price, and the only mechanism left is molecular diffusion across the channel. That takes a time $w^2/D$, so the length of channel required to mix is proportional to $Uw^2/D$ — which is to say proportional to the Péclet number, and at a Péclet number of $10^5$ the channel has to be tens of thousands of widths long. Nobody can etch that.

The standard answer is a staggered herringbone floor: shallow grooves cut at an angle across the bottom of the channel. A grooved surface offers less resistance along the grooves than across them, so the flow near the floor is deflected sideways, and continuity turns that into a transverse circulation in the cross-section — the streamlines become helices, and the flow is three-dimensional while remaining perfectly steady.

One helix alone does not mix; it wraps the interface round a single axis and the wrapping saturates. The trick is the staggering: every half-cycle the groove pattern’s asymmetry is reversed, which moves the centre of the transverse rotation to the other side of the channel. A parcel therefore experiences one rotation, then a different one about a different centre, then the first again — and two rotations about different centres do not commute, which is precisely the blinking vortex’s mechanism, obtained here from a fixed geometry rather than from a clock.

What it buys is a change of exponent rather than a constant. Chaotic advection stretches the interface between the two fluids exponentially with distance along the channel, so the diffusion distance falls exponentially, and the length required to mix grows only as $\ln(\mathrm{Pe})$. Against the diffusive channel’s $\mathrm{Pe}$, at $10^5$ that is about twelve against a hundred thousand.

The same principle at a larger scale is an ordinary static mixer: a pipe with fixed helical elements of alternating handedness, no moving parts, used throughout process engineering. Both devices are shaped entirely by the observation this essay is about — that a steady flow with a third dimension in it has streamlines that need not lie on surfaces, and a designer who arranges for them not to gets mixing for the price of some machining.

What is not in the velocity field #

The third dimension — which is an odd thing to say about a velocity field, and it is exactly right.

Everything the two-dimensional proof needs is a property of $\mathbf u$: it is steady, it is divergence-free, its trajectories are its streamlines. All three hold here. What fails is a property of the space: that a curve in it separates it. That is not in the velocity field, it is not in the equations of motion, and it is not something a longer simulation or a finer grid would reveal.

The rule the essays around this one are written to asks each to name what its answer needs beyond the flow, and this is the most geometric answer in the field. The flow is the same object in two dimensions and three; the container is not.

What this does and does not say about turbulence #

Three things worth separating, because this result is often carried further than it goes.

It does say that steadiness is not a barrier to mixing, and that a laminar, steady, exactly-solved flow can spread a tracer through a volume. Industrial mixers exploit exactly this: a static mixer has no moving parts and no unsteadiness, and it works.

It does not say that the ABC flow is turbulent. It has no cascade, no range of scales, no dissipation, and its energy spectrum is three delta functions. Turbulence is a different object and the confusion between chaotic advection and turbulence is worth avoiding.

And it does not say that the flow is unstable. Whether the ABC flow is a stable solution of the Navier–Stokes equations is a separate question with its own answer — it is not, above a Reynolds number of order one — and the chaos discussed here is in the trajectories of a fixed field, not in the field’s own evolution. A flow can have chaotic streamlines and be perfectly stable, and this collection’s blinking vortex is such a case.

Why the Beltrami property matters here #

It would be easy to construct a three-dimensional field with tangled streamlines and prove nothing, because a field that solves no equation is not a flow.

The Beltrami condition is what closes that gap. $\boldsymbol\omega = \mathbf u$ makes the Lamb vector vanish identically, so the steady Euler equations reduce to $\nabla(p/\rho + \tfrac12 u^2) = 0$ and the pressure is whatever makes that true — a constant total head. There is nothing left to satisfy.

That also means the flow has maximal helicity, $\int\mathbf u\cdot\boldsymbol\omega$, which is the same integral that obstructs a Clebsch representation. The two facts are the same fact: a flow with non-zero helicity has no pair of scalars whose level sets are its streamlines, and a flow whose streamlines are not level sets has no reason for them to lie on surfaces.

So the counterexample is not an accident of a clever choice of coefficients. Helicity is the obstruction, and the flow with the most of it is the one that mixes.

Where the third dimension comes from in practice #

Real flows are three-dimensional almost always, so the question is why two-dimensional intuition survives at all.

It survives where a symmetry enforces it: a long cylinder, a shallow layer, a rotating fluid at low Rossby number. In each of those the third dimension is suppressed by something, and the suppression is what makes the flow behave as the proof says.

It fails wherever a small out-of-plane component appears — and a small one is enough, because the argument above is topological rather than quantitative. The ABC flow with $C = 0.05$ has a weak third component and its sections already show a thin chaotic layer between the islands. There is no threshold below which three-dimensionality is safely negligible for mixing; there is only a rate.

That is the practical warning. A calculation that treats a flow as two-dimensional because the out-of-plane velocity is small will get the forces right, the pressure right and the streamline pattern nearly right, and will get the mixing qualitatively wrong.

The model limit #

Two, and both are about what the flow is being asked to represent.

The ABC flow has no boundaries. It is periodic, so there is no wall, no boundary layer and no place for vorticity to enter. That is what makes it exactly solvable and it is also what makes it unlike any flow in a laboratory.

And the coverage measurement depends on the grid. Forty by forty is stated on the figure because a fine enough grid makes every finite orbit sparse: with four thousand points and a two-hundred-square grid both trajectories would cover a few per cent and neither would be growing. What the comparison rests on is not the absolute coverage but that one number is flat under longer running and the other is not, and that comparison is grid-independent.