The count a pattern cannot break

Every picture of a flow on this site has a few special points in it: places where the velocity is zero. The nose and tail of a body, the middle of an eddy, the point where a separated flow closes behind an obstacle.

They look like details of the particular flow. They are not: their kinds and their number are constrained by an integer, and the constraint has nothing to do with the Navier–Stokes equations, the Reynolds number, or whether the fluid is viscous.

Two kinds, and only two #

Near a point where the velocity vanishes, the field is its velocity gradient — everything else is higher order — so the pattern is decided by a 2×2 matrix. The classification is the standard one from linear systems: real eigenvalues of the same sign give a node, complex ones with a real part give a spiral, real ones of opposite sign give a saddle, and purely imaginary ones give a centre.

Now impose incompressibility. Mass conservation in the plane is

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,$$

which is exactly the statement that the matrix is trace-free. The two eigenvalues therefore sum to zero, and that eliminates half the list at once: they are either real and opposite — a saddle — or purely imaginary — a centre.

A plane incompressible flow has no nodes and no spirals, ever. A published sketch showing a streamline spiralling into a point is not a slightly inaccurate picture of a vortex; it is a picture of a flow with a sink in it, and one line of arithmetic says so before any physics is consulted.

This is the same decomposition that splits a velocity gradient into stretch, shear and spin, used for a different purpose: there the trace was the volumetric expansion and here it is the thing that has to vanish.

The theorem, and how it is checked #

The Poincaré index theorem gives the count. Assign each critical point an index: +1 for a centre, −1 for a saddle. Then for any closed curve avoiding them,

$$\text{winding number of the velocity direction round the curve} = \sum \text{indices inside}.$$

The left-hand side is measured by walking round the curve and accumulating how far the velocity vector rotates, which comes out an integer for any curve that avoids the zeros. The right-hand side needs every critical point found and classified.

The two routes share no arithmetic — one differentiates nothing and the other solves for zeros with Newton’s method — and flowcheck requires them to agree on four fields. That check earned its keep immediately, in a way worth recording.

The first version of the scan found nothing at all in a field whose loop wound once. The reason was a triviality with a general moral: the scan looked for sign changes among the corners of each cell, Math.sign(0) is zero, and a symmetric flow puts its critical points exactly on the symmetry axis — where a symmetric grid puts its corners. Rounding then made the corner value −1.1 × 10⁻¹⁶, which is not a sign change either. The scan now asks a different question — whether a Newton step from the cell’s centre is shorter than the cell — and the index theorem is what caught the original.

The lattice, where the count can be done by eye #

The cellular flow is the cleanest case because the answer is countable by hand. It is also the flow the material-loop essay uses, and this is a second thing to know about it: it is a lattice of centres and saddles, and the saddles are exactly where the differential rotation that wound that loop up comes from.

Points are born in pairs #

The index is conserved in a stronger sense than the theorem states, and the reason is continuity.

Change a parameter of the flow slowly. The winding number round a distant loop is an integer, and an integer cannot change continuously, so it cannot change at all as long as no critical point crosses that loop. Whatever happens to the critical points inside must leave their sum alone, which means they can only appear or disappear in pairs whose indices cancel: a centre with a saddle.

That figure is the standard example and the merge is exact: the surface stagnation points are at $\sin\theta = -\Gamma/4\pi Ua$, which has no solution once the ratio passes one. Two points did not vanish separately, they merged — and the point that emerges into the fluid inherits the pair’s arithmetic.

The two branches meeting at a point is the signature of the event, and the vertical tangent there is worth noticing: the stagnation points move slowly while the circulation is small and then race together, so a bearing or a rotor operating near the critical value has a pattern that is unusually sensitive to the spin.

The same shape of event appears wherever a flow pattern changes: a separation bubble appearing behind a cylinder as the Reynolds number rises is a saddle and a centre being born together on the surface, twice over. The pattern cannot change gradually; it changes by events, and each event conserves the count.

Why a wake has to balance #

The separation pattern’s total of zero is not a coincidence of that model, and the argument is worth following because it applies to every wake there is.

Far from any body the flow is uniform: the velocity vector points the same way everywhere, so a loop drawn out there sees it turn through no angle at all and the winding number is zero. The theorem then says that the indices of everything inside sum to zero — so any pattern of critical points in the fluid behind a body must contain exactly as many saddles as centres.

That is a real constraint on what a wake can look like. A single recirculating cell with nothing else is forbidden. Two counter-rotating cells with one saddle between them is forbidden. What is allowed — and what every visualisation of a steady separation bubble shows — is two cells and two saddles: one where the dividing streamline leaves, one where it closes.

The same reasoning explains a fact about vortex streets. A row of alternating vortices has a centre in each and a saddle between each neighbouring pair, so a street of n vortices carries n centres and n − 1 saddles between them — index sum 1, which is what a loop enclosing a net circulation sees. The street this site cannot draw is nevertheless a pattern this site can count.

What the count is good for #

It is fair to ask what any of this buys, given that it computes no force and predicts no number. Three things, and all of them are about pictures.

It checks a visualisation. An oil-flow or smoke picture is a pattern of critical points, and counting them is the standard way to decide whether a proposed reading of one is possible. A reading that gives the wrong total has missed a point — very often a saddle, because saddles are inconspicuous and sit where nothing seems to be happening.

It checks a computation. A steady solution’s critical-point count is a cheap invariant to evaluate, and a solver whose pattern sums to the wrong number has either a source in it or a zero that it has not resolved. This is the numerical version of the same test that caught the scan bug in this essay.

It bounds what a change of parameter can do. Because the count cannot change continuously, a pattern cannot deform smoothly into another with a different total. Something has to happen: a pair is born, a pair merges, or a critical point crosses the boundary of the region. The vocabulary of “the flow changes character at this Reynolds number” is, in the pattern’s own terms, exactly this.

Bodies, and the version of the rule that engineers use #

Everything above is for critical points in the open fluid. A body changes the bookkeeping, because a body is a hole in the plane and the theorem is about regions without holes.

The version used in experimental fluid mechanics — Hunt’s rule, from the work on flow visualisation in the 1970s — puts the half-nodes and half-saddles that sit on a surface into the same sum with a weight of one half, and relates the total to the number of bodies in the section. It is the tool for checking an oil-flow visualisation or a surface streak pattern: count the points on the surface, add up the halves, and see whether the total is what the geometry allows.

This site does not compute that version, and the honest reason is that the pattern on a surface is a skin-friction field rather than a velocity field, and nothing here solves for one on a three-dimensional body. What is asserted is the plane theorem on open fluid, which is what the four checks above are.

Reading a photograph with the count #

The most common use of this arithmetic is in front of a picture, and it is worth walking through what that looks like.

A surface-flow visualisation — oil on a wing, or dust on a wind-tunnel floor — shows a pattern of lines with a few special points in it. Some are obvious: a clear focus where the lines spiral in, a node where they radiate. The saddles are the hard ones, because a saddle looks like nothing much: lines passing by in two directions with a gap between them, in a region where the eye sees no event.

The count is what forces them to be looked for. If the visible nodes and foci add to more than the geometry allows, saddles have been missed — and the arithmetic says how many. That is a genuinely useful instruction for reading a picture, because it converts “look carefully” into “there are two more of something, and it is a saddle”.

This is the one place in the subject where a topological theorem does experimental work, and it is why the technique spread through experimental aerodynamics in the 1970s rather than through theory.

Why an oil-flow picture may spiral when a streamline picture may not #

There is a tension between two statements in this essay that a careful reader will have noticed. A plane incompressible flow has no foci — that is proved above, from the trace — and a surface-flow visualisation routinely shows lines spiralling into a point. Both are correct, and reconciling them identifies what those spirals actually are.

The theorem’s premise is that the field being examined is divergence-free in the plane it is drawn in. A slice through a three-dimensional flow is not: fluid leaves the slice, so the in-plane divergence is whatever the out-of-plane gradient makes it, and the trace is free to be anything.

The skin-friction field is the same case in its most useful form. What an oil film shows is the direction of the wall shear stress — the limiting direction of the velocity as the surface is approached — and that is a two-dimensional vector field on a curved surface with no incompressibility constraint of its own, because the fluid immediately above it can move away from the wall. So the full classification returns: nodes, foci and saddles are all admissible, and each of them means something.

A focus is the important one. Lines spiralling into a point on a surface are the footprint of fluid leaving that surface — a vortex lifting off, taking with it the fluid the skin-friction lines are converging towards. A focus in an oil-flow picture is direct evidence of three-dimensional separation, and it is the signature by which such separation is identified: the leading-edge vortex of a delta wing leaves one on the upper surface, and so does the horseshoe vortex at the base of any obstacle standing in a boundary layer.

The count comes back too, in the form the three-dimensional theorem gives. On a closed body of the topology of a sphere, the Poincaré–Hopf theorem requires

$$\sum (\text{nodes and foci}) - \sum (\text{saddles}) = 2,$$

the Euler characteristic of the surface. So a body must have critical points in its skin-friction pattern — a smooth, everywhere non-zero surface flow over a closed body is topologically impossible, which is the fluid-mechanical form of the theorem about combing a hairy ball. A sphere in a stream satisfies it with the two obvious ones, a front node and a rear node and nothing else; at a Reynolds number where the rear separates, a focus and a saddle are born together and the sum is unmoved.

And the topology of the body enters, which is a genuinely surprising place to find it. A body with a hole through it — a duct, an annular cowl, an aerofoil with a slot — has an Euler characteristic of zero rather than two, so its surface pattern is required to sum to nothing rather than to two. The arithmetic an experimenter checks an oil-flow picture against is therefore different for a nacelle than for a fuselage, and it differs by exactly the hole.

Which is the cleanest statement of what this essay is about. The count is not a property of the flow, the fluid or the speed. It is a property of the surface the pattern is drawn on, and the flow is obliged to arrange itself to satisfy it.

What the model does not contain #

The fields in these figures are patterns, not solutions. The separation bubble is a uniform stream plus two cored vortices, chosen to reproduce the arrangement of critical points behind a body at a Reynolds number of a few tens. It is not a solution of the Navier–Stokes equations and the captions say so. That is legitimate here in a way it would not be elsewhere on this site, because the subject is what arrangements are possible — but a figure implying it was a computed wake would be claiming a resolution nothing here has.

The scan can miss a pair. Two critical points closer together than the grid spacing are invisible to it, and nothing about the method rules that out. What the index theorem gives is a detector: if the winding number and the sum disagree, something was missed. It cannot tell what.

Only isolated, non-degenerate points. A critical point where the velocity gradient is singular — which is what happens exactly at a bifurcation — has no well-defined type, and the classifier reports it as degenerate rather than guessing.

Two dimensions throughout. In three dimensions the classification is by three eigenvalues, the constraint from incompressibility is that they sum to zero rather than that two of them do, and the index theorem becomes the Poincaré–Hopf theorem, whose right-hand side is the Euler characteristic of the surface. A sphere’s is 2, which is the origin of the rule that the skin-friction pattern on a three-dimensional body has two more nodes than saddles.

No time. All of this is instantaneous. In an unsteady flow the critical points move, appear and annihilate, and the count holds at every instant without saying anything about the sequence.

Who found it, and when #

The index of a vector field is Poincaré’s, from the 1880s, and it belongs to the same work on the three-body problem that produced the sections the mixing essay uses. Fluid mechanics adopted it late. The application to flow patterns was pushed in the 1950s and 60s by Legendre in France and Lighthill in Britain — Lighthill’s chapter on attachment and separation in Laminar Boundary Layers (1963) is the one usually cited — and by the 1970s it had become the standard way of reading a surface flow visualisation.

The reason it arrived so late is worth a sentence. Nobody needs it to solve a flow, and it says nothing about any number an engineer wants. What it does is say when a picture is impossible, and pictures only became central to the subject when flow visualisation did.

Where the ladder goes next #

Both kinematic essays so far have taken the velocity field as given and asked what it does to particles and to patterns. The next question is what decides the field itself when a new physical effect is added at the smallest scale — surface tension, which does nothing at all to a large body of water and completely governs a small one, and whose length is a number every fluid has.