A stagnation point can leave the flow through the wall, and the count is kept by halves

The sign a stagnation point carries in space gave every stagnation point of a three-dimensional flow an index — the sign of its velocity gradient’s determinant, +1 or −1 — and showed that the indices inside a closed surface add up to the number of times the field’s direction wraps the sphere as the surface is covered. On a periodic box, which has no outside, the eight stagnation points of an exact Euler flow summed to zero and could disappear only in pairs of opposite sign.

It closed on the case every real flow has and that box did not: a wall. The count computed on a body had already counted the skin-friction pattern on a closed surface, where the nodes minus the saddles equal two, and the classification literature adds a rule for sections cut through a flow above a wall, in which a point lying on the wall is counted as half a point. Nothing in either says why half, or whether a half behaves like a half when the flow changes. The question left open was whether the halves really keep the count, and whether a point in the fluid can disappear by reaching the wall instead of meeting a partner.

Both answers are yes, and the reason for the half is not a convention. A flow with a no-slip wall is half of a flow without one, and the points on the wall are the points that are their own reflection. An exact flow shows a point in the fluid leaving through the wall, alone, with the wall’s own point changing kind to keep the count — and shows that a section through the flow, the picture people actually draw, cannot see the exchange at all.

A wall is a sheet of zeros #

The trouble a wall makes is immediate. No slip means the velocity is zero at every point of the wall, so in the language of the index the whole wall is a surface of stagnation points. None of them is isolated, none has an index, and any closed surface that touches the wall passes through zeros, where the field has no direction to count.

The usual way round is to look at the skin friction instead: the tangential shear the flow exerts on the wall, which is the slope of the tangential velocity there. Close to the wall the tangential velocity grows as the height times the skin friction, and the normal velocity, by continuity, as the height squared. The skin friction is a perfectly good vector field on the wall, with isolated zeros — nodes where the shear lines radiate, foci where they spiral, saddles where they cross — and those are the points oil-flow pictures of separated flows are read by. What is missing is a single field that carries both the skin friction’s points and the fluid’s.

Dividing by the height supplies it. Let $V = u/z$, with $z$ the height above the wall. Above the wall $V$ points exactly where $u$ does, since dividing by a positive number changes no direction; so it has the same stagnation points in the fluid, with the same indices. On the wall its tangential part is the skin friction and its normal part, the normal velocity over the height, is zero. Its zeros on the wall are the skin friction’s isolated points. And each of those now has a three-dimensional index: the product of its two-dimensional index in the skin-friction pattern and the sign of the normal rate — positive where the flow leaves the wall, at a separation, negative where it arrives, at an attachment.

Reflecting doubles the fluid and not the wall #

Now reflect $V$ through the wall: keep the tangential part and reverse the normal one below. The reflected field is continuous across the wall, because the normal part is zero there. Every stagnation point in the fluid has an image below of the same index — a reflection reverses one direction and the field’s normal component together, so the determinant’s sign is kept. And every point on the wall is its own image.

Take a large closed surface round the region and its mirror image. The degree of the reflected field over it counts every point inside, so

$$2\sum_{\text{fluid}} \text{index} + \sum_{\text{wall}} \text{index} = \text{degree over the doubled surface},$$

which is the half-point rule: the fluid’s indices plus half the wall’s equal half the degree. The half is not a weighting chosen to make a sum come out. It is the statement that a wall point is one point of the doubled flow where a fluid point is two.

A swirling flow onto a plate #

The rule wants a flow on which it can be tested, with points in the fluid, points on the wall, and a parameter that moves them. The kinematics alone suffice — the count is about a field, not about the equations that produced it — so the flow can be written down rather than solved for, provided it is exactly incompressible and exactly no-slip.

Take a stagnation flow onto a plate, with a swirl about the axis and a radial rate that changes with height:

$$u_h = z\left[p(z)\,(x, y) + \Gamma\,(-y, x)\right], \qquad w = -z^2 q(z),$$

$$p(z) = \sigma + p_1 z + p_2 z^2, \qquad q(z) = \sigma + \tfrac23 p_1 z + \tfrac12 p_2 z^2.$$

The tangential velocity grows as $z$ and the normal as $z^2$, so there is no slip; $q$ is fixed by $p$ so that the divergence is zero identically, which the check below confirms by finite differences to two parts in ten billion. The skin friction on the wall is $\sigma(x, y) + \Gamma(-y, x)$: a focus on the axis that spreads outward when $\sigma > 0$ and gathers inward when $\sigma < 0$. So $\sigma$ is the skin friction’s spreading rate, and its sign is the sign of attachment or separation.

The stagnation points in the fluid lie on the axis, where the horizontal velocity vanishes, at the heights where $q$ does. With $p_1 = -1.5$ and $p_2 = 1$, which are the values used below, they are at $z = 1 \pm \sqrt{1 - 2\sigma}$. On the axis below the lower point the flow runs down to the wall; between the two it runs up; above the upper one it runs down again. That is a bubble of reversed axial flow sitting on the axis of a swirling stream — the shape of a vortex-breakdown bubble, whose existence the swirl that holds a wave still traces to a swirl strong enough to hold a wave against the stream. Here it is only a shape; nothing claims this field is what a real breakdown looks like, and the waves that decide where a real bubble sits are the subject of a vortex is a waveguide.

The swirl $\Gamma$ matters to the count in one way: it keeps the horizontal part of every Jacobian on the axis non-singular, because its determinant is $p^2 + \Gamma^2$. Without it, the point where the two interior points meet would also be a height at which $p$ vanishes, and the whole horizontal plane there would degenerate.

One point leaves through the wall, two meet in the fluid #

The first figure follows the points as $\sigma$ changes. For $\sigma$ between zero and a half there are two in the fluid: the lower one, at $1 - \sqrt{1 - 2\sigma}$, of index +1, and the upper one, at $1 + \sqrt{1 - 2\sigma}$, of index −1. The wall’s point is an attachment focus, of index −1 in space. In the half-point count that is $+1 - 1 + \tfrac12(-1) = -\tfrac12$.

At $\sigma = \tfrac12$ the two interior points meet at a height of one and vanish together, as points of opposite index in open space must. Beyond it the fluid has no stagnation points at all, the wall’s attachment remains, and the count is $\tfrac12(-1) = -\tfrac12$.

At $\sigma = 0$ something the periodic box could not show happens. As $\sigma$ falls towards zero the lower point descends — its height is $\sigma$ to first order — and at zero it reaches the wall. Below zero it is gone, and it has not met anything: the upper point is still in the fluid, far above, with its index of −1. What has changed is the wall. The skin-friction focus, which spread outward from the axis, now gathers inward, and the flow leaves the wall on the axis instead of arriving there. The focus has the same two-dimensional index, +1, on both sides of zero; the normal rate has changed sign, and so its index in space has gone from −1 to +1. The count below zero is $-1 + \tfrac12(+1) = -\tfrac12$.

The interior point, of index +1, has been paid for by a change of two in the wall’s index, counted as half. That is the trade the open question asked about. In open space a point can only disappear with a partner of opposite sign. Against a wall it has a second way out: it can be absorbed by the wall, and the price is a change of kind in the wall’s own point — here, attachment turning into separation. In the bubble’s language, the reversed-flow bubble on the axis has come down onto the plate, and the plate’s centre, which the stream used to strike, now sends fluid up into the bubble.

The two exits are two boundaries of one region #

The second figure places the two events on a map. In the plane of the spreading rate $\sigma$ and the radial rate’s first layering $p_1$, with $p_2 = 1$, the flow has one interior point wherever $\sigma < 0$, two between $\sigma = 0$ and the curve $\sigma = 2p_1^2/9$ (for negative $p_1$), and none beyond it. The two-point region is bounded by both kinds of event: its left edge is the wall exit, its right edge the meeting in the fluid.

So the same pair of points, starting at $\sigma = 0.2$ and $p_1 = -1.5$, can lose its lower member in two ways depending only on which way the parameters move. Raise $\sigma$ and the lower point rises to meet the upper one and both vanish; lower it and the lower point sinks into the wall and the upper one survives. The topology does not prefer either. What it forbids is a third way — the lower point vanishing alone in the fluid — and it forbids it because its index has nowhere to go.

The count, done three ways #

The third figure does the counting the long way, without using the formulas for where the points are. For each value of $\sigma$ it covers a box above the wall, from just above the plate to well above the bubble, and sums the solid angle the field’s direction sweeps over the box’s surface. That degree is the sum of the interior indices: −1 below zero, where only the upper point exists; 0 between zero and a half, where the two cancel; 0 beyond, where there are none. It covers the wall point with a small sphere of the reflected field: +1 below zero, −1 above. And it covers a box and its mirror image with the reflected field, which counts the interior twice and the wall once: −1 at every value of $\sigma$, on both sides of both events, to six decimal places.

That flat line is the test. The interior count jumps at zero, the wall count jumps at zero by twice as much in the opposite direction, and the doubled count does not move. The halves are halves, measured rather than assumed: at the wall event the fluid loses an index of +1, the wall gains two, and two times the first equals the second. Each point’s index was also checked separately against the degree of a small sphere round it, at four values of $\sigma$, and agreed in every case.

A section sees only saddles #

The pictures people draw of separated flows are mostly sections: a vertical plane through the flow, its in-plane velocity drawn as streamlines, the critical points marked as nodes, foci and saddles, and the wall’s points counted as halves. The fourth figure is that section for this flow, through the axis at $\sigma = 0.2$.

All three points in it are saddles. The upper one is a saddle in space too, in the sense that its index is −1. The lower one, of index +1 in space, is also a saddle in the section. The reason is short. A section through the axis sees the radial rate $p$ and the axial rate, and not the swirl, which is normal to the plane. At a root of $q$, the relation between $p$ and $q$ makes $p = (z/2)\,q'$, so the radial rate always has the sign of $q'$ and the axial rate, $-zq'$, always the opposite sign. In the section one direction is always inflow and the other outflow: a saddle, every time. In space the lower point has two inflowing directions — the radial one twice over, wound into a spiral by the swirl — and one outflowing, which makes its index +1. The section loses the second radial direction, and with it the sign.

So the section’s count cannot be conserved, and the third figure’s dashed line shows it is not: the reflected section count is −3 with one interior point, −5 with two and −1 with none. At the wall event the section loses a saddle in the fluid and its wall half-point stays a saddle; at the meeting in the fluid two saddles vanish together, which in a genuine plane flow is impossible. In each section, at each instant, the half-point rule holds for what the section shows; between instants, nothing ties the totals together, because the section is not a flow. A section’s critical points can be counted, but they cannot be tracked: the conservation that makes the count useful belongs to the three-dimensional field, and a section that passes through a point changes the point’s type whenever the direction the section discards is the one that decided it.

That is the kinematic version of a warning familiar from oil-flow interpretation: the count a pattern cannot break holds for a plane flow’s own critical points, and two kinds is a plane flow’s privilege explained why a plane flow has only saddles and centres. A section of a three-dimensional flow is not a plane flow, and its saddles may be anything — one more case of the picture belonging to whoever is watching, with the watcher here a plane rather than a moving frame.

The skin friction on the wall #

The fifth figure is what an oil-flow picture of the plate would show. The swirl winds the shear lines into a spiral round the axis; the sign of $\sigma$ decides whether they wind outward, from an attachment, or inward, to a separation. As a two-dimensional pattern the two are the same kind of point, a focus of index +1, and the pattern on the wall alone does not record which way the flow is going normal to it. The direction of winding does: an oil streak drawn outward from the centre says attachment, and one drawn in says separation.

That is the whole of the wall event as seen from the wall. Dialled through zero the focus reverses its sense, and nothing else in the picture changes. The rule that a closed body’s skin friction has two more nodes than saddles is unaffected, because the wall pattern’s two-dimensional index is the same; what has changed is the three-dimensional index, which the oil cannot show and the half-point count needs.

What the flow leaves out #

It is kinematic. The field is exactly incompressible and no-slip, and that is all the count requires; it is not a solution of the Navier–Stokes equations, and its radial layering is a polynomial chosen to put points where they can be followed. A real swirling flow onto a plate — a vortex touching down on the ground, a swirling jet striking a wall — would have its points determined by the dynamics, and the claim here is only that whatever those points do, they must do it inside this bookkeeping.

The wall is a plane, and the flow grows without bound. The degree over a large box is −1 because of how the polynomial field behaves far away; a bounded flow over a closed body would fix the doubled degree by the body’s topology instead, which is how the closed-surface rule arises. The half-point rule itself does not depend on either.

The events are the simplest ones. The points here move on the axis, by symmetry. A flow without that symmetry can have a point reach the wall off the axis, where it must land on a skin-friction line and split it, creating a saddle and a node on the wall together — a different trade with the same total.

Who counted halves first #

The rule that points on a wall count as halves was stated by Hunt, Abell, Peterka and Woo in 1978, for sections through three-dimensional separated flows around obstacles, alongside the rule for a closed body’s skin friction; Tobak and Peake extended it to the bifurcations of surface and section patterns in the 1980s. The doubling argument that explains it is the standard device for extending the Poincaré–Hopf theorem to a field tangent to a boundary. What this essay adds is the test on a flow where the exchange between the fluid and the wall can be watched happen, and the measured fact that a section through the axis shows the exchange as something the section’s own count cannot conserve. Related reading on flows that are three-dimensional and steady and still behave unexpectedly is in steady, three-dimensional, and mixing anyway.

How it was checked #

The divergence, computed by central differences at three points with three different parameter sets, is below two parts in ten billion, which is the differencing’s own error; the velocity on the wall is exactly zero. Every interior and wall point’s index, computed from the sign of its Jacobian’s determinant, equals the degree of the Gauss map over a small sphere round it at four values of $\sigma$ on both sides of both events. The degree over the doubled box is −1 to six decimals at every $\sigma$ drawn, and equals twice the interior indices plus the wall’s, each counted from the formulas. The section’s winding is counted with a step-halving loop that refuses to pass through a zero. The field refuses a layering that would turn the outer flow back onto the wall and a swirl of zero, the case in which its axis points can degenerate.

Still open: a point that lands off the axis #

Everything here moves on the axis, so a point reaching the wall always lands on the wall’s one critical point and changes its kind. In a flow without the symmetry, an interior point descends somewhere else, onto an ordinary stretch of skin-friction line, and the count says it cannot vanish there alone: it must create a pair on the wall — a node and a saddle of the skin friction, whose indices in space must add to twice its own. The next calculation would break the swirl’s symmetry with a cross-flow, follow an interior point down to the wall off the axis, and identify the pair it leaves: which of them is the attachment and which the separation, and whether the separation line that oil-flow pictures show running out of such a saddle is the trace of the point that came down.