A stagnation point off the axis can land only on a saddle or a node, and turns it inside out

A stagnation point can leave the flow through the wall found the trick that makes a wall countable. A no-slip wall is a sheet of zeros of the velocity, so the index count that holds in open space has nothing to count there. Divide the velocity by the height and the wall’s zeros shrink to the skin friction’s nodes, foci and saddles; reflect the result through the wall and each of those is counted once while every point in the fluid is counted twice. In a swirling stagnation flow onto a plate, a point in the fluid could then sink into the wall without a partner, and the wall’s focus, directly beneath it on the axis, changed from attachment to separation to keep the count.

Everything in that flow happened on the axis. The essay ended by asking about a flow without the symmetry, in which an interior point descends somewhere else — onto, as it put it, an ordinary stretch of skin-friction line — and guessed that the count would then force a pair of new points into being on the wall, a node and a saddle, whose indices add to twice the point’s own. The guess is wrong in an instructive way. There is no ordinary stretch of skin-friction line for a point to land on.

Why there is nowhere else to land #

The argument is one line once the division by height is in hand. Write $\mathbf V = \mathbf u/z$. Above the wall it points the same way as the velocity, so it has the same stagnation points with the same indices; at the wall it is the skin friction, with a normal part that vanishes. And it is continuous down to the wall: the velocity’s tangential part grows in proportion to the height, so dividing by the height leaves something finite and smooth.

An interior stagnation point at height $z_0$ is a zero of $\mathbf V$. Let it descend. As $z_0 \to 0$ the zero of $\mathbf V$ tends to a point of the wall, and by continuity $\mathbf V$ vanishes there in the limit — which is to say the skin friction vanishes there. A stagnation point can only reach the wall at a point where the skin friction is already zero: a node, a focus or a saddle of the skin-friction pattern. It cannot land on a stretch of skin-friction line, because along such a stretch $\mathbf V$ points along the line, nonzero, and a zero cannot approach it.

That changes what the count has to say. In the reflected field the interior point, of index $\iota$, is counted twice; when it reaches the wall it disappears, and the wall point it lands on must absorb $2\iota$. A wall point’s three-dimensional index is its two-dimensional skin-friction index times the sign of the rate at which the flow leaves or approaches the wall there. The skin-friction pattern is smooth through the landing, so its two-dimensional index cannot change; the only thing that can change is the sign of the normal rate. The wall point turns from separation to attachment, or the other way, and its index changes by two. One point, flipped, keeps the count — no pair is needed, and none appears. That is exactly what happened on the axis, and the symmetry turns out to have been irrelevant to it.

A flow to watch it in #

The claim is about any no-slip, incompressible flow, and a flow that has no axis of symmetry is easy to write down. Take

$$\mathbf V = \Big(x^2 - \mu + a z,\; y,\; -\tfrac12(2x + 1)\,z\Big), \qquad \mathbf u = z\,\mathbf V.$$

Its velocity vanishes on the wall, and it is divergence-free: $z(\partial_x V_x + \partial_y V_y) + \partial_z(zV_z) = z(2x + 1) - (2x + 1)z = 0$. It is a local model, the kind of field the topology of the count a pattern cannot break is a statement about, rather than a solution of any particular body’s flow; the claim needs nothing more.

Its skin friction is $(x^2 - \mu, y)$: for any positive $\mu$ a saddle at $x = -\sqrt\mu$ and a node at $x = +\sqrt\mu$. Its one interior stagnation point is where all three components vanish above the wall: $y = 0$, $x = -\tfrac12$ from the vertical component, and then $z_0 = (\mu - \tfrac14)/a$ from the first. For $\mu$ above a quarter the point is in the flow, at a height that falls as $\mu$ falls; at $\mu = \tfrac14$ it reaches the wall, and at that moment the saddle has moved to $x = -\sqrt{1/4} = -\tfrac12$, directly underneath it. Nothing arranged the coincidence; continuity did.

The saddle turns inside out #

The normal rate at the saddle — the vertical velocity’s growth with height there, divided by the height — is $(2\sqrt\mu - 1)/2$. It is positive for $\mu$ above a quarter, so the flow rises off the wall at the saddle: a saddle of separation. It is negative below a quarter: the flow falls onto the wall there, a saddle of attachment. The figure shows that the sign changes at exactly the value of $\mu$ at which the interior point reaches the wall, as the argument requires. The node, meanwhile, is untouched: its normal rate is $-(2\sqrt\mu + 1)/2$, negative at every $\mu$, an attachment node throughout.

The indices bear it out, each computed independently from the field on a small sphere round the point. Above a quarter the interior point has index +1, counted twice; the separation saddle has $-1 \times +1 = -1$; the attachment node $+1 \times -1 = -1$; the sum is zero. Below a quarter the interior point has gone, the saddle is an attachment saddle, $-1 \times -1 = +1$, the node is $-1$ still, and the sum is zero. The degree of the reflected field’s direction over a box enclosing everything, a Gauss-map integral that knows nothing about where the points are, is zero at every value of $\mu$. The interior point did not vanish; it went into the wall and became the change in the saddle’s sign.

What the wall shows, and what it does not #

Here is the part that matters for anyone who reads a wall. The skin-friction pattern — what oil or china clay on a model shows after a wind tunnel run — is $(x^2 - \mu, y)$, and it is the same pattern for every positive $\mu$: a saddle and a node, the lines from far away converging on the saddle’s outgoing separatrix and spreading from the node. The saddle’s outgoing separatrix is exactly the kind of line oil flow is read as separation from: neighbouring skin-friction lines run into it from both sides, the oil piles up along it, and in a photograph it is a dark streak running out of a saddle.

But whether the flow actually leaves the wall along that line depends on the normal rate, and the pattern does not contain it. With the interior point hanging above the saddle the flow does leave: it rises off the wall at the saddle, passes beneath the interior point and turns away. With the point landed it does not: the flow comes down onto the wall at the saddle and spreads along it. The same converging line is a separation line in one case and an attachment line in the other, and the picture on the wall is identical. Lighthill’s rule of 1963, that separation lines begin at saddles of the skin friction, is a statement about the pattern; whether a given saddle is a separation is a statement about the flow above it, and the interior stagnation point is the thing that decides.

That is the answer to the last part of the earlier essay’s question — whether the separation line oil-flow pictures show running out of such a saddle is the trace of the point that came down. It is the other way round. The line is the skin friction’s, and it is there whether or not the point is above. The point is what makes the line a separation: while it hangs over the saddle, the dividing surface between the flow that leaves the wall and the flow that does not passes through it and comes down to the wall along the line, and once it has landed there is no dividing surface to come down.

What the symmetric flow could not show #

On the axis of the swirling flow the landing and the flip looked like two facts that happened to coincide: a point sank, and the focus beneath it changed. The asymmetric flow separates them enough to see that they are one fact. Here the saddle moves as the parameter changes — from $x = -0.77$ at $\mu = 0.6$ to $-0.55$ at $0.3$ — while the interior point stays at $x = -\tfrac12$ and only sinks. The two travel independently and arrive together, because the point’s height and the saddle’s normal rate are both the same function of the flow near the wall, read in two places. In the swirling flow both lived on the axis, so their meeting looked like geometry; here it is plainly dynamics of the field, and it would happen wherever the point and the saddle started.

The independence also says what an experimenter would see while it happens. Watching the wall, the saddle drifts smoothly and nothing marks the instant; watching the flow above it, a stagnation point sinks and vanishes. The two records only agree on a date if they are read together, and a smoke picture of the plane above the saddle and an oil picture of the wall beneath, taken at the same instant, are the measurement that would see the landing directly.

The surface the point was holding up #

The interior point is not an isolated curiosity. In the flow above a separation saddle, the fluid divides: some of it, arriving along the wall, rises off at the saddle and leaves; some passes over it. The surface between them — the separation surface — comes down to the wall along the saddle’s converging line and is pinned, above the saddle, by the interior stagnation point, which is a saddle of the three-dimensional flow lying in it. As the point sinks the separation surface flattens onto the wall, and when the point lands the surface has no height left: the fluid that was rising off now stays, and the line it rose from becomes a line it settles onto.

That is why a separation, in this sense, is a property of the flow rather than of the wall. The skin friction records where the near-wall fluid goes along the surface; the separation surface and the stagnation points on it record whether any of that fluid leaves. A designer reading an oil-flow photograph of a wing or a car is reading the first and inferring the second, and this small flow is a counter-example to the inference: every line in the photograph the same, and the separation present in one case and absent in the other.

The long argument about open separation #

The distinction has been argued over for decades, and this small flow sits at the heart of it. Lighthill’s topological picture made separation a property of the skin friction: a separation line is a convergence line out of a saddle, and every one must start and end at critical points. Wang’s measurements on bodies of revolution at incidence in the 1970s found convergence lines that start nowhere — “open separation”, lines along which the skin-friction lines merge without a saddle at their ends — and the debate since has been about whether a convergence line without a saddle can be a separation. Tobak and Peake’s synthesis of 1982 kept the topological rules and called the open kind a separation in a weaker sense.

The flow here adds a third possibility: a convergence line with a saddle at its end, satisfying every topological rule, which is not a separation at all, because nothing leaves the wall along it. The skin friction is necessary and not sufficient. What makes a separation is a surface in the flow that comes down to the wall, and the stagnation points in the flow — counted with the wall’s points by halves — are where such surfaces are anchored. A wall’s pattern is a record of the flow’s footprints, not of which of them belong to fluid leaving.

Where the landing could go instead #

The model put the descending point over a saddle. A variant that placed it over the node instead would land it there, and the node would turn from attachment to separation — a point from which the flow rises off the wall in every direction, the foot of a vortex standing on the wall. The count works the same way with the signs exchanged: whatever the interior point’s index, the wall point it lands on changes by twice that. Which of the skin friction’s zeros the point lands on is decided by which one is under it when it arrives, and that is the whole of the freedom the asymmetric flow has that the axisymmetric one did not.

The landing also cannot be on a focus that is not already of the right kind — a focus’s two-dimensional index is +1 like a node’s, and it behaves the same way, which is what the swirling flow on the axis showed. A wall’s critical points are, in this sense, the only doors in it: a stagnation point in the fluid can leave only through one of them, and opening a door turns it round.

The convention, and what the model leaves out #

Indices are signs of Jacobian determinants, computed as the degree of the field’s direction over a small sphere; a wall point’s index is computed for the reflected field, whose tangential part is even in the height and normal part odd. The normal rate at a wall point is the limit of the vertical velocity over the height squared, the sign of which is separation (positive, flow leaving) or attachment. The model is kinematic: it is incompressible and satisfies no slip, and it is not a solution of the Navier–Stokes equations for any particular body. That does not weaken the argument, which uses only continuity and the count, and the field’s polynomial form lets every point be found exactly. A real flow would have its points in less tidy places, and a stagnation point might approach the wall over a long range of a parameter rather than linearly, but the landing could only be on a skin-friction zero and could only flip it.

Who counted the wall #

Lighthill’s chapter of 1963 in Rosenhead’s book set out the rules relating a wall’s skin-friction pattern to separation; Hunt, Abell, Peterka and Woo gave the counting rules with halves for flows over obstacles in 1978. Wang argued for open separation in a series of papers from 1972, and Tobak and Peake reconciled the views in 1982. The reflected-field count with interior points twice and wall points once is the earlier essay’s; the off-axis landing and its flip are computed here.

How it was checked #

The field’s divergence, by central differences at three points, is zero to $4 \times 10^{-12}$, and its velocity on the wall is zero. Every point’s index is the degree of the field’s direction over a sphere of radius a ten-thousandth of the scale round it, refined until neighbouring directions differ by less than a sixth of a turn; those agree with the signs the Jacobians predict. The reflected box’s degree, over a box enclosing all the points and their images, is zero at μ = 0.15, 0.2, 0.3, 0.4, 0.5 and 0.6, so the count is kept on both sides of the landing. The model refuses a non-positive μ, which would remove the skin friction’s saddle and node, and a non-positive height coefficient.

Still open: a point that arrives at a degenerate place #

Every landing here is on a simple saddle or node. A real body’s skin friction has places where two of its critical points are about to merge — a saddle and a node meeting and annihilating as the incidence changes — and a stagnation point in the flow can arrive at such a place. The next calculation would take a field in which a skin-friction saddle and node approach each other as one parameter changes and an interior point descends as another does, and map the two-parameter plane: whether the interior point’s landing and the wall pair’s annihilation can happen together, what the count requires when they do, and whether the codimension-two point where they meet is where the large-scale change in a body’s separated flow — the appearance of a pair of lee vortices, say — actually begins.