The swirl that holds a wave still

Put a swirl on a pipe flow and something changes that has no analogue in a straight one. The flow becomes able to carry axisymmetric waves — standing rings of expansion and contraction of the stream tubes — and those waves travel both with the flow and against it.

Against it is the interesting direction. If a wave can run upstream, then something happening at the far end of the pipe can be felt at the near end, and the flow arriving is not free to be whatever it likes. If no wave can run upstream, it is.

Benjamin called the two states subcritical and supercritical, and the words are not borrowed by analogy. They are open-channel flow’s own words, and the number that really is one is the essay about the version of this statement in a channel: at $\mathrm{Fr} = 1$ a disturbance stops being able to travel upstream, the specific energy is least and the equations change type, and the threshold does not move when the tolerance does.

This essay computes the swirl at which the same three things happen, and then shows that it is not a number.

Bragg–Hawthorne, which is the whole apparatus #

A steady axisymmetric flow of an inviscid fluid satisfies

$$\psi_{xx} + r\frac{\partial}{\partial r}\left(\frac{1}{r}\psi_r\right) = r^2 H'(\psi) - K(\psi)K'(\psi),$$

where $\psi$ is the Stokes stream function, $K = rV$ the circulation function and $H$ the total head. That is the Bragg–Hawthorne equation, and it is the axisymmetric relative of the statement inviscid does not mean irrotational makes: a rotational flow is perfectly permitted in the exact theory, and its vorticity is carried along streamlines as a function of the stream function.

Two things make it usable here. Both $H$ and $K$ are constant on streamlines, so a columnar flow specifies them once and for all; and the equation is linear in $\psi_{xx}$, so a small axial perturbation of a columnar state satisfies a linear ordinary differential equation across the pipe.

Writing $\psi = \psi_0(r) + \phi(r)\cos(kx)$ and linearising gives

$$\phi'' - \frac{\phi'}{r} - \big(k^2 + G(r)\big)\phi = 0, \qquad G = r^2 H''(\psi_0) - K'(\psi_0)^2 - K(\psi_0)K''(\psi_0),$$

with $\phi \sim r^2$ at the axis — the radial velocity is $\phi/r$ and must be finite — and $\phi(R) = 0$ at the wall.

A standing long wave is a non-trivial solution at $k = 0$. The critical swirl is the swirl at which one appears.

The case with a closed form #

For uniform axial velocity $W$ and solid-body rotation $\Omega$, $H'$ and $K'$ are both constant, so $G$ is the constant $-4\Omega^2/W^2$ and the equation is Bessel’s. Its regular solution is $\phi = rJ_1(\lambda r)$ with $\lambda^2 = 4\Omega^2/W^2 - k^2$, and the wall condition is $J_1(\lambda R) = 0$.

At $k = 0$ that reads $2\Omega R/W = j_{1,1}$, so the critical swirl number is

$$S^\star = \frac{\Omega R}{W} = \frac{j_{1,1}}{2}.$$

Neither number is quoted here. $j_{1,1}$ is found by Newton’s method on the power series for $J_1$, at 3.8317059702, with $J_1$ there coming out at $10^{-16}$; half of it is 1.9158530.

The same number is then obtained the hard way, by shooting the perturbation equation from the axis and bisecting on the swirl until $\phi(R) = 0$. It agrees with the closed form to 2.5 parts in ten thousand million, which is the check that the shooting machinery is solving the problem the closed form solves — because the shooting machinery is what every other profile in this essay is done with.

Reading the shot, which is where the number comes from #

The shooting picture is worth dwelling on because it is the mechanism, and because it makes the eigenvalue condition concrete rather than algebraic.

Start at the axis with $\phi = r^2$ and $\phi' = 2r$, which is the only behaviour that keeps the radial velocity finite there, and integrate outward. What happens next depends entirely on the sign and size of $G$. Where $G$ is negative the equation is oscillatory and $\phi$ curves back towards the axis; where it is positive the equation is exponential and $\phi$ runs away. Solid-body rotation makes $G$ the negative constant $-4\Omega^2/W^2$, so the whole solution is oscillatory and the swirl controls how fast it oscillates.

At a low swirl the oscillation is slow: $\phi$ rises across the pipe and is still rising at the wall. At a high swirl it is fast: $\phi$ has already turned over and crossed zero before the wall. Somewhere between the two there is a swirl at which the first zero lands exactly on the wall, and that is the standing wave. The bisection is doing nothing cleverer than finding it.

The same picture explains why every profile has a critical swirl and why they are all different. The quantity that decides is not the swirl number but the integral of $G$ across the pipe, weighted by where the solution has amplitude — and concentrating the circulation near the axis puts a large $G$ where $\phi$ is small.

The two wave speeds, and a Galilean argument #

The criticality condition gives the critical swirl. It also gives the wave speeds, and for a uniform axial velocity it gives them rigorously and in one line.

A wave moving at $c$ is a standing wave in the frame moving at $c$. In that frame the axial velocity is $W - c$ and the swirl profile is unchanged — a boost along the axis does not touch the azimuthal velocity — so the criticality condition applies with $W$ replaced by $W - c$:

$$\frac{2\Omega R}{|W - c|} = j_{1,1} \quad\Longrightarrow\quad c_\pm = W\left(1 \pm \frac{2S}{j_{1,1}}\right).$$

The upstream-running root is zero at $S = j_{1,1}/2$, exactly, and negative above it. Below the critical swirl both waves run downstream and nothing can reach back; above it one of them runs upstream and everything can.

That is the whole of the criticality statement, and it is the same sentence as the channel’s.

What it means physically, which is a jump #

Benjamin’s reading of it is the one that makes vortex breakdown intelligible.

A hydraulic jump takes a supercritical stream to a subcritical one: the flow arrives too fast for its own waves, meets a downstream condition it cannot ignore, and adjusts through a short, dissipative region to a conjugate state carrying the same flow force. That is what the shock in a river computes — conjugate depths, half the arriving energy destroyed, and the loss matching the closed form to nine figures.

Vortex breakdown is the same transition in a swirling column: a supercritical vortex meets an adverse condition downstream, cannot pass the information upstream, and adjusts abruptly through a bubble or a spiral to a subcritical state with a much larger core. The abruptness is not viscosity; it is the change of type.

Three statements at one number, which is the test of a threshold #

The collection has a standing test for whether a dimensionless number’s critical value means anything: does more than one thing happen there? A group whose value of one is merely where somebody chose to put a coefficient fails it; a group at which several independent statements coincide passes.

The Froude number passes, and the number that really is one is the essay that says so: at $\mathrm{Fr} = 1$ the characteristic speeds $U \pm \sqrt{gh}$ change sign, the specific energy is least, and the equations change type — three statements, one number, no tolerance anywhere.

The critical swirl passes the same test. At $S = j_{1,1}/2$ a standing long wave first exists, the upstream-running wave speed crosses zero, and the linear operator’s first eigenvalue passes through zero so that the two-point problem becomes solvable. All three are the same fact seen from three sides, which is what a genuine threshold looks like from the inside.

What the swirl number does not share with the Froude number is universality of the value, and the rest of this essay is about that.

And then it stops being a number #

Everything above is exact for one profile: uniform axial flow with solid-body rotation all the way to the wall. Real vortices are not that, and the eigenvalue problem does not care what a reader would prefer.

Take a Rankine vortex — solid-body rotation inside a core, potential flow outside — and shrink the core. Define the swirl number the natural way, as the core’s own $\Omega R/W$. The critical value computed by shooting rises from 1.916 at a core filling the pipe to 4.917 at a core a quarter of the radius, a factor of two and a half.

Define it the other natural way, as the swirl at the wall, and the same family runs the other way: from 1.916 down to 0.307.

And a q-vortex with its circulation concentrated near the axis gives values from 1.739 down to 0.403 as the concentration is tightened.

Across twelve perfectly ordinary columnar vortices, each with a swirl number defined in a way somebody uses, the critical value spans a factor of sixteen. That is the refutation this essay carries, and it is not about exotic profiles. It is about the two most common vortex models in the subject.

What the collapse is claiming #

A dimensionless group is a claim that a second axis does not matter, and this collection has spent a a run of essays on what happens when the claim is false. A limit nothing reaches is about incomplete similarity, where the residual dependence is a power law nobody can remove. Counting what matters is about how many groups a problem has in the first place.

The swirl number’s case is simpler and worse. It is a single number standing in for a function — the whole radial distribution of circulation — and the eigenvalue problem is a functional of that distribution. There is no reason for a single moment of a function to determine an eigenvalue of an operator built from it, and it does not.

What is universal is the structure: there is a critical state, the upstream wave speed crosses zero there, and the transition through it is a jump. What is not universal is where.

The one thing every profile agrees about #

There is a genuinely profile-independent statement here and it should not be lost in the scatter.

At the critical state, and only there, the slowest wave stands still. That is what makes the state detectable without solving anything: run a computation or an experiment, watch a disturbance introduced downstream, and see whether it propagates upstream. The swirl number at which it stops is that profile’s critical swirl, whatever anybody’s formula says.

This is the same discipline the collection applies to the number that is not a number — a critical Reynolds number that depends on what is being disturbed and by how much — and to five numbers one name, where one Richardson number turns out to be five different quantities that coincide in one flow.

Two conventions, one flow, two answers #

The clearest way to see how little the number carries is to notice that a single flow has more than one of them.

Take the Rankine vortex with a core at a quarter of the pipe radius. Its core rotates at $\Omega$, so its swirl number by the core convention is $\Omega R/W$. Its azimuthal velocity at the wall is $\Omega r_c^2/R$, so its swirl number by the wall convention is $\Omega R r_c^2/R^2 \cdot 1/W$, which is smaller by $(r_c/R)^2$ — a factor of sixteen at that core size.

Both conventions are in use, both are dimensionless, both reduce to the same thing when the core fills the pipe, and they differ by sixteen on a perfectly ordinary vortex. The critical value quoted in each convention differs by the same factor, which is why the sweep above shows one family going up and the same family going down.

There is nothing wrong with either convention. What is wrong is quoting a critical value without saying which one is meant, and the fact that 1.9 is quoted in both literatures without qualification is the reason this essay exists.

The same threshold, designed for and designed against #

Crossing the critical swirl is usually described as a failure, and there is a large class of machines built around crossing it on purpose.

Take a burner past the threshold and the breakdown produces a recirculation zone on the axis — a pocket of fluid that returns upstream instead of going with the flow. That pocket carries hot combustion products back to meet the incoming fresh mixture, and it anchors a flame without any physical body in the stream to hold it. Essentially every gas-turbine combustor and industrial burner is swirl-stabilised in exactly that way, and the swirl number is a design setting rather than a limit.

On a delta wing the identical phenomenon is a hazard. The leading-edge vortices supply a large part of the lift, and as incidence rises their breakdown moves forward over the wing; the lift they were carrying goes with it, and the pitch-up that follows is what bounds the usable envelope.

One threshold, one mechanism, and two machines on opposite sides of it — which is the sharpest argument available that the number is a property of the flow rather than of anybody’s intentions for it.

Where the analogy stops #

The channel analogy is exact in the two places used above and inexact everywhere else, so it is worth saying where it stops.

A channel has one wave speed pair because it has one degree of freedom, the depth. A swirling column has an infinite family: the $k = 0$ condition used here is the first radial mode, and there are higher ones at $j_{1,2}$, $j_{1,3}$ and so on, each with its own critical swirl. A flow can therefore be subcritical with respect to the first mode and supercritical with respect to the second, which a channel cannot be.

And the conjugate state is not computed here. Benjamin’s theory pairs each columnar flow with another of the same flow force, the way conjugate depths are paired, and the transition between them is what the breakdown is. Finding it needs a nonlinear two-point problem with the same $H(\psi)$ and $K(\psi)$ and a different radial distribution, and that is not attempted.

Limits recorded rather than smoothed over #

The Galilean argument for the wave speeds needs a uniform axial velocity. With $W$ varying across the pipe a boost changes the profile’s shape, the criticality condition is not simply translated, and the wave speeds have to come from a full eigenvalue problem with $c$ in it. Every wave speed quoted here is for the uniform case.

The flow is inviscid, columnar and axisymmetric. Real breakdown is often non-axisymmetric — the spiral form is a helical mode — and the whole apparatus here is blind to it.

The critical swirl is a property of a state, not a prediction of breakdown. A flow can be subcritical and never break down, because nothing downstream is asking it to. Criticality says the transition is possible; what makes it happen is a boundary condition.

And the profiles here are models. A Rankine vortex has a discontinuous vorticity and a q-vortex is a fitted form; neither is a measurement. What the sweep establishes is that the answer moves a great deal across ordinary choices, which is a statement about the collapse rather than about any particular vortex.

What is worth remembering #

Three things, in order of how portable they are.

The structure is a theorem. A columnar swirling flow has a critical state; below it no standing long wave exists and no information travels upstream; above it both do; and the transition between the two is abrupt and dissipative for the same reason a hydraulic jump is. Nothing about a profile enters any of that.

The value for one profile is exact and beautiful. Uniform axial flow with solid-body rotation gives $j_{1,1}/2$, and $j_{1,1}$ is the first zero of a Bessel function because the perturbation equation is Bessel’s. That number is worth knowing and worth being able to derive.

And the value for any other profile has to be computed. It is a two-point eigenvalue problem, it is half a page of shooting, and the answer moves by a factor of sixteen across the vortices a reader is likely to meet. A calculation that assumes 1.9 for a concentrated vortex is out by a factor of two and a half in one direction or five in the other, depending on whose convention it inherited.

The limit and its residue #

The swirl number is what remains when a radial distribution is collapsed onto one quantity. The collapse is a limit in the same sense as every other one in this run of essays: something is thrown away, an answer survives, and the question is what the answer has lost.

Here it has lost the whole profile, and the profile is what sets the number. The structure survives — subcritical, supercritical, a wave that stops, a jump — and the value does not.

That is a better position than it sounds. A structure that survives every profile is a theorem; a value that survives one is a calibration. The essay’s business is knowing which of the two is in hand.