The one rotational solution anybody can write down

The previous rung established that a steady inviscid flow can carry vorticity, that the vorticity has to be a function of the streamfunction, and that the function is not determined by anything in the problem. That is a licence to write down solutions, and the question it leaves is which ones anybody has ever managed to write.

The answer is: very few. Almost every exact solution of the steady Euler equations with vorticity in it is either a parallel shear, which is trivial, or a member of one small family. The famous member is Hill’s, from 1894, and it is the one worth knowing because it is exactly the case the following rung shows to be the only one a real fluid can approach.

The construction #

Work in spherical polars with the Stokes streamfunction, which is the axisymmetric relative of the plane one: $u_r = (r^2\sin\theta)^{-1}\partial_\theta\psi$ and $u_\theta = -(r\sin\theta)^{-1}\partial_r\psi$.

Outside the sphere, take the flow past a sphere that every course computes:

$$\psi_{\text{out}} = \tfrac12 U r^2\sin^2\theta\left(1 - \frac{a^3}{r^3}\right).$$

Inside, take

$$\psi_{\text{in}} = -\tfrac34 U r^2 \sin^2\theta\left(1 - \frac{r^2}{a^2}\right).$$

Both vanish on $r = a$, so the surface is a streamline of each. Differentiating both in $r$ at the surface gives $\tfrac32 U a \sin^2\theta$ from each side, so the tangential velocity is continuous — which is the only place the construction could fail, and the check that says it does not.

Applying the axisymmetric operator to the interior gives a vorticity

$$\omega_\varphi = -\frac{15U}{2a^2}\,r\sin\theta,$$

proportional to the distance from the axis and to nothing else. Outside it is zero, to eight decimal places, measured by differencing the velocity field rather than by reading it off the formula it was constructed from.

Why that vorticity distribution and not another #

Because it is the one that makes the flow steady.

Steady axisymmetric Euler requires $\omega_\varphi/\varpi$ to be a function of $\psi$ alone, where $\varpi = r\sin\theta$ is the distance from the axis. Hill’s choice is the simplest possible one: that function is a constant. So $\omega_\varphi = A\varpi$ throughout the interior, the governing equation is linear, and its solution is the quadratic above.

Choose a different function and there is a different vortex. Norbury computed the whole one-parameter family in 1973 — from a thin ring of small core all the way to Hill’s sphere, which is the fat end of it, the member where the core has swelled until it fills the whole bubble. Every one of them is an exact steady solution and Hill’s is the only one with a closed form.

The fluid that overtakes it #

The interior motion is the part that surprises, and the numbers are worth stating in the frame that makes them physical.

In the lab frame the vortex travels at $U$ through fluid that is at rest far away. On the axis inside it, the fluid moves forwards at $2.5U$ — two and a half times the speed of the thing containing it. Round the outside of the interior it comes back, and the whole interior recirculates in a closed loop that never exchanges a particle with the outside.

That last clause is what makes a vortex ring the transport mechanism it is. The fluid inside is carried bodily, not entrained and not diffused. A smoke ring keeps its smoke; a squid’s jet pulse carries its own momentum to where it is going; a heart’s mitral inflow forms a ring that carries blood into the ventricle without mixing it with what is there. In every case the mechanism is the closed streamline surface above.

Every quantity it has is finite #

This is worth spelling out because the standard alternative — a ring of concentrated vorticity — has several that are not.

Its kinetic energy is finite, which a thin-cored ring’s is not: a filament’s energy diverges logarithmically as the core shrinks, so a ring’s speed depends on a core size that has to be supplied from outside the model. Hill’s vortex has no such parameter.

Its impulse is finite and exact, and is $2\pi\rho a^3 U$ — the added-mass impulse of a sphere plus the momentum of the fluid it carries, a distinction that has to be made carefully.

And its vorticity is bounded everywhere, with no singular filament in it. The whole solution can be differentiated as many times as anybody wants, in the interior and in the exterior, with one discontinuity in the vorticity at the surface and none in the velocity.

The energy and the impulse, and the relation between them #

The two integral quantities are worth having side by side, because their ratio is the one number that characterises a vortex of this kind and is what a dimensional argument would have to guess.

For a sphere of radius $a$ translating at $U$, the impulse is $2\pi\rho a^3 U$ and the kinetic energy — interior and exterior together — is $\tfrac{10}{7}\pi\rho a^3 U^2$. So

$$\frac{E}{I U} = \frac{5}{7},$$

a pure number with no geometry left in it. Compare a solid sphere, where the same ratio would be $\tfrac12$, and a thin-cored ring, where it depends logarithmically on the core size and therefore is not a number at all.

The reason to care is that $E$ and $I$ are what survive when the details do not. A real vortex ring loses its exact structure within a few diameters of travel; what it keeps, until viscosity has had time to act on the whole of it, is its impulse — because impulse is conserved and structure is not. Everything predictive about a real ring is a statement about $E$ and $I$, and a solution’s value is largely in supplying the relation between them.

The family has an end, and something stops there #

The section above put $E/IU = 5/7$ beside a thin ring’s logarithm and called the difference a parameter removed. There is a stronger reason to care about that ratio, and it decides how large a vortex ring can be made.

Order the Norbury family by how fat the core is and follow the energy each member carries at a given impulse and circulation. It falls monotonically as the core swells, and Hill’s sphere — the fat end, where the core has filled the bubble — is the member with the least of it. There is nothing past that; the family stops, because a core cannot be fatter than the whole vortex.

Now consider how a ring is actually made. A piston pushes a slug of fluid out of a tube, the shear layer at the lip rolls up, and the growing ring is fed circulation, impulse and energy in whatever proportions the jet happens to deliver. Those proportions are fixed by the geometry of the orifice and by nothing about the ring. So the jet supplies energy per unit impulse at some rate, and the ring can only accept it while a steady ring of that description exists. Push for long enough and the jet’s ratio falls below what the family’s last member can carry, and at that point the ring cannot grow any further: it detaches from the feeding jet and travels on, leaving the rest of the discharge behind as a trailing column that never rolls up.

The stroke length at which that happens, divided by the orifice diameter, is the formation number, and it is close to four for a remarkable range of generators — round orifices, nozzles, different piston programmes, different Reynolds numbers. A slug longer than four diameters does not make a bigger ring. It makes the same ring, followed by a jet.

That is a genuinely predictive consequence of a nineteenth-century exact solution, and it is one the solution’s author had no interest in. It also settles the biological cases the section above listed as uses of the closed streamline surface. Squid, jellyfish and the human heart all operate at stroke ratios near four, measured rather than assumed, and they do so because there is no advantage in going further: past the formation number the extra fluid ejected is not being packaged into a travelling vortex, so it carries its momentum away as a jet instead — which is a worse deal, because a jet’s momentum per unit energy is lower than a ring’s.

The mitral inflow into a healthy left ventricle sits just under the number, and it moves away from it in several kinds of heart disease, which is why the ratio has become a clinical measurement. A cardiologist reading it is reading how far along the Norbury family the patient’s filling vortex gets before it pinches off.

The argument that makes all of this rigorous is Kelvin’s, sharpened by Benjamin: a steadily translating vortex ring is not merely a solution of Euler’s equations but an extremum of energy among all flows with the same impulse and circulation. A variational characterisation is what turns a family of solutions into a selection rule, and it is the same manoeuvre as the one the next rung uses — asking not what the equations permit but what a limit of them selects.

It is worth noting what the formation number is not. It is not a limit on how much impulse a generator can deliver, and not a limit on the momentum a squid can produce — a longer stroke ejects more fluid and produces more thrust. What it limits is how much of that can be packaged into a single travelling vortex, which is a statement about efficiency rather than about capacity.

The atmosphere it belongs to #

A vortex ring is one of the few flows in this subject that a person can make in a kitchen and that a theory can compute exactly, and the gap between those two facts is where most of the interest is.

The exact solution is steady and a real ring is not: it entrains, it spreads, it decelerates, and after enough time it becomes unstable to azimuthal waves and breaks up. The solution above is the zeroth term of that story — the state a ring approaches from below as viscosity becomes unimportant, and departs from as time passes.

And it is axisymmetric, which removes the mechanism that dominates three-dimensional vortex dynamics: stretching. A vortex line in Hill’s solution is a circle about the axis and stays one. Nothing tilts it, nothing stretches it, and nothing in this flow can amplify its own vorticity — which is why the axisymmetric problem is soluble and the general one is not.

Two dimensions has no counterpart #

The plane analogue of Hill’s vortex is worth asking about, because its absence says something about the axisymmetric case.

In the plane, the steady condition is that $\omega$ itself is a function of $\psi$, and the simplest choice is again a constant: a patch of uniform vorticity. Kirchhoff found the exact solution in 1876 — an elliptical patch rotating steadily — and the circular case is the Rankine vortex, which does not translate at all.

There is no plane vortex that travels steadily through still fluid the way Hill’s does. A counter-rotating pair does, but a pair is two singularities rather than one region, its energy diverges logarithmically, and it has no closed body of carried fluid unless the cores are given a size. The third dimension is what allows a self-propelled lump of fluid to exist, and the reason is the same one that makes three dimensions kinder throughout this subject: a disturbance decays faster, so an isolated structure can be genuinely isolated.

What viscosity does to it, in order #

The solution is inviscid and every real ring is not, and the departures arrive in a definite sequence worth knowing because each one is a different physical mechanism.

First the surface diffuses. The vorticity jump at $r = a$ is a vortex sheet, and a sheet in a viscous fluid thickens as $\sqrt{\nu t}$ immediately. Within that layer fluid that was inside begins to be outside, which is the process called detrainment and is what makes a smoke ring gradually lose its smoke to the fluid it is travelling through.

Then the whole vortex slows. The impulse is conserved but the energy is not, so as energy is dissipated at fixed impulse the vortex must grow and slow — radius up, speed down, in the combination that keeps $\rho a^3 U$ fixed. A real ring’s radius grows roughly as $t^{1/4}$ and its speed falls as $t^{-3/4}$, which is what a laminar ring does and is measurable in a tank.

And finally it goes unstable. Azimuthal waves grow on the core — Widnall’s instability — and the ring develops the wavy, then lumpy, then broken structure that ends its life as a vortex ring and begins it as a patch of turbulence.

None of that is in the solution, and the solution is nevertheless the right starting point, because all three are perturbations of it.

What the picture cannot show #

The frame. Every figure is drawn in the frame that moves with the vortex, where the flow is steady and the picture is legible. In the lab frame the streamlines look completely different — the exterior fluid moves aside and closes up, the interior loop becomes a set of forward-moving loops, and nothing about the picture is steady. Neither frame is more correct, and a reader who does not know which is being drawn will get the interior speed wrong by $U$.

The surface is a vortex sheet. The vorticity jumps from $A\varpi$ to zero across it, so the tangential velocity gradient does not exist there even though the velocity does. In an inviscid fluid that is permitted. In a real one it diffuses immediately, thickening into a shear layer whose growth is what eventually destroys the vortex, and the figures draw a discontinuity that lasts for no time at all.

And nothing is drawn about stability. Hill’s vortex is a solution and it is not a robust one: subjected to a general disturbance it deforms, and the axisymmetric family is only neutrally stable to some modes. What is drawn is a state, not an attractor.

Who found it, and when #

M. J. M. Hill published it in 1894 in the Philosophical Transactions, as a two-page note. Kelvin had already established the vortex-ring picture and Helmholtz’s laws had been in place since 1858; what was missing was an exact solution with a finite core, and Hill’s is the only one with a closed form to this day. Norbury computed the family it belongs to in 1973, numerically, and the numerical answer at the fat end is Hill’s formula.

The surprising connection is with the way the solution is used. Hill’s vortex is the standard model for a thermal — the rising bubble of warm air that a glider circles in and that a cumulus cloud is the top of — and for the head of a starting plume, and for the puff of fluid at the front of a jet that has just been switched on. In each case what is being borrowed is not the vorticity distribution, which is almost certainly wrong, but the closed streamline surface: a bounded region of fluid that travels without exchanging its contents. That is the feature the solution has and a filament model does not, and it is why a nineteenth-century exact solution is still the first thing anybody draws.

That is the last thing worth taking from this rung. An exact solution’s value is rarely the numbers it produces; it is that it removes a parameter. A ring model needs a core size and gets its speed from a logarithm of it. Hill’s vortex needs nothing, and the price of needing nothing is that it describes exactly one member of a family whose other members are what a real fluid usually contains.

Where the ladder goes next #

Below this rung is the general statement that a steady Euler flow can have vorticity in it and that the distribution is a free function.

Above it is the question this rung answers by fiat: which member of the family a real fluid picks, and why the answer comes from a limit rather than from the equations.

Beside it are the ring at the other end of the same family and why three dimensions are kinder than the plane this solution has no counterpart in.