The bubble that hammers

A vapour cavity forms on the low-pressure part of a propeller blade, is swept aft, and reaches a region where the pressure is ordinary again. What happens next is not that it shrinks quietly.

It collapses. The liquid rushing inwards has nothing to slow it — the cavity is empty, so there is no gas to compress — and the inward speed accelerates without limit as the radius shrinks. In the last microsecond the wall is moving at hundreds of metres per second and the pressure in the liquid just outside it is several thousand atmospheres.

That is what removes metal from a propeller.

The equation, from energy alone #

Rayleigh’s model is as simple as a model gets, and its simplicity is what makes the result exact.

A spherical cavity of radius $R$ in an infinite incompressible liquid, with pressure $p_\infty$ far away and nothing at all inside. The liquid is inviscid and irrotational, so the flow outside is a radial source field: continuity gives the velocity at radius $r$ as $\dot{R}\,R^2/r^2$, and the whole kinetic energy of the liquid is a single integral,

$$T = 2\pi\rho R^3 \dot{R}^2$$

which is a finite number even though the liquid is infinite, because the velocity falls as $r^{-2}$.

Equating that to the work done by the ambient pressure as the cavity shrinks from $R_0$ to $R$ gives

$$\dot{R}^2 = \frac{2\Delta p}{3\rho}\left(\frac{R_0^3}{R^3} - 1\right)$$

and everything in this essay is a consequence of that one line.

The kinetic energy of the liquid outside a moving boundary is the same object this site computes when a body starts from rest — the added mass of a cylinder is the same integral over the same kind of field. A collapsing cavity is an added-mass problem in spherical symmetry, run to its conclusion.

The comparison is worth pushing one step, because it explains the runaway. A body accelerating through a fluid carries a fixed amount of liquid with it, so its added mass is a constant and the force required is proportional to the acceleration. A collapsing cavity carries an amount that shrinks as $R^3$, so the effective inertia is disappearing while the driving pressure stays the same. The same force on a vanishing mass is the whole of it, and it is why the pressure integral over the surface of a moving body — an entirely inviscid, entirely reversible calculation — ends here in something that destroys bronze.

The collapse time, and why the constant was computed #

Integrating $\mathrm{d}t = -\mathrm{d}R/|\dot{R}|$ from $R_0$ down to zero gives a definite integral with a closed form in gamma functions:

$$\tau = R_0\sqrt{\frac{3\rho}{2\Delta p}}\int_0^1 \frac{\mathrm{d}y}{\sqrt{y^{-3}-1}} = 0.9146813565\,R_0\sqrt{\frac{\rho}{\Delta p}}$$

Every textbook prints 0.91468. This site computed it, for a reason worth recording.

The quadrature is done after a substitution that removes the endpoint singularity, and it resolves the tenth figure. Comparing it against a pasted 0.91468 would have been a check on the rounding rather than on the integral — an assertion whose tolerance was set by how many digits somebody chose to print in 1917. So the closed form is evaluated here too, via the beta-function identity that the substitution $y^3 = \sin^2\psi$ produces, with a Lanczos gamma function good to $10^{-15}$.

The two agree to a part in $10^9$: quadrature $9.146814\times10^{-5}$ s, closed form $9.146814\times10^{-5}$ s, for a millimetre cavity at one bar.

Ninety-one microseconds. That is the whole event, and the reason cavitation is heard as a hiss rather than as a series of pops.

The shape of the curve matters as much as the number. The bubble spends most of the ninety-one microseconds barely moving — it is at half its radius after four-fifths of the time — and then disappears. That is because the driving pressure is the same throughout while the mass of liquid being accelerated shrinks, so the collapse is a runaway rather than a decay, and almost everything that matters happens in the last few per cent.

Two things follow from the runaway that are easy to miss. The first is that the initial size hardly matters to the violence: the peak pressure depends on $(R_0/R)^3$, so a bubble is dangerous when it has shrunk by a given factor, not to a given size, and small bubbles are as destructive as large ones per event. The second is that anything which slows the last stage — a little gas inside, a neighbouring bubble, a nearby wall — changes the outcome completely, while anything that changes the early stages changes almost nothing. Every refinement of this model since 1917 has been a refinement of the last microsecond.

What the pressure reaches #

The collapse is violent, and the violence can be computed without leaving the model.

The pressure field around the cavity follows from the same radial solution:

$$\frac{p(r)}{p_\infty} = 1 + \frac{R}{3r}(z-4) - \frac{R^4}{3r^4}(z-1), \qquad z = \left(\frac{R_0}{R}\right)^3$$

It is zero at the wall — the cavity is empty — rises to a maximum a little over one and a half radii out, and settles to ambient far away. The maximum is found here by scanning the field rather than by quoting Rayleigh’s expression for it, so the two can be printed side by side.

The maximum sits at $r/R = 1.587$, which is $(4)^{1/3}$ in the large-$z$ limit, and the peak pressure grows as $z$ — that is, as the inverse cube of the radius. Nothing about a bubble at its original size is dangerous. Everything about one at three per cent of it is.

Five thousand atmospheres is 500 megapascals. The yield strength of a manganese bronze propeller alloy is around 200 MPa. The pressures the model produces are more than twice what the metal can take, which is the essay’s refutation stated as an inequality rather than as an assertion about chemistry.

The model predicting its own failure #

Rayleigh’s derivation assumes the liquid is incompressible. That assumption is what makes the velocity field a simple $r^{-2}$ source and what makes the kinetic-energy integral converge.

And the same derivation says $\dot{R}$ grows as $R^{-3/2}$, without bound. So there is necessarily a radius at which the model is driving the liquid faster than sound travels in it, at which point the incompressible assumption is not slightly strained — it is gone.

The crossing is at

$$\frac{R}{R_0} = \left(1 + \frac{3\rho c^2}{2\Delta p}\right)^{-1/3} = 0.03121$$

computed rather than estimated, and asserted to be small: a model whose own assumption survived all the way down would be self-consistent, and this one is not.

That is an unusually clean example of something this site asks of every figure — say where the model stops — because here the model says it itself. No external knowledge is needed to know that Rayleigh’s calculation fails; the calculation produces the number at which it fails, from its own output.

What is beyond it is where all the interesting physics lives: a compression wave radiating outward as the collapse rebounds, temperatures inside a real gas-filled bubble high enough to produce light, and — nearest a solid surface — an asymmetric collapse that forms a microjet through the middle of the bubble, aimed at the wall, at a few hundred metres per second. That jet, rather than the spherical pressure pulse, is now thought to do most of the damage. It requires the asymmetry a nearby wall provides, which this spherically symmetric model has thrown away by construction.

That universality is worth noticing. The collapse time is proportional to $R_0$ and to $\sqrt{\rho/\Delta p}$, and the shape of the curve depends on nothing. A one-millimetre cavity at one bar takes 91 µs; a ten-micrometre one takes 0.91 µs; a cavity at ten bar takes a third as long as one at one bar. Everything scales, which is the mark of a model with no material property in it beyond a density.

Where these bubbles came from, and where they die #

The two rungs of this anchor are a single story told at two scales, and it is worth putting the ends of it together.

A parcel of liquid crosses the leading edge, is accelerated, has its pressure pulled below $p_v$, grows a cavity, is carried aft, meets the pressure recovery on the back of the section, and the cavity collapses. The first half of that is an ideal-flow calculation this site has been doing since its foundation phase. The second half is this essay. The whole cycle takes a few milliseconds, and it happens a few thousand times a second on every blade of a cavitating propeller.

What makes it destructive is the asymmetry between the two halves. Growth is gentle, driven by a pressure difference of at most a bar and spread over a distance of chord. Collapse is driven by the same pressure difference and concentrated into a volume shrinking as the cube of the radius, so the energy density rises without limit while the total energy stays the same. It is a focusing problem, not an energy problem, and no amount of reducing the energy involved changes the character of the focus.

That focusing is why the collapse ends in a compression wave rather than in a gentle stop.

The measured acoustic emission from a collapsing cavitation bubble is a sharp pulse a fraction of a microsecond long, which is exactly what a shock radiating from a point looks like. Its existence is implied by this model’s failure and is not describable within it, and what a shock costs is the argument that says the energy it carries away is genuinely lost rather than returned to the bubble — which is why the rebounds get smaller.

What it takes to make it stop #

The engineering conclusion follows from the two rungs together and is unusually blunt.

Cavitation damage is not managed by materials. Harder alloys — nickel-aluminium bronze, stainless steels, cavitation-resistant weld overlays — extend the life of a surface by a useful factor and none of them is immune, because the pressures involved exceed every candidate’s yield strength. What works is preventing inception, which is the previous rung’s calculation: flatten the pressure roof, unload the tip, run deeper, run slower, or accept the erosion and plan to repair it.

The same logic applies away from propellers. Cavitation destroys pump impellers, control valves, spillway aprons, ship rudders and diesel cylinder liners — the last of these by a route worth naming, since the liner vibrates against the coolant rather than moving through it, and produces cavitation without anything travelling anywhere.

A spillway is the case where cavitation and open-channel flow meet. Water on a long chute reaches speeds where the pressure over any small surface irregularity — a construction joint, a lifted panel, a patch of erosion — falls to vapour pressure, and the resulting cavitation removes the concrete downstream of the irregularity. The remedy in modern designs is an aerator: a step that admits air into the flow, so the cavities collapse into a compressible air–water mixture instead of into pure liquid. That works precisely because it destroys the assumption this essay’s model rests on. An incompressible liquid focuses the collapse; a compressible one cushions it, and the same distinction between the two decides everything in the compressible field.

The same trick appears on ships as air lubrication and on propellers as deliberate ventilation, and in every case the engineering move is to give the collapse something to compress.

The same collapse, wanted #

Everything above treats the collapse as a nuisance to be prevented. The identical mechanism, driven on purpose, is the working principle of several machines — and the reason is the essay’s own arithmetic: the energy is conserved and the volume it occupies shrinks as the cube of the radius, so the collapse is a focusing device and focusing is useful whenever the focus is aimed at something.

An ultrasonic cleaning bath is a controlled erosion machine. A transducer drives the liquid’s pressure below the vapour pressure once per cycle at a few tens of kilohertz, so cavities grow and collapse tens of thousands of times a second, and the ones that collapse against a surface jet at it and strip whatever is stuck there. The frequency chooses the violence, because it chooses the bubble size: a higher frequency grows smaller cavities and cleans more gently, which is why delicate work is done at the top of the band.

Lithotripsy breaks a kidney stone with focused pressure pulses, and a substantial part of the breaking is done by the cavitation those pulses induce at the stone’s surface rather than by the pulse itself.

And sonochemistry exploits the interior. The gas in a real bubble is compressed almost adiabatically in that last microsecond, reaching temperatures of thousands of kelvin in a volume of a few cubic micrometres — hot enough to drive reactions, and in some conditions hot enough to emit light.

Where the model stops, listed #

The cavity is empty. A real bubble contains vapour and non-condensable gas, which cushions the final stages, prevents the singularity and produces the rebound. The Rayleigh–Plesset equation adds that content, along with surface tension and viscosity, and gives a collapse that stops at a finite minimum radius and bounces.

The liquid is incompressible. Located above, at $R/R_0 = 0.0312$.

The collapse is spherical. A bubble near a wall collapses asymmetrically and jets. That is probably the dominant damage mechanism and this model cannot represent it at all.

And there is one bubble. Real cavitation is a cloud, the bubbles interact, and the collapse of a cloud is more violent than the sum of its parts because the outer bubbles focus energy inward.

Nor is there any viscosity, surface tension or heat transfer. Each is negligible for a millimetre cavity at one bar and none is negligible at every scale: surface tension dominates below about a micrometre, and the vapour inside a real bubble has to condense at the wall for the collapse to proceed at all, which is a heat-transfer problem the model does not contain. That last is why the model behaves quite differently in a liquid near its boiling point, where the vapour cannot condense fast enough and the collapse is cushioned — a thermal effect with no analogue in the site’s other discontinuity, and one of the reasons cavitation in hot water is less erosive than the pressure calculation alone would suggest.

Four absences, and the honest summary is that this model gets the time scale right, gets the order of magnitude of the pressure right, and gets the mechanism of the damage wrong in detail while getting its character — mechanical, not chemical — exactly right.

Who found it, and when #

Lord Rayleigh published the calculation in 1917, in the Philosophical Magazine, at the request of the Royal Navy: Parsons’ cavitation tunnel had shown what was happening and nobody could say why the damage was so severe. The paper is four pages long and contains the collapse time, the pressure field and the observation that the pressures involved are enormous.

Milton Plesset added the bubble’s contents, surface tension and viscosity in 1949. The microjet was seen by Benjamin and Ellis in 1966 with high-speed photography — half a century after Rayleigh, and the first time anybody could look fast enough.

Rayleigh appears elsewhere on this site for a stability criterion about an inflection point and for the threshold at which a heated layer starts to convect, and the three have a family resemblance worth noticing: each is a drastically simplified system carried to an exact answer, and in each case the exact answer is the one that turned out to matter.

The sequence is a good one for this site’s habits. An exact calculation on a drastically simplified system produced the right order of magnitude and the right explanation in 1917; the refinements since have changed the mechanism in detail and not the conclusion; and the thing that eventually showed what was really happening was a camera, not a theory.

Where the field goes #

This anchor closes the field’s engagement with a body in a liquid, and it closes it on a model that computes the point at which it stops being true — which is as good an ending as this field has.

One anchor remains, and it goes back to a machine. Euler’s turbomachinery equation says that the work a rotor does per unit mass is the blade speed times the change in swirl, and nothing else: no blade shape, no pressure, no gas properties, no efficiency. It is the same equation for a pump, a compressor, a turbine and a fan, it follows from angular momentum on a control volume, and it closes the field on the note it opened with — a box drawn round a machine that does not need to know what is inside it.