A threshold that is also a duration

When a body tears the water computes the cavitation number and the minimum-pressure envelope of a section: inception is expected where the local pressure first reaches vapour pressure, and the number that decides it is a pressure divided by a dynamic head. The bubble that hammers then takes a cavity that already exists and collapses it.

Between those two there is a step that neither of them takes, and it is the step where the physics stops being a comparison of pressures. A bubble does not appear because a pressure was reached. It grows, and growing takes time.

The bubble that is already there #

Water does not tear at its theoretical strength. Pure water pulled on carefully withstands something like a thousand bar of tension; tap water gives out at a fraction of one. The difference is nuclei — microscopic pockets of gas, stabilised on particles or in crevices, that are already there before anything happens.

So the object to study is not a liquid but a bubble, of some small radius, containing some gas, sitting in equilibrium. Take five microns of air in water at one atmosphere as the case.

Its equilibrium is a balance: the gas inside plus the vapour pushing out, surface tension pulling in, the far-field pressure pushing in. Lower the far-field pressure and the bubble grows a little, which lowers the gas pressure and lowers the surface-tension term too, and the two do not fall at the same rate.

The equilibrium curve has a minimum. To the left of it, a bubble pushed outwards is pushed back; to the right, a bubble pushed outwards keeps going. The minimum sits at 18.19 microns for this bubble — 3.64 times its rest radius — and the pressure there is the static threshold: 1.043 bar of tension below atmospheric. Below that far-field pressure no equilibrium exists at any radius and the bubble grows without limit.

That number is Blake’s threshold, and it is the correct answer to the question “how much tension does this bubble need?” — provided the tension is applied for long enough.

The word “long” is doing all the work #

Here is the thing the static picture cannot say. Getting from five microns to eighteen is not a change of state; it is a journey, and the bubble has to make it before the tension is taken away.

A propeller blade passing a point, an ultrasound cycle, a valve’s cavitating jet: none of them holds the pressure down indefinitely. Each provides a pulse of tension of some amplitude for some duration, and whether the bubble runs away depends on whether it got past 18.19 microns before the pulse ended.

The figure is the whole argument in one picture. The same bubble, the same 1.5 bar of tension, twice. Given six microseconds it crosses the critical radius and is gone. Given 1.2 microseconds the tension is released while it is still at eight microns, and it comes back. The amplitude did not decide this.

The curve #

Bisecting on amplitude at each of eight pulse lengths gives the threshold curve.

At a hundred microseconds it takes 1.055 bar — 1.1 per cent above Blake’s static value, which is the right amount of agreement to expect from a dynamic calculation approaching a static limit slowly. At ten microseconds it is 1.057. At five, 1.11. At two, 1.82. At one, 4.65. At half a microsecond, 16.1. At three tenths, 43.2.

A factor of forty-one, for one bubble in one liquid, with nothing changed but how long the tension lasts.

The shape is a floor and a wall. The floor is the static threshold and it is nearly flat over two decades of duration. The wall is inertial: at short durations the required amplitude climbs roughly as the inverse square of the time available, because a bubble accelerated harder covers more distance in a given time in proportion to the acceleration, and the acceleration is set by the tension.

There is no single number here to call the cavitation threshold. There is a curve, and which point on it applies is a property of the machine rather than of the liquid.

Why the wall has the slope it has #

The steep end of the curve can be read off without the integration, and doing so is worth a paragraph because it says what the wall is made of.

Under a large tension the gas and surface-tension terms are negligible and the bubble is a Rayleigh cavity being pulled open: the equation reduces to a balance of the tension against the inertia of the liquid being pushed aside, and the bubble’s wall accelerates at roughly the tension over the density times the radius. Integrating twice, the distance covered in a time goes as the tension times the time squared. To cover a fixed distance — the gap between the rest radius and the critical one — in a time that is halved therefore needs four times the tension.

That is the inverse-square wall, and the computed curve follows it: from 0.5 to 0.3 microseconds the duration falls by a factor of 1.67 and the threshold rises by 2.68, against the 2.78 the square would give. The small shortfall is the gas term, which is not quite negligible even there.

The wall is inertia and nothing else, which is why it does not care about surface tension, and why it is the same wall for a bubble of any size once the distance is measured in its own critical radii.

Which of the bubble’s own times sets the corner #

The corner between floor and wall is at a few microseconds for this bubble. The question worth asking is what sets it, because the answer generalises and the curve itself does not.

Run the whole calculation for bubbles of two, five and twenty microns. The floors barely move: the static threshold is 1.17, 1.04 and 1.00 bar, because surface tension matters less to a big bubble and the atmospheric term dominates. The corners move by a factor of thirty-two, over a tenfold range in radius.

Thirty-two is not ten, so the corner is not simply proportional to the bubble’s size. The obvious candidate time — the inertial or Rayleigh time, the radius times the square root of density over pressure — is proportional to the radius, and dividing the corner by it leaves a spread of 3.2 across the three bubbles. That candidate is wrong.

The right time is the one that matches the physics: not how fast the bubble responds, but how long it takes to arrive. The critical radius is 2.7, 3.6 and 6.6 times the rest radius for the three bubbles — it grows as the square root of the radius, because the gas content does — so the journey is longer for a bigger bubble in more than proportion. The time to make it at the Rayleigh speed is the critical radius times the square root of density over the driving tension, and dividing the corner by that leaves 0.91, 0.98 and 1.09.

A spread of twenty per cent across a factor of ten in bubble size and thirty-two in the corner itself. That is a collapse, and it identifies the mechanism: the threshold is a duration because the bubble has a distance to cover.

What the solver computed, and how it was checked #

Rayleigh–Plesset with gas, vapour, surface tension and viscosity, integrated with a fourth-order step whose size follows the bubble’s own inertial time, and bisected on amplitude to twenty-six halvings.

Blake’s static threshold for this bubble is 1.043 bar and its critical radius is 18.19 µm, 3.64 times its rest radius.

Against Blake. The long-pulse threshold must approach the static value computed from the equilibrium curve, and it lands 1.1 per cent above it. Those are two independent calculations — one an algebraic maximisation, the other a bisection on an integration — and their agreement is the main check.

Against monotonicity. The threshold must not rise as the pulse is lengthened. It is checked to be non-increasing across all eight durations, which catches the failure mode where the integrator is manufacturing growth out of a step-size resonance.

Against a collapse. The corners must agree in units of the growth time better than in units of the inertial time. They do, by a factor of 2.7, and the assertion is written that way round rather than as a bare tolerance — a check that only says “within 20 per cent” would pass if both candidates were bad.

And a refusal. The march is required to fail rather than continue if the radius goes non-positive, which is what a step size chosen badly during a collapse produces.

The same shape, in a gas #

The structure here — a threshold that is really a curve because the system has a journey to make — has an exact counterpart elsewhere in this collection.

A gas that has not decided to react yet computes an induction time: a shocked mixture does not ignite when it reaches the ignition temperature, it ignites some time afterwards, and whether it ignites at all depends on whether it is held hot for longer than that. The ignition temperature quoted in tables is the long-hold limit of a curve, in exactly the sense that Blake’s threshold is the long-pulse limit of this one.

The two are the same kind of object and they are not the same physics: the bubble’s delay is inertial and the gas’s is chemical. What they share is the failure mode of the number that gets tabulated. A threshold measured under a long hold and applied to a short one is not conservative; it is wrong by whatever factor the corner supplies, and here that factor is forty.

The general form is the subject of every memory number is one time over another, and this bubble’s group is the pulse duration over the growth time.

What happens after it crosses #

Crossing the critical radius is where this essay’s question is answered and where the interesting part of the event starts.

Past the critical radius the growth is explosive: with the gas pressure falling as the cube of the radius there is nothing left to resist, and the bubble runs to whatever size the pulse’s remaining duration allows — hundreds of microns for an ultrasonic drive. Then the pressure recovers and the liquid comes back in, and the bubble that hammers computes what that costs: a collapse whose duration is set by the size it reached, and pressures inside it that the incompressible model itself predicts it cannot survive.

So the growth phase computed here is the loading of the event and the collapse is the discharge. The significance of the duration threshold is that it decides how many nuclei get loaded at all, and the count is what erosion rate scales with.

The same two-stage structure appears in a pipe rather than at a blade. Stopping water costs more than moving it records that the negative half of a surge wave is clipped at vapour pressure, which is a cavity opening; and the part of the closure a pipe cannot see notes that the collapse of that cavity is what produces the second and larger spike. The cavity there is metres long rather than microns and it opens for the same reason.

What it changes about an inception prediction #

The standard inception calculation takes a computed pressure field, finds where it drops below vapour pressure, and predicts cavitation there. The correction this essay implies has a direction and a size.

A nucleus passing through a short low-pressure region may not cavitate at all. The residence time in the suction peak of a fast section can be tens of microseconds, and for the smallest nuclei that is near the corner. This raises the predicted inception speed, and it is part of why inception is often observed later than predicted.

Nucleus size decides which nuclei this protects. Big nuclei have low static thresholds and long growth times; small ones have high thresholds and short growth times. So a short pulse selects for small nuclei with high thresholds, and a water sample’s nucleus population interacts with the pulse duration rather than merely setting a single threshold.

And it explains a scale effect that is otherwise awkward. Geometrically similar bodies at the same cavitation number have residence times proportional to their length over their speed, so a model and a ship do not present their nuclei with the same pulse. Nothing in the cavitation number carries that, and the corner is where it enters.

Where this leaves the cavitation number #

Not nowhere, and it is worth being precise about what survives.

The cavitation number is still the right first question. It compares the available suction with the available tension and it decides whether the pressure field can get anywhere near vapour pressure. If it cannot, none of this matters.

What it cannot do is decide inception once it is close. At that point the answer depends on the nuclei present, on how long each of them spends in the suction, and on the curve above. Those are three pieces of information the cavitation number does not contain, and the standard practice of treating inception as a fixed value of it is a fit to a particular water quality and a particular size.

The desinent number is the practical acknowledgement of this. Cavitation observed while reducing speed disappears at a different number from the one at which it appeared, and a hysteresis of that kind is the signature of a process with a state in it. The state here is the bubble’s own radius.

What the picture cannot show #

The threshold curve is drawn for a rectangular pulse, which no machine produces. A rounded pulse of the same peak and duration is weaker, because the bubble spends part of the pulse under less than the peak tension, and the correction is largest near the corner where the shape matters most.

The radius traces stop at the critical radius, because the question asked is whether the bubble crosses it. What happens afterwards — explosive growth to hundreds of microns, then a collapse — is the subject of the essay one rung below this one, and it is where the damage comes from.

And every figure here is one bubble. A real event involves a population, and the observable — the noise, the erosion rate, the loss of thrust — is an integral over that population that this calculation does not attempt.

Where the model stops #

Rayleigh–Plesset assumes spherical symmetry, and a bubble near a wall or in a shear does not stay spherical. The asymmetry is what produces the damaging microjet, and it is entirely absent here.

The gas is polytropic with a fixed exponent, which is a stand-in for the heat transfer between the gas and the liquid. Near the corner the growth is fast enough to be nearly adiabatic and at the floor it is nearly isothermal, so a single exponent is wrong at one end whichever value is chosen.

No mass transfer. Vapour is treated as arriving instantly at its equilibrium pressure, and gas is treated as neither entering nor leaving. Over many cycles gas does diffuse in, which is rectified diffusion, and it is how a bubble that survives one pulse becomes one that does not survive the next.

And the liquid is incompressible, so the growth cannot radiate sound. That is a good approximation during growth and a bad one during collapse.

Two of those four are the same class of omission that the drag that integrates a whole history is about: a term that carries the history of the motion rather than its present state. The Basset force on a particle and the thermal history of the gas in a bubble are both convolutions, and both are the first thing dropped.

Who found it, and when #

Blake’s threshold is of 1949 and the equilibrium argument behind it is Rayleigh’s, from 1917. The recognition that a finite pulse needs more than the static threshold is standard in the acoustic cavitation literature — it is why sonochemistry is described by an amplitude and a frequency — and Neppiras and Noltingk had the dynamic calculation in the 1950s.

What is less standard is stating it as an engineering correction to inception prediction on bodies, where the cavitation number is still normally treated as a threshold on its own. Brennen’s treatment makes the residence-time point explicitly, and it remains outside most practice.

Limits recorded rather than smoothed over #

One nucleus, one liquid. Everything is a 5 micron air bubble in clean water at 20 degrees except where the size sweep says otherwise. Nothing here predicts what nuclei a given water contains, which is the input the whole calculation is most sensitive to.

The corner is defined at twice the floor, which is a convention. A different definition moves all three corners together and does not change the collapse, which was checked by taking the ratio rather than the values.

The short-pulse end is a search limit. Below about 0.2 microseconds the bisection finds no amplitude in a range extending to eighty times the static threshold, and rather than report an extrapolated number the sweep starts at 0.3.

And the approach to Blake is slow by construction. Near the static threshold the growth is critically slow, so a pulse of any finite length reads a little above the static value. The 1.1 per cent at a hundred microseconds is not a numerical error; it is the physical statement that an infinite hold is not available.