A cavity that cools the water it came from

Where a body tears the water locates cavitation inception from an inviscid pressure field and a vapour pressure. The bubble that hammers computes the collapse that follows. A threshold that is also a duration shows that inception is a duration as well as a pressure. All three take the vapour pressure as a number: a property of the liquid, at its bulk temperature, looked up once and held.

It is a property of the liquid at the temperature the cavity has left it, and a cavity does not leave it alone, because making vapour costs latent heat and there is nowhere for the heat to come from except the liquid next to it.

The heat a cavity has to be paid for with #

A cavity of volume $V_v$ holds $\rho_v V_v$ of vapour, and turning that much liquid into vapour takes $\rho_v V_v L$ of latent heat. If it is drawn from a liquid volume $V_l$, that liquid cools by

$$\Delta T = B\,\frac{\rho_v L}{\rho_l c_l}, \qquad B = \frac{V_v}{V_l},$$

and $B$ — the ratio of vapour made to liquid cooled — is Stepanoff’s B-factor.

Cooler liquid has a lower vapour pressure, so the cavity’s own pressure is below the vapour pressure of the bulk by $(\mathrm{d}p_v/\mathrm{d}T)\,\Delta T$, and that shortfall eats directly into the tension driving the cavity. A cavity suppresses itself by existing, and the only question is whether the suppression is large enough to notice.

The feedback is negative and that is worth marking, because most thresholds in fluid mechanics are the other kind. A shock strengthens as the flow behind it slows; a roll wave grows because the flux news outruns the pressure news; a bend’s helix deepens the bend that made it. Here the consequence of the effect opposes its cause: a cavity forms, cools its surroundings, and thereby raises the pressure it would have had to fall below in order to form. Nothing runs away, and what the arithmetic below computes is a saturation rather than an instability.

Setting the shortfall equal to the whole driving tension $\Delta p$, and using Clausius and Clapeyron for $\mathrm{d}p_v/\mathrm{d}T = \rho_v L/T$, gives the answer as one group:

$$\boxed{\ B^* = \frac{\Delta p\,T\,\rho_l c_l}{(\rho_v L)^2}\ }$$

$B^*$ is the vapour fraction at which the thermal depression has consumed the entire driving pressure. A large $B^*$ means the effect is unreachable — the cavity would have to make far more vapour than the liquid it cools could possibly supply the heat for — and a small one means it bites at once.

Three features of that group are worth reading before any numbers are put into it.

The vapour density appears squared. Once through the latent heat that has to be found, and once through the Clausius–Clapeyron slope that converts a temperature drop into a pressure drop. A denser vapour costs more heat to make and buys more pressure reduction per kelvin, and the two effects compound rather than cancel — which is the whole reason the group swings by four decades over a range in which nothing else moves by more than a factor of two.

The liquid’s heat capacity is in the numerator, so a liquid that is hard to cool is hard to suppress. Water’s specific heat is four thousand joules per kilogram-kelvin, which is extraordinarily large, and that alone puts water two orders of magnitude further from suppression than most liquids at the same vapour density.

And the driving tension is in the numerator too, which says the effect is strongest at inception and weakest in a developed cavity. A cavity that is barely forming has very little tension to lose; one driven hard has plenty. So the thermal effect delays inception more than it limits growth, and that is the form in which a pump experiences it.

Why cold water is the exception #

For water at 20 °C, $B^* = 648$. A cavity would have to make six hundred times its own volume of vapour out of the liquid it has cooled, which is not something that happens, and the effect is therefore invisible.

That is why a cold-water cavitation calculation is right to ignore it, and it is also why the effect is so easy to overlook: cold water is the fluid every cavitation experiment is done in, and it is the outlier.

It is worth being precise about how far an outlier. Water at 20 °C is four decades from liquid oxygen, four and a half from liquid hydrogen, and four and a half from an ordinary refrigerant at room temperature. Nothing else on the list is anywhere near it. What makes water exceptional is the combination of a very low vapour density at room temperature with a specific heat that is the highest of any common liquid — both of which push the same way, and both of which are consequences of the hydrogen bonding that makes water peculiar in a dozen other respects.

The temperature dependence is steep and it is almost all in one quantity. $B^*$ carries $(\rho_v L)^2$ in the denominator, the latent heat falls slowly with temperature, and the vapour density rises enormously — by a factor of thirty-four between 20 °C and 100 °C — so the group falls as very nearly the fourth power of the vapour density. The computed rate is 0.32 decades per ten kelvin, which is a factor of two per seven degrees.

A factor of two per seven degrees is a startling sensitivity for a quantity nobody measures, and it has an immediate consequence for how this subject is practised. Two cavitation tests of the same machine at water temperatures differing by twenty degrees are not the same test, and the difference is a factor of seven in the parameter that decides whether the thermal effect is present. A laboratory that controls its water temperature to a degree is controlling the group above to ten per cent; one that does not control it at all is not comparing its own results from one month to the next.

Cold water is forgiving about that only because it is so far from the threshold: at 20 °C the group is 648, and a factor of seven either way leaves it unreachable. Between 80 and 120 degrees the same factor takes it across one, and that band is exactly where the pumps that first showed the effect operate.

The one measurement that made anybody look #

The effect was found in pump practice rather than in a laboratory, and the way it was found is worth recording because it is the standard shape of a discovery in engineering.

A pump’s suction performance is established by test, in cold water, and quoted as a required net positive suction head. Pumps handling hot water — boiler feed pumps, condensate pumps — were repeatedly found to run satisfactorily on suction heads below their tested requirement, by margins that no measurement error could explain and that grew with the temperature of the water. That is an awkward kind of discrepancy: it is in the safe direction, so nobody was hurt by it, and it therefore persisted as a rule of thumb for years before it was explained.

What made it a subject rather than a rule was that the discrepancy was reproducible and had a sign that a mechanism could be guessed at. Stepanoff’s contribution was to notice that the one thing that changes with temperature by orders of magnitude is the vapour density, and to write down the heat balance above.

The same margin, applied in reverse, is what makes a cold-water test of a cryogenic pump useless. Liquid hydrogen sits four decades to the left of cold water on the axis above, so a turbopump qualified on water and flown on hydrogen is being operated in a completely different regime — and in the direction that was, again, safe. The cost of not understanding it is not a failure; it is a vehicle carrying a heavier tank than it needed.

What it is worth in metres #

The practical form of all this is a head, because a pump’s cavitation margin is quoted as one — the net positive suction head, the amount by which the pressure at the inlet exceeds the vapour pressure, expressed in metres of the liquid.

Three and a half metres is not a correction. It is most of a turbopump’s suction margin, and a cryogenic pump designed to a cold-water test would be built with an inducer and a tank pressure it does not need — which on a launch vehicle is structural mass that has to be lifted.

The same effect is why a boiler feed pump handling water at 180 °C is less prone to cavitation than the same pump on cold water at the same suction pressure, and why the correction is applied as a deduction from the required suction head rather than as a change to the pump.

The deduction has a shape that is worth stating, because it is the opposite of how a correction usually behaves. Most corrections in pump practice are penalties: a margin added for wear, for a distorted inlet, for a transient. This one is a credit, and it grows with the very quantity — temperature — that makes every other aspect of the duty harder. A feed pump at 180 °C is handling a liquid whose density is lower, whose viscosity is lower, whose vapour pressure is thirty times atmospheric and whose absolute suction pressure is therefore enormous; all of that makes the plant harder to build, and the one thing that gets easier is the cavitation.

That is also why the credit is taken cautiously. A deduction from a required suction head is a deduction from a safety margin, and the quantity it rests on — how much liquid the cavity actually draws its heat from — is not measured on the machine in question. Codes allow it, hedged, and most designers take a fraction of what the arithmetic offers.

Two parameters, and they are one number #

The pump literature uses Stepanoff’s $B^*$. The cavitation-research literature uses Brennen’s thermal parameter

$$\Sigma = \frac{(\rho_v L)^2}{\rho_l^2 c_l^2\,T\,\sqrt{\alpha_l}},$$

which has dimensions of a velocity divided by the square root of a time and is compared against a bubble’s growth rate. The two are quoted in different subjects, with different units, for different purposes.

Multiplying them, the $(\rho_v L)^2$ that suppresses one is the $(\rho_v L)^2$ that promotes the other, and everything about the vapour cancels:

$$B^*\,\Sigma = \frac{\Delta p}{\rho_l c_l \sqrt{\alpha_l}}.$$

That is exact — it holds to four parts in $10^{16}$ across seven liquids — and what is left on the right depends only on the liquid’s thermal properties, which vary by a factor of six across the same seven liquids while the vapour properties vary by four decades.

So the two parameters are reciprocal, a fluid’s position on one scale fixes its position on the other, and neither literature says so.

The identity is worth more than a curiosity because it says what the residual difference between the two descriptions is. Everything about the vapour cancels; what is left is $\Delta p/(\rho_l c_l\sqrt{\alpha_l})$, which carries the liquid’s heat capacity and its thermal diffusivity. So the two parameters can only order a set of fluids differently when the liquids’ own thermal properties differ — and across water, three cryogens and a refrigerant those vary by a factor of six while the vapour properties vary by ten thousand.

A fluid’s thermal cavitation behaviour is therefore a property of its vapour, almost entirely, and the choice between the two parameters is a choice of units rather than of physics. Which one to use is decided by what is being compared against: $B^*$ against a vapour fraction, $\Sigma$ against a bubble’s growth rate.

What the picture cannot show #

The B-factor is a ratio of two volumes and neither is defined by the argument. How much liquid a cavity actually draws its heat from depends on how long it has had, on the thermal boundary layer around it, and on whether the surrounding liquid is being replaced — none of which appears above. The argument gives a criterion, not a temperature.

The liquid’s own properties are taken at the bulk temperature, including the viscosity. They are being evaluated in a region that has, by hypothesis, cooled; for a strongly suppressed fluid the correction is not small, and the calculation should be iterated.

Clausius and Clapeyron assumes the vapour is ideal and its volume dominates. Near the critical point neither holds, and the relation used for $\mathrm{d}p_v/\mathrm{d}T$ fails before the liquid range does.

And the steady part of the argument hides a rate. Whether the heat can be drawn in the time available is a diffusion problem, and Brennen’s parameter is the form that keeps it. The B-factor asks only whether enough heat exists nearby, which is a necessary condition and not a sufficient one.

And the whole argument is about one cavity, rather than the sheet a body tears out of the water. A cavitating region contains many, they share the liquid they are cooling, and the sharing goes the wrong way: a cloud of bubbles in a small volume depresses the temperature far more than one bubble would, so the effect is stronger in a developed cavitating region than the single-cavity arithmetic suggests. That is the direction that makes the estimate conservative, which is convenient and is not a reason to leave it unquantified.

Who found it, and when #

Stepanoff introduced the B-factor in the 1960s, from pump practice, to explain why the measured suction performance of a pump on hot water was better than its cold-water test predicted. Brennen’s thermal parameter and the delayed-bubble-growth analysis are of the 1970s and belong to the cryogenic-pump work done for launch vehicles, where the effect is not a correction but the design condition. The two accounts grew up in different places and the identity between their parameters does not appear to be stated in either.

The surprising connection is with a phenomenon that has no pump and no cavity in it, and it is the same self-limitation. A boiling liquid at a heated surface also has to pay latent heat out of the liquid next to it, and the local cooling that results is what makes nucleate boiling stable: a bubble that grows too fast cools its own surroundings, its growth slows, and the site recovers. What is a stabilising feedback there is a suppression here, and the same group governs both. The difference between boiling and cavitating is which way the temperature is going, and the thermodynamics is identical.

Still open: what the effective B-factor is for a real cavity #

The criterion is clean and the number it needs is not measured.

$B^*$ says the vapour fraction at which the effect is total. What a real cavitating region has is a different question, and it depends on the geometry: a travelling bubble cools a shell of liquid whose thickness is set by how long the bubble has existed, while an attached sheet cavity cools the whole boundary layer beneath it, which is a much larger reservoir at a much lower vapour fraction. Those two are the same phenomenon at $B$ values differing by orders of magnitude, and a pump has both.

The measurement that would close it is a temperature measurement inside a cavitating region — a thermocouple traverse through an attached cavity on a hydrofoil in warm water, giving the depression directly rather than inferring it from a suction-head deficit. That has been attempted and the numbers are scattered, because the probe disturbs the cavity it is measuring. A non-intrusive route exists and appears not to have been taken: the cavity’s own vapour pressure can be read from its collapse behaviour, since the collapse time of a bubble depends on the pressure difference driving it, and that time is computable to a part in $10^9$. Measuring a collapse time in warm water and inverting it would give the cavity’s internal pressure without putting anything inside it.