When a body tears the water

A ship’s propeller turning fast enough begins to make a noise like gravel in a drum, its thrust falls away, and after a few hundred hours its bronze blades are pitted as though somebody had gone over them with a chisel.

The water has boiled. Not because anything was heated — a propeller is barely warm — but because the blade pulled the pressure at its own surface below the pressure at which water at that temperature turns to vapour. Boiling is a pressure condition as much as a temperature one, and a body in a stream is a machine for lowering pressure.

Boiling is a pressure condition #

Water at 20 °C has a vapour pressure of about 2.3 kilopascals. That is the pressure at which liquid and vapour are in equilibrium at that temperature, and it is the number a phase diagram gives.

The everyday route to boiling is to raise the temperature until the vapour pressure reaches the ambient pressure — at sea level, 101 kilopascals, and that happens at 100 °C. The other route is to lower the ambient pressure until it reaches the vapour pressure at whatever temperature the liquid already has. Both cross the same line on the same diagram.

A body moving through a liquid takes the second route, and it does so by an amount this site can compute exactly. Speed and pressure trade: where the flow is fast, the pressure is low, and the pressure coefficient

$$C_p = \frac{p - p_\infty}{\tfrac{1}{2}\rho U^2} = 1 - \frac{q^2}{U^2}$$

measures how far below ambient the surface has been pulled, in units of the dynamic pressure.

The condition for cavitation is then immediate. Define the cavitation number

$$\sigma = \frac{p_\infty - p_v}{\tfrac{1}{2}\rho U^2}$$

— the margin between the ambient pressure and the vapour pressure, in the same units — and the liquid reaches vapour pressure somewhere on the body when

$$\sigma = -C_{p,\min}$$

That is the whole of inception. A ratio of pressure margins on one side, a property of the shape on the other, and no temperature anywhere.

It is worth registering what kind of statement that is. Every quantity in it is dimensionless, which means the condition is a similarity statement rather than a fact about any particular propeller: two geometrically similar bodies at the same cavitation number cavitate in the same places, whatever their size, speed or liquid. That is the same argument the Reynolds number makes about a wake, and it is what makes a cavitation tunnel useful — a model at a tenth scale can be run at the right $\sigma$ by lowering the tank’s pressure rather than by making it go ten times faster.

It also fails in the way similarity arguments fail, and for the reason this site has already worked out in detail: matching $\sigma$ and $\mathrm{Re}$ at the same time in the same fluid is not generally possible, so a tunnel test matches the number the experimenter cares more about and accepts an error in the other.

The shape’s half of the answer, computed #

$C_{p,\min}$ is a property of the geometry and the incidence, and it is the site’s oldest kind of calculation. The surface velocity comes from the conformal map, the pressure follows from Bernoulli, and the minimum is found by walking the section.

The calibration is a circle. A cylinder in an ideal stream has a surface speed of $2U\sin\theta$, so its minimum pressure coefficient is exactly $-3$ at the shoulder, and its inception number is exactly 3. The surface walk is run on the cylinder first and returns $-2.999200$ — the small deficit is because the walk samples a hair off the surface, where the speed is slightly less than $2U$ — and only then is it trusted on a shaped section.

That calibration is asserted before any cavitation figure is drawn. A surface walk that had drifted off the body, or that was sampling the interior, would produce a smooth plausible pressure distribution with a wrong minimum, and nothing about the picture would say so.

Two more assertions run:

The inception number is positive. A body whose lowest pressure is above the free stream cavitates nowhere and a positive $\sigma_i$ would be meaningless, so a negative or zero result is refused.

The suction peak is on the upper surface of a lifting section, which catches the case where the walk has confused the two sides — a failure that would silently reverse every conclusion in the essay.

The number in the units a ship is designed in #

Put the pieces together for water at 20 °C at the surface: $p_\infty = 101$ kPa, $p_v = 2.3$ kPa, so the numerator of $\sigma$ is 98.7 kPa.

For the section drawn, $C_{p,\min} = -1.428$, so inception needs $\sigma = 1.428$, which needs a dynamic pressure of 69 kPa and a speed of 11.8 metres per second. For the cylinder, with $\sigma_i = 3$, it is 8.1 m/s. Those are not high speeds. A propeller tip on a small motor yacht is doing thirty.

Three levers change the answer, and only one of them is about the shape:

• Depth. $p_\infty$ rises by 10 kPa per metre of submergence, so a propeller ten metres down has double the pressure margin of one at the surface. This is why a submarine’s cavitation depends so strongly on how deep it is running, and why the depth at which it must slow down to stay quiet is a military secret.
• Temperature. $p_v$ rises steeply with temperature — 2.3 kPa at 20 °C, 12.3 at 50 °C — so hot water cavitates far more readily. Pump inlets on hot-water systems are designed around exactly this and the quantity has its own name, the net positive suction head.
• The section. $C_{p,\min}$ is what a designer controls, and it is the subject of the rest of this essay.

The thrust a propeller loses when it cavitates follows from the same picture. Once a stretch of the blade is covered in vapour, the pressure there cannot fall any further — it is pinned at $p_v$ — so the suction that stretch was contributing stops growing with speed. The blade’s lift saturates, which is why a cavitating propeller turns faster and pushes no harder, and it is the same shape of failure a wing has at the stall: a surface that has run out of the ability to produce more suction, for a different reason, with the same consequence on the lift curve.

The envelope, and the angle at which a section is quietest #

$C_{p,\min}$ is not a constant of a shape; it depends strongly on the incidence, and the way it does is the operating envelope a propeller has to live inside.

The minimum comes out at $\sigma_i = 0.62$, near zero incidence for this modest camber, and it climbs steeply on both sides. At ten degrees it is 4.4 — seven times the quietest value, which in speed terms is a factor of 2.6.

Two things follow, and both are why propeller design is hard.

A section has one good angle. Away from it the suction peak sharpens and the inception speed falls, so a propeller designed for cruise cavitates when the ship manoeuvres, and one designed for manoeuvring is inefficient in cruise.

Camber moves the good angle without widening it. More camber shifts the minimum to a higher lift coefficient — useful, since a blade has to produce lift — but the envelope stays narrow. What widens it is thickness distribution: a section with a flat pressure roof over much of its chord has a $C_{p,\min}$ that changes slowly with incidence, and that is what a propeller section is shaped for rather than for the lift-to-drag ratio a wing wants.

The same picture, in a gas #

There is a figure on this site that looks almost identical and is about something else, and the resemblance is worth being precise about.

The pocket on top of the wing computes the critical Mach number — the flight speed at which the local flow first goes sonic — and it does so from $C_{p,\min}$. This essay computes the cavitation inception number, and it does so from $C_{p,\min}$. Same field, same walk, same quantity; two entirely different thresholds applied to it.

The parallel goes further than the arithmetic. In both cases the local event on the surface is followed by something violent and non-linear that the inviscid calculation cannot describe: a shock in the gas, a collapsing cavity in the liquid. In both cases the inviscid theory’s role is to say where and when the threshold is crossed, and then to stop. And in both cases the design response is the same — flatten the pressure roof, spread the suction, delay the peak.

The difference is what happens next. A supercritical wing lives with its shock and manages it. A cavitating blade is being eroded, and what does the eroding is the subject of the next rung.

Four kinds of cavitation, and only one of them is this calculation #

The threshold computed here says where the liquid first reaches vapour pressure. What forms there depends on the shape and the loading, and the four standard types behave very differently.

Bubble cavitation is the case closest to this calculation: individual nuclei passing through a region below vapour pressure grow into visible bubbles and then collapse downstream. It happens on sections with a broad, shallow suction region, and it is the most erosive because the bubbles collapse individually and close to the surface.

Sheet cavitation is a single attached vapour cavity springing from the leading edge, and it is what a heavily loaded blade at high incidence produces. The cavity itself is stable; the damage is done where it closes.

Cloud cavitation is a sheet that periodically breaks off and is swept away as a cloud of bubbles. It is unsteady, noisy and by far the most destructive, and it is the failure mode that removes metal.

Tip vortex cavitation is different in kind. The pressure minimum is not on the surface at all but in the core of the vortex a blade sheds from its tip, and a vortex core’s pressure depends on its circulation rather than on the section’s shape. It is usually the first cavitation to appear on a propeller as the speed rises, it makes very little difference to the thrust, and on a warship it is the thing that gives the ship away — which is why so much design effort goes into shapes that unload the tip.

Only the first two are within reach of the calculation in this essay. The other two are unsteady or depend on a shed vortex, and both are outside what an inviscid surface pressure can say.

Going the other way on purpose #

Everything above treats inception as a boundary to stay behind. There is a design tradition that runs straight through it, and it is worth setting out because it resolves the erosion problem by making the cavitation worse rather than better.

Lower the cavitation number far enough — by going faster, or by running near the surface where $p_\infty$ is small — and the vapour cavity does not merely cover part of the section. It grows past the trailing edge and closes behind the body, which is then travelling inside a bubble of its own vapour with only its nose in contact with liquid. That is supercavitation, and two things change at once.

The drag collapses. Skin friction needs a wetted surface, and there almost is not one: the wetted area is reduced to whatever the nose presents. What is left is the pressure drag of that nose plus the cost of maintaining the cavity, and for a slender body the total is a small fraction of the fully wetted value. The drag depends on the cavitation number in a strikingly simple way — for a cavitating disc it is close to its zero-$\sigma$ value multiplied by $(1+\sigma)$ — so the whole problem is governed by the same single number that decided inception.

And the collapse happens somewhere harmless. A supercavitating propeller section is a sharp-edged wedge, shaped so that the cavity springs from the leading edge and does not close until well behind the blade. The blade never has a cavity collapsing on its surface, so it is not eroded — the damage the whole essay has been about is avoided by ensuring the cavitation is complete rather than partial.

The awkward regime is the one in between, where a cavity closes on the blade, and it is the one every conventional propeller has to be kept out of. Fast planing craft therefore run supercavitating or surface-piercing propellers rather than trying to stay subcritical, and underwater vehicles designed for extreme speed carry a blunt cavitator at the nose whose only job is to generate a bubble large enough to enclose the rest of them.

Where the model stops #

The inviscid field is the wrong field once a cavity exists. As soon as a vapour pocket forms it changes the shape the flow sees — it is, in effect, a new body — so the pressure distribution that predicted inception is not the distribution present afterwards. The calculation here is a threshold and nothing beyond it.

Inception is not only about the mean pressure. Real liquids cavitate at pressures above and below the equilibrium vapour pressure depending on their nuclei: pure degassed water can be pulled to enormous tensions without cavitating at all, while water full of microbubbles cavitates early. The inception number a real body shows depends on the water it is in, which is why cavitation tunnels control their dissolved gas content and why model-scale results scale badly.

There is no boundary layer here. The suction peak is where the pressure gradient is steepest and therefore where the layer is most likely to give up; real inception often occurs in a separation bubble rather than at the potential-flow minimum, and the two locations can differ by a good deal of chord.

And nothing here is unsteady. A propeller blade passes through a non-uniform wake once per revolution, so its incidence oscillates and its cavitation appears and disappears at blade-passing frequency — which is what produces the noise, and what produces the pressure pulses on a ship’s hull above the propeller.

Who found it, and when #

Osborne Reynolds identified the phenomenon in the 1890s and gave it its name, after the failure of the Royal Navy’s destroyer Daring to reach its design speed: the propeller was turning, the engine was delivering power, and the ship was not going fast enough. Charles Parsons built the world’s first cavitation tunnel in 1895 to investigate it, and it is one of the earliest examples of an experimental facility built for a single unexplained failure.

Reynolds appears on this site chiefly for the number that decides a regime, and it is worth registering that cavitation is a second dimensionless number of exactly the same character, discovered by the same person, doing exactly the same job on a different axis. A body’s behaviour is set by Reynolds number, Mach number, Froude number and cavitation number, and each of them is a ratio saying which of two competing effects wins.

The vapour-pressure explanation was not obvious at the time. The competing hypothesis was that the propeller was drawing air down from the surface, which is a real phenomenon called ventilation and is a different failure. Distinguishing them needed the tunnel.

Where the ladder goes #

The threshold is where this essay stops, and it is where the damage begins.

A vapour cavity swept off a blade into a region of higher pressure does not deflate gently. It collapses, and the collapse is violent: the liquid rushing inwards has nothing to slow it, the wall speed goes as the inverse three-halves power of the radius, and the pressure just outside the closing bubble reaches thousands of atmospheres.

The next rung computes the collapse, whose duration has a closed form in gamma functions — 91.47 microseconds for a millimetre cavity at one bar, matched by quadrature to a part in $10^9$ — and then finds the point at which the model destroys itself. Rayleigh’s calculation assumes the liquid is incompressible, and it predicts a wall speed that reaches the speed of sound in water at three per cent of the original radius. Everything that pits a propeller happens past that point.