THE PHYSICAL NATURE OF CAVITATION

AND THE MECHANISM OF CAVITATION DAMAGE

I. Metter

INTRODUCTION

At the end of the last century, shipbuilding engineers and designers of turbines and turbopumps first encountered a peculiar destruction of the metal parts of hydraulic machines and of propeller blades. The destruction occurred whenever the pressure in the flow, in the region of the future damage, remained for a long time reduced to the value of the elasticity of water vapor; instead of a homogeneous flow, the metal surface was washed by a mixture of water and water vapor. The formation, under these conditions, of bubbles filled with the vapor of the liquid was called cavitation (“cavitas”—cavity).

Fig. 1. Cavitation damage to propeller blades.

Cavitation, as a rule, led to two consequences: first, the efficiency of the hydraulic mechanism—the turbine, turbopump, or ship’s propeller—decreased, and, second, along with the decrease in efficiency, in all cases without exception damage was found in individual places on turbine blades or propeller blades (Fig. 1). Because of cavitation damage it was sometimes necessary

replace a turbine runner after several weeks of operation, and a ship’s propeller after several months of sailing. This occurred, for example, on such large ships as the Lusitania and the Mauretania. The damage to their propellers covered an area of a quarter of a square meter and penetrated into the blade to a depth of 5–7 cm. Cases are known of catastrophic destruction of propellers due to cavitation after several hours of operation—on high-speed vessels, for example on torpedo boats. It should be noted at once that the damage is not similar to that which arises as a result of purely chemical or electrochemical action of water on metal.

In combating cavitation in the design of modern super-powerful turbines and centrifugal pumps, special attention must also be paid. At the DneproGES, it was necessary in its time to carry out extensive auxiliary hydraulic-engineering works in order to eliminate or weaken the harmful effects accompanying cavitation.

Unfortunately, for a long time the problem of cavitation did not receive sufficient attention. Systematic laboratory investigations of the phenomena accompanying cavitation began only in 1922. Since then about one hundred works have been published and several collections devoted to cavitation have appeared.

At first attempts were made to approach the problem of cavitation by a purely hydrodynamic route, and only external circumstances were taken into account—the velocity of the flow, the pressure in it, and the dependence of both on time. Later the necessity became clear of taking account of the physical conditions in the liquid—temperature, density, surface tension, viscosity, the amount of gas dissolved in it, etc.

These factors, each in its own way, influence the process of bubble formation, their stability, and the activity of their action on a solid surface near which, or on which, they arise.

In the present survey we shall consider chiefly those works devoted to elucidating the physical nature of cavitation and the mechanism of cavitation damage. As for the cavitation resistance of materials, we shall confine ourselves at the end to only the most general indications.

1. CONDITIONS FOR THE OCCURRENCE OF CAVITATION

In 1850 Berthelot¹, apparently for the first time, succeeded in “stretching” a liquid. A tube from which the air had previously been pumped out was overturned into a cup of mercury. Then the tube, with the mercury that had entered it, was heated until the mercury “adhered” to the glass. After this Berthelot slowly cooled it. The mercury did not detach from the glass and, consequently, was stretched by the negative pressure that arose during cooling. Worthington² in 1892 used Berthelot’s idea and constructed an apparatus (“tonometer”) for stretching and measuring the stretching in a column of liquid. Worthington used ethyl alcohol

were “stretched” by a pressure of 17 atmospheres. Later it proved possible to stretch water to 34 atmospheres, and ether to 72 atmospheres.

Reynolds also carried out experiments on stretching a liquid. He rotated a U-shaped tube containing liquid and produced stretching by the action of centrifugal forces.

By first creating considerable negative pressures in all such experiments, it was then possible to cause a “rupture” of the liquid column into two parts, with the formation between them of a cavity filled with the vapor of this liquid. In this way a cavitation cavity can be produced by artificially stretching a liquid.

Under natural conditions, in a flow, we encounter stretching of a liquid and the appearance of ruptures in it much more often. The motion of a steady flow obeys the well-known Bernoulli law: for all cross sections of a horizontal flow the sum must remain constant

\[ p+\frac{\rho v^{2}}{2}=\mathrm{const}. \]

It is obvious that at places where the flow narrows, where the velocity increases, the pressure \(p\) inside the liquid may fall to 0 and even assume a negative value. In these places the liquid will be subjected to stretching and will exhibit discontinuities of continuity—cavitation bubbles, i.e., bubbles filled with vapor, will appear in it.

Reynolds’ classical experiment is well known: water, flowing through a narrow section of a tube, “boils” in it at room temperature; at this place there is vigorous formation of cavitation bubbles. They accumulate in such quantity that the jet becomes cloudy-white because of them.

Photographing a stream of water flowing through a special cavitation tube (a tube with a constriction, a diffuser, or a nozzle), at various flow velocities—up to 180 km/hour—was carried out in a number of laboratories\(^{3,3a}\) (1914–1926). It was established that cavitation bubbles may be very small—fractions of a cubic millimeter—but may also occupy a volume of several liters. Near the propeller of a ship, for example, the volume of cavitation cavities reaches several cubic meters.

In 1932, Gaines\(^{4}\) developed a new method for producing cavitation bubbles, extremely convenient for laboratory investigations. It consists in the following: in a metal rod 25 cm long, immersed in a liquid, magnetostrictive longitudinal oscillations with a frequency of 9–10 thousand Hz are produced by means of a special generator. The generator power is 250 W. The amplitude of the rod oscillations is tenths or hundredths of a millimeter. It is easy to verify that the acceleration of the end of the rod in Gaines’ experiments reached 3000 \(g\). As a result of such oscillations of the rod, an intense wave of condensations and rarefactions was created in the liquid. If in the half-period of rar-

if the hydrostatic pressure drops to the vapor pressure or below, i.e., if the liquid proves to be stretched, then a cavitation bubble may be born in it. In the following half-period it will be compressed, and, if the bubble is not crushed in the process, it must begin to pulsate in time with the acoustic oscillations of the generator.

In 1935 Henseiker\(^5\) filmed the behavior of bubbles with a motion-picture camera and a centrifuge. It turned out that the bubbles really do oscillate—contracting and expanding—inside the liquid.

Gaines’s experiments showed that after several minutes of operation of the generator the end surface of a nickel rod proved to be damaged; moreover, the character of the damage exactly resembled ordinary cavitation damage.

It is interesting to note that the first destructions under the action of cavitation produced by acoustic oscillations were discovered on the membranes of underwater telegraphy apparatus. Sometimes, after several minutes of operation at a frequency of 7–8 thousand cycles, the membranes went out of order.

Fig. 2. Diagram of a diffuser for studying cavitation:
\(a\) — protruding screw causing cavitation;
\(b\) — specimen under test \((60 \times 30 \times 8\ \mathrm{mm})\);
\(c\) — throttle.

Thus, with the occurrence of cavitation bubbles we encounter either purely mechanical stretching of a liquid, or suitable hydromechanical conditions in a flow, or, finally, the action of acoustic oscillations in the volume of the liquid.

2. METHODS OF STUDYING CAVITATION

Four different methods are known for studying cavitation and the consequences it produces:

a) Investigations in cavitation tubes or diffusers\(^3, 6, 7\).
b) Impact of a jet or drops\(^8, 9, 10\).
c) The shock-wave method\(^ {11}\).
d) Gaines’s method\(^4\).

Let us give a brief characterization of these methods.

a) Experiments in diffusers with a large head height (up to 350 m) were carried out beginning in 1932 by Schröter\(^6\). Schröter constructed a diffuser (Fig. 2) in which, by means of a simple device—the extension of a screw—it was possible to change the cross-section of the flow in the diffuser itself. This made it possible to create numerous cavitation bubbles, which were carried by the flow over the fixed-

...made in the tube in the form of a specimen of the material under test. Behind the specimen there was a throttle which, by creating back pressure, regulated the discharge of water from the apparatus. With the aid of such a throttle it was possible, at will, to change the pressure distribution in the tube and thereby to shift the region of bubble condensation inside the tube. The investigations showed that destruction of the surface of the specimen is always observed in the zone of bubble condensation, i.e., where they collapse, and not at the place where they originate (where the pressure is minimal)—see Fig. 3. The screw which caused cavitation and was all the time in the zone of reduced pressures (zone \(ab\) in Fig. 3) never suffered any damage.

Fig. 3. Pressure variation along the length of the diffuser and location of the damage zone.

Hunsaker\(^5\), conducting investigations in a cavitation tube by means of a stroboscope, discovered the periodic character of cavitation. He observed the occurrence, growth, and weakening of cavitation, with the whole cycle occupying thousandths of a second.

As regards the external appearance of specimens damaged in diffusers or in cavitation tubes, their surface resembled a target after inaccurate shooting: at first several scattered hits appear, and then the surface becomes pitted with traces of cavitation impacts.

It is convenient to estimate the significance of the damage arising from cavitation impacts by the loss of weight of a specimen subjected to cavitation action. Curves of weight loss as a function of time are shown in Fig. 4\(^ {12}\). We see that in all cases an “incubation period” is required (sometimes exceeding 10–20 hours) before damage appears that affects the loss of weight. After the “incubation period” the loss of weight follows a linear law, and the tangent of the angle of inclination may serve as a measure of the rate of damage.

b) Exactly the same conclusions were also obtained in experiments with a jet of water directed at a metal\(^ {8,9,10,13}\). The specimen was fastened in the form of a rod on the rim of a wheel brought into rapid rotation. A narrow jet of water was directed perpendicular to the plane

wheels. At each revolution the specimen crossed the jet. At a relative velocity of the jet and the specimen of \(200\ \text{m/sec}\) and a jet diameter of \(1.5\ \text{mm}\), the hardest materials exhibited a characteristic erosion, completely similar to the damage produced by cavitation.

The curves of weight loss of specimens subjected to impacts by jets have exactly the same form as in tests in a cavitation

Fig. 4. Weight loss of specimens made of various materials, caused by cavitation damage, as a function of cavitation time.

pipe or diffuser \(^{3a,7}\). Here too, before the first damage appears, a certain time must elapse (the “incubation period”). Only after millions of impacts of the jet does the specimen begin to lose weight (Fig. 5).

c) Ackeret and Galler \(^{11}\) in 1938 investigated the destruction of metals by shock waves. The apparatus consisted of a thick-walled steel cylinder, tapering downward into a cone and filled with water. In the cylinder there was a piston fitting tightly against the surface of the water. A special device struck the piston 16 times per second. At the base of the cone there was placed the specimen under investigation, or piezoquartz for measuring the pressure. The water in the cylinder was under a pressure of \(7\ \text{atm}\), which, in the authors’ opinion, eliminated the possibility of cavitation arising. At each blow on the piston the shock wave was transmitted through the liquid to the specimen, and after 10–12 hours of operation, i.e. after 60,000 blows of such a “hydraulic hammer,” damage distinctly similar to cavitation damage was observed.

Piezocuarz in these experiments registered comparatively small pressures—precisely those that should arise during hydraulic impact.

A microscopic examination of a cast-iron specimen tested in this apparatus showed that graphite inclusions of size \(0.001\ \mathrm{mm}\) were knocked out of the surface after several thousand impacts. We note that high pressures are not needed to knock out graphite grains—the pressures developing in the apparatus during hydraulic impact are quite sufficient.

It is likely that, after microscopic cracks form at the “weak spots” of the surface, further damage is facilitated: the water in the depressions acts like a wedge.

Fig. 5. Weight loss of specimens made of various materials, as a function of the number of jet impacts. (Relative velocity \(77\ \mathrm{m/sec}\), jet diameter \(8\ \mathrm{mm}\).)

Experiments with other metals, carried out in this apparatus, give a less clear picture of the damage. But in these cases as well one may expect that the most insignificant surface defects, which do not reveal themselves in rough fatigue tests, play the role of “weak spots” under the action of the shock wave. It is precisely from these places that the finest particles are knocked out (as a result of several thousand impacts), after which all subsequent impacts become more effective. It should be pointed out that some authors call into question the absence of cavitation in the apparatus of Ackermann and Galler: in the rarefaction half-period of the shock wave, the inception of cavitation is quite possible.

Thus, for all the methods considered for investigating cavitation damage, the periodicity of the processes accompanying cavitation is essential. For a diffuser or cavi-

tion tube the frequency of these processes is of the order of hundreds of hertz; for jet impacts the periodicity is set by the number of revolutions of the wheel; in the case of shock waves in Akkeret’s apparatus the frequency was 16 hertz; in Geynes’ apparatus, as has already been said, frequencies of the order of many thousands of hertz can be produced.

c) Especially convenient for studying the purely physical features of cavitation is the method of Geynes^4. The duration of an experiment is short. The apparatus requires a small quantity of water and little power, while at the same time making it easy to photograph the entire process as a whole and to observe the behavior of individual bubbles. Transition from one liquid to another is carried out very simply. The dependence on

Fig. 6. Formation of bubbles on the surface of an oscillating specimen: a—at a small amplitude of oscillation; b—at a large amplitude (~0.015 mm).

temperature, pressure, etc., can be studied without particular difficulties.

Finally, it makes it possible to compare the pattern of bubble formation at the surface of the cavitating specimen with the nature of the surface damage. All this has been the reason why, in recent times, Geynes’ method has found wide application for the investigation of cavitation processes.

Let us first trace, following the work of Kornfeld and Suvorov^13, how the amplitude of oscillation of the specimen affects the behavior of cavitation bubbles. At an oscillation amplitude of the order of 0.005 mm, air bubbles appear on the surface of the specimen and slowly move over the surface. If the generator is switched off, the bubbles remain visible in the liquid; this indicates that they are filled with air.

At larger amplitudes, beginning with 0.01 mm, a whitish cloud appears on the surface of the oscillating rod. It consists of an accumulation of rapidly moving bubbles (Fig. 6). The cloud disappears completely as soon as the generator ceases operating. This indicates the cavitation origin of the bubbles. Experiments have shown

... that such a cloud, 20–30 min after its appearance, causes damage to the surface precisely at the place above which it had formed. If the cloud of bubbles is moved over the surface, the place of damage to the specimen is correspondingly displaced.

With a still greater increase of the amplitude, to 0.015 mm, the cloud becomes still more distinct; in its center there appears a white bubble 1–2 mm in diameter. It should be noted that the formation of bubbles at this amplitude is accompanied by a sharp, unpleasant whistle*).

Cavitation under such a mode of generator operation produces the greatest damage. After 30–40 seconds (and not minutes, as before) a hole 1.5 mm in diameter and up to 1 mm deep is formed in the center of the aluminum specimen. Similar results were obtained by Kornfeld and Suvorov for other metals, glass, and rubber.

In some cases one cannot judge damage by loss of weight, since near a depression on the surface of the metal a mound in the form of a crater is formed: damage is clearly present, while the weight has not only not decreased, but sometimes has increased owing to the formation of a metal-oxide film.

Therefore, along with loss of weight, in what follows we shall also take into account the external appearance of the surface and the results of microscopic and X-ray analysis.

3. DAMAGE IN WATER, AQUEOUS SOLUTIONS, AND ORGANIC LIQUIDS

A specimen oscillating in water at room temperature exhibits a region of damage in the form of a circle around the center of the specimen. It was observed that cavitation bubbles occupy a region which, as the temperature of the water is raised, shifts from the center of the specimen toward its periphery. Correspondingly, the damage, concentrated at low water temperature in the center, at 50–60° acquires an annular character; at still higher temperatures it weakens, and at 100° disappears completely. This picture does not depend on the material of the specimen. Damage at various water temperatures is shown in Figs. 7^14 and 8^12.

Analysis of the photographs leads to the conclusion^14 that the pattern of surface damage is always in agreement with the character and activity of cavitation directly above the damaged spot. When the rate of bubble formation is insufficient, or when they are completely absent, no damage is observed at all.

*) The noise and whistling accompanying the collapse of bubbles also occur when water boils. Reynolds explained by this the “singing” of a kettle placed on a fire. For the same reason, apparently, hand-basins and the “singing” of water pipes rumble. See Raynolds Scient. Papers, 2, 578 (1894).

THE PHYSICAL NATURE OF CAVITATION

Studies of cavitation in aqueous solutions of common salt, hydrogen peroxide (3%), and caustic potash (0.01 normal) did not show any substantial difference between its character and the conditions of its occurrence and the character and conditions of occurrence of cavitation in distilled water or in water from the tap. This is understandable, since the vapor pressure of a weak solution differs little from the vapor pressure of the solvent; therefore the character of bubble formation is similar in both cases. The same applies also to the character of the damage (Fig. 9)12, 14. The total loss of weight after 10 min of operation of the generator proved to be greatest at a temperature of about 50°, as in pure water.

In order to investigate the influence of the physical properties of a liquid on the character of cavitation in it, it was necessary to turn to liquids that differ greatly in vapor pressure, viscosity, and surface tension. For this purpose experiments14 were carried out in gasoline, benzene, and ethyl ether at a temperature of 26°. Aluminum served as the specimen—for it the duration of the experiment proved to be the shortest. The results are shown in Fig. 10.

A complete analogy with the results presented in the preceding photographs is observed: the regions of damage on the specimens are correspondingly similar to those obtained in the case of water at temperatures of 20, 60, and 90°. It should be noted that at these temperatures the vapor pressures of water are 17, 150, and 525 mm, while at a temperature of 26° gasoline has a vapor pressure of 16 mm, benzene 100 mm, and ethyl ether 500 mm.

In ethyl ether the specimen for a long time showed no macroscopic damage, but under the microscope traces of a weak action on the metal were noticeable. Here we have a picture the same as in water at a temperature close to the boiling point.

Experiments with organic liquids14 showed that at different temperatures, but at one and the same vapor pressure, the outward appearance of damage in different liquids is similar: a small damage at the center of the specimen at pressures up to 100 mm expands as the vapor pressure increases and takes on the form of a ring encircling the center of the specimen. On approaching the boiling temperature for each liquid, the cavitational action on the metal weakens. We are once again convinced that the primary processes responsible for cavitation damage require the presence of a twofold phase: the liquid and its vapor must exist simultaneously. Moreover, the formation of stable bubbles must take place, seated at definite places on the specimen or slowly moving over its surface. From this point of view, the following experiment14 is of interest. Along the surface of a cavitation specimen, with the aid of a wire, a bubble 0.5 mm in diameter is slowly moved. In exact correspondence with the path of motion of the bubble, damage is found on the surface of the metal. Observations of cavitation bubbles in Haines’s apparatus showed that the bubbles

Fig. 7. Cavitation damage to aluminum at various water temperatures. Cavitation time—1 min.

Fig. 7 (continued). Cavitation damage to aluminum at various water temperatures. Cavitation time—1 min.

70°

80°

90°

100°

Fig. 8. Dependence of the weight loss of a magnesium specimen on the water temperature. Cavitation time—10 min. Amplitude—0.06 mm.

Vertical axis: Weight loss in mg

Horizontal axis: Water temperature

0 20 40 60 80 100°C

they themselves travel over the surface, the motion occurring in jumps. This is especially clearly seen in liquids of high viscosity (as, however, also in water at sufficiently small oscillation amplitudes). The rate of displacement, as observations have shown, is small—several centimeters per second.

Fig. 9. Cavitation damage to aluminum in aqueous solutions: a—in a 30% hydrogen peroxide solution, cavitation time—3 min.; b—in a NaCl solution (0.01 normal), cavitation time—1 min.

Fig. 10. Damage to aluminum after 17 min. of cavitation: a—in gasoline, b—in benzene, c—in ethyl ether.

Microscopic examination of the surfaces of samples cavitated in water sometimes reveals a linear character of the damage: traces of damage are arranged almost along straight lines[^14] (Figs. 11, 12). This agrees well with the jump-like character of the motion of bubbles over the surface. It is interesting that also in

in experiments with a cavitation tube, noticeable traces of the displaced bubbles are seen on the specimen (Fig. 13).

One may try to calculate the velocity of displacement, using the photograph presented in Fig. 11, and proceeding from the assumption

Fig. 11. Microdamages of brass after short-term cavitation in water, having a linear character; a—magnified 400 times, b—magnified 1200 times.

Fig. 12. Linear damages of the tin surface during cavitation in water.

Fig. 13. Linear microdamages of high-alloy steel after cavitation in a diffuser.

that the bubble, contracting and expanding in time with the oscillations of the generator, acts on the surface of the metal at each contraction. Such an assumption, as we shall see, has sufficient grounds.

We shall assume that the distance \(\Delta S\) between two neighboring traces of the bubble in the photograph represents the path of the bubble during one oscillation. Then its velocity is

\[ v=\frac{\Delta S}{\Delta T}=\Delta Sf, \]

where \(f\) is the frequency of the generator. Taking into account the 400-fold enlargement of the photograph, we have \(\Delta S=0.0003\ \text{cm}\). The frequency of the generator is \(10^4\ \text{Hz}\). Consequently,

\[ v=3\cdot 10^{-4}\cdot 10^4=3\ \text{cm/sec}. \]

4. DEPENDENCE ON EXTERNAL PRESSURE AND TEMPERATURE

Let us turn to a description of Novotny’s experiments \({}^{14}\) at various pressures and temperatures. First of all, let us consider the results of experiments with brass in water at \(80^\circ\), \(90^\circ\), and \(100^\circ\text{C}\), at a pressure of \(2\ \text{atm}\). The author of the work points out the following similarity: the specimen after cavitation at

Fig. 14. Surface of aluminum after 10 min. of cavitation in water at various external pressures.

\(100^\circ\) and \(2\ \text{atm}\) looks the same as after cavitational action at \(60\)—\(70^\circ\) and at atmospheric pressure. Similarly, the photograph corresponding to \(80^\circ\text{C}\) and a pressure of \(2\ \text{atm}\) is similar to the photograph at room temperature. Novotny emphasizes the decisive role played by the ratio between the external pressure and the elasticity of the vapor.

To verify this assertion he carried out investigations with a cavitating aluminum rod in water at room temperature, but at reduced pressure. The results are presented in Fig. 14.

At \(P=760\ \text{mm}\) the picture is identical with that shown in Fig. 7 at \(20^\circ\). The damage at a pressure \(P=380\ \text{mm}\), however, recalls the damage at atmospheric pressure but at a temperature of \(70\)—\(80^\circ\).* Thus, the character of the action of cavitation bubbles on the specimen depends substantially on the difference \((P_{\text{external}}-P_{\text{vapor}})\).

The rule according to which, when \(P_{\text{external}}-P_{\text{vapor}}=0\), there is no damage, is also fulfilled here (see the far-right photograph in Fig. 14).

We can now understand why in the cavitation tube damage was registered not in those places where the pressure decreases

\[ \text{* The difference in intensity is explained by the fact that the experiments at reduced pressure lasted 10 min., while at atmospheric pressure they lasted 1 min.} \]

to the vapor pressure (i.e., where bubbles form intensively), while at the place where the pressure increases somewhat (see Fig. 3): only where the pressure difference \(P_{\text{external}}-P_{\text{vapor}}\) is sufficiently large are conditions created for the successful development of the primary processes leading to damage.

By itself, the temperature of the liquid in which cavitation occurs does not make it possible to draw conclusions about the expected damage. It is important to know, as we have seen, what the difference is between the external pressure and the vapor pressure. However, the dependence between the character of the damage and this difference is complicated by a number of secondary factors and could not be revealed in pure form. Among these factors the most important role, apparently, is played by surface tension.

5. THE INFLUENCE OF VISCOSITY AND SURFACE TENSION

In order to assess the role of surface tension, liquids were taken with strongly differing coefficients of surface tension \(\sigma\) and with similar viscosities and vapor pressures. Table 1 shows that the influence of the surface tension of a cavitating liquid on the process of destruction of the specimen is very great.

Table 1

The losses in weight of the specimen as a function of the magnitude of \(\sigma\) are presented on the curve in Fig. 15. We see that with an increase in \(\sigma\) the loss in weight also increases.

Estimating the influence of viscosity is incomparably more difficult, since most viscous liquids have a low vapor pressure, while the losses depend substantially on the difference \((P-P_{\text{vapor}})\). The results of experiments\({}^{14}\), presented in Fig. 16, allow one to think that, in assessing the activity of cavitation action, viscosity is a comparatively insignificant factor.

6. Possible Mechanism of Damage

The first attempt to construct a theory of the destructive action of cavitation belongs to Cook[^15]. His idea was as follows: when a cavitation bubble collapses, the water directly strikes the surface of the metal and thereby damages the surface. Cook calculated the pressure that should arise on the surface of a metallic sphere of a given radius located at the center of collapse

Fig. 15. Dependence of the weight loss of a specimen on the coefficient of surface tension of the liquid.

Fig. 16. Weight loss of aluminum after 20 min of cavitation in a glycerin–water mixture at 28° C as a function of glycerin concentration (curve I, left scale). Curve II (right scale) shows the corresponding change in the vapor elasticity of the mixture. The numbers on curve II indicate the viscosity of the mixture.

of a cavitation bubble.

ing bubble at the moment when the water, “flowing” into the bubble, reaches the surface of the sphere (Fig. 17). But the case considered by Cook is unreal. In fact, a cavitation bubble either sits on the surface or is located close to it (Fig. 18). In such a case the hydraulic impact will occur when the bubbles close completely. Cook’s formulas are unsuitable for this case—at \(r \to 0\) they lead to infinite pressure values.

Rayleigh\(^{16}\), in 1917, attempted to remove the difficulties of Cook’s theory by showing that considerable pressures must arise in the liquid itself near the closing cavitation bubble. According to Rayleigh, there is no need for an impact of the liquid against the metal surface. It is sufficient if the contracting bubble is located near a solid surface.

Fig. 17.

Assuming that the liquid is incompressible, neglecting its viscosity, and considering that the bubble contains no gas, Rayleigh arrives at the following results: the velocity of contraction of the bubble surface is equal to

\[ u=\sqrt{\frac{2}{3}\frac{P}{\rho}\left(\frac{R_0^3}{R^3}-1\right)}. \]

Here \(R_0\) is the initial radius of the bubble, \(R\) the radius at the given instant of time, \(P\) the pressure in the liquid, and \(\rho\) its density. The time of complete closure of the bubble (“contraction to a point”), according to Rayleigh, is given by the formula

\[ \tau=0.91R_0\sqrt{\frac{\rho}{P}}. \]

Fig. 18.

It is easy to verify that, for an initial bubble radius \(R_0=1\) mm, the time of its closure is \(10^{-4}\) sec. The pressure arising in the vicinity of the bubble reaches a maximum at a distance \(1.57R_0\) from the center and there is equal to

\[ P_{\max}\approx 0.163\left(\frac{R_0}{R}\right)^3P. \]

Table II illustrates the results of Rayleigh’s theory;

One can make use of Rayleigh’s values for the velocity of contraction of a bubble and substitute it into Cook’s formula for the pressure arising when a ball enclosed in the bubble strikes a solid surface:

\[ P_{\text{according to Cook}}=u\sqrt{\rho\beta} =\sqrt{\frac{2}{3}\,P\beta\left(\frac{R_0^3}{R^3}-1\right)}, \]

where \(\beta\) is the bulk modulus of water. The fourth column in Table II gives the values of \(P_{\text{according to Cook}}\) calculated by this formula.

The numerical values of velocities and pressures given in Table II undoubtedly exceed the true values. Rayleigh’s theory does not take into account the viscosity and compressibility of the liquid. It is also necessary to take into account the presence of vapor and gas inside the bubble. The latter circumstance was taken into account by Rayleigh, who assumed adiabatic

Table II

compression of the gas contained in the bubble, which led to a certain reduction in the rate of contraction of the bubble, and consequently also of the pressure arising near it. However, such an assumption is extremely crude. Thus, Rayleigh’s theory, while indicating the possibility of large pressures arising near contracting bubbles, is not capable of giving a concrete value of the pressure with all the physical factors that play a role in cavitation taken into account.

Ackret \(^{17}\) in 1930 proceeded from the assumption that a bubble, having originated in a rapid flow of liquid and subsequently entering a region of increased pressure, undergoes sharp compression. The pressure and temperature of the noncondensing gas inside the bubble must then increase considerably. In order to calculate them, it is necessary to know how the condensation of hot vapors proceeds during rapid compression of the bubble, and also to have data on heat transfer through the surface of the bubble. Approximate estimates, under the assumption of adiabatic compression of the bubble, were made by Ackret: for one case he obtained a pressure of 2500 atm and a temperature inside the bubble of \(2000^\circ\) K.

Marinesco \(^{18}\) made an attempt to measure directly the temperature of bubbles, for which he used a set of explosive powders not wetted by the liquid. Powders with various, but known, ignition temperatures were added to the cavitating liquid, and

that the temperature of the bubbles was determined from the flash of the powder. It proved to be equal to 230° C.

Several works have been devoted to the study of the luminescence of cavitation bubbles[^19][^20][^21]. Frenkel[^21] showed that the luminescence is apparently a consequence of the balloelectric effect: it is caused by an electric discharge that takes place in the “atmosphere” of the cavitation bubble at the moment of its formation.

Smith[^22] theoretically investigated the question of the possibility of the destructive action of gas bubbles pulsating in the field of an intense sound wave. When such a wave travels in a liquid, part of the gas dissolved in it is released in the form of small bubbles. Moving irregularly, they increase in size partly owing to adsorption of gas from the liquid and partly owing to coalescence with other bubbles. At one stage of their growth the bubbles pass through a short period of great activity—they move especially sharply and rapidly in the liquid. Then, after becoming somewhat calmer, they reach 1–2 mm in diameter, rise to the surface, and escape into the atmosphere.

Smith showed that small gas bubbles in a liquid in the field of a sound wave oscillate as a resonant mechanical system possessing one degree of freedom. Since the bubble is small compared with the length of the sound wave, the variable pressure in the wave acts almost uniformly on the entire surface of the bubble; therefore it oscillates almost radially. Taking into account the action of inertia forces and “radiation friction,” Smith comes to the conclusion that, when the radii of the bubbles are somewhat smaller than the resonant radii, the pressure near the surface of the bubble may increase up to 15,000 atm. As for the values of the resonant radii, they are given in Table III (according to Smith).

Table III

Probably the period of the most active behavior of the bubbles coincides with the moment when their radii are very close to the resonant ones. Objects in contact with such bubbles may be subjected to great stresses and become damaged.

Smith’s conclusions concerning the possibility of the occurrence of damage as a result of the pulsation of air bubbles of resonant size, it would seem, could also be used to explain the destructive action of cavitation. Indeed, in a liquid under any condi-

... the gas dissolved in liquids, as a result of which the cavitation bubbles pulsating, say, in a Geynes apparatus above the cavitating surface are filled with a mixture of vapor and gas. With a sharp contraction of the bubble the vapor condenses, while the gas is compressed.

But the point is that an oscillating bubble, as shown by the experiments of Kornfeld and Suvorov^13, while contracting, loses stability of form. It retains the shape of a sphere only until the radius has decreased by 3–5 times, after which the bubble is sharply deformed. In particular, if the bubble sits on the surface of a metal, it may divide into parts. A schematic representation of the successive stages of this division is given in Fig. 19.

Fig. 19. Successive changes in the shape of a cavitation bubble during its compression.

The violation of the spherical form of the bubble may be given the following qualitative explanation. The spherical form of the bubble is usually ensured by the predominant action of surface-tension forces. At comparatively large amplitudes of oscillation the hydrodynamic forces may exceed the surface-tension forces, and the bubble may exhibit an entirely unexpected change of form.

Thus, Smith’s conclusions, based on the assumption of preservation of the bubble’s form down to dimensions of 0.003 mm, prove untenable.

From this same point of view the mechanism of destruction according to Rayleigh is also improbable: if one can speak only of a fivefold decrease in the radius of the bubble, then, according to Rayleigh’s theory, one should expect the appearance of pressures of only a few tens of atmospheres, which are clearly insufficient to explain the destructive action of cavitation.

Kornfeld and Suvorov^13 further point out that, at the moments of flattening of cavitation bubbles pulsating on the solid surface of a specimen, repeated impacts of water against the surface of the specimen occur, and these cause damage. This is all the more plausible because, as we have seen, there is a clear similarity between cavitation damage and that produced by the impact of a jet of water. In Fig. 19 the direction of the water impacts is indicated by arrows.

Kornfeld and Suvorov^13 repeated the experiments with crossing jets, at a specimen rotation speed of about 50 m/sec. It took 400,000 revolutions of the disk for noticeable damage, in the form of small depressions, to appear on the surface of the aluminum. The appearance and size of the depressions are very similar to those which appear at the center of an oscillating aluminum specimen (at an amplitude of 0.015 mm), one minute after the appearance of cavitation bubbles, i.e., after approximately 400,000 oscillations of the specimen.

Proceeding from this, Kornfeld and Suvorov believe that, in order to investigate cavitation resistance or the strength of a material, devices of the type of Schroter’s diffuser or complex magnetostriction apparatus are not needed; since the cause of destruction is the impacts of water, a sufficiently rotating wheel with a specimen and a jet of water is enough.

7. CHEMICAL REACTIONS DURING CAVITATION

The oldest notion is that the causes of damage during cavitation have a purely chemical nature. At the boundary between the metal and the liquid, chemical or electrochemical processes continuously take place, causing corrosion. Cavitation accelerates these processes, giving rise to gas evolution and to an increase in temperature and pressure. The role of the flow, from this point of view, is reduced merely to the removal of corrosion products.

Experiments in a cavitation tube\(^{14}\) do indeed show that on the surface of specimens, in those places where periodic closure of cavitation bubbles occurs, clearly expressed “temper colors” appear. This has been found on bronze, brass, cast-iron, and steel specimens. In some cases, oxide and hydroxide films on the metal could be seen with the unaided eye. They were of such thickness that they gave the surface an interference blue coloration. With a short duration of the experiment the film could easily be separated from the surface. With a longer cavitation action the oxide film proved to be firmly bound to the metal. The external appearance of the surface after such experiments resembled the surface of a metal heated in air to a high temperature.

This picture appears especially vividly on cadmium specimens. Around the damaged place, which rises on the surface in the form of a crater, a brown border is clearly visible, representing a thin layer of cadmium oxide. X-ray analysis confirms this conclusion.

However, the view of chemical processes as the principal cause of damage during cavitation does not find confirmation. Chemical processes only accompany the main mechanism of action on the material, preparing the surface of the specimen for subsequent, easier damage and thereby accelerating the process of destruction itself. It is known, for example, that a specimen of pure magnesium, during cavitation for less than 0.5 sec., reveals strong plastic deformations at individual points of the surface. Even on steel, which is considerably more resistant than magnesium, damage is visible after such short intervals of time that secondary effects, in particular chemical ones, cannot have time to develop seriously. Here, evidently, damage occurs with each individual impact. The significance of the damage,

inflicted by a single impact depends on a whole series of causes, including the physical properties of the material.

In particular, one should point here to Fater’s experiment²⁴. A special device, ejecting drops of liquid at high velocity, was directed at a copper specimen, and after \(1/30\) sec. damage appeared on its surface with a depth of \(0.07\) mm and a diameter of \(0.2\) mm. Clearly, no considerations of chemical processes or of “fatigue” of the material can be invoked to explain the destructive action under these conditions: the damage was detected after one or, in any case, only a few impacts of drops.

Finally, the results of experiments³˒¹⁴ with chemically passive materials also argue against a purely chemical point of view: for example, experiments involving the destruction of agate, concrete, gold, etc. The study of microphotographs leads to the conclusion that, in cavitation, local stresses and temperature “peaks” (“primary processes”) first arise at individual microscopic areas, and only later, with continued cavitation action, do mechanical damages appear—cracks are formed. Weight loss, naturally, is not detected at the first stage—this explains the presence of an “incubation period” on the weight-loss curves. Chemical processes at this stage play an auxiliary role: they promote the formation on the surface of the specimen of such sites at which cavitation bubbles become fixed, causing damage by their activity.

Only at later stages of cavitation action on a surface do two new factors acquire serious significance—corrosion (especially in the case of chemically active materials) and “fatigue” of the metal.

8. CAVITATION RESISTANCE

AND MECHANICAL PROPERTIES OF MATERIALS

On the basis of experiments in a cavitation tube, attempts were first made to place the loss of weight (or volume) of a specimen in direct dependence on the ultimate strength, which is known from ordinary tests of materials.

The table IV presented below shows how untenable such a point of view proved to be¹⁴.

Thus, the customary characteristics of materials prove to be completely inadequate when one attempts, on their basis, to evaluate the resistance of a material to cavitation damage (the so-called “cavitation resistance of the material”). That is why Haller’s conclusion was erroneous: he initially believed that the higher the hardness of a material, the higher its cavitation resistance. That this does not correspond to reality is seen from the experiments of Mousson²⁵ and Kerr²⁶.

Musson showed that for cast iron, for example, with increasing hardness the loss of volume increases, i.e., the cavitation resistance of cast iron decreases.

Table IV

Decrease in the volume of cast iron and alloyed cast iron under cavitation

resistance. In carbon steel, an increase in hardness does not cause a noticeable change in weight loss. On the other hand, in high-alloy steel an increase in hardness is always accompanied by a decrease in losses, i.e., by an increase in cavitation resistance.

It is interesting to note that after heat treatment the resistance to cavitation action increases with increasing hardness. Thus, for example, for stainless steel containing 0.12% C and 12.25% Cr, Musson found the following decrease in the volume of the specimen as a function of Brinell hardness:

Table V

Metals with only a slight difference in chemical composition, but with almost the same hardness, show a considerable difference in cavitation resistance. For example, molybdenum steel (0.27% C, 0.52% Mo), at a Brinell hardness of 192, showed a 35% greater loss than Cr—Ni—Mo steel (0.28% C, 1.37% Ni, 0.6% Cr, 0.25% Mo) at almost the same hardness ($H_B = 179$) and tensile strength.

* The figures in the last column give, in arbitrary units, the decrease in the volume of the specimen as a result of cavitation damage.

As for experiments with nonmetallic specimens, the behavior of glass\(^ {14}\) is of particular interest. Fig. 20 shows a photograph, enlarged 220 times, of a round glass plate after several minutes of cavitation in water. It is easy to see on it damage from single impacts in the form of separate, numerous small holes in the glass. Only rarely are they connected by thin cracks. This photograph may serve as a convincing illustration of how small are the regions where the “primary” processes (peaks of temperature and pressure) take place. If one also recalls that glass is an extremely brittle material, it becomes clear how rapidly these processes must proceed in order for the pattern of damage to have the appearance shown in the photograph.

Fig. 20. Cavitation damage to a glass plate (enlarged approximately 200 times).

In experiments with single crystals of common salt (cavitating in benzene) and cadmium (in water)\(^ {14}\), on individual areas of the surface it was possible to distinguish traces of local melting under the influence of sharp maxima of temperature and pressure (Fig. 21). Experiments with rubber lead to the same conclusions. After several hours of operation, such large internal stresses appeared in it that, according to Haller, it became liquid inside.

It should be pointed out that attempts were made to find a connection between the melting temperature and cavitation resistance. Table VI shows how successful this idea is.

Table VI

It is hardly possible to hope to establish a simple dependence on only one of the physical characteristics of a material. Microphotographs and X-ray photographs of damaged materials give grounds for asserting that, after a brief cavitational action on a material, we are dealing with structural changes in the crystalline grains and in the crystalline bonds, with fragmentation of crystalline grains, with disorientation of crystallites, with the appearance of slip traces and cracking, with obvious consequences of plastic deformation on small areas of the specimen surface \((10^{-6}—10^{-2}\ \text{cm}^2)^{6,14,27}\). The static stresses that would be needed for such deformations would reach \(9000\ \text{kg}/\text{cm}^2\)!

Fig. 21. Hole melted in a cadmium single crystal during cavitation in water.

The features of the crystalline structure, in particular the sizes of the crystalline grains, play an important role in evaluating the cavitational resistance of a material. Experiments have shown, for example, that a fine-grained structure proved to be more resistant to cavitational damage. Strongly porous cast irons and steels are damaged especially easily.

In order once again to note the distinctive character of cavitational action on a material and its difference from purely mechanical damage, let us give a brief description of one peculiar research method that was used in connection with the study of cavitation. We have in mind work with sand-blast apparatuses \(^{28}\). From a special “sand box” a jet of sand was directed at a rotating specimen—as in experiments with a jet of water. From the loss of weight or volume it was possible to judge the character and magnitude of the damage. No analogy with the behavior of specimens in Gaine’s apparatus or in the cavitation tube was found. In experiments with a jet of sand, the ordinary strength characteristics of materials do not lose their force: the higher the hardness and the tensile strength, the more resistant to the impacts of the sand jet the material proves to be.

The large number of works devoted to cavitation now makes it possible to distinguish two directions for research:

1. The study of the mechanism of cavitational action on a material—this is a field dealt with by physics and hydrodynamics;

1. Study of the cavitation resistance of materials, which belongs to materials science.

Our review has been devoted mainly to the first question. But in conclusion we shall add one further observation concerning the results of research in the second direction.

In any mechanical tests of a material, as is known, there exist certain stress limits below which the material is not damaged at all (elastic limit, strength limit, etc.).

In cavitation, however, as in the case of corrosion, we are dealing with a continuous process of wear of the material: if cavitation has begun, any material will sooner or later be damaged. The hardest and toughest materials are not exceptions, even under the weakest cavitation.

REFERENCES

1. Berthelot, Ann. de Chemie 30, 232 (1850).
2. Worthington, Proc. Roy. Soc. 50, (1891); Phill. Trans. of Roy. Soc. A183, 355 (1892).
3. H. Fottinger, “Hydraulische probleme”, VDI-Verlag, Berlin, 1926, p. 14.
   3a. H. Fottinger, “Hydromechanische Probleme des Schiffsantriebs”, Vortr., Hamburg, 1932, Berlin, 241 pp.
4. N. Gaines, Physics 3, 209 (1932).
5. J. Hunsaker, Mech. Engng. 57, 211 (1935); Trans. A. S. M. E., 57, 423 (1935).
6. H. Schroter, Z. VDI 76, 511, (1932); 77, 865 (1934); 78, 349 (1934); 80, 479 (1936).
7. P. Haller, Schweiz. Bauztg. 101, 243 (1933). Handb. der Werkstoffkunde II Bd., Berlin (1939), 471.
8. E. Honegger, B. B. C.-Mit. 14, 74 (1927).
9. J. Ackeret and P. Haller: Schweiz. Bauztg. 108, 105 (1936).
10. M. Vater, Z. VDI 81, 1305 (1937); 82, 672 (1938); 83, 725 (1939).
11. J. Ackeret and P. Haller, Schweiz. Arch. angew. Wiss. Techn. 4, 293 (1938).
12. W. Schumb, H. Peters, K. Milligen, Metals and Alloys 8, 126 (1937).
13. M. Kornfeld and L. Suvorov, J. of Physics, 6, 75 (1942).
14. H. Nowotny, “Werkstoffzerstörung durch Kavitation”, Berlin, 1943.
15. S. Cook, Engineering 107, 501, 515 (1919); Prcc. Roy. Soc. A119, 481 (1928).
16. J. Rayleigh, Phil. Mag. (VI), 34, 94 (1917).
17. J. Ackeret, Techn. Mech. Thermodyn. (Forsch. Ing. Wes.), 1, 1—72 (1930); “Hydromech. Probleme des Schiffsantriebs”, 1932, 227.
18. Marinesco, C. R. 201, 1187 (1935).
19. Zhimakov, Izv. Acad. Sci. USSR 3, 450 (1934).
20. Frenzel, Zeits. f. Phys. Chemie, 27B, 421, 1934; Chambers, J. chem. Phys. 5, 290, 1937.

1. Ya. I. Frenkel, Acta Physico-Chimica 12, 317 (1940).
2. F. Smith, Phil. Mag. 19, 1147 (1935).
3. H. Müller, “Hydromech. Probleme des Schiffsantrieb,” Vortrag, Hamburg, 1932, Berlin, 1932, 311 pp.
4. M. Vater and W. Sorberger, Z. VDI 83, 725 (1939).
5. J. Mousson, Trans. A. S. M. E. 59, 399 (1937), Z. VDI 82, 397 (1938).
6. S. Kerz, Trans. A. S. M. E. 59, 373 (1937).
7. H. Böttcher, Z. VDI 80, 1499 (1936).
8. M. Vater, Z. VDI 82, 836 (1938).
9. V. Pozdyunin, DAN SSSR 39, 8 (1943); Izv. O. T. N. AN SSSR No. 1–2 (1944), No. 1–2 (1945); Izv. O. T. N. AN SSSR 2, 2 (1946).