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Cavitation and the generation of tension in liquids

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1984 J. Phys. D: Appl. Phys. 17 2139

(http://iopscience.iop.org/0022-3727/17/11/003)

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J. Phys. D : Appl. Phys., 17 (1984) 2139-2164. Printed in Great Britain

REVIEW ARTICLE

Cavitation and the generation of tension in liquids

D H Trevena Department of Physics, University College of Wales, Penglais, Aberystwyth, Dyfed SY23 3BZ, UK

Received 1 March 1984

Abstract. The phenomenon of cavitation occurs when a tension, applied to a liquid. exceeds a certain critical value known as the breaking tension or cavitation threshold. The present review article considers various ways in which liquids have been subjected to tension and how the accompanying cavitation has been studied. The importance of these matters in medicine and botany is also discussed.

1. Introduction

The word cavitation was used for the first time in 1895 as a result of some trials on H.M.S. Daring, a new destroyer. It was found that the propellers were not developing sufficient thrust because they created voids and clouds of bubbles in the water when the pressure near the blades fell to a negative value (tension) of about half an atmosphere. The term cavitation to describe this effect was suggested by R E Froude, a distinguished naval architect (Hutton 1972). Since then, in the Engineering literature, it is usually stated that cavitation occurs when the pressure in a liquid falls to the vapour pressure of the liquid. This is misleading: it is more correct to say that cavitation occurs when tiny bubbles are observed to form as a consequence of pressure reduction, usually to negative pressures. However, the fact that a liquid can sustain a tension before it eventually cavitates was known earlier to Donny (1846) and to Berthelot (1850).

In the present paper we shall be concerned with the recent use of Berthelots original method together with other ways of subjecting a liquid to tension. Inextricably bound up with this application of tension to a liquid is the occurrence of cavitation. In some cases the inducing of cavitation at the breaking tension of the liquid is the desired goal. In other cases it is the suppression of cavitation, as in gear pumps, which is the objective.

Broadly speaking, cavitation can start in one of three ways at a suitable nucleus when the pressure is reduced to a negative value. It is thought that there are three types of such nuclei. First, in water, there is evidence for the existence of a large number of minute spherical gas bubbles some of which may be stabilised against gaseous diffusion by a skin of organic impurity (Iyengar and Richardson 1958, Hayward 1970). Secondly, there may be solid particles (motes) in the liquid with gas trapped in crevices in these particles (Apfell970, Crum 1979). In the third place, gas can be trapped in tiny crevices in the walls of the vessel containing the liquid (Winterton 1977, Overton and Trevena 1980).

0022-3727/84/112139 + 27 $02.25 @ 1984 The Institute of Physics 2139 Dll-C

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The present paper will be devoted to a survey of the various ways in which a liquid can be subjected to tension. In some of these cases the tension is not always large enough to cavitate the liquid whilst in other cases the breaking tension (or cavitation threshold) is reached. Bubble studies, involving the growth of a bubble and its subsequent time- history, will also be described. The cavitation erosion produced on a solid is a well- known phenomenon and its occurrence and production are discussed. Finally, the relevance of these effects in other disciplines, such as botany and medicine, is summarised.

2. Static application of tension to a liquid

2.1. The Berthelot tube method

In his original experiments Berthelot used a sealed cylindrical glass tube which was almost completely filled with water. On heating the tube the water expanded until it completely filled the tube at a certain temperature Tf but, when it was subsequently allowed to cool, the adhesion of the liquid to the walls of the tube prevented the liquid from contracting at a greater rate than the internal volume of the tube. In this way a progressively increasing tension F was set up in the liquid as the temperature fell below Ti until it eventually broke at a lower temperatureTb. As the liquid broke there was a sudden increase AVin the external volume of the tube because of the release of tension in the liquid at the same instant. In his experiments Berthelot found that the fractional increase AV/V in the volume of the water was 1/120 just before the break occurred and he calculated that the tension in the water was 50 atm at the same instant (Trevena 1978).

Since Berthelot performed his original work, the Berthelot tube method has been reported intermittently in the literature. Three important papers on the method by Temperley and Chambers (1946) and by Temperley (1946, 1947) describe how the original experiments of Berthelot were repeated. Their conclusion was that the strength of water in the presence of glass is between 30 and 50 atm. These tensions were calculated from the measured value of AV/V and various elastic data for the tube.

In the Berthelot tube work described so far it was only possible to measure experi- mentally the actual breaking tension when the enclosed liquid fractured. A considerable step forward was made when a transducer method of monitoring continuously the growth of this tension was developed by Trevena and his co-workers (Chapman et a1 1975). This was done by attaching an unbonded strain gauge pressure transducer to a steel Berthelo: tube in such a way that the pressure sensing diaphragm of the transducer formed one of the end walls of the tube. Using this technique, Richards and Trevena (1976) were able to plot afamily of (tension, temperature) curves for water; it is believed that this is the first time that such curves had been reported for a liquid in the negative pressure range. These curves extended to negative pressures of about -30 atm; it was not possible to follow the curves for any lower values of these pressures because the liquid in the tube broke at this value of the negative pressure.

This work led naturally to the development of a further new type of Berthelot tube, designed so as to enable the tube to be evacuated thoroughly before the introduction of the test liquid (Jones et a1 1981). By this means it was hoped that much of the air existing in crevices in the tube wall would be removed and so enable the test liquid to withstand a larger tension before breaking. In the actual experiments this was found to be the case: tensions of up to 46 atm (at the time, an all-time high for water in steel) were obtained

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before cavitation occurred. Later, using the same design of tube, Overton et a1 (1982) describe experiments in which water was degassed before introducing it into an evacu- ated tube. With this process of prior degassing, potential cavitation nuclei in the liquid were removed and much larger breaking tensions of up to 68 atm were obtained.

A second electrical modification of the original Ber:helot tube method has been reported by Sedgewick and Trevena (1978). In this work pressure changes in the liquid were monitored by means of a semiconductor strain gauge attached to the outside of a steel tube so that it measured the accompanying circumferential strain. While the method is perfectly reliable it has not been used to the same extent as the transducer method.

Mention must also be made of the work of Evans (1979) who used a glass tube shaped into a spiral; this method was based on an idea introduced by Meyer (1911 j . A long glass tube was formed into a spiral consisting of an odd number of half-circles with the two ends of the tube parallel and pointing in the same direction. Internal pressure would cause the coil to unwind slightly (as in a Bourdon pressure gauge) so that the two ends separate; conversely, internal tension would bring the two ends closer together. By monitoring changes in the separation of the tube ends with a distance meter a record of pressure (or tension) against temperature could be made. After this technique had been established it was used to study some factors which influence nucleation, in particular the nucleation gas bubbles in the supersaturated diver (Evans 198lj. This is a good example of the use of the Berthelot tube technique in medical physics.

2.2. Theoretical aspects of the Berthelot tube method

In the previous section we dealt with the development of the experimental procedure of the Berthelot tube method. In recent years there has been a parallel development in the theory of this fascinating experiment. Some of these theoretical aspects will now be considered.

The theory of the method as used by Temperley (1947) and all previous workers was limited in that the various equations used yielded only the actual breaking value of the tension. A more comprehensive theory was derived by Williams and Trevena (1977) which made it possible to calculate the values of both the pressure and the tension at any stage during a Berthelot tube experiment and not merely that of the breaking tension. These calculated values agree closely with the experimental values given by the experi- mental isochores obtained by Richards and Trevena (1976).

At a later stage this theory was extended still further by Jones and Trevena (1980) and a fairly detailed summary of their work will now be given. They started by considering a typical (pressure, temperature) curve for water in a steel tube as obtained by Richards and Trevena. The general shape of such a curve is shown in figure 1; in this figure, as is usual in Berthelot tube work, the values of the pressure plotted are those relative to the external ambient pressure of one atmosphere. At A , where the temperature T = To, both the tube and enclosed liquid are in an unstressed state and the absolute pressure in the liquid is 1 atm at this stage. Suppose that the largest experimental value obtained for the tension is at the point B corresponding to some lower temperature T = T I (

2142 D H Trevena

The authors show that the slope of the curve in figure 1 is given by

dP/d 2- = (PL - PS)/(KL + Ks) (1) where pL, pS are the expansivities of the liquid and the steel and KL, Ks are their isothermal compressibilities. Also from equation (1) we see that the maximum tension which can be generated in the system occurs when dp/d T = 0, that is, when PL = PS. This occurs at the point R in figure 1 at a temperature of T = T, = 6 "C. So, as we follow the curve from A the tension F will increase as the temperature falls to T = T, and will thereafter decrease. For this reason this temperature T, at the pressure minimum is referred to as the 'reversal temperature.' We shall return to this point in a moment.

Figure. 1. General shape of the (pressure, temperature) curve for water in a steel Berthelot tube.

The p ' s and K ' s in equation (1) will vary with temperature and this variation can be obtained from tables; this enables one to plot the graph of dp/d T against T. The value of the maximum tension F,,, at the point R in figure 1 is given by

and, similarly, the tension F a t a temperature T ( T, < T < To) is r T

p = -F = J (dp/dT)dT. T@

Thus F,,, and F can readily be obtained from the appropriate areas under the (dp/d T, T ) curve. For example for To = 30 "C, F,, = 73 atm.

In later experiments, two of the isochores obtained for water in a steel tube exhibited the reversal effect at around 5 "C (Jones et a1 1981). This was an exciting experimental result.

2.3. The centrifugal method

We now consider another 'static' method in which a gradually increasing tension is applied to a liquid column until it breaks.

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We first give the theory of the method by considering a Z-shaped capillary tube, open at both ends, and containing a liquid. If such a tube is mounted horizontally and spun in its plane with angular velocity w about a vertical axis through its centre, then a pressure gradient of p d r is generated in the outward direction at a distance r from the axis. The pressure is always atmospheric at the two ends of the tube and decreases as we approach the axis. As w is made to increase, the pressure eventually becomes negative and when wis sufficiently large the liquid ruptures at the centre of the tube. From a knowledge of this critical value w, the value of the breaking tension is easily calculated.

Some experiments using distilled water in a Pyrex glass capillary were described by Briggs (1950) in a much-quoted paper. Briggs (who, at the time, was head of the US National Bureau of Standards) emphasises the need for scrupulous cleanliness in such measurements. In the centrifugal field one half of the liquid column was pulling against the other until, at W = W , , the column broke at its centre. Briggs found the breaking tension of distilled water to be 277 atm at 10 "C, the highest experimental value ever reported. He also studied the effect of temperature on breaking tension. This tension decreased from a maximum of 277 atm at 10 "C to 217 atm at 50 "C. This decrease is to be expected because at the critical temperature the tension must be zero. As the temperature fell from 5 "C to 0 "C the breaking tension fell very rapidly and this rep- resents another anomaly in the behaviour of water in this temperature range. Briggs emphasises that it was not possible to say whether the rupture of the water column occurred at the wall of the capillary (loss of adhesion) or in the body of the liquid (loss of cohesion).

Strube and Lauterborn (1970) developed the method to study cavitation nuclei at the interface between quartz glass and pure degassed water. The maximum breaking tension observed was 175 atm and they found that the gas content of the water does not play a discernible part (thus implying that 'loss of cohesion' is not the important factor in the experiment). Their overall conclusions were that hydrophobic impurities and cavities at the wall played the major role in the formation of nuclei for triggering off cavitation.

3. Dynamic stressing of a liquid

Broadly speaking, there are two ways in which a liquid can be subjected to tension under dynamic stressing. In the first of these a compressional pulse is first produced in the liquid and is later converted into a tension pulse by reflection at a suitable boundary. The second method is somewhat more direct in that a tension pulse is generated ab initio in the liquid. Some of the work based on these two methods will now be described.

3. l . Dynamic stressing based on the repection principle

The classic example of this type occurs in work using underwater explosions. The resulting upward-going pressure wave is reflected as a tension wave at the free surface of the water. Much of this work was carried out during the Second World War (Trevena 1967). A small-scale experiment of this kind, in which an explosive charge was detonated a small distance below the surface of a liquid, has been described by Wilson et a1 (1975). The basic idea was to obtain high-speed motion photographs of the spray dome formed above the original undisturbed free surface as a result of detonating the charge. The

2144 D H Trevena

authors show that the initial spray dome velocity V0 is given by

where p is the maximum pressure of the explosion, F is the maximum tension in the reflected pressure wave, pis the density of the liquid and Uis the pressure wave velocity. In the actual experiments a small electrically actuated detonator containing 0.1 g of explosive was placed at depths R varying between 2.5 and 12.7 cm below the free surface. A high-speed camera was used to photograph the initial displacement of the spray dome and from this sequence of pictures V. was found. A graph of V. against R was then drawn and extrapolation of this graph to V0 = 0 corresponded to F = 2p (see equation (2)). This value R 0 of R turned out to be 31 cm. A graph was then drawn of the peak pressure p against the so-called 'similarity parameter' for explosives, W1I3/R, where Wis the mass of the charge. The peak pressure p was obtained from the particle velocity by means of tables of the Rankine-Hugoniot equations. From the extrapolated value of W"3/R corresponding to V0 = 0 (that is, to R = Ro) a value of p , and thus of F. was obtained. The value of Ffor the cavitation threshold of ordinary water was 8.0 atm.

The bullet-piston method, used over a number of years by Trevena and his col- leagues, is also based on the reflection of a pressure pulse at the free surface of a liquid (Trevena 1975). In this method the liquid is contained in a vertical cylindrical tube fitted with a steel piston at its lower end. A pressure pulse is then generated in the liquid by firing a lead bullet to strike the lower end of the piston normally at its centre. The (pressure, time) curve of this pulse rises rapidly in about 50 ,m to a maximum value p and then decays more gradually over a few hundred microseconds. For a pulse in water p could be up to about 300 atm and the total effective duration Tof the pulse is typically 500 , p . When this pulse reaches the upper free surface of the liquid it is reflected as a tension pulse. At a sufficiently large depth 1 below the free surface, such that 21/c > 5, the incident and reflected pulses will not overlap in time (c being the velocity of propa- gation of the pulses in the liquid). Piezolectric transducers were used at a depth 1 to measure the peak amplitudes, p and F , of the incident and reflectedpulses respectively. Many records are taken in which p is gradually increased by using different kinds of bullets and pistons of varying masses. For each record the values of p and F are noted and the (F .p ) curve drawn, This curve shows that the value of Fdoes not increase linearly with p. as one might expect on a simple reflection principle, but levels off at a constant limiting value F , This limiting or 'plateau' value clearly represents the maximum tension the liquid can stand under these experimental conditions and it represents the breaking tension (Couzens and Trevena 1969,1974).

This 'plateau' method was later extended by Sedgewick and Trevena (1976) to show that the effect of boiling and deionisation increased the ability of water to withstand tension before cavitation occurred. The various breaking tensions which they observed are summarised in table 1.

Table 1. Breaking tensions for four types of water.

Type of water Breaking tension F (atrn)

Ordinary tap-water Deionised water Boiled tap-water Boiled deionised water

9.0 10.0 11.5 14.5

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They also studied the effect of repeated stressing by subjecting a single specimen of tap-water to stressing at regular 3 min intervals. (The limitations of the experimental method dictated a minimum interval of 3 min between successive shots.) The values of Pwere then plotted for each successive shot and the results, shown as curve A in figure 2, show that P increased with repeated failure of the test liquid. The conclusion is that some of the air nuclei (dissolved gas, or, more likely, microscopic air bubbles) are 'cleaned up' by each successive shot until eventually fi itself tends to some upper limit of about 11.0 atm. Further support for this conclusion was obtained by leaving the test specimen to stand for 24 h before subjecting it to further repeated stressing (see curve B of figure 2).

Figure 2. Graph showing the variation of breaking tension with successive shots (after Sedgewick and Trevena 1976).

Sedgewick and Trevena also studied the way in which F for tap-water varied with temperature. Their graph ofpagainst temperature showed asharp peak at a temperature of 4 "C corresponding to the maximum density of water.

Recently, Overton and Trevena (1982) have used the bullet-piston method to show that the dynamic breaking tension of a liquid depends on the stressing rate: a higher stressing rate leads to a higher breaking tension and vice versa.

Carlson and Henry (1973) also used the reflection principle to convert a pressure pulse into a tension pulse. In their work this was achieved by allowing the pressure pulse to be reflected, not at the free surface of the liquid, but at a suitable liquid-solid interface. Furthermore, their tensions were applied at stress rates which were about 10' times higher than those in the work of Wilson et a1 and Trevena et al. The essential part of their apparatus is shown in figure 3. The test liquid L was confined to a cell between a solid plate S and a stretched aluminised Mylar film M. A pulsed electron beam generator was used to produce a stress pulse in the solid plate. This compressional stress wave generated in the solid travelled into the liquid and, since the acoustic impedances ( p c ) of the solid and liquid were chosen to be virtually equal, this stress wave was only very slightly affected in crossing the solid-liquid boundary. This compressional wave, after travelling through the liquid? was reflected as a tension wave at the interface M. In this work the solid plate was made of a composite organic material (Astrel360) and its thickness was 2 mm. The test liquid was glycerol and the sample had a thickness of 3 mm, while the stretched Mylar film was only 6 pm in thickness. The free-surface motion of the Mylar film, caused by the reflection on it of the incident stress wave from the liquid, was monitored by means of a velocity interferometer using light from a He-Ne laser. From

2146 D H Trevena

an observation of fringe shift data the (velocity, time) records of the Mylar film could be obtained and hence a value of the negative stress that was generated in the liquid. From a series of experiments Carlson and Henry found the breaking tension of glycerol to be 600 atm. They also make the point that the glycerol did break in each of the experiments and that this failure occurred at a distance of about 0.2 mm away from the Mylar film, that is, within the body of the liquid itself.

P u l s e d

beam

Figure 3. The apparatus of Carlson and Henry (1973).

Carlson and Levine (1974) extended the cell method to study the variation in the breaking tension Pof glycerol with its viscosity, q. They did this by measuring$ over the temperature range 220 to 350 K ; over this temperature range q varies widely between

and lo5 Pa S . A log-log plot of E against q showed that the experimental results could be divided into two distinct regions. For viscosities up to 19 Pa S , F could be described by the empirical relation of Couzens and Trevena (1974)

E= k q x with x = 0.30. For values of q from 19 to lo5 Pa s , F was virtually constant at 2500 atm. This experiment is one of the few in which the breaking tension has been correlated with viscosity. In a third use of the method Carlson (1975) found the breaking tension of mercury to be 19 000 atm.

3.2. Ab initio methods of generating a tension pulse Lackme (1978) used a very direct and novel way of applying a tension pulse to a liquid. A test specimen of water was containedin asealedcylindrical thick-walled tube, mounted with its axis vertical. The axial length of the enclosed liquid column was 4 cm and its radius 1 cm. At its lower end the liquid column was supported by a vertical cylindrical aluminium bar and a similar bar rested flush on top of the liquid. A tension pulse was applied to the liquid at its lower end. This was done by fixing a circular metal plate to the bottom end of the lower bar and then causing a ring-shaped weight to fall on to this plate. The impact of the weight caused a stress wave to travel up the lower bar and produce a tension pulse at the base of the liquid column. This pulse, after travelling up the liquid, was transmitted into the upper bar. A strain gauge mounted on the lower bar recorded the incident pulse entering the lower end of the liquid and a similar gauge on the upper bar recorded the transmitted pulse. The duration of the incident pulse was about 100 ps and its amplitude could be varied by changing the height of fall of the ring-shaped weight. With the experimental set-up, tensions of up to 5 atm could be transmitted through the water column. Lackme emphasises that this value was not that of the breaking tension but rather that of the maximum tension that could be generated in his apparatus. The method, however, seems to be a very promising one.

A second method is the tube-arrest method used by Overton and Trevena (1981). They used a cylindrical Perspex tube, of length 100 cm and internal diameter 2.5 cm, which was half filled with water and mounted vertically. The tube could be pulled down

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against rubber-tensioned supports and then released suddenly so that it rose through a vertical distance z (of - 5 cm) before being arrested by a Tufnol buffer. After the impact the water tended to continue its upward motion thus causing a tension pulse, together with cavitation, to occur near the base of the liquid column. The resulting pressure changes in the liquid were monitored by means of a piezoelectric transducer mounted axially into the bottom of the tube and photographic records of the accompanying oscilloscope traces were made. By varying the distance z , it was possible to vary the velocity U of the tube just before the arrest and hence the amplitude F of the initial tension pulse at the bottom of the liquid column.

The cavitation occurring took the form of a small cluster of tiny cavities, within the main body of the liquid, at a height of 1 or 2 cm above the base of the liquid column. A typical oscillogram showed the following features. An initial tension pulse of amplitude 15 atm was followed, after an interval of 12 ms, by a pressure pulse of amplitude 9 atm; thereafter the record was of an oscillatory nature showing tension and pressure pulses of progressively decreasing amplitudes. Such pressure-tension cycles had been reported by earlier workers and are accompanied by collapse-growth in the cavitation.

This tube-arrest technique was later developed into a reliable method for measuring the cavitation thresholds of various liquids (Williams et a1 1982). This work involved a repeated stressing of the test liquid at regular time intervals (e.g. every 15 S ) , the impact velocity Ubeing increased after each stressing (by increasing z ) . The amplitude Fof the initial tension pulse was recorded each time and the (F ,U) graph plotted. This graph followed a fairly steep linear rise at first up to a certain velocity U = Uc: at this stage there was an abrupt change of slope with the curve following a more gradual linear rise. For velocities U < U, no cavitation occurred in the tube after impact whereas for L > Uc cavitation was always present. Thus the tension corresponding to U = V , represents the cavitation threshold. This value of U , was found to be very repeatable and this makes the method a very reliable one for measuring the breaking tension.

3.3. The use of a water shock tube Water shock tube methods have been used in cavitation studies in recent years. In such work a tension wave is produced in a liquid either directly or by converting a positive pulse by a suitable reflection method.

Some experiments of the former kind have been described by Fujikawa and Aka- matsu (1978). They used a sealed vertical tube, most of which contained degassed tap water whilst the space above the water contained a mixture of air and helium under pressure, the upper end of the tube being sealed by means of a diaphragm. Hydrogen and oxygen bubbles were generated in the bottom of the tube by electrolysis and the diaphragm was then broken. This caused a rarefaction wave to propagate down the helium-air chamber and down through the water column. This down-going rarefaction wave in the water was reflected as a rarefaction at the closed lower end of the tube. Transducers were used to measure the tension generated in the liquid. It was also possible to study photographically the growth, collapse and rebound of bubbles near the base of the water column; this work will be considered further in D 4.4. It should be emphasised, however, that these were not true cavitation bubbles (in the sense that they were actually produced by the tension pulse) but ready-made ones produced by electrolysis.

A water shock tube method, but based on a reflection principle, has been described by Richards er a1 (1980). The test liquid (deionised water) was contained in the lower part of a vertically mounted combustion-driven shock tube. When an oxyacetylene

DI1-D

2148 D H Treuena

mixture, contained in the top part of the tube, was ignited with a suitable source, a detonation wave travelled down the tube and was normally reflected at the water interface; this caused a strong compression wave to be generated in the liquid column. When this compressional wave reached the bottom of the column, which was supported by a thin Mylar diaphragm, it was reflected and travelled upwards as a wave of tension. The pressure changes occurring in the column were followed by transducers mounted in portholes in the walls of the tube. A window section was incorporated in the lower half of the tube so as to enable streak schlieren photography to be used to record both the incident compressional and reflected tension waves. Such records provided infor- mation on the rate of growth and the behaviour of the bubble cloud which was formed when the amplitude of the tension wave reached a critical value, which, in this case, was 12 atm. The most interesting feature of the streak photograph was the presence of the strong acoustic field radiated at the collapse of the bubbles.

4. Bubble studies 4. l . Introductory remarks A vast literature exists on the topic of bubble studies. In the present section an attempt will be made to summarise the main results and the various experimental methods, Most of the experimental work has involved either (a) the growth, under an applied tension, of bubbles from microbubbles already initially present in the liquid or ( b ) the creation of bubbles by a concentrated injection of energy at a point in the liquid, as in laser and spark work (see 4.3). Sometimes a bubble after growing to a certain maximum size will collapse violently; such a transient cavity was first studied by Lord Rayleigh as long ago as 1917. There is also the pulsating bubble which grows and contracts in decreasing cycles over a relatively long period before finally disappearing. Of course, cavitation usually occurs as a large burst of bubbles but, in the main, it has proved more profitable to study the growth and collapse of a single cavity.

Differential equations describing the dynamics of a single cavity have been widely used and these will be discussedmore fully in 4.5. Essentially they tell us how the radius R of the cavity varies with time. The basic physics of the problem is however a very complex matter. In the first place there is the question of the composition of the gas- vapour mixture inside the cavity; that is, how much of it is gas (air) and how much is liquid vapour? The answer to this question is not known and various assumptions have been made in order to deal mathematically with the model bubble. In one case it is assumed that the relative amounts of gas and vapour stay constant as R varies. In another it is assumed that, as R changes, either evaporation or condensation occurs so as to maintain the vapour pressure at its equilibrium value, determined by the temperature of the surrounding liquid. There are also a number of other factors to be considered. In the main these are: energy losses (involved in damped oscillations of a cavity), heat conduction, viscosity, compressibility and surface tension. Also there is mass transfer by diffusion (although its effect is not very marked in cavity dynamics) and finally there is temperature discontinuity at the phase interface.

Two comprehensive accounts of these topics have been given by Oldenziel (1979) and by Godefroy and Oldenziel(l981).

4.2. Ultrasonic work

This section will be kept brief for two reasons. The first is the fact that a great deal has already been written about ultrasonics (in both papers and books) (e.g. Flynn 1964) and

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it would therefore be superfluous to write at length here. The second is that various applications of ultrasonics will be described in 5 of the present article.

The processes occurring in ultrasonic stressing are briefly as follows. When a high- frequency sound wave is passed through a liquid the latter is subjected to a compression and a tension during the positive and negative half-periods, respectively, of the pressure cycle. One would therefore expect cavitation to occur during the negative half-cycle and this is found to be the case provided the wave amplitude is sufficiently high (that is, a tension of sufficient magnitude is generated). During the next compressional half-cycle the cavities collapse (partially) and re-grow in the next rarefaction half-cycle. This steady growth and collapse of cavities gives rise to a continuous hissing sound called cavitation noise. We now summarise some recent papers.

Lauterborn and his colleagues have described their recent work on the spectra of cavitation noise (Lauterborn and Cramer 1981a, b , 1982). Their spectra consisted of sharp lines on a noise background. Both lines and background were produced by a cloud of oscillating bubbles generated and sustained by the drivingsound field. The mechanism needing explanation was the one by which the energy in the driving sound wave of frequencyfo is transformed into the different frequency components of the spectrum. It was found that the frequencies of the lines in the spectrum were related arithmetically to fo in that they consisted of its harmonics nfo (n = 2.3, , . .) and its subharmonics fo/m ( m = 2,3, . . .) and their harmonics. The harmonics nfo were explained in terms of forced nonlintar bubble oscillations and the noise background was attributed to the shock waves emitted by collapsing bubbles. A satisfactory explanation for the presence of the subharmonics was, however, not known.

The method used by Lauterborn and his colleagues to set up their cavitation bubble field was to generate a high-intensity acoustic field in water. Instead of using a sound beam focused into a point, as in laser work (see 4.3), they used a sound field produced by a hollow cylinder of piezoelectric material (PZT-4) which was totally submerged in the water. The cylinder had a resonance frequency of 23.56 kHz and when driven in its first radial mode the maximum sound pressure was generated along its axis. At sufficient intensity, streamers of tiny bubbles moving towards the axis appeared and these coalesced on the axis to form a smaller number of much larger bubbles. The sound emitted by the bubble field was recorded by a microphone immersed in the water. Both the generation of the bubble field and the recording of the cavitation noise were con- trolled by a minicomputer. The test liquid was ordinary tap-water left standing for a few hours so as to allow the removal of large bubbles. The transducer cylinder was driven at resonance Cfo = 23.56 kHz) and the voltage driving it was increased at a steady rate from zero to some maximum value. In one case this voltage was increased steadily from 0 to 60 V over 120 ms and thereafter kept constant for the rest of the experiment. It was found that there were only two thresholds for the onset of broad-band noise (20 and 55 V). As the voltage increased from zero the spectrum first exhibited five definite lines (nfo, n = 1 to 5 ) and there was no noise continuum. Just before the voltage reached the first threshold (20 V) a noise spectrum appeared in a small frequency range between 70 and 130 kHz. Above this first threshold the spectrum reverted to being definite lines again, their frequencies now beingfo/2, fo, 3fd2, 2f0, 5fd2 etc. Later, just below the second threshold at 55 V the harmonics nfo (n = 2 , 3 , . . .) became relatively stronger. Finally, at the fully developed noise cavitation, noise again occurred in addition to the distinct lines; this noise occurred mainly between the frequencies 100 and 200 kHz and near the line 3fd2. In fact the line 3fd2 was always the strongest line in the fully developed cavitation noise spectrum.

2150 D H Trevena

In this type of spectrum the sharp lines and the broad-band noise are clearly related to the dynamical behaviour of the cloud of cavitation bubbles. But to find a satisfactory model in terms of such a large number of bubbles is clearly a formidable task. As discussed elsewhere in the present paper, work has been done on a single spherical bubble subjected to a sinusoidal driving pressure of increasing amplitude. Such a math- ematical model is a highly nonlinear differential equation of second order for the bubble radius as a function of time; it includes the effects of surface tension, viscosity and the compressibility of the liquid (Cramer 1980). A still more comprehensive model needs to be developed which will include the interaction of the bubbles, that is, the fact that they all couple via their sound radiation. This theory must also explain why the ampli- tudes of the lines in the spectrum change as the driving pressure changes: some decrease while others increase so that energy flows back and forwards between the lines. Lauter- born and Cramer (1982) end their paper thus: A complete theory of acoustic cavitation noise spectra demands a self-consistent set of equations where the noise output also acts as input to drive the bubbles. We are far from achieving this goal.

4.3. Laser work When a test liquid is irradiated by a focused high-power laser there is a large injection of energy into the liquid at the focal point. This intense concentration of energy at this point usually produces one large cavity which grows and later collapses. There are, in addition. small solid impurities (motes) and also dissolved gases in the focal region and these absorb some of the injected energy. So liquid breakdown, that is, the production of cavities in the liquid, can occur as a result of (a) the laser energy entering the focal point and (6) the energy absorbed by the impurities which then become hot spots on which these cavities can form. There is considerable evidence that these hot spots play a big role in the initial liquid breakdown. A cavity is formed around each spot as centre and this cavity then grows rapidly at first; later its radial growth is slowed down and during this deceleration a rarefaction wave travels out of the centre. This results in a tension field in the focal region, which, in turn, will cause further cavities to form. These tension-induced bubbles are the more familiar or true type of cavitation bubble. Filtered and unfiltered tap-water has been used in laser experiments and it is found that the filtered water has considerably fewer hot spots.

Lauterborn (1972, 1974) initially used high-speed photography to study laser- induced breakdown in liquids. In this work intense light from a ruby laser was brought to a focus at some interior point in the test liquid (both water and silicone oils were used). The laser energy was injected into this point as a pulse of duration 3C-50 ns. High-speed photography, using frame rates of up to 850 000 s-l, was used to study the formation, growth, collapse and rebound of cavities. Although this work yielded a good deal of information about cavitation bubble dynamics, a major step forward was made when high-speed holography, rather thanphotography, was employed (Lauterborn and Ebel- ing 1977). Two ruby lasers were used, one for producing the breakdown and one for holography. The laser for breakdown emitted pulses of 30-50 ns duration and energy 0.1 J . The advantage of holography over photography is this. A great deal of white light is emitted during the breakdown process and this makes for difficulties with ordinary photography. However, when holography is used to record the breakdown, this light constitutes an incoherent background on the holographic plate and does not appear on the reconstruction of the events in the liquid. This makes it far easier to observe simultaneously the bubble formation and the accompanying wave emission.

Lauterborn and his colleagues have since investigated laser breakdown in great detail

Cavitation 2151

and are probably the best-known research school in this field. Lauterborn (1979) intro- duced the term 'optic cavitation' for the formation of cavities in a liquid by light. Haussmann and Lauterborn (1980) describe a method for digital 3-D image processing from hologram reconstructions; in this way they analysed automatically fast moving bubble fields.

Hentschel and Lauterborn (1982) report on studies on the dynamics of laser-pro- duced single bubbles and the sound and pressure waves radiated by them. For this purpose they used a microphone to monitor the (pressure, time) curves. This microphone was placed at a distance of 10-20 mm from the point of breakdown so that it picked up the sound emitted by the cavity. A typical (pressure, time) curve showed that a strong pressure wave was radiated on breakdown; there were pressure waves of progressively lower strengths radiated at each successive cavity minimum. From their data they were able to calculate the energy lost during one cycle of a cavity oscillation (which is the difference in potential energy between two successive maxima) and they showed that the acoustically radiated energy was only a small part of this energy loss. They concluded that other significant factors must be present. They state that: 'It is conjectured that in addition to the known damping mechanisms of heat conduction and diffusion, damping may also be provided by the nonsphericity of the collapse'. This is an important observation.

4 . 4 . Water shock tube experiments Japanese workers have produced some very elegant work in this field (Fujikawa and Akamatsu 1978, 1980a, b). Their experimental method, in which they injected a tension pulse directly into the top of a column of water, has already been described in Q 3.3. This down-going tension wave was reflected as a tension wave at the closed lower end of the tube. This up-going tension wave was subsequently reflected at the water-gas interface as a compressional wave, and so on.

Studies were made using bubbles produced by electrolysis. For this purpose platinum electrodes situated at the bottom of the tube were used to produce a single hydrogen bubble, of radius 0.15 mm, and this bubble then rose through the water with a velocity of about 15 mm S - ' . When the bubble reached a suitable height an electromagnetic plunger broke the diaphragm and the tension wave entered the top of the water column. The bubble grew under this tension wave to some maximum radius and then collapsed under the following compressional wave. This growth, collapse and rebound of bubbles at various horizontal distances from the wall of the tube was studied by means of high- speed photography and in-line Fraunhofer holography using a pulsed dye laser.

The growth of a single bubble situated relatively far from the tube wall is shown in figure 4, which is the normalised (radius, time) curve for the bubble. The bubble grew to its maximum radius under the tension wave and then collapsed rapidly under the succeeding compressional wave. A second situation which they considered was that of a bubble relatively near to the tube wall. They found that the nearer the bubble was to the wall, it grew and collapsed in increasingly asymmetric forms. During growth, the bubble became elongated in a direction parallel to the solid boundary, but on later collapse it became elongated in a direction normal to the boundary. Also, as this growth-collapse behaviour proceeded the bubble moved nearer to the boundary and formed a water-jet. Thirdly, they studied the behaviour of twin bubbles during collapse and rebound. They found that the density of water in between the bubbles changed due to the collision of the spherical shock waves radiated from the two bubbles. Sometimes the twin bubbles formed water-jets directed towards each other.

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JCompress!onal wave 1 0 -

4p Q

0 5 - Tension wave

0 ( Jl 1 2 Time 3 Imsl 4 Figure4. Normalised (radius, time) curve for a bubble (after Fujikawa and Akamatsu 1978).

Perhaps the most important result reported in these three papers is the light shed on how cavitation damage on a surface occurs. Previously, two explanations for such damage had been put forward. One is that small liquid jets, formed on bubbles, impinge on the surface. The second explanation is that a shock wave is radiated into the liquid when the inward, collapsing motion of a bubble is arrested and that this impulsive pressure is the one responsible for cavitation damage. The work in these three papers suggests that it is the second explanation which is the valid one.

Another similar water shock tube arrangement has been described by Matsumoto and Shirakura (1980) in which a heated wire was used to produce ready made bubbles. This wire was a thick platinum wire near the bottom of the water column; it was electrically heated and its temperature could be varied. This heating caused weak spots to form in the water around the wire. When the diaphragm was ruptured as before a tension wave was generated and was propagated down the water column. Under the influence of this tension wave cavitation bubbles which formed at these weak spots grew and their subsequent dynamic behaviour was studied using high-speed photography. In this work the gas content of the water was varied between 1.4 PPM and 15.0 PPM and the initial temperature of the wire varied from 30 to 100 C. An interesting result observed was this: if no bubble had been formed around the wire when the tension wave first arrived, the water actually sustained a transitory tension of about 0.1 MPa (1 atm). This tensile strength occurred when both the gas content of the water and the wire temper- ature were low; it did not occur when either this gas content or the wire temperature was high.

The photographic results were also quite spectacular. With low gas content the bubbles appeared in the growth phase as a string of pearls on the wire. When the gas content was high the bubbles coalesced and grew to become one long cylindrical bubble whose axis was the wire itself. The corresponding collapse phases also showed differ- ences. The importance and role of the gas content of the water are clearly brought out in this work.

4.5. Mathematical theory of bubble dynamics In the search for a mathematical model of a cavitation bubble it is usually assumed that such a bubble starts as avapour or gas nucleus already present in the liquid. Subsequently, if the liquid is subjected to a tension, the bubble will grow to some maximum radius and collapse rapidly; it may then rebound and repeat the cycle a number of times. If, instead, the liquid is stressed ultrasonically the nucleus may pulsate linearly about its equilibrium radius or it may oscillate in a nonlinear motion.

We first consider the so-called Rayleigh cavity-the simplest model of all, but one which is still used a great deal. Once the spherical cavity has been created it starts to

Cavitation 2153

collapse. Rayleigh obtained the differential equation governing the radius R(t) of this cavity for given values of the pressurePo at infinity and the pressurepi inside the cavity. In this case p0 is the ambient pressure for the cavity and is equal to the pressure in the liquid (assumed to be incompressible) at the position of the cavity and in the absence of the cavity. pi is taken as the vapour pressurep, at all stages of the collapse (except when R is very small). In Rayleighs model the spherical cavity collapses from some maximum initial radius R, from rest with ( p o - p,) remaining constant. Rayleigh obtained the

for the speed of the interface at a smaller radius R; p is the density of the liquid. From this equation the time taken for the cavity to collapse completely turns out to be

the Rayleigh collapse time. The non-dimensional relationship between t/tand R/R, has been found to agree well

with experimental observations of a collapsing cavity. Let us, however, repeat that the Rayleigh model assumes that the cavity contains

vapour only and that the vapour pressure remains constant during all stages of the collapse. We now look further into these two assumptions which represent a grossly oversimplified picture of the true state of affairs.

In most cases the contents of the bubble will be a mixture of gas and vapour and. as the bubble collapses rapidly the constant internal pressure assumption is hardly valid. To make progress, let us first assume that we have a single bubble, containing vapour only, at its maximum radius. As the bubble subsequently collapses, the condensation process sets in and, at a later stage, the vapour fails to condense rapidly enough and thus becomes compressed as the bubble continues to contract. In other words, the vapour will behave like a non-condensable gas (a non-equilibrium effect). The elastic properties of both this non-condensable gas and water, together with the inertia of the water, provide the necessary conditions for an oscillatory system. The compression of this gas causes a bounce or a cushioning effect until the internal gas pressure reverses the sign of dR/dt at the minimum bubble radius. Thereafter the bubble grows or rebounds and at the same time a pressure pulse is radiated into the liquid. If we further assume that the bubble contains gas (air) as well as vapour then at the minimum bubble radius, the presence of such gas will enhance the cushioning effect just described.

So, clearly, Rayleighs original model must be modified to take account of ali sorts of extra factors. Such modifications have been undertaken by various workers, each adding various new factors into the simple Rayleigh model. It is impossible to mention all these modified attempts at describing bubble dynamics, but the paper by Fujikawa and Akamatsu (1980b) contains a good example of the modified approach to the problem. Their model takes into account the effects of compressibility of the liquid, non-equilib- rium condensation of the vapour, heat conduction inside the bubble and in the surround- ing liquid and the temperature discontinuity at the phase interface.

They formulate the problem by taking a spherical bubble of initial radius R. containing both vapour and non-condensable gas in a viscous, compressible liquid. At time t = 0 they consider the ambient pressure to be raised to some value plf; subsequently the bubble begins to collapse and this is accompanied by phase change (condensation) and heat conduction through the bubble wall. Before they write down their basic equations

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the following assumptions are made:

(1) The bubble always remains spherical. (2) Liquid compressibility and viscosity do not affect each other. (3) Gravity and diffusion effects are negligible. (4) The pressure is uniform throughout the inside of the bubble. (5) The vapour and gas in the bubble are inviscid and obey the perfect gas law. (6) The temperatures of this vapour and gas are equal. (7) The thermal boundary layers both inside and outside the bubble are thin com-

(8) There is a thin but finite non-equilibrium region at the phase interface because

(9) The physical properties of the liquid and gases are constant.

pared with the bubble radius.

of the continual change of phase there.

All this is a far cry from Rayleigh's simple model! On the basis of these assumptions the authors derive three sets of equations for : ( a )

the external region occupied by the liquid; (6) the inside of the bubble occupied by the mixture of vapour and gas; and (c) the phase interface. The solution of these equations is a rather formidable and lengthy procedure and details may be found in the original paper. The main results obtained may be summarised as follows. It was found that a pressure wave was radiated into the liquid at the instant of rebound of the collapsing bubble; the results show that such a pressure wave would also occur at the rebound of a bubble which includes vapour only. It was also possible to obtain histories of the bubble radius. the bubble wall velocity, the temperatures of the bubble contents and the surrounding liquid, and the vapour pressure and gas pressure inside the bubble. The radial pressure distributions in the liquid outside the bubble at various times were also calculated.

It is also necessary for us to consider briefly the motion of a single spherical cavity under the influence of a sound field. The cavity usually pulsates and various differential equations have been used to describe the motion of such a cavity. The wavelength of the sound is always assumed to be large compared with the cavity radius. In the differential equations the effects of various factors are taken into account: these are heat conduction. viscosity, sound radiation, compressibility and surface tension. When the effects of heat conduction, sound radiation and viscosity are included the solutions describe so-called dissipative motion. These three factors cause irreversible transformations of energy and these lead to damped motions of the bubble. It is found that thermal conduction is by far the largest of these three dissipative effects. For a further discussion the reader is referred to an article by Flynn (1964).

The dynamics of cavity clusters (as distinct from that of a single cavity) has been considered by Hansson er a1 (1982). They discuss the case of water containing air in the form of microbubbles, their initial radii ranging from to lo-' m and the volume concentration of free gas from loT6 to In other words they used a two-phase mathematical model-a bubbly liquid-rather than use a one-phase model (that is. the liquid only). They were able to calculate the pressure and cavitation development in such a 'real' liquid in two experimental situations, namely, when the liquid was (a ) between a vibrating horn and a stationary solid surface placed at a small distance from the horn and (6) in a tube which was accelerated by axial impact on the tube. Both of these experimental situations are essentially one-dimensional systems. The results of the analysis show that the two-phase model explains satisfactorily the formation and collapse of cavity clusters in real liquids.

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5. Cavitation erosion

5.1. Cavitation damage and cavitation erosion When cavitation bubbles collapse near the surface of a solid in contact with the liquid, the surface is damaged. This process is referred to as cavitation damage or cavitation erosion. It is useful, however, to make a distinction between the two terms, although we shall probably not always be consistent in this respect. For example, consider the tube-arrest method described in 9 3.2. It was often found that the bubble collapse would actually crack the tube containing the liquid and this somewhat catastrophic event is clearly cavitation damage. On the other hand in both cavitation tunnel and vibratory cavitation tests the test surface i s gradually eaten away or pitted as time proceeds. This is what we shall mean by cavitation erosion.

5.2. Water tunnel work

Some interesting work on the cavitation erosion which occurs in various hydrodynamic flow situations has been reported by Selim and Hutton (1981). The water tunnel used was of the small, variable pressure, closed circuit type. The rectangular cross-section of the parallel-sided test section was 20 X 40 mm and the water was circulated by a 19 kW centrifugal pump and bypass control to give velocities varying between 15 and 45 m S - . The pressure could be varied independently by a pneumatic controller over a range of 0 to 10 bar and the total air content of the water was measured with a van Slyke apparatus (Williams 1954).

Experiments were performed with three shapes of body spanning the 20 mm direction (see figure 5 ) . In figure 5 ( a ) a two-dimensional Venturi-liner is shown; in figure 5(b) we have a circular cylinder and in figure 5(c) a 60 wedge with its apex upstream. The dimensions of these three bodies were such that there was the same throat area in each of the three cases. Hence, for the same flow-rate, the three throat velocities were also the same. The authors define a local cavitation number, U. for the flow at the throat as

U = - P -P. t p U 2

where p , U are respectively the static pressure and mean velocity at the throat, p v the vapour pressure corresponding to the bulk water temperature and p the density of water. In this work it was difficult to measure U directly and so it was calculated from the upstream cavitation number a0 which was directly measurable. This was defined as

wherepo, U0 are respectively the pressure and mean velocity upstream of the body. The specimen tested for erosion consisted of a rectangular plate mounted on the

side-wall downstream of the throat. In most cases the specimen was made from 99% pure aluminium; in one case cast iron was used. The aim of the work was to study the cavitation erosion caused on a specimen by various flow situations. The erosion was assessed by measuring the progressive loss in weight of the specimen plate. This plate was weighed initially and thereafter after every hour or so of exposure to cavitation. The weight loss rate (WLR) was then A W / ( T - TO), where AW is the weight lost by a time T after beginning the tests (the exposure time) and TO is the so-called incubation time during which there is no measurable weight loss. This incubation time has been previously discussed by Trevena (1982).

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IC I \ S Idem1 I speclrnen

Figure 5 . The three shapes of body used in the water tunnel: ( a ) a two-dimensional Venturi-liner: ( b ) circular cylinder; (c) 60" wedge with apex upstream (after Selim and Hutton 1981).

In the actual experiments, weight loss measurements were carried out not only on the side-wall plates but also on the three geometrical bodies shown in figure 5 . Two main types of measurement were made, namely, the variation of the WLR (a) with velocity at constant a and ( b ) with a at constant velocity. In case (a ) the general trend was for the side-wall erosion to be greater for the same upstream cavitation conditions, Furthermore the variation of WLR with velocity followed a power law, that is

WLR X U".

The value of n , for aluminium, varied between 3.07 (for the Venturi-liner) and 7.13 (for the cylinder). In case ( b ) the situation was quite different and there were no clear trends such as power law relations. The WLR rose as a decreased and reached a maximum somewhere between a = 0.025 and U = 0.055; thereafter it decreased as adecreased.

It is also interesting to compare the type of break-up for the pure aluminium specimen with that for the grey cast iron specimen. In the case of aluminium a pit or crater of roughly hemispherical shape with a raised lip would be formed, showing clearly the plastic deformation caused by a collapsing bubble. Once two of these lips overlapped, the material common to both tended to be extruded in a thin sliver which stood out from the plate's surface. These slivers then broke off, probably due to flow forces, and thus produced weight loss in the specimen. Prior to that there had been considerable plastic deformation and changes in the surface but with no measurable weight loss. For the cast iron the method of break-up was quite different. Electron microscopy revealed an initial removal of graphite flakes and lumps, thus leaving holes in the surface. At a later stage, grains were removed by cracking around the grain boundaries. The surface was not raised as in the case of the aluminium.

Lush (1979) also studied the surface deformation produced on aluminium by a cavitating flow. For this purpose a series of aluminium plates was placed in the side-wall of a cavitation tunnel and subjected to the cavitating flow produced in a convergent- divergent section similar to that in figure 5(a) . The resulting surface deformation was analysed using a Talysurf surface finish measuring system. The results were compared with those predicted by a theoretical model for the formation of a single cavitation pit by the microjet produced by a collapsing bubble. This model predicted a threshold velocity below which no pitting can occur and this predicted velocity agreed closely with experiment. In other experiments using a cavitation tunnel Lush et a1 (1979) investigated the relation between cavitation noise and erosion.

Cavitation 2157

5.3. Vibratory cavitation erosion techniques

In 1963 a standard technique for vibratory cavitation testing was produced at the UK National Engineering Laboratory (NEL) and subsequent methods have, very largely, been a development of this technique. At that time, this technique involved the use of a 20 kHz water-cooled, magnetostrictive transducer driven by a 500 W ultrasonic gen- erator. The vibrating unit had a peak-to-peak amplitude of 50 pm and was situated above a disc-shaped test specimen. The tip of the vibrating unit and the test-piece specimen were immersed in distilled water and the acoustic pressure field generated resulted in the formation of a cavitation bubble field in the film separating the vibrating tip from the specimen. In some arrangements the specimen itself was caused to vibrate as well; in others it was stationary. Again, in the earlier work, the liquid was contained in an open beaker but, more recently, a closed container has been used. The advantage of this closed container, as compared with the open beaker, is that it enables the effects of elevated temperatures and pressures, and also those of dissolved gas content, to be studied. An excellent summary of the development and standardisation of this type of erosion testing has been given by Hobbs (1976) of the NEL. The ultrasonic vibrator is now an established tool of cavitation erosion research and has the great advantage that a large amount of data can be obtained relatively quickly for a wide range of liquids, materials and test conditions.

Some interesting erosion tests using this technique have been described by Singer and Harvey (1979a). The essential part of the apparatus is shown in figure 6. The vibrator used was a 150 W ultrasonic drill unit with a stellite tip attached to the free end of the horn driven at 20 kHz with a peak-to-peak amplitude of 50 pm. The upper 'test' surface of the specimen was at a separation distance of about 1 mm away from the tip. Both specimen and the end of the horn were immersed in an open bath containing about 3 litres of distilled water whose temperature was controlled by a heat exchanger; a stirrer (not shown) was used to maintain a uniform bulk temperature within the bath. The cavitation cloud was generated in the film of liquid separating the stellite tip and the specimen. The gas content was measured with a van Slyke apparatus.

The authors examined the erosion produced on stationary specimens of annealed high conductivity copper. The weight losses were measured using an analytical balance. Various types of test were carried out. Firstly, tests were performed at varying gas content levels and it was found that the maximum rate for the copper increased from 0.73 to 4.35 mil h" as the gas content rose from 7.5 to 18.4 m1 litre" (1 mil = 25.4 pm). Photographic studies and surface measurements showed also that the form of the erosion pattern changed with gas content. Secondly, tests were made to ascertain the effect of the mean bulk temperature on the erosion rate for temperatures between 17 and 75 "C. For temperatures up to 50 "C this rate was not very sensitive to temperature while for temperatures above 50 "C it decreased quite markedly with temperature. Furthermore. as the temperature increased, a central portion of the eroded area of the specimen became less damaged; this effect became more pronounced with increasing temperature and it was also confirmed by the surface measurements. Figure 7 of the original paper is reproduced in figure 7 (plate).

Singer and Harvey (1979b) also reported some erosion tests using stationary Plasti- cine specimens and it was found that this resulted in several deep pits being formed on the surface. Furthermore, where there was a scratch mark on the smoothed Plasticine surface this would, after exposure to cavitation, result in a string of pits within the scratch. A similar effect had been previously described by Brunton (1970). In a later

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Stell lte . t l P

Ccvltatlar c ! o u d

T e s t s p e c i m e n

Distilled water

Figure 6 . Apparatus for vibratory erosion tests (after Singer and Harvey 1979a).

paper Singer and Harvey (1981) describe further work on the occurrence of this string of pits which occur on these surface scratches in Plasticine. The apparatus was the same as that shown in figure 7 except that the liquid was tap-water whose gas content was about 18.5 m1 litre-. To obtain a Plasticine specimen a piece of brass was bored out and filled with Plasticine. The excess was removed with a straight edge so that it was flush with the surface of the brass and then, where appropriate, lightly smoothed with the finger. In one set of experiments Plasticine specimens, in which only half the surface had been smoothed, were used. After being subjected to cavitation for 30 S it was found that the smoothed area was virtually undamaged while the unsmoothed one was severely damaged. A closer examination of the unsmoothed surface, prior to cavitation, showed that this surface had a sponge-like matrix appearance. The authors suggest that this porous matrix had been an air-trap from which bubbles became detached from the surface in the accoustic pressure field with the subsequent formation of microjets and their resulting damage to the surface. This explanation was further supported by the fact that whenspecimens were prepared, as before, with bothsmoothed andunsmoothed regions and then placed in boiling water for some 3 min so that the surface voids had been degassed, it was found that, after exposure to cavitation, the unsmoothed area did not show a large number of pits. This work is, in many ways, not unlike that of Overton and Trevena (1980) on the role of surface nucleation sites on the inner wall of a Berthelot tube.

Research carried out by the UK National Coal Board (NCB) has been concerned with the cavitation erosion produced by fire-resistant hydraulic fluids. The two types of fluid most commonly used are water-containing emulsions. One type is a dilute emulsion of about 5 % oil in water and the other an invert emulsion of about 40% water in oil. Work on cavitation erosion in the underground equipment used by the NCB has shown that this erosion occurs more in equipment using dilute emulsions than in that using invert emulsions. Talks (1983) has reported on work in which a comparison was made between the cavitation erosion properties of the two types of emulsion and those of the two constituents (mineral oil and water) which made up these emulsions. The test material chosen was the type of brass used in some of the underground equipment. The method employed was the open beaker vibratory test method using a 20 kHz vibratory cavitation apparatus. It was found that the addition of up to 10% of emulsifying oil to distilled water had no effect on erosion rates while similar additions to hard water

Figure7. Photographsand surface measurementsof the erosion patterns obtained for copper stationary specimens at different bulk temperatures (after Singer and Harvey 1979a) After 1 h exposure: (a ) 25 "C; ( b ) 45 "C; (c) 75 "C.

[facing page 2158)

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reduced rates by up to one-third. Furthermore, the erosion rates for the dilute emulsion varied with temperature in a manner similar to that of water while the invert emulsion behaved like a mineral oil.

5.4. Erosion testing with a cavitating jet The pioneering work using this technique has been carried out by Lichtarowicz (1972) and his colleages. The method, which is now an established one for the erosion testing of minerals, is briefly as follows. A cavitating jet is supplied from a constant pressure source (pressure = p u ) through a long orifice type nozzle which discharges into a test cell where the ambient pressure is kept at some value Pd (the downstream pressure). A circular cylindrical specimen (the target) is mounted coaxially with the jet and the separation l between the nozzle entry edge and the target can be set to any desired value. The cavitation can be observed and photographed through windows on two opposite sides of the test chamber.

Under these conditions the flow rate depends only on the upstream pressure p u and Lichtarowicz (1979) uses a cavitation number a = pd/pu which. so defined, is the ratio of the forces suppressing cavitation to those producing i t . The target specimen erodes as a result of the collapse of the cavitation bubbles in the incident jet and this erosion is quantified by measuring the weight lost in a given time. The results can be presented as a graph of the cumulative erosion rate (CER) versus time: CER is defined as the total weight loss divided by total elapsed time (Lichtarowicz 1981). Tests carried out at constant cavitation number at the value of I at which CER is a maximum show that this peak CER p [ . where the index n 4 for hydraulic oil as test liquid. As aincreases. n decreases. For water there is a similar relation between peak CER and pu. with n decreasing from 5 to 3 as awas increased (Lichtarowicz and Kay 1983).

Using high-speed flash photographs, Lichtarowicz (1981) found that the cavitation occurs in bursts which appear like separate clouds travelling with the jet. Furthermore. a vortex filament. looking like a corkscrew, can be seen around the jet. This spiralling vortex core is present both near the nozzle and near the target specimen.

This jet method has many advantages over some of the other methods used for erosion testing. The apparatus is small and uses flow effects to produce cavitation; it thus offers all the advantages of venturi-type and tunnel methods without their main drawbacks of size and long testing times. In the jet method the testing times can readily be adjusted by choosing a suitable upstream pressure and the results can then be scaled up or down easily provided the cavitation number is kept constant.

5.5. Experiments with spark-induced bubbles

Lush et a1 (1983) have studied the effect of a single bubble collapse on the surface of a test specimen of 99% pure aluminium. Such a single bubble was produced by discharging a 0.01 ,uF capacitor at a potential of 15 kV between two tungsten electrodes which were submerged in a tank of deionised water. The spark gap between the tips of the two electrodes was 1 mm and the energy in the spark was about 1 J . The bubble produced was not a true cavitation bubble, that is, a bubble produced by tension in the liquid, but a ready made one created by the discharge of the spark. (Other examples of ready- made bubbles were described in 4.4 of the present paper). When a true cavitation bubble subsequently collapses near the surface of the solid a microjet is formed and a shock wave is later produced when the cavity rebounds. However, in the case of a cavity produced by a spark, an additional shock wave (also produced by the spark) is radiated

2160 D H Trevena

into the surrounding liquid ahead of the expanding cavity. The effect of this extra initial shock wave is to cause extra cavitation damage to the surface.

The aluminium material was chosen as the test specimen because it has suitable plastic flow stress properties and is fairly soft and should therefore show a large amount of damage. In fact the damage in this case appeared as a pit or pits and these were of two types. One type consisted of a relatively shallow pit and the other was considerably deeper. The profiles of these pits were obtained by photographing them under an interference microscope, which utilised the green line of mercury? and the maximum depth of each pit was found by counting the fringes. The authors also give a theoretical prediction of the type of damage to be expected on a surface by ( a ) a microjet and ( b ) the shock wave from the collapse centre. As a result of this theory it was concluded that the deeper type of pit was caused by the microjet formed when the bubble collapses while the shallow type was due to the shock wave radiated from the spark.

Earlier spark gap work by Jones and Edwards (1960) had been concerned with the effect of the collapse of a single cavity in water on a duralumin surface. The changes in the dimensions of the cavity were also studied by schlieren and spark shadow photography.

5.6. Cavitation damage in large concrete structures

We now turn from laboratory conditions to consider cavitation damage to concrete elements in large hydraulic structures such as hydroelectric power stations and similar installations. Two papers, one by Kenn and Garrod (1981) and the other by Kenn (1983) contain an account of the Tarbela Tunnel collapse of 1974.

The cavitation damage in these situations is attributed to the presence of severely- sheared water flows within the hydraulic structures and its seems that the onset of this damage starts when the velocity differential across the sheared flow reaches 30 m S- or so (the threshold velocity). Above this threshold value the intensity of the damage increases very severely. Such sheared flows are produced when submerged high-velocity water streams, generated by high pressure heads, separate from solid boundaries- particularly discontinuities in these boundaries such as buttress walls, partly-opened gates, construction joints and any hollows and protuberances. Strong eddies are gen- erated along the planes of shear, and pressures inside these eddies can fall to the cavitation threshold value with the production of clouds of tiny vapour cavities. The eddies and cavities are then swept downstream into regions of higher local pressure where the cavities collapse and disappear. If they collapse near a solid boundary the cavitation damage to the solid as a result of a large number of such bubble collapses can be very severe indeed.

Cavitation damage from sheared flows occurred in tunnels at Tarbela in 1974. In one tunnel the centre intake gate was fully opened while both side gates were shut, the flow being under high head and very high water velocities. Vorticity-induced cavitation was generated in the two planes of sheared water flows leaving the inner walls of the adjacent piers. Many of these cavities later collapsed in the regions of higher pressure further downstream, thus causingerosion of the concrete tunnel walls. As this damage continued a hole was eventually formed ir the tunnel lining; later this hole became enlarged and this led to a collapse of the tunnel. For a full account of these events reference should be made to the two papers referred to above.

These field-situation patterns of sheared-flow cavitation have also been reproduced and studied using small-scale models in the laboratory (Kenn 1983). All such small-scale

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models are subject to errors (scale effects) since factors such as viscous forces, surface tension, elastic forces and air-content will, of necessity, be incorrectly scaled. Even so, a small-scale model, if tested in a water tunnel at full-scale heads and velocities, can predict the patterns of cavitation and the cavitation erosion to be expected in the full- scale structure. Kenn used a model of the Tarbela intake tunnel, tested under full-scale pressures and velocities and with the centre gate open and both side gates shut. He was able to reproduce the cavitating eddies generated in the highly-sheared zones between the central water flow and the adjacent, relatively still water on either side. Furthermore the pattern of the cavitation damage in the model bore a strong resemblance to that in the actual Tarbela tunnel.

6. Other related work

We now consider some related work in the field of cavitation. Firstly, let us take tribonucleation (nucleation by rubbing). This is the process of

initiating cavitation by rubbing together two solid surfaces inside a liquid under tension. Some experiments on the tribonucleation of bubbles in liquids have been described by Hayward (1967). In this work a previously denucleated liquid (that is, one free of gaseous cavitation nuclei) was contained under a tension of 0.15 atm in a glass test chamber. A small magnet was held against the upper wall of this chamber by a second magnet clamped to the outside. The tension in the liquid was maintained as long as the first magnet was held immobile but as soon as it was moved by the smallest possible distance (< 1 mm) a bubble was nucleated. Another example of the nucleation of bubbles by gentle rubbing was obtained by rolling a very small steel ball slowly over the horizontal glass base of the same test chamber. This very slow rolling did not produce nucleation but. as soon as the speed of rolling was increased, a bubble was nucleated because some sliding then occurred as well as rolling.

The occurrence of tribonucleation is one factor responsible for cavitation in hydro- machinery. The liquid in such a machine will cavitate wherever one working part rubs on another; for example, where a shaft passes through a seal.

Secondly, we consider the occurrence of cavitation in lubricating films in bearings. This matter has been studied extensively by Dowson and his colleagues and a compre- hensive account has been given by Dowson and Taylor (1979). In such lubricating films both gaseous and vaporous cavitation can occur but the former type is the more common. The authors make the point that many machine elements present to the lubricating film a clearance space which has a convergent-divergent section. Such machine elements include journal bearings, gears and rolling-element bearings. Osborne Reynolds showed that, under conditions of slow viscous flow, a wedge-shaped lubricant film with a slight convergence in the direction of motion was essential if load-supporting pressures were to be generated in the lubricant. The necessary convergent part of the clearance space is. as already stated, present in most machine elements. In the accompanying divergent part of the clearance space subambient pressures are developed and these pressures give rise to cavitation. The authors describe and illustrate the appearance of cavities in four different forms of sliding bearing. In some cases the gas present split up into a number of discrete finger-shaped bubbles separated by narrow continuous streams of oil. A photograph is also given of a beautiful flower-like gas-filled cavity surrounded by oil between a separating steel sphere and plane. For further details reference should be made to the original paper.

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A finger-joint is a biological bearing and this topic has been discussed by Dowson er a1 (1971). In their work, load-separation curves were obtained for the metacarpo- phalangeal joint of the middle finger (a ball and socket joint containing synovial fluid). A special machine, described in the paper. was designed and constructed to obtain these curves. The load was increased in steps and an x-ray exposure taken at each separate load value. It was found that, as the load increased, the bone separation at first increased gradually and fairly linearly until, at a certain load, a sudden jump in the separation took place at the instant of cracking of the joint. Furthermore, at the same instant a dark bubble in the synovial fluid appeared on the x-ray plate. The joint separation then gradually returned to its pre-cracking value in about 20 min. If the load was next increased in steps as before another crack would then be produced, but not before this 20 min period had elapsed. The authors conclude that joints therefore crack because of cavitation in the synovial fluid, while the failure of some joints to crack is due to either lax ligaments or non-conforming joint surfaces.

Thirdly. in botany, we know that columns of sap transport water from the roots of a tree to the uppermost leaves. The atmosphere will support a column of water of only 10.4 m in height and, since some trees are far taller than this, the only way in which sap can be drawn up to these greater heights is by the existence of a negative pressure in the sap column. Some recent work has shown that negative pressures of -50 atm exist in mangrove trees; even higher pressures of up to -80 atm have been measured in desert plants in their efforts to suck up every drop of water from their dry environment. Not all the vessels which carry these columns are filled with water. When under tension some of the columns of water tend to break and cavitation, in the form of air and water vapour, occurs. A sensitive microphonic probe can detect this cavitation acoustically. When a column of water breaks, the walls of the vessels, which have been pulled inwards by the tension, relax and tend to vibrate. These vibrations, when amplif