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The Dynamics and Acoustics of Travelling Bubble Cavitation S. Ceccio, C. Brennen (California Institute of Technology, USA) ABSTRACT Individual travelling cavitation bubbles gener- ated on two axisymmetric headforms were detected using a surface electrode probe. The growth and col- lapse of the bubbles were studied photographically, and these observations are related to the pressure fields and viscous flow patterns associated with each head- form. Measurements of the acoustic impulse generated by the bubble collapse are analyzed and found to cor- relate with the maximum volume of the bubble for each headform. These results are compared to the ob- served bubble dynamics and numerical solutions of the Rayleigh- Plesset equation. Finally, the cavitation nu- clei flux was measured and predicted cavitation event rates and bubble maximum size distributions are com- pared with the measurements of these quantities. 1. INTRODUCTION Though the dynamics and acoustics of travelling bubble cavitation have been extensively studied both experimentally and theoretically, the behavior of nat- urally occurring cavitation bubbles near surfaces has not been examined in great detail. It has been known for some time that cavitation bubbles generated near surfaces are usually not spherical (as often assumed in theory) but hemispherical caps (Knapp and Hollan- der (1948) and Parkin (1952~), and a cavitation bubble collapsing near a solid boundary may produce a micro- jet of fluid which has been speculated to cause surface cavitation damage (Benjamin and Ellis (1966), Plesset and Chapman (1970), Lauterborn and Bolle (1975), Kimoto (1987) and, for a review, Blake and Gibson (1987~. The complex shapes that travelling bubbles assume will clearly be influenced by macroscopic flow phenomena such as pressure gradients, boundary lay- ers, separation, and turbulence. Researchers have at- tempted to study these effects by observing cavita- tion bubbles induced in a venturi (Kling and Hammitt (1972~) or above a surface (Chahine et. al. (1979), van der Meulen (1989~. Yet detailed, systematic studies of hydrodynamically-produced cavitation bubbles are almost non-existent. The random nature of naturally occurring cavitation is the primary reason why investi- gators have focused on integral measurements in their study of cavitating flows, leaving the detailed behavior of individual cavitation bubbles unexamined. Analyses of cavitation noise have generally been based on the theoretical behavior of single, spherical bubbles following the work of Fitzpatrick and Stras- berg (1956~. From this data base, researchers have synthesized the acoustic emission from cavitating flows 331 with multiple events (Blake (1986~. Many experi- ments have attempted to extract the actual behavior of individual bubbles from the integral measurement of the noise produced by cavitation (Mellon (1956), Blake, Wolpert, and Geib (1977), Hamilton (1981), Hamilton, Thompson, and Billet (1982), and Marboe, Billet, and Thompson (1986~. Although trends are seen in the measured spectra which may be related to theoretical predictions, the difficulty of obtaining free field acoustic spectra in the confines of most water tunnels has always made interpretation of experimen- tal spectra problematic. Researchers have also attempted to treat cav- itation as a stochastic process. The spectral emis- sion of a cavitating flow will depend not only on the noise produced by single bubbles but also on the cav- itatior~ rate and event statistics (Morozov (1969) and Baiter (1986~. Furthermore, cavitation noise scaling like that suggested by Blake, Wolpert, and Geib (1977) will be significantly influenced by changes in the cavi- tation event rate. As the number of cavitation events increase, bubble interactions will affect individual bub- ble volume histories and their acoustic emission (e.g. Morch (1982), Arakeri and Shanmuganathan (1985), and d'Agostino, Brennen, and Acosta (1988~. Analy- ses of multiple bubble effects depend upon a knowledge of the nuclei distribution in the flow and the dynamics causing the nuclei to cavitate. Yet, the effect of nuclei number distribution on the total cavitation process is poorly understood, and this is due largely to the difficulty of accurately mea- suring this quantity. In fact, most cavitation studies neglect to include any measure of the nuclei number distribution. As we shall demonstrate, the number and size distribution of cavitation bubbles, and the resulting noise emission, can vary substantially over the course of an experiment, even at a nominally fixed operating point. Although the mean cavitation event rate may be approximately determined by the acous- tic pulse rate (Marboe, Billet, and Thompson (1986~), cavitation bubble size distributions have only been de- termined in very rough form (Baiter (1974) and Meyer, Billet, and Holl (1989~. Although knowledge of the cavitation rate and bubble size distribution is essen- tial, no simple method has been found to count and measure cavitation bubbles. The above observations indicate a need to study the dynamics and acoustic emission of individual cavi- tation bubbles. A method of detecting and measuring cavitation bubbles was needed, and this paper presents

data obtained through the use of a new electrical probe developed for this purpose. Using this new instrument experiments were performed to study individual cavi- tation events and their statistics in an attempt to ad- dress the above issues. 2. NOMENCLATURE A (RO) streamtube capture area for given nuclei Cp pressure coefficient, (P-Po) / ( ~ pU2) CPM minimum pressure coefficient on body surface f frequency I measured acoustic impulse I* dimensionless acoustic impulse N (RO) free stream nuclei distribution PrO (RAT) max. bubble volume distribution associated with nuclei of size RO Pr (RM) maximum bubble volume distribution PA acoustic pressure PO freestream pressure Pv water vapor pressure r acoustic path length R (t) calculated bubble radius Rc critical nuclei radius RB headform radius at CPM RH headform radius RL bubble radius along trajectory RM cavitation bubble maximum radius RMR cavitation bubble maximum reduced radius RO nuclei radius Re Reynolds number, UD/u S water surface tension SP acoustic pressure spectral coefficients t time integration limits for experimental impulse acoustic pulse duration calculated dimensionless pulse duration free stream velocity calculated bubble volume Weber number, pU2RH/S constant, pulse width relationship constant, nuclei stability relationship headform radius of curvature at CPM water viscosity water density cavitation number, (P-Pv) / (2pU2) bubble cavitation inception index attached cavity formation index cavitation event rate to ~ t2 T T* U V (t) We ct IJ p ai ~ac 3. EXPERIMENTAL SETUP The experiments were conducted in the Caltech Low Turbulence Water Tunnel (LTWT),a full descrip- tion of the facility is presented by Gates (1977~. For all experiments, the test section free stream velocity was set and the tunnel static pressure lowered until the desired cavitation number was reached. The oper- ating air content was generally between 6 - 8ppm, and the tunnel water was well filtered. The free stream nuclei number distribution of the upstream fluid was measured using in-line pulsed holography. A detailed description of the holographic system is presented by Katz (1981~. Two axisymmetric headforms were used in the present experiments. The first was a Schiebe head- form with an ultimate diameter of 5.08cm. (Gates et al (1979~; the second, which has a modified ellip- soida1 shape with a diameter of 5.59cm, is known as the tJ '_ 0.2 z ~O O -o .2 ~ - O . 4 - u' -0.6 c -0.8 _ r-~) I.T.T.C. BODY 1 2.0 X/R 1 o.8 0.4 I ' I 10 0.6 o.a o.`n ; , j , , , , , 1 2.2 SCHIEBE BODY I ~ 2.c 0.2 ~ 1 0 1 C -o - 0. 0 . 6 -0.8 1 ~ 40.8 ~' C P~n ' - 0 ?S 0 0.2 o.. 0.6 0.8 %/R Figure 1. Surface pressure distributions and profiles of the I.T.T.C body alla the Schiebe body. I.T.T.C. headform (Lindgren and Johnsson (1966)). Surface pressure distributions for the Schiebe body (Gates et al (1979)) and the I.T.T.C. headform (Hoyt (1966)) are availa61e in the literature. The headform contours and surface pressure distributions are pre- sented in Figure 1. The headforms were fabricated out of lucite, a material whose acoustic impedance is a fair match to that of water. The hollow interior of both bodies was filled with water in which a hydrophore was placed. The hydrophore, an ITC-1049, has a relatively flat response out to 80kHz. Except for ultralow frequen- cies (<< lHz), the hydrophore signal was not filtered. All acoustic signals were digitized at a sampling rate of lMHz. Because of the relatively good acoustic impedance match between lucite and water, the in- terior hydrophore allows the noise generated by the cavitation bubbles to reach the hydrophore relatively undistorted; reflected acoustic signals from other parts of the water tunnel only make their appearance after the important initial signal has been recorded. In addition to the hydrophore, each headform was provided with novel equipment developed from instrumentation which had previously been used to measure volume fractions in multiphase flows (Bernier (1981~. This instrumentation consisted of a series of electrodes arrayed on the headform surface which were used to detect and measure individual cavitation bubbles. A pattern of alternating electric potentials is applied to the electrodes and the electric current from each is monitored. When a bubble passes over one of the electrodes the impedance of the local conducting medium is altered, causing a change in the current from the electrode. This change, which is detected and recorded, permits the position and volume of the bubble to be monitored. 332

One specific electrode geometry consisted of patches arrayed in the flow direction to cover the ma- jor extent of the cavitating region. Another consisted of electrodes which encircled the entire circumference of the headform in the region of maximum bubble growth. These two electrode geometries were used for different purposes. Signals from the patch electrodes indicated cavitation at a specific location on the head- form, and, by electronically triggering flash photog- raphy, simultaneous plan and profile photographs of individual bubbles could be taken at a prescribed mo- ment in the bubble history. Thus, a whole series of bubbles could be inspected at the same point in their trajectory. Furthermore, by simultaneously recording the acoustic signal from the hydrophore, one could correlate the noise with the geometry of the bubbles. The circular geometry was used to detect the oc- currence of every cavitation bubble at a particular lo- cation on the headform. This position was chosen to be near the location of maximum bubble volume, and for relatively moderate event rates only one bubble would occur over the electrode at any given time. Because almost all the cavitation bubbles maintain the same distance above the electrodes (this will be discussed below), the output of the circular electrode system is directly proportional to the area covered by the bub- ble, and the peal; of the signal is proportional to the major diameter of the bubble base. This system was calibrated photographically and found to be quite lin- ear. The volume of the bubbles was then determined from a measure of the base diameter using a functional relationship derived through the photographic study of many individual bubbles (Ceccio (1990~. Two kinds of experiments were performed with the circular elec- trode system. The first involved the measurement of event statistics and bubble maximum size distribu- tions. In the second, the acoustic emission of indi- vidual cavitation bubbles was analyzed and the result correlated with the bubble maximum volume. 4. OBSERVATIONS OF SINGLE CAVITA- TION BUBBLES PROFILE VIEW . v ~ ~ ~ "PYRAMID / SHAPE ' ~ " WE DGE " - ~ ~ SHAPE ~ HEADFORM SURFACE PLAN VIEW - 5 mrn r sussLE / FISSION ~ 000 - ROUGH ~ SURFACE " BUBBLE WAVE DIRECTION OF FLOW Figure 2. Schematic diagram of typical bubble evo- lution on the Schiebe headform. Cavitation bubbles were observed on both the Schiebe and I.T.T.C. headforms over a range of cav- itation numbers. The cavitation number was var- ied between the traveling bubble cavitation inception value, hi, and the value at which attached cavitation occurred, Sac. The inception index on both bodies was strongly dependent on the ambient nuclei num- ber distribution (Ooi (1981~. Inception occurred on the Schiebe body at cavitation numbers as high as Hi = 0.65, and on the I.T.T.C. body at Hi = 0.58 for tunnel water of 6 - 7ppm air content. However on both bodies the inception index was reduced to about Hi = 0.50 immediately after deaeration. Any definition of the bubble cavitation inception index must there- fore be associated with a particular free stream nuclei number distribution. The attached cavitation forma- tion index for the Schiebe body was aaC = 0.40 and for the I.T.T.C. body arc = 0.41. These values were al- most constant over the fairly narrow range of Reynolds numbers of the experiments (Re = 4.4 x 105 - 4.S x 105 ). Before detailing the results from each headform one observation can be made for both geometries. For a given tunnel velocity and cavitation number, the maximum bubble volumes were quite uniform. Al- though the incoming nuclei diameter ranged over al- most three orders of magnitude, the maximum cav- itation bubble volume varied over only one order of magnitude. The reason for this is given below. For both headforms, the growth phase of the nu- clei was very similar to that described in the original observations of Knapp and Hollander (1948) and El- lis (1952~. For most of their evolution, the bubbles take on a hemispherical or "cap" shape and move ex- tremely close to the headform surface; only very oc- casionally would quasi-spherical bubbles be observed at a distance above the surface. Small waves could be observed on the bubble surface in many instances. As the bubbles reach their maximum volume they become somewhat elongated in the direction normal to their motion while their thickness normal to the surface re- mains relatively constant. At this point, the difference in the flows around the two bodies begins to cause differences in the bubble dynamics. The Schiebe body was designed to suppress lam- inar separation in the region of cavitation (Schiebe (1972~. It possesses a sharp pressure drop with a min- imum pressure coefficient of-0.75 (Figure 1~. Figure 2 represents a schematic of the typical bubble evolu- tion, and Figure 3 consists of a series of photographs of bubbles at various stages during this process. After the bubble has reached its maximum volume, it begins to lose its cap-like shape and becomes elongated progress- ing into a pyramid-lilte shape; the bubble thickness normal to the headform surface consistently decreases after reaching its maximum. The bubble then collapses rapidly and develops an elongated shape. The elonga- tion of the bubble and the formation of tubes is proba- bly due to rotation of the bubbles caused by the shear in the boundary layer. As the bubble collapses it may fission into two or three tubes of collapsing vapor, and the residual gas in these tubes may cause a rebound to produce a rough bubble or group of bubbles after collapse. The I.T.T.C. headform has a relatively smooth pressure drop with a minimum pressure coefficient of -0.62. A distinguishing feature of this headform is that, unlike the Schiebe body, it possess a laminar 333

-A - ~ - L . . _ 11~ 1 11' ~ _ 1h 1 ~ ~ Twirl ::: warty _~_~ ~ ~ - _ ~ __ ~ ~ ~ ~ ~ ~ ~ ~ . l ~ __ ~ ~ ~ ~ ~ ~ ~ ~ ~ Id_ ~ ~ ~:~ ~. :~ ~ ~ ~ ~ ~ ~ ~ ,, ~ . ~ ~ _ 1 fir :_ ~ _: _, . - . ~ i.... - ' r . ...~... a. . : it_ it_ . ~ . . ... Profile View _] Plan View Figure 3. Series of photographs cletailing typical bubble evolution on the Schiebe hea(lform, U 9m/s and ~ 0.45. 334

PROFILE VIEW "SNOUT" \ SHAPE ~ "WEDGE" ~ \ SHAPE ~ ~ =-~ ~ _ 14FAnFORM RII~FAr~ DIRECTION OF FLOW ~- PLaN VIEW r ROUGH \ UNDERSIDE ~ SURFACE \ \ WAVES (D o ROUGH 'TRAILING BUBBLE STREAM E R - 5 non ~ 1 Figure 4. Schematic diagram of typical bubble evo- lution on the I.T.T.C. headform. separation region (Figure 1~. Figure 4 is a schematic of the typical bubble evolution, and Figure 5 presents a series of photographs of bubbles at various stages of this development. The bubble has a cap-like shape until it reaches its maximum volume where it then becomes further elongated evolving into the wedge- lil;e shape. However, unlike the bubbles on the Schiebe body, the cavity starts to lift off the surface and begins to roll up into a snout-like shape. This may be due to recirculating flow associated with the separation region or the stretching of the bubble in the velocity gradient. As it collapses, the "snout" continues to role up into a vapor tube eventually collapsing to produce a rough bubble after collapse. On both the Schiebe and I.T.T.C. headforms the rough bubble or group of bubbles which is formed after collapse is sheared by the surface flow and usually disperses into smaller bubbles on the order of 50pm, although a second collapse and rebound is not uncom- mon. The mean lifetime of a bubble depends upon the tunnel velocity, cavitation number, and initial nuclei size, but, for most of the observed bubbles on both headforms, it is approximately 3ms. The laminar separation on the I.T.T.C. body has been carefully studied in the context of its effect on attached cavitation (Arakeri and Acosta (1973~. Clearly, the separated flow also influences bubble cav- itation for cavitation bubbles were observed riding over the separation "bubble". As seen in Figures 4 and 5 the underside of the bubbles become roughened as they pass over the region of turbulent reattachment. These local flow disturbance seem to shear vapor off the un- derside of the bubble, leaving a trail of much smaller bubbles. This phenomenon was not observed on the Schiebe body. Furthermore, some bubbles were seen to cause local attached cavitation. When the operating cavita- tion number was close to the attached cavity formation index, trailing "streamers" were often observed down stream of the cavitation bubble (Figure 6~. These streamers were generally associated with the larger bubbles on the I.T.T.C. body (and occasionally on the Schiebe body) and were seen to develop gradually at the location of the laminar separation point (Arakeri and Acosta (1973~. As the bubble is swept down- stream, the streamers continue to grow, and in may cases persist even after the bubble has collapsed. Why these bubbles cause the attached cavitation streamers at the lateral extremities of the bubble is unclear. This phenomena has also been observed with travelling bub- ble cavitation on hydrofoils (van der Meulen (1980) and Rood (1989~. The process could be considered an inception mechanism for attached cavities. The classic observations of Knapp and Hollan- der (1949) may be compared those of this study. Both experiments revealed that bubbles travelling near sur- faces are cap shaped' and the gross characteristics of growth and collapse are similar. However, the pres- sure distribution on the ogive of Knapp and Hollander generated a long and steady growth, and the bubbles often retained a quasi-spherical shape even near the final stages of collapse. These bubbles would often re- bound many times maintaining their quasi- spherical shape after each collapse. The bubbles observed in this study usually rebounded only once and lost most of their coherent shape after the first collapse. This difference may be explained by noting that the water tunnel facility used by Knapp and Hollander was not equipped with any deaeration system, and extremely bubbly flows were used to increase the odds of pho- tographing a cavitation event. Consequently, the cav- itating nuclei observed by Knapp and Hollander were large, containing more undissolved gas. Increasing the amount of residual gas reduces the violence of the bub- ble collapse making coherent rebounds possible. On the other hand, the nuclei populations of the present study were quite small, and the cavitation bubbles ob- served were almost entirely vaporous. Such bubbles collapse violently and therefore coherent rebounds are less likely. Photographs of bubbles presented by Ellis (1952) show many of the features in the present study. Prin- cipally, bubbles formed close to the headform also pro- gressed from a cap shape to a wedge shape before col- lapse, although the collapse mechanism is difficult to distinguish in Ellis' silhouette images. He observed that the bubble surface profile approximately coin- cided with lines of constant pressure for bubbles near the point of maximum volume. This accounts for the wedge shape of the bubble. Examination of the iso- baric lines computed for flow around the Schiebe body (Schiebe (1972~) also show the bubbles observed in this study are being shaped by the pressure gradients close to the surface. Returning to the present study, the collapse mechanisms for bubbles on both headforms were dis- cerned through the study of many photographs. A composite mechanism is presented in Figure 7 for the Schiebe body with sample photographs in Figure 8. For the I.T.T.C. body similar results are included in Figures 9 and 10. Previous researchers have noted the generation of a liquid microjet in bubbles collapsing near a solid surface (Lauterborn and Bolle (1975) and Kimoto (1987), for example), and this microjet is of- ten identified as the main cause of cavitation erosion damage. Although many photographs were taken dur- ing the present investigation, a reentrant microjet was 335

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not observed in any of the present photographs of bub- ble collapse, although the jet may have occurred too rapidly to be detected. The observed bubbles lack the compact geometry we might expect to be associated with coherent microjet formation. 5. MEASUREMENT OF THE ACOUSTIC EMISSION OF SINGLE CAVITATION BUB- BLES The detailed relationship between the collapse mechanism of hydrodynamic cavitation bubbles and the resulting noise generation is not completely clear but some features are suggested by the present work. First, as other investigators have concluded (for ex- ample Harrison (1952) and Chahine, Courbiere, and Garnaud (1979~), the majority of the noise is gener- ated by the violence of the first collapse; the growth phase contributed no measurable noise signal. The rebound produces a rough bubble which may also col- lapse to produce a second noise pulse of lesser mag- nitude. However, noise was not necessarily generated by every bubble collapse. Smaller bubbles would often collapse without an acoustic pulse, and larger bubbles would sometimes produced a muted collapse. Figure 11 presents two examples of the initial noise pulse generated by the collapse of a bubble on the I.T.T.C. headform. The first pulse has only one peak, but the second trace is an example of a multiple peal; event. Multiple peaks suggest bubble fission prior to collapse, and the photographs presented in the previous section reveal that many bubbles have undergone fission. Although some researchers have used the peak acoustic pressure to characterized cavitation noise in- tensity (e.g. Van der Meulen (1989~), in this study the magnitude of acoustic pulses will be characterized by the acoustic impulse defined as It2 I= PAdt (1) Jt1 PROFILE VIEW OR \ r Bt~ BUBB' F COLLAPSES REBOUND \ \ REBOUNDS NEAR SURFACE ~ ~ -~( _ ~ HEADFORM SURFACE Pl;AN VIEW _ 5 mm BUBBLE FISSION ~ O ROUGH BUBBLE MAY REBOUND DIRECTION OF FLOW Figure 7. Schematic diagram of typical bubble col- lapse mechanism on the Schiebe headform. PROFILE VIEW AFTER -, BEFORE BUBBLE LIDS REBOUND/ r REBOUND ~ OFF SURFACE ~ ~ ~ ~-W~ ~ ~ HEADFORM SURFACE Plan VIEW DIRECTION OF FLOW / _ ~_ C: ~ Al ~ ~ _ VAPOR DISSIPATES ROUGH BUBBLE ~ SHEARED MAY REBOUND VAPOR Figure 9. Schematic diagram of typical bubble col- lapse mechanism on the I.T.T.C. headform. The times to and t2 were chosen to exclude the shallow pressure rise before collapse and the reverbera- tion produced after the collapse. Experimentally mea- sured impulses for the Schiebe body at a tunnel veloc- ity of U = 9m/s and cavitation numbers of a = 0.55 and a = 0.42 are presented in Figure 12 and 13. The data all appear to lie below an envelope which passes through the origin. The existence of this well-defined impulse envelope suggests that a collapsing bubble can generate, for a certain maximum volume, a specific impulse if it collapses in some particular but unknown u) 15- 10 5F O . -5 _ -10 _ _ TIME (AS ) 1 5 10 _ 5 _ O - 5 _ - 1 0 _ o TIME (AS) 500 Figure 11. Two examples of typical cavitation ini- tial noise pulses. The bubbles were generated on the I,T.T.C. headform at a = 0.45 and U = 8.7m/s. 338

- a ~ i:: ~ ~:~:~: i::: :: :: ~ ::~: l 'I _~ 1 =~: ~ ~ ~ ~ ~ ~ - ~ ~ ~ ~ ~ ~ - ~ 1 _ 1 , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~. 1 r. ~ ~ ~ ~ I.'' ~ [~ . 1~ ~ ~ ~ ,~ [' ~ i~§ - I~ ~ I~ - : i~ ~ ~ ~:~} ~ ~ : ::::: ::::::::: I::: :::: - :~ ~ ~ ~:~: ~: :: :: :::: :: ~ ::::: : ::~:::: : ~ _ ~ ~ ~ ~ ~ ~ ~ ~ 1 ~ ~ A::.: ~ ::: :: :::: ~:::~::~: ::::::: ~ ~ -,,,0 ::::::: :::::::::: - ~ - ~ - ~ ::: A:::: - - ::: ~ :: ~ :~; - ~ :: : :: ~ :~::::~:::~: ~ ~ ~ ~ ~ ~ - ~ ~ - ~: - - - ~ ~ ~ :3F~ ~ : : : ~i::: ant: i: i:::::: :~:: :: ::: ::: ~ - ~.~1 Figure 8. Series of photographs detailing typical bubble collapse mechanism on the Schiebe headform, U-9m/s and <:r 0.45. 339

~ ~ - ~ ~ ~ ` l - ~ ~ ~ ~ ~:~; ~:: ~1 ~`~ ~ 1 ~ ~_ - ~A:: - ~ -: ~- :: :~ - ~:: --~- ~- ~ - ~ - - ~ - - ~ - ~- ~: ~ : : ~I: ::::: :: : :: i::: ::::: :~ ~ :::::::: : :~T~ i: :~ ~ _I : - ~ - - .~ - it ~ - ~ - ~ ~ ~ - Am - ~ ::~: ~ ~ #a ;:: - 3~ _ ~....11.1 : ~- ~ : - ~ - - - ~ma - a~a, ~ma= ~- - - - Figure 10. Series of photographs detailing typical bubble collapse mechanism on the I.T.T.C. heaciform, U 8.7m/s and a 0.45. 340

0.20 L In 0.10 0.00 0.1 l l l 0.20 SO BODY Number of U -9 - k Peaks a - Q42 + O ~ x 2 ~. · O>2 x x x ~Y X ~X ~ ./ b ° , A. 6!. . x x At.' · As 11! . .. .: . .: . ~I ·;l i- I I 0.00 0 40.00 80.00 120.00 160.00 0 MAGNUM VOLUME. 3 Figure 12. Acoustic impulse plotted against the maximum bubble volume for the Schiebe body at U = 9m/s and a = 0.42. way. It can, however, produce less than this maximum impulse if it collapses in other ways. The different symbols represent the different number of acoustic peaks which are generated upon collapse. As shown in Figure 12, the probability that a collapse will produce multiple peaks increases for larger bubbles. Yet, even as the number of peaks in- creases, the impulse often reaches its maximum possi- ble value implying that, in some collapse mechanisms, fission does not decrease the total stored energy avail- able to produce noise. Other large bubbles collapse to produce almost no acoustic impulse. The production of noise upon collapse is the result of violent changes in bubble volume near the point of minimum bubble volume, but- larger bubbles may be sheared apart and dissipate thus losing their organized shape and pre- venting a coherent and concentrated collapse. Fur- thermore, larger bubbles may contain more contam- inant gas (as a result of dissolution) and this would cushion the collapse and reduce the acoustic emission. At higher cavitation numbers such as that of Fig- ure 13 the number of larger bubbles is reduced, and most bubbles collapse to produce only one acoustic pulse. However a large number of very small bub- bles will collapse and produce no significant impulse, and these cases are represented by the "0" symbols. Mute events are generally not examples of "pseudo- cavitation" as observed by Dreyer (1987) but distinct cavitation events with a near-silent collapse mecha- nism. The general trends in the data for the Schiebe body are also evident in the results from the I.T.T.C. headform. Significantly, however, the average acous- tic impulse is about three times larger than that of the Schiebe body. This will be discussed further be- low. Furthermore, as the cavitation number is low- ered to near the attached cavitation inception index of the I.T.T.C. body, the impulse data changes sig- nificantly. Figure 14 presents an example of data from the I.T.T.C. body taken at a tunnel velocity of U = 8.7m/s and a cavitation number of a = 0.42 at near the attached cavitation formation index. The impulses generated by smaller bubbles are much more uncertain, and, for many larger bubbles, no significant impulse is generated. Since these larger bubbles gen scHIEs 0 Number of E B DY Peaks u -9m/s a ~ 0.55 + I x 2 0 >2 x . .. x xx ~,.~. t... 0 40.00 80.00 MAXIMUM VOLUME, mm3 Figure 13. Acoustic impulse plotted against the maximum bubble volume for the Schiebe body at U = 9m/s and a = 0.55. erally have trailing streamers, it would seem that the streamers interfere with the collapse in a way which decreases or eliminates the noise. The average number of peaks for a given average diameter is plotted in Figure 15 for both headforms. For smaller bubbles, the average is less than unity, re- flecting the influence of muted bubbles, and for larger bubbles, multiple peaking produces an average above unity. For the case of the I.T.T.C. body, however, the muting effect of the trailing streamers causes a reduc- tion in the average number of peaks for the data set with the largest average volume. This data set occurs at the lowest cavitation number, near the attached cav- itation inception point. 6. COMPARISON WITH ANALYTICAL RE- SULTS In order to place the above experimental results in some analytical perspective, calculations were made of the bubble sizes and acoustic impulses predicted by integration of the Rayleigh-Plesset equation starting `,0.40- l LT.T.C BODY U-U-h o-~ ~ x 0.20 ~O . x A .lK . 0~00 N~ of Cot O . ~ X 2 O >2 A O o o · ...... ............... I .. 0 40.00 80.00 1 20.00 1 60.00 hWOM~ VOWb~. ~, Figure 14. Acoustic impulse plotted against the maximum bubble volume for the I.T.T.C. body at U = 8.7m/s and ~ = 0.42. 341

with various sizes of freestrea~n nuclei. The known surface pressure distributions for both headforms were employed to construct the pressure-time history which a nucleus would experience while passing near the headform. No slip between the bubble and the liquid and a small offset from the stagnation streamline are assumed. Calculations were performed with various free stream velocities, cavitation numbers, and offsets from the stagnation streamline. Figure 16 provides an example of the dependence of the maximum bub- ble radius on the original nucleus size for the I.T.T.C. headform and various cavitation numbers. Note that nuclei below a certain size (which depends on the cavi- tation number) hardly grow at all and would therefore not contribute visible cavitation bubbles. This critical size is predicted by the stability analysis of Johnsson and Hsieh (1966) and Flynn (1964~. Bubbles below the critical size grow quasistatically, whereas larger bub- bles grow explosively. A bubble is critically unstable if RL, > 8 5 1 RH 3 pRHU2 ~-a-CPM) where CPA.! is the minimum pressure coefficient (-0.62 for the I.T.T.C. headform) and Ret is the local bubble size. The computations show that so long as the bubble remains stable, then Rat; is somewhere in the range RO < RI, < 2Ro for the common circumstances of interest here. Consequently, the critical nucleus size Rc is given by RC > 8 /3S ~ (~3) where ~ is a constant. The results of this simple expression are presented in Figure 17 along with data on the critical nucleus size obtained from the Rayleigh- Plesset solutions. The qualitative agreement is excel- lent and suggests a value of ,B slightly greater that 0.5. Note that the higher the velocity, U. the smaller the critical size, and therefore the larger the number of nuclei that will be involved in cavitation. u' o 111 I' ~ Iii to ·. ^~° 1 - O ~ ~ 0 c, . · . . . 0 · I1lC o ° Schiebe 0 . · . , . I . , 0 10 20 30 40 50 AVE. MAX. BUBBLE VOLUME (mm3) Figure 15. Average number of peaks as a function of average maximum bubble volume for bubbles gen- erated on the Schiebe body and the I.T.T.C. body. ~ ~ 0 o IL LO 1 0; u, 10 _~ '' · :1 n \ 04 ~: ~ ~ ~ ~ ~ R.,. = Ro 1 0 , ~ I' 1 ~ I 0~ 2 1 0-1 NUCLEI RADIUS/ HEADFORM RADIUS, Ro/RH Figure 16. Numerical calculation of the bubble max- imum radius as a function of nucleus radius for nuclei passing near the I.T.T.C. headform. The other feature of Figure 16 which is impor- tant to note is that virtually all nuclei greater than the critical nucleus size grow to approximately the same maximum size. The asymptotic growth rate of an un- stable cavitating bubble is a function only of the pres- sure and not the initial nucleus size. Consequently the maximum size achieved will be approximately in- dependent of the nucleus size. This accounts for the uniformity of cavitation bubbles observed experimen- tally. Similar calculations were performed for nuclei experiencing the Schiebe body pressure distribution, and the results were qualitatively similar to those of the I.T.T.C. body. The above calculations yield the volume-time history for a cavitating bubble, and the acoustic pres- sure generated by the bubble may be approximately given by PAtr,t) = 4P dt2 (4~) u, - c~ o is I i_ _ C) fir _ 2 0.0004 ~ S/PLRHU { 0.000036 ~ 0 0.1 CAVITATION NUMBER, 0- Figure 17. Crital nuclei radius as a function of flow parameters for nuclei passing near the I T.T.C. head- form. 342

0.4 0.3 - J 0.2 In U) At lo In a.. - S/pRHU2- 0.0004. 0.000036 / a= 0.5 / 0.~ I.T.T.C. ~,/ ^~' SCEIIE~E ~ ~ MY 0 0.04 0.05 0.06 ~UIUUIU BUBBLE VOLUIVIE / RH3 Figure 18. Numerical calculation of the acoustic impulse as a function of the maximum bubble volume for bubbles generated on the Schiebe body and the I.T.T.C. body. were V (t) is the bubble volume, p is the fluid density, and r is the distance from the center of the bubble. This relationship is valid in the acoustic far- field and for subsonic wall velocities. The acoustic impulses, I, were calculated from the definition (1) where to and t2 were taken to be the times when d2V/dt2 = 0 before and after the first collapse. For those nuclei which become unstable and ex- plosively cavitate the non-dimensional impulse, I*, is defined as I*= R U (5) where we have assumed r = RH since this is the location of the hydrophore in the experiments. The impulse I* is plotted in Figure 18 against the maximum volume of the bubbles non-dimensionalized by RH. A number of investigators (i.e. Fitzpatrick and Strasberg (1956) and Hamilton et.al. (1982~) have suggested that the magnitude of the acoustic signal should be related to the maximum size of the bubble, and this is born out in Figure 18 where the data for a range of cavitation numbers and two Weber numbers, We, are contained within a fairly narrow envelope. The median line was converted to dimensional values and is plotted in Figure 19 where it is com- pared with data sets from the Schiebe and I.T.T.C. experiments. It is strilting to note that the envelope of the maximum impulse from the experiments is within a factor of two of the Rayleigh- Plesset calculation for the I.T.T.C. body and within a factor of six for the Schiebe body. This suggests that, despite the depar- ture from the spherical shape during collapse, the in- compressible Rayleigh-Plesset solutions correctly pre- dict the order of magnitude of the noise impulse gen- erated by individual bubbles. It is not surprising that the predicted impulse is greater than the experimental value. In fact, the theoretical impulse may be considered the maximum 0.40 0.00 impulse possible for a given bubble volume since a spherically symmetric collapse is probably the most efficient noise producing mechanism. The difference between the measured impulses and the theoretical impulse is an indication of the inefficiency of the ac- tual collapse mechanism. Furthermore the average impulses are closer to the theoretically predicted val- ues for the I.T.T.C. body than for Schiebe body, and this is consistent with the photographic evidence that the I.T.T.C. collapse mechanism is more compact than that on the Schiebe body. The duration of the impulse (as opposed to the magnitude) is much better understood. Here, the du- ration is defined as T = to-t2. This time is simply related to the total collapse time derived by Rayleigh (1917) which is used by many authors (e.g. Blake, Wolpert, and Geib (1977) and Arakeri and Shanmu- ganathan (1985~. Like the collapse time, it will be approximated by U (a) (6) where or is some constant of order unity. It follows that the dimensionless impulse duration T* = TU/RH should be primarily a function of RM/RH, and this is confirmed by the results of the Rayleigh- Plesset solutions shown in Figure 20. Also plotted are typical experimental data from the Schiebe body. Note that the calculated results lie within a narrow envelope for a range of cavitation numbers and that the slope of the narrow envelope is close to unity. The experimental data is about one third the predicted magnitude. Note, however, that the definitions of to and t2 are somewhat arbitrary. Figure 21 presents spectra of the noise measured in the experiments. A series of individual acoustic pulses were recorded at a particular velocity and cav- itation number. The resulting spectra were averaged to produce the composite spectra in the figure; the 0.60- ~ / / ~ at/ T.C. ] MAY ~ ~ ~ ~ ~ I.T.T.C DATA U .8.7 As o -0.45 ic~oo~fic: ~SHE DATA U -90 ~O .0.42 0.00 40.00 _ 80.00 t 20.00 160 .00 MAXIMUM VOLUME, mm3 Figure 19. Comparison of theoretically predicted and experimentally measured acoustic impulse as a function of the maximum bubble volume for bubbles generated on the Schiebe body and I.T.T.C. body. Experimental data for a = 0.45 and U = 9m/s for the Schiebe body and U = 8.7m/s for the I.T.T.C. body. 343

- o' - - - signals were not altered to remove the erects of tun- nel reverberation. Such a composite spectrum will be equivalent to the spectrum derived from a measure- ment of a long series of cavitation noise pulses, pro- vided the cavitation events occur randomly (Morozov (1969~. The measured spectral shape varies little with cavitation number; only the overall spectral magnitude changes. A decrease of approximately-12dB/decade is noted until about 100kHz where a sharp falloff oc- curs. This cut-off frequency corresponds to the fre- quency response limit of the hydrophore. Asymptotic analyses of the Rayleigh-Plesset equation (Blake (1986~) predict a spectral shape of f-2/5 for frequencies in the range of 10~^Hz to 100kHz. The experimental spectrum has a shape of approxi mately f-3/5 which is similar but not identical to the predicted trend. Hamilton (1981), on the other hand, observed an almost completely flat spectrum in this range based on his integral measurement of bubble cavitation noise. The high frequency roll-off associ- ated with fluid compressibility was not observed below 100kHz, and this is consistent with the observations of Hamilton (1981) and Barker (1975~. 7. OBSERVATIONS OF CAVITATION EVENT RATES AND BUBBLE MAXIMUM SIZE DISTRIBUTIONS Experiments were performed to measure the cav- itation event rate and bubble maximum size distribu- tion on both headforms along with the freestream nu- clei number distribution. Furthermore, an analytical model was derived to study the relationship between the nuclei flux and the resulting cavitation statistics. The cavitation event rate and bubble maximum size distribution were measured for several thousned events at various operating conditions, and examples of these measurements for the Schiebe headform are given in Figure 22. Note that the bubble maximum sizes are presented as reduced radii. The reduced bub- ble radius is the radius of a sphere of volume equal to the measured bubble volume. Although the four bub- ble size distributions presented are all at the same cav- itation number and tunnel velocity, their event rates and size distributions are quite different. Since the 200.00 /\-TIEORY 150.001 / _ 1 oo.oo so.oo o.oo o. SEE BODY U-9m/t O-Q42 0.10 tar 0.05 o Cl UJ · . . . . ~ 1 . · ,· ~· · cn i: . . ~O. ~ 5 ~! . in Fat . . · · ~ .... . . . . r - I- - I 10 40.00 80.00 120.00 160 00 BUBBLI? MAXIMUM VOLUME, mm3 Figure 20. Comparison of theoretically predicted and experimentally measured pulse width as a function of the maximum bubble volume for bubbles on the Schiebe body at U-9m/s and ~ = 0.45. 30 10 _ - E L - L o~ 0.. 0.03 1 0'3 ; ~- ~o jo 1 to Frequency t Adz] ~oO Figure 21. Averaged acoustic spectra derived from acoustic pulses generated by bubbles on the Schiebe body at average U = 8.7m/s and ~ = 0.45,0.50, and 0.56m/~. cavitation bubble maximum volume distribution is di- rectly related to the incoming nuclei number distribu- tion these results clearly indicate that the nuclei num- ber distribution can be quite different for the same tunnel operating conditions. Weak control of the num- ber of nuclei was affected through deaeration and nu- clei injection. But, as Figure 22 indicates, the nuclei number distribution is a highly variable factor which influences travelling bubble cavitation and cavitation noise. The time between cavitation events was Pois- son distributed, as would be expected for randomly distributed nuclei. Consequently, the total noise spec- tra produced by these flows should be equivalent to the composite specrtra presented in Figure 21. A relationship between the nuclei flux and the resulting cavitation event rate and bubble maximum size distribution can be developed as follows. Whether a nucleus cavitates or not is strongly determined by the local minimum pressure it experiences. On the surface of the headform, this pressure is given by the minimum pressure coefficient. On streamlines above ~ q ~ n 1 2 BUBBLE MAXIMUM REDUCED RADIUS, R&.4R (mm) 3 4 Figure 22. Example of four bubble maximum size distributions for a particular free stream velocity and cavitation number for cavitation on the Schiebe body. 344

\ id' ~ - z 1010 o m - in 1' UJ Jo t, 1 0. Expanmental polats with best fit line . V 'I '1 \ 10 100 200 NUCLEI RADIUS, R (p m) Figure 23. Example measurement of the free stream nuclei number distribution, U = 9m/s and a = 0.45. the body surface, the fluid pressure may still be low enough to cause a nucleus to cavitate provided that the minimum pressure it experiences is below the crit- ical pressure, derived from Equation (3~. An incorn- ing streamtube may therefore be defined for a nucleus of specific size such that the nucleus will always en- counter a pressure low enough to cause it to cavitate during its flow around the body. The fluid capture area of this streamtube will be a function of the nuclei radius, RO, the free stream cavitation number, and the flow geometry. By assuming that the pressure gradient normal to the surface corresponds to the centrifugal pressure gradient caused by the radius of curvature a, of the surface at the minimum pressure point, and by assuming no slip between the nuclei and the fluid the following expression for the nuclei capture area A (RO), may be readily obtained (Ceccio (1990~: A(RO) = RBC ~-a-CpM) (1 RO) where RO is the original nuclei radius, RB is the headform radius at the point of minimum pressure, and RC is the minimum cavitatable nucleus given by Equation (3~. Equation (7) may be rewritten as ( RO ) (8) where Av is the capture area enclosing all streamlines which involve pressures less than vapor pressure; note that Av is a function only of the flow geometry and free stream conditions. Finally, the to tal flux of cavitatable nuclei or total cavitation event rate, E), is Ioo O= A(Ro)N(Ro)UdRo (9) RC where N (RO) is the free stream nuclei number distribution. Now consider the distribution of bubble maxi- mum sizes which this process will produce. This distri- bution is the result of different nuclei trajectories and sizes. Cavitating nuclei travelling on streamlines far- ther away from the headform will not grow to the same maximum volume as those travelling near the surface. Consequently, a flux of uniform nuclei, RO, will yield a probability distribution distribution of bubble maxi- mum sizes, RM, denoted by Pro (\RM). Because of the slight dependence of bubble maximum size upon nu- cleus size, Pro is a function of RO. A flux of nuclei rep- resented by the nuclei number distribution, N (RO), will therefore produce a distribution of maximum bub- ble sizes, Pr, given by Pi (Ro) = (~' ~ ProA(Ro) N (Ro)UdRo (10) If no relationship existed between nuclei size and the maximum bubble size, Pr would be independent of the nuclei number distribution; changes in N (RO) would merely change the total event rate. The exper- imental data indicate, however, that the bubble max- imum size distributions are influenced by the nuclei number distribution. The varying event rates reported in Figure 22 indicate different nuclei populations, and each example is accompanied by a unique bubble size distribution. The small influence of nuclei size upon the maximum bubble size will ultimately have a sig- nificant influence upon the bubble maximum size dis- tribution. We shall now compare the measured cavitation event rates and bubble maximum size distributions with the predicted quantities based on holographically- determined free stream nuclei number distributions. The nuclei populations were measured at the same time tliat the cavitation statistics were recorded, and the smallest nucleus which could be detected with cer- tainty was approximately 20,um in diameter. An ex- ample nuclei distribution is presented in Figure 23. Table 1 presents the measured event rates and the predicted event rates based on Equations (7) and (9~. The measured event rates fall within the range of the predicted values, with the uncertainty in the predicted event rates resulting from uncertainty in the measured 1 PREDICTED (3 (events/sec) 128 ~ 25 164 st 25 1 47 t 25 MEASURED (events/sec) l 156 147 162 1 _ _ _ Table 1. Comparison of measured and predicted cavitation event rates for cavitation generated on the I.T.T.C. body at U = 9m/s and a = 0.45. 345

0 . 1 0 o 3 G - ~ G r~ m 0~50 - cnp UJ m m 3 CALCULATED EVENT RATE . 128 events / see MEASURED EVENT RATE _ 156 evens / see CALCUlATED ...... MEASIRED BUBBLE MAX. REDUCED RADIUS, RM (mm) Figure 24. Calculated and measured event rate and bubble maximum size distribution for cavitation on the I.T.T.C. headform at U = 9m/s and ~ = 0.45. nuclei number distributions. The close match between the predicted and measured event rates indicates that the nucleus stability criteria from Equation (3) ad- equately models the actual cavitation process. The ninimum cavitatable nucleus for this flow is calculated to be approximately 20pm in radius, and the measured nuclei number distribution indicate that most of the cavitating nuclei are in the range 20 to 100,um. The success of the model suggests that the quantities Av and Rc may be used to adequately characterize the nuclei capture area for flows over more complicated bodies. The calculated bubble maximum volume distri- butions, however, depart substantially from the mea- sured size distribution in terms of its details. Fig- ure 24 presents a measured bubble maximum size dis- tribution along with the predicted distribution based on Equations (7), (9), and (10), the results of Figure 16, and the measured free stream nuclei distribution. The calculated size distribution departs substantially from the measured distribution in its details. The pre- dicted bubble size range is about twice the observed size range, and the number of larger bubbles predicted is much smaller than the observed percentage. These discrepancies may be the result of several phenomena. First, the maximum size achieved by a nucleus sub- jected to a specific pressure history may not be ad- equately predicted by the Rayleigh- Plesset equation since bubble growth may be limited by the positive pressure gradients above the headform surface. Once the bubble has grown sufficiently, the mean pressure on the bubble surface will be larger than the surface pres- sure used in the Rayleigh-Plesset calculation, reducing the driving force for bubble growth. Furthermore, the experimental bubble maximum size distributions often show several maxima which were repeatable for nom- inally fixed operating conditions. These distributions cannot be simulated with simple, smooth nuclei dis- tributions with several well defined peaks. It seems likely that these maxima are the result of a compli- cated nuclei number distribution. Such detail could not be ascertained using the current holographic nu- clei distribution methodology; its existence was only revealed by the electrode system which permits very large quantities of data on bubble size distributions. 8. CONCLUSION Although theories of individual bubble cavita- tion abound, this study demonstrates that a great deal may still be learned through the observation of naturally occurring cavitation bubbles, especially bub bles formed in flows near surfaces. Cavitation bubbles are significantly affected by the viscous flow near sur- faces, and this in turn effects their noise production and possibly their damage potential. Yet, numerical integration of the Rayleigh-Plesset provided a reason- able base for comparison with the experimentally mea- sured data. The relationship between the nuclei flux and the resulting cavitation was successfully predicted based upon simple parameters derived from the non- cavitating flow around the body, although estimation of the bubble maximum size distribution was more dif- ficult. By combining the results of this study, cavitation noise may systematically be synthesized. Analysis of cavitation event statistics and size distributions can relate the freestream nuclei distribution to the cavita- tion process. And, once the number and size of the cavitation events are known, the total noise emission may be estimated based on the single bubble measure- ments. The results presented here are useful for the case of limited cavitation, but multiple bubble effects must be included to characterize flows in which the bubbles interact with one another. The importance of the nuclei number distribution as a parameter in cavitation studies cannot be overemphasized, although simple and accurate methods are still needed to mea- sure this quantity with speed, ease, and precision. ACKNOWLEDGEMENTS The authors would like to thank Professor Allan Acosta for his advice and considerations. We would also like to acknowledge the assistance of Sanjay Ku- mar and Douglas Hart. This work was supported by the Office of Naval Research under contract number N-00014-85-I(-0397. 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Rood, E. P. 1989. Mechanisms of Cavitation Incep- tion. International Syrup. on Cavitation Incep- tion, ASME FED Vol. 89, pp. 1-22. Schiebe, F. R. 1972. Measurement of the Cavitation Susceptibility of Water Using Standard Bodies. Rep. No. 118, St. Anthony Falls Hydraulic Laboratory, University of Minnesota. Van der Menlen, J. H. J. 1980. Boundary Layer and Cavitation Studies of NACA 16-012 and NACA 4412 Hydrofoils. Thirteenth Symposium on Naval Hydrodynamics, Tokyo. Van der Meulen, J. H. J. and van Renesse, R. L. 1989. The Collapse of Bubbles in a Flow Near a Boundary. Seventeenth Symposium on Naval Hydrodynamics, The Hague. DISCUSSION William B. Morgan David Taylor Research Center, USA This paper presents a very interesting investigation of cavitation acoustics and the authors are congratulated for such a fine and thorough piece of work. I have one question concerning Fig. 19. This figure shows a significant difference between the acoustic impulses from the ~I.T.T.C.. headform and the ~SchiebeW headform. Do the authors feel this difference is due to the difference in the way the bubbles collapse relative to the headform or do you think there would be an actual difference in the radiated noise? AUTHORS' REPLY The authors would like to thank Dr. Morgan for pointing out this phenomena. The significant difference in the average acoustic impulse measured for the two headforms prompted the authors to investigate several factors which could explain the difference. Care was taken to accurately measure the true bubble maximum volume, since bubbles on the I.T.T.C. body were often larger than those on the Schiebe body. Yet, bubbles of equal maximum volume on the two headforms were found to produce significantly different impulses. In fact, a listener standing near the tunnel could easily detect the difference in the acoustic emission between the two headforms. Consequently, the authors have concluded that different acoustic impulses generated by bubbles of equal maximum volume result from the significant difference in the bubble collapse mechanisms, in turn influences the radiated noise. 348