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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Influence of acoustic interaction in noise generating cavitation Jan Haliander (SSPA Sweden AB, Sweden) Goran Bark (Chalmers University of Technology, Sweden) ABSTRACT The primary aim was to conduct a preliminary study of the role of interaction between a few cavities on the generation of high frequency cavitation noise. Differ- ent cavitation processes, statistical distributions of parameters and initial conditions for interacting cavi- ties were observed using high-speed films from experi- ments. The numerical study concerns acoustic interaction between two spherical cavities. Monte Carlo simulations with random input data were carried out. Except at the highest frequencies, the interaction between cavities is not critical for most predictions of cavitation noise. The sizes, the number of cavities and the pressure forcing the collapse must still be well sim- ulated. However, the interaction is important at the highest frequencies. 1 INTRODUCTION High frequency cavitation noise generated during the collapse of vapour cavities, caused for example by a marine propeller (Fig. 1), is investigated in this paper. Although the discussion in the paper has primarily to do with the sheet cavity, it is valid in principle also for vortex cavities. During part of the cavitation cycle, the sheet cavity typically breaks up into cloud-like sub- cavities, that may later collapse close to each other. These "break-offs" usually start with the growth of dis- turbances due to re-entrant jets in the sheet cavity. These disturbances make parts of the sheet grow thicker, while the visual appearance changes from glossy to a white bubbly structure. The broken-off white cloud cavities are usually believed to be clusters of a vast number of small vapour bubbles. The final part of the collapse is often very violent, generating high and narrow pressure pulses (Fig. 1~. These pulses have a broadband (continuous) spectrum and dominate Figure 1: Model propeller behind a ship model in the SSPA cavitation tunnel. Noise signals from three hydrophores in the near field of the cavitating propeller. The sharp pulses are generated by collapsing cavities. the high frequency part of the cavitation noise spec- trum. The highest pulses can also contribute to erosion of the propeller. The flow field and its characteristics determine the break up of the sheet cavity and its disintegration. This disintegration determines the initial conditions for the following collapses. Important initial conditions are the sizes of the resulting cavities, the space between them and other cavities or boundaries. The disintegrated parts are convected downstream by the flow, i.e. they are transformed to travelling cavities with the potential to be noisy. The local flow field partly determines the surrounding pressure and the start of the collapses. The collapse is forced by the difference between the surrounding pressure (time dependent) and the pressure of the vapour and gas inside the cavity. Hence, the surrounding pressure is a key parameter; it is com- posed of the pressure in the undisturbed flow, the pres- sure disturbance associated with the local non- cavitating flow, the possible pressure disturbance from stationary cavities and, finally, the pressure disturbance from the motion of neighbouring cavities. This last contribution, approached here primarily as an acoustic interaction arising from the motion of neighbouring cavities, is the main subject of the present paper. That
such interaction between collapsing and expanding cavities can result in significantly higher pressure pulses than the collapse of independent cavities was first demonstrated theoretically by van Wijngaarden (1964) and others. It is generally believed today that this interaction plays a key role in erosion. For the ero- sion problem, it is thought that a large number of small cavities close together (cloud cavitation) and close to a solid surface is of relevance; however, for noise gener- ation, a few rather large cavities may dominate. The main engineering question underlying this study is whether it is necessary to account for interac- tion between individual cavities in a model of the type suggested by Matusiak (1992) for prediction of cavita- tion noise. A preliminary analysis of the main effects, made by applying a simple model, was intended to yield some guidelines for the possible need of develop- ment and application of more elaborate methods. Simi- larly, scale effects in the distribution of cavities on a model scale propeller in a cavitation tunnel may, under the influence of interaction, also lead to scale effects that affect the prediction of full scale noise. This pre- liminary analysis was conducted as a Ph.D. project at Chalmers University of Technology and the results are presented in Hallander (2002~. 2 SYNCHRONIZATION AND INTERACTION IN COLLAPSE PROCESSES Synchronization and interaction are key concepts used frequently in the paper. Synchronization of a number of cavities in a collapse means here the process by which a pressure field of a given spatial extent and temporal duration causes the cavities to collapse more or less simultaneously. The synchronizing pressure field can be generated by a global flow, for example, over a pro- peller blade or by a significant local flow process such as the collapse of a nearby large cavity. Although the cavities can collapse independently of each other, they can also begin to interact acoustically and hydrody- namically during the collapse. When the interaction becomes strong, the importance of synchronization by the global flow gradually declines, and the collapse of some cavities can finally be forced by a pressure aris- Synchronisation and interaction are most probable when cavities are generated by disintegration of a mother cavity, so that they appear close together in time and space. Typical behaviours in different flows are discussed in Sections 5 and 7. 3 CHOICE OF METHODS, PRIORITIES, APPROXIMATIONS AND LIMITATIONS The choice of cavity configurations in this study corre- sponds to characteristic configurations among the large voids (but not the very largest) that take part in collapse events where interaction between the cavities is likely. To find representative configurations with noisy cavita- tion, high speed films were studied. That significant interaction between small and closely spaced cavities within collapsing clouds (that generate the very highest frequencies) can be expected has been shown in numerical studies by March (1980) and Chahine and Duraiswami ( 1992~. In this study it was instead a prior- ity to study the influence of the interaction between the larger sub-cavities taking part in fast collapses. By studying this problem, it is possible to estimate typical lower frequencies at which the influence of interaction can be expected. However, it was not a goal to deter- mine the lowest frequency limit for interaction effects. The complexity of the cavitation process is a major problem in this work: for example, it was hard to select a model simple enough to be manageable but still useful in describing the effects. In the present study it was important to vary several parameters of interacting cavities. A code based on the most advanced model for the interaction of many cavities of general shape can be very time consuming, which is why it was disregarded at this stage. After an initial study (Hallander 1995), a model describing two spheri- cal cavities only was selected. Although two cavities are of course the very lowest limit, they can neverthe- less reflect basic properties of the processes studied on film (which normally have three to five sub-cavities of significant size in addition to the main cavity). The number of high pressure pulses observed are often rela- tively few (see Fig. 1), which also supports the choice of a model with a few cavities. For spherical cavities, process by which the pressure radiated from one cavity the model is rather complete, which permits acceptable adds to the pressure forcing the collapse of a neigh- studies of parameter influences and adaptation to bouring cavity. When the interaction is strong, this experimental observations. A key aspect was also the pressure can be the dominating pressure forcing the chance to make extensive Monte Carlo simulations collapse. using random parameter values for the model. ing from the interaction between nearby cavities. In this context, (acoustic) interaction refers to the / 2
4 THEORETICAL MODEL 4.1 Fujikawa's two bubble equations with modifi- cations The bubble model selected was originally used by Fujikawa (1984) and Fujikawa and Takahira (1986~. The interaction is simulated with coupled equations of radial motion for two spherical cavities: R (t)R (t) 1- " + 3R2`t' 1 _4R ~ ~ (1) [ Cw ] 2 [ 3cw] + ~ Spa ~ (t) + Pw 7 (R2~2)R2~2) + 2R2~2~) R.(t). ~ -P7~(R~(t) t)- Pl~(R~(t) t)] = 0 _ 2' ~ + 3R2(t) 1 - [ Cw ] 2 [ 3cw ] (2) + ~ ~Pa2(t) + Pw ~ (R~)R~) + 2R~) -Pr2(R2(t),t)- C Pt2(R2(t)'t)] = 0 where Rifts is the radius of cavity number i (Ci), CW is the velocity of sound in water, Pw is the density of water and l is the distance between the cavity centres. The main geometry is illustrated in Fig. 2. The exter- nally applied pressure Pai~t) is further discussed in sec- tion Section 4.3. The retarded time (i is (i = t - (I - Ri(<i)~/Cw · `3y The pressure in the liquid at the wall of Ci is PLi'Ri~t) t) = PVi fty + p i~t' _ 2SW _ 4`awR'( ~ (4 where PVitt) is the vapour pressure, Pgitt) is the gas pressure, sw is the surface tension of water and ,UW is the dynamic viscosity of water. The influence of the viscosity term in Eqn. (4) has been shown to be very small in comparison with those of the compressibility of the liquid and the interaction between bubbles (Fujikawa and Takahira 1986) and is often disregarded. ~_ Figure 2: Main geometry for two cavities, C1 and C2, with radii Ri and spacing 1. The pressure generated by the cavities is calculated at the distance of ri from the centre of each cav- iW. Artificial damping To obtain reasonable pressure pulses, needed for a meaningful simulation of the inter- action effects, some additional damping was needed. Cavities observed on propeller blades and hydrofoils appear to have much greater dissipation than the damp- ing provided by the last term in Eqn. (4~. Hence, an artificial damping term was introduced to simulate losses due to phenomena not fully accounted for by the model. Examples of such phenomena are lack of spher- ical symmetry of real cavities, continued disintegration and reshaping of the cavities during the collapse, and imperfections in the modelling of vapour and gas con- tent. Accordingly, Eqn. (4) was replaced by PLi'Ri't', t) = Pvift) + Pgitt) - Ri(~) Kd Rj¢~/R (5) where Kd is the artificial damping parameter. An artifi- cial damping term of the form used in Eqn. (5) is inde- pendent of the initial cavity radius (Roi); the pressure amplitudes generated (without interaction) remain pro- portional to the initial radius. Other parameters that affect the damping of the collapse are the initial gas pressure and the accommodation coefficient. For parameter values, see Section 4.5. Cavity shape Due to interaction with inhomogeni- ties, such as boundaries or other cavities, or the growth of initial disturbances, the shape of a cavity generally changes during a collapse. In principle, these deforma- tions can be taken into account, for example by using a boundary element method, Chahine et al. (1992~. As regards the deformation during the collapse, it is indi- cated in Bark and van Berlekom (1978) and further conf1rmed in Bark (1986), that cavities of almost any initial shape often roughly approach a spherical shape towards the end of a collapse. A key mechanism in this transformation is the disintegration of the cavity into parts. Based on such observations, studies of spherical 3
cavities can be justified, at least as a first approxima- tion and when the primary purpose is to study interac- tion and radiation dominated by the monopole behaviour. In fact, it should be kept in mind that the volume variation corresponding to spherically collapsing cavi- ties (monopoles) is a more effective source of noise than deformation motions (multipoles), so the pressure generated is usually over-estimated in the model used here. An effect of deformation is that it can be expected to result in more viscous dissipation and thus less vol- ume acceleration than for spherical cavities. This latter effect is simulated in the computations by the artificial damping term, Eqn. (5~. However, the selected value of the damping parameter in this study tends to result in too much damping of the motion during the final col- lapses; the deformation losses occur earlier in the proc- ess and should be compensated for in a different way. Relative motion between the two cavities Although the translational motions of the centres of the cavities sometimes are critical, they are not taken into account in the present study. This may be an acceptable approximation for interaction where the dis- tance between the cavities is relatively large. However, when the forced cavities are generated at the boundary of the forcing (main) cavity, as shown in Schoon and Bark (1998b), this approximation can be expected to lead to a significant underestimation of the interaction. During a collapse, the flow induced by one cavity usually results in a translation of the neighbouring cav- ity. Experiments by Testud-Giovanneschi et al. (1990) show that the cavities move towards each other, and they deviate from their spherical shape. The closer the cavities are to each other, the larger these effects become. They also show that this influence is strongest on the smallest cavity. Thus, the smaller cavity is closer than the initial interdistance l to the larger one when it collapses. However, support for the present approximation was obtained from high speed films of cavities on foils, e.g. Fig. 6, where no significant trans- lation between the cavities is observed for distances and diameters similar to the ones used in the present simulations. 4.2 Gas and vapour pressures inside a cavity It is often supposed that the vapour pressures inside the cavities, PVift), increase during a very fast collapse, because there is an upper limit of the rate of condensa- tion of the vapour. The use of a non-equilibrium vapour pressure allows a smaller amount of permanent (non- condensable) gas, PgO, at the start of the collapse; thus, the cavity will return to a smaller and more realistic equilibrium radius after the collapse. The equations of mass conservation and states for the gas and vapour within the cavities give the rates of change of gas pressure and vapour pressure within the cavities (Tomita and Shima 1979~: Pgi~t) = Pgi (t) (Ti (t) Ri (t) ) Pvitt) = Pvitt) {TimeR. `~:Rift)_ i ~v: Lei Pvi(~) 42~/ ,4~) PVitt))- ft1 (6) (7) where A is the accommodation coefficient for conden- sation and evaporation, Kv is the gas constant of the vapour and Taint) is the temperature of the liquid at the wall of the cavity Ci. The saturation vapour pressure PLei~t) iS a function of TLift) and Ri~t). An expression can be found in Sato et al. (1996) or Atkins (1995~: PI,ei (t) = P e {2sw/[Ri(~)pwKvT~;(~)] } (8) where PVe is the equilibrium vapour pressure of the liq- uid far from the cavities. The temperature Ti~t) of the gas-vapour mixture within Ci is given by Tomita and Shima (1979~: Titt) = R. (t)~(K DPvi(~) + (Kv1)Pgitt)] {(KV - 1 )(Pgi ft) + PVi ft))Ritt) + APi, e i tt) ( Ti f t) - TL i tt) ) ~ } - (9) where Kg and KV are the specific heat ratios of the non- condensable gas and vapour, respectively. The temper- ature in the liquid at the wall of Ci is approximated by the temperature in the liquid far from the cavities, Sato et al. (1996~: T~ift)~ Too 4 (10)
This approximation assumes that Trif t) is much smaller than Titt), meaning that the heat transfer is negligible. Although still approximate, this procedure yields a more realistic behaviour of the interacting cav- ities than the use of a constant vapour pressure. To take the variation of temperature in the liquid with time into account requires solution of the energy equation (Tom- ita and Shima 1979, Fujikawa and Akamatsu 1980~. Gas diffusion The effect of gas diffusion is dis- cussed by Watanabe and Prosperetti (1994~. The effect of this mass transfer is that the gas content of the cavi- ties is higher when they reach their final equilibrium radii after several collapses and rebounds than it was in the original cavitation nuclei. A cavity may also cap- ture cavitation nuclei or other cavities during its life- time. However, mass transfer of gas is not included in the model above. 4.3 The externally applied pressure on a hydrofoil or propeller blade The externally applied pressure Paints in equations (1) and (2) can be an arbitrary function of time (a slight modification of Fujikawa's original equations). The spatial pressure distribution on a cavitating foil or pro- peller blade section is not obvious; with cavitation the pressure distribution changes. The pressure equals the vapour pressure for the part covered by the sheet cav- ity. After a transition zone, the pressure returns to that of a non-cavitating body, according to van Oossanen (19744. If there is a stagnation point behind the sheet cavity, the pressure gradient can be very steep. The steepness of the pressure gradient in this zone (Fig. 3) has a large influence on the collapse times and, thus, on the violence of the collapses. This is demonstrated in Hallander (1999~. The time history of the externally applied pressure is also affected by the time variation of the inflow and, for an oscillating foil, also by the oscillation of the foil (Schoon 2000~. Thus, the gradi- ent of Part) may be somewhat steeper than estimated above. A preliminary way to study this effect is to imple- ment the time history of the pressure increase as an increase from PVe at the point of disintegration to the undisturbed pressure POO at the end of the foil, as Matu- siak (1992) did. If the cavity is assumed to move with the undisturbed flow velocity, UOO' after disintegration at time, to, this motion in the spatial pressure distribu- tion on a foil or propeller blade gives the external pres- sure as a function of time (Fig. 3~. An observation made by Schoon (2000, Fig. 6.3, p. 54) supports the theory that a bubble just outside the boundary layer 16 12 10 8 6 4 2 ~ Poo / ' Adopted pressure increase Lip - Linear pressure increase ve tOi+tinci Figure 3: The solid line shows the time history of the exter- nally applied pressure (Pa(t)) used in the simulations. The collapse of a cavity starts at toi. The dashed line shows a lin- ear pressure gradient, which was used by Matusiak (1992) for example. moves with a velocity close to UOO- In some experimen- tal observations, due to local flow variations (e.g. vorti- ces), the cavities do not move during the collapse. Typical collapse times of the cavities studied are about 2 to 4 ms for the first collapse. This is shorter than the time of the pressure increase from Pee to POO assumed above (which is 8.4 ms). The cavities thus collapse under a pressure of less than POOP and the motion decays before they reach the trailing edge. 4.4 The numerical solution Fujikawa's equations, (1) and (2), were solved together with equations (5), and (6H9) for each cavity. It was assumed that the cavities were initially at rest at a uni- form pressure and that each cavity starts its collapse from an initial radius, Roi at time toi when it is exposed to the externally applied pressure, Paid). These initial conditions could have been reached either by the growth of a cavitation nucleus or by the disintegration of a larger cavity into parts. The system of coupled ordinary differential equa- tions (ODE) was solved with a MATLAB - SIMULINK model. A variable order solver of the Adams-Bashforth-Moulton type ("odell3") was used. The time-stepping is critical when a cavity reaches its minimum; an adaptive solver which varies both step size and order is strongly recommended. The retarded time, Eqn. (3), was approximated by {i = t - (I - Rift)~/cw (11) in the SIMULINK system.
The pressure generated by the two cavities, pity =pltrl,t) +p2(r2,t), was calculated at the distance of ri = 1 m from the centre of each cavity, Fig. 2. This distance is much greater than the distance between the cavities (ri>>l) in the configurations studied. The finite speed of propagation is also disregarded when the generated pressures are computed, since it is merely a comparison at the same distance. The gener- ated pressure is then: pi~ri,t) = p(R' (t)Ritt) + 2Ri~t)R' (t)~/ri . (12) For each simulation, the peak pressure amplitude defined as Pma¢~` = maximum~p~t99 was recorded. The average power density spectrum G69 for each example (in the Monte Carlo simulations) was esti- mated by the following process. The signal pity from each simulation was interpo- lated with a time step of 2 10-8 s (fs = 50 MHz). This time step was found small enough to give a good representation of the signal. The interpolated signal was decimated (i.e. low- pass filtered and down-sampled) tofu= 1 MHz. · A windowed periodogram G( )~ was computed for the decimated signal from each simulation: N- ~ _ G(k)¢f' = ~ ~ p~n~w~njej21lin (13) n = 0 k= 0,1,...,K-1 where pan) is the decimated sample sequence of length N from each simulation and K is the number of simulations in the example. The sample sequence was padded with zeros so N became the next power of two. The window function won) was a Hanning window of length N. The average power spectral density (PSD) G69 was computed as the sum of all the periodograms divided by the number of periodograms (K) and the norm squared of the window function (U9: K- ~ (f) KU ~ (A (14) k = 0 where N- ~ U = N ~ W2(n). (15) n = 0 4.5 Choice of model constants The physical constants used in the simulations where chosen to correspond to cavitation tunnel conditions (fresh water, T.= = 20 °C), and are shown in Table 1. The values of the initial gas content, Pgoi, the accom- modation coefficient, A, and the damping parameter, Kd, are constants that strongly affect the model behav- iour. They are selected (tuned) to give the model rea- sonable behaviour in comparison with experimental data. In Hallander (1995) it was shown that a lower artificial damping or a lower initial gas pressure makes the simulated pulses higher and narrower. An increase of the accommodation coefficient also has a similar effect. A systematic variation of the accommodation coefficient, carried out by Fujikawa and Akamatsu (1980, p. 507, Table 2), shows that a higher value of A causes a cavity collapse to a smaller minimum radius and generates higher pressure. Table 1: Physical constants and others that affect the model behaviour used in the simulations. Parameter Value Too 20 °C cw Pw . sw V . Kg Pve A Pgoi Kd 1483 m/s . 998.2 kg/m3 . 0.07061 N/m 1 13 461.9 J/(kgK) 1.40 i.33 ~- 2.337 kPa _- 0.015 0.010 Pa 2.337 kPa 50 Pas/m The damping parameter is an artificial constant (see page 3), while the other two parameters have a physical interpretation. However, it is not possible to experimentally determine their values. Values of A as low as 0.01 have been suggested for evaporation in some studies, according to Brennen (1995~. Fujikawa and Akamatsu (1980) refer to a study where 0.04 is suggested. The comparison of an average measured sequence at unsteady cavitation with one of the highest simulated ones, Section6.2, indicates that the total 6
(a) 1~ \~ / (b) Figure 4: Schematic cavitation model explaining the nomenclature. A sheet cavity CO disintegrates into a main cloud formation (first generation sub-cavity) Cat. During the disappearance of CO and Cat, a number of (second generation) sub-cavities C2 are generated by farther disintegration of CO and Cat. 3. model damping chosen was somewhat too high. Con- sidering this, a value slightly higher than 0.015 would have been a more appropriate choice. The initial gas pressure is assumed to be the same within all cavities in the present study. This simplified assumption may be justified by arguing that all of the cavities in a simulation are generated by disintegration of the same main cavity; thus, they have the same ini- tial gas pressure. In practice, the gas pressure is unknown for a cavity which, after a complex history (see "Gas diffusion" on page 5), is about to start its collapse. However, for a cavity growing from a nucleus of known diameter, the initial gas pressure can be esti- mated from the pressure balance expressed by Eqn. (4~. 5 EXPERIMENTAL OBSERVATIONS OF CAVITY DISINTEGRATION AND STATISTICAL PROPERTIES Use of experimental data Experimental data were used at different stages of the study. 1. Choice of bubble model: Observations of the number of larger cavities and the number of sharp pulses were used in the choice of theoretical model, Section 3. Choice of geometrical configurations: Observa- tions of sizes of cavities and spacing between them. Statistical data for the Monte Carlo simulations (Section 6.2) were estimated from the experi- ments. Observations from earlier experiments Based on basic studies (Hallander 1995, Hallander and Bark 1997) and the examination of high speed films, the time displacement, space between cavity centres, and initial radii of the cavities at the start of the collapses were believed to be key parameters; hence, they were treated as stochastic variables. High speed film obser- vations revealed that the statistical properties of the cavitation process (especially the disintegration of larger cavities into parts) depend on the flow condi- tions. In Hallander (1998), two (somewhat idealised) types of processes were identified. The first type, shown in Fig. 4, was to simulate unsteady cavitation which occurs when, for example, the wake behind a ship, has large gradients in time and space. This pro- duces strongly correlated disintegrations that generate closely situated cavities which are likely to interact during a rather synchronized collapse. The second type of process was to simulate almost steady cavitation which can occur in a wake with small gradients in time and space. This results in weakly correlated disintegra- tions and less synchronized collapses. Estimates of dis- tribution parameters for the stochastic variables were made for these two model processes. These distribution parameters were then used to generate random input data to Monte Carlo simulations with the system of two spherical cavities, Section 6.2. An improved experimental study In cooperation with another project (Schoon 2000), a new method for the generation of unsteady cavitation on a stationary foil was developed and tested in the SSPA cavitation tunnel (Hallander and Bark 1998, Hallander 1999~. The properties of cloud cavitation for four conditions, ranging from steady to highly unsteady, were analysed. Input parameters for numerical studies were then esti- mated from two of these conditions. An unsteady inflow was generated by an oscillat- ing foil (pitching around its mid-chord axis) positioned upstream from a stationary test foil. This has advan- tages over a conventional arrangement with an oscillat- ing test foil. Although an oscillating test foil could generate approximately the desired motion, it tends to overemphasize the synchronization of bubbles, as well as the collapse-forcing pressure, both due to unwanted pressure terms caused by the pitching motion (Kruppa 1986 and Schoon and Bark 1998a). The new arrange- ment used was intended to generate a more propeller- like cavitation development. As expected, the new experimental set-up did not generate as violent and coherent collapses as the earlier experiments with one oscillating foil did. 7
(alt= 100.7 ms (b~t=213.5ms (a) t= 450.1 ms (b) t = 466.0 ms c) t = 216.6 ms (d) t = 240.8 ms Figure 5: Quasi-steady cavitation on a foil with oscillating inflow (2 Hz), Uoo=5.0 m/s and cavitation number ~=1.0. The re-en~ant jets have time to develop fitlly and to disintegrate parts of the sheet cavity several times during one oscillation cycle. When the inflow to the test foil is stationary or varying slowly, disintegrations of a steady or quasi- steady character occur, as shown in Fig. 5. The re- entrant jets have time to fully develop and are con- stantly breaking off parts from the aft section of the sheet cavity in a boiling-like pattern. This pattern is influenced by the length and thickness of the sheet cav- ity. As this cavity grows longer and thicker, the disinte- grated cavities also become larger. The formation of first generation sub-cavities appears to be dispersed quite well in time, as well as in space, although the times at which they form are somewhat correlated by the re-entrant jet mechanism. When the oscillation of the inflow to the test foil is further increased, the disintegrations become unsteady, as shown in Fig. 6. More coherent cloud formations are broken off from the sheet. These main, first generation sub-cavities (typically about four) disintegrate further into second generation sub-cavities which are close together. The collapse of these coherent clouds gener- ated significantly higher pulses (about ten times) than the more random collapses under steady and quasi- steady conditions. An extreme example is that of downstream mov- ing collapses at highly unsteady cavitation, see Fig. 7, where a fast collapsing glossy sheet generates a large number of surrounding cavities (Schoon and Bark 1998b). This results in a very violent, synchronized and almost simultaneous collapse of all the cavities. Summary of experimental observations From ear- lier experiments, it is concluded that re-entrant jets can significantly influence the disintegration of a sheet 8 c) t= 467.3 ms (d) t = 467.7 ms Figure 6: Unsteady cavitation on a foil with oscillating inflow (lOHz), Uoo=5.0 m/s and a=1.0. Disturbances arise in the sheet cavity (a), which break up into cloud-like sub-cavities that collapse (b - d). (a) t= 231.7 ms (b) t = 234.9 ms c~t=235.9ms (d~t=236.4ms Figure 7: Highly unsteady cavitation (with a downstream moving collapse) on a foil with oscillating inflow (15 Hz), Uoo=5.0 m/s and ~=0.78. cavity. From the present experiments with foils in unsteady flow, it follows that the disintegration of the sheet into parts can also be significantly influenced by the variations of the unsteady inflow. Due to this unsteady inflow, the sheet can disappear from the upstream edge, which transforms a part of the sheet into a travelling cavity. These two main mechanisms and the balance between them influence the sub-cavity distribution in time and space, and consequently, the amount of acoustic interaction. Apart from re-entrant jets and variation of inflow, the motion of a wavy cav- ity surface towards the blade surface results, for rea- sons of geometry, in further disintegration.
Some behaviours, which depend on the size and spacing of the cavities, can be identified. 1. Classical cloud cavitation with a vast amount of small bubbles close to each other can undergo a highly synchronized collapse, as described by March (1980) and others. This type of collapse is not included in the present study. Instead, attention is given to larger and more sparsely distributed voids, which can be parts of a disintegrating cloud. The aim was to investigate the significance of the interaction between these larger voids. 2. When the variation of the inflow to a foil or pro- peller blade is slow, the sheet cavity approaches a steady state behaviour; re-entrant jets develop and the sheet disintegrates into multiple, rather small bubble formations, distributed over a relatively large area. Consequently, the probability that many bubbles would collapse simultaneously and close together is rather limited. It is also signifi- cant that most of the bubbles are of about the same size. This behaviour is referred to as weakly syn- chronized collapse, in which interaction can also be expected to be rather weak, since the cavities are relatively separated in time as well as in space. When the inflow varies rapidly, the re-entrant jet and the related cloud formation may not develop completely; specifically, all the cloud formations it generates may be almost simultaneously exposed to the pressure forcing the collapse. In this case interaction can be supposed to be more important. 6 NUMERICAL STUDIES 6.1 Basic studies The behaviour of the equation system, describing two spherical, interacting cavities (Section4.1), was stud- ied by varying the input data. An example of time his- tories of radii and pressure for such a system is shown in Fig. 8. The second collapse of C2 is significantly amplified here by the pressure generated from the first collapse of Car. Figure 9 shows the time histories for the same system without the interaction. The effects of parameter variations on the interac- tion between two cavities were demonstrated in Hal- lander (1995~. Examples of parameters varied included the space between the cavities, the initial gas pressure and the artificial damping. The initial gas pressure within the cavities was observed to strongly affect the behaviour of the cavities and the pressure pulses gener- ated. The results show that higher initial gas pressures or higher artificial damping reduce the pressures gener- ated, which results in less interaction. The maximum was found to occur when Cat generates first a positive pressure during the collapse of C2 followed by a nega- tive one during the rebound. This implies that the time displacement between the cavities is of great impor- tance, as was also shown by Sato et al. (1994~. The time displacement between the two collapsing cavities was further investigated in Hallander and Bark (1997~; the effect of a systematic variation of the time displacement is given in Fig. 10. The maximum pres- sure amplitudes generated by each cavity and the total energy radiated by both cavities were studied as func- tions of the time displacement. Although significantly higher pressure pulses were found, the increase occurred only within a quite narrow band of time dis- placements, as shown in Fig. 10; this may reduce the effective influence of interaction for statistically dis- tributed cavities. The interaction influences the smaller cavity much more than the larger one. According to Fujikawa and Takahira (1986), this influence increases when Ro2/Ro~ decreases. Interaction and overlap were observed to increase the total energy radiated at the highest frequencies. The total energy radiated reaches its maximum when the pressure peaks generated by the two cavities overlap; it sinks to a minimum when they are in the opposite phase. When the distance between the cavities is decreased, the interaction becomes stronger, and both the generated pressures and the total energy radiated increase. Sometimes a "capture" of the smaller cavity was observed. This means that the motion (expansion and collapse) of the smaller cavity is strongly influ- enced by the pressure from a larger neighbour cavity. Such a capture is shown in Fig. 8. This capture gives a small expansion of C2 and prolongs the time to its first collapse. The analysis of two interacting cavities was, by an approximation, extended to the interaction of a few more cavities in Hallander and Bark (2000~. This gave an increase of approximately 3 dB to the energy spec- tral density above 20 kHz. 6.2 Monte Carlo simulations Basic studies (Section 6.1) showed that the interaction between the cavities was sensitive to several parame- ters. The following questions can then be raised: How often does maximum interaction occur and what is its average effect in more realistic cavitation processes? To facilitate describing the generation and collapse of cavity distributions, high speed films were analysed and distribution parameters were estimated. Monte
7k 61 5 1 2 O rat 0.12 0.10 0.08 0.06 - 0.04 0.02 O. 1 2 \ \ \ - 1 -0.02 0 1 2 3 t [ms] 4 5 6 Figure 8: Time history of radii (upper) and generated pres- sure (lower) of two interacting cavities: alto ~ 0.44 ms, I ~ 11 mm, Rol ~ 6.4 mm and Ro2 ~ 3.0 mm. The second eol- lapse of C2 is amplified significantly here by the pressure generated from the first collapse of Cl. 6 5 2 O 0 1 0.12 0.10 0.08 0.06 0.04 0.02 OF - 1 -0.02 o . . 1 1 7 - _ _ - C ~" _C, 4 3 - \\/ \ ~_~ 3 4 5 6 t [ms] 2 \ \ ~- \ \ / \ \/ ~ ~ . J. \'_ 3 4 5 t [ms] 6 2 3 t [ms] 4 s 6 Figure 9: Time history of radii (upper) and generated pres- sure (lower) for the same initial configuration as in Fig. 8 but computed without interaction. 0.25 Carlo simulations were used to investigate the average influence of interaction between statistically distrib- 0.20 uted cavities (Hallander 1998~. In a new series of simulations (Hallander 1999), distribution parameters estimated from two sets of new experimental conditions were used to produce 2000 sets of random input data. These distributions of input data are shown in Figures 1 1 and 13. The distributions of peak pressure amplitude (resulting from the Monte Carlo simulations) are shown in Figures 12 and 14. Average power density spectra were also calculated, see Fig. 15. The new estimates of distribution parame- ters, in combination with a new model for the exter- nally applied pressure (Section 4.3), gave smaller effects of interaction on average than those used previ- ously. Nevertheless, the simulations indicate that for strongly coherent collapses, the interaction can result in some very high pressure pulses. 10 0.15 0.10 0.05 o -2 -1 0 1 Ato [ms] Figure 10: Maximum pressure amplitude (at r = 1 m) gener- ated by each cavity as a function of difference in starting time: Rol = 8.0 mm, Ro2 = 4.0 mm and I = 32 mm.
0.8 , 0.6 0.4 0.2 c' o 0.25 ~ 0.2 5 0.15 , 0.1 0.05 O _ o 0.25 ~ 0.2 `~5 0. 15 , 0.1 ._ ~ 0.05 , h . . 1 ~ ( ) 2 4 6 8 0 let = t - t [ms] J( _ . _ 0.25 ~ 0.2 `~= 0. 15 :> 0. 1 0.05 _ O ~ 2 4 6 ~ 0 Rn1 [mm] Figure 11: Distributions of random input data from simula- tions of unsteady cavitation. 0.12, 0.1 c> =3 0.08 0.06 c, ;> ._ V 0 04 0.02 O- \J I /. ~ 1 C) ~ 0.~s 5 A: 0.06 ._ V 0 04 0.02 ,2 x10-3 1 Ill llm ~ 0.04 0.06 ~ 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 p lP max 00 o 0 0.02 0.04 0.06 0.08 0.1 0.12 p lP max of Figure 12: Distribution of peak amplitudes from simulations of unsteady cavitation with interaction (upper) and without interaction (lower). 0.25 cat 0.2 0.15 ;> 0.1 c, 0.05 O _ 0.25 ~ 0.2 Co.ls , 0.1 ~ 0.05 _ ~ O 20 40 60 -4 -2 0 2 4 no I [mm] At =t -t [ms] 0.25 cat ~ 0.2 Co.ls , 0.1 . - 0.05 O 20 40 60 I [mm] 0.25 ~ 0.2 =0.15 , 0.1 _ 0.05 O 2 4 6 8 0 2 4 6 8 0 2 4 6 8 Ro2 [mm] Ro1 [mm] Ro2 [mm] Figure 13: Distributions of random input data from simula- tions of quasi-steady cavitation. 0.1 t3 0.08 0.06 c' ._ c' 0 04 0.02 O O. O - IL 004 0.064~0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 p lP manic on 0.12 c' ~ 0.08 3 G 0.06 <,, 0.04 0.02 O _ 0 0.02 0.04 0.06 0.08 0.1 0.12 p lP max of Figure 14: Distribution of peak amplitudes from simulations of quasi-steady cavitation with interaction (upper) and with- out interaction (lower). 11
~ 100 m 80 - 60 $ c 40 o O 20 120 =~ I - without interaction I ~ . ~ . . . . . . . . . . . . . . ..... . . . . .... 105 lo6 103 104 f [Hz] 80 60 40 20 O I with interaction I I - without interaction I . . . . . . . . . . . . ..... . . . . .... lot lo2 103 104 lob lo6 f [Hz] Figure 15: Average power density spectra from simulations of unsteady cavitation (upper) and quasi-steady cavitation (lower). The expectation of the initial radii are larger in the quasi-steady ease, which results in higher spectral levels. When comparing some time series of pressure sig- nals, the simulated peak pressure amplitudes appear higher than measured ones. However, after decimation of the simulated signal (to a sampling frequency of similar size), the order of magnitude is approximately the same, as shown in Fig. 16. The upper graph shows the same signal as in Fig. 8, but here it is decimated to fs 250 kHz. The lower graph shows an average sequence of a measured signal (from the unsteady experimental condition, Fig. 6~. The decrease of the simulated signal after decimation indicates that the interaction has the most influence well above 100 kHz in this example. Comparison of simulated spectra with and without interaction, Fig. 15, shows that the average increase of the PSD due to interaction is less than 3 dB below 60 kHz in model scale (the difference increases with frequency). The conclusion drawn from these observations is that the interaction does not signifi- cantly influence the radiated noise for lower frequen- c~es. 0.030 0.025 0.020 0.015 0.010 0.005 Ot...... -1 -0.005- 2.5 0.03O 0.025 0.020 0.015 0.010 0.005 ~~- ............. / 3 3.5 4 t [ms] - 2 11 - 1: 1 ~ ~1 ~ . . - -0.005 2275.5 2276 2276.5 2277 t [ms] Figure 16: Comparison of a simulated signal of two inter- aeting cavities (upper) with a measured signal where the highest peaks are dominated by only a few cavities (lower). The simulated signal is the same as in Fig. 8, but decimated tof5 250 kHz. The measured signal is sampled atf5 262 kHz. The expectation of the initial radii are larger in the quasi-steady example, which results in higher spectral levels, Fig. 15. This is so because only parts of the cav- itation processes are taken into account when investi- gating the importance of interaction between cavities on different scales. In more complete models of cavita- tion processes, the number of collapses of different types involved in each process must be taken into con- sideration. Further comparison of the two signals in Fig. 16 shows that they behave similarly, but the simulated one has wider pulses. Although the measured sequence shown is a quite average one, the simulated one is among the highest found in the Monte Carlo simula- tions. The conclusion can be drawn from this that the total model damping used in the simulation studies was too high. The choice of these parameters was discussed in Section 4.5. However, the two sequences show that the model is able to generate reasonably realistic sig- nals. 12
6.3 Influence of interaction on cavity motions, sig- nals and spectra: Summary of observations 1. From an acoustical point of view, the pressure fields generated by each cavity are scattered by neighbouring ones. In particular, if the time dis- placement of the motions of the cavities is favour- able, the pressure pulse from a large cavity changes substantially the motion of a small cavity, while the influence of the small one on the large one can be quite limited. The collapse of the small cavity often becomes more violent, while the col- lapse of the large one is slowed a bit. (These ~en- eral observations have been made by several authors.) 2. The amplitude of the pulse from the large cavity decreases a bit because of the interaction, while the pulse from the small one becomes significantly higher, although of very short duration. Here' the energy spectral density of the sum of the pulses, when the interaction is taken into account, increases considerably at the highest frequencies, while the levels at low frequencies mostly remain relatively unchanged. In real cavitation processes, involving many cavities that behave partly randomly, the trends indicated above still apply, however the details are much influenced by the statistical properties of the cavitation processes. FORMATION OF PRESSURE FIELDS THAT LEAI) TO INTERACTION AND SYNCHRONIZATION The present foil experiments, with varying strengths and periodicity of the gusts entering the test foil, con- f~rm earlier indications that the variations of the global flow in time and space can generate differing bubble distributions and influence the synchronization of the collapses. The synchronization of the collapses of bub- bles increases, i.e. all of the bubbles tend to collapse more nearly simultaneously, with increasing amplitude and frequency of the gusts in the inflow. Hence, the general conclusion can be drawn that the variation in time and space of the global pressure that corresponds to the inflow to the foil is significant; the degree depends on the amplitude, duration and spatial extent of the temporal variations of the pressure. These cir- cumstances can significantly influence noise, vibra- tions and erosion as well. Synchronization can occur on multiple spatial scales. In experiments with foils in unsteady flow, con- ditions can be found in which collapses are undoubt- edly synchronized by the global flow. However, towards the end of the process, the collapses seem also to be influenced by strong interaction which gives rise to a second synchronization, this time within a smaller region. Although these configurations are of primary interest for erosion, they can also be expected to influ- ence noise at the very highest frequencies. Due to this complexity, it is concluded that simulation of cavita- tion noise by model experiments is still the most relia- ble method; this can be expected to remain so, possibly for a long time. However, even experimental methods can be tricky. The processes involved in the disintegration of the main cavity (a sheet, for example) determine the distri- bution in time and space of large as well as small sub- cavities. Large sub-cavities, which later on become the forcing cavities in the interaction process, are specifi- cally generated by two mechanisms: the early action of break-off due to re-entrant jets or the possible increase of the pressure (above the level of cavitation pressure) at the upstream cavitation edge which can result in a transient or "downstream moving collapse". The small sub-cavities are formed by minor break-off processes in the remaining sheet or by further disintegration of the large sub-cavities mentioned above. All such cavi- ties are then, more or less synchronously, forced to col- lapse by the globally increasing pressure (generated by the motion of the propeller in the wake). The numerical interaction model predicts higher individual pulses as well as higher mean spectra (PSD) at high frequencies, when using statistical input from the unsteady experimental condition in which the sub- cavities collapse in the most synchronized way. This trend could be anticipated and it should be noted that its influence at the highest frequencies could make some difference, although the model can be expected to underestimate the interaction effect. An important conclusion drawn from the comparison of the simu- lated examples is that the simulations, by using the sta- tistical distributions, demonstrate the effect of large scale synchronization of sub-cavities. This synchroni- sation emanates from the global flow field, which determines the development of re-entrant jets, down- stream moving collapses, etc. In summary, two conceptual synchronization mechanisms can be identified for a collective collapse of cavities. 1. The first and, usually, most global synchronization is due to the increase of the environmental pres- sure, resulting from the decrease of the angle of attack of a propeller blade or foil. This starts the collapse of bubbles in a relatively large region. 13
Second, a large and violently collapsing cavity can, by acoustic interaction, synchronize the col- lapse of nearby small cavities. Supposing that the pressure pulse from the large cavity has roughly spherical spreading, it follows that this synchroni- zation is usually more limited in space than the first one. When the second type of synchronization is added to the first one, small cavities close to the forcing one can collapse very violently. 8 SOME FURTHER COMMENTS ON THE INFLUENCE OF ACOUSTIC INTERACTION ON THE NOISE SPECTRA GENERATED Some questions, about the contribution of acoustic interaction to the noise spectra generated, which have not been addressed elsewhere in this study, are dis- cussed below. The influence of small, non-interacting cavities It is evident that cavities smaller than the ones studied in the present simulations occur in real cavitation proc- esses. Even if every pulse from the small cavities may be weak, the number of small pulses can be high, and overlapping of the pulses is possible. A general answer to this question may not exist. Interacting as well as non-interacting cavities of different sizes may exist in a specific cavitation process. The two types of processes identified in Section 6.2 indicate that the relation between interacting and non-interacting collapses can shift and is controlled by the large scale development of the cavity in time and space. It is thus possible, at least in cavitation processes where the interaction is weak and random, that the high frequency levels can be dominated by a large number of small cavities which collapse without significant interaction. However, the fact remains that noisy cavitation events in many instances are dominated by relatively few (about three to five) very high and sharp pulses per cavitation cycle. Frequency range of acoustic interaction What the present study shows is that interaction between cavities can, for a given group of cavities, extend the frequency range upwards by tenfold and generate some extremely high pulses. However, analysis of the linear scaling of cavitation noise, Hallander and Bark (2000), indicates that the formulas based on linear acoustics can be regarded as approximations for frequencies at which the spectral levels are influenced by interaction. As to the energy transferred to higher frequencies due to interaction, it cannot be excluded that the total energy radiated by the cavities is lowered by this. One reason is that when a cavity collapses to a smaller min- imum radii, the viscous dissipation increases. Another 1 reason is that the development of shock waves with higher dissipation may increase due to the extremely high and sharp pulses generated. This loss in shock waves is indicated in Levkovskii's scaling theory (Lev- kovskii 1968~. As noted above, the choice of cavity sizes implies that typical low frequencies (but not the very lowest) influenced by interacting cavities have been estimated. However, there are some other possible ways that inter- action effects can be found at lower frequencies. The behaviour of the main (forcing) cavity is dis- turbed (slowed down) by the driven cavity. The driven cavity is sometimes observed to expand early in the process due to entrapment in the low pressure generated by the forcing cavity. The effect of this is probably weak due to low radiation efficiency for the forced cavity. 3. There might be cavities present that are larger than the ones being studied. If the smallest bubbles take part in the interaction, this will determine the levels at the highest frequencies. The interaction increases by the ratio Ro~/Ro2, which means that all cavities smaller than the Ro2 used in the simulations are exposed to even more interaction; this raises the high frequency range. An exception to this may arise if the smaller cavities are much further away from the main cavity than the one being studied. In this situation it is possible that the high frequency level could be dominated by many non-interacting cavities, as discussed above. The probability of synchronization becomes higher when the ratio Ro~/Ro2 increases. In extreme cases, such as the one in Schoon and Bark (1998b), the probability of powerful synchronization of the col- lapses is high and superposition of the pulses from the driven cavities is also likely. Consequently, there are reasons to believe that interaction often dominates the high frequency spectrum, despite the possible excep- tions mentioned in the discussion above. 9 CONCLUSIONS 1. A main engineering conclusion drawn from the numerical simulations is that, provided the levels at the very highest frequencies are disregarded, the acoustic interaction between medium sized dominating structures does not make much differ- ence. This implies that if a numerical method, for exam- ple of the type presented by Matusiak (1992), can generate an acceptable size and time distribution of 14
cavities and a realistic collapse forcing pressure, it can also be expected to generate realistic noise lev- els up to medium high frequencies (i.e. up to the order of some hundred times the blade-rate). Since these requirements are rather demanding, methods of this type are not yet standard engineering proce- dures. When making predictions with model tests, the implication is that, except for the very highest fre- quencies, the distribution in space and time of nearby cavities is not very critical. However, the number and size of the cavities have to be reasona- bly well simulated, particularly if the mean value spectrum over all pulses is of interest. When an accurate estimate of the noise at the very highest frequencies is desired, the numerical simu- lations indicate that an accurate simulation of both the acoustic interaction and the statistical proper- ties of cloud cavitation is required. As the time displacements between cavities are very critical, this problem is very difficult to solve in numerical simulations. In model testing, there are still serious problems, but some success can be expected with a proper global flow. In extreme cases, for exam- ple with small cavities close to a large one, a good simulation of interaction can be anticipated according to observations. The statements in points 1 and 2 hold for interac- tion between small to medium sub-cavities (clouds or single voids) typically generated by a disintegrating sheet. These sub-cavities can be expected to determine the lower, if not the very lowest, end of the frequency range at which interaction effects appear. ACKNOWLEDGEMENTS Most of the work for this paper was carried out within the "Doctoral Student Programme" in underwater acoustics and underwater technology. The programme was funded by the Swedish Armed Forces (FM) and managed by the Swedish Defence Research Agency (FOI). REFERENCES Atkins, P.W., Physical Chemistry. 5th ea., Oxford University Press, Oxford, UK, 1995, pp. 963-964. Bark, G. "Development of violent collapses in propeller cavitation." Cavitation and Multiphase Flow Noise. ASME Winter Annual Meeting, Anaheim, CA, USA. ASME, New York, NY, USA, FED Vol. 45, 1 986. Bark, G & van Berlekom, W.B., "Experimental investigations of cavitation dynamics and cavitation noise." 12th Symposium on Naval Hydrodynamics Washington, DC, USA. National Academy Press, Washington, DC, USA, 1978. Brennen, C.E., Cavitation and bubble dynamics. Oxford University Press, New York, NY, USA, 1995, pp. 54-56. Chahine, GL. & Duraiswami, R., "Dynamical interactions in a multi-bubble cloud." ASME. J. Fluids Engng. Vol. 114, No. 4, 1992, pp. 680-686. Chahine, GL., Duraiswami, R. & Rebut, Ma "Analytical and numerical study of large bubble/ bubble and bubble/flow interactions." 1 9th Symposium on Naval Hydrodynamics, Seoul, Korea. National Academy Press, Washington, DC, USA, 1992. Fujikawa, S., "Weak interactions between two spherical bubbles in a compressible liquid." 1 0th International Symposium on Nonlinear Acoustics Kobe, Japan. ISNA Teikohsha Press, Kadoma, Japan, 1984. Fujikawa, S. & Akamatsu, T., "Effects of the non- equilibrium condensation of vapour on the pressure wave produced by the collapse of a bubble in a liquid." J. Fluid Mech. Vol. 97, No. 3, 1980, pp. 481-512. Fujikawa, S. & Takahira, H., "A theoretical study on the interaction between two spherical bubbles and radiated pressure waves in a liquid." Acustica Vol. 61, No. 3, 1986, pp. 188-199. Hallander, J., "An introductory study of the interaction between cavities in the generation of cavitation noise." Licentiate thesis, Report CHA/NAV/R-95/0037, Chalmers University of Technology, Goteborg, Sweden, 1995. Hallander, J. and Bark, G. "Influence of time displacement and other parameters on the interaction between neighbouring cavities in the generation of propeller cavitation noise." ASME Fluids Engineering Division Summer Meeting, Vancouver, BC, Canada. ASME, New York, NY, USA, 1997. Hallander, J., "On the influence of stochastic processes on the generation of cavitation noise from mutually interacting cavities." Third International Symposium on Cavitation, Grenoble, France, 1998. Hallander, J. & Bark, G. "Some statistical properties of cloud cavitation on a foil in unsteady flow." ASME Fluids Engineering Division Summer Meeting, Washington, DC, USA. ASME, New York, NY, USA, FED Vol. 245, 1998. 15
Hallander, J., "Cloud cavitation on a foil in unsteady flow. Experiments and numerical simulation of statistical properties." Report CHA/NAV/R-99/0068, Chalmers University of Technology, Goteborg, Sweden, 1999. Hallander, J. and Bark, G. "Interaction between collapsing cavities Influence on noise generation and scaling." ASME 2000 Fluids Engineering Division Summer Meeting, Boston, MA, USA. ASME, New York, NY, USA, FED Vol. 251, 2000. Hallander, J., "Influence of acoustic interaction between cavities that generate cavitation noise." Ph.D. Thesis, Chalmers University of Technology, Goteborg, Sweden, 2002. Kruppa, C.F.L., "On the relevance of cavitation inception tests with oscillating foils." International Symposium on Cavitation, Sendai, Japan, 1986. Levkovskii, Y.L., "Modelling of cavitation noise." Sov. Phys. Acoust. Vol. 13, No. 3, 1968, pp. 337-339. Matusiak, J., "Pressure and noise induced by a cavitating marine screw propeller." Ph.D. Thesis, Helsinki University of Technology, Espoo, Finland, 1992. March, K.A., "On the collapse of cavity clusters in flow cavitation." 1 st International Conference on Cavitation and Inhomogeneties in Underwater Acoustics, Gottingen, Germany. Springer-Verlag, Berlin, Germany, Series in Electrophysics Vol. 4, 1980. van Oossanen, P., "Calculation of performance and cavitation characteristics of propellers including effects of non-uniform flow and viscosity." NSMB Publ. No. 457. Netherlands Ship Model Basin, Wageningen, The Netherlands, 1974. Sato, H., Sun, X.W., Odagawa, M., Maeno, K. ~ Honma, H., "An investigation on the behavior of laser induced bubble in cryogenic liquid nitrogen." ASME. J. Fluids Engng. Vol. 118, No. 4, 1996, pp. 850-856. Sato, K., Tomita, Y. & Shima, A., "Interaction of two bubbles produced with time difference." Cavitation and Multiphase Flow. ASME Fluids Engineering Division Summer Meeting, Lake Tahoe, NV, USA. ASME, New York, NY, USA, FED Vol. 194, 1994. Schoon, J., "A method for the study of unsteady cavitation. Observations on collapsing sheet cavities." Ph.D. Thesis, Chalmers University of Technology, Goteborg, Sweden, 2000. Schoon, J. & Bark, G. "An experimental method for generating unsteady cavitation on a stationary wing." Third International Symposium on Cavitation Grenoble, France. Vol. 2, 1998a. Schoon, J. & Bark, G. "Some observations of violent collapses of sheet cavities and subsequent cloud cavitation on a foil in unsteady flow." ASME Fluids Engineering Division Summer Meeting, Washington, DC, USA. ASME, New York, NY, USA, FED Vol. 245, 1998b. Testud-Giovanneschi, P., Alloncle, A.P. & Duiresne, D. "Collective effects of cavitation: Experimental study of bubble-bubble and bubble-shock wave interactions." J. Appl. Physics Vol. 67, No. 8, 1990, pp. 3560-3564. Tomita, Y. & Shima, A., "The effects of heat transfer on the behaviour of a bubble and the impulse pressure in a viscous compressible liquid." Z. angew. Math. Mech. Vol. 59, 1979, pp. 297-306. Watanabe, M. & Prosperetti, A., "The effect of gas diffusion on the nuclei population downstream of a cavitation zone." Cavitation and gas-liquid flow in fluid machinery and devices. ASME Fluids Engineering Division Summer Meeting, Lake Tahoe, NV, USA. ASME, New York, NY, USA, FED Vol. 190 1994. van Wijngaarden, L., "On the collective collapse of a large number of gas bubbles in water." 11th International Congress of Applied Mechanics, Munich, Germany. Springer-Verlag, Berlin, Germany, 1964. 16
DISCUSSION J. Matusiak Helsinki University of Technology, Finland I would like to congratulate the author for a very interesting paper. The author has successfully demonstrated the effect of the interaction of cavitation bubbles on noise. I was particularly happy with the conclusion that although interaction may result in high pressure pulses, it does not significantly affect pressure spectrum except at very high frequencies. This was my simplifying assumption in the method of evaluating the pressure generated by a cavitating propeller (Matusiak 1 992a, 1 992b). I found particularly interesting your experimentally derived information on the distributions of cavitation bubbles radii (Figures 1 1 and 13) used in the Monte Carlo simulations. Did you find a correlation of the radii and total volume of cavitation bubbles with the sheet cavitation geometry and a disintegration of fixed cavitation? I had a simple model (Matusiak 1 992a, 1 992b) relating these quantities. I am curious whether your observations support this model. AUTHORS' REPLY We did not try to find a correlation of the radii and total volume of cavitation bubbles with the sheet cavitation geometry. The estimates of distribution parameters were made from high- speed film recordings of two experimental conditions, Figures S and 6. The geometry of these sheet cavities are relatively similar, but the cavity dynamics are very different (quasi-steady and unsteady). In fact, all experimental parameters except the oscillation frequency were the same for these two conditions. The time displacement, initial radii and distance between the cavities were supposed to be stochastic variables. Data for these variables were sampled from the high-speed film recordings by identifying start times of collapses and by measuring the corresponding initial radii and distances on the screen (Hallander 2002: Paper IV). Distribution parameters were estimated by fitting probability distributions to the data. Since the high-speed film recordings were limited to 1.3 s, the number of samples used for the data fitting was small. The estimated distribution parameters were finally used to generate random input data to the Monte Carlo simulations, Figures 11 and 12.
