C.Brennen, T.Colonius, Y.C.Wang, A.Preston (California Institute of Technology, USA)
This paper describes investigations of the dynamics and acoustics of clouds of cavitation bubbles. Recent experimental and computational findings show that the collapse of clouds of cavitating bubbles can involve the formation of bubbly shock waves and that the focussing of these shock waves is responsible for the enhanced noise and damage in cloud cavitation. The recent experiments and computations of Reisman et al. (1) complement the work begun by Mørch and Kedrinskii and their coworkers (2,3,4) and demonstrate that the very large impulsive pressures generated in bubbly cloud cavitation are caused by shock waves generated by the collapse mechanics of the bubbly cavitating mixture. Here we describe computational investigations conducted to explore these and other phenomena in greater detail as part of an attempt to find ways of ameliorating the most destructive effects associated with cloud cavitation.
Understanding such bubbly flow and shock wave processes is important because these flow structures propagate the noise and produce the impulsive loads on nearby solid surfaces in a cavitating flow. How these shocks are formed and propagate in the much more complex cloud geometry associated with cavitating foils, propeller or pump blades is presently not clear. However, the computational investigations reveal some specific mechanisms which may be active in the dynamics and acoustics of these more complex flows.
a 
Amplitude of plate motion 
A 
Radius of the bubble cloud 
A_{0} 
Initial radius of the bubble cloud 
C_{p} 

C_{pmin} 
Minimum pressure coefficient 
D 
Reference body size 
k 
Polytropic constant for gas inside the bubble 
p 
Fluid pressure (Pa) 
p_{0} 
Upstream reference pressure (Pa) 
p_{υ} 
Vapor pressure 
R 
Bubble radius 
R_{0} 
Initial radius of the bubble 
t 
Time 
T 
Period of plate oscillation 
U 
Reference velocity of the flow (m/s) 
α 
Void fraction of the bubbly mixture 
α_{0} 
Initial void fraction of bubbly mixture 
α_{c} 
Critical void fraction 
β 

ρ 
Density of the liquid 
σ 
In many flows of practical interest, clouds of cavitation bubbles are periodically formed and then collapse. This temporal periodicity may occur naturally as a result of the shedding of bubblefilled vortices or it may be the response to a periodic disturbance imposed on the flow. Common examples of imposed fluctuations are the interaction between rotor and stator blades in a pump or turbine and the interaction between a ship’s propeller and the nonuniform wake created by the hull. When the density of cavitation events increases in space or time and bubbles therefore begin to interact, a whole new set of phenomena may be manifest
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Cloud Cavitation Phenomena
C.Brennen, T.Colonius, Y.C.Wang, A.Preston (California Institute of Technology, USA)
ABSTRACT
This paper describes investigations of the dynamics and acoustics of clouds of cavitation bubbles. Recent experimental and computational findings show that the collapse of clouds of cavitating bubbles can involve the formation of bubbly shock waves and that the focussing of these shock waves is responsible for the enhanced noise and damage in cloud cavitation. The recent experiments and computations of Reisman et al. (1) complement the work begun by Mørch and Kedrinskii and their coworkers (2,3,4) and demonstrate that the very large impulsive pressures generated in bubbly cloud cavitation are caused by shock waves generated by the collapse mechanics of the bubbly cavitating mixture. Here we describe computational investigations conducted to explore these and other phenomena in greater detail as part of an attempt to find ways of ameliorating the most destructive effects associated with cloud cavitation.
Understanding such bubbly flow and shock wave processes is important because these flow structures propagate the noise and produce the impulsive loads on nearby solid surfaces in a cavitating flow. How these shocks are formed and propagate in the much more complex cloud geometry associated with cavitating foils, propeller or pump blades is presently not clear. However, the computational investigations reveal some specific mechanisms which may be active in the dynamics and acoustics of these more complex flows.
1. NOMENCLATURE
a
Amplitude of plate motion
A
Radius of the bubble cloud
A0
Initial radius of the bubble cloud
Cp
Cpmin
Minimum pressure coefficient
D
Reference body size
k
Polytropic constant for gas inside the bubble
p
Fluid pressure (Pa)
p0
Upstream reference pressure (Pa)
pυ
Vapor pressure
R
Bubble radius
R0
Initial radius of the bubble
t
Time
T
Period of plate oscillation
U
Reference velocity of the flow (m/s)
α
Void fraction of the bubbly mixture
α0
Initial void fraction of bubbly mixture
αc
Critical void fraction
β
ρ
Density of the liquid
σ
2. INTRODUCTION
In many flows of practical interest, clouds of cavitation bubbles are periodically formed and then collapse. This temporal periodicity may occur naturally as a result of the shedding of bubblefilled vortices or it may be the response to a periodic disturbance imposed on the flow. Common examples of imposed fluctuations are the interaction between rotor and stator blades in a pump or turbine and the interaction between a ship’s propeller and the nonuniform wake created by the hull. When the density of cavitation events increases in space or time and bubbles therefore begin to interact, a whole new set of phenomena may be manifest
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both in the dynamics and the acoustics. In many of these cases the coherent collapse of the cloud of bubbles can cause a substantial increase in the radiated noise and potential for damage. Much recent interest has focused on the dynamics and acoustics of finite clouds of cavitation bubbles because of these very destructive effects (see, for example, Knapp (5), Bark and van Berlekom (6), Soyama et al. (7)). The purpose of the computational investigations outlined in this paper is to explore the various phenomena which can occur in the dynamics anfd acoustics of these bubbly cavitating flow and, where possible, to compare these with experimental observations.
1. Some Recent Experiments on Cloud Cavitation
Numerous investigators (for example, Knapp (5), Wade and Acosta (8), Bark and van Berlekom (6), Shen and Peterson (9,10), Bark (11), Franc and Michel (12), Hart et al. (13), Kubota et al. (14,15), Le et al. (16), de Lange et al. (17), Kawanami et al. (18)) have studied the complicated flow patterns involved in the production and collapse of cloud cavitation on a hydrofoil. The basic features of the cyclic process of cloud formation and collapse (whether on a stationary or oscillating foil) are as follows. The growth phase usually involves the expansion of a single attached cavity, at the end of which a reentrant jet penetrates the cavity from the closure region. This penetration breaks the cavity into a bubbly cloud which collapses as it is convected downstream.
The radiated noise all occurs during this bubbly part of the cycle. It consists of pressure pulses of very short duration and large magnitude. The pulses have been observed and measured by a number of investigators including Bark (11), Bark and van Berlekom (6), Le et al. (16), Shen and Peterson (9,10), McKenney and Brennen (19) and Reisman et al. (20). More recently, Reisman et al. (1) have made measurements of the impulsive pressures on the suction surface of a hydrofoil (within the cloud cavitation), have simultaneously measured the radiated pulses and have correlated these pressure traces with images from highspeed movies. Very large pressure pulses were recorded by the surface transducers, with typical magnitudes as large as 10bar and durations of the order of 10−4s. These help to explain the enhanced noise and cavitation damage associated with cloud cavitation. For example, the large impulsive surface loadings due to these pulses could be responsible for the foil damage reported by Morgan (21), who observed propeller blade trailing edges bent away from the suction surface and toward the pressure surface.
Reisman et al. (1) also correlated highspeed movies of the clouds with the pressure measurements and found that the pressure pulses recorded (both on the foil surface and in the far field) were associated with specific structures (more precisely, the dynamics of specific structures) which are visible in the movies. Indeed, it appears that several types of propagating structures (shock waves) are formed in a collapsing cloud and dictate the dynamics and acoustics of collapse. One type of shock wave structure is associated with the coherent collapse of a welldefined bubble cloud which separates from the rear of the cavitation zone and is convected downstream. This type of structure causes the largest impulsive pressures and radiated noise. The pulses it produces are termed global pulses since they are recorded almost simultaneously by all transducers. The highspeed motion pictures showed that the cavitation cloud undergoes a rapid and coherent collapse and that these collapses generate the global pulses. It was noted that such cloud collapses do not involve large and rapid changes in the volume of the cloud. Rather, they involve rapid and propagating changes in the void fraction distribution within the cloud.
Unexpectedly, two other types of structures were observed. Typically, the pulses produced by these structures were registered by only one transducer and these events were therefore termed local pulses. They are recorded when an ephemeral, localized and transient low void fraction structure forms in the bubbly cavitating cloud and happens to pass over the face of the transducer. While these local events are smaller and therefore produce less radiated noise, the pressure pulse magnitudes are almost as large as those produced by the global events. How and why these low void fraction structures (shock waves) form in the flow is not at all clear.
Finally, we note that injection of air into the cavitation on the suction surface can substantially reduce the magnitude of the pressure pulses produced (Ukon (22), Arndt et al. (23), Reisman et al. (24)). However Reisman et al. (25) have shown that the bubbly shock wave structures still occur; but with the additional air content in the bubbles, the pressure pulse magnitudes are greatly reduced.
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2. On Calculations of Cloud Cavitation
The experimental and practical observations of cloud cavitation exemplified by the preceding discussion have generated much interest in modelling the dynamics and acoustics of these flows. These efforts began with the work of van Wijngaarden (26) who first attempted to model the behavior of a collapsing layer of bubbly fluid next to a solid wall. The essence of his approach was to couple conventional continuity and momentum equations for the compressible mixture to a RayleighPlesset equation for the bubble dynamics (see, for example, Brennen (27)) which would provide the neccessary relation connecting the local pressure with the bubble size and therefore the local mixture density. This is the basis for most of the computational investigations which have been carried out subsequently. Later investigators explored numerical methods which incorporate the individual bubbles (Chahine (28)) and continuum models which, for example, analyzed the behavior of shock waves in a bubbly liquid (Noordzij and van Wijngaarden (29), Kameda and Matsumoto (30)) and identified the natural frequencies of spherical clouds of bubbles (d’Agostino and Brennen (31)). Indeed the literature on the linearized dynamics of clouds of bubbles has grown rapidly (see, for example, d’Agostino et al. (32,33), Omta (34), Prosperetti (35)). However, apart from some weakly nonlinear analyses (Kumar and Brennen (36,37,38)) only a few papers have addressed the highly nonlinear processes which are an inevitable consequence of the nonlinearities in the RayleighPlesset equation and are most evident during the collapse of a cloud of bubbles. Chahine and Duraiswami (39) have conducted numerical simulations using a number of discrete bubbles and demonstrated how the bubbles on the periphery of the cloud develop inwardly directed reentrant jets. However, most clouds contain many thousands of bubbles and it therefore is advantageous to examine the nonlinear behavior of continuum models.
Another perspective on the subject of collapsing clouds was that introduced by Mørch and Kedrinskii and their coworkers (Mørch (2,3), Hanson et al. (4)). They surmised that the collapse of a cloud of bubbles involves the formation and inward propagation of a shock wave and that the geometric focusing of this shock at the center of cloud creates the enhancement of the noise and damage potential associated with cloud collapse. We begin the review of computational efforts by describing studies of the dynamics of a spherical cloud of bubbles, a simple example which illustrates these processes of shock wave formation, propagation and focussing.
3. Computations: [1] Dynamics of a Spherical Cloud
Wang and Brennen (40,41,42) (see also Reisman et al. (1)) used a mixture model comprising the continuity, momentum and RayleighPlesset equations (for specifics see, for example, d’Agostino et al. (31,32,33)) to study the nonlinear growth and collapse of a spherical cloud of bubbles. A finite cloud of nuclei is subjected to an episode of low pressure which causes the cloud to cavitate; the pressure then returns to the original level causing the cloud to collapse. Wang and Brennen used the computational model to study the various cloud dynamics and acoustics exhibited in various parametric regimes. Key parameters are the cavitation number, σ, which characterizes the the initial pressure level, the magnitude of the low pressure episode characterized by a minimum pressure coefficient, Cpmin, and by a duration, D/U. The initial radius and void fraction of the cloud are denoted by A0 and α0 respectively and the initial radius of the bubbles within the cloud is denoted by R0.
This parametric exploration revealed that the dynamics and acoustics depended in an important way on the “cloud interaction parameter”, β, defined as
(1)
Note that, while the initial void fraction, α0, is very small, the ratio, A0/R0, may be very large so that β could be small or large compared with unity. Earlier linear and weakly nonlinear studies of cloud dynamics (d’Agostino & Brennen (31,32,33), Kumar & Brennen (36,37,38)) showed that the cloud natural frequency is strongly dependent on this parameter. If β is small, the natural frequency of the cloud is close to that of the individual bubbles in the cloud. In other words, the bubbles in the cloud tend to behave as individual units in an infinite fluid and the bubble/bubble interaction effects are minor. On the other hand bubble interaction effects dominate when the value of β is greater than order one. Then the collective oscillation of bubbles in the cloud results in a cloud natural frequency which is lower than the natural frequency of individual bubbles.
In all of the computations of Wang and Bren
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nen, the bubbles in the interior of the cloud are shielded by the outer shell of bubbles and grow to a smaller maximum size. This shielding effect is typical of the bubble/bubble interaction phenomenon in cavitating cloud dynamics (d’Agostino & Brennen (31,33); Omta (34); Smereka & Banerjee (48); Chahine & Duraiswami (39)).
The nonlinear computations showed that when β≫1, spherical cloud collapse began on the cloud surface and propagated inwards as a collapse front. As a result of the bubble collapse within this front, a large pressure pulse or shock wave becomes an integral part of the propagating collapse front. The structure of this shock is very similar to those in the bubbly flows investigated by Noordij and van Wijngaarden (29) and other investigators (see, for example, Brennen (27), Kameda and Matsumoto (30)); the shock is comprised of a series of rebounds and secondary collapses. The locations with small bubble size represent regions of low void fraction and higher pressure due to the local bubble collapse. The magnitude of the pressure pulse grows due to geometric focussing as the front progresses inward and reaches very large magnitudes when it reaches the center of the cloud. At this instant a very large, positive pressure pulse is radiated away from the cloud into the farfield; all the earlier or later radiated noise is almost insignificant by comparison. Another characteristic of the collapse is the relatively small change which occurs in the cloud radius, A. The essential collapse process is unlike that of a single bubble and involves the propagation of void fraction waves within the cloud rather than radical volumetric change.
In contrast, Wang and Brennen found that the dynamics and acoustics when β≪1 were quite different. Then collapse began with the bubbles at the cloud center and the collapse front propagated outward producing a much more benign process and much reduced radiated noise.
[Parenthetically we note that all of the preceding computations were done with a cloud consisting of a uniform dispersion of identical bubbles. Real clouds will have a distribution of bubble sizes. Wang (43) has recently developed a computational methodology to deal with this more complex circumstance. Moreover, he observes some significant differences in the dynamics and acoustics of these nonmonodisperse clouds.]
When compared with the experimental observations of Reisman et al. (1) and others, the results of the computations of Wang and Brennen reveal many parallels. First note that the value of β in a real flow will increase as the cavitation number is decreased and more and more nuclei are activated. Thus the critical value of β of about unity may be manifest in practice as a critical cavitation number below which cloud cavitation begins.
Secondly, we note that shock waves characterized by large, positive pressure pulses and zones of low void fraction may form during the cloud collapse process. They will then propagate through the cloud and can grow to very large magnitudes due to geometric focussing. Such structures were observed by Reisman et al. (1).
Finally we note that there may be important implications for model testing in water tunnels. Since α0 and R0 tend to be similar at the model and full scales but A0 could be radically different, it follows that the model tests might be conducted at much smaller values of β than pertain at full scale. Consequently there is a real risk that violent cloud cavitation could occur at full scale and not at the model scale.
4. Computations: [2] Harmonic Cascading
We now shift attention to a different set of computational investigations in order to discuss two other nonlinear acoustic phenomena which have been identified from the calculations of cloud dynamics. Both of these emerge from calculations of the response of a liquid layer (laced with cavitation nuclei or bubbles but at small void fraction) to the vibration of an infinite flat plate in a direction normal to its surface. Kumar and Brennen (36,38) present both the linear and a weakly nonlinear solution to this problem utilizing the same set of model equations described above. While a multitude of nonlinear effects occur in the weakly nonlinear solution (in which terms quadratic in the oscillation amplitude are retained but all higher order terms are neglected) we highlight just one here because of its practical significance. This emerges from calculations using typical size distributions of bubbles; each size, of course, has its own single bubble natural frequency with the larger number of smaller bubbles having a much higher natural frequency than the fewer, larger bubbles.
Kumar and Brennen (37) identify a phenomenon they call harmonic cascading. In this process the larger bubbles, responding mostly to oscillations
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at their relatively low natural frequency, produce higher harmonics due to nonlinear effects. These higher frequencies in turn excite a much larger number of smaller bubbles because that higher harmonic corresponds to the basic natural frequency of those smaller bubbles. The measured spectra of cavitation reported by Mellen (44) and by Blake et al. (45) contain peaks which may well be due to harmonic cascading. In any case, it seems clear that nonlinear effects combined with a typical bubble number distribution provide a mechanism for the cascading of acoustic energy from low frequencies to high frequencies where the attenuation is, in turn, more effective in damping the noise.
5. Computations: [3] Acoustic Saturation
A second notable phenomenon emerges from fully nonlinear calculations of the same physical problem of a liquid layer and will be referred to as acoustic saturation. In this case a distribution of bubble sizes is not essential and so results are presented for nuclei of just one uniform equilibrium size. Colonius et al. (46) and Brennen et al. (47) demonstrate what happens as the amplitude of the plate oscillation is increased from the linear regime up to the highly nonlinear regime in which true cavitation bubble collapses occur. The bubbles next to the wall behave as expected as the amplitude is increased as illustrated in figure 1. This shows the typical development of a RayleighPlesset solution with increasing amplitude. We note that in this and the other results presented in this section, that the bubble damping has been adjusted to a value which is consistent with the experimentally observed spatial attenuation of acoustic waves in bubbly liquids. This also produces a more realistic number of rebounds than the much smaller value used in the nozzle calculations of the next section (see Colonius et al. (46)).
More unexpected is the behaviour of the bubbles further away from the plate. At low amplitudes the sinusoidal pressure oscillations produced by the oscillating plate simply propagate away with an amplitude directly proportional to the amplitude of motion of the plate. But, as nonlinear effects begin to become significant, the radiated signal is attenuated relative to the expected linear magnitude and, eventually, reaches an asymptotic value. This phenomenon is illustrated in figures 2, 3 and 5. Figure 2 presents the typical pressure signals occurring at a distance from the plate and the
Figure 1. Typical radii/time histories for the bubbles at the surface of the plate for plate velocity amplitudes, a, equal to 0.05 (solid line), 0.075 (dots), 0.1 (dashdot), 0.2 (dashes) and 0.3 (solid line). The initial void fraction and cavitation number are respectively 0.01 and 0.475 and other parameters are given in Colonius et al. (46).
Figure 2. The fluid velocity (normalized by the velocity amplitude at the plate surface) at the plate surface (solid line) and at a distance of 400 initial bubble radii into the fluid as a function of reduced time for plate velocity amplitudes, a, equal to 0.0001 (dots), 0.001 (dashdot) and 0.01 (dashes) initial bubble radii. The initial void fraction and cavitation number are respectively 0.01 and 0.475 and other parameters are given in Colonius et al. (46).
essentially linear behavior at low amplitudes (the vertical coordinate is the instantaneous fluid velocity normalized by the magnitude of the wall velocity oscillations). On the other hand, figure 3 presents the typical behaviour at large amplitudes; the nonnormalized signals at a distance from the plate are almost independent of the plate amplitude. Thus further increase in the plate oscillation amplitude produces no change in the radiated noise. This
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Figure 3. As figure 2 but larger amplitude plate motions, a, equal to 0.05 (solid line), 0.075 (dots), 0.1 (dashdot), 0.2 (dashs) and 0.3 (solid line) initial bubble radii. Note that, unlike figure 2, the vertical scale is not normalized by a. From Colonius et al. (46).
is, in part, caused by the formation of a highly active layer adjacent to the plate where the bubbles grow large enough that their momentary natural frequency is no longer large compared to the plate oscillation frequency. Thus acoustic energy is trapped and dissipated in this layer near the plate (see also Smereka and Banerjee (48)). Increasing the plate amplitude simply results in greater dissipation and in no increase in the radiated noise. The nature of the layer for this particular case is further illustrated in figure 4. Note that the layer decreases in thickness from about 200 initial bubble radii at an amplitude of a=0.025 to 120 initial bubble radii at a=0.3. The figure clearly shows how the response increases with amplitude within the layer but is independent of ampltitude outside the layer.
The acoustic saturation phenomenon in this case is summarized in figure 5 which plots the magnitude of the bubble radius oscillations some distance from the plate as a function of the amplitude of the plate oscillations. This clearly demonstrates the acoustic saturation phenomenon. It could be a contributing factor to the frequently made experimental observation that, as the pressure in a flow is decreased and the cavitation increases, the noise often plateaus out and may even decrease after a certain point.
Finally, we remark that the description of the acoustic saturation phenomenon given above is pertinent to the subresonant behaviour manifest when the plate oscillation frequency is substan
Figure 4. Snapshots at one moment in time (t/T=76) of the variation in bubble radius with distance from the plate for the same case as figure 1.
Figure 5. The amplitude of the bubble radius oscillations for the computations of figures 2 and 3 as a function of the plate amplitude, a. The circles are the bubble radius amplitudes at the wall and the crosses are the bubble radius amplitudes at a distance from the wall of 400 initial bubble radii.
tially smaller than the natural frequency of the individual bubbles in the fluid. Other nonlinear effects are exhibited when the plate frequency approaches or exceeds the bubble natural frequency. Then, as anticipated by Smereka and Banerjee (48) (see also Lauterborn and Koch (49)) and confirmed by Brennen et al. (47), the nonlinear effects produce a cascade of bifurcations leading, eventually, to a chaotic response of the bubbly fluid. This superresonant regime is not, however, of great practical interest for, other than in the context of ultrasonic cavitation, it is uncommon to en
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counter circumstances in which the imposed excitation reaches such a high frequency.
6. Computations: [4] Steady Flow through a Nozzle
All of the above computations involve bubbly liquids which are not flowing in the mean. It is more challenging to devise numerical methodologies for cases in which the bubbly liquid is flowing and particularly difficult when dealing with a two or threedimensional flow. In this section we examine a case of a onedimensional cavitating flow, namely that through a nozzle. When the same approach is applied to such a bubbly cavitating flow the solutions exhibit several interesting features which may provide insight into more complex two and three dimensional flows such as those over hydrofoils or propeller blades. Using the same basic equations previously discussed, namely a continuity equation, an inviscid momentum equation and the RayleighPlesset equation to connect the local pressure with the local mixture density or bubble size, Wang and Brennen (50) present results for the onedimensional, bubbly, frictionless cavitating flow through a convergent/divergent nozzle. The following typical flow will illustrate the phenomena manifest in the nozzle computations. A bubbly liquid, composed of air bubbles (k=1.4) in water at 20°C (liquid density, 1000 kg/m, surface tension, 0.073 N/m), flows through a nozzle whose total length is 500 initial upstream bubble radii. The minimum or throat pressure coefficient for incompressible flow is −1.0 and the present example is for an upstream cavitation number, σ, of 0.8. The Reynolds number is also important since it determines the damping of the bubble oscillations (see Wang and Brennen (50)). It could be chosen, as suggested by Chapman and Plesset (51), so as to accommodate other, nonviscous contributions to the bubble damping. Five different upstream void fractions, α0, of the order of 10−6 are used in the computation and the results are shown in figures 6, 7, 8 and 9. Figures 6 and 7 respectively present the axial variations of the mixture velocity and the mixture pressure coefficient in this typical calculation.
The case of α0=0 corresponds to the incompressible pure liquid flow. It is notable that even for an upstream void fraction as small as 2×10−6, the characteristics of the flow are radically changed from the case without bubbles. Radial pulsation of bubbles results in the downstream fluctuations of
Figure 6. The nondimensional mixture velocity distribution as a function of the nondimensional position in the flow for five different upstream void fractions, α0, from Wang and Brennen (50). The dimensionless length of the nozzle is 500 with the throat located at 250; the cavitation number, σ=0.8, the throat pressure coefficient for incompressible flow is −1.0 and other parameters are as given in the original reference. Note that, for this case, the critical value of the void fraction is 2.862×10−6.
Figure 7. The mixture pressure coefficient corresponding to figure 6. From Wang and Brennen (50).
the flow. The amplitude of the velocity fluctuation is 10% of that of the incompressible flow in this case. As α0 increases further, the amplitude as well as the wavelength of the fluctuations increase. However, the velocity does eventually return to the upstream value. In other words, the flow is still “quasistatically stable.” However, as α0 increases to a critical value, αc (αc≈2.862×10−6 in the present example), a bifurcation occurs. Wang and Brennen (50) show that a critical state is reached downstream of the nozzle when the instantaneous bubble radius, Rc, reaches a value given approxi
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Figure 8. The void fraction distribution corresponding to figure 6. From Wang and Brennen (50).
mately by
(2)
where R0 is the radius of the bubbles (assumed monodisperse) in the upstream flow. This implies that when the instantaneous bubble size exceeds Rc, the flow becomes unstable and flashes. In the context of an experiment in which only the upstream void fraction, α0, is increased while the other parameters are held fixed, this implies that the flow will flash at a critical value of the upstream void fraction, αc, which in the present example yields (σ/2αc)1/3≈51. Figure 9 demonstrates the veracity of the above expression for Rc/R0.
The flow becomes quasistatically unstable and flashes to vapor if the radius of the cavitating bubbles is greater than Rc. In this circumstance, the growth of bubbles increases the mixture velocity due to mass conservation of the flow. The velocity increase then causes the mixture pressure to decrease according to the momentum equation. The decrease of the pressure is fed back to the RayleighPlesset equation and results in further bubble growth. In this case the velocity and void fraction of the mixture increase and the pressure coefficient of the flow decreases significantly below the upstream values and the flow flashes to vapor. On the other hand, if the bubbles do not grow beyond Rc, the flow is quasistatically stable and is characterized by large amplitude spatial fluctuations downstream of the throat.
Figure 8 illustrates the void fraction distribution in the flow. When the flow becomes quasistatically unstable, the bubble void fraction quickly
Figure 9. The nondimensional bubble radius distribution corresponding to figure 6. From Wang and Brennen (50).
approaches unity. This means that the flow is flashing to vapor. However, it should be cautioned that when α becomes large, the present model loses validity since it is limited to flows with small void fraction. Figure 9 indicates the nondimensional bubble radius distribution in the flow. Due to time lag during the bubble growth phase, bubbles reach the maximum size downstream of the throat. With increase in the upstream void fraction, the maximum size of the bubbles increases and is shifted further downstream. The bubbles grow without bound after reaching a critical radius, Rc, at which flashing begins.
Note from figure 7, that the downstream mixture pressure does not return to the upstream value except in the case of the pure liquid flow. Since viscous effects are neglected in the global mixture flow and the only dissipation present is that in the RayleighPlesset equation representing bubble dynamic damping, the pressure and energy losses are caused by the radial motion of bubbles and are therefore “cavitation losses.”
In addition to the two different flow regimes, another important feature in the quasistatically stable flow is the typical frequency associated with the downstream periodicity. This “ringing” will result in acoustic radiation at frequencies corresponding to this wavelength. How this ring frequency relates to the upstream flow condition remains to be studied.
In summary, it is found that the nonlinear bubble dynamics coupled with the equations of motion of the mixture strongly affect the structure of the
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flow even for very small bubble populations. Two different flow regimes, distinguished by the parameter Rc=R0(σ/2αc)1/3, are revealed in the steady state solutions. The results also imply that there exists a domain of pressure drops and void fractions for which no steady state flow solution exists. It remains to be determined whether those conditions lead to the unsteady, oscillatory cavitating flows which are observed in practice.
7. Computations: [5] Two and Threedimensional Cavitating Bubbly Flows
It is more challenging to devise numerical methodologies for two and threedimensional bubbly cavitating flows. In this regard we should mention several recent efforts to develop approximate methods for twodimensional flows. Both Song (52) and Merkle and Feng (53) have modified compressible gas dynamic codes, introducing some artificial algebraic equation of state to relate the mixture density to the pressure. While these results are of some qualitative value they omit the essential bubble dynamic effects which would result from a more appropriate differential “equation of state” such as implied by the RayleighPlesset equation. Perhaps Kubota et al. (15) come closest to a true twodimensional methodology; however, by not permitting the bubbles to collapse below the original nuclei size, they exclude the formation of the large pressure perturbations and shock waves which are such an important part of cloud cavitation.
8. Computations: [5] Example: Encounter of Low Pressure Pulse with a Cylindrical Cloud
We illustrate some of the multidimensional phenomena by presenting preliminary computational results for a particularly simple twodimensional calculation, namely the response of a cylindrical bubble cloud to an incident, planar pressure pulse (specifically a low pressure pulse) travelling through the liquid (which is assigned some compressibilty so the progress of the wave may be recorded). The initial conditions are shown in figure 10. In this particular example the initial bubble cloud has a Gaussian radial distribution of void fraction reaching a maximum of 0.5% at the center of the cloud. The gray scale of the righthand image depicts the void fraction distribution while that in the lefthand image shows the liquid density and, by implication, the pressure and scattered waves. The incident wave moves from left to right in the sequence of images, figures 10 through 15, and is diffracted and scattered by the low sound speed inside the cloud. The pressure in the wave is low enough to cause cavitation and the bubbles within the cloud grow to nearly 100 times their original volume before collapsing violently. The bubbles on the lefthand side of the cloud grow first and collapse first, the latter process initiating a collapse front which propagates from left to right through the cloud. Sufficient damping is included in the RayleighPlesset equation so that only a few bubble rebounds occur, but these are evident in the scattered sound (see figure 15).
9. Concluding Comments
In this paper we have summarized some of the recent advances in our understanding of bubbly cloud cavitation. It is becoming clear that effects of the interaction between bubbles may be crucially important especially when they give rise to the phenomenon called cloud cavitation. Calculations of the growth and collapse of a spherical cloud of cavitating bubbles show that when the cloud interaction parameter (β) is large enough, collapse occurs first on the surface of the cloud. The inward propagating collapse front becomes a bubbly shock wave which grows in magnitude due to geometric focussing. Very large pressures and radiated impulses occur when the shock reaches the center of the cloud.
Of course, actual clouds are far from spherical. And, even in a homogeneous medium, gasdynamic shock focussing can be quite complex and involves significant nonlinear effects (see, for example, Sturtevant and Kulkarny (54)). Nevertheless, it seems evident that once collapse is initiated on the surface of a cloud, the propagating shock will focus and produce large local pressure pulses and radiated acoustic pulses. It is not, however, clear exactly what form the foci might take in the highly nonuniform, threedimensional bubbly environment of a cavitation cloud on a hydrofoil, for example.
Experiments with hydrofoils experiencing cloud cavitation have shown that very large pressure pulses occur within the cloud and are radiated away from it during the collapse process. Within the cloud, these pulses can have magnitudes as large as 10bar and durations of the order of 10−4s. This suggests a new perspective on cavitation damage and noise in flows which involve large collec
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Figures 9 through 15. A series of snapshots in time of a planar pressure pulse impinging on a cylindrical cloud. The image on the right shows the fluid density (the white circle is the initial location of the cloud) while that on the right shows the void fraction distribution. Note that the image on the right is a closeup view compared with that on the left.
Figure 10. Initial conditions with low pressure pulse (vertical) moving from left to right and about to impinge on the cylindrical bubble cloud.
Figure 11. Expansion (or cavitation) of cloud.
tions of cavitation bubbles with a sufficiently large void fraction (or, more specifically, a large enough β) so that the bubbles interact and collapse coherently. This view maintains that the cavitation noise and damage is generated by the formation and propagation of bubbly shock waves within the collapsing cloud. The experiments reveal several specific shock wave structures. One of these is the mechanism by which the large coherent collapse of a finite cloud of bubbles occurs. A more unexpected result was the discovery of more localized bubbly shock waves propagating within the bubbly mixture in several forms, as crescentshaped regions and as leading edge structures. These seem to occur when the behavior of the cloud is less coherent. They produce surface loadings which are within an order of magnitude of the more coherent events and could also be responsible for cavitation damage. However, because they are more localized, the radiated noise they produce is much smaller than that due to global events.
The phenomena described are expected to be important features in a wide range of cavitating flows. However, the analytical results clearly suggest that the phenomena may depend strongly on the cloud interaction parameter, β. If this is the case, some very important scaling effects may occur. It is relatively easy to envision a situation in which the β value for some small scale model experiments is too small for cloud effects to be important but in which the prototype would be operating at a much larger β due to the larger cloud size, A0 (assuming the
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Figure 12. Scattered wave. Collapse beginning on left side of cloud.
Figure 13. Collapse front moves through cloud from left to right.
Figure 14. Collapse produces second radiated pulse.
void fractions and bubble sizes are comparable). Under these circumstances, the model would not manifest the large cloud cavitation effects which could occur in the prototype.
Computational methods will play a key role in these developing studies. Not only will such methods be needed for the prediction of these flows in practical applications (particularly to predict the noise and damage potential) but they are almost essential in building our understanding of simpler
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Fig. 1. Geometry and coordinate system used to model the interaction among two gas bubbles and a free surface.
Fig. 2. Level 1 and 8 of the triangulation of a sphere.
Fig. 3. Level 16 of the triangulation of a free surface.
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Fig. 4 Evolution of a gas bubble above a wall for ε=100, δ=0.75, γr=1.0: (a) 3D model results, (b) axisymmetrical model results. The dimensionless times t corresponding to shapes are (1) 2.019, (2) 2.273.
Fig. 5. Evolution of a gas bubble beneath a free surface for ε=100, δ=0.5, γf=1.0: (a) 3D model results, (b) axisymmetrical model results. The times t corresponding to successive shapes are (1) 0.711, (2) 1.360.
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Fig. 6. The evolution of a gas bubble above a wall at an angle of 45° to the horizontal characterised by ε=100, δ=0.5, and γr=1.0. The dimensionless times t corresponding to successive shapes are: (1) 0.000, (2) 0.300, (3) 0.995, (4) 1.619, (5) 1.907, (6) 2.066, (7) 2.1171, (8) 2.215, respectively.
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Fig. 7. The evolution of a gas bubble near a vertical wall characterised by ε=100, δ=0.5, γr=1.0. The dimensionless times t corresponding to successive shapes are: (1) 0.000, (2) 0.289, (3) 0.976, (4) 1.301, (5) 1.873, (6) 2.099, (7) 2.207, (8) 2.311, respectively.
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Fig. 8. The evolution of a gas bubble under a wall at an angle of 45° to the horizontal characterised by ε=100, δ=0.5, and γr=1.0. The dimensionless times t corresponding to successive shapes are: (1) 0.000, (2) 0.313, (3) 0.651, (4) 1.317, (5) 1.598, (6) 1.878, (7) 2.092, (8) 2.195, respectively.
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Fig. 9. Evolution of two gas bubbles beneath a free surface for ε=100, δ=0.5, and γf=1.0. The dimensionless distance between the centres of the two bubbles at inception is 2.0. The dimensionless times t corresponding to shapes are (1) 0.000, (2) 0.260, (3) 1.119, (4), 1.565, (5) 1.670 and (6) 1.789.
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Fig. 10. Evolution of two gas bubbles beneath a free surface for ε=100, δ=0.5, and γf=1.0. The dimensionless distance between the centres of the two bubbles at inception is 2.0. The dimensionless times t corresponding to shapes are (1) 0.000, (2) 0.236, (3) 1.030, (4) 1.240, (5) 1.450, and (6) 1.562.
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Fig. 11. Evolution of two gas bubbles beneath a free surface for ε=100, δ=0.5, and γf=1.0. The dimensionless distance between the centres of the two bubbles at inception is 2.0. The dimensionless times t corresponding to shapes are (1) 0.000, (2) 0.236, (3) 0.716, (4) 0.939, (5) 1.444, and (6) 1.402.
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DISCUSSION
Y.Cao
University of New Orleans, USA
This is an interesting paper. The authors have presented a 3D numerical method to simulate the dynamics of gas bubbles in a liquid, which is a very difficult problem. For axissymmetrical case, the results obtained using the 3D code agree well with those obtained using an axissymmetrical code.
I have the following comments and questions:
1. The authors use a triangular mesh for the surface discretization. The initial mesh starts with an icosahedron. The icosahedron consists of 20 equalsized equilateral triangles with all the 12 vertices lying on a sphere. The mesh is refined by farther dividing each triangle into smaller equilateral triangles and projecting them onto the sphere. The process is repeated until the desired resolution is achieved. The connectivity of the mesh is also built during the process. The mesh (e.g., its nodes) is then timestepped using an EulerLagrange’s method. The mesh deforms in the time stepping procedure. However, no regridding of the surface is performed, and the mesh connectivity remains unchanged. As is known, for violent surface motion such as the gas bubble expansion, collapse and water jet formation, the mesh density will change dramatically. This can also be seen in the figures given in the paper. The drastic nonuniform mesh density usually would cause numerical instability and breakdown of simulation. It is surprising that the authors do not seem to have this difficulty in their simulations.
2. For a gas bubble, the bubble surface is continuously smooth until a water jet is formed. Even when the water jet is formed, the surface is not smooth only in a small region near the water jet where the surface slope changes rapidly. Most of the surface remains smooth. As known, for a point on a smooth surface, the theoretical solid angle is 2π. The numerical solid angle computed based on the discretized surface should not be significantly different from the theoretical value 2π and approaches the theoretical value as the mesh becomes finer. It is my opinion that it is not necessary to replace the theoretical value with the numerical solid. This will represent some CPU time saving. Of course, the actual solid angle should be used where the surface is not smooth and has a shape corner or edge. Have the authors examined the influence of the solid angle on the final results and the CPU time?
3. For the cases with the free surface, the computational free surface domain used seems too small. One would expect a strong nonphysical reflection from the free surface truncation unless some sort of open boundary condition (or treatment) is applied. What kind of treatment is used in the paper?
AUTHORS’ REPLY
We would like to thank Dr. Cao for his comments. The authors agree that the numerical instability may happen for the simulation of gas bubble collapse and water jet formulation, as happened in the past studies. However, smooth and stable solutions were obtained in our simulations.
In principle, the solid angle is close to 2π as the mesh becomes very fine. However, the difference between the actual solid angle and 2π may not be small in a discretized modeling. Take a spherical bubble as an example: the difference is about 10% when it is meshed into n=1280 triangles. Its influence on the final results should be of the same order. Moreover, the authors found that the accurate solid angles are crucial for the stability of the numerical modeling. The calculation of solid angles is of O(n) and it is not a substantial part of the CPU time of the modeling.
The computational domain of the free surface was around 10 Rm (maximum radius of the bubble) in radius. Only a small part of the computational domain was shown in the paper. We have compared with a larger computational domain around 15 Rm in radius and found the results agreed up to first four significant figures.
DISCUSSION
K.Thiagarajan
Australian Maritime Engineering, Australia
Viscous effects and surface tension are important considerations when looking at interactions of bubbles with free surface. Do the authors plan to introduce these effects into their simulation program?
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AUTHORS’ REPLY
The surface tension may be important for very small bubbles, which can be included in the present program. Because the shear stress between the gas/liquid is much less than liquid/liquid, very little vorticity is produced over there and the flow remains as irrotational. Thus, the viscous effects are small and negligible.
DISCUSSION
U.Bulgarelli
Istituto Nazionale per Studi ed Esperienze di Architettura Navale, Italy
Is there any relationship between the time step and partial steps for stability?
When the bubble increases you put more panel on the surface, you discretize each time of the surface?
When there is a finger in the bubble you lose the existence of the potential as in toroidal bubbles?
AUTHORS’ REPLY
1. In the paper, the time step is controlled by setting a limit for the maximum change in potential on the boundary for each time step. However, the time step may also be controlled by the maximum spatial step of the nodes on the boundary as you suggested, which may be relative to the local element size.
2. The mesh deforms in the time stepping without regridding. However, the grids around the jet automatically become dense as required by the modeling.
3. The bubble surface is discretized and described as a set of triangles. We have not used a surface equation to express the bubble surface. Even if there are fingers on the bubble surface, it can be described as a set of triangles. There is no multivalue problem. Whereas, for toroidal bubbles, the flow field is doubly connected. The solution of the Laplace equation in a doublyconnected domain is not unique. Special modeling is needed to make the solution unique (Wang, 1998).
References
Wang, Q.X. 1998. The numerical analysis of the evolution of a gas bubble near an inclined wall. Theoret. and Comput. Fluid Dynamics 12, 29–51.
Wang, Q.X., et al. Toroidal bubbles near a rigid boundary. Theoret. and Comput. Fluid Dynamics (in press).
DISCUSSION
R.Tong
University of Birmingham, USA
The surface shapes are well resolved and the interaction of two bubbles with a free surface is an interesting case. The computation of normal directions and tangential velocities at nodes on the bubble surface is an aspect which requires careful handling. The use of a weighted averaging described in this paper appears to overcome the difficulties which can lead to a breakdown of the solution. There are three areas which I think are worthy of further mention: the particular context for which solutions are sought and the suitability of the boundary integral method (BIM); the underlying rationale for different choices of surface approximation methods and their relationship to the equations being solved; the causes of numerical instabilities in BIM for unsteady interface motion.
The BIM does not allow for the breakup of the bubble surface or for compressible effects in the liquid. For an underwater explosion near a free surface, experiments show the surface breaking into a plume of spray rather than forming a smooth jet and for cases where there is a jet, it is likely that surface tension aids the maintenance of stability. There are thus particular flow regimes where BIM provides accurate information (see Blake et al., 1998) and others where it is not applicable.
The methods of surface approximation chosen depend on the nature of the underlying equation Smoothness of the liquid surface is implied by the BIM formulation and convergence to the smooth surface can be attained as the number of planar elements is increased. Some anomalies occur in the convergence of linear approximations as noted in Blake et al. (1995) and Zinchenko et al. (1997) which should be compared with the method presented in the present paper. It is also possible to improve the
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approximation of the surface by using higher order basis functions (see the literature on hp approximation in Finite and Boundary Elements). The radial functions discussed in Tong (1997) can give spectrally accurate approximation to smooth surfaces which aids computational efficiency and can be necessary for representation of the modes of bubble oscillation.
The occurrence of numerical instabilities in other methods seems exaggerated in the present paper. The breakdown of the computation reported in Blake et al. (1995) in due to the jet tip reaching the opposite bubble surface and the computation goes as far as equivalent axisymmetric cases. Some discussion of the causes of instability is given in Baker & Nachbin (1998).
References
Baker, G. and Nachbin, A. 1998 Stable methods for vortex sheet motion in the presence of surface tension. SIAM J. Sci. Comput. 19, 1737–1786.
Blake, J.R., Tomita, Y., and Tong, R.P. 1998 The art, craft and science of modelling jet impact in a collapsing cavitation bubble. Appl. Sci. Res. 58, 77–90.
Tong, R.P. 1997 A new approach to modelling an unsteady free surface in boundary integral methods with application to bubblestructure interactions. Math. Comput. Sim. 44, 415–426.
Zinchenko, A.Z., Rother, M.A., and Davis, R.H. 1997 A novel boundaryintegral algorithm for viscous interaction of deformable drops. Phys. Fluids 9, 1493–1511.
AUTHORS’ REPLY
The free surface breaks into a water spike covered by a spray for an underwater explosion near it. The phenomena cannot be fully simuated by BIM, but it is the first step toward more accurate modeling. For very small bubbles, the surface tension may help to maintain the smoothness of the jet. However, the smooth jet may occur for cases whereby the surface tension is small and negligible. Some examples can be found in the experiments of Blake & Gibson (1981).
A highorder localsurfacefitting is suitable for a surface when the surfacefitting behavior matches with the surface. It is hard to provide a highorder localsurfacefitting that is good for an arbitrary surface. It is hard to compare the interpolation scheme on how accurate they are for an arbitrary shape. Besides, the high order panel methods are known to be more susceptible to numerical instability. Therefore, the authors prefer global linear interpolation coupled with the weighted averaging approach.
The jets were not smooth at the end of collapse phase, and sometimes broke in the middle stage of jetting in the previous simulations (Blake et al. 1995, 1997). In the present work, the smooth jets were simulated nearly until the jet impacts upon the opposite bubble surface. The very sharp jets were simulated in the present work that had not been simulated in the previous papers.
Reference
Blake & Gibson 1981 J. Fluid Mech. 111, 124–140.