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On the effect of microbubble injection on cavitation bubble dynamics in liquid mercury
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 600 (2009) 367–375
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
On the effect of microbubble injection on cavitation bubble dynamics in liquid mercury Masato Ida a,, Takashi Naoe b, Masatoshi Futakawa b a b
Center for Computational Science and E-systems, Japan Atomic Energy Agency, Higashi-Ueno, Taito-ku, Tokyo 110-0015, Japan J-PARC Center, Japan Atomic Energy Agency, Tokai-mura, Ibaraki-ken 319-1195, Japan
a r t i c l e in fo
abstract
Article history: Received 19 March 2008 Received in revised form 19 August 2008 Accepted 27 November 2008 Available online 7 December 2008
The effect of microbubble injection has been studied numerically to clarify the role of injected bubbles in the experimentally observed suppression of cavitation in liquid mercury. Recently, we attempted to inject gas microbubbles into liquid mercury in order to mitigate cavitation damage on mercury vessels, a critical issue in spallation neutron sources. From an experimental study using an electromagnetically driven impact test machine and a bubble generator, we found that by injecting microbubbles, the magnitude of the negative pressure generated in liquid mercury is slightly decreased and cavitation damage is remarkably reduced. In this paper, we have performed a numerical study using a multibubble model and experimentally obtained pressure–time curves in order to thoroughly explain the experimental findings. We have found that the observed slight change in negative pressure has a strong impact on cavitation bubble dynamics and was caused by the positive pressure wave that the injected bubbles radiated. Also, we have examined whether the injected microbubbles can cause significant erosion, and found that their collapse intensity is much smaller than that of cavitation bubbles since their expansion ratio is relatively small. Additionally we have examined high-frequency pressure pulses observed experimentally only when microbubbles were injected, and clarified that they are due to the free oscillation of injected bubbles. & 2008 Elsevier B.V. All rights reserved.
Keywords: Spallation neutron source Cavitation Liquid mercury Pressure wave Bubble Erosion
1. Introduction The Japan Atomic Energy Agency (JAEA) and the High Energy Accelerator Research Organization (KEK) are currently promoting a R&D project on the development and use of a MW-class proton accelerator, called the J-PARC (Japan Proton Accelerator Research Complex) project [1–3]. One of the aims of the project is to provide a high-power spallation neutron source to be used for leading-edge researches in materials and life sciences. In the neutron source being developed, liquid mercury flowing inside a target vessel is bombarded by high-intensity proton beams to produce high neutron fluxes. The accelerated proton beams have an energy of 3 GeV and are repeatedly injected into the mercury at a repetition rate of 25 Hz. In the development of the neutron source, cavitation in liquid mercury and the resulting erosion are now significant problems. From several different experiments [4–10], it was suggested that high-intensity pressure waves produced by the enormous energy release due to spallation reactions cause cavitation in liquid mercury and it will significantly reduce the lifetime of the target vessel by causing cavitation erosion. In Refs. [4,5], using a split Corresponding author. fax: +81 3 5246 2537.
E-mail address: [email protected] (M. Ida). 0168-9002/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.11.124
Hopkinson pressure bar technique, the JAEA target team first found erosion damage (i.e., damage pits) on the surface of a metal chamber filled with liquid mercury. The authors of the paper assumed that the erosion damage was caused by cavitation bubbles emerging in liquid mercury, but could not provide a definitive evidence of the origin of the observed erosion. In a following study by a group at the Spallation Neutron Source, Oak Ridge National Laboratory [6,7], it was demonstrated that the same problem is caused by actual proton beams. In Ref. [8], using an electromagnetically driven impact test machine which reproducibly produces pressure pulses in liquid mercury by mechanical impact, the erosion damage was systematically evaluated under mechanical impacts of up to over 10 million cycles. In Refs. [9,10], using the same impact test system, but with an image monitoring apparatus, the image of cavitation bubbles emerging in liquid mercury was recorded, which was the first definitive evidence of the occurrence of cavitation, and the dynamics of the cavitation bubbles was investigated numerically and theoretically to provide basic knowledge (e.g., expansion velocity in a negative pressure condition) of bubble dynamics in liquid mercury. Given these findings, we and collaborators are now performing various investigations to overcome this critical issue [11–15]. Our present aim is to propose a technique to reduce the erosion damage or suppress the cavitation itself.
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One of the mitigation techniques that we have examined is microbubble injection [13–15]. In this technique, gas bubbles (e.g., helium bubbles) are injected into flowing mercury through a bubble generator, and hence the mercury becomes a bubbly liquid even before cavitation occurs. Since gas bubbles in a liquid are known to change dynamic and acoustic properties of the host liquid (e.g., sound speed, acoustic attenuation and dispersion) in diverse ways, we thought that they would play some role in the cavitation process in liquid mercury. Though the effectiveness of this approach has not yet been fully clarified, our recent theory [14] and experiments [15] suggested that it can suppress cavitation inception and significantly reduce cavitation damage at least for a limited parameter range accessible to our impact test machine. The theory presented in Ref. [14] showed that the injected bubbles radiate a positive pressure wave when they expand in response to the action of negative pressure, and thereby suppress cavitation inception in certain situations. The experiments performed in Ref. [15], where a large number (5000) of mechanical impacts were imposed on pure and bubbly mercury, demonstrated clearly that the number of damage pits are drastically reduced by injecting gas bubbles at least for the pressure pulses produced in the impact test, whose rise time and duration are on the order of 1 ms. In this paper, we further examine the effect of microbubble injection numerically and attempt to explain several qualitative features found in the result of the impact test. From the previous experiments, we found that the magnitude of the negative pressure produced in mercury by mechanical impact is slightly decreased (i.e., the tensile stress generated in mercury is reduced) by the injected bubbles. We perform here a numerical study of bubble dynamics in the experimental condition to show that the slight change in negative pressure has a strong impact on cavitation bubble dynamics. Also, we investigate the pressure wave radiated by injected bubbles to clarify the mechanism of how the injected bubbles changed the negative pressure value and cavitation bubble dynamics. Though such an investigation of the bubble-radiated pressure wave was already done in Ref. [14] using an idealized pressure history in mercury, we reexamine it by using experimentally obtained pressure–time curves in both singlephase (mercury alone) and bubbly flow cases. Furthermore, we
examine the dynamics of injected bubbles to provide an answer to a legitimate question whether the injected bubbles themselves cause significant erosion. To this end we study numerically the correlation between the expansion ratio and the collapse velocity of injected bubbles. Also, we examine the dynamics of injected bubbles in the positive pressure period of the mechanically induced pressure pulse, and attempt to clarify the origin of highfrequency pressure pulses which have sometimes been observed experimentally and have a fundamental frequency higher than that of the mechanically induced pressure pulse. The rest of this paper is organized as follows: In Section 2, the experiment using the impact test machine and several important qualitative features of the experimental result are briefly reviewed. In Section 3, the model equations used in this numerical study, a Rayleigh–Plesset type nonlinear system of equations that takes bubble–bubble interaction into account and a simple formula for the bubble-radiated pressure, are introduced. Section 4 presents numerical results and discussions, and Section 5 summarizes this paper.
2. Brief review of experiments The experimental setup of the impact test is illustrated in Fig. 1. In the shallow stainless-steel cylinder, filled with mercury (not degassed), cavitation event is reproduced by imposing mechanical impacts from the bottom using an electromagnetic coil. The image of cavitation bubbles is monitored using a high-speed camera (NAC, Memrecam fx RX6) through a clear acrylic window placed on the upper surface of the mercury. A straight flow channel of 33 mm width and 12 mm height runs through the cylinder, one end of which is connected to a bubbler tube (bubble generator) made of sintered porous tungsten. When the effect of microbubble injection is examined, helium gas is injected through the bubbler tube into flowing mercury, which will form microbubbles through pinch-off processes. Pure or bubbly mercury flows in the flow channel at a constant mean velocity of 0.3 m/s. In the bubbly flow case, the flow rate of the injected helium gas was set to 6.48 ml/min and the void fraction was thus about 0.1%. Since liquid mercury is an opaque liquid, we can see only bubbles in
Expansion tank
Helium gas
Acrylic window
P
Flow meter
Bubbler P
Shallow cylinder E.M.force
MIMTM
Mercury
PM-pump Fig. 1. Experimental setup of the impact test. The acrylic window on the shallow cylinder is replaced with a pressure transducer when the pressure change in liquid mercury is measured. Helium microbubbles are injected into flowing mercury using a bubbler tube.
ARTICLE IN PRESS M. Ida et al. / Nuclear Instruments and Methods in Physics Research A 600 (2009) 367–375
0.7
0.7 Pulses 0.6 Pressure change [MPa]
contact with the acrylic window. The average radius of the injected bubbles deduced from captured images taken through the acrylic window was about 50 mm, and a small number of much larger bubbles (over 150 mm in radius) were observed. When the pressure change in the shallow cylinder was measured, the acrylic window was replaced with a pressure transducer (Entran, EPXH). For more detail of the experimental setup, see Ref. [15]. In the experiment, three cases were considered: the singlephase static case where single-phase mercury at rest is used, the single-phase flow case where single-phase mercury is forced to flow, and the bubbly flow case where microbubbles are injected into flowing mercury. From the experimental investigation, it was found that both mercury flow and microbubble injection are effective to reduce cavitation damage. Since we could not find any significant difference between the pressure–time curves in the single-phase static and single-phase flow cases, we hypothesized that the damage reduction found in the single-phase flow case was caused by the deformation of cavitation bubbles due to mercury flow. Though further discussion is needed on the singlephase flow case, in the present numerical study we do not consider it because the theoretical model used cannot treat bubble deformation. Typical pressure changes generated by a single mechanical impact with (the solid line) and without (the dashed line) applying microbubble injection are shown in Fig. 2. In the present study, compression impacts were imposed on the mercury, which produced a positive pressure pulse followed by a negative pressure tail. As shown in our previous papers [9,10], the negative pressure in the single-phase case triggers cavitation in liquid mercury, which will cause serious erosion damages when a large number of mechanical impacts are applied [8]. We can find an important difference between the two observed pressure changes, although they are qualitatively the same. The maximum magnitude of the negative pressure in the bubbly flow case is slightly smaller than that in the single-phase case; that is, the injected microbubbles decreased the magnitude of negative pressure. Because, when microbubbles were injected, cavitation bubbles have never been found on the upper surface of mercury and only injected bubbles (which could be seen since before impacting and underwent volume change after impacting) have been recorded, this slight (but reproducible) change should have a considerable influence on cavitation bubble dynamics. In Sections 4.1 and 4.2, we clarify the impact and origin of this slight change.
369
0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
1.5 2 Time [ms]
2.5
3
3.5
Fig. 3. High-frequency pressure pulses sometimes observed when microbubble injection was applied. The solid and dashed lines, respectively, are for the cases with and without microbubble injection. In the positive pressure period of the pressure change in the case with microbubbles, several pressure pulses are observed. As can be seen by comparing the solid lines in Figs. 2 and 3, the details of the pressure–time curve in bubbly flow cases changed depending on the case. However, in any case the magnitude of negative pressure tended to decrease.
In bubbly flow cases, we have sometimes observed highfrequency pressure pulses in the positive pressure period, which have a non-negligible amplitude and a fundamental frequency significantly higher than that of the mechanically induced pulse; see Fig. 3. The origin of those high-frequency pulses are discussed in Section 4.4. We should note here that the present experimental setup does not allow us to reproduce some remarkable numerical predictions. In Ref. [13], using a direct numerical simulation (DNS) technique, Okita et al. showed that gas microbubbles absorb thermal expansion of liquid mercury, which is caused by the spallation reactions, and thus significantly reduce the amplitude of pressure waves. Since our impact test uses mechanical impacts to induce pressure waves, the absorption effect of injected bubbles cannot be reproduced and examined. In Ref. [16], using a different DNS technique, Lu et al. showed that when gas bubbles are injected, the pressure oscillation in mercury decays more than 10 times faster, while its maximum amplitude becomes slightly larger, than in the pure mercury case. Such a large change in decay time could not be found in our impact test. These points are worth examining but a detailed discussion is not given here since the aim of the present paper is to explain the findings from the impact test.
Pressure change [MPa]
0.6 0.5
3. Model equations
0.4 0.3 0.2 0.1 0 0
0.5
1
1.5 2 Time [ms]
2.5
3
3.5
Fig. 2. Typical pressure changes in liquid mercury generated by a single mechanical impact. The solid and dashed lines, respectively, are for the cases with and without microbubble injection.
Key factors in the present numerical study are the pressure wave radiated by the injected bubbles and the interaction between bubbles through the bubble-radiated pressure waves. As known well, bubbles in a pressure field undergo volume change and radiate a pressure wave into the surrounding liquid. The bubble-radiated pressure wave causes acoustic interaction between nearby bubbles. This interaction is known to change the dynamics of bubbles and lead to rich physics [17–22]. As mentioned above, in Ref. [14] we found that the pressure waves radiated by injected microbubbles can in some cases cause the suppression of cavitation inception. That is, the interaction between the injected bubbles and cavitation bubbles is of great importance in our investigation. To take the interaction into consideration, we use a Rayleigh–Plesset type multibubble model.
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The multibubble model used in this study is the coupled Keller–Miksis equation [14], which reads ! R_ i 1 Ri R€ i þ c
! ! R_ i 3 R_ i _ 2 1 R d p 1þ R ¼ ps;i þ i 2 2c i r c rc dt s;i
ps;i pb;i
pb;i ¼
2 N X 1 dðRj R_ j Þ D dt j¼1;jai ij
2s 4mR_ i pex ðtÞ P0 Ri Ri
2s P0 þ Ri0
3
R3i0 hi
3
R3i hi
(1)
(2)
!ki (3)
4. Numerical study of bubble dynamics in the impact test We now consider the dynamics of cavitation and injected bubbles in the experimental condition. First, we examine the effect of the observed slight change in negative pressure on cavitation bubble dynamics using a single-bubble model (Eq. (1) with Dij ! 1), and then discuss the underlying mechanism of the slight change using the multibubble model. In the multibubble study, we try to explain the experimental result in the bubbly flow case by using only the pressure change in the single-phase case. Last, we investigate the high-frequency pulses using the singlebubble model. 4.1. Single-bubble study on the impact of the slight change in negative pressure
for i ¼ 1; 2; . . . ; N where Ri ¼ Ri ðtÞ and Ri0 are the time dependent radius and initial radius, respectively, of bubble i, c ¼ 1450 m=s, r ¼ 13528 kg=m3 , and m ¼ 1:52 103 Pa s are the sound velocity, density, and viscosity, respectively, of liquid mercury, s ¼ 0:47 N=m is the surface tension, Dij is the center-to-center distance between bubbles i and j, P 0 ¼ 0:1013 MPa is the atmospheric pressure, pex ðtÞ is the external pressure which drives bubble dynamics, hi (¼ Ri0 =11:26 for mercury and Ri0 =9:81 for helium) and ki are the hard-core radius and polytropic exponent, respectively, of the gas inside bubble i, N is the total number of bubbles, and the overdots denote the time derivative. This system of nonlinear differential equations describes the dynamics of N coupled spherical bubbles subjected to a pressure change. Eq. (1) governs the radial motion of interacting bubbles. The last term of this equation describes the acoustic interaction between the bubbles through the pressure waves radiated by the bubbles themselves, and couples the equations for bubbles 12N. Eq. (3) describes the pressure inside bubble i, where we assumed a van-der-Waals type equation of state which takes into account the excluded volume effect, a real gas effect. Since the liquid mercury used in the experiment was not degassed, we assumed that the content of cavitation bubbles is a gas–vapor mixture. The vapor pressure of mercury (0.28 Pa) was, however, neglected, because it is much smaller than the absolute value of the negative pressure observed in the impact test and hence it has no significant influence on cavitation bubble dynamics. The injected bubbles were assumed to be helium bubbles as in the experiment. The value of ki is known to depend on the dynamics and thermal property of bubbles. In the present study, we used the following simple determination [14]: (
ki ¼
1
for Ri ðtÞ4bRi0
g otherwise
(4)
where g is the specific heat ratio of the gas inside the bubbles (53 for both mercury and helium) and we assumed b ¼ 0:2. This formula approximately describes the adiabatic behavior (e.g., Ref. [23]) of violently collapsing bubbles. The pressure radiated by bubble i is determined by the following simple formula [14,18] with Ri and its time derivatives given by Eq. (1): pi ¼
r dðR2i R_ i Þ ri
dt
(5)
where r i is the distance from the center of bubble i. This equation can be derived from the Euler and continuity equations of fluid flow, and corresponds to the last term of Eq. (1). This equation is used to examine how the injected bubbles affect the negative pressure value in liquid mercury.
The dynamics of single cavitation bubbles for different initial radii computed using Eq. (1) with Dij ! 1 are shown in Fig. 4. We used here the two recorded pressure changes in Fig. 2 as the external pressure pex ðtÞ. Interestingly, the dynamics of cavitation bubbles for different pressure changes are significantly different from each other. For the pressure change in the bubbly flow case, the cavitation bubbles could not grow significantly, whereas, for the pressure change in the single-phase case, the cavitation bubbles, except for very small ones, expanded explosively. This result is consistent with the experimental observation that cavitation bubbles, which were observed without microbubble injection, were never found in the bubbly flow case. The above result implies that the negative pressure in the bubbly flow case does not exceed the (dynamic) Blake threshold pressure of the cavitation bubbles, while that in the single-phase case does. The Blake threshold pressure of a cavitation bubble is the liquid pressure at which the cavitation bubble begins to expand explosively, and has in general a negative value. When the liquid pressure becomes lower than the threshold, the pressure inside the bubble, which is always positive, can no longer balance the liquid pressure through the pressure jump due to surface tension, and hence the bubble will expand without bound until the liquid pressure turns into positive. The above numerical result suggests that the injected bubbles reduced the magnitude of negative pressure in liquid mercury to a level that does not exceed the Blake threshold pressure of the cavitation bubbles. From the above result, we can conclude that the microbubble injection significantly affects the dynamics of cavitation bubbles by altering the negative pressure value. The above numerical result, however, does not say anything about how the injected bubbles modified the liquid pressure. In the next subsection, we address this question by employing the multibubble model. 4.2. Multibubble study on how the injected bubbles change the liquid pressure and cavitation bubble dynamics In the experiment, a large number of gas bubbles were injected into liquid mercury, which leads to the interaction between nearby injected bubbles and also between injected bubbles and cavitation bubbles (or cavitation nuclei). Here we first show a result of a reduced problem, a nine-body problem (i.e., the interaction of a cavitation bubble with eight injected bubbles), and discuss basic roles of the injected bubbles. We then consider an extended problem where a large number of bubbles are assumed to be homogeneously distributed, and perform a parametric study to clarify the dependence of the suppression effect on the initial radii of injected bubbles. In the nine-body problem, we assume that eight injected bubbles are identical and arranged as a cube, and a cavitation
ARTICLE IN PRESS M. Ida et al. / Nuclear Instruments and Methods in Physics Research A 600 (2009) 367–375
371
Radius [µm]
1000
100 20
20 10
10
10
5
5
Liquid pressure [MPa]
1 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0
1
2 Time [ms]
3
0
1
2 Time [ms]
3
Fig. 4. Radius–time curves of a single cavitation bubble for the two pressure changes in Fig. 2. The numbers shown in the upper panels indicate the initial radii ðmmÞ of the corresponding bubble, and the lower panels show the liquid pressure. We stopped the computation just after the bubble has violently collapsed, because after that moment the deformation of the bubble should not be negligible. This treatment is also applied in the following examples.
boundary [10]. Also, half of the eight injected bubbles can be considered as the mirror images of four real injected bubbles originating from the sound reflection at the window surface [10,21]. Under these assumptions, Eq. (1) is reduced to ! ! ! 3 R_ 1 _ 2 1 R1 d R_ 1 R_ 1 1 p 1þ R1 R€ 1 þ R1 ¼ ps;1 þ 2 2c c r c rc dt s;1
1
! R_ 2 R2 R€ 2 þ c
3 3 1 þ pffiffiffi þ h 2h 2D12
2 h pffiffiffi D12 3
Fig. 5. Arrangement of bubbles. Eight injected bubbles, denoted by the larger spheres, are arranged as a cube, and a cavitation bubble, the smaller sphere, is placed at the mass center of the cube. Note that the sizes of the presented spheres do not accurately reflect the actual sizes of the bubbles in the numerical study.
bubble is placed at the mass center of the cube; see Fig. 5. Due to the symmetry of arrangement, in numerical computation this example can be treated as a two-body problem of a cavitation bubble (called hereafter bubble 1) and an injected bubble (bubble 2). The cavitation bubble is assumed to be a hemispherical bubble on the acrylic window but is treated as a spherical bubble, because the thickness of the viscous boundary layer forming on the window surface is much smaller than the radii of expanding bubbles and hence the window surface only acts as a mirror
(6)
! ! R_ 2 3 R_ 2 _ 2 1 R2 d p 1þ R2 ¼ ps;2 þ 2 2c r c rc dt s;2
b
8 dðR22 R_ 2 Þ D12 dt
1 dðR21 R_ 1 Þ dðR22 R_ 2 Þ b D12 dt dt
(7)
(8)
(9)
where D12 is the distance between the centers of bubbles 1 and 2, and h is the length of a side of the cube, which corresponds to the center-to-center distance between nearest neighbor injected bubbles. For the external pressure pex ðtÞ, as mentioned above we use only the observed pressure change in the single-phase case (the dashed curve in Fig. 2). Thus, in this investigation, only the bubble-radiated pressure waves can be the source of pressure modification. The radius–time curves for R10 ¼ 10 mm and R20 ¼ 50 mm with D12 ¼ 1 and 750 mm are shown in Fig. 6. For D12 ¼ 1 (i.e., when the bubbles are isolated), the cavitation bubble expanded explosively and then collapsed violently as observed in the experiment without microbubble injection [9,10]. On the other hand, for a sufficiently small D12 (Fig. 6(b)), the explosive expansion was completely suppressed by the surrounding gas bubbles (and also the dynamics of the injected bubbles were altered by the interaction between them). This result seems to be
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and low deceleration (note that dðR2i R_ i Þ=dt in Eq. (5) can be 2 decomposed into 2Ri R_ i þ R2i R€ i ). These positive pressures decreased the magnitude of the negative pressure at the position of the cavitation bubble and consequently suppressed the explosive expansion of the cavitation bubble. This explanation is consistent with the experimentally observed change in negative pressure value (Fig. 2) and the previous theory [14]. Next we consider a more realistic case where a large number of cavitation and injected bubbles interact. In this investigation, in order to simplify the coupled Keller–Miksis equation, we employ the local homogeneous model proposed by Kubota et al. [24], which is a sort of mean field approximation. In that model, bubbles in a large bubble cluster are assumed to be spherical and locally identical, and the bubble population is described by a locally defined number density. This model allows one to approximately solve the dynamics of a large bubble cluster by a simple and efficient technique. We here extend the model so that a bubble cluster containing two types of bubbles can be treated. Assuming that the radii of cavitation and injected bubbles in a locally homogeneous cluster are R1 ¼ R3 ¼ ¼ RN1 and R2 ¼ R4 ¼ ¼ RN , respectively (here N is an even number), and their local number densities ðnÞ are the same (see Fig. 7), the summation term in Eq. (1) is approximated as follows:
600 D12 = ∞
Radius [μm]
500 400 300
Injected
Cavitation
200 100 0
D12 = 750 μm
Radius [μm]
500 400
Injected 300 200 100
2 N X 1 dðRj R_ j Þ ¼ D dt j¼1;jai ij
N=2 X dðR21 R_ 1 Þ 1 dðR22 R_ 2 Þ þ D dt D dt j¼1;2j1ai i;2j1 j¼1;2jai i;2j " # Z DD 2_ 2_ 1 dðR1 R1 Þ dðR2 R2 Þ 4pr 2 dr þ n r dt dt 0 " # 2_ 2_ dðR1 R1 Þ dðR2 R2 Þ þ ¼ 2pnDD2 , (10) dt dt
Cavitation 0
Driving pressure [MPa]
0.4 0.3 D12 = ∞
0.2 0.1 0
D12 = 750 μm -0.1 0
1
2
3
Time [ms] Fig. 6. Radius–time curves in the nine-bubble case for (a) D12 ¼ 1 and (b) 750 mm ð¼ 15R20 Þ, and (c) driving pressure acting on the cavitation bubble in the two cases. In the case of D12 ¼ 750 mm, the magnitude of the negative pressure acting on the cavitation bubble was decreased by the nearby gas microbubbles, as observed experimentally (Fig. 2). The strong pressure pulse seen in the case of D12 ¼ 750 mm is due to the collapse of the injected bubbles (such strong pressure pulses have sometimes been observed in the experiment as well, though no pressure pulses can be seen in the selected pressure history because the time resolution of the pressure transducer (about 5:6 ms) is not enough to fully resolve the pressure pulses whose width is typically a few microseconds or less).
N=2 X
1
where DD is the cutoff distance due to the finite sound speed and acoustic shielding. This approximation of the summation term allows us to solve the problem as a two-body problem of bubbles 1 and 2. We introduce here the mean distance between nearestneighbor bubbles of the same kind, Dmean . Using this quantity, the number density is given as n ¼ 1=D3mean . As suggested in Ref. [25], we assume DD to be Dmean -dependent. However, we use a larger value ðDD 1:5Dmean Þ than that used in Ref. [25] ðDD ¼ Dmean Þ because the expansion ratio of injected bubbles for DD ¼ Dmean was almost the same as in the nine-bubble case although a larger number of bubbles are considered (a larger DD gives a smaller
Cavitation bubbles Injected bubbles
consistent with the experimental observations [15] that no cavitation bubbles were observed and cavitation damage was significantly reduced when injecting microbubbles. To clarify the mechanism of the change in cavitation bubble dynamics, we examine the pressure acting on the cavitation bubble. Fig. 6(c) shows the time history of the driving pressure on bubble 1, which was determined by the sum of pex ðtÞ and the pressures radiated by the eight injected bubbles. As can be seen clearly in this figure, the magnitude of negative pressure for D12 ¼ 750 mm is smaller than that for D12 ¼ 1, meaning that the injected bubbles decreased the magnitude of the driving pressure on the cavitation bubble. This result can be explained as follows: In the early stage of expansion, the injected bubbles radiate positive pressure waves because of their high expansion velocity
R1 R2
Fig. 7. Local homogeneity assumption used in the present study. A large threedimensional bubble cluster that consists of cavitation and injected bubbles is considered. In the large cluster, both cavitation and injected bubbles are spherical and bubbles of the same kind have, locally, the same radius. The number densities of both bubbles are locally the same.
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expansion ratio). As in the previous example, the observed pressure change in the single-phase case is used as the external pressure pex ðtÞ (that is, we roughly assume that the external pressure acting on the homogeneous cluster is spatially uniform). The expansion ratios of the bubbles, maxRj ðtÞ=Rj0 , as functions of the initial radius of the injected bubble ðR20 Þ are plotted pffiffiffi in Fig. 8(a). Here, we assumed R10 ¼ 10 mm and Dmean ¼ ð2= 3Þ750, the later of which corresponds to the distance between nearestneighbor injected bubbles in the nine-bubble case. The expansion ratios in the present case are overall smaller than those in the nine-bubble case (Fig. 8(b)) due to the interaction of a larger number of bubbles, but the results of both cases are qualitatively the same.
373
As can be seen in the figure, when the injected bubbles are larger than the cavitation bubbles ðR20 410 mmÞ, the explosive expansion of the cavitation bubbles is suppressed and their expansion ratio is thus very small. However, when R20 4R10 but R20 is not so large ð10 mmoR20 o30 mmÞ, the expansion ratio of the injected bubbles is very large, implying the possibility that the injected bubbles themselves cause erosion even though cavitation inception is successfully suppressed. In contrast, when R20 is much larger than R10 , the expansion ratios of both bubbles are small. This result could explain the experimental observations that the injection of large gas bubbles ðR20 X50 mmÞ suppressed cavitation inception and significantly reduced erosion. The next subsection discusses how the change of the expansion ratio affects the bubbles’ collapse intensity.
25 4.3. Single-bubble study on why the injected bubbles did not cause erosion
20
15
10
Cavitation
5
0 3
10
30
100
R20 [µm] 25
Injected
30 15 25 10 Expansion ratio
Expansion ratios
20
As proved in Ref. [14] and in the above subsection, the expansion ratio of sufficiently large bubbles decreases as their initial radii increase. This is an important key to understanding why the injected bubbles did not cause erosion. Here, we examine the correlation between the expansion ratio and the maximum collapse velocity of the bubbles. In this investigation we neglect bubble–bubble interaction to know the maximum collapse intensity that the bubbles may have (bubble–bubble interaction decreases bubbles’ expansion velocity [10]). Fig. 9 shows the expansion ratio and the maximum collapse velocity (normalized by the sound velocity of liquid mercury, maxjdRðtÞ=dtj=c) of single injected bubbles as functions of their initial radii R20 . Here we used the observed pressure change in the single-phase condition as pex ðtÞ. This result confirms that the collapse velocity correlates very well with the expansion ratio. For 10 mmoR20 o25 mm, the collapse velocity exceeds the sound velocity of liquid mercury, indicating that the bubbles have a very large collapse intensity. For larger R20, the collapse velocity decreases monotonically with increasing R20 and finally becomes much smaller than the sound velocity. This figure also indicates that cavitation bubbles, if they expand explosively, must have a very strong collapse intensity since they have a very large expansion ratio.
Cavitation 5
20
2
15
1.5
10
1
max|dR/dt| / c
Expansion ratios
Injected
0 30
10
100
R20 [µm] Fig. 8. (a) Expansion ratios of cavitation and injected pffiffiffi bubbles in a locally homogeneous cluster for R10 ¼ 10 mm and Dmean ¼ ð2= 3Þ750, as functions of the initial radius of the injected bubble. In (b), the expansion ratio for the nine-bubble case is shown as reference (results for R20 450 mm are not shown because bubble collision was observed during the computation). In the range of 10 mmoR20 o20 mm, the bubbles’ dynamics changes sensitively with R20 because this range is close to the cavitation threshold, near which a small fluctuation (in, e.g., the pressure change) can result in a large change in bubble dynamics.
0.5
5
0
0 3
10
30
100
300
Initial radius R20 [µm] Fig. 9. Expansion ratio and maximum collapse velocity of single injected bubbles as functions of their initial radii. Here the collapse velocity is normalized by the sound velocity of liquid mercury.
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Based on the above numerical results, we can explain the experimental findings from the impact test as follows. In the single-phase case, many cavitation bubbles emerged in liquid mercury and expanded explosively. They then collapsed violently and caused significant erosion. In the bubbly flow case, cavitation inception was suppressed and only injected bubbles, which had much larger initial radii than cavitation bubbles, expanded. Since the expansion ratio of injected bubbles is small, their collapse velocity and erosion intensity were much smaller than those of explosively expanding cavitation bubbles. Therefore, cavitation erosion was greatly reduced by injecting gas bubbles.
0.7 0.6 Pressure change [MPa]
374
5. Conclusion Based on pressure–time curves obtained from the impact test and relatively simple theoretical models, we have studied numerically the effect of microbubble injection which we are attempting to mitigate cavitation damage in spallation neutron sources. The important findings presented in this paper are summarized as follows: (1) The slight decrease of the magnitude of negative pressure due to microbubble injection has a strong impact on cavitation bubble dynamics in liquid mercury. For the negative pressure in the case without microbubble injection, cavitation bubbles undergo explosive expansion which leads to a subsequent violent collapse. On the other hand, for the negative pressure in the case with microbubble injection, cavitation bubbles do not expand significantly.
r1 = 40 R10
0.4 0.3 0.2 0.1
4.4. Single-bubble study on the experimentally observed highfrequency pulses
Experiment
0
0.7
Pressure change [MPa]
0.6 r1 = 20 R10
0.5 0.4 0.3 0.2 0.1
Experiment
0
1200 1000 Radius [µm]
Lastly, we discuss the high-frequency pressure pulses in the positive pressure phase. In this investigation, we used the singlebubble model with Eq. (5) and, for pex ðtÞ, the observed pressure change in the single-phase case. Here, we assume that the pressure pulses are the pressure wave radiated by an injected bubble passing near the pressure transducer. Figs. 10(a) and (b) show the computed liquid pressure at different distances r 1 from a single injected bubble (bubbles 1), determined by the sum of pex ðtÞ and Eq. (5). The initial radius of the bubble was assumed to be 450 mm, which is relatively larger than that in the previous example. The numerical results reproduce, at least qualitatively, important characteristics of the pulses such as frequency and amplitude. The pressure pulses found in the numerical study are due to the free oscillation of the injected bubble triggered by the sudden rise of liquid pressure. The radius–time curve of the bubble is shown in Fig. 10(c). The oscillatory behavior in the positive pressure period is the free oscillation of the bubble at its eigenfrequency. The eigenfrequency of a pulsating bubble is known to depend on the bubble’s radius, and a smaller bubble has a higher eigenfrequency. The bubble radius assumed in the above example was deduced from the pulse interval. We should note here that the actual radius of the bubble that caused the experimentally observed pulses might be different from the radius assumed above. It is known that bubble–bubble and bubble–wall interactions can change the eigenfrequency of bubbles (see, e.g., Refs. [18–21,26,27]). When, for example, two bubbles pulsate in phase with each other, the eigenfrequencies of the bubbles are decreased. Since in the experiment many bubbles interact with each other in the vicinity of an acrylic window, the actual radius should be not equal to 450 mm (probably smaller than this). However, we argue that the origin of the pressure pulses was clearly identified by this numerical study.
0.5
800 600 400 200 0 0.5
1
1.5 Time [ms]
2
2.5
Fig. 10. (a, b) Liquid pressures at different distances from a single bubble and (c) radius of the injected bubble as functions of time. In (a) and (b), the solid, dashed, and dotted curves, respectively, denote the numerical results (for r 1 ¼ 40R10 (a) and 20R10 (b)), an experimental result in the bubbly flow case (already shown in Fig. 3), and the external pressure assumed in the computation. The oscillatory behavior of the gas bubble shown in (c) is the origin of the pressure pulses found in (a) and (b).
(2) The slight change in negative pressure is caused by the positive pressure waves that the injected bubbles radiate when they expand in response to negative pressure. The positive pressure waves reduce the magnitude of negative pressure to a level that does not exceed the Blake threshold pressures of cavitation bubbles. That is, the bubble-radiated pressure waves suppress cavitation inception. (3) Erosion intensity of the injected bubbles should be much smaller than that of cavitation bubbles since their expansion
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0.02
However, the effectiveness of microbubble injection in more realistic conditions (i.e., when actual proton beams are used) is still unclear. Further efforts are needed to assess the true effectiveness of the technique.
0.01
Acknowledgments
0.03
Pressure change [MPa]
375
0 -0.01 Threshold (arbitrary)
-0.02 -0.03 1.6
1.8
2 Time [ms]
2.2
2.4
Fig. 11. Magnified view of the two experimentally obtained pressure changes in the negative pressure period. The solid straight line denotes the Blake threshold pressure of an arbitrary cavitation bubble. Our numerical results suggested that by injecting gas microbubbles, the negative pressure value changed across the threshold pressure to a level that does not exceed the threshold, as indicated by an arrow.
ratio and collapse velocity are much smaller than those of cavitation bubbles. This finding can (at least partly) explain why the injected bubbles did not cause erosion and remarkably reduced cavitation damage. (4) The high-frequency pressure pulses observed experimentally in the positive pressure period of the mechanically induced pulse are due to injected bubbles. In this period, the injected bubbles undergo free oscillation triggered by the sudden pressure rise, and radiate high-frequency pressure waves. These findings clarify the role of gas microbubbles injected into liquid mercury and the underlying mechanism of the damage reduction observed in the impact test [15]. The present results suggest that the observed slight change in negative pressure value took place across the Blake threshold pressures of cavitation bubbles; that is, the injected bubbles worked to raise pressure to a level that does not exceed the threshold; see Fig. 11. This is consistent with the experimental observation that no cavitation bubbles were seen when microbubble injection was applied.
This work was partly supported by the Ministry of Education, Culture, Sports, Science, and Technology of Japan through a Grantin-Aid for Young Scientists (B) (no. 17760151) and by the Japan Society for the Promotion of Science through a Grant-in-Aid for Scientific Research (no. 17360085). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
S. Nagamiya, J. Nucl. Mater. 343 (2005) 1. Y. Ikeda, J. Nucl. Mater. 343 (2005) 7. S. Nagamiya, Nucl. Phys. A 774 (2006) 895. M. Futakawa, H. Kogawa, R. Hino, J. Phys. IV 10 (2000) 237. M. Futakawa, H. Kogawa, R. Hino, H. Date, H. Takeishi, Int. J. Impact Eng. 28 (2003) 123. B.W. Riemer, J.R. Haines, J.D. Hunn, D.C. Lousteau, T.J. McManamy, C.C. Tsai, J. Nucl. Mater. 318 (2003) 92. J.D. Hunn, B.W. Riemer, C.C. Tsai, J. Nucl. Mater. 318 (2003) 102. M. Futakawa, T. Naoe, H. Kogawa, C.-C. Tsai, Y. Ikeda, J. Nucl. Sci. Technol. 40 (2003) 895. T. Naoe, M. Futakawa, T. Koyama, H. Kogawa, J. JSEM 6 (2006) 301 (in Japanese). M. Ida, T. Naoe, M. Futakawa, Phys. Rev. E 75 (2007) 046304. M. Futakawa, T. Naoe, H. Kogawa, M. Teshigawara, Y. Ikeda, J. Nucl. Mater. 356 (2006) 168. A. Fujiwara, K. Okita, Y. Matsumoto, M. Futakawa, S. Hasegawa, H. Kogawa, Y. Ikeda, in: Proceedings of the 17th Meeting of the International Collaboration on Advanced Neutron Sources (ICANS-XVII), Los Alamos National Laboratory, LA-UR 06–3904, Santa Fe, New Mexico, vol. II, 2006, p. 552. K. Okita, A. Fujiwara, S. Takagi, Y. Matsumoto, M. Futakawa, in: Proceedings of the Sixth International Symposium on Cavitation, CAV2006, Wageningen, The Netherlands 2006, p. 1-10. M. Ida, T. Naoe, M. Futakawa, Phys. Rev. E 76 (2007) 046309. T. Naoe, M. Ida, M. Futakawa, Nucl. Instr. and Meth. A 586 (2008) 382. T. Lu, R. Samulyak, J. Glimm, J. Fluids Eng. 129 (2007) 595. A.A. Doinikov, J. Fluid Mech. 444 (2001) 1. M. Ida, Phys. Lett. A 297 (2002) 210. M. Ida, Phys. Rev. E 67 (2003) 056617. M. Ida, Phys. Rev. E 72 (2005) 036306. E.M.B. Payne, S.J. Illesinghe, A. Ooi, R. Manasseh, J. Acoust. Soc. Am. 118 (2005) 2841. A.A. Doinikov (Ed.), Bubble and Particle Dynamics in Acoustic Fields: Modern Trends and Applications, Trivandrum, India, Research Signpost, 2005. K. Yasui, Phys. Rev. E 56 (1997) 6750. A. Kubota, H. Kato, H. Yamaguchi, J. Fluid Mech. 240 (1992) 59. M. Arora, C.D. Ohl, D. Lohse, J. Acoust. Soc. Am. 121 (2007) 3432. A. Shima, Trans. ASME J. Basic Eng. 93 (1971) 426. M. Ida, J. Phys. Soc. Jpn 71 (2002) 1214.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
On the effect of microbubble injection on cavitation bubble dynamics in liquid mercury Masato Ida a,, Takashi Naoe b, Masatoshi Futakawa b a b
Center for Computational Science and E-systems, Japan Atomic Energy Agency, Higashi-Ueno, Taito-ku, Tokyo 110-0015, Japan J-PARC Center, Japan Atomic Energy Agency, Tokai-mura, Ibaraki-ken 319-1195, Japan
a r t i c l e in fo
abstract
Article history: Received 19 March 2008 Received in revised form 19 August 2008 Accepted 27 November 2008 Available online 7 December 2008
The effect of microbubble injection has been studied numerically to clarify the role of injected bubbles in the experimentally observed suppression of cavitation in liquid mercury. Recently, we attempted to inject gas microbubbles into liquid mercury in order to mitigate cavitation damage on mercury vessels, a critical issue in spallation neutron sources. From an experimental study using an electromagnetically driven impact test machine and a bubble generator, we found that by injecting microbubbles, the magnitude of the negative pressure generated in liquid mercury is slightly decreased and cavitation damage is remarkably reduced. In this paper, we have performed a numerical study using a multibubble model and experimentally obtained pressure–time curves in order to thoroughly explain the experimental findings. We have found that the observed slight change in negative pressure has a strong impact on cavitation bubble dynamics and was caused by the positive pressure wave that the injected bubbles radiated. Also, we have examined whether the injected microbubbles can cause significant erosion, and found that their collapse intensity is much smaller than that of cavitation bubbles since their expansion ratio is relatively small. Additionally we have examined high-frequency pressure pulses observed experimentally only when microbubbles were injected, and clarified that they are due to the free oscillation of injected bubbles. & 2008 Elsevier B.V. All rights reserved.
Keywords: Spallation neutron source Cavitation Liquid mercury Pressure wave Bubble Erosion
1. Introduction The Japan Atomic Energy Agency (JAEA) and the High Energy Accelerator Research Organization (KEK) are currently promoting a R&D project on the development and use of a MW-class proton accelerator, called the J-PARC (Japan Proton Accelerator Research Complex) project [1–3]. One of the aims of the project is to provide a high-power spallation neutron source to be used for leading-edge researches in materials and life sciences. In the neutron source being developed, liquid mercury flowing inside a target vessel is bombarded by high-intensity proton beams to produce high neutron fluxes. The accelerated proton beams have an energy of 3 GeV and are repeatedly injected into the mercury at a repetition rate of 25 Hz. In the development of the neutron source, cavitation in liquid mercury and the resulting erosion are now significant problems. From several different experiments [4–10], it was suggested that high-intensity pressure waves produced by the enormous energy release due to spallation reactions cause cavitation in liquid mercury and it will significantly reduce the lifetime of the target vessel by causing cavitation erosion. In Refs. [4,5], using a split Corresponding author. fax: +81 3 5246 2537.
E-mail address: [email protected] (M. Ida). 0168-9002/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.11.124
Hopkinson pressure bar technique, the JAEA target team first found erosion damage (i.e., damage pits) on the surface of a metal chamber filled with liquid mercury. The authors of the paper assumed that the erosion damage was caused by cavitation bubbles emerging in liquid mercury, but could not provide a definitive evidence of the origin of the observed erosion. In a following study by a group at the Spallation Neutron Source, Oak Ridge National Laboratory [6,7], it was demonstrated that the same problem is caused by actual proton beams. In Ref. [8], using an electromagnetically driven impact test machine which reproducibly produces pressure pulses in liquid mercury by mechanical impact, the erosion damage was systematically evaluated under mechanical impacts of up to over 10 million cycles. In Refs. [9,10], using the same impact test system, but with an image monitoring apparatus, the image of cavitation bubbles emerging in liquid mercury was recorded, which was the first definitive evidence of the occurrence of cavitation, and the dynamics of the cavitation bubbles was investigated numerically and theoretically to provide basic knowledge (e.g., expansion velocity in a negative pressure condition) of bubble dynamics in liquid mercury. Given these findings, we and collaborators are now performing various investigations to overcome this critical issue [11–15]. Our present aim is to propose a technique to reduce the erosion damage or suppress the cavitation itself.
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One of the mitigation techniques that we have examined is microbubble injection [13–15]. In this technique, gas bubbles (e.g., helium bubbles) are injected into flowing mercury through a bubble generator, and hence the mercury becomes a bubbly liquid even before cavitation occurs. Since gas bubbles in a liquid are known to change dynamic and acoustic properties of the host liquid (e.g., sound speed, acoustic attenuation and dispersion) in diverse ways, we thought that they would play some role in the cavitation process in liquid mercury. Though the effectiveness of this approach has not yet been fully clarified, our recent theory [14] and experiments [15] suggested that it can suppress cavitation inception and significantly reduce cavitation damage at least for a limited parameter range accessible to our impact test machine. The theory presented in Ref. [14] showed that the injected bubbles radiate a positive pressure wave when they expand in response to the action of negative pressure, and thereby suppress cavitation inception in certain situations. The experiments performed in Ref. [15], where a large number (5000) of mechanical impacts were imposed on pure and bubbly mercury, demonstrated clearly that the number of damage pits are drastically reduced by injecting gas bubbles at least for the pressure pulses produced in the impact test, whose rise time and duration are on the order of 1 ms. In this paper, we further examine the effect of microbubble injection numerically and attempt to explain several qualitative features found in the result of the impact test. From the previous experiments, we found that the magnitude of the negative pressure produced in mercury by mechanical impact is slightly decreased (i.e., the tensile stress generated in mercury is reduced) by the injected bubbles. We perform here a numerical study of bubble dynamics in the experimental condition to show that the slight change in negative pressure has a strong impact on cavitation bubble dynamics. Also, we investigate the pressure wave radiated by injected bubbles to clarify the mechanism of how the injected bubbles changed the negative pressure value and cavitation bubble dynamics. Though such an investigation of the bubble-radiated pressure wave was already done in Ref. [14] using an idealized pressure history in mercury, we reexamine it by using experimentally obtained pressure–time curves in both singlephase (mercury alone) and bubbly flow cases. Furthermore, we
examine the dynamics of injected bubbles to provide an answer to a legitimate question whether the injected bubbles themselves cause significant erosion. To this end we study numerically the correlation between the expansion ratio and the collapse velocity of injected bubbles. Also, we examine the dynamics of injected bubbles in the positive pressure period of the mechanically induced pressure pulse, and attempt to clarify the origin of highfrequency pressure pulses which have sometimes been observed experimentally and have a fundamental frequency higher than that of the mechanically induced pressure pulse. The rest of this paper is organized as follows: In Section 2, the experiment using the impact test machine and several important qualitative features of the experimental result are briefly reviewed. In Section 3, the model equations used in this numerical study, a Rayleigh–Plesset type nonlinear system of equations that takes bubble–bubble interaction into account and a simple formula for the bubble-radiated pressure, are introduced. Section 4 presents numerical results and discussions, and Section 5 summarizes this paper.
2. Brief review of experiments The experimental setup of the impact test is illustrated in Fig. 1. In the shallow stainless-steel cylinder, filled with mercury (not degassed), cavitation event is reproduced by imposing mechanical impacts from the bottom using an electromagnetic coil. The image of cavitation bubbles is monitored using a high-speed camera (NAC, Memrecam fx RX6) through a clear acrylic window placed on the upper surface of the mercury. A straight flow channel of 33 mm width and 12 mm height runs through the cylinder, one end of which is connected to a bubbler tube (bubble generator) made of sintered porous tungsten. When the effect of microbubble injection is examined, helium gas is injected through the bubbler tube into flowing mercury, which will form microbubbles through pinch-off processes. Pure or bubbly mercury flows in the flow channel at a constant mean velocity of 0.3 m/s. In the bubbly flow case, the flow rate of the injected helium gas was set to 6.48 ml/min and the void fraction was thus about 0.1%. Since liquid mercury is an opaque liquid, we can see only bubbles in
Expansion tank
Helium gas
Acrylic window
P
Flow meter
Bubbler P
Shallow cylinder E.M.force
MIMTM
Mercury
PM-pump Fig. 1. Experimental setup of the impact test. The acrylic window on the shallow cylinder is replaced with a pressure transducer when the pressure change in liquid mercury is measured. Helium microbubbles are injected into flowing mercury using a bubbler tube.
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0.7
0.7 Pulses 0.6 Pressure change [MPa]
contact with the acrylic window. The average radius of the injected bubbles deduced from captured images taken through the acrylic window was about 50 mm, and a small number of much larger bubbles (over 150 mm in radius) were observed. When the pressure change in the shallow cylinder was measured, the acrylic window was replaced with a pressure transducer (Entran, EPXH). For more detail of the experimental setup, see Ref. [15]. In the experiment, three cases were considered: the singlephase static case where single-phase mercury at rest is used, the single-phase flow case where single-phase mercury is forced to flow, and the bubbly flow case where microbubbles are injected into flowing mercury. From the experimental investigation, it was found that both mercury flow and microbubble injection are effective to reduce cavitation damage. Since we could not find any significant difference between the pressure–time curves in the single-phase static and single-phase flow cases, we hypothesized that the damage reduction found in the single-phase flow case was caused by the deformation of cavitation bubbles due to mercury flow. Though further discussion is needed on the singlephase flow case, in the present numerical study we do not consider it because the theoretical model used cannot treat bubble deformation. Typical pressure changes generated by a single mechanical impact with (the solid line) and without (the dashed line) applying microbubble injection are shown in Fig. 2. In the present study, compression impacts were imposed on the mercury, which produced a positive pressure pulse followed by a negative pressure tail. As shown in our previous papers [9,10], the negative pressure in the single-phase case triggers cavitation in liquid mercury, which will cause serious erosion damages when a large number of mechanical impacts are applied [8]. We can find an important difference between the two observed pressure changes, although they are qualitatively the same. The maximum magnitude of the negative pressure in the bubbly flow case is slightly smaller than that in the single-phase case; that is, the injected microbubbles decreased the magnitude of negative pressure. Because, when microbubbles were injected, cavitation bubbles have never been found on the upper surface of mercury and only injected bubbles (which could be seen since before impacting and underwent volume change after impacting) have been recorded, this slight (but reproducible) change should have a considerable influence on cavitation bubble dynamics. In Sections 4.1 and 4.2, we clarify the impact and origin of this slight change.
369
0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
1.5 2 Time [ms]
2.5
3
3.5
Fig. 3. High-frequency pressure pulses sometimes observed when microbubble injection was applied. The solid and dashed lines, respectively, are for the cases with and without microbubble injection. In the positive pressure period of the pressure change in the case with microbubbles, several pressure pulses are observed. As can be seen by comparing the solid lines in Figs. 2 and 3, the details of the pressure–time curve in bubbly flow cases changed depending on the case. However, in any case the magnitude of negative pressure tended to decrease.
In bubbly flow cases, we have sometimes observed highfrequency pressure pulses in the positive pressure period, which have a non-negligible amplitude and a fundamental frequency significantly higher than that of the mechanically induced pulse; see Fig. 3. The origin of those high-frequency pulses are discussed in Section 4.4. We should note here that the present experimental setup does not allow us to reproduce some remarkable numerical predictions. In Ref. [13], using a direct numerical simulation (DNS) technique, Okita et al. showed that gas microbubbles absorb thermal expansion of liquid mercury, which is caused by the spallation reactions, and thus significantly reduce the amplitude of pressure waves. Since our impact test uses mechanical impacts to induce pressure waves, the absorption effect of injected bubbles cannot be reproduced and examined. In Ref. [16], using a different DNS technique, Lu et al. showed that when gas bubbles are injected, the pressure oscillation in mercury decays more than 10 times faster, while its maximum amplitude becomes slightly larger, than in the pure mercury case. Such a large change in decay time could not be found in our impact test. These points are worth examining but a detailed discussion is not given here since the aim of the present paper is to explain the findings from the impact test.
Pressure change [MPa]
0.6 0.5
3. Model equations
0.4 0.3 0.2 0.1 0 0
0.5
1
1.5 2 Time [ms]
2.5
3
3.5
Fig. 2. Typical pressure changes in liquid mercury generated by a single mechanical impact. The solid and dashed lines, respectively, are for the cases with and without microbubble injection.
Key factors in the present numerical study are the pressure wave radiated by the injected bubbles and the interaction between bubbles through the bubble-radiated pressure waves. As known well, bubbles in a pressure field undergo volume change and radiate a pressure wave into the surrounding liquid. The bubble-radiated pressure wave causes acoustic interaction between nearby bubbles. This interaction is known to change the dynamics of bubbles and lead to rich physics [17–22]. As mentioned above, in Ref. [14] we found that the pressure waves radiated by injected microbubbles can in some cases cause the suppression of cavitation inception. That is, the interaction between the injected bubbles and cavitation bubbles is of great importance in our investigation. To take the interaction into consideration, we use a Rayleigh–Plesset type multibubble model.
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The multibubble model used in this study is the coupled Keller–Miksis equation [14], which reads ! R_ i 1 Ri R€ i þ c
! ! R_ i 3 R_ i _ 2 1 R d p 1þ R ¼ ps;i þ i 2 2c i r c rc dt s;i
ps;i pb;i
pb;i ¼
2 N X 1 dðRj R_ j Þ D dt j¼1;jai ij
2s 4mR_ i pex ðtÞ P0 Ri Ri
2s P0 þ Ri0
3
R3i0 hi
3
R3i hi
(1)
(2)
!ki (3)
4. Numerical study of bubble dynamics in the impact test We now consider the dynamics of cavitation and injected bubbles in the experimental condition. First, we examine the effect of the observed slight change in negative pressure on cavitation bubble dynamics using a single-bubble model (Eq. (1) with Dij ! 1), and then discuss the underlying mechanism of the slight change using the multibubble model. In the multibubble study, we try to explain the experimental result in the bubbly flow case by using only the pressure change in the single-phase case. Last, we investigate the high-frequency pulses using the singlebubble model. 4.1. Single-bubble study on the impact of the slight change in negative pressure
for i ¼ 1; 2; . . . ; N where Ri ¼ Ri ðtÞ and Ri0 are the time dependent radius and initial radius, respectively, of bubble i, c ¼ 1450 m=s, r ¼ 13528 kg=m3 , and m ¼ 1:52 103 Pa s are the sound velocity, density, and viscosity, respectively, of liquid mercury, s ¼ 0:47 N=m is the surface tension, Dij is the center-to-center distance between bubbles i and j, P 0 ¼ 0:1013 MPa is the atmospheric pressure, pex ðtÞ is the external pressure which drives bubble dynamics, hi (¼ Ri0 =11:26 for mercury and Ri0 =9:81 for helium) and ki are the hard-core radius and polytropic exponent, respectively, of the gas inside bubble i, N is the total number of bubbles, and the overdots denote the time derivative. This system of nonlinear differential equations describes the dynamics of N coupled spherical bubbles subjected to a pressure change. Eq. (1) governs the radial motion of interacting bubbles. The last term of this equation describes the acoustic interaction between the bubbles through the pressure waves radiated by the bubbles themselves, and couples the equations for bubbles 12N. Eq. (3) describes the pressure inside bubble i, where we assumed a van-der-Waals type equation of state which takes into account the excluded volume effect, a real gas effect. Since the liquid mercury used in the experiment was not degassed, we assumed that the content of cavitation bubbles is a gas–vapor mixture. The vapor pressure of mercury (0.28 Pa) was, however, neglected, because it is much smaller than the absolute value of the negative pressure observed in the impact test and hence it has no significant influence on cavitation bubble dynamics. The injected bubbles were assumed to be helium bubbles as in the experiment. The value of ki is known to depend on the dynamics and thermal property of bubbles. In the present study, we used the following simple determination [14]: (
ki ¼
1
for Ri ðtÞ4bRi0
g otherwise
(4)
where g is the specific heat ratio of the gas inside the bubbles (53 for both mercury and helium) and we assumed b ¼ 0:2. This formula approximately describes the adiabatic behavior (e.g., Ref. [23]) of violently collapsing bubbles. The pressure radiated by bubble i is determined by the following simple formula [14,18] with Ri and its time derivatives given by Eq. (1): pi ¼
r dðR2i R_ i Þ ri
dt
(5)
where r i is the distance from the center of bubble i. This equation can be derived from the Euler and continuity equations of fluid flow, and corresponds to the last term of Eq. (1). This equation is used to examine how the injected bubbles affect the negative pressure value in liquid mercury.
The dynamics of single cavitation bubbles for different initial radii computed using Eq. (1) with Dij ! 1 are shown in Fig. 4. We used here the two recorded pressure changes in Fig. 2 as the external pressure pex ðtÞ. Interestingly, the dynamics of cavitation bubbles for different pressure changes are significantly different from each other. For the pressure change in the bubbly flow case, the cavitation bubbles could not grow significantly, whereas, for the pressure change in the single-phase case, the cavitation bubbles, except for very small ones, expanded explosively. This result is consistent with the experimental observation that cavitation bubbles, which were observed without microbubble injection, were never found in the bubbly flow case. The above result implies that the negative pressure in the bubbly flow case does not exceed the (dynamic) Blake threshold pressure of the cavitation bubbles, while that in the single-phase case does. The Blake threshold pressure of a cavitation bubble is the liquid pressure at which the cavitation bubble begins to expand explosively, and has in general a negative value. When the liquid pressure becomes lower than the threshold, the pressure inside the bubble, which is always positive, can no longer balance the liquid pressure through the pressure jump due to surface tension, and hence the bubble will expand without bound until the liquid pressure turns into positive. The above numerical result suggests that the injected bubbles reduced the magnitude of negative pressure in liquid mercury to a level that does not exceed the Blake threshold pressure of the cavitation bubbles. From the above result, we can conclude that the microbubble injection significantly affects the dynamics of cavitation bubbles by altering the negative pressure value. The above numerical result, however, does not say anything about how the injected bubbles modified the liquid pressure. In the next subsection, we address this question by employing the multibubble model. 4.2. Multibubble study on how the injected bubbles change the liquid pressure and cavitation bubble dynamics In the experiment, a large number of gas bubbles were injected into liquid mercury, which leads to the interaction between nearby injected bubbles and also between injected bubbles and cavitation bubbles (or cavitation nuclei). Here we first show a result of a reduced problem, a nine-body problem (i.e., the interaction of a cavitation bubble with eight injected bubbles), and discuss basic roles of the injected bubbles. We then consider an extended problem where a large number of bubbles are assumed to be homogeneously distributed, and perform a parametric study to clarify the dependence of the suppression effect on the initial radii of injected bubbles. In the nine-body problem, we assume that eight injected bubbles are identical and arranged as a cube, and a cavitation
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Radius [µm]
1000
100 20
20 10
10
10
5
5
Liquid pressure [MPa]
1 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0
1
2 Time [ms]
3
0
1
2 Time [ms]
3
Fig. 4. Radius–time curves of a single cavitation bubble for the two pressure changes in Fig. 2. The numbers shown in the upper panels indicate the initial radii ðmmÞ of the corresponding bubble, and the lower panels show the liquid pressure. We stopped the computation just after the bubble has violently collapsed, because after that moment the deformation of the bubble should not be negligible. This treatment is also applied in the following examples.
boundary [10]. Also, half of the eight injected bubbles can be considered as the mirror images of four real injected bubbles originating from the sound reflection at the window surface [10,21]. Under these assumptions, Eq. (1) is reduced to ! ! ! 3 R_ 1 _ 2 1 R1 d R_ 1 R_ 1 1 p 1þ R1 R€ 1 þ R1 ¼ ps;1 þ 2 2c c r c rc dt s;1
1
! R_ 2 R2 R€ 2 þ c
3 3 1 þ pffiffiffi þ h 2h 2D12
2 h pffiffiffi D12 3
Fig. 5. Arrangement of bubbles. Eight injected bubbles, denoted by the larger spheres, are arranged as a cube, and a cavitation bubble, the smaller sphere, is placed at the mass center of the cube. Note that the sizes of the presented spheres do not accurately reflect the actual sizes of the bubbles in the numerical study.
bubble is placed at the mass center of the cube; see Fig. 5. Due to the symmetry of arrangement, in numerical computation this example can be treated as a two-body problem of a cavitation bubble (called hereafter bubble 1) and an injected bubble (bubble 2). The cavitation bubble is assumed to be a hemispherical bubble on the acrylic window but is treated as a spherical bubble, because the thickness of the viscous boundary layer forming on the window surface is much smaller than the radii of expanding bubbles and hence the window surface only acts as a mirror
(6)
! ! R_ 2 3 R_ 2 _ 2 1 R2 d p 1þ R2 ¼ ps;2 þ 2 2c r c rc dt s;2
b
8 dðR22 R_ 2 Þ D12 dt
1 dðR21 R_ 1 Þ dðR22 R_ 2 Þ b D12 dt dt
(7)
(8)
(9)
where D12 is the distance between the centers of bubbles 1 and 2, and h is the length of a side of the cube, which corresponds to the center-to-center distance between nearest neighbor injected bubbles. For the external pressure pex ðtÞ, as mentioned above we use only the observed pressure change in the single-phase case (the dashed curve in Fig. 2). Thus, in this investigation, only the bubble-radiated pressure waves can be the source of pressure modification. The radius–time curves for R10 ¼ 10 mm and R20 ¼ 50 mm with D12 ¼ 1 and 750 mm are shown in Fig. 6. For D12 ¼ 1 (i.e., when the bubbles are isolated), the cavitation bubble expanded explosively and then collapsed violently as observed in the experiment without microbubble injection [9,10]. On the other hand, for a sufficiently small D12 (Fig. 6(b)), the explosive expansion was completely suppressed by the surrounding gas bubbles (and also the dynamics of the injected bubbles were altered by the interaction between them). This result seems to be
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and low deceleration (note that dðR2i R_ i Þ=dt in Eq. (5) can be 2 decomposed into 2Ri R_ i þ R2i R€ i ). These positive pressures decreased the magnitude of the negative pressure at the position of the cavitation bubble and consequently suppressed the explosive expansion of the cavitation bubble. This explanation is consistent with the experimentally observed change in negative pressure value (Fig. 2) and the previous theory [14]. Next we consider a more realistic case where a large number of cavitation and injected bubbles interact. In this investigation, in order to simplify the coupled Keller–Miksis equation, we employ the local homogeneous model proposed by Kubota et al. [24], which is a sort of mean field approximation. In that model, bubbles in a large bubble cluster are assumed to be spherical and locally identical, and the bubble population is described by a locally defined number density. This model allows one to approximately solve the dynamics of a large bubble cluster by a simple and efficient technique. We here extend the model so that a bubble cluster containing two types of bubbles can be treated. Assuming that the radii of cavitation and injected bubbles in a locally homogeneous cluster are R1 ¼ R3 ¼ ¼ RN1 and R2 ¼ R4 ¼ ¼ RN , respectively (here N is an even number), and their local number densities ðnÞ are the same (see Fig. 7), the summation term in Eq. (1) is approximated as follows:
600 D12 = ∞
Radius [μm]
500 400 300
Injected
Cavitation
200 100 0
D12 = 750 μm
Radius [μm]
500 400
Injected 300 200 100
2 N X 1 dðRj R_ j Þ ¼ D dt j¼1;jai ij
N=2 X dðR21 R_ 1 Þ 1 dðR22 R_ 2 Þ þ D dt D dt j¼1;2j1ai i;2j1 j¼1;2jai i;2j " # Z DD 2_ 2_ 1 dðR1 R1 Þ dðR2 R2 Þ 4pr 2 dr þ n r dt dt 0 " # 2_ 2_ dðR1 R1 Þ dðR2 R2 Þ þ ¼ 2pnDD2 , (10) dt dt
Cavitation 0
Driving pressure [MPa]
0.4 0.3 D12 = ∞
0.2 0.1 0
D12 = 750 μm -0.1 0
1
2
3
Time [ms] Fig. 6. Radius–time curves in the nine-bubble case for (a) D12 ¼ 1 and (b) 750 mm ð¼ 15R20 Þ, and (c) driving pressure acting on the cavitation bubble in the two cases. In the case of D12 ¼ 750 mm, the magnitude of the negative pressure acting on the cavitation bubble was decreased by the nearby gas microbubbles, as observed experimentally (Fig. 2). The strong pressure pulse seen in the case of D12 ¼ 750 mm is due to the collapse of the injected bubbles (such strong pressure pulses have sometimes been observed in the experiment as well, though no pressure pulses can be seen in the selected pressure history because the time resolution of the pressure transducer (about 5:6 ms) is not enough to fully resolve the pressure pulses whose width is typically a few microseconds or less).
N=2 X
1
where DD is the cutoff distance due to the finite sound speed and acoustic shielding. This approximation of the summation term allows us to solve the problem as a two-body problem of bubbles 1 and 2. We introduce here the mean distance between nearestneighbor bubbles of the same kind, Dmean . Using this quantity, the number density is given as n ¼ 1=D3mean . As suggested in Ref. [25], we assume DD to be Dmean -dependent. However, we use a larger value ðDD 1:5Dmean Þ than that used in Ref. [25] ðDD ¼ Dmean Þ because the expansion ratio of injected bubbles for DD ¼ Dmean was almost the same as in the nine-bubble case although a larger number of bubbles are considered (a larger DD gives a smaller
Cavitation bubbles Injected bubbles
consistent with the experimental observations [15] that no cavitation bubbles were observed and cavitation damage was significantly reduced when injecting microbubbles. To clarify the mechanism of the change in cavitation bubble dynamics, we examine the pressure acting on the cavitation bubble. Fig. 6(c) shows the time history of the driving pressure on bubble 1, which was determined by the sum of pex ðtÞ and the pressures radiated by the eight injected bubbles. As can be seen clearly in this figure, the magnitude of negative pressure for D12 ¼ 750 mm is smaller than that for D12 ¼ 1, meaning that the injected bubbles decreased the magnitude of the driving pressure on the cavitation bubble. This result can be explained as follows: In the early stage of expansion, the injected bubbles radiate positive pressure waves because of their high expansion velocity
R1 R2
Fig. 7. Local homogeneity assumption used in the present study. A large threedimensional bubble cluster that consists of cavitation and injected bubbles is considered. In the large cluster, both cavitation and injected bubbles are spherical and bubbles of the same kind have, locally, the same radius. The number densities of both bubbles are locally the same.
ARTICLE IN PRESS M. Ida et al. / Nuclear Instruments and Methods in Physics Research A 600 (2009) 367–375
expansion ratio). As in the previous example, the observed pressure change in the single-phase case is used as the external pressure pex ðtÞ (that is, we roughly assume that the external pressure acting on the homogeneous cluster is spatially uniform). The expansion ratios of the bubbles, maxRj ðtÞ=Rj0 , as functions of the initial radius of the injected bubble ðR20 Þ are plotted pffiffiffi in Fig. 8(a). Here, we assumed R10 ¼ 10 mm and Dmean ¼ ð2= 3Þ750, the later of which corresponds to the distance between nearestneighbor injected bubbles in the nine-bubble case. The expansion ratios in the present case are overall smaller than those in the nine-bubble case (Fig. 8(b)) due to the interaction of a larger number of bubbles, but the results of both cases are qualitatively the same.
373
As can be seen in the figure, when the injected bubbles are larger than the cavitation bubbles ðR20 410 mmÞ, the explosive expansion of the cavitation bubbles is suppressed and their expansion ratio is thus very small. However, when R20 4R10 but R20 is not so large ð10 mmoR20 o30 mmÞ, the expansion ratio of the injected bubbles is very large, implying the possibility that the injected bubbles themselves cause erosion even though cavitation inception is successfully suppressed. In contrast, when R20 is much larger than R10 , the expansion ratios of both bubbles are small. This result could explain the experimental observations that the injection of large gas bubbles ðR20 X50 mmÞ suppressed cavitation inception and significantly reduced erosion. The next subsection discusses how the change of the expansion ratio affects the bubbles’ collapse intensity.
25 4.3. Single-bubble study on why the injected bubbles did not cause erosion
20
15
10
Cavitation
5
0 3
10
30
100
R20 [µm] 25
Injected
30 15 25 10 Expansion ratio
Expansion ratios
20
As proved in Ref. [14] and in the above subsection, the expansion ratio of sufficiently large bubbles decreases as their initial radii increase. This is an important key to understanding why the injected bubbles did not cause erosion. Here, we examine the correlation between the expansion ratio and the maximum collapse velocity of the bubbles. In this investigation we neglect bubble–bubble interaction to know the maximum collapse intensity that the bubbles may have (bubble–bubble interaction decreases bubbles’ expansion velocity [10]). Fig. 9 shows the expansion ratio and the maximum collapse velocity (normalized by the sound velocity of liquid mercury, maxjdRðtÞ=dtj=c) of single injected bubbles as functions of their initial radii R20 . Here we used the observed pressure change in the single-phase condition as pex ðtÞ. This result confirms that the collapse velocity correlates very well with the expansion ratio. For 10 mmoR20 o25 mm, the collapse velocity exceeds the sound velocity of liquid mercury, indicating that the bubbles have a very large collapse intensity. For larger R20, the collapse velocity decreases monotonically with increasing R20 and finally becomes much smaller than the sound velocity. This figure also indicates that cavitation bubbles, if they expand explosively, must have a very strong collapse intensity since they have a very large expansion ratio.
Cavitation 5
20
2
15
1.5
10
1
max|dR/dt| / c
Expansion ratios
Injected
0 30
10
100
R20 [µm] Fig. 8. (a) Expansion ratios of cavitation and injected pffiffiffi bubbles in a locally homogeneous cluster for R10 ¼ 10 mm and Dmean ¼ ð2= 3Þ750, as functions of the initial radius of the injected bubble. In (b), the expansion ratio for the nine-bubble case is shown as reference (results for R20 450 mm are not shown because bubble collision was observed during the computation). In the range of 10 mmoR20 o20 mm, the bubbles’ dynamics changes sensitively with R20 because this range is close to the cavitation threshold, near which a small fluctuation (in, e.g., the pressure change) can result in a large change in bubble dynamics.
0.5
5
0
0 3
10
30
100
300
Initial radius R20 [µm] Fig. 9. Expansion ratio and maximum collapse velocity of single injected bubbles as functions of their initial radii. Here the collapse velocity is normalized by the sound velocity of liquid mercury.
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Based on the above numerical results, we can explain the experimental findings from the impact test as follows. In the single-phase case, many cavitation bubbles emerged in liquid mercury and expanded explosively. They then collapsed violently and caused significant erosion. In the bubbly flow case, cavitation inception was suppressed and only injected bubbles, which had much larger initial radii than cavitation bubbles, expanded. Since the expansion ratio of injected bubbles is small, their collapse velocity and erosion intensity were much smaller than those of explosively expanding cavitation bubbles. Therefore, cavitation erosion was greatly reduced by injecting gas bubbles.
0.7 0.6 Pressure change [MPa]
374
5. Conclusion Based on pressure–time curves obtained from the impact test and relatively simple theoretical models, we have studied numerically the effect of microbubble injection which we are attempting to mitigate cavitation damage in spallation neutron sources. The important findings presented in this paper are summarized as follows: (1) The slight decrease of the magnitude of negative pressure due to microbubble injection has a strong impact on cavitation bubble dynamics in liquid mercury. For the negative pressure in the case without microbubble injection, cavitation bubbles undergo explosive expansion which leads to a subsequent violent collapse. On the other hand, for the negative pressure in the case with microbubble injection, cavitation bubbles do not expand significantly.
r1 = 40 R10
0.4 0.3 0.2 0.1
4.4. Single-bubble study on the experimentally observed highfrequency pulses
Experiment
0
0.7
Pressure change [MPa]
0.6 r1 = 20 R10
0.5 0.4 0.3 0.2 0.1
Experiment
0
1200 1000 Radius [µm]
Lastly, we discuss the high-frequency pressure pulses in the positive pressure phase. In this investigation, we used the singlebubble model with Eq. (5) and, for pex ðtÞ, the observed pressure change in the single-phase case. Here, we assume that the pressure pulses are the pressure wave radiated by an injected bubble passing near the pressure transducer. Figs. 10(a) and (b) show the computed liquid pressure at different distances r 1 from a single injected bubble (bubbles 1), determined by the sum of pex ðtÞ and Eq. (5). The initial radius of the bubble was assumed to be 450 mm, which is relatively larger than that in the previous example. The numerical results reproduce, at least qualitatively, important characteristics of the pulses such as frequency and amplitude. The pressure pulses found in the numerical study are due to the free oscillation of the injected bubble triggered by the sudden rise of liquid pressure. The radius–time curve of the bubble is shown in Fig. 10(c). The oscillatory behavior in the positive pressure period is the free oscillation of the bubble at its eigenfrequency. The eigenfrequency of a pulsating bubble is known to depend on the bubble’s radius, and a smaller bubble has a higher eigenfrequency. The bubble radius assumed in the above example was deduced from the pulse interval. We should note here that the actual radius of the bubble that caused the experimentally observed pulses might be different from the radius assumed above. It is known that bubble–bubble and bubble–wall interactions can change the eigenfrequency of bubbles (see, e.g., Refs. [18–21,26,27]). When, for example, two bubbles pulsate in phase with each other, the eigenfrequencies of the bubbles are decreased. Since in the experiment many bubbles interact with each other in the vicinity of an acrylic window, the actual radius should be not equal to 450 mm (probably smaller than this). However, we argue that the origin of the pressure pulses was clearly identified by this numerical study.
0.5
800 600 400 200 0 0.5
1
1.5 Time [ms]
2
2.5
Fig. 10. (a, b) Liquid pressures at different distances from a single bubble and (c) radius of the injected bubble as functions of time. In (a) and (b), the solid, dashed, and dotted curves, respectively, denote the numerical results (for r 1 ¼ 40R10 (a) and 20R10 (b)), an experimental result in the bubbly flow case (already shown in Fig. 3), and the external pressure assumed in the computation. The oscillatory behavior of the gas bubble shown in (c) is the origin of the pressure pulses found in (a) and (b).
(2) The slight change in negative pressure is caused by the positive pressure waves that the injected bubbles radiate when they expand in response to negative pressure. The positive pressure waves reduce the magnitude of negative pressure to a level that does not exceed the Blake threshold pressures of cavitation bubbles. That is, the bubble-radiated pressure waves suppress cavitation inception. (3) Erosion intensity of the injected bubbles should be much smaller than that of cavitation bubbles since their expansion
ARTICLE IN PRESS M. Ida et al. / Nuclear Instruments and Methods in Physics Research A 600 (2009) 367–375
0.02
However, the effectiveness of microbubble injection in more realistic conditions (i.e., when actual proton beams are used) is still unclear. Further efforts are needed to assess the true effectiveness of the technique.
0.01
Acknowledgments
0.03
Pressure change [MPa]
375
0 -0.01 Threshold (arbitrary)
-0.02 -0.03 1.6
1.8
2 Time [ms]
2.2
2.4
Fig. 11. Magnified view of the two experimentally obtained pressure changes in the negative pressure period. The solid straight line denotes the Blake threshold pressure of an arbitrary cavitation bubble. Our numerical results suggested that by injecting gas microbubbles, the negative pressure value changed across the threshold pressure to a level that does not exceed the threshold, as indicated by an arrow.
ratio and collapse velocity are much smaller than those of cavitation bubbles. This finding can (at least partly) explain why the injected bubbles did not cause erosion and remarkably reduced cavitation damage. (4) The high-frequency pressure pulses observed experimentally in the positive pressure period of the mechanically induced pulse are due to injected bubbles. In this period, the injected bubbles undergo free oscillation triggered by the sudden pressure rise, and radiate high-frequency pressure waves. These findings clarify the role of gas microbubbles injected into liquid mercury and the underlying mechanism of the damage reduction observed in the impact test [15]. The present results suggest that the observed slight change in negative pressure value took place across the Blake threshold pressures of cavitation bubbles; that is, the injected bubbles worked to raise pressure to a level that does not exceed the threshold; see Fig. 11. This is consistent with the experimental observation that no cavitation bubbles were seen when microbubble injection was applied.
This work was partly supported by the Ministry of Education, Culture, Sports, Science, and Technology of Japan through a Grantin-Aid for Young Scientists (B) (no. 17760151) and by the Japan Society for the Promotion of Science through a Grant-in-Aid for Scientific Research (no. 17360085). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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