Numerical simulations of acoustic cavitation noise with the temporal fluctuation in the number of bubbles

Ultrasonics Sonochemistry 17 (2010) 460–472

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Ultrasonics Sonochemistry journal homepage: www.elsevier.com/locate/ultsonch

Numerical simulations of acoustic cavitation noise with the temporal fluctuation in the number of bubbles Kyuichi Yasui *, Toru Tuziuti, Judy Lee, Teruyuki Kozuka, Atsuya Towata, Yasuo Iida National Institute of Advanced Industrial Science and Technology (AIST), 2266-98 Anagahora, Shimoshidami, Moriyama-ku, Nagoya 463-8560, Japan

a r t i c l e

i n f o

Article history: Received 30 June 2009 Received in revised form 11 August 2009 Accepted 21 August 2009 Available online 28 August 2009 PACS: 43.35.Ei 43.35.Vz 43.30.Nb 43.50.Ed 43.25.Yw 78.60.Mq Keywords: Acoustic cavitation noise Broad-band noise Numerical simulation Transient cavitation Stable cavitation Bubble number density

a b s t r a c t Numerical simulations of cavitation noise have been performed under the experimental conditions reported by Ashokkumar et al. (2007) [26]. The results of numerical simulations have indicated that the temporal fluctuation in the number of bubbles results in the broad-band noise. ‘‘Transient” cavitation bubbles, which disintegrate into daughter bubbles mostly in a few acoustic cycles, generate the broadband noise as their short lifetimes cause the temporal fluctuation in the number of bubbles. Not only active bubbles in light emission (sonoluminescence) and chemical reactions but also inactive bubbles generate the broad-band noise. On the other hand, ‘‘stable” cavitation bubbles do not generate the broad-band noise. The weaker broad-band noise from a low-concentration surfactant solution compared to that from pure water observed experimentally by Ashokkumar et al. is caused by the fact that most bubbles are shape stable in a low-concentration surfactant solution due to the smaller ambient radii than those in pure water. For a relatively high number density of bubbles, the bubble–bubble interaction intensifies the broad-band noise. Harmonics in cavitation noise are generated by both ‘‘stable” and ‘‘transient” cavitation bubbles which pulsate nonlinearly with the period of ultrasound. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction When a liquid is irradiated by a strong ultrasonic wave, many tiny gas bubbles appear. The phenomenon is called acoustic cavitation [1–3]. There are two categories of acoustic cavitation bubbles. One is ‘‘stable” cavitation bubbles which are shape stable and have relatively long lifetimes. The other is ‘‘transient” cavitation bubbles which are shape unstable and disintegrate into daughter bubbles mostly in a few acoustic cycles. The term ‘‘transient cavitation” is sometimes used for cavitation associated with high-energy bubble collapse which results in the light emission (sonoluminescence (SL)) or chemical reactions inside or outside bubbles (sonochemical reactions) [1–7]. In the present paper, however, the term ‘‘transient cavitation bubbles” is used for shape-unstable cavitation bubbles irrespective of whether they are active or inactive in SL and sonochemical reactions. The cavitation bubbles pulsate according to the pressure oscillation of ultrasound. When the ambient bubble radius, which * Corresponding author. Tel.: +81 52 736 7218; fax: +81 52 736 7400. E-mail address: [email protected] (K. Yasui). 1350-4177/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ultsonch.2009.08.014

is defined as the radius of a bubble when ultrasound is absent, is in a certain range, a bubble dramatically expands during the rarefaction phase of ultrasound [8]. Subsequently, at the compression phase of ultrasound, a bubble collapses very violently due to the inertia of the surrounding liquid, which is sometimes called high-energy bubble collapse [1]. At the end of the bubble collapse, the temperature and pressure inside a bubble increases dramatically to thousands of Kelvin and thousands of bars, respectively, which is the cause of SL and sonochemical reactions [9–11]. For ‘‘stable” cavitation bubbles, there are two categories as in the case of ‘‘transient” cavitation bubbles. One is low-energy stable bubbles which are inactive in SL and sonochemical reactions. The other is high-energy stable bubbles which are active in SL or sonochemical reactions. The high-energy stable bubbles are sometimes called ‘‘repetitive transient bubbles”, where the term ‘‘transient” is used for the high-energy bubble collapse [1]. A high-energy stable bubble has been widely known in the experiment of single-bubble SL which is from a single-bubble stably trapped at the pressure antinode of a standing ultrasonic wave [4–6]. Every pulsating bubble emits a secondary acoustic wave. The acoustic emission from acoustic cavitation bubbles is called

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acoustic cavitation noise [12]. Just after the end of the violent collapse of a bubble, a shock-wave is emitted from the bubble. Matula et al. [13] detected an acoustic signal from a bubble just after the end of the violent collapse by a hydrophone in a single-bubble system, which is due to the shock-wave. The emission of a shock-wave from a bubble has been optically confirmed both in single-bubble and multibubble systems [14–16]. The frequency spectra of acoustic cavitation noise have been reported since 1952 [1–3,17–22]. Depending on the experimental conditions, they consist of some or all of the following components; the peak at the driving frequency (f0), harmonics (nf0, where n is the natural number), half-order subharmonic (f0/2), ultraharmonics ((2n + 1)f0/2), and the broad-band noise. For the mechanism behind the generation of the broad-band noise, two ideas have been proposed [1]. One is the acoustic signals due to shockwaves emitted from bubbles [23,24]. The shock-waves are detected by a hydrophone as pulses which result in the broad-band component of the frequency spectrum (cf. the frequency spectrum of the delta-function is broad-band). The other is the acoustic waves radiated from non-periodically pulsating bubbles (chaotically pulsating bubbles) [25]. Non-periodically emitted acoustic waves including shock-waves result in the broad-band component of the frequency spectrum (cf. the frequency spectrum of a nonperiodic (chaotic) function is broad-band). In the present paper, it will be shown that shock-waves emitted from bubbles do not result in the broad-band noise if the bubbles are shape stable. Furthermore, it will be shown that the contribution of non-periodically pulsating bubbles to the generation of the broad-band noise is minor at least under the experimental condition of Ashokkumar et al. [26] for which the present numerical simulations have been performed. Instead, the temporal fluctuation in the number of bubbles results in the broad-band noise according to the present study. Next we will discuss the origin of harmonics. Any temporally periodic function is expressed by the Fourier series which consists of the fundamental frequency and its harmonics. As the deviation from the sinusoidal function of the fundamental frequency increases, the intensity of harmonics increases. Two ideas have been proposed on the deviation from the sinusoidal form. One is the acoustic emission from cavitation bubbles including shock-wave emissions. The other is the nonlinear propagation of ultrasound [27]. The instantaneous local sound velocity increases as the degree of compression (the instantaneous local density) increases. Furthermore, a portion of an acoustic wave which has larger instantaneous local-particlevelocity propagates faster [28]. As a result, an acoustic wave is gradually steepened and finally becomes a sawtooth wave. Harmonics component of the acoustic wave is gradually intensified as the acoustic wave is gradually steepened. This nonlinear propagation is usually observed in megahertz range of ultrasound [29]. In bubbly liquids, on the other hand, nonlinear propagation is due to the acoustic emissions from bubbles [30–36]. As the mechanism of the generation of the half-order subharmonic and ultraharmonics, several ideas have been proposed. One is the acoustic waves including shock-waves radiated from bubbles which pulsate periodically with the doubled acoustic period [25]. Another is the acoustic waves radiated by the surface waves of relatively large bubbles [1,37]. The other is due to the nonlinear propagation of ultrasound in bubbly liquids [1]. In the present paper, the first idea has been studied, while the third idea is included in it as explained before. For the second idea, more studies will be required on non-spherical bubble dynamics [38]. The numerical simulations in the present paper have been performed under the experimental conditions reported by Ashokkumar et al. [26]. They measured the acoustic signal by using a cavitation sensor developed at the National Physical Labora-

tory, UK [39,40]. According to them [26], the sensor has a flat, usable response up to 10 MHz. They measured the acoustic signals in pure water and aqueous sodium dodecyl sulfate (SDS: surfactant) solutions of various concentrations. The frequency and intensity of ultrasound are 515 kHz and 2.2 W/cm2, respectively. This intensity is far above the threshold for SL and sonochemical reactions which is 0.54 W/cm2. If the ultrasound is a traveling wave, 2.2 W/cm2 corresponds to the acoustic amplitude of 2.6 bar according to the following relationship [41]: I ¼ P 2a =ð2qcÞ, where I is the acoustic intensity (W/m2), Pa is the acoustic amplitude (Pa), q is the liquid density, and c is the sound velocity. However, in the experiment, the ultrasound was not a pure traveling wave but a damped standing wave [26,41]. Thus, numerical simulations have been performed not only for the acoustic amplitude of 2.6 bar but for those in the range of 0–3 bar. With regard to pure water, the frequency spectrum of the acoustic signal consists of the peak at the driving frequency (f0 = 515 kHz), the peaks at harmonics (nf0), broadband noise, and weaker peaks at the second-order subharmonic (f0/2) and ultraharmonics ((2n + 1)f0/2). On the contrary, in lowconcentration SDS solutions (for the concentration range of 0.5– 2 mM), they consist only of the peaks at the driving frequency and harmonics. The intensity of the broad-band noise and that of ultraharmonics were 1 or 2 orders of magnitude lower than those in pure water. However, for concentration of SDS higher than 3 mM, the frequency spectrum of acoustic cavitation noise is similar to that from pure water. One of the aims of the present study is to unveil the mechanism behind the difference of the acoustic emission spectrum in lowconcentration SDS solution from that in pure water.

2. Model The pressure (p) of an acoustic wave radiated from a pulsating bubble is expressed by Eq. (1) when the time delay due to the finite velocity of acoustic-wave propagation is neglected [42].

p¼

 q  2€ R R þ 2RR_ 2

ð1Þ

r

where q is the liquid density, r is the distance from a bubble, R is the instantaneous bubble radius, and the dot denotes the time derivative (d/dt). It should be noted that the effect of the time delay just shifts the time series of the pressure of an acoustic wave and does not affect its frequency spectrum. Here we consider a bubble cloud in which bubbles are spatially uniformly distributed. Furthermore, the ambient bubble radius is assumed to be the same for all the bubbles. Then, the pressure (P) of acoustic waves radiated from bubbles in the bubble cloud is expressed by Eq. (2).

P¼

N X i¼1

N   X 1 € þ 2RR_ 2 pi ¼ q R2 R r i¼1 i

! ð2Þ

where the sum is for all the bubbles in the bubble cloud, N is the number of bubbles in the bubble cloud, pi is the pressure of an acoustic wave radiated from a bubble numbered i, and ri is the distance from the bubble numbered i. When the observation point (the position of a hydrophone) is in the bubble cloud, Eq. (2) is expressed by using the ‘‘coupling strength” (S) of the bubble cloud introduced by the authors in Ref. [43].

S¼

N X 1 r i¼1 i

  € þ 2RR_ 2 P ¼ Sq R2 R

ð3Þ

ð4Þ

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Here we will review the meaning of the ‘‘coupling strength” (S) of the bubble cloud. The velocity field (w) around a pulsating bubble is expressed as Eq. (5) under the incompressible liquid approximation [42].

w¼

R2 R_ r2

ð5Þ

where r is the distance from the center of the bubble. As a first step, a pair of bubbles will be considered. The equation for the bubble pulsation taking into account the effect of the bubble–bubble interaction has been derived as follows using Eq. (5) [43,44].

! ! _ _ _ R_ m € þ 3 R_ 2 1  R þ 2m RR þ c1 c1 qL;i 3c1 3c1 qL;i 2 !    R_ 1 R pB  pS t þ ¼ 1þ  p1 c1 c1 qL;1 ! € _ mR m R_ þ 1 þ c1 c1 qL;i qL;i

1

_ _ R_ _ dqL;i R dqL;i m m mR  þ  R_ þ qL;i 2qL;i 2c1 qL;i qL;i dt c1 q2L;i dt  R dpB 1  _ 2 €b þ 2Rb Rb þ R2b R  c1 qL;1 dt d

þ

_ m

!

ð6Þ

_ is the rate of evaporation where c1 is the sound velocity in liquid, m of water vapor at the bubble wall (negative value means condensation), qL;i is the liquid density at the bubble wall, qL;1 is the liquid density far from a bubble, pB is the liquid pressure at the bubble wall, pS ðtÞ is the instantaneous pressure of ultrasound at time t, pS ðtÞ ¼ pa sin xt, pa is the pressure amplitude of ultrasound (acoustic amplitude), x is the angular frequency of ultrasound, p1 € ¼ ddtm_ , d is the distance between two bubbles is the static pressure, m (the distance from the center of a spherical bubble to that of the other spherical bubble), and Rb is the instantaneous radius of the other bubble. The last term is the influence of the other bubble on the bubble pulsation. Without the last term, Eq. (6) is identical to the modified Keller equation described in Refs. [45–47]. It should be noted that the last term is an approximate one as in Ref. [44] _ € neglecting the terms containing cR1b , cRb RR_b , etc. and the effect of the 1 b time delay due to the finite speed of sound propagation [48,49]. In order to solve Eq. (6), the corresponding equation for the other bubble should be simultaneously solved in order to calculate Rb . When the number of bubbles is N, the last term in Eq. (6) should be replaced by the term (7).



N1  X 1  _2 €i 2Ri Ri þ R2i R di i¼1

ð7Þ

where the summation is for the other N  1 bubbles, di is the distance between the bubble and the bubble i, and Ri is the instantaneous radius of the bubble i. In order to calculate the term (7), N  1 equations for the pulsation of N  1 bubbles should be simultaneously solved to calculate Ri. Thus, the number of equations to be solved simultaneously is N. With the same assumption used in the derivation of Eq. (2) from Eq. (1), the number of equations to be solved can be reduced to 1. Here, the last term of Eq. (6) is replaced by the term (8).

!



 X 1  € 2R_ 2 R þ R2 R d i i

ð8Þ

where the coefficient of the term has been defined as the ‘‘coupling strength” (S) of the bubble cloud (Eq. (3)) because it indicates the strength of the bubble–bubble interaction. Note that the term (8) equals P=q according to Eq. (4). It is consistent with the modified

Keller equation because the term (8) can be regarded as the extra acoustic pressure due to cavitation noise. Although there are more sophisticated theories [50,51] of the bubble–bubble interaction, the term (8) has been used in the present study for simplicity. Furthermore, the modified Keller equation with the term (8) has been validated through the study of the bubble–bubble interaction under an ultrasonic horn [43]. It should also be noted that the effect of the bubble–bubble interaction (the term (8)) is nearly negligible under most of the conditions studied in the present paper. Now we consider the condition of the experiment by Ashokkumar et al. [26]. According to Ref. [39], the cavitation sensor used in the experiment is a hollow, open-ended cylindrical cavity (Fig. 1). The inner cylindrical surface of the cavitation sensor is formed from 110-l piezoelectric film. The sensor can measure the cavitation noise in the cylindrical cavity by the piezoelectric film. The inner diameter of the cylinder is 30 mm. Although the height of the cylinder is 22 mm that is about 7.5 times of the wavelength of ultrasound in water at 515 kHz, we will consider here only a single anti-nodal plane of a standing ultrasonic wave for simplicity because bubbles at around each anti-nodal plane behave in a similar manner (Fig. 1). When bubbles are distributed at around an antinodal plane of acoustic pressure, the ‘‘coupling strength” (S) at the inner surface of the cylindrical cavity is estimated as follows.

S¼ ¼

k 4

Z

aD 0

pkn

2pnr pkn dr ¼ ar 2

Z 0

aD

r dr ar

ða þ D  aloge D þ aloge aÞ

2

ð9Þ

where k is the wavelength of ultrasound, a is the inner radius of the cylindrical cavity, D is the mean distance from the inner surface of the cylindrical cavity to a nearest bubble, n is the number density of bubbles, and r is the radius from the center of the cylindrical cavity. In Eq. (9), it has been assumed that bubbles are spatially uniformly distributed at around an anti-nodal plane of acoustic pressure and that the thickness of the cylindrical anti-nodal region equals k/4. The mean distance from the surface of the cylindrical cavity to a pffiffiffi nearest bubble is estimated by D ¼ 1= 3 n. On the other hand, the number (N) of bubbles in the cylindrical anti-nodal region is estimated by Eq. (10).

N¼

pkna2

ð10Þ

4

In Fig. 2, the ‘‘coupling strength” (S) has been shown as a function of the number of bubbles (N) for a = 15 mm at 515 kHz (k = 2.9 mm). Now, we will discuss the influence of the distribution of ambient bubble radius on the acoustic cavitation noise. The whole system of bubbles with various ambient radii can be divided into groups of bubbles according to the ambient radius. When the effect of the bubble–bubble interaction is negligible on the bubble pulsation, the frequency spectrum of acoustic cavitation noise of the

30 mm Anti-nodal plane

22 mm λ/4=0.728 mm

Fig. 1. The system that is modeled in the present paper. A thin cylindrical region around an anti-nodal plane of thickness of a quarter wavelength (k/4) has been considered in a hollow cylindrical cavitation sensor.

K. Yasui et al. / Ultrasonics Sonochemistry 17 (2010) 460–472

V=514 mm3

5

S (m-1)

10

4

10

3

10

0

100

200

300

400

500

Number of bubbles Fig. 2. The ‘‘coupling strength” (S) as a function of the number of bubbles (N) in the thin cylindrical region of 514 mm3 in volume shown in Fig. 1 (Eqs. (9) and (10)).

whole system is constructed just by the linear algebraic sum of the frequency spectra of all the groups. In other words, numerical simulations can be performed for each group separately. On the other hand, when the bubble pulsation is strongly influenced by the bubble–bubble interaction (for a relatively high number density of bubbles), complicated calculations are required taking into account the bubble–bubble interaction among all the bubbles of different ambient radii. In the present paper, however, most of the cases studied are in the former category (the effect of the bubble–bubble interaction is nearly negligible). It justifies the assumption of a single ambient radius of bubbles as a step to construct the frequency spectrum of acoustic cavitation noise for the whole system. In order to construct the cavitation noise spectrum for the whole system, the information on the distribution of ambient bubble radius as well as that of the acoustic amplitude is required [52]. The latter is not known at present, while the former for active bubbles in SL has been reported [53]. Now, we consider the relationship between fragmentation of bubbles and the temporal variation in the number of bubbles. Again, the whole system of bubbles is divided into groups of bubbles according to the ambient radius. Consider a bubble in a group. When this bubble emits a daughter bubble due to its shape instability, which is a case of fragmentation of a bubble, the ambient radius of the original bubble considerably decreases. As a result, number of bubbles in the group decreases by one. On the other hand, number of bubbles in another group with the ambient radius of a daughter bubble increases by one. In this way, the number of bubbles in each group temporally varies by fragmentation of bubbles. Coalescence of bubbles also results in the temporal fluctuation in the number of bubbles. In the present study, the temporal fluctuation in the number of bubbles is taken into account by the temporal fluctuation in the ‘‘coupling strength” (S) as follows. The temporal fluctuation in the number of bubbles takes place mostly at the bubble collapse because fragmentation of bubbles takes place mostly at the bubble collapse. Thus, a significant variation in the number of bubbles is once an acoustic cycle. It is modeled by Eq. (11).

Sðt þ TÞ ¼ SðtÞ þ ðDSÞrn

ð11Þ

where S(t + N) and S(t) are the ‘‘coupling strength” at time t + N and t, respectively, T is the acoustic period, DS is the maximum amplitude of the temporal variation of S per acoustic cycle, and rn is a random number generated by a computer from 1 to 1. The

463

change of S by Eq. (11) is once an acoustic cycle. rn is a uniform random number because fragmentation of bubbles is a random process. Although fragmentation of a bubble is predictable, random nucleation/coalescence of bubbles and random bubble motion due to secondary Bjerknes force among bubbles make fragmentation of bubbles an unpredictable random process [1]. When S becomes negative in the numerical simulations, it is replaced by 0. The maximum amplitude of the temporal fluctuation (DS) is related to the lifetime (L) of a bubble as DS = S0/L, where S0 is the initial coupling strength and L is the lifetime of a bubble in acoustic cycle. Temporal variation of the ‘‘coupling strength” (S) directly influences both the intensity of acoustic emissions through Eq. (4) and the bubble pulsation through the term (8) in the modified Keller equation. In most of the present numerical simulations, the former effect is dominant. The lifetime (L) of a bubble is estimated by the numerical simulation of the shape oscillation of a bubble. A bubble is assumed to disintegrate into daughter bubbles when the amplitude of the nonspherical component of the bubble shape exceeds the mean bubble radius [5,54,55]. The amplitude of the non-spherical component of the bubble shape is calculated as follows [54,55]. A small distortion of the spherical surface is described by RðtÞ þ an ðtÞY n , where R(t) is the instantaneous mean radius of a bubble at time t, Yn is a spherical harmonic of degree n, and an ðtÞ is the amplitude of the non-spherical component. The dynamics for the amplitude of non-spherical component an ðtÞ is given by

€n þ Bn ðtÞa_ n  An ðtÞan ¼ 0 a

ð12Þ

where the overdot denotes the time derivative (d/dt),

 € b r  R d 2lR_ An ðtÞ ¼ ðn  1Þ  n 3  ðn  1Þðn þ 2Þ þ 2nðn þ 2Þðn  1Þ R qR R R3 ð13Þ and

Bn ðtÞ ¼

  3R_ d 2l þ ðn þ 2Þð2n þ 1Þ  2nðn þ 2Þ2 R R R2

ð14Þ

where bn ¼ ðn  1Þðn þ 1Þðn þ 2Þ, r is the surface tension, l is the liquid viscosity, and d is the thickness of the thin layer where fluid flows

d ¼ min

rffiffiffiffiffi  l R ; x 2n

ð15Þ

where x is the angular frequency of ultrasound. In the present numerical simulations, only n = 2 and 3 modes have been considered because they are the dominant modes for most cases. Next, we will discuss the response of the hydrophone (the cavitation sensor). In the experiment of Ashokkumar et al. [26], the sensor has a flat, usable response up to 10 MHz. According to Luther et al. [42], such hydrophone characteristics can be approximately modeled by a low-pass filter expressed by Eq. (16).

€ þ 2cpfc U_ þ 4p2 f 2 U ¼ PðtÞ þ p ðtÞ U S c

ð16Þ

where U is the hydrophone signal, c is the coefficient for damping, fc is the characteristic frequency of the hydrophone, P(t) is the instantaneous pressure of acoustic waves radiated from bubbles given by Eq. (4), and pS(t) is the instantaneous pressure of the driving ultrasound. The characteristic frequency (fc) of hydrophone in Eq. (16) is related to a cut-off frequency since the sensitivity of the hydrophone above fc is lower than that below fc. In the present numerical simulations, the characteristic frequency and the coefficient for damping have been assumed as fc = 5 MHz and c = 1, respectively. For a larger value of fc, the intensity of high frequency component becomes stronger. For a larger value of c, the frequency cut-off becomes sharper.

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a

b

0.006

7

0.004

6

Bubble radius (μm)

0.005

0.003

a2 / R

8

0.002 0.001 0

4 3 2 1

-0.001 -0.002

5

0

20

40

60

80

0 680

100

685

Time (μs)

c

690

695

700

695

700

Time (μs)

d

200

400 300

Hydrophone signal (a.u.)

Acoustic emission (bar)

150

100

50

0

-50 680

685

690

695

700

200 100 0 -100 -200 -300 680

685

690

Time (μs)

Time (μs)

e

515 kHz, 2.6 bar, R0=1.5 μm S=104 m-1, ΔS=0 7

Hydrophone signal (a.u.)

10

6

10

5

10

4

10

0

2000

4000

6000

8000

10000

Frequency (kHz) Fig. 3. The result of the numerical simulation with a constant ‘‘coupling strength” of 104 m1 (without the temporal fluctuation in the number of bubbles). The ambient bubble radius is 1.5 lm, which is typical in low-concentration SDS solutions [53]. The frequency and pressure amplitude of ultrasound are 515 kHz and 2.6 bar, respectively. The ‘‘coupling strength” of 104 m1 corresponds to the number density of bubbles of 0.13 mm3and N = 65 in the thin cylindrical region of Fig. 1. The time axis is the same for (b)–(d) for 680–700 ls. (a) The amplitude of non-spherical component (n = 2) of the bubble shape relative to the instantaneous mean bubble radius (R) for the initial 100 ls. (b) The radius of a bubble (R). (c) The pressure of acoustic waves radiated from bubbles (P). (d) The hydrophone signal (U) in arbitrary unit. (e) The frequency spectrum of the hydrophone signal (U) with the logarithmic vertical axis.

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Next, the present model of the bubble dynamics will be discussed. The temperature and pressure are assumed to be spatially uniform inside a bubble except at the thermal boundary layer near the bubble wall [46]. In the model, the following effects have been taken into account; non-equilibrium evaporation and condensation of water vapor at the bubble wall, and thermal conduction inside a bubble. The details of the model have been described in Ref. [46]. It should be noted that in the present numerical simulations the effects of thermal conduction outside a bubble, chemical reactions, and ionization of gases inside a bubble have been neglected. (The liquid temperature at the bubble wall is assumed to be equivalent to the ambient liquid temperature in the present simulations.) The present numerical simulations have been performed for air bubbles in water or aqueous SDS solution at 20 °C. Finally, the bubble dynamics model in aqueous SDS solution will be discussed. In the present model, there are two physical properties which are different from those of pure water. One is surface tension and the other is surface dilatational viscosity. Surface tension (r) of a pulsating bubble in aqueous SDS solution is given by Eq. (17) [56,57].

r ¼ r0 1 

E0

R20

r0 R2

!! 1

ð17Þ

where r0 is surface tension at equilibrium in aqueous SDS solution (r0 ¼ 6:0  102 N=m in 1.5 mM SDS solution [57], while r ¼ 7:28  102 N=m in pure water), E0 is Gibbs elasticity (E0 ¼ 1:7  103 N=m in 1.5 mM SDS solution [57]), and R0 is the ambient bubble radius. Both surface tension and surface dilatational viscosity are used in the calculation of the instantaneous liquid pressure at the bubble wall (pB(t)) in the modified Keller equation [58,59].

_ 2r 4l _ m R pB ðtÞ ¼ pðtÞ   R R qL;i

! _2 m

1

qL;i



1

!

qg



4R_ jS R2

ð18Þ

where p(t) is the pressure inside a bubble, qg is the density inside a bubble, and jS is surface dilatational viscosity (jS ¼ 5  107 N s=m in 1.5 mM SDS solution [57] and 0 in pure water). It should be noted here that both Gibbs elasticity and surface dilatational viscosity in aqueous SDS solution have been experimentally reported to vary widely with impurities and methods of measurements [57,58,60].

a

250

SDS solution

7

SDS solution

pure water 200

Acoustic emission (bar)

Bubble radius (μm)

The numerical simulations have been performed under the condition of the experiment by Ashokkumar et al. [26]. They measured the acoustic signals in pure water and SDS aqueous solutions at 515 kHz. According to Lee et al. [53], the ambient radii of SL bubbles in pure water range from 2.8 to 3.7 lm at 515 kHz, while in 1.5 mM SDS solution they range from 0.9 to 1.7 lm. In Fig. 3, the results of the numerical simulation for the ambient bubble radius of 1.5 lm, which is typical in 1.5 mM SDS solution, have been shown when the ultrasonic frequency and the pressure amplitude of ultrasound are 515 kHz and 2.6 bar, respectively. The ‘‘coupling strength” (S) is constant as 104 m1 which corresponds to the number density of bubbles of about 0.13 mm3 and the number of bubbles of about 65 under the present configuration. The constant ‘‘coupling strength” is caused by the fact that a bubble is shape stable as the amplitude of nonspherical component of the bubble shape is always less than 1 in this case (Fig. 3a). In other words, the lifetime of a bubble is infinite (L ¼ 1) and the maximum amplitude of the temporal variation of the ‘‘coupling strength” is 0. In order to obtain the sufficient resolution in the frequency spectrum, the calculation was performed for 700 ls, which corresponds to 360.5 acoustic cycles. A bubble expands during the rarefaction phase of ultrasound and collapses violently at the compression phase (Fig. 3b). After a violent collapse, two smaller expansions and milder collapses take place. Then, a bubble expands again and the process is accurately repeated. The strong acoustic emission takes place at the end of each violent collapse, which is due to the shock-wave emission (Fig. 3c). The maximum pressure of the emitted acoustic waves is the same for each acoustic cycle because the bubble pulsation is accurately temporally periodic and the ‘‘coupling strength” is assumed to be constant. The hydrophone signal consists of both the sinusoidal-like waveform originated from the driving ultrasound and the sharp peaks originated from shock-waves emitted by bubbles (Fig. 3d). The frequency spectrum of the hydrophone signal consists of the peak at the driving frequency and the peaks at its harmonics (Fig. 3e). It is quite similar to the experimentally observed spectrum from low-concentration SDS solutions (0.5– 2 mM) [26]. It is concluded that stable cavitation bubbles do not generate broad-band noise although shock-waves are emitted from bubbles.

b

8

6 5 pure water 4 3 2

150 100 50 0

1 0 47

3. Results and discussions

47.5

48

Time (μs)

48.5

49

-50 47

47.5

48

48.5

49

Time (μs)

Fig. 4. Comparison of the calculated result for pure water with that for 1.5 mM SDS (surfactant) solution. The condition for the numerical simulation is the same as that in Fig. 3. The time axis is the same for (a) and (b) from 47 to 49 ls (for nearly one acoustic cycle). (a) The bubble radius (R). (b) The pressure of an acoustic wave radiated from bubbles (P).

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K. Yasui et al. / Ultrasonics Sonochemistry 17 (2010) 460–472

a

b

S0=104 m-1, ΔS=S0/4

25000

14 12

Bubble radius (μm)

S (m-1)

20000

15000

10000

10 8 6 4

5000 2 0

0

100

200

300

400

500

600

0 680

700

685

Time (μs)

c

d

250

Hydrophone signal (a.u.)

Acoustic emission (bar)

695

700

695

700

300 200

200 150 100 50 0 -50 680

690

Time (μs)

685

690

695

700

100 0 -100 -200 -300 680

685

690

Time (μs)

Time (μs)

e

515 kHz, 2.6 bar, R0=3 μm S0=104 m-1, ΔS=S0/4

Hydrophone signal (a.u.)

107

6

10

105

104

0

2000

4000

6000

8000

10000

Frequency (kHz) Fig. 5. The result of the numerical simulation with the temporal fluctuation in the number of bubbles (‘‘coupling strength” (S)) for the ambient bubble radius of 3 lm which is typical in pure water [53]. The frequency and pressure amplitude of ultrasound are the same as those in Fig. 3. The initial ‘‘coupling strength” is S0 = 104 m1 and the maximum amplitude of the variation of the ‘‘coupling strength” per acoustic cycle is S0/4. The time axis is the same for (b)–(d). (a) The randomly varying ‘‘coupling strength” (S) as a function of time generated by a computer using Eq. (11). The time axis is from 0 to 700 ls. (b) The radius of a bubble (R). (c) The pressure of acoustic waves radiated from bubbles (P). (d) The hydrophone signal (U) in arbitrary unit. (e) The frequency spectrum of the hydrophone signal (U) with the logarithmic vertical axis.

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Avvaru and Pandit [22] analyzed the frequency spectra of acoustic cavitation noise using the linear resonance frequencies of bubbles. They assumed that a peak in the frequency spectrum is originated from bubbles for which the linear resonance frequency coincides with the frequency at the peak. However, for relatively high acoustic amplitudes, bubble pulsation is strongly nonlinear and a bubble radiates acoustic waves of not only the driving frequency but also its harmonics as shown in Fig. 3e. Thus, the method of Avvaru and Pandit [22] is not applicable to cavitation under high intensity ultrasound. The numerical simulation in Fig. 3 has been performed for pure water although the ambient bubble radius of 1.5 lm is a typical one in a low-concentration SDS solution [53]. Here, we will discuss the difference of the bubble dynamics in low-concentration SDS solution from that in pure water. The bubble expansion is slightly larger in 1.5 mM SDS solution than that in pure water due to smaller surface tension according to the present numerical simulation

without the bubble-bubble interaction 7

Hydrophone signal (a.u.)

10

6

10

5

10

4

10

0

2000

4000

6000

8000

10000

Frequency (kHz) Fig. 6. The calculated frequency spectrum of the acoustic cavitation noise (hydrophone signal) when the effect of the bubble–bubble interaction is neglected. The condition for the numerical simulation is the same as that in Fig. 5.

a

without the bubble-bubble interaction,

(Fig. 4a). As a result, the bubble collapse is slightly more violent and the acoustic emission is slightly stronger in 1.5 mM SDS solution (Fig. 4b). Nevertheless, the frequency spectrum of the acoustic cavitation noise (hydrophone signal) in 1.5 mM SDS solution has no significant difference from that in pure water although it is not shown here. Furthermore, Ashokkumar et al. [61] experimentally reported that SDS did not significantly affect the radial dynamics of a single-bubble although the concentration was 0.03 mM in the experiment which is much smaller than that in the other experiment by Ashokkumar et al. [26] and the present numerical simulation. Thus, in the following discussions on the spectra of acoustic cavitation noise, the difference in the radial dynamics of a bubble in low-concentration SDS solution from that in pure water is neglected. Next, a numerical simulation is performed for the ambient bubble radius of 3 lm, which is typical for a SL bubble in pure water at 515 kHz [53]. It should be noted that in low-concentration SDS solution bubbles of 3 lm in ambient radius are nearly absent according to the experimental observation by Lee et al. [53]. When the ultrasonic frequency, the pressure amplitude of ultrasound, and the ‘‘coupling strength” are the same as those in Fig. 3, the lifetime of a bubble of 3 lm in ambient radius is four acoustic cycles according to the present numerical simulation. It should be noted that this lifetime is for a steady-state pulsation of a bubble and is different from the lifetime of six acoustic cycles for the initial transient pulsation. The bubble pulsation reaches a steady-state after the initial transient pulsation [1]. The initial transient pulsation will be discussed later. Due to the relatively short lifetime of a bubble, the number of bubbles should temporally fluctuate. The temporally fluctuating ‘‘coupling strength” (S) has been generated by a computer using Eq. (11) with S0 = 104 m1 and DS = S0/4 (Fig. 5a). The radius–time curve is not affected significantly by the temporal fluctuation of the ‘‘coupling strength” (S) because for these values of S the effect of the bubble–bubble interaction is negligible (Fig. 5b). On the other hand, the intensity of acoustic emission (shock-waves) temporally fluctuates significantly as it is directly proportional to S (Eq. (4)) (Fig. 5c). As a result, the intensity of the peaks in the hydrophone signal due to shock-waves temporally fluctuates (Fig. 5d), resulting in the broad-band component in the frequency spectrum of the acoustic cavitation noise (hydrophone signal) (Fig. 5e). Thus, it is

b

with the bubble-bubble interaction

S0=105 m-1, ΔS=S0/4

S0=105 m-1, ΔS=S0/4

7

7

10

Hydrophone signal (a.u.)

Hydrophone signal (a.u.)

10

6

10

5

10

4

10

0

2000

4000

6000

Frequency (kHz)

8000

10000

6

10

5

10

4

10

0

2000

4000

6000

8000

10000

Frequency (kHz)

Fig. 7. The calculated frequency spectra of the acoustic cavitation noise (hydrophone signal) when the initial coupling strength (S0 = 105 m1) is one order of magnitude larger than that in Figs. 5 and 6. The other conditions are the same as those in Figs. 5 and 6. (a) Without the bubble–bubble interaction. (b) With the bubble–bubble interaction.

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concluded that the temporal fluctuation in the number of bubbles results in the broad-band noise. Next we will discuss the effect of the bubble–bubble interaction on the frequency spectrum of the cavitation noise (hydrophone signal). The bubble–bubble interaction can be removed in the numerical simulations solely by omitting the term (8) from the modified Keller equation. The frequency spectrum of the cavitation noise without the bubble–bubble interaction shown in Fig. 6 is similar to that with the bubble–bubble interaction shown in Fig. 5e. Thus, for these relatively small values of the ‘‘coupling strength” shown in Fig. 5a, the effect of the bubble–bubble interaction is negligible. It justifies the assumption of a single ambient radius of bubbles as explained in II Model. However, for an order of magnitude larger value of the ‘‘coupling strength”,

R0=5 μm

a

the effect of the bubble–bubble interaction is significant (Fig. 7) (S0 = 105 m1 which corresponds to the number density of bubbles of about 0.85 mm3 and the number of bubbles of about 435 in the present configuration shown in Fig. 1). By the bubble–bubble interaction, the intensity of the broad-band noise significantly increases. Furthermore, the half-order subharmonic (f0/2) and ultraharmonics ((2n + 1)f0/2) are generated by the bubble–bubble interaction. For this case, bubbles of different ambient radii affect the pulsation of each bubble in a complex way due to the bubble–bubble interaction and numerical simulations should be simultaneously performed for the whole system. For the ‘‘coupling strength” (S) assumed in the present numerical simulations except that in Fig. 7, however, the effect of the bubble–bubble interaction is nearly negligible.

R0=5 μm

b

S0=104 m-1, ΔS=S0/4

S0=104 m-1, ΔS=S0/4

16

7

10

Hydrophone signal (a.u.)

Bubble radius (μm)

14 12 10 8 6 4 2 0 680

685

690

695

6

10

5

10

4

10

0

700

2000

4000

6000

8000

10000

Frequency (kHz)

Time (μs)

Fig. 8. The result of the numerical simulation when the ambient bubble radius is R0 = 5 lm. The maximum amplitude of the temporal variation of the ‘‘coupling strength” per acoustic cycle is DS = S0/4. The other conditions are the same as those in Fig. 5. (a) The radius–time curve. (b) The frequency spectrum of the acoustic cavitation noise (hydrophone signal).

a

b

R0=6 μm

R0=6 μm S0=104 m-1, ΔS=S0/6

S0=104 m-1, ΔS=S0/6

20

7

Hydrophone signal (a.u.)

Bubble radius (μm)

10 15

10

5

0 650

660

670

680

Time (μs)

690

700

6

10

5

10

4

10

0

2000

4000

6000

8000

10000

Frequency (kHz)

Fig. 9. The result of the numerical simulation when the ambient bubble radius is R0 = 6 lm. The maximum amplitude of the temporal variation of the ‘‘coupling strength” per acoustic cycle is DS = S0/6. The other conditions are the same as those in Fig. 5. (a) The radius–time curve. (b) The frequency spectrum of the acoustic cavitation noise (hydrophone signal).

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a

2.5

nf0/4

nf0/2

Stable 2 (strong nf0) 1.5

chaotic

nf0

0

Stable (weak nf0)

Stable (weak nf0)

0.5

0

nf0

Transient (broad-band)

1

5

10

15

20

Ambient bubble radius (μm) Fig. 11. The regions for ‘‘transient” cavitation bubbles and ‘‘stable” cavitation bubbles in the parameter space of ambient bubble radius (R0) and the acoustic amplitude (pa). The ultrasonic frequency is 515 kHz and the ‘‘coupling strength” is assumed to be constant as S = 104 m1. Here ‘‘stable” (‘‘transient”) cavitation bubbles are defined as those which are shape stable (unstable). The thickest line is the border between the region for ‘‘stable” cavitation bubbles and that for ‘‘transient” ones. The type of bubble pulsation has been indicated as chaotic (non-periodic) or periodic with the acoustic period (nf0), doubled acoustic period (nf0/2), or quadrupled acoustic period (nf0/4). The summary has been shown in Table 1.

few hundred acoustic cycles according to the present numerical simulations. Even at lower acoustic amplitude than 1 bar, there is a range of ambient radius for ‘‘transient” bubbles (Fig. 11). The range is nearly centered at the linear resonance radius (Rres) of 6.9 lm at 515 kHz (Eq. (19)) [62].

x2 ¼

  2r 3 j p þ ð3 j  1Þ 1 Rres qR2res 1

ð19Þ

where x is the angular frequency of ultrasound (x = 2pf), q is the liquid density, j is the polytropic exponent (it is 1.4 for an air

b

R0=7 μm 20

515 kHz

3

Acoustic amplitude (bar)

Next, we will discuss the influence of the ambient bubble radius on acoustic cavitation noise. With the same ultrasonic frequency, the acoustic amplitude, and the ‘‘coupling strength” as those of Figs. 3–5, the lifetime of a bubble for a steady-state pulsation is four and six acoustic cycles for the ambient bubble radius (R0) of 5 lm and 6, 7 lm, respectively, according to the present numerical simulations. Thus, the number of bubbles should temporally fluctuate for these cases. Without the bubble–bubble interaction, the bubble pulsation is periodic with doubled acoustic period for R0 = 5 lm. Due to the effect of the bubble–bubble interaction, the bubble pulsation is not strictly periodic as shown in Fig. 8a because the ‘‘coupling strength” temporally fluctuates. The resultant frequency spectrum of the cavitation noise (hydrophone signal) consists of the broad-band component, peaks at the driving frequency (f0), harmonics, the half-order subharmonic and ultraharmonics (nf0/2) (Fig. 8b). The broad-band noise is originated from the temporal fluctuation of the ‘‘coupling strength”. For R0 = 6 lm, the bubble pulsation is periodic with quadrupled acoustic period when the effect of the bubble–bubble interaction is neglected. With the bubble–bubble interaction, the bubble pulsation is not strictly periodic (Fig. 9a). The frequency spectrum of the cavitation noise (hydrophone signal) consists of the broad-band component, peaks at the driving frequency (f0), harmonics, the half-order subharmonic, and ultraharmonics (nf0/4) (Fig. 9b). For R0 = 7 lm, the bubble pulsation is non-periodic (chaotic) irrespective of whether the effect of the bubble–bubble interaction is taken into account (Fig. 10a). As a result, the frequency spectrum of the cavitation noise (hydrophone signal) consists of the peak at the driving frequency and the broad-band component. For this case, the broad-band noise originates not only from the temporal fluctuation in the number of bubbles but also from non-periodic pulsation of bubbles. For the ambient radius larger than 5 lm, bubbles are inactive in SL according to Lee et al. [53]. Thus, it is also concluded that not only active bubbles in SL but also inactive bubbles generate the broad-band noise. In the present study, numerical simulations have been performed for various ambient bubble radii and acoustic amplitudes at 515 kHz. The results have been summarized in Fig. 11 and Table 1. The lifetime of the ‘‘transient” bubbles ranges from a few to a

R0=7 μm S0=104 m-1, ΔS=S0/6

S0=104 m-1, ΔS=S0/6 7

Hydrophone signal (a.u.)

Bubble radius (μm)

10 15

10

5

0 680

685

690

Time (μs)

695

700

6

10

5

10

4

10

0

2000

4000

6000

8000

10000

Frequency (kHz)

Fig. 10. The result of the numerical simulation when the ambient bubble radius is R0 = 7 lm. The maximum amplitude of the temporal variation of the ‘‘coupling strength” per acoustic cycle is DS = S0/6. The other conditions are the same as those in Fig. 5. (a) The radius–time curve. (b) The frequency spectrum of the acoustic cavitation noise (hydrophone signal).

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bubbles pulsate periodically with the acoustic period at least in the parameter space shown in Fig. 11. For ‘‘transient” bubbles, there are four types in the bubble pulsation in the parameter space. One is non-periodic (chaotic) pulsation. The others are periodic pulsations with the acoustic period, or doubled acoustic period, or quadrupled acoustic period. These are summarized in Table 1. ‘‘Transient” bubbles generate the broad-band noise because there should be temporal fluctuation in the number of bubbles due to a finite lifetime of a bubble. On the other hand, ‘‘stable” bubbles do not generate the broad-band noise. Much weaker broad-band noise in low-concentration SDS solution compared to that in pure water experimentally reported by Ashokkumar et al. [26] is caused by the fact that most bubbles are ‘‘stable” in low-concentration SDS solution due to their smaller ambient radii as seen in Fig. 3e. It should be noted that smaller ambient radii are due to the inhibition of coalescence of bubbles covered with the charged surfactant (SDS) [63]. For higher concentration of SDS than 3 mM, however, the electric charge of SDS on the bubble surface is partly neutralized by the dissociated SDS molecules acting as excess electrolyte and the coalescence of

Table 1 The relationship between the type of cavitation bubbles and that of the cavitation noise spectrum in the parameter space shown in Fig. 11. ‘‘Chaotic (initial transient)” means non-periodic pulsation only at the initial transient stage although the pulsation becomes periodic at the steady-state. Type

Pulsation

Noise spectrum

Stable Low energy High energy

Periodic (period T) Periodic (period T)

Weak nf0 Strong nf0

Periodic (period T) Periodic (period 2T) Periodic (period 4T) Chaotic (steady-state) Chaotic (initial transient)

nf0 + broad-band nf0/2 + broad-band nf0/4 + broad-band Broad-band Broad-band

Transient

bubble), p1 is the static pressure, and r is the surface tension. The range of the ambient radius for chaotic (non-periodic) pulsation at relatively high acoustic amplitude is also nearly centered at the linear resonance radius. Both low-energy and high-energy ‘‘stable”

a

b

initial transient pulsation

15

10

5

0

steady state pulsation 20

Bubble radius (μm)

Bubble radius (μm)

20

0

5

10

15

15

10

5

0 80

20

85

90

95

100

Time (μs)

Time (μs)

c

d

515 KHz, 2 bar, R0=13 μm

initial transient pulsation

S=104 m-1, ΔS=0

0.8 6

10

Hydrophone signal (a.u.)

Acoustic emission (bar)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6

0

5

10

Time (μs)

15

20

5

10

initial transient 4

10

steady state

3

10

0

2000

4000

6000

8000

10000

Frequency (kHz)

Fig. 12. The result of the numerical simulation when the ambient bubble radius is R0 = 13 lm. The ultrasonic frequency and the acoustic amplitude are 515 kHz and 2 bar, respectively. The ‘‘coupling strength” is assumed to be constant as S = 104 m1. (a) The radius–time curve for the initial transient pulsation. (b) The radius–time curve for the steady-state pulsation. (c) The pressure (P) of acoustic waves radiated from bubbles for the initial transient pulsation. The time axis is the same as that in (a). (d) The comparison of the frequency spectrum of the acoustic cavitation noise (hydrophone signal) for the initial transient pulsation with that for the steady-state pulsation.

K. Yasui et al. / Ultrasonics Sonochemistry 17 (2010) 460–472

bubbles takes place to a larger extent, which results in the presence of larger bubbles than those in low-concentration SDS solution [64,65]. This is the reason for the stronger broad-band noise in higher-concentration SDS solution than 3 mM [26]. Harmonics originate not only from ‘‘stable” bubbles but also from ‘‘transient” bubbles which pulsate temporally periodically during their lifetimes. The half-order subharmonic and ultraharmonics ((2n  1)f0/2) originate from ‘‘transient” bubbles which pulsate periodically with doubled acoustic period as there are no such ‘‘stable” bubbles at least in the parameter space in Fig. 11. In the experimental data of the frequency spectra of acoustic cavitation noise by Ashokkumar et al. [26], there are no peaks at ultraharmonics of degree (2n  1)f0/4. This suggests that there is negligible population of bubbles in the parameter space for nf0/4 in Fig. 11. This suggests that there is negligible population of bubbles in the parameter space for chaotic pulsation in Fig. 11 because the area for chaotic pulsation in the parameter space is even smaller than that for nf0/4. Therefore, the main origin of the broad-band noise is not the chaotic pulsation of bubbles but the temporal fluctuation in the number of bubbles at least under the condition of the experiment by Ashokkumar et al. [26]. Finally, we will discuss the initial transient pulsation of a bubble. The bubble pulsation reaches a steady-state after the initial transient pulsation. Here the term transient is different from that used in ‘‘transient cavitation” and means ‘‘before the steady-state”. The time required for reaching a steady-state depends on the initial condition of the bubble pulsation (R and dR/dt at t = 0). In Fig. 12a, a part of the initial transient pulsation has been shown when the ambient bubble radius is 13 lm and the acoustic amplitude is 2 bar. The initial condition is at rest at the ambient size(R = R0 and dR/dt = 0 at t = 0). The bubble pulsation reaches the steadystate after about 40 acoustic cycles. The steady-state pulsation has been shown in Fig. 12b. Although the steady-state pulsation is periodic, the initial transient one is non-periodic (chaotic). Thus, the acoustic emission during the initial transient pulsation is nonperiodic (chaotic) (Fig. 12c). As a result, the broad-band noise is generated for the initial transient pulsation (Fig. 12d). It should be noted that the broad-band component for the steady-state pulsation seen in Fig. 12d is not a real broad-band noise but just a result of a short time series of the hydrophone signal for the Fourier analysis. The initial transient pulsation can contribute to the broad-band noise even if the bubble pulsation is periodic in the steady-state. It should be noted that under this condition a bubble is ‘‘transient” (shape unstable) because a bubble has a finite lifetime of 19.9 and 31.1 ls for the initial transient pulsation and the steady-state one, respectively.

tion noise are generated not only by ‘‘stable” cavitation bubbles but also by ‘‘transient” cavitation bubbles when their pulsation is periodic during their lifetimes. For a relatively high number density of bubbles, the bubble–bubble interaction intensifies the broad-band noise. The initial transient pulsation of bubbles can generate the broad-band noise even if the pulsation is periodic at the steady-state.

Acknowledgement The authors would like to thank M. Ashokkumar for his valuable comments.

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[8]

[9] [10]

[11] [12] [13]

[14] [15] [16]

[17] [18] [19]

[20]

4. Conclusion [21]

Numerical simulations of acoustic cavitation noise have been performed under the conditions of the experiment reported by Ashokkumar et al. [26]. It has been shown that the temporal fluctuation in the number of bubbles (‘‘coupling strength” (S) of the bubble cloud) generates the broad-band noise. ‘‘Transient” cavitation bubbles generate the broad-band noise if ‘‘transient” cavitation bubbles are defined as those which disintegrate into daughter bubbles mostly in a few acoustic cycles because they cause the temporal fluctuation in the number of bubbles. In lowconcentration SDS (surfactant) solutions, most cavitation bubbles are shape stable according to the present numerical simulations due to their small ambient radii [53]. Thus the broad-band noise is much weaker than that from pure water as ‘‘stable” cavitation bubbles do not generate the broad-band noise. Not only active bubbles in SL or sonochemical reactions but also inactive bubbles generate the broad-band noise. Harmonics in the acoustic cavita-

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