Nonlinear ultrasonic waves in bubbly liquids with nonhomogeneous bubble distribution: Numerical experiments

Ultrasonics Sonochemistry 16 (2009) 669–685

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Nonlinear ultrasonic waves in bubbly liquids with nonhomogeneous bubble distribution: Numerical experiments Christian Vanhille a,*, Cleofé Campos-Pozuelo b a b

ESCET, Universidad Rey Juan Carlos, Tulipán, s/n. 28933 Móstoles, Madrid, Spain Instituto de Acústica, CSIC, Spain

a r t i c l e

i n f o

Article history: Received 4 February 2008 Received in revised form 18 November 2008 Accepted 18 November 2008 Available online 10 December 2008 Keywords: Nonlinear acoustics Ultrasonic waves Bubbly liquids Bubbly layers Numerical acoustics Sonochemistry Cavitation

a b s t r a c t This paper deals with the nonlinear propagation of ultrasonic waves in mixtures of air bubbles in water, but for which the bubble distribution is nonhomogeneous. The problem is modelled by means of a set of differential equations which describes the coupling of the acoustic field and bubbles vibration, and solved in the time domain via the use and adaptation of the SNOW-BL code. The attenuation and nonlinear effects are assumed to be due to the bubbles exclusively. The nonhomogeneity of the bubble distribution is introduced by the presence of bubble layers (or clouds) which can act as acoustic screens, and alters the behaviour of the ultrasonic waves. The effect of the spatial distribution of bubbles on the nonlinearity of the acoustic field is analyzed. Depending on the bubble density, dimension, shape, and position of the layers, its effects on the acoustic field change. Effects such as shielding and resonance of the bubbly layers are especially studied. The numerical experiments are carried out in two configurations: linear and nonlinear, i.e. for low and high excitation pressure amplitude, respectively, and the features of the phenomenon are compared. The parameters of the medium are chosen such as to reproduce air bubbly water involved in the stable cavitation process. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction The effect of bubbles on the propagation of ultrasonic waves in liquid is of definitive importance in many applications. In particular, in the framework of sonochemistry, the understanding of the behaviour of ultrasonic waves in bubbly water is necessary [1]. Many applications of high-power ultrasounds are based on the presence of bubbles in liquids or on cavitation. Linear and nonlinear propagation through bubbly liquids is a current and active area of research in acoustics [2,3]. This is the case in therapeutic and/or diagnostic medical applications of ultrasound [4–7], and in underwater acoustics [8,9]. Gong and Hart [10] studied the chemical effects of the collapse of one bubble by coupling chemical kinetics in the bubble with the bubble dynamics. In sonochemistry, each bubble cannot be considered as if it was isolated. The global behaviour of the whole sonoreactor has been scarcely studied. Some complex models exist in the linear range, and they are usually used for design and development in laboratories [11–13]. Yasui et al. [14] analyzed the static spatial acoustic distribution in a three-dimensional resonator through a finite element model. The wall vibration was taken into account. However, dispersion and nonlinear effects due to the bubbles were not considered, but their presence induced an effect on attenua* Corresponding author. Tel.: +34 91 664 74 82. E-mail address: [email protected] (C. Vanhille). 1350-4177/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ultsonch.2008.11.013

tion. Other models that take into account the presence of bubbles also exist [15–18]. Nevertheless, these models are based on linear acoustic pressure and linear bubble vibration. Among them, the models of Refs. [16–18] also consider the geometrical complexity of the system and the Bjerknes forces. However, the effects we search for in many applications of high-power ultrasounds, and especially in sonochemistry, appear only above an intensity threshold that induces a nonlinear behaviour of the pressure field. Colonius et al. [19] studied, via a numerical model, the saturation effect of acoustic pressure in a liquid with homogeneous density excited by a harmonic wall for which the oscillation frequency is much smaller than the bubble natural frequency. The homogeneous spatial distribution of small bubbles induces an acoustic behaviour different from that of a homogeneous liquid (decrease of phase speed, dispersive properties, increase of losses, high compressibility and nonlinear effects) [2,20,21]. Several works consider nonhomogeneous spatial distributions of bubbles. In particular, the nonlinear response and resonance effects of a bubbly layer to a harmonic wave were studied by Karpov et al. [8], in the framework of parametric generation of a low-frequency signal. This study was based on a numerical model. The system bubble – acoustic field was coupled and the dynamics of the bubble vibration did not contain the adiabatic restriction. However, only one or two frequencies were considered in the incident wave and only thin bubble layer were described. Sutin et al. [9] developed an analytical model for the nonlinear scattering.

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Lo et al. [22] studied the shielding effect of a bubble layer experimentally. They described a technique of spatial control therapy by means of the generation of walls of gas bubbles. Dähnke and Keil [23] studied the nonhomogeneous distribution of bubbles by means of a linear pressure theory. Macpherson [24] characterized a bubbly liquid via the production of bubbly layers, by measuring the response to pulses and determining the attenuation. Leighton et al. [25] proposed an inversion technique to measure the size distributions of gas bubbles in the ocean. Ref. [22] justifies the need of understanding the nonlinear ultrasonic behaviour of waves in nonhomogeneous distribution of bubbles in liquids for applications in medicine. In sonochemistry, bubbles gather forming bubbly layers in the liquid. This study is especially interesting in this framework. However, we have to note that in the context of sonochemistry, the model will have to account for many other phenomena, such as Bjerknes forces attraction, the compressibility of liquid and entropic variations in the bubble equation. It must be noticed that Bjerknes forces could change the shape of the layer because of the standing waves that are formed in the layer. In this paper we analyze the linear and nonlinear propagation of ultrasonic waves in a nonhomogeneous distributed bubbly liquid. Air bubbly layers are assumed to be placed in water. The mathematical model describing the bubble vibration and acoustic field system is presented in Section 2. A second-order equation written in a volume formulation is considered for bubbles vibration and coupled with the linear non-dissipative wave equation (bubble vibration and acoustic field calculations are coupled). The physical hypotheses and limitations of the system are given. The numerical code chosen to solve the problem is cited as well. Section 3 presents the results and the corresponding discussions. Section 4 exposes the conclusions of this work. 2. Model The mathematical model chosen here is the one used in Refs. [20,2,26,21]. We consider plane ultrasonic waves propagating in an air–water mixture. Air bubbles are spherical, adiabatic, and all have the same size. Acoustic attenuation and nonlinear effects are only due to the bubbles oscillations. Considering an adiabatic behaviour of air in bubbles in the model (no heat transfer between air in bubbles and surrounding water) overestimates the nonlinear effects corresponding to the inertia of the bubble vibration. In the bubble equation, the compressibility of water is neglected and implies less damping in the bubble vibration and less attenuation of the associated nonlinear effects. Some other restrictive features of the model are: bubbles are monopole; their pulsation is radially symmetric and assumed to be small; their surface tension is neglected. In the semi-infinite space domain, bubbles are not uniformly distributed but concentrated in some regions. The density of bubbles depends on the one-dimensional space coordinate, i.e. Ng is a function of x: Ng(x) (the spatial distribution is not homogeneous). For numerical purpose, the semi-infinite space domain is limited to the studied domain X = [0,L]. The system is formed by the Rayleigh–Plesset (second-order equation for bubble dynamics) and linear non-dissipative wave equations. It is written: o2 p ox2

 c12

o2 m ot 2

  2 þ dx0g ootm þ x20g m þ gp ¼ aðmÞ2 þ b 2m oot2m þ ðootm Þ2 x 2 X; t 2 T

0l

o2 p ot 2

2

¼ q0l Ng ðxÞ oot2m x 2 X; t 2 T

ð1Þ

ð2Þ where t is the time independent variable, T is the (bounded) time domain of the study corresponding to X, c0l is the low amplitude sound speed of water, q0l is the density of water at the equilibrium

state, pðx; tÞ is the acoustic pressure, mðx; tÞ ¼ Vðx; tÞ  m0g is the bubble volume variation, where V and m0g ¼ 4pR30g =3 are the current and equilibrium state bubble volumes, respectively, R0g is the equidamping coeflibrium bubble radius, d ¼ 4ml =ðx0g R20g Þ is the viscousq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ficient,

ml is the cinematic viscosity of water, x0g ¼

3cg p0g =q0l R20g

corresponds to the lowest resonance frequency of the bubble, p0g ¼ q0g c20g =cg is the atmospheric pressure of air, cg is the specific heats ratio of air, q0g is the density of air at the equilibrium state, c0g is the low amplitude sound speed of air, g ¼ 4pR0g =q0l , a ¼ ðcg þ 1Þx20g =2m0g , and b ¼ 1=ð6m0g Þ. At the outset the following initial conditions are applied (rest state):

pðx–0; t ¼ 0Þ ¼ 0 x 2 X

ð3Þ

op ðx–0; t ot

ð4Þ

¼ 0Þ ¼ 0 x 2 X

mðx; t ¼ 0Þ ¼ 0 x 2 X

ð5Þ

om ðx; t ot

ð6Þ

¼ 0Þ ¼ 0 x 2 X

m does not need boundary conditions. The acoustic field (plane waves) is generated by a continuous time-function pressure source of amplitude p0 at x ¼ 0, and its open-field (progressive) character is imposed at x ¼ L via the other boundary condition on p [27]:

pðx ¼ 0; tÞ ¼ p0 sinðxf tÞ t 2 T

ð7Þ

op ðx ox

ð8Þ

¼ L; tÞ ¼ 1 c 0l

op ðx ot

¼ L; tÞ t 2 T

where xf ¼ 2pf is the excitation pulsation. The description of the terms of this set of equations can be found in Ref. [21]. The Simulation of Nonlinear Waves – Bubbly Liquid (SNOW-BL) code was developed to solve Eqs. (1)–(8), but with a constant value N g , i.e. in the case of homogeneous bubble distribution. The description of the numerical model can be found in Vanhille and Campos-Pozuelo [21]. It was based on the finite-difference method in space and time (implicit and second-order scheme) [28]. Here this code is adapted to the possible variations of N g with space in order to solve the problem treated in this paper, Eqs. (1)–(8). In particular, the case for which we are interested with is now able to be simulated, i.e. bubbly layers can be introduced into the domain X. In a layer, the characteristics of the medium have almost not changed, i.e. this is almost the same fluid, but its acoustic ones have changed hugely. The main effect of bubbles is to increase compressibility, even for low void fraction, and therefore, to decrease sound velocity. Based on the finite-difference method, the code requires the data of the discretization in space and time. Once these data and all the physical parameters are defined and introduced into the model, the SNOW-BL code gives the nonlinear solution of the problem. 3. Numerical experiments, results and discussion Section 3 is devoted to analyse the effects caused on the (linear and nonlinear) propagation in water of plane ultrasonic waves by layers of air bubble. The position and thickness of the bubbly layer, as well as the excitation pressure amplitude are changed to carry out this analysis. The shape of the layer is also changed to consider bubbly clouds. Several layers are also considered in the last paragraph of the section. We are especially interested in investigating the influence of the layers on the nonlinear characteristic of the propagation by comparing different excitation amplitude cases. The parameters of the medium are chosen here such as to reproduce air bubbly water in the stable cavitation process: q0l ¼ 1000 kg=m3 , c0l ¼ 1500 m=s, and ml ¼ 1:4  106 m2 =s for water; q0g ¼ 1:29 kg=m3 , c0g ¼ 340 m=s, and cg ¼ 1:4 for air. The

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initial radius of all the bubbles is R0g ¼ 4:5  106 m [29]. This data means that x0g ¼ 4:70  106 Hz, i.e. f0g ¼ 7:48  105 Hz, and d ¼ 6:02  102 . Fig. 1 shows an analysis of the importance of the different terms in the differential equation, Eq. (2), via their evolution with the frequency, for different amplitudes. They are presented in relative units: we show the ratio of each term to the inertial term ðx2f  x20g Þ, which strongly depends on the frequency. Only for very high frequency the viscous term becomes important. The nonlinear term, nevertheless, becomes important very quickly, but it depends on the amplitude. In particular, for pressure values of the order of 3  104 Pa, the nonlinear term is of the order of the top solid line in the bottom sketch of the figure (the red line in the web version of this article). Bubbles act in the right-hand side of Eq. (1). Their presence there implies a change of sound velocity, which appears implicitly in the model via the coupling of both equations. The sound velocity provokes distortion due to dispersion. The nonlinearity, as well as the dissipation, of the bubbly medium is exclusively due to the coupling of the bubble equation, Eq. (2), which includes nonlinear and dissipative terms, and the acoustic equation via the right-hand side of Eq. (1). We place this study in the context of sonochemistry [1]. In this sense, we consider continuous waves, at a constant harmonic excitation frequency usual in sonochemistry applications: 0.7

Viscosity-inertial terms ratio Nonlinear-inertial terms ratio

0.5 0.4 0.3 0.2

3.1. One resonant layer: effect of the resonant water domain

0.1

0

0.5

1

1.5

0.4

2 2.5 3 Frequency (Hz)

3.5

4

4.5

5 5

x 10

Viscosity-inertial terms ratio Nonlinear-inertial terms ratio

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

f ¼ 24500 Hz, i.e. xf ¼ 1:54  105 Hz and klf ¼ 6:12  102 m (in water). We have to note that the excitation frequency is well below the resonance frequency of the bubbles, i.e. xf =x0g ¼ 3:28  102 . First, the effect of one layer is studied. The influence of its dimension and position is analysed. The effect of its density and shape (cloud) is also studied. Afterwards, the effect of several consecutive layers is analysed. In all the following experiments, linear results refer to the excitation pressure amplitude p0 ¼ 100 Pa, while the nonlinear one corresponds to p0 ¼ 2:5  104 Pa, except in Section 3.5. The layers used in this Section 3 are now defined. The bubble density distribution is such that N g ¼ 0 m3 out of the layer and N g ¼ 2  1012 m3 inside the layer (see figures) (the bubble density nulls out of the layers and is N g ¼ 2  1012 m3 inside the layers), i.e. a quite high volumetric void fraction is considered inside the layer 0.08%. This means that [21]: c0L ¼ 423:74 m=s in the bubbly layer. By taking into account the value of the frequency f ¼ 24500 Hz also in the bubbly layer, we have kLf ¼ 1:73 102 m. The thickness of each layer W L is calculated in relation to kLf , and its position is evaluated with respect to klf (pure water wavelength) and given by the coordinate of the left edge: xp . The space between x ¼ 0 and xp is called the pre-layer water domain (PRWD), and the space placed beyond the layer is called the post-layer water domain (POWD). The layers are (see figures): L1. W L ¼ 8:65  103 m, i.e. W L ¼ 0:5kLf at xp ; L2. W L ¼ 1:73  102 m, i.e. W L ¼ kLf at xp ; L3. W L ¼ 3:46  102 m, i.e. W L ¼ 2kLf at xp ; L4. W L ¼ 1:73  101 m, i.e. W L ¼ 10kLf at xp ; L5. W L ¼ 9:78  102 m, i.e. W L ¼ 5:65kLf at xp ; L6. Cloud (see the top part of Fig. 9); L7. W L ¼ 1:49  101 m, i.e. W L ¼ 8:64kLf at xp ; SL2. W L ¼ 1:73  102 m, i.e. W L ¼ kLf at two xp ; SL3. W L ¼ 1:73  102 m, i.e. W L ¼ kLf at three xp .

0.6

0

671

0

1

2

3

4 5 6 Frequency (Hz)

7

8

9

10 4

x 10

Fig. 1. Terms in Eq. (2) for different amplitudes. Evolution with frequency in relative units.

In this paragraph, one bubbly layer is considered in water. Figs. 2, 3a, 4, and 5 correspond to L1, L2, L3, and L4, respectively. Five positions of the layer are contemplated for L1, L2, and L3: xp = (i) 0 m; (ii) klf =4; (iii) klf =2; (iv) klf ; (v) 2klf . Only one position is considered for L4: (v) xp = 2klf . Each position of the layer corresponds to one row in each corresponding figure. The left part of each row indicates the N g ðxÞ function: the bubble density ðm3 Þ as a function of the space coordinate. The central part of each row shows the representation of the acoustic pressure field in the linear configuration. The nonlinear acoustic pressure field is represented in the right part of each row. These representations are given as functions of space coordinate (horizontal axis) and time (vertical axis); colours mean amplitude. In theses characteristic curves representations, the larger is the slope of the ðx; tÞ graph, the lower is the wave speed. In Fig. 5 are also shown waveforms at some particular points indicated by the lines (red lines in the web version of this article). Note that the horizontal scale in Fig. 4 is not the same in all the displays. In the linear case, first the effect to be noted is the huge decrease of phase speed in the bubbly layer. Nevertheless, because of the small size of the layer, the effect of this deceleration of the wave has a very small effect on the global phase speed, i.e. it arrives after the same time almost to the same point that the corresponding wave propagating in a completely homogeneous fluid (water). However, this narrow layer produces important effects. It induces reflections of the wave on its left face (out of the layer, in the PRWD between the first interface and the source) and later on its right face (inside the layer, between both interfaces). A

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Fig. 2. L1 layer. Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours). Middle column: linear case; right column: nonlinear case. Left column: definition and position of the layer.

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standing wave is created inside the layer. In cases (ii), (iii), (iv), and (v), a standing wave is created in the PRWD between x ¼ 0 and xp . These two phenomena imply a screen effect by the layer, i.e. acoustic energy transmitted beyond the layer to the POWD is weak.

673

Nevertheless, in case (ii) the acoustic pressure does not increase very much in the PRWD (water domain preceding the layer), which is the opposite of what occurs for cases (iii), (iv), and (v) for which it strongly increases.

Fig. 3. L2 layer. (a) Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours). Middle column: linear case; right column: nonlinear case. Left column: definition and position of the layer. (b) Waveform at x ¼ 6:12  104 m and x ¼ 5:51  103 m in configuration (iii) ðxp ¼ klf =2Þ, in the linear (left) and nonlinear (right) cases.

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Fig. 3 (continued)

It must be noted that, in case (ii), the biggest screen effect occurs for L3 (see Fig. 4). This behaviour indicates the low impedance of the bubbly layer in comparison with pure water: amplitudes increase for the freewall resonator case, i.e. the effective impedance of the bubbly layer is much smaller than the one of water. In the nonlinear case, the nonlinear characteristics are demonstrated (see for example the right column of Fig. 5) by a strong asymmetry of the acoustic wave (mean pressure does not null), as well as an important harmonic distortion and the related nonlinear attenuation. It must be noted that this nonlinear feature is weakened here because not much energy is transmitted due to reflections and standing waves are turning on. It can be noticed here that the nonlinear standing waves formed in the PRWD and inside the layer are absolutely different from the ones obtained in a homogeneous fluid (see for example [30,31]). An analytical solution has been written in a linear case presented above giving very similar results about steady amplitudes. Note that a study of nonlinear ultrasonic standing waves in a bubbly liquid with homogeneous bubble density was given in [32]. The characteristic impedance of water is Z 0l ¼ 15  105 kg=m2 s. The initial radius and bubble density used in the case of L4 (Fig. 5) induce c0L ¼ 423:74 m=s and q0L ¼ 999:24 kg=m3 in the bubbly layer, and the characteristic impedance is Z 0L ¼ 4:23 105 kg=m2 s. Thus, Z 0L =Z 0l ¼ 0:28 and the values of the reflection ðRÞ and transmission ðTÞ coefficients at both interfaces of the layer are: Rxp ¼ 0:56, T xp ¼ 0:44, Rxp þW L ¼ 0:56, T xp þW L ¼ 1:56. At the left-hand side of the left interface of the layer ðx p Þ, the backward wave is 180° phase shifted with respect to the forward wave (these two waves rest themselves). This agree with the theory

[33], because Rxp is negative. The transmission loss at this left interface is TLxp ¼ 1:63 dB. However, Rxp þW L is positive and the two waves at the left-hand side of the right interface of the layer ðxp þ W  L Þ have the same phase (these two waves sum themselves). The transmission loss at this right interface is TLxp þW L ¼ 1:63 dB. It can be seen at the end of the top waveforms in this Fig. 5 that the PRWD is affected by the transmitted backward wave at xp (which is nonlinear due to the bubbles and then creates an excitation source of the PRWD that contains several harmonics) after its reflection on the right face of the layer. Thus, this effect allows us to have at x ¼ 0 some nonlinear information coming from the bubbles, once this transmitted wave has reached x ¼ 0. This fact is clear in Fig. 3a, in cases (ii) and (iii) for instance. For configuration (iii), Fig. 3b shows the acoustic pressure waveform at two points very close to x ¼ 0: x ¼ 6:12  104 m and x ¼ 5:51  103 m, in the linear (on the left) and nonlinear (on the right) cases. One indeed can realize that the wave is reached by the nonlinear effects from t ’ 1:6  104 s, while in the linear case this is just a question of linear interaction. This method may be useful for nonlinear characterization of the layer from the origin (source). 3.2. One resonant layer: effect of the non-resonant water domain In this paragraph, one bubbly layer is considered in water. Two configurations are assumed (Fig. 6): L2 at xp ¼ 0:65klf and L4 at xp ¼ 1:65klf . In this figure are also shown waveforms at some particular points indicated by the lines (red lines in the web version of this article). By comparing this figure with Figs. 3 and 5, we can conclude that resonance of the PRWD is the predominant phenomenon in

C. Vanhille, C. Campos-Pozuelo / Ultrasonics Sonochemistry 16 (2009) 669–685

675

Fig. 4. L3 layer. Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours). Middle column: linear case; right column: nonlinear case. Left column: definition and position of the layer.

the linear case (the most important feature occurring in the PRWD). In addition, the strongly nonlinear and dispersive character of the layer becomes evident in the water domain, in which interferences between the two waves containing nonlinear distortion produce a special pattern (nonlinear sketch of Fig. 6a). This

phenomenon is due to the fact that the different harmonics travels at different phase speeds in the layer (due to dispersion), and get the PRWD at different moments and with different phases, which causes this non-classical interferences pattern. This phenomenon can also be seen in other figures.

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Fig. 5. L4 layer. Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours). Left column: linear case; right column: nonlinear case. Top sketch: definition and position of the layer. Waveforms at the points given by the red lines. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

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677

3.3. One layer with resonant water domain: effect of the bubble density In this paragraph, we consider one bubbly layer in water: L2 at (iv) xp ¼ klf and we change the bubble density. W L ¼ kLf only when N g ¼ 2  1012 m3 . For all the other densities the layer is not resonant. We study here how this fact affects the reflected field between the excitation and the left edge of the layer, the field inside the layer, and the transmitted signal after the layer. Fig. 7a shows the results for (top to bottom) N g ¼ 2  1012 m3 , N g ¼ 1:1  1012 m3 , N g ¼ 2  1011 m3 , and N g ¼ 2  1010 m3 . Fig. 7b displays the results obtained with bubbles homogeneously distributed in the whole domain. The bubble density is chosen so that the total number of bubbles in the domain matches the one in the layer for the first row of Fig. 7a, which yields N g ¼ 9:42  1010 m3 . In Fig. 7c are shown the waveforms at x ¼ 0:1 m corresponding to the first row of Fig. 7a and b. In the first rows of Fig. 7a we see that the effect of the layer is strong at high density in the layer. The last row of Fig. 7a shows that the layer has no effect on the field at low density in the layer. We note that the phase speed changes inside the layer (Fig. 7a): a lot if the layer is dense, not much if the layer is not dense. This speed variation (first row of Fig. 7a) is not the same as the one obtained in a homogeneous bubbles distribution case (Fig. 7b and c). By comparing the linear results (and the nonlinear ones) at N g ¼ 2  1012 m3 and at N g ¼ 1:1  1012 m3 (first two rows of Fig. 7a), we note that the amplitude is higher in the case N g ¼ 1:1  1012 m3 . This behaviour is due to the fact that, since the layer is not resonant at this density anymore, energy is less confined inside the layer and more energy is given back to the PRWD, which implies higher amplitude. However, the nonlinear effects are more important in the case for which the density is higher, in the layer but also in the PRWD and the POWD. This last point is due to the strong accumulation of the effects derived from the bubbles (more numerous at higher density). The nonlinear behaviour of the wave is much more marked in the homogeneous distribution case (Fig. 7b) than in the nonhomogeneous distribution case (first row of Fig. 7a): the spatial accumulation effect of the harmonic distortion as well as the generation and amplitude of the frequency due to the resonance frequency of the bubbles has to be noted. The asymmetry is much bigger in the homogeneous case (Fig. 7c) (note the off-center position of the zero in the colorbar of the right graph of Fig. 7b). In both cases of bubbles homogeneously distributed (Ng ¼ 9:42  1010 m3 ), and concentrated in a layer (N g ¼ 2  1012 m3 ), the (equilibrium global effective) density of the fluid is not modified in relation to pure water (see Table 1). However, the acoustic parameters are very strongly modified: harmonic distortion exists; the global effective phase speed, compressibility, attenuation parameter (in linear and nonlinear cases), and nonlinearity parameter are very different (see Table 1). In this table, the attenuation parameter is based on rms acoustic pressure and the nonlinearity parameter is defined as the ratio of the second to the first harmonic amplitude at x ¼ 0:21 m. 3.4. One non-resonant layer: effect of the resonance and nonresonance of the water domain

Fig. 6. Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours). Left column: linear case; right column: nonlinear case. Top sketch: definition and position of the layer. (a) L2 layer. (b) L4 layer. Waveforms at the points given by the red lines. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

In this paragraph, one bubbly layer is considered in water. Two configurations are assumed (Fig. 8): L5 at (iii) xp ¼ klf =2 (resonant water domain) and at xp ¼ 0:36klf (non-resonant water domain). In this figure are also shown waveforms at some particular points indicated by the lines (red lines in the web version of this article). By comparing Fig. 8a and b, it is clear that the resonant PRWD keeps acoustic energy (the amplitude of the standing wave is high) and prevents it not to pass to the layer. The nonlinearity of the

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Fig. 7. Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours). Left column: linear case; right column: nonlinear case. (a) L2 layer. Top sketch: definition and position of the layer. From top to bottom: density of bubbles is decreasing. (b) Homogeneous distribution of the same number of bubbles as for the first row of Fig. 7a. (c) Waveforms at x ¼ 0:1 m corresponding to two cases: the first row of Fig. 7a and Fig. 7b.

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Fig. 7 (continued)

Table 1 Global effective parameters for three kinds of medium. Global (in space) effective parameter

Water

Bubbly liquid with homogeneous density distribution ðN g ¼ 9:42  1010 m3 Þ

Bubbly liquid with layer ðN g ¼ 2  1012 m3 Þ

Density (kg/m3) Phase speed (m/s) Compressibility (ms2/kg) Attenuation parameter (m1) (linear case) Attenuation parameter (m1) (nonlinear case) Nonlinearity parameter (%) (nonlinear case)

1000 1500 4.4  1010 0

999.96 1212.7 6.8  1010 0.08

999.99 1317 5.77  1010 1.52

0

1.02

1.56

0

46.3

24.48

layer clearly influences the behaviour of the POWD, but its backward wave also influences the PRWD which is excited at the interface xp by a function containing several harmonics: the pattern is strongly modified (see Sections 3.1 and 3.2). 3.5. One cloud: effect of its shape In this paragraph, one bubbly cloud is considered in water. Two configurations are assumed (Fig. 9a and b): L6 and L7 at xp ¼ 1:79klf . The PRWD is not resonant in both cases. When the bubble distribution in the layer is smooth, no reflection even occurs and all energy is transmitted to the layer and the POWD. When the layer is not smooth, reflections and standing waves occur in the PRWD, inducing a loss of energy crossing through the layer to the POWD and nonlinear effects are then smaller in the POWD. The nonlinearity is much more marked when a lot of energy has the possibility to cross through the layer (case of Fig. 9a).

Fig. 9c shows the harmonic decomposition, as a function of the space coordinate, for the L6 layer with the amplitude excitation p0 ¼ 5  103 Pa. One can see how the highest harmonics increase when the wave passes through the cloud. The value of the bubble density gradient at the interface between the PRWD and the layer is very important and strongly determines the shielding effect of the layer (it does not only depend on the existence of resonances in the layer and the PRWD). For layers or clouds appearing in practical applications, this tool may be used to assess the influence of the bubble density gradient on the shielding. One-layer partial conclusions:  Then, from the results seen in Sections 3.1–3.4, it seems that in the linear or nonlinear cases, the preponderant parameter is the resonance of the PRWD. According to this parameter, the screen effect is present or not. Indeed:

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Fig. 8. Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours). Left column: linear case; right column: nonlinear case. Top sketch: definition and position of the layer. Waveforms at the points given by the red lines. (a) L5 layer with resonant water domain. (b) L5 layer with non-resonant water domain. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 8 (continued)

j

when the PRWD is resonant, acoustic energy is trapped (a standing wave is created there and pressures increase) and does not enter much into the layer (resonant or not) and the POWD;

j

when the PRWD is not resonant, energy is transmitted to the layer, and there, the behaviour depends on the resonance or non-resonance of the layer.

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Fig. 9. Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours). Left column: linear case; right column: nonlinear case. Top sketch: definition and position of the layer. (a) L6 layer. (b) L7 layer. (c) Harmonic decomposition for the L6 layer at p0 ¼ 5  103 Pa (space coordinate and frequency: horizontal axis; pressure amplitude: vertical axis).

 Anyway, the nonlinearity is getting more important when the bubbly layer is bigger and, in the POWD, when energy that crosses through the layer is more important.

 For any of the precedent domains (Sections 3.1–3.4), if the layer is resonant, energy is trapped therein, more energy is given to bubbles, and more nonlinear is the behaviour of the

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Fig. 10. SL2 layers. Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours) with zoom. Left column: linear case; right column: nonlinear case. Top sketch: definition and position of the layers.

wave. However, the dispersive character of the medium can strongly act against nonlinearity. The harmonics are in general not resonant, and then less energetic. Moreover, these harmonics approach the bubble resonance and are affected by attenuation which follows a particular dependence on frequency. Distortion can then be small.  The nonlinear distortion strongly depends on the size of the layer, but also on its position, because of energy that reaches the layer after the reflections. Two factors participate in this sense: the wider is the layer, higher is the nonlinear distortion; if the layer is resonant, more energy is trapped.  The nonlinearity is more important beyond the layer when more energy can cross the layer.  The shielding effect of the layer is strongly determined by the bubble density gradient at the left interface.

3.6. Effect of several resonant layers In Section 3.6, we suppose the existence of several consecutive layers in water. Two layers SL2 are firstly considered (Fig. 10), each one at: (i) xp ¼ 0 m and (iii) klf =2. The left pattern represents the linear case (global space results (top) and zoom (bottom)) and the right pattern represents the nonlinear case. Three layers SL3 are secondly considered (Fig. 11), each one at: (i) xp ¼ 0 m, (iii) klf =2, and (iv) klf . Left patterns correspond to the linear case (from top to bottom: acoustic pressure (global space and zoom), bubble volume variation (last row)) while right patterns correspond to the nonlinear case. The presence of several layers induces the rising of the complexity of the acoustic pressure pattern. The resonance of the layers appears clearly in the figures.

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Fig. 11. SL3 layers. Acoustic pressure field (space coordinate: horizontal axis; time: vertical axis; amplitude: colours) with zoom, and bubble volume variation field. Left column: linear case; right column: nonlinear case. Top sketch: definition and position of the layers.

Nonlinear effects are cumulative when several layers are present, and the signal obtained after the last layer is very nonlinear. They affect the bubble volume variation as well. With several layers, nonlinear attenuation is extremely much higher (it plays a preponderant role in the very strongly nonlinear cases) than in the linear case (the main phenomenon in this case is

reflection). This result demonstrates the possibility for creating efficient acoustic filters for high amplitudes by means of the collocation of bubbly layers. It may be difficult to construct big and efficient resonators because, due to Bjerknes forces, bubbles gather together and form thin layers [34] which could create screens when the acoustic field intensity rises.

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4. Conclusions This paper deals with the nonlinear propagation of ultrasonic waves in mixtures of air bubbles in water, with nonhomogeneous bubble density, i.e. layers or clouds. The acoustic field and bubbles vibration are coupled by means of a set of differential equations. The SNOW-BL code solves this problem in the time domain. The effect of the layers on the continuous ultrasonic waves and bubble vibrations is analysed, especially the nonlinear ones. Different positions, densities, sizes, and shapes of the layers generate several behaviours. These features depend on the resonance of the layers, of the domain between the source and the layers, and on the intensity of the pressure wave. Nonlinear screen effects are studied. The preponderant parameter is the resonance of the PRWD, and depending on this parameter, the screen effect is present or not. However, nonlinearity is more important when the bubbly layer is wider and even more when it is resonant. But the dispersive character of the medium can act against nonlinearity, especially when nonlinear harmonics are near the bubble resonance. Nonlinear distortion also depends on the position of the layer. Comparisons with effective parameters of homogeneous density and water are established. Several layers cumulate nonlinear effects. Another important effect is that the shielding effect of the layer is strongly determined by the bubble density gradient at its left interface. Acknowledgements This work is supported by the research projects DPI2005-00894, CM-URJC-CEF-091-4-M263, HA2005-0151, and DPI2008-01429. References [1] T.J. Mason, J.P. Lorimer, Applied Sonochemistry: The Uses of Power Ultrasound in Chemistry and Processing, Wiley-VCH, Weinheim, 2002. [2] M.F. Hamilton, D.T. Blackstock (Eds.), Nonlinear Acoustics, Academic Press, San Diego, 1998. [3] S.G. Kargl, J. Acoust. Soc. Am. 111 (2002) 168–173. [4] Y. Matsumoto, J.S. Allen, S. Yoshizawa, T. Ikeda, T. Kaneko, Exp. Therm. Fluid Sci. 29 (2005) 255–265. [5] C.C. Church, E.L. Carstensen, Ultrasound Med. Biol. 27 (2006) 1435–1437.

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