⇦Chinese Physics Letters, 2017, Vol. 34, No. 3, Article code 034302⇨ Localization in an Acoustic Cavitation Cloud * Bo-Ya Miao(苗博雅), Yu An(安宇)** Affiliations Department of Physics, Tsinghua University, Beijing 100084 Received 7 December 2016 ^*Supported by the National Natural Science Foundation of China under Grant No 11334005, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No 20120002110031, and the Tsinghua Fudaoyuan Foreigh Visiting Support Project.
^**Corresponding author. Email: anyuw@mail.tsinghua.edu.cn Citation Text: Miao B Y and An Y 2017 Chin. Phys. Lett. 34 034302 Abstract Using a nonlinear sound wave equation for a bubbly liquid in conjunction with an equation for bubble pulsation, we theoretically predict and experimentally demonstrate the appearance of a gap in the frequency spectrum of a sound wave propagating in a cavitation cloud comprising bubbles. For bubbles with an ambient radius of 100 μm, the calculations reveal that this gap corresponds to the phenomenon of sound wave localization. For bubbles with an ambient radius of 120 μm, this spectral gap is related to a forbidden band of the sound wave. In the experiment, we observe the predicted gap in the frequency spectrum in soda water. However, in tap water, no spectral gap is present because the bubbles are much smaller than 100 μm. DOI:10.1088/0256-307X/34/3/034302 PACS:43.25.Yw, 43.35.Ei, 72.15.Rn, 73.20.Fz © 2017 Chinese Physics Society Article Text When a high-amplitude ultrasonic wave passes through a liquid, many tiny bubbles are generated that violently and nonlinearly pulsate, undergoing periodic expansion and compression. These bubbles form a 'cavitation cloud', which can display various patterns and also emit faint glow (i.e., sonoluminescence).[1,2] Many cavitation phenomena arise through the nonlinear response of oscillating bubbles to the sound field. Here we reveal a new phenomenon caused by the nonlinearity of the ultrasonic wave itself. Although this phenomenon has not been predicted or observed previously in a cavitation cloud, it may be related to sound wave localization. Sound wave localization has been observed in phonon crystals.[3] Local resonance in certain media can also block the transmission of photons or phonons at particular frequencies,[4] resulting in photon or phonon localization. Similarly, numerical simulation of sound wave propagation in a bubbly liquid predicted a forbidden band slightly above the Minnaert resonance frequency[5] because of multiple scattering by bubbles.[6-8] However, to date, there has been no report on the experimental observation of such a phenomenon in real bubbly liquid. In the present work, we theoretically predict the frequency-dependent transmission of sound waves through a cavitation cloud, and then validate these predictions experimentally by observing cavitation clouds in tap and soda water. In a bubbly liquid such as soda water, the cavitation bubbles are usually much larger and the bubble number density is much higher than that in regular tap water. Using mean field theory of an intense ultrasonic wave propagating in a bubbly liquid, under the assumption that all bubbles have an ambient radius of 100 μm, we numerically predict a gap in the frequency spectrum. After trying many different liquids, we observe the predicted phenomenon in soda water. The experimental setup is simple, comprising a cylindrical ultrasonic horn tip with a diameter of 2 cm (UH-800 A, Autoscience Instrument Co. Ltd.) that was slightly immersed into liquid. A hydrophone (CS-5) was placed in the liquid at a fixed distance from the horn tip to measure the acoustic signals, which was displayed on a Tektronix DPO 2024 oscilloscope. Although the frequency of the ultrasonic wave was 20 kHz, nonlinear bubble oscillations in the cavitation cloud generated a broadband frequency signal,[9] equivalent to the range of the frequency sweep used in the present experiment. Through simple analysis of the frequency spectrum, we gained an understanding of the frequency dependence of sound wave transmission in the cavitation cloud. The fast Fourier transforms (FFT) of the measured sound wave signals are shown in Figs. 1(a) and 1(b) for tap water and soda water, respectively. In soda water, although the ultrasound intensity was still high, it was reduced in comparison to that in tap water to ensure the cavitation cloud was confined to the region surrounding the horn tip and to avoid excessive bubble nucleation throughout the liquid. We observed a spectral gap in the low-frequency region for soda water between 40 and 200 kHz (inset in Fig. 1(b)), but no gap was observed for tap water. Thus sound waves with frequencies in this range are not transmitted through the cavitation cloud. To further investigate the differences in the frequency spectrum between tap and soda water, we imaged the cavitation cloud immediately after switching off the ultrasound source using a digital microscope (3R-MSTVUSB273, Anyty). During cavitation, bubbles pulsate violently and move chaotically. When the sound source is switched off, the bubbles exist in a transient steady state at their ambient size. From many images, we estimated the ambient bubble radius. As shown in Fig. 2, the average ambient radii in soda water and tap water were 0.10 and 0.020 mm, respectively. The larger bubble size and bubble number density in soda water indicate that the spectral gap is unlikely to be predicted[6-8] using the multiple scattering theory. In the multiple scattering theory, neither the intensity of the ultrasonic field nor the bubble size are important. However, the conditions of the experiment are close to those set in our numerical calculation. Moreover, the experimentally observed results are similar to our prediction by numerically solving the nonlinear sound wave equation in the cavitation cloud.

Fig. 1. Measured frequency spectrum of cavitation clouds in (a) tap water and (b) soda water. The spectrum was obtained by FFT of the time-domain signal of 200 acoustic cycles at the edge of the cavitation cloud. For soda water, a large spectral gap is present from 40 to 200 kHz. The peaks around 250 and 750 kHz are perturbation signals due to the hydrophone's resonant response. The sensitivity of the hydrophone decreases at high frequency and no correction was made for this in the present measurement.

Fig. 2. Microscopic snapshots in (a) tap water and (b) soda water captured by a portable digital microscope. The scale units are millimeters. The ambient bubble radius in tap water and soda water are about 0.020 and 0.10 mm, respectively. In addition, the bubble number density in soda water is higher than that in tap water.

The nonlinear sound wave equation in a bubbly liquid[10] is $$\begin{align} \nabla ^2p-\frac{1}{c_{\rm l}^2}\frac{\partial ^2p}{\partial t^2}=-4\pi \rho _{\rm l} N(2R\dot {R}^2+R^2\ddot {R}),~~ \tag {1} \end{align} $$ where $p$ is the acoustic pressure, $\rho _{\rm l}$ is the liquid density, $c_{\rm l}$ is the sound speed in the liquid at ambient temperature and pressure (1 atm), $R$ is the radius of the bubble with the ambient radius $R_{0}$, and $N$ is its number density. The individual bubble radius can be calculated by a following modified equation[11] $$\begin{align} &(1-M)R\ddot {R}+\frac{3}{2}\Big(1-\frac{M}{3}\Big)\dot {R}^2\\ =\,&(1+M)\frac{p_{\rm l} -p}{\rho _{\rm l}}+\frac{t_R}{\rho _{\rm l}}\dot {p}_{\rm l},~~ \tag {2} \end{align} $$ where $M\equiv \dot {R}/c_{\rm l}$ is the bubble-wall Mach number, $p_{\rm l} =p_{\rm g}(R,t)-4\eta \dot {R}/R-2\sigma /R$ is the pressure on the liquid side of the bubble wall, $p_{\rm g}(R, t)$ is the pressure on the gas side of the bubble wall, $\eta$ is the shear viscosity, $\sigma$ is the surface tension coefficient of the liquid, and $t_{R}\equiv R/c_{\rm l}$. The isothermal process of the gas inside the bubble is a good approximation, i.e., the pressure inside a bubble is assumed as $p_{\rm g} =\frac{\mu _{\rm g} \bar {R}T}{V_R -b}+p_{\rm v}$, where $\mu_{\rm g}$ is the gas mole number, $\bar {R}$ is the gas constant, $T$ is ambient temperature, $V_{R}$ is the bubble volume, $b$ is the van der Waals hard core volume, and $p_{\rm v}$ is the vapor pressure of the surrounding liquid. Using Eqs. (1) and (2), we can describe how a high-amplitude ultrasonic wave produces a cavitation cloud from a bubbly liquid, and how the sound wave propagates through the cloud.[9] For simplicity, we have assumed that all bubbles are identical and homogeneously distributed in the liquid. Because the horn tip (i.e., the ultrasonic source) is cylindrical, we use an axisymmetric approximation to simplify the calculation. The driving frequency and geometric dimensions of the sound source are the same as those used in the experiment. According to the numerical simulation, under the condition that the number density of bubbles $N$ is about 1$\times$10$^{9}$ m$^{-3}$, the gap in the frequency spectrum is only pronounced when the ambient bubble radius is about 80–120 μm. If the ambient bubble radius is smaller than 80 μm, a less pronounced frequency gap might be observed in experiments. Thus the smaller the bubble size is, the less clear the gap in the frequency spectrum is. For ambient radii greater than 120 μm, as the bubble size increases, the gas fraction increases and the bubbly liquid becomes closer to a gas phase. As a consequence, all frequencies were blocked by the cavitation cloud with the blocking effect becoming more pronounced as the bubble size increased. A typical frequency spectrum for an ambient radius of 20 μm is shown in Fig. 3(a), which may represent the case of tap water. No gap is observed, similar to the experimental observation. If we assume that the ambient bubble radius is 100 μm and the number density of bubbles $N$ is $1 \times10^{9}$ m$^{-3}$ (Fig. 3(b)), a gap appears in approximately the same frequency range (i.e., 40–200 kHz) as the experimental observation shown in Fig. 1(b). The Minnaert resonance frequency, which can be calculated by $f=\frac{1}{2\pi a}({\frac{3\gamma p_0}{\rho}})^{1/2}$, is about 30 kHz for an ambient radius of 100 μm. Thus the gap we observed is different from that predicted by the multiple scattering theory. By contrast, the nonlinear sound wave equation considers the secondary radiation of bubbles, which is different from multiple scattering.

Fig. 3. Calculated frequency spectrum 31 mm from the surface of the horn tip at the edge of the cavitation cloud. The bubble number density $N$ is $1\times10^{9}$ m$^{-3}$ and the ambient bubble radii are (a) 20 μm and (b) 100 μm. The data of 200 acoustic cycles are used.

Fig. 4. Spatial distribution of the sound pressure along the symmetrical axis of the horn tip corresponding to the three points labeled in Fig. 3(b). (a) Point A 20 kHz, (b) point B 180 kHz, and (c) point C 1050 kHz.

In general, a spectral gap in the low-frequency region signifies the occurrence of wave localization.[4] To investigate the frequency dependence of ultrasonic wave transmission, we select three different frequencies that are labeled by A, B, and C in Fig. 3(b), for which we calculate the spatial distribution of the sound pressure along the symmetrical axis of the horn tip. In the calculation, the thickness of the cavitation cloud is assumed to be 30 mm. Point A (20 kHz) is the fundamental frequency of the ultrasonic source, and the sound pressure corresponding to this frequency as a function of the distance to the horn tip surface is shown in Fig. 4(a). Most of the acoustic energy at 20 kHz is confined within the cavitation cloud. At distances greater than 30 mm (i.e., outside the cavitation cloud), the sound pressure decreases, meaning that some ultrasonic waves at 20 kHz are transmitted through the cavitation cloud. The sound pressure as a function of the distance from the surface of the horn tip corresponding to point B in Fig. 3(b) is shown in Fig. 4(b). Point B corresponds to 180 kHz, which is within the spectral gap. At this frequency, the sound wave is localized within the cavitation cloud and is not transmitted. Point C is in the region of broadband noise. Sound waves at this frequency are transmitted through the cavitation cloud with slight attenuation (Fig. 3(c)). Thus the gap in the frequency spectrum corresponds to the localization of the sound wave. However, this argument is valid for 100-μm bubbles driven by an intense ultrasonic wave, but not for the weak wave case.

Fig. 5. The frequency spectrum (a) and spatial distribution (b) observed at $z=30$ mm point under very weak broadband frequency driving. The driving pressure consists of continuous frequencies from 10 kHz to 2 MHz (100 Hz every interval) with equal amplitude ($p_{\rm a}=0.001$ bar) to simulate the broadband frequencies. The radius of cavitation cloud bubble and number density are $R_{0}=100$ μm, $N=10^{9}$ m$^{-3}$, respectively.

For a weak driving case, we apply a cylindrical ultrasonic horn tip with diameter of 2 cm to emit very weak ($p_{\rm a}=0.001$ bar) sound wave consisting of broadband frequencies from 10 kHz to 2 MHz (100 Hz every interval) into the bubbly liquid with 100-μm 10$^{9}$ m$^{-3}$ bubbles in calculation. The calculated results show that the spectral gap appears as in the case that the intense sound wave is applied (see Fig. 5(a)). However, for the frequency at the gap (we choose 150 kHz), the sound wave in the bubbly liquid decays monotonically as the penetrating depth (see Fig. 5(b)), which means that the gap is more likely the forbidden band rather than the sound wave localization. In this mean, we may deduce that the sound wave localization in the case of Fig. 4(b) is the effect of the nonlinearity of the violent bubble pulsation.

Fig. 6. (a) Calculated frequency spectrum 31 mm from the surface of the horn tip at the edge of the cavitation cloud. In the calculation, the bubble number density $N$ is 1$\times$10$^{9}$ m$^{-3}$ and the ambient bubble radius is 120 μm. The data for 200 ultrasonic cycles are used. (b) Spatial distribution of the sound pressure along the symmetrical axis of the horn tip corresponding to point A in panel (a).

Fig. 7. (a) Calculated frequency spectrum 31 mm from the surface of the horn tip surface at the edge of the cavitation cloud. In the calculation, we consider the bubbles with different ambient radii: $R_{0}=100$ μm with number density $N_{100\,μ{\rm m}}$ of $9.5\times10^{8}$ m$^{-3}$ and $R_{0}=120$ μm with $N_{120\,μ{\rm m}}=0.5\times10^{8}$ m$^{-3}$. The data of 200 ultrasonic cycles are used. (b) Spatial distribution of the sound pressure along the symmetrical axis of the horn tip corresponding to point A in panel (a).

For an ambient radius of 120 μm, a spectral gap can still be observed, as shown in Fig. 6(a). For point A in Fig. 6(a), which is within the spectral gap, the sound pressure as a function of distance from the surface of the horn tip is shown in Fig. 6(b). Numerical fitting of the curve reveals that the sound pressure decays exponentially. Thus sound waves at frequencies within the spectral gap cannot propagate in the cavitation cloud. This also means that the spectral gap observed for 120-μm bubbles with a number density $N$ of $1\times10^{9}$ m$^{-3}$ probably corresponds to a forbidden band of the sound wave. The question therefore arises whether the spectral gap observed in Fig. 1(b) is due to the localization described in Fig. 4(b) or to the sound wave being blocked by the cavitation cloud, as shown in the case in Fig. 6(b). In reality, cavitation clouds comprise bubbles with different ambient radii. For simplicity, we perform the calculations for a mixture of 95% 100-μm bubbles and 5% 120-μm bubbles, the results are shown in Fig. 7. The frequency spectrum is very similar to that of 120-μm bubbles (Fig. 6(a)). However, for the point in the spectral gap, the spatial distribution of the sound pressure along the symmetrical axis of the horn tip no longer decays exponentially; rather, it fluctuates, which reflects wave localization. This mixed bubble case may represent more closely the experimental conditions than the case with bubbles of a single size. If the calculations are performed for bubbles with a distribution of sizes, the calculated result should match the experimental observation even more closely. In summary, we have employed a theoretical method based on a nonlinear ultrasonic wave equation and a bubble dynamics equation to simulate sound wave propagation in a cavitation cloud. The calculations predict a gap in the frequency spectrum for bubbles with an ambient radius of about 100 μm. Experimentally, in soda water, we observe the predicted phenomenon. Further calculations reveal that for a cavitation cloud comprising 100-μm bubbles, the spectral gap is due to sound wave localization. Since the bubbles pulsate violently in cavitation cloud, this localization is the effect of strong nonlinearity. By contrast, for 120-μm bubbles, the spectral gap is attributed to forbidden bands that prevent sound wave propagation. In the experiment, it is possible that the bubbles have a wide distribution of ambient radii and that the spectral gap results from sound wave localization. In contrast to that in soda water, the cavitation cloud in tap water consists of bubbles much smaller than 100 μm; therefore, in tap water, no spectral gap is observed. References Bubble dynamics, shock waves and sonoluminescenceInhibited Spontaneous Emission in Solid-State Physics and ElectronicsLocally Resonant Sonic MaterialsXVI. On musical air-bubbles and the sounds of running waterAcoustic Localization in Bubbly Liquid MediaAir Bubbles in Water: A Strongly Multiple Scattering Medium for Acoustic WavesAcoustic localization in weakly compressible elastic media containing random air bubblesNonlinear bubble dynamics of cavitationBubble oscillations of large amplitude

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