Theoretical prediction of the scattering of spherical bubble clusters under ultrasonic excitation

Due to the nonlinear vibration of ultrasound contrast agent bubbles, a nonlinear scattered sound field will be generated when bubbles are driven by ultrasound. A bubble cluster consists of numerous bubbles gathering in a spherical space. It has been noted that the forward scattering of a bubble cluster is larger than its backscattering, and some studies have experimentally found the angular dependence of a bubble cluster’s scattering signal. In this paper, a theory is proposed to explain the difference of acoustic scattering at different directions of a bubble cluster when it is driven by ultrasound, and predicts the angular distribution of scattered acoustic pressure under different parameters. The theory is proved to be correct under circumstances of small clusters and weak interactions by comparing theoretical results with numerical simulations. This theory not only sheds light on the physics of bubble cluster scattering, but also may contribute to the improvement of ultrasound imaging technology, including ultrasonic harmonic imaging and contrast-enhanced ultrasonography.     

超声波作用下泡群的共振声响应

从球状泡群气泡动力学方程出发, 考虑泡群间次级声辐射的影响, 得到了声场中两泡群共同存在时气泡振动的动力学方程, 并以此为基础探讨声波驱动下双泡群振动系统的共振响应特征. 由于泡群间气泡间的相互作用, 系统存在低频共振和高频共振现象, 两不同共振频率的数值与泡群内气泡的本征频率相关. 泡群内气泡的本征频率又受到初始半径、泡群大小和泡群内气泡数量的影响. 气泡自由振动和驱动声波的耦合激起泡群内气泡的受迫振动, 气泡初始半径、气泡数密度和驱动声波频率等都会影响泡群内气泡的振动幅值和初相位. 关键词： 气泡群 共振 声响应 超声空化     

球状泡群内气泡的耦合振动

振动气泡形成辐射场影响其他气泡的运动, 故多气泡体系中气泡处于耦合振动状态. 本文在气泡群振动模型的基础上, 考虑气泡间耦合振动的影响, 得到了均匀球状泡群内振动气泡的动力学方程, 以此为基础分析了气泡的非线性声响应特征. 气泡间的耦合振动增加了系统对每个气泡的约束, 降低了气泡的自然共振频率, 增强了气泡的非线性声响应. 随着气泡数密度的增加, 振动气泡受到的抑制增强; 增加液体静压力同样可抑制泡群内气泡的振动, 且存在静压力敏感区(1–2 atm, 1 atm=1.01325×10⁵ Pa); 驱动声波对气泡振动影响很大, 随着声波频率的增加, 能够形成空化影响的气泡尺度范围变窄. 在同样的声条件、泡群尺寸以及气泡内外环境下, 初始半径小于5 μm 的气泡具有较强的声响应. 气泡耦合振动会削弱单个气泡的空化影响, 但可延长多气泡系统空化泡崩溃发生的时间间隔和增大作用范围, 整体空化效应增强.     

含气泡液体中气泡振动的研究

研究了含气泡液体中单个气泡在驱动声场一定情况下的振动过程. 让每次驱动声场作用的时间特别短, 使气泡半径发生微小变化后再将其变化反馈到气泡群对驱动声场的散射作用中去, 从而可以得到某单个气泡周围受气泡散射影响后的声场, 接着再让气泡在该声场作用下做短时振动, 如此反复. 通过这样的方法, 研究了液体中单个气泡的振动情况并对其半径变化进行了数值模拟, 结果发现, 在液体中含有大量气泡的情况下, 某单个气泡的振动过程明显区别于液体中只有一个气泡的情况. 由于大量气泡和驱动声场的相互作用, 使气泡半径的变化存在多种不同的振动情况, 在不同的气泡大小和含量的情况下, 半径变化过程分别表现为: 在平衡位置附近振荡的过程; 周期性的空化过程; 一次空化过程后保持某一大小振荡的过程; 增长后维持某一大小振荡的过程等. 所以, 对于含气泡液体中气泡振动的研究, 在驱动声场一定的情况下, 必须考虑气泡含量的因素. 关键词： 含气泡液体 超声空化 散射 数值模拟     

非球形效应对强声场中次Bjerknes力的影响

考虑了非球形气泡在声场中的形状振动,推导了非球形气泡和球形气泡之间的次Bjerknes力方程,数值模拟了声场中非球形气泡和球形气泡之间的次Bjerknes力和两个球形气泡之间的次Bjerknes力,并对非球形气泡和球形气泡之间的次Bjerknes力的影响因素进行了分析讨论.研究结果表明:当驱动声压振幅大于非球形气泡的Black阈值且又能使得非球形气泡稳定振动时,在第一个声驱动周期内,非球形气泡和球形气泡之间的次Bjerknes力和两个球形气泡的次Bjerknes力方向差异较大,在大小上是两个球形气泡次Bjerkens力的数倍,且有着更长的作用距离.非球形气泡和球形气泡之间的次Bjerknes力取决于非球形气泡的形状模态、两个气泡初始半径的比值、驱动声压振幅、气泡间距和两个气泡的相对位置.     

Oscillation and Migration of Bubbles within Ultrasonic Field

The oscillation and migration of bubbles within an intensive ultrasonic field are important issues concerning acoustic cavitation in liquids.We establish a selection map of bubble oscillation mode related to initial bubble radius and driving sound pressure under 20 kHz ultrasound and analyze the individual-bubble migration induced by the combined effects of pressure gradient and acoustic streaming.Our results indicate that the pressure threshold of stable and transient cavitation decreases with the increasing initial bubble radius.At the pressure antinode,the Bjerknes force dominates the bubble migration, resulting in the large bubbles gathering toward antinode center,whereas small bubbles escape from antinode.By contrast,at the pressure node,the bubble migration is primarily controlled by acoustic streaming,which effectively weakens the bubble adhesion on the container walls,thereby enhancing the cavitation effect in the whole liquid.     

Interaction of a bubble and a bubble cluster in an ultrasonic field

Using an appropriate approximation, we have formulated the interacting equation of multi-bubble motion for a system of a single bubble and a spherical bubble cluster. The behavior of the bubbles is observed in coupled and uncoupled states. The oscillation of bubbles inside the cluster is in a coupled state. The numerical simulation demonstrates that the secondary Bjerknes force can be influenced by the number density, initial radius, distance, driving frequency, and amplitude of ultrasound. However, if a bubble approaches a bubble cluster of the same initial radii, coupled oscillation would be induced and a repulsive force is evoked, which may be the reason why the bubble cluster can exist steadily. With the increment of the number density of the bubble cluster, a secondary Bjerknes force acting on the bubbles inside the cluster decreases due to the strong suppression of the coupled bubbles. It is shown that there may be an optimal number density for a bubble cluster which can generate an optimal cavitation effect in liquid for a stable driving ultrasound.     

气泡线性振动时近海面气泡群的声散射

海洋中的不同成因的气泡群是常见的水下声学目标及声呐混响源,因此对水下气泡群进行声学建模意义重大。利用有效媒质理论描述气泡群内部的相速度及声衰减变化,并考虑到海洋中气泡群往往产生于不同界面附近,进一步利用球面波叠加原理描述海面对气泡群散射声波的再辐射,导出了平海面作用下气泡群声散射截面的一般表达式,建立了其声散射模型,研究了单一尺寸及混合尺寸气泡群的声学特性。数值分析表明,气泡群的谐振频率会随其半径或孔隙率增加而降低;由于海面的存在,气泡群声散射截面会随频率进行周期性变化,且随气泡群远离海面,这一变化逐渐加剧。此外,若气泡的黏滞阻尼项在全部阻尼项中占比较高,气泡群声散射强度会在谐振频率附近存在起伏振荡。该模型可为近海面鱼群、气泡羽流及海底泄漏的甲烷气体的声学建模提供一定的理论基础。     

Interaction of two bubbles with distortion in an acoustic field

Expression of the secondary Bjerknes force of two bubbles is obtained by considering the distrotion of two bubbles. The secondary Bjerknes forces in different acoustic fields are simulated, and the influence factors are analyzed and discussed. It is shown that the distortion of a bubble has an important influence on the interaction of two bubbles. The strength and even the directions of the secondary Bjerknes force of two bubbles with distortion differ considerably from the predictions of the sherical symmetry theory. The results show that when two bubbles oscillated stably in an acoustic field, the secondary Bjerknes force of two bubbles with distortion is several times more than that of two spherical bubbles in the same condition. The secondary Bjerknes force of two bubble with distortion has more interaction distance than that of two spherical bubbles. The secondary Bjerknes force of two bubbles with distortion depends on the distance of two bubbles, the shape mode of two bubbles, the equilibrium radii of two bubbles and the driving acoustic filed. The nonspherical distortion effects of the secondary Bjerknes has an importance on understanding the structure formation of bubbles and evolution process of bubble group in an acoustic field.     

耦合双泡声空化特性的理论研究

当双泡中心间距足够小时,由于气泡间辐射压力波的存在,作用在气泡上的压力不等于外部驱动压力.通过考虑双泡之间的辐射压力波,利用改进的Keller-Miksis方程,分别计算了不同大小、不同间距、含不同惰性气体的双泡在声空化过程中半径的变化、次Bjerknes力的变化和双泡内温度的变化.计算结果表明,当双泡大小不同时,小气泡受到的抑制作用较强,温度变化也比较大.随着双泡间距离从100μm增大到1 cm时,气泡间的次Bjerknes力的数量级从10~(-4)N减小到10~(-8)N.含不同惰性气体的耦合双泡在回弹阶段表现出明显不同的振荡规律.     

方波驱动下双气泡的动力学行为*

探索方波驱动下双气泡的脉动规律,能够促进方波在声空化工程中的实际应用。本文通过数值求解双气泡耦合方程组,研究了方波驱动下双气泡的动力学行为,得到了多种条件下不同时刻两个气泡半径的数值,并以此计算出气泡间的次Bjerknes力。研究表明,增大驱动频率会使得两个气泡膨胀时能达到的最大半径和次Bjerknes力减小。当两个气泡的平衡半径不同时,其中一个气泡的剧烈收缩会使得另一个气泡产生一个振动方向相反的声脉冲。随着两个气泡平衡半径差距的增加,气泡收缩的时间间隔增大。此外,当驱动声压幅值逐渐增大时,气泡脉动规律也会发生很大的变化。     

The effects of coupling and bubble size on the dynamical-systems behaviour of a small cluster of microbubbles

This study endeavours to apply a theoretical model for predicting the dynamics of a bubble cluster of various sizes, within which each bubble may assume different initial conditions from other bubbles in the cluster. The resulting system of coupled Keller-Miksis-Parlitz equations are solved numerically, and the effects of coupling and bubble size on bubble cluster dynamics are examined for a given set of ultrasound parameters. It has been found that the effects of coupling are significant, and a bubble cluster's bifurcation characteristics and route to chaos can be altered by inter-bubble interactions. This gives rise to the possibility of suppressing the chaotic oscillations of microbubbles by varying bubble cluster size. Small equilibrium radii bubbles have little influence on the dynamics of neighbouring bubbles in a cluster via coupling. Furthermore, a bubble system consisting of smaller-sized bubbles transitions from order to chaos at lower driving pressure amplitudes.     

Ultrasonic scattering cross sections of shell-encapsulated gas bubbles immersed in a viscoelastic liquid: first and second harmonics

The oscillations of gas bubbles, without shell, immersed in viscoelastic liquids and driven by an acoustic wave have been the subject of several investigations. They demonstrate that the viscosity coefficient and the spring constant of the liquid have significant influence on the scattering cross section of the gas bubble. For shell-encapsulated gas bubbles, the investigations have been concentrated to bubbles immersed in a pure viscous liquid. This present work computes the ultrasonic scattering cross section, first and second harmonics, of shell-encapsulated gas bubbles immersed in a viscoelastic liquid. The theoretical model of the bubble oscillation is based on the generalized Rayleigh-Plesset equation of motion of a spherical cavity immersed in a viscoelastic liquid represented by a three-parameter linear Oldroyd model. The scattering cross section is computed for Albunex type of bubble (shell thickness=15 nm, shell shear viscosity=1.77 Pas, shell modulus of rigidity=88.8 MPa) irradiated by a 3.5 MHz ultrasonic pressure wave with an amplitude of 30 kPa. The results demonstrate that encapsulated bubbles respond independently of the surrounding liquid being pure viscous or viscoelastic as long as the surrounding liquid shear viscosity is as low as 10(-3) Pas. Nevertheless, for higher shear viscosities, the bubble responds differently if the surrounding liquid is pure viscous or viscoelastic. In general, the scattering cross sections of first and second harmonics are larger for the viscoelastic liquid.     

Enhancement of oscillation amplitude of cavitation bubble due to acoustic wake effect in multibubble environment

Acoustic cavitation occurs in ultrasonic treatment causing various phenomena such as chemical synthesis, chemical decomposition, and emulsification. Nonlinear oscillations of cavitation bubbles are assumed to be responsible for these phenomena, and the neighboring bubbles may interact each other. In the present study, we numerically investigated the dynamic behavior of cavitation bubbles in multi-bubble systems. The results reveal that the oscillation amplitude of a cavitation bubble surrounded by other bubbles in a multi-bubble system becomes larger compared with that in the single-bubble case. It is found that this is caused by an acoustic wake effect, which reduces the pressure near a bubble surrounded by other bubbles and increases the time delay between the bubble contraction/expansion cycles and sound pressure oscillations. A new parameter, called “cover ratio” is introduced to quantitatively evaluate the variation in the bubble oscillation amplitude, the time delay, and the maximum bubble radius.     

两种气泡混合的声空化

将非线性声波方程和改进的Rayleigh-Plesset方程联立可以描述空化环境中的声场及相应的气泡动力学特征. 用时域有限差分方法模拟了圆柱形容器内两种气泡相互混合时的空化情况. 在烧杯内的稳态背景声场形成过程中, 瓶壁耗散吸收扮演了重要的角色. 在稳态背景声场的基础上, 分析了混合气泡与声场的相互作用、气泡之间的相互作用、混合情况下的频谱特性. 结果表明: 两种气泡平衡半径都不太大时, 气泡与声场的相互作用不强, 声场及气泡的行为也比较规律; 相反, 当其中一种气泡平衡半径相对比较大时, 声场与气泡具有较强的非线性相互作用, 声场及气泡的行为表现出复杂的特性.     

Interactions of bubbles in acoustic Lichtenberg figure

The evolution of acoustic Lichtenberg figure (ALF) in ultrasound fields is studied using high-speed photography. It is observed that bubbles travel along the branch to the aggregation region of an ALF, promoting the possibility of large bubble or small cluster formation. Large bubbles move away from the aggregation region while surrounding bubbles are attracted into this structure, and a bubble transportation cycle arises in the cavitation field. A simplified model consisting of a spherical cluster and a chain of bubbles is developed to explain this phenomenon. The interaction of the two units is analyzed using a modified expression for the secondary Bjerknes force in this system. The model reveals that clusters can attract bubbles on the chain within a distance of 2 mm, leading to a bubble transportation process from the chain to the bubble cluster. Many factors can affect this process, including the acoustic pressure, frequency, bubble density, and separation distance. The larger the bubble in the cluster, the broader the attraction region. Therefore, the presence of large bubbles might enhance the process in this system. Local disturbances in bubble density could destroy the ALF structure. The predictions of the model are in good agreement with the experimental phenomena.     

Theoretical estimation of sonochemical yield in bubble cluster in acoustic field

In order to learn more about the physical phenomena occurring in cloud cavitation, the nonlinear dynamics of a spherical cluster of cavitation bubbles and cavitation bubbles in cluster in an acoustic field excited by a square pressure wave are numerically investigated by considering viscosity, surface tension, and the weak compressibility of the liquid.The theoretical prediction of the yield of oxidants produced inside bubbles during the strong collapse stage of cavitation bubbles is also investigated. The effects of acoustic frequency, acoustic pressure amplitude, and the number of bubbles in cluster on bubble temperature and the quantity of oxidants produced inside bubbles are analyzed. The results show that the change of acoustic frequency, acoustic pressure amplitude, and the number of bubbles in cluster have an effect not only on temperature and the quantity of oxidants inside the bubble, but also on the degradation types of pollutants, which provides a guidance in improving the sonochemical degradation of organic pollutants.     

Oscillating bubble concentration and its size distribution using acoustic emission spectra

New method has been proposed for the estimation of size and number density distribution of oscillating bubbles in a sonochemical reactor using acoustic emission spectra measurements. Bubble size distribution has been determined using Minnaert's equation [M. Minnaert, On musical air bubbles and sound of running water, Philanthr. Mag. 16 (1933) 235], i.e., size of oscillating bubble is inversely related to the frequency of its volume oscillations. Decomposition of the pressure signal measured by the hydrophone in frequency domain of FFT spectrum and then inverse FFT reconstruction of the signal at each frequency level has been carried out to get the information about each of the bubble/cavity oscillation event. The number mean radius of the bubble size is calculated to be in the range of 50-80mum and it was not found to vary much with the spatial distribution of acoustic field strength of the ultrasound processor used in the work. However, the number density of the oscillating bubbles and the nature of the distribution were found to vary in different horizontal planes away from the driving transducer surface in the ultrasonic bath. A separate set of experiments on erosion assessment studies were carried out using a thin aluminium foil, revealing a phenomena of active region of oscillating bubbles at antinodal points of the stationary waves, identical to the information provided by the acoustic emission spectra at the same location in the ultrasonic bath.     

Dynamics of a bubble cluster in an acoustic field

A mathematical model describing the dynamics of clustered gas bubbles under the effect of an acoustic field is presented. The proposed model is used as the basis for an analytical study of small bubble oscillations in monodisperse and polydisperse clusters and for a numerical study of nonlinear bubble oscillations under high-amplitude external pressures. The following effects are found to occur in a polydisperse cluster: a synchronization of the collapse phases of bubbles with different radii and a collapse intensification for bubbles of one size in the presence of bubbles of another size. These effects are explained by the interaction between bubbles of different radii in the cluster.     

Investigation of a mutual interaction force at different pressure amplitudes in sulfuric acid

This paper investigates the secondary Bjerknes force for two oscillating bubbles in various pressure amplitudes in a concentration of 95% sulfuric acid.The equilibrium radii of the bubbles are assumed to be smaller than 10 μm at a frequency of 37 kHz in various strong driving acoustical fields around 2.0 bars (1 bar=10 5 Pa).The secondary Bjerknes force is investigated in uncoupled and coupled states between the bubbles,with regard to the quasi-adiabatic model for the bubble interior.It finds that the value of the secondary Bjerknes force depends on the driven pressure of sulfuric acid and its amount would be increased by liquid pressure amplitude enhancement.The results show that the repulsion area of the interaction force would be increased by increasing the driven pressure because of nonlinear oscillation of bubbles.