Numerical simulations of three-dimensional nonlinear acoustic waves in bubbly liquids

This paper presents three-dimensional simulations of nonlinear propagation of ultrasonic waves through bubbly liquids, which represent the continuity of our previous works included in the numerical tool SNOW-BL. The behavior of three-dimensional nonlinear acoustic waves in bubbly liquids is analyzed by means of numerical predictions. Nonlinearity, attenuation, and dispersion due to the presence of bubbles in the liquid are taken into account. The numerical solution to the differential problem is obtained by means of a finite-difference scheme. The simulations we present here consider a homogeneous distribution of bubbles in the liquid. Results compare high and low-amplitude waves to detect the nonlinear effects of the bubbles. Results are shown for radiation and enclosure problems.     

声波在含气泡液体中的线性传播

为了探讨含气泡液体对声波传播的影响, 研究了声波在含气泡液体中的线性传播. 在建立含气泡液体的声学模型时引入气泡含量的影响,建立气泡模型时引用 Keller的气泡振动模型并同时考虑气泡间的声相互作用,得到了经过修正的气泡振动方程. 通过对含气泡液体的声传播方程和气泡振动方程联立并线性化求解,在满足 (ω R₀)/c << 1 的前提下,得到了描述含气泡液体对声波传播的衰减系数和传播速度. 通过数值分析发现,在驱动声场频率一定的情况下,气泡含量的增加及气泡的变小均会导致衰减系数增加和声速减小;气泡的体积分数和大小一定时, 驱动声场频率在远小于气泡谐振频率的情况下,声速会随驱动频率的增加而减小; 气泡间的声相互作用对声波传播速度及含气泡液体衰减系数的影响不明显.最终认为气泡的大小、 数量和驱动声场频率是影响声波在含气泡液体中线性传播的主要因素. 关键词： 含气泡液体 线性声波 声衰减系数 声速     

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     

Acoustic cavitation mechanism: a nonlinear model

During acoustic cavitation process, bubbles appear when acoustic pressure reaches a threshold value in the liquid. The ultrasonic field is then submitted to the action of the bubbles. In this paper we develop a model to analyze the cavitation phenomenon in one-dimensional standing waves, based on the nonlinear code SNOW-BL. Bubbles are produced where the minimum rarefaction pressure peak exceeds the cavitation threshold. We show that cavitation bubbles appear at high amplitude and drastically affect (dissipation, dispersion, and nonlinearity) the ultrasonic field. This paper constitutes the first work that associates the nonlinear ultrasonic field to a bubble generation process.     

A simple model of ultrasound propagation in a cavitating liquid. Part I: Theory, nonlinear attenuation and traveling wave generation

The bubbles involved in sonochemistry and other applications of cavitation oscillate inertially. A correct estimation of the wave attenuation in such bubbly media requires a realistic estimation of the power dissipated by the oscillation of each bubble, by thermal diffusion in the gas and viscous friction in the liquid. Both quantities and calculated numerically for a single inertial bubble driven at 20 kHz, and are found to be several orders of magnitude larger than the linear prediction. Viscous dissipation is found to be the predominant cause of energy loss for bubbles small enough. Then, the classical nonlinear Caflish equations describing the propagation of acoustic waves in a bubbly liquid are recast and simplified conveniently. The main harmonic part of the sound field is found to fulfill a nonlinear Helmholtz equation, where the imaginary part of the squared wave number is directly correlated with the energy lost by a single bubble. For low acoustic driving, linear theory is recovered, but for larger drivings, namely above the Blake threshold, the attenuation coefficient is found to be more than 3 orders of magnitude larger then the linear prediction. A huge attenuation of the wave is thus expected in regions where inertial bubbles are present, which is confirmed by numerical simulations of the nonlinear Helmholtz equation in a 1D standing wave configuration. The expected strong attenuation is not only observed but furthermore, the examination of the phase between the pressure field and its gradient clearly demonstrates that a traveling wave appears in the medium.     

Interaction between moving bubbles and an acoustic field: Controlling flows and nonlinear acoustic vision

Results from studying the interaction between gas bubbles and the field of a flow-through acoustic resonator, and the Raman scattering of acoustic waves by moving bubbles, are presented. The structure of the distribution of bubble concentration in the resonator is studied. It is shown that nonlinear scattering by moving bubbles can be used to image bubble objects.     

Acoustic Localization in Weakly Compressible Elastic Media Permeated with Air Bubbles

The propagation of longitudinal acoustic waves in weakly compressible elastic media permeated with air bubbles is investigated on the basis of the radial pulsation equation of a single bubble. The multiple scattering of waves in such media is rigorously described by using a self-consistent approach. Theoretical results show that there exists strong acoustic localization in a range of frequency slightly above the bubble resonance frequency, even for a very small volume fraction of bubbles. Further study reveals that the localization is in fact attributed to collection behaviour of bubbles, allowing for an efficient cancellation of propagating waves. This is essentially consistent with the known conclusions recently drawn for bubbly liquid by Kou et al. [2003 Appl. Phys. Left. 83 4247]     

Nonlinear increase in bubbles radii caused by sound in a bubbly liquid

The nonlinear interaction of acoustic and entropy modes in a bubbly liquid is considered. The reasons for interaction are both nonlinearity and dispersion. In the field of intense sound, a decrease in the mixture density is predicted. That corresponds to the well-established growth of bubbles volumes due to rectified diffusion. The nonlinear interaction of modes as a reason for a bubble to grow due to sound, is discovered. The example considers variation in the mixture density and bubbles radii caused by acoustic soliton.     

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.     

含气泡液体中的非线性声传播

考虑到分布在液体中的气泡是声波在含气泡液体中传播时引起非线性的一个很重要的因素,本文研究了声波在含气泡液体中的非线性传播.将气体含量的影响引入到声波在液体中传播的方程中,从而得到声波在气液混合物中传播的数学模型.通过对该模型进行数值模拟发现,气体含量、驱动声场声压幅值及驱动声场作用时间均会影响到气液混合物中的声场分布及声压幅值大小.液体中的气泡会"阻滞"液体中声场的传播并将能量"聚集"在声源附近.对于连续大功率的驱动声场来说,液体中的气泡会"阻滞"气液混合物中声场及其能量的传播.     

Nonlinear oscillation and acoustic scattering of bubbles

The scattered acoustic pressure and scattered cross section of bubbles is studied using the scattered theory of bubbles. The nonlinear oscillations of bubbles and the scattering acoustic fields of a spherical bubble cluster are numerically simulated based on the bubble dynamic and fluid dynamic. The influences of the interaction between bubbles on scattering acoustic field of bubbles are researched. The results of numerical simulation show that the oscillation phases of bubbles are delayed to a certain extent at different positions in the bubble cluster, but the radii of bubbles during oscillation do not differ too much at different positions. Furthermore, directivity of the acoustic scattering of bubbles is obvious. The scattered acoustic pressures of bubbles are different at the different positions inside and outside of the bubble cluster. The scattering acoustic fields of a spherical bubble cluster depend on the driving pressure amplitude, driving frequency, the equilibrium radii of bubbles, bubble number and the radius of the spherical bubble cluster. These theoretical predictions provide a further understanding of physics behind ultrasonic technique and should be useful for guiding ultrasonic application.     

A two-dimensional nonlinear model for the generation of stable cavitation bubbles

Bubbles appear by acoustic cavitation in a liquid when rarefaction pressures attain a specific threshold value in a liquid. Once they are created, the stable cavitation bubbles oscillate nonlinearly and affect the ultrasonic field. Here we present a model developed for the study of bubble generation in a liquid contained in a two-dimensional cavity in which a standing ultrasonic field is established. The model considers dissipation and dispersion due to the bubbles. It also assumes that both the ultrasonic field and the bubble oscillations are nonlinear. The numerical experiments predict where the bubbles are generated from a population of nuclei distributed in the liquid and show how they affect the ultrasonic field.     

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.     

A simple model of ultrasound propagation in a cavitating liquid. Part II: Primary Bjerknes force and bubble structures

In a companion paper, a reduced model for propagation of acoustic waves in a cloud of inertial cavitation bubbles was proposed. The wave attenuation was calculated directly from the energy dissipated by a single bubble, the latter being estimated directly from the fully nonlinear radial dynamics. The use of this model in a mono-dimensional configuration has shown that the attenuation near the vibrating emitter was much higher than predictions obtained from linear theory, and that this strong attenuation creates a large traveling wave contribution, even for closed domain where standing waves are normally expected. In this paper, we show that, owing to the appearance of traveling waves, the primary Bjerknes force near the emitter becomes very large and tends to expel the bubbles up to a stagnation point. Two-dimensional axi-symmetric computations of the acoustic field created by a large area immersed sonotrode are also performed, and the paths of the bubbles in the resulting Bjerknes force field are sketched. Cone bubble structures are recovered and compare reasonably well to reported experimental results. The underlying mechanisms yielding such structures is examined, and it is found that the conical structure is generic and results from the appearance a sound velocity gradient along the transducer area. Finally, a more complex system, similar to an ultrasonic bath, in which the sound field results from the flexural vibrations of a thin plate, is also simulated. The calculated bubble paths reveal the appearance of other commonly observed structures in such configurations, such as streamers and flare structures.     

On the physical origin of conical bubble structure under an ultrasonic horn

The cavitation field generated by an ultrasonic horn at low frequency and high power is known to self-organize into a conical bubble structure. The physical mechanism at the origin of this bubble structure is investigated using numerical simulations and acoustic pressure measurements. The thin bubbly layer lying at horn surface is shown to act as a nonlinear thickness resonator that amplifies acoustic pressure and distorts acoustic waveform. This mechanism explains the self-stabilization of the conical bubble structure as well as the generation of shock wave and the focusing at very short distance.     

Numerical simulations of stable cavitation bubble generation and primary Bjerknes forces in a three-dimensional nonlinear phased array focused ultrasound field

We present a model developed for studying the generation of stable cavitation bubbles and their motion in a three-dimensional volume of liquid with axial symmetry under the effect of finite-amplitude phased array focused ultrasound. The density of bubbles per unit volume is determined by a nonlinear law which is a threshold-dependent function of the negative acoustic pressure reached in the liquid, in which nuclei are initially distributed. The nonlinear mutual interaction of ultrasound and bubble oscillations is modeled by a nonlinear coupled differential system formed by the wave and a Rayleigh-Plesset equations, for which both the pressure and the bubble oscillation variables are unknown. The system, which accounts for nonlinearity, dispersion, and attenuation due to the bubbles, is solved by numerical approximations. The nonlinear acoustic pressure field is then used to evaluate the primary Bjerknes force field and to predict the subsequent motion of bubbles. In order to illustrate the procedure, a medium-high and a low ultrasonic frequency configurations are assumed. Simulation results show where bubbles are generated, the nonlinear effects they have on ultrasound, and where they are relocated. Despite many physical restrictions and thanks to its particularities (two nonlinear coupled fields, bubble generation, bubble motion), the numerical model used in this work gives results that show qualitative coherence with data observed experimentally in the framework of stable cavitation and suggest their usefulness in some application contexts.     

Modelling of nonlinear effects and the response of ultrasound contrast micro bubbles: simulation and experiment

The propagation of diagnostic ultrasonic imaging pulses in tissue and their interaction with contrast micro bubbles is a very complex physical process, which we assumed to be separable into three stages: pulse propagation in tissue, the interaction of the pulse with the contrast bubble, and the propagation of the scattered echo. The model driven approach is used to gain better knowledge of the complex processes involved. A simplified way of field simulation is chosen due to the complexity of the task and the necessity to estimate comparative contributions of each component of the process. Simulations are targeted at myocardial perfusion estimation. A modified method for spatial superposition of attenuated waves enables simulations of low intensity pulse pressure fields from weakly focused transducers in a nonlinear, attenuating, and liquid-like biological medium. These assumptions enable the use of quasi-linear calculations of the acoustic field. The simulations of acoustic bubble response are carried out with the Rayleigh-Plesset equation with the addition of radiation damping. Theoretical simulations with synthesised and experimentally sampled pulses show that the interaction of the excitation pulses with the contrast bubbles is the main cause of nonlinear scattering, and a 2-3 dB increase of second harmonic amplitude depends on nonlinear distortions of the incident pulse.     

Nonlinear scattering of acoustic waves by natural and artificially generated subsurface bubble layers in sea

The paper describes nonlinear effects due to a biharmonic acoustic signal scattering from air bubbles in the sea. The results of field experiments in a shallow sea are presented. Two waves radiated at frequencies 30 and 31-37 kHz generated backscattered signals at sum and difference frequencies in a bubble layer. A motorboat propeller was used to generate bubbles with different concentrations at different times, up to the return to the natural subsurface layer. Theoretical consideration is given for these effects. The experimental data are in a reasonably good agreement with theoretical predictions.     

The effect of nearby bubbles on array gain

The coherent processing of signals from multiple hydrophones in an array offers improvements in angular resolution and signal-to-noise ratio. When the array is steered in a particular direction, the signals arriving from that direction are added in phase, and any signals arriving from other directions are not. Array gain (AG) is a measure of how much the signal arriving from the steering direction is amplified relative to signals arriving from all other directions. The subject of this paper is the manner in which the AG of an acoustic array operating in water that contains air bubbles is affected by scattering from nearby bubbles. The effects of bubbles on acoustic attenuation and dispersion are considered separately from their effects on AG. Acoustic measurements made in bubbly water using the AB Wood tank at the Institute of Sound and Vibration Research, University of Southampton, in June 2008 show that as bubble density increases, relative phase shifts in individual hydrophone signals increase and signal correlation among the hydrophones is reduced. A theory and numerical simulation linking bubble density at the hydrophone to the AG is in good agreement with the measurements up to the point where multiple scattering becomes important.