Numerical models for the study of the nonlinear frequency mixing in two and three-dimensional resonant cavities filled with a bubbly liquid

The objective of this work is to develop versatile numerical models to study the nonlinear distortion of ultrasounds and the generation of low-ultrasonic frequency signals by nonlinear frequency mixing in two and three-dimensional resonators filled with bubbly liquids. The interaction of the acoustic field and the bubble vibrations is modeled through a coupled differential system formed by the multi-dimensional wave equation and a Rayleigh-Plesset equation. The numerical models we develop are based on multi-dimensional finite-volume techniques and a time discretization carried out by finite differences. Numerical experiments are performed for complex modes in many different cavities considering different kinds of boundary conditions and taking advantage of the dispersive character of the bubbly fluid to match specific resonances of the cavities. Results show the distribution of fundamental and harmonics for single frequency excitation and difference-frequency component for two-frequency excitation that are promoted by the strong nonlinearity of the bubbly medium. The numerous simulations analyzed suggest that the new numerical models developed and proposed in this paper are useful to understand the behavior of ultrasounds in bubbly liquids for sonochemical processes and applications of nonlinear frequency mixing.     

Generation of subharmonics in acoustic resonators containing bubbly liquids: A numerical study of the excitation threshold and hysteretic behavior

In this paper we study the generation and behavior of subharmonics in a bubbly liquid confined in an acoustic resonator, through numerical simulations carried out at finite-amplitude acoustic pressure. Several configurations in terms of resonator length and driving frequency are considered here. Our results show that these frequency components, created from a higher-frequency signal at the source (ultrasound), are due to the nonlinearity of the medium at high acoustic-pressure amplitude and to the configuration of the resonator (geometry and boundaries). We also show that they have an amplitude-threshold dependence, which is in concordance with the literature. The response of these subharmonics to different sequences of pressure amplitudes also reveals the hysteretic nature of the bubbly liquid.     

Numerical simulation of the nonlinear ultrasonic pressure wave propagation in a cavitating bubbly liquid inside a sonochemical reactor

We investigate the acoustic wave propagation in bubbly liquid inside a pilot sonochemical reactor which aims to produce antibacterial medical textile fabrics by coating the textile with ZnO or CuO nanoparticles. Computational models on acoustic propagation are developed in order to aid the design procedures. The acoustic pressure wave propagation in the sonoreactor is simulated by solving the Helmholtz equation using a meshless numerical method. The paper implements both the state-of-the-art linear model and a nonlinear wave propagation model recently introduced by Louisnard (2012), and presents a novel iterative solution procedure for the nonlinear propagation model which can be implemented using any numerical method and/or programming tool. Comparative results regarding both the linear and the nonlinear wave propagation are shown. Effects of bubble size distribution and bubble volume fraction on the acoustic wave propagation are discussed in detail. The simulations demonstrate that the nonlinear model successfully captures the realistic spatial distribution of the cavitation zones and the associated acoustic pressure amplitudes.     

声波在含气泡液体中传播特性及产热效应*

该文对含气泡液体中的声波方程采用线性分析方法,研究了超声波在含气泡液体中的传播特性以及产热效应。当声波在含气泡液体中传播时,气泡的存在会影响声波的传播,在声波频率接近气泡共振频率的频段内,声信号在液体中传播时剧烈衰减,而在声波频率远远高于或低于气泡共振频率时,声波的传播基本不受影响。在接近气泡共振的频段内,声波耗散的能量最终转化为热能。同时液体中的气泡会在声波驱动下径向振动并辐射声波,伴随气泡壁在液体中的粘滞振动,热量随之产生。结果表明,两种产热机制分别在不同频段起主导作用。     

Properties of sound attenuation around a two-dimensional underwater vehicle with a large cavitation number

Due to the high speed of underwater vehicles,cavitation is generated inevitably along with the sound attenuation when the sound signal traverses through the cavity region around the underwater vehicle.The linear wave propagation is studied to obtain the influence of bubbly liquid on the acoustic wave propagation in the cavity region.The sound attenuation coefficient and the sound speed formula of the bubbly liquid are presented.Based on the sound attenuation coefficients with various vapor volume fractions,the attenuation of sound intensity is calculated under large cavitation number conditions.The result shows that the sound intensity attenuation is fairly small in a certain condition.Consequently,the intensity attenuation can be neglected in engineering.     

Room temperature multiwavelength erbium-doped fiber ring laser using a highly nonlinear photonic crystal fiber

A multiwavelength laser source is demonstrated with a high power erbium-doped fiber amplifier as the gain medium. A highly nonlinear photonic crystal fiber (PCF) is inserted in the ring cavity to provide nonlinear gain by four-wave mixing. A Sagnac loop is incorporated in the ring cavity serving as a comb-like multichannel filter. The comparison between fiber ring laser without PCF and with PCF shows that the highly nonlinear PCF can generate a larger number of excited wavelengths and help stabilize the output power.     

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 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.     

Creating a collimated ultrasound beam in highly attenuating fluids

We have devised a method, based on a parametric array concept, to create a low-frequency (300-500 kHz) collimated ultrasound beam in fluids highly attenuating to sound. This collimated beam serves as the basis for designing an ultrasound visualization system that can be used in the oil exploration industry for down-hole imaging in drilling fluids. We present the results of two different approaches to generating a collimated beam in three types of highly attenuating drilling mud. In the first approach, the drilling mud itself was used as a nonlinear mixing medium to create a parametric array. However, the short absorption length in mud limits the mixing length and, consequently, the resulting beam is weak and broad. In the second improved approach, the beam generation process was confined to a separate “frequency mixing tube” that contained an acoustically non-linear, low attenuation medium (e.g., water) that allowed establishing a usable parametric array in the mixing tube. A low-frequency collimated beam was thus created prior to its propagation into the drilling fluid. Using the latter technique, the penetration depth of the low frequency ultrasound beam in the drilling fluid was significantly extended. We also present measurements of acoustic nonlinearity in various types of drilling mud.     

Peculiarities of Acoustic Wave Reflection from a Boundary or Layer of a Two-Phase Medium

A mathematical model is presented for determining the oblique incidence of an acoustic wave at both a boundary and layer of a gas–drop mixture or a bubbly liquid of finite thickness. The basic wave reflection and transmission patterns are established for the incidence of a low-frequency acoustic wave at an interface between a pure gas and a gas–drop mixture, as well as between a pure and bubbly liquid. A range of varying volume fractions for a drop is determined, for which the zero value of the reflection coefficient is possible for low frequencies at oblique incidence. It is shown that the reflection coefficient will never be zero at angles of incidence above 24.5° from a gas–drop mixture at a pure gas boundary; however, when a wave is incident from a pure gas at a gas–drop mixture boundary, a zero reflection coefficient is possible for nonzero angles of incidence and the volume fraction of inclusions. The results of calculating reflection of an acoustic wave from a two-phase layer of a medium with a finite thickness are presented. It is established that the minimum reflection coefficient is possible depending on the perturbation frequency for a certain range of angles of incidence for the boundary or the layer of the gas–drop mixture, which is governed mainly by difference in densities between it and the pure gas.     

Refraction and reflection of sound at the boundary of a bubbly liquid

Reflection and refraction of acoustic waves at the interface between pure water and bubbly water are investigated for the case of oblique incidence. From an analysis of analytic solutions, it is concluded that, for a wave incident on the interface from the side of a bubbly liquid, a critical angle of incidence, which depends on the frequency and the parameters of the disperse system, exists, so that, at angles of incidence exceeding the critical one, the wave is totally reflected from the interface.

     

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.     

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.     

Influence of rigid wall on the nonlinear pulsation of nearby bubble

This paper mainly focuses on the nonlinear pulsation of a bubble near the rigid wall. Dynamics of near-wall bubble and free bubble are discussed and compared in details. Investigation reveals as the driving acoustic pressure amplitude increases, nonlinear pulsation of bubble becomes intense gradually. Besides, decreasing the viscosity of host liquid is advantageous for the nonlinear pulsation of bubble. Bifurcation diagrams of bubble radius show acoustic reflection of the rigid wall makes the initial bifurcation appear at low driving acoustic amplitude and on bubble with small ambient radius, and makes the bifurcation still exist for bubble in high-viscosity liquids. That indicates the rigid wall will produce enhancement on the nonlinearity of nearby bubble. As the bubble approaches the wall, the enhancement becomes strong. Moreover, research on the influence of driving frequency shows the rigid wall makes the frequency band corresponding to chaos around the resonant frequency of free bubble shift downward.     

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.     

Perturbation method for the second-order nonlinear effect of focused acoustic field around a scatterer in an ideal fluid

Two nonlinear models are proposed to investigate the focused acoustic waves that the nonlinear effects will be important inside the liquid around the scatterer. Firstly, the one dimensional solutions for the widely used Westervelt equation with different coordinates are obtained based on the perturbation method with the second order nonlinear terms. Then, by introducing the small parameter (Mach number), a dimensionless formulation and asymptotic perturbation expansion via the compressible potential flow theory is applied. This model permits the decoupling between the velocity potential and enthalpy to second order, with the first potential solutions satisfying the linear wave equation (Helmholtz equation), whereas the second order solutions are associated with the linear non-homogeneous equation. Based on the model, the local nonlinear effects of focused acoustic waves on certain volume are studied in which the findings may have important implications for bubble cavitation/initiation via focused ultrasound called HIFU (High Intensity Focused Ultrasound). The calculated results show that for the domain encompassing less than ten times the radius away from the center of the scatterer, the non-linear effect exerts a significant influence on the focused high intensity acoustic wave. Moreover, at the comparatively higher frequencies, for the model of spherical wave, a lower Mach number may result in stronger nonlinear effects.