Primary Bjerknes force in a three-dimensional nonlinear resonator: Numerical simulations

The effect of the primary Bjerknes force caused by a three-dimensional nonlinear standing ultrasonic wave on a population of nonlinear oscillating bubbles is studied in this paper by analyzing the results obtained from simulations performed with a numerical model at low and moderate pressure amplitudes. Small air bubbles are evenly distributed in a water filled cavity excited at resonance for which axial symmetry is assumed. Both the bubble oscillation variable and the pressure variable are unknown in the nonlinear set of coupled differential equations that describes the interaction of ultrasound and bubbles. Simulation results show that the three-dimensional primary Bjerknes force field is strongly amplitude dependent. We also analyze whether taking the term in the differential system that defines the nonlinear behavior of the pressure field into account is determinant or not on the computation of the force field. The corresponding results corroborate the one-dimensional conclusions on the fundamental importance of considering this nonlinear acoustic term to obtain an accurate approximation of the force in a cavity.     

Forced oscillations of two gas bubbles in a fluid in the vicinity of bubble contact

For a theoretical derivation of bubble coalescence conditions, nonlinear forced oscillations of two closely spaced spherical bubbles subjected to the action of a periodic external pressure field are considered. The equations, asymptotic with respect to a small distance between the bubble surfaces, are derived to describe the approach of the bubbles under the action of (i) the Bjerknes attraction force averaged over the oscillation period and (ii) the viscous drag. It is shown that due to nonlinear interaction of the viscous drag with the radial and translational oscillations of the bubbles a unidirectional repulsive force is generated, which prevents the approach of the bubbles. The coalescence of the bubbles is possible when the nondimensional parameter combined from the amplitude and frequency of the external pressure field, the bubble radius, and the fluid viscosity is greater than a certain critical value. The obtained coalescence condition is qualitatively confirmed by experiments.     

Influence of heat transfer on the dynamic response of a spherical gas/vapour bubble

The standard approach to analyse the bubble motion is the well known Rayleigh–Plesset equation. When applying the toolbox of nonlinear dynamical systems to this problem several aspects of physical modelling are usually sacrificed. Particularly in vapour bubbles the heat transfer in the liquid domain has a significant effect on the bubble motion; therefore the nonlinear energy equation coupled with the Rayleigh–Plesset equation must be solved. The main aim of this paper is to find an efficient numerical method to transform the energy equation into an ODE system, which, after coupling with the Rayleigh–Plesset equation can be analysed with the help of bifurcation theory. Due to the strong nonlinearity and violent bubble motions the computational effort can be high, thus it is essential to reduce the size of the problem as much as possible. In the first part of the paper finite difference, Galerkin and spectral collocation methods are examined and compared in terms of efficiency. In the second part free and forced oscillations are analysed with an emphasis on the influence of heat transfer. In the case of forced oscillations the unstable branches of the amplification diagrams are also computed.     

A New Method for Approximate Analytical Solutions to Nonlinear Oscillations of Nonnatural Systems

This paper deals with nonlinear oscillations of a conservative,nonnatural, single-degree-of-freedom system with odd nonlinearity. Bycombining the linearization of the governing equation with the method ofharmonic balance, we establish approximate analytical solutions for thenonlinear oscillations of the system. Unlike the classical harmonicbalance method, the linearization is performed prior to proceeding withharmonic balancing thus resulting in linear algebraic equations insteadof nonlinear algebraic equations. Hence, we are able to establish theapproximate analytical formulas for the exact period and periodicsolution. These approximate solutions are valid for small as well aslarge amplitudes of oscillation. Two examples are presented toillustrate that the proposed formulas can give excellent approximateresults.     

三维气泡与刚性壁面的相互作用研究

基于势流理论建立水下爆炸气泡运动三维模型,采用边界积分法求解拉普拉斯方程,得到气泡的变形及位置,并在计算过程中引入弹性网格技术,避免了因网格扭曲而导致的数值发散,进而模拟了刚性壁面附近三维气泡的动态特性。在数值模拟过程中,将本文计算值与实验数据进行对比分析,结果表明,计算值与实验数据吻合良好。在此基础上,分别模拟了弱浮力、强Bjerknes力,强浮力、弱Bjerknes力以及浮力与Bjerknes力相当时壁面附近气泡的运动特征,并将各种工况的计算结果与基于开尔文冲量理论(Kelvin Impulse)的Blake准则进行对比分析与讨论,得到了不同参数下气泡的运动特征。     

考虑气动力作用的载流覆冰导线主共振及谐波共振分析

针对载流导线的非线性振动问题,在以往只考虑安培力的载流导线振动方程中引入了气动荷载。在此基础上进一步引入了受迫激励荷载,以研究动态风或相邻档导线对载流覆冰导线非线性振动特征的影响,建立了一种新的气动力-安倍力-受迫激励联合作用下的载流覆冰导线系统。推导出非线性振动方程,利用Galerkin方法将该振动方程转变为有限维度的常微分方程,采用多尺度法求解得到系统的非线性受迫主共振和亚谐波共振的幅-频响应函数。通过数值计算,分析了参数变化对系统受迫共振响应的影响以及受迫主共振定常解的稳定性。结果表明,考虑气动力的振动幅值和系统非线性较未考虑气动力时更小和更弱;线路参数的变化对导线的响应幅值和系统的非线性都有一定程度的影响;主共振和亚谐波共振的响应幅值随着激励幅值的增大而增大,共振峰值向着调谐参数σ的负值方向偏移,呈现出软弹簧特征并伴随着多值和跳跃现象;主共振时,随着调谐参数的变化,响应幅值则出现同步和失步现象。     

Bubble dynamics: a review and some recent results

Several aspects of small-amplitude oscillations of bubbles containing gas, vapor, or a gas-vapor mixture are discussed. An application to pressure-wave propagation in a bubbly liquid is described. Nonlinear forced oscillations are considered in the light of recent research on forced oscillations of nonlinear systems. The growth of vapor bubbles, an extension of the Rayleigh-Plesset equation to non-Newtonian liquids and appreciable mass transfer at the interface, and a boundary integral numerical method for nonspherical cavitation bubble dynamics are also briefly discussed.     

Nucleate pool boiling in microgravity: Recent progress and future prospects

Pool boiling on flat plates in microgravity has been studied for more than 50 years. The results of recent experiments performed in sounding rocket are presented and compared to previous results. At low heat flux, the vertical oscillatory motion of the primary bubble is responsible for the increase in the heat transfer coefficient in microgravity compared to ground experiments. The effect of a non-condensable gas on the stabilisation of the large primary bubble on the heater is pointed out. Experiments on isolated bubbles are also performed on ground and in parabolic flight. The effect of a shear flow on the bubble detachment is highlighted. A force balance model allows determining an expression of the capillary force and of the drag force acting on the bubble.     

Forced vibration analysis of the milling process with structural nonlinearity,internal resonance,tool wear and process damping effects

In this paper, forced vibration analysis of an extended dynamic model of the milling process is investigated, in the presence of internal resonance. Regenerative chatter, structural nonlinearity, tool wear and process damping effects are included in the proposed model. Taking into account the average and first order expansion of Fourier series for cutting force components; their closed form expressions are derived. Moreover, in the presence of large vibration amplitudes, the loss of contact effect is included in this model. Analytical approximate response of the nonlinear system is constructed through the multiple-scales approach. Dynamics of the system is studied for two cases of primary and super-harmonic resonance, associated with the internal resonance. Under steady state motion, the effects of structural nonlinearity, cutting force coefficients, tool wear length and process damping are investigated on the frequency response functions of the system. In addition, existence of multiple solutions, jump phenomenon and energy transfer between vibration modes are presented and compared for tow cases of primary and super-harmonic resonances.     

A further investigation about a simple model of the hunting behavior of some old locomotives

This study is concerned with forced damped purely nonlinear oscillators and their behaviour at different excitation frequencies. First, their dynamics is considered numerically for the response determined in the vicinity of a backbone curve with the aim of detecting coexisting responses that have not been found analytically so far. Both the cases of low and high excitation amplitudes are investigated. Second, the angular excitation frequency is lowered significantly for different powers of nonlinearity, and the system’s behaviour is examined qualitatively, which has not been considered previously related to a general class of purely nonlinear oscillators. It is illustrated that the response at a low-valued angular excitation frequency has a form of bursting oscillations, consisting of fast oscillations around a slow flow. Finally, approximate analytical solutions are presented for the slow and fast flow for a general class of purely nonlinear oscillators.     

Bubbling behavior of cohesive particles in a two-dimensional fluidized bed with immersed tubes

Fluidization hydrodynamics are greatly influenced by inter-particle cohesive forces. This paper studies the fluidization of large cohesive particles in a two-dimensional fluidized bed with immersed tubes using “polymer coating” to introduce cohesive force, to gain better understanding of bubbling behavior when particles become cohesive and its effect on chemical processes. The results show that the cohesive force promotes bubble splitting in the tube bank region, thereby causing an increase in the number and a decline in the aspect ratio of the bubbles. As the cohesive force increases within a low level, the bubble number increases and the bubble diameter decreases, while the aspect ratio exhibits different trends at different fluidization gas velocities. The difference in the evolution of bubble size under various cohesive forces mainly takes place in the region without tubes. When the cohesive force is large enough to generate stable agglomerates on the side walls of the bed, the bubble number and the bed expansion sharply decrease. The tubes serve as a framework that promotes the agglomeration, thus accelerating defluidization. Finally, the bubble profile around tubes was studied and found to greatly depend both on the cohesive forces and the location of tubes.     

Chaotic behavior of a single spherical gas bubble surrounded by a Giesekus liquid: A numerical study

In the present work, nonlinear oscillations of a spherical, acoustically driven gas bubble in a Giesekus liquid are examined numerically. A novel approach based on the Gauss–Laguerre quadrature (GLQ) method is implemented to solve the integro-differential equation governing bubble dynamics in a Giesekus liquid. It is shown that, using this robust method, numerical results could be obtained at very high amplitudes and frequencies typical of ultrasound applications. The GLQ method also enabled obtaining results at very high Deborah and Reynolds numbers over prolonged dimensionless times not reported previously. Based on the results obtained in this work, it is concluded that the GLQ method is well suited for bubble dynamics studies in viscoelastic liquids. It is also concluded that the extensional-flow behavior of the liquid surrounding the bubble (as represented by the mobility factor in the Giesekus model) has a strong effect on the chaotic behavior of the bubble, and this is particularly so at high Deborah numbers, high amplitudes and/or high frequencies of the acoustic field. A period-doubling bifurcation structure is predicted to occur for certain values of the mobility factor.     

均匀电场中气泡上升特性的实验研究

电场中的气泡对于强化传热有显著的作用, 对于电场中气泡动力学特性的研究对增强换热器效率, 提高能源利用率有重要意义. 为了获得外加电场作用下气泡的动力学特性, 设计与搭建了可视化实验平台. 采用50 kV高压直流电源构建均匀电场, 高清摄像机拍摄实验图像. 引入电场强度、气泡体积与溶液介电常数作为变量, 探究其对于气泡动力学特性的影响. 观测了竖直与水平均匀电场中气泡的上升过程, 分析了不同变量下气泡变形状况与上升速度的变化. 引入气泡长宽比L/D用于表示气泡拉伸变形程度, 截取单个气泡上升过程分时段图像展示形态变化过程. 研究结果表明, 气泡沿电场方向伸长, 且电场强度越大, 变形越明显; 竖直电场中气泡伸长导致上升速度增大, 而水平电场中气泡上升速度减小. 气泡尺寸增大, 浮升力作用增强, 气泡上升速度增大. 溶液介电常数增加, 电场力作用明显增加, 气泡变形更加明显.      

Oscillations of a gas-vapor bubble in an acoustic field

It is shown that at large vapor contents, as a result of the combined action of phase transitions and capillary effects, the small radially symmetric oscillations of gas-vapor bubbles in an acoustic field are unstable in amplitude. The critical vapor concentration in the bubble separating regions of qualitatively different bubble behavior in the acoustic field is determined. Expressions are obtained for the decay rate of the radial oscillations of the gas-vapor bubble and the growth rate characterizing the rate of increase of oscillation amplitude in the region of instability. It is shown that adding only a slight amount of gas to the vapor bubble leads to a marked decrease in the growth rate. It is found that in the particular case of a vapor bubble the tine growth rate characterizing the development of the instability is of the same order as the second resonance frequency of the vapor bubble. This may serve to explain why in the case of vapor bubble oscillations the second resonance effect, which has been established in a number of theoretical studies and is widely discussed in the literature, has not yet been experimentally confirmed. The problem of spherically symmetrical processes around gasvapor bubbles was posed in [1], and their small oscillations are investigated in detail in [2–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 79–33, May–June, 1986.The authors are grateful to R. I. Nigmatulin for useful discussions.     

Machine learning models for the secondary Bjerknes force between two insonated bubbles

The secondary Bjerknes force plays a significant role in the evolution of bubble clusters.However,due to the complex dependence of the force on multiple paramet...     

Evolution of a Small Distortion of the Spherical Shape of a Gas Bubble under Strong Expansion-Compression

The evolution of a small distortion of the spherical shape of a gas bubble which undergoes strong radial expansion-compression upon a single oscillation of the ambient liquid pressure under a harmonic law are analyzed by numerical experiments. It is assumed that the distortions of the spherical bubble shape are axisymmetric and have the form of individual spherical surface harmonics with numbers of 2–5. Bubble-shape oscillations prior to the beginning of expansion are taken into account. Generally, the distortion value during bubble expansion-compression depends on the phase of bubble-shape oscillation at the beginning of the expansion (initial phase). Emphasis is placed on the dependence of the maximum distortions in the initial phase at certain characteristic times of bubble expansion-compression on the amplitude of the external excitation, liquid viscosity, and distortion mode (harmonic number). The parameters of the problem are typical of the stable periodic sonolumiescence of an individual air bubble in water at room temperature. An exception is the liquid pressure oscillation amplitude, which is varied up to values that are five times the static pressure. That large excitation amplitudes are beyond the stability threshold of periodic oscillations of spherical bubbles. Their consideration is of interest from the point of view of increasing the compression ratio of the bubble gas, i. e., increasing the maximum temperature and density achievable in the final compression stage.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 17–28, July–August, 2005.     

Free Vibration in a Class of Self-Excited Oscillators with 1:3 Internal Resonance

Free vibration of a two degree of freedom weakly nonlinear oscillator is investigated. The type of nonlinearity considered is symmetric, it involves displacement as well as velocity terms and gives rise to self-excited oscillations in many engineering applications. After presenting the equations of motion in a general form, a perturbation methodology is applied for the case of 1:3 internal resonance. This yields a set of four slow-flow nonlinear equations, governing the amplitudes and phases of approximate motions of the system. It is then shown that these equations possess three distinct types of solutions, corresponding to trivial, single-mode and mixed-mode response of the system. The stability analysis of all these solutions is also performed. Next, numerical results are presented by applying this analysis to a specific practical example. Response diagrams are obtained for various combinations of the system parameters, in an effort to provide a complete picture of the dynamics and understand the transition from conditions of 1:3 internal resonance to non-resonant response. Emphasis is placed on identifying the effect of the linear damping, the frequency detuning and the stiffness nonlinearity parameters. Finally, the predictions of the approximate analysis are confirmed and extended further by direct integration of the averaged equations. This reveals the existence of other regular and irregular motions and illustrates the transition from phase-locked to drift response, which takes place through a Hopf bifurcation and a homoclinic explosion of the averaged equations.     

Interactions of multiple spark-generated bubbles with phase differences

This paper aims to study the complex interaction between multiple bubbles, and to provide a summary and physical explanation of the phenomena observed during the interaction of two bubbles. High-speed photography is utilized to observe the experiments involving multiple spark-generated bubbles. Numerical simulations corresponding to the experiments are performed using the Boundary Element Method (BEM). The bubbles are typically between 3 and 5 mm in radius and are generated either in-phase (at the same time) or with phase differences. Complex phenomena are observed such as bubble splitting, and high-speed jetting inside a bubble caused by another collapsing bubble nearby (termed the ‘catapult’ effect). The two-bubble interactions are broadly classified in a graph according to two parameters: the relative inter-bubble distance and the phase difference (a new parameter introduced). The BEM simulations provide insight into the physics, such as bubble shape changes in detail, and jet velocities. Also presented in this paper are the experimental results of three bubble interactions. The interesting and complex observations of multiple bubble interaction are important for a better understanding of real life applications in medical ultrasonic treatment and ultrasonic cleaning. Many of the three bubble interactions can be explained by isolating bubble pairs and classifying their interaction according to the graph for the two bubble case. This graph can be a useful tool to predict the behavior of multiple bubble interactions.