II. Recirculation flow induced by a bubble stream rising in a viscous liquid

The recirculation flow induced by the rising motion of a bubble stream in a viscous fluid within an open-top rectangular enclosure is studied. The three-dimensional volume averaged conservation equations are solved by a control-volume method using a hybrid finite differencing scheme to describe the liquid phase hydrodynamics. The momentum exhange between the bubbles and the liquid phase is modeled with a source term equals to the volumetric buoyancy force acting on the gas in the bubble stream. The volumetric buoyancy force accounts for in line interactions between bubbles through the average gas volume fraction in the gas liquid column which depends on the size and the rising velocity of bubbles. The fluid flow within an open-top rectangular enclosure is further investigated by particle image velocimetry for a bubble stream rising in a water-glycerol solution. The measured fluid velocities in a vertical plane are compared with the predictions of the numerical model over a wide range of fluid viscosity (43 mPa s-800 mPa s) and gas flow rates. Finally, the recirculation flows resulting from the interaction of two neighbouring vertical bubble streams are studied. Received: 23 July 1997 / Revised: 19 December 1997 / Accepted: 11 May 1998     

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.     

流体黏性及表面张力对气泡运动特性的影响

基于气泡边界层理论,引入黏性修正,采用边界积分法,考虑黏性效应和表面张力在单气泡以及双气泡耦合作用过程中的影响.首先将建立的数值模型与Rayleigh-Plesset的解析解进行对比,发现二者符合良好,验证了数值模型的有效性;在此基础上,建立考虑流体弱黏性效应的双气泡耦合模型,研究流体黏性和表面张力作用下,气泡表面变形、射流速度、流场能量转换等物理量的变化规律;最后研究雷诺数和韦伯数对于气泡脉动特性的影响规律.结果表明,流体黏性会抑制气泡脉动和气泡射流发展,降低气泡半径和射流速度;表面张力不改变气泡脉动幅值,但缩短了脉动周期,提升气泡势能.     

Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method

The rise of bubbles in viscous liquids is not only a very common process in many industrial applications, but also an important fundamental problem in fluid physics. An improved numerical algorithm based on the front tracking method, originally proposed by Tryggvason and his co-workers, has been validated against experiments over a wide range of intermediate Reynolds and Bond numbers using an axisymmetric model [J. Hua, J. Lou, Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 22 (2007) 769–795]. In the current paper, this numerical algorithm is further extended to simulate 3D bubbles rising in viscous liquids with high Reynolds and Bond numbers and with large density and viscosity ratios representative of the common air–water two-phase flow system. To facilitate the 3D front tracking simulation, mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh. On the latter mesh, the governing Navier–Stokes equations for incompressible, Newtonian flow are solved in a moving reference frame attached to the rising bubble. Specifically, the equations are solved using a finite volume scheme based on the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm, and it appears to be robust even for high Reynolds numbers and high density and viscosity ratios. The 3D bubble surface is tracked explicitly using an adaptive, unstructured triangular mesh. The numerical model is integrated with the software package PARAMESH, a block-based adaptive mesh refinement (AMR) tool developed for parallel computing. PARAMESH allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer. Further, Peskin distribution function is applied to interpolate the variable values between the front and the background meshes. Detailed sensitivity analysis about the numerical modeling algorithm has been performed. The current model has also been applied to simulate a number of cases of 3D gas bubbles rising in viscous liquids, e.g. air bubbles rising in water. Simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities. In addition, we applied this model to simulate the interaction between two bubbles rising in a liquid, which illustrated the model’s capability in predicting the interaction dynamics of rising bubbles.     

Scaling and instabilities in bubble pinch-off

We have used a 100 000 frame-per-second video to analyze the pinch-off of nitrogen gas bubbles in fluids with a wide range of viscosity. If the external fluid is highly viscous (eta(ext)>100 cP), the bubble neck radius is proportional to the time before break, tau, and decreases smoothly to zero. If the external fluid has low viscosity (eta(ext)<10 cP), the radius scales as tau(1/2) until an instability develops in the gas bubble, which causes the neck to rupture and tear apart. Finally, if the viscosity of the external fluid is in an intermediate range, an elongated thread is formed, which breaks apart into micron-sized bubbles.     

Flow of an electrically conducting bubble liquid in the field of an electromagnetic force

Physical processes accompanying the flow of a conducting bubble liquid in crossed electric and magnetic fields are considered. Based on the general equations of mechanics of multiphase media, we develop a one-dimensional model of the flow of and heat exchange in a compressible bubble liquid when the phases are not in thermal and velocity equilibrium. The model is numerically investigated. It is demonstrated that, when the bubble liquid flows along the electromagnetic force vector, the bubbles lag behind the carrying flow and are compressed and warmed up. This causes oscillations of the bubble volume, as well as oscillations of the parameters of both the disperse and carrying phase. In particular, the compression of the bubbles reduces the volumetric gas content, as well as increases the effective conductivity of the flow and the electromagnetic force in the downstream direction. This sets conditions for crisis of the bubble flow when the electromagnetic force expels the bubbles against the main stream. On the basis of the solutions obtained, the efficiency of a gas compressor is calculated.     

DNS using CLSVOF method of single micro-bubble breakup and dynamics in flow focusing

Numerical simulations are performed to investigate the breakup of air bubble in flow focusing configuration; the CLSVOF (coupled level set with volume of fluid) method is employed to track the interface, which allows a better identification of the liquid–gas interface via a function called level set. The CFD simulations showed that the velocity ratio, the interfacial tension, the outer channel diameter, the continuous phase viscosity, the orifice width and length play an important role in the determination of the air bubble’s size and shape. However, at low capillary number, increasing the flow velocity ratio gives a smaller bubble size in shorter time, while the increase in interfacial tension leads to a bigger bubble. Moreover, the carrier fluid is found to slightly affect the bubbling mechanism, while the smallest bubbles were obtained with the smallest orifice size. In addition, three breakup regimes are observed in this device: disc-bubble (DB), elongated bubble (EB) and the slug bubble (SB) regime flows. This work also demonstrates that the CLSVOF is an effective method to simulate the bubbles breakup in flow focusing geometry. In addition, a comparison of our computational simulations with available experimental results reveals reasonably good agreement.

Graphic abstract

     

Numerical Simulation of Vapor Bubble Growth and Heat Transfer in a Thin Liquid Film

A mathematical model is developed to investigate the dynamics of vapor bubble growth in a thin fiquid film, movement of the interface between two fluids and the surface heat transfer characteristics. The model takes into account the effects of phase change between the vapor and liquid, gravity, surface tension and viscosity. The details of the multiphase flow and heat transfer are discussed for two cases： （1） when a water micro-droplet impacts a thin liquid film with a vapor bubble growing and （2） when the vapor bubble grows and merges with the vapor layer above the liquid film without the droplet impacting. The development trend of the interface between the vapor and liquid is coincident qualitatively with the available literature, mostly at the first stage. We also provide an important method to better understand the mechanism of nucleate spray cooling.     

Tracer dispersion in power law fluids flow through porous media: Evidence of a cross-over from a logarithmic to a power law behaviour

An analytical model is presented to describe the dispersion of tracers in a power-law fluid flowing through a statistically homogeneous and isotropic porous medium. The model is an extension of Saffman's approach to the case of non-Newtonian fluids. It is shown that an effective viscosity depending on the pressure gradient and on the characteristics of the fluid, must be introduced to satisfy Darcy's law. An analytical expression of the longitudinal dispersivity is given as a function of the Peclet number Pe and of the power-law index n that characterizes the dependence of the viscosity on the shear rate . As the flow velocity increases the dispersivity obeys an asymptotic power law: . This asymptotic regime is achieved at moderate Peclet numbers with strongly non-Newtonian fluids and on the contrary at very large values when n goes to 1 ( for n=0.9). This reflects the cross-over from a scaling behaviour for towards a logarithmic behaviour predicted for Newtonian fluids (n=1). Received: 22 July 1997 / Revised and Accepted: 2 July 1998     

Fingering instability in a water-sand mixture

The temporal evolution of a water-sand interface driven by gravity is experimentally investigated. By means of a Fourier analysis of the evolving interface the growth rates are determined for the different modes appearing in the developing front. To model the observed behavior we apply the idea of the Rayleigh-Taylor instability for two stratified fluids. Carrying out a linear stability analysis we calculate the growth rates from the corresponding dispersion relations for finite and infinite cell sizes. Based on the theoretical results the viscosity of the suspension is estimated to be approximately 100 times higher than that of pure water, in agreement with other experimental findings. Received: 12 December 1997 / Revised: 23 March 1998 / Accepted: 28 April 1998     

Experimental Study on Physical Mechanism of Drag Reduction of Hydrophobic Materials in Laminar Flow

We experimentally study the physical mechanism of the drag reduction of hydrophobic materials in the macroscopic scale. The experiment includes the drag and velocity measurements of laminar boundary layer flow over flat plates, and the observation of air bubbles on the surfaces. The plate surfaces have different wetting and roughness properties. In the drag measurements, the plates with bubbles on the surfaces lead to drag reduction, but not for those without bubbles. Velocity measurement confirms that the flow is laminar and gives apparent fluid slip on the plate wall with bubbles. In observation, air bubbles in macroscopic size emerge and enlarge on hydrophobic surfaces but not on hydrophilic surfaces. Therefore, the drag reduction of hydrophobic materials is explained by the generation of air bubbles of macroscopic size that cause the apparent velocity slip.     

水中上浮气泡动态特性研究

针对水中的上浮气泡,计入表面张力的作用,然后将气泡边界层内黏性效应分为法向和切向两部分,其中附加法向黏性应力通过Young-Laplace关系考虑;附加切向黏性力是基于黏性耗散能量等效原理,引入黏性修正压力代替.首先建立了轴对称和三维上浮气泡边界元模型,将数值结果与理论值和实验值进行对比分析,有良好的符合度,验证了数值模型的有效性;然后针对毫米量级上浮气泡的平衡速度与形态,讨论了气泡初始条件、表面张力和黏性对气泡上浮过程中动力学行为的影响;最后,提出了一种处理三维上浮气泡融合的数值方法,计算结果与实验现象符合良好,并且能够反映气泡融合后的复杂现象细节.     

Numerical simulation of gas bubble motion in a flow-through resonator

The problem of gas bubble motion in an acoustic resonator with a fluid flow is solved using numerical methods. It is shown that the distribution of the bubble concentration, which is nonuniform over the resonator length, is formed upon homogeneous introduction of bubbles. The problem on the bubble concentration distribution the along the resonator axis (with fluctuations of the bubble introduction period taken into account) is considered, and the fluctuation parameters are determined at which the periodic structure of the concentration distribution is preserved. The distribution of bubbles with different sizes over the resonator length is determined. It is shown that a resonator with a fluid flow accomplishes bubble selection by size (the average bubble concentration in the resonator increases with an increase in bubble size). The field in the resonator was calculated taking into account the effect of bubbles on sound velocity and damping.     

Transient settling of a dilute spherical suspension droplet at low Reynolds number

The transient settling in a viscous incompressible fluid of a spherical dilute cloud of particles starting from rest under the influence of a small constant applied force is studied in a continuum model on the basis of the linearized Navier-Stokes equations. Explicit expressions are derived for the motion of the cloud and for the flow velocity and pressure of the fluid. Equations of transient Stokesian dynamics are formulated that allow numerical study of the motion of a dilute cloud of particles of arbitrary initial configuration.     

气泡体积分数对沙质沉积物低频声学特性的影响

由于有机物质分解等原因,实际的海底沉积物中存在气泡,气泡的存在会显著影响沉积物低频段的声学特性,因此研究气泡对沉积物低频段声速的影响机理具有重要意义.考虑到外场环境的不可控性,在室内水池中搭建了大尺度含气非饱和沙质沉积物声学特性获取平台,在有界空间中应用多水听器反演方法首次获取了含气非饱和沙质沉积物300—3000 Hz频段内的声速数据(79—142 m/s),并同时利用双水听器法获取了同一频段的数据(112—121 m/s).在声波频率远低于沉积物中最大气泡的共振频率时,根据等效介质理论,将孔隙水和气泡等效为一种均匀流体,改进了水饱和等效密度流体近似模型.模型揭示了气泡对沉积物低频段声学特性的影响规律,理论上解释了沉积物中声速的降低.通过分析模型预报声速对模型参数的敏感性,根据测量得到的声速分布反演得到了沉积物不同区域的气泡体积分数,气泡体积分数从1.07%变化到2.81%.改进的模型为沉积物中气泡体积分数估计提供了一种新方法.     

黏性液体中单个气泡上升的形状特性

采用基于Level Set方法的直接数值模拟技术对黏性液体中单个气泡的上升运动进行三维模拟.数值模拟采用拟单相流模型处理气泡内外的气液两相流动,应用Level Set方法捕捉运动气泡的变形.针对Eo数从O(0)～O(2),Mo数从O(-11)～O(2)的流动范围,重点研究了上升气泡的形状特性,并与经典的气泡形状图谱进行了比较.模拟结果表明,上升气泡的形状与无量纲参数(Eo、Mo和Re)密切相关.在高Re的扁椭球区域,数值发现了气泡形状的周期性振荡行为.     

Air bubbles under vertical vibrations

This paper reports on an experimental study of the splitting instability of an air bubble a few centimetres in diameter placed in a sealed cylindrical cell filled with liquid and submitted to vertical oscillations. The response of the bubble to the oscillations is observed with a high-speed video camera. It is found that the bubble dynamics is closely associated with the acceleration of the cell Γ. For small acceleration values, the bubble undergoes minor shape deformations. With increasing acceleration values, these deformations are amplified and for sufficiently large Γ the bubble becomes toroidal. The bubble may then become unstable and split into smaller parts. The onset of bubble division is studied and its dependency on physical parameters such as the fluid viscosity, the fluid surface tension and the initial size of the bubble is presented. It is found that the criterion for the bubble splitting process is associated with a threshold based on the acceleration of the oscillations. Above this threshold, the number of bubbles present in the cell is observed to grow until a final steady state is reached. Data analysis reveals that the final bubble size may be characterized in terms of Bond number.     

Anisotropic elastic properties of microtubules

The present study aims at providing insight into the acceleration mechanism of a bubble chain rising in shear-thinning viscoelastic fluids. The experimental investigation by the Particle Image Velocimetry (PIV), birefringence visualisation and rheological simulation shows that two aspects are central to bubble interactions in such media: the stress creation by the passage of bubbles, and their relaxation due to the fluids memory forming an evanescent corridor of reduced viscosity. Interactions between bubbles were taken into account mainly through a linear superposition of the stress evolution behind each bubble. An analytical approach together with the rheological consideration was developed to compute the rise velocity of a bubble chain in function of the injection period and bubble volume. The model predictions compare satisfactorily with the experimental investigation.     

Induced bubble shape oscillations and their impact on the rise velocity

When a 2-4 mm diameter bubble rising with constant velocity hits a thin wire, bubble shape oscillations can be induced. As a consequence also the bubble rise velocity strongly oscillates. With the help of a force balance we show that these velocity oscillations are an added-mass effect. Received 9 April 2002 / Received in final form 11 July 2002 Published online 14 October 2002 RID="a" ID="a"e-mail: lohse@tn.utwente.nl