DISCRETE MODELLING OF TWO-DIMENSIONAL LIQUID FOAMS

Liquid foam is a dense random packing of gas or liquid bubbles in a small amount of immiscible liquid containing surfactants. The liquid within the Plateau borders, although small in volume, causes considerable difficulties to the investigation of the spatial structure and physical properties of foams, and the situation becomes even more complicated as the fluid flows. To solve these problems, a discrete model of two-dimensional liquid foams on the bubble scale is proposed in this work. The bubble surface is represented with finite number of nodes, and the liquid within Plateau borders is discretized into lattice particles. The gas in bubbles is treated as ideal gas at constant temperatures. This model is tested by choosing an arbitrary shape bubble as the initial condition. This then automatically evolves into a circular shape, which indicates that the surface energy minimum routine is obeyed without calling external controlling conditions. Without inserting liquid particle among the bubble channels, periodic ordered and disordered dry foams are both simulated, and the fine foam structures are developed. Wet foams are also simulated by inserting fluid among bubble channels. The calculated coordination number, as a function of liquid fractions, agrees well with the standard values.     

Controlled foam generation using cyclic diphasic flows through a constriction

Numerous industrial and academic applications of liquid foams require a fine control over their bubble size distribution and their liquid content. A particular challenge remains the generation of foams with very small bubbles and low liquid content. A simple technique which fulfils these different criteria, the “double-syringe technique”, has been exploited for decades in hospital applications. In this technique, the foaming liquid and gas are pushed repeatedly back and forth through the constriction that connects two syringes. After having motorised the technique we investigate here the influence of the different processing conditions on the obtained foam properties in a quantitative manner. We show that this technique is unique in producing foams with the same characteristic bubble size distributions over a wide range of processing conditions (tubing, fluid velocities,...), making it an ideal tool for controlled foam generation. In contrast to other techniques, the liquid fraction in the double-syringe technique can be varied without impacting the bubble size distribution. Using high-speed imaging we show that bubbles are dispersed in the aqueous phase at two different places in the device via a hitherto unreported fragmentation mechanism. We put in evidence that the obtained bubble size distributions are largely independent of most processing parameters with the exception of the geometry of the constriction and the foam formulation. We put forward a first analysis of the non-dimensional numbers of the flow and compare our results with bubbles size distributions obtained from fragmentation processes. Future work on simplified model systems is required to explain the observed mechanisms.     

Surfactant Concentration and End Effects on Foam Flow in Porous Media

Foaming injected gas is a useful and promising technique for achieving mobility control in porous media. Typically, such foams are aqueous. In the presence of foam, gas and liquid flow behavior is determined by bubble size or foam texture. The thin-liquid films that separate foam into bubbles must be relatively stable for a foam to be finely textured and thereby be effective as a displacing or blocking agent. Film stability is a strong function of surfactant concentration and type. This work studies foam flow behavior at a variety of surfactant concentrations using experiments and a numerical model. Thus, the foam behavior examined spans from strong to weak.Specifically, a suite of foam displacements over a range of surfactant concentrations in a roughly 7m², one-dimensional sandpack are monitored using X-ray computed tomography (CT). Sequential pressure taps are employed to measure flow resistance. Nitrogen is the gas and an alpha olefin sulfonate (AOS 1416) in brine is the foamer. Surfactant concentrations studied vary from 0.005 to 1wt%. Because foam mobility depends strongly upon its texture, a bubble population balance model is both useful and necessary to describe the experimental results thoroughly and self consistently. Excellent agreement is found between experiment and theory.     

Prediction of mono-disperse gas–liquid turbulent flow in a vertical pipe

A two-fluid model in the Eulerian–Eulerian framework has been implemented for the prediction of gas volume fraction, mean phasic velocities, and the liquid phase turbulence properties for gas–liquid upward flow in a vertical pipe. The governing two-fluid transport equations are discretized using the finite volume method and a low Reynolds number $k-\varepsilon$ model is used to predict the turbulence field for the continuous liquid phase. In the present analysis, a fully developed one-dimensional flow is considered where the gas volume fraction profile is predicted using the radial force balance for the bubble phase. The current study investigates: (1) the turbulence modulation terms which represent the effect of bubbles on the liquid phase turbulence in the k–ε transport equations; (2) the role of the bubble induced turbulent viscosity compared to turbulence generated by shear; and (3) the effect of bubble size on the radial forces which results in either a center-peak or a wall-peak in the gas volume fraction profiles. The results obtained from the current simulation are generally in good agreement with the experimental data, and somewhat improved over the predictions of some previous numerical studies.     

Trapped Gas Fraction During Steady-State Foam Flow

Trapped or stationary gas contributes significantly to the extent of gas mobility reduction for aqueous foams. Simultaneous measurements of effluent bubble sizes and trapped gas saturation in sandstone are reported for the first time. Roughly 80% of the gas saturation in an aqueous foam is stationary at steady state in this permeable porous medium. The experiments show that as gas velocity increases, the trapped gas fraction decreases. Similarly, as injected gas–liquid ratio increases, the trapped gas fraction decreases. Hence, the absolute velocities of gas and aqueous surfactant solution are fundamental to foamed-gas mobility reduction for they help determine in situ foam texture. Effluent foam bubbles range in size from 60 to 120 μm in diameter. The smaller the effluent bubble, the smaller is the fraction of mobile gas. Scaling laws from network percolation theory are used to engender a mechanistic understanding of the various parameters identified as important in the experimental program. The closed form approimation predicts that the trapped gas fraction is a weak function of pressure gradient, foam-bubble size, and the permeability of the porous medium. Moreover, the theory reproduces well the newly obtained experimental data.     

A numerical strategy for the direct 3D simulation of the expansion of bubbles into a molten polymer during a foaming process

In the framework of the foam process modelling, this paper presents a numerical strategy for the direct 3D simulation of the expansion of gas bubbles into a molten polymer. This expansion is due to a gas overpressure. The polymer is assumed to be incompressible and to behave as a pseudo‐plastic fluid. Each bubble is governed by a simple ideal gas law. The velocity and the pressure fields, defined in the liquid by a Stokes system, are subsequently extended to each bubble in a way of not perturbing the interface velocity. Hence, a global velocity–pressure‐mixed system is solved over the whole computational domain, thanks to a discretization based on an unstructured first‐order finite element. Since dealing with an Eulerian approach, an interface capturing method is used to follow the bubble evolution. For each bubble, a pure advection equation is solved by using a space–time discontinuous‐Galerkin method, coupled with an r‐adaptation technique. Finally, the numerical strategy is achieved by considering a global mesh expansion motion, which conserves the amount of liquid into the computational domain during the expansion. The expansion of one bubble is firstly considered, and the simulations are compared with an analytical model. The formation of a cellular structure is then investigated by considering the expansion of 64 bubbles in 2D and the expansion of 400 bubbles in 3D. Copyright © 2007 John Wiley & Sons, Ltd.     

台阶式并行微通道内气泡群自组装行为及其对气泡生成的反馈效应

台阶式微通道乳化装置因易于高通量生产均一性的气泡及液滴而受到关注.本文利用高速摄像仪研究了台阶式并行微通道装置空腔内的气泡群复杂行为及其对气泡生成的反馈效应.实验设计的操作变量为气液相进出口位置、气相流速和液相流速. 在实验操作范围内,共发现了气泡的单管生成模式和多管生成模式.研究了空腔内气泡群复杂行为随操作条件的变化趋势. 发现在受限空间内,气泡在水平面内发生挤压堵塞能够自组装成具有特定几何特点的二维晶格,分别为有序的行三角晶格、有序的竖三角晶格和无序的三角晶格.晶格结构与气相压力密切相关; 同时, 气泡界面能量随着气相压力的增大而增大.运用介尺度、能量和活化等概念分析了气泡群复杂行为对气泡生成方式的影响,充分阐释了受限空间内气泡群的介尺度效应.以变异系数CV来表示气泡的均匀性特征, 考察了气泡晶格自组装行为的控制因素.结果表明: 气泡的自组装路径由气泡尺寸及其分布决定,有序的三角晶格变异系数小于5%, 无序的三角晶格变异系数大于5%.      

Mixtures of foam and paste: suspensions of bubbles in yield stress fluids

We study the rheological behavior of mixtures of foams and pastes, which can be described as suspensions of bubbles in yield stress fluids. Model systems are designed by mixing monodisperse aqueous foams and concentrated emulsions. The elastic modulus of the bubble suspensions is found to depend on the elastic capillary number $\textit{Ca}_{_G}$ , defined as the ratio of the paste elastic modulus to the bubble capillary pressure. For values of $\textit{Ca}_{_G}$ larger than $\simeq 0.5$ , the dimensionless elastic modulus of the aerated material decreases as the bubble volume fraction $\phi $ increases, suggesting that bubbles behave as soft elastic inclusions. Consistently, this decrease is all the sharper as $\textit{Ca}_{_G}$ is high, which accounts for the softening of the bubbles as compared to the paste. By contrast, we find that the yield stress of most studied materials is not modified by the presence of bubbles. This suggests that their plastic behavior is governed by the plastic capillary number $\textit{Ca}_{\tau_y}$ , defined as the ratio of the paste yield stress to the bubble capillary pressure. At low $\textit{Ca}_{\tau_y}$ values, bubbles behave as nondeformable inclusions, and we predict that the suspension dimensionless yield stress should remain close to unity, in agreement with our data up to $\textit{Ca}_{\tau_y}=0.2$ . When preparing systems with a larger target value of $\textit{Ca}_{\tau_y}$ , we observe bubble breakup during mixing, which means that they have been deformed by shear. It then seems that a critical value $\textit{Ca}_{\tau_y}\simeq 0.2$ is never exceeded in the final material. These observations might imply that, in bubble suspensions prepared by mixing a foam and a paste, the suspension yield stress is always close to that of the paste surrounding the bubbles. Finally, at the highest $\phi $ investigated, the yield stress is shown to increase abruptly with $\phi $ : this is interpreted as a “foamy yield stress fluid” regime, which takes place when the paste mesoscopic constitutive elements (here, the oil droplets) are strongly confined in the films between the bubbles.     

Wall shear stress modified by bubbles in a horizontal channel flow of silicone oil in the transition region

We performed laboratory experiments on bubbly channel flows using silicone oil, which has a low surface tension and clean interface to bubbles, as a test fluid to evaluate the wall shear stress modification for different regimes of bubble migration status. The channel Reynolds numbers of the flow ranged from 1000 to 5000, covering laminar, transition and turbulent flow regimes. The bubble deformation and swarms were classified as packing, film, foam, dispersed, and stretched states based on visualization of bubbles as a bulk void fraction changed. In the dispersed and film states, the wall shear stress reduced by 9% from that in the single-phase condition; by contrast, the wall shear stress increased in the stretched, packing, and foam states. We carried out statistical analysis of the time-series of the wall shear stress in the transition and turbulent-flow regimes. Variations of the PDF of the shear stress and the higher order moments in the statistic indicated that the injection of bubbles generated pseudo-turbulence in the transition regime and suppressed drag-inducing events in the turbulent regime. Bubble images and measurements of shear stress revealed a correlated wave with a time lag, for which we discuss associated to the bubble dynamics and effective viscosity of the bubble mixture in wall proximity.     

Modelling large-bubble formation at submerged orifices

The formation of large gas bubbles at submerged orifices is investigated numerically with a two-dimensional, transient, finite difference model using a volume fraction specification to track the movement of the gas-liquid interface. Experimentally observed features of large-bubble formation such as the initial toroidal shape of the bubbles and the penetration of liquid down the pipe centreline are well predicted by the model. The expected oscillatory nature of growth is also observed. The bubble departure volume corresponds to experiments and to the model of Davidson and Schuler. At present the simulations do not extend far enough to investigate multiple-bubble ejection and important bubble-to-bubble interactions during growth and after departure.     

Direct 2D simulation of small gas bubble clusters: from the expansion step to the equilibrium state

A numerical strategy, based on an adaptive finite element method, is proposed for the direct two‐dimensional simulation of the expansion of small clusters of gas bubbles within a Newtonian liquid matrix. The velocity and pressure fields in the liquid are first defined through the Stokes equations and are subsequently extended to the gas bubbles. The liquid–gas coupling is imposed through the stress exerted on the liquid by gas pressure (ruled by an ideal gas law) and by surface tension. A level set method, combined with a mesh adaptation technique, is used to track liquid–gas interfaces. Many numerical simulations are presented. The single bubble case allows to compare the simulations to an analytical model. Simulations of the expansion of small clusters are then presented showing the interaction and evolution of the gas bubbles to an equilibrium state, involving topological rearrangements induced by Plateau's rule. Copyright © 2006 John Wiley & Sons, Ltd.     

多孔介质中稳定泡沫的流动计算及实验研究

将多孔介质简化为一簇变截面毛管束,根据多孔介质的颗粒直径、颗粒排列方式、孔喉尺度比及束缚水饱和度,计算出变截面毛细管的喉道半径和孔隙半径. 在考虑多孔介质喉道和孔隙中单个气泡的受力和变形基础上,利用动量守恒定理,推导出单个孔隙单元内液相的压力分布和孔隙单元两端的压差计算公式,最终得到多孔介质的压力分布计算公式. 利用长U型填砂管对稳定泡沫的流动特性进行了实验研究. 研究结果表明:稳定泡沫流动时多孔介质中的压力分布呈线性下降,影响泡沫在多孔介质中流动特性的因素包括:多孔介质的孔喉结构、泡沫流体的流量和干度、气液界面张力、气泡尺寸,其中孔喉结构和泡沫干度是影响泡沫封堵能力的主要因素.关键词: 稳定泡沫;多孔介质;变截面毛管;流动;表观粘度;压力分布;实验研究      

On the numerical study of bubbly flow created by ventilated cavity in vertical pipe

The dispersion of bubbles into a down-liquid flow in a vertical pipe is investigated. At low flow rates, the intended design of a swarm of discrete bubbles is achieved. At high flow rates, a ventilated cavity is nonetheless formed, which is attached close to the gas sparger. Behind this ventilated cavity, three different flow regimes characterize the complex bubbly flow field downstream of the down-liquid flow: vortex region with high void fraction, transitional region and pipe flow region. In this study, a numerical model that solved the entire development of the gas–liquid flow including the extended single-phase liquid region upstream to the wall-jet and recirculating-vortex zones in order to allow a more realistic determination of the boundary conditions of the down-liquid flow was adopted. Coupling with the Eulerian–Eulerian two-fluid model to solve the respective gas and liquid phases, a population balance model was also applied to predict the bubble size distribution in the wake right below the cavity base as well as further downstream in the transitional and fully-developed pipe flow regions. The numerical model was evaluated by comparing the numerical results against the data derived from theoretical, numerical and experimental approaches. Prediction of the Sauter mean bubble diameter distributions by the population balance approach at different axial locations confirmed the dominance of breakage due to the high turbulent intensity below the ventilated cavity which led to the generation of small gas bubbles at high void fraction. Further downstream, the coalescence effect dominated leading to merging of the small bubbles to form bigger bubbles.     

垂直上升圆管内等温泡状流的数值模拟

介绍了描述等温状态下垂直气泡流流动的总体平衡方法(population balanceapproach). 将平均气泡数密度控制方程引入到双流体模型中实现总体平衡. 介绍了描述气泡聚并与破裂机制的Yao和Morel理论模型以及Hibiki等人的实验方法. 利用商业计算流体力学软件ANSYSCFX10, 对圆管特定位置上的5个基本变量:气含率、气泡平均直径、相间表面积浓度以及气体和液体速度沿半径分布进行了模拟计算. 预测结果与实验数据的比较表明,预测结果和实测数据之间有很好的一致性.      

The interaction of background ocean air bubbles with a surface ship

A two-fluid model suitable for the calculation of the two-phase flow field around a naval surface ship is presented. This model couples the Reynolds-averaged Navier–Stokes (RANS) equations with equations for the evolution of the gas-phase momentum, volume fraction and bubble number density, thereby allowing the multidimensional calculation of the two-phase flow for monodisperse variable size bubbles. The bubble field modifies the liquid solution through changes in the liquid mass and momentum conservation equations. The model is applied to the case of the scavenging of wind-induced sea-background bubbles by an unpropelled US Navy frigate under non-zero Froude number boundary conditions at the free surface. This is an important test case, because it can be simulated experimentally with a model-scale ship in a towing tank. A significant modification of the background bubble field is predicted in the wake of the ship, where bubble depletion occurs along with a reduction in the bubble size due to dissolution. This effect is due to lateral phase distribution phenomena and the generation of an upwelling plume in the near wake that brings smaller bubbles up to the surface. © 1998 John Wiley & Sons, Ltd.     

The Effect of Bubbly Oil on the THD Lubrication of Tilting-pad Journal Bearings Subjected to Rotating Unbalance Load

Little work has been published on the contribution of entrained air and/or gas bubbles on the lubricant viscosity and hence on the hydrodynamic performance of bearings. In this work, a thermo-hydrodynamic analysis of the performance of dynamically loaded tilting-pad journal bearing lubricated by bubbly oil is carried out. The non-steady Reynolds equation for compressible fluid and the energy equation are solved iteratively using finite difference method to study the effect of air bubbles on the bearing performance characteristics. The effect of bubbles content on the lubricant viscosity is considered.The results showed that the value of maximum pressure slightly increases with increasing the bubble ratio up to 0.2, while the film thickness deteriorates at very high bubble content. The pressure peak moves in the downstream direction with the increase of bubble ratio ₁, which would be of great importance in deciding the pivot location in tilting-pad bearings.     

Shock structure in bubbly liquids: comparison of direct numerical simulations and model equations

Shock wave structure in a bubbly mixture composed of a cluster of gas bubbles in a quiescent liquid with initial void fractions around 10% inside a 3D rectangular domain excited by a sudden increase in the pressure at one boundary is investigated using the front tracking/finite volume method. The effects of bubble/bubble interactions and bubble deformations are, therefore, investigated for further modeling. The liquid is taken to be incompressible while the bubbles are assumed to be compressible. The gas pressure inside the bubbles is taken uniform and is assumed to vary isothermally. Results obtained for the pressure distribution at different locations along the direction of propagation show the characteristics of one-dimensional unsteady shock propagation evolving towards steady-state. The steady-state shock structures obtained by the present direct numerical simulations, which show a transition from A-type to C-type steady-state shock structures, are compared with those obtained by the classical Rayleigh–Plesset equation and by a modified Rayleigh–Plesset equation accounting for bubble/bubble interactions in the mean-field theory.      

Pressure wave breakdown of a gas film in a liquid-filled slit

In [1–4] the results of investigating the breakdown of gas bubbles by medium-intensity pressure waves are presented and various bubble breakdown mechanisms are proposed. It is shown that breakdown may occur as a result of the formation of a cumulative jet on the boundary of the bubble or as a result of instability due to the relative motion of the bubble in the wave. In [5] experimental data on the pressure wave breakdown of a gas film in a liquid on a solid wall are reported. It is shown that at wave amplitudes p/p₀1 a liquid jet is formed at the edge of the gas film. The jet, traveling along the wall, strips off the film and carries it into the surrounding liquid. Below we investigate the pressure wave behavior of a gas film in a liquid-filled slit.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 175–178, September–October, 1992.     

Bubbly jets in stagnant water

Air–water bubbly jets are studied experimentally in a relatively large water tank with a gas volume fraction, C_o, of up to 80% and nozzle Reynolds number, Re, ranging from 3500 to 17,700. Measurements of bubble properties and mean axial water velocity are obtained and two groups of experiments are identified, one with relatively uniform bubble sizes and another with large and irregular bubbles. For the first group, dimensionless relationships are obtained to describe bubble properties and mean liquid flow structure as functions of C_o and Re. Measurements of bubble slip velocity and estimates of the drag coefficient are also provided and compared to those for isolated bubbles from the literature. The study confirms the importance of bubble interactions to the dynamics of bubbly flows. Bubble breakup processes are also investigated for bubbly jets. It was found that a nozzle Reynolds number larger than 8000 is needed to cause breakup of larger bubbles into smaller bubbles and to produce a more uniform bubble size distribution. Moreover, the Weber number based on the mean water velocity appears to be a better criteria than the Weber number based on the bubble slip velocity to describe the onset of bubble breakup away from the nozzle, which occurs at a Weber number larger than 25.     

基于气泡形态影响的水下气幕对冲击波衰减效果分析

气泡帷幕是水下爆炸冲击波防护的重要手段,对其作用机理及技术参数的深入研究对水下爆破安全与应用具有重要意义。采用高速摄影技术对室内小型水下气泡帷幕模型拍摄发现气幕在形成过程和与水下爆炸冲击波相互作用过程中均具有高度非连续性和非均匀性,且气幕区域内气体与液体混杂,界面轮廓复杂多样。在此基础上,考虑气泡形状及界面影响下,通过LS-DYNA有限元软件自带的APDL语言进行编程,实现了在设定的气幕区域内,通过设定气泡直径变化范围及气泡直径之间的最小差异值随机投放一定数量不同直径的气泡来模拟真实气幕中气泡的分布,并通过改变固定区域内气泡个数来模拟不同气压值工况下的气幕效果。分析发现该方法能够更加真实反映气幕在冲击波防护过程中的防护机理,随着单位区域内气泡数量的增大,防护效果越明显,但当气泡数量达到一定数量后气幕整体连续性及稳定性基本固定,防护效果也趋于稳定。