Lattice gas simulations of two-dimensional liquid foams

Liquid foam is a dense random packing of gas bubbles in a small amount of immiscible liquid containing surfactants. The liquid within the Plateau borders, although small in volume, causes considerable difficulties to investigations of the physical properties of foams, and the situation becomes even more complicated if the flow of the liquid through the foam is considered too. Here we propose a fresh approach to tackling these issues by introducing a discrete two-dimensional hybrid lattice gas model of liquid foams. While lattice gas models have been used to model two-phase liquids in the past, their application to the study of liquid foams is novel and proves promising. We represent bubble surfaces by a finite number of nodes, and model the surrounding liquid as a lattice gas (with a finite number of liquid particles). The gas in the bubbles is treated as an ideal gas at constant temperature. The model is tested by choosing an arbitrarily shaped bubble that evolves into a circular shape in agreement with Laplaces law. The model is then employed to simulate periodic ordered and disordered dry and wet foams. Since our model is specifically designed to handle wet foams up to a critical liquid fraction of 0.16 (void fraction of random packing of disks), we are able to compute the variation in coordination number (average number of neighbours of a bubble) over the whole range of liquid fractions, and we find it to be a linear function of the shear modulus.This paper was presented at the first Annual European Rheology Conference (AERC) held in Guimarães, Portugal, 11–13 September 2003.     

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.     

液态泡沫渗流的机理研究进展

液态泡沫是具有高度自组织结构的非平衡系统. 泡沫中的微量液体在重力与毛细管力作用下, 在由薄膜、柏拉图通道以及交汇点形成的通道网络内的流动称为泡沫渗流(foamdrainage), 它直接影响泡沫结构的稳定性. 本文从泡沫物理学角度对液态泡沫的结构首先做了简单介绍, 并对泡沫材料与多孔介质之间的区别和联系做了简单介绍. 文章分析了由泡沫渗流、气泡粗化和液膜破裂而引起的泡沫结构演化规律, 着重介绍了目前泡沫渗流研究中对柏拉图通道边界条件的处理方法、相应的渗流模型及其在一维泡沫渗流中的应用, 并对泡沫渗流实验检测手段及微重力条件下的泡沫渗流实验和理论研究做了综述.      

Liquid distribution and electrical conductivity in foam

Experiments were conducted with aqueous foam generated by bubbling nitrogen through anionic, cationic and nonionic surfactant solutions. The ratio of the electrical conductivity of the foam to that of the liquid was found to increase monotonically with the volumetric liquid fraction in the foam. The choice of surfactant as well as the degree of inhomogeneity in bubble size were found to be without effect on the relationship. However, for a fixed liquid fraction, it was found that decreasing the mean bubble size can decrease the conductivity ratio somewhat, as well as accelerate the approach toward Lemlich's limit for low density foam. This effect was attributed to increased suction within the Plateau borders.     

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      

Elongated bubble shape in inclined air-water slug flow

The shape of elongated bubbles in upward inclined air-water slug flow was studied experimentally by quantitative measurements of the cross sectional distribution of the phases within the pipe, using a wire mesh sensor. Ensemble-averaged shapes of elongated bubbles were determined for a wide range of gas and liquid flow rates, as well as for different pipe inclination angles. The elongated bubble nose can be characterized by an annular domain where liquid is present above the gas. The effect of gas and liquid flow rates, as well as of the pipe inclination angle on the bubble shape (front and tail) is studied. A simplified theoretical model is proposed to determine the bubble front shape. The model predictions compare favorably with the experimental results.     

Numerical study of macroscopical drainage process in fabricating foamed aluminum using microscopical method

The velocity field in a single Plateau border（PB） of the aluminum foam in the drainage process is studied using a mathematical model for the flow inside a microchannel.We show that the liquid/gas interface mobility characterized by the Newtonian surface viscosity has a substantial effect on the velocity inside the single PB.With the same liquid/gas interfacial mobility and the same radius of the curvature,the maximum velocity inside an exterior PB is about 6～8 times as large as that inside an interior PB.We also find a critical value of the interfacial mobility in the interior PB.For the values greater and less than this critical value,the effects of the film thickness on the velocity in the PB show opposite tendencies.Based on the multiscale methodology,with the coupling between the microscale and the macroscale and the results obtained from the microscopical model,a simplified macroscopical drainage model is presented for the aluminum foams.The comparisons among the computational results obtained from the present model,the experimental data quoted in the literature,and the results of the classical drainage equation show a reasonable agreement.The computational results reveal that the liquid holdup of the foams is strongly dependent on the value of the mobility and the bubble radius.     

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.     

Experimental investigation of horizontal air–water bubbly-to-plug and bubbly-to-slug transition flows in a 3.81 cm ID pipe

The present study seeks to investigate horizontal bubbly-to-plug and bubbly-to-slug transition flows. The two-phase flow structures and transition mechanisms in these transition flows are studied based on experimental database established using the local four-sensor conductivity probe in a 3.81 cm inner diameter pipe. While slug flow needs to be distinguished from plug flow due to the presence of large number of small bubbles (and thus, large interfacial area concentration), both differences and similarities are observed in the evolution of interfacial structures in bubbly-to-plug and bubbly-to-slug transitions. The bubbly-to-plug transition is studied by decreasing the liquid flow rate at a fixed gas flow rate. It is found that as the liquid flow rate is lowered, bubbles pack near the top wall of the pipe due to the diminished role of turbulent mixing. As the flow rate is lowered further, bubbles begin to coalesce and form the large bubbles characteristic of plug flow. Bubble size increases while bubble velocity decreases as liquid flow rate decreases, and the profile of the bubble velocity changes its shape due to the changing interfacial structure. The bubbly-to-slug transition is investigated by increasing the gas flow rate at a fixed liquid flow rate. In this transition, gas phase becomes more uniformly distributed throughout the cross-section due to the formation of large bubbles and the increasing bubble-induced turbulence. The size of small bubbles decreases while bubble velocity increases as gas flow rate increases. The distributions of bubble size and bubble velocity become more symmetric in this transition. While differences are observed in these two transitions, similarities are also noticed. As bubbly-to-plug or bubbly-to-slug transition occurs, the formation of large elongated bubbles is observed not in the uppermost region of bubble layer, but in a lower region. At the beginning of transitions, relative differences in phase velocities near the top of the pipe cross-section to those near the pipe center become larger for both gas and liquid phases, because more densely packed bubbles introduce more resistance to both phases.     

电场作用下气泡上升行为特性的数值计算研究

利用电场控制气泡形态及运动,强化气液相间传热传质是电流体动力学的重要研究内容之一. 然而目前多数研究集中在非电场下的气泡动力学上,对于电场下的气泡行为特性及电场的作用机制仍需开展深入研究. 本研究对电场作用下单个气泡在流体中上升过程的动力学行为进行了数值模拟研究. 在建立二维模型的基础上求解电场方程与Navier-Stokes方程,并采用水平集方法捕捉了上升气泡的位置及形状. 模拟结果的准确性与有效性通过与前人实验和数值结果进行对比得到了验证. 通过改变雷诺数、邦德数和电邦德数等不同参数研究了电场下液体黏度、表面张力和电场力对气泡运动变形的影响. 计算结果表明,电场对气泡的动态特性有显著影响. 非电场情况下液体黏度和表面张力较大时气泡基本维持球状,反之气泡发生变形并逐步达到稳定状态. 此外,电场作用使气泡在初始上升阶段发生剧烈形变,随着不断上升,气泡形变程度不断减小,且气泡的上升速度和长径比均出现振荡. 垂直电场使气泡的上升速度有较大的提高,且随着电邦德数的增大,难以达到相对稳定的状态.      

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.     

Numerical study on the hydrodynamic behavior of the dielectric fluid around an electrical discharge generated bubble in EDM

In the process of EDM, due to the electrical current, very small bubbles are created within the gap. These bubbles are connected to each other and generate a single bubble. The vapor bubble continues to grow until it finally collapses to small bubbles. The bubble behavior can be ascertained on the distribution of the pressure in the dielectric fluid around the bubble. In this paper, velocity fields and pressure distribution in the dielectric fluid around the bubble that is generated in the process of EDM are investigated numerically. The tool and the workpiece are assumed as two parallel rigid boundaries with dielectric liquid between them. The boundary integral equation method is applied for the numerical solution of the problem. This study can lead to better understanding of the bubble importance in the performance of the electrical discharge machining process.     

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.     

Investigation of surfactant effect on the bubble shape and mass transfer in a milli-channel using high-resolution microfocus X-ray imaging

In this paper we present an experimental study on the influence of surface active agents (surfactants) on Taylor bubble flow in a vertical millimeter-size channel. Moreover we give a short review on the subject and previous investigations. We investigated the shape and dissolution rate of individual elongated carbon dioxide Taylor bubbles, which were hydraulically fixed in a downward flow of water. Bubble shape and dissolution rate was determined from microfocus X-ray radiographs. From the shrinking rate we calculated the liquid side mass transfer coefficient.The results show that the presence of surfactants causes a change of the bubble shape and leads to a slight increase of the liquid film thickness around the bubble and as a result the elongation of contaminated bubbles. In addition, the comparison of clean and contaminated bubbles indicate that presence of surfactant has a more significant impact on the dissolution rate of small bubbles. Furthermore, applying different concentrations of surfactant reveals that in our case, where surface coverage ratio of surfactant on the bubbles is high, increase of contamination does not have a noticeable influence on the mass transfer coefficient of bubbles.     

近自由面的多个水下爆炸气泡相互作用研究

将气泡运动阶段周围的流场假设为无黏、无旋、不可压缩的理想流体,运用边界积分法模拟流场中气泡的运动,并开发了三维计算程序,计算值与实验值吻合较好.用该方法模拟了近自由面多气泡之间的相互作用,包括同相气泡和异相气泡. 通过计算发现,气泡的周期随两气泡中心的距离减小而增大,这是由于多气泡之间存在抑制作用,特别是对异相气泡,这种抑制作用更加明显,称之为多气泡之间的抑制效应. 无论有、无自由面存在,多气泡之间均存在抑制效应,由于抑制效应导致同相与异相气泡相互耦合作用的动态特性存在巨大的差异,这些现象可为将来研究多个同时或延时产生的水下爆炸气泡的威力提供参考.      

Pool boiling heat transfer of ultra-light copper foam with open cells

High speed visualizations and thermal performance studies of pool boiling heat transfer on copper foam covers were performed at atmospheric pressure, with the heating surface area of 12.0 mm by 12.0 mm, using acetone as the working fluid. The foam covers have ppi (pores per inch) from 30 to 90, cover thickness from 2.0 to 5.0 mm, and porosity of 0.88 and 0.95. The surface superheats are from −20 to 190 K, and the heat fluxes reach 140 W/cm². The 30 and 60 ppi foam covers show the periodic single bubble generation and departure pattern at low surface superheats. With continuous increases in surface superheats, they show the periodic bubble coalescence and/or re-coalescence pattern. Cage bubbles were observed to be those with liquid filled inside and vented to the pool liquid. For the 90 ppi foam covers, the bubble coalescence takes place at low surface superheats. At moderate or large surface superheats, vapor fragments continuously escape to the pool liquid.     

Quasi-static bubble in a yield stress fluid: elasto-plastic model

The influence of the yield stress of Carbopol® gel dispersions on the behaviour of quasi-static bubbles was investigated. Many fluids, from many different industrial fields, have yield stress behaviour. Most of them contain gas bubbles. To study bubble behaviour in such suspensions, a transparent model fluid (dispersion of Carbopol® in water) was used. The experimental device allowed to quasi-statically increase bubble internal pressure with small pressure step to reach a maximum target internal pressure and the pressure setpoint was inverted to return to the initial pressure. Hysterical behaviour of the bubbles was highlighted as they did not regain their initial shape because of yield stress. We show that the rheological behaviour is related to the internal pressure, bubble geometry and yield stress in quasi-static conditions. A modification of the Laplace law depending on the yield stress of the fluid and bubble sphericity was proposed and validated.     

Wake properties of a single gas bubble in three-dimensional liquid-solid fluidized bed

Experiments were performed to investigate the wake properties of a single gas bubble in a three-dimensional liquid-solid fluidized bed via a video camera moving at the same speed as the bubble. The solids holdup in the fluidized bed varied up to around 10%. The bubble size varied from 5 to 20 mm with corresponding bubble Reynolds numbers ranging from 1000 to 6500. The bubble was observed to have two types of wake configurations depending on the bubble size: the asymmetric/helical vortex wake for small bubbles and the symmetric wake for large bubbles. The bubble shape and relative rise velocity in the fluidized bed can be well-represented by correlations developed for single bubbles in liquid media, although the bubble shape in liquid-solid media is slightly more flattened compared to that in liquid media. The bubble rocking frequency was found to be independent of particle properties and to correspond in magnitude to the vortex shedding frequency in a two-dimensional liquid-solid fluidized bed. The average primary wake size in three dimensions is comparable to that in two dimensions.     

三维气泡与自由表面相互作用的直接数值模拟

采用VOF中的PLIC界面重构方法数值模拟了三维气泡在液流中上升并与自由表面相互作用的运动.分别考察了不同初始高度,有无来流及有无再生气泡对气泡上升高度、上升速度、压力及与自由表面相互作用等的影响.结果表明:气泡初始位置越低,顶端上升的高度越大,自由面隆起的范围更广.越靠近自由表面,底部射流横向发展越窄,而向上的压力梯度,气泡上升速度,底部射流上升高度越大,反之则反;但如果底部射流均在接近自由表面以前已横向发展充分,则差别不大.气泡外形、上升高度、破裂时间以及上升速度与来流无关.产生再生气泡后,原生气泡与再生气泡相吸,相互加速对方的上升;自由表面抬升的高度增幅较大,范围拓宽,上升速度也大大增加,且再生气泡越多,自由表面隆起的范围越大.