Numerical simulations of droplet formation in a cross‐junction microchannel by the lattice Boltzmann method

An immiscible liquid–liquid multiphase flow in a cross‐junction microchannel was numerically studied using the lattice Boltzmann method. An improved, immiscible lattice BGK model was proposed by introducing surface tension force based on the continuum surface force (CSF) method. Recoloring step was replaced by the anti‐diffusion scheme in the mixed region to reduce the side‐effect and control the thickness of the interface. The present method was tested by the simulation of a static bubble. Laplace's law and spurious velocities were examined. The results show that our model is more advantageous for simulations of immiscible fluids than the existing immiscible lattice BGK models. Computational results of multiphase flow in a cross‐junction microchannel were obtained and analyzed based on dimensionless numbers. It is found that the flow pattern is decided mostly by the capillary number at a small inlet flux. However, at the same capillary number, a large inlet flux will lead to much smaller droplet generation. For this case, the flow is determined by both the capillary number and the Weber number. Copyright © 2007 John Wiley & Sons, Ltd.     

Interface‐fitted moving mesh method for axisymmetric two‐phase flow in microchannels

A boundary‐fitted moving mesh scheme is presented for the simulation of two‐phase flow in two‐dimensional and axisymmetric geometries. The incompressible Navier‐Stokes equations are solved using the finite element method, and the mini element is used to satisfy the inf‐sup condition. The interface between the phases is represented explicitly by an interface adapted mesh, thus allowing a sharp transition of the fluid properties. Surface tension is modelled as a volume force and is discretized in a consistent manner, thus allowing to obtain exact equilibrium (up to rounding errors) with the pressure gradient. This is demonstrated for a spherical droplet moving in a constant flow field. The curvature of the interface, required for the surface tension term, is efficiently computed with simple but very accurate geometric formulas. An adaptive moving mesh technique, where smoothing mesh velocities and remeshing are used to preserve the mesh quality, is developed and presented. Mesh refinement strategies, allowing tailoring of the refinement of the computational mesh, are also discussed. Accuracy and robustness of the present method are demonstrated on several validation test cases. The method is developed with the prospect of being applied to microfluidic flows and the simulation of microchannel evaporators used for electronics cooling. Therefore, the simulation results for the flow of a bubble in a microchannel are presented and compared to experimental data.     

MOVEMENT AND SPLASHING OF A DROPLET IMPACTING ON A WET WALL

壁面液体层的存在对液滴撞击壁面的运动具有重要的影响。采用气液两相流动相界面追踪的水平集和流体体积复合方法和壁面润湿模型，实现了液滴撞击湿润壁面运动的数值求解；在此基础上，开展了液滴撞击湿润壁面运动的研究。研究结果表明：液滴以不同速度撞击湿润壁面时，会呈现出黏附铺展、波动运动、皇冠几何体运动以及飞溅运动等几种不同的运动形态，液滴撞击湿润壁面后的压力分布是不同运动形态形成的主要原因；飞溅运动是一定条件下皇冠几何体运动的一种特殊形态，液滴从皇冠几何体侧壁顶端的飞溅分离满足毛细破碎理论；撞击速度对分离液滴的运动方向影响较小，而对壁面液体层厚度的影响则较大；撞击速度和壁面液体层厚度对分离液滴形态、飞溅分离位置、飞溅速度以及飞溅发生时刻等都具有一定的影响。     

Simulations of free surface flows with implementation of surface tension and interface sharpening in the two-fluid model

Two-fluid model used for free surface flows with large characteristic scales is improved; the smeared interface is sharpened with conservative level set method and the surface tension force with wetting angle is implemented. Surface tension force is split between two phases with several models. Detailed analysis showed the splitting of surface tension force with volume averaging as the most appropriate. The improved two-fluid model with interface sharpening and implemented surface tension is validated on several test cases. The pressure jump over a droplet interface test case showed that the pressure jump in simulation converges with grid refinement to the analytical one. The parasitic currents in simulation are one order of magnitude larger than in simulation with volume of fluid model. In the oscillating droplet test case the time period of oscillating droplet with initially ellipsoid or square shape is similar to the analytical time period. In the rising bubble test case, the rising bubble position, terminal velocity, and circularity are similar to the one observed in simulations with level set model. The wetting angle is implemented in the two-fluid model with interface sharpening and surface tension force. Model is tested in the simulation of droplet in contact with wall with different wetting angles.     

液滴撞击具有浸润性分布的倾斜固壁铺展行为的格子Boltzmann研究

采用基于单组分多相伪势模型的格子Boltzmann方法,模拟了三维液滴撞击左右两侧浸润性不同的倾斜固壁的铺展过程,获得了液滴在壁面两侧的铺展因子、相对铺展宽度、相对高度和液滴运动速度随时间的变化情况,研究了壁面浸润性分布和壁面倾斜角度对液滴铺展过程的影响．结果表明,液滴在倾斜壁面的铺展过程受到重力和表面力的综合作用,重力影响液滴的铺展和沿壁面向下的滑动,壁面浸润性分布影响液滴向壁面亲水侧横向移动．     

A filter‐based,mass‐conserving lattice Boltzmann method for immiscible multiphase flows

Some issues of He–Chen–Zhang lattice Boltzmann equation (LBE) method (referred as HCZ model) (J. Comput. Physics 1999; 152 :642–663) for immiscible multiphase flows with large density ratio are assessed in this paper. An extended HCZ model with a filter technique and mass correction procedure is proposed based on HCZ's LBE multiphase model. The original HCZ model is capable of maintaining a thin interface but is prone to generating unphysical oscillations in surface tension and index function at moderate values of density ratio. With a filtering technique, the monotonic variation of the index function across the interface is maintained with larger density ratio. Kim's surface tension formulation for diffuse–interface method (J. Comput. Physics 2005; 204 :784–804) is then used to remove unphysical oscillation in the surface tension. Furthermore, as the density ratio increases, the effect of velocity divergence term neglected in the original HCZ model causes significant unphysical mass sources near the interface. By keeping the velocity divergence term, the unphysical mass sources near the interface can be removed with large density ratio. The long‐time accumulation of the modeling and/or numerical errors in the HCZ model also results in the error of mass conservation of each dispersed phase. A mass correction procedure is devised to improve the performance of the method in this regard. For flows over a stationary and a rising bubble, and capillary waves with density ratio up to 100, the present approach yields solutions with interface thickness of about five to six lattices and no long‐time diffusion, significantly advancing the performance of the LBE method for multiphase flow simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

An approach to control the spurious currents in a multiphase lattice Boltzmann method and to improve the implementation of initial condition

Multiphase lattice Boltzmann methods are known to generate spurious or parasitic currents at the fluid–fluid interfaces. This nonphysical phenomenon has to be avoided, or at least controlled, in order to achieve reliable solutions. In this article, a method to control these fictitious velocities via lattice refinement is proposed, which is based on interface thickness control for which both the spurious currents and the physical fluid–fluid interface thickness vanishes as the spatial resolution increases. It has been found that a proper interface thickness adjustment is required as the lattice refinement is applied, or an increase in spurious currents, instead of a reduction, can occur. By combining the new method with an appropriate multiphase flow initialization, the overall stability for high density O(1000) and viscosity O(100) ratios is greatly improved. Although this research has been conducted with a Rothman and Keller type lattice Boltzmann model, it is believed that other types of multiphase lattice Boltzmann models could benefit from the basic ideas underlying this research. Copyright © 2015 John Wiley & Sons, Ltd.     

Lattice-Boltzmann model for compressible perfect gases

We present an adaptive lattice Boltzmann model to simulate supersonic flows. The particle velocities are determined by the mean velocity and internal energy. The adaptive nature of particle velocities permits the mean flow to have high Mach number. A particle potential energy is introduced so that the model is suitable for the perfect gas with arbitrary specific heat ratio. The Navier-Stokes equations are derived by the Chapman-Enskog method from the BGK Boltzmann equation. As preliminary tests, two kinds of simulations have been performed on hexagonal lattices. One is the one-dimensional simulation for sinusoidal velocity distributions. The velocity distributions are compared with the analytical solution and the measured viscosity is compared with the theoretical values. The agreements are basically good. However, the discretion error may cause some non-isotropic effects. The other simulation is the 29 degree shock reflection. The project supported by the National Natural Science Foundation of China (Grant Nos. 19672030 and 19972037) and by the Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry     

气液非混相驱替过程中的卡断机理及模拟研究

研究气液非混相驱替过程中的相界面卡断机理及其影响因素在气驱, 气水交替及泡沫驱等提高油气采收率领域具有重要意义. 本文在原始伪势格子玻尔兹曼模型的基础上, 改进流体-流体作用力格式, 添加流-固作用力, 耦合RK状态方程, 并采用精确差分方法将外力添加到LBM框架中. 通过校准模型的热力学一致性以及模拟测试界面张力, 静态平衡接触角及液相在角隅的滞留等一系列两相体系验证模型的准确性. 基于改进的伪势格子玻尔兹曼模型, 在孔-喉-孔系统中开展气液非混相驱替模拟, 结果表明: 卡断现象与驱替压差, 孔喉长度比及孔喉宽度比有关, 只有当驱替压差处于一定范围内时, 气液两相驱替过程中才会发生卡断现象; 当驱替压差大于临界驱替压差上限时, 即使达到了经典静态准则所预测的卡断条件, 卡断也会被抑制; 当驱替压差小于临界驱替压差下限时, 无法克服毛管“钉扎”作用, 形成无效驱替. 对于固定孔喉宽度比的孔-喉-孔结构, 随着孔喉长度比的增大, 发生卡断现象的驱替压差范围增大; 对于固定孔喉长度比的孔-喉-孔结构, 随着孔喉宽度比的减小, 发生卡断现象的驱替压差范围增大.      

Hybrid lattice Boltzmann finite‐difference simulation of axisymmetric swirling and rotating flows

The axisymmetric flows with swirl or rotation were solved by a hybrid scheme with lattice Boltzmann method for the axial and radial velocities and finite‐difference method for the azimuthal (or swirl) velocity and the temperature. An incompressible axisymmetric lattice Boltzmann D2Q9 model was proposed to solve the axial and radial velocities through inserting source terms into the two‐dimensional lattice Boltzmann equation. Present hybrid scheme was firstly validated by simulations of Taylor–Couette flows between two concentric cylinders. Then the benchmark problems of melt flow in Czochralski crystal growth were studied and accurate results were obtained. Numerical experiment demonstrated that present axisymmetric D2Q9 model is more stable than the previous axisymmetric D2Q9 model (J. Comp. Phys. 2003; 186 (1):295–307). Hence, compared with the previous model, present numerical method provides a significant advantage in simulation melt flow cases with high Reynolds number and high Grashof number. Copyright © 2006 John Wiley & Sons, Ltd.     

A multi‐component lattice Boltzmann model with non‐uniform interfacial tension module for the study of blood flow in the microvasculature

This study aims at analyzing the red blood cell (RBC) deformation and velocity while streaming through venules and through capillaries whose diameters are smaller than the RBC size. The characteristics of the RBC shape change and velocity can potentially help in diagnosing diseases. In this work, the RBC is considered as a surfactant‐covered droplet. This is justified by the fact that the cell membrane liquefies under pressure in the capillaries, and this allows the marginalization of its mechanical properties. The RBC membrane is in fact a macro‐colloid containing lipid surfactant. When liquefied, it can be considered as a droplet of immiscible hemoglobin covered with lipid surfactant in a plasma surrounding. The local gradient in the surface tension due to non‐uniform local interface surfactant distribution is neglected here, and a non‐uniform zonal‐averaged value of surface tension representative of the surfactant bulk zonal concentration is rather implemented. The interplay between the surface tension geometry and the hydrodynamic conditions determines the droplet shape by affecting a change in its Weber number, and influences its velocity. The Gunstensen lattice Boltzmann model for immiscible fluids is used here since it provides independent adjustment of the local surface tension, and allows the use of fluids with viscosity contrast. The proposed concept was used to investigate the dynamic shape change of the RBC while flowing through the microvasculature, and to explore the Fahraeus and the Fahraeus–Lindqvist effects. Copyright © 2010 John Wiley & Sons, Ltd.     

Interfacial flows in corrugated microchannels: Flow regimes,transitions and hysteresis

We report simulations of gas–liquid two-phase flows in microchannels periodically patterned with grooves and ridges. A constant effective body force is applied on both fluids to simulate a pressure-driven creeping flow, and a diffuse-interface model is used to compute the interfacial evolution and the contact line motion. Depending on the body force, capillary force and the level of liquid saturation, a number of flow regimes may appear in the corrugated microchannel: gas flow, blockage, liquid flow, bubble–slug flow, droplet flow, annular flow and annular-droplet flow. A map of flow regimes is constructed for a set of geometric and flow parameters starting from a prescribed initial configuration. Some of the regimes are new, while others have been observed before in straight tubes and pipes. The latter are compared with previous experiments in terms of the regime map and the holdup ratio. The transition among flow regimes shows significant hysteresis, largely owing to the pinning of the interface at sharp corners in the flow conduit. Hysteresis is reduced if the sharp corners are rounded. Under the same operating conditions, different flow regimes can be realized from different initial conditions. The roles of geometry and wettability of the channel walls are also elucidated.     

Modeling surface tension of a two‐dimensional droplet using smoothed particle hydrodynamics

Surface tension plays a significant role at the dynamic interface of free‐surface flows especially at the microscale in capillary‐dominated flows. A model for accurately predicting the formation of two‐dimensional viscous droplets in vacuum or gas of negligible density and viscosity resulting from axisymmetric oscillation due to surface tension is solved using smoothed particle hydrodynamics composed of the Navier‐Stokes system and appropriate interfacial conditions for the free‐surface boundaries. The evolution of the droplet and its free‐surface interface is tracked over time to investigate the effects of surface tension forces implemented using a modified continuous surface force method and is compared with those performed using interparticle interaction force. The dynamic viscous fluid and surface tension interactions are investigated via a controlled curvature model and test cases of nonsteady oscillating droplets; attention is focused here on droplet oscillation that is released from an initial static deformation. Accuracy of the results is attested by demonstrating that (i) the curvature of the droplet that is controlled; (ii) uniform distribution of fluid particles; (iii) clean asymmetric forces acting on the free surface; and (iv) nonsteady oscillating droplets compare well with analytical and published experiment findings. The advantage of the proposed continuous surface force method only requires the use of physical properties of the fluid, whereas the interparticle interaction force method is restricted by the requirement of tuning parameters.     

Size prediction of κ-carrageenan droplets formed in co-flowing immiscible liquid

The formation of κ-carrageenan droplets in channel emulsification was experimentally investigated. The dispersed phase was vertically injected into co-flowing immiscible palm oil in the direction of gravity. This study focused on predicting κ-carrageenan drop size using force balance analysis. The force balance model considers the interfacial tension to be the solitary attaching force, while a combination of the drag force from the co-flowing palm oil and the body force of the extruding κ-carrageenan liquid act as the detaching forces. The conventional model gave poor predictions for droplet size, with an average relative deviation of 23%. This large deviation could be attributed to necking phenomena and an underestimation of the drag force generated on the shear-thinning κ-carrageenan solution. By incorporating correction factors, the average relative deviation of the force balance model dropped to 4%.     

A multiphysics model for charged liquid droplet breakup in electric fields

A computational multiphysics model for simulating the formation and breakup of droplets from axisymmetric charged liquid jets in electric fields is developed. A fully-coupled approach is used to combine two-phase flow, electrostatics, and transport of charged species via diffusion, convection, and migration. A conservative level-set method is shown to be robust and efficient for interface tracking. Parametric simulations are performed across a range of fluid properties corresponding to commonly used liquids in inkjet printing and spray applications to examine their role in jet evolution and droplet formation. Specifically, the effects of electric potential drop, surface tension, viscosity, and mobility are investigated. Droplet velocity and size distributions are calculated, and the corresponding mean values are found to increase and decrease respectively with increasing electric field strength. The variations in droplet velocity and size are quantified, and droplet size and charge levels agree well with experimental values. Increasing mobility of charged species is found to enhance jet velocity and accelerate droplet formation by shifting charge from the liquid interior to the interface.     

Algebraic coupled level set-volume of fluid method for surface tension dominant two-phase flows

In this study, an Algebraic Coupled Level Set-Volume of Fluid (A-CLSVOF) method is proposed for the simulation of interfacial capillary flows. The Volume of Fluid (VOF) method is utilized to incorporate one-way coupling with the Level Set (LS) function in order to improve the accuracy of the surface tension force calculation and to reduce the presence of parasitic currents. In this method, both VOF and LS functions are transported where the new volume fraction determines the interface seed position utilized by the reinitialization procedure for the LS field. Computational efficiency is enhanced through the use of the advected LS field, serving as an initial condition for the reinitialization procedure. The Hamilton–Godunov function is used with a second order (in space and time) discretization scheme to produce a signed distance function. In order to evaluate the performance of the methodology implemented here for capillary dominant flows, four different cases were considered: 1. static droplet; 2. capillary wave relaxation; 3. Rayleigh–Taylor instability and 4. droplet impact on a liquid pool. The simulations demonstrated reduction in spurious currents, accurately predicted capillary pressure and demonstrated overall improvements in efficiency and error reduction owing to the addition of LS advection. The present methodology is tested against previously published experimental results for a droplet impact on a deep liquid pool. Simulation results demonstrate excellent agreement with measured interface height up to and beyond the formation of a Rayleigh–Jet and break-up formation of a secondary daughter drop.     

Numerical simulation of internal gravity waves using a lattice gas model

Internal waves are modelled in two different circumstances: in a continuously stratified fluid and at the interface between two immiscible fluids. This is done using the lattice gas approach. The standard single phase model and an immiscible two-phase model are both modified to incorporate gravitational interactions. Standing internal waves are set up in both models and are seen to oscillate under the action of the gravitational interaction. The results obtained suggest that the lattice gas approach can be a useful tool in the modelling of such phenomena. © 1998 John Wiley & Sons, Ltd.     

基于改进LBM的气液自发渗吸过程中动态润湿效应模拟

微通道内气液自发渗吸是广泛发生在自然界及诸多工业领域的物理现象,而动态接触角是影响整个渗吸过程的关键因素.针对该问题,本文使用改进的伪势多相流格子玻尔兹曼方法 (LBM),直接捕捉微通道内气液自发渗吸过程中的实时接触角,并分析接触角的动态变化特性及其对渗吸长度的影响.首先,本文在原始的伪势多相流LBM的基础上耦合Peng-Robinson (PR)状态方程,改进流体-流体作用力以及流-固作用力格式,并采用精确差分方法将外力添加至LBM框架中.然后,通过校准模型的热力学一致性,模拟测试界面张力,静态平衡接触角等界面现象验证了模型的准确性.最后,基于建立的模拟方法,在水平方向上模拟微通道内气液自发渗吸过程.结果表明:渗吸过程中的接触角呈现动态变化特征,在渗吸初期,因受到惯性力的影响存在较大波动;随着渗吸距离的增大,其逐渐减小并趋近于静态平衡接触角.渗吸过程中的接触角与微通道尺寸及静态接触角有关,随着微通道宽度增大,实时的动态接触角与静态接触角相差大;随着静态接触角增大,实时的动态接触角与静态接触角的相差增大.此外,忽略动态接触角的Lucas-Washburn (LW)方程所预测的弯液面位置...     

A numerical study of droplet motion/departure on condensation of mixture vapor using lattice Boltzmann method

Droplet motion/departure, which is governed by external force acceleration coefficient, droplet radius and surface wettability on solid surfaces under external forces such as gravitational force, play a significant role in characterizing condensation heat transfer, especially when high fractional non-condensable gases (NCG) present. However, due to the challenge in visualizing the vapor/steam velocity field imposed by droplet motion/departure, the detailed mechanism of droplet motion/departure on condensing surfaces has not been completely investigated experimentally. In this study, droplet motion/departures on solid surfaces under external forces and their interactions with steam flow are simulated using two dimensional (2D) multiphase lattice Boltzmann method (LBM). Large external force acceleration coefficient, droplet radius and contact angle, lead to large droplet deformation and high motion/departure velocity, which significantly shortens the droplet residual time on the solid surface. Our simulation shows that steam vortices (lateral velocity) induced by droplet motion/departure can greatly disturb the vapor flow and would be intensified by increasing external force acceleration coefficient, droplet radius, and contact angle. In addition, the location of vortex center shifts in the ascending direction with increase of these factors. The average lateral velocities induced by droplet motion/departure at various conditions are obtained. The mass transfer resistance is substantially reduced owing to the droplet motion/departure, leading to an enhanced heat flux. The experimental results are compared to validate the influence of droplet motion/departure on condensation heat transfer performance, especially for steam–air mixture with the presence of high fractional NCG.     

A γ‐DGBGK scheme for compressible multi‐fluids

The paper presents a Discontinuous Galerkin γ‐BGK (γ‐DGBGK) method for compressible multicomponent flow simulations by coupling the discontinuous Galerkin method with a γ‐BGK scheme based on WENO limiters. In this γ‐DGBGK method, the construction of the flux in the DG method is based on the kinetic scheme which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation at cell interfaces. WENO limiters are used to obtain uniform high‐order accuracy and sharp non‐oscillatory shock transition, and time accuracy obtained by integration for the flux function at the cell interface. Numerical examples in one and two space dimensions are presented to illustrate the robust and accuracy of the present scheme. Copyright © 2010 John Wiley & Sons, Ltd.