A collapsing bubble-induced microinjector: an experimental study

In this study, a new drop-on-demand actuation mechanism, which uses an oscillating bubble as actuator is proposed and its feasibility is investigated via the consideration of two important design parameters, namely, bubble distance to free surface and nozzle dimension. The droplet ejection process captured using high-speed photography technique shows that such an actuation mechanism has interesting features and perhaps some advantages over the conventional ones (thermal bubble, piezoelectric, etc.) employed in inkjet printers such as the ejection of a droplet free of satellite droplets and others.     

Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions

In this paper we present a three‐dimensional Navier–Stokes solver for incompressible two‐phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties. Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third‐order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second‐order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first‐order convergence in the smoothing region. We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three‐dimensional results with those of quasi‐two‐dimensional and two‐dimensional simulations. This comparison clearly shows the need for full three‐dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd.     

A control volume finite element method for three-dimensional three-phase flows

A novel control volume finite element method with adaptive anisotropic unstructured meshes is presented for three-dimensional three-phase flows with interfacial tension. The numerical framework consists of a mixed control volume and finite element formulation with a new P₁DG-P₂ elements (linear discontinuous velocity between elements and quadratic continuous pressure between elements). A “volume of fluid” type method is used for the interface capturing, which is based on compressive control volume advection and second-order finite element methods. A force-balanced continuum surface force model is employed for the interfacial tension on unstructured meshes. The interfacial tension coefficient decomposition method is also used to deal with interfacial tension pairings between different phases. Numerical examples of benchmark tests and the dynamics of three-dimensional three-phase rising bubble, and droplet impact are presented. The results are compared with the analytical solutions and previously published experimental data, demonstrating the capability of the present method.     

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.     

A numerical and experimental study of a collapsing bubble-induced droplet ejector

This paper aims to study a novel drop-on-demand droplet generation mechanism in which the oscillation and deformation of a non-equilibrium bubble in close proximity to a free surface induce an axisymmetric liquid spike on the free surface. The evolution of the liquid spike and its deformation due to the effect of surface tension force lead to the formation of a droplet. The free surface can be accorded by either a circular hole on a horizontal flat plate or by the top opening/nozzle of a vertical cylinder. A high-speed camera capable of obtaining images at a frame rate of 15,000 fps is utilized to observe the droplet formation process. Numerical simulations corresponding to the experiments are performed using the boundary integral spatial solution coupled with the time integration, i.e., a mixed Eulerian–Lagrangian method. In the experiments the bubble is generated using a very low voltage (only 55 V) in contrast to the relatively much higher voltages usually employed in reported works. This is very attractive from a safety viewpoint and accords great simplification of the setup. A comparison is made between the numerical and experimental results. A reasonable agreement has been found. The influences of the main design parameters, namely, the bubble-free surface distance and the dimension of the hole/nozzle on the bubble dynamics and on the droplet formation process are discussed and the conditions of the bubble dynamics under which a satellite-free droplet can be generated are sought. Furthermore, the effects of different geometries, namely, the horizontal flat plate and the vertical cylinder on the bubble dynamics and on the droplet features are examined. One important feature of the proposed actuation mechanism is the capability of producing droplets much smaller than the nozzle size. The possible applications of this mechanism are those where the accurate direction of the ejected droplet is of great importance such as inkjet printing.      

A pressure correction‐volume of fluid method for simulation of two‐phase flows

A pressure correction method coupled with the volume of fluid (VOF) method is developed to simulate two‐phase flows. A volume fraction function is introduced in the VOF method and is governed by an advection equation. A modified monotone upwind scheme for a conservation law (modified MUSCL) is used to solve the solution of the advection equation. To keep the initial sharpness of an interface, a slope modification scheme is introduced. The continuum surface tension (CST) model is used to calculate the surface tension force. Three schemes, central‐upwind, Parker–Youngs, and mixed schemes, are introduced to compute the interface normal vector and the gradient of the volume fraction function. Moreover, a height function technique is applied to compute the local curvature of the interface. Several basic test problems are performed to check the order of accuracy of the present numerical schemes for computing the interface normal vector and the gradient of the volume fraction function. Three physical problems, two‐dimensional broken dam problem, static drop, and spurious currents, and three‐dimensional rising bubble, are performed to demonstrate the efficiency and accuracy of the pressure correction method. Copyright © 2010 John Wiley & Sons, Ltd.     

Fringe element reconstruction for front tracking for three‐dimensional incompressible flow analysis

Fringe element reconstruction technique for tracking the free surface in three‐dimensional incompressible flow analysis was developed. The flow field was calculated by the mixed formulation based on a four‐node tetrahedral element with a bubble function at the centroid (P1+/P1). Since an Eulerian approach was employed in this study, the flow front interface was advected by the flow through a fixed mesh. For accurate modelling of interfacial movement, a fringe element reconstruction method developed can provide not only an accurate treatment of material discontinuity but also surface tension across the interface. The effect of surface tension was modelled by imposing tensile stress directly on the constructed surface elements at the flow front interface. To verify the numerical approach developed, the developed algorithm was applied to two examples whose solutions are available in references. Good agreement was obtained between the simulation results and these solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

A method for overcoming the surface tension time step constraint in multiphase flows II

A method for overcoming the surface tension time step constraint is presented. The algorithm presented in this work is an improvement on the work presented by Sussman and Ohta (SIAM J Sci Comput 2009). In this work, the method of Sussman and Ohta is extended in order to treat problems with contact angle dynamics. Furthermore, this work presents a more efficient method for computing volume‐preserving motion by mean curvature than the method presented previously. The new method is tested on the following four 2D problems: (1) 3D axisymmetric (r?z) surface tension driven zero gravity droplet oscillation, (2) measurement of the magnitude of parasitic currents for a droplet on a substrate initialized in static equilibrium, (3) relaxation of a 2D droplet on a substrate to static shape, and (3) relaxation of a 2D bubble on a substrate to static shape. Copyright © 2011 John Wiley & Sons, Ltd.     

The intersection marker method for 3D interface tracking of deformable surfaces in finite volumes

Currently, the majority of computational fluid dynamics (CFD) codes use the finite volume method to spatially discretise the computational domain, sometimes as an array of cubic control volumes. The Finite volume method works well with single‐phase flow simulations, but two‐phase flow simulations are more challenging because of the need to track the surface interface traversing and deforming within the 3D grid. Surface area and volume fraction details of each interface cell must be accurately accounted for, in order to calculate for the momentum exchange and rates of heat and mass transfer across the interface. To attain a higher accuracy in two‐phase flow CFD calculations, the intersection marker (ISM) method is developed. The ISM method is a hybrid Lagrangian–Eulerian front‐tracking algorithm that can model an arbitrary 3D surface within an array of cubic control volumes. The ISM method has a cell‐by‐cell remeshing capability that is volume conservative and is suitable for the tracking of complex interface deformation in transient two‐phase CFD simulations. Copyright © 2015 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.     

Numerical study on the oscillation of a transient bubble near a confined free surface for droplet generation

In the present work, the oscillation of a spark-created bubble near a confined water–air interface and the ensuing droplet generation and ejection are studied numerically using the boundary element method. The interface is accorded by the top opening of either one of the following symmetrical configurations, which are distinguished by the value of angle between their vertical symmetry axis and lateral wall (i.e., θ): (i) a centrally perforated horizontal flat plate (θ = 90°) and (ii) vertically placed cylinder (θ = 0°), nozzle (θ > 0°) and diffuser (θ < 0°). Furthermore, the influences of the effective parameters such as the strength parameter (i.e., the intensity of local energy input), the bubble-free surface distance (standoff distance) and the nozzle size on the bubble dynamics and droplet formation and ejection processes are investigated. It was found that the moment at which the bubble attained its maximum volume was advanced as θ increased. In addition, by decreasing θ the attraction of the bubble toward the free surface during its expansion phase and its migration from the free surface during its contraction phase became stronger. Furthermore, for the nozzle case, by increasing θ, the volume of the droplet was increased. It was also found that by increasing the strength parameter, the volume of the droplet increased and its pinch-off happened earlier. Finally, as the standoff distance was increased, the volume of the droplet increased and its pinch-off was delayed.     

Critical evaluation of CFD codes for interfacial simulation of bubble‐train flow in a narrow channel

Computational fluid dynamics (CFD) codes that are able to describe in detail the dynamic evolution of the deformable interface in gas–liquid or liquid–liquid flows may be a valuable tool to explore the potential of multi‐fluid flow in narrow channels for process intensification. In the present paper, a computational exercise for co‐current bubble‐train flow in a square vertical mini‐channel is performed to investigate the performance of well‐known CFD codes for this type of flows. The computations are based on the volume‐of‐fluid method (VOF) where the transport equation for the liquid volumetric fraction is solved either by the methods involving a geometrical reconstruction of the interface or by the methods that use higher‐order difference schemes instead. The codes contributing to the present code‐to‐code comparison are an in‐house code and the commercial CFD packages CFX, FLUENT and STAR‐CD. Results are presented for two basic cases. In the first one, the flow is driven by buoyancy only, while in the second case the flow is additionally forced by an external pressure gradient. The results of the code‐to‐code comparison show that only the VOF method with interface reconstruction leads to physically sound and consistent results, whereas the use of difference schemes for the volume fraction equation shows some deficiencies. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical Simulation of Three-Dimensional Bubble Oscillations by a Generalized Vortex Method

A numerical method is implemented for simulating the simultaneous three-dimensional volume and shape oscillations of a compressible vapor or gas bubble suspended in an inviscid ambient fluid in the presence of interfacial tension. The flow generated by the bubble expansion, contraction, and deformation is represented by an interfacial distribution of potential dipoles supplemented by a point source situated inside the bubble, accounting for changes in the bubble volume. The mathematical formulation is completed by setting the strength of the point source proportional to the integral of the density of the double-layer potential over the interface. The motion of marker points distributed over the interface is computed using a boundary-element implementation of Baker's generalized vortex method in which the normal component of the interfacial velocity is computed in terms of tangential derivatives of the vector potential associated with the dipoles, whereas the tangential component of the interfacial velocity is computed in terms of the surface gradient of the scalar harmonic potential. The density of the double-layer distribution is computed by solving an integral equation of the second kind using an iterative method, while the evolution of the interfacial distribution of the harmonic potential is computed using Bernoulli's equation for irrotational flow. The onset of interfacial irregularities due to numerical instabilities is prevented by truncating the Fourier–Legendre spectrum of the interfacial distribution of the harmonic potential. With smoothing implemented, the numerical method is capable of describing simultaneous volume and shape oscillations for an indefinite period of time. Received 7 September 2001 and accepted 30 April 2002 Published online 30 October 2002 RID="*" ID="*" This research was supported by a grant provided by NASA. Communicated by J.R. Blake     

An improved three-dimensional model for interface pressure calculations in free-surface flows

A three-dimensional method for the calculation of interface pressure in the computational modeling of free surfaces and interfaces is developed. The methodology is based on the calculation of the pressure force at the interfacial cell faces and is mainly designed for volume of fluid (VOF) interface capturing approach. The pressure forces at the interfacial cell faces are calculated according to the pressure imposed by each fluid on the portion of the cell face that is occupied by that fluid. Special formulations for the pressure in the interfacial cells are derived for different orientations of an interface. The present method, referred to as pressure calculation based on the interface location (PCIL), is applied to both static and dynamic cases. First, a three-dimensional motionless drop of liquid in an initially stagnant fluid with no gravity force is simulated as the static case and then two different small air bubbles in water are simulated as dynamic cases. A two-fluid, piecewise linear interface calculation VOF method is used for numerical simulation of the interfacial flow. For the static case, both the continuum surface force (CSF) and the continuum surface stress (CSS) methods are used for surface tension calculations. A wide range of Ohnesorge numbers and density and viscosity ratios of the two fluids are tested. It is shown that the presence of spurious currents (artificial velocities present in case of considerable capillary forces) is mainly due to the inaccurate calculation of pressure forces in the interfacial computational cells. The PCIL model reduces the spurious currents up to more than two orders of magnitude for the cases tested.

Also for the dynamic bubble rise case, it is shown that using the numerical solver employed here, without PCIL, the magnitude of spurious currents is so high that it is not possible to simulate this type of surface tension dominated flows, while using PCIL, we are able to simulate bubble rise and obtain results in close agreement with the experimental data.     

A cost‐effective curvature calculation approach for interfacial flows on unstructured meshes

We present a simple and cost‐effective curvature calculation approach for simulations of interfacial flows on structured and unstructured grids. The interface is defined using volume fractions, and the interface curvature is obtained as a function of the gradients of volume fractions. The gradient computation is based on a recently proposed gradient recovery method that mimicks the least squares approach without the need to solve a system of equations and is quite easy to implement on arbitrary polygonal meshes. The resulting interface curvature is used in a continuum surface force formulation within the framework of a well‐balanced finite‐volume algorithm to simulate multiphase flows dominated by surface tension. We show that the proposed curvature calculation is at least as accurate as some of the existing approaches on unstructured meshes while being straightforward to implement on any mesh topology. Numerical investigations also show that spurious currents in stationary problems that are dependent on the curvature calculation methodology are also acceptably low using the proposed approach. Studies on capillary waves and rising bubbles in viscous flows lend credence to the ability of the proposed method as an inexpensive, robust, and reasonably accurate approach for curvature calculation and numerical simulation of multiphase flows.     

单液滴内空化气泡的生长及溃灭研究

超空化燃油射流使得喷雾中部分燃油分裂液滴内含有空化气泡;空化气泡的生长及溃灭对液滴的分裂与雾化具有重要影响. 基于VOF 方法首次对超空化条件下燃油液滴内空化气泡的生长及溃灭过程进行了数值模拟. 通过研究发现,单液滴内空化气泡的生长过程可以按控制机理划分为表面张力控制阶段、综合竞争阶段和惯性力控制阶段;在第I 阶段,空泡的生长主要受表面张力的控制作用,惯性力对空泡生长的促进作用及黏性力对空泡生长的抑制作用可以忽略;在第II 阶段,空泡的生长受表面张力、惯性力及黏性力三者的综合作用,空泡的生长速率是促进空泡生长的惯性力和抑制空泡生长的表面张力及黏性力相互竞争、共同作用的结果;在第III 阶段,空泡的生长主要受惯性力的控制作用,抑制空泡生长的表面张力及黏性力的作用基本可以忽略. 单液滴内空化气泡的溃灭过程由多个溃灭阶段和反弹阶段构成,类似于有阻尼弹簧振子的振动过程;根据每个溃灭周期结束时空泡半径随时间的变化历程,可以将空泡的溃灭分为快速溃灭期、缓慢溃灭期以及稳定期;溃灭初期空泡溃灭压力的变化非常剧烈,但空泡溃灭体积的变化则要相对平缓得多;空泡反弹压力随时间的变化与空泡反弹体积随时间的变化基本对应.      

竖直流道宽度对气泡运动行为影响的数值模拟

用数值方法模拟了竖直通道宽度对气泡在液体中的非定常运动、变形以及传热特性的影响。在这个模拟中,界面跟踪采用了VOF方法,并采用PL IC进行界面重构。主流场计算采用有限容积方法将控制方程离散,其中扩散项采用中心差分格式,对流项采用一阶迎风格式。用成熟的S IM PLE算法求解N-S方程的速度与压力的耦合问题。引入CSF模型处理运动界面的表面张力。利用所编制的程序计算了竖直流道中的单个气泡的形状、运动特性以及气泡内外流场与传热特性,并对竖直通道宽度在不同情况下,对气泡的形状、运动特性以及传热特性进行了进一步的研究。得到了一系列有价值的结果,并与实验结果比较。表明数值模拟结果与实验结果吻合的较好。     

Development of a hybrid particle‐mesh method for two‐phase flow simulations

A hybrid particle‐mesh method was developed for efficient and accurate simulations of two‐phase flows. In this method, the main component of the flow is solved using the constrained interpolated profile/multi‐moment finite volumemethod; the two‐phase interface is rendered using the finite volume particle (FVP) method. The effect of surface tension is evaluated using the continuum surface force model. Numerical particles in the FVP method are distributed only on the surface of the liquid in simulating the interface between liquid and gas; these particles are used to determine the density of each mesh grid. An artificial term was also introduced to mitigate particle clustering in the direction of maximum compression and sparse discretization errors in the stretched direction. This enables accurate interface tracking without diminishing numerical efficiency. Two benchmark simulations are used to demonstrate the validity of the method developed and its numerical stability. Copyright © 2016 John Wiley & Sons, Ltd.     

A new algorithm for surface tension model in moving particle methods

A new algorithm for the surface tension model was developed for moving particle methods. The algorithm is based on the link‐list search algorithm and the continuum surface tension (CST) model. The developed algorithm with the CST model was implemented to a kind of moving particle approach, the finite volume particle (FVP) method. The FVP method with the new algorithm was tested by oscillatory behaviour of a two‐dimensional droplet. The oscillatory period agrees well with analytical one, and the transient shape of the droplet is also in good agreement with that obtained by other numerical methods. The droplet impact on a liquid surface was also studied using the new algorithm. The deposition and splashing phenomena were clearly reproduced. Simulated spread radius of the splashing phenomena was consistent with a power law. Copyright © 2007 John Wiley & Sons, Ltd.