Continuum Surface Force Research Articles

A path-conservative finite volume scheme for compressible multi-phase flows with surface tension

The accurate simulation of compressible multi-phase flows with surface tension effects is currently still one of the most challenging problems in computational fluid dynamics (CFD). The basic difficulties are the capturing of the correct interface dynamics between the two fluids as well as the computation of the interface curvature. In this paper, we present a novel path-conservative finite volume discretization of the continuum surface force method (CSF) of Brackbill et al. to account for the surface tension effect due to curvature of the phase interface. This is achieved in the context of a diffuse interface approach, based on the seven equation Baer–Nunziato model of compressible multi-phase flows. Such diffuse interface methods for compressible multi-phase flows including capillary effects have first been proposed by Perigaud and Saurel. In the CSF method, the surface tension effect is replaced by a volume force, which is usually integrated as a classical volume source term. However, since this source term contains the gradient of a color function that is convected with the flow velocity, we propose to integrate the CSF source term as a non-conservative product and not simply as a source term, following the ideas on path-conservative finite volume schemes put forward by Castro and Parés.For that purpose, we use the new generalized Osher-type Riemann solver (DOT), recently proposed by Dumbser and Toro and compare it with a path-conservative Roe and Rusanov scheme. Via numerical evidence we can show that if the curvature computation is exact, our new scheme is well-balanced for a steady circular bubble in equilibrium according to the Young–Laplace law. This means that the pressure jump term across the interface in the momentum equation is exactly balanced with the surface tension force.We apply our scheme to several one- and two-dimensional test problems, including 1D Riemann problems, an oscillating droplet, as well as a deforming droplet initially attached to a wall and which is subject to gravity and surface tension forces. Finally, we also check the rising of a gas bubble due to buoyancy forces. We compare our numerical results with those published previously in the literature.

Numerical investigation of the moving liquid column coalescing with a droplet in triangular microchannels using CLSVOF method

The dynamic behavior of the moving liquid column coalescing with a sessile droplet in triangular microchannels is numerically investigated by using coupled volume of fluid with level set interface tracking method implemented in ANSYS Fluent 14.5 in conjunction with the continuum surface force model. It is found that for both hydrophobic and hydrophilic microchannels, the coalescence between the moving liquid column and droplet can accelerate the original liquid column movement as a result of the induced curvature that lowers the liquid pressure at the interface. As compared to the rectangular microchannel with the same hydraulic diameter, the triangular microchannel exhibits smaller velocity increment ratio because of stronger viscous effect. Simulation results also reveal that the velocity increment ratio increases with the contact angle in hydrophobic microchannels, but it is reverse in the hydrophilic microchannels. The effects of the droplet size, lengthways and transverse positions are also investigated in this work. It is shown that larger droplet and smaller distance between the droplet and inlet or the substrate center can result in larger velocity increment ratio as a result of higher surface energy and lower viscous dissipation energy, respectively. The results obtained in this study create a solid theoretical foundation for designing and optimizing microfluidic devices encountering such a typical phenomenon.

A numerical study of microfluidic droplet transport in a parallel-plate electrowetting-on-dielectric (EWOD) device

The fluid dynamics of microwater droplet transport in a parallel-plate electrowetting-on-dielectric (EWOD) device have been investigated via a numerical model. The transient governing equations are solved by a finite volume scheme with a two-step projection method on a fixed computational domain. The interface between liquid and gas is tracked by a coupled level set and volume-of-fluid method. A continuum surface force model is employed to evaluate surface tension at the interface. A simplified model is adopted for the viscous stresses exerted by the parallel plates at the gas–liquid interface in conjunction with contact angle hysteresis implemented as a crucial element in EWOD modeling. Excellent agreement has been achieved between the numerical and published experimental results. Special attention has been focused on some localized areas near the ON/OFF electrode border where the transport process is primarily influenced. A dimensionless curvature has been introduced and a critical value has been identified beyond which the droplet would split during the transport. A parametric study has been performed in which the effects of several crucial parameters including initial droplet shape, static contact angles, contact angle hysteresis, viscous stress, channel height and electrode size on the transport process have been revealed.

Effects of the evaporator and condenser length on the performance of pulsating heat pipes

A two-dimensional single loop closed-loop pulsating heat pipe (CLPHP) is investigated by CFD in present work. The volume of fluid (VOF) method is used to investigate the start-up characteristic and thermal performance of CLPHP with different ratios of the evaporator length to that of the condenser. Considering the action of wall adhesion and the effect of surface tension, the continuum surface force (CSF) model is adopted. The investigation is carried out by changing input power (from 10 W to 40 W) and filling ratio (from 30% to 60%). The numerical results are compared with available experimental results at the same condition. It's found that the start-up time and the thermal resistance of CLPHP reduces with the increase of input power. At the same filling ratio and input power condition, reducing the length of condenser within a suitable range would accelerate the start-up process and decrease the thermal resistance while the dry-out occurred easily at low filling ratio. Two types of start-up, sudden temperature increasing starts at lower input power and the smooth temperature increasing starts at higher input power are observed. The bubbles creation and growth motions are observed inside the CLPHP with vapor plugs oscillation and circulation motions.

Experimental and Numerical Analysis of Filament Front Deformation for Direct-Print

The purpose of this research is to perform an investigation of continuously dispensed material (or filament) deformation during dispensing. Depending on the material properties, and the process parameters the deformation behavior of the filament changes, so as the formation of the filament front (the front face of the filament, i.e., dispensing material). The focus of this investigation is the study of the evolution of the filament shape for Newtonian fluid experimentally and computationally. The experimental analysis has been performed with commercially available monomers with the help of a screw driven micro dispensing system installed on a high precision xyz translation stage. The imaging system consists of a high resolution CMOS camera. The developed computational model utilizes an adaptive quadtree spatial discretization with piecewise–linear geometrical volume–of–fluid (VOF) method for calculating the volume fraction for this multiphase problem. The model employs the continuum–surface–force model for formulating the surface tension, whereas the height function (HF) to estimate the curvature for tracking the evolution of the filament shape during the deformation. The computational model has been developed using an open source solver, Gerris Flow Solver. The considered governing and process parameters for this investigation are Froude number (Fr), Reynolds number (Re), gap ratio (GR), and velocity ratio (VR). The VR is the ratio of travel velocity to dispensing velocity. The GR is the ratio of filament height to filament diameter. Results have been presented as the interface contour for the filament front. The investigation shows that the results found from the developed model have a good agreement with experimental results, and the deformation phenomena is greatly influenced by the variation of the governing parameters.

Lattice Boltzmann simulation of the thermocapillary flow in an annular two liquid layers system with deformable interface

A hybrid model, coupling a finite difference method with a multiple-relaxation-time lattice Boltzmann method, integrating continuum surface force model and phase-field method, for axisymmetric two-phase thermocapillary flow with a deformable interface is developed. Thermocapillary flow, originating from an unbalanced surface tension along the interface of two immiscible liquids in an annular cavity with a horizontal temperature gradient, is simulated numerically. The dynamic behavior of the interface is captured using the phase-field method, and no a priori assumption is made regarding the interface shape and deformation. The continuum surface force model is adopted to add the unbalanced surface tension. The flow field is simulated by multiple-relaxation-time lattice Boltzmann method and both phase-field equation and the energy equation are solved by finite difference method. The dependence of fluid convection and interface deformation on the ratio of physical properties between the two liquid layers is investigated.

Numerical Modeling of Multiple Bubbles Condensation in Subcooled Flow Boiling

The understanding of multiple bubbles condensation is of significant importance in developing continuum models for the large-scale subcooled flow boiling. The computational fluid dynamics (CFD) modeling for multiple bubbles condensation is developed with the volume of fluid (VOF) method in this work. An explicit transient simulation is performed to solve the governing equations including the source terms for heat and mass transfer due to condensation. The geometric reconstruction scheme, which is a piecewise linear interface calculation (PLIC) method, is employed to keep the interface sharp. The surface tension is modeled by the continuum surface force (CSF) approach, which is taken into account in the numerical model. Numerical simulations predict the dynamical behavior of the actual condensing bubbles. The results show that the condensation rate of a single bubble is influenced by the velocity of the fluid flow and the temperature difference between the bubble and fluid. For multiple bubbles, the effect of bubble–bubble interaction on their condensation process is analyzed based on the numerical predictions. The condensation rate of lower bubbles increases due to the random perturbation induced by other bubbles. The influence of other bubbles on the condensation rate can be neglected if the distances between the bubbles are large enough.

A Numerical Study of Droplet Splitting and Merging in a Parallel-Plate Electrowetting-on-Dielectric Device

Microwater droplet splitting and merging in a parallel-plate electrowetting-on-dielectric (EWOD) device have been studied numerically. The transient governing equations for the microfluidic flow are solved by a finite volume scheme with a two-step projection method on a fixed computational domain. The interface between liquid and gas is tracked by a coupled level set (LS) and volume-of-fluid (CLSVOF) method. A continuum surface force (CSF) model is employed to model the surface tension at the interface. Contact angle hysteresis which is an essential component in EWOD modeling is implemented together with a simplified model for the viscous stresses exerted by the two plates at the solid–liquid interface. The results of the numerical model have been validated with published experimental data and the physics of droplet motion within the EWOD device has been examined. A parametric study has been performed in which the effects of channel height and several other parameters on the fluid motion have been studied.

A CSF-SPH method for simulating drainage and imbibition at pore-scale resolution while tracking interfacial areas

We present a conservative smoothed particle hydrodynamics (SPH) model to study the flow of multiple, immiscible fluid phases in porous media using direct pore-scale simulations. Particular focus is put on continuously tracking the evolution of interfacial areas, which are considered to be important morphological quantities affecting multiphase transport in porous media. In addition to solving the Navier–Stokes equations, the model accounts for the effects of capillarity at interfaces and contact lines. This is done by means of incorporating the governing interfacial mass and momentum balances using the continuum surface force (CSF) method, thus rendering model calibration routines unnecessary and minimizing the set of constitutive and kinematic assumptions. We address the application of boundary conditions at rigid solid surfaces and study the predictive capability of the model as well as optimal choices for numerical parameters using an extensive model validation procedure. We demonstrate the applicability of the model to simulate multiphase flows involving partial wettability, dynamic effects, large density ratios (up to 1000), large viscosity ratios (up to 100), as well as fragmentation and coalescence of fluid phases. The model is used to study the evolution of fluid–fluid interfacial areas during saturation-controlled primary drainage and main imbibition of heterogeneous pore spaces at low capillary numbers. A variety of pore-scale effects, such as wetting phase entrapment and fragmentation due to snap-off, are observed. Specific fluid–fluid interfacial area is observed to monotonically increase during primary drainage and hysteretic effects are apparent during main imbibition.

Numerical simulation of bubble behavior in subcooled flow boiling under velocity and temperature gradient

In this paper, numerical simulation of the bubble condensation in the subcooled boiling flow is performed. The interface between two-phase is tracked via the volume of fluid (VOF) method with continuous surface force (CSF) model, implemented in the open source OpenFOAM CFD package. In order to simulate the condensing bubble with the OpenFOAM code, the original energy equation and mass transfer model for phase change have been modified and a new solver is developed. The Newtonian flow is solved using the finite volume scheme based on the pressure implicit with splitting of operators (PISO) algorithm. Comparison of the simulation results with previous experimental data revealed that the model predicted well the behavior of the actual condensing bubble. The bubble lifetime is almost proportional to bubble initial size and is prolonged by increasing the system pressure. In addition, the initial bubble size, subcooling of liquid and velocity gradient play an important role in the bubble deformation behavior. Velocity gradient makes the bubble move to the higher velocity region and the subcooling rate makes it to move back to the lower velocity region.

Effect of orifice size and bond number on bubble formation characteristics: A CFD study

The volume of fluid with the continuum surface force (VOF‐CSF) model has been used in the present numerical study to investigate bubble formation and shapes in a bubble column. The effects of orifice sizes ranging from 0.5–1.5 mm on the bubble formation stages (i.e., expansion, elongation, and pinch‐off), bubble contact angle, departure diameter, time, and shape of bubble are investigated under a constant inlet velocity (0.2 m/s) boundary condition (BC). It is found that the initial formation of a bubble is dependent on the orifice diameter. Consequently, the formation of the bubble's hemispherical shape at the orifice is faster for a smaller orifice diameter than for a larger orifice diameter which forms a bigger bubble. A leading bubble requires a longer time to detach itself from an orifice in comparison to the next bubble (the second bubble), but interestingly the third bubble detaches much faster than the second. This model was also used to investigate the effect of the Bond number (Boσ). An increase of the Bond number (Boσ) from 0.047 to 0.16 speeds up the bubble pinch‐off, but a further increase in the Boσ (e.g. 0.17–0.47) slows it down. Based on the simulated cases used in this study, the findings demonstrate the capacity and accuracy of the CFD method in predicting bubble formation characteristics. The findings may be useful in the design of spargers for bubble column reactors.

A Comprehensive Computational Fluid Dynamics Study of Droplet‐Film Impact and Heat Transfer

AbstractThe Eulerian multiphase model and continuum surface force (CSF) are employed to simulate the liquid droplet impinging onto a solid wall with a pre‐existing thin film of the same liquid. The numerical results are compared with the experimental data reported in the literature, indicating a reasonable matching. The flow field and splashing behavior of a droplet impinging onto a liquid film are analyzed. The reason for the edge of the crown to eject into secondary drops is found. The splashing behavior can be influenced by the impacting velocity and fluid properties. The effects of impact velocity, droplet diameter, depth of film, liquid property, and droplet and wall temperature on the heat removal are investigated. Numerical results demonstrate that an increase in impact velocity, droplet diameter, film depth, cooling droplet, and wall temperature enhances the dissipated heat. These results can provide a reference for designing spray‐cooling systems.

Numerical simulation of wetting phenomena by a meshfree particle method

Simulations of wetting phenomena by a meshfree particle method are presented. The incompressible Navier–Stokes equations are used to model the two-phase flow. The continuous surface force model is used to incorporate the surface tension force. Chorin’s projection method is applied to discretize the Navier–Stokes equations. The different fluid phases are identified by assigning different colors and different material properties (density, viscosity) to the particles that remain unchanged throughout a simulation. Two-phase flow is captured by a one-fluid model via using weighted averages of the density and viscosity in a region around the fluid–fluid interface. The differential operators at each particle are computed from the surrounding cloud of particles with the help of the least-squares method. The numerical results are compared with specific analytical solutions, but also with previously considered test cases involving wetting of a container and sessile drops. A good overall agreement is found.

A contoured continuum surface force model for particle methods

A surface tension model is essential to simulate multiphase flows with deformed interfaces. This study develops a contoured continuum surface force (CCSF) model for particle methods. A color function that varies sharply across the interface to mark different fluid phases is smoothed in the transition region, where the local contour curvature can be regarded as the interface curvature. The local contour passing through each reference particle in the transition region is extracted from the local profile of the smoothed color function. The local contour curvature is calculated based on the Taylor series expansion of the smoothed color function, whose derivatives are calculated accurately according to the definition of the smoothed color function. Two schemes are proposed to specify the smooth radius: fixed scheme, where 2×re (re = particle interaction radius) is assigned to all particles in the transition region; and varied scheme, where re and 2×re are assigned to the central and edged particles in the transition region respectively. Numerical examples, including curvature calculation for static circle and ellipse interfaces, deformation of square droplet to a circle (2D and 3D), droplet deformation in shear flow, and droplet coalescence, are simulated to verify the CCSF model and compare its performance with those of other methods. The CCSF model with the fixed scheme is proven to produce the most accurate curvature and lowest parasitic currents among the tested methods.

Modeling and Parametric Study of the Heat Transfer Inside a Lithium Bromide Flow Confined by Wire Screen

The present study deals with the numerical analysis of heat transfer inside a lithium bromide-water solution flowing down between finely meshed plastic wire screens. These screens confine the flow through capillary action while allowing the water vapour transfer inside an innovative absorber technology. The complex menisci shape formed on the confinement grid level, where the surface tension forces are of first importance, are reconstructed by a volume-of-fluid model. A continuum surface force model is used to account for the surface tension force. A static contact angle is used to define the wall adhesion. A new algorithm, consisting to set an unique constant temperature at the liquid/vapour interface and to determine the evolution of heat transfer characteristics over the simulation domain, has been implemented and validated by analytical solution. A parametric study has been conducted to determine the effect of the geometry, the contact angle and the shape of the wire on the heat transfer.

Implementation of a height function method to alleviate spurious currents in CFD modelling of annular flow in microchannels

The use of the continuum surface force (CSF) model in surface tension modelling using the volume of fluid (VOF) method is known to introduce unphysical velocities, called parasitic or spurious currents arising mostly from poor calculation of interface curvature. In this paper, a modified 2-D axisymmetric height function (HF) method is proposed and implemented in parallelised computations. The default CSF method in the commercial software ANSYS Fluent 15.0 is replaced using User-Defined Functions (UDFs) of ANSYS Fluent and the improvements obtained are evaluated in test cases with stationary and moving interfaces. The application of the HF method to annular flow modelling has shown massive improvements in interface modelling and potential for research on annular flow hydrodynamics and heat transfer.

A multiple marker level-set method for simulation of deformable fluid particles

A novel multiple marker level-set method is introduced for Direct Numerical Simulation of deformable fluid particles (bubbles and droplets), which is integrated in a finite-volume framework on collocated unstructured grids. Each fluid particle is described by a separate level-set function, thus, different interfaces can be solved in the same control volume, avoiding artificial and potentially unphysical coalescence of fluid particles. Therefore, bubbles or droplets are able to approach each other closely, within the size of one grid cell, and can even collide. The proposed algorithm is developed in the context of the conservative level-set method, whereas, surface tension is modeled by the continuous surface force approach. The pressure–velocity coupling is solved by the fractional-step projection method. For validation of the proposed numerical method, the gravity-driven impact of a droplet on a liquid–liquid interface is studied; then, the binary droplet collision with bouncing outcome is examined, and finally, it is applied on simulation of gravity-driven bubbly flow in a vertical column.

Dynamic Behavior of the Liquid Flow Coalescing with a Droplet in Hydrophobic Microchannels.

In this study, the dynamic behavior of the moving liquid column coalescing with a sessile droplet in a hydrophobic microchannel under pressure driven flow conditions is numerically investigated using coupled Volume of Fluid with Level Set (CLSVOF) interface tracking method implemented in ANSYS-Fluent 14.5 in conjunction with the continuum surface force (CSF) model. Numerical result reveals that the coalescence between the moving liquid column and droplet can accelerate the original liquid column movement. Effects of the wettability, head pressure, and droplet size and position are also investigated. It is found that the velocity increment ratio increases with increasing the contact angle and decreasing the head pressure. Larger droplet and smaller distance between the droplet and inlet can result in a larger velocity increment ratio as a result of higher surface energy and lower viscous dissipation energy. The maximum velocity increment ratio of 0.17 is obtained with a 10000-µm3 droplet that is positioned at 200 µm in a microchannel with 100 µm in width and 300 µm in length and contact angle of 120°.

Dynamics of double emulsion break-up in three phase glass capillary microfluidic devices

Pinch-off of a compound jet in 3D glass capillary microfluidic device, which combines co-flowing and countercurrent flow focusing geometries, was investigated using an incompressible three-phase axisymmetric Volume of Fluid–Continuum Surface Force (VOF–CSF) numerical model. The model showed good agreement with the experimental drop generation and was capable of predicting formation of core/shell droplets in dripping, narrowing jetting and widening jetting regimes. In dripping and widening jetting regimes, the presence of a vortex flow around the upstream end of the necking thread facilitates the jet break-up. No vortex flow was observed in narrowing jetting regime and pinch-off occurred due to higher velocity at the downstream end of the coaxial thread compared to that at the upstream end. In all regimes, the inner jet ruptured before the outer jet, preventing a leakage of the inner drop into the outer fluid. The necking region moves at the maximum speed in the narrowing jetting regime, due to the highest level of shear at the outer surface of the thread. However, in widening jetting regime, the neck travels the longest distance downstream before it breaks.

Double emulsion production in glass capillary microfluidic device: Parametric investigation of droplet generation behaviour

A three-phase axisymmetric numerical model based on Volume of Fluid–Continuum Surface Force (VOF–CSF) model was developed to perform parametric analysis of compound droplet production in three-phase glass capillary devices that combine co-flow and countercurrent flow focusing. The model predicted successfully generation of core–shell and multi-cored double emulsion droplets in dripping and jetting (narrowing and widening) regime and was used to investigate the effects of phase flow rates, fluid properties, and geometry on the size, morphology, and production rate of droplets. As the outer fluid flow rate increased, the size of compound droplets was reduced until a dripping-to-jetting transition occurred. By increasing the middle fluid flow rate, the size of compound droplets increased, which led to a widening jetting regime. The jetting was supressed by increasing the orifice size in the collection capillary or increasing the interfacial tension at the outer interface up to 0.06N/m. The experimental and simulation results can be used to encapsulate CO2 solvents within gas-permeable microcapsules.