Continuum Surface Force Research Articles

Density-Scaled Balanced Continuum Surface Force Model with a Level Set Based Curvature Interpolation Technique

We examine the recently-proposed density-scaled balanced continuum surface force (CSF) model with a level set-based curvature interpolation technique. The density-scaled balanced CSF model is combined with a numerical framework which is based on the coupled level set and volume-of-fluid (CLSVOF) method, the tangent of hyperbola for interface capturing/weighted line interface calculation (THINC/WLIC) scheme, the constrained interpolation profile conservative semi-Langrangian with rational function (CIP-CSLR) and the volume/surface integrated average-based multi-moment method (VSIAM3). The present CSF model is examined for various bench mark problems such as drop–drop collisions and drop splashing. Comparisons of the present model results with experimental observations and those from the other CSF models show that the present CSF model can minimize spurious current and capture complicated fluid phenomena with minimizing floatsam.

Validation of the S‐CLSVOF method with the density‐scaled balanced continuum surface force model in multiphase systems coupled with thermocapillary flows

SummaryThree numerical methods, namely, volume of fluid (VOF), simple coupled volume of fluid with level set (S‐CLSVOF), and S‐CLSVOF with the density‐scaled balanced continuum surface force (CSF) model, have been incorporated into OpenFOAM source code and were validated for their accuracy for three cases: (i) an isothermal static case, (ii) isothermal dynamic cases, and (iii) non‐isothermal dynamic cases with thermocapillary flow including dynamic interface deformation. Results have shown that the S‐CLSVOF method gives accurate results in the test cases with mild computation conditions, and the S‐CLSVOF technique with the density‐scaled balanced CSF model leads to accurate results in the cases of large interface deformations and large density and viscosity ratios. These show that these high accuracy methods would be appropriate to obtain accurate predictions in multiphase flow systems with thermocapillary flows. Copyright © 2016 John Wiley & Sons, Ltd.

Numerical Study of the Forced Convective Condensation on a Short Vertical Plate

ABSTRACTNumerical investigation was conducted on the effects of gravity, surface tension, and wall adhesion upon condensation on a short vertical plate. The volume of fluid method was applied to model the interaction between the liquid and vapor phases and to capture the interface. The surface tension was implemented by employing the method of continuum surface force model. A modified phase-change model, derived from basic equations related to the kinetic gas theory, was proposed and verified based on the cases of Nusselt film condensation of water vapor on a vertical flat plate, the forced convection film condensation on a horizontal flat plate, and the capillary blocking due to condensation in a horizontal miniature circular tube. The predicted results showed that a laminar capillary wavy flow regime exists and the waves enhance the heat transfer of condensation on the plate. The mean film thickness increases and the heat transfer performance becomes worse with decrease of gravity. A high value of surface tension or contact angle, representing a large surface free energy difference, leads to an enhancement of heat transfer on the plate with large-amplitude waves.

Modeling the phase change process for a two-phase closed thermosyphon by considering transient mass transfer time relaxation parameter

The method with minimal errors considering transient mass transfer time relaxation parameter for the phase change process is proposed in this study. A model coupled Volume of Fluid (VOF) model, phase change model and continuum surface force model is developed to simulate heat transfer characteristics and phase change process for a two-phase closed thermosyphon. The mass transfer process is implemented by adding User Define Function (UDF) to FLUENT code. The results obtained from this study show that the model with transient mass transfer time relaxation parameter has smaller relative errors (0.27–0.73%) for absolute temperature distributions along the wall of a two-phase closed thermosyphon than the model without transient mass transfer time relaxation parameter (2.01–2.97%). The model with transient mass transfer time relaxation parameter has smaller relative errors for the thermal resistances at evaporation and condensation sections (3.21–4.23% and 2.45–6.78%) than the model without transient mass transfer time relaxation parameter (18.31–21.74% and 15.34–28.25%), respectively. The model developed in this study can also detail the phase change process and therefore can be used to predict heat performance better for different types of heat pipes.

Numerical study of multiphase droplet dynamics and contact angles by smoothed particle hydrodynamics

In this paper, an efficient method is developed for droplet simulation with high density ratios using smoothed particle hydrodynamics (SPH). The new approach is straightforward compared to previous studies and is applied to both static and dynamic droplet problems. Regarding the continuum surface force method which needs the normal vector and surface curvature in surface tension calculation, a color function formulation is introduced for tracking and correcting color function of boundary particles. This leads to a stable and accurate droplet simulation which reflects the reality of surface-tension force and droplet profile without the need for setting the desired unit normal vectors directly. The developed model is validated for droplet oscillation, static contact angles, liquid droplet evolution, dynamic wetting angles on inclined surfaces and droplets collision. The results demonstrates that the developed formulation can capture the equilibrium contact angle and handle droplets motion over inclined surfaces with different inclination angles and gravitational accelerations accurately.

Volume-of-fluid simulations of bubble dynamics in a vertical Hele-Shaw cell

Bubbles in confined geometries serve an important role for industrial operations involving bubble-liquid interactions. However, high Reynolds number bubble dynamics in confined flows are still not well understood due to experimental challenges. In the present paper, combined experimental and numerical methods are used to provide a comprehensive insight into these dynamics. The bubble behaviour in a vertical Hele-Shaw cell is investigated experimentally with a fully wetting liquid for a variety of gap thicknesses. A numerical model is developed using the volume of fluid method coupled with a continuum surface force model and a wall friction model. The developed model successfully simulates the dynamics of a bubble under the present experimental conditions and shows good agreement between experimental and simulation results. It is found that with an increased spacing between the cell walls, the bubble shape changes from oblate ellipsoid and spherical-cap to more complicated shapes, while the bubble path changes from only rectilinear to a combination of oscillating and rectilinear; the bubble drag coefficient decreases and this results in a higher bubble velocity caused by a lower pressure exerted on the bubble; the wake boundary and wake length evolve gradually accompanied by vortex formation and shedding.

High fidelity anisotropic adaptive variational multiscale method for multiphase flows with surface tension

We propose in this work an adaptive variational multiscale method for two-fluid flows with surface tension. A level set function is used to provide a precise position of the interfaces. The implementation of the surface tension in the context of the Continuum Surface Force is proposed to circumvent the capillary time step restriction. The obtained system is then solved using a new derived Variational Multiscale stabilized finite element method designed to handle the abrupt changes at the interface and large density and viscosity ratios. Combined with an a posteriori error estimator, we show that anisotropic mesh adaptation provides highly stretched elements at the interfaces and thus yield an accurate modeling framework for two-phase incompressible isothermal flows. Stable and accurate results are obtained for all two- and three-dimensional numerical examples. To the best of our knowledge, these are the first simulation results for representative time-dependent three-dimensional two-fluid flow problems using an implicit treatment of the surface tension and a dynamic unstructured anisotropic mesh adaptation.

A two-phase solver for complex fluids: Studies of the Weissenberg effect

In this work a new two-phase solver is presented and described, with a particular interest in the solution of highly elastic flows of viscoelastic fluids. The proposed code is based on a combination of classical Volume-of-Fluid and Continuum Surface Force methods, along with a generic kernel-conformation tensor transformation to represent the rheological characteristics of the (multi)-fluid phases. Benchmark test problems are solved in order to assess the numerical accuracy of distinct levels of physical complexities, such as the interface representation, the influence of advection schemes, the influence of surface tension and the role of fluid rheology. In order to demonstrate the new features and capabilities of the solver in simulating of complex fluids in transient regime, we have performed a set of simulations for the problem of a rotating rod inserted into a container with a viscoelastic fluid, known as the Weissenberg or Rod-Climbingeffect. Firstly, our results are compared with numerical and experimental data from the literature for low angular velocities. Secondly, we have presented results obtained for high angular velocities (high elasticity) using the Oldroyd-B model which displayed very elevated climbing heights. Furthermore, above a critical value for the angular velocity, it was observed the onset of elastic instabilities driven by the combination of elastic stresses, interfacial curvature and secondary flows, that to the authors best knowledge, were not yet reported in the literature.

Characterization of the velocity fields generated by flow initialization in the CFD simulation of multiphase flows

Recent improvements in numerical algorithms for Eulerian-based CFD simulations of multiphase flows can now recover the exact balance between the pressure and surface tension terms within the momentum equation. This has led to the minimization of the so called “spurious velocities”: unphysical flows appearing at interfaces when using implementations of the Continuum Surface Force technique to model surface tension forces. Nevertheless, non-null velocity fields are still observed in the proximity of interfaces despite the absence of external forces, and it will be shown here that they can still considerably influence the accuracy of two-phase flow simulations. This study demonstrates that the evolution of these velocity fields is due to the generation of a capillary wave at the beginning of the simulation. This arises from the inability of the surface tension algorithm to evaluate a constant interface curvature from the initial color function field, although this is obtained by mapping an exact circumference on the computational grid, which then generates an initial perturbation that evolves as a capillary wave. A linear stability analysis of a two-dimensional circular static droplet subjected to initial small amplitude perturbations is here developed and compared with the results of CFD simulations to corroborate this thesis. This methodology provides a novel insight on the nature and behavior of spurious currents and their influential parameters, and it yields results which are not only in line with previous observations but also unify some contrasting trends reported in the literature. It is observed that the velocity magnitude induced by the capillary wave scales with the surface tension coefficient and the inverse of the fluid density and droplet radius. Furthermore, it decreases when refining the computational mesh only if the interface curvature estimations converge to the exact value, while otherwise showing the opposite trend.

Numerical simulation of bubble condensation using CF-VOF

In present study, subcooled boiling is simulated using color function volume of fluid (CF-VOF) method. For this purpose, energy equation and Tanasawa mass transfer model accompanied with some suitable source terms are implemented in OpenFOAM solver (interFoam). The surface tension between vapor–liquid phases is considered using continuous surface force (CSF) method. In order to reduce spurious current near interface, a smoothing filter is applied to improve curvature calculation. The variation of saturation temperature in vapor bubble with local pressure is considered with Clausius–Clapeyron relation. The numerical model is validated with one-dimensional Stefan problem.The shape and life time history of single vapor bubble condensation are verified with existing experimental data. Computational study shows bubble life time is nearly proportional with bubble size and it is prolonged at bubble swarm motion. The present study reveals some fundamental characteristics of single and multiple vapor bubble condensation and is expected to be instructive for further applications.

Simulation on Thermocapillary-Driven Drop Coalescence by Hybrid Lattice Boltzmann Method

A hybrid two-phase model, incorporating lattice Boltzmann method (LBM) and finite difference method (FDM), was developed to investigate the coalescence of two drops during their thermocapillary migration. The lattice Boltzmann method with a multi-relaxation-time (MRT) collision model was applied to solve the flow field for incompressible binary fluids, and the method was implemented in an axisymmetric form. The deformation of the drop interface was captured with the phase-field theory, and the continuum surface force model (CSF) was adopted to introduce the surface tension, which depends on the temperature. Both phase-field equation and the energy equation were solved with the finite difference method. The effects of Marangoni number and Capillary numbers on the drop’s motion and coalescence were investigated.

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

SUMMARYA 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.

On the physically based modeling of surface tension and moving contact lines with dynamic contact angles on the continuum scale

The description of wetting phenomena is a challenging problem on every considerable length-scale. The behavior of interfaces and contact lines on the continuum scale is caused by intermolecular interactions like the Van der Waals forces. Therefore, to describe surface tension and the resulting dynamics of interfaces and contact lines on the continuum scale, appropriate formulations must be developed. While the Continuum Surface Force (CSF) model is well-engineered for the description of interfaces, there is still a lack of treatment of contact lines, which are defined by the intersection of an ending fluid interface and a solid boundary surface. In our approach we use a balance equation for the contact line and extend the Navier-Stokes equations in analogy to the extension of a two-phase interface in the CSF model. Since this model depicts a physically motivated approach on the continuum scale, no fitting parameters are introduced and the deterministic description leads to a dynamical evolution of the system.As verification of our theory, we show a Smoothed Particle Hydrodynamics (SPH) model and simulate the evolution of droplet shapes and their corresponding contact angles. Modeling of dynamic, advancing and receding contact angles.Introduction of the Contact Line Force model (CLF).Leap-Frog time integration scheme for Incompressible Smoothed Particle Hydrodynamics (ISPH).Direct Numerical Simulation (DNS) of two-phase flow including surface tension.

Numerical modeling of microscale droplet dispensing in parallel-plate electrowetting-on-dielectric (EWOD) devices with various reservoir designs

Water droplet dispensing in microfluidic parallel-plate electrowetting-on-dielectric (EWOD) devices with various reservoir designs has been numerically studied. The Navier–Stokes equations are solved using a finite-volume formulation with a two-step projection method on a fixed grid. The free surface of the liquid is tracked by a coupled level set and volume-of-fluid method with the surface tension force determined by the continuum surface force model. Contact angle hysteresis which is an indispensible element in EWOD modeling has been implemented. A simplified model is adopted for the viscous stresses exerted by the parallel plates at the solid–liquid interface. Good agreement has been achieved between the numerical results and the corresponding experimental data. The dispensing mechanism has been carefully examined, and droplet volume inconsistency for each design has been investigated. It has been discovered that the pressure distribution on the cutting electrode at the beginning of the cutting stage is of considerable significance for the inconsistency of droplet volumes. Several key elements which directly affect the pressure distribution and volume inconsistency have been identified.

Simulation of Convective Heat Transfer of Gas–Liquid Bubble Train Flow in Wet Microtube

In this paper, a direct numerical simulation of a two‐phase incompressible gas–liquid flow for simulation of bubble motion and convective heat transfer in a microtube is presented. The microtube radius is 10 μm. The interface between the two phases is tracked by the volume of fluid method with the continuous surface force model. Newtonian flows are solved using a finite volume scheme based on the PISO algorithm. Numerical simulation is done on an axisymmetric domain with a periodic boundary condition for different values of pressure gradient, void fraction, and bubble period. Mean pressure gradient is fixed for each simulation. The superficial Reynolds numbers of gas and liquid phases studied are 0.3 to 7 and 5 to 210, respectively. Numerical results are coincident with the Serizawa regime map, and there is a linear relation between the void fraction and gas flow ratio. Simulation shows local Nusselt number increases in the presence of a gas bubble.

Numerical Modelling of Wetting Phenomena During Melting of PCM

Latent heat thermal energy storage (LHTES) units using phase change material (PCM) exhibit a high thermal capacity, but also a low charging and discharging power. Macro-encapsulation of PCM is one way of enhancing the heat transfer rate in thermal storage units. During encapsulation, the cavity left to reduce mechanical stresses in the macro capsule, locks air. This makes the capsule a multiphase system consisting of immiscible phases namely PCM and air.Numerical modelling of a PCM capsule enables a detailed understanding of the phase change process of the PCM under the influence of air. In this article, two immiscible fluids PCM and air have been modeled using a continuum surface force (CSF) model in the open source computational fluid dynamics (CFD) software, OpenFOAM. The wetting of the melted PCM on the capsule wall is taken into account by implementing a contact angle boundary condition. The surface tension of the PCM with air is just contributed to the liquid phase of the PCM. The nonlinear enthalpy-temperature relation during the phase change is taken into consideration by a source-based fictitious method. The flow in the solid phase of the PCM is ramped down by considering a high solid viscosity. Experimental results are employed to validate the overall model.The results obtained from the complete numerical model have shown a great acceptance when compared with experiments. Despite several small deviations in the results, the numerical modelling is a potential tool to optimize the efficiency of thermal storage units.

Numerical Investigation of Water Entry of Half Hydrophilic and Half Hydrophobic Spheres

A numerical simulation to investigate the water entry of half-half sphere which is hydrophobic on one hemisphere and hydrophilic on the other is performed. Particular attention is given to the simulation method based on solving the Navier-Stokes equations coupled with VOF (volume of fluid) method and CSF (continuum surface force) method. Numerical results predicted experimental results, validating the suitability of the numerical approach to simulate the water entry problem of sphere under different wetting conditions. Numerical results show that the water entry of the half-half sphere creates an asymmetric cavity and “cardioid” splash, causing the sphere to travel laterally from the hydrophobic side to the hydrophilic side. Further investigations show that the density ratio and mismatch of asymmetric in wetting condition affect the trajectory, velocity, and acceleration of the half-half sphere during water entry. In addition, the total hydrodynamic force coefficient is investigated as a result of the forces acting on the sphere during water entry dictated by the cavity formation.

A Q2Q1 integrated finite element method with the semi‐implicit consistent CSF for solving incompressible two‐phase flows with surface tension effect

SummaryAn integrated finite element method (FEM) is proposed to simulate incompressible two‐phase flows with surface tension effects, and three different surface tension models are applied to the FEM to investigate spurious currents and temporal stability. A Q2Q1 element is adopted to solve the continuity and Navier–Stokes equations and a Q2‐iso‐Q1 to solve the level set equation. The integrated FEM solves pressure and velocity simultaneously in a strongly coupled manner; the level set function is reinitialized by adopting a direct approach using interfacial geometry information instead of solving a conventional hyperbolic‐type equation. In addition, a consistent continuum surface force (consistent CSF) model is utilized by employing the same basis function for both surface tension and pressure variables to damp out spurious currents and to estimate the accurate pressure distribution. The model is further represented as a semi‐implicit manner to improve temporal stability with an increased time step. In order to verify the accuracy and robustness of the code, the present method is applied to a few benchmark problems of the static bubble and rising bubble with large density and viscosity ratios. The Q2Q1‐integrated FEM coupled with the semi‐implicit consistent CSF demonstrates the significantly reduced spurious currents and improved temporal stability. The numerical results are in good qualitative and quantitative agreements with those of the existing studies. Copyright © 2015 John Wiley & Sons, Ltd.

Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow

In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface force (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter via Cahn–Hilliard equation which is solved in the frame work of LBE. The scalar convection–diffusion equation for temperature field is resolved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two/three dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results.

A fourth-order accurate curvature computation in a level set framework for two-phase flows subjected to surface tension forces

We propose an accurate and robust fourth-order curvature extension algorithm in a level set framework for the transport of the interface. The method is based on the Continuum Surface Force approach, and is shown to efficiently calculate surface tension forces for two-phase flows. In this framework, the accuracy of the algorithms mostly relies on the precise computation of the surface curvature which we propose to accomplish using a two-step algorithm: first by computing a reliable fourth-order curvature estimation from the level set function, and second by extending this curvature rigorously in the vicinity of the surface, following the Closest Point principle. The algorithm is easy to implement and to integrate into existing solvers, and can easily be extended to 3D. We propose a detailed analysis of the geometrical and numerical criteria responsible for the appearance of spurious currents, a well known phenomenon observed in various numerical frameworks. We study the effectiveness of this novel numerical method on state-of-the-art test cases showing that the resulting curvature estimate significantly reduces parasitic currents. In addition, the proposed approach converges to fourth-order regarding spatial discretization, which is two orders of magnitude better than algorithms currently available. We also show the necessity for high-order transport methods for the surface by studying the case of the 2D advection of a column at equilibrium thereby proving the robustness of the proposed approach. The algorithm is further validated on more complex test cases such as a rising bubble.