Continuum Surface Force Model Research Articles

A modified continuum surface force (M-CSF) model for two-phase flow problems in smoothed particle hydrodynamics

ABSTRACT This work proposes a modified interfacial tension model based on the continuum surface force (M-CSF) in the context of smoothed particle hydrodynamics. This correction aims to enhance computational stability, improve force evaluation precision, and increase symmetry around the interface. Several two-phase flow benchmarks are solved using both the conventional and proposed CSF models, and the results are compared with each other and with the available literature. The results indicate that the modified model can efficiently increase the force evaluation accuracy in the pressure field at the interface. For instance, the relative error in pressure calculation using the proposed and conventional CSF models is 0.05% and 3.5%, respectively, when compared to the analytical solution, with both models having the same particle resolutions for the droplet deformation problem. In predicting the critical surface tension for hydrodynamic instabilities, such as the Rayleigh-Taylor instability, the M-CSF methodology exhibits much better alignment with existing theories, showing less than 5% deviation, while conventionally used CSF models can deviate up to 15% for the same problem. These findings confirm the superiority of the proposed methodology in evaluating interfacial forces, even in complex hydrodynamic instabilities.

An Enhanced Mapping Interpolation ISPH–FVM Coupling Method for Simulating Two-Phase Flows with Complex Interfaces

In this paper, an enhanced mapping interpolation ISPH–FVM coupling method is developed to improve the accuracy and stability of two-phase flows simulation. Similar to the original ISPH–FVM coupling method, the coupling framework between the numerical schemes of ISPH and FVM is established at the overlapping region of the ISPH particles and the FVM grids. Although the original ISPH–FVM coupling method can effectively predict movement and deformation of the two-phase interface, it is difficult to ensure the mass conservation in the flow domain during the information transfer between ISPH particles and FVM grids, thus weakening the numerical accuracy of interaction between different fluid phases. To improve the mapping interpolation accuracy and ensure the mass conservation, a volume fraction correction scheme combined with the particle approximate interpolation technique is introduced to form an enhanced mapping interpolation ISPH–FVM coupling method. The effect of surface tension is assessed by the continuum surface force (CSF) model. Several 2D/3D numerical cases are given to verify the efficiency and robustness of the present ISPH–FVM coupling method. Numerical results demonstrate that the enhanced mapping interpolation developed in the ISPH–FVM coupling method can reproduce the two-phase flow with complex interfaces effectively.

A reduction-consistent phase field model for non-isothermal multiphase flows of N immiscible incompressible fluids

In this study, we developed a reduction-consistent phase field model for non-isothermal incompressible N-phase flows. The model is based on the high-order spectral element method. To account for thermocapillary effects, we reformulated the continuum surface force model specifically for N-phase flows. In order to enhance computational efficiency, time-independent coefficient matrices for all variables involved were reconstructed, and an unsymmetrized multifrontal LU factorization was employed to solve the linear algebraic equations, which is derived by discretization. To verify the model effectiveness in calculating surface tension and describing three-phase interfacial dynamics, we conducted several experiments, including the stationary two-droplet example, the three-phase droplet spreading, and the equilibrium morphology of double emulsion droplet. Through these experiments, both the reduction-consistency and robustness of our model were demonstrated. Moreover, we validated the proposed model's applicability to non-isothermal three-phase flows by investigating the thermocapillary migration of single droplet and two droplets of different phases. Notably, we explored the thermocapillary migration of three-phase double droplets, focusing particularly on how the encapsulation process affects overall thermocapillary motion. Our findings indicate that the interaction between the two vortices near the interfaces of the two droplets strongly influences their migration behavior.

Numerical simulations of thermal capillary migration of a droplet on a temperature gradient wall with smoothed particle hydrodynamics method

Thermal capillary migration is a phenomenon due to the Marangoni effect, which refers to the spontaneous motion of a liquid on a non-isothermal surface. Numerical simulation of thermal capillary migration of a droplet is extremely difficult due to the multi-physics field coupling and the violent motion of the droplet surface. In this paper, an improved smooth particle hydrodynamics (SPH) method is developed for simulating thermal capillary migration of a droplet driven by thermal gradient. In improved SPH model, an improved continuous surface force model is proposed to enhance the accuracy and stability of surface tension force calculation by introducing an improved surface tangential force. The contact angle model is utilized to model the surface wettability. The SPH method for simulating the thermal fluid flow is developed based on the continuum, momentum, and energy equations. In addition, kernel gradient correction and particle shifting technique are utilized to improve the accuracy and stability of the SPH method. The correctness and effectiveness of the improved SPH method are verified by numerical examples. Moreover, the motions of a droplet driven by thermal gradient under different conditions are investigated. Comparing with the results obtained by experiments and other resources, we can conclude that the improved SPH model is effective in modeling the thermal capillary migration of a droplet.

Comparison of surface tension models for the simulation of two-phase flow in an ISPH-FVM coupling method

In the simulation of two-phase flow dominated by surface tension, accurate surface tension modeling is beneficial to better reproduce and understand the mechanism of interphase flow. In this paper, based on an ISPH-FVM coupling framework, three different surface tension models are implemented and tested respectively, including the generally used continuum surface force (CSF) model, the continuous surface stress (CSS) model and the height function (HF) model. In the present ISPH-FVM coupling framework, the ISPH particle approximate interpolation technique combined with a volume fraction correction scheme is employed to ensure the volume conservation in the computational domain during the information transfer between particles and grids. Meanwhile, the three surface tension models are discretized and calculated by the volume fraction defined on the FVM grid. The volume fraction of the FVM grid is obtained by approximate interpolation of ISPH particles within the grid support domain. Several benchmark cases are tested to verify the performance of three surface tension models in the ISPH-FVM coupling method. The results show that the CSF model and CSS model have less spurious currents and better robustness than HF model under the present coupling method. In addition, the CSF model and CSS model can simulate the flow regime involving complex interface topology changes more accurately than HF model.

A fully explicit incompressible smoothed particle hydrodynamics approach for modeling transient heat transfer and thermo-capillary flows

This work presents a Lagrangian meshless method for modeling transient heat transfer and thermo-capillary effects at the interface of two-phase incompressible fluids. To simulate the thermo-capillary effect, we add the Marangoni force to the continuum surface force (CSF) model, by considering the forces caused by the surface tension gradient which acts tangentially to the interface position. In the current study, we extend our previously proposed model which uses a fully explicit incompressible smoothed particle hydrodynamics (EISPH) approach along with corrected SPH and VKF kernel function, to obtain stable and accurate results in non-isothermal single and multi-phase problems.The model is validated for several test cases, including transient heat conduction, natural convection heat transfer, and thermo-capillary droplet migration, against conventional numerical methods. The validated model is used to study the effects of several dimensionless parameters on thermo-capillary droplet migration. The results show that the proposed EISPH method can model accurately complex heat transfer problems such as thermo-capillary induced motion.

Study of melt pool dynamics and porosity forming mechanism of laser beam oscillation welding of titanium and aluminum

In this study, a numerical model of oscillation weld butt joint is developed to investigate the welding of titanium alloy with aluminum alloy. Three oscillation paths, namely, straight, sine, and circular, are used to study the distribution of force in the molten pool, the welding temperature field, and the formation and evolution of porosity within the weld. A 3D Gaussian heat source is used to represent the laser beam. The volume of the fluid method is employed to track the gas-liquid free surface, and the gas-liquid interface force is transformed by using the continuous surface force model. The mechanism of keyhole collapse and pore formation was examined along with the fluid flow, surface tension, and recoil pressure on the molten pool. The results confirmed that the highest welding quality is acquired by using a laser welding circular path. Notably, numerical simulation results are validated through experimental data, and circular oscillating laser welding significantly reduced weld seam porosity in the welding of Ti–Al dissimilar alloys. The circular oscillation path with an offset of 0.6 mm and an oscillation amplitude of 0.6 mm is identified as the optimal approach for suppressing pores in the weld joint. This research provides valuable insights into the fundamental mechanisms of keyhole collapse and pore formation in laser welding, which contributes to the advancement of effective welding strategies for dissimilar alloys.

Non-local modelling of multiphase flow wetting and thermo-capillary flow using peridynamic differential operator

AbstractInterfaces in multiphase flows are affected by surface tension, and when temperature gradients occur in the flow domain, tangential surface tensions along the interface also arise. As the behaviour of fluids contacting on a solid surface is also governed by surface tension, the description of the wetting phenomenon is challenging. Peridynamic differential operator (PDDO) can express partial differentials of any order by integral equations. Therefore, the governing equations for multiphase fluid motion, such as the Navier–Stokes equations and energy equations, can be reformulated in terms of integral equations. In this study, a novel non-local method is developed for modelling the multiphase fluid flow motion using the PDDO, and the thermal effect on surface tension force is considered. To describe the surface tension forces in the normal and tangential directions, the non-local form of the continuum surface force (CFS) model is presented. Besides, to overcome the inaccuracy of the unit normal vectors at the three-phase flow intersection region, an additional treatment for this region is presented. Finally, several benchmark multiphase fluid flow cases, such as square droplet deformation, surface wetting, and droplet migration in thermo-capillary flow are presented and validated. The results demonstrate that the developed non-local model can accurately capture the surface tension effect in multiphase fluid flow motion.

A modified dynamic Lee model for two-phase closed thermosyphon (TPCT) simulation

A modified model consisting of a dynamic Lee model, volume of fluid model, and continuum surface force model is developed. The modified model investigates heat transfer characteristics of the vapor–liquid phase change process and details of the two-phase flow during operation of a two-phase closed thermosyphon. The mass transfer time relaxation parameters for the Lee phase change model are the most critical coefficients which determine the rate of the vapor–liquid phase change. A dynamic adjustment of the mass transfer time relaxation parameters for the Lee phase change model is realized based on the amount of mass transfer between the vapor and liquid phases and the values of the mass transfer time relaxation parameters become stabilized. The relative error between the modified model and experimental data for the temperature distribution is 5%, representing an acceptable agreement. Compared with the original model, the maximum thermal resistance errors in evaporation section and condensation section are reduced by 19.3% and 107.1%, respectively. These results indicate that the modified model can provide good corrections with high accuracy.

Investigation of electrostatically tunable adhesion and instability of flying head slider

The interfacial adhesion between microstructures is inevitable in a micro-electro-mechanical system (e.g., hard disk drive (HDD)), which may lead to complicated microtribodynamics problems. This research has investigated the effect of surface potential on the interfacial adhesion and microtribodynamics of the head–disk interface (HDI) in an HDD. A dynamic continuum surface force model, where the electrowetting is considered, is proposed to evaluate the interfacial interaction, and then employed into a two-degree-of-freedom (2DOF) model to theoretically analyze the potential influence mechanism on the microtribodynamics. The results confirm that the elimination of potential can effectively repress the adhesion retention, which is further proved by the measured slider response with a laser Doppler vibrometer (LDV). Moreover, the effect of the potential on the adhesion-induced instability is also analyzed through the phase portrait. It tells that the critical stable flying height can be lowered with the elimination of potential.

Bubble-assisted matte transportation into slag phase: A numerical investigation

Copper loss into the slag phase during the smelting process can be minimized by controlling gas bubble size and hydrodynamics. Here, bubble rising characteristics from copper matte to slag phase were investigated using the volume-of-fluid modelling approach, in which the continuous surface force model was adopted to define the surface tension parameters. The simulated results for bubble terminal velocities in bulk water and the literature data showed relative errors up to 4.51 %. The obtained qualitative and quantitative data from simulation at the liquid–liquid interface agreed well with the literature experimental results. The simulation setup was applied to investigate three different sizes of bubble rising dynamics from matte to slag phase with the bubble Reynolds number (Re) up to 1734 and the Eotvos number (Eo) up to 3.2. For low Re and Eo, the bubble rising velocity decreased before the matte film rupture and the bubble required a long time to complete that film drainage due to the balance of interfacial tension force and bubble buoyancy. No matte residue associated with the bubble rise in the slag phase was obtained because of restoring initial, non-deformed shape of bubbles. For the higher Re and Eo, quicker matte film drainage was obtained around the upper bubble surface while the entrained matte contribution to the slag phase was strongly dependent on the bubble-shape deformation, which was discussed with the well-known Grace diagram. The decreasing viscosity and increasing surface tension of the slag phase significantly affected the matte film height more than the film rupture time. The film height and rupture time increased and decreased with increasing Eo, respectively. The outcomes of our studies are useful for the design and operation of smetling furnaces in pyrometallurgical purification of metal concentrates.

An improved continuum surface tension model in SPH for simulating free-surface flows and heat transfer problems

The continuum surface force (CSF) model in conjunction with the smoothed particle hydrodynamics (SPH) method is commonly used in the simulation of multi-phase flows with surface tension effect. However, it cannot be directly applied for modeling free-surface flows since the normalization requirement, which states that the integral of the surface delta function through the interface should be equal to one, is violated due to the particle deficiency issue. In this work, we propose an improved CSF model based on the SPH method with the aim of recovering the normalization requirement. The particle deficiency issue at the free surface is tackled by introducing the jump in color function into the calculation of the surface delta function. As a result, the accuracy of the surface tension measurement is significantly improved without using the virtual particles. Besides, this improved CSF model also allows an accurate and efficient implementation of the Robin boundary condition which is frequently encountered in heat transfer problems. Several benchmark problems are investigated to validate the performance of the proposed method. The numerical results verify that the improved CSF model in SPH method is well capable of simulating free-surface flows and heat transfer problems.

A three-dimensional ISPH-FVM coupling method for simulation of bubble rising in viscous stagnant liquid

The bubble rising problem has always been one of the research hotpots in the field of multiphase flow simulation due to the complex topological evolution of the phase interface. In this paper, an approach of coupling incompressible smoothed particle hydrodynamic (ISPH) and finite volume method (FVM) in three-dimensional space (abbreviated as 3D ISPH-FVM coupling method) is developed to simulate the bubble rising in viscous stagnant liquid. In the 3D ISPH-FVM coupling method, the coupling framework between the two numerical schemes is established through the information exchange at the overlapping regions. The effect of surface tension is evaluated by implementing the 3D continuum surface force model (CSF). To verify the interface capture accuracy of the 3D ISPH-FVM coupling method, several benchmark cases are tested. Then, the single bubble rising in viscous fluid is simulated by 3D ISPH-FVM coupling method, in which the results are compared with the experimental data and other numerical results in the literatures. To further explore the ability of 3D ISPH-FVM coupling method in bubble rising simulation, the effect of some parameters (including numerical and physical parameters) on single bubble rising is investigated. Afterwards, the dynamics phenomena of two bubbles coalescence are also studied. Numerical results validate the robustness and stability of the present coupling model in simulating the bubble rising.

Applying surface tension as pressure boundary condition in free surface flow analysis by moving particle simulation method

A model that introduces surface tension as a pressure boundary condition, named the surface tension as pressure (STP) model, was developed for free surface flow analyses by the moving particle simulation (MPS) method. The STP model assigns to surface particles the liquid pressure of Laplace’s formula. The model is an alternative to previous models that apply surface tension as volume force such as the continuum surface force model. Problems that appeared when using the volume force models, such as the dependencies of calculation results on particle resolution and pressure gradient accuracy, were solved by using the STP model. Calculations predicted the theoretical values of the internal pressure of a 3D spherical droplet and the oscillation period of a 2D elliptic droplet over a wide range of surface tension coefficients and droplet sizes with errors less than 10%. Since the STP model is easy to implement, does not increase computation cost from previous models, and does not require surface reconstruction or additional marker particles, the model is suitable for practical and large-scale free surface flow problems that involve violent deformation of the liquid surface such as liquid atomization.

Numerical analysis of flow characteristics and bubble behavior inside a two-phase closed thermosyphon under various temperature difference

In the present work, a two-dimensional axisymmetric two-phase closed thermosyphon model (TPCT) filled with refrigerant R-134a is developed based on the volume of fluid method, Lee model, and continuum surface force model in the Fluent to simulate the two-phase flow characteristics and bubble behaviors inside the TPCT. The effects of increasing the temperature difference (ΔT) between the evaporator and condenser on the inner wall temperature, liquid film distribution and bubble growth process are highlighted. The results show that at different ΔT and FR, the inner wall temperature of the evaporator section increases and then decreases in inhomogeneity with increasing temperature difference at 0.5 and 0.75 FR, while at 0.05 and 0.25 FR, almost uniform inhomogeneity is observed. Bubble behavior is varied and affected by different ΔT at 0.75 FR, the comprehensive bubble incubation and growth is achieved by observing video.

Interfacial-dominated torque response in liquid–liquid Taylor–Couette flows

Immiscible and incompressible liquid–liquid flows are considered in a Taylor–Couette geometry and analysed by direct numerical simulations coupled with the volume-of-fluid method and a continuum surface force model. The system Reynolds number $Re \equiv r_i \omega _i d / \nu$ is fixed to $960$ , where the single-phase flow is in the steady Taylor vortex regime, whereas the secondary-phase volume fraction $\varphi$ and the system Weber number $We \equiv \rho r_i^2 \omega _i^2 d / \sigma$ are varied to study the interactions between the interface and the Taylor vortices. We show that different Weber numbers lead to two distinctive flow regimes, namely an advection-dominated regime and an interface-dominated regime. When $We$ is high, the interface is easily deformed because of its low surface tension. The flow patterns are then similar to the single-phase flow, and the system is dominated mainly by advection (advection-dominated regime). However, when $We$ is low, the surface tension is so large that stable interfacial structures with sizes comparable to the cylinder gap can exist. The background velocity field is modulated largely by these persistent structures, thus the overall flow dynamics is governed by the interface (interface-dominated regime). The effect of the interface on the global system response is assessed by evaluating the Nusselt number $Nu_{\omega }$ based on the non-dimensional angular velocity transport. It shows non-monotonic trends as functions of the volume fraction $\varphi$ for both low and high $We$ . We explain how these dependencies are closely linked to the velocity and interfacial structures.

Energy-conserving formulation of the CSF model for the simulation of surface tension at fluid-fluid interfaces with smoothed particle hydrodynamics

Numerical modeling of surface tension requires an accurate description of fluid-fluid interfaces as well as an efficient discretization of the governing equations. As a Lagrangian method, smoothed particle hydrodynamics (SPH) is therefore well-suited for modeling of surface tension, since any movement of the interface is naturally incorporated. However, current surface-tension formulations based on the Continuum Surface Force (CSF) model suffer from spurious currents near the interface which lead to mixing of fluids and can cause instabilities, especially for low viscosities. In this work, we propose a new discretization of the CSF model which conserves the total energy of the fluid, suppressing the aforementioned spurious currents successfully. In numerical tests, the new discretization is compared to analytical solutions and current formulations of the CSF model, showing the improved accuracy and convergence and, hence, stability of the new method as a consequence of the reduced spurious currents.

Improvement of surface tension discrete model in the ISPH-FVM coupling method

In the simulation of two-phase incompressible fluid involving surface tension, the approximate calculation of surface tension plays an important role in the accuracy of phase interface tracking and reproduction. The ISPH-FVM coupling method was proposed to simulate two-phase incompressible flow, which combines the advantages of the incompressible smoothed particle hydrodynamics (ISPH) in interface tracking and the finite volume method (FVM) in the calculation of flow field. In the original ISPH-FVM coupling method, the continuous surface force model (CSF) is used to estimate the surface tension between phase interfaces, which is discretized by smoothed particle approximation (ISPH-FVM-S). In the calculation of ISPH-FVM-S, the accuracy of surface tension method will be decreased through the direct particle interpolation when the interface changes complex. To improve the simulation accuracy of the ISPH-FVM coupling method, this paper proposes two improved discrete models for surface tension calculation, one is inspired by VOF method (ISPH-FVM-V) and the other is inspired by Level Set method (ISPH-FVM-L). Several benchmark cases are tested to illustrate the effectiveness of two improved surface tension discrete models. The results show that ISPH-FVM-V model and ISPH-FVM-L model have better accuracy and stability than ISPH-FVM-S in two-phase flow problems with complex interface topology changes. The work of this paper can further expand the application of ISPH-FVM coupling method in simulation of complex two-phase flow involving surface tension.

Assessment of an energy-based surface tension model for simulation of two-phase flows using second-order phase field methods

Second-order phase field models have emerged as an attractive option for capturing the advection of interfaces in two-phase flows. Prior to these, state-of-the-art models based on the Cahn-Hilliard equation, which is a fourth-order equation, allowed for the derivation of surface tension models through thermodynamic arguments. In contrast, the second-order phase field models do not follow a known energy law, and deriving a surface tension term for these models using thermodynamic arguments is not straightforward. In this work, we justify that the energy-based surface tension model from the Cahn-Hilliard context can be adopted for second-order phase field models as well and assess its performance. We test the surface tension model on three different second-order phase field equations; the conservative diffuse interface model of Chiu and Lin [1], and two models based on the modified Allen-Cahn equation introduced by Sun and Beckermann [2]. Additionally, we draw the connection between the energy-based model with a localized variation of the continuum surface force (CSF) model. Using canonical tests, we illustrate the lower magnitude of spurious currents, better accuracy, and superior convergence properties of the energy-based surface tension model compared to the CSF model, which is a popular choice used in conjunction with second-order phase field methods, and the localized CSF model. Importantly, in terms of computational expense and parallel efficiency, the energy-based model incurs no penalty compared to the CSF models.

Numerical study of satellite droplet formation in dripping faucet

• Dynamic relation between the liquid/gas interface, pressure and velocity profile has been investigated. • Physics of capillary wave generation in liquid filament has been analyzed. • Satellite drop generation depends on the acceleration and distribution of the axial velocity at the cone-filament junction. • Satellite drop is less likely to be formed at high inlet velocity and low surface tension coefficient. Dynamics of primary and satellite drop formation has wide engineering applications. Drop formation from a vertical capillary tube in air goes through various stages that include necking, bifurcation, recoil, wave generation and secondary necking and bifurcation. In this study, the problem has been studied numerically where the free surface of the pendant drop is tracked by a coupled level set and volume-of-fluid method with the surface tension force determined by the continuum surface force model. The evolution of the complete droplet separation process has been simulated. The numerical results have been compared with published experimental data. The mechanism for the recoiling of the liquid filament after droplet detachment and the subsequent formation of a satellite droplet has been studied and dynamic correlation among liquid/gas interface topology, velocity and pressure field has been investigated. Bell shaped distribution of axial velocity profile and fast acceleration at the cone-filament junction has been found to lead to satellite drop generation. Parametric studies on the effects of inlet velocity, surface tension coefficient and viscosity have been performed. Satellite droplet is less likely to be formed at high inlet velocity and low surface tension coefficient. Viscosity, on the other hand, has been found to play an important role on the shape of liquid filament and satellite droplet. These findings can help better understanding of drop-on-demand liquid dispensing, inkjet printing, spray combustion and many more engineering applications.