Continuum Surface Force Research Articles

Simplified conservative discretization of the Cahn-Hilliard-Navier-Stokes equations

In this paper we construct a novel discretization of the Cahn-Hilliard equation coupled with the Navier-Stokes equations. The Cahn-Hilliard equation models the separation of a binary mixture. We construct a very simple time integration scheme for simulating the Cahn-Hilliard equation, which is based on splitting the fourth-order equation into two second-order Helmholtz equations. We combine the Cahn-Hilliard equation with the Navier-Stokes equations to simulate phase separation in a two-phase fluid flow in two dimensions. The scheme conserves mass and momentum and exhibits consistency between mass and momentum, allowing it to be used with large density ratios. We introduce a novel discretization of the surface tension force from the phase-field variable that has finite support around the transition region. The model has a parameter that allows it to transition from a smoothed continuum surface force to a fully sharp interface formulation. We show that our method achieves second-order accuracy, and we compare our method to previous work in a variety of experiments.

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.

A one-domain pore-resolved approach for multiphase flows in porous media

Modeling multiphase flow in porous structures remains a challenge due to the complexity of handling multiple interfaces. This paper presents a one-domain pore-resolved simulation approach for immiscible two-phase flows in porous media, using a monolithic fluid–solid coupling framework to implicitly consider the existence of solid objects, with the fluid–fluid interfaces captured through solving an algebraic volume of fluid equation. Fluid interfacial tension is considered by integrating a continuum surface force, and the wall wettability condition is imposed by modifying the contact angle of the fluid interface at the embedded solid surface. The resulting equations are simple and stable, as there are no empirical models or parameters involved for the interface representation. This approach has been validated through performing a series of test-case simulations, including capillary-dominated flow, capillary rise with gravity, Taylor film formation, and finally two-phase flow in a heterogeneous porous structure. The numerical approach is demonstrated to be well suited for investigating pore-scale two-phase flows in realistic porous media.

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 investigation of flow boiling characteristics of R134A in a smooth horizontal tube

Three-dimensional transient simulations using ANSYS Fluentʼs mixture multiphase model was conducted to investigate the impact of heat flux, mass flux, and saturation temperature variations on pressure drop behaviour and heat transfer characteristics during flow boiling of R134a refrigerant in smooth horizontal tubes of different diameters. The simulations employ an implicit numerical method to solve the governing equations, with relevant source terms for mass and energy transfer, incorporating the continuum surface force (CSF) and shear stress transport (SST) κ-ω models to account for surface tension and flow turbulence respectively. The results are presented in both dimensional (heat transfer coefficient and pressure drop) and non-dimensional (Nusselt number and two-phase frictional factor) forms. The Nusselt number shows a direct relationship with the Reynolds number, while the two-phase friction factor exhibits different behaviour. At higher vapor quality, mass flux significantly affects heat transfer performance and pressure drop. Heat flux plays a crucial role in heat transfer, but its impact on pressure drop is negligible. Higher saturation temperatures enhance heat transfer while reducing pressure gradients. The obtained results demonstrate good agreement with existing correlations, with mean absolute deviations (MAD) as low as 1.2 % for the heat transfer coefficient and 23.7 % for the pressure drop behaviour. This highlights the model's accuracy in predicting flow boiling behaviour of R134a in smooth horizontal channels, providing valuable insights for thermal analysis. The findings contribute to the understanding of flow boiling processes and aid in designing and optimizing efficient heat exchangers for various engineering applications.

The Numerical Simulation of Marangoni Problems Employing Multi-phase Parallel SPH Method

When facing microscale problems, the phenomenon of droplet movement caused by temperature gradients is usually attributed to the Marangoni effect. This study first constructed a multiphase smoothed particle fluid dynamics (SPH) system structure, integrating continuous surface force (CSF) model and multiphase model to explore the behavior of droplets under Marangoni effect. To further improve computational efficiency, this study also utilized the Open Multi Processing (OpenMP) parallel computing framework to perform parallel optimization on the SPH algorithm. Subsequently, the effectiveness of the constructed SPH framework in addressing thermal capillary phenomena was verified by simulating the Marangoni migration phenomenon of silicone oil droplets in fluorine solutions. In addition, this study also investigated the polymerization process of two droplets in microchannels under the Marangoni effect, and obtained the complete dynamic changes of droplets from motion to polymerization. During this period, we conducted a detailed analysis of the variation process of the velocity vector field, the spatial distribution of the temperature field, and the evolution characteristics of the surface tension gradient during the droplet movement process, thereby comprehensively revealing how the Marangoni effect drives the phenomenon of droplet movement and aggregation.

Numerical investigation of water entry of hydrophobic spheres

We perform numerical simulations to study the dynamics of the entry of hydrophobic spheres in a pool of water. To track the air-water interface during the translation of the sphere in the pool of water, we use the volume of fluid model. The continuum surface force method computes the surface tension force. To represent the hydrophobic surface properties, we use wall adhesion in terms of a static contact angle. We perform simulations with different diameters and impact speeds of the sphere. Our simulations capture the formation of different types of air cavities, pinch-offs of these cavities, and other finer details similar to the experiments performed at the same parameters. Finally, we compare the drag force among the different hydrophobic cases. We further perform simulations of hydrophilic spheres impacting the pool of water and compare the drag force with the analogous hydrophobic cases. We conclude that the spheres with hydrophobic surfaces encounter a lower drag than their hydrophilic counterparts. This lower drag of the hydrophobic spheres is attributed to the formation of the air cavity by the hydrophobic surfaces while translating through the pool of water, which reduces the area of the sphere in contact with water. In contrast, no such air cavity forms in the case of hydrophilic spheres.

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.

Numerical study of Two-Phase flow behavior during dryout in a Two-Phase closed thermosyphon under thermal boundary conditions

This paper presents a two-dimensional axisymmetric two-phase closed thermosyphon (TPCT) model filled with R-134a refrigerant using the Volume of Fluid (VOF) method, Lee model, and Continuum Surface Force (CSF) method within Computational Fluid Dynamics (CFD) and simulates the flow characteristics inside the TPCT during the occurrence of dry out. Specifically, it discusses the two-phase flow in the condensation and evaporation sections when dry out occurs. The results indicate that the condensation section exhibits varying forms of vapor–liquid interface waves at different times. Phenomena such as liquid carryover and entrainment become increasingly prominent over time, contributing significantly to refrigerant blockages and dryout in the evaporation section. The mechanism behind dryout in the evaporation section is revealed to be the result of the initially continuous falling film being disrupted by fluctuations in the liquid–vapor interface, leading to the premature drying of thin liquid films in the troughs of these waves.

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.

A Vaporization Model for Continuous Surface Force Approaches and Subcooled Configurations

The integration of phase change phenomena through an interface is a numerical challenge that requires proper attention. Solutions to properly ensure mass and energy conservation were developed for finite difference and finite volume methods, but not for Finite Element methods. We propose a Finite Element phase change model based on an Eulerian framework with a Continuous Surface Force (CSF) approach. It handles both momentum and energy conservation at the interface for anisotropic meshes in a light an efficient way. To do so, a model based on the Level Set method is developed. A thick interface is considered to fit with the CSF approach. To properly compute the energy conservation, heat fluxes are extended through this interface thanks to the resolution of a transport equation. A dedicated pseudo compressible Navier–Stokes solver is added to compute velocity jumps with a source term at the interface in the velocity divergence equation. Several 1D and 2D benchmarks are considered with increasing complexity to highlight the performances of each feature of the framework. This stresses the capacity of the model to properly tackle phase change problems.

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.

Numerical investigation of the influence of surface wettability on water entry of spheres

The water-entry problem is a complex multiphase hydrodynamic problem that is directly related to many engineering applications and natural phenomena, such as torpedo airdrops, seaplane landings, and ship slamming. Therefore, studying the influence of the microscopic properties of the object surface on the macroscopic phenomenon during water entry is necessary. In this study, the volume of fluid model and continuum surface force models are coupled to establish a multiphase flow numerical method for the water entry of objects considering surface wettability. The effect of surface wettability on the evolution of the cavity, multiphase flow-field structure, and hydrodynamic force characteristics are analyzed in detail. The results show that the movement of liquid film formed on the surface of the sphere at the early stage is the key to the formation of the cavity. For hydrophobic spheres, the liquid film separates near the equator of the sphere, and air enters it to form a cavity. At the moment of pinch-off, the pressure in the lower cavity increases, which generates a force that pushes the sphere to accelerate the fall, and this force is higher for spheres with a smaller density ratio. The flow-field structure shows that both rotational and shear effects play a dominant role in the evolution of the flow field in the cavity. For hydrophilic spheres, the liquid film follows the contact line along the surface of the sphere and converges at the top to form an upward jet.