Phase-field Lattice Boltzmann Method Research Articles

Alternative wetting boundary condition for binary fluids based on phase-field lattice Boltzmann method

Alternative wetting boundary condition for binary fluids based on phase-field lattice Boltzmann method

A numerical model for solitary wave breaking based on the phase-field lattice Boltzmann method

This study presents a numerical investigation of a solitary wave breaking over a slope by using the phase-field lattice Boltzmann method. The incompressible two-phase flow equations are solved by using a velocity-based formulation of the two-phase lattice Boltzmann method with a central-moment collision model to accurately simulate wave breaking problems. For interface capture, a phase-field lattice Boltzmann method that ensures mass conservation is employed. The validity of the proposed method is confirmed through solitary wave propagation and transformation problems, and the obtained results are in good agreement with the experimental and calculated results. The proposed method is then employed to analyze wave breaking on a slope, demonstrating strong concordance with experimental data and existing computational findings. By analyzing the instantaneous flow characteristics and the temporal evolution of the variation in kinetic, potential, and total energy from deep to shallow water, the model can reveal the macroscopic characteristics of solitary wave breaking. Because the phase-field model effectively simulates wave breaking and air entrainment, it can depict wave energy dissipation more accurately than the single-phase lattice Boltzmann method with free surface tracking.

A novel phase-field lattice Boltzmann framework for diffusion-driven multiphase evaporation

Heat transfer and phase change phenomena, particularly diffusion-driven droplet evaporation, play pivotal roles in various industrial applications and natural processes. Despite advancements in computational fluid dynamics, modeling multiphase flows with large density ratios remains challenging. In this study, we developed a robust and stable conservative Allen–Cahn-based phase-field lattice Boltzmann method to solve the flow field equations. This method is coupled with the finite difference discretization of vapor species transport equation and the energy equation. The coupling between the vapor concentration and temperature field at the interface is modeled by the well-known Clausius–Clapeyron correlation. Our approach is capable of simulations under real physical conditions and is compatible with graphics processing unit architecture, making it ideal for large-scale industrial simulations. Three validation test cases are conducted to demonstrate the consistency of the presented model, including simulations of Stefan flow, the evaporation of suspended droplets containing water, acetone, and ethanol in the air, and the evaporation of a water sessile droplet on a flat surface. The results show that the model is able to predict the behavior and characteristics of each case accurately. Notably, our numerical results exhibit a maximum relative error of approximately 1% in simulations of Stefan flow. In the case of suspended droplet evaporation, the observed maximum difference between the calculated wet bulb temperatures and those derived from psychrometric charts is approximately 0.9 K. Moreover, our analysis of the sessile droplet reveals a good agreement between the results obtained by our model for the evaporative mass flux and those obtained from the existing models in the literature for different contact angles.

Integration of Hertz–Knudsen–Schrage phase change in phase-field lattice Boltzmann method: Validation and parametric studies

The phase transition between liquid and vapor is of vital importance in daily life and industry. Given the importance of the lattice Boltzmann method (LBM), in particular the phase field method, in the simulation of two-phase flows, a robust LBM phase transition model is essential. This study introduces a novel approach by integrating the widely used, Hertz–Knudsen–Schrage (HKS) phase change rate into a conservative phase-field LBM. The phase-field and momentum equations are solved using the Boltzmann distribution function, whereas the energy equation is solved using the finite difference method. Once the necessary parameters for the calculation of the phase change rate are obtained, the corresponding source terms are incorporated into each equation. The model's validation is performed through a series of benchmark problems, including the one-dimensional Stefan problem, Nusselt's film condensation, bubble detachment, centered droplet evaporation, and sessile droplet evaporation. The results demonstrate favorable agreement between the LBM solution and analytical or empirical data. Furthermore, this study highlights the model's ability to approximate steady-state phenomena with minimal reliance on the phase change coefficient of the HKS theory. It also underscores the model's capacity to accurately capture transient phenomena by appropriately selecting values for this coefficient. In addition, parametric studies are conducted to investigate evaporation problems using the HKS theory for recognizing the effect of superheat, contact angle, and droplet size on evaporation. Finally, this model not only can detect trends and behaviors of phenomena but also can adapt empirical and analytical results with good agreement.

Lattice Boltzmann simulation of breaking waves

Water wave breaking at the ocean’s surface is a turbulent phenomenon and encompasses a wide range of physical processes such as generation of bubbles/drops with different spatio-temporal scales. In the present study, breaking of periodic waves in deep water regime has been simulated using phase-field lattice Boltzmann method (LBM). Compared to direct Navier–Stokes equation based numerical models, the present LB model is Poisson-free and explicit in time integration. Both the flow field and interface are solved using LBM with two sets of distribution functions. The normalized pressure-velocity formulation is used for the flow field, whereas the conservative Allen–Cahn equation is used to capture the interface. An impulsively started wave approach with Stokes third order wave profile has been used to initiate the breaking process. Through systematic variation of Weber number and initial steepness, the effect of surface tension on the kinematic, geometric, and energy dissipation characteristics has been studied for three breaker types, viz., spilling, weak plunging, and plunging breakers. The results are in good agreement with the past experimental observations and direct numerical simulation. In particular, the obtained breaking strength for different Weber numbers agrees well with the semi-empirical relation used in a wind-wave forecasting models.

Droplet dynamics passing through the flexible constriction in the channel

We conducted a numerical investigation into droplet dynamics within a flexible constriction using the phase-field lattice Boltzmann method. Our study focused on constriction bending stiffness, Weber number, and constriction-to-droplet diameter ratio. Flexibility impedes droplet passage at low Weber numbers but significantly facilitates it at moderate and high Weber numbers. Passage times decrease with increasing Weber numbers and are proportional to the droplet's maximum deformed length. An anomalous phenomenon is observed: “more haste, less speed.” The underlying mechanics arising from the interaction between the flexible constriction and the droplet are elucidated. The findings enhanced our understanding of droplet behavior in constrained environments.

Implementation of contact line motion based on the phase-field lattice Boltzmann method.

This paper proposes a strategy to implement the free-energy-based wetting boundary condition within the phase-field lattice Boltzmann method. The greatest advantage of the proposed method is that the implementation of contact line motion can be significantly simplified while still maintaining good accuracy. For this purpose, the liquid-solid free energy is treated as a part of the chemical potential instead of the boundary condition, thus avoiding complicated interpolations with irregular geometries. Several numerical testing cases, including droplet spreading processes on the idea flat, inclined, and curved boundaries, are conducted, and the results demonstrate that the proposed method has good ability and satisfactory accuracy to simulate contact line motions.

Modeling coupled growth and motion of solid-air dendrite induced by convection in liquid hydrogen using phase-field lattice Boltzmann method

The safety hazard arising from the growth, condensation and accumulation of trace air in liquid hydrogen deserves attention. In order to predict the microstructural growth evolution and oxygen solute segregation distribution characteristics during solid-air dendrite motion induced by solution convection, a partially saturated bounce-back lattice Boltzmann coupled non-isothermal phase field method (PSLBM-PF) is proposed in this study. The partially saturated bounce-back lattice Boltzmann method (PSLBM) is employed to calculate the flow field and the forces and moments between the fluid and solid phases, and the phase field method (PF) is utilized for the computation of the growth of solid-air dendrites and the distribution of solute segregation. The motion of solid-air dendrites is determined by the momentum equation. The accuracy of the calculated interaction forces between fluid and solid phases was validated through the sedimentation process of circular particles in the liquid, and the capability of the present model to preserve the shape during solid-air dendrite motion (translational and rotational) was confirmed. Subsequently, this model was employed to investigate the growth and sedimentation motion of solid-air dendrites with different growth orientations in the solution. During dendrite settlement, the growth of the lower dendrite arms is promoted, while the settlement velocity and the solid phase fraction are almost unaffected by the growth orientation. Then, the differences in morphology evolution and solute distribution under conditions of forced convection for both stationary and motion are comparatively investigated, utilizing the free growth of solid-air dendrites under convection-free conditions as a reference. Finally, the model was utilized to study the growth of individual solid-air dendrites as well as translational and rotational motion under the action of shear flow. The presence of convection significantly alters the solute and temperature distributions at the solid-liquid interface, thereby influencing the dendrite growth process. The simulation results indicate that this model can predict the solid-air dendrites microstructural growth evolution and oxygen solute segregation distribution accompanying the convection and motion.

Computational study of a thin film flow over a topographical feature using phase-field lattice Boltzmann method

The study employs the phase-field lattice Boltzmann method (PFLBM) to explore the dynamics of a thin film flowing over a topographical feature such as a mound or a trench. The mesoscopic nature of PFLBM makes it a suitable technique for problems involving tracking the evolution of a liquid–air interface. PFLBM simulation results are validated with experimental and analytical results confirming the viability of the numerical approach for such problems. The effect of changing the topographical height, aspect ratio, viscosity ratio, and presence of multiple mounds on the film profiles are systematically analyzed. It is observed that a steady-state solution could not be obtained for large height topographical features. The transition from a steady-state interfacial pattern to an unsteady-steady state is found to depend on the width of the topography. Geometry-based condition is employed to deal with the contact points present in the film dynamics beyond rupture. For large contact angles, the unsteady cases result in film rupture and form a continuous array of droplets of equivalent dimensions at a periodic interval. Increasing the aspect ratio reduces the width of the capillary ridge formed above the topographical feature, while the viscosity ratio reduces the maximum height of the ridge. The shapes of the capillary ridges formed over multiple mounds in the flow direction are independent if the separation between the successive mounds is beyond a critical value. This critical value strongly depends on the capillary number and is independent of the dimensions of the mound.

Wetting and Spreading Behavior of Axisymmetric Compound Droplets on Curved Solid Walls Using Conservative Phase Field Lattice Boltzmann Method.

Compound droplets have received increasing attention due to their applications in many several areas, including medicine and materials. Previous works mostly focused on compound droplets on planar surfaces and, as such, the effects of curved walls have not been studied thoroughly. In this paper, the influence of the properties of curved solid wall (including the shape, curvature, and contact angle) on the wetting behavior of compound droplets is explored. The axisymmetric lattice Boltzmann method, based on the conservative phase field formulation for ternary fluids, was used to numerically study the wetting and spreading of a compound droplet of the Janus type on various curved solid walls at large density ratios, focusing on whether the separation of compound droplets occurs. Several types of wall geometries were considered, including a planar wall, a concave wall with constant curvature, and a convex wall with fixed or variable curvature (specifically, a prolate or oblate spheroid). The effects of surface wettability, interfacial angles, and the density ratio (of droplet to ambient fluid) on the wetting process were also explored. In general, it was found that, under otherwise identical conditions, droplet separation tends to happen more likely on more hydrophilic walls, under larger interfacial angles (measured inside the droplet), and at larger density ratios. On convex walls, a larger radius of curvature of the surface near the droplet was found to be helpful to split the Janus droplet. On concave walls, as the radius of curvature increases from a small value, the possibility to observe droplet separation first increases and then decreases. Several phase diagrams on whether droplet separation occurs during the spreading process were produced for different kinds of walls to illustrate the influences of various factors.

Two-phase flow dynamics study in the trapezoidal gas channel of PEM fuel cell based on lattice Boltzmann model

ABSTRACT Water management is a critical challenge in ensuring the performance and durability of low-temperature proton exchange membrane fuel cells (PEMFCs). This study utilizes the phase-field lattice Boltzmann method to conduct three-dimensional numerical simulations, investigating the dynamics of two-phase flow in the cathode gas channel (GC) of PEMFC with a trapezoidal cross-section. The effects of the water inlet location, the surface wettability of GC, the open angle of GC, and the air velocity on liquid water distribution and discharge are analyzed. The results indicate that positioning the water inlet in the middle of the GC significantly enhances PEMFC performance while having minimal impact on average pressure drop. A wall contact angle of 70° or 110°Can optimize fuel cell performance from different directions. Setting a wall contact angle at 70° optimizes gas flow stability and maintains a low-pressure drop value for the GC. A wall contact angle of 110° focuses on optimizing gas reactant diffusion ability into porous electrodes and GC drainage rate. An opening angle of 55° for the trapezoidal gas channel improves overall fuel cell performance. Increasing air velocity facilitates the film flow formation of liquid water on top wall surfaces within the GC.

Numerical simulation of factors in charge of dendrite growth in zinc-nickel single flow batteries

There is a potential danger of battery internal short-circuiting and shortened life span in zinc-nickel single flow batteries which is usually caused by the formation of dendritic crystals as a result of non-uniform electrodeposition. For the purpose of restraining dendrite growth and prolonging the battery service life, it is necessary to investigate the mechanism of dendrite growth that occurs on the surface of zinc anode. Here, a model for two-dimensional zinc dendrite growth is established by the phase field-lattice Boltzmann method (PF-LBM) to simulate how the morphology of zinc dendrites evolves as well as the corresponding distribution of the concentration field, potential field and flow field, and to analyze the influences of the applied voltage, electrolyte flow rate, anisotropy strength and exchange current density on dendrite growth. As the results show, higher exchange current density and applied voltage can promote dendrite growth. By choosing a suitable anisotropic strength, it is possible to change the morphology of the dendrites and decrease the formation of dendrites. Enlarging the electrolyte flow rate may make the distribution of ion concentration more uniform, which can inhibit the dendrite growth as well as affect the morphology of dendrites.

A regularized lattice Boltzmann model with filter for multiphase flow with diffusion-dominated mass transfer considering two-film theory

Complex flow, considering the interfacial mass transfer with the two-film theory, is always encountered in critical industrial processes. The phase-field lattice Boltzmann method (PFLBM) coupling with the revised Fick's law mass transfer convection–diffusion equation (CDE) is a practical approach to predict the bulk concentration distribution in two-phase flows. However, solutions of concentration have oscillations and even diverge near the sharp gradient when the relaxation time of governing equations is close to 0.5 (i.e., diffusion-dominated). In this paper, an integrated PFLBM model considering two-phase flow and interfacial mass transfer with a new filtering algorithm and collision operator was built to extend the wider range of the existing model for the two-film CDE with an extremely low diffusion coefficient. First, the two-film mass transfer model from our team was furthermore developed with the second-order formation to meet the high precision of concentration on two-phase interfaces. Then, directional filtering algorithms and regularized-finite-difference (rLBM-FD) collision operator were introduced to improve the numerical stability and limit the numerical diffusion. Four common collision operators were implemented and thoroughly tested in two cases to verify the robustness and accuracy of our new model. In conclusion, the combination of the rLBM-FD with standard non-linear filter reaches the highest robustness, mass-conservativeness, and limitation on numerical diffusion. The directional non-linear filter has the lowest computational cost of any microscopic variable filter and can increase the robustness by two times. Macro-variable filtering is not appropriate for treating the two-film equilibrium because the mass loss and robustness are unacceptable.

Investigation on interphase mass transfer coefficient of a deformable bubble with density contrast using phase-field lattice Boltzmann method

The gas–liquid mass transfer process with density contrast is modeled in this paper using a phase-field Lattice Boltzmann (LB) method. In this method, the phase-field model used to calculate the flow field and the continuous species transfer model used to calculate the concentration distribution are coupled by introducing the divergence condition of the velocity in the presence of mass transfer into the Allen-Cahn equation. For a deformable bubble rising in stagnant liquid, a two-dimensional parameter dependence study is conducted and the results indicate that an increase in initial diameter, as well as a decrease in liquid diffusion coefficient or liquid viscosity, can improve the steady-state value for the Sherwood number, whereas the distribution coefficient has a little impact on this steady-state value. Furthermore, a correlation originally proposed by Takemura and Yabe (1998) modified with a shape factor can be used to predict the steady-state Sherwood number in the numerical results.

A three-dimensional fully threaded tree adaptive mesh phase-field lattice Boltzmann method for gas–liquid phase change problems

A fully threaded tree adaptive mesh lattice Boltzmann method based on the phase-field model with the conservative Allen–Cahn equation is presented for the simulation of multiphase flows and heat transfer, especially the gas–liquid phase change problems in three dimensions. The presented model incorporates the conservative Allen–Cahn equation for interface tracking and employs hydrodynamics and temperature evolution D3Q19 lattice Boltzmann equations to recover the corresponding Navier–Stokes equations and energy equations. The gas–liquid phase change at the phase interface can be reflected with introducing the mass production rate in the lattice Boltzmann evolution equations. With the fully threaded tree adaptive mesh implemented to capture the phase interface, the computational efficiency can obviously be raised while ensuring the accurate capture of gas–liquid interface. The present method is used to reproduce several classical benchmarks, namely, the droplet evaporation in superheated gas, the buoyancy-driven bubble rising in viscous liquid, the 3-dimensional Rayleigh Taylor instability problem, the nucleate boiling on a wall with constant temperature, and the film boiling on superheated bottom.

Numerical Simulation on Insoluble Surfactant Mass Transfer on Deformable Bubble Interface in a Couette Flow by Phase-Field Lattice Boltzmann Method-Finite-Difference Method Hybrid Approach.

Elaborate management on bubble shape and transportations depends on the balance between multiple physical behaviors for two-phase flow with Marangoni stress and the interface mass transfer. In this paper, a new model combining PFLBM (phase-field lattice Boltzmann method) and FDM ( finite-difference method) coupling with the ghost-cell method was built. The PFLBM-FDM was validated for the high accuracy, less computational cost, and low mass loss compared to other methods. Based on the PFLBM-FDM, a surfactant-laden bubble deformed and transported in a laminar Couette flow was investigated. The deformation ratio and transportation velocity were explored with different density ratios, surface tensions, shear velocities, and diffusion coefficients. The numerical results showed that the equilibrium state of the bubble deformation was decided only by the dimensionless numbers when the Sh number was higher than 100. Moreover, the transportation velocity of the bubble can be controlled by the balance between the Marangoni stress and shear velocity. When the Sh is lower than 100, the Marangoni stress from the surfactant is not a long-range force, which only works at the early flow. Otherwise, the Marangoni stress will be a long-range force that provides a persistent force to accelerate the bubble by ∼10%. Increasing ReH will further intensify the effect. Based on all the data, a correlation of the bubble deformation including with the densities of two fluids was obtained and the error range is less 5%.

A hybrid axisymmetric conservative phase-field lattice Boltzmann method for hollow droplet migration

A hybrid axisymmetric conservative phase-field lattice Boltzmann method is applied to investigate the influence of Marangoni number (Ma), density ratio (ρr), and radius ratio (Rr) on thermocapillary migration of a deformable hollow droplet with difference in variable fluid properties, where ρr (Rr) is the density (radius) ratio of the hollow part of the droplet. The isotherms show that heat transfer around the hollow droplet is changed from conduction to convection with the increase in Ma. However, the temperature gradient across the hollow droplet decreases with Ma, which induces a small magnitude of migration velocity. When ρr is increased, the isotherms are accumulated around the hollow droplet front with a large temperature gradient, which enhances the hollow droplet migration, while the migration velocity is decreased with the increase in Rr. It is observed that thermocapillary migration of the hollow droplet finally becomes a pure droplet with the influence of aforementioned parameters, and it experiences interface breaking and coalescing, which causes a large transient variation in migration velocity. The magnitude of this transient variation in migration velocity is not obviously affected by Ma but significantly affected by ρr and Rr. The measured evolution of d (the dimensionless distance between inner and outer fronts of the hollow droplet) demonstrates that ρr has a significant influence on the reduction rate of d in comparison with the influence of Ma and Rr. Similar influences on the relative migration velocity between the fluid of the hollow part inside the droplet and the sealed fluid of the droplet are observed.

A phase-field Lattice Boltzmann method for liquid-vapor phase change problems based on conservative Allen-Cahn equation and adaptive treegrid

A phase-field Lattice Boltzmann method (LBM) based on Allen-Cahn (A-C) equation with a modified adaptive mesh refinement (AMR) method is presented to simulate the liquid-vapor phase change problem. Three Lattice Boltzmann equations are solved: one for the conservative Allen-Cahn equation, one for the incompressible Navier-Stokes equations, and one for the energy equation. These three equations are written in the form of multiple-relaxation-time, together with the equilibrium and forcing terms using the set of Hermite polynomials. The adaptive mesh refinement technique with quadtree grid data structure is implemented to capture the liquid-vapor interface in simulations, in which the mesh can be locally refined by subdividing blocks identified by user-defined criterion. The Lax-Wendroff finite-difference scheme is adopted to calculate the whole computational domain continuously under nonuniform grids. The present method is validated by the classical Stefan problem and then performed to simulate the film boiling heat transfer. The numerical experiments verify the ability of the proposed method to capture the correct evolution of the interface, guarantee the numerical accuracy and remarkably save the computational cost.

A new three dimensional cumulant phase field lattice Boltzmann method to study soluble surfactant

Surfactants play a critical role in the physics of paint and coating formulations, affecting key rheological properties such as viscosity, yield stress, and thixotropy. This paper proposes a new three-dimensional phase-field model that uses the cumulant lattice Boltzmann method (LBM) to simulate soluble surfactants. Although current phase-field models commonly use Langmuir's relationship, they cannot calculate interfacial tension analytically, or the LBM models used are unstable when viscosities are low. However, the proposed method overcomes these limitations through two main features. First, the main parameters for modeling and controlling the surfactant's strength and interaction with other phases are directly obtained from a given initial interfacial tension and bulk surfactant, eliminating the need for trial-and-error simulations. Second, a new equilibrium distribution function in the moment space that includes diagonal and off diagonal elements of the pressure tensor is used to minimize Galilean invariance violation. Additionally, there is no need to use an external force to recover multiphase flows, which could break mass conservation. Furthermore, this method has significant potential for parallelization since only one neighbor's cell is used for discretization. The method shows Langmuir relation behavior and is validated with analytical solutions for various interfacial tensions and surfactant concentrations. Moreover, the paper demonstrates the influence of interfacial tension and surfactants on spurious velocities, indicating the method's stability at low viscosities. The dynamics of droplets in the presence of the surfactants is studied in spinodal decomposition and under various external forces. The method accurately simulates the breaking-up and coalescence for these cases. Furthermore, the method successfully simulates the breakage of a liquid thread at a high viscosity ratio.

Interaction between a rising bubble and a stationary droplet immersed in a liquid pool using a ternary conservative phase-field lattice Boltzmann method.

When a stationary bubble and a stationary droplet immersed in a liquid pool are brought into contact, they form a bubble-droplet aggregate. Its equilibrium morphology and stability largely depend on the combination of different components' surface tensions, known as the "spreading factor." In this study, we look at the interaction between a rising bubble and a stationary droplet to better understand the dynamics of coalescence and rising and morphological changes for the bubble-droplet aggregate. A systematic study is conducted on the interaction processes with various bubble sizes and spreading factors in two dimensions. The current simulation framework consists of the ternary conservative phase-field lattice Boltzmann method (LBM) for interface tracking and the velocity-pressure LBM for hydrodynamics, which is validated by benchmark cases such as the liquid lens and parasitic currents around a static droplet with several popular surface tension formulations. We further test our LBM for the morphology changes of two droplets initially in contact with various spreading factors and depict the final morphologies in a phase diagram. The separated, partially engulfed, and completely engulfed morphologies can be replicated by systematically altering the sign of the spreading factors. The rising bubble and stationary droplet interaction are simulated based on the final morphologies obtained under stationary conditions by imposing an imaginary buoyancy force on the rising bubble. The results indicate that the bubble-droplet aggregate with double emulsion morphology can minimize the distortion of the bubble-droplet aggregate and achieve a greater terminal velocity than the aggregate with partially engulfed morphology.