Phase-field Lattice Boltzmann Model Research Articles

Numerical study of magnesium dendrite microstructure under convection: Change of dendrite symmetry

Besides diffusion and capillary, convection which is unavoidable under terrestrial condition has remarkable effects on the microstructure evolution during solidification. In this study, a phase-field lattice-Boltzmann model, accelerated by state-of-the-art parallel-adaptive mesh refinement algorithm, is solved to investigate the morphological evolution of the Mg-Gd dendrite under convection. The lengths of the dendrite primary arms are quantified to analyze the asymmetric dendrite patterns under convection. The effects of the multiple factors including the orientation angle, the flow intensity, and the undercooling are elucidated, and the relation between the length ratios and the three independent factors is established through multiple regression analysis. The upstream-downstream arm length difference and the included angle between the primary arms are characterized to illustrate the effect of convection on the evolution of the Mg-Gd dendrite. The 3D morphological selection, together with algorithm performance tests, is further discussed to elucidate the change of morphological symmetry under different growth conditions and to demonstrate the robustness of the numerical scheme. Deep understanding of the synergy between convection-induced solute transport and undercooling-driven growth, which largely determines the morphological selection, can assist guidance for the prediction and control of the magnesium alloy microstructures.

Dynamics of droplet adsorption by liquid film on a grooved surface

Understanding the dynamics of droplet adsorption by liquid film on a grooved surface is of great significance for the possible manipulation of dropwise condensation on the grooved surface. In this study, an improved phase-field lattice Boltzmann model is proposed to describe the process of droplet adsorption from the ridge to the liquid film within the channel. The results indicate that the leading edge of the droplet undergoes two accelerations during the adsorption process, obeying the power law of and , respectively. The adsorption process between droplets with different sizes and the liquid film exhibits self-similarity characteristics including the same first peak velocity, the similar droplet displacement-time curve and the equal dimensionless spreading length of . Decreasing the contact angle of the droplet from to accelerates the displacement of the leading edge and extends the spreading length. These findings may help reveal the mechanics of droplet adsorption by the liquid film on the grooved surface and thus manipulate the condensation behavior for the heat transfer enhancement.

Examining a Conservative Phase-Field Lattice Boltzmann Model for Two-Phase Flows

In this study, a conservative phase-field lattice Boltzmann model is examined for simulating immiscible two-phase flows with large density and viscosity ratios. This model is built upon the Allen–Cahn equation, incorporating a filtered collision operator and high-order corrections in the equilibrium distribution functions. The numerical model is also integrated with the volumetric boundary scheme and the immersed boundary method to achieve various stationary and moving boundary conditions on arbitrary geometries for different application scenarios. A comprehensive evaluation of this phase-field solver is conducted through a range of academic and industrial cases. First, droplet splashing cases at different Reynolds numbers are studied. The phase-field LB model effectively captures the surface wave motion and splashing of the droplet, although some smearing of small structures is observed due to the diffused interface property of the numerical model. Second, a tuned liquid damper is simulated, accurately capturing sloshing forces on solid walls. In the simulation of a dam-breaking case, predicted gauge pressure and free-surface elevation time histories compare well with experimental data. Lastly, a gearbox case under dip lubrication is simulated, with both the gearbox churning loss and oil distribution being well predicted across a range of rotating speeds. The validation studies demonstrate the effectiveness of the present phase-field lattice Boltzmann model in simulating immiscible two-phase flows with large and realistic density and viscosity ratios. The strengths, limitations, and weaknesses of the conservative phase-field LB model are discussed, and suggestions for the development and application of this numerical model are provided.

Lattice Boltzmann simulation of plunging breakers

ABSTRACT In the present work, the conservative phase-field lattice Boltzmann model is applied to simulate plunging breakers in shallow waters. The solver is computationally efficient as it is explicit in time stepping and does not require a solution for the pressure Poisson equation. Nevertheless, the solver can simultaneously handle a large Reynolds number ( = 10 4 ) and a high-density ratio of 1000. Simulations are performed by varying initial steepness and dispersion parameters, which influence the nonlinearity degree of wave (or the growth speed of instability) and the oscillatory motion (limited by the bottom) on the breaking characteristics. Detailed analyses of kinematics, air entrainment, vorticity, and energy dissipation are presented.

Long-distance settling simulation of equiaxed dendrite by a moving-frame algorithm: phase-field lattice Boltzmann study with parallel-GPU AMR

In large-ingot castings, the settling of equiaxed dendrites often results in distinct cone-shaped negative segregation in the lower region of the ingot. To accurately predict and control such macrosegregation, it is important to understand the kinetic behavior of equiaxed dendrites in the melt. The phase-field lattice Boltzmann (PF-LB) model is powerful for simulating dendrite growth with melt convection and solid motion. However, it is computationally expensive and represents only the short-distance motion of dendrites in three-dimensional (3D) simulations. For an efficient 3D evaluation of the effect of dendrite motion and rotation on growth behavior, we introduce the moving frame algorithm to PF-LB simulations. Here, the computational domain tracks the settling dendrite to express long-distance settling without restricting the domain size. The PF-LB simulations were accelerated by parallel computing using a combination of multiple GPUs and adaptive mesh refinement (AMR), also referred to as parallel GPU-AMR. The moving-frame algorithm was modified to adapt to AMR. From the simulation results, we demonstrate that the proposed method helps evaluate the effect of dendrite rotation on the settling and growth velocities of equiaxed dendrites in 3D.

Phase-field lattice Boltzmann model with singular mobility for quasi-incompressible two-phase flows.

In this paper, a lattice Boltzmann for quasi-incompressible two-phase flows is proposed based on the Cahn-Hilliard phase-field theory, which can be viewed as an improved model of a previous one [Yang and Guo, Phys. Rev. E 93, 043303 (2016)2470-004510.1103/PhysRevE.93.043303]. The model is composed of two LBE's, one for the Cahn-Hilliard equation(CHE) with a singular mobility, and the other for the quasi-incompressible Navier-Stokes equations(qINSE). Particularly, the LBE for the CHE uses an equilibrium distribution function containing a free parameter associated with the gradient of chemical potential, such that the variable (and even zero) mobility can be handled. In addition, the LBE for the qINSE uses an equilibrium distribution function containing another free parameter associated with the local shear rate, such that the large viscosity ratio problems can be handled. Several tests are first carried out to test the capability of the proposed LBE for the CHE in capturing phase interface, and the results demonstrate that the proposed model outperforms the original LBE model in terms of accuracy and stability. Furthermore, by coupling the hydrodynamic equations, the tests of double-stationary droplets and droplets falling problems indicate that the proposed model can reduce numerical dissipation and produce physically acceptable results at large time scales. The results of droplets falling and phase separation of binary fluid problems show that the present model can handle two-phase flows with large viscosity ratio up to the magnitude of 10^{4}.

A second-order phase field-lattice Boltzmann model with equation of state inputting for two-phase flow containing soluble surfactants

In this paper, we propose a phase field-lattice Boltzmann (LB) model with an equation of state (EOS) inputting for two-phase flow containing soluble surfactants. In this model, both the order parameter for the phase field and the surfactant concentration are described by second-order partial differential equations, along with Navier–Stokes equations for the flow field. Changes in surfactant concentration do not affect the order parameter distribution; hence, an unwanted sharpening effect cannot arise. Most importantly, in the existing models, the EOS of surface tension is determined by posterior simulation tests instead of being directly set as an input parameter before the simulations. Hence, it is difficult to determine the model parameters in practical applications. To address this issue, we systematically develop a fully analytical EOS for surface tension based on the Gibbs–Duhem equation. Subsequently, an approximate explicit form for EOS is provided by utilizing the Jacobi–Gauss quadrature rule. Furthermore, a multiple-relaxation-time LB scheme is utilized to numerically solve the governing equations of three physical fields. Two benchmark examples are simulated to validate the accuracy of the present model. The consistency between the numerical results and the analytical EOS is verified. Moreover, the dynamics of droplets with surfactant in simple shear flow is investigated, unveiling the profound impact of various factors, such as surfactant bulk concentration, capillary number, and viscosity ratio, on single droplet deformation and two equal-sized droplets interaction. A detailed exploration of the fluid mechanism involved in two-phase flow with soluble surfactants is presented.

Dynamic behaviors and heat transfer characteristics of impacting droplets on heated superhydrophobic surfaces with randomly distributed rough structures: Numerical simulation and theoretical analysis

Due to the extensive usage of superhydrophobic surfaces, numerous studies have investigated the dynamics and heat transfer of a droplet impacting on superhydrophobic surfaces. However, previous studies did not consider the actual superhydrophobic surfaces with randomly distributed rough structures, especially so in heat transfer of droplet impacting on superhydrophobic surfaces with random rough structures. To address this issue, a multiple distribution function phase-field lattice Boltzmann model was developed to model the droplet impacting on randomly rough surfaces. After experimental validation, this model was used in the analysis of the impacting dynamics and heat transfer characteristics. The differences between actual randomly rough surface and ideal smooth surface were revealed. Herein, random rough structures significantly influence the wetting state and heat transfer of impacting droplet on solid surface. Moreover, the effects of roughness on droplet bouncing ability and the total transferred heat were evaluated. The results show that the random rough surface with smaller skewness, a kurtosis of 3.0, and a standard deviation of 0.3 μm could simultaneously promote impacting droplet to rebound from the surface and to reduce the total transferred heat. Subsequently, the predictive correlation for total transferred heat was proposed based on the roughness parameters. This work provides novel insights into the design of rough structures on functional superhydrophobic surfaces.

Bouncing modes and heat transfer of impacting droplets on textured superhydrophobic surfaces

Superhydrophobic modifications could effectively minimize heat exchange at the interfaces of impacting droplets and solid surfaces. Previous studies have lacked numerical explorations regarding the effect of bouncing modes on the heat transfer characteristics during droplets impacting on textured superhydrophobic surfaces with micro-pillars. To address this issue, a multiple distribution function phase-field lattice Boltzmann model is developed to numerically study dynamic behaviors and heat transfer during droplet impact. Comparisons between the simulations and previous experimental results validate the computation model. Subsequently, the dynamic behaviors of impacting droplets and the effects on the heat transfer were studied using the proposed model. The effects of the textured surface structural parameters on the dynamics and heat transfer are discussed in detail. The numerical results indicate four possible bouncing modes of the impacting droplets: Cassie bouncing, partially penetrated bouncing, pancake bouncing and Wenzel bouncing. These modes depend on the surface energy stored in the penetrating droplet in the microstructures cavities of the surface. Moreover, the synergistic effects of contact time and contact area affect the heat transfer performance. Further, the developed theoretical model to predict the total transferred heat is based on the identified droplet dynamics. Finally, the effects of roughness parameters on total transferred heat are studied, and the design principles of textured superhydrophobic surfaces for heat transfer suppression are given for two application scenarios. The results demonstrate that the control of microstructures would be crucial for the dynamics and heat transfer of impacting droplets on textured superhydrophobic surfaces.

Simplified wetting boundary scheme in phase-field lattice Boltzmann model for wetting phenomena on curved boundaries.

In this work, a simplified wetting boundary scheme in the phase-field lattice Boltzmann model is developed for wetting phenomena on curved boundaries. The proposed scheme combines the advantages of the fluid-solid interaction scheme and geometric scheme-easy to implement (no need to interpolate the values of parameters exactly on solid boundaries and find proper characteristic vectors), the value of contact angle can be directly prescribed, and no unphysical spurious mass layer-and avoids mass leakage. Different from previous works, the values of the order parameter gradient on fluid boundary nodes are directly determined according to the geometric formulation rather than indirectly regulated through the order parameters on ghost solid nodes (i.e., ghost contact-line region). For this purpose, two numerical approaches to evaluate the order parameter gradient on fluid boundary nodes are utilized, one with the prevalent isotropic central scheme and the other with a local gradient scheme that utilizes the distribution functions. The simplified wetting boundary schemes with both numerical approaches are validated and compared through several numerical simulations. The results demonstrate that the proposed model has good ability and satisfactory accuracy to simulate wetting phenomena on curved boundaries under large density ratios.

A three-dimensional non-orthogonal multiple-relaxation-time phase-field lattice Boltzmann model for multiphase flows at large density ratios and high Reynolds numbers

This study proposes a three-dimensional non-orthogonal multiple-relaxation-time (NMRT) phase-field multiphase lattice Boltzmann (PFLB) model within a recently established unified lattice Boltzmann model (ULBM) framework [Luo et al., Phil. Trans. R. Soc. A 379, 20200397, 2021]. The conservative Allen-Cahn equation and the incompressible Navier-Stokes (NS) equations are solved. In addition, a local gradient calculation scheme for the order parameter of the Allen-Cahn equation is constructed with the non-equilibrium part of the distribution function. A series of benchmark cases are conducted to validate the proposed model, including the two-phase Poiseuille flow, Rayleigh-Taylor instability, binary liquid/metal droplet collision, and a bubble rise in water. The present simulation results are in good agreement with existing simulation and experimental data. In the simulation of the co-current two-phase Poiseuille flow, the present model is proven to resolve the discontinuity at the phase interface and provide accurate results at extremely high density ratios (i.e., up to 106). Finally, the proposed model is adopted to simulate two challenging cases: (1) water droplet splashing during its impacting on a thin liquid film and (2) liquid jet breakup. The simulation results demonstrate an excellent agreement with previous experimental results, both qualitatively and quantitatively. In these simulations, the Weber number and Reynolds number reach 105 and 6000, respectively, and the viscosity can be as low as ∼10−4, in the lattice unit.

Phase-field lattice Boltzmann model of growth and buoyancy driven motion of dendritic particles: the effect of Coriolis force

A phase-field lattice Boltzmann (PFLB) model we developed recently (Rátkai L, Pusztai T and Gránásy L 2019 npj Comput. Mater. 5 113) was adapted to investigate the effect of Coriolis force on the solidification morphology forming in a centrifuge. The moving particles are handled using an overlapping multi-grid scheme, assigning a separate moving grid to each individual particle, with attached local fields. Relying on this approach, solidification problems, in which solute diffusion, melt flow, solid motion, the effect of gravity and collision of the particles occur simultaneously can be modelled. Using relevant properties of an Al-Ti alloy, the interaction of columnar and equiaxed solidification was studied. It was found that the Coriolis force influences the path of the falling dendritic particles, which in turn governs the position, at which the transition between columnar and equiaxed morphologies takes place.

Modeling of droplet dynamics with soluble surfactant by multi-relaxation-time phase-field lattice Boltzmann method

The multiphase fluid system in the presence of surfactant is frequently encountered in numerous scientific and engineering applications. Developing a model for accurately simulating such a complex system is of great significance. In this work, we propose a multi-relaxation-time phase-field lattice Boltzmann model for simulating droplet dynamics with soluble surfactants. The accuracy and validity of the model are verified by benchmark cases including static droplet and Rayleigh–Taylor instability tests. The effects of surfactant, capillary number, and density ratio on single-droplet deformation and two-droplet interaction under shear flow are investigated. Simulation results indicate that the Marangoni stress generated by the inhomogeneous distribution of surfactant at the interface plays the role of promoting droplet deformation and hindering droplet coalescence. Within the studied range, it tends to be much easier for droplets to deform with the decrease in density ratio. The increase in the capillary number and surfactant concentration is conducive to promoting the deformation and breakup of droplets. In addition, a higher surfactant concentration is found to result in greater liquid film thickness between droplets, which would hinder the coalescence of the droplets.

Phase-field Lattice Boltzmann model for liquid bridges and coalescence in wet granular media

In this paper a phase-field based Lattice Boltzmann model is formulated to simulate the formation and coalescence of capillary bridges between spherical solid particles as well as the associated capillary forces. To capture the air–water interface, Allen–Cahn equation is coupled with two-phase Navier–Stokes equation through a surface tension term. The capillary force arising from the water interaction with a curved solid surface is formulated along with a numerical integration scheme. Two benchmark examples are considered to validate the proposed model, namely: the spreading of a drop on a spherical particle and capillary rise in a narrow tube. Then, the capillary forces due to isolated and coalesced liquid in two and three spherical particle configurations are computed to illustrate the pendular regime and its transition to the funicular regime. Numerical simulation results are found to be in good agreement with available experimental and numerical data for wetting in a doublet and a triplet of particles, respectively. The numerical results indicate that the proposed model provides a viable framework within which the complex capillary interface evolution during the wetting of a large assembly of particles could be captured. • The proposed model is used to simulate capillary bridges in an air–water system. • Transition of capillary bridges from pendular to funicular regimes is handled. • A numerical procedure is proposed to compute capillary forces. • Validation on small granular assemblies composed of two and three particles.

Data assimilation with phase-field lattice Boltzmann method for dendrite growth with liquid flow and solid motion

Integrating phase-field simulations and in situ observation experiments is a promising approach to better understand dendrite growth during alloy solidification. To integrate simulations and experiments, we developed a data assimilation system using an ensemble Kalman filter for dendrite growth with liquid flow and solid motion. In this system, we used the phase-field lattice Boltzmann (PF–LB) model as the simulation model. Through twin experiments in which an isolated equiaxed dendrite grows with sedimentation owing to the density difference between the solid and liquid phases, we validated that the developed data assimilation system can effectively incorporate the observation data into the PF–LB simulation. In addition, we discuss the accuracy of data assimilation in different conditions by comparing the results of columnar dendrite growth with natural convection.

Wetting boundary schemes in modified phase-field lattice Boltzmann method for binary fluids with large density ratios

In this work, wetting boundary schemes in a modified phase-field lattice Boltzmann (LB) method are developed to simulate the contact-line motion and contact angle hysteresis of binary fluids with large density ratios. In the modified phase-field LB method, two multiple-relaxation-time (MRT) LB equations are utilized, one with a nondiagonal relaxation matrix used to solve the conserved Allen-Cahn (A-C) equation and the other with a modified discrete source term used to solve the incompressible Navier-Stokes (N-S) equations. Based on the idea of geometric formulation, two wetting boundary schemes corresponding to the modified phase-field LB model are proposed. In the first scheme, a layer of virtual nodes is utilized to realize the geometric formulation. The values of the order parameter at the virtual nodes are determined by the central difference method. In the second scheme, the geometric formulation is realized through a local scheme for the gradient terms of the order parameter where the nonequilibrium parts of the distribution function are utilized. Compared with the first wetting boundary scheme and some previous models, the second wetting boundary scheme does not require the virtual nodes and simplifies the numerical implementation. The developed two schemes are validated through a series of benchmark simulations, including droplet spreading on idea substrate, capillary intrusion and droplet shearing on nonideal substrate. It is found that satisfactory accuracy can be achieved by both two schemes, and the second scheme can obtain better stability while the first scheme shows better performance in simulating contact angle hysteresis.

Phase-field lattice Boltzmann model for two-phase flows with large density ratio.

In this work, a lattice Boltzmann (LB) model based on the phase-field method is proposed for simulating large density ratio two-phase flows. An improved multiple-relaxation-time (MRT) LB equation is first developed to solve the conserved Allen-Cahn (AC) equation. By utilizing a nondiagonal relaxation matrix and modifying the equilibrium distribution function and discrete source term, the conserved AC equation can be correctly recovered by the proposed MRT LB equation with no deviation term. Therefore, the calculations of the temporal derivative term in the previous LB models are successfully avoided. Numerical tests demonstrate that satisfactory accuracy can be achieved by the present model to solve the conserved AC equation. What is more, the discrete force term of the MRT LB equation for the incompressible Navier-Stokes equations is also simplified and modified in the present work. An alternative scheme to calculate the gradient terms of the order parameter involved in the discrete force term through the nonequilibrium part of the distribution function is also developed. To validate the ability of the present LB model for simulating large density ratio two-phase flows, series of benchmarks, including two-phase Poiseuille flow, droplet impacting on thin liquid film, and planar Taylor bubble are simulated. It is found that the results predicted by the present LB model agree well with the analytical, numerical, and experimental results.

Acceleration of dendrite growth computation with thermosolutal convection by parallel-GPU mother-leaf AMR method

Dendrite morphology formed during solidification of alloys is largely affected by thermosolutal convection. Although the phase-field lattice Boltzmann (PF-LB) model has made it possible to reproduce dendrite growth with themosolutal convection, the large computational cost is a major problem. In this study, to accelerate the PF-LB simulation with thermosolutal convection, the parallel-GPU mother-leaf AMR method is implemented. We confirmed that the developed scheme can significantly speed up PF-LB simulations.

Magnetic field-induced control of a compound ferrofluid droplet deformation and breakup in shear flow using a hybrid lattice Boltzmann-finite difference method

The deformation and breakup dynamics of a compound ferrofluid droplet under shear flow and uniform magnetic field are numerically studied in this paper. Utilizing magnetic field provides the possibility to obtain better control over the compound droplet morphology and breakup in a simple shear flow. To solve the governing equations for interfaces motion and hydrodynamics, the conservative phase field lattice Boltzmann model is employed, and a finite difference approach is applied for calculating the magnetic field. To verify the accuracy of present simulations, the results are validated with those of four relevant benchmarks including liquid lens between two stratified fluids, three-phase morphology diagram, droplet under simple shear flow fluid flow, and ferrofluid droplet elongation under a uniform magnetic field. The magnetic field lines and magnetic force in the presence of ferrofluid are visualized. The effects of capillary and magnetic Bond numbers on the deformation and inclination angle of compound droplet under magnetic field at the angles of 45∘ and 135∘ are investigated when the inner or outer droplet is ferrofluid. The results show that by increasing capillary number, the shear force becomes greater and deforms the outer interface, whereas its effect on the inner interface is negligible and its small deformation is due to the squeezing force by the outer interface. As the magnetic Bond number increases, so does the magnetic force, and it leads to both of inner and outer interfaces elongation in the direction of the imposed magnetic field. Finally, the regime maps of compound droplet morphology and breakup in terms of capillary and magnetic Bond numbers are presented when the outer droplet is ferrofluid and the magnetic field is applied at the directions of 45∘ and 135∘, respectively. The border between breakup and non-breakup regions is illustrated, and according to the equilibrium shape of the compound droplet, donut-shaped, egg-shaped, eye-shaped, and peach-shaped regimes are observed in the non-breakup region.

Stability assessment of the phase-field lattice Boltzmann model and its application to Taylor bubbles in annular piping geometries

This study parametrically assessed the stability of the phase-field lattice Boltzmann model (PFLBM) before applying it to analyze the effect of annular piping geometry on the flow of Taylor bubbles. The impacts of both eccentricity and pipe diameter ratio were examined, providing an insight into the behavior of these bubbles as well as the applicability and shortcomings in current prediction methodologies. A recently developed PFLBM was implemented into the open-source simulation framework, waLBerla, for this analysis. The stability properties of the code were investigated in detail by assessing various lattice discretizations and relaxation kernels applied to the Rayleigh–Taylor benchmark problem and a Rayleigh–Taylor instability in a tubular geometry, with gravitational Reynolds numbers of up to 30 000 and 10 000, respectively. This paper makes three contributions relating to the stability and usage of the PFLBM as well as the flow of Taylor bubbles in annular pipes. First, the work numerically explored the stability properties of the velocity-based, PFLBM and concluded the impact of various collision models and lattice discretizations on simulation results. Second, it provided a flexible open-source code that the interested researcher can use interactively for practical flow problems as well as the analysis of numerical properties of various lattice Boltzmann algorithms. Finally, it quantified the effect of pipe eccentricity and diameter ratio on the propagation of a Taylor bubble inside a water-filled annular pipe, concluding that a previously defined closure model captured the diameter ratio for the cases examined. To extend this work, future studies aim to analytically investigate the stability properties parametrically observed in this study and apply the findings to simulate the interaction of multiple Taylor bubbles.