Phase-field Lattice Boltzmann Model Research Articles

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.

Numerical Study on Bubble Rising in Complex Channels Saturated with Liquid Using a Phase-Field Lattice-Boltzmann Method

Packed bed reactors have been widely applied in industrial production, such as for catalytic hydrogenation. Numerical simulations are essential for the design and scale-up of packed beds, especially direct numerical simulation (DNS) methods, such as the lattice-Boltzmann method (LBM), which are the focus of future researches. However, the large density difference between gas and liquid in packed beds often leads to numerical instability near phase interface when using LBM. In this paper, a lattice-Boltzmann (LB) model based on diffuse-interface phase-field is employed to simulate bubble rising in complex channels saturated with liquid, while the numerical problems caused by large liquid-to-gas density ratio are solved. Among them, the channel boundaries are constructed with regularly arranged circles and semicircles, and the bubbles pass through the channels accompanied by deformation, breakup, and coalescence behaviors. The phase-field LB model is found to exhibit good numerical stability and accuracy in handing the problem of the bubbles rising through the high-density liquid. The effects of channel structures, gas-liquid physical properties, and operating conditions on bubble deformation, motion velocity, and drag coefficient are simulated in detail. Moreover, different flow patterns are distinguished according to bubble behavior and are found to be associated with channel structure parameters, gravity Reynolds number (ReGr), and Eötvös number (Eo).

Phase-field lattice Boltzmann model for interface tracking of a binary fluid system based on the Allen-Cahn equation.

A lattice Boltzmann (LB) model is proposed to track the interface of binary fluid system based on the conservative-form Allen-Cahn (A-C) equation for phase field. Utilizing an equilibrium distribution function and a modified LB equation, this model is able to correctly recover the conservative A-C equation through the Chapman-Enskog analysis. A series of two-dimensional (2D) and three-dimensional (3D) phase-capturing benchmark tests have been conducted for validation, which include the diagonal translation of a circular interface, the rigid-body rotation of a Zalesak disk, and the deformation of 2D circular interface and 3D spherical interface in shear flows, all illustrating better accuracy and stability of the proposed model than the previous models tested. By coupling the incompressible hydrodynamic equation, a stationary droplet, a spinodal decomposition, and the Rayleigh-Taylor instability are simulated as well, showing the satisfying performance of the model in dealing with complex interfaces of binary fluid systems.

Three-dimensional phase field modeling of columnar to equiaxed transition in directional solidification of Inconel 718 alloy

Understanding the columnar to equiaxed transition (CET) is vital in production of materials with superior properties using casting, welding, and additive manufacturing processes. In this study, a three dimensional (3D) phase field-lattice Boltzmann (PF-LB) model was developed to simulate the CET in directional solidification of Inconel 718 alloy. The phase field (PF) method was used to determine the solid/liquid transition. The solute diffusion equation was solved by the lattice Boltzmann method (LBM), due to its suitability for parallel processing and its simplicity in terms of modeling complex boundaries. A CET solidification map was developed for different temperature gradients and growth rates and the evolution of dendrites for equiaxed, columnar, and mixed regimes was studied. The resulting microstructure for different growth regimes was demonstrated and discussed in terms of grain size and orientation. In addition, the variation of maximum solute concentration along the sample height was investigated. It was found that while initial grain size does not affect the average grain size for equiaxed growth, it affects the PDAS for columnar growth significantly. A model was proposed to predict the primary dendrite arm spacing for columnar growth in a wide range of temperature gradient, solidification rate, and initial grain size. The novelty of this model is in the inclusion of the effect of initial grain size, which is important in processes that involve melting and solidification on a preexisting substrate, such as welding and additive manufacturing.

Generalized conservative phase field model and its lattice Boltzmann scheme for multicomponent multiphase flows

In this paper a generalized conservative phase field-lattice Boltzmann model is proposed for immiscible incompressible flows with any number of components. In this phase field model, the each phase field variable satisfies a generalized eikonal equation and the volume fraction constraint. By introducing a quadratic functional, the dynamics of phase field variables can be treated as a gradient flow method. The phase field equations are reformulated based on the mass conservative law and the reduction-consistency conditions. Moreover, based on the generalized free energy functional, a family of generalized continuous surface tension force formulations is deduced by using the virtual work method. A lattice Boltzmann model (LBM) is developed to solve the generalized conservative phase field equations and the hydrodynamics equations. In the numerical simulation section, the motion of circular interfaces and Poiseuille flow with four phases are simulated first. The convergence rate and reduction-consistent of present model for the phase field equations are validated. The computational accuracy of present model for the hydrodynamics equations is also validated. Then serval two-dimensional (2D) examples with pairwise surface tension effect, including the droplets immersed in another fluid with four fluid phases, spreading of liquid lenses and spinodal decomposition with four fluid phases, are simulated. The obtained numerical results are in good agreement with the analytical solutions. Four continuous surface tension force formulations are also comparative studied. Finally, as a practical application, a three-dimensional (3D) bubble rising in a three-layer liquid is investigated.