Multirelaxation-time interaction-potential-based lattice Boltzmann model for two-phase flow

Improved lattice Boltzmann model for immiscible multicomponent systems with high viscosity gradients at the interface

We propose alternative discretization schemes for improving the lattice Boltzmann pseudopotential model for incompressible multicomponent systems, with the purpose of modeling the flow of immiscible fluids with a large viscosity ratio. Compared to the original model of Shan-Chen [Phys. Rev. E 47, 1815 (1993)1063-651X10.1103/PhysRevE.47.1815], the present discretization schemes consider: (i) an explicit force term, (ii) a second-order discretization of the stream term, (iii) a moments-based model for the kinetic nonequilibrium distributions, and (iv) a high-order discretization of the spatial derivative terms. To verify the accuracy of the proposed model, the effects of varying the viscosity ratio as well as both fluid's viscosities on spurious currents and capillary number are investigated for the problems dealing with a static bubble, two-component Poiseuille flow, and immiscible fluid-fluid displacement. The resulting algorithm maintains the simplicity of the pseudopotential model while allowing an easy implementation for multicomponent systems. The results of the model herein proposed show improved control of the interface region and interfacial tension, relatively smaller magnitudes of spurious current values with increasing viscosity ratio, and also a significantly wider stability range with respect to the previously best results in the literature.

Theoretical and numerical study on the well-balanced regularized lattice Boltzmann model for two-phase flow

In the multiphase flow simulations based on the lattice Boltzmann equation (LBE), the spurious velocity near the interface and the inconsistent density properties are frequently observed. In this paper, a well-balanced regularized lattice Boltzmann (WB-RLB) model with Hermite expansion up to third order is developed for two-phase flows. To this end, the equilibrium distribution function and the modified force term proposed by Guo [Phys. Fluids 33, 031709 (2021)1070-663110.1063/5.0041446] are directly introduced into the regularization of the transformed distribution functions when considering the LBE with trapezoidal integral. First, to give a detailed comparison of the well-balanced lattice Boltzmann equation (WB-LBE), WB-RLB, and second-order mixed difference scheme (SOMDS) proposed by Lee and Fischer [Phys. Rev. E 74, 046709 (2006)1539-375510.1103/PhysRevE.74.046709], the theoretical analyses on the force balance of LBE with two different gradient operators, isotropic central scheme (ICS) and SOMDS, as well as the numerical simulations of the stationary droplet are carried out. The force analysis shows that SOMDS can achieve a higher accuracy than ICS for the force balance, which has been validated in the simulations of stationary droplet cases. For the stationary droplet cases, all three models (WB-LBE, WB-RLB, and SOMDS) can capture the physical equilibrium state even at a large density ratio of 1000. Also, the numerical investigations of the WB-RLB model with third-order expansion (WB-RLB3) demonstrate that adjusting the relaxation parameters of the third-order moment can further improve the accuracy and stability of the WB-RLB model. Then, both the droplet coalescence and the phase separation cases are investigated with considering the effect of different interface thickness, which demonstrates that the performance of the WB-RLB for the two-phase dynamic problems is still quite well, and it exhibits better numerical stability when compared with the WB-LBE. In addition, the contact angle problem is investigated by the present WB-RLB model; the numerical results show that the predicted values of the contact angles agree well with the analytical solutions, but the well-balance property is not validated, especially near the three-phase junction. Overall, the present WB-RLB model exhibits excellent numerical accuracy and stability for both static and dynamic interface problems.

The Surfactant Role on a Droplet Passing through a Constricted Microchannel in a Pressure-Driven Flow: A Lattice Boltzmann Study

The role of surfactants in the flow of a droplet driven by a pressure gradient through a constricted microchannel is simulated by using our recently developed lattice Boltzmann method. We first study the surfactant role on a droplet flowing through a microchannel with a shrunken square section under different surfactant concentrations and capillary numbers (i.e., imposed pressure gradients). As the surfactant concentration increases, the droplet flow regime first changes from the flow regime I of the droplet getting stuck at the entrance of the constricted channel to the flow regime II of the droplet flowing through the constricted channel with breakup, and then to the flow regime III of the droplet flowing through the constricted channel without breakup. As the capillary number increases, the surfactant role on the number of mother droplets breaking up and the time of mother droplets completely flowing through the constricted section tend to decrease, suggesting that the surfactant effects are gradually weakened. Then, a phase diagram describing how the surfactant concentration and capillary number affect the droplet flow regime is presented. As the surfactant concentration increases, the critical capillary number that distinguishes droplet flow regimes I from II gradually decreases, while the critical capillary number that distinguishes droplet flow regimes II from III first increases and then decreases.

Wall wettability effect on process of collapse of single cavitation bubbles in near-wall region using pseudo-potential lattice Boltzmann method

This study investigates the effect of wall wettability on cavitation collapse based on a large-density-ratio lattice Boltzmann method (LBM) pseudo-potential model. The validity and superiority of the proposed model in simulation of cavitation under complex conditions are confirmed by comparing with theories, experiments, and numerical results by other models. Our simulations indicate that wall wettability has a significant influence on near-wall cavitation of an order no less than the effect of the initial bubble distance. A criterial initial distance exists in near-wall cavitation within which the micro-jet will direct toward the wall. This criterial distance is shown to be positively correlated with the contact angle by a cosine function. Within this distance, the lifetime of the bubble decreases by up to 50%, and the increase of the maximum micro-jet velocity and collapse pressure are up to 131% and 65%, respectively, when the contact angle increases from the hydrophilic 53° to the hydrophobic 113°. Without considering the shock-wave mechanism, the impact pressure transmitted to the hydrophilic wall is of the same order as the maximum collapse pressure while the impact velocity is an order smaller than the maximum micro-jet velocity. Wall wettability affects collapse through the Bjerknes force and the pressure around the bubble. Preliminary analysis also suggests that the relation between the pressure difference and the intensity of collapse exhibits more patterns than we have assumed, which fits a logistic curve well, and appears not changing with the contact angle or the initial bubble distance.

Modified radius-weighted lattice Boltzmann model to address singularities in axisymmetric multiphase flows

The radius-weighted lattice Boltzmann model has achieved great success in the simulation of axisymmetric flows. However, severe spurious currents near the axis are observed when this model is extended to simulate axisymmetric multiphase flows. In this study, to determine the origin of this singularity, we conducted a truncation error analysis based on high-order Taylor series expansion and identified the leading error terms through dimensionless analysis. By neglecting the error terms in proportion to the radius, we obtained the final forms of the singular terms in the axisymmetric lattice Boltzmann model. We proposed a modified model by including an additional correction term, to remove the singularity at the third order. We validated the proposed model using numerical tests for flat and spherical interfaces. Results showed that the present modified model reduced the spurious currents near the axis by two orders of magnitude compared with the original model. This modified model also has been successfully applied to predict bubble dynamics in an air-water system. Our numerical results are in excellent agreement with available experimental observations in terms of bubble shapes and terminal velocities.

Rayleigh-Plateau Instability of a Particle-Laden Liquid Column: A Lattice Boltzmann Study

Colloidal particles known to be capable of stabilizing fluid-fluid interfaces have been widely applied in emulsion preparation, but their precise role and underlying influencing mechanism remain poorly understood. In this study, a perturbed liquid column with particles evenly distributed on its surface is investigated using a three-dimensional lattice Boltzmann method, which is built upon the color-gradient two-phase flow model but with a new capillary force model and a momentum exchange method for particle dynamics. The developed method is first validated by simulating the wetting behavior of a particle on a fluid interface and the classic Rayleigh-Plateau instability and is then used to explore the effects of particle concentration and contact angle on the capillary instability of the particle-laden liquid column. It is found that increasing the particle concentration can enhance the stability of the liquid column and thus delay the breakup, and the liquid column is most stable under slightly hydrophobic conditions, which corresponds to the lowest initial liquid-gas interfacial free energy. Due to different pressure gradients inside and outside the liquid column and the capillary force being directed away from the neck, hydrophobic particles tend to assemble in a less compact manner near the neck of the deformed liquid column, while hydrophilic particles prefer to gather far away from the neck. For hydrophobic particles, in addition to the influence of the initial liquid-gas interfacial free energy, the self-assembly of particles in a direction opposite to the liquid flow also contributes to opposing the rupture of the liquid column.

Coupled lattice Boltzmann-large eddy simulation model for three-dimensional multiphase flows at large density ratio and high Reynolds number

A coupled lattice Boltzmann-large eddy simulation model is developed for modeling three-dimensional multiphase flows at large density ratios and high Reynolds numbers. In the framework of the lattice Boltzmann method, the model is proposed based on the standard Smagorinsky subgrid-scale approach, and a reconstructed multiple-relaxation-time collision operator is adopted. The conservative Allen-Cahn equation and Navier-Stokes equations are solved through the lattice Boltzmann discretization scheme for the interface tracking and velocity field evolution, respectively. Relevant benchmark cases are carried out to validate the performance of this model in simulating multiphase flows at a large density ratio and a high Reynolds number, including a stationary droplet, the process of spinodal decomposition, the Rayleigh-Taylor instability, the phenomenon of a droplet splashing on a thin liquid film, and the liquid jet breakup process. The maximum values of density ratio and Re number are 1000 and 10 240, respectively. The capability and reliability of the proposed model have been demonstrated by the good agreement between simulation results and the analytical solutions or the previously available results.

Lattice Boltzmann simulation of three-phase flows with moving contact lines on curved surfaces

A numerical method for simulating three-phase flows with moving contact lines on arbitrarily complex surfaces is developed in the framework of the lattice Boltzmann method. In this method, the immiscible three-phase flow is modeled through a multiple-relaxation-time color-gradient model, which not only allows for a full range of interfacial tensions but also produces stable outcomes for a wide range of viscosity ratios. A characteristic line model is introduced to implement the wetting boundary condition, which is not only easy to implement but is also able to handle arbitrarily complex boundaries with prescribed contact angles. The developed method is first validated by the simulation of a Janus droplet resting on a flat surface, a perfect Janus droplet deposited on a cylinder, and the capillary intrusion of ternary fluids for various viscosity ratios. It is then used to study a compound droplet subject to a uniform incoming flow passing through a multipillar structure, where three different values of surface wettability are considered. The simulated results show that the surface wettability has significant impact on the droplet dynamic behavior and final fluid distribution.

Pore-Scale Modeling of Anode Catalyst Layer Tolerance upon Hydrogen Sulfide Exposure in PEMFC

A pore-scale contamination model is developed to resolve the physicochemical processes in the anode catalyst layer for a deeper insight into the hydrogen sulfide (H2S) contamination mechanism. The present model is based on lattice Boltzmann method (LBM) and a novel iteration algorithm is coupled to overcome the time-scale issue in LBM which can extend its application. The microstructure of CL is stochastically reconstructed considering the presence of carbon, Pt, ionomer, and pores. The proposed model is validated by comparing the experimental data and can accurately predict the effect of H2S contamination on performance with time. The results show that the fuel cell performance is not sensitive to the anode Pt loading under the clean fuel condition as the hydrogen oxidation reaction is easy to activate. However, higher Pt loading can effectively prolong the operation time under the H2S contamination by providing a larger buffer reactive area and a lower H2S concentration condition. Furthermore, the H2S contamination in the fuel gas should be strictly restricted as it directly affects the poisoning rate and significantly affects the operation time.

Graphical abstract

Physicochemical processes in the ACL with reactant transport through micro porous layer (MPL) to active Pt sites

Shape Optimization of a Microhole Surface for Control of Droplet Wettability via the Lattice Boltzmann Method and Response Surface Methodology

The chief aim is to explore the wetting state on a microhole surface and to optimize the shape parameters of a microhole surface. A two-dimensional pseudopotential model was established, and the effects of shapes on the wetting behavior were explored. The shape parameters were optimized via the response surface methodology. The results reveal that the microhole surface can achieve a superhydrophobic state. When the diameter varies from 25 to 200 μm, the droplet is gradually lifted. However, when the diameter of the microhole is too large, the contact angle decreases rapidly. When the microhole diameter increases, relative radii of the x- and y-directions exhibit increasing trends. With the increase of the spacing, the gaps between the microholes are gradually filled with the droplet. When spacing increases, relative radii of x- and y-directions exhibit decreasing trends. The largest contact angle of 171.246° at the diameter of 76 μm and the spacing of 48 μm is observed.

Dissolution process of a single bubble under pressure with a large-density-ratio multicomponent multiphase lattice Boltzmann model

A large-density-ratio and tunable-viscosity-ratio multicomponent multiphase pseudopotential lattice Boltzmann model is used to study the dissolution process of a bubble under pressure. The multi-relaxation-time collision operator, exact-difference-method external force scheme, and scaling coefficient k are applied to ensure the numerical stability of the model. The influence of k in the equation of state (EOS) and intermolecule interaction strength on the stationary bubble evolution process are discussed, and the effect of k on thermodynamic consistency is also analyzed. The results indicate that adjusting the scaling coefficient in the EOS changes the surface tension and interface thickness, and that the gas-liquid interface width w is proportional to 1/sqrt[k]. Considering the effect of k on the surface tension, interface thickness, and thermodynamic consistency, the scaling coefficient should be between 0.6 and 1. Furthermore, the dissolution process of a single bubble under pressure is studied using the developed model, and it is found that the dissolution mass and concentration of dissolved gas increase linearly with increases in the pressure difference, and that the concentration of dissolved gas is proportional to the gas pressure after the fluid system reaches equilibrium. These results are consistent with Henry's law.

Lattice-Boltzmann simulation of free nematic-isotropic interfaces

We use a hybrid method of lattice Boltzmann and finite differences to simulate flat and curved interfaces between the nematic and isotropic phases of a liquid crystal described by the Landau-de Gennes theory. For the flat in¬terface, we measure the interfacial velocity at different temperatures around the coexistence. We show that the interface is completely static at the coexistence temperature and that the profile width is in line with the theoretical predictions. The interface is stable in a range of temperatures around coexistence and dis¬appears when one of the two phases becomes mechanically unstable. We stabi¬lize circular nematic domains by a shift in temperature, related to the Laplace pressure, and estimate the spurious velocities of these lattice Boltzmann simu¬lations.

Central-Moments-Based Lattice Boltzmann for Associating Fluids: A New Integrated Approach

Dynamic and thermodynamic behaviors of associating fluids play a crucial role in various science and engineering disciplines. Cubic plus association equation of state (CPA EOS) is implemented in a central-moments-based lattice Boltzmann method (LBM) in order to mimic the thermodynamic behavior of associating fluids. The pseudopotential approach is selected to model the multiphase thermodynamic characteristics such as reduced density of associating fluids. The priority of central-moments-based approach over multiple-relaxation-time collision operator is highlighted by performing double shear layers. The integration of central-moments-based LBM and CPA EOS is useful to simulate the dynamic and thermodynamic characteristics of associating fluids at high flow rate conditions, which is extended to high-density ratio scenarios by increasing the anisotropy order of gradient operator. In order to increase the stability of the model, a higher anisotropy order of the gradient operator is implemented; about 34 present reduction in spurious velocities is noticed in some cases. The type of gradient operator considerably affects the model thermodynamic consistency. Finally, the model is validated by observing a straight line in the Laplace law test. Prediction of thermodynamic behaviors of associating fluids is of significance in various applications including biological processes as well as fluid flow in porous media.

Thermodynamic consistency of a pseudopotential lattice Boltzmann fluid with interface curvature

Thermodynamic consistency of pseudopotential lattice Boltzmann models is a major topic that needs comprehensive evaluations. When interface is flat, pseudopotential models can give density-pressure isotherms in excellent agreement with those from equation of state. When interface is curved, thermodynamic equilibriums are affected by interface curvature, and consistency of pseudopotential models has not been systematically evaluated. In this study, we show that the effect of Laplace pressure on phase equilibrium is quantitatively consistent with Kelvin equation at high reduced temperatures (≥0.7). At low temperatures, inconsistency that can be attributed to the effect of orientation of the interface was noted, and it can be improved by tuning of the pseudopotential. By relating interfacial tension of a simulated fluid to that of a real fluid, the lattice spacing of pseudopotential model is found to be on the order of several molecular diameters, the typical range of intermolecular interactions. Interfacial thickness at different temperatures in pseudopotential model compared well with experiments and molecular dynamics simulations, which confirms that the calculated length scale is reasonable. Evaluation of a free energy lattice Boltzmann model indicate that it is consistent with Kelvin equation at high temperatures. The free energy model, however, is not as accurate as the tested pseudopotential model, and discrepancies may come from the relative inaccuracies in the predictions of vapor densities and the thinner interfaces.

Hybridized method of pseudopotential lattice Boltzmann and cubic-plus-association equation of state assesses thermodynamic characteristics of associating fluids

It is crucial to properly describe the associating fluids in terms of phase equilibrium behaviors, which are needed for design, operation, and optimization of various chemical and energy processes. Pseudopotential lattice Boltzmann method (LBM) appears to be a reliable and efficient approach to study thermodynamic behaviors and phase transition of complex fluid systems. However, when cubic equations of state (EOSs) are incorporated into single-component multiphase LBM, simulation results are not well matched with experimental data. This study presents the utilization of cubic-plus-association (CPA) EOS in the LBM structure to obtain more accurate modeling results for associating fluids. An approach based on the global search optimization algorithm is introduced to find the optimal association parameters of CPA EOS for water and primary alcohols in the lattice units. The thermodynamic consistency is verified by the Maxwell construction and is also improved by the forcing scheme of [Q. Li, K. H. Luo, and X. J. Li, Phys. Rev. E 86, 016709 (2012)10.1103/PhysRevE.86.016709]. The spurious velocity is reduced with increasing isotropy in the gradient operator. Furthermore, an extended version of CPA EOS is introduced, which increases the system stability at low reduced temperatures. There is a very good match between the LBM results and experimental data, confirming the reliability of the model developed in the present study. The introduced approach has potential to be employed for simulating transport phenomena and interfacial characteristics of associating fluids in porous systems.

Active nematic-isotropic interfaces in channels

We use numerical simulations to investigate the hydrodynamic behavior of the interface between nematic (N) and isotropic (I) phases of a confined active liquid crystal. At low activities, a stable interface with constant shape and velocity is observed separating the two phases. For nematics in homeotropic channels, the velocity of the interface at the NI transition increases from zero (i) linearly with the activity for contractile systems and (ii) quadratically for extensile ones. Interestingly, the nematic phase expands for contractile systems while it contracts for extensile ones, as a result of the active forces at the interface. Since both activity and temperature affect the stability of the nematic, for active nematics in the stable regime the temperature can be tuned to observe static interfaces, providing an operational definition for the coexistence of active nematic and isotropic phases. At higher activities, beyond the stable regime, an interfacial instability is observed for extensile nematics. In this regime defects are nucleated at the interface and move away from it. The dynamics of these defects is regular and persists asymptotically for a finite range of activities. We used an improved hybrid model of finite differences and the lattice Boltzmann method with a multi-relaxation-time collision operator, the accuracy of which allowed us to characterize the dynamics of the distinct interfacial regimes.

Hybrid Wettability-Induced Heat Transfer Enhancement for Condensation with NonCondensable Gas

Heat transfer enhancement in dropwise condensation is widely investigated on a superhydrophobic surface with the advances in surface engineering, but the influence of a large amount of noncondensable gas (NCG) has not been clarified. In this work, the condensation heat transfer with a large amount of NCG is investigated by developing a multiphase lattice Boltzmann model for a multicomponent system. First, the condensation of a single droplet on a hydrophobic surface with NCG is simulated, demonstrating the capacity of the present model to capture the behaviors of different components during phase change and predict the significant influence of even a small fraction of the NCG on heat transfer. Then, solid surfaces with mixed wettability are built by introducing a fraction of hydrophilic parts to enhance heat transfer. It is found that there exists an optimized proportion which could maximize the condensation heat transfer efficiency corresponding to a specific mass fraction of NCG. Furthermore, the mechanism of this optimized proportion is revealed by examining the dynamic behaviors of condensation in a typical case, as a balance between a promotion of the nucleation rate and a put off of transition to filmwise condensation.

Quantification of Uncertainty and Best Practice in Computing Interfacial Curvature from Complex Pore Space Images

Recent advances in high-resolution three-dimensional X-ray CT imaging have made it possible to visualize fluid configurations during multiphase displacement at the pore-scale. However, there is an inherited difficulty in image-based curvature measurements: the use of voxelized image data may introduce significant error, which has not-to date-been quantified. To find the best method to compute curvature from micro-CT images and quantify the likely error, we performed drainage and imbibition direct numerical simulations for an oil/water system on a bead pack and a Bentheimer sandstone. From the simulations, local fluid configurations and fluid pressures were obtained. We then investigated methods to compute curvature on the oil/water interface. The interface was defined in two ways; in one case the simulated interface with a sub-resolution smoothness was used, while the other was a smoothed interface extracted from synthetic segmented data based on the simulated phase distribution. The curvature computed on these surfaces was compared with that obtained from the simulated capillary pressure, which does not depend on the explicit consideration of the shape of the interface. As distinguished from previous studies which compared an average or peak curvature with the value derived from the measured macroscopic capillary pressure, our approach can also be used to study the pore-by-pore variation. This paper suggests the best method to compute curvature on images with a quantification of likely errors: local capillary pressures for each pore can be estimated to within 30% if the average radius of curvature is more than 6 times the image resolution, while the average capillary pressure can also be estimated to within 11% if the average radius of curvature is more than 10 times the image resolution.

Numerical Study of Droplet Dynamics on a Solid Surface with Insoluble Surfactants

Surfactants are widely used in many industrial processes, where the presence of surfactants not only reduces the interfacial tension between fluids but also alters the wetting properties of solid surfaces. To understand how the surfactants influence the droplet motion on a solid surface, a hybrid method for interfacial flows with insoluble surfactants and contact-line dynamics is developed. This method solves immiscible two-phase flows through a lattice Boltzmann color-gradient model and simultaneously solves the convection-diffusion equation for surfactant concentration through a finite difference method. In addition, a dynamic contact angle formulation that describes the dependence of the local contact angle on the surfactant concentration is derived, and the resulting contact angle is enforced by a geometrical wetting condition. Our method is first used to simulate static contact angles for a droplet resting on a solid surface, and the results show that the presence of surfactants can significantly modify surface wettability, especially when the surface is more hydrophilic or more hydrophobic. This is then applied to simulate a surfactant-laden droplet moving on a substrate subject to a linear shear flow for varying effective capillary number ( Ca_e), Reynolds number ( Re), and surface wettability, where the results are often compared with those of a clean droplet. For varying Ca_e, the simulations are conducted by considering a neutral surface. At low values of Ca_e, the droplet eventually reaches a steady deformation and moves at a constant velocity. In either a clean or surfactant-laden case, the moving velocity of the droplet linearly increases with the moving wall velocity, but the slope is always higher (i.e., the droplet moves faster) in the surfactant-laden case where the droplet exhibits a bigger deformation. When Ca_e is increased beyond a critical value ( Ca_e,c), the droplet breakup would happen. The presence of surfactants is found to decrease the value of Ca_e,c, but it shows a non-monotonic effect on the droplet breakup. An increase in Re is able to increase not only droplet deformation but also surfactant dilution. The role of surfactants in the droplet behavior is found to greatly depend upon the surface wettability. For a hydrophilic surface, the presence of surfactants can decrease the wetting length and enables the droplet to reach a steady state faster; while for a hydrophobic surface, it increases the wetting length and delays the departure of the droplet from the solid surface.

Phase-field method based on discrete unified gas-kinetic scheme for large-density-ratio two-phase flows

In this paper, a phase-field method under the framework of discrete unified gas-kinetic scheme (DUGKS) for incompressible multiphase fluid flows is proposed. Two kinetic models are constructed to solve the conservative Allen-Cahn equation that accounts for the interface behavior and the incompressible hydrodynamic equations that govern the flow field, respectively. With a truncated equilibrium distribution function as well as a temporal derivative added to the source term, the macroscopic governing equations can be exactly recovered from the kinetic models through the Chapman-Enskog analysis. Calculation of source terms involving high-order derivatives existed in the quasi-incompressible model is simplified. A series of benchmark cases including four interface-capturing tests and four binary flow tests are carried out. Results compared to that of the lattice Boltzmann method (LBM) have been obtained. A convergence rate of second order can be guaranteed in the test of interface diagonal translation. The capability of the present method to track the interface that undergoes a severe deformation has been verified. Stationary bubble and spinodal decomposition problems, both with a density ratio as high as 1000, are conducted and reliable solutions have been provided. The layered Poiseuille flow with a large viscosity ratio is simulated and numerical results agree well with the analytical solutions. Variation of positions of the bubble front and spike tip during the evolution of Rayleigh-Taylor instability has been predicted precisely. However, the detailed depiction of complicated interface patterns appearing during the evolution process is failed, which is mainly caused by the relatively large numerical dissipation of DUGKS compared to that of LBM. A high-order DUGKS is needed to overcome this problem.

Lattice Boltzmann simulation of immiscible three-phase flows with contact-line dynamics

A multiphase lattice Boltzmann method is developed to simulate immiscible three-phase flows with contact-line dynamics. In this method, the immiscible three-phase flow is modeled by a multiple-relaxation-time color-gradient model, which not only allows for a full range of interfacial tensions but also can produce viscosity-independent results especially when the fluid-surface interactions are considered. To achieve the desired contact angles, a weighted contact angle model is utilized to obtain a relatively smooth transition of contact angle for each fluid, which is enforced through a geometrical wetting condition. This method is first validated by simulations of a Janus droplet resting on a surface for various contact angles and fluid properties and dynamic capillary filling of ternary fluids with different viscosity ratios. It is then used to simulate a Janus droplet on a substrate subject to Poiseuille flow. Results show that the droplet may undergo three typical modes, namely, two stable deformation modes and breakup mode, which depend not only on the inlet velocity but also on the fluid viscosity. The terminal velocity of moving droplet increases linearly with the inlet velocity in both stable modes only when three fluids do not differ much in their viscosities.

Single-component multiphase lattice Boltzmann simulation of free bubble and crevice heterogeneous cavitation nucleation

This work serves as an important extension of previous work on cavitation simulation [Sukop and Or, Phys. Rev. E 71, 046703 (2005)10.1103/PhysRevE.71.046703]. A modified Shan-Chen single-component multiphase lattice Boltzmann method is used to simulate two different heterogeneous cavitation nucleation mechanisms, the free gas bubble model and the crevice nucleation model. Improvements include the use of a real-gas equation of state, a redefined effective mass function, and the exact difference method forcing scheme. As a result, much larger density ratios, better thermodynamic consistency, and improved numerical accuracy are achieved. In addition, the crevice nucleation model is numerically investigated using the lattice Boltzmann method. The simulations show excellent qualitative and quantitative agreement with the heterogeneous nucleation theories.

Color-gradient lattice Boltzmann model with nonorthogonal central moments: Hydrodynamic melt-jet breakup simulations

We develop a lattice Boltzmann (LB) model for immiscible two-phase flow simulations with central moments (CMs). This successfully combines a three-dimensional nonorthogonal CM-based LB scheme [De Rosis, Phys. Rev. E 95, 013310 (2017)2470-004510.1103/PhysRevE.95.013310] with our previous color-gradient LB model [Saito, Abe, and Koyama, Phys. Rev. E 96, 013317 (2017)2470-004510.1103/PhysRevE.96.013317]. Hydrodynamic melt-jet breakup simulations show that the proposed model is significantly more stable, even for flow with extremely high Reynolds numbers, up to O(10^{6}). This enables us to investigate the phenomena expected under actual reactor conditions.

Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows

In this paper, we present a simple and accurate lattice Boltzmann (LB) model for immiscible two-phase flows, which is able to deal with large density contrasts. This model utilizes two LB equations, one of which is used to solve the conservative Allen-Cahn equation, and the other is adopted to solve the incompressible Navier-Stokes equations. A forcing distribution function is elaborately designed in the LB equation for the Navier-Stokes equations, which make it much simpler than the existing LB models. In addition, the proposed model can achieve superior numerical accuracy compared with previous Allen-Cahn type of LB models. Several benchmark two-phase problems, including static droplet, layered Poiseuille flow, and spinodal decomposition are simulated to validate the present LB model. It is found that the present model can achieve relatively small spurious velocity in the LB community, and the obtained numerical results also show good agreement with the analytical solutions or some available results. Lastly, we use the present model to investigate the droplet impact on a thin liquid film with a large density ratio of 1000 and the Reynolds number ranging from 20 to 500. The fascinating phenomena of droplet splashing is successfully reproduced by the present model and the numerically predicted spreading radius exhibits to obey the power law reported in the literature.

Transition point prediction in a multicomponent lattice Boltzmann model: Forcing scheme dependencies

Pseudopotential-based lattice Boltzmann models are widely used for numerical simulations of multiphase flows. In the special case of multicomponent systems, the overall dynamics are characterized by the conservation equations for mass and momentum as well as an additional advection diffusion equation for each component. In the present study, we investigate how the latter is affected by the forcing scheme, i.e., by the way the underlying interparticle forces are incorporated into the lattice Boltzmann equation. By comparing two model formulations for pure multicomponent systems, namely the standard model [X. Shan and G. D. Doolen, J. Stat. Phys. 81, 379 (1995)JSTPBS0022-471510.1007/BF02179985] and the explicit forcing model [M. L. Porter et al., Phys. Rev. E 86, 036701 (2012)PLEEE81539-375510.1103/PhysRevE.86.036701], we reveal that the diffusion characteristics drastically change. We derive a generalized, potential function-dependent expression for the transition point from the miscible to the immiscible regime and demonstrate that it is shifted between the models. The theoretical predictions for both the transition point and the mutual diffusion coefficient are validated in simulations of static droplets and decaying sinusoidal concentration waves, respectively. To show the universality of our analysis, two common and one new potential function are investigated. As the shift in the diffusion characteristics directly affects the interfacial properties, we additionally show that phenomena related to the interfacial tension such as the modeling of contact angles are influenced as well.

Lattice Boltzmann modeling and simulation of liquid jet breakup

A three-dimensional color-fluid lattice Boltzmann model for immiscible two-phase flows is developed in the framework of a three-dimensional 27-velocity (D3Q27) lattice. The collision operator comprises the D3Q27 versions of three suboperators: a multiple-relaxation-time (MRT) collision operator, a generalized Liu-Valocchi-Kang perturbation operator, and a Latva-Kokko-Rothman recoloring operator. A D3Q27 version of an enhanced equilibrium distribution function is also incorporated into this model to improve the Galilean invariance. Three types of numerical tests, namely, a static droplet, an oscillating droplet, and the Rayleigh-Taylor instability, show a good agreement with analytical solutions and numerical simulations. Following these numerical tests, this model is applied to liquid-jet-breakup simulations. The simulation conditions are matched to the conditions of the previous experiments. In this case, numerical stability is maintained throughout the simulation, although the kinematic viscosity for the continuous phase is set as low as 1.8×10^{-4}, in which case the corresponding Reynolds number is 3.4×10^{3}; the developed lattice Boltzmann model based on the D3Q27 lattice enables us to perform the simulation with parameters directly matched to the experiments. The jet's liquid column transitions from an asymmetrical to an axisymmetrical shape, and entrainment occurs from the side of the jet. The measured time history of the jet's leading-edge position shows a good agreement with the experiments. Finally, the reproducibility of the regime map for liquid-liquid systems is assessed. The present lattice Boltzmann simulations well reproduce the characteristics of predicted regimes, including varicose breakup, sinuous breakup, and atomization.

Lattice Boltzmann model for a steady radiative transfer equation

A complete lattice Boltzmann model (LBM) is proposed for the steady radiative transfer equation (RTE). The RTE can be regarded as a pure convection equation with a source term. To derive the expressions for the equilibrium distribution function and the relaxation time, an artificial isotropic diffusion term is introduced to form a convection-diffusion equation. When the dimensionless relaxation time has a value of 0.5, the lattice Boltzmann equation (LBE) is exactly applicable to the original steady RTE. We also perform a multiscale analysis based on the Chapman-Enskog expansion to recover the macroscopic RTE from the mesoscopic LBE. The D2Q9 model is used to solve the LBE, and the numerical results obtained by the LBM are comparable to the results obtained by other methods or analytical solutions, which demonstrates that the proposed model is highly accurate and stable in simulating multidimensional radiative transfer. In addition, we find that the convergence rate of the LBM depends on the transport properties of RTE: for diffusion-dominated RTE with a large optical thickness, the LBM shows a second-order convergence rate in space, while for convection-dominated RTE with a small optical thickness, a lower convergence rate is observed.

Lattice Boltzmann modeling of contact angle and its hysteresis in two-phase flow with large viscosity difference

Contact angle hysteresis is an important physical phenomenon omnipresent in nature and various industrial processes, but its effects are not considered in many existing multiphase flow simulations due to modeling complexity. In this work, a multiphase lattice Boltzmann method (LBM) is developed to simulate the contact-line dynamics with consideration of the contact angle hysteresis for a broad range of kinematic viscosity ratios. In this method, the immiscible two-phase flow is described by a color-fluid model, in which the multiple-relaxation-time collision operator is adopted to increase numerical stability and suppress unphysical spurious currents at the contact line. The contact angle hysteresis is introduced using the strategy proposed by Ding and Spelt [Ding and Spelt, J. Fluid Mech. 599, 341 (2008)JFLSA70022-112010.1017/S0022112008000190], and the geometrical wetting boundary condition is enforced to obtain the desired contact angle. This method is first validated by simulations of static contact angle and dynamic capillary intrusion process on ideal (smooth) surfaces. It is then used to simulate the dynamic behavior of a droplet on a nonideal (inhomogeneous) surface subject to a simple shear flow. When the droplet remains pinned on the surface due to hysteresis, the steady interface shapes of the droplet quantitatively agree well with the previous numerical results. Four typical motion modes of contact points, as observed in a recent study, are qualitatively reproduced with varying advancing and receding contact angles. The viscosity ratio is found to have a notable impact on the droplet deformation, breakup, and hysteresis behavior. Finally, this method is extended to simulate the droplet breakup in a microfluidic T junction, with one half of the wall surface ideal and the other half nonideal. Due to the contact angle hysteresis, the droplet asymmetrically breaks up into two daughter droplets with the smaller one in the nonideal branch channel, and the behavior of daughter droplets is significantly different in both branch channels. Also, it is found that the contact angle hysteresis is strengthened with decreasing the viscosity ratio, leading to an earlier droplet breakup and a decrease in the maximum length that the droplet can reach before the breakup. These simulation results manifest that the present multiphase LBM can be a useful substitute to Ba et al. [Phys. Rev. E 88, 043306 (2013)PLEEE81539-375510.1103/PhysRevE.88.043306] for modeling the contact angle hysteresis, and it can be easily implemented with higher computational efficiency.

Assessment of interaction potential in simulating nonisothermal multiphase systems by means of lattice Boltzmann modeling

The pseudopotential lattice Boltzmann model (PP-LBM) is a very popular model for simulating multiphase systems. In this model, phase separation occurs via a short-range attraction between different phases when the interaction potential term is properly chosen. Therefore, the potential term is expected to play a significant role in the model and to affect the accuracy and the stability of the computations. The original PP-LBM suffers from some drawbacks such as being capable of dealing with low density ratios only, thermodynamic inconsistency, and spurious velocities. In this paper, we aim to analyze the PP-LBM with the view to simulate single-component (non-)isothermal multiphase systems at large density ratios and in spite of the presence of spurious velocities. For this purpose, the performance of two popular potential terms and of various implementation schemes for these potential terms is examined. Furthermore, the effects of different parameters (i.e., equation of state, viscosity, etc.) on the simulations are evaluated, and, finally, recommendations for a proper simulation of (non-)isothermal multiphase systems are presented.

Contact angles in the pseudopotential lattice Boltzmann modeling of wetting

In this paper we investigate the implementation of contact angles in the pseudopotential lattice Boltzmann modeling of wetting at a large density ratio ρ_{L}/ρ_{V}=500. The pseudopotential lattice Boltzmann model [X. Shan and H. Chen, Phys. Rev. E 49, 2941 (1994)10.1103/PhysRevE.49.2941] is a popular mesoscopic model for simulating multiphase flows and interfacial dynamics. In this model the contact angle is usually realized by a fluid-solid interaction. Two widely used fluid-solid interactions, the density-based interaction and the pseudopotential-based interaction, as well as a modified pseudopotential-based interaction formulated in the present paper are numerically investigated and compared in terms of the achievable contact angles, the maximum and the minimum densities, and the spurious currents. It is found that the pseudopotential-based interaction works well for simulating small static (liquid) contact angles θ<90^{∘}, however, it is unable to reproduce static contact angles close to 180^{∘}. Meanwhile, it is found that the proposed modified pseudopotential-based interaction performs better in light of the maximum and the minimum densities and is overall more suitable for simulating large contact angles θ>90^{∘} as compared with the two other types of fluid-solid interactions. Furthermore, the spurious currents are found to be enlarged when the fluid-solid interaction force is introduced. Increasing the kinematic viscosity ratio between the vapor and liquid phases is shown to be capable of reducing the spurious currents caused by the fluid-solid interactions.

Pattern formation in liquid-vapor systems under periodic potential and shear

In this paper the phase behavior and pattern formation in a sheared nonideal fluid under a periodic potential is studied. An isothermal two-dimensional formulation of a lattice Boltzmann scheme for a liquid-vapor system with the van der Waals equation of state is presented and validated. Shear is applied by moving walls and the periodic potential varies along the flow direction. A region of the parameter space, where in the absence of flow a striped phase with oscillating density is stable, will be considered. At low shear rates the periodic patterns are preserved and slightly distorted by the flow. At high shear rates the striped phase loses its stability and traveling waves on the interface between the liquid and vapor regions are observed. These waves spread over the whole system with wavelength only depending on the length of the system. Velocity field patterns, characterized by a single vortex, will also be shown.

Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows

In this paper, a phase-field-based multiple-relaxation-time lattice Boltzmann (LB) model is proposed for incompressible multiphase flow systems. In this model, one distribution function is used to solve the Chan-Hilliard equation and the other is adopted to solve the Navier-Stokes equations. Unlike previous phase-field-based LB models, a proper source term is incorporated in the interfacial evolution equation such that the Chan-Hilliard equation can be derived exactly and also a pressure distribution is designed to recover the correct hydrodynamic equations. Furthermore, the pressure and velocity fields can be calculated explicitly. A series of numerical tests, including Zalesak's disk rotation, a single vortex, a deformation field, and a static droplet, have been performed to test the accuracy and stability of the present model. The results show that, compared with the previous models, the present model is more stable and achieves an overall improvement in the accuracy of the capturing interface. In addition, compared to the single-relaxation-time LB model, the present model can effectively reduce the spurious velocity and fluctuation of the kinetic energy. Finally, as an application, the Rayleigh-Taylor instability at high Reynolds numbers is investigated.

Interaction pressure tensor for a class of multicomponent lattice Boltzmann models

We present a theory to obtain the pressure tensor for a class of nonideal multicomponent lattice Boltzmann models, thus extending the theory presented by X. Shan [Phys. Rev. E 77, 066702 (2008)] for single-component fluids. We obtain the correct form of the pressure tensor directly on the lattice and the resulting equilibrium properties are shown to agree very well with those measured from numerical simulations. Results are compared with those of alternative theories.

Lattice Boltzmann modeling of multiphase flows at large density ratio with an improved pseudopotential model

Owing to its conceptual simplicity and computational efficiency, the pseudopotential multiphase lattice Boltzmann (LB) model has attracted significant attention since its emergence. In this work, we aim to extend the pseudopotential LB model to simulate multiphase flows at large density ratio and relatively high Reynolds number. First, based on our recent work [Q. Li, K. H. Luo, and X. J. Li, Phys. Rev. E 86, 016709 (2012)], an improved forcing scheme is proposed for the multiple-relaxation-time pseudopotential LB model in order to achieve thermodynamic consistency and large density ratio in the model. Next, through investigating the effects of the parameter a in the Carnahan-Starling equation of state, we find that the interface thickness is approximately proportional to 1/√a. Using a smaller a will lead to a wider interface thickness, which can reduce the spurious currents and enhance the numerical stability of the pseudopotential model at large density ratio. Furthermore, it is found that a lower liquid viscosity can be gained in the pseudopotential model by increasing the kinematic viscosity ratio between the vapor and liquid phases. The improved pseudopotential LB model is numerically validated via the simulations of stationary droplet and droplet oscillation. Using the improved model as well as the above treatments, numerical simulations of droplet splashing on a thin liquid film are conducted at a density ratio in excess of 500 with Reynolds numbers ranging from 40 to 1000. The dynamics of droplet splashing is correctly reproduced and the predicted spread radius is found to obey the power law reported in the literature.

Multicomponent interparticle-potential lattice Boltzmann model for fluids with large viscosity ratios

This work focuses on an improved multicomponent interparticle-potential lattice Boltzmann model. The model results in viscosity-independent equilibrium densities and is capable of simulating kinematic viscosity ratios greater than 1000. External forces are incorporated into the discrete Boltzmann equation, rather than through an equilibrium velocity shift as in the original Shan and Chen (hereafter, SC) model. The model also requires the derivation of a momentum conserving effective velocity, which is substituted into the equilibrium distribution function and applies to both the single- and multiple-relaxation-time formulations. Additionally, higher-order isotropy is used in the calculation of the fluid-fluid interaction forces to reduce the magnitude of spurious currents (i.e., numerical errors) in the vicinity of interfaces. First, we compare the model to the SC model for static bubble simulations. We demonstrate that the model results in viscosity-independent equilibrium bubble densities for a wide range of kinematic viscosities, which is not the case for the SC model. Furthermore, we show that the model is capable of simulating stable bubbles for kinematic viscosity ratios greater than 1000 (when higher-order isotropy is used), whereas the SC model is known to be limited to kinematic viscosity ratios on the order of 10. Next we verify the model for surface tension via Laplace's law and show that the model results in the same surface tension values for a range of kinematic viscosities and kinematic viscosity ratios of 10, 100, and 1000. The model is also verified for layered cocurrent flow though parallel plates. We show that the simulated velocity profiles preserve continuity at the interface for kinematic viscosity ratios ranging from 0.001 to 1000 and that the model accurately predicts nonwetting and wetting phase relative permeability for kinematic viscosity ratios of 0.01 to 100.

Forcing scheme in pseudopotential lattice Boltzmann model for multiphase flows

The pseudopotential lattice Boltzmann (LB) model is a widely used multiphase model in the LB community. In this model, an interaction force, which is usually implemented via a forcing scheme, is employed to mimic the molecular interactions that cause phase segregation. The forcing scheme is therefore expected to play an important role in the pseudoepotential LB model. In this paper, we aim to address some key issues about forcing schemes in the pseudopotential LB model. First, theoretical and numerical analyses will be made for Shan-Chen's forcing scheme [Shan and Chen, Phys. Rev. E 47, 1815 (1993)] and the exact-difference-method forcing scheme [Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)]. The nature of these two schemes and their recovered macroscopic equations will be shown. Second, through a theoretical analysis, we will reveal the physics behind the phenomenon that different forcing schemes exhibit different performances in the pseudopotential LB model. Moreover, based on the analysis, we will present an improved forcing scheme and numerically demonstrate that the improved scheme can be treated as an alternative approach to achieving thermodynamic consistency in the pseudopotential LB model.

Effect of the forcing term in the multiple-relaxation-time lattice Boltzmann equation on the shear stress or the strain rate tensor

In this work, the effect of the forcing term (or external force) in the multiple-relaxation-time lattice Boltzmann equation (MRTLBE) on the shear stress or the strain rate tensor is studied theoretically and numerically. Through a Chapman-Enskog analysis and numerical simulations, we show that the shear stress (or the strain rate tensor) derived from the MRTLBE is second-order accurate in space. We then examine the influence of the forcing term on the shear stress or the strain rate tensor, and demonstrate that the forcing term effect must be included when the shear stress or the strain rate tensor is computed with the nonequilibrium part of the distribution function.

Additional interfacial force in lattice Boltzmann models for incompressible multiphase flows

The existing lattice Boltzmann models for incompressible multiphase flows are mostly constructed with two distribution functions: one is the order parameter distribution function, which is used to track the interface between different phases, and the other is the pressure distribution function for solving the velocity field. In this paper, it is shown that in these models the recovered momentum equation is inconsistent with the target one: an additional force is included in the recovered momentum equation. The additional force has the following features. First, it is proportional to the macroscopic velocity. Second, it is zero in every single-phase region but is nonzero in the interface. Therefore it can be interpreted as an interfacial force. To investigate the effects of the additional interfacial force, numerical simulations are carried out for the problem of Rayleigh-Taylor instability, droplet splashing on a thin liquid film, and the evolution of a falling droplet under gravity. Numerical results demonstrate that, with the increase of the velocity or the Reynolds number, the additional interfacial force will gradually have an important influence on the interface and affect the numerical accuracy.

Forcing term in single-phase and Shan-Chen-type multiphase lattice Boltzmann models

Numerous schemes have been proposed to incorporate a bulk forcing term into the lattice Boltzmann equation. In this paper we present a simple and straightforward comparative analysis of five popular schemes [Shan and Chen, Phys. Rev. E 47, 1815 (1993); Phys Rev Lett. 81, 1618 (1998); He et al., Phys. Rev. E 57, R13 (1998); Guo et al., Phys. Rev. E 65, 046308 (2002); Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)] in which their differences and similarities are identified. From the analysis we classify the schemes into two groups; the behaviors of the schemes in each group are proven to be identical up to second order. Numerical test simulating the two-dimensional unsteady Taylor-Green vortex flow problem demonstrate that all five schemes are of comparable accuracy for single-phase flow. However, for two-phase flow the situation is different, which is demonstrated by incorporating these schemes into different Shan-Chen-type multiphase models. The forcing scheme in the original Shan-Chen (SC) multiphase model turns out to be inaccurate in terms of the resulting surface tension for different density ratios and relaxation times. In the numerical tests, a typical equation of state and interparticle interactions including next-nearest neighbors were incorporated into the SC model. Our results confirm that the surface-tension values obtained from the original SC lattice Boltzmann method (LBM) simulation depend on the value of the relaxation time τ. For τ<0.7Δt, the surface tension agree well with the analytical solutions. However, when τ>0.7Δt, the surface tension turns out to be systematically larger than the analytical one, exceeding it by more than a factor of 2 for τ=2Δt. In contrast, with the application of the scheme proposed by He et al., the SC LBM produces very accurate surface tensions independent of the value of τ. We also found that the densities of the coexisting liquid and gas can be adjusted to match those at thermodynamic equilibrium if the particle interaction term includes next-nearest-neighbor contributions. The obtained results will be useful for further studies of two-phase flow with high density ratios using the SC LBM approach.