Lattice Boltzmann model for simulating flows with multiple phases and components

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.

Pore-Level Multiphase Simulations of Realistic Distillation Membranes for Water Desalination

Membrane distillation (MD) is a thermally driven separation process that is operated below boiling point. Since the performance of MD modules is still comparatively low, current research aims to improve the understanding of the membrane structure and its underlying mechanisms at the pore level. Based on existing realistic 3D membrane geometries (up to 0.5 billion voxels with 39nm resolution) obtained from ptychographic X-ray computed tomography, the D3Q27 lattice Boltzmann (LB) method was used to investigate the interaction of the liquid and gaseous phase with the porous membrane material. In particular, the Shan and Chen multi-phase model was used to simulate multi-phase flow at the pore level. We investigated the liquid entry pressure of different membrane samples and analysed the influence of different micropillar structures on the Wenzel and Cassie-Baxter state of water droplets on rough hydrophobic surfaces. Moreover, we calculated the liquid entry pressure required for entering the membrane pores and extracted realistic water contact surfaces for different membrane samples. The influence of the micropillars and flow on the water-membrane contact surface was investigated. Finally, we determined the air-water interface within a partially saturated membrane, finding that the droplet size and distribution correlated with the porosity of the membrane.

Thermal lattice Boltzmann model for liquid-vapor phase change

The lattice Boltzmann method is adopted to solve the liquid-vapor phase change problems in this article. By modifying the collision term for the temperature evolution equation, a thermal lattice Boltzmann model is constructed. As compared with previous studies, the most striking feature of the present approach is that it could avoid the calculations of both the Laplacian term of temperature [∇·(κ∇T)] and the gradient term of heat capacitance [∇(ρc_{v})]. In addition, since the present approach adopts a simple linear equilibrium distribution function, it is possible to use the D2Q5 lattice for the two-dimensional cases considered here. Thus, the present model is more efficient than previous models in which the lattice is usually limited to the D2Q9. The proposed model is first validated by the problems of droplet evaporation in open space and droplet evaporation on a heated surface, and the numerical results show good agreement with the analytical results and the finite difference method. Then it is used to model the nucleate boiling problem, and the relationship between detachment bubble diameter and gravitational acceleration obtained with the present approach fits well with previous works.

Analysis and reconstruction of the multiphase lattice Boltzmann flux solver for multiphase flows with large density ratios

The multiphase lattice Boltzmann flux solver (MLBFS) has been proposed to tackle complex geometries with nonuniform meshes. It also has been proven to have good numerical stability for multiphase flows with large density ratios. However, the reason for the good numerical stability of MLBFS at large density ratios has not been well established. The present paper reveals the relation between MLBFS and the macroscopic weakly compressible multiphase model by recovering the macroscopic equations of MLBFS (MEs-MLBFS) with actual numerical dissipation terms. By directly solving MEs-MLBFS, the reconstructed MLBFS (RMLBFS) that involves only macroscopic variables in the computational processes is proposed. The analysis of RMLBFS indicates that by combining the predictor step, the corrector step of MLBFS introduces some numerical dissipation terms which contribute to the good numerical stability of MLBFS. By retaining these numerical dissipation terms, RMLBFS can maintain the numerical stability of MLBFS even at large density ratios.

Color-gradient lattice Boltzmann model for immiscible fluids with density contrast

We present a color-gradient-based lattice Boltzmann model for immiscible fluids with a large density contrast. The model employs the velocity-based equilibrium distribution function, initially proposed for the phase-field-based model by Zu and He [Phys. Rev. E 87, 043301 (2013)1539-375510.1103/PhysRevE.87.043301], with a modification necessary to satisfy the kinematic condition at the interface. Different from the existing color-gradient models, the present model allows to specify interface mobility that is independent of the fluid density ratio. Further, we provide a unified framework, which uses the recursive representation of the lattice Boltzmann equation, to derive the governing equations of the system. The emergent color dynamics thus obtained, through an analysis of the segregation operator, is shown to obey the locally conservative Allen-Cahn equation. We use a series of benchmarks, which include a stationary drop, a layered Poiseuille flow, translation of a drop under a forced velocity field, the Rayleigh-Taylor instability, and the capillary intrusion test to demonstrate the model's ability in dealing with complex flow problems.

Free-Energy-Based Discrete Unified Gas Kinetic Scheme for van der Waals Fluid

The multiphase model based on free-energy theory has been experiencing long-term prosperity for its solid foundation and succinct implementation. To identify the main hindrance to developing a free-energy-based discrete unified gas-kinetic scheme (DUGKS), we introduced the classical lattice Boltzmann free-energy model into the DUGKS implemented with different flux reconstruction schemes. It is found that the force imbalance amplified by the reconstruction errors prevents the direct application of the free-energy model to the DUGKS. By coupling the well-balanced free-energy model with the DUGKS, the influences of the amplified force imbalance are entirely removed. Comparative results demonstrated a consistent performance of the well-balanced DUGKS despite the reconstruction schemes utilized. The capability of the DUGKS coupled with the well-balanced free-energy model was quantitatively validated by the coexisting density curves and Laplace's law. In the quiescent droplet test, the magnitude of spurious currents is reduced to a machine accuracy of 10-15. Aside from the excellent performance of the well-balanced DUGKS in predicting steady-state multiphase flows, the spinodal decomposition test and the droplet coalescence test revealed its stability problems in dealing with transient flows. Further improvements are required on this point.

Asymmetric Condensation Characteristics during Dropwise Condensation in the Presence of Non-condensable Gas: A Lattice Boltzmann Study

In this work, the condensation characteristics of droplets considering the non-condensable gas with different interaction effects are numerically studied utilizing a multicomponent multiphase thermal lattice Boltzmann (LB) model, with a special focus on the asymmetric nature induced by the interaction effect. The results demonstrate that for isolated-like growth with negligible interactions, the condensation characteristics, that is, the concentration profile, the temperature distribution, and the flow pattern, are typically symmetric in nature. For the growth regime in a pattern, the droplet has to compete with its neighbors for catching vapor, which leads to an overlapping concentration profile (namely the interaction effect). The distribution of the condensation flux on the droplet surface is consequently modified, which contributes to the asymmetric flow pattern and temperature profile. The condensation characteristics for droplet growth in a pattern present an asymmetric nature. Significantly, the asymmetric condensation flux resulting from the interaction effect can induce droplet motion. The results further demonstrate that the interaction strongly depends on the droplet's spatial and size distribution, including two crucial parameters, namely the inter-distance and relative size of droplets. The asymmetric condensation characteristics are consequently dependent on the difference in the interaction intensities on both sides of the droplet. Finally, we demonstrate numerically and theoretically that the evolution of the droplet radius versus time can be suitably described by a power law; the corresponding exponent is kept at a constant of 0.50 for isolated-like growth and is strongly sensitive to the interaction effect for the growth in a pattern.

Consistent and conservative phase-field-based lattice Boltzmann method for incompressible two-phase flows

In this work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method, and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and conservative phase-field model. Additionally, if the divergence of the extra momentum flux is seen as a force term, the extra force in the present LB method would include another term which has not been considered in the previous LB methods. To quantitatively evaluate the incompressibility and the consistency of the mass conservation, two statistical variables are introduced in the study of the deformation of a square droplet, and the results show that the present LB method is more accurate. The layered Poiseuille flow and a droplet spreading on an ideal wall are further investigated, and the numerical results are in good agreement with the analytical solutions. Finally, the problems of the Rayleigh-Taylor instability, a single rising bubble, and the dam break with the high Reynolds numbers and/or large density ratios are studied, and it is found that the present consistent and conservative LB method is robust for such complex two-phase flows.

Simulation of Pressure-Driven and Channel-Based Microfluidics on Different Abstract Levels: A Case Study

A microfluidic device, or a Lab-on-a-Chip (LoC), performs lab operations on the microscale through the manipulation of fluids. The design and fabrication of such devices usually is a tedious process, and auxiliary tools, such as simulators, can alleviate the necessary effort for the design process. Simulations of fluids exist in various forms and can be categorized according to how well they represent the underlying physics, into so-called abstraction levels. In this work, we consider simulation approaches in 1D, which are based on analytical solutions of simplified problems, and approaches in 2D and 3D, for which we use two different Computational Fluid Dynamics (CFD) methods-namely, the Finite Volume Method (FVM) and the Lattice-Boltzmann Method (LBM). All these methods come with their pros and cons with respect to accuracy and required compute time, but unfortunately, most designers and researchers are not aware of the trade-off that can be made within the broad spectrum of available simulation approaches for microfluidics and end up choosing a simulation approach arbitrarily. We provide an overview of different simulation approaches as well as a case study of their performance to aid designers and researchers in their choice. To this end, we consider three representative use cases of pressure-driven and channel-based microfluidic devices (namely the non-Newtonian flow in a channel, the mixing of two fluids in a channel, and the behavior of droplets in channels). The considerations and evaluations raise the awareness and provide several insights for what simulation approaches can be utilized today when designing corresponding devices (and for what they cannot be utilized yet).

Multiphase curved boundary condition in lattice Boltzmann method

The boundary treatment is fundamental for modeling fluid flows especially in the lattice Boltzmann method; the curved boundary conditions effectively improve the accuracy of single-phase simulations with complex-geometry boundaries. However, the conventional curved boundary conditions usually cause dramatic mass leakage or increase when they are directly used for multiphase flow simulations. We find that the principal reason for this is the absence of a nonideal effect in the curved boundary conditions, followed by a calculation error. In this paper, incorporating the nonideal effect into the linear interpolation scheme and compensating for the interpolating error, we propose a multiphase curved boundary condition to treat the wetting boundaries with complex geometries. A series of static and dynamic multiphase simulations with large density ratio verify that the present scheme is accurate and ensures mass conservation.

Study on the Ways to Improve the CO2-H2O Displacement Efficiency in Heterogeneous Porous Media by Lattice Boltzmann Simulation

To improve the efficiency of CO₂ geological sequestration, it is of great significance to in-depth study the physical mechanism of the immiscible CO₂-water displacement process, where the influential factors can be divided into fluid-fluid and fluid-solid interactions and porous media characteristics. Based on the previous studies of the interfacial tension (capillary number) and viscosity ratio factors, we conduct a thorough study about the effects of fluid-solid interaction (i.e., wettability) and porous media characteristics (i.e., porosity and non-uniformity of granule size) on the two-phase displacement process by constructing porous media with various structural parameters and using a multiphase lattice Boltzmann method. The displacement efficiency of CO₂ is evaluated by the breakthrough time characterizing the displacement speed and the quasi-steady state saturation representing the displacement amount. It is shown that the breakthrough time of CO₂ becomes longer, but the quasi-steady state saturation increases markedly with the increase in CO₂ wettability with the surface, demonstrating an overall improvement of the displacement efficiency. Furthermore, the breakthrough time of CO₂ shortens and the saturation increases significantly with increasing porosity, granule size, and non-uniformity, showing the improvement of the displacement efficiency. Therefore, enhancing the wettability of CO₂ with the surface and selecting reservoirs with greater porosity, larger granule size, and non-uniformity can all contribute to the efficiency improvement of CO₂ geological sequestration.

Study on Interparticle Interaction Force Model to Correct Saturation Density of Real Cryogenic Fluid for LBM Simulation

Cryogenic liquefaction energy storage is an important form of storage for sustainable energy liquid hydrogen and other gases. The weighting parameter A in the parameter-adjusted two-phase LBM model is important for the deviation of simulation results. The aim of this paper is to discover the appropriate parameter to eliminate the deviation, and to solve the problem of large deviation between the theoretical solution and the simulated value that is caused by using different equations of state in LBM simulation. The modified PT equation of state, which is suitable for cryogenic fluids, is combined with the parameter-adjustable two-phase model to simulate the saturation density at different temperatures. Four typical cryogenic fluids—nitrogen, hydrogen, oxygen, and helium—are exploratively simulated to find the suitable parameters to eliminate errors by analyzing the results with theoretical solutions. This is an efficient solution to the deviation between the simulated value and the theoretical solutions, which is caused by the different equation of state in LBM. The optimal A-value of the model based on the PT equation of state was obtained as −0.21, while droplets and bubbles were set into the calculation region, and an inverse relationship between the interface density gradient and temperature was analyzed. The analysis and comparison of the simulation results under the optimal value and the experimental values have laid an important foundation for the phase change simulation of the real cryogenic fluids at the mesoscopic scale.

Lattice Boltzmann Modeling of Spontaneous Imbibition in Variable-Diameter Capillaries

Previous micro-scale studies of the effect of pore structure on spontaneous imbibition are mainly limited to invariable-diameter capillaries. However, in real oil and gas reservoir formations, the capillary diameters are changing and interconnected. Applying the lattice Boltzmann color gradient two-phase flow model and the parallel computation of CPUs, we simulated the spontaneous imbibition in variable-diameter capillaries. We explored the reasons for the nonwetting phase snap-off and systematically studied the critical conditions for the snap-off in spontaneous imbibition. The effects of pore-throat aspect ratio, throat diameter, and the pore-throat tortuosity of the capillary on spontaneous imbibition were studied. Through analyzing the simulated results, we found that the variation in the capillary diameter produces an additional resistance, which increases with the increase in the pore-throat ratio and the pore-throat tortuosity of a capillary. Under the action of this additional resistance, the snap-off phenomenon sometimes occurs in the spontaneous imbibition, which makes the recovery efficiency of the non-wetting phase extremely low. In addition, the main factors affecting this phenomenon are the pore-throat ratio and the pore-throat tortuosity, which is different from the conventional concept of tortuosity. When the snap-off does not occur, the spontaneous imbibition velocity increases when the throat diameter increases and the pore-throat aspect ratio is fixed, and when the period increases, i.e., the diameter changing rate decreases, the spontaneous imbibition velocity also increases. In addition, when the capillary throat diameter is fixed, a bigger pore diameter and a smaller period of sine function both inhibit the speed of spontaneous imbibition.

Lattice Boltzmann modeling of two-phase electrohydrodynamic flows under unipolar charge injection

In this work, a two-dimensional droplet confined between two parallel electrodes under the combined effects of a nonuniform electric field and unipolar charge injection is numerically investigated using the lattice Boltzmann method (LBM). Under the non-Ohmic regime, the interfacial tension and electric forces at the droplet surface cooperate with the volumetric Coulomb force, leading to complex deformation and motion of the droplet while at the same time inducing a bulk electroconvective flow. After we validate the model by comparing with analytical solutions at the hydrostatic state, we perform a quantitative analysis on the droplet deformation factor D and bulk flow stability criteria T_{c} under different parameters, including the electric capillary number Ca, the electric Rayleigh number T, the permittivity ratio ɛ_{r}, and the mobility ratio K_{r}. It is found that the bulk flow significantly modifies the magnitude of D, which in turn decreases T_{c} of the electroconvective flow. For a droplet repelled by the anode, ɛ_{r}>1, an interesting linear relationship can be observed in the D-Ca curves. However, for a droplet attracted to the anode, ɛ_{r}<1, the system is potentially unstable. After first evolving into a quasisteady state, the droplet successively experiences steady flow, periodic flow, second steady flow, and oscillatory flow with increasing T. Moreover, discontinuities can be observed in the D-T curves due to the transitions of bulk flow.

Pore-Scale Study on Convective Drying of Porous Media

In this work, a numerical model for isothermal liquid-vapor phase change (evaporation) of the two-component air-water system is proposed based on the pseudopotential lattice Boltzmann method. Through the Chapman-Enskog multiscale analysis, we show that the model can correctly recover the macroscopic governing equations of the multicomponent multiphase system with a built-in binary diffusion mechanism. The model is verified based on the two-component Stefan problem where the measured binary diffusivity is consistent with theoretical analysis. The model is then applied to convective drying of a dual-porosity porous medium at the pore scale. The simulation captures a classical transition in the drying process of porous media, from the constant rate period (CRP, first phase) showing significant capillary pumping from large to small pores, to the falling rate period (FRP, second phase) with the liquid front receding in small pores. It is found that, in the CRP, the evaporation rate increases with the inflow Reynolds number (Re), while in the FRP, the evaporation curves almost collapse at different Res. The underlying mechanism is elucidated by introducing an effective Péclet number (Pe). It is shown that convection is dominant in the CRP and diffusion in the FRP, as evidenced by Pe > 1 and Pe < 1, respectively. We also find a log-law dependence of the average evaporation rate on the inflow Re in the CRP regime. The present work provides new insights into the drying physics of porous media and its direct modeling at the pore scale.

Exploring the Role of Initial Droplet Position in Coalescence-Induced Droplet Jumping: Lattice Boltzmann Simulations

Coalescence-induced droplet jumping on superhydrophobic surfaces with different initial positions was numerically simulated using the 2D multi-relaxation-time (MRT) Lattice Boltzmann method (LBM). Simulation results show that for coalesced droplets with radii close to the structure length scale, the change of initial droplet positions leads to a significant deviation of jumping velocity and direction. By finely tuning the initial droplet positions on a flat-pillared surface, perpendicular jumping, oblique jumping, and non-jumping are successively observed on the same structured surface. Droplet morphologies and vector diagrams at different moments are considered. It is revealed that the asymmetric droplet detachment from the structured surface leads to the directional transport of liquid mass in the droplet and further results in the oblique jumping of the coalesced droplet. In order to eliminate the influence of initial droplet position on droplet jumping probability, a surface with pointed micropillars is designed. It is demonstrated that compared to flat-topped micropillars, a surface with pointed micropillars can suppress the initial droplet position effects and enhance droplet jumping probability. Furthermore, the effect of droplet/structure scale on droplet jumping is investigated. The influence of initial positions on coalescence-induced droplet jumping from the refined surface can be ignored when the droplet scale is larger than three times the structure scale. This study illustrates the role of initial droplet position in coalescence-induced droplet jumping and provides guidelines for the rational design of structured surfaces with enhanced droplet self-shedding for energy and heat transfer applications.

Toward a Lattice Boltzmann Method for Solids—Application to Static Equilibrium of Isotropic Materials

This work presents a novel method for simulating the behavior of solid objects with the Lattice Boltzmann Method (LBM). To introduce and validate our proposed framework, comparative studies are performed for computing the static equilibrium of isotropic materials. Remembering that the LBM has strong theoretical foundations in the Boltzmann equation; this latter is firstly adjusted to solid motions, through its Boltzmann-Vlasov special case. This is indeed the case when combined with a suitable mean-field external force term to set a reliable solid framework. Secondly, a library is built and plugged on the top of the well-known Parallel Lattice Boltzmann Solver (PaLaBoS) library. Numerical implementations based on the previous equation of motion for solids are led in a non-intrusive manner so as to present results with an easy and flawless reproducibility. A newly designed Lattice Boltzmann Method for Solids (LBMS) is exhibited through a few key algorithms, showing the overall operation plus the major improvements. Efficiency, robustness and accuracy of the proposed approach are illustrated and contrasted with a commercial Finite Element Analysis (FEA) software. The obtained results reveal considerable potential concerning static and further dynamic simulations involving solid constitutive laws within the LBM formalism.

Comparative investigation of a lattice Boltzmann boundary treatment of multiphase mass transport with heterogeneous chemical reactions

Multiphase reactive transport in porous media is an important component of many natural and engineering processes. In the present study, boundary schemes for the continuum species transport-lattice Boltzmann (CST-LB) mass transport model and the multicomponent pseudopotential model are proposed to simulate heterogeneous chemical reactions in a multiphase system. For the CST-LB model, a lattice-interface-tracking scheme for the heterogeneous chemical reaction boundary is provided. Meanwhile, a local-average virtual density boundary scheme for the multicomponent pseudopotential model is formulated based on the work of Li et al. [Li, Yu, and Luo, Phys. Rev. E 100, 053313 (2019)10.1103/PhysRevE.100.053313]. With these boundary treatments, a numerical implementation is put forward that couples the multiphase fluid flow, interfacial species transport, heterogeneous chemical reactions, and porous matrix structural evolution. A series of comparison benchmark cases are investigated to evaluate the numerical performance for different pseudopotential wetting boundary treatments, and an application case of multiphase dissolution in porous media is conducted to validate the present models' ability to solve complex problems. By applying the present LB models with reasonable boundary treatments, multiphase reactive transport in various natural or engineering scenarios can be simulated accurately.

Unified lattice Boltzmann method with improved schemes for multiphase flow simulation: Application to droplet dynamics under realistic conditions

As a powerful mesoscale approach, the lattice Boltzmann method (LBM) has been widely used for the numerical study of complex multiphase flows. Recently, Luo et al. [Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 379, 20200397 (2021)10.1098/rsta.2020.0397] proposed a unified lattice Boltzmann method (ULBM) to integrate the widely used lattice Boltzmann collision operators into a unified framework. In this study, we incorporate additional features into this ULBM in order to simulate multiphase flow under realistic conditions. A nonorthogonal moment set [Fei et al., Phys. Rev. E 97, 053309 (2018)10.1103/PhysRevE.97.053309] and the entropic-multi-relaxation-time (KBC) lattice Boltzmann model are used to construct the collision operator. An extended combined pseudopotential model is proposed to realize multiphase flow simulation at high-density ratio with tunable surface tension over a wide range. The numerical results indicate that the improved ULBM can significantly decrease the spurious velocities and adjust the surface tension without appreciably changing the density ratio. The ULBM is validated through reproducing various droplet dynamics experiments, such as binary droplet collision and droplet impingement on superhydrophobic surfaces. Finally, the extended ULBM is applied to complex droplet dynamics, including droplet pancake bouncing and droplet splashing. The maximum Weber number and Reynolds number in the simulation reach 800 and 7200, respectively, at a density ratio of 1000. The study demonstrates the generality and versatility of ULBM for incorporating schemes to tackle challenging multiphase problems.

Characterization of condensation on nanostructured surfaces and associated thermal hydraulics using a thermal lattice Boltzmann method

The dynamics of the condensation process on nanostructured surfaces can be modulated substantially by tuning the surface architecture. Present study uses the mesoscopic framework of lattice Boltzmann method (LBM) to explore the role of surface morphology and cold spot temperature in determining the visual state of the condensate droplet, mode of nucleation, and associated rates of energy and mass interactions. A multiple relaxation time-(MRT)-based LBM solver, coupled with pseudopotential model, has been developed to simulate a rectangular domain of saturated vapor, housing a cold spot on the bottom rough surface. Superhydrophobicity has been achieved for certain combinations of surface parameters, with the intercolumn spacing being the most influential one. Gradual increase in the spacing modifies the nucleation mode from top through side to bottom, while the droplet changes from Cassie to Wenzel state. The Cassie drop in top nucleation mode exhibits the largest contact angle and least rate of surface heat transfer. Both types of Wenzel drops display large rate of condensation and two peaks in heat transfer, along with very short nucleation time in comparison with Cassie drops. Couple of phase diagrams have been developed combining all four scenarios of condensation predicted by the present model. One important novelty of the present study is the consideration of nonisothermal condition within LB structure. Enhancement in the degree of subcooling at the cold spot encourages greater condensation and Cassie-to-Wenzel transition.

How Heterogeneous Pore Scale Distributions of Wettability Affect Infiltration into Porous Media

Wettability is an important parameter that significantly determines hydrology in porous media, and it especially controls the flow of water across the rhizosphere—the soil-plant interface. However, the influence of spatially heterogeneous distributions on the soil particles surfaces is scarcely known. Therefore, this study investigates the influence of spatially heterogeneous wettability distributions on infiltration into porous media. For this purpose, we utilize a two-phase flow model based on Lattice-Boltzmann to numerically simulate the infiltration in porous media with a simplified geometry and for various selected heterogeneous wettability coatings. Additionally, we simulated the rewetting of the dry rhizosphere of a sandy soil where dry hydrophobic mucilage depositions on the particle surface are represented via a locally increased contact angle. In particular, we can show that hydraulic dynamics and water repellency are determined by the specific location of wettability patterns within the pore space. When present at certain locations, tiny hydrophobic depositions can cause water repellency in an otherwise well-wettable soil. In this case, averaged, effective contact angle parameterizations such as the Cassie equation are unsuitable. At critical conditions, when the rhizosphere limits root water uptake, consideration of the specific microscale locations of exudate depositions may improve models of root water uptake.

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.

Dynamics of two-dimensional liquid bridges

We have simulated the motion of a single vertical, two-dimensional liquid bridge spanning the gap between two flat, horizontal solid substrates of given wettabilities, using a multicomponent pseudopotential lattice Boltzmann method. For this simple geometry, the Young-Laplace equation can be solved (quasi-)analytically to yield the equilibrium bridge shape under gravity, which provides a check on the validity of the numerical method. In steady-state conditions, we calculate the drag force exerted by the moving bridge on the confining substrates as a function of its velocity, for different contact angles and Bond numbers. We also study how the bridge deforms as it moves, as parametrized by the changes in the advancing and receding contact angles at the substrates relative to their equilibrium values. Finally, starting from a bridge within the range of contact angles and Bond numbers in which it can exist at equilibrium, we investigate how fast it must move in order to break up.

Improved three-dimensional thermal multiphase lattice Boltzmann model for liquid-vapor phase change

Modeling liquid-vapor phase change using the lattice Boltzmann (LB) method has attracted significant attention in recent years. In this paper, we propose an improved three-dimensional thermal multiphase LB model for simulating liquid-vapor phase change. The proposed model has the following features. First, it is still within the framework of the thermal LB method using a temperature distribution function and therefore retains the fundamental advantages of the thermal LB method. Second, in the existing thermal LB models for liquid-vapor phase change, the finite-difference computations of the gradient terms ∇·u and ∇T usually require special treatment at boundary nodes, while in the proposed thermal LB model these two terms are calculated locally. Moreover, in some of the existing thermal LB models, the error term ∂_{t_{0}}(Tu) is eliminated by adding local correction terms to the collision process in the moment space, which causes these thermal LB models to be limited to the D2Q9 lattice in two dimensions and the D3Q15 or D3Q19 lattice in three dimensions. Conversely, the proposed model does not suffer from such an error term and therefore the thermal LB equation can be constructed on the D3Q7 lattice, which simplifies the model and improves the computational efficiency. Numerical simulations are carried out to validate the accuracy and efficiency of the proposed thermal multiphase LB model for simulating liquid-vapor phase change.

Droplet evaporation in finite-size systems: Theoretical analysis and mesoscopic modeling

The classical D^{2}-Law states that the square of the droplet diameter decreases linearly with time during its evaporation process, i.e., D^{2}(t)=D_{0}^{2}-Kt, where D_{0} is the droplet initial diameter and K is the evaporation constant. Though the law has been widely verified by experiments, considerable deviations are observed in many cases. In this work, a revised theoretical analysis of the single droplet evaporation in finite-size open systems is presented for both two-dimensional (2D) and 3D cases. Our analysis shows that the classical D^{2}-Law is only applicable for 3D large systems (L≫D_{0}, L is the system size), while significant deviations occur for small (L≤5D_{0}) and/or 2D systems. Theoretical solution for the temperature field is also derived. Moreover, we discuss in detail the proper numerical implementation of droplet evaporation in finite-size open systems by the mesoscopic lattice Boltzmann method (LBM). Taking into consideration shrinkage effects and an adaptive pressure boundary condition, droplet evaporation in finite-size 2D/3D systems with density ratio up to 328 within a wide parameter range (K=[0.003,0.18] in lattice units) is simulated, and remarkable agreement with the theoretical solution is achieved, in contrast to previous simulations. The present work provides insights into realistic droplet evaporation phenomena and their numerical modeling using diffuse-interface methods.

Coolant Wetting Simulation on Simplified Stator Coil Model by the Phase-Field Lattice Boltzmann Method

Stator coils of automobiles in operation generate heat and are cooled by coolant poured from above. The flow characteristic of the coolant depends on the coil structure, flow condition, solid–fluid interaction, and fluid property, which has not been clarified due to its complexities. Since straight coils are aligned and layered with an angle at the coolant-touchdown region, the coil structure is simplified to a horizontal square rod array referring to an actual coil size. To obtain the flow and wetting characteristics, two-phase fluid flow simulations are conducted by using the phase-field lattice Boltzmann method. First, the flow onto the single-layered rod array is discussed. The wetting area is affected both by the rod gap and the wettability, which is normalized by the gap and the averaged boundary layer thickness. Then, the flow onto the multi-layered rod arrays is investigated with different rod gaps. The top layer wetting becomes longitudinal due to the reduction of the flow advection by the second layer. The wetting area jumps up at the second layer and increases proportionally to the below layers. These become remarkable at the narrow rod gap case, and finally, the dimensionless wetting area is discussed at each layer.

Numerical Studies on Cellulose Hydrolysis in Organic-Liquid-Solid Phase Systems with a Liquid Membrane Catalysis Model

The catalytic hydrolysis of cellulose to produce 5-hydroxymethylfurfural (HMF) is a powerful means of biomass resources. The current efficient hydrolysis of cellulose to obtain HMF is dominated by multiphase reaction systems. However, there is still a lack of studies on the synergistic mechanisms and component transport between the various processes of cellulose hydrolysis in a complex multiphase system. In this paper, a liquid membrane catalytic model was developed to simulate the hydrolysis of cellulose and its further reactions, including the adsorption of the liquid membrane on cellulose particles, the consumption of cellulose solid particles, the complex chemical reactions in the liquid membrane, and the transfer of HMF at the phase interface. The simulations show the synergistic effect between cellulose hydrolysis and multiphase mass transfer. We defined an indicator () to characterize the sensitivity of HMF yield to the initial liquid membrane thickness at different reaction stages. decreased gradually when the glucose conversion increased from 0 to 80%, and increased with the thickening of the initial liquid membrane thickness. It was shown that the thickening of the initial liquid membrane thickness promoted the HMF yield under the same glucose conversion. In summary, our results reveal the mechanism of the interaction between multiple physicochemical processes of the cellulose liquid membrane reaction system.

Mesoscale perspective on the Tolman length

We demonstrate that the multiphase Shan-Chen lattice Boltzmann method (LBM) yields a curvature dependent surface tension σ as computed from three-dimensional hydrostatic droplets and bubbles simulations. Such curvature dependence is routinely characterized, at first order, by the so-called Tolman length δ. LBM allows one to precisely compute σ at the surface of tension R_{s} and determine the Tolman length from the coefficient of the first order correction. The corresponding values of δ display universality for different equations of state, following a power-law scaling near the critical temperature. The Tolman length has been studied so far mainly via computationally demanding Molecular Dynamics simulations or by means of Density Functional Theory approaches playing a pivotal role in extending Classical Nucleation Theory. The present results open a hydrodynamic-compliant mesoscale arena, in which the fundamental role of the Tolman length, alongside real-world applications to cavitation phenomena, can be effectively tackled. All the results can be independently reproduced through the "idea.deploy" framework.

Shaping the equation of state to improve numerical accuracy and stability of the pseudopotential lattice Boltzmann method

It has recently been shown that altering the shape of the metastable and unstable branches of an equation of state (EOS) can substantially improve the numerical accuracy of liquid and vapor densities in the pseudopotential lattice Boltzmann method [Peng et al., Phys. Rev. E 101, 063309 (2020)2470-004510.1103/PhysRevE.101.063309]. We found that this approach reduces stability of the method in nonequilibrium conditions and is unstable for bubbles at low reduced temperatures. Here we present an improved method for altering the shape of the metastable and unstable branches of the EOS which remains stable for both equilibrium and nonequilibrium situations and has no issues with bubbles. We also performed a detailed study of the stability of the methods for a droplet impact on a liquid film for reduced temperatures down to 0.35 with Reynolds number of 300. Our approach remained stable for a density ratio of up to 3.38×10^{4}.

A lattice Boltzmann study of particle settling in a fluctuating multicomponent fluid under confinement

We present mesoscale numerical simulations based on the coupling of the fluctuating lattice Boltzmann method for multicomponent systems with a wetted finite-size particle model. This newly coupled methodologies are used to study the motion of a spherical particle driven by a constant body force in a confined channel with a fixed square cross section. The channel is filled with a mixture of two liquids under the effect of thermal fluctuations. After some validations steps in the absence of fluctuations, we study the fluctuations in the particle's velocity at changing thermal energy, applied force, particle size, and particle wettability. The importance of fluctuations with respect to the mean settling velocity is quantitatively assessed, especially in comparison with unconfined situations. Results show that the expected effects of confinement are very well captured by the numerical simulations, wherein the confinement strongly enhances the importance of velocity fluctuations, which can be one order of magnitude larger than what expected in unconfined domains. The observed findings underscore the versatility of the proposed methodology in highlighting the effects of confinement on the motion of particles in the presence of thermal fluctuations.

Asymmetric invasion in anisotropic porous media

We report and discuss, by means of pore-scale numerical simulations, the possibility of achieving a directional-dependent two-phase flow behavior during the process of invasion of a viscous fluid into anisotropic porous media with controlled design. By customising the pore-scale morphology and heterogeneities with the adoption of anisotropic triangular pillars distributed with quenched disorder, we observe a substantially different invasion dynamics according to the direction of fluid injection relative to the medium orientation, that is depending if the triangular pillars have their apex oriented (flow aligned) or opposed (flow opposing) to the main flow direction. Three flow regimes can be observed: (i) for low values of the ratio between the macroscopic pressure drop and the characteristic pore-scale capillary threshold, i.e., for Δp_{0}/p_{c}≤1, the fluid invasion dynamics is strongly impeded and the viscous fluid is unable to reach the outlet of the medium, irrespective of the direction of injection; (ii) for intermediate values, 1<Δp_{0}/p_{c}≤2, the viscous fluid reaches the outlet only when the triangular pillars are flow-opposing oriented; (iii) for larger values, i.e., for Δp_{0}/p_{c}>2, the outlet is again reached irrespective of the direction of injection. The porous medium anisotropy induces a lower effective resistance when the pillars are flow-opposing oriented, suppressing front roughening and capillary fingering. We thus argue that the invasion process occurs as long as the pressure drop is larger then the macroscopic capillary pressure determined by the front roughness, which in the case of flow-opposing pillars is halved. We present a simple approximated model, based on Darcy's assumptions, that links the macroscopic effective permeability with the directional-dependent front roughening, to predict the asymmetric invasion dynamics. This peculiar behavior opens up the possibility of fabrication of porous capillary valves to control the flow along certain specific directions.

Simulations of surface acoustic wave interactions on a sessile droplet using a three-dimensional multiphase lattice Boltzmann model

This study reports the development of a three-dimensional numerical model for acoustic interactions with a microscale sessile droplet under surface acoustic wave (SAW) excitation using the lattice Boltzmann method (LBM). We first validate the model before SAW interactions are added. The results demonstrate good agreement with the analytical results for thermodynamic consistency, Laplace law, static contact angle on a flat surface, and droplet oscillation. We then investigate SAW interactions on the droplet, with resonant frequencies ranging 61.7-250.1 MHz. According to our findings, an increase in wave amplitude elicits an increase in streaming velocity inside the droplet, causing internal mixing, and further increase in wave amplitude leads to pumping and jetting. The boundaries of wave amplitude at various resonant frequencies are predicted for mixing, pumping, and jetting modes. The modeling predictions on the roles of forces (SAW, interfacial tension, inertia, and viscosity) on the dynamics of mixing, pumping, and jetting of a droplet are in good agreement with observations and experimental data. The model is further applied to investigate the effects of SAW substrate surface wettability, viscosity ratio, and interfacial tension on SAW actuation onto the droplet. This work demonstrates the capability of the LBM in the investigation of acoustic wave interactions between SAW and a liquid medium.

Lattice Boltzmann and Jones matrix calculations for the determination of the director field structure in self-propelling nematic droplets

Nematic droplets immersed in aqueous surfactant solutions can show a self-propelled motion induced by a Marangoni flow in the droplet surface. In addition to the self-propulsion, the Marangoni flow induces within the droplet a convective flow which considerably influences the nematic director field of the droplet. We report numerical simulations aiming at the determination of the director field in the self-propelling droplet. The convective flow and the resulting structure of director field are described by a lattice Boltzmann model. The reliability of the obtained structures is proved by subsequent Jones matrix calculations which enable the direct comparison of experimental polarizing microscopy images of self-propelling droplets with calculated images based on the simulated structures.

A unified lattice Boltzmann model and application to multiphase flows

In this work, we develop a unified lattice Boltzmann model (ULBM) framework that can seamlessly integrate the widely used lattice Boltzmann collision operators, including the Bhatnagar-Gross-Krook or single-relation-time, multiple-relaxation-time, central-moment or cascaded lattice Boltzmann method and multiple entropic operators (KBC). Such a framework clarifies the relations among the existing collision operators and greatly facilitates model comparison and development as well as coding. Importantly, any LB model or treatment constructed for a specific collision operator could be easily adopted by other operators. We demonstrate the flexibility and power of the ULBM framework through three multiphase flow problems: the rheology of an emulsion, splashing of a droplet on a liquid film and dynamics of pool boiling. Further exploration of ULBM for a wide variety of phenomena would be both realistic and beneficial, making the LBM more accessible to non-specialists. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Lattice Boltzmann simulations of drying suspensions of soft particles

The ordering of particles in the drying process of a colloidal suspension is crucial in determining the properties of the resulting film. For example, microscopic inhomogeneities can lead to the formation of cracks and defects that can deteriorate the quality of the film considerably. This type of problem is inherently multiscale and here we study it numerically, using our recently developed method for the simulation of soft polymeric capsules in multicomponent fluids. We focus on the effect of the particle softness on the film microstructure during the drying phase and how it relates to the formation of defects. We quantify the order of the particles by measuring both the Voronoi entropy and the isotropic order parameter. Surprisingly, both observables exhibit a non-monotonic behaviour when the softness of the particles is increased. We further investigate the correlation between the interparticle interaction and the change in the microstructure during the evaporation phase. We observe that the rigid particles form chain-like structures that tend to scatter into small clusters when the particle softness is increased. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Modified phase-field-based lattice Boltzmann model for incompressible multiphase flows

Based on the phase-field theory, a multiple-relaxation-time (MRT) lattice Boltzmann model is proposed for the immiscible multiphase fluids. In this model, the local Allen-Chan equation is chosen as the target equation to capture the phase interface. Unlike previous MRT schemes, an off-diagonal relaxation matrix is adopted in the present model so that the target phase-field equation can be recovered exactly without any artificial terms. To check the necessity of removing those artificial terms, comparative studies were carried out among different MRT schemes with or without correction. Results show that the artificial terms can be neglected at low March number but will cause unphysical diffusion or interface undulation instability for the relatively large March number cases. The present modified model shows superiority in reducing numerical errors by adjusting the free parameters. As the interface transport coupled to the fluid flow, a pressure-evolution lattice Boltzmann equation is adopted for hydrodynamic properties. Several benchmark cases for multiphase flow were conducted to test the validity of the present model, including the static drop test, Rayleigh-Taylor instability, and single rising bubble test. For the rising bubble simulation at high density ratios, bubble dynamics obtained by the present modified MRT lattice Boltzmann model agree well with those obtained by the FEM-based level set and FEM-based phase-field models.

Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxation-time finite-difference lattice Boltzmann method (SRT-FDLBM). Then, a simplified version of MRT-FDLBM (SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schrödinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., Appl. Math. Comput. 309, 334 (2017)10.1016/j.amc.2017.04.015], and the lattice Boltzmann method (LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

Fluid-wall interactions in pseudopotential lattice Boltzmann models

Designing proper fluid-wall interaction forces to achieve proper wetting conditions is an important area of interest in pseudopotential lattice Boltzmann models. In this paper, we propose a modified fluid-wall interaction force that applies for pseudopotential models of both single-component fluids and partially miscible multicomponent fluids, such as hydrocarbon mixtures. A reliable correlation that predicts the resulting liquid contact angle on a flat solid surface is also proposed. This correlation works well over a wide variety of pseudopotential lattice Boltzmann models and thermodynamic conditions.