Lattice Boltzmann model for simulating flows with multiple phases and components

Hydrodynamics of immiscible binary fluids with viscosity contrast: a multiparticle collision dynamics approach

We present a multiparticle collision dynamics (MPC) implementation of layered immiscible fluids A and B of different shear viscosities separated by planar interfaces. The simulated flow profile for imposed steady shear motion and the time-dependent shear stress functions are in excellent agreement with our continuum hydrodynamics results for the composite fluid. The wave-vector dependent transverse velocity auto-correlation functions (TVAF) in the bulk-fluid regions of the layers decay exponentially, and agree with those of single-phase isotropic MPC fluids. In addition, we determine the hydrodynamic mobilities of an embedded colloidal sphere moving steadily parallel or transverse to a fluid-fluid interface, as functions of the distance from the interface. The obtained mobilities are in good agreement with hydrodynamic force multipoles calculations, for a no-slip sphere moving under creeping flow conditions near a clean, ideally flat interface. The proposed MPC fluid-layer model can be straightforwardly implemented, and it is computationally very efficient. Yet, owing to the spatial discretization inherent to the MPC method, the model can not reproduce all hydrodynamic features of an ideally flat interface between immiscible fluids.

Implementation of lattice Boltzmann free-surface and shallow water models and their two-way coupling

•A detailed, practical description of a 2D lattice Boltzmann (LB) free-surface model and its coupling with a 1D LB shallow water model is provided.•A Python code is provided, that implements the Gaussian droplet benchmark of the research article (Thorimbert et al., 2019) corresponding to this method article.•Particular attention is given to the details of the free-surface implementation which, in the literature, vary among authors. These ambiguities must be addressed in order to build a reproducible scheme, as well as the exact implementation and parameters of the coupling model proposed in the associated research article.

Alternative wetting boundary condition for the chemical-potential-based free-energy lattice Boltzmann model

The free-energy lattice Boltzmann (LB) method is a multiphase LB approach based on the thermodynamic theory. Compared with traditional free-energy LB models, which employ a nonideal thermodynamic pressure tensor, the chemical-potential-based free-energy LB model has attracted much attention in recent years as it avoids computing the thermodynamic pressure tensor and its divergence. In this paper, we propose an improved wetting boundary condition for the chemical-potential-based free-energy LB model. Different from the original wetting boundary condition in the literature, the improved wetting boundary condition utilizes a surface chemical potential that is compatible with the chemical potential of the fluid domain. Accordingly, the thermodynamic consistency of the chemical-potential-based free-energy LB model can be retained by the improved wetting boundary condition. Numerical simulations are performed for droplets resting on flat and cylindrical surfaces with different contact angles. The numerical results show that the improved wetting boundary condition yields more reasonable results and the maximum spurious velocities are found to be smaller by 2 ∼ 3 orders of magnitude than those produced by the original wetting boundary condition.

Lattice Boltzmann modeling of interfacial mass transfer in a multiphase system

In the present study, a numerical model based on the lattice Boltzmann method (LBM) is proposed to simulate multiphase mass transfer, referred to as the CST-LB model. This model introduced continuum species transfer (CST) formulation by an additional collision term to model the mass transfer across the multiphase interface. The boundary condition treatment of this model is also discussed. In order to verify the applicability, the CST-LB model is combined with the pseudopotential multiphase model to simulate a series of benchmark cases, including concentration jump near the interface, gas dissolution in a closed system, species transport during drainage in a capillary tube, and multiphase species transport in the porous media. This CST-LB model can also be coupled with other multiphase LBMs since the model depends on the phase fraction field, which is not explicitly limited to specified multiphase models.

Respiratory droplets interception in fibrous porous media

We investigate, by means of pore-scale lattice Boltzmann simulations, the mechanisms of interception of respiratory droplets within fibrous porous media composing face masks. We simulate the dynamics, coalescence, and collection of droplets of the size comparable with the fiber and pore size in typical fluid-dynamic conditions that represent common expiratory events. We discern the fibrous microstructure into three categories of pores: small, large, and medium-sized pores, where we find that within the latter, the incoming droplets tend to be more likely intercepted. The size of the medium-sized pores relative to the fiber size is placed between the droplet-to-fiber size ratio and a porosity-dependent microstructural parameter L ϵ * = ϵ / ( 1 - ϵ ) , with ϵ being the porosity. In larger pores, droplets collection is instead inhibited by the small pore-throat-to-fiber size ratio that characterizes the pore perimeter, limiting their access. The efficiency of the fibrous media in intercepting droplets without compromising breathability, for a given droplet-to-fiber size ratio, can be estimated by knowing the parameter L ϵ * . We propose a simple model that predicts the average penetration of droplets into the fibrous media, showing a sublinear growth with L ϵ * . Permeability is shown also to scale well with L ϵ * but following a superlinear growth, which indicates the possibility of increasing the medium permeability at a little cost in terms of interception efficiency for high values of porosity. As a general design guideline, the results also suggest that a fibrous layer thickness relative to the fiber size should exceed the value L ϵ * in order to ensure effective droplets filtration.

Structure and isotropy of lattice pressure tensors for multirange potentials

We systematically analyze the tensorial structure of the lattice pressure tensors for a class of multiphase lattice Boltzmann models (LBM) with multirange interactions. Due to lattice discrete effects, we show that the built-in isotropy properties of the lattice interaction forces are not necessarily mirrored in the corresponding lattice pressure tensor. This finding opens a different perspective for constructing forcing schemes, achieving the desired isotropy in the lattice pressure tensors via a suitable choice of multirange potentials. As an immediate application, the obtained LBM forcing schemes are tested via numerical simulations of nonideal equilibrium interfaces and are shown to yield weaker and less spatially extended spurious currents with respect to forcing schemes obtained by forcing isotropy requirements only. From a general perspective, the proposed analysis yields an approach for implementing forcing symmetries, never explored so far in the framework of the Shan-Chen method for LBM. We argue this will be beneficial for future studies of nonideal interfaces.

Lattice Boltzmann Modeling of Drying of Porous Media Considering Contact Angle Hysteresis

Drying of porous media is governed by a combination of evaporation and movement of the liquid phase within the porous structure. Contact angle hysteresis induced by surface roughness is shown to influence multi-phase flows, such as contact line motion of droplet, phase distribution during drainage and coffee ring formed after droplet drying in constant contact radius mode. However, the influence of contact angle hysteresis on liquid drying in porous media is still an unanswered question. Lattice Boltzmann model (LBM) is an advanced numerical approach increasingly used to study phase change problems including drying. In this paper, based on a geometric formulation scheme to prescribe contact angle, we implement a contact angle hysteresis model within the framework of a two-phase pseudopotential LBM. The capability and accuracy of prescribing and automatically measuring contact angles over a large range are tested and validated by simulating droplets sitting on flat and curved surfaces. Afterward, the proposed contact angle hysteresis model is validated by modeling droplet drying on flat and curved surfaces. Then, drying of two connected capillary tubes is studied, considering the influence of different contact angle hysteresis ranges on drying dynamics. Finally, the model is applied to study drying of a dual-porosity porous medium, where phase distribution and drying rate are compared with and without contact angle hysteresis. The proposed model is shown to be capable of dealing with different contact angle hysteresis ranges accurately and of capturing the physical mechanisms during drying in different porous media including flat and curved geometries.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11242-021-01644-9.

A Novel Approach of Unit Conversion in the Lattice Boltzmann Method

The lattice Boltzmann method (LBM) is an alternative method to the conventional computational fluid dynamic (CFD) methods. It gained popularity due to its simplicity in coding and dealing with a complex fluid flow such as the multiphase flow. The method is based on the kinetic theory, which is mesoscopic scale. Hence, applying the LBM method for macroscopic problems requires a proper conversion from the physical scale (conventional units) to the mesoscopic scale (lattice units) and vice versa. The Buckingham π theorem and the principle of corresponding states are the popular methods used for data reductions and unit conversion processes in the LBM. Nevertheless, those methods have some issues, such as difficulty in converting specific quantities, such as thermo-physical properties. The current work uses a novel dimensional analysis method systematically for mapping properties’ units between scales. Moreover, the approach has the flexibility in selecting parameters to ensure the stability of the method of solution. Several benchmark examples are used to evaluate the feasibility and accuracy of the proposed approach. In conclusion, the proposed approach showed the flexibility of the mapping between meso-scale to macro-scales and vice versa on solid bases rather than ad-hoc methods.

Discovery of Dynamic Two-Phase Flow in Porous Media Using Two-Dimensional Multiphase Lattice Boltzmann Simulation

The dynamic two-phase flow in porous media was theoretically developed based on mass, momentum conservation, and fundamental constitutive relationships for simulating immiscible fluid-fluid retention behavior and seepage in the natural geomaterial. The simulation of transient two-phase flow seepage is, therefore, dependent on both the hydraulic boundaries applied and the immiscible fluid-fluid retention behavior experimentally measured. Many previous studies manifested the velocity-dependent capillary pressure–saturation relationship (Pc-S) and relative permeability (Kr-S). However, those works were experimentally conducted on a continuum scale. To discover the dynamic effects from the microscale, the Computational Fluid Dynamic (CFD) is usually adopted as a novel method. Compared to the conventional CFD methods solving Naiver–Stokes (NS) equations incorporated with the fluid phase separation schemes, the two-phase Lattice Boltzmann Method (LBM) can generate the immiscible fluid-fluid interface using the fluid-fluid/solid interactions at a microscale. Therefore, the Shan–Chen multiphase multicomponent LBM was conducted in this study to simulate the transient two-phase flow in porous media. The simulation outputs demonstrate a preferential flow path in porous media after the non-wetting phase fluid is injected until, finally, the void space is fully occupied by the non-wetting phase fluid. In addition, the inter-relationships for each pair of continuum state variables for a Representative Elementary Volume (REV) of porous media were analyzed for further exploring the dynamic nonequilibrium effects. On one hand, the simulating outcomes reconfirmed previous findings that the dynamic effects are dependent on both the transient seepage velocity and interfacial area dynamics. Nevertheless, in comparison to many previous experimental studies showing the various distances between the parallelly dynamic and static Pc-S relationships by applying various constant flux boundary conditions, this study is the first contribution showing the Pc-S striking into the nonequilibrium condition to yield dynamic nonequilibrium effects and finally returning to the equilibrium static Pc-S by applying various pressure boundary conditions. On the other hand, the flow regimes and relative permeability were discussed with this simulating results in regards to the appropriateness of neglecting inertial effects (both accelerating and convective) in multiphase hydrodynamics for a highly pervious porous media. Based on those research findings, the two-phase LBM can be demonstrated to be a powerful tool for investigating dynamic nonequilibrium effects for transient multiphase flow in porous media from the microscale to the REV scale. Finally, future investigations were proposed with discussions on the limitations of this numerical modeling method.

Kinetic-Based Multiphase Flow Simulation

Multiphase flows exhibit a large realm of complex behaviors such as bubbling, glugging, wetting, and splashing which emerge from air-water and water-solid interactions. Current fluid solvers in graphics have demonstrated remarkable success in reproducing each of these visual effects, but none have offered a model general enough to capture all of them concurrently. In contrast, computational fluid dynamics have developed very general approaches to multiphase flows, typically based on kinetic models. Yet, in both communities, there is dearth of methods that can simulate density ratios and Reynolds numbers required for the type of challenging real-life simulations that movie productions strive to digitally create, such as air-water flows. In this article, we propose a kinetic model of the coupling of the Navier-Stokes equations with a conservative phase-field equation, and provide a series of numerical improvements over existing kinetic-based approaches to offer a general multiphase flow solver. The resulting algorithm is embarrassingly parallel, conservative, far more stable than current solvers even for real-life conditions, and general enough to capture the typical multiphase flow behaviors. Various simulation results are presented, including comparisons to both previous work and real footage, to highlight the advantages of our new method.

Mesoscopic Lattice Boltzmann Modeling of the Liquid-Vapor Phase Transition

We develop a mesoscopic lattice Boltzmann model for liquid-vapor phase transition by handling the microscopic molecular interaction. The short-range molecular interaction is incorporated by recovering an equation of state for dense gases, and the long-range molecular interaction is mimicked by introducing a pairwise interaction force. Double distribution functions are employed, with the density distribution function for the mass and momentum conservation laws and an innovative total kinetic energy distribution function for the energy conservation law. The recovered mesomacroscopic governing equations are fully consistent with kinetic theory, and thermodynamic consistency is naturally satisfied.

Algorithmic augmentation in the pseudopotential-based lattice Boltzmann method for simulating the pool boiling phenomenon with high-density ratio

The pseudopotential-based lattice Boltzmann method (LBM), despite enormous potential in facilitating natural development and migration of interfaces during multiphase simulation, remains restricted to low-density ratios, owing to inherent thermodynamic inconsistency. The present paper focuses on augmenting the basic algorithm by enhancing the isotropy of the discrete equation and thermodynamic consistency of the overall formulation, to expedite simulation of pool boiling at higher-density ratios. Accordingly, modification is suggested in the discrete form of the updated interparticle interaction term, by expanding the discretization to the eighth order. The proposed amendment is successful in substantially reducing the spurious velocity in the vicinity of a static droplet, while allowing stable simulation at a much higher-density ratio under identical conditions, which is a noteworthy improvement over existing Single Relaxation Time (SRT)-LBM algorithms. Various pool boiling scenarios have been explored for a reduced temperature of 0.75, which itself is significantly lower than reported in comparable literature, in both rectangular and cylindrical domains, and also with micro- and distributed heaters. All three regimes of pool boiling have aptly been captured with both plain and structured heaters, allowing the development of the boiling curve. The predicted value of critical heat flux for the plain heater agrees with Zuber correlation within 10%, illustrating both quantitative and qualitative capability of the proposed algorithm.

Hysteresis in spreading and retraction of liquid droplets on parallel fiber rails

Wetting and spreading of liquids on fibers occur in many natural and artificial processes. Unlike on a planar substrate, a droplet attached to one or more fibers can assume several different shapes depending on geometrical parameters such as liquid volume and fiber size and distance. This paper presents lattice Boltzmann simulations of the morphology of liquid droplets on two parallel cylindrical fibers. We investigate the final shapes resulting from spreading of an initially spherical droplet deposited on the fibers and from retraction of an initial liquid column deposited between the fibers. We observe three possible equilibrium configurations: barrel-shaped droplet, droplet bridges, and liquid columns. We determine the complete morphology diagram for varying inter-fiber spacings and liquid volumes and find a region of bistability that spans both the column regime and the droplet regime. We further present a simulation protocol that allows one to probe the hysteresis of transitions between different shapes. The results provide insights into energies and forces associated with shape transformations of droplets on fibers that can be used to develop fiber-based materials and microfluidic systems for manipulation of liquids at a small scale.

Influence of Wetting on Viscous Fingering Via 2D Lattice Boltzmann Simulations

We present simulations of two-phase flow using the Rothman and Keller colour gradient Lattice Boltzmann method to study viscous fingering when a “red fluid” invades a porous model initially filled with a “blue” fluid with different viscosity. We conducted eleven suites of 81 numerical experiments totalling 891 simulations, where each suite had a different random realization of the porous model and spanned viscosity ratios in the range $$M\in [0.01,100]$$ M ∈ [ 0.01 , 100 ] and wetting angles in the range $$\theta _w\in [180^\circ ,0^\circ ]$$ θ w ∈ [ 180 ∘ , 0 ∘ ] to allow us to study the effect of these parameters on the fluid-displacement morphology and saturation at breakthrough (sweep). Although sweep often increased with wettability, this was not always so and the sweep phase space landscape, defined as the difference in saturation at a given wetting angle relative to saturation for the non-wetting case, had hills, ridges and valleys. At low viscosity ratios, flow at breakthrough is localized through narrow fingers that span the model. After breakthrough, the flow field continues to evolve and the saturation continues to increase albeit at a reduced rate, and eventually exceeds 90% for both non-wetting and wetting cases. The existence of a complicated sweep phase space at breakthrough, and continued post-breakthrough evolution suggests the hydrodynamics and sweep is a complicated function of wetting angle, viscosity ratio and time, which has major potential implications to Enhanced Oil Recovery by water flooding, and hence, on estimates of global oil reserves. Validation of these results via experiments is required to ensure they translate to field studies.

Consistent evaporation formulation for the phase-field lattice Boltzmann method

A consistent evaporation model is developed for the conservative Allen-Cahn-based phase-field lattice Boltzmann method that uses an appropriate source term to recover the advection-diffusion equation for the specific humidity. To evaluate the accuracy of the proposed scheme, simulations are conducted of a steady-state one-dimensional Stefan flow for a flat interface and a three-dimensional evaporating sessile droplet on a flat substrate for a curved interface. It is confirmed that the results for the evaporative mass flux of the Stefan flow agree well with those obtained from the analytical solution within a specific humidity range of 0.8 or less at the liquid-gas interface. For the sessile droplet case, the results for the dependence of the contact angle on the evaporative mass flux and its profile show good agreement with those obtained from the model of Hu and Larson [J. Phys. Chem. B 106, 1334 (2002)JPCBFK1520-610610.1021/jp0118322].

A Review on the Hydrodynamics of Taylor Flow in Microchannels: Experimental and Computational Studies

Taylor flow is a strategy-aimed flow to transfer conventional single-phase into a more efficient two-phase flow resulting in an enhanced momentum/heat/mass transfer rate, as well as a multitude of other advantages. To date, Taylor flow has focused on the processes involving gas–liquid and liquid–liquid two-phase systems in microchannels over a wide range of applications in biomedical, pharmaceutical, industrial, and commercial sectors. Appropriately micro-structured design is, therefore, a key consideration for equipment dealing with transport phenomena. This review paper highlights the hydrodynamic aspects of gas–liquid and liquid–liquid two-phase flows in microchannels. It covers state-of-the-art experimental and numerical methods in the literature for analyzing and simulating slug flows in circular and non-circular microchannels. The review’s main objective is to identify the considerable opportunity for further development of microflows and provide suggestions for researchers in the field. Available correlations proposed for the transition of flow patterns are presented. A review of the literature of flow regime, slug length, and pressure drop is also carried out.

Regimes of motion of magnetocapillary swimmers

The dynamics of a triangular magnetocapillary swimmer is studied using the lattice Boltzmann method. We extend on our previous work, which deals with the self-assembly and a specific type of the swimmer motion characterized by the swimmer's maximum velocity centred around the particle's inverse viscous time. Here, we identify additional regimes of motion. First, modifying the ratio of surface tension and magnetic forces allows to study the swimmer propagation in the regime of significantly lower frequencies mainly defined by the strength of the magnetocapillary potential. Second, introducing a constant magnetic contribution in each of the particles in addition to their magnetic moment induced by external fields leads to another regime characterized by strong in-plane swimmer reorientations that resemble experimental observations.

Optimal surface-tension isotropy in the Rothman-Keller color-gradient lattice Boltzmann method for multiphase flow

The Rothman-Keller color-gradient (CG) lattice Boltzmann method is a popular method to simulate two-phase flow because of its ability to deal with fluids with large viscosity contrasts and a wide range of interfacial tensions. Two fluids are labeled red and blue, and the gradient in the color difference is used to compute the effect of interfacial tension. It is well known that finite-difference errors in the color-gradient calculation lead to anisotropy of interfacial tension and errors such as spurious currents. Here, we investigate the accuracy of the CG calculation for interfaces between fluids with several radii of curvature and find that the standard CG calculations lead to significant inaccuracy. Specifically, we observe significant anisotropy of the color gradient of order 7% for high curvature of an interface such as when a pinchout occurs. We derive a second order accurate color gradient and find that the diagonal nearest neighbors can be weighted differently than in the usual color-gradient calculation such that anisotropy is minimized to a fraction of a percent. The optimal weights that minimize anisotropy for the smallest radius of curvature interface are found to be w=(0.298,0.284,0.275) for diagonal nearest neighbors for the cases of the interface smoothing parameter β=(0.5,0.7,0.99), somewhat higher than the w=0.25 value derived by Leclaire et al. [Leclaire, Reggio, and Trepanier, Computers and Fluids 48, 98 (2011)CPFLBI0045-793010.1016/j.compfluid.2011.04.001] based on obtaining isotropic errors to second order. We find that use of these optimal w values yields over a factor of 10 decrease in anisotropy and over a factor of 30 decrease in mean anisotropy relative to using the standard w=1 value. And we find a factor of about 2 decrease in the anisotropic error and up to factor 15 decrease in mean anisotropic error relative to the choice of w=0.25 for small radius of curvature interfaces. The improved CG calculations will allow the method to be more reliably applied to studies of phenomenology and pore scale processes such as viscous and capillary fingering, and droplet formation where surface-tension isotropy of narrow fingers and small droplets plays a crucial role in correctly capturing phenomenology. We present an example illustrating how different phenomena can be captured using the improved color-gradient method. Namely, we present simulations of a wetting fluid invading a fluid filled pipe where the viscosity ratio of fluids is unity in which droplets form at the transition to fingering using the improved CG calculations that are not captured using the standard CG calculations. We present an explanation of why this is so which relates to anisotropy of the surface tension, which inhibits the pinchouts needed to form droplets.

Impact of the prequench state of binary fluid mixtures on surface-directed spinodal decomposition

Using lattice Boltzmann simulations we investigate the impact of the amplitude of concentration fluctuations in binary fluid mixtures prior to demixing when in contact with a surface that is preferentially wet by one of the components. We find a bicontinuous structure near the surface for an initial, prequench state of the mixture close to the critical point where the amplitude of concentration fluctuations is large. In contrast, if the initial state of the mixture is not near the critical point and concentration fluctuations are relatively weak, then the morphology is not bicontinuous but remains layered until the very late stages of coarsening. In both cases, it is the morphology of a depletion layer rich in the nonpreferred component that dictates the growth exponent of the thickness of the fluid layer that is in direct contact with the substrate. In the early stages of demixing, we find a growth exponent consistent with a value of 1/4 for a prequench state away from the critical point, which is different from the usual diffusive scaling exponent of 1/3 that we recover for a prequench state close to the critical point. We attribute this to the structure of a depletion layer that is penetrated by tubes of the preferred fluid, connecting the wetting layer to the bulk fluid even in the early stages if the initial state is characterized by concentration fluctuations that are large in amplitude. Furthermore, we find that in the late stages of demixing the flow through these tubes results in significant in-plane concentration variations near the substrate, leading to dropletlike structures with a concentration lower than the average concentration in the wetting layer. This causes a deceleration in the growth of the wetting layer in the very late stages of the demixing. Irrespective of the prequench state of the mixture, the late stages of the demixing process produce the same growth law for the layer thickness, with a scaling exponent of unity usually associated with the impact of hydrodynamic flow fields.

Modeling ternary fluids in contact with elastic membranes

We present a thermodynamically consistent model of a ternary fluid interacting with elastic membranes. Following a free-energy modeling approach for the fluid phases, we derive the governing equations for the dynamics of the ternary fluid flow and membranes. We also provide the numerical framework for simulating such fluid-structure interaction problems. It is based on the lattice Boltzmann method for the ternary fluid (Eulerian description) and a finite difference representation of the membrane (Lagrangian description). The ternary fluid and membrane solvers are coupled through the immersed boundary method. For validation purposes, we consider the relaxation dynamics of a two-dimensional elastic capsule placed at a fluid-fluid interface. The capsule shapes, resulting from the balance of surface tension and elastic forces, are compared with equilibrium numerical solutions obtained by surface evolver. Furthermore, the Galilean invariance of the proposed model is proven. The proposed approach is versatile, allowing for the simulation of a wide range of geometries. To demonstrate this, we address the problem of a capillary bridge formed between two deformable capsules.

Effects of Cracks and Geometric Parameters on the Flow in Shale

In recent years, shale oil/gas has become increasingly important in global energy. The natural pores of shale are mainly of micro-nano sizes and have the cross-scale characteristics, which makes the traditional method difficult and impractical in studying the seepage of shale. In order to obtain the characteristics of seepage of the crack-pore-throat system, the lattice Boltzmann method and dimensional analysis were used to study the seepage in an idealized crack-pore network. The influences of the geometric factors, including crack location, crack opening, and interval between two vertical neighbor throats and boundary conditions on the seepage were studied. The results show that the slip boundary conditions enhance the seepage rate. The enhancement with slip coefficients is nonuniform. The total flux is nearly equal when the crack is near either the inlet or outlet, but larger than that when the crack is located in the middle of the model. The flux ratio between the main throats when the crack is located near the outlet is the greatest. When the crack is near the outlet, the water channel is the largest possible while it is not easy to form when the crack is in the middle. With increase in the opening ratio of the crack-to-throat, the total flow of the system increases. The increase degree decreases with the increasing opening ratio. When the opening ratio is greater than 9, the increase in flux becomes very small. If the crack-pore-throat system is very uniform or even symmetric, the flow rate in the vertical throat/crack is very small. Hence, it is not beneficial to the gas/oil production and gas/oil displacement.

Modeling Immiscible Fluid Displacement in a Porous Medium Using Lattice Boltzmann Method

The numerical investigation of the interpenetrating flow dynamics of a gas injected into a homogeneous porous media saturated with liquid is presented. The analysis is undertaken as a function of the inlet velocity, liquid–gas viscosity ratio (D) and physical properties of the porous medium, such as porous geometry and surface wettability. The study aims to improve understanding of the interaction between the physical parameters involved in complex multiphase flow in porous media (e.g., CO2 sequestration in aquifers). The numerical simulation of a gaseous phase being introduced through a 2D porous medium constructed using seven staggered columns of either circular- or square-shaped micro-obstacles mimicking the solid walls of the pores is performed using the multiphase Lattice Boltzmann Method (LBM). The gas–liquid fingering phenomenon is triggered by a small geometrical asymmetry deliberately introduced in the first column of obstacles. Our study shows that the amount of gas penetration into the porous medium depends on surface wettability and on a set of parameters such as capillary number (Ca), liquid–gas viscosity ratio (D), pore geometry and surface wettability. The results demonstrate that increasing the capillary number and the surface wettability leads to an increase in the effective gas penetration rate, disregarding porous medium configuration, while increasing the viscosity ratio decreases the penetration rate, again disregarding porous medium configuration.

Lucas-Washburn Equation-Based Modeling of Capillary-Driven Flow in Porous Systems

Fluid flow in porous systems driven by capillary pressure is one of the most ubiquitous phenomena in nature and industry, including petroleum and hydraulic engineering as well as material and life sciences. The classical Lucas-Washburn (LW) equation and its modified forms were developed and have been applied extensively to elucidate the fundamental mechanisms underlying the basic statics and dynamics of the capillary-driven flow in porous systems. The LW equation assumes that fluids are incompressible Newton ones and that capillary channels all have the same radii. This kind of hypothesis is not true for many natural situations, however, where porous systems comprise complicated pore and capillary channel structures at microscales. The LW equation therefore often leads to inaccurate capillary imbibition predictions in such situations. Numerous studies have been conducted in recent years to develop and assess the modifications and extensions of the LW equation in various porous systems. Significant progresses in computational techniques have also been attained to further improve our understanding of imbibition dynamics. A state-of-the-art review is therefore needed to summarize the recent significant models and numerical simulation techniques as well as to discuss key ongoing research topics arising from various new engineering practices. The theoretical basis of the LW equation is first introduced in this review and recent progress in mathematical models is then summarized to demonstrate the modifications and extensions of this equation to various microchannels and porous media. These include capillary tubes with nonuniform and noncircular cross sections, discrete fractures, and capillary tubes that are not straight as well as heterogeneous porous media. Numerical studies on the LW equation are also reviewed, and comments on future works and research directions for LW-based capillary-driven flows in porous systems are listed.

Digital blood in massively parallel CPU/GPU systems for the study of platelet transport

We propose a highly versatile computational framework for the simulation of cellular blood flow focusing on extreme performance without compromising accuracy or complexity. The tool couples the lattice Boltzmann solver Palabos for the simulation of blood plasma, a novel finite-element method (FEM) solver for the resolution of deformable blood cells, and an immersed boundary method for the coupling of the two phases. The design of the tool supports hybrid CPU-GPU executions (fluid, fluid-solid interaction on CPUs, deformable bodies on GPUs), and is non-intrusive, as each of the three components can be replaced in a modular way. The FEM-based kernel for solid dynamics outperforms other FEM solvers and its performance is comparable to state-of-the-art mass-spring systems. We perform an exhaustive performance analysis on Piz Daint at the Swiss National Supercomputing Centre and provide case studies focused on platelet transport, implicitly validating the accuracy of our tool. The tests show that this versatile framework combines unprecedented accuracy with massive performance, rendering it suitable for upcoming exascale architectures.

Lattice Boltzmann Solver for Multiphase Flows: Application to High Weber and Reynolds Numbers

The lattice Boltzmann method, now widely used for a variety of applications, has also been extended to model multiphase flows through different formulations. While already applied to many different configurations in low Weber and Reynolds number regimes, applications to higher Weber/Reynolds numbers or larger density/viscosity ratios are still the topic of active research. In this study, through a combination of a decoupled phase-field formulation—the conservative Allen–Cahn equation—and a cumulant-based collision operator for a low-Mach pressure-based flow solver, we present an algorithm that can be used for higher Reynolds/Weber numbers. The algorithm was validated through a variety of test cases, starting with the Rayleigh–Taylor instability in both 2D and 3D, followed by the impact of a droplet on a liquid sheet. In all simulations, the solver correctly captured the flow dynamics andmatched reference results very well. As the final test case, the solver was used to model droplet splashing on a thin liquid sheet in 3D with a density ratio of 1000 and kinematic viscosity ratio of 15, matching the water/air system at We = 8000 and Re = 1000. Results showed that the solver correctly captured the fingering instabilities at the crown rim and their subsequent breakup, in agreement with experimental and numerical observations reported in the literature.

Revisiting the Homogenized Lattice Boltzmann Method with Applications on Particulate Flows

The simulation of surface resolved particles is a valuable tool to gain more insights in the behaviour of particulate flows in engineering processes. In this work the homogenized lattice Boltzmann method as one approach for such direct numerical simulations is revisited and validated for different scenarios. Those include a 3D case of a settling sphere for various Reynolds numbers. On the basis of this dynamic case, different algorithms for the calculation of the momentum exchange between fluid and particle are evaluated along with different forcing schemes. The result is an updated version of the method, which is in good agreement with the benchmark values based on simulations and experiments. The method is then applied for the investigation of the tubular pinch effect discovered by Segré and Silberberg and the simulation of hindered settling. For the latter, the computational domain is equipped with periodic boundaries for both fluid and particles. The results are compared to the model by Richardson and Zaki and are found to be in good agreement. As no explicit contact treatment is applied, this leads to the assumption of sufficient momentum transfer between particles via the surrounding fluid. The implementations are based on the open-source C++ lattice Boltzmann library OpenLB.

Electroosmotic flow: From microfluidics to nanofluidics

Electroosmotic flow (EOF), a consequence of an imposed electric field onto an electrolyte solution in the tangential direction of a charged surface, has emerged as an important phenomenon in electrokinetic transport at the micro/nanoscale. Because of their ability to efficiently pump liquids in miniaturized systems without incorporating any mechanical parts, electroosmotic methods for fluid pumping have been adopted in versatile applications—from biotechnology to environmental science. To understand the electrokinetic pumping mechanism, it is crucial to identify the role of an ionically polarized layer, the so‐called electrical double layer (EDL), which forms in the vicinity of a charged solid–liquid interface, as well as the characteristic length scale of the conducting media. Therefore, in this tutorial review, we summarize the development of electrical double layer models from a historical point of view to elucidate the interplay and configuration of water molecules and ions in the vicinity of a solid–liquid interface. Moreover, we discuss the physicochemical phenomena owing to the interaction of electrical double layer when the characteristic length of the conducting media is decreased from the microscale to the nanoscale. Finally, we highlight the pioneering studies and the most recent works on electro osmotic flow devoted to both theoretical and experimental aspects.

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.

Force methods for the two-relaxation-times lattice Boltzmann

The two-relaxation-times collision benefits the steady lattice Boltzmann method by yielding viscosity-independent numerical errors. We present in an intuitive way how to incorporate popular force methods into the two-relaxation-times collision. We subsequently rewrite force methods into a generic equation to reveal commonalities and differences. We prove that force methods with a second-order velocity moment of the force break the viscosity independence. A force method with only a first-order velocity moment of the force averts this breakage. We validate our proof numerically.

Inhomogeneous Distribution of Polytetrafluorethylene in Gas Diffusion Layers of Polymer Electrolyte Fuel Cells

Polymer electrolyte fuel cells require gas diffusion layers that can efficiently distribute the feeding gases from the channel structure to the catalyst layer on both sides of the membrane. On the cathode side, these layers must also allow the transport of liquid product water in a counter flow direction from the catalyst layer to the air channels where it can be blown away by the air flow. In this study, two-phase transport in the fibrous structures of a gas diffusion layer was simulated using the lattice Boltzmann method. Liquid water transport is affected by the hydrophilic treatment of the fibers. Following the assumption that polytetrafluorethylene is preferably concentrated at the crossings of fibers, the impact of its spatial distribution is analyzed. Both homogeneous and inhomogeneous distribution is investigated. The concentration of polytetrafluorethylene in the upstream region is of advantage for the fast transport of liquid water through the gas diffusion layer. Special attention is given to the topmost fiber layer. Moreover, polytetrafluorethylene covering the fibers leads to large contact angles.

Optimal Wetting Angles in Lattice Boltzmann Simulations of Viscous Fingering

We conduct pore-scale simulations of two-phase flow using the 2D Rothman–Keller colour gradient lattice Boltzmann method to study the effect of wettability on saturation at breakthrough (sweep) when the injected fluid first passes through the right boundary of the model. We performed a suite of 189 simulations in which a “red” fluid is injected at the left side of a 2D porous model that is initially saturated with a “blue” fluid spanning viscosity ratios$$M = \nu _\mathrm{r}/\nu _\mathrm{b} \in [0.001,100]$$M=νr/νb∈[0.001,100]and wetting angles$$\theta _\mathrm{w} \in [ 0^\circ ,180^\circ ]$$θw∈[0∘,180∘]. As expected, at low-viscosity ratios$$M=\nu _\mathrm{r}/\nu _\mathrm{b} \ll 1$$M=νr/νb≪1we observe viscous fingering in which narrow tendrils of the red fluid span the model, and for high-viscosity ratios$$M \gg 1$$M≫1, we observe stable displacement. The viscous finger morphology is affected by the wetting angle with a tendency for more rounded fingers when the injected fluid is wetting. However, rather than the expected result of increased saturation with increasing wettability, we observe a complex saturation landscape at breakthrough as a function of viscosity ratio and wetting angle that contains hills and valleys with specific wetting angles at given viscosity ratios that maximize sweep. This unexpected result that sweep does not necessarily increase with wettability has major implications to enhanced oil recovery and suggests that the dynamics of multiphase flow in porous media has a complex relationship with the geometry of the medium and the hydrodynamical parameters.

Achieving thermodynamic consistency in a class of free-energy multiphase lattice Boltzmann models

The free-energy lattice Boltzmann (LB) model is one of the major multiphase models in the LB community. The present study is focused on a class of free-energy LB models in which the divergence of thermodynamic pressure tensor or its equivalent form expressed by the chemical potential is incorporated into the LB equation via a forcing term. Although this class of free-energy LB models may be thermodynamically consistent at the continuum level, it suffers from thermodynamic inconsistency at the discrete lattice level owing to numerical errors [Guo et al., Phys. Rev. E 83, 036707 (2010)10.1103/PhysRevE.83.036707]. The numerical error term mainly includes two parts: one comes from the discrete gradient operator and the other can be identified in a high-order Chapman-Enskog analysis. In this paper, we propose an improved scheme to eliminate the thermodynamic inconsistency of the aforementioned class of free-energy LB models. The improved scheme is constructed by modifying the equation of state of the standard LB equation, through which the discretization of ∇(ρc_{s}^{2}) is no longer involved in the force calculation and then the numerical errors can be significantly reduced. Numerical simulations are subsequently performed to validate the proposed scheme. The numerical results show that the improved scheme is capable of eliminating the thermodynamic inconsistency and can significantly reduce the spurious currents in comparison with the standard forcing-based free-energy LB model.

Investigating the effect of porosity on the soil water retention curve using the multiphase Lattice Boltzmann Method

The soil water retention curve (SWRC) is the most commonly used relationship in the study of unsaturated soil. In this paper, the effect of porosity on the SWRC is investigated by numerically modeling unsaturated soil using the Shan-Chen multiphase Lattice Boltzmann Method. The shape of simulated SWRCs are compared against that predicted by the van Genuchten model, demonstrating a good fit except at low degrees of saturation. The simulated SWRCs show an increase in the air-entry value as porosity decreases.

Macroscopic Lattice Boltzmann Method

The lattice Boltzmann method (LBM) is a highly simplified model for fluid flows using a few limited fictitious particles. It has been developed into a very efficient and flexible alternative numerical method in computational physics, demonstrating its great power and potential for resolving more and more challenging physical problems in science and engineering covering a wide range of disciplines such as physics, chemistry, biology, material science and image analysis. The LBM is implemented through the two routine steps of streaming and collision using the three parameters of the lattice size, particle speed and collision operator. A fundamental question is if the two steps are integral to the method or if the three parameters can be reduced to one for a minimal lattice Boltzmann method. In this paper, it is shown that the collision step can be removed and the standard LBM can be reformulated into a simple macroscopic lattice Boltzmann method (MacLAB). This model relies on macroscopic physical variables only and is completely defined by one basic parameter of the lattice size δx, bringing the LBM into a precise “lattice” Boltzmann method. The viscous effect on flows is naturally embedded through the particle speed, making it an ideal automatic simulator for fluid flows. Three additional advantages compared to the existing LBMs are that: (i) physical variables can directly be retained as the boundary conditions; (ii) much less computational memory is required; and (iii) the model is unconditionally stable. The findings are demonstrated and confirmed with numerical tests including flows that are independent of and dependent on fluid viscosity, 2D and 3D cavity flows and an unsteady Taylor–Green vortex flow. This provides an efficient and powerful model for resolving physical problems in various disciplines of science and engineering.

Capillary interactions between soft capsules protruding through thin fluid films

When a suspension dries, the suspending fluid evaporates, leaving behind a dry film composed of the suspended particles. During the final stages of drying, the height of the fluid film on the substrate drops below the particle size, inducing local interface deformations that lead to strong capillary interactions among the particles. Although capillary interactions between rigid particles are well studied, much is still to be understood about the behaviour of soft particles and the role of their softness during the final stages of film drying. Here, we use our recently-introduced numerical method that couples a fluid described using the lattice Boltzmann approach to a finite element description of deformable objects to investigate the drying process of a film with suspended soft particles. Our measured menisci deformations and lateral capillary forces, which agree well with previous theoretical and experimental works in case of rigid particles, show that the deformations become smaller with increasing particle softness, resulting in weaker lateral interaction forces. At large interparticle distances, the force approaches that of rigid particles. Finally, we investigate the time dependent formation of particle clusters at the late stages of the film drying.

Simulation of collapsing cavitation bubbles in various liquids by lattice Boltzmann model coupled with the Redlich-Kwong-Soave equation of state

A computational technique based on the pseudo-potential multiphase lattice Boltzmann method (LBM) is employed to investigate the collapse dynamics of cavitation bubbles of various liquids in the vicinity of the solid surface with different wettability conditions. The Redlich-Kwong-Soave equation of state (EoS) that includes an acentric factor is incorporated to consider the physical properties of water (H_{2}O), liquid nitrogen (LN_{2}), and liquid hydrogen (LH_{2}) in the present simulations. Accuracy and performance of the present multiphase LBM are examined by simulation of the homogenous and heterogeneous cavitation phenomena. The good agreement of the results obtained based on the present solution algorithm in comparison with the available data confirms the validity and capability of the multiphase LBM employed. Then, the cavitation bubble collapse near the solid wall is studied by considering the H_{2}O, LN_{2}, and LH_{2} fluids, and the wettability effect of the surface on the collapse dynamics is investigated. The obtained results demonstrate that the collapse phenomenon for the H_{2}O is more aggressive than that of the LH_{2} and LN_{2}. The cavitation bubble of the water has a shorter collapse time with an intense liquid jet, while the collapse process in the LN_{2} takes a longer time due to the larger radius of its bubble at the rebound. Also, this study demonstrates that the increment of the hydrophobicity of the wall causes less energy absorption by the solid surface from the liquid phase around the bubble that leads to form a liquid jet with higher kinetic energy. Therefore, the bubble collapse process occurs more quickly for hydrophobic surfaces, regardless of the fluids considered. The present study shows that the pseudopotential LBM with incorporating an appropriate EoS and a robust forcing scheme is an efficient numerical technique for simulation of the dynamics of the cavitation bubble collapse in different fluids.

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

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

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

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

Equilibrium Orientation and Adsorption of an Ellipsoidal Janus Particle at a Fluid–Fluid Interface

We investigate the equilibrium orientation and adsorption process of a single, ellipsoidal Janus particle at a fluid–fluid interface. The particle surface comprises equally sized parts that are hydrophobic or hydrophilic. We present free energy models to predict the equilibrium orientation and compare the theoretical predictions with lattice Boltzmann simulations. We find that the deformation of the fluid interface strongly influences the equilibrium orientation of the Janus ellipsoid. The adsorption process of the Janus ellipsoid can lead to different final orientations determined by the interplay of particle aspect ratio and particle wettablity contrast.