Free-energy Lattice Boltzmann Method Research Articles

Study of single-component two-phase free energy lattice Boltzmann models using various equations of state

In single-component two-phase flow simulations by the lattice Boltzmann method (LBM), one key topic of interest is the equation of state (EOS), which has been studied more in the pseudo-potential approach but much less in the free energy approach. Here, we conducted fair assessment of several popular non-ideal gas EOS in free energy LBM. The inability to independently tune the surface tension and interface thickness was identified for existing free energy formulation using most EOS. A simple modification was proposed and incorporated into three free energy LBM models, including the pressure tensor model, Lee–Fischer's model, and Guo's well-balanced model. Good compatibility with all of them was demonstrated by numerical simulations. In addition, a piecewise-linear (PWL) EOS proposed in pseudo-potential LBM was found to be also feasible in free energy LBM. Extensive numerical experiments on diverse problems (some containing contact lines) revealed that the influence of EOS was rather limited when key parameters are nearly the same (easily achievable with the proposed modification). The capability of the Lee–Fischer and well-balanced models to reduce the spurious currents down to machine accuracy (for static drop tests) was found to be independent of the EOS (including the customized PWL EOS). Overall, the pressure tensor LBM is associated with larger spurious currents and deviations in the density and chemical potential, thus less accurate than the other two. However, it is less sensitive to the relaxation parameter and grid number, thus more robust.

Binary fluid flow simulations with free energy lattice Boltzmann methods

We use free energy lattice Boltzmann methods to simulate shear and extensional flows of a binary fluid in two and three dimensions. To this end, two classical configurations are digitally twinned, namely a parallel-band device for binary shear flow and a four-roller apparatus for binary extensional flow. The free energy lattice Boltzmann method and the test cases are implemented in the open-source parallel C++ framework OpenLB and evaluated for several non-dimensional numbers. Characteristic deformations are captured, where breakup mechanisms occur for critical capillary regimes. Though the known mass leakage for small droplet-domain ratios and large Cahn numbers is observed, suitable mesh sizes show good agreement to analytical predictions and reference results.

An accelerated process-based method for the accurate computation of relative permeability from direct simulations of two-phase flow on micro-computed tomography images of porous media

Pore-scale forces have significant effects on the macroscopic behavior of multi-phase flow through porous media. We develop a robust and accurate accelerated process-based method for the computation of relative permeability from direct simulations of pore-scale two-phase flow on micro-computed tomography images. In the pressure drop calculation, we take advantage of an existing analysis that establishes a relationship between pore-scale forces and Darcy-scale pressure drops using an energy-conservation approach. We establish a thermodynamically consistent approximation of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each flowing phase for the first time within the context of a free-energy lattice Boltzmann method (LBM). In addition, we propose and test a new computationally efficient partial-mirror periodic boundary condition for a fully coupled visco-capillary pore-scale flow simulator based on a free-energy LBM. The new boundary condition is imposed only in the main flow direction and significantly reduces the computational cost of the process-based relative-permeability computation protocol at a small compromise on accuracy.We first compute primary-drainage and subsequent imbibition relative permeability curves for a reservoir sandstone rock sample. We use this real-reservoir dataset to validate the pressure drop computation method and the partial-mirror periodic boundary condition. We then simulate the entire drainage and imbibition cycle using an extensively studied Berea sandstone dataset. We quantitatively demonstrate that pore-scale direct numerical simulation-based relative permeability curves computed with our novel process-based method agree well with experimental steady-state relative permeability measurements. We also quantitatively demonstrate that the new partial-mirror periodic boundary condition accelerates the relative permeability computations 4 to 13 times.

Numerical simulation of a drying colloidal suspension on a wettable substrate using the lattice Boltzmann method

A computational method based on the lattice Boltzmann method (LBM) is developed to simulate a drying colloidal suspension on a wettable substrate. In this method, solid particles moving in the fluid are treated with an improved smoothed-profile LBM, and the liquid–vapor system is described by the free-energy LBM. To represent solvent evaporation dominated by density gradients, the density at the boundary of the simulation domain is specified to be lower than the equilibrium vapor density. We simulated the drying of a colloidal suspension on a wettable substrate and found that particles with higher wettability aggregated more slowly, but the aggregation proceeded more significantly. This result can be explained based on capillary interactions (i.e., capillary flotation force and capillary immersion force) between the particles. The proposed method can effectively simulate the particle arrangement process induced by the evaporation of the solvent from a colloidal suspension on a wettable substrate.

Dynamic behavior investigation of capillary rising at various dominant forces using free energy lattice Boltzmann method

Dynamic behavior investigation of capillary rising at various dominant forces using free energy lattice Boltzmann method

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.

A new criterion of coalescence-induced microbubble detachment in three-dimensional microfluidic channel

This work is motivated by an experiment of microbubble transport in a polymer microfluidic gas generation device where coalescence-induced detachment exhibits. We numerically study three-dimensional microbubble coalescence using the graphics processing unit accelerating free energy lattice Boltzmann method with cubic polynomial boundary conditions. The focus is on the coalescence-induced microbubble detachment (CIMD) in microfluidics. From the experimental observation, we identified that size inequality between two-parent bubbles and the size of the father (large) bubble are key factors to determine if a CIMD will occur. First, the analytical relationship between equilibrium contact angle and dimensionless wetting potential and experimental results of coalescence with and without CIMD are employed for the verification and validation, respectively. From eighteen experimental and computational cases, we derive a new criterion for CIMD: CIMD occurs when the two-parent bubbles are (nearly) equal with a relatively large radius. The underlying mechanism behind this criterion is explored by the time evolution of the velocity vector field, vorticity field, and kinetic energy in the entire coalescence. It is found that the symmetric capillary force drives the formation of vertical flow stream to the horizontal alignment of parent bubbles and the blockage of the downward stream due to the solid interface promotes the intensity of the upward stream. Meanwhile, large-sized parent bubbles transfer a large amount of kinetic energy from the initial free surface energy, which is essential to lead a CIMD in the post-coalescence stage. Such a new criterion is expected to impact the design and optimization of microfluidics in various applications.

Single drop breakup in turbulent flow

The breakup process of a single drop in homogeneous isotropic turbulence was studied using direct numerical simulations. A diffuse interface free energy lattice Boltzmann method was applied. The detailed visualization of the breakup process confirmed breakup mechanisms previously outlined such as initial, independent, and cascade breakups. High‐resolution simulations allowed to visualize another drop breakup mechanism, burst breakup, which occurs when the mother drop has a large volume, and the flow is highly turbulent. The simulations indicate that the type of the breakup mechanism is a strong function of mother drop size and energy input. Large mother drops in highly turbulent flow fields are more likely to burst, producing a large number of drops of the size close to the Kolmogorov length scale. Small drops in moderate turbulence tend to break only once (initial breakup). The interfacial energy of a drop was tracked as a function of time during drop deformation and breakage. The maximum energy level of the deformed mother drop was compared to commonly used estimates of critical energy necessary to break a drop. Our results show that these reference levels of critical energy are usually underestimated. Moreover, in some cases even if the critical energy level was exceeded, the drop did not break because the time of the interaction between the drop and the eddies was not enough to finish the breakup. The numerical insight presented here can be used as a guideline for the selection of assumptions and simplifications behind breakup kernels.

Liquid drop breakup in homogeneous isotropic turbulence

Purpose This paper aims to demonstrate the capabilities of a diffuse interface free energy lattice Boltzmann method to perform direct numerical simulations of liquid–liquid dispersions in a well-controlled turbulent environment. The goal of this research study is to develop numerical techniques that can visualize and quantify drop interaction with the turbulent vortices. The obtained information will be used for the development of sub-models of drop breakup for multi-scale simulations. Design/methodology/approach A pure binary liquid system is considered that is subject to fully developed statistically stationary turbulent flow field in a cubic fully periodic box with the edge size of 300 lattice units. Three turbulent flow fields with varying energy input are examined and their coherent structures are visualized using a normalized Q-criterion. The evolution of the liquid–liquid interface is tracked as a function of time. The detailed explanation of the numerical method is provided with a highlight on a choice of the numerical parameters. Findings Drop breakup mechanisms differ depending on energy input. Drops break due to interaction with the vortices. Quantification of turbulent structures shows that the size of vortices increases with the decrease of energy input. Drop interacts simultaneously with multiple vortices of the size comparable to or smaller than the drop size. Vortices of the size smaller than the drop size disturb drop interface and pinch off the satellites. Vortices of the size comparable to the drop size tend to elongate the drop and tear it apart producing daughter drops and satellites. Addition of the second phase enhances turbulent dissipation at the high wavenumbers. To obtain physically realistic two-phase energy spectra, the multiple-relaxation-time collision operator should be used. Originality/value Detailed information of drop breakup in the turbulent flow field is crucial for the development of drop breakup sub-models that are necessary for multi-scale numerical simulations. The improvement of numerical methods that can provide these data and produce reliable results is important. This work made one step towards a better understanding of how drops interact with the turbulent vortices.

On the interaction between bubbles and the free surface with high density ratio 3D lattice Boltzmann method

The bubbles rise up and burst at the free surface is a complex two-phase process. A free energy lattice Boltzmann method (LBM) model is adopted in this paper to study this phenomenon. The interface capturing technique [Zheng et al., 2006] is used to deal with the high density ratio problem. The Laplace law and the air-water interface capturing ability are validated for the multiphase model. The interaction between the single bubble or multiple bubbles and the free surface are studied by the multiphase model. The force acting on the bubble and the evolution of the free surface is studied. Meanwhile, effect of the initial distance between two adjacent bubbles on interaction effects of multiple bubbles is investigated as well.

Effects of initial conditions on the coalescence of micro-bubbles

The effects of initial conditions on the coalescence of two equal-sized air micro-bubbles ( R0) in water are studied using the lattice Boltzmann method. The focus is on effects of two initial set-ups of parent bubbles, separated by a small distance d and connected with a neck bridge radius r0, on the neck bridge growth at the early stage of the bubble coalescence. A sophisticated free energy lattice Boltzmann method model based on the Cahn-Hilliard diffuse interface approach is employed. This lattice Boltzmann method model has been demonstrated suitable for handling a large density ratio of two fluids up to 1000 and capable of minimizing the nonphysical spurious current. In both initial scenarios, the neck bridge evolution exhibits a half power-law scaling, [Formula: see text] after a development time. The half power-law agrees with the recent analytical prediction and experimental results. It has been found that smaller initial separation distance or smaller initial neck bridge radius results in faster growth of neck bridge and bubble coalescence, which is similar to the effects of these two initial scenarios on droplet coalescence. The physical mechanism behind each behavior has been explored. For the initial connected case, faster neck growth and longer development time corresponding to smaller initial neck radius is due to the significant bias between the capillary forces contributed by the meniscus curvature and the neck bridge curvature, whereas in the case of initial separated scenario, faster growth and shorter development time corresponding to shorter separation distance is due to the formation of elongated neck bridge. The prefactor A0 that represents the growth of neck bridge radius at the characteristic time ti captured in each case is in good agreement with the experimental results.

Lattice-Boltzmann flow simulation of an oil-in-water emulsion through a coalescing filter: Effects of filter structure

The permeation of an oil-in-water (O/W) emulsion through a coalescing filter was numerically studied using the lattice Boltzmann method (LBM). A numerical simulation model for the coalescing phenomena was developed based on the free-energy LBM. We investigated the effects of the wettability of fibers, filter porosity, and fiber diameter on the coalescing behaviors by performing two-dimensional permeation simulations for the O/W emulsions through modeled fibrous filters. We mainly focused on hydrophilic filters because they did not generate small secondary droplets during oil droplet detachment from the filter, and this is preferred for precise separation of oil and water. Our simulations demonstrated that filters with larger pore spacings enable formation of larger droplets but allow more droplets to pass without coalescing. To solve this problem, we designed bi-layered filters composed of a small-pore filter to accurately catch the droplets and a large-pore filter to enlarge the droplets; we demonstrated the effectiveness of the bilayer structure for membrane coalescence.

Self-propelled dropwise condensation on a gradient surface

A model of vapor condensation on a solid surface is developed and numerically analyzed using the free-energy lattice Boltzmann method. Based on the model, the condensation phase change on hydrophobic, hydrophilic and gradient surfaces are simulated with a particular focus on the condensation on a gradient surface. The droplet nucleation, growth, deformation, coalescence and motion during the condensation on a gradient surface are investigated. The present simulation reproduces the self-propelled dropwise condensation on a gradient surface, the film condensation on a hydrophilic surface and the conventional dropwise condensation on a hydrophobic surface. The results indicate that the condensed droplets on a gradient surface can be swept in time to provide a favorable condition for the subsequent condensation. On a smooth gradient surface, owing to the unbalanced wetting force, the vapor condenses into a thin film firstly and then fractures into droplet nucleation as the condensation process goes on. The larger wettability gradient results in a larger amplitude oscillation of condensation rate and a slighter variation of surface coverage.

Investigating the effect of interfacial tension and contact angle on capillary pressure curve, using free energy Lattice Boltzmann Method

Two phase flow of gas and oil through a porous medium was simulated by Free Energy model of Lattice Boltzmann Method. Constant pressure boundary condition for Free Energy model was derived and then, capillary pressure curve was determined for high density ratio flow (gas and oil flow) through porous media. A validation test was carried out to calculate pore size distribution from determined capillary pressure curve. Good agreement between this pore size distribution and real pore size distribution of porous medium was seen. Finally, the effect of interfacial tension and contact angle on shape of capillary pressure curve was investigated. Comparison of these results with experimental results of literature in this field shows good consistency between trends of capillary pressure curves. The results confirm that Free Energy lattice Boltzmann method is capable of determining acceptable capillary pressure curve for gas and oil flow in porous media which is supposedly the first usage of Free Energy model for this objective.

Evaluation the Radius Dependency of Surface Tension in Nano-Droplets by a Diffuse-Interface Lattice Boltzmann

In this paper, the dependency of surface tension and droplet radius is investigated by determination of pressure jump across the interface of a nano-scale droplet. For this goal, a gasliquid system consists of a nano-droplet and surrounding vapour is modelled using a diffuseinterface, free energy lattice Boltzmann method. The density and pressure distribution inside and outside the droplet can be obtained from LBM. Consequently, the surface tension is calculated by Laplace equation. Surface tension is found initially for carbon dioxide and then the results are extended for different materials and conditions. Interface thickness can be calculated as a function of material properties and conditions. The results show that, when the ratio of droplet radius to interface thickness (R* = R / h) is greater than 8.0, the surface tension is not a function of radius and it will be equal to the flat surface tension (constant regime). But if this ratio be less than 6.0, the surface tension decreases with radius in a linear pattern (linear regime). It is also shown that for a wide range of properties and conditions the quantitative behaviour of surface tension relative to droplet radius is the same.

Effects of dispersed phase viscosity on drop deformation and breakup in inertial shear flow

The deformation and breakup of a single liquid drop subjected to simple shear flow is studied numerically using a diffuse interface free energy lattice Boltzmann method. The effect of dispersed phase viscosity on the behavior of the drop at a drop Reynolds number Re=10 is investigated over the range of viscosity ratios λ=0.1−2 (dispersed phase viscosity over continuous phase viscosity) with a focus on λ<1. For every λ the critical capillary number Cac for breakup is determined. For the range of λ considered, Cac decreases as λ increases. Both the extent of deformation and the breakup mechanism depend on the viscosity ratio and the capillary number. At the highest subcritical capillary number, the drop becomes less elongated and more inclined towards the vertical axis as the viscosity ratio increases. The changes in the drop breakup process are examined as the capillary number increases from the lowest supercritical Ca~Cac, to 1.2, 1.5 and 2Cac. Drops break by the end-pinching mechanism, except for λ=2 at Ca=2Cac where the drop undergoes capillary wave breakup.

Lattice Boltzmann simulations of drop deformation and breakup in shear flow

The behavior of a single liquid drop suspended in another liquid and subjected to simple shear flow is studied numerically using a diffuse interface free energy lattice Boltzmann method. The system is fully defined by three physical, and two numerical dimensionless numbers: a Reynolds number Re, a capillary number Ca, the viscosity ratio λ, an interface-related Peclet number Pe, and the ratio of interface thickness and drop size (the Cahn number Ch). The influence of Pe,Ch and mesh resolution on accuracy and stability of the simulations is investigated. Drops of moderate resolution (radius less than 30 lattice units) require smaller interface thickness, while a thicker interface should be used for highly resolved drops. The Peclet number is controlled by the mobility coefficient Γ. Based on the results, the simulations are stable when Γ is in the range 1–15. In addition, the numerical tool is verified and validated in a wide range of physical conditions: Re=0.0625-50,λ=1,2,3 and a capillary number range over which drops deform and break. Good agreement with literature data is observed.

Lattice Boltzmann simulations of a single n-butanol drop rising in water

The motion of an n-butanol drop in water under the influence of gravity was numerically studied using a diffuse interface free energy lattice Boltzmann method. A pure two-liquid system without mass transfer between the phases was considered. A range of drop diameters of 1.0–4.0 mm covered the flow conditions. Most calculations were carried out in a moving reference frame. This allowed studying of long-term drop behavior in a relatively small computational domain. The capability of the method to capture the drop shape especially in the oscillating regime was demonstrated. For each drop diameter the evolution of the drop velocity in time, the terminal rise velocity and drop's shape were determined. The results were compared to experimental and numerical results and to semi-empirical correlations. The deviation of the simulated terminal velocity from other results is within 5% for smaller drops and up to 20% for large oscillating drops. It was shown that beyond the onset of shape oscillations the binary system converges towards a constant capillary number of 0.056.

Boundary condition considerations in lattice Boltzmann formulations of wetting binary fluids

We propose a new lattice Boltzmann numerical scheme for binary-fluid surface interactions. The new scheme combines the existing binary free energy lattice Boltzmann method [Swift et al., Phys. Rev. E 54 (1996)] and a new wetting boundary condition for diffuse interface methods in order to eliminate spurious variations in the order parameter at solid surfaces. We use a cubic form for the surface free energy density and also take into account the contribution from free energy in the volume when discretizing the wetting boundary condition. This allows us to eliminate the spurious variation in the order parameter seen in previous implementations. With the new scheme a larger range of equilibrium contact angles are possible to reproduce and capillary intrusion can be simulated at higher accuracy at lower resolution.

Lattice Boltzmann simulations of water transport in gas diffusion layer of a polymer electrolyte membrane fuel cell

The effect of wettability on water transport dynamics in gas diffusion layer (GDL) is investigated by simulating water invasion in an initially gas-filled GDL using the multiphase free-energy lattice Boltzmann method (LBM). The results show that wettability plays a significant role on water saturation distribution in two-phase flow in the uniform wetting GDL. For highly hydrophobicity, the water transport falls in the regime of capillary fingering, while for neutral wettability, water transport exhibits the characteristic of stable displacement, although both processes are capillary force dominated flow with same capillary numbers. In addition, the introduction of hydrophilic paths in the GDL leads the water to flow through the hydrophilic pores preferentially. The resulting water saturation distributions show that the saturation in the GDL has little change after water breaks through the GDL, and further confirm that the selective introduction of hydrophilic passages in the GDL would facilitate the removal of liquid water more effectively, thus alleviating the flooding in catalyst layer (CL) and GDL. The LBM approach presented in this study provides an effective tool to investigate water transport phenomenon in the GDL at pore-scale level with wettability distribution taken into consideration.