Eliminating parasitic currents in the lattice Boltzmann equation method for nonideal gases

Color-gradient-based phase-field equation for multiphase flow

In this paper, the underlying problem with the color-gradient (CG) method in handling density-contrast fluids is explored. It is shown that the CG method is not fluid invariant. Based on nondimensionalizing the CG method, a phase-field interface-capturing model is proposed which tackles the difficulty of handling density-contrast fluids. The proposed formulation is developed for incompressible, immiscible two-fluid flows without phase-change phenomena, and a solver based on the lattice Boltzmann method is proposed. Coupled with an available robust hydrodynamic solver, a binary fluid flow package that handles fluid flows with high density and viscosity contrasts is presented. The macroscopic and lattice Boltzmann equivalents of the formulation, which make the physical interpretation of it easier, are presented. In contrast to existing color-gradient models where the interface-capturing equations are coupled with the hydrodynamic ones and include the surface tension forces, the proposed formulation is in the same spirit as the other phase-field models such as the Cahn-Hilliard and the Allen-Cahn equations and is solely employed to capture the interface advected due to a flow velocity. As such, similarly to other phase-field models, a so-called mobility parameter comes into play. In contrast, the mobility is not related to the density field but a constant coefficient. This leads to a formulation that avoids individual speed of sound for the different fluids. On the lattice Boltzmann solver side, two separate distribution functions are adopted to solve the formulation, and another one is employed to solve the Navier-Stokes equations, yielding a total of three equations. Two series of numerical tests are conducted to validate the accuracy and stability of the model, where we compare simulated results with available analytical and numerical solutions, and good agreement is observed. In the first set the interfacial evolution equations are assessed, while in the second set the hydrodynamic effects are taken into account.

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

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

General wetting energy boundary condition in a fully explicit nonideal fluids solver

We present an explicit finite-difference method to simulate the nonideal multiphase fluid flow. The local density and momentum transport are modeled by the Navier-Stokes equations and the pressure is computed by the van der Waals equation of the state. The static droplet and the dynamics of liquid-vapor separation simulations are performed as validations of this numerical scheme. In particular, to maintain the thermodynamic consistency, we propose a general wetting energy boundary condition at the contact line between fluids and the solid boundary. We conduct a series of comparisons between the current boundary condition and the constant contact angle boundary condition as well as the stress-balanced boundary condition. This boundary condition alleviates the instability induced by the constant contact angle boundary condition at θ≈0 and θ≈π. Using this boundary condition, the equilibrium contact angle is correctly recovered and the contact line dynamics are consistent with the simulation by applying a stress-balanced boundary condition. Nevertheless, unlike the stress-balanced boundary condition for which we need to further introduce the interface thickness parameter, the current boundary condition implicitly incorporates the interface thickness information into the wetting energy.

Phase-field lattice Boltzmann equation for wettable particle fluid dynamics

In this paper a phase-field based lattice Boltzmann equation (LBE) is developed to simulate wettable particles fluid dynamics together with the smoothed-profile method (SPM). In this model the evolution of a fluid-fluid interface is captured by the conservative Allen-Cahn equation (CACE) LBE, and the flow field is solved by a classical incompressible LBE. The solid particle is represent by SPM, and the fluid-solid interaction force is calculated by direct force method. Some benchmark tests including a single wettable particle trapped at the fluid-fluid interface without gravity, capillary interactions between two wettable particles under gravity, and sinking of a horizontal cylinder through an air-water interface are carried out to validate present CACE LBE for fluid-fluid-solid flows. Raft sinking of multiple horizontal cylinders (up to five cylinders) through an air-water interface is further investigated with the present CACE LBE, and a nontrivial dynamics with an unusual nonmonotonic motion of the multiple cylinders is observed in the vertical plane. Numerical results show that the predictions by the present LBE are in good agreement with theoretical solutions and experimental data.

Electrowetting lattice Boltzmann method for micro- and nano-droplet manipulations

Electrowetting has become a widely used tool for manipulating tiny amounts of liquids on surfaces. This paper proposes an electrowetting lattice Boltzmann method for manipulating micro-nano droplets. The hydrodynamics with the nonideal effect is modeled by the chemical-potential multiphase model, in which the phase transition and equilibrium are directly driven by chemical potential. For electrostatics, droplets in the micro-nano scale cannot be considered as equipotential as macroscopic droplets due to the Debye screening effect. Therefore, we linearly discretize the continuous Poisson-Boltzmann equation in a Cartesian coordinate system, and the electric potential distribution is stabilized by iterative computations. The electric potential distribution of droplets at different scales suggests that the electric field can still penetrate micro-nano droplets even with the screening effect. The accuracy of the numerical method is verified by simulating the static equilibrium of the droplet under the applied voltage, and the results show the apparent contact angles agree very well with the Lippmann-Young equation. The microscopic contact angles present some obvious deviations due to the sharp decrease of electric field strength near the three-phase contact point. These are consistent with previously reported experimental and theoretical analyses. Then, the droplet migrations on different electrode structures are simulated, and the results show that droplet speed can be stabilized more quickly due to the more uniform force on the droplet in the closed symmetric electrode structure. Finally, the electrowetting multiphase model is applied to study the lateral rebound of droplets impacting on the electrically heterogeneous surface. The electrostatic force prevents the droplets from contracting on the side which is applied voltage, resulting in the lateral rebound and transport toward the side.

Diffuse interface model for a single-component liquid-vapor system

We elucidate the theoretical relationships among fundamental physical concepts that are involved in the diffuse interface modeling for an isothermal single-component liquid-vapor system, which cover both the equation of state (EOS) and the surface tension force. As an example, a flat surface at equilibrium is discussed both theoretically and numerically by using two different approaches. Particularly, the force structure in the transition region is clearly presented, which demonstrates that the capillary contributions due to the density gradients can suppress the mechanical instability of the thermodynamic pressure and lead to constant hydrodynamic pressure (and chemical potential). Then, by comparing with the van der Waals (vdW) EOS for a flat interface at equilibrium, it is shown that applying the double-well approximation can give qualitative predictions for relatively high density ratio (ρ_{l}/ρ_{g}=7.784) and satisfactory results for relatively low density ratio (ρ_{l}/ρ_{g}=1.774). The main cause for this observation is attributed to the nonlinear variation of the generalized coefficient function in the double-well formulation at different density ratios. In addition, for the latter case, we simulate a droplet impact on a hydrophilic wall by using a recently proposed well-balanced discrete unified gas kinetic scheme (WB-DUGKS), which justifies the applicability of the double-well approximation to complex interfacial dynamics in the low-density-ratio limit. Furthermore, the reason for the inconsistency between the coefficients of the mean-field force expressions in the existing literature is explained.

Interaction between a rising bubble and a stationary droplet immersed in a liquid pool using a ternary conservative phase-field lattice Boltzmann method

When a stationary bubble and a stationary droplet immersed in a liquid pool are brought into contact, they form a bubble-droplet aggregate. Its equilibrium morphology and stability largely depend on the combination of different components' surface tensions, known as the "spreading factor." In this study, we look at the interaction between a rising bubble and a stationary droplet to better understand the dynamics of coalescence and rising and morphological changes for the bubble-droplet aggregate. A systematic study is conducted on the interaction processes with various bubble sizes and spreading factors in two dimensions. The current simulation framework consists of the ternary conservative phase-field lattice Boltzmann method (LBM) for interface tracking and the velocity-pressure LBM for hydrodynamics, which is validated by benchmark cases such as the liquid lens and parasitic currents around a static droplet with several popular surface tension formulations. We further test our LBM for the morphology changes of two droplets initially in contact with various spreading factors and depict the final morphologies in a phase diagram. The separated, partially engulfed, and completely engulfed morphologies can be replicated by systematically altering the sign of the spreading factors. The rising bubble and stationary droplet interaction are simulated based on the final morphologies obtained under stationary conditions by imposing an imaginary buoyancy force on the rising bubble. The results indicate that the bubble-droplet aggregate with double emulsion morphology can minimize the distortion of the bubble-droplet aggregate and achieve a greater terminal velocity than the aggregate with partially engulfed morphology.

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.

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

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

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.

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.

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.

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.

Pinning-depinning of the contact line during drop evaporation on textured surfaces: A lattice Boltzmann study

The evaporation of the liquid droplet on a structured surface is numerically investigated using the lattice Boltzmann method. Simulations are carried out for different contact angles and pillar widths. From the simulation for the Cassie state, it is found that the evaporation starts in a pinned contact line mode. Then, when the droplet reaches the receding state, the contact line jumps to the neighboring pillar. Also, the depinning force decreases with increasing the contact angle or the pillar width. In the Wenzel state, the droplet contact line remains on the initial pillar for all of its lifetime.

Impact of collision models on the physical properties and the stability of lattice Boltzmann methods

The lattice Boltzmann method (LBM) is known to suffer from stability issues when the collision model relies on the BGK approximation, especially in the zero viscosity limit and for non-vanishing Mach numbers. To tackle this problem, two kinds of solutions were proposed in the literature. They consist in changing either the numerical discretization (finite-volume, finite-difference, spectral-element, etc.) of the discrete velocity Boltzmann equation (DVBE), or the collision model. In this work, the latter solution is investigated in detail. More precisely, we propose a comprehensive comparison of (static relaxation time based) collision models, in terms of stability, and with preliminary results on their accuracy, for the simulation of isothermal high-Reynolds number flows in the (weakly) compressible regime. It starts by investigating the possible impact of collision models on the macroscopic behaviour of stream-and-collide based D2Q9-LBMs, which clarifies the exact physical properties of collision models on LBMs. It is followed by extensive linear and numerical stability analyses, supplemented with an accuracy study based on the transport of vortical structures over long distances. In order to draw conclusions as generally as possible, the most common moment spaces (raw, central, Hermite, central Hermite and cumulant), as well as regularized approaches, are considered for the comparative studies. LBMs based on dynamic collision mechanisms (entropic collision, subgrid-scale models, explicit filtering, etc.) are also briefly discussed. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Reduction-consistent phase-field lattice Boltzmann equation for N immiscible incompressible fluids

In this paper, we develop a reduction-consistent conservative phase-field method for interface-capturing among N (N≥ 2) immiscible fluids, which is governed by conservative Allen-Cahn equation (CACE); here the reduction-consistent property is that if only M (1≤M≤N-1) immiscible fluids are present in a N-phase system, the governing equations for N immiscible fluids must reduce to the corresponding M immiscible fluids system. Then we propose a reduction-consistent lattice Boltzmann equation (LBE) method for solving N immiscible incompressible fluids with high density and viscosity contrasts. Some numerical simulations are carried out to validate the present LBE such as stationary droplets, spreading of a liquid lens, and spinodal decomposition together with the reduction-consistent property, and the numerical results predicted by present LBE are in good agreement with the analytical solutions/other numerical results, which also demonstrate the reduction-consistent property by present LBE.

General power-law temporal scaling for unequal-size microbubble coalescence

We systematically study the effects of liquid viscosity, liquid density, and surface tension on global microbubble coalescence using lattice Boltzmann simulation. The liquid-gas system is characterized by Ohnesorge number Oh≡η_{h}/sqrt[ρ_{h}σr_{F}] with η_{h},ρ_{h},σ, and r_{F} being viscosity and density of liquid, surface tension, and the radius of the larger parent bubble, respectively. This study focuses on the microbubble coalescence without oscillation in an Oh range between 0.5 and 1.0. The global coalescence time is defined as the time period from initially two parent bubbles touching to finally one child bubble when its half-vertical axis reaches above 99% of the bubble radius. Comprehensive graphics processing unit parallelization, convergence check, and validation are carried out to ensure the physical accuracy and computational efficiency. From 138 simulations of 23 cases, we derive and validate a general power-law temporal scaling T^{*}=A_{0}γ^{-n}, that correlates the normalized global coalescence time (T^{*}) with size inequality (γ) of initial parent bubbles. We found that the prefactor A_{0} is linear to Oh in the full considered Oh range, whereas the power index n is linear to Oh when Oh<0.66 and remains constant when Oh>0.66. The physical insights of the coalescence behavior are explored. Such a general temporal scaling of global microbubble coalescence on size inequality may provide useful guidance for the design, development, and optimization of microfluidic systems for various applications.

Numerical simulation of dissolved air flotation using a lattice Boltzmann method

In this paper, the behavior of a bubble and droplet rising in a system, namely, a dissolved air flotation system, is investigated under different conditions. A lattice Boltzmann model which is based on the Cahn-Hilliard equations for ternary flows is implemented. This model can handle high density and viscosity ratios, remove parasitic currents, and capture partial and total spreading conditions. Two classical problems, such as spreading of a liquid lens and the Rayleigh-Taylor instability are used to determine the accuracy of the model. As a practical application, three-component flow in a tank is studied and the dynamics of bubble and droplet under different conditions is investigated. We then concentrate on the dimensionless average velocity and locations of bubble and droplet at different density ratios, viscosity ratios, and diameter ratios. Also, total spreading and partial spreading conditions are compared. The numerical results are justifiable physically and show the ability of this model to simulate three-component flows.

Lattice Boltzmann model for weakly compressible flows

We present an energy conserving lattice Boltzmann model based on a crystallographic lattice for simulation of weakly compressible flows. The theoretical requirements and the methodology to construct such a model are discussed. We demonstrate that the model recovers the isentropic sound speed in addition to the effects of viscous heating and heat flux dynamics. Several test cases for acoustics and thermal and thermoacoustic flows are simulated to show the accuracy of the proposed model.

Multiphase flows of N immiscible incompressible fluids: Conservative Allen-Cahn equation and lattice Boltzmann equation method

In this paper, we develop a conservative phase-field method for interface-capturing among N (N≥2) immiscible fluids, the evolution of the fluid-fluid interface is captured by conservative Allen-Cahn equation (CACE), and the interface force of N immiscible fluids is incorporated to Navier-Stokes equation (NSE) by chemical potential form. Accordingly, we propose a lattice Boltzmann equation (LBE) method for solving N (N≥2) immiscible incompressible NSE and CACE at high density and viscosity contrasts. Numerical simulations including stationary droplets, Rayleigh-Taylor instability, spreading of liquid lenses, and spinodal decompositions are carried out to show the accuracy and capability of present LBE, and the results show that the predictions by use of the present LBE agree well with the analytical solutions and/or other numerical results.

Thermodynamic consistency of a pseudopotential lattice Boltzmann fluid with interface curvature

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

Spontaneous shrinkage of droplet on a wetting surface in the phase-field model

Phase-field theory is widely used to model multiphase flow. The fact that a drop can shrink or grow spontaneously due to the redistribution of interface and bulk energies to minimize the system energy may produce ill effects on the simulation. In this Rapid Communication, the spontaneous behavior of a drop on a partially wetting surface is investigated. It is found that there exists a critical radius dependent on the contact angle, the domain size, and the interface width, below which the drop will eventually disappear. In particular, the critical radius can be very large when the surface becomes very hydrophilic. The theoretical prediction of the critical radius is verified numerically by simulating a drop on a surface with various contact angles, the domain sizes, and the interface widths.

Phase-field-theory-based lattice Boltzmann equation method for N immiscible incompressible fluids

From the phase field theory, we develop a lattice Boltzmann equation (LBE) method for N (N≥2) immiscible incompressible fluids, and the Cahn-Hilliard equation, which could capture the interfaces between different phases, is also solved by LBE for an N-phase system. In this model, the interface force of N immiscible incompressible fluids is incorporated by chemical potential form, and the fluid-fluid surface tensions could be directly calculated and independently tuned. Numerical simulations including two stationary droplets, spreading of a liquid lens with and without gravity and two immiscible liquid lenses, and phase separation are conducted to validate the present LBE, and numerical results show that the predictions by LBE agree well with the analytical solutions and other numerical results.

Hydrodynamic behavior of the pseudopotential lattice Boltzmann method for interfacial flows

The lattice Boltzmann method (LBM) is routinely employed in the simulation of complex multiphase flows comprising bulk phases separated by nonideal interfaces. The LBM is intrinsically mesoscale with a hydrodynamic equivalence popularly set by the Chapman-Enskog analysis, requiring that fields slowly vary in space and time. The latter assumptions become questionable close to interfaces where the method is also known to be affected by spurious nonhydrodynamical contributions. This calls for quantitative hydrodynamical checks. In this paper, we analyze the hydrodynamic behavior of the LBM pseudopotential models for the problem of the breakup of a liquid ligament triggered by the Plateau-Rayleigh instability. Simulations are performed at fixed interface thickness, while increasing the ligament radius, i.e., in the "sharp interface" limit. The influence of different LBM collision operators is also assessed. We find that different distributions of spurious currents along the interface may change the outcome of the pseudopotential model simulations quite sensibly, which suggests that a proper fine-tuning of pseudopotential models in time-dependent problems is needed before the utilization in concrete applications. Taken all together, we argue that the results of the proposed paper provide a valuable insight for engineering pseudopotential model applications involving the hydrodynamics of liquid jets.

Liquid membrane catalytic model of hydrolyzing cellulose into 5-hydroxymethylfurfural based on the lattice Boltzmann method

Conversion of cellulose to 5-hydroxymethylfurfural (HMF) is an important means of biomass utilization.

Phase-field lattice Boltzmann modeling of boiling using a sharp-interface energy solver

The main objective of this paper is to extend an isothermal incompressible two-phase lattice Boltzmann equation method to model liquid-vapor phase change problems using a sharp-interface energy solver. Two discrete particle distribution functions, one for the continuity equation and the other for the pressure evolution and momentum equations, are considered in the current model. The sharp-interface macroscopic internal energy equation is discretized with an isotropic finite difference method to find temperature distribution in the system. The mass flow generated at liquid-vapor phase interface is embedded in the pressure evolution equation. The sharp-interface treatment of internal energy equation helps to find the interfacial mass flow rate accurately where no free parameter is needed in the calculations. The proposed model is verified against available theoretical solutions of the two-phase Stefan problem and the two-phase sucking interface problem, with which our simulation results are in good agreement. The liquid droplet evaporation in a superheated vapor, the vapor bubble growth in a superheated liquid, and the vapor bubble rising in a superheated liquid are analyzed and underlying physical characteristics are discussed in detail. The model is successfully tested for the liquid-vapor phase change with large density ratio up to 1000.

Analysis of force treatment in the pseudopotential lattice Boltzmann equation method

In this paper, different force treatments are analyzed in detail for a pseudopotential lattice Boltzmann equation (LBE), and the contribution of third-order error terms to pressure tensor with a force scheme is analyzed by a higher-order Chapman-Enskog expansion technique. From the theoretical analysis, the performance of the original force treatment of Shan-Chen (SC), Ladd, Guo et al., and the exact difference method (EDM) are ɛ_{Ladd}<ɛ_{Guo}<ɛ_{EDM}≤ɛ_{SC} with the relaxation time τ≥1, while ɛ_{Ladd}<ɛ_{Guo}<ɛ_{SC}<ɛ_{EDM} with τ<1; here ɛ is a parameter related to the mechanical stability and the subscripts are the corresponding force scheme. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to show that the predictions of modified force treatment of the pseudopotential LBE are all in good agreement with the analytical solution and other predictions.

Shear viscosity of a two-dimensional emulsion of drops using a multiple-relaxation-time-step lattice Boltzmann method

An extended Benzi-Dellar lattice Boltzmann equation scheme [R. Benzi, S. Succi, and M. Vergassola, Europhys. Lett. 13, 727 (1990)EULEEJ0295-507510.1209/0295-5075/13/8/010; R. Benzi, S. Succi, and M. Vergassola, Phys. Rep. 222, 145 (1992)PRPLCM0370-157310.1016/0370-1573(92)90090-M; P. J. Dellar, Phys. Rev. E 65, 036309 (2002)1063-651X10.1103/PhysRevE.65.036309] is developed and applied to the problem of confirming, at low Re and drop fluid concentration, c, the variation of effective shear viscosity, η_{eff}=η_{1}[1+f(η_{1},η_{2})c], with respect to c for a sheared, two-dimensional, initially crystalline emulsion [here η_{1} (η_{2}) is the fluid (drop fluid) shear viscosity]. Data obtained with our enhanced multicomponent lattice Boltzmann method, using average shear stress and hydrodynamic dissipation, agree well once appropriate corrections to Landau's volume average shear stress [L. Landau and E. M. Lifshitz, Fluid Mechanics, 6th ed. (Pergamon, London, 1966)] are applied. Simulation results also confirm the expected form for f(η_{i},η_{2}), and they provide a reasonable estimate of its parameters. Most significantly, perhaps, the generality of our data supports the validity of Taylor's disputed simplification [G. I. Taylor, Proc. R. Soc. London, Ser. A 138, 133 (1932)1364-502110.1098/rspa.1932.0175] to reduce the effect of one hydrodynamic boundary condition (on the continuity of the normal contraction of stress) to an assumption that interfacial tension is sufficiently strong to maintain a spherical drop shape.

Pseudopotential lattice Boltzmann equation method for two-phase flow: A higher-order Chapmann-Enskog expansion

In this paper, a higher order Chapmann-Enskog expansion technique is applied to pseudopotential lattice Boltzmann equation (LBE), and the contribution of third order error terms to pressure tensor is analyzed in detail. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to validate the LBE, and the results show that the predictions of the present LBE agree well with the analytical solution and other predictions.

Cascaded lattice Boltzmann method with improved forcing scheme for large-density-ratio multiphase flow at high Reynolds and Weber numbers

A recently developed forcing scheme has allowed the pseudopotential multiphase lattice Boltzmann method to correctly reproduce coexistence curves, while expanding its range to lower surface tensions and arbitrarily high density ratios [Lycett-Brown and Luo, Phys. Rev. E 91, 023305 (2015)PLEEE81539-375510.1103/PhysRevE.91.023305]. Here, a third-order Chapman-Enskog analysis is used to extend this result from the single-relaxation-time collision operator, to a multiple-relaxation-time cascaded collision operator, whose additional relaxation rates allow a significant increase in stability. Numerical results confirm that the proposed scheme enables almost independent control of density ratio, surface tension, interface width, viscosity, and the additional relaxation rates of the cascaded collision operator. This allows simulation of large density ratio flows at simultaneously high Reynolds and Weber numbers, which is demonstrated through binary collisions of water droplets in air (with density ratio up to 1000, Reynolds number 6200 and Weber number 440). This model represents a significant improvement in multiphase flow simulation by the pseudopotential lattice Boltzmann method in which real-world parameters are finally achievable.

Simulation of two-phase liquid-vapor flows using a high-order compact finite-difference lattice Boltzmann method

A high-order compact finite-difference lattice Boltzmann method (CFDLBM) is extended and applied to accurately simulate two-phase liquid-vapor flows with high density ratios. Herein, the He-Shan-Doolen-type lattice Boltzmann multiphase model is used and the spatial derivatives in the resulting equations are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient two-phase flow solver. A high-order spectral-type low-pass compact nonlinear filter is used to regularize the numerical solution and remove spurious waves generated by flow nonlinearities in smooth regions and at the same time to remove the numerical oscillations in the interfacial region between the two phases. Three discontinuity-detecting sensors for properly switching between a second-order and a higher-order filter are applied and assessed. It is shown that the filtering technique used can be conveniently adopted to reduce the spurious numerical effects and improve the numerical stability of the CFDLBM implemented. A sensitivity study is also conducted to evaluate the effects of grid size and the filtering procedure implemented on the accuracy and performance of the solution. The accuracy and efficiency of the proposed solution procedure based on the compact finite-difference LBM are examined by solving different two-phase systems. Five test cases considered herein for validating the results of the two-phase flows are an equilibrium state of a planar interface in a liquid-vapor system, a droplet suspended in the gaseous phase, a liquid droplet located between two parallel wettable surfaces, the coalescence of two droplets, and a phase separation in a liquid-vapor system at different conditions. Numerical results are also presented for the coexistence curve and the verification of the Laplace law. Results obtained are in good agreement with the analytical solutions and also the numerical results reported in the literature. The study shows that the present solution methodology is robust, efficient, and accurate for solving two-phase liquid-vapor flow problems even at high density ratios.

Coalescence of bubbles and drops in an outer fluid

When two liquid drops touch, a microscopic connecting liquid bridge forms and rapidly grows as the two drops merge into one. Whereas coalescence has been thoroughly studied when drops coalesce in vacuum or air, many important situations involve coalescence in a dense surrounding fluid, such as oil coalescence in brine. Here we study the merging of gas bubbles and liquid drops in an external fluid. Our data indicate that the flows occur over much larger length scales in the outer fluid than inside the drops themselves. Thus, we find that the asymptotic early regime is always dominated by the viscosity of the drops, independent of the external fluid. A phase diagram showing the crossovers into the different possible late-time dynamics identifies a dimensionless number that signifies when the external viscosity can be important.

Improved forcing scheme in pseudopotential lattice Boltzmann methods for multiphase flow at arbitrarily high density ratios

The pseudopotential lattice Boltzmann method has been widely used to simulate many multiphase flow applications. However, there still exist problems with reproducing realistic values of density ratio and surface tension. In this study, a higher-order analysis of a general forcing term is derived. A forcing scheme is then constructed for the pseudopotential method that is able to accurately reproduce the full range of coexistence curves. As a result, multiphase flow of arbitrarily high density ratios independent of the surface tension can be simulated. Furthermore, the interface width can be tuned to allow for grid refinement and systematic error reduction. Numerical results confirm that the proposed scheme enables independent control of density ratio, surface tension, and interface width simultaneously.

Interface-capturing lattice Boltzmann equation model for two-phase flows

In this work, an interface-capturing lattice Boltzmann equation (LBE) model is proposed for two-phase flows. In the model, a Lax-Wendroff propagation scheme and a properly chosen equilibrium distribution function are employed. The Lax-Wendroff scheme is used to provide an adjustable Courant-Friedrichs-Lewy (CFL) number, and the equilibrium distribution is presented to remove the dependence of the relaxation time on the CFL number. As a result, the interface can be captured accurately by decreasing the CFL number. A theoretical expression is derived for the chemical potential gradient by solving the LBE directly for a two-phase system with a flat interface. The result shows that the gradient of the chemical potential is proportional to the square of the CFL number, which explains why the proposed model is able to capture the interface naturally with a small CFL number, and why large interface error exists in the standard LBE model. Numerical tests, including a one-dimensional flat interface problem, a two-dimensional circular droplet problem, and a three-dimensional spherical droplet problem, demonstrate that the proposed LBE model performs well and can capture a sharp interface with a suitable CFL number.

Lattice Boltzmann equation method for the Cahn-Hilliard equation

In this paper a lattice Boltzmann equation (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard equation (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010)]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results.

Consistent lattice Boltzmann equations for phase transitions

Unlike conventional computational fluid dynamics methods, the lattice Boltzmann method (LBM) describes the dynamic behavior of fluids in a mesoscopic scale based on discrete forms of kinetic equations. In this scale, complex macroscopic phenomena like the formation and collapse of interfaces can be naturally described as related to source terms incorporated into the kinetic equations. In this context, a novel athermal lattice Boltzmann scheme for the simulation of phase transition is proposed. The continuous kinetic model obtained from the Liouville equation using the mean-field interaction force approach is shown to be consistent with diffuse interface model using the Helmholtz free energy. Density profiles, interface thickness, and surface tension are analytically derived for a plane liquid-vapor interface. A discrete form of the kinetic equation is then obtained by applying the quadrature method based on prescribed abscissas together with a third-order scheme for the discretization of the streaming or advection term in the Boltzmann equation. Spatial derivatives in the source terms are approximated with high-order schemes. The numerical validation of the method is performed by measuring the speed of sound as well as by retrieving the coexistence curve and the interface density profiles. The appearance of spurious currents near the interface is investigated. The simulations are performed with the equations of state of Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, and Carnahan-Starling.

Consistent simulation of droplet evaporation based on the phase-field multiphase lattice Boltzmann method

In the present article, we extend and generalize our previous article [H. Safari, M. H. Rahimian, and M. Krafczyk, Phys. Rev. E 88, 013304 (2013)] to include the gradient of the vapor concentration at the liquid-vapor interface as the driving force for vaporization allowing the evaporation from the phase interface to work for arbitrary temperatures. The lattice Boltzmann phase-field multiphase modeling approach with a suitable source term, accounting for the effect of the phase change on the velocity field, is used to solve the two-phase flow field. The modified convective Cahn-Hilliard equation is employed to reconstruct the dynamics of the interface topology. The coupling between the vapor concentration and temperature field at the interface is modeled by the well-known Clausius-Clapeyron correlation. Numerous validation tests including one-dimensional and two-dimensional cases are carried out to demonstrate the consistency of the presented model. Results show that the model is able to predict the flow features around and inside an evaporating droplet quantitatively in quiescent as well as convective environments.

Timescales of emulsion formation caused by anisotropic particles

Particle stabilized emulsions have received much interest in the recent past, but our understanding of the dynamics of emulsion formation is still limited. For simple spherical particles, the time dependent growth of fluid domains is dominated by the formation of droplets, particle adsorption and coalescence of droplets (Ostwald ripening), which eventually can be almost fully blocked due to the presence of the particles. Ellipsoidal particles are known to be more efficient stabilizers of fluid interfaces than spherical particles and their anisotropic shape and the related additional rotational degrees of freedom have an impact on the dynamics of emulsion formation. In this paper, we investigate this point by means of simple model systems consisting of a single ellipsoidal particle or a particle ensemble at a flat interface as well as a particle ensemble at a spherical interface. By applying combined multicomponent lattice Boltzmann and molecular dynamics simulations we demonstrate that the anisotropic shape of ellipsoidal particles causes two additional timescales to be of relevance in the dynamics of emulsion formation: a relatively short timescale can be attributed to the adsorption of single particles and the involved rotation of particles towards the interface. As soon as the interface is jammed, however, capillary interactions between the particles cause a local reordering on very long timescales leading to a continuous change in the interface configuration and increase of the interfacial area. This effect can be utilized to counteract the thermodynamic instability of particle stabilized emulsions and thus offers the possibility to produce emulsions with exceptional stability.

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

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

High-accuracy approximation of high-rank derivatives: isotropic finite differences based on lattice-Boltzmann stencils

We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are derived directly from this basis. Furthermore, in this formal development, we extend the theory of Hermite polynomial tensors in the case of discrete spaces and present expressions for the discrete inner products between monomials and Hermite polynomial tensors. In addition, we prove an equivalency between two approaches for constructing lattice-Boltzmann stencils. For the numerical verification of the presented finite differences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils.

Free-energy-based lattice Boltzmann model for the simulation of multiphase flows with density contrast

A free-energy-based phase-field lattice Boltzmann method is proposed in this work to simulate multiphase flows with density contrast. The present method is to improve the Zheng-Shu-Chew (ZSC) model [Zheng, Shu, and Chew, J. Comput. Phys. 218, 353 (2006)] for correct consideration of density contrast in the momentum equation. The original ZSC model uses the particle distribution function in the lattice Boltzmann equation (LBE) for the mean density and momentum, which cannot properly consider the effect of local density variation in the momentum equation. To correctly consider it, the particle distribution function in the LBE must be for the local density and momentum. However, when the LBE of such distribution function is solved, it will encounter a severe numerical instability. To overcome this difficulty, a transformation, which is similar to the one used in the Lee-Lin (LL) model [Lee and Lin, J. Comput. Phys. 206, 16 (2005)] is introduced in this work to change the particle distribution function for the local density and momentum into that for the mean density and momentum. As a result, the present model still uses the particle distribution function for the mean density and momentum, and in the meantime, considers the effect of local density variation in the LBE as a forcing term. Numerical examples demonstrate that both the present model and the LL model can correctly simulate multiphase flows with density contrast, and the present model has an obvious improvement over the ZSC model in terms of solution accuracy. In terms of computational time, the present model is less efficient than the ZSC model, but is much more efficient than the LL model.

Shrinkage of bubbles and drops in the lattice Boltzmann equation method for nonideal gases

One characteristic of multiphase lattice Boltzmann equation (LBE) methods is that the interfacial region has a finite (i.e., noninfinitesimal) thickness known as a diffuse interface. In simulations of, e.g., bubble or drop dynamics, for problems involving nonideal gases, one frequently observes that the diffuse interface method produces a spontaneous, nonphysical shrinkage of the bubble or drop radius. In this paper, we analyze in detail a single-fluid two-phase model and use a LBE model for nonideal gases in order to explain this fundamental problem. For simplicity, we only investigate the static bubble or droplet problem. We find that the method indeed produces a density shift, bubble or droplet shrinkage, as well as a critical radius below which the bubble or droplet eventually vanishes. Assuming that the ratio between the interface thickness D and the initial bubble or droplet radius r0 is small, we analytically show the existence of this density shift, bubble or droplet radius shrinkage, and critical bubble or droplet survival radius. Numerical results confirm our analysis. We also consider droplets on a solid surface with different curvatures, contact angles, and initial droplet volumes. Numerical results show that the curvature, contact angle, and the initial droplet volume have an effect on this spontaneous shrinkage process, consistent with the survival criterion.

Multicomponent lattice Boltzmann equation method with a discontinuous hydrodynamic interface

In the multicomponent lattice Boltzmann equation simulation method (MCLB), applied to the continuum regime of fluid flow, the finite width of the fluid-fluid interface introduces unphysical scales. We present a practical, robust, computationally efficient, and easy to implement solution to this problem which needs only low order interpolation to be stable and accurate and is applicable to any MCLB variant which uses a continuous phase field to distinguish between immiscible fluids with arrested coalescence. Our method extends the ideas of Kim and Pitsch, [Phys. Fluids 19, 108101 (2007)] and uses no external force distribution whatsoever to generate continuum interfacial physics, i.e., the Laplace law and no traction conditions on interfacial stresses. As such, it is amenable to the simplest form of Chapman-Enskog analysis used for lattice Boltzmann models. We assess our method and proceed to compare key results obtained with it against other equivalent data, obtained using the established continuum regime MCLB technique based upon the work of Lishchuk, Care, and Halliday, [Phys. Rev. E 67, 036701 (2003)] and Halliday, Hollis, and Care, [Phys. Rev. E 76, 026708 (2007)], quantifying performance in terms of the minimum feasible capillary available to simulation using that technique.

Approach and coalescence of liquid drops in air

The coalescence of liquid drops has conventionally been thought to have just two regimes when the drops are brought together slowly in vacuum or air: a viscous regime corresponding to the Stokes-flow limit and a later inertially dominated regime. Recent work [Proc. Natl. Acad. Sci. 109, 6857 (2012)] found that the Stokes-flow limit cannot be reached in the early moments of coalescence, because the inertia of the drops cannot be neglected then. Instead, the drops are described by an "inertially limited viscous" regime, where surface tension, inertia, and viscous forces all balance. The dynamics continue in this regime until either viscosity or inertia dominate on their own. I use an ultrafast electrical method and high-speed imaging to provide a detailed description of coalescence near the moment of contact for drops that approach at low speed and coalesce as undeformed spheres. These measurements support a description of coalescence having three regimes. Signatures both before and after contact identify a threshold approach speed for deformation of the drops by the ambient gas.