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Pool boiling heat transfer enhancement on structured surfaces with columns is investigated by a thermal two-phase lattice Boltzmann model. The effects of geometrical parameters, including column height (H), width (W) and gap spacing (D), are discussed in detail. The size of columns is comparable to the diameter of a typical detached bubble ( d b ). It is found that the heat transfer performance mainly depends on two factors: the heated surface area A and local convective flow field. When W and D are properly chosen as about W = D ≈ 4 d b , increasing H enhances the heat transfer solely due to the increase of the surface area. When W and D are small, it is observed that the top of the columns and the channels are mostly covered by a layer of vapor, respectively, which weaken the convection. Under the circumstances, although surface area A increases, heat transfer enhancement is not so significant. For the surface tension effect, the enhancement of the structured surface decreases with the increase of capillary number compared to that of the plain surface. The mechanism is that at larger capillary number, the convection close to the channel may be partially blocked, which hinders the heat transfer. An optimal enhancement could be achieved when A is as large as possible and meanwhile the convection is not hindered by the bubbles behaviors. The finding here may shed some light on mechanism of heat transfer enhancement on structured surfaces with columns.

The effective viscosities of dilute and semi-dilute suspensions in a two-dimensional shear flow are studied using the lattice Boltzmann method. The suspensions contain non-Brownian hard circular buoyant porous particles. Here a more accurate formula for intrinsic viscosity as a function of Darcy number (Da) for the whole Da regime is proposed through our numerical result. The effects of fluid inertia, permeability of the particle, and confinement of the bounding walls are investigated. It is found that for the cases with a small Da, the effective viscosity significantly increases with confinement and fluid inertia. However, for the cases with a large Da, the confinement ratio and fluid inertia have very minor effect. Moreover, for semi-dilute suspensions, the permeability of the particle weakens the effect of the hydrodynamic interactions between particles on the relative viscosity ηr and makes ηr decrease. The above phenomena can be well understood through quantifying the disturbance of the porous particle to the flow.

We develop a semi-analytical model to calculate the entry pressure in rough capillaries based on an energy balance principle and wetting theory on rough surfaces. During the drainage process, we assume fluids form arc menisci at corners of capillary cross-sections depending on the apparent angle, and that there is a wetting phase film adsorbed on the rough surface depending on the wetting condition. A Logistic model is proposed to explain the contact angle dependence on the structure of rough surfaces. Then the capillary entry pressure is calculated by extended MS-P (Mayer, Stowe and Princen) method. We verify the model using the pseudopotential Lattice Boltzmann model and obtain good agreement between the analytical and simulated pressures. Taking capillaries with triangular sections with pillar pillar-type rough surfaces as example, we discuss how rough structures and contact angles influence capillary behaviors. The results reveal that both the wetting phase saturation and entry pressure of rough capillaries are larger than those in smooth capillaries under the same conditions. Moreover, the entry pressure is not sensitive to the roughness factor under strong wetting conditions and is much more sensitive to the pillar asperity height than the other structure parameters.

The complete growth process of a single bubble in quiescent liquid is simulated using a three-dimensional hybrid thermal lattice Boltzmann model. The non-equilibrium extrapolation pressure boundary condition is extended to handle the thermal multiphase flow. Unfavorable spurious currents are usually generated in the vicinity of curved interfaces when two-phase lattice Boltzmann methods are applied. Here a level-set scheme is incorporated into the simulations to accurately represent interfacial dynamics. The phase change is controlled by an equation of state automatically instead of any artificial phase change model. Hence the present simulation is more accurate and thermodynamically consistent. The temperature, velocity fields during the bubble growth are consistent with relevant theories. The bubble growth rate obtained from the lattice Boltzmann simulations agree well with the analytical solutions. The result shows that the present scheme is able to simulate the relevant thermal bubble dynamics quantitatively.

The motion and rotation of an ellipsoidal particle inside square tubes and rectangular tubes with the confinement ratio R/a∈(1.0,4.0) are studied by the lattice Boltzmann method (LBM), where R and a are the radius of the tube and the semi-major axis length of the ellipsoid, respectively. The Reynolds numbers (Re) up to 50 are considered. For the prolate ellipsoid inside square and rectangular tubes, three typical stable motion modes which depend on R/a are identified, namely, the kayaking mode, the tumbling mode, and the log-rolling mode are identified for the prolate spheroid. The diagonal plane strongly attracts the particle in square tubes with 1.2≤R/a

The dynamics of a capsule in general flows is studied analytically and numerically. The capsule is modeled as a liquid-filled drop enclosed by a membrane. We adapted the Keller-Skalak (KS) theory and Skotheim-Secomb model to the case of general flow, the governing equations are derived. It is found that when viscosity ratio λ = 1 , the capsule dynamics in general flows is controlled by two dimensionless parameters, the ratio of vorticity to strain rate of the flow and the ratio of the elastic force to fluid stress. In the literature, the transition between swinging (SW) and tumbling (TU) is always one way (TU to SW). As far as we know, it is the first time that the TU-SW-TU transition has been identified, i.e., the transition may also transfer from SW to TU after the transition (TU to SW) occurs under some circumstances. The possible mechanism is that the rotation of the flow suppresses the deformation along the vorticity direction of the capsule. The shape dynamics of a capsule is studied in detail and the rheology of dilute capsule suspension is also investigated briefly.

We propose an alternative method to implement the contact angle boundary condition on a solid wall and apply it in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces in which the flow equations are solved by the lattice-Boltzmann method and the interface equations are solved by the finite-difference method. Using the hyperbolic tangent profile of the order parameter across an interface in phase-field theory, we were able to obtain its unknown value at a ghost point from the information at only one point in the fluid domain. This is in contrast with existing approaches relying on interpolations involving several points. The special feature allows it to be more easily implemented on immersed surfaces cutting through the grid lines. It was properly incorporated into the framework of the hybrid lattice-Boltzmann finite-difference simulation, and applied successfully for several problems with different levels of complexity. First, the equilibrium shapes of a droplet on a sphere with different contact angles and radii were studied under cylindrical geometry and a good agreement with theoretical predictions was found. Preliminary studies on a three-dimensional droplet spreading on a sphere were also performed and the results agreed well with the corresponding axisymmetric results. Second, the spreading of a two-dimensional drop on an embedded inclined wall with a given contact angle was investigated and the results matched those on a flat wall at the domain boundary under the same condition. Third, capillary filling in a cylindrical tube was studied and the speed of the interface in the tube was found to follow Washburn's law. Fourth, a droplet impacting on a sphere was investigated and several different outcomes were captured depending on the Reynolds number, the viscosity ratio, and the wettability and radius of the sphere. Finally, the proposed method was shown to be capable of studying even more complicated problems involving the interaction between a droplet and multiple objects of different sizes and contact angles.

Sorting compliant capsules is an interesting research topic. In this paper, a simple bifurcated micro-channel is used to sort the particles with different rigidities. The behavior of a compliant particle inside the channel is investigated numerically. The fluid flow and the particle’s deformation are solved by Lattice Boltzmann Method (LBM) and Lattice Spring Model (LSM), respectively. The fluid and solid solvers are coupled through interpolated bounce-back scheme. Two benchmark problems are used to validate our method. One is the motion of a compliant capsule in a channel and the other is the deformation of a capsule inside a simple shear flow. The results are quantitatively consistent with those in literature. By taking advantage of the rotating of capsules in shear flow, a simple distinguished bifurcated micro-channel is proposed to sort capsules with different rigidities. In this micro-channel, the initial offset and shear stress induce the rotating and lateral migration of the capsule and flux ratio is determined by the outlet pressures. The competition between the effect of initial offset and flux ratio contributes to the sorting mechanism. Compared to other micro-channels with different geometrical models, present one is more convenient and may be more efficient to screen the microcapsule we want.

A multiphase lattice Boltzmann flux solver (MLBFS) is proposed in this paper for incompressible multiphase flows with low- and large-density-ratios. In the solver, the flow variables at cell centers are given from the solution of macroscopic governing differential equations (Navier-Stokes equations recovered by multiphase lattice Boltzmann (LB) model) by the finite volume method. At each cell interface, the viscous and inviscid fluxes are evaluated simultaneously by local reconstruction of solution for the standard lattice Boltzmann equation (LBE). The forcing terms in the governing equations are directly treated by the finite volume discretization. The phase interfaces are captured by solving the phase-field Cahn-Hilliard equation with a fifth order upwind scheme. Unlike the conventional multiphase LB models, which restrict their applications on uniform grids with fixed time step, the MLBFS has the capability and advantage to simulate multiphase flows on non-uniform grids. The proposed solver is validated by several benchmark problems, such as two-phase co-current flow, Taylor-Couette flow in an annulus, Rayleigh-Taylor instability, and droplet splashing on a thin film at density ratio of 1000 with Reynolds numbers ranging from 20 to 1000. Numerical results show the reliability of the proposed solver for multiphase flows with high density ratio and high Reynolds number.

Interaction of vortex rings with solid is an important research topic of hydrodynamic. In this study, a multiple-relaxation time (MRT) lattice Boltzmann method (LBM) is used to investigate the flow of a vortex ring impacting spheroidal particles. The MRT-LBM is validated through the cases of vortex ring impacting a flat wall. The vortex evolution due to particle size, the aspect ratio of a prolate particle, as well as Reynolds (Re) number are discussed in detail. When the vortex ring impacting a stationary sphere, the primary and secondary vortex rings wrap around each other, which is different from the situation of the vortex ring impacting a plate. For the vortex ring impacting with a prolate spheroid, the secondary vortex ring stretches mainly along the long axis of the ellipsoid particle. However, it is found that after the vortex wrapping stage, the primary vortex recovers along the short axis of the particle faster than that in the long axis, i.e., the primary vortex ring stretches mainly along the short axis of the particle. That has never been address in the literature.

In many lattice Boltzmann studies about bubble rising, mass conservation is not satisfactory and the terminal bubble rising shape or velocity is not so consistent with experimental data as those obtained through other CFD techniques. In this paper, based on the multiphase model (He et al., 1999 [1]), a mass-conserving axisymmetric multiphase lattice Boltzmann model is developed. In the model, a mass correction step and an effective surface tension formula are introduced into the model. We demonstrate how the macroscopic axisymmetric Cahn–Hilliard equation and Navier–Stokes equation are recovered from the lattice Boltzmann equations through Chapman–Enskog expansion. The developed model is applied to simulate the bubble rising in viscous fluid. The mass correction step in our scheme significantly improves the bubble mass conservation. The surface tension calculation successfully predicts the terminal bubble shapes and reproduces the effect of initial bubble shape. The terminal bubble rising velocities are very consistent with experimental and numerical data in the literature. Qualitatively, the wakes behind the bubbles also agree well with experimental data. This model is useful for predicting the axisymmetric two-phase flows.

This study presents model extensions for a lattice Boltzmann (LB) approach to thermal axisymmetric flow including swirl or rotation. An incompressible axisymmetric lattice Boltzmann D2Q9 model was applied to solve the axial and radial velocities through inserting source terms into the two-dimensional lattice Boltzmann equation. The equations governing azimuthal (or swirling) velocity and the temperature were also solved by the LBM. It is found that this scheme is much more stable and consistent compared to previous hybrid schemes. It provides a significant advantage in simulation of melt flows with high Reynolds number and high Grashof number. The present scheme was validated by comparing the LB results with benchmark solutions for melt flow in Czochralski crystal growth. Unsteady flows with high Grashof numbers were studied in detail. The critical Grashof number for the onset of the oscillation is found to be about 2.5 × 10 6 . The oscillation amplitude ψ max is proportional to ( Gr - Gr c ) 0.5 for 2.5 × 10 6 Gr 6 × 10 6 . The frequencies and flow patterns of the unsteady flows are also analyzed. The distributions of the mean quantities of the temperature and rms of temperature at Grashof number as high as 6 × 10 7 is found to be similar to those obtained by 3D simulations.

The recently proposed (Dash et al., 2013) flexible forcing immersed boundary (IB)–lattice Boltzmann method (LBM) is improved in this work to simulate three dimensional fluid–solid interaction problems. In conventional IB–LBM schemes, explicit calculation of the force density may produce unsatisfactory no-slip boundary condition. This further leads to unphysical streamline penetrations and improper force/torque calculations for the stationary/moving spheres. Following an implicit calculation of the force density, a single Lagrangian velocity correction term is prescribed here to satisfy the no-slip condition within a convergence limit. This formulation is mathematically simple and with the help of flexible number of sub-iteration/forcing, the computational time is significantly reduced. The proposed new version of IB–LBM model is validated with three-dimension flow past stationary and moving spheres, and the results are found in excellent agreement with the data obtained from in-house experiments and previous literatures.

A multiple-relaxation-time (MRT) Rothman and Keller (R–K) lattice Boltzmann model is presented for two phase flows with kinematic viscosity contrast. For two-phase flows in porous media, the numerical stability may be reduced due to the presence of complex wall boundaries. The MRT R–K model is shown to be able to ensure better numerical stability and reduce spurious currents significantly. The non-equilibrium bounce back scheme is extended to handle the pressure and velocity boundary condition in twophase flow simulations. Immiscible displacement in complex heterogeneous media is investigated and three typical flow patterns are obtained, stable displacement, viscous fingering and capillary fingering. Cases with both capillary number Ca and viscosity ratio M ranging from 10 � 3 to 10 3 are simulated. The three typical flow patterns correspond to the three domains in the M–Ca phase-diagram. The boundaries that separate the three domains in the model results are qualitatively consistent with previous experimental studies. The MRT R–K model coupled with the developed boundary condition is a good tool for the study of two-phase flows in porous media.

[1] Lattice Boltzmann flow simulations provide a physics-based means of estimating intrinsic permeability from pore structure and accounting for inertial flow that leads to departures from Darcy's law. Simulations were used to compute intrinsic permeability where standard measurement methods may fail and to provide better understanding of departures from Darcy's law under field conditions. Simulations also investigated resolution issues. Computed tomography (CT) images were acquired at 0.8 mm interscan spacing for seven samples characterized by centimeter-scale biogenic vuggy macroporosity from the extremely transmissive sole-source carbonate karst Biscayne aquifer in southeastern Florida. Samples were as large as 0.3 m in length; 7–9 cm-scale-length subsamples were used for lattice Boltzmann computations. Macroporosity of the subsamples was as high as 81%. Matrix porosity was ignored in the simulations. Non-Darcy behavior led to a twofold reduction in apparent hydraulic conductivity as an applied hydraulic gradient increased to levels observed at regional scale within the Biscayne aquifer; larger reductions are expected under higher gradients near wells and canals. Thus, inertial flows and departures from Darcy's law may occur under field conditions. Changes in apparent hydraulic conductivity with changes in head gradient computed with the lattice Boltzmann model closely fit the Darcy-Forchheimer equation allowing estimation of the Forchheimer parameter. CT-scan resolution appeared adequate to capture intrinsic permeability; however, departures from Darcy behavior were less detectable as resolution coarsened.

Sedimentation behaviors of an oblate ellipsoidal particle inside narrow [R/a∈(1.2,2.0)] infinitely long circular tubes are studied by the lattice Boltzmann method, where R and a are the radius of the tube and the length of the semimajor axis of the ellipsoid, respectively. The Archimedes numbers (Ar) up to 70 are considered. Four periodic and two steady sedimentation modes are identified. It is the first time that the anomalous mode has been found in a circular tube for an ellipsoidal particle. The phase diagram of the modes as a function of Ar and R/a is obtained. The anomalous mode is observed in the larger R/a and lower-Ar regime. Through comparisons between the anomalous and oscillatory modes, it is found that R/a plays a critical role for the anomalous mode. Some constrained cases with two steady modes are simulated. It is found that the particle settles faster in the unconstrained modes than in the corresponding constrained modes. This might inspire further study on why the particle adopts a specific mode under a certain circumstance.

The lattice Boltzmann method (LBM) has been applied to electrohydrodynamics (EHD) in recent years. In this paper, Shan–Chen (SC) single-component multiphase LBM is developed to study large-density-ratio EHD problems. The deformation/motion of a droplet suspended in a viscous liquid under an applied external electric field is studied with three different electric field models. The three models are leaky dielectric model, perfect dielectric model and constant surface charge model. They are used to investigate the effects of the electric field, electric properties of liquids and electric charges. The leaky dielectric model and the perfect dielectric model are validated by the comparison of LBM results with theoretical analysis and available numerical data. It shows that the SC LBM coupled with these electric field models is able to predict the droplet deformation under an external electric field. When net charges are present on the droplet surface and an electric field is applied, both droplet deformation and motion are reasonably predicted. The current numerical method may be an effective approach to analyze more complex EHD problems.

A free energy (FE) model, the Shan–Chen (S–C) model, and the Rothman and Keller (R–K) model are studied numerically to evaluate their performance in modeling two-dimensional (2D) immiscible two-phase flow in porous media on the pore scale. The FE model is proved to satisfy the Galilean invariance through a numerical test and the mass conservation of each component in the simulations is exact. Two-phase layered flow in a channel with different viscosity ratios was simulated. Comparing with analytical solutions, we see that the FE model and the R–K model can give very accurate results for flows with large viscosity ratios. In terms of accuracy and stability, the FE model and the R–K model are much better than the S–C model. Co-current and countercurrent two-phase flows in complex homogeneous media were simulated and the relative permeabilities were obtained. Again, it is found that the FE model is as good as the R–K model in terms of accuracy and efficiency. The FE model is shown to be a good tool for the study of two-phase flows with high viscosity ratios in porous media.

In this paper, the Shan–Chen-type (SC) multiphase lattice Boltzmann model was used to study the viscous coupling effects for immiscible two-phase flow in porous media. In the model, any typical equation of state can be incorporated and different contact angles of the gas–liquid interface at a solid wall can be obtained easily through adjusting the ‘density of wall’ (Benzi et al., Phys. Rev. E 2006; 74(2):021509). The viscous coupling effects due to capillary number, the viscosity ratio and the wetting angle were investigated. The two-phase flows with density ratio as high as 56 in porous media were simulated. For different viscosity ratios and wettability, two-phase flow patterns and relative-permeability curves as a function of wetting saturation were obtained. It is observed that when the wetting phase is less viscous and covers the solid surface, the relative permeability of the non-wetting phase may be greater than unity. Here, the SC model is demonstrated as a suitable tool to study the immiscible two-phase flow in porous media because it is simple, easy to get the desired contact angle and able to simulate immiscible phase flow with high-density ratio. Copyright © 2008 John Wiley & Sons, Ltd.

Isothermal gas flow in microtubes with a sudden expansion or contraction is studied numerically by lattice Boltzmann method. An axisymmetric D2Q9 model is used to simulate gas slip flow in micro-circular pipes. With the boundary condition combined specular and bounce-back schemes, the computed results are in excellent agreement with analytical solution for straight microtube. For the gas flow in the expanded or constricted tubes, we carried out simulations of several Knudsen numbers with inlet/outlet pressure ratio 3. It is found the pressure drop in each section can be predicted well by the theory of straight tubes. For smaller Knudsen number, flow separation in the expanded tube is observed. While for large Knudsen number, there is no vortex at corner and the streamlines are attached to boundary. In the constricted tube, the vortex at corner is very weak. These results are consistent with some experimental conclusions.

AU›o®ƒa®EAÝ : LBM, lattice Boltzmann method; MPA, Marshall’s probabilistic approach; PCC, probabilistic capillary connectivity. SO›‘®ƒ½ S›‘a®EA : FUƒ‘aƒ½Ý We derived two new expressions for the intrinsic permeability (k) of fractal porous media. The fi rst approach, the probabilisƟ c capillary connecƟ vity (PCC) model, is based on evaluaƟ ng the expected value of the cross-secƟ onal area of pores connected along various fl ow paths in the direcƟ on in which the permeability is sought. The other model is a modifi ed version of Marshall’s probabilisƟ c approach (MPA) applied to random cross matching of pores present on two parallel slices through a fractal porous medium. The Menger sponge is a three-dimensional mass fractal that represents the complicated pore space geometry of soil and rock. PredicƟ ons based on the analyƟ cal models were compared with esƟ mates of k derived from laƫ ce Boltzmann method (LBM) simulaƟ ons of saturated fl ow in virtual representaƟ ons of Menger sponges. Overall, the analyƟ cally predicted k values matched the k values from the LBM simulaƟ ons with

PurposeThis paper aims to examine how using lattice Boltzmann method (LBM) aids the study of the isothermal‐gas flow with slight rarefaction in long microtubes.Design/methodology/approachA revised axisymmetric lattice Boltzmann model is proposed to simulate the flow in microtubes. The wall boundary condition combining the bounce‐back and specular‐reflection schemes is used to capture the slip velocity on the wall. Appropriate relation between the Knudsen number and relax‐time constant is defined.FindingsThe computed‐slip velocity, average velocity and non‐linear pressure distribution along the microtube are in excellent agreement with analytical solution of the weakly compressible Navier‐Stokes equations. The calculated‐friction factors are also consistent with available experimental data. For simulations of slip flow in microtube, LBM is more accurate and efficient than DSMC method.Research limitations/implicationsThe laminar flow in circular microtube is assumed to be axisymmetric. The present LBM is only applied to the simulation of slip flows (0.01 Kn0Practical implicationsLattice‐BGK method is a very useful tool to investigate the micro slip flows.Originality/valueA revised axisymmetric D2Q9 lattice Boltzmann model is proposed to simulate the slip flow in axisymmetric microtubes.

The axisymmetric flows with swirl or rotation were solved by a hybrid scheme with lattice Boltzmann method for the axial and radial velocities and finite-difference method for the azimuthal (or swirl) velocity and the temperature. An incompressible axisymmetric lattice Boltzmann D2Q9 model was proposed to solve the axial and radial velocities through inserting source terms into the two-dimensional lattice Boltzmann equation. Present hybrid scheme was firstly validated by simulations of Taylor–Couette flows between two concentric cylinders. Then the benchmark problems of melt flow in Czochralski crystal growth were studied and accurate results were obtained. Numerical experiment demonstrated that present axisymmetric D2Q9 model is more stable than the previous axisymmetric D2Q9 model (J. Comp. Phys. 2003; 186(1):295–307). Hence, compared with the previous model, present numerical method provides a significant advantage in simulation melt flow cases with high Reynolds number and high Grashof number. Copyright © 2006 John Wiley & Sons, Ltd.

In this paper an accurate axisymmetric lattice Boltzmann D2Q9 model is proposed to simulate the steady and pulsatile flows in circular pipes. Present model is derived from an incompressible D2Q9 model and some errors in a previous model [Halliday et al., Phys. Rev. E64, 011208 (2001)] are revised. In the previous model, some terms relative to the radial velocity are missing and would lead to large error for constricted or expanded pipe flows. Present model is validated by cases of laminar steady flow through constricted tubes and 3D Womersley flow. Comparing with the previous model, our model is much more accurate for steady flow in constricted circular pipes. For 3D Womersley flow, it is also observed that the present model can reduce the compressibility effect in previous model.

In this paper, a recent curved non-slip wall boundary treatment for isothermal Lattice Boltzmann equation (LBE) [Z. Guo, C. Zheng and B. Shi, Phys. Fluids14(6) (2002)] is extended to handle the thermal curved wall boundary for a double-population thermal lattice Boltzmann equation (TLBE). The unknown distribution population at a wall node which is necessary to fulfill streaming step is decomposed into its equilibrium and non-equilibrium parts. The equilibrium part is evaluated according to Dirichlet and Neumann boundary constraints, and the non-equilibrium part is obtained using a first-order extrapolation from fluid lattices. To validate the thermal boundary condition treatment, we carry out numerical simulations of Couette flow between two circular cylinders, the natural convection in a square cavity, and the natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder. The results agree very well with analytical solution or available data in the literature. Our numerical results also demonstrate that the TLBE together with the present boundary scheme is of second-order accuracy.

The free locomotion of a two-dimensional flapping flexible plate near the flat ground is studied by the lattice Boltzmann method for fluid flow and a finite-element method for the plate motion. The fluid flow and plate deformation are coupled through the immersed boundary scheme. When the leading edge of the plate is forced to oscillate sinusoidally near the ground, the plate may move freely in the horizontal direction due to the fluid-structure interaction. The mechanisms underlying the ground effect are elucidated. Besides a moderate rigidity, it is found that an appropriate density ratio between the plate and surrounding fluid (M) can improve the propulsive efficiency of the plate. When M is relatively small, the lateral force is enhanced, and the input work is increased when the plate is near the ground; when M is large, the deformation of the plate is inhibited and the input work is decreased when the plate is close to the ground. Usually the closer the plate flapping is to the wall, the more efficie...

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.

In this paper, a scheme for specifying contact angle and its hysteresis is incorporated into a multiphase lattice Boltzmann method. The scheme is validated through investigations of the dynamic behaviors of a droplet sliding along two kinds of walls: a smooth (ideal) wall and a rough or chemically inhomogeneous (nonideal) wall. For an ideal wall, the wettability of solid substrates is able to be prescribed. For a nonideal wall, arbitrary contact angle hysteresis can be obtained through adjusting advancing and receding angles. Significantly different phenomena can be recovered for the two kinds of walls. For instance, a droplet on an inclined ideal wall under gravity is impossible to stay stationary. However, the droplet on a nonideal wall may be pinned due to contact angle hysteresis. The steady interface shapes of the droplet on an inclined nonideal wall under gravity or in a shear flow quantitatively agree well with the previous numerical studies. Besides, the complex motion of a droplet creeping like an inchworm could be simulated. The scheme is found suitable for the study of contact line problems with and without contact angle hysteresis.

Zhang and Chen proposed a lattice Boltzmann method for simulations of liquid-vapor thermal flows [Zhang and Chen, Phys. Rev. E 67, 066711 (2003)]. In their numerical test, the authors show simulation results with a temperature ratio of approximately 1.2 for a thermal two-phase flow with $Tl0.8{T}_{c}$. However, we believe this to be erroneous and demonstrate the error to originate from a wrong choice of a parameter of critical importance in the Carnahan-Starling equation of state used in their work. Our numerical tests show that using the correct parameter choice, the scheme is only valid and robust to simulate temperature ratios below 1.10 and the lowest temperature in the flow field should be $0.85{T}_{c}$.

Numerous schemes have been proposed to incorporate a bulk forcing term into the lattice Boltzmann equation. In this paper we present a simple and straightforward comparative analysis of five popular schemes [Shan and Chen, Phys. Rev. E 47, 1815 (1993); Phys Rev Lett. 81, 1618 (1998); He et al., Phys. Rev. E 57, R13 (1998); Guo et al., Phys. Rev. E 65, 046308 (2002); Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)] in which their differences and similarities are identified. From the analysis we classify the schemes into two groups; the behaviors of the schemes in each group are proven to be identical up to second order. Numerical test simulating the two-dimensional unsteady Taylor-Green vortex flow problem demonstrate that all five schemes are of comparable accuracy for single-phase flow. However, for two-phase flow the situation is different, which is demonstrated by incorporating these schemes into different Shan-Chen-type multiphase models. The forcing scheme in the original Shan-Chen (SC) multiphase model turns out to be inaccurate in terms of the resulting surface tension for different density ratios and relaxation times. In the numerical tests, a typical equation of state and interparticle interactions including next-nearest neighbors were incorporated into the SC model. Our results confirm that the surface-tension values obtained from the original SC lattice Boltzmann method (LBM) simulation depend on the value of the relaxation time τ. For τ0.7Δt, the surface tension agree well with the analytical solutions. However, when τ0.7Δt, the surface tension turns out to be systematically larger than the analytical one, exceeding it by more than a factor of 2 for τ=2Δt. In contrast, with the application of the scheme proposed by He et al., the SC LBM produces very accurate surface tensions independent of the value of τ. We also found that the densities of the coexisting liquid and gas can be adjusted to match those at thermodynamic equilibrium if the particle interaction term includes next-nearest-neighbor contributions. The obtained results will be useful for further studies of two-phase flow with high density ratios using the SC LBM approach.

The forcing term in the lattice Boltzmann equation (LBE) is usually used to mimic Navier-Stokes equations with a body force. To derive axisymmetric model, forcing terms are incorporated into the two-dimensional (2D) LBE to mimic the additional axisymmetric contributions in 2D Navier-Stokes equations in cylindrical coordinates. Many axisymmetric lattice Boltzmann D2Q9 models were obtained through the Chapman-Enskog expansion to recover the 2D Navier-Stokes equations in cylindrical coordinates [I. Halliday et al., Phys. Rev. E 64, 011208 (2001); K. N. Premnath and J. Abraham, Phys. Rev. E 71, 056706 (2005); T. S. Lee, H. Huang, and C. Shu, Int. J. Mod. Phys. C 17, 645 (2006); T. Reis and T. N. Phillips, Phys. Rev. E 75, 056703 (2007); J. G. Zhou, Phys. Rev. E 78, 036701 (2008)]. The theoretical differences between them are discussed in detail. Numerical studies were also carried out by simulating two different flows to make a comparison on these models' accuracy and $\ensuremath{\tau}$ sensitivity. It is found all these models are able to obtain accurate results and have the second-order spatial accuracy. However, the model C [J. G. Zhou, Phys. Rev. E 78, 036701 (2008)] is the most stable one in terms of $\ensuremath{\tau}$ sensitivity. It is also found that if density of fluid is defined in its usual way and not directly relevant to source terms, the lattice Boltzmann model seems more stable.

In this paper, the 3D Navier–Stokes (N–S) equation and Cahn–Hilliard (C–H) equations were solved using a free-energy-based lattice Boltzmann (LB) model. In this model, a LB equation with a D3Q19 velocity model is used to recover continuity and N–S equations while another LB equation with D3Q7 velocity model for solving C–H equation (Int. J. Numer. Meth. Fluids, 2008; 56:1653–1671) is applied to solve the 3D C–H equation. To avoid the excessive use of computational resources, a moving reference frame is adopted to allow long-time simulation of a bubble rising. How to handle the inlet/outlet and moving-wall boundary conditions are suggested. These boundary conditions are simple and easy for implementation. This model's performance on two-phase flows was investigated and the mass conservation of this model was evaluated. The model is validated by its application to simulate the 3D air bubble rising in viscous liquid (density ratio is 1000). Good agreement was obtained between the present numerical results and experimental results when Re is small. However, for high-Re cases, the mass conservation seems not so good as the low-Re case. Copyright © 2009 John Wiley & Sons, Ltd.

A parallel implementation of the three-dimensional Shan-and-Chen multicomponent, multiphase lattice Boltzmann method (LBM) was used to simulate the equilibrium distributions of two immiscible fluids in porous media. The simulations were successfully validated against cone-beam x-ray microtomographic data on the distribution of oil (decane), water, and air phases in a 5-mm cube of porous medium composed of packed quartz sand grains. The results confirm that LBM models allow for the straightforward incorporation of complex pore space geometry determined from x-ray microtomography measurements and that simulated wetting and nonwetting phase distributions are consistent with x-ray observations on both macroscopic and microscopic scales.

We propose a method for approximating the adhesion parameters in the Shan and Chen multicomponent, multiphase lattice Boltzmann model that leads to the desired fluid-solid contact angle. The method is a straightforward application of Young's equation with substitution of the Shan and Chen cohesion parameter and a density factor for the fluid-fluid interfacial tension, and the adhesion parameters for the corresponding fluid-solid interfacial tensions.

A novel flexible forcing hybrid immersed boundary-lattice Boltzmann model (IB-LBM) is introduced in the present paper for solving moving boundary problems. In conventional IB-LBM schemes, explicit formulations of force density term may not ensure no-slip boundary condition accurately, which leads to inaccurate force and torque calculations for moving object. Following an implicit force density calculation, a single Lagrangian velocity correction term is advised in this study. The formula for this correction term is much simpler and with the help of flexible number of sub-iteration/forcing, the computational time is significantly saved. The no-slip boundary condition is achieved accurately within a convergence limit. The proposed algorithm shows advantages for unsteady and moving boundary problem, where boundary convergence is satisfied consistently at every time step. In particular, a 2D particulate flow case is simulated in a constricted channel. Interesting observations and results are discussed in this article.

Sedimentation behaviours of an ellipsoidal particle in narrow and infinitely long tubes are studied by a multi-relaxation-time lattice Boltzmann method (LBM). In the present study, both circular and square tubes with 12/13 ⩽ D/A ⩽ 2.5 are considered with the Galileo number (Ga) up to 150, where D and A are the width of the tube and the length of major axis of the ellipsoid, respectively. Besides three modes of motion mentioned in the literature, two novel modes are found for the narrow tubes in the higher Ga regime: the spiral mode and the vertically inclined mode. Near a transitional regime, in terms of average settling velocity, it is found that a lighter ellipsoid may settle faster than a heavier one. The relevant mechanism is revealed. The behaviour of sedimentation inside the square tubes is similar to that in the circular tubes. One significant difference is that the translation and rotation of ellipsoid are finally constrained to a diagonal plane in the square tubes. The other difference is that the anomalous rolling mode occurs in the square tubes. In this mode, the ellipsoid rotates as if it is contacting and rolling up one corner of the square tube when it settles down. Two critical factors that induce this mode are identified: the geometry of the tube and the inertia of the ellipsoid.

A novel flexible forcing immersed boundary and lattice Boltzmann method (IB-LBM) is introduced in this paper. In the conventional IB-LBM scheme, explicit calculation of the force density term may not guarantee exact satisfaction of the no-slip boundary condition. This produces unphysical streamline penetration into the solid object. In this study, an implicit approach is followed where the force density is obtained from the unknown velocity correction with a unique single Lagrangian velocity correction term and flexible number of forcing steps. The proposed algorithm is computationally efficient and easy for new code development. Also the algorithm is found advantageous for unsteady flow case where no-slip condition is accurately satisfied. Flow past circular cylinder is studied for both steady and unsteady flow cases to validate the proposed scheme.

Originally, the color-gradient model proposed by Rothman and Keller (R–K) was unable to simulate immiscible two-phase flows with different densities. Later, a revised version of the R–K model was proposed by Grunau et al. [D. Grunau, S. Chen and K. Eggert, Phys. Fluids A: Fluid Dyn. 5, 2557 (1993).] and claimed it was able to simulate two-phase flows with high-density contrast. Some studies investigate high-density contrast two-phase flows using this revised R–K model but they are mainly focused on the stationary spherical droplet and bubble cases. Through theoretical analysis of the model, we found that in the recovered Navier–Stokes (N–S) equations which are derived from the R–K model, there are unwanted extra terms. These terms disappear for simulations of two-phase flows with identical densities, so the correct N–S equations are fully recovered. Hence, the R–K model is able to give accurate results for flows with identical densities. However, the unwanted terms may affect the accuracy of simulations significantly when the densities of the two fluids are different. For the simulations of spherical bubbles and droplets immersed in another fluid (where the densities of the two fluids are different), the extra terms may not be important and hence, in terms of surface tension, accurate results can be obtained. However, generally speaking, the unwanted term may be significant in many flows and the R–K model is unable to obtain the correct results due to the effect of the extra terms. Through numerical simulations of parallel two-phase flows in a channel, we confirm that the R–K model is not appropriate for general two-phase flows with different densities. A scheme to eliminate the unwanted terms is also proposed and the scheme works well for cases of density ratios less than 10.

We propose a hybrid lattice-Boltzmann finite-difference method to simulate axisymmetric multiphase flows. The hydrodynamics is simulated by the lattice-Boltzmann equations with the multiple-relaxation-time (MRT) collision model and suitable forcing terms that account for the interfacial tension and axisymmetric effects. The interface dynamics is captured by the finitedifference solution of the convective Cahn–Hilliard equation. This method is applied to simulate a quiescent drop, an oscillating drop, a drop spreading on a dry surface and a drop accelerated by a constant body force. It is validated through comparisons of the computed results for these problems with analytical solutions or numerical solutions by other different methods. It is shown that the MRT-based method is able to handle more challenging cases than that with the single-relaxation-time collision model for axisymmetric multiphase flows due to its improved stability.

Numerous lattice Boltzmann (LB) methods have been proposed for solution of the convection?diffusion equations (CDE). For the 2D problem, D2Q9, D2Q5 or D2Q4 velocity models are usually used. When LB convection?diffusion models are used to solve a CDE coupled with Navier?Stokes equations, boundary conditions are found to be critically important for accurately solving the coupled simulations. Following the idea of a regularized scheme (Latt et al 2008 Phys.?Rev.?E 77 056703), a regularized boundary condition for solving a CDE is proposed. A simple extrapolation scheme is also proposed for the Neumann boundary condition. Spatial accuracies of three existing and the proposed boundary conditions are discussed in details. The numerical evaluations are based on simulations of steady and unsteady natural convection flows in a cavity and an unsteady Taylor?Couette flow. Our studies show that the simplest D2Q4 model with terms of O(u) in the equilibrium distribution function is capable of obtaining results of equal accuracy as D2Q5 or D2Q9 models for the CDE. A slightly revised LB equation for solving a CDE that is used to cancel some unwanted terms does not seem to be necessary for incompressible flows. The regularized boundary condition for solving the CDE has second-order spatial accuracy and it is the best one in terms of the spatial accuracy. The regularized scheme and non-equilibrium extrapolation scheme are applicable to handle both the Dirichlet and Neumann boundary conditions. For the Neumann boundary condition with zero flux, all the five boundary conditions are applicable to give accurate results and the bounce-back scheme is the simplest one.

In this paper, the viscous coupling effects for immiscible two-phase (gas-liquid) flow in porous media were studied using the Shan–Chen-type single-component multiphase lattice Boltzmann model. Using the model, the two-phase flows in porous media with density ratio as high as 56 could be simulated and the contact angle of the gas-liquid interface at a solid wall is adjustable. To investigate viscous coupling effects, the co- and countercurrent steady-state two-phase flow patterns and relative permeabilities as a function of wetting saturation were obtained for different capillary numbers, wettabilities, and viscosity ratios. The cocurrent relative permeabilities seem usually larger than the countercurrent ones. The opposing drag-force effect and different pore-level saturation distributions in co- and countercurrent flows may contribute to this difference. It is found that for both co- and countercurrent flows, for strongly wet cases and viscosity ratio M>1, knw increase with the driving force and the vis...