Simulation of multiphase and multi component flows by lattice boltzmann method

2.2 General framework 29 2.5 Generation of single speed models from the platform 40 3.4.2 The implementation by lattice Boltzmann method 69 Chapter 4 Lattice Boltzmann Interface Capturin

Flows by Lattice Boltzmann Method

Zheng Hongwei

(B Sc., M Sc.)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

2007

To My Parents

I would like to express my sincere gratitude to my advisors, Professor Shu Chang and

Professor Chew Yong Tian, for their invaluable guidance and advice, and encouragement

throughout the course of this thesis Besides, I also want to take this opportunity to acknowledge

and appreciate the National University of Singapore for the scholarship they have provided

In addition, I would like to give my special thanks to my parents and wife whose patient

love and support enabled me to complete this work

Zheng Hongwei

Chapter 2 A General Platform for Developing a New Lattice Boltzmann Model

2.2 General framework 29

2.5 Generation of single speed models from the platform 40

3.4.2 The implementation by lattice Boltzmann method 69

Chapter 4 Lattice Boltzmann Interface Capturing Method for Incompressible

Multiphase Flows

Appendix A Derivation of the new interface capturing method 105

Appendix B Galilean invariance of interface capturing method 108

Chapter 5 Three-Dimensional Applications of the New Interface Capturing

Method

5.2.1 Cahn-Hilliard equation by lattice Boltzmann method 120

5.2.2 The direction split flux corrected transport (FCT)’s VOF 123

Chapter 7 Three-Dimensional Applications of the New Lattice Boltzmann Model

for Multiphase Flows

7.2 Implementation of lattice Boltzmann method 173

7.3.1 Inteface only problem under the vortex velocity field 176

8.3.3 Three suspension components under shear flow 202

9.1.2 Development of a lattice Boltzmann interface capturing model 214

9.1.3 Development of a lattice Boltzmann model for multiphase flows with large

Numerical simulation of multiphase flows has drawn the attention by of many researchers due to their many practical applications in both nature and industry When compared to the traditional computational fluid dynamics (CFD) solvers, lattice Boltzmann method (LBM) has become a promising tool due to its simplicity, since it only involves a series of collision and streaming processes Consequently, with these advantages, it is natural to apply it to model multiphase and multi-component flows

In fact, the main aim of the thesis is to develop new lattice Boltzmann (LB) models for these complex flows

Since many theoretical derivations are involved in the LB modeling of these flows, a general platform is developed to simplify the process It serves as a general guide for the researchers to develop their own LB models easily The object of this platform is

to answer the two questions: one is that under which conditions, a discrete velocity model can recover the Navier-Stokes equation and another question is how to construct a velocity model which will satisfy these conditions Based on the platform,

we can easily determine the equilibrium distribution functions for not only all the published models but also the new LB models according to different applications

The model of the multiphase and multi-component flows consists of interface capturing and surface tension modeling For the interface modeling, a new interface capturing method is proposed to recover the Cahn Hilliard equation without additional terms It can also keep the Galilean invariance In this model, the modified LB equation (Lamura and Succi, 2002) is adopted to remove the time derivative related term In addition, it does not require fourth order tensor of the discrete velocity model

Q5 means five velocities) and the D3Q7 discrete velocity model can be used in two, and three-dimensional applications, respectively As a result, the computational efficiency is greatly improved

For the surface tension modeling, it employs a thermodynamically consistent form of the surface tension term There are two forms in modeling the surface tension term: the potential form and stress form Due to the different discretization error of these two forms, we choose the potential form which is useful to eliminate parasite current

(Jamet et al., 2002) Based on the general platform and the choice of potential form, a

LB model for multiphase flows with large density ratio is developed The two-

dimensional results agree well with numerical data of Takada et al (2001) The three- dimensional results agree well with the experimental findings of Clift et al (1978) for

a real bubble rising under buoyancy with density ratio of 1000 Besides simulation of multiphase flows, the non-sticking multi-component flows are also investigated

In summary, four main parts are involved in this thesis A general platform for developing new LB models, a LB model for interface capturing, a LB model for multi-phase flows with large density ratio and a LB model for non-sticking multi-component flow were presented The simulation of multiphase flows includes modeling of the interface capturing and the surface tension term This model recovers the Chan-Hilliard equation without any additional terms and reduces the spurious current by choosing the potential form of surface tension

Table 2.1 Comparison of vortex centers by different lattice models (Re=1000) 50

Table 2.2 Comparison of primary vortex centers by different lattice models and

Table 6.2 Parameters for the simulation of a bubble rising under buoyancy 164 Table 6.3 Terminal velocity of a bubble rising under buoyancy 164

Figure 2.1 Configuration of D2S9 and D2S11 Models 52

Figure 2.3 Normalized U velocity profile along central line 53

Figure 3.5 The verification of Laplace law81

Figure 4.3 Order parameter profile along the normal direction of the interface as a

function of the distance from the center of the disk 110

Figure 4.5 The results of simple translation by different methods 111

Figure 4.6 The results of the Hirt-Nichol’s VOF method (rotation case) 111

(a) T/4, (b) T/2, (c) 3T/4, (d) T

Figure 4.7 The results of the Level Set method (rotation case) 112

(a) T/4 (b) T/2, (c) 3 T /4, (d) T

Figure 4.9 Some disturbances generated by the method of Inamuro et al (2004) 113

Figure 4.10 The results of the present method (Shear flow) 113

Figure 4.14 The result of the Hirt-Nichol’s VOF method (Elongation) 115

Figure 5.2 The initial configuration for the rotation case 137

(a) Elongation (b) Shear Flow

Figure 5.4 The results of the direction split FCT VOF method (Rotation) 138

Figure 6.2 The verification of Laplace law165

(a) case 1 (b) case 2 (c) case 3 (d) case 4

Figure 6.6 The flow pattern of a bubble rising under buoyancy with density ratio of

Figure 7.2 The result of FCT VOF method188

a) t=0.5s b) t=1.0s c) t=1.5s d) t=2.0s

a) t=0.5s b) t=1.0s c) t=1.5s d) t=2.0s

Figure 7.4 The verification of Laplace law190

Figure 7.5 The initial configuration of capillary wave in three directions 190

Figure 7.6 Bubble rising with a spherical shape under low Reynolds number

Figure 7.7 Bubble rising with a dimpled shape under large Eo number (Eo=53.946)

Figure 8.2 Bifurcation (minimum density and maximum density vs |g| ) 206

a) Step=100 b) Step=500 c) Step=2000 d) Step=10000

Figure 8.5 Results of non-sticking case (g j k, =0.13) 209

a) Step=100 b) Step=500 c) Step=5000 d) Step=10000

Figure 8.6 The critical value g*a b, (vertical axis) versus g 0,a (horizontal axis) 209

Figure 8.8 Results of sticking case (g j k, =0.08) under shear flow 211

a) Step=200 b) Step=1000 c) Step=5000

Figure 8.9 Results of non-sticking case (g j k, =0.13) under shear flow 212

a) Step=1000 b) Step=5000 c) Step=8000

DnQm Lattice notation (n = dimension, m = number of particle velocities)

DnSm Single speed lattice notation (n = dimension, m = number of particle

f Distribution function of the i-th particle in the m-th sublattice

i, j, k The index of grid points in x, y, z direction, respectively

n Unit normal vector

NX, NY, NZ The number of grid points in x, y, z direction, respectively

u , v, w The Cartesian components of the velocity

W The interface thickness

α

w Weights (Generalized lattice tensors)

x , y, z The Cartesian coordinates

Greek Letters

, Particle vectors

ℑ Free energy density

k The coefficient which is used to control the surface tension

Γ The coefficient which is used to control the mobility

ψ Bulk free energy density

Collision term for interface capturing equation

ζ The coordinate which is perpendicular to the interface

CFD Computational fluid dynamics

DSFCT Direction split flux corrected transport

DV Discrete velocity

FCT Flux corrected transport

FLAIR Flux Line-segment model for Advection and Interface Reconstruction LBE Lattice Boltzmann equation

LBM Lattice Boltzmann method

LGCA Lattice gas cellular automata

LSM Level set method

N-S Navier-Stokes

ODEs Ordinary differential equations

TLLBM Taylor-series-expansion- and Least-square-based lattice Boltzmann

method VOF Volume of fluid method

Chapter 1 Introduction

1.1 Background

Since the fluid is ubiquitous in the world, fluid dynamics has wide applications in

many areas of industry Among these areas, the dynamics of multiphase flow has

drawn many attentions from the scientists for a long time The term multiphase flow

here is used to refer to any fluid flow consisting of more than one phase or component

The dynamics of multiphase flows has many practical applications in engineering,

such as oil-water flow in porous media, boiling fluids, liquid metal melting and

solidification Because of the complexity of these problems, it is often necessary to

utilize computational (or numerical) methods to solve the complex equations that

describe them In fact, there are many computational methods developed in both the

traditional computational fluid dynamics (CFD) and the lattice Boltzmann method

(LBM) Among them, LBM has become a promising tool since 1980s due to its

simplicity It is very simple in the sense that it only involves a series of iterative steps

which consist of the collision and the stream step The collision step is a local

operation which is conductive to parallel computing Besides, unlike the traditional

computational fluid dynamics (CFD), it does not solve the non-linear partial

differential equation (PDE) Furthermore, the pressure is calculated from the state

equation instead of solving the Poisson equation in CFD For these reasons, it has

become a promising solver for incompressible flows with low Reynolds numbers

Consequently, with these advantages, it is natural to apply it to model multiphase

flows

1.2 Overview of lattice Boltzmann method

To understand the lattice Boltzmann method, it is necessary for us to review some

basic aspects of LBM first In history, the lattice Boltzmann method comes from

Lattice Gas cellular automata (LGCA) which is firstly proposed by Hardy, Pomeau

and de Pazzis (1973) The particle of LGCA is considered as a virtual particle which

consists of many real molecules These particles have the same mass and are

in-distinguishable They propagate under a certain discrete velocity (DV) model They

exchange momentum while conserving the mass and momentum at each site Besides,

each site should obey the exclusion principle that it may be empty or occupied by at

most one particle in every site This Boolean feature will lead to Fermi-Dirac type of

equilibrium distributions (Wolf-Gladrow, 2000)

The algorithm of LGCA can be divided into advection (or propagation or stream) step

and collision step During the advection step, the virtual particles propagate to the

nearest-neighboring sites with the constraint of the DV model This can be

implemented simply by using the CShift procedure (under Visual Fortran or Visual

C++ environment) In contrast, the collision step is completely local The collision

step is the most time-consuming because it is related to the recalculation of the

distribution function Due to the locality and time consuming feature of the collision

step, the computing requires nearly no communications Hence, LGCA is very

suitable to parallel computing because it can produce high computation to

communication ratio

Despite of these good features, there are two main shortcomings of the LGCA One

major drawback of the method is the statistical noise in the computed hydrodynamic

fields This shortcoming is a direct consequence of the single particle Boolean

dynamics To obtain accurate results for the hydro-dynamical fields, averaging over a

large number of time steps and/or lattice points are often required and therefore the

simulations may be rather inefficient

The second drawback is the lack of Galilean invariance In the area of hydrodynamics,

it should also be consistent with the Navier-Stokes equation The Navier-Stokes

equation has the feature that it is invariant when the frame is transformed to a

different inertial frame under the Galilean rule Galilean invariance requires the

coefficient of the nonlinear advection term to be one It can be easily found that the

momentum flux tensor in the first order is related to the advection term This

momentum flux tensor is defined as

=

=

δρ

i i eq

i

u g c u gu c

c

with

( )

d

d D

D

g

−

−+

=

1

212

Here, b is the number of discrete velocities

The first term of the momentum flux tensor is responsible for the advection term

u

uf∇f in the Navier-Stokes equation; the second term is associated with fluid pressure

To make advection Galilean invariant we need g to be equal to one as stated before

The pressure term is also weird because it consists of a velocity dependent term This

is different from the density dependent pressure term in the Navier-Stokes equation

Besides, if you write out the momentum flux tensor in the second order, you will find

that the viscosity is also a function of density These lead to the lack of Galilean

invariance

To remove these drawbacks of the LGCA, the lattice Boltzmann method is proposed

The Boolean variables and the Fermi-Dirac equilibrium distribution function in

LGCA are replaced by real type variables which are ensemble averaged quantity and

Maxwell equilibrium distribution function in LBM Then, the statistical noise is

reduced and the Galilean invariance is recovered in the incompressible limit

Furthermore, the collision operator was replaced by the single relaxation time

approximation, introduced initially by Bhatnagar, Gross, and Krook (BGK, 1954)

The collision and stream steps are kept in LBM Thus, LBM inherits the simplicity

and other advantages of LGCA while it overcomes most of the disadvantages of

LGCA

However, the requirement of only Galilean invariance and continuous distribution

function are insufficient For the Newtonian fluids, the viscous stress tensor of

Navier-Stokes is isotropic We know that the momentum flux tensor in the second

order represents the viscous stress tensor The equilibrium distribution function of

LBM can be expanded as a function of discrete velocity of virtual particles Thus, the

stress tensor can be expressed as the combination of lattice tensors up to 5th rank

Because of this, lattice tensor up to 5th rank should be isotropic if we want LBM to be

consistent with hydrodynamics

Although Qian et al (1992) listed many discrete velocity models, it is not an easy task

to construct a DV model A good DV model not only meets these isotropic

requirements, but also should be consistent with Galilean invariance together with the

provided distribution function Many people feel that these models are like ad hoc

models (Wolf-Gladrow, 2000) It seems that these models are created inexplicably in

the literature To solve this problem, we presented a platform for people to construct

new models in Chapter 2

In conclusion, LBM is easy to implement due to the stream-collision algorithm and

bounce back rule for non-slip boundary conditions The pressure is related to the

density So, people do not need to obtain the pressure through the Poisson equation as

most of the traditional CFD methods do Thus, the calculation is relatively cheap

1.3 Literature review of multi-phase flow modeling

The dynamics of flows with interfaces (for example, multiphase flows) has received

much attention for years since they are ubiquitous in both nature and industry In

general, they are modeled by the Navier-Stokes equations with interfaces In the bulk

of the phases, most of the traditional numerical methods, such as finite elements,

finite volumes, and finite differences, can be applied However, there are specific

problems due to the presence of the interface: location of the discontinuity and

computation of surface stresses or surface tensor force Thus, two main issues, surface

tension force modeling and interface modeling, have to be considered Once these two

modelings are set, the problem will turn to numerical scheme for solving the

Navier-Stokes equations with interfaces In fact, a variety of methods are developed to

deal with the interface, depending on the physical modeling and the numerical

methods

1.3.1 Traditional CFD methods

To obtain the exact locations of these interfaces is very important because they are not

known a priori in most of the practical cases The modeling of the interface depends

on the viewpoint of the interface You can regard it as either a thin or a thick region of

space Different viewpoints contribute to different modelings In literature, there are

many ways to characterize the interface, for example, particle in cell (PIC), boundary

integral methods, volume of fluid (VOF), front-tracking, arbitrary

Lagrangian-Eulerian (ALE), level set, immersed boundary, and immersed interface

method, etc They can be categorized as three basic types: interface tracking or

interface capturing or a combination of the interface tracking and capturing Here, we

concentrate on the review of the first two types on a fixed mesh The first technique

requires tracking the interface explicitly by marking it with special marker points, or

by attaching it to a mesh surface In the second technique, either mass-less particles,

or an indicator function or an order parameter marks the different phases or

components on either side of the interface

1.3.1.1 Interface tracking methods

In interface tracking methods, the indicator (or marker or color) particles are often

used to track the interfaces According to the distribution location of these virtual

particles, it can be further divided into three categories: moving mesh methods,

surface tracking methods, and volume tracking methods

For moving-mesh methods, the interfacial region is regarded as an infinitely thin or

sharp dividing surface The interface mesh points are tracked explicitly and move

with the interface in a Lagrangian manner The equations of motion are solved in

separate domains and the appropriate boundary conditions are applied at the interface

To prevent the mesh from tangling, a dynamic data structure is used so the moving

mesh points can change their nearest neighbors during the calculation Also, new

points are added when the mesh becomes sparse or a new interface appears, and mesh

points are removed when they are denser than necessary for an accurate calculation

Obviously, it will need more effort and is likely to be incorrect if applied to those

cases where the length scale is comparable to the thickness of the interface Moreover,

interface fitting grids are required in the simulation This is impractical for flows

involving coalescing or splitting phases

In contrast, surface tracking methods (Glimm et al., 1981a and 1981b; Unverdi et al.,

1992) track the location of the interface by interpolating between ordered marker

particles along the interface Because this is a lower dimensional problem, the

additional effort to accurately resolve small sub-grid scale structure in the interface is

usually small as compared with the overall computation time In general, the surface

tracking technique requires the information of geometry For example, Unverdi et al

(1992) tracks interfaces by following the movement of control points These points

mark the center of the interfaces The interfaces can be reproduced by connecting the

control points using curves (2D) or triangular surfaces (3D) Surface tension forces

must be transformed from the surface force into a volumetric force at the control

points And then, it will spread into the fixed grids The main problems are that it has

to handle topological changes and it does not conserve mass or volume Besides, it is

difficult to be used in the 3D calculations because of the need to utilize adaptive

surface grids

For volume tracking methods (such as the marker and cell, volume of fluid methods

etc), the regions are implicitly reconstructed from marker points, or alternatively, the

fractional volume One of the earliest volume tracking methods for material interfaces

is the marker and cell (MAC) method (Welch et al., 1966) Marker particles are

scattered initially to identify each material region in the calculation These particles

are transported in a Lagrangian manner along with the materials The interface is

reconstructed using the marker particle densities in the mixed cells with marker

particles of two or more materials The interface reconstruction scheme may also use

the density of particles in the surrounding cells to reconstruct a more accurate

interface location To improve the efficiency of the MAC method, the initial marker

particles are scattered more densely (or only) near the interfaces Besides the low

efficiency, numerical errors in transporting the marker particles can cause an artificial

numerical diffusive mixing near the interface To reduce the artificial numerical

diffusive mixing, far more marker particles than computational cells are needed,

leading to increase computational cost In addition, it does not satisfy mass

conservation To overcome these drawbacks, the fractional marker volume methods

(usually called the volume of fluid, VOF) were developed (Hirt and Nichols, 1981;

Ashgriz and Poo 1991; Rider and Kothe 1995; Rudman 1997, 1998; Rider and Kothe

1998; Gueyffier et al, 1998; Ubbink O., 1999; Pilliod and Puckett 2004) The

fractional marker represents the fractional volume of each material occupied in each

computational cell, for example, zero means that no material is there and one means

that it is completely filled The fractional volume of each solution region located in

each computational cell is calculated and advanced during the computation by solving

an auxiliary scalar transport equation Thus, two major problems arise: one is how to

identify the exact surface location and the other is how to advect the surface To

identify the location, a reconstruction step of the interface from these fractional

volumes is employed Unlike the surface tracking methods, very little sub-grid scale

structure is retained during the calculation Within each cell, the volume regions can

be represented by unions of rectangles, triangles, or regions bounded by

piecewise-polynomial surfaces Although the more complicated methods yield a

better approximation to the position of the interface, they produce more numerical

diffusion Besides the interface reconstruction, a suitable advection algorithm must be

applied to solve the transport equation A common one is the so-called donor-acceptor

technique This technique is based on describing a surface orientation and then

moving the surface with the velocity normal to that orientation In the original

donor-acceptor technique, the surface cell is assumed to be either horizontal or

vertical The decision regarding the orientation is made based on studying the

neighboring cells Once the surface orientation is identified, different techniques can

be used for its advection The donor-acceptor technique, which is used in the VOF

method, emphasizes control of interface diffusion rather than control of the liquid

fraction in a cell Therefore, ad hoc techniques (for example, limiter) must be

designed to remove “bad” points For example, cells having liquid fractions either less

than zero or greater than unity are corrected by redistributing liquid around them The

principal advantage of VOF methods is their inherent volume conserving property

Nevertheless, spurious bubbles and drops may be created The reconstruction of the

interface from the volume fractions and the computation of geometric quantities such

as curvature are typically less accurate than other methods discussed here since the

curvature and normal vectors are obtained by differentiating a nearly discontinuous

function (volume fraction)

For these methods stated above, another important issue other than the interface

recording aspects is the accurate representation of the surface tension force Usually,

the surface tension term is computed either with the continuum surface force (CSF)

model or with the continuum surface stress (CSS) formulation The CSF is firstly

introduced by Brackbill et al (1992) and represents the surface tension effects in a

form of a smoothly varied volumetric force The local surface tension force is set

equal to the product of local gradient of the continuum variable, its field curvature and

the surface tension Thus, the total force on the fluid through an interface is

proportional to the interface’s curvature Then, the surface tension is added to the

momentum equation and it is applied to a thin volume near the interface In addition,

it can be implemented so as to conserve mass or volume The model has been applied

by coupling with the interface description by either volume-of-fluid (Lafaurie et al.,

1994) or level-set (Osher et al., 1988) methods

1.3.1.2 Interface capturing methods

For interface capturing methods such as level-set (LSM) and phase field methods

(PFM), the interface is implicitly captured by a contour of a particular scalar function

The level set method is firstly introduced by Osher and Sethian (1988) The basic idea

is to use a smooth function (level set function) defined on the whole solution domain

to represent the interface LSM overcomes the inaccuracies because the level set

function is smooth at physical discontinuities and allows ones to maintain a maximum

numerical accuracy there This smooth function is the main difference from the other

methods which may use discrete representation of the discontinuous step function

(Kothe et al., 1998) Thus, highly accurate numerical solutions to the scalar transport

equation are possible In free surface flow, this level set function is transported by the

flow velocity In contrast to the volume fraction, it is just an indicator and has no

physical meaning Thus, the function needs not satisfy the conservation law It only

needs to consider the differentiation of the convection term The problem is that,

when large topological changes occur around the interface, the level set method

requires a re-initialization procedure to keep the distance property This may violate

mass conservation for each phase or component (Kothe et al., 1998) The extension to

three-dimensional one is also not easy

Another interface capturing method is the phase field method (Rowlinson and Widom,

1982; Anderson and McFadden et al., 1998; Verschueren et al., 2001) It is becoming

a popular choice for modeling the motion of multiphase fluids The basic idea is to

introduce a conserved order parameter (e.g., mass concentration) that varies

continuously over thin interfacial layers and is mostly uniform in the bulk phases The

interfacial region is then identified by a range of contours and no interface tracking is

required It can be proved that the method is equivalent to the singular interface

theory in the asymptotic limits (Emmerich, 2003) The equation of motion which is

modified to account for the presence of thin layer can be applied over the entire

domain For example, Navier-Stokes equations were modified to include a capillary

tensor accounting for interfacial forces The capillary tensor which accounts for

capillary forces was derived by using reversible thermodynamic arguments To be

consistent with the sharp interface theory, the surface tension can be given in terms of

the excess internal energy which is distributed throughout a three-dimensional layer

rather than being defined on a two-dimensional surface The evolution of order

parameter is described by Cahn-Hilliard Equation This equation is modified to

account for hydro-dynamic transport for the chemical potential The advantages of the

phase-field method are: (1) topological changes are automatically described; (2) the

composition field has a physical meaning not only near the interface but also in the

bulk phases; (3) complex physics (for example, liquid crystal or polymer) can easily

be incorporated into the framework The main drawback is that the discretization of

fourth order derivative is required in the calculation and is not easy to be extended to

multi-component flow

1.3.2 Lattice Boltzmann methods

Due to the limitations in traditional CFD, that is naturally a demand to develop a

lattice Boltzmann method to model multiphase flows In fact, several methods have

been developed to model multiphase flows by LBM during the last twenty years

(Do-Quang et al., 2000; Nourgaliev et al., 2003) They are the color method (Rothman

and Keller, 1988; Gunstensen et al., 1991), the potential method (Shan et al., 1993),

the free-energy based method (Swift et al., 1995 and 1996; Buick and Greated, 1998;

Lamura and Gonnella, 2001; Cristea and Sofonea, 2003; Inamuro et al., 2004) and the

discrete Boltzmann with the interaction force method (He et al., 1998) All these

models regard the interface as a transition layer instead of a thin curve and do not

need to track the interface position

The first lattice gas model for immiscible two-phase flow was proposed by Rothman

and Keller (1988) Two kinds of colored particles and distribution functions are

introduced to represent the two phases After each collision, a recolor scheme is

applied to obtain the new distribution function value The local flux and color gradient

are then calculated By minimizing the work, the surface tension emerges from the

model and drives the flow into the immiscible two phases They also extended the

color model to ternary phase flow (Rothman and Keller, 1991) However, the color

model is not so practical because it inherits the noisy nature and other defects of the

lattice gas method

Gunstensen et al (1991) developed a two-phase Lattice Boltzmann Model based on

the color model As Swift et al (1996) pointed out, it is also not consistent with the

free-energy functional Besides, although the non-physical properties like the lack of

Galilean invariance and statistical noise are overcome, the pressure is still velocity

dependent Third, another drawback of this model is that it needs a time consuming

numerical maximisation of the local work Fourth, the perturbation step with the

redistribution of colored distribution functions causes an anisotropic surface tension

that induces nonphysical vortices near interfaces D’ortona et al (1994) modified the

model to solve this problem In their model, the re-coloring step is replaced by an

evolution equation

The second kind of LBE models for two phase flow introduces the non-local

inter-particle interaction potential (Shan and Chen, 1993, 1994) The effect of

inter-molecular attraction and repulsion is represented by additional momentum

change at each lattice site and at each iterative step of the recursion equations In this

model, the net momentum is not conserved by the BGK collision operator at each site,

but the total momentum of the system is conserved if no net momentum exchange

occurs at the boundary is assumed (Shan and Chen, 1994) In this potential model, an

extreme case is that the two fluids are immiscible When the pre-factor g is smaller

than 0.2, the two fluids are miscible This is the main advantage of this model The

main drawback of this approach is that the equilibrium state is not thermodynamically

consistent: there is no underlying free energy and well-defined temperature What is

more, the momentum is not conserved locally This gives rise to the significant

interface spurious velocity

The third kind of LBE models was developed by Swift et al (1995, 1996) This model

is thermodynamically consistent, with well-defined temperature and free-energy

function The equilibrium is obtained by minimizing the free energy Then one can

construct an Euler-Lagrange equation The evaluation of this equation will lead to the

reversible pressure tensor and chemical potential for either phases or components

Finally, the Navier-Stokes equation is modified so that the pressure tensor is replaced

by the summation of reversible pressure tensor and traditional irreversible pressure

tensor This makes the method a rather general tool to study the dynamics of systems

with a given free energy The chemical potential is used to describe the growth of the

interface Another feature is that the model conserves the local momentum Therefore,

the interface spurious velocity is considerably smaller than that generated by the

interaction potential models The coefficients of the Chapman–Enskog expansion are

tuned in order to produce Navier-Stokes equations from the first two moments of the

lattice Boltzmann equation

The major drawback of this approach for liquid-gas multiphase flow is that it lacks

Galilean invariant (Swift et al., 1996) It must be pointed out that not all free-energy

based models lack the Galilean invariant In fact, the binary fluid model is Galilean

invariance Because of that, transient or steady state flow simulations may lead to

reduced accuracy in the calculation of the flow field or in the shape and velocity of

moving droplets Another drawback of original binary fluid model is the limit of

density ratio and viscosity ratio

The fourth model is directly derived from the lattice Boltzmann equation of the

non-ideal dense gases (Luo, 1998) In fact, it concentrates on the modeling of the

force term in LBM In the original paper, there is no numerical verification of this

model available Besides, as pointed out by Do-Quang et al.(2000), this model is only

correct to first order (inviscid equations), but incorrect to second order (viscous

equations) Furthermore, the continuity equation contains density diffusion terms, and

the momentum equation has nonphysical terms somewhat similar to the ones

appearing in the model of Swift and collaborators Thus, it is not Galilean invariant

when there is density gradient

In He et al (1998; 1999), it starts from analyzing the Boltzmann equation with a force

field which is similar to the kinetic equation in Luo’s method The only difference

from Luo’s method is the form of the forcing term This modification does improve

the method as stated by Do-Quang et al (2000) For example, unlike Luo’s method, it

does not have the diffusion term in the mass conservation equation and the

nonphysical viscous term in the momentum conservation term Besides, they claimed

that this form of forcing term contributes to the instability of the method if the force is

large By introducing a new distribution function which is used to calculate the

pressure, the governing equation is modified with an additional term which is

proportional to the velocity difference They claimed that it can be used to model the

large force term However, we can not obtain the density from this distribution Thus,

another distribution function is added and used to capture the interface as well as

calculate the density by a simple interpolation algorithm This interpolation is the

same as that in the VOF or LSM Nevertheless, the lattice Boltzmann equation for this

distribution function recovers a convection diffusion equation It is not a scalar

transport equation but that with some additional terms Thus, the interpolation will

result in the violation of mass conservation law

In summary, the color method and the potential method regard the interface as those

regions where the density gradient is non-zero The free-energy based method and the

force method explicitly capture the interface by solving an interface capturing

equation In the potential method, the physics of the interface capturing equation is

not clear Besides, the mobility is fixed Thus, the mobility can not be chosen as the

physical mobility of the correspondent fluid However, in the free-energy method, the

interface capturing equation is the Cahn-Hilliard equation (a special

convection-diffusion equation) This equation is thermodynamically consistent

Besides, it explicitly involves the immiscible property of the different phases and the

mobility can be flexibly modified according to the physical one Thus, it is more

valuable than the other interface capturing methods in both LBM and traditional CFD

(such as VOF or LS method) On the other hand, it should be noted that, the original

free-energy based LBM does not completely recover the lattice Boltzmann equation

to the Cahn-Hilliard equation

Besides, the original free-energy based method (Swift et al., 1996) also shows some

deficiencies One of the main drawbacks of this method for multiphase flows is that it

lacks Galilean invariant (Swift et al., 1996) Because of that, transient or steady state

flow simulations may lead to reduced accuracy in the calculation of the flow field or

in the shape and velocity of moving droplets To overcome this problem, Inamuro et

al (2000) and Kalarakis et al (2002) presented a Galilean invariant model for liquid

gas problem using D2Q9 and D2Q7 models Inamuro et al used an asymptotic

analysis to restore Galilean invariance in the nine-bit model It greatly improves

performance during simulation of Couette flow, and shear and translation of a droplet

In the work of Kalarakis et al (2002), an improvement of the single-component

two-phase seven-bit model is proposed and used to simulate droplet formation and

droplet motion under an external flow field They also show that Galilean invariance

can be restored to the second-order accuracy by using a new formulation of the

zero-order momentum flux tensor