Lattice Boltzman Book

8/18/2019 Lattice Boltzman Book

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1

Chapter 1

Introduction

1.1 Description of Fluid System at Different Scales

Fluids, such as air and water, are frequently met in our daily life.

Physically, all fluids are composed of a large set of atoms or molecules

that collide with one another and move randomly. Interactions among

molecules in a fluid are usually much weaker than those in a solid, and a

fluid can deform continuously under a small applied shear stress. Usually,

the microscopic dynamics of the fluid molecules is very complicated and

demonstrates strong inhomogeneity and fluctuations. On the other hand,

the macroscopic dynamics of the fluid, which is the average result of the

motion of molecules, is homogeneous and continuous. Therefore, it is

can be expected that mathematical models for fluid dynamics will be

strongly dependent on the length and time scales at which the fluid is

observed. Generally, the motion of a fluid can be described by three

types of mathematical models according to the observed scales, i.e.

microscopic models at molecular scale, kinetic theories at mesoscopicscale, and continuum models at macroscopic scale.

1.1.1 Microscopic description: molecular dynamics

In microscopic models, the motion of each molecule is tracked so that its

position and momentum can be obtained. The collective dynamics of the

whole fluid system can then be computed through some statisticalmethods. Usually, the molecular dynamics of the fluid is described by

Newton’s second law,

i i m =r F , (1.1)


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2 Lattice Boltzmann Method and Its Applications in Engineering

where m is the mass of the fluid molecule, r i is the position vector of the

molecule i, the dots represent time derivatives, and F i is the total force

experienced by the molecule, which usually includes two parts,

1,

N

i ij i

j j i = ≠

= +∑F f G , (1.2)

where f ij is the force exerted by molecule j , N is the number of molecules

in the system, and G i is the external force such as gravity. The inter-

molecular force can be expressed in terms of an interaction potential φ(r ),

(| |)ij ij φ= −∇f r , (1.3)

where rij is the displacement between molecules i and j. In microscopic

models, the inter-molecular potential plays a vital role. A widely used

model is the Lennard-Jones 12-6 potential,

12 6

( ) 4r

r r

σ σ φ ε

= −

, (1.4)

where ε characterizes the interaction strength and σ represents the

interaction range.

By solving Eq. (1.1) we can obtain the position and velocity of each

molecule at every time, and then the macroscopic variables of the fluid,

such as density, velocity, and temperature, can be measured from the

microscopic information: A a = 〈 〉. Here A is a macroscopic quantity and

a is the corresponding microscopic quantity; the symbol 〈⋅〉 represents

the ensemble average of a microscopic variable. The transport

coefficients of the fluid (viscosity, thermal conductivity, diffusivity, etc.)

can also be measured according to the linear response theory, which

indicates that each transport coefficient can be obtained through the

Einstein expression or the Green-Kubo relation [Rapaport, 2004].

An alternative description of the fluid is to consider the Hamiltonian

of the N -body system H H( , , )t = q p , where ( , )q p are the generalizedcoordinates that constitute the phase space of the system, i.e. q = (q 1, q 2,

…, q N ) are the 3N spatial coordinates of the N molecules, and p = (p 1, p 2,

…, p N ) are the 3N conjugate momenta. H is the total energy of the system

including the kinetic energy and the potential energy due to molecular


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Introduction 3

interactions. In terms of the Hamiltonian of the system, the motion of the

fluid molecules can be expressed as

H H,i i

i i

∂ ∂= = −

∂ ∂ q p

p q , i =1, 2, …, N . (1.5)

In either the Newtonian formulation (1.1) or the Hamiltonian

formulation (1.5), the numbers of unknowns are very large (6N ). Even

for a small volume of fluid in practice, the number of molecules is so

large (~1023) that it is impractical to describe the whole system with a

molecular model. Actually, even with the most advanced computerresources, molecular dynamics method is still limited to sub-micrometer

systems.

1.1.2 Mesoscopic description: kinetic theory

A coarser description of the N -body fluid system is to make use of the

concept of probability distribution function (pdf) ( , , )N f t q p in the phasespace, which determines the probability N f d d q p that the state of the

system falls in the infinitesimal volume [ , ] [ , ]d d + × +p p p q q q in the 6N dimensional phase space. From f N , all of the statistical properties of the

molecular dynamics can be extracted. The evolution of f N follows the

Liouville equation [Harris, 1971],

1

0N

N N N N j j

j j j

df f f f

dt t =

∂ ∂ ∂ = + ⋅ + ⋅ =

∂ ∂ ∂ ∑ p q

p q

, (1.6)

or

H H

1

0N

N N N

j j j j j

f f f

t =

∂ ∂ ∂ ∂ ∂ − − = ∂ ∂ ∂ ∂ ∂ ∑

q p p q . (1.7)

From f N , we can define some reduced distribution functions,

1 1 1 1 1 1( , , , , ) ( , , , , )s s s N N N s s N N F f d d d d + += ∫ q p q p q p q p q p q p , (1.8)

which is termed as the s-particle probability distribution function. We

can derive a chain of evolution equations for F s (1 ≤ s ≤ N ) from the

Liouville equation, which is usually called as the BBGKY (Bogoliubov–

Born–Green–Kirkwood–Yvon) hierarchy. This hierarchy is identical to


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the original Liouville equation completely. In this chain, the first

equation for the single-particle pdf contains the two-particle pdf, the

second equation for the two-particle pdf contains the three-particle pdf,

and generally the s -th equation for the s -particle pdf contains ( 1)s + -thpdf. Therefore, the BBGKY hierarchy is fully coupled, and the

difficulties of solving it are the same as solving the Liouville equations.

However, under some assumptions, the BBGKY hierarchy can be

truncated at certain orders so that a smaller set of equations can be

obtained to approximate the original chain. Actually, this approach has

served as a common strategy for developing kinetic equations in manyapplications of kinetic theory. Particularly, the approximation equation

truncated at the first order leads to the well known Boltzmann equation

for the velocity distribution function f , which is defined as

ξ 1 1 1( , , ) ( , , ) f t mNF t =x q p , (1.9)

where we have changed the notations from phase space to physical space:

x = q 1 is the position of a particle and 1/m = p ξ is its velocity. TheBoltzmann equation, which plays the central role in kinetic theory of

gases, describes the transportation of f :

ξ ( , ) f

f f f t

∂+ ⋅ ∇ = Ω

∂, (1.10)

where Ω represents the rate of change in f due to binary molecularcollisions. When the velocity distribution function is obtained, the fluid

density ρ , velocity u , and internal energy e can be determined from its

moments:

ξ ξ ξ ξ 2

2, , ( , ) C f d f d e t f d ρ ρ ρ= = =∫ ∫ ∫ u x , (1.11)

where C is the magnitude of the particular velocity = −C u ξ (In thisbook, the magnitude of a vector a will be always denoted by a ). The

pressure tensor and heat flux, can also be determined from f

:ξ ξ ξ ξ

2

2( , , ) , ( , , )C f t d f t d = =∫ ∫ P CC x q C x . (1.12)

The evolution equations for the fluid density, velocity, and energy can be

derived from the Boltzmann equation under some assumptions.


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Introduction 5

The collision operator Ω in the Boltzmann equation (1.10) is a bi-linear integral function of f , and conserves mass, momentum, and energy:

ξ ξ ξ ( ) ( , , ) 0i f t d ψ =∫ x , (1.13)

where ξ 21, , / 2C ψ = (or 2ξ ) are called summational or collisional invariants. From the Boltzmann equation, it can be shown that the

H -function, defined by ξ ( ) lnH t f f d d = ∫ x , will always decrease withtime ( H -theorem), i.e.,

0

dH

dt ≤ . (1.14)

The equality holds only and if only the system reaches its equilibrium

state, whence the distribution function is a Maxwellian one,

ξ ξ

2( )

3/2

( )( , , ) exp

(2 ) 2eq f t

RT RT

ρ

π

− = −

u x , (1.15)

where /B R k m

= is the gas constant with k B the Boltzmann constant andm the molecular mass. At equilibrium, the collision does not take neteffects, i.e., ( ) ( )( , ) 0eq eq f f Ω = . With this knowledge, the collision operatorcan be approximated with some simplified models [Harris, 1971], among

which the BGK (Bhatnagar-Gross-Krook) model [Bhatnagar et al., 1954]

is the most popular one,

( )1 eq BGK

c

f f

τ

Ω = − − , (1.16)

where c τ is the relaxation time. This model reflects the overall effect of

intermolecular collisions, i.e., the distribution function relaxes to the

equilibrium state with collisions. It can be easily verified that the BGK

collision model conserves mass, momentum, and energy, and satisfies

the H -theorem. However, because only one relaxation time is used to

characterize the collision effects, this model also suffers from some

limitations. For example, the Prandtl number (Pr), which reflects thedifference between momentum exchange and energy exchange during

the collision process, is fixed at 1 in the BGK model, while the

full Boltzmann collision operator gives Pr = 2/3. Some modified

models have been proposed to overcome this problem, such as the


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Ellipsoidal Statistical BGK model [Holway, 1966] and the Shakhov

model [1968].

1.1.3 Macroscopic description: hydrodynamic equations

At macroscopic scale, a fluid is treated as a continuum regardless of

its molecular structure and interaction; the fluid is assumed to be

continuously distributed throughout the domain of interest, having its

own properties such as density, velocity, and temperature. The molecular

properties are reflected by the transport coefficients of the fluid such asthe viscosity and thermal conductivity. A good criterion to determine if a

continuum model is acceptable is to check whether the Knudsen number

Kn of the fluid system, which is defined as the ratio of mean-free-path of

molecules (λ ) to the characteristic length (L) of the flow region, is

sufficiently small. The mean free path is the average distance of the fluid

molecules between two successive collisions. For an ideal gas composed

of hard-sphere molecules, it can be expressed as2 1

( 2 )nd λ π −

= , wheren is the number density of the molecules and d is the diameter of the gas

molecule. The characteristic length is usually taken as the size of the

flow domain or objects in the flow, such as the diameter of an object in

the flow. In some cases, it is more suitable to define L based on the scale

at which a flow variable changes, e.g. /| |L ρ ρ= ∇ .

In continuum theory, the motion of fluid is described by a set of

partial differential equations (PDEs), which actually describes the

conservation laws of the fluid [Batchelor, 1967]:

( ) 0t

ρρ

∂+ ∇ ⋅ =

∂u , (1.17)

( )( ) p

t

ρρ

∂+ ∇ ⋅ = −∇ + ∇ ⋅

∂τ

u uu , (1.18)

( )( ) :

e e p

t

ρρ

∂+ ∇ ⋅ = −∇ ⋅ − ∇ ⋅ + ∇

∂

τ u q u u , (1.19)

where p is the pressure, τ is the deviatoric stress tensor, and q is the heat

flux. This set of equations are not closed because the variables p, τ , and q

are yet unknown. The pressure can be modeled by an equation of state,


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Introduction 7

i.e. p =p(e ), while the deviatoric stress tensor can be modeled by a

stress-strain constitution equation,

o2

2 ( ) 2 ( )3

µ µ µ µ µ ′ ′= + ∇ ⋅ = + + ∇ ⋅

τ S u I S u I , (1.20)

where µ and µ′ are the first and second dynamic viscosities, I is thesecond-order unit tensor, and S is the strain-rate tensor defined by

[ ( ) ] / 2T = ∇ + ∇S u u . Here, a tensor with a notation “ ° ” denotes thetraceless part of the original tensor. The coefficient 2 /3η µ µ′= + is

also called bulk viscosity in some textbooks, which is usually assumed to

be zero (Stokes’ hypothesis). The heat flux in Eq. (1.19) is usually

related to the temperature gradient following Fourier’s law,

κ= − ∇q T , (1.21)

where κ is the thermal conductivity. The set of equations (1.17)-(1.19)

with the constitution relations (1.20) and (1.21) are the widely used

Navier-Stokes-Fourier equations, which are also called Navier-Stokesequations sometimes. This set of equations can be simplified in some

special cases. If the viscosity and thermal conductivity are neglected, the

Navier-Stokes equations then reduce to the Euler equations; If the fluid

density does not change with motion, i.e.,

0d

dt t

ρ ρρ

∂= + ⋅ ∇ =

∂ u , (1.22)

the fluid is said to be incompressible, and the continuity equation (1.17)

becomes 0∇ ⋅ =u , and the stress tensor can be simplified as 2µ=τ S .The compressibility of a fluid can be characterized by the Mach number

of the flow, / s M U c = , where U is the characteristic velocity of the flowand c s is the sound speed. Usually, the fluid flow can be regarded as

incompressible when M < 0.3.

Although the conservation equations are developed phenomeno-

logicallly, it is shown that they can also be derived from the Boltzmannequation. Actually, by multiplying the Boltzmann equation (1.10) by the

collisional invariants i ψ and integrating over the molecular velocity

space, one can get Eqs. (1.17) ~ (1.19) after identifying that p= − τ P I .


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The stress tensor τ and the heat flux q can be approximated via some

asymptotic methods (e.g., Hilbert or Chapman-Enskog expansions).

In most of practical problems, the continuum model works very well.

However, with the increasing interests in nano/micro and multi-scale

problems, the continuum model becomes inadequate due to the relative

large Knudsen number of the system. In viewing that the Navier-Stokes

equations can be derived from the Boltzmann equation, some extended

hydrodynamic models beyond the Navier-Stokes model have been

developed from the Boltzmann equation from different viewpoints, such

as the Burnett equations, super-Burnett equations, Grad’s 13-momentequations (see a recent comparison of these hydrodynamic models by

Lockerby et al. [2005]).

1.2 Numerical Methods for Fluid Flows

The mathematical models for fluid flows, either the Newtonian equation

for the vast number of molecules, or the Boltzmann equation for thesinglet distribution function, or the Navier-Stokes equations for the

macroscopic flow variables, are all extremely difficult to solve

analytically, if not impossible. Accurate numerical methods, however,

have been proven to be able to provide satisfying approximate solutions

to these equations. Particularly, with the rapid development of computer

hardware and software technology, numerical simulation has become an

important methodology for fluid dynamics.The most successfully and popular fluid simulation method is the

Computational Fluid Dynamics (CFD) technique, which is mainly

designed to solve the hydrodynamic equations based on the continuum

assumption. In CFD, the flow domain is decomposed into a set of sub-

domains with a computational mesh, and the mathematical equations are

discretized using some numerical schemes such as finite-volume, finite-

element, or finite-difference methods, which results in an algebraic

system of equations for the discrete flow variables associated with the

computational mesh. Computation can then be carried out to find the

approximate solutions on a computer by solving the algebraic system

using some sequential or parallel algorithms. CFD has developed into a


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Introduction 9

branch of fluid mechanics since 1960s, and many textbooks on different

topics in this field are available [e.g., Anderson, 1995; Ferzige and

Peric, 2002; Patankar, 1980]. There are also many free and commercial

softwares for both fundamental researches and practical engineering

applications.

With the increasing interests in micro and nano scale science and

technology, molecular dynamic simulation (MDS) techniques have also

received much attention in modern fluid mechanics. In MDS, the

movements of individual atoms or molecules of the fluid are recorded by

solving the Newtonian equations on a computer. The main advantage ofMDS is that macroscopic collective behaviors of the fluid can be directly

connected with the molecular behaviors, in which the molecular structure

and microscopic interactions can be described in a straightforward

manner. Therefore, MDS is very helpful for understanding the

fundamental microscopic mechanism of macroscopic fluid phenomena.

However, due to the vast number of atoms of the fluid, the computational

cost of MDS is rather expensive and this disadvantage limits it tosystems with a temporal scale of picoseconds and spatial scale of

nanometers.

Besides the macroscopic CFD and microscopic MDS methods,

another kind of numerical methods for fluid system is developed based

on mesoscopic models. Basically, mesoscopic methods can be classed

into two types. The first type is to solve the kinetic equations (e.g., the

Boltzmann equation) with some numerical schemes. Such methods

include the finite difference discretization of the linearized Boltzmann

equation [Kanki and Iuchi, 1973], the elementary method [Cercignani,

1988], the discrete velocity or discrete ordinate methods [Aristov, 2001;

Sone et al., 1989], the finite difference Monte Carlo method [Cheremisin,

1991], gas kinetic scheme [Xu, 1993], and lattice Boltzmann equation

(LBE) method [Benzi et al., 1992]. Another type of mesoscopic method

is to construct numerical models to simulate the physical process of some

virtual fluid particles directly. The most well-known method of such typeis the Direct Simulation Monte Carlo (DSMC) method [Bird, 1994], and

other methods of this type include the lattice gas automata (LGA)

method [Rothman and Zaleski, 1997] and the dissipative particle

dynamics (DPD) method [Hoogerbrugge and Koelman, 1992].


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Because the underlying mathematical models have different physical

bases, the numerical methods also have their own application ranges.

CFD is very successful in continuum flows, and MDS is most suitable

for nanoscale flows. With the increasing interests in multiscale flows, the

mesoscopic method has received particular attentions in recent years,

among which the lattice Boltzmann equation (LBE) method, or lattice

Boltzmann method), is perhaps the most active topic in this field due to

some distinctive features. In addition to a large amount of journal papers

in different research fields, LBE is also a hot topic in many international

conferences on fluid dynamics, and some courses on this topic have beenset up in some universities. In the rest part of this book, we will give a

systemic introduction of this method.

1.3 History of LBE

Historically, the LBE evolved from the LGA method, which is an

artificial microscopic model for gases, and later it was shown that LBEcould also be derived from the Boltzmann equation following some

standard discretization. From the first viewpoint, LBE can be regarded as

a fluid model, while the second viewpoint indicates that LBE is just a

special numerical scheme for the Boltzmann equation. Despite of this

conceptual difference, either approach demonstrates that LBE is a

method which is quite different from the traditional CFD algorithms.

1.3.1 Lattice gas automata

LGA is the precursor of LBE, which aims to simulate fluid flows with

simple fluid models. In LGA, the fluid is treated as a set of simulated

particles residing on a regular lattice with certain symmetry properties,

where they collide and stream following some prescribed rules that

satisfy some necessary physical laws. The philosophy behind LGA is

that fluid behaviors at macroscale are nothing but statistical collective

results of the micro-dynamics of fluid molecules, and are insensitive to

the detailed information of the individual molecules. In other words,

fluids with different micro structures and interactions may have the


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Introduction 11

same macroscopic phenomena. Therefore, it is possible to simulate

macroscopic flows with a fictitious micro fluid model which has simple

micro-dynamics but satisfies some necessary physical laws. LGA is just

one such fluid model, and the key requirement of a LGA model is that

the mass, momentum, and energy must be conserved during the particle

collision and streaming processes.

The first LGA model was attributed to three French scientists, Hardy,

Pomeau, and de Pazzis, which is called the HPP model after the authors

[Hardy et al., 1973a, 1973b]. This model utilizes a two-dimensional

square lattice on which the gas particles at a node can move to any of thefour nearest neighboring nodes along the lattice lines (see Fig. 1.1). The

collision of the HPP model follows the so called head-on rule, namely,

when two particles move to one same node with opposite velocities, their

velocities will turn around 90° after the collision; In any other cases, no

collision occurs and the particle velocities remain unchanged.

Mathematically, the motion of the particles in HPP model can be

described by the following discrete kinetic equation,( , ) ( , ) ( ( , ))i i t t i i n t n t C n t δ δ + + = +x c x x , (1.23)

where ( , ) 0i n t =x or 1 representing the number of particles moving withdiscrete velocity c i at node x and time t , δ t is the time step, C i is the

collision operator representing the influence of particle collisions. The

1

2

3

4

Fig. 1.1. Lattice and discrete velocities of the HPP model.


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discrete velocity set of the HPP model is given by c i = c e i with e 1 = (1,

0), e 2 = (0, 1), e 3 = (−1, 0), and e 4 = (0, −1), and c = δ x / δ t is the lattice

speed where δ x is the lattice spacing. The collision operator describing

the head-on rule can be expressed as

1 3 2 1 3 2(1 )(1 ) (1 )(1 )i i i i i i i i i C n n n n n n n n ⊕ ⊕ ⊕ ⊕ ⊕ ⊕= − − − − − , (1.24)

where “⊕” represents the modulo 4 addition. It can be verified that C i

conserves mass, momentum, and energy:

0i i

C =∑ , 0i i i C =∑c ,2

02i i

i

c C =∑ . (1.25)

The evolution of the fictitious particles can also be decomposed into two

sub-processes, i.e,

Collision: ( , ) ( , ) ( ( , ))i i i n t n t C n t ′ = +x x x , (1.26)

Streaming: ( , ) ( , )i i t t i n t n t δ δ ′+ + =x c x . (1.27)

The flow variables such as the density, velocity, and temperature canbe obtained from the ensemble average (distribution function) of the

Boolean number i i f n = 〈 〉 ,

2, , ( )2

i i i i i i

i

mf f e m

m RT f ρ ρ ρ ρ= = = = −∑ ∑ ∑u c u c , (1.28)

where m is the molecular mass of the gas, and will be assumed to be 1

without loss of generality. It is not straightforward to compute theensemble average from n i , and usually it can be replaced by temporal or

spatial (or both) average in applications.

Although the microdynamics of the HPP model satisfies the basic

conservation laws, the hydrodynamics variables do not satisfy the

continuum equations due to the insufficient symmetry of the square

lattice. Actually, the HPP model was designed as a micro fluid model

rather than a computational method for hydrodynamic flows. Even

though, the basic idea behind the HPP model opens a new way for flow

computations.

The symmetry requirement on the lattice was first discovered in 1986

when Frisch, Hasslacher, and Pomeau proposed their hexagonal LGA


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Introduction 13

model [Frisch et al., 1986] (and simultaneously by Wolfram [1986]),

which is now known as FHP model after the authors. This model uses a

triangular lattice, and each node has six nearest neighbours (Fig. 1.2).

The discrete velocities can be expressed as c i = c (cos θi , sin θi ) with θi =

(i − 1)π/3 for i =1 ~ 6. Like in the HPP model, the state of FHP model

can be described by six Boolean numbers n i that represent the number of

particles moving with velocity c i . The collision rule of the FHP model

includes five different cases, as shown in Fig. 1.2.

It is noted that in some cases, one input state may correspond to two

possible output states (cases a and d shown in Fig. 1.2). In such cases,each output state is chosen randomly with equal probability. As such, the

mathematical formulation of the collision operator can be written as

1 3 5 2 4 1 3 5 2 4

1 4 2 3 5 2 5 1 3 4

3 1 2 4 5

(1 )

,

i i i i i i i i i i i i i

i i i i i i i i i i i i

i i i i i i

r

C n n n n n n n n n n n n

rn n n n n n n n n n n n

n n n n n n

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

⊕ ⊕ ⊕ ⊕ ⊕

= −

+ −+

−

(1.29)

where “⊕” represents the modulo 6 addition and r is a random number

distributed uniformly in [0,1].

It can be shown that the collision operator of the FHP model leads to

a Fermi-Dirac distribution at equilibrium for the distribution function

[Frisch et al., 1987], i.e.,

( ) / 6

1 exp[ ]

eq i

i

f

A B

ρ=

+ + ⋅c u

, (1.30)

where A and B are the Lagrangian multipliers, which can be determined

by expanding ( )eq i f into a series of u and imposing the mass and

momentum constraints (for isothermal flows) given by Eq. (1.28). The

final expression of ( )eq i f for the FHP model can be written as [Frisch

et al., 1987],

( )2 4:1 ( )6 2

eq i i i

s s

f G c c

ρ ρ

⋅ = + +

c u Q uu , (1.31)

where 2 2/2s c c = is usually called sound speed in LGA (and LBE),

G (ρ)=(6 − 2ρ)/(6 − ρ), and 2i i i s c = −Q c c I . With this expanded equilibrium


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distribution function (EDF), it can be shown that the fluid density and

velocity of the FHP model satisfy the following hydrodynamic equations

in the incompressible limit (ρ → ρ0):

0∇ ⋅ =u , (1.32)

Fig. 1.2. FHP model and the collision rule.


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Introduction 15

(0) 20

0

( )( )p

g t

ρ ν ρ

∂ ∇+ ⋅ ∇ = − + ∇

∂

u

u u u , (1.33)

where2

2

21s p c g

c

= −

u

ρ , (0) 23

0

1 1

(1 ) 4s t c

s ν δ

ρ

= − −

,

with s = ρ0/6, g = (ρ0 − 3)/(ρ0 − 6). In general g ≠ 1 and this leads to the

breakdown of the Galilean invariance. However, the Galilean invariance

can be restored by rescaling the time, i.e., t → t /g (ρ0), and the momentum

equation becomes

2P t

∂+ ⋅ ∇ = −∇ + ∇

∂u u u

u

ν , (1.34)

where P = p/ρ0g and ν = ν (0)/g . Equation (1.34) resembles the incom-

pressible Navier-Stokes equation very closely, except that the rescaled

pressure P depends on the velocity. This is somewhat unphysical, but as

the Mach number is small, the pressure p satisfies an equation of state of

ideal gas, 2s p c ρ= .

Both the HPP and FHP models are designed for two-dimensional

(2D) flows. It is not an easy task to find a 3D lattice with sufficient

symmetries. On the other hand, it was shown that the 4D Face-Centered-

Hyper-Cube (FCHC) lattice, which contains 24 discrete velocities with

magnitude 2c , has the desired properties [d'Humières et al., 1986]. By

projecting back onto three-dimensional, one can obtain a set of 3D

discrete velocities that have the required symmetries. However, thenumber of possible states of the collision rule in the FCHC model is still

very huge (224!), so it is a tough job to design an efficient collision rule

for it, and this was also the target of many subsequent works. The

theoretical foundation of the FCHC model was accomplished by

Wolfram [1986] and Frisch et al. [1987], and after that, many

applications have been conducted to complex flows such as multiphase

systems and flows in porous media.The implementation of a LGA model is straightforward, just a

following of the collision-streaming paradigm. Particularly, the Boolean

representation of the basic variable in LGA means that the computation

can be realized with pure Boolean operations without round-off errors,


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and the computation will be unconditional stable. Furthermore, because

the update of the state at each node is completely local, the algorithm

exhibits natural parallelism and thus is very suitable for parallel

computations. Despite these advantages, the LGA method also has some

disadvantages, such as the statistical noise arising from the Boolean

variables, the violation of the Galilean invariance, and the dependence of

velocity of the pressure. Historically, it is these unexpected features that

motivate the invention of the lattice Boltzmann method.

1.3.2 From LGA to LBE

The LBE first appeared in the analysis of the hydrodynamic behaviors of

the LGA model [Frisch et al., 1987]. But it was McNamara and Zanetti

who first proposed using LBE as a computation method [1988]. In order

to reduce the statistical noise in LGA, they replaced the Boolean variable

n i by the real-valued distribution function f i directly, and the collision

rule for f i is the same form as that for n i . The evolution equation of thismethod (MZ model) can be written as

( , ) ( , ) ( ( , ))i i i i t t f t f t f t δ δ + + − = Ωx c x x , (1.35)

where the term on the right hand is the collision operator, ( )i f Ω = ( ) ( ) ( )i i i C n C n C f 〈 〉 ≈ 〈 〉 = .Although the MZ model can remove the statistical noise effectively,

the collision operator is still rather complicated. Soon after the MZ

model, Higuera and Jimenez proposed an improved version (HJ model)

[1989]. They showed that the collision operator in the MZ model could

be approximated by a linearized one by assuming that f i is close to its

equilibrium,( ) ( )eq neq

i i i f f f = + , (1.36)

where ( )eq i f is the expansion of the Fermi-Dirac distribution function with

a similar formulation as Eq. (1.31), and( )neq

i f is the nonequilibrium part.Expanding i Ω around

( )eq i f leads to a linearized collision operator,

( )( ) ( )eq i ij j j f K f f Ω = − , (1.37)


17/34

Introduction 17

where /ij i j K f = ∂Ω ∂ is the collision matrix. Here the fact that( )( )eq i i f Ω 0= has been used. The use of the linearized collision operator

significantly simplified the HJ model, although both K ij and( )eq

i f are still

dependent on the collision operator of the underlying LGA model.

A LBE model utilizing a collision matrix independent of any LGA

models was developed by Higuera et al. (HSB model) [1989], but the

EDF still comes from the LGA:

( )0 02

0

:( )eq i i i i

f d D G d bd c

ρ

⋅= + +

Q c u

uu , (1.38)

where D is the spatial dimensional, b is the number of the discrete

velocities, and bd 0 =ρ0 is the mean density, while the function G and the

tensor Q i are given by

2 20

0 40

(1 2 )( ) ,

2 (1 ) i i i

i

D d c G d

c d D

−= = −

−Q c c I . (1.39)

Because the collision matrix K ij in the HSB model can be constructedindependently from the LGA models, it is possible to derive the desired

hydrodynamic equations when the elements are setup appropriately.

In around 1991, the collision matrix is further simplified by several

groups independently [Chen et al., 1991; Koelman, 1991; Qian et al.,

1992], 1τ −=K I , where τ is a nondimensional parameter. As such the

collision operator can be simply expressed as

( )1( ) eq i i i f f f τ

Ω = − − , (1.40)

which just has the same formulation as the BGK model in kinetic theory

(see Eq. (1.16)). Usually a LBE with such a collision operator is called

lattice BGK (LBGK) or single-relaxation-time (SRT) model. Unlike

previous LBE models, the EDF in LBGK model is completely

independent of LGA method, and can be constructed with some

freedoms such that the corresponding hydrodynamic equations satisfy thedesired ones. Therefore, the choice of the equilibrium distribution

function is one of the key problems in LBGK method. For isothermal

flows, the BGK collision operator should conserve mass and momentum,


18/34


i.e.,

( ) ( )

,eq eq

i i i i i i i i i i f f f f ρ ρ= = = =∑ ∑ ∑ ∑c c u . (1.41)

The use of the BGK collision operator enhances greatly the

computational efficiency of LBE, and makes the implementation of the

collision process much easier than other models. The LBGK model is

perhaps the most popular one in the LBE community.

1.3.3 From continuous Boltzmann equation to LBE

As shown above, LBE was originated from the LGA method. On the

other hand, it can show that LBE can also be derived from the continuous

Boltzmann equation. For simplicity and without loss of generality, we

take the isothermal LBGK model as an example. The starting point is the

Boltzmann equation with the BGK approximation,

( ) ( ) ( ) ( )ξ

ξ ξ ξ ξ , ,

1, , , , , ,eq c

f t f t f t f t

t τ

∂ + ⋅ ∇ = − − ∂

x x x x , (1.42)

where ( )eq f is the Maxwellian distribution function. The first step is to

discretize the velocity space of ξ into a finite set of velocities { }i c

without affecting the conservation laws. To do so, ( )eq f is first expanded

into a Taylor series in terms of the fluid velocity,

ξ ξ ξ 2 2( )2

2

/

(

exp 1(2 ) 2

)

2 2eq

D f R

u

T RT RT T RT R

ρ

π

⋅⋅ = − + + −

u u

. (1.43)

It should be born in mind that this expansion can only be used for low

Mach number flows, i.e. | |/ 1RT u . In order to obtain the correct

Navier-Stokes equations in the limit of low Mach number, the discrete

velocity set should be chosen so that the following quadratures of the

expanded EDF hold exactly

ξ ξ ( ) ( )( ), 0 3k eq k eq i i i i

f d w f k = ≤ ≤∑∫ c c (1.44)

where w i and c i are the weights and points of the numerical quadrature

rule. Based on the formulation of the expanded EDF given by Eq. (1.43),

it is natural to choose a Gaussian quadrature with at least fifth-order.


19/34

Introduction 19

Once the quadrature rule is chosen, we can define a discrete distribution

function, ( , ) ( , , )i i i f t w f t =x x c , which satisfies the following equation

( )1 eq i i i i i

c

f f f f

t τ

∂ + ⋅ ∇ = − − ∂ c , (1.45)

where ( ) ( )( , ) ( , , )eq eq i i i i f t w f t =x x c . Obviously, the fluid density and velocitycan be obtained from the discrete distribution function,

,i i i i i

f f ρ ρ= =∑ ∑u c . (1.46)

Integrating equation (1.45) from t to t t δ + along the characteristic lineand assuming that the collision term is constant during this interval, we

can obtain

( )( , ) ( , ) ( , ) ( ,1

)eq i i t t i i i f t f t f t f t δ δ τ

+ + − = − − x c x x x , (1.47)

where /c t τ τ δ = is the dimensionless relaxation time. Clearly, this is justthe LBGK model.

1.4 Basic Models of LBE

In this section we will list some widely used LBE models for isothermal

flows, and then show how to derive macroscopic equations from LBGK

models using an asymptotical method. LBE models for thermal flows

will be detailed in one specific chapter.

1.4.1 LBGK models

The LBGK models are the most popular LBE method and have been

widely applied in variety of complex flows. Among the available models,

the group of Dn Qb (n -dimensional b-velocity) models proposed by Qian

et al. are the most representative ones [1992]. In Dn Qb, the discrete EDFcan be expressed as

2 2( )

2 4 2

( )1

2 2

i eq i i i

s s s

u f

c c c ω ρ

⋅ = + + −

⋅

u c c u , (1.48)


20/34


where ω i is the weight associated with the velocity c i , and the sound

speed c s is model dependent. In Tab. 1.1 several popular Dn Qb models

are presented (here the lattice speed c is assumed to be 1).

The lattice velocities of the Dn Qb models can form certain lattice

tensors with different ranks. The n -th rank lattice tensor is defined as

1 2 1 2... ...n n i i i i L c c c α α α α α α= ∑ . (1.49)

Consequently, we have the 1st, 2

nd, 3

rd and 4

th rank lattice tensors as

, , ,

.

i i i i i i

i i i

i i i i

i

L c L c c L c c c

L c c c c

= = =

=

∑ ∑ ∑

∑

α α αβ α β αβγ α β γ

αβγζ α β γ ζ

(1.50)

A tensor of n -th rank is called isotropic if it is invariant with respect toarbitrary orthogonal transformations (rotations and reflections). The most

general isotropic tensors up to 4th rank are provided by the following

theorem.

Table 1.1. Parameters of some Dn Qb models.

Model Lattice vector i c Weight i w 2

s c D1Q3

0,

±1

2/3,

1/61/3

D1Q5

0,

±1,

±2

6/12,

2/12,

1/12

1

D2Q7

(0,0),

(±1/2, ± 3 2 )1/2,

1/121/4

D2Q9(0,0),

(±1,0),(0, ±1),

(±1, ±1)

4/9,1/9,

1/36

1/3

D3Q15

(0,0,0),

(±1,0,0), (0, ±1,0),(0,0, ±1),

(±1, ±1, ±1)

2/9,

1/9,

1/72

1/3

D3Q19

(0,0,0),

(±1,0,0), (0, ±1,0),(0,0, ±1),

(±1, ±1,0), (±1,0, ±1),(0, ±1, ±1)

1/3,

1/18,

1/36

1/3


21/34

Introduction 21

• There are no isotropic tensors of rank 1 (vectors).

• An isotropic tensor of rank 2 is proportional to the 2nd

Kronecker

delta tensor αβ δ .

• An isotropic tensor of rank 3 is proportional to αβγ ε with 123ε =

231 312 1ε ε= = , 132 213 321 1ε ε ε= = = − and zero others.• There are three different (linear independent) tensors of rank 4, αβ γζ δ δ ,

αγ βζ δ δ , and αζ βγ δ δ , which can be combined to give a general form,

L a b c αβγζ αβ γζ αγ βζ αζ βγ δ δ δ δ δ δ = + + , where α, β , γ and ζ are arbitraryconstants.

The lattice tensors can also be generalized to include the weights. A

n-th rank generalized lattice tensor is defined as

1 2 1 2... ...

n n i i i i

i

T c c c α α α α α αω= ∑ , (1.51)

where the ωi's are the weights appearing in the discrete equilibrium

function. In the LBE context, the lattice tensors of odd rank are usually

zero. With the knowledge about the lattice tensors, it is easy to find thefirst several velocity moments of the discrete EDF ( )eq i f ,

( )eq i

i

f ρ=∑ , ( )eq i i i

f ρ=∑c u , ( )eq i i i i

f pρ= +∑ c uu c I ,

δ( ) 2 2[ ] ( )eq i i i i s s

i

f c c u u c u c c α β γ αβγ α βγ β αγ γ αβ ρ ρ δ δ δ = + +≡∑ u . (1.52)

These relations are helpful in the derivation of the hydrodynamic

equations from the Dn Qb LBGK models.

1.4.2 From LBE to the Navier-Stokes equations:

Chapman-Enskog expansion

The macroscopic dynamics of a fluid can be seen as the result of the

collective behavior of microscopic particles in the system and it is well

described by the Navier-Stokes equations. The derivation of themacroscopic Navier-Stokes equations from the LBE runs under the

Chapman-Enskog expansion, which is a multi-scale analysis developed

by Chapman and Enskog. As an example, in what follows we will

perform an analysis of the D2Q9 model. First the following multiscale


22/34


expansions are introduced,

(0) (1) (2)2i i i i f f f f ε ε= + + + , 0 1

2t t t ε ε∂ = ∂ + ∂ , 0α αε∂ = ∂ , (1.53)

where ε is a small number proportional to Kn, and t ∂ and α∂ are theshort notations of / t ∂ ∂ and / x α∂ ∂ , respectively. In this expansion, t 0 isthe fast convective scale, and t 1 is the slow diffusive scale. The second

order Taylor series expansion of the LBGK equation (1.47) yields

2 21 ( ) ( )

2

eq i i i i i i t

t

t

D f D f f f O δ

τδ

δ + = − − + , (1.54)

where i t i t i D c α α= ∂ + ⋅ ∇ = ∂ + ∂c . Here the Einstein’s summation ruleis used, i.e., two repeated index means taking the summation over this

index. Substituting the expansions (1.53) into Eq. (1.54), and equating

the coefficients of each order of ε, one can obtain that

ε0: (0) ( )eq i i f f = , (1.55)

ε1: ( ) ( )0 1(0) 1t

i i i D f f τδ

= − , (1.56)

ε2: ( ) ( ) ( ) ( )1

0 1 0 2(0) (0) 2 1[ ]2

t i t

i i i i i

t

f D f D f f δ

δ

τ ∂ + + = − , (1.57)

or

1

(0) (0) (1) (2)1 112

t i i i i

t

f D f f τ τδ

∂ + − = − , (1.58)

where0

(0)0i t i D = ∂ + ⋅ ∇c . From Eq. (1.55), together with Eqs. (1.46)

and (1.52), we can obtain that

( ) 0k i i

f =∑ , ( )k i i i

f =∑ 0c for 0k > . (1.59)

Multiplying Eq. (1.56) by 1 and c i , respectively, and taking

summation over i , we can obtain the mass and momentum conservation

equations at order of ε,

0 0 ( ) 0t ρ ρ⋅∂ + ∇ =u , (1.60)

π 0 0

(0)( ) 0t ρ∂ + ∇ ⋅ =u , (1.61)


23/34

Introduction 23

where (0) (0)i i i c c f u u pα β α β αβ αβ π ρ δ = = +∑ is the zeroth-order momentumflux tensor, with 2s p c ρ= . Here the following properties of the generalizedlattice tensors of the D2Q9 model have been used:

0i i i i i i i i

c c c c α α β γ ω ω= =∑ ∑ , 2i i i s i

c c c α β αβ ω δ =∑ ,

4i i i i i s

i

c c c c c α β γ θ αβγθω = ∆∑ , (1.62)

where αβγζ αβ γζ αγ βζ αζ βγ δ δ δ δ δ δ ∆ = + + . Equations (1.60) and (1.61) arethe Euler equations, just the same as those from the continuous

Boltzmann equation in kinetic theory.

Similarly, the zeroth and first order moments of Eq. (1.58) leads to

the conservation equations at order of ε2,

10t ρ∂ = , (1.63)

11

2π

1

(1)0( ) 0t

τ ρ

− ∂ ⋅∇

+ =u , (1.64)

where (1) (1)i i i i c c f α β αβ π = ∑ . In order to evaluate(1)αβ π , we multiply Eq. (1.56)

by i i c c α β and take summation over i :

( )

0

0

0 0

0

(1) (0) (0)0

2

20

2

0 02

0 0 0

20 0

1=

=

( )

( ) ( )

i i i t i i i i i i

i i i t

t s

s

s t t

t s

s

c c f c c f c c c f

u u c

c u u u

c u u u p

u u p c u u

c u u

α β α β γ α β γ

α β αβ

γ α βγ β αγ γ αβ

γ γ αβ β α α

α β β α β β α

α β β α

τδ

ρ ρδ

ρ δ δ δ

ρ ρ δ ρ

ρ ρ

ρ

− ∂ + ∂

∂ ++ ∂ + +

∂ + ∂ + ∂ + ∂ + ∂ + ∂ + ∂ + ∂

=

=

∂ + ∂

∑ ∑ ∑

0

2 30 0

( )

( ) ,s

u u u

c u u O M

γ α β γ

α β β α

ρ

ρ

− ∂ = ∂ + ∂ +

(1.65)

where M is the Mach number. Here the equations on the first order of ε ,

(1.60) and (1.61) have been used to evaluate the time derivatives.

Specifically, the following relations have been applied:

0 0 ( )t u β β ρρ∂ ∂−= , (1.66)

0

(0)0 0 0( ) ( )t u p u u αα β α β α β β ρ π ρ= − ∂−∂ = −∂ ∂ , (1.67)

0 0 0t u p u u α α β β αρρ∂ = ∂ ∂− − , (1.68)


24/34


where the last equation is derived from the first two equations. Equation

(1.65) gives then that (1) 0 0( )p t u u α β β ααβ π τ δ = − ∂ + ∂ after neglecting the

terms of O (M 3).

Combining the mass and momentum conservation equations on both

the ε and ε2 scales, we can get the hydrodynamic equations correspond-

ing to the D2Q9 model:

( ) 0t ρ ρ∂ + ∇ ⋅ =u , (1.69)

( )

( )( ) T p

t

ρρ ρν

∂ + ∇ = −∇ + ∇ ⋅ ∇ + ∇ ∂

u

uu u u , (1.70)

where ν is kinematic viscosity given by

2 1

2s t c ν τ δ

= − , (1.71)

and / 3s c c = for the D2Q9 model. In small Mach number limit, the

density variation can be negligible. Thus one can further obtain the

incompressible Navier-Stokes equations

0∇ ⋅ =u , (1.72)

21

pt

ν ρ

∂+ ⋅ ∇ = − ∇ + ∇

∂

u

u u u . (1.73)

It is noted that low Mach number assumption has been employed in both

the expansion of the discrete EDF from the Maxwellian distributionfunction and the derivation given above. As such, the LBGK models are

only suitable for low speed flows in principle. On the other hand, it is

known that the density variation of a flow is proportional to the square of

the Mach number, which means that the LBGK method is actually an

artificial compressibility method for the incompressible Navier-Stokes

equations. Therefore, besides the temporal and spatial discretization

errors, LBGK method also suffers from an additional error from the

compressibility when applied to incompressible flows. Several

incompressible LBGK models have been developed to overcome the

compressibility error from different viewpoints [Zou et al., 1995; He and

Luo, 1997; Chen and Ohashi, 1997; Guo et al., 2000].


25/34

Introduction 25

Now we discuss the accuracy of the LBGK models. First it is noted

that the LBGK equation (1.47) is a first-order finite-difference scheme

for the discrete velocity Boltzmann equation (1.45). If we apply the

Chapman-Enskog analysis to the Eq. (1.45) directly, we can also derive

the isothermal or athermal Navier-stokes equations (1.70), but with a

kinematic viscosity 2 2s c t s c c ν τ δ τ = = which contains no temporal andspatial discretization errors. In the above analysis, however, the

expansion of the LBGK equation (1.47) was truncated up to O( 2t δ ) and

O( 2x δ ), which means that LBGK method can be viewed as a second-order

scheme for the compressible Navier-Stokes equations after absorbing thenumerical viscosity 2 /2t n s c ν δ = , which comes from the term 2 2( /2)t i i D f δ in Eq. (1.54), into the physical one. This is an appealing feature of the

LBGK method for complex flows. Also, because the underlying lattices

of the LBGK models posses better rotational symmetry than classical

second-order finite-difference/finite-volume schemes, LBGK and other

LBE models may yield more accurate results than classical CFD methods

with the same temporal and spatial accuracies. Actually, somecomparative studies indicate that the results of LBE are even comparable

to spectral methods for turbulent flows [Martinez et al., 1994].

1.4.3 LBE models with multiple relaxation times

Like the BGK model in kinetic theory, the LBGK model uses a

relaxation process with a single relaxation time to characterize the

collision effects, which means that all of the modes relax to theirequilibria with the same rate. However, physically, these rates should be

different during the collision process. In order to overcome this

limitation, a collision matrix with different eigenvalues or multiple

relaxation times can be used. Actually, such models were proposed by

d'Humèriers almost at the same time when the LBGK method was

developed [d'Humières, 1992]. Such MRT-LBE models have been

attracting more and more attentions recently due to some inherentadvantages [Lallemand and Luo, 2000].

The LBE with a MRT collision operator can be expressed as,

( )( , ) ( , ) , 0 1eq i i t t i ij j j j

f t f t f f i b + + − = − Λ − = − ∑x c x ∼δ δ , (1.74)


26/34


or

Λ ( )( , ) ( , ) ( )eq i t t t t δ δ + + − = − −f x c f x f f , (1.75)

where b is the number of discrete velocities, and ΛΛΛΛ is the collision matrix.

Equation (1.74) or (1.75) describes the evolution of the population

0 11( , , , )T

b f f f −= f in the velocity space V. On the other hand, theevolution process can also be described in a moment space M. To see

how this is realized, we first define b moments based on f ,

k k m φ= ⋅f , k = 0 ∼ b −1, (1.76)

where k φ is a vector with b elements, each of which is a polynomial of

the discrete velocities. These b vectors are independent and thus form a

basis of the velocity space. The relation between the moments and the

distribution functions can also be expressed in a concise form,

0 1 1( , , , )T

bm m m −= = m Mf , (1.77)

where the invertible transformation matrixM

is composed of the vectorsk φ . The space where the moments take values is just the moment space.

From Eq. (1.75), we can obtain the moment evolution equation as,

( )( , ) ( , ) ( )eq i t t t t δ δ + + − = − −m x c m x S m m , (1.78)

where Λ 1−=S M M is usually a diagonal matrix, i.e. S = diag (s 0, s 1, …,s b−1), and

( ) ( )eq eq =m Mf is the equilibria in the moment space.In practical applications, however, the MRT-LBE usually combines

the evolutions in the velocity space and moment space. That is, the

collision step is executed in the moment space while the streaming step is

still performed in the velocity space just like the LBGK method.

Therefore, the basic flowchart of a MRT-LBE model can be described as

follows,

• Transforming the distribution functions f to moments m according to

Eq. (1.77);

•

Colliding in moment space: ( )( )eq ′ = − −m m S m m ;• Transforming the post-collision moments m' back to the post-

collision distribution function: 1−′ ′=f M m ;•

Streaming in velocity space: ( , ) ( , )i i t t i f t f t δ δ ′+ + =x c x .


27/34

Introduction 27

Therefore, the evolution of the MRT-LBE can be effectively expressed

as

1 ( )( , ) ( , ) ( )eq i t t t t δ δ −

+ + − = − −f x c f x M S m m . (1.79)

In MRT-LBE, the relaxation rates s i , or the relaxation times τ i = 1/s i can

be tuned with some freedom. This means that these parameters can be

optimized to enhance the performance of the algorithm [Lallemand and

Luo, 2000]. It is also noted that as s i =1/τ , the MRT-LBE reduces to the

LBGK models. Nonetheless, the involvement of the transforms between

the velocity and moment spaces will increase some computational costs.However, the increase is not significant because M is usually an

orthogonal matrix. Some numerical tests demonstrate that the increase in

time is within 20% [d'Humières et al., 2002].

The transformation matrix M , or the base vectors k φ , can be

constructed from the polynomials m n l ix iy iz c c c (m , n , l ≥ 0) through the

Gram-Schmidt orthogonalization [Bouzidi et al., 2001]. Generally, the

basis includes the following vectors corresponding to the mass andmomentum:

0

1 2

(1 , 1 , , 1)

( , , , ) , 1 , 2 , , .

T b

bT

bc c c D α α α α α

= ≡

= =

1

φ

φ

(1.80)

Another key point of the MRT-LBE model is the equilibria in moment

space. In principle m (eq ) can be constructed with some freedoms. But a

more convenient way is to compute them from the discrete EDFs f i(eq)

of

the Dn Qb models once the transformation matrix M is determined.

Several 2D and 3D MRT-LBE models are listed below. For simplicity,

we assume that c = 1.0 in all cases, otherwise the base vectors can be

normalized by c and the results are similar.

(1) Standard D2Q9 model

The lattice is the same as the D2Q9 LBGK model, i.e, a two-dimensionalsquare lattice. The corresponding moments are

( ), , , , , , , ,T

x x y y xx xy e j q j q p pρ ε=m ,


28/34


while the equilibria in the moment space are

( )( ) 2 2 2 21, 2 3 , , , , , , , T eq x x y y x y x y u u u u u u u u u u ρ α β = − + + − − −m ,

where α and β are two free parameters, and when α = 1 and β = −3, theequilibria are consistent with those of the D2Q9 LBGK model. The

transformation matrix is

1 1 1 1 1 1 1 1 14 1 1 1 1 2 2 2 2

4 2 2 2 2 1 1 1 10 1 0 1 0 1 1 1 10 2 0 2 0 1 1 1 10 0 1 0 1 1 1 1 10 0 2 0 2 1 1 1 10 1 1 1 1 0 0 0 00 0 0 0 0 1 1 1 1

− − − − − − − − − − − − − − −= − − − − − − − − − −

M .

The relaxation rates corresponding to the moments are

(0 , , , 0 , , 0 , , , )e q q s s s s s s ε ν ν =S ,

and the kinematic viscosity and bulk viscosity are related to the

relaxation rates se and sν ,

2 21 1 1 1

,2 2

s t s t

e

c c s s ν

ν δ ζ δ = − = −

,

where 2 1/3s c = .

(2) Rectangular D2Q9 model

An advantage of MRT-LBE is that a nonstandard mesh can be used as

the underlying lattice due to the freedoms in choosing the equilibria and

the relaxation times. For example, a D2Q9 MRT-LBE model using a

rectangular lattice has been developed. The lattice vectors are given by

0 1 0 1 0 1 1 1 1

0 0 0a a a a a a

− − − = − − − c ,


29/34

Introduction 29

where /y x a δ δ = is the ratio of the lattice spacing and here δ x is set to beunity. The nine moments are

( ), , , , , , , 3 , T

x x y y xx xy e j q j q p pρ ε=m ,

and the corresponding equilibria are given by

( )

( ) 2 2 2 2 21 4

2 2 2 2 2 24

41

1 , 2(3 ) 3 , , 1 3 , , ( 4 2 ,

, , (

)/

/2 /3 2 )/3 + , ,

eq s x x

T

y y s x y x y

c r u u u u u r a

u u r r c a a u u u u a a

ρ α β γ

γ

= − + + − −− − −

m

where c s , α, β , and γ are free parameters. The transformation matrix is

1 2 3 2 3 1 1 1 1

4 5 6 5 6 4 4 4 4

1 1 1 1 1 1 1 1 1

2

4 2 2 2 2 1 1 1 1

0 1 0 1 0 1 1 1 1

0 2 0 2 0 1 1 1 1

0 0 0

0 0 2 0 2

2

0 0 0 0 0 1 1 1 1

r r r r r r r r r

a a a a a a

a a a a a a

r r r r r r r r r

− − − − − − − − − − −= − − − − − − − − −

M ,

where 21 1r a = + , 22 1 2r a = − , 23 2r a = − + , 24 1r a = − + , 25=2+r a ,and 2

6 1 2r a = − − . The relaxation rates are similar to those of the

standard D2Q9 model,

(0, , , 0, , 0, , , )e q q s s s s s s ε ν ν =S .

The shear and bulk viscosities are

2 24 1 1 7 3 12 1 1,

6 2 12 2s

t t

e

a c

s s ν

γ γ ν δ ζ δ

+ + + − = − = − .

As expected, if a = 1, 2 1/3s c = , and γ = −2, this rectangular modelreduces to the standard D2Q9 model. It should be noted that the

corresponding rectangular D2Q9 LBGK model does not exist.


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(3)

D3Q13 model

D3Q13 MRT-LBE model uses the following discrete velocities,

0 1 1 1 1 1 1 0 0

0 1 1 1 1

0 0 0 1

0 0 1 1

0 0 1 1 1 1

1 1 1 1

0 0

1 1 0 0 1

− −−

−

− − = − − − − − −

c .

The corresponding moments are

( ), , , , , 3 , , , , , , , T

x y z xx ww xy yz zx x y z j j j e p p p p p t t t ρ=m ,

with the following equilibrium moments,

()

3 13( ) 2 2 2 2 2 2

2 21, , , , (13 8) , 3 , ,

, , , 0, 0, 0 .

eq x y z s x y z

T

x y y z z x

u u u c u u u u u

u u u u u u

ρ= − + − −m

The corresponding transformation matrix is

1 1 1 1 1 1 1 1 1 1 1 1 1

0 1 1 1 1 0 0 1 1 1 1 0 0

0 1 1 0 0 1 1 1 1 0 0 1 1

0 0 0 1 1 1 1 0 0 1 1 1 1

12 1 1 1 1 1 1 1 1 1 1 1 1

0 1 1 1 1 2 2 1 1 1 1 2 2

0 1 1 1 1 0 0 1 1 1 1 0 0

0 1 1 0 0 0 0 1 1 0 0 0 0

0 0 0 0 0 1 1 0 0 0 0 1 10 0 0 1 1 0 0 0 0 1 1 0 0

0 1 1 1 1 0 0 1 1 1 1 0 0

0 1 1 0 0 1 1 1 1 0 0 1 1

0 0 0 1 1

− − − −− − − −

− − − −−

− − − −− − − −=

− −

− −− −

− − − −− − − −

−

M

1 1 0 0 1 1 1 1

− − −

,

and the relaxation rates are

(0, 0, 0, 0, , , , , , , , , )e t t t s s s s s s s s s ν ν ν ν ν ′ ′ ′=S ,

while the shear and bulk viscosities are ( 2 1/3s c = )

21 1 1 1 1 1 2 3 1 1,

4 2 2 2 3 2

s t t t

e

c

s s s ν ν ν δ δ ζ δ

− = − = − = − ′ .


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Introduction 31

(4) D3Q15 model

This model shares the same discrete velocities as the D3Q15 LBGK

model. The fifteen moments are

( ), , , , , , , , , 3 , , , , ,T

x x y y z z xx ww xy yz zx xyz e j q j q j q p p p p p t ρ ε=m ,

with the corresponding equilibria,

( ) 2

2 2 2 2

7 7 71, 1, 1, , , , , , ,

3 3 3

3 , , , , , 0 .

eq x x y y z z

T

x y z x y y z z x

u u u u u u u

u u u u u u u u u u

ρ= − − − − − − −

m

The transformation matrix is given as follows,

1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1

16 4 4 4 4 4 4 1 1 1 1 1 1 1 10 1 1 0 0 0 0 1 1 1 1 1 1 1 10 4 4 0 0 0 0 1 1 1 1 1 1 1 10 0 0 1 1 0 0 1 1 1 1 1 1 1 10 0 0 4 4 0 0 1 1 1 1 1 1 1 10 0 0 0 0 1 1 1 1 1 1 1 1 1 10 0 0 0 0 4 4 1 1 1 1 1 1 1 10 2 2 1 1 1 1 0 0 0 0 0 0 0 0

− − − − − − −− − − − − −

− − − − −− − − − −

− − − − −− − − − −

− − − − −=− − − − −

− − − −

M

0 0 0 1 1 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 1 1 1 1 10 0 0 0 0 0 0 1 1 1 1 1 1 1 10 0 0 0 0 0 0 1 1 1 1 1 1 1 10 0 0 0 0 0 0 1 1 1 1 1 1 1 1

− −

− − − − − − − − − − − − − − − −

.

The relaxation rates are set as follows,

(0, , , 0, , 0, , 0, , , , , , , )e q q q t s s s s s s s s s s s =S ε ν ν ν ν ν ,

which are related to the viscosities as follows,

,1 1 1 2 1 1

3 2 9 2t t

e s s ν ν δ ζ δ

= − = − .


32/34


(5) D3Q19 model

The discrete velocity set of the D3Q19 MRT model is the same as that of

the LBGK model. The corresponding moments are

( ), , , , , , , , , 3 , 3 , , , , , , , ,T

x x y y z z xx xx ww ww xy yz zx x y z e j q j q j q p p p p p t t t ρ ε π π=m ,

while the corresponding equilibrium moments are

( ) 2 2

( ) ( )2 2 2 2

2 2 21, 11 19 , , , , , , , ,

3 3 3 3 , , , , , , , 0, 0, 0 ,

eq x x y y z z

T eq eq

xx ww x y z x y y z z x

m u u u u u u u u

p pu u u u u u u u u u

= − + + − − − − −

ρ α β

γ γ

ρ ρ

where α, β , and γ are free parameters. As α = 3, β = −11/2, and γ = −1/2,the equilibria reduce to those of the D3Q19 LBGK model. The

transformation matrix of this model is

.

The relaxation rates of this model are

(0, , , 0, , 0, , 0, , , , , , , , , , , )e q q q t t t s s s s s s s s s s s s s s s ε ν π ν π ν ν ν =S ,

while the shear and bulk viscosities as

,1 1 1 2 1 1

3 2 9 2t t

e s s ν ν δ ζ δ

= − = − .


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Introduction 33

It can be shown that with the equilibrium moments described above,

the Navier-Stokes equations can be obtained either by the Chapman-

Enskog expansion method [d'Humières, 1992; d'Humières et al., 2002]

or a linear analysis [Lallemand and Luo, 2000].

1.5 Summary

In this chapter we have presented an overview of the LBE method. It can

be seen that the LBE is a novel approach for fluid dynamics, quite

different from the classical CFD techniques that are based on the solution

of Navier-Stokes equations. Firstly, although LBE aims to solve

continuum flows at macroscopic scales, it is developed based on

microscopic models (LGA) or kinetic theory (Boltzmann equation). This

feature suggests that the LBE has the potential to model nonequilibrium

flows which cannot be described by the Navier-Stokes equations.

Secondly, the kinetic nature of LBE makes it suitable for describing the

internal interactions among fluid particles and those between the fluidand external environment, and this brings great advantages in simulating

complex flows such as multi-component/phase flows and flows in porous

media. Finally, the clear evolution picture of LBE makes it quite easy to

code; Particularly, the collision process involves only local computations

and the streaming process can be realized by simple index shift

operations, which makes LBE well suited for massive parallel computers.

These distinctive features have made LBE an attractive numericaltool for modeling and simulating complicated fluid flows and associated

phenomena. Actually, LBE can find its applications in simulation of

laminar and turbulent flows, interfacial dynamics of multiphase flows,

flows in porous media, particulate and suspension systems, bio-fluid

mechanics, electro-kinetic flows, magneto-hydrodynamics, quantum and

relativistic flows, nano/micro fluidics, and multiscale flows. The

developments of LBE at different stages have been summarized in

several excellent review papers [Benzi et al., 1992; Chen and Doolen,

1998; Yu et al., 2003; Aidun and Clausen, 2010], and a number of books

on both general LBE methods [Wolf-Gladrow, 2000; Succi, 2001] and

specific topics [Sukop and Thorne, 2006; Zhou, 2004] are also available.


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The past two decades have witnessed the rapid development and great

success of LBE in fundamental theory and applications in a variety of

related areas, and nowadays the research in this field is still very active

and exciting. The LBE has been becoming a topic of great interest in

many international conferences, and even two specific conferences on

this subject, DSFD ( International Conference on the Discrete Simulation

of Fluid Dynamics) and ICMMES ( International Conference for

Mesoscopic Methods in Engineering and Science), have been established.

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