boltzmann - national institute of aerospace
TRANSCRIPT
Lattice Boltzmann Methods for Computational Fluid
Dynamics
Li-Shi Luo
National Institute of Aerospace144 Research Drive
Hampton, Virginia 23666, USA
Email: [email protected]: http://research.nianet.org/~luo
August 7 { 12, 2003Institut f�ur Computeranwendungen im Bauingenieurwesen (CAB)
Technischen Universit�at BraunschweigPockelsstra�e 3, 38106 Braunschweig, Germany
Contents
1 From Lattice Gas Automata to Lattice Boltzmann Equation | A
Historic Review 1
1.1 Lattice Gas Automata . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 evolution of lattice gas automata . . . . . . . . . . . . . . . . 11.1.2 collision operator . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Chapman-Enskog Analysis . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Lattice Gas Automata Hydrodynamics . . . . . . . . . . . . . . . . . 61.4 Lattice Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . 71.5 Lattice BGK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 LBE Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Mathematical Derivation of the Lattice Boltzmann Models 10
2.1 Integral Solution of the Boltzmann Equation . . . . . . . . . . . . . 102.2 Passage to Lattice Boltzmann Equation . . . . . . . . . . . . . . . . 11
2.2.1 low Mach number expansion of the equilibrium distributionfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 discretization and conservation laws . . . . . . . . . . . . . . 112.2.3 nine-velocity LBE model on a square lattice in two dimensions 12
2.3 Kinetic Theory for Dense Gases . . . . . . . . . . . . . . . . . . . . . 132.3.1 Boltzmann's Theory for Rare�ed Gases (1890') . . . . . . . . 132.3.2 Enskog's Theory for Dense Gases (1917) . . . . . . . . . . . . 132.3.3 Enskog Closure of Two-Particle Distribution Function f2 . . 142.3.4 Non-Local Collision Terms in the Enskog Equation . . . . . . 142.3.5 Normal Solutions of the Enskog Equation . . . . . . . . . . . 152.3.6 The Navier-Stokes Equations . . . . . . . . . . . . . . . . . . 152.3.7 Incompressible and Isothermal Fluids . . . . . . . . . . . . . 16
2.4 Lattice Boltmzann Model for Dense Gases . . . . . . . . . . . . . . . 162.4.1 Discretization of Continuous Enskog Equation . . . . . . . . 162.4.2 The External Forcing . . . . . . . . . . . . . . . . . . . . . . 172.4.3 The Lattice Boltzmann Model for Nonideal Gases . . . . . . 17
2.5 Some Recent Progress . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 A Critical Review of Existing Lattice Boltzmann Models for Nonideal
Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1
2.6.1 General ideas the LBE Models for Nonideal Gases (1990') . . 182.6.2 Model with Interaction Potential . . . . . . . . . . . . . . . . 192.6.3 Modi�ed Model with Interaction Potential . . . . . . . . . . . 192.6.4 Model with Free Energy . . . . . . . . . . . . . . . . . . . . . 202.6.5 Equivalence of Hamiltonian and Free Energy Approach . . . 21
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 The Generalized Lattice Boltzmann Equation 23
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 The Lattice Boltzmann Equation in Moment Space . . . . . . . . . . 243.3 The Generalized BGK Approximation in Moment Space . . . . . . . 253.4 The Equilibria in Moment Space . . . . . . . . . . . . . . . . . . . . 263.5 The Generalized Lattice Boltzmann Equation . . . . . . . . . . . . . 273.6 The Linearized Lattice Boltzmann Equation . . . . . . . . . . . . . . 273.7 Eigenvalue Problem of the Linearized Lattice Boltzmann Equation . 283.8 Determination of the Adjustable Parameters . . . . . . . . . . . . . . 283.9 Behaviors of Eigenvalues of the Linearized Collision Operator L . . . 293.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Boundary Conditions in LBE Method and Applications 33
4.1 Bounce-Back Boundary Conditions . . . . . . . . . . . . . . . . . . . 334.2 Bounce-Back Boundary Conditions with Interpolations . . . . . . . . 344.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Epilogue 39
2
List of Figures
1.1 Collisions of FHP LGA model. Note that the �gure does not includedthose collisions that can be obtained by applying rotations of multiple�=3 to input and output states simultaneously. . . . . . . . . . . . . 2
1.2 Evolution of FHP LGA model. Solid and hollow arrows representparticles with velocities corresponding to times t and t + 1, respec-tively. That is, the hollow arrows are the �nal con�gurations of theinitial con�gurations of solid arrows after one cycle of collision andadvection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Discrete velocity set on a square lattice in two dimensions. . . . . . . 13
3.1 Logarithmic eigenvalues of the nine-velocity model. The values of theparameters are �2 = �8, �3 = 4, c1 = �2, 1 = 3 = 2=3, 2 = 18,and 4 = �18. The relaxation parameters are: s2 = 1:64, s3 = 1:54,s5 = s7 = 1:9, and s8 = s9 = 1:99. The streaming velocity V isparallel to k with V = 0:2, and k is along the x axis. (a) Re(ln z�)and (b) Im(ln z�). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Stability of the generalized LBE model vs. the BGK LBE model inthe parameter space of V and s8 = 1=� . (left) max[Re(ln z�)] forgiven V . (right) Stability region of GLBE vs. LBGK model. . . . . 30
3.3 k-dependence of viscosities and g-factor. The solid lines, dotted lines,and dashed lines correspond to � = 0, �=8, and �=4, respectively.Three LBE model tested: (a) with no interpolation, (a) with centralinterpolation, and (c) with upwind interpolation. . . . . . . . . . . 30
3.4 Decay of discontinuous shear wave velocity pro�le uy(x; t). (left) Thelines and symbols (�) are theoretical and numerical results, respec-tively. Only the positive half of each velocity pro�le is shown. LBEmodel (a) with no interpolation, (b) with the central interpolationand r = 0:5. (right) Decay of uy(x; t) at a location close to the dis-continuity x = 3Nx=4. The solid lines and dashed lines are analyticand numerical results, respectively. The time is rescaled as r�2�k2t. 31
3
3.5 Decay of discontinuous shear wave velocity pro�le uy(x; t) with a con-stant streaming velocity Vx = 0:08 = U0. The solid lines and symbols(�) are theoretical and numerical results, respectively. The dashedlines in (b) and (c) are obtained by setting gn = 1. (a) no interpola-tion, (b) central interpolation and r = 0:5, (c) upwind interpolationand r = 0:5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1 Layout of the regularly spaced lattices and curved wall boundary. Thecircles (�), discs (�), shaded discs (�), and diamonds (3) denote uidnodes, boundary locations (xw), solid nodes which are also boundarynodes (xb) inside solid, and solid nodes, respectively. . . . . . . . . . 35
4.2 Illustration of the boundary conditions for a rigid wall located arbi-trarily between two grid sites in one dimension. The thin solid linesare the grid lines, the dashed line is the boundary location situatedarbitrarily between two grids. Shaded discs are the uid nodes, andthe discs (�) are the uid nodes next to boundary. Circles (�) are lo-cated in the uid region but not on grid nodes. The square boxes (2)are within the non- uid region. The thick arrows represent the tra-jectory of a particle interacting with the wall, described in Eqs. (4.3a)and (4.3b). The distribution functions at the locations indicated bydiscs are used to interpolate the distribution function at the locationmarked by the circles (�). (a) � � jxj � xwj=�x = 1=2. This is theperfect bounce-back scheme with no interpolations. (b) � < 1=2. (c)� � 1=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4
List of Tables
1.1 Collision table for the FHP-I six-velocity model. . . . . . . . . . . . 4
5
Abstract
The Lecture Notes are used for the a short course on the theory and applicationsof the lattice Boltzmann methods for computational uid dynamics taugh by theauthor at Institut f�ur Computeranwendungen im Bauingenieurwesen (CAB), Tech-nischen Universit�at Braunschweig, during August 7 { 12, 2003. The lectures coverthe basic theory of the lattice Boltzmann equation and its applications to hydrody-namics. Lecture One brie y reviews the history of the lattice gas automata and thelattice Boltzmann equation and their connections. Lecture Two provides an a priori
derivation of the lattice Boltzmann equation, which connects the lattice Boltzmannequation to the continuous Boltzmann equation and demonstrates that the latticeBoltzmann equation is indeed a special �nite di�erence form of the Boltzmann equa-tion. Lecture Two also includes the derivation of the lattice Boltzmann model fornonideal gases from the Enskog equation for dense gases. Lecture Three studiesthe generalized lattice Boltzmann equation with multiple relaxation times. A sum-mary is provided at the end of each Lecture. Lecture Four discusses the uid-solidboundary conditions in the lattice Boltzmann methods. Applications of the latticeBoltzmann mehod to particulate suspensions, turbulence ows, and other ows arealso shown. An Epilogue on the rationale of the lattice Boltzmann method is given.Some key references in the literature is also provided.
Lecture 1
From Lattice Gas Automata to Lattice Boltzmann Equation | A
Historic Review
1.1 Lattice Gas Automata
1.1.1 evolution of lattice gas automata
The lattice gas automaton (LGA) model proposed by Frisch, Hasslacher, and Pomeau[14], and Wolfram [65] evolves on a two-dimensional triangular lattice space. Theparticles have momenta that allow them to move from one site on the lattice to an-other in discrete time steps. A particular lattice site is occupied by either no particleor one particle with a particular momentum pointing to a nearest neighboring site.Therefore, at the most a lattice site can be simultaneously occupied by six particles,hence this model is called the six-velocity model or FHP-I model. The evolution ofthe LGA model consists of two steps: collision and advection. The collision processis partially described in Fig. 1.1. For example, two particles colliding with oppositemomenta will rotate their momenta 60� clockwise or counter-clockwise with equalprobability. In Fig. 1.1, we do not list those con�gurations which can be obtainedby rotational transformation, and which are invariant under the collision process.The particle number, the momentum, and the energy are conserved in the collisionprocess locally and exactly. (Because the FHP-I model has only one speed, theenergy is no longer an independent variable, it is equivalent to the particle number.However, for multi-speed models, the energy is an independent variable.)
The evolution of the lattice gas automata is very simple. The collision steponly involves local information, and the advection step is uniform. Both collisionand advection processes at each lattice site can be executed synchronously. Theevolution equation of the lattice gas automata can be written as
n�(xi + e�; t+ 1) = n�(xi; t) + C�(fn�g) ; (1.1)
where n�(xi; t) is the (Boolean) particle number of particles with velocity e�, n� 2f0; 1g; the subscript � and � denote discrete velocities, �; � 2 f1; 2; � � � ; bg asillustrated in Fig. 1.1, where b is the total number of the discrete velocities in theset fe�j� = 1; 2; : : : ; bg; and the discrete velocities in the FHP-I model are givenby
e� � (cos[(�� 1)�=3]; sin[(� � 1)�=3]) ; � = 1; 2; : : : ; 6; (1.2)
1
Input State Output State Input State Output State
1
2
6
3
4
5
1
2
6
3
4
5
1
2
6
3
4
5
1
2
6
3
4
5
1
2
6
3
4
5
1
2
6
3
4
5
1
2
6
3
4
5
1
2
6
3
4
5
1
2
6
3
4
5
1
2
6
3
4
5
Figure 1.1: Collisions of FHP LGA model. Note that the �gure does not included those
collisions that can be obtained by applying rotations of multiple �=3 to input and output states
simultaneously.
and C�(n) is the collision operator, and C� 2 f�1; 0; 1g for any Boolean LGAmodels. The local hydrodynamic quantities, such as density � and momentum �uare related to fn�g by
�(xi; t) = mX�
n�(xi; t) ; (1.3a)
�u(xi; t) = mX�
e�n�(xi; t) : (1.3b)
Fig. 1.2 illustrates the evolution of the system in one time step from t to t+�t. Inthis �gure, solid and hollow arrows represent particles with corresponding velocityat time t and t + �t, respectively. The system evolves by iteration of the collisionand advection processes. Evolution from t (solid arrows) to t+ 1 (hollow arrows):
1.1.2 collision operator
The collision operator is constructed such that the local conservation laws of mass,momentum, and energy are exactly preserved, i.e.,X
�
C� = 0 ; (1.4a)
X�
e�C� = 0 ; (1.4b)
X�
e2�C� = 0 : (1.4c)
2
Figure 1.2: Evolution of FHP LGA model. Solid and hollow arrows represent particles with
velocities corresponding to times t and t + 1, respectively. That is, the hollow arrows are the
�nal con�gurations of the initial con�gurations of solid arrows after one cycle of collision and
advection.
For the FHP model, the collision operator can be written in general as the following
C�(fn�(x; t)g) =Xs; s0
(s0� � s�) �ss0
bY�=1
ns�� (1� n�)(1�s�)
=Xs; s0
(s0� � s�) �ss0
bY�=1
�n�s� (1.5)
where s � fs1; s2; : : : ; sbg and s0 � fs01; s02; : : : ; s0bg are possible pre-collision andpost-collision Boolean state, respectively. The Boolean random number �ss0 mustsatisfy the following conditions (normalization, semi-detailed balance, and isotropyunder the discrete symmetry rotational group G):X
s0
�ss0 = 1 ; 8 s ; (1.6a)
Xs
�ss0 = 1 ; 8 s0 ; (1.6b)
�g(s)g(s0) = �ss0 ; 8 g 2 G ; and 8 s; s0; (1.6c)
and the conservation laws of mass, momentum, and energy:X�
(s� � s0�)h�ss0i = 0 ; (1.7a)
X�
(s� � s0�)e�h�ss0i = 0 ; (1.7b)
X�
(s� � s0�)e2�h�ss0i = 0 : (1.7c)
3
Input State Output State
010010001001
100100
010101 101010
001011 100110
110110011011
101101
Table 1.1: Collision table for the FHP-I six-velocity model.
For example, the two-body collision term in the six-velocity FHP model is given by
C(2)� = +�(2)R n�+1n�+4�n��n�+2�n�+3�n�+5
+�(2)L n�+2n�+5�n��n�+1�n�+3�n�+4 (1.8)
�[�(2)R + �(2)L ]n�n�+3�n�+1�n�+2�n�+4�n�+5 ;
where �n� � (1�n�) is the Boolean complement of n� �(2)
R and �(2)L the Boolean ran-dom numbers which determine a head-on two-body collision to rotate 60� clockwise(L) or counter-clockwise (R), as illustrated in Fig. 1.1, and they must satisfy theisotropic condition: h�(2)R i = h�(2)L i, and
In practice, the collision can be implemented with various algorithms. One caneither use logical operations, or by table-lookup. The collision rules shown in Fig. 1.1can also be represented by a collision table, as shown by Table 1.1. In Table 1.1,each bit in a binary number represents a particle number n�, � = 1; 2; : : : ; 6; fromright to left. The limitation of table lookup is the size of the table, which is 2b, whereb is the number of discrete velocities in the model. Both logic operations and tablelookup can be extremely fast on digital computers, and especially so on dedicatedcomputers [60].
1.2 Chapman-Enskog Analysis
The Chapman-Enskog analysis is a procedure to solve the Boltzmann equation bymeans of (asymptotic) perturbation technique [4]. Through the solution of theBoltzmann equation, the hydrodynamic equations and transport coe�cients can bederived from macroscopic dynamics. Based on dimensional analysis, a dimensionlessparameter " can be introduced in the collision term in the Boltzmann equation:
@tf + � �rf =1
"C(f; f); " � Kn =
l
L; (1.9)
4
where the perturbation parameter " is the Knudsen number, which is the ratiobetween microscopic and macroscopic characteristic lengths, l and L. The normalsolution of the Boltzmann equation satis�es the following ansatz:
f(x; �; t) = f(x; �; �; u; T ) : (1.10)
That is, the time dependence of the normal solution is through its dependence onthe local hydrodynamic (conserved) moments �, u, and T . The distribution functionf(x; �; t) can be expanded in terms of "
f =1Xn=0
"nf (n); (1.11)
with the constraints
Zd�f (0)
24 1
�
(� � u)2
35 = �
24 1
u
D�
35 ; (1.12)
Zd�f (n)
24 1
�
(� � u)2
35 = 0 ; n � 1 ; (1.13)
i.e., the higher order, non-equilibrium parts of f do not contribute to the hydro-dynamic (conserved) moments. However they do contribute to the gradients of thehydrodynamic moments. The collision term can also be expanded in terms of "
C(f; f) =
1Xn=0
"nC(n) ; C(n) =X
k+l=n
C(f (k); f (l)) : (1.14)
The normal solution can be obtained by solving the equations successively in theorder of ":
O("�1) : C(f (0); f (0)) = 0; (1.15a)
O("0) : @tf(0) + � �rf (0) = 2C(f (0); f (1)): (1.15b)
The O("�1) equation yields the Maxwellian equilibrium distribution function:
f (0) =�
(2��)D=2exp
��(� � u)2
2�
�: (1.16)
In general it is laborious to obtain the �rst order solution f (1) [4]. However, for theBoltzmann equation with Bhatnagar-Gross-Krook approximation [1],
@tf + � �rf = � 1
�[f � f (0)] ; (1.17)
5
the �rst order solution is easy to obtain:
f (1) = ��(@tf (0) + � �rf (0)) ; (1.18)
The hydrodynamic equations are obtained by evaluating the moments of theBoltzmann equation with the normal solutions:
Zd� (@tf + � �rf)
24 1
�12(� � u)2
35 = 0 : (1.19)
The above equation leads to the Euler equations for f = f (0), and to the Navier-Stokes equations for f = f (0) + f (1).
1.3 Lattice Gas Automata Hydrodynamics
The ensemble average of the LGA evolution equation 1.1 with the assumption ofrandom phase leads to the following equation [15]:
f�(xi + e�; t+ 1) = f�(xi; t) + �(f) ; (1.20)
where f�(xi; t) is the single particle distribution function with discrete velocity e�,f� = hn�i 2 [0; 1]. The lattice Boltzmann collision operator �(f) = hC�(n)i 2[�1; 1] is given by:
�(f) =Xs;s0
(s0� � s�) h�ss0ibY
�=1
f s�� (1� f�)(1�s�) ; (1.21)
where C�(n) is the LGA collision operator, and the random phase (molecular chaos)assumption is used to obtain �(f):
hf�f� � � � f i = hf�ihf�i � � � hf i (1.22)
The local hydrodynamic moments are computed from ff�g as the following:
�(xi; t) =X�
f�(xi; t) ; (1.23a)
�u(xi; t) =X�
e�f�(xi; t) : (1.23b)
Due to the Boolean nature of the lattice-gas automata, the equilibrium distribu-tion, which is the solution of �(f) = 0, is a Fermi-Dirac distribution:
f (0)� =1
1 + exp(a+ b � e�) ; (1.24)
6
where a and b are functions of the conserved moments and cannot be obtainedanalytically in general. Usually, a and b are obtained perturbatively as Taylor seriesof � and u in the limit of low Mach number (small u).
By applying the Chapman-Enskog analysis in the hydrodynamic limit (longwave-length and low frequency) to the lattice Boltzmann equation (1.20), the follow-ing macroscopic equation can be derived from the Frisch-Hasslacher-Pomeau (FHP)lattice-gas automaton model [14, 65, 15] in the low March number limit:
@t(�u) +r�[g(�)�uu] = �rP + �r2(�u) + �rr�(�u): (1.25)
The FHP-I (six-velocity) lattice-gas model have some obvious shortcomings:
� The LGA simulations are intrinsically noisy due to large uctuation of parti-cle number n�. A spatial or temporal average is required to obtain smoothmeasurements;
� The LGA models are lack of Galilean invariance because the velocity spaceis discrete and �nite. This is re ected by the fact that g(�) 6= 1 in the LGAhydrodynamic equation;
� The viscosity � of the LGA models are determined by their collision mech-anisms. It is di�cult to increase the Reynolds number Re due to the lowerbound of �;
� The equation of state (for the FHP-I six-velocity model)
P = c2s�
�1� g(�)
u2
c2
�(1.26)
is unphysical because its dependence on u2.
� There exist (unphysical) spurious conserved quantities due to the simplicity ofspatial-temporal dynamics of the LGA systems.
Much of research e�ort has been to overcome the artifacts of the lattice gasautomata. There are two approaches to remedy the shortcomings of the FHP-Ilattice-gas automata. One is to construct more complicate lattice gas model withmore discrete velocities. The other is to use the lattice Boltzmann equation.
1.4 Lattice Boltzmann Equation
Historically, the lattice Boltzmann equation is directly obtained from the latticegas automata by taking ensemble average of Eq. (1.1) [48]. However, such latticeBoltzmann scheme is di�cult to be generalized in three-dimensions, and it is com-putationally ine�cient due to the cumbersome collision operator.
7
To improve the computational e�ciency, one can use linearized collision operator[30]. The collision operator linearized about the equilibrium is given by
L�� =@�
@f�
����f=f(0)
[f� � f (0)� ] : (1.27)
With the linearized collision operator, the LBE computation is greatly simpli�ed.The collision operator becomes a b� b matrix of constant matrix elements.
1.5 Lattice BGK Model
The linearized lattice Boltzmann equation can be further simpli�ed by using theBhatnagar-Gross-Krook approximation of single relaxation time [1]. Thus the colli-sion process in the lattice BGK model [5, 55] is characterized by a relaxation time� :
�(f) = �1
�[f� � f (eq)
� ] : (1.28)
The equilibrium distribution function is generally in the form of
f (eq)� = w��
�1 +A (e� � u) +B (e� � u)2 + C u2
�; (1.29)
where w�, A, B, and C are determined by conservation laws. The lattice BGKmodel [55, 5] is described by the following equation
f�(xi + e�; t+ 1) = f�(xi; t)� 1
�
�f�(xi; t)� f (eq)
� (xi; t)�: (1.30)
1.6 LBE Hydrodynamics
In the limits of Kn; �x; �t ! 0, the Navier-Stokes equation can be derived from thelattice BGK equation:
� @tu+ �ur�u = �rP + � �r2u ; (1.31)
with the isothermal equation of state and the viscosity given by
P = c2s � ; (1.32a)
� = c2s
�� � 1
2
�; (1.32b)
where cs �pkBT=m =
p� is the sound speed, depending on the discrete velocity
set. For the FHP six-velocity model on a triangular lattice in two dimensions,cs = 1=
p2, and for the nine-velocity model on a square lattice n two dimensions,
cs = 1=p3.
8
Obviously, the lattice Boltzmann equation overcomes some the shortcomings ofthe lattice gas automata. First, the severe uctuation is eliminated. Secondly, theviscosity is easy to adjust. Thirdly, Galilean invariance is restored up to a certainorder in wave number k. And the equation of state has no unphysical dependenceon u2. However, the lattice Boltzmann equation cannot completely eliminate thespurious invariant quantities. In addition, the roundo� error causes instability inthe lattice Boltzmann equation.
1.7 Summary
The lattice-gas automata have the following features:
� Mimicking (simpli�ed) molecular dynamics of structureless particles;
� Exactly preserving conservation laws;
� Processing basic symmetries to simulate hydrodynamics;
� Including spurious conserved quantities due to simplicity of spatial-temporaldynamics;
� Large uctuations is intrinsic to the system;
� LGA computation is Boolean in nature (integer or logic or table-lookup algo-rithms).
In contrast, the lattice Boltzmann equation has the following features:
� Simulate hydrodynamics based on a drastically simpli�ed Boltzmann equationwith a small set of discrete velocities;
� Overcoming some defects of the lattice-gas automata, such as large uctua-tions, in exibility to adjust the viscosity, non-Galilean invariant, and unphys-ical equation of state.
� Better numerical e�ciency in compared with the lattice gas automata undercertain conditions.
In conclusion, the LGA and LBE methods can simulate hydrodynamics.
9
Lecture 2
Mathematical Derivation of the Lattice Boltzmann Models
2.1 Integral Solution of the Boltzmann Equation
For the sake of simplicity without losing generality, we study the continuous Boltz-mann equation with Bhatnagar-Gross-Krook (BGK) approximation [1]:
@tf + � �rf = � 1
�[f � f (0)] ; f � f(x; �; t) ; (2.1)
where f (0) is the Boltzmann-Maxwellian equilibrium distribution function:
f (0) = � (2��)�D=2 exp
��(� � u)2
2�
�; (2.2)
where � = kBT=m, kB , T , andm are the Boltzmann constant, temperature, and par-ticle mass, respectively. The macroscopic quantities are the hydrodynamic momentsof f or f (0):
� =
Zfd� =
Zf (0)d� ; (2.3a)
�u =
Z�fd� =
Z�f (0)d� ; (2.3b)
D
2�� =
1
2
Z(� � u)2fd� =
1
2
Z(� � u)2f (0)d� : (2.3c)
Rewrite the Boltzmann BGK Equation in the form of ODE:
Dtf +1
�f =
1
�f (0) ; Dt � @t + � �r ; (2.4)
and integrate Eq. (2.4) over a time step �t along characteristics, we have:
f(x+ ��t; �; t+ �t) = e��t=� f(x; �; t) (2.5)
+1
�e��t=�
Z �t
0et
0=� f (0)(x+ �t0; �; t+ t0) dt0 :
By Taylor expansion, and with � � �=�t, we obtain:
f(x+ ��t; �; t+ �t)� f(x; �; t) = �1
�[f(x; �; t)� f (0)(x; �; t)] +O(�2t ) : (2.6)
Note that a �nite-volume scheme or higher-order schemes can also be formulatedbased upon the integral solution.
10
2.2 Passage to Lattice Boltzmann Equation
There are three necessary steps to obtain the lattice Boltzmann equation fromEq. (eqn:bgk-integral):
1. Low Mach number expansion of the equilibrium distribution function;
2. Discretization of velocity space � to obtain necessary and minimum numberof discrete velocities f��g;
3. Discretization of x space according to f��g and �t.
2.2.1 low Mach number expansion of the equilibrium distribution func-
tion
Low Mach number (u � 0) expansion of the equilibrium distribution function f (0)
up to O(u2) is su�cient to derive the Navier-Stokes equations:
f (eq) =�
(2��)D=2exp
���
2
2�
��1 +
� � u�
+(� � u)22�2
� u2
2�
�+O(u3) : (2.7)
It should be noted that many defects of the lattice Boltzmann method are relatedto the above low Mach number expansion of the equilibrium function. However, thisexpansion is necessary to make the lattice Boltzmann method a simple and explicit
scheme.
2.2.2 discretization and conservation laws
The conservation laws are preserved exactly, if the hydrodynamic moments (�, �u,and ��) are evaluated exactly:
I =
Z�mf (eq)d� =
Zexp(��2=2�) (�)d�; (2.8)
where 0 � m � 3, and (�) is a polynomial in �. The above integral can be evaluatedby quadrature:
I =
Zexp(��2=2�) (�)d�=
Xj
Wj exp(��2j =2�) (�j) (2.9)
where �j and Wj are the abscissas and the weights. Then
� =X�
f (eq)� =
X�
f�; �u =X�
��f(eq)� =
X�
��f�; (2.10)
where f� � f�(x; t) � W�f(x; ��; t), and f (eq)� � W�f
(eq)(x; ��; t). The keymessage is that the quadrature must preserve the conservation laws exactly .
11
2.2.3 nine-velocity LBE model on a square lattice in two dimensions
In two-dimensional Cartesian (velocity) space, set
(�) = �mx �ny ;
the integral of the moments can be given by
I = (p2�)(m+n+2)ImIn; Im =
Z +1
�1
e��2�md�; (2.11)
where � = �x=p2� or �y=
p2�. The second-order Hermite formula (k = 2) is the
optimal choice to evaluate Im for the purpose of deriving the nine-velocity model ona two-dimensional square lattice, i.e.,
Im =3X
j=1
!j�mj :
Note that the above quadrature is exact up to m = 5 = (2k+1). The three abscissasin momentum space (�j) and the corresponding weights (!j) are:
�1 = �p3=2 ; �2 = 0 ; �3 =
p3=2 ;
!1 =p�=6 ; !2 = 2
p�=3 ; !3 =
p�=6 :
(2.12)
Then, the integral of moments becomes:
I = 2�
"!22 (0) +
4X�=1
!1!2 (��) +
8X�=5
!21 (��)
#; (2.13)
where
�� =
8<:
(0; 0) � = 0;
(�1; 0)p3�; (0; �1)p3�; � = 1 { 4;
(�1; �1)p3�; � = 5 { 8:
(2.14)
Identifying c � �x=�t =p3�, or c2s = � = c2=3,
W� = (2� �) exp(�2�=2�)w� ; (2.15)
where �x is the lattice constant, then we have
f (eq)� (x; t) = W� f
(eq)(x; ��; t)
= w� �
�1 +
3(e� � u)c2
+9(e� � u)2
2c4� 3u2
2c2
�; (2.16)
12
0 1
2 56
3
7 4 8
Figure 2.1: Discrete velocity set on a square lattice in two dimensions.
where weight coe�cient w� and discrete velocity e� are:
w� =
8<:
4=9;1=9;1=36;
e� = �� =
8<:
(0; 0); � = 0 ;(�1; 0) c; (0; �1) c; � = 1 { 4;(�1; �1) c; � = 5 { 8:
(2.17)
With fe�j� = 0; 1; : : : ; 8g, a square lattice structure is constructed in the physicalspace, as shown in Fig. 2.1. (This model is also denoted as D2Q9 model.) Similarly,other LBE models in either two-dimensions (D2Q6 and D2Q7) or three-dimensions(D3Q27) can be derived [24].
2.3 Kinetic Theory for Dense Gases
2.3.1 Boltzmann's Theory for Rare�ed Gases (1890')
The Boltzmann equation
@tf + � �rf + a �r�f =
Zd�1 [f
0f 01 � ff1] (2.18)
is valid in the Boltzmann Gas Limit (BGL):
Particle Number N !1 ; (2.19a)
Interaction Range r0 ! 0 ; (2.19b)
Mean-Free-Path l � (Nr20 )�1 ! Constant ; (2.19c)
Interaction Volume Nr30 ! 0 : (2.19d)
Because of Nr30 ! 0, the Boltzmann equation can only retain the thermodynamicsof ideal gases.
2.3.2 Enskog's Theory for Dense Gases (1917)
For hard spheres of radius r0, the Boltzmann equation is modi�ed for dense gasesas follows (by Enskog):
@tf + � �rf + a�r�f = J ; (2.20)
13
where a is the acceleration due to external �eld, and J is the Enskog collision term
J =
Zd�1 [g(x+ r0r̂)f
0f 01(x+ 2r0r̂)� g(x� r0r̂)ff1(x� 2r0r̂)] ; (2.21)
g is the radial distribution function, r̂ is the unit vector in the direction from thecenter of the second particle of f(x; �1) to the center of the �rst particle of f(x; �) atthe instant of contact during a collision, and �1 is the collisional space of the secondparticle of f(x; �1). The Enskog collision term J can be expanded as a Taylor seriesin space:
J = J (0) + J (1) + J (2) + � � � ; (2.22a)
J (0) = g
Zd�1[f
0f 01 � ff1] ; (2.22b)
J (1) = r0
Zd�1r̂ �rg[f 0f 01 + ff1] ; (2.22c)
J (2) = 2r0g
Zd�1r̂ � [f 0rf 01 + frf1] : (2.22d)
It is only necessary to retain terms up to the �rst order gradient.The essence of Enskog's theory is to explicitly consider the volume exclusion
e�ect in real gases consisting particles of �nite sizes. The collision is non-localfor particles of �nite sizes. One rami�cation of the non-local collision is that theconservation laws only hold globally, but not locally. It should be noted that, similarto the Boltzmann equation, the Enskog equation has anH-Theorem and a consistentthermodynamics.
2.3.3 Enskog Closure of Two-Particle Distribution Function f2
The essential assumption in Enskog's theory is the factorization of the two particledistribution function | the Enskog closure:
f2(x1; �1; x2; �2; t) = g(jx1 � x2j)f1(x1; �1; t)f1(x2; �2; t) ; (2.23)
where g is the radial distribution function (pairwise correlation). In contrast, theBoltzmann closure is
f2(x1; �1; x2; �2; t) = f1(x1; �1; t)f1(x2; �2; t) :
2.3.4 Non-Local Collision Terms in the Enskog Equation
With the approximation f � f (0), which is consistent with the Chapman-Enskoganalysis, we have
J (1) = �f (0) b � �0 �rg ; (2.24a)
14
J (2) = �f (0) b � g
�2�0 �r ln�+
2
(D + 2)
�0i �0j @iuj�
+
�1
(D + 2)
�20�� 1
�r�u
+1
2
�D
(D + 2)
�20�� 1
��0 �r ln �
�; (2.24b)
where f (0) is the Maxwellian equilibrium distribution function given by
f (0)(�; u; �) = �(2��)�D=2 exp
��(� � u)2
2�
�; (2.25)
and �0 = (��u) is the peculiar velocity, � = kBT=m is the normalized temperature,and b = V0=m = 4�r30=3m is the second virial coe�cient.
2.3.5 Normal Solutions of the Enskog Equation
The �rst and second order normal solution of the Enskog equation, obtained via
Chapman-Enskog analysis, are:
f (0) = � (2��)�D=2 exp
��(� � u)2
2�
�; (2.26)
f (1) = �f (0) 1g
��1 +
2
(D + 2)b�g
�A �r ln �
+
�1 +
4
D(D + 2)b�g
�Bij@iuj
�: (2.27)
With b = 0 and g = 1, the solutions reduce to that of the Boltzmann equation.Note that r� does not appear in f (1), it appears in f (2) | the Burnett solution
(1935). This is consistent with the dimensional analysis of the Navier-Stokes equa-tion, r� is in the order of O(K2
n), where Kn is the Knudsen number, which is alsothe small expansion parameter in the Chapman-Enskog analysis.
2.3.6 The Navier-Stokes Equations
The Navier-Stokes equations derived from the Enskog equation are
@t�+r� (�u) = 0 ; (2.28a)
@tu+ u �ru = �1
�rP+ a ; (2.28b)
@t� + u �r� = �1
�r� q � 1
�Pij@iuj + a � u ; (2.28c)
15
where
Pij =
Zd� �0i�0jf = [��(1 + b�g)� �2r� u] �ij
�"2
g
�1 +
4
D(D + 2)b�g
�2
�1 +2D
(D + 2)�2
#Sij ; (2.29a)
Sij =1
2[@iuj + @jui]� 1
Dr� u �ij ; (2.29b)
q =
Zd�
1
2�20�0f = �
�1
g(1 + b�g)2�+
D
2�2
�r� : (2.29c)
2.3.7 Incompressible and Isothermal Fluids
Because for incompressible and isothermal uids, r�u = 0 and r� = 0, therefore,
J (1) + J (2) � �f (0) b� (� � u)�[rg + gr ln �2]
= �f (0) b� g (� � u)�r ln(�2g) = J 0 : (2.30)
The modi�ed Boltzmann equation, with BGK approximation, is
@tf + � �rf + a �r�f = � g�[f � f (0)]� f (0) b� g (� � u)�r ln(�2g) : (2.31)
The equation of state derived from the above modi�ed Boltzmann equation (obtainedby computing the �rst moment of the non-local collision term) is
P = � � [1 + b� g] ; g = g(b�) : (2.32)
It should be noted that the non-ideal gas e�ects come from the non-local collisionterm, which is the manifestation of the volume exclusion e�ect, or other inter-particleinteractions. It cannot be a result due to a body force.
2.4 Lattice Boltmzann Model for Dense Gases
2.4.1 Discretization of Continuous Enskog Equation
Similar to the previous Section, we can rewrite the Enskog BGK Equation in theform of ODE:
df
dt+g
�f =
g
�f (0) + J 0 ;
d
dt� @
@t+ � �r+ a�r� : (2.33)
Integration of Eq. (2.33) over a time step �t along characteristic line leads to
f(x+ ��t +1
2a�2t ; � + a�t; t+ �t) = e��tg=� f(x; �; t) (2.34)
+e��tg=�Z �t
0dt0 et
0g=�h g�f (0) + J 0
i(x+�t0+ 1
2at02; �+at0; t+t0)
:
16
By Taylor expansion, and with � � �=�t, we obtain:
f(x+ ��t; �; t+ �t)� f(x; �; t) = �g�[f(x; �; t)� f (0)(x; �; t)]
+�J 0 � a�r�f
��t +O(�2t ) : (2.35)
This completes the temporal discretization. The discretization of phase space (x; �)can be accomplished in the same manner as discussed in the previous Lecture. How-ever, there are two extra terms needed to be dealt with here: the forcing term andthe Enskog collision term.
2.4.2 The External Forcing
The forcing term must satisfy the following moment constraints:Zd� a�r�f = 0 ; (2.36a)Zd� � a�r�f = ��a ; (2.36b)Zd� �i�j a�r�f = �� (aiuj + ajui) : (2.36c)
Similar to the equilibrium, the forcing term is expanded in term of u as the following
a �r�f = �� exp(��2=2�)�1
�(� � u) +
(� �u)�2
�
�� a : (2.37)
Note that in the above expansion, only the terms up to the �rst order in u havebeen retained, because there is an overall factor of �t in the forcing term. If thesecond-order moment constraint Eq. (2.36) is ignored:
a �r�f = �� exp(��2=2�) 1�� � a : (2.38)
2.4.3 The Lattice Boltzmann Model for Nonideal Gases
The LBE model for non-ideal gases derived from the Enskog equation for dense gasesis given as the following:
f�(x+ e��t; t+ �t)� f�(x; t) = �g�[f� � f (eq)
� ] + (J 0� + F�) �t (2.39)
where
J 0� = �f (eq)� b�g (e� � u) �r ln(�2g) ; (2.40a)
F� = w��
�3
c2(e� � u) +
9
c4(e� � u)e�
�� a : (2.40b)
17
The equation of state and the viscosity of the model are given by
P = ��[1 + b�g] ; (2.41a)
� =1
3
��
g� 1
2
�c �x =
��
g� 1
3
�� �t : (2.41b)
Note that the viscosity depends on the radial distribution function g. This depen-dence of � on g can be eliminated by setting the relaxation parameter to 1=� . Forhard-spheres, the radial distribution function is [4]
g = 1 +5
8b�+ 0:2869(b�)2 + : : : :
2.5 Some Recent Progress
Once it is realized that the lattice Boltzmann equation is related to a partial di�er-ential equation, the Boltzmann equation, many numerical techniques used to solvePDEs can be immediately applied to solve the lattice Boltzmann equation.
First, one can abandon the regular lattice of the LBE method by decouplingthe discretizations between space, time and momentum space, and using inter-polation/extrapolation techniques [26, 27]. With the freedom of using interpola-tion/extrapolation techniques, both body-�tted mesh [22, 23] and grid re�nement[13] can be implemented in the LBE method.
Secondly, one can devise implicit method [62] or multi-grid technique to solvethe lattice Boltzmann equation for steady state calculations. These techniques canaccelerate the computational speed by two orders of magnitude.
2.6 A Critical Review of Existing Lattice Boltzmann Models for Non-
ideal Gases
2.6.1 General ideas the LBE Models for Nonideal Gases (1990')
The usual practice to achieve nonideal gas e�ect by using a lattice Boltzmann modelconsists the following two tricks:
� Use of interaction potential, or \brute force";
� Creation of \new" and \nonideal" equilibrium distribution function.
That is, given a hydrodynamic equation with an (almost) arbitrary stress tensor �:
�@tu+ �u�ru =r�
Then, employ the following tricks:P� e�;ie�;jf
(eq)�
P� e�;ie�;jF�
9=;)r�
18
2.6.2 Model with Interaction Potential
Given a potential U [56, 57], �a = �r�U , then consider
�u =X�
e�f� + �a�t =X�
e�f� + ��u (2.42)
In the Navier-Stokes equation, rewrite [56, 57]
�rP + �a = �r(�� + �U) ; �t = 1 : (2.43)
Therefore, e�ectively:P = ��[1 + U=�] (2.44)
Equivalent to [up to O(�t)]:f (0)� = f (0)
� (u+ �u) (2.45)
Defects of this model are:
� Lack of equilibrium thermodynamics;
� Incorrect heat transfer (�a � u does not a�ect heat ux q).
2.6.3 Modi�ed Model with Interaction Potential
With a few crude approximations, such as f = f (0) [28], make
a �r�f � �1
�f (0)(� � u) � a (2.46)
and assume the forcing:
a = �rV � b��gr ln(�2g) (2.47)
where potential V accounts for the attractive component of the intermolecular pair-wise potential. The \equation of state" for this model is:
P = �� (1 + b�g) + V (2.48)
This model shares the same defects of the previous one conceptually. In addition,V is redundant | its e�ect can be included in g.
19
2.6.4 Model with Free Energy
Use the free energy inspired by Cahn-Hilliard's model
=
Zdxh�2kr�k2 + (�)
i; (2.49)
the the pressure tensor is given by [59, 58]:
P = ��
��� = p� ��r2�� �
2kr�k2 ; (2.50a)
Pij = P �ij + � @i� @j� : (2.50b)
The equation of state isp = � 0 � : (2.51)
The equilibrium distribution function is obtained by imposing the following con-straint: X
�
e�;ie�;jf(eq)� = Pij (2.52)
The equilibrium distribution function satis�ed the above constraint is
f (eq)� =1
3�
�1 + (e� � u) + 2(e� � u)2 � 1
2u2
�(2.53)
+�
3
�(e2�;x � e2�;y)
�(@x�)
2 � (@y�)2�+ 2e�;xe�;y@x� @y�
��3�r2�+
1
3[� 0(�)� (�)� �] :
There are three parts in the above equilibrium distribution. The �rst term in brack-ets [ ] is nothing but the usual equilibrium distribution function of the seven-velocityFrisch-Hasslacher-Pomeau model [15, 25]. The second term in bracket f g is an ex-pression of the tensor Eij � (e�;i@i�)(e�;j@j�) written in terms of a traceless and ano�-diagonal part with correct symmetry such that all the terms proportional to �reduce to the term �[�r2�+ kr�k2=2] in the diagonal part of the pressure tensor,given by Eq. (2.50a). This term is directly taken from Cahn-Hilliard's model and itinduces surface tension due to density gradient in addition to the part due to the(nonideal gas) equation of state, but it does not contribute to the hydrodynamicpressure (or the equation of state). The nonideal gas part in the equation of stateis contained in the last part of the above equation, [� 0(�)� (�)� �]=3, which canbe written in a density expansion in general [41, 43]. This term can be viewed as anequivalent forcing term as the following:
F� = �[� 0(�)� (�)] = e� �rU : (2.54)
20
The connection between the free-energy model and the interaction model becomesexplicit and obvious now. The di�erence among the two lies in their numericalimplementations.
Defects of the free-energy model are:
� Non-Navier-Stokes: lost of Galilean invariance;
� Inconsistent with the Chapman-Enskog analysis;
� Inconsistent thermodynamics: temperature depends on r�, etc.;
� Incorrect heat transfer.
2.6.5 Equivalence of Hamiltonian and Free Energy Approach
Given a Hamiltonian
H =
NXi=1
�1
2m�2i + U(ri)
�+Xi<j
Vij(jri � rj j) (2.55)
the partition function can be constructed
Z =
ZdrNd�N exp(�H=kBT ) (2.56)
then the free energy can be obtained
F = �kBT lnZ : (2.57)
Therefore the formalisms of H and F are equivalent | there is gain or lost ofinformation by using one formalism or the other. The di�erence is that the Hamil-tonian H is local whereas the free energy functional F is global. Furthermore, thefree-energy density (e.g., (�)) is local.
2.7 Summary
The conclusions which we can draw are:
� The lattice Boltzmann equation can be directly derived from partial di�erentialequation without being referenced to its historic predecessor | the lattice gasautomata.
� The lattice Boltzmann equation is a special �nite di�erence form of the Boltz-mann equation. Phase space and time are discretized in a coherent mannersuch that physical space becomes a lattice space.
21
� In the lattice Boltzmann equation, the conservation laws are preserved rigor-ously with discrete momentum space.
� Many numerical techniques for solving PDEs can be used to improve the latticeBoltzmann method.
For the LBE models for nonideal gases, we have discussed three existing LBEmodels for nonideal gases. We concluded that:
� Lattice Boltzmann for nonideal gases can be derived from the Enskog equation;
� Most existing LBE models are ad hoc, and conceptually incorrect;
� Numerics is very important in the interface dynamics.
22
Lecture 3
The Generalized Lattice Boltzmann Equation
3.1 Motivation
As it is shown, the lattice Boltzmann equation is a special �nite di�erence form ofthe Boltzmann equation. The most drastic approximation made in the derivationof the lattice Boltzmann equation is the discretization of momentum space � into avery small set of discrete velocities f��j� = 1; : : : ; bg. The discretization of phasespace and time inevitably introduces truncation error and numerical artifacts. Itis highly desirable to reduce the e�ect of the artifacts. In the lattice Boltzmannmethod, the artifacts due to the following factors are to be analyzed:
� Dissipation due to hyperviscosity. The higher order truncation error wouldresult to hyperviscosty in the LBE hydrodynamics. The hyperviscosity hasthe following form in general
�(k) = �0 � �1k2 + �2k
4 + � � �+ (�1)n�nk2n + � � � : (3.1)
� Galilean Invariance. For any �nite di�erence scheme, Galilean invariance canonly be satis�ed up to a certain extend. In a reference frame with velocity V ,the phase of a plain wave is adjusted accordingly:
exp[i(k�x� !t)] =) exp[i(k�x� !t)� ig(k)k�V t] (3.2)
where g is the Galilean coe�cient, which can be expressed as the following:
g(k) = g0 � g1k2 + g2k
4 + � � � + (�1)ngnk2n + � � � : (3.3)
If the system is Galilean invariant, then we must have g = 1.
� Isotropy. Discretization causes anisotropic e�ect, that is, the transport coe�-cients, the shear viscosity �(k), the bulk viscosity �(k), the sound speed cs(k),and the Galilean coe�cient g(k), all depend on the direction of wave vector k.
We propose to studies the generalized hydrodynamics [39, 21] of the lattice Boltz-mann equation by a systematic analysis of the linearized dispersion equation of thelattice Boltzmann equation [37]. The analysis would help us to optimize the latticeBoltzmann method in terms of reducing its a�ects.
23
3.2 The Lattice Boltzmann Equation in Moment Space
The lattice Boltzmann equation in particle velocity space is
f�(xi + e�; t+ 1) = f�(xi; t) + �(f) : (3.4)
The hydrodynamic moments are computed from ff�g as follows:
�(xi; t) =X�
f�(xi; t) ; (3.5a)
�u(xi; t) =X�
e�f�(xi; t) : (3.5b)
The lattice Boltzmann equation (3.4) can be rewritten in a concise vector form:
jf(xi + e�; t+ 1)i = jf(xi; t)i+ j(f)i (3.6)
where the Dirac notations of bra h�j for row vector and ket j�i for column vector areused,
jf(xj)i � (f0; f1; : : : ; f8)T ; (3.7a)
jf(xj + e�; t+ 1)i � (f0(xj + e0; t+ 1); : : : ; f8(xj + e8; t+ 1))T ; (3.7b)
j(f)i � (0(f); 1(f); : : : ; 8(f))T ; (3.7c)
and T is the transpose operator.Given a set of b discrete velocities, fe�j� = 0; 1; : : : ; (b�1)g, with corresponding
distribution functions, ff�j� = 0; 1; : : : ; (b�1)g, one can construct a b-dimensionalvector space V = R
b based upon the discrete velocity set. One can also construct aspace M = R
b based upon the (velocity) moments of ff�g. Obviously, there are bindependent moments for the discrete velocity set. The reason in favor of using themoment representation over the distribution function representation is somewhatobvious. It is well understood in the context of kinetic theory that various physicalprocesses in uids, such as viscous transport, can be approximately described bycoupling or interaction among `modes' (of the collision operator), and these modesare directly related to the moments (e.g., the hydrodynamic modes are linear combi-nations of mass, and momenta moments). Thus the moment representation providesa convenient and e�ective means by which to incorporate the physics into the LBEmodels. Because the physical signi�cance of the moments is obvious (hydrodynamicquantities and their uxes, etc.), the relaxation parameters of the moments are di-rectly related to the various transport coe�cients. This mechanism allows us tocontrol each mode independently. This also overcomes some obvious de�ciencies ofthe usual BGK LBE model, such as a �xed Prandtl number, which is due to a singlerelaxation parameter of the model.
24
For the nine-velocity LBE model on a square lattice in two-dimensions, thefollowing mapping
M �
0BBBBBBBBBBBB@
h�jhejh"jhjxjhqxjhjy jhqyjhpxxjhpxyj
1CCCCCCCCCCCCA�
0BBBBBBBBBBBB@
1 1 1 1 1 1 1 1 1�4 �1 �1 �1 �1 2 2 2 24 �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
1CCCCCCCCCCCCA
(3.8)
� (j�i; jei; j"i; jjxi; jqxi; jjyi; jqyi; jpxxi; jpxyi)T
uniquely maps a vector jfi in discrete velocity space V = Rb to a vector j%i inmoment space M = R
b , and vice versa, that is
j%i = Mjfi ; jfi = M�1j%i : (3.9)
Note that the row vectors in M have explicit physical signi�cances related to themoments of ff�g in discrete velocity space: j�i is the density mode; jei is theenergy mode; j"i is related to energy square; jjxi and jjyi correspond to the x andy components of momentum (mass ux); jqxi and jqyi correspond to the x andy components of energy ux; and jpxxi and jpxyi correspond to the diagonal ando�-diagonal components of the stress tensor.
The moments for the nine-velocity model are:
Order Quantity Moment
0 Density: � = h�jfi = hf j�i ;2 Energy: e = hejfi = hf jei4 Energy Square: " = h"jfi = hf j"i ;1 x-Momentum: jx = hjxjfi = hf jjxi ;3 x-Heat Flux: qx = hqxjfi = hf jqxi ;1 y-Momentum: jy = hjyjfi = hf jjyi ;3 y-Heat Flux: qy = hqyjfi = hf jqyi ;2 Diagonal stress: pxx = hpxxjfi = hf jpxxi ;2 O�-diagonal stress: pxy = hpxyjfi = hf jpxyi :
3.3 The Generalized BGK Approximation in Moment Space
Instead of the single relaxation time approximation, one can use multiple relaxationtimes approximation. This is a generalization of the BGK approximation [31, 8].
25
For the nine-velocity LBE model, at most there can be six independent relaxationparameters because there are only six kinetic (non-conservative) modes in the model.Therefore, the relaxation process can be cast as the following0BBBBBBBBBBBB@
���e�"�jx�qx�jy�qy�pxx�pxy
1CCCCCCCCCCCCA
=
0BBBBBBBBBBBB@
0 0 0 0 0 0 0 0 00 �s2 0 0 0 0 0 0 00 0 �s3 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 �s5 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 �s7 0 00 0 0 0 0 0 0 �s8 00 0 0 0 0 0 0 0 �s9
1CCCCCCCCCCCCA
0BBBBBBBBBBBB@
���e�"�jx�qx�jy�qy�pxx�pxy
1CCCCCCCCCCCCA
(3.10)
where �%� denotes the change of the moment %� due to collision (or relaxation),while �%� denotes the deviation from the equilibrium. In vector notation, we have
j�%i � j%i � j%(eq)i ; (3.11a)
j�%i = S j�%i : (3.11b)
3.4 The Equilibria in Moment Space
The equilibrium distribution functions depend only upon conserved moments:
e(eq) =1
hejei��2 h�j�i �+ 2 (hjxjjxij2x + hjyjjyij2y)
�(3.12a)
=1
4�2 �+
1
6 2 (j
2x + j2y) ;
"(eq) =1
h"j"i��3 h�j�i �+ 4 (hjxjjxij2x + hjyjjyij2y )
�(3.12b)
=1
4�3 �+
1
6 4 (j
2x + j2y) ;
q(eq)x =hjxjjxihqxjqxic1jx =
1
2c1jx ; (3.12c)
q(eq)y =hjyjjyihqyjqyic1jy =
1
2c1jy ; (3.12d)
p(eq)xx = 11
hpxxjpxxi (hjxjjxij2x � hjyjjyij2y) =
3
2 1(j
2x � j2y ) ; (3.12e)
p(eq)xy = 3
phjxjjxihjy jjyihpxxjpxxi (jxjy) =
3
2 3(jxjy) : (3.12f)
There are seven adjustable parameters in the model: �2, �3, c1, 1, 2, 3, 4. Theseparameters will be determined by the analysis of the linearized dispersion equation[37].
26
3.5 The Generalized Lattice Boltzmann Equation
The generalized lattice Boltzmann equation with multiple relaxation parameters iswritten as [8, 37]
jf(xi + e�; t+ 1)i = jf(xi; t)i+M�1S [j%(xi; t)i � j%(eq)(xi; t)i] : (3.13)
The computation of the generalized lattice Boltzmann equation involves the follow-ing steps (after the initialization of jfi):
� Project jfi to moments by j%i = Mjfi, and compute the equilibrium j%(eq)i;� Collision in moment space (multiple-parameter relaxation)
j�%i � S [j%i � j%(eq)i] ;
� Project j%i back to jfi = M�1j%i to perform advection step in velocity space:
jf(xi + e�; t+ 1)i = jf(xi; t)i+M�1j�%i :
The computational overhead due to the projections between V and M is not heavy.It is generally about 10 { 20% of the of the LBGK algorithm.
3.6 The Linearized Lattice Boltzmann Equation
Suppose the system in uniform state of � and V = (Vx; Vy), and
jfi = jf (0)i+ j�fi : (3.14)
The lattice Boltzmann equation can be linearized:
j�f(xj + e�; t+ 1)i = j�f(xj ; t)i+M�1CMj�f(xj ; t)i : (3.15)
In Fourier space, the above linearized LBE becomes
Aj�f(k; t+ 1)i = j�f(k; t)i+M�1CMj�f(k; t)i ; (3.16)
where the linearized collision operator C and the advection operator A are given by:
C�� =h%�j%�ih%� j%�i
@
@%�[%� � %
(eq)� ] ; (3.17a)
A�� = exp(ie� � k) ��� : (3.17b)
The linearized lattice Boltzmann equation can be written as the following:
j�f(k; t+ 1)i = Lj�f(k; t)i ; (3.18)
where L is the linearized evolution operator:
L = A�1[I+M�1CM] : (3.19)
27
3.7 Eigenvalue Problem of the Linearized Lattice Boltzmann Equation
The solution of the linearized lattice Boltzmann equation is equivalent to an eigen-value problem of the linearized evolution operator:
det[L� zI] = 0 : (3.20)
Hydrodynamic modes (corresponding to z� = 1) of L at k = (k cos �; k sin �) ! 0
are one transverse (shear) mode and two longitudinal (sound) modes:
j%T i = cos �jjxi � sin �jjyi � jjT i ; (3.21a)
j%�i = j�i � (cos �jjxi+ sin �jjyi) � j�i � jjLi : (3.21b)
When k 6= 0, we have
j%T (t)i = zTt j%T (0)i = exp[�ik(gV cos�)t] exp(��k2t)j%T (0)i ; (3.22a)
j%�(t)i = z�t j%�(0)i
= exp[�ik(cs � gV cos�)t] exp[�(�=2 + �)k2t]j%�(0)i : (3.22b)
Transport coe�cients �, �, cs, and g are functions of wavevector k, and the ad-justable parameters in the model [see Eqs. (3.12)]. By optimizing the isotropy ofthe transport coe�cients and minimizing the non-Galilean e�ect of the model, thevalues of the adjustable parameters can be determined.
3.8 Determination of the Adjustable Parameters
The Galilean invariance of the phases in the transverse mode (zT ) and the soundmodes (z�) up to k leads to:
1 = 3 =2
3; (3.23a)
2 = 18 : (3.23b)
Isotropy of the attenuation of the transverse mode (zT ) and the Galilean invarianceof the attenuation of the sound modes (z�) lead to
c1 = �2 ; (equivalent to cs = 1=p3) ; (3.24a)
�2 = �8 : (3.24b)
The remaining adjustable parameters are: �3 and 4 (in �(eq)). If we chose
�3 = 4 ; (3.25a)
4 = �18 ; (3.25b)
and s� = 1=� , the generalized lattice Boltzmann equation reduces to a lattice BGKmodel.
28
3.9 Behaviors of Eigenvalues of the Linearized Collision Operator L
The behaviors of eigenvalues of the linearized collision operator L determine thelocal stability of the LBE model: If Re(ln z�) > 0, then the corresponding modej%�i is unstable. Figure 3.1 shows the real and imaginary parts of the nine rootsof L with a given set of parameters. It shows that when k = �, one kinetic modebecomes \quasi-conservative," because the corresponding eigenvalue is equal to 1when k = �. This is the mode which generates the checker-board pattern andinstigates instability at short wave length.
Figure 3.1: Logarithmic eigenvalues of the nine-velocity model. The values of the parameters
are �2 = �8, �3 = 4, c1 = �2, 1 = 3 = 2=3, 2 = 18, and 4 = �18. The relaxation
parameters are: s2 = 1:64, s3 = 1:54, s5 = s7 = 1:9, and s8 = s9 = 1:99. The streaming
velocity V is parallel to k with V = 0:2, and k is along the x axis. (a) Re(ln z�) and (b)
Im(ln z�).
The adjustable parameters in our model can be used to alter the properties of themodel. The stability of the BGK LBE model and our model is compared in Fig. 3.2.In this case we choose the adjustable parameters in our model to be the same asthe BGK LBE model, but maintain the freedom of di�erent modes to relax withdi�erent relaxation parameters s�. Figure 3.2 shows that for each given value of V ,there exists a maximum value of s8 = 1=� (which determines the shear viscosity)below which there is no unstable mode. The values of other relaxation parametersused in our model are s2 = 1:63, s3 = 1:14, s5 = s7 = 1:92, and s9 = s8 = 1=� .Figure 3.2 clearly shows that our model is more stable than the BGK LBE modelin the interval 1:9 � s8 = 1=� � 1:99. Therefore, we can conclude that by carefullyseparating the kinetic modes with di�erent relaxation rates, we can indeed improvethe stability of the LBE model signi�cantly.
The behaviors of transport coe�cients are also determined by the eigenvalues of
29
Figure 3.2: Stability of the generalized LBE model vs. the BGK LBE model in the parameter
space of V and s8 = 1=� . (left) max[Re(ln z�)] for given V . (right) Stability region of GLBE
vs. LBGK model.
L. Figure 3.3 shows the k-dependence of the viscosity and the Galilean coe�cient g.It shows that in small scales (large k), the LBE model becomes more anisotropic andnon-Galilean invariant. In addition, the use of interpolation signi�cantly increasesthe hyperviscosity and the e�ect of non-Galilean invariance.
Figure 3.3: k-dependence of viscosities and g-factor. The solid lines, dotted lines, and dashed
lines correspond to � = 0, �=8, and �=4, respectively. Three LBE model tested: (a) with no
interpolation, (a) with central interpolation, and (c) with upwind interpolation.
The artifacts of the LBE method can also be analyzed. We study an interestingand revealing case in which the initial velocity �eld contains shocks. Consider aperiodic domain of size Nx � Ny = 84 � 4. At time t = 0, we take a shear wave
30
Figure 3.4: Decay of discontinuous shear wave velocity pro�le uy(x; t). (left) The lines and
symbols (�) are theoretical and numerical results, respectively. Only the positive half of each
velocity pro�le is shown. LBE model (a) with no interpolation, (b) with the central interpolation
and r = 0:5. (right) Decay of uy(x; t) at a location close to the discontinuity x = 3Nx=4. The
solid lines and dashed lines are analytic and numerical results, respectively. The time is rescaled
as r�2�k2t.
uy(x; 0) of rectangular shape (discontinuities in uy at x = Nx=4 and x = 3Nx=4):
uy(x; 0) = U0; 1 < x � Nx=4; 3Nx=4 < x � Nx;
uy(x; 0) = �U0; Nx=4 < x � 3Nx=4:
The initial condition ux(x; 0) is set to zero everywhere. We consider two separatecases with and without a constant streaming velocity V .
Figures 3.4(a) and 3.4(b) show the decay of the rectangular shear wave simulatedby the normal LBE scheme and the LBE scheme with second-order central interpo-lation (with r = 0:5, where r is the ratio between advection length �x and grid size�x), respectively. The lines are theoretical results with �(kn) obtained numerically.The times at which the pro�le of uy(x; t) (normalized by U0) shown in Fig. 3.4 aret = 100, 200, . . . , 500. The numerical and theoretical results agree closely with eachother. The close agreement shows the accuracy of the theory. In Fig. 3.4(b), theovershoots at early times due to the discontinuous initial condition are well capturedby the analysis. This overshoot is entirely due to the strong k-dependence of �(k)caused by the interpolation. This phenomena is an artifact due to discretization,and is not connected to any physical e�ect. This artifact is also commonly observedin other CFD methods involving interpolations.
Similarly to Fig. 3.4, Fig. 3.5 shows the evolution of uy(x; t) for the same times asin Fig. 3.4. The solid lines and the symbols (�) represent theoretical and numericalresults, respectively. Shocks move from left to right with a constant velocity Vx =0:08. Figures 3.5(a), 3.5(b), and 3.5(c) show the results for the normal LBE schemewithout interpolation, the scheme with second-order central interpolation, and thescheme with second-order upwind interpolation, respectively. In Figs. 3.5(b) and
31
Figure 3.5: Decay of discontinuous shear wave velocity pro�le uy(x; t) with a constant stream-
ing velocity Vx = 0:08 = U0. The solid lines and symbols (�) are theoretical and numerical
results, respectively. The dashed lines in (b) and (c) are obtained by setting gn = 1. (a) no
interpolation, (b) central interpolation and r = 0:5, (c) upwind interpolation and r = 0:5.
3.5(c), the dotted lines are the results obtained by setting gn = 1. Clearly, thee�ect of g(k) is signi�cant. For the LBE scheme with central interpolation, theresults in Fig. 3.5(b) with g(k) = 1 underpredict the overshooting at the leadingedge of the shock and overpredict the overshooting at the trailing edge, whereas theresults in Fig. 3.5(c) for the LBE scheme with upwind interpolation overpredict theovershooting at the leading edge of the shock and underpredict the overshooting atthe trailing edge.
3.10 Summary
We have constructed a generalized LBE model with multiple relaxation times. Thehydrodynamic behavior of the model is obtained by analyzing the linearized disper-sion equation. Based on our analysis, we can draw the following conclusions:
� The generalized LBE is superior than the Lattice BGK model in terms ofstability, isotropy, and Galilean invariance. The Prandtl number of the gener-alized LBE can be arbitrary;
� Analysis of the linearized dispersion equation is equivalent to the Chapman-Enskog analysis, while Chapman-Enskog analysis is not valid for situation of�nite wavevector k;
� Analysis of the linearized dispersion equation is also applicable to complex uids (e.g., viscoelastic uids).
We also realize the limitations of the dispersion equation analysis. It cannot dealwith nonlocal e�ects (gradients) and the boundary conditions. A detailed treatmentof the generalized lattice Boltzmann equation is provided in Ref. [37].
32
Lecture 4
Boundary Conditions in LBE Method and Applications
Fluid-solid boundary condition, and other types of boundary conditions are im-portant in CFD simulations. In the LBE method, the primitive variables are thedistribution functions ff�g, but not the hydrodynamic variables �, u, and T . Ac-cordingly, the boundary conditions in the LBE simulations are imposed on the dis-tribution functions ff�g, and only on the hydrodynamic variables indirectly . Thisis one of the most distinctive features of the LBE methods, as compared to the con-ventional CFD methods which are based on the discretizations of the Navier-Stokesequations. We shall discuss one of most popular uid-solid boundary conditions inthe LBE method: bounce-back boundary conditions.
4.1 Bounce-Back Boundary Conditions
The bounce-back boundary conditions were �rst used in the lattice-gas to mimic theno-slip boundary conditions on a uid-solid surface. The idea is very simple andintuitive as the name suggests: when a particle hits on a solid surface, it simplyreverses its momentum.
The bounce-back boundary conditions can be easily adopted in the LBE method.Suppose a lattice site xb is designated as a solid site on a boundary, the bounce-backboundary conditions can be simply expressed as
f�(xj ; �) = f��(xb; �); 8 xj; fxj jxj + e��t = xbg: (4.1)
The time argument in the above formula is deliberately omitted because the bounce-back collision process can be thought to take place either simultaneously or in onetime step. This leads two implementations of the bounce-back boundary conditions,with the di�erence of one time step between the outgoing (from uid to solid) andincoming (from solid to uid) states. This makes no di�erence for steady-statesituations. However, the spatial-temporal symmetry of these two implementationsare di�erent: one (no time lag) destroys and the other (with one time step lag)preserves the checker-board spatial symmetry, and this may a�ect the numericalstability of the LBE methods near the boundary [37].
The implementations of the bounce-back boundary conditions in the LBE meth-ods can have some variations. For instance, one can relax the distribution functions
33
at the boundary nodes to the equilibria with the velocity imposed by the ow bound-ary conditions [29].
One most important feature of the bounce-back boundary conditions is that the ow boundary conditions cannot be satis�ed on the last uid nodes next to the solidboundaries (nodes). The ow boundary conditions are satis�ed some where betweena uid node and a solid node. For simple ows, such as the Couette ow or Poiseuille ow, the exact location where the ow boundary conditions are satis�ed are knownanalytically [39, 16, 17, 40, 29], if the boundaries are aligned with the lattice.
Despite of its simplicity, the bounce-back boundary conditions are su�cientlyaccurate, i.e., it is second-order accurate. However, the bounce-back scheme ap-proximates any continuous curve/surface with a zig-zag staircase. This practicedegrades the geometric integrity of a continuous boundary. We shall discuss how toimprove the preservation of the geometric integrity of a continuous boundary next.
4.2 Bounce-Back Boundary Conditions with Interpolations
In the LBE method, the uid simulation is conducted on a Eulerian Cartesian mesh,which is �xed in space and time. If a body moving in uid has to be considered,then the Lagrangian dynamics has to be used to describe the motion of the movingbody, i.e., the coordinate system associated with the body is moving with respect tothe Eulerian coordinate system for the uids. Thus, there are two di�erent method-ologies in general: (1) body-�tted meshes and (2) Cartesian meshes with immersedbody method (IBM); and the LBE treatment of solid-boundaries belongs to the lat-ter methodology. Figure 4.1 depicts the basic ideas in the LBE-IBM treatment of acurved boundary.
In order to improve the quality of the bounce-back boundary treatment, theprecise boundary location must be resolved within one grid spacing. The simplestway to accomplish this is to introduce interpolations at the boundary.
With the picture for the simple bounce-back scheme in mind, let's �rst considerthe situation depicted in Fig. 4.2b in detail for the case of � < 1=2. At time t thedistribution function of the particle with velocity pointing to the wall (e1 in Fig. 4.2)at the grid point xj (a uid node) would end up at the point xi located at a distance(1�2�)�x away from the grid point xj, after the bounce-back collision, as indicatedby the thin bent arrow in Fig. 4.2b. Because xi is not a grid point, the value off3 at the grid point xj needs to be reconstructed. Noticing that f1 starting frompoint xi would become f3 at the grid point xj after the bounce-back collision withthe wall, we construct the values of f1 at the point xi by a quadratic interpolationinvolving values of f1 at the three locations: f1(xj); f1(xj0) = f1(xj � e1�t), andf1(xj00) = f1(xj � 2e1�t). In a similar manner, for the case of � � 1=2 depicted inFig. 4.2c, we can construct f3(xj) by a quadratic interpolation involving f3(xi) thatis equal to f1(xj) before the bounce-back collision, and the values of f3 at the nodesafter collision and advection, i.e., f3(xj0), and f3(xj00). Therefore the interpolations
34
e
e e
e e e
e e e e e
@@@@@@@@
@@@@@@@@
@@@@@@@@
@@@@@@@@
@@@@@@@@
@@@@@@@@
@@@@@@@@
��
��
��
��
u u u u
u u u
u u
u
u
u
u
u
u
u
u
uuu
u
u
u
xb
xw
xj
xj0
@@@R@
@@I
e�
e��
?
6
��x
?
6
�x
Figure 4.1: Layout of the regularly spaced lattices and curved wall boundary. The circles (�),
discs (�), shaded discs (�), and diamonds (3) denote uid nodes, boundary locations (xw), solid
nodes which are also boundary nodes (xb) inside solid, and solid nodes, respectively.
are applied di�erently for the two cases:
� For � < 1=2, interpolate before propagation and bounce-back collision;
� For � � 1=2, interpolate after propagation and bounce-back collision.
We do so to avoid the use of extrapolations in the boundary conditions for the sakeof numerical stability. This leads to the following interpolation formulas (where thenotations f̂� and f� denote the post-collision distribution functions before and afteradvection):
f��(xj ; t) = � (1 + 2�)f̂�(xj ; t) + (1� 4�2)f̂�(xj0 ; t)
��(1� 2�)f̂�(xj00 ; t) + 3w�(e� � uw) ; � <1
2; (4.2a)
f��(xj ; t) =1
� (2� + 1)f̂�(xj ; t) +
(2�� 1)
�f��(xj0 ; t)
�(2�� 1)
(2� + 1)f��(xj00 ; t) +
3w�
�(2� + 1)(e� � uw) ; � � 1
2; (4.2b)
w� =
�2=9; � = 1 { 4;2=36; � = 5 { 8;
(4.2c)
In practice, it is more e�cient to combine collision and advection into one step.Because advection simply corresponds to shifts of indices labeling spatial nodes of
35
(a) � = 1=2
u ue e ��
uxj00 xj0 xj xsxw
wall
�x=2��x� -
(b) � < 1=2
u ue e u��
��
exj00 xj0 xj xsxwxi
wall
��x��x� -
e� -e3 e1
(c) � � 1=2
u ue e u
�x� - ��x� -
��
xj00 xj0 xj xsxwxi
e
wall
Figure 4.2: Illustration of the boundary conditions for a rigid wall located arbitrarily between
two grid sites in one dimension. The thin solid lines are the grid lines, the dashed line is the
boundary location situated arbitrarily between two grids. Shaded discs are the uid nodes, and
the discs (�) are the uid nodes next to boundary. Circles (�) are located in the uid region
but not on grid nodes. The square boxes (2) are within the non- uid region. The thick arrows
represent the trajectory of a particle interacting with the wall, described in Eqs. (4.3a) and
(4.3b). The distribution functions at the locations indicated by discs are used to interpolate the
distribution function at the location marked by the circles (�). (a) � � jxj � xwj=�x = 1=2.
This is the perfect bounce-back scheme with no interpolations. (b) � < 1=2. (c) � � 1=2.
ff�g, the actual formulas used in simulations are:
f��(xj ; t) = � (1 + 2�)f�(xj + e��t; t) + (1� 4�2)f�(xj ; t)
��(1� 2�)f�(xj � e��t; t) + 3w�(e� � uw) ; � <12; (4.3a)
f��(xj ; t) =1
� (2� + 1)f�(xj + e��t; t) +
(2�� 1)
�f��(xj � e��t; t)
�(2�� 1)
(2� + 1)f��(xj � 2e��t; t) +
3w�
�(2�+ 1)(e� � uw) ; � � 1
2:(4.3b)
The above formulas are implemented as follows. The collision-advection process
36
including the bounce-back collision with boundaries are divided in the following �vesteps:
� Step 1: Compute moments at all uid nodes;
� Step 2: Relax the non-conserved moments at all uid nodes, add momentumchange if an external force is presented;
� Step 3: Compute the post-collision distributions ff�g at all uid nodes;
� Step 4: Advect ff�g throughout the system, including all uid and non- uidnodes. Some distributions will be advected from uid nodes to non- uid nodes,and some from non- uid nodes to uid nodes.
� Step 5: Recompute the distributions that are advected from non- uid nodes to uid nodes according to Eq. (4.3a) or (4.3b), depending on the precise locationof the boundary (i.e., the value of �).
When boundaries lie in between two nodes of two grid points, the above imple-mentation is equivalent to the so-called \node implementation" of the bounce-backscheme, i.e., the bounce-back collision does not take time.
Two formulae (4.3a) and (4.3b) can be compressed to a single formula if the�rst-order interpolations are used [3, 63].
4.3 Applications
In the section we will mention concisely the areas in which the LBE method hasbeen successfully applied, and provide some key references.
Flow through porous media. The key issue in the LBE direct numerical simula-tions (DNS) of ow through porous media is to accurately capture complicatedboundary geometry even with very low resolutions. To this end, the latticeBoltzmann method has been successful [47, 61, 50]. This is due to the factthat the LBE method can accurately compute the mass/momentum transfersat boundaries [3, 49, 63].
Suspensions in uids. The LBE method has been very successful in the directnumerical simulations of particulate and colloidal suspensions in uids [34, 35,51, 36, 52, 53, 54]. The LBE method has been e�ective in the simulations ofa single particle in uid [51, 52, 54] or the rheology of particulate suspensions[34, 35, 36, 53].
Mutli-component and interfacial dynamics. We mention in passing that themulticomponent LBE schemes can also be derived from the continuous Boltz-mann equations for mixtures [44, 45]. The LBE method has applied to simulateinterfacial phenomena (e.g., [18, 6]).
37
Direct numerical simulations (DNS) of turbulence. Recent results show thatthe LBEmethod can reproduce well-known results for the homogeneous isotropicturbulence ow (e.g., [46, 64]). And the LBE method can be vary competitivewhen compared with pseudo-spectral method.
Large eddy simulations (LES). The LBE-LES for turbulent ows is one of newareas. Most LBE-LES schemes use the Smagorinsky sub-grid model, whichcan be easily implemented in the LBE method [12, 11, 32, 33].
Non-Newtonian uids. The LBE models for linear viscoelastic uids have devel-oped [19, 20, 38] and applied in simulations [18].
38
Epilogue
In �ve lectures I tried to give an overview on the theory of the lattice Boltzmannequation. The emphasis of these lectures is on the mathematical justi�cation of thelattice Boltzmann method. Unfortunately I did not have the time to discuss theapplications of the method, which I shall refer the readers to a recent review on thelattice gas and lattice Boltzmann method [43]. Here I would like to a few words onthe rationale, or philosophy of the lattice gas and lattice Boltzmann methods.
It is a well known fact that a uid is a discrete system with a large number(� 1023) of particles (molecules). A system of many particles can be described byeither molecular dynamics (MD) or a hierarchy of kinetic equations (the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy), and these two descriptions are equivalent.With the molecular chaos assumption due to Boltzmann, the BBGKY hierarchycan be closed with a single equation: the Boltzmann equation for the single particledistribution function. On the other hand, a uid can also be treated as a continuumdescribed by a set of partial di�erential equations for uid density, velocity, andtemperature: the Navier-Stokes equations. It should be stressed that the continuumtreatment of uid is an approximation. This approximation works extremely wellunder many circumstances.
It is usually convenient to use the Navier-Stokes equations to some uid prob-lems. Unfortunately these equations can be very di�cult or even impossible to solveunder some circumstances such as inhomogeneous multiphase or multicomponent ows, or granular ows. In the case of multiphase or multicomponent ows, inter-faces between di�erent uid components (e.g. oil and water) or phases (e.g. vaporand water) cause the numerical di�culties. Computationally, one might be ableto track a few, but hardly very many interfaces in a system. Realistic simulationsof uid systems with density or composition inhomogeneities by direct solution ofthe Navier-Stokes equations is therefore impractical. We can also look at the prob-lem from a di�erent perspective: interfaces between di�erent components or phasesof a uid system are thermodynamic e�ects which result from interactions amongmolecules. To solve the Navier-Stokes equations, one needs to know the equation ofstate, which is usually unknown at an interface. It is therefore di�cult to incorporatethermodynamics into the Navier-Stokes equations in a consistent or a priori fashion.Hence we encounter some fundamental di�culties. In the case of granular ow, thesituation is even worse: it is not even clear that there exists a set partial di�eren-tial equations analogous to the Navier-Stokes equations which correctly model suchsystems. Instead, granular ow is usually modeled by equations completely lacking
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the fundamental validity of the Navier-Stokes equations.Although the Navier-Stokes equations are inadequate in some circumstances,
neither molecular dynamics nor the Boltzmann equation are practical alternativesbecause solutions of molecular dynamics or the Boltzmann equation pose formidabletasks which demand much more computational e�ort than the solution of the Navier-Stokes equations. Thus, we face the following predicament: although the Navier-Stokes equations are inadequate, molecular dynamics or the Boltzmann equationare much too di�cult to solve and are even unnecessarily complicated if we are onlyinterested in the macroscopic behaviors of a system. It is within this context thatthe lattice-gas automata (simpli�ed molecular dynamics) and the lattice Boltzmannequation (simpli�ed Boltzmann equation) become alternatives. It has been realizedthat hydrodynamics is insensitive to the details of the underlying microscopic ormesoscopic dynamics | the Navier-Stokes equations are merely statements of con-servation laws, which re ect the same conservation laws in microscopic dynamics,and constitutive relations, which re ect the irreversible nature of the macroscopicdynamics. Di�erent inter-molecular interactions would only result in di�erent nu-merical values of the transport coe�cients. Since the details of the microscopicdynamics are not important if only the hydrodynamic behavior of system is of in-terest, one may ask the following question: What constitutes a minimal microscopicor mesoscopic dynamic system which can provide desirable physics at the macro-scopic level (hydrodynamics, thermodynamics, etc.). It turns out that the essentialelements in such a microscopic or mesoscopic dynamic system are the conservationlaws and associated symmetries. It is based upon this rationale that the lattice gasand the lattice Boltzmann models were constructed as reduced models to numericallysimulate various complex systems.
It should also be pointed out that kinetic theory is valid for a wide range ofdensities covering gases, liquids, and even solids. It is within the framework andupon the foundation of kinetic theory that the lattice gas and lattice Boltzmannmethods are formulated as consistent and e�ective simulation tools.
Since kinetic theory plays such an important role in the lattice gas and the latticeBoltzmann methods, it would be perhaps appropriate to end this series of lecturesby quoting Professor E.G.D. Cohen [7] and Boltzmann [2] on their views on kinetictheory.
\. . . I note that the objection is often raised that kinetic theory is restricted in its
applications, is very complicated and that more general results can be much easier
obtained by using hydrodynamic-like theories. All this is certainly true. However,
if one is not just interested in obtaining new results, but also wants to understand
their foundation in the molecular structure of matter and see the connection between
various, at �rst sight, very di�erent phenomena from a uni�ed point of view, kinetic
theory has proved to be an indispensable means to achieve this goal. In addition, it
has led to new results, in spite of its complicated structure.
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\It seems to me then that kinetic theory �nds itself at the end of the twentieth
century again in an impasse and in a situation not too di�erent from that in which it
found itself at the end of the nineteenth century. It might therefore be appropriate
to justify this survey by quoting some words Boltzmann used in 1898 . . . "E.G.D. Cohen (1993)
\It was just at this time that attacks on the theory of gases began to increase. I am
convinced that these attacks are merely based on a misunderstanding, and that the
role of gas theory in science has not yet been played out. . . .
\In my opinion it would be a great tragedy for science if the theory of gases were
temporarily thrown into oblivion because of a momentary hostile attitude toward
it, as was for example the wave theory because of Newton's authority.
\I am conscious of being only an individual struggling weakly against the stream
of time. But it still remains in my power to contribute in such a way that, when
the theory of gases is again revived, not too much will have to be rediscovered.
. . .When consequently parts of the argument become somewhat complicated, I must
of course plead that a precise presentation of these theories is not possible without
a corresponding formal apparatus. . . . "L. Boltzmann (1898)
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