Improved treatments for general boundary conditions in the lattice Boltzmann method for convection-diffusion and heat transfer processes

Abstract

In spite of the increasing applications of the lattice Boltzmann method (LBM) in simulating various flow and transport systems in recent years, complex boundary conditions for the convection-diffusion and heat transfer processes in LBM have not been well addressed. In this paper, we propose an improved bounce-back method by using the midpoint concentration value to modify the bounced-back density distribution for LBM simulations of the concentration field. An accurate finite-difference scheme in the normal boundary direction has also been introduced for gradient boundary conditions. Compared with existing boundary methods, our method has a simple algorithm and can easily deal with boundaries with general geometries, motions, and surface conditions (the Dirichlet, Neumann, and mixed conditions). Carefully designed simulations are performed to examine the capacity and accuracy of this proposed boundary method. Simulation results are compared with those from theory and a representative boundary method, and an improved performance is observed. We have also simulated the effect of reference velocity on global accuracy to examine the performance of our model in preserving the fundamental Galilean invariance. These boundary treatments for concentration boundary conditions can be readily applied to other processes such as heat transfer systems.

Full text

PHYSICAL REVIEW E 88, 033304 (2013)
Improved treatments for general boundary conditions in the lattice Boltzmann method
for convection-diffusion and heat transfer processes
Qing Chen,1,2Xiaobing Zhang,1,*and Junfeng Zhang2,†
1School of Energy and Power Engineering, Nanjing University of Science and Technology, Jiangsu 210094, China
2Bharti School of Engineering, Laurentian University, 935 Ramsey Lake Road, Sudbury, Ontario P3E 2C6, Canada
(Received 11 April 2013; revised manuscript received 9 August 2013; published 26 September 2013)
In spite of the increasing applications of the lattice Boltzmann method (LBM) in simulating various flow and
transport systems in recent years, complex boundary conditions for the convection-diffusion and heat transfer
processes in LBM have not been well addressed. In this paper, we propose an improved bounce-back method by
using the midpoint concentration value to modify the bounced-back density distribution for LBM simulations
of the concentration field. An accurate finite-difference scheme in the normal boundary direction has also been
introduced for gradient boundary conditions. Compared with existing boundary methods, our method has a
simple algorithm and can easily deal with boundaries with general geometries, motions, and surface conditions
(the Dirichlet, Neumann, and mixed conditions). Carefully designed simulations are performed to examine the
capacity and accuracy of this proposed boundary method. Simulation results are compared with those from
theory and a representative boundary method, and an improved performance is observed. We have also simulated
the effect of reference velocity on global accuracy to examine the performance of our model in preserving
the fundamental Galilean invariance. These boundary treatments for concentration boundary conditions can be
readily applied to other processes such as heat transfer systems.
DOI: 10.1103/PhysRevE.88.033304 PACS number(s): 47.11.−j, 47.10.ab, 44.05.+e
I. INTRODUCTION
The lattice Boltzmann method (LBM) has been gradually
accepted as a useful simulation method for fluid flows and
other associated phenomena in the past two decades [1–3].
Unlike conventional computational methods, which are based
on the numerical discretization of macroscopic governing
equations, LBM originates from the classical kinetic theory.
Due to such kinetic particulate nature, LBM possesses several
attractive features, such as the simple algorithm formulation,
relative easiness in dealing with complex geometries, and
potential efficiency with parallel computation. Applications of
LBM simulations can be found in various flow situations, for
example, multiphase flows, porous flows, compressible flows,
particulate flows, biological flows, and microfluidics [1–4].
It have been shown that, via the Chapman-Enskog
expansion, the macroscopic continuity and momentum
(Navier-Stokes) equations can be obtained from the collision-
propagation processes of particle density distributions in
LBM [2]. Therefore, from a pure mathematical point of
view, the LBM algorithm can be considered as a numerical
solver of partial differential equations. Following this train
of thought, the original LBM algorithm for fluid flows has
been tuned to solve the governing differential equations for
various processes and phenomena, including the electric fields,
magnetic fields, porous flows, and axisymmetric flows [3].
In addition, convection-diffusion and heat transfer processes
are commonly encountered in various natural and industrial
situations. Early efforts in LBM simulations of such systems
can be traced back to the pioneering work by Dawson et al.
[5]. Two sets of particle distribution functions (PDFs) were
*zhangxb680504@163.com
†jzhang@laurentian.ca
employed there, with one set for the solute density and another
for the solvent density. He et al. [6]haveimprovedthis
double-PDF model by assigning one set of PDFs for the solute
concentration, and another set for the flow dynamics of the
total fluid (including the solute and solvent). This approach
has become the standard for LBM convection-diffusion simu-
lations. In addition, models for the multiple-relaxation-time
(MRT) LBM algorithm [7], irregular lattice structures [8],
and anisotropic diffusion [9,10] have also been reported. Shi
et al. [11] have also proposed a more accurate treatment for
the source term. Similar double-PDF thermal LBM models for
heat transfer have also been developed and utilized in various
applications [12–15]. In these models, temperature is actually
considered a scalar property, and its evolution is governed
by a convection-diffusion-type differential equation. For this
reason, in the following sections in this paper, we will focus
on the convection-diffusion processes, and all analysis and
discussions are readily applicable to heat transfer systems.
As with any other numerical methods, correct and accurate
boundary treatments play a crucial role in LBM simulations.
However, unlike the extensive efforts in the boundary meth-
ods for LBM flow simulations, LBM boundary conditions
for convection-diffusion systems have not been addressed
adequately. A review on these existing boundary conditions
can be found in a recent publication [16]. As one can see
there, most of these existing boundary methods are limited to
flat surfaces [6], stationary surfaces [17,18], or certain types
of boundary conditions (i.e., the boundary-value Dirichlet or
boundary-gradient Neumman boundary conditions) [19,20].
There are also several boundary methods proposed for
general curved boundary geometries [21–23]; however, the
complicated algorithms make them difficult to be utilized for
systems with complex boundaries, such as transport processes
in particulate and porous flows. In addition, these methods
typically do not consider the effect of boundary movement
033304-1
1539-3755/2013/88(3)/033304(12) ©2013 American Physical Society

QING CHEN, XIAOBING ZHANG, AND JUNFENG ZHANG PHYSICAL REVIEW E 88, 033304 (2013)
on the concentration field. By extending the classical bounce-
back scheme in LBM for fluid dynamics [24] to convection-
diffusion simulations, Zhang et al. [16] recently have proposed
a boundary method capable of dealing with general boundary
situations, including the Dirichlet, Neumann, and mixed Robin
boundary conditions, boundary velocity, and nonflat surfaces.
Chen et al. [25] have argued that this bounce-back scheme
is only suitable for static boundaries; however, more rigorous
analysis and validations might be necessary to examine this
concern in depth and details.
The half-way bounce-back scheme in LBM suffers a
low resolution in representing the actual smooth boundary
geometry [24]. In this paper, we improve the spatial resolution
of the bounce-back scheme by Zhang et al. [16] with a recently
developed midpoint bounce-back technique for flow [26] and
an accurate finite-difference scheme for electric field [27]
simulations. The key idea is that the bounced-back PDFs
should be adjusted according to the velocity and concentration
values at the midpoint of a boundary lattice link, instead of
those on the physical boundary. Such midpoint values can
be obtained via an interpolation or extrapolation along the
boundary lattice direction. In addition, for the Neumann and
Robin boundary conditions, when evaluating the boundary
concentration value from the boundary gradient, the finite-
difference scheme employed in Ref. [16] has neglected the
possible surface heterogeneity (i.e., boundary condition vari-
ation over the surface). Such possible boundary heterogeneity
has also been considered in our improved finite-difference
scheme along the boundary normal direction. Several carefully
designed systems have been simulated, and the results have
been compared with those from theoretical solutions and
the original method by Zhang et al. [16]. The Galilean
invariance from our LBM model has also been examined
by studying the effect of reference velocity on global error.
These simulations and comparisons demonstrate that our
boundary treatments have improved numerical accuracy and
can be employed in convention-diffusion systems with various
boundary conditions. The method described here can also
be readily extended to MRT LBM models and heat transfer
processes.
II. MODELS AND METHODS
For simplicity and convenience, in this paper, the model
description and demonstration simulations are presented
using the classical lattice Bhatnagar-Gross-Krook (LBGK)
model over a D2Q9 (two-dimensional, 9-velocity) lattice
structure. Extending this present work to three-dimensional
systems, other lattice structures, and/or MRT schemes is
straightforward.
A. The lattice Boltzmann method for
convection-diffusion processes
To simulate the convection-diffusion process in an in-
compressible flow, two sets of PDFs are employed: one
to solve the flow dynamics and another one for the con-
centration field. Their evolution is governed by the follow-
ing well-known lattice Boltzmann equations (LBEs) with
the single-relaxation-time approximation for the collision
operator [1–3],
fi(x+eiδt,t +δt)−fi(x,t )=−1
τffi(x,t)−feq
i(x,t),
(1)
gi(x+eiδt,t +δt)−gi(x,t )=−1
τggi(x,t)−geq
i(x,t),
(2)
where fi(x,t) and gi(x,t) are the PDFs for, respectively, the
flow and concentration fields at location xand time tand the
subscript iindicates the lattice direction. The lattice velocities
eifor the D2Q9 lattice structure adopted in this study are
given as
e0=(0,0); (3)
ei=cos (i−1)π
2,sin (i−1)π
2δx
δt ,i=1−4; (4)
ei=cos (2i−1)π
4,sin (2i−1)π
4√2δx
δt ,i=5−8.
(5)
Here δx is the lattice grid unit and δt is the time step. The
relaxation parameters τfand τgare related to, respectively,
the macroscopic fluid kinematic viscosity νand the diffusivity
Din a similar fashion as
ν=(2τf−1)δx2
6δt ,(6)
D=(2τg−1)δx2
6δt .(7)
The equilibrium distributions feq
iand geq
iare typically
expressed as
feq
i=ωiρ1+ei·u
c2
s+(ei·u)2
2c4
s−u·u
2c2
s,(8)
geq
i=ωiC1+ei·u
c2
s+(ei·u)2
2c4
s−u·u
2c2
s.(9)
For the D2Q9 lattice structure, the lattice sound speed cs=
√3δx/δt, and the lattice weight factors are ω0=4/9, ω1−4=
1/9, and ω5−8=1/36. The macroscopic properties, including
the fluid density ρ, flow velocity u, flow pressure P, and
concentration C, are readily available from the PDFs as
ρ=
i
fi,u=
i
fiui/ρ,
(10)
P=c2
sρ, C =
i
gi.
It can be shown that macroscopic flow and transport equations
∂ρ
∂t +∇·(ρu)=0,(11)
∂u
∂t +(u·∇)u=−1
ρ∇P+μ
ρ∇2u,(12)
∂C
∂t +∇·(Cu)=∇·(D∇C),(13)
033304-2

IMPROVED TREATMENTS FOR GENERAL BOUNDARY ... PHYSICAL REVIEW E 88, 033304 (2013)
can be derived from the above formulations via the Chapman-
Enskog analysis [2].
B. The midpoint bounce-back scheme for boundary
velocity and boundary concentration
Different from other numerical methods, in LBM the
boundary requirements for macroscopic properties are imple-
mented by specifying the PDFs entering the simulation region
across the boundary. As we typically have more unknown
incoming PDFs to determine and less macroscopic constraints
available, assumptions are required to make the equation
system complete. Next we briefly describe the midpoint
bounce-back boundary method for the velocity boundary
condition recently developed by Yin and Zhang [26] and then
extend this scheme to the Dirichlet concentration boundary
condition.
For the convenience of our description, we consider the
solid surface in the D2Q9 lattice space as illustrated in
Fig. 1(a). A boundary lattice link, with two end nodes xsin the
solid domain and xfin the fluid domain, intersects with the
boundary surface at the boundary node xb. The midpoint of
this lattice link is denoted by xm. After the collision step at the
fluid node xf, the PDF f∗
i[i=7inFig.1(a)]leavesxf, and
is then assumed to be bounced-back at the midpoint xmin the
reversed direction and with a modified magnitude as f¯
i[¯
i=5
in Fig. 1(a)]:
f¯
i=f∗
i−2ωiρ
c2
s
um·ci,(14)
where umis the midpoint velocity at xmto be determined.
Different from the classical Ladd bounce-back method [24],
here we are using the midpoint velocity uminstead of the
boundary velocity ub.
To find out the midpoint velocity um, we denote the fraction
of the boundary lattice link in the solid domain as , i.e.,
=
xs−xb
xs−xf
,(15)
and 0 1. For 1/2, the midpoint xmlocates
between xband xf, and the midpoint velocity umcan be readily
obtained with a linear interpolation:
um=
1
2ub+1
2−uf
1−,(16)
where ubis the imposed boundary velocity at the intersection
point xb, and ufis the flow velocity calculated at the fluid node
xf.For>1/2, the midpoint xmis in the solid domain and
therefore an extrapolation is needed to obtain velocity um.For
a better numerical stability as in previous studies [28,29], we
use uff =u(xff ), the velocity at the second inner fluid node
xff (instead of the first fluid node xf) from the boundary 
[Fig. 1(a)],
um=
3
2ub−−1
2uff
2−.(17)
For the particular case with =1/2, either Eq. (16) or
Eq. (17) yields um=ub, and our treatment simply reverts
back to the original Ladd method.
From the above description, we see that, like in the original
Ladd method [24], here the physical boundary surface 
actually is represented by a virtual stairwise surface [dot-
dashed lines in Fig. 1(a)] in the LBM calculation. However, we
use the estimated midpoint velocity umon virtual surface ,
instead of the boundary velocity ubon the physical surface ,
to modify the bounced-back PDF magnitude in Eq. (14); and,
hence, the discrepancy between the bounce-back process at
the midpoint and the PDF modification using the boundary
velocity in the original Ladd method has been removed.
Numerical results have shown that this midpoint bounce-back
method has a good numerical accuracy [26].
For the Dirichlet concentration boundary condition, re-
cently Zhang et al. [16] have extended the original Ladd
approach [24] to modify the boundary PDFs gi,
g¯
i=−g∗
i+2ωiCb1+(ei·ub)2
2c4
s−ub·ub
2c2
s,(18)
where Cbis the concentration value on the boundary surface.
Again, since the physical surface is replaced be a stairwise
f
xff
xmfi*
fi
xs
x
b
x
solid
Γ’
Γ
fluid
(a)
*
4
x
A
A
A
A
12
43
n
xb
solid
fluid
xx
xx
1
2
3
(b)
FIG. 1. (Color online) Schematic illustrations for the midpoint bounce-back scheme (a) and the normal-direction finite-difference estimation
of boundary value from boundary gradient (b). See text for details.
033304-3

QING CHEN, XIAOBING ZHANG, AND JUNFENG ZHANG PHYSICAL REVIEW E 88, 033304 (2013)
surface along lattice link midpoints, the spatial resolution of
the boundary geometry is low, especially for surface with
high curvatures (for example, particles and porous structures).
To improve the spatial accuracy, the midpoint bounce-back
method described above for flow field can be easily applied to
the Dirichlet concentration boundary condition by replacing
the concentration Cband velocity ubon the physical boundary
with the concentration Cmand velocity umat the midpoint xm
as follows:
g¯
i=−g∗
i+2ωiCm1+(ei·um)2
2c4
s−um·um
2c2
s,(19)
and Cmcan be obtained in a similar way as that for umvia an
appropriate interpolation or extrapolation,
Cm=
1
2Cb+1
2−Cf
1−,1/2,(20)
Cm=
3
2Cb−−1
2Cff
2−,>1/2.(21)
C. The finite difference scheme for boundary
concentration gradient
In chemical reaction and heat transfer systems, the gradient
of the interesting macroscopic variable (Chere) is often
involved in defining the boundary condition [6]. However, in
our above formulation for the boundary treatment, the bound-
ary concentration value is required to adjust the boundary
PDFs. A finite-difference approximation is thus necessary to
obtain the required concentration value from the available
concentration gradient at the boundary. In Ref. [16], the
following finite-difference scheme is employed:
∂C
∂n b≈Cb−Cf
(δx/2)n·ei
,(22)
where nis the unit vector in the normal direction on the
boundary, pointing into the simulated domain. Again here it is
assumed the boundary locates at the midpoint of the boundary
lattice link (indicating be the δx/2 in the denominator). In
addition to this spatial inaccuracy, this scheme is conducted
along the boundary lattice link direction. It should be no-
ticed that the boundary gradient is defined in the boundary
normal direction, which often differs from the boundary
lattice direction (at one boundary location, there is only one
normal direction but, very likely, several boundary lattice
links), as shown in Fig. 1(b). The finite-different scheme in
Eq. (22) is actually only correct for flat boundaries with a
uniform boundary gradient.
To address these issues, here we adopt the finite-difference
scheme recently developed by Oulaid and Zhang [27]. This
method has considered the real boundary location, and the
finite-difference approximation is performed along the bound-
ary normal direction. Consider the situation in Fig. 1(b).To
obtain the concentration value at the boundary point xb,first
we find the concentration value at the location x∗, which is
of a distance δfrom xbinto the simulated domain along the
outward normal direction n(i.e., x∗=xb+δn), for example,
via a bilinear interpolation as
C(x∗)≈4
i=1AiC(xi)
δx2,(23)
with xithe four nearest lattice nodes and Aithe corresponding
fractional areas in the lattice cell [Fig. 1(b)]. The boundary
value can then be related to the boundary gradient by
∂C
∂n b≈C(x∗)−C(xb)
δ.(24)
For the Neumann boundary condition with specified
concentration gradient, the boundary concentration value is
directly available from Eq. (24),
C(xb)≈C(x∗)−δ∂C
∂n b
,(25)
and this estimated boundary value then can be utilized in
the evaluation of midpoint concentration value for the PDF
modification. As for the general mixed Robin boundary
condition
a1
∂C
∂n +a2Cb=a3,(26)
where a1,a2, and a3are prescribed constants, the expression
of the boundary value can be solved as
C(xb)≈a1C(x∗)−a3δ
a1−a2δ.(27)
In simulations, the value of δshould be appropriately selected:
An unnecessarily large δcan induce larger finite-difference
errors; while a too-small δvalue may result in an interpolation
point x∗too close to the solid boundary even with some of its
neighboring lattice nodes xiin the solid domain [see Fig. 1(b)
inset], and this will damage the correct interpolation of C(x∗)
in Eq. (23). In this study, we use δ=1.5δx in all simulations
with boundary concentration gradient involved.
III. DEMONSTRATION SIMULATIONS AND DISCUSSION
To examine the capacity and performance of the above
described boundary treatments, we have carefully designed
several test systems, including the convection-diffusion sys-
tems between two horizontal plates, the pure diffusion situ-
ations between inclined plates and coaxial circular surfaces,
and the convection-diffusion problem of a cylinder moving in
a straight channel. The results are presented in nondimensional
LBM units.
A. Spatial resolution: Convection-diffusion between horizontal
plates with Dirichlet conditions
Following the numerical tests in Ref. [16], we first consider
the convection-diffusion systems between two horizontal
plates, both aligned along the lattice grid lines. The distance
between these two plates is H. The constant concentration
values on the upper and lower plates are, respectively, Cuand
Cl. Two flow situations are considered, the classical Poiseuille
flow between stationary walls driven by a body force and the
Couette flow with transboundary injections,
u(y=0) =0,u(y=H)=U0,(28)
v(y=0) =v(y=H)=V0,(29)
033304-4

IMPROVED TREATMENTS FOR GENERAL BOUNDARY ... PHYSICAL REVIEW E 88, 033304 (2013)
where U0in the shear velocity on the top plate and V0is
the injection velocity. The respective flow and concentration
solutions for these systems are [16]
u=4U0y
H−y
H2,v=0,(30)
C=Cl+(Cu−Cl)y
H,(31)
for the Poiseuille flow (U0here is the maximum velocity at the
channel centerline, and it is related to the applied body force
Galong the channel direction via U0=GH 2/8ν) and
u=U0
eRe y
H−1
eRe −1,v=V0,(32)
C=Cl+(Cu−Cl)ePe y
H−1
ePe −1,(33)
for the injection Couette flow. The Reynolds and Peclet
numbers are defined as Re =U0H/ν and Pe =V0H/D,
respectively. The concentration distribution actually is not
affected by the Poiseuille flow in Eq. (31), since the flow
direction (horizontal) is exactly perpendicular to the concentra-
tion gradient direction (vertical), and thus the convection term
∇·(ρu) is 0 in the convection-diffusion equation [Eq. (13)].
Nevertheless, this system can still serve as a simple case to
validate the program and examine the model performance.
Our simulations are conducted over a rectangular D2Q9
domain with periodic boundary conditions applied at the left
and right domain edges. The channel width is H=6forthe
Poiseuille flow simulations, and it is H=20 for the injection
Couette flow calculations. The injection flow velocity V0is
set to be the same as the top plate shear velocity U0in
the injection Couette flows. The relaxation parameters are
set as τf=τg=1 for all simulations in this work, and
their effects have been examined in Ref. [16]. The body
force for the Poiseuille flow and boundary velocities for the
injection Couette flow are calculated to have Re =Pe =10,
as employed in Ref. [16]. In addition, the channel centerline
location relative to the underlying lattice grid lines is adjusted
to test the result sensitivity to boundary locations. The vertical
distance between the channel centreline and the closest grid
line below is defined as the centerline offset α. Three different
offset values are considered with α=0.2, 0.5, and 0.7 for
both the Poiseuille and injection Couette flows. At α=0.5,
both the upper and lower plates locate exactly along the
midplane between two lattice grid lines. For comparison,
both the original bounce-back scheme in Ref. [16] and the
midpoint bounce-back method presented in this paper are
utilized.
The calculated velocity and concentration distributions
across the channel are displayed in Fig. 2. The transverse
velocity vis not shown there since it is uniform and 0 in the
Poiseuille flows and V0in the injection flows. Overall, both
boundary methods have successfully reproduced the general
variation trends (black lines). However, when we vary the
relative position of the channel centerline from α=0.2, 0.5,
to 0.7, the original bounce-back scheme does not respond to
such boundary location change: The same value is produced at
a particular lattice node even when the surface has been shifted
upward or downward. On the other side, with our improved
midpoint method, the simulated profiles correctly follow the
theoretical prediction based on the current centerline position.
The still relatively evident deviations from the theoretical
curves are due to the narrow channel width H=6forthe
Poiseuille flows and the large nonlinear slopes in velocity
0.0
0.2
0.4
0.6
0.8
1.0
u / U0
−3 −2 −1 0 1 2 3
1.00
1.02
1.04
1.06
1.08
1.10
y
C
−10 −5 0 5 10
y
8 9 10
0.4
0.6
0.8
1.0
8 9 10
1.04
1.06
1.08
1.10
(b)
(a) (c)
(d)
FIG. 2. (Color online) The velocity (upper panels) and concentration (lower panels) distributions for the Poiseuille (left panels) and injection
Couette (right panels) flows. The black lines are theoretical solutions, and the symbols are results from LBM simulations: open symbols for the
original bounce-back scheme [16], and filled symbols for our improved midpoint method. The symbol shapes and colors are used to indicate
the different centerline offsets: red circles for α=0.2, blue squares for α=0.5, and black diamonds for α=0.7. The range of 8 <y<10 in
the right panels have been enlarged in the insets to show more details.
033304-5

QING CHEN, XIAOBING ZHANG, AND JUNFENG ZHANG PHYSICAL REVIEW E 88, 033304 (2013)
and concentration near the top plate in the injection Couette
flows. For offset α=0.5, the results from the original and
the midpoint methods are identical, and the open squares are
completely overlapped by the filled squares in Fig. 2.This
identity is expected since with α=0.5wehavexb=xmfor
all boundary lattice links.
B. Boundary gradient approximation: Diffusion with
sinusoidal concentration gradient
To test the improvement of our finite-difference approxima-
tion in the normal boundary direction in Eq. (24), we consider
the pure diffusion problem between the two horizontal plates
with H=20 and α=0.5. A constant-value Dirichlet bound-
ary is applied on the lower plate, and a sinusoidal concentration
gradient is imposed on the upper plate,
C(y=0) =C0,(34)
∂C
∂y y=H=Csin βx, (35)
where Cis the gradient variation amplitude and βis the wave
number. The flow velocity is 0 everywhere by setting the two
plates stationary and applying no body force. This makes the
system analytically solvable, and the exact solution of this
problem is
C(x,y)=C0+C
β
sinh βy
cosh βH .(36)
In our calculation, we use C0=1, C=0.01, and β=
π/24. The rectangular simulation domain has a width of
48 lattice units to include one entire period of the gradient
variation, and thus the periodic boundary condition can be
applied in the horizontal direction. Moreover, both the original
[Eq. (22)] and our improved [Eq. (24)] finite-difference
schemes have been considered. The distance δfor our
normal-direction finite-difference approximation is selected
as 1.5 lattice units to avoid any possible involvement of a
solid node in the bilinear interpolation for the concentration
value at x∗.
Figure 3displays the LBM results (color symbols) in
comparison with the analytical solutions (black lines) at
several representative horizontal and vertical locations. Un-
surprisingly, our finite-difference scheme along the boundary
normal direction exhibits a much better performance than that
along the boundary lattice link, particularly in the region
near the top plate on which the nonuniform concentration
gradient is imposed. For example, the theoretical value at
location (x=12, y=19.5) is 1.0707, that from our normal-
direction scheme is 1.0687, and that from the lattice-direction
finite-difference scheme is 1.0555 (7.6 times larger in relative
errors). Since the centerline offset αis 0.5 in these simulations,
0 5 10 15 20
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
y
C
0 12 24 36 48
x
(b)
(a)
FIG. 3. (Color online) The concentration distributions between a constant-value and a sinusoidal gradient boundaries. Calculated results at
five xlocations (x=12, 18, 24, 30, and 36, from top to bottom and in the order of squares, upward triangles, circles, downward triangles, and
diamonds, left panel) and one ylocation (y=19.5, the first layer of lattice nodes from the top plate, right panel) are selected to display the
concentration field over the two-dimensional domain. The theoretical solutions are plotted as black curves, and results from LBM simulations
are displayed as symbols (red open symbols for results from the finite-difference scheme along the boundary lattice link [16], and the blue
filled symbols for those from our finite-difference scheme in the boundary normal direction).
033304-6

IMPROVED TREATMENTS FOR GENERAL BOUNDARY ... PHYSICAL REVIEW E 88, 033304 (2013)
the difference observed here must be from the different
finite-difference schemes.
It is interesting to note that at x=24, both finite-difference
schemes yield identical results [circles in Fig. 3(a)]. We
have expected to see the most significant difference between
the two schemes at this location, since this is the place
where the steepest gradient variation occurs. Through a careful
examination, we find the reason of these identical results is
due to the antisymmetric gradient variation at this particular
location. For the upper boundary lattice node at x=24 and
y=19.5 (the upper wall locates at y=20), there are three
boundary PDFs, g∗
2,g∗
5, and g∗
6, to be bounced back from
the top surface, and they return to this same node as g4,g7,
and g8, respectively. For the g∗
2→g4bouncing-back process,
the lattice direction is just the boundary normal direction,
and therefore no difference exists for the two different
finite-difference schemes. For the g∗
5→g7and g∗
6→g6
processes, the lattice directions deviated from the boundary
normal direction; however, the inaccuracy by neglecting the
gradient variation (antisymmetric here) from these two sides
has the same magnitude and opposite signs. When these two
PDFs g7and g8arrive the boundary node, the concentration
value there is calculated by Eq. (22), and their individual
inaccuracy will be simply canceled out. Similar analysis can be
applied to the upper boundary nodes at x=12 and 36, where
the symmetric gradient variations take place, and therefore
the largest errors from the lattice-direction finite-difference
scheme are observed.
C. General conditions on curved boundaries:
Diffusion between circular surfaces
As reviewed in the Introduction, previous boundary meth-
ods for convection-diffusion LBM models are typically limited
to flat surfaces or certain boundary conditions or suffer low
geometry resolutions. Here we simulate the pure diffusion
situation between two circular surfaces with radii Rin and Rout,
respectively. Again the fluid flow is excluded for simplicity.
The inner surface is assigned a constant concentration value of
Cin (Dirichlet condition), while three different situations are
considered for the outer surface:
(1) Case A: the Dirichlet condition with constant concen-
tration value Cout;
(2) Case B: the Neumann condition with constant concen-
tration gradient C
out;
(3) Case C: the mixed Robin condition as given in Eq. (26).
The respective solutions for these axisymmetric systems are
C(r)=Cin +(Cout −Cin)ln(r/Rin)
ln(Rout/Rin )for Case A, (37)
C(r)=Cin −RoutC
out ln(r/Rin)forCaseB,(38)
and
C(r)=Cin +a2Cin −a3
a2ln(Rin/Rout )−a1/Rout
×ln(r/Rout) for Case C. (39)
In our simulation, the boundary radii are Rin =40 and
Rout =80, and the inner boundary concentration value is Cin =
1.5. For Case A the boundary value on the outer surface is set
as Cout =0.5, and for Case B the boundary gradient on the out
surface is C
out =0.018. The parameter in the Robin condition
on the outer surface for Case C are a1=18, a2=0.5, and
a3=−0.1. Such a particular set of parameters generate similar
concentration distributions for these three different boundary
combinations. The simulated concentration profiles from the
boundary treatments presented in this paper are displayed
in Fig. 4. A good agreement between the simulated and
theoretical distributions, for all cases considered, can be
observed there. Slightly larger errors are noticed near the outer
surface for Case B due to the strong dependence on boundary
gradient via a first-order finite-difference approximation. This
set of simulations demonstrates that our boundary method is
capable of dealing with various boundary conditions on curved
surfaces.
D. Numerical accuracy: Flow and diffusion between
inclined plates and circular surfaces
As for typical LBM boundary models, we now study the
numerical accuracy of our boundary treatment. Two systems
40 50 60 70 80
0.4
0.6
0.8
1.0
1.2
1.4
1.6
r
C
40 50 60 70 80
r
40 50 60 70 80
r
(b)
(a) (c)
FIG. 4. (Color online) The calculated concentration distributions between two circular surfaces with the Dirichlet condition on the inner
surface and difference conditions on the outer surface: (a) the Dirichlet condition, (b) the Neumann condition, and (c) the mixed Robin
condition. The symbols are from our LBM simulations and the black lines are from the analytical solutions.
033304-7

QING CHEN, XIAOBING ZHANG, AND JUNFENG ZHANG PHYSICAL REVIEW E 88, 033304 (2013)
are considered here: the simple Couette shear flows between
two inclined parallel plates and the circular Couette flows
between two coaxial circular surfaces. For the inclined plate
system, the inclination angle is θ=tan−15/8[16], and a pe-
riodic domain is employed [26]. The boundary concentrations
are Cl=0 on the lower, stationary plate, and Cu=1.0onthe
upper, moving plate with a shear velocity U0=10−4.Forthe
circular Couette flows, we maintain the outer radius as twice of
the inner radius, i.e., Rout =2Rin. The boundary concentration
values are Cin =1.5 on the inner, stationary surface and
Cout =1.0 on the outer, rotating surfaces. The rotation speed
of the outer surface is adjusted with the system size such
that Re =R2
in/ν =5[26,29]. These boundary parameters
are held constant, while different plate separations Hfor
the inclined-plate system and annular gaps Rout −Rin =Rin
(since Rout =2Rin in this work) for the circular system are
utilized to examine the numerical accuracy.
The global relative error is defined as
EA=(ALBM −Ath)2
A2
th 1/2
,(40)
where both summations are performed over all lattice nodes
in the simulated domain. The property Acould be the
concentration Cor the velocity magnitude U=(u2+v2)1/2,
and the subscripts LBM and th indicate the calculated and
theoretical values.
The calculated global errors are plotted in Fig. 5, and
linear fittings are conducted in these logarithmic graphs, with
the line slopes also displayed in Fig. 5. The fitting slope
is usually considered as the accuracy order of a numerical
model. Here, similarly to those observed in typical LBM
boundary methods, slopes for both flow and concentration
from the circular system are about 2, indicating a second-order
overall accuracy for the circular system. However, for the
system with inclined plates, the accuracy orders are much
lower: 1.383 for flow and 1.009 for concentration. The
slope of 1.009 for concentration is similar to that from the
original boundary method by Zhang et al. (1.03 there) [16,30];
however, the error magnitude is orders smaller (∼10−9from
our improved method vs ∼10−3from the original method),
indicating more accurate results have been produced from our
improved method. It is interesting to note that different slopes
(accuracy orders) can be observed from different systems with
the same numerical method. A qualitative understanding can
be established by considering the fact that the global errors
actually include the inaccuracy from the LBM algorithm as
well as the assumptions in the boundary method. For the
inclined-plates system, since the LBM algorithm mathemati-
cally can be considered a second-order finite-difference solver
of differential equations [16,31,32], the LBM solutions for the
linear flow and concentration fields between two parallel plates
should be perfectly exact (except the inevitable computer
round-off errors). However, any boundary inaccuracy will
affect the entire linear distribution and yield an incorrect
slope between the two plates, as shown in Ref. [31]. Thus,
the relative error between the theoretical line (connecting the
real boundary values with the correct slope) and the simulated
line (connecting the shifted boundary values due to boundary
inaccuracy with a slightly different slope) is proportional to
the original error at boundary, which is directly related to the
boundary resolution. This has been confirmed by our further
experiments. For the H=16 separation, when we change the
tolerance value in the calculation of the boundary node location
from its current value 10−7to 10−6(i.e., to set the boundary
location less precisely), the global error in concentration
10−7
10−6
EU
20 40 80 160
10−10
10−9
H
EC
10−4
10−3
10−2
EU
8 16 32 64
10−5
10−4
10−3
Rin
EC
(b) slope: −1.009
(a) slope: −1.383 (c) slope: −1.994
(d) slope: −2.046
FIG. 5. (Color online) The global relative errors for flow (upper panels) and concentration (lower panels) between inclined plates (left
panels) and circular surfaces (right panels) with different separations. The black lines are from linear fittings of the relative errors in the
logarithmic scale. The slope of the fitting line is also shown in the lower-left corner in each panel.
033304-8

IMPROVED TREATMENTS FOR GENERAL BOUNDARY ... PHYSICAL REVIEW E 88, 033304 (2013)
increases from 1.002 ×10−9to 1.259 ×10−8. On the other
hand, in the circular system, the flow and concentration dis-
tributions are neither exactly linear (first-order) nor parabolic
(second-order). It appears that the error from the second-order
LBM algorithm in describing the nonlinear and nonparabolic
flow or concentration fields suppresses the relatively small
boundary inaccuracy and dominates the overall global error
behavior. As a result, the error-resolution relationship yields
an approximately second-order accuracy. Accuracy orders
between 1 and 2 have also been reported from other LBM
boundary methods [22,27,33], indicating the combined effects
from boundary and LBM algorithm.
In addition to the different slopes, the absolute error
magnitudes from the inclined-plate system are orders lower
than those from the circular system. This might be mainly
due to the particular interpolation or extrapolation schemes
employed to estimate the midpoint velocity and concentration
values. For the inclined-plate system, the velocity and con-
centration distributions are linear, and the linear interpolation
or extrapolation schemes can yield excellent predictions of
the midpoint values. This will not be the case for the
circular system, where neither the flow nor the concentration
distributions is linear. Moreover, compared to the flat walls in
the inclined-plate system, the curved surfaces in the circular
system could also introduce larger numerical errors. At last,
we also notice that the relatively larger error magnitudes for
flow fields than those for concentration fields, in both the
inclined-plate and circular systems, are mainly due to the
particular formulation for the relative error in Eq. (40).
The velocity magnitude varies from 0 to U0=10−4in the
inclined-plate system or from 0 to 0.02 in the circular system,
while the concentration values cover a range of [0, 1] in the
inclined-plate system and a range of [1,1.5] in the circular
system. These relatively larger Ath values in the denomina-
tor for concentration calculations cause the corresponding
relative errors to be much smaller than those of the flow
calculations.
E. Galilean invariance: Accuracy with
different reference velocities
So far our simulations are performed in relatively simple
systems, and all the boundary surfaces are not moving. At
last, we consider a more general system with a circular
cylinder of radius Rmoving at a constant velocity U0in a
horizontal channel with width Halong the centerline. Constant
concentrations are applied on the cylinder surface (Cc) and
the channel surfaces (Cs). In addition, we also consider the
system with the cylinder stationary but with the channel
walls moving in the opposite direction −U0. According to the
Galilean invariance principle, these two approaches should
yield the same concentration field, and the flow velocities
should have a difference of U0between them. The parameters
we use here are H=160, R=40, U0=5×10−3,Cc=1.1,
and Cs=1.0. For the moving-cylinder system, the cylinder
surface is moving in space, and some fluid lattice nodes in
front of the cylinder can be covered, as well as some solid
lattice nodes at the back of the cylinder can be released.
Here we adopt the treatment described in Ref. [34], for
both the flow and concentration PDFs at these nodes. The
PDFs for the covered nodes will be simply ignored, while
the PDFs for a newly released node xnare assigned via
a weighted extrapolation from existing neighboring fluid
nodes,
fi(xn)=jωj[2fi(xn+ejδt)−fi(xn+2ejδt)]
jωj
,(41)
gi(xn)=ωj[2gi(xn+ejδt)−gi(xn+2ejδt)]
jωj
,(42)
where the lattice index iruns from 0 to 8, and the summations
in the right-hand side are taken over all lattice directions jat
xnpointing to existing fluid nodes.
The results from these two simulations are displayed in
Fig. 6. The pressure and velocity fields between such two
systems have been examined in our previous publication [34],
and it is confirmed again in this study that the pressure
distributions are similar, while the velocity fields have a U0
difference. The concentration fields also appear identical from
the contour lines and the centerline profiles have excellent
agreement with each other [Figs. 6(a)–6(c)]. The asymmetric
distribution about the cylinder reflects the convection effect on
the diffusion process. For a more quantitative comparison,
similar to Eq. (40), here we define a global difference in
concentration between such two systems as
EC=(Cmc −Cmw)2
C2
mw 1/2
,(43)
where Cmc and Cmw are, respectively, the simulated con-
centration values from the moving-cylinder and moving-wall
systems. The moving-wall system has been selected as the
comparison reference in the denominator, since there the
curved cylinder surface is stationary and only the flat wall
surfaces are moving tangentially, and the simulated results are
supposed to be closer to the real values.
The calculated differences by use of Eq. (43) with different
relative velocity U0are displayed in Fig. 6(d). At the extreme
situation with U0=0, the two systems become identical,
and a zero difference is expected. When a relative motion
is introduced, a linear relationship between the difference
ECand relative velocity U0(fitting slope =0.1852) can be
seen there. The increase of ECwith U0is not surprising. In
general, there are three sources that are responsible for the
numerical error in our LBM simulations: the second-order
LBM algorithm which may not be able to describe the flow and
concentration distributions exactly; the linear interpolations or
extrapolations in evaluating the midpoint values umand Cm;
and the assumptions employed in the bounce-back boundary
scheme (see the appendix). As a matter of fact, a large relative
velocity U0produces larger velocity and concentration gra-
dients around the cylinder surface; and these larger gradients
will affect all the error sources negatively. However, detailed
analysis of the E2∼U0relationship for this cylinder-channel
system could be difficult, since the imposed velocity U0affects
the concentration field both directly (larger U0→stronger
convection effect →larger concentration gradient in front
of cylinder →less accurate LBM concentration results)
and indirectly [larger U0→larger velocity gradient near
cylinder →less accurate LBM flow results →less accurate
LBM concentration results via Eq. (9)].
033304-9

QING CHEN, XIAOBING ZHANG, AND JUNFENG ZHANG PHYSICAL REVIEW E 88, 033304 (2013)
y
0
100
200
50
150
0 200 400 600 800 1000
1.00
1.05
1.10
x
Cy
0
100
200
150
50
Moving Walls
Moving Cylinder
(c)
(b)
(a)
00.0025 0.005 0.0075 0.01
0
0.5
1
1.5
2
2.1 x 10−3
U0
EC
10−4 10−310−2
10
−4
10
−3
U0
Ec
linear fitting
Simulations
(d)
FIG. 6. (Color online) The flow (arrows) and concentration (contour lines) distributions for the moving-cylinder (a) and moving-wall
(b) systems with the wall-cylinder relative velocity U0=5×10−3. The concentration variations along the channel centerline from these
two systems in (a) and (b) are also plotted in (c) for comparison. The global differences in concentration between the moving-cylinder and
moving-wall systems with different relative velocity U0are plotted in (d) (the inset shows the same data but in logarithmic scales).
For simplicity, below we consider the Case A in Sec. III C,
however, with both inner and outer surfaces moving at a
same velocity U0. The flow velocity should be constant U0
everywhere, and the concentration field should not be affected
by this uniform velocity and can still be described by Eq. (37).
Here we use Rin =40, Rout =80, and τf=τg=1. With
different boundary velocities U0, two boundary concentration
conditions are adopted to examine the effect of concentration
gradient on the global error E2:(1)Cin =Cout =1.0for
a uniform concentration distribution and (2) Cin =1.5 and
Cout =0.5 for a nonuniform concentration field. First, we look
at the uniform concentration case with Cin =Cout =1.0. For
this situation, all the error sources have been eliminated to
our best: The LBM algorithm can describe the uniform flow
and concentration fields exactly; the midpoint values can be
evaluated with linear approximation along boundary lattice
links accurately; and all approximations with the bounce-
back scheme (see the appendix) are perfectly satisfied. Our
calculations show that the surface velocity U0has not evident
influence on the simulation accuracy, and the relative error E2
varies in a range of 10−17 ∼10−14 (black squares in Fig. 7). We
believe such tiny differences between the simulated and theo-
retical concentration values are from the inevitable numerical
errors associated with the finite precision in computers. This
is confirmed by looking at the simulated fluid velocity, and we
find that the difference between the LBM velocity and the im-
posed boundary velocity U0is typically of 10−15 ∼10−18.On
the other hand, for the nonuniform system with Cin =1.5 and
Cout =0.5, several-order-larger errors are observed (blue cir-
cles in Fig. 7). An error of E2=5.06 ×10−5exists even when
the surfaces are not moving at all (U0=0). Clearly, this error
is directly related to the concentration variation in the space.
When the surface motion is introduced, again we see that the
global error increases with U0approximately linearly, except in
the low-velocity region U0<10−3. The constant and uniform
boundary and fluid velocity U0, in principle, should has no
effects on accuracies of the LBM algorithm and midpoint value
evaluations (i.e., U0does not change the spatial concentration
distribution). However, the flow velocity near the boundary
is directly involved in Eqs. (A7)–(A9), and a larger velocity
U0will make these approximations less accurate, especially
the difference in the linear velocity terms in Eq. (A7).
0 0.002 0.004 0.006 0.008 0.01
0
1
2
3
4
5
x 10−4
U0
E2
10−4 10−3 10−2
10−4
10−3
U0
E2
Cin=1.5, Cout=0.5
Cin=Cout=1.0
FIG. 7. (Color online) The global relative errors for concentration
between two coaxial cicular surfaces moving at reference velocity U0
with same concentration field (Cin =Cout =1.0, black squares) or
different concentration field (Cin =1.5andCout =0.5, blue circles).
The inset shows the same data of the latter case but in logarithmic
scales.
033304-10

IMPROVED TREATMENTS FOR GENERAL BOUNDARY ... PHYSICAL REVIEW E 88, 033304 (2013)
Therefore, the nonuniform concentration distribution is mainly
responsible for the increase in E2with the boundary velocity
U0in Fig. 7.
The above simulations and analysis indicate that, as the
overall accuracy discussed in Sec. III D, the performance
of a LBM model in preserving the fundamental Galilean
invariance is a complicated phenomenon. Several factors,
including the LBM algorithm accuracy, the boundary method,
and the particular flow and concentration distributions being
simulated, are all playing roles, and their effects are highly
interconnected. For the LBM algorithm and boundary method
employed in this study, the Galilean invariance performance is
generally satisfactory. The global relative error could become
significant for a larger relative or reference velocity U0;
however, such a large U0is not favorable for incompressible
flows [2] and should be avoided in practical applications.
In addition, our up-to-date literature search indicates that,
in previous LBM simulations of convection-diffusion or
heat transfer processes, the system boundaries are typically
stationary, or with some boundary velocities (e.g., shearing
velocity, injection velocity, or rotating velocity) imposed at
fixed spatial boundary locations, and no physical boundary
movement and displacement have been considered. Our simu-
lation of the moving cylinder demonstrates that our boundary
treatments (including those for the boundary velocity, the
boundary concentration, and the node status change) can be
a good choice for simulating systems with physical boundary
movement, such as the convection-diffusion and heat transfer
processes in particulate flows.
IV. SUMMARY
We have extended the recent midpoint boundary method for
LBM flow simulations to the convection-diffusion processes.
We have also proposed to evaluate the concentration boundary
value from the concentration gradient via a finite-difference
scheme in the normal direction to consider nonuniform bound-
ary conditions. Compared with other existing LBM boundary
methods, our boundary treatments described in this paper
can work with arbitrary curved geometry, boundary velocity
and motion, and various surface conditions (the Neumann,
Dirichlet, or mixed Robin conditions). To examine the capacity
and performance of the present boundary methods in describ-
ing flow and concentration fields, several carefully designed
simulations have been conducted. The simulation results have
been compared with analytical solutions and numerical results
from a representative boundary method. These simulations and
comparisons show that our methods have good performances
in dealing with complex boundary situations, including the pre-
cise boundary location, the accurate solution near the boundary
with a nonuniform gradient distribution, the capability to work
for different boundary conditions on curved surfaces, and an
improved numerical accuracy. In addition, we have examined
the performance of our method in preserving the Galilean
invariance, and the effect of reference velocity on numerical
accuracy has been investigated. The moving-cylinder simula-
tion also demonstrates the potential usefulness of our methods
in simulating systems with physical boundary movement. The
relatively simple algorithm is particularly advantageous for
transport processes in particulate and porous systems.
Last, we point out that, although the model description as
well as the simulations presented in this paper are all two-
dimensional with the D2Q9 lattice structure and the single-
relaxation-time LBGK algorithm, extending these methods to
three-dimensional situations, other lattice structures, and/or
MRT LBM models should be straightforward and present
no technical difficulty. The present methods are also readily
applicable to other processes and phenomena that can be
described by convection-diffusion-type differential equations,
for example, heat transfer and convection-diffusion-reaction
processes. More complicated boundary conditions such as
chemical reactions at boundary can also be implemented [6].
ACKNOWLEDGMENTS
This work was supported by the Natural Science and Engi-
neering Research Council of Canada (NSERC), the Laurentian
University Research Fund (LURF), the Research Fund for the
Doctoral Program of Higher Education, and the Natural Sci-
ence Foundation of Jiangsu Province, China (BK20131348).
J.Z. is grateful to T. Zhang, B. Shi, and Z. Guo at Huazhong
University of Science and Technology (Wuhan, China) for
clarifying the global error calculation in Ref. [16]. Q.C. ac-
knowledges the visiting studentship from Nanjing University
of Science and Technology for his study in Laurentian Univer-
sity. This work was made possible by the facilities of Westgrid
(www.westgrid.ca) and SHARCNet (the Shared Hierarchical
Academic Research Computing Network, www.sharcnet.ca).
APPENDIX
In this appendix, we discuss how the boundary PDF
g¯
iin Eq. (18) can be derived and the assumptions and
approximations involved. Here we follow Zhang et al. [16]
and only consider the situation that the boundary locates at
the midposition between two vertical lattice lines [Fig. 8(a)].
After the collision step at the boundary fluid node xf,the
density distribution g∗
i(xf) leaves toward the surface wall,
and it is then bounced back at the wall xb(xwin Ref. [16])
with a modified magnitude as g¯
i(xf), which will be utilized
in the next collision operation at xf.InFig.8,wehavethe
boundary lattice directions i=1 and ¯
i=3. On the other hand,
we can consider the solid node at xsas a ghost fluid node. The
postcollision distribution g∗
¯
i(xs) will move to xfduring the
regular propagation process [Fig. 8(b)],
g¯
i(xf)=g∗
¯
i(xs).(A1)
To obtain an estimate of g¯
i(xf)org∗
¯
i(xs), several approxi-
mations are necessary. First, we assume that all postcollision
PDFs are close to their individual equilibrium values [2], i.e.,
g∗
i≈geq
i,(A2)
and applying Eq. (9) to nodes xfand xs, respectively, yields
g∗
i(xf)≈ωiCf1+ei·uf
c2
s+(ei·uf)2
2c4
s−uf·uf
2c2
s,(A3)
g∗
¯
i(xs)≈ω¯
iCs1+e¯
i·us
c2
s+(e¯
i·us)2
2c4
s−us·us
2c2
s.(A4)
Here subscripts fand shave been adopted to indicate node
locations for flow velocity uand concentration C. Adding
033304-11

QING CHEN, XIAOBING ZHANG, AND JUNFENG ZHANG PHYSICAL REVIEW E 88, 033304 (2013)
xδx
xfxbxs
xf
()
xf
()
gi
−
i
*
δ
fluid solid
0.5
(a)
wall
g
xδx
xfxs
xf
()
xf
()
gi
xs
gi()
xs
gi
−
*()
xb
wall
*
−
δ
fluid solid
0.5
(b)
gi
FIG. 8. Schematics showing (a) the bouncing-back process of
density distribution at a solid surface and (b) the propagation process
between the ghost (solid) and fluid nodes across the boundary. Filled
arrows are used to indicate postcollision density distributions leaving
lattice nodes, and open arrows are for the precollision distributions
arriving at nodes after one time step.
Eqs. (A3) and (A4), and noticing ei=−e¯
iand ωi=ω¯
i, one
can have
g∗
i(xf)+g∗
¯
i(xs)≈ωi(Cf+Cs)+ei·(Cfuf−Csus)
c2
s
+Cf(ei·uf)2+Cs(ei·us)2
2c4
s
−Cfuf·uf+Csus·us
2c2
s.(A5)
Since Csand usat the ghost node xsare not available, we
assume slow variations in concentration Cand velocity u,as
well as any product combinations from them, between xfand
xs. We then can approximate the right-hand side terms in the
above equation with the twice of the corresponding midpoint
values for addition and zeros for subtraction as follows:
Cf+Cs≈2Cb,(A6)
Cfuf−Csus≈0,(A7)
Cf(ei·uf)2+Cs(ei·us)2≈2Cb(ei·ub)2,(A8)
Cfuf·uf+Csus·us≈2Cbub·ub,(A9)
where Cband ubare, respectively, the concentration and
velocity at the boundary. Substituting these approximations
back to Eq. (A5),wehave
g∗
i(xf)+g∗
¯
i(xs)≈ωi2Cb+Cb(ei·ub)2
c4
s−Cbub·ub
c2
s.
(A10)
At last, we replace g∗
¯
i(xs) with g¯
i(xf) according to the
propagation operation in LBM Eq. (A1), and move the left-
hand side term g∗
i(xf) to the right,
g¯
i(xf)≈−g∗
i(xf)+2ωiCb1+(ei·ub)2
2c4
s−ub·ub
2c2
s,
(A11)
which is identical to Eq. (18).
[1] S. Chen and G. D. Doolen, Annu. Rev. Fluid Mech. 30, 329
(1998).
[2] S. Succi, The Lattice Boltzmann Equation (Oxford University
Press, Oxford, 2001).
[3] J. Zhang, Microfluidics Nanofluidics 10, 1 (2011).
[4] Q. Chen and X. Zhang, Int. J. Mod. Phys. C 23, 1250074 (2012).
[5] S. P. Dawson, S. Chen, and G. D. Doolen, J. Chem. Phys. 98,
1514 (1993).
[6] X. He, N. Li, and B. Goldstein, Mol. Simul. 25, 145 (2000).
[7] I. Rasin, S. Succi, and W. Miller, J. Comput. Phys. 206, 453
(2005).
[8]R.G.M.vanderSmanandM.H.Ernst,J. Comput. Phys. 160,
766 (2000).
[9] X. Zhang, A. G. Bengough, J. W. Crawford, and I. M. Young,
Adv. Water Resour. 25, 1 (2002).
[10] I. Ginzburg, Adv. Water Resour. 28, 1171 (2005).
[11] B. Shi, B. Deng, R. Du, and X. Chen, Comput. Math. Appl. 55,
1568 (2008).
[12] J. Eggels and J. Somers, Int. J. Heat Fluid Flow 16, 357 (1995).
[13] X. He, S. Chen, and G. Doolen, J. Comput. Phys. 146, 282
(1998).
[14] Z. Guo, C. Zheng, B. Shi, and T. S. Zhao, Phys.Rev.E75,
036704 (2007).
[15] J. Wang, D. Wang, P. Lallemand, and L. Luo, Comput. Math.
Appl. 65, 262 (2013).
[16] T. Zhang, B. Shi, Z. Guo, Z. Chai, and J. Lu, Phys. Rev. E 85,
016701 (2012).
[17] Q. Kang, D. Zhang, S. Chen, and X. He, Phys.Rev.E65, 036318
(2002).
[18] Q. Kang, P. C. Lichtner, and D. Zhang, Water Resour. Res. 43,
W12S14 (2007).
[19] S. D. C. Walsh and M. O. Saar, Phys.Rev.E82, 066703 (2010).
[20] H. Yoshida and M. Nagaoka, J. Comput. Phys. 229, 7774 (2010).
[21] I. Ginzburg, Adv. Water Resour. 28, 1196 (2005).
[22] L. Li, R. Mei, and J. F. Klausner, J. Comput. Phys. 237, 366
(2013).
[23] S. Chen, K. H. Luo, and C. Zheng, J. Comput. Phys. 231, 8278
(2012).
[24] A. J. C. Ladd, J. Fluid Mech. 271, 285 (1994).
[25] L. Chen, Q. Kang, B. A. Robinson, Y. He, and W. Tao, Phys.
Rev. E 87, 043306 (2013).
[26] X. Yin and J. Zhang, J. Comput. Phys. 231, 4295 (2012).
[27] O. Oulaid and J. Zhang (unpublished).
[28] R. Mei, L. Luo, and W. Shyy, J. Comput. Phys. 155, 307 (1999).
[29] Z. Guo, C. Zheng, and B. Shi, Phys. Fluids 14, 2007 (2002).
[30] T. Zhang, B. Shi, Z. Guo, Z. Chai, and J. Lu, Phys. Rev. E 88,
029903(E) (2013).
[31] G. Le and J. Zhang, Phys.Rev.E79, 026701 (2009).
[32] X. He, Q. Zou, L. Luo, and M. Dembo, J. Stat. Phys. 87, 115
(1997).
[33] D. Hlushkou, D. Kandhai, and U. Tallarek, Int. J. Numer.
Methods Fluids 46, 507 (2004).
[34] X. Yin, G. Le, and J. Zhang, Phys.Rev.E86, 026701
(2012).
033304-12