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Journal for Multiscale Computational Engineering, 12 (3): 177–192 (2014)
HYBRID LATTICE BOLTZMANN AND FINITE
VOLUME METHODS FOR FLUID FLOW PROBLEMS
Zheng Li,1,2 Mo Yang,2& Yuwen Zhang1,∗
1Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia,
Missouri 65211, USA
2College of Energy and Power Engineering, University of Shanghai for Science and Technology,
Shanghai 200093, China
∗Address all correspondence to Yuwen Zhang, E-mail: zhangyu@missouri.edu
Two hybrid methods that combine the lattice Boltzmann method (LBM) and finite volume method (FVM) are proposed
in this paper. The D2Q9 model is used in LBM, while the SIMPLE algorithm with QUICK scheme is chosen for
FVM. The hybrid methods are fulfilled by dividing the computational domain into the LBM and FVM zones, with
a public message-passing zone in between, and only macroscopic velocities are needed to be passed between the two
separated zones. A nonequilibrium extrapolation scheme and the finite difference velocity gradient method are used to
pass information between the SIMPLE and LBM zones. The proposed methods are used to solve problems with the lid-
driven flow, and the results obtained using pure LBM, pure SIMPLE, and the hybrid methods with different schemes
agreed with each other very well in different Reynolds numbers.
KEY WORDS: finite volume method, lattice Boltzmann method, Navier-Stokes, viscous flows, incompress-
ible flow
1. INTRODUCTION
Both the lattice Boltzmann method (LBM) (Succi, 2001; Zou et al., 1995; Chen and Doolen, 1998) and finite volume
method (FVM) (Patankar, 1980) are widely used methods in computational fluid dynamics (CFD) and each has their
own advantages. The LBM developed from lattice gas automata (LGA) (Inamuro and Sturtevant, 1990) is growing
to be versatile in many fields (Yu et al., 2003) because of its three important features (Chen and Doolen, 1998): (1)
the nonlinear convection term in Navier-Stokes (NS) equations is linear in the phase space; (2) the incompressible
NS equations can be regained by Chapman-Enskog expansion (Chapman and Cowling, 1970); and (3) a minimal set
of velocities in phase space is needed in LBM. Meanwhile, FVM is successful because its discrete schemes always
satisfy conservation laws and can be adopted easily due to its clear physical meaning (Tao, 2001). The semi-implicit
method for pressure linked equation (SIMPLE) (Patankar, 1980) algorithm is widely used in FVM to solve fluid flow
and heat-transfer problems. Combination of the LBM and FVM will allow us to combine the advantages of both of
them.
Meanwhile, development of CFD involves a very wide variation of scales (Succi et al., 2001), ranging from nano-
/micro-, meso-, to macroscales (Abraham, 2000). Molecular dynamics (MD) (Grest and Kremer, 1986) is applicable
to nano- and microscale problems, and LBM is a typical mesoscopic scale method (Pomeau, 2007); FVM, on the other
hand, falls into the category of a macroscale approach (Tao, 2001). It is impossible to solve a multiscale problem using
any single-scale method. For example, MD simulation cannot be used in the entire simulation domain, and FVM is
not suitable for the microscopic region; LBM requires several times more computational time than the FVM to obtain
the results with the same accuracy in the macroscopic problem (Kandhai et al., 1999). So there is a demand to develop
multiscale methods to take advantage of different methods. There are some existing multiscale methods reported in
1543–1649/14/$35.00 c
⃝2014 by Begell House, Inc. 177
178 Li, Yang, & Zhang
NOMENCLATURE
alattice acceleration Vvelocity
Blattice body force u0lid velocity (m/s)
clattice speed usound sound speed (m/s)
cslattice sound speed Ωcollision operator
eavelocity in every direction Γcoefficient in SIMPLE
fdensity distribution ρdensity (kg/m3)
Fnondimensional time µdynamic viscosity (N s/m2)
Kcoefficient in Chapman-Enskog expansion νkinematic viscosity (m2/s)
mdimension of the problem τ0relaxation time
ppressure (N/m2)δtmagnitude of the time step (s)
Pnondimensional pressure
rlattice location vector Superscript
Re Reynolds number eq equilibrium
Ssource term
ttime Subscript
uhorizontal velocity (m/s) adirection of the discretized velocity
vvertical velocity (m/s) LLBM
Unondimensional horizontal velocity SSIMPLE
Vnondimensional vertical velocity 1, 2 order for Chapman-Enskog expansion
the literature: MD-FVM (Nie and Chen, 2004; Flekkoy et al., 2000; O’Connell and Thompson, 1995), LBM-MD
(Dupuis et al., 2007; Werder et al., 2005; Fedosov and Karniadakis, 2009), the LBM-finite difference method (FDM)
(Christensen and Graham, 2010; Albuquerque et al., 2004; Latt, 2007), and LBM-FVM (Mondal and Mishra, 2008;
Joshi et al., 2010; Mondal and Mishra, 2009).
Two hybrid methods to combine LBM and FVM with the SIMPLE algorithm are proposed in this paper. The
key point of the hybrid method is to pass the information on the interface between LBM and FVM. It is difficult
to transfer velocity obtained from FVM into node populations that are needed in LBM, and density distribution
function reconstruction was used to solve the interface problem in the hybrid method (Luan et al., 2011, 2012). A
nonequilibrium extrapolation scheme (Guo et al., 2002) and the finite difference velocity gradient method (Latt and
Chopard, 2008) are used in this work to pass information between the FVM and LBM zones. The lid-driven flow
problem is solved to test the proposed methods.
2. ALGORITHM
Statistical behaviors of a fluid that is not in thermodynamic equilibrium can be described by the Boltzmann equation:
∂f
∂t +ε·∂f
∂r +a·∂f
∂ε= Ω (f)collision (1)
where fis the density distribution, and Ωis the collision operator that is dictated by the collision rules. There are
many efforts to simplify the Boltzmann equation. The Bhatnagar-Gross-Krook (BGK) model that uses the Maxwell
equilibrium distribution feq will be used in this work:
feq =n1
(2πRgT)m/2exp −(ε−u)2
2RgT(2)
Journal for Multiscale Computational Engineering
Hybrid Lattice Boltzmann and Finite Volume Methods 179
where mis the dimension of the problem. Equation (2) describes the situation that the system has reached the final
equilibrium. The BGK model assumes that the collision term is the time relaxation from density distribution to the
Maxwell equilibrium distribution. Assuming the relaxation time is τ0, the Boltzmann equation under the BGK model
(LBGK) can be expressed as ∂f
∂t +ε·∂f
∂r +a·∂f
∂ε=−1
τ0
(f−feq)(3)
The LBM used in this paper is a special scheme of LBGK. Only limited numbers of directional derivatives are applied
to Eq. (3), and there must be enough information to obtain the macroscopic governing equation. The D2Q9 model
is used, and nine directions are selected in the two-dimensional (2D) problem shown in Fig. 1. The velocity in every
direction is
ea=
(0,0) a= 1
c−cos aπ
2,−sin aπ
2a= 2,3,4,5
√2c−cos (2a+ 1)π
4,−sin (2a+ 1)π
4a= 6,7,8,9
(4)
where cis a constant in the lattice unit. The density distribution fain the fixed direction can be obtained by integrating
Eq. (3):
fa(r+eaδt, t +δt)−fa(r, t) = −1
τ0
(feq −fa) + δtBa(5)
where δtis the magnitude of the time step, while Bais the body force in the fixed direction.
When the velocity is low, feq can be simplified as
feq =ρωa1 + ea·u
RgT+(ea·u)2
2R2
gT2−u2
RgT(6)
ωa= (2πRgT)−m/2exp −e2
a
2RgT(7)
where feq can be further simplified with regard to cs, the speed of sound in the lattice unit:
feq =ρωa1 + ea·u
c2
s
+(ea·u)2
2c4
s−u2
2c2
s(8)
FIG. 1: Nine directions in the D2Q9 model
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180 Li, Yang, & Zhang
where
ωa=
4
9a= 1
1
9a= 2,3,4,5
1
36a= 6,7,8,9
(9)
Applying the following Chapman-Enskog expansion equations,
∂
∂r =K∂
∂r1(10)
∂
∂t =K∂
∂t1
+K2∂
∂t2(11)
fa=f0
a+Kf 1
a+K2f2
a(12)
to Eq. (5), the macroscopic governing equations can be obtained from LBM:
∂ρ
∂t +∇ · (ρV) = 0 (13)
∂(ρV)
∂t +∇ · (ρV V ) = −∇p+∇ · ρν ∇V+ (∇V)T−ν
c2
s∇ · (ρVVV) (14)
ν=c2
sτ0−1
2∆t(15)
where ∆tis the time step in LBM. Equation (14) differs from the macroscopic momentum conservation due to the
presence of the term
∇ · ρν −ν
c2
s∇ · (ρV V V )
Fortunately, it can be neglected when the Mach number is low, which is the case under consideration. Thus the LBM
satisfies the same macroscopic governing equations as the macroscopic method, which meets the requirement to
combine LBM with FVM.
To obtain the macroscopic parameter, the following two additional equations are needed:
ρ=
9
a=1
fa(16)
ρV=
9
a=1
eafa(17)
where eais the vector representing the velocity in every discrete direction, and Vis the vector of the macroscopic
velocity. The locations of the computing nodes are shown in Fig. 2.
Meanwhile, SIMPLE is a very popular FVM algorithm (Patankar, 1980) that solves the general equations in
macroscopic scale based on the control volume shown in Fig. 3. For a 2D problem in the Cartesian coordinate system,
the general equations can be expressed as
∂Φ
∂t +∂(ρuΦ)
∂x +∂(ρvΦ)
∂y =∂
∂x Γ∂Φ
∂x +∂
∂y Γ∂Φ
∂y +S(18)
The SIMPLE algorithm with QUICK scheme (Tao, 2001) is employed to solve Eq. (18). Staggered grids are used
in SIMPLE, and the locations of the SIMPLE variables are shown in Fig. 4. It can be seen that the locations of
macroscopic parameters are different in SIMPLE and LBM, even if the grids are the same.
Journal for Multiscale Computational Engineering
Hybrid Lattice Boltzmann and Finite Volume Methods 181
FIG. 2: Computational nodes in LBM
FIG. 3: Control volume in 2D FVM
FIG. 4: Variable locations in SIMPLE
Volume 12, Number 3, 2014
182 Li, Yang, & Zhang
3. HYBRID LBM AND FVM METHOD
3.1 Interface of the Hybrid Method
In order to combine LBM and SIMPLE in the same problem, the computational domain is divided into two zones as
shown in Fig. 5. In most cases, the wider the message-passing zone, the better the accuracy. However, an enlarged
message-passing zone also increases the computational time in every time step. Meanwhile, although the grid is uni-
form in the whole domain, the locations of velocity on the grid are different for LBM and SIMPLE, as discussed
above. So, additional interpolation, which may require the information on the nearby nodes, is needed in the informa-
tion sharing process. Figure 6 shows a more clear view of this in one control volume in the message-passing zone. It
is necessary to point out that it is almost impossible to transfer the pressure in LBM to that in SIMPLE because they
have different ways to obtain the pressure. In LBM, the ideal gas law is used to obtain the pressure, while SIMPLE
solves the pressure correction equation based on the conservation of mass. The example in this paper does not need
to transfer the pressure information in the message-passing zone shown in Fig. 5 due to the nature of method used to
combine LBM and SIMPLE. The interface between LBM and the SIMPLE zone is treated as a fixed velocity bound-
ary problem at every time step. There is no need for any special treatment to the SIMPLE zone other than setting the
boundary velocity in the program. On the other hand, it is different for the LBM zone because its original variable is
density distribution on the computational nodes. The three density distributions f4,f8,f9and density ρare unknown,
as shown in Fig. 7, while there are only three equations:
FIG. 5: Computational domain for LBM and SIMPLE
FIG. 6: Details on one control volume in the message-passing zone
Journal for Multiscale Computational Engineering
Hybrid Lattice Boltzmann and Finite Volume Methods 183
FIG. 7: Boundary condition in LBM at the interface
f4+f8+f9=ρ−f1−f2−f3−f5−f6−f7(19)
f4+f8+f9=−ρuL+f2+f6+f7(20)
f8−f9=ρvL+f5+f6−f3−f7(21)
3.2 Selected LBM Boundary Conditions
There are several methods to solve this problem on the boundary (Tao and Chopard, 2008) in LBM. The nonequilib-
rium extrapolation scheme (Guo et al., 2002) and finite difference velocity gradient method (Tao and Chopard, 2008)
are used in this paper to replace all the density distributions on the boundary.
For the nonequilibrium extrapolation scheme (hybrid method 1),
(fa−feq
a)boundary = (fa−feq
a)inner a= 1,2···9(22)
This scheme has second-order accuracy and can meet the requirement of the combining method.
On the other hand, the finite difference velocity gradient method (hybrid method 2) is used to replace all the
density distributions on the boundary. Equation (12) can be rewritten as
fa=f(0)
a+Kf (1)
a+O(K2)(23)
The following equations can be obtained depending on different order of K:
K0:f(0)
a=feq (24)
K1:∂
∂t1
+ea· ∇1feq +1
τδt
f(1)
a= 0 (25)
K2:∂f eq
∂t2
+∂
∂t1
+ea· ∇11−1
2τf(1)
a+1
τδt
f(2)
a= 0 (26)
Then f(1)
acan be obtained from Eq. (25):
f(1)
a=−τδt∂
∂t1
+ea· ∇1feq (27)
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184 Li, Yang, & Zhang
The hydrodynamic variables also have the relation with density distributions:
Π =
9
i=1
eaeafa(28)
where Πis the moment of order 2. Assuming that
Q=eaea−c2
sI(29)
where Iis the identity tensor, one obtains
Kf (1)
a=−τωi
c2
sQa:ρ∇u−ea∇:ρuu +1
2c2
s
(ea· ∇) (Qa:ρuu)(30)
Substituting Eq. (24) to Eqs. (17) and (28) and approximating Kf (1) by the first term only (Tao and Chopard, 2008),
the density distribution can be expressed as
fa=feq −τωi
c2
s
Qa:ρ∇u(31)
where Qais defined by the following equation:
Q=eiei−c2
sI(32)
So the density distribution can be related to the strain rate tensor Sdue to the symmetry of Qa:
fa=feq −τωa
c2
s
Qa:S(33)
where
S=1
2∇u+ (∇u)T(34)
Thus the information of the strain rate tensor on l2is needed to be transferred to the LBM zone from the FVM zone
in hybrid method 2.
3.3 Implementation of the Hybrid Method
The boundaries of the LBM and SIMPLE zones are l1and l2, as shown in Fig. 5. Meanwhile, they are the inner nodes
in both SIMPLE and LBM zones. In addition, it can also be seen that there is no difference between boundaries and
inner nodes for LBM, as can be seen from Fig. 2. Therefore, transferring information from SIMPLE to LBM, one
only needs the information on l1from the nearby nodes in the SIMPLE zone:
ui,j,L =(ui+1,j+1,S +ui+1,j,S )
2(35)
vi,j,L =(vi+1,j+1,S +vi,j+1,S)
2(36)
The procedure to transfer information from LBM to SIMPLE is similar, except the locations of nodes are different for
inner nodes and boundary nodes as shown in Fig. 4. The following equations can be used in this transfer process:
ui,j,S =(ui−1,j−1,L +ui−1,j,L)
2(37)
vi,j,S =vi,j−1,L (38)
Journal for Multiscale Computational Engineering
Hybrid Lattice Boltzmann and Finite Volume Methods 185
After transferring the information between the two zones, the hybrid method turns to fixed velocity boundary problem
on the interfaces l1and l2, as shown in Fig. 5 in LBM and SIMPLE. In addition, more information needs to be
transferred from the SIMPLE zone to the LBM zone besides the velocity for the hybrid method 2. The velocity
gradients (∂u)/(∂x),(∂u)/(∂y),(∂v)/(∂x), and (∂v)/(∂y)on l1are also needed in order to get the strain rate tensor.
While (∂u)/(∂y),(∂v)/(∂x), and (∂u)/(∂x)can be approximated by a central difference scheme, the conservation
of mass requires that Fedosov and Karniadakis (2009)
∂v
∂y =−∂u
∂x (39)
The following steps should be taken to transfer information between the two zones in the hybrid method to combine
LBM and SIMPLE (see Fig. 5):
1. Assume the velocity on x=l2.
2. Use SIMPLE to solve the velocity in the SIMPLE zone.
3. Transfer the information on x=l1from the SIMPLE zone to the LBM zone.
4. Use the LBM to solve the velocity in the LBM zone.
5. Transfer the information on l2from the LBM zone to the SIMPLE zone.
6. Go back to step 2 until the velocity is converged.
4. VALIDATION OF LID-DRIVEN FLOW
4.1 Problem Statement
Lid-driven flow is used to test the hybrid methods to combine LBM and FVM. No-slip boundary conditions are
applied to this 2D problem, and the flow is driven by a constant lid velocity u0on the top of the square cavity while
the velocities on all other boundaries are zero, as shown in Fig. 8.
This problem can be described by the following governing equations:
∂ρ
∂t +∂(ρu)
∂x +∂(ρv)
∂y = 0 (40)
Fig. 8
FIG. 8: Lid-driven flow
Volume 12, Number 3, 2014
186 Li, Yang, & Zhang
∂(ρu)
∂t +∂(ρuu)
∂x +∂(ρvu)
∂y =−∂p
∂x +µ∂2u
∂x2+∂2u
∂y2(41)
∂(ρv)
∂t +∂(ρuv)
∂x +∂(ρvv)
∂y =−∂p
∂y +µ∂2v
∂x2+∂2v
∂y2(42)
which are subject to the following boundary conditions:
x= 0 : u= 0, v = 0 (43)
x=h:u= 0, v = 0 (44)
y= 0 : u= 0, v = 0 (45)
y=h:u=u0, v = 0 (46)
In addition, the Reynolds number is defined with the constant lid velocity on the top:
Re =u0h
ν(47)
4.2 Selection of Units
The lattice unit is applied to LBM, while the SIMPLE algorithm is based on nondimensional governing equations.
A 160 ×160 uniform grid is applied to the entire computational domain after grid testing. Three shared grids are
selected after testing, and the grid in LBM is 80 ×160 while that in SIMPLE is 83 ×160. As discussed above, eahas
different values in different directions in the lattice unit. In order to combine these two methods with different units,
they must be used to describe the same situation in the actual unit.
In the LBM unit conversion process, which changes all the properties into the lattice unit, the speed of sound cs
and time step ∆tare fixed so that the density distributions are on the computational nodes shown in Fig. 2:
cs= 1/√3,∆t= 1 (48)
Assuming that the real speed of sound is usound and the number of nodes in the ydirection is n+ 1, the dimensionless
velocities in LBM are
UL=u
√3usound
(49)
VL=v
√3usound
(50)
and the time step is
∆t=HL/n
√3cs
= 1 (51)
Thus the coordinates in lattice unit become
XL=nx
h(52)
YL=nx
h(53)
In order to allow the boundary velocity in the lattice unit in LBM to be same as the dimensionless velocity in SIMPLE,
it is assumed that the lid velocity and the speed of sound have the followingrelationship:
10u0=√3usound (54)
which requires that 10u0be used in the nondimensional process. The dimensionless lid velocity in the LBM then
becomes
U0= 0.1(55)
Journal for Multiscale Computational Engineering
Hybrid Lattice Boltzmann and Finite Volume Methods 187
At this point, the only unknown parameter in the lattice unit is the kinematic viscosity νLand it can be obtained from
the Reynolds number:
νL=U0HL
ReL(56)
To satisfy Eq. (15), the relaxation time τ0can be obtained to fulfill the LBM:
τ0= 3νL+ 0.5(57)
To obtain the same boundary dimensionless velocity as that in lattice unit, the following nondimensional variables are
defined:
Us=u
10u0
, Vs=v
10u0
, Xs=x
h, Ys=y
h
F=t
h/(10u0), P =p
ρ(10u0)2,Res=u0h
ν
(58)
The governing equations (40)–(46) can be nondimensionalized as
∂Us
∂Xs
+∂Vs
∂Ys
= 0 (59)
∂Us
∂F +∂(UsUs)
∂Xs
+∂(VSUS)
∂Ys
=−∂P
∂Xs
+1
10Res∂2US
∂X 2
s
+∂2US
∂Y 2
s(60)
∂Vs
∂F +∂(UsVs)
∂Xs
+∂(VSVS)
∂Ys
=−∂P
∂Ys
+1
10Res∂2VS
∂X 2
s
+∂2VS
∂Y 2
s(61)
Xs= 0 : Us= 0, Vs= 0 (62)
Xs= 1 : Us= 0, Vs= 0 (63)
Ys= 0 : Us= 0, Vs= 0 (64)
Ys= 1 : Us= 0.1, Vs= 0 (65)
In order to meet the requirement for describing the same situation in the actual unit in LBM and SIMPLE, the following
two additional equations are needed: Res=ReL(66)
∆F=∆t
n(67)
where n+ 1 is the number of nodes in the ydirection. By following the above nondimensionalization procedures for
LBM and SIMPLE, the nondimensional lid velocities in both methods are 0.1. So the same nondimensional velocities
are reached at the same real location for the same Reynolds number in every time step. Therefore the nondimensional
velocities can be transferred directly.
4.3 Results and Discussions
The lid-driven flow is widely used as a benchmark solution to test the accuracy of numerical methods. To assess the
hybrid methods, the results of Ghia et al. (1982) are used for comparison. The lid-driven flow is solved for three
different Reynolds numbers at 100, 400, and 1000, respectively. A pure SIMPLE with QUICK scheme, LBM with a
nonequilibrium extrapolation boundary method, and the two methods combining SIMPLE and LBM are applied to
solve this problem. As discussed above, all these four methods use the same grid of 160 ×160, while the grid in the
message-passing zone in the hybrid method is 3 ×160. Figures 9–11 show the streamlines at three Reynolds numbers
obtained from the three methods, while Figs. 12 and 13 are the horizontal velocity profiles in the middle of the x
direction and vertical velocity profiles in the middle of the ydirection compared with that in Ghia et al. (1982). It
Volume 12, Number 3, 2014
188 Li, Yang, & Zhang
(a) SIMPLE (b) LBM
(c) Hybrid method 1 (d) Hybrid method 2
FIG. 9: Streamlines at Re = 100
(a) SIMPLE (b) LBM
(c) Hybrid method 1 (d) Hybrid method 2
FIG. 10: Streamlines at Re = 400
Journal for Multiscale Computational Engineering
Hybrid Lattice Boltzmann and Finite Volume Methods 189
(a) SIMPLE (b) LBM
(c) Hybrid method 1 (d) Hybrid method 2
FIG. 11: Streamlines at Re = 1000
should be pointed out that the velocities in the reference need to be multiplied by 0.1 due to different nondimensional
procedures.
The streamlines obtained from the combined method are highly similar to those obtained from the pure SIMPLE
and pure LBM, as shown in Figs. 9–11. The positions of the centers of the primary vortices are (0.6125, 0.7375) by
SIMPLE, (0.61875, 0.74375) by LBM, and (0.6172, 0.7344) in Ghia et al. (1982) at Re = 100. And the three locations
from these three methods are (0.5500, 0.60625), (0.55625, 0.6125), and (0.5547, 0.6055) for Re = 400, and (0.5250,
0.55625), (0.53125, 0.56875), and (0.5313, 0.5625) for Re = 1000, respectively. The differences of the locations of
the centers of the primary vortices are insignificant in the three Reynolds numbers from SIMPLE and LBM, and the
streamlines obtained from the two methods are highly similar. In addition, as shown in Figs. 12 and 13, the horizontal
velocity profiles in the middle of the xdirection and vertical velocity profiles in the middle of the ydirection obtained
by SIMPLE and LBM are also very close to that in the reference. In other words, the SIMPLE and LBM methods
used in this paper are reliable. Therefore the streamlines obtained from SIMPLE can be treated as standard results. It
can be concluded that the SIMPLE and LBM used in this paper are reliable and the accuracy of the combined method
depends only on the solution of the interface itself.
On the other hand, the streamlines obtained from hybrid methods 1 and 2 for Reynolds numbers at 100, 400,
and 1000, shown in Figs. 9–11, are almost the same as those obtained from SIMPLE and LBM. The positions of the
centers of the primary vortices obtained from the hybrid method for three different Reynolds numbers are (0.6125,
07375), (0.5547, 0.6055), and (0.5313, 0.5625) for hybrid method 1, and (0.6000, 071875), (0.5750, 0.60625), and
(0.54375, 0.56875) for hybrid method 2, respectively. Thus the differences from the reference are still at the same
level as those from pure SIMPLE and LBM. The hybrid method can reach the same flow pattern as the other two
methods. Notably, the results at the message-passing zone did not show any instability.
Volume 12, Number 3, 2014
190 Li, Yang, & Zhang
(a) Re = 100
(b)Re = 400
(a) Re = 100 (b) Re = 400
(c) Re = 1000
(c) Re = 1000
FIG. 12: Horizontal velocity profiles
(a) Re = 100
FIG. 13: Vertical velocity profiles
Journal for Multiscale Computational Engineering
Hybrid Lattice Boltzmann and Finite Volume Methods 191
With regard to the details of the fluid field, these two hybrid methods show different accuracies in different
Reynolds numbers. The horizontal velocity profiles in the middle of the xdirection and vertical velocity profiles in
the middle of the ydirection obtained by hybrid method 1 agree with the result from Ghia et al. (1982) for Reynolds
numbers equal to 100 and 400. Meanwhile, the horizontal velocity profiles in the middle of the xdirection and vertical
velocity profiles in the middle of the ydirection obtained by hybrid method 2 satisfy the result from Ghia et al. (1982)
for Reynolds numbers equal to 400 and 1000. The nonequilibrium extrapolation scheme in hybrid method 1 obtains
the unknown density distribution by assuming the nonequilibrium part of the density distribution is equal to that on
the nearby inner nodes. So the larger the velocity gradient, the worse the accuracy. By contrast, the finite difference
velocity gradient method in hybrid method 2 obtains the unknown density distribution by relating the nonequilib-
rium part of the density distribution with the velocity distribution. So the larger the velocity gradient, the better the
accuracy. Since the velocity gradient on the interface will become more valid with increasing Reynolds number, hy-
brid method 1 is suitable for the case when the Reynolds number is low, while hybrid method 2 is suitable for the
higher-Reynolds-number cases.
For the incompressible problem, LBM needs several times more computational time than that of the SIMPLE
(Kandhai et al., 1999) algorithm, while LBM saves a great deal of computational time in the complex fluid flow prob-
lems (Yoshino et al., 2004). The total computational time of any hybrid method always depends on the computational
time of the slower one. The extra time consumption of message passing can be neglected compared with the total
computational time. Thus the computational time in the lid-driven flow depends on that in the LBM zone when these
two zones have the same grids. The computational efficiency of the hybrid methods are between those of SIMPLE
and LBM. The main purpose of solving the lid-driven flow with the hybrid methods, which reach the same accuracy
with more time consumption compared with SIMPLE, is to demonstrate that these two hybrid methods can build a
relation between LBM and FVM to solve the fluid flow problem together. These hybrid methods have the potential to
save time with the same problem-solving accuracy, including several parts that can take advantage of both SIMPLE
and LBM in their subdomains.
5. CONCLUSIONS
Two new methods that combine the lattice Boltzmann method (LBM) and finite volume method (FVM) are proposed.
The computational domain is divided into two zones with a message-passing zone between them. LBGK is used
in LBM, while the SIMPLE algorithm with a QUICK scheme is used for FVM. At the interface, a nonequilibrium
extrapolation scheme and finite difference velocity gradient method are utilized to transfer the macrovelocities to a
density distribution fiin the LBM zone from the SIMPLE zone, while the density distribution can be advanced into
macrovelocities directly. The capabilities of the two hybrid methods are demonstrated by simulating the lid-driven
flow for different Reynolds numbers and comparing the results with those of the benchmark solution. The results
show that hybrid method 1 is suitable for the case when the Reynolds number is low, while hybrid method 2 is
suitable for the high-Reynolds-number case for the incompressible fluid flow problem.
ACKNOWLEDGMENTS
Support for this work by the U.S. National Science Foundation under Grant No. CBET-1066917 and the Chinese
National Natural Science Foundation under Grants No. 51129602 and 51076105 are gratefully acknowledged.
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