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Abstract—In this paper, a lattice BGK model for thermal
flows is used to simulate laminar natural convection in an
infinite horizontal channel, partially heated by Lattice
Boltzmann method. We applied the model based on the
double-distribution function approach in two dimensional.
Velocity and temperature distributions as well as Nusselt
number were obtained and analyzed for the Rayleigh number
ranging from 2.103 to 5.104 with the Prandtl number equal 0.71.
Index Terms—Double distribution function LBGK,
horizontal channel, natural convection, thermal lattice
Boltzmann method.
I. INTRODUCTION
Natural convection represents an extremely interesting
subject due to the coupling between fluid flow and energy
transport. This area of research is attractive because of its
potential in practical engineering applications such as thermal
design of buildings, furnace design, electronic equipment and
others. Previously, many investigators have studied
convection in various geometries. The development of
theoretical models, numerical algorithms and experimental
approaches constitutes a solid base for advancement of
knowledge in this field.
The lattice Bhatnagar–Gross–Krook (LBGK) method, a
novel kinetic-based numerical approach for simulating fluid
flows and associated transport phenomena, has developed
rapidly since its emergence. Many studies have made great
strides in constructing its theoretical foundation [1]-[4] and
improving its numerical performance [5]–[9] over the last
decade. Unlike conventional numerical schemes, which
discretize the macroscopic governing equations directly, the
LBGK method solves the kinetic equation at the mesoscopic
scale, i.e. the Boltzmann equation with the BGK assumption
[1], [2]. Historically, the LBGK method originated from the
lattice-gas cellular automata method (LGCA) [8]-[10], a
microscopic model for fluid systems where the imaged fluid
particles collide and move on a regular lattice, and indeed it is
very similar to the LGA method, except that particles residing
on the lattice are replaced by the corresponding distribution
Manuscript received December 29, 2013; revised March 20, 2014.
S. Houat is with the Mechanical Engineering Department, University of
Abdelhamid Ibn Badis University of Mostaganem, Bp300, 27000
Mostaganem, Algeria, and Laboratory of Numerical and experimental
modeling of mechanical phenomena (e-mail: sa_houat@ yahoo.fr).
N. Saidi is with the Department of Biology, University of Abdelhamid
Ibn Badis University of Mostaganem, Bp300, 27000 (e-mail:
n_saidi@yahoo.fr).
functions and the collision operator is approximated by the
BGK assumption. But later it was realized that LBGK could
also be viewed as a special finite difference scheme of the
continuous Boltzmann equation on a regular lattice [1], [2],
which also defines the associated discrete particle velocities.
From this viewpoint, discretization for the particle velocity
can be decoupled from the spatial discretization, since the
particle velocity in the Boltzmann equation is independent of
the particle position [5]. This implies that we can discretize
the continuous velocity space into a set of discrete velocities
with sufficient symmetry (physical symmetry).
Currently, a few thermal lattice Boltzmann models have
been proposed. The earliest model which is known as multi
speed model [11], uses the same distribution function in
defining the macroscopic temperature. However, this model is
reported to suffer numerical instability [12] and has a demerit
that it can simulate thermal fluid flows only at fixed Prandtl
number [13]. As an alternative approach, Shan proposed the
so-called passive-scalar model [14]. This model suggests that
the flow fields (velocity and density) and the temperature are
represented by two different distribution functions. The
macroscopic temperature is assumed to satisfy the same
evolution equation as a passive scale, which is advected by the
flow velocity but does not affect the flow field.
The work of Luo and He [15] demonstrated that the
isothermal lattice Boltzmann equation can be directly
obtained by properly discretizing the continuous Boltzmann
equation in both time and space phases. Following the same
procedure, He [16] proposed the double-distribution function
model, where the thermal lattice Boltzmann evolution
equation can be derived by discretizing the continuous
Boltzmann equation for the internal energy distribution. It has
been shown that this model is simple and applicable to
problems with different Prandtl numbers [17]. More
importantly, this model requires low order moment and thus
provides higher numerical stability than the passive-scalar
model.
In this paper, we use two dimensional simulation for
natural convection heat transfer in an infinite horizontal
channel partially heated developed new five velocity lattice
models of the internal energy density distribution function for
incompressible flow.
II. DESCRIPTION OF PHYSICAL PROBLEM
The studied configuration, sketched in Fig. 1, is an infinite
channel discretely heated from below where adiabatic
partitions are regularly placed at the center of the adiabatic
The Study of the Natural Convection in an Infinite
Horizontal Channel Partially Heated by Lattice
Boltzmann Method
S. Houat and N. Saidi
International
Journal of Materials, Mechanics and Manufacturing, Vol. 2, No. 3, August 2014
251
DOI: 10.7763/IJMMM.2014.V2.137
surfaces. The lower wall is partially heated and maintained at
constant temperature at T2=1. The remaining portions of the
lower boundary are adiabatic. The upper wall of the channel,
placed at a height H from the lower one, is also maintained at
constant but cold temperature at T1=0.
The periodic nature of the system and the associated
boundary conditions permits the subdivision of the channel
into finite simple domains (SD) of length L. The study can be
conducted in a SD, limited by the fictive boundaries P1 and
P2 (Fig. 1). Such technique of subdivision was used in the
past by [18] and [19] to study the natural convection in a
channel provided, respectively, with adiabatic and heated
portions on its lower wall. In references [19] and [20] have
shown that although the SD is a representative entity of the
studied configuration, its use allows obtaining only solutions
verifying the periodic conditions imposed by P1 and P2. They
have, thus, established the limitations of the SD by
considering a calculation domain twice as long and called a
double domain (DD) of length 2L. Consequently, in the
present study, the simulations were performed in a DD,
limited by the space between the fictive boundaries P1 and
P3.
III. NUMERICAL METHOD
A. Double Population Thermal Lattice Boltzmann Method
The physical space is divided into a regular lattice and the
velocity space is discretized into a finite set of velocities {c
},
the Boltzmann equation with Bhatnagar-Gross-Krook (BGK)
approximation [21] can be discretized as [17], [22]:
(Δ Δ ) ( )
( ) ( ) Δ.
αiα α i
eq
αiαi
α
v
f x +c t,t + t f x ,t =
f x ,t f x ,t + t F
τ
(1)
where, t and ct are time and space increments,
respectively. f
is the single-particle velocity distribution
function along the
th direction. f
eq is the equilibrium
distribution function,
is the single relaxation time and F
is
the external force.
There are different types of lattice for LBM. For simplicity
and without loss of generality, we consider the
two-dimensional square lattice with nine velocities, the D2Q9
model:
(0,0) 0,
11
[cos( ),sin( )] 1,2,3,4,
22
55
2 [cos( ),sin( )] 5,6,7,8,
2 4 2 4
α α
α=
c = ce = α α
cπ π α =
α π α π
cπ+π+α=
(2)
The equilibrium distribution function for D2Q9 model is
given by:
2
2 4 2
3 9 3
1 ( )
2c 2c
eq
α α α α
f=ρw + c u+ c u u u
c
(3)
where w0 = 4/9, w1 = w2 = w3 = w4 = 1/9, and w5 = w6 = w7 =w8
=1/36. The macroscopic density
and velocity u are related
to the distribution function by:
9
1=α
α
f=ρ
(4)
α
=α
αcf=ρu
9
1
(5)
Using the Chapman-Enskog expansion, the equation (1)
can recover the Navier-Stokes equation to the second order of
accuracy, with the kinematic viscosity given by:
2
( 0.5) Δ
3
v
τct
ν=
(6)
The LBGK evolution equation for the temperature is [17]:
( ( )
(Δ Δ ) ( ) eq
αiαi
αiα α iT
g x ,t) g x ,t
g x +c t,t + t g x ,t = τ
(7)
For the evolution of g, given its simplified equilibrium
distribution function, a D2Q5 lattice is preferred [21]. In the
D2Q5 topology, the velocities v are:
(0,0) 0,
11
[cos( ),sin( )] 1,2,3,4,
22
α
α=
v= α α
cπ π α =
(8)
Fig. 1. Geometry of physical problem.
The associated weights wTa are wT0 = 1/3, wT1 = wT2 = wT3 =
wT4 = 1/6. The equilibrium distribution function for D2Q5
model is given by:
uv
c
+wT=g α
T
α
eq
α2
3
1
(9)
At each lattice node, the macroscopic temperature is defined
as:
5
1=α
α
g= T
(10)
And the thermal diffusivity (in lattice units) is related to the
relaxation time is :
3
0.5 2Δt)c(τ
=κT
(11)
B. Heat Transfer
The thermal conditions applied on the two parallel
stationary walls, the hot (bottom) and the cold (top)
introduces a temperature gradient in a fluid, and the
consequent density difference what induces a fluid motion
that is, convection. In the simulation, the Boussinesq
approximation is applied to the buoyancy force term [14],
[16], [17], [22]:
International Journal of Materials, Mechanics and Manufacturing, Vol. 2, No. 3, August 2014
252
00
gρG=ρβ T T j
(12)
where
is the thermal expansion coefficient, g0 is the
acceleration due to gravity, T0 is the average temperature and
j is the vertical direction opposite to that of gravity. So the
external force in Eq. (1) is
eq
ffuc=F 3G
(13)
The dynamical similarity depends on two dimensionless
parameters: the Prandtl number, Pr and the Rayleigh number,
Ra defined as,
Pr= ν
κ
and
νκ
βΔTHg
=Ra 3
0
(14)
where T is the wall temperature difference, H is the distance
between the walls.
The Nusselt number, Nu is one of the most important
dimensionless numbers in describing the convective transport.
The Nusselt number for the hot wall is defined as the ratio
between the heat transports by convection to the heat
transmission due to conduction:
κΔT
H>Tu
+=Nu y
<
1
(15)
Here <uy T> denotes the average over the convection layer.
IV. RESULTS AND DISCUSSIONS
This kind of the study is a classical benchmark on the
thermal models defined by The Rayleigh-Benard convection
flow. The fluid is enclosed between two parallel stationary
walls, the hot (bottom) and the cold (top), and experiences the
gravity force. Density variations caused by the temperature
variations drive the flow, while the viscosity will counteract to
equilibrate it.
In all our simulations, the study is stationary, the Prandtl
number Pr=0,71 and Rayleigh number is varied between
2.103 and 5.104. For the boundary conditions at the top and the
bottom walls, we applied the bounce back condition is applied
for the fluid distribution. Different numbers of nodes were
tested and the results were similar. The results presented here
are Nx×Ny = 121×61 for the periodic side boundary
conditions.
A. Validation Model
The configuration used for validation of the model is a
horizontal chanel totally heated in bottom wall. The geometry
configuration is similar in Fig. 1 except that b/L ratio equal 1
(i.e. 100% heated). After simulation for different Rayleigh
numbers, Extrapolating the obtained values of Nusselt
number, an estimate of the final converged solution can be
done (Nu). In Fig. 2, the isotherms of Ra = 5.103 and Ra = 104
are plotted for the case of 121 grid nodes in the y-direction. In
Fig. 3, the contours of the stream function of the
incompressible flow field for Ra = 5.103 and Ra = 104 are
plotted.
Fig. 2. Contour plot with iso-temperature lines for Ra = 5.103 (top) and
Ra = 104 (bottom).
Fig. 3. Contour plot with the stream function for Ra = 5.103 (top) and
Ra = 104 (bottom).
What showed the appearance of two Benard rollers against
rotative with height and width equal H (height of the channel).
The extrapolated converged values of Nusselt number at
various Rayleigh numbers are plotted in Fig. 4, and compared
with a empirical power law [23] and the standard reference
data [24]. The critical Rayleigh number is defined by
Rac=1707.8. The present results obtained by TLBM model is
found to be in good agreement with [23] and [24].
103104105
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
Present work
Reference
__ Ra=1,56 (Ra/Racr)0,296
Nusselt number
Rayleigh number
Fig. 4. Nusselt number vs. Rayleigh number for b/L=1.
Square: The current LB model; Triangles: Reference data of Ref. [24];
Line: Empirical power law Nu = 1.56(Ra/Rac)0.296 [23].
B. Simulation
In this study, was used to predict the natural convection in a
channel partially heated by studying the variation of the
Rayleigh number. The geometry configuration is shown in Fig.
1. The portion heated in bottom wall is varied b/L ratio
International Journal of Materials, Mechanics and Manufacturing, Vol. 2, No. 3, August 2014
253
isvaried between 0,2 and 0,8 (i.e. 20%, 40%, 60%, 80%
heated). The boundary conditions is similar at applied in
validation case, except that between the heated portions we
consider the adiabatic wall boundary in second order.
The heat transfer is described by the Nusselt number Nu,
defined in (15), as the ratio between convective heat transport
to the heat transport due to temperature conduction. For the
computation of the Rayleigh number, the Nusselt number
results is shown in Fig. 4.
Fig. 5. Contour plot with iso-temperature line for Ra = 104.
The top to bottom b/L=0,8; 0,6; 0,4 and 0,2.
103104105
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
b/L=0,20
b/L=0,40
b/L=0,60
b/L=0,80
b/L=1
Nusselt number
Rayleigh number
Fig. 7. Nusselt number vs. Rayleigh number for
b/L=1; 0.8; 0.60; 0.40; 0.20
The results obtained showed clearly that the phenomenon of
transfer decrease according to the geometry heated. The
isothermes and streamlines for the fluid flow is illustred in Fig.
2 and Fig. 3 what showed the appearance of four Benard rollers
against rotative with height H (height of the channel) and width
L/2 for all configurations.
Fig. 6. Contour plot with streamlines for Ra=104.
The top to bottom b/L=0,8; 0,6; 0,4 and 0,2.
V. CONCLUSION
We have presented a lattice Boltzmann thermal model for
convection heat transfer in an infinite horizontal channel
partially heated in bottom wall. In this model, the temperature
field is modeled by a new lattice Boltzmann equation, while
the velocity field is simulated by the lattice Boltzmann
isothermal model for flows. The present model has all the
advantages, including good numerical stability and the ability
to handle convection heat transfer problems. The numerical
results of the used problem for validation in a channel totally
heated by bottom wall, have demonstrated the accuracy and
reliability of the used LBM, and the good agreement between
the results obtained and the results in the literature.
This numerical model was used to predict and to study the
natural convection in a channel partially heated by studying
the variation of the Rayleigh number and the heated
geometrical portion.
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S. Houat was born in Oran in 1965. He has been a
senior lecturer since 1991 at the Department of
mechanics engineering in Mostaganem, Algeria. He
graduated from the University of Science and
Technology, in Oran, Algeria in 1988 with a BEng
degree, in 1991 with a magister degree and in 2007
with a PhD degree. He held several teaching
responsibility and the post head of Department of
Mechanical Engineering at the University of
Mostaganem until 2011. Currently he is responsible for a recent group of
researchers working on the application of the lattice Boltzmann method in
transport phenomena.
Dr. Houat currently focuses on studies and analysis of incompressible
fluid flows accompanied by heat transfer using the thermal lattice Boltzmann
method.
Author’s formal
photo
International Journal of Materials, Mechanics and Manufacturing, Vol. 2, No. 3, August 2014
255
