Natural Convection and Entropy Generation in Γ-Shaped Enclosure Using Lattice Boltzmann Method

1
Trans. Phenom. Nano Micro Scales, 1(1): 1-18, Winter - Spring 2013
DOI: 10.7508/tpnms.2013.01.001
ORIGINAL RESEARCH PAPER .
Natural Convection and Entropy Generation in Γ-Shaped Enclosure
Using Lattice Boltzmann Method
E. Fattahi1
, M. Farhadi1,*
, K. Sedighi1
Faculty of Mechanical Engineering, Babol University of Technology Babol, Iran
Abstract
This work presents a numerical analysis of entropy generation in Γ-Shaped enclosure that was submitted to the
natural convection process using a simple thermal lattice Boltzmann method (TLBM) with the Boussinesq
approximation. A 2D thermal lattice Boltzmann method with 9 velocities, D2Q9, is used to solve the thermal
flow problem. The simulations are performed at a constant Prandtl number (Pr = 0.71) and Rayleigh numbers
ranging from 103
to 106
at the macroscopic scale (Kn = 10-4
). In every case, an appropriate value of the
characteristic velocity i.e. y
V g THbD ‫؛‬ is chosen using a simple model based on the kinetic theory. By
considering the obtained dimensionless velocity and temperature values, the distributions of entropy generation
due to heat transfer and fluid friction are determined. It is found that for an enclosure with high value of
Rayleigh number (i.e., Ra=105
), the total entropy generation due to fluid friction and total Nu number increases
with decreasing the aspect ratio.
Keywords: Entropy Generation; Lattice Boltzmann Method; Natural Convection; Γ-Shaped enclosure
1. Introduction
The lattice Boltzmann (LB) method is a powerful
approach to hydrodynamics, with applications
ranging for vast Reynolds numbers and modeling the
physics in fluids [1–4]. Various numerical
simulations have been performed using different
thermal LB models or Boltzmann-based schemes to
investigate the natural convection problems [5–11].
The lattice Boltzmann equation (LBE) is a minimal
form of the Boltzmann kinetic equation, and the result
is a very elegant and simple evolution equation for a
number of distribution functions, which represent the
number of fluid particles moving in these discrete
__________
*
Corresponding author
Email Address: mfarhadi@nit.ac.ir
with speed ci .In LBM the domain is discretized in
uniform Cartesian cells which each one holds a fixed
directions. With respect to the more conventional
numerical methods commonly used for the study of
fluid
flow situations, the kinetic nature of LBM
(Lattice Boltzmann Method) introduces several
advantages, including easy implementation of
boundary conditions and fully parallel algorithms. In
addition, the convection operator ( .c f‫ر‬
rr
) is linear,
no Poisson equation for the pressure must be solved
and the translation of the microscopic distribution
function into the macroscopic quantities consists of
simple arithmetic calculations. The phenomenon of
natural convection in enclosures has attracted
increasing attention in recent years.

E. Fattahi et al./ TPNMS 1 (2013) 1-18
2
Nomenclature
hT Hot temperature ( )K
ic Discrete lattice velocity in direction (i) cT
Cold temperature ( )K
sc Speed of sound in Lattice scale 0T Bulk temperature (K), (T0= (Th+Tc)/2)
iF External force in direction of lattice
velocity
v,u Horizontal and vertical components of
velocity
eq
if Equilibrium distribution 1
( . )m s−
yg Acceleration due to gravity, 2
( . )m s −
kw Weighting factor
H Height of enclosure ( )m w
non-dimensional length of step, (w′/ H)
h Non-dimensional height of step, (h′/H)
k
Thermal conductivity 1 1
( . . )W m K− −
Greek symbols
uN Mean Nusselt number β Thermal expansion coefficient ( )1
K −
yx NuNu , Local Nusselt number along surfaces µ Molecular viscosity 1 1
( . . )kg m s− −
Pr Prandtl number ( / )ν α ϕ Irreversibility distribution ratio
Ra
Rayleigh number 3
( / )g THβ αν∆ ρ Density 3
( . )kg m−
Kn Knudsen number τ Lattice relaxation time
genS ′′′ Total volumetric entropy generation rate t∆ Lattice time step
( )3 1
. .W m K− −
PS ′′′ Volumetric entropy generation rate due to
Subscript
friction ( )3 1
. .W m K
− −
C cold
TS ′′′ Volumetric entropy generation rate due to h hot
heat transfer ( )3 1
. .W m K
− −
i discrete lattice directions
Applications extending from the double paned
windows in buildings to the cooling of electronic
systems are examples of natural convection systems.
In natural convection processes, the thermal and the
hydrodynamic are coupled and both are, according to
Bejan [12], strongly influenced by the fluid thermo-
physical characteristics, the temperature differences
and the system geometry. The comprehensive
reviews of articles on natural convection were made
by Catton [13], Ostrach [14] and Kakac and Yener
[15]. In addition to the studies [16-20], Lage and
Bejan [21] investigated numerically the natural
convection in a square enclosure heated and cooled in
the horizontal direction in the Prandtl number range
0.01– 10 and the Rayleigh number range 102
–1011
.
Notable researches have been done to investigate
importance of entropy generation in thermal systems.
Entropy generation and its minimization were
investigated widely with Bejan [22-24]. Natural
convection in enclosure was summarized in
rectangular coordinates by Davis [25]. In his study,
he made a review about numerical studies and
investigated the effect of various non-dimensional
numbers and boundary conditions on natural
convection heat transfer. Additionally, an analysis of
the entropy generation in rectangular cavities was

3
performed by Oliveski et al. [26]. They found that for
the same aspect ratio, the entropy generation due to
the viscous effects increases with the Rayleigh
number and, for a certain Rayleigh number, the
entropy generation due to the viscous effects also
increases with the aspect ratio.
Ha and Jung [27] used LBM to investigate the
steady, three-dimensional, conjugate heat transfer of
natural convection and conduction in a vertical cubic
enclosure within which a centered, cubic, heat-
conducting body generates heat. They found that the
fluid flow and temperature distribution show very
complex three-dimensional pattern. Mezrhab et al.
[28] studied the radiation-natural convection
interactions of a square heat-conduction body within
a differentially heated square cavity. Dagtekin et al.
[29] dealt with the prediction of entropy generation
of natural convection in a Γ-shaped enclosure using
FDM (Finite Difference Method). They found that
the main entropy generation is formed due to heat
transfer for Ra<105
, while the contribution due to
fluid friction becomes stronger for Ra>105
.
In the present study natural convection and entropy
generation was simulated numerically in the Γ-shaped
enclosure using LBM. As the horizontal walls are
insulated perfectly, vertical walls heats. An in house
lattice BGK (Bhatnagar–Gross–Krook) scheme
FORTRAN code was used to simulate the present
problem. The contribution of this work is the analyses
of the variation of entropy generation in relation to
Rayleigh number, aspect ratio at fixed irreversibility
coefficient(φ=10-6
) in Γ-shaped enclosure. The results
are displayed graphically in term of the streamlines,
isotherms and local entropy generation contours to
show the effect of aspect ratio (AR) and Rayleigh
number. To calculate the entropy generation, a new
model [30] was used to determine the dimensionless
velocity. The results of the present study show that
this model is a suitable for calculating the entropy
generation in the natural convection problems.
2. Numerical Procedure
2.1 The Lattice Boltzmann Method
In investigating the natural convection problems,
the effect of viscous heat dissipation can be neglected
for applications in incompressible flow [10]. This
assumption can be used to simulate the natural
convection by LBM. The LB model used here is the
same as that employed in [9-11]. The thermal LB
model utilizes two distribution functions, f and g, for
the flow and the temperature field, respectively. It
uses modeling of movement of fluid particles to
capture macroscopic fluid quantities such as velocity,
pressure and temperature. In this approach the fluid
domain is discretized in uniform Cartesian cells.
Each cell holds a fixed number of distribution
functions, which represent the number of fluid
particles moving in specified discrete directions. For
this work, the most popular model for the 2D case,
the D2Q9 model, which consists of 9 distribution
functions, has been used (Fig. 1). The values of
0
4 9w = for 0
0c = (for the static particle),
1 4
1 9w −
= for 1 4
1c −
= and 5 9
1 36w −
= for
5 9
2c −
= are assigned for this model.
The density and distribution functions i.e. the f
and g (temperature distribution function), are
calculated by solving the Lattice Boltzmann equation
(LBE), which is a special discretization of the kinetic
Boltzmann equation. After introducing BGK
approximation, the general form of lattice Boltzmann
equation with external force can be written as:
(1)
( , )
( , ) ( , ) ( , )
.
i i
eq
ii i
i i
f x c t t t
t
f x t x t f x tf
tc F
υτ
+ ∆ +∆ =
∆  + − 
+∆
For the flow field and
(2)
( , ) ( , )
( , ) ( , )
ii i
eq
ii
D
x c t t t x tg g
t
x t g x tg
τ
+ ∆ +∆ =
∆  + −  
For the temperature field
Where t∆ denotes lattice time step, i
c is the discrete
lattice velocity in direction k , i
F is the external
force in direction of lattice velocity, υ
τ and D
τ
denotes the lattice relaxation time for the flow and
temperature field. The kinetic viscosity υ and the
thermal diffusivity α, are defined in terms of their
respective relaxation times, i.e. 2
( 1 2)s
c υυ τ= − and
2
( 1 2)s Dcα τ= − , respectively. The speed of sound
( s
c ) is a lattice-dependent quantity, which has the
value of 1 3 for the D2Q9 model. Note that the

4
limitation 0.5 < τ should be satisfied for both
relaxation times to ensure that viscosity and thermal
diffusivity are positive. Furthermore, the local
equilibrium distribution functions are calculated with
equations (3, 4) for the flow and temperature fields,
respectively.
(3)
( )2
2 4 2
.. 1 1 .
. . 1
2 2
ρ
 
 = + + −
 
 
eq ii
ii
s s s
c uc u u u
wf
c c c
(4)2
.
. ( , ). 1
 
= + 
 
eq i
ii
s
c u
T x tg w
c
Where i
w a weighting factor, u is velocity vector
andρ is the lattice fluid density.
In order to incorporate buoyancy force in the
model, the force term in the equation (1) need to be
calculated as below in vertical direction (y) [31]:
(5)3i i yF w g βθ=
The Boussinesq approximation is applied and
radiation heat transfer is negligible. θ is non-
dimensional temperature.
To ensure that the code works in near
incompressible regime, the characteristic velocity of
the flow y
V g THβ≡ ∆ must be small compared
with the fluid speed of sound. In the present study,
the characteristic velocity was selected as 0.1 of
speed of sound.
Finally, the following macroscopic variables can
be calculated in terms of these variables, with the
following formula.
(6)Flow density: i
i
fρ = ∑
(7)Momentum: i j ji
j
u f cρ = ∑
(8)
Temperature: i
i
T g= ∑
2.2 Entropy Generation Calculation
The following dimensionless variables (primed
quantities are dimensional) are used:
(9)
( )
( , ) ( , ) , , ( , ) / ,
, , /c h c
x y x y H u v u v H
T T T T T T T Pr
α
υ α
′ ′ ′ ′= =
′= − ∆ ∆ = − =
Fig. 1. Discrete velocity vectors for the D2Q9 model of
LBM
Volumetric entropy generation due to heat
transfer, T
S ′′′ , and friction, P
S ′′′ , are calculated as
below:
(10)
2'''
2
( )= ∇T
k
TS
T
(11)'''
PS
T
µ
ψ=
where ψ is defined by:
(12)
u ju ui i
x x xj i j
ψ
 ∂∂ ∂ = +  ∂ ∂ ∂ 
And the total volumetric entropy generation can
be obtained by:
(13)SSS PTgen
'''''''''
+=
The dimensionless form of Eq. (13) is called Ns
(local EG number) which is defined as follow:
(14)
22
2 22
2
T
P
T T
Ns
x y
S
u v v u
x y x y
S
φ
   ∂ ∂  = +     ∂ ∂   
      ∂ ∂ ∂ ∂      + + + +           ∂ ∂ ∂ ∂       

5
The irreversibility distribution ratio, φ (the ratio
of entropy generated due to fluid friction to heat
transfer), is written as follows:
(15)
2
0T
k H T
µ α
φ
 =   ∆
0
T is bulk temperature ( ( )0
/ 2h c
T T T= + (K). The
dimensionless total entropy generation is the integral
volume of the computational domain:
(16)s
v
S N dV= ∫
2.3 Curved Boundary Treatment
Consider Fig. 2 is a part of an arbitrary curved
wall geometry, where the black small circles on the
boundary w
x , the open circles represent the boundary
nodes in the fluid region f
x and the grey circles
indicate those in the solid region b
x .In the boundary
condition ( , ), ( , )b b
f t g tx x are needed to perform the
streaming steps on fluid nodes f
x .
The fraction of an intersected link in the fluid
region ∆ is defined by:
f
f b
w−
∆ =
−
x x
x x
(17)
The standard (half-way) bounce back no-slip
boundary condition always assumes a delta value of
0.5 to the boundary wall (Fig.3a). Due to the curved
boundaries, delta values in the interval of (0, 1] are
now possible. Fig.3b shows the bounce back
behavior of a surface with a delta value smaller than
0.5 and Fig.3c shows the bounce back behavior of a
wall with delta bigger than 0.5. In all three cases, the
reflected distribution function ( , )f t tα + ∆x at f
x is
unknown. Since the fluid particles in the LBM are
always considered to move one cell length per time
step, the fluid particles would come to rest at an
intermediate node i
x . In order to calculate the
reflected distribution function in node f
x , an
interpolation scheme has to be applied. For treating
velocity field in curved boundaries, the method is
based on the method reported in [32]. For
temperature field, the method is based on an
extrapolation method of second-order accuracy
applied in [33].
Fig. 2. Schematic view of Γ-shaped enclosure
Fig. 3. Layout of the regularly spaced lattices and curved
wall boundary
2.3.1 Velocity in curved boundary condition
To calculate the distribution function in the solid
region ( , )b
f tα x based upon the boundary nodes in
the fluid region, the bounce-back boundary
conditions combined with interpolations including a
one-half grid spacing correction at the boundaries [3,
34].

6
Then the Chapman–Enskog expansion for the
post-collision distribution function on the streaming
step is conducted as:
2
( , ) (1 ) ( , )
( , )
3
2 ( , ) .
b f
b
f w
f t t f t t
f t t
w t t
c
α α
ο
α
α α
λ
λ
ρ
+∆ = − +∆
+ + ∆
− + ∆
x x
x
x e u
(18)
Where
( )2
( , ) ( , )
3
( , ) .
eq
b f
f bf f
f t t f t t
w t t
c
ο
α α
α αρ
+ ∆ = + ∆
+ + ∆ −
x x
x e u u (19)
,
2 1 1
, 0
2 2
bf ff
m
ifλ
τ
=
∆ −
= < ∆ ≤
−
u u
(20a)
3 3
(1 ) ,
2 2
2 1 1
, 1
1 2
2
bf f w
m
ifλ
τ
= − +
∆ ∆
∆ −
= < ∆ ≤
+
u u u
(20b)
w
u denotes the velocity of solid wall, bf
u is the
imaginary velocity for interpolations and α α≡ −e e .
Temperature in curved boundary condition
For temperature field in curved boundary this
study use the method is based on the method reported
in [34]. Distribution function for temperature divided
two parts, equilibrium and non equilibrium
( , ) ( , ) ( , )eq neq
b b b
g t g t g tα α α= +x x x (21)
By substituting Eq.21 into temperature streaming
step, we have
( , ) ( , )
1
(1 ) ( , )
eq
b b
neq
b
T
g t t g t
g t
α α
α
τ
+ ∆ =
+ −
x x
x
(22)
Obviously to calculate ( , )b
g t tα + ∆x , both
( , )eq
b
g tα x and ( , )neq
b
g tα x are required.
Equilibrium and non equilibrium parts of Eq.21
are define as:
*
2
3
( , ) 1 .eq
b b bg t w T
c
α α α
 
= + 
 
x e u (23)
*
b
T is determined by linear extrapolation using either:
*
1, if 0.75b bT T= ∆ ≥ (24a)
*
1 2(1 ) , if 0.75b b bT T T= + − ∆ ∆ ≤ (24b)
Where ∆ is the fraction of the intersected link in the
fluid region (Eq. 17), which is illustrated in Fig. 3
and:
1 [ ( 1) ]/b w fT T T= + ∆ − ∆ (25a)
2 [2 ( 1) ]/(1 )b w ffT T T= + ∆ − + ∆ (25b)
Where Tf and Tff denote the fluid temperatures in
node xf and xff, respectively. The extrapolation
scheme is the same as Ref. [35].
The next task is to determine the ( , )neq
b
g tα x .
Second-order approximation is also used.
( , )neq
b
g tα x is evaluated as:
( , ) ( , )
(1 ) ( , )
neq neq
b f
neq
ff
g t g x t
g x t
α α
α
= ∆
+ − ∆
x
(26)
From the Chapman-Enskog analysis,
( , )neq
g tα x
can be expressed as:
1
( , ) ( , ) .neq
g t g t xα α δ=x x (27)
Where
0
( , )g tα x is the same order as ( , )eq
g tα x .
Since
1 1
( , ) ( , ) ( )g t g t O xα α δ− =w fx x ,
2
( , ) ( , ) ( )neq neq
g t g t O xα α δ− =w fx x .
By the same token, it can be proved that
2
( , ) ( , ) ( )neq neq
g t g t O xα α δ− =w ffx x (28)
That means the approximation ( , )neq
b
g tα x is of
second order in space which is in consistent with
thermal lattice Boltzmann equation.

7
3.2. Computational Domain and Numerical
Details
Figure 4 shows the Γ-Shaped enclosure analyzed
in the present study. h, w, AR are the height of the
step, the width of the step and the aspect ratio (h/w),
respectively. The step heater kept at a constant
temperature of Th and the vertical walls of the
enclosure are fixed at low isothermal of Tc. The top
and bottom walls are insulated.
No-slip boundary condition was imposed on all
the walls of the cavity. Dirichlet type boundary
conditions have been used that the insulated
boundary conditions were simulated by converting
them to Dirichlet type by using a second order
accurate finite-difference approximation. The
boundary conditions have been implemented by
using the counter-slip approach such as Dixit and
Babu [8]. Although the suitability of the counter slip
approach has only been established for the
hydrodynamic boundary conditions [36], but Dixit
and Babu [8] have shown that this technique is useful
for modeling the thermal boundary conditions to
simulate the flow and heat transfer in the cavity.
They found that “the traditional implementation of
the Dirichlet boundary condition gave rise to a
spurious gradient in the temperature between the wall
and the first point in the fluid. This gradient was a
steep drop in the temperature near the hot wall and
the cold wall and it persisted even after decreasing
the mesh spacing by a factor of 8. This clearly
implies that the source of this phenomenon is the
implementation of the boundary condition and not
the grid spacing especially for a lower value of
Rayleigh number. This, in turn, would yield incorrect
values of wall Nusselt number. To overcome this
difficulty, the counter-slip approach has been used
for simulating the Dirichlet boundary condition also”.
This method was investigated carefully in detail by
Dixit [37]. As mentioned above, this boundary
condition was used on the walls of the cavity.
The Rayleigh number (Ra) and Nusselt number
(Nu) for the current problem are defined as follow:
(29)3
Ra g THβ αυ= ∆
Local Nusselt numbers are defined on front and
top face of the step as Nux, Nuy
(30)
,x y
h w
T T
Nu Nu
x y
  ∂ ∂  =− = −     ∂ ∂ 
The average Nusselt number is calculated by
integrating the local Nusselt number along surface of
step as:
(31)
0 0
1
h w
y xNu Nu dy Nu dx
h w
 
 = + +   
∫ ∫
Determination of characteristic velocity is
necessary to simulate the natural convection in LBM.
This velocity is defined as y
V g THβ≡ ∆ . The
kinetic viscosity and thermal diffusivity are
calculated from the characteristic velocity through
the following relationships, respectively:
(31)
2 2
2 Pr
Pr
V H
Ra
and
υ
υ
α
=
=
The relaxation times, υ
τ and D
τ , for flow and
temperature LB equations given in equation (1) and
(2) can then be determined. It implies that kinetic
viscosity (υ) and thermal diffusivity (α) can not be
considered as constants in LBM simulations if the
characteristic velocity (V) is kept constant. Kao et al.
[38] developed a new model for determining an
appropriate characteristic velocity value based on the
principle of kinetic theory. From the kinetic theory
[39, 40], the Knudsen number is defined as [41]:
(32)
Re
.
2
Ma
Kn
πγ
=
Where Ma is the Mach number, Re is the
Reynolds number, and the heat capacity ratio to be γ
= 5/3 for a monatomic ideal gas and γ = 7/5 for a
diatomic gas according to the definition of mean free
path,
2 s
c
πγ υ
λ = , the Knudsen number for this
geometry can be written as [38]:
(33)Hc
v
H
Kn
s .
.
2
πγλ
=≡
From the definition of Rayleigh number and Eq
(17):
(34)
3
where,
2Pr
.. 2
2
22
2 c
c
cKnRa
V s
s
==
πγ
By this definition, the characteristic velocity is a
function of the Rayleigh number, Knudsen number,

8
Prandtl number and the value of c, that all are
specified as a given values in LBM simulations
which includes the macroscopic and mesoscopic
scales at natural convection problems.
For grid independency, the average Nusselt
number over the step was calculated at high Ra
numbers for different grid points. As seen in table 1
for grid points passing from 80×80 and 100×100 for
Ra = 105
and 106
, respectively, no considerably
change in the average Nusselt number was observed
(maximum variation is less then 0.16%). According
to the table 1, the 80×80 grid points was used for Ra
≤ 105
and 100×100 grid points was used only for Ra
= 106
.
(a)
(b)
(c)
Fig. 4. Illustration of the bounce-back boundary
conditions. (a) ∆ = 1/2, the ‘‘perfect” bounce-back without
interpolation. (b) ∆ < 1/2, the bounce-back with
interpolations before the collision with the wall located at
xw. (c) ∆ > 1/2, the bounce-back with interpolations after
the collision with the wall
5. Results and discussion
To validate the numerical simulation, the results
of natural convection in square and Г-shaped
enclosure were compared with previous works ([25]
and [29]).
In the square cavity, flow was heated from the left
wall, while the right wall was maintained at a
constant low temperature. Meanwhile, the upper and
bottom walls were assigned adiabatic boundary
conditions. A vertical gravitational effect was applied
in the y-direction. Regarding the flow field, the
square cavity was assumed to be closed and the no-
slip boundary conditions were imposed at each of the
four solid walls. In this simulations, appropriate
values of V were obtained using the model presented
in Eq. (23) with a fixed Knudsen number of Kn = 10-
4
at the macroscopic scale and Rayleigh numbers of
Ra = 105
and 106
, respectively.
The results of the present simulation in
comparison with the previous study ([25]) are shown
in the table 2. The results of the present study show a
good agreement (maximum variation is less than 2%)
with the previous study.
Figure 5 shows the temperature contours and
streamlines in the Г-shaped enclosure in comparison
with the results of the FDM [29]. The results show a
good accuracy of present simulation. As mentioned
above, this model is a suitable technique for
simulation of fluid flow and heat transfer in natural
convection problems using LBM in the cavity. On
the other hand, to validate the calculated entropy
generation by this model, the results of the entropy
generation in the cavity was compared with the work
of the Oliveski et al. [26] in figure 6. This figure
shows the total entropy generation in the cavity
verses Rayleigh number at different irreversibility
distribution ratio, φ. Results show a good agreement
with the previous results that shows the ability of this
model to calculate of the entropy generation. Effect
of the aspect ratio and Rayleigh number over the
temperature distribution, flow field and entropy
generation in Γ-shaped enclosure has been analyzed
numerically using Lattice Boltzmann method at
constant irreversibility coefficient(ψ=10-6
). In this
simulation the Rayleigh number and aspect ratio are
changed from 103
to 106
and 0.25 to 3 respectively.
Results are presented as a form of streamlines,
temperature contours, entropy generation contours,
mean Nusselt number and total entropy generation.
Figure 7 shows the streamlines and temperature
contours for AR=1(h=0.5, w=0.5) at different
Rayleigh number. There exist two rotating cells in
the enclosure. The main and the larger rotating cell
on the left occupies region between left vertical wall
and front face of the hot step which circulates
counter clockwise (CCW). The small and secondary

E. Fattahi
Fig. 5. Streamlines and temperature contours for different Rayleigh numbers in comparison with the FDM [29]
9
Ref [29],Ra=10
Present study, Ra=10
Ref [29], Ra=10
Streamlines and temperature contours for different Rayleigh numbers in comparison with the FDM [29]
Ref [29],Ra=104
Ref [29], Ra=105
Streamlines and temperature contours for different Rayleigh numbers in comparison with the FDM [29]

10
Fig. 6. Validation of total entropy generation in the square cavity in comparison with Ref [26]
Table 1.
The averaged Nusselt Number on the step's wall of Γ-shaped enclosure for different grid points
Mesh size
Ra 40×40 80×80 100×100 110×110
105
18.52 18.66 18.69 -
106
26.93 27.15 27.23 27.26
Table 2.
The averaged Nusselt Number on the vertical Boundary at x=0 in comparison with the previous study
Nu Ref[25] Present study Ref[25] Present study Present study
Ra 40×40 40×40 80×80 80×80 100×100
105
4.487 4.449 4.523 4.508 4.512
106
8.798 8.623 8.928 8.883 8.891
cell forms over the step and circulates clockwise.
At Ra=103
, the heat transfer is mainly controlled by
conduction. The isotherms are nearly parallel to the
vertical wall which indicates domination of
conduction heat transfer in this case. At Ra=105
, the
intensity of the circulation becomes stronger, which
Ra
S
103
104
105
10610-1
10
0
101
102
103
104
ϕ=10−3
ϕ=10−4
ϕ=10−5
ϕ=10
−3
ϕ=10
−4
ϕ=10−5


Ref. [36]
Present
study

11
implies that the convection heat transfer begins
dominating the thermal flow field in the cavity. As
the Rayleigh number becomes large (Ra=106
), the
crowded streamlines and isothermal lines indicate
that the hydrodynamic and thermal boundary layers
have been developed along the hot and cold walls,
respectively, reflecting rigorous heat transfer rate
occurred .
Variation of the flow field and temperature
distribution has a different effect over the entropy
generation. Figure 8 shows the contours of local
entropy generation due to heat transfer and fluid
friction at different Rayleigh number for
AR=1(h=0.5, w=0.5) at ϕ =10-6
. A comparison of the
entropy generation maps for heat transfer and fluid
friction (Fig. 8a.) at Ra=103
confirms the less effect
of the fluid flow on entropy generation since the
strength of recirculation is relatively low. The
entropy generation due to fluid friction and heat
transfer is considerably high close to the step’s hot
walls. At higher Rayleigh number, buoyancy force
and consequently the fluid flow effects increases,
hence, the values of entropy generation due to fluid
friction increase. This phenomenon can be observed
near the walls of cavity with brighter color of the
figure 8(c, d).One of the main parameters that have a
major effect over the flow field and heat transfer
distribution in the enclosure is the aspect ratio (AR).
The effect of AR and length of horizontal and
vertical walls of the enclosure over the flow field and
temperature contours are shown in figures 9 and 10.
It is observed that when the isotherm lines are
parallel together, the main mechanism of the heat
transfer is conduction (Fig 9a.). This phenomenon
can be observed for the high length of the horizontal
hot wall (h). If the distance between the hot step and
cold wall increases, the convective effect on heat
transfer rate will increase. By decreasing the length
of vertical wall, the second cell (over the step) gets
bigger, so the heat transfer rate increases (Fig. 10c.).
It is observed from the temperature contours which
show the more diffusion of the heat to the main flow.
It should be mentioned that the constant aspect ratio
can be created by different length of the horizontal
and vertical walls of the enclosure. This variation on
the height of the walls has a special effect on heat
transfer rate which will be explain as follows.
The total entropy generation at various Rayleigh
numbers for different aspect ratio is plotted in figure
11. For all cases, the total entropy generation
increases with increasing Rayleigh number, though
the rate of increase is different as seen in the figure
11. It is observed that, for Rayleigh number lower
than 105
the effect of the aspect ratio on total entropy
generation is insignificant but for greater Rayleigh
number the total entropy generation is increased
considerably. It is due to the formation of the
recirculation areas and their speed which have a
direct effect over the entropy generation due to fluid
friction. The effect of the secondary recirculation
area over the entropy generation due to the fluid
friction is more important than the first bubble at
high Rayleigh number. This is occurred at lower
aspect ratio and length of the vertical hot wall (h).
One of the main parameters in natural convection in
the cavity is the rate of the heat transfer. In this study,
the variation of the mean Nusselt number over the step
walls of the cavity verses aspect ratio is plotted at
different Rayleigh number. With increasing the AR,
the size of the secondary recirculation over the
horizontal wall of the step reduces and fills the gap
between the horizontal surface of the step and
insulated wall of the cavity. This phenomenon
reduces the effect of the convection so the Nusselt
number decreases which is observed for all size of the
h (Fig. 12). On the other hand, at lower AR, the first
recirculation propagates in the cavity and fills
commonly with second bubble the gap between
thehorizontal surface of the step and insulated wall.
This phenomenon increases the effect of the
convective heat transfer in comparison with the
conduction and subsequently causes to increase the
Nusselt number. With increasing the Rayleigh
number, the speed of the flow in the cavity increases,
therefore the convective heat transfer increases. This
increase in the rate of the heat transfer is shown at the
mean Nusselt number curve. It should be mentioned
that the best heat transfer was observed for AR less
than unity at h=0.25. It is due to formation of the
bubbles in the cavity which was explained above.
With increasing the h form 0.25 to 0.75, the Nusselt
number decreases approximately 72% at all Rayleigh
numbers for AR=1. This result shows that the effect of
the h on heat transfer is more important than the
aspect ratio.

Fig. 7.Streamlines (top) and temperature contours (bottom) for h = w = 0.5 at different Rayleigh numbers
12
Streamlines (top) and temperature contours (bottom) for h = w = 0.5 at different Rayleigh numbers
(a)Ra= 103
(b)Ra=104
(c)Ra=105
(d)Ra= 106
Streamlines (top) and temperature contours (bottom) for h = w = 0.5 at different Rayleigh numbers

E. Fattahi
Fig. 8. Entropy generation due to heat transfer (on the top) and fluid friction (on the bottom) for h = w = 0.5 at
different Rayleigh number
13
Entropy generation due to heat transfer (on the top) and fluid friction (on the bottom) for h = w = 0.5 at
(a)Ra= 103
(b)Ra=104
(c)Ra=105
(d)Ra= 106
Entropy generation due to heat transfer (on the top) and fluid friction (on the bottom) for h = w = 0.5 at

14
6. CONCLUSION
In this study the effect of aspect ratio and
Rayleigh number on the heat transfer and entropy
generation in the Γ-Shaped enclosure was
investigated numerically using Lattice Boltzmann
Method. The results of this study show that new
model of the Kao et al. [32] for determining an
appropriate characteristic velocity value based on the
principle of kinetic theory is not only a suitable
model for calculating the heat transfer characteristic
but also a very useful technique to calculate the
entropy generation in the natural convection
problems. The accuracy of the present study for
calculating the entropy generation in comparison
with previous result [26] in the square cavity was
very well. In Γ-Shaped enclosure, the numerical
results show that increasing the Rayleigh number
causes to increase the Nusselt number for all cases.
At high aspect ratio, the formation of the bubbles
causes to increase the diffusion heat transfer in
comparison with convective heat transfer so Nusselt
number decreases for all cases. The best heat transfer
performance was observed at low AR. Also it was
observed that the lowest size of the vertical wall of
the step has a best heat transfer rate for low AR. The
entropy generation is independent of the AR for low
Rayleigh number which is due to the low speed of the
fluid and subsequently low entropy generation due to
the fluid friction. In natural convection in the cavity,
the main part of the entropy generation was created
by fluid friction so it was high at high Rayleigh
number. The size of the vertical wall of the step does
not have any observable effect on total entropy
generation except for high Rayleigh number.
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E. Fattahi
Fig. 10. Streamlines (on the top) and temperature contours (on the bottom) at Ra =10
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Streamlines (on the top) and temperature contours (on the bottom) at Ra =105
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17
Fig. 11. Variation of the dimensionless total entropy generation with Ar at different Rayleigh number and φ=10-6
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Fig. 12.Variation of mean Nusselt number with Ar at different Rayleigh number. () h=0.25, ( ) h=0.5 and (О) h=0.75.
Ar
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101
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Ar
Nu
0.5 1 1.5 2 2.5 3
20
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60
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Ra= 10
4
Ra=105
Ra= 10
6

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