In this paper, we consider a two-dimensional numerical study of laminar conjugate natural convection in a square enclosure. The left vertical wall of the enclosure is thick, while the other three walls are taken to be of zero thickness. The enclosure is subjected to horizontal temperature gradient. The right vertical wall has constant temperature θc, while the outer surface of the left vertical wall is partially heated in the top θh, with θh>θc. The remaining left portion in the vertical wall and the two horizontal walls are considered adiabatic. The governing equations are solved by finite volume method using SIMPLER algorithm and Hybrid scheme. The parameters considered are: Rayleigh number (500≤ Ra ≤106), Prandtl number (Pr=0.71), wall to fluid thermal conductivity ratio (0.1≤ Kr ≤100), the ratio of wall thickness to its heigh (0.2≤ D≤ 0.5) and wall to fluid thermal diffusivity ratio (R=1). The main focus of the study is on examining the effect of Ra, Kr and D on the flow and heat transfer in the enclosure. A comparison with the special isothermal wall case is also studied. The obtained results show that for a given thickness, either increasing the Rayleigh number and the thermal conductivity ratio, can increase the average Nusselt number, the interface temperature and the flow velocity. Generally it is found that the increase of the thickness of the bounded wall can decrease , especially for Kr > 1 and Ra < 104   increases with the increase of D. The wall-fluid interface temperature is found to be quite non- uniform. This non uniformity tends to make the flow pattern in the enclosure asymmetric. For low Rayleigh number and low conductivity ratio, the flow velocity is neglected and conduction heat transfer is dominated in the enclosure

BELAZIZIA, Abdennacer

I-Revues

21
ème
 Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
Numerical study of conjugate natural convection in a square 
enclosure with top active vertical wall 
A. BELAZIZIA
a,b
, S. ABBOUDI
a
 , S. BENISSAAD
b
  
a.institut IRTES,Laboratoire M3M, site de Sévenans, UTBM, 90010 Belfort Cedex France 
Phone : 33(3)84583036, Email : nacer.belazizia@gmail.com 
b. Laboratoire d’Energétique Appliquée et de la Pollution. Université Mentouri Constantine 
Abstract 
In this paper, we consider a two-dimensional numerical study of laminar conjugate natural convection in a 
square enclosure. The right vertical wall has constant temperature θc, while the outer surface of the left 
vertical wall is partially heated in the top θh, with θh>θc. The remaining left portion in the vertical wall and 
the two horizontal walls are considered adiabatic. The obtained results show that for a given thickness, D 
either increasing the Rayleigh number Ra and the thermal conductivity ratio Kr, can increase the average 
Nusselt number Nu , the interface temperature and the flow velocity. Generally it is found that the increase 
of the thickness of the bounded wall can decrease Nu , especially for Kr >1 and Ra<10
4
 Nu  increases with 
the increase of D. The wall-fluid interface temperature is found to be quite non- uniform. This non 
uniformity tends to make the flow pattern in the enclosure asymmetric.  
Keywords: conjugate natural convection, square enclosure, top active vertical wall, finite volumes 
method, thermal conductivity ratio.  
Résumé 
Dans cet article, nous présentons une étude bidimensionnelle sur la convection naturelle laminaire 
conjuguée dans une enceinte carrée. La paroi droite est soumise à la température θc , tandis que la surface 
extérieure de la paroi gauche est partiellement chauffée sur la moitié supérieure θh, avec θh>θc. La moitié 
inférieure et les autres parois horizontales sont considérées adiabatiques. Les résultats obtenus montrent 
que pour une épaisseur donnée D, l’accroissement des paramètres nombre de Rayleigh Ra et le rapport des 
conductivités thermiques Kr entrainent un accroissement du nombre de Nusselt moyen Nu , de la 
température de l’interface et de la vitesse de l’écoulement. Généralement il a été trouvé que l’augmentation 
de l’épaisseur de la paroi solide réduit  Nu sauf pour le cas Kr >1 et Ra<104 Nu croit avec l’accroissement 
de D. la température de l’interface solide-fluide est non uniforme. Cette non uniformité rend la structure de 
l’écoulement non symétrique. 
Mots clés: convection naturelle conjuguée, enceinte carrée, paroi chauffée en haut, volumes finis, 
rapport des conductivités thermique.  
1. Introduction 
Natural convection in enclosures is a topic of considerable engineering interest. Applications range from 
thermal design of buildings, to cryogenic storage, furnace design, nuclear reactor design, and others. In 
many studies, the walls of the enclosure are assumed to be of zero thickness and conduction in the walls is 
not accounted for. In addition convection heat transfer is due to the imposed temperature gradient between 
the opposing walls of the enclosure taking the entire vertical wall to be thermally active. However, in many 
practical situations, especially those concerned with the design of thermal insulation. It is only a part of the 
wall which is thermally active and conduction in the walls can have an important effect on the natural 
convection flow in the enclosure [1-12]. 
This article presents a numerical study of steady laminar conjugate natural convection in a square 
enclosure. The outer surface of the thick vertical wall is partially heated in the top.   The main focus is on 
examining the effect of conduction in the wall, Rayleigh number, and wall thickness on heat transfer and 
fluid flow.  
21
ème
 Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
 
Nomenclature 
D           dimensionless wall thickness,   Hd  
Kr          thermal conductivity ratio, 
f
kwkKr   
L         cavity length,            m 
H       wall height,               m 
 h         height of the active part of the wall   m  
Nu       local Nusselt number,         Eq(7) 
Nu       average Nusselt number,    Eq(7) 
rP        Prandtl number of the fluid,        
Ra        Rayleigh number,    cThTHg 
3
    
t            dimensionless time,  2* Ht                      
U , V    dimensionless velocity components, 
 Hu  ,    Hv   
P         dimensionless pressure,    2Hp    
X , Y   non-dimensional Cartesian coordinates Hx , 
Hy  
Greek symbols 
          thermal diffusivity,                  m2s-1  
  
*
       thermal diffusivity ratio,     
fw
    
            non-dimensional temperature 
   cThTcTT   
      non-dimensional stream function, YU    
Subscripts 
 f        fluid  
  w       wall 
  wf     solid-fluid interface 
2. Problem geometry 
The geometry of the problem is shown in fig.1. The left vertical wall has a thickness d , the other walls are 
assumed to be of zero thickness. The outer vertical surface of the left wall is partially heated (h=H/2) in the 
top Th. The right wall is maintained at cooled temperature Tc such that hT > cT , while the horizontal walls 
are insulated. The problem is solved for air with 700.rP .  
 
 
 
 
 
                                                      0


x
T
 
 
 
 
Fig 1: Physical configuration. 
3. Governing equations 
The flow in the enclosure is assumed to be two-dimensional. All fluid properties are constant. The fluid is 
considered to be incompressible and Newtonian. The Boussinesq approximation is applied. Viscous 
dissipation, heat generation and radiation effects are neglected. The dimensionless form of the governing 
equations can be written as: 
- at 0t ; 0 fwVU   
- For t  > 0 
●  fluid part  
 0





Y
V
X
U
  (1) 
    
H 
0


y
T
 
air 
d 
g 
0


y
T
 
Th 
Tc 
y 
x 
  
L 
21
ème
 Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
 
























2
2
2
2
Y
U
X
U
P
X
P
Y
U
V
X
U
U
t
U
r  (2) 
 .. rr PRa
Y
V
X
V
P
Y
P
Y
V
V
X
V
U
t
V


























2
2
2
2
 (3) 
 






















2
2
2
2
YXY
V
X
U
t
fffff 
 (4) 
● solid part  
 
















2
2
2
2
YXt
www 
 *   (5) 
Where the subscript f  for fluid layer and w  for the wall, Ra  is the Rayleigh number, 
* is the diffusivity 
ratio and rP  is the Prandtl number. 
The boundary conditions are: 
                                   101 ,,,, XUXUYUYDU  ;        101 ,,,, XVXVYVYDV   (6a) 
                     10 Yw , ;   01 Yf , ;
 
0
0



Y
Xw , , 
 
0
0



Y
Xf ,
 ;
 
0
1



Y
Xw ,   0
1



Y
Xf ,
           (6b)  
                                           YDf ,  =  YDw , ; 
 
solide
w
fluide
f
X
YD
Kr
X
YD




 ,),( 
                           (6c) 
Where fw kkKr    the thermal conductivity ratio, and D is the dimensionless wall thickness of the solid 
part.  
The local and average Nusselt numbers are defined by: 
 
1,DXX
Nu











; dY NuNu 
1
0
 (7) 
4. Numerical method  
Equations (1) to (5) subjected to the boundary conditions (6) are integrated numerically using the finite 
volume method described by Patankar [13]. A uniform mesh of (90x90) is used in X and Y directions. A 
hybrid scheme and first order implicit temporally discretisation are used. The iteration process is terminated 
under the following conditions: 
   
ji ji
n
ji
n
ji
n
ji
, ,
,,, /
51 10  
For wall side DXX NuNu  0     ; For fluid side  1  XDX NuNu  
Where   represents: U , V  and   ; n  denotes the iteration step. 
6. Numerical validation 
In order to validate our results a comparison was made with the results obtained by Kaminski and Prakash 
[1] (see table 1). A good agreement between the obtained and reported results can be observed. 
Table 1: Comparison of Nu  with  Kaminski solution [1]. 
Ra  Kr  Kaminski  Present Study  
 
7.10
2 
 
1 
5 
10 
∞ 
0.87 
1.02 
1.04 
1.06 
0.867 
1.017 
1.040 
1.063 
 
7.10
4 
1 
5 
10 
∞ 
2.08 
3.42 
3.72 
4.08 
2.084 
3.417 
3.719 
4.076 
21
ème
 Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
7. Results and discussion  
 The tests The results are generated for different values of governing parameters: 1*  ; 5.02.0  D , 
610500  Ra  and  10010 Kr. . 
7.1. Streamlines plots 
For Ra=10
5,
 Fig. 2 show the effect of both wall thickness D and thermal conductivity ratio Kr , on fluid 
motion in the enclosure. The circulation pattern is in clockwise direction, with flow upward at the hot left 
solid-fluid interface and downward at the cold right wall.  
Panel (a) represents a poorly conducting wall ( 1.0Kr ), Panel (b) equal fluid/wall conductivity ( 1Kr ) 
and panel (c) a high wall conductivity ( 10Kr ). We can see for Kr=0.1 and Kr=10 that the increase of wall 
thickness D  leads to reduce the maximum values of the dimensionless stream function max . The 
convection in this case becomes weaker because of the temperature drop in the wall. In contrast to that, for 
Kr=10 max increase with the increase of wall thickness. It can be seen also from Fig. 2 that the strength of 
the fluid circulation is increasing with the increase of the thermal conductivity ratio.  As a result the 
convection becomes more important because of the increase of the effective temperature difference driving 
the flow.      
                                                                                           
                                                                                           
                                                                                           
Fig 2: Streamlines: from left to right D 0.2, 0.3, 0.4, 0.5.  (a) Kr 0.1; (b) Kr 1; (c) Kr 10. 
7.2. Isotherms plots 
The isotherms are shown in fig.3, for different values of wall thickness D  and constant Rayleigh number 
(Ra=10
5
). A comparison is made for three cases of thermal conductivity ratio: 1.0Kr , 1 and 10. 
For poor conductive wall (Panel a 1.0Kr ), the average Nusselt number have low values comparing with 
those in panel (b) and (c). This is a logical result since reducing the thermal conductivity of the wall leads to 
the increase in thermal resistance of the overall system and therefore reducing the Nusselt number. Heat 
transfer is mainly by heat conduction for all the values of D . 
As Kr increases (Kr=1 Panel b), conduction in the solid wall is more important and so that for convection 
heat transfer in fluid part. The average Nusselt Nu  becomes higher than that in case (a).   
The third case (Panel c) represents a high and good conductive wall ( 10Kr ). The convection heat transfer 
is very important comparing with cases (a) and (b). The values of  Nu  are decreasing with the increase of 
wall thickness, and increasing with the increase of Kr. 
 
(a)  
 
max =3.64 max =3.05  max =2.32   
(b)  
 
max =5.77 max =5.76 
 max =5.49   
(c)  
max =8.15 max =8.41  max =8.84   
21
ème
 Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
                                     
                           Nu =0.273                     Nu =0.197                      Nu =0.158                     Nu =0.134 
                                     
                             Nu =1.56                      Nu =1.30                       Nu =1.13                        Nu =1.01 
                                     
                            Nu =3.65                       Nu  =3.60                       Nu =3.55                      Nu =3.49 
Fig 3: Isothermes: from left to right D 0.2, 0.3, 0.4, 0.5.  (a)  Kr 0.1. (b) Kr 1. (c) Kr 10. Ra=105  
7.3. Interface temperature 
The variation of wall/fluid interface temperature for different values of the thermal conductivity ratio and at 
constant D=0.2 and 
710Ra is shown in Fig.4. A comparison is made with the standard enclosure 
( 0D , Kr ). Natural convection inside the fluid part is driven by the temperature difference between 
the interface and the cold wall.
 
This difference is lower for walls with poor thermal conductivity. It becomes 
more important with the increase of Kr, and lead to increase the average Nusselt number. For high values of 
Kr ( Kr ) the interface temperature distribution tend to become uniform which represents the standard 
enclosure with isothermal vertical walls ( 1 ) and zero wall thickness ( 0D ). 
As shown in fig.4 the temperature profile across the solid/fluid interface is quite non uniform. This non 
uniformity has a noticeable effect on the flow field. The flow structure is asymmetric as shown in fig.2. 
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Kr=100
Kr=10
Kr=5
Kr=1
Kr=0.5
Kr=0.1

wf
H
D=0.2
 
Fig 4: Variation of wall-fluid interface temperature. Ra 105. 
7.4. Average Nusselt Number 
Fig.5 shows the effect of Rayleigh number, thermal conductivity ratio and wall thickness on the rate of heat 
transfer across the enclosure. For the three cases of Kr, we note that Nu  increases with the increase of Ra. 
For low conductive wall (Kr=0.5), where the solid part is an insulate material, Nu  has small values 
comparing with those with Kr=1 and Kr=5. In this case temperature difference driving the flow between the 
(a) 
(b) 
(c) 
21
ème
 Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
interface and the cold boundary is very small, so most of heat transfer is by heat conduction. As the 
conductivity ratio increases 1Kr , reducing the wall thickness leads to enhance heat transfer by natural 
convection and so that for Nu . In addition for Ra<103, Nu is almost constant so no effect of wall thickness 
in this case.  For high conductive walls ( 10Kr ) and low values of Ra  ( Ra <104) we found that Nu  is 
reducing with reducing the wall thickness. This means that the thermal resistance of the wall is less than that 
of fluid medium for highly conductive walls and Ra <104. 
1000 10000 100000 1000000
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Nu
Ra
Kr=0.5
 D=0.2
 D=0.3
 D=0.4
 D=0.5
1000 10000 100000 1000000
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Nu
Ra
Kr=1
 D=0.2
 D=0.3
 D=0.4
 D=0.5
1000 10000 100000 1000000
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Nu
Ra
Kr=5
 D=0.2
 D=0.3
 D=0.4
 D=0.5
 
Fig 5: Variation of Nu  with  Rayleigh number Ra . 
8. Conclusion 
A numerical model was employed to analyze the flow and heat transfer of air filled in a square enclosure 
with partially heated and conductive vertical wall. The following conclusions are drawn: conduction and wall 
thickness make a strong effect on natural convection in the fluid part. It is found that the rate of heat transfer 
increases and the fluid moves with greater velocity when the values of Rayleigh number and conductivity 
ratio increase. It is observed that the temperature difference between the interface and the cold boundary is 
reducing with increasing the wall thickness and therefore reducing the average Nusselt number. It is found 
that as the wall thickness increases the average Nusselt number decreases and the maximum value of the 
dimensionless stream function in the fluid part are higher with thin walls. It is found for special cases at low 
Ra  ( Ra<104) and high conductive walls ( 10Kr ), the values of  Nu  are increasing with the increase of 
the wall thickness. For low values of Kr  (poor conductive wall), where the solid wall is insulation material, 
Nu  has low values with those at high values of Kr  because of the increase in the thermal resistance of the 
overall system and vas-versa. The heat is transferred mainly by conduction in both wall and fluid layer for 
small values of Ra  and the average Nusselt is approximately constant.   
REFERENCES  
[1] D.A. Kaminski, C. Prakash, Conjugate natural convection in a square enclosure: effect of conduction in one of the 
vertical walls, Int. J. Heat Mass transfer 12(1986)1979-1988. 
[2] D.M. Kim, R. Viskanta, Effect of wall heat conduction on natural convection heat transfer in a square enclosure, J. 
Heat Transfer 107(1985)139-146   
[3] A. Liaqat, A.C. Baytas, Conjugate natural convection in a square enclosure containing volumetric sources, Int. J. 
Heat Mass transfer. 44(2001)3273-3280. 
[4] Shao-Feng Dong, Yin-Tang Li, Conjugate of natural convection and conduction in a complicated  enclosure, Int. J. 
Heat Mass transfer 47(2004)2233-2239. 
[5] Manab Kumar Das, K. Saran Kumar Reddy, Conjugate natural convection heat transfer in an inclined square cavity  
containing a conducting block, Int. J. Heat Mass transfer 49(2006)4987-5000. 
[6] Abdullatif Ben-Nakhi, Ali J.Chamkha, Conjugate natural convection in a square enclosure with inclined thin fin of 
arbitrary length, Int. J. Thermal sciences 46(2007)467-478. 
[7] Nawaf H.Saeid, Conjugate natural convection in a porous enclosure: effect of conduction in one of the vertical 
walls, Int. J. Thermal sciences 46(2007)531-539. 
[8] Moghtada Mobedi, Conjugate natural convection in a square cavity with finite thickness horizontal walls, Int. 
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[9] H. Nouanegue, A. Muftuoglu, E. Bilgen, Heat transfer by natural convection, conduction and radiation in an 
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Numerical study of conjugate natural convection in a square enclosure with top active vertical wall.

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