Thermal Science & Engineering Vol.14 No.3 (2006)

Laminar Natural Convection Heat Transfer in an Air Filled Square Cavity with∗
Two Insulated Baffles Attached to its Horizontal Walls
Himsar AMBARITA†, Kouki KISHINAMI†, Mashasi DAIMARUYA†, Takeo SAITOH‡,
Hiroshi TAKAHASHI†, and Jun SUZUKI†

Abstract
This paper attempts to study numerically a differentially heated square cavity, which is formed by horizontal
adiabatic walls and vertical isothermal walls. Two perfectly insulated baffles were attached to its horizontal walls at
symmetric position. Heat transfer by natural convection of dry air was studied by solving mass, momentum and energy
equations numerically. Streamlines and isotherms are produced and heat transfer is calculated. A parametric study is
carried out using following parameters: Rayleigh number from 104 to 108, non-dimensional thin baffles length are 0.6,
0.7, and 0.8, non-dimensional baffle positions Sb from 0.2 to 0.8. It was observed that the two baffles trap some fluid in
the cavity and affect the flow fields. The flow for cavities with Sb <0.5 at low Ra tends to circulate as a primary vortex
strangled by the baffles while at high Ra it tends to separate into two different vortexes. For the cavities with Sb >0.5 ittends to separate into two different vortexes at low Ra while at high Ra tends to circulate as a primary vortex strangled
by the baffles. It is found that Nusselt number is an increasing function of Ra, a decreasing one of baffle length, andstrongly depends on Sb. Another interesting phenomenon of the typical cavity is that a particular case is the opposite of
the other case as long as the sum of Sb is equal to 1. Thus, the typical cavity can allow the heat flow in one direction but
significantly blocks it in the opposite direction. The typical cavity can be proposed as a heat version of a diode. The heat
may be transferred up to 42% from one direction but blocked up to 98% in the opposite direction by using a particular
cavity with Lb =0.7 and Sb=0.4 at Ra=108.
Key Words: Cavity with baffles, Natural convection, Heat version of diodeNomenclatureg:
Gravitational acceleration[m/s2]
H:
Cavity height[m]k:Thermal conductivity[W/(m⋅K)]
K:
Heat flow coefficient defined in Eq. (3)l:Baffle length[m]L:Non-dimensional baffle length, = l H
Nu:
Nusselt number defined in Eq. (10)p:
Pressure[Pa]
P:
Non-dimensional pressure
Pr:
Prandtl number, = υ αRa:Rayleigh number, = gβ ( Th − Tc )H 3 αν

θ
υ
ρ
ψ

s
S
T
U,V

1 Introduction

α
β

:
:
:
:
:
:

Baffle position[m]
Non-dimensional baffle position, = s H
Temperature[K]
Non-dimensional velocities
Thermal diffusivity[m2/s]
Coefficient of thermal expansion
[K-1]

:
:
:
:

Non-dimensional temperature
Kinematic viscosity
Density
Stream function

Sub/Superscriptsb:related to bafflec:related to cold surfaceh:related to hot surfacel:related to left vertical wallr:
related to right vertical wall:
averageNatural convection in an air filled cavity withvertical walls that are heated and cooled while its
horizontal walls are adiabatic has received a great
consideration because many of the industrial applications∗ Received: February 27, 2006, Editor: Mamoru SENDA†

Department of Mechanical System Engineering, Muroran Institute of Technology (27-1 Mizumoto cho, Muroran050-8585, JAPAN)‡
School of Environmental Studies, Tohoku University (01 Aoba-yama, Sendai 980-8579 JAPAN)- 35 -[m2/s][kg/m3]© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.3 (2006)

employ this concept as a prototype. Noticeable examples
include heating and ventilating of rooms, solar collector
systems, electronic cooling devices, and cooling of
nuclear reactors. In many applications, for some reasons,the cavity is partitioned by attaching baffle(s) to its
vertical and (or) to its horizontal wall(s). Recently,
studies of heat transfer and fluid flow characteristics ofthe partitioned cavity has come under scrutiny both
numerically and experimentally [1-17]. Based on thebaffle(s) position, the published works can be segregatedinto two groups. The baffle(s) are attached to the verticalwall(s) [1-11] and the baffle(s) are attached to thehorizontal wall(s) [12-17] of the cavity. Since this paper
considers a square cavity with a baffle attached to eachof its horizontal walls, only the typical problems will bebriefly reviewed.
The cavity with baffle(s) attached to its horizontal
walls received less attention than the cavity withbaffle(s) attached to its vertical wall(s). After an
extensive literature study, we have found threepublications about this particular cavity. Bajorek and
Llyod [14] studied experimentally a differentially heated
air filled square partitioned cavity for Rayleighnumbers 1.25 ×10 5 − 2.16 ×10 6 . The insulated baffle isattached to the horizontal walls at positions in the middle.
Non-dimensional baffle length is 0.25. The effect of thebaffle positions was not considered. It was found that thebaffles significantly influence the heat transfer rate andthe average Nusselt numbers reduced to approximately
15 % compared to the non-partitioned cavity.
Observations of the interferometric fringes indicated thatthe core region is unsteady beginning at Rayleighnumbers 3.5 × 105 . Jetli et al. [15] studied numerically adifferentially heated air filled square partitioned cavity
for Rayleigh numbers 10 4 − 3.55 × 105 with three different
combinations of the baffle positions. The first case, thebottom baffle is on the hot side and the top baffle is on
the cold side, the second case both baffles are in the
middle, and the third case the bottom baffle is on thecold side and the top baffle is on the hot side.
Non-dimensional baffles length is fixed at 0.33. The
results clearly demonstrate that the baffles positions havea significant effect on the heat transfer and flow
characteristics of the fluid. For all baffles locations, theaverage Nusselt number is smaller than thecorresponding value in a cavity without baffle.
Recently, Bilgen [16] studied numerically thedifferentially heated air filled partitioned shallow cavity(aspect ratio 0.3 to 0.4) for Rayleigh number from 104 to
1011. Non-dimensional baffle length was varied from 0 to
0.15. The baffle position was varied in the middle orslightly farther away from the heated side
(non-dimensional positions are 0.5-0.6) and its
conductivity is 20 relative to the air conductivity (forusual construction materials). It was found that the flow
regime was laminar for Rayleigh numbers up to 108
thereafter turbulent. The heat transfer was reduced when
two baffles were used instead of one, aspect ratio wasmade smaller, and positions of the baffles were farther- 36 -

away from the hot wall.
The literature review cited above shows that the
study of a partitioned cavity with non-dimensional
baffle’s length more than 0.5 have not been reported. It
was reported that the length of the baffle strongly affects
the heat transfer and flow characteristics in the cavity but
no trapped fluid phenomenon was captured.
In this paper we present a numerical study of naturalconvection in an air filled partitioned square cavity. The
cavity was differentially heated and to its top and its
bottom walls a baffle was attached. The baffles are thin,perfectly insulated and non-dimensional length of morethan 0.5 in order to capture the trapped fluid. The
positions of the top baffle from the right wall and thebottom baffle from the left wall are the same, so theconfiguration of the baffles in the cavity is always
symmetrical. The literature review shows that thisproblem has not been addressed [22]. The objective of
this paper is to make clear the effects of the long baffles
on the flow and temperature fields and on the heattransfer characteristics of the cavity. The result can be
expected from this paper is to show that the cavity withbaffles are attached to its horizontal walls can beproposed as a heat version of diode which allows theheat to flow in one direction but significantly blocks it in
the opposite direction.
2 Problem Definition
The typical cavity with the boundaries and its
coordinates system are depicted in Figure 1. In this study
only the square cavity is considered, its height and width
are denoted by H . The cavity is differentially heated, left
and right walls are isothermal at Th and Tc
respectively (Th > Tc ) and horizontal walls are adiabatic.
Two thin baffles with non-dimensional length Lb ,perfectly insulated, were attached to the top and thebottom wall. The non-dimensional position of the bottom
baffle from the left wall and top baffle from the rightwall are the same and denoted by S b .
The flow is assumed to be two-dimensional and insteady state condition. The compressibility, radiation
heat exchange, and dissipations are negligible. All of thethermal properties are constant, except density in thebuoyancy force. Boussinesq approximation is used tomodel the buoyancy force. The governing equations
are converted into the non-dimensional form by defining
the non-dimensional variables.
X =

θ=

x
y
uH
vH
pH 2
, Y=
, U=
, V=
, P=
,
H
H
α
α
ρα 2

T − TcTh − Tc(1)

The pressure p in the above equation is the reducedpressure which is defined as local pressure subtracted byhydrostatic pressure. Based on these non-dimensional© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.3 (2006)y

3 Numerical Procedure

adiabaticsbg

All of the governing equations were transformed
into sets of algebraic equations based on the finite
volume method. The staggered grid system was used. Inorder to handle convective and diffusion terms the powerlaw scheme was employed. The sets of algebraic
equations were solved by using the line by line method
which is combined with the Thomas algorithm. The
SIMPLE algorithm [21] was used to couple the velocity,pressure, and temperature fields.
In the SIMPLE algorithm under-relaxation factor isan essential problem and there is no special rule indetermining the proper under-relaxation factor. A large
under-relaxation factor, especially for momentum
equations, will lead the calculation into divergencehowever a very small under-relaxation factor leads to alengthy calculation. After some calculations, employing
an under-relaxation factor larger than 0.2 for momentum
equations (especially for cases Ra higher then 106) will
not reach the convergence. In order to get theconvergence for all problems, the small under-relaxation
factors were employed. Considering both computational
cost and convergence the under-relaxation factor for
momentum equations is 0.01, for pressure it is 0.2, andfor temperature it is 0.8. Based on this procedure the
FORTRAN codes to solve all of the governing equations
have been developed.

T = TcT = Th

lb

lb

sb
x
adiabatic

Fig. 1 Schematic of the square cavity with thin insulated
baffles attached to the horizontal walls
variables, the governing equations are obtained asfollows.
∂U ∂V
+
=0
∂X ∂Y

(2)

U

 ∂ 2U ∂ 2 U 
∂U
∂U
∂P

+V
=−
+ Pr 
+
2
∂X
∂Y
∂X
∂Y 2 
 ∂XU

 ∂ 2V ∂ 2V ∂V∂V∂P
 + Ra Pr θ (4)+V=−+ Pr 
+
2
∂X
∂Y
∂Y
∂Y 2  ∂XU∂θ∂θ  ∂ 2θ ∂ 2θ 

+V
=
+
∂X
∂Y  ∂X 2 ∂Y 2 

3.1 Grid Independence Test
In this study, non-uniform grid spacing was
employed. However near the walls and the baffles evensmaller grid spacing was used. In order to determine theproper grid numbers due to accuracy and computationaltime, a grid independence test was conducted for aparticular case. Four different grid sizes were tested in
the case where Lb=0.6 and Sb=0.3. N u and the maximum
absolute stream function are used as the sensitivity
measure of the accuracy of the solution, the results arepresented in Table 1.

(5)

The boundary conditions are:On the left wall:
U = V = 0, θ = 1
On the right wall: U = V = 0 , θ = 0
On the top and bottom walls: U = V = 0 ,

(3)

(6)
(7)
∂θ
=0
∂Y

(8)

On the baffles: U = V = 0(9)In order to compare total heat transfer rate, Nusselt
number is used. The local and average Nusselt numbers
are defined as follows.
∂θ ∂X X =0
Nu y = −(10)(θ h − θ c )Table 1 Comparison parameters for grid sensitive test

1

Nu =

∫ Nu dY

(11)

y

0

The comparison parameter for flow characteristic isrepresented by the stream function, and defined asfollow.
∂ψ
∂ψ
U =−
, V=
(12)
∂Y

Grid
numbersNu

60 × 60
80 × 80
100 × 100
120 × 1200.75430.81280.84380.8532

ψ max

8.51
8.67
8.749
8.783

∆N u

(%)
7.75
3.81
1.11
-

∆ψ max

(%)
1.88
0.91
0.39
-

Comparison of the both parameters values amongfour different cases suggest that two grids sizes 100 × 100
and 120 × 120 give nearly the same results (deviation lessthan 1.5 %). Considering both the accuracy and thecomputational time, the 100 × 100 grid size was
employed for all calculations.

∂X- 37 -© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.3 (2006)

3.2 Validation of the CodeIn order to make sure that the developed codes arefree of error coding, a validation test was conducted.
Calculations for an air filled square cavity without baffle
for Ra=104 to 108 were carried out and the results areshown in Table 2. The results of the previous
publications for the same problem are also presented in
the table. Data from the table shows that the results ofthe code, even though there are some differences, doagree very well with the previous works results. Thosedifferences are not essential, the maximum difference is
1.7 %, and probably caused by the different grids sizes
and round-offs in the computational process. Based on
this successful validation, the problem is solved by using
the code.

0.9

0.8

0.7
0.6
0.5

Ra=104ψ max = 5.07

Ra=105ψ max = 9.61

0.4

0.3
0.2

0.1

0.9
0.8
0.7
0.6

0.5

0.4
0.3
0.2

Table 2 Comparison of the present result and theprevious works result

0.1

0.9
0.8

Reference

Average Nusselt numbers, N u

0.7
0.6

4

Ra=10

5

Ra=10

2.234
2.44684.51
5.5231

8.798
8.8359--2.2415
2.244
2.24
2.247
2.245
2.228

4.5236
4.51
4.532
4.521
4.514

8.71268.8554
8.82
8.893
8.8
8.804

16.93516.62916.52

31.52030.48

Ra=10Davish[18]Hortmann[19]
Saitoh[20]Collins[3]Nag[2]
Shi[1]
Bilgen[7]Present

6

Ra=10

7

Ra=10

0.5

8

Ra=106

ψ max = 16.80

Ra=107

ψ max = 30.24

0.4
0.3
0.2
0.1

0.9
0.8

0 .7
0 .6

0.5
0.4
0.3

Flow and temperature fields for the square cavity
without baffle are presented in Fig. 2 for reference. Fig.
2(a) shows the isotherms. As natural convection
strengthened, temperature contours show slight deviationfrom the pure conduction case with the isotherms
becoming skewed. Under high Rayleigh numbers
conditions, the degree of distortion from the pureconduction case is very marked and the contour lines
become almost horizontal lines around the center of thecavity. Fig. 2(b) shows the streamlines. The rise of thefluid due to heating on the left wall and consequentfalling of the fluid on the right wall creates a clockwise
rotating vortex, referred to as the primary vortex.
Another feature of these streamline patterns is thatthe streamlines become more packed next to the sidewall as the Rayleigh number increases. This suggests thatthe flow moves faster as natural convection is intensified.
The maximum absolute value of the stream function in
the cavity is also shown in Fig. 2. Therefore, themaximum absolute value of the stream function can beviewed as a measure of the intensity of naturalconvection in the cavity. As the Rayleigh number
increases the maximum absolute value of the stream
function increases. This means the intensity of naturalconvection in the cavity increases as the Rayleighnumber increases.

- 38 -

0.2
0.1

0.9
0.8

0.7

Ra=1080.6
0.5

ψ max = 54.32

0.4
0.3
0 .2
0.1

Fig. 2 Isotherms (a) and streamlines (b) of naturalconvection in a square cavity without baffle
4 Results and Discussion
In order to understand the flow and temperaturefields and heat transfer characteristics of the typicalcavity a total of 90 cases were considered. To study theeffects of the baffle position, the non-dimensional baffle
positions Sb= 0.2, 0.3, 0.4, 0.6, 0.7, and 0.8 wereconsidered. To study the effects of baffle length, thenon-dimensional baffle lengths Lb=0.6, 0.7, and 0.8 wereconsidered. The fluid inside the cavity is dry air with Pr= 0.7 and Rayleigh number varied from 104 to 108. Flowand temperature fields and Nusselt number are
examined.
© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.3 (2006)

4.1 Flow and Temperature Fields
Flow fields for the typical cases with nondimensional baffle length Lb = 0.6 are presented in Fig. 3.
The plots are arranged going from left to right with theascending of Rayleigh numbers and from top to bottom
with the ascending of Sb values. The maximum value ofthe absolute stream function for each case is alsopresented in the figure. We note that for all Ra theappearance of the flow fields and maximum value of theabsolute stream function in all cases is totally different tothe square cavity without baffle. This is already expected
because the presence of the two baffles with Lb more than
0.5 inside the cavity totally modifies the flow field
patterns. A glance at Fig. 3 shows that the flow fields canbe divided into two different patterns. The first pattern isthe fluid circulates and creates a large clockwise primaryvortex strangled by the baffles. The second pattern is thefluid separated into two different vortexes. The two
baffles create fluid trapping phenomena in the cavity.
The first row of Fig. 3 is the flow fields for the casewhen S b = 0.2 . Since S b < 0.5 , the bottom baffle is on the
hot side and the top baffle is on the cold side of thecavity. At a low Rayleigh number Ra=104 there are two
trapped fluids in the cavity, hot trapped fluid and cold
trapped fluid. The hot trapped fluid exists between thebottom baffle and the hot wall and the cold trapped fluid
between the top baffle and cold wall. This is because at
Ra=104 natural convection is too weak to make thetrapped fluids moving and also the space of the trapped
fluids is limited due to a small S b .
The natural convection creates a primary vortex
strangled by these two trapped fluids. For Ra=105 asintensity of natural convection increases, some of thetrapped fluids are flowing and the primary vortex starts
dividing into two vortexes. For Ra=106 naturalconvection is strong enough to make all of the trapped
fluids flowing.
In order to satisfy the continuity, the circulation on
the hot side is separated from the circulation on the coldside. The primary vortex has been divided into two
different vortexes, cold vortex and hot vortex. These two
vortexes are separated by one trapped fluid which isstagnant between the two baffles. For cases with Ra=107
and Ra=108 the trapped fluid can be seen clearly. This isbecause the vortex on each side becomes more vigorous
showed by the more packed streamlines.
The second row of Fig. 3 is the flow fields whenSb=0.3. The figure shows that even for a low Rayleighnumber Ra=104 the primary vortex starts dividing into
two vortexes. When Ra=105 is reached the primaryvortex has been divided into two different vortexes. Thisis because the space between each baffle and the nearest
vertical wall is larger compared to the correspondingcases when Sb=0.2. For Rayleigh numbers higher than
105 each vortex becomes more vigorous and the trapped
fluid between the two baffles can be seen clearly. The
effects of the S b value can be seen clearly by observing
the case with Sb=0.4. In this case even for a low Rayleigh- 39 -

number Ra=104 the primary vortex has been divided into
two separated vortexes. This is because the spaces
between each baffle and the nearest vertical wall arelarger than the latter cases.
The forth row until the sixth row of the Fig. 3 are thecases when S b > 0.5 . In these cases the bottom baffle ison the cold side and the top baffle is on the hot side of
the cavity. The flow fields for case with S b = 0.6 areshown in the forth line of the figure. At Ra=104 and
Ra=105 there are two vortexes in the cavity and a trapped
fluid exists between the two baffles. This is because at
the low Ra, the strength of the flowing fluids is notstrong enough to penetrate the stagnant fluid between thetwo baffles. Since the spaces between each baffle and thenearest vertical wall are wide enough for the flowing
fluid the circulations in both sides, cold side and hot side,
are not mixed.
For Ra=106 the fluid moves faster as the Rayleighnumber increases, the two vortexes penetrate thestagnant fluid then merge to become a primary vortex.
The natural convection can not move all of the fluid in
the entire cavity and some of the fluids are trapped in the
hot side and the cold side. The hot trapped exists
between the top baffle and the hot wall. While the cold
trapped fluid exists between the bottom baffle and thecold wall. The primary vortex circulates in a single cellstrangled by these two trapped fluids.
At higher Rayleigh numbers Ra=107 and Ra=108
the fluid moves faster, indicated by the contour of thestream line more packed, and make the trapped fluids
more clear. The fifth and sixth rows of the Fig. 3 are flow
field for cases with S b = 0.7 and S b = 0.8 respectively.
The figure shows that even for Ra=104 the flow field has
become a primary vortex strangled by two trapped fluids.
This is because the space of each trapped fluid is smaller
compared to the case with S b = 0.6 .
Similar observations were made (not shown) for
cases with Lb = 0.7 and Lb = 0.8 . In these cases the flow
fields show the same pattern with the corresponding casewith Lb = 0.6 .
Temperature fields when Lb = 0.6 are presented inFig. 4. The plots are arranged going from left to rightwith the ascending of the Rayleigh number and from top
to bottom with the ascending of the S b value. The
contour level increments for each case are kept constant
at 0.1. Indeed in comparison to the square cavity without
baffles the appearance of the temperature fields isstrongly modified due to the presence of the two
insulated baffles.
The first row of Fig. 4 is for S b = 0.2 . At Ra=104
when the natural convection is weak the isotherms takeplace only in the areas where the fluid moves. The
isotherms show slight deviations from the pureconduction case with the contour lines becoming skewed.
In the trapped fluid areas heat transfer is inactive due topresence of the insulated baffles and stagnant fluids.

© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.3 (2006)

ψ max = 0.62

ψ max = 2.32

ψ max = 6.93

ψ max = 19.56

ψ max = 38.43

ψ max = 0.42

ψ max = 2.54

ψ max = 8.75

ψ max = 21.36

ψ max = 37.32

ψ max = 0.40

ψ max = 2.64

ψ max = 9.32

ψ max = 18.18

ψ max = 31.57

ψ max = 0.46

ψ max = 2.49

ψ max = 13.45

ψ max = 45.82

ψ max = 107.40

ψ max = 0.53

ψ max = 6.07

ψ max = 23.28

ψ max = 45.26

ψ max = 93.69

ψ max = 1.93

ψ max = 10.21

ψ max = 23.10
4

ψ max = 82.30

ψ max = 42.22
5

6

Fig. 3 Streamlines for Lb = 0.6 : the first column: Ra=10 , second: Ra=10 , third: Ra=10 , fourth: Ra=107, fifth: Ra=108.
The first row: Sb = 0.2 , second: Sb = 0.3 , third: Sb = 0.4 , fourth: Sb = 0.6 , fifth: Sb = 0.7 , sixth: Sb = 0.8- 40 -© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.3 (2006)

=104

0.8

=105

0.9

0.7

0.9

=106

=107

=108

0.8

0.9
0.7

0.6

0.8

0.7

0.6

0.4

0.4
0.4

0.6
0.4

0.5

0.5

0.9

0.3

0.3

0.9
0.7

0.8

0.7

0.6

0.5

0.5

0.1

0.3
0.1

0.2

0.2

0.3

0.8
0.6
0.4

0.5

0.2

0.1
0.3

0.9

0.2

0.1

0.2

0.1

0.8

0.9

0.7

0.8

0.7

0.6

0.3

0.9

0.9

0.8

0.6

0.5

0.4
0.3

0.4

0.9

0.8

0.7

0.6

0.5

0.5

0.4

0.3

0.5
0.3

0.4

0.2

0.2

0.9

0.8
0.6
0.4
0.2

0.7

0.6
0.1

0.7
0.5
0.3

0.1

0.1

0.8
0.7
0.6
0.5
0.4
0.3
0.2

0.8
0.7 0.6
0.5
0.4
0.3
0.2

0.2

0.2

0.1

0.1

0.8

0.9

0.7
0.6
0.5

0.4
0.3

0.8
0.7
0.6
0.5
0.4
0.3
0.2

0.9
0.7

0.9

0.8

0.6
0.5
0.4
0.3
0.2

0.9

0.1

0.1

0.1
0.1

0.2
0.1

0.6

0.6
0.5

0.2

0.2

0.1

0.1
0.3

0.3

0.4
0.5

0.4

0.6

0.6

0.4
0.3

0.5

0.5

0.4
0.2

0.9

0.1

0.9

0.1

0.6

0.3 0.2
0.1

0.8
0.6

0.5

0.7

0.4

0.3

0.8

0.7

0.8

0.7

0.7

0.5

0.2
0.9

0.7

0.5

0.6

0.9

0.8

0.5

0.8

0.9

0.4

0.4

0.6
0.3

0.2
0.6
0.1

0.4

0.3
0.2

0.4

0.4
0.3

0.1

0.5

0.9

0.5

0.9

0.6

0.1

0.7
0.6

0.8

0.6

0.5

0.3

0.2 0.3

0.2

0.8

0.7

0.1

0.5

0.9

0.2

0.9

0.5

0.1

0.8

0.8

0.7

0.7

0.9
0.8

0.7
0.4

0.4

0.6
0.3

0.2
0.1

0.6
0.2

0.6

0.4
0.6

0.5

0.1

0.9 0.8

0.5

0.1

0.9

0.8

0.8
0.7

0.3
0.2

0.5

0.1

0.7

0.4

0.9

0.3

0.2
0.3

0.4
0.9

0.7
0.7

0.3
0.5

0.6
0.9

0.2

0.5

0.8
0.7

0.1

0.4

0.4

0.8

Fig. 4 Temperature fields for Lb = 0.6 : the first column: Ra=104, second: Ra=105, third: Ra=106, the fourth: Ra=107,
fifth: Ra=108. The first row: Sb = 0.2 , second: Sb = 0.3 , third: S b = 0.4 , fourth: S b = 0.6 , fifth: S b = 0.7 ,sixth: S b = 0.8- 41 -© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.3 (2006)

At Ra=105 the isotherms in the top left and bottom
right of the cavity become more skewed but almost
horizontal between the two baffles. This is because
natural convection is more vigorous in the top left andbottom right of the cavity but becomes weak between thetwo baffles due to separation of the primary vortex. For
Ra=106, we note that the primary vortex has separated
completely. Isotherms diminished on vortex areas but
were more packed on the trapped fluid between thebaffles. This is due to temperature difference on thevortex areas is small due to fluid circulation but it ishigh on the trapped fluid. For high Rayleigh numbers
Ra=107 and Ra=108 the isotherms become more packed
in the trapped fluid area but almost disappeared on thevortex areas. This is due to the vortexes becoming morevigorous and results in a very small temperature
difference. The temperature difference on the vortex
areas is only about 0.2 for Ra=107 and is only about 0.1
for Ra=108. The trapped fluid blocks the convection heattransfer from the hot wall to the cold wall andconduction heat transfer is only considered.
The
second row and the third row of Fig. 4 are thetemperaturefields
for
cases
when
bafflepositions S b = 0.3 and S b = 0.4 . The figure shows that theappearance of the temperature fields of these cases issimilar to the cases when S b = 0.2 .
The fourth row is for case when S b = 0.6 . At Ra=104
where the flow is still separated into two vortexes and atrapped fluid exists between the two baffles theisotherms are skewed on the vortex areas but almost
horizontal in the trapped fluid area. At Ra=105 theisotherms are more skewed in the vortex areas but lesshorizontal in the trapped fluid area since intensity of thenatural convection increases. At Ra=106 when the two
vortexes have merged the isotherms are totally modified.
The trapped fluids in the top left and bottom right of thecavity are inactive with respect to convective andconductive heat transfer due to the presence of stagnant
fluid and insulated baffles, in Fig. 4 showed by theempty isotherms areas. These trapped fluids reduce theend-to-end heat transfer on the vertical walls. The
isotherms show that the natural convection takes places
only on uncovered area of the vertical walls. For higher
Rayleigh numbers Ra=107 and Ra=108 naturalconvection becomes more vigorous and the fluid moves
faster and the isotherms become more packed next to
uncovered vertical walls. The fifth and the sixth rows ofFig. 4 are the temperature fields for cases when the baffle
positions S b = 0.7 and S b = 0.8 . The figure shows that theappearance of the temperature fields of the latter cases
are quite similar with the corresponding casewhen S b = 0.6 . Similar observations were also made (notshown) for Lb = 0.7 and Lb = 0.8 , which show thesimilarity with the corresponding cases when Lb = 0.6 .

Ra=10 , the average is 87.9%. At low Ra, the circulation
is not perfectly separated and causes a relatively lower
decrease compared to the higher Ra. For Sb=0.3, N u isdecreased by 83% at Ra=104 and is 96% at Ra=108, theaverage is 90.5%. These values suggest that the case with
Sb=0.3 blocks the heat transfer rate more effectively thanthe case with Sb=0.2. Indeed since Sb < 0.5 N u
correlates with conduction heat transfer through thetrapped fluid. Thus N u is trapped fluid’s dimension
dependent that increases as its width increases but
decreases as its length increases. Moving the bafflesfrom S b = 0.2 to S b = 0.3 will reduce the width of the

4.2 Heat Transfer
In order to evaluate how the presence of the two

trapped fluid from 0.6 to 0.4 and causes N u to decreases.
At a high Ra where the two vortexes are vigorous,
related to small temperature difference on the vortex

baffles affects the heat transfer rate through the cavity,
the average Nusselt number will be discussed. The
average Nusselt numbers for all cases as a function ofRayleigh number, Ra and for various parameters, Lb = 0.6 ,Lb = 0.7 , and Lb = 0.8 are presented in Fig. 5, Fig. 6, andFig. 7 respectively. In the figures N u for the cavity
without baffle is also presented by a dashed line as areference. Generally, as expected N u is an increasing
function of Ra and a decreasing function of Lb . The
figures show that the appearance of the N u lines forcases when S b < 0.5 is totally different from S b > 0.5 .
Nusselt numbers for cases when S b < 0.5 are lower thanthe cases when Sb > 0.5 especially at high Ra regions.
This is because in these cases, the trapped fluid between
the two baffles totally blocks the convective heat transfer.
However for the cases with S b > 0.5 , total heat transfer
rate correlates with the natural convection on uncovered
vertical walls. There is an exception, in the case whenisdifferent
from
the
cases
S b = 0.6
S b = 0.7 and S b = 0.8 at low Ra regions. This is because inthese regions the circulation is still separated by thetrapped fluid.
100

SSb=0.2b
SSb=0.6bSSb=0.3bSSb=0.7bSSb=0.4bSSb=0.8
b

Nu

10

Nu
1

0.1
1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Ra
Ra

Fig. 5 Nusselt number as a function of Ra for the case
Lb = 0.6 with S b as a parameter
Fig. 5 shows the case with Lb=0.6. For S b = 0.2 ,4N u is decreased by 80% at Ra=10 and is 94.2% at
8- 42 -© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.3 (2006)

smaller, equal to 0.2, N u is decreased by 88% at Ra=104
and is 98% at Ra=108, the average is 94.2%. These
values suggest that the case with S b = 0.4 blocks the heattransfer rate more effectively than the latter cases. This isbecause the width of the trapped fluid becomes smaller.
The
width
ratio
of
the
trappedfluid
for S b = 0.3 and S b = 0.4 is 0.4/0.2=2, this ratio is almost
the same with N u ratio that is 1.91 for Ra=106, 1.97 for
Ra=107 and 2.0 for Ra=108. These results reveal that thechange in N u is trapped fluid’s width dependent which isrelated to the baffle positions.
100

SSb=0.2b
SSb=0.6bSSb=0.3bSSb=0.7bSSb=0.4bSSb=0.8
b

Nu

10

Nu
1

0.1
1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Ra
Ra

Fig. 6 Nusselt number as a function of Ra for the case
Lb = 0.7 with S b as a parameter
For Sb=0.6, N u is decreased by 88% at Ra=104,
92% at Ra=105, 63% for Ra=106, and less than 50% for
Ra>106. At low Ra regions this case blocks the heattransfer significantly (>85%) because the two vortexes
have not merged. For Ra>106 the heat transfer rate isblocked less than 50% because the two separated
vortexes have merged. For Sb=0.7, N u is decreased
by 83% at Ra=104, 63% at Ra=105, less than 50% for
Ra>105. And for Sb=0.8, N u is decreased by 74% at
Ra=104, 62% at Ra=105, about 50% for Ra>105. These
results are in accordance with the observations that are
made in Fig. 3 and Fig. 4 that the trapped fluid in the top
left and bottom right of the cavity cover some verticalwalls and inactive respect to convection and conduction
heat transfer. The uncovered vertical walls still have asignificant effect on the total heat transfer rate.
The graph also shows that for cases when S b > 0.5 athigh Ra regions N u is only a function of Ra and it is
not affected by the baffle positions. This is because totalheat transfer rate correlates with the uncovered verticalwalls only. Another feature is that at high Ra regions thelines of N u for the cases are parallel with the cavity
without baffle. This is because at high Ra regions theheat transfer characteristics for these cases will besimilar in comparison to the cavity without baffle. The
only difference is in the length of the uncovered verticalwalls
Fig. 6 is for Lb=0.6. The graph shows that theappearance of N u lines pattern is similar with thecorresponding case when Lb = 0.6 . The only differences
are in the quantity of N u . This is because the length of
the baffles changes dimension of the trapped fluid andfinally affects the quantity of N u . For cases with Sb < 0.5
where N u depends on the dimension of the trapped
fluid between the two baffles, the longer baffles are
related to the longer trapped fluid. The length of thetrapped fluid for the case with Lb = 0.6 is 0.2 and for thecase with Lb=0.7 is 0.4, the ratio is 0.5. Since N u istrapped fluid’s dimension dependent that increases as its
width increases but decreases as its length increases,
N u for Lb=0.7 is about 0.5 times lower than Lb=0.6. For
cases with S b > 0.5 where the total heat transfer rate
correlates with the natural convection on the uncovered
vertical walls, the baffles length show significant effectin the quantity of N u . The baffle with Lb=0.7 covers thevertical wall 70% (the uncovered wall is 30%). Since theheat transfer rate correlates with natural convection on
the uncovered vertical walls then the cases with longer
uncovered areas reveal the higher N u . This results inN u for case with Lb=0.7 being lower than Lb=0.6.
Fig. 7 is for Lb = 0.8 . It was observed, again, that theappearance of the N u lines is similar to the latter
cases Lb = 0.6 and Lb = 0.7 . As expected since the baffles
are longer than in the latter cases the total heat transfer
rate for this case is lower. The figure shows Nusselt
numbers for cases with S b < 0.5 always less than 1 andfor cases with S b > 0.5 always less than 10. The reasonsare already discussed in the aforementioned paragraph.
100

SSb=0.2b
SSb=0.6bSSb=0.3bSSb=0.7bSSb=0.4bSSb=0.8b

10
Nuareas, the ratio N u will almost be the same as the ratio oftrapped fluid’s width. For example at Ra=108 N u =1.76
for Sb=0.2 and =1.21 for Sb=0.3, the ratio is 1.46, while
the width ratio is 0.6/0.4=1.5. Furthermore, forS b = 0.4 where the width of the trapped fluid becomes

Nu
1

0.1
1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Ra
Ra

Fig. 7 Nusselt numbers as a function of Ra for the case
Lb = 0.8 with S b as a parameter
4.3

Heat Version of Diode
Another interesting phenomenon of the typicalcavity is that a particular case is the opposite of theother case as long as the sum of S b is equal to 1. The
case when Sb=0.2 is the opposite of the case when Sb=0.8,- 43 -© 2006 The Heat Transfer Society of Japan

K=

θ =0

θ = 0.8

θ =0

θ = 0.2

θ =0

θ = −0.5

θ =0

θ =0

θ = −1

θ =0

θ =0

θ = 0.5
θ = −0.2

θ = −0.8

right wall N u at Ra=108 is 1.21 (blocks the heat 96%) but
if the right wall is hotter than the left wall (the same casewith Sb=0.7) N u at Ra=108 is 15.9 (passes the heat
52.3%) compared to the cavity without baffle. Thismeans that the same cavity can significantly block theheat from the left to the right wall but can also allow it toflow from the right to the left wall.
An additional calculation for a particular cavity with
7
S b = 0.3 and Lb = 0.6 at Ra=10 when θ l = 1 was carriedout. The right wall was fixed at temperature θ = 0 and theleft wall temperature was varied from θ l = 1 , 0.8, 0.5, 0.2,
-0.2, -0.5, -0.8, and -1.
The flow fields are presented in Fig. 8. The figure
shows that for θ l = 1 , 0.8, 0.5, and 0.2 the trapped fluid
exists between the two baffles, separates the primaryvortex and switches off the convective heat transfer. This
results in a very low N u or blocks the heat transfer
significantly. For θ l = −0.2 , -0.5, -0.8, -1 the separated
vortexes have been merging and create a counterclockwise primary vortex. The fluids in the bottom left
and top right of the cavity become trapped due topresence of the baffles and inactive respect to convective
and conductive heat transfer. The heat transfer rate is still
significant due to natural convection on the uncovered
vertical walls.
Nusselt number for this particular case as a functionof the left wall temperature is shown in Fig. 9. As areference, we note that N u =16.75 for the cavity without
baffle. The figure shows that if the temperature of the left
wall is hotter than the right wall N u is very low andonly about 1 (the heat is blocked up to 94%). If thetemperature of the left wall is colder than the right wall
N u is 8.23 (the heat is transferred up to 49%).
Inorder to evaluate the performance of the cavity totransfer and to block the heat flow a parameter has beenpromoted and named as a heat flow coefficient. The heatflow coefficient is defined as a ratio of N u when thefluid flow creates a primary vortex and N u when it isblocked. The heat flow coefficient is calculated by using
equation (13).

θ =1

the case when Sb=0.3 and Sb=0.7, and the case whenSb=0.4 and Sb=0.6. For example, a pair Sb=0.3 and Sb=0.7,
the case when Sb=0.3 will reveal the same results (flowfields, temperature fields and N u ) with Sb=0.7 if its
vertical walls temperature are reversed with each other.
In this cavity ( S b = 0.3 ), if the left wall is hotter than theθ =0

Thermal Science & Engineering Vol.14 No.3 (2006)

Fig. 8 Flow fields for a particular cavity with S b = 0.3
and Lb = 0.6 when the right wall temperature was
fixed at θ = 0 while the left wall was varied
from θ = 1 to θ = −1

10

Nu

1

N ucirculating(13)
N ublocked
By using this equation the heat flow coefficient for theparticular cavity with S b = 0.3 and Lb = 0.6 at Ra=107 is
K=8.23/1=8.23. It does mean that the particular cavity
can pass the heat 8.23 times more than the blocked heat.
The heat flow coefficient for all cases at Ra=106,
Ra=107, and Ra=108 are presented in Fig. 10. The figure

0.1
-1

-0.5

0

0.5

1

θl
Fig. 9 Nusselt numbers for a particular cavity with
S b = 0.3 and Lb = 0.6 as a function of the left wall
temperature with the right wall fixed at θ r = 0- 44 -© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.3 (2006)shows that for higher Rayleigh numbers K is relatively
bigger. It can be seen that the best combination of thebaffle dimension and position is for baffle length
Lb = 0.7 and its position Sb = 0.4 . This combination gives
K=7.21 at Ra=106, K=20.24 at Ra=107, and K=35.6 at
Ra=108.
40

LLb=0.6b35

LLb=0.7b

LLb=0.8
b

30
25

K 20
15
10
5
0
0.1

0.2

0.3

0.4

0.5

Sb

Fig. 10 Heat flow coefficient of the typical cavity forRayleigh numbers Ra=106, Ra=107, and Ra=108
In fluids, a check valve is a mechanical device, avalve that normally allows fluid to flow through it in
only one direction. Furthermore in electronics, a diode is
a component that restricts the direction of movement ofcharge carriers. It allows an electric current to flow in
one direction, but essentially blocks it in the opposite
direction. Thus the diode can be thought of as anelectronic version of a check valve. It was shown that the
typical cavity allows the heat to flow in one direction but
significantly blocks it in the opposite direction. Since thetypical cavity acts like a check valve or a diode it will bereferred to as a heat version of a diode. We are convincedthat the characteristics of the cavity as a heat version of adiode will be significantly affected by the aspect ratio of
the cavity. However in this paper only the square cavity
is considered. This is because we intend to only provide
the information that the cavity with baffles attached tothe horizontal walls can be considered as a heat version
of a diode. The full investigation of this cavity as a heatversion of a diode remains to be investigated.

the flow and temperature fields compared to thecorresponding cavity without baffle. Two different flow
field patterns were observed. The first pattern is flow
fields with two different vortexes separated by a trapped
fluid between the baffles and the second pattern is flow
field with a primary vortex strangled by two trapped
fluids. The flow field pattern is non-dimensional baffle
positions and Rayleigh numbers dependent. For cases
when S b < 0.5 at low Rayleigh numbers the flow tends
to circulate as a single primary vortex but at highRayleigh numbers tends to separate into two different
vortexes. For cases with Sb > 0.5 at low Rayleighnumbers the flow tends to separate into two different
vortexes but at high Rayleigh numbers tends to circulate
as a primary vortex. Nusselt number is an increasing
function of Ra and a decreasing function of baffle’s
length. For cases when Sb <0.5, N u is trapped fluid’s
dimension dependent that increases as its width increases
and decreases as its length increases especially for highRa regions. For cases when Sb >0.5, N u is the uncovered
vertical walls lengths dependent. Another interesting
phenomenon of the typical cavity is that a particular caseis the opposite of the other case as long as the sumof S b is equal to 1. It was shown that the typical cavity
allows the heat to flow in one direction but significantly
block it in the opposite direction. Since the typical cavity
acts like a check valve or a diode it will be referred to asa heat version of a diode. In order to evaluate theperformance of the cavity to transfer and to block theheat flow a parameter has been promoted and named as aheat flow coefficient. The heat flow coefficient is defined
as a ratio of N u when the heat is transferred and N uwhen it is blocked. The best combination of the baffle
position and its length in order to block and to pass theheat flow is the particular cavity with Lb = 0.7 and
Sb = 0.4 .
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Convection Heat Transfer in a DifferentiallyHeated Square Cavity Due to a Thin Fin on the
Hot Wall”, Journal of Heat Transfer, 125 (2003),
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[2] A.Nag, A. Sarkar, and V. M.K. Sastri, “Natural
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[4] F. Ampofo, “Turbulent natural convection in an air
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5 Conclusions
Heat transfer by natural convection in adifferentially heated square cavity with two thininsulated baffles has been numerically studied. The
cavity was performed by vertical isothermal walls andadiabatic horizontal walls. Two thin insulated baffles
were attached to its horizontal walls at symmetricpositions. Its non-dimensional length, Lb was varied from
0.6, 0.7, and 0.8 and its non-dimensional
positions, S b from 0.2 to 0.8. Rayleigh number ranged
from 104 to 108.
The conclusions can be drawn from this study are as
follows. The presence of the two baffles with
non-dimensional length greater than 0.5 totally modifies- 45 -© 2006 The Heat Transfer Society of Japan

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- 46 -

two-dimensional partially partitioned cavity”, Int.
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© 2006 The Heat Transfer Society of Japan