Numerical Heat Transfer, Part A: Applications
An International Journal of Computation and Methodology
ISSN: 1040-7782 (Print) 1521-0634 (Online) Journal homepage: http://www.tandfonline.com/loi/unht20
Mixed Convection in Lid-Driven Trapezoidal
Cavities with an Aiding or Opposing Side Wall
Muneer A. Ismael & Ali J. Chamkha
To cite this article: Muneer A. Ismael & Ali J. Chamkha (2015) Mixed Convection in Lid-Driven
Trapezoidal Cavities with an Aiding or Opposing Side Wall, Numerical Heat Transfer, Part A:
Applications, 68:3, 312-335, DOI: 10.1080/10407782.2014.986001
To link to this article: http://dx.doi.org/10.1080/10407782.2014.986001
Published online: 22 Apr 2015.
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Date: 27 October 2015, At: 03:57
Numerical Heat Transfer, Part A, 68: 312–335, 2015
Copyright # Taylor & Francis Group, LLC
ISSN: 1040-7782 print=1521-0634 online
DOI: 10.1080/10407782.2014.986001
MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL
CAVITIES WITH AN AIDING OR OPPOSING SIDE WALL
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Muneer A. Ismael1 and Ali J. Chamkha2
1
Mechanical Engineering Department, Engineering College,
University of Basrah, Basrah, Iraq
2
Mechanical Engineering Department, Prince Mohammad Bin Fahad
University, AlKhobar, Kingdom of Saudi Arabia
Mixed convection heat transfer and fluid flow fields inside a lid-driven trapezoidal cavity
were studied numerically. The cavity horizontal walls were thermally insulated while the
inclined side walls were maintained isothermally at different temperatures. Forced convection was induced by moving the hotter right inclined side wall. The problem is formulated
using the stream function–vorticity procedure. Together with the established boundary conditions on the right moving wall, the problem is solved by the finite difference method. The
Richardson number Ri (0.01–10) and inclination angle of the side walls U (66–80) were
considered as pertinent parameters and investigated in two lid-driven cases: aiding and
opposing directions. The results show that the behavior of Nusselt number is different from
Richardson number depending on the direction of the lid. The inclination angle of the side
walls was found to have a significant effect on Nusselt number when Ri was relatively low
(1); otherwise, a negligible effect of U on Nusselt number was recorded.
1. INTRODUCTION
The mixed-flow surveyor is surely encountered in large studies on lid-driven
flow within cavities or enclosures, for two reasons. The first is pertinent to industrial
engineering applications such as solar power collection, cooling of electronic devices,
lubrication, nuclear reactors, food processing, and even the dissipation of waste heat
from steam–electric generating plants [1]. The second relates to the mathematician
interest in such engineering problems to test the validation of new analytical and
numerical methods [2–8].
Mixed convection (forced and free) flow in a lid-driven cavity or enclosure is
generated by two mechanisms. The first relates to shear flow caused by the movement of one (or both) of the cavity wall(s) while the second relates to buoyancy flow
induced by thermal, non-homogeneous cavity boundaries. These two mechanisms
may act either in concert or opposition depending on the direction(s) of the moving
wall(s). Different aspect ratio cavities with moving walls were the subject of an early
Received 17 February 2014; accepted 6 November 2014.
Address correspondence to Muneer A. Ismael, Mechanical Engineering Department, Engineering
College, University of Basrah, P.O. Box 810, Basrah 61004, Iraq. E-mail: muneerismael@yahoo.com
Color versions of one or more of the figures in the article can be found online at www.tandfonline.
com/unht.
312
MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
313
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NOMENCLATURE
g
Gr
H
k
n
N
Nu
Nuav
Pr
q
Re
Ri
S
T
u
U
V
v
gravitational field, m s2
Grashof number, Gr ¼ gbDTH3=n2
cavity height, m
thermal conductivity, W=m K
normal vector to the inclined side wall
number of grids
local Nusselt number
average Nusselt number
Prandtl number, Pr ¼ n=a
velocity of the moving wall, m=s
Reynolds number, Re ¼ jqj H=n
Richardson number, Ri ¼ Gr=Re2
dimensionless segment along the side wall
temperature, K
velocity component along
x-direction, m s1
dimensionless velocity component
along x-direction
dimensionless velocity component
along y-direction
velocity component along
y-direction, ms1
x, y
X,Y
a
b
m
n
H
K
U
w
W
x
X
Cartesian coordinates, m
dimensionless Cartesian coordinates
thermal diffusivity, m2 s1
thermal expansion coefficient, K1
dynamic viscosity, Pa s
kinematic viscosity, m2 s1
dimensionless temperature
parameter used in Eq. (9) and defined
in Eq. (10)
inclination angle of the side
walls, stream function, m2 s1
dimensionless stream function
vorticity, s1
dimensionless vorticity
Subscripts
av
average
c
cold
h
hot
L
left
R
right
study by Torrance et al. [9], who demonstrated a marked influence of buoyancy at
larger aspect ratio at Grashof number, Gr ¼ 106. Iwatsu et al. [10] conducted a
comprehensive numerical solution for 3D flow within a cubic cavity with a moving
upper surface. They expanded the flow description through another work [11] to
ascertain the relative importance of natural and forced convection inside a 2D cavity
with a moving top wall. Rather than the viscous fluid considered in [11], Khanafer
and Chamkha [12] studied mixed convection inside a fluid-saturated porous media
cavity. Aydin [13] investigated the effect of a moving vertical wall for aiding and
opposing modes. He found that the mixed convection range of Richardson number
for the opposing-buoyancy case was wider than that of the aiding-buoyancy case.
Chamkha [14] expanded upon the work in [13] to include the effect of the presence
of internal heat generation or absorption and an external magnetic field. He found
that increasing the magnetic field strength significantly reduced average Nusselt
number for both aiding and opposing cases, while the presence of the internal
heat-generation effects was found to decrease average Nusselt number significantly
in aiding flow, and to increase it for opposing flow. Al-Amiri et al. [15] studied
the combined buoyancy effects of thermal and mass diffusion in a square cavity with
a moving upper wall. They demonstrated that heat transfer and mass transfer characteristics inside the cavity were enhanced for low values of Richardson number due
to the dominant influence of the mechanical effect induced by the moving lid. Sharif
[16] conducted a study on mixed convection in an inclined, rectangular, shallow cavity (aspect ratio ¼ 10). He concluded that the average or overall Nusselt number
increased slightly with cavity inclination when forced convection predominated
(Ri ¼ 0.1), while it increased much more rapidly with inclination angle when natural
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314
M. A. ISMAEL AND A. J. CHAMKHA
convection predominated (Ri ¼ 10). Waheed [17] studied two cases of mixed convection in a rectangular cavity with movement of either the top or bottom wall. His
results showed that an increase in Richardson and Prandtl number enhanced fluid
flow and energy distribution, respectively, within the enclosure, and heat flux on
the heated wall, while an increase in the aspect ratio suppressed it. Due to the complexity of the actual representation of a cavity with a moving wall perpendicular to a
stationary wall, very little experimental work is to be found in the literature (but see
Koseff and Street [18] and Mohammad and Viskanta [19]). Mixed convection in
lid-driven cavities can be enhanced by handling nanofluid media, as demonstrated
by Tiwari and Das [20] who considered three combinations of double lid-driven vertical walls under aiding and opposing modes. Many other studies regarding mixed
convection with nanofluids can be found in [21–24]. Mixed convection in a cavity
with an internal blockage to simulate heat transfer control was investigated by
Sun et al. [25] using conductive triangular fins, or to simulate heat exchangers as
in Rahman et al. [26], Wang et al. [27], and Islam et al. [28].
All the above studies (and many others not mentioned here for brevity) are
concerned with rectangular or square cavities. However, non-rectangular cavity geometries are also studied in many situations to extend the applications of lid-driven
mixed convection. Novel hydrodynamic simulations were performed by Migeon
et al. [29] to investigate and compare three standard manufactured cavity shapes
(square, rectangular, and semicircular) where the lid was created by a vertical wall.
They found that flow within a semicircular cavity had a much more homogeneous
and uniform recirculation, with no secondary flow zones. Mercan and Atlik [30] considered the lid-driven aspect in arc-shaped cavities of different cross section with high
Reynolds number. Saha et al. [31] investigated mixed convection within a parallelogram cavity using the penalty finite element method. They reported that maximum
heat transfer was obtained for an inclination angle of the cavity side walls of 30 .
A lid-driven triangular enclosure filled with nanofluids was studied by Rahman
et al. [32]. In an experimental study of natural convection in a box-type solar cooker
by Kumar [33], the trapezoidal enclosure was proven to posses a major advantage –
the absorption of a higher fraction of incident solar radiation falling on the aperture
at a large incidence angle. However, many published works regarding natural convection in a trapezoidal enclosure are found in the literature but few dealing with
mixed convection. Mamun et al. [34] conducted a numerical study of 2D mixed convection heat transfer in a trapezoidal cavity with a constant heat flux from the bottom wall while the top wall was maintained isothermally and moving in the
horizontal direction. Battachharya et al. [35] analyzed the flow structure and temperature patterns during mixed convection within a lid-driven trapezoidal enclosure
with a cold top moving wall and the bottom wall heated by two different
modes – isothermally and non-isothermally.
From the present literature survey, it can be concluded that all mixed convection in lid-driven cavities is due to the movement of either the horizontal or vertical
walls (i.e., lid movement coincided with one of the Cartesian axes). To the best of the
authors’ knowledge, a lid driven by an inclined side wall of a trapezoidal enclosure
has never been studied. This may be due to the complexity of the boundary conditions inherent in such a problem. Therefore, the present study focuses on the
numerical analysis of mixed convection within a trapezoidal cavity, with the side wall
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MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
315
representing the moving lid in both aiding and opposing flow modes, using the
vorticity–stream function formulation. The aim of the present study is to discover
the ability of the vorticity–stream function formulation to deal with such problems,
and also to illustrate the novel features of flow circulation and temperature fields in
such a geometry. The present geometry can be counted as a simulation of industrial
applications that has a solid material motion inside a trapezoidal chamber, such as
float glass manufacturing and continuous reheating furnaces.
It is worth mentioning that the present paper is part of an ongoing project to
establish mixed convection within fluid-saturated porous media and nanofluids confined to side wall lid-driven trapezoidal cavities.
2. MATHEMATICAL MODEL
A two-dimensional trapezoidal cavity is considered for the present study, with
the physical dimensions as shown in Figure 1. The horizontal walls are fixed and
thermally insulated, the left side wall is fixed and maintained isothermally at Tc,
and the right side wall moves with a velocity vector q and is isothermally maintained
at Th, where Th > Tc. The direction of right wall movement may be either upward
(aiding) or downward (opposing). The inclination angle of the side walls is U in such
a way that when U equals 90 , the cavity becomes a square of length H. The flow is
assumed to be steady and laminar, and viscous dissipation is neglected. Except for
density, the properties of the fluid are taken to be constant. It is further assumed that
Figure 1. Physical domain and coordinates system.
316
M. A. ISMAEL AND A. J. CHAMKHA
the Boussinesq approximation is valid for buoyancy force. Taking into account these
assumptions, the governing equations (continuity, momentum, and energy) in a 2D
Cartesian coordinate system can be expressed in terms of the stream function–
vorticity formulation as follows:
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q2 w q2 w
þ
¼ x
q x 2 q y2
qw
qy
ð1Þ
!
qx
q w qx
q2 x q2 x
qT
¼n
þ 2 þ gb
2
qx
qx
qy
qx
qy
qx
q w qT
q w qT
¼a
qy
qx
qx
qy
q2 T q 2 T
þ
q x2 q y 2
ð2Þ
!
ð3Þ
The usual form of definition of the stream function, w and vorticity x, is u ¼ qw
qy ,
qv
qu
v ¼ qw
qx , and x ¼ qx qy.
The boundary conditions for the problem under consideration are
1. on the left side wall:
T ¼ Tc ; w ¼ 0;
!
q2 w q2 w
x¼
þ
q x2 q y 2
2. on the right side wall:
qw
T ¼ Th ;
¼ v ¼ jqj sin U;
qx
!
q2 w q2 w
þ
x¼
q x2 q y 2
ð4Þ
3. on the top (y ¼ H) and on the bottom (y ¼ 0) walls:
qT
q2 w
¼ 0; w ¼ 0; x ¼ 2
qy
qy
where u and v are the velocity components along the axes x and y, respectively; q ¼ iu
þjv is the velocity vector of the moving wall (i and j being the unit vectors in the
x and y directions, respectively); g is acceleration due to gravity; n is the kinematic
viscosity; and b is the thermal expansion coefficient.
MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
317
By introducing the following nondimensional parameters:
X¼
x
;
H
y
w
u
; W¼
; U¼ ;
H
H j qj
j qj
T Tc
Hx
; X¼
H¼
DT
j qj
Y¼
V¼
v
j qj
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(where jqj is the magnitude of the velocity vector) into Eqs. (1)–(4), the final
nondimensional governing equations become
q2 W q2 W
þ
¼ X
q X2 q Y2
qW
qY
qW
qY
qX
qX
qW
qX
qH
qX
qW
qX
qX
qY
1
¼
Re
qH
qY
ð5Þ
q2 X
q2 X
þ
q X2 q Y2
1
¼
Pr Re
!
þ Ri
q2 H q2 H
þ
q X2 q Y2
qH
qX
ð6Þ
!
ð7Þ
with the dimensionless boundary conditions:
1. on the left side wall X ¼ 0:
q2 W q2 W
H ¼ 0; W ¼ 0; X ¼ þ
q X2 q Y2
!
2. on the right side wall X ¼ 1:
qW
q2 W q2 W
¼ sin U; X ¼ H ¼ 1;
þ
qX
q X2 q Y2
!
ð8Þ
3. on the top (y ¼ 1) and on the bottom (y ¼ 0) walls:
qH
q2 W
¼ 0; W ¼ 0; X ¼ qY
q Y2
where Pr ¼ n=a is Prandtl number, Ri ¼ Gr=Re2 is Richardson number, Gr ¼
gbDTH3=n2 is Grashof number, and Re ¼ jqj H=n is Reynolds number.
3. NUMERICAL FORMULATION
The governing partial differential equations set (5)–(7), along with the boundary conditions (8), were solved using the second-order finite difference method. In
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318
M. A. ISMAEL AND A. J. CHAMKHA
order to obtain a solution for this nonlinear set of equations, an iterative procedure
was followed. To overcome the problem of an unstable solution, the convective
terms appearing in the momentum and energy equations were treated by the upwind
scheme. A successive under relaxation (SUR) procedure was also used to speed up
the convergence of the solution with less computational effort.
When one uses the stream function–vorticity formulation, it is strongly recommended that the boundary conditions of the vorticity must be carefully selected.
However, in the present geometry, extreme caution is required because there is an
inclined moving wall. When we refer to the classical lid-driven square cavity with
top wall moving to the right, the formula for vorticity on the moving wall is given
by Jensen’s formula as cited in Fletcher [36]:
X0 ¼ 0:5
3K
ð8 W 1 W 2 Þ 2
DY
DY
ð9Þ
where subscript 0 refers to the points on the wall, 1 refers to the points adjacent to
the wall, and 2 refers to the second line of points adjacent to the wall and K:
qW
K¼
ð10Þ
qY 0
Implementing and extending this formula to verify the vorticity boundary conditions
on the trapezoidal cavity walls, we obtain the following formulas:
1. on the horizontal bottom wall:
X¼
8 Wi;jþ1 þ Wi;jþ2
2DY 2
ð11Þ
X¼
8 Wi;j1 þ Wi;j2
2DY 2
ð12Þ
8 Wiþ1;j þ Wiþ2;j 8 Wi;jþ1 þ Wi;jþ2
þ
2DX 2
2DY 2
ð13Þ
2. on the horizontal top wall:
3. on the inclined left wall (fixed):
X¼
4. on the inclined right wall (moving):
X¼
8 Wi1;j þ Wi2;j
3 sinðUÞ 8 Wi;jþ1 þ Wi;jþ2
3 cosðUÞ
þ
d
þd
2DX 2
2DY 2
DX
DY
ð14Þ
where d ¼ 1 for upward movement and d ¼ þ1 for downward movement of the
wall.
MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
319
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The Gauss–Seidel iteration procedure was employed in the numerical solution.
Between each successive iteration steps, the difference in the variables W, X, and H is
monitored. The iteration process is terminated when the maximum percentage difference in each variable satisfies the following:
nþ1
v ði; jÞ vn ði; jÞ 106
ð15Þ
max
vn ði; jÞ
where v denotes any variable – W, X, or H – and n is the iteration step.
The physical quantities of interest are the local and average Nusselt numbers
on the left and right inclined walls, NuL,R and Nuav, respectively, calculated on the
inclined heated side wall of the cavity. The local Nusselt number is defined as
Nu ¼
H Qw
k DT
ð16Þ
where heat transfer from the inclined right hot wall, Qw is given by
qT
Qw ¼ k
qn
Using Eqs. (16) and (17) and the nondimensional definitions, we get
qH
Nu ¼ qn
ð17Þ
ð18Þ
where on the left wall
qH qH
qH
¼
sin ðUÞ þ
cos ðUÞ
qn qX
qY
and on the right wall
qH
qH
qH
¼
sin ðUÞ þ
cos ðUÞ
qn
qX
qY
Hence, the average Nusselt number is given by
1
Nuav ¼ 1=sin U
1=sin
Z U
qH
dS
qn
ð19Þ
0
where dS is a segment along the inclined wall: dS ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dX 2 þ dY 2
3.1. Grid Independency and Validation
The solution within a trapezoidal cavity was achieved by a rectangular discretization, Nx Ny with the inclined walls adjusted to pass exactly through boundary
nodes (see Figure 2). When the inclination angle, U is chosen, the ratio of number of
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320
M. A. ISMAEL AND A. J. CHAMKHA
Figure 2. Numerical finite difference method (FDM) grid.
divisions is given by
tan U ¼
DY Nx 1
¼
DX Ny 1
ð20Þ
Three values of U (66 , 73 , and 80 ) were studied. Table 1 presents three
groups of mesh size for each value of U. The selected grid sizes (226 101,
Table 1. Grid independency test for aiding case, Ri ¼ 10, Re ¼ 100, and Pr ¼ 6.26
U
66
73
80
Grid size Nx Ny
Nuav along the left side wall
Nuav along the right side wall
64 21
68 31
91 41
113 51
136 61
158 71
181 81
203 91
226 101
248 111
66 21
99 31
132 41
165 51
197 61
230 71
263 81
114 21
171 31
228 41
285 51
341 61
398 71
455 81
8.150
9.330
10.004
10.409
10.689
10.913
11.078
11.200
11.306
11.43
9.150
10.084
10.590
10.899
11.080
11.218
11.200
10.213
10.897
11.012
11.156
9.700
10.100
10.150
6.750
7.960
9.030
9.730
10.208
10.586
10.855
11.045
11.213
11.400
7.220
8.823
9.780
10.370
10.740
11.009
11.220
8.490
9.720
10.133
10.807
11.780
11.790
12.380
MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
321
Table 2. Comparison to Tiwari and Das [20]: square cavity with two vertical sides being the lid. Pr ¼ 6.2,
clear water
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Case A
Ri (Re)
0.1 (316.23)
1 (100)
10 (31.623)
Case B
Tiwari and Das [20]
Present
% error
Tiwari and Das [20]
Present
% error
30.67
17.96
10.19
29.22
17.43
9.95
4.7
2.95
2.3
11.54
6.65
4.25
11.15
6.61
4.22
3.38
0.6
0.7
230 71, and 285 51) for U ¼ 66 , 73 , and 80 , respectively, are shown as the
shaded cells in Table 1. The criteria behind invoking these grid sizes are the convergence of Nusselt number values between successive mesh refinements and the time
required by the processor for each mesh size. The marked discrepancy noted in
the mesh refinement at 80 configuration is attributed to the relatively large ratio
of the discretization intervals of the X, Y coordinates (DX and DY, respectively). This
ratio, for boundary fit purposes, diverges from unity with increasing U as shown in
Eq. (20) (i.e., when the numerical cells become rectangular and elongated vertically).
However, when the mesh is refined excessively, we expect error propagation which
leads to oscillation in Nusselt number on the left wall. However, the convergence
of the numerical solution does not necessarily mean that the correct solution has
been obtained. Therefore, it is strongly recommended that the numerical code be
compared to results obtained by other numerical methodologies. Tiwari and Das
[20] solved a square cavity with fixed horizontal walls and moving vertical walls.
Two different cases of the direction of the vertical walls, both for clear fluid (with
no nanoparticles), were compared by setting U ¼ 90 with a grid size of 71 71.
The results are shown in Table 2, and they enhance our confidence in the results
of the present numerical treatments, which were coded using FORTRAN.
4. RESULTS AND DISCUSSION
The theory behind this study is based on fixing the Prandtl number at Pr ¼ 6.26
(water) and Reynolds number at Re ¼ 100. The pertinent parameters that are
expected to affect mixed convection inside such a trapezoidal cavity are the inclination angle of the side walls, U (66 , 73 , and 80 ) and Richardson number, Ri
(0.01, 0.1, 1, and 10), which emphasizes the importance of buoyancy-driven natural
convection relative to lid-driven forced convection. The ranges of U and Ri were
tested for two cases: aiding mixed convection (when the right hot side wall moves
upward) and opposing mixed convection (when the right hot side wall moves downward). The results are displayed in the following figures by streamlines, isotherms,
and local and average Nusselt numbers.
322
M. A. ISMAEL AND A. J. CHAMKHA
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Figures 3–5 depict contour maps of the streamlines and isotherms for the aiding case. The effects of the inclination side wall U for Ri ¼ 0.1 are presented in
Figure 3, which shows the single cell vortex formation of the streamlines. Due to
the imposed thermal boundary conditions together with the upward movement of
the right side wall, the direction of the vortex is counterclockwise, with congregated
streamlines close to the moving wall and some small perturbations appearing in the
Figure 3. Streamlines (left) and isotherms (right) for Ri ¼ 0.1 (aiding).
323
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MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
Figure 4. Streamlines (left) and isotherms (right) for Ri ¼ 1 (aiding).
lower part of the cavity. These perturbations are due to the mechanical friction
resulting from wall movement. The perturbations become more clear for U ¼ 80 .
Because of the predominance of forced over natural convection (Ri ¼ 0.1), the zone
on the left hand side of the cavity appears stagnant because it is distant from the
source of forced convection. From the values of maximum stream function labeled
M. A. ISMAEL AND A. J. CHAMKHA
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324
Figure 5. Streamlines (left) and isotherms (right) for Ri ¼ 10 (aiding).
on each streamline contour map, the vortex becomes stronger with increasing values
of U from 66 to 80 . The strengthening of the streamlines with U can be seen clearly
in the isotherms congregated close to the left side wall. The congregation of the isotherms can also be seen close to the lower part of the inclined right moving wall and
exactly within the perturbation regions of the streamlines. However, a substantial
zone in the middle of the cavity appears to be isothermal.
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MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
325
For equivalent natural and forced convection (Ri ¼ 1), the streamlines
(Figure 4) tend to be expanded throughout the cavity and hence minimize the stagnant zone observed for Ri ¼ 0.1 (Figure 3). Generally, single cellular behavior with
counterclockwise rotation still holds with a slight increase in vortex strength for
U 73 . No significant contrast can be seen for the isotherms except that at
U ¼ 80 (Figure 4) when they tend to be horizontal at the lower part of the cavity,
an indication of induced natural convection.
Figure 5 shows the predominance of natural convection over forced convection
(i.e., Ri ¼ 10). The streamlines are expanded to almost occupying the whole cavity,
with an extended core toward the lid-driven action. As can be seen from this figure
and especially when U ¼ 73 and 80 , the vortex core is very stretched in a manner
that predicts the possibility of breaking the vortex core into two, showing double-eye
cell behavior which is expected to be a feature of streamlines when Ri is increased to
100. However, the maximum vortex strength is lower than the two previous convection modes (Ri ¼ 0.1 and 1). The parallel isotherms banding close to the left fixed
wall decrease while the main feature within the cavity is the horizontal streamlines,
which is the result expected due to the predominance of natural convection.
The streamline and isotherm contour maps for the opposing lid-driven case are
shown in Figures 6–8. Figure 6 shows the behavior of the streamlines and isotherms
for Ri ¼ 0.1. It is clearly seen from this figure that the action of forced convection is
completely predominant, in which there is a single clockwise (negative stream function) rotation cell with the core being close to the moving inclined right wall. The
perturbations in this case are localized to the upper right corner. Due to the relatively
weak natural convection, the clockwise single vortex becomes stronger with increasing values of U, and this is attributed to the provision of more space in the lower part
of the cavity. As previously seen in the aiding lid-driven case, there is a stagnant zone
localized at the upper left part of the cavity. The isotherm contours indicate a thermal boundary layer close to the upper part of the moving right inclined side wall and
mostly horizontal lines in the upper part of the cavity. Another thermal boundary
layer is observed close to the lower part of the fixed left side wall. This layer becomes
thicker with increasing values of U.
For equivalent natural and forced convection modes (Ri ¼ 1), the streamlines
and isotherms are manifested by different appearances as shown in Figure 7. The
streamlines are formed as counter-rotating double vortices, a clockwise example
adjusting to the right moving side wall (due to forced convection) while the other
is counterclockwise (due to natural convection) and localized at the left part of
the cavity. Due to the extra space available by increasing the value of U, the counterclockwise vortex is elongated toward the bottom of the cavity, becomes stronger, and
tends to squeeze the clockwise vortex toward the right inclined moving wall. The
strength of the clockwise vortex increases when U is increased from 66 to 73 but
decreases again when it is further increased to 80 , because of the squeezing effect
of the counterclockwise vortex. Random distribution of the isotherms is recorded
with the thin thermal boundary layer close to the lower part of the fixed left inclined
wall (due to forced convection). This boundary layer is shifted toward the middle of
the bottom wall with increasing values of U. When Ri is increased to 10, the counterclockwise vortex becomes more predominant and occupies most of the cavity, and it
strongly squeezes the clockwise example to curtail it with a small portion adjacent to
M. A. ISMAEL AND A. J. CHAMKHA
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326
Figure 6. Streamlines (left) and isotherms (right) for Ri ¼ 0.1 (opposing).
the moving right wall, as shown in Figure 8. The strength of the predominant
vortex increases with increasing values of U, while the strength of the squeezed
vortex decreases slightly. The double-eye appearance of the dominant vortex
again appears with increasing values of U. Again, if Ri is set high (e.g., 100), the
double-eye counterclockwise vortex will appear. The isotherms are aligned along
327
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MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
Figure 7. Streamlines (left) and isotherms (right) for Ri ¼ 1 (opposing).
mostly horizontal lines within the core of the cavity due to the predominance of
natural convection, while close to the inclined side walls the isotherms are aligned
parallel.
M. A. ISMAEL AND A. J. CHAMKHA
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328
Figure 8. Streamlines (left) and isotherms (right) for Ri ¼ 10 (opposing).
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MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
329
Local Nusselt number distribution along both the left wall, NuL and the right
moving wall, NuR are plotted in Figure 9 for the aiding lid-driven case and for three
values of U. The sudden changes (like peaks) in the distribution of NuR (solid lines)
close to the lower right corner refer to the perturbations are indicated there. These
peaks (or perturbations) can be attributed to the singularities resulting from the
meeting of stationary and moving walls, and also to the fluid falling on the bottom
wall suddenly being driven by shear friction resulting from the moving wall. A close
examination of Figures 9a–9c indicates that the effect of U on the distribution of
NuR (solid lines) and NuL (dashed lines) is more significant when Ri < 10. For the
opposing lid-driven case (Figure 10), the peaks of NuR are transferred to the upper
right corner where streamline perturbation exists. This figure also indicates that
when Ri ¼ 10, the effect of U seems to be less significant. In general, due to the shear
effect imposed by the right moving wall, most disturbances of local Nusselt number
are indicated there.
The overall heat transfer represented by average Nusselt number on the right hot
moving inclined wall (Eq. 19) is depicted for the aiding lid-driven case in Figure 11,
and for the opposing lid-driven case in Figure 12. Figure 11 shows that the average
Nusselt number, Nuav increases according to different functions with Ri in the aiding
lid-driven case. Moreover, it is clear that the effect of U on Nuav is more significant at
low values of Ri. This is because the flow circulation is mainly due to the moving wall
(forced convection) and therefore the decrease in U results in a faster-moving boundary, and hence mechanical shear friction becomes the predominant force. When Ri is
increased, the effect of U is severely limited at Ri ¼ 10 (see Figure 11).
For the opposing lid-driven case, the behavior is completely different
(Figure 12), where Nuav for U ¼ 66 decreases with Ri, whereas for the other two
values of U, it decreases up to Ri ¼ 1 then increases. This behavior is attributed to
the fact that when Ri ¼ 0.1, convection is mainly due to the opposed moving boundary and when Ri is increased, natural convection is switched to compete with forced
convection, but in an opposing manner, to Ri ¼ 1. Beyond Ri ¼ 1, natural convection
begins to be predominant over the mechanical effect for U ¼ 73 and 80 , but for
U ¼ 66 the opposing effects of forced and natural convection still decrease the overall heat transfer. This is because of the large volume of fluid driven by the greater
surface available with U ¼ 66 . Generally, as mentioned in the aiding case, Nuav
increases with decreasing values of U. Apart from the above-mentioned reasons,
decrease in U means an increase in the amount of heat transferred to the adjacent
fluid which, in turn, increases overall heat transfer. However, it is worth mentioning
that the maximum values of Nuav are recorded at Ri ¼ 10 for the aiding case, and at
Ri ¼ 0.1 for the opposing case.
Finally, the least square method is used to derive a correlation between average
Nusselt number and Richardson number (Ri) and the sidewall angle (U) for both
aiding and opposing cases. For the aiding case, a power correlation is found to fit
the data with a correlation r-factor of 0.9336:
Nuav ¼ 190:078 Ri0:1113 U0:736
ð21Þ
For the opposing case, and due to the complex behavior of average Nusselt number
with Ri and U, the best fit correlation is found to be governed by a
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330
Figure 9. Local Nusselt number distribution (aiding).
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MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
Figure 10. Local Nusselt number distribution (opposing).
331
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332
M. A. ISMAEL AND A. J. CHAMKHA
Figure 11. Average Nusselt number over the right moving wall (aiding).
polynomial=simplified quadratic correlation with a correlation r-factor of 0.9559:
Nuav ¼ 7:5818 1:9918 Ri þ 0:1719 U þ 0:1786 Ri2 0:0025 U2
Figure 12. Average Nusselt number over the right moving wall (opposing).
ð22Þ
MIXED CONVECTION IN LID-DRIVEN TRAPEZOIDAL CAVITIES
333
5. CONCLUSIONS
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Mixed convection inside a trapezoidal cavity was studied numerically. The
horizontal walls of the cavity were kept adiabatic while the inclined side walls were
maintained isothermally at different temperatures. The right inclined wall was driven
either upward (aiding) or downward (opposing). The stream function–vorticity
formulation was invoked with careful boundary conditions on the inclined side
walls. The following concluding remarks were drawn from studying the effects of
Richardson number and the angle of the side wall inclination, for both aiding and
opposing cases.
1. The stream function–vorticity formulation, with careful choice of boundary conditions especially on the moving inclined side wall, demonstrated the possibility of
a numerical solution of such problems even with a basic finite difference method.
2. The opposite lid-driven case was characterized by the feature of a clockwise single
vortex rotation at very low Richardson number or by double cellular rotation
vortices for higher Richardson number (Ri > 0.1).
3. The inclination angle of the side walls played a key role in overall heat transfer at
lower Richardson number (Ri 1), while at higher Richardson number (Ri > 1.0)
the effect of U was negligible.
4. In the aiding lid-driven case, average Nusselt number was an increasing function
of Richardson number. Function speed was faster for higher values of the
inclination angle of the side walls.
5. In the opposing lid-driven case, average Nusselt number was a decreasing function of Ri for U ¼ 66 and a decreasing–increasing function for U ¼ 73 and 80 .
6. In general, Nusselt number was predicted to be higher for lower values of the
inclination angle of the side walls.
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