International Journal of Heat and Mass Transfer 52 (2009) 2824–2833

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier.com/locate/ijhmt

Visualization of heat flow due to natural convection

within triangular cavities using Bejan’s heatline concept
Tanmay Basak a, G. Aravind a, S. Roy b,*
a
Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India
b
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India

a r t i c l e i n f o a b s t r a c t

Article history: Natural convection and flow circulation within a cavity has received significant attention in recent times.
Received 15 April 2008 The wide range of applicability of flow inside a cavity (food processing industries, molten metal indus-
Received in revised form 6 October 2008 tries, etc.) requires thorough understanding for cost efficient processes. This paper is based on compre-
Available online 11 February 2009
hensive analysis of heat flow pattern using Bejan’s heatline concept. The key parameters for our study
are the Prandtl number, Rayleigh number and Nusselt number. The values of Prandtl number (0.015,
Keywords: 0.026, 0.7 and 1000) have been chosen based on wide range of applicability. The Rayleigh number has
Heatlines
been varied from 102 to 105. Interesting results were obtained. For low Rayleigh number, it is found that
Heatfunction
Inverted triangular cavity
the heatlines are smooth and perfectly normal to the isotherms indicating the dominance of conduction.
Streamlines But as Ra increases, flow slowly becomes convection dominant. It is also observed that multiple second-
Streamfunctions ary circulations are formed for fluids with low Pr whereas these features are absent in higher Pr fluids.
Natural convection Multiple circulation cells for smaller Pr also correspond multiple cells of heatlines which illustrate less
thermal transport from hot wall. On the other hand, the dense heatlines at bottom wall display enhanced
heat transport for larger Pr. Further, local heat transfer (Nul, Nut) are explained based on heatlines. The
comprehensive analysis is concluded with the average Nusselt number plots. A correlation for average
heat transfer rate and Ra has been developed and the range of Rayleigh number is also found, to depict
the conduction dominant heat transfer.
Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction ous applications. Poulikakos and Bejan [14] have carried out exten-
sive analysis on natural convection in an attic space. Holtzman
The relevance of buoyancy induced circulations causing trans- et al. [15] and Del Campo et al. [16] did numerical study of natural
port of heat and mass is significant in various physical systems. convection in triangular enclosures. Later, Kent et al. [17] and Omri
Especially, the applicability of natural convection inside triangular et al. [18] carried out numerical study on right-angled and isosce-
enclosure, in wide range of engineering processes from energy re- les triangular cavities, respectively. Varol et al. [19] did the study of
lated to geophysical and material processing industry is very well natural convection in a triangular enclosure with flush mounted
known [1–6]. In particular, convective heat transfer is widely used heater on the wall. Recently, Sieres et al. [20] carried out analysis
in material processing industries like food industries, molten salt of convection within a triangular enclosure for cavities with vari-
application (e.g. fuel cell technology), molten metal applications, able aperture. A few other recent investigations on natural convec-
etc. [7,8]. Various other studies involving sterilization, solidifica- tion within triangular cavities for various applications have been
tion of food, food separation processes and other natural convec- carried out by earlier researchers [21–26]. However, a comprehen-
tion based processes have also been reported by earlier sive analysis on natural convection flows in complex enclosures is
researchers [9–13]. The analysis of flow for such systems is impor- yet to appear in the literature. It is essential to study the heat
tant for a complete understanding of the problem. Numerical mod- transfer characteristics in complex geometries to obtain optimal
eling may be employed to understand and analyze these systems. design of the processes for improving the product quality.
The advantage of numerical simulations is that the expensive Although a number of numerical investigations [17–26] has
experimental costs can be reduced. been carried out in triangular cavities, the detailed analysis of heat
Numerical and experimental studies on natural convection in flow was poorly understood. The motivation for this work arises
triangular cavities have received significant attention due to vari- from the fact that there is a lack on visualization of heat flow to
analyze the optimal thermal mixing and temperature distribution
* Corresponding author. within triangular enclosures. In view of various applications of
E-mail addresses: tanmay@iitm.ac.in (T. Basak), sjroy@iitm.ac.in (S. Roy). thermal processes, a comprehensive understanding of heat transfer

0017-9310/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijheatmasstransfer.2008.10.034
 T. Basak et al. / International Journal of Heat and Mass Transfer 52 (2009) 2824–2833 2825

Nomenclature

g acceleration due to gravity (m s2) Greek symbols

k thermal conductivity (W m1 K1) a thermal diffusivity (m2 s1)
L height of the triangular cavity (m) b volume expansion coefficient (K1)
N total number of nodes c penalty parameter
Nu Nusselt number C boundary
Nu average Nusselt number h dimensionless temperature
p pressure (Pa) m kinematic viscosity (m2 s1)
P dimensionless pressure q density (kg m3)
Pr Prandtl number U basis functions
R residual of weak form w dimensionless streamfunction
Ra Rayleigh number P dimensionless heatfunction
T temperature (K)
Th temperature of hot inclined wall (K) Subscripts
Tc temperature of cold top wall (K) i residual number
u x component of velocity (m s1) k node number
U x component of dimensionless velocity l left wall
v y component of velocity (m s1) r right wall
V y component of dimensionless velocity t top wall
X dimensionless distance along x-coordinate
x distance along x-coordinate (m)
Y dimensionless distance along y-coordinate
y distance along y-coordinate (m)

and flow circulations within triangular cavities is very much essen- tions. Typically, Pr = 0.015 corresponds to molten metals and
tial for industrial development. Current work attempts to analyze Pr = 988.24 corresponds to olive oil. Non-orthogonal grid genera-
heat transfer, correlations and energy distributions using heatline tion is done with iso-parametric mapping as given in Appendix A.
approach.
The heatline concept was first introduced by Kimura and Bejan 2. Governing equations and solution procedure
[27] and Bejan [28]. Heatline is the best tool to analyze and under-
stand the heat flow in 2D convective transport processes and this 2.1. Momentum and energy formulation
concept is similar to streamline which is important to analyze fluid
motion. Heatlines represent heatflux lines which represent the tra- The fluid properties are assumed to be constant except the den-
jectory of heat flow and they are normal to the isotherms for con- sity in the body force term which was determined according to the
ductive heat transfer. It may be noted that heatfunctions are Boussinesq approximation. This approximation is used in the field
mathematical representations of heatlines and each heatline con- of buoyancy driven flows and it is based on the fact that density in
tour corresponds to constant heatfunction. Various applications the body force term varies linearly with temperature. Under these
using heatlines were studied by Bello-Ochende [29], Costa [30– assumptions, the governing equations for steady two dimensional,
33], Mukhopadhyay et al. [34,35] and Deng and Tang [36]. Re- laminar, incompressible flows can be written in dimensionless
cently, Dalal and Das [37] have used heatline method for the visu- form as:
alization of flow in a complicated cavity. However, a
oU oV
comprehensive analysis on heat flow during natural convection þ ¼ 0; ð1Þ
in a triangular cavity with the heatline approach is yet to appear oX oY
in the literature. !
The aim of this paper is to study the circulations and tempera- oU oU oP o2 U o2 U
U þV ¼ þ Pr þ ; ð2Þ
ture distribution and to analyze the flow of heat due to natural oX oY oX oX 2 oY 2
convection in an isosceles right angled inverted triangular enclo-
sure with an aspect ratio of 2:1, involving hot inclined walls and Y, V
cold top wall. The geometry of this enclosure with boundary con-
ditions is shown in Fig. 1. Numerical results are presented in terms
of isotherms, streamlines and heatlines along with the local and
average heat transfer rates. Galerkin finite element method with
penalty parameter has been used to solve the non-linear coupled
partial differential equations of flow and temperature fields. To = 0
solve the Poisson equation for streamfunctions and heatfunctions, C B g
Galerkin method is also used. It may be noted that Galerkin meth-
od has been used to evaluate heatfunction for the first time in this
work. The jump discontinuity in Dirichlet type of wall boundary
=1 =1
conditions for temperature at the corner points correspond to com-
putational singularities. This problem is tackled by considering the
average temperature of the two walls at the corner and keeping the
adjacent grid nodes at the respective wall temperature similar to O X, U
A
earlier works. We have considered Prandtl number from low to
high range (0.015–1000) for fluids of various industrial applica- Fig. 1. Schematic diagram of the physical system with the boundary conditions.
2826 T. Basak et al. / International Journal of Heat and Mass Transfer 52 (2009) 2824–2833

!
oV oV oP o2 V o2 V o2 w o2 w oU oV
U þV ¼ þ Pr þ þ Ra Pr h; ð3Þ 2
þ ¼  : ð12Þ
oX oY oY oX 2 oY 2 oX oY 2 oY oX
The sign convention is that, positive sign of w denotes anti-
oh oh o2 h o2 h clockwise circulation and clockwise circulation is represented by
U þV ¼ þ ; ð4Þ
oX oY oX 2 oY 2 negative sign of w. The no-slip condition is valid at all boundaries
as there is no cross flow, hence w = 0 is used for boundaries.
where
Streamfunctions corresponding to fluid velocities are obtained
x y uL vL T  Tc using finite element method as discussed earlier [26,38,39].
X¼ ; Y¼ ; U¼ ; V¼ ; h¼
L L a a Th  Tc The heat flow within the enclosure is displayed using ohthe oh
heat-

pL2 m gbðT h  T c ÞL3 Pr function (P) obtained from conductive heat fluxes  oX ;  oY as
P¼ ; Pr ¼ ; Ra ¼ : ð5Þ well as convective heat fluxes (Uh, Vh). The heatfunction satisfies
qa 2 a m2
the steady energy balance equation [Eq. (4)] [27] such that
In Eq. (5), X and Y are the dimensionless distances along x- and
oP oh
y-coordinate, respectively, L is vertical depth of the cavity, i.e. per- ¼ Uh  ;
pendicular distance of the bottom corner from top horizontal wall,
oY oX ð13Þ
oP oh
U and V are the corresponding velocity components along the coor-  ¼ Vh  ;
oX oY
dinate axes, P denotes the dimensionless pressure whereas Pr and
Ra denote Prandtl number and Rayleigh numbers, respectively. No which yield a single equation
slip conditions are assumed at all the walls and the boundary con-
o2 P o2 P o o
ditions for the velocity components are 2
þ ¼ ðUhÞ  ðVhÞ: ð14Þ
oX oY 2 oY oX
UðX; 1Þ ¼ VðX; 1Þ ¼ 0 on BC;
The sign convention for heatfunction is as follows. The positive
UðX; YÞ ¼ VðX; YÞ ¼ 0 on AC; ð6Þ sign of P denotes anti-clockwise heat flow and clockwise heat flow
UðX; YÞ ¼ VðX; YÞ ¼ 0 on AB: is represented by negative sign of P. Heatfunctions are obtained
via finite element method similar to the procedure for evaluation
The boundary conditions for temperature are
of streamfunctions.
hðX; 1Þ ¼ 0 on BC; In order to obtain an unique solution of Eq. (14), following
hðX; YÞ ¼ 1 on AC; ð7Þ boundary conditions are implemented. Neumann boundary condi-
tions for P are obtained due to isothermal (hot or cold) wall based
hðX; YÞ ¼ 1 on AB:
on Eq. (13) and for isothermal (hot or cold) wall
The continuity equation (1) is used as a constraint due to mass
n  rP ¼ 0: ð15Þ
conservation and this constraint can be used to obtain the pressure
distribution. The momentum and energy balance equations [Eqs. The following are the Dirichlet boundary conditions:
(2)–(4)] are solved using Galerkin finite element method. In order pffiffiffi
P¼ 2Nul at X ¼ 0; Y ¼ 1; ð16Þ
to solve Eqs. (2) and (3), penalty finite element method has been
employed to eliminate the pressure P with a penalty parameter c
P ¼ 0 at X ¼ 1; Y ¼ 0; ð17Þ
and the incompressibility criteria given by Eq. (1) via following
relationship: and
  pffiffiffi
oU oV P ¼  2Nur at X ¼ 2; Y ¼ 1: ð18Þ
P ¼ c þ : ð8Þ
oX oY It may be noted that, Nul and Nur are average Nusselt numbers
Typically c = 107 yields consistent solutions. Applying Eq. (8), at the left and right walls, respectively. The details on evaluation of
the momentum balance equations [Eqs. (2) and (3)], are reduced to Nusselt numbers are discussed next.
  ! The heat transfer coefficient in terms of local Nusselt number
oU oU o oU oV o2 U o2 U (Nu) is defined by
U þV ¼c þ þ Pr þ ; ð9Þ
oX oY oX oX oY oX 2 oY 2 oh
Nu ¼  : ð19Þ
and on
  ! Here n denotes the normal direction of the plane. The local Nus-
oV oV o oU oV o2 V o2 V selt numbers at top wall(Nut), left wall(Nul) and right wall(Nur) are
U þV ¼c þ þ Pr þ þ RaPrh: ð10Þ
oX oY oY oX oY oX 2 oY 2 defined as

The system of equations [Eqs. (9), (10) and (4)] with appropriate X
9
oUi
Nut ¼  hi ; ð20Þ
boundary conditions are solved using Galerkin finite element oY
i¼1
method as discussed in earlier references [26,38,39].  
X9
1 oUi 1 oUi
Nul ¼ hi pffiffiffi þ pffiffiffi ; ð21Þ
i¼1 2 oX 2 oY
2.2. Streamfunction and heatfunction
and
The fluid motion is displayed using the streamfunction, w,  
X9
1 oUi 1 oUi
obtained from velocity components U and V. The relationships Nur ¼ hi  pffiffiffi þ pffiffiffi : ð22Þ
between streamfunction, w and velocity components for i¼1 2 oX 2 oY
two-dimensional flows are
The average Nusselt numbers at the top and side walls are
ow ow R2 Z
U¼ and V ¼  ; ð11Þ Nut dX 1 2
oY oX Nut ¼ 0
¼ Nut dX ð23Þ
R2
dX 2 0
which yield a single equation 0
 T. Basak et al. / International Journal of Heat and Mass Transfer 52 (2009) 2824–2833 2827

and number with distance and Rayleigh number, and also, variation
pffiffi of average Nusselt number vs. Rayleigh number have been shown
Z 2
1 to illustrate heat transfer rates. Detailed explanation is given in
Nul ¼ Nur ¼ pffiffiffi Nul dS: ð24Þ
2 0 various succeeding sections.
Here dS denotes the elemental length along inclined sides of the
triangular cavity. 3.2. Isotherms, streamlines and heatlines

3. Results and discussion Due to the temperature gradient imposed by hot side walls and
cold top wall and the buoyancy force, hot (lighter) fluid tends to
3.1. Numerical tests move near the top wall and cold (heavier) fluid tends to move to-
wards bottom. Hot fluid along the inclined walls moving towards
The computational domain in n–g-coordinates (see Appendix A) the top wall and cold fluid from the center of top wall tending to
consists of 20  20 bi-quadratic elements which correspond to move towards the bottom wall lead to two oppositely circulating
41  41 grid points. Note that, the computational grid in the trian- rolls in the system. It is observed that the symmetric flow and tem-
gular domain is generated via mapping the triangular domain into perature patterns occur for Ra = 102–105 with all representative Pr
square domain in n–g-coordinate system as shown in Fig. 2 and the values based on symmetric thermal boundary conditions at in-
procedure is outlined in Appendix A. The bi-quadratic elements clined side walls. Based on the physical systems, the left half of
with lesser number of nodes smoothly capture the non-linear vari- the axis of symmetry gives clockwise circulation pattern. The sym-
ations of the field variables which are in contrast with finite differ- metric solutions for the parameter ranges have been obtained
ence/finite volume solutions available in the literature [22]. based on solutions of governing equations within the entire do-
In the current investigation, Gaussian quadrature based finite main. The non-symmetric solutions even with some specific sym-
element method provides the smooth solutions at the interior do- metric boundary conditions in rectangular domains were found for
main including the corner regions as evaluation of residuals de- some other parameter ranges [40] and the investigations on non-
pends on the interior Gauss points and thus the effect of corner symmetric solutions for the triangular domains are the subject of
nodes are less profound in the final solution. In general, the Nusselt future research.
numbers for finite difference/finite volume based methods are cal- Firstly, flow and thermal dynamics for Pr = 0.015 are reported
culated at any surface using some interpolation functions which for various Rayleigh numbers (see Figs. 3–5). At low Rayleigh num-
are now avoided in the current work. The present finite element ber (Ra = 102), the isotherms are smooth and monotonic and the
method based approach offers special advantage on evaluation of magnitude of streamlines are quite small (see Fig. 3a and b). This
local Nusselt number at the left, right and top walls as the element shows that at small Ra, heat transfer is mostly conduction domi-
basis functions have been used here to evaluate the heatflux [39]. nant. The isotherms span the entire enclosure and they are sym-
Our simulation studies on isotherm and streamline have also been metric with respect to the vertical center line.
compared with earlier studies [22] and the results are in well The heatlines are constructed based on heat flux boundary con-
agreement. In this study, Prandtl number is varied from 0.015 to ditions and the corner edges of the top wall are maintained at aver-
1000 covering wide range of applications. Also, Rayleigh number age Nusselt number ðP ¼ Nul Þ as the bottom edge is maintained at
effects with Ra = 102–105 have been studied. Variation of Nusselt P = 0. Therefore, the large values of P at the edges of the top wall

a η
Y

Mapping

X ξ

b Y η

6
3 9

3 6 9
2 8
5
Mapping
2 5 8

1 4 7

X 1 ξ
4 7
Global co−ordinate system Local co−ordinate system

Fig. 2. (a) The mapping of triangular domain to a square domain in n–g-coordinate system and (b) the mapping of an individual element to a single element in n–g-coordinate
system.
2828 T. Basak et al. / International Journal of Heat and Mass Transfer 52 (2009) 2824–2833

TEMPERATURE, STREAMFUNCTION,
0.1 .001 −0 0.001
0.2 −0 .00
8 3 05
0.4 0.3
−0 .00 0.0 0.0
1
0.5
0.6

1
.0
0.7 08

−0
0.8
−0
.00 0.0
5 0 3
0.0
0.9
0.95

HEATFUNCTION,

−1.4
1.2

−0.6 −0
0.9

−0.4 −0.3

−0.
0 .6

−0.2
0 .4

7
.5
−0.1
0.2
0.1

−0.04
0.04
0
Fig. 3. (a) Temperature (h), (b) streamfunction (w), and (c) heatfunction (P) contours with Pr = 0.015 and Ra = 102. Clockwise and anti-clockwise flows are shown via negative
and positive signs of streamfunctions and heatfunctions, respectively.

TEMPERATURE, STREAMFUNCTION,
−0.1

0.01
0.1
−0.

0.2 0.3
−0.1

0.1
6
−0.
01

0.6 3 1
0. 0.7 −1

1.
.3

6
0.4
0.8

−1.
0.5

1.3
6

−1

−0 0.6
.3
0.1
0.9

HEATFUNCTION,
−0.

−1.7
1.7

0.8

−0.8
2
0.

−0
−1.3
1.3

.1 1
0. 0.4
1

−1

−0.4
0.6
.6

0.
−0

4
2
−0. 04 0.1
−0 .

Fig. 4. (a) Temperature (h), (b) streamfunction (w), and (c) heatfunction (P) contours with Pr = 0.015 and Ra = 104. Clockwise and anti-clockwise flows are shown via negative
and positive signs of streamfunctions and heatfunctions, respectively.

are mainly due to the cold wall being directly in contact with hot and the deformation is due to the presence of significant convec-
inclined walls. The basis of the sign convention is that heat flows tion in the system (Fig. 4a). It is also observed that the intensity
from hot to cold wall and the positive heatfunction corresponds of buoyancy driven circulations inside the cavity increases as seen
to anti clockwise heat flow. Fig. 3c illustrates that the heat flow oc- from greater strength of streamfunctions. Intensity of circulations
curs mainly due to the conduction as the heatlines are nearly per- are greater near the center and least at the wall due to no slip
pendicular to the isotherm lines as well as the walls. An important boundary conditions. Secondary circulations are also developed
point to note is that the heatlines with greater strength are clus- near the intersection of inclined walls with the cold top wall. The
tered around the top portion of the inclined wall and as we move heatlines illustrate that, convection dominant effect plays critical
down along the inclined wall, the strength of heatline goes to as role on larger heat to flow from the bottom portion of inclined
low as 0.04. This means that major amount of heat flux or transport walls to the top wall (Fig. 4c). It is observed that the heatlines
occurs near the cold wall. Thus, relatively less heat flow occurs are less dense near the top corner points where heat transport is
from the bottom edge of the enclosure. It is also illustrated that conduction dominant as the intensity of fluid circulation is less
the top cold wall receives most of the heat from the upper half as seen from streamfunctions. Although the infinite heat transfer
of hot inclined walls during conduction dominant heat transfer. occurs at the top corner points, the heatlines are less dense near
As Rayleigh number is increased to 104 (Fig. 4a–c), isotherms those points due to absence of convection. It is interesting to ob-
tend to deform but they are symmetric to the vertical central line serve that the heatlines are quite dense near the bottom portions
 T. Basak et al. / International Journal of Heat and Mass Transfer 52 (2009) 2824–2833 2829

TEMPERATURE, STREAMFUNCTION,
0.1 1 0.1

−0.
0.1 −1

−0
0.2 0.3

−0.3
0.3

.1
−2.5
0.5 1 2.5

−1
−2.5
0.9

0.8
−6

0.6

4
2.5
0.7 −1

0.95

HEATFUNCTION,
2.3 −2.3

−0.1
0.4

1.6

−1.6
.1
.4 −0
−0.1
0.1
1 −1

−0
−2.3
.5
−3

3.
5
2.5
−0.4
0.1

Fig. 5. (a) Temperature (h), (b) streamfunction (w), and (c) heatfunction (P) contours with Pr = 0.015 and Ra = 105. Clockwise and anti-clockwise flows are shown via negative
and positive signs of streamfunctions and heatfunctions, respectively.

of inclined walls and that specifies more heat transfer from the re- from the bottom portion of hot walls and thus enhanced convec-
gime near the bottom corner to the top cooled wall. Also, it is seen tive heat transfer occurs as seen from dense heatline contours.
that the shape of heatlines near the core is identical to that of Therefore, the temperature gradient is less due to enhanced ther-
streamlines signifying the convection dominant heat flow due to mal mixing near bottom potion of the central regime. In contrast,
large intensity of circulations (large values of streamfunctions). the heatlines were found to be less dense near the horizontal wall
Therefore, the temperature gradients as seen from Fig. 4a are less where the heat transport is mainly due to conduction.
near the center of each half due to large heat distribution resulting Next, the effect of change in Prandtl number has been investi-
from convective heat transfer. gated. Fig. 6 illustrates distribution for Pr = 0.026 and Ra = 105. As ex-
As Rayleigh number is increased to 105 (Fig. 5a–c), multiple and pected the qualitative trends in flow and thermal characteristics are
stronger circulations appear and those result in more deformations identical to those in Fig. 5a–c. However, we do see that the number of
in the isotherms. It may be noted that the magnitude of stream- multiple circulations as illustrated by the streamlines has slightly
functions are larger for Ra = 105 signifying the larger intensity of been decreased especially towards the corners of the top wall. The
circulations. Multiple circulations greatly influence the heatline heatlines show identical quantitatively features for Pr = 0.015 and
patterns and heat distributions as seen in Fig. 5c. Similar to previ- Pr = 0.026 (Figs. 5c and 6c). The interesting comparison may be illus-
ous case (Fig. 4), the top portions of the inclined hot walls do not trated as Prandtl Number is increased to 0.7 (Fig. 7c). Streamlines
distribute much heat to the cold wall. Note that, the intensity of illustrate that for Pr = 0.7, all the multiple circulations that were ob-
fluid circulations is found to be much stronger near the bottom served at Pr 6 0.026, have been totally absent (Fig. 7b). The stream-
corner for Ra = 105. This strong convection cell distributes heat lines are now elliptical towards the center and smooth triangular

TEMPERATURE, STREAMFUNCTION,
0.1 −0.1 −2 0.1 −0.1 1
1

0.2 .5 .3
0.

0.3
0.4 −0
2.5

5 −1 −1 1
0.
−4

−0
4
−6

.1
6

0.6 2.
5
0.7

−0
0.8 .3
0.95

HEATFUNCTION,
2.5
−0.1

−2
4
1.6

−0
0.4

−0.

.4
1 0.1 −0
−0.
4

−0.4 .4
0.

1.2
−3
−1.2

1.2
.5
3.5

−3
1.2
−0.4
0.1

Fig. 6. (a) Temperature (h), (b) streamfunction (w), and (c) heatfunction (P) contours with Pr = 0.026 and Ra = 105. Clockwise and anti-clockwise flows are shown via negative
and positive signs of streamfunctions and heatfunctions, respectively.
2830 T. Basak et al. / International Journal of Heat and Mass Transfer 52 (2009) 2824–2833

TEMPERATURE, STREAMFUNCTION,
0.1 −0.1 −1 0.1
2.5 10
−10

0.6
0.8

0.3
0.7

0.3

15
−15
−12

0.4

12
0.5

8
−4
−1

1
0.9

HEATFUNCTION,
−0.4

−3.5
3.5
1.2

2.5

−2.5
−1.

−3
2 2.5
−3

−6

6
−5

5
−1
.2
0.4

Fig. 7. (a) Temperature (h), (b) streamfunction (w), and (c) heatfunction (P) contours with Pr = 0.7 and Ra = 105. Clockwise and anti-clockwise flows are shown via negative
and positive signs of streamfunctions and heatfunctions, respectively.

curves near the walls. The intensity of the flow has been tremen- for Pr = 0.7, 1000. It is interesting to observe that heat transport in a
dously increased. The strength of the streamfunction has been in- large regime at the core is due to convection. The large regime of
creased to almost double near the center regime. The heatlines are convection is due to the large amount of heat transport from the
similar to streamlines at the core signifying convective heat flow inclined walls associated with large intensity of circulations (Figs.
and a large amount of heat flow occurs from the bottom portion of 7c and 8c). The convective heat transport has been suppressed
inclined wall as seen from dense heatlines. for lower Pr due to the presence of multiple circulations.
Fig. 8a–c illustrates the profiles for large Prandtl number
(Pr = 1000) with Ra = 105. It is noteworthy to mention that Figs. 7 3.3. Heat transfer rate: Nusselt numbers
and 8 have almost the similar trend for isotherms, streamlines
and heatlines. The streamlines become smooth elliptical towards Fig. 9a shows the variation of heat transfer rate (Nur, Nul) along
the corner points of the top wall. All the multiple circulations that the inclined wall. A wavy distribution pattern for the heat transfer
were seen in previous cases (Figs. 5b and 6b) have been completely rate is observed for Pr = 0.015 and Ra = 105. The heat transfer rate
disappeared into a single circulation with a central core. The is quite small till the distance being 0.2, as this region is near the
strength of streamfunction at the central regime is larger com- intersection of hot walls, resulting in low heat transfer between
pared to the previous cases. Identical trend in heatlines is observed the fluid. Thereafter heat transfer rate steadily increases until the
based on Figs. 5 and 6c and Figs. 7 and 8c. The absence of multiple distance being 0.4. The same pattern is observed again after distance
heat circulations in the system is observed and a very intense heat 0.6 until up to 0.8. The wavy pattern occurs due to the multiple cir-
flow occurs across the inclined walls represented by dense heatline culations cells with low Prandtl numbers for Ra = 105 (see Figs. 5b

TEMPERATURE, STREAMFUNCTION,
−0.1 −1 0.1
0.2 −2.5 2.5
0.8 −6 −10 8
0.7

12
0.3

0.
6

−15
0.5

15
−1

10
0.4

0
2

−1
−6 4
−1 1
0.9

HEATFUNCTION,
−0
.8
−3.5

−0.
5

−2

4
2.

−1.2 1.6
.5
3.5

.5 3.5
−3 5
−6

−5
2.5
.6
−1

−0.8
0.4

Fig. 8. (a) Temperature (h), (b) streamfunction (w), and (c) heatfunction (P) contours with Pr = 1000 and Ra = 105. Clockwise and anti-clockwise flows are shown via negative
and positive signs of streamfunctions and heatfunctions, respectively.
 T. Basak et al. / International Journal of Heat and Mass Transfer 52 (2009) 2824–2833 2831

Inclined wall Top Wall Inclined Wall

Local Nusselt Number, Nur or Nul

7.5

Average Nusselt Number, Nut

Average Nusselt Number, Nul

25
8
20 6
Pr=0.7, Ra= 105
Pr=0.015, Ra= 105
15 4.5 Pr=1000
6 Pr=1000

10 Pr=1000, Ra= 105

3
Pr=0.015
5 Pr=0.015
4 1.5
0 103 104 105 103 104 105
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Distance Rayleigh Number, Ra Rayleigh Number, Ra

Fig. 10. Variation of average Nusselt number with Rayleigh number for (a) top wall
Top Wall and (b) inclined wall for Pr = 0.015 and 1000.

25
Local Nusselt Number, Nut

lowest at the center attributed by the less dense heatlines (Figs.

20 7c and 8c).
Pr=0.7, Ra= 105 Fig. 10a and b shows the distributions of the average Nusselt
15 number of top and inclined walls, respectively, vs. the logarithmic
Pr=1000, Ra= 105 Rayleigh number. The average Nusselt number is obtained via
10 Pr=0.015, Ra= 105 Simpson’s one third rule [see Eqs. (23) and (24)]. General observa-
tion is that the average Nusselt number increases with Rayleigh
5 numbers. Fig. 10a illustrates that the average Nusselt number re-
mains almost constant till Ra = 4  103 for Pr = 1000. Thereafter
0 the average Nusselt number for Pr = 1000 increases rapidly to
0 0.3 0.6 0.9 1.2 1.5 1.8 reach a very high value at Ra = 105 whereas for Pr = 0.015, the aver-
Distance, X
age Nusselt number does increase but rate of the increase is very
Fig. 9. Variation of local Nusselt number with distance for various Pr low. The smaller variation for Nut is due to secondary and multiple
(0.015, 0.7, 1000) with Ra = 105 at (a) inclined wall and (b) top wall. circulations and heat transfer in Pr = 0.015 (see Fig. 5c) compared
to single symmetric highly intense circulation in Pr = 1000 (see
Fig. 8c). Similar pattern is observed in Fig. 10b for the inclined
and 6b). Each circulation cell is characterized by an intense central
walls. Basedpffiffiffi
on energy balance, the average Nusselt number of
core region and a less intensified outer region. As a result, the heat
top wall is 2 times that of the inclined wall. This result is well
transfer starts to decrease at the zone of intersection of two circula-
matched in all our cases verifying the energy conservation within
tion cells. The heatlines are also less dense at the zone between two
the system. A correlation has been developed for a generalized
circulation cells (Fig. 5c). In contrast, the absence of wavy pattern for
relationship between Nut or Nul and Ra in a convection dominated
Nusselt number with higher Prandtl numbers (Pr = 0.7  1000) is
regime. A correlation for Nut or Nul and Ra was observed for
due to absence of multiple circulations. A local maxima of Nul or
Pr = 0.7–1000, however, correlation could not be obtained for
Nur occurs at the distance being 0.3 for Pr = 0.7 and 1000 and this lo-
Pr = 0.015 as the overall heat transfer rate is small and conduction
cal maxima is attributed to the dense heatlines as seen in Figs. 7c and
dominant regime is observed at higher Ra. The limit of Ra is ob-
8c. After distance being 1, heat transfer rate increases rapidly and at
served as 4  103 for Pr = 1000 and convection is found to be signif-
the edge, the rate is infinite, i.e. at the point of intersection of cold and
icant for Ra P 4  103 and the following correlation is obtained:
hot walls for all ranges of Pr. The heat transport near the top corner pffiffiffi
regimes is predominantly by conduction and thereafter there is a Nut ¼ 2Nul ¼ 1:8332Ra0:1288 ; Pr ¼ 1000; 4  103 6 Ra 6 105 :
sharp fall of Nul or Nur from the top corner along the inclined wall. ð25Þ
The sharp decrease of Nul or Nur is also attributed to the less dense
heatlines as seen in Figs. 5c, 6c, 7c and 8c.
Fig. 9b shows the variation of Nusselt number (Nut) with the 4. Conclusion
distance along the top wall. It may be noted that at X = 0 and
X = 2 the heat transfer rate is infinite at all ranges of Pr. This is ex- The objective of this paper is to understand a physical as well as
pected at these points, as the hot wall intersects the cold wall computational insight due to heat flow for natural convection within
exhibiting maximum heat transfer. The heat transfer rate sharply a complex enclosure. The system considered here is an inverted tri-
falls when distance is further increased from X = 0 till X = 0.3 and angular cavity which has wide range of applicability in industries as
heat transfer rate decreases thereafter till X = 0.6 especially for discussed earlier. The key controlling parameters for our analysis are
Pr = 0.015. This is due to the multiple circulation cells for Rayleigh number and Prandtl number which govern the overall heat
Pr = 0.015 (see Fig. 5b). In addition, the less heat transfer rate is transfer rate, i.e. Nusselt number. The motivation is to understand
due to highly dispersed heatlines, as seen in Fig. 5c. The heat trans- the effect of each of these parameters on the heat flow process. In
fer rate remains almost constant and low till X = 1.4 for Pr = 0.015. addition, the values of Prandtl numbers (0.015, 0.7, 1000) have been
It is observed that, the heatlines concentrate within X 6 0.3 and chosen such that the system depicts wide range of commonly used
heatlines are well dispersed within 0.6 6 X 6 1.4. The qualitative applications. The visualization of heat flow inside any cavity is
trend is nearly the same for Pr = 0.7, 1000 with Ra = 105. It is impor- incomplete unless we know about the heat flow and hence we have
tant to note that Nut is larger except at the central regime for high- introduced the heatlines concept in the triangular cavity, which en-
er Pr due to dense heatlines with enhanced circulation cells (Figs. ables us to understand the heat flow trajectory. During conduction
7c and 8c) whereas, the heat transfer rate (Nut) for Pr = 0.015 is dominant heat transfer, it is observed that the isotherms, stream-
2832 T. Basak et al. / International Journal of Heat and Mass Transfer 52 (2009) 2824–2833

lines and heatlines are found to be monotonic and smooth curves. The above basis functions are used for mapping the triangular
Also, the heatlines will be perfectly normal to the isotherm during domain or elements within the triangle into square domain and
conduction dominant heat transfer. However, as Ra increases to the evaluation of integrals of residuals.
104, convection is initiated, and the flow patterns get distorted with
initiation of secondary circulation cells. The strong dominance of
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