A Mixed Convection Study in Inclined Channels with Discrete Heat Sources

19
plays a considerable role on the second module heating. For instance, what has just been
said happens in the case where Re = 1000 and d = 1, 2, and 3.

Gr =10
3
γ (degree)
0 20406080
Nu
1
2
3
4
5
6
7
Re=1
Re=5
Re=10
Re=50
Re=100

(a)
Gr = 10
4
γ (degree)
020406080
N

8
9
10
11
12
Re = 1
Re = 5
Re = 10
Re = 50
Re =100
Re = 200
Re = 500

(c)
Fig. 12. Average Nusselt vs γ: 1 ≤ Re ≤ 500 and (a) Gr = 10
3
, (b) Gr = 10
4
, and (c) Gr = 10
5


A Mixed Convection Study in Inclined Channels with Discrete Heat Sources

21

0
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2
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.
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1
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.
0
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0
.
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2
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.
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4
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0
.
0
4
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0
.
0
4
5
0
.
0
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0.045

0
.
0
8
9
0
.
0
8
9
0
.
1
1
2
0
.
1
3
4
0
.
1
5
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0
.
1
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7

0
0
.
0
3
9
0.059
0
.
0
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9
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0
.
0
9
8
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.
1
1
8
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.
1
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5
0
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6
5
0
.
0
8
2
0
.
0
9
8
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1
1
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1
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1
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3
8
0
.
0
5
3Fig. 14. Isotherms for Re = 1, 10, 100, 1000, γ = 0° , and Gr = 10
5
, Δθ = 0.02
Re = 1
d = 1
Re = 10
d = 1
Re = 50
d = 1
Re = 1
d = 2
Re = 1000
d = 1
Re = 100
d = 1
Re = 100
d = 2
Re = 50
d = 2
Re = 10
d = 2

d=3
Gr=10
3
- Heater 2
1 10 100 1000
Nu
0
2
4
6
8
10
12
d=1
d=2
d=3

Gr=10
4
- Heater 1
1 10 100 1000
0
2
4
6
8
10
12
14
d=1

14
d=1
d=2
d=3
Gr=10
5
- Heater 2
Re
1 10 100 1000
NU
0
2
4
6
8
10
12
d=1
d=2
d=3

Fig. 15. Nu for Re = 1, 10, 10
2
, 10
3
, d = 1, 2, 3, Gr = 10
5
, 10
4
, and 10

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources

23
Re = 100, Gr = 10
5
Heater 1
θ
0,00
0,05
0,10
0,15
0,20
0,25
d = 1
d = 2
d = 3
Re = 1000, Gr = 10
5
Heater 1
θ
0,00
0,05
0,10
0,15
0,20
0,25
d = 1
d = 2
d = 3
Re = 100, Gr = 10

Fig. 16. Temperature on modules 1 and 2 for d = 1, 2, 3; Re = 100, 1000, and Gr = 10
5

module is submitted to higher heat transfer since it is constantly been bombarded with cold
fluid from the forced convection. On the other hand, it can be seen again that a flow wake
from the first source reaches the second one and this is responsible for the bifurcation of the
Nusselt number curves. Here, one can note the time spent by the hot fluid coming from the
first source and traveling to the second one. For example, for Re = 100 and d = 1, 2, and 3,
the time shots are, respectively, around t = 1.4, 3.0, and 4.0. However, the converged values
for these last cases are almost the same. As seen earlier, periodic oscillations appear for
Re = 10.
5.3 Case with three heat sources
The results presented here are obtained using the finite element method (FEM) and a
structured mesh with rectangular isoparametric four-node elements in which ΔX = 0.1 and
ΔY = 0.05. A mesh sensibility analysis was carried out (Guimaraes, 2008). The temperature
distributions for Reynolds numbers Re = 1, 10, 50, and 100, Grashof number Gr = 10
5
, and
inclination angles γ = 0° (horizontal), 45°, and 90° (vertical) are available in Fig. (18). For
Re = 1 and γ = 0° and 45°, there is a formation of thermal cells which are localized in regions
close to the modules. When Re = 1, the flow is predominantly due to natural convection. As
Re is increased, these cells are stretched and hence forced convection starts to be
characterized. By keeping Re constant, the inclination angle variation plays an important
role on the temperature distribution. The effect of γ on temperature is stronger when low
velocities are present. For example, when Re = 10 and γ = 0°, 45° and 90°, this behavior is
noted, that is, for γ = 0° and Re = 10, a thermal cell is almost present, however, for
γ = 45°and Re = 10, those cells vanish. This is more evident when Re =1 and γ = 45° and 90°. Convection and Conduction Heat Transfer

NU
5
10
15
20
25
30
Heater 1
Heater 2
Re = 10, d = 2
t
5101520
NU
5
10
15
20
25
30
Heater 1
Heater 2
Re = 100, d = 2
t
5101520
NU
5
10
15
20
25

NU
5
10
15
20
25
30
Hea ter 1
Hea ter 2
Re = 1000, d = 3
t
5101520
NU
5
10
15
20
25
30
Heater 1
Heater 2

Fig. 17. Average Nusselt number vs Time: Gr = 10
5
, Re = 10, 10
2
, 10
3
, d =1, 2 , 3, Heater 1, 2


6
0
.
0
8
0
.
0
2
0
.
0
4
0
.
0
6
0
.
0
8
0
0.5
1
0.02
0.04
0
.
0
6

0
6
0
.
0
8
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.
1
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1
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2
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0
4
0
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0
.

6
0
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.
0
6
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1
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6
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0
.
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1
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0
4
0
.
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0
.
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8
0
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0.08
0
.
0
6
0
.
0
8
0
.
1
0
0
.
1
2
0
.
1
0
0
.
0
8
0
.
1
0
0

0
.
0
6
0
.
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8
0
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1
0
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1
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0
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4
0
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0
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0
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0
6
0

6
0
.
0
6
0
.
0
8
0
.
0
8
0
0.5
1
Re = 1
γ = 0
0

Re = 10
γ
= 0
0

Re = 50
γ = 0
0

Re = 100

0

Re = 100
γ = 90
0Fig. 18. Isotherms for Gr = 10
5
, Re = 1, 10, 50, 100 and γ = 0°, 45°, 90°

Convection and Conduction Heat Transfer

26
It is worth observing that, the fluid heated in the first heater reaches the second one, and
then the third one. Thus, this process of increasing temperature provides undesirable
situations when cooling is aimed.

Re = 10, γ = 0°
Re = 100, γ = 0°
Re = 10, γ = 45°
Re = 100, γ = 45°
Re = 10, γ = 90°
Re = 100, γ = 90°

Fig. 19. Velocity vectors for Gr = 105, Re = 10 and 100, and γ = 0°, 45°and 90°
Figure (19) depicts the velocity vectors for Re = 10 and 100 and Gr = 10
5
for γ = 0°, 45°, and
90°. It can be noted that for Re = 10 and γ = 0°, 45°, and 90°, recirculations are generated by

analyzing each graphic separately, it can be observed that NU
H1
tends to become more
distant from NU
H2
and NU
H3
as Reynolds number is increased, starting from an initial
value for Re = 1 which is almost equal to NU
H2
and NU
H3
. This agreement at the beginning
means that the heaters are not affecting one another. Here, it can be better perceived that
behavior found in Fig. (13), where a heater is affected by an upstream one. That is the
reason why NU
H1
shows higher values. The only case in which the heaters show different
values for Re = 1 is when Gr = 10
5
and γ = 90°. Overall, the strongest average Nusselt
number variation is between 0° and 45°. Practically in all cases, NU
H1
, NU
H2
,and NU
H3

increase in this angle range, 0° and 45°, while for Gr = 10
5

Gr=10
3
, γ = 0°
Re
10
0
10
1
10
2
10
3
NU
0
2
4
6
8
10
12
14
16
NUH!
NUH2
NUH3
Gr=10
3
, γ = 45°
Re
10

10
3
NU
0
2
4
6
8
10
12
14
16
NUH1
NUH2
NUH3
Gr=10
4
, γ = 0°
Re
10
0
10
1
10
2
10
3
NU
0
2

14
16
NUH1
NUH2
NUH3
Re
10
0
10
1
10
2
10
3
NU
0
2
4
6
8
10
12
14
16
NUH1
NUH2
NUH3
Gr=10
5
, γ = 0°

0
10
1
10
2
10
3
NU
0
2
4
6
8
10
12
14
16
NUH1
NUH2
NUH3
Gr=10
5
, γ = 90°

Fig. 20. Average Nusselt number vs Reynolds number for Gr = 10
3
, 10
4
, 10
5

, 10
5
and γ = 90°. In the
beginning, all three Nusselt numbers on H
1
, H
2
, and H
3
have the same behavior and value.
These numbers tend to converge to different values as time goes on. However, before they
do so, they bifurcate at a certain point. This denotes the moment when a heated fluid wake
from a previous source reaches a downstream one.

Re=10, Gr=10
5
, γ=0°
θ
0,00
0,05
0,10
0,15
0,20
0,25
0,30
Sour ce 1
Sour ce 2
Sour ce 3
Re=10, Gr=10
5

Re=100, Gr=10
5
, γ=0°
Sour ce
θ
0,00
0,05
0,10
0,15
0,20
0,25
0,30
Sour ce1
Sour ce2
Sour ce3
Re=100, Gr=10
5
, γ=45°
Sour ce
θ
0,00
0,05
0,10
0,15
0,20
0,25
0,30
Source1
Source2
Source3

Re=1000, Gr=10
5
, γ=45°
Sour ce
θ
0,00
0,02
0,04
0,06
0,08
0,10
0,12
Source1
Source2
Source3
Re=1000, Gr=10
5
, γ=90°
Sou rce
θ
0,00
0,02
0,04
0,06
0,08
0,10
0,12
Sour ce 1
Sour ce 2
Sour ce 3

4
6
8
10
H1
H2
H3
Re = 10, Gr = 10
4
, γ = 90°
t
0 5 10 15 20 25
NUM
2
4
6
8
10
H1
H2
H3
Re = 100, Gr = 10
4
, γ = 90°
t
0 10203040506070
NUM
2
4
6

H1
H2
H3

Fig. 22. Module average Nusselt number in time: Re = 10 e 100 , Gr = 10
3
, 10
4
, 10
5
, γ = 90°
6. Conclusions
1. The mixed convection was studied in a simple channel considering the effect of the
inclination angle and some physical parameters. The ranges performed were as follows:
1 ≤ Re ≤ 500, 10
3
≤ Gr ≤ 10
5
, and 0° ≤ γ ≤ 90°. The set of governing equations were
discretized and solved using the Galerkin finite element method (FEM) with the Penalty
formulation in the pressure terms and the Petrov-Galerkin perturbations in the
convective terms in all throughout the chapter. 5980 four-noded elements were used to
discretize the spatial domain. Comparisons were performed to validate the
computational code. It was observed from the results of the present problem that the
effect of the inclination angle on the velocity and temperature distributions plays an
important role on the heat transfer for low Re and high Gr. For high Re, the effect of the
orientation was negligible. One must understand that when the words ‘low’ and ‘high’
were mentioned here, it meant low and high compared to the limits considered in this
work. In general, it was also discovered that an inclination angle around 60° and 75°
provided a slight most desirable work conditions when cooling is aimed. It was said

is that there was an optimum distance in which two sources could be placed apart from
each other, that is, d = 3, although d = 2 did not present a significant change in
temperature either. Further investigations are encouraged taking into consideration
more heaters and different arrays.
3. In this work, mixed convection heat transfer study in an inclined rectangular channel
with three heat sources on the lower wall was carried out using the same. Effects on the
Nusselt number along the heat sources as well as the velocity vectors in the domain
were verified by varying the following parameters: γ = 0°, 45°, 90°, Re = 1, 10, 50, 100,
1000, Gr = 10
3
, 10
4
, 10
5
. In general, the inclination angle had a stronger influence on the
flow and heat transfer since lower forced velocities were present, especially when the
channel was between 0° and 45°. It could be noted that in some cases some heat sources
were reached by a hot wake coming from a previous module, thus, increasing their
temperatures. Primary and secondary recirculations and reversal flow were present in
some situations such as Re = 10, γ = 45° and 90°. In problems where heat transfer
analysis on electronic circuits is aimed, cases with the lowest temperatures, and hence,
the highest Nusselt numbers, are the most suitable ones. Therefore, the channel
inclination angles 45° and 90° were the best ones with little difference between them.
An exception was the case with Gr = 10
5
and Re = 1000, where γ = 0° was the ideal
inclination.
The authors found an interesting behaviour in all three cases with one, two or three heat
sources: There is a moment in all three cases studied when oscillations in time near the heat
sources appear, featuring their initial strength and then reaching its maximum and,

heat transfer in a rectangular channel with discrete heat sources at the top and at
the bottom,
International Communications in Heat and Mass Transfer, Vol. 32, pp.
1244–1252.
Binet, B. & Lacroix, M. (2000). Melting from heat sources flush mounted on a conducting
vertical wall,
Int. J. of Numerical Methods for Heat and Fluid Flow, Vol. 10, pp. 286–306.
Baskaya, S.; Erturhan, U.; Sivrioglu, M. (2005). An experimental study on convection heat
transfer from an array of discrete heat sources,
Int. Comm. in Heat and Mass Transfer,
Vol. 32, pp. 248–257.
Da Silva, A.K.; Lorente, S.; Bejan, A. (2004). Optimal distribution of discrete heat sources on
a plate with laminar forced convection,
International Journal of Heat and Mass
Transfer
, Vol. 47, pp. 2139–2148.
Heinrich, J. C. & Pepper, D. W. (1999). Intermediate Finite Element Method. Ed. Taylor &
Francis, USA.
Bae, J.H. & Hyun, J.M. (2003). Time-dependent buoyant convection in an enclosure with
discrete heat sources,
Int. J. Thermal Sciences.
Madhavan, P.N. & Sastri, V.M.K. (2000). Conjugate natural convection cooling of protruding
heat sources mounted on a substrate placed inside an enclosure: a parametric
study,
Comput. Methods Appl. Mech Engrg., Vol. 188, pp. 187-202.
Choi, C.Y. & Ortega, A. (1979). Mixed Convection in an Inclined Channel With a Discrete
Heat Source,
International Journal of Heat and Mass Transfer, Vol. 36, pp. 3119-
3134.
Bercovier, M. & Engelman, M. (1979). A Finite Element for the Numerical Solution of

Petrov-Galerkin technique and the penalty, Brazil.
Guimarães, P. M. (2008). Combined free and forced coonvection in an inclined channel with
discrete heat sources, Int. Communications of Heat and Mass Transfer, Vol. 35, pp.
1267-1274.
0
Periodically Forced Natural Convection Through
the Roof of an Attic-Shaped Building
Suvash Chandra Saha
School of Engineering and Physical Sciences, James Cook University
Australia
1. Introduction
Buoyancy-induced fluid motions in cavities have been discussed widely because of the
applications in nature and engineering. A large body of l iterature exists on the forms of
internal and external forcing, various geometry shapes and temporal conditions (steady or
unsteady) of the resulting flows. Especially for the classic cases of rectangular, cylindrical or
other regular geometries, many authors have investigated imposed temperature or boundary
heat fluxes. R eviews of these research can be found in Ostrach (1988) and Hyun (1994).
The rectangular cavity is not an adequate m odel for many geophysical situations where a
variable (or sloping) geometry has a significant effect on the system. However, the convective
flows in triangular shaped enclosures have received less attention than those in rectangular
geometries, even though the topic has been of interest for more than two decades
Heat transfer through an attic space into or out of buildings is an important issue for attic
shaped houses in both hot and cold climates. The heat transfer through attics is mainly
governed by a natural convection process, and affected by a number of factors including
the geometry, the interior structure and the insulation etc. One of the important objectives
for design and construction of houses is to provide thermal comfort for occupants. In the
present energy-conscious society, it is also a requirement for houses to be energy efficient,
i.e. the energy consumption for heating or air-conditioning of houses must be minimized. A
small number of publications are devoted to laminar natural convection in two dimensional
isosceles triangular cavities in the vast literature on convection h eat transfer.

(1988) examined the entire isosceles triangular cavities for seven possible combinations of hot
wall, cold wall and insulated wall using the finite element method based on a stream function
or vorticity formulation. A two dimensional right triangular cavity filled with air and water
with various aspect ratios and Rayleigh numbers are also examined by Salmun (1995a).
The stability of the reported single-cell steady state solution was re-examined by Salmun
(1995b) who applied the same procedures developed by Farrow & Patterson (1993) for
analysing the stability of a basic flow solution in a wedge-shaped geometry. Later
Asan & Namli (2001) carried out an investigation to examine the details of the transition from
a single cell to multi cellular structures. Haese & Teubner (2002) investigated the phenomenon
for a large-scale triangular enclosure for night-time or winter day conditions with the effect of
ventilation.
Holtzmann et al. (2000) modelled the buoyant airflow in isosceles triangular cavities with a
heated bottom base and symmetrically cooled top sides for the aspect ratios of 0.2, 0.5, and
1.0 with various Rayleigh numbers. They conducted flow visualization studies with smoke
injected into the cavity. The main objective of their research was to validate the existence of the
numerical prediction of the symmetry-breaking bifurcation of the heated air currents that arise
with gradual increments in Rayleigh number. Ridouane & Campo (2006) has also investigated
the numerical prediction of the symmetry-breaking bifurcation. The author reported that as
Ra is gradually increased, the symmetric plume breaks down and fades away. Thereafter, a
subcritical pitchfork bifurcation is created giving rise to an asymmetric plume occurring at
a critical Rayleigh number, Ra
= 1. 42 × 10
5
. The steady state laminar natural convection in
right triangular and quarter circular enclosures is investigated by Kent et al. (2007) for the case
of winter-day temperature condition. A number of aspect ratios and Rayleigh numbers have
been chosen to analyse the flow field and the heat transfer.
Unlike night-time conditions, the attic space problem under day-time (heating from above)
conditions has received very limited attention. This may due to the fact that the flow structure
in the attics subject to the daytime condition is relatively simple. The flow visualization

Fig. 1. A very few studies for diurnal heating and cooling effect on the attic space are reported
in the literature (Saha et al., 2010b; 2007). The authors discussed a general flow structure and
heat transfer due to the effect of periodic thermal forcing. A detailed explanation of choosing
the period for the model attic is required as the 24-hour period for the field situation is not
applicable here.
In this study, numerical simulations of natural convection in an attic space subject to diurnal
temperature condition on t he sloping wall have been carried out. An explanation of choosing
the period of periodic thermal effect has been given with help of the scaling analysis which is
available in the literature. Moreover, the effects of the aspect ratio and Rayleigh number on
the fluid flow and heat transfer have been discussed in details as well as the formation of a
pitchfork bifurcation of the flow at the symmetric line of the enclosure.
2. Formulation of the problem
The physical system is sketched in Fig. 2, which is an air-filled isosceles triangular cavity of
variable aspect ratios. Here 2l is the length of the base or ceiling, T
0
is the temperature applied
on the base, T
A
is the amplitude of temperature fluctuation on the inclined surfaces, h is the
height of the enclosure and P is the period of the thermal forcing.
35
Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building
4 Will-be-set-by-IN-TECH

T
=
T
0

,

S



0
P
/
2
sin
0 

v
u
t
T
T
T
A
S D

E

2
u
∂x
2
+
∂
2
u
∂y
2

(2)
∂v
∂t
+ u
∂v
∂x
+ v
∂v
∂y
= −
1
ρ
∂p
∂y
+ ν

∂
2
v

+
∂
2
T
∂y
2

(4)
where u and v are the velocity components a long x
− and y−directions, t is the time, p is
the pressure, ν, ρ, β and κ are kinematic viscosity, density of the fluid, coefficient of thermal
expansion and thermal diffusivity respectively, g is the acceleration due to gravity and T is
the fluid temperature.
The boundary conditions for the present numerical simulations are also shown in Fig. 2. Here,
the temperature of the bottom wall of the cavity is fixed at T
= T
0
. A periodic temperature
boundary condition is applied to the two inclined walls. The Rayleigh number for the periodic
boundary condition has been defined based on the maximum temperature difference between
the inclined surface and the bottom over a cycle a s
Ra
=
2gβT
A
h
3
κν
Three aspect ratios 0.2, 0.5 and 1.0, four Rayleigh numbers, 1.5
× 10

t
s
=
(
1 + Pr)
1/2
(1 + A
2
)
1/2
ARa
1/2
Pr
1/2

h
2
κ

,(5)
and the heating-up or cooling-down time scale for the enclosure to be filled with hot or cold
fluid under the same boundary conditions as in Saha et al. (2010c) is
t
f
=
(
1 + Pr)
1/4
A
1/2

1/3
A
2/3
Ra
1/3
Pr
1/3

t
p
h
2
/κ

1/3

h
2
κ

,(7)
and the h eating-up or cooling-down time scale of the enclosure under the same boundary
conditions from Saha et al. (2010a) is
t
fr
=

h
(1 + A
2

Ra
1/4
(1 + A
2
)
1/4
h
2
t
p

1/2
⎤
⎦
,(9)
However, if the cavity is filled with cold fluid before the ramp is finished then the filling up
time is given in Saha et al. (2010a) as
t
fq
∼
h
8/7
t
3/7
p
κ
4/7
Ra
1/7
A

Steady state time (t
s
) for
sudden heating/cooling
Quasi-steady time (t
sr
) for
ramp heating/cooling (t
p
=
1000s)
Ra = 1.5u10
6

Ra = 7.2u10
3

Ra = 1.5u10
6

Ra = 7.2u10
3

A=0.2
8.15s
-
40.51s
-
A=0.5
2.54s

h(1 + A
2
)
1/4
Ra
1/4
A
1/2

t
p
t

1/4
. (11)
and
u
∼ Ra
1/2
κ
h

t
t
p

1/2
. (12)
respectively. When the hot fluid travels through the boundary layer and reaches the top tip of
the cavity then it has no choice but to move downward along the symmetry line of the cavity.

aspect ratios, A
= 0.2 and 1.0, the filling-up times are 213.32s and 122.67s respectively for
Ra
= 1 × 10
6
. However, the filling-up time for ramp boundary conditions depends on the
length of the ramp time. If the ramp time is 200s then the filling-up time for the lowest
38
Convection and Conduction Heat Transfer
Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 7
Aspect
ratio

Filling-up time (t
f
) for
sudden heating/cooling
Filling-up time (t
fr
) for
ramp heating/cooling (t
p
=
1000s)
Ra = 1.5u10
6

Ra = 7.2u10
3


Equations (1) - (4) are solved along with the initial and boundary conditions using the SIMPLE
scheme. The finite volume method has been chosen to discretize the governing equations,
with the
QUICK scheme (see Leonard & Mokhtari, 1990) approximating the advection term.
The diffusion terms are discretized using central-differencing with second order accurate. A
second order implicit time-marching scheme has also been used for the unsteady term. An
extensive mesh and time step dependence tests have been coonducted in Saha et al. (2010a;b;c)
5. Flow response to the periodic thermal forcing
The flow response to the periodic thermal forcing and the heat transfer through the sloping
boundary are discussed for the case with A
= 0.5, Pr = 0.72 and Ra = 1.5 × 10
6
in this section.
5.1 General flow response to diurnal heating and cooling
Since the initial flow is assumed to be isothermal and motionless, there is a start-up process
of the flow response. In order to minimize the start-up effect, three full thermal forcing cycles
are calculated in the numerical simulation before consideration of the flow. It is found that
the start-up effect for the present case is almost n egligible, and the flow response in the third
cycle is identical to that in the previous cycle. In the following discussion, the results of the
third cycle are presented.
Fig. 3 shows snapshots of streamlines and the corresponding isotherms and vector field at
different stages of the cycle. The flow and temperature structures, shown in Fig. 3 at t
= 2.00P,
represent those at the beginning of the daytime heating process in the third thermal forcing
cycle. At this time, the inclined surfaces and the bottom surface of the enclosure have the same
temperature, but the temperature inside the enclosure is lower than the temperature on the
boundaries due to the cooling effect in the previous thermal cycle. The residual temperature
structure, which is formed in the previous cooling phase, is still present at t
= 2.00P.The
39

(t
= 2.70P, Fig. 3), the cold-air layer under the inclined surfaces becomes unstable. At
the same time, the hot-air layer above the bottom surface also becomes unstable. As a
consequence, sinking cold-air plumes and rising hot-air plumes are visible in the isotherm
contours and a c ellular flow pattern is formed in the corresponding stream function contours.
It is also noticeable that the flow is symmetric about the geometric symmetry plane at this
time. However, as time increases the flow becomes asymmetric about the symmetric line (see
isotherms at t
= 2.95P). The large cell from the right hand side of the centreline, which is still
growing, pushes the cell on the left of it towards the left tip. At the same time this large cell
also changes its position and attempts to cross the centreline of the cavity and a small cell next
to it moves into its position and grows.
At t
= 2.975P, the large cell in the stream lines has crossed the centerline and the cell o n
the right of it grows and becomes as large as it is after a short time (for brevity figures not
included). The flow is also asymmetric at this time. However, it returns to a symmetric flow
at the time t
= 3.00P which is the same as that at t = 2.00P, a nd similar temperature and flow
structures t o those at the beginning of the forcing cycle are formed. The above described flow
development is repeated in the next cycle.
The horizontal velocity profiles (velocity parallel to the bottom surface) and the corresponding
temperature profiles evaluated along the line DE shown in Fig. 2 at different time instances of
the third thermal forcing cycle are depicted in Fig. 4. At the beginning of the cycle (t
= 2.00P)
the velocity is the highest near the roof of the attic (see Fig. 4a), which is the surface driving the
flow. At the same time, the body of fluid residing outside the top wall layer moves fast toward
the bottom tips to fill up the gap. As time progresses the vertical velocity increases and the
horizontal temperature decreases (see t
= 2.05P). A three layer structure in the velocity field
40

temperature of the top surface increases (t
= 2.05P) while the bottom surface temperature
42
Convection and Conduction Heat Transfer
Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 11
remains the same. It is noteworthy that the top surface reaches its peak temperature at
t
= 2.25P (for brevity the profile is not included). After this time the top surface temperature
starts to decrease which can be seen at time t
= 2.40P. By comparing the temperature profiles
at t
= 2.05P and t = 2.45P shown in Fig. 4(b), it is clear that the temperatures at both
the top and bottom surfaces are the same for these two time instances. However, d ifferent
temperature structures are seen in the interior region. The same phenomenon has been found
at the times t
= 2.50P and t = 2.00P.
In Fig. 4(c), the velocity profiles at the same location during the night-time cooling phase are
displayed. In this phase the flow structure is more complicated. At t
= 2.55P the velocity near
the bottom surface is slightly higher than that near the top. Again a three layer structure of the
velocity field appeared which is seen at t
= 2.65P,2.75P and 2.85P. The maximum velocity
near the ceiling occurs at t
= 2.75P when the cooling is at its maximum. After that it decreases
and the flow reverses completely at t
= 3.0P. The corresponding temperature profiles for the
night-time condition are shown in Fig. 4(d). It is seen that the temperature lines are not as
smooth as those observed for the daytime condition. At t
= 2.55P, the temperature near the
bottom surface decreases first and then increases slowly with the height and again decreases

heat transfer coefficient instead of a changing temperature difference, which would give an
undefined value of the heat transfer c oefficient at particular times.
Fig. 5 shows the calculated average Nusselt number on the inclined and bottom surfaces of
the cavity. The time histories of the calculated Nusselt number on the inclined surfaces exhibit
certain significant features. Firstly, it shows a periodic behaviour in response to the periodic
thermal forcing. Secondly within each cycle of the flow response, there is a time period
with weak heat transfer and a period with intensive heat transfer. The weak heat transfer
corresponds to the daytime condition when the flow is mainly dominated by conduction
and the strong heat transfer corresponds to the n ight-time condition. At night, the boundary
layers adjacent to the inclined walls and the bottom are unstable. Therefore, sinking and rising
plumes are formed in the inclined and horizontal boundary layers. These plumes dominate
43
Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building