
Research Article


Natural Convection in a Horizontal Wavy Enclosure 

Slimani Abdelkader,
Rebhi Mebrouk,
Belkacem Abdellah
and
Bouhadef Khadidja



ABSTRACT

A numerical analysis is performed to study the influence of the amplitude b of the sinusoidal undulation, of bottom wall, of an horizontal enclosure on the heat transfers by twodimensional natural laminar convection. The hot wavy bottom wall and the cold straight top wall are kept isothermal at temperature equal, respectively to T_{h} and T_{c}. And the two vertical straight walls (right and left) are considered symmetrical. Results are presented in the form of local and global Nusselt numbers distribution for a selected range of Raleigh (10^{3}10^{6}) and for four values (0.00, 0.15, 0.25, 0.35) of aspect ratios A = 2.b/a. Air of Pr = 0.71 in an enclosure of non dimensional length As = L/H equal to 4 is chosen as the flow model to examine the influences of the aspect ratios at various Raleigh Number. The natural convection equations are discretised, using an implicit finite difference method, based on the finite volume approach. The SIMPLE algorithm, assumes the linkage between velocities and pressure fields. The variation of the Raleigh Number has allows the obtaining of several types of flows and several forks between these flows. The flow in enclosure is characterized with recirculation zones over summits and in hollows, where the local Nusselt numbers is always minimal. The numerical results show that the flow and the heat transfer are strongly affected by the amplitude b. In a general the transfers developed within an enclosure by not flat topography are lower than those obtained in a horizontal and uniform enclosure of the same length.





INTRODUCTION
Natural convection heat transfer wallfluid has many significant engineering
applications experimentally and analytically; for example, doublewall thermal
insulation, solarcollectors, electric machinery, cooling systems of microelectronic
devises, natural circulation in the atmosphere, etc. A considerable number of
published articles are available that deal with flow characteristics, heat transfer,
flow and heat transfer instability, design aspects, etc. On the other hand,
studies dealing with convection problems inside more complex geometries have
been rather limited. Complex geometry covers different types of geometries configurations,
namely Lshaped cavities (Mahmud, 2002), trapezoïdal cavities (Peric, 1993),
arc shaped enclosures (Chen and Cheng, 2002), wavy cavities (Mahmud et al.,
2002), etc. The complex fluid dynamic behaviour as well as their geometrical
complexity and its effect on flow phenomenon have motivates many researchers
to perform experimental or numerical work on this topic. In present study (Mahmud
et al., 2002), have shown the effect of surface waviness on natural convection
heat transfer and fluid flow inside a vertical wavy walled enclosed for a range
of wave ratio (0.00≤λ≤0.4) and aspect ratio (1.0≤A≤2.0). They
observed that aspect ratio is the most important parameter for heat and fluid
flows and higher heat transfer is obtained at a lower aspect ratio for a certain
value of Grashof number. Das and Mahmud (2003) analyzed the free convection
inside both the bottom and the ceiling, wavy and the isothermal enclosure. They
indicated that, only at the lower Grashof number, the heat transfer rate rises
when the amplitude wave length ratio changes from zero to other values. Mahmud
and Islam (2003) solved the laminar free convection and entropy generation inside
an inclined wavy enclosure using SIP (Strongly Implicit Procedure) solver on
a nonstaggered grid arrangementThey obtained that the inclination angle of
cavity affects the entropy generation due to the heat transfer and fluid friction.
Among them, Kumar (2000), presented the parametric results of flow and thermal
fields inside a vertical wavy enclosure with porous media. Hadjadj and Kyal
(1999) numerically examined the effect of sinusoidal protuberances on fluid
flow and heat transfer inside an annular space using a nonorthogonal coordinate
transformation. We have reported that both the local and average heat transfer
increase with the increase of protuberance amplitude and Rayleigh number and
decreasing Prandtl number. Kumar and Shalini (2003) investigated the effects
of surface undulations on natural convection in porous enclosure with global
cumulative heat flux boundary conditions for different undulation numbers and
thermal stratification level. They indicated that the local Nusselt numbers
are very sensitive to thermal stratification. Kumar et al. (1998) solved
the free convection problem in an enclosure with bottom wave wall heated and
cooled from the ceiling using finite element method. He indicated that separation
zones starts around Ra = 50 in the single wave case and around Ra = 25 in the
case with six waves per unit length. Murthy et al. (1997) analyzed the
effect of the surface undulations on the natural convection heat transfer from
an isothermal surface in a Darcian fluidsaturated porous enclosure by using
the finite element method on a graded nonuniform mesh System. They found that
the flowdriving Rayleigh number Ra together with geometrical parameters
of wave amplitude, wave phase and the number of waves considered in the horizontal
dimensions of the cavity influenced the flow and heat transfer process in the
enclosure. Gao et al. (2000) solved the natural convection inside the
wavy and inclined solar collector but they did not interest in flow behavior.
Adjiout et al. (2002) performed and solved a similar problem with Gao
and others, but in their case, the left vertical wall is flat and cold, whereas
the right wall is wavy and hot. They indicated that the mean Nusselt number
is decreased when it is compared with the square cavity. Dalal and Das (2005)
made a numerical solution to investigate the inclined right wall wavy enclosure
with spatially variable temperature boundary conditions. Oztop (2005) applied
the elliptic grid generation to obtained sinusoidal duct geometry to enhance
the forced convection heat transfer. Varol and Oztop (2006) investigated the
effects of inclination angle on the laminar natural convection heat transfer
and fluid flow in a wavy solar collector in steady state régime. They
observed that the inclination angle is the most important and effective parameter
on heal transfer which can be used to control the heat transfer inside the collector.
Saidi et al. (1987) presented numerical and experimental results of flow
over and heat transfer from, a sinusoidal cavity. They reported that the total
heat exchange between the wavy wall of the cavity and the flowing fluid was
reduced by the presence of vortex. The vortex plays the role of a thermal screen,
which creates a large region of uniform temperature in the bottom of the cavity.
Wang and Vanka (1995) presented heat transfer and flow characteristics inside
a wavy walled charnel. Nishimura et al. (1984) investigated flow characteristics
such as flow pattern, pressure drop and wall shear stress in a channel with
symmetry; sinusoidal wavy wall. Asako and Faghri (1987) gave a finitevolume
prediction of heat transfer and fluid flow characteristics inside a wavy walled
duct and tube, respectively. Aydin et al. (1999), Eisherbiny (1996),
Hamady et al. (1989) and Ozoe et al. (1975) presented results
of heat transfer characteristics inside the rectangular enclosures at different
aspect ratios and orientations without surface waviness.
The main purpose of the present study is to provide the effect of the amplitude
of the sinusoidal oundulation, of bottom wall, of a shallow and horizontal wavy
enclosure on the heat transfers by twodimensional natural laminar convection
natural. To the best of the authors' knowledge; free convection in the shallow
wavy enclosure has not yet been investigated.
The rate of heat transfer in terms of local and global Nusselt numbers are presented for different aspect ratios and Rayleigh numbers; respectively. Flow and thermal fields are analyzed by parametric presentation of streamlines and isothermal lines. PROBLEM STATEMENT
Consider a twodimensional wavy wall horizontal enclosure of length L and height H as shown in Fig. 1. The hot wavy bottom wall and the cold straight top wall are kept isothermal at temperature equal respectively to T_{h} and T_{c} and the two vertical straight walls (right and left) are considered symmetrical. The fluid is assumed to be of constant properties and the Boussinesq approximation is employed for the gravity terms. Stresswork are neglected. In the present study, a, a^{2}/α and α/a are used, respectively, as the length, time and velocity scales, where α denotes the thermal diffusivity. The dimension less temperature function is defined as θ = (TT_{r})/ΔT, in which ΔT = (T_{h}T_{c}) stands for the temperature difference of the two isothermal walls and T_{r} = T_{c} is the fluid temperature at reference state.
The wavy wall function and governing equations in dimension less form can be cast into the following form respectively.

Fig. 1: 
Schematic representation of the computational geometry under
consideration 
where A = (2.b/a)v is the aspect ratio of the enclosure; Ra = g.β.a^{3}.(T_{h}T_{c})/v.α
is the Rayleigh number, Pr = v/α is the Prandtl number, β is the thermal
expansion coefficient and v is the kinematics viscosity. The two opposite walls
(bottom and top) are kept at the uniform temperatures T_{h} and T_{c},
respectively. The boundary conditions for the system of Eq. 25
are
U = ∂V/∂X = ∂θ/∂X = 0; at two side walls
U = V 0, θ = 1; at hot wavy bottom wall
U = V = θ = 0; at cold straight top wall
HEAT TRANSFER
Heat transfer rate is measured by local (Nux) and average (Nu) Nusselt numbers. Following equations arc used to calculate the local and average Nusselt numbers:
Where the gradient term in (∂θ/∂N)_{w} Eq.
(6) is the temperature gradient normal to the hot wavy wall.
NUMERICAL PROCEDURE
The system of Eq. (25) with the boundary
conditions stated above is solved by using finite volume method and the S1MPLE
algorithm, Patanker (1980). Since the flow fields for the parameter range considered
lie in the steady flow regime. The value of Δτ = 10^{3} is
used through the course of the computation.

Fig. 2: 
Streamlines (right) and isotherms (left) for Ra=3800, Pr =
5580 and As = 1, a) Present prediction b) Soong et al. (1996) prediction 

Fig. 3: 
Streamlines (bottom) and isotherms (top) for Ra = 5000, Pr
= 0.71 and As = 4, a) Present prediction b) Soong et al. (1996) prediction 

Fig. 4: 
Streamlines (bottom) and isotherms (top) for Ra = 20000, Pr
= 0.71 and As = 4, a) Present prediction b) Soong et al. (1996) prediction 
As. the maximum relative deviation of the mean Nusselt numbers between two
successive timesteps is less than the value of 10^{4} or the maximum
value of the relative deviation of' the temperature less than 10^{6}
the procedure is regarded as converged.
To cheek the validity of the present numerical procedure, thermally driven flows in a square and rectangular cavity were solved. For code validation; our numerical resultants (for aspect ratio of the enclosure A = 0) are compared with the resultants of Soong et al. (1996) solutions (for inclination angle γ = 0). Comparisons arc lines in Fig. 24. Present prediction shows a very good agreement with the result of Soong et al. (1996). ACCURACY
In the present study, five combinations (30x113; 46x177; 62x241; 74x289 and
82x321) of control volumes are used to test the effect of grid size on the accuracy
of the predicted results. Figure 5 shows the distribution
of average Nusselt numbers of the hot wall as a function of grid sizes for four
different Rayleigh numbers.

Fig. 5: 
Variation of average Nusselt number as a Function of control
volume for A = 0.25 and As = 4 
It is clear from the figure that at lower Rayleigh number, average Nusselt
number is almost independent of grid sizes. At higher Rayleigh numbers, the
two higher grid sizes (74x289) and 82x321) give almost the same result (Fig.
5). It is well known that the high mesh refinement always provides better
result in the finitevolume method. The main disadvantage in taking higher mesh
number is the increase in calculation time, which can be reduced by using a
higher speed of Pentium processor. Our goal was to get better results. Thus;
throughout this study, the results are presented for 74x289 CVs' for better
accuracy.
RESULTS AND DISCUSSION
Flow and thermal field: The flow pattern inside the wavy enclosure and
the temperature profile are presented in terms of streamlines and isothermal
lines in Fig. 6(ad) for A = 0.25; As = 4 at four selected
Rayleigh numbers. At low Rayleigh number when convection current inside the
cavity is comparatively weak fluid stream near the hot wall tends to move towards
the centerline or crest of the cavity were two streams from the opposite direction
mix and rise upwards.
Two symmetric counterrotating vortices are observed at top and bottom of the
wave wall on the flow field due to the uniform temperature gradient. At low
Rayleigh number Fig. 6(a, b) circulation inside the cavity
is very weak. Convection is less prominent at this Rayleigh number and heat
transfer is mainly dominated by conduction. Isothermal lines are nearly parallel
to each other and follow the geometry of the wavy surfaces. A further increase
of Rayleigh numbers increase the circulation strength inside the enclosure.

Fig. 6: 
Streamlines (bottom) and isotherms (top) for different Rayleight
number at A = 0.25; As = 4 and Pr = 0.71 

Fig. 7: 
Streamlines (bottom) and isotherms (top) for different aspect
ratio at As = 4; Pr = 0.71 and Ra = 10^{5} 
Here, uniform temperature profile is changed and three high gradient spot is
observed at top and bottom of the hot wall due to rapid circulation of fluid
inside the cavity. Isotherms turn up (convective distortion) towards die cold
wall due to the strong influence of the convection current.
Figure 7 shows the effect of aspect ratio (A) on flow field
at constant Rayleigh number (Ra = 1.10^{5}). Whatever the value of A,
flow inside the cavity is characterized by same number of cell multicellular
pattern. An increase in the value of A decreases the size and the maximum values
of vortices respectively.

Fig. 8: 
Variation of local Nusselt number along the wave wall; for
As = 4; Pr = 0.71 and for different aspect ratios at different Rayleigh
number 
Local heat transfer distribution: Local Nusselt number distribution
at the bottom wall is shown in Fig. 8(ag) for As = 4 and
A = 0.15; 0.25 and 0.35 at seven selected Rayleigh numbers. The profile of the
distribution of the local Nusselt number present a symmetry with regard to hollows
and a periodicity with regard to summits.
For the low values of the Rayleigh number (Ra≤10^{4}) the thermal
exchange makes essentially by conduction, the distribution of the local Nusselt
number possesses maximums (peaks) at the level of summits (Fig.
8a, b). For Ra>10^{4}, the thermal exchange by convection is
dominant, the peaks observed at summits ease with the increase of the Rayleigh
number (Fig. 8ch).
Average heat transfer distribution: The effect of aspect ratio of the
enclosure on average heat rate transfer is shown in Table 1.

Fig. 9: 
Variation of average Nusselt number for As = 4; Pr = 0.71
and for different aspect ratios at different Rayleigh number 
Table 1: 
Average Nusselt number and maximum values of ψ (ψ_{max})
at different Rayleigh number for different aspect ratios 

At lower Rayleigh number the effect of aspect ratio is significant. However,
at higher Rayleigh number this effect is very small. At higher aspect ratio
heat transfer rate is higher at lower Rayleigh number when aspect ratio increase
from zero to other values and after then, further increases of aspect ratio
shows a negligible effect on average heat transfer rate (Fig.
9).
CONCLUSIONS
Laminar steady natural convection heat transfer and fluid flow in shallow wavy enclosure are investigated numerically. The main conclusions that are drawn the present study are provided below.
• 
Two circulation cells are obtained at the enclosure cavity
of wave in different directions especially at the small aspect ratio. 
• 
Flow and thermal fields are affected by geometrical parameters
and Rayleigh number 
• 
Heat transfer is increased with the increasing of aspect ratio. 
• 
Local Nusselt numbers show wavy variation and maximum Nusselt
number values are obtained on the top of the wave at low Rayleigh number
(Ra = 1.10^{4}). For Ra>10^{4}, the peaks observed at
summits ease with the increase of the Rayleigh number. 
ACKNOWLEDGMENTS
This study was realized in the Institute of Mechanical Engineering of Béchar University, in collaboration with the L.T.P.M.P Laboratory, Faculty of Mechanical and Processes Engineering, Houari Boumediene University of Sciences and Technology. Algiers, Algeria.
NOMENCLATURE
A 
aspect ratio, A = 2.b/a () 
a 
wave length of the wavy wall; (m) 
As 
non dimensional length, As = L/h () 
b 
amplitude height of the wavy wall; (m) 
g 
gravitational acceleration, (m sec^{2}) 
H 
height of the enclosure; (m) 
h 
heat transfer coefficient, (W m^{2}.K) 
L 
length of the enclosure, (m) 
Nu 
mean Nusselt number 
Nux 
local Nusselt number 
p, P 
dimensional pressure and pressure (N m^{2}) 
Pr 
Prandtl number, Pr = v/α 
Ra 
Rayieigh number, Ra = g.β (T_{p}  T_{o}).h^{3}/α.v 
t 
time, (sec) 
u, v 
dimensional velocity components, (m sec^{1}) 
U, V 
dimensionless velocity components 
x, y 
cartesian coordinates; (m) 
X, Y 
dimensionless coordinates 
Greek symbols
α 
thermal diffusivity.(m^{2} sec) 
β 
thermal expansion coefficient,(K^{1}) 
λ 
Thermal conductivity, (w.m.^{1} K.^{1}) 
v 
Kinemtic viscosity, (m^{2} sec^{1}) 
θ 
Dimensionless temperature function
θ = (TT_{c})/(T_{h}T_{c}) 
ρ 
Density of the fluid(kg m^{3}) 
Ψ 
Dimensionless stream function 
τ 
Dimensionless time, τ = α.t/a^{2} 
Subscripts

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