INTRODUCTION
The twodimensional laminar natural convection flow of air in a vertical channel made object of many theoretical and experimental work because of its implication in a number of natural phenomena and industrial processes as varied as the habitat, drying, the heat exchangers, electronics, etc. We distinguish the three following cases:
• 
Twodimensional laminar natural convection in a vertical channel
formed by two parallel plates (Elenbaas, 1942; Ostrach, 1952; Bodoia and
Osterle, 1962; Engel and Mueller, 1967; Aung et al., 1972; Aung,
1972; Hugot, 1972; Aihara, 1973; Miyatake and Fujii, 1972; Akbari and Borges,
1979; Dalbert et al., 1981; Wirtz and Stutzman, 1982; Sparrow et
al., 1984; Sparrow and Azevedo, 1985; Webb and Hill, 1989). 
• 
Laminar natural convection flow in a twodimensional vertical channel,
but with an imposed particular condition, either in the configuration of
the channel, or in the way of tackling the problem (Dyer and Fowler, 1966;
Kettleborough, 1972; Dyer, 1975, 1978; Nakamura et al., 1982; Sparrow
and Tao, 1982; Agonaffer and Watkins, 1985; Azevedo and Sparrow, 1986; Sparrow
et al.,1988; Sparrow and Ruiz, 1988). 
• 
Laminar and twodimensional mixed convection flow in a vertical channel
(Dalbert et al., 1980; Penot and Dalbert, 1983; Aung and Worku, 1986a,
b; Habchi and Acharrya, 1986). 
Most of these studies relate to the case where the walls are maintained at
a constant temperature and are based on a parabolic formulation of the equations
in which the axial diffusion is neglected. The conditions of uniform heat flux
on the walls were initially studied by Engel and Mueller, 1967 followed by Aung
et al. (1972), Dalbert et al. (1981), Writz and Stutzman (1982),
Dyer (1975), Penot and Dalbert (1983).
BarCohen and Resenhow (1984) worked out starting from the experimental and numerical results their predecessors, a whole of correlations characterizing the heat transfer in a channel whose walls are symmetrically or asymmetrically heated by a uniform heat flux or maintained at temperatures.
Naylor et al. (1991) were among the first to solve the complete NavierStokes equations and to show the influence with low numbers Rayleigh of one extension to the entry of channel.
It can be observed that the motion pressure at the entry of the channel was considered a long time as equal to that reigning outside. This assumption neglects the influence of the pressure at the entry due to the fluid acceleration. Aihara (1973) is one of the first which studied the influence of inlet conditions of pressure and the velocity profile within an isothermal vertical channel. Dalbert et al. (1981) confirms the influence of the inlet conditions of pressure within an vertical channel asymmetrically heated with uniform heat flux, while being based on the results of the limits cases (Aung, 1972).
Recently, Marcondes and Maliska (1999) numerically demonstrated in an asymmetrically heated channel, that using the inlet motion pressure P_{m} = 0.5Q^{2 }instead of P_{m} = 0, has no noticeable influence on the mean Nusselt number but substantially decreases the mass flow rate.
The natural convection in vertical channel with roughness elements has been less frequently reported in the literature, even though it is encountered in several technological applications. In such circumstances the natural convective flow may be significantly altered by roughness elements mounted on the channel surfaces.
Habchi and Acharya (1986) numerically investigated the laminar mixed convection of air in a vertical channel with a single rectangular rib. They demonstrated that for a symmetrically heated channel the mean Nusselt numbers are smaller then the corresponding smooth channel Nusselt numbers, i.e. for an unobstructed channel. Mehrotra and Acharya (1993) conducted an experimental study of natural convection heat transfer in smooth and ribbed UWT and UHF vertical channels. Said and Krane (1990) preformed a numerical and experimental investigation of natural convection in a vertical channel with a single semicircular obstacle. They showed that the location of the rib along the wall affects the rate of heat transfer. A free convection flows in vertical channel with two rectangular obstructions on opposite walls was studied numerically by Viswatmula and Ruhul Amin (1995). They found that the average Nusselt numbers for obstructed channels are less than those of the smooth channel.
Desrayaud and Fichera (2002) numerically investigated the laminar natural convective flows in a vertical channel with two symmetrical isothermal or adiabatic ribs. They found that the mean Nusselt number at high Rayleigh number is very similar for isothermal and adiabatic ribs whatever their location.
Hadjadj and Kyal (1999) numerically investigated the effect of sinusoidal protuberances on heat transfer and fluid flow inside an annular space using a nonorthogonal coordinate transformation. They reported that the both local and average heat transfer increase with the increase of protuberances amplitude and Rayleigh number and decreasing Prandtl number.
Obviously, further numerical studies of the problem of the natural convection in obstructed channel with different geometries are necessary. Thus, the objective of this study is to numerically investigate the natural convection in an asymmetrically heated vertical channel with a sinusoidal obstruction and to study the effect of obstruction amplitude, location of the obstruction and aspect ratio of the channel on heat transfer and fluid flow.
MATHEMATICAL MODELING
In the proposed study, we consider a vertical two dimensional channel formed by two plates of height H separated by a distance d. Left plate of the channel is flat and adiabatic; the right plate is maintained at uniform temperature and obeys at the following law (Fig. 1):
The wavy obstruction is located at a distance L1 downstream of the channel inlet. The local distance X along the channel is measured from the leading edge of the left plate of the channel.
The flow is considered steadystate, laminar and incompressible with no internal heat generation and the Boussinesq approximation has been applied. Other valid assumptions are that the compression work, viscous dissipation and radiative transport are negligibly small.
The governing equations for the flow in a vertical channel are parabolic in nature. But the presence of recirculating flows in the vertical obstruction channel makes these governing equations elliptic. With these assumptions, the governing equations can be written in the following dimensionless form:
Conservation of mass:

Fig. 1: 
Schematic diagram of the problem under consideration 
Conservation of momentum:
Conservation of energy:
The mass flow rate is calculated at the entrance of the channel as:
With
Where, the dimensionless variables are:
Boundary and initial conditions: Figure 1 shows the
geometry under consideration in the present investigation with axis system.
The wall at the left side is adiabatic and at the right side is maintained at
uniform temperature T_{w}. The fluid is assumed to enter into the channel
at ambient temperature T_{0}, ambient pressure and the mass flow rate
given by Eq. (6). At the upper boundary, the streamwise variations
of the velocity and temperature are neglected while the pressure is assumed
to be equal to the ambient pressure. If the fluid enters by the top of the channel
because of a recirculation, it is supposed to be at the ambient temperature.
The gravity acceleration g acts vertically downwards. No slip boundary condition
is applied for velocity components at vertical walls.
Boundary conditions can be summarized by the following Eq:
As the initial condition, a motionless state and uniform temperature are taken:
Heat transfer: The local heat transfer evaluated at the wall surface of the channel is calculated in terms of local Nusselt number using the following equation:
Local Nusselt numbers based on the channel width are integrated to calculate the global or average value of Nusselt number according to
Solution methodology: The numerical technique used in the present study is similar to that of Issa (1985) and Issa et al. (1991) based on the finitevolume method as described by Patankar (1980). The solution domain is first subdivided into finite number of control volumes (CV). Body fitted, nonorthogonal grids are used. Grids are oriented in such a way that the number of CV is higher near the wall where higher gradients of variable values are expected (Fig. 1). A collocated variable arrangement is used in the present investigation. All variable are calculated at the center of each CV. PISO (Pressure Implicit with Splitting of Operators) algorithm is used to solve the coupled system of governing equations. First the momentum equations (Eq. 3 and 4) are discretized and linearized. Convective fluxes are approximated using UDS scheme, which is second order accurate. Central differencing scheme (CDS) approximated diffusive fluxes and linear interpolation is used to compute cellface diffusion coefficients based on cellcenter values. Discretized momentum equations lead to an algebraic equation system for velocity components U and V where pressure, temperature, fluid properties are taken from the previous iteration except the first iteration where initial conditions are applied. These linear equations systems are solved iteratively (inner iteration) to obtain an improved estimate of velocity. The improved velocity field is then used to estimate new mass fluxes, which satisfy the continuity equation. Pressurecorrection equation is then solved using the same linear equation solver and to the same tolerance. Energy equation is then solved in he same manner to obtain better estimate of new solution. This competes one outer iteration and is repeated until the residual is less than or equal to 10^{5}. In this study the incomplete factorisation method (also called the incomplete LU decomposition) is used to solve the linear equation systems. To avoid divergence, underrelaxation parameter 0.7 is used for velocity, 0.3 for pressure and 0.8 for temperature.
Accuracy: For reasonably high values of Ra^{*}(≥100), any region outside the channel can be neglected, upstream thermal diffusion through the inlet section being negligible. Thus, because the fluid is not preheated before entering, the computational domain can be restricted to the space between the plates.
In the present investigation a nonorthogonal, nonuniform and nonstaggered grid system is used. Five combinations (25x100, 30x100, 40x100, 40x120 and 50x140) of control volumes are used to test the effect of grid size on the accuracy of the predicted results. Figure 2 shows the distribution of average Nusselt numbers of the hot wall as a function of grid sizes for four different channel Rayleigh numbers. It is evident from the figure that for lower channel Rayleigh numbers the Nusselt number is independent of the grid sizes studied here and for the higher channel Rayleigh numbers the three higher grid sizes give the same result. Thus, throughout this study the results are presented for 40x100 CVs’.
Predicted results of average Nusselt numbers for a smooth channel with the same boundary conditions are compared with the correlation of Churchill and Usagi (Taine and Petit, 1989). This correlation, valid for low and high values of the channel Rayleigh number, is as follows:

Fig. 2: 
Average Nusselt number as a function of number of control
volume for A = 5, a = 0.450, L_{0} = 1, L_{1} = 2 

Fig. 3: 
Variation of the average Nusselt number versus channel Rayleigh
number for Pr = 0.71 
Figure 3 shows this comparison. Predicted results show a very good agreement with correlation (14). Thus, this can give us confidence in the numerical simulations reported in what follows.
RESULTS AND DISCUSSION
The focus of this study is to examine the influences upon flow and heat transfer to changes in the geometrical parameters of a single sinusoidal obstruction in an asymmetrically heated vertical channel. Although our theoretical model has an academic character, one can imagine an extension being able to present an interest at the level of practical applications such as the heat exchangers with fins, the applications in electronics such as the girdles of cooling presenting protuberances on their external surfaces , the solar collectors as well as the systems of stocking energy.
The computations were carried out for six values of the sinusoid amplitude describing the obstruction, a = 0.0 to 0.5. Three different aspect ratios were analyzed, A = 3, 5, 10 for a channel Rayleigh number range, 10^{2}≤Ra^{*}≤2.10^{5}. This Rayleigh number is defined as the ratio of the Rayleigh number to the aspect ratio of the channel. All the numerical computations were performed for air (Pr = 0.71). The results are presented as isothermal contours and streamlines. The local and average heat transfer coefficients are also calculated.
Flow and thermal field: Figure 4 and 5
show the streamlines and isothermal contours at Ra^{*} = 10^{3}
and 10^{5} for six different amplitudes of the sinusoidal obstruction
(a = 0.5 to 0).
As the flow approaches the obstruction, the streamlines are deflected towards the adiabatic wall and therefore, in the partially obstructed region, the streamlines are more densely packed.
Beyond the obstruction, the flow separates and reattaches further downstream on the hot plate.
The isotherm distribution expectedly shows a monotonic decrease of temperature from the hot wall to the adiabatic surface.
Local heat transfer: The local Nusselt number along the hot wall is
presented in Fig. 6 for four different channel Rayleigh numbers,
Ra^{*} = 10^{2}, 10^{3}, 10^{4} and 10^{5},
for a constant amplitude of the sinusoidal obstruction a = 0.375, aspect ratio
A = 5 and L_{1} = 2.0. It is seen from this figure that the local Nusselt
number increases with distance up the wall, then decreases and then increases
again for all the range of channel Rayleigh numbers investigated. The lowest
values of local Nusselt numbers are at the two intersections between the obstruction
and the wall. As the velocity of the flow increases in the vicinity of the obstruction,
the heat transfer coefficient increases to a maximum value and then decreases
further up the channel where the velocity decreases to a minimum value and then
increases again.

Fig. 4: 
Isothermal lines (left) and streamlines (right) for A = 5,
L_{0} = 1, L_{1} = 2 and Ra* = 10^{3} for (a) a
= 0.500, (b) a = 0.450, (c) a = 0.375, (d) a = 0.250, (e) a = 0.125, (f)
a = 0.000 
The local Nusselt number is amplified when the channel Rayleigh numbers are
increased.
Effect of sinusoidal obstruction: The effect of the obstruction on heat
transfer is shown in Fig. 7 and 8 by the
curve of local Nusselt numbers along the wall. The profiles of the local Nusselt
numbers are presented for four values of the amplitude of the sinusoidal obstruction
(a = 0.0, a = 0.125, a = 0.25, a = 0.45) for Ra^{*} = 10^{3}
and Ra^{*} = 10^{5}. Since the heat flux transferred to the
air decreases with the temperature difference between the wall and the air and
the flow velocity is maximum in the portion of the channel where the section
is restricted, the local Nusselt number presents a peak at a Xcoordinate corresponding
to the top of the obstruction. This peak decrease with the amplitude of the
obstruction. The flow velocity is more marked for high Rayleigh numbers and
consequently the local Nusselt number increases with the amplitude of the obstruction.
The Nusselt number falls in the recirculation zones with the increase of the
Rayleigh number.

Fig. 5: 
Isothermal lines (left) and streamlines (right) for A = 5,
L_{0} = 1, L_{1} = 2 and Ra* = 10^{5} for (a) a
= 0.500, (b) a = 0.450, (c) a = 0.375, (d) a = 0.250, (e) a = 0.125, (f)
a = 0.000 

Fig. 6: 
Local Nusselt number variation with distance along the left
wall at different channel Rayleigh number for A = 5, a = 0.375, L_{0}
= 1, L_{1} = 2 
Figure 9 shows the variation of the average Nusselt numbers
as a function of the channel Rayleigh number for various values of the amplitude
of the sinusoidal obstruction, while the values of the inlet mass flow rate
reported in Fig. 10.

Fig. 7: 
Local Nusselt number variation with distance along the left
wall at different amplitudes of the sinusoidal obstruction for A = 5, Ra*
= 10^{3}, L_{0} = 1, L_{1} = 2 

Fig. 8: 
Local Nusselt number variation with distance along the left
wall at different amplitudes of the sinusoidal obstruction for A = 5, Ra*
= 10^{5}, L_{0} = 1, L_{1} = 2 
It is obvious that the average Nusselt numbers are largely below those of
smooth channel, although the heated surface is greater. The increase of the
amplitude (a = 0.0 to 0.5) lead to a loss of average heat transfer about 23%.
Clearly, the inlet flow rate of the obstructed channels is significantly lower
than that of the unobstructed channel. Increasing the channel Rayleigh number
intensifies these trends.
Figure 11 and 12 present the changes
in the average Nusselt number and in the mass flow rate as the amplitude of
the sinusoidal obstruction increases from 0 to 0.5 for a fixed obstruction length
of 1.

Fig. 9: 
Variation of the average Nusselt number versus channel Rayleigh
number for obstructed channels (L_{0} = 1, L_{1} = 2, A
= 5) 

Fig. 10: 
Variation of the inlet mass flow rate versus channel Rayleigh
number for obstructed channels (L_{0} = 1, L_{1} = 2, A
= 5) 
The value of Q for all cases decrease while showing the same trend, a sharp
decrease for lowest amplitude for the obstruction and a linear decrease from
a = 0.25 to 0.5. This decrease is mush greater for greatest value of the channel
Rayleigh number.
Obstruction allows having a better local heat transfer (at the level of the
obstruction). This transfer decreases when obstruction is moved towards the
exit of the canal (Fig. 13). For the large values of Rayleigh
number the transfer increases but in an almost identical way for the three locations
of the obstruction (Fig. 14).

Fig. 11: 
Variation of the average Nusselt number as a function of the
amplitude of the sinusoidal obstruction for three different channel Rayleigh
numbers (A = 5, L_{0} = 1, L_{1} = 2) 

Fig. 12: 
Variation of the mass flow rate as a function of the amplitude
of the sinusoidal obstruction for three different channel Rayleigh numbers
(A = 5, L_{0} = 1, L_{1} = 2) 
This is due to the recirculation zone which appears after the obstruction
for these Rayleigh numbers.
Effect of aspect ratio: To study the effect of aspect ratio on the heat
transfer rate, the average Nusselt number of the three different geometries
is plotted in Fig. 15.The Nusselt number is an increasing
function of the Rayleigh number and the aspect ratio A.

Fig. 13: 
Local Nusselt number profile along the left wall for three
different locations of the sinusoidal obstruction with Ra* = 10^{3},
A = 5, a = 0.375, L_{0} = 1 

Fig. 14: 
Local Nusselt number profile along the left wall for three
different locations of the sinusoidal obstruction with Ra* = 10^{5},
A = 5, a = 0.375, L_{0} = 1 
The exchange surface between the hot wall and the ambient environment is especially
important than the aspect ratio is high. The heat transfer between the fluid
and the wall being related to this exchange surface, it results from it that
the Nusselt number is all the more high as the aspect ratio is important. The
average heat transfer at A = 3 is over 34% lower than that for A = 5, while
for A = 5 is about 41 % less compared with that for A = 10.
Besides, the vertical velocity believes with A (Fig. 16) Therefore, the thermal exchange, which is proportional at this velocity, is all the more high as the aspect ratio is large.

Fig. 15: 
Variation of average Nusselt number with aspect ratio of the
channel for a = 0.250, L_{0} = 1, L_{1} = 2 

Fig. 16: 
Variation of the inlet mass flow rate with aspect ratio of
the channel for a = 0.250, L_{0} = 1, L_{1} = 2 
Correlation: Although the interest of the present study is to put in evidence the different behavior of the flow and the heat transfer according to various geometrical parameters of the obstructed channel, the results obtained by our numerical simulation can be exploited and presented in a form of correlations which can interest practical environment.
For illustrated well this and by means of the leastsquare regression analysis,
the average Nusselt number ,
for aspect ratio of the channel A = 5 with center location of the obstruction
can be correlated well with the channel Rayleigh number (Ra^{*}) and
amplitude of the sinusoidal obstruction (a) of the following form:
Where the coefficients a_{1}, b_{1} and c_{1} are listed
in table 1 for Ra^{* } = 10^{2} – 2.10^{5}
and A = 5.

Fig. 17: 
Comparison of computed and correlated results for average
Nusselt number versus channel Rayleigh number for obstructed channels. (L_{0}
= 1, L_{1} = 2, A = 5) 
It was found that the correlation must be bifurcated about the amplitude of
the sinusoidal obstruction to provide a good fit with the numerical results.
Figure 17 shows the typical behavior of .
The evolution of local Nusselt number was not able to be fitted by using the least squares technique because of its irregular variation due to the presence of the obstruction. To surmount this difficulty, it would have been desirable that the surface of the canal is constituted by sinusoidal obstructions so that the local Nusselt number varies in a regular way. This configuration which can be treated by our simulation code is not the object of this study.
CONCLUSIONS
We deferred some results concerning a numerical study of the influence of a sinusoidal obstruction on the transfers in a channel delimited by two vertical plates for a wide range of Ra. The presence of obstruction in the channel causes better local heat transfer rates due to increments in both the vertical flow and surface area on the obstructed wall. The numerical results show in particular that the influence of the obstruction on the transfers is all the more significant as its amplitude is high. For high values of the Rayleigh number, it appears a recirculation zone located in the higher part of the channel downstream from the obstruction. The local Nusselt number is maximum at the top of the obstruction, zone where the convectifs transfers are most intense. However, the average Nu decreases when compared to that for the unobstructed channel owing to the reduction of mass flow rate and the existence of stagnant region caused by the obstruction.
The performance of the methodology applied in the simulations has been also illustrated in terms of grid independency, computational effort and convergence rates. Nevertheless, the transient behavior at Rayleigh numbers higher than those considered in this work as well as the study of the influence of the aspect ratio of the channel lower than those studied here on the thermal and flow patterns need to be explored in further research. The logical future studies of this topic would consist in:
• 
To consider a threedimensional case. 
• 
To study the mixed convection case. 
• 
To study also, the case where the wall of the canal is completely covered
with sinusoidal obstructions. 
Nomenclature
a 
amplitude of the sinusoidal obstruction. a_{0} /d 
A 
aspect ratio of the channel. A = H/d 
d 
channel width, m 
F(X) 
geometric function 
g 
gravitational acceleration, ms^{2} 
h 
heat transfer coefficient, Wm^{2}K^{1} 
H 
channel height, m 
L_{0} 
obstruction length. l_{0}/d 
L1 
obstruction location from channel inlet. l_{1}/d 
Nu 
local Nusselt number 

average Nusselt number 
P_{m} 
dimensionless motion pressure 
Pr 
Prandtl number 
Q 
dimensionless mass flow rate 
Ra 
Rayleigh number 
Ra^{*} 
channel Rayleigh number (Ra^{*} = Ra/A) 
T_{0} 
ambient temperature, K 
U, V 
dimensionless velocity components 
X,Y 
Dimensionless coordinates 
Greek symbols
α 
thermal diffusivity, m^{2}.s^{1} 
β 
thermal expansion coefficient of fluid, k^{1} 
θ 
dimensionless temperature 
ρ 
density of fluid 
λ 
thermal conductivity of fluid, W.m^{1}.K^{1} 
v 
kinematic viscosity of fluid, m^{2}.s^{1} 
∆T 
temperature difference , K 
Subscript
0 
at ambient air 
w 
at wall 