NUMERICAL TWO-DIMENSIONAL NATURAL CONVECTION IN AN AIR FILLED SQUARE ENCLOSURE, TILTED (25° AND 65°) IN RELATION TO THE HORIZONTAL PLANE, HEATED FROM TWO OPPOSITE SIDES FOR RAYLEIGH NUMBERS RANGING BETWEEN 10 AND 2.10

Steady and transient laminar two-dimensional natural convection of a Newtonian fluid in an inclined square enclosure was numerically investigated. The enclosure was heated on the opposite sides while it was cooled on the other two sides. The inclined angles were 25° and 65° to the horizontal plane. The effect of Rayleigh numbers ranging between 10 and 2.10 on the flow development and heat transfer was studied. It was found that Nusselt number increases with the increase of Rayleigh number. Under low Rayleigh numbers the numerical studies predict the onset of stationary bicellular flow. The study showed that when the Rayleigh number was increased, an overcritical Hopf bifurcation transformed the fixed point to a limit cycle and the steady-state flow becoming oscillatory.


INTRODUCTION
Natural convection in rectangular enclosures has been widely studied both numerically and experimentally due to its applications in numerous natural phenomena such as field temperature prediction in buildings and in industrial processes such cooling of electronics fittings.Natural convection steps as in thermal insulation of buildings with hollow bricks and doubling glazing, flat-plate collectors, cooling by natural radiation.Studies are numerical or experimental.Important reviews of such heat and mass transfer have been presented and discussed by Ostrach S, Bejan A., Yang K. T. and Berger P.
The most studied cases are rectangular cavities with one wall heated, the opposite side maintained cold and the two remaining sides assumed perfectly insulated.Muzishima et al have studied natural convection in a rectangular cavity at titled angle 0° and 1° in relation to the horizontal plane and differentially heated.They obtained a pitchfork bifurcation.Chen J. C. et al studying natural convection in a rectangular cavity heating from below and cooling from ceiling with adiabatic sides, showed that at low Grashof Gr numbers (Gr ≤ 10 3 ) the conductive heat transfer dominates while at high Grashof numbers (Gr ≥ 10 4 ) the convective heat transfer dominates.
Skouta A. and Skoua R. studied the convective heat transfer in a square enclosure titled at 45° in relation to the horizontal plane, heated from the opposite sides and cooled on the other.They showed that the larger the Rayleigh number is, the more sensitive the attractor becomes to time steps and mesh grids.The attractor bifurcates from a limit point to a limit cycle via an overcritical Hopf bifurcation for a Rayleigh number value between 1,11.10 5 and 1,12.10 5 .For tilt angle of 25° or 65°, the attractor bifurcates from a limit point to a limit cycle via an overcritical Hopf bifurcation for a Rayleigh number value equal 2.10 6 .Hamady F. J. et al studied numerically and experimentally the local natural convection in an air-filled differentially heated inclined enclosure for Rayleigh number between 10 4 and 10 6 .Measurements of local and mean Nusselt numbers are obtained at various inclination angles ranging between 0° (heated from above) and 180° (heated from below).They showed that the heat flux at the hot and cold boundaries had a strong dependence on the angle of inclination and the Rayleigh number.
In this investigation, natural convection in enclosure with aspect ratio 1 at inclined angle 25° and 65° were solved numerically by formulation the equations of transfer, of the vorticity and the stream function by central finite-difference (CFD) discretization, which are then solved by using an alternate direction implicit method (ADI).

MATHEMATICAL FORMULATIONS
The system under study is an air-filled square enclosure with vertical square section.Figure1 depicts its transversal section along the Cartesian coordinates (A, x, y).This enclosure is assumed to be very elongated along the horizontal Az direction and perpendicular to the right section.Its sides are inclined at an angle α = 25° or α = 65° with the horizontal plane.
Initially, the system is in thermodynamic equilibrium at temperature T c .At an initial time t 0 , two opposite walls are raised to a warm temperature T h while the sides BC and AD were maintained at a temperature T c with T h > T c .We assume that the fluid is Newtonian and incompressible.All the physical properties of the fluid are constant except the density in the buoyancy term, witch obeys the Boussinesq approximation so that the Prandlt number of air is fixed to 0.71.Radiation, viscous dissipation and pressure effects in the heat transfer equation were negligible.Under the above assumptions, the dimensionless unsteady governing equations in terms of temperature (T), vorticity (ω) and stream function (ψ), using the Cartesian coordinate system are: -Heat equation:

NUMERICAL RESOLUTION METHOD
The differential system (1 -6) is solved using finite difference method.The discretization scheme used is centred for the space derivative and first order forward for the time derivatives.The wall condition on the vorticity function is evaluated by extrapolation on the internal nodes according to the technique of Woods.The discretized forms of the temperature and vorticity equations are solved by means of an implicit method with alternate directions (A.D.I.) associated with the Gauss elimination method.The stream function is obtained by solving the equation (3) using a successive over relaxation method (S.O.R.) and the velocity field is inferred from stream function.
At each iteration, the test of onvergence is A same criterion has been imposed on temperature and vorticity.
For temperature the test of convergence is and for vorticity the test of convergence is:

Choice of space mesh and time step
At first time we have made calculations to test the sensibility of solutions to space mesh and time step for various Rayleigh numbers.Table 2 and figure 2 show the influence of space mesh for time step dt = 10 -5 . Table 2 assembles the values of global Nusselt number Nu h on hot walls, on cold walls Nu c , maximal stream function Psi max , minimal stream function Psi min , central stream function Psi m and central temperature T m for different space mesh for time step dt = 10 -5 while table 3 and figure 3 show the influence of time step on the same variables.The calculations show that when the Rayleigh number increases, the results are sensitive to the choice of space mesh and time step.When the solution is stationary, figures 2 and 3 as tables 2 and 3 show that the space mesh and the time step can be chosen respectively equal to 141 x141 and 9.10 -6 .

Fixed point
Even at very low Rayleigh numbers, when the heat transfer is essentially conductive, there is a movement due to the tilt of the walls.The dynamic parameters evolve to a stationary asymptotic limit.
For Ra ≤ 1,95.10 6 , the thermodynamic parameters evolve at long time towards a stationary asymptotic limit.This behaviour is illustrated on figure 4 which represents the temporal variation of maximal stream function (a), the trajectory in the phase plane (Psimax, Psi min ) (b), the streamlines (c) and the isotherms (d) for Ra = 1,8.10 6.We see that the flow involves two cells which rotate in the opposite direction (figure 4 (c)).One cell rotates clockwise (negative) while the other rotates counter clockwise (positive).Temporal variations of all parameters converge on a point.All thermodynamic parameters evolve according amortized oscillations before becoming stable at their average value at long time.If one builds up the trajectories of phase of these different parameters, one obtains spirals which start from exterior and lead each one to a point and the attractor is a fixed point.
When Rayleigh number increases from Ra = 1,958.106 , the oscillations of temporal curves dump down more and more with difficulty.
We have also tested the sensibility of the attractor fixed point to initial conditions.Figures 5 (a Progressively that Rayleigh number increases, natural convection expands and thermodynamic parameters increase also except central temperature Tm who decreases.Both temperatures Tm are symmetrical with regard to central temperature T 0 = 0.5 (Figure 6).
With complementary inclination angle of 65°, there is symmetry between the two results concerning the streamlines for example (figure 7).In case of inclination angle of 65°, one cell rotates clockwise (positive) while the other rotates counter clockwise (negative).These results have been obtained also by Skouta R.

Hopf bifurcation
For Ra > 1,958.10 6 , mesh grid was 151 x 151 and dt = 9.10 -6 .These results have been obtained as we have done in the fixed point case.When Rayleigh number increases from Ra > 1,958.10 6 , the flow expands and the computing time necessary to obtain the attractor is increasingly large.There is thus a critical Rayleigh number Ra c situated in interval [1,958.10 6; 1,96.10 6 ] from which the attractor is periodic as one sees on figure 8 who represents temporal evolution (a), the amplitude spectrum (b) of Tm and the trajectory in the phase plane (Tm, Psim) (c) for Ra = 1,96.10 6.The plotting of the amplitude spectrum obtained by Fast Fourier Transform (Figure 9 (c)) corroborates the existence of a limit cycle.There is a critical value of the Rayleigh number above which the attractor is periodic and independent of initial conditions (figure 9 (d)).
To determine the nature of the phenomenon corresponding to the transition from fixed point to limit cycle, we have studied both the variations of oscillatory amplitude versus the square root of the gap between (Ra.10 -6 -Ra c .10 -6 ) 1/2 in witch Ra c is the critical Rayleigh number and also the variations of the fundamental frequency in vicinity of bifurcation point.Figure 9 represents the variations of oscillatory amplitude versus (Ra.10 -6 -1,958.10-6 ).It shows that the amplitude of cycle is proportional to this gap, witch increases as the square root of the gap at bifurcation point.There two characteristics allow us to conclude that the bifurcation is an over critical Hopf bifurcation.the inclined (25°) enclosure.The effect of inclination angle on Nusselt number is more pronounced when the Rayleigh number increases.This is due to the fact that when the Rayleigh number increases the convection is the dominant mode of the heat transfer and the orientation of gravity vector with respect to the density gradient in the enclosure influences the convection and heat transfer more.

CONCLUSION
The present study has presented a numerical data showing the influence of inclination on thermodynamic parameters as global Nusselt numbers, stream function and temperatures distributions in the cavity for 25° and 65° tilt angles and Rayleigh number included in the interval [1.10 3 , 2.10 6 ].The implicit centred finite difference method used, allowed us to find with an excellent agreement the results of the literature concerning problems similar to that considered here when the attractor is a fixed point or limit cycle.Generally, the results are more sensitive to space and time steps when Rayleigh number increases.
It has been shown that when Rayleigh number increases, there is a critical value of it included in the interval [1,958.10 6, 1,96.10 6 ] where the flow undergoes a Hopf bifurcation.
Theoretically, the routes toward Hopf bifurcation for complementary angles (25 ° and 65° for example) are identical.If the cavity elements of symmetry are considered, it is sufficient to solve the Boussinesq equations for tilt angles included in the interval [0°, 45°].
In general, heat transfer increases with increase in Rayleigh number.From the predicted results, simple correlation for mean Nusselt number as a function of Rayleigh number is obtained for design applications.

334F
. P. Kieno, Unit of Formation and Research in Exact and Applied Sciences, University of Ouagadougou, 03 BP 7021 Ouagadougou 03, Burkina Faso A. Ouédraogo, Unit of Formation and Research in Exact and Applied Sciences, University of Ouagadougou, 03 BP 7021 Ouagadougou 03, Burkina Faso

.Figure 1 :
Figure 1: Schematic representation of the system section in the Cartesian frame (A, x, y) at 25° angle tilt.

Figure 2 :
Figure 2: Variations of thermodynamic variables for Ra = 1,8.10 6and dt = 1.10 -5 ) and (b) represent respectively temporal signal of Tm and the trajectory in the phase plane (Psimax, Tm).One shows that the two branches of solutions for figure 5 (a) and the three branches of solutions for figure 5 (b), corresponding to initial conditions of very different temperatures at starting, meet at long time.

Figure 4 :Figure 5 :
Figure 4: Representation of time evolutions of the maximal stream function (a), of the trajectory in the phase plane (Psim, dPsim/dt), (b), of the streamlines (c) and of the isotherms (d) for Ra = 1,8.106

Table 1 :
Results obtained by G. De Vahl Davis and those of present calculations.

Table 2 :
Variations of thermodynamic variables for different space mesh when the Rayleigh number Ra = 1,8.10 6and time step dt = 1.10 -5 .In brackets, the number indicates the relative gap.

Table 3 :
Variations of thermodynamic variables for different values of time step when Ra = 1,8.10