Numerical Study of 2-D Natural Convection in a Square Porous Cavity : Effect of Three Mode Heating

The work we present in this paper is a continuation of a series of studies on the numerical study of natural convection in a square porous cavity saturated by a Newtonian fluid. The left vertical wall is subjected to a temperature varying sinusoidally in time while the right vertical wall is either at a constant temperature, or varying sinusoidally in time. The upper and lower horizontal walls are thermally adiabatic. Darcy model is used, it is also assumed the fluid studied is incompressible and obeys the Boussinesq approximation. The focus is on the effect of the modulation frequency (10 100 ω ≤ ≤ ) on the structure of the flow and transfer thermal. The results show that the extremal stream functions ( max ψ et min ψ ), the average Nusselt number at the hot ( h T ) and cold ( c T ) walls respectively Nuh and Nuc are periodic in the range of parameters considered in this study. In comparison with the constant heating conditions, it is found that the variable heating causes the appearance of secondary flow, whose amplification depends on the frequency of modulation of the imposed temperature but also of the heating mode. The results are shown in terms of streamlines and isotherms during a flow cycle.


ω ≤ ≤
) on the structure of the flow and transfer thermal.The results show that the extremal stream functions ( max ψ et min ψ ), the average Nusselt number at the hot ( h T ) and cold ( c T ) walls respectively Nuh and Nuc are periodic in the range of pa- rameters considered in this study.In comparison with the constant heating conditions, it is found that the variable heating causes the appearance of secondary flow, whose amplification depends on the frequency of modulation of the imposed temperature but also of the heating mode.The results are shown in terms of streamlines and isotherms during a flow cycle.

Introduction
The natural convection in confined porous media saturated by fluid is fundamental in the fields of engineering and physics.This interest arises from the importance of this heat transfer mode in various engineering fields such as storage of the thermal energy, solar energy collectors, thermal design for the buildings, cooling of electronic components (Alves and Altemani [1], Kuznetsov and Sheremet [2], Sheremet and Pop [3]), the underground spread of pollutants (Bagchi and Kulacki [4]).The literature on the convective flow in porous media is abundant.An excellent review of most of these studies is in the books (Nield and Bejan [5], Pop and Ingham.[6], Vafai [7], Ingham and Pop [8]) that give a complete overview of the current state of research in this area.However, most of these studies, theorical and experimental, on side convection, consider boundary conditions thermal constant (temperature or flux constant heat).Excellent review articles including one written by Baytas and Pop [9], give detailed accounts results obtained with these boundaries conditions.However, these boundary conditions do not reflect what is encountered in many practical situations where the temperature gradient is a function of time.This is the case of electronic components that dissipate power intermittently during operation on/of.Mastering the gradient behavior heat in these real situations can be used to control the flow convective.For example, it may be used to control the quality and structure of a solid resulting from solidification of an alloy by influencing the process of transport.The effect of the modulation of temperature on natural convection in cavity fluid medium was several time studied.For example, Schaladow et al.
[10] studied the case of a cavity subjected to a temperature which increases linearly each time.Both considered simulations (numerical and experimental) show that the flow and temperature field are very little affected by these boundary conditions.For their part, Kazmierczak and Chinoda [11] have numerically studied the effect of temperature varying sinusoidally in time on fluid flow and heat transfer in a square cavity.These authors showed that the heat transfer means in time is substantially insensitive to the periodic change of the wall temperature.Lage and Bejan [12], studied enclosures with one sidewall heated using a pulsating heat flux and the other sidewall cooled at constant temperature.They showed that at high Rayleigh numbers, the buoyancy-driven flow has the tendency to resonate to the periodic heating that has been supplied from the side.
Lage et al. [13], have studied the phenomenon of flow interference caused by discrete, solid objects placed inside fluid saturated enclosures under natural convection.Their analysis confirms and quantifies the predominance of the horizontal interference when the enclosure becomes shallow ( 1 A > ).Seyf and Rassoulinejad-Mousavi [14] proposed a new analytical solution for 2D Darcy-Brinkman equations in porous channels filled with porous media subjected to various boundary conditions at walls.The obtained ODE is solved analytically using homotopy perturbation method.It was shown that there is an excellent agreement between the presented models and the results of the CFD and previous works.Rassoulinejad-Mousavi and Yaghoobi [15] studied the effect of form drag term on the viscous dissipation through a parallel plate channel packed with a porous medium with isoflux or isothermal condition at the walls.Abourida et al. [16], have examined the effect of imposed sinusoidal temperatures and the thermophysical ones on the fluid flow and heat transfer within the cavity.They showed that all the obtained solutions concerning the fluid flow are periodic, with a period identical to that imposed to the variable temperatures.Note that many of these works consider cavities in which the temperature of cold wall is maintained constant.However, very little work is done on natural convection in a cavity filled with a porous medium with boundary conditions thermal periodic in time and using the Darcy model.Some of the documents dealing with this problem are, Saeid [17], Malomar et al. [18].The main objective of the present study is to contribute to the enrichment of the kind of problem by examining the effect of the imposed sinusoidal temperatures parameter (frequency) on the structure of the flow and transfer thermal.We study numerically natural convection unsteady in a square cavity, filled with a porous medium and the vertical walls are subjected at least to a temperature varying sinusoidally with time.The horizontal walls are thermally adiabatic and Darcy model was used.

Basic Equations
A schematic geometry of the problem is shown in Figure 1, where x′ and y′ are the cartesian coordinates and H ′ is the size of the walls.This is a square porous cavity two-dimensional and the horizontal walls are assumed to be thermally adiabatic.All walls of the cavity are assumed to be impermeable.At the same, the vertical walls, left wall (hot h T ) and right wall (cold c T ) are subjected at least to a temperature varying sinusoidally with time such that: ( ) ( ) where R T is the reference temperature (in our study R c T T′ = ), modulation, φ′ the phase angle and ω′ the modulation frequency.The fol- lowing three cases are considered: 1) only the left wall's temperature is modulated, the right wall is held at constant temperature; 2) the temperatures modulation are out-of-phase, i.e. π φ′ = ; 3) the temperatures modulation are in-phase, i.e. 0 φ′ = .The Darcy-Boussinesq approximation is employed.Iso- tropy, homogeneity and local thermal equilibrium in the porous medium are assumed.Under these assumptions, the equations governing the problem is the equation continuity, the equation of motion and the energy equation, respectively (see Nield and Bejan [5]): where u′ and v′ are the velocity components along x′ and y′ , T ′ is the fluid temperature, t′ is the time, ( ) The quantities σ and α are defined by ( ) where m k is the thermal conductivity (solid phase + fluid phase).
Equations ( 3)-( 5) are subject to the following boundary conditions and initial: One can introduce a stream function ψ ′ defined by u y 3) is satisfied.Then the governing Equations (3)-( 5) can be rewritten with the following dimensionless variables: τ ′ represents the dimensional period and is connected to the modulation fre- quency dimensional by the following equation 2π Substituting (7) in ( 3)-( 5) we obtain the following dimensionless governing equations: associated with initial and boundary conditions where ( ) represents the Rayleigh number.
At each time t the average Nusselt numbers at the vertical walls (hot and cold) are defined by, respectively:

Numerical Method
The equations of motion (8) and energy ( 9) associated with the boundary conditions (10) are discretized by a finite difference scheme, centred and accurate to the second order.The energy equation is then solved by the implicit method of alternating directions (ADI).The linear discretized equations were solved by Thomas algorithm.For equation of motion, the obtained linear discretized equation was solved by the sucessive over-relaxation method.Uniform grids have been selected in both the x and y direction.We have developed a numerical code with Fortran 95.
The calculation stops when between two time steps, the following condition is satisfied by the stream function: Preliminary tests on the influence of the mesh have allowed us to retain a uniform mesh size of 120*120.The time step used is 4 10 − .The present nu- merical code have been validated against the works of Walker and Homsy [19], Bejan [20], Beckerman et al. [21], Moya et al. [22]; Manole and Lage [23], Baytas and Pop [9] for the steady state natural convection in a square porous cavity with isothermal vertical and adiabatic horizontal walls.Table 1 shows the values of the average Nusselt number computed for various Rayleigh numbers in the range 3 10 10 − in comparison with other authors.Note that all numerical simulations are initialized by considering a conductive state and constant heating conditions.When steady regime is established, we introduce the excitatory temperatures and expecte the establishment of a periodic regime.

Influence of the Modulation Frequency
To highlight the effect of ω frequency modulation, we present the temporal evolution of the functions max ψ , min ψ , Nuc and Nuh for the three types of oscillatory heating under the following conditions 0.8 a = ;    creases when ω increases.We also note that the nature of sinusoidal oscil- lations is kept in Nuh .Figure 2(e) shows that for 30 ω = , Nuh takes nega- tive values during a brief portion of the cycle.
In the case where the temperatures excitatory evolve in phase opposition,  direction and of which intensity decreases when min ψ tends to 0. Unlike the case where only the temperature of the hot wall is variable, the sinusoidal nature of the excitation is not conserved for the different functions.Figures 3(d)-(e) show that during part of the cycle, the average Nusselt numbers Nuc and Nuh take negative values.These negative values found, are justified by the fact that for amplitudes ( 0.5 a > ) the cold wall acquires temperatures higher than the wall supposed to be hot.This then results in a transfer of heat from the cold wall inwardly of the cavity as soon as this wall becomes hotter than the surrounding fluid.Thereafter the fluid surrounding the hot wall becomes hotter than this wall it results a heat transfer from the fluid outwardly the cavity via the hot wall.
For the last mode of heating studied where excitatory temperatures are in because even for this range frequency, important differences qualitative and quantitative are observed.

Streamlines and Isotherms
To understand the details of flow and heat transfer in the cavity, we produced streamlines and isotherms for each of the three modes heating, during a flow cycle for 0.8 a = ; Before discussing the results for the oscillatory regime, we produced streamlines and isotherms in steady state ( 0 a = ) for 3 10 Ra = . Note that in the case of constant heating ( 0 a = ), the flow in the cavity remains monocell, consisting of cell rotating in clockwise direction and having a symmetry relative to the center of the cavity (see Figure 5).
In the case where only the hot temperature ( h T ) is modulated, Figures

Conclusions
The numerical study of unsteady natural convection in a porous cavity square whose side walls are subjected at least to a temperature varying sinusoidally with time was investigated.The mathematical model used is that of Darcy in the Boussinesq approximation.The algorithm was validated by direct comparison with previously published work and the results were considered in good agreement.Streamlines and isotherms were produced for heat to the outside environment.The evolution of temperatures in phase opposition is the best way to remove the heat to the outside environment.When oscillatory regime is imposed, important differences are noted in terms flow structure and of transfers heat unlike the case of constant heating; depending on the purpose, these differences can be exploited by the modeller.

Figure 1 .
Figure 1.Schematic diagram of the physical model and coordinate system.
Figure 2(a), Figure 3(a) and Figure 4(a) show the temporal evolutions of h T and c T for 30 ω = .Thus Figures 2(b)-(e) respectively show the temporal evolution of the

Figure 2 .
Figure 2. Effect of ω for 0.8 a = and

Figure 3 .
Figure 3.Effect of ω for 0.8 a = and

Figures 3 (
Figures 3(b)-(e) show that all the solutions obtained are periodic, periods equal to those the imposed excitatory temperatures.In the frequency range 10 30 ω ≤ ≤ , the amplitudes of the functions presented remain very close except for

Figure 4 .
Figure 4. Effect of ω for 0.8 a = and

phase, Figures 4
(b)-(e) show that only functions Nuc and Nuh retain the sinusoidal look and the period imposed on excitatory temperatures while max ψ and min ψ are periodic, with a period which is half that of excitatory temperatures.These figures also show that the amplitudes of functions increase when o mega increases except Nuc where the amplitudes are very close to the range of frequencies considered.However, we note that the most important effects of frequency modulation ω are rated for high values of this parameter ( 100ω >).But for convenience we chose to study the functions for 10 30 ω ≤ ≤ Figure 5. Streamlines (left) and isotherms (right) for

Table 1 .
Comparison of the average Nusselt number of the hot wall.