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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≤ ω≤100) on the structure of the flow and transfer thermal. The results show that the extremal stream functions (
Ψ_{max} et Ψ_{min}), the average Nusselt number at the hot (
T_{h}) and cold (
T_{c}) 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.

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 [

A schematic geometry of the problem is shown in

where

modulation,

where

Equations (3)-(5) are subject to the following boundary conditions and initial:

One can introduce a stream function

Substituting (7) in (3)-(5) we obtain the following dimensionless governing equations:

associated with initial and boundary conditions

where

At each time t the average Nusselt numbers at the vertical walls (hot and cold) are defined by, respectively:

The equations of motion (8) and energy (9) associated with the boundary con- ditions (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

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

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.

To highlight the effect of

Thus Figures 2(b)-(e) respectively show the temporal evolution of the

Paper | Ra = 10 | Ra = 100 | Ra = 1000 |

Walker and Homsy [ | - | 3.10 | 12.96 |

Bejan [ | - | 4.20 | 15.80 |

Beckerman et al. [ | - | 3.11 | - |

Moya et al. [ | 1.065 | 2.80 | - |

Manole and Lage [ | - | 3.12 | 13.64 |

Baytas and Pop [ | 1.079 | 3.16 | 14.06 |

Prsent result | 1.078 | 3.12 | 15.60 |

functions

In the case where the temperatures excitatory evolve in phase opposition, Figures 3(b)-(e) show that all the solutions obtained are periodic, periods equal to those the imposed excitatory temperatures. In the frequency range

direction and of which intensity decreases when

For the last mode of heating studied where excitatory temperatures are in phase, Figures 4(b)-(e) show that only functions

To understand the details of flow and heat transfer in the cavity, we produced streamlines and isotherms for each of the three modes of heating, during a flow cycle for

Before discussing the results for the oscillatory regime, we produced stream- lines and isotherms in steady state (

In the case where only the hot temperature (

mum of the absolute value of

of center-left of the cavity and compared to the constant heating case (

of the cavity (

isotherms show a spacing between isotherms at the hot wall, which results in significant reduction in heat transfer, demonstrated by the average Nusselt numbers are near their minimum values. Once that the temperature of the hot wall again believes in leaving its minimum value, the secondary cell rotating in counterclockwise direction decreases in intensity and eventually disappear in favor of the formation of a cell rotating in clockwise direction, near the hot wall and which is added to the cell rotating in clockwise direction already in the vicinity of the cold wall (

In the case where temperatures evolve in phase opposition, Figures 7(a)-(f) show streamlines (left) and isotherms (right) at times corresponding to the letters a, b, c, d, e and f in

walls (

In case where the two temperatures are changing in phase, Figures 8(a)-(l) show the streamlines (left) and isotherms (right) corresponding to the instants a, b, … and l in

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 agree- ment. Streamlines and isotherms were produced for

Based on the results found in this study, we remark that the oscillatory 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. In addition the imposed oscillatory heating improves thermal transfer compared to the constant heating and constitutes the best way to remove the

heat to the outside environment. The evolution of temperatures in phase oppo- sition is the best way to remove the heat to the outside environment. When oscillatory regime is imposed, important differences are noted in terms of flow structure and of transfers heat unlike the case of constant heating; depending on the purpose, these differences can be exploited by the modeller.

We thank the Editor and the referee for their comments.

Malomar, G.E.B., Mbow, C., Tall, P.D., Gueye, A., Traore, V.B. and Beye, A.C. (2017) Numerical Study of 2-D Natural Convection in a Square Porous Cavity: Effect of Three Mode Heating. Open Journal of Fluid Dynamics, 7, 89-104. https://doi.org/10.4236/ojfd.2017.71007

('), Dimensional variables

(-), Average values

c, Cold

h, Hot

f, Fluid

min, Minimum value

max, Maximum value