Comparative Study between the DarcyBrinkman Model and the Modified NavierStokes Equations in the Case of Natural Convection in a Porous Cavity
Razli MEHDAOUI^{*}, Mohammed ELMIR^{*}, Belkacem DRAOUI^{*}, Omar IMINE^{**}, Abdelkader MOJTABI^{***}
^{*}Vibration & Thermal Laboratory, University of Bechar, PO Box 417, Bechar, Algeria.
^{**}Faculty of Mechanical Engineering, U.S.T.Oran, Algeria.
^{***}IMFT, UPS, 118 route de Narbonne, 31062, Toulouse cedex, France.
Email(s): ramehd2002@yahoo.fr, elmir@mail.univbechar.dz, bdraoui@yahoo.fr, Imine_Omar@yahoo.fr, mojtabi@imft.fr
Abstract
The objective of this study is the comparison between the model of DarcyBrinkman and the NavierStokes equations modified, in the case of the natural convection. The study is made in a porous vertical square cavity saturated by a Newtonian fluid. A cylindrical heat source maintained at a uniform heat flux is introduced into porous medium. The equations which describe the thermal transfer and the hydrodynamic flow of the two models are solved numerically by means of the software package Femlab 3.2 based on the finite element method. The results obtained are in the form of average kinetic energy per unit mass, the local and the average Nusselt numbers, the pressure and the viscous force per unit area.
Keywords
Natural convection; Porous; Saturated; DarcyBrinkman; Femlab 3.2.
Introduction
Natural convection in porous media is of practical interest in several sciences, engineering, agriculture; energy’s stocking system, building, geothermal science, medical and biological sciences. We note that in the theoretical domain, several works related to natural convection in porous media was been examined. We quote those of Nield [1], Cheng [2], Combarnous and Bories [3]. We also find the problems of thermal sources flooded in a porous media studied by Bejan [4], Hickos [5] and Polikakos & Bejan [6]. An important number of studies of natural convection in confined and semiconfined porous media, especially those of Bejan and Khair [7], Weber [8], Masuouka and al. [9]. Convective processes of fluid flow and associated heat transfer in porous cavities have been studied extensively [1012]. These studies focus on the thermal convection performance within a heated porous cavity for different geometrical parameters (aspect ratio), heating mode (isothermal) [13]. The two dimensional free convection within a porous square cavity heated on one vertical side and cooled on the opposite side, while the horizontal walls are adiabatic, is currently considered a reference or benchmark solution for verifying other solution procedures. For a list of the basic references concerning this subject, we refer to the review articles by Ingham and Pop [14], Vafai [15], Pop and Ingham [16], Bejan and Kraus [17], Ingham and al. [18], Bejan and al. [19]. The Brinkmanextended Darcy model has been considered by Tong and Subramanian [20], and Lauriat and Prasad [21] to examine the buoyancy effects on free convection in a vertical cavity. Eungsoo and al. [22] have examined the nonDarcy effects of the natural convection in porous cavity.
In this fundamental study, we compare the two models (DarcyBrinkman and NavierStokes equations modified) and to visualize their influences on the natural convection in porous square cavity whose walls are isothermal. A cylindrical heat source maintained at a isoflux is introduced into porous media.
Physical and mathematical models
In the presence of a concentric cylindrical source of diameter D, submerged in porous media subjected to a uniform heat flow. We suggest studying the natural convection in a square cavity (Fig. 1) of side length H, in presence of concentric cylindrical source of diameter D, submerged in porous media subjected to a uniform heat flux q. The walls of cavity are kept at uniform temperature T_{w}.
Figure 1. Physical model
We suppose that the porous media is saturated, the flow is laminar, twodimensional and the fluid is Newtonian. The properties of the fluid and the porous media are constants. The viscous dissipation and the radiation are neglected. The Boussinesq’s approximation is valid.
_{} 
The flow field is governed by the DarcyBrinkman equation, the NavierStokes equation modified by using the extension of DarcyBrinkman [23], and the thermal field by the energy equation.
The volumeaveraged conservation dimensionless equations of mass, DarcyBrinkman, NavierStokes modified and energy are:
_{} 
(1) 
_{} 
(2) 
_{} 
(3) 
_{} 
(4) 
The governing equations are made dimensionless by adopting the following dimensionless quantities:
_{} 
In the above system, the following dimensionless numbers appear:
_{} 
k represents the thermal conductivity of porous media
_{}, where f is fluid properties and s is solid properties 
The hydrodynamic boundary conditions at express the impermeability and the no slip on the rigid walls and the contour of cylindrical source of the cavity. The external walls are maintained at uniform temperature all are adiabatic and the cylindrical source is maintained at uniform temperatures. These dimensionless boundary conditions are represented in table1:
Table 1. Boundary conditions
Faces 
V 
T 
External walls 
0 
0 
Concentric circular source 
0 
∂T/∂n=1 
The dimensionless average kinetic energy per unit mass is given by:
_{} 
(5) 
where S is the dimensionless domain.
The average Nusselt number, on the circumference source is defined by
_{} 
(6) 
where c is the dimensionless arclength.
The dimensionless viscous stress, on the circumference source is determined by the relation:_{}, where _{} is the tensor of the constraints and _{} the tensor of the rates of deformation.
Numerical method
Numerical results are obtained by solving the system of differential equations (1)–(4), with appropriate boundary conditions, using the Galerkin finiteelement method implemented through the software package Femlab 3.2. The twodimensional spatial domain is divided into triangular elements (unstructured mesh) and a Lagrangequadratic interpolation has been chosen. The mesh is refined near the boundaries and we have adopted 818 elements and the number of freedom degree is equal to 5681 (Fig.2). A nonlinear solver has been used and the nonlinear tolerance has been set to 10^{6}.
Figure2. Mesh grid
Results and Discussion
To describe the flow structure of natural convection in the porous cavity, using the two models, the following parameters are fixed:
A=0.25, Da=0.002, Pr=0.71, e=0.513
The effective viscosity in the Brinkman’s term is equal to the fluid viscosity m_{eff}=m what corresponds to R_{n} =1.
Ra^{*}
Figure 3. Evolution of the average kinetic energy per unit of mass vs.
modified Rayleigh number
Figure 3 represents the evolution of the average kinetic energy per unit of mass in whole domain, for various values of modified Rayleigh number Ra^{*}, by using the two models: DarcyBrinkman and modified NaviersStokes.
The two curves evolve in the same way and believe with the Ra^{*} increase. A light variation appears on the values between the two models for Ra^{*} >10^{5}, which justifies the results of table1. For Ra^{*} <10^{4}, the average kinetic energy E_{ka} is almost null, thus giving a static pseudo flow that favours the conductive transfer mode. Beyond these values, the flow is carried out with velocities important enough, thus giving a dominating convective transfer mode.
The results obtained for the two models are in good agreement thus making it possible to neglect the influence of convective term (V.Ñ)V in the NaviersStokes equations (See Table 2).
Table 2. Average kinetic energy
Ra^{*} 
3e2 
3e3 
3e4 
3e5 

E_{ka} 
DaBr 
19.193 
243.70 
1724.6 
10305 
N.S.m 
19.217 
245.45 
1737.4 
9905.0 

Error (%) 
0.12 
0.72 
0.74 
3.95 
Figure 4.a. Evolution of the local Nusselt number along the circumference of heat source  DaBr model
Figure 4 (a and b) represents the evolution of the local Nusselt number along the circumference of heat source, for various values of Ra^{*} and that for two models. The two curves evolve in the same way between two extremums (max, min). Nu_{l} increases with the Ra^{*} increase. A bifurcation appears for Ra^{*} > 3e3^{ }with the position (Arclength=0.6). With this position and for Ra^{*}=3e5, the precision on local Nusselt number between the two models is about 10%. This precision decreases with the Ra^{*} decrease.

Figure 4.b. Evolution of the local Nusselt number along the circumference of heat source  N.S.m model
Ra^{*}
Figure 5. Evolution of the average Nusselt number vs. modified Raleigh number
Figure 5 represents the variation of the average Nusselt number Nu_{m }with various values of Ra^{*} on the circumference of heat source. For the two models, Nu_{m} evolves in the same way. For Ra^{*}>1e4, a light variation on the values is distinguished. The values of Nu_{m} increases with the Ra^{*} increase allowing the increase in the rate of the convective transfer mode. The analysis of the values of Nu_{m} deferred in table2 enables us to conclude that convective term (V.Ñ)V, in the NaviersStokes equation, does not have almost any influence.
Table 3. Average Nusselt number
Ra^{*} 
3e2 
3e3 
3e4 
3e5 

Nu_{m} 
DaBr 
3.646 
6.498 
10.61 
19.92 
N.S.m 
3.647 
6.474 
10.47 
16.72 

Error (%) 
0.03 
0.40 
1.40 
1.20 
Ra*
Figure 6. Evolution of the maximum pressure in absolute value vs. modified Raleigh number
Figure 6 represents the variation of the maximum pressure in absolute value │P│_{max}, according to Ra^{* }on whole geometrical domain. Same remarks raised on the preceding figures. Except for Ra^{*} >3e4, a difference in pressure about 2e5 is noted (see Table3).
Table 4. Maximum absolute pressure
Ra^{*} 
3e2 
3e3 
3e4 
3^{e}5 

│P│_{max} 
DaBr 
0.55e5 
2.83e5 
1.80e5 
11.9e5 
N.S.m 
0.55 e5 
2.80 e5 
1.69 e5 
10.3 e5 

Error (%) 
0.18 
1.09 
6.11 
13.7 
Figure7 (a and b) represents the evolution of the viscous stress along the circumference of heat source for various Ra^{*} values. The pace of the evolution is the same one for the two models. For low Ra^{*} values, the viscous stress are almost null, what confirms nature pseudostatics of the flow thus giving a conductive mode transfer.
Figure 7. Evolution of the viscous stress along the circumference of heat source
(a) DaBr model  (b) N.S.m model
The curves present extremums (max, min) on the remarkable points of the circumference. The rate of deformation is important at the points (Arclength=0.4; 0.8) where the shearing of the fluid is important. For high Ra^{* }values the viscous stress are symmetrical in the DarcyBrinkman model, on the other hand they lose their symmetry in the modified NavierStokes model. For the various Ra^{*} values, the precision on the viscous stress is about 17%.
Conclusions
The natural convection in a porous vertical square cavity saturated by a Newtonian fluid is considered in this paper. A cylindrical heat source maintained at a uniform heat flux is introduced into porous medium. The walls of cavity are kept at uniform temperature T_{w}. The dimensionless forms of the continuity, DarcyBrinkman equation, modified NavierStokes equations and the energy equation are solved numerically using the Galerkin finiteelement method implemented through the software package Femlab 3.2.
From the results obtained in the form of average kinetic energy, average Nusselt number and pressure for various modified Rayleigh number values, the relative error between the two models does not exceed 6%. Except for Ra^{*}=3e5 which corresponds with Ra=1.5e8 (limit of the laminar flow), the error on the pressure can reach 13%. One can conclude that the convective term (V.Ñ)V does not have almost any influence on the results of the two models.
For the grid considered and by using a CPU of processor 1.7 GHz and 2 Go of RAM. The solution time for Ra^{*}=3e5 is of 82 seconds for modified NavierStokes model. On the other hand the solution time is of 65.5 seconds for the DarcyBrinkman model. Therefore, the DarcyBrinkman model allows a saving of time of 25% compared to that of modified NavierStokes model. It is preferable to use the DarcyBrinkman model.
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Nomenclature
A D Da DaBr E_{ka} H K k N.S.m Nu_{l} Nu_{m} OM^{*} P^{*} P* 
Aspect ratio; A = D/H [] Diameter of the cylinder source[m] Darcy number [] DarcyBrinkman Average kinetic enrgy [] Width of the square cavity [m] Permeability [m^{2}] Thermal conductivity [w(mK)^{1}] Modified Navierstokes Local Nusselt Number [] Average Nusselt Number [] Vector position [m] Pressure [N.m^{2}] Pressure dimensionneless [] 
Pr Ra Ra^{*} R_{n} T^{*} T V^{*} V
a e n μ (r_{f}C_{p}) 
Prandlt number [] Rayleigh number [] Modified Rayleigh number [] Viscosity ratio; R_{n} = m_{eff}/m_{f} [] Temperature [K] Temperature dimensionless [] Vector velocity [m.s^{1}] Vector velocity dimensionless [] Greek symbols: Thermal diffusivity [m^{2}.s^{1}] Porosity [] Kinematic viscosity [m^{2}.s^{1}] Dynamic viscosity [m^{1}.s^{1}] Heat capacity [j.m^{3}.K^{1}] 