Nomenclature

Effects of the bounding solid walls are examined numerically for slow flow overregular, square arrays of circular cylinders between two parallel plates. A local magnitudeof the rate of entropy generation is used effectively to determine the flow region affected bythe presence of the solid boundary. Computed axial pressure gradients are compared to thecorresponding solution based on the Darcy-Brinkman equation for porous media in whichthe effective viscosity appears as an additional property to be determined from the flowcharacteristics. Results indicate that, between two limits of the Darcian porous medium andthe viscous flow, the magnitude of μ (the ratio of the effective viscosity to the fluid ˆviscosity) needs to be close to unity in order to satisfy the non-slip boundary conditions atthe bounding walls. Although the study deals with a specific geometric pattern of the porousstructure, it suggests a restriction on the validity of the Darcy-Brinkman equation to modelhigh porosity porous media. The non-slip condition at the bounding solid walls may beaccounted for by introducing a thin porous layer with μ = 1 near the solid walls. ˆ

less than unity to as high as ~10 < for high porosity porous media [3,5,[8][9][10].The validity of the Darcy-Brinkman equation has also been a subject of investigation, particularly in relation to the boundary conditions at the solid-as well as fluid-interface [11,12].In the present study our subject of interest is flow over regular square arrays of circular cylinders bounded at the top and the bottom by solid plates.The flow in the absence of the bounding walls is studied in detail by Sangani and Acrivos [15].The analysis which solves the Navier-Stokes equations rather than the Darcy equation, yields a relation between the permeability of the regular array structure and the porosity (volume fraction occupied by the flow), confirming that the Darcy equation is valid for flow through regular structures over the whole spectrum of the porosity.Therefore, quantitative relations between the wall effects and the Darcy-Brinkman equation may be examined in a more focused manner through the present investigation.One of the objectives of the present analysis is to examine flow structure near the bounding walls.It is generally accepted that the effects of a solid boundary is confined within a thin boundary layer [12].Our second objective is to test the feasibility of addressing the thin region by using the Darcy-Brinkman equation.The Darcy-Brinkman equation in recent years is employed in biomedical hydrodynamic studies [7], including its use in modeling a thin fibrous surface layer coating blood vessels (endothelial surface layer) as it is a highly permeable, high porosity porous medium [13,14,17].A better understanding of the characteristics of the Darcy-Brinkman equation, therefore, is an important part of more practical problems; thus forming a motivation of the present report.

Analyses
As shown in Figure 1 where p = pressure, s u = superficial velocity, K = permeability of porous medium.In terms of non-dimensional variables defined as, / y y H = , / s m u u u = ( m u = mean velocity = / 2 q H , q = volume flow rate per channel width), the formulation and its solution become, where ˆ/ e μ μ μ = (viscosity ratio), / Da H K = .(5) The viscous flow limit corresponds to the case of ˆ1 μ = in the equation above.
The Darcian limit of flow through a porous medium is recovered as For the case of our interest, the porous medium consisting of regular square arrays of cylinders, the permeability, K , may be expressed as, (2 ) ( ) where C (solid fraction) = 1 φ − (φ = porosity), 2l = side length of a single square unit.(See Figure 1.)The function, ( ) f C , is given in an explicit form for high porosity arrays and in a graphical form for low porosity arrays in [15].8).
(b) Typical computational grid for a unit cell.
Flow, conditions of which lie between the two limits of Eqs.(5,6), is investigated numerically.Referring to Figure 2, governing equations and boundary conditions are, and on cylinder surface, (4.8b) Two commercially-available computational programs for hydrodynamic analyses are used in our numerical experiments, FLUENT (by Fluent Inc.) and FEMLAB (by COMSOL Group).The former is used as the main program; while, the latter is employed to confirm results from the former.Eq.( 8) is solved computationally as the porosity, φ , and the length ratio, / H l , are varied systematically.The porosity represents a fraction of the flow field in the cross sectional area of a ( 2 2  l l × ) unit cell; while, the ratio, / H l (integer), indicates a number of cell layers over the channel depth.Figure 3 is presented to depict the range of the porosity of our investigation for the case of / 10  H l = .For the high porosity channel flow of 99.99%The computational procedure is outlined below: 1. Solve Eq.( 8) and find the x-direction pressure gradient, / dp dx − ), for a specified value of the porosity at the length ratio, / 1 H l = (that is, the case of a single unit cell over the channel).(Actual computations are performed over multiple longitudinal cell columns (5 -10) to ensure that the periodic conditions at the cross-sectional cell boundaries are satisfied accurately.)

Increase the value of /
H l by one, and repeat the computation.
3. In the case of / 1 H l = , the solid wall affects the entire flow field.As the magnitude of / H l increases, cells near the center of the channel (y = 0 in Figure 1) become less sensitive to the presence of the solid walls.Computation for a fixed value of porosity is terminated when the size of the wall-affected region becomes independent of / H l .
Sangani and Acrivos [15] reports solutions of the velocity field in a single unit cell (i.e.flow through square arrays of cylinders without solid bounding plates) by applying the least square collocation method [4] to a series solution, which satisfies a part of the required boundary conditions exactly.We used this solution for confirmation of the validity of our computational results as well as for determination of the required numerical conditions (number and size of the computational mesh as well as the conversion criteria).The validity of the computational results is confirmed by recovering the permeability, K , (Eq.( 7)) reported in [15] over the range of 0.215 ( min imum φ (cylinders in contact with each other) 1 φ < < .When a unit cell becomes completely unaffected by the presence of the bounding walls, the velocity field should be identical to that of the solution in [15] everywhere in the cell.However, it is not easy to find the identity for the two-dimensional velocity field we are analyzing.Instead, the identity of the rate of entropy generation over a unit cell is used to find a degree of the wall effects on the flow field in the respective unit cell.

Results and Discussion
Figure 4 shows the axial velocity ( u ) profile over the channel half-depth at a cross sectional boundary between two lateral cell columns (with the number of cylinders per lateral column = four), as the porosity is varied.The centers of circular cylinders are located at x = l , 3l , 5l and 7l .The parabolic profile is computationally recovered for 100%  Table 1 lists computational results of the rate of entropy generation of each cell for the case of ten cells across the entire channel depth.Even at a very high porosity of 99.99% φ = , the plate effects are limited to the region within three cells from the plate wall with the affected region becoming even smaller as the porosity is decreased., the viscous effect of the flow around the cylinder remains important with the minimum value of 2 Da being limited to ~1.5 .(The computational lower limit is due to difficulties of the computational grid generation that is small in size and large in numbers sufficiently for accurate resolution of the velocity field near the cylinder.)The lowest value of 2 Da corresponds to the case of / 1 H l = (a single cylinder over the channel width) for each set of numerical results with a fixed value of φ .Also, results may not be presented in a continuous curve as an incremental change in the number of cylinders in the lateral direction leads to a step change in 2 Da .Figure 6 indicates that, between the Darcian and the viscous flow limit, all numerical results may be recovered by setting ˆ1., and that the effect of the bounding walls on the flow structure, which is confined to a narrow region near the walls, may be approximated by a porous medium with ˆ1 μ = .Figure 7  By setting / 0 L H = and 1 in Eq.( 10), Eq.( 4) is recovered for a porous layer with ˆI μ μ = and with ˆ1 μ = , respectively.The left hand side of Eq.( 10) is the ratio of the non-dimensional pressure gradient of the three layer model to that of a single layer (= the Darcian limit of / 0 L H = ).For a common pressure gradient, therefore, the mean velocity, m μ , of the three layer model is higher than that of the single layer.Greater the effective viscosity of the middle layer is relative to the fluid viscosity, more flow is channeled through the top (bottom) layer.The length scale of the top layer depth, L , is of the order of 6l in Figure 2, implying that the depth ratio, / L H , depends on the characteristic length scale of the porous structure (such as l in Figure 2) as well as on the macroscopic scale of the channel depth, H. Figure 8 shows Eq.( 10) as Da is varied with / L H as third parameter for the case of ˆ10 I μ = .For a case in which the wall layer thickness, L , is ~1/1000 of the channel half-depth, H , an increase of ~10% in the mean velocity across the channel is predicted at 2 3 10 Da = due to a higher flow velocity in the wall layer.Although a substantial reduction of the wall effect is observed as the magnitude of / L H is reduced from side of a unit cell ( Figure 1(b)) L depth of the top and the bottom layer in three layer model (Figure 7) p pressure q volume flow rate per channel depth / , G cell S rate of entropy generation over a unit cell per channel depth /// G S rate of entropy generation per channel width T temperature [K] u x-direction velocity of viscous flow u / the top and the bottom layer is Figure 7 (=1) ˆI Introduction The Darcy-Brinkman equation is a governing equation for flow through a porous medium with an extra Laplacian (viscous) term (Brinkman term) added to the classical Darcy equation.The equation has been used widely to analyze high-porosity porous media.The dynamic viscosity, e μ , associated with the Brinkman term is referred to as the effective viscosity.Studies in the past yielded varying results for the magnitude of the viscosity ratio, μ ( / e μ μ = with μ = fluid viscosity) between slightly , we consider a steady, incompressible, fully-developed, very slow ( Re (Reynolds number) 0 → ) flow across regular square arrays of circular cylinders, bounded by parallel plates.Governing equations based on the Darcy-Brinkman equation for porous media and the boundary conditions are,

Figure 1 .
Figure 1.(a) Flow between two parallel plates filled with regular square arrays of circular cylinders, (b) Regular square arrays of circular cylinders.

,Figure 3 .
Figure 3. Examples of flow field over a channel half-depth.

(
Figure 4(a)) as viscous flow between two parallel plates.Even at a very low solid fraction ( 4 (b)), the presence of cylinders is seen to affect the velocity profile substantially near the middle region of the channel; while, the velocity profile retains parabolic characteristics near the plate.It should also be noted that the effects of the cylinder adjacent to the wall (with its center at x l = ) are more significant for the low porosity cases.

Table 1 .
Variation of / , G cell S over Channel Half-Width.(Cell I in contact with the bounding plate at y=H in Fig 2(a))

Figure 6 .
Figure6summarizes the computational results of the (non-dimensional) axial pressure gradient vs 2 Da , Eq.(4), covering the porosity range up to 4).Our numerical range is limited to / 20 H l ≤.Classical Darcy's law is valid under the assumption of very slow ( Re 0 → ) flow through a layer of a porous medium in which the local (microscopic) length scale ( l in the present analysis) is sufficiently small compared to the overall (macroscopic) length scale ( H ). For the two cases of (a) 50% for Darcian limit.A corresponding thermal problem had been analyzed[2,6,16] for stability of a fluid saturated porous medium between two horizontal plates heated from below.Results, obtained under an assumption of e μ μ = ( ˆ1 μ = ), indicate that the Darcian limit is reached in terms of the critical Rayleigh number for the onset of convection if Our computational results along with Figure6indicate that the Darcy-Brinkman equation with the viscosity ratio substantially different from unity fails to satisfy the non-slip condition at bounding walls, particularly under the high porosity condition of

1 =
is a sketch of a three layer model, proposed to accommodate the wall effects in the analyses of a Brinkman porous medium with ˆ1 μ ≠ .It consists of two layers near the bounding walls with ˆB μ (viscosity ratio of the top and the bottom layer) , and a middle (interior) layer with ˆI μ (viscosity ratio of the middle layer) 1 ≠ .A solution for a parallel flow through the three layers may be sought from the Darcy-Brinkman equation for the top ( H L y H − ≤ ≤ ) as well as for the middle layer ( 0 y H L ≤ ≤ − ) under the conditions of the symmetry at 0 y = , the non-slip condition at y H = , and the velocity-and the shear stress-continuity at y H L = − .

Figure 7 .
Figure 7. Schematic Diagram of Three Layer Model.