Free convection in a wavy cavity filled with a porous

Free convection in a porous wavy cavity based on the Darcy-Brinkman-Forchheimer extended model

X.B. Chen, P. Yu, S.H. Winoto, and H.T. Low* Department of Mechanical Engineering National University of Singapore, Singapore 117576 *

Division of Bioengineering

Correspondence Author: Assoc Prof H.T. Low Address: Division of Bioengineering, National University of Singapore, Singapore 117576 Tel: (65) – 6874-2225 Fax: (65) - 67791459 Email: [email protected]

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Free convection in porous wavy cavity

Abstract: A numerical investigation is carried out for steady, free convection inside a cavity filled with a porous medium. The cavity has vertical wavy walls which are isothermal. The top and bottom horizontal straight walls are kept adiabatic. The numerical method is based on the finite volume method with body-fitted and non-orthogonal grids. A generalized model, which includes Brinkman term, Forcheimmer term and non-linear convective term, is used. Studies are carried out for a range of wave ratio

λ = 0 to 1.8, aspect ratio A = 1 to 5, Darcy number Da= 10−1 to 10−6 ,

and Darcy-Rayleigh number Ra ∗ = 10 to 105 . Results are presented in the form of streamlines, isotherms, and local and average Nusselt numbers. The generalized model which considers viscous, inertia and convective effects enables results to be obtained for a wider range of Darcy and Rayleigh numbers.

Key Words: Natural Convection; Porous Flow; Wavy Wall; Porous Medium Model

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X.B. Chen, P. Yu, S.H. Winoto and H.T. Low

NOMENCLATURE A

aspect ratio

a

amplitude of wave

Da

Darcy number

F

overall flux

K

permeability of porous medium, m 2

L

cavity height, m

Lref

characteristic dimension

Nu

local Nusselt number

Nua

average Nusselt number

Pr

fluid Prandtl number

p, p f

dimensionless local average and intrinsic average pressure

Ra

clear fluid Rayleigh number

Ra*

Darcy-Rayleigh number (=Ra Da)

S

source term

T

fluid temperature, K

Tc

temperature of cold wavy-wall (left), K

Th

temperature of the hot wavy -wall (right), K

T0

characteristic temperature of porous medium = TC , K

u, v

dimensionless velocity components along x- and y- axes, respectively

W

average width of cavity

x, y

dimensionless Cartesian coordinates

Greek symbols

α

thermal diffusivity

β

coefficient of volumetric thermal expansion

3


ε

porosity

εc

convergence error

γ

kinematic viscosity

λ

surface waviness

ρ

fluid density

ϕ

general dependent variable

Subscripts B

buoyancy source term

D

Darcy term

F

Forchheimer term

f

fluid

ref

reference

Superscript

∗

non-dimensional quantities

c

convection effect

d

diffusion effect

m

iteration time

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INTRODUCTION Free convection in porous media enclosure has many engineering applications, such as flow past porous scaffolds in bioreactors, drying process, electronic cooling, ceramic processing, overland flow during rainfall. In thermal insulation engineering, an appreciable insulating effect is derived by placing porous material in the gap between the cavity walls, and in multishield structures of nuclear reactors between the pressure vessel and the reactor. In practice, the cavity wall may be roughened or have protrusions attached to it to enhance the heat transfer process. Thus it is of interest to study the effects of surface undulations or waviness in porous cavities. There have been many previous studies on natural convection in porous enclosures [1, 2, 3, 4, 5]. The models and equations for single-phase flow through porous media have been reviewed by Hamdan [6], which include those of Darcy, Darcy-Lapwood, Brinkman, Darcy-Forchheimer, Darcy-Lapwood-Brinkman,

Darcy-Forchheimer-Brinkman,

Darcy-Lapwood

Forchheimer-Brinkman, and Rudraiah. The non-Darcian effects in natural convection are presented systematically in the works of Lauriat and Prasad [7], Ettefagh et al. [8], and Karimi-Fard et al. [9]. Non-Darcian effects represent deviations from the Darcy law, including the viscous and inertial effects, porosity variation effect, and the convective effect. They concluded that Darcy law is only valid when the pore Reynolds number is of order unity; and if the Rayleigh number is large enough, Darcy law will over predict the actual flow motion and the other effects cannot be neglected. Heat transfer in irregular geometry is also of interest recently. Kumar et al [10, 11], Murthy et. al [12] and Kumar and Shalini [13] have studied natural convection heat transfer in a cavity, enclosing a porous medium, which has one wavy vertical wall. Subsequently Mahmud and Fraser [14] considered a wavy cavity, consisting of two wavy-vertical isothermal walls and two horizontal adiabatic walls, which enclosed a viscous and incompressible fluid (clean fluid) inside. Recently, Misirlioglu et al. [15] extended the study of natural convection in the wavy cavity by considering that it enclosed a porous medium. Their numerical results were based on Darcy law and the stream-function method was used to solve the governing equations. Flow and heat transfer characteristics (isothermal, streamlines and local and average Nusselt numbers) were investigated in their study.

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Free convection in porous wavy cavity The objective of the present paper is to continue the study on a wavy cavity enclosing a porous medium by using the Darcy-Brinkman-Forchheimer extended model [16]. This extended model can depict the flow inside the porous region more generally by including the non-Darcian effects like viscous, intertia, and convective terms. It enables studies at higher Reynolds and Rayleigh numbers. A numerical method based on finite volume method with a collocated variable arrangement is developed, using body-fitted and non-orthogonal grids. The results are shown in terms of local and average Nusselt numbers, isotherms and streamlines at various aspect ratio, surface waviness and porosity. A wider range of Rayleigh number and Darcy number is considered.

GOVERNING EQUATIONS AND BOUNDARY CONDITIONS The flow and heat transfer characteristics inside a wavy cavity made of two straight horizontal adiabatic walls and two bent vertical isothermal walls are considered here, shown as in Figure 1. The height is L, interval spacing is W, and amplitude of the wavy bent wall is a. The porous medium temperature is initially T0 . The cavity is heated by the right wall at temperature TH and cooled by the left wall at temperature TC . The porous media is assumed to be rigid, homogeneous and isotropic; and saturated with the same single-phase fluid. The porous medium flow in this study is based on the Darcy-Brinkman-Forchheimer extended model [17, 18], which includes the convective, viscous and inertial effects. The governing equations, in non-dimensional forms, can be expressed as: Continuity equation:

∂u ∂v + =0 ∂x ∂y

(1)

X-momentum equation:

1

ε

{u

∂ u ∂ u ( ) + v ( )} = ∂x ε ∂y ε

1 ∂ Pr 1.75 (u 2 + v 2 )1/ 2 u Pr ∂ 2u ∂ 2u − u− + + (ε p f ) − ( ) 3/ 2 ε ∂x ε ∂x 2 ∂y 2 Da Da ε 150

(2)

Y-momentum equation:

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X.B. Chen, P. Yu, S.H. Winoto and H.T. Low 1

ε

{u

∂ v ∂ v ( ) + v ( )} = ∂x ε ∂y ε

1 ∂ Pr 1.75 (u 2 + v 2 )1/ 2 v Pr ∂ 2 v ∂ 2 v − v− + + (ε p f ) − ( ) + Ra Pr T 3/ 2 Da ε ∂y ε ∂x 2 ∂y 2 Da ε 150

(3)

Energy equation:

∂T ∂T ∂ 2T ∂ 2T +v = + u ∂x ∂y ∂x 2 ∂y 2

(4)

where ε is the porosity, T is the temperature of the porous medium, x, y are the dimensionless Cartesian coordinates, and u, v are the dimensionless velocity components along x- and y- axes, respectively. p f is the intrinsic average pressure. The local average and intrinsic average can be linked by the Dupuit-Forchheimer relationship, for example, p = ε p f . The following scales have been used for non-dimensionalisation,

x Lref

x* =

p* = Ra =

,

p ρα / L2ref 2

g β ΔTL3ref

γα

y Lref

y* =

,

,

T* =

, Pr =

u* =

u α / Lref

,

v* =

v α / Lref

T − T0 TH − TC

(5)

κ γ , Da = 2 Lref α

where γ is the fluid kinematic viscosity, K is the permeability, β is the coefficient of the thermal expansion, T0 = TC is the characteristic temperature, diffusivity,

Lref

α is the effective thermal

is the characteristic dimension, and ΔT = TH − TC . In practice, the

permeability K for porous medium ranges from 10 −4 to 10−8 . Note that the asterisks have been omitted from the dimensionless governing equations for the sake of convenience. In the momentum Eqs. (2) and (3), the second, third and fourth terms on the right hand side are called Darcy, Forchheimer and Brinkman terms respectively. For natural convection in porous media, another important parameter is the Darcy-Rayleigh number [9], defined as,

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Ra∗ = Ra Da =

gK βΔTLref

(6)

αγ

The boundary conditions are as follows,

∂T = 0, on y=0, 0 ≤ x ≤ W ∂n ∂T = 0, on y=L, 0 ≤ x ≤ W ∂n

(7)

T=0, on 0 ≤ y ≤ L , x = λ [1 − cos(2π y / L )] T=1.0, on 0 ≤ y ≤ L , x = 1 + λ[1 − cos(2π y / L )] where

λ = a / W is the surface waviness of the wavy cavity. The aspect ratio of the wavy cavity

is A = L/W.

NUMERICAL FORMULATION The discretization method for Finite Volume Method can be found in preivous literatures [19, 20, 21]. Here the discretization method for the momentum equations is given out, and similar procedure can be obtained for the energy equation. A typical control volume is shown in Figure 2. For a general dependent variable ϕ , a final discrete form over the control volume can be written as:

Fe + Fw + Fn + Fs = S

(8)

where Fe , Fw , Fn and Fs are the overall fluxes (including both convection and diffusion) of

ϕ

at faces e, w, n, s, which denote east, west, north, and south of the control volume; and S is

the source term. The detailed numerical methodology for obtaining the convective flux ( Fec , Fwc , Fnc , and Fsc ) and diffusive flux ( Fed , Fwd , Fnd , and Fsd ) are given in Ferziger and Perić [22]. To avoid the non-orthogonal effect, the midpoint rule with the deferred correction term [23] applied to the integrated diffusive flux is given by:

⎡⎛ ∂ϕ ⎞ ⎛ ∂ϕ ⎞ ⎤ ⎛ ∂ϕ ⎞ Pr ⎛ ∂ϕ ⎞ Pr Pr F = ( )e ⎜ ⎟ Se = ( ) e Se ⎜ ⎟ −⎜ ⎟ + ( ) e S e ⎢⎜ ⎟ ⎥ ε ⎝ ∂n ⎠e ε ε ∂ n ⎝ ⎠e ⎝ ∂ξ ⎠e ⎦⎥ ⎝ ∂ξ ⎠e ⎣⎢ d e

old

(9)

where Se is the surface area of face e; and φe is the value of φ at the center of the cell face. me and

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X.B. Chen, P. Yu, S.H. Winoto and H.T. Low Se can be calculated as:

Se =

(S ) + (S ) x 2 e

y 2 e

(10)

where Sx and Sy are the surface vector components. If an implicit flux approximation of the term

⎛ ∂ϕ ⎞ ⎜ ⎟ is applied, the final expression of Eq. (9) then becomes, ⎝ ∂ξ ⎠ e Fed = (

Pr

ε

ϕE − ϕP

)e Se

LPE

+(

Pr

ε

) e S e ( gradϕ )e

old

G G ⋅ ( n − iξ )

(11)

The different methods to approximate the value of φ and its derivative at the cell face result in different interpolation schemes. In the present study, the central difference scheme (CDS) is used. Then the cell-face values of the variables are approximated as:

ϕe ≈ ϕe′ = λeϕ E + (1 − λe ) ϕ P for face e where the interpolation factor

(12)

λe is defined as:

G G re − rP λe = G G G G re − rP + rE − re

(13)

G

where re is the position vector. Eq. (13) is a second-order approximation at the location e′ on the straight line connecting nodes P and E (Figure 2). If the cell-face center e does not coincide with the location e′ , a correction term needs to be added in Eq. (13) to restore the second-order accuracy, which can be expressed as follows:

G

G

ϕe ≈ ϕe′ + ( gradϕ )e′ ⋅ ( re − re′ )

(14)

To obtain the deferred derivatives at the cell face, they are calculated first at the control volume centers and then interpolated to the cell faces. By using the Gauss’ theorem, the derivative at the CV centers can be approximated by the average value over the cell:

⎛ ∂ϕ ⎞ ⎜ ⎟ ≈ ⎝ ∂xi ⎠ P

∫

Ω

∂ϕ dΩ G G ∂xi = ∫ ϕ ii ⋅ ndS ≈ ∑ ϕc Sci ， c = e, n, w, s S ΔΩ c

(15)

Then the cell-center derivatives can also be interpolated to the cell-face centers using the same interpolation as that described by Eq. (12) to (14).

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Free convection in porous wavy cavity The convective flux at a cell face is,

Fec =

1

ε∫

Se

uG G u ⋅ ndS ≈ meue ε e2

(16)

ε

where u and v are the velocity components in the x and y directions; me is the mass flux cross the surface e; me can be calculated as:

me = ( S x u + S y v )

(17)

e

The volume integral of the pressure gradient term is,

Qup, P =

⎛ ∂ (ε p f ⎜ − ε ∫Ω ⎜⎝ ∂x 1

) ⎞⎟

⎛ δ pf ⎞ dΩ ≈ − ⎜ ⎟ ΔΩ ⎟ δ x ⎠P ⎝ ⎠P

(18)

For the Darcy term in Eqs. (2) and (3), the volume integral can be expressed as:

⎛ Pr ⎞ ⎛ Pr ⎞ QDu = ∫ − ⎜ u ⎟ dΩ = − ⎜ ⎟ ΔΩ ⋅ u p Ω ⎝ Da ⎠ p ⎝ Da ⎠ p

(19)

For the Forchheimer term, the volume integral is given by:

⎛ 1.75 QFu = ∫ − ⎜ Ω ⎜ ⎝ 150

⎛ 1.75 u 2 + v2 u ⎞ Ω = − d ⎟ ⎜ 3/ 2 ⎜ 150 Da ε ⎟⎠ ⎝ p

u 2 + v2 ⎞ ⎟ ΔΩ ⋅ u p Daε 3/ 2 ⎟⎠

(20)

p

And in the Y-momentum equation, the buoyancy source term is given by,

QBu = ∫ ( Ra Pr T ) p d Ω = ( Ra Pr ) p ΔΩ ⋅ Tp Ω

(21)

It is convenient to treat the Darcy and Forchheimer terms as source terms. However, Eqs. (19) and (20) indicate that, after integrating, both terms become a product of Darcy velocity component and a coefficient. The two coefficients can be added into the coefficients of the algebraic u

equation Ap , which will accelerate the convergence rate. Then the final discrete form of the X-momentum equation is:

⎛δ p ⎞ APu uP + ∑ Alu ul = Qu*, P − ⎜ ⎟ ΔΩ ⎝ δ x ⎠P l

(22)

where P is the index of an arbitrary node; the index l denotes the four neighboring points E, W, S, N;

the coefficients APϕ , AEϕ , AWϕ , ANϕ , ASϕ are those of the resultant algebraic equations; *

and Qu , P is the integral of the source term contributed by other forces. In the present study, SIMPLEC method [21] is applied to couple the velocity and pressure. To avoid oscillations in the pressure or velocity, the interpolation proposed by Rhie and Chow [24] is

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X.B. Chen, P. Yu, S.H. Winoto and H.T. Low adopted:

⎛ 1 ⎜ m m ue = ( u ) − ΔΩ e ⎜ u u e ⎜ AP + ∑ Al l ⎝

⎞ ⎡ ⎟ ⎢⎛⎜ δ ( ε p f ⎟ ⎢⎜ δ x ⎟ ⎣⎝ ⎠e

) ⎞⎟ − ⎛⎜ δ (ε p f ⎟ ⎜ δx ⎠e ⎝

) ⎞⎟ ⎤⎥ ⎟ ⎥ ⎠e ⎦

m −1

(23)

RESULTS AND DISCUSSION Various computations were carried out for the following ranges of parameters: Darcy-Rayleigh number from 10 to 105 , aspect ratio A from 1 to 5, and surface waviness

λ from 0 to 1.8. During

the computation, a non-orthogonal, non-uniform and non-staggered grid system was implemented. Considering the computational cost and accuracy, a grid independence survey (detail not given here) shows that it is adequate to use a mesh composed of 3200 control volumes and 3444 nodes, with 40 control volumes along the width of the cavity and 80 control volumes along the height. To check whether the steady state solution is achieved, the variation of velocity and temperature distribution is observed between the consecutive time steps according to the following equation:

∑ϕ

m +1 i, j

− ϕim, j / ∑ ϕim, j+1 ≤ ε c

(24)

where ε c = 10 −6 is set here. The physical quantities of interest are the local Nusselt number, Nu, which is defined as,

Nu =

∂T ∂n

(25) wall

and the average Nusselt number, Nuα , which is obtained by integration of local Nusselt numbers over the wall as, s

1 Nua = ∫ Nuds S0

(26)

Here S is the wall where the Nusselt number is evaluated. For the validation of the numerical method used in this study, the results are compared with reference [25] and [26] for natural convective heat transfer in a porous fluid cavity, and a 80x80 mesh is used here. In Table 1, the prediction for

ε = 0.9999 , Pr=0.72 and Da = 107 (to model non-porous case)

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Free convection in porous wavy cavity are compared with benchmark results on natural convection inside a non-porous rectangular cavity by de Vahl Davis [25] and a porous rectangular cavity by Nithiarasu etc. [26]. In Table 2, our results for ε = 0.4, 0.6, 0.9 with different Rayleigh numbers and Darcy numbers are compared with the general porous model [26]. It is seen that the present values of Nua are in very good agreement with those obtained by previous authors, which validates the present numerical code. Figures 3a show the typical isotherm and flow streamline for aspect ratio A = 1 and the Rayleigh-Darcy number is varied from Ra∗ = 10 to 105 . It can be seen that hot fluid moves up along the hot wall (right wall) and turns to the left at the top adiabatic wall, then move down along the cold wall. This causes the circulation inside the cavity. Figures 3b and 3c show results for other aspect ratios of 3 and 5 respectively, also from Ra∗ = 10 to 105 . The other parameters are fixed at

λ = 0.5 , ε = 0.4 , and Da = 10−2 . Note that the present Darcy number is much higher

than those which assumed Darcy law, which is valid for Da ≤ 10 −4 [9, 15]. At low Darcy-Rayleigh number (Figure 3a), the flow circulation is very weak, which means a conduction-dominant flow and heat transfer. The result shows similar behaviours with those in a previous study [15]. However, some differences also exist especially near top and bottom regions at the right hand side. The present streamlines do not have so large curvatures as those based on Darcy law especially at Ra∗ = 105 . This is because Darcy model does not consider the convective, inertia, and viscous terms and thus the flow is able to turn abruptly at the corners, especially those at the right hand side of the cavity. At higher aspect ratio A=3 (Figures 3b), it is interesting that there is a main recirculating flow near the central region and two smaller recirculations near the top and bottom regions. The flow near the axial region is unable to recirculate to the top or bottom regions, due to the viscous effect. Had Darcy law, which has no viscous term, been assumed then there will just be a single main recirculation. It is noted that near the walls, where the natural convection is strong enough, the fluid is able to recirculate to the top unlike those near the axis. When the Ra* reaches 103 (Figure 3b middle), the convective effect is strong enough for the whole flow domain to have a single

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X.B. Chen, P. Yu, S.H. Winoto and H.T. Low recirculation. At large Ra∗ = 105 (Figures 3a, b and c, streamline diagrams on right) there are two recirculation loops at the middle region, instead of just one. The convection caused by the hot and cold walls is so large that each wall generates a recirculation flow. It is noted that near the middle region, along the x direction from cold to hot walls, the temperature gradient is negative near the axis. This is different from those at lower Ra* in which the temperature gradient is positive throughout the region from cold to hot walls. This explains the distortion of the recirculation streamline at the middle region. Figure 4 show the surface waviness effect on the flow and thermal fields at Darcy-Rayleigh number of Ra∗ = 10 to 105 . At low Darcy-Rayleigh number (Figure 4a), there is one main recirculation region if the cavity is less wavy (see streamline diagram on left).; however at large waviness value

λ = 0.6 there are smaller recirculation regions at the top and bottom in addition

to the main one at the middle region The flow near the axial region is unable to recirculate to the top or bottom regions, due to the viscous effect. However, the fluid near the walls is able to recirculate to the top, as the natural convection near the walls is stronger than those near the axis. When the Ra* reaches 105 (Figures 4a, b and c, streamline diagrams on right), the convective effect is strong enough for the whole flow domain to have a single recirculation. The effect of waviness seems to be similar to that caused by aspect ratio which has been discussed earlier. Both waviness and aspect ratio increase the length of the cavity and hence the viscous effect. However when the Darcy-Rayleigh number becomes large, it dominates over the viscous effect, so that the recirculation flow from the middle region is able to reach the top and bottom. The natural convection at large Darcy-Rayleigh number becomes so strong that it generates two recirculation regions at the middle as discussed earlier. The above results have been presented in terms of aspect ratio, waviness and Darcy- Rayleigh number. However, the Darcy number and porosity are also specified because the present model is not based on Darcy equation in which the permeability effect is included in the Darcy-Rayleigh number. The dependence on Darcy number and porosity will now be examined. To examine the dependence on Darcy number, the local Nusselt number along the cold wall is presented in Figures 5a and b. It is seen that for low Darcy-Rayleigh number (Figures 5a), the

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Free convection in porous wavy cavity local Nusselt number has negligible dependence on Darcy number which is thus not needed to be specified as another independent parameter. However at large Darcy-Rayleigh number (Ra*=103 in Figures 5b) there is a strong dependence on Darcy number. The Darcy number characterizes the permeability effect and Rayleigh number characterizes the natural convection effect. However the combined effects are not characterized by the Darcy-Rayleigh number when it is of large values. In addition to the Darcy-Rayleigh number, the Darcy number is needed as another independent parameter. It is seen that the Darcy number seems to change the local Nusselt number greatly, around an order of magnitude. This is because when Darcy number is changed, the Rayleigh number also changes in order to keep the Darcy-Rayleigh number constant. That is, the effects of permeability and natural convection are not isolated in Figure 5b. To see the real effects of Darcy number, instead of setting the Darcy-Rayleigh number constant, the Rayleigh number is made constant in Figures 6a and b. it is seen that at low Ra of 103 the effect of Da is negligible. However at high Ra of 105 there is obvious effect of Da. At larger Da, or larger permeability, there is smaller flow resistance through the porous medium, leading to larger flow convection and heat transfer. Thus the local Nusselt number increases with Da. However, when Da is less than 10 −4 , the Nu changes little with Da (see Figure 6b). This range of low Darcy number less than 10 −4 is called the Darcy flow region [9, 15]. The effect of porosity is examined next, as presented in Figures 7a and b. At larger porosity, with permeability constant, there is larger flow through the porous medium, leading to larger flow convection and heat transfer. Thus the local Nusselt number increases with porosity. The dependence is rather similar to that of Darcy number considered above. There is negligible dependence on porosity at low Darcy-Rayleigh number (Figure 7a) but when Ra* becomes large the dependence is noticeable (Figure 7b). The porosity, like Darcy number, is needed as another independent parameter. The dependence on Darcy number and porosity can be seen from the generalized momentum Eq. (3). The Darcy number is present in the Darcy and inertia terms; and the porosity is in the convective, inertia and viscous terms. However, when the Darcy equation is assumed, there is no dependence on Darcy number and porosity; also, the effect of Darcy number is included in the

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X.B. Chen, P. Yu, S.H. Winoto and H.T. Low Darcy-Rayleigh number. In Figure 8 the effect of aspect ratio on local Nusselt number is shown. For the case shown, a larger aspect ratio reduces the local Nusselt number. When the aspect ratio is small, A=1, the local Nusselt number along the middle region approaches zero, which indicates that the temperature gradient there is small (see Figure 3a isotherm) due to the shape effect. The near-zero Nusselt number is more obvious in Figure 9(a) in which the shape effect is more pronounced, with A=0.5 and

λ up to 1.8. It is interesting that at λ =1.4, slightly negative

Nusselt numbers are found, which indicates negative temperature gradient. Negative Nusselt numbers were also observed by Misirlioglu et al. [15]. However positive Nusselt number is always found at another aspect ratio, A = 4 (Figure 9b) in which the temperature gradient is large (see Figure 4b isotherm). The values of the average Nusselt number Nua at the cold wall (left) are shown in Table 3 for different Darcy-Rayleigh number Ra*, aspect ratio A and wave amplitude λ . In general, Nua increases with Darcy-Rayleigh number due to stronger natural convection. For each Darcy-Rayleigh number, the aspect ratio or wave amplitude has effect on the Nusselt number. Whether the Nusselt number increases or decreases may be due to the different flow regimes inside the cavity as explained by Mahmut and Fraser [14]. There may be an optimum aspect ratio or wave amplitude to achieve a peak Nusselt number which enhances the heat transfer.

CONCLUSIONS The steady free-convection inside a porous cavity, with wavy vertical walls, is numerically studied in this paper. The Darcy-Brinkman-Forchheimer extended model is used, which includes the viscous, inertia and convective effects. Different from the Darcy law, the generalised model is also valid for large Darcy-Rayleigh number. A body fitted, non-orthogonal grid is used with SIMPLEC method to solve the governing equations. Isotherms and stereamlines are presented at different aspect ratio, waviness, and Darcy-Rayleigh number. The present results show some interesting differences as compared with the Darcy law results due to its over prediction of the velocity field. The present flow field does not have so large curvature near the top and bottom regions. At low Darcy-Rayleigh number another recirculation zone may appear at both the top and

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Free convection in porous wavy cavity bottom regions, which are additional to the main recirculation at the middle. At large Darcy-Rayleigh number, the top and bottom recirculations vanished and the middle recirculation was distorted into two. The results of local Nusselt number show that the dependence on Darcy number and porosity is not small at large Darcy-Rayleigh number.

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REFERENCES 1. P. Vasseur, C.H. Wang, M. Sen, Brinkman Model for Natural Convection in a Shallow Porous Cavity with Uniform Heat Flux, Numer. Heat Transfer, Part A, 15, pp. 221-242, 1989. 2. S.W. Hsiao, C.K. Chen, Natural Convection Heat Transfer from a Corrugated Plate Embedded in an Enclosed Porous Medium, Numer. Heat Transfer, Part A, 25, pp. 331-345, 1994 3. P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Double-diffusive Natural Convection in an Enclosure Filled with Fluid-saturated Porous Medium: A Generalized non-Darcy Approach, Numer. Heat Transfer, Part A, 30, pp. 413-426, 1996 4. P. Bera, V. Eswaran, P. Singh, Numerical Study of Heat and Mass Transfer in an Anisotropic Porous Enclosure due to Constant Heating and Cooling, Numer. Heat Transfer, Part A, 34, pp. 887-905, 1998 5. E. Holzbecher, Free Convection in Open-top Enclosures Filled with a Porous Medium Heated from Below, Numer. Heat Transfer, Part A, 46, pp. 241-254, 2004 6. M.H. Hamdan, Single-Phase Flow through Porous Channels: A Review of Flow Models and Channel Entry Conditions, Applied Mathematics and Computation, 62, pp. 203-222, 1994 7. G. Lauriat and V. Prasad, Non-Darcian Effects on Natural Convection in a Vertical Porous Enclosure, Int. J. Heat Mass Transfer, Vol. 32, No. 11, pp. 2135-2148, 1989 8. J.Ettefagh, K. Vafai, and S.J. Kim, Non-Darcian Effects in Open-Ended Cavities Filled with a Porous Medium, ASME J. Heat Mass Transfe,r vol. 113, pp. 747-756, 1991 9. M.Karimi-Fard, M.C.Charrier-Mojtabi, and K.Vafai, Non-Darcian Effects on Double Diffusive Convection within a Porous Medium, Numer. Heat Transfer, Part A, 31, pp. 837-852, 1997 10. B. V. Ratish Kumar, P. Singh, and P.V.S.N. Murthy, Effects of Surface Undulations on Natural Convection in a Porous Square Cavity, ASME. J. Heat Transfer, 119, pp.848-851, 1997 11. B.V. Ratish Kumar, P.V.S.N. Murthy, and P.Singh, Free Convection Heat Transfer From an Isothermal Wavy Surface in a Porous Enclosure, Int. J. Numer. Meth. Fluids, 28, pp.633-661, 1998 12. P.V.S.N. Murthy, B.V. Rathish Kumar, and P. Singh, Natural Convection Heat Transfer From a Horizontal Wavy Surface in a Porous Enclosure, Numer. Heat Transfer Part A, 31, pp.207-221,1997

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Free convection in porous wavy cavity 13. B.V. Rathish Kumar, and Shalini, Free Convection in a Non-Darcian Wavy Porous Enclosure, Int. J. Engng. Sci., 41, pp.1827-1848, 2003 14. S. Mahmud, and R.D. Fraser, Free Convection and Entropy Generation inside a Vertical Inphase Wavy Cavity, Int. Commun. Heat Mass Transfer, 31 pp.455-466, 2004 15. A. Aydin Misirlioglu, Cihat Baytas, and I. Pop, Free Convection in a Wavy Cavity Filled with a Porous Medium, Int. J. Heat Mass Transfer, 48 pp.1840-1850, 2005 16. P. Yu, T. S. Lee, Y. Zeng and H.T. Low, A Numerical Method for Flows in Porous and Homogenous Fluid Domains Coupled at the Interface by Stress Jump. Int. J. Numer. Meth. Fluids,, in press, 2007 17. CT Hsu and P Cheng, Thermal Dispersion in a Porous Medium, Int. J. Heat Mass Transfer, vol. 33, pp. 1587-1597, 1990. 18. P Nithiarasu, KN Seetharamu and T Sundararajan, Finite Element Modelling of Flow, Heat and Mass Transfer in Fluid Saturated Porous Media, Arch. Comput. Meth. Engng., vol. 9, pp. 3-42, 2002. 19. S. V. Pantankar, and D.B.Spalding, A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic flows, Int. J. Heat Mass Transfer, vol. 15, pp. 1787-1806, 1972. 20. S. V. Pantankar, A Calculation Procedure for Two-Dimensional Elliptic Situations, Numer. Heat Transfer, vol. 4, pp. 409-425, 1981. 21. J. P. van Doormaal and G. D. Raithby, Enhancement of SIMPLE Method for Predicting Incompressible Fluid Flows, Numeri. Heat Transfer, vol. 7, pp. 147-163, 1984. 22. JH Ferziger and M Perić, Computational Methods for Fluid Dynamics, 2nd ed., pp. 222-233, Springer, Berlin, 1999. 23. S Muzaferija, Adapative Finite Volume Method for Flow Predictions Using Unstructured Meshes and Multigrid Approach. PhD Thesis, University of London, 1994. 24. CM Rhie and WL Chow, Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation, AAIA J., vol. 21, pp. 1525-1532, 1983. 25. de Vahl Davis, Natural Convection of Air in a Square Cavity: a Bench Mark Numercal Solution. Int. J. Numer. Meth. Fluids,, 3, 249-264, 1983. 26. P. Nithiarasu, K.N. Seetharamu and T. Sundararajan, Natural Convective Heat Transfer in a

18

X.B. Chen, P. Yu, S.H. Winoto and H.T. Low Fluid Saturated Variable Porosity Medium. Int. J. Heat Mass Transfer., Vol. 40, no. 16, pp. 3955-3967, 1997

19


Table 1. Comparison of present results (Pr=0.72) with [25], [26]

Rayleigh number

Nusselt number[25]

Nusselt number[26]

Present

103

1.116

1.127

1.117

10 4

2.238

2.245

2.248

105

4.509

4.521

4.531

106

8.817

8.800

8.877

20


Table 2. Comparison of Nusselt number along the hot wall (Pr=1.0) with [26]

ε = 0.4

ε = 0.6

ε = 0.9

Darcy

Rayleigh

Number

number

[26]

Present

[26]

Present

[26]

Present

10−2

103

1.01

1.01

1.015

1.012

1.023

1.020

10 4

1.408

1.362

1.530

1.500

1.64

1.63

105

2.983

2.990

3.555

3.445

3.91

3.92

105

1.067

1.064

1.071

1.070

1.072

1.071

106

2.55

2.60

2.725

2.714

2.740

2.801

107

7.81

7.86

8.183

8.648

9.202

9.49

107

1.079

1.078

1.079

1.078

1.08

1.08

108

2.97

3.05

2.997

3.081

3.00

3.08

109

11.46

12.39

11.79

12.98

12.01

13.30

10−4

10−6

21


Table 3. Average Nusselt number at the cold wall for different aspect ratios and wave amplitude (Da=0.01, ε = 0.4 )

Nusselt number for cold wall

A=4

λ = 0.5

Ra∗ = 101

Ra∗ = 103

Ra∗ = 105

λ =0

1.000

1.614

4.920

λ = 0.3

1.038

1.561

5.468

λ = 0.4

1.068

1.526

5.358

λ = 0.5

1.107

1.498

5.229

λ = 0.6

1.155

1.487

5.091

A=1

1.274

1.659

6.796

A=2

1.219

1.768

6.001

A=3

1.153

1.645

5.292

A=4

1.107

1.498

5.229

A=5

1.077

1.377

4.333

22


Figures Captions Figure 1. Schematic diagram of the porous cavity…………………………….………………...24 Figure 2. A typical two-dimensional control volume…………………………………….……...25 Figure 3. Isotherms (top) and streamlines (bottom) at different Darcy-Rayleigh number Ra∗ = 10 ,

103 , 105 (left to right); with λ = 0.5 , Da = 0.01, ε = 0.4 ; at a) A = 1, b) A = 3, c) A =5...26-28 Figure 4. Isotherms (top) and streamlines (bottom) at different waviness ratio λ = 0, 0.4, 0.6 (left to right); with A = 4, Da = 0.01,

ε = 0.4 ; at a) Ra∗ = 10 , b) Ra∗ = 103 ,

c) Ra∗ = 105 …………………………………………………………………….…………….29-31 Figure 5. Local Nusselt number along the cold wall and its dependence on Darcy number at a)

Ra∗ = 10 , b) Ra∗ = 103 ; other parameters are ε = 0.4 , A=4, λ = 0.5 ……………………..32-33 Figure 6. Local Nusselt number along the cold wall and its dependence on Darcy number at a)

Ra =103 , b) Ra = 105 ; other parameters are ε = 0.4 , A=4, λ = 0.5 ……………………..34-35 Figure 7. Local Nusselt number along the cold wall and its dependence on porosity at

Ra∗ = 10

b)

Ra∗ = 103 ;

other

parameters

are

fixed

at

a)

Da= 10 −2

,

A=4, λ = 0.5 ………………………………………………………………………….….…..36-37 Figure 8. Effect of different values of aspect ratio local Nusselt number along the cold walls; at

Ra∗ = 103 , Da= 10−2 , ε = 0.4 , λ = 0.5 ……………………………………………………….38 Figure 9. Effect of waviness on local Nusselt number along the cold walls; at Ra ∗

= 103 ,

Da= 10−2 , ε = 0.4 ; a) A=0.5, b)A=4…………………………………………………………39-40

23


2a

TH

L

TC

W Figure 1. Schematic diagram of the porous cavity.

24


y

G n ●N

● NW

n

G n ● W

G ey

● SW

G ex

ne

nw P

w sw

s

● NE

G n

e ● se

● E

e′

●

S ●

G n

SE x

Figure 2. A typical two-dimensional control volume.

25


Figure 3a Figure 3. Isotherms (top) and streamlines (bottom) at different Darcy-Rayleigh number Ra ∗ = 10 , 103 , 105 (left to right); with

λ = 0.5 , Da = 0.01, ε = 0.4 ;

at a) A = 1, b) A = 3, c) A =5.

26


Figure 3b Figure 3. Isotherms (top) and streamlines (bottom) at different Darcy-Rayleigh number Ra ∗ = 10 ,

103 , 105 (left to right); with λ = 0.5 , Da = 0.01, ε = 0.4 ; at a) A = 1, b) A = 3, c) A =5.

27


Figure 3c Figure 3. Isotherms (top) and streamlines (bottom) at different Darcy-Rayleigh number Ra ∗ = 10 ,

103 , 105 (left to right); with λ = 0.5 , Da = 0.01, ε = 0.4 ; at a) A = 1, b) A = 3, c) A =5.

28


Figure 4a Figure 4 Isotherms (top) and streamlines (bottom) at different waviness ratio λ = 0, 0.4, 0.6 (left to right); with A = 4, Da = 0.01,

ε = 0.4 ; at a) Ra ∗ = 10 , b) Ra ∗ = 103 , c) Ra ∗ = 105 . 29


Figure 4b Figure 4. Isotherms (top) and streamlines (bottom) at different waviness ratio λ = 0, 0.4, 0.6 (left to right); with A = 4, Da = 0.01,

ε = 0.4 ; at a) Ra ∗ = 10 , b) Ra ∗ = 103 , c) Ra ∗ = 105 . 30


Figure 4c Figure 4. Isotherms (top) and streamlines (bottom) at different waviness ratio λ = 0, 0.4, 0.6 (left to right); with A = 4, Da = 0.01,

ε = 0.4 ; at a) Ra ∗ = 10 , b) Ra ∗ = 103 , c) Ra ∗ = 105 . 31


1.5

Nu

1.2

0.9

0.6 Da=1e-1 Da=1e-2 Da=1e-4 Da=1e-6

0.3

0

0

1

2

3

4

Y Figure 5a Figure 5. Local Nusselt number along the cold wall and its dependence on Darcy number at a) Ra ∗ = 10 , b) Ra ∗ = 103 ; other parameters are ε = 0.4 , A=4, λ = 0.5 .

32


30

25 Da=1e-1 Da=1e-2 Da=1e-4 Da=1e-6

Nu

20

15

10

5

0

0

1

2

3

4

Y Figure 5b Figure 5. Local Nusselt number along the cold wall and its dependence on Darcy number at a) Ra ∗ = 10 , b) Ra ∗ = 103 ; other parameters are ε = 0.4 , A=4, λ = 0.5 .

33


1.6

Nu

1.2

0.8

0.4

0

Da=1.0e-1 Da=1.0e-2 Da=1.0e-4 Da=1.0e-6 0

1

2

3

4

Y Figure 6a Figure 6. Local Nusselt number along the cold wall and its dependence on Darcy number at a) Ra=103 , b) Ra = 105 ; other parameters are ε = 0.4 , A=4, λ = 0.5 .

34


5

Nu

4

Da=1.0e-1 Da=1.0e-2 Da=1.0e-4 Da=1.0e-6

3

2

1

0

0

1

2

3

4

Y Figure 6b Figure 6. Local Nusselt number along the cold wall and its dependence on Darcy number at a) Ra=103 , b) Ra = 105 ; other parameters are ε = 0.4 , A=4, λ = 0.5 .

35


1.5

Nu

1.2

0.9

0.6

ε =0.2 ε =0.4 ε =0.6 ε =0.9

0.3

0

0

1

2

3

4

Y Figure 7a Figure 7. Local Nusselt number along the cold wall and its dependence on porosity at a) Ra ∗ = 10 b) Ra ∗ = 103 ; other parameters are fixed at Da= 10 −2 , A=4, λ = 0.5 .

36


6

ε =0.2 ε =0.4 ε =0.6 ε =0.9

5

Nu

4

3

2

1

0

0

1

2

3

4

Y Figure 7b Figure 7. Local Nusselt number along the cold wall and its dependence on porosity at a) Ra ∗ = 10 b) Ra ∗ = 103 ; other parameters are fixed at Da= 10 −2 , A=4, λ = 0.5 .

37


10

A=1 A=2 A=3 A=4 A=5

8

Nu

6

4

2

0

0

1

2

3

4

5

Y Figure 8. Effect of different values of aspect ratio local Nusselt number along the cold walls; at

Ra ∗ = 103 , Da= 10−2 , ε = 0.4 , λ = 0.5 .

38


12

λ =0.5 λ =1.0 λ =1.4 λ =1.8

10

Nu

8

6

4

2

0 0

0.2

0.4

0.6

0.8

1

Y Figure 9a Figure 9. Effect of waviness on local Nusselt number along the cold walls; at Ra ∗

= 103 ,

Da= 10 −2 , ε = 0.4 ; a) A=0.5, b)A=4.

39


5

λ =0 λ =0.3 λ =0.4 λ =0.5 λ =0.6

4

Nu

3

2

1

0

0

1

2

3

4

Y Figure 9b Figure 9. Effect of waviness on local Nusselt number along the cold walls; at Ra ∗

= 103 ,

Da= 10 −2 , ε = 0.4 ; a) A=0.5, b)A=4.

40