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International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx

ICHMT-02872; No of Pages 7

Contents lists available at ScienceDirect

International Communications in Heat and Mass Transfer

j ourna l homepage: www.e lsev ie r .com/ locate / ichmt

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Turbulent free convection in a porous square cavity using the thermalequilibirum model☆

Paulo H.S. Carvalho, Marcelo J.S. de Lemos ⁎Departamento de Energia — IEME, Instituto Tecnológico de Aeronáutica— ITA, 12228-900 São José dos Campos— SP, Brazil

O

☆ Communicated by Dr. W.J. Minkowycz.⁎ Corresponding author.

E-mail address: delemos@ita.br (M.J.S. de Lemos).

0735-1933/$ – see front matter © 2013 Published by Elsehttp://dx.doi.org/10.1016/j.icheatmasstransfer.2013.10.00

Please cite this article as: P.H.S. Carvalho, Mmodel, Int. Commun. Heat Mass Transf. (201

Oa b s t r a c t

a r t i c l e i n f o

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Available online xxxx 18

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Keywords:Turbulence modelingPorous mediaHeat transferNatural convection

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RThis work investigates the influence of porosity and thermal conductivity ratio on the Nusselt number of a cavityfiled with a fluid saturated porous substrate. The flow regime considered intra-pore turbulence and amacroscopic k-ε model was applied. Heat transfer across the cavity assumed the hypothesis of thermalequilibrium between the solid and the fluid phases. Transport equations were discretized using the control-volume method and the system of algebraic equations was relaxed via the SIMPLE algorithm. Results showedthat when using the one energy equation model under the turbulent regime, simulated with a High Reynoldsturbulence model, the cavity Nusselt number is reduced for higher values of the ratio ks/kf as well as when thematerial porosity is increased. In both cases, conduction thorough the solid material becomes of a greaterimportance when compared with the overall transport that includes both convection and conductionmechanisms across the medium.

© 2013 Published by Elsevier Ltd.

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1. Introduction

Thermal convection in porous media and the parameters that affectheat transfer across a heterogeneous medium have been studiedextensively in recent years. There are several applications in industryfor this type of technology. Examples are studies on grain storage,optimization of solar collectors design, safety of nuclear reactors anddesign of porous burners for industrial furnaces, to mention a few.Traditionally, modeling of macroscopic transport for incompressibleflows in porous media has been based on the volume-average meth-odology [1–4]. Additionally, if the flow fluctuates in time, the literaturepresents a number of time- and volume-averaging techniques thatfollow distinct sequences when applying both averaging operators[5–11]. Recently, a concept named double decomposition [12] showedthat the sets of macroscopic mass transport equations are equivalent,regardless of the order of application of the averaging operators.

When buoyancy forces are of concern, natural convection occurs inenclosures as a result of gradients in densities which, in turn, are dueto variations in temperature or mass concentration within themedium.

For clear cavities, the first turbulence model introduced forcalculating buoyant flows was proposed by Markatos and Pericleous[13]. They performed steady 2-D simulations for Ra up to 1016 andpresented a complete set of results. Ozoe et al. [14], in the light of thesame model adopted by [13], applied it to 2D calculations up to Ra=1011. Henkes et al. [15] compared two different turbulence models for2D calculations, namely the standard High Reynolds k-ε closure as

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.J.S. de Lemos, Turbulent free3), http://dx.doi.org/10.1016

well as the Low–Reynolds number form of the model. Further, Fusegiet al. [16] presented 3D calculations for laminar flow for Ra up to 1010

in a cube. The results revealed that the behaviors of the flow andcomparisons were made with 2D simulations. The differences werereported considering heat transfer correlation between Nu and Ra for2D and 3D cases. Later, Barakos et al. [17] also studied the problem ofnatural convection flow in a clean square cavity. The k-ε model hasbeen used for modeling turbulence with and without wall functions.

For cavities fitted with a porous material, the problem of freeconvection in enclosures with distinct temperatures applied on eachside of the cavity has been shown to represent a number of engineeringsystems of practical relevance. The monographs of Nield and Bejan [18]and Ingham and Pop [19] fully document natural convection in porousmedia. In addition, several articles published in the literature madeimportant contributions to the understanding of this problem [20–26].Baytas and Pop [27] considered a numerical study of steady freeconvection flow in rectangular and oblique cavities, filled with homo-geneous porous media using a nonlinear axis transformation. TheDarcy momentum and energy equations were numerically solvedusing the (ADI) method.

In the work of Braga and de Lemos (2004) [28], an approximatecritical Rayleigh was proposed comparing the behavior of Laminar andHigh Reynolds turbulence model solutions. The geometry thereinvestigated was a square cavity totally filled with a porous material,which was heated from the left and cooled from the opposing side.Also worth to mention is that the work in [28] was based on the localthermal equilibrium (LTE) hypothesis, which considers one uniquetemperature for both the fluid and the solid porous material. Othercases not involving gravity driven motion [29] have also been analyzedwith the laminar version of the LTE model detailed in [12]. Further, in

convection in a porous square cavity using the thermal equilibirum/j.icheatmasstransfer.2013.10.003

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T1:1 Nomenclature

T1:2 Latin charactersT1:3 cF Forchheimer coefficientT1:4 c′s Non-dimensional turbulence model constantsT1:5 cp Specific heatT1:6 D Deformation rate tensor, D=[∇u+(∇u)T]/2T1:7 Da Darcy number, Da ¼ K

H2

T1:8 D Particle diameter, DT1:9 g Gravity acceleration vectorT1:10 Gi Generation rate of ⟨k⟩i due to the action of the porousT1:11 matrixT1:12 Gβ

i Generation rate of ⟨k⟩i due to buoyant effectsT1:13 h Heat transfer coefficientT1:14 H Cavity heightT1:15 I Unit tensorT1:16 K Permeability, K ¼ D2ϕ3

144 1−ϕð Þ2T1:17 k Turbulent kinetic energy per unit mass, k ¼ u′ � u′=2T1:18 kf Fluid thermal conductivityT1:19 ks Solid thermal conductivityT1:20 Kdisp Conductivity tensor due to thermal dispersionT1:21 Kdisp,t Conductivity tensor due to turbulent thermal dispersionT1:22 Kt Conductivity tensor due to turbulent heat fluxT1:23 Ktor Conductivity tensor due to tortuosityT1:24 L Cavity widthT1:25 Nu Nusselt number, Nu ¼ hL

�keff

T1:26 Pi Production rate of ⟨k⟩i due to gradients of uD

T1:27 Pr Prandtl numberT1:28 Raf Macroscopic Fluid Rayleigh number, Raf ¼ gβϕH

3ΔTv f αeff

T1:29 Ram Darcy–Rayleigh number, Ram=Raf ⋅Da= gβϕHΔTKν f αeff

T1:30 Racr Critical Rayleigh numberT1:31 ReD Reynolds number based on the particle diameter, ReD ¼T1:32 ρ uDj jD

μ f

T1:33 T TemperatureT1:34 u Microscopic velocityT1:35 uD Darcy or superficial velocity (volume average of u)T1:36T1:37 Greek charactersT1:38 α Thermal diffusivityT1:39 β Thermal expansion coefficientT1:40 ΔV Representative elementary volumeT1:41 ΔVf Fluid volume inside ΔVT1:42 ε ε ¼ μ∇u′ : ∇u′ð ÞT=ρ, Dissipation rate of kT1:43 μ Dynamic viscosityT1:44 μt Microscopic turbulent viscosityT1:45 μtϕ Macroscopic turbulent viscosityT1:46 ν Kinematic viscosityT1:47 ρ DensityT1:48 σ′s Non-dimensional constantsT1:49 ϕ ϕ ¼ ΔV f

�ΔV , Porosity

T1:50T1:51 Special charactersT1:52 φ General variableT1:53 φ Time averageT1:54 φ′ Time fluctuationT1:55 ⟨φ⟩i Intrinsic averageT1:56 ⟨φ⟩v Volume averageT1:57

iφ Spatial deviationT1:58 |φ| Absolute value (Abs)T1:59 φ General vector variableT1:60 φeff Effective value of φ, φeff=ϕφf+(1−ϕ)φs

T1:61 φs,f solid/fluidT1:62 φH,C Hot/coldT1:63 φϕ Macroscopic valueT1:64 ()T TransposeT1:65

2 P.H.S. Carvalho, M.J.S. de Lemos / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx

Please cite this article as: P.H.S. Carvalho, M.J.S. de Lemos, Turbulent freemodel, Int. Commun. Heat Mass Transf. (2013), http://dx.doi.org/10.1016

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[28] it was also shown that lowDarcynumbers impact in higher averageNusselt numbers at the hot wall. However, in reference [28] simulationswere limited to a single solid-to-fluid thermal conductivity ratio,ks/kf=1, and a single porosity value, ϕ=0.8.

Motivated by the foregoing work, the contribution of this work is toextend the findings in [28] varying now the ratio ks/kf and the porosityφ. The turbulence model here adopted is the macroscopic k-ε withwall function in addition to the Low Reynolds number version of themodel. The findings herein broaden the simulations presented earlierin [28] since a greater number of heterogonous systems are nowinvestigated, leading to the analysis and optimization of a wider rangeof practical engineering systems.

2. The problem under consideration

The problem considered is showed schematically in Fig. 1a andrefers to a square cavity with sides L=H=1m completely filled witha porous medium. The cavity is isothermally heated from the left, TH,and cooled from the opposing side, TC. The other twowalls are thermallyinsulated. These boundary conditions are widely applied when solvingbuoyancy-driven cavity flows. The porous medium is considered to berigid and saturated by an incompressible fluid. The modified Rayleighnumber, Ram, is a dimensionless parameter used in porous media

ED P

RTH TC

dT/dy=0

dT/dy=0g

yx

H

L

a

b

Fig. 1. a) Geometry under consideration; b) 80 × 80 stretched grid.

convection in a porous square cavity using the thermal equilibirum/j.icheatmasstransfer.2013.10.003

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t1:2

t1:3

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t1:5

t1:6

t1:7

t1:8

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Table 2 t2:1

t2:2Comparison of laminar results for average Nusselt at hot wall, Nu Nuw , with Ram varyingt2:3from 10 until 104, Da=10−7, ϕ=0.8 and ks/kf=1.

t2:4Ram

t2:510 102 103 104

t2:6Walker and Homsy [20] – 3.097 12.96 51.0t2:7Bejan [21] – 4.2 15.8 50.8t2:8Beckermann et al. [23] – 3.113 – 48.9t2:9Gross et al. [24] – 3.141 13.448 42.583t2:10Manole and Lage [25] – 3.118 13.637 48.117t2:11Moya et al. [26] 1.065 2.801 – –

t2:12Baytas and Pop [27] 1.079 3.16 14.06 48.33t2:13Braga and de Lemos [28] 1.090 3.086 12.931 38.971t2:14Present results 1.087 3.093 13.041 39.288

3P.H.S. Carvalho, M.J.S. de Lemos / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx

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analysis and it is defined as Ram = RafDa, where Da = K/H2, αeff ¼keff�

ρ cpð Þ f . D is the particle diameter used to calculate the permeability,

K, and is here given by D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi144K 1−φð Þ2

φ3

r.

3. Governing equations

For turbulent buoyant flows, macroscopic governing equations areobtained by taking both volumetric and time averaging of the entireequation set. The final forms of the equations considered here aregiven in detail in [12] and for this reason their derivation need not berepeated. They read:

Continuity:

∇ � uD ¼ 0 ð1Þ

Momentum:

ρ∂uD

∂t þ∇ � uDuD

ϕ

� �� �¼ −∇ ϕ ph ii

� þ μ∇2uD þ∇ � −ρϕ u′u′

D Ei� �

−ρβϕgϕ T �i−Tref

� − μφ

KuD þ cFϕρ juDjuDffiffiffiffi

Kp

� �ð2Þ

Turbulent kinetic energy:

ρ∂∂t ϕ kh ii�

þ∇ � uD kh ii� � �

¼ ∇ � μ þμtϕ

σk

� �∇ ϕ kh ii� � �

þ Pi þ Gi

þ Giβ−ρϕ εh ii ð3Þ

Dissipation rate of turbulence kinetic energy:

ρ∂∂t ϕ εh ii�

þ∇ � uD εh ii� � �

¼ ∇ � μ þμ tϕ

σε

� �∇ ϕ εh ii� � �

þ c1Pi εh iikh ii

þc2εh iikh ii G

i þ c1c3Giβ

εh iikh ii −c2 f 2ρϕ

εh ii2kh iið4Þ

where the Dupuit–Forchheimer relationship,uD ¼ φ uh ii, has been usedand uh ii identifies the intrinsic (liquid) average of the local velocityvector u, ⟨k⟩i is the intrinsic average for k, ⟨ε⟩i is the intrinsic dissipationrate of k, K is the medium permeability, Gi ¼ ckρϕ kh ii juDj=

ffiffiffiffiK

pis the

generation rate of ⟨k⟩i due to the action of the porous matrix, Giβ ¼ φ

μtϕσ t

βkϕg �∇ T

�i is the generation rate of ⟨k⟩i due to buoyant effects, βϕ ¼ρβ T−Trefð Þh ivρϕ Th ii−Trefð Þ is the macroscopic thermal expansion coefficient and βk

ϕ ¼β u0T ′ �vϕ u0T ′

f

�i is the macroscopic thermal coefficient appearing in the Gβi

term. Here, for simplicity, we assume βϕk =βϕ=β (see [12] for details).

Further, Pi ¼ −ρ u′u′D Ei

: ∇uD is the production rate of ⟨k⟩i due to

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Table 1Damping functions and constants for High and Low Reynolds turbulence models.

High Reynolds modelproposed by Launderand Spalding [30]

Low Reynolds model proposed by Abe et al. [31]

fμ 1.01− exp − νεð Þ0:25y

14ν

h in o21þ 5

k2=νεð Þ0:75 exp − k2=νεð Þ200

� �2" #( )

f2 1.01− exp − νεð Þ0:25y

3:1ν

h in o21−0:3 exp − k2=νεð Þ

6:5

� �2" #( )

σk 1.0 1.4σε 1.33 1.3c1 1.44 1.5c2 1.92 1.9

Please cite this article as: P.H.S. Carvalho, M.J.S. de Lemos, Turbulent freemodel, Int. Commun. Heat Mass Transf. (2013), http://dx.doi.org/10.1016

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OFgradients of uD , the c's are constants and f2 is a damping function to

be commented upon later. The term −ρϕ u′u′D Ei

is known as themacroscopic Reynolds stress tensor (MRST) and is given by:

−ρϕ u′u′D Ei ¼ μtϕ

2 D �v−2

3ϕρ kh iiI ð5Þ

where

D �v ¼ 1

2∇ ϕ uh ii�

þ ∇ ϕ uh ii� h iTh i

ð6Þ

is the macroscopic deformation rate tensor. The macroscopic turbulentviscosity μ tϕ

is modeled as,

μ tϕ¼ ρ cμ f μ

kh ii2εh ii ð7Þ

where cμ is a constant and fμ is another damping function to bepresented below.

In a similar way, applying both time and volumetric average to themicroscopic energy equation and invoking the Local Thermal EquilibriumHypothesis, which considers as mentioned T f

�i ¼ Ts �i ¼ T

�i, a

modeled form for the macroscopic energy equation reads (see [12]),

ρcp�

fϕþ ρcp

� s1−ϕð Þ

� ∂ T �i∂t þ ρcp

� f∇ � uD T

�i� ¼ ∇ � Keff �∇ T

�in oð8Þ

where, Keff, given by:

Keff ¼ ϕkf þ 1−ϕð Þksh i

Iþ Ktor þ Kt þ Kdisp þ Kdisp;t ð9Þ

is the effective conductivity tensor. In order to be able to apply Eq. (8), it isnecessary to determine the conductivity tensors in Eq. (9), i.e. Ktor, Kt, Kdisp

and Kdisp,t. The turbulent heat flux and turbulent thermal dispersionterms, Kt and Kdisp,t are modeled such that,

−ϕ ρcp�

fu′ T ′

f

D Ei ¼ ϕ ρcp�

f

νtϕ

Prt∇ T f

D Ei ¼ Kt þ Kdisp;t

� �∇ T f

D Eið10Þ

Table 3 t3:1

t3:2Effect of simulation model on average Nusselt at hot wall, Nu Nuw , with Da=10−7, ϕ=t3:30.8, and ks/kf=1.

t3:4Ram

t3:5103 104 105 106

t3:6Braga and de Lemos [28] (HR) 13.032 40.614 101.647 237.546t3:7High Re turbulence model 13.271 41.792 101.503 234.930t3:8Low Re turbulence model, yW+=0.767 13.132 40.602 96.918 –

t3:9Laminar model 13.041 39.288 88.238 172.491

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where the symbolνtϕexpresses themacroscopic kinematic eddy viscosity

such that μtϕ¼ ρ f νtϕ

and Prt is a constant known as turbulent Prandtlnumber, which is often represented in the literature by σt. A furthersimplification is that, under the conditions here studied, Ktor and Kdisp

are negligible when compared to Kt Kt and Kdisp [28].

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3.1. Wall treatment and boundary conditions

In this work, two forms of the k-ε model are employed, namely theHigh Reynolds (Launder and Spalding [30]) and Low Reynolds number(Abe et al. [31]) turbulence models. The constants and formulae usedas damping functions are shown in Table 1. Boundary conditions aregiven by:

On the solid walls (Low Reynolds turbulence model):

u ¼ 0; k ¼ 0; ε ¼ ν∂2k∂y2

ð11Þ

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a

b

Fig. 2. Average Nusselt at hot wallNuw forϕ=0.8,D=1.06×10−3m,Da=10−7 and 80×80 stturbulence model, varying ks/kf.

Please cite this article as: P.H.S. Carvalho, M.J.S. de Lemos, Turbulent freemodel, Int. Commun. Heat Mass Transf. (2013), http://dx.doi.org/10.1016

OF

On the solid walls (High Reynolds turbulence model):

uuτ

¼ 1κ

ln yþE�

; k ¼ u2τ

c1=2μ

; ε ¼ c3=4μ k3=2w

κyw; qw ¼

ρcp�

fc1=4μ k1=2w T−Tw

� �Prtκ

ln yþw�

þ cQ Prð Þ� �

ð12Þ

with, uτ ¼ τwρ

� 1=2; yþw ¼ ywuτ

ν ; cQ ¼ 12:5Pr2=3 þ 2:12 ln Prð Þ−5:3 forPrN0:5 where, Pr and Prt are, as mentioned, the Prandtl and turbulentPrandtl numbers, respectively, qw is the wall heat flux, uτ is the wall-friction velocity, yw is the non-dimensional coordinate normal to wall,K is the von Kármán constant, and E is a constant that depends on theroughness of the wall.

4. Numerical method and solution procedure

The numerical method employed for discretizing the governingequations is the control-volume approach. Hybrid schemes, upwind

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retched grid: a) Laminar andHigh Reynolds turbulencemodels, ks/kf=1, b) High Reynolds

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differencing scheme (UDS) and central differencing scheme (CDS), areused for interpolating the convection fluxes. The well-establishedSIMPLE algorithm [32] is applied for handling the pressure–velocitycoupling. Algebraic equation sets for each variable were solved by theSIP procedure of [33]. In addition, the concentration of nodal pointsclose to the walls aims at capturing the boundary layers close to thesolid surfaces.

5. Results and discussion

To guarantee a grid independent solution, runs were performed usingstretched grids with 60 × 60, 80 × 80, 100 × 100 control volumes andRam=105. In these three cases, the average Nusselt number at the hotwall showed that the grid 80 × 80 is refined enough to capture theboundary layers at vertical surfaces, since the difference in the resultswas less than 2% when compared with similar simulations obtained

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a

b

Fig. 3. Effects of porosity on average Nusselt at hot wall, Nuw, using the HR turbu

Please cite this article as: P.H.S. Carvalho, M.J.S. de Lemos, Turbulent freemodel, Int. Commun. Heat Mass Transf. (2013), http://dx.doi.org/10.1016

F

with a finer grid. For example, for Ram=105 and for a grid with 80×80nodes (Fig. 1b), the average Nusselt at hot wall was calculated as101.503, whereas for a mesh of size 100×100, the computed value was103.38. As such, a grid of size 80×80was chosen for all simulations herein.

Further, the local Nusselt number on the hot wall for the squarecavity at x=0 is defined as,

Nu ¼ hL=keff∴Nu ¼ qWkeff

!x¼0

LTH−TC

ð13Þ

and the average Nusselt number is given by,

Nuw ¼ 1H

ZH0

Nudy: ð14Þ

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lent model, D=1.06× 10−3m and varying Ram: a) ks/kf=1, b) ks/kf=10.

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6. Laminar model solution

In order to calibrate the solution, runs were performed using thestretched grid shown in Fig. 1b. Results are presented shown inTable 2 and compared with the literature for φ=0.8 and Da=10−7.In all results, the Prandtl Number, thermal conductivity ratio betweensolid and fluid phases, fluid density and specific heat considered weretaken as unity. As one can see in Table 2, the present results agreewell with those reported in the literature.

7. Turbulent model solution

7.1. High Reynolds turbulence model

As Braga and de Lemos [28] pointed out, it is important to emphasizethat the main objective of this work is not to simulate the transitionmechanism from laminar regime to fully turbulent flow, but ratheridentify the ranges of validity of each model. Here, a 80×80 stretched

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a

b

Fig. 4. Overall heat flux along hot wall using the HR turbulent model, D=

Please cite this article as: P.H.S. Carvalho, M.J.S. de Lemos, Turbulent freemodel, Int. Commun. Heat Mass Transf. (2013), http://dx.doi.org/10.1016

F

grid has been used and the results are compared with those in [28] inTable 3 for Ram ranging from 10 to 106. Table 3 also presents resultsusing the Laminar model. As can be seen in Table 3 and Fig. 2a, thecritical Rayleigh number, Racr, is assumed to be the value when thetwo solutions deviate from each other. Here, the computed Racr alsoagrees with that proposed in Braga and de Lemos [28] as Racr = 104

for ks/kf=1.

7.2. Low Reynolds turbulence model

It is known that Low Reynolds turbulence models make use ofdamping functions and require that the first grid node close to thewall corresponds to a non-dimensional wall distance of about yw+≈ 1.Going back to Table 3, one can note that for Ram up to 104, calculatedNusselt are similar, regardless of the model used, namely HR, LR orLaminar models. After such value, here considered as a critical value,Racr, the role of the model used becomes important when obtainingNuw.

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1.06 × 10−3m, Ram=105 and varying ϕ: a) ks/kf=1, b) ks/kf=10.

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8. Effect of thermal conductivity ratio ks/kf

The ratio between the solid and fluid thermal conductivity plays animportant role in the study of energy transport across saturated porousmedia. To the best of the authors' knowledge, in all published materialsthe ratio ks/kf has been set to unit. However, for real engineeringproblems, such ratio may attain higher values and, for that, a study forobtaining Nu Nuw with a varying ratio ks/kf is here included.

Fig. 2b shows values for Nuw when ks/kf is varied from 1 to 20 fora range for Ram covering 10 to 106 and using the High Reynoldsturbulence model. As the ks/kf increases, the average Nusselt decreases.As conduction mechanism through the solid material becomes of ahigher importance, the relative contribution of the transport due toconvective currents is reduced, leading to reduction on the numericalvalue Nuw. In the limiting case, when ks/kf → ∞, the cavity Nusseltnumber would tend to attain unity reflecting the fact that conductionheat transfer, in such limiting case, would be the dominant mechanismof heat transport across the cavity.

9. Effect of porosity φ

Porosity is another important parameter to be consideredwhen heattransport across the cavity is investigated. For a cavity as in Fig. 1a, Nuwrepresents the ratio between the sum of convection and conductiontransport mechanisms over conduction transport alone. Therefore, fora purely conductive heat transfer process across the cavity, Nuw wouldattain the unity valid, as defined by Eqs. (13) and (14).

Fig. 3a indicates that as porosity decreases, Nuw increases anddifferences became higher for higher values of Ram. All runs in Fig. 3were performed using the High Reynolds turbulence model, the80×80 stretched grid in Fig. 1b and ks/kf=1. The same effect also occursfor ks/kf=10(Fig. 3b). One explanation for this behavior is that althoughthe convective effects increase as the porosity increases, on the overall,including the conduction through the solid, the total heat transfer acrossthe cavity is reduced, which, in turn, reduces Nuw (see the definition ofNuw by inspecting Eqs. (13) and (14)). Also, for the same value for Ram,comparing Fig. 3a and b shows that Nuw is reduced as ks/kf increases, aresult that is coherent with those shown in Fig. 2b where, for a largersolid-to-fluid thermal conductivity ratio, the relative importance ofconvective heat transport across the cavity is reduced, reducing thenNuw.

Finally, Fig. 4 shows that indeed a higher porosity implies a loweroverall wall heat flux qw along the hot wall at x=0, which, in light ofthe definition of Nuw by Eqs. (13) and (14), explains why the Nusseltnumber is reduced when ϕ increases. In spite of having more voidspace for the fluid to flow, the enhancement of the overall heattransport across the cavity does not occur as ϕ increases.

The reduction of Nuw when ϕ increases can be better understood byinspecting again Fig. 4a that is plotted for ks/kf = 1. In this case, theeffective conductivity, which is given by keff = ϕ kf + (1 − ϕ)ks orkeff=kf[ϕ+(1−ϕ)(ks/kf)], will always give keff/kf for ks/kf=1 regardlessof the value ofϕ. Therefore, the reduction on qw asϕ increases implies ina reduction of Nuw by means of its definition (Eqs. (13) and (14)). Asimilar reasoning applies when examining Fig. 4b for ks/kf=10.

10. Conclusion

Computations for laminar and turbulent flowswith themacroscopick-ε model with wall function for natural convection in a square cavityfully filled with porous medium were performed. Results indicate thatwhen using the one energy equationmodel under the turbulent regimesimulated with a High Reynolds turbulence model, the cavity Nusseltnumber is reduced for higher values of the ratio ks/kf as well as whenthe material porosity in increased. In both cases, conduction thoroughthe solid material becomes of a greater importance when comparedwith the overall transport that includes also convective heat transport

Please cite this article as: P.H.S. Carvalho, M.J.S. de Lemos, Turbulent freemodel, Int. Commun. Heat Mass Transf. (2013), http://dx.doi.org/10.1016

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due to fluid motion. The work herein might benefit the solution ofindustrial problems of practical relevance.

Acknowledgments

The authors are thankful to CNPq and CAPES, Brazil, for theirinvaluable financial support during the course of this research.

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