1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
554
A NUMERICAL ANALYSIS OF THREE-DIMENSIONAL DARCY
MODEL IN AN INCLINED RECTANGULAR POROUS BOX USING S A
R TECHNIQUE
Dr. R. P. Sharma
Dept. of Mechanical Engineering, Birla Institute of Technology, Mesra, Ranchi, 835215
India
ABSTRACT
In this paper, numerical studies on three- dimensional natural convection in an
inclined differentially heated porous box employing Darcy flow model are presented. The
relative effects of inertia and viscous forces on natural convection in porous media are
examined. The governing equations for the present studies are obtained by setting Da=0 and
Fc/Pr = 0 in the general governing equations for Darcy flow description. The system is
characterized by Rayleigh number (Ra), two aspect ratios (ARY, ARZ), and angle of
inclination (φ). Numerical solutions have been obtained by employing the S A R scheme for
different values of Rayleigh number (Ra), aspect ratio and angle of inclination. It is found
that for the Darcy flow model Nusselt number for 3-D and 2-D are same. Also there exists a
critical angle of inclination of the porous box at which the average Nusselt number become
maximum i.e. 30° in Darcy flow model.
Keywords: Darcy, SAR, Porous, critical angle of inclination etc.
1.0 INTRODUCTION
Owing to the relevance in many physical system of interest, investigation on fluid
flow and heat transfer in porous media are being widely reported in the literature. Gill [1]
reported on the stability aspects of an infinite vertical porous layer subjected to a temperature
difference. Gill employed the Darcy flow model and assumed constant fluid properties except
for density variation in evaluating the buoyancy force. This feature of accounting for the
variation in density in evaluating the buoyancy force only is commonly referred to as
Boussinesq approximation. The study [1] concluded that the thermal convection generated
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ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
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2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
555
flow is always stable. Experimental studies along with integral solutions for the rectangular
porous slabs have been reported by Klarsfeld [2]. Numerical results for Darcy flow model
also have been reported by Vlasuk [3]. Bories and Combarnous [4] reported experimental
results for Rayleigh numbers ranging from 100 to 1000. According to the numerical studies
of Bankvall [5], onset of boundary layer type flow occurred when the Rayleigh number is
greater than 200. Weber [6] developed an Oseen linearized solution for the boundary layer
regime for high aspect ratios. Weber’s study accounted for the variation in viscosity using an
average of hot and cold values. Seki, Fukusako and Inaba [7] correlated the average Nusselt
number with Ra, Pr and AR from the experimental results obtained with different solid fluid
combinations in the range 1 < Ra < 105, 1< Pr < 200 {Pr(= ν/α) is the Prandtl number} for
AR = 5,10 and 26. Comparison with the numerical results of Vlasuk [3] established that the
Darcy model is adequate to describe the fluid flow for Ra < 1000. Bejan and Tien [8] and
Walker and Homsy [9] obtained analytical results for small aspect ratios employing Darcy
flow model. Walker and Homsy [9] also presented numerical solutions for boundary layer
equations. Isothermal boundary conditions are considered, where two opposite walls are kept
at constant but different temperatures and the other two are thermally insulated. Three main
convective modes are found, conduction single and multiple cell convection and their
features described in detail. Local and global Nusselt numbers are presents as function of the
external parameters. Moya, et, al., [10] analyzed two-dimensional natural convective flow in
a titled rectangular porous material saturated with fluid by solving numerically the mass,
momentum and energy balance equations using Darcy's law and the Boussinesq
approximation. Sharma R P & Sharma R V has worked on modelling &simulation of three –
dimensional natural convection in a porous box and concluded that three-dimensional
average Nusselt values are lower than two-dimensional values. [11]
2.0 MATHEMATICAL MODELLING
2.1 Governing Equation
2.2
Fig. 1 Physical model and co-ordinate system

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
556
The physical model is shown in Fig. 1 is a parallelepiped box of length L, width B and height
H filled with fluid saturated porous medium.
Governing dimensionless equations for natural convection in the porous box comprising of
conservation of mass, momentum and energy are as follows:
0=
∂
∂
+
∂
∂
+
∂
∂
Z
W
Y
V
X
U
(1)
U+ φρ
β
sin||
Pr T
Ra
X
P
UV
Fc
∆
−
∂
∂
−= + Da 





∂
∂
+
∂
∂
+
∂
∂
2
2
2
2
2
2
Z
U
Y
U
X
U
(2)
V+ φρ
β
cos||
Pr T
Ra
Y
P
VV
Fc
∆
−
∂
∂
−= + Da 





∂
∂
+
∂
∂
+
∂
∂
2
2
2
2
2
2
Z
V
Y
V
X
V
(3)
W+
Z
P
WV
Fc
∂
∂
−=||
Pr
+ Da 





∂
∂
+
∂
∂
+
∂
∂
2
2
2
2
2
2
Z
W
Y
W
X
W
(4)
2
2
2
2
2
2
ZYXZ
W
Y
V
X
U
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
+
∂
∂ θθθθθθ
(5)
ρ = 1 - β∆T (θ-0.5) (6)
Non-dimensional parameters Ra, the Rayleigh number, Fc, the Forchheimer number, Pr, the
Prandtl number and Da, the Darcy number are defined by –
να
β TKgL
Ra
∆
= ;
L
K
Fc
′
= ;
α
v
=Pr ; 2
L
K
Da = (7)
Hydrodynamic Boundary Conditions
(i) Without Brinkman terms
U = 0 at X = 0, 1 for 0 ≤ Y ≤ ARY and 0 ≤ Z ≤ ARZ
V = 0 at Y = 0, ARY for 0 ≤ X ≤ 1 and 0 ≤ Z ≤ ARZ (8)
W = 0 at Z = 0, ARY for 0 ≤ X ≤ 1 and 0 ≤ Y ≤ ARY
Thermal Boundary Conditions
θ = 0 at X = 0, for 0 ≤ Y ≤ ARY and 0 ≤ Z ≤ ARZ
θ = 1 at X = 1, for 0 ≤ Y ≤ ARY and 0 ≤ Z ≤ ARZ (9)
0
Y
=
∂
θ∂
at Y = 0, ARY for 0 ≤ X ≤ 1 and 0 ≤ Z ≤ ARZ
0
Z
=
∂
θ∂
at Z = 0, ARZ for 0 ≤ X ≤ 1 and 0 ≤ Y ≤ ARY
Where ARY, the vertical aspect ratio and ARZ the horizontal aspect ratio are defined as

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
557
ARY = H/L (10)
ARZ = B/L (11)
Boundary conditions on temperature (θ) are the same as given by Eq. (9). The average
Nusselt number based on the characteristic length, L of the box is defined as,
k
Lh
Nu = (12)
The average Nusselt number at X = 0 and X = 1 is obtained by numerical integration
according to,
∫ ∫ =∂
∂
−=
Y ZAR AR
xZY
h
XARAR
Nu
0 0
0
1 θ
dY dz (13)
∫ ∫ =∂
∂
−=
Y ZAR AR
xZY
c
XARAR
Nu
0 0
1
1 θ
dY dZ (14)
In order to obtain the numerical solution of the above equations along with boundary
conditions, the Successive Accelerated Replacement (S A R) scheme has been employed and
results are obtained.
The successive accelerated replacement (S A R) scheme is a point iterative scheme.
The basic philosophy of the S A R scheme is to guess a profile for each variable which
satisfies the boundary conditions. Each dependent variable is associated with one governing
equation. It is natural to associate the equation for a variable which contains the highest order
derivative of that variable. For example, conservation of energy equation is associated for
temperature. The non-dimensionalised form of the governing equations with boundary
conditions is written in finite difference form using central differencing scheme. Let the finite
difference equation governing a variable φ be given by J.,IΦ = 0 at any mesh point (I, J)
corresponding to (X, Y) position. The guessed profile will not in general satisfy the equation.
The error arising out of the guessed profile is evaluated. Let the error in the equation at (I, J)
and at the nth iteration be n
J,IΦ . The (n+l)th
approximation to the variable φ is obtained from,
J,I
n
J,I
~
n
J,I
~
n
J,I
ln
J,I
/ φ∂Φ∂
Φ
ω−φ=φ +
(15)
where, ω is an acceleration factor which varies between 0 and 2. The procedure of correcting
the variable at every mesh point in the entire region is repeated until a set of convergence
criteria are satisfied. The criterion is,
ln
J,I
n
J,I
ln
J,I
+
+
φ
φ−φ
< ∈
where, ∈, the error tolerance limit is a prescribed small positive number. The feature of using
the corrected value of the variable immediately upon becoming available is inherent in this
method.

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
558
The average Nusselt number at X = 0 and X = 1 is obtained by numerical integration
according to,
∫ ∫ =∂
∂
−=
Y ZAR AR
xZY
h
XARAR
Nu
0 0
0
1 θ
dY dZ (16)
∫ ∫ =∂
∂
−=
Y ZAR AR
xZY
c
XARAR
Nu
0 0
1
1 θ
dY dZ (17)
RESULTS & DISCUSSION
Figs. 2 and 3 show iso-vector-potential lines (Ψz) and Figs. 4 and 5 show isotherms
(ө) for inclined porous box. For ϕ= - 45°, a multi-cellular flow is observed (see Fig. 2). For
positive angle of inclination (ϕ = 45°), unicellular flow exists, (Fig. 3). Compared to vertical
porous box, there is drastic change in temperature field particularly for negative angle of
inclination (Fig. 4). It can be seen from Fig. 4 that isotherms are confined around diagonal in
the X-Y plane of the porous box. Fig. 6 shows the variation of average Nusselt number (Nu)
with vertical and horizontal aspect ratios (ARY and ARZ) for ϕ = 45° at Ra=1000. It can be
noted from the above surface plots that average Nusselt number is independent of horizontal
aspect ratio (ARz). Hence, for inclined porous box, the heat transfer is two-dimensional like
the vertical porous box. Flow field, temperature field and heat transfer for natural convection
in porous box depends on the angle of inclination. There exists a critical angle of inclination
at which the average Nusselt number value is maximum (+30°) and then decreases for
Ra=200, 500, 1000 and 2000 (Fig. 7). When the Rayleigh number increases, the buoyancy
forces increases. The physical mechanism behind it is that when a horizontal layer of a
viscous fluid is heated, a temperature gradient is set up. Convective motions will commence
as soon as the vertical temperature gradient is greater than a given critical value.
7.3 CONCLUSIONS
Numerical solutions to the equations governing natural convection heat transfer in an
inclined porous box have been obtained using the SAR scheme. The flow description is
within the framework of Darcy’s law. The results obtained within the framework of Darcy
flow description are the same as the corresponding two-dimensional system. The non-
dimensional parameters needed to describe the system are Ra, ARy, ARz and ϕ. The
numerical solutions obtained include flow and temperature field and average Nusselt
number.Iso-vector potential lines and isotherms are similar to streamlines and isotherms for
two-dimensional system. Average Nusselt number values are independent of horizontal
aspect ratio (ARz). There exists a critical angle of inclination at which the average Nusselt
number is maximum. The critical angle of inclination is independent of Rayleigh number and
it depends on vertical aspect ratio (ARy). As ARy increases, the critical angle of inclination
(ϕ) decreases. For ARy=1.0, critical angle of inclination is 30°.

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
559
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Fig. 2: Iso-vector-potential ( ) for Ra=1000, Fig. 3: Iso-vector-potential ( ) for Ra=1000,
ARY=1.0, ARZ=1.0 and ϕ= -450 ARY=1.0, ARZ=1.0
Fig. 4: Isotherms for Ra=1000, ARY=1.0, Fig. 5 : Isotherms for Ra=1000, ARY=1.0,
ARZ=1.0 and ϕ= -450 ARZ=1.0 and ϕ= 450

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
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Fig. 6: Variation of Nu with ARY and Fig. 7: Variation of Nu with ϕ for
ARZ for Ra=1000 and ϕ= -450 ARY=1.0 and ARZ=1.0
REFERENCES
[1] A.E. Gill, A proof that convection in a porous vertical slab is stable, Journal of Fluid
Mechanics,Vol. 35 pp.545-547, 1969.
[2] S. Klarsfeld, Champs de temperature associes aux mouvements de convection
naturelle dans un milien poreux limite, Revue Gen, Thermique, Vol. 9 pp.1403-
1424, 1970.
[3] M.P. Vlasuk, Transfer de chaleur par convection dans une couche poreuse, In
Proccedings, 4th
All-Union Heat and Mass Transfer Conference, Minsk, USSR,
1972.
[4] S.A. Bories and M.A. Combarnous, Natural convection in a sloping porous layer,
Journal of Fluid Mechanics, Vol. 57 pp.63-79,1973.
[5] C.G. Bankvall, Natural convection in vertical permeable space, Warme-and
Stoffubertagung, Vol. 7 pp.22-30, 1974.
[6] J.E. Weber, The boundary layer regime for convection in a vertical porous layer,
International Journal of Heat and Mass Transfer, Vol. 18 pp.569-573, 1975.
[7] N.Seki,S. Fukusako, and H. Inaba, Heat transfer in a confined cavity packed with
porous media, International Journal of Heat and Mass Transfer, Vol. 21 pp.985-989,
1978.
[8] A. Bejan and C. L. Tien, Natural convection in a horizontal porous medium subjected
to end-to-end temperature difference, ASME Trans., Journal of Heat Transfer,
Vol.100 pp.191-198, 1978.
[9] K. L. Walker and G.M. Homsy, Convection in a porous cavity, Journal of Fluid
Mechanics, Vol. 97 pp.449-474, 1978.
[10] Sara. L. Moya, Eduardo Ramos and Mihir Sen, Numerical study of natural convection
in a tilted rectangular porous material , International Journal of Heat and Mass
Transfer, Vol. 30 pp.741-755, 1987.

8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
561
[11] R.P. Sharma, R.V. Sharma, “Modelling & simulation of three-dimensional natural
convection in a porous media”, International Journal of Mechanical Engineering and
Technology (IJMET), Volume 3, Issue 2, 2012, pp. 712-721, ISSN Print:
0976 – 6340, ISSN Online: 0976 – 6359.
[12] Dr. R. P. Sharma and Dr. R. V. Sharma, “A Numerical Study of Three-Dimensional
Darcybrinkman-Forchheimer (Dbf) Model in a Inclined Rectangular Porous Box”,
International Journal of Mechanical Engineering & Technology (IJMET), Volume 3,
Issue 2, 2012, pp. 702 - 711, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.