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
259
A NUMERICAL STUDY OF THREE-DIMENSIONAL DARCY-
FORCHHEIMER (D-F) MODEL IN AN INCLINED RECTANGULAR
POROUS BOX
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- Forchheimer (D-F) flow model
are presented. The effects of non-linear inertia forces on natural convection in porous media
are examined. When nonlinear inertial terms are included in the momentum equation, no-slip
conditions for velocity at the walls are satisfied. The governing equations for the present
studies are obtained by setting Da=0 and Fc/Pr ≠ 0 in the general governing equations for D-F
flow description. The system is characterized by Rayleigh Number (Ra), two aspect ratios
(ARY, ARZ), ratio of Forchheimer number to Prandtl number (Fc/Pr) and the angle of
inclination φ. Numerical solutions have been obtained employing S A R scheme for 200< Ra
< 2000, 0.2 < ARY < 5, 0.2 < ARZ < 5 10-5
< Fc/Pr < 10-2
and –60o
≤ φ < 60o
.
Keywords: non-linear inertia, natural convection, Forchheimmer number& Prandtl number
efc
1.0 INTRODUCTION
Poulikakos and Bejan [1] employed Forchheimer extended Darcy flow model and
obtained boundary layer solutions for a tall cavity and the agreement with the numerical
results for Ra = 2000 and AR = 2 has been found to be good. Based on these studies,
Poulikakos and Bejan classified the flow regimes as Darcy, intermediate and non-Darcy.
Tong and Subramaniam [2] developed boundary layer solutions to Brinkman extended Darcy
flow model based on the modified Oseen technique. Beckermann et al. [3] demonstrated that
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 4, Issue 3, May - June (2013), pp. 259-265
© IAEME: www.iaeme.com/ijmet.asp
Journal Impact Factor (2013): 5.7731 (Calculated by GISI)
www.jifactor.com
IJMET
© I A E M E

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
260
the inclusion of both the inertia and boundary effects is important for natural convection in a
rectangular cavity packed with spherical particles. Prasad and Tuntomo [4] reported
numerical results for Forchheimer extended Darcy flow model for AR > 1. The study [4]
concluded that the inclusion of Forchheimer inertial term does not render boundary layer type
simplification and the influence on the flow field and heat transfer is significant. Satyamurty
and Rao [5] examined the relative influence of variable fluid properties and Forchheimer
extended Darcy flow model on average Nusselt number. Zebib and Kassoy [6] analytically
studied, using weakly nonlinear theory, the possibility of two and three-dimensional flow
patterns to exist in a rectangular parallelepiped. Holst [7] numerically solved transient, three-
dimensional natural convection in a porous box. They concluded that, compared to the two-
dimensional values, the three-dimensional heat flow under certain instances is higher at high
Rayleigh numbers Horne [8] investigated the tendency of the flow to be two or three-
dimensional. Chan and Bannerjee [9] studied three-dimensional natural convection using a
finite difference scheme based on the simplified marker and cell technique Dawood and
Burns [10] studied steady three-dimensional convective heat transfer in a porous box with
side heating numerically using multigrid method, which accelerated the convergence.
Dawood and Burns [10] studied steady three-dimensional convective heat transfer in a porous
box with side heating numerically using multigrid method, which accelerated the
convergence. Dawood and Burns reported numerical results have been reported for horizontal
and vertical aspect ratios, 0.5 < ARz, ARy < 20 at a Rayleigh number of 200. Vasseur et al.,
[11] studied analytically and numerically the thermally driven flow in a thin, inclined,
rectangular cavity filled with a fluid saturated porous layer. 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. [14]
2.0 MATHEMATICAL MODELLING
2.1 Governing Equation
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
261
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. The dimensionless conservation of mass,
momentum and energy are derived and non-dimensional parameters Ra, the Rayleigh
number, Fc, the Forchheimer number, Pr, the Prandtl number and Da, the Darcy number are
defined by -
ρ = 1 - β∆T (θ-0.5) ;
να
β TKgL
Ra
∆
= ;
L
K
Fc
′
= ;
α
v
=Pr ; 2
L
K
Da = (1)
After eliminating the pressure term from the derived non-dimensional momentum equations,
vector potential formulation equations are as follows –
x
2
xx Dacos
Z
Ra|V|
Z
|V|
V
Y
|V|
W
Pr
Fc
Ω∇+φ
∂
θ∂
−=








Ω+
∂
∂
−
∂
∂
+Ω (2)
y
2
yy Dasin
Z
Ra|V|
X
|V|
W
Z
|V|
U
Pr
Fc
Ω∇+φ
∂
θ∂
=








Ω+
∂
∂
−
∂
∂
+Ω (3)
z
2
zz
Dacos
X
sin
Y
Ra|V|
Y
|V|
U
X
|V|
V
Pr
Fc
Ω∇+





φ
∂
θ∂
−φ
∂
θ∂
−=








Ω+
∂
∂
−
∂
∂
+Ω
Energy equation can be rewritten as, (4)
2
2
2
2
2
2
ZYXZ
W
Y
V
X
U
∂
θ∂
+
∂
θ∂
+
∂
θ∂
=
∂
θ∂
+
∂
θ∂
+
∂
θ∂ (5)
The velocity components U, V and W are related to the components of the vector-potential
by,
ZY
U yz
∂
ψ∂
−
∂
ψ∂
= (6)
XZ
V zx
∂
ψ∂
−
∂
ψ∂
= (7)
YX
W xy
∂
ψ∂
−
∂
ψ∂
= (8)
Boundary Conditions on Vector Potential Ψ( )
Boundary condition on vector-potential Ψ( ) due to Hirasaki and Hellums [12] are given by,
0
X
zy
x
=Ψ=Ψ=
∂
Ψ∂ at X = 0, 1
0
Y
xz
y
=Ψ=Ψ=
∂
ψ∂ at Y = 0, ARY
0
Z
yx
z
=Ψ=Ψ=
∂
ψ∂ at Z = 0, ARZ (9)

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
262
Boundary Conditions on Vorticity-Vector (Ω )
(i) Vorticity vector (Ω ) at the walls for no-slip condition due to Aziz and Hellums [13] are
given by,
xΩ = 0; ;
X
W
y
∂
∂
−=Ω
X
V
z
∂
∂
=Ω at X = 0, 1
;
Y
W
x
∂
∂
=Ω ;0y =Ω
Y
U
z
∂
∂
−=Ω at Y = 0, ARY
Z
V
x
∂
∂
−=Ω ;
Z
U
y
∂
∂
=Ω 0x
=Ω at Z = 0, ARZ (10)
(ii) Vorticity vector Ω( ) at the walls when velocity slip is allowed are given by,
;
Z
V
Y
W
x
∂
∂
−
∂
∂
=Ω ;
X
W
y
∂
∂
−=Ω
X
V
z
∂
∂
=Ω at X = 0,1
;
Y
W
x
∂
∂
=Ω ;
X
W
Z
U
y
∂
∂
−
∂
∂
=Ω
Y
U
z
∂
∂
−=Ω at Y = 0, ARY
;
Z
V
x
∂
∂
−=Ω ;
Z
U
y
∂
∂
=Ω
Y
U
X
V
z
∂
∂
−
∂
∂
=Ω at Z = 0, ARZ (11)
Thermal boundary conditions on temperature (θ) are applied and 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,
∫ ∫
=∂
θ∂
−=
Y ZAR
0
AR
0
0xZY
h
XARAR
1
Nu dY dZ (13)
∫ ∫
=∂
θ∂
−=
Y ZAR
0
AR
0
1xZY
c
XARAR
1
Nu dY dZ (14)
RESULTS & DISCUSSION
When the non-linear inertial terms due to Forchheimer are included in the porous box
inclined at φ = -30o
, the strength of convection increases as can be seen from Fig. 2. There is
significant change in isotherms particularly in the core region which gives rise to higher heat
transfer. Variation of average Nusselt number with angle of inclination of porous box (φ) is
shown in Figs. 3 and 4.In the above plots as the angle of inclination increases from –60o
to +
60o
, the average Nusselt number first increases attains a peak (φ = -30o
) and then decreases.
For a given angle as Fc/Pr increases, average Nusselt number decreases. The change in
average Nusselt number is more pronounced when the angle of inclination is negative (-60o
to
0o
). For positive angle of inclination, the effect of Fc/Pr is negligible. As expected, the
average Nusselt number (Nu) increases with increasing Ra but the effect of Fc/Pr is just the
reverse due to increasing effect of inertial forces which increase with increase in Fc/Pr. In

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
263
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. 4, as the angle of inclination (φ) increases from –60o
to 60o
, the average Nusselt number
(Nu) values first increases and reaches peak at φ = -30o
and then decreases. The physical
reason/mechanism can be explained on the basis of flow fields and temperature fields shown
in Fig. 2 for φ = -30o
. It can be seen from the flow and temperature field that the natural
convection is more pronounced in case of φ = -30o
. The velocity gradient and temperature
gradient near the hot and cold walls are steeper for φ = -30o
. When φ > φc, the effect of
viscous forces becomes predominating and Nu values decrease. At very large φ, the effect
becomes significant and the mode of heat transfer becomes primarily conductive leading to a
constant Nu for all values of Fc/Pr [4].
Fig. 2: Iso-vector-potential ( xΨ ) for Ra = 1000, ARY =ARZ =1.0, Fc/Pr = 10-2
and φ = -30o
-60 -40 -20 0 20 40 60
0
2
4
6
8
10
12
14
16
18
20
22
Nu
φ
Fc/Pr=10
-2
Fc/Pr=10
-3
Fc/Pr=10
-5
Fig.3: Variation of Nu with ϕ for Fig. 4 : Variation of Nu with ϕ for
Ra=2000, ARy=1.0 and ARz=1.0 Ra=1000, ARy=0.5 and ARz=1.0

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
264
7.3 CONCLUSIONS
Numerical solutions to the equations governing natural convection heat transfer in an
inclined porous box have been obtained employing SAR scheme. The flow description is
within the framework of Darcy-Forchheimer flow model. The flow and temperature fields for
Darcy-Forchheimer flow description are similar to that of two-dimensional Darcy flow
model. The strength of natural convection is lower than the Darcy flow model due to inertia
effect. The average Nusselt number values are independent of horizontal aspect ratio (ARZ)
i.e. the case of two-dimensional flow. For Fc/Pr < 10-5
, the average Nusselt number values
are close that of Darcy flow model. The critical angle of inclination is –30o
that is same as
Darcy-Brinkman flow model
REFERENCES
[1] S. Ergun, Fluid flow through packed columns, Chemical Engineering Progress, pp.
89-94, 1952.
[2] Poulikakos and A. Bejan, The departure from Darcy flow in natural convection in a
vertical porous layer, Physics of Fluids, Vol. 28 pp.3477-3484, 1985.
[3] T.W. Tong and E. Subramanian , A boundary layer analysis for naural convection in
porous enclosure use of Brinkman extended Darcy flow model, International Journal
of Heat and Mass Transfer, Vol. 28 pp.563-571, 1985.
[4] V. Prasad and A. Tuntomo, Inertia effects on natural convection in a vertical porous
cavity, Numerical Heat Transfer, Vol. 11 pp.295-320, 1987.
[5] V.V.Satyamurthy and M.D. Rao, Relative effects of variable fluid properties and
non-Darcy flow on convection in porous media, ASME trans. HTD-96 ,pp.618-627,
1988.
[6] A Zebib and D.R. Kassoy, Three-dimensional natural convection motion in a confined
porous medium, Physics of Fluids, Vol.21 pp.1-3, 1978.
[7] P.H. Holst, Transient three-dimensional natural convection in confined porous media,
International Journal of Heat and Mass Transfer, pp. 73-90, 1972.
[8] R.N. Horne, 3-D natural convection in a confined porous medium heated from below,
Journal of Fluid Mechanics, Vol. 92, pp. 751-766, 1979.
[9] Y.T. Chan and S. Bannerjee, Analysis of transient three-dimensional natural
convection in a porous media, ASME Trans. Journal of Heat Transfer, Vol.103
pp.242-248, 1981.
[10] Amir S. Dawood and P.J. Burns, Steady three-dimensional convective heat transfer in
porous box via multigrid, Numerical Heat Transfer, Part A, Vol. 22 pp.167-198,
1992.
[11] P.Vasseur, M.G. Satish and L. Robillard, Natural Convection in a thin, Inclined
Porous Layer exposed to a constant Heat Flux , International Journal of Heat and
Mass Transfer , Vol. 3 pp.537-549, 1987.
[12] G.J. Hirasaki and J.D. Hellums, A general formulation of the boundary conditions on
vector potential in three-dimensional hydrodynamics. Q. Appl. Math., Vol. 16, pp.
331-342, 1968.
[13] K. Aziz and J.D. Hellums, Numerical solution of the three-dimensional equations of
motion for laminar natural convection. Physics of Fluids, Vol. 10, pp. 314-324, 1967.

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
265
[14] 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.
[15] 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.