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ASME-ATI-UIT 2015 Conference on Thermal Energy Systems: Production, Storage, Utilization and the Environment
17 – 20 May, 2015, Napoli, Italy
Keywords: Benchmark solutions, Finite element method,
Stability analysis, Laminar free convection, Time-periodic
oscillating flow field, Heat transfer coefficient calculation.
INTRODUCTION
Natural convection in fluid-saturated porous and partially
porous cavities is of great interest in many engineering
applications. In the scientific literature, few works are
available about heat transfer analysis in tall, partly porous
cavities in presence of natural convection [1,2]. The main
objective of the present paper consists of a better insight of
the transient buoyancy driven heat and fluid flow
phenomenon in porous and partly porous tall cavities. To this
purpose, the transient pressure, velocity and temperature
fields have been calculated by employing the Artificial
Compressibility (AC) version of the Characteristic Based
Split (CBS) algorithm for the resolution of the generalized
model for heat and fluid flow through saturated porous media.
The AC-CBS was first introduced by Chorin [3] and then
further developed [4,5]. The proposed numerical scheme has
been successfully stabilized by the authors and applied to the
simulation of both forced and natural convection steady-state
problems in free-fluid, porous and partly porous domains [5-
10]. The interest in the AC version of the CBS scheme for
incompressible fluid dynamic studies has increased thanks to
the advantages offered in terms of computational efficiency
with respect to implicit or semi-implicit procedure [11-16],
besides its easy implementation. To the authors knowledge,
numerical procedures adopted in the scientific literature for
the resolution of transient buoyant flows in porous or partly
porous tall cavities are only implicit or semi-implicit. These
procedures are very demanding from the computational point
of view especially in presence of a porous layer, that requires
a very small time step size applied to all nodes. On the other
hand, a proper stabilized matrix inversion free procedure, like
the one employed in the present AC-CBS algorithm, employs
an adaptive local time step, that allows to speed up the
solution process even in presence of large source terms
(porous domains, buoyant forces, etc.). In order to obtain a
transient solution, a dual time stepping approach is employed
[12, 16, 17], consisting in the splitting of a transient problem
into several instantaneous steady state successive steps, by
adding the so called true transient term to the governing
partial differential equations. This term is discretized to get a
third order of approximation over the real time step size. The
additional term has been stabilized by analysing the order of
magnitude of each term of the governing PDEs. The stabilized
AC-CBS algorithm is here applied for the first time to the
simulation of transient buoyancy driven flows in porous and
partly porous tall cavities. In this paper, the effects caused by
the presence of a porous side on the buoyancy driven flow in
a tall cavity are analysed in detail. In particular, the influence
of parameters such as the Ra and the Da numbers, and the
geometrical AR on the flow characteristics has been
investigated. The proposed model has been first validated
against data available in the scientific literature showing an
excellent agreement. Then, the transient AC-CBS algorithm
has been employed to investigate the dependence of Nu
number on Ra and Da numbers for a 4:1 and 8:1 partly porous
cavity. The Ra and Da numbers ranges of variation are
16
10 3.4 10
and 52
10 10
, respectively. The paper is
organized as follows: the next section presents the equations
TRANSIENT INCOMPRESSIBLE FLOW IN A PARTIALLY POROUS
BUOYANCY DRIVEN TALL CAVITY
F. Arpino*, G. Cortellessa*, M. Dell’Isola*, G. Ficco*
A. Carotenuto°, N. Massarotti°
* Dipartimento di Ingegneria Civile e Meccanica, Università degli Studi di Cassino e del Lazio
Meridionale, Via G. Di Biasio 43, 03043 Cassino (FR), Italy
° Dipartimento di Ingegneria, Università di Napoli “Parthenope”, Isola C4, Centro Direzionale di
Napoli, 80143 Napoli, Italy
ABSTRACT
In this paper the authors numerically investigate dynamic heat transfer related to free convection in porous and partially porous
cavities, paying particular attention to the dependence of velocity and temperature fields on Rayleigh (Ra) number, porous layer
permeability, and cavity aspect ratio (AR), that is the ratio between its height and width. Heat and fluid flow inside the cavity have
been described by using the generalized porous medium model and the governing equations have been numerically solved using
the Artificial Compressibility (AC) version of the Characteristic Based Split (CBS) scheme. The proposed model has been firstly
applied to the simulation of heat and fluid flow in a square porous cavity, and an excellent agreement has been found between the
obtained results and the data available in the literature. Then, the analysis has been extended, here for the first time, to the transient
natural convection in partly porous cavities with AR equal to 4:1 and 8:1, for different Ra and Darcy (Da) numbers. The mean
Nusselt (Nu) number has been calculated as function of cavity geometrical characteristics, porous layer permeability and fluid
properties.
of the generalized porous medium model; in section 3, the
obtained results for porous square and partially porous
vertical cavities are reported. Some conclusions are drawn in
the last section.
GOVERNING EQUATIONS
The generalized model describing flow and energy transport
in a saturated porous medium with constant porosity, filled
with a single phase incompressible fluid consists of a set of
conservation equations. It can be derived by averaging the
Navier-Stokes equations over a representative elementary
volume, using the well-known volume averaging procedure
[18, 19]. Assuming local thermal equilibrium and uniform
properties of the porous medium, the non-dimensional form
of the generalized model can be written for the natural
convection case as follows:
Mass conservation equation
12
12
0
uu
xx
(1)
x1 momentum conservation equation
111
12
22
12
22
11
1
22
112
11 1
Pr 1 Pr
uuu
uu
txx
uu
pJ u
Ra Da Ra
xxx
(2)
x2 momentum conservation equation
222
12
22
122
22
22
2
22
12
11 1
Pr 1 Pr
uuu
p
uu
txxx
uu
JuT
Ra Da Ra
xx
(3)
Energy conservation equation
22
12 22
12 12
Pr
TTT TT
uu
txx xx
Ra
(4)
Where the buoyancy effects are incorporated by invoking the
Boussinesq approximation:
fref reffref
g
gTT
(5)
The scales and the parameters used to derive the above non-
dimensional equations for natural convection are:
***
**
3
*
2
*
*
;;;
2
Ra ; Pr ;
1
;
;Da ;
;;
;J ; ;
hc
ihc
r
ir
fh c f
ff f
pp
fs
pf
eff
f
fefff
ff
ff
pf
x
TT
TT
xT T
LTT
gTTL
cc
c
u
u
L
g
LT
gLT p
tt p
LgLT
c
(6)
The generalized model reduces to the Navier Stokes equations
when the porous medium disappears, i.e. when 1
and
Da while, as the porosity 1
and Da ,
convection disappears and equation (4) describes heat
conduction in a solid body. Therefore, the above procedure
can be used to describe coupled conduction-convection
problems as well as interface problems in which a saturated
porous medium interacts with a single-phase fluid, with no
additional matching conditions required at the free fluid –
porous domains interface.
The PDEs constituting the generalized porous medium model
have been numerically solved by the transient AC-CBS
algorithm. The governing equations have been discretized in
time along the characteristics, and in space using the standard
Galerkin procedure [20]. In order to reach the convergence of
the solution a stability analysis has been considered. It is
based on the order of magnitude analysis of the different
terms of the discretized governing PDEs. This approach has
been described in detail in recent publications of the authors
[7, 8] and is here applied. Transient problems have been
modelled using a dual time-stepping technique and adding a
true transient term to the first or the third step of the AC-CBS
algorithm. The true transient term is discretized over the real
time to the third order of approximation, using the following
relation:
23
11 18 9 2
6
nn n n
nii i i
iuu u u
u
(7)
The explicit nature of the true transient term introduces a
limitation on the pseudo time step size to ensure the required
stability of the method. On the basis of the order of magnitude
analysis [21-23] it has been derived that 320t
for a
third order of approximation.
RESULTS
Transient incompressible flow in a square buoyancy
driven porous cavity
In the present section, transient heat and fluid flow in a square
porous cavity is numerically investigated. In order to validate
the proposed numerical model, the obtained results have been
compared with the analytical and numerical data available in
reference [24]. The computational domain, the imposed
boundary conditions and the used mesh are available in
Figure 1. Simulations have been conducted using a
computational grid composed by 40401 nodes and 80000
triangular elements, chosen on the basis of a mesh sensitivity
analysis. The initial condition employed to perform transient
simulations consists of an isothermal fluid at rest, with a non-
dimensional temperature equal to zero. The gravity vector is
directed in the negative y-coordinate direction. As concerns
the boundary conditions, it is assumed that the left and right
vertical walls are maintained at a constant dimensionless
temperature of +0.5, and -0.5, respectively. The top and
bottom walls are considered adiabatic.
Data in reference [24] are produced for the following three
values of the porous Rayleigh number,
/
P
Ra g K TL RaDa
: 102, 103 and 104 and for a
Prandtl number (Pr) equal to 1. Saeid and Pop [24] neglected
the viscous drag and inertia terms in the governing equations,
given the low values of Darcy and porous Rayleigh numbers
considered. Unlike the assumptions proposed by Saeid and
Pop [24], in the present work the viscous drag and inertia
terms in the governing equations have not been neglected.
Figure 2a shows the mean Nusselt number (Num) at the cavity
hot wall as a function of the Darcy number (Da) for the three
different porous Rayleigh numbers considered. From the
analysis of this figure it can be evidenced that when low
buoyant forces are considered (RaDa=102), the mean Nusselt
number at the hot wall is almost constant with the Darcy
number. When RaDa=104 the mean Nusselt number strongly
depends on the Darcy number. In order to reproduce the flow
condition indicated by Saeid and Pop [24], simulations have
been conducted employing a Darcy number equal to 10-8 and
for the following three values of the Rayleigh number: 1010,
1011, 1012. Porosity in equations (2)-(3) has been set to 1.
The transient behaviour of the proposed numerical scheme for
the description of heat and fluid flow in a porous square
cavity has been assessed by comparing the variation over time
of the mean Nusselt number at the hot wall with the numerical
results available in [24]. This comparison is available in
Figure 2b for
2
10RaDa
,
3
10RaDa
, and
4
10RaDa
,
showing an excellent agreement with reference data. The
decreasing trend of the average Nusselt number during the
first time intervals indicates that heat transfer by conduction
dominates the problem, while the following positive trend
evidences that the amount of heat transferred by convection is
increasing. Such aspect has been evidence in Figure 3, where
the temperature contours and the streamlines obtained for
3
10RaDa
, for a real time value of 250 (top), 1000 (middle)
and 2500 (bottom). In the first instants of the simulations, the
heat transfer is dominated by conduction. Convective heat
transfer contribution suddenly increases becoming dominant
for a real time value larger than 250.
Figure 1. Square cavity filled with a porous medium. Problem definition and boundary conditions employed (left); computational
grid composed by 40401 nodes and 80000 triangular elements (right).
Figure 2. Nusselt number (Num) at the hot wall as a function of the Darcy number (Da) for three values of the porous Rayleigh
number (left) and variation with time of the mean Nusselt number at the hot wall (right).
Figure 3. Square cavity filled with a porous medium. Temperature contours (left) and streamlines (right) at a real time of 250 (top),
1000 (middle) and 2500 (bottom) for Da=10-8 and Ra=1011.
Table 1. Natural convection in a square porous cavity: comparison of the steady state mean Nusselt number (Num) at the hot wall
with reference data from the scientific literature.
Nu
m
Reference Ra=108 Ra=109 Ra=1010
Walker and Homsy (Ref. 11 of [24]) 3.097 12.960 51.000
Bejan (Ref. 12 of [24]) 4.200 15.800 50.800
Gross et al. (Ref. 13 of [24]) 3.141 13.448 42.583
Manole and Lage (Ref. 14 of [24]) 3.118 13.637 48.117
Baytas (Ref. 10 of [24]) 3.160 14.060 48.330
Saeid and Pop [24] 3.002 13.726 43.953
Present results 3.101 13.576 45.353
In Table 1, the obtained results have been validated by
comparing the mean Nusselt number at the hot wall with the
data available in the scientific literature at steady-state
condition, for the three different values of the Ra considered.
The proposed results are in good agreement with the reference
data from the scientific literature, and represent a further
assessment of the effectiveness of the proposed numerical
scheme.
Transient incompressible flow in a partially porous
buoyancy driven tall cavity
In this section, the numerical results obtained for transient
natural convection in a partly porous tall cavity with an
Aspect Ratio (AR) of 8:1 and 4:1 are presented. To the
authors’ knowledge, a fully explicit numerical procedure is
here used for the first time to numerically investigate heat and
fluid flow in partially porous tall cavities. The computational
domain and the boundary conditions employed are reported in
Figure 4. In particular, zero velocity (u1=u2=0) is assumed at
the four cavity walls. The left and right cavity sides are kept at
a dimensionless temperature value of 0.5 and -0.5,
respectively. The top and bottom walls are assumed to be
adiabatic. The initial condition employed to perform transient
simulations consists of an isothermal fluid at rest, with a non-
dimensional temperature equal to zero. The mesh in the figure
refers to the cavity with AR equal to 4:1 and is composed by
5717 nodes and 10120 triangular elements. The mesh
employed for the 8:1 AR case consists of 7379 nodes and
13956 triangular elements. All the grids are refined near the
walls and in correspondence of the interface between free
fluid and porous region, and have been chosen on the basis of
a mesh sensitivity analysis. The fluid is assumed to be
incompressible and Newtonian. The porous medium is
assumed to be isotropic and homogeneous.
The numerical investigations have been carried out assuming:
Prandtl (Pr) number equal to 0.71; Rayleigh number (Ra)
ranging between 10 and the value at which oscillations start in
absence of the porous layer inside the cavity (3.4×106 for the
4:1 cavity and 3.4×105 for the 8:1 cavity) [25]; uniform and
constant porosity equal to 0.5; Darcy (Da) number ranging
between 10-5 and 102. The simulations have been performed
employing a real time step size equal to 0.1.
Figure 5 shows the temperature contours obtained at a real
time level of 500, for an AR equal to 8-1 and for a Darcy
number ranging from 10-4 to 100. In particular, results in
Figure 5 have been obtained for a Ra of 3.4×105. When
Da=10-4, the heat transfer in the porous layer is dominated by
conduction. As Da number increases, convective heat transfer
mechanism becomes dominant. The porous layer weakly
influences the temperature fields when Da=100, and periodic
oscillations appear in correspondence of the left and right top
corners (points 1 and 4 in Figure 4, respectively). From the
analysis of such figure, it can be observed that oscillations are
much more pronounced in correspondence of the top corners,
while are hardly distinguishable in the remaining zones of the
computational domain.
The practical interest of heat and fluid flow in partially porous
tall cavities and the poor information available for such
problem in the scientific literature pushed the authors to
investigate the two considered ARs in terms of dependence of
the mean Nusselt number (Num) at the hot wall on the Da and
Ra numbers. Figure 6 shows, for the two investigated cavities,
the variation of Num with Ra and Da. The obtained results
showed that Nu number is an increasing function of both Ra
and Da numbers. The Num decreases as the Ra decreases and
becomes independent on the Ra number when Ra is lower
than about 103, indicating that conductive heat transfer
mechanism dominates. Such aspect has been observed for
both 4:1 and 8:1 cases. In fact, Num is constant and about
equal to 0.9 when Ra≤103 regardless of the Da considered. As
regards the dependence of Num on the Da number, it increases
as the Da increases, even though it can be observed that Num
becomes independent on the considered Darcy number if Da
is larger than about 10-1. Results coincide to the fluid cavity
case when Da≥102 for both the considered ARs. Table 2
reports the quantitative results obtained for the Da and Ra
numbers considered in the present investigations. From the
obtained results, it can be concluded that heat transfer in a
partially porous tall cavity with an AR of 4:1 or 8:1 is
independent on the Ra number if this parameter is lower than
about 103, and is independent on the Da number if this
parameter is larger than about 10-1. The range of practical
interest, characterized by a solution influenced by both Ra and
Da, is for Ra≥103 and Da≤10-1.
CONCLUSIONS
In the present paper, the Artificial Compressibility (AC)
version of the Characteristic Based Split (CBS) algorithm has
been employed to solve transient natural convection in porous
and partially porous rectangular cavities. The finite element
based numerical tool, together with the stabilization analysis
developed by the authors, has been successfully applied to
solve the governing equations of the generalized porous
medium model for buoyancy driven flows. In order to obtain
a transient solution, a dual time stepping approach has been
employed, and the true transient term has been discretized
over the real time to the third order of approximation.
The transient numerical procedure has been validated
comparing the present results with the analytical and
numerical data available in the literature for natural
convection in a square porous cavity, observing an excellent
agreement. Then, the stabilized AC-CBS algorithm has been
applied, here for the first time, to the simulation of transient
buoyancy driven flows in partly porous cavities with an
aspect ratio (AR) equal to 4:1 and 8:1. In particular, the
influence of Rayleigh (Ra) number, Darcy (Da) number and
AR on the transient thermo fluid dynamic phenomena has
been investigated. The porous layer strongly influences the
transient temperature and velocity fields. In fact, periodic
oscillations reach steadiness in presence of a porous matrix
with low value of the Da number, for both the ARs
considered. The dependence of the Nusselt (Nu) number on
the Ra and Da numbers has been also analysed. The obtained
results have shown that the Nu number is an increasing
function of Ra and Da numbers for both the ARs considered.
The authors believe that the present algorithm has revealed a
powerful tool to study transient natural convection in partly
porous tall cavities, and the obtained results could be useful
for several industrial applications.
Figure 4. Natural convection in a partially porous tall cavity. Problem definition with boundary conditions employed (left); 4:1
computational grid, composed by 5717 nodes and 10120 triangular elements (right).
Figure 5. Temperature contours in a 8:1 buoyancy driven partly porous tall cavity for different values of the Darcy (Da) number.
Real time level =500, Ra=3.4×105.
Figure 6. Variation of Num with Ra (a-c) and Da (b-d) for a 8:1 and for a 4:1 partially filled porous cavity.
Table 2. Natural convection in partially porous 8:1 and 4:1 cavities: mean Nusselt number (Num) at the hot wall for different Ra
and Da numbers.
8:1 cavity
Rayleigh number, Ra
4:1 cavity
Rayleigh number, Ra
103 104 105 3.4x105 103 104 105 3.4×105 106 3.4×106
Darcy
number, Da
102 1.03 1.56 2.97 4.02 1.05 1.83 3.39 4.62 6.03 8.10
1 1.03 1.56 2.97 4.02 1.05 1.82 3.39 4.62 6.02 8.10
10-2 1.01 1.37 2.81 3.90 1.02 1.63 3.26 4.51 5.95 8.10
10-3 1.00 1.08 2.03 3.16 1.00 1.15 2.50 3.85 5.41 7.69
10-4 1.00 1.02 1.27 1.70 1.00 1.03 1.41 2.04 3.21 5.37
10-5 1.00 1.01 1.16 1.33 1.00 1.02 1.24 1.42 1.62 2.25
ACKNOWLEDGMENT
The present work is partly supported by the jrp sib64 metefnet,
jointly funded by the emrp participating countries within
euramet and the european union. one of the authors gratefully
acknowledges the support of a emrp reg granting scheme.
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