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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

i

uu 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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