MAC simulation of thermosolutal natural convection in a ...used finite volume methods. Roy et al....

International Journal of Mathematics Trends and Technology (IJMTT) – Volume 49 Number 5 September 2017

ISSN: 2231-5373 http://www.ijmttjournal.org Page 291

MAC simulation of thermosolutal natural

convection in a porous enclosure with

opposing temperature and concentration

gradients B Md Hidayathulla Khan

1, V Ramachandra Prasad

2, R Bhuvana Vijaya

3

1Research Scholar, Department of Mathematics, Jawaharlal Nehru Technological University Ananthapur

Anantapuramu – 515 002, India 2Professor, Department of Mathematics, Madanapalle Institute of Technology and Science, Madanapalle

Madanapalle – 517 325, India 3Associate Professor, Department of Mathematics, Jawaharlal Nehru Technological University Ananthapur

Anantapuramu – 515 002, India

Abstract— A numerical study is conducted on

thermosolutal natural convective flow inside a

porous mixture of a rectangular enclosure with

aspect ratio four. The flow enhancement is observed

due to rising of Rayleigh number and Darcy number

with buoyancy parameter. The transport equations

for continuity, momentum, energy and species

transfer are solved numerically with Marker and

Cell (MAC) method. Two dimensional computational

visualization results illustrating the influence of fluid

parameters such as Darcy number, Rayleigh number

and buoyancy ratio on contour maps of the

Streamlines, Isotherms, and Iso-concentrations as

well as the mid-section of velocity of the cavity are

reported and discussed. The local Nusselt number

and Sherwood numbers are increasing along the

vertical walls for increasing Darcy number.

Keywords— Porous medium, Isothermal walls,

Double Diffusive Natural Convection, Aspect ratio,

Rectangular Enclosure, MAC method.

I. INTRODUCTION

The combination of temperature and

concentration gradients generates thermosolutal or

two- component convection due to thermal and

species (solutal) buoyancy effects. Thermosolutal

convection in a porous medium has a wide range of

engineering applications in numerous technological

areas including materials engineering, solar

collectors, microbial fuel cells, enclosure ventilation,

fire dynamics, geological nuclear waste storage,

crystal growth processes etc. The seminal reviews

elaborating on these and many other applications are

the works of Gebhart et al. [1] and earlier that of

Ostrach [2]. The large body of experimental

literature in enclosure heat and mass transfer is

complimented by an every growing quantity of

investigations into theoretical and computational

simulations. In such studies a key focus is the

assessment of the competition between the two

buoyancy forces which may act either in either the

opposite or the same direction. These differences can

profoundly influence momentum, thermal and

species diffusion characteristics. Recently Zhang [3]

considered combined combined heat and mass

transfer in hollow fibre membrane air humidification

systems. Teamah et al. [4] used a Patankar finite

volume technique to simulate coupled thermosolutal

convection in shallow titled enclosures with a

moving lid, identifying the generation of secondary

vortex structures in the core of the enclosure with

specific values of the buoyancy ratio parameter,

which is disintegrated at critical title angle. Naik and

Bhattacharyya [5] computed the viscous Newtonian

flow, heat and mass transfer in a cubical lid-driven

cavity, noting the departure from two-dimensional

patterns and the presence of a significant transverse

velocity at higher Reynolds number which perturbs

the vortex patterns. Chen et al. [6] studied

bifurcation phenomena in two-sided lid-driven

cavity thermosolutal convection flows for different

aspect ratios using Keller’s continuation method,

emphasizing the emergence of a stable asymmetric

flow state at high aspect ratio and low Reynolds

number. Many other studies in double diffusive

cavity convection have been communicated with a

variety of computational algorithms including Xin et

al. [7], Cibik and Kaya [8] who employed finite

element methods and also considered nonlinear

porous media and Serrano-Arellano et al. [9] who

used finite volume methods. Roy et al. [10]

considered thermo-diffusion and stability aspects. Bettaibi et al. [11] deployed a hybrid Lattice

Boltzmann-multiple relaxation time method for heat

and mass diffusion and a finite difference technique

for fluid dynamics computation of thermosolutal

mixed convection in rectangular enclosure. Ren and

Chan [12] studied cross diffusion effects in

enclosure transport phenomena with a lattice

Boltzmann technique. Al-Amiri et al. [13]

performed a numerical study of double-diffusive

mixed convection in a lid-driven square cavity,

examining a wide range of buoyancy ratio and

Richardson numbers. Bhargava et al. [14] utilized a

variational Galerkin method to compute the double-

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ISSN: 2231-5373 http://www.ijmttjournal.org 92

diffusive convection in a square enclosure filled with

porous media saturated with micropolar fluid. Bég et

al. [15] applied both FTCS (forward time centred

space) and network electrothermal numerical codes

to compute the radiative-convective thermosolutal

transport in an annular non-Darcy porous media

enclosure as a simulation of hybrid solar collectors.

Ali Mchirgui et al. [16] computed the entropy

generation rate in double-diffusive convection in an

inclined porous cavity for various aspect ratios,

observing that it is minimized at aspect ratios of 0.5

and 1 for Darcy numbers, Da=10-2

and 10-4

.

Chamkha and Hameed [17] employed the Blottner

finite difference implicit method to simulate two-

dimensional laminar thermosolutal convective flow

in an inclined enclosure filled with porous medium

and binary gas mixture, evaluating the influence of

inverse Darcy number and inclination angle on

streamlines, isotherms, isoconcentrations and density

contours. Al-Farhany and Turan [18] studied double-

diffusive natural convection in a tilted rectangular

porous cavity utilizing the finite volume approach

and the pressure velocity coupling via a SIMPLER

algorithm. They observed the governing parameters

with different aspect ratios from 2 to 5, inclination

angles ranging from 00 to 85

0 and buoyancy ratio

between -5 to 5. Jena et al. [19] simulated double-

diffusive buoyancy opposed natural convection in a

square cavity filled with porous medium having

partially active side walls with an improved marker

and cell (MAC) and a second order gradient

dependent consistent hybrid upwind scheme

(GDCHUSSO) for convective term discretization to

analyze the influence of Darcy number, buoyancy

ratio and active thermal and solutal locations on rate

of heat and mass transfer. Xu et al. [20] used a

Lattice Boltzmann code to analyse composite natural

convection heat and mass transfer inside a porous

enclosure with oscillation characteristics, noting that

with an increase in Darcy number from 10-4

to 10-2

,

the flow experiences a steady-state, quasi-periodic

oscillation, unsteady doubling periodic oscillation

and non-periodic oscillation.

In the present work we extend the above

simulations to consider in more detail thermo-solutal

convection in a rectangular enclosure containing a

saturated porous medium with large aspect ratio

with a marker and cell (MAC) computational code.

The influence of buoyancy ratio N, Darcy number

Da and Rayleigh Number Ra on heat and mass

transfer characteristics are depicted in detail.

Validation with previous studies is included. The

investigation is relevant to solar cell convection

flows and electronic cooling technologies.

II. MATHEMATICAL FORMULATION

Fig. 1 shows the physical model for the current

two-dimensional rectangular enclosure problem. The

enclosure is of the width W and height H (aspect

ratio Ar = (H/W)). Gravity (g) acts downwards in the

enclosure which contains an isotropic, homogenous

porous medium.

Fig. 1 Schematic diagram of the rectangular

cavity

The left wall of the enclosure is fully heated and

solute rich and maintained at uniform temperature

and concentration. The right wall of the enclosure is

maintained at low temperature and concentration and

the remaining walls of the enclosure are adiabatic

and impermeable. Furthermore all thermophysical

properties of the fluid such as thermal conductivity,

specific heats, viscosity, thermal expansion

coefficients and permeability are taken to be

constant except for the fluid density variation in the

buoyancy term. An incompressible Newtonian fluid

is considered. Viscous dissipation and cross-

diffusion effects are neglected. Darcy’s law is

implemented for low-Reynolds number porous

media hydrodynamics. The regime contains a non-

reactive species which also generates a buoyancy

effect owing to concentration differences. The

density is calculated by utilizing the Boussinesq

approximation as follows:

0 1 T c c cT T C C (1)

where 0 is the fluid density at ambient

conditions. The unsteady transport flow model based

on the above assumptions i.e. the governing

conservation equations for double-diffusive natural

convection flow (conservation of mass, momentum,

energy and species concentration) can be written in

vector notation as:

. 0 v (2)

1.

c c y

pt

g T T g C C eK

vv v v

v

(3)




.( )T

T Tt

v (4)

.( )C

C D Ct

v (5)

where ye

is the unit vector along the vertical axis.

We consider time-dependent, two-dimensional flow,

and invoke the following dimensionless variables to

simplify the physics:

2

2

2

,, , , ,

, ,c c

h c h c

x yt uW vWX Y U V

W W

pW T T C CP

T T C C

(6)

The emerging governing equations (2)-(5) in an

(X,Y) coordinate system therefore reduce to non-

dimensional form:

0U V

X Y

(7)

2 2

2 2

PrPr

U U U P U UU V U

X Y X X Y Da

(8)

2 2

2 2Pr

Pr.Pr

V V V P V VU V

X Y Y X Y

V Ra NDa

(9)

2 2

2 2U V

X Y X Y

(10)

2 2

2 2

1U V

X Y Le X Y

(11)

Here

3

2

PrPr , ,

h c c h c

T h c

g T T W C CRa N

T T

and LeD

which denote respectively the Prandtl

number, Rayleigh number, buoyancy ratio parameter

and Lewis Number. The dimensionless horizontal

and vertical direction coordinates are X and Y and

the dimensionless velocity components along X- and

Y-directions are U and V respectively. is the

kinematic viscosity of the fluid, and D are the

thermal and mass (species) diffusivities. To simulate

air the Prandtl number is taken as 0.71 (Pr=0.71).

The appropriate dimensionless initial and boundary

conditions (no-slip is considered) are:

0: 0U V for all walls

0 :U VX X

at Y=0,4

0, 1U V at X=0

0, 0U V at X=1 (12)

The fluid motion is represented using the stream

function evaluated with dimensionless velocity

components U and V. The relationship between the

velocity components and stream function is

determined via the classical Cauchy-Riemann

equations – see Basak et al. [21]:

UY

and VX

(13)

The local Nusselt and Sherwood numbers (i.e.

wall heat and mass transfer rates) are computed with

the following relations:

0,1X

NuX

and

0,1X

ShX

(14)

The average Nusselt and Sherwood Numbers of

the side walls are evaluated using:

0

1Ar

Nu NudyAr

and 0

1Ar

Sh ShdyAr

(15)

III. MAC COMPUTATIONAL SOLUTION

The normalized governing partial differential

equations (7)-(11) subject to the imposed

dimensionless boundary conditions (12) are solved

numerically. The discretization of the time

derivative is performed by the MAC (marker and

cell) method which is an implicit scheme using a

staggered grid. The velocity–pressure gradient

causes strong coupling in the continuity and the

projection method [22] is implemented for the

primary and secondary momentum equations (8) and

(9). The special finite difference scheme of

Ambethkar et al. [23] and Harlow and Welch [24] is

implemented to minimize the numerical diffusion for

the advection terms. The Poisson equation for

pressure gradient is solved using an accelerated full

multigrid method, while the central finite difference

discretized equations are solved using the SOR

(Successive Over Relaxation) method [25] with

careful selection of relaxation factors. Finally, the

convergence of the numerically computed results is

established at each time step according to the

following criterion: 1 6

, ,

,

10k ki j i j

i j

(16)

In eqn. (16) the generic variable designates

, , ,U V P , and k denotes the iteration time levels.

In the above inequality the subscripts i,j represent

the space coordinates X and Y respectively. The

numerical computations have been conducted for

20 80 to 80 320 grids by discretizing a rectangular

enclosure with aspect ratio four (Ar=4). It has been

observed that for the grid size 40 160 , there is no

tangible modification in the average Nusselt number

Nu for the hot wall with N=1, Pr=0.71, Ra=103,

Le=1 and Da=10-4

. The grid independence test is

summarized in Table 1. A 40 160 grid is therefore

deployed to compute all solutions for various

thermal parameters and achieves sufficient accuracy.

To verify the accuracy of the MAC numerical

method employed for the solution of the problem

under consideration, it has been validated (after

making the necessary modifications) against the

control-volume based finite-difference discretization

solutions presented in an earlier study by Mahapatra

et al. [26] who considered double-diffusive natural

convection in a lid-driven cavity.




S.No. Grid Size (W X H) Nu 1 20X80 0.999968

2 30X120 1.000012

3 40X160 1.000017

4 50X200 1.000016

5 60X240 1.000017

6 70X280 1.000018

7 80X320 1.000020

Table.1 Grid independence test for rectangular

cavity (Ar=4).

Present Results Mahapatra et al.[26]

Fig. 2 Comparision of the streamlines, isotherms and

isoconcentrations for uniformly heated and

uniformly concentrated left and bottom walls with

N=50, Pr=0.7 and Ra=103

The comparisons for the streamlines, isotherms, iso-

concentration contours are depicted in fig. 2 for the

cases of uniformly heating and concentration with

N 50,Pr 0.7 and 310Ra . The correlation between

the present MAC solutions and Mahapatra et al. [26]

is evidently very good, lending high confidence to

the accuracy of the present MAC numerical code.

IV. RESULTS AND DISCUSSION

The governing parameters in the present

simulations of thermosolutal natural convection in a

rectangular porous enclosure with large aspect ratio

are the Prandtl number, Pr; the buoyancy ratio, N;

the Rayleigh number, Ra; the Darcy number, Da. In

this study we have considered the working fluid as

air (Pr=0.71), the aspect ratio of the rectangular

enclosure, Ar=4, Lewis number, Le=1 (momentum

and species diffusivity are equal), and buoyancy

ratio, N=1 (thermal and species buoyancy forces are

the same). The numerical computations are

performed for thermal Rayleigh number 104 and 10

5

and the Darcy number varying from 10-4

to 10-1

. The

numerical results are visualized in the form of

streamlines, isotherms and isoconcentration contours

for different values of Darcy number with Rayleigh

number 104 and 10

5 in Figs 3- 5.

Da=0.0001 Da=0.001 Da=0.01 Da=0.1

Figs. 3(a) Streamlines (Top), isothermal (Middle)

and iso-concentration (Bottom) contours for Ra=104,

Pr=0.71, Le=1, N=1.

Fig. 3a, b illustrate the effect of Darcy number on

the streamlines, isotherms and iso-concentration

contour evolution with Pr = 0.71, N=1, Le=1, for

two different thermal Rayleigh numbers, i.e. Ra=104

and Ra = 105, respectively. These Rayleigh numbers

imply very strong thermal buoyancy relative to

viscous hydrodynamic force and will generate

significant convection current in the regime and the

impact of thermal buoyancy effect is tenfold greater

in fig 3b compared with fig. 3a. In Fig. 3(a) the

upper row of plots shows the modification in

streamline plots with increasing Darcy parameter.




The flow in a rectangular enclosure consists of a

weak rotating eddy, the circulations move from

centre of the enclosure to the walls of the enclosure.

The flow circulations are changed gradually with

increasing of Darcy parameter. The increase in

Darcy number implies an increase in permeability of

the porous medium. This reduces the Darcian drag

forces in both the primary and secondary momentum

equations i.e. –Pr/Da U and - Pr/Da V, respectively.

The impedance of solid fibers in the regime is

therefore reduced and the flow is accelerated. This

results in a more intense and symmetric vortex

structure in the cavity and the elimination of

skewness which is computed for lower values of

Darcy number. In all cases however a single

circulation is formed around the centre of the

enclosure. Variation in Darcy number significantly

modifies the structure in particular near the upper

and lower walls but does not generate any further

circulation zones. A similar pattern of behaviour has

been reported by Zhang et al. [27] and Lin [28]

although they employed a generalized Darcy-

Rayleigh number, as did Bhargava et al. [14]. The

temperature (row 2) and iso-concentration (row 3)

patterns are almost parallel lines for very low Darcy

number (Da = 0.0001). The opposing temperature

and concentration gradients imposed at walls

encourage heat transfer by thermal conduction. The

isothermal cold wall receives more heat than the

isothermal hot wall. Therefore, the temperature

gradient of uniform hot wall is less than that of the

uniform cold wall. The parallel line of isotherms and

isoconcentration contours are transformed into

inclined contours (with an ever-growing sigmoidal

structure from the lower bottom left of the enclosure

to the upper right corner of the enclosure) with

increasing Darcy number. The contribution of

convection is noticeable at high Darcy number as

evident by the departure of the isotherms and

isoconcentration from the parallel pattern and the

mechanism of heat transfer is gradually shifted to

natural convection domination. Increasing

permeability therefore encourages thermal

convection in the regime and aids in the diffusion of

heat and species and this is further assisted by the

equivalence of thermal and species buoyancy forces

(N=1). In Fig. 3(b) the greater thermal buoyancy

force (Rayleigh number 510Ra ) exerts a profound

alteration in streamline plots (upper row). The vortex

structure is similar to that for fig. 3a with

Da = 0.0001 and 0.001; however with subsequent

elevation in Darcy number there is a departure from

the generally smooth single circulation zone.

Da=0.0001 Da=0.001 Da=0.01 Da=0.1

Fig. 3(b) Streamlines, isotherms and iso-

concentration contours for Ra=105, Pr=0.71, Le=1.0,

N=1.0

At lower Darcy numbers the vertical circulation

rotates in clockwise direction within the enclosure.

However it becomes distorted and warped towards

the upper left corner and lower right corner of the

enclosure with greater permeability. The reduction

in damping effect associated with Darcian drag

forces is therefore manifested in a strongly

asymmetric vortex pattern which is further distorted

at very high Darcy number (Da = 0.1). There is also

constriction in the circulation pattern at the upper

and lower cavity walls and an expansion in the

central core region at high Darcy number. A marked

transition in temperature and iso-concentration

contours (middle and lower rows in fig. 3b) is also

observed with increasing Darcy number. The

contours of isotherms and iso-concentrations which

are parallel to the vertical walls in fig 3a (Ra = 104)

are now distorted in fig. 3b with greater Rayleigh

number Ra = 105) for the same low value of Darcy

number (Da = 0.0001). A more significant

modification in isotherms and iso-concentrations is

observed with increasing Darcy number and

although the sigmoidal pattern is computed at

Da = 0.001 and 0.1, this structure is further evolved

throughout the enclosure at highest Da (0.1) i.e.

much stronger bending of contour patterns arises.

The asymmetric distribution is amplified and




inclination of contours is enhanced with increasing

Darcy parameter.

Fig. 4(a) shows the influence of Darcy parameter

on dimensionless velocities of the mid-height and

mid-width of enclosure for Ra=104, Pr=0.71, Le=1,

N=1. It is observed that the horizontal velocity

component increases with greater Darcy number i.e.

flow acceleration is induced with greater

permeability of the porous medium.

Fig. 4(a) Effect of Darcy parameter (Da) on mid-

section velocity profiles in enclosure for Ra=104,

Pr=0.71, Le=1, N=1.

This behaviour is evident in the upper half of the

cavity, while, in the lower half, a reflection of this

behaviour is apparent, which is associated with

momentum re-distribution in the regime. Vertical

velocity raises near the heated wall with increase in

Darcy number and the fluid temperature also

increases near the cold wall. The Darcy number

influence on mid-height (Y=2) and mid-width (X=0.5)

velocity profiles of U and V – velocity components

for fixed parameters Ra=105, Pr=0.71, Le=1, N=1

are displayed in fig. 4(b). It is observed that the

velocity component increases with Darcy number

and this indicates that natural (free) convection

dominates at high Darcy parameter. The mid-height

velocity is more near the horizontal walls. In

addition the velocity is elevated with increasing

Darcy parameter. The mid-width velocity is high

near the isothermal hot and isothermal cold walls.

Fig. 5 depicts the effect of buoyancy ratio on

local Nusselt and Sherwood number along the side

walls (isothermal hot and cold walls). The local

Nusselt and Sherwood number of hot wall

monotonically increases with increasing buoyancy

ratio (i.e. greater species buoyancy force relative to

thermal buoyancy force). The temperature and

concentration of fluid particles are decreased at the

enclosure end corner. The opposite mechanism is

observed for the cold wall. The rate of heat and mass

transfer on the left isothermal hot wall decreases to 1

from (0, 0) to (0, 2.3) and then also decreases

between (0, 2.3) to (0, 4). It observed that the rate of

heat and mass transfer on right isothermal cold wall

is boosted to 1 from (1, 0) to (1, 1.7) and in the

remaining zones of the enclosure heat and mass

transfer rates are increased with increasing buoyancy

ratio. A domineering species buoyancy force relative

to thermal buoyancy force therefore dramatically

modifies heat and mass transfer in the cavity.

(a) (b)

(c) (d)

Fig. 5 Variation of Local Nusselt and Sherwood

number along the side walls with buoyancy ratio (N)

for Ra=104, Pr=0.71, Da=10

-4

V. CONCLUSIONS

A numerical study has been performed for

nonlinear thermosolutal natural convection heat and

mass transfer in a two-dimensional air-filled porous

rectangular enclosure with opposing temperature and

concentration gradients. The governing coupled non-

linear partial differential equations have been non-

dimensionalized and solved by a MAC (Marker and

Cell) computational algorithm. Verification with

previous studies has been conducted and furthermore

a grid-independence study undertaken. Graphical

results for flow patterns, isotherms and

isoconcentration contours and mid-section velocity

profiles for different parameters have been presented.

It has been shown that flow, thermal and solutal

transfer characteristics inside the rectangular

enclosure depended strongly on Darcy number,

buoyancy ratio and thermal Rayleigh number effects.

In general, increasing Darcy number (i.e. greater

permeability of the porous medium) produces an

enhancement in the rate of heat and mass transfer in

certain zones of the enclosure. The local Nusselt and

Sherwood number increase for the major part of the

left vertical wall and decrease for the remainder of

the hot wall with increasing buoyancy ratio; the

converse response is exhibited at the isothermal cold

wall. The heat and mass transfer rates at the cavity

walls are elevated with increasing Rayleigh number.

Greater heat transfer rates are observed at the bottom

of the left wall and at top portion of the right wall

with increasing Darcy number as well as buoyancy

ratio with various values of Rayleigh number and

also it is observed that the heat and mass seems to be

constant at (0, 2.3) and (1, 1.7). With higher

Rayleigh number the streamline distributions are

more severely distorted at higher Darcy numbers and

asymmetry in isotherm and iso-concentration

profiles is enforced at lower Darcy number.




The present work has considered Newtonian

viscous flow. Future investigations will address

nanofluid and non-Newtonian (micropolar) liquids

in double-diffusive enclosure flows and will be

communicated imminently.

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