Simulation of Natural Convection in a Complicated ... · Simulation of natural convection in a complicated enclosure 2835 details of solution procedure. Finally, we evaluate the computational

Applied Mathematical Sciences, Vol. 6, 2012, no. 57, 2833 - 2842

Simulation of Natural Convection in a Complicated

Enclosure with Two Wavy Vertical Walls

Pensiri Sompong

Department of Mathematics, Faculty of ScienceKhon Kaen University, Khon Kaen 40002, Thailand

Centre of Excellence in MathematicsCHE, Si Ayutthaya Rd., Bangkok 10400, Thailand

ys [email protected]

Supot Witayangkurn1

Department of Mathematics, Faculty of ScienceKhon Kaen University, Khon Kaen 40002, Thailand

supot [email protected]

Abstract

Natural convection in a square enclosure having two wavy verticalwalls is studied. An enclosure is filled with fluid-saturated porous me-dia. A cavity is heated spatially on bottom wall while remaining wallsare maintained at lower temperature. The objective of this study isto examine the flow field, temperature distribution and heat transferinside the cavity when values of Darcy number, Rayleigh number andwave amplitude are changed. To analyze the performance, the governingdifferential equations and boundary conditions are coded into FlexPDE6.17 Professional which is a finite element model builder and numericalsolver. From the study results, decreasing Darcy and Rayleigh num-bers decrease the strength of convection. In addition, the magnitudesof temperature and heat distributions become smaller. In the case of in-creasing wave amplitude, the flow intensity is obstructed by barricadesresulting from extending the amplitude.

Keywords: Finite Element Method, Natural Convection, Porous Media

1Corresponding Author.

2834 P. Sompong and S. Witayangkurn

1 Introduction

The study of natural convection in a two-dimensional enclosure with fluid-saturated porous media has received significant attention and it plays an im-portant role in many applications. Many researchers study the different shapesof enclosure with various boundary conditions. Most of them are cylindrical,rectangular, trapezoidal and triangular. Due to a large number of technicalapplications such as geophysics, geothermal reservoirs, insulation of building,crude oil production, separation processes in industries, etc., the convectionmotion in a complicated cavity is studied to understand these systems.

In a cavity filled with porous media, Basak et al. [1] studied the naturalconvection flows in a trapezoidal enclosure and influence of uniform and non-uniform heating of bottom wall such that two vertical walls are maintainedat constant cold temperature while the top wall is insulated. Basak et al. [2]investigated the effects of uniform and non-uniform heating of inclined wallon natural convection flows within an isosceles triangular enclosure. Theirsolution procedures are carried out by a penalty finite element analysis withbi-quadratic elements. Koca et al. [3] considered the triangular domain whichbottom wall is heated partially whereas the temperature of inclined wall ismaintained at lower uniform temperature than heated wall. In addition, tri-angular enclosure can be used in the application of roof structure to studyheat transfer inside it. Asan and Namli [4],[5] investigated the effects of high-base ratio, Rayleigh number and heat transfer rate for the convection in apitched roof of triangular cross-section under summer and winter conditionsby using control volume integration solution technique. Pensiri and Supot [6]also analyzed the flow field and temperature distribution in an attic space ofright-triangular enclosure such that the inclined wall is heated and the ceilingroom is cooler.

In a complicated cavity, Dalal and Das [7],[8],[9] studied the natural con-vection in a square enclosure having three flat walls and a wavy right verticalwall consisting of one, two and three undulations. Furthermore, Oztop et al.[10] studied the natural convection heat transfer in a wavy-walled enclosurecontaining internal heat sources at different wave ratios by using finite-volumemethod. Interesting parameters are the internal and external Rayleigh num-bers and the amplitude of wavy walls.

This paper is the study of natural convection in a square enclosure suchthat two vertical walls are wavy. The fluid-saturated porous media is containedinside it. Bottom wall is heated spatially while remaining walls are maintainedat lower temperature. The objective is to investigate the convection motion in-side the cavity when Darcy number, Rayleigh number and wave amplitude arevaried by using finite element method. In section 2, the problem characteristicsand notations used throughout this paper are described. Section 3 involves the

Simulation of natural convection in a complicated enclosure 2835

details of solution procedure. Finally, we evaluate the computational resultsand the conclusions are discussed.

2 Notations and Problem Characteristics

The following notations are applied throughout the paper.

AR aspect ratio, AR = H/L Da Darcy numberg acceleration due to gravity (m s−1) H height of the enclosure (m)L length of the enclosure (m) Nu local Nusselt numberp pressure (Pa) P dimensionless pressurePr Prandtl number Ra Rayleigh numberT temperature (C) TH temperature of hot wallTC temperature of cold wall n number of undulationsλ wave amplitude γ penalty parameterθ dimensionless temperature ν kinematic viscosity (m2 s−1)ρ density (kg m−3) ψ stream functionK permeability of the porous medium φ heaet functionx,y Cartesian coordinates (m)u,v x,y components of velocityU ,V x,y components of dimensionless velocityX,Y dimensionless distance along x,y coordinate

The objective of this paper is to investigate the flow field, temperature dis-tribution and heat transfer due to natural convection inside a square enclosurehaving two wavy vertical walls. The fluid-saturated porous media consideredas incompressible and Newtonian is contained inside an enclosure. Here, fluidis air, Prandtl number is kept constant at 0.71, and it is assumed to be lami-nar. Because the enclosure has two wavy vertical walls, the expression of leftand right wavy walls are given by (1) and (2), respectively.

f1(y) = 1 − λ+ λ cos(2πn(1 − y)) (1)

f2(y) = 1 − λ+ λ cos(2πny). (2)

Different values of temperature are assigned to each boundary. Bottom wall isheated spatially, TH , while remaining walls are maintained at lower tempera-ture, TC . The temperature expression on the bottom is written as

T (x) = TC +TH − TC

2

[1 − cos

(2πx

L

)]. (3)

Velocity on all solid walls are zero (u = v = 0). Aspect ratio illustratesproportion of domain size. It can be defined as the ratio of the height of


H

u = v = T = 0

u = v = 0 , T = Sinusoidal

L

Figure 1: A cavity with two wavy vertical walls and boundary conditions.

vertical wall to the length of bottom wall (AR = H/L). A physical model ofan enclosure discussed above can be shown in Fig.1.

3 Governing Equation

Natural convection can be expressed by differential equations of conserva-tion of mass, momentum and energy ([11]). The flow is simulated on twodimensional and the physical properties of fluid within an enclosure are as-sumed to be constant except the density in the buoyancy force. The governingequations for steady two-dimensional natural convection flow in the porouscavity are:

∂u

∂x+∂v

∂y= 0, (4)

u∂u

∂x+ v

∂u

∂y= −1

ρ

∂p

∂x+ ν

(∂2u

∂x2+∂2u

∂y2

)− ν

Ku, (5)

u∂v

∂x+ v

∂v

∂y= −1

ρ

∂p

∂y+ ν

(∂2v

∂x2+∂2v

∂y2

)− ν

Kv + gβ (T − TC) , (6)

u∂T

∂x+ v

∂T

∂y= α

(∂2T

∂x2+∂2T

∂y2

). (7)

with following boundary conditons:u(f1(y), y) = u(f2(y), y) = u(x, 0) = u(x, y) = 0,

v(f1(y), y) = v(f2(y), y) = v(x, 0) = v(x, y) = 0,

T (f1(y), y) = T (f2(y), y) = T (x, y) = TC = 0,

T (x, 0) = TH = TC +TH − TC

2

[1 − cos

(2πx

L

)].


The above governing equations are transformed to non-dimensional form byusing the following change of variables:

X =x

L, Y =

y

L, U =

uL

α, V =

vL

α, θ =

T − TCTH − TC

P =pL2

ρα2, P r =

ν

α, Ra =

gβ (TH − TC)L3Pr

ν2, Da =

K

L2.

The equation (4)-(7) in terms of dimensionless are:

∂U

∂X+∂V

∂Y= 0, (8)

U∂U

∂X+ V

∂U

∂Y= − ∂P

∂X+ Pr

(∂2U

∂X2+∂2U

∂Y 2

)− Pr

DaU, (9)

U∂V

∂X+ V

∂V

∂Y= −∂P

∂Y+ Pr

(∂2V

∂X2+∂2V

∂Y 2

)− Pr

DaV +RaPrθ, (10)

U∂θ

∂X+ V

∂θ

∂Y=

(∂2θ

∂X2+∂2θ

∂Y 2

). (11)

Since the present work is to study the flow field, the fluid motion is displayedby using the stream function which is defined as U = ∂ψ

∂Yand V = − ∂ψ

∂X. Thus,

Eq.(8) is changed to Eq.(12)

∂2ψ

∂X2+∂2ψ

∂Y 2=∂U

∂Y− ∂V

∂X. (12)

To eliminate the pressure P , we use the penalty finite element method with apenalty parameter γ such that P = −γ

(∂U∂X

+ ∂V∂Y

)([12]). Substituting P into

Eq.(9) and (10) yield Eq.(13) and (14).

U∂U

∂X+ V

∂U

∂Y= γ

∂

∂X

(∂U

∂X+∂V

∂Y

)+ Pr

(∂2U

∂X2+∂2U

∂Y 2

)− Pr

DaU, (13)

U∂V

∂X+ V

∂V

∂Y= γ

∂

∂Y

(∂U

∂X+∂V

∂Y

)+ Pr

(∂2V

∂X2+∂2V

∂Y 2

)− Pr

DaV +RaPrθ.

(14)The transformed boundary conditions are:

U(f1(Y ), Y ) = U(f2(Y ), Y ) = U(X, 0) = U(X, Y ) = 0,

V (f1(Y ), Y ) = V (f2(Y ), Y ) = V (X, 0) = V (X, Y ) = 0, (15)


θ(f1(Y ), Y ) = θ(f2(Y ), Y ) = θ(X, Y ) = 0,

θ(X, 0) = 12(1 − cos(2πX)) .

To visualize the heat transfer by fluid flow, heatfunction for a two-dimensionalconvection problem in dimensionless form can be defined as [13]

∂2φ

∂X2+∂2φ

∂Y 2=∂(Uθ)

∂Y− ∂(V θ)

∂X. (16)

With following boundary conditions:

φ(X, 0) = π cos(πX)

φ(f1(Y ), Y ) = φ(f2(Y ), Y ) = φ(X, Y ) = 0. (17)

4 Results and Discussion

To analyze the numerical solutions, the governing equations mentioned pre-viously are code into FlexPDE 6.17 Professional. FlexPDE is a software pack-age which performs the operations necessary to turn a description of a partialdifferential equations system into a finite element model. Then, it solves thesystem and present graphical and tabular outputs of the results. In this ex-periment, the interested parameters are Darcy number, Rayleigh number andwave amplitude. The results obtained from varying value of these parametersare displayed by streamlines, isotherms and heatlines in which graphical out-puts from FlexPDE are modified to show numerical values.

In this study, the experiments are categorized according to the differentvalues of Da, Ra and λ. Computations have been carried out for variousDa = 10−4 − 10−2 and Ra = 10 − 105 covering wide range of applications([1],[9]). Also, the range of wave amplitude from 0.01 − 0.1 have been studied([9]). The aspect ratio,AR, and numbers of undulations are fixed at 1 and2, respectively. Boundary conditions of walls and heatfunction are given byEq.(15),(16) and (17).

Streamlines, isotherms and heatlines obtained from varying interested pa-rameters are shown for some cases to illustrate fluid circulation and heattransfer inside the cavity. Fig.2. shows the results of decreasing Da in whichFig.2(a)-2(c) are the results forDa = 10−2 and Fig.2(d)-2(f) are forDa = 10−4.Streamlines illustrate that fluid circulations are symmetric with respect to thevertical center line ψ = 0.0 such that the left half is positive value and theright half is negative. The opposite sign indicates a direction motion in such away that positive and negative signs give anti-clockwise and clockwise circula-tion patterns, respectively. The main flow moves down to the bottom and theflow strength decreases when Da is decreased. The maximum value of streamfunction in Fig.2(a) is ψmax = 7.5 and continually decreases to ψmax = 0.16 as


shown in Fig.2(d). The temperature distribution can be seen from isothermpattern. In Fig.2(b), the isotherms disperse the entire enclosure while themagnitude of distribution becomes smaller at low Da as shown in Fig.2(e).Decreasing Da also plays a role on heat transfer. Heatlines in Fig.2 showsthat heat flows from the bottom portion to the top wall and heat distributionbecomes smaller similar to isotherm pattern.

a

b

c

d e

f

g

hi

j

k

l

m

n

o

pp

p

p

q

r

s

t

u

v

wx

yz

A

BC

D

Eox

E : 7.50D : 7.00C : 6.50B : 6.00A : 5.50z : 5.00y : 4.50x : 4.00w : 3.50v : 3.00j : -3.00i : -3.50h: -4.00g: -4.50f : -5.00e: -5.50d :-6.00c : -6.50b :-7.00a : -7.50

(a)

b

c

d

e

fg h

i

j

k

l

mn o

pr

t

o

x

u : 1.00t : 0.95s : 0.90r : 0.85q : 0.80p : 0.75o : 0.70n : 0.65m : 0.60l : 0.55k : 0.50j : 0.45i : 0.40h : 0.35g : 0.30f : 0.25e : 0.20d : 0.15c : 0.10b : 0.05a : 0.00

(b)

abd

d

e

f

gh

i

jk

l

m

n

o

p

q

r sox

v : 3.30u : 3.00t : 2.70s : 2.40r : 2.10q : 1.80p : 1.50o : 1.20n : 0.90m : 0.60j : -0.30i : -0.60h : -0.90g : -1.20f : -1.50e : -1.80d : -2.10c : -2.40b : -2.70a : -3.00

(c)

a

b

c

de

f

g

h

i i

i

i

i

i

i

i

i

j

k

l

m

nop

qo

x

q : 0.16p : 0.14o : 0.12n : 0.10m : 0.08l : 0.06k : 0.04j : 0.02i : 0.00h :-0.02g :-0.04f : -0.06e :-0.08d :-0.10c : -0.12b :-0.14a : -0.16

(d)

bc

d

e

fg

h

ij

k

lmn

pq rs


(e)

abce fg

h

i

j

k

l

m

n opqrstu

v : 3.30u : 3.00t : 2.70s : 2.40r : 2.10q : 1.80p : 1.50o : 1.20n : 0.90m : 0.60l : 0.30k : 0.00j : -0.30i : -0.60h : -0.90g : -1.20f : -1.50e : -1.80d : -2.10c : -2.40b : -2.70a : -3.00

(f)

Figure 2: Streamlines (left), isotherms (middle) and heatlines (right) for Ra = 105,λ = 0.05, Da = 10−2 (above), Da = 10−4 (bottom).

As Ra is decreased to 103 (Fig.3), it is observed that the intensity of circu-lation is less than the previous case with Ra = 105 (Fig.2(a)). Similar to thecase of decreasing Da, the magnitudes of temperature and heat distributionsbecome smaller when Ra is reduced.

The results obtained from varying wave amplitude are displayed by stream-line (Fig.4). It is noted that the flow field near two vertical wavy walls is dis-torted due to the shape of the enclosure. Moreover, the flow strength decreaseswith increasing values of wave amplitude. For λ = 0.02, the maximum valueof stream function is ψmax = 9.0 and ψmax = 5.5 when λ is increased to 1.0.


ab

cd

e

f

g

h

i

j

j j

j

j

jj

j

k

l

m

n

op

q

r

sox

s : 4.50r : 4.00q : 3.50p : 3.00o : 2.50n : 2.00m : 1.50l : 1.00k : 0.50j : 0.00i : -0.50h : -1.00g : -1.50f : -2.00e : -2.50d : -3.00c : -3.50b :-4.00a : -4.50Scale = e-2

(a)

b

c

d

e

f

gh

i

j kl

m

no

pq

r s

o

x


(b)

cdeg

h

i

j

kl

mn

op

q rs

v : 3.30u : 3.00t : 2.70s : 2.40r : 2.10q : 1.80p : 1.50o : 1.20n : 0.90m : 0.60l : 0.30k : 0.00j : -0.30i : -0.60h : -0.90g : -1.20f : -1.50e : -1.80d : -2.10c : -2.40b : -2.70a : -3.00

(c)

Figure 3: Streamlines (left), isotherm (middle) and heatlines (right) for Da = 10−2,λ = 0.05, Ra = 103.

a

bcd

e

f

g

h

i

jj

j

j

j

j

j

j

j

k

l

m

n

o

pq

r

so

x

s : 9.00r : 8.00q : 7.00p : 6.00o : 5.00n : 4.00m : 3.00l : 2.00k : 1.00j : 0.00i : -1.00h :-2.00g :-3.00f : -4.00e : -5.00d :-6.00c : -7.00b :-8.00a : -9.00

(a)

a

b

c

d

e

f

g

h

i

j

k

l

m

no o

o

p

q

r

s

t

u

v

w

x

y

z

A

B

C

ox

C : 7.00A : 6.00y : 5.00w : 4.00u : 3.00s : 2.00r : 1.50q : 1.00p : 0.50o : 0.00n : -0.50m :-1.00l : -1.50k : -2.00i : -3.00g : -4.00e : -5.00c : -6.00a : -7.00

(b)

a

b

c

d

e

f

g

h

i

jk

l

l

l

l

l l

m

n

o

p

q

r

s

tu vw

ox

w : 5.50v : 5.00t : 4.00r : 3.00q : 2.50p : 2.00o : 1.50n : 1.00m : 0.50l : 0.00k : -0.50j : -1.00i : -1.50h :-2.00g :-2.50f : -3.00d :-4.00b :-5.00a : -5.50

(c)

Figure 4: Streamlines for different wave amplitude for Ra = 105, Da = 10−2,λ = 0.02, 0.06, 1.0, respectively.

Since there is no a big effect for isotherms and heatlines, the results are notshown here.

5 Conclusion

The present work investigates the natural convection in a square enclosuresuch that two vertical walls are wavy. The fluid-saturated porous media is con-tained inside it. Here, the considered fluid is air. The aim of this study is toexamine the effects of Darcy number, Rayleigh number and wave amplitude onnatural convection expressed by differential equation of conservation of mass,


momentum and energy. To analyze the performance, the governing equationsare coded into a software package FlexPDE 6.17 Professional. Interesting re-sults are obtained and displayed by streamlines, isotherms and heatlines.

The values of Rayleigh and Dacy numbers have been chosen based on widerange of applicability. The Ra has been varied from 10−105. In the enclosurefilled with porous media, Da is used to describe the flow inside it. In order toobserve permeability, the range of Da is chosen between 10−4 − 10−2. Stream-lines shown in previous section illustrate that the decrease of Da decreasedthe strength of convection. In addition, the magnitudes of isotherms and heat-lines patterns become smaller with low Da. In the case of increasing Ra, itis equivalent to increase buoyancy driven flow. It is observed that adding Raenhances the strength of heat and fluid flow. The increase of wave amplitudeaffects the flow intensity inside the cavity, that is, the flow is obstructed bybarricades resulting from extending the amplitude. The wavy wall has no abig effect for temperature and heat distribution.

ACKNOWLEDGEMENTS. This research is (partially) supported byCentre of Excellence in Mathematics, the Commission on Higher Education,Thailand. The authors would like to thank Department of Mathematics, Fac-ulty of Science, Khon Kaen University (Thailand) for computational resourcesnecessary for this work.

References

[1] T. Basak, S. Roy, A. Singh and I. Pop, Finite element simulation of naturalconvection flow in a trapezoidal enclosure filled with porous medium dueto uniform and non-uniform heating, International Journal of Heat andMass Transfer, 52 (2009), 70 - 78.

[2] T. Basak, S. Roy, B. Krishna and A.R. Balakrishnan, Finite element anal-ysis of natural convection flow in a isosceles triangular enclosure due touniform and non-uniform heating at the side walls, International Journalof Heat and Mass Transfer, 51 (2008), 4496 - 4505.

[3] A. Koca, H.A. Oztop and Y. Barol, The effects of Prandtl number onnatural convection in triangular enclosure with localized heating frombelow, International Journal of Heat and Mass Transfer, 34 (2007), 511- 519.

[4] H. Asan and L. Namli, Laminar natural convection in a pitched roof oftriangular cross-section: summer day boundary conditions, Energy andBuilding, 33 (2000), 69 - 73.


[5] H. Asan and L. Namli, Numerical simulation of buoyant flow in a roofof triangular cross-section under winter day boundary conditions, Energyand Building, 33 (2000), 753 - 757.

[6] P. Sompong and S. Witayangkurn, Numerical study of natural convectionin a triangular roof in which the temperature outside the roof is hot, LaosJournal on Applied Science, 2 (2011), 391 - 397.

[7] A. Dalal and M.K. Das, Laminar natural convection in an inclined compli-cated cavity with spatially variable wall temperature, International Jour-nal of Heat and Mass Transfer, 48 (2005), 3833 - 3854.

[8] A. Dalal and M.K. Das, Natural convection in a cavity with a wavy wallheated from below and uniformly cooled from the top and both sides,ASME Journal of Heat Transfer, 128 (2006), 717 - 725.

[9] A. Dalal and M.K. Das, Heatline method for the visualization of naturalconvection in a complicated cavity, International Journal of Heat andMass Transfer, 51 (2008), 263 - 272.

[10] H.F. Oztop, E. Abu-Nada, Y. Varol and A. Chamkha, Natural convectionin wavy enclosures with volumetric heat sources, International Journal ofThermal Sciences, 50 (2011), 502 - 514.

[11] M. Sathiyamoorthy, T. Basak, S. Roy and I. Pop, Steady natural con-vection flow in a square cavity filled with a porous medium for linearlyheated side wall(s), International Journal of Heat and Mass Transfer, 50(2007), 1892 - 1901.

[12] J.N. Reddy, An introduction to the finite element method, McGraw-Hill,New York, 1993.

[13] T. Basak, G. Aravind and S. Roy, Visualization of heat flow due to nat-ural convection within triangular cavities using Bejan’s heatline concept,International Journal of Heat and Mass Transfer, 52 (2009), 2824 - 2833.

Received: January, 2012

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Simulation of Natural Convection in a Complicated ... · Simulation of natural convection in a complicated enclosure 2835 details of solution procedure. Finally, we evaluate the computational