Bejan's heatlines and numerical visualization of convective heat flow in differentially heated...

lable at ScienceDirect

Energy 64 (2014) 69e94

Contents lists avai

Energy

journal homepage: www.elsevier .com/locate/energy

Bejan’s heatlines and numerical visualization of convective heat flowin differentially heated enclosures with concave/convex side walls

Pratibha Biswal, Tanmay Basak*

Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

a r t i c l e i n f o

Article history:Received 4 January 2013Received in revised form7 October 2013Accepted 11 October 2013Available online 5 December 2013

Keywords:Natural convectionCurved enclosuresStreamlinesHeatlinesNusselt numberThermal management

* Corresponding author. Tel.: þ91 44 2257 4173; faE-mail address: [email protected] (T. Basak).

0360-5442/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.energy.2013.10.032

a b s t r a c t

Numerical simulation for natural convection flow in fluid filled enclosures with curved side walls iscarried out for various fluids with several Prandtl numbers (Pr ¼ 0.015, 0.7 and 1000) in the range ofRayleigh numbers (Ra ¼ 103e106) for various cases based on convexity/concavity of the curved side wallsusing the Galerkin finite element method. Results show that patterns of streamlines and heatlines arelargely influenced by wall curvature in concave cases. At low Ra, the enclosure with highest wall con-cavity offers largest heat transfer rate. On the other hand, at high Ra, heatline cells are segregated andthus heat transfer rate was observed to be least for highest concavity case. In convex cases, no significantvariations in heat and flow distributions are observed with increase in convexity of side walls. At high Raand Pr, heat transfer rate is observed to be enhanced greatly with increase in wall convexity. Resultsindicate that enhanced thermal mixing is observed in convex cases compared to concave cases.Comparative study of average Nusselt number of a standard square enclosure with concave and convexcases is also carried out. In conduction dominant regime (low Ra), concave cases exhibit higher heattransfer rates compared to square enclosure. At high Ra, low Pr, concave cases with P1P01 ¼ 0:4 is ad-vantageous based on flow separation and enhanced local heat transfer rates.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Natural convection is the mode of heat transfer where buoyancyforces govern the fluid motion. Study of natural convection inconfined cavities as well as in open channels has received signifi-cant attention due to versatile natural, industrial and engineeringapplications, especially those associated with conservation of en-ergy. Some of the important industrial and engineering applica-tions of natural convection are cooling of electronic devices [1],thermal storage tanks [2], nuclear reactors [3], proper ventilationand thermal comfort in buildings [4,5], efficient design of buildings[6], analyzing thermal storage characteristics of aluminum [7],drying of fruit and vegetables using solar dryer [8], enhanced gasrecovery [9] etc. Also, knowledge of natural convection plays asignificant role in the design strategy of solar energy collectorswhich involves efficient collection, storage and distribution of solarenergy [10]. A good amount of research has been carried outto understand the role of natural convection in solar energy field[11e14]. In above mentioned applications, natural convection playsa vital role in conservation of energy through efficient heat transfer

x: þ91 44 2257 0509.

All rights reserved.

process. In order to analyze heat efficiency in a system, it is requiredto assess the amount of heat transfer. Thus, study of natural con-vection in enclosure with various geometric configuration, espe-cially enclosure with curved walls may be relevant in many heattransfer applications. In energy point of view, it is important toshow the direction and intensity of energy flow during naturalconvection in an enclosure. Knowledge of direction and intensity ofenergy flow during natural convection in complicated cavities asconsidered in current work may be used in design of variousthermal equipments with better thermal performance.

Investigations of natural convection in a confined cavity inpresence of various boundary conditions with different parametershave been carried out by researchers for various energy relatedapplications [15,16]. Several studies on buoyancy driven flowwithin triangular enclosures have been carried out by researchersfor proper design and ventilation in rooms [17e19]. Natural con-vection in air filled 2D tilted square cavities and parallelogrammiccavities for various geometrical and thermal configurations wasinvestigated by Bairi [20,21]. All above mentioned works have beencarried out to understand natural convective heat flow withinregular and irregular enclosures with flat wall. In contrast, there arelimited number of studies with curved walled enclosures becauseof complexity of flow inside the enclosure due to complicated ge-ometry. Wall curvature of the curved wall is considered to be one of

Delta:1_given name

Delta:1_surname

mailto:[email protected]

http://crossmark.crossref.org/dialog/?doi=10.1016/j.energy.2013.10.032&domain=pdf

www.sciencedirect.com/science/journal/03605442

http://www.elsevier.com/locate/energy

http://dx.doi.org/10.1016/j.energy.2013.10.032



P. Biswal, T. Basak / Energy 64 (2014) 69e9470

the important parameters for assessment of heat flow, fluid flowand thermal characteristics of the fluid within the enclosure.

Natural convection heat transfer within enclosures withcurved or wavy surfaces is very often used in microelectronicindustries [22], solar energy systems [23], solidification process[24] etc. Cooling process of water in a circular enclosure sub-jected to non-uniform boundary conditions is studied by Ala-wadhi [25]. Further, Alawadhi [26] also investigated solidificationprocess of water inside an elliptical enclosure with various aspectratios and inclination angles. Gao et al. [27] conducted a nu-merical study on natural convection inside channel between theflat-plate cover and the sine-wave absorber in a cross-corrugatedsolar air heater.

Investigations on natural convection as mentioned above, havebeen analyzed based on conventional streamline and isothermconcepts which are not adequate to understand direction andintensity of convective heat transfer. Current work is devoted toanalyze the distribution of heat flow associated with naturalconvection within an enclosure with curved side walls usingheatline method. Heatlines concept was first introduced byKimura and Bejan [28]. Mathematically, magnitudes of heatlinesare represented by heatfunctions which are further related to theNusselt number. A unified approach for streamline, heatline andmassline methods to visualize two dimensional transport phe-nomena is implemented by Costa [29]. An extensive review onBejan’s heatlines and masslines for convection heat transfer andmass transfer visualization, respectively was also presented byCosta [30]. Further, Dalal and Das [31] studied heatline method forthe visualization of natural convective heat flow in a complicatedcavity.

Several enclosures with various geometrical orientations sub-jected to different boundary conditions have been considered forheatline based heat flow visualization of natural convection. Kaluriand Basak [32] analyzed Bejan’s heatlines and thermal mixingduring natural convection in a square enclosure with distributedheat sources. In addition, earlier numerical investigations on heatflow visualization during natural convection within the squareenclosure [33], tilted square enclosure [34,35], triangular enclosure[36], rhombic enclosure [37] and trapezoidal enclosure [38] withvarious uniform and non-uniform heating of the wall are found inthe literature. In all the above mentioned works, the Galerkin finiteelement method is employed to solve the governing equations forvarious boundary conditions. Earlier works were carried out withthe heatline analysis of natural convection in enclosures involvingregular geometries with straight walls. The finite element methodhas also an advantage to study natural convection problems withincomplicated domains with curved walls involving heatfunctionevaluation via automatically built-in complex boundary conditionsvia Galerkin weighted residual. The complete understanding onheat flow visualization with thermal management via heatlinemethod in various cavities with convex or concave side walls hasindustrial relevance and these analyses are reported for first time inthis paper.

The aim of the present paper is to provide a complete under-standing about the fluid flow, heat flow and thermal characteristicswithin cavities with curved side walls. The concave and convex sidewalls are chosen to investigate efficient thermal management viaheatline approach and the major objective of this work is to studythe distribution of heat and fluid flow within the cavity. Currentphysical configurations with various curved side walls can be usedin design of various heat transfer devices such as solar collectors,heat exchangers, storage and cooling devices. The cavity is sub-jected to various boundary conditions such as isothermally hot leftwall, isothermally cold right wall and adiabatic horizontal walls.The Galerkin finite element method with penalty parameter [39] is

used to solve the nonlinear coupled partial differential equationsgoverning the flow and thermal fields and the finite elementmethod is further used to solve the Poisson equation for stream-function and heatfunction. Numerical results are obtained todisplay the streamlines, heatlines and isotherms for various Ray-leigh numbers (103 � Ra � 106), Prandtl numbers (Pr ¼ 0.015, 0.7and 1000) and wall curvatures. The heat transfer rates along thecurved side walls are presented in terms of local and averageNusselt numbers.

2. Mathematical modeling and simulations

The physical domain in three dimensional form is shown inFig. 1(aeb). The computational domain is shown in Fig. 1(c) basedon semi-infinite approximation along Z direction. Thermo-physicalproperties of the fluid in the flow field are assumed to be constantexcept density. The variation of density with temperature can becalculated using Boussinesq approximation. In this way, the tem-perature field and flow fields are coupled. Under these assump-tions, governing equations for steady two-dimensional naturalconvection flow in the enclosure with curved side walls usingconservation of mass, momentum and energy can be written withfollowing dimensionless variables or numbers:

X ¼ xL; Y ¼ y

L; U ¼ uL

a; V ¼ vL

a; q ¼ T � Tc

Th � Tc

P ¼ pL2

ra2; Pr ¼ n

a; Ra ¼ gbðTh � TcÞL3Pr

n2:

Here x and y are the distances measured along the horizontaland vertical directions, respectively; u and v are the velocitycomponents in the x and y directions, respectively; T denotes thetemperature; n and a are kinematic viscosity and thermal diffu-sivity, respectively; b denotes the volume expansion coefficient; pis the pressure and r is the density; Th and Tc are the temperaturesat hot left wall and cold right wall, respectively and L is the heightor length of the base of the cavity. Note that, X and Y are dimen-sionless coordinates varying along horizontal and vertical di-rections, respectively; U and V are dimensionless velocitycomponents in the X and Y directions, respectively; q is thedimensionless temperature; P is the dimensionless pressure andRa and Pr are Rayleigh and Prandtl number, respectively. Thegoverning equations in dimensionless forms for continuity [Eq.(1)], momentum balance [Eqs. (2) and (3)] and energy balance [Eq.(4)] are as follows:

vUvX

þ vVvY

¼ 0; (1)

UvUvX

þ VvUvY

¼ �vPvX

þ Pr

v2UvX2 þ v2U

vY2

!; (2)

UvVvX

þ VvVvY

¼ �vPvY

þ Pr

v2VvX2 þ v2V

vY2

!þ RaPrq; (3)

and

Uvq

vXþ V

vq

vY¼ v2q

vX2 þv2q

vY2 (4)

and the governing equations [Eqs. (2)e(4)] are subjected to thefollowing boundary conditions;

ADIABATIC

(a)

(b)

ADIABATIC

ADIABATIC

CONVEX SIDE WALLS

CONCAVE SIDE WALLS

X

Y

Z

Y

XZ

ADIABATIC

COLD

WALL

COLD

WALL

WALLHOT

WALLHOT

A

B0

1

B

11 DD

CC

Y,VY,V

X,UA

0

X,U1

θ=1LLAWLLAW

θ=1P PP 2 P P

)3−1(SESAC)3−1(SESACSLLAWEDISXEVNOCSLLAWEDISEVACNOC

P2

ADIABATIC WALL

LLAWCITABAIDALLAWCITABAIDA

ADIABATIC WALL

COLDHOT

WALL

HOTCOLD

WALL

θ=0θ=0

P ’’’’ P2 2111 1

(c)

Fig. 1. (a) Schematic diagram of the physical system with (a) convex side walls and (b) concave side walls, (c) schematic diagram of the computational domain for enclosure withconcave/convex side walls.

P. Biswal, T. Basak / Energy 64 (2014) 69e94 71

U ¼ V ¼ 0; vqvY ðX;0Þ ¼ 0; for Y ¼ 0 on wall AB

U ¼ V ¼ 0; q ¼ 0; for X ¼ aY2 þ bY þ c on wall DA

U ¼ V ¼ 0; vqvY ðX;1Þ ¼ 0; for Y ¼ 1 on wall CD

U ¼ V ¼ 0; q ¼ 1; for X ¼ a0Y2 þ b0Y þ c0 on wall BC(5)

Note that, a, b, c, a0, b

0and c

0are dimensionless numbers. The

coefficients of the quadratic equation are calculated by Cramer’srule using three coordinates for various convex and concave curvesas shown in Fig. 1(a), (b). The values of the dimensionless numbersfor all the considered cases are given in Table 1.

3. Solution procedure and evaluation of dimensionlessquantities

3.1. Simulations for flow and temperature fields

Themomentum and energy balance equations (Eqs. (2)e(4)) aresolved using the Galerkin finite element method. The continuityequation [Eq. (1)] has been used as a constraint due to mass con-servation and this constraint may be used to obtain the pressuredistribution. In order to solve Eqs. (2) and (3), we use the penaltyfinite element method where the pressure, P is eliminated by apenalty parameter, g and the incompressibility criteria given by[Eq. (1)] which results in

YÞ

Table 1Values of P1P01 for left wall and P2P02 for right wall for all concave and convex cases; values of a, b, c for left wall and a0 , b0 , c0 for right wall as calculated by Cramer’s rule usingcoordinates of the curve (A; P01;D for left wall and B; P02; C for right wall) [see Fig. 1(c)]. Length of curved wall is denoted by Smax.

Concave Convex Smax

P1P01 P2P02 a b c a0 b0 c0 P1P01 P2P02 a b c a0 b0 c0

Case 1 0.1 �0.1 �0.4 0.4 0 0.4 �0.4 1 �0.1 0.1 0.4 �0.4 0 �0.4 0.4 1 1.026Case 2 0.2 �0.2 �0.8 0.8 0 0.8 �0.8 1 �0.2 0.2 0.8 �0.8 0 �0.8 0.8 1 1.098Case 3 0.4 �0.4 �1.6 1.6 0 1.6 �1.6 1 �0.4 0.4 1.6 �1.6 0 �1.6 1.6 1 1.333


P ¼ �g

�vUvX

þ vVvY

�(6)

The continuity equation [Eq. (1)] is automatically satisfied forlarge values of g. Typical values of g that yield consistent solutionsare 107. Using Eq. (6), the momentum balance equations [Eqs. (2)and (3)] reduce to

UvUvX

þ VvUvY

¼ gv

vX

�vUvX

þ vVvY

�þ Pr

v2UvX2 þ v2U

vY2

!(7)

and

UvVvX

þVvVvY

¼ gv

vY

�vUvX

þ vVvY

�þPr

v2VvX2 þ

v2VvY2

!þRa Pr q (8)

The system of equations [Eqs. (4), (7) and (8)] with boundaryconditions [Eq. (5)] are solved using the Galerkin finite elementmethod [39]. Expanding the velocity components (U, V) and tem-perature (q) using basis set fFkgNk¼1 as,

UzXNk¼1

UkFkðX; YÞ; VzXNk¼1

VkFkðX; YÞ and qzXNk¼1

qkFkðX;

(9)

The Galerkin finite element method yields nonlinear residualequations for Eqs. (4), (7) and (8), at nodes of internal domain U.

Rð1Þi ¼XNk¼1

Uk

ZU

" XNk¼1

UkFk

!vFk

vXþ XN

k¼1

VkFk

!vFk

vY

#FidXdY

þ g

264XN

k¼1

Uk

ZU

vFi

vXvFk

vXdXdY þ

XNk¼1

Vk

ZU

vFi

vXvFk

vYdXdY

375

þ PrXNk¼1

Uk

ZU

�vFi

vXvFkvX

þ vFi

vYvFkvY

�dXdY

(10)

Rð2Þi ¼XNk¼1

Vk

ZU

" XNk¼1

UkFk

!vFk

vXþ XN

k¼1

VkFk

!vFk

vY

#FidXdY

þg

264XN

k¼1

Uk

ZU

vFi

vYvFk

vXdXdY þ

XNk¼1

Vk

ZU

vFi

vYvFk

vYdXdY

375

þPrXNk¼1

Vk

ZU

�vFi

vXvFkvX

þ vFi

vYvFkvY

�dXdY

�RaPrZU

XNk¼1

qkFk

!FidXdY

(11)and

Rð3Þi ¼XNk¼1

qk

ZU

" XNk¼1

UkFk

!vFkvX

þ XN

k¼1

VkFk

!vFkvY

#FidXdY

þXNk¼1

qk

ZU

�vFi

vXvFkvX

þ vFi

vYvFkvY

�dXdY

(12)

Bi-quadratic basis functions with three point Gaussian-quadrature are used to evaluate the integrals in the residualequations except the second term in Eqs. (10) and (11). In Eqs. (10)and (11), the second term containing the penalty parameter (g) isevaluated with two point Gaussian-quadrature (reduced integra-tion penalty formulation [39]). The non-linear residual equations[Eqs. (10)e(12)] are solved using NewtoneRaphson method todetermine the coefficients of the expansions in Eq. (9). The detailedsolution procedure is given in an earlier work [33].

3.2. Streamfunction, Nusselt number and heatfunction

3.2.1. StreamfunctionThe fluid motion is displayed using the streamfunction (j) ob-

tained from velocity components U and V. The relationships be-tween streamfunction, (j) and velocity components for twodimensional flows are

U ¼ vj

vY; V ¼ �vj

vX(13)

which yield a single equation

v2j

vX2 þv2j

vY2 ¼ vUvY

� vVvX

(14)

Using the above definition of the streamfunction, the positivesign of j denotes anticlockwise circulation and the clockwisecirculation is represented by the negative sign of j. Expanding

the streamfunction (j) using the basis set fFkgNk¼1 as

j ¼ PNk¼1 jkFkðX;YÞ and the relation for U, V from Eq. (9), the

Galerkin finite element method yields the following linear residualequations for Eq. (14).

Rsi ¼XNk¼1

jk

ZU

�vFi

vXvFkvX

þ vFi

vYvFkvY

�dXdY

þXNk¼1

Uk

ZU

FivFk

vYdXdY �

XNk¼1

Vk

ZU

FivFk

vXdXdY (15)

The no-slip condition is valid at all boundaries as there is nocross flow, hence j¼ 0 is used as residual equations at the nodes forthe boundaries. The bi-quadratic basis function is used to evaluatethe integrals in Eq. (15) and j’s are obtained by solving the N linearresidual equations [Eq. (15)] [33].


3.2.2. Nusselt numberThe heat transfer coefficient in terms of the local Nusselt

number (Nu) is defined as

Nu ¼ �vq

vn(16)

where n denotes the normal direction on a plane. The normal de-rivative is evaluated using the bi-quadratic basis set. The localNusselt numbers at left wall (Nul) and rightwall (Nur) are defined as

Nul ¼X9i¼1

qi

�sin4

vFi

vX� cos4

vFi

vY

�(17)

and

Nur ¼X9i¼1

qi

�esin4

vFi

vXþ cos4

vFi

vY

�(18)

The average Nusselt numbers at the side walls are

Nul ¼

ZS0

Nulds0

ZS0

ds0

(19)

and

Nur ¼

ZS0

Nurds0

ZS0

ds0

(20)

Here S is the length and ds0 is the small elemental length along thecurved wall.

3.2.3. HeatfunctionThe heat flow within the enclosure is displayed using the

heatfunction (P) obtained from conductive heat fluxes (�vq/vX, �vq/vY) as well as convective heat fluxes (Uq, Vq). The heat-function satisfies the steady energy balance equation [Eq. (4)] [28]such that

vPvY ¼ Uq� vq

vX

�vPvX ¼ Vq� vq

vY

(21)

which yield a single equation

v2P

vX2 þ v2P

vY2 ¼ v

vYðUqÞ � v

vXðVqÞ (22)

Expanding the heatfunction (P) using the basis set fFkgNk¼1 asP ¼ PN

k¼1 PkFkðX;YÞ and the relationship for U, V and q from Eq.(9), the Galerkin finite element method yields the following linearresidual equations for Eq. (22).

Rhi ¼XNk¼1

Pk

ZU

�vFi

vXvFkvX

þ vFi

vYvFkvY

�dXdY

�ZG

Fin,VPdGþXNk¼1

Uk

ZU

XNk¼1

qkFk

!FivFk

vYdXdY

þXNk¼1

qk

ZU

XNk¼1

UkFk

!FivFk

vYdXdY

�XNk¼1

Vk

ZU

XNk¼1

qkFk

!FivFkvX

dXdY

�XNk¼1

qk

ZU

XNk¼1

VkFk

!FivFk

vXdXdY

(23)

The above residual equation [Eq. (23)] is further supplementedwith various Dirichlet and Neumann boundary conditions in orderto obtain a unique solution. Neumann boundary conditions ofP areobtained for isothermal (hot or cold) walls as derived from Eq. (21)and are specified as follows:

n,VP ¼ 0 ðisothermal hot left wallÞ¼ 0 ðisothermal cold right wallÞ (24)

The top and bottom insulated walls may be represented byDirichlet boundary condition as obtained from Eq. (21) which issimplified into vP/vX ¼ 0 for an adiabatic wall. A reference value ofP is assumed as 0 at X ¼ 0, Y ¼ 0 and henceP ¼ 0 is valid for Y ¼ 0,cX. The value of P at the top wall is obtained as, P ¼ SmaxNul forY ¼ 1, cX. Note that, Smax is length of the curved wall. It may benoted that, the unique solution of Eq. (22) is strongly dependent onthe non-homogeneous Dirichlet conditions. At the top horizontalwall BC, boundary condition for P may be obtained by integratingEq. (21).

Pð0;1Þ ¼ Pð0;0Þ þZSmax

0

�vP

vs0

�ds0

¼ Pð0;0Þ þ SmaxNul

¼ SmaxNul ¼ Pð1;1Þ ¼ PðX;1Þ

(25)

4. Results and discussion

4.1. Numerical tests and parameters

The computational domain in x�h coordinate system consists of28 � 28 bi-quadratic elements which correspond to 57 � 57 gridpoints. The computational grid with curved side walls is generatedbymapping the curved domain into a regular square domain in x�h

coordinate system [36]. The bi-quadratic elements with lessernumber of nodes smoothly capture the nonlinear variations of thefield variables, which are in contrast with the finite-difference/finite-volume solutions. In the current investigation, theGaussian-quadrature-based finite element method providessmooth solutions at the interior domain including the corner re-gions as evaluation of residuals depends on interior Gauss pointsand thus the effect of corner nodes is less pronounced in the finalsolution. In general, Nusselt numbers are calculated at any surfaceusing some interpolation functions in the finite difference/finitevolume based methods which are avoided in the present work. On

Fig. 2. Streamlines (first column), heatlines (second column) and isotherms (third col-umn) for a square enclosure with hot left wall, cold right wall and adiabatic horizontalwalls at Pr¼ 0.71 for (a) Ra¼ 103, (b) Ra¼ 104, (c) Ra¼ 105 and (d) Ra¼ 106 for the presentwork (top panels) and as reported by Deng and Tang [40] (bottom panels). (Lower panelsof (a)e(d) reproduced from Deng and Tang [40] with permission from Elsevier).


the other hand, the finite element approach offers special advan-tage on evaluation of local Nusselt number at the left and rightwalls as the element basis functions are used to evaluate the heatflux. Benchmark studies were carried out for the square cavity witha hot left wall and a cold right wall in presence of adiabatic hori-zontal walls, similar to the case reported by Deng and Tang [40] andWan et al. [41]. The results in terms of streamlines, isotherms andheatlines are in excellent agreement with the earlier work [40] [seeFig. 2]. In addition, benchmark studies were also carried out interms of average Nusselt number ðNurÞ and maximum stream-function ðjjjmaxÞ value for a square enclosure as reported by Dengand Tang [40] for various Ra [see Table 4]. Also, the results in termsof average Nusselt number as reported by Wan et al. [41] arecompared with the current work [see Table 4] and the results are ingood agreement with earlier works [40,41]. Current solutionscheme produces grid invariant results for Nur and jjjmax as shownin Tables 2 and 3, respectively for all the concave and convex cases.Also, illustrations for grid dependence test are shown in Figs. 3and 4 for various grid sizes. It is found that spurious flow circula-tion cells and non-smooth isotherms are the typical characteristicsfor smaller grid points whereas 24 � 24 and 28 � 28 meshes giveidentical flow and isotherm characteristics.

Detailed computations are carried out for all concave/convexcases for a range of Rayleigh numbers (Ra ¼ 103e106) at various Pr(Pr¼ 0.015, 0.7 and 1000). Intensity of fluid flow is veryweak at lowRa and heat transfer is mainly due to conduction and isotherms aresmooth vertical curves. At higher Ra, stronger fluid circulation cellsare formed due to significant effect of convective force. Higherbuoyancy forces also cause the isotherms to be distorted. Also,closed loop heatlines are observed within the enclosure indicatinghigh thermal mixing during high convective effect. At lower Pr,fluid circulation cells are circular in shape whereas circulation cellsstrongly follow the shape of the enclosure at higher Pr. At lowercurvature of concave wall, the intensity of fluid flow as well as theconvective heat circulation is high compared to that at high cur-vature of side walls. In contrast, for convex cases, larger convectiveeffect is seen in the enclosure with higher curvature. Detailed ex-planations of the results with particular emphasis on the effect ofwall curvature and various parameters (Ra and Pr) on the iso-therms, streamlines and heatlines for a few test cases are presentedin following sections.

4.2. Isotherms, streamlines and heatlines in the enclosure withconcave side walls

The enclosure with concave side walls is considered with threecases based on three different curvatures of the concave wall. Theoriginal square enclosure is modified to a curved walled enclosureby shifting the mid points of side walls, P1 and P2 in the inwarddirection to P01 and P02, respectively, such that AP01D and BP02C formcurves which obey quadratic equations; X ¼ aY2 þ bY þ c andX¼ a

0Y2þ b

0Yþ c

0, respectively [see Fig.1(c) and Table 1]. The values

of P1P01 or P2P02 are assumed to be L/10, 2L/10 and 4L/10 for case 1,case 2 and case 3, respectively, where L is the height or length ofbase of the cavity. Figs. 5e10 display the streamlines, heatlines andisotherms for various values of Pr (Pr ¼ 0.015, 0.7 and 1000) and Ra(Ra¼ 103e106) within the cavity with various concavewalls. Due tothe imposed thermal boundary conditions, hot fluid rises up alongthe left wall and flows down along the right wall forming clockwisecirculation cell within the cavity.

Fig. 5 shows the effect of wall curvature on fluid flow, temper-ature and heat flow inside the enclosure at low Ra and Pr (Ra ¼ 103

and Pr ¼ 0.015). The magnitudes of streamline are very smalldepicting weak fluid flow which signifies conduction dominantheat transfer for all cases. As the wall curvature increases from case

Table 2Comparison of average Nusselt number on right wall ðNurÞ for various grid systems at Ra ¼ 105 and Pr ¼ 1000 with various wall curvatures in presence of uniform heating onthe left wall, cold right wall and adiabatic horizontal walls.

Concave Convex

8 � 8 16 � 16 20 � 20 24 � 24 28 � 28 8 � 8 16 � 16 20 � 20 24 � 24 28 � 28

Case 1 4.99 4.64 4.59 4.57 4.55 5.38 5.03 4.92 4.86 4.83Case 2 4.91 4.63 4.59 4.58 4.57 5.84 5.39 5.26 5.18 5.13Case 3 4.72 4.69 4.69 4.68 4.68 6.96 6.22 6.05 5.95 5.89

Table 3Comparison of maximum streamfunction values ðjjjmaxÞ for various grid systems at Ra¼ 105 and Pr¼ 1000 with various wall curvatures in presence of uniform heating on theleft wall, cold right wall and adiabatic horizontal walls.

Concave Convex

8 � 8 16 � 16 20 � 20 24 � 24 28 � 28 8 � 8 16 � 16 20 � 20 24 � 24 28 � 28

Case 1 10.68 10.64 10.64 10.63 10.63 12.13 12.15 12.14 12.14 12.14Case 2 10.34 10.30 10.30 10.30 10.30 13.13 13.05 13.04 13.04 13.04Case 3 4.96 4.97 4.99 4.99 4.99 15.16 14.64 14.64 14.63 14.63


1 to case 2, size of fluid circulation cell as well as magnitude ofstreamfunction decrease. This is due to the fact that the availablearea for fluid circulation decreases with increase in wall curvature.The fluid circulation is almost absent in the corner region of thecavity for all cases. The fluid circulation cells adjacent to the wallattain shape of the enclosure in case 3 [see Fig. 5(c)]. In the middleportion of the enclosure, two clockwise rotating fluid circulationcells divide the cavity into two symmetric parts with respect to thecenterline (Y ¼ 0.5) [see Fig. 5(c)]. The intensity of fluid circulationis weaker in case 3 compared to case 1 and case 2(jjjmax ¼ 0:87;0:49 and 0.08 for case 1, case 2 and case 3,respectively). The direction and magnitude of heat flow in theenclosure can be demonstrated via heatlines and heatfunctions,respectively. As the intensity of fluid flow reduces, at low Ra, end-to-end heatlines connecting two side walls are observed in thecavity. Slightly distorted heatlines are observed in the core regionfor case 1 and case 2 due to comparatively stronger convective ef-fect in those cases. Heatlines are smooth and almost straight linesat core in case 3 depicting highly conductive heat transfer. Overall,the heat transfer in the cavity is conduction dominant in all thecases and highest heat transfer rate is observed in case 3 (Note that,jPjmax ¼ 1:23, 1.46 and 3.02 for case 1, case 2 and case 3, respec-tively). Consequently, it is observed that the smooth isothermsoccur in parallel to each other and to the side walls for case 1 andcase 2. At higher concavity of the side walls, the isotherms stronglyfollow the geometry of the enclosure as seen in Fig. 5(c). Also, theisotherms are found to be compressed in the central portion of theenclosure in case 3 based on high heatfunction gradient. This canalso be explained from the heatline contours where dense end-to-end heatlines are observed in the central portion of the enclosure incase 3.

As Ra increases to 104, comparatively higher buoyancy forcescause the streamline cells to be elongated in the diagonal directions

Table 4Comparisons of present results with benchmark solution [40] for natural convectionin square cavity in presence of air (Pr ¼ 0.71), at various Ra.

Present work Deng and Tang [40] Wan et al. [41]

jjjmax Nu jjjmax Nu jjjmax Nu

Ra ¼ 103 1.17 1.118 1.17 1.118 e 1.117Ra ¼ 104 5.07 2.248 5.04 2.254 e 2.254Ra ¼ 105 9.61 4.564 9.50 4.557 e 4.598Ra ¼ 106 16.78 9.215 16.32 8.826 e 8.976

in all the cases [see Fig. 6]. As a result, secondary circulation cells,however small, tend to form near the corner region of the cavity incase 1 and case 2. It is found that both end-to-end as well as closedloop heatlines appear within the cavity for case 1 and case 2. On theother hand, except near the bottom portion of the enclosure, end-to-end heatlines connecting the left and right wall are observedin case 3 [Fig. 6(c)]. This is due to the strength of fluid circulationcells, which are quite low (jjjmax ¼ 0:8 for case 3) resulting inweak convective heat transport in case 3. Heat is mainly trans-ported from the lower portion of the hot wall and that is received atthe top portion of cold right wall for case 1 and case 2. Thus, iso-therms are highly compressed along the lower portion of left walland upper portion of right wall. Due to convective heat transportvia convective closed loop heatlines, thermal mixing was found tooccur leading to dispersed isotherms at the center in case 1 andcase 2. On the other hand, dense heatlines at the center of the wallslead to dense isotherms at the center as seen in case 3.

Fig. 7 presents the streamline, heatline and temperature distri-butions for the enclosure with concave side walls at Pr ¼ 0.015 forhigher Ra (Ra ¼ 105). At Ra ¼ 105, buoyancy force increasessignificantly leading to enhanced convection in the cavity. The in-tensity of primary fluid flow circulation cell is found to be signifi-cantly higher compared to previous cases with lower Ra. Secondaryfluid circulation cells withmuch smaller magnitude are observed inthe corner region of the cavity for all the cases. In contrast to theprevious case with lower Ra, circular streamline cells are observedin the upper and lower halves of the cavity in case 3 due to veryhigh convective effect in the presence of high curvature [seeFig. 7(c)]. The detailed analysis on flow cells and separation of flowwill be discussed in Section 4.3. Note that, jjjmax ¼ 8:47;7:29 and3.87 for case 1, case 2 and case 3, respectively. As a result of highthermal mixing in case 1, convective heat transfer rate is highest incase 1 (jPjmax ¼ 3:53;2:93 and 4.32 for case 1 case 2 and case 3,respectively). It is observed that the heatlines are denser at themiddle portion of left wall, indicating large amount of heat beingdrawn from the middle portion of the wall for all the cases. Sparseheatlines are observed near the lower right corner of the cavitydepicting very less amount of heat being delivered to those regions.Also, tiny secondary heat circulation cells were found to appearnear the corner regions of the cavity for all the cases. In case 1 andcase 2, the central portion of the cavity is filled with closed loopheatlines whereas dense end-to-end heatlines are observed in thecentral portion in case 3. The intensity of closed loop heatline cellsis higher in the upper portion of the enclosure compared to the


lower portion [see Fig. 7(c)]. Isotherms are found to be highly dis-torted in the core and are compressed towards the top right andbottom left corner for low wall curvature cases (case 1 and case 2)due to the presence of closed loop heatlines near the core anddense end-to-end heatlines near the middle portion of the rightwall andmiddle portion of the left wall in case 1 and case 2. As seenin Fig. 7(c), isotherms are compressed in the central zone due to thepresence of dense end-to-end heatlines in case 3. In addition to thecentral zone, compressed isotherms are also observed near thelower portion of the left wall and the upper portion of the right wallin case 3.

As Pr increases to 0.7 at Ra ¼ 105, secondary fluid circulationcells disappear in the enclosure for all cases [see Fig. 8]. Similar toprevious cases, flow separation occurs for case 3. Due to diagonaldetachment of the fluid circulation cells at core, two streamlinescells are observed at the core only in case 1. In case 2, the streamlinecells get elongated, but only one fluid flow circulation cell occurs atthe core. Due to constriction in the side walls, the fluid circulationcells get segregated at the core and two clockwise rotatingstreamline cells are found in the top and bottom halves of theenclosure in case 3. In contrast to low Pr case [Fig. 7], sparseheatlines are not observed at the corner region of the enclosure dueto comparatively higher momentum diffusivity. Common to allcases, closed loop heatline cells expand resulting in end-to-endheatlines to be compressed in the top portion. As a result, heat-lines are denser in the present case with Pr ¼ 0.7, depictingcomparatively larger heatfunction gradient at the side walls in allcases. Trends of heat circulation cell at the core are found to bequalitatively similar to the streamline cells depicting dominance ofconvective heat transfer. Due to presence of dense heatlines, iso-therms are highly compressed towards the lower half of the leftwall and upper half of the right wall.

At Pr ¼ 0.7 and Ra ¼ 106, the convective effect is enhancedsignificantly and the heat transfer in the enclosure occurs primarilydue to convection [see Fig. 9]. Intensity of fluid flow is found to bevery high and streamlines are elongated in the horizontal directionin cases 1 and 2. Shape of the streamline circulation cells stronglyfollows the shape of the enclosure, especially near the solid walls.This is due to high buoyant effect at high Ra, which causescompression of streamlines towards the side walls in all the cases.As a result, fluid circulation cells at the core of the cavity elongate inthe horizontal direction and gets detached in case 1 and case 2which is in contrast to Ra ¼ 105 case. On the other hand, thestreamline cells are separated in vertical direction as horizontalelongation is not possible due to highly concave side walls in case 3.Unlike previous case with Ra ¼ 105, the magnitudes of stream-function are found to be almost identical in all the cases. Note that,jjjmax ¼ 16:46; 16.30 and 15.72 for case 1, case 2 and case 3,respectively. As a result of high magnitude of streamfunction,closed loop heatlines with very high values are also found in theenclosure for all the cases. Large closed loop heatlines span almostwhole area of the cavity except the top portion. Consequently,heatlines are compressed and densely spaced heatlines are found atthe middle portion of left and right walls. Heatfunction gradient isvery high along the side walls signifying large heat transfer ratefrom the left wall to the right wall. The isotherms are highly dis-torted due to high thermal mixing at the core. The thickness ofthermal boundary layer near the lower portion of the left wall andtop portion of the right wall is very less due to high heat transferrate at larger Ra (Ra ¼ 106).

Fig. 10 illustrates streamline, heatline and temperature distri-butions for fluid with high viscosity or low thermal conductivity(Pr ¼ 1000) and Ra ¼ 105 within the enclosure with concave sidewalls. Multiple circulations do not occur for higher Pr due to highmomentum diffusivity corresponding to a single circulation cell

which expands and spans the entire enclosure for case 1 and case 2.However, fluid circulation cells adjacent to the wall take shape ofthe enclosure and circulation cells near the core region get dividedinto two parts due to high curvature of the walls in case 3. Thedetails of flow separation are discussed in Section 4.3. The intensityof fluid flow circulation cells becomes significantly stronger athigher Pr as seen from themagnitudes of streamfunction. Note that,jjjmax ¼ 10:63, 10.29 and 4.99 for case 1, case 2 and case 3,respectively. Almost similar behavior in the heatline contours isobserved in the present case as in low Pr (Pr ¼ 0.7) case [see Figs. 8and 10]. Enhanced convection causes the isotherms to be com-pressed towards the top portion of the right wall and lower portionof the left wall for case 1 and case 2. It is also observed that theisotherms are compressed near the middle portion of the enclosureas well as near the bottom portion of the left wall and top portion ofthe right wall indicated by high heatfunction gradient in thoseregions in case 3 [Fig. 10(c)].

4.3. Effect of curvature on vertical velocity profiles for concave cases

To illustrate the effect of wall concavity on the flow character-istics, vertical velocity profiles for all concave cases are presented inFigs. 11e13. The line plots of vertical velocity are drawn for Ra¼ 103

and 105 with Pr¼ 0.015 at various locations in Y direction includingthe top, bottom and throat sections. Common to all Y locations,vertical velocity, V is positive along left half while that is negativealong right half which is in accordancewith streamlines patterns asseen in Figs. 5 and 7. Fig. 11(a) and (b) presents vertical velocityprofiles at various locations in Y direction for case 1 at Ra¼ 103 and105, respectively for Pr ¼ 0.015. At Y ¼ 0.053 and 0.946 [see top andbottom panels, (i) and (v) of Fig. 11(a) and (b)], motion of fluid ismostly in X direction as seen from streamlines which are parallel tothe horizontal walls [Figs. 5(a) and 7(a)]. Thus, the verticalcomponent of velocity, V is almost negligible as seen in the top andbottom panels of Fig. 11(aeb). It is also found that the vertical ve-locity, V is zero at the middle plane, X ¼ 0.5 for all Y values irre-spective of Ra and Pr and these are evident from tangent ofstreamlines at X ¼ 0.5. At the top and bottom halves of the enclo-sure, for Y¼ 0.696 [see Fig.11(aeb)(ii)] and Y¼ 0.303 [see Fig.11(aeb)(iv)], the streamlines tend to become more curved and are notparallel to the horizontal walls. Thus, tangent to the streamlinesleads to significant velocity in both X and Y directions and as aresult, magnitude of V is observed to be comparatively larger forY ¼ 0.696 and 0.303. At the centerline of the enclosure [seeFig. 11(aeb)(iii), for Y ¼ 0.5], intense streamline circulation cellstake almost circular shape and at each point, U and Vmay be foundto be significant. As observed from Fig. 11(aeb)(iii), V is observed tobe largest at the middle portion of the enclosure. Based on velocityprofiles at Ra ¼ 103 and Ra ¼ 105 [see Fig. 11(aeb)], it is observedthat the magnitude of velocity component in Y direction is largerfor high Ra. This is due to the fact that at high Ra, intense buoyancyforce leads to circular streamlines with larger magnitudes [seeFig. 7(a)].

The vertical velocity (V) profiles for case 2 at various locations atPr¼ 0.015 are illustrated in Fig. 12(a) and (b). Qualitative features ofV are observed to be almost similar in the present case as in case 1[see Figs. 11 and 12]. Magnitude of vertical velocity, V is found to belesser in case 2 compared to case 1. This can also be inferred fromthe streamline contours, where magnitude of streamfunction islower in case 2 compared to case 1 for both Ra ¼ 103 and 105 [seeFigs. 5(b) and 7(b)]. At Y¼ 0.053 and 0.946, velocity in Y direction isvery less, which can also be explained from the streamline con-tours, which are almost parallel to the top and bottom wallsespecially at low Ra [Fig. 5(b)]. Thus, the tangent to the streamlinesresults in X velocity component and the velocity in Y direction is

ψ(a) Π θ

(b)

(c)

(d)

(e)

(f)

(g)

Fig. 3. Streamlines (j), heatlines (P) and isotherms (q) for Pr ¼ 1000 and Ra ¼ 105 as a representative test case of concave side walls (case 3) for various grid sizes: (a) 4 � 4, (b)8 � 8, (c) 12 � 12, (d) 16 � 16, (e) 20 � 20, (f) 24 � 24 and (g) 28 � 28.


negligible which may be clearly inferred from the velocity profilesat low Ra [see Fig. 12(a)(i) and (v)]. Further, almost ellipticalstreamline cells with larger magnitude are observed near the core(Y ¼ 0.303 and 0.696), away from the solid walls at low Ra [seeFig. 5(b)] resulting in comparatively larger magnitude of V. Thus,velocity in Y direction is found to be comparatively larger atY ¼ 0.696 and 0.303 [see Fig. 12(b)(ii) and (v)] than those atY¼ 0.946 and 0.053. Similar to case 1, magnitude of V is observed tobe largest at the core for Y ¼ 0.5 and this trend is clearly observed

from the streamline cells at the core [see Fig. 5(b)]. At all locations,for various values of Y, magnitude of V is larger for Ra ¼ 105

compared to Ra¼ 103. At Ra ¼ 105, as seen from Fig. 12(b)(i) and (v)[Y ¼ 0.053 and 0.946], magnitude of V is found to be largercompared to low Ra as prominent secondary fluid flow cells areseen in the corner regions [see Fig. 7(b)]. The tangents to the sec-ondary streamline cells at the corner region indicate significantmagnitude of velocity in Y direction, although the magnitude islesser and thus the velocity profile is found to be slightly wavy [see

ψ(a) Π θ

(b)

(c)

(d)

(e)

(f)

(g)

Fig. 4. Streamlines (j), heatlines (P) and isotherms (q) for Pr ¼ 1000 and Ra¼ 105 as a representative test case of convex side walls (case 3) for various grid sizes: (a) 4 � 4, (b) 8 � 8,(c) 12 � 12, (d) 16 � 16, (e) 20 � 20, (f) 24 � 24 and (g) 28 � 28.


Fig.12(b)(i) and (v)]. As seen from the strength of streamline cells athigh Ra, it is evident that fluid circulateswith very high velocity andvelocities along both X and Y direction are very high for Y ¼ 0.303and 0.696 [see Fig. 12(b)(ii) and (iv)]. At the core, fluid circulationcells take fully circular shape and attain highest magnitude, thus,based on tangent to streamlines, the velocity components at thisregion are also observed to be largest [Fig. 12(b(iii))].

In case 3, many interesting trends in velocity profile areobserved at each section of the enclosure due to highly curved sidewalls. At Ra ¼ 103, magnitude of V is found to be lesser at Y ¼ 0.053and 0.964 compared to case 1 and case 2. This can also be explainedfrom the streamline contours in Fig. 5(c), where the magnitude ofstreamfunction is less in case 3 compared to case 1 and case 2. At

the corner regions of the enclosure, fluid is almost stagnant at lowRa [Ra ¼ 103, Fig. 5(c)] and fluid velocity is almost negligible inthose regions as seen from Fig. 13(a)(i) and (v). Fluid velocity (V) isslightly larger at the core compared to that at corner regions as seenfrom the streamline cells [Fig. 13(a)(i) and (v)]. At Y ¼ 0.303 and0.696, magnitude of V is quite large which can also be clearlydemonstrated from the streamline plots [Fig. 5(c)], where two cir-cular fluid circulation cells are observed at the top and bottomhalves of the enclosure. The tangents to the streamline cells resultin significant values of V near the walls. Note that, flow separationoccurs only at the inner circulations whereas, separation does notoccur at the outer streamline cells [Fig. 5(c)]. It is interesting to notethat at Y ¼ 0.5, near the core, vertical fluid velocity is lesser

−0.001

−0.2−0.6−0.8

ψ(a)1.1

0.7

0.2

0.05

Π

0.9

0.7

0.5

0.3 0.1

θ

−0.01

−0.08

−0.3

−0.45

(b)1.41.2

0.7

0.30.20.1

0.9

0.7 0.50.3 0.1

−0.08

−0.05

−0.01

0.002

(c)32.82.52

10.4

0.05

0.90.7

0.50.3 0.1

Fig. 5. Streamlines (j), heatlines (P) and isotherms (q) for (a) case 1, (b) case 2 and (c) case 3 for enclosure with concave side walls at Pr ¼ 0.015 and Ra ¼ 103. Clockwise and anti-clockwise fluid flows are shown via negative and positive signs of streamfunction, respectively and direction of heat flows are demonstrated via heatlines with arrows.


compared to Y¼ 0.303 and 0.696, which is in contrast of cases 1 and2 where the vertical velocities, V are largest at Y ¼ 0.5. This trend ofV is observed due to highly curved side walls, which results inseparation of streamline cells [see Fig. 5(c)]. It is observed that,unlike case 1 and case 2, streamline cells with lesser magnitude are

−4−3−1.5

−0.05

0005

ψ(a)

−1−0.50.05

0.7

1.6

1.951.3

Π

−2.5−1

−0.2

0.01

−0.01

0.01−0.01−0.05

−0.05

(b)

−0.0.04

00.3

1.2

1.651.75

−0.5

−0.3

−0.1

−0.02

0.001

(c)3

2.6

2.2

21.0.60.2

0.02

Fig. 6. Streamlines (j), heatlines (P) and isotherms (q) for (a) case 1, (b) case 2 and (c) caseclockwise fluid flows are shown via negative and positive signs of streamfunction, respecti

observed at core (Y¼ 0.5), depicting that fluid velocity is very less atthe core. It is also observed that, fluid velocity is significantly lesserat the core (Y ¼ 0.5), compared to Y ¼ 0.303 and 0.696, and thatresults in separation of fluid circulation cells [see Figs. 5(c) and13(a)]. On the other hand, it is observed that magnitude of V is
0.9
0.7

0.5

0.3

0.1

θ

3

.09

0.9

0.6

0.3

0.1

3

0.9 0.7

0.10.30.5

3 for enclosure with concave side walls at Pr ¼ 0.015 and Ra ¼ 104. Clockwise and anti-vely and direction of heat flows are demonstrated via heatlines with arrows.

−8.4−7

−2

−0.01

−0.01

−0.1

ψ(a)

−2.5

−2−0.5

3.3

2.72

0.7

0.5

0.1

Π

0.9

0.6

0.4

0.2

θ

−0.1

−0.1

−6

−2.5

−0.1 −1

−0.01

(b)

−2

−10.10.10.005

2.5 2.852

10.5

0.9

0.70.5

0.3

0.1

−3

−2

−1−0.1

−0.005

−0.00

(c)4.33.82.8

1.6

2.4

1.6

0.50.1−0.6

0.9 0.7

0.10.3

0.7



largest at the core for case 1 and case 2, which resulted singleconcentric fluid circulation cell at the core Y ¼ 0.5 [see Figs. 5(aeb),11 and 12(a)].

At high Ra, magnitude of V is significantly larger as seen from themagnitudes of streamline cells at high Ra [see Figs. 7(c) and 13(b)].

−8−6−2−0.1

−9 −9

ψ(a)

−2.4

−1.

0.05

2.63.6

0.05

Π

−9−4−1

−1

−0.01

(b)

−2

−1−

0.1

4.49

4.231.40.

−4

−3

−1

−0.1

(c)

−0

1.61.6

244.6


Multiple fluid circulation cells are observed near the horizontalwalls [Fig. 7(c)]. This is clearly reflected in the wavy distributions ofV profiles at high Ra for Y ¼ 0.053 and 0.946 [see Fig. 13(b)(i) and(v)]. At Y ¼ 0.053 and 0.964, vertical velocity, V is very less at thecorner regions, which can be explained from the small secondary

3

0.9

0.7

0.5

0.3

0.1

θ

0.3

005 0.9

0.7

0.5

0.3

0.1

.8

0.8 0.9

0.7

0.10.3

3 for enclosure with concave side walls at Pr ¼ 0.7 and Ra ¼ 105. Clockwise and anti-vely and direction of heat flows are demonstrated via heatlines with arrows.

−16−14−10−6

−2ψ(a)

−3.7

−3

−1

−22 58

Π

0.9

0.7

0.5

0.2

θ

−15−12−6 −4

−16

−14

(b)

−3.7 −3.5

−2−1

−21467 0.9

0.7

0.50.3

0.1

−4−8−14

−15

−10

−4

(c)

−3−28

40.1

−3

0.9

0.7

0.5

0.3

0.1



circulation cells at the corner regions [Fig. 7(c)]. Compared to thecorner region, magnitude of V is significant at X ¼ 0.4 and 0.6,which is observed for all Y. This can also be inferred from thestreamline cells, where tangent to larger circulation leads to large Vat X¼ 0.4 and 0.6. On comparing magnitude of V at Y¼ 0.303, 0.696and 0.5, it is observed that jV j is significantly less at Y¼ 0.5, which is

−0.1−3−7−10

−10.5

ψ(a)

−2.8

−−1

−

4.53 1−0.3

Π

−10

−7−1 −3

(b)

−2−1.5

4.531.5

−4

−3

−2−1

−0.1

(c)

0

1

2

2

44.7

−0.


in contrast to case 1 and case 2 at high Ra. As magnitude of verticalvelocity is observed to be very high at Y ¼ 0.696 and 0.303ðjV jmax ¼ 23:28Þ, the streamlines are perfectly circular as seen inthe upper and lower halves of the enclosure which is a resultant ofsignificant X and Y component velocity (U, V). In case 1 and case 2, itis seen that streamline cells are circular and single circulation cell

2

0.3

0.9

0.7

0.5

0.3

0.1

θ

.5

−0.5

0.9

0.7

0.5

0.3

0.1

.1

.3

5

0.9

0.7

0.10.3

3 for enclosure with concave side walls at Pr ¼ 1000 and Ra ¼ 105. Clockwise and anti-vely and direction of heat flows are demonstrated via heatlines with arrows.

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

V

Y=0.964(i)(a)

Ra=103

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

V

Y=0.696(ii)

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

V

Y=0.5(iii)

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

V

Y=0.303(iv)

0 0.2 0.4 0.6 0.8 1X

-4

-2

0

2

4

V

Y=0.053(v)

0 0.2 0.4 0.6 0.8 1-60

-30

0

30

60

V

Y=0.964(i)

Ra=105

(b)

0 0.2 0.4 0.6 0.8 1-60

-30

0

30

60

V

Y=0.696(ii)

0 0.2 0.4 0.6 0.8 1-60

-30

0

30

60

V

Y=0.5(iii)

0 0.2 0.4 0.6 0.8 1-60

-30

0

30

60

V

Y=0.303(iv)

0 0.2 0.4 0.6 0.8 1X

-60

-30

0

30

60

V

Y=0.053(v)

Fig. 11. Distribution of vertical velocity component (V), along various Y at Pr ¼ 0.015 with (a) Ra ¼ 103 and (b) Ra ¼ 105 for case 1 of the enclosure involving concave side walls.


with very high magnitude is observed at the core of the enclosure[see Fig. 7(aeb)]. In contrast, case 3 exhibits a pair of disconnectedstreamline cells at the top and bottom halves and the flow sepa-ration occurs at the centerline (Y ¼ 0.5). This trend is clearly illus-trated in the line plots of vertical velocity. An overall comparison ofV for all cases [Figs. 11e13] shows that, magnitude of V is largest atthe core (Y ¼ 0.5) for case 1 and case 2, depicting that fluid motionis highly intensified at the core of the enclosure. In contrast, due tohighly curved side walls in case 3, magnitude of V is very less atthroat region (Y ¼ 0.5), compared to other zones (especiallyY ¼ 0.303 and 0.696). Thus, vertical velocity component is

negligible at Y ¼ 0.5, and that ultimately causes flow separation forcase 3. It is interesting to note that, the fluid velocity graduallybecomes less intense at the core as Ra increases for case 3 signifyingflow separation at Ra ¼ 105.

4.4. Isotherms, streamlines and heatlines in the enclosure withconvex side walls

The enclosure with convex side walls is considered with threecases based on three different curvatures of the side walls. Similarto the concave case, the square enclosure is modified to a curved

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

V

Y=0.964(i)

Ra=103

(a)

0.2 0.4 0.6 0.8-4

-2

0

2

4

V

Y=0.696(ii)

0.2 0.4 0.6 0.8-4

-2

0

2

4

V

Y=0.5(iii)

0.2 0.4 0.6 0.8-4

-2

0

2

4

V

Y=0.303(iv)

0 0.2 0.4 0.6 0.8 1X

-4

-2

0

2

4

V

Y=0.053(v)

0 0.2 0.4 0.6 0.8 1-60

-30

0

30

60

V

Y=0.964(i)

Ra=105

(b)

0.2 0.4 0.6 0.8-60

-30

0

30

60

V

Y=0.696(ii)

0.2 0.4 0.6 0.8-60

-30

0

30

60

V

Y=0.5(iii)

0.2 0.4 0.6 0.8-60

-30

0

30

60

V

Y=0.303(iv)

0 0.2 0.4 0.6 0.8 1X

-60

-30

0

30

60

V

Y=0.053(v)



walled enclosure by shifting the point P1 and P2 to P01 and P02,respectively in the outward direction such that AP01D and BP02C formcurves which obey quadratic equations; X ¼ aY2 þ bY þ c andX¼ a

0Y2þ b

0Yþ c

0, respectively [see Fig.1(c) and Table 1]. The values

of P1P01 or P2P02 are same as in the concave casewhich are L/10, 2L/10

and 4L/10 for case 1, case 2 and case 3, respectively. Figs. 14e17illustrate the streamline, heatline and isotherm contours forvarious values of Pr (Pr ¼ 0.015, 0.7 and 1000) and Ra (Ra ¼ 103e106) within enclosures with convex side walls. The imposed ther-mal boundary conditions are similar to concave cases which involvehot left wall, cold right wall and adiabatic horizontal walls.

Fig. 14 displays the distributions at Ra ¼ 103 and Pr ¼ 0.015within the enclosure with convex side walls for all the three cases.Similar to the cases with concave walls, clockwise circulations offluid are induced in the cavity as low density fluid rises up along thehot left wall and flows down along the cold right wall. Fluid flow isweak as seen from the small magnitudes of streamfunction in allthe three cases of convexities. Smooth and circular fluid circulationcells span the entire enclosure in all cases. Largest magnitude ofstreamfunction is observed in the core which is almost similar forall cases (jjjmax ¼ 1:25, 1.28 and 1.25 for case 1, case 2 and case 3,respectively). Comparative study of concave and convex cases

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

V

Y=0.964(i)

Ra=103

(a)

0.3 0.4 0.5 0.6 0.7-4

-2

0

2

4

V

Y=0.696(ii)

0.35 0.4 0.45 0.5 0.55 0.6 0.65-4

-2

0

2

4

V

Y=0.5(iii)

0.3 0.4 0.5 0.6 0.7-4

-2

0

2

4

V

Y=0.303(iv)

0 0.2 0.4 0.6 0.8 1X

-4

-2

0

2

4

V

Y=0.053(v)

0 0.2 0.4 0.6 0.8 1-60

-30

0

30

60

V

Y=0.964(i)

Ra=105

(b)

0.3 0.4 0.5 0.6 0.7-60

-30

0

30

60

V

Y=0.696(ii)

0.35 0.4 0.45 0.5 0.55 0.6 0.65-60

-30

0

30

60

V

Y=0.5(iii)

0.3 0.4 0.5 0.6 0.7-60

-30

0

30

60

V

Y=0.303(iv)

0 0.2 0.4 0.6 0.8 1X

-60

-30

0

30

60

V

Y=0.053(v)



shows that the geometry of the enclosure is one of the importantfactors which affects the strength of the fluid circulation as well asheat flow within the enclosure. In the concave case, the enclosureoffers lower area for fluid circulation compared to the convex case.Consequently, themagnitude of fluid circulation is higher in convexcases compared to concave cases [see Figs. 5 and 14]. As wall con-vexity increases, no remarkable change is observed in the nature offluid circulation cells in convex cases whereas in concave casesignificant decrease in the magnitude of streamlines was observed[see Fig. 5]. Both end-to-end and closed loop heatline circulationcells are observed in the enclosure which is in contrast to theconcave case where only end-to-end heatlines were observed for

low Ra and Pr [see Fig. 5]. However, magnitudes of end-to-endheatlines are higher than closed loop heatlines indicating domi-nance of conductive heat transport within the enclosure. High heatflux is observed near top portion of the enclosure where straightheatlines with highmagnitudes are found (jPjmax ¼ 1:02, 0.97 and0.93 for case 1, case 2 and case 3, respectively). Qualitative nature ofheatlines is similar for all wall curvatures. Isotherms are almostparallel curves and are evenly distributed throughout the enclosurein all three cases as strength of convection is low and heatlines arealmost parallel curves. Isotherms are found to be distorted at thecore due to convective heat transfer as seen from closed loopheatline cells.

−1.2−0.6

−0.2

−0.4

−0.05

−1

ψ(a)1

0.8

0.4 0.1

−0.1

−0.01

Π

0.9

0.7 0.6

0.30.5

0.1

θ

−1.2−0.8

−0.4−0.05

−0.005

(b)

−0.1

−0.01

0.05

0.2

0.6

0.970.8

0.9

0.7 0.50.30.1

−1.2−0.8

−0.4

−0.05

−0.005

(c)

−0.05

0.080.3

0.50.7

0.9

0.9

0.7

0.5 0.3

0.1

Fig. 14. Streamlines (j), heatlines (P) and isotherms (q) for (a) case 1, (b) case 2 and (c) case 3 for enclosure with convex side walls at Pr ¼ 0.015 and Ra ¼ 103. Clockwise and anti-clockwise fluid flows are shown via negative and positive signs of streamfunction, respectively and direction of heat flows are demonstrated via heatlines with arrows.


As Ra increases to 105, many interesting features are observed instreamlines, heatlines and isotherms distributions [see Fig. 15].Streamlines are observed to be significantly more intense than thatof Ra ¼ 103 case. The maximum magnitudes of streamlines arefound to be jjjmax ¼ 7:1 for case 1 and case 2 and jjjmax ¼ 7:49 forcase 3. In case 1, tiny secondary circulation cells tend to form in thecorner regions. In contrast to lower Ra case, the fluid circulationcells tend to form elliptic shape in the core. In addition, at the core,the streamline cells get elongated diagonally and divided into threecells. In contrast to the streamline cells, unique closed loop heatlinecells are found at the core and they are compressed towards the topleft corner of the enclosure. Similar to the previous case with lowerRa, both end-to-end and closed loop heatlines are found in theenclosure. However, in the present case, the magnitude of closedloop heatline circulation cells is significantly higher than lower Racase. Higher intensity of closed loop heatlines depicts recirculationof heat energy due to enhanced thermal mixing in the core athigher Ra. Themaximummagnitudes of heatlines are also observedto be higher for the present case compared to previous case withlow Ra (jPjmax ¼ 3:38 for case 1, jPjmax ¼ 3:47 for case 2 andjPjmax ¼ 3:72 for case 3). Dense heatlines emanating from the leftwall illustrate high convective heat transfer from the entire left wallto the entire right wall in all the cases. Comparative studies ofconcave and convex cases depict that the convective heat transportis more dominant in convex case for the considered parameters(Ra ¼ 105 and Pr ¼ 0.015). Enhanced thermal mixing causes theheatlines to get pushed towards the bottom left corner and topright corner [see Fig. 15]. Isotherms are compressed towards thosezones of high heat transfer due to enhanced convection effect.Isotherms are also found to be highly dispersed at the core toillustrate significant thermal mixing due to enhanced convectiveheat transport.

To illustrate the effect of high Ra (Ra ¼ 106) on the trends ofstreamlines, heatlines and isotherms at Pr ¼ 0.7, results for a testcase in convex enclosure are shown in Fig. 16(aec). The fluid cir-culation cells get elongated in the horizontal direction, towards thetop portion of the left wall and bottom portion of the right wall. As aresult, the streamline cell gets detached at the core and localized

fluid flow circulation cells are found in the left and right half of thecore of the enclosure. Intense closed loop heatline cells depict veryhigh recirculation of thermal energy at the core. Dense end-to-endheatlines along the left wall signify very high heat transfer from theentire left wall. The qualitative trends of closed loop heatlines arealmost similar to those of streamlines. The isotherms are parallel tothe horizontal walls due to very high thermal mixing in theenclosure. In addition, isotherms are compressed towards the leftand right walls depicting low boundary layer thickness corre-sponding to high heat transfer rates similar to situation withconcave side walls at high Ra.

Distributions of streamlines, isotherms and heatlines forPr ¼ 1000 and Ra ¼ 105 are displayed in Fig. 17. Due to highermomentum diffusivity, single primary clockwise fluid circulationcell occupies the entire cavity in all the cases. The flow circulationcell takes the shape of the enclosure and further, the circulation cellelongates and gets divided into two parts of the same magnitudesin the core region. In contrast to low Pr cases, the degree of flowseparation is weak at higher Pr. As the wall curvature increasesfrom case 1 to case 3, the strength of fluid circulation cells alsoincreases (jjjmax ¼ 12:14 for case 1, jjjmax ¼ 13:04 for case 2 andjjjmax ¼ 14:63 for case 3). The closed loop heatlines take the shapeof the enclosure and they occupy around 80% of the cavity. A largeamount of heat is drawn from the entire left wall, as indicated bydense heatlines with large magnitudes. Note that, jPjmax ¼ 5:11for case 1, jPjmax ¼ 5:42 for case 2 and jPjmax ¼ 6:18 case 3. Dueto the presence of dense heatlines near the bottom portion of theleft wall and the top portion of the right wall, isotherms are foundto be compressed in those regions.

4.5. Local and average Nusselt numbers

The variation of local heat transfer rates along the right wall(Nur) with distance for Pr ¼ 0.015, 0.7 and 1000 at Ra ¼ 103 withboth concave and convex cases is presented in Fig. 18(aef). It isinteresting to observe that, due to the symmetric thermal boundarycondition, the trend of Nur is observed to be the mirror reflection ofNul (Figure not shown). Note that, the amount of heat delivered to

−7

−7.08

−7.08

−7

−6−4−

0.05

ψ(a)

−2 −1.8−1.5

−0.5

0.5

1.5

3.2Π

0.9 0.7

0.5

0.3 0.1

θ

−7.08 −7−7.08

−7

−5

−3−2

(b)

−2 −1.8

−1.5

0.05

−0.5

1

2.53.4

0.9 0.7

0.5

0.30.1

−7.47−7.4

−7.47

−7

−4−0.01

−2−6

(c)

−2.1

−1.9 −1.5

−1−0.4

3.51.5

0.1

0.9 0.7

0.5

0.3 0.2

Fig. 15. Streamlines (j), heatlines (P) and isotherms (q) for (a) case 1, (b) case 2 and (c) case 3 for enclosure with convex side walls at Pr ¼ 0.015 and Ra ¼ 105. Clockwise and anti-clockwise fluid flows are shown via negative and positive signs of streamfunction, respectively and direction of heat flows are demonstrated via heatlines with arrows.


the right wall is maximum in case 3 for both concave and convexside walled enclosures irrespective of Pr. Maximum Nur occurs atthe middle portion of the wall for enclosure with concave sidewalls. On the other hand, for convex walls, local rate of heat transferis largest near the top edge of the wall.

Fig. 18(a) illustrates variation of local Nusselt numbers along theconcave right wall (Nur) for all the three cases of wall curvatures atRa ¼ 103 and Pr ¼ 0.015. Qualitatively similar features are observedin the distribution of Nur for case 1 and case 2. Local heat transferrate for the right wall (Nur) increases slowly from the bottom edgetill the distance being S ¼ 0.69 and S ¼ 0.65 for case 1 and case 2,

−17

−17

−16.5

−15−11−9

−5 −1ψ(a)

−4

−3

−1

−14

Π

−18

−18

−17

−14−12−6 −4

(b)

−4.5

−3.5

−1

9

−20

−20

−20−18−

14−8 −4

(c)

−5

−1−

−1 3

Fig. 16. Streamlines (j), heatlines (P) and isotherms (q) for (a) case 1, (b) case 2 and (c) casclockwise fluid flows are shown via negative and positive signs of streamfunction, respecti

respectively. Further, Nur decreases with distance along the wall tillthe top edge is reached. However, the amount of heat transferred tothe right wall is slightly larger in case 2 compared to case 1. Thismay also be explained based on the heatline contours, where highlydense end-to-end heatlines with comparatively high value ofheatfunction gradient are observed in case 2 [see Fig. 5(a) and (b)].In case 3, rapid increase in Nur is observed from the bottom edge tillthe middle portion of the wall, where Nur attains maximum value.Occurrence of maxima in the Nur distribution for case 3 is clearlydepicted in the heatline contours in Fig. 5(c). Short end-to-endheatlines with high heatfunction gradient are found to be

.4

−3.4−3

7 9.5 0.9

0.7

0.5

0.2

θ

−3.5

−2

−236 0.9

0.7

0.5

0.30.2

−4

−4

3

6 9 0.90.7

0.5

0.30.2

e 3 for enclosure with convex side walls at Pr ¼ 0.7 and Ra ¼ 106. Clockwise and anti-vely and direction of heat flows are demonstrated via heatlines with arrows.

−11.7 −11.7−11

−5

−8

−0.5

ψ(a)

−3.5−3

−1−0.1

−2−12 −0.1

44.5Π

0.9 0.7

0.5

0.30.1

θ

−0.5−4−8 −10

−12−12.8

(b)

−3.5 −2

−0.1

−2−0.13 15

0.9 0.7

0.5

0.30.1

−0.5−5−11 −13

−14 −14

(c)64 2 0.1 −2−3−4

−0.5−2

0.9 0.7

0.5

0.30.1

Fig. 17. Streamlines (j), heatlines (P) and isotherms (q) for (a) case 1, (b) case 2 and (c) case 3 for enclosure with convex side walls at Pr ¼ 1000 and Ra ¼ 105. Clockwise and anti-clockwise fluid flows are shown via negative and positive signs of streamfunction, respectively and direction of heat flows are demonstrated via heatlines with arrows.


clustered in the middle portion of the wall representing large heattransfer rates. At low Ra (Ra ¼ 103), heatlines are parallel depictingconduction dominant heat transport. Due to the sudden contrac-tion at the throat area in case 3, largely dense heatlines occur due tolarger thermal gradient at the throat area. The heatlines aredisperse and parallel representing less jPj values due to smallerthermal gradient at the top and bottom portions.

The local heat transfer rates along right wall of the enclosurewith convex side walls at Ra ¼ 103 for Pr ¼ 0.015 are displayed inFig. 18(b). It is found that, Nur decreases from the bottom edge andremains almost constant till S¼ 0.98, 1.05 and 1.30 for case 1, case 2and case 3, respectively. This is due to presence of sparse heatlineswith less heatfunction gradients as seen in Fig. 14(aec). A sharpincrease in Nur is observed near the top portion of the right walldue to dense heatlines near the top portion. The heatlines arelargely compressed at the top portion due to occurrence of heatlinecirculation cells. As seen from Fig. 18(a), in the concave cases, therate of heat transfer was maximum near the middle portion of thewall. In contrast, in the present case (convex side walls), the heattransfer rate is found to be minimum near the middle portion andthat is maximum near the top portion of the wall [see Fig. 18(a)].The variations of local heat transfer rate (Nur) for Ra ¼ 105 andPr¼ 0.7 in both concave and convex sidewall cases are qualitativelysimilar to that of Pr ¼ 0.015 and a similar explanation follows [seeFig. 18(cef)].

Variation of local Nusselt number with distance along the rightwall for both concave and convex cases in presence of convectiondominant heat transport (Ra ¼ 105) is presented in Fig. 19(aef). Itmay be noted that at higher Ra, local Nusselt number showsinteresting patterns for various concavities of the side walls asillustrated in Fig. 19(a), (c) and (e). The local heat transfer rate alongthe right wall (Pr ¼ 1000) increases from the bottom portion of thewall and attains maximum value near S ¼ 0.7 for case 1 and case 2[see Fig. 19(a)]. The maxima in the distribution of Nur occur due todense heatlines at the upper middle portion of the right wall[Fig. 7(aeb)]. In contrast to case 1 and case 2, wavy pattern isobserved in the distribution of Nur for case 3 as seen in Fig. 19(a).The oscillatory nature of Nur in case 3 can be explained based on

various zones of dense heatlines along the curved right wall [seeFig. 7(c)]. Due to high concavity, the lower and upper portions willhave disconnected heat circulation and heat transfer zones. Also,due to heatline circulation cells occurring along the lower portionand upper portion, the heatlines are pushed along 0.45 � S � 0.55and S ¼ 0.8. Thus Nur shows maxima at these portions. Heatlinesare strongly coupledwith flowprofiles at Ra¼ 105. In general,U andV values are larger near the top and bottom portions of the throatarea and flow separation occurs at the throat area in case 3. Also,due to highly concave walls in case 3 [see Fig. 7(c)], the obstructionin fluid flow causes detachment in the fluid circulation cell in thethroat area. As a result, relatively sparse heatlines are found nearthe upper middle portion of the right wall near at 0.6� S� 0.8. Thesparse heatlines correspond to less heatfunction gradients whichfurther result in less Nusselt number at those regions. Overall, thepresence of various zones of dense and sparse heatlines on the rightwall in case 3 results in a wavy pattern in the local Nusselt numberdistribution. It is found that Nur is less at the top portion of the walldue to sparse heatlines. Also, as seen from Fig. 7(c), almost stagnantzone is observed in the bottom right portion of the enclosure incase 3. This results in very high thermal boundary layer thickness asseen from very sparsely spaced heatlines with less magnitude nearthe bottom portion of the right wall. As a consequence, the localNusselt number along the right wall is also very less at the bottomportion of the wall.

The distributions of Nur for convex cases for Ra ¼ 105 andPr ¼ 0.015 are presented in Fig. 19(b). It is observed that, Nur de-creases from the bottom edge and that remains almost constantdue to presence of sparse heatlines throughout the right wallexcept the top portion [see Fig. 15]. Due to high convective effect,the size of the closed loop heat circulation cell is large and that fillsthe entire enclosure in all the cases [Fig. 15]. This results in almostconstant magnitude of heatfunction and consequently very lessheatfunction gradient occurs along the right wall. However, due tohigher convective effect in case 3 as seen from the higher magni-tude and larger size of closed loop heatlines, the end-to-endheatlines take longer path to deliver heat to the right wall in case3 [see Fig. 15(c)]. Thus, the density of heatlines on the right wall in

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

2

4

6

Loc

al N

usse

lt N

umbe

r, N

u r

(a)Concave

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

2

4

6

Loc

al N

usse

lt N

umbe

r, N

u r

Pr=0.015

(b)Convex

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

2

4

6

Loc

al N

usse

lt N

umbe

r, N

u r

(c)

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

2

4

6

Loc

al N

usse

lt N

umbe

r, N

u r

Pr=0.7

(d)

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

2

4

6

Loc

al N

usse

lt N

umbe

r, N

u r

(e)

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

2

4

6

Loc

al N

usse

lt N

umbe

r, N

u r

Pr=1000

(f)

Fig. 18. Variation of local Nusselt number with distance on cold right wall (Nur) at Ra ¼ 103 for enclosure with concave side walls [(a) Pr ¼ 0.015, (c) Pr ¼ 0.7, (e) Pr ¼ 1000] andconvex side walls [(b) Pr ¼ 0.015, (d) Pr ¼ 0.7, (f) Pr ¼ 1000]; square enclosure (� � �), case 1 (///), case 2 (e e e e e) and case 3 ( ).


case 3 is less compared to case 1 and case 2 [see Fig. 15(c)]. As aresult, the local rate of heat transfer to the right wall (Nur) in case 3is lesser throughout the right wall except the top portionwhere themagnitude of heatfunction is larger in case 3. Wavy nature isobserved in the distribution of Nur in all the cases along a largerportion of right wall. Also, as the effect of the closed loop heatlinesis less pronounced in the top portion of the enclosure, dense end-to-end heatlines are observed in the top portion of the wall for allthe cases. This results in sharp increase in Nur near the top edge ofthe enclosure. It is observed that, maximum Nur occurs in case 3, atthe top edge of the right wall.

Fig. 19(c) and (d) illustrates local heat transfer rates (Nur) forconcave and convex walls, respectively at high Pr and Ra (Pr ¼ 0.7and Ra ¼ 105). It is found that, Nur increases till a distance beingS ¼ 0.8 followed by a decreasing trend in the upper portion for case1 and case 2 [Fig. 19(c)]. It may be noted that dense heatlinesemanate from the middle portion of the left wall connecting to the

middle portion of the right wall, illustrating that large amount heatis transferred to the middle portion of the right wall in case 1 andcase 2 and that results in maxima in Nur distribution [Figure notshown]. Similar to Pr ¼ 0.015 case, case 3 shows wavy trends in thedistributions of Nur even at high Pr [Fig. 19(c)]. Initially, local heattransfer rate for case 3 (Nur) increases with distance and reaches toa local maxima at S ¼ 0.5 and thereafter that decreases to attainlocal minima at S ¼ 0.7. Due to segregation of closed loop heatlinecells in case 3, the local heat transfer rate along the right wall is lessin the upper middle portion due to occurrence of sparse heatlines[see Fig. 10(c)]. Further, Nur increases and attains another maximanear the upper middle portion of the wall (S ¼ 1) due to thepresence of highly dense heatline circulation cell in that region. Thetrends of Nur for convex cases are almost similar to that of the lowerPr case, but the magnitude of Nur is larger for higher Pr [seeFig. 19(d)]. The variations for Pr ¼ 1000 are similar to those ofPr ¼ 0.7 for both concave and convex walls and hence the results

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

4

8

12

Loc

al N

usse

lt N

umbe

r, N

u r

(a)Concave

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

4

8

12

Loc

al N

usse

lt N

umbe

r, N

u r

(b)ConvexPr=0.015

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

4

8

12

Loc

al N

usse

lt N

umbe

r, N

u r

(c)

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

4

8

12

Loc

al N

usse

lt N

umbe

r, N

u r

(d)Pr=0.7

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

4

8

12

Loc

al N

usse

lt N

umbe

r, N

u r

(e)

0 0.2 0.4 0.6 0.8 1 1.2Distance

0

4

8

12

Loc

al N

usse

lt N

umbe

r, N

u r

(f)Pr=1000

Fig. 19. Variation of local Nusselt number with distance on cold right wall (Nur) at Ra ¼ 105 for enclosure with concave side walls [(a) Pr ¼ 0.015, (c) Pr ¼ 0.7, (e) Pr ¼ 1000] andconvex side walls [(b) Pr ¼ 0.015, (d) Pr ¼ 0.7, (f) Pr ¼ 1000]; square enclosure (� � �), case 1 (///), case 2 (e e e e e) and case 3 ( ).


are not explained for the brevity of manuscript [see Fig. 19(e) and(f)].

In addition to concave and convex cases, local Nusselt numberfor a square enclosure is also presented for various Ra and Pr inFigs. 18(aef) and 19(aef). Local Nusselt number distributions forconcave cases are presented and compared with square enclosurein Fig.18(a) at Ra¼ 103 and Pr¼ 0.015. As seen in concave cases, Nurincreases till the middle portion and decreases further till the topportion of the right wall. It may be noted that Nur for a square cavityremains almost constant till S ¼ 0.4, followed by a slight increasenear the top portion. As the heatlines are denser at the middleportion of concave walls (all cases), Nur is larger for cases 1e3(concave) compared to square enclosure [see Fig. 18(a)]. A peak inthe distribution on Nur is observed at the middle portion of theenclosure for case 3, where Nur is significantly larger compared tocase 1, case 2 and square enclosure. On the other hand, Nur forconvex cases at Pr ¼ 0.015 and Ra ¼ 103 decreases from the bottomportion of the wall and attains a minimum value near the middle

portion [Fig. 18(b)]. Further, Nur increases and maximum value oflocal heat transfer is observed at the top portion of the wall. It isobserved that, Nur for square enclosure is larger than enclosurewith convex walls based on larger heatfunction gradient. However,at the top and bottom corners of the right wall, local heat transferrate is larger in convex cases compared to square enclosure. At highPr (Pr ¼ 0.7 and 1000), qualitative trends of Nur are found to bealmost identical as in low Pr, for concave, convex and squareenclosures.

It is found that, Nur for all concave cases shows wavy trend dueto presence of multiple heatline circulation cells at high Ra(Ra ¼ 105) and Pr ¼ 0.015 [see Fig. 19(a)]. Especially in case 3, ahighly wavy trend in Nur is observed based on the regions of denseand sparse heatlines as discussed earlier. At high Ra and Pr ¼ 0.015,local Nusselt number at right wall (Nur) of a square enclosure alsoshows wavy trend [see Fig. 19(a)]. As seen in Fig. 19(a), Nur forsquare enclosure increases till S¼ 0.6 and further, that decreases tillthe top portion of right wall (S ¼ 1). A maxima of Nur occurs at


S ¼ 0.65, which corresponds a maxima due to presence of denseheatlines at that region. Qualitative trend of Nur for case 1 and case2 (concave) is almost identical to that of square enclosure. Quan-titatively, local heat transfer rate for square enclosure is larger thancase 1 and case 2 except for 0.5� S� 0.7, whereNur is larger in case1 and case 2 compared to square enclosure. On the other hand,there are twomaxima of Nur in case 3 and local heat transfer rate isfound to be smaller at the middle portion for case 3 (concave)compared to the square enclosure and smaller heat transfer rate isdue to flow separation caused by the highly concave side walls.

At Pr ¼ 0.015 and Ra ¼ 105, Nur increases with distance and amaxima is seen at S ¼ 0.6 for case 1, case 2 which are in contrast tocase 3 of convex enclosure [Fig. 19(b)]. Square and convex enclo-sures (case 1 and case 2) exhibit almost similar qualitative trendswith distance as seen from Fig. 19(b). Value of Nur at the maxima islargest for square enclosure compared to all convex cases [seeFig. 19(b)]. Similar to lower Ra cases, Nur is larger for convex casescompared to square enclosure near the top and bottom portions ofthe right wall.

At larger value of Pr (Pr ¼ 0.7), and Ra ¼ 105, due to significantmomentum diffusivity compared to thermal diffusivity

103 104 105

Rayleigh Number, Ra

0

2

4

6

Ave

rage

Nus

selt

Num

ber,

Nu r P1P1’=0

P1P1’=0.1

P1P1’=0.2

P1P1’=0.4

Concave

Pr=0

(a)

103 104 105

Rayleigh Number, Ra

0

2

4

6

Ave

rage

Nus

selt

Num

ber,

Nu r P1P1’=0

P1P1’=0.1

P1P1’=0.2

P1P1’=0.4

Concave

Pr=1

(c)

Fig. 20. Variation of average Nusselt number ðNurÞ with Rayleigh number (Ra) for enclosuPr ¼ 0.015, (d) Pr ¼ 1000]; square enclosure (� � �), case 1 (///), case 2 (e e e e e) an

corresponding to enhanced convection transport, wavy nature ofNur is not observed in square as well as case 1 and case 2 of concaveenclosure [see Fig. 19(c)]. An increasing trend of Nur with distanceexcept at the top portion is observed for cases 1, 2 (concave) andsquare enclosures. Also, local heat transfer rate is larger in thesquare enclosure compared to case 1 and case 2 (concave cases)throughout the length. Similar to Pr ¼ 0.015 case, two maximaseparated by a minima at center for Nur occurs for case 3. Note that,Nur for square enclosure is larger compared to that of case 3,throughout the length except at S ¼ 0.5. In convex case [seeFig. 19(d)], Nur for square enclosure is larger than that of case 1 andcase 2 throughout the length except the top and bottom portions ofthe enclosure. Significantly lesser value of Nur may be observed incase 3 than that of square enclosure except only at the bottomportion of the enclosure, where Nur is slightly larger in case 3. AtPr ¼ 1000, the qualitative and quantitative trends of Ra ¼ 105 arealmost similar to that of Pr ¼ 0.7 for all cases including squareenclosure [see Fig. 19(e) and (f)].

Fig. 20(aed) shows the variation of average Nusselt number ofthe right wall (Nur) with Ra for all the cases of concave and convexside walls at Pr ¼ 0.015 and 1000. It may be noted that the average

.015

103 104 105

Rayleigh Number, Ra

0

2

4

6

Ave

rage

Nus

selt

Num

ber,

Nu r P1P1’=0

P1P1’=-0.1

P1P1’=-0.2

P1P1’=-0.4

Convex(b)

000

103 104 105

Rayleigh Number, Ra

0

2

4

6

Ave

rage

Nus

selt

Num

ber,

Nu r P1P1’=0

P1P1’=-0.1

P1P1’=-0.2

P1P1’=-0.4

Convex(d)

re with concave side walls [(a) Pr ¼ 0.015, (c) Pr ¼ 1000] and convex side walls [(b)d case 3 ( ).


Nusselt number on the right wall ðNurÞ refers to the amount of heatreceived by the right wall. The thermal equilibrium has beenverified for all the considered cases, such that, Nul ¼ Nur. Thesemi-log linear plot is obtained with a number of data sets. A leastsquare curve is fitted and the overall error is within 5%.

The average Nusselt number for case 1 increases very slowly forRa� 104 and thereafter larger increasing trend is found for Ra� 104

[see Fig. 20(a)]. This is attributed to the presence of sparse heatlinesat low Ra (Ra � 104) and dense heatlines at high Ra (Ra � 104) [seeFigs. 5e7]. As seen from Fig. 20(a), the trends of average Nusseltnumber for case 2 and 3 are found to be almost invariant with Ra forRa � 104 in the enclosure with concave side walls. This may beexplained from the heatline contours, as no remarkable change inmagnitude of heatlines is observed as Ra increases from 103 to 104.Based on comparison of all the cases of wall curvatures, at Ra ¼ 106

and 104, highly dense heatlines with very high heatfunction gra-dients are observed in case 3. As a result, [see Figs. 5(c) and 6(c)] theglobal rate of heat transfer is highest in case 3 at low Ra (Ra ¼ 103

and 104). At higher Ra, the heat flow pattern is highly sensitive withconcavity of side walls. As seen earlier [see Fig. 7], the flow and heatcirculation cells are segregated in case 3 for Ra ¼ 105. The localheatline circulation cells are decoupled and thus overall heattransfer is not enhanced for case 3. Also, the local rate of heattransfer (Nur) is significantly less in the bottom, upper middle andupper portion of the right wall for case 3 at low Pr and higher Ra[see Fig. 19(a)]. Consequently, case 3 offers slightly lesser rate ofheat transfer compared to case 1 at convection dominant regime(Ra ¼ 105). It may also be noted that, due to comparatively higherheatfunction gradient at Ra ¼ 105, overall rate of heat transfer isobserved to be slightly higher in case 3 compared to case 2.

As Pr increases to 1000, the heat transfer rate for the right wallin case 1 and case 2 in presence of dominant convection mode(Ra¼ 105) is much higher than that of lower Pr case [see Fig. 20(c)].At higher Pr (Pr ¼ 1000), dramatic increase in the average Nusseltnumber with Ra is observed especially for case 1 and case 2 withconcave side walls. Similar to low Pr case, Nur is maintained con-stant up to Ra¼ 104 and that increases with Ra for Ra � 104 for case3. In addition, Nur is found to be maximum in case 3 for a range oflow Ra (103 � Ra � 104). This is due to higher heat transfer rate incase 3 (concave) at low Ra which is clearly illustrated in localNusselt number plots [see Fig. 18(e)]. As seen from Fig. 19(e), athigher Ra (Ra¼ 105), Nur shows maxima near the top portion of thewall for case 1 and case 2 due to occurrence of densely spacedheatlines in those regions [see Fig.10(a) and (b)]. On the other hand,local minima are observed near the middle portion of the wall forcase 3 which is attributed to the sparse heatlines [see Fig. 10(c)].Thus, larger value of Nur is observed in case 1 and case 2 comparedto case 3 for higher Ra at Pr ¼ 1000.

In case of convex side walls, variation of Nur with Ra showsalmost similar qualitative features for all the cases at Pr¼ 0.015 [seeFig. 20(b)]. It is observed that Nur increases with Ra for all the casesdue to enhanced convection effect in the enclosure with convexside walls. Similar to concave walls, maximum value of Nur isobserved at Ra¼ 105 due to high heatfunction gradient at higher Ra.However, Nur is highest for case 1 and lowest for case 3 throughoutthe range of Ra. Local heat transfer rate is much lesser for case 3with high convexity [see Figs. 18(b) and 19(b)] and consequentlyNur is much lower for case 3. In case of highly viscous fluids(Pr¼ 1000), almost similar qualitative and quantitative features areobserved in the distribution of Nur in all the cases of convex wall[see Fig. 20(d)]. It is also interesting to note that, enhancement ofNur with Ra is observed for convex wall cases at high Pr. Moreover,due to comparatively significant heatfunction gradient at high Prand Ra in convex cases [see Figs. 17(aec) and 10(aec)], local heattransfer rate is found to be higher in convex cases and the larger

rate occurs near the top portion of the right wall. As a result, theoverall rate of heat transfer to the right wall is larger for convexwallat Pr¼ 1000 and Ra¼ 105. Comparison of concave and convex casesshows that, the wall curvature of the concave wall significantlyaffects the overall rate of heat transfer. On the other hand, forconvex cases, qualitative trends of the global heat transfer rates arealmost identical for all wall curvatures. In addition, at higher Ra andPr, Nur for convex cases are found to be higher than concave cases,which can also be explained based on local Nusselt number dis-tributions, as the local heat transfer rates for convexwalls are foundto be higher compared to those for concavewalls. Also, as seen fromthe contours in Figs. 10 and 17, the gradients of heatfunction inconvex enclosure cases are observed to be larger than those inconcave cases.

Comparative study on average heat transfer rates for concaveand convex cases with a standard square cavity is also illustrated inFig. 20(a)e(d). At both Pr (0.015 and 1000), lesser heat transfer rateis observed in square enclosure compared to concave cases for lowRa [Ra ¼ 103e104, Fig. 20(a), (c)]. At low Ra, the buoyancy force isless and conductive heat transfer is very high for case 3 of concavecase. This is due to the least distance between highly concave sidewalls, which results in very high heat transfer from the left wall toright wall, especially at the middle portion of side walls. This canalso be explained from the local Nusselt number plots [Fig. 18(a)],where Nur for case 3 (concave) is significantly higher compared tosquare enclosure throughout the length except the top and bottomcorner region. Also, at the middle portion of the side wall, a sharppeak with very high value of Nur is observed. As a result, Nur is veryhigh for case 3 (concave) compared to that of square enclosure atlow Ra. Similarly, for Ra ¼ 103e2 � 103, due to dominance ofconductive heat transfer, average heat transfer rate in case 1 andcase 2 (concave) is slightly larger than that of square enclosure. Thistrend of Nur can also be inferred from the local Nusselt numberplots [see Fig. 18(a)], where Nur for case 1 and case 2 (concave) islarger than that of square enclosure throughout the wall except thetop and bottom portions.

As Ra increases, heat received by the right wall is found to behighest in square case compared to all concave cases. At high Ra(Ra¼ 104e105), due to increase in buoyancy force and separation offlow field in case 3 (concave), convective heat transfer is very highat top and bottom portions of the wall. On the other hand, due toflow separation (case 3), heat transfer rate is very less at the middleportion of the wall near the throat section which may be clearlyobserved fromminima in the local Nusselt number plot at distance0.8 [see Fig. 19(a)]. It is interesting to observe that the local Nusseltnumber, Nur for case 3 (concave) is higher compared to cases 1e2(concave) at the other two zones corresponding to two maxima.Thus, Nur for case 3 is large at Ra¼ 105, but that is smaller than Nurfor square enclosure at Ra ¼ 105 [see Fig. 20(a)]. At higher Pr(Pr ¼ 1000) almost similar trend in Nur may be observed as in lowPr for case 3 as seen in Fig. 20(a) and (c). It is observed that, forRa � 104, Nur for cases 1e2 and square enclosure is found to besignificantly higher compared to case 3. This is due to the fact that,at large Ra, convective force becomes dominant especially in case 1,case 2 (concave) and square enclosure due to the larger areaavailable for fluid flow. The flowing fluid carries large amount ofheat from the entire hot surface to the entire cold surface asinferred from large value of Nur in case 1, case 2 (concave) andsquare enclosure. In case 3 (concave), at high Ra, due to constrictionat Y ¼ 0.5, streamline cell gets separated and compressed towardsthe top and bottom adiabatic walls. Due to separation of flow field,heat is mostly transferred locally for case 3 (concave). This can alsobe visualized from dense heatlines at the top and bottom halvesand mostly sparse heatlines at the throat region of the enclosure.On the other hand, at the throat region, although the gradient of


temperature is highest, due to separation of flow field, heat transferrate is low. As a result, at the middle portion of the enclosure(Y ¼ 0.5), significant drop in local heat transfer is observed [seeFig. 19(c) and (e)]. Thus, Nur at high Ra is found to be least in case 3for Pr ¼ 1000 [Fig. 20(c)]. Comparing trends of Nur for Pr ¼ 0.015and 1000, it is observed that at high Pr (Pr ¼ 1000), Nur for case 3 issignificantly lesser than that of cases 1 and 2 (concave) especially atlarge Ra, which is in contrast to low Pr case (Pr ¼ 0.015). Althoughflow separation occurs for both Pr ¼ 0.015 and 1000 (case 3), thelesser heat transfer rate ðNurÞ for case 3 at Pr¼ 1000 is due to lesserheatfunction gradient at the right wall.

As seen in Fig. 20(b) and (b), qualitative trends of averageNusselt number are identical in convex cases and square enclosure.This is due to the fact that, the fluid flow pattern and trend of localNusselt number [Figs.18 and 19] are almost similar in all the convexcases (cases 1e3) and square enclosure. However, magnitude ofoverall heat transfer is highest in square enclosure for the entirerange of Ra for Pr ¼ 0.015. This can be clearly observed fromFigs. 18(b) and 19(b), where Nur is larger in square enclosurecompared to convex cases throughout the length except the bottomportion of the right wall. At high Pr (Pr ¼ 1000), average Nusseltnumber at the right wall ðNurÞ in the square enclosure is identical tothose of case 1 and case 2 of convex enclosures in the entire rangeof Ra. Slightly lower value ofNur is observed in case 3 of convex case[Fig. 20(d)].

Fig. 21(aeb) illustrates the overall comparison of Nur forconcave, convex and square enclosures for various Ra (103 and 105)and Pr (0.015 and 1000). Average Nusselt numbers are shown fordistances, P1P01, which vary from �0.4 to 0.4. It may be noted that,negative value of P1P01 corresponds to convex side walls and posi-tive value of P1P01 signifies enclosure with concave side walls [seeFig. 1(c) and Table 1]. It may also be noted that, P1P01 ¼ 0 depicts

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4P1P1′

0

2

4

6

Ave

rage

Nus

selt

Num

ber

Convex Concave

(a) Pr=0.015

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4P1P1′

0

2

4

6

Ave

rage

Nus

selt

Num

ber

Convex Concave

(b) Pr=1000

Fig. 21. Variation of average Nusselt number ðNurÞ with P1P01 for convex cases(negative value of P1P01; left half), square enclosure ðP1P01 ¼ 0Þ and concave cases(positive value of P1P01; right half) at Ra ¼ 103 (e e e e e) and 105 ( ) for (a)Pr ¼ 0.015 and (b) Pr ¼ 1000.

square enclosure. At low Ra and Pr (Pr ¼ 0.015), Nur increases asP1P01 increases from�0.4 to 0.4 as seen in Fig. 21(a). At low Ra, heattransfer rate is largest for P1P01 ¼ 0:4 (concave: case 3) which is dueto high conductive heat transfer mainly at the throat region ofenclosures as discussed earlier. It may be noted that, averageNusselt number ðNurÞ is observed to be larger throughout the rangeof P1P01, at high Ra (Ra¼ 105) compared to that of low Ra (Ra¼ 103).At high Ra and Pr ¼ 0.015, heat transfer rate increases as P1P01 in-creases from �0.4 (highly convex) to 0 (square). This can also beexplained from the heatline contours, where the heatlines near themiddle portion of the wall become sparse as the wall curvatureincreases from case 1 ðP1P01 ¼ �0:1Þ to case 3 ðP1P01 ¼ �0:4Þ,depicting lesser heatfunction gradient in case 3 compared to that ofcase 1, case 2 and square enclosure. Further, as wall concavity in-creases ðP1P01 ¼ 0e0:4Þ, Nur decreases and attains a minimumvalue at P1P01 ¼ 0:3 followed by a slightly increasing trend tillP1P01 ¼ 0:4 [right half of Fig. 21(a)]. This is due to the fact that atlow Pr and high Ra, size and magnitude of secondary heatline cellsat the corner regions increase as P1P01 increases from 0 to 0.3 andthat results in sparse heatlines which further leads to decrease inheatfunction gradient at the wall. At P1P01 ¼ 0:4 (case 3, concave),flow separation occurs and secondary heatline cells are not found atthe corner regions and heatfunction gradients are larger at the wallcompared to those of P1P01 ¼ 0:2 and 0.3 [see Fig. 7(aec)]. As aresult, slightly increasing trend in Nur distribution is observed athigh Ra only for concave cases for high wall concavity ðP1P01 ¼ 0:4Þ.Overall comparison of convex (left half: P1P01 ¼ �0.1 to �0.4) andconcave (right half: P1P01 ¼ 0.1e0.4) cases reveals that Nur is largerin convex cases.

The qualitative and quantitative trends of Nur are found to bealmost similar for low Ra at Pr ¼ 0.015 and 1000 [see Fig. 21(a) and(b)]. At low Ra, the fluid flow is less intense and heat transfer ismainly due to conduction for all cases even at high Pr. Similar tolower Pr, Nur increases as P1P01 increases with Ra ¼ 103. Hence, heattransfer rate is largest in case 3 of concave case at low Ra. AtRa ¼ 105 and high Pr (Pr ¼ 1000), Nur increases as P1P01 increasesfrom�0.4 (highly convex) to 0 (square), which is also similar to thatat Pr ¼ 0.015. In contrast to low Pr case, at high Pr (Pr ¼ 1000), Nurdecreases till P1P01 ¼ 0:4 (concave cases) [right panel of Fig. 21(b)].This can be explained based on heatlines at high Pr, where theheatfunction gradient at the lower portion of right wall decreasesas P1P01 increases from 0 to 0.4. The is due to the stagnant zone ofcold fluid at lower portion of the right wall leading to sparseheatlines in all concave cases signifying less heat transfer rate to thelower portion of right wall [see Fig. 10]. As P1P01 increases, area ofthe stagnant zone near the bottom portion of the right wall alsoincreases, which leads to decrease in heatfunction gradientresulting in lesser heat transfer rate [see Fig.10]. As expected, due tolarger momentum diffusivity, at high Pr (Pr ¼ 1000), convectiveheat transfer is significant andNur is significantly larger than that oflow Pr for all P1P01 at Ra ¼ 105. Overall, it may be concluded that, atlow Ra (Ra ¼ 103), larger heat transfer rate is observed in concavecases compared to that of convex and square enclosures for both Pr(Pr ¼ 0.015 and 1000). Note that, largest heat transfer rate isobserved for case 3 at low Ra (concave, P1P01 ¼ 0.4) and that can alsobe clearly inferred from the local Nusselt number plots, where amaxima with very high value of Nur is found in the middle portionof the wall [see Fig. 18]. Also, this effect can also be clearly depictedfrom distribution of Nur, where heat transfer rate in case 3(concave, P1P01 ¼ 0:4) is larger compared to all other cases for bothPr at low Ra (Ra ¼ 103) [see Fig. 20]. On the other hand, at high Ra,due to significant convective effect, heat transfer rate is found to belarger in convex cases than that of concave cases for all Pr, which isalso observed in the average Nusselt number distribution [seeFig. 20].


5. Conclusion

Current study deals with the analysis of natural convection in anenclosure with curved side walls involving isothermally hot leftwall, cold right wall and adiabatic horizontal walls. Bejan’s heat-lines concept is successfully implemented to investigate the di-rection and intensity of energy flow in the enclosure. The penaltyfinite element method has been used to obtain smooth solutions interms of streamlines, heatlines and isotherms, local and averageNusselt number distributions for a selected range of Rayleighnumber (103�105) for different fluids (Pr ¼ 0.015,0.7 and 1000)with various wall curvatures (concave and convex). Results indicatethat fluid flow and heat flow distributions inside the enclosure arestrongly affected by Ra, Pr and wall curvatures. The main resultsmay be summarized as follows:

� In concave cases, size and magnitude of fluid flow circulationcells are greatly influenced by the wall curvature. In case 1 andcase 2, almost identical qualitative features of fluid flow distri-butions are obtained, whereas for high wall curvature (case 3),fluid flow circulation cells are found to be separated in coreregion. Case 1 offers highest area for fluid movement and as aresult, magnitudes of streamlines are largest in case 1 followedby case 2 and case 3. Similar to the streamlines, heatlines arealso largely influenced by thewall curvature in concave case. It isfound from the results that, at low Ra, case 3 offers very highheat transfer rate due to dense end-to-end heatlines throughoutthe enclosure. On the other hand, at high Ra, heatline cells aresegregated and thus heat transfer rate was observed to be leastin case 3. Note that, case 1 offers larger heating effect at high Ra.

� Streamline, isotherm and heatline contours show almost similarqualitative features for all cases of convexities at low Ra and Pr. Athigh Ra and Pr, convective effect is observed to be enhancedgreatly with increase inwall convexity. As a result, the magnitudeof heatfunction in the core is observed to be highest in case 3.

� Comparative studies of concave and convex cases show thatheat and flow distributions are found to be affected significantlyby the wall curvature for concave cases. In convex cases, notmuch variations in the contours may be observed with wallcurvature. At low Ra, only end-to-end heatlines are observed inthe enclosure for concave cases, whereas in convex cases bothend-to-end and closed loop heatlines are observed even at lowRa. As wall curvature increases, magnitude of streamlines andintensity of closed loop heatlines are found to be decreasing inconcave cases. On the other hand, increase in wall curvature inconvex cases results in enhancement of magnitudes of stream-lines and closed loop heatlines. At Ra ¼ 105 and Pr ¼ 0.015,secondary fluid and heat flow circulation cells are observedwithin the enclosure with concave wall. In contrast, single fluidflow as well as heat flow circulation cell except at the cornerzones is found in the enclosure for all cases of convexities atRa ¼ 105 and Pr ¼ 0.015. The intensity of heatfunctions in thecavity (for both concave and convex) increases significantly as Princreases to 0.7 and 1000. Single streamline and heatline cellsare found and they tend to take the shape of the enclosure nearthe wall, whereas two fluid and heat flow circulation cells arefound at the core at high Pr.

� At low Ra, the local heat transfer to the right wall (Nur) is highestnear the middle portion of the wall in concave cases for allconcavities. In case 3 (high wall concavity), a peak is observed inthe distribution of Nur at the middle portion of concave wall.Local heat transfer rate is lesser than that of all concave casesthroughout the length except at the lower corner of the wall forall Pr. At high Ra and low Pr,Nur showswavy pattern irrespectiveof degree of concavity of the side walls. Local Nusselt number of

square enclosure is found to be lesser than that of all concavecases, throughout the length only except at the bottom corner ofenclosure at low Ra. At high Ra, trends of Nur for case 1, case 2and square enclosures are almost similar for all Pr. In case 3,wavy trend of Nur is observed and that deviates to a largerextent than that of square enclosure.

� Variations of Nur with distance for all the cases of convex sidewalls show almost similar qualitative trends. In contrast toconcave cases, maximum Nur occurs at the top portion of thewall in all three cases. Trends of Nur are almost invariant with Prat low Ra for all the cases of convexities. Distributions of Nur forsquare enclosure are found to be qualitatively similar to that ofconvex enclosure except at the top and bottom corners of theenclosure at low Ra for all Pr. At high Ra, oscillatory nature isobserved in the trend of Nur in all the cases of convexities at lowPr, which is also found in square cavity. Also, for Pr ¼ 0.7 and1000, distributions of Nur do not showwavy trends in all convexand square enclosures. Local heat transfer rate for squareenclosure is larger than that of enclosure with convex wallsthroughout the length except at the top and bottom portions forall Ra and Pr.

� Variation of average Nusselt number with Ra depicts that enclo-sure with highly concave side wall (case 3) exhibits larger Nurcompared to all other cases and square enclosure at low Ra for allPr. Also, at low Ra, square enclosure exhibits lowest heat transferrate compared to all concave cases. At high Ra (Ra ¼ 105), Nur isfound to be highest in case 1 (least concavity) compared to allother cases for all Pr. Further, square enclosure performs betterthan the concave enclosures at higher values of Ra.

� In convex side wall case, overall rate of heat transfer is observedto be highest in case 1 with less convexity, irrespective of Ra andPr. In addition, large enhancement of Nur is observed with Ra.Average Nusselt number of square enclosure is found to beslightly larger than case 1 of convex case at low Pr for all Ra. Athigher Pr, almost identical values of Nur are found for squareenclosure and enclosures with convex walls (case 1 and case 2).

� At low Ra, largest heat transfer rate is observed for concave cases(P1P01 ¼ 0.1e0.4) for all Pr. Average Nusselt number for squareenclosure ðP1P01 ¼ 0Þ is larger than that of convex cases andlesser than that of concave cases for all Pr at low Ra. At high Ra,heat transfer rate in convex cases (P1P01 ¼ �0:4 to �0.1), arefound to be significantly larger than that of all concave cases(P1P01 ¼ 0.1e0.4) for all Pr. Also, heat transfer rate for squareenclosure is significantly larger than that of concave cases andslightly larger than convex cases at high Ra.

Overall, the heatline approach is found to be the most effectivetool to visualize convective heat transfer in order to decide variousgeometric and thermal parameters to achieve efficient heat transferrates. At high concavity, due to flow separation within the enclo-sure, heat flows locally whereas, at higher convexity, no heat flowseparation is observed. Thus, influence of various concavities andconvexities on the heat transfer rate during natural convection iswell understood and effectively examined by the heatlineapproach. Results show that, to process all most all industrial fluidsat less Ra, concave cases may be chosen over square enclosure. Forenclosure with curved side walls, various concave/convex wallcurvatures can be used to process fluids with locally and globallyenhanced heat transfer rates.

Acknowledgment

Authors would like to thank anonymous reviewers for criticalcomments and suggestions, which improved the quality of themanuscript.


References

[1] da Silva AK, Lorente S, Bejan A. Constructal multi-scale structures for maximalheat transfer density. Energy 2006;31:620e35.

[2] Papanicolaou E, Belessiotis V. Transient development of flow and temperaturefields in an underground thermal storage tank under various charging modes.Sol Energy 2009;83:1161e76.

[3] Barthels H, Rehm W, Jahn W. Theoretical and experimental investigations intothe safety behavior of small HTRs under natural convection conditions. Energy1991;16:371e80.

[4] Kheireddine AS, Sanda MH, Chaturvedi SK, Mohieldin TO. Numerical predic-tion of pressure loss coefficient and induced mass flux for laminar naturalconvective flow in a vertical channel. Energy 1997;22:413e23.

[5] Xaman J, Ortiz A, Alvarez G, Chavez Y. Effect of a contaminant source (CO2 ) onthe air quality in a ventilated room. Energy 2011;36:3302e18.

[6] Antar MA. Thermal radiation role in conjugate heat transfer across a multiple-cavity building block. Energy 2010;35:3508e16.

[7] Sobrinho PM, Carvalho JA, Silveira JL, Magalhaes P. Analysis of aluminumplates under heating in electrical and natural gas furnaces. Energy 2000;25:975e87.

[8] El-Sebaii AA, Aboul-Enein S, RamadanMRI, El-Gohary HG. Empirical correlationsfor drying kinetics of some fruits and vegetables. Energy 2002;27:845e59.

[9] Singh AK, Goerke UJ, Kolditz O. Numerical simulation of non-isothermalcompositional gas flow: application to carbon dioxide injection into gas res-ervoirs. Energy 2011;36:3446e58.

[10] Hami K, Draoui B, Hami O. The thermal performances of a solar wall. Energy2012;39:11e6.

[11] Reddy KS, Kumar NS. An improved model for natural convection heat lossfrom modified cavity receiver of solar dish concentrator. Sol Energy 2009;83:1884e92.

[12] Kumar NS, Reddy KS. Numerical investigation of natural convection heat lossin modified cavity receiver for fuzzy focal solar dish concentrator. Sol Energy2007;81:846e55.

[13] Horta P, Henriques JCC, Collares-Pereirac M. Impact of different internalconvection control strategies in a non-evacuated CPC collector performance.Sol Energy 2012;86:1232e44.

[14] Gonzaleza MM, Palafox JH, Estradaa CA. Numerical study of heat transfer bynatural convection and surface thermal radiation in an open cavity receiver.Sol Energy 2012;86:1118e28.

[15] Dalal R, Naylor D, Roeleveld D. A CFD study of convection in a double glazedwindow with an enclosed pleated blind. Energy Build 2009;41:1256e62.

[16] Sigey JK, Gatheri FK, Kinyanjui M. Numerical study of free convection turbu-lent heat transfer in an enclosure. Energy Convers Manag 2004;45:2571e82.

[17] Wang SM, Shen ZG, Gu LX. Numerical simulation of buoyancy-driven turbu-lent ventilation in attic space under winter conditions. Energy Build 2012;47:360e8.

[18] Saha SC, Patterson JC, Lei CW. Natural convection in attics subject to instanta-neous and ramp cooling boundary conditions. Energy Build 2010;42:1192e204.

[19] Saha SC. Scaling of free convection heat transfer in a triangular cavity forPr � 1. Energy Build 2011;43:2908e17.

[20] Bairi A. Nusselt-Rayleigh correlations for design of industrial elements:experimental and numerical investigation of natural convection in tiltedsquare air filled enclosures. Energy Convers Manag 2008;49:771e82.

[21] Bairi A. Transient free convection in passive buildings using 2D air-filledparallelogram-shaped enclosures with discrete isothermal heat sources. En-ergy Build 2011;43:366e73.

[22] Koki K, Hakaru S, Ikuo T. An experimental study on natural convective heattransfer from a vertical wavy surface heated at convex/concave elements. ExpTherm Fluid Sci 1990;3:305e15.

[23] Varol Y, Oztop HF. Buoyancy induced heat transfer and fluid flow inside atilted wavy solar collector. Build Environ 2007;42:2062e71.

[24] Kashani S, Ranjbar AA, Abdollahzadeh M, Sebti S. Solidification of nano-enhanced phase change material (NEPCM) in a wavy cavity. Heat MassTransf 2012;48:1155e66.

[25] Alawadhi EM. Cooling process of water in a horizontal circular enclosuresubjected to non-uniform boundary conditions. Energy 2011;36:586e94.

[26] Alawadhi EM. A solidification process with free convection of water in anelliptical enclosure. Energy Convers Manag 2009;50:360e4.

[27] Gao WF, Lin WX, Lu ER. Numerical study on natural convection inside thechannel between the flat-plate cover and sine-wave absorber of a cross-corrugated solar air heater. Energy Convers Manag 2000;41:145e51.

[28] Kimura S, Bejan A. The heatline visualization of convective heat transfer.J Heat Transf T ASME 1983;105:916e9.

[29] Costa VAF. Unification of the streamline, heatline and massline methods forthe visualization of two-dimensional transport phenomena. Int J Heat MassTransf 1999;42:27e33.

[30] Costa VAF. Bejan’s heatlines and masslines for convection visualization andanalysis. Appl Mech Rev 2006;59:126e45.

[31] Dalal A, Das MK. Heatline method for the visualization of natural convectionin a complicated cavity. Int J Heat Mass Transf 2008;51:263e72.

[32] Kaluri RS, Basak T. Analysis of distributed thermal management policy forenergy-efficient processing of materials by natural convection. Energy2010;35:5093e107.

[33] Basak T, Roy S. Role of ‘Bejan’s heatlines’ in heat flow visualization andoptimal thermal mixing for differentially heated square enclosures. Int J HeatMass Transf 2008;51:3486e503.

[34] Singh AK, Roy S, Basak T. Analysis of Bejan’s heatlines on visualization of heatflow and thermal mixing in tilted square cavities. Int J Heat Mass Transf2012;55:2965e83.

[35] Singh AK, Roy S, Basak T. Visualization of heat transport during natural con-vection in a tilted square cavity: effect of isothermal and non-isothermalheating. Numer Heat Transf A Appl 2012;61:417e41.

[36] Kaluri RS, Anandalakshmi R, Basak T. Bejan’s heatline analysis of naturalconvection in right-angled triangular enclosures: effects of aspect-ratio andthermal boundary conditions. Int J Therm Sci 2010;49:1576e92.

[37] Anandalakshmi R, Basak T. Heat flow visualization for natural convection inrhombic enclosures due to isothermal and non-isothermal heating at thebottom wall. Int J Heat Mass Transf 2012;55:1325e42.

[38] Basak T, Ramakrishna D, Roy S, Matta A, Pop I. A comprehensive heatlinebased approach for natural convection flows in trapezoidal enclosures: effectof various walls heating. Int J Therm Sci 2011;50:1385e404.

[39] Reddy JN. An introduction to the finite element method. 2nd ed. New York:McGraw-Hill; 1993.

[40] Deng QH, Tang GF. Numerical visualization of mass and heat transport forconjugate natural convection/heat conduction by streamline and heatline. IntJ Heat Mass Transf 2002;45:2373e85.

[41] Wan DC, Patnaik BSV, Wei GW. A new benchmark quality solution for thebuoyancy-driven cavity by discrete singular convolution. Numer Heat Transf B2001;40:199e228.

Glossary

g: acceleration due to gravity, m s�2

L: height of enclosure, mN: total number of nodesNu: local Nusselt numberNu: average Nusselt numberp: pressure, PaP: dimensionless pressurePr: Prandtl numberR: residual of weak formRa: Rayleigh numberS: dimensionless distance along the wallSmax: length of the curved walls0: dummy variableT: temperature, KTh: temperature of hot right wall, KTc: temperature of cold left wall, Ku: x component of velocityU: x component of dimensionless velocityv: y component of velocityV: y component of dimensionless velocityx: distance along x coordinateX: dimensionless distance along x coordinatey: distance along y coordinateY: dimensionless distance along y coordinate

Greek symbols

a: thermal diffusivity, m2 s�1

b: volume expansion coefficient, K�1

g: penalty parameterq: dimensionless temperaturen: kinematic viscosity, m2 s�1

r: density, kg m�3

F: basis functionsP: dimensionless heatfunction4: angle made by tangent of curved wall with positive x axisj: dimensionless streamfunctionU: two dimensional domainx: horizontal coordinate in a unit squareh: vertical coordinate in a unit square

Subscripts

k: node numberl: left wallr: right wall

http://refhub.elsevier.com/S0360-5442(13)00884-0/sref1





















































































































































Bejan's heatlines and numerical visualization of convective heat flow in differentially heated...

Documents

Transcript of Bejan's heatlines and numerical visualization of convective heat flow in differentially heated...