Entropy generation based approach on natural convection in enclosures with concave/convex side walls

International Journal of Heat and Mass Transfer 82 (2015) 213–235

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Entropy generation based approach on natural convection in enclosureswith concave/convex side walls

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.10.0360017-9310/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (P. Biswal), [email protected]

(T. Basak).

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 a b s t r a c t

Article history:Received 21 April 2014Received in revised form 13 October 2014Accepted 14 October 2014Available online 5 December 2014

Keywords:Natural convectionConcave surfaceConvex surfaceGalerkin finite element methodEntropy generationNusselt number

Computational study of natural convection within differentially heated enclosures with curved (concave/convex) side walls is carried out via entropy generation analysis. Numerical simulation has been carriedout for various Prandtl numbers (Pr ¼ 0:015 and 1000) and Rayleigh numbers ð103

6 Ra 6 105Þ with dif-ferent wall curvatures. Results are presented in terms of isotherms ðhÞ, streamlines ðwÞ, entropy genera-tion due to heat transfer ðShÞ and fluid friction ðSwÞ. The effects of Rayleigh number on the total entropygeneration ðStotalÞ, average Bejan number ðBeav Þ and global heat transfer rate ðNurÞ are examined for all thecases. Maximum values of Sh (Sh;max) are found at the middle portion of the side walls for concave cases,whereas, Sh;max is observed near the top right and bottom left corner of the cavity for convex cases. On theother hand, Sw;max is seen near the solid walls of the cavity for all concave and convex cases. At all Ra andlow Pr, largest heat transfer rate and lesser entropy generation is found for case 3 (highly concave case).Overall, for convex case, case 1 or case 2 (lesser convex cases) are efficient for all Ra and Pr. On the otherhand, case 3 of concave case (highly concave) offers larger heat transfer rate and lesser entropy genera-tion compared to less concave and all convex cases at low Ra and all Pr. At high Ra and low Pr, case 3 (con-cave) may be the optimal case whereas, at high Ra and high Pr, case 1 (less concave) may berecommended based on higher heat transfer rate. A comparative study of the concave and convex casesalso revealed that the concave cases with high concavity (case 3) may be chosen as the energy efficientcase at high Ra and high Pr.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Natural convection, also termed as buoyancy induced flow playsa vital role in various engineering, industrial and natural applica-tions. Typical applications of natural convection include solarponds [1], geothermal plants and heat exchangers [2], fuel cells[3], processing of electronic equipments [4], melting process[5,6], food processing [7,8], crystal growth [9], cooling process[10] etc. Investigation of natural convection within confined enclo-sures (internal natural convection) are particularly complex whereboth heat and fluid flow distributions are highly influenced by var-ious process and geometrical parameters. Analysis of internal flowproblems lead to complex physics due to coupling between themomentum and thermal transport properties. Also, the heat andfluid flow distributions are dependent on the complexity of solidboundaries of the enclosure. The problem of natural convection

becomes further complicated when the flow is considered in anenclosure with complicated geometrical configuration.

The shapes and geometrical configurations of enclosed cavitiesplay a vital role in various applications such as thermal processingof materials. Enclosures with flat solid walls (square, rectangular,trapezoidal, triangular, etc.) are the most common types of geo-metrical configuration as reported by various researchers [11–16]. Experimental investigation for laminar natural convectionfor air in square cavity with a partition on the top wall is carriedout by Wu and Ching [11]. In another work, Basak et al. [12] per-formed numerical simulation of natural convection and examinedthe effect of temperature boundary conditions on heat transfercharacteristics in a square cavity. Saravanan and Sivaraj [13] ana-lyzed natural convection in an air filled square enclosure with alocalized non-uniform heat source that is mounted on the centralregion of the bottom wall. Study of natural convection during melt-ing of a phase change material (PCM) in a rectangular enclosure ispresented by Qarnia et al. [14]. Sieres et al. [15] carried outnumerical study of laminar natural convection with and withoutthe presence of surface-to-surface radiation within right-angledtriangular cavities filled with air. Investigation of several physical

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Nomenclature

g acceleration due to gravity, m s�2

L height or length of base of the enclosure, mLl dimensionless distance along left wallLr dimensionless distance along right wallN total number of nodesNu local Nusselt numberNu average Nusselt numberp pressure, PaP dimensionless pressurePr Prandtl numberR Residual of weak formRa Rayleigh numberS dimensionless entropy generationSh dimensionless entropy generation due to heat transferSw dimensionless entropy generation due to fluid frictionStotal dimensionless total entropy generations0 dummy variableT temperature, KT0 bulk 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 coordinate

X dimensionless distance along x coordinatey distance along y coordinateY dimensionless distance along y coordinate

Greek symbolsa thermal diffusivity, m2 s�1

b volume expansion coefficient, K�1

c penalty parameterh dimensionless temperaturem kinematic viscosity, m2 s�1

q density, kg m�3

U basis functions/ irreversibility distribution ratiou angle made by the tangent of curved wall with positive

X axisw dimensionless streamfunctionX two dimensional domainn horizontal coordinate in a unit squareg vertical coordinate in a unit square

Subscriptsk node numberl left wallr right wall

214 P. Biswal, T. Basak / International Journal of Heat and Mass Transfer 82 (2015) 213–235

and geometric parameters for natural convection in trapezoidalcavities with two internal baffles was carried out numerically byda Silva et al. [16] using the element based finite volume method.

Although analysis of natural convection in simple enclosureshas been the topic of great interest, but many industrial or processapplications involve cavities with irregular shapes where complex-ity of walls influences the characteristics of flow and heat transferrates which are often far from being simple. A few recent studieson natural convection within enclosures with curved/wavy wallswere reported by earlier researchers [17–22]. Although significantnumber of research works on internal natural convection in enclo-sures with complicated geometries appear in literature [17–22],but analysis with energy efficient approach is yet to appear andcurrent work is an attempt on analysis of natural convective heattransfer within complex cavities via energy efficient approach forthermal engineering applications.

The thermal efficiency of a system can be assessed based on thelaws of thermodynamics, but all the above mentioned works arebased on first law of thermodynamics. An efficient system maybe defined based on minimization of exergy loss or loss due toirreversibilities for flow and heat transport. The irreversibilitiesare quantified based on second law of thermodynamics via analysisof entropy generation. The quantity ‘‘exergy’’ represents the ‘‘use-ful energy’’ of a system and the destroyed exergy or exergy loss isproportional to the entropy generation. During natural convection,the entropy generation or exergy loss in thermal processing shouldbe minimized to achieve an optimal processing situation with min-imum irreversibilities. The optimum condition can be assessed viaentropy generation minimization (EGM). Comprehensive discus-sion on analysis of entropy generation for various physical systemswith many practical and engineering applications was provided byBejan [23–26]. A few earlier works on entropy generation duringnatural convection within enclosures with various geometries arereported in the literature and are discussed briefly next.

A few studies based on entropy generation during natural con-vection within square or rectangular cavities with various thermal

and process parameters have been reported in the literature [27–31]. Numerical investigation on entropy generation during naturalconvection in vertical channel for symmetrically and uniformlyheated wall was carried out by Andreozzi et al. [27]. The influenceof Rayleigh number and irreversibility distribution ratio heat trans-fer and fluid friction irreversibility during natural convectionwithin square cavities is examined numerically by Magherbiet al. [28]. Erbay et al. [29] performed a computational study toaccess entropy generation for transient laminar natural convectionin a square cavity in presence of completely or partially hot leftwall and cold right wall. Analysis of buoyancy driven convectionvia entropy generation was carried out for C-shaped enclosuresby Dagtekin et al. [30]. In another study, Famouri and Hooman[31] investigated the entropy generation during free convectionin a partitioned square cavity, with adiabatic horizontal and iso-thermally cold vertical walls. They concluded that fluid frictionirreversibility has very less contribution to total entropy genera-tion and heat transfer irreversibility increases with dimensionlesstemperature difference and Nusselt number. In addition, research-ers also worked on entropy generation for natural convection intriangular enclosures and inclined enclosures [32–35]. Entropygeneration during natural convection within non-uniformly heatedporous isosceles triangular cavities with various positions wasstudied by Varol et al. [32]. Recently, analysis of entropy genera-tion during natural convection in a porous right angled triangularenclosure is carried out by Basak et al. [33]. Baytas [34,35] ana-lyzed thermodynamic optimization and entropy generation duringnatural convection in inclined enclosures filled with fluid or porousmedia. Also, an extensive review of second law analysis of thermo-dynamics in enclosures due to natural and mixed convection flowfor energy systems was performed by Oztop and Al-Salem [36].

Most of the earlier investigations of entropy generation duringnatural convection are concerned on the heat transfer in enclosureswith flat wall. As reported in the literature, a few recent investiga-tions were focused on the natural convection and entropy genera-tion in enclosures with more complex geometric configurations

P. Biswal, T. Basak / International Journal of Heat and Mass Transfer 82 (2015) 213–235 215

such as enclosures with wavy walls [37–39]. The analysis of freeconvection in an inclined wavy enclosure using the entropy gener-ation concept was carried out by Mahmud and Islam [37]. Theyconcluded that, for each Rayleigh number, average entropy gener-ation rate decreases with inclination angle and the minimum valueis observed at an angle of 90�. Also, it was found that, at this incli-nation (90�), entropy generation is almost identical for all Rayleighnumbers. In another study, Mahmud and Fraser [38] examinedheat transfer and entropy generation due to natural convectionin an enclosure with straight horizontal walls and wavy verticalwalls. They observed that at a specific surface waviness and atlow Rayleigh number, entropy generation is less and remainsalmost constant with Ra, whereas at high Ra, entropy generationincreases rapidly with Ra. They also found that total entropy gen-eration is almost identical for all the cases of surface wavinessand cavities with high aspect ratio (tall cavity) show least totalentropy generation. Ziapour and Dehnavi [39] focused their atten-tion on entropy generation due to incompressible natural convec-tion within C-shaped enclosures with circular corners. It wasfound that, at higher range of Rayleigh number, the average Nus-selt number values increase linearly. As evident from the literaturesurvey, a generalized analysis on natural convection via entropygeneration within enclosures with various concave and convexside walls filled with various viscous Newtonian fluid has not beenattempted till date. Natural convection heat transfer within com-plex enclosure is frequently encountered in various engineeringapplications such as heat exchangers, cooling of electronic devices,double glazed windows, furnaces, solar collectors, etc. Due tonumerous energy related applications of complex enclosures, thestudy of entropy generation during natural convection in compli-cated cavities with various curved walls may be important toachieve the optimal situation of a thermal system.

The prime objective of the present work is to analyze the effectof wall curvature on entropy generation during natural convectionwithin differentially heated enclosure with curved (concave/con-vex) side walls. The coupled partial differential equations aresolved via Galerkin finite element method [40] with penaltyparameter. Further, finite element approach offers special advan-tage over finite difference or finite volume methods as elementalbasis functions are employed for accurate evaluation of gradientsor derivatives of velocity and temperature in entropy generationequation. Note that, for complex cavities with curved wall(s), cal-culation of derivative terms in entropy generation problembecomes cumbersome by finite difference/finite volume approach.Current work is the first attempt for the investigation of entropygeneration during natural convection using elemental basis func-tions via Galerkin finite element method for curved domains.Numerical simulations are carried out for a range Rayleigh number(Ra ¼ 103 � 105) and various Pr, such as Pr ¼ 0:015 (molten met-als), 0.7 (air), 1000 (olive/engine oils). Results are presented interms of contours of temperature ðhÞ, entropy generation due toheat transfer ðShÞ, streamlines ðwÞ and entropy generation due tofluid friction ðSwÞ. In addition, the effects of Ra on total entropy gen-eration (Stotal), average Bejan number (Beav ) and average heat trans-fer rate (Nur) are also presented.

2. Mathematical modeling and simulations

The three dimensional representation of the physical domainfor the enclosure with concave and convex side walls are illus-trated in the left column of Fig. 1(a) and (b), respectively. Semi-infi-nite approximation is assumed for the domain on Z direction. Thecomputational domain is shown in the right column of Fig. 1(a) and(b) for both concave and convex side wall cases. Thermophysicalproperties of the fluid are assumed to be constant except the

density as natural convection arises due to density difference.The relation between change in density and temperature variationis given by Boussinesq approximation and hence, the temperatureand flow fields are coupled. Fluid is assumed to be incompressibleand Newtonian and the flow field is laminar. In addition, no slipboundary condition is assumed near the solid boundaries. Underthe above mentioned assumptions, governing equations using con-servation of mass, momentum and energy for steady two-dimen-sional natural convection flow can be expressed with followingdimensionless variables and numbers:

X ¼ xL; Y ¼ y

L; U ¼ uL

a; V ¼ vL

a; h ¼ T � Tc

Th � Tc

P ¼ pL2

qa2 ; Pr ¼ ma; Ra ¼ gbðTh � TcÞL3Pr

m2

Here x and y are the distances measured along the horizontal andvertical directions, respectively; u and v are the velocity compo-nents in the x and y directions, respectively; T denotes the temper-ature; m and a are kinematic viscosity and thermal diffusivity,respectively; p is the pressure and q is the density; Th and Tc arethe temperatures at hot left wall and cold right wall, respectively;g is the acceleration due to gravity; L is the height or length ofthe base of the cavity, P is the dimensionless pressure and b denotesthe volume expansion coefficient. Note that, X and Y are dimension-less coordinates varying along horizontal and vertical directions,respectively; U and V are dimensionless velocity components inthe X and Y directions, respectively; h is the dimensionless temper-ature; Ra and Pr are Rayleigh and Prandtl numbers, respectively.The governing equations in dimensionless forms for continuity(Eq. (1)), momentum balance (Eq. (2) and (3)) and energy balance(Eq. (4)) are as follows:

@U@Xþ @V@Y¼ 0 ð1Þ

U@U@Xþ V

@U@Y¼ � @P

@Xþ Pr

@2U

@X2 þ@2U

@Y2

!ð2Þ

U@V@Xþ V

@V@Y¼ � @P

@Yþ Pr

@2V

@X2 þ@2V

@Y2

!þ RaPrh ð3Þ

and

U@h@Xþ V

@h@Y¼ @2h

@X2 þ@2h

@Y2 ð4Þ

and the governing equations (Eqs. (2)–(4)) are subjected to the fol-lowing boundary conditions;

U ¼ V ¼ 0;@h@YðX;0Þ ¼ 0; for Y ¼ 0 on wall AB ð5aÞ

U ¼ V ¼ 0; h ¼ 1; for X ¼ aY2 þ bY þ c on wall DA ð5bÞ

U ¼ V ¼ 0;@h@YðX;1Þ ¼ 0; for Y ¼ 1 on wall CD ð5cÞ

U ¼ V ¼ 0; h ¼ 0; for X ¼ a0Y2 þ b0Y þ c0 on wall BC ð5dÞ

Note that, a; b; c; a0; b0 and c0 are coefficients of quadratic equations(Eq. (5b) and (5d)), which are calculated by Cramer’s rule usingthree coordinates for various convex and concave curves. The valuesof a� c and a0 � c0 are given in Table 1 for three representative cases(cases 1–3). The schematics of the computational domain with var-ious concave and convex cavities are shown in Fig. 1(a) and (b).

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

Table 1The values of a; b; c for left wall and a0 ; b0 ; c0 for right wall as calculated by Cramer’s rule using the coordinates of the curve (A; P01 ; D for left wall and B; P02 ; C for the right wall)(see Fig. 1). The length of the curved wall is denoted by Li (where i ¼ l or r).

Concave Convex Li

a b c a0 b0 c0 a b c a0 b0 c0

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


3. Solution procedure and evaluation of dimensionlessquantities

3.1. Simulations for flow and temperature fields

Galerkin finite element method is employed to solve momen-tum and energy balance equations (Eqs. (2)–(4)). It may be notedthat, continuity equation (Eq. (1)) has been used as a constraintfor penalty optimization finite element method. In order to solveEqs. (2) and (3), we use the penalty finite element method wherethe pressure, P is eliminated by a penalty parameter, c and theincompressibility criteria given by Eq. (1) which results in

P ¼ �c@U@Xþ @V@Y

� �ð6Þ

The continuity equation (Eq. (1)) is automatically satisfied for largevalues of c. Typical values of c which yield consistent solutions are107. Using Eq. (6), the momentum balance equations (Eqs. (2) and(3)) reduce to

U@U@Xþ V

@U@Y¼ c

@

@X@U@Xþ @V@Y

� �þ Pr

@2U

@X2 þ@2U

@Y2

!ð7Þ

and

U@V@Xþ V

@V@Y¼ c

@

@Y@U@Xþ @V@Y

� �þ Pr

@2V

@X2 þ@2V

@Y2

!þ RaPr h ð8Þ

The system of equations (Eqs. (4), (7) and (8)) with boundary con-ditions (Eq. 5(a–d)) are solved using Galerkin finite element method


[40]. Expanding the velocity components ðU; VÞ and temperatureðhÞ using basis set fUkgN

k¼1 as,

U �XN

k¼1

Uk UkðX;YÞ; V �XN

k¼1

Vk UkðX;YÞ and h �XN

k¼1

hk UkðX;YÞ

ð9Þ

Galerkin finite element method yields nonlinear residual equationsfor Eqs. (7), (8) and (4), at nodes of internal domain X. The detailedsolution procedure is given in an earlier work [41].

3.2. Streamfunction, entropy generation and Nusselt number

3.2.1. StreamfunctionThe fluid motion is displayed using the streamfunction (w)

obtained from velocity components (U and V). The relationshipsbetween streamfunction (w) and velocity components for twodimensional flows are

U ¼ @w@Y

; V ¼ � @w@X

ð10Þ

which yield a single equation

@2w

@X2 þ@2w

@Y2 ¼@U@Y� @V@X

ð11Þ

Using the above definition of the streamfunction, the positive signof w denotes anticlockwise circulation and the clockwise circulationis represented by the negative sign of w. Expanding the streamfunc-tion (w) using the basis set fUkgN

k¼1 as w ¼PN

k¼1wkUkðX;YÞ and therelation for U; V from Eq. (9), the Galerkin finite element methodyields the linear residual equations for Eq. (11) and the detailedsolution procedure to obtain w s at each node points are given inan earlier work [41].

3.2.2. Entropy generationEntropy generation per unit volume may be calculated from

second law of thermodynamics for an open system using entropybalance [23] as follows:

Entropy generation ¼ Entropy transfer associated with heat transfer

þ Entropy convected in and out of the system

þ Rate of accumulation of entropy

ð12Þ

During natural convection, entropy generation occurs due toheat transfer irreversibility and fluid friction irreversibility. Consid-ering a control volume which allows both mass and energy trans-port with the surrounding and assuming local thermodynamicequilibrium, the total local entropy generation from Eq. (12) for atwo-dimensional natural convection system [23] in cartesian coor-dinates may be written as follows:

_S000gen ¼k

T20

@T@x

� �2

þ @T@y

� �2" #

þ lT0

2@u@x

� �2

þ @v@y

� �2 !

þ @v@xþ @u@y

� �" #ð13Þ

The first term (square bracketed) in Eq. (13) is the entropy gen-eration due to heat transfer due to temperature gradient and sec-ond term (square bracketed) represents the entropy generationdue to fluid friction caused by velocity gradient. As seen from Eq.(13), amount of entropy generation for natural convection flowsis a positive and finite quantity as long as temperature and velocitygradients exist in the system. Also, the entropy generation due tofluid friction and heat transfer are strongly correlated with geo-metric characteristics the system. The dimensionless form of Eq.

(13) based on individual terms of entropy generation due to heattransfer (Sh) and entropy generation due to fluid friction (Sw) maybe written as follows:

Sh ¼@h@X

� �2

þ @h@Y

� �2" #

; ð14Þ

Sw ¼ / 2@U@X

� �2

þ @V@Y

� �2" #

þ @U@Yþ @V@X

� �2( )

ð15Þ

In above equation (Eq. (15)), / is called irreversibility distributionratio, defined as:

/ ¼ lTo

ka

LDT

� �2ð16Þ

In the current study, / is considered as 10�4 based on previous studiesby Ilis et al. [42]. In addition, a larger value for / was taken into con-sideration by few earlier researchers [43]. Several other researchershave also worked on the effect of / on total entropy generation todecide the optimum value of / [28,30,42]. It may be noted that, / isobtained after non-dimensionalizing the entropy balance equation.The irreversibility distribution ratio (/) is a parameter based on thethermal and physical properties of the fluids (see Eq. (16)). Asreported by many researchers, the value of / for natural convectionin fluid media is taken as 10�4. Ilis et al. [42] have investigated theeffect of / on the total entropy generation within a range of Ra. Theyobserved that total entropy generation is significantly large andmainly due to the fluid friction irreversibility for all Ra at / ¼ 10�2.On the other hand, at / ¼ 10�4 the heat transfer and fluid frictionirreversibilities were found to be dominant at low and high Ra,respectively. Thus, the earlier work [42] presented the spatial repre-sentation of entropy generation due to heat transfer and fluid frictiondistribution at/ ¼ 10�4 to bring out the individual effects of the irrev-ersibilities associated with natural convection for all Ra. Dagtekinet al. [30] also studied on entropy generation and illustrated the effectof / on the total entropy generation for various Ra. They concludedthat, at lower values of / ð/ < 10�4Þ, the magnitudes of total entropygeneration were very less even for convection dominant regime withhigh Ra. At higher/ (/ > 10�4), the total entropy generation increasesrapidly and significantly larger value of total entropy generation isobserved even at very less values of Ra. A similar study was carriedout by Magherbi et al. [28] to demonstrate the effect of / on the totalentropy generation during natural convection. They also observedthat at low / (/ < 10�4), the total entropy generation is significantlylesser and remains almost constant with Ra whereas, rate of increaseof total entropy generation with Ra is significantly larger for higher /(/ > 10�4). Thus, based on previous studies, the value of irreversibil-ity distribution ratio is considered as 10�4 in the present work. Anorder of magnitude analysis also gives a similar result. For example,the properties of a gaseous fluid (air, Pr ¼ 0:7) at 298 K are in therange of l � Oð10�5Þ; k � Oð10�2Þ;a � Oð10�5Þ and for a representa-tive case with To=DT2 � Oð10Þ; / is obtained as Oð10�4Þ. Here, To isthe bulk temperature, evaluated as ðTh þ TcÞ=2.

Based on velocity and temperature fields at each node, theknown temperature and velocity fields are incorporated in thederivative terms of the expression for entropy generation. As men-tioned earlier, the derivatives terms are calculated using the ele-mental basis set which is based on finite element method. Eachelement with nine node bi-quadratic elements is mapped fromX—Y to a unit square n—g domain using iso-parametric mapping[33]. The domain integrals in residual equations are evaluated inthe n—g domain using nine node bi-quadratic basis functions. Thederivative of any function f over an element e can be written as:

@f e

@n¼X9

k¼1

f ek@Ue

k

@nð17Þ


Here, f ek is the value of the function at local node k in the element e

and Uek is the value of basis function at local node k in the element e.

Each node is shared by four elements in the interior domainwhereas, each node is shared by two elements along the boundary.Hence, for the node shared by two or four elements, the value of thederivative of any function at the global node number (i), is averagedover those shared elements (Ne), i.e.,

@f i

@n¼ 1

Ne

XNe

e¼1

@f e

@nð18Þ

Thus, local entropy generation due to thermal ðSh;iÞ and fluid frictionðSw;iÞ irreversibilities at each node are given by,

Sh;i ¼@hi

@X

� �2

þ @hi

@Y

� �2" #

ð19Þ

Sw;i ¼ / 2@Ui

@X

� �2

þ @Vi

@Y

� �2" #

þ @Ui

@Yþ @Vi

@X

� �2( )

ð20Þ

Note that, the derivative terms in Eqs. (19) and (20)@hi@X ;

@hi@Y ;

@Ui@X ;

@Ui@Y ;

@Vi@X ;

@Vi@Y

h iare evaluated following Eq. (18). The total

entropy generation due to natural convection flow (Stotal) in the cav-ity is given by the summation of total entropy generation due toheat transfer (Sh;total) and fluid friction (Sw;total) irreversibility. Fur-ther, Stotal is obtained via integrating the local entropy generationrates (Sh;i and Sw;i) over the considered domain X.

Stotal ¼ Sh;total þ Sw;total ð21Þ

where,

Sh;total ¼Z

X

@

@X

XN

k¼1

hkUk

!" #2

þ @

@Y

XN

k¼1

hkUk

!" #28<:

9=;dX dY ð22Þ

Sw;total ¼ /Z

X2

@

@X

XN

k¼1

UkUk

!" #2

þ 2@

@Y

XN

k¼1

VkUk

!" #28<:

þ @

@Y

XN

k¼1

UkUk

!þ @

@X

XN

k¼1

VkUk

!" #29=;dX dY ð23Þ

The integrals in Eqs. (22) and (23) are evaluated via three-point ele-ment-wise Gaussian quadrature integration method. To study therelative dominance of heat transfer and fluid friction irreversibility,a dimensionless parameter, Bejan number (Beav ) is used in the cur-rent work. Bejan number is defined as,

Beav ¼Sh;total

Sh;total þ Sw;total¼ Sh;total

Stotalð24Þ

In accordance to the above definition, Beav > 0:5 implies dominanceof heat transfer irreversibility and Beav < 0:5 implies dominance offluid friction irreversibility.

3.3. Nusselt number

The heat transfer coefficient in the outward direction from asolid surface can be written in terms of the local Nusselt numberðNuÞ which is defined as

Nu ¼ � @h@n

ð25Þ

where n denotes the normal direction on a plane. The normal deriv-ative is evaluated by the bi-quadratic basis set based on finite ele-ment method. The local Nusselt numbers at left wall Nulð Þ andright wall Nurð Þ are defined as

Nul ¼X9

i¼1

hei sin u

@Uei

@X� cos u

@Uei

@Y

� �ð26Þ

and

Nur ¼X9

i¼1

hei � sin u

@Uei

@Xþ cos u

@Uei

@Y

� �ð27Þ

where hei is the value of h at local node i of element e and Ue

i arebasis sets of an element e. The average Nusselt numbers at the sidewalls are

Nul ¼R Ll

0 Nulds0R Ll0 ds0

ð28Þ

and

Nur ¼R Lr

0 Nur ds0R Lr

0 ds0ð29Þ

Here Ll and Lr are the lengths along left and right walls, respectivelyand ds0 is the small elemental length along the curved wall.

4. Results and discussion

4.1. Numerical tests and parameters

The mesh for the computational domain in n—g coordinate sys-tem consists of 28� 28 bi-quadratic elements. Note that, the28� 28 bi-quadratic elements correspond to 57� 57 grid pointsin the n—g domain. The computational grid with curved side wallsis generated by mapping the curved domain into a regular squaredomain in n—g coordinate system [44]. In contrast to the finite-dif-ference or finite-volume solutions, bi-quadratic elements with alesser number of nodes in finite element method smoothly capturethe non-linear variations of the field variables. Thus, finite elementmethod based on Gaussian quadrature offers smooth solutions atthe interior domain in addition to the corner regions as evaluationof residuals is influenced by interior Gauss points. Consequently,the effect of corner nodes is less pronounced in the final solution.The present results in terms of entropy generation due to heattransfer and fluid friction (Sh and Sw) are compared with a bench-mark problem as reported by Ilis et al. [42] for a differentiallyheated, air-filled (Pr ¼ 0:71) square cavity (see Fig. 2). Currentresults are observed to be in excellent agreement with the previouswork [42] as seen from Fig. 2.

In internal natural convection problems, Nusselt numbers arecalculated at the outer surface using some interpolation functionsin finite difference/finite volume based methods which are avoidedin the present work. Current work employs finite elementapproach that offers special advantage on evaluation of local Nus-selt number at the solid surfaces using element basis functions.Grid invariant results are presented for the present solutionscheme in terms of Nur; Beav and Stotal as shown in Table 2–4,respectively for all the concave and convex cases. Detailed discus-sion on variation of local entropy generation rate due to heat trans-fer (Sh) and fluid friction (Sw), with isotherms (h) and flow fields (w),average Bejan number (Beav) and average heat transfer rate interms of Nusselt number (Nur) for various Ra; Pr and wall curva-tures is presented in following sections.

An analysis is carried out in the present work to study the effectof / on total entropy generation rates for various Ra and Pr for allconcave cases (see Fig. 3). As seen from Fig. 3(a), Stotal is almost sim-ilar throughout the range of Ra for / ¼ 10�6 and 10�5. The totalentropy generation is found to be very insignificant for lower val-ues of / (/ ¼ 10�6 and 10�5) throughout the range of Ra. This is due

0.6

0.9

0.9

0.6

1.3

1.9

1.3

1.9

(a)S

θ

0.02

0.1

0.05

0.05

0.05

0.05

Sψ

510

40

1020

50(b)

10

200

200

10

1050

Fig. 2. Local entropy generation due to heat transfer (Sh) and fluid friction (Sw) for a square enclosure with hot left wall, cold right wall and adiabatic horizontal walls forPr ¼ 0:71, (a) Ra ¼ 103 (top figure: present work, lower figure: work reported by Ilis et al. [42]) and (b) Ra ¼ 105 (top figure: present work, lower figure: work reported by Iliset al. [42]). (Lower figures of Figs. (a) and (b) reproduced from Ilis et al. [42] with permission from Elsevier).

Table 2Comparison of average Nusselt number on the right wall (Nur) 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

12� 12 16� 16 20� 20 24� 24 28� 28 12� 12 16� 16 20� 20 24� 24 28� 28

Case 1 4.63 4.53 4.48 4.45 4.44 5.07 4.94 4.80 4.72 4.71Case 2 4.28 4.21 4.18 4.17 4.16 5.12 4.91 4.79 4.67 4.63Case 3 3.54 3.52 3.51 3.51 3.51 4.89 4.66 4.53 4.41 4.40


to the fact that, at low /, the contribution of fluid friction irrevers-ibility on Stotal is significantly less. In addition, only Sh contributes tothe major part of Stotal for / < 10�4. Thus, the total entropy

generation occurs only due to heat transfer irreversibility and issignificantly less even at larger values of Ra for / < 10�4. At/ ¼ 10�4, the total entropy generation remains almost constant

Table 3Comparison of average Bejan number values (Beav ) for various grid systems at Ra ¼ 105 and Pr ¼ 1000 with various wall curvatures in presence of uniform heating on the left wall,cold right wall and adiabatic horizontal walls.

Concave Convex

12� 12 16� 16 20� 20 24� 24 28� 28 12� 12 16� 16 20� 20 24� 24 28� 28

Case 1 0.19 0.19 0.19 0.19 0.19 0.18 0.18 0.18 0.18 0.18Case 2 0.2 0.2 0.2 0.2 0.2 0.18 0.18 0.18 0.17 0.17Case 3 0.26 0.26 0.26 0.26 0.26 0.17 0.17 0.17 0.17 0.17

Table 4Comparison of total entropy generation values (Stotal) for various grid systems at Ra ¼ 105 and Pr ¼ 1000 with various wall curvatures in presence of uniform heating on the leftwall, cold right wall and adiabatic horizontal walls.

Concave Convex

12� 12 16� 16 20� 20 24� 24 28� 28 12� 12 16� 16 20� 20 24� 24 28� 28

Case 1 24.03 24.06 24.07 24.08 24.08 26.70 26.92 26.97 27.00 27.00Case 2 23.24 23.24 23.24 23.24 23.24 28.83 29.07 29.12 29.14 29.14Case 3 18.35 18.35 18.35 18.35 18.35 33.93 34.10 34.11 34.13 34.13

103 104 105

Ra

0

50

100

Sto

tal

Pr=0.015

Case 1(a)

103 104 105

Ra

0

50

100

Sto

tal

Case 2

103 104 105

Ra

0

50

100

Sto

tal

Case 3

103 104 105

Ra

0

50

100

Sto

tal

Pr=0.7

(b)

103 104 105

Ra

0

50

100

Sto

tal

103 104 105

Ra

0

50

100

Sto

tal

103 104 105

Ra

0

50

100

Sto

tal

Pr=1000

(c)

103 104 105

Ra

0

50

100

Sto

tal

103 104 105

Ra

0

50

100

Sto

tal

Fig. 3. Variation of Stotal with Ra for all concave cases (left column: case 1, middle column: case 2 and right column: case 3) at various / [(}—}—}): / ¼ 10�6, (+–+–+):/ ¼ 10�5, (—): / ¼ 10�4, (––––): / ¼ 10�3 and (� � �): / ¼ 10�2] for (a) Pr ¼ 0:015, (b) Pr ¼ 0:7 and (c) Pr ¼ 1000.


for Ra ¼ 103 � 104 depicting conduction dominant heat transfer.Further, increasing trend of total entropy generation is observedas Ra increases till 105. It may be noted that, the heat transfer irre-versibility as well as the fluid friction irreversibility enhance with

Ra, which is clearly visible from the distribution of Stotal for/ ¼ 10�4. In contrast, at high / (/ ¼ 10�3 and 10�2), the entropygeneration is larger even at lower values of Ra depicting larger Sw

even at a lesser value of Ra. As Ra increases further, the total

0.9

0.7

0.5

0.3 1.0

θ(a)

1.21.6

1.6

1.2

0.7

2

2

0.7

Sθ

−0.005

−0.3−0.6

−0.8

ψ

0.01

0.03

0.01

0.01

Sψ

Sψ,max=0.69

Sψ,max=0.69

Sθ,max=2.71Sθ,max=2.71

0.9

0.7

0.5

0.30.1

θ(b)1.5

2.5

2

3 3.5

1

2

2.51.5

Sθ

−0.01

−0.1

−0.25

−0.45

−0.05−0.2

ψ

0.007

0.007 0.05

Sψ

Sθ,max=3.99

Sθ,max=3.99

Sψ,max=0.6

Sψ,max=0.6

0.90.7

0.50.3 0.1

θ(c)

10

15

4

4

Sθ

−0.01

−0.05

−0.03

−0.07

ψ

0.01

Sψ

Sθ,max=30.4

Sθ,max=30.4

Sψ,max=0.09Sψ,max=0.09

Fig. 4. Isotherms (h), local entropy generation due to heat transfer (Sh), streamlines(w), and local entropy generation due to fluid friction (Sw) for (a) case 1 (b) case 2and (c) case 3 in concave cases for Pr ¼ 0:015 and Ra ¼ 103.


entropy generation increases in a remarkable rate and significantlylarge value of Stotal is seen at both moderate and higher range of Rafor / ¼ 10�3 and 10�2. A similar trend is observed for all the casesand all Pr. Overall, the total entropy generation is very small forlower / (/ < 10�4) and significantly larger values of total entropygeneration are found for higher values of / (/ > 10�4). Thus,/ ¼ 10�4 can be used to evaluate the entropy generation. Also,the competition between the heat transfer and fluid friction irre-versibility can be clearly addressed for / ¼ 10�4. It may be notedthat, the value of / (/ ¼ 10�4) is also in accordance to the previousworks [28,30,42].

4.2. Isotherms, streamlines and entropy generation maps for theenclosure with concave side walls

Three cases based on three different curvatures of the concaveside walls are considered in order to emphasize the curvatureeffect on the fluid flow and heat flow distributions. The originalsquare enclosure is modified to a curved walled enclosure by shift-ing the mid points of the side walls, P1 and P2 in the inward direc-tion to P01 and P02, respectively, such that AP01D and BP02C formcurves which obey the quadratic equations; X ¼ aY2 þ bY þ c andX ¼ a0Y2 þ b0Y þ c0, respectively (see Fig. 1(a) and Table 1). The val-ues of P1P01 or P2P02 are assumed to be L=10;2L=10 and 4L=10 forcases 1, 2 and 3, respectively, where L is the height or length ofbase of the cavity. Figs. 4–7 show isotherms (h), entropy generationdue to heat transfer (Sh), streamlines (w) and entropy generationdue to fluid friction (Sw) for various fluids (Pr ¼ 0:015 and 1000)with Ra ¼ 103 � 105 within the enclosure with concave side walls.It is observed that the entropy generation due to fluid flow irrever-sibilities is maximum near the walls due to high velocity gradientin those region. It is also observed that the entropy generation dueto heat transfer is significant in the regions of high heat transferrate. In addition, significant entropy generation due to flow irrever-sibilities occur at the interface of multiple circulation cells.

Fig. 4(a)–(c) illustrate isotherms (h), streamlines (w) and entropygeneration maps (Sh and Sw) for concave cases (cases 1–3) at low Ra(Ra ¼ 103) and Pr (Pr ¼ 0:015). The isotherms are smooth verticallines perpendicular to the adiabatic horizontal walls indicating con-duction dominant heat transfer. As the wall curvature increases fromcase 1 to case 3, the boundary layer thickness gradually becomes lar-ger at the corner regions of the enclosure. Also, with increase in cur-vature of the side walls, the hot and cold walls approach closer forcase 3. As a result, conductive heat transfer at the throat region isintense in case 3 and that can also be explained based on compressedisotherms near the middle portion of the enclosure. However, heatdistribution is inadequate in the corner regions of the enclosure incase 3 as seen from the presence of hot stagnant fluid at the left cor-ner region and cold stagnant fluid at the right corner region. It isfound that Sh is observed to be negligible at the corner region ofthe enclosure, especially in case 3 due to very less heat transfer rateat those regions. The corresponding entropy generation map alsodepicts that entropy generation due to heat transfer is maximumnear the middle portion of the side walls for all the cases. Significantentropy generation in case 3 is attributed to very high conductiveheat transfer at the middle portion of side walls based on com-pressed isotherms due to high thermal gradients. Also, it is observedthat the local entropy generations at various interior locations of theenclosure (except the corner region) are found to be significant incase 3, compared to cases 1 and 2. Consequently, overall entropygeneration due to heat transfer (Sh) is larger in case 3 throughoutthe enclosure, especially at the core compared to that of cases 1and 2. Note that, Sh;max ¼ 2:71; 3:99 and 30.4 occur for cases 1, 2and 3, respectively.

It is observed that at lower Ra and Pr, single and smooth fluidcirculation cells span the entire enclosure for cases 1 and 2

(Fig. 4(a) and (b)). Due to the imposed thermal boundary condition,fluid from the lower part of the left wall gets heated up and movesupward due to buoyancy force and flows down along the cold rightwall. Fluid circulation cell gets segregated at the core and twoclockwise rotating loops are observed in case 3 due to the effectof highly concave side walls. The strength of circulation cells arefound to be higher in the core and least near the walls due to

0.9

0.7

0.4

0.2

θ(a)

1

24

4

711

8 8

Sθ

−4−3−2

−1.5

−0.5 −0.05

ψ

11

11

Sψ

Sθ,max=12

Sθ,max=12

Sψ,max=52.09

Sψ,max=52.09

0.9

0.7

0.5 0.3 0.1

θ(b)

1

1

2

3

4

6

8

Sθ

−0.02

−0.02−0.02

−0.6−1.4

−2.62

ψ

0.1

0.1

0.1

Sψ

Sθ,max=11.6

Sθ,max=11.6


0.9

0.7

0.5

0.3

0.1

θ(c)210

2

Sθ

−0.05−0.6

−0.8

0.008

ψ

0.5

0.5

Sψ


Sψ=5.59

Sψ=5.14



0.9

0.6

0.4

0.1

θ(a)

0.5

0.510

10

54

Sθ

−8.4

−7 −5−0.1

−0.01

−0.1

ψ

44

4 4

4 4

Sψ

Sθ,max=48.42

Sθ,max=48.42

Sψ,max=1340

Sψ,max=1340

Sψ=19.3

Sψ=19.3

0.9 0.7

0.3

0.1

0.5

θ(b)

1

110

10

Sθ

−0.1

−0.03−0.03

−7

−4−2−0.5

−0.5

ψ10

5

5

5

Sψ


Sψ,max=1396

Sψ,max=1396

Sψ=26.4

Sψ=26.4

0.9

0.10.4

0.7

θ(c)

5

5

10

10

25 25S

θ

−0.05

−3

−3.8−0.5−2

−0.5

ψ

20

20

20

20

SψSψ=257

Sψ=257

Sψ,max=822 Sψ,max=822

Sθ,max=60.23

Sθ,max=60.23



no-slip boundary condition at the solid wall. Comparative study ofall the cases shows that the strength of fluid circulation cell is high-est in case 1 and least in case 3. As the flow circulation cells areweaker at low Ra, entropy generation due to flow irreversibilitiesis also smaller for all the cases. Common to all the cases, velocity

gradient between the rotating fluid and solid wall is very high,thus, entropy generation due to fluid friction is significant near

0.9

0.7

0.5

0.3

0.1

θ(a)

1

1

0.1

0.1

0.40.4

Sθ

−10.5−7−3−0.5

−5−9

ψ

11

10

10

10

10

Sψ

Sθ,max=67.46

Sθ,max=67.46

Sψ,max=690.54

Sψ,max=690.54

Sψ=111

Sψ=110

0.9

0.7

0.4

0.2

θ(b)1

1

5

525

15

Sθ

−10

−8−5−3−0.5

ψ10 1

10

10

10

Sψ


Sψ,max=701

Sψ,max=701

Sψ=104

Sψ=107

0.9

0.70.5

0.2

θ(c)

5

25

5

Sθ

−1 −4

−2

ψ

20

20

Sψ

Sψ=99.99

Sψ=98.08

Sψ,max=494Sψ,max=494

Sθ,max=46.91

Sθ,max=46.91

Fig. 7. Isotherms (h), local entropy generation due to heat transfer (Sh), streamlines(w), and local entropy generation due to fluid friction (Sw) for (a) case 1 (b) case 2and (c) case 3 in concave cases for Pr ¼ 1000 and Ra ¼ 105.


the solid walls. On the other hand, at the core of the enclosure, thevelocity gradient between the rotating fluid layers is quite less.Therefore, the entropy generation due to fluid friction is negligiblenear the central regime compared to that near solid walls for all

the cases. It may be noted that, Sw;max is highest for less concavecase (case 1) compared to cases 2 and 3. This is due to larger avail-able area for fluid circulation in case 1 which in turn results incomparatively high magnitude of streamfunction which furtherleads to larger velocity gradient at the solid walls. In case 3, dueto highly curved side walls, the available area for fluid flow is veryless and jwjmax is least due to very less velocity gradient. Note that,Sw;max ¼ 0:69; 0:6 and 0.09 for cases 1, 2 and 3, respectively. Due toless velocity gradient at the core, Sw is very less at the core for allthe cases. At low Ra, conductive heat transport takes commandand entropy generation due to heat transfer is significantly largerthan that of entropy generation due to fluid friction throughoutthe cavity in all the cases as seen from Fig. 4(a)–(c).

As Ra increases to 104, slight distortion in the trend of isothermsmay be observed in the core of the cavity for cases 1 and 2 due toenhanced convective effect (Fig. 5(a)–(c)). Significant amount ofheat is transferred from the bottom portion of the hot left wall tothe top portion of the cold right wall and this can also be explainedbased on the compressed isotherms at those regions in cases 1 and2. As a result, maximum value of entropy generation due to heattransfer irreversibility (Sh;max) occurs near the bottom portion ofthe left wall and top portion of the right wall for cases 1 and 2.In contrast, isotherms are compressed at the throat region due tohighly concave walls in case 3. The heat transfer rate from thehot left wall to the cold right wall is significant near the middleportion of the wall in case 3. The compression of isotherms resultsin larger thermal gradient near the middle portion of the sidewalls. Consequently, Sh;max is observed near the middle portion ofthe side walls in case 3. It is interesting to observe that the signif-icant values of Sh are concentrated near the lower portion of leftwall and upper portion of right wall for cases 1 and 2, whereas lar-ger Sh values are observed near the throat region due to higherthermal gradient for case 3. At the interior zone, Sh for case 3(Sh � 10) is larger compared to that of cases 1 (Sh � 1) and 2(Sh � 2) due to high thermal gradient in case 3. It is also found thatentropy generation due to heat transfer is negligible in all the cor-ner regions of the enclosure in case 3. Eventhough the qualitativenature of isotherms and entropy generation maps are similar tothose for low Ra case, the magnitude of Sh increases with Ra inall the cases. However, the increase in Sh with Ra for case 3 is notsignificant unlike cases 1 and 2. Note that, Sh ¼ 12:1, 11.6 and31.79 occur for cases 1, 2 and 3, respectively.

At Ra ¼ 104, due to enhanced buoyancy effect, comparativelystronger intensity of fluid circulation cells occur in all the cases(Fig. 5(a)–(c)). As a result, magnitudes of streamlines are foundto be comparatively higher than those of lower Ra case for cavitieswith concave and convex surfaces. Due to high convective effect,the streamline cells are elongated in the diagonal direction in allthe cases and the effect is more prominent in case 2. Also, tiny sec-ondary fluid circulation cells with very less magnitude form nearthe corner regions of the enclosure especially for cases 1 and 2.At high wall curvature, the streamlines cells are segregated andtwo circulation cells are seen at the top and bottom halves of thecavity for case 3. Unlike cases 1 and 2, secondary streamline cellsare suppressed and not prominent at the corner regions in case3. Due to intense fluid circulation cells, large velocity gradientstend to develop near the solid walls and larger entropy generationdue to fluid friction is observed at the solid walls for all the cases.Maximum Sw is observed at the middle portions of side walls withSw;max ¼ 52:09 and 14.08 for cases 1 and 2, respectively. In contrast,significant fluid friction irreversibility is observed at top and bot-tom portions of side walls as well as middle portions of horizontalwalls for case 3. Due to the high velocity gradient, Sw;max

(Sw;max ¼ 13:96) is observed at the upper portion of left wall andlower portion of right wall in case 3. Also, Sw is significant at themiddle portion of top and bottom walls in case 3 with Sw ¼ 5:14.


As the velocity gradient is less at the top and bottom walls, Sw atthose regions are found to be negligible for cases 1 and 2. A circularpattern in distribution of Sw is observed at the interior region ofcases 1 and 2 with Sw � 1 and 0.1 for cases 1 and 2, respectively.In contrast, two circular patterns in Sw distribution (Sw � 0:5) areobserved at top and bottom halves of the cavity in case 3. Overallcomparison of Sw and Sh depicts that, Sw is larger compared to thatof Sh due to comparatively larger convective effect for cases 1 and2. On the other hand, conductive heat transfer is still dominant andSh is larger than Sw throughout the enclosure for case 3.

As Ra further increases to 105, several interesting features areobserved in the trends of isotherm (h) and entropy generation(Sh) maps as seen in Fig. 6(a)–(c). Isotherms at the core are highlydistorted and compressed along bottom portion of the left wall andtop portion of the right wall especially in cases 1 and 2, signifyingdominance of convection at high Ra. The thicknesses of thermalboundary layer at the bottom portion of left wall and top portionof right wall are greatly reduced compared to that of low Ra cases.As wall curvature increases to case 3, isotherms are observed to becompressed towards the entire left and right walls, except the topleft and bottom right corners. Also, due to highly curved side wallsin case 3, compression of isotherms along the left and right wallsare more pronounced compared to that of cases 1 and 2. As a con-sequence, larger thermal gradient in case 3 results in higherentropy generation due to heat transfer in case 3. Maximumentropy generation due to heat transfer (Sh;max) is almost identicalbetween cases 1 and 2, as there is no significant variation in ther-mal gradients. Note that, Sh;max ¼ 48:42 and 48.37 occur for cases 1and 2, respectively. Location of Sh;max is observed to be near thelower portion of left wall and upper portion of right wall for case1, whereas for case 2, Sh;max occurs almost at the middle portionsof side walls. In contrast, Sh;max is observed at the top portion ofthe left wall and bottom portion of the right walls withSh;max ¼ 60:23 in case 3. Due to significant temperature uniformityin case 1, thermal gradient is less and thus, Sh (Sh � 0:5) is less atthe interior region. Heat transfer irreversibility at the core isslightly larger in case 2 than that in case 1 due to comparativelyhigher temperature gradient in case 2 (Sh � 1 for case 2). On theother hand, Sh is significantly larger at the interior region for case3 with 5 6 Sh 6 30 due to very high thermal gradient at the throatregion. As a result of larger boundary layer thickness, Sh is negligi-ble at the corner regions in all the cases and this effect is moreprominent in case 3.

At Ra ¼ 105, enhanced buoyancy force results in stronger fluidcirculation cells as seen from the magnitude of streamfunctionsin all the cases (see Fig. 6(a)–(c)). Also, prominent multiple fluidcirculation cells are observed near the corner regions of the enclo-sure especially in cases 1 and 2. Due to larger magnitude of stream-lines, larger velocity gradient exists near the side walls comparedto that of low Ra cases. Consequently, the magnitudes of entropygeneration due to fluid friction (Sw) are comparatively higher inall the cases compared to low Ra cases. Fluid friction is significantat the middle portions of the side walls for cases 1 and 2. Thus,Sw;max for cases 1 and 2 are observed at the middle portions ofcurved side walls. The complex fluid flow pattern in case 3 resultsin higher velocity gradient near the top and bottom portions of theside walls in contrast to cases 1 and 2. Consequently, Sw;max for case3 is observed near the top portion of left wall and bottom portionof right wall. Comparative study of all the cases shows that due tohigher velocity gradient at the side walls in case 2, Sw;max is highestin case 2 followed by case 1 and case 3. It may be noted that,Sw;max ¼ 1340, 1396 and 822 for cases 1, 2 and 3, respectively.Active zones of Sw also occur at the middle portions of the horizon-tal walls in case 1 (Sw ¼ 19:3), case 2 (Sw ¼ 26:4) and case 3(Sw ¼ 257) due to large velocity gradients in those regions. It is alsointeresting to note that, fluid flow irreversibility is negligible at the

throat portion of the side walls in case 3, which is in contrast tocases 1 and 2. Circular patterns in the Sw contours are observedat the core for cases 1 and 2 with Sw � 4 and 10 for cases 1 and2, respectively. Due to effect of segregated streamline cells at thetop and bottom halves, two circular patterns of Sw are observedat the top and bottom halves of the enclosure in case 3 withSw � 20. Note that, velocity gradient is larger at the solid wallsbased on comparatively stronger streamline cells and that resultsin larger Sw;max for cases 1 and 2 than that for case 3. On the otherhand, due to constriction at the throat region, the friction betweenthe adjacent fluid layers are larger for cases 3. Thus, local entropygeneration due to fluid friction at interior zone is larger for case 3compared to that of cases 1 and 2.

At high Pr (Pr ¼ 1000) and Ra ¼ 105, isotherms are stronglycompressed towards major portions of left and right walls asobserved in Fig. 7(a)–(c). This results in larger heat transfer rateas well as larger entropy generation due to heat transfer irrevers-ibility at those regions. The magnitude of Sh;max is observed to becomparatively larger than the previous case with Pr ¼ 0:015 dueto larger heat transfer irreversibility based on higher temperaturegradient at high Pr in all the cases (see Figs. 6 and 7). Entropy gen-eration due to heat transfer is significant at the lower half of leftwall and upper half of right wall as seen from dense Sh contoursin those regions for all the cases. In contrast, due to very high wallcurvature in case 3, isotherms are compressed at the throat and asa result, entropy generation due to thermal gradient is significantat the throat region. Due to highly compressed isotherms at thebottom portion of left wall and top portion of right wall based onintense convection, temperature gradient is large and that resultsin largest Sh;max in case 1 (Sh;max ¼ 67:46; 62:58 and 46.91 for cases1, 2 and 3, respectively). Thermal mixing is larger at the core of thecavity mainly for cases 1 and 2 due to high convective effects. Thus,isotherms are highly distorted at the core for cases 1 and 2. Dis-torted isotherms at the core lead to very less thermal gradientsand that results in insignificant Sh in cases 1 and 2. In contrast,due to larger temperature gradient at the core, heat transfer irre-versibility is significant at the core with Sh � 25 for case 3.

At Pr ¼ 1000 and Ra ¼ 105, magnitude of streamfunction isfound to be significantly larger and streamline cells take shape ofenclosure in all the cases (Fig. 7). Comparative studies on Figs. 6and 7 show that as Pr increases from 0.015 to 1000, secondarystreamline cells are not found and a sharp increase in the magni-tude of streamfunction is observed at high Pr for all the cases ofwall curvatures. Due to higher momentum diffusivity at high Pr,fluid flow circulation cells take the shape of the enclosure in cases1 and 2. On the other hand, fluid flow circulation cells adjacent tothe boundary walls take the shape of the enclosure and cells nearthe core are found to be segregated due to high wall curvature incase 3. At high Pr, the bulk fluid circulates with very high velocityand reaches to the corner regions of the enclosure in all the casesdue to high momentum diffusivity. Entropy generation is signifi-cant along the left and right walls and Sw;max is found to occur atthe middle portions of side walls in all the cases. Eventhough theintensity of fluid flow is larger at the core for high Pr, velocity gra-dients are lesser for Pr ¼ 1000, especially at the walls. Thus, Sw;max

is lesser for Pr ¼ 1000 with Sw;max ¼ 690;701 and 494 occurring forcases 1, 2 and 3, respectively. Significant amount of entropy gener-ation is also observed at the top and bottom walls for all the casesand Sw is larger in case 1. Note that, Sw ¼ 109, 104 and 98.08 occurat the middle portion of top and bottom walls for cases 1, 2 and 3,respectively. Almost entire left and right walls act as active zonesof Sw at high Pr in all the cases. At the core, Sw contours strongly fol-low the shape of the enclosure in cases 1 and 2 with Sw 6 10, whichis in contrast to low Pr cases, where Sw contours at the core werecircular. Also, in contrast to Pr ¼ 0:015 case, multiple segregatedzones with Sw � 20 are observed at the interior region for case 3

0.9

0.7

0.5 0.3

0.1

θ(a)

0.5

11

Sθ

Sθ,max=11.6


(see Figs. 6(c) and 7(c)). At high Pr; Sw active zones are observed atthe corner regions for all the cases. Comparative study of heattransfer and fluid friction irreversibilities illustrates that, at bothhigh Ra and Pr; Sh, due to heat transfer irreversibility is lesser asconvection is dominant and Sw is larger than Sh.

10.5

−0.05

−0.4

−0.8−1.2

ψ

0.01

0.01

0.05

0.05

Sψ

Sψ=0.13

Sψ,max=0.58

Sψ,max=0.58

Sθ,max=11.6

0.9 0.7

0.5

0.3 0.1

θ(b)

0.5

1.5

0.51

Sθ

−0.09

−0.5

−1−1.28

ψ

0.01

0.01

0.05

Sψ

Sθ,max=23.51

Sψ,max=0.58

Sθ,max=23.51Sψ,max=0.580.9

0.7 0.5

0.3 0.1

θ(c)

0.50.5 0.

1

0.5 1

1

Sθ

−0.01

−0.3 −0.8

−1.25

ψ

0.010.01

0.01

Sψ

Sθ,max=60.95

Sψ,max=0.61

Sψ,max=0.61

Sθ,max=60.95

Fig. 8. Isotherms (h), local entropy generation due to heat transfer (Sh), streamlines(w), and local entropy generation due to fluid friction (Sw) for (a) case 1 (b) case 2and (c) case 3 in convex cases for Pr ¼ 0:015 and Ra ¼ 103.

4.3. Isotherms, streamlines and entropy generation maps in theenclosure with convex side walls

The enclosure with convex side walls is considered with threecases based on three different curvatures of side walls. Similar tothe concave case, the square enclosure is modified to a curvedwalled enclosure by shifting the point P1 and P2 to P01 and P02,respectively in the outward direction such that AP01D and BP02Cform curves which obey the quadratic equations;X ¼ aY2 þ bY þ c and X ¼ a0Y2 þ b0Y þ c0, respectively (seeFig. 1(b) and Table 1). The values of P1P01 or P2P02 are same as inthe concave and they are L=10; 2L=10 and 4L=10 for cases 1, 2and 3, respectively.

At Ra ¼ 103 for Pr ¼ 0:015, isotherms are smooth curves illustrat-ing dominance of conductive heat transfer for all the convex cases(see Fig. 8(a)–(c)). It is observed that, the qualitative trends of iso-therms are almost similar in all convex cases. Isotherms are slightlycompressed at the top portion of right wall and bottom portion of leftwall in all the cases signifying comparatively larger heat transfer rateat those regions. Hence, zones for maximum entropy generation(Sh;max) are found near the top portion of right wall and bottom por-tion of left wall of the enclosure, which is in contrast to the concavecases where maximum entropy generation due to heat transfer wasobserved near the middle portion of the side walls (see Figs. 4 and 8).Due to larger convective effect for case 3, thermal gradients are lar-ger for case 3. Thus, Sh;max is larger for case 3. As wall convexityincreases, qualitative nature of Sh is observed to be similar, but themagnitude of Sh;max increases (Sh;max ¼ 11:6, 23.51 and 60.95 for cases1, 2 and 3, respectively). Comparative study of concave and convexcases shows that, due to larger convective effect in convex cases,fluid flow is more intense and isotherms are largely compressedfor convex cases (see Figs. 4 and 8). Thus, Sh;max is larger for all convexcases compared to concave cases. Due to less thermal gradient neartop and bottom adiabatic walls, Sh is very less at those regions in allthe cases. Due to almost similar patterns of isotherms and identicaltemperature gradients at the interior region, Sh 6 1 is observed in allthe cases. At the core, Sh is lesser for convex cases compared to that ofconcave cases due to lesser temperature gradient, which can also beexplained based on the slightly distorted isotherms for convex cases(Figs. 4 and 8).

Due to similar thermal boundary conditions with the concavecases, fluid near the hot left wall with comparatively less densitymoves upward and relatively heavy fluid near cold right wallmoves downward forming a clockwise fluid flow circulation cell(Fig. 8(a)–(c)). As a consequence of low buoyancy force at low Raand Pr ¼ 0:015, fluid flow is weak and the magnitudes of stream-function are found to be less. Similar to concave cases, it isobserved that the velocity gradient is larger at the solid walls com-pared to that of core in all the cases. In contrast to concave cases,Sw;max is observed at the middle portion of top and bottom wallsin all the cases and that can be explained based on the velocity gra-dient which is found to be very high near the top and bottom walls.Maximum entropy generation due to fluid friction is almost similarin cases 1 and 2 (Sw;max ¼ 0:58) and that is slightly larger for case 3(Sw;max ¼ 0:61). Magnitudes of Sw contours for cases 1–3 are almostsimilar at the interior region with Sw � 0:01 (see Fig. 8(a)–(c)).Comparison of Sh and Sw distribution at low Ra and Pr reveals thatheat transfer irreversibility is dominant in total entropy generationat low Ra.

Fig. 9(a)–(c) display isotherms, streamlines and entropy gener-ation maps due to heat transfer and fluid friction (Sh and Sw) atRa ¼ 105 and Pr ¼ 0:015 for all convex cases. At high Ra, enhancedconvective effect is observed and that results in larger thermalmixing at the core for all the cases. Hence, isotherms are highly dis-torted at the core and compressed towards the large portions ofleft and right walls. As a result, Sh is significant throughout the leftand right walls for all the cases. Larger compression of isothermstowards the top right and bottom left corners results in larger ther-

0.9 0.7

0.5

0.3 0.1

θ(a)

0.50.5

3

205

10

Sθ

−7−6−4−2

−0.5

ψ

0.5

1

1

15 5

Sψ

Sθ,max=50.96

Sθ,max=50.96Sψ,max=526

Sψ,max=526

Sψ=251

0.9

0.8

0.6

0.4

0.2

0.1

θ(b)

0.5

0.5 1

10

105

Sθ

−7−6−5−2

−3

−0.5

ψ

0.5

10.5

1

1

Sψ

Sθ,max=139

Sθ,max=139

Sψ,max=538

Sψ,max=538

Sψ=141

0.9

0.6

0.4

0.1

θ(c)

15 5

5

0.1

Sθ

−7.4−7.4

−6−3

−0.1

ψ

0.1 1

0.5

10

10

Sψ

Sθ,max=441

Sθ,max=441

Sψ,max=484

Sψ,max=484

Sψ=235

Fig. 9. Isotherms (h), local entropy generation due to heat transfer (Sh), streamlines(w), and local entropy generation due to fluid friction (Sw) for (a) case 1 (b) case 2and (c) case 3 in convex cases for Pr ¼ 0:015 and Ra ¼ 105.


mal gradient which further results in maximum Sh at those regionsin all the cases. Also, magnitude of heat transfer irreversibility islarger for higher Ra due to stronger compression of isotherms atthe bottom left and top right corners (see Fig. 9). As wall curvatureincreases from case 1 to case 3, heat transfer irreversibility at thetop right corner and bottom left corner increases. Note that,Sh;max ¼ 50:96, 139 and 441 occur for cases 1, 2 and 3, respectively.It is observed that, due to larger available area for fluid circulationin convex cases, fluid flows with high velocity, and isotherms are

largely compressed resulting in larger Sh;max compared to that ofall concave cases (see Figs. 6 and 9). Heat transfer irreversibilitiesat the core are observed to be almost similar in all the cases withSh � 0:5. Due to lesser temperature gradients at the core for convexcases, Sh at the core is lesser for convex cases compared to that ofconcave cases (see Figs. 6 and 9). Active zones for Sh are observed atthe corner regions of the cavity for all convex cases which are incontrast to concave cases, where Sh was very less at the top rightand bottom left corners.

At Ra ¼ 105, the fluid flow intensity is high and that results inenhanced convective transport inside the cavity for all cases (seeFig. 9). Tiny secondary fluid circulation cells tend to form nearthe top right portion and bottom left corner of the cavity for allthe cases. As the fluid flow is more intense compared to low Ra, lar-ger velocity gradients are observed near the solid walls and thatresults in higher entropy generation due to fluid friction. Maxi-mum entropy generation due to fluid friction is largest in case 2(Sw;max ¼ 538) compared to that of case 1 (Sw;max ¼ 526) and case3 (Sw;max ¼ 484), which occurs near the left portion of top walland right portions of bottom wall in all the cases, which is in con-trast to concave cases. Velocity gradient between the fluid layers atthe core decreases with increase in wall curvature from case 1 tocase 3, hence Sw at the core is least for case 3. Note that, at the core,Sw � 0:5 for cases 1 and 2, whereas Sw � 0:1 is seen for case 3. Atthe core, Sw is smaller in convex cases than that in concave cases,which can be clearly inferred from the magnitudes of Sw. Due tosignificant convective effect, the heat transfer irreversibility (Sh)is lesser than that of fluid friction irreversibility (Sw) in all the cases.

As Pr increases to 1000, the isotherms are strongly compressedtowards almost entire left and right walls for all the cases due tovery high momentum diffusivity (see Fig. 10). As a result, both leftand right walls act as active regions for entropy generation due toheat transfer in all the cases. Similar to lower Pr cases, maximummagnitudes of Sh are found at the top right corner and bottom leftcorner of the enclosure for all the cases due to high temperaturegradient based on densely clustered isotherms for all the cases.Due to larger convective effect in case 3, isotherms are largely com-pressed towards the bottom left and top right corner and thatresults in highest value of Sh;max in case 3, compared to cases 1and 2. Magnitude of Sh;max is significantly larger at high Pr com-pared to that of low Pr cases. It may be noted that, Sh;max ¼ 279,673 and 1628 occur for cases 1, 2 and 3, respectively. Thermal mix-ing is found to be enhanced at the core for all the cases and thusisotherms are highly distorted. Hence, Sh is negligible at the coreand at the horizontal walls based on very less temperature gradi-ents in all the cases. Due to significant temperature uniformity atthe core in all the cases, isotherms are largely distorted and tem-perature gradient is negligible at the core. Thus, very less entropygeneration due to heat transfer is observed at the core of the cavitywith Sh � 1 for all the cases. It is interesting to note that, atRa ¼ 105 and Pr ¼ 1000, Sh;max is significantly larger in the caseswith convex surfaces than those of corresponding cases with con-cave surfaces due to larger convective effect for convex surfaceswhich lead to very high thermal gradient near the walls (see Figs. 7and 10). On the other hand, Sh is larger at the interior region of thecavity in the concave cases compared to that in convex cases (seeFigs. 7 and 10).

At higher Ra and Pr, single flow circulation cells span the entireenclosure and attain the shape of the cavity based on enhancedconvection due to higher momentum diffusivity over thermal dif-fusivity (see Fig. 10). This is in contrast to lower Pr case where sec-ondary flow circulation cells tend to form near the top right cornerand bottom left corner of the enclosure. Active zones of frictionalirreversibilities are found near all four walls due to the effect ofstrong fluid flow cells resulting in large velocity gradients. In con-trast to low Pr, maximum entropy generation due to fluid friction is

0.9 0.7

0.4

0.2

θ(a)1

0.5

10 10

Sθ

−0.09−2−6−8

−11−12

ψ

115

5 10

Sψ

Sθ,max=279

Sθ,max=279

Sψ,max=646

Sψ,max=646

Sψ=125

0.90.7

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0.30.1

θ(b)

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−1−5−9−12

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ψ

1105

Sψ

Sθ,max=673

Sθ,max=673

Sψ=132

Sψ,max=609

Sψ,max=609

0.90.6

0.30.1

θ(c)

0.1

1

1

0.1

Sθ

−14

−10−6

−2−0.1ψ

1

1

5

5

Sψ

Sθ,max=1628

Sθ,max=1628

Sψ,max=542

Sψ,max=542

Fig. 10. Isotherms (h), local entropy generation due to heat transfer (Sh), stream-lines (w), and local entropy generation due to fluid friction (Sw) for (a) case 1 (b) case2 and (c) case 3 in convex cases for Pr ¼ 1000 and Ra ¼ 105.


observed at the upper portion of right wall and bottom portion ofleft wall for all the cases (see Figs. 7 and 10). This is due to the hor-izontal elongation to fluid circulation cells in all the cases at high Prand that further results in compression of streamline cells towardsthe left and right walls. Thus, frictional irreversibility is quite highat the left and right walls. Although magnitudes of streamlines aresmaller in case 1, Sw;max is found to be larger in case 1 (Sw;max ¼ 646)compared to case 2 (Sw;max ¼ 609) and case 3 (Sw;max ¼ 542). This isdue to the less area available fluid circulation in case 1, whichresults in larger velocity gradient at the side walls (seeFig. 10(a)). It may be noted that, due to comparatively lesser veloc-

ity gradients, Sw;max is lesser for cases 1 and 2 of convex cases thanthat of corresponding concave cases (see Figs. 7(a)–(b) and 10(a)–(b)). On the other hand, due to high convective effect for case 3 ofconvex cavity, velocity gradients and eventually Sw are larger forcase 3 (convex) than those for case 3 (concave) (see Figs. 7(c)and 10(c)). Due to less velocity gradients at the core, fluid frictionirreversibility is less and that is found to be almost identical for allthe cases with Sw � 1. It may be noted that, Sw is smaller at the coreof the cavity for all convex cases compared that of the concavecases (see Figs. 7 and 10). This is due to larger convective effectat the core of convex cases that leads to lesser velocity gradientsat the core.

4.4. Average Nusselt number, total entropy generation and averageBejan number

The variations of total entropy generation due to heat transferand fluid friction irreversibilities ðStotalÞ, average Bejan numberðBeavÞ and average Nusselt number for the right wall ðNurÞ vs log-arithmic Rayleigh number (Ra) for concave cases are presented inbottom, middle and top panels, respectively of Fig. 11(a) and (b)for various Prandtl numbers ðPr ¼ 0:015 and 1000).

Fig. 11(a) represents distributions of Stotal, Beav and Nur forPr ¼ 0:015 for all concave cases. As Ra increases from 103 to105; Stotal increases for all the cases and largest value of Stotal isobserved for high Ra. At low Ra, conductive transport is dominantand fluid friction is lesser for all the cases. It was found that, Sh;total

remains significant over Sw;total for 1036 Ra 6 104. Intensity of fluid

flow is less leading to very less velocity gradients and Sw;total is neg-ligible at low Ra. Thus, total entropy generation in the cavityremains almost constant until Ra 6 104 for all the cases (see bot-tom panel of Fig. 11(a)). Note that, Stotal for case 3 is found to be lar-ger compared to cases 1 and 2 although Sw;total is least in case 3 for1036 Ra 6 104. This is due to the fact that, the heat transfer irre-

versibility is significantly larger for case 3 due to the presence ofcompressed isotherms at the throat (see Figs. 3 and 4). Note that,at Ra ¼ 103; Sh;total ¼ 1:22 and Sw;total ¼ 0:02 for case 1; andSh;total ¼ 1:45 and Sw;total ¼ 0:01 for case 2 and Sh;total ¼ 3:02 andSw;total ¼ 0:0027 for case 3. As Ra increases to 104; Sh;total ¼ 1:98and Sw;total ¼ 0:74 for case 1; and Sh;total ¼ 1:82 and Sw;total ¼ 0:45for case 2 and Sh;total ¼ 3:11 and Sw;total ¼ 0:20 for case 3. Convectionheat transfer is observed to be initiated at Ra ¼ 8� 103 for case 1and case 2, and at Ra ¼ 2� 104 for case 3. Due to gradual increaseof both fluid friction and heat transfer irreversibilities at higherRa; Stotal increases rapidly for Ra P 104. It may be noted that Stotal

for cases 2 and 3 attain almost similar value at Ra ¼ 105, whereasStotal or case 1 attains the highest value. This can be explained basedon the individual values of Sh;total and Sw;total for case 1 at high Ra. Itis observed that, at high Ra, intensity of convection is significantlylarger in case 1 (jwjmax ¼ 8:47) compared to cases 2 (jwjmax ¼ 7:29)and 3 (jwjmax ¼ 4:31) leading to larger velocity gradients in case 1.Thus, Sw;total is significantly larger for case 1 that results in overallincrease in Stotal. At Ra ¼ 105, it is found that, Sh;total ¼ 3:43 andSw;total ¼ 11:90 occur for case 1; Sh;total ¼ 2:92 and Sw;total ¼ 7:74occur for case 2 and Sh;total ¼ 4:31 and Sw;total ¼ 6:11 occur for case 3.

Distribution of average Bejan number (Beav) indicates the dom-inance of entropy generation due to heat transfer or fluid frictionirreversibilities during natural convection. As mentioned in Sec-tion 3.2.2, Beav P 0:5 indicates entropy generation due to heattransfer and Beav 6 0:5 indicates entropy generation due to fluidfriction. A generalized decreasing trend in Beav with Ra may beobserved for all the cases (see the middle panel plots ofFig. 11(a)). The maximum value for Beav (Beav ¼ 1) is observed atlow Ra, which signifies that the entropy generation in the cavityis mainly due to heat transfer irreversibility. This can also beexplained based on the magnitudes of Sh;total and Sw;total at low Ra

103 104 105

Rayleigh number

0

10

20

Sto

tal

(a) Pr = 0.015

0

0.5

1

Be a

v2

4

6

Nu r

103 104 105

Rayleigh number

0

10

20

Sto

tal

Pr = 1000 (b)

0

0.5

1

Be a

v

2

4

6

Nu r

Fig. 11. Variations of total entropy generation (Stotal: bottom panel), average Bejan number (Beav : middle panel), and average Nusselt number at the right wall (Nur: top panel)with Ra for the concave cases (case 1 (-----), case 2 (––––) and case 3 ( )) at Pr ¼ 0:015 and Pr ¼ 1000.


signifying dominance of Sh;total. As Ra increases, Beav decreases forall the cases depicting dominance of fluid flow irreversibility overheat transfer irreversibility. Due to larger convective effect, veloc-ity gradient is larger at high Ra and Sw;total dominates over Sh;total asseen from the value of Beav (Beav ¼ 0:22 and 0.27 at for cases 1 and2, respectively) at Ra ¼ 105 (see middle panel plots of Fig. 11(a)).Due to highly compressed isotherms and lesser velocity gradientsfor case 3, it is evident that Sh;total still dominates over Sw;total andSh;total ¼ 4:02 and Sw;total ¼ 3:88 occur even at Ra ¼ 7� 104 for case3. At Ra ¼ 105, heat transfer rate is found to be increased in addi-tion to fluid friction irreversibility for case 3. Thus, Sw;total

(Sw;total ¼ 6:11) is found to be slightly larger than Sh;total

(Sh;total ¼ 4:31) and Beav ¼ 0:41 occurs at Ra ¼ 105 for case 3. Notethat, for all Ra; Sh;total is larger for case 3 than that for cases 1 and2 due to larger thermal gradients based on compressed isothermsat the neck region of case 3. In addition, due to less area availablefor fluid flow, magnitudes of streamline are smaller and Sw;total isless for case 3 compared to cases 1 and 2 for all Ra. As a result,Beav is observed to be larger for case 3 throughout the range of Ra.

Available energy for heating process decreases because entropygeneration increases as some part of the available energy is utilizedto remove irreversibilities. Total entropy generation in the cavity(Stotal) is maintained almost constant with Ra for 103

6 Ra 6 104

as Sw;total is negligible over Sh;total for all cases. In addition, Sh;total isalmost constant for 103

6 Ra 6 104 due to conduction dominance.Thus, the heat transfer rate in terms of average Nusselt number forright wall (Nur) is also maintained constant with Ra for1036 Ra 6 104. At 103

6 Ra 6 104, heat transfer is mainly due totemperature difference (conduction dominant) and very lessamount of energy is used to overcome the fluid friction irreversibil-ities (Sw;total) for all the cases. Further, as Ra increases, the totalentropy generation increases and this can also be explained basedon significant values of both Sw;total and Sh;total for Ra P 104. Due to

larger convective effect, the isotherms are compressed signifyinglarger thermal gradients across the cavity at high Ra. It may benoted that, the heat transfer due to temperature gradient is muchlarger than the losses due to frictional irreversibility. Consequently,the heat transfer rate (Nur) increases rapidly for Ra P 104 and thatattains largest value at Ra ¼ 105 for all the cases (see top panel ofFig. 11(a)). Available area for fluid circulation is larger that resultsin larger convective effects for case 1 for Ra P 104. Thus, highvelocity gradients result in significant values of Sw;total for case 1especially in the range of Ra P 104. Eventhough Sw;total is larger,Nur is also larger for case 1 compared to that of cases 2 and 3 forRa P 104. This is due the fact that, the heat transfer due to temper-ature gradients is sufficiently larger to overcome the fluid frictionirreversibility for case 1. At low Ra, both Nur and Stotal are foundto be largest for case 3 than that of cases 1 and 2 while, at highRa, case 3 exhibits larger heat transfer rate at a lowest Stotal. Even-though the irreversibility is high, the amount of heat transport islarge for case 3 at low Ra. Thus, case 3 may be optimal based onhigher heat transfer rate with lesser entropy generation comparedto cases 1 and 2.

As Pr increases to 1000, the distributions of Stotal are qualita-tively similar with Pr ¼ 0:015 at low Ra (103

6 Ra 6 104) for allthe cases (see lower panel of Fig. 11(a) and (b)). Due to dominantconductive effect at low Ra; Sh;total dominates over Sw;total for all thecases. Note that, at Ra ¼ 103; Sh;total ¼ 1:23 and Sw;total ¼ 0:02 occurfor case 1; Sh;total ¼ 1:46 and Sw;total ¼ 0:01 occur for case 2 andSh;total ¼ 3:02 and Sw;total ¼ 0:002 occur for case 3. It may also benoted that, at Ra ¼ 104; Sh;total ¼ 2:28 and Sw;total ¼ 0:99 occur forcase 1; Sh;total ¼ 2:28 and Sw;total ¼ 0:91 occur for case 2 andSh;total ¼ 3:12 and Sw;total ¼ 0:26 occur for case 3. As Sh;total is signifi-cant for case 3 due to high conductive heat transport based onlarge temperature gradient, Stotal is larger for case 3 withRa ¼ 103 � 104. At higher Ra (Ra P 104), rapid increase in Stotal is


observed for all the cases. Note that, Stotal for cases 1 and 2 are lar-ger than that of case 3 for Ra P 104. At high Ra, total entropy gen-eration due to heat transfer is almost similar in all the cases withSh;total ¼ 4:64, 4.65 and 4.71 for cases 1, 2 and 3, respectively. Onthe other hand, due to large convective effect for cases 1 and 2,Sw;total is significantly high for cases 1 and 2. It may be noted that,Sw;total ¼ 19:43, 18.59 and 13.63 for cases 1, 2 and 3, respectivelyoccur at Pr ¼ 1000 and Ra ¼ 105. As a result of larger Sw;total in cases1 and 2, compared to that of case 3, total entropy generation (Stotal)is larger in cases 1 and 2 at high Ra and Pr. It may also be notedthat, due to higher momentum diffusivity, Sh;total and Sw;total are lar-ger for high Pr compared to that for low Pr in all the cases.

The distribution of Beav at Pr ¼ 1000 is observed to be qualita-tively similar to that of Pr ¼ 0:015 case for Ra 6 104 (middle panelof Fig. 11(a) and (b)). As Ra increases further (Ra P 104), steepdecrease in Beav is observed at high Pr for all the cases. This isdue to the rapid increase in Sw;total with Ra at Pr ¼ 1000 due to highmomentum diffusivity. Cases 1 and 2 exhibit almost identical trendof Beav throughout the range of Ra due to almost similar magnitudeof Sh;total and Sw;total for all Ra. Due to high conductive heat transferbased on high temperature gradients, Sh;total is larger for case 3 thanthat of cases 1 and 2 with all Ra. Also, due to constriction at theneck region, magnitude of flow field is smaller and Sw;total is smallerfor case 3. Thus, Beav is larger for case 3 throughout the range of Ra.

Due to high momentum diffusivity at high Pr, rate of increase ofStotal as well as Nur is found to be larger than that of low Pr for all Ra(see Fig. 11(b)). As Ra increases from 103 to 104;Nur for cases 1 and2 increase very slowly whereas that of case 3 remains constant. Atlow Ra (103 � 104), heat transfer is mainly due to conduction for allthe cases although the conductive transport is more prominent forcase 3. Due to presence of dense isotherms at the throat region forcase 3, Nur is larger while Sh;total is also significantly larger com-pared to cases 1 and 2. Fig. 11(b) shows that, Nur as well as Stotal

is larger for case 3 than that for cases 1 and 2 for Ra ¼ 103 � 104.At high Ra (Ra P 104), fluid flow irreversibilities are larger andtotal entropy generation (Sh;total and Sw;total) increases rapidly withRa due to significant Sw;total. Overall, at high Ra, rate of heat trans-port due to temperature gradient is found to be even larger thanlost energy due to irreversibilities for all the cases. Consequently,Nur increases exponentially for Ra P 104 in all the cases. Due tosignificant convective effect for cases 1 and 2, largely compressedisotherms are seen near the right wall illustrating larger tempera-ture gradient. Thus, the heat transport rate is larger for cases 1 and2 for Ra P 104 eventhough substantial amount of energy is utilizedto overcome significant fluid friction and heat transfer irreversibil-ities. Hence, rapid increase in the heat transfer rate is observed forcases 1–2 and Nur is observed to be larger for cases 1–2 than thatfor case 3 at Ra P 104. Although the temperature gradients arehigh at the neck, the gradient is less in other portions due to lesserconvective effect in case 3 for Ra P 104. Thus, at high Ra, insignif-icant thermal gradients at the corner regions of the cavity in case 3result in lesser Nur for case 3 compared to cases 1 and 2. Compar-ative studies of cases 1–3 show that at low Ra, case 3 is preferredover cases 1 and 2 due to larger heat transfer rate. On the otherhand, Nur is largest for case 1 with slightly larger value of Stotal athigh Ra. Thus, at high Ra, case 1 is preferred over cases 2 and 3based on high heat transfer rate with reasonable entropy genera-tion rate.

Fig. 12(a) and (b) show the distributions of Stotal; Beav and Nur

for Pr ¼ 0:015 and 1000 for all the cases of convex side walls. AtPr ¼ 0:015, qualitative and quantitative trends of Stotal vs Ra arealmost similar for all the cases (see bottom panel of Fig. 12(a)).At low Ra (Ra 6 104), heat transfer within the cavity is conductiondominant that can be inferred from the smooth and monotonic iso-therms in Fig. 8(a)–(c) which are due to weak flow circulation cellsinside the cavity. Thus, velocity gradients are significantly less

intense compared to temperature gradients for conduction heattransfer. Consequently, Sh;total is significant and Sw;total is negligibleat low Ra. It may be noted that, Sh;total ¼ 1:01, 0.96 and 0.87 occurfor cases 1, 2 and 3, respectively whereas Sw;total ¼ 0:03 occurs forall the cases at Ra ¼ 103. At Ra ¼ 104, it is found that,Sh;total ¼ 1:83 and Sw;total ¼ 0:74 occur for case 1; Sh;total ¼ 1:65 andSw;total ¼ 0:66 occur for case 2; Sh;total ¼ 1:40 and Sw;total ¼ 0:57 occurfor case 3. The total entropy generation in the cavity is mainly dueto the contribution of Sh;total, while contribution of Sw;total is negligi-ble. Thus, the rate of increase in Stotal with Ra is very less forRa 6 104. As seen from the temperature and streamline contoursin Fig. 8(a)–(c), the gradients of velocity and temperature contrib-ute to almost similar magnitudes of Stotal for all convex cases atRa 6 104. As Ra increases further (Ra P 104), enhanced convectiveeffect in the cavity results in significant increase of both Sw;total andSh;total. Thus, a rapid increase in the trend of Stotal is observed forRa P 104 with all convex cases. In contrast to low Ra; Sw;total is sig-nificantly larger than that of Sh;total for Ra P 104 as observed for allthe cases due to larger velocity gradients. At Ra ¼ 105, totalentropy generation is slightly larger for case 3 compared to cases1 and 2, due to larger values of Sh;total and Sw;total for case 3. At lowRa (Ra 6 104), total entropy generation is lesser for convex casescompared to all concave cases (see Figs. 11(a) and 12(a)). Totalentropy generation for all convex cases are found to be almostidentical to that of case 1 of concave case at high Ra (Ra P 104).It may be noted that at Ra ¼ 105, total entropy generation for allconvex cases is larger than those for cases 2 and 3 of concave cases(see Figs. 11(a) and 12(a)).

The qualitative trends of Beav vs Ra are identical for all convexcases at Pr ¼ 0:015 (middle panel of Fig. 12(a)). At low Ra, maxi-mum value for Beav , (Beav ¼ 0:96) is observed for all the cases indi-cating that magnitudes of Sh;total are significantly larger comparedto those of Sw;total. As Ra increases further, Beav decreases for allthe cases depicting dominance of fluid flow irreversibility overheat transfer irreversibility due to larger convective transport. AtRa ¼ 105, velocity gradient is significantly larger and Sw;total domi-nates over Sh;total. Thus, Beav ¼ 0:23 occurs for all convex cases atRa ¼ 105 (see middle panel plots of Fig. 12(a)). As the shape (sidewall curvature) has negligible effect on the characteristics of heatflow and fluid flow as seen from Figs. 8 and 9, the temperatureand velocity gradients lead to similar magnitudes of Sh;total andSw;total for all the cases resulting in identical Beav . In addition, asthe area available for fluid flow is larger for convex cases, magni-tudes of streamlines are larger and Sw;total is larger for all convexcases compared to concave cases. As a result, Beav is observed tobe smaller for convex cases compared to all concave cases through-out the range of Ra (middle panel of Fig. 11(a) and 12(a)).

The distribution of Nur with Ra for all convex cases at Pr ¼ 0:015is displayed in the top panel of Fig. 12(a). As Ra increases from 103

to 104, total entropy generation in the cavity (Stotal) is maintainedalmost constant with Ra as Sw;total is negligible over Sh;total for all con-vex cases. This is due to conduction dominant transport at low Rabased on less fluid velocity leading to negligible fluid flow irrevers-ibility (Sw;total) in the cavity. Thus, for 103

6 Ra 6 104, average heattransfer rate is mainly due to temperature difference (conductiondominant) and thus, amount of available energy used to removethe fluid friction irreversibilities (Sw;total) is less for all the cases.Therefore, the overall heat transfer rate to the right wall (Nur) isfound to be almost invariant with Ra for 103

6 Ra 6 104 in all thecases. As Ra increases further, there is an exponential increase inStotal as seen from bottom panel plots of Fig. 12(a) for all the cases.This can be explained based on enhanced convection in the cavitythat further results in significant values of both Sw;total and Sh;total forRa P 104 for all the cases. At high Ra (Ra ¼ 105), isotherms are lar-gely compressed towards the top half of the right wall due to sig-nificant convective effect. Thus, larger thermal gradients along the

103 104 105

Rayleigh number

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Pr = 0.015 (a)

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6

Nu r

Fig. 12. Variations of total entropy generation (Stotal: bottom panel), average Bejan number (Beav : middle panel), and average Nusselt number at the right wall (Nur: top panel)with Ra for the convex cases (case 1 (-----), case 2 (––––) and case 3 ( )) at Pr ¼ 0:015 and Pr ¼ 1000.


right wall are observed at high Ra for all the cases. Overall, the heattransport due to temperature gradient along the right wall is muchlarger than the amount of energy lost due to fluid friction and heattransfer irreversibilities. The heat transfer rate (Nur) increases rap-idly for Ra P 104 and that attains largest value at Ra ¼ 105 for allthe cases (see top panel of Fig. 12(a)). Case 1 exhibits larger heattransfer rate for all Ra and Sh;total is found to be lowest for case 1.As the area available for fluid flow is larger for case 3, convectiveeffect is also larger and the isotherms are largely compressed forcase 3 resulting in largest Sh;total. Larger amount of available energyis used to overcome the heat transfer irreversibility for case 3 andsimultaneously lowest value of Nur occurs for case 3 with all Ra.Based on high heat transfer rate, case 1 is the optimal choice com-pared to cases 2 and 3 at Pr ¼ 0:015 at all Ra.

At high Pr (Pr ¼ 1000), the qualitative trends of Stotal for all con-vex cases are similar with Pr ¼ 0:015 (see lower panel of Fig. 12(a)and (b)). At low Ra, dominant conductive transport results in largerSh;total over Sw;total for all the cases. Note that, atRa ¼ 103; Sh;total ¼ 1:04, 0.99 and 0.95 occur for cases 1, 2 and 3,respectively whereas, Sw;total ¼ 0:04 occurs for all the cases. It mayalso be noted that, at Ra ¼ 104, Sh;total ¼ 2:30 and Sw;total ¼ 1:19 occurfor case 1; Sh;total ¼ 2:38 and Sw;total ¼ 1:31 occur for case 2 andSh;total ¼ 2:58 and Sw;total ¼ 1:57 occur for case 3. As Sh;total andSw;total are almost similar for all the cases at Ra 6 8� 103; Stotal

exhibits an identical trends for all the cases at Ra 6 8� 103. Duringthe onset of convection at Ra ¼ 104; Stotal is slightly larger for case3 as Sh;total and Sw;total are found to be higher for case 3. Further, rapidincrease in Stotal is observed for Ra P 104 for all the cases. As theconvective effect is significantly larger for case 3 due to larger areaavailable for fluid motion, the heat transfer as well as fluid frictionirreversibility is larger for case 3. At high Ra (Ra ¼ 105),Sh;total ¼ 4:87, 5.11 and 5.73 occur for cases 1, 2 and 3, respectivelywhereas, at Pr ¼ 1000 and Ra ¼ 105; Sw;total ¼ 22:12, 24.13 and

28.36 occur for cases 1, 2 and 3, respectively. Note that, Sw;total issignificantly larger for case 3 compared to case 1 and 2. Hence,for Ra P 104; Stotal is observed to be larger for case 3. It may alsobe noted that, due to higher momentum diffusivity, Sh;total andSw;total are larger for high Pr compared to that for low Pr in all thecases. Comparison of distribution of Stotal for concave and convexcases depicts that the total entropy generation is larger for convexcases than that of concave cases for all Ra at Pr ¼ 1000 (see bottompanel of Figs. 11(b) and 12(b)).

As expected, a generalized decreasing trend of Beav vs Ra isobserved for all convex cases for Pr ¼ 1000. At high Pr, the rateof decrease of Beav is observed to be high for Pr ¼ 1000 in all thecases and the effect is more clearly visible for Ra P 104. This isdue to high momentum diffusivity at high Pr that results in therapid increase in Sw;total with Ra at Pr ¼ 1000. Note that, slightly lar-ger value of Beav is observed for case 1 followed by cases 2 and 3within the entire range of Ra. This is due to the fact that fluid veloc-ity as well as Sw;total is lesser for case 1 compared to cases 2 and 3 forall Ra. Eventhough the distribution of Beav at Pr ¼ 1000 followsqualitatively similar trend to that of Pr ¼ 0:015 for all the cases,magnitude of Beav for Pr ¼ 1000 is lower than that of Pr ¼ 0:015especially at high Ra. Comparison of Beav for concave and convexcases depicts that the value of Beav for convex cases are lesser forall Ra than that of concave cases. This is due to the largely intenseconvective transport for convex cases resulting in larger Sw;total forall convex cases compared to concave cases.

Rate of increase of overall heat transfer rate (Nur) is significantlylarger for all convex cases at Pr ¼ 1000 than that at Pr ¼ 0:015 (seetop panel of Fig. 12(b)). Unlike Pr ¼ 0:015; Nur increases withmoderate rate for all the cases as Ra increases from 103 to 104. Heattransfer is mainly due to conduction for all the cases at low Ra(Ra ¼ 103). As Ra increases from 103 to 104 at Pr ¼ 1000, convec-tive effect starts to dominate and significantly larger value of Nur


is observed at Ra ¼ 104 compared to that of Ra ¼ 103. At high Ra(Ra P 104), fluid flow irreversibilities are significantly larger andtotal entropy generation (Sh;total and Sw;total) increases rapidly withRa. This is due to high convective effects at high Ra resulting in lar-ger velocity as well as temperature gradients for all the cases. Lar-gely compressed isotherms at the top half of the right wall signifythat rate of heat transport due to temperature gradient is much lar-ger than lost energy due to fluid friction and heat transfer irrever-sibilities for all the cases. Consequently, Nur increases rapidly forRa P 104 for all convex cases. Largely compressed isotherms attop portion of right wall are seen due to larger temperature gradi-ent for case 1 and 2. As a result, the heat transport rate is larger forcases 1 and 2 for all Ra although some amount of energy is utilizedto overcome irreversibilities. Note that, larger available area forfluid flow leads to larger velocity gradients and thus, the fluid fric-tion irreversibility is much larger for case 3 compared to that forcases 1 and 2 and the overall heat transfer rate is lesser for case3, as significant amount of available energy is being used to over-

-0.4 -0.2 0 0.2 0.4P1P1′

0

2

4

6

Nu r

Convex Concave

(a) Pr=0.015

-0.4 -0.2 0 0.2 0.4P1P1′

0

0.5

1

Be a

v

-0.4 -0.2 0 0.2 0.4P1P1′

0

5

10

15

20

S tot

al

0.4 0.2 0 -0.2 -0.4P2P2′

-0.4 -0.2P1

0

2

4

6

Nu r

Convex

(b) Pr=

-0.4 -0.2P1

0

0.5

1

Be a

v

-0.4 -0.2P1

0

6

12

18

24

30

S tot

al

0.4 0.2P2

Fig. 13. Variation of total entropy generation (Stotal: bottom panel), average Bejan numbewith P1P01 for convex walls (�0:4 6 P1P01 6 0), straight wall (square enclosure) (P1P01 ¼ 0)for (a) Pr ¼ 0:015, (b) Pr ¼ 0:7 and (c) Pr ¼ 1000.

come fluid friction irreversibility. At low Ra (Ra 6 104), Nur for allconvex cases are almost similar to that of cases 1 and 2 for concavecase. On the other hand, case 3 of concave case exhibits larger heattransfer rate than all convex cases for low Ra due to very high tem-perature gradients for case 3 (concave). At high Ra;Nur for all con-vex cases are larger compared to all concave cases for Pr ¼ 1000.

Comparison of various cases of concavities (0 6 P1P01 6 0:4) andconvexities (�0:4 6 P1P01 6 0) are carried out based on totalentropy generation (Stotal), average Bejan number (Beav) and aver-age heat transfer (Nur) as illustrated in Fig. 13. In addition, perfor-mances of all test cases are compared with a standard squarecavity corresponding to P1P01 ¼ P2P02 ¼ 0. The optimum case withminimum entropy generation and larger heat transfer rate is iden-tified for processing of various fluids (Pr ¼ 0:015, 0.7 and 1000) atvarious Ra (103

6 Ra 6 105).Fig. 13(a) demonstrates total entropy generation rate (Stotal), aver-

age Bejan number (Beav ) and average Nusselt number (Nur) for var-ious convex (�0:4 6 P1P01 6 0) and concave (0 6 P1P01 6 0:4) cavities

0 0.2 0.4P1′

Concave

0.7

0 0.2 0.4P1′

0 0.2 0.4P1′

0 -0.2 -0.4P2′

-0.4 -0.2 0 0.2 0.4P1P1′

0

2

4

6

Nu r

Convex Concave

Pr=1000(c)

-0.4 -0.2 0 0.2 0.4P1P1′

0

0.5

1

Be a

v

-0.4 -0.2 0 0.2 0.4P1P1′

0

5

10

15

20

25

30

35

S tot

al

0.4 0.2 0 -0.2 -0.4P2P2′

r (Beav : middle panel) and average Nusselt number on the right wall (Nur: top panel)and concave walls (0 6 P1P01 6 0:4) at Ra ¼ 103 (� � �), 104 (––––) and 105 ( )


in addition to a square cavity (P1P01 ¼ 0) with Pr ¼ 0:015. As seenfrom the bottom panel of Fig. 13(a), Stotal is sufficiently larger forRa ¼ 105 than those for Ra ¼ 103 and 104. At low Ra (Ra ¼ 103 and104), the total entropy generation increases very slowly with P1P01and attains a maxima for the square cavity (P1P01 ¼ 0). Thus, thesquare cavity with P1P01 ¼ 0 exhibits maximum Stotal compared to

that of convex cavities at low Ra. At Ra ¼ 103; Stotal further increaseswith P1P01 and the largest value is found for highly concave case withP1P01 ¼ 0:4. This is due to the very high temperature gradients in the

cavity for the concave cases (P1P01 > 0) especially at Ra ¼ 103. On the

other hand, at Ra ¼ 104; Stotal remains constant from P1P01 ¼ 0 toP1P01 ¼ 0:3 and further, that increases as P1P01 increases from 0.3 to0.4. It may be noted that, Stotal is dominated by Sh for lower Ra

(Ra ¼ 103 � 104) especially for concave cavities where Sh increasesas P1P01 increases. Thus, the maxima in the distribution of Stotal occurs

at P1P01 ¼ 0:4 for Ra ¼ 103 � 104. On the other hand, Stotal has signif-

icant contribution of Sw for Ra ¼ 105 for all the cases. In addition, dueto larger convective effect, the thermal gradients also increase withRa for all the cases. Thus, as mentioned earlier, larger Stotal is observedfor Ra ¼ 105 for all the cases. However, due to the competitionbetween Sh and Sw, an interesting wavy trend in the Stotal distribution

is observed at Ra ¼ 105. At Ra ¼ 105, a decreasing trend in the Stotal

distribution is seen from P1P01 ¼ �0:4 (highly convex) to P1P01 ¼ 0(square). Thus, a local minima in the distribution is seen for thesquare cavity. Further, P1P01 P 0 corresponds to concave cavitiesand a wavy trend in the distribution of Stotal is seen with P1P01 forthe concave cases. Note that, a local maxima of Stotal is found forP1P01 ¼ 0:1 that corresponds to the case 1 of concave case. Further,Stotal decreases and minimum values of Stotal occur for intermediateconcavity of concave cases (P1P01 ¼ 0:2). Additionally, slight increasein Stotal is seen with increase in wall concavity from case 2(P1P01 ¼ 0:2) to case 3 (P1P01 ¼ 0:4) at Ra ¼ 105. Overall, at low Ra

(103 and 104), the entropy generation is minimized for highly convexcase with P1P01 ¼ �0:4. On the other hand, at high Ra (Ra ¼ 105), theleast value of entropy generation rate is observed for the enclosurewith concave side walls corresponding to P1P01 ¼ 0:2� 0:3.

Middle panel of Fig. 13(a) presents variation of average Bejannumber (Beav) with P1P01 for convex (�0:4 6 P1P01 6 0), concave(0 6 P1P01 6 0:4) and square cavities (P1P01 ¼ 0) with Pr ¼ 0:015. Itmay be noted that, the larger value of Beav is observed forRa ¼ 103 followed by Ra ¼ 104 and 105 for all the enclosures. Atlow Ra (Ra ¼ 103), Beav remains almost constant with concavity/convexity due to almost similar thermal and negligible velocitygradients for all the cases. Also, due to larger heat transfer irrevers-ibility, Beav � 1 is observed for all the cases of concavity, convexityand square at low Ra. At Ra ¼ 104; Beav for convex cases remainsconstant as P1P01 increases from �0.4 (highly convex) to �0.1 (lessconvex). Also, the magnitudes of Beav for all convex cases(P1P01 < 0) are found to be similar to that for the square cavity(P1P01 ¼ 0) for Ra ¼ 104. It may be noted that Beav � 0:7 occurs forall the convex cases as well as the square cavity for Ra ¼ 104. Thisis due to the onset of convection at Ra ¼ 104 for all the convexcases and the square cavity. Thus, due to a larger value of Sw andalmost similar Sh (with Ra ¼ 103), Beav � 0:7 occurs and that is alsolesser than the concave cases for Ra ¼ 104. Further, for the concavecase (P1P01 > 0), as wall concavity increases, the distance betweenhot and cold walls decreases. Thus, the thermal gradients as wellas heat transfer irreversibility increase with P1P01 for P1P01 > 0. Asa result, an increasing trend of Beav with P1P01 is observed forRa ¼ 104. At Ra ¼ 105; Beav remains constant (Beav � 0:25) withP1P01 for all the cases of convexities and square cavity(�0:4 6 P1P01 6 0). Further, a slightly wavy and increasing trend

in the distribution of Beav is seen as P1P01 increases from 0 to 0.4(concave cases). This is due to the fact that, as P1P01 increases, var-ious zones of compressed isotherms are observed in the cavity andthat results in increase in Sh at Ra ¼ 105. On the other hand, as thefluid velocity decreases with the wall concavity, Sw also decreasesfor Ra ¼ 105. Overall, the average Bejan number increases withthe concavity (P1P01 > 0) at Ra ¼ 105. It is interesting to note that,values of Beav are similar for all concave, convex and square cavi-ties Ra ¼ 103. On the other hand, concave case with high wall con-cavity (P1P01 ¼ 0:4) exhibits maximum value of Beav for larger Ra(104 and 105). The larger value of Beav for larger wall concavityindicates larger Sh with lesser Sw. Thus, the enclosure with highwall concavity may be foretasted as a preferred configuration ofhigher Nur as discussed next.

Global heat transfer rates in terms of average Nusselt numbers(Nur) for convex (�0:4 6 P1P01 6 0), concave (0 6 P1P01 6 0:4) andsquare (P1P01 ¼ 0) cavities are presented in the top panel ofFig. 13(a) with Pr ¼ 0:015. As expected, due to larger buoyancyforce at high Ra, the average Nusselt number is larger forRa ¼ 105 compared to that of 103 and 104 for all cavities. Note that,Nur increases very slowly with wall convexity (P1P01 < 0) tillP1P01 ¼ 0 for all Ra. Thus, the larger value of Nur is found for squarecavity with P1P01 ¼ 0 compared to that of convex cavities withP1P01 < 0 for all Ra. Further, as P1P01 increases from 0 to 0.4 (concavecavities), a monotonic increasing trend of Nur similar to Beav isobserved at Ra ¼ 103. Thus, the largest overall heat transfer rateis observed for the cavity with higher concavity (P1P01 ¼ 0:4) atRa ¼ 103. This is due to the fact that, at less Ra, the conductive heattransfer is larger and the effect is more pronounced in the cavitywith very high concave walls. The maxima of Nur corresponds tothe maxima in the distribution of Stotal which is observed for thecavity with highly concave walls at Ra ¼ 103. At high Ra (104 and105), Nur remains almost constant as P1P01 increases fromP1P01 ¼ 0 (square) to 0.1 (less concavity). Further increase in P1P01from 0.1 to 0.3 reduces Nur for both Ra ¼ 104 and 105. As the wallconcavity increases (P1P01 ¼ 0:1 to 0.3), the thermal gradients alsoincrease resulting in enhanced heat transfer irreversibility. As aresult, larger amount of available energy is used to overcome theheat transfer irreversibility. Thus, a decreasing trend in Nur is seenas P1P01 increases from 0.1 to 0.3. Further, increase in wall concavityresults in very high temperature gradients and the heat transferrates due to thermal gradients are sufficiently higher to overcomethe heat transfer irreversibility at P1P01 ¼ 0:4. Consequently,increase in P1P01 from 0.3 to 0.4 results in increase in Nur for bothRa ¼ 104 and 105. It may be concluded that, the heat transfer rateis largest for case 3 (highly concave) of concave case compared toall convex and square cavity especially at low Ra. At high Ra, largerNur may be observed for cases 1 and 3 of the concave cases (veryhigh and very less concavity). Although the total entropy genera-tion is largest for highly concave case, based on high heating effect,the enclosure with highly concave enclosure is favorable for theprocessing of fluids with less Pr.

The bottom panel of Fig. 13(b) illustrates distributions of Stotal

with P1P01 for various convex, concave and square cavities atPr ¼ 0:7. At low Ra (Ra ¼ 103), similar to Pr ¼ 0:015 case, Stotal

increases with P1P01 at a very less rate. In contrast to less Ra; Stotal

decreases as P1P01 increases from �0.4 to 0.4 for Ra ¼ 104 and 105.However, the decreasing rate of Stotal is significantly larger forRa ¼ 105 case. It may be noted that, Stotal decreases with P1P01throughout the range of P1P01 for Ra ¼ 104 at Pr ¼ 0:7 which is in con-trast to Pr ¼ 0:015 case where slightly wavy trend in the Stotal distri-bution was observed (the bottom panels of Fig. 13(a) and (b)). At highRa; Stotal decreases with P1P01 for P1P01 < 0:2 with a significantly largerrate compared to that for Pr ¼ 0:015 (Fig. 13(a)). Also, in contrast tolow Pr case, the highly wavy distribution is not observed for Pr ¼ 0:7case for P1P01 > 0 at high Ra. The magnitude of Stotal is also signifi-


cantly larger for high Ra (Ra ¼ 105) compared to less Ra (Ra ¼ 103

and 104) for all the cases of concavity and convexity. At Ra ¼ 103

and 104, the Stotal is found to be minimum for the cavity with highlyconvex side walls involving P1P01 ¼ �0:4. At Ra ¼ 105; Stotal decreasesrapidly with P1P01 (�0:4 6 P1P01 6 0) for convex cases. This is attrib-uted to the significant convective effect in the cavity with highly con-vex walls which results in highly compressed isotherms as well assignificant fluid velocity. Thus, the thermal and velocity gradientsare very high for the cavity with high convex walls and that furtherdecreases with the decrease in wall convexity. As a result, the com-parative study of the convex and square cavity (P1P01 ¼ 0) shows thatthe minimum value of entropy generation rate is found for thesquare cavity with P1P01 ¼ 0. At the concave zone (0 6 P1P01 6 0:4),steep decreasing trend in Stotal is observed till P1P01 ¼ 0:2. This canalso be explained based on the magnitudes of the fluid friction irre-versibility. As the wall concavity increases (P1P01 ¼ 0:1 to 0.2), thearea available for fluid flow decreases and further, the fluid frictionirreversibility also reduces. Thus, a reduction in the magnitude ofStotal is seen as P1P01 increases from 0.1 to 0.2. Further, Stotal increasesslightly for P1P01 ¼ 0:2� 0:3 and thereafter, Stotal decreases as P1P01increases from 0.3 to 0.4. The increasing trend in Stotal forP1P01 ¼ 0:2� 0:3 can be explained based on the magnitudes of Sw

which increase in this zone of P1P01 (P1P01 ¼ 0:2� 0:3). Further, atvery high concavity (P1P01 ¼ 0:3� 0:4), Sh increases slightly whereasSw decreases significantly. Thus, Stotal decreases with P1P01 for highlyconcave cases (P1P01 ¼ 0:3� 0:4). The cavity with highly convex sidewalls (case 3 of convex case) offers minimum Stotal for lesser range ofRa (Ra ¼ 103) at Pr ¼ 0:7. At Ra ¼ 104 and 105; Stotal is minimum forthe enclosure with highest wall concavity (case 3 of concave case)for Pr ¼ 0:7.

As seen from the average Bejan number plot (middle panel ofFig. 13(b)), Beav increases very slowly as P1P01 increases from�0.4 (highly convex) to 0.4 (highly concave) at Ra ¼ 103. This isdue to the larger fluid friction irreversibility in the convex casescompared to the concave cases at low Ra (Ra ¼ 103). The trendsof Beav for Pr ¼ 0:015 and 0.7 are similar especially at Ra ¼ 103.Further, as expected, lesser values of Beav are observed atRa ¼ 104 compared to Ra ¼ 103 for all cavities similar toPr ¼ 0:015 case. At Ra ¼ 104; Beav increases slowly with P1P01 forP1P01 6 0:2 which is in contrast to Pr ¼ 0:015 where constant Beav

was observed. Note that, for �0:1 6 P1P01 6 0:2,0:63 6 Beav 6 0:71 occurs at Pr ¼ 0:7 and Beav ¼ 0:71 occurs atPr ¼ 0:015 for Ra ¼ 104. It may be noted that, the magnitudes ofSh as well as Sw are almost similar for all the convex cases andthe square cavity at Ra ¼ 104. Thus, the rate of increase in Beav isvery slow for �0:4 6 P1P01 6 0 at Ra ¼ 104. In addition, as the cavitydeviates from a square cavity to the cavity with less concave wall(case 1: P1P1 ¼ 0:1), Beav increases with a very negligible rate. Fur-ther, Beav increases with a larger rate as P1P01 increases from 0.2 to0.4 (concave cavity) for Ra ¼ 104 and Pr ¼ 0:7. The increasing trendof Beav can be explained based on the magnitudes of Sh for highlyconcave cases, which is significantly larger due to high thermalgradients at Ra ¼ 104. Also, the thermal gradients are larger forthe concave cases that results in larger Beav in the concave casescompared to the convex cases. It may be noted that, the magni-tudes of Beav are comparatively lesser for Pr ¼ 0:7 compared tothose for Pr ¼ 0:015 at Ra ¼ 104 for all cavities. This is due to thefact that, based on larger momentum diffusivity at high Pr, the fluidfriction irreversibility is larger and eventually the magnitudes of Sw

are also larger compared to those for Pr ¼ 0:015. A similar effect isobserved for larger Ra (Ra ¼ 105), where the magnitude of Beav isremarkably lesser for Pr ¼ 0:7 compared to that for Pr ¼ 0:015.At high Ra (Ra ¼ 105), significantly lesser magnitude of Beav isobserved for all the cavities due to the dominance of Sw over Sh.It may be noted that, Beav remains almost constant till P1P01 ¼ 0:3and further a slightly increasing trend in the distribution of Beav

is seen for P1P01 > 0:3. Overall, larger Beav is found for highly con-cave case (case 3) for all Ra.

Average Nusselt number is larger for Ra ¼ 105 followed by 104

and 103 for all the cavities with Pr ¼ 0:7 similar to Pr ¼ 0:015 (thetop panel of Fig. 13(b)). At Ra ¼ 103; Nur increases linearly with asteady rate as P1P01 increases from�0.4 (highly convex) to 0.4 (highlyconcave). A similar trend was also seen for Pr ¼ 0:015 case. The tem-perature gradients are lesser for the cavity with highly convex sidewalls and that slowly increases as the convexity decreases(P1P01 ¼ �0:4 to 0.1) for Ra ¼ 103. On the other hand, the thermal gra-dients as well as the heat transfer rates increase as the wall concavityincreases for low Ra. It may be noted that, as the heat transfer andfluid friction irreversibilities are less for low Ra (the bottom panelsof Fig. 13(b)) and very less amount of energy is utilized to overcomethe irreversibilities for all the cases. As the wall convexity decreasesfrom case 3 (P1P01 ¼ �0:4) to case 1 (P1P01 ¼ �0:1), similar to the lowRa case, Nur increases with a very lesser rate at moderate Ra(Ra ¼ 104). Further, Nur remains constant with P1P01 for concavecases till P1P01 ¼ 0:2. In the low Pr case, a peak in the Nur distributionwas seen for the square cavity with P1P01 ¼ 0, which is in contrast tothe present case. It may be noted that, Nur remains almost constantas P1P01 increases from 0 to 0.1 for Ra ¼ 104. Further, Nur decreasesand attains a local minima at P1P01 ¼ 0:3 (moderate concave case)for Ra ¼ 104. Thereafter, due to very high thermal gradients in thecavity with larger concavity, Nur increases as P1P01 increases from0.3 to 0.4 for Ra ¼ 104. It may be noted that, Stotal is minimum for case3 of concave enclosure (P1P01 ¼ 0:4) for Ra ¼ 104 as seen from thebottom panel of Fig. 13(b). Thus, least amount of the available energyis utilized to overcome the irreversibilities and maximum Nur isobserved for the enclosure with highly concave side walls(P1P01 ¼ 0:4) at Ra ¼ 104. At Ra ¼ 105, the heat transfer rate increasesas P1P01 increases from�0.4 (highly convex) to�0.1 (less convexity).This can also be explained based on the decreasing trend of Stotal withP1P01 for P1P01 < 0. As the lost available energy decreases with P1P01,the overall heat transfer rates increase for P1P01 < 0 at Ra ¼ 105.However, the rate of increase in Nur with P1P01 (P1P01 < 0) is lesserfor the present case with Pr ¼ 0:7 compared to that of the previouscase with Pr ¼ 0:015 for Ra ¼ 105. Further, Nur decreases monoton-ically as P1P01 increases from �0.1 (less convex) to 0.4 (highly con-cave). The temperature gradients along the walls are lesser for theconcave cases due to lesser convective effect compared to the convexcases especially at Ra ¼ 105. Also, due to the segregation of thestreamline cells for the highly concave case (P1P01 ¼ 0:4) at high Ra(Ra ¼ 105), the thermal gradients are less and thus, the heat transferrate is minimum. Consequently, the heat transfer rate for concavecases (P1P01 > 0) is found to be lesser than the convex and squarecases especially at higher Ra. It may be concluded that, heat transferrate is largest for the highly concave case (P1P01 ¼ 0:4) at low Ra withmoderate value of Stotal and largest Beav . The larger heat transfer rateis observed for less convex case (P1P01 ¼ �0:1) at higher values of Rawith largest Stotal. As inferred from Fig. 13(c), the variations ofStotal; Beav and Nur with P1P01 are similar for Pr ¼ 0:7 and 1000. Thus,a similar explanation follows and discussions for Pr ¼ 1000 arealluded for the brevity of the manuscript.

In order to achieve efficient heat transfer rate, the natural con-vection heating system has to be optimized based on lesserentropy generation rate and larger heat transfer rate and averageBejan number. Based on largest heating effect with high Beav , thehighly concave case is favorable for all Pr at low Ra. At a moderatevalue of Ra (Ra ¼ 104), the square cavity may be useful to processfluids with all ranges of Pr due to larger heat transfer and moderateentropy generation rates. The minimum entropy generation withlarger average Bejan number and heat transfer rates occur for case3 of concave case (P1P01 ¼ 0:4) at Ra ¼ 105 and Pr ¼ 0:015. Thus, toprocess fluids with less Pr at convection dominant regime (highRa), case 3 (P1P01 ¼ 0:4) may be chosen as the optimized case.


Although the heat transfer rate is slightly lesser for case 3 (highlyconcave: P1P01 ¼ 0:4), based on minimum entropy generation andlarger average Bejan number, case 3 (highly concave) is the bestchoice for high Ra and also at high Pr (Pr ¼ 0:7 and 1000).

5. Conclusion

In the present study, investigation of natural convection viaentropy generation due to heat transfer and fluid friction irreversi-bilities is carried out in differentially heated curved (concave/con-vex) walled enclosures. The flow and temperature distributions areobtained for various process fluids ðPr ¼ 0:015 and 1000) with arange of Rayleigh numbers ð103

6 Ra 6 105Þ and three representa-tive cases of wall curvatures. The dimensionless entropy genera-tion due to heat transfer (Sh) and fluid friction (Sw)irreversibilities are obtained via finite element post processingfor various Ra; Pr and wall curvatures. The elemental basis set isinvoked to calculate the derivative at each node in the domain,which is further evaluated based on the function values of adjacentelements that share the node. Current approach offers accurateestimation of Sh and Sw over a curved domain. Important resultsof this study are summarized as follows:

5.1. Isotherms and streamlines

� Due to conduction dominant heat transfer at low Ra (Ra ¼ 103),isotherms are smooth and monotonic whereas, distorted iso-therms are observed for higher Ra (Ra ¼ 105) especially at thecore due to convection dominant heat transfer. The isothermsare highly distorted for convex cases compared to those of con-cave cases at high Ra depicting high convective effect for convexcases. At high Ra, isotherms are compressed along the top por-tion of the right wall and bottom portion of left wall for all thecases except case 3 of concave case, where compressed iso-therms were seen at the throat region.� Flow circulations are weak at low Ra and magnitude of stream-

function increases with Ra and Pr for all the cases. Streamlinecells are almost circular in shape and multiple circulation cellsare observed at corner regions for lower Pr ðPr ¼ 0:015Þ andhigh Ra for cases 1 and 2 (concave). Due to constriction at theneck region, the streamlines are elongated and segregated forcase 3 (concave). Thus, pair of clockwise streamline cells areobserved for case 3 (concave). Due to dome shaped geometryof convex enclosures, streamlines are elongated in the horizon-tal direction. Separation of streamline cells are observed at thecore of the cavity for high Ra and Pr for all convex cases. Overallmagnitude of streamline cells are found to be larger for convexcases compared to that of concave cases.

5.2. Entropy generation vs heat transfer rate

� The entropy generation due to the heat transfer are higheralmost near the middle portion of the side walls due to hightemperature gradient at those regions irrespective of Ra andPr for all concave cases. Due to larger temperature gradientsat the throat region, heat transfer irreversibility is large for case3 (concave). Thus, as wall concavity increases, Sh increasesthroughout the cavity. On the other hand, significant Sh isobserved near the top right and bottom left corner of the cavityfor all convex cases for all Ra and Pr. In addition, increase in wallconvexity results in increase in Sh.� The fluid friction irreversibility is observed to be larger mainly

near the regions where moving fluid element is in contact withthe solid wall. Due to significant velocity gradients at the mid-dle portion of the side walls, Sw;max occurs at the middle portions

of side walls for cases 1 and 2 (concave). Due to separation offluid flow cells, larger Sw;max is seen near the top half of left walland bottom half of the right wall for case 3 (concave). Due tolarger velocity gradients at the middle portions of the horizontalwalls, Sw;max is observed at those zones for all convex cases.� The entropy generation due to fluid friction and heat transfer is

observed to be significantly influenced by the wall curvature inconcave cases, whereas qualitative trends of Sh and Sw arealmost similar for all convex cases.� Largest heat transfer rate (Nur) with moderate Stotal is observed

for case 3 (concave) at low Ra for all Pr. At high Ra, largest Nur isobserved for cases 1 and 3 for Pr ¼ 0:015 where, Stotal is largestfor case 1 and lowest for case 3. Thus, throughout the range ofRa, case 3 may be chosen over cases 1 and 2 based on highNur and less Stotal for the processing of fluid with low Pr(Pr ¼ 0:015). At Pr ¼ 1000 and low Ra (Ra 6 104), case 3 (con-cave) is energy efficient based on larger heat transfer rate andBejan number with lesser total entropy generation. At high Rawith Pr ¼ 1000;Nur is quite low for case 3, compared to cases1 and 2 with larger Beav and lesser Stotal. Thus, the geometricalshape as in case 3 (highly concave) is energy efficient due to les-ser total entropy generation for all Ra and Pr.� At low Pr (Pr ¼ 0:015), rate of total entropy generation are

almost similar for all the cases and overall heat transfer rateis larger for case 1 (convex) for all Ra. At Pr ¼ 1000, larger heattransfer is observed for case 1 with minimum total entropy gen-eration for all Ra. Thus, based on larger heat transfer rate andless entropy generation, case 1 or case 2 (convex) is efficientfor the thermal processing of fluids with wide range of Pr.

Higher heat transfer rate with lower values of Stotal or higher val-ues of Beav may be maintained for efficient heat transfer process-ing. Based on test studies for concave cases, case 3 is efficient forall Pr and Ra. On the other hand, for convex cases, less convex cases(case 1 or case 2) are better for all Ra and Pr. Comparison of con-cave and convex cases shows that at low Ra, based on significantheating effect, highly concave case (case 3) is efficient comparedto less concave (cases 1 and 2) and all convex cases (cases 1–3)for all Pr. The highly concave cases (case 3) exhibit minimumentropy generation rate and maximum average Bejan number.Thus, highly concave case may also be recommended for the opti-mum case at high Ra and all Pr.

Conflict of interest

None declared.

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Entropy generation based approach on natural convection in enclosures with concave/convex side walls

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