International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

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International Journal of Heat and Mass Transfer

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Analysis of Bejan’s heatlines on visualization of heat flow and thermal mixingin tilted square cavities

Abhishek Kumar Singh a, S. Roy a, Tanmay Basak b,⇑a Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, Indiab Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

a r t i c l e i n f o

Article history:Received 22 June 2011Received in revised form 21 December 2011Available online 22 March 2012

Keywords:Penalty finite element methodNatural convectionTilted square cavityStreamlinesIsothermsHeatlines

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2012.02.002

⇑ Corresponding author. Tel.: +91 44 2257 4173; faE-mail addresses: abhishek.iitm1@gmail.com (A

(S. Roy), tanmay@iitm.ac.in (T. Basak).

a b s t r a c t

This article analyzes the detailed heat transfer phenomena during natural convection within tilted squarecavities with isothermally cooled walls (BC and DA) and hot wall AB is parallel to the insulated wall CD. Apenalty finite element analysis with bi-quadratic elements has been used to investigate the results interms of streamlines, isotherms and heatlines. The present numerical procedure is performed over a widerange of parameters (103

6 Ra 6 105,0.015 6 Pr 6 1000,0� 6 u 6 90�). Secondary circulations cells areobserved near corner regions of cavity for all u’s at Pr = 0.015 with Ra = 105. Two asymmetric flow circu-lation cells are found to occupy the entire cavity for u = 15� at Pr = 0.7 and Pr = 1000 with Ra = 105. Heat-lines indicate that the cavity with inclination angle u = 15� corresponds to large convective heat transferfrom the wall AB to wall DA whereas the heat transfer to wall BC is maximum for u = 75�. Heat transferrates along the walls are obtained in terms of local and average Nusselt numbers and they are explainedbased on gradients of heatfunctions. Average Nusselt number distributions show that heat transfer ratealong wall DA is larger for lower inclination angle (u = 15�) whereas maximum heat transfer rate alongwall BC occur for higher inclination angle (u = 75�).

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The phenomenon of natural convection in closed cavities hasreceived considerable attention due to its practical relevance invarious applications, such as geophysics simulation, food engineer-ing, room heating and ventilation design, solar energy collectiondesign, cooling of computer systems and other electronic equip-ments, etc. [1–6]. There are two ways to describe the phenomenonof natural convection such as, experimental methods and numeri-cal simulations. Analysis with numerical methods is more prefera-ble as high cost is involved in experimental methods. The fluid flowand heat transfer characteristics of such systems are predicted bythe mass, momentum and energy conservation equations withappropriate boundary conditions. Processes involving natural con-vection flows within cavities are important for rectangular andnonrectangular enclosures. A number of studies on natural convec-tion within enclosures with various shapes (square, triangular,trapezoidal, etc.) have also been performed and the solutions forboth flow and thermal fields have been reported in literature [7–9].

A few earlier investigations on natural convection flows withincavities for various thermal boundary conditions may be outlined

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x: +91 44 22570509..K. Singh), sjroy@iitm.ac.in

as follows. Ganzarolli and Milanez [10] studied natural convectionflow in rectangular enclosure heated from below and cooled alonga single side or both sides using a streamfunction–vorticity formu-lation and presented the numerical results for various Rayleighnumbers (103

6 Ra 6 107), Prandtl numbers (0.7 6 Pr 6 7) and as-pect ratios (0.66 6 AR 6 8). Corcione [11] studied numerical inves-tigation on natural convection in air-filled, 2D rectangularenclosures heated from below and cooled from above with widevariety of thermal boundary conditions at the side walls. Raviet al. [12] studied the structure of steady, laminar natural convec-tion in a square enclosure for high Rayleigh numbers with inclina-tion angle, u = 90�. The effect of a simultaneous differential heatingof both the horizontal and vertical walls of a square cavity was ana-lyzed by Shiralkar and Tien [13]. November and Nansteel [14] per-formed analytical and numerical studies on natural convectionheat transfer in a square, water filled enclosure heated from belowand cooled on one vertical side. The numerical study of naturalconvection in a square enclosure for hot bottom wall and varioushot/cold side walls with insulated top wall was carried out byRoy and Basak [15]. A number of studies on the convection pat-terns in the square enclosure are carried out by several investiga-tors such as Patterson and Imberger [16], Hyun and Lee [17],Lage and Bejan [18], Xia and Murthy [19] and Nicolette et al. [20].

The brief literature survey as mentioned above, provides vari-ous studies of flow and heat transfer characteristics within the

Nomenclature

g acceleration due to gravity (m s�2)L side of the tilted square cavity (m)N total number of nodesn normal vector to the planeNu local Nusselt numberNu average Nusselt numberp pressure (Pa)P dimensionless pressurePr Prandtl numberR residual of weak formRa Rayleigh numberT temperature of the fluid (K)Th temperature of hot wall (K)Tc temperature of cold wall (K)u x component of velocity (m s�1)U x component of dimensionless velocityv y component of velocity (m s�1)V y component of dimensionless velocityX dimensionless distance along x coordinatex distance along x coordinate (m)Y dimensionless distance along y coordinate

y distance along y coordinate (m)

Greek symbolsa thermal diffusivity (m2 s�1)b volume expansion coefficient (K�1)c penalty parameterC boundary of two dimensional domainh dimensionless temperaturem kinematic viscosity (m2 s�1)q density (kg m�3)u inclination angle with the positive direction of X axisU basis functionsw dimensionless streamfunctionP dimensionless heatfunction� error in heat balance within the cavity

Subscriptsi residual numberk node number

2966 A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

square cavity. Due to various applications of natural convectionwithin tilted cavity such as, solar energy collectors, solar passivedesign, double-glazed windows, car batteries, cooling of electronicequipment and industrial processes such as crystal growth andelectrochemical metal refining, the heat transfer by natural con-vection within the tilted cavity in passive solar systems constitutesa major area of recent researches. It is the recognition of these ef-fects on the performance of passive systems that necessitates toinvestigate the heat energy flow within tilted square cavity to de-sign of efficient thermal systems. During the past two decades, fewexperimental and numerical studies have been carried out forunderstanding fluid flow and associated heat transfer during natu-ral convection within tilted cavities. These studies were focused tounderstand the effect of various Rayleigh numbers (Ra), Prandtlnumbers (Pr) and tilted angles (us) on fluid flow and heat transfer.

The effect of inclination angle on natural convection withintilted cavities has been discussed by few investigators [21–35].Baez and Nicolas [21] numerically analyzed the natural convectionheat transfer within a rectangular cavity filled with porous mediafor various inclination angle. Natural convection within tiltedsquare cavity with one side cooled at uniform temperature andthe partially heated at the opposite wall is investigated by Corcioneand Habib [22]. Experimental and numerical values of the Nusseltnumber for heat transfer in a tilted square cavity with uniformtemperature on one inclined side and at a lower uniform temper-ature on the opposite side were studied by Ozoe et al. [23].Hamady et al. [24] analyzed the effect of inclination angle on theheat transfer characteristics in an air-filled differentially heatedenclosure. They found the strong dependence of the heat flux onthe inclination angles (us) and Rayleigh numbers (Ra). Bairi et al.[25] investigated numerically and experimentally the natural con-vection in rectangular inclined enclosures for high Rayleigh num-bers. They obtained a correlation between Nusselt numbers andRayleigh numbers and they found the minimal value for Nusseltnumbers at inclination angle, u = 270�. Rasoul and Prinos [26]studied the effect of inclination angle on laminar natural convec-tion in a square cavity for inclination angles (40� 6u 6 160�), Ray-leigh numbers (103

6 Ra 6 106) and Prandtl numbers(0.02 6 Pr 6 4000). Schinkel and Hoogendoorn [27] carried out

numerical and experimental study on natural convection withininclined air filled cavities with aspect ratios (1 6 AR 6 18) at Ray-leigh numbers (Ra) varying within 2 � 104 � 5 � 105 and tilting an-gles of the enclosure (40� 6 u 6 90�). Cianfrini et al. [28] studiednatural convection in tilted square enclosures with two adjacentwalls heated and the two opposite walls cooled. They have pre-sented flow patterns and temperature distributions with Rayleighnumbers (104

6 Ra 6 106) and inclination angles of cavity(0� 6 u 6 360�). Oztop [29] numerically investigated the naturalconvection in a partially cooled and inclined rectangular enclosurefilled with saturated porous medium. It is found that the inclina-tion angle is dominant parameter on heat transfer and flow circu-lations. Aydin et al. [30], Aounallah et al. [31] and Lee and Lin [32]made different applications for fluid filled inclined enclosures andfound that inclination angle can be used as a control parameter ofheat transfer and fluid flow. The effect of inclination angle on nat-ural convection is also analyzed by Catton et al. [33], Strada andHeinrich [34], Dropkin and Somercales [35] for different Rayleighnumbers based on arbitrary angle of inclinations.

The detailed analysis on heat flow via heatlines is important forvarious tilted cavities. A generalized formulation on heatlines forany tilt angle is developed for the first time in this work. Alsothe effect of inclination of the cavity to the direction of gravityhas many engineering applications. In the case of tilted cavity,enclosures are inclined to the direction of gravity. Hence, buoyancyforces have both components relative to the walls of the enclosurewhich influence strongly the flow structure and the heat transfertherein. The aim of the present investigation is to analyze the nat-ural convection flow within the tilted square cavity for hot wall ABand cold side walls (BC and DA) in the presence of insulated wallCD with tilting angles of the enclosure (0� 6 u 6 90�). The heatingstrategy within the inclined cavity also needs the complete under-standing of thermal mixing and visualization of heat flow whichforms the basis of the present work.

Streamlines and isotherms are generally used for numericalanalysis of fluid flow and temperature distribution. Streamlinesare useful to visualize the fluid flow whereas isotherms determineonly temperature distribution which may not be suitable to visual-ize the direction and intensity of heat transfer. In order to visualize

(a)

(b)

(c)

Fig. 1. Schematic diagram of the computation domain with the boundary condi-tions. The dashed line represents the line of geometric symmetry. The wall CD ismaintained adiabatic for cases (a)–(c).

A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983 2967

the heat flow, heatlines which are analogous to streamlines may beused. The heatline concept was introduced by Kimura and Bejan[36] which can be used to visualize the path of heat flow, its mag-nitude and zones of high heat transfer. Mathematically, heatlinesare represented by heatfunctions and proper dimensionless formsof heatfunction are closely related the Nusselt numbers. Costa [37–39] analyzed heatline and massline for the natural convection tovisualize two dimensional transport phenomena. The heatlinesconcept has been further studied by Zhao et al. [40], Liu et al.[41], Dalal and Das [42], Aggarwal and Manhapra [43], Kim andJang [44], Zhao et al. [45], Mobedi and Oztop [46] and Ochende[47]. It may be noted from the literature that no attempt has beenmade to study the natural convection within the tilted square cav-ity using Bejan’s heatline concept. Therefore as a step towards theeventual development of studies on natural convection flows with-in a tilted square cavity, heatline concept is used for the first timeto visualize the heat energy flow in a tilted square cavity.

The objective of the present investigation is to analyze the effectof the inclination angle on the fluid flow and heat flow due to nat-ural convection and thermal mixing within tilted square cavities interms of streamfunctions, isotherms and heatfunctions. The Galer-kin finite element method [48] with penalty parameter is used tosolve the nonlinear coupled partial differential equations govern-ing the flow and thermal fields and the method is further used tosolve the Poisson equation for streamfunction and heatfunction.Simulations are performed for a range of parameters,Ra = 103 � 105, Pr = 0.015 � 1000 [0.015 (molten metals), 0.7 (airor gaseous substances) and 1000 (olive/engine oils)] andu = 0� � 90�.

2. Mathematical formulation and solution procedure

2.1. Governing equations, boundary conditions and simulationstrategy

The physical domain of tilted square cavity inclined at an angleu = 15�, 45� and 75� with X axis is shown in Fig. 1(a)–(c), respec-tively. The wall AB of the cavity is maintained hot and side walls(BC and DA) are maintained cold in presence of adiabatic wallCD. Physical properties are assumed to be constant except the den-sity in buoyancy term, where change in density due to temperaturevariation is calculated using Boussinesq approximation. Theflow and thermal fields inside the cavity are governed by theNavier–Stokes and the energy balance equations, respectively.The governing equations for steady natural convection flow usingconservation of mass, momentum and energy in dimensionlessform can be written as:

oUoXþ oV

oY¼ 0; ð1Þ

UoUoXþ V

oUoY¼ � oP

oXþ Pr

o2U

oX2 þo2U

oY2

!; ð2Þ

UoVoXþ V

oVoY¼ � oP

oYþ Pr

o2V

oX2 þo2V

oY2

!þ Ra Pr h; ð3Þ

UohoXþ V

ohoY¼ o2h

oX2 þo2h

oY2 ; ð4Þ

where

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 : ð5Þ

In Eq. (5), X and Y are the dimensionless distances along x-y coordi-nate, respectively; U and V are the dimensionless velocity compo-

nents along X and Y directions, respectively; P denotes thedimensionless pressure; Th and Tc are the temperatures at hot andcold walls, respectively; L is the length of each side of the tiltedsquare cavity; h is the dimensionless temperature; Pr and Ra denotePrandtl number and Rayleigh number, respectively. No slip condi-tions are assumed at all the walls of tilted square cavity and theboundary conditions for the velocity components and temperatureare as follows:

UðX;YÞ ¼ 0; VðX;YÞ ¼ 0; h ¼ 1 along AB;UðX;YÞ ¼ 0; VðX;YÞ ¼ 0; h ¼ 0 along BC;UðX;YÞ ¼ 0; VðX;YÞ ¼ 0; h ¼ 0 along DA;UðX;YÞ ¼ 0; VðX;YÞ ¼ 0; n:rh ¼ 0 along CD:

ð6Þ

The continuity Equation [Eq. (1)] is used as a constraint due tomass conservation and this constraint can be used to obtain thepressure distribution. The momentum and energy balance equa-tions (Eqs. (2)–(4)) are solved using Galerkin finite element meth-od [48]. In order to solve Eqs. (2) and (3), penalty finite elementmethod has been employed to eliminate the pressure, P with apenalty parameter, c and the incompressibility criteria given byEq. (1) via following relationship:

P ¼ �coUoXþ oV

oY

� �: ð7Þ

Typically c = 107 yields consistent solutions. Applying Eq. (7), themomentum balance equations (Eqs. (2) and (3)), are reduced to:

2968 A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

UoUoXþ V

oUoY¼ c

o

oXoUoXþ oV

oY

� �þ Pr

o2U

oX2 þo2U

oY2

!; ð8Þ

and

UoVoXþ V

oVoY¼ c

o

oYoUoXþ oV

oY

� �þ Pr

o2V

oX2 þo2V

oY2

!þ Ra Pr h: ð9Þ

The system of equations (Eqs. (4), (8) and (9)) with boundaryconditions (Eq. (6)) are solved by using Galerkin finite elementmethod. Expanding the velocity components (U,V) and tempera-ture (h) 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: ð10Þ

The Galerkin finite element method yields the following nonlin-ear residual equations for Eqs. (8), (9) and (4), respectively, atnodes of internal domain X:

Rð1Þi ¼XN

k¼1

Uk

�Z

X

XN

k¼1

UkUk

!oUk

oXþ

XN

k¼1

VkUk

!oUk

oY

" #UidXdY

þ cXN

k¼1

Uk

ZX

oUi

oXoUk

oXdXdY þ

XN

k¼1

Vk

ZX

oUi

oXoUk

oYdXdY

" #

þ PrXN

k¼1

Uk

ZX

oUi

oXoUk

oXþ oUi

oYoUk

oY

� �dXdY ;

ð11Þ

Rð2Þi ¼XN

k¼1

Vk

�Z

X

XN

k¼1

UkUk

!oUk

oXþ

XN

k¼1

VkUk

!oUk

oY

" #UidXdY

þ cXN

k¼1

Uk

ZX

oUi

oYoUk

oXdXdY þ

XN

k¼1

Vk

ZX

oUi

oYoUk

oYdXdY

" #

þ PrXN

k¼1

Vk

ZX

oUi

oXoUk

oXþ oUi

oYoUk

oY

� �dXdY � RaPr

�Z

X

XN

k¼1

hkUk

!Ui dXdY;

ð12Þ

and

Rð3Þi ¼XN

k¼1

hk

�Z

X

XN

k¼1

UkUk

!oUk

oXþ

XN

k¼1

VkUk

!oUk

oY

" #Ui dXdY

þXN

k¼1

hk

ZX

oUi

oXoUk

oXþ oUi

oYoUk

oY

� �dXdY : ð13Þ

Bi-quadratic basis functions with three point Gaussian quadra-ture are used to evaluate the integrals in residual equations exceptthe second term in Eqs. (11) and (12). In Eqs. (11) and (12), the sec-ond term containing the penalty parameter (c) is evaluated withtwo point Gaussian quadrature method. The non-linear residualequations (Eqs. (11)–(13)) are solved using Newton–Raphson

method to determine the coefficients of the expansions in Eq.(10). Fig. 2 shows the grid generation for (x,y) and (n,g) coordinatesvia the following relationship:

X ¼X9

k¼1

Ukðn;gÞ xk; Y ¼X9

k¼1

Ukðn;gÞ yk; ð14Þ

where (xk,yk) are the X, Y coordinates of the k nodal points as seen inFig. 2 and Uk(n,g) is the basis function. The basis functions are givenin Appendix A.

2.2. Streamfunction, Nusselt number and heatfunction

2.2.1. StreamfunctionThe fluid motion is displayed using the streamfunction (w) ob-

tained from velocity components (U and V). The relationships be-tween streamfunction (w) and velocity components (U and V) fortwo dimensional flows are:

U ¼ owoY

and V ¼ � owoX

; ð15Þ

which yield a single equation:

o2w

oX2 þo2w

oY2 ¼oUoY� oV

oX: ð16Þ

The sign convention is as follows. Positive sign of w denotesanti-clockwise circulation and clockwise circulation is representedby negative sign of w. Expanding the streamfunction (w) using ba-sis set fUkgN

k¼1 as:

w �XN

k¼1

wkUkðX;YÞ; ð17Þ

and the relationship for U and V from Eq. (10), the Galerkin finiteelement method yields the following linear residual equations forEq. (16).

RðwÞi ¼XN

k¼1

wk

ZX

oUi

oXoUk

oXþ oUi

oYoUk

oY

� �dXdY �

ZCUin:rwdC

þXN

k¼1

Uk

ZX

UioUk

oYdXdY �

XN

k¼1

Vk

ZX

UioUk

oXdXdY: ð18Þ

The no-slip condition is valid at all boundaries and there is no crossflow, hence w = 0 is used for boundaries. The bi-quadratic basisfunction is used to evaluate the integrals in Eq. (18) and w’s are ob-tained by solving the N linear residual equations [Eq. (18)].

2.2.2. Nusselt numberThe heat transfer coefficient in terms of local Nusselt number

(Nu) is defined by

Nu ¼ � ohon; ð19Þ

where n denotes the unit normal direction on a plane. The localNusselt numbers at wall AB (NuAB), at the wall DA (NuDA) and atthe wall BC (NuBC) are defined as:

NuAB ¼ �X9

i¼1

hi sinðuÞ oUi

oX� cosðuÞ oUi

oY

� �; ð20Þ

NuDA ¼X9

i¼1

hi cosðuÞ oUi

oXþ sinðuÞ oUi

oY

� �; ð21Þ

and

NuBC ¼ �X9

i¼1

hi cosðuÞ oUi

oXþ sinðuÞ oUi

oY

� �: ð22Þ

Mapping

Mapping

η

ξ

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

Local co−ordinate systemGlobal co−ordinate system

η

Y

X

Y

X ξ

(b)

(a)

Fig. 2. (a) The mapping of tilted square domain to a square domain in n � g coordinate system and (b) the mapping of an individual element to a single element in n � gcoordinate system.

A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983 2969

The average Nusselt numbers at any wall (AB, BC or DA) are givenas:

Nus ¼Z 1

0NusdS; ð23Þ

where dS denotes the small elemental length along sides of thetilted square cavity. Note that, NuDA þ NuBC ¼ NuAB whereasNuDA – NuBC due to tilted geometry with respect to gravity. Percent-age of error in heat balances within the cavity can be calculatedusing average Nusselt number at the wall AB, wall BC and wallDA as follows:

� ¼jNuAB � NuDA þ NuBC

� �j

min½NuAB; NuDA þ NuBC� �

�� 100: ð24Þ

2.2.3. HeatfunctionThe heat flow within the enclosure is displayed using the heat-

function (P) obtained from conductive heat fluxes � ohoX ;� oh

oY

� �as

well as convective heat fluxes (Uh,Vh). The heatfunction satisfiesthe steady energy balance equation (Eq. (4)) such that:

oPoY¼ Uh� oh

oX;

� oPoX¼ Vh� oh

oY;

ð25Þ

which yield a single equation:

o2P

oX2 þo2P

oY2 ¼o

oYðUhÞ � o

oXðVhÞ: ð26Þ

The sign convention for heatfunction is as follows. The positive signof P denotes anti-clockwise heat flow and clockwise heat flow isrepresented by negative sign of P. Expanding the heatfunction(P) using basis set fUkgN

k¼1 as:

P �XN

k¼1

Pk UkðX;YÞ; ð27Þ

and the relationship for U, V and h from Eq. (10), the Galerkin finiteelement method yields the following linear residual equations forEq. (26).

RðhÞi ¼XN

k¼1

Pk

ZX

oUi

oXoUk

oXþ oUi

oYoUk

oY

� �dXdY �

ZCUin:rPdC

þXN

k¼1

Uk

ZX

XN

k¼1

hkUk

!Ui

oUk

oYdXdY þ

XN

k¼1

hk

ZX

XN

k¼1

UkUk

!

�UioUk

oYdXdY �

XN

k¼1

Vk

ZX

XN

k¼1

hkUk

!Ui

oUk

oXdXdY

�XN

k¼1

hk

ZX

XN

k¼1

VkUk

!Ui

oUk

oXdXdY : ð28Þ

2970 A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

In order to obtain a unique solution of Eq. (26), various Dirichletand Neumann boundary conditions are implemented. Neumannboundary conditions for P are obtained due to isothermal (hot orcold) wall based on Eq. (25) and the normal derivatives (n �rP)are specified as follows:

(a) for cold walls BC and DA:

n � rP ¼ 0: ð29Þ

(b) for hot wall AB:

n � rP ¼ 0: ð30Þ

The top insulated wall CD may be represented by the Dirichletboundary condition obtained from Eq. (25) which is simplified intooPon ¼ 0 for an adiabatic wall CD. A reference value of P is assumedas 0 at D(� sin u, cos u) on the adiabatic wall CD. It may be notedthat the unique solution of Eq. (26) is strongly dependent on thenonhomogeneous Dirichlet conditions. The boundary conditionsat junction of hot and cold wall are obtained by integrating Eq.(25) along boundaries from the reference point until the end pointof the junction. Dirichlet conditions for P have been obtainedbased on integration of local values of heatfunctions as discussedin Appendix B. Hence, following Dirichlet boundary conditionsare obtained as:

P ¼ NuDA at edge A; ð31Þ

and

P ¼ �NuBC at edge B: ð32Þ

3. Results and discussion

3.1. Numerical procedure and validation

The computational domain consists of 28 � 28 bi-quadratic ele-ments which correspond to 57 � 57 grid points. The computationalgrid in the tilted square domain is generated via mapping of tiltedsquare domain into square domain in n � g coordinate system asshown in Fig. 2 and the procedure is outlined in Appendix A. Thebi-quadratic elements with lesser number of nodes smoothly cap-ture the non-linear variations of the field variables which are incontrast with finite difference or finite volume solutions. In orderto assess the accuracy of numerical procedure, we have testedour algorithm based on the grid size (57 � 57) for a tilted squareenclosure filled with air (Pr = 0.71) subject to hot wall DA and coldwall BC in the presence of remaining adiabatic walls (walls AB andCD) for inclination angles u = 0�, 30� and 50� at Ra = 104 and 105

similar to earlier work [26] and results are in good agreement.The results based on the current simulation studies areshown in Figs. 3 and 4. Table 1 shows that average Nusselt num-bers ðNuÞ agree with the results of earlier researchers [24,26] fora range of Rayleigh numbers (Ra = 104 � 105) and inclination an-gles (u = 0� � 90�).

In the current investigation, numerical simulations are carriedout for various values of Ra(Ra = 103 � 105), Pr(Pr = 0.015 � 1000)and inclination angles (u = 15�, 45� and 75�). The jump discontinu-ities in Dirichlet type of wall boundary conditions at the cornerpoints (junction of hot and cold walls) correspond to computa-tional singularities. The grid size dependent effect for the temper-ature discontinuity at corner points upon the local and averageNusselt numbers tend to increase as the mesh spacing at the corneris reduced. The problem is resolved by specifying the average tem-perature of the hot and cold walls at the junction and keeping theadjacent grid-nodes at the respective wall temperatures [49,50].

The present finite element approach offers special advantage onevaluation of average Nusselt number at the wall AB, wall BCand wall DA (see Fig. 1) as the element basis functions are usedto evaluate the heat flux. Current solution scheme with uniformheating of wall AB, isothermally cold side walls (BC and DA) andadiabatic wall CD produces grid invariant results based on28 � 28 elements (57 � 57 grid points) as shown in Table 2 andthe percentage error for average Nusselt number is found to beminimum for 28 � 28 elements. The benchmark results for heat-lines are already described in a earlier work [49] and thus the de-tails are omitted for the brevity of the manuscripts. Detailedexplanation of the results with particular emphasis on the effectof inclination angle on the streamlines, isotherms, heatlines, localand average Nusselt numbers is given in various succeedingsections.

3.2. Streamlines, isotherms and heatlines

Figs. 5–9 illustrate the streamlines, isotherms and heatlines forvarious Pr = 0.015 � 1000 and Ra = 103 � 105 with tilted angles,u = 15�, 45� and 75� in presence of isothermally hot wall AB, coldside walls (BC and DA) and adiabatic wall CD. For the case of hor-izontal square (u = 0), with isothermally hot wall AB and cold sidewalls (BC and DA), fluid rises from the center of the hot wall AB dueto buoyancy, directed against the gravity and rolls down along thecold walls (BC and DA), forming two symmetric rolls with clock-wise and anticlockwise rotations inside the cavity [49,50]. In thecase of inclined square, the effects of both the tangential and nor-mal components of buoyancy force relative to hot wall AB play crit-ical role for both flow and thermal characteristics. As tilted angleincreases, buoyancy force along the wall AB gradually increasesleading to stronger anticlockwise fluid circulations. This is furtherdue to increase in tangential component of buoyancy force whichis zero in the case of horizontal square (u = 0). Results indicate thatthe fluid circulations, isotherms and heatlines are strongly depen-dent on the inclination angle of the cavity.

Fig. 5(a)–(c) display streamlines, isotherms and heatlines forPr = 0.015 and Ra = 103 with inclination angles u = 15�, 45� and75�. It is observed that two asymmetric rolls consisting of anti-clockwise strong primary circulation cells and clockwise weak sec-ondary circulation cells are formed for small inclination angle(u = 15�). Due to inclination of the cavity, the hot fluid tends to risealong the tilted wall AB and the right circulation cells tend to grad-ually diminish (see Fig. 5(a)). In this case, the magnitudes ofstreamfunctions are considerably smaller due to conduction dom-inant heat transfer. The isotherms are smooth and monotonicwhich illustrate that heat transfer is primarily due to conduction.The isotherms with h = 0.1 occur symmetrically near the side walls(BC and DA) of the enclosure and other isotherms with h P 0.2 aresmooth curves symmetric with respect to geometric symmetry linefor u = 15� (see Fig. 5(a)). As u increases from 15� to 45�, left circu-lation cell tends to become stronger. Also, left circulation cellgrows in size and magnitude whereas right circulation cells disap-pear (see Fig. 5(b)). This is due to increase in tangential componentof buoyancy force, which is stronger in the anticlockwise direction.Similar to previous case (u = 15�), isotherms are almost symmetricnear side walls (BC and DA) for h = 0.1 and other isotherms withh P 0.2 are nearly symmetric with respect to geometric symmetryline (see Fig. 5(a)–(b)). The magnitudes of streamfunctions areslightly increased and isotherms follow the similar pattern as pre-vious cases for u = 75�. It is found that, jwjmax = 0.4, 0.76 and 0.93for u = 15�, 45� and 75�, respectively. It is interesting to observethat the intensity of fluid flow increases with the inclination angle(u) due to increase in tangential component of buoyancy forcealong the tilted wall AB. It may be noted that the temperature pro-files are almost invariant with respect to inclination angles during

ψ

−5−3

−0.5−0.1(a)

A B

CDθ

0.9 0.8

0.6 0.5

0.4 0.30.2 0.

1

A B

CD

−6.5

−5

−2

−0.5

−0.1

(b)

A

B

C

D 0.9 0.8

0.6 0.5

0.4

0.3

0.2

0.1

A

B

C

D

−7.4−6 −2

−0.1

−0.5

(c)

A

B

C

D

0.9 0.7

0.6

0.5

0.4 0.3

0.20.1

A

B

C

D

Fig. 3. Streamfunction (w) and temperature (h) contours with boundary conditions, h(X,Y) = 1 (wall DA), h(X,Y) = 0 (wall BC) and ohon ¼ 0 (walls AB and CD) of tilted square with

Pr = 0.71 and Ra = 104 for (a) u = 0�, (b) u = 30�, (c) u = 50�, where n is unit outward normal direction of wall AB and CD [26]. Clockwise and anti-clockwise flows are shownvia negative and positive signs of streamfunctions.

A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983 2971

conduction dominant heat transfer at lower Rayleigh number(Ra = 103) (see Fig. 5(a)–(c)). Common to all cases, the magnitudesof streamfunctions are found to be small except at the core of vor-tex and heat transfer is primarily due to conduction dominantmode and that has been illustrated based on heatlines as follows.The heat flow distribution may be further explained based onheatlines.

It may be noted that the signs of heatfunction are dependent ontwo corner boundary conditions (Eqs. (31) and (32)) of the wall AB.The sign convention is based on the fact that the positive heatfunc-tion corresponds to anticlockwise heat flow and negative heatfunc-tion corresponds to clockwise heat flow. It is observed thatmagnitudes of heatfunction (jPj) are small implying small heatflow due to weak flow circulation at lower Rayleigh number(Ra = 103). The heatfunction (jPj) varies within 0.005 � 2.95 alongthe wall BC and that results in higher thermal gradient along thewall BC for u = 15�. It is also observed that, the upper portion ofthe wall BC receives lesser heat from the wall AB as the magnitudesof heatfunction varying within 0.005 6 jPj 6 0.2 whereas the low-er portion of the wall BC receives significant heat as magnitudes ofheatfunction varies within 0.2 6 jPj 6 2.95 (see Fig. 5(a)). As a re-sult, thermal boundary layer thickness is small near to lower por-tion of the wall BC compared to the upper portion of the wall BC.On the other hand, jPj is found to vary within 0.005 � 2.83 alongthe wall DA whereas 0.005 6 jPj 6 2.95 is observed along the wall

BC (see Fig. 5(a)). Therefore, thermal boundary layer thicknessalong the wall DA is slightly larger than that along the wall BC. Itis very interesting to note that the thermal boundary layer thick-ness increases as we move from bottom to top portion of the sidewalls (BC and DA) due to low heatfunction gradient in the upperportion of the cavity. This is further due to less heat transfer fromthe wall AB to upper portion compared to lower portion of the sidewalls (BC and DA). It may be noted that the heatlines are quitedense near the middle portion of the wall AB and this implies en-hanced heat transfer from that regime. It is also observed thatsome heatlines directly start from the wall AB and end on cold sidewalls (BC and DA) near to small region of corners of wall AB due toconductive heat transport in that region. Heat transfer rate is foundto be quite larger at the corners of the wall AB due to junction ofhot and cold walls, and low at the top edge. The top portion ofthe cavity is maintained cooled, h � 0.1 as jPj varies within0.005 � 0.1. The dense heatlines near the middle portion of wallAB result in large thermal gradient at the middle portion of thewall AB and hence h varies within 0.5 � 0.9 along the central por-tion of the wall (see Fig. 5(a)).

As u increases to 45�, the heat flow from wall AB to the wall BCincreases due to anticlockwise circulations in the cavity (seeFig. 5(b)). The hot fluid rises along the wall AB and further, thatmoves along wall BC and hence a significant amount of heat trans-fer occurs from the hot fluid to wall BC. Dense heatlines occur near

(a)

(b)

(c)

Fig. 4. Streamfunction (w) and temperature (h) contours with boundary conditions, h(X,Y) = 1 (wall DA), h(X,Y) = 0 (wall BC) and ohon ¼ 0 (walls AB and CD) of tilted square with

Pr = 0.71 and Ra = 105 for: (a) u = 0�, (b) u = 30�, (c) u = 50�, where n is unit outward normal direction of wall AB and CD [26]. Clockwise and anti-clockwise flows are shownvia negative and positive signs of streamfunctions.

Table 1Comparison of present results with the benchmark solutions of Hamady et al. [24]and Rasoul and Prinos [26] for natural convection in a tilted square cavity filled withair (Pr = 0.71). Current study employs 28 � 28 bi-quadratic elements (57 � 57 gridpoints).

u Ra Current work Hamady et al. [24] Rasoul and Prinos [26]Nu Nu Nu

15� 105 3.8907 4.0863 –

30� 105 4.0862 3.9092 –

40� 104 2.1201 – 2.3190105 4.1663 3.9662 3.9430

60� 104 2.1183 – 2.3266105 4.2888 4.1277 4.1487

75� 105 4.1113 4.3600 –

90� 104 1.8952 – 2.5054105 4.2034 4.0503 4.0859

2972 A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

the middle portion of wall AB and those lines are directly con-nected to the wall BC. This further signifies enhanced heat trans-port to the wall BC. Therefore, magnitude of heatfunctions (jPj)

along the wall BC increases and that varies as 0.005 6 jPj 6 3.04.Consequently, thermal boundary layer thickness near the wall BCis slightly less compared to previous case with u = 15�. It is inter-esting to observe that, small zone near the wall BC corresponds toparallel heatlines which are also perpendicular to the isothermssignifying conduction dominant zone. On the other hand, 90% ofthe wall DA corresponds to dispersed parallel heatlines (seeFig. 5(b)). The magnitudes of heatfunction (jPj) along the wallDA vary as 0.004 6 jPj 6 2.76 whereas 0.005 6 jPj 6 3.04 is ob-served along the wall BC. Thus, thermal boundary layer thicknessnear to wall DA is larger than that along the wall BC. Similar tou = 15�, the isotherms near to wall AB are highly compresseddue to dense heatlines. It is very interesting to observe that thedense heatlines emanating from the middle portion of the wallAB are slightly compressed along the right side of the cavity dueto enhanced buoyancy force along the wall AB (see Fig. 5(b)). Sim-ilar trend in heatlines is observed for the higher inclination angle,u = 75� (see Fig. 5(c)). However, dense heatlines emanating fromthe middle portion of the wall AB are more inclined towards thewall BC compared to u = 15� and u = 45� (see Fig. 5(a)–(c)). There-fore, these dense heatlines depicting enhanced heat transport tothe wall BC. This is due to strong convective flow due to enhanced

Table 2Comparison of average Nusselt number ðNuABÞ for various grid systems at Ra = 105

and Pr = 0.7 with various inclination angles (us) in presence of uniform heating ofwall AB, isothermally cold walls (BC and DA) and adiabatic wall CD. Percentage oferror (�) in the heat balance within the tilted square cavity is calculated using Eq. (24).

u Grid points

49 � 49 53 � 53 57 � 57

NuAB Error (�) (%) NuAB Error (�) (%) NuAB Error (�) (%)

15� 8.76 0.44 8.85 0.39 8.95 0.3445� 8.73 0.82 8.81 0.73 8.90 0.6475� 8.62 0.34 8.71 0.32 8.80 0.29

A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983 2973

tangential component of buoyancy force along the wall AB. Conse-quently, the isotherms are progressively compressed along thewall BC and the isotherms tend to be asymmetric for 75�. The over-all amount of heat transfer along the wall BC increases with incli-nation angle and that decreases along the wall DA with increase ininclination angle as the magnitudes of heatfunction (jPj) on BCvary as 0.005 6 jPj 6 2.95 for u = 15�, 0.005 6 jPj 6 3.04 foru = 45� and 0.004 6 jPj 6 3.07 for u = 75� whereas0.005 6 jPj 6 2.83 for u = 15�, 0.004 6 jPj 6 2.76 for u = 45� and0.004 6 jPj 6 2.74 for u = 75� are found along the wall DA (seeFig. 5(a)–(c)).

Fig. 6(a)–(c) shows streamfunction, temperature and heatfunc-tion contours for Pr = 0.015 and Ra = 104 for various tilt angles. AsRayleigh number increases to 104, the convection starts to play adominant role and the streamline cells grow in magnitude andthey follow completely circular patters in contrast to Ra = 103 irre-spective of inclination angle (u). At u = 15�, the maximum value ofstreamfunction jwjmax is found to be 5 whereas jwjmax = 0.4 withinprimary circulation cell for Ra = 103. The smaller amount of fluid

ψ

0.40.30.20.10.02 −0.015

−0.02

(a)

A

B

C

D

θ

0.1

0

0.2

0.40.0

A

D

0.760.6

0.25

0.1

0.005(b)

A

B

C

D 0.1

0

0D

A

BC

D

0.930.8

0.35

0.1

0.005(c)

0.1

0.1

0.2

C

D

Fig. 5. Streamfunction (w), temperature (h) and heatfunction (P) contours with boundarytilted square with Pr = 0.015 (molten metals) and Ra = 103 for: (a) u = 15�, (b) u = 45�, (signs of streamfunctions and heatfunctions, respectively.

recirculates near to the right corner of the wall AB and top cornerof the wall BC, while more hot fluid reaches from wall AB to coldside walls (BC and DA) with anticlockwise circulations. Hence pri-marily circulation cells occupy almost entire cavity and a numberof secondary circulation cells are also observed near to cornerpoints of the wall BC for u = 15� (see Fig. 6(a)). The strong anti-clockwise circulations occur due to increase in the magnitude ofthe tangential component of buoyancy force along the hot wallAB corresponding to Ra = 104.

At u = 45�, the strength of the flow circulations increases as thethermal driving force induced by the hot wall AB progressively in-creases. It may be noted that jwjmax = 5.5 for u = 45� whereasjwjmax = 5 is observed for u = 15�. Similar to previous case(u = 15�), the path of fluid is completely circular and secondary cir-culations appear near the top and bottom corners of the cavity foru = 45� (see Fig. 6(a) and (b)). As u increases to 75�, the strong pri-mary circulation and very weak secondary circulation on the topcorners of wall CD are still observed (see Fig. 6(c)). The intensityof flow circulations is represented with jwjmax is 5, 5.5 and 5.3for u = 15�, 45� and 75�, respectively. Overall, the intensity of thefluid flow is larger for u = 45� as compared to u = 15� andu = 75�. Common to all, primary fluid circulation near to the coreregion becomes stronger as compared to previous case withRa = 103, and they occupy almost the entire part of the cavity dueto convection effect.

At Ra = 104, heatlines illustrate complex heat flow patternswhich are depicted by the secondary heatline cells occurring inthe cavity. The heatlines are not smooth as in the previous case(Ra = 103) and they are distorted due to dominance of convection.It may be noted that two types of heatlines appear within the cav-ity (see Fig. 6(a)). One is due to heat transfer directly from wall ABto cold side walls (BC and DA) and another is due to heat flow

.1

6.8 B

CΠ

0.01 −0.01

0.1−0.1

0.2−0.2

0.3

−0.3

0.5

−0.5

0.9

−1

A

B

C

D

.1

.20.

30.50.70.9

A

B

C

0.01

0.1

0.2

0.30.5

−0.0

1

−0.1

−0.2

3.0− −0.5

−0.8

A

B

C

D

0.3

0.4

0.6

0.8

A

B

0.01

0.1 0.2

0.30.5

−0.01

−0.1 −0.2−0.4 −0.7

−1

A

BC

D

conditions, h(X,Y) = 1 (wall AB), h(X,Y) = 0 (walls BC and DA) and ohon ¼ 0 (wall CD) of

c) u = 75�. Clockwise and anti-clockwise flows are shown via negative and positive

ψ

541

0.1

−0.0

45(a)

A

B

C

D

θ

0.1

0.2

0.3

0.4

0.6

0.9

0.2 0.1

A

B

C

D

Π

1.31.1

0.02

0.2

0.4

0.5 0.4 −0.2 −1.6−1

−0.5

−0.2

−0.02

A

B

C

D

5.54

0.5

0.1

−0.03(b)

A

B

C

D 0.1

0.2

0.3

0.4

0.5

0.7

0.1

A

B

C

D 1.41.30.02

0.20.4

0.5 0.4 −0.1

−1.4

−0.02

−0.2 −0.4

−1−1

.6

A

B

C

D

5.34

2

0.1

−0.03(c)

A

BC

D0.1

0.1

0.2 0.

30.4 0.

60.

8

A

BC

D

0.02 0.2

0.4 0.4

−0.2−1.6−1.2

−0.7

−0.4

−0.1

−0.02

1.30.8

A

BC

D

Fig. 6. Streamfunction (w), temperature (h) and heatfunction (P) contours with boundary conditions, h(X,Y) = 1 (wall AB), h(X,Y) = 0 (walls BC and DA) and ohon ¼ 0 (wall CD) of

tilted square with Pr = 0.015 (molten metals) and Ra = 104 for: (a) u = 15�, (b) u = 45�, (c) u = 75�. Clockwise and anti-clockwise flows are shown via negative and positivesigns of streamfunctions and heatfunctions, respectively.

ψ

2015

0.5

−0.1

−0.5

(a)

A

B

C

D

θ

0.1

0.2

0.3 0.4

0.6

0.2 0.1

A

B

C

D

Π

0.05

0.3

1

1.31.11

−2.8

−0.05−0

.2−16.2

A

B

C

D

19.5

15

5

−0.1

0.01

(b)

A

B

C

D 0.1

0.2

0.3

0.4

0.6

0.20.1

A

B

C

D 0.05

0.25

11.11

1.11

−2

−0.05 −0.2

−0.3

5.6

A

B

C

D

1513

5

−0.01

−0.1 0.01

A

BC

D

(c)

0.1

0.2

0.3

0.40.

50.

70.2

0.1

A

BC

D 0.030.3

1 1

−2.8

−0.0

3 −0.2 −0.5

4

−2

A

BC

D

Fig. 7. Streamfunction (w), temperature (h) and heatfunction (P) contours with boundary conditions, h(X,Y) = 1 (wall AB), h(X,Y) = 0 (walls BC and DA) and ohon ¼ 0 (wall CD) of

tilted square with Pr = 0.015 (molten metals) and Ra = 105 for: (a) u = 15�, (b) u = 45�, (c) u = 75�. Clockwise and anti-clockwise flows are shown via negative and positivesigns of streamfunctions and heatfunctions, respectively.

2974 A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

ψ

13.5

115 −14

−9−30.5

−0.5(a)

A

B

C

D

θ

0.10.2

0.3

0.4

0.5

0.7

0.40.

3 0.20.1

A

B

C

D

Π

0.1

1.3

22.5

5.8

2.5

−2.5

−2.5

−0.1−1−1.8

−6

A

B

C

D

209

5

−0.5

A

B

C

D

(b)

0.1 0.20.

30.4 0.

5

0.7

0.2 0.1

A

B

C

D 0.1

0.5

1.31.5

−0.1 −0.25 −0.3−1

−2.5

1.5

7.2

A

B

C

D

17.515

1 (c)

A

BC

D 0.10.2

0.3

0.4 0.5

0.70.2

0.1

A

BC

D 0.1 0.7

1

−0.1 −0.5 −1 −2

5.3

1

A

BC

D

Fig. 8. Streamfunction (w), temperature (h) and heatfunction (P) contours with boundary conditions, h(X,Y) = 1 (wall AB), h(X,Y) = 0 (walls BC and DA) and ohon ¼ 0 (wall CD) of

tilted square with Pr = 0.7 (air) and Ra = 105 for: (a) u = 15�, (b) u = 45�, (c) u = 75�. Clockwise and anti-clockwise flows are shown via negative and positive signs ofstreamfunctions and heatfunctions, respectively.

(a)

(b)

(c)

Fig. 9. Streamfunction (w), temperature (h) and heatfunction (P) contours with boundary conditions, h(X,Y) = 1 (wall AB), h(X,Y) = 0 (walls BC and DA) and ohon ¼ 0 (wall CD) of

tilted square with Pr = 1000 (olive/engine oil) and Ra = 105 for: (a) u = 15�, (b) u = 45�, (c) u = 75�. Clockwise and anti-clockwise flows are shown via negative and positivesigns of streamfunctions and heatfunctions, respectively.

A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983 2975

2976 A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

circulation as results of strong fluid flow circulation (jwjmax = 5) incentral region of the cavity. It is interesting to note that, denseheatlines correspond to jPj = 0.02 � 1 emanating from the 30% ofthe wall AB delivers heat to the wall BC. Hence, significant amountof heat is distributed to the wall BC based on the heatfunction var-iation as 0.01 6 jPj 6 3.64 (see Fig. 6(a)). As a result, isothermswith h 6 0.4 are largely compressed towards the lower portion ofthe wall BC and thus, thermal boundary layer thickness is smallin that regime. It may be noted that, at the onset of convection, iso-therms break into two type of discrete contours which move to-wards the side walls. The isotherms with h 6 0.2 are shiftedtowards the side walls (BC and DA) and other isotherms withh P 0.3 are found as highly distorted asymmetry curves. The heatflow is more towards the upper portion of side wall BC as com-pared to the previous case (Ra = 103). It may be noted that jPj var-ies as 0.01 6 jPj 6 0.5 along the upper portion of the wall BC inpresent case whereas 0.005 6 jPj 6 0.2 is observed in that regimefor Ra = 103. Thus thermal boundary layer thickness along theupper portion of wall BC for Ra = 104 is less than that the previouscase (Ra = 103). The heat flow is lesser towards the wall DA as jPjvarying within 0.01 6 jPj 6 2.74 whereas 0.01 6 jPj 6 3.64 is ob-served along the wall BC. Thus, thermal boundary layer thicknessis large along wall DA than that along the wall BC. It is interestingto see that thermal boundary layer thickness near to side walls (BCand DA) is reduced compared to previous case with lower Rayleighnumber (Ra = 103). It may be noted that the lower portion of theside walls receive larger heat compared to top portion of the sidewalls. Thus, thermal boundary layer thickness is more compressedtowards the lower portion of side walls (see Fig. 6(a)). It is interest-ing to see that, heatlines take longer paths from the wall AB toreach the cold wall DA as seen from the heatlines contours (seeFig. 6(a)). Due to strong convective circulation cells at the core,closed loop heatlines are found near central portion of the cavityunlike in previous case with Ra = 103. Thus the temperature gradi-ent in this zone is found to be less primarily due to convectiveheatline cells with jPjmax = 1.3. Therefore a large central region ofthe cavity is maintained to the uniform temperature(h = 0.1 � 0.4) (see Fig. 6(a)). It may be noted that lower portionof the cavity is hotter compared to the upper portion of the cavity.This is clearly illustrated by the temperature distribution in thewhole cavity, as 70% of the upper portion is maintained ath = 0.1 � 0.3 and remaining 30% lower portion is maintained ath = 0.3 � 1 (see Fig. 6(a)). It is observed that dense heatlines corre-spond to � 0.5 6P 6 2.74 along the left portion of the wall ABindicating large heatfunction gradient in that regime comparedto the right portion of the wall AB. Therefore, isotherms withh P 0.4 are largely compressed towards the left portion of the wallAB whereas isotherms with h P 0.4 are less compressed towardsthe right portion of the wall AB (see Fig. 6(a)). Thus, thermalboundary layer thickness is less near to left portion compared toright portion of the wall AB.

As u increases to 45�, the heat flow from the wall AB to wall BCincreases due to increase in tangential component of buoyancyforce (see Fig. 6(b)). It may be noted that dense heatlines emanat-ing from the middle portion wall AB delivers heat to the lower por-tion of the wall BC. In addition, the heatlines near to right corner ofwall AB also delivers heat to wall BC. Hence, significant amount ofheat is distributed near to wall BC as heatfunction (jPj) varieswithin 0.01 6 jPj 6 3.81 along the wall BC for the present casewhereas 0.01 6 jPj 6 3.64 is observed along the wall BC foru = 15�. Based on the dense heatlines towards the wall BC, iso-therms with h 6 0.4 are more compressed near to wall BC of thecavity as compared to u = 15� (see Fig. 6(a) and (b)). Thus, thermalboundary layer thickness near to that regime slightly decreasescompared to the previous case with u = 15�. On the other hand,there is no remarkable change in magnitude of the heatfunction

along the wall DA as 0.01 6 jPj 6 2.72 occurs for the present case(u = 45�) whereas 0.01 6 jPj 6 2.74 is observed at u = 15�. Thus,the thermal boundary layer thickness along the wall DA is almostsimilar as the previous case (u = 15�). The qualitative trends of thestreamlines and heatlines are same at the core of the cavity. Similarto the previous case with u = 15�, thermal boundary layer thick-ness along the lower portion of wall DA is larger than that alongthe lower portion of wall BC at u = 45�. This is due to less heattransfer from the wall AB to wall DA compared to the wall BC.Closed loop heatline cells grow in magnitude (jPjmax = 1.4) signify-ing enhanced thermal mixing during enhanced convection whichresult in recirculation of heat at the core and thus large portionof the core of the cavity is maintained at h = 0.2 � 0.5 (see Fig. 6(b)).

As inclination angle (u) further increases to 75�, the dense heat-lines are still observed near to corner B of the cavity depicting moreheat being received by the region (see Fig. 6(c)). Hence, significantamount of heat is distributed near to lower portion of the wall BCbased on 0.5 6 jPj 6 3.84. The isotherms as well as heatlines showqualitatively similar features to those of previous case (u = 45�). Atthe higher inclination angle (u = 75�), thermal boundary layerthickness near to lower portion of wall BC is slightly reduced com-pared to u = 15� and u = 45� based on magnitude of the heatfunc-tions jPj varying on BC as 0.01 6 jPj 6 3.64 for u = 15�,0.01 6 jPj 6 3.81 for u = 45� and 0.01 6 jPj 6 3.84 for u = 75�whereas 0.01 6 jPj 6 2.74 for u = 15�, 0.01 6 jPj 6 2.72 foru = 45� and 0.01 6 jPj 6 2.68 for u = 75� are observed along thewall DA. Overall, the heat transfer rate for the wall BC increaseswith angle of inclination (u) but heat transfer rate for the wallDA decreases with the increase of angle of inclination (u).

As Ra increases to 105, buoyancy forces dominate over viscousforces leading to enhanced convection in the cavity and intensityof circulation cells are further intensified as seen from the largemagnitudes of the streamfunction (see Fig. 7(a)–(c)). It may benoted that trend of the fluid flow patterns via streamlines are com-pletely circular and qualitatively similar for Ra = 104 irrespective ofinclination angle (u). At u = 15�, the maximum intensity ofstreamfunction (jwjmax) is found to be 20, which is an increase ofabout 300% as Ra increases from 104 to 105 (see Fig. 7(a)). Second-ary circulations are also observed with the strong primary circula-tion cells in top and bottom portion of the wall BC. Similar toprevious case with Ra = 104, the streamline cells form circular pri-mary circulation cells which occupy almost the entire cavity.

As u increases to higher inclination angles, fluid circulations atthe core region of cavity gradually become weaker as seen basedon the magnitudes of the streamfunction varying as jwjmax = 20for u = 15�, jwjmax = 19.5 for u = 45� and jwjmax = 15 for u = 75�(see Fig. 7(a)–(c)). It is interesting to observe that secondary circu-lations appear at the bottom and top corner regions of the wall BCfor u = 15� whereas secondary circulations are found to occur at allthe corners of the cavity for the higher inclination of the cavitywith u = 45� and u = 75� (see Fig. 7(b) and (c)). The streamline pro-files are qualitative similar for all the inclination angles (us) espe-cially for u = 45� and u = 75�. It is interesting to observe that theintensity of fluid flow increases with the angle of inclination (u)at lower Rayleigh number (Ra = 103) whereas intensity of fluid flowdecreases with increase of angle of inclination (u) at higher Ray-leigh number (Ra = 105) (see Fig. 5(a)–(c) and Fig. 7(a)–(c)).

Enhanced heat flow occurs due to increase in convection dom-inance which is seen by distorted heatlines corresponding to largermagnitudes of the heatfunction at the higher Rayleigh number(Ra = 105). The streamlines and heatlines are almost identicaland they take circular shape due to enhanced thermal mixing ef-fects. The magnitudes of heatfunction (jPj) varying within0.01 6 jPj 6 4.8 along the wall BC results in higher thermal gradi-ent along the wall BC for u = 15� (see Fig. 7(a)). The isotherms withh 6 0.6 are highly compressed near to the bottom portion of the

A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983 2977

wall BC due to enhanced heat transfer along the wall BC (seeFig. 7(a)). It may be noted that the lower portion of wall BC receivesignificant heat from the wall AB as magnitudes of heatfunctionvarying as 1 6 jPj 6 4.8 whereas 0.01 6 jPj 6 1 is observed alongthe upper portion of wall BC. Therefore, thermal boundary layerthickness is small near to lower portion compared to upper portionof the wall BC (see Fig. 7(a)). It is interesting to note that the ther-mal boundary layer thickness along the wall BC is reduced at high-er Rayleigh number (Ra = 105), compared to previous case withRa = 104. This is due to the fact that, heatfunction (jPj) varies as0.01 6 jPj 6 4.8 along the wall BC for the present case whereas0.01 6 jPj 6 3.64 is observed along the wall BC for Ra = 104 whenu = 15� (see Fig. 7(a)). On the other hand jPj is found to be varyas 0.01 6 jPj 6 3.34 along the wall DA whereas 0.01 6 jPj 6 4.8for the wall BC for u = 15�. Therefore thermal boundary layerthickness along the wall DA is larger than that along the wall BC.Overall, isotherms with h 6 0.4, are more compressed near thelower portion of wall BC indicating large amount of heat(0.01 6 jPj 6 4.8) being transferred from wall AB to wall BCwhereas isotherms with h 6 0.3 are less compressed near to wallDA indicating less heat (0.01 6 jPj 6 3.34) transport near to wallDA. Due to enhanced heat transport to the lower portion of the sidewalls (BC and DA), the boundary layer thickness is reduced andh = 0.1 � 0.2 is maintained near the walls BC and DA. The closedloop heatlines cells are formed at the core of the cavity indicatingthe recirculation of heat energy or enhanced thermal mixing at thecenter (jPjmax = 6.2) and heatline cells also signify convection heattransport due to strong heat circulation cells for u = 15�. As a re-sult, a large regime of central portion of cavity remains at uniformtemperature (h = 0.3 � 0.5) (see Fig. 7(a)). The isotherms withh P 0.4 are highly compressed towards the left half portion of wallAB due to dense heatlines corresponding to � 1 6P 6 1.11 ema-nating from that regime. The thermal boundary layer thickness islarge near to right half portion of the wall AB due to dispersedheatline in that regime (see Fig. 7(a)).

Similar trend in heatlines is observed for higher inclination an-gles (u = 45� and 75�). However, it may be noted that, the thermalboundary layer thickness along the wall BC is reduced withincreasing inclination angle as magnitude of heatfunctions jPj var-ies within 0.01 6 jPj 6 4.8 for u = 15�, 0.01 6 jPj 6 4.97 for u = 45�and 0.01 6 jPj 6 5.0 for u = 75� at Ra = 105. Similar to the caseswith Ra = 104, the thermal boundary layer thickness along the wallBC is smaller than that along the wall DA for all inclination anglesat Ra = 105 based on the magnitudes of heatfunctions (jPj). Thepath of streamfunction and heatfunction contours are quite similarto circular path for all the inclination angles (us). It may be notedthat, as u increases the convective heat transport at the core of thecavity gradually becomes weaker for Ra = 105. Note thatjPjmax = 6.2 for u = 15�, jPjmax = 5.6 for u = 45� whereas jPjmax = 4for u = 75� at the core of the cavity (see Fig. 7(a)–(c)). Therefore,enhanced thermal mixing occurs at the core of cavity for lowerinclination angle (u = 15�) with Ra = 105. Due to local thermal mix-ing, the lower half of the cavity is maintained at higher tempera-ture with h P 0.4 whereas large portion of the upper half ismaintained as h = 0.2 � 0.4 at the higher inclination angle(u = 75�). Overall, the isotherms with h 6 0.2 are shifted near theside walls and larger portion of central regime corresponds toh = 0.3 � 0.4 irrespective of inclination angle due to enhanced ther-mal mixing at core regime of the cavity.

Distributions of streamlines, isotherms and heatlines for Pr = 0.7 and Ra = 105 are displayed in Fig. 8(a)–(c). It is observed thatquite symmetric vortex patterns along the geometric symmetryline with eye near the center of each half occur for Pr = 0.7 in con-trast to Pr = 0.015. The intensity of fluid circulation is found to beless as compared to Pr = 0.015. The maximum value of streamfunc-tion (jwjmax) is found to be 14 whereas jwjmax = 20 with Pr = 0.015

for u = 15�. It is interesting to observe that the center of left vortexfor the fluid circulations is pushed towards the wall AB and centerof right vortex is pushed towards the wall CD due to enhancedbuoyancy force along the wall AB. Reduction in strength of primarycirculation cells and simultaneously increases in the strength ofsecondary circulation cells is found as Pr increases from 0.015 to0.7 for u = 15� (see Fig. 7(a) and Fig. 8(a)). The enhanced circulationcells denote the larger amount of convective flow as seen from thelarger magnitudes of streamfunctions at the each core of the circu-lation. It is also found that secondary circulations cells near thecorner region are completely absent for Pr = 0.7 with Ra = 105

(see Fig. 8(a)).As u increases to 45�, strong primary anticlockwise circulation

cells occupy almost the entire part of cavity except top corner (seeFig. 8(b)). This is due to enhanced tangential components of buoy-ancy force along the wall AB which tend to impose an anticlock-wise fluid circulation as mentioned earlier in previous case(Pr = 0.015). However, clockwise weak secondary circulation cellsare found near top portion of the wall BC. The flow circulation cellsare almost circular in shape near the core and further, they expandand take the shape of the cavity near cavity walls due to intenseconvection unlike in previous case with Pr = 0.015. As inclinationangle (u) further increases to 75�, the primary circulation cellsspan the entire cavity whereas secondary circulation cells com-pletely disappear (see Fig. 8(c)). This is due to strong convectiveflux along the wall AB due to enhanced buoyancy force along wallAB. It may be noted that the maximum magnitude of the stream-function (jwjmax) are around 14, 20 and 17.5 at u = 15�,u = 45�and u = 75�, respectively (see Fig. 8(a)–(c)).

Heatlines illustrate the larger heat flow during enhanced con-vection at Ra = 105 for Pr = 0.7. The dense heatlines correspondingto 2.5 P jPjP 0 in the left portion of wall AB and 0 6 jPj 6 2.5in the right portion of wall AB rise high up to the adiabatic wallCD and then descend downwards to the cold regimes of the wallsBC and DA. Therefore, large amount of heat is transferred from thewall AB to walls BC and DA and near to top wall CD. It may benoted that the heatlines are symmetric along the geometric sym-metry line in contrast to lower Prandtl number (Pr = 0.015). Highermagnitudes of heatfunctions are observed along the side walls (BCand DA) as jPj varies within 0.06 6 jPj 6 4.41 along the wall BCand 0.07 6 jPj 6 4.49 along the wall DA, due to larger heating ef-fects. Isotherms with h 6 0.3 are observed to be compressed alongthe top portion of side walls (BC and DA) for u = 15� (see Fig. 8(a)).The boundary layer thickness near top portion of the side walls islargely reduced due to high heat flow as represented by heatlinesof high magnitude of heatfunctions in that region compared toPr = 0.015 (see Fig. 7(a) and Fig. 8(a)). It may also be noted that, iso-therms with h P 0.5 are confined within a small regime near thewall AB due to dense heatlines corresponding to �2.5 6P 6 2.5(see Fig. 8(a)). The magnitude of the clockwise heatline cells isslightly higher than those to the anticlockwise heatline cells.Strong thermal mixing with jPjmax = 6 occurs at core of clockwiseheatline cells due to recirculation of heat energy. Therefore, a largeportion of the right half of the cavity is maintained with uniformtemperature as h varying as 0.2 � 0.4 compared to left half of thecavity. The enhanced heatlines circulation cells denote the largeamount of convective heat transport as seen from the large magni-tudes of heatfunctions at the each core of the circulation. There-fore, the central top portion of the cavity corresponds toh = 0.3 � 0.5.

As u increases to 45�, the strength of anticlockwise heatline cir-culation cells increases and streamlines and heatlines are identicalexcept near the corner and side walls (see Fig. 8(b)). It may benoted that, the dense heatlines emanate from the middle portionof the wall AB and end on the wall BC. Similar to previous case withPr = 0.015, isotherms are compressed near to left portion of the

2978 A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

wall AB due to high heat transfer from that regime to the walls BCand DA for u = 45�. Thus, the thermal boundary layer thickness issmall near left portion of wall AB. Significant amount of heat is dis-tributed to the wall BC based on 0.01 6 jPj 6 5.38. The small regionconfined by isotherms with h P 0.5 is shifted towards the lowerportion of wall BC due to dense heatlines along that regime (seeFig. 8(b)). Consequently, thermal boundary layer thickness nearlower portion of wall BC is smaller compared to the previous casewith u = 15�. The heat flow is less towards the upper portion of thewall BC as compared to the previous case with u = 15�. It may benoted that jPj varies as 0.01 6 jPj 6 0.5 along the upper portionof the wall BC in present case whereas 0.06 6 jPj 6 2 is observedin that regime for u = 15�. Thus thermal boundary layer thicknessalong the upper portion of wall BC for u = 45� is more than thatprevious case with u = 15�. On the other hand, thickness ofboundary layer near to wall DA is increased slightly compared tou = 15� due to less heat flux in that region as represented bymagnitude of heatfunction (0.03 6 jPj 6 3.44) for the presentcase with u = 45� whereas 0.07 6 jPj 6 4.49 is observed atu = 15� (see Fig. 8(a) and (b)). Large portion of core of the cavityremains at uniform temperature (h = 0.3 � 0.5) due to enhancedthermal mixing, illustrated by highly intense primary heatlinecirculation cells (jPjmax = 7.2) as seen from the heatfunctioncontours in Fig. 8(b).

At u = 75�, the isotherms and heatlines are found to be qualita-tively similar to those of u = 45� (see Fig. 8(b) and (c)). The denseheatlines emanating from the middle portion of the wall AB aremore inclined towards the wall BC compared to u = 45�. This isdue to strong convective flow due to enhanced tangential compo-nent of buoyancy force along the wall AB. Similar to the previouscase with Pr = 0.015, the overall amount of heat transfer alongthe wall BC increases with inclination angle and that decreasesalong the wall DA with increase in inclination angle. Note that,the magnitude of heatfunction (jPj) on BC vary as 0.06 6 jPj 6 4.41for u = 15�, 0.01 6 jPj 6 5.38 for u = 45� and 0.01 6 jPj 6 5.82 foru = 75� whereas 0.07 6 jPj 6 4.49 for u = 15�, 0.03 6 jPj 6 3.44for u = 45� and 0.02 6 jPj 6 2.94 for u = 75� are found along thewall DA. Therefore, the thermal boundary layer thickness is smallerfor wall BC as compared to wall DA for all the inclination angles(see Fig. 8(a)–(c)). The isotherms with h P 0.4 are highly com-pressed towards left half side of the wall AB due to dense heatlinesin that regime. Therefore, boundary layer thickness along the wallAB is small in left half portion of the wall AB (see Fig. 8(c)). Iso-therms with h P 0.4 are less compressed towards the right halfof the wall AB. Thus, thermal boundary layer thickness is less nearto left portion compared to right portion of the wall AB. Closed loopconvective heatline cells (jPjmax = 5.3) lead to enhanced thermalmixing and thus a large central portion of the cavity is maintaineduniform temperature as h = 0.3 � 0.4 (see Fig. 8(c)).

Fig. 9(a)–(c) illustrates the flow and heat characteristics for lar-ger Prandtl number (Pr = 1000) with Ra = 105. As Pr increases to1000, the intensity of clockwise flow circulation increases due tohigh momentum diffusivity for Pr = 1000. But, the qualitativetrends of the streamlines are similar to that of Pr = 0.7 irrespectiveof inclination angle (see Fig. 8(a)–(c) and Fig. 9(a)–(c)). At u = 45�,the primary circulation cells occupy almost 75% of the cavity due toenhanced buoyancy force along the wall AB. The weak clockwisecirculation cells are also observed near to the corner C of the cavity.As u increases to 75�, primary fluid circulation cells grow bigger insize with large magnitude and they occupy entire part of the cavitydue to enhanced convection effect and buoyancy force whereassecondary circulation cells completely disappear (see Fig. 8(c)).The intensity of streamfunction is found to be high compared tothat of Pr = 0.7 for u = 15� whereas that is less for the higher incli-nation angles (u = 45� and u = 75�). Note that, jwjmax = 17.5, 15.5and 17 for u = 15�, u = 45� and u = 75�, respectively at Pr = 1000

whereas jwjmax = 14, 20 and 17.5 for u = 15�, u = 45� and u = 75�,respectively at Pr = 0.7 (see Fig. 8(a)–(c) and Fig. 9(a)–(c)).

Heatlines follow similar qualitative trend as found in earliercase (Pr = 0.7) for Pr = 1000. The dense heatlines corresponding to3 P jPjP 0 in left portion of wall AB and 0 6 jPj 6 3 in right por-tion of wall AB emanate from the wall AB and end on the walls BCand DA. Therefore, large amount of heat is transferred from thewall AB to walls BC and DA (see Fig. 9(a)). Higher magnitudes ofheatlines are observed along the walls BC and DA as jPj varies as0.08 6 jPj 6 4.95 along the wall BC and that varies as0.08 6 jPj 6 4.71 along the wall DA for u = 15�. It is interestingto observe that the isotherms with h 6 0.3 are more compressednear to upper portion as compared to lower portion of side walls(BC and DA). This is due to dense heatlines corresponding to0.08 6 jPj 6 2.5 along that regime. Therefore, thermal boundarylayer thickness near top portion of the side walls is smaller as com-pared to lower portion of the side walls (see Fig. 9(a)). It may benoted that the isotherms are highly compressed along the wallAB due to large magnitudes of heatfunction. Dense heatlines(�3 6P 6 3) near the bottom wall AB indicate smaller thicknessof thermal boundary layer in that regime and also isotherms withh P 0.6 are confined within a small regime near the wall AB (seeFig. 9(a)). Similar to the previous case with Pr = 0.7, the magnitudeof the clockwise heatline cells is higher than those to the anticlock-wise heatline cells. Hence, strong thermal mixing with jPjmax = 8occurs at core of clockwise heatline cells due to recirculation ofheat energy. As a result, large portion of the right half of the cavityis maintained with temperature (h) varying as 0.2 � 0.5 comparedto left half of the cavity (see Fig. 9(a)).

As u increases to 45�, the heat flow from the wall AB to the wallBC increases due to strong anticlockwise circulation in the cavity.This is further due to strong convective flux along the wall AB. Sim-ilar to previous case with u = 15�, heatline cells are identical withflow circulation cells. In contrast to Pr = 0.7, one additional heatcirculation cell is formed near top portion of the wall BC foru = 45� due to convection induced by flow circulation (seeFig. 9(b)). It may be noted that larger portion of wall AB deliversheat to the lower portion of the wall BC based on dense heatlinesin that regime as jPj varies as 1 6 jPj 6 4.6 along the lower portionof the wall BC for u = 45�. Thus, the thermal boundary layer thick-ness is small near lower portion of wall BC. It is also found thatboundary layer thickness near top portion of the wall BC atPr = 1000 is larger than that at Pr = 0.7 for u = 45� due to less heat-function gradient based on the presence of secondary heat circula-tion cells in that regime (see Fig. 9(b)). On the other hand, jPj isfound to vary as 2 6 jPj 6 4.07 along the lower portion of wallDA whereas 1 6 jPj 6 4.6 for the lower portion wall BC atu = 45�. Therefore thermal boundary layer thickness near to lowerportion of the wall DA is larger than that at the lower portion of thewall BC. The heatlines are dense near the wall AB and hence iso-therms with high magnitude h P 0.6 are highly compressed alongthe wall AB. Thermal boundary layer thickness is small near to wallAB based on dense heatlines (see Fig. 9(b)). The closed loop heat-lines cells are formed at the core of the cavity indicating the recir-culation of heat energy or enhanced thermal mixing at the center.As a result, a large regime of central portion of cavity is maintainedwith uniform temperature (h) varies as 0.4–0.6 (see Fig. 9(b)).

As u further increases to 75�, the heatlines exhibit cells similarto streamline at the core (see Fig. 9(c)). Highly dense heatlines areseen along the wall AB. Dense heatlines are also seen at the lowerportion of wall BC. The isotherms are largely compressed along thewall AB and near to the wall BC. Also, thermal boundary layerthickness near to the lower portion of the wall BC is highly reduceddue to increase in heat flow in that regime (see Fig. 9(c)). It may benoted that the magnitude of heatfunction (jPj) along the wall BCvaries as 0.01 6 jPj 6 5.74 whereas 0.04 6 jPj 6 4.6 is observed

A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983 2979

along the wall BC in previous case with u = 45�. Similar to the pre-vious case with Pr = 0.7, the heat transfer rate for the wall DA de-creases with the increase of angle of inclination as jPj varieswithin 0.08 6 jPj 6 4.71 for u = 15�, 0.06 6 jPj 6 4.07 for u = 45�and 0.03 6 jPj 6 3.05 for u = 75� (see Fig. 9(a)–(c)). The closed loopconvective heatline cells are found at the central zone of the cavityindicating enhanced thermal mixing in that region. Large part ofcentral portions of the cavity is maintained at uniform temperatureat around h = 0.4 � 0.6 and h = 0.3 � 0.6 for u = 45� and u = 75�,respectively. As u increases, the convective heat transport in thecentral portion of the cavity gradually becomes weaker. Note that,jPjmax = 8 for u = 15�, jPjmax = 6.5 for u = 45� and jPjmax = 6 foru = 75�.

3.3. Heat transfer rates: local Nusselt numbers

The variation of local heat transfer rates along wall AB (NuAB),wall DA (NuDA) and wall BC (NuBC) vs distance for Pr = 0.015,Pr = 0.7 and Pr = 1000 at Ra = 105 are presented in the Fig. 10(a)–(i). Fig. 10(a)–(c), (d)–(f) and (g)–(i) show the variations forPr = 0.015, Pr = 0.7 and Pr = 1000, respectively. It is observed thatNuAB is found to be infinite at the corners of the wall AB for all val-ues of Pr and u. This may be explained based on gradients of heat-function which are also related to local Nusselt number. Theinfinite value of local Nusselt number at the corner is due to largervalue of heatfunction gradient which is clearly illustrated via heat-lines with the corresponding high magnitude of jPj.

It may be noted that the variation of the local Nusselt number isindependent of the inclination angle for the lower Prandtl number(Pr = 0.015). It is found that NuAB decreases up to the distance beingX 6 0.18 irrespective of u (see Fig. 10(a)) due to decrease in heat-function gradient within 0 6 X 6 0.18 where jPj varies as3.34 P jPjP 0.80 for u = 15�, 3.29 P jPjP 0.73 for u = 45� and3.17 P jPjP 0.70 for u = 75� (see Fig. 7(a)–(c)). Further, NuAB in-creases up to X = 0.35, thereafter that decreases up to X = 0.78. Itis observed that, NuAB distribution attains local maxima within0.3 6 X 6 0.4 along the wall AB due to heatlines corresponding to0.13 6 jPj 6 0.66 for u = 15�, 0.09 6 jPj 6 0.8 for u = 45� and0.08 6 jPj 6 0.91 for u = 75� on that zone (see Fig. 7(a)–(c)). Itmay be noted that NuAB also exhibits local minima within0.7 6 X 6 0.9 due to sparse heatlines corresponding to2.21 6 jPj 6 2.67 for u = 15�, 2.46 6 jPj 6 2.92 for u = 45� and2.54 6 jPj 6 3.07 for u = 75� within 0.7 6 X 6 0.9. Finally, NuAB

starts to increase and reach the maximum value at the corner pointB (see Fig. 10(a)) due to increase in heatfunction gradient as jPjvaries 2.67 6 jPj 6 4.8 for u = 15�, 2.92 6 jPj 6 4.97 for u = 45�and 3.07 6 jPj 6 5.0 for u = 75� within 0.9 6 X 6 1 (see Fig. 7(a)–(c)).

It is observed that heat transfer rate for wall DA (NuDA) is infi-nite at the corner point A of the cavity irrespective of u (seeFig. 10(b)). Thereafter, NuDA decreases up to Y = 0.28 for u = 15�,Y = 0.32 for u = 45� and Y = 0.28 for u = 75� due to sharp decreasein heatfunction as jPj varies 3.34 P jPjP 1.16 within 0 6 Y 6 0.28for u = 15�, 3.29 P jPjP 1.11 within 0 6 Y 6 0.32 for u = 45� and3.17 P jPjP 1.02 within 0 6 Y 6 0.28 for u = 75� (see Fig. 7(a)–(c)). Further, NuDA increases up to Y = 0.6 for all inclination anglesdue to heatlines corresponding to 0.69 6 jPj 6 1.16 for u = 15�,0.63 6 jPj 6 1.11 for u = 45� and 0.62 6 jPj 6 1.02 for u = 75�within 0.36 6 Y 6 0.6 (see Fig. 7(a)–(c)). Similar to the NuAB, varia-tion of NuDA is also independent of the inclination angle. It is ob-served that NuDA distribution attains local maxima within0.55 6 Y 6 0.65 along the wall DA due to heatlines correspondingto 0.77 P jPjP 0.56 for u = 15�, 0.79 P jPjP 0.55 for u = 45�and 0.69 P jPjP 0.54 for u = 75� on that zone (see Fig. 7(a)–(c)).Thereafter NuDA distribution is found to be almost constant within0.8 6 Y 6 1. This is due to sparse heatlines corresponding to

0.23 P jPjP 0.01 for u = 15�, 0.22 P jPjP 0.01 for u = 45� and0.21 P jPjP 0.01 for u = 75� within 0.8 6 Y 6 0.98. It is interest-ing to note that, NuDA tends to almost zero at the corner D (seeFig. 10(b)) which is clearly represented by small magnitudes ofheatfunction at that corner for all the inclination angles (seeFig. 7(a)–(c)).

The variation of NuBC is qualitatively similar to that of NuDA forPr = 0.015 (see Fig. 10(b) and (c)). It is found that NuBC shows itsmaximum at the corner point B of the cavity irrespective of u(see Fig. 10(c)). It may be noted that NuBC distribution decreaseswithin 0 6 Y 6 0.25 for all the inclination angles due to sharp de-crease in heatfunction gradient as jPj varies 4.8 P jPjP 2.11 foru = 15�, 4.97 P jPjP 2.20 for u = 45� and 5.0 P jPjP 2.15 foru = 75� within 0 6 Y 6 0.25 (see Fig. 7(a)–(c)). Further NuBC in-creases up to Y = 0.28 and thereafter, that decreases slowly up toY = 1 due to sparse heatline in that regime (see Fig. 7(a)–(c)).

The local heat transfer rate for Pr = 0.7 is shown in theFig. 10(d)–(f). It is found that NuAB for u = 15� decreases up tothe distance being X = 0.58 (see Fig. 10(d)) due to sharp decreasein heatfunction as jPj varies 4.49 P jPjP 0.51 within0 6 X 6 0.58 for u = 15� (see Fig. 8(a)). Thereafter, NuAB increasesup to X = 1 due to the presence of highly dense heatline circulationcells with higher magnitude. This is further due to increase in theheatfunction gradient as jPj varies 0.51 6 jPj 6 4.41 within0.58 6 X 6 1. It is interesting to observe that NuAB distribution foru = 15� exhibits local minima within 0.5 6 X 6 0.6 along the wallAB due to heatlines corresponding to 0.33 6 jPj 6 0.56 foru = 15� on that zone. Note that, for u = 45�, NuAB decreases up toX = 0.9 and thereafter a sharp rise is observed (see Fig. 10(d)). Fur-ther, NuAB reaches the maximum value at the X = 1. It is observedthat NuAB for u = 75� is qualitatively similar to that for u = 45�. Itmay be noted that the magnitude of NuAB at u = 15� is more thanthat of u = 45� and u = 75� within 0.7 6 X 6 1 along the wall ABdue to heatlines corresponding to 0.91 6 jPj 6 4.41 for u = 15�whereas those correspond to 3.01 6 jPj 6 5.38 for u = 45� and3.45 6 jPj 6 5.82 for u = 75� within 0.7 6 X 6 1 (see Fig. 8(a)–(c)).The local minima of NuAB is attained for u = 45� and u = 75� within0.75 6 X 6 0.85 due to heatlines corresponding to3.20 6 jPj 6 3.44 for u = 45� and 3.66 6 jPj 6 3.91 for u = 75� onthat zone (see Fig. 8(b) and (c)).

Fig. 10(e) illustrates the local heat transfer rates along the wallDA for Pr = 0.7. It is found that NuDA decreases within 0 6 Y 6 0.18for u = 15�, 0 6 Y 6 0.18 for u = 45� and 0 6 Y 6 0.25 for u = 75�(see Fig. 10(e)). This is due to sharp decrease in heatfunction asjPj varies as 4.49 P jPjP 2.67 for u = 15� within 0 6 Y 6 0.18,3.44 P jPjP 1.69 for u = 45� within 0 6 Y 6 0.18 and2.94 P jPjP 1.16 for u = 75� within 0 6 Y 6 0.25 (see Fig. 8(a)–(c)). Thereafter, NuDA increases up to Y = 0.9 but decreases slightlyat the later regime, 0.9 6 Y 6 1 for all the inclination angles (seeFig. 10(e)). Maximum local heat transfer rate (NuDA) is observedfor u = 15� followed by u = 45� and u = 75� within 0.2 6 Y 6 1. Thisis clearly illustrated based on the heatfunction gradient as jPj var-ies as 2.66 P jPjP 0.07 for u = 15� whereas 1.68 P jPjP 0.03 foru = 45� and 1.18 P jPjP 0.02 for u = 75� within 0.2 6 Y 6 0.98(see Fig. 8(d)–(f)). It may be noted that local maxima in NuDA dis-tribution for u = 15� occurs at Y = 0.82 due to heatlines corre-sponding to 0.85 P jPjP 0.37 within 0.8 6 Y 6 0.9 along the wallDA.

Fig. 10(f) illustrates that NuBC for u = 15� sharply decreases upto Y = 0.18 due to heatlines corresponding to 4.41 P jPjP 2.5within 0 6 Y 6 0.18. Thereafter, NuBC increases up to Y = 0.8 dueto dense heatlines corresponds to 0.88 6 jPj 6 2.5. Further, NuBC

remains almost constant in the later regime 0.8 6 Y 6 1 (seeFig. 10(f)). It is found that NuBC sharply decreases up to distancebeing Y = 0.8 for the higher inclination angles (u = 45� andu = 75�) (see Fig. 10(f)) due to sharp change in heatfunction within

0 0.2 0.4 0.6 0.8Distance

-5

0

5

10

15

Nu A

B

Wall AB

Pr

=0.

015

(a) (b) (c)

0 0.2 0.4 0.6 0.8Distance

-5

0

5

10

15

Nu D

A

Wall DA

0 0.2 0.4 0.6 0.8Distance

-5

0

5

10

15

Nu B

C

Wall BC

0 0.2 0.4 0.6 0.8Distance

-5

0

5

10

15

Nu A

B

Wall AB

Pr

=0.

7

(d) (e) (f)

0 0.2 0.4 0.6 0.8Distance

-5

0

5

10

15

Nu D

A

Wall DA

0 0.2 0.4 0.6 0.8Distance

-5

0

5

10

15

Nu B

C

Wall BC

0.2 0.4 0.6 0.8Distance

-5

0

5

10

15

Nu A

B

Wall AB

Pr

=10

00

(g) (h) (i)

0 0.2 0.4 0.6 0.8Distance

-5

0

5

10

15

Nu D

A

Wall DA

0 0.2 0.4 0.6 0.8Distance

-5

0

5

10

15

Nu B

C

Wall BC

Fig. 10. Variation of local Nusselt number (NuAB, NuDA and NuBC) with distance for u = 15�(. . ..), u = 45�(– – –) and u = 75�(�) with Pr = 0.015 ((a) NuAB, (b) NuDA, (c) NuBC),Pr = 0.7 ((d) NuAB, (e)NuDA, (f) NuBC) and Pr = 1000 ((g)NuAB, (h) NuDA, (i) NuBC).

2980 A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

0 6 Y 6 0.8 where jPj varies as 5.38 P jPjP 0.24 for u = 45� and5.82 P jPjP 0.21 for u = 75� within 0 6 Y 6 0.8. Thereafter valueof NuBC remains almost constant within 0.8 6 Y 6 1. The variationof local heat transfer rates (NuAB,NuDA and NuBC) for Pr = 1000 isqualitatively similar to that of Pr = 0.7 and a similar explanationfollows (see Fig. 10(d)–(f) and Fig. 10(g)–(i)). The only exceptionmay be noted as the quantitative difference is slightly more forPr = 1000 between the magnitudes of NuBC for u = 45� andu = 75� as compared to the results (NuBC) for Pr = 0.7 (seeFig. 10(f) and Fig. 10(i)). The physical reason may be explainedbased on heatfunction gradient.

3.4. Overall Heat transfer rates and average Nusselt numbers

The overall effects on heat transfer rates via average Nusseltnumber for wall AB ðNuABÞ, wall DA ðNuDAÞ and wall BC ðNuBCÞ vslogarithmic Rayleigh number for Pr = 0.015, Pr = 0.7 and Pr = 1000are displayed in Fig. 11(a)–(i). It may be noted that the averageNusselt number on cold walls BC and DA ðNuBC and NuDAÞ refer tothe amount of heat received by those cold walls while averageNusselt number on hot wall AB ðNuABÞ refers to the amount of heattaken away from the hot wall. The average Nusselt numbersðNuAB;NuDA and NuBCÞ are obtained using Eq. (23), where the inte-gral is evaluated using Simpson’s 1

3 rule. Fig. 11(a)–(f) illustratesthat average Nusselt number ðNuAB; NuDA and NuBCÞ increases withRayleigh number irrespective of u and Pr. It may be noted that theaverage heat transfer rate ðNuAB;NuDA and NuBCÞ is high at Ra = 105

compared to Ra = 103 due to higher heatfunction gradients at high-er Ra (Ra = 105). A least square curve is fitted and the overall erroris estimated within 1%. Tables 3–5 show average Nusselt numbercorrelation along hot wall AB and cold walls BC and DA atPr = 0.015 and Pr = 1000.

It is observed that NuAB distribution is almost similar for all theinclination angles with Pr = 0.015. Note that, NuAB distribution re-mains almost constant up to Ra = 3 � 103 irrespective of inclinationangle (see Fig. 11(a)–(c)). The constant value of NuAB is clearly ob-served from the heatfunction values corresponding to0.006 6 jPj 6 2.95 for u = 15�, 0.006 6 jPj 6 3.04 for u = 45� and0.004 6 jPj 6 3.07 for u = 75� at Ra = 103 whereas those corre-spond to 0.004 6 jPj 6 3.15 for u = 15�, 0.02 6 jPj 6 3.33 foru = 45� and 0.006 6 jPj 6 3.37 for u = 75� at Ra = 3 � 103. Further,NuAB increases within 3 � 103

6 Ra 6 105 (see Fig. 11(a)) due to in-crease in the heatfunction gradients because jPj varies as0.01 6 jPj 6 4.80 for u = 15�, 0.04 6 jPj 6 4.97 for u = 45� and0.04 6 jPj 6 5.0 for u = 75� at Ra = 105 (see Fig. 7(a)–(c)).

The variation of NuDA is qualitatively similar to that of NuAB forPr = 0.015 (see Fig. 11(a) and (b)). It may be noted that NuDA is al-most constant up to Ra = 8 � 103 irrespective of inclination angle.Thereafter, increasing trend in NuDA is observed due to larger gra-dients of heatfunction corresponding to 0.01 6 jPj 6 3.34 foru = 15�, 0.01 6 jPj 6 3.29 for u = 45� and 0.01 6 jPj 6 3.17 foru = 75� at Ra = 105 whereas those correspond to 0.01 6 jPj 6 2.70for u = 15�, 0.01 6 jPj 6 2.68 for u = 45� and 0.19 6 jPj 6 2.65 foru = 75� at Ra = 8 � 103. It is also found that NuBC is a increasing

103 104 105

Rayleigh Number

6

8

10

Nu A

B

Pr

=0.

015

(a) Wall AB

103 104 105

Rayleigh Number

2

3

4

5

6

Nu D

A

(b) Wall DA

103 104 105

Rayleigh Number

2

3

4

5

6

Nu B

C

(c) Wall BC

103 104 105

Rayleigh Number

6

8

10

Nu A

B

Pr

=0.

7

(d) Wall AB

103 104 105

Rayleigh Number

2

3

4

5

6

Nu D

A

Wall DA(e)

103 104 105

Rayleigh Number

2

3

4

5

6

Nu B

C

(f) Wall BC

103 104 105

Rayleigh Number

6

8

10

Nu A

B

Pr

=10

00

(g) Wall AB

103 104 105

Rayleigh Number

2

3

4

5

6

Nu D

A

(h) Wall DA

103 104 105

Rayleigh Number

2

3

4

5

6

Nu B

C

(i) Wall BC

Fig. 11. Variation of average Nusselt number ðNuAB;NuDA and NuBCÞ with Rayleigh number for u = 15�(. . ..), u = 45�(– – –) and u = 75�(�) with Pr = 0.015 ((a) NuAB , (b) NuDA ,(c) NuBC ), Pr = 0.7 ((d) NuAB , (e) NuDA , (f) NuBC ) and Pr = 1000 ((g) NuAB , (h) NuDA , (i) NuBC ).

Table 3Correlations for average Nusselt number along wall AB ðNuABÞ.

u NuAB

Pr = 0.015 Pr = 1000

15� 2.4829Ra0.1028 Ra P 6 � 103 1.4394Ra0.1658 Ra P 7 � 103

45� 2.6495Ra0.0985 Ra P 5 � 103 2.1006Ra0.1229 Ra P 2 � 104

75� 2.7640Ra0.0938 Ra P 5 � 103 2.2590Ra0.1183 Ra P 6 � 103

Table 4Correlations for average Nusselt number along wall BC ðNuBC Þ.

u NuBC

Pr = 0.015 Pr = 1000

15� 1.2276Ra0.1178 Ra P 3 � 103 0.6607Ra0.1752 Ra P 8 � 103

45� 1.3582Ra0.1122 Ra P 2 � 103 0.9598Ra0.1360 Ra P 5 � 104

75� 1.3889Ra0.1108 Ra P 2 � 103 1.0768Ra0.1452 Ra P 3 � 103

Table 5Correlations for average Nusselt number along wall DA ðNuDAÞ.

u NuDA

Pr = 0.015 Pr = 1000

15� 1.2953Ra0.0817 Ra P 7 � 103 0.7823Ra0.1559 Ra P 7 � 103

45� 1.3145Ra0.0793 Ra P 7 � 103 0.5451Ra0.1752 Ra P 2 � 104

75� 1.4167Ra0.0696 Ra P 7 � 103 1.1785Ra0.0824 Ra P 2 � 104

A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983 2981

function within 1036 Ra 6 105 for Pr = 0.015 as seen from the

Fig. 11(c). The increasing trend in NuBC is observed due to the largergradients of heatfunctions corresponding to 0.01 6 jPj 6 4.80for u = 15�, 0.01 6 jPj 6 4.97 for u = 45� and 0.01 6 jPj 6 5.0 foru = 75� at Ra = 105 whereas those vary as 0.005 6 jPj 6 2.95for u = 15�, 0.005 6 jPj 6 3.04 for u = 45� and 0.004 6 jPj 6 3.07for u = 75� at Ra = 103 (see Fig. 5(a)–(c) and Fig. 7(a)–(c)).

The trend on variation of NuAB at Pr = 0.7 is qualitatively similarto that of NuAB with Pr = 0.015 (see Fig. 11(a) and Fig. 11(d)). It maybe noted that NuDA remains constant up to Ra = 3 � 103 for u = 15�due to heatfunction gradients corresponding to 0.005 6 jPj 6 2.84for u = 15� at Ra = 103 and 0.01 6 jPj 6 2.79 at Ra = 3 � 103. Fur-ther, NuDA increases up to Ra = 105 due to increase in heatfunctioncorresponding to 0.07 6 jPj 6 4.49 at Ra = 105 whereas0.01 6 jPj 6 2.79 is observed at Ra = 3 � 103 for u = 15�. Similarly,it is observed that NuDA is almost constant up to Ra = 104 and there-after increases to reach maxima at Ra = 105 for higher inclinationangles (u = 45� and u = 75�). Note that jPj varies as 0.03 6 jPj6 3.44 for u = 45� and 0.02 6 jPj 6 2.94 for u = 75� at Ra = 105

(Fig. 8(b) and (c)) whereas 0.01 6 jPj 6 2.58 for u = 45� and0.01 6 jPj 6 2.75 for u = 75� occurred at Ra = 104 (Figure notshown).

Fig. 11(f) illustrates the distribution of average Nusselt numberfor wall BC. It is found that NuBC slowly increases from Ra = 103 toRa = 104 for u = 15�, based on the heatfunction values correspond-ing to 0.027 6 jPj 6 3.24 at Ra = 104 and 0.005 6 jPj 6 2.94 is ob-served at Ra = 103. Further, NuBC increases up to Ra = 105 due tolarge heatfunction gradient corresponding to 0.06 6 jPj 6 4.41 at

2982 A.K. Singh et al. / International Journal of Heat and Mass Transfer 55 (2012) 2965–2983

Ra = 105 and 0.027 6 jPj 6 3.24 at Ra = 104. It is observed that NuBC

increases from Ra = 103 to Ra = 105 for higher inclination angles(u = 45� and u = 75�). The increasing trend in NuBC for higher incli-nation angles is observed due to larger gradients of heatfunctioncorresponding to 0.01 6 jPj 6 5.38 for u = 45� and 0.01 6 jPj6 5.82 for u = 75� at Ra = 105 whereas those correspond to0.005 6 jPj 6 3.04 for u = 45� and 0.005 6 jPj 6 3.09 for u = 75�at Ra = 103. The variation of average Nusselt numbersðNuAB; NuDA and NuBCÞ for Pr = 1000 is qualitatively similar to thatof Pr = 0.7 and a similar explanation follows (see Fig. 11(d)–(f) andFig. 11(g)–(i)). The only exception may be noted as trend of NuBC

for Pr = 1000 is slightly different from that Pr = 0.7. It is observedthat NuBC for u = 45� follow wavy pattern due to secondary heat-line circulation cells at higher Prandtl number (Pr = 1000).

4. Conclusions

The streamlines and heatlines are used to analyze the fluid flowand heat flow characteristics during natural convection insidetilted square cavities to find the role of inclination angle (u) forenhancing the heat distribution and thermal mixing within theenclosure. The effects of both the tangential and normal compo-nents of buoyancy force relative to hot inclined wall, AB play crit-ical role for both flow and thermal characteristics. Bejan’s heatlineconcept is used to visualize the heat flow inside the cavity andimportant results are listed as follows:

� Strong anticlockwise circulation cells with weak clockwise cir-culation cells are found to occur for u = 15� whereas clockwisecirculation cells are absent and anticlockwise circulation cellsspan almost the entire cavity for the higher us (u = 45� and75�) at Ra = 103 irrespective of Pr.� At convection dominant region (Ra = 105), secondary flow circu-

lation cells occur near corner regions of cavity for all u s atPr = 0.015. The anticlockwise circulation cells tend to becomestrong and clockwise circulation cells tend to disappear onincreasing of inclination angle at higher Pr (Pr = 0.7 and 1000).� The closed loop heatline cells are observed near central portion

of the cavity due to onset of convection and that results in recir-culation of heat energy at that regime irrespective of u forRa � 104 at Pr = 0.015. Stronger closed loop heatline cells leadto enhanced thermal mixing for Ra = 105 and thus central por-tion of the cavity is maintained at h = 0.3 � 0.4.� The heat transfer rate along the wall BC increases with inclina-

tion angle whereas that decreases along the wall DA with theincrease in inclination angle irrespective of Pr and Ra.� The variation of average Nusselt numbers shows that maximum

heat transfer occurs from the wall AB to wall BC (maximumNuBCÞ for u = 75� whereas maximum heat transfer rate fromthe wall AB to wall DA (maximum NuDAÞ is observed foru = 15� irrespective of Pr.

Acknowledgments

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

Appendix A

The name ‘‘isoparametric’’ derives from the fact that the sameparametric function describing the geometry may be used forinterpolating spatial variable within an element. Fig. 2 shows thetransformation between (x,y) and (n,g) coordinates can be defined

by X ¼P9

k¼1Ukðn;gÞ xk and Y ¼P9

k¼1 Ukðn;gÞyk where (xk,yk) arethe X, Y coordinates of the k nodal points as seen in Fig. 2 andUk(n,g) is the basis function. The nine basis functions are:

U1 ¼ ð1� 3nþ 2n2Þð1� 3gþ 2g2Þ;U2 ¼ ð1� 3nþ 2n2Þð4g� 4g2Þ;U3 ¼ ð1� 3nþ 2n2Þð�gþ 2g2Þ;U4 ¼ ð4n� 4n2Þð1� 3gþ 2g2Þ;U5 ¼ ð4n� 4n2Þð4g� 4g2Þ;U6 ¼ ð4n� 4n2Þð�gþ 2g2Þ;U7 ¼ ð�nþ 2n2Þð1� 3gþ 2g2Þ;U8 ¼ ð�nþ 2n2Þð4g� 4g2Þ;U9 ¼ ð�nþ 2n2Þð�gþ 2g2Þ;

The above basis functions are used for mapping the tilted squaredomain or elements within the tilted square into square domainand the evaluation of integrals of residuals.

Appendix B

B.1. Derivation of boundary conditions for heatfunction

In order to obtain unique solution of the Eq. (26), the residualequation, Eq. (28) is solved with various Dirichlet and Neumannboundary conditions may be represented by Dirichlet boundarycondition obtained from the Eq. (25). A reference value of P is as-sumed as 0 at D(� sin u, cos u) and hence P = 0 is valid for adiabaticwall CD. The tangent vector of walls DA and BC are given as TDA = i sinu � j cos u and TBC = �i sin u + j cos u where dP =rP.T ds.

(a) At D(� sin u, cos u):

Pð� sinu; cos uÞ ¼ 0:

At A(0,0)

PðAÞ ¼ PðDÞ þZ A

DrP:TDAds ð33Þ

¼Z A

Dsin u

oPoX� cos u

oPoY

� �ds

¼Z A

Dcos u

ohoXþ sinu

ohoY

� �ds ¼ NuDA: ð34Þ

(b) At C(cos u � sin u, cos u + sin u):

P cos u� sinu; cos uþ sinuð Þ ¼ 0:

At B(cos u, sin u):

PðBÞ ¼ PðCÞ þZ B

CrP � TBCds ð35Þ

¼ �Z C

B� sinu

oPoXþ cos u

oPoY

� �ds

¼ �Z C

B� cos u

ohoX� sin u

ohoY

� �ds ¼ �NuBC : ð36Þ

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