A complete heatline analysis on mixed convection within a square cavity: Effects of thermal boundary conditions via thermal aspect ratio

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International Journal of Thermal Sciences 57 (2012) 98e111

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International Journal of Thermal Sciences

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A complete heatline analysis on mixed convection within a square cavity: Effectsof thermal boundary conditions via thermal aspect ratio

D. Ramakrishna a, Tanmay Basak b, S. Roy a, I. Pop c,*

aDepartment of Mathematics, Indian Institute of Technology Madras, Chennai - 600036, IndiabDepartment of Chemical Engineering, Indian Institute of Technology Madras, Chennai - 600036, Indiac Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

a r t i c l e i n f o

Article history:Received 23 May 2011Received in revised form12 January 2012Accepted 13 January 2012Available online 6 March 2012

Keywords:StreamlinesHeatlinesMixed convectionSquare cavityThermal aspect ratioPenalty finite element method

* Corresponding author. Tel: þ40 264 594315; fax:E-mail addresses: [email protected] (T. Basak

[email protected] (I. Pop).

1290-0729/$ e see front matter � 2012 Elsevier Masdoi:10.1016/j.ijthermalsci.2012.01.010

a b s t r a c t

In this article, numerical investigation is carried out for mixed convection heat transfer within squarecavities for various thermal boundary conditions on bottom and side walls based on thermal aspect ratio(A). A penalty finite element analysis with bi-quadratic elements has been used to investigate the resultsin terms of isotherms, streamlines, heatlines and average Nusselt numbers for a wide range of param-eters (1 � Re�100, 0.015 � Pr�10, 103 � Gr�105). A detailed analysis of flow pattern shows that naturalconvection or forced convection depends on both parameters: Ri (Ri ¼ Gr/Re2) and Pe (Pe ¼ Re$Pr).Results indicate that, at low Pr (Pr ¼ 0.015) with low Gr (Gr ¼ 103), isotherms are decoupled with flowprofile and conduction dominant heat transfer is observed irrespective of Re, due to low Peclet number.At Gr ¼ 103, lid-driven force dominates and the non-symmetric flow distribution occurs irrespective of Re(1,10 and 100), Pr (0.015, 0.7 and 10) and thermal aspect ratio (0.1, 0.5 and 0.9). At Gr ¼ 105 with Re ¼ 1,natural convection dominates the flow irrespective of Pr and A. Considerably smaller dominance of lid-driven force is observed over buoyancy force at Gr ¼ 105 with Re ¼ 10 irrespective of Pr for A ¼ 0.1 and0.5, whereas strong effect of lid-driven force is found at Gr ¼ 105 with Re ¼ 100 irrespective of Pr and A.Multiple circulations are found in streamlines and heatlines especially for A ¼ 0.5 and 0.9 at high Rey-nolds number (Re ¼ 100) with Pr ¼ 10 and Gr ¼ 105. It is found that, streamlines and heatlines circulationcells follow qualitatively similar pattern for higher Pr (Pr � 0.7) at Gr ¼ 105 irrespective of Re. Thermalgradient is found to be high at the center of the bottom wall for A ¼ 0.1 due to highly dense heatlines atthat zone whereas that is low for A ¼ 0.9 irrespective of Re, Pr and Gr. It is also found that, as thermalaspect ratio increases, the average Nusselt number decreases for Pr ¼ 0.015 and Pr ¼ 0.7 irrespective ofRe. Finally, it is concluded that overall heat transfer rates are higher for A ¼ 0.1 as compared to otherthermal aspect ratios (A ¼ 0.5, A ¼ 0.9) irrespective of Pr (0.015 � Pr�10), Re (1 � Re�100) and Gr(103 � Gr�105).

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1. Introduction

The study of mixed convection heat transfer within closedcavities has been regarded as one of the most important researchtopics due to its wide spectrum of scientific and engineeringapplications such as chemical vapour deposition processes [1],electronic cooling devices [2e4], polymer processing [5], green-house covering [6], energy extraction [7], manufacturing processes[8], material processing [9], wedge flow processes [10], stretchingsheet extraction processes [11] etc. In mixed convection flows, the

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forced convection and free convection effects are of comparablemagnitudes. The non-dimensional parameters which influence theheat transfer during mixed convection are Grashof number (Gr),Reynolds number (Re) and Prandtl number (Pr) where Gr and Rerepresent the strength of the natural convection and forcedconvection, respectively. Another dimensionless parameter,Richardson number Ri ¼ Gr/Ren, also involves in the analysis ofmixed convection which represents the importance of naturalconvection relative to the forced convection. The exponent n,depends on the geometry, thermal boundary condition and thefluid. The natural convection is negligible as Ri/0 and forcedconvection is negligible as Ri/N.

Extensive investigations were carried out numerically todescribe mixed convection heat transfer in lid-driven cavities. Al-Amiri et al. [12] analyzed the mixed convection heat transfer in

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X, U

Y, V

U=1, V=0 Adiabatic Wall

θ= A(1−Y) θ= A(1−Y)

πA+(1−A) Sin( X) θ =

g

T=Th

T=TcT=Tc

T=Th

T=TH

Fig. 1. Schematic diagram of the physical system.

D. Ramakrishna et al. / International Journal of Thermal Sciences 57 (2012) 98e111 99

lid-driven cavity with sinusoidal wavy bottom surface where thevertical walls are insulated and the wavy bottom surface is main-tained at a uniform temperature which is higher than that of thetop lid. Mohamad and Viskanta [13] investigated transientconvective motion and heat transfer in a square cavity driven bycombined temperature gradient and imposed lid shear where thehorizontal walls are adiabatic and the vertical walls are maintainedat different but constant temperatures. Further, Mohamad andViskanta [14] studied the effect of lid motion on flow structure andheat transfer in stably stratified laminar flow in a shallow cavityheated from above and cooled from below. Cheng [15] analyzed thecharacteristics of mixed convection flows and heat transfer detailsin a lid-driven square cavity when both Gr and Re are increasedsimultaneously from the steady, laminar flow regimes to a condi-tion where Hopf bifurcation occurs. The bottom wall of the squarecavity is hot and top moving wall is maintained at constant coldtemperature whereas the vertical side walls are insulated. Maha-patra et al. [16] investigated opposing mixed convection ina differentially heated partitioned square enclosure. The left wall ofthe enclosure is hot and right wall is maintained at constant coldtemperature whereas horizontal walls are insulated. Some inves-tigations related to mixed convection conduction-convectionconjugate heat transfer problems for different types of fins whichcan be applied to chemical, mechanical and electrical industrialapplications are also available in the literature. Hsiao and Hsu [17]studied conjugate mixed convection heat transfer of a second-grade visco-elastic fluid past a triangular fin fluid with variousconduction-convection parameters. Hsiao [18] also proposed a newand effective improved semi-analytic and semi-empirical formulato solve a conjugate free convection problem in the incompressiblelaminar boundary layer flow and heat conduction in a solid wall forthe flow passing a flat plate fin.

A few other applications on mixed convection involve variouscomplex situations. Aydin [19] investigated the transport mecha-nism of laminar mixed convection in a shear and buoyancy drivencavity where two orientations of thermal boundary conditions atthe cavity walls are considered to simulate the aiding and opposingbuoyancy mechanisms. Reima et al. [20] studied the flow ofa viscous thermally-stratified fluid in a square container where theflow is driven by the top lid of the container, which executestorsional oscillations. Further, Reima et al. [21] numerically studiedthe effect of oscillating lid for a viscous thermally-stratified fluid ina square container. Reima and Hyun [22] also studied threedimensional flows in a cubical container with a vertical tempera-ture gradient where flows are driven by the top lid, which slides inits own plane at a constant speed. The top wall is maintained athigher temperature than the bottom wall, whereas the end wallsand side walls are thermally insulated. Oztop and Dagtekin [23]analyzed mixed convection heat transfer in a vertical two sidedlid-driven differentially heated square cavity where moving sidewalls are maintained at different constant temperatures while thehorizontal walls are thermally insulated.

However, most of the above investigations have been analyzedbased on streamlines and isotherms and the detailed analysis ofheat flow was poorly understood. Streamlines adequately depictfluid flow whereas isotherms indicate only temperature distribu-tionwhichmay not be very useful as heat flowwould nomore be innormal direction to isotherms for a convection dominant regime.The heatline technique is the best way to visualize the heat transferoccurring in a two-dimensional convective transport process. Theconcept of ‘heatlines’was introduced by Kimura and Bejan [24]. Theheatlines, analogous to streamlines, are trajectories of flow of heatenergy and thus, they are useful to visualize the total energy flow.Heatlines are mathematically represented by heatfunctions andproper dimensionless form of heatfunctions are closely related to

Nusselt numbers. The heatline concept since then was applied andextended to various problems [25e40]. A comprehensive review ofheatlines with various applications was reported by Costa [32]. Adetailed study on influence of thermal aspect ratio during mixedconvection heat transfer is yet to appear in the literature.

The present study deals with mixed convection within squarecavity based on thermal aspect ratio on bottom and sidewalls whenthe top wall is adiabatic. The important non-dimensional param-eter, thermal aspect ratio (A) is considered for thermal boundaryconditions. By varying thermal aspect ratio (A) from 0 to 1, variousthermal boundary conditions are imposed on the bottom and sidewalls, such as A ¼ 0 corresponds to non-uniformly heated bottomwall and cold side walls whereas A ¼ 1 corresponds to uniformlyheated bottom wall and linearly heated side walls.

In the current study, the non-linear coupled partial differentialequations of flow and temperature fields have been solved usingGalerkin finite element method. Finite element method is furtheremployed to solve the Poisson equation for streamfunctions andheatfunctions. The advantage of this method is that the homoge-neous Neumann boundary conditions are automatically built in theformulations. Note that, solving heatfunction with various non-homogeneous boundary conditions is cumbersome and wepropose a generic finite element post processing for heatfunctionequation. The heat transfer characteristics are studied by analyzingaverage Nusselt numbers. Also, various qualitative features ofaverage Nusselt numbers are adequately explained based onheatlines. The heating and thermal mixing have been studied forvarious lid velocities in terms of Re ranging from 1 to 100. Also,various fluids of scientific and industrial importance have beenchosen for the study, namely molten metals (Pr ¼ 0.015), air(Pr ¼ 0.7) and aqueous chemical solution (Pr ¼ 10).

2. Mathematical formulation and simulation

2.1. Governing equations and boundary conditions

A schematic diagram of two-dimensional square cavity with thephysical dimensions as shown in Fig. 1. The top wall is assumed tomove from left to right with a uniform velocity U0. The flow isassumed to be laminar and the fluid properties are assumed to beconstant except for the density variation which is assumed tofollow Boussinesq approximation while viscous dissipation termsare considered to be negligible. The viscous incompressible flowand the temperature distribution inside the cavity are governed bythe NaviereStokes and the energy equations, respectively. The aim

D. Ramakrishna et al. / International Journal of Thermal Sciences 57 (2012) 98e111100

of the present study is to investigate the steady solutions andhence, we have considered the time independent differentialequations. The governing equations for steady mixed convectionflow in square cavity using conservation of mass, momentum andenergy in dimensionless form can be written as:

vUvX

þ vVvY

¼ 0; (1)

UvUvX

þ VvUvY

¼ �vPvX

þ 1Re

v2UvX2 þ v2U

vY2

!; (2)

UvVvX

þ VvVvY

¼ �vPvY

þ 1Re

v2VvX2 þ v2V

vY2

!þ GrRe2

q; (3)

Uvq

vXþ V

vq

vY¼ 1

RePr

v2q

vX2 þv2q

vY2

!: (4)

with following boundary conditions

U ¼ 0;V ¼ 0; q ¼ Aþ ð1� AÞsinðpXÞcY ¼ 0;0 � X � 1;

U ¼ 0;V ¼ 0; q ¼ Að1� YÞ;cX ¼ 0;0 � Y � 1;

U ¼ 0;V ¼ 0; q ¼ Að1� YÞ;cX ¼ 1;0 � Y � 1;

U ¼ 1;V ¼ 0;vq

vY¼ 0;cY ¼ 1;0 � X � 1:

(5)

The dimensionless variables and parameters are defined asfollows:

X ¼ xL;Y ¼ y

L;U ¼ u

U0;V ¼ v

U0; q ¼ T � Tc

TH � Tc;A ¼ Th � Tc

TH � Tc

P ¼ pL2

ra2;Pr ¼ n

a;Re ¼ U0L

n; Gr ¼ gbðTH � TcÞL3

n2

(6)

Here x and y are the distances measured along the horizontaland vertical directions, respectively; u and v are the velocitycomponents in the x and y directions, respectively; T denotes thetemperature; p is the pressure and r is the density; Th is thetemperature at the bottom edges of the side walls, Tc is thetemperature at the top edges of the side walls and TH is thetemperature at the center of the bottomwall; L is the length of theside of the square cavity; U0 is the velocity of the top wall; X and Yrepresent dimensionless horizontal and vertical coordinates,respectively; U and V represent the velocity components in the Xand Y directions, respectively; A is the thermal aspect ratio; q is thedimensionless temperature; P is the dimensionless pressure; Re, Prand Gr are Reynolds, Prandtl and Grashof numbers, respectively.

2.2. Numerical solution procedure

The momentum and energy balance equations (Eqs. (2)e(4)which are elliptic system of the equations have been solved usingthe Galerkin finite element method. The continuity equation (Eq.(1)) is used as a constraint due to mass conservation and thisconstraint may be used to obtain the pressure distribution. In orderto solve Eqs. (2)e(3), we use the penalty finite element methodwhere the pressure P is eliminated by a penalty parameter, g andthe incompressibility criteria given by Eq. (1) results in

P ¼ �g

�vUvX

þ vVvY

�: (7)

The continuity equation (Eq. (1)) is automatically satisfied forlarge values of g. Typical values of g that yield consistent solutionsare 107. Using Eq. (7), the momentum balance equations (Eqs.(2)e(3) reduce to

UvUvX

þ VvUvY

¼ gv

vX

�vUvX

þ vVvY

�þ 1Re

v2UvX2 þ v2U

vY2

!; (8)

UvVvX

þ VvVvY

¼ gv

vY

�vUvX

þ vVvY

�þ 1Re

v2VvX2 þ v2V

vY2

!þ GrRe2

q:

(9)

The system of equations [Eqs. (4), (8) and (9)] with boundaryconditions [Eq. (5)] are solved by using Galerkin finite elementmethod [41]. Since the solution procedure is explained in an earlierwork [42,43], the detailed description is not included in this paper.The numerical solutions are obtained in terms of velocity compo-nents (U, V).

2.3. Streamfunction, Nusselt number and heatfunction

2.3.1. StreamfunctionThe fluid motion is displayed using the streamfunction (j)

obtained from velocity components U and V. The relationshipsbetween streamfunction, (j) and velocity components for two-dimensional flows are

U ¼ vj

vYand V ¼ �vj

vX; (10)

which yield a single equation

v2j

vX2 þv2j

vY2 ¼ vUvY

� vVvX

: (11)

The no-slip condition is valid at all boundaries and there is nocross flow. Eq. (11) has been solved by finite element method asdiscussed in earlier work [44,45]. The sign convention is as follows.Positive sign of j denotes anti-clockwise circulation and clockwisecirculation is represented by negative sign of j.

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

number (Nu) is defined by

Nu ¼ �vq

vn; (12)

where n denotes the outward normal direction on a plane. The localNusselt numbers at bottom wall (Nub), at the left wall (Nul) and atthe right wall (Nur) are defined as

Nub ¼X9i¼1

qivFi

vY; (13)

Nul ¼X9i¼1

qivFi

vX; (14)

Nur ¼ �X9i¼1

qivFi

vX(15)

In general, the average Nusselt numbers at the bottom, left andright walls in absence of any distributed heat sources are given by

Table 1Comparison of average Nusselt number for bottom wall with the benchmark solu-


Z1Nu dX

tions of Moallemi and Jang [47] for mixed convection in a lid-driven square cavity forvarious Grashof numbers (Gr) and Prandtl (Pr) at Re ¼ 500. Current study employs28 � 28 bi-quadratic elements (57 � 57 grid points).

Pr Gr Current work Nu Moallemi and Jang [47] Nu

0.01 104 1.0431 1.01670.01 105 1.0721 1.03680.1 104 2.3815 2.23820.1 105 2.8704 2.62901 104 5.5695 5.60891 105 6.3313 6.2118

Nub ¼ 0

b

Xj10¼Z10

NubdX; (16)

Nul ¼

Z10

NuldY

Y j10¼Z10

NuldY; (17)

and

Nur ¼

Z10

NurdY

Y j10¼Z10

NurdY : (18)

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

heatfunction (H) obtained fromconductive heatfluxes�� vq

vX,�vq

vY

�as well as convective heat fluxes (Uq, Vq). The steady energy balanceequations [Eq. (4)] can be arranged as

v

vX

�Uq� 1

Re Pr

�vq

vX

��þ v

vY

�Vq� 1

Re Pr

�vq

vY

��¼ 0: (19)

The heatfunction satisfies the above equation such that

vHvY

¼ Uq� 1Re Pr

vq

vX

�vHvX

¼ Vq� 1Re Pr

vq

vY

(20)

which yield a single equation

v2HvX2 þ v2H

vY2 ¼ v

vYðUqÞ � v

vXðVqÞ: (21)

The sign convention for heatfunction is as follows. The positivesign of H denotes anti-clockwise heat flow and clockwise heat flowis represented by negative sign of H. Expanding the heatfunction(H) using the basis set F as H ¼ PN

k¼1HkFkðX;YÞ. Eq. (21) can besolved using Galerkin finite element method as discussed in earlierworks [44e46]. The boundary conditions of heatfunction are asfollows. Neumann boundary conditions for H are obtained due toisothermal (hot or cold) wall based on Eq. (20) and they are spec-ified as follows:

(a) for left wall

n$VH ¼ ARe Pr

(22)

(b) for bottom wall

n$VH ¼ pð1� AÞcosðpXÞRePr

(23)

(c) for right wall

n$VH ¼ � ARePr

(24)

and

(d) for top wall

H ¼ constant ðadiabatic wallÞThe top insulated wall may be represented by Dirichlet

boundary condition as obtained from Eq. (20) which is simplifiedinto vH/vX ¼ 0 for an adiabatic wall. A reference value of H isassumed as 0 at X¼ 0,Y¼ 1 and hence H¼ 0 is valid for Y¼ 1,cX. Itmay be noted that, the unique solution of Eq. (21) is stronglydependent on the non-homogeneous Dirichlet conditions. Thefollowing non-homogeneous Dirichlet boundary conditions areemployed to obtain the solution for Eq. (21).

Hð0;0Þ ¼ 1Re Pr

Nul (25)

Hð1;0Þ ¼ � 1Re Pr

Nur (26)

3. Results and discussion

3.1. Numerical tests

The computational domain consists of 28 � 28 bi-quadraticelements which correspond to 57 � 57 grid points. The bi-quadratic elements with lesser number of nodes, as thatsmoothly capture the non-linear variations of the field variableswhich are in contrast with finite difference or finite volume solu-tions available in the literature [23]. In the current investigation,Gaussian quadrature based finite element method provides thesmooth solutions at the interior domain including the cornerregions as evaluation of residuals depends on the interior Gausspoints and thus the effect of corner nodes are less profound in thefinal solution. The present finite element method based approachoffers special advantage on evaluation of local Nusselt number atthe left, right and bottomwalls as the element basis functions havebeen used here to evaluate the heat flux.

The simulations on heatfunction for natural convection withinsquare cavities are compared with Deng and Tang’s work [35] andthe results were found to be good agreement. The comparisons areshown in an earlier work [44] and thus the comparisons are notshown here for the brevity of the manuscript. Validation of currentnumerical strategy for mixed convection problem has been carriedout via comparing with earlier simulations studies [47]. Table 1shows that average Nusselt number ðNubÞ agrees with the results


of earlier work [47] for a range of Grashof number (Gr ¼ 104�105)and Prandtl number (Pr ¼ 0.01, 0.1 and 1) at Re ¼ 500.

In the current study, numerical solution on flow and tempera-ture fields are obtained for visualization of heatlines with variousvalues of parameters A (A ¼ 0.1e0.9), Gr (Gr ¼ 103�105), Pr(Pr ¼ 0.015e10) and Re (Re ¼ 1�100). The solution for the currenttest case is strongly dependent on a non-homogeneous Dirichletboundary condition and the sign of the heatfunction is governed bythe sign of a non-homogeneous Dirichlet boundary condition. Forthe current situation, negative sign of heatlines represent clockwiseflow of heat while positive sign refers anti-clockwise flow. Fluid isvirtually stagnant and the heat transfer is conduction dominant forlow values of the governing parameters (Gr, Pr, Re). Under theseconditions, heatlines essentially represent ‘heat flux’ lines, whichare commonly used for conductive heat transport [44,45]. It may benoted that the magnitude of heatfunction refers to the dimen-sionless amount of heat that is being transferred. The heat flux ata zone is equal to the gradient of heatfunctions which is illustratedby heatlines. It is observed that heatlines emanate from hot surfaceand end on the cold surface and they are perpendicular to theisothermal surfaces, similar to the heatlines, during conductiondominant regime. As heatlines approach the adiabatic wall, theyslowly bend and become parallel to surface. The detailed discussionon heat flow and fluid flow characteristics is discussed in followingsections. The heat transfer rates based on average Nusselt numbersare also evaluated along bottom and side walls and they areexplained based on heatlines.

3.2. Effect of thermal aspect ratios on flow and thermalcharacteristics

Fig. 2 illustrates the streamlines, isotherms and heatlines forRe ¼ 1, Pr ¼ 0.015, Gr ¼ 103 and A ¼ 0.1e0.9. It is observed that the

ψ

0.13

−0.17

0.1−0.14

−0.16

−0.1

0.070.03

−0.07

−0.03

−0.03

0.01

0 0.5 10

0.5

1

−0.13

−0.1−0.05−0.01

−0.07

0.07

0.05

0.030.01−0.05

0 0.5 10

0.5

1

0.025

0.01

0.020.005 −0.13

−0.1−0.07

−0.03

−0.05

−0.01

0 0.5 10

0.5

1

θ

0.0.0.0.0.

0.1

0 00

0.5

1

0.00.0.0.

0.20.1

0 00

0.5

1

0

0.

0.0.0.30

.2

0 00

0.5

1

a

b

c

Fig. 2. Streamfunction (j), temperature (q) and heatfunction (H) contours w

effect of lid-driven flow does not influence the temperature profileat Pr ¼ 0.015 and Gr ¼ 103. The isotherms are smooth andsymmetric, indicating conduction dominant heat transfer due tolow Peclet number (Pe ¼ 0.015). Due to the drag force induced bymotion of upper wall, a small quantity of fluid is pulled towards theleft corner (Fig. 2aec). Two asymmetric rolls with clockwise andanti-clockwise circulations are observed (see Fig. 2aec). Theclockwise circulations are found to be more intense compared tothe anti-clockwise circulations. It is observed that, the flow char-acteristics are qualitatively similar for A ¼ 0.1, 0.5 and 0.9(Fig. 2aec). The magnitudes of streamfunctions are very low andisotherms are smooth and symmetric signifying conductiondominant heat transfer. The maximum value of streamfunctionðjjjmaxÞ is 0.17 for A ¼ 0.1. The isotherms with q � 0.2 occurssymmetrically along the vertical symmetric line at center forA ¼ 0.1 (Fig. 2a). But, for A ¼ 0.5, the isotherms with q � 0.2 occursymmetrically near the top portion of side walls of the enclosure(Fig. 2b) whereas q� 0.3 are smooth symmetric curves with respectto the vertical symmetric line and jjjmax is found to be 0.13. Athigher thermal aspect ratio (A¼ 0.9), the isotherms with q� 0.3 arefound to be symmetric near the side walls and q � 0.6 are almostparallel to bottom wall (Fig. 2c). The maximum value of stream-function ðjjjmaxÞ is observed as 0.13 for A ¼ 0.9.

Heat flow distribution inside the cavity is illustrated by heat-lines. Common to all thermal aspect ratios at Pr ¼ 0.015, Gr ¼ 103

and Re¼ 1, the heatlines appear to be symmetric and they are foundto be orthogonal to the isotherms signifying the conductiondominant heat transfer (Fig. 2aec). At A ¼ 0.1, the heatlines arefound to be dense near the central zone of the bottom wall indi-cating larger heat flow and consequently, temperature variation ishigh at that zone. As A increases to 0.5, eventhough heatlines aredense near the central zone of the bottomwall, the intensity of heatflow is less compared to that of A ¼ 0.1 and thus, the temperature

975

32

0.1

.5 1

9.7

543

0.20.1

.5 1

.9

7

54

0.30.2

.5 1

H0.5 −0.

52 −26 −610 −10

18 −1828 −2844 −44

0 0.5 10

0.5

1

0.4 −0.42 −2

6 −610 −10

16 −16

24 −24

38 −38

0 0.5 10

0.5

1

1 −14 −410 −10

18 −18

24

−24

28

−28

0 0.5 10

0.5

1

ith Re ¼ 1, Pr ¼ 0.015, Gr ¼ 103 and (a) A ¼ 0.1 (b) A ¼ 0.5 (c) A ¼ 0.9.


variation becomes smaller at that zone. At A ¼ 0.9, the isothermswith q � 0.4 are layered along the bottom wall due to the verticalheatlines at conduction dominant heat transfer and the tempera-ture variation is very low. At A ¼ 0.1, heatlines in the top region ofthe side walls are sparsely distributed with low magnitude whichsignifies less heat is being received from the bottomwall. Boundarylayer is found to be developed at the edges of the side walls and thethickness is largest at the top portion due to less heat flow in thatregion. At A ¼ 0.5, uniform thermal gradient occurs along the sidewalls due to almost parallel heatlines along the side walls as seen inFig. 2b. The heatlines are found to be converged near the topportion of the sidewalls for A¼ 0.9 and thus, isothermswith q� 0.3vary in a small zone.

At higher Gr (Gr ¼ 105) with Re ¼ 1, Pr ¼ 0.015 the streamlinesbecome symmetric due to the dominance of natural convection(figure not shown). The flow characteristics are qualitatively similarfor A ¼ 0.1, 0.5 and 0.9. The temperature distribution and heatenergy distribution are almost qualitatively similar to previouscase. As Pr increases to 0.7 at Re ¼ 1 with Gr ¼ 105, convectionsignificantly compress the temperature profile for all A’s (figure notshown) and weak secondary circulations appear near the bottomcorners of the cavity for A ¼ 0.9. The isotherms with q ¼ 0.1e0.4,q ¼ 0.1e0.5 and q ¼ 0.1e0.6 occur symmetrically near the sidewalls of the cavity for A ¼ 0.1, 0.5 and 0.9, respectively, whereas theother isotherms with q � 0.5, q � 0.6 and q � 0.7 are smoothsymmetric curves with respect to vertical symmetric line at thecenter for A¼ 0.1, 0.5 and 0.9, respectively. The flow characteristics,isotherms and heat energy distributions with various aspect ratiosfor Pr ¼ 0.7 and Pr ¼ 10 at Re ¼ 1 with Gr ¼ 105 are found to bequalitatively similar. A detailed discussion on results for Pr ¼ 10follow next.

Fig. 3 shows the streamfunction, temperature and heatfunctioncontours for various thermal aspect ratios with Re ¼ 1, Pr ¼ 10 and

ψ

4 −4

2 −2

31 −1

−3

a

b

c

0 0.5 10

0.5

1

4.2

2 −2

−4.2

3 −31 −1

0 0.5 10

0.5

1

2.8 −2.8

2 −2−0.50.5

−1 1

−0.5 0.5

0 0.5 10

0.5

1

00

0.30.2

0.4

0 00

0.5

1

0

0

0.5

0.4

0 00

0.5

1

0.

0.0.6

0.5

0 00

0.5

1

Fig. 3. Streamfunction (j), temperature (q) and heatfunction (H) contours

Gr¼ 105. It may be noted that the natural convection dominates theflow and the circulations are found to be symmetric (Fig. 3aec). Thestrength of the fluid flow is decreased as compared to previous case(Pr ¼ 0.7). The maximum intensity of flow (jjjmax) is representedwith 4, 4.2 and 2.8 for A ¼ 0.1, 0.5 and 0.9, respectively at Pr ¼ 10(Fig. 3) whereas that was represented with 14.5, 13 and 11 forA ¼ 0.1, 0.5 and 0.9, respectively at Pr ¼ 0.7 (figure not shown).Similar to previous case, secondary circulation cells are found toappear near the bottom corners of the cavity at A¼ 0.9 (Fig. 3c). Theisotherms with q � 0.3 for A ¼ 0.1 and q � 0.4 for A ¼ 0.5 arecompressed towards the central and top portion of the side wallswhereas the isotherms with q � 0.5 are compressed towards thetop portion of the side walls (Fig. 3aec). It may be noted that, theisotherms with q� 0.6 cover a large portion of the cavity for A¼ 0.1and 0.5, whereas small portion near the side walls is maintainedwith q � 0.6 for A ¼ 0.9 (see Fig. 3aec).

Heat flow distribution inside the cavity is illustrated by heat-lines. The heatline cells are found to be qualitatively similar tostreamlines which indicate convection dominant heat transfer athigher Prandtl number (Pr ¼ 10, Fig. 3aec). At low A (A ¼ 0.1), themaximum intensity of heat flow may be observed as jHjmax ¼ 1:5at the core (see Fig. 3a). Thus, a large central regime is maintainedwith q ¼ 0.3e0.4 in each half of the cavity. On the other hand, theheat flow at the core is represented with jHjmax ¼ 2:2 at A ¼ 0.5and thus, q varies within 0.5e0.6 at large upper portion of the cavity(see Fig. 3b). At A ¼ 0.9, the secondary heatline circulations pushthe primary circulations towards the upper portion of the cavityand jHjmax is found to be 1.5, which is confined within upperportion. It is observed that temperature (q) varies as 0.6e0.7 at thecore of the enclosure (Fig. 3c). The isothermswith q� 0.5 for A¼ 0.1and q � 0.6 for A¼ 0.5 are found to be highly compressed along thebottom wall compared to previous case due to highly dense heat-lines at that zone (see Fig. 3a and b). Thus, highly thin thermal

θ

.5

.70.3 0.2

0.4

.5 1

.6

.8

0.5

0.4

.5 1

7

8 0.6

0.5

.5 1

H

1.5 −1.5

0.7 −0.7

1 −10.2 −0.2

0 0.5 10

0.5

1

2.2 −2.2

1 −0.5

1.5 −1.50.3 −0.3

0 0.5 10

0.5

1

1.5 −1.5

1 −1

−0.7 0.7

0.1 −0.1

0 0.5 10

0.5

1

with Re ¼ 1, Pr ¼ 10, Gr ¼ 105 and (a) A ¼ 0.1 (b) A ¼ 0.5 (c) A ¼ 0.9.


boundary layer is observed along the middle portion of the bottomwall (Fig. 3a and b). At higher thermal aspect ratio (A ¼ 0.9),thermal gradient is less compared to A ¼ 0.1 and 0.5 and thus,thermal boundary layer thickness is larger along the bottom wall.

Fig. 4 illustrates the streamlines, isotherms and heatlines forRe ¼ 10, Pr ¼ 0.7, Gr ¼ 103 with A ¼ 0.1, 0.5 and 0.9. It is interestingto observe that at low thermal aspect ratio (A ¼ 0.1), the primarycirculations occupy most of the cavity except a small region con-sisting of secondary cells occupying near the bottom portion of theleft wall (Fig. 4a). Asymmetric profile in temperature distribution isobserved. As the anti-clockwise circulation cells in the right half arestronger, the isothermwith q ¼ 0.1 is found to be clustered towardsthe right wall for A ¼ 0.1. As A increases to 0.5, the secondarycirculations near the left portion of the bottom wall tend to vanishand the primary circulations almost occupy entire cavity (Fig. 4b).The temperature contours for q � 0.2 are stretched and get asym-metrically shifted towards the top corners of the side walls. Athigher A (A ¼ 0.9), the secondary circulations vanish and theprimary circulations occupy the entire cavity (Fig. 4c). Theisotherms with q � 0.3 are stretched and found to be compressedalong the top corners of the side walls.

Temperature profile and heat distributions are further explainedbased on heatlines. Common to all thermal aspect ratios, dominantconvective heat transfer is observed based on the distorted heat-lines (Fig. 4aec At low thermal aspect ratio A (A ¼ 0.1), largerthermal gradient is found at the center of the bottom wall due todense heatlines are converged at that regime (Fig. 4a). The thermalgradient decreases as A increases from 0.1 to 0.9, due to less denseheatlines at higher thermal aspect ratios. It is interesting to observethat, single cell with jHjmax ¼ 0:04 is observed near the top portionof the right wall at A¼ 0.9. The thermal boundary layer thickness isfound to be larger near upper portion of the side walls for A ¼ 0.1

ψ

−0.09

−0.05

−0.01

−0.07

0.0050.0010 0.5 10

0.5

1

−0.09

−0.05

−0.010.0010 0.5 10

0.5

1

−0.1

−0.05

−0.01

0 0.5 10

0.5

1

θ

0.0.

0.30.

0.1

0 00

0.5

1

0.

0.

00.3

0.20.1

0 00

0.5

1

0.

0.

0.5

0.0.30.2

0 00

0.5

1

a

b

c


comparing to that of 0.5 and 0.9 due to sparse heatlines observed inthose regions at A ¼ 0.1 (Fig. 4aec).

As Gr increases to 104 at Re ¼ 10 with Pr ¼ 0.7, the secondarycirculations become larger compared to previous case due to theenhancement of buoyancy force (figure not shown). Similar toprevious cases, thermal gradient is high along the central zone ofbottomwall for A¼ 0.1 and that decreases as A increases from 0.1 to0.9. Heatlines are distorted due to dominant convective heattransfer for A ¼ 0.1e0.9 and primary heatline circulation cellsappear in right half of the cavity. It is also found that bottom walldelivers more heat to the left wall compared to that of right wallirrespective of A.

Fig. 5 illustrates the streamlines, isotherms and heatlines forRe ¼ 10, Pr ¼ 0.7, Gr ¼ 105. As Pr increases to 0.7, the strength of theprimary and secondary circulations increases and almostsymmetric pattern in streamlines is observed especially for A ¼ 0.1and 0.5 (Fig. 5a and b). The maximum value of streamfunctionðjjjmaxÞ is found to be 1.5 and 1.4 for A ¼ 0.1 and 0.5, respectively.The isotherms with q � 0.3 for A ¼ 0.1 and q � 0.4 for A ¼ 0.5 arehighly compressed towards the sidewalls and other isothermswithq � 0.5 for A ¼ 0.1 and q � 0.6 for A ¼ 0.5 are more compressedtowards the bottom wall. Asymmetric trends are also found instreamfunction and temperature profiles at higher A (A ¼ 0.9). Theright circulation cells are found to be larger with jjjmax ¼ 1:4(Fig. 5c). On the other hand, a weak circulation cell withjjjmax ¼ 0:01 is observed at the bottom right corner of the cavity.Due to strong circulations along the right wall, the isotherms withq� 0.4 are stretched and they are found to be attracted towards theright corner of the top wall. Also, the other isotherms with q � 0.5are attracted towards the upper portion of the right wall (Fig. 5c).

The effect of enhanced convection can be explained effectivelybased on heatline distributions. It is interesting to note that,

85

2

0.1

.5 1

9

6

.4

0.20.1

.5 1

9

7

4

0.30.2

.5 1

H0.001

0.030.05

0.12 −0.1−0.05−0.03

−0.015

−0.005

0 0.5 10

0.5

1

0.001

0.03

0.05

0.09

−0.035

−0.05−0.07

−0.025−0.01

0 0.5 10

0.5

1

0.070 080.06

0.001

−0.04

−0.04

−0.05

0 0.5 10

0.5

1

ith Re ¼ 10, Pr ¼ 0.7, Gr ¼ 103 and (a) A ¼ 0.1 (b) A ¼ 0.5 (c) A ¼ 0.9.

ψ

1.4

1.2

0.9

0.30.1

0.60.1

−1.5−1.2−0.6−0.1

−1.4−0.9−0.3

0 0.5 10

0.5

1

1.2−1.4

0.9

0.6

0.3

0.1

−1.2

−0.9−0.3

−0.6−0.1

0 0.5 10

0.5

1

0.90.60.3

0.10.01

0.4

−1.4

−1.2

−0.9−0.3

−0.6−0.1

0.01

0 0.5 10

0.5

1

θ

0.80.60.5

0.4 0.4

0.3 0.3

0.20.2

0 0.5 10

0.5

1

0.90.70.6

0.5 0.5

0.40.4

0.3

0.30 0.5 10

0.5

1

0.90.80.7

0.6

0.50.40.3

0.4

0.30.2

0 0.5 10

0.5

1

H

0.5

0.40.3

−0.6−0.5−0.4−0.3

0.25

0.2

0.05

−0.25−0.2

−0.1

0 0.5 10

0.5

1

0.6

0.40.3

0.2

−0.7−0.5−0.4−0.2−0.3

−0.05

−0.2

−0.10 0.5 10

0.5

1

0.50.30.20.15

0.40.05

0.1

−0.9−0.7

−0.8

−0.1

−0.5−0.3

−0.6

−0.2

0 0.5 10

0.5

1

a

b

c

Fig. 5. Streamfunction (j), temperature (q) and heatfunction (H) contours with Re ¼ 10, Pr ¼ 0.7, Gr ¼ 105 and (a) A ¼ 0.1 (b) A ¼ 0.5 (c) A ¼ 0.9.


heatline circulation cells are similar to streamline cells at higherPrandtl number (Pr ¼ 0.7) with higher Grashof number (Gr ¼ 105),illustrating dominant convection heat transfer (Fig. 5aec), whereasthey were not similar for lower Grashof numbers (Fig. 4aec). AtA ¼ 0.1, the maximum value of heatfunction ðjHjmaxÞ is 0.6 at thecore as seen in Fig. 5a. In addition, dense heatlines occur at thecenter of the cavity. Therefore, temperature (q) varies as 0.4e0.5along the central vertical line. As thermal aspect ratio (A)increases to 0.5, the maximum intensity of heat flow is representedwith jHjmax ¼ 0:6 and 0.7 along the left and right portions,respectively and thus, a large central regime is maintained withq ¼ 0.5e0.6 within the cavity (Fig. 5b). At higher thermal aspectratio (A ¼ 0.9), the heatline circulation cells in the left and rightportions correspond to jHjmax ¼ 0:5 and 0.9, respectively and q

varies within 0.4e0.5 in the top central portion of the cavity(Fig. 5c). Due to weak circulation cells near the left portion, thethermal mixing at the top portion is not enhanced for A ¼ 0.9.

Note that, due to dense heat flow the isotherms with q� 0.5 andq � 0.6 are compressed along the central zone of bottom wall forA ¼ 0.1 and 0.5 respectively. Thus, the thickness of thermalboundary layer is also found to be small along the bottom wall. Athigher thermal aspect ratio (A¼ 0.9), the less dense heatlines alongthe bottom wall results in less thermal gradient (Fig. 5c). It isobserved that heat flow occurs from the bottom wall and the heatflow trajectory passes along central vertical line to reach side wallsfor A ¼ 0.1 and 0.5, whereas that passes along an inclined pathtowards right wall to reach side walls for A ¼ 0.9. It may be notedthat left and right walls receive almost same amount of heat due toalmost symmetric heatlines for A ¼ 0.1 and 0.5, whereas right wallreceives more heat compared to that of left wall for A ¼ 0.9.

Fig. 6 displays the streamlines, isotherms and heatlines forRe ¼ 10, Pr ¼ 10, Gr ¼ 105 and A ¼ 0.1e0.9. At higher Pr, the

isotherms tend to deform largely due to the presence of strongconvective heat transfer and the isotherms are non-symmetric dueto dominant effect of lid velocity (see Fig. 6aec). The primarycirculation cell is stronger than the anti-clockwise secondary cellswhich occur near the left half of the cavity due to the lid-drivenforce. At low A, the maximum strength ðjjjmaxÞ of the circulationsare observed as 0.36 and 0.4 at left and right portions, respectively(Fig. 6a). Hence the isotherms with q � 0.3 are graduallycompressed towards the side walls and q � 0.5 are symmetricallycompressed towards the bottom wall. As thermal aspect ratio (A)increases to 0.5 (see Fig. 6b), the magnitude of streamfunctionincreases as jjjmax ¼ 0:37 and 0.45 at left and right portions,respectively. The isotherms with q � 0.4 are gradually compressedtowards the side walls whereas the isotherms with q � 0.6 arefound to be symmetric along the vertical central line and they arehighly compressed towards the bottom wall. At higher A (A ¼ 0.9),multiple circulations are formed inside the cavity withjjjmax ¼ 0:08, 0.27 and 0.05 at left, right and bottom portions ofthe cavity, respectively due to the enhanced convection effects(Fig. 6c). The isotherms with q � 0.3 are clustered around the topcorner of the side walls.

Heatline patterns are found to be similar to the streamlines dueto convective heat transfer in the cavity at higher Prandtl number(Pr ¼ 10). At low thermal aspect ratio (A ¼ 0.1), a large zone ismaintainedwith uniform temperature (q) varyingwithin 0.3e0.4 atthe left half of the cavity due to the anti-clockwise heat flowcirculations correspond to jHjmax ¼ 0:15. Similarly, a large zone isobserved with q ¼ 0.3e0.4 at the right half of the cavity due toclockwise heat flow circulations corresponding to jHjmax ¼ 0:16(Fig. 6a). At A ¼ 0.5, a zone with q ¼ 0.4e0.5 is observed at left andright portions of the cavity due to the anti-clockwise heat flowcirculations corresponding to jHjmax ¼ 0:15 and clockwise

ψ

0.36

0.3

0.1

0.1

−0.4

−0.35−0.1

−0.2

0 0.5 10

0.5

1

0.37

0.3

−0.45

−0.30.1

−0.10 0.5 10

0.5

1

0.08 −0.27

0.01 0.05

−0.1

−0.2

0 0.5 10

0.5

1

θ

0.40.4

0.50.3 0.30.2

0.2

0 0.5 10

0.5

1

0.50.5

0.4 0.40.6

0.80 0.5 10

0.5

1

0.3

0.3

0.4

0.5 0.6 0.80.9

0 0.5 10

0.5

1

H

0.15 −0.16

0.1 −0.1

0.05 −0.1−0.05

0 0.5 10

0.5

1

0.2 −0.23

0.1

0.15−0.2

−0.05

−0.1

0 0.5 10

0.5

10.05

0.018

−0.15

−0.05

−0.1

0.032

0 0.5 10

0.5

1

a

b

c

Fig. 6. Streamfunction (j), temperature (q) and heatfunction (H) contours with Re ¼ 10, Pr ¼ 10, Gr ¼ 105 and (a) A ¼ 0.1 (b) A ¼ 0.5 (c) A ¼ 0.9.


circulations correspond to jHjmax ¼ 0:23, respectively (Fig. 6b). Athigher thermal aspect ratio (A ¼ 0:9), temperature (q) varieswithin 0.3e0.4 in a large portion of the upper half of the cavity dueto the primary circulation with jHjmax ¼ 0:15 (Fig. 6c). It is inter-esting to note that, multiple circulations in heatlines occur withjHjmax ¼ 0:05, 0.15 and 0.03 at the left, right and bottom portionsof the cavity, respectively.

It may be noted that, the isotherms with q � 0.5 and q � 0.6 arehighly compressed along the bottom wall for A ¼ 0.1 and 0.5respectively, due to large dense heatlines within 0.2� X� 0.8 alongthe bottom wall (Fig. 6a and b). Thus, the thermal gradient is largenear central zone of the bottomwall. At higher thermal aspect ratio(A¼ 0.9), the temperature gradient is smaller along the bottomwallcompared to A ¼ 0.1 and 0.5 due to less dense heatlines along thebottomwall (Fig. 6c). It is observed that heatlines starting from thebottom wall travels along an inclined path to reach side walls forA ¼ 0.1 and A ¼ 0.5, whereas those follow a complicated path forA ¼ 0.9. It is found that upper portion of right wall receives moreheat from bottom wall compared to that of upper portion of leftwall irrespective of A.

Fig. 7 illustrates the streamfunction (j), temperature (q) andheatfunction (H) contours for Re ¼ 100, Pr ¼ 0.015, Gr ¼ 103 andA ¼ 0.1e0.9. Common to all thermal aspect ratios (A ¼ 0.1e0.9),a single primary circulation span the entire cavity due to domi-nant lid-driven force (see Fig. 7aec). Qualitatively similar pattern instreamlines is found for A ¼ 0.1, 0.5 and 0.9. The magnitude of thestreamfunction is very low as jjjmax ¼ 0:09 and the isotherms aresmooth and monotonic indicating conduction dominant heattransfer. The isotherms are decoupled with streamlines andconduction dominant heat transfer is observed due to low Pe(Pe ¼ 1.5). It may be noted that, the temperature and the heatlinedistribution patterns are almost qualitatively similar to Re ¼ 1 with

same other parameters and similar explanation that of Fig. 2follows.

Simulations are also done for the streamlines, isotherms andheatlines for Re ¼ 100, Pr ¼ 0.015, Gr ¼ 105 and A ¼ 0.1e0.9 (figurenot shown). The secondary circulation cells are found to appear atthe left portion of the cavity irrespective of A. In addition, tertiarycirculations also occur for A ¼ 0.9. The isotherms and heatlines forGr ¼ 105 are qualitatively similar to the previous case and Fig. 2.

Fig. 8 illustrates the streamlines, isotherms and heatlines forRe ¼ 100, Pr ¼ 0.7 and Gr ¼ 105 for various thermal aspect ratios. Atlow thermal aspect ratio (A ¼ 0.1), the primary circulations occupymost of the cavity except at top and bottom corners of the left wallconsisting of secondary and tertiary circulations for Pr ¼ 0.7(Fig. 8a). The isotherms with q � 0.2 are clustered towards thebottom and left walls for A ¼ 0.1. The secondary and tertiarycirculations tend to suppress near the top corner at A¼ 0.5 (Fig. 8b).The isotherms with q � 0.3 are clustered towards the top corners ofthe cavity. It is observed that, multiple circulations occur withjjjmax ¼ 0:01, 0.001 and 0.29 at top left, bottom right corners andcentral portions of the cavity, respectively at A¼ 0.9. The isothermswith q � 0.3 are pushed towards the top corners of the cavity.

Temperature profile and heat distributions inside the cavity arefurther illustrated based on heatlines. It may be noted that, heatlinecells and streamline cells are similar at higher Prandtl number(Pr¼ 0.7, Fig. 8aec). It is observed that, the heat flow characteristicsare qualitatively similar for all thermal aspect ratios (Fig. 8aec). Atlow thermal aspect ratio (A ¼ 0.1), heatline cells withjHjmax ¼ 0:075 are observed at the core signifying enhancedthermal mixing (Fig. 8a). As a result, q varies within 0.3e0.4 at thecore of the cavity. As A increases to 0.5, the intensity of heat flowincreases as jHjmax ¼ 0:1 near the core (Fig. 8b). Thus, thetemperature (q) at the core varies within 0.4e0.5. At higher thermal

ψ

−0.09−0.05

−0.07

−0.01−0.001

0 0.5 10

0.5

1

−0.09−0.05

−0.01−0.001

−0.07

−0.03

0 0.5 10

0.5

1

−0.09−0.05−0.03

−0.01−0.001

−0.07

0 0.5 10

0.5

1

θ

0.80.60.40.30.2

0.1

0.1

0 0.5 10

0.5

1

0.90.7

0.50.40.3

0.20.1 0.20.1

0 0.5 10

0.5

1

0.90.80.70.60.50.4

0.30.20.30.2

0 0.5 10

0.5

1

H0.0010.03 −0

.030.07

−0.070.1−0.10.15

−0.150.2

−0.2

0.4 −0.4

0 0.5 10

0.5

1

0.01 −0.010.05

−0.050.1

−0.10.15

−0.15

0.2

−0.20.3

−0.30.4 −0.4

0 0.5 10

0.5

1

0.01 −0.010.07 −0.05

0.15 −0.1

0.2 −0.15

−0.2

−0.250.250.3

0 0.5 10

0.5

1

a

b

c


ψ

−0.24

−0.19

−0.12

−0.05

0.001

0.01

−0.01

0 0.5 10

0.5

1

−0.24

−0.19

−0.12

−0.05

0.001

0.001

−0.01

0 0.5 10

0.5

1

−0.24

−0.19

−0.12−0.05−0.01 0.0

01

.001

0 0.5 10

0.5

1

θ

0.70.60.5

0.4 0.3

0.2

0.20.1

0 0.5 10

0.5

1

0.80.70.6

0.5 0.4 0.3

0.30.2 0.2

0 0.5 10

0.5

1

0.90.80.70.6

0.5

0.40.3

0.3

0.2

0 0.5 10

0.5

1

H

−0.075

−0.07

−0.06

−0.05

−0.03

−0.020.03

0.0010.005

0.01

0.02

−0.01

0 0.5 10

0.5

1

−0.1

−0.08

−0.06

−0.04

−0.02

−0.01

0.001

0.025

0.01

0 0.5 10

0.5

1

0.001

−0.12−0.09

−0.11

−0.050.02

0.001

0.01

−0.07−0.03

−0.01

0 0.5 10

0.5

1

a

b

c




aspect ratio (A ¼ 0.9), the maximum intensity of heat flow ðjHjmaxÞis observed as 0.12 and q varies within 0.3e0.4 in a large portion ofthe cavity (Fig. 8c).

The isothermswith q� 0.4 and q� 0.5 are compressed along thecentral and right portion of bottom wall for A ¼ 0.1 and 0.5respectively, due to dense heat flow at that zone. But, the thermalgradient is quite large and the thickness of thermal boundary layeris found to be small for A ¼ 0.1 comparing to that of A ¼ 0.5 withinthat zone (Fig. 8a and b). The thermal gradient is lesser for A ¼ 0.9along the bottom wall compared to other thermal aspect ratios(A ¼ 0.1 and 0.5). It is interesting to observe that, heat flow occursfrom bottomwall to side walls along a circular path. It is found thatbottomwall delivers more heat to the left wall compared to that ofright wall.

Fig. 9 illustrates the distribution of streamfunction (j),temperature (q) and heatfunction (H) contours for Re¼ 100, Pr¼ 10,Gr ¼ 105 for various thermal aspect ratios (A). The dominant lid-driven force is observed at A ¼ 0.1 as a single primary circulationspans the entire cavity (Fig. 9a). The maximum intensity of fluidflow ðjjjmaxÞ is found to be 0.12 and the isotherms with q � 0.2 aremore compressed towards the bottom and left wall. At A ¼ 0.5, themultiple circulations are observed in lower half of the cavity. Theisotherms are found to be deformed based on multiple circulationsand the isothermwith q� 0.1 is clustered in a small regionwhereasthe isotherms with q � 0.3 occupy 55% of the cavity. At A ¼ 0.9,multiple circulation cells are found with jjjmax ¼ 0:08, 0.04 and0.01 at upper, bottom right and bottom left portion, respectively(Fig. 9c). The isotherms with q � 0.2 are pushed towards the rightwall and a small zone of upper portion of left wall whereas theisotherms with q � 0.3 occupy 50% of the cavity.

At the onset of convection, many interesting features of heat-lines have been observed for A ¼ 0.1e0.9 (Fig. 9aec). It is also

ψ

−0.12

−0.07

−0.010 0.5 10

0.5

1

−0.09−0.05

0.02−0.005

−0.01

0 0.5 10

0.5

1

−0.08−0.03

0.04

0.01−0.01

0 0.5 10

0.5

1

θ0.2

0.3

0.50 00

0.5

1

0.2

0.50.6

0 00

0.5

1

0.3

0.6

0.4

0.3

0 00

0.5

1

a

b

c


observed that heatline circulations are qualitatively similar tostreamlines at higher Prandtl number (Pr ¼ 10, Fig. 9aec). At lowthermal aspect ratio (A ¼ 0.1), the maximum intensity of heat flownear the core is observed with jHjmax ¼ 0:022 and almost 80% ofthe cavity is maintained with uniform temperature as q varieswithin 0.1e0.2 (Fig. 9a). It is observed that, multiple circulations inheatlines are observed in lower half of the cavity at A ¼ 0.5 and 0.9(see Fig. 9b and c). At A ¼ 0.5, the maximum intensity of heat flowðjHjmaxÞ circulation is found to be 0.013 at the upper half of thecavity (Fig. 9b). Almost 45% of the cavity is maintained withuniform temperature at the upper half of the cavity. At higherthermal aspect ratio (A ¼ 0.9), a uniform thermal zone is observedin upper half of the cavity with q varying as 0.2e0.3 due to heatcirculation cells corresponding to jHjmax ¼ 0:017 at the upper halfof the cavity.

At low thermal aspect ratio (A ¼ 0.1), the isotherms with q � 0.2are highly compressed with larger thermal gradient along thebottom wall within 0.35 � X � 0.7 due to highly dense heatlinesalong the bottom wall (Fig. 9a). At A ¼ 0.5, the isotherms withq � 0.5 are highly compressed near the central and right portion ofthe cavity. At higher thermal aspect ratio (A ¼ 0.9), thermalgradient is low due to sparse heatlines along the bottom wall. Itmay be noted that large amount of heat flow occurs from bottomwall to left wall as seen from dense heatlines for A¼ 0.1 and A¼ 0.5(see Fig. 9a and b).

3.3. Overall heat transfer rates and average Nusselt numbers

The overall effects on heat transfer rates are shown in Fig. 10aand b where the distributions of the average Nusselt number atbottom and side walls vs. logarithmic Grashof number are shownfor Pr ¼ 0.015e10, Re ¼ 1 and 100. It is observed that, heat transfer

0.1

.5 1

0.30.4

0.5.5 1

0.7

0.8

0.2

.5 1

H

−0.022

−0.015−0.005

0.0001

0.00450 0.5 10

0.5

1

−0.013

0.007

0.003 −0.005

−0.003

−0.001

0 0.5 10

0.5

1

−0.015−0.004

−0.017

0.0220.01

−0.004

0 0.5 10

0.5

1

ith Re ¼ 100, Pr ¼ 10, Gr ¼ 105 and (a) A ¼ 0.1 (b) A ¼ 0.5 (c) A ¼ 0.9.

0

2

4

6

8

10

Nu b

Re = 1

A=0.1

A=0.5

A=0.9

0

2

4

Nu l A=0.1

A=0.5

A=0.9

103 104 105

Grashof Number, Gr

0

2

4

Nu r

A=0.1 A=0.5

A=0.9

0

2

4

6

8

10

Nu b

Re = 100

A=0.1A=0.5

A=0.9

0

2

4

6

Nu l

A=0.1A=0.5

A=0.9

103 104 105

Grashof Number, Gr

-1

0

1

2

3

Nu r

A=0.1A=0.5

A=0.9

a b

Fig. 10. Variation of average Nusselt number with Grashof number for (a) Re ¼ 1 (b) Re ¼ 100 with Pr ¼ 0.015(..), Pr ¼ 0.7(��) and Pr ¼ 10(�) respectively. In each plot upperpanel corresponds to bottom wall, middle panel corresponds to left wall and lower panel corresponds to right wall.


rates (Nub,Nul andNur) are constant for the entire range of Gr for allthermal aspect ratios (A) at Pr ¼ 0.015 and Re ¼ 1 signifyingconduction dominant heat transfer (Fig. 10a). The constant valuesare observed due to almost same heatfunction gradients for the

0

3

6

9

12

Nu b

Re = 1

0

2

4

6

Nu l

0 0.2 0.4 0.6 0.8 1Thermal aspect ratio, A

0

2

4

6

Nu r

a

Fig. 11. Variation of average Nusselt number with thermal aspect ratio for (a) Re ¼ 1

entire range of Gr. It may be noted that the heat transfer rates (Nub,Nul and Nur) are almost constant at lower Rayleigh numbers,whereas they are increasing functions at higher Rayleigh numbersfor Pr¼ 0.7, irrespective of A (Fig. 10a). Note that, heat transfer rates

0

3

6

9

12

Nu b

Re = 100

0

2

4

6

Nu l

0 0.2 0.4 0.6 0.8 1Thermal aspect ratio, A

0

1

2

3

Nu r

b

and (b) Re ¼ 100 with Gr ¼ 105; Pr ¼ 0.015(..), Pr ¼ 0.7(��) and Pr ¼ 10(�).


(Nub, Nul and Nur) increase within 103 � Gr�105 at Pr ¼ 10 irre-spective of A except for Nul at A ¼ 0.9. The increasing trend isobserved due to larger gradients of heatfunctions at Gr ¼ 105

comparing to that of Gr ¼ 103. It is observed that average Nusseltnumber distributions (Nub, Nul and Nur) at Re ¼ 100 are qualita-tively similar to that of Re ¼ 1, irrespective of Pr and A, except atPr¼ 10 with A¼ 0.5 and 0.9 (Fig. 10a and b). Note that, Nub, Nul andNur distributions at Re ¼ 100 with Pr ¼ 10, A ¼ 0.5 exhibit oscil-latory variation at higher Grashof numbers (Fig. 10b) due to largergradients of heatfunctions at various Gr ’s.

Fig. 11a and b illustrate the distributions of the average Nusseltnumber at bottom and side walls vs. thermal aspect ratio (A¼ 0�1)for Pr ¼ 0.015e10, Gr ¼ 105 with Re ¼ 1 and 100, respectively. It isfound that heat transfer rates (Nub, Nul and Nur) decrease as Aincreases from 0 to 1 for Pr ¼ 0.015 and Pr ¼ 0.7 irrespective of Re(see Fig. 11a and b). Note that, heat transfer rates (Nub, Nul and Nur)at Re¼ 1 with Pr ¼ 10 exhibits oscillations at higher thermal aspectratios (Fig. 11a) due to larger gradients of heatfunctions at variousthermal aspect ratios. The oscillations are also observed for heattransfer rates (Nub, Nul and Nur) within 0.2 � A�0.6 at Re ¼ 100with Pr ¼ 10 (Fig. 11b) due to the larger gradients of heatfunctionsat various thermal aspect ratios within 0.2 � A�0.6.

4. Conclusion

The objective of this paper is to analyze the effect of variousthermal boundary conditions based on thermal aspect ratio (A) onthe flow and heat transfer characteristics due to lid-driven mixedconvection flows within a square cavity for various Prandtlnumbers (Pr ¼ 0.015, 0.7, 10), Re (Re ¼ 1, 10, 100) for Gr ¼ 103�105with thermal aspect ratio (A ¼ 0.1e0.9). The governing equationsare solved using Galerkin finite element method with penaltyformulation in terms of heatlines, streamlines and isotherms.Important features on heating and flow characteristics based onheatlines are outlined below for various test studies.

� It is found that, conduction dominant heat transfer is observedvia smooth and symmetric isotherms at Pr ¼ 0.015 irrespectiveof Gr (103� Gr�105), Re (1, 10 and 100) and A (0.1� A�0.9), dueto low Peclet number.

� Lid-driven effect is dominant over buoyancy force at Gr ¼ 103

irrespective of Re (1, 10 and 100), Pr (0.015, 0.7 and 10) and A(0.1, 0.5 and 0.9). Comparably smaller dominance of lid-driveneffect over buoyancy force is observed at Re¼ 10 with Gr¼ 105,irrespective of Pr (0.015, 0.7 and 10) for A ¼ 0.1 and 0.5. Theinfluence of lid-driven force is found to be larger for A ¼ 0.9compared to that of A ¼ 0.1 and 0.5 at Re ¼ 10 with Gr ¼ 105

irrespective of Pr (0.015, 0.7 and 10). Stronger influence of lid-driven force is found at Re ¼ 100 with Gr ¼ 105, irrespective ofPr (0.015, 0.7 and 10) and A (0.1, 0.5 and 0.9), due to lowerRichardson number (Ri ¼ 10).

� Natural convection heat transfer is found via symmetricisotherms and heatlines at Re¼ 1 with Gr ¼ 105, irrespective ofPr and A, due to higher Richardson number (Ri ¼ 105).

� Heatlines and streamlines vary identically near the core forconvection dominant flow at higher Prandtl numbers (Pr�0.7)with Gr ¼ 105 irrespective of Re.

� Larger thermal gradients are observed at the center of thebottomwall at A ¼ 0.1 due to large dense heatlines at that zoneand that decreases as A increases from 0.1 to 0.9 irrespective ofRe and Pr.

� Heat transfer rates (Nub, Nul, Nur) are constant for the entirerange of Gr for all thermal aspect ratios at lower Pr (Pr¼ 0.015),signifying conduction dominant heat transfer.

� It is found that, the overall heat transfer rate is high at A ¼ 0.1compared to that of A¼ 0.5 and A¼ 0.9 irrespective of Re and Pr.

� The variation of average Nusselt number with thermal aspectratio exhibits that, overall heat transfer rate decreases asthermal aspect ratio (A) increases from 0 to 1 for Pr¼ 0.015 and0.7 irrespective of Re. Oscillatory pattern is found in averageNusselt number distribution within 0.6 � A�1 for Re ¼ 1 withPr ¼ 10 whereas that is observed within 0.2 � A�0.6 forRe ¼ 100 with Pr ¼ 10.

Acknowledgement

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

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Nomenclature

A: aspect ratio of temperature difference (Th�Tc)/(TH�Tc) or thermal aspect ratiog: acceleration due to gravity,m s�2

Gr: Grashof numberH: heatfunctionk: thermal conductivity, W m�1 K�1

L: length of the side of the square cavity, mNu: local Nusselt numberp: pressure, PaP: dimensionless pressurePe: Peclet numberPr: Prandtl numberR: Residual of weak formRa: Rayleigh numberRi: Richardson numberT: temperature, KTh: temperature at the bottom edges of the side walls, KTc: temperature at the top edges of the side walls, KTH: temperature at the center of the bottom wall, Ku: x component of velocityU: x component of dimensionless velocityv: y component of velocityV: y component of dimensionless velocityX: dimensionless distance along x coordinateY: dimensionless distance along y coordinate

Greek symbolsa: thermal diffusivity, m2 s�1

b: volume expansion coefficient, K�1

g: penalty parameter4: angle of inclinationq: dimensionless temperaturen: kinematic viscosity, m2 s�1

r: density, kg m�3

F: basis functionsj: streamfunctionx: horizontal coordinate in a unit squareh: vertical coordinate in a unit square

Subscriptsb: bottom walll: left wallr: right walls: side wall

Documents

A complete heatline analysis on mixed convection within a square cavity: Effects of thermal boundary conditions via thermal aspect ratio