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Numerical Heat Transfer, Part B:Fundamentals: An International Journalof Computation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unhb20

Effect of Various Thermal BoundaryConditions on Natural Convection in aTrapezoidal Cavity with Linearly HeatedSide Wall(s)E. Natarajan a , S. Roy a & Tanmay Basak ba Department of Mathematics , Indian Institute of TechnologyMadras , Chennai, Indiab Department of Chemical Engineering , Indian Institute ofTechnology Madras , Chennai, IndiaPublished online: 04 Oct 2007.

To cite this article: E. Natarajan , S. Roy & Tanmay Basak (2007) Effect of Various Thermal BoundaryConditions on Natural Convection in a Trapezoidal Cavity with Linearly Heated Side Wall(s), NumericalHeat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 52:6,551-568, DOI: 10.1080/10407790701563623

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EFFECT OF VARIOUS THERMAL BOUNDARYCONDITIONS ON NATURAL CONVECTIONIN A TRAPEZOIDAL CAVITY WITH LINEARLYHEATED SIDE WALL(S)

E. Natarajan and S. RoyDepartment of Mathematics, Indian Institute of Technology Madras,Chennai, India

Tanmay BasakDepartment of Chemical Engineering, Indian Institute of Technology Madras,Chennai, India

A penalty finite-element-based study has been carried out for natural-convection flow in a

trapezoidal cavity with uniformly heated bottom wall and linearly heated and/or cooled

vertical wall(s) in the presence of an insulated top wall. For linearly heated side walls, sym-

metry in flow patterns is observed, whereas secondary circulation is observed for the linearly

heated left wall and cooled right wall. The local Nusselt number indicates reversal of heat

flow at the side walls or the left wall. The average Nusselt number versus Rayleigh number

illustrates that the overall heat transfer rate at the bottom wall is larger for the linearly

heated left wall and cooled right wall.

1. INTRODUCTION

In recent years, an ever-increasing awareness of thermally driven flows hasbeen reflected in considerable interest in fluid motions and transport processesgenerated or altered by buoyancy force for practical applications in many fields ofscience and technology. Consequently, a significant amount of research is being car-ried out in diverse areas of meteorology, geophysics, energy storage, fire control,studies of air movement in attics and greenhouses, solar distillers, growth of crystalsin liquids, etc. The essential coupling of transport properties of flow and thermalfields leads to an added complexity in buoyancy-driven flows. A comprehensivereview by Ostrach [1] explains that internal natural-convection flow problems aremore complex than external ones. Further, Gebhart [2] and Hoogendoorn [3]emphasized various aspects of natural-convection flow in a square cavity. Externalbuoyancy-driven flow problems are considerably simpler than internal buoyancy-driven flow problems. The physical reason is that at large Rayleigh number, classicalboundary-layer theory yields the same simplifications for external problems, namely,

Received 17 December 2006; accepted 15 June 2007.

Address correspondence to Tanmay Basak, Department of Chemical Engineering, Indian Institute

of Technology Madras, Chennai 600036, India. E-mail: tanmay@iitm.ac.in

551

Numerical Heat Transfer, Part B, 52: 551–568, 2007

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790701563623

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the region exterior to the boundary layer is unaffected by the boundary layer.In contrast, for confined natural convection, boundary layers form near the wallsand the region exterior to them is enclosed by the boundary layers and forms a coreregion. Since the core is partially or fully encircled by the boundary layers, the coreflow is not readily determined from the boundary conditions but depends on theboundary layer, which in turn is influenced by the core. The interactions betweenthe boundary layer and core constitute a major complexity in the problem and areinherent to all confined convection configurations, namely, that the flow patterncannot be predicted a priori from the given boundary conditions and geometry. Infact, the situation is even more intricate because it often appears that more thanone global core flow is possible and flow subregions, such as cells and layers, maybe embedded in the core. This physical complexity in confined convection is not onlya topic for analysis but has equal significance for numerical and experimental inves-tigations. The extensive studies using various numerical simulations reported byPatterson and Imberger [4], Nicolette et al. [5], Hall et al. [6], Hyun and Lee [7],Fusegi et al. [8], Lage and Bejan [9, 10], and Xia and Murthy [11] ensure that severalattempts have been made to acquire a basic understanding of natural-convectionflow and heat transfer characteristics in an enclosure.

The majority of works dealing with convection in enclosures is restricted to thecases of simple geometry, e.g., rectangular, square, cylindrical, and spherical cavities.However, the configurations of actual containers occurring in practice are often farfrom being simple. Iyican et al. [12] investigated natural-convective flow and heattransfer within a trapezoidal enclosure with parallel cylindrical top and bottom wallsat different temperatures and plane adiabatic side walls. The flow features in trap-ezoidal enclosures are predicted based on data for rectangular enclosures. A criticalRayleigh number is presented depending on the tilting angle, where unicellularconvection is observed. Karyakin [13] reported two-dimensional laminar natural

NOMENCLATURE

g acceleration due to gravity, m=s2

H height of the trapezoidal cavity

k thermal conductivity, W=m K

Nu local Nusselt number

p pressure, Pa

P dimensionless pressure

Pr Prandtl number

Ra Rayleigh number

T temperature, K

Tc temperature of cold vertical wall, K

Th temperature of hot bottom wall, K

u x component of velocity

U x component of dimensionless

velocity

v y component of velocity

V y component of dimensionless

velocity

X dimensionless distance along x

coordinate

Y dimensionless distance along y

coordinate

a thermal diffusivity, m2=s

b volume expansion coefficient, K�1

c penalty parameter

g vertical coordinate in a unit square

h dimensionless temperature

n kinematic viscosity, m2=s

n horizontal coordinate in a unit square

q density, kg=m3

U basis functions

u angle of inclination of the left wall

w stream function

Subscripts

b bottom wall

l left wall

r right wall

s side wall

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convection in enclosures of arbitrary cross section. This study is based on transientnatural convection in an isosceles trapezoidal cavity inclined at angle u to the verti-cal plane, and a single circulation region is found in the steady-state case. The heattransfer rate is found to increase with increasing angle u. Peric [14] studied naturalconvection in trapezoidal cavities with a series of systematically refined grids from10� 10 to 160� 160 and observed the convergence of results for grid-independentsolutions. Kuyper and Hoogendoorn [15] investigated laminar natural-convectionflow in trapezoidal enclosures to study the influence of the inclination angle onthe flow and also the dependence of the average Nusselt number on the Rayleighnumber. Thermosolutal heat transfer within a trapezoidal cavity heated at the bot-tom and cooled at the inclined top part was investigated by Boussaid et al. [16].The convective heat transport equation was solved by the alternating directionimplicit (ADI) method combined with a fourth-order compact Hermitian method.It is evident from the literature that no attempt has been made to perform detailedcalculations of local and average Nusselt numbers on a natural-convection flowwithin a trapezoidal enclosure for various thermal boundary conditions. Therefore,as a step toward the eventual development of natural-convection flows within closedenclosures, it is interesting to pursue a complete understanding of heat transfer ratesfor many engineering applications such as cooling of computer systems and otherelectronic equipment.

The present study deals with a natural-convection flow within a trapezoidalenclosure where the bottom wall is heated and vertical walls are linearly heated orcooled whereas the top wall is well insulated. The consistent penalty finite-elementmethod [17] has been used to solve the nonlinear coupled partial differentialequations for flow and temperature fields.

2. GOVERNING EQUATIONS

Consider a trapezoidal cavity of length L and height H with the left wallinclined at an angle u ¼ 30� to the y axis as shown in Figure 1. The velocity

Figure 1. Schematic diagram of the physical system.

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boundary conditions are considered as no-slip on solid boundaries. The fluid isconsidered as incompressible and Newtonian and the flow is assumed to be laminar.For the treatment of the buoyancy term in the momentum equation, the Boussinesqapproximation is employed to account for the variations of density as a function oftemperature, and to couple in this way the temperature field to the flow field.

Using the following transformation of variables,

X ¼ x

HY ¼ y

HU ¼ uH

aV ¼ vH

aP ¼ pH2

qa2h ¼ T � Tc

Th � Tcð1Þ

the governing equations in nondimensional form for steady natural-convection flowusing conservation of mass, momentum, and energy can be written as

qU

qXþ qV

qY¼ 0 ð2Þ

UqU

qXþ V

qU

qY¼ � qP

qXþ Pr

q2U

qX 2þ q2U

qY 2

!ð3Þ

UqV

qXþ V

qV

qY¼ � qP

qYþ Pr

q2V

qX 2þ q2V

qY 2

!þRa Pr h ð4Þ

UqhqXþ V

qhqY¼ q2h

qX 2þ q2hqY 2

ð5Þ

with the boundary conditions (see Figure 1)

U ¼ 0 V ¼ 0 h ¼ 1 on AB

U ¼ 0 V ¼ 0 h ¼ 1� Y or h ¼ 0 on BC

U ¼ 0 V ¼ 0 h ¼ 1� Y on AD

U ¼ 0 V ¼ 0qhqY¼ 0 on CD

ð6Þ

Here X and Y are dimensionless coordinates varying along the horizontal andvertical directions, respectively; U and V are dimensionless velocity componentsin the X and Y directions, respectively; h is the dimensionless temperature; P is thedimensionless pressure; and Ra and Pr are Rayleigh and Prandtl numbers, respectively.

3. SOLUTION PROCEDURE AND POSTPROCESSING

The momentum and energy balance equations [Eqs. (3)–(5)] are solved usingthe Galerkin finite-element method. The continuity equation [Eq. (2)] will be usedas a constraint due to mass conservation, and this constraint may be used to obtainthe pressure distribution [17–19]. In order to solve Eqs. (3) and (4), we use the

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penalty finite-element method where the pressure P is eliminated by a penalty para-meter c and the incompressibility criteria given by Eq. (2) (see Reddy [17]) results in

P ¼ �cqU

qXþ qV

qY

� �ð7Þ

The continuity equation [Eq. (2)] is automatically satisfied for large values of c.Typical values of c that yield consistent solutions are 105�107 [17–19].

Using Eq. (7), the momentum balance equations [Eqs. (3) and (4)] reduce to

UqU

qXþ V

qU

qY¼ c

qqX

qU

qXþ qV

qY

� �þ Pr

q2U

qX 2þ q2U

qY 2

!ð8Þ

and

UqV

qXþ V

qV

qY¼ c

qqY

qU

qXþ qV

qY

� �þ Pr

q2V

qX 2þ q2V

qY 2

!þRa Pr h ð9Þ

The system of equations [Eqs. (5), (8), and (9)] with boundary conditions [Eq. (6)]is solved by using Galerkin finite-element method [17–19]. The detailed descriptionof the solution procedure may be found in earlier works [18, 19]. The numerical solu-tions are obtained in terms of the velocity components (U , V ) and temperature (h).The flow circulations are represented by stream function (w), where the streamfunction (w) is defined in the usual way as U ¼ qw=qY and V ¼ �ðqw=qX Þ [20].It may be noted that a positive sign on w denotes anticlockwise circulation and theclockwise circulation is represented by a negative sign on w. The no-slip condition isvalid at all boundaries as there is no cross flow, hence w ¼ 0 is used for the boundaries.

The heat transfer coefficient in terms of the local Nusselt number (Nu) isdefined by

Nu ¼ � qhqn

ð10Þ

where n denotes the normal direction on a plane. The normal derivative is evaluatedby the bi-quadratics basis set in the n�g domain. The local Nusselt numbers at thebottom wall ðNubÞ, left wall ðNulÞ, and right wall ðNurÞ are defined as

Nub ¼ �X9

i¼1

hiqUi

qYð11Þ

Nul ¼ �X9

i¼1

hi cos uqUi

qXþ sin u

qUi

qY

� �ð12Þ

and

Nur ¼ �X9

i¼1

hi cos uqUi

qX� sin u

qUi

qY

� �ð13Þ

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Note that, u ¼ 30� for the present work. The average Nusselt numbers at thebottom, left, and right walls are

Nub ¼R 1

0 Nub dX

X j10¼Z 1

0

Nub dX ð14Þ

Nul ¼ cos uZ 1= cos u

0

Nul ds1 ð15Þ

and

Nur ¼ cos uZ 1= cos u

0

Nur ds2 ð16Þ

where ds1 and ds2 are the small elemental lengths along the left and right walls.

4. NUMERICAL TESTS

Numerical solutions are obtained for various values of Ra ¼ 103–106 andPr ¼ 0.07–100 with uniform heating of the bottom wall, side walls linearly heatedor cooled, and the top wall well insulated. The jump discontinuity in Dirichlet-typewall boundary conditions at the right corner point (see Figure 1) may correspond toa computational singularity. In particular, the singularity at the right corner node ofthe bottom wall needs special attention. The grid size-dependent effect of the tem-perature discontinuity at the corner point on the local (and the overall) Nusselt num-bers tends to increase as the mesh spacing at the corner is reduced. One of the waysfor handling the problem is assuming the average temperature of the two walls at thecorner and keeping the adjacent grid nodes at the respective wall temperatures.Alternatively, based on earlier work by Ganzarolli and Milanez [21] and Corcione[22], this procedure is still grid-dependent unless a sufficiently refined mesh is imple-mented. Accordingly, once any corner formed by the intersection of two differentlyheated boundary walls is assumed at the average temperature of the adjacent walls,the optimal grid size obtained for each configuration corresponds to the mesh spa-cing over which further grid refinements lead to grid-invariant results in both heattransfer rates and flow fields.

A grid invariance test on temperature and flow fields also has been carried out,and the computational domain consisting of 20� 20 bi-quadratic elements or 41� 41grid points in the n�g domain as seen in Figure 2a is found to give accurate results,with an error limit within �3%. In addition, the bi-quadratic elements with fewernodes smoothly capture the nonlinear variations of the field variables, in contrastto the finite-difference=finite-volume solutions available in the literature [14, 16].The complete grid invariance analysis is not shown for the sake of brevity.

In the current investigation, the Gaussian quadrature-based finite-elementmethod provides smooth solutions at the interior domain including the cornerregions, as evaluation of residual depends on interior Gauss points and thus theeffect of corner nodes is less pronounced in the final solution [17]. In general, theNusselt numbers for finite-difference=finite-volume-based methods are calculatedat any surface using some interpolation functions [14, 15], which are now avoided

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in the current work. The present finite-element approach also offers special advan-tage in the evaluation of local Nusselt number at the bottom and side walls, asthe element basis functions are used to evaluate the heat flux.

5. RESULTS AND DISCUSSION

5.1. Effects of Rayleigh Number

5.1.1. Case I: Linearly heated side walls. Figures 3–9 illustrate the streamfunction and isotherm contours for various values of Ra ¼ 103–105 andPr ¼ 0.07–100 with uniformly heated bottom wall and linearly heated side wallswhile the top wall is well insulated. A representative illustration is also shown forhigher Ra (Ra ¼ 106). As expected, due to the linearly heated vertical walls andthe uniformly heated bottom wall, fluids rise up from the middle portion of the bot-tom wall and flow down along the two vertical walls, forming two symmetric rolls

Figure 2. (a) Mapping of trapezoidal domain to a square domain in n–g coordinate system. (b) Mapping

of an individual element to a single element in n–g coordinate system.

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with clockwise and anticlockwise rotations inside the cavity. Since there is nosingularity at the corners of the bottom wall for the present prescribed temperaturedistributions, the temperature contours are smoothly distributed inside the cavity,without any singularity. At Ra ¼ 103 and Pr ¼ 0.07, the magnitudes of stream func-tions are considerably lower and the heat transfer is purely due to conduction, asseen in Figure 3. During conduction-dominant heat transfer, the temperature con-tours with h � 0:7 are smooth curves which span the entire enclosure, and theyare symmetric with respect to the vertical symmetric line at the center. The conduc-tion-dominant heat transfer will be illustrated later via average Nusselt numberversus Rayleigh number distribution in order to demonstrate the significant effectof convective heat transfer.

For Pr ¼ 0.07, the conduction-dominant heat transfer mode will occur up toRa < 5� 103. At Ra ¼ 5� 103 with Pr ¼ 0.07, the circulation near the centralregimes gradually becomes stronger, and consequently, the temperature contourwith h ¼ 0:7 starts shifting toward the side wall and breaks into two symmetric con-tour lines (see Figure 4). As Ra increases to 105, the buoyancy-driven circulationinside the cavity also increases, as seen from the greater magnitudes of the streamfunctions. It may be noted that the maximum value of stream function is found tobe 9 for Ra ¼ 105. Consequently, at Ra ¼ 105, the temperature gradients near boththe bottom and side walls tend to be significant to develop the thermal boundarylayer, and the isotherm lines with values h � 0:7 cover almost 70% of the cavity(Figure 5). It is also interesting to note that the temperature contours (h� 0.7) arepushed toward the upper portion of the side walls, which may result in enhanced

Figure 3. (a) Temperature and (b) stream function contours for linearly heated side walls, with Pr ¼ 0.07

and Ra ¼ 103. Clockwise and anticlockwise flows are shown via negative and positive signs of stream

functions, respectively.

Figure 4. (a) Temperature and (b) stream function contours for linearly heated side walls, with Pr ¼ 0.07

and Ra ¼ 5� 103. Clockwise and anticlockwise flows are shown via negative and positive signs of stream

functions, respectively.

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heat transfer rates at the side walls as discussed later. A representative case for higherRa (Ra ¼ 106) has been illustrated in Figure 6. For Ra ¼ 106, the temperature con-tours with h < 0:7 are strongly compressed toward the side walls due to enhancedcirculation at the core. It may be noted that the maximum value of w is 9 forRa ¼ 105, whereas the maximum value of w is 21 for Ra ¼ 106. It is interesting toobserve that multiple secondary circulations are also developed near the centralregime of the bottom wall as well as near the bottom corners, and therefore, the tem-perature contours with h ¼ 0.8–0.9 show some oscillations.

Comparative studies of Figures 5–9 show that as Pr increases from 0.07 to 100,the values of stream functions in the core cavity increase, i.e., the flow rate increases.Figures 5 and 6 show that streamlines are almost circular for small Prandtl numbers(Pr ¼ 0.07). On the other hand, as Pr increases from 0.07 to 100, the streamlines nearthe walls of the enclosure take the shape of the enclosure because of the high flowrate, especially for Pr ¼ 100. It may be noted that the maximum value of w is 9for Pr ¼ 0.07, whereas it is around 14 for Pr ¼ 100 at Ra ¼ 105. The isotherm lineswith h � 0:7 are gradually compressed toward the side walls for higher Pr. ForPr � 10, the isotherm lines with h � 0:6 cover almost 85% of the entire cavity(Figures 8 and 9).

5.1.2. Case II: Linearly heated left wall with cooled right wall. Streamfunction and isotherm contours are displayed in Figures 10–14 for Ra ¼ 103–105

and Pr ¼ 0.07–10 when the bottom wall is uniformly heated and the left wall is lin-early heated while the right wall is cooled and the top wall is well insulated. Similarto case I, a representative case has also been shown for Ra ¼ 106 (Figure 12). As seen

Figure 5. (a) Temperature and (b) stream function contours for linearly heated side walls, with Pr ¼ 0.07

and Ra ¼ 105. Clockwise and anticlockwise flows are shown via negative and positive signs of stream

functions, respectively.

Figure 6. (a) Temperature and (b) stream function contours for linearly heated side walls, with Pr ¼ 0.07

and Ra ¼ 106. Clockwise and anticlockwise flows are shown via negative and positive signs of stream

functions, respectively.

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in Figures 10–14, due to the uniformly heated bottom wall and cooled right wall, asingularity appears at the right bottom edge of the cavity. The formation of the ther-mal boundary layer along the left wall of the cavity is weak, whereas isotherm con-tours with h � 0:5 are compressed along the right wall of the cavity, forming a strongthermal boundary layer. The symmetrical circulation pattern which was observed forthe linearly heated side walls case is absent in this case because of the nonsymmetricthermal boundary conditions. Streamlines closer to the top left wall show secondarycirculations, whereas streamlines closer to the right wall show stronger circulations.The physical reason is that the uniformly heated bottom wall and linearly heated leftwall cause the fluids to move with less circulation along the left wall whereas, due tothe cold right wall, a larger amount of fluid flows along the right wall. As a result,strong circulation patterns are formed on the right side of the cavity, whereas second-ary circulation patterns appear on the left side of the cavity.

It is seen from Figure 10 that the temperature contours are almost parallel, theheat transfer is dominated by conduction, and the stream function contours showsmall values for Ra ¼ 103. As Rayleigh number increases, the temperature contourlines become distorted and tend to break as a result of the higher intensity of the cir-culation. For Ra ¼ 105 with Pr ¼ 0.07, the values of stream functions become largerand the temperature contours are compressed near the cold wall, as seen in Figure 11.The secondary circulation cell also shows a neck formation near the bottom and leftwalls. As Ra increases to 106 (see Figure 12), the temperature contour lines on thecold right wall become strongly compressed due to enhanced convection. AtRa ¼ 106, the maximum value of w is found to be 30. An additional secondary cell

Figure 8. (a) Temperature and (b) stream function contours for linearly heated side walls, with Pr ¼ 10

and Ra ¼ 105. Clockwise and anticlockwise flows are shown via negative and positive signs of stream

functions, respectively.

Figure 7. (a) Temperature and (b) stream function contours for linearly heated side walls, with Pr ¼ 0.7

and Ra ¼ 105. Clockwise and anticlockwise flows are shown via negative and positive signs of stream

functions, respectively.

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is also formed near the left wall, and this additional cell is the result of the gradualformation of neck in the secondary circulation pattern as seen in Figure 11.

Comparative studies of Figures 11–14 for Pr ¼ 0.07–10 have also been shown.As Pr increases, the values of stream function increase. In addition, a circularstreamline pattern occurs for Pr ¼ 0.07, whereas for Pr ¼ 10, streamlines take theshape of the container. It is interesting to observe that the secondary circulation cellsnear the left wall tend to have neck formation at smaller Pr, whereas the larger pri-mary circulation cells suppress the area of secondary circulation at larger Pr. It maybe interesting to note that the isotherms are largely compressed near the bottom andright walls for Pr� 0.7, as seen in Figures 13 and 14, whereas the isotherms arestrongly compressed near both side walls for case I with Pr� 0.7.

5.2. Heat Transfer Rates: Local and Average Nusselt Numbers

5.2.1. Case I: Linearly heated side walls. Figures 15a and 15b displaythe effects of Ra and Pr on the local Nusselt numbers at the bottom and side walls(Nub, Nus) for linearly heated side walls. At the edges of the bottom wall, the heattransfer rate Nub is 1 because of the linearly heated side walls (see Figure 15a). Also,because of symmetry in the temperature field, the heat transfer rate is symmetric withrespect to the midlength (X ¼ 1=2), with the minimum value at the center. ForRa ¼ 105 with Pr ¼ 0.7, a sinusoidal type of heat transfer rate symmetric with respectto the midlength X ¼ 1=2 is observed. The sinusoidal type of heat transfer rate is dueto increased intensity of circulation occurring symmetrically near the bottom wall,

Figure 10. (a) Temperature and (b) stream function contours for linearly heated left wall and cooled right

wall with Pr ¼ 0.07 and Ra ¼ 103. Clockwise and anticlockwise flows are shown via negative and positive

signs of stream functions, respectively.

Figure 9. (a) Temperature and (b) stream function contours for linearly heated side walls, with Pr ¼ 100

and Ra ¼ 105. Clockwise and anticlockwise flows are shown via negative and positive signs of stream

functions, respectively.

TRAPEZOIDAL CAVITY WITH HEATED SIDE WALL(S) 561

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which results in compression of isotherms at two locations near the bottom wall. AsPr increases from 0.7 to 100, this sinusoidal type of heat transfer rate becomes muchstronger. The temperature contours are compressed around the intermediate zonesbetween the corner and the vertical line of symmetry, and the local Nusselt numberis maximum at around X ¼ 0:3 and 0.7. Variations of Prandtl numbers fromPr ¼ 10 to 100 at Ra ¼ 105 show only a slight increase in the heat transfer rate.

Figure 15b illustrates the heat transfer rate at the side walls. Due to the sym-metry in the boundary condition, the average Nusselt number is identical along bothside walls. For Ra ¼ 103, Pr ¼ 0.07, due to weak circulations, the heat transfer rateslowly increases along the side wall. For Ra ¼ 105, due to stronger circulations, theheat transfer rate decreases in the lower halves of the side walls and an increasingtrend of heat transfer is observed in the upper halves of the side walls. The localNusselt number becomes negative, which implies that part of the heat gained fromthe lower positions of the wall will return back to it at the higher positions. Further,it is observed that the temperature contours are compressed toward the side wallsaway from the corner points of the bottom wall, as seen in Figures 5–9.

The overall effects on the heat transfer rates are displayed in Figures 17a and17b, where the distributions of the average Nusselt number of the bottom and sidewalls are plotted versus the logarithmic Rayleigh number. The average Nusseltnumbers are obtained using Eqs. (14)–(16), where the integral is evaluated usingSimpson’s one-third rule. The values of the average Nusselt numbers along the sidewalls are less compared to the bottom wall. This is due to the fact that the heattransfer to the fluid from the bottom wall is more compared to the side wall. It is

Figure 11. (a) Temperature and (b) stream function contours for linearly heated left wall and cooled right

wall with Pr ¼ 0.07 and Ra ¼ 105. Clockwise and anticlockwise flows are shown via negative and positive

signs of stream functions, respectively.

Figure 12. (a) Temperature and (b) stream function contours for linearly heated left wall and cooled right

wall with Pr ¼ 0.07 and Ra ¼ 106. Clockwise and anticlockwise flows are shown via negative and positive

signs of stream functions, respectively.

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also observed that the average Nusselt numbers for both bottom and side wallsremain constant up to Ra ¼ 5� 103. The dependence of average Nusselt numberon the Prandtl number is also found to be significant. The average Nusselt numbers(Nub, Nus) also increase with increase in Prandtl number because of the enhancedheat transfer rate at higher Pr.

5.2.2. Case II: Linearly heated left wall with cooled right wall. Figures16a–16c display the effects of Ra and Pr on the local Nusselt number at the bottom,left, and right walls (Nub, Nul , Nur) for linearly heated left wall and cooled rightwall. The heat transfer rate Nub is minimum at the left edge of the bottom walldue to the linearly heated left wall and maximum at the right edge of the bottomwall due to the cooled right wall for Ra varying within 103 to 105 (see Figure 16a).For all the cases, the isotherm contours near the cold right wall are highlycompressed due to the discontinuity present in the right edge, resulting in highthermal gradient near the right edge of the bottom wall.

Figure 16b illustrates the heat transfer rate (Nul) at the bottom edge of the leftwall and it is seen that Nul is negative, indicating that the heat loss from the left wallto the cooler fluid and its magnitudes increase from the bottom edge to the top edgeof the left wall. At Ra ¼ 103 with Pr ¼ 0.07, the local Nusselt number is a monoto-nically increasing function toward the top edge of the left wall. For Ra ¼ 105, thelocal Nusselt number shows oscillations except near the top edge for Pr ¼ 0.7 and10. The oscillations are due to the presence of secondary circulations. Further, the

Figure 13. (a) Temperature and (b) stream function contours for linearly heated left wall and cooled right

wall with Pr ¼ 0.7 and Ra ¼ 105. Clockwise and anticlockwise flows are shown via negative and positive

signs of stream functions, respectively.

Figure 14. (a) Temperature and (b) stream function contours for linearly heated left wall and cooled right

wall with Pr ¼ 10 and Ra ¼ 105. Clockwise and anticlockwise flows are shown via negative and positive

signs of stream functions, respectively.

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secondary circulations push the temperature contours at the top edge of the left walland these result in large local Nusselt number. Figure 16c illustrates the heat transferrate (Nur) at the right wall. The heat transfer rate at the right wall is maximum at thebottom edge of the right wall due to the presence of a singularity in the thermalboundary condition. For Ra ¼ 103 with Pr ¼ 0.07, the local Nusselt number is foundto be a monotonically decreasing function of Rayleigh number. For Ra ¼ 105 withPr ¼ 0.07–10, the local Nusselt number exhibits oscillatory behavior. This is due tothe fact that larger primary circulations push the temperature contours near themiddle portion of the right wall.

Figure 15. Variation of local Nusselt number with distance along (a) bottom wall and (b) side wall for case

I: linearly heated side walls.

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The overall effects of Ra and Pr on the average Nusselt numbers at the bottomand side walls are displayed in Figures 17c and 17d. The average Nusselt numbers forthe bottom, left, and right walls remain constant up to Ra ¼ 3� 103. It is observedthat the average Nusselt numbers for all the walls increase smoothly with Ra,whereas for Pr ¼ 0.07, Nul decreases due to the presence of secondary circulationin a larger portion of the left wall. For Pr ¼ 0.07, the average Nusselt numberdecreases up to Ra ¼ 104 and then increases up to Ra ¼ 105, whereas forPr ¼ 0.7, 10, and 100, the average Nusselt numbers increase smoothly for all Rabecause of the increased convection. In contrast, the average Nusselt numbers atthe right wall are found to increase with Prandtl numbers, as discussed for the caseof linearly heated side walls.

Figure 16. Variation of local Nusselt number with distance along (a) bottom wall, (b) left wall, and

(c) right wall for case II: linearly heated left wall and cooled right wall.

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6. CONCLUSION

In this article, the phenomenon of natural-convection heat transfer in trap-ezoidal enclosures heated from below and linearly heated and cooled from the sideshas been studied in detail. The penalty finite-element method is advantageous forobtaining smooth solutions in terms of stream function and isotherm contours forwide ranges of Ra and Pr. For the case of linearly heated side walls, temperaturecontours are compressed toward the side walls, whereas temperature contours arestrongly compressed toward the cold right wall and weakly compressed toward

Figure 17. Variation of average Nusselt number with Rayleigh number for linearly heated side walls

[(a) and (b)] and linearly heated left wall and cooled right wall [(c) and (d)]. The inset of (d) shows plot

of average Nusselt number versus Rayleigh number for right wall.

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the linearly heated left wall for the case of linearly heated left wall and cooled rightwall. The stream function contours show two symmetric circulations inside the cav-ity for the case of linearly heated side walls. Stronger circulation rolls near the coldright wall and secondary circulation rolls near the linearly heated left wall areobserved for the case of linearly heated left wall and cooled right wall. The con-duction-dominant heat transfer modes have been observed for Ra � 5� 103 withthe case of linearly heated side walls, and for Ra � 3� 103 with the case of linearlyheated left wall and cooled right wall.

For the case of linearly heated side walls and linearly heated left wall withcooled right wall, the local Nusselt numbers at the side walls and left wall becomenegative, which means that part of the heat gained from the lower positions of thewall will return back to it at the higher positions. For the case of linearly heated leftwall and cooled right wall, the local Nusselt number at the left wall exhibits oscillat-ory behavior due to the presence of secondary circulation toward the linearly heatedleft wall. The local Nusselt number at the right wall is found to decrease with dis-tance for conduction-dominant heat transfer, whereas for convection mode, the localNusselt number is found to increase. The effect of Prandtl number in the variation ofaverage Nusselt number is found to be more significant for Pr between 0.07 and 0.7than between 10 and 100. Average Nusselt number versus Rayleigh number illus-trates that the overall heat transfer rate at the bottom wall is greater for the linearlyheated left wall and cooled right wall.

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