A Complete Heatline Analysis on Visualization of Heat Flow and

ARTICLE pubs.acs.org/IECR

A Complete Heatline Analysis on Visualization of Heat Flow and Thermal Mixing during Mixed Convection in a Square Cavity with Various Wall Heating Tanmay Basak,*,† P. V. Krishna Pradeep,† and S. Roy‡ †

Department of Chemical Engineering and ‡Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India ABSTRACT: A wide range of applications involving mixed-convection studies can be found in various engineering processes such as thermal discharge of water bodies, float glass production, heat exchangers, nuclear reactors, and crystallization process. The present study focuses on understanding the thermal mixing scenarios for mixed-convection lid-driven flow in a square cavity using heatlines. Thermal mixing is analyzed for four different thermal boundary conditions, and heat flow patterns in mixed convection are analyzed using Bejan’s heatlines concept for wide ranges of parameters (Pr = 0.015
7.2, Re = 1
100, and Gr = 103
105, where Pr, Re, and Gr denote the Prandtl, Reynolds, and Grashof numbers, respectively). The results indicate that, at low Pr values (Pr = 0.015), the transport is conduction-dominant irrespective of the values of Gr and Re. The trends of heatlines and streamlines are identical near the core for high-Re cases. A single circulation cell was observed in the streamlines for any Pr g 0.7 at high Re and low Gr values for uniform heating of the bottom surface with cold side walls. It was observed that thermal mixing increased significantly with subsequent rises in Gr for high-Pr fluids. Patterns of heatlines and multiple circulation cells of heatlines were found to lead to enhanced thermal mixing, with the thermal boundary layer much compressed toward the walls for linearly heated side walls. The heat-transfer rates along the walls are illustrated by the local Nusselt number distribution based on gradients of heatfunctions for the first time in this work. Nusselt numbers with infinitely large magnitudes were observed at hot
cold junctions, illustrating high heattransfer rates. An oscillatory distribution in the local heatfunction rate was observed as a result of sinusoidal heating of the bottom surface for high-Pr fluids. Negative heat-transfer rates or local Nusselt numbers were observed along the side walls when side wall(s) was/were linearly heated, as explained based on negative heatfunction gradients. Also, the effect of Gr on the local and average Nusselt numbers in different cases can be adequately explained based on heatlines. Dense heatlines signifying higher overall heattransfer rates along the bottom surface and side walls were observed for uniform bottom surface heating, whereas lower heat-transfer rates were observed for sinusoidal heating. Nonmonotonic distributions in overall heat-transfer rates along the bottom surface and left wall were observed when both walls were linearly heated, whereas a smooth and exponential increase was observed when the right wall was isothermally cooled.

1. INTRODUCTION The fundamental problem of heat transfer in a closed cavity resulting from combined forced and natural convection has received considerable attention from researchers. This situation is commonly referred to as mixed convection. In mixed-convection flows, the effects of both forced and free convection are of comparable magnitudes. The lid-driven cavity, along with the thermal nonhomogeneity, gives rise to a buoyancy force that, in turn, impacts the coupled fields of velocity and temperature in the cavity. The governing dimensionless parameters for mixed convection are the Grashof number (Gr), Reynolds number (Re), and Prandtl number (Pr). Note that Gr and Re represent the strengths of the effects of the natural and forced convection flows, respectively. An important dimensionless number, the Richardson number (Ri), which is defined as Ri = Gr/Ren, characterizes the mixed-convection flow such that Ri = ¥ denotes natural convection and Ri = 0 represents forced convection. The exponent n depends on the geometry, the thermal boundary conditions, and the fluid. The relative importance of forced convection over thermal diffusion is denoted by the Peclet number (Pe = Re 3 Pr). The mixed-convection problem for a lid-driven cavity in an enclosure finds a wide range of applications in various fields of r 2011 American Chemical Society

engineering and science such as float glass production,1 hydraulics of nuclear reactors,2 dynamics of lakes,3 heat transfer in vertical tubes,4
6 and thermal discharge in water bodies.7 The lid-driven-cavity problem has been extensively used as a benchmark case for the evaluation of numerical solution algorithms.8,9 A few earlier investigations involved detailed analyses of convective transport for material processing.10
14 Some previous studies involved applications with mixed convection in lid-driven square/rectangular cavities of various fluids. Convective motion and heat transfer driven by a combined temperature gradient and imposed lid shear in a square cavity filled with a low-Prandtl-number fluid (Pr = 0.005) involving a stably stratified fluid was studied by Mohamad and Viskanta.15,16 This study was extended in determining the velocity profiles and frictional pressure drop for shear-thinning materials in a rectangular cavity.17 Al-Amiri et al.18 analyzed the effects of mixedconvection heat transfer in a lid-driven cavity with a sinusoidal wavy bottom surface. Turki et al.19 carried out a numerical Received: December 20, 2010 Accepted: April 19, 2011 Revised: April 1, 2011 Published: April 19, 2011 7608

dx.doi.org/10.1021/ie102530u | Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research investigation to analyze forced and mixed convection in a horizontal channel with a built-in heated square cylinder. Guo and Sharif20 analyzed mixed convection in rectangular cavities at various aspect ratios with moving isothermal side walls and a constant-flux heat source. Bhoite et al.21 studied mixed convection in a shallow enclosure with a series of heat-generating equipment. Al-Amiri et al.22 studied steady mixed convection in a square lid-driven cavity under the combined buoyancy effects of thermal and mass diffusion. Conjugate-mixed-convection heat transfer in a lid-driven enclosure with a thick bottom surface was studied by Oztop et al.23 Mixed-convection studies have also been carried out for various power-law fluids involving various applications.24
27 Although a number of numerical investigations based on liddriven closed enclosures have been carried out, the detailed analysis of heat flow paths during mixed convection is not yet well understood. The current work is based on heat flow visualization during mixed convection in square cavities. The heat flow is visualized using heatfunctions that are derived first time in this work for mixed-convection problems. Isotherms are useful for detecting hot and cold regions in the domain subject to various boundary conditions. However, isotherms fail to establish the heat flow patterns in the presence of various realistic boundary conditions. The direction of heat flow is governed by the heat flux, and heat flux lines can easily be derived as being perpendicular to the isotherms in the presence of conductive heat transport. The heat flux lines will be highly nontrivial when the isotherms are strongly coupled with convective transport. In general, streamlines are helpful for visualizing flow trajectories, flow separations, and multiple circulations, among other phenomena. Consequently, heat transport due to complex flow patterns has to be analyzed to quantify the role of the heat distribution for various hot/cold zones in the domain. Therefore, “heatlines” analogous to streamlines are derived in this work to display heat flow trajectories for mixed convection. Various boundary conditions represent heating patterns that can be employed in various applications and heat flow with such boundary conditions involving complex flow patterns are analyzed with heatlines. Heatlines will be shown to be a useful numerical tool for explaining isotherms involving hot/cold zones in the cavity. Thermal management for efficient operation in cavities with various patterns of wall heating with movement of the top adiabatic lid is an important issue that is explained using heatlines in this work. The heatline method is employed to visualize the heat transfer in two-dimensional convective transport process.28,29 Heatfunctions are the mathematical representation of heatlines, and each heatline corresponds to a constant heatfunction. The proper dimensionless forms of heatfunctions are closely related to Nusselt numbers. The heatline concept was first introduced by Kimura and Bejan.28 Various applications using heatlines were further studied in analyzing thermal convection and heat flow in electroconductive melts.30,31 Further, the concept has been extended to visualize mass transfer using masslines.32
34 The study of heat flow using the heatline approach has also been extended to cylindrical enclosures.35 Heat flow visualization in a complicated cavity was studied by Dalal and Das36 using the heatline concept. The effects of wall-located heat barriers on conjugate conduction/natural-convection heat transfer and fluid flow in enclosures were studied using heatlines by Hakyemez et al.37 Recently, energy flux vectors were also employed to visualize heat flow.38,39

ARTICLE

The heatline concept has also been applied for analyzing heat transfer involving forced convection.40 Studies have also been carried out in mixed convective ventilation systems using the heatline concept. Deng and Tang41 applied the heatline approach to visualize the heat transport in a mixed convective ventilation system. This study was extended to analyze the fluid-, heat-, and contaminant-transport structures of laminar double-diffusive mixed convection in a two-dimensional ventilated enclosure by Deng et al.42 A correlation between the two convection parameters Re and Gr was also obtained. Zhao et al.43,44 used the heatline method for the visualization of heat flow in the presence of mixed convection in an enclosure with ventilation ports. However, a detailed investigation of heat flow in mixed-convection lid-driven cavities subject to generalized boundary conditions has yet to appear in the literature. The aim of the current study was to analyze the heat flow due to mixed convection in a square cavity filled with an incompressible fluid for various thermal boundary conditions as a first attempt. The main objective of the present study was to examine the extent of thermal mixing at the core of the cavity in the presence of a moving top surface. First, a square cavity with a uniformly heated bottom surface and cold side walls in the presence of an insulated moving top surface was considered. This study was also extended for a nonuniformly heated bottom surface. Further, the influence of linearly heated side wall(s) with a uniformly heated bottom surface was studied. A penalty finite-element approach using the Galerkin method was applied to solve the nonlinear coupled equations for flow and temperature fields. The Galerkin method was further employed to solve the Poisson equation for streamfunctions and heatfunctions. A finite discontinuity was found to exist at the junction of the hot and cold walls, leading to a mathematical singularity. The solution of the heatfunction for this type of situation demands implementation of exact boundary conditions. The heatlines and thermal mixing were studied for various lid velocities in terms of Re values ranging from 1 to 100. Each case was studied for commonly used fluids with Pr = 0.015
7.2 for various industrial applications.

2. MATHEMATICAL FORMULATION AND SIMULATION The physical domain consists of a square cavity with the physical dimensions shown in Figure 1. The top surface is assumed to move with a uniform velocity of U0. Four cases were considered in the present study as follows: In case 1, the bottom surface is isothermally heated, and the side walls are isothermally cooled. In case 2, the bottom surface is nonisothermally heated, and the side walls are isothermally cooled. In case 3, both of the side walls are linearly heated and the bottom surface is isothermally hot. Finally, in case 4, the left wall is linearly heated, the right wall is isothermally cooled, and the bottom surface is isothermally hot. In all cases, the top moving surface is well insulated. The flow is assumed to be laminar, and the fluid properties are assumed to be constant, except for the body force term, for which the density variation is assumed to follow Boussinesq approximation, and the viscous dissipation terms, which are considered to be negligible. Under these assumptions, the dimensionless governing equations can be written as DU DV þ ¼0 DX DY 7609

ð1Þ

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

The momentum and energy balance equations (eqs 2
4) were solved using the Galerkin finite-element method. The continuity equation (eq 1) was used as a constraint due to mass conservation, and this constraint can be used to obtain the pressure distribution. A penalty finite-element approach was used to solve eqs 2 and 3, where the pressure P was eliminated using a penalty parameter (γ)45 and the incompressibility criteria given by eq 1, resulting in the expression   DU DV þ ð7Þ P ¼
γ DX DY The continuity equation is satisfied for large values of γ. A typical value of γ yielding consistent solutions is 107. Using eq 7, the momentum balance equations (eqs 2 and 3) reduce to !   DU DU D DU DV 1 D2 U D2 U þV ¼γ þ þ þ U DX DY DX DX DY Re DX 2 DY 2

Figure 1. Schematic diagram of the physical system.

U

DU DU DP 1 D2 U D2 U þV ¼
þ þ DX DY DX Re DX 2 DY 2

DV DV DP 1 D2 V D2 V þV ¼
þ U þ DX DY DY Re DX 2 DY 2

U

ð8Þ

! ð2Þ

! þ

Dθ Dθ 1 D2 θ D2 θ þV ¼ þ DX DY Re 3 Pr DX 2 DY 2

Gr θ Re2

and

!   DV DV D DU DV 1 D2 V D2 V Gr þV ¼γ þ þ 2θ U þ þ DX DY DY DX DY Re DX 2 DY 2 Re

ð3Þ

ð9Þ

ð4Þ

The system of equations (eqs 4, 8, and 9) with boundary conditions (eq 5) were solved using the Galerkin finite-element method. Expanding the velocity components (U, V) and temperature (θ) using the basis set {Φk}N k=1 as

!

with the boundary conditions

N

U

∑ UkΦk ðX, Y Þ, k¼1

θ

∑ θk Φk ðX, Y Þ k¼1

V 

N

∑ Vk Φk ðX, Y Þ, k¼1

and

U ¼ 0, V ¼ 0, θ ¼ 1 or sinðπXÞ,

"Y ¼ 0, 0 e X e 1

U ¼ 0, V ¼ 0, θ ¼ 0 or 1
Y ,

"X ¼ 0, 0 e Y e 1

U ¼ 0, V ¼ 0, θ ¼ 0 or 1
Y , Dθ ¼ 0, U ¼ 1, V ¼ 0, DY

"X ¼ 1, 0 e Y e 1

ð10Þ

"Y ¼ 1, 0 e X e 1

the Galerkin finite-element method yields the following nonlinear residual equations for eqs 8, 9, and 4, respectively, at nodes of internal domain Ω

ð5Þ The dimensionless variables and parameters are defined as

ð1Þ Ri

x y u v T
Tc X ¼ ,Y ¼ ,U ¼ ,V ¼ , θ ¼ , L L U0 U0 Th
Tc ð6Þ p ν U0 L gβðTh
Tc ÞL3 , Re ¼ , Gr ¼ P¼ , Pr ¼ FU0 2 R ν ν2 where x and y are the distances along the horizontal and vertical directions, respectively; u and v are the velocity components in the x and y directions, respectively; T denotes the temperature; p is the pressure; F is the density; Th and Tc are the temperatures at the hot and cold surfaces, respectively; L is the length of the side of the square cavity; U0 is the velocity of the top surface; X and Y represent the dimensionless horizontal and vertical coordinates, respectively; U and V represent the dimensionless velocity components in the X and Y directions, respectively; θ is the dimensionless temperature; P is the dimensionless pressure; and Re, Pr, and Gr are the Reynolds, Prandtl, and Grashof numbers, respectively.

N

# DΦk Φi dX dY ¼ Uk U k Φk V k Φk DY Ω k¼1 k¼1 k¼1 " # Z Z N N DΦi DΦk DΦi DΦk dX dY þ dX dY þγ Uk Vk Ω DX DY Ω DX DY k¼1 k¼1  Z  1 N DΦi DΦk DΦi DΦk dX dY þ Uk ð11Þ þ DY DY Re k ¼ 1 Ω DX DX N

∑

Z "

N

∑

!

DΦk þ DX

∑

N

∑

!

∑

∑

ð2Þ Ri

# DΦk Φi dX dY ¼ Vk U k Φk Vk Φ k DY Ω k¼1 k¼1 k¼1 " # Z Z N N DΦi DΦk DΦi DΦk þγ Uk Vk dX dY þ dX dY Ω DY DX Ω DY DY k¼1 k¼1  Z  1 N DΦi DΦk DΦi DΦk þ Vk þ dX dY DY DY Re k ¼ 1 Ω DX DX ! Z N Gr θk Φk Φi dX dY ð12Þ
2 Re Ω k ¼ 1 N

∑

Z "

N

∑

∑

!

DΦk þ DX

N

∑

!

∑

∑

∑

7610

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research and ð3Þ

Ri

¼

N

∑ k¼1

þ

Z " θk

Ω

N

∑ k¼1

! Uk Φk

DΦk þ DX

N

∑ k¼1

! Vk Φk

ARTICLE

DΦk Φi dX dY DY

 Z  1 N DΦi DΦk DΦi DΦk θk þ dX dY Re 3 Pr k ¼ 1 Ω DX DX DY DY

∑

and

#

ð13Þ

The set of nonlinear algebraic equations (eqs 11
13) was solved using reduced integration technique45,46 and the Newton
Raphson method as discussed in an earlier work.47 The numerical solutions were obtained in terms of the velocity components (U, V). 2.1. Streamfunction, Nusselt Number, and Heatfunction. 2.1.1. Streamfunction. The fluid motion can be visualized using the streamfunction (ψ) obtained from velocity components U and V. The relationships between the streamfunction and velocity components for two-dimensional flows are48 Dψ U ¼ DY

and

Dψ V ¼
DX

D2 ψ D2 ψ DU DV
ð15Þ þ 2 ¼ 2 DX DY DY DX Using this definition of the streamfunction, a positive value of ψ denotes anticlockwise circulation, and a negative value of ψ denotes clockwise circulation. Expanding the streamfunction N (ψ) using the basis set {Φk}N k=1 as ψ = ∑k=1ψkΦk(X,Y) and the relationships for U and V from eq 10, the Galerkin finiteelement method yields the following linear residual equations for eq 15  Z  N DΦi DΦk DΦi DΦk s þ ψk dX dY Ri ¼ DY DY Ω DX DX k¼1 Z Z N DΦk dX dY U k Φi
Φi n 3 rψ dΓ þ ð16Þ DY Γ Ω k¼1 Z N DΦk
dX dY Vk Φi DX Ω k¼1

∑

∑

∑

No-slip conditions are valid at all boundaries as there is no cross-flow; hence, ψ = 0 was used as the residual equation at the nodes for the boundaries. The biquadratic basis function was used to evaluate the integrals in eq 16, and the ψ functions were obtained by solving the N linear residual equations (eq 16). 2.1.2. Nusselt Number. The heat-transfer coefficient in terms of the local Nusselt number (Nu) is defined by Dθ Dn

and

9

DΦi DY

9

DΦ

∑ θi i¼1

Nul ¼

i θi ∑ DX i¼1

Z

1

Z

Nus dY Nus ¼

ð20Þ

0

Y j10

¼

1

Nus dY

ð22Þ

0

where Nus can refer to Nul and Nur for the left and right walls, respectively. 2.1.3. Heatfunction. The heat flow in the enclosure can be visualized using the heatfunction Π obtained from conductive heat fluxes (
∂θ/∂X,
∂θ/∂Y) as well as convective heat fluxes (Uθ, Vθ). The steady energy balance equation (eq 4) can be rearranged as " "  #  # D 1 Dθ D 1 Dθ Uθ
Vθ
þ ¼ 0 ð23Þ DX Re 3 Pr DX DY Re 3 Pr DY The heatfunction satisfies this equation such that DΠ 1 Dθ ¼ Uθ
DY Re 3 Pr DX DΠ 1 Dθ ¼ Vθ

DX Re 3 Pr DY

ð24Þ

which yields the single equation D2 Π D2 Π D D ðUθÞ
ðV θÞ ð25Þ þ ¼ DX 2 DY 2 DY DX Using the above definition of the heatfunction, a positive value of Π denotes anticlockwise heat flow, and a negative value of Π denotes clockwise heat flow. Expanding the heatfunction (Π) using N the basis set {Φk}N k=1 as Π = ∑k=1ΠkΦk(X,Y) and using the relationship for U, V, and θ from eq 10, the Galerkin finite-element method yields the following linear residual equations for eq 25 Rih

 Z  DΦi DΦk DΦi DΦk þ dX dY ¼ Πk DY DY Ω DX DX k¼1 ! Z Z N N DΦk dX dY
Φi n 3 rΠ dΓ þ Uk θk Φk Φi DY Γ Ω k¼1 k¼1 ! Z N N DΦk þ dX dY θk Uk Φk Φi DY Ω k¼1 k¼1 ! Z N N DΦk dX dY
Vk θk Φk Φi DX Ω k¼1 k¼1 ! Z N N DΦk
dX dY θk Vk Φk Φ i DX Ω k¼1 k¼1 N

∑

∑

ð17Þ

where n denotes the normal direction on a plane. The local Nusselt numbers at the bottom surface (Nub), left wall (Nul), and right wall (Nur) are defined as Nub ¼

DΦ

9

i θi ∑ DX i¼1

Note that the Nusselt numbers were evaluated using nine basis functions for a biquadratic element, and thus, nine terms were used in the summation for evaluation of the local Nusselt numbers (Nul, Nur, and Nub). The average Nusselt numbers at the bottom surface and side walls are Z 1 Nub dX Z 1 ð21Þ Nub ¼ 0 ¼ Nub dX Xj10 0

ð14Þ

which yield the single equation

Nu ¼


Nur ¼


ð18Þ

ð19Þ

∑

∑

∑

∑

∑

∑

∑

ð26Þ 7611

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

The residual equation (eq 26) is further supplemented with various Dirichlet and Neumann boundary conditions to obtain unique solution of eq 25. Neumann boundary conditions are obtained from eq 24 for isothermally heated or cooled walls and are specified as follows n 3 rΠ ¼ 0 n 3 rΠ ¼

ðuniformly heated=cooled wallÞ

π cosðπXÞ Re 3 Pr

n 3 rΠ ¼


1 Re 3 Pr

Table 1. Comparison of the Average Nusselt Number for the Bottom Surface with the Benchmark Solutions of Moallemi and Jang51 for Mixed Convection in a Lid-Driven Square Cavity for Various Grashof (Gr) and Prandtl (Pr) Numbers at Re = 500a

ð27Þ

Nu

ðsinusoidal heating, bottom surfaceÞ ð28Þ ðlinearly heated right wallÞ

ð29Þ

and

Pr

Gr

current work

Moallemi and Jang51

0.01

104

1.0431

1.0167

0.01 0.1

105 104

1.0721 2.3815

1.0368 2.2382

0.1

105

2.8704

2.6290

1

104

5.5695

5.6089

1

105

6.3313

6.2118

Current study employed 28  28 biquadratic elements (57  57 grid points).

a

n 3 rΠ ¼

1 Re 3 Pr

ðlinearly heated left wallÞ

ð30Þ

The top insulated surface can be represented by Dirichlet boundary condition as obtained from eq 24, which is simplified into ∂Π/∂X = 0 for an adiabatic surface. A reference value of Π is assumed as 0 at X = 0, Y = 1, and hence, Π = 0 is valid for Y = 1, "X. It can be noted that the unique solution of eq 25 is strongly dependent on the nonhomogeneous Dirichlet conditions. The following nonhomogeneous Dirichlet boundary conditions are employed to obtain the solution for eq 25 1 Nul Re 3 Pr 1 Πð1, 0Þ ¼
Nur Re 3 Pr Πð0, 0Þ ¼

ð31Þ

3. RESULTS AND DISCUSSION 3.1. Numerical Tests. The computational domain consists of 28  28 biquadratic elements, corresponding to 57  57 grid points. The biquadratic elements with fewer nodes smoothly capture the nonlinear variations of the field variables, in contrast to finite-difference or finite-volume solutions. The present finiteelement-based approach offers special advantages in the evaluation of the local Nusselt numbers at the left, right, and bottom surfaces as the element basis functions used here to evaluate the heat flux. It can be noted that the corner points of the square cavity have temperature and velocity singularities. Both corner points of the bottom surface are intersections of hot and cold isothermal surfaces for case 1, and a similar situation occurs for the right corner of the bottom surface for case 4. The jump discontinuities in Dirichlet-type or isothermal wall boundary conditions at the corner points correspond to computational singularities. In particular, the singularities at the corner nodes of the bottom surface require special attention. The grid size-dependent effects of the temperature discontinuity at the corner points on the local (and overall) Nusselt numbers tend to increase as the mesh spacing at the corner is reduced. One way to handle this problem is to assume the average temperature of the two walls at the corner and keep the adjacent grid nodes at the respective wall temperatures, as suggested by Ganzarolli and Milanez.49 This procedure is still grid-dependent unless a sufficiently refined mesh is implemented. Once any corner formed by the

intersection of two differently heated boundary walls is assumed as the average temperature of the adjacent walls, the optimal grid size obtained for each configuration corresponds to the mesh spacing over which further grid refinements lead to grid-invariant results in both heat-transfer rates and flow fields. Similar observations were also reported by Corcione.50 It can also be noted that there are velocity or stress singularities at the corners of the top surface as the uniformly moving top surface intersects with two stationary side walls. Fluid particles are assumed to be stagnant at the top corner points, as was also assumed by earlier investigators.51 Note that the velocity Dirichlet boundary conditions are defined such that the specification of stresses cannot be required at the boundary walls. In the current investigation, the Gaussian quadrature-based finite-element method provides smooth solutions in the interior domain, including the corner regions, as the evaluation of the residuals depends on the interior Gauss points and, therefore, the effects of the corner nodes are less profound in the final solution. Further, the overall heat balance was verified for all test cases with validation studies, and it was found that the average Nusselt number of a hot isothermal wall is equal to the sum of the average Nusselt numbers of the cold wall. To assess the accuracy of the present numerical approach, we tested our algorithm based on the grid size (57  57) for a square enclosure with a hot bottom surface and a cold top surface in the presence of insulated side walls with uniformly moving top surface similar to the earlier work.51 The streamlines and isotherms are in reasonable agreement, and these comparisons are not shown for brevity. Table 1 reports detailed comparisons of average Nusselt numbers from the present work and earlier results51 for various Pr and Gr values at Re = 500. The average Nusselt numbers based on present computations are in good agreement with the earlier results51 for the entire parameter range. To validate heatfunction contours, we carried out simulations for all the cases with a range of Rayleigh numbers (Ra = 0, 10, 100, 103) at Re = 0, which corresponds to natural convection. Because of the unavailibility of previous works on heatfunctions for mixed-convection problems, the heatfunctions or heatlines of the present work were validated with natural convection problems as reported in an earlier work.28 Comparisons were found to be in good agreement, and the detailed comparisons were already shown in an earlier work.52 The solution is strongly dependent on inhomogeneous Dirichlet boundary conditions, 7612

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research and the sign of heatfunction is governed by the sign of the inhomogeneous Dirichlet conditions. In the current situation, a negative sign of heatlines represents a clockwise flow of heat, whereas a positive sign refers to an anticlockwise flow. A detailed discussion on heat transport based on heatlines for various cases is presented in later sections. In this study, numerical solutions of flow and temperature fields were obtained for visualization of heatlines with various values of Pr (Pr = 0.015
7.2), Re (Re = 1
100), and Gr (Gr = 103
105). The fluid is virtually stagnant, and the heat transfer is conduction-dominant for low values of the governing parameters (Re, Pr, and Gr). Under these conditions, heatlines essentially represent heat flux lines, which are commonly used for conductive heat transport.53 A few interesting features of heatlines are noted as follows: The heatlines are observed to emanate from a hot surface and end on a cold surface and are perpendicular to the isotherms, similarly to heat flux lines, for conductiondominant heat transfer. In cases 3 and 4 for a linearly heated wall, some heatlines are found to emanate from the hot portion of the wall and end on a relatively cold portion of the same wall at high Grashof numbers. As the heatlines approach the adiabatic wall, they slowly bend and become parallel to the surface. A detailed discussion on heat transport based on heatlines for various cases is presented in the following sections. 3.2. Case 1: Isothermally Hot Bottom Surface and Cooled Side Walls. Figure 2 shows the streamlines, isotherms, and heatlines for Re = 1, Pr = 0.015, and Gr = 103
105 with uniform heating of the bottom surface in the presence of isothermally cold side walls. The dimensionless parameter Ri = Gr/Re2 can be introduced to explain natural convection versus mixed convection. It is observed from the plots that the fluid circulation is strongly dependent on Gr. Because the side walls are cold, hot fluid at the bottom surface tends to rise up and flow down along the cold vertical walls. Here, the effect of the moving top surface is seen from the two asymmetric rolls for Gr = 103 or Ri = 103

ARTICLE

(see Figure 2a). A small amount of fluid is pulled toward the left corner because of inertia induced by the moving wall. The small magnitude of the streamfunction signifies the dominant conductive heat transfer, which is also represented by the almost symmetric isotherms. As Gr increases to 104 (Ri = 104), the effect of the moving surface tends to disappear, and the circulation becomes symmetric (see Figure 2b). Note that a larger value of Ri denotes natural convection-dominant flow. Here, the isotherms are also found to be smooth symmetric curves. At high Gr (Gr = 105), the strength of buoyancy is found to be enhanced, and the circulation becomes symmetric. The natural convection is found to be dominant at Ri = 105, and the boundary layer thickness along the side walls is found to be lower. The heat flow distribution inside the cavity is illustrated by heatlines. Common to all Gr values at Pr = 0.015 and Re = 1, the heatlines appear to be symmetric, and they are also perpendicular to the isotherms, which signifies conduction-dominant heat transfer (Figure 2a
c). The magnitudes of the heatlines are very high at the bottom corners, and the heat flux is high at these corners, as the cold wall is in direct contact with the hot surface. Denser heatlines also occur near the bottom corners, and |Π| varies in the range of 24
45 near the bottom corners. On the other hand, the top portion of the side walls receives mainly heat from the center of the bottom surface, and the disperse or less dense heatlines at the top portion of side walls signify less heat absorption. Thus, the boundary layer thickness is larger at the top portion. Figure 3 shows the streamlines, isotherms, and heatlines for Re = 1 and Pr = 7.2 for various Grashof numbers. At high Pr values, the isotherms tend to deform because of the presence of convective heat transfer at Gr = 103, and the isotherms are nonsymmetric because of the dominant effect of the lid velocity. It is interesting to observe that the dominant effect of the motion of the lid is still observed at low Gr (103). As Gr increases, the isotherms with θ e 0.4 are gradually compressed toward the side

Figure 2. Streamfunction (ψ), temperature (θ), and heatfunction (Π) contours for case 1 with Re = 1; Pr = 0.015; and Gr = (a) 103, (b) 104, and (c) 105. 7613

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

Figure 3. Streamfunction (ψ), temperature(θ), and heatfunction (Π) contours for case 1 with Re = 1; Pr = 7.2; and Gr = (a) 103, (b) 104, and (c) 105.

walls, illustrating the dominant convection effect unlike cases for low Pr values with identical Gr values (see Figures 2b and 3b). Although the nonsymmetric circulation cells occur at Gr = 103, the streamline circulation cells become symmetric because of the dominance of natural convection at high Gr values (Gr = 105 or Ri = 105). The larger intensity of flow enhances thermal mixing, which results in a uniform temperature distribution over a larger portion in the central core. This regime corresponds to θ varying in the range of 0.4
0.5 (see Figure 3c). The larger intensity of flow also causes a smaller thickness of the boundary layer at the top portion of side walls (Figure 3c). The heat distribution has some interesting features at large Pr values (Pr = 7.2) for various Gr values. The magnitude of the heatfunction (|Π|) varies in the range of 0
0.14 along the side walls except near the corners for low Gr (Gr = 103). The heatlines starting from the right portion of the bottom surface travels a longer path to reach the top portion of the right wall. Thus, the right portion receives more heat, and the boundary layer thickness is small near the right wall. It is also observed that some heatlines directly start from the hot bottom surface and end at the cold side wall near a small region of the bottom corners because of the conduction-dominant heat transfer. It is also interesting to note that the clockwise circulation for the heatlines is stronger, implying that the right half receives more heat than the left portion of the cavity for Gr = 103 (Figure 3a). At Gr = 104 (see Figure 3b), the effect of the lid velocity is less important. However, a large number of denser heatlines that emanate from the bottom surface end up in the top portion of the vertical walls. The vertical wall regions with 0.45 e Y e 1 correspond to |Π| = 0
0.3, illustrating the significant heat flow in this regime. It is interesting to observe that the top portion of the side walls receives larger heat than the bottom portion of the walls, as the bottom portion of the side walls consists of highly dispersed heatlines. Therefore, the larger thickness of the boundary layer occurs near the bottom portion of the side walls, and dense

heatlines in the top portion result in a compressed boundary layer. A compressed boundary layer thickness is observed within 5% of the side walls for Gr = 105 (Figure 3c). On the other hand, the boundary layer thickness is larger at the top portion of the side walls for smaller Gr values (Gr = 103). It is observed that the effect of the lid velocity disappears and the heatlines are symmetric for high Gr (105), with two symmetric rolls corresponding to |Π|max = 2.27 (see Figure 3c). It can be noted that dense heatlines occurring in the central regime of the cavity signify enhance thermal mixing. Thus, a large regime at the central region corresponds to θ = 0.4
0.5. It is observed from the streamlines that the effect of buoyancy becomes weaker as compared to that of the lid-driven force, and a primary circulation cell occupies 75% of the cavity because of the enhanced effect of inertia induced by the moving lid, at low Gr (Gr = 103) for Pr = 0.7 and Re = 10 (figure not shown). The heatlines are perpendicular to the side walls, but they are not symmetric at low Gr (Gr = 103). They are more inclined toward the left wall, and the thermal boundary layer formed is more compressed toward the left wall compared to the right wall. However, the distributions of streamlines, isotherms, and heatlines for higher Gr values (Gr = 104 and 105) are qualitatively similar to those for Pr = 7.2 (Figure 4), and the detailed discussions follow as for Figure 4. Figure 4 shows the streamlines, isotherms, and heatlines for Re = 10 and Pr = 7.2 for various Grashof numbers or Ri varying in the range Ri = 10
103. Lower Ri values imply dominant forced convection, which can be observed from the single circulation in the streamlines that span the entire cavity for Gr e 103 (see Figure 4a). Only a small secondary circulation can be seen near the bottom left corner of the cavity. The isotherms are not symmetric, and an isothermal zone exists in the right half of the cavity because of the weak clockwise primary circulation. The cold fluid tends to flow down along the right wall. A large regime near the right wall is maintained at θ = 0.1 because of decreased 7614

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

Figure 4. Streamfunction (ψ), temperature(θ), and heatfunction (Π) contours for case 1 with Re = 10; Pr = 7.2; and Gr = (a) 103, (b) 104, and (c) 105.

thermal mixing, as seen in Figure 4a, which will be explained later based on heatlines. The temperature near the core of the cavity varies from 0.1 to 0.2. Enhanced convective effects are observed at Ri = 102, and enhanced secondary circulation to the bottom left portion of the cavity is observed (see Figure 4b). Also, an isothermal zone corresponding to θ = 0.4
0.5 is observed at the core of the cavity. The dominant effect of the moving lid can still be found for higher Gr values (Figure 4c). The increase in Gr enhances the effect of the natural convection, which can be observed from the streamlines, as the span of the secondary circulation has increased. It is interesting to observe that the flow circulation cells are almost of same size. Similarly to previous cases, a large central regime is maintained at θ = 0.4
0.5. The temperature profile and heat distribution are further illustrated based on heatlines. It is interesting to note that heatlines that emanate from 0 e X e 0.8 of the bottom surface end at the left wall only for Gr = 103 (see Figure 4a). Thus, the left wall receives more heat than the right wall, leading to a higher temperature gradient near the left wall. Therefore, the thermal boundary layer formed along the left wall is thinner than that formed along the right wall, as seen from heatlines, and the heatfunction varies in the range |Π| = 0.001
0.025 along the left wall. The right wall with 0.7 e Y e 1 corresponds to Π = 0
0.001, illustrating the lower flow in this regime, and the rest of the region does not receive significant heat, as seen from the heatlines. Thus, the right wall is maintained around θ e 0.1 (Figure 4a). It can be noted that heatlines that emanate from 0 e X e 0.58 of the bottom surface end toward the left wall for Gr = 104 (Figure 4b). Similarly to Figure 4a, the boundary layer thickness on the left wall is lower, and the top part receives more heat, with Π varying in the range of 0
0.025. On the other hand, a small zone (0.5 e X e 0.6) of the bottom surface transfers heat to the top portion of the right wall. A relatively cooler region is observed at the bottom portion of the right wall, as there are dispersed heatlines in that zone. Because of the convective heat

flow circulation near the top portion, θ varies in the range of 0.4
0.5 (Figure 4b). The dominance of convective heat transport is clearly illustrated by the heatlines for higher Gr values, as the heatlines are similar to the streamlines (Figure 4c). Although the effect of the moving lid is observed from the two asymmetric rolls, the two walls receive the same amounts of heat. It is observed that |Π|max for the heatline cells changes from 0.02 to 0.2 as Gr increases from 103 to 105, signifying enhanced thermal mixing. The dense heatlines near the central regime of the bottom surface also denote enhanced heat transfer from the bottom surface. Thus, the isotherms are more compressed toward the bottom surface and side walls, signifying a large isothermal region in the cavity where θ varies in the range of 0.4
0.5 for Gr = 105 (see Figure 4c). Figure 5 shows the streamlines, isotherms, and heatlines for Re = 100, Pr = 7.2, and Gr = 103
105 or Ri = 0.1,1,10. Lower Ri values correspond to larger lid-driven effects, and thus, the dominant effect of the moving lid is clearly seen from the streamlines irrespective of Gr value for Re = 100, Pr = 7.2, and Gr = 103
105 or Ri = 0.1,1,10. Nonsymmetric distorted isotherms are observed because of the dominant lid-driven force, the isotherms are compressed toward the bottom surface and left walls, and it is observed that a large region near the right half becomes isothermally cooled. At a larger Gr value (Gr = 105), the isotherms are highly compressed near the left and bottom surfaces. It is interesting to observe that forced convection is dominant enough to maintain θ = 0.1
0.3 in nearly 75% of the cavity (see Figure 5c). A strong thermal boundary layer develops near the bottom surface and left wall at higher Gr values, which is illustrated based on heatlines next. At low Gr (Gr = 103) (Figure 5a), most of the heatlines from the bottom surface end up on the left wall, as in Figure 4a. Thus, a higher thermal gradient exists near the bottom portion of the left wall compared to the right wall, and the thermal boundary layer is more compressed toward the left wall. It is interesting to observe 7615

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

Figure 5. Streamfunction (ψ), temperature(θ), and heatfunction (Π) contours for case 1 with Re = 100; Pr = 7.2; and Gr = (a) 103, (b) 104, and (c) 105.

that |Π| varies in the range of 0
0.001 for 0.6 e Y e 1 on the left wall, whereas |Π| is observed to vary in the range of 0.001
0.008 along the bottom portion of the left wall. Thus, a higher gradient of |Π| signifying a smaller thickness of the thermal boundary layer occurs near the bottom portion of the left wall (Figure 5a). Significant variation in |Π| is not observed along the right wall as |Π| varies in the range of 0.002
0.0025 along a large region on the right wall. Thus, a lower thermal gradient exists near the right wall, and the thickness of the thermal boundary layer is high. The strong primary circulation cell draws heat from the heat flow path that connects the bottom surface and left wall. The circulation cell is quite large, and the heat drawn from the path is also circulated through the left cold wall. It is also observed that |Π|max = 0.01 in the core of the circulation cell for Gr = 104. Therefore, a large portion near the right wall is maintained at θ = 0.1 (Figure 5b). As Gr increases to 105 (Figure 5c), because of the enhanced heatline circulation cells, the right wall also receives a significant amount of heat. It is observed that the heatlines that emanate from 0 e X e 0.6 on the bottom surface end on the left wall. Thus, higher thermal gradients are observed near the left wall corresponding to 0.001 e |Π| e 0.01, and the thermal boundary layer is more compressed toward the left wall. It is also interesting to observe that |Π| varies in the range from 0 to 0.004 near the right wall. Thus, higher thermal gradients, which signify smaller thicknesses of the thermal boundary layer occur near the right wall, whereas the thickness of the thermal boundary layer near the right wall was found to be large for lower Gr values. The intensity of the heatline circulation cell is higher at Gr = 105, with |Π|max = 0.03, whereas |Π|max is around 0.008
0.012 for Gr = 103 and 104. Overall, an isothermal zone is observed at the core of the cavity, with θ varying in the range of 0.1
0.3 for Gr = 105 (see Figure 5c). 3.3. Case 2: Nonisothermal Hot Bottom Surface and Isothermally Cooled Side Walls. The streamlines, isotherms,

and heatlines in the case of nonuniform heating of the bottom surface with a sinusoidal variation in the temperature were also studied. In the case of uniform heating of the bottom surface, a finite discontinuity in the Dirichlet boundary conditions for the temperature distribution occurs at the edges of the bottom surface. The mathematical singularity at the edges of the bottom surface is removed by nonuniform heating, and a smooth temperature distribution is provided in the entire cavity. The distributions of streamlines, isotherms, and heatlines are qualitatively similar to those of case 1 under identical parameters, and the maximum value of the streamfunction is found to be almost same for all Grashof numbers. Thus, illustrative figures are not shown for brevity, and similarly to case 1, qualitative explanation can be drawn for identical parameters (Gr, Pr, Re). 3.4. Case 3: Linearly Heated Side Walls with Isothermal Hot Bottom Surface. Figures 6
8 display the streamlines, isotherms, and heatlines for various Pr and Re values at Gr = 103
105 when the side walls are linearly heated whereas the bottom surface is isothermally hot. Figure 6a shows that the effect of the lid-driven force is significant for Gr = 103 (Ri = 103), as the fluid layers rising from the core are being pulled toward the top left corner of the cavity. The primary circulation cell is stronger than the anticlockwise secondary cell, which occurs near the left corner because of the buoyancy force. Overall, the strength of the flow is weak, as |ψ|max = 0.1. Hence, the temperature contours for θ e 0.3 occur symmetrically near the corners of the top surface, and the other temperature contours are smooth parallel curves that span the entire cavity and are symmetric with respect to a vertical symmetric line representing conduction-dominant heat transfer. At Gr = 104 (Ri = 104), natural convection becomes dominant, as seen from the almost symmetric circulation cells (Figure 6b). The isotherms for θ e 0.4 occur symmetrically near the corners of the top surface. The rest of the isotherms are not smooth parallel curves, and they are pulled along the top surface, signifying convective heat transfer. As Gr increases to 105, two 7616

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

Figure 6. Streamfunction (ψ), temperature(θ), and heatfunction (Π) contours for case 3 with Re = 1; Pr = 0.7; and Gr = (a) 103, (b) 104, and (c) 105.

Figure 7. Streamfunction (ψ), temperature(θ), and heatfunction (Π) contours for case 3 with Re = 1; Pr = 7.2; and Gr = (a) 103, (b) 104, and (c) 105.

secondary circulations are formed inside the cavity (Figure 6c). Although the fluid in contact with the upper part of the left wall is cold, a significant amount of cold fluid is dragged by the motion of the upper lid. The primary circulation cell with |ψ|max = 10 is primarily due to the lid velocity. Because of the strong circulation cells occupying the top portion of the side walls, the isotherms are compressed along these portions. The isotherms were also found to be compressed along the right portion of the bottom

surface. Because of the intense circulation at the top portion of the cavity, θ varies in the range of 0.6
0.7. Dominant conductive heat transport is observed based on heatlines that are perpendicular to the hot bottom surface for low Grashof numbers (Gr = 103). Heatlines are also found to be parallel, vertical with respect to the bottom surface, and the gradient of the heatfunctions is uniform. Thus, isotherms are parallel to the bottom surface except near the top corners where 7617

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

Figure 8. Streamfunction (ψ), temperature(θ), and heatfunction (Π) contours for case 3 with Re = 100, Pr = 7.2 and Gr = (a) 103, (b) 104, and (c) 105.

the heatlines are found to be converging. Therefore, isotherms are dense near the top corner points (Figure 6a). As Gr increases to 104, because of the stronger flow circulation, strong convective heatline cells are also observed along the center of both the left and right halves. Thus, the isotherms are pulled toward the top surface. It is observed that the heatlines are perpendicular to the bottom surface and also isotherms with θ g 0.9 are parallel to the bottom surface. The heatline path also indicates that a significant amount of heat is being transported to the top portion of the side walls. Thus, the temperature gradient is high in a small regime of the top portion (Figure 6b). At Gr = 105, two heatline circulation cells similar to streamline cells are observed because of the dominant convection heat transfer as seen from the streamfunctions. The dense heatlines along the junction of the two heatline cells result in enhanced thermal mixing, and θ varies in the range of 0.6
0.7 in the top portion of the central region of the cavity. The heatlines are dispersed along the bottom portion of both the left and right walls. Thus, the fluid is warm near the bottom surface, and θ varies in the range of 0.7
0.9. On the other hand, the heatlines are dense along the top portion of the cavity. Hence, the isotherms are dense near the top portion of the side walls in Figure 6c. Figure 7 shows the streamlines, isotherms, and heatlines for Re = 1, Pr = 7.2, and Gr = 103
105 in the presence of linearly heated side walls. It can be noted that an increase in the Prandtl number enhances convective heat transfer. The dominant liddriven effect is illustrated by the strong primary circulation in the streamlines with |ψ|max = 0.32 inside the cavity. The isotherms with θ g 0.5 are slightly distorted and pulled toward the top surface along the central vertical line with an inclination toward the right wall. Because of the strong clockwise circulation, isotherms with θ e 0.4 are more compressed toward the top right corner of the cavity. Natural convection dominates over the lid-driven force, as observed from the almost symmetric streamline cells (see Figure 7b). The primary circulation cell gets

deformed toward the middle portion of the cavity, and the strength of secondary circulation near the top left corner increases. The isotherms with θ e 0.6 occur symmetrically near the corners of the side walls. It is interesting to observe multiple circulation cells within the cavity at Gr = 105 (Figure 7c). The secondary circulation cells push the primary circulations toward the upper portion of the cavity because of the enhanced convection from the hot lower half of the cavity. A pair of symmetric circulations with hot and cold regimes appear distinctly within the cavity, because of the positive and negative temperature gradients with respect to the center along the vertical walls. Because of the enhanced convection, the circulation near the bottom corners of the cavity increases, and the isotherms with θ g 0.5 occupy more than 70% of the cavity. The isotherms are distorted because of the multiple circulation cells, and a large isothermal zone with θ varying in the range of 0.5
0.6 is observed at the top portion of the cavity. The dominant convective heat transfer is observed based on the single primary heatline circulation cell that spans more than 50% of the cavity with |Π|max = 0.18. The heatline path illustrates that, because of the heatline circulation, a significant amount of heat is transported to the top portion of the side walls, where |Π| varies in the range of 0.01
0.05 along the right wall and in the range of 0.01
0.095 along the left wall. Thus, isotherms with θ e 0.4 are more compressed toward the top right corner of the cavity (see Figure 7a). At Gr = 104, symmetric heatline circulation cells similar to the streamlines are observed because of the enhanced buoyancy forces. The dense heatlines along the junction of the heatline cells result in enhanced thermal mixing, and θ varies in the range of 0.6
0.7 at the top portion of the cavity. Dense heatlines that emanate from the bottom surface end toward the top portion of the side walls, where |Π| varies in the range of 0.01
0.1. It is also observed that the strength of primary circulation cells is greater, spanning the top potion of the cavity. Thus, higher thermal gradients are observed at the top 7618

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research corners of the cavity, and the thermal boundary layer is compressed toward the top portion (see Figure 7b). It is interesting to observe multiple heatline circulation cells similar to streamline cells at higher Gr values. Dense heatlines are observed along the junction of the heatline cells, and isothermal zones are also observed in both the right and left halves of the bottom portion of the cavity because of the heatline circulation with |Π|max = 1.5. Hence, θ varies in the range of 0.7
0.8. Intense primary heatline circulation is also observed in the top portion of the cavity, where θ varies in the range of 0.5
0.6 (see Figure 7c). The dominant effect of the lid-driven force is observed as only a single streamline circulation in the cavity for Gr = 103 and Re = 10, similar to that for Re = 1 with Pr = 7.2. It is observed that heatlines that start from more than 80% of the bottom surface end toward the left wall, signifying larger heat transport toward the left wall. At higher Gr values (Gr = 104), isotherms tend to be distorted because of the secondary circulation in heatlines at the bottom portion of the cavity (figure not shown). Multiple circulation cells in streamlines and heatlines are observed because of the larger natural convection effect at Gr = 105. The trends in streamlines, isotherms, and heatlines are qualitatively similar to those in Figure 7c, and a similar explanation follows. Figure 8 shows the streamlines, isotherms, and heatlines for Re = 100, Pr = 7.2, and Gr = 103
105 (Ri = 0.1
10) for linearly heated side walls. The dominant lid-driven force is observed for Gr = 103, as a single primary circulation spans the entire cavity (see Figure 8a). It is observed that isotherms with θ e 0.3 are more compressed toward the side walls because of the lid-driven effect. The rest of the isotherms are also distorted and are compressed toward the bottom surface and the side walls (Figure 8a). As Gr increases to 104, secondary circulation cells are observed in the bottom left corner of the cavity, because of the enhanced natural convection at larger Pr values. Because of the high lid velocities, hot fluid rising from the bottom is dragged to the top, and the fluid layers slide along with the moving wall.

ARTICLE

Thus, isotherms are strongly compressed at the top portion of the left wall, as well as the bottom portion of the cavity, and are parallel to the bottom surface (Figure 8b). Isotherms are found to be compressed along the side walls as well as along the bottom surface of the cavity. Common to both Gr = 103 and 104, a large regime of the top portion is maintained at θ e 0.3 (isothermal zone). The span of the secondary circulation is increased because of the enhanced buoyancy effects at Gr = 105. It is interesting to observe the dominant lid-driven force in the top half of the cavity and the dominant buoyancy forces in the bottom half of the cavity. At larger Gr values (Gr = 105), an isothermal zone with θ = 0.3 is observed in the top portion, whereas another isothermal zone with θ = 0.5 is also observed in the bottom portion of the cavity (Figure 8c). Heatline patterns are found to be similar to streamlines, which represent convection-dominant heat transfer in the cavity. A single heatline circulation cell with |Π|max = 0.026 is observed at low Gr, and more than 50% of heatlines that emanate from the bottom surface end toward the top portion of the left wall. Thus, high thermal gradients exists near the top portion of the left wall. A large region near the top portion has the heatline circulation cells signifying enhanced thermal mixing and θ = 0.3 is maintained in that regime (Figure 8a). At Gr = 104, two heatline circulation cells similar to streamline cells are observed. The secondary heatline circulation cell represents convective heat transfer due to secondary flow circulation. A largely compressed isotherm near the bottom left corner is due to dense heatlines. Similarly to the Gr = 103 case, large heatline circulation cells promote thermal mixing, resulting in a large isothermal zone with θ = 0.3 (Figure 8b). Two heatline circulation cells similar to flow circulations are observed at Gr = 105. The heatline circulation cells with uniform intensities illustrate the isothermal zone near the top and bottom surfaces. It is interesting to observe the larger heatline path covering the circulation cells, and this illustrates larger heat distribution to the top portion of the cavity (Figure 8c).

Figure 9. Streamfunction (ψ), temperature(θ), and heatfunction (Π) for case 4 with Re = 1, Pr = 0.015 and Gr = (a) 103, (b) 104, and (c) 105. 7619

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research 3.5. Case 4: Linearly Heated Left Wall and Isothermally Cooled Right Wall. Figures 9
11 show the streamlines, iso-

therms, and heatlines for various Pr and Re values at Gr = 103
105 with a linearly heated left wall, an isothermally cooled right wall, and a hot bottom surface. Because the left wall is linearly heated and the bottom surface is uniformly heated, hot fluid from the bottom rises to the top along the left wall and flows down along the cooled right wall. Thus, strong primary

ARTICLE

circulation cells result in unidirectional flow for these boundary conditions. The effect of the lid-driven force is dominant for low Grashof numbers (Gr = 103) (see Figure 9a) as the fluid rises along the left wall and flows down the cold right wall, forming a primary circulation cell with |ψ|max = 0.6, spanning the entire cavity. A small zone near the top portion of the left wall corresponds to θ e 0.2, isotherms with θ e 0.2 are also compressed along the right wall, and other isotherms with

Figure 10. Streamfunction (ψ), temperature(θ), and heatfunction (Π) contours for case 4 with Re = 10, Pr = 0.7 and Gr = (a) 103, (b) 104, and (c) 105.

Figure 11. Streamfunction (ψ), temperature (θ), and heatfunction (Π) contours for case 4 with Re = 10, Pr = 7.2 and Gr = (a) 103, (b) 104, and (c) 105. 7620

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research θ g 0.3 are smooth curves spanning the entire cavity. As Gr increases to 104, stronger circulation cells are observed with |ψ|max = 5.2 and similar qualitative distributions of isotherms are observed for Gr = 103 and 104. A weak secondary circulation to the top left corner of the cavity is observed at Gr = 105 (Figure 9c). The dominant lid-driven force is observed at higher Gr values as strong primary circulation cells span more than 98% of the cavity with |ψ|max = 52. Because of the stronger convective heat transport, isotherms with θ e 0.3 are pushed along the side walls. Convection-dominant heat transfer within the cavity is clearly illustrated by the heatlines for Gr = 103 and 104. Although the heatlines appear to be parallel, a smooth monotonic distribution is not observed, and the heatlines are distorted, implying the presence of convection heat transfer. High heat flux exists at the bottom right corner of the cavity, as the hot bottom surface is in direct contact with isothermally cold right wall. This is illustrated by the heatlines with |Π| varying in the range of 35
55 at the bottom right corner of the cavity (Figure 9a,b). Heatlines are also parallel to the left side of the cavity. Thus, the isotherms are parallel to the bottom surface in the left portion of the cavity, whereas they appear to be converging significantly from the right portion of the bottom surface. As Gr increases to 105, the heatlines are much distorted, and a single primary circulation with |Π|max = 32.5 is observed near the bottom right portion of the cavity. This is due to dominant convective flow as |ψ|max = 50 (Figure 9c). It can be observed that the dense heatlines that start from the bottom surface (near X = 0.5) reach the top portion of the side walls, and a distorted path consisting of dense heatlines signifies convection-dominant heat transfer in the specific zone. The isotherms are also compressed along the left wall because of the convective heat flow near the bottom corner, as illustrated by the heatline circulation cell. Figure 10 shows the streamlines, isotherms, and heatlines for Re = 10, Pr = 0.7,and Gr = 103
105. The effect of the lid-driven flow is dominant for low Gr as observed from the streamlines. The isotherms with θ e 0.2 are pushed toward the right wall and a small regime to the upper portion of the left wall (Figure 10a). As Gr increases to 104
105 (see Figure 10b,c), the strength of the streamline circulations is increased, and the isotherms are largely compressed along the right wall. The heat flow distribution inside the cavity is illustrated by heatlines. Dominant convective heat transfer is observed from the distorted heatlines at low Gr (Figure 10a). However, no heatline circulation cells are observed within the cavity. As Gr increases to 104, enhanced thermal mixing is observed based on heatline circulations. The intensity of the heatlines (|Π|) ending toward the top portion of the right wall varies in the range of 0.01
0.1. Consequently, isotherms with θ e 0.4 are more compressed toward the right wall. Similarly, the top portion of the left wall corresponds to a large gradient of heatfunctions that leads to a large gradient of isotherms (θ e 0.6) (see Figure 10b). Enhanced circulation in the heatlines are observed at the core of the cavity for Gr = 105, signifying enhanced thermal mixing. The maximum magnitude of heatlines (|Π|max) is found to be 1.2 at the center of the core. The temperature in the core varies in the range of θ = 0.5
0.6. It is observed that the intensity of the heatlines that end on the top portion of the right wall is much larger than those ending toward the top portion of the left wall. Thus, the isotherms are more compressed toward the top portion of the right wall and the boundary layer thickness is less. The isotherms are also pushed toward the left corner of the bottom

ARTICLE

surface, as dense heatlines are found to occur in that zone for Gr = 105. It is also observed that the primary heatline circulation spans more than 80% of the cavity. Figure 11 shows the streamlines, isotherms, and heatlines for Re = 10, Pr = 7.2, and Gr = 103
105 or Ri = 1
103. A single circulation in the streamlines is observed at Gr = 103, because of the larger lid-driven effects at low Ri (Ri = 1). However, the isotherms are significantydistorted implying convective heat transfer in the cavity (Figure 11a). As Gr increases to 104, the strength of streamlines is found to be increased because of the enhanced buoyancy forces and the isotherms for θ e 0.5 tend to move toward the left wall and isotherms with θ e 0.4 are largely compressed along the right wall (Figure 11b), signifying the dominance of natural convection at larger Ri. Secondary flow circulation in the streamlines is found near the top left corner of the cavity for Gr = 105 (see Figure 11c). The isotherms are highly compressed toward the right portion of the bottom surface and cooled right wall. The temperature of the major part of the cavity lies in the range θ = 0.4
0.5. The heatlines indicate that streamlines and heatline circulations are similar at the core for all Gr. The thermal mixing through heatline circulation is seen near the top portion where θ = 0.3
0.4 is observed for Gr = 103 (see Figure 11a). The heatline circulation cells grow larger at Gr = 104, and thus a large central regime has θ=0.4
0.5 at the core (Figure 11a). Dense heatlines are also seen connecting the bottom surface and the top portion of the left and right wall. Thus, the isotherms are highly compressed along the bottom surface and top portions of side walls. Enhanced circulation cells occur near the bottom and top portions and a large regime has θ = 0.4
0.5 for Gr = 105 (Figure 11c). Similar to lower Gr values, isotherms are highly dense along the bottom surface and side walls at Gr = 105. The distributions in streamlines, isotherms, and heatlines for Re = 100, Pr = 7.2 and Gr = 103
105 are qualitatively similar to those for case 1 under identical parameters, as seen in Figure 5, and hence, the detailed discussions are omitted for brevity. 3.6. Heat-Transfer Rates: Local Nusselt Numbers. The distributions of local Nusselt numbers illustrate the conduction-dominant mode for low governing parameters (Re = 1, Gr = 103, Pr = 0.015 and 0.7) for Gr e 104. The natural convection-dominant mode is observed for Re = 10 for higher Pr and Gr values, and thus, a symmetric distribution is observed in Nub. Also, the distributions of Nul and Nur are qualitatively similar to those for Re = 10. Therefore, we discuss here the test cases for Re = 100 (Ri = 0.1
10) for varying Pr values at Gr = 103 and 105. Mixed convective effects are clearly observed at Re = 100 and Gr = 105. Also, negative heat-transfer rates along the side walls are observed for Pr = 0.7 at Re = 100 in cases 3 and 4. 3.6.1. Case 1: Isothermal Hot Bottom Surface and Isothermally Cooled Side Walls. The upper plots in Figure 12a
c show the local Nusselt number as a function of distance along the bottom surface and side walls for Pr = 0.015
7.2 with Gr = 103 and 105 at Re = 100 for case 1. The solid and dotted lines represent heat-transfer rates for Gr = 103 and Gr = 105, respectively. The upper plot in Figure 12a shows that the heattransfer rate (Nub) is very high at the edges and is gradually reduced toward the center of the bottom surface. This is due to the large heat flux at the hot
cold junction, which is illustrated by the higher magnitude of the heatfunction near the bottom corners of the cavity. The local Nusselt number distribution at low Pr values (Pr = 0.015) is nearly uniform irrespective of Gr along a large portion of the bottom surface, except at the 7621

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

Figure 12. Variation of the local Nusselt number with distance at the (a) bottom surface, (b) right wall, and (c) left wall with Re = 100 and Gr = 103 (
) and 105 ( 3 3 3 ). In each plot, the upper panel corresponds to case 1, and the lower panel corresponds to case 2.

hot
cold junctions. This trend arises because conduction-dominant heat transfer is observed from the smooth and parallel heatlines along the bottom surface for Pr = 0.015 irrespective of the Gr value (figure not shown). The local Nusselt number distribution for Pr = 0.7 at Gr = 103 is similar to the distribution for Pr = 0.015 except in the range of 0.4 e X e 0.9 along the bottom surface, where higher heat-transfer rates (Nub) are observed because of the dense heatlines corresponding to 0.001 e |Π| e 0.012 in this zone (figure not shown) for Pr = 0.7. A similar variation of Nub is observed for Pr = 7.2 and Gr = 103, but larger values of Nub are observed along the bottom surface at this Pr value compared to lower Pr values because of enhanced convective transport. It is observed that, at low Gr (Gr = 103), |Π| varies in the range of 0.0001
0.004 along 0.2 e X e 0.8 of the bottom surface whereas |Π| varies in the range of 0.004
0.0045 along 0.1 e X e 0.2 of the bottom surface for Pr = 7.2 (Figure 5a). Therefore, the heat-transfer rate along the bottom surface first reaches a minimum at X = 0.1
0.2, and that gradually increases to X = 1. Based on the heatfunctions, Nub is symmetric with respect to the center line for Pr = 0.015, whereas it is not symmetric for Pr = 0.7 and 7.2 for Gr = 103. It is interesting to observe that the variation of Nub at Gr = 105 is similar to that at lower Gr values because of the conductiondominant heat transfer for Pr = 0.015. It can also be noted that,

for Pr = 0.7, the heat-transfer rate (Nub) increases with X along 0.2 e X e 0.9 of the bottom surface. Similarly to the previous case with Gr = 103, higher heat-transfer rates are observed along the corners, and thus, Nub reaches a sudden maximum near the corners. Higher values of Nub are found for Pr = 7.2 because of the enhanced heat transfer. As for Pr = 0.7, Nub first reaches a minimum at X = 0.1 that increases along the range X = 0.1
0.9 of the bottom surface and that suddenly reaches a maximum at X = 1. Dense heatlines corresponding to |Π| = 0
0.005 are observed along 0.5 e X e 0.9 of the bottom surface, signifying higher heat-transfer rates in this zone, and |Π| is observed to vary in the range of 0.005
0.007 along 0.2 e X e 0.5 of the bottom surface (see Figure 5c). Thus, Nub increases along the length of the bottom surface for Pr = 7.2. Also, larger gradients in heatfunctions with 0.007 e |Π| e 0.015 are observed at the bottom left corner, signifying larger values in Nub. The upper plot of Figure 12b shows the local heat-transfer rates along the right wall for Pr = 0.015
7.2 with Gr = 103 and 105 at Re = 100 for case 1. Because of the presence of the moving adiabatic wall, the local Nusselt numbers are not similar along the two walls, even though symmetry in the thermal boundary conditions exists. However, the local Nusselt number is infinitely large at the point of intersection of the hot and cold surfaces, and consequently, the variation of heatfunction is also large in these 7622

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research intersection points for all Pr values. Thereafter, the Nusselt number decreases sharply from the bottom edge along both side walls. Note that the Nusselt number (Nur) decreases continuously because of the less intense heatlines, with 0 e |Π| e 0.005 occurring in the upper half zone of the side walls for Pr = 0.7 at Gr = 103 (figure not shown). A similar qualitative trend is observed for Pr = 0.015 irrespective of the Gr value because of the conduction-dominant regime. Lower heat-transfer rates (Nur) along 0.1 e X e 0.5 of the right wall are observed for Pr = 7.2, and Nur reaches a secondary maximum at X = 1 of the right wall. This trend is observed because of the dense heatlines at the top portion of the right wall (see Figure 5a). It can be noted that less intense heatlines with 0.0012 e |Π| e 0.0018 are observed along 0.1 e X e 0.5 of the right wall, signifying lower heat-transfer rates (Nur) in this zone. On the other hand, a maximum in Nur is observed at X = 1 because of the dense heatlines for 0 e |Π| e 0.001 at the top portion of the right wall. Higher heat-transfer rates along the right wall (Nur) because of the larger heating effects are observed at Gr = 105 for Pr = 0.7 and 7.2, but the variation of the heat-transfer rates with distance is qualitatively similar to that for lower Gr values. It is interesting to observe that, for Pr = 0.7 and Gr = 105, a secondary local maximum occurs in Nur corresponding to |Π| = 0
0.01 along 0.8 e Y e 1 of the right wall, whereas |Π| varies in the range of 0.015
0.025 along 0.1 e Y e 0.9 of the right wall. Thus, Nur reaches a sudden maximum at Y = 1 because of the higher heattransfer rates near Y = 1. A similar qualitative trend occurs with Pr = 7.2, but higher heat-transfer rates are observed because of the enhanced convective effects based on the heatlines at Gr = 105. The upper plot of Figure 12c shows the local heat-transfer rates along the left wall for Pr = 0.015
7.2 with Gr = 103 and 105 at Re = 100 for case 1. The variation in Nul for Pr = 0.015 is similar to that along the right wall (Nur) because of the conduction dominance and symmetric heatline distribution for Gr = 103 (figure not shown). Higher heat-transfer rates (Nul) along 0.1 e X e 0.4 of the left wall are observed compared to the right wall (Nur) for Pr = 0.7 and Gr = 103, corresponding to heatlines with 0.001 e |Π| e 0.004 in this zone along the left wall (figure not shown). Further, Nul gradually decreases and reaches a minimum at Y = 1 as less intense heatlines with |Π| =0
0.001 occur in this zone. A similar qualitative trend occurs with Pr = 7.2 and Gr = 103, but larger Nul values are observed because of the larger convective effects based on dense heatlines (|Π| = 0
0.005) for high-Pr fluids. A sudden minimum in Nul is observed at Y = 1 because of the less dense heatlines with |Π| e 0.0001 near the adiabatic wall (see Figure 5a). It is observed that, for Pr = 0.7 and Gr = 105, Nul suddenly decreases at Y = 0.1, whereas it reaches a local maximum (Nul = 5.37) at X = 0.2
0.3. Further, Nul gradually decreases along the length until Y = 1. The minimum at X = 1 occurs because of the less intense heatlines corresponding to 0 e |Π| e 0.005 along 0.7 e Y e 1 of the left wall (figure not shown). It is interesting to observe that, for Pr = 7.2 and Gr = 105, Nul decreases along the length and suddenly reaches a maximum at Y = 0.1
0.15, thereafter reaching a minimum at Y = 0.7. Further, Nul reaches a secondary maximum at Y = 0.9 and suddenly reaches a minimum at Y = 1. Dense heatlines corresponding to 0.005 e |Π| e 0.007 are observed at Y = 0.1
0.15 on the left wall, corresponding to a maximum in Nul (16.6). Lower gradients in the heatfunctions corresponding to 0.001 e |Π| e 0.002 are observed along 0.5 e Y e 0.75 of the left wall. Thus, a minimum in Nul is observed in

ARTICLE

this zone. A secondary maximum is observed near Y = 1 corresponding to dense heatlines with 0.0001 e |Π| e 0.001 near Y = 0.98. A sudden minimum at Y = 1, similar to that observed at lower Gr values, occurs because of the less intense heatlines corresponding to |Π| e 0.0001 in this zone. Also, heatlines that emanate from 60% of the bottom surface end toward the left wall, signifying higher heat-transfer rates along the left wall compared to the right wall. 3.6.2. Case 2: Nonisothermal Hot Bottom Surface and Isothermally Cooled Side Walls. The lower plots of Figure 12a
c display the effects on the local Nusselt numbers at the bottom surface and side walls of Gr (103 and 105) and Re = 100 for varying Pr values (0.015
7.2). The nonuniform heating provides a sinusoidal type of local heat-transfer rate that is symmetric with respect to the vertical center line of the bottom surface. It is observed that the heat-transfer rate (Nub) is 0 near the junction of the hot and cold surfaces and that it increases slowly to X = 0.5. Further, it decreases symmetrically until X = 1 for Pr = 0.015 irrespective of the Gr value. The maximum in Nub at the center is due to the dense heatlines corresponding to 0.01 e Π e 0.5 (figure not shown). It is interesting to observe a local maximum in Nub at X = 0.6 for Pr = 0.7 and 7.2 at Gr = 103. The variation in the heat-transfer rate (Nub) is nonsymmetric because of the dominant lid-driven force in the cavity, as illustrated by the strong primary circulation in the heatlines and the denser heatlines corresponding to 0.5 e X e 0.7 of the bottom surface with |Π| varying in the range of 0.004 e |Π| e 0.005 for Pr = 0.7 (figure not shown) and in the range of 0 e |Π| e 0.0015 for Pr = 7.2 (figure not shown). Sinusoidal variations of Nub for Pr = 0.7 and 7.2 are observed at higher Gr values. A local minimum in Nub at X = 0.2
0.3 and a maximum in Nub at X = 0.6
0.7 (Nub = 8.65) are observed for Pr = 0.7 and Gr = 105. Heatlines corresponding to |Π| = 0.021
0.025 are observed along X = 0.2
0.3, signifying lower heat-transfer rates in this regime. Thus, a local minimum is observed at X = 0.3. Very intense heatlines corresponding to |Π| = 0
0.02 are observed in the range X = 0.6
0.7, illustrating the larger Nub value in this regime for Pr = 0.7 (figure not shown). The high value of Nub (15.5) for Pr = 7.2 compared to that for Pr = 0.7 on the bottom surface indicates that more heat is transferred for Pr = 7.2 at Gr = 105. A sinusoidal variation is clearly observed from the heat-transfer rates, with a maximum at X = 0.6
0.7 and a local minimum at X = 0.2
0.25. Less intense heatlines corresponding to |Π| = 0.005
0.006 are observed along 0 e X e 0.3 of the bottom surface. Also, a weak circulation in the heatline cells is observed near the bottom left corner of the cavity, where lower heating effects are observed. Dense heatlines with |Π| varying in the range of 0
0.003 at X = 0.6
0.65 indicate higher heat-transfer rates in this zone. Thus, a maximum in Nub is observed in this zone. The lower plot in Figure 12b show the local heat-transfer rates (Nur) for the cold right wall. The qualitative trend of Nur for case 2 is similar to that for case 1 except near the distance Y = 0. The lower plot in Figure 12c show the local heat-transfer rates for the cold left wall for various governing parameters. It is observed that the variation of Nul is qualitatively similar to that for case 1 along 0.1 e Y e 1 on the left wall and, similarly to Nur, Nul reaches a maximum at Y = 0. Thus, detailed discussions of the distributions of Nur and Nul based on heatlines are omitted for brevity. 3.6.3. Case 3: Linearly Heated Side Walls with Isothermal Hot Bottom Surface. The upper plots in Figure 13a
c show the local Nusselt number as a function of distance along the bottom surface and side walls for Gr = 103 and 105 with Pr = 0.015
7.2 at 7623

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

Figure 13. Variation of the local Nusselt number with distance at the (a) bottom surface, (b) right wall, and (c) left wall with Re = 100 and Gr = 103 (
) and 105 ( 3 3 3 ). In each plot, the upper panel corresponds to case 3, and the lower panel corresponds to case 4.

Re = 100. Note that the heat-transfer rate (Nub) is 1 at the edges of the bottom surface, because of the linearly heated side walls. Because of the symmetry in the temperature field, the heattransfer rate is symmetric along the bottom surface and is almost constant because of the conduction-dominant heat transfer irrespective of the Gr value for Pr = 0.015 (figure not shown). It is seen that Nub for Pr = 0.7 is almost the same as that for Pr = 0.015 in the range of 0 e X e 0.22 because of the conductiondominant heat transfer. Convective heat transport is dominant for Pr = 7.2 and heat flow is found to occur from the bottom surface as well as the bottom portion of the side walls for Gr = 103 (Figure 8a). Significantly dense heatlines with |Π| = 0.0001
0.002 are observed at X = 0.5
0.6, whereas |Π| varies in the range of 0.002
0035 along X = 0
0.1 and in the range of 0.0001
0.001 at X = 0.8
1. Thus, higher heat-transfer rates are observed near X = 0.6 along the bottom surface, and a maximum in Nub is observed in this zone. At higher Gr values (Gr = 105), because of the enhanced convection effects, higher heat-transfer rates (Nub) are observed at X = 0.5
0.6 for Pr = 0.7. The high values of Nub occur at X = 0.5 for Gr = 105 because of the very intense heatlines corresponding to 0.001 e |Π| e 0.01. It is interesting to observe a maximum in Nub at X = 0.25 and, thereafter, a gradual decrease along the length for Pr = 7.2 and Gr = 105. This maximum is due to the dense heatlines

corresponding to 0.001 e |Π| e 0.02 occurring in the range X = 0.25
0.3 (see Figure 8c). Note that Nub is almost constant in the range of 0.8 e X e 0.9, and thereafter, it decreases to 1 at the right corner. Also, |Π| varies in the range of 0.001
0.0015 for 0.8 e X e 0.9, so the heat-transfer rate is constant in this zone. Lower gradients in heatfunction are also observed at X = 1, so Nub reaches a minimum at X = 1. The upper plot in Figure 13b shows the heat-transfer rates along the right wall. The heat-transfer rate at Y = 0 for the side walls is 0 because of the linear heating of the walls. The heattransfer rate (Nur) is almost constant throughout the bottom surface for Pr = 0.015, as seen in upper plot in Figure 13b. This is due to the disperse and parallel heatlines with |Π| varying in the range of 0.001
0.025 along the right wall, irrespective of the Gr value. It is interesting to observe that the heat-transfer rate (Nur) for low Gr values (Gr = 103) is negative over a large region on the right wall for Pr = 0.7 and 7.2 because of the negative gradients in the heatfunctions, which are represented through heatlines |Π| = 0.001
0.01 for Pr = 0.7 and |Π| = 0.0012
0.002 for Pr = 7.2 (see Figure 8a). Thereafter, the Nur distribution increases slowly along the side wall (Figure 13b, upper panel) because of the dense heatlines corresponding to 0.001 e |Π| e 0.005 toward the top portion of the side wall (see Figure 8c), with a maximum at Y = 1. 7624

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research It is interesting to observe the oscillatory trend in heat-transfer rate for Pr = 0.7 and 7.2 at higher Gr = 105. It is observed that, for Pr = 0.7, Nur increases to a maximum value at Y = 0.2, thereafter decreasing to 0 at Y = 0.38, because of the less intense heatlines corresponding to 0 e |Π| e 0.005 in this zone (figure not shown). Further, Nur is negative and reaches a minimum at Y = 0.5, as observed from the heatlines with 0.005 e |Π| e 0.01 that start and end on the same wall in this zone (figure not shown). Note that negative heat-transfer rates arise because dense heatlines start and end on the right wall and, thus, ∂Π/∂Y changes sign as explained earlier by the sign convention for the heatfunctions. Higher heat-transfer rates with increasing degrees of oscillation are observed at higher Pr values (Pr = 7.2) for Gr = 105. It is interesting to observe that Nur shows a local maximum at Y = 0.4. This is due to the very intense heatlines corresponding to 0 e |Π| e 0.01 along the right wall. Dense heatlines corresponding to |Π| = 0.001
0.005 are observed to end along Y = 0.4 (Figure 8c), indicating a maximum in Nur. However, the variation in Nur for Pr = 7.2 is qualitatively similar to that for Pr = 0.7, and negative heat-transfer rates are observed in the range of 0.6 e Y e 0.85; this can be explained similarly to the observations for Pr = 0.7. The upper plot in Figure 13c shows the heat-transfer rates along the left wall for case 3. It is found that the variation of heattransfer rates along the left wall (Nul) is similar to that of the right wall for Pr = 0.015, because of the largely dispersed heatlines (figure not shown). It is also observed that Nul monotonically increases along the left wall and reaches a maximum at Y = 0.9. Thereafter, it remains almost constant for Pr = 0.7 and Gr = 103. However, it can be noted that negligible heat-transfer rates are observed along 0 e Y e 0.4 of the left wall for Pr = 0.7. This trend arises because very few heatlines with low gradients of the heatfunctions emanate from the hot bottom portion of the left wall, as observed for the right wall (figure not shown). Thus, higher values of Nur are observed compared to Nul. A similar qualitative trend in Nul is observed for Pr = 7.2 and Gr = 103. However, a maximum in Nul is observed at Y = 0.95. This is due to the very intense heatlines corresponding to 0.0001 e |Π| e 0.002 along 0.9 e Y e 0.95 of the left wall (see Figure 8c). It can be noted that higher heat-transfer rates for Pr = 0.7 are observed at higher Gr values, but the heat-transfer rate is similar to that for Gr = 103 in the range Y = 0.8
1. It is interesting to note that, for Pr = 7.2, Nul is negative near the bottom portion of the wall, but it reaches to maximum and then reaches a minimum suddenly near the top portion of the left wall. Negative heat-transfer rates are observed because of the less intense heatlines that emanate and end on the bottom portion of the left wall with |Π| = 0
0.002. Further, Nul increases gradually, corresponding to very intense heatlines (|Π| = 0
0.005) along 0.2 e Y e 1 of the left wall. 3.6.4. Case 4: Linearly Heated Left Wall and Isothermally Cooled Right Wall. The lower plots in Figure 13a
c show the local Nusselt number as a function of distance along the bottom surface and side walls, respectively for Pr = 0.015
7.2 with Gr = 103 and 105 at Re = 100 for case 4. Figure 13a illustrates that the heat-transfer rate (Nub) is very high at the right edge and thereafter Nub decreases toward the center of the bottom surface. The maximum of Nub is due to large heat flux at the hot
cold junction as similar to case 1. The local Nusselt number distribution at low Pr (Pr = 0.015) is nearly uniform along the bottom surface except near the right wall. Conduction-dominant heat transfer is observed as the heat-transfer rate remains same irrespective of Gr as depicted by the heatlines (figure not shown).

ARTICLE

The local Nusselt number distributions for Pr = 0.7 shows a similar trend of the maximum at the hot
cold junction. The variation of Nub for Pr = 0.7 at Gr = 103 is similar to the distribution for Pr = 0.015 and are almost constant in the range of 0.2 e X e 0.8 because of the less intense heatlines corresponding to 0 e |Π| e 0.01 within this regime (figure not shown). It is observed that variation of Nub for Pr = 7.2 in the range of 0.2 e X e 1 is similar to that of case 1. Very intense heatlines with |Π| = 0
0.003 are observed along the 0.6 e X e 0.8 of the bottom surface and a large as well as uniform variation in Nub is observed in this zone (figure not shown). Larger variation in Nub due to larger heating effects is observed at Gr = 105. Higher heat-transfer rates corresponding to 0.001 e |Π| e 0.02 are observed along 0.1 e X e 0.6 of the bottom surface whereas |Π| varies in the range of 0.02 e |Π| e 0.03 along 0 e X e 0.1 of the bottom surface for Pr = 0.7 (figure not shown). Thus, Nub is observed to be increasing in this zone. At higher Gr (Gr = 105), the variation in Nub in the range of 0.1 e X e 1 for Pr = 7.2 is qualitatively similar to that of case 1 and a similar explanation based on heatlines follows. Figure 13b (lower panel) shows the heat-transfer rates along the right wall. Because the right wall is isothermally cold, larger heat-transfer rates exist near the hot
cold junctions similar to case 1. Conduction-dominant heat transfer is observed for Pr = 0.015. However, variation in Nur is qualitatively similar to that of case 1 (see Figure 12b, upper panel). Note that Nusselt number decreases continuously because of the less gradients in heatfunctions with 0.01 e |Π| e 0.015 along 0.1 e Y e 0.7 of the right wall for Pr = 0.7 and Gr = 103 (figure not shown). Similar trend is observed with Pr = 7.2, but larger values of Nur are observed corresponding to larger gradients in heatfunctions (|Π| = 0.02
0.004) occurring in this region compared to Pr = 0.7. It is interesting to observe that Nur reaches a minimum at Y = 0.1, thereafter that increases slowly up to Y = 0.8. Further, that suddenly reaches a maximum at Y = 1. This trend on the variation at higher Gr is similar to case 1 and can be explained based on heatlines as discussed earlier. Figure 13c (lower panel) shows the heat-transfer rates along the left wall. It is interesting to observe that Nul increases and reaches a maximum near Y = 1 for Pr = 0.015 at Gr = 105. This trend is observed because of the onset of convective heat transfer in the cavity as illustrated by the distorted heatlines to the right half of the cavity (figure not shown). The distribution in Nul is qualitatively similar to that in case 3 for low and higher Gr with Pr = 0.7, and thus, detailed explanations based on heatlines are omitted. Similar qualitative trend is observed in Nul irrespective of Gr. But, larger values of Nul are observed for Pr = 7.2 at Gr = 105, compared to lower Gr. It can be noted that Nul suddenly increases to a maximum at Y = 0.97 corresponding to very intense heatlines in this zone (figure not shown) and thereafter decreases to a minimum at Y = 1, similarly to the case for Gr = 103. Smaller gradients in |Π| (|Π| = 0
0.001) are observed at Y = 1 of the left wall, whereas larger gradients in heatfunctions corresponding to |Π| = 0
0.005 are found to occur on the right wall in this zone. Thus, Nul reaches a sudden minimum at Y = 1, whereas Nur reaches a maximum in this zone. 3.7. Overall Heat Transfer and Average Nusselt Numbers. The overall effects on the heat-transfer rates are shown in Figures 14 and 15, where the distributions of the average Nusselt numbers at the bottom surface and side walls are shown as functions of the logarithmic Grashof number for cases 1
4. 7625

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research

ARTICLE

Figure 14. Variation of the average Nusselt number with Grashof number (Gr) for cases 1 (a,b) and 2 (c,d) with Re = 100 and Pr = 0.015, 0.7, and 7.2.

Figure 15. Variation of the average Nusselt number with Grashof number (Gr) for cases 3 (a,b) and 4 (c,d) with Re = 100 and Pr = 0.015, 0.7, and 7.2.

The average Nusselt numbers were obtained using eqs 21 and 22, and the integrals were evaluated using Simpson’s 1/3 rule. 3.7.1. Case 1: Isothermal Hot Bottom Surface and Isothermally Cooled Side Walls. The overall effects on the heat-transfer rates along the bottom surface and left wall, for uniform bottomsurface heating (case 1), are displayed in parts a and b, respectively, of Figure 14. The inset of Figure 14b shows the average Nusselt number distributions for the right wall. It is observed that the average Nusselt numbers (Nub, Nul, and Nur) are invariant with Gr for Pr = 0.015, signifying conduction-dominant heat transfer. The constant and low values of Nub also occur as the heatlines are observed to be perpendicular to the bottom surface and side walls (figure not shown). It can be noted that Nub is constant for Pr = 0.7 up to Gr = 5  104 and thereafter increases exponentially to reach a maximum at Gr = 105. It is observed that |Π| varies in the range of 0.001 e |Π| e 0.018 along the bottom surface for Gr = 103, whereas it varies in the range of 0.001 e |Π| e 0.05 along the bottom surface for Gr = 105. Thus, larger values of Nub are found with larger values of Gr. Based on a similar qualitative trend in the heatfunctions, Nub is found to be an increasing function of Gr for Pr = 7.2. The trend in the variation of Nul is similar to that in the variation of Nub for Pr = 0.015 because of the conductiondominant mode. Note that Nul is constant up to Gr = 6  103 and thereafter gradually increases during the convection-dominant mode for Pr = 0.7. Lower values of Nul are observed because of the less intense heatlines corresponding to 0 e |Π| e 0.02 at Gr = 103 compared to larger Gr values, which corresponds to

dense heatlines (|Π| = 0
0.06). This can also be explained based on lower the distribution in Nul at Gr = 103 compared to that at Gr = 105 (see Figure 12c, upper panel). It can be noted from eq 31 that the nonhomogeneous Dirichlet boundary condition employed is Π(0,0) = (1/Re 3 Pr)Nul, and thus, an increase in Pr leads to a decrease in the values of the heatfunctions. Thus, lower gradients in the heatfunctions are observed for Pr = 7.2, implying larger Nul values compared to those for Pr = 0.7, for otherwise identical parameters. It is interesting to observe that larger values in Nul occur over the range of Gr values for Pr = 7.2 compared to lower Pr. It can also be noted that larger values of Nub occur at Gr = 105, corresponding to intense heatlines with |Π| = 0
0.006 along the bottom surface. This can also be explained based on the larger local heat-transfer rates along the left wall (Nul) for Pr = 7.2 at higher Gr values (see Figure 12c, upper panel). Similarly to Nul, Nur is invariant with Gr at Pr = 0.015, because of the conduction-dominant mode. However, it is interesting to observe the magnitudes of Nur for Pr = 0.7. Specifically, the distribution in Nur in the range Gr e 2  104 is less for Pr = 0.7 than for Pr = 0.015. This is because smaller values of Nur occur for Pr = 0.7 than for Pr = 0.015 in a large region on the left wall for lower Gr values (Figure 12b, upper panel). However, the Nur distribution is larger for Pr = 0.7 than for lower Pr values at higher Gr values. It is observed that the distribution in Nur for Pr = 7.2 is qualitatively similar to that for Pr = 0.7 over the range of Gr values. Note that larger Nur values are observed because of the larger values of Nur for Pr = 7.2 compared to lower Pr values (see Figure 12b, upper panel). 7626

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research It can be noted that heatfunctions vary as |Π| = 0
0.7 in the conduction-dominant regime along the right wall at Gr = 103 for Pr = 0.015, whereas they vary in the ranges of 0 e |Π| e 0.01 for Pr = 0.7 and 0 e |Π| e 0.002 for Pr = 7.2. However, smaller gradients in the heatfunctions for larger Pr values, under otherwise identical conditions, implies higher heat-transfer rates (as seen from eq 31); thus, larger values of Nur are observed for Pr = 7.2 even at low Gr. A similar trend for larger Nur values with Pr = 7.2 is also observed at higher Gr values, and this can be similarly explained based on heatlines. Larger gradients in the heatfunctions (|Π| = 0
0.01) are observed at Gr = 105, compared to lower Gr (Gr = 103) (|Π| e 0.002) for Pr = 7.2. Thus, Nur increases monotonically with Gr to reach a maximum at Gr = 105. Overall, larger distributions in the average Nusselt numbers are observed along the bottom surface compared to the side walls based on the overall heat balance Nub = Nur þ Nul, as verified based on Figure 14a,b. 3.7.2. Case 2: Nonisothermal Hot Bottom Surface and Isothermally Cooled Side Walls. The overall effects on the heat-transfer rates along the bottom surface and left wall, for nonuniform heating of the bottom surface (case 2), are displayed in parts c and d, respectively, of Figure 14. The inset in Figure 14d shows the average Nusselt number distributions for the right wall. Similar to the previous case, the average Nusselt number distributions (Nub, Nul, and Nur) remain constant throughout the entire range of Gr values for Pr = 0.015 because of the conduction-dominant heat transfer, which can also be explained based on lower local Nusselt number distributions (Figure 12, lower plots). However, the distributions of Nub, Nul and Nur increase monotonically with Grashof number for Pr = 0.7 and 7.2. This is due to the similar types of streamline and heatline distributions observed. Lower values of the heat-transfer rates are observed because of the smaller heating effects due to the sinusoidal temperature distribution along the bottom surface. It can be noted that the distributions in the overall heat-transfer rates for the bottom surface and side walls are qualitatively similar to those for case 1, and thus, detailed discussions are omitted for brevity. It can also be noted that lower overall heat-transfer rates are observed in case 2 because of the lower magnitudes of the heatlines compared to those in case 1 under identical parameters. This can also be explained based on the lower distributions in local heat-transfer rates along the bottom surface and side walls, compared to those in case 1 (see Figure 12c, upper and lower panels). 3.7.3. Case 3: Linearly Heated Side Walls with Isothermal Hot Bottom Surface. Parts a and b, respectively, of Figure 15 display the overall effects of the heat-transfer rates (Nub and Nul) for the case of linearly heated side walls. The inset in Figure 15b shows the variation of the average Nusselt number for the right wall (Nur). The average heat-transfer rates (Nub, Nul, and Nur) for Pr = 0.015 are invariant with Gr because of the dominant effect of conduction, as observed by the heatlines perpendicular to the hot bottom surface (figure not shown). This can also be explained based on the low and uniform variation of Nub, as seen from upper plot in Figure 13a. Figure 15a shows that Nub is constant up to Gr = 8 103 and thereafter increases for Pr = 0.7, because of the dense heatlines corresponding to larger heat-transfer rates along the bottom surface at higher Gr values (figure not shown). The larger values of Nub can also be explained based on the larger distribution of Nub values at higher Gr compared to Gr = 103 (Figure 13a, upper panel). It is interesting to observe an inflection point in Nub at

ARTICLE

Gr = 104
2  104 for Pr = 7.2. This is due to the less intense heatlines observed near the central portion of the bottom surface corresponding to 0.001 e |Π| e 0.002. However, very intense heatlines (|Π| = 0
0.005) as seen in Figure 8c corresponding to higher heat-transfer rates are observed for higher Gr values. This is also seen from larger magnitudes of Nub, with a maximum at X = 0.3 (Figure 13a, upper panel). Consequently, Nub increases monotonically to a maximum as Gr increases to 105. The distribution in Nul is almost uniform over the range of Gr values for Pr = 0.015 because of the conduction-dominant heat transfer. A similar qualitative trend is observed with Pr = 0.7. However, Nul increases slightly for Gr g 7  103 at Pr = 0.7. A few interesting features are observed for the overall Nusselt number distribution along the left wall (Nul) for Pr = 7.2. It is seen that Nul is almost constant up to Gr = 7  103, thereafter reaching a minimum at Gr = 104
2  104 and further monotonically increasing up to Gr = 105. Note that Nul shows a kink for Pr = 7.2 and Gr = 104, because of the nonmonotonic behavior for Nul. This is also because negative values of Nul are observed and the overall heat-transfer rate (Nul) decreases. It can be noted that larger gradients in the heatfunctions corresponding to |Π| = 0
0.005 are observed at lower Gr values compared to those (|Π| = 0
0.004) at Gr = 104 (see Figure 8a,b). Thus, overall lower heat-transfer rates (Nul) are observed at Gr = 104. The further increase in Gr corresponds to higher heat-transfer rates, and thus, very intense heatlines, compressed toward the left wall, are observed at Gr = 105 (|Π| = 0
0.0055) (see Figure 8c). It can also be noted that the gradients in the magnitudes of Nul at Gr = 105 and Gr = 103 are much less unlike those in other cases where a large gradient in Nul is observed over the range of Gr values. Even though larger gradients in heatfunctions are observed at Gr = 105, compared to Gr = 104, it can be noted that |Π| varies in the range of 0
0.0055 for Gr = 105 whereas it varies in the range of 0
0.005 for Gr = 103 along the left wall. Thus, larger magnitudes are not observed at Gr = 105 compared to Gr = 103. It is interesting to observe that the average heat-transfer rate (Nur) is 0 over the range of Grashof numbers for Gr e 104 with Pr = 0.7. This is because less intense heatlines (|Π| = 0
0.002) that start from a large portion on the right wall (Y e 0.8) end toward the top portion of the right wall (0.8 e Y e 1) (figure not shown). This trend is explained based on the negative local heat-transfer rates at low and high Gr values as observed from the upper plot in Figure 13b. Also, a lower distribution in Nur (Nur ≈ 0) is observed for Pr = 0.7 compared to Pr = 0.015 over a large range of Gr values (Gr e 8  104) (Figure 13b, upper panel). This is because large negative heatfunction gradients are observed for Pr = 0.7 over a large zone on the right wall for Gr e 8  104, and thus, the overall heat-transfer rates remains low. This is also explained by the smaller values of Nur for Pr = 0.7 along the right wall for Gr = 103. Oscillatory variation in Nur is observed and consequent positive and negative magnitudes are observed with a maximum at Gr = 105. Thus, larger Nur values are observed at higher Gr values for Pr = 0.7. Negative overall heat-transfer rates (Nur) are observed on the right wall for Gr e 9  103 for Pr = 7.2. This is based on the fact that negative Nur values are observed over a large region on the right wall for lower Gr values (see Figure 13b, upper panel, Pr = 7.2). An overall lower distribution in Nur is observed compared to that in Nul for Pr = 7.2, similar to the case for lower Pr values. This is because larger negative gradients in heatfunctions are observed toward the right wall, signifying lower Nur values, compared to Nul. 7627

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research 3.7.4. Case 4: Linearly Heated Left Wall and Isothermally Cooled Right Wall. The overall effects on the heat-transfer rates along the bottom surface and left wall for case 4 are displayed in parts c and d, respectively, of Figure 15. The inset of Figure 15d shows the average Nusselt number distributions for the right wall. Similar to the previous cases, the average Nusselt number distributions (Nub, Nul, and Nur) for Pr = 0.015 are uniform over the range of Gr values because of the conduction-dominant regime. It can be observed that the trend in the variations of Nub and Nur are qualitatively similar to those for cases 1 and 2. Larger Nub values are observed in case 1 compared to case 4. This is because of the larger gradients in the heatfunctions in case 1 along the bottom surface compared to those in case 4. Also, larger values of Nub are observed in case 4 compared to case 3. This can be explained based on the larger gradients in the heatfunctions in case 4 corresponding to |Π| =0
0.05, whereas lower gradients corresponding to 0 e |Π| e 0.02 occur along the right half of the bottom surface for Pr = 0.7 at Gr = 105. A similar qualitative explanation can be made for Pr = 7.2. The trend in the variation of Nul is qualitatively similar to that in case 3 for smaller Pr values. A smooth and monotonic distribution in Nul is observed for Pr = 7.2, over the entire range of Gr values, unlike in case 3. This is due to the uniform distribution of heatlines along the bottom surface over the range of Gr values in case 4. A larger distribution in Nur is observed compared to cases 1 and 3, irrespective of the Pr values. This is due to the larger distribution in Nur for case 4, which can also be explained by dense heatlines toward the right wall.

4. CONCLUSIONS The prime objective of this article was to study the effects of various thermal boundary conditions on the flow and heattransfer characteristics due to lid-driven mixed-convection flows within a square cavity based on lower and higher Pr values (Pr = 0.015, 0.7 and 7.2) and Re values (Re = 1, 10, 100) for Gr = 103
105. The Galerkin penalty finite-element method was employed to obtain smooth solutions in terms of heatlines, streamlines, and isotherms. The heat flow inside the cavity was visualizaed using the heatline concept, which enables an understanding of the heat flow trajectory. Heatlines are uniquely determined from heatfunctions, which are obtained by solving the Poisson equation. Important features of the heating and flow characteristics based on heatlines are outlined below for various test studies. Isothermal Cold Side Walls and Isothermal/Nonisothermal Hot Bottom Surface (Cases 1 and 2). Larger gradients in

heatfunctions (|Π|) are observed along the left wall compared to right wall at high Re and Pr values, signifying higher thermal gradients. The heatlines clearly demonstrate that convective heat flow determines the temperature pattern at high Re. The heatlines and streamlines behave identically near the core for convection-dominant flow. It was found that the effect of heating is more pronounced near the bottom surface and left walls as the formation of thermal boundary layers is restricted near the bottom surface and left wall for Pr = 7.2. In contrast, symmetric heating patterns with symmetric circulation cells are observed at smaller Re values for higher values of Gr. Various interesting features in the heat-transfer rates are found at high Re. The heat-transfer rates are gradually enhanced with increasing Gr for higher Pr values. It was found that Nub reaches a maximum at the corners and attains a minimum at X = 0.15
0.2

ARTICLE

because of the less intense heatlines occurring in that zone (case 1). For uniform heating, an additional maximum in Nur is observed at Y = 1 because of the dense heatlines, whereas a minimum in Nul is observed, corresponding to less intense heatlines at Y = 1. An oscillatory variation in Nub is observed for high-Pr fluids, with larger Nub values in the right half of the bottom surface for sinusoidal heating of the bottom surface (case 2). The distributions of Nul and Nur are qualitatively similar except at the junction of hot and cold surfaces for uniform/ nonuniform heating of the bottom surface. Exponential increases in Nub, Nul, and Nur are observed with Gr for higher Pr values. The smaller values of Nur for Pr = 0.7, compared to those for Pr = 0.015, for low Gr values can also be justified based on the smaller values of Nur for Gr = 103. Variations in average Nusselt numbers are qualitatively similar for uniform/nonuniform heating except that lower values in the average Nusselt numbers are observed in the nonuniform heating case because of the smaller heating effects. Linearly Heated Side Wall(s) and Isothermal Hot Bottom Surface (Cases 3 and 4). Heatlines that start and end on the same walls are observed, because of the linear heating of the side walls. The heatline circulation cells stretch diagonally to the column for low Re for Pr = 0.7. The thickness of the thermal boundary layer is less toward the top portion of the right wall. Enhanced convective effects are observed based on multiple circulations in the heatline cells for Pr g 0.7 at Gr = 105 (case 3). On the other hand, the heat-transfer rate is quite large at the right corner of the bottom surface for case 4. The pattern of multiple circulations in heatline cells are observed at Pr = 7.2, Gr = 105, and Re g 10 (case 4). The thermal boundary layer develops near the bottom edges, and the thickness of boundary layer is also smaller at the top portion of the cold right wall, signifying high heat transfer to the top portion at higher Gr values. Overall, the heat transport is more toward the right wall for all Gr values irrespective of the Pr value in case 4 compared to case 3. Larger values of Nub are observed at the right half of the bottom surface for low Gr values because of the very intense heatlines, whereas a maximum in Nub is observed in the left portion of the bottom surface at higher Gr values in case 3. A large portion along the right wall corresponds to negative Nur values at low Gr, whereas an oscillatory trend is observed at higher Gr values for Pr = 0.7 and 7.2. On the other hand, larger values of Nur are observed for case 4 at the corners of the right wall irrespective of the Gr value for higher Pr values. Also, larger values of Nur are observed along the right wall in case 4 compared to case 3. Although the variations in the local heat-transfer rates are qualitatively similar for cases 3 and 4, larger values of Nul are observed at higher Gr values in case 4 compared to those in case 3 at higher Pr values. It can also be noted that distribution of Nul is invariant with Gr near the top portion of the left wall in case 3, whereas larger Nul values, because of very intense heatlines, are observed in case 4 at higher Gr values. Nonmonotonic variations in Nub and Nul are observed for Pr = 7.2 (case 3), because of the negative heat-transfer rates at Gr = 104. Smaller gradients in Nul over the range of Gr values are observed in case 3, whereas larger values of Nub and Nul with exponential increases with Gr are observed in case 4. Negative overall heat-transfer rates because of the largely negative local Nur values are observed for Pr = 0.7 and 7.2 at lower Gr values (case 3), whereas larger values of Nur are observed in case 4, signifying higher heat-transfer rates compared to case 3. The overall Nusselt number distribution in case 4 is qualitatively 7628

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research similar to that in case 2 along the bottom surface and side walls. However, higher heat-transfer rates are observed along right wall for case 4 compared to case 2 because of the dense heatlines and larger Nur distribution.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The authors thank the anonymous reviewers for critical comments and suggestions that improved the quality of the article. ’ NOMENCLATURE g = acceleration due to gravity, m s
2 J = Jacobian of residual equations k = thermal conductivity, W m
1 K
1 L = height of the square cavity, m N = total number of nodes Nu = local Nusselt number Nu = average Nusselt number p = pressure, Pa P = dimensionless pressure Pr = Prandtl number R = residual of weak form Re = Reynolds number Gr = Grashof number T = temperature, K Tc = temperature of cold wall, K Th = temperature of hot bottom surface, 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 the x coordinate Y = dimensionless distance along the y coordinate Greek Symbols

r = thermal diffusivity, m2 s
1 β = volume expansion coefficient, K
1 γ = penalty parameter θ = dimensionless temperature ν = kinematic viscosity, m2 s
1 F = density, kg m
3 Φ = basis functions ψ = streamfunction Π = heatfunction Subscripts

b = bottom surface i = residual number k = node number s = side wall Superscripts

n = Newton iterative index

’ REFERENCES (1) Pilkington, L. A. B. Review lecture: The float glass process. Proc. R. Soc. London A 1969, 314, 1–25.

ARTICLE

(2) Ideriah, F. J. K. Prediction of turbulent cavity flow driven by buoyancy and shear. J. Mech. Eng. Sci. 1980, 22, 287–295. (3) Imberger, J.; Hamblin, P. F. Dynamics of lakes, reservoirs, and cooling ponds. Ann. Rev. Fluid Mech. 1982, 14, 153–187. (4) Joye, D. D. Design criterion for the heat-transfer coefficient in opposing flow, mixed convection heat transfer in a vertical tube. Ind. Eng. Chem. Res. 1996, 35 (7), 2399–2403. (5) Saylor, P. E.; Joye, D. D. Hydrostatic correction and pressuredrop measurement in mixed convection heat transfer in a vertical tube. Ind. Eng. Chem. Res. 1991, 30 (4), 784–788. (6) Joye, D. D.; Bushinsky, J. P.; Saylor, P. E. Mixed convection heat transfer at high Grashof number in a vertical tube. Ind. Eng. Chem. Res. 1989, 28 (12), 1899–1903. (7) Cha, C. K.; Jaluria, Y. Recirculating mixed convection flow for energy extraction. Int. J. Heat Mass Transfer 1984, 27, 1801–1812. (8) Schreiber, R.; Keller, H. B. Driven cavity flows by efficient numerical techniques. J. Comput. Phys. 1983, 49, 310–333. (9) Thompson, M. C.; Ferziger, J. H. An adaptive multigrid technique for the incompressible Navier
Stokes equations. J. Comput. Phys. 1989, 82, 94–121. (10) Al-Nimr, M. A.; Hader, M. A. MHD free convection flow in openended vertical porous channels. Chem. Eng. Sci. 1999, 54, 1883–1889. (11) Wang, L. B.; Wakayama, N. I. Control of natural convection in non- and low-conducting diamagnetic fluids in a cubical enclosure using inhomogeneous magnetic fields with different directions. Chem. Eng. Sci. 2002, 57 (11), 1867–1876. (12) Nazridoust, K.; Ahmadi, G. Computational modeling of methane hydrate dissociation in a sandstone core. Chem. Eng. Sci. 2007, 62, 6155–6177. (13) Ganguli, A. A.; Pandit, A. B.; Joshi, J. B. Numerical predictions of flow patterns due to natural convection in a vertical slot. Chem. Eng. Sci. 2007, 62 (16), 4479–4495. (14) Makayssi, T.; Lamsaadi, M.; Naimi, M.; Hasnaoui, M.; Raji, A.; Bahlaoui, A. Natural double-diffusive convection in a shallow horizontal rectangular cavity uniformly heated and salted from the side and filled with non-Newtonian power-law fluids: The cooperating case. Energy Convers. Manage. 2008, 49 (8), 2016–2025. (15) Mohamad, A. A.; Viskanta, R. Transient low Prandtl number fluid convection in a lid driven cavity. Numer. Heat Transfer A: Appl. 1991, 19, 187–205. (16) Mohamad, A. A.; Viskanta, R. Flow and heat transfer in a lid driven cavity with a stably stratified fluid. Appl. Math. Model. 1995, 19 (8), 465–472. (17) Sun, K. H.; Pyle, D. L.; Baines, M. J.; Hall-Taylor, N.; Fitt, A. D. Velocity profiles and frictional pressure drop for shear thinning materials in lid-driven cavities with fully developed axial flow. Chem. Eng. Sci. 2006, 61, 4697–4706. (18) Al-Amiri, A.; Khanafer, K.; Joseph, B.; Pop, I. Effect of sinusoidal wavy bottom surface on mixed convection heat transfer in a lid driven cavity. Int. J. Heat Mass Transfer 2007, 50, 1771–1780. (19) Turki, S.; Abbassi, H.; Ben Nasrallah, S. Two-dimensional laminar fluid flow and heat transfer in a channel with a built-in heated square cylinder. Int. J. Therm. Sci. 2003, 42 (12), 1105–1113. (20) Guo, G. H.; Sharif, M. A. R. Mixed convection in rectangular cavities at various aspect ratios with moving isothermal sidewalls and constant flux heat source on the bottom wall. Int. J. Therm. Sci. 2004, 43 (5), 465–475. (21) Bhoite, M. T.; Narasimham, G. S. V. L.; Murthy, M. V. K. Mixed convection in a shallow enclosure with a series of heat generating components. Int. J. Therm. Sci. 2005, 44 (2), 121–135. (22) Al-Amiri, A. M.; Khanafer, K. M.; Pop, I. Numerical simulation of combined thermal and mass transport in a square lid-driven cavity. Int. J. Therm. Sci. 2007, 46 (7), 662–671. (23) Oztop, H. F.; Sun, C.; Yu, B. Conjugate-mixed convection heat transfer in a lid-driven enclosure with thick bottom wall. Int. Commun. Heat Mass Transfer 2008, 35 (6), 779–785. (24) Kleinstreuer, C.; Wang, T. Y. Mixed convection heat and surface mass transfer between power-law fluids and rotating permeable bodies. Chem. Eng. Sci. 1989, 44 (12), 2987–2994. 7629

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630

Industrial & Engineering Chemistry Research (25) Rao, P. K.; Sahu, A. K.; Chhabra, R. P. Flow of newtonian and power-law fluids past an elliptical cylinder: A numerical study. Ind. Eng. Chem. Res. 2010, 49 (14), 6649–6661. (26) Srinivas, A. R.; Bharti, R. P.; Chhabra, R. P. Mixed convection heat transfer from a cylinder in power-law fluids: Effect of aiding buoyancy. Ind. Eng. Chem. Res. 2009, 48 (21), 9735–9754. (27) Soares, A. A.; Anacleto, J.; Caramelo, L.; Ferreira, J. M.; Chhabra, R. P. Mixed convection from a circular cylinder to power law fluids. Ind. Eng. Chem. Res. 2009, 48 (17), 8219–8231. (28) Kimura, S.; Bejan, A. The heatline visualization of convective heat transfer. J. Heat Transfer 1983, 105, 916–919. (29) Bejan, A. Convective Heat Transfer; Wiley: New York, 1984. (30) Bello-Ochende, F. L. A heat function formulation for thermal convection in a square cavity. Int. Commun. Heat Mass Transfer 1988, 15, 193–202. (31) Morega, A. M. The heat function approach to the thermomagnetic convection of electroconductive melts. Rev. Roum. Sci. Tech. Electrotech. Energ. 1988, 33 (4), 155–156. (32) Trevisan, O. V.; Bejan, A. Combined heat and mass transfer by natural convection in a vertical enclosure. J. Heat Transfer 1987, 109, 104–112. (33) Costa, V. A. F. Unification of the streamline, heatline and massline methods for the visualization of two-dimensional transport phenomena. Int. J. Heat Mass Transfer 1999, 42, 27–33. (34) Costa, V. A. F. Bejan’s heatlines and masslines for convection visualization and analysis. Appl. Mech. Rev. 2006, 59, 126–145. (35) Aggarwal, S. K.; Manhapra, A. Use of heatlines for unsteady buoyancy-driven flow in a cylindrical enclosure. J. Heat Transfer 1989, 111, 576–578. (36) Dalal, A.; Das, M. K. Heatline method for the visualization of natural convection in a complicated cavity. Int. J. Heat Mass Transfer 2008, 51 (1
2), 263–272. (37) Hakyemez, E.; Mobedi, M.; Oztop, H. F. Effects of wall-located heat barrier on conjugate conduction/natural-convection heat transfer and fluid flow in enclosures. Numer. Heat Transfer A: Appl. 2008, 54, 197–220. (38) Hooman, K.; Gurgenci, H.; Dincer, I. Heatline and energy flux vector visualization of natural convection in a porous cavity occupied by a fluid with temperature dependent viscosity. J. Porous Media 2009, 12, 265–275. (39) Hooman, K. Energy flux vectors as a new tool for convection visualization. Int. J. Numer. Methods Heat Fluid Flow 2010, 20, 240–249. (40) Morega, A. M.; Bejan, A. Heatline visualization of forced convection laminar boundary layers. Int. J. Heat Mass Transfer 1993, 36 (16), 3957–3966. (41) Deng, Q. H.; Tang, G. F. Numerical visualization of mass and heat transport for mixed convective heat transfer by streamline and heatline. Int. J. Heat Mass Transfer 2002, 45 (11), 2387–2396. (42) Deng, Q. H.; Zhou, J.; Mei, C.; Shen, Y. M. Fluid, heat and contaminant transport structures of laminar double-diffusive mixed convection in a two-dimensional ventilated enclosure. Int. J. Heat Mass Transfer 2004, 47 (24), 5257–5269. (43) Zhao, F. Y.; Liu, D.; Tang, G. F. Inverse determination of boundary heat fluxes in a porous enclosure dynamically coupled with thermal transport. Chem. Eng. Sci. 2009a, 64, 1390–1403. (44) Zhao, F. Y.; Liu, D.; Tang, L.; Ding, Y. L.; Tang, G. F. Direct and inverse mixed convections in an enclosure with ventilation ports. Int. J. Heat Mass Transfer 2009b, 52, 4400–4412. (45) Reddy, J. N. An Introduction to Finite Element Analysis; McGrawHill: New York, 1993. (46) Taylor, R. L.; Zienkiewicz, O. C.; Too, J. M. Reduced integration technique in general analysis of plates and shells. Int. J. Numer. Methods Eng. 1971, 3, 275–290. (47) Basak, T.; Roy, S.; Balakrishnan, A. R. Effect of thermal boundary conditions on natural convection flows in a square cavity. Int. J. Heat Mass Transfer 2006, 49, 4525–4535. (48) Batchelor, G. K. Introduction to Fluid Dynamics; Cambridge University Press: New York, 1993.

ARTICLE

(49) Ganzarolli, M. M.; Milanez, L. F. Natural convection in rectangular enclosures heated from below and symmetrically cooled from the sides. Int. J. Heat Mass Transfer 1995, 38, 1063–1073. (50) Corcione, M. Effects of the thermal boundary conditions at the sidewalls upon natural convection in rectangular enclosures heated from below and cooled from above. Int. J. Therm. Sci. 2003, 42, 199–208. (51) Moallemi, M. K.; Jang, K. S. Prandtl number effects on laminar mixed convection heat transfer in a lid-driven cavity. Int. J. Heat Mass Transfer 1992, 35, 1881–1892. (52) Basak, T.; Roy, S. Role of ’Bejan’s heatlines’ in heat flow visualization and optimal thermal mixing for differentially heated square enclosures. Int. J. Heat Mass Transfer 2008, 51, 3486–3503. (53) Eckert, E. R. G.; Drake, R. M. Analysis of Heat and Mass Transfer; McGraw-Hill: New York, 1972.

7630

dx.doi.org/10.1021/ie102530u |Ind. Eng. Chem. Res. 2011, 50, 7608–7630