Mixed convection characteristics in rectangular enclosure containing ...

International Journal of Thermal Sciences 130 (2018) 100–115

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Mixed convection characteristics in rectangular enclosure containing heated elliptical block: Effect of direction of moving wall

T

Krunal M. Gangawane∗,1, Siddharth Gupta Department of Chemical Engineering, School of Engineering, University of Petroleum and Energy Studies, Dehradun-248007, Uttrakhand, India

A R T I C LE I N FO

A B S T R A C T

Keywords: Elliptical block Mixed convection Reynolds number Nusselt number Grashof number Rectangular enclosure

The present research numerically explores the hydrodynamic and thermal characteristics due to mixed convection in the rectangular enclosure with one of the vertical wall moving and containing centrally placed heated elliptical block for steady state condition. In particular, the influence of different flow pertaining parameters, such as, the direction of moving vertical wall (either in positive –y or negative –y coordinate directions), aspect ratio of an elliptical cylinder (Er = 0.5, 1 and 2) are studied for the broad range of dimensionless parameters (Reynolds number: 1 ≤ Re ≤ 5000; Prandtl number: 1 ≤ Pr ≤ 100 and Grashof number: 0 ≤ Gr ≤ 105) by using finite volume method (FVM) and SIMPLE numerical approach. The analysis of physical insights of an enclosure is accomplished by systematic evaluation of the streamlines as well as isotherms profiles. In order to compare mean heat transfer from an elliptical cylinder with that of circular shape, normalized Nusselt number (NuN) is estimated. Heat transfer characteristics are more predominantly affected due to the direction of moving lid than the aspect ratio of the elliptical cylinder. Also, the enclosure with moving wall along positive –y direction tends to enhance heat transfer rate. Rather than a pure circular shape, the enclosure with heated elliptical block (Er≠1) shows a higher rate of convective heat transfer.

1. Introduction Convective heat transfer from cylinders of an elliptical cross section has received huge consideration in recent years because of its numerous engineering applications as well as pragmatic relevance. The development of high-performance heat exchanger devices for making effective use of energy is a very crucial and pressing problem nowadays. For an instance, the process heat transfer equipment, such as heat exchangers made of tubes of circular cross-section are widely used in many chemical/process, automobile or like industries (probes and sensors, hot wire anemometry, RTM process of producing fiber composites, filters of aerosol, screens of filtration process, etc.). The flow circulation over such tubes, however, is not necessarily normal to the tube axis which makes the tube cross-section in the direction of flow to have an elliptical shape. In general, the elliptical geometry can represent the circular cylinder as well as very thin plate depending on the value of the axis ratio [1,2]. The heat transfer characteristics from an elliptical tube/ block depend on its geometry (i.e. aspect ratio, surface roughness, etc.), fluid properties, flow approaching condition, and the block/tube surface temperature variation. Moreover, the study of heat transfer from an elliptical body inside a channel/enclosure is of great theoretical

significance as elliptical shape can be considered as a prototype for modeling heat transfer from a broad range of bluff bodies (as it can used to the elaborate the influence of both angle of attack and thickness). The different modes of convective heat transfer modes may significantly take place ranging from forced convection dominated regime to natural convection one inside above mentioned systems. The mixed convection (combined free and forced) heat transfer occurs when both inertial and buoyancy forces are of comparable magnitude. The basic parameters which govern the forced and natural convection are Reynolds number (Re) and Grashof number (Gr), respectively [1–3]. On the account of fundamental and pragmatic implication, the experimental and numerical investigations within square/rectangular enclosures have been extensively studied and very well documented. The reason for the overwhelming popularity of such constellations is due to the simple domain with the ability to explore the broad varieties of fluid flow and heat transfer fundamentals (multiplicity of steady solutions, boundary layer, vortex size and location, circulation of fluid, etc.). Typical examples which can be idealized as an enclosure are cooling systems of electronic gadgets, high performance building insulation, multi-shed structures, furnace, food processing (heating of various foodstuffs like beans, carrot and potato chips, etc.), lubrication

∗

1

Corresponding author. E-mail addresses: [email protected], [email protected] (K.M. Gangawane). Current address: Department of Chemical Engineering, National Institute of Technology Rourkela, Rourkela 769008, Orissa, India.

https://doi.org/10.1016/j.ijthermalsci.2018.04.010 Received 22 November 2016; Received in revised form 15 January 2018; Accepted 9 April 2018 1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.


K.M. Gangawane, S. Gupta

Nomenclature ER Cp CWT Gr H h k L Nu Nu NuN P Pe Pr qw Ra Re Ri T Uw

ux , u y ux, uy x, y x,y Δθ

aspect ratio of elliptical cylinder (Er = radius along x-direction/radius along –y-direction), dimensionless heat capacity, J/(kg.K) constant wall temperature, K Grashof number, dimensionless height of cavity (Characteristic length), m heat transfer coefficient, W/(m2.K) thermal conductivity, W/(m.K) width of cavity, m local nusselt number, dimensionless average nusselt number, dimensionless normalized nusselt number, dimensionless Pressure, N/m2 Peclet number, dimensionless Prandtl number, dimensionless heat flux, W/m2 Rayleigh number, dimensionless Reynolds number, dimensionless Richardson number, dimensionless Temperature, K velocity of moving wall, m/s

velocity components, m/s velocity components, dimensionless co-ordinates, m co-ordinates, dimensionless normalized temperature difference, dimensionless

Greek μ α β θ ν ρ ρ Ψ Ω

dynamic viscosity, N.s/m2 thermal diffusivity, m2/s coefficient of thermal expansion, 1/K normalized temperature, dimensionless kinematic viscosity, m2/s density, kg/m3 average density determined at Tref, K stream function, m2/s angular rotational speed, RPM

Subscripts c w ref

technologies, fluidized bed drying of fibrous substances, solar heat collectors, drying, etc. [4–11]. Moreover, there are few industrial processes (crystal growth, solidification, etc.) in which convection currents should be minimized/restricted. This can be accomplished by placing obstacle/object/cylinder (of any shape circular, elliptical square, triangular, etc.) is used for controlling fluid flow due to convection [12] in enclosure. Ample literature is available for convection heat transfer in lid-driven enclosures [13–18]. Most of the literature have explored the lid-driven cavity with one of the walls exposed to higher temperature/ heat source. Few studies have covered enclosure with one or two-sided moving lid. However, very limited information on the lid-driven rectangular enclosure containing obstacle for a wide range of Prandtl numbers is documented. Few studies [19–21] have provided the benchmarking of results for hydrodynamics in the lid-driven enclosure. Many studies [22–30] have revealed the natural/forced/mixed convection characteristics in enclosure with adiabatic/isothermal obstacle of varying shape. Abu-Nada et al. [31] delineated the influence of the presence of horizontal circular cylinder on mixed convection characteristics for nanofluids. Oztop et al. [32] investigated the fluid flow due to mixed convection in the lid-driven enclosure containing twodimensional body for a wide range of Richardson numbers. They reported the presence of a circular body can be considered as a control parameter for heat and fluid flow in enclosures or likes structures. Billah et al. [33] presented the numerical study of combined convection in the lid-driven cavity having heated circular hollow cylinder. They found the significant impact of the cylinder diameter on overall fluid flow and heat transfer behavior. Subsequently, Rahman et al. [34] studied mixed convection in the double lid-driven cavity under the influence of magnetic field having heat generating a square block for a wide range of Ri. Moreover, Błasiak and Kolasinski [35] computationally modeled two-dimensional mixed convection in the lid-driven cavity with a constant heat flux boundary condition for a wide range of pertinent parameters. Increase in Ri causes the decrease in average surface averaged Nu. Recently, natural convection from the heated hexagonal block for non-Newtonian power-law fluids is reported by Gangawane and Manikandan [36]. They observed remarkable influence of thermal boundary conditions on the overall structure of fluid flow in the cavity. More recently, Gangawane [37] identified the critical Reynolds number in top lid-driven containing heated triangular block for the laminar condition due to mixed convection. Convection rate increases only up

cold wall or surface reference value

to Recr = 190–220 for lower Grashof number values (Gr ≤ 102). In recent years, the studies delineating heat and fluid flow characteristics from heated elliptical cylinder have seen remarkable growth. For an instance, Badr [38] presented numerical study of mixed convection from straight elliptical tube for laminar range of Re (Re = 50–200) and Grashof number (Gr = 0-106). It is observed that, for given Re, augmentation in Gr tends to suppress the vortex shedding. Kondjoyan and Daudin [39] studied the effect of free stream turbulence intensity (1.5–40%) on the heat and mass transfer from the surface of elliptical as well as the circular cylinder. Later, the heat transfer including an optimization study for comparison of the staggered circular and elliptical tubes for forced convection is reported by Matos et al. [40]. This research concluded that the elliptical arrangement can enhance heat transfer up to 13%. Chen [41] presented the numerical simulation study for pressure melting ice around the horizontal elliptical cylinder by solving Reynolds equation ( μ∂2u/ ∂s 2 = dp / dh , where ‘s’ and ‘h’ are the distances in normal and tangential directions, respectively). Meng et al. [42] experimentally analyzed the convective heat transfer in an alternating elliptical axis tubes. The cross-sectional change in the alternating elliptical axis tubes strongly affects the heat transfer characteristics. Subsequently, Faruquee et al. [43] elucidated the influence of axis ratio on heat and fluid flow properties around the elliptical cylinder for laminar flow condition. The wake size and the drag coefficient show linear variation with an axis ratio of the cylinder. Later, Bharti et al. [44] presented an extensive study of forced convection heat transfer from channel built-in an elliptical cylinder for non-Newtonian power-law fluids. They reported the similar influence of elliptical cylinder on flow governing parameters as that of the circular one. Cheng [45] numerically investigated the natural convection boundary layer flow with temperature-dependent viscosity from a horizontal elliptical cylinder with constant surface heat flux thermal condition. Local Nusselt number of the elliptical cylinder increases with the Prandtl number. Chandra and Chhabra [46] presented detailed analysis of forced convection from semi-circular cylinder in unconfined flow condition. This study identified the critical values of Reynolds number for wake formation and onset of vortex shedding. Subsequently, the heat transfer characteristics across a pair of confined elliptical cylinders in the line array for non-Newtonian power-law fluids is studied by Nejat et al. [47] for different aspect ratio of elliptical cylinder (Er = 0.25–2). They observed that the elliptical cylinder with Er = 0.5 can be more efficient 101



either at uy = +Uw or -Uw, while all other walls are stationary. Top (y = 1) and bottom (y = 0) walls are maintained at the adiabatic thermal condition. The right vertical wall (x = 2) is maintained at ambient temperature (θc). The enclosure contains heated elliptical block with either of aspect ratio, Er = 0.5, 1 or 2, at its center with constant temperature, θw (θw > θc). Moreover, physical properties of the working fluid are assumed to be independent of the temperature. The viscous dissipation effects and radiation heat transfer are neglected. The buoyancy force term is expressed by gyβ(θw-θc). The extent of the variation of density with temperature varies from one fluid to another. In order to represent the variation of density with temperature, wellknown Boussinesq approximation is widely used in natural/mixed convection studies [8–11]. The continuity, momentum and energy equations (in non-dimensional form) after mathematical rearrangements are expressed as follows [36,37]:

than the circular cylinder. Mehrizi et al. [48] carried out lattice Boltzmann simulation of natural convection in an elliptical-triangular annulus for different Rayleigh numbers (Ra = 103-105). Nusselt number value decreases with increase in aspect ratio of the elliptical cylinder (Er). Recently, Patel and Chhabra [49] numerically studied heat transfer from the heated elliptical cylinder for Bingham fluids for a wide range of aspect ratios of the cylinder (Er = 0.1–10). It is found that the influence of aspect ratio of the elliptical block on heat transfer rate is found to be significant for Er < 1. More recently, Sivasankaran et al. [50] presented detailed discussions on the effect of driven-wall on heat transfer characteristics due to mixed convection in lid-driven cavity with sinusoidal heating. They reported more significant effect on heat transfer at low Ri and -Ulid. Moreover, the combined effect of mixed convection in the lid-driven cavity under the effect of magnetic field containing different blockages has been reported recently [51–54]. Therefore, in contrary to the all-encompassing studies documented on the mixed convection in lid-driven cavities with or without adiabatic/isothermal body/bodies of varying shapes (circular, square, rectangular, triangular, etc.), the hydrodynamic and thermal analysis of lid-driven rectangular enclosure containing elliptical block in its infancy. Most of the studies revealing the heat transfer characteristics from the elliptical cylinder are, channel flow studies [38–49]. No prior study (as much known to Author) is available, presenting detailed flow and thermal characteristics of the rectangular cavity containing elliptical block for the range of governing (Re, Pr, Gr, etc.) as well geometric parameters (Er, the direction of moving wall, etc.). The objective of the present research is to fulfill the lapse found in the literature. Therefore, the present work is conducted to explore the influence of varying parametric conditions, such as, direction of moving lid (+Uw or -Uw), aspect ratio of elliptical block (Er = 0.5, 1, 2), Reynolds number (1 ≤ Re ≤ 5000), Prandtl number (1 ≤ Pr ≤ 100) and Grashof number (0 ≤ Gr ≤ 105), thereby, resultant Richardson number range of 0 ≤ Ri ≤ 105 are illustrated for mixed convection in rectangular cavity (with Newtonian fluid) containing heated elliptical block. The problem definition, description of geometry along with mathematical formulation for this work is presented in succeeding section.

∂x u x + ∂ y u y = 0

(1)

(u x ∂x u x + u y ∂x u x ) = −∂x p + Re−1 [∂2 x u x + ∂2 y u x ]

(2)

Gr (u x ∂x u y + u y ∂y u y ) = −∂y p + Re−1 [∂2 x u y + ∂2 y u y] + ⎛ 2 ⎞ θ ⎝ Re ⎠

(3)

(u x ∂x θ + u y ∂y θ) = ∂ ; ∂x

(Re×Pr)−1 [∂2

xθ

+

∂2

y θ]

(4)

∂ , ∂y

∂y = (x, y), (ux, uy), p, ρ, ν and θ represent space where, ∂x = gradients, Cartesian directions, velocity components, pressure, density, kinematic viscosity, and temperature, respectively. Governing field equations in dimensionless form are obtained by following dimensionless parameters. y x ;y= H H uy u ux = U x ; u y = U ; lid lid

x=

θ=

p=

p 2 ; ρUlid

T − Tc Tw − Tc

The variables with cap (¯ ) are dimensional one. The description of the physical realistic boundary conditions for present problem is shown in Table 1. All walls are no-slip. The elliptical block (θw) and right vertical wall (θc) are specified with constant wall temperature, while the top and bottom walls are assumed to be thermally insulated. Different dimensionless groups appearing in above equations are represented as follows:

2. Problem definition and governing equations The schematic of the problem under consideration along with the definition of the aspect ratio of an elliptical block is shown in Fig. 1 (I and II). A 2D, steady, laminar, mixed convection in an incompressible, Newtonian fluid in a rectangular enclosure (AR = L/H = 2, where L and H are cross-sectional length and height of the enclosure). The left vertical wall of the enclosure is moving with uniform vertical velocity

• Reynolds number (Re) = • Prandtl number (Pr) =

HUw ρ μ Cp μ ν = α k

Fig. 1. (I) Schematic representation of problem under consideration, (II) Aspect ratio of elliptical block (Er). 102



Table 1 Description of hydrodynamic and thermal boundary conditions. Zone

Dimensions

Nu =

−

∂θ dθ ∂ns

(6)

Thermal

3. Numerical methodology The present research used the finite volume method (FVM) for discretization of the governing partial differential equations to yield a set of linear algebraic equations. The field governing equations are solved by using the ANSYS FLUENT (version 15) commercial CFD solver. The ‘triangular’ mesh of non-uniform grid spacing is generated with denser crowding of the nodes around the block. The mesh structure generated for the rectangular enclosure with elliptical block (Er = 0.5) is depicted in Fig. 2. The semi-implicit method for the pressure linked equations (SIMPLE) scheme is used to solve the pressure-velocity decoupling. The developed system of algebraic equations the by using the Gauss–Siedel (G–S) point-by-point iterative method. Further, the upwind differencing method a for the formulation of the convection contribution of the coefficients to finite-volume equations. The momentum and energy terms the by means of third order accurate QUICK (Quadratic Upwind Interpolation for Convection Kinetics). It uses 3-point point upstream weighted quadratic interpolation for calculation of cell face values. The iterative process is terminated for convergence criteria of 10−6 based on the normalized residuals for each field equation. For the general variable of ζ, the generalized convergence equation at nodem’ is expressed as follows [36,37]:

u x = 0; uy = +1 or −1

T = Tc = 0

Bottom wall Top wall

0 ≤ x ≤ 2; y=0 0 ≤ x ≤ 2; y=1

u x = 0; uy = 0

∂T

Elliptical block

Center of block: x=y=0.5 Er=a/b=0.5, 1 and 2

u x = 0; uy = 0

T = Tw = 1

• Grashof number (Gr) = • Richardson number (Ri) =

∂y

=0

gβ Δ θH3 (μ / ρ)2 Gr Re2

=

gβ Δ TH 2 Uw

The mass, momentum and energy equations (Eqs. (1)–(4)) in conjunction with boundary conditions (Table 1) yield the primitive variable fields (ux, uy, P, and θ). These variables are further processed to obtain the parameter of scientific or engineering importance, such as streamlines, vorticity, nusselt number, etc. These parameters as expressed below;

• Stream function (ψ): It is estimated from velocity field as, •

2π

Hydrodynamics x=0; 0 ≤ y ≤ 1 x=1; 0 ≤ y ≤ 1

∫ ux dy

∫0

Boundary conditions

Left wall Right wall

ψ=

1 2π

Γ mζ m = (5)

∑nm Γ nmζ nm + λ

(8)

Further, the scaled residual can be given as below. Nusselt number: The average Nusselt number on the surface of the heated elliptical block is estimated by the following expression [47]:

RΓ =

∑Cells ∑nλ Γ nbζ nb + λ − Γ mζ m

where, m,n,

∑Cells Γ mζ m Γ nb

and b are the node, neighboring node, influence

Fig. 2. Schematic of the grid in the domain (bottom) and in the region close to the elliptical cylinder (top) with Er = 0.5. 103

(9)



coefficients for the neighboring nodes and constant part, respectively. In numerical studies, proper choice of grid and domain sizes is a very important step in the accuracy of results is dependent on these parameters. Therefore, the accurate selection of these parameters is very important. Subsequent section examines the grid independence test as well as validation of adopted numerical simulation procedure.

Table 2 Comparison of the average Nusselt number (Nu) at the top wall with previous studies [55–57] at Pr = 0.71 and Gr = 102. Source

Re

Present Biswas et al. [55]

3.1. Grid independence test and validation The authenticity, as well as, precision of the numerical procedure is naturally dependent upon a judicious selection of few parameters such as sizes of the computational domain and computational grid. In this work, the size of the computational domain is itself defined by the geometry of problem. The grid independence study is conducted by varying number of nodes over the elliptical block. Therefore, a thorough grid independence study has been carried out by using four different numbers of nodes over the elliptical block (G1, G2, G3, and G4). The influence of different node sizes on average Nusselt number values for the elliptical block with an aspect ratio of Er=2 for Re = 1, 1000; Pr = 1, 100 and Gr = 0, 105 is shown in Fig. 3. It is observed that deviation between G1 to G5 is 0.068% (Gr = 0,Pr = 1,Re = 1), 2.2% (Gr = 0,Pr = 1,Re = 1000), 0.22% (Gr = 0,Pr = 100,Re = 1), 0.22% (Gr = 0,Pr = 100,Re = 1000), 0.18% (Gr = 105,Pr = 1,Re = 1), 0.09% (Gr = 105,Pr = 1,Re = 1000), 1.62% (Gr = 105,Pr = 100,Re = 1) and 4.2% (Gr = 105,Pr = 100,Re = 1000). The average difference between G1 to G5 is estimated to be 1.15%. It can also be clearly seen that the effect of the change in the number of nodes after G2 is insignificant with an enormous increase in the computational time as well as space. Changing grid size after G2, the computational time and number of iterations increase by almost two-three times. Thus, keeping in mind the factors such as, accuracy, computational time and space, grid size G3 (Nodes: 150) is found to be optimum and have been used for simulation in present work.

Waheed [56] Roy et al. [57]

1

500

1000

1.005 1.000 (0.5%) 1.000 (0.5%) 0.998 (0.7%)

4.532 4.522 (0.2%) 4.537 (0.1%) 4.509 (0.5%)

6.445 6.431 (0.2%) 6.484 (0.6%) 6.345 (1.5%)

(Note: Bracketed values represent % relative deviation from present simulation values). Table 3 Convergence history indicating number of Iterations required for achieving convergence (10−6) for different Re, Pr Gr and same initial guess. Er

Re

Pr = 1

Pr = 100

Gr = 0

Gr = 10

5

Gr = 0

Gr = 105

0.5

1 1000

1547 3457

5654 8975

1676 4005

5987 9245

1

1 1000

3312 9824

8912 10125

4167 10731

7452 11457

2

1 1000

3568 10067

9012 10875

6142 12003

7945 13567

Convergence time: One iteration ≈1 s.

factors, the present results reported herein are believed to be reliable and accurate well within 2%. It is well acknowledged that the purpose of the numerical studies is to mimic the real phenomenon with acceptable level of accuracy. In order to achieve this, the comparison of numerical results with experimental observation is mandatory. As the present problem statement has not been performed experimentally till a date (as much known to the author), the validation of present numerical strategy is performed for the limiting problem of considered one, i.e., experimental investigation of mixed convection in top lid-driven arc shaped cavity as given by Chen and Cheng [58]. The top-moving wall of cavity is exposed to ambient thermal condition; while arc is heated. The validation is performed for Re = 500, Gr = 106 and Pr = 0.71. Fig. 4 shows the comparison of streamline patterns given in Chen and Cheng [58]. The careful examination of Fig. 4 (a, b) reveals that the present results

3.2. Validation of simulation procedure In this work, validation is based on the model of [55–57]. For validation purpose (based on average Nu of the heated wall), the simple problem of top heated and the bottom cooled cavity is simulated for Re = 1, 500 and 1000, Gr = 102 and Pr = 0.71. Table 2 shows the comparison of Nu values with selected studies [55–57]. The analysis of Table 3 reveals the excellent similitude of present results with selected previous studies. The minimum and maximum deviations between present and literature values are found to be 0.1% and 1.5%, respectively. The average relative error is observed to be 0.5%. Such deviations in results are quite common in modeling studies, which are resultant due to the accuracy of the numerical method, modeling error, discretization (or linearization) error, numerical errors (due to iteration, round up and programming) etc. Therefore, considering all these

Fig. 3. Grid independence results for Re = 1,1000; Pr = 1100 at Gr = 0 and Er=2. 104



which complicates the fluid flow and heat transfer phenomenon arising due to moving lid as well as buoyancy-driven flow. Having acquired the assurance in the present numerical procedure, the subsequent section presents new results obtained to elucidate the influence of flow governing parameters (i.e., Prandtl number, Reynolds number, Grashof number, aspect ratio of elliptical block, direction of moving lid, etc.) on the detailed mixed convection flow phenomenon in lid-driven enclosure containing centrally placed hexagonal block in terms of the streamline and isotherm patterns, average Nusselt numbers and Colburn factor. 4. Results and discussion This research is intended to explore the mixed convection heat transfer in a rectangular enclosure with one of the vertical wall moving (with either of +Uw or -Uw). In particular, the influence of the presence of elliptical block for either of three aspect ratio (Er = 0.5, 1 and 2) with constant temperature thermal condition on the convection characteristics have the for pertinent flow governing parameters, such as Reynolds, Grashof, and Prandtl numbers. The Reynolds number is varied as Re = 1, 100, 1000 and 5000, Grashof number as Gr = 0, 102, 103, 104 and 105 thus covering a wide range of Richardson number (0 ≤ Ri ≤ 105) for Pr = 1 and 100. Extensive results encompassing the influences of the Re, Pr, and Gr on the local and global heat transfer characteristics (such as streamline and isotherm contours, the as well as average Nusselt numbers, normalized Nusselt number, etc.) are presented and discussed herein the ensuing section.

Fig. 4. Comparison of streamline profiles between (a) Chen and Cheng [58] and (b) present simulation results for the problem of top lid-driven arc shaped cavity.

matches well with experimental observation [58]. The formation of eddy, slightly off-center in the cavity is visible. Therefore, the comparison shown in Fig. 4 boosts Authors' confidence in the accuracy of the numerical solver, therefore the present numerical methodology and grid structure can be used to obtain new results. In order to give an idea about the convergence time of numerical simulations, Table 4 showing number of iterations to attain convergence for extreme range of dimensionless parameters considered herein with the same initial guess is also included. It can be observed that higher computational time is required to attain convergence for higher Re and Gr values, which is due to mixed convection dominance;

4.1. Streamline and isotherm contours In lid-driven enclosure (without blockage), the fluid movement is achieved by moving lid in such a way that the fluid in the vicinity of moving wall gets pulled along with the moving wall (due to the shear force exerted by the wall on fluid). It initiates the fluid movement in the

Table 4 Nusselt number averaged over elliptical block for range of considered parameters. AR

Re

Lid direction: Negative Y 0.5 1 100 1000 5000

Pr = 1

Pr = 100

Gr = 0

Gr = 103

Gr = 105

Gr = 0

Gr = 103

Gr = 105

0.069782 0.070465 0.075446 0.108196

0.071988 0.07057 0.073829 0.113681

0.25581473 0.25461346 0.27618942 0.34114975

0.070472 0.185195 0.280551 0.335326

0.256401 0.187046 0.30129 0.355546

0.834383 0.802135 1.087656 1.328091

1

1 100 1000 5000

0.065972 0.066717 0.07238 0.101632

0.067999 0.066639 0.073346 0.103585

0.22131559 0.22018835 0.24172673 0.31733763

0.066715 0.179673 0.219482 0.468758

0.221848 0.181427 0.330032 0.405918

0.672778 0.657891 0.950562 1.236041

2

1 100 1000 5000

0.070386 0.071633 0.117688 0.31854

0.072541 0.070833 0.111175 0.319002

0.24565955 0.24480309 0.25958085 0.39381465

0.071617 0.208759 0.517276 1.539919

0.246265 0.193786 0.460505 1.542038

0.771212 0.746287 1.011527 2.147023

Lid direction: Positive Y 0.5 1 100 1000 5000

0.069782 0.070466 0.0755 0.108086

0.072024 0.074219 0.080431 0.104289

0.25581843 0.25528571 0.26620546 0.23060927

0.070472 0.185304 0.280263 0.336268

0.256671 0.300341 0.418174 0.429943

0.834402 0.791006 1.003781 1.059566

1

1 100 1000 5000

0.065972 0.066717 0.072513 0.101541

0.068032 0.070077 0.074715 0.101456

0.22131889 0.22090076 0.22981969 0.23003446

0.066715 0.179697 0.205478 0.470443

0.222148 0.273714 0.380636 0.512055

0.672812 0.646584 0.845405 1.065141

2

1 100 1000 5000

0.070386 0.071636 0.117689 0.318568

0.072594 0.075892 0.124037 0.318095

0.24566339 0.24497981 0.27522492 0.25083232

0.071617 0.208824 0.5172 1.542067

0.246256 0.305558 0.558045 1.540011

0.771241 0.731943 1.001545 1.442695

105



remains occupied with nearly stagnant fluid due to the dominant shear force of wall. The augmentation in Gr causes natural convection to be significant and thereby, increasing the strength of buoyancy-driven flow in the enclosure. For higher Gr (Gr ≥ 103), due to the enhancement in the buoyancy-driven flow from the block, the flow circulation nearly occupies the enclosure (due to buoyancy driven convection between heated block and cold-vertical walls). At Re = 1, Gr = 103 (Ri = 103), the convection is mainly dominated due to buoyancy-driven flow. The influence of moving lid is found to be insignificant in this case. The fluid flow then redirects from moving lid (left wall) to right (ambient) wall. Flow circulation takes place from heated block towards the ambient wall. The fluid close to block gets heated and thereby, becoming lighter. These lighter fluid elements then approach the top horizontal wall and later towards the left ambient wall with a gradual drop in fluid temperature (which increases the density of elements). These heavier elements start moving towards the heated block, thus completing fluid circulation in a clockwise manner. As the flow circulation is quite limited to the walls and block, the region in the middle of the block and the right wall becomes quasi-motionless. This region is also called as convection cell or eddy. Due to increase in Gr, the size of eddy elongates. This phenomenon is visible only for low Re (Re ≤ 100). For higher Re, as the forced convection becoming dominant than natural convection, such flow behavior is not observed and fluid flow remains confined to the left horizontal half of enclosure. Fig. 5(b) depicts the streamline variations for lid having + Uw for otherwise identical conditions. It is clearly evident from Fig. 5(b) that the flow circulation gets significantly influenced due to change in the direction of moving wall. The flow movement is observed beyond the block also, indicating higher fluid circulation than the enclosure of wall direction, –Uw. Therefore, the moving wall in the direction positive vertical direction found to enhance the flow circulation than wall moving along gravity field. Similar flow characteristics are observed for Er = 1 (Fig. 6a–b). Very minor discrepancies are noted in streamlines behavior with that of Er = 0.5 for –Uw (Fig. 6(a)). For enclosure wall

enclosure along the of moving wall. Therefore, the circulation of fluid in the enclosure is experienced in the direction of moving wall. The direction of moving wall shapes the orientation/circulation of fluid in an enclosure. When the wall is moving along negative –y direction (–Uw), the circulation of fluid is counter-clockwise and co-current for wall movement along positive –x direction (+Uw). On the other hand, the buoyancy-driven flow from heated block initiates as the fluid elements near the heated block gain energy becoming lighter and approaches the ambient wall with a gradual drop in temperature and with augmentation in density. Comparatively, the weighty fluid particles, further, get attracted towards the heated wall, thus completing the fluid circulation in the clockwise or circular manner [36,37]. Figs. 5(a)–10(b) depict the variation of the streamlines (Figs. 5(a)–7(b)) as well as the isotherms (Figs. 8(a)–10(b)) patterns for range of Grashof number (Gr = I-0,II-103, III-105), Reynolds number (Re = a-1, b-100, c-1000, d-5000). Figures are arranged in the increasing order of Er, i.e., Er = 0.5 (Figs. 5 and 8), Er = 1 (Figs. 6 and 9) and Er = 2 (Figs. 7 and 10), where ‘a’ and ‘b’ represents the direction (or velocity) of moving lid as –Uw and + Uw, respectively. Due to space constraints, the contour figures are shown only for Pr = 1. The careful observation of Fig. 5(a) illustrates the effect of Grashof number and Reynolds number on flow field for Er = 0.5 and –Uw. At Re = 1, due to lower inertial forces exerted by the lid on fluid flow, the fluid movement remains significant from moving wall only up to elliptical block. For pure forced convection case (Gr = 0), the flow circulation remains more or less limited to the left horizontal half of enclosure (Fig. 5(a)) with the formation of a central eddy. As the enclosure shape is rectangular, due to a low force exerted by moving wall over fluid, flow movement confines to the left horizontal half of enclosure. Only the size of eddy (created due to the clockwise circulation of fluid) gets affected with Re at Gr = 0. In particular, eddy size decrease with Re (increase in Re, makes rapid fluid circulation). The effect of Gr is seen to be significant only up to Re ≤ 100. For higher Re (Re ≥ 1000), flow remains confined towards the moving wall and right horizontal half of enclosure

Fig. 5a. Streamline patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 0.5 with -Uw. 106



Fig. 5b. Streamline patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 0.5 with +Uw.

Fig. 6a. Streamline patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 1 with -Uw.

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Fig. 6b. Streamline patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 1 with +Uw.

Fig. 7a. Streamline patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 2 with -Uw.

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Fig. 7b. Streamline patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 2 with +Uw.

Fig. 8a. Isotherm patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 0.5 with -Uw.

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Fig. 8b. Isotherm patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 0.5 with +Uw.

Fig. 9a. Isotherm patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 1 with -Uw.

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Fig. 9b. Isotherm patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 1 with +Uw.

Fig. 10a. Isotherm patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 2 with -Uw.

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Fig. 10b. Isotherm patterns for range of Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr = 1) at Er = 2 with +Uw.

Fig. 11. Variation of Nusselt number values over the elliptical block ( A → B → A ) for different Re, Pr, Gr and (a) Er = 0.5, (b) Er = 2 as well as Uw.

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seen in right horizontal half of cavity. Also, due to higher flow circulation in the cavity, the temperature gradients along the block increases (as boundary layer thickness decreases). The significant impact of the direction of moving wall is seen on the isotherms orientation at Re = 5000. For wall moving in –Uw, the isotherm gets slightly elongated towards the bottom wall and the reverse is observed for wall moving with +Uw. Somewhat same temperature profiles are observed for Er = 0.5 (Fig. 9(a) and (b)) and 1 (Fig. 10(a) and (b)). Only minor influence is observed on isotherm clustering around the block. Therefore, Er can be considered as limiting case of a circular cylinder. But the remarkable difference is observed in isotherms patterns for Er = 2 (Fig. 10(a) and (b)). As per earlier discussion, elliptical block with Er = 2 enhances the fluid circulation initiated by moving the vertical wall. Therefore, the higher impact of convection heat transfer is attained than other two cases (Fig. 10(a) and (b)). The major difference in temperature profiles is observed for higher Re (≥1000) and lower Gr (≤103). Isotherms are observed to be more inclined and stretched towards the top wall (moving wall direction -y) or bottom wall (moving wall direction + y). Fluid circulation in the enclosure is evident for Gr = 0 and Re = 5000 for an elliptical block of Er = 2, which is not observed in Er = 0.5 and 1. For Re < 1000 and Gr = 105, the isotherm structures remains more or less similar to Er = 0.5 and 1. Therefore, it can be concluded that the flow and heat transfer behavior in the enclosure is predominantly defined by the direction of moving wall followed by Er, Gr, and Re for otherwise similar conditions. After presenting and discussing detailed flow and temperature characteristics in the cavity, it is mandatory to illustrate the heat transfer characteristics delineated by using the Nusselt number and Colburn factor.

moving with Uw, streamline patterns are observed to be more uniformly distributed for higher Gr values (due to natural convection dominance and thereby movement of fluid from block towards the right wall), i.e., Gr ≥ 103. The increase in aspect ratio to Er = 2 (Fig. 7(a) and (b)) facilities the fluid circulation as acts as a streamlined body (as the cylinder elongated horizontally). The size of eddies formed is found to be highest than other chosen Er, implying higher flow circulation in the enclosure. For Re = 5000, the flow splits into three zones (one near moving wall, second above block and third along the ambient wall). With an increase in Gr (due to increasing mixed convection effect), the size of formed eddies elongates. Therefore, the aspect ratio of the block as well as moving wall velocity is found to have a dominant effect on controlling overall structure of fluid flow in the enclosure than governing dimensionless numbers. The temperature distribution in an enclosure is depicted in Figs. 8(a)–10(b) for mentioned parametric conditions. The analysis of Figs. 8(a)–10(b) reveals that for Gr ≤ 103, the isotherm profiles are found to be parallel to the ambient wall (right wall) for all considered Re. This is due to conduction dominant heat transfer region resulting in the weak buoyancy-driven flow. Also, the formation of an oval-shaped hot fluid region surrounding to block is sighted (due to higher viscous force than inertial force caused by moving wall) for Re ≤ 1000 and Gr ≤ 103. For Re = 5000, and lower Gr values (Gr = 0, 103), isotherms looks somewhat pulled in the direction of moving lid. All in all, the heat transfer is mainly due to conduction as visible due to the parallel lines of isotherms along the ambient wall. But the remarkable difference is observed on temperature profiles at Gr = 105 due to increase in the strength of buoyancy flow. Crowding of isotherms from block towards the ambient wall is observed. At Re = 1. At higher Re, due to higher flow circulation from moving wall, the isotherm crowding can only be

Fig. 12. Variation of Normalized Nusselt number (NuN) for range of Re, Pr, Gr and Er = 0.5 and 2.

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cavity is observed due to the significant effect created by buoyancydriven flow from the block for Gr ≥ 103. Also, for Re = 5000, the flow circulation limits towards the moving wall of the rectangular enclosure. The heat transfer remains conduction dominant for all considered Re at Gr = 0 for Er = 0.5 and 1. The heat transfer rate due to convection augments with Re, Pr, and Gr for all other considered conditions. Higher Nu values are estimated for enclosure moving wall at + y than –y. Elliptical block with Er = 2 found to accelerate the fluid flow and thereby heat transfer in the enclosure. In order to provide the comparison of Nu between different Er, normalized Nusselt number (NuN) is evaluated. It is observed that higher heat transfer rate can the by using an elliptical block rather than a pure circular cylinder.

4.2. Heat transfer characteristics The heat transfer characteristics in the enclosure with the heated elliptical block are explained by means of local (Nu) as well as averaged Nusselt number or heat flux (Eq. (6)) calculated over the elliptical block for parametric conditions considered herein. Fig. 11 shows the variation of localized values of the Nusselt number along with the elliptical block (Er = 0.5 and 2) for the considered range of pertaining parameters. The variation of Nu is over block is indicated over the zones as, A → B → A . Top two rows represent results for uy = +Uw; while bottom two shows for uy = -Uw for different Grashof numbers. It can be observed that regardless of the direction of moving wall, higher temperature gradients are obtained for an elliptical block with Er = 2. Further, higher peaks of Nu are obtained near B part of the elliptical block (B part is facing moving wall). The gathering of higher clusters of temperature gradients along region B of the block of aspect ratio 2 is due to its capability to accelerate fluid flow due to moving wall as well as buoyancy flow from the block. Also, higher values of Nu are obtained for higher intensity buoyancy-driven flow, therefore, increasing the flow circulation. Similar influence with remaining parameters (Reynolds and Prandtl numbers), as discussed in the previous section, is observed on Nu patterns. Nu shows proportional dependence with Re and Pr for otherwise similar conditions. Therefore, higher heat transfer rate can be achieved for the cavity with uy = +Uw and Er = 2. Further, estimated local Nusselt number values are further averaged over the elliptical block to yield average Nusselt number (Nu ) for the range of governing parameters as well Er and direction of moving wall are tabulated in Table 4. It can be seen that Nu increases in Gr and Re for other parameters considered herein. Highest convective heat transfer rate can be achieved for the enclosure with elliptical block of aspect ratio Er = 2 followed by Er = 0.5 and finally by circular block (Er = 1). As per earlier discussions, elliptical block with Er = 0.5 can be seen as limiting case for Er = 1 (due to nearly similar streamlines and isotherm profiles). Therefore, the enclosure with Er = 0.5 can be used to achieve higher heat transfer rate by convection than the circular one, without much altering fluid flow in the enclosure. Otherwise, the elliptical block with Er = 2 should be the best choice for attaining high heat transfer due to convection. In order to explain the advantages of using elliptical block over circular one, normalized Nusselt numbers are evaluated for similar conditions by using the following expression:

NuN =

Nu Er ≠ 1 Nu Er = 1

Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx. doi.org/10.1016/j.ijthermalsci.2018.04.010. References [1] T. Ota, H. Nishiyama, Y. Taoka, Heat transfer and flow around an elliptic cylinder, Int, J. Heat and Mass Transfer 27 (l) (1984) 1771–1779. [2] T. Ota, S. Aiba, T. Tsuruta, M. Kaga, Forced convection heat transfer from an elliptical cylinder with axis ratio 1:3, Bulletin of the JSME 26 (212) (1983) 262–267. [3] M.K. Moallemi, K.S. Jang, Prandtl number effects on laminar mixed convection heat transfer in a lid-driven cavity, Int J Heat Mass Tran 35 (8) (1992) 1881–1892. [4] S.U.S. Choi, D.A. Siginer, H.P. Wang (Eds.), Enhancing thermal conductivity of fluids with nanoparticles development and applications of non-Newtonian flows, vol 231, The American Society of Mechanical Engineers, New York, 1995, pp. 99–105 (66). [5] H. Liang, S. You, H. Zhang, Comparison of three optical models and analysis of geometric parameters for parabolic trough solar collectors, Energy 96 (2016) 37–47. [6] M. Bortolato, S. Dugaria, D.D. Col, Experimental study of a parabolic trough solar collector with flat bar-and-plate absorber during direct steam generation, Energy 116 (2016) 1039–1050. [7] G.H.R. Kefayati, Simulation of double diffusive MHD (magnetohydrodynamic) natural convection and entropy generation in an open cavity filled with power-law fluids in the presence of Soret and Dufour effects (Part I: study of fluid flow, heat and mass transfer), Energy 107 (2016) 889–916. [8] K.M. Gangawane, R.P. Bharti, S. Kumar, Lattice Boltzmann analysis of natural convection in a partially heated open ended enclosure for different fluids, J. Taiwan Insti of Chem Engi 49 (2015) 27–39. [9] K.M. Gangawane, R.P. Bharti, S. Kumar, Lattice Boltzmann analysis of effect of heating location and Rayleigh number on natural convection in partially heated open ended cavity, Kor J Chem Eng 32 (8) (2015) 1498–1514. [10] K.M. Gangawane, R.P. Bharti, S. Kumar, Two-dimensional lattice Boltzmann simulation of natural convection in differentially heated square cavity: effect of Prandtl and Rayleigh Numbers, Can J Chem Eng 93 (2015) 766–780. [11] K.M. Gangawane, R.P. Bharti, S. Kumar, Effects of heating location and size on natural convection in partially heated open ended enclosure by using lattice Boltzmann method, Heat Tran Eng 36 (7) (2015) 507–522. [12] J.M. House, C. Beckermann, T.F. Smith, Effect of a centered conducting body on natural convection heat transfer in an enclosure, Numer Heat Tran 18 (1990) 213–225. [13] R. Gutt, T. Gro, On the lid-driven problem in a porous cavity. A theoretical and numerical approach, Appl Math Comput 266 (2015) 1070–1082. [14] A.M. Rashad, M.A. Ismael, A.J. Chamkha, M.A. Mansour, MHD mixed convection of localized heat source/sink in a nanofluid-filled lid-driven square cavity with partial slip, J Taiwan Inst Chem Eng 68 (2016) 173–186 https://doi.org/10.1016/j.jtice. 2016.08.033. [15] Z.-Y. Li, Z. Huang, W.Q. Tao, Three-dimensional numerical study on fully-developed mixed laminar convection in parabolic trough solar receiver tube, Energy 113 (2016) 1288–1303. [16] K. Ghachem, L. Kolsi, C. Maatki, A. Alghamdi, H.F. Oztop, M.N. Borjini, et al., Numerical simulation of three-dimensional double diffusive convection in a liddriven cavity, Int J Therm Sci 110 (2016) 241–250. [17] M. Roy, S. Roy, T. Basak, Role of various moving walls on energy transfer rates via heat flow visualization during mixed convection in square cavities, Energy (2014) 1–22 xxx https://doi.org/10.1016/j.energy.2014.11.059. [18] G.H.R. Kefayati, R.R. Huilgol, Lattice Boltzmann Method for simulation of mixed convection of a Bingham fluid in a lid-driven cavity, Int J Heat Mass Tran 103 (2016) 725–743. [19] U. Ghia, K.N. Ghia, C.T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J Comput Phys 48 (1982) 387–411. [20] D. Mansutti, G. Graziani, R. Piva, A discrete vector potential model for unsteady incompressible viscous flows, J Comput Phys 92 (1991) 161–184. [21] P.N. Shankar, M.D. Deshpande, Fluid mechanics in the driven cavity, Annu Rev Fluid Mech 136 (2000) 93–136.

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Fig. 12 presents the variation of NuN for different Re, Pr, Gr and Er = 0.5, 1 as well as the direction of moving wall (+y and -y). For NuN > 1, higher heat transfer rate is achieved and when NuN < 1, heat transfer deterioration is observed. For both elliptical blocks, higher values of NuN (> 1) are obtained. Therefore, using an elliptical block rather than circular one can enhance the heat transfer considerably in devices such as heat exchangers. 5. Concluding remarks Numerical investigation of two-dimensional, steady, laminar, mixed convection in rectangular cavity with built-in elliptical heated block is performed for broad range of pertinent flow parameters such as, Reynolds number (1 ≤ Re ≤ 1000), Prandtl number (1 ≤ Pr ≤ 100) and Grashof number (0 ≤ Gr ≤ 105) for different aspect ratios of elliptical block (Er = 0.5,1 and 2) as well as the moving wall direction (+Uw or -Uw). The numerical validation of present work with previous works shows excellent agreement with previous similar studies. In the rectangular enclosure with a heated elliptical block, the intensity of fluid circulation is found to be more with left wall velocity of +Uy due to the higher impact of shear force exerted by moving the wall on the fluid in its vicinity. Formation of an eddy in right horizontal half of 114



[40] R.S. Matos, J.V.C. Vargas, T.A. Laursen, F.E.M. Saboya, Mixed convection from a straight isothermal tube of elliptic cross-section, Int J Heat Mass Tran 44 (2001) 3953–3961. [41] W. Chen, H. Li, M. Gao, Z. Liu, F. Sun, The pressure melting of ice around a horizontal elliptical cylinder, Heat Mass Tran 42 (2005) 138–143. [42] Ji-An Meng, Xin-Gang Liang, Ze-Jing Chen, Zhi-Xin Li, Experimental study on convective heat transfer in alternating elliptical axis tubes, Exp Therm Fluid Sci 29 (2005) 457–465. [43] Z. Faruquee, D.S.-K. Ting, A. Fartaj, R.M. Barron, R. Carriveau, The effects of axis ratio on laminar fluid flow around an elliptical cylinder, Int J Heat Fluid Flow 28 (2007) 1178–1189. [44] R.P. Bharti, P. Sivakumar, R.P. Chhabra, Forced convection heat transfer from an elliptical cylinder to power-law fluids, Int J Heat Mass Tran 51 (2008) 1838–1853. [45] C.-Y. Cheng, Natural convection boundary layer flow of fluid with temperaturedependent viscosity from a horizontal elliptical cylinder with constant surface heat flux, Appl Math Comput 217 (2010) 83–91. [46] A. Chandra, R.P. Chhabra, Flow over and forced convection heat transfer in Newtonian fluids from a semi-circular cylinder, Int J Heat Mass Tran 54 (2011) 225–241. [47] A. Nejat, E. Mirzakhalilia, A. Aliakbari, M.S.F. Niasar, K. Vahidkhah, NonNewtonian power-law fluid flow and heat transfer computation across a pair of confined elliptical cylinders in the line array, J Non-Newtonian Fluid Mech 171–172 (2012) 67–82. [48] A.A. Mehrizi, K. Sedighi, M. Farhadi, M. Sheikholeslami, Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, Int Commun Heat Mass Tran 48 (2013) 164–177. [49] S.A. Patel, R.P. Chhabra, Heat transfer in Bingham plastic fluids from a heated elliptical cylinder, Int J Heat Mass Tran 73 (2014) 671–692. [50] S. Sivasankaran, H.T. Cheong, M. Bhuvaneswari, P. Ganesan, Effect of moving wall direction on mixed convection in an inclined lid-driven square cavity with sinusoidal heating, Numer Heat Tran, Part A: Applications 69 (6) (2016) 630–642. [51] D. Chatterjee, P. Halder, Magnetoconvective transport in a lid-driven square enclosure with two rotating circular cylinders, Heat Tran Eng 37 (2) (2016) 198–209. [52] S. Bansal, D. Chatterjee, Magneto-convective transport of nanofluid in a vertical liddriven cavity including a heat conducting rotating circular cylinder, Numer Heat Tran, Part A: Applications 68 (2) (2015) 411–431. [53] S.K. Gupta, D. Chatterjee, B. Mondal, Investigation of Mixed Convection in a ventilated cavity in the presence of a heat conducting circular cylinder, Numer Heat Tran, Part A: Applications 67 (1) (2015) 52–74. [54] D. Chatterjee, S.K. Gupta, Hydromagnetic mixed convective transport in a nonisothermally heated lid-driven square enclosure including a heat-conducting circular cylinder, Ind Eng Chem Res 53 (51) (2014) 19775–19787. [55] N. Biswas, N.K. Manna, P.S. Mahapatra, Enhanced thermal energy transport using adiabatic block inside lid-driven cavity, Int J Heat Mass Tran 100 (2016) 407–427. [56] M.A. Waheed, Mixed convective heat transfer in rectangular enclosures driven by a continuously moving horizontal plate, Int J Heat Mass Tran 52 (2009) 5055–5063. [57] M. Roy, S. Roy, T. Basak, Role of various moving walls on energy transfer rates via heat flow visualization during mixed convection in square cavities, Energy 82 (2015) 1–22. [58] Chin-Lung Chen, Chin-Hsiang Cheng, Experimental and numerical study of mixed convection and flow pattern in a lid-driven arc-shape cavity, Heat Mass Tran 41 (2004) 58–66.

[22] M. Jami, A. Mezrhab, M. Bouzidi, P. Lallemand, Lattice Boltzmann method applied to the laminar natural convection in an enclosure with heat cylinder conducting body, Int J Therm Sci 46 (2007) 38–47. [23] D. Angeli, P. Levoni, G.S. Barozzi, Numerical predictions for stable buoyant regimes within a square cavity containing a heated horizontal cylinder, Int J Heat Mass Tran 51 (2008) 553–565. [24] G. Cesini, M. Paroncini, G. Cortella, M. Manzan, Natural convection from a horizontal cylinder in a rectangular cavity, Int J Heat Mass Tran 42 (1999) 1801–1811. [25] S.F. Dong, Y.T. Li, Conjugate of natural convection and conduction in a complicated enclosure, Int J Heat Mass Tran 24 (2004) 2233–2239. [26] J.M. House, C. Beckermann, T.F. Smith, Effect of a centered conducting body on natural convection heat transfer in an enclosure, Numer Heat Tran 18 (1990) 213–225. [27] J.Y. Oh, M.Y. Ha, Y.S. Kim, Numerical study of heat transfer and flow of natural convection in an enclosure with a heat-generating conducting body, Numer Heat Tran 31 (1997) 289–304. [28] M.Y. Ha, M.J. Jung, Y.S. Kim, A numerical study on transient heat transfer and fluid flow of natural convection in an enclosure with a heat-generating conducting body, Numer Heat Tran 35 (1999) 415–434. [29] M.Y. Ha, M.J. Jung, A numerical study on three-dimensional conjugate heat transfer of natural convection and conduction in a differentially heated cubic enclosure with a heat-generating cubic conducting body, Int J Heat Mass Tran 43 (2000) 4229–4248. [30] A. Mezrhab, H. Bouali, C. Abid, Modelling of combined radiative and convective heat transfer in an enclosure with a heat-generating conducting body, Int J Comput Meth 2 (3) (2005) 431–450. [31] A. Abu-Nada, K. Ziyad, M. Saleh, Y. Ali, Heat transfer enhancement in combined convection around a horizontal cylinder using nanofluids, J Heat Tran 130 (2008) 1–4. [32] H.F. Oztop, Z. Zhao, B. Yu, Fluid flow due to combined convection in lid-driven enclosure having a circular body, Int J Heat and Fluid Flow 30 (2009) 886–901. [33] M.M. Billah, M.M. Rahman, U.M. Sharif, N.A. Rahim, R. Saidur, M. Hasanuzzaman, Numerical analysis of fluid flow due to mixed convection in a lid-driven cavity having a heated circular hollow cylinder, Int Commun Heat Mass Tran 38 (2011) 1093–1103. [34] M.M. Rahman, M.M. Billah, N.A. Rahim, Finite element simulation of magnetohydrodynamic mixed convection in a double-lid driven enclosure with a square heat-generating block, J Heat Tran 134 (2012) 1–8. [35] P. Błasiak, P. Kolasinski, Modelling of the mixed convection in a lid-driven cavity with a constant heat flux boundary condition, Heat Mass Tran 52 (3) (2016) 595–609. [36] K.M. Gangawane, B. Manikandan, Laminar natural convection characteristics in an enclosure with heated hexagonal block for non-Newtonian power law fluids, Chin J Chem Eng (2016), http://dx.doi.org/10.1016/j.cjche.2016.08.028. [37] K.M. Gangawane, Computational analysis of mixed convection heat transfer characteristics in lid driven cavity containing triangular block with constant heat flux: effect of Prandtl and Grashof numbers, Int J Heat Mass Tran (2016), http://dx.doi. org/10.1016/j.ijheatmasstransfer.2016.09.061. [38] H.M. Badr, Mixed convection from a straight isothermal tube of elliptic cross-section, Int J Heat Mass Tran 37 (15) (1994) 2343–2365. [39] A. Kondjoyan, J.D. Daudin, Mixed convection from a straight isothermal tube of elliptic cross-section, Int J Heat Mass Tran 38 (10) (1994) 1735–1749.

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