Das2016.pdf

International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Review

Studies on natural convection within enclosures of various (non-square) shapes – A review Debayan Das a, Monisha Roy b, Tanmay Basak a,⇑ a b

Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India

a r t i c l e

i n f o

Article history: Received 20 January 2016 Received in revised form 5 August 2016 Accepted 12 August 2016 Available online xxxx Keywords: Natural convection Fluid or porous media Triangular Trapezoidal Parallelogrammic Curved walls

a b s t r a c t Natural convection in an enclosure (internal convection) is an important problem due to its significant practical applications. In energy related applications, natural convection plays a dominant role in transport of energy for the proper design of enclosures in order to achieve higher heat transfer rates. This review summarizes the studies on natural convection heat transfer in triangular, trapezoidal, parallelogrammic enclosures and enclosures with curved and wavy walls filled with fluid or porous media. In addition, this review also summarizes the natural convection studies in the nanofluid filled enclosures. Studies have been performed for the enclosures subjected to different thermal boundary conditions. A number of the studies demonstrated that the variation of the aspect ratio and base angle of the triangular and rhombic/parallelogrammic enclosures had a wide influence on the flow distribution pattern. In the trapezoidal enclosure, the aspect ratio of the cavity as well as the presence of the baffles along the walls played a significant role in the temperature and flow distribution. The flow patterns within the complex enclosures were found to be largely dependent on the amplitude-wavelength ratio and number of undulations of the wavy walls. In addition, the researchers have also studied the effect of the various parameters such as the Rayleigh numbers, Prandtl numbers, Darcy numbers, Darcy–Rayleigh number, irreversibility distribution ratios, volume fraction of the nanoparticles, etc. Overall, the current review paper presents an useful insight into the potential strategies for enhancing the convection heat transfer performance. Ó 2016 Elsevier Ltd. All rights reserved.

Contents 1.

2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Natural convection: internal and external flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Natural convection: internal flows in various geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling and simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Fluid media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Nanofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Heat transfer rates: Nusselt numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Heat flow visualization: heatlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Entropy generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Fluid media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. Porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3. Nanofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

02 02 02 04 04 04 05 06 07 07 07 08 08 08 09

⇑ Corresponding author. E-mail addresses: [email protected] (D. Das), [email protected] (M. Roy), [email protected] (T. Basak). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

2

D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

3.

4.

5.

6.

7.

2.6. Entropy generation number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Numerical simulations and post processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triangular enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Fluid media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Nanofluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trapezoidal enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Fluid media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Nanofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parallelogrammic and rhombic enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Fluid media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Fluid media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Nanofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Triangular enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Trapezoidal enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Rhombic/parallelogrammic enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Complex enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction 1.1. Natural convection: internal and external flows Natural convection is one of the major modes of heat transfer. Based on the flow pattern and geometry, natural convection can be classified as external or internal. During the internal flow arrangement, the flowing fluid is surrounded by the solid boundaries, whereas a solid object is surrounded or covered by the flowing fluid during the external flow arrangement. The flow through pipes and ducts are the primary examples of the internal flows. Flows over the flat plate, cylinders and spheres are examples of the external flows. However, natural convective flows are complex because of the essential coupling of the hydrodynamic and thermal flow fields. In particular, the internal flow problems are more complex than the external flow situations. The external flow problems can be modeled using the classical boundary layer theory by assuming that the region outside the boundary layer is unaffected by the solid boundary [1]. In contrast, for the internal convection, the interactions between the boundary layer and core constitute a major complexity in the problem. Based on the applied thermal boundary conditions, the internal convective systems can be classified into two types: (a) enclosures heated from side walls (temperature gradient is orthogonal to the direction of gravity) and (b) enclosures heated from bottom wall (temperature gradient is parallel to the direction of gravity) [1]. Natural convection in a differentially heated enclosure is an example of type-(a) and Rayleigh–Benard convection between two infinite horizontal plates is an example of type-(b). In addition, other thermal boundary conditions may involve the combination of differential and Rayleigh–Benard heating at various segments of walls etc. Various types of heating patterns during the internal convection involve engineering applications such as electronic equipment cooling [2], lubrication systems [3], heat exchangers [4], solar energy collectors [5], electric ovens [6], solar desalination systems [7], melting and solidification processes [8,9], etc. 1.2. Natural convection: internal flows in various geometries Natural convection in rectangular and square enclosures have been extensively studied in the literature and these studies primar-

09 09 09 09 17 21 21 21 25 28 29 29 32 34 34 40 44 46 47 47 48 48 48

ily focused on the effect of boundary conditions, aspect ratios and medium of heat transfer on natural convection. A number of such studies are summarized by Ostrach [10,11], Hoogendoorn [12] and Fusegi and Hyun [13]. The effect of aspect ratios on the flow pattern and energy transport within rectangular enclosures with the isothermally hot side wall and cooled ceiling using the streamfunc tion–vorticity formulation has been reported by Aydin et al. [14]. Later, Sarris et al. [15] investigated numerically natural convection in a rectangular enclosure with a sinusoidal temperature profile on the upper wall and adiabatic conditions on the bottom and side walls. They reported that the fluid circulation intensity and the thermal penetration depth increases with the aspect ratio. Basak et al. [16] studied the effect of thermal boundary conditions on natural convection flows within a square cavity using the penalty finite element method. They reported that the overall heat transfer rates are lower for the non-uniform heating case compared to that for the uniform heating case. Recently, Kaluri and Basak [17] studied the effect of the distributed heating on natural convection in a square cavity via the heatline approach. Their study shows that the heat distribution and thermal mixing in a cavity is greatly enhanced in the distributed heating case compared to the isothermal hot bottom wall case. A number of studies reported the wide variety of applications of convection within the porous enclosures, which include thermal insulation [18], grain storage and drying [19], thermal energy storage systems [20], geological storage of CO2 [21] etc. Studies on natural convection in square or rectangular type enclosures filled with the porous medium can be found in the earlier works [22–26]. Trevisan and Bejan [22] reported an analytical and numerical study of natural convection heat and mass transfer through a vertical slot filled with the porous medium subjected to the uniform heat flux. They developed the overall heat and mass transfer correlations for the porous media with Le = 1 and the buoyancy effect governed by both the temperature and concentration variations. Later, Lage and Bejan [23] studied the effect of the pulsating heat input on natural convection in a porous square enclosure. They conducted the numerical experiments for Prandtl numbers varying within 0.01–7, heat flux Rayleigh number (heat flux Rayleigh number is proportional to Rayleigh number based on the average instantaneous side-to-side temperature difference) within 103
109, and non-dimensional frequency range 0–0.3. Song and Viskanta [24]

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

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D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

conducted the experimental and theoretical study of natural convection flow and heat transfer within a rectangular enclosure partially filled with an anisotropic porous medium. They used volumeaveraged conservation equations to model the effect of the anisotropic flow characteristics of the porous medium on the flow and heat transfer. Kim et al. [25] studied natural convection in a porous square enclosure using the Brinkman-extended Darcy model and reported that in the conduction dominant regime, the porous region acts as a heat-generating solid block. The results indicate that there exists an asymptotic convection regime where the flow is nearly independent of the permeability and conductivity of the porous medium. Hossain and Wilson [26] reported the unsteady laminar natural convection flow in a fluid-saturated porous rectangular enclosure with the hot bottom wall, non-isothermal left wall and cold top and right walls. Their study shows that, the volumetric flow rate of the fluid and the heat transfer rate at the walls are reduced with the increase of the porosity of the medium. A number of studies on natural convection heat transfer in enclosures with complex shapes besides regular geometries such as square or rectangle, were reported in the literature due to their applications in various engineering problems [27–29]. The coupling between the hydrodynamic and thermal fields in a complex geometry via the buoyancy force makes the mathematical model rather challenging. Thus, researchers have carried out significant studies on natural convection in the non-rectangular enclosures

(a)

with the inclined, curved and wavy side walls during the last two decades. A few of earlier studies on natural convection in various non-rectangular enclosures are mentioned below. Philip [27] first developed the exact solutions for the small Rayleigh number during the free convection in various enclosures with various shapes (rectangular, elliptical and triangular enclosures) due to the uniform temperature gradient normal to the gravitational field. The effect of the enclosure aspect ratio and orientation on the flow fields has been studied and it was concluded that at the low Rayleigh number, convective flows are independent of the cavity orientation. Lee [28] presented the numerical and experimental studies of the fluid motion and heat transfer in a differentially heated non-rectangular enclosure. The effect of the Rayleigh number, aspect ratios and inclination of the enclosure on the flow and thermal characteristics have been reported in their study. One of the interesting conclusions was that the maximum value of the average Nusselt number occurs at an inclination angle of 180 and minimum value at 270 . Hyun and Choi [29] studied the transient natural convective heat transfer in a parallelogram-shaped enclosure at large Rayleigh numbers using the finite-difference method. They reported the possibility of utilizing the parallelogram-shaped enclosure as a transient thermal diode, by means of controlling the tilt angle of the partition walls of the enclosure. Iyican et al. [30] studied natural convective flow and heat transfer within a trapezoidal enclosure with the parallel

(b)

Hot o r or Ad Cold iabati c

r Cold Hot o batic ia or Ad

Hot or Cold or Adiabatic

Adiabatic Adiabatic

Hot or Cold or Adiabatic

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ld

batic

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batic

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or A

Hot or Cold

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ld Co

ld

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atic

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H or ot o Ad r C iab old ati c

H or ot o Ad r C iab ol ati d c

old c r C ati t o iab o H Ad or

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old ic r C at t o iab Ho Ad or

ld Co tic or ba ot ia H r Ad o

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(d) Cold or Adiabatic

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tic H d or ot or Adi aba

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c ld ati Co iab Ad

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Cold or Adiabatic

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Adiabatic

Fig. 1. Schematic diagram of the physical geometries with various thermal boundary conditions, (a) triangular enclosures, (b) trapezoidal enclosures, (c) parallelogramic and rhombic enclosures, and (d) complex enclosures.

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

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D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

cylindrical top and bottom walls at different temperatures and plane adiabatic side walls. Karyakin [31] reported two dimensional laminar natural convection in isosceles trapezoidal cavities. Varol et al. [32] studied natural convection within trapezoidal enclosures partially cooled from the inclined wall. A comprehensive review on natural convection in triangular enclosures is reported by Kamiyo et al. [33]. Their review focused on the whole range of the buoyancy-induced flow-regimes in triangular enclosures. In addition, the effects of the pitch angle, Rayleigh number and various thermal boundary conditions on the fluid and heat flow fields were also examined in detail. Later, Saha and Khan [34] reviewed natural convection heat transfer through the atticshaped space. They also reported the studies on attics subjected to the localized heating and attics filled with the porous media. The review report of research and future aspects on various irregular shape cavities will be important for analyzing various applications such as energy intensive processes, material processing etc and these studies are yet to appear in literature. Thus, the aim of this review is to present a summery of past and recent studies on natural convection in triangular, trapezoidal, parallelogramic/ rhombic and enclosures with the curved and wavy walls filled with the fluid and porous media. This review also addresses the flow and heat transfer details on natural convective systems with a detailed discussion about the effects of the enclosure orientation and the thermal boundary conditions along with the efficiency of heat transfer via heatlines and entropy generation as a first attempt. Natural convection within enclosures of various shapes and different thermal boundary conditions have wide ranges of applications in cooling processes [35], solar energy [36,37], fuel cells [38], heat exchangers [39], melting process [40], food technology [41]. The present review may be helpful for the future researchers in choosing the realistic enclosures and suitable process parameters for a convective system.

2. Modelling and simulations Fig. 1(a–d) represent the triangular, trapezoidal, parallelogrammic, rhombic and curved wall enclosures with possible thermal boundary conditions based on various case studies in the literature. The mathematical model is developed for the natural convective systems based on the following assumptions.  Fluid is assumed to be incompressible and Newtonian.  The no-slip boundary condition is assumed at the solid boundaries.  The fluid flow is assumed to be laminar and two dimensional.  The thermo-physical properties of the fluid except the density variation in the buoyancy term are considered to be constant. The Boussinesq approximation is invoked to relate the variation of density with temperature in the body force term.  The temperature of the fluid phase is equal to the temperature of the solid phase in the case of the porous bed and the local thermal equilibrium (LTE) is applicable.  Radiation heat transfer is negligible.  In the case of convection in nanofluids, the nanoparticles have a uniform size and shape. The nanoparticles are assumed to be well dispersed within the base fluid. The base fluid and nanoparticles are in thermal equilibrium and have the same flow velocity. 2.1. Governing equations 2.1.1. Fluid media Conservations of mass, momentum and energy balance equations for the incompressible fluid in the steady state are as follows

@u @ v þ ¼ 0; @x @y

ð1aÞ

! @u @u @p @2u @2u qu þ qv ¼
þ l þ ; @x @y @x @x2 @y2 ! @v @v @p @2v @2v þ gbqðT h
T c Þ; qu þ qv ¼
þ l þ @y @x @y @x2 @y2     @ðuTÞ @ðv TÞ @ @T @ @T þ qc p ¼ k þ k : qcp @x @y @x @x @y @y

ð1bÞ ð1cÞ ð1dÞ

Here x and y are the distances measured 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; g denotes the acceleration due to gravity; m and a are the kinematic viscosity and thermal diffusivity, respectively; p is the pressure and q is the density; T h and T c are the temperatures at hot left wall and cold right wall, respectively. The dimensionless variables or numbers have been used for defining the governing equations for the steady two dimensional natural convection flow. The dimensionless variables are enlisted as follows:

x X¼ ; L P¼

pL2

qa2

y Y¼ ; L ;

Pr ¼

U¼

uL

a

;

V¼

vL a

;

h¼

T
Tc ; Th
Tc

m gbðT h
T c ÞL3 Pr ; Ra ¼ : a m2

ð2Þ

Here L is the height or length of the base of the cavity; X and Y are the dimensionless coordinates varying along the horizontal and vertical directions, respectively; U and V are the dimensionless velocity components in the X and Y directions, respectively; h is the dimensionless temperature; P is the dimensionless pressure; Ra and Pr are the Rayleigh and Prandtl numbers, respectively. Using the above relations (Eq. (2)), the steady state governing equations for pressure velocity formulation or streamfunction–vor ticity formulation or vorticity-velocity formulation are as follows: 2.1.1.1. Pressure velocity formulation [16].

@U @V þ ¼ 0; @X @Y ! @U @U @P @2U @2U ; þ þV ¼
þ Pr U @X @Y @X @X 2 @Y 2 ! @V @V @P @2V @2V U þ Ra Pr h; þV ¼
þ Pr þ @X @Y @Y @X 2 @Y 2 U

@h @h @ 2 h @ 2 h þV ¼ þ : @X @Y @X 2 @Y 2

ð3aÞ ð3bÞ ð3cÞ ð3dÞ

2.1.1.2. Streamfunction–vorticity formulation [14,15]. A streamline is a line that is tangential to the instantaneous velocity vector of the flow. The streamfunction (w) can be used to plot the streamlines, which represent the trajectories of particles in a steady flow. The positive sign of w denotes the anticlockwise circulation and negative sign represents the clockwise circulation. The streamfunction is defined in the following manner based on the continuity equation

U¼

@w @w and V ¼
; @Y @X

ð4Þ

The combination of the above relationships yields a following single equation

@2w

@2w

@X

@Y 2

þ 2

¼

@U @V
: @Y @X

ð5Þ

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

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D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

The no-slip condition (w ¼ 0) is valid at all the boundaries as there is no cross flow. The steady state streamfunction–vorticity formulation for the fluid velocities is given as:

!

@w @ X @w @ X @2X @2X @h
Ra Pr þ ;
¼ Pr @Y @X @X @Y @X @X 2 @Y 2 2

w¼ ð6bÞ

where vorticity is denoted by X and defined as

X¼


@2w

@2w

@X

@Y


2

: 2

ð7Þ

2.1.1.3. Vorticity-velocity formulation [4,42]. The non-dimensional vorticity-velocity formulation of the governing equations are as follows:

@X @ X @ 2 X @ 2 X Ra @ X þ þ þV ¼ ; @X @Y @X 2 @Y 2 Pr @X ! @X @X 1 @2h @2h ; þ þV ¼ U @X @Y Pr @X 2 @Y 2

U

ð8aÞ ð8bÞ

2.1.2. Porous media The development of various models on natural convection in porous media is well illustrated by Nield and Bejan [43] and Ingham and Pop [44]. Various forms of the momentum equations for porous media are as follows: 2.1.2.1. Darcy model. The continuity equation for porous media can be written as follows [45]:

@uD @ v D þ ¼ 0: @x @y

ð9Þ

Here, uD and v D are the seepage velocities along the x and y directions, respectively, p denotes the intrinsic fluid pressure, lf denotes the viscosity of the fluid, K represents the permeability of the medium, q is the density of the medium and g represents acceleration due to the gravity. Assuming the Boussinesq approximation, q ¼ q0 ½1
bðT
T 0 Þ in the body force term, the momentum equation for the two dimensional fluid flow through an isotropic porous medium based on the Darcy model [46,47] is as follows:

@uD @ v D Kgbq0 @T :
¼ @y @x lf @x

ð10Þ

Here b is the coefficient of volumetric thermal expansion and q0 is the density of the fluid at some standard temperature T 0 . Also, the energy balance equation for the isotropic porous media is as follows:

! @T @T @2T @2T : þ vD ¼ aeff uD þ @x @y @x2 @y2

ð11Þ

Here aeff is the effective thermal diffusivity. The streamfunction-velocity formulation of Eqs. (10) and (11) are 2

0

2

0

@ w @ w Kgbq0 @T ; þ 2 ¼ @x2 @y lf @x

ð12aÞ !

@w0 @T @w0 @T @2T @2T þ
¼ aeff : @x2 @y2 @y @x @x @y

x X¼ ; L

ð6aÞ

2

@w @h @w @h @ h @ h
¼ þ ; @Y @X @X @Y @X 2 @Y 2

Using the following dimensionless variables,

ð12bÞ

w0

aeff

y Y¼ ; L ;

Pr ¼

uD ¼

@w0 ; @y

vD ¼


@w0 ; @x

h¼

T
Tc ; Th
Tc

m gbðT h
T c ÞL3 Pr K ; Ra ¼ ; Da ¼ 2 ; aeff m2 L

ð13Þ

the steady state dimensionless forms of mass, momentum and energy balance equations (Eqs. (10) and (11)) [48] are as follows:

@U D @V D þ ¼ 0; @X @Y 2 2 @ w @ w @h ; þ ¼
RaDa @X @X 2 @Y 2 2 @w @h @w @h @ h @ 2 h
¼ þ ; @Y @X @X @Y @X 2 @Y 2

ð14aÞ ð14bÞ ð14cÞ

2.1.2.2. Darcy extended Forchheimer model. The Darcy law is valid for the viscous flow involving the less intense flow and less Reynold’s number. The measured relationship between the pressure gradient and volume averaged velocity may be correlated by Forchheimer’s modification of Darcy’s model [49]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  lf @p cF q  2 2  ¼
uD
pffiffiffiffi0   ðuD þ v D ÞuD ; @x K K qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  lf @p cF q  2 2  ¼
v D
pffiffiffiffi0   ðuD þ v D Þv D
gbq0 ðT
T 0 Þ: @y K K

ð15aÞ ð15bÞ

Here cF is the dimensionless inertia parameter which varies with the nature of the porous medium. Based on the Ergun model [50], cF ¼ 1:75 , where  denotes the porosity. In Eqs. (15a) and 3 1=2 ð150 Þ

(15b), the first term of RHS is linear drag term introduced by Darcy and the second term is the non-linear drag term (Forchheimer term). 2.1.2.3. Brinkman model. Brinkman [51] modified the Darcy flow model involving the transition from the Darcy flow to the highly viscous flow (without porous matrix) in the limit of extremely high permeability.

lf @p ¼
uD þ leff r2 uD ; @x K lf @p ¼
v D þ leff r2 v D ; @y K

ð16aÞ ð16bÞ

where leff represents the effective viscosity of the fluid. Based on the Brinkman model, the momentum and energy balance equations [52,53] in terms of the intrinsic velocities (uI and v I ) are as follows:

!   @uI @uI @p lf @ 2 uI @ 2 uI q0 uI þ 2 ; þ vI ¼

uI þ lf @x @x @y K @x2 @y !   @v @v @p l @2v I @2v I q0 uI I þ v I I ¼

f v I þ lf þ @y @x @y K @x2 @y2 þ q0 gbðT
T 0 Þ;

    @ðu TÞ @ðv TÞ @ @T @ @T keff þ keff : qcp I þ qcp I ¼ @x @y @x @x @y @y

ð17aÞ

ð17bÞ ð17cÞ

Note that, the intrinsic and Darcy velocities are related by uD ¼ uI and v D ¼ v I and keff is the effective thermal conductivity of the fluid saturated porous media. Using the dimensionless variables,

x X¼ ; L

y Y¼ ; L

UI ¼

uI L

aeff

;

VI ¼

vIL aeff

;

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

6

D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

h¼

T
Tc ; Th
Tc

Dam ¼

K

L2

P¼

pL2

q0 a2eff

;

aeff ¼ 3

;

Ram ¼

keff

q0 cp

gbðT h
T c ÞL q

2 0 Pr

l2f

;

Prm ¼

f

@U I @V I þ ¼ 0; @X @Y

lf ; q0 aeff

;

ð18Þ

@U I @V I þ ¼ 0; @X @Y ! @U I @U I @P Pr m @2 UI @2 UI
U I þ Pr m þ þ VI ¼
; UI @X Dam @X @Y @X 2 @Y 2 ! @V I @V I @P Pr m @2V I @2V I V I þ Prm þ
UI þ VI ¼
@Y Dam @X @Y @X 2 @Y 2

ð19aÞ ð19bÞ

UI ð19cÞ

þ Ram Prm h; UI

2

@h @h @ h @ h þ VI ¼ þ : @X @Y @X 2 @Y 2

ð19dÞ

Note that, U I and V I are the dimensionless intrinsic velocity components in the X and Y directions, respectively; Pr m ; Ram and Dam are the modified Prandtl number, modified Rayleigh number and modified Darcy number, respectively. 2.1.2.4. Generalized model. The generalized form of the momentum balance equations based on the Brinkman-Forchheimer extended Darcy model [55] are as follows:

!   lf @ 2 u D @ 2 u D @uD @uD @p lf
u v þ þ þ ¼
u D D @x K @y  @x2 @y2 2 D @x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:75q0 ffi uD u2D þ v 2D ; ð20aÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 150K 3 !   lf @ 2 v D @ 2 v D q0 @v D @v D @p lf ¼

vD þ uD þ þ vD 2 @y K @x @y  @x2 @y2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:75q0 ffi v D u2D þ v 2D þ q0 gðT
T 0 Þ: ð20bÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 150K 3

q0

Using the dimensionless variables,

x X¼ ; L P¼

y Y¼ ; L

pL2

q0 a2eff

;

Pr ¼

UD ¼

uD L

aeff

;

VD ¼

@U I @U I @P Pr m @2 UI @2 UI U I þ Prm þ
þ VI ¼
@X Dam @X @Y @X 2 @Y 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1:75 U I U I þ V I pffiffiffiffiffiffiffiffiffi ;
pffiffiffiffiffiffiffiffiffi Dam 150 ! @V I @V I @P Prm @2V I @2V I
V I þ Pr m þ UI þ VI ¼
@Y Dam @X @Y @X 2 @Y 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1:75 V I U I þ V I pffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffi þ Ram Pr m h; Dam 150 UI

the dimensionless forms of mass, momentum and energy equations (Eqs. (17a)–(17c)) in terms of the non-dimensional intrinsic velocities are as follows [54]:

2

ð23aÞ !

v DL aeff

;

h¼

ð21Þ

the nondimensional forms of momentum Eqs. (20a) and (20b) [56] in terms of the Darcy velocities are as follows:

!   @U D @U D @P Pr Pr @ 2 U D @ 2 U D
UD þ U þ þ VD ¼
@X Da @Y  @X 2 @Y 2 2 D @X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2D þ V 2D U D 1:75 pffiffiffiffiffiffi pffiffiffiffiffi ; ð22aÞ
pffiffiffiffiffiffiffiffiffi Da 150 3 !   1 @V D @V D @P Pr Pr @ 2 U D @ 2 U D
VD þ UD þ þ VD ¼
2 @X Da @X @Y  @X 2 @Y 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2D þ V 2D V D 1:75 pffiffiffiffiffi þ RaPrh: pffiffiffiffiffiffi ð22bÞ
pffiffiffiffiffiffiffiffiffi Da 150 3 1

Recently, Singh et al. [57] presented the governing equations in terms of the intrinsic velocities (U I ; V I ). The nondimensional forms of governing equations (using Eq. (18)) are:

ð23cÞ ð23dÞ

2.1.3. Nanofluids Nanofluids are defined as the liquid dispersions of submicron solid particles or nanoparticles in pure fluids. The term ‘nanofluid’ was first proposed by Choi [58]. Nanofluid has gained significant attention in the recent years due to the easy production methods and inexpensive price. Also, the thermal conductivity of nanofluids relative to the base fluids is high. Thus, nanofluids can be applied in many energy related systems such as the cooling of electronics, cooling and heating in buildings, medical applications [59], radiators [60], solar collectors [61]. Because of the nanoparticles in fluid, the additional physical parameters such as the effective thermal conductivity (keff ), effective viscosity (leff ), solid volume fraction (/), etc are incorporated in the governing equations. The modified governing equations based on the effective properties are as follows [62–66]:

@u @ v þ ¼ 0; @x @y !   @u @u @p @2u @2u ; qnf u þ v þ ¼
þ lnf @x @y @x @x2 @y2 !   @v @v @p @2v @2v qnf u þ v þ ¼
þ lnf @y @x @y @x2 @y2 þ g½ð1
/ÞðqbÞbf þ /ðqbÞs ðT
T c Þ; ! @T @T @2T @2T ; þv ¼ anf þ u @x @y @x2 @y2

T
Tc ; Th
Tc

m gbðT h
T c ÞL3 Pr K ; Ra ¼ ; Da ¼ 2 ; aeff m2 L

@h @h @ 2 h @ 2 h þ : þ VI ¼ @X @Y @X 2 @Y 2

ð23bÞ

ð24aÞ ð24bÞ

ð24cÞ ð24dÞ

Here,

qnf ¼ ð1
/Þqbf þ /qs ; ðqcp Þnf ¼ ð1
/Þðqcp Þbf þ /ðqcp Þs ; ðqbÞnf ¼ ð1
/ÞðqbÞbf þ /ðqbÞs ; keff ¼ kstatic þ kBrownian ;

lBrownian

kBrownian

lstatic ¼

anf ¼

keff ; ðqcp Þnf

leff ¼ lstatic þ lBrownian ;

sffiffiffiffiffiffiffiffiffiffiffiffi jT f ðT; /Þ; ¼ 5  10 b1 /qbf 2qs Rs 1 4

sffiffiffiffiffiffiffiffiffiffiffiffi jT f ðT; /Þ; ¼ 5  10 b1 /qbf ðcp Þbf 2qs Rs 2 4

lbf ð1
/Þ2:5

;

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

7

D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx



 ðks þ2kbf Þ
2/ðkbf
ks Þ kstatic ¼ kbf ðoriginal Maxwell modelÞ; ðks þ2kbf Þþ/ðkbf
ks Þ " # 3 ðkeq þ2kbf Þþ2/ðkeq
kbf Þð1þ rÞ ðmodified Maxwell modelÞ; ¼ kbf ðkeq þ2kbf Þ
/ðkeq
kbf Þð1þ rÞ3

" keq ¼ ks c1

#

2ð1
c1 Þ þ ð1 þ 2c1 Þð1 þ rÞ3


ð1
c1 Þ þ ð1 þ 2c1 Þð1 þ rÞ

:

3

ð25Þ

Note that, qs ; qbf and qnf are the densities of the nanoparticles, base fluid and nanofluid respectively, lbf ; lnf represent the dynamic viscosities of the base fluid and nanofluid respectively, mbf ; mnf represent the kinematic viscosities of the base fluid and nanofluid respectively. Here / denotes the volume fraction of nanoparticles in the base fluid, bbf and bs are thermal expansion coefficients of the base fluid and solid, respectively; abf and anf denote the thermal diffusivity of the base fluid and nanofluid respectively. Also, r is the ratio of the thickness of the nano-layer to the original radius of nanoparticles, kbf and ks are the thermal conductivities of the base fluid and nanoparticles, respectively, keq is the equivalent thermal conductivity of nanoparticles and their layers, c1 is the ratio of the thermal conductivity of the nanolayers (knl ) to the thermal conductivity of the nanoparticles (ks ), f 1 and f 2 are the modeling functions, Ra and Pr are Rayleigh and Prandtl numbers, respectively. Note that, keff ¼ kstatic and leff ¼ lstatic in the absence of Brownian motion of the nanoparticles. Using the dimensionless variables enlisted below,

x X¼ ; L

y Y¼ ; L

U¼

uL

abf

;

V¼

vL

T
Tc h¼ ; Th
Tc

;

abf

gbbf ðT h
T c ÞL3 Pr mbf ; Pr ¼ ; Ra ¼ ; P¼ 2 abf qbf abf m2bf pL

2

ð26Þ

the dimensionless forms of mass, momentum and energy Eqs. (24b)–(24d) are as follows: 2.1.3.1. Pressure velocity formulation [62].

@U @V þ ¼ 0; @X @Y qbf @P mnf @U @U þV ¼
þ Pr U @X @Y qnf @X mbf U

qbf @P mnf @V @V þV ¼
þ Pr @X @Y qnf @Y mbf

@2U @X

2

@2V @X

2

þ þ

@2U

!

@Y 2 ! @2V @Y

;

@X

2




@2w @Y 2

:

or

@2w ; @x2

ð29aÞ

Adiabatic wall : rT ¼ 0; Isothermal walls : T ¼ T h or T c ; Sinusoidally heated vertical walls : T ¼ T h
ðT h
T c Þsin

py L

Sinusoidally heated horizontal walls : T ¼ T h
ðT h
T c Þsin y Linearly heated vertical walls : T ¼ T h
ðT h
T c Þ ; L

x Linearly heated horizontal walls : T ¼ T h
ðT h
T c Þ : L

þ Tc;

px L

þ Tc

ð29bÞ

2.3. Heat transfer rates: Nusselt numbers The dimensionless heat flux in terms of the local Nusselt number (Nu) is defined as

Nu ¼

hL @h ¼
; k @n

ð30Þ

where h is the convection heat transfer coefficient. For the nanofluids, the local Nusselt number ðNuÞ is modified as

Nu ¼

  knf @h hL : ¼
kf kbf @n

ð31Þ

The average Nusselt number (NuS ) represents the overall heat transfer rate along the wall and is defined as

1 S

Z

S

Nuds;

ð32Þ

0

2.4. Heat flow visualization: heatlines The heatlines, which are analogous to the streamlines, may be used to visualize the path and intensity of heat flow within cavities. The trajectories of heatlines are useful to visualize the flow of heat from the hot to cold regime within the cavity and the distribution of heat in a cavity during the two-dimensional convective transport process was studied with the help of heatlines [68]. Mathematically, the heatlines are represented by the heatfunctions. The heatfunction (h) can be used to plot the heatlines and  is obtained from the conductive heat fluxes
@T as well ;
@T @x @y

as the convective heat fluxes (uT; v T). The heatfunction ðhÞ satisfies the steady energy balance equation for the fluid media or homogeneous porous medium:

ð28aÞ

! @T @T @2T @2T ; þv ¼a u þ @x @y @x2 @y2

ð33Þ

such that,

ð28bÞ

bf

@2w

@2w ; @y2

and several thermal boundary conditions (Fig. 1(a–d)) have been used as follows:

where S is the arc length of the wall.

2.1.3.2. Streamfunction–Vorticity formulation [67].

X¼


X¼

ð27bÞ

ð27dÞ

! knf @w @h @w @h @2h @2h kbf i ; þ
¼h ðqc Þ @Y @X @X @Y @X 2 @Y 2 ð1
/Þ þ / ðqcppÞ s

Stationary walls : u ¼ v ¼ 0;

NuS ¼

ð27cÞ

!     @ @w @ @w @ lnf @ X
¼ X X @X @Y @Y @X @X lbf @X !! @ lnf @ X Pr þ @Y lbf @Y ½ð1
/Þ þ / qqs  bf ! @h bs þ RaPr / þ ð1
/Þ ; @X bbf

The no-slip boundary conditions at all the solid boundaries during the natural convection process are as follows

ð27aÞ

2

ð1
/ÞðqbÞbf þ /ðqbÞs ; þ RaPrh qnf bnf ! @h @h anf @ 2 h @ 2 h : þ þV ¼ U @X @Y abf @X 2 @Y 2

2.2. Boundary conditions

ð28cÞ

@h @T ¼ qcp uðT
T 0 Þ
k ; @y @x @h @T
¼ qcp v ðT
T 0 Þ
k : @x @y

ð34aÞ ð34bÞ

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

8

D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

The dimensionless form of the heatfunction (P) using the nondimensional variables (see Eq. (2)) are as follows:

@P @h ; ¼ Uh
@X @Y @P @h ;
¼ Vh
@Y @X

ð35aÞ ð35bÞ

@ P @X 2

@ P 2

þ

@Y 2

¼

@ @ ðUhÞ
ðVhÞ: @Y @X

ð36Þ

adiabatic wall : P ¼ 0 or a reference value; ð37Þ

Note that, the boundary conditions for P involving the nonisothermal walls (Eq. (29b)) may be obtained for various cases using Eqs. (35a) and (35b). The Dirichlet boundary conditions for P at the corner nodes of two dimensional domain are obtained via the integration of Eqs. (35a) and (35b) [69,70]. 2.5. Entropy generation Natural convection in the closed cavities become a very rich area of the investigation due to its versatile applications in the thermal engineering, design and optimization of a thermal system. However, the efficiency of the thermal system is affected due to the irreversibilities of the system. Thus, in order to improve the efficiency of the system, the irreversibilities of the system are required to be minimized. Although several advanced optimization methods such as the artificial neural network, genetic algorithms may be adequate to present the physical explanation behind the optimized situation, the entropy generation minimization (EGM) is an efficient optimization method based on the concepts of Thermodynamics as reported by several researchers. The entropy generation during natural convection can be studied via combining the irreversibilities due to temperature gradient (heat transfer irreversibility) and irreversibilities due to velocity gradient (fluid friction irreversibility) [71,72]. Entropy generation minimization (EGM) was first introduced by Bejan [73]. 2.5.1. Fluid media Based on the thermodynamic equilibrium of linear transport theory, the volumetric entropy generation for fluid media in the two dimensional Cartesian coordinate system is given as:

S_ 000 gen

"

 2 # k @T @T ¼ 2 þ @x @y T0 "   2 #  2 !  2 lf @u @v @ v @u þ : þ 2 þ þ @x T0 @y @x @y

2



@h þ @Y

2

2

of U was taken as 10
2 [74] or 10
4 [76,77] for the fluid media.

The positive sign of P denotes the anti-clockwise circulation and negative sign represents the clockwise circulation. The boundary conditions of the heatfunction for natural convection in the bounded enclosure [69,70] are as follows (see Fig. 1(a–d)):

isothermal hot or cold wall : n:rP ¼ 0:

@h Stotal ¼ @X

lT 0 a Here U ¼ kL is the irreversibility distribution ratio. The value 2 DT 2

which can be written in a single equation 2

" (    )  2 # 2 2 @U @V @V @U þU 2 þ þ ; ð39aÞ þ @X @Y @X @Y 2 3 ! ! 2 2  2  2 @h @h @2w @2w @2w 5 þ þ U44 þ 2
2 Stotal ¼ : ð39bÞ @X @Y @X@Y @ Y @ X 

2.5.2. Porous media There are various forms of the entropy generation during natural convection in the fluid saturated porous media [43]. The entropy generation due to the fluid friction depends on the viscous dissipation models and the detailed discussion on various viscous dissipation models, their limits of applicability and other important issues on various aspects of modeling viscous dissipation in porous media is available in the open literature [78–81]. In a fluid saturated porous media, based on the assumption of the local thermal equilibrium and the Darcy model, the entropy generation equation for natural convection is [82]:

"   2 # 2 lf 2 @T @T þ ðu þ v 2D Þ: þ @x @y KT 0 D

k S_ 000 gen ¼ 2 T0

ð40Þ

The dimensionless form of the total entropy generation in terms of velocity and streamfunction using dimensionless variables (Eq. (13)) are



2  @h þ U U 2D þ V 2D ; @Y "   2  2  2 # 2 @h @h @w @w : ¼ þ þU þ @X @Y @Y @X

Stotal ¼ Stotal

@h @X

2



þ

lf T 0 a2eff

Here U ¼ k

eff K DT

2

ð41aÞ ð41bÞ

is the irreversibility distribution ratio for the

porous media and U ¼ 10
2 is considered by earlier researchers [82,83]. Al-Hadhrami et al. [79] proposed the viscous dissipation model which includes the correct asymptotic behavior in both the fully Darcy and Newtonian fluid flow limits. Based on the model (obtained from the momentum equation known as Brinkman model) suggested by Al-Hadhrami et al. [79], the entropy generation for natural convection is

"   2 # 2 lf 2 @T @T þ þ ðu þ v 2D Þ @x @y KT 0 D (  2  2 !  2 ) lf @uD @v D @ v D @uD þ þ : 2 þ þ T 0 @x @y @x @y

keff S_ 000 gen ¼ T 20

2

ð42Þ

The dimensionless form of the total entropy generation in the porous media using the dimensionless variables (Eq. (21)) is

ð38Þ

Here S_ 000 gen denotes the entropy generation in the fluid media and T 0 denotes the temperature. Note that, the first term in the right hand side of the above equation corresponds to the rate of the entropy generation associated with the heat transfer which is due to temperature difference. The entropy generation due to the fluid friction (velocity gradient) is presented in the second term of the right hand side of Eq. (38). The dimensionless form of the total entropy generation [74,75] in terms of velocity temperature formulation and streamfunction–vorticity formulation, respectively are obtained using the dimensionless variables or numbers (Eq. (2)) as follows:

" (  2  2 2  2 ! @h @h @U D @V D þ þ U ðU 2D þV 2D ÞþDa 2 þ @X @Y @X @Y  2 )# @V D @U D þ þ : ð43Þ @X @Y 

Stotal ¼

T o lf a2eff

Here U ¼ k

2 eff ðDTÞ K

is the irreversibility distribution ratio and


2

U ¼ 10 is considered by earlier researchers [84,85]. Recently, Basak and coauthors [57,84] used the Darcy–Brinkman model and also the Darcy–Brinkman extended Forchheimer model for the momentum balance equations and the dimensionless form of the entropy generation equation due to the heat transfer ðSh Þ and fluid friction ðSw Þ using Eq. (18) is as follows:

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Stotal ¼

2 h i @h þ U 2I þ V 2I @Y "  2  2 !  2 # @U I @V I @U I @V I þ : þ þ þ Dam 2 @X @Y @Y @X @h @X

2



3. Triangular enclosures

þ

Here U ¼

lf T o



ffiffiffim pa K DT

3.1. Fluid media

ð44Þ

2

is the irreversibility distribution ratio for pffiffiffi the porous media and am ¼ aeff . As mentioned earlier,

U ¼ 10


2

keff

is considered by earlier studies [84,57].

2.5.3. Nanofluids The volumetric total entropy generation for the nanofluids is given by [86]: knf S_ 000 gen ¼ 2 T0

"

@T @x

2



@T þ @y

2 # þ

lnf

"

T0

 2

@u @x

2



@v þ @y

2 # 2 !  @ v @u þ þ ; @x @y ð45Þ

and the dimensionless form using the relationships in Eq. (25) is given as " Stotal ¼

@h @X

2

 þ

@h @Y

2 #

" þU 2



@U @X

2 þ

 2 !  2 # @V @V @U þ : þ @Y @X @Y ð46Þ

Here U ¼

2 nf T 0 anf knf L2 DT 2

l

is the irreversibility distribution ratio for

entropy generation in nanofluids.

2.6. Entropy generation number The relationship between the volumetric total entropy generation and dimensionless entropy generation number is given as follows:

Ns ¼

S000 gen S000 0

:

9

ð47Þ

Here, N s is the dimensionless entropy generation number, S000 gen is the volumetric entropy generation rate and S000 0 is the characteristics transfer rate [71].

2.7. Numerical simulations and post processing A number of works have been carried out to develop the numerical schemes in order to solve the coupled governing equations (mass balance, momentum balance and energy balance equations for the pure fluid media or fluid saturated porous media or nanofluid) with various velocity and thermal boundary conditions for natural convection in the closed enclosures. A comprehensive review of the previous studies shows that, the study of natural convection in cavities was carried out using the finite difference, finite volume and finite element methods. Further, the equation of heatlines with appropriate heatline boundary conditions were solved by researchers using several numerical techniques to visualize the energy flow during natural convection within the closed cavities. In addition, the finite difference, finite volume and finite element methods are used to evaluate derivative terms of the entropy generation equations in the post processing part to calculate the irreversibilities in the thermal systems.

Natural convection and fluid flow in the triangular enclosures have been analyzed extensively in versatile applications. Salmun [42] reported convection patterns in a triangular enclosure filled with air (Pr ¼ 0:72) or water (Pr ¼ 7:1) and 102 6 Ra 6 105 for var¼ 0:1
1) in the presence of the hot ious aspect ratios (A ¼ height base bottom wall, cold hypotenuse and adiabatic vertical wall. The time dependent, vorticity-velocity formulation of the governing equations (Eqs. (3a), (8a) and (8b)) were solved numerically using two different schemes to validate the available results in the literature. Note that, the streamfunction equation in both schemes is solved by the successive over relaxation method (SOR). It was observed that, at the low Ra, the changes in the aspect ratio had the negligible effect on the streamfunction and isotherms within the enclosure. However, the changes in the aspect ratio do affect the flow pattern and temperature fields significantly at the high Ra (Fig. 2(i)). It was found that, as the aspect ratio decreases, the multiple circulation cells are formed within the triangular enclosure, at the high Ra. The fluid motion is found to be more intense in the right half of the enclosure and hence, the size of streamline circulation cells is observed to increase in size near those regions. This further results in the periodic isotherm distribution throughout the enclosure. In addition, the variation of the average Nusselt number with time was studied in details. It was observed that, the average Nusselt number at the bottom wall rises abruptly and thereafter, that slowly increases the steady-state value, for all aspect ratios (Fig. 2(ii)). The flow within a triangular enclosure representing an attic with the hot base, cooled inclined surface and insulated vertical side wall, has been modeled by Haese and Teubner [4]. The unsteady version of the vorticity-velocity formulation of the governing Eqs. (8a) and (8b) were solved numerically using the finite different method. The inflow/outflow system on the temperature distribution of the attic has been studied in order to eradicate the temperature-related problems in buildings. Studies were carried out for fluids with Pr ¼ 0:71 for various values of slopes of the inclined wall and Ra (710, 7100, 71,000 and 710,000). At the high Ra, for larger aspect ratios, the secondary circulation cells of the higher magnitude are observed at the top and left portions of the triangular enclosure with a large primary circulation cell, occupying the central core region. As the aspect ratio decreases, the primary circulation cells weaken and the secondary circulation cells expand in size and intensifies. This results in the wavy distribution of the isotherms throughout the enclosure at the high Ra and the distribution becomes qualitatively similar to that observed in the earlier work [42]. Further, the air temperature within the attic was found to be increased by adding more inflow points to the enclosure. Also, the inflow velocity plays the significant role in enhancing the temperature of fluid near the base of the attic. Joudi et al. [5] studied the performance of a prism shaped storage solar collector with a right triangular cross sectional area for the hot inclined wall and well insulated bottom and vertical walls. The governing equations (Eqs. (1a)–(1d)) are solved by the Galerkin finite element method using ANSYS software. The temperature distribution of water (Pr ¼ 7) in the storage solar collector at various times was observed via isotherms for the typical winter and summer days. It was found that, early in the day, the temperature distribution is symmetric due to the low velocity, which is insufficient to circulate the fluid (water) within the system (Fig. 3). However, as the day progressed, the convective effects become more prominent leading to the distortion of isotherms. This also resulted in the formation of multiple fluid circulation cells during the

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Fig. 2. (i) Steady-state streamfunction ðwÞ (left column) and isotherm ðhÞ (right column) with Prandtl number, Pr ¼ 0:72, Grashof number, Gr ¼ 105 and height-base aspect ratio, (a) A ¼ 1:0, (b) A ¼ 0:5, (c) A ¼ 0:3, (d) A ¼ 0:15 and (e) A ¼ 0:1 for hot bottom wall, cold hypotenuse and adiabatic vertical walls [42]; (ii) the average Nusselt number (Nu) vs time (t) for numerical experiments with Pr ¼ 0:72; Gr ¼ 105 for various A [42]. (figure is reproduced from Salmun [42] with permission from Elsevier)

evening time (Fig. 3). During the last hours, the whole tank content was found to be nearly at the same temperature because of the diminishing solar radiation and high velocity in the tank. Further, it was found that the insertion of a horizontal partition within the storage collector enhances stratification of the water and renders higher mean tank temperature and higher stored energy. This enhancement was found to be maximum for the height ratio 0:2 (ratio of the height of the partition and height of the triangular cav-

ity) and length ratio 0:6 (ratio of the length of the partition and base of the triangular cavity). Ridouane et al. [87] investigated the laminar natural convection in the air (Pr ¼ 0:7) filled right-angled triangular enclosure with the hot vertical wall, cold inclined wall and adiabatic horizontal wall. They performed the computational analysis using the finite volume method and reported the effect of the apex angle (5 6 a 6 63 ) on heat and fluid flows for various height based

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Fig. 3. Temperature (Kelvin) and velocity (m/s) distribution of water (Pr ¼ 7:2) in the storage solar collector for hot inclined wall and adiabatic bottom and side walls during the operation period at 1st of January [5]. (figure is reproduced from Joudi et al. [5] with permission from Elsevier).

Rayleigh numbers (RaH ), 103 6 RaH 6 106 . The convective term in the governing equations (Eqs. (1a)–(1d)) was discretized using the QUICK scheme whereas the pressure–velocity coupling was handled by the SIMPLE scheme. The mean wall heat flux at the vertical heated wall (qw ) has been evaluated and illustrated in this work [87]. At the low RaH (the height based Rayleigh number), the single vortex with the clockwise rotation was found to occur at the center of the enclosure, irrespective of the apex angles. However, at the high RaH (RaH ¼ 106 ), the vortex for the larger apex angles was observed to move downward to the bottom corners of the enclosure whereas the vortex for the smaller apex angle was found to be unaffected. In addition, the strength of the vortex is also enhanced with RaH as seen via the magnitude of the streamfunction gradient. It was also observed that the local Nusselt number attains the largest value a ¼ 15 whereas the smallest value occurs for a ¼ 63 . Note that the average heat transfer rate increases with RaH for each fixed apex angle. Also, the average heat transfer rate diminishes with the apex angles for each fixed RaH . Finally it was found that, the heat transfer rate within the enclosure enhances largely with the decrease in both a and RaH . Thus, it may be concluded that the inclination angle and RaH play the significant role in natural convection heat transfer within a triangular enclosure.

Oztop et al. [88] examined the convective heat transfer and fluid flow in a shed roof with or without eave for the summer boundary conditions. The non-dimensional streamfunction–vorticity form of the governing equations (Eqs. (6a)–(7))) were solved using the successive under relaxation (SUR) method. The boundary condition indicates that the inclined surface and eave were kept at the hot isothermal condition and the bottom wall of the roof was cooled with the adiabatic vertical wall. Numerical simulations were carried out involving various governing parameters such as the aspect ratio, eave length and Rayleigh number for Pr ¼ 0:7. It is interesting to observe that, the single fluid circulation cell occurs within the triangular enclosure at 103 6 Ra 6 105 , irrespective of the eave length and aspect ratio (AR). Also, the isotherms were found to be mostly parallel to the bottom wall and perpendicular to the insulated vertical wall, irrespective of the aspect ratios (AR), for the same range of Ra. Further, it is found that, at Ra ¼ 106 and AR ¼ 0:5, the large single primary circulation cell breaks up into multiple circulation cells due to the higher convective force. It is interesting to observe that, the formation of single fluid circulation cell takes place even at Ra ¼ 106 for aspect ratios other than AR ¼ 0:5, irrespective of the eave lengths considered. However, distorted isotherms occur within the enclosure with the increase in Ra, for all the aspect ratios (AR) and eave lengths. Further, it was observed that the heat trans-

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fer rate from the inclined wall to the bottom wall increases as the eave length increases. Also, it was found that the presence of the eave in the shed roof, increases the heat transfer rate. Overall, it may be concluded that the eave length and aspect ratio are the two most effective parameters to control the heat transfer rates for the identical Ra. Kent [89] carried out the studies on laminar natural convection in an enclosure of the isosceles triangular cross-section with the cold base and hot inclined walls for different base angles ð15 6 c 6 75 Þ and various Ra (103
105 ) for Pr ¼ 0:71. The governing equations (Eqs. (3a)–(3d)) were discretized using the finite volume method and solved using the SIMPLE algorithm. Base angles were found to have the significant influence on the fluid flow and heat transfer within the triangular enclosure. The uniform heating of the side walls results in the formation of two symmetric circulation cells. It is interesting to note that the qualitative trends of the streamlines remain similar for all Ra and base angles. As the base angle decreases, the size of the upper isothermal region reduces and the isotherms are found to be compressed towards the bottom wall. As a result, the high temperature gradient was found to occur in the lower section of the triangular region, especially near the edges of the bottom wall indicating a considerable proportion of the heat transfer rates in those regions. Thus, the triangular enclosures with the lower base angles are found to be more effective to achieve the higher heat transfer rate. Kent et al. [90] further numerically investigated the phenomenon of natural convection in the right angled triangular domain filled with air (Pr ¼ 0:71). The studies were carried out for various aspect ratios (0:1 6 A 6 4) and Rayleigh numbers (103 6 Ra 6 105 ) at different thermal boundary conditions: (i) case 1: the hypotenuse is cold, horizontal wall is adiabatic and vertical wall is hot (A ¼ 1), (ii) case 2: the hypotenuse is adiabatic, horizontal wall is hot and vertical wall is cold (A ¼ 1), (iii) case 3: the hypotenuse is cold, horizontal wall is hot and vertical wall is adiabatic (A ¼ 0:1; 0:5; 1; 2 and 4), (iv) case 4: the hypotenuse is cold, horizontal and vertical walls are hot (A ¼ 1), and (v) case 5: the quarter circle arc is cold, horizontal wall is adiabatic and vertical wall is hot (A ¼ 1). The finite volume method was used for discretization of the governing equations (Eqs. (3a)–(3d)) and subsequently, the equations were solved using the Fluent 6:0:12 commercial software. Results for the various cases were also compared with those of a quarter circular enclosure. It was observed that, for the higher aspect ratios (A), the effect of the hot bottom wall at the upper portion of the triangle is negligible as fluid forms a single circulation cell which covers only the middle and bottom portions of the triangular enclosure for 103 6 Ra 6 105 . As a result, the isotherms are observed to be smooth throughout the enclosure, depicting the dominance of conduction dominant regime, at the higher aspect ratios (A), throughout the range of Ra (Ra ¼ 103
105 ). However, the multiple streamline circulation cells are formed for the triangles with the lower aspect ratios, throughout the domain as Ra increases from 103 to 105 . In addition, the mean Nusselt number along the hot wall of the triangular enclosure for the lower aspect ratios was found to be higher which may be attributed to the high convective effect on the heat transfer in the triangle with the lowest aspect ratio. Overall, the mean Nusselt number for the hot wall was observed to increase with Ra for various thermal boundary conditions, signifying the enhanced convective heat transfer. Saha et al. [91] analyzed natural convection in an air filled (Pr ¼ 0:71) tilted triangular enclosure with the discrete bottom heating. The remaining parts of the bottom wall are insulated and the other two walls are isothermally cold. The dimensionless form of governing equations (Eqs. (1a)–(1d)) are modified for the inclined cavity and they are solved using the finite element based

adaptive meshing technique. Numerical tests were conducted with various aspect ratios (0.5–1) and inclination angles (0°–60°). The Grashof number based on the enclosure height was varied from 103 to 106 . As the aspect ratio increases, the primary circulating cell gradually expands within the enclosure space, resulting in higher recirculation strength. The tiny secondary circulation cells are observed near the right corners of the triangular enclosure and those gradually decrease in size as Gr increases. Also, the isotherms are mostly compressed near the heater and they are also largely distorted in the core, at higher Gr. For the lower aspect ratios (A ¼ 0:5) with the inclination angle, u ¼ 0 , the symmetric circulation cells are observed for Gr ¼ 103
106 . It is interesting to observe that for A ¼ 0:5, the secondary circulation cells in the right bottom corners decrease in size with the inclination angle. As a result, the magnitude of the primary circulation cell increases and isotherms are strongly compressed, resulting in a large temperature gradient near the right wall. Further, the size of the heater was found to have the significant effect on the heat transfer rate within the enclosure. The average Nusselt number is found to decrease with Gr whereas that is observed to increase with the inclination angle. The optimum thermal performance was achieved with the small heater size at the higher Grashof number and larger inclination angles. Basak et al. [92,93] implemented the penalty finite element method with the pressure velocity formulation (Eqs. (3a)–(3d)) in order to study the effect of various thermal boundary conditions during natural convection in the right-angled and isosceles triangular enclosures. Both the studies involve two different cases of thermal boundary conditions. Basak et al. [92] considered the following cases within the right angled triangle enclosure: (i) the uniform or linear heating of the vertical wall with isothermally cold inclined wall (ii) the uniform or linear heating of the inclined wall with isothermally cold vertical wall. Further, Basak et al. [93] also investigated the effect of two different thermal boundary conditions in the isosceles triangle enclosure: (i) case 1: the uniformly heating of the inclined walls with cold isothermal bottom wall (ii) case 2: the non-uniform heating of the inclined walls with cold isothermal bottom wall. Studies were carried out over the wide range of Ra (103 6 Ra 6 105 ) and Pr (0:026 6 Pr 6 1000) numbers. At the onset of convection dominant mode, the isotherms are compressed towards the hot wall. However, at the higher Ra, the isotherms are highly distorted and confined to only some portion of the triangle. The conduction dominant heat transfer was observed up till Ra 6 6  104 , during the uniform and non-uniform heating of the side walls with Pr ¼ 0:026 [93]. Further, it was found that the overall heat transfer rate is lower for the linear heating cases. Overall, it was concluded that the Rayleigh number (Ra) has the stronger effect on the heat transfer rate for all Prandtl numbers (Pr). Koca et al. [94] investigated the effect of Prandtl number on natural convection in air (Pr ¼ 0:71) filled triangular enclosures with the localized heating from below. The finite-difference technique was used to solve the governing equations (Eqs. (6a)–(7)). The bottom wall of the triangle was heated partially with the heater while the inclined wall was maintained at a lower uniform temperature. The remaining portion of the bottom wall and the entire vertical wall were kept insulated. The detailed analysis was carried out for the dimensionless heater locations (0:15
0:95), dimensionless heater length (0:1
0:9), Prandtl number (0:01 6 Pr 6 15) and Ra (103 6 Ra 6 106 ). The results were presented via streamlines and isotherms involving the aspect ratio of 1 for various heater locations and heater lengths. At Ra ¼ 103 and 104 , the fluid rises from the middle portion of the heater and falls down along the side walls forming two fluid circulation cells (Fig. 4(i)(a and b)). Due to the inclined wall, lesser fluid movement occurs in the right half

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and consequently the size of the fluid circulation cell in that region decreases, in contrast to the circulation cell present in the left half. Note that, at Ra ¼ 103 , smooth isotherms are observed throughout the enclosure depicting the dominance of the viscous force over buoyancy force (Fig. 4(i)(a)). However, at Ra ¼ 104 , the isotherms are distorted and they are largely compressed towards the inclined wall and heater (Fig. 4(i)(b)). At Ra ¼ 105 , the strength of the fluid is found to be increased and a large primary circulation cell is formed in the counter clockwise direction (Fig. 4(i)(c)). However, the multiple cells with the large intensity are formed at the convection regime with the higher Ra (Fig. 4(i)(c and d)). It was also found that the heat transfer rate is increased with the heater length, Rayleigh number and Prandtl number. Also, the heat trans-

13

fer rate was found to be maximum when the heater was present near the right corner of the triangular enclosure (Fig. 4(ii)(a–c)). Varol et al. [95] carried out numerical studies on natural convection heat transfer in a triangular enclosure with a flush mounted heater on the vertical wall. The remaining portion of the vertical wall was maintained adiabatic. In addition, the bottom wall was also maintained adiabatic whereas, the inclined wall was isothermally cooled. The finite difference method has been used to solve the governing equations (Eqs. (6a)–(8b)) and linear algebraic equations were solved via the Successive Under Relaxation (SUR) technique. The results were obtained in terms of streamlines and isotherms for various Rayleigh numbers (104 6 Ra 6 106 ) with Pr ¼ 0:71. The effect of the location of heaters and aspect ratio of

Fig. 4. (i) Streamfunction ðwÞ and isotherm ðhÞ for Prandtl number, Pr ¼ 0:71, dimensionless heater length, w ¼ 0:5, dimensionless heater location, s ¼ 0:5, aspect ratio, A ¼ 1 for various Rayleigh numbers, (a) Ra ¼ 103 (b) Ra ¼ 104 (c) Ra ¼ 105 and (d) Ra ¼ 106 [94]; (ii) variation of mean Nusselt number (Nu) with Rayleigh number (Ra) for various values of Pr (Pr ¼ 0:01
15) and A ¼ 1:0 (a) w ¼ 0:2 and s ¼ 0:5, (b) w ¼ 0:8 and s ¼ 0:5 and (c) w ¼ 0:2 and s ¼ 0:2 [94]. (figures are reproduced from Koca et al. [94] with permission from Elsevier).

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the triangle on the mean Nusselt number (Nu) were also studied. Fig. 5(i) illustrates the streamlines and isotherms for the case involving the central positioning of the heaters along the vertical wall for various Ra. A single clockwise fluid circulation cell is observed for all Ra (Fig. 5(i)). As Ra increases, the vortex of the fluid circulation cell shifts to the top corner of the triangle (Fig. 5(i)(a– c)). Isotherms demonstrate a semi-circle shaped distribution at the low Ra due to the lesser intensity of fluid flow. As Rayleigh number increases, isotherms are found to be more compressed towards the heaters and upper half of the inclined wall due to the high thermal mixing near the core (Fig. 5(i)(a–c)). It is found that the mean Nusselt number (Nu) is almost invariant with Ra for the case involving the heater positioning along the upper half of the left wall (Fig. 5(ii)(a–c)). On the other hand, Nu is found to increase monotonically with Ra for the cases involving the location of the heaters along the lower and central portion of the vertical wall (Fig. 5(ii)(a–c)). It may also be noted that the overall Nu

decreases with the aspect ratio for all the heater positions along the vertical wall. Sojoudi et al. [96] carried out the numerical simulations in order to study the unsteady air flow and heat transfer in a partitioned triangular cavity which was differentially heated from the left inclined wall. Also, an additional heat source was placed at the bottom wall of the triangular cavity. The finite volume numerical method was employed to solve the governing equations (Eqs. (1a)–(1d)). The effect of various parameters on the fluid flow and heat transfer was studied including Rayleigh number (103 6 Ra 6 106 ), heater size (hs ¼ 0:2
0:6) and aspect ratio (A ¼ 0:1
1) with Pr ¼ 0:72. It was observed that less time is needed at higher Ra to reach the steady state. At higher Ra, isotherm distribution depicts the formation of higher thermal gradient adjacent to the left inclined wall and the effect was found to be more prominent at Ra ¼ 105 . At Ra ¼ 103 ; 104 and 106 , a single large fluid circulation cell is observed within each left and right

Fig. 5. (i) Streamfunction (w) and isotherm (h) for the central positioning of the heater (P2) involving the aspect ratio, A ¼ 1, along the vertical wall for (a) Ra ¼ 104 , (b) Ra ¼ 105 and (c) Ra ¼ 106 for Pr ¼ 0:71 (ii) Variation of mean Nusselt number (Nu) with different Rayleigh number (Ra) for various aspect ratios, (a) A ¼ 0:3, (b) A ¼ 0:6, (c) A ¼ 1 and heater positions (P1: heater located along the upper half of the left wall, P2: heater centrally located along the left wall, P3: heater located along the lower half of the left wall) [95]. (figures are reproduced from Varol et al. [95] with permission from Elsevier).

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halves of the cavities whereas, at Ra ¼ 105 , the single fluid circulation cell in the right half of the cavity is split into smaller cells under the influence of dominant natural convection. For the greater heater size (hs P 0:5), the multiple circulation cells in the right half of the cavity is found to expand whereas a single fluid circulation cell still occurs in the left half of the cavity. In addition, it is also found that the thermal boundary layer thickness is increased along the left wall for the greater heater size (hs P 0:5). For the higher values of A, the number of vortices is reduced and a single large cell is formed in each left and right halves of the cavity. The isotherms depict that the left half of the cavity is almost maintained hot at higher A. It is found that the variation of Ra; A and hs does not have any significant effect on the heat transfer rate of the left wall (Nu). It is also observed that Nu of the left wall decreases with time. Basak and co-workers [70,97,98] also investigated natural convection phenomena with the heat flow visualization in triangular enclosures via the heatline (Eq. (36)) approach. Basak et al. [70] analyzed the heat flow pattern in an inverted triangular cavity with the hot inclined walls and cold horizontal wall using Bejan’s heatline concept. Their study focuses on the effect of the Prandtl number (Pr ¼ 0:015
1000) and Rayleigh number ðRa ¼ 102
105 ) for the heat flow visualization during natural convection. It was reported that at the low Ra, the heatlines are smooth and perfectly normal to the isotherms indicating the dominance of conduction. In addition, the multiple secondary circulation cells were formed for the fluid with the low Pr and they gradually disappear with Pr (Pr P 0:7). Also, the overall heat transfer rate shows the wavy pattern due to the multiple circulation cells at Pr ¼ 0:015 and 0:026 and this pattern disappears at the high Pr. Kaluri et al. [97] carried out the detailed analysis on the effects of the aspect ratio and thermal boundary conditions for the fluid and heat flow within right angled triangular enclosures for a wide range of fluids at various Ra (Ra ¼ 103
105 ) via the heatline approach. The aspect ratio of the triangular domain was adjusted by varying the top angle, which represents the angle between the vertical and inclined walls. It was observed that, the maximum heat flux at the top vertex decreases and the thermal mixing within the cavity increases with the top angles. For the lower top angles, the heatlines in the upper zone remain nearly parallel even at the higher Ra, depicting the lesser heat transfer rate to the upper portion of the enclosure. However, as the top angle increases, the circular heatline cells cover the major portions of the cavity, depicting the convection dominant mode. It is also found that, the fluid in the lower corners is adequately heated in the presence of the hot right wall in contrast to the cold left wall. Overall, the isotherms and streamfunctions are found to be qualitatively similar to the previous works [92,93] irrespective of Ra and Pr. Further, the heat transfer characteristics, in terms of the average Nusselt numbers, indicate that the isothermal heating cases exhibit the exponential decrease in Nusselt number with Ra whereas the linear heating cases show the local intermediate maxima. Later, Basak et al. [98] investigated the indirect heat recovery through the entrapped triangular enclosures involving material processing applications. The parametric study was performed for a wide range of fluids ðPr ¼ 0:015
1000Þ to analyze the heat transfer process. At the low Ra (Ra ¼ 103 ), heat transfer was mostly conduction dominant especially for Pr ¼ 0:015. As Ra increases to 104 , the secondary circulation cells occur in the lower triangle and the primary circulation cells are found to be slightly distorted due to the the onset of convection. At the high Ra (Ra ¼ 105 ), the magnitude of streamfunctions and heatfunctions are found to be intense and that further depicts the strong dominance of convection heat transfer process (Fig. 6(a)). As Pr increases to 0:7, the fluid

Fig. 6. Isotherm ðhÞ, streamfunction ðwÞ and heatfunction ðPÞ at Rayleigh number, Ra ¼ 105 involving various Prandtl numbers (a) Pr ¼ 0:015 and (b) Pr ¼ 1000 [98]. (figures are reproduced from Basak et al. [98] with permission from Elsevier).

and heat flow is found to be intensified. However, the multiple circulations cells still occur in the lower triangle similar to Pr ¼ 0:015. The heat transport within the enclosure is found to be convection dominant based on the multiple flow and heat circulation cells for Pr ¼ 0:015 and Pr ¼ 0:7 fluids. On the other hand, for high Pr (Pr ¼ 1000) fluids, the multiple circulations disappear and heatlines are found to be dense near the bottom portion of the inclined walls within the lower triangular cavity (Fig. 6(b)) depicting larger heat transfer rate along that zone. Overall, the optimum heat transfer rate was observed for large Pr fluids (Pr ¼ 1000) in the lower triangular cavity and for all Pr fluid ðPr ¼ 0:015
1000Þ in the upper triangular cavity. Analysis of the irreversibility based on the entropy generation [74,75] within triangular cavities have been carried out in the past few years. Varol et al. [74] carried out the numerical study on free convection in an air filled (Pr ¼ 0:71) isosceles triangular enclosure partially heated from the below and cooled from the sloping walls.

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The momentum and energy balance equations (Eqs. (6a) and (6b)) associated with the entropy generation equation (Eq. (39b)) are solved using the finite difference method. The effect of various governing parameters such as the Rayleigh number, length of heater, location of heater and inclination angle of triangle have been studied in details. Note that, the isotherms were largely compressed towards the left inclined wall and bottom wall, depicting large temperature gradients in those regions (Fig. 7(a–c)). It is also observed that the conduction mode of heat transfer is effective for small values of the inclination angle of the sloping wall due to the short distance between the hot and cold walls. At the high Ra (Ra P 105 ), the multiple fluid circulation cells and plume type temperature distributions were formed due to the strong convection (Fig. 7(b and c)). The length of the heater was also found to be more effective at higher values of inclination angles of the slop-

ing wall due to the dominance of convection. Also, the entropy generation due to the fluid friction was found to increase with Ra and it was found in the regions with high velocity gradients. It may be noted that the location of heaters play significant role in the entropy generation due to the fluid friction along the side walls. It was observed that the central positioning of the heaters along the bottom wall results in the formation of a large fluid circulation cells in the right half whereas the small fluid circulation cell is observed along the left half of the triangular enclosure. Therefore, the zones with the large entropy generation due to the fluid friction are observed along the middle portion of the right wall, lower half of the left wall and middle portion of the bottom wall. As the heaters are shifted towards the right corner of the bottom wall, the large primary fluid circulation cell occupies almost the entire enclosure and consequently, the entropy generation

Fig. 7. Streamfunction (w), isotherm (h), entropy generation due to heat transfer (Sh ) and entropy generation due to fluid friction (Sw ) for sloping wall inclination angle of 0 cavity, b ¼ 60 , dimensionless location of heater, c0 ¼ c=L ¼ 0:25, dimensionless length of the heater, l ¼ l=L ¼ 0:50 for various Rayleigh numbers (a) Ra ¼ 104 , (b) Ra ¼ 105 and (c) Ra ¼ 8:8  105 at Pr ¼ 0:71 [74]. (figure is reproduced from Varol et al. [74] with permission from Elsevier).

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zones are shifted towards the upper left and right walls of the triangular enclosure. Note that, the entropy generation due to the heat transfer is found to be maximum near the edges of the heater, due to large heat flux in those regions. Overall, it was concluded that, the entropy generation depends largely on the sloping wall inclination angle and thus the sloping wall angle can be used as a parameter to control the heat transfer, energy saving and entropy generation. Recently, Basak et al. [75] studied the role of the entropy generation within the right angled triangular enclosure during natural convection. Four different heating strategies (case 1: isothermal hot left wall and isothermal cold right wall, case 2: linearly heated left wall and isothermal cold right wall, case 3: isothermal cold left wall and isothermal hot right wall, case 4: isothermal cold left wall and linearly heated right wall) were considered in their numerical study (Eqs. (3a)–(3d) and (39a)). Computations were carried out for a wide range of Rayleigh numbers (Ra ¼ 103
105 ) during the thermal processing of various fluids (Pr ¼ 0:025 - 1000) at various top angles (u ¼ 15 and 45 ). Note that, the isotherms and streamfunctions observed for the present work are qualitatively similar (Fig. 6(a and b)) to the earlier works [92,93]. At the low Rayleigh number (Ra ¼ 103 ), the heat transfer is primarily due to conduction and Stotal (Eq. (39a)) is found to be higher for the inclination angle, / ¼ 15 for all Pr due to high Sh;total and that is further due to the high thermal gradient. As Ra increases to 105 , the fluid flow as well as thermal energy transport is intensified due to the enhanced convection. Consequently, the maximum value of the entropy generation due to the fluid friction (Sw;max ) also increases for all u’s. It was found that the maximum value of the entropy generation due to the heat transfer (Sh;max ) occurs near the vertex of the enclosures for the cases 1 and 3, near the corner between the left wall and bottom wall for the case 2 and near the lower portion of the right wall for the case 4. On the other hand, the maximum value of the entropy generation due to the fluid flow (Sw;max ) occurs near the side walls for all the cases depicting the presence of high velocity gradients in those regions. Finally, it was concluded that, the triangular enclosure with the top angle, / ¼ 15 gives the high average heat transfer rate and minimum entropy generation for all the fluids at Ra ¼ 105 . 3.2. Porous media A number of studies on natural convection in porous triangular enclosures [48,99–103] have been carried out by the earlier investigators. Varol et al. [48] studied the effect of adiabatic thin fin on buoyancy driven convection and fluid flow in a right angled triangular enclosure filled with the porous media. The vertical wall of the enclosure was insulated while both the hot bottom wall and the cold inclined walls were kept at isothermal temperatures. The Darcy’s model (Eqs. (14a)–(14c)) was used to define the governing equations and the model was solved using the finite difference technique. The studies were carried out by varying the thin fin position (0:2
0:6), height-base aspect ratio of the triangular enclosure (A ¼ 0:25
1), Darcy–Rayleigh number (RaD ¼ Ra  Da ¼ 100
1000) and dimensionless height of the fin (h ¼ 0:1
0:4). It is observed that, the isotherms on the right side of the fin show a stratified flow at the convection dominant regime (at the high RaD ) (Figure not shown). However, the presence of the adiabatic fin causes the lower heat transfer at the left part of the fin compared to the right part and the fin behaves as a curtain between the hot and cold walls at the high RaD . Note that, the multiple fluid circulation cells are formed within the right angled triangular enclosure when the adiabatic fin is located in the left half of the bottom wall. It is interesting to observe that, changing the fin position from the left half to the right half along the bottom wall

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results in the formation of a large flow circulation cell encompassing the entire left half whereas the small fluid circulation cells are confined near the right corners of the triangular enclosure (Figure not shown). Also, changing the fin position results in the lesser distortion of isotherms near the central core whereas the strong periodic behavior of isotherms are observed near the right corner of the triangular enclosure (Figure not shown). As the fin length increases, the enlargement of the primary circulation cell in the left half occurs. However the qualitative feature of the multiple circulation cells in the right half does not vary with the fin length. It was reported that the average Nusselt number is an increasing function of Darcy–Rayleigh number (RaD ), but the average Nusselt number was observed to be constant at very small RaD due to the dominance of the quasi-conductive heat transfer regime. Also, both the average and local Nusselt numbers were found to decrease with the aspect ratio. Varol et al. [99] also performed the numerical tests to analyze natural convection in porous right angled triangular enclosures with various sets of temperature boundary conditions on the vertical, bottom, and inclined walls, (i) case 1: insulated vertical wall, hot bottom wall and cold hypotenuse, (ii) case 2: insulated vertical wall, cold bottom wall and hot hypotenuse, (iii) case 3: hot vertical wall, insulated bottom wall and cold hypotenuse, (iv) case 4: cold vertical wall, insulated bottom wall and hot hypotenuse, (v) case 5: cold vertical wall, hot bottom wall and insulated hypotenuse, and (vi) case 6: hot vertical wall, cold bottom wall and insulated hypotenuse. The aspect ratio (0:25
2) and Darcy–Rayleigh number (RaD ¼ 100
1000) were varied in order to study the flow and temperature distributions within the triangular enclosure. The solutions of the governing equations (Darcy model) (Eqs. (14a)– (14c)) and linear algebraic equations were carried out using the central difference and the successive under relaxation method (SUR), respectively. At the high RaD , it is interesting to observe that, multiple vortices occur for the case with the cold hypotenuse wall, hot bottom wall and insulated vertical wall (case 1). This may be due to the dominance of buoyancy forces and the larger slope of the inclined wall, which results in the rapid development of the multiple circulation cells, even at the corner regions. Also, the wavy and distorted isotherms are seen due to the intense convection at the higher RaD . However, at the low RaD , the formation of a large single circulation cell takes place almost for all the test cases. It was found that the local heat transfer rate is an increasing function of RaD for all the cases. Also, the effect of the aspect ratio on the local heat transfer rate strongly depends on the thermal boundary conditions. Further, Varol et al. [100] presented the comprehensive analysis on the effect of natural convection, due to various inclination angles (0 to 360 ) of a porous triangular enclosure relative to the gravity, using the Darcy model (governing Eqs. as given in (14a)–(14c) are modified for the inclined cavity). The vertical wall of the inverted right angled triangular cavity was hot and horizontal top wall was cold with the adiabatic hypotenuse. The heat transfer rate was observed to increase with Darcy–Rayleigh number (RaD ) for all values of the inclination angle. It was found that the maximum and minimum values of the average Nusselt number at the hot wall occur for the inclination angles 330 and 210 , respectively. In addition, for the inclination angle of 210 , the isothermal walls resulted in the streamline cells to circulate separately in the triangular enclosure, without mixing and thus, the fluid flow was found to be stable. However, it was also found that, for inclination angles within 0
90 and RaD > 1000, the fluid flow field becomes unstable and consequently, the high mixing occurs in the triangular enclosure. Isotherms are also found to be distorted for the inclination angles within 0
90 at RaD > 1000. However, for the inclination angles greater than 90 , the thermal mixing is found to be lesser in a large portion of the enclosures.

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Overall, it was concluded that the inclination angle can be considered as a control parameter for the heat transfer and flow field especially at the high RaD numbers. Basak et al. [101] studied the effect of the uniform and non-uniform heating of the bottom wall on natural convection in a porous isosceles triangular cavity using the Darcy–Brinkman model (Eqs. (19a)–(19d)). The governing parameters are considered as Darcy number (10
5 6 Da 6 10
3 ), Rayleigh number (103 6 Ra 6 106 ) and Prandtl number (0:026 6 Pr 6 1000). It was found that at the low Da (Da ¼ 10
5 ), the heat transfer is mainly due to conduction irrespective of Ra and Pr. Similar to the previous study [93], at the onset of convection dominant mode, isotherms are strongly compressed towards the bottom wall and the compression is found to be stronger for the uniform heating case. At the high Ra, the compression of isotherms near the corners of the bottom wall also results in the multiple circulation cells for Da ¼ 10
3 and Pr ¼ 0:026. Similar to the earlier study [93], the streamfunction contours are observed to be circular in shape at the low Pr (Pr ¼ 0:026) whereas at the high Pr (Pr ¼ 1000), they resemble the shape of the cavity. Finally, it was concluded that the average heat transfer rate is an increasing function of Da and average Nusselt number follows the power law variations with Ra for various Da and Pr involving the convection dominant regime. Oztop et al. [102] examined the effects of the density variation and aspect ratio on natural convection in right angled triangular porous enclosure for the isothermally hot bottom wall, cold vertical wall and well insulated hypotenuse. The Darcy model based Eqs. (9) and (11) and the Non-Boussinesq model based momentum equations are solved using the finite difference method. The calculations were performed for various Darcy–Rayleigh numbers (RaD ¼ Ra  Da) (50 6 RaD 6 1000) based on the Non-Boussinesq models and height-base aspect ratio ð0:25 6 A 6 1:0Þ. It was found that the multiple circulation cells are formed at all RaD due to the effect of the density inversion in a porous triangular enclosure. Isotherms also deviate from their directions due to the effect of the high flow velocity. Note that, as RaD increases, the S-shaped isotherms were formed and consequently the distortion occurs in isotherms at the high RaD . In addition, the overall heat transfer rates become identical for all the aspect ratios (A) at the high Darcy–Rayleigh number (RaD ¼ 1200). Varol [103] reported the effect of the conducting body on natural convection in the porous right angled triangular enclosure with the hot bottom wall, cold hypotenuse and insulated vertical wall using the Darcy model (Eqs. (14a)–(14c)). The center of the conducting body was located onto the center of the gravity of the right-angled triangular cavity. The numerical studies were carried out by varying the aspect ratio of the conducting body at various values of RaD (100
1000). The thermal conductivity ratio is assumed as unity since the thermal conductivity of fluid and solid are equal. At the low RaD (RaD ¼ 100), two fluid circulation cells were observed near the right bottom corner of the triangle and the top of the square body. Note that, the flow strength is stronger near the right bottom corners due to a high temperature difference between the hot and cold surfaces. At the highest value of the Darcy–Rayleigh number (RaD ), multiple circulation cells were observed and the isotherms were highly distorted around the solid object. It is also interesting to observe that the change in the width and height of the solid object drastically influenced the fluid flow pattern within the enclosure. However, the multiple circulation cells were observed for almost all the cases considered, at the high RaD (RaD ¼ 500), depicting the dominance of convection. The heat transfer rate was observed to increase with the thermal conductivity of the body and RaD . Note that, the heat transfer rate was observed to be almost constant throughout the regime for the high

values of thermal conductivity. Finally, it was concluded that, the position of the conducting body within the enclosure has significant effect on the heat transfer and flow behavior compared to the thickness of the body especially at the high RaD . A few studies on natural convection in the porous media filled triangular enclosures based on the heatline approach have been performed by Varol et al. [104] and Basak and coauthors [105– 107]. Varol et al. [104] analyzed the heat flow in a porous right angled triangular enclosure via the heatline method (Eq. (36)). The Darcy model based momentum and energy balance equations (Eqs. (14a)–(14c) and Eq. (36)) are solved by the finite difference technique. Three different thermal boundary conditions were applied for the vertical and inclined walls (case 1: non-uniform heating of the bottom wall with the cold inclined and vertical walls, case 2: non-uniform heating of the bottom wall with the cold inclined wall and adiabatic vertical wall, case 3: non-uniform heating of the bottom wall with the cold vertical wall and insulated inclined wall). The numerical studies were carried out for the various values of the Darcy–Rayleigh number (RaD ¼ 100
1000) and aspect ratio (0:25 6 A 6 1). Due to the maximum heating near the central zone of the bottom wall, the fluid rises from the middle portion of the bottom wall and flows down along the side walls (Fig. 8 (a–d)). Note that, for the case 1, the size and intensity of the fluid circulation in the left half is greater than the one on the right half due to the large space available for the fluid movement in the left half (Fig. 8(a–d)). For the cases 2 and 3, the primary fluid circulation cell occupies nearly the entire cavity with small secondary cells at the bottom corner regions (Figures not shown). As RaD increases to 1000, the primary fluid circulation cell on the left half spreads largely towards the top half and compresses the secondary fluid circulation cell to the bottom right corners of the enclosure due to the intense convection in the case 1 (Fig. 8(a–d)). In addition, the strength of the fluid circulation cells largely increases for all the cases due to the enhancement of buoyancy forces. Isotherm contours represent the plume type distributions in the case 1 (Fig. 8 (a–d)). Due to the presence of the inclined wall, the isotherms are observed to have skewness towards the left vertical wall of the enclosure. The magnitude of the heatfunction increases with Darcy–Rayleigh number (RaD ) as seen from the heatline contours (Figures not shown). This shows the heat transfer rate from the bottom wall increases with RaD . Further, it was seen that the decrease in the aspect ratio (A) led to the formation of Benard cells and the isotherms show the wavy variation for the case 1 (Fig. 8(a–d)). Four vortices were also observed based on the heatfunction contours. It was found that the qualitative distributions of streamlines, isotherms and heatlines for the case 2 are similar to the case 1 for the cavity aspect ratio, A ¼ 0:5. In contrast to the cases 1 and 2, Benard cells did not form for the case 3 at the low aspect ratio (A ¼ 0:25). Finally, the average heat transfer rate was found to be an increasing function of Darcy–Rayleigh number (RaD ), irrespective of the cases. Also, the maximum average heat transfer rate was observed for the case 3 type boundary condition. Basak et al. [105] analyzed the natural convection (Eqs. (19a)– (19d)) in an inverted porous triangular cavity with the hot inclined walls and cold top wall. The effects of Prandtl numbers (Pr ¼ 0:015
1000) and Darcy numbers (Da ¼ 10
5
10
3 ) on natural convection have been investigated in this study. Further, Basak et al. [106] studied the heat recovery and heat transfer rates in entrapped porous triangular cavities. Also, Basak and coauthors [107] carried out the numerical studies on heat distribution and thermal mixing during the steady laminar natural convective flow within a porous right-angled enclosure. They examined the influence of various thermal boundary conditions and inclination angles on the evaluation of complex heat flow patterns and reported that the isothermal heating of walls enhances the heat

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Fig. 8. Streamfunction (w), isotherm (h) and heatfunction (P) at RaD ¼ 1000 for various aspect ratios, A (a) A ¼ 1:0, (b) A ¼ 0:75, (c) A ¼ 0:50 and (d) A ¼ 0:25 for case 1 (cold vertical and inclined walls with non-uniformly heated bottom wall) at Pr ¼ 0:71 [104]. (figures are reproduced from Varol et al. [104] with permission from Elsevier).

distribution and thermal mixing. Common to all the studies, it was observed that, at the small Darcy number (Da ¼ 10
5 ), the magnitudes of streamfunction are small and the heatlines are orthogonal to the isotherms, indicating the conduction dominant heat transfer. As Da increases to 10
4 , convection is initiated within the enclosures. Dense heatlines are observed near the bottom portion of the inclined wall for Da ¼ 10
4 . As Da increases to 10
3 , the strong influence of convection is observed based on the higher strength of the streamfunctions and heatfunctions. As Pr increases from 0:015 to 1000, the intensity of the dense heatlines increases. The enhanced thermal mixing occurs near the core within the triangle for the fluids with higher Pr as the heat transfer at the core is dominated by convection as seen from the heatlines. The overall heat transfer rate at the active walls remains almost invariant at the low Da due to the conductive heat transfer irrespective of Ra and

Pr. However the overall heat transfer rate shows increasing trend at the high Da. The number of studies on the entropy generation during natural convection in triangular enclosures filled with the fluid-saturated porous media is very limited. Varol et al. [83] numerically studied the entropy generation during natural convection in an isosceles inverted triangular enclosure with various inclination angles filled with the fluid-saturated porous medium. The inclined walls (equal length) of the triangular enclosure were maintained adiabatic whereas the top wall was non-uniformly heated. The finite difference formulation was used for the dimensionless governing equations [basic governing equations as given in Eqs. (14a)–(14c) are modified for the inclined cavity], and entropy generation equations (Eq. (41b)) for the present problem. Computations were performed for various Darcy–Rayleigh numbers (RaD ¼ Ra  Da ¼ 102
103 )

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and inclination angles (/ ¼ 0
180 ). As the top wall is sinusoidally heated, the middle portion of the top wall receives maximum heat. Thus, the fluid near those regions rotates in the opposite direction towards the side walls and forms two symmetric circulation cells for / ¼ 0 and 180 . On the other hand, for / ¼ 45 and 135 , the asymmetric pair and for / ¼ 90 , multiple circulation cells are observed. Note that, the active zones of the fluid friction irreversibilities occur where the fluid circulation cells undergo friction with the walls of the enclosure (Fig. 9(a–d)). On the other hand, the maximum entropy generation due to the heat transfer takes place near the non-isothermally heated wall, due to high thermal gradients in those regions (Fig. 9(a–d)). The intensity of the fluid flow increases with RaD irrespective of / due to the enhanced convection. Consequently, the heat transfer rate along with fluid friction and heat transfer irreversibilities also increase. It is interesting to observe that, as RaD increases, the fluid friction irreversibility dominates over the heat transfer irreversibility for all /. Note that, the minimum entropy generation due to the heat transfer as well as the fluid friction occurs for / ¼ 180 at the high RaD . Basak et al. [84] have analyzed the entropy generation due to natural convection (Eqs. (19a)–(19d) and Eq. (44)) in the right-angled triangular enclosures saturated with the porous media for four different thermal boundary conditions (case 1: isothermal hot left wall and isothermal cold right wall, case 2: linearly heated left wall and isothermal cold right wall, case 3: isothermal cold left wall and isothermal hot right wall, case 4: isothermal cold left wall and linearly heated right wall) similar to the earlier work [107]. The Galer-

kin finite element method has been used to solve the governing equations (Eqs. (19a)–(19d)). In addition, the entropy generation terms (Eq. (44)) have also been evaluated via finite element basis functions. Analysis has been done for Pr ¼ 0:025 and Pr ¼ 1000 with Da ¼ 10
5
10
3 involving two different top angles / ¼ 15 and 45 . It was found that, the maximum heat transfer irreversibility (Sh;max ) is high near the vertex of the cavity for the case 1. On the other hand, Sh;max was significant at the junction between the linearly heated left wall and adiabatic bottom wall for the case 2 due to the large thermal gradient. The fluid friction irreversibility (Sw ) was found to be larger at the middle portions of the side walls for both the cases 1 and 2 and the generation of entropy due to the fluid friction increases at the high Da (Da P 10
4 ) for all the inclination angles. Note that, Sh;max and Sw;max were observed to be higher for the case 1 compared to the case 2 for all the inclination angles and Pr with Da ¼ 10
5
10
3 . The maximum heat transfer rates with minimum total entropy generation and higher cup-mixing temperature were obtained for / ¼ 15 at Da ¼ 10
3 irrespective of Pr. The active zones of Sh and Sw for the cases 3 and 4 were similar to the cases 1 and 2, respectively. Similar to the cases 1–2, Sw increases for Da P 10
4 for all / and the maximum values of Sh and Sw are larger for the case 3 compared to the case 4 irrespective of Pr; / and Da. The maximum heat transfer rate as well as minimum Stotal was found for / ¼ 15 cavities at Da ¼ 10
3 for all Pr except at Pr ¼ 0:025 in the case 3 where the total entropy generation is maximum for / ¼ 15 cavities. In addition, the cup mixing temperature was higher for

Fig. 9. Entropy generation due to fluid friction irreversibility (Sw ) and heat transfer irreversibility (Sh ) for adiabatic side walls with non uniformly heated top wall for Darcy– Rayleigh number, RaD ¼ 500 for different inclination angles, (a) / ¼ 45 (b) / ¼ 90 , (c) / ¼ 135 , (d) / ¼ 180 [83]. (figure is reproduced from Varol et al. [83] with permission from Elsevier).

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/ ¼ 45 cavities compared with / ¼ 15 cavities in the case 3 at Da ¼ 10
3 for all Pr. 3.3. Nanofluid The numerical study on natural convection in a right triangular enclosure filled with a water–CuO nanofluid, with a heat source on its vertical wall was studied by Ghasemi and Aminossadati [63]. Their study was based on the effect of the solid volume fraction (/), heat source location, Rayleigh number ðRaÞ, height-base aspect ratio (A) and Brownian motion on the overall heat transfer rate. The Brownian motion is incorporated into the governing equations (Eqs. (27a)–(27d)) and they are solved by the control volume method. In order to study the effect of Ra, comparative studies were carried out for both nanofluids (water–CuO) and conventional fluid (water). The isotherms and streamlines show similar qualitative features for the clear fluid and nanofluid at various Ra. The position of the heater plays the significant role on the heat transfer enhancement. During the conductive mode of heat transfer at the low Ra, the local heat transfer rate increases as the heater moves upwards. At the high Ra (Ra ¼ 106 ), the convective cell becomes weaker as the heater moves upwards and thus the heat transfer rate decreases. It may be noted that, the local heat transfer rate is significantly affected for different aspect ratios (A) for the fixed position and length of the heater, Ra and solid volume fraction. It was found that, the heat transfer rate decreases with the aspect ratios at the low Ra whereas at the high Ra (Ra ¼ 105 ), an increasing trend of heat transfer rate with the aspect ratio is observed. It is worthwhile to mention that, when the Brownian motion is incorporated, the solid volume fraction has diverse effects on the heat transfer rate for various Ra. At the low Ra, the heat transfer rate was observed to increase with the solid volume fraction, whereas, at the high Ra, an optimum solid volume fraction leads to a maximum heat transfer rate. Interestingly, the heat transfer rate was found to increase with the solid volume fraction for all Ra, when the Brownian motion was not considered. Further, Aminossadati and Ghasemi [64] studied and compared the buoyancy flow and heat transfer in a partially heated isosceles triangular enclosure filled with Ethylene Glycol-Copper (Eg-Cu) nanofluid and pure ethylene glycol. The modified Maxwell model is used to determine the effective thermal conductivity of nanofluid ðkeff Þ (Eqs. (27a)–(27d)). The bottom wall was thermally insulated while the two sides of the triangle were isothermally cold and a heater was positioned on the bottom wall. Their study focused on the effect of the heater location, Rayleigh number, apex angle on the overall heat transfer rate of the cavity. It was observed that the intensity of streamfunction is found to be higher for the Ethylene Glycol-Copper nanofluid compared to the pure Ethylene Glycol especially at the high Ra and consequently, isotherms were found to be strongly compressed near the hot zones for the Ethylene Glycol-Copper nanofluid case. The overall heat transfer rate from the heat source increases with the solid volume fraction at the high Ra due to the enhanced effective thermal conductivity of nanofluid. It is also observed that the heater location plays a vital role on the fluid behavior within the enclosure. In the case of the shorter heat source length, the maximum heat transfer rates are achieved for the lowest heat source height. On the other hand, for the longer heat source lengths, the heat transfer rate decreases initially and thereafter that increases as the height of the heater increases. Finally it was concluded that, the heat transfer rates obtained based on the modified Maxwell model are usually higher than those obtained based on the original Maxwell model. Rezaiguia et al. [108] examined natural convection in a tilted isosceles triangular enclosure filled with the water–Cu nanofluid combined with an heat source at the base wall. The left and right

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walls of the enclosure were maintained at the constant cold temperatures, while the remaining parts of the bottom wall are insulated. The SIMPLE algorithm was used to calculate the pressure (governing equations (Eqs. (27a)–(27d)) are modified for the inclined cavity) and the QUICK scheme was used for discretization of the convection terms in the dimensionless momentum and energy equations. The effect on thermal and fluid flow for various Rayleigh number (104 6 Ra 6 106 ), heater length (0:2
0:8), solid volume fraction, (0
0:06) and inclination angle (0
45 ) are investigated similar to the earlier work [64]. Similar to the earlier work [64], their result shows that the heat transfer rate increases in the presence of nanoparticles especially at the low values of Rayleigh numbers (Ra). It is also interesting to observe that, the solid volume fraction has the significant effect in the conduction dominant regime (low Ra) in contrast to the convection regime (high Ra). It is also found that, the temperature of the fluid in contact with the heater is observed to be maximum at the high Ra for an inclination angle, d ¼ 15 (critical angle).

4. Trapezoidal enclosures 4.1. Fluid media A number of works have been carried out on natural convection in trapezoidal enclosures. The numerical investigation on natural convection was carried out within a partially divided trapezoidal cavity by Moukalled and Darwish [109]. The dimensionless governing equations (Eqs. (3a)–(3d)) were solved using the control volume method. Two different thermal boundary conditions were considered, case 1: the hot left wall and cold right wall and case 2: the hot right wall and cold left wall. The effects of various parameters such as Rayleigh number (Ra ¼ 103
106 ), Prandtl number (Pr ¼ 0:7; 10 and 130), baffle height and baffle locations, on the heat transfer rate were studied. It may also be noted that, the baffle was located vertically on the bottom wall. At the high Ra ðRa P 103 Þ, the isotherms are highly distorted and they are stratified at the middle portion of the cavity due to the enhanced convection. Further, the presence of the baffle on the bottom wall results in the formation of two asymmetric fluid circulation cells in the left and right sides of the baffle at Ra ¼ 103 . However, as Ra increases to 105 , the two fluid circulation cells tend to merge with each other, depicting large convective effects within the trapezoidal enclosure (Fig. 10(i)(a–d)). In addition, the height of the baffle also has a profound effect on the fluid flow behavior and temperature distribution within the enclosure (Fig. 10(i)(a– d)). It was observed that the large primary fluid circulation cell breaks into two cells with the increase of the baffle height. For both the cases 1 and 2, the total heat transfer rate increases with Ra due to the intense convection at the high Ra for a fixed baffle height. Finally, it was observed that the presence of the baffle decreases the overall heat transfer rate in the trapezoidal cavity, irrespective of the position and height of the baffle. The maximum decrease in the heat transfer rate was found to occur for the baffle placed near to the left wall, irrespective of the baffle height. Moukalled and Darwish [110] further extended their study to investigate the effects of the height and position of the baffle protruding out from the inclined top wall, on the fluid flow and heat transfer in a trapezoidal enclosure with the similar thermal boundary conditions as mentioned in the earlier work [109]. The computations in their study were carried out for four Rayleigh numbers (Ra ¼ 103
106 ) and three Prandtl numbers (Pr ¼ 0:7; 10 and 130). It is interesting to observe that the inclusion of baffles from the top wall, results in the formation of two clockwise rotating vortices, at the low Ra (Fig. 10(ii)(a)). As Ra increases, the interaction

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Fig. 10. Streamfunction (w) and isotherm (h) for (i) Ra ¼ 105 , baffle location, Lb ¼ L=3 and various baffle heights (Hb ) [L is the width of the cavity; H is the height of the short vertical wall; H is the height of the cavity at the location of baffle] (a) no baffle (b) Hb ¼ H =3 (c) Hb ¼ 2H =3 (d) Hb ¼ H for Pr ¼ 0:7 [109] (ii) Lb ¼ L=3 and Hb ¼ 2H =3 at various Rayleigh numbers, (a) Ra ¼ 103 , (b) Ra ¼ 104 , (c) Ra ¼ 105 and (d) Ra ¼ 106 for Pr ¼ 0:7 [110]. (figures are reproduced from Moukalled and Darwish [109,110] with permission from Taylor & Francis).

between the fluid circulation vortices increases and the two vortices gradually merge with each other (Fig. 10(ii)(b–d)). Moreover, the eye of the vortex in the right-hand portion of the domain moves upward and to the left in the lower right portion of the enclosure with the gradual increase of Ra (Fig. 10(ii)(b–d)). Further, the increase in the height of the baffle was found to lead to the separation of the large fluid circulation cell into two circulations as seen in the previous work (Fig. 10(i)(a–d)). Similar to the earlier case [109], it was found that the overall heat transfer rate in the trapezoidal cavity is greatly reduced in the presence of the baffle, irrespective of the size and position of the baffle. Arici and Sahin [111] studied the effect of the divider on natural convection in a partially divided trapezoidal enclosure with the summer (the left vertical and top inclined walls are hot, bottom wall is cold and right vertical wall is adiabatic) and winter (the left vertical and top inclined walls are cold, bottom wall is hot and the right vertical wall is adiabatic) boundary conditions. The finite volume method was used to solve the governing equations (Eqs. (3a)– (3d)). A horizontal divider was included in such a way that the entire domain is divided into a square region and a right triangular region. Three different configurations were considered based on the position of the divider, (i) configuration 0: no divider (ii) configuration I: the divider attached to the insulated vertical wall and (iii) configuration II: the divider attached to the point of intersection between the vertical and inclined walls. Their study demonstrated that there was no significant influence of the divider

on the temperature distribution and consequently on the heat transfer rates during the summer boundary conditions for Ra ¼ 104
107 . In contrast, the heat transfer rate is reduced due to the presence of the divider during the winter boundary conditions. It was observed that, as the horizontal divider was placed to oppose buoyancy, the flow strength became much weaker and there was a formation of two separate uniform temperature zones for configuration I for the winter boundary conditions. On the other hand, the buoyancy-assisting effect was observed and the uniform temperature field was observed within the whole enclosure for the configuration II. The overall heat transfer rate was found to be higher for the configuration 0 (without baffle) compared to the configurations I and II (with baffle). Finally, it was found that, the heat loss from the enclosure is less for the configuration I. On the other hand, in order to maintain the uniform temperature distribution within the enclosure, the configuration II is ideal. Silva et al. [112] studied the effect of the inclination angle of the top wall on the heat transfer rate in a trapezoidal cavity with two baffles placed on the cavity’s horizontal surface. The element based finite volume method has been used to numerically solve the nondimensional governing equations (Eqs. (1a)–(1d)). The effect of three inclination angles of the upper surface as well as the effect of the Rayleigh number (Ra), Prandtl number (Pr), and baffle’s height (Hb ) on the streamfunctions, temperature profiles, and local and average Nusselt numbers has been numerically analyzed and illustrated. At the low Ra, three internal vortices exist within the

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cavity for all the inclination angles (Fig. 11(i)). It is also interesting to observe that the isotherms are parallel lines at the low Ra [see Fig. 11(i)]. It may be noted that, the temperature varies almost linearly within the cavity, due to the conduction dominant heat transfer at the low Ra (Fig. 11(i)). It may also be noted that the temperature gradient near to the active walls increases smoothly with the inclination angle of the top surface. However, the qualitative trends of isotherms remain identical throughout the enclosures, irrespective of the inclination angles. The three internal vortices are merged into a single cell at the high Ra (Ra ¼ 106 ) for all the inclination angles (Fig. 11(ii)). It is also interesting to observe that, the isotherms are more distorted and the convection dominant heat transfer occurs at the high Ra. At Ra ¼ 106 , the thermal stratification is found near to the lower and upper surfaces of the cavity. It is also found that, the temperature and velocity gradients within the enclosure decrease with the height of the baffles. Finally, it was concluded that for a fixed height of baffles, the overall heat transfer rate increases with the inclination angle.

23

The investigation on the fluid and heat flow patterns in the presence of natural convection in the trapezoidal enclosure with various thermal boundary conditions has been extensively performed by Basak and co-workers [69,113,114]. Natural convection in trapezoidal enclosures for the uniformly heated bottom wall, linearly heated vertical wall(s) (case 1) and linearly heated left wall or the isothermally cold right wall (case 2) in the presence of the insulated top wall have been investigated numerically by Basak et al. [113]. The penalty finite element method has been used to solve the velocity and thermal fields (Eqs. (3a)–(3d)). Parametric studies for the wide range of Rayleigh numbers (Ra ¼ 103
105 ) and Prandtl numbers (Pr ¼ 0:7
1000) with various tilt angles of the side walls (/ ¼ 45
0 ) have been performed. At the low Pr (Pr ¼ 0:7
10) and high Ra (Ra ¼ 105 ), the secondary circulations are observed near the bottom corners for / ¼ 0 compared to / ¼ 30 and 45 for the case 1. As Pr increases from 0:7 to 1000, fluid tends to take the shape of the container and the streamlines near the boundary of the cavity are parallel to the side walls

Fig. 11. Streamfunction (w) and isotherm (h) for Prandtl number Pr ¼ 0:7, baffle height (Hb ¼ HI =3; HI is the height of the enclosure where the baffle is located) at different inclination angles (a) of the top wall of the cavity for Rayleigh number, (i) Ra ¼ 103 [(a) a ¼ 10 , (b) a ¼ 15 and (c) a ¼ 20 ] and (ii) Ra ¼ 106 [(a) a ¼ 10 , (b) a ¼ 15 and (c) a ¼ 20 ] [112]. (figures are reproduced from Silva et al. [112] with permission from Elsevier).

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signifying the intense convection within the cavity. The isotherms along the side walls were observed to be highly compressed and the thickness of the thermal boundary layer was reduced at Pr ¼ 1000 and Ra ¼ 105 for the case 1. In contrast to the case 1, stronger secondary circulations were observed near the top portion of the left wall for the case 2 involving all the inclination angles. It was found that, the secondary circulations were found to be stronger for / ¼ 30 and 45 and the secondary circulations were even stronger at the higher Pr (Pr ¼ 1000) for the case 2. Since the thermal gradient was high near the cold right wall, isotherms were largely compressed towards the right wall compared to the left wall for the case 2. Multiple circulation cells for the square enclosure plays the important role for the non-monotonic trend of the average Nusselt numbers profiles for the case 2. The overall heat transfer rate at the bottom wall was found to be significantly larger for the case 2 compared to the case 1, irrespective of Pr for all the inclination angles. It was also observed that the average heat transfer rate at the left wall was significantly high for the square enclosure compared to the trapezoidal enclosures for Ra P 104 . Basak and co-workers [69] carried out the heatline analysis within a trapezoidal enclosure for the uniformly and nonuniformly heated bottom wall, insulated top wall and cold side walls. It is found that at the low Pr and high Ra, the symmetric multiple circulations cells appear near the top corners for u ¼ 30 and 45 for the case with the uniformly heated bottom wall and cold side walls [69]. On the other hand, the weak secondary circulation are observed near the center of the side walls for the square cavity. The shape of heatlines near the core is identical with the streamlines, indicating the convection dominant heat transfer at Ra ¼ 105 . Due to the enhanced buoyancy convection at the high Ra, the large amount of heat flow occurs from the bottom wall to the top portion of the vertical wall resulting in a large isothermal regime at the top portion of the cavity for u ¼ 45 and 30 . At Pr ¼ 0:7 and 1000, the multiple fluid and heat circulation cells disappear and the strength of convective cells is increased. It is also found that, due to the lesser heating effects, the intensities of the streamfunction and heatfunction are observed to be considerably lower for the nonuniform bottom wall heating case [69]. The average heat transfer rate at the bottom wall was found to be higher for the square cavity (u ¼ 0 ) with the uniformly heating bottom wall whereas in the case of the non-uniformly heated bottom wall, the average heat transfer rate is nearly identical for all the tilt angles, irrespective of Pr at the high Ra. Also, the magnitudes of average Nusselt numbers are observed to be higher for the uniform heating cases compared to the non-uniform heating cases [69]. Basak et al. [114] extended the heatline analysis for natural convection in the trapezoidal enclosure involving the inclination angles (the angle between the right wall and X-axis), / ¼ 45
60 (trapezoidal enclosures) and 90 (square enclosure) with similar thermal boundary conditions as mentioned in the earlier work [113]. Analysis was carried out for Rayleigh numbers, Ra ¼ 103
105 and Prandtl numbers Pr ¼ 0:015
1000. The symmetric and end to end heatlines were observed for the case 1 (linearly heated left and right walls). The compression of isotherms with higher magnitudes near the side walls is prominent for the trapezoidal enclosures at the higher Ra (Ra ¼ 105 ). It may be noted that, the intensity of heatline circulation cell was lower for the square cavity compared to the trapezoidal cavities irrespective of Pr and Ra. In the case of the linearly heated left wall and cold right wall (case 2), the compression of isotherms is observed along the bottom and right walls due to the larger intensity of primary heatline circulations near the right wall. Also, the compression of isotherms leads to the stronger thermal boundary layer along the right wall (case 2). Similar to the previous work [113], the average heat transfer rate was found to be maximum for the square enclo-

sure compared to the trapezoidal enclosures, in both the cases 1 and 2. Till date, limited studies have been carried out on the entropy generation via natural convection in trapezoidal cavities. Basak et al. [115] analyzed the entropy generation during natural convection (Eqs. (3a)–(3d) and Eq. (39a)) in a trapezoidal cavity with various inclination angles (u ¼ 45
90 ) for thermal processing of various fluids (Pr ¼ 0:015
1000) over a wide range of Rayleigh numbers (Ra ¼ 103
105 ) for the isothermal (case 1) and nonisothermal (case 2) heating at the bottom wall. It is observed that, at Ra ¼ 103 , the heat transfer rate in the cavity is primarily due to conduction and the total entropy generation (Stotal ) is higher for the case 1, irrespective of Pr and u. Common to all Pr and u, the total entropy generation within the cavity is highly influenced by the heat transfer irreversibility (Sh ) which is also indicated by high values of Beav at the low Ra. As Ra increases to 105 , the enhanced convection causes the large heat transport from the heat sources. As a result, the entropy generation due to the fluid friction (Sw ) also increases for all the inclination angles at Ra ¼ 105 for all Pr. The active zones of Sh are found near the lower portions of the trapezoidal cavity due to the larger temperature gradient whereas the active zones of Sw appear near the regions where the walls of the enclosure are in direct contact with the adjacent circulation cells, irrespective of Pr and the inclination angles. In addition, the friction between the circulation cells also contributes to the fluid friction irreversibilities. Note that, the active zones of the entropy generation due to the fluid friction are mostly observed in the bottom half of the side walls for the higher inclination angles (u ¼ 45 ). On the other hand, the zones of fluid friction gradually move upwards along the side walls for the lower inclination angles (u ¼ 30 and 0 ), irrespective of Ra and Pr. However, the magnitudes of Sw between the circulation cells are comparatively lesser than those obtained from Sw between the circulation cells and the walls of the trapezoidal enclosure. The total entropy generation is found to be an increasing function of Ra and Pr for both the isothermal (case 1) and non-isothermal (case 2) heating cases. Overall, it was concluded that the non-isothermal heating strategy (case 2) is more energy efficient than the isothermal heating for all the inclination angles. Ramakrishna et al. [116] studied the natural convection flows involving the entropy generation within trapezoidal cavities for the isothermally hot left wall, cold right wall and adiabatic horizontal walls. The governing equations as mentioned in the earlier work [115] were solved using the finite element method and the results have been presented in terms of the streamlines, heatlines, isotherms, entropy generation due to the fluid friction, entropy generation due to the heat transfer, average Bejan number, total entropy generation and average Nusselt number. As seen in the earlier studies for the hot and cold walls, the heat transfer within the cavity is primarily due to conduction as seen from the low magnitudes of the streamlines and heatlines at Ra ¼ 103 . As Ra increases to 105 , the enhanced convection is observed based on larger magnitudes of the streamlines and heatlines. The maximum values of the entropy generation due to the fluid friction (Sw;max ) and heat transfer (Sh;max ) are lower at Ra ¼ 103 and higher at Ra ¼ 105 due to the enhanced fluid flow at the higher Ra, irrespective of the inclination angles and Pr. It is also found that Sh;max occurs at the left edge of the bottom wall for the trapezoidal enclosure whereas that occurs near the lower portion of the left wall and the upper portion of the right wall for the square enclosure, irrespective of Pr and Ra. In addition, the fluid friction irreversibility (Sw ) was found to contribute significantly to the Stotal especially for the high Pr fluids. Overall, it was concluded that, the trapezoidal cavity with u ¼ 60 is the optimal shape for the thermal processing

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of fluids with Pr ¼ 0:015 whereas the square cavity is the optimal design for the thermal processing of the fluids with Pr ¼ 7:2 based on the low values of the total entropy generation (Stotal ) and high values of the average heat transfer rate at the hot left wall (Nul ). 4.2. Porous media Baytas and Pop [117] carried out numerical investigations to study natural convection in an inclined porous trapezoidal enclosure with the cold top cylindrical wall, hot bottom cylindrical wall and adiabatic non-parallel walls. The alternating Direction Implicit (ADI) method was used to solve the polar form of the unsteady state momentum (based on the Darcy model) and energy equation (Eqs. (9)–(11)). The investigation was performed for various tilt angles (/ ¼ 15
165 ) and Darcy–Rayleigh number, RaD ¼ 100
1000 (RaD ¼ Ra  Da). Similar to the earlier studies, it was observed that the conduction-dominant heat transfer occurs at the low RaD , irre-

25

spective of the inclination angles. It is interesting to observe that the multiple circulation cells occur at / ¼ 15 and 105
165 whereas the unicellular cells occur for 45 6 / 6 105 at RaD ¼ 900 (Fig. 12). Consequently, the isotherms exhibit waviness for the lower / values whereas the stratification of isotherms is observed at / P 135 (Fig. 12). It was observed that the average Nusselt number (Nus ) is an increasing function of RaD for all the angles, / except / ¼ 15 and for / ¼ 15 case, Nus decreases with RaD for 400 6 RaD 6 600. Also, it was found that Nus is maximum for / ¼ 45 and minimum for / ¼ 165 at RaD ¼ 900. Kumar and Kumar [118] studied the effect of natural convection in a porous trapezoidal cavity for various inclination angles (/ ¼ 65 ; 72 ; 78 and 85 ), Darcy–Rayleigh number (RaD ¼ 25
250) and modified Grashof number, (Gr  ¼ 0
2). Darcian (Eqs. (9)–(11)) and non-Darcian (Eqs. (15a) and (15b)) assumptions on the porous model are considered to construct the momentum equations. The non-linear partial differential

Fig. 12. Isotherm (h) and streamfunction (w) for k ¼ 30 (k is the angle between the side walls), A ¼ 2 (A is the cavity aspect ratio) and Darcy–Rayleigh number, RaD ¼ 900 involving various tilt angles (/) [117]. (figure is reproduced from Baytas and Pop [117] with permission from Elsevier).

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equations (governing equations) are solved using the finite element method (FEM) for various values of the flow and geometric parameters involving both the Darcian and non-Darcian models. Both the inclined walls of the trapezoidal enclosure were maintained at isothermal conditions whereas the horizontal walls were maintained at adiabatic conditions. It is observed that the simultaneous isothermal heating of the left and right walls results in the formation of a single fluid circulation cells. Note that, the intensity of the fluid circulation increases with RaD , leading to the large amount of thermal mixing near the central core of the trapezoidal enclosure. In addition, large thermal gradients are observed along the top portion of the right side wall, depicting the high heat transfer rates near those regions. It is interesting to observe that the inclination angle of the side walls of the trapezoidal enclosure also influences the fluid flow and heat transfer rate within the trapezoidal enclosure. The average Nusselt number was also reported to increase with RaD . Further, the average Nusselt number was also found to increase with the inclination angle of the walls. It was found that, the overall heat transfer rate is higher for the nonDarcian model (Darcy–Forchheimer model) compared to the Darcian model (Gr  ¼ 0) at the identical RaD and /. Varol et al. [32] carried out the numerical study to investigate the steady free convection flow in a two-dimensional porous right-angle trapezoidal enclosure with the hot left vertical wall, partially cold inclined wall and insulated horizontal walls. The Darcy model based momentum equations along with the energy equation (Eqs. (14a)–(14c)) were solved by using the finite difference method. Three cases were considered in this study based on the position of the cold zone along the inclined wall: (i) case 1: the cold zone is located at the upper half of the inclined wall (adjacent to the top wall), (ii) case 2: the cold zone is located in the middle of the inclined wall and (iii) case 3: the cold zone is located at the bottom half of the inclined wall (adjacent to the bottom wall). Note that, the portions other than the cold section on the inclined wall are maintained at adiabatic conditions. Computations were carried out for various Darcy–Rayleigh numbers (100 6 RaD 6 1000) and aspect ratios (0:25 6 A 6 0:75). Due to the heating of the left vertical wall and simultaneous partial cooling of the inclined wall, the fluid within the trapezoidal enclosure forms a large circulation cell in the clockwise direction. It is interesting to observe that the magnitude of the fluid circulation cell increases with RaD and that gradually takes the shape of the enclosure at the high RaD . It is worthwhile to mention that the fluid flow patterns are observed to be dissimilar for three different cases. Note that, the eye of the primary vortex tends to shift downward with the gradual shifting of the cooler towards the bottom portions, along the right inclined wall. The isotherms also display interesting features for all the cases. It is observed that, the positioning of the cooler near the top portion of the inclined wall, results in the occurrence of large cold zones in the bottom portion of the trapezoidal enclosure. Similarly, the presence of the cooler near the bottom portions of the inclined wall results in the formation of cold spots in the upper zone of the trapezoidal enclosure. In addition, the variation of the aspect ratio of the trapezoidal enclosures is found to have significant effect on the fluid flow behavior and the temperature distribution within the enclosure. Overall, it was also concluded that Nusselt number (Nu) and the strength of fluid flow increase with RaD . Also, both the local and mean Nusselt numbers were found to be the lower for the case 3 compared to the cases 1 and 2. Recently, Varol [119] analyzed the heat transfer and fluid flow within two entrapped porous trapezoidal cavities involving the cold inclined walls and hot horizontal walls. The Darcy model based governing equations (Eqs. (14a)–(14c)) were solved using the finite difference method for various aspect ratios of two

entrapped trapezoidal cavities, Darcy–Rayleigh number ðRaD Þ and the thermal conductivity ratio between the middle horizontal wall and fluid medium. Due to the isothermal heating of the bottom wall, fluid near the center of the bottom wall rises up for the lower trapezoidal cavity whereas due to the isothermal heating at the top wall, the fluid near the inclined wall flows down for the upper trapezoidal cavity. Thus a pair of symmetric circulations in opposite directions were found in both the cavities (Fig. 13(a–d)). At the low ðRaD ¼ 100 and 250), the isotherms for the upper trapezoidal cavity are found to be smooth whereas for the lower trapezoidal cavity, the isotherms are distorted near the cold walls and at the central regime (Fig. 13(a and b)). As RaD increases, the isotherms are largely distorted and the oscillatory trend was found in the lower cavity (Fig. 13(c and d)). The plume shaped isotherms were observed near the bottom walls for both the upper and lower cavities. On the other hand, for the upper trapezoidal enclosure, stratification zones of isotherms have been observed near the top wall signifying that RaD has negligible effect on the flow field for the upper trapezoidal cavity. In addition, it was also concluded that the thermal conductivity ratio and the aspect ratio of the trapezoidal enclosures plays a significant role in the heat transfer enhancement within the entrapped porous enclosure. Moreover, the average Nusselt number was observed to be higher for the lower aspect ratio at the higher RaD . Basak et al. [120] performed the numerical investigation of natural convection in a porous trapezoidal enclosure via the heatline approach, for identical thermal boundary conditions as mentioned in the earlier work [69]. The Darcy–Brinkman based governing equations (Eqs. (19a)–(19d) and (36)) were solved using the finite element method. The numerical solutions were studied in terms of the streamlines, isotherms, heatlines, local and average Nusselt numbers for a wide range of parameters, Da ð10
5
10
3 Þ; Pr ð0:015
1000Þ, Ra ð103
106 Þ and inclination angles (/ ¼ 45
90 ). It may be noted that the qualitative features of the heatlines and streamlines are almost similar to the work carried out by Basak et al. [69] for the fluid media. Overall, it was concluded that the low Da corresponds to the conduction dominant regime whereas the high Da signifies the convection dominant regime. It was observed that, the average heat transfer rates at the bottom and side walls are the increasing functions of Ra and Da, irrespective of Pr and /. Similar to the fluid media case [69], the overall heat transfer rates at the active walls were observed to be higher for the square cavity compared to the trapezoidal cavities irrespective of Pr and Da. Ramakrishna et al. [121] reported the investigations on natural convection using the concept of heatline (Eqs. (19a)–(19d) and (36)) in porous trapezoidal enclosures based on two different thermal boundary conditions: (i) case 1: the linearly heated side walls and (ii) case 2: the linearly heated left wall and cold right wall. In both the cases, the bottom wall was isothermally hot and top wall was adiabatic. The studies were carried out for various Darcy number (Da), Prandtl number (Pr) and Rayleigh number (Ra) involving various inclination angles ð/Þ for the identical parameter ranges as discussed in the previous work [120]. The fluid flow and heat flow features within the enclosures during the conduction and convection dominant regimes are qualitatively similar to the results obtained by Basak et al. [114] for the fluid media (Fig. 14(i–iv)). In the present work, it is observed that the wall effect is larger within the enclosures and that contrasts the flow features obtained in the work carried out by Basak et al. [114]. Note that, at the high Da (Da ¼ 10
3 ), the effect of the porous media was found to be considerably lesser and the flow features were almost identical to the fluid media [114] (Fig. 14(iii–iv)). Similar to the earlier work for the fluid media [114], the average Nusselt number at the bottom wall

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Fig. 13. Streamfunction (w) and isotherm (h) for height-base aspect ratio, A ¼ 0:3, thermal conductivity ratio, k ¼ 1 and various Darcy–Rayleigh numbers (a) RaD ¼ 100, (b) RaD ¼ 250, (c) RaD ¼ 500, and (d) RaD ¼ 1000 [119]. (figure is reproduced from Varol [119] with permission from Elsevier).

is higher for the square cavity than the trapezoidal cavities at the high Da and Ra with the low Pr for the case 1. The number of studies of the entropy generation in a porous trapezoidal enclosure is limited in the literature. The entropy generation analysis during natural convection in a porous trapezoidal structure of various inclination angles with the isothermal (case 1) and non-isothermal (case 2) hot bottom wall was numerically investigated by Basak et al. [122]. The governing equations (Eqs. (19a)–(19d)) are solved using the finite element method as discussed in the earlier work [115]. Simulations were carried out for various ranges of modified Darcy number (Dam ), Prandtl number (Pr m ) at the modified Rayleigh number (Ram ¼ 106 ). The inclination angle, u is defined as the angle between the left inclined wall and the Y-axis for three different inclination angles; u ¼ 45 ; 60 and 90 . It may be noted that the fluid flow and temperature distributions within the enclosures are found to be qualitatively similar to the previous work carried out by Basak et al. [115]. It is observed that the maximum heat transfer irreversibility ðSh Þ occurs near bottom corners for the case 1 and near the middle part of the left and right portions of the bottom wall for the case 2 at Dam ¼ 10
3 and Ram ¼ 106 irrespective of Pr m . It was found that, the maximum value of the fluid friction irreversibility ðSw Þ occurs along the top wall at Prm ¼ 0:015, and at the interface between the counter rotating circulation cells near the bottom wall at Pr m ¼ 1000 with

u ¼ 45 and 60 ; Dam ¼ 10
3 and Ram ¼ 106 , for the cases 1–2, similar to the fluid media [115]. However, for u ¼ 90 (square cavity), the maximum fluid friction irreversibility occurs near the side walls for all Prm at Dam ¼ 10
3 and Ram ¼ 106 , irrespective of the cases. The Stotal is found to be remarkably lower for the case 2 compared to the case 1 during the convection dominant regime for all Pr and this feature is similar to those observed in the work carried out by Basak et al. [115]. It is found that, the high heat transfer rate, optimal thermal mixing and minimum entropy generation rate occur for 7  10
5 6 Dam 6 4  10
4 at all the inclination angles. It is also recommended that the case 2 (non-isothermal heating) is preferable for the efficient processing involving the fluids with the high Pr m and the case 1 (isothermal heating) is preferable for the low Pr m fluids at Dam ¼ 10
3 . Finally it is concluded that, the square cavities followed by trapezoidal cavities with u ¼ 60 may be the optimal geometries for the efficient thermal processing involving cases 1 and 2 at all Pr m and Dam . Later, Ramakrishna et al. [123] numerically investigated the thermal management within a porous trapezoidal enclosure during the natural convection using the heatlines and entropy generation. The left wall was maintained isothermally hot whereas the right wall was maintained isothermally cold. The horizontal walls were kept adiabatic. Similar to the previous works [115,122], Galerkin finite element method has been used to

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Fig. 14. Streamfunction (w), isotherm (h) and heatfunction (P) at Rayleigh number Ra ¼ 106 (i) uniformly heated bottom wall with linearly heated side walls at Pr ¼ 0:015 and Da ¼ 10
5 ; (ii) uniformly heated bottom wall with linearly heated left wall and cold right wall at Pr ¼ 0:015 and Da ¼ 10
5 ; (iii) uniformly heated bottom wall with linearly heated side walls at Pr ¼ 988:24 and Da ¼ 10
3 , and (iv) uniformly heated bottom wall with linearly heated left wall and cold right wall at Pr ¼ 988:24 and Da ¼ 10
3 [121]. (figures are reproduced from Ramakrishna et al. [121] with permission from Taylor & Francis).

analyze streamlines, isotherms, heatlines, entropy generation due to fluid friction and heat transfer over wide range of

4.3. Nanofluids

parameters (10
5 6 Da 6 10
3 ; 0:015 6 Ra 6 1000, 30 6 u 6 90 )

Saleh et al. [124] numerically investigated and compared natural convection heat transfer in a trapezoidal enclosure filled with Cu–nanofluid, water–Al2O3 and the base fluid (water). The governing equations in terms of streamfunction–vorticity formulation (Eqs. (28a)–(28c)) were solved numerically using the finite difference method. The left and the right inclined walls were maintained isothermally hot and cold, respectively while the horizontal walls of the trapezoidal enclosure were adiabatic. The effects of Grashof number (Gr), inclination angle of the sloping wall ðuÞ, solid volume fraction ð/Þ on the fluid and heat flow fields have been investigated. It is interesting to observe that the fluid circulation pattern within the trapezoidal enclosure is similar for both the Cu–nanofluid and base fluid. Note that, the qualitative features of streamlines within the enclosures filled with the base fluid is similar to the results obtained by Ramakrishna et al. [116]. However, the intensity of the streamfunctions was found to be higher for the nanofluid compared to the base fluid. It was found that the average heat transfer rate increases for the Cu-nanofluid in contrast to the base fluid. It was also observed that the intensity of

at Ra ¼ 106 . For all u, the heat circulation cells were found to intensify with Da. Consequently, the thermal mixing was found to be enhanced as Da increases from 10
5 to 10
3 . The active zones of Sw were found to occur along the left wall for u ¼ 30 and 90 irrespective of Pr whereas that occurs along the right wall for

u ¼ 60 during the convection dominant regime (Da ¼ 10
3 ). The thermal irreversibility zones were observed at the left edge of the bottom wall irrespective of Pr and Da for u ¼ 30 and 60 whereas that occurs at the left edge of bottom wall and right edge of top wall for u ¼ 90 for all Da. Overall, the total entropy generation was found to be higher for Pr ¼ 1000 compared to that of Pr ¼ 0:015 at the higher Da. It is also found that the trapezoidal cavities with u ¼ 60 and 90 correspond to the less entropy generation with significant heat transfer rates at Da ¼ 10
3 for Pr ¼ 0:015 and Pr ¼ 1000 and thus the trapezoidal cavities with u ¼ 60 may be the optimal design for the thermal processing of fluids at Pr ¼ 0:015 and 1000.

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streamfunctions increases with the solid volume fraction of the nanoparticles in the base fluid. The overall heat transfer rate was observed to increase with the solid volume fraction. Consequently the overall heat transfer rate were observed to be higher for the Cunanofluid compared to water–Al2O3 for the equal concentration level. Further, Saleh and co-authors [125] studied the fluid and thermal flow characteristics in the trapezoidal enclosure filled with water–Al2O3 nanofluids for the differentially heated side walls and adiabatic top and bottom walls. The non-dimensional forms of the governing equations (Eqs. (27a)–(27d)) were solved numerically using the finite difference method. It was observed that Rayleigh number ðRaÞ, the base angle (angle formed by the bottom wall of the trapezoidal enclosure with the X-coordinate), volume fraction and size of the nano-particles play pivotal role in natural convection heat transfer within the trapezoidal enclosure. The fluid flow and temperature distribution within the enclosure is found to be similar to the results obtained in the previous work [124]. Note that, the fluid flow within the trapezoidal enclosure is intensified as Ra increases, for a constant volume fraction of nanoparticles in the base fluid and a constant base angle. In addition, the gradual increase in Ra results in the distortion of isotherms in contrast to the low Ra. It was also interesting to observe that, the magnitudes of streamfunction are found to be decreased as the base angle and volume fraction of the nanoparticles in the base fluid increase. It was concluded that the trapezoidal shaped cavity is more efficient than the square cavity for the enhanced heat transfer. Moreover, it was found that, there is no significant improvement in the convective heat transfer rate using nanofluids compared to the base fluid [124]. The parametric study on natural convection phenomena in a trapezoidal cavity filled with water–Cu nanofluid was studied by Nasrin and Parvin [126]. The horizontal walls of the enclosure were insulated while the left and right inclined walls were isothermally hot and cold, respectively. The governing equations (Eqs. (27a)– (27d)) were solved numerically based on the finite element technique with Galerkin’s weighted residual simulation. Solutions were obtained for a wide range of aspect ratio (0:65 6 A 6 2) and Prandtl number (1:47 6 Pr 6 8:81) at the constant Rayleigh number (Ra ¼ 105 ) and solid volume fraction (/ ¼ 0:05). In contrast to the previous works [124,125], the effect of the aspect ratio of the trapezoidal enclosures was studied in details for both the nanofluids and base fluid. For the trapezoidal enclosures with the lower aspect ratio, there exists a large dissimilarity between the base fluid and nanofluid, as seen from the streamline contours. In addition, short distances between the hot and cold surfaces in the trapezoidal enclosure with the higher aspect ratio, results in the increase in rate of heat transfer, in contrast to the other values of A. Similar to the previous works [124,125], the strength of the flow circulation is observed to be much higher in the presence of nanoparticles for the high Pr values. Consequently, the isotherms are compressed towards the side walls and large thermal gradients are observed along the left inclined surface of the enclosure. Overall, Cu nanoparticles with the highest Pr fluid is concluded to be most effective in enhancing the performance of the heat transfer rate within the trapezoidal enclosures with the low aspect ratio (A). 5. Parallelogrammic and rhombic enclosures 5.1. Fluid media In the recent past, extensive studies have been carried out in the rhombic and parallelogrammic shaped enclosures as discussed next. Yuncu and Yamac [127] reported the laminar natural convection in an air (Pr ¼ 0:71) filled parallelogrammic cavity with the

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horizontal isothermal walls and insulated inclined walls. They compared the results with those of a rectangular cavity with similar boundary conditions. The governing equations (Eqs. (6a)–(7)) are expressed in terms of the dimensionless streamfunction, vorticity and temperature. The finite difference technique has been implemented and the difference equations were obtained using the integration cell approach. Studies were carried out for two different aspect ratios (A ¼ 1 and 2) and three different inclination angles (d ¼ 90 , 45 and 26:6 ). Since the left wall is maintained at a higher temperature compared to the right wall, the fluid near the hot left wall rises and falls down along the cold right wall. Thus the single circulation cell is formed at the low Ra (Ra ¼ 1733) for A ¼ 1 and d ¼ 45 . In addition, distorted isotherms, parallel to the side walls, are observed within the parallelogrammic enclosure for the identical parameters. Also, in order to find the influence of the direction of the heat flow on the heat transfer rate, the forward average heat transfer coefficient (the temperature on the front surface of the enclosure is higher compared to the back surface) and backward average heat transfer coefficient (the temperature on the front surface of the enclosure is lower compared to the back surface) were measured. It is observed that the rectification ratio, R (the ratio of the average Nusselt number in the forward and backward directions) increases sharply with Ra for the lower aspect ratio (A ¼ 1) and lesser inclination angles (d). However, the rate of increase of R is lesser for the enclosures with the higher aspect ratio (A) and lower inclination angles (d). It is also found that R is invariant with Ra for the highest inclination angle irrespective of the aspect ratio (A). Finally, it was concluded that, a parallelogrammic cavity transmits more heat in the forward direction than a rectangular cavity of the same aspect ratio. In addition, it was also inferred that the average Nusselt number in the forward direction of the parallelogrammic cavity is higher compared to the backward direction for almost all the cases. Moukalled et al. [128] studied the effect of the enclosure gap ratios and parallelogrammic angles on the laminar natural convection in a horizontal parallelogrammic annulus. Their study depicts that the increase in the enclosure gap ratio and parallelogram angle increases the flow strength and convective heat transfer rate. In addition, Moukalled and Acharya [129] examined the effect of the vertical eccentricity on natural convection in a parallelogrammic annulus. The effect of key parameters such as enclosure gap values (Er ¼ 0:5
0:75), eccentricity values ( ¼
0:25 to 0:25), and Rayleigh number (Ra ¼ 104 -108 ) were studied. It was found that the negative values of eccentricity lead to an increase in heat transfer rate. Also, it was concluded that at the low Ra, there is a critical positive eccentricity value at which the flow strength was found to be the maximum. At the high Ra, the higher positive eccentricity factor resulted in the decrease of the flow strength and heat transfer rate. Naylor and Oosthuizen [130] reported two-dimensional natural convective flow in a differentially heated parallelogram shaped enclosure. The hot left and cold right walls of the enclosure were kept isothermal and inclined at an angle b with respect to the gravity. The top and bottom walls of the enclosure were horizontal and adiabatic. The computations were carried out for a wide range of Rayleigh number (103 6 Ra 6 105 ) and the height base aspect ratio (0:5 6 A 6 3) at different inclination angles (
60 6 b 6 60 ) of the side walls with the bottom wall. The inclination angle (b) is defined as the angle between the side walls and the Y-axis. The results were presented for Pr ¼ 0:7. Note that, the fluid within the parallelogrammic enclosure forms a primary fluid circulation cell with the single vortex for
60 6 b 6 60 whereas multiple vortices are observed within the enclosure for
30 6 b 6 30 . It is interesting to observe that the intensity of convection is higher for the positive inclination angles compared to the negative inclination

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angles, irrespective of Ra and A (aspect ratio). In addition, it is found that the fluid circulation cells are elongated along the horizontal direction for b ¼
60 and the penetration of fluid is deeper along the bottom corner regions. Note that, the average heat transfer rate increases monotonically with Ra for all the inclination angles (
60 6 b 6 60 ). Overall, it was concluded that the positive values of the inclination angles result in more uniform distribution of heat transfer rates compared to the negative inclination angles. Aldridge and Yao [131] studied natural convection in a parallelogrammic enclosure with the hot left wall, cold right wall and adiabatic horizontal walls. The finite volume method was used to solve the unsteady governing equations [see the steady balance equations: Eqs. (1a)–(1d)]. Simulations were carried out for Pr ¼ 6:8 (water) and Pr ¼ 3300 (silicone oil), at various Rayleigh numbers, Ra ¼ 4:3  104
2:92  107 . At the high Pr and low Ra, the weak convection dominance occurs throughout the enclosure and the boundary layers were absent, as the viscous forces are dominant over the buoyancy forces. Due to the buoyancy forces and imposed temperature gradient between the hot isothermal left wall and cold isothermal right wall, fluid rises near the hot left wall and flows down along the cold right wall, forming a single clockwise circulation cell within the parallelogrammic cavity at Pr ¼ 3300. As Ra increases to 3:2  105 , the greater buoyancy force results in the formation of the thin boundary layers along the side walls for Pr ¼ 3300. In contrast to the low Ra, multiple vortices are observed near the core of the parallelogrammic enclosure, depicting the intense convection especially near the core of the enclosure at the high Ra. At Pr ¼ 6:8 and Ra ¼ 2:55  106 , the secondary vortices as observed for the high Pr fluids, shift towards the walls and appear as an elliptical pattern with the large ellipticity. The average heat transfer rate of the hot left wall was found to increase sharply with Ra. Bairi [132] studied natural convection in closed cavities of the parallelogram section filled with air (Pr ¼ 0:71) and suggested a new definition of the Nusselt number. The two dimensional parallelogrammic enclosure was filled with air and consisted of the hot isothermal left wall, cold isothermal right wall, along with the insulated top and bottom walls. It may be noted that the adiabatic top and bottom walls are inclined at an angle a with respect to the horizontal X-axis. In most of the studies, the Nusselt number is based on either the height of the active walls or the distance between them. Although this definition correctly accounts for the heat exchange phenomena in cavities of the rectangular section, Bairi [132] proposed that the original definition of the Nusselt number is insufficient for the case of the cavities with inclined walls (parallelogrammic enclosures) as the heat transfer in the parallelogrammic enclosures does not occur only by convection between the active walls, but also by radiation and conduction through the passive walls. Thus, a modified definition of the Nusselt number was proposed for the parallelogrammic enclosures following a 2D numerical simulations based on the boundary element method. In another study, Bairi et al. [133] analyzed the transient natural convection in the parallelogrammic enclosure and examined the model for the different inclination angles of the cavity. Similar to the previous work [132], the left and right vertical walls of the enclosure were kept as isothermally hot and cold, respectively and the adiabatic top and bottom walls were tilted at an angle a with respect to the horizontal X-axis. The motion pressure (motion pressure is the sum of intrinsic pressure and hydraulic pressure) was incorporated in the momentum equations (Eqs. (1b) and (1c)) and the viscous dissipation term was incorporated in the energy balance equation (Eq. (1d)). The governing equations were solved using the finite volume method coupled with the SIMPLE algorithm. The study was carried out for a wide range of Rayleigh

numbers (105 6 Ra 6 109 ) at Pr ¼ 0:7. In addition, the numerical simulations were performed for five specific angles (a ¼ 0 ; 30 and 60 ) and three Fourier numbers (Fo ¼ 0:0033; 0:0133 and 0:1333). It is interesting to observe that, at the low Fo (Fo ¼ 0:0033) and a ¼ 0 , the major fluid movement occurs along the boundaries of the enclosure and the central core of the enclosure remains essentially stagnant based on the weak multiple vortices. This feature can also be verified from the isotherms which clearly depict that the variation in temperature occurs only along the boundaries and a large central core of the enclosure is maintained at an uniform temperature (Fig. 15). As Fo increases, the thermal diffusivity towards the core of the enclosure also increases and hence the fluid movement was found to be more intense. It is also observed that the multiple vortices occurring at the low Fo, merge and form the fluid circulation cell with the single vortex. In addition, the temperature stratification is observed near the core of the enclosure, at the low Fo (Fig. 15). For positive angles, stagnation zones are observed along the corners of the cavity and similar to a ¼ 0, majority of the fluid movement is observed along the walls connecting the hot and cold plates through the peripheral path (Fig. 15). On the other hand, for negative angles, the fluid flow between the isothermal walls becomes gradually disconnected as the angle increases. This further results in the formation of the tiny circulation cells in the proximity of each wall separately. Note that, this effect is more significant at the high Fo. The maximum Nusselt number and maximum velocity near the hot wall were determined for all the angles ðaÞ and they were compared to the cavity with a ¼ 0 . Finally, the simulation results in this study were validated with the experimental data and the comparison is in the reasonable agreement. Costa et al. [134] carried out a numerical study for the laminar natural convection heat transfer occurring in a vertical stack of parallelogrammic partial enclosures and reported the thermal diode effect on the thermal performance of the enclosure. The left wall of the stack is maintained at higher temperature compared to the right wall and the horizontal top and bottom walls are assumed to be adiabatic. The momentum equations were obtained in terms of the driving pressure (driving pressure is the sum of intrinsic pressure and hydraulic pressure). Further, the momentum (Eqs. (1b) and (1c)) and energy balance equations (Eq. (1d)) are solved using the control volume based finite element method. The length of each partition was less than the width of the main enclosure. The thermal diode effect offered by the geometry was thoroughly analyzed in terms of the partition’s inclination angle and materials for the various thermal boundary conditions. The fluid flow and heat transfer within the enclosure, were analyzed via isotherms, streamlines and heatlines. The results were obtained for the fluids with Pr ¼ 0:73 at Ra ¼ 107 for various inclination angles of the shutter with the horizontal plane (a ¼ 0 ; 30 and 60 ) and two different aspect ratios (L=H ¼ 0:2 and 0:5). It is interesting to observe that in the absence of the shutters, the isotherms are distorted lines, originating from the bottom wall and ending on the top wall. Streamlines and heatlines demonstrate a single clockwise vortex in the center of the main enclosure in the absence of the shutters (a ¼ 0 ). However, in the presence of the shutters, the fluid and heat circulation cells are observed in each partition as seen in Fig. 16. In addition, the heat flow within the enclosure is also demonstrated by the end to end convective heatlines which emanate from the lower portion of the left wall and end on the right wall in each panel (Fig. 16). It is interesting to observe that isotherms with h ¼ 0:4
0:6 originate near the corners of the bottom wall and follow a zig-zag path, before ending near the corners of the top wall (Fig. 16). Note that, as the inclination angle between the shutter and horizontal plane increases, the inclined fluid and heat circulation cells of higher magnitudes are observed in each

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Fig. 15. Isotherms (h) and streamlines (w) for inclination angle a ¼ 0; 30 , 60 at different Fourier numbers F o and Rayleigh number Ra ¼ 1:2  108 [133] for Pr ¼ 0:7. (figures are reproduced from Bairi et al. [133] with permission from Elsevier).

partition. As the aspect ratio (L=H) increases to 0:5, concentrated isotherms were also observed near the lower half of the hot wall and at the upper half of the cold wall. The cases which involved high thermal conductivity of the shutters compared to the fluid demonstrated the maximum heat transfer rate irrespective of Ra and a. Anandalakshmi and Basak [135] analyzed the energy distribution and thermal mixing in the steady laminar natural convective flow through the rhombic enclosures using the heatline concept and also extended this study in order to find the efficiency [136] of this system. The momentum and energy balance equations (Eqs. (3a)–(3d)) are solved using the Galerkin finite element method in both the studies. They presented their results for the various regimes of Prandtl and Rayleigh numbers with various

inclination angles ðu ¼ 30 ; 40 and 75 Þ and reported that the enhanced thermal mixing occurs at u ¼ 75 for both the isothermal and non-isothermal heating of the bottom wall. Isotherms, streamlines and heatlines are mostly smooth curves for all the inclination angles during the conduction dominant heat transport at Ra ¼ 103 . At the higher Ra (Ra ¼ 105 ), multiple flow circulations are observed for the low Pr values (0:015 and 0:7) involving all the inclination angles. On the other hand, multiple circulation cells are suppressed and two asymmetric flow circulation cells are found to occupy the entire cavity for the higher Pr fluids (7:2 and 1000) at the higher inclination angles. Interesting features on the heatlines further demonstrate that the major heat transport to the left wall occurs directly from the bottom wall whereas the right wall receives heat based on the convective cell at the smaller inclination

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Fig. 16. Isotherm (h), streamfunction (w) and heatfunction (P) for the enclosure with shutters of inclination angle a ¼ 30 , thermal conductivity ratio Rc ¼ 100, main enclosure aspect ratio, L=H ¼ 0:2 and Rayleigh number Ra ¼ 107 [134] for Pr ¼ 0:73. (figures are reproduced from Costa et al. [134] with permission from Elsevier).

angles [135]. On the other hand, the heat flow is evenly distributed to the side walls at larger u’s and larger Pr [135]. At the low Ra (Ra ¼ 103 ), the entropy generation (Eq. (39a)) within the cavity was found to be dominated by Sh for all the inclination angles 5

(u), irrespective of Pr. However, at the high Ra (Ra ¼ 10 ), the fluid flow intensifies and Sw also increases for all u, irrespective of Pr. The entropy generation due to heat transfer is found to be significant in the lower portions of the cavity due to the large temperature gradients in those regions [136] whereas the entropy generation due to the fluid friction is significant corresponding to large velocity gradient, especially at the zone where the solid wall is in contact with the adjacent circulation cells. The comparison of the magnitudes indicates that the maximum entropy generation due to the heat transfer is identical for both Ra ¼ 103 and Ra ¼ 105 , whereas the entropy generation due to the fluid friction is lower for Ra ¼ 103 and that is higher for Ra ¼ 105 due to the higher fluid friction, for all the inclination angles and Pr. Overall, the total entropy generation (Stotal ) is found to be low for the lower inclination angles for all Pr values at Ra ¼ 105 [136]. In another study, Anandalakshmi and Basak [137] carried out the entropy generation studies within the rhombic enclosure which was subjected to the differential heating (case 1) and Rayleigh–Benard convection (case 2). The streamline features for the case 1 exhibit the single fluid circulation cell at the low Ra whereas the multiple fluid circulation cells and single fluid circulation cell with multiple vortices are observed at the higher Ra especially for the larger inclination angles (u) and high Pr. The streamline features of the case 2 at Ra ¼ 105 demonstrate that the multiple flow circulation cells occur for the low Pr (Pr ¼ 0:015) whereas the strong fluid circulation cell with a single vortex is observed

for the high Pr (Pr ¼ 1000). Similar to the previous work [136], the active zones of Sh and Sw are found to occur near the isothermal walls for all the inclination angles (u) irrespective of Pr in both the cases at Ra ¼ 105 . In addition, the active zones of Sw are also found to occur near the adiabatic walls of the cavity for all u’s irrespective of Pr in both the cases at Ra ¼ 105 . Also, the regions between the fluid layers of primary circulation cells act as the strong active zones of Sw for all u’s in the case 2 at Pr ¼ 0:015 and Ra ¼ 105 . The total entropy generation (Stotal ) and maximum heat transfer rates (Nu) are found to be significantly low for the lower u in both the cases at Ra ¼ 105 irrespective of Pr. The average heat transfer rate was found to be higher for the Rayleigh–Benard convection (case 2) compared to the differential heating case (case 1). However, the total entropy generation was found to be higher for the case 2 compared to the case 1. Thus, it was finally concluded that Rayleigh– Benard convection (case 2) is not an energy efficient process compared to the differential heating case (case 1) due to the large fluid friction irreversibilities.

5.2. Porous media A few studies are available on natural convection in the porous parallelogrammic enclosures. Baytas and Pop [138] presented the detailed numerical simulations for natural convection in a differentially heated porous parallelogrammic enclosure with the isothermal inclined walls and adiabatic horizontal walls. For simplification, the computational domain was mapped onto a rectangular shaped cavity using a non-linear axis transformation. The Darcy model based unsteady form of the governing equations (Eq. (12b)) are expressed in the new coordinate system and solved

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numerically using the ADI (Alternative Direction Implicit) finite difference method. The numerical computations have been carried out for various inclination angles (/ ¼ 0 ; 15 , 30 ; 45 and 60 ) at two different values of Darcy–Rayleigh numbers (RaD ¼ 103 and 104 ). At / ¼ 0 (square enclosure) and RaD ¼ 103 , a large primary circulation cell of lesser intensity is observed within the enclosure. In addition, at the low RaD (¼ 103 ), weak thermal boundary layers are also observed near the top portion of the left wall and bottom portion of the cold right wall, depicting lesser heat transfer rates to those regions. As RaD increases to 104 , the thicknesss of the thermal boundary layers especially at the top portion of left wall and bottom portion of the right wall are greatly reduced. It may be noted that the isotherms in the central region of the enclosure are observed to be nearly parallel to the horizontal walls especially for the lower inclination angles even at RaD ¼ 104 . The distributions of the streamlines and isotherms within the parallelogrammic enclosures for the inclination angles 0 to 45 , are qualitatively similar to those observed for the square enclosure, at RaD ¼ 103 and 104 . However, the primary fluid

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circulation cells are reduced in size with the increase in the inclination angle. At the higher RaD , a pair of asymmetric cells are observed only for the parallelogrammic enclosure with / ¼
60 , in contrast to the other enclosures. Finally, it was concluded that, for all the enclosures, the rate of heat transfer is higher for the high RaD . Moukalled and Darwish [139] presented a numerical study on the laminar natural convection in a porous rhombic annulus. Dar cy–Brinkman–Forchheimer model based momentum and energy balance equations (Eqs. (9), (11) (20a) and (20b)) were solved using the pressure-based finite volume method. Numerical simulations were carried out for a wide range of Rayleigh number (Ra ¼ 104
107 ), Darcy number (Da ¼ 10
5
10
1 ), porosity ( ¼ 0:3; 0:6 and 0:9), enclosure gap (Eg ¼ 0:875; 0:75; 0:5, and 0:25) and Prandtl numbers (Pr ¼ 0:7 and 5). It was found that the intensity of the fluid flow and convection heat transfer increase with Ra; Da; Eg and . Fig. 17(a-l) demonstrate the distributions of streamlines and isotherms for the half domain of the parallelogrammic enclosure. It is interesting to observe that the strength

Fig. 17. Streamlines (w) and isotherms (h) for the enclosure between two isothermal concentric cylinders of rhombic cross-sections (half domain) for different values of porosity ðÞ at Rayleigh number, Ra ¼ 106 , enclosure gap ratio, Eg ¼ 0:875, and Darcy number, Da ¼ 0:1 [139] for Pr ¼ 0:7 and 5. (figures are reproduced from Moukalled and Darwish [139] with permission from Taylor & Francis).

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of the streamfunction decreases with the porosity () at Ra ¼ 106 and Eg ¼ 0:875 for all Pr (see Fig. 17(a-f)). It may also be noted that the flow circulation cells within the enclosures are unicellular at the high Eg (Eg ¼ 0:875) irrespective of  and Pr. However, as enclosure gap ratio (Eg ) decreases (Eg ¼ 0:25), the flow circulation cells tend to segregate near the core of the enclosure and eventually forms a pair of fluid circulation cells. The isotherms of Fig. 17(g– l) demonstrate that the right half of the enclosure is mostly maintained cold with h ¼ 0:1 at Eg ¼ 0:875 for all  and Pr. In addition, it is also observed that the isotherms are largely compressed towards the hot zones and they are distorted near the central core region of the enclosure especially for high  and Pr (Fig. 17(g–l)). For low values of Eg , isotherms are found to be parallel lines and the heat transfer within the enclosure is conduction dominant. It was also observed that the average heat transfer rate was higher when the enclosure gap ratio (Eg ) is lesser irrespective of Pr. The average heat transfer rate also indicates the existence of the critical Ra for a fixed Pr; Da;  and Eg . Finally, it was concluded that the critical Ra decreases with Da and  and the critical Ra increases as Eg decreases. Anandalakshmi and Basak [140] analyzed the energy distribution and thermal mixing in the steady laminar natural convective flow through the porous rhombic cavities using the Darcy–Brinkman model (Eqs. (19a)–(19d)). The effect of the Darcy number (Da) and the role of the inclination angles on the energy distribution and thermal mixing within the porous rhombic cavities with the isothermal (case 1) and non-isothermal (case 2) hot bottom walls were demonstrated via the heatlines. They concluded that the cup mixing temperature was higher for the case 1 compared to the case 2. Isotherms, streamlines, and heatlines are monotonic and smooth curves for all the inclination angles at Da ¼ 10
5 irrespective of the heating strategies. The left cold wall is found to receive more heat from the large portion of the hot bottom wall for the lower inclination angles. As the inclination angle increases, the heat distribution is found to be even throughout the enclosure, as indicated by the heatlines at Da ¼ 10
5 . Further, the distortion of the heatlines indicates that the onset of convection occurs at

and consequently, Stotal decreases with the thermal aspect ratio (Fig. 18(ii)). Further, it was observed that, the total entropy generation (Stotal ) is found to be lower for the lower inclination angles and higher for the higher inclination angles irrespective of Pr and Da (Fig. 18(i and ii)). It was concluded that the minimum entropy generation (Stotal ) with the higher heat transfer rate and reasonable heat transfer rate occur for Pr ¼ 0:015 and Pr ¼ 1000, respectively at u ¼ 30 cavities with all the thermal aspect ratios irrespective of Da. Similar to the studies within fluid media [137], Anandalakshmi and Basak [142] analyzed the entropy generation within the rhombic cavity filled with fluid saturated porous media for the similar thermal boundary conditions as mentioned in the other study [137]. The simulations were performed for Da ¼ 10
5
10
3 and Pr ¼ 0:015
7:2 at a fixed Rayleigh number (Ra ¼ 106 ) in order to show the effect of Da. The results were presented in terms of the isotherms (h), streamlines (w), entropy generation maps due to heat transfer (Sh ), and entropy generation maps due to fluid friction (Sw ). The active zones of Sh and Sw are observed near the junction of adiabatic and isothermal walls, at Da ¼ 10
5 in both the cases, irrespective of Pr and the inclination angles. At Da ¼ 10
3 , active zones of heat transfer and fluid friction irreversibilities occur over a wide region along the isothermal walls, for the higher inclination angles. It was found that, at the lower inclination angles, Sh;max decreases with Da in the case 1 for all Pr. However, for the higher inclination angles Sh;max shows increasing trend with Da. In contrast to the case 1, Sh;max increases with Da in the case 2, irrespective of Pr and the inclination angles. Note that, in contrast to Da ¼ 10
5 , the maximum entropy generation due to fluid friction (Sw;max ) is higher at Da ¼ 10
3 due to the enhanced fluid flow irrespective of heating situations (case 1 and case 2), inclination angles, and Pr. Overall, similar to the fluid media [137], the Rayleigh–Benard convection (case 2) is not an energy efficient process compared to the differential heating case (case 1) for the porous media. 6. Complex geometries


4

Da ¼ 10 due to an increase in the permeability of the porous media, and the heatlines from the hot bottom wall take a longer path to reach the cold side walls of the cavity for all the inclination values. Multiple flow circulations are observed for the fluids with the low Pr at Da ¼ 10
3 . The overall heat transfer rate from the bottom wall to the left wall was found to be higher for u ¼ 30 for the case 1 irrespective of Pr. On the other hand, the maximum heat transfer rate from the bottom wall occurs for u ¼ 75 for the case 2 at the high Da and Pr. A few earlier works [141,142] have extended natural convection studies for the entropy generation (Eq. (44)) in the porous rhombic cavities. Anandalakshmi and Basak [141] analyzed natural convection in porous rhombic enclosures for the various thermal aspect ratios via the entropy generation approach. The effect of the thermal aspect ratio and inclination angle (u) for the various governing parameters (Darcy number, Prandtl number) were illustrated via the heat transfer irreversibility (Sh ) and fluid friction irreversibility (Sw ). In their study, they observed that the entropy generation due to the heat transfer and fluid friction are significant on the bottom and left walls, especially at the high Da for the lower inclination angles. However, for the higher inclination angles, all the walls act as the active zones of the entropy generation due to heat transfer and fluid friction at Da ¼ 10
3 . It was found that at the lower thermal aspect ratio (Fig. 18(i)), the entropy generation in the cavity is dominated by both Sh and Sw for all u’s, irrespective of the Darcy (Da) and Prandtl numbers (Pr). As the thermal aspect ratio increases, Sh as well as Sw decreases for all the inclination angles

6.1. Fluid media A few studies are found in the literature on the natural convection in the enclosures with curved and wavy walls [143–154] as shown in Fig. 1(d). Varol and Oztop [143] numerically investigated natural convection heat transfer and fluid flow within a rectangular wavy walled and inclined solar collector. Flat ceiling was considered as the cover (cold temperature) of the solar collector and wavy wall as the absorber (hot temperature). The governing equations [basic governing equations as given in Eqs. (1a)–(1d) are modified for the inclined cavity] were solved using the CFD software package. The working fluid was considered as air (Pr ¼ 0:71) within the collector. The numerical computations were carried out within the enclosure for various parameters such as, Rayleigh number (Ra ¼ 106
8  106 ), inclination angle of the insulated wall with the horizontal axis (0o
90o ), aspect ratio (A ¼ 2
3), and wavelength (1 6 L 6 4). As the wavy wall is kept at a higher temperature, fluid within the cavity rises up due to the buoyancy forces and impinges on the top wall, before falling in the downward direction. This results in the formation of the clockwise and anti-clockwise fluid circulation cells within the cavity. Rectangular shaped fluid circulation cells are observed near the ends of the enclosure at the low Ra. Note that, the fluid between the cells are found to be stagnant which can be further verified from the smooth isotherms. As Ra increases, the flow intensity increases at the middle of the enclosure. The isotherms show the swirling pattern at the high Ra and that is intensified to the cavity

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Fig. 18. Isotherms (h), local entropy generation due to heat transfer (Sh ), streamlines (w) and local entropy generation due to fluid friction (Sw ) at Prandtl number, Pr ¼ 0:015, Darcy number, Da ¼ 10
3 ; Ra ¼ 106 for (i) thermal aspect ratio A ¼ 0:1 at inclination angles (a) / ¼ 30 , (b) / ¼ 45 and (c) / ¼ 75 and (ii) A ¼ 0:9 at inclination angles (a) / ¼ 30 , (b) / ¼ 45 and (c) / ¼ 75 [141]. (figures are reproduced from Anandalakshmi and Basak [141] with permission from Elsevier).

of the wavy wall. Overall, the isotherms are found to be highly concentrated near the wavy wall at both the high and low Ra. It was observed that the length of the circulation cells increases with the inclination angle of the enclosure with the horizontal axis. In addition, the aspect ratio of the enclosure was also found to largely affect the thermal and flow fields within the enclosure. Overall, it was concluded that the heat transfer rate increases with the aspect ratio and Rayleigh number. It was also found that the maximum heat transfer rate was obtained for the maximum value of aspect ratio (A ¼ 4) and inclination angle (90o ). Adjlout et al. [144] numerically studied the effect of the hot wavy wall on the laminar natural convection in an inclined square cavity using the controlling parameters such as the inclination angles and number of undulations. The right hot wavy wall was maintained at a constant temperature, T h while the left cold wall was kept at constant temperature, T c (see Fig. 19). All the other walls were insulated. Two different geometrical configurations

were used in the present study involving one undulation (Fig. 19 (i)) and three undulations (Fig. 19(ii)) of the right wall. The governing equations [basic governing equations as given in Eqs. (1a)–(1d) are modified for the inclined cavity and obtained in the vorticityvelocity form] were solved using the finite difference method. The tests were carried out for different inclination angles (0 6 u 6 180 ), undulation amplitudes (0:05 6 k 6 0:08) and Rayleigh numbers (103 6 Ra 6 106 ) at Pr ¼ 0:71. It is interesting to observe that a pair of the symmetric fluid circulation cells appear in the upper and lower halves of the enclosure for the cavities with one undulation at the higher Ra. The isotherms are also found to be perpendicular to the insulated walls and almost all the isotherms are parallel to each other depicting large zones of stratification (Fig. 19(i)). As the inclination angle between the cavity and Xaxis increases, the fluid circulation cells in the top and bottom half of the enclosure combine and form a large primary fluid circulation cell with multiple vortices. This results in the slight distortion of

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Fig. 19. Streamlines (w) and isotherms (h) at Rayleigh number, Ra ¼ 105 and Prandtl number, Pr ¼ 0:71 for various angles (/) (i) one undulation (ii) three undulations [144]. (figures are reproduced from Adjlout et al. [144] with permission from Elsevier).

the isotherms, near the central region of the complex enclosure. In addition, the primary fluid circulation cell with single vortex is observed at the high Ra (Fig. 19(i)). This promotes considerable thermal mixing near the core of the enclosure as seen from the large zone of uniform temperature near the core (Fig. 19(i)). As the number of undulations along the right wall increases to 3, three pairs of the fluid circulation cells are observed in the top, middle and bottom halves of the enclosure (Fig. 19(ii)). The vertical and wavy isotherms are observed within the enclosure, at very low inclination angle (Fig. 19(ii)) for the greater number of undulations (Fig. 19(ii)). Note that, the waviness of the isotherms near the wavy right wall is higher than those observed near the left wall for the geometry with three undulations on the right wall (Fig. 19(ii)). For the maximum inclination angle and three undulations, a primary fluid circulation cell is observed in the enclosure, along with the tiny secondary circulation cells near the bottom left corners of the enclosure. Note that, the presence of the secondary circulation

cells near the left halves results in the stronger compression of isotherms along the left wall and hence, the larger thermal gradients are observed along the left wall especially for the higher inclination angles (u ¼ 180 ). Based on the flow features and temperature distributions in the enclosure, it was finally concluded that an increase in the undulation number on the hot wall increases the heat transfer rate for the higher inclination angle. Ridouane and Campo [145] performed the numerical tests to analyze natural convection within the air (Pr ¼ 0:71) filled circular cavity inscribed in a square cavity (enclosure 2). In addition, a square cavity (enclosure 1) and a circular cavity (enclosure 3) have also been included in the analysis. The finite volume method was used to solve the governing equations (Eqs. (1a)–(1d)). The left and right vertical walls of the cavities were maintained at temperature, T h and T c , respectively. The top and bottom walls of the enclosure were considered to be adiabatic. Note that, at the low Ra, the isotherms near the walls are parallel, whereas the iso-

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therms near the core of the enclosures are distorted. It is interesting to observe that at the high Ra (Ra ¼ 106 ), the single vortex as observed at the low Ra has been replaced with two vortices, in the lower right and upper left corners in the enclosures, depicting the enhanced buoyancy force at the high Ra. Note that, the isotherms are nearly horizontal in the middle region of the enclosure depicting the dominance of convection at the high Ra. Overall, it was concluded that for all the Rayleigh numbers (Ra), the circular cavities provide the higher heat transfer rates compared to the square cavities. Dalal and Das [146] presented the effect of the non-uniform heating of the bottom wall on natural convection in a twodimensional cavity with a wavy right vertical wall. The bottom wall was heated by a spatially varying temperature and the other three walls were kept at a constant lower temperature. A semiimplicit method was used to couple the momentum and continuity equations (Eqs. (3a)–(3d)) and the QUICK scheme was implemented in approximating the convective terms for both the momentum and energy equations. The numerical studies were carried out for three different undulations and different Rayleigh numbers (10 6 Ra 6 106 ) for a constant Prandtl number (Pr ¼ 0:71). The presence of the undulation in the right wall was found to affect both the local heat transfer rate and flow field. Due to the conduction dominant heat transfer, a minor variation in flow patterns is observed for Ra ¼ 1
1000. On the other hand, the significant variation in the fluid circulation pattern is observed at the higher Ra (Ra P 1000) as the buoyancy forces are enhanced and the heat transfer within the enclosure is mostly convection dominant. Due to the waviness of the right wall, the asymmetric fluid circulation cells are observed and the intensity of streamfunctions is found to be different for both the circulation cells. It is also observed that the size of the right fluid circulation cell increases with Ra whereas the left fluid circulation cell squeezes towards the left halves of the enclosure. In addition, the isothermal lines

37

the left top corner and that circulates in the counter-clockwise direction. The other cell turns in the clockwise direction and its shape fills the entire geometry. It is interesting to observe that the isotherms are distorted in the regions where the primary fluid circulation cell is present whereas the isotherms are parallel to the left wall, near the region where the weak fluid circulation cell was present. As the amplitude of the wavy wall increases from 0.8 to 1.1, the intensity of the fluid circulation cells increases (Figure not shown). In addition, the multi-cellular fluid circulation cell slowly transforms into a unicellular fluid circulation cell, with the amplitude of the wavy wall (Figure not shown). It is also interesting to observe that the flow rate increases at the throat due to the narrow region in the central portion of the enclosure and the isotherms are almost diagonal at the throat, at lower amplitude of the wavy wall (a). In addition, it was also observed that fluid flow separates near the central region of the enclosures, at the high RaI =RaE . Overall, it was concluded that the thermal gradients along the side walls are higher for the enclosure with the larger wavy-wall amplitude and hence, the heat transfer rates are found to be higher for the case with high wavy-wall amplitude (Figure not shown). It was also inferred that the direction of the heat transport depends strongly on the ratio of the internal Rayleigh number (RaI ) to the external Rayleigh numbers (RaE ). Das and Mahmud [148] investigated the effect of the amplitude-wavelength ratio on the fluid flow and temperature distribution within a isothermal wavy walled enclosure. The top wavy wall was maintained at a higher temperature compared to the bottom wavy wall whereas the vertical walls were considered as adiabatic. The integral forms of the governing equations (Eqs. (1a)– (1d)) were solved numerically using the finite volume method. The computations were carried out for various Grashof numbers (103
107 ) and amplitude-wavelength ratios (0
0:15). Note that, the Prandtl number of the fluid was considered as 1 and the aspect ratio of the enclosure was maintained at 4 for all the cases. At the

are found to be distorted at Ra ¼ 105 and Ra ¼ 106 due to the influence of increased convection current. It is observed that during the

high Gr (Gr P 104 ), the core of the fluid circulation cells is found to be elongated and the intensity of the streamfunction is also found

conduction dominant regime (Ra ¼ 103 ), the average Nusselt number increases with the number of undulations. However, as the convection mode of heat transfer becomes dominant with Ra, the average Nusselt number decreases within the undulated enclosures. Further, Oztop et al. [147] reported the effects of the volumetric heat sources on natural convection heat transfer and flow structures in a wavy-walled enclosure filled with air (Pr ¼ 0:71). The vertical walls of the enclosure were assumed to be differentially heated whereas the top and bottom wavy walls were considered to be adiabatic. The streamfunction–vorticity formulation (Eq. (6a)) and the energy equation (Eq. (6b)) with the internal heat generation are solved using the finite volume method. The diffusion term in the vorticity and energy equations is approximated by a second-order central difference scheme. Both the internal (RaI ), external Rayleigh numbers (RaE ) and the amplitude of the wavy sinusoidal walls (a ¼ 0:8
1:1) were the effective parameters. It is interesting to observe that the ratio of RaI =RaE (RaI =RaE ¼ 1 and 10) has the considerable influence on the flow and temperature distributions within the enclosures for the various amplitudes

to increase. At Gr ¼ 104 , due to the onset of the convection, isotherms tend to swirl and distort throughout the enclosure. Note that, two hot spots are observed at the bottom wall whereas three hot spots are observed near the top wall. In addition, multiple fluid circulation cells with four vortices are observed within the complex enclosure, due to the comparatively higher buoyancy effects. The strength of the fluid circulation cells also increases signifying the enhanced convection within the enclosure at the high Gr. It may also be noted that, the multi-cellular flow patterns turns into

(a) of the sinusoidal walls. At RaI ¼ 104 and RaE ¼ 104 , the fluid within the enclosure forms a large primary flow circulation cell with multiple vortices and the intensity of streamfunction is also lesser. The isotherms on the side walls display larger thermal gradients along those regions. On the other hand, at RaI ¼ 105 and RaE ¼ 104 , the internal heat generation dominates over the external heat heat generation and two separate fluid circulation cells are observed within the enclosure. A weak cell is located near

a bi-cellular flow at Gr ¼ 106 . It is noteworthy to mention that the high convection current shifts the core of each vortex towards the adiabatic wall of the cavity. Consequently, the periodic distortion features of the isotherms disappear and hot spots are observed to occupy nearly the entire bottom wall. Hence, a large zone in the middle portions of the enclosure is maintained at an uniform temperature. It was also reported that at the lower Gr, the effect of the amplitude-wavelength ratio is significant. However, at the higher Gr, this effect is very small. The heat transfer rate is higher at the lower Gr when the amplitude-wavelength ratio increases from zero to higher values. However, further increase of the amplitude-wavelength ratio shows the negligible effect on the average heat transfer rate. Ashjaee et al. [149] experimentally and numerically studied the free convection along an isothermal vertical wavy surface. Experiments were carried out at three different amplitude-wavelength ratios (a = 0:05; 0:1 and 0:2) and for various Ra (2:9  105
5:8  105 ). The right wall and right half of the bottom wall of the enclosure were maintained adiabatic whereas the left and

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top walls were maintained at isothermally cold conditions in the presence of the hot wavy wall. A finite-volume based code was developed in order to solve the basic governing equations (Eqs. (1a)–(1d) are modified for the inclined cavity) and the simulation results were also verified with the experimental data. At the high Ra, the periodic behavior of the local heat transfer coefficient is observed and the periodicity is identical to the wavelength of the sinusoidal surface. It is also observed that the locations of the crest and trough of the local heat transfer coefficients for the different amplitude-wavelength ratio (a) are the same. However, they are observed to shift upstream of the crests and troughs of the wavy surface. It can also be seen that the heat transfer rates are large near the portions of the wavy surface facing downward the flow as the velocity gradients along those regions are significantly large. In contrast, the heat transfer coefficients are found to be lesser for the portions of the surface facing upward the flow since the velocity gradient is significantly smaller. Overall, it was concluded that the average heat transfer coefficient was found to decrease with the amplitude-wavelength ratio. Finally, the experimental data were correlated with a single equation which gave the local Nusselt number along the surface as a function of the amplitudewavelength ratio and Rayleigh number. Triveni and Panua [150] carried out laminar natural convection studies in an isosceles right-angled triangular cavity consisting of a caterpillar (C)-curve shape wavy isothermally hot bottom wall. Three boundary conditions were considered in the present work (Case 1: cold vertical wall and adiabatic inclined wall, case 2: cold inclined wall and adiabatic vertical wall, case 3: cold inclined and vertical walls). The governing equations (Eqs. (1a)–(1d)) along with the corresponding boundary conditions are solved by finite volume method. The effect of the various parameters such as the number of undulations of the bottom wall (2 6 N 6 5) and Rayleigh number (105 6 Ra 6 107 ) on the fluid flow and temperature distribution within the cavity were numerically studied using FLUENT 6.3. At Ra ¼ 105 and N ¼ 5, a single rotating fluid circulation cell is formed for the cases 1 and 2 whereas, two fluid circulation cells are observed for the case 3 (see Fig. 20(i)). It is also found that the magnitude of the fluid circulation cell is found to be higher for the case 3 compared to the cases 1 and 2 (see Fig. 20(i)). The isotherms at Ra ¼ 105 and N ¼ 5 demonstrate that they originate from the left bottom corner of the cavity and spread throughout the cavity for the case 1 while, the isotherms are initiated from the right corner of the cavity for the case 2 (see Fig. 20(i)). In contrast, the isotherms are plume shaped and they are dispersed throughout the cavity for the case 3 (see Fig. 20(i)). As N reduces to 4, the strength of the fluid circulation cell is enhanced and two fluid circulations are observed in contrast to N ¼ 5 (see Fig. 20(ii)). At N 6 3, the intensity of the fluid circulation cell decreases and an unicellular fluid circulation cell encompasses the entire triangular enclosure. The isotherm distribution for the various values of N depict that the thermal mixing is larger near the core and hence, a large zone near the central core is maintained at uniform temperature in all the cases (see Fig. 20(ii)). This further leads to the isotherms to be largely compressed towards the hot bottom and cold side walls of the cavity for all the cases (see Fig. 20(ii)). As Ra increases from 105 to 107 , the strength of the fluid circulation cell is largely enhanced common to all the cases and hence, thermal mixing near the core further increases. Note that, the vortex of the fluid circulation cell expands horizontally for the higher values of Ra. Consequently, the Nusselt number illustrates that the heat transfer rate increases steeply with Ra. On the other hand, Nu is found to be almost invariant with the number of undulations (N), especially at higher Ra. The heatline studies for the visualization of natural convection within a complicated cavity were also carried out by Dalal and

Das [151]. The boundary conditions in this work are similar to the earlier work [146]. The integral forms of the governing equations (Eqs. (3a)–(3d) and (36)) are solved numerically using the finite-volume method. The results were obtained for Ra ¼ 100
106 ; Pr ¼ 0:71, and undulation amplitude of 0
0:1. At the low Ra, the streamfunctions are qualitatively similar to those observed in an earlier work [146], irrespective of the undulation amplitude. In addition, at the high undulation amplitude and Ra, the upper portion of the fluid circulation cell in the right half of the enclosure is found to be elongated towards the wavy wall similar to the earlier work [146]. Note that, as the relative size of the cells changes with the undulation amplitude, the isotherms are observed to have a skewness towards the left vertical wall as the amplitude of the wavy wall is increased. In contrast to the earlier work [146], the heat flow visualization within the complicated cavity has been carried out for the present work. Note that, similar to the streamlines, asymmetric heat circulation cells are observed irrespective of the undulation-amplitude ratio. As the undulation-amplitude increases, the size of the heat circulation cell occurring at the left halves reduces whereas the size of the cells at the right halves of the enclosure increases. Overall, it was concluded that the heat transfer rate by the left and top walls increase with Ra. Also, the heat transfer rate was found to be slightly affected by the presence of undulations on the right wall. In another study, Biswal and Basak [152] carried out the numerical computations within the differentially heated enclosures involving curved side walls at various Prandtl numbers (Pr ¼ 0:015
1000) for various Rayleigh numbers (Ra ¼ 103
106 ) involving various cases based on the convexity/ concavity of the curved side walls. The left wall of the enclosure was maintained isothermally hot and the right wall was kept isothermally cold along with insulated horizontal wall. The mass, momentum and energy balance equations (Eqs. (3a)–(3d) and (36)) with boundary conditions are solved using the Galerkin finite element method. Comparative studies of the concave and convex cases show that the heat and flow distributions are found to be affected significantly by the wall curvature for the concave cases. In convex cases, the significant variations in the thermal and flow characteristics are not observed for various wall curvatures. At the low Ra, only end-to-end heatlines are observed in the enclosure for the concave cases, whereas in the convex cases both the end-toend and closed loop heatlines are observed even at the low Ra. As the wall curvature increases, the magnitude of streamlines and intensity of the closed loop heatlines are found to be decreasing in the concave cases. On the other hand, the increase in the wall curvature in convex cases results in the enhancement of magnitudes of streamlines and closed loop heatlines. At Ra ¼ 105 and Pr ¼ 0:015, the secondary fluid and heat flow circulation cells are observed within the enclosure along the concave wall. In contrast, the single fluid flow and heat flow circulation cells are observed throughout the enclosure except at the corner zones for all the cases of convexities at Ra ¼ 105 and Pr ¼ 0:015. At the low Ra, the largest heat transfer rate is observed for the concave cases whereas the heat transfer rate is found to be significantly larger in the convex cases at the high Ra for all Pr. Mahmud and Islam [153] investigated natural convection and entropy generation numerically within a differentially heated wavy enclosure. The finite-volume method was used to solve the governing differential equations [basic governing equations as given in Eqs. (1a)–(1d) are modified for the inclined cavity] and to evaluate the entropy generation terms (Eq. (38)). The numerical computations were carried out for the different values of wave ratio [defined as amplitude/average width (k ¼ 0
0:4)], heightbase aspect ratio (A ¼ 1:0
2:0), Rayleigh number (Ra ¼ 1
107 ) for fluids with Pr ¼ 0:7. The angle of the inclination (u) of the wavy

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Fig. 20. (i) Effect of various thermal boundary conditions (Case 1: cold vertical wall and adiabatic inclined wall, case 2: cold inclined wall and adiabatic vertical wall, case 3: cold inclined and vertical walls) on the streamlines (w) and isotherm (h) distribution within the right angled triangular enclosure with number of undulations of the bottom wall, N ¼ 5; d ¼ 0:05 (d ¼ h=w; d is the aspect ratio of the curvature) and Ra ¼ 105 for Pr ¼ 7:2; (ii) Effect of the number of undulations of the bottom wall (N) on the streamline and isotherm distribution within the right angled triangular enclosure at Ra ¼ 106 for Pr ¼ 7:2. (figures are reproduced from Triveni and Panua [150] with permission from Elsevier).

enclosure was also varied between 0
360 . The heat transfer irreversibility and fluid friction irreversibility were numerically evaluated using the Bejan number. Note that, a single fluid circulation cell is observed for all the inclination angles except 90 and 

5

270 . The convection current is sufficiently strong at Ra ¼ 10 and that causes the convective distortion of isotherms. Note that, the active zones of the entropy generation due to heat transfer occur along the lower portion of the hot wall and upper portion of the cold wall and these observations are similar to the results obtained by Ilis et al. [76] within a square domain. Consequently, the heat transfer rate is also found to be high near those regions. Note that, for the enclosures with the inclination angle of

u ¼ 90 , a pair of symmetric fluid circulation cells are observed in the upper and lower halves of the enclosure. However, the isotherms display parallel lines which depict conduction dominant heat transfer, even at Ra ¼ 105 . Note that, the high velocity gradients are observed along the left and right regions of the wavy walls for u ¼ 90 and therefore, those regions act as the zones of the high entropy generation due to the fluid friction. On the other hand, an unstable condition in fluid is achieved for the inclination angle, u ¼ 270 and the total entropy generation along the wavy walls were found to be negligible. The average heat transfer rate displayed a wavy variation with the angle of inclination of the cavity and the heat transfer rate was found to be minimum for u ¼ 90 .

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Overall, the entropy generation rate is lesser and the heat transfer rate is optimum for the inclination angle corresponding to u ¼ 45 and 135 . Mahmud and Fraser [154] examined natural convection and entropy generation [basic governing equations as given in Eqs. (1a)–(1d) and (39a) are modified for the inclined cavity] within a wavy enclosure with adiabatic horizontal walls and side wavy walls following the cosine profile. The right wall is maintained isothermally hot whereas the left wall is maintained at isothermally cold conditions. Simulations were carried out for a wide range of wave ratio (0
0:6), aspect ratio (1
4) and Rayleigh number (10
107 ) for Pr ¼ 0:71. The fluid and temperature distribution features within the enclosure are qualitatively similar to the results obtained in the previous work [145]. The multiple vortices are observed within the enclosure at the lower Ra. As Ra increases from 102 to 104 , the buoyancy effect increases within the enclosure and multiple vortices tend to merge into a single crescent shaped vortex. Note that, the crescent shaped vortex shifts downwards as Ra increases to 105 and the vortex tends to shift towards the left wall leading to the large velocity gradients. At the higher Ra, isotherms are distorted near the central region of the enclosure and they are found to be more intense near the wavy walls. Thus, the large temperature gradients are observed in the vicinity of the wavy walls depicting the entropy generation due to heat transfer. As the aspect ratio and wave ratio of the enclosure increase, the asymmetric core of the circulation occurring for the lower aspect ratio and wave ratio change into a symmetric shaped core. Further, the entropy generation number (the ratio of local entropy generation rate and characteristic entropy transfer rate) is plotted as a function of Ra. It is found that in the conduction dominant regime, the entropy generation number is nearly independent of Ra. The entropy generation number increases sharply with Ra and the rate of increase in the entropy generation number is found to be higher in the regime beyond Ra P 102 . The Nusselt number plots display that the heat transfer rate is invariant till Ra ¼ 103 and that increases sharply in the convection dominant regime. Overall, the enclosures without wavy walls were found to display the high heat transfer rate with the minimum entropy generation rate. Dagtekin et al. [155] performed a numerical study to investigate the second law analysis due to the buoyancy-induced flow in a Cshaped enclosure. The right upper and left vertical walls were maintained isothermally cold whereas the bottom and top horizontal walls were maintained adiabatic. Note that, the lower vertical and horizontal walls [see Fig. 21] are maintained isothermally hot (Fig. 21(i)(a–c)). The finite-volume method was used to solve the governing differential equations (Eqs. (3a)–(3d)) and to evaluate the entropy generation terms (Eq. (39a)). The effects of the geometrical ratio and Rayleigh number on the entropy generation were thoroughly investigated in their study. Streamlines, isotherms, entropy generation due to heat transfer and fluid friction were illustrated for the various values of Ra (103
106 ), irreversibility distribution ratio (/D ¼ 10
5
10
3 ), step height (0:25 6 h 6 0:75) and step width (0:25 6 w 6 0:75) for Pr ¼ 0:71. Due to the uniform heating of the step, an anti-clockwise fluid circulation cell of lesser intensity is observed over the forward step (Fig. 21(i)(a–c)). On the other hand, the large clockwise recirculation cell occupies the region between the left vertical walls and front face of the step (Fig. 21(i)(a–c)). As Ra increases to 105 and 106 , the intensity and size of the fluid circulation cell increases and isotherms are found to be largely compressed along the isothermal walls resulting in large thermal gradients. The intense convection also results in a large zone in the upper half of the enclosure to be maintained at an uniform temperature. The entropy generation is considerably high close to the isothermal

walls due to the fluid friction and heat transfer (Fig. 21(i)(a–c)). Note that, the adiabatic walls are inactive for entropy generation due to heat transfer. Further, it was observed that the change in step height (h) and step width (w) largely affects the flow and thermal characteristics within the enclosure. It was also found that at low irreversibility ratios (/D ¼ 10
5 and 10
4 ), the entropy generation increases very slowly over a wide range of Ra. On the other hand, the overall entropy generation increases rapidly for the higher irreversibility ratios (/D ¼ 10
3 ). The overall heat transfer rate was found to be highest for the case with h ¼ 0:75 and w ¼ 0:25. 6.2. Porous media Studies on the porous media filled wavy enclosures can be found in a few earlier works [56,156–159]. Kumar et al. [156] solved the coupled streamfunction-temperature equations governing the Darcian flow and convection process (Eqs. (9)–(11)) in a fluid-saturated porous wavy enclosure with an isothermal sinusoidal bottom surface using the Bubnov Galerkin finite element method. Note that, the bottom wavy wall is isothermally hot whereas the top wall is maintained isothermally cold, along with the adiabatic side walls. Numerical computations were carried out in order to study the effect of the parameters such as wave amplitude (a), number of waves per unit length (N), wave phase (W p ), aspect ratio (A) and Rayleigh number (RaD ). At the high RaD , the flow separation and reattachment on the wavy wall is observed due to the change in the pressure gradient resulting from the asymmetric geometric configuration. It is also observed that the separation and reattachment points move closer to the leading and trailing edges, respectively at the high RaD . In addition, several distinct circulation zones also appear in the core of the domain due to the local fluid movement near those regimes and consequently, the wavy isotherms are observed near the core and bottom regions of the enclosure. Note that, the secondary fluid circulation zones hinder the heat transfer process within the enclosures. It is interesting to observe that, as the amplitude of the wavy wall (a) increases, the number of fluid circulation zones remains the same, but the flow separates and reattaches on the bottom surface leading to the formation of separate fluid circulation cells in the wavy domains. It is also observed that at small amplitudes, the flow in the right half of the domain is more intense and at higher amplitudes, the flow in the central circulation region is more intense. Isotherms display extreme wavy nature, as the amplitude of the undulation increases. Further, the global heat transfer rate within the system was found to decrease with the amplitude and number of waves per unit length. The marginal changes were observed in the global heat flux with RaD and W p . It was finally concluded that the sinusoidal bottom surface undulations of the isothermal wall of a porous enclosure reduce the heat transfer rate in the system. Misirlioglu et al. [157] numerically investigated the steadystate free convection within a porous cavity involving two horizontal straight walls and two vertical bent-wavy walls. The wavy walls are assumed to follow a profile of the cosine curve. The horizontal walls are maintained adiabatic, whereas the right undulated wall was maintained at a hotter temperature compared to the left undulated wall. The Darcy model based unsteady governing equations (Eqs. (14a)–(14c)) are solved numerically using the Galerkin finite element method. The flow and heat transfer characteristics (isothermal, streamlines and local and average Nusselt numbers) within the enclosure are illustrated for a wide range of the Rayleigh numbers (RaD ¼ 10
103 ), cavity aspect ratio (A ¼ 1
5) and surface waviness parameter (k ¼ 0
0:6). It is interesting to observe that the vortices of the primary fluid circulation cell for k ¼ 0 are stretched near the core of the enclosure at the high RaD . As the sur-

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Fig. 21. (i) Streamlines (w), isotherms (h), local entropy generation due to heat transfer (Sh ), and local entropy generation due to fluid friction (Sw ) for dimensionless height (h) and length (w) of step h ¼ w ¼ 0:5 (a) Ra ¼ 103 , (b) Ra ¼ 104 and (c) Ra ¼ 105 [155]; (ii) isotherms (h) for various dimensionless height of the step (h) and width of the step ðwÞ [(a) h ¼ 0:5 and w ¼ 0:75, (b) h ¼ 0:75 and w ¼ 0:5, (c) h ¼ 0:5 and w ¼ 0:25, and (d) h ¼ 0:75 and w ¼ 0:25] at Gr ¼ 104 , ratio of minimum to maximum nanoparticles, R ¼ 0:001 in vertical annulus [169]. (figures are reproduced from Dagtekin et al. [155] and Dehnavi and Rezvani [169] with permission from Elsevier).

face waviness (k) increases, the detachment of the vortex leads to the formation of multi cellular fluid circulation cells within the enclosure (Fig. 22). As RaD increases, the strength of the fluid circulation cell increases and streamlines near the core become more elliptic in shape with increase in k due to enhanced buoyancy effects. As a result, the isotherms are squeezed towards the side walls and hence, large thermal gradients are observed in those regions. It is interesting to observe that at the high RaD , irrespective of k, the thickness of the thermal boundary layer is found to increase towards the left bottom and right top corners of the enclosure. This further depicts the formation of cold and hot spots near the left bottom and right top corners of the enclosure, respectively. Overall, it was concluded that the flow and thermal structures

were found to be highly dependent on the surface waviness and cavity aspect ratio especially at the high Rayleigh numbers (RaD ). Further, Misirlioglu et al. [158] used the unsteady Darcy flow model with the Boussinesq approximation [basic governing equations as given in Eqs. (14a)–(14c) are modified for the inclined cavity] in order to investigate numerically natural convection within an inclined wavy cavity filled with a porous medium. The bottom wavy wall is hot while the top wavy wall is maintained isothermally cold in the presence of the adiabatic side walls. The numerical computations were carried out for various values of inclination angle (0 6 u 6 90 ), Rayleigh number (10 6 RaD 6 103 ), aspect ratio (1 6 A 6 3) and surface waviness parameter (0 6 k 6 0:3). The streamline patterns display that, at an inclination angle greater

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Fig. 22. Isotherms (h) and streamlines (w) at Rayleigh number RaD ¼ 10 for aspect ratio, A ¼ 4 and surface waviness, k ¼ 0; 0:3; 0:4; 0:5, and 0:6 (left to right) [157]. (figures are reproduced from Misirlioglu et al. [157] with permission from Elsevier).

than 45 , stable and unicellular fluid circulation cells are observed within the enclosure for all the cases. For the lower inclination angles (u 6 30 ), the flow behavior highly depends on the surface waviness parameter (k). It is interesting to observe that, the unicellular structure prevails for the low waviness values (k ¼ 0 and 0:1) whereas the flow behavior is observed to be complicated for the higher surface waviness values (k ¼ 0:2 and 0:3). Note that, asymmetric fluid circulation cells occur for the horizontal enclosure (u ¼ 0 ) at the low RaD . It is observed that, the number of fluid circulation rolls increases with RaD for the flat walls (u ¼ 0 ). Note that, three parallel fluid circulation zones occur at the high RaD within the horizontal enclosure for the flat walls. In addition, the fluid circulation rolls are found to be distorted and disoriented with respect to the horizontal walls at the high RaD . Overall, it was concluded that the flow and thermal structures were found to be highly dependent on the surface waviness for the inclination angles less than 45 , especially at RaD ¼ 103 . Chen et al. [56] studied the free convection within a wavy porous enclosure with isothermal wavy walls and adiabatic horizontal

walls. The wavy right wall is maintained at isothermally hotter temperature compared to the left wall. They analyzed the flow characteristics within the cavity by solving the Darcy–Brinkman– Forchheimer model based governing equations (Eqs. (23a)–(23d)) using the finite volume method. The studies were carried out for a range of surface waviness (k ¼ 0
1:8), aspect ratio (A ¼ 1
5) and Darcy–Rayleigh number (RaD ¼ 10
105 ). At the higher aspect ratio, it is interesting to observe a main recirculating flow in the central region and two smaller recirculations at the top and bottom regions of the enclosure. On the other hand, at the high RaD , a pair of recirculation loops is observed near the middle regime due to the intense convection. It is also observed that the isotherms are distorted near the middle regime of the enclosure and they are compressed towards the side walls depicting large thermal gradients at the high RaD . Note that, the distortion of isotherms and the flow structure at the high RaD is significant especially for high surface waviness values (k). Thus, similar to the previous works [118,157,158], the surface-waviness parameter (k) is found to have profound effect on the flow and temperature distributions within

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the wavy enclosure. Overall, among all the parameters, RaD was found to affect significantly the streamline and temperature distributions within the enclosures. Later, Khanafer et al. [159] reported the non-Darcian effects on natural convection in a porous wavy enclosure. The left wavy wall is maintained at a hotter temperature compared to the right vertical wall whereas the top and bottom horizontal walls were maintained adiabatic. Note that, the porous medium is considered to be homogeneous, isotropic and is saturated with fluid which is in local thermodynamic equilibrium with the solid matrix of the porous medium. The transport equations (Eqs. (9), (22a) and (22b)) were solved using the finite element formulation based on the Galerkin method of weighted residuals. Different flow models for porous media such as the Brinkman-extended Darcy (Eqs. (17a) and (17b)), Forchheimer-extended Darcy (Eqs. (15a) and (15b)), and the other generalized flow models were also considered. The computations were carried out to study the impact of amplitude of the wavy surface (0 6 A 6 0:25), Rayleigh number (104 6 Ra 6 106 ), and number of undulations (0 6 N 6 3) on the flow structure and heat transfer characteristics. Note that, all the results have been obtained for Da ¼ 10
2 . Due to the isothermal heating of the left wall, fluid forms a primary fluid circulation cell which encompasses the entire enclosure. It is interesting to observe that the fluid circulation near the core of the enclosure is elongated along the horizontal direction at the low A. As A increases, the elongation of the fluid circulation cell near the core decreases. In addition, the fluid flow is found to be negligible towards the top and bottom corners along the left wall and thus, the hot spots are formed near those regions. Due to the higher velocity gradients near the wavy surface, the higher local heat flux variation is observed for higher values of the amplitude of the wavy surface and this subsequently increases the heat transfer rate. Similar to the previous work [56], Ra is found to have large effects on the flow and temperature distributions within the enclosure. It is interesting to observe that the heat flux profile is highly oscillating for the higher values of N whereas the trend of the heat flux profile is found to be sluggish for the lower values of N. Note that, the heat flux is maximum near the bottom of the vertical wall as the clockwise rotating fluid carries energy from the left wavy wall leading to lesser thermal gradients. Further, it was also concluded that the average Nusselt number profile for the Darcy–Brinkman model and the generalized model are nearly similar. However, the Darcy–Brinkmanship–Forchheimer model was found to overestimate the average Nusselt number in contrast to the other two models. Basak et al. [160] analyzed natural convection in a porous right angled triangular enclosure with a convex/concave hypotenuse using the Bejan’s heatline concept. The cavity was subjected to an isothermal cold left wall, isothermal hot curved right wall, and an adiabatic bottom wall. The Darcy–Brinkman model based governing equations (Eqs. (23a)–(23d)) and heatfunction equation (Eq. (36)) are solved by the Galerkin finite element method. The nature of the fluid flow and thermal behavior within the wavy enclosure has been illustrated via streamlines, isotherms and heatlines. The numerical studies were carried out for a wide range of

Darcy

number

(Da ¼ 10
5
10)

and

Prandtl

number

(Pr ¼ 0:01
7:2) at a fixed Rayleigh number (Ra ¼ 106 ). The strength of the fluid circulation cell is higher for the triangular enclosure with the convex hypotenuse compared to the concave hypotenuse, irrespective of Da and Pr. Further, the heatlines give an in depth view of the energy flow within the enclosure. The end-to-end heatlines are observed to be smooth and parallel to the bottom wall indicating the conduction dominant heat transfer within the cavity at the low Da for all the cases. The heat flow circulation cells with very less magnitudes are found in the bottom portion of the enclosure for all the cases at the low Da. In all the

43

cases, the magnitudes of heatfunction are significantly high at the top edge of the enclosure due to the temperature singularity at the hot and cold junction. At the high Da, the intensity and size of the heatline circulation cell increase. The intense heat circulation cells occupy almost the entire part of the cavity for all the cases indicating the high thermal mixing. The local heat transfer rate increases along the left wall and attains highest value near the top edge of the enclosure for all the cases. The overall heat transfer rate from the hot wall increases with both Ra and Da and can be further explained based on the dense heatlines near the hot regimes especially at the high Da. Mahmud and Fraser [161] numerically investigated the nature of heat transfer and entropy generation for natural convection in a two-dimensional circular section enclosure filled with the porous media. The Darcy momentum equations and entropy generation equation (Eqs. (14a)–(14c), (40) and (47)) are used to model the porous media and Eqs. (14a)–(14c) have been solved using the control volume based finite volume method. In this study, the effect of RaD on the entropy generation in the porous media is studied in details. The overall results are illustrated in terms of the streamlines and Bejan number contours. At the low RaD , the streamlines are concentric circles except near the center region of the cavity, where the elliptic core is observed. As RaD increases, the convection current develops within the cavity and the isothermal lines start to swirl. In addition, the streamlines near the core of the circular enclosure are found to be elongated towards the horizontal direction. This results in the compression of the isotherms and streamlines towards the side walls. Hence, the large velocity and temperature gradients are found along the walls. This can be further clarified from the local Bejan number contours which clearly signify the large heat transfer and fluid friction irreversibilities along the side walls at the high RaD . Further, it was found that for the conduction regime, both the average Nusselt number and entropy generation number are independent of the Rayleigh number. In the convection dominant regime, these parameters show an increasing trend with the Rayleigh number (RaD ). Finally, it was inferred that the magnitude of the overall entropy generation rate is higher near the walls in contrast to the central core regions of the cavity. Natural convection in a two-dimensional porous right-angled triangular enclosure with an undulated left wall has been analyzed numerically by Bharadwaj and co-workers [162–164]. In all the studies, the bottom wall was kept at sinusoidal temperature. It may be noted that the undulated wall and the inclined walls were maintained at isothermal cold temperature in the first two studies [162,163] whereas the inclined wall was maintained adiabatic and the undulated wall was maintained at isothermal cold temperature in the last study [164]. The unsteady forms of the governing equations (Eqs. (17a)–(17c)) were solved using the finite-difference technique. The effect of the various parameters such as Rayleigh number (Ra ¼ 103
106 ), Darcy number (Da = 10
4 -10
2 ) and the undulations on the left wall has been investigated in detail. It is observed that at the low Da (Da ¼ 10
4 ) and Ra (Ra ¼ 103
104 ), the heat transfer irreversibilities ðSh Þ dominate over fluid fiction irreversibilities ðSw Þ whereas at the high Ra (Ra ¼ 105
106 ), the irreversibility due to heat transfer and fluid friction are nearly similar and the total entropy generation increases with Da within the cavity [162–164]. Also, the fluid friction irreversibility increases rapidly with Da at constant Ra. At the high Da ðDa ¼ 10
2 Þ, the total entropy generation due to the fluid friction dominates within the cavity for the high Ra (Ra ¼ 105 and 106 ) in contrast to the low Ra (Ra ¼ 103 and 104 ) (Fig. 23(i)(a–d)). It is observed that, the heat transfer rate increases significantly for the undulated left wall compared to the non-undulated wall irrespective of Ra and Da (Fig. 23(ii)). Also, Sw increases abruptly for higher Ra and Da in

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Fig. 23. (i) Local entropy generation due to heat transfer (Sh ) and fluid flow (Sw ) for irreversibility distribution ratio, / ¼ 10
2 , Darcy number, Da ¼ 10
4 and Pr ¼ 0:71 at various Rayleigh number, (a) Ra ¼ 103 , (b) Ra ¼ 104 , (c) Ra ¼ 105 , (d) Ra ¼ 106 , (ii) comparison of the maximum Nusselt number (Numax ) for undulation case () and noundulation case (N) for Da ¼ 10
2 on left wavy wall for Pr ¼ 0:71 [162]. (figures are reproduced from Bharadwaj and Dalal [162] with permission from Elsevier).

the undulated wall case compared to the no-undulation case. In contrast, the undulations on the left wall do not have the significant effect on the heat transfer irreversibility. 6.3. Nanofluids Abu-Nada et al. [165] investigated the effect of various nanofluids on the heat transfer enhancement during natural convection in a horizontal concentric annuli. The effect of the various parameters such as Ra (Ra ¼ 103
105 ), / (0 6 / 6 0:1) and L=D (L=D ¼ 0:2
0:8) ratio was studied in detail. Note that, Pr of the base fluid was maintained constant at 6:2 for all the case studies. The water-based nanofluid containing various volume fractions of the Al2O3 nanoparticles was used in the present study. The governing equations (Eqs. (24a)–(24d)) were solved using the finite volume method. It is observed that the higher heat transfer rate is achieved around the inner cylinder surface except in the plume region at the high values of L=D ratio. For the low Ra, the addition of the Al2O3 nanoparticles results in the increase in the Nusselt

number around the inner cylinder surface. As Ra increases, the effect of the Al2O3 nanoparticles on the heat transfer rate within the cavities is less pronounced. In addition, the magnitude of the Nusselt number is observed to decrease for L=D ¼ 0:4 and the Al2O3 nanoparticles show the reverse effect on the Nusselt number, especially for L=D ¼ 0:4. It was also observed that the increase in volume fraction of the Al2O3 nanoparticles results in the enhanced heat transfer at the core where the intensity of the plume was observed to be higher. Abouali and Falahatpisheh [166] numerically investigated natural convection of the Al2O3 nanofluid in a vertical annuli. The vertical walls are maintained at the constant temperature and the horizontal walls are adiabatic. The effect of nanofluids on natural convection is investigated as a function of the geometrical and physical parameters for various particle fractions (/ ¼ 0
0:06), aspect ratios (1 6 A 6 5) and Gr (103 6 Gr 6 105 ). The cylindrical coordinates representations of the governing equations (Eqs. (24a)–(24d)) are integrated over each control volume to obtain a set of linear algebraic equations which were further solved by

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D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

the SIMPLE algorithm. The final results were illustrated in terms of the Nusselt number and they were also validated with the experimental data. The variations of the Nusselt number for various volume fractions of nanoparticles vs aspect ratios illustrate that the maximum Nusselt number occurs at the aspect ratio, 1:8 and 1:2 for Gr ¼ 103 and 104 , respectively. In addition, the Nusselt number is a decreasing function of the aspect ratio for Gr P 103 . It may be noted that for the higher Grashof number, the decrease of the Nusselt number due to the nanofluid compared to the base fluid is more pronounced. A correlation for the ratio of the local Nusselt number (Nubf ) for the base fluid and local Nusselt number for the nanofluid (Nunf ) demonstrates that the ratio follows a monotonically and linearly decreasing function of the particle fraction. Arefmanesh et al. [167] studied the heat transfer and flow characteristics in a two-square duct annuli filled with TiO2-water nanofluid with an isothermally hot inner boundary and cold outer boundary. The outer duct is maintained at a constant cold temperature while the inner duct is kept at a differentially higher constant temperature. The governing equations (Eqs. (24a)–(24d)) in terms of the primitive variables are solved using the finite volume method and the SIMPLER algorithm. The effects of the Rayleigh number (103 6 Ra 6 106 ), aspect ratio of the annulus (0:25 6 A 6 0:75), and volume fraction of the nanoparticles (/ ¼ 0
0:04) on the fluid flow and heat transfer are investigated in detail. The fluid is gradually heated by the sides of the inner square and consequently, the fluid starts to expand as it moves upward. Subsequently, the fluid is cooled by the sides of the outer square and contracts as it moves downward. Hence, the counterrotating anti-clockwise and clockwise fluid circulations, are formed in the left and the right halves of the annulus, respectively. Due to the symmetry of the square geometry, these counter-rotating eddies are symmetric with respect to the vertical centerline of the squares. At the low A and Ra, the streamlines are found to be evenly distributed throughout the enclosure. As Ra increases, the streamlines become more densely packed adjacent to the sides of the inner and the outer squares and the strength of the fluid circulation cell also increases in contrast to the low Ra. Distinct thermal boundary layers are formed around the inner square as well as along the sides of the outer square at the high Ra. As A increases, the streamlines and isotherms become more densely packed within the annulus. It may be noted that the volume fraction of the nanoparticles (/) has the negligible effect on the fluid flow and temperature distribution at the low Ra whereas the increase in / significantly affects the fluid flow and temperature distribution within the enclosure at the high Ra. The average Nusselt number increases monotonically with Ra irrespective of the volume fraction. However, the magnitude of the Nusselt number decreases with the volume fraction. Natural convective heat transfer in a C type enclosure filled with Cu–water nanofluid was investigated by Mahmoodi and Hashemi [168]. The top, left and bottom walls of the C-shaped enclosure were maintained at a high temperature whereas the square rid was maintained at a cold temperature. The small vertical wall joining the rid and the bottom and top walls are maintained at adiabatic conditions. The governing equations (Eqs. (24a)–(24d)) were solved using the control volume approach. The rate of heat transfer within the C-shaped enclosure was thoroughly investigated for the various parameters such as Rayleigh number (Ra ¼ 103
106 ), the aspect ratio of the C-shaped enclosure (A ¼ 0:2
0:8) and the volume fraction of the Cu nanoparticles (0 6 / 6 1). At the low A, the fluid gets heated by the hot walls and expands as it moves upward. Consequently, the fluid is cooled by the cold rib and compressed as it moves downward. Hence, a clockwise fluid circulation cell is established encompassing the whole portion of the enclosure. As A increases, the area of the

45

square rib increases and the primary fluid circulation cell in the left half of the C-shaped enclosure gets compressed. Consequently, the small amount of fluid gets detached and forms weak secondary fluid circulation cell under the cold rib. As A increases further, the multicellular flow pattern is absent under the cold rib due to the existence of small gap between the hot bottom wall and the cold rib which limits the flow movement. It may be noted that, as the aspect ratio of the cavity (A) increases, the isotherms are evenly distributed on the top of the cold rib. However, the isotherms are found to be highly distorted near the left half of the enclosure for all the aspect ratios. For the lower aspect ratio (A ¼ 0:2) and higher Ra, the core of the flow cell moves downward resulting in more densely packed streamlines at the bottom of the enclosure. In addition, distorted parabolic shaped isotherms are observed at the high Ra and low A. On the other hand, a single fluid circulation cell of the high intensity is observed in the left half of the enclosure for the higher A and Ra, whereas flow circulations are absent in the right half. Note that, isotherms are smooth and parallel in the right half of the enclosure whereas they are found to be distorted in the left half of the enclosure. It was further observed that the average Nusselt number increases with the nanoparticles volume fraction for all Ra and A. In addition, the effect of the nanoparticles on the enhancement of heat transfer at the low Ra is more significant than that at the high Ra. Fluid flow and heat transfer in a C shaped enclosure for the differentially heated side walls was studied by Dehnavi and Rezvani [169] using Al2O3 as the nanoparticles. The step of the C-shaped enclosure is maintained as isothermally hot and the other two vertical isothermal walls were maintained cold while the top and bottom walls were maintained adiabatic. The heat transfer and fluid flow within the enclosure are thoroughly investigated for various parameters such as the non-uniform nanoparticles size, mean nanoparticle diameter, nanoparticle volume fraction, Grashof number (Gr), dimensionless height of the step (h), and dimensionless width of the step (w). The momentum and energy balance equations (Eqs. (24a)–(24d)) have been solved numerically using the finite volume approach. The streamline and isotherm features are qualitatively similar to the work carried out by Dagtekin et al. [155] irrespective of Gr. Note that, the isotherms show slightly greater distortion, depicting the strong and dominant convection at the high Gr (Fig. 21(ii)(a–d)). However, the strength of the convection decreases with the cavity aspect ratio (A ¼ h=w) and thus, the conduction dominant heat transfer occurs for A ¼ 2 (h ¼ 0:5 and w ¼ 0:25) and 3 (h ¼ 0:75 and w ¼ 0:25) (Fig. 21(ii) (c and d)) in contrast to A ¼ 0:66 (h ¼ 0:5 and w ¼ 0:75) and 1:5 (h ¼ 0:75 and w ¼ 0:5) (Fig. 21(ii)(a and b)). It is also interesting to observe as that distance between the hot wall and left cold wall increases, the heat transfer rate increases. Further, it was also observed that the heat transfer rate increases by almost 21% and 17% for Gr ¼ 103 and Gr ¼ 105 , respectively as the mean diameter of nanoparticles decreases from 200 nm to 5 nm. On the other hand, the heat transfer rate was found to increase by 17% and 13% for Gr ¼ 103 and Gr ¼ 105 , respectively as the mean nanoparticle volume fraction increases from 0:01 to 0:05. Cho et al. [170] performed a numerical investigation based on natural convection heat transfer characteristics within a complicated cavity filled with nanofluid (Al2O3-water). The left wavy wall was maintained at higher temperature compared to the right wavy wall and the horizontal walls were maintained adiabatic. The governing equations (Eqs. (24a)–(24d)) are formulated using the Boussinesq approximation and the complex-wavy-surface is modeled as the superimposition of two sinusoidal functions. The effect of the various parameters such as Rayleigh number (Ra ¼ 100
106 ), amplitude of wavy surface (a ¼ 0 to 0:75), wavelength of wavy surface (k ¼ 1
8) and volume fraction of nanopar-

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D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

ticles (0 6 / 6 0:1) were analyzed in detail. The results were illustrated in terms of streamlines, isotherms and Nusselt number distribution within the cavity. Due to the uniform heating of the left wall, a clockwise slanted fluid circulation cell is found to encompass the entire convex enclosure, irrespective of Ra. It is observed that the circulation structure expands and becomes more complex at high Rayleigh number (Ra). The significant twisting of the isotherms are also observed signifying the dominance of convection. In addition, it is seen that the isotherms are concentrated near the crest region of the hot wavy-wall and the trough region of the cold wavy-wall. Consequently, the large temperature gradients are formed in the crest and trough regions of the hot and cold wavy walls, respectively. It is interesting to observe that the mean Nusselt number increases with the wave amplitude (a) at the low values of Rayleigh number. However, at the higher values of the Rayleigh number (Ra P 104 ), the mean Nusselt number decreases with the wave amplitude. It is also observed that the mean Nusselt number increases as the wavelength is increased from k ¼ 1 to 4 irrespective of the Rayleigh number. However, a decreasing trend in the mean Nusselt number is observed as the wavelength of the wavy surface (k) increases from 4 to 8. Further, it is also found that the heat transfer rate increases to a great extent with the volume fraction of the nanoparticles. Very few studies have been reported till date on the entropy generation studies within a nanofluid filled complex enclosures. Esmaeilpoura and Abdollahzadeh [86] investigated the effects of the Grashof number and volume fraction of Cu–water nanofluid on natural convection heat transfer and fluid flow within a twodimensional differentially heated (hot left wall and right cold wall along with horizontal adiabatic walls) wavy enclosure. The finitevolume method with non orthogonal body fitted collocated grid arrangement is used to solve the governing differential equations (Eqs. (24a)–(24d)). The computations were performed for various Grashof numbers (Gr ¼ 104 to 106 ), nanoparticles volume fraction (0% to 10%) and surface waviness (0
0:4) for different patterns of the wavy enclosure. Four different patterns based on the projection of the side walls (case 1: rectangular, case 2: inward left inward right walls, case 3: inward left - outward right walls, case 4: outward left and outward right walls) were considered for the study [86]. In the absence of the nanoparticles, the streamlines depict that an unicellular flow circulation cell exists for the cases 1 and 3, whereas multi-cellular fluid circulation cells were observed for the cases 2 and 4 for Gr ¼ 104 . The increase in Grashof number results in the vortex for the cases 1 and 3 to grow and break-up into smaller vortices. On the other hand, the multiple vortices are observed for the cases 2 and 4 merge and form a single vortex. The isotherm contours display large zones of stratified temperature along the central core region in all the enclosures for all Gr. However, the comparison of the patterns for the cases 2 and 3 at the middle of the cavity shows that the thickness of thermal boundary layer for the case 3 pattern is more than the case 2 pattern. It was found that the entropy generation increases with the Grashof number and that also decreases with the increasing surface waviness. Note that, the actives zones of the entropy generation were observed along the walls of the enclosure at the higher Gr. It was also observed that the average entropy generation is lower in the case of nanofluids compared to the pure fluid. It was also inferred from the studies that the average Nusselt number increases with the Grashof number and that decreases with surface waviness, irrespective of the volume fractions. It was finally concluded that among all wavy enclosures filled with nanofluids, the case 4 pattern exhibit maximum value of average Nusselt number while the case 2 pattern has the minimum value of the average entropy generation. Saidi and Karimi [171] carried out the numerical studies on natural convection cooling in an L-shape enclosure filled with copper–

water nanofluid. The bottom wall and the left vertical wall were maintained isothermally hot whereas the opposite walls were maintained isothermally cold. Note that, the walls connecting the hot and cold walls were maintained adiabatic. The modified governing equations (Eqs. (27a)–(27d)) were solved numerically using the finite volume approach. The effects of the Rayleigh number (103 6 Ra 6 106 ), aspect ratio of the L-shaped enclosure (A ¼ 0:2 and 0:4) and the volume fraction of the Cu nanoparticles (0 6 / 6 0:1) on the heat transfer coefficient, temperature and flow fields were illustrated in details. Similar to the previous works [167,170], it was observed that the multiple fluid circulation cells intensify with Ra. In addition, the isotherms were found to be largely distorted due to the intense convection especially at Ra ¼ 106 . For the higher aspect ratio (A), intense fluid circulation cells were observed to encompass the entire enclosure whereas the flow circulation cells were found to be prevalent only near the vertical zone of the L-shaped enclosure for the low A. As a result, the heat transfer coefficients were found to be significantly higher for the higher A. The flow and temperature distributions were found to be almost qualitatively similar for all / irrespective of A. However, the intensity of the streamfunction was found to increase with / and hence the heat transfer coefficient was found to be higher for the high values of /. In another work, Cho et al. [172] numerically investigated natural convection heat transfer performance and entropy generation of Al2O3-water nanofluid in an inclined wavy-wall cavity. The shape of the wavy-wall cavity, the boundary conditions and the governing equations (Eqs. (24a)–(24d)) were similar to that considered in the previous work [170]. The streamlines, isotherms and entropy maps were achieved for two different Ra (102 and 106 ) and various inclination angles (0 6 u 6 315 ). At Ra ¼ 102 , the buoyancy effect is weak as indicated by the weak fluid circulation cells and hence, the isotherms follow the geometry of the enclosure, irrespective of u. As u increases, the fluid circulation cell forms multiple vortices which eventually separates into secondary fluid circulation cells near the upper and lower halves of the enclosure, especially for u ¼ 90 and 270 . Note that, the lower thermal mixing is indicated by the clustered isotherms near the central core of the enclosure at Ra ¼ 102 , irrespective of u. High entropy generation occurs near the lower-left crest region and upper-right trough region of the cavity at Ra ¼ 102 . As Ra increases to 106 , the buoyancy effect within the cavity becomes stronger, and hence, the heat transfer is dominated by convection. As a result, more complex circulation structures are induced and the isotherms are more twisted. It is seen that the isotherms are closely packed near the lower left crest region and upper-right trough region of the cavity. Consequently, the greater local entropy generation occurs over these extended regions of the cavity. At Ra ¼ 106 , fluid circulation cell with multiple vortices are observed in the vertical direction for u ¼ 45 and 90 whereas, fluid circulation cells formed multiple vortices horizontally for u P 135 . Due to the dominance of conduction at Ra ¼ 102 , the mean Nusselt number has a low and approximately constant value for all inclination angles. However, at the high Ra, the convection mechanism dominates the heat transfer performance and the buoyancy effect depends on the orientation of the cavity. As a result, the mean Nusselt number varies steeply with the increase in the inclination angle. 7. Conclusion Current review is a comprehensive outlook on the research progress made on natural convection in various non-square enclosures (triangular, trapezoidal, rhombic or parallelogrammic and compli-

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cated enclosures). In the present era, the enclosures with nonsquare shapes have drawn wide attention due to its extensive applications in various food processing and material processing industries. The enclosures with the straight walls are much easier to model and the hydrodynamic and thermal flow patterns are less complex compared to the enclosure with the complex structure such as the inclined, curved and wavy surfaces. Subsequently, a number of works on the non-square enclosures are found. The fluid flow, temperature distribution and heat flow within the enclosures are comprehensively illustrated using the streamlines, isotherms and heatlines, respectively. As discussed by earlier workers, the heatline approach was found to be the most effective tool to visualize the convective heat transfer in order to decide the various geometric and thermal parameters to achieve efficient heat transfer rates. In addition, many studies on the entropy generation within the enclosures have also been listed in order to achieve an energy efficient approach for enhanced thermal processing of the materials. Based on natural convection in various enclosures, the following observations are listed below: 7.1. Triangular enclosures  Conduction dominant heat transfer occurs at the low Pr and Ra based on the parallel isotherms and less intense flow. On the other hand, the stronger influence of convection is observed at the high Ra based on the multiple circulations of streamlines or heatlines and distorted isotherms. Note that, the aspect ratio [4,42,88,90,91,97] and base angle [74,75,87,89,91,97] of the various triangular enclosures play the pivotal role in the fluid flow distributions within the enclosures. Natural convection simulation results with the discrete heating [74] in the triangular enclosure demonstrate that the location of the heaters along the central regime of the bottom wall results in the minimum entropy generation due to the fluid friction along the side walls.  Darcy number (permeability) plays a significant role in the fluid flow, heat transfer and entropy generation within the porous triangular enclosures during natural convection. The results have been illustrated in details for various models such as Darcy model, Darcy–Brinkman model and Darcy–Brinkman–Forchhei mer model. Note that, the Darcy number and Rayleigh number have been incorporated into a Darcy–Rayleigh number (RaD ¼ Ra  Da) and the effect of RaD on the flow field in the triangular enclosures has been studied in details [48,83,99,100,102–104]. At the small Darcy–Rayleigh number (RaD ), the magnitudes of streamfunction are small and the heatlines [104–107] are orthogonal to the isotherms, indicating conduction dominant heat transfer. On the other hand, at the high RaD , convection is initiated, the isotherms are gradually distorted and thermal mixing is observed in almost all the cases. A number of earlier works are devoted on convection with the explicit role of Rayleigh and Darcy numbers within the triangular enclosures [84,101,105–107]. Similar to the fluid media, the base angles and aspect ratio of the triangular enclosures influence the streamline and heatline distribution within the porous triangular enclosures. Further, heat transfer and fluid friction irreversibilities within the triangular enclosures have been quantified using the entropy generation by few researchers.  The addition of nanoparticles in the base fluid plays the crucial role in enhancing the heat transfer rate within the triangular enclosures. As a result, the effect of various nanofluids such as Copper–water [63] and Ethylene Glycol–Copper–water [64] nanofluid within the triangular enclosures have been investigated in details. In addition, the results have also been demonstrated for the different values of the Rayleigh number, solid volume fraction, heat source location, enclosure aspect ratio

47

and Brownian motion on the flow and temperature fields within the various triangular enclosures are examined in details [63,64,108]. For all values of the solid volume fraction, the higher heat transfer rate with Ra occurs due to strengthening of the buoyancy forces [63,64]. The location of the heat source along the walls of the triangular enclosures significantly affects the heat transfer rate at various Rayleigh numbers [64,108]. It was also observed that the heat transfer rate within the nanofluid filled triangular enclosure continuously increases with the solid volume fraction at all Rayleigh numbers when the Brownian motion is neglected [63]. Overall, it was inferred that the effective heat transfer rate within the nanofluid filled enclosure was found to be higher compared to the regular fluid filled enclosure.

7.2. Trapezoidal enclosures  A number of earlier works [109–112] involve natural convection in trapezoidal cavities in conjunction with internal baffles attached to the walls and internal baffles are shown to demonstrate the significant effect on flow and heat transfer characteristics. The effects of the Rayleigh number, Prandtl number, aspect ratio, baffle height, and baffle location on the flow and temperature fields were investigated in details for various boundary conditions. The presence of baffles was found to largely affect the flow and temperature distribution within the trapezoidal enclosures [109,110]. In addition, the inclination angles of the side walls of the trapezoidal enclosure with the vertical axis also play a significant role in the fluid and heat flow distributions inside the enclosures [69,112–116]. In most of the cases, it is concluded that the trapezoidal enclosure with higher inclination angle is the optimal shape for the thermal processing irrespective of Prandtl number (Pr ¼ 0:015
1000).  A few studies on natural convection in porous trapezoidal enclosures have also been presented. Studies have been carried out for various values of flow parameters such as modified Rayleigh (RaD ) numbers [32,117–119] and various geometric parameters (inclination angles and aspect ratio) with Darcian and non-Darcian assumptions on the porous model within the trapezoidal enclosures. It was observed that partial cooling along the side walls of the porous trapezoidal enclosure has the significant effect on the flow field within the porous trapezoidal enclosures [32]. Also, it was inferred that the trapezoidal enclosure with the higher inclination angles corresponds to the highest heat transfer rate along the hotter wall, irrespective of RaD and imposed thermal boundary conditions [120–122]. The thermal mixing within the porous trapezoidal enclosures was clearly illustrated by the dense heatline distributions and intense heat circulation cells [32,120–122]. Further, the entropy generation studies within the porous trapezoidal enclosures illustrate that the high heat transfer rate, optimal thermal mixing and minimum entropy generation rate are achievable only at the lower RaD .  The problem of natural convection heat transfer in a trapezoidal enclosure filled with nanofluids has been studied numerically by various researchers. The structure of the fluid flow and heat transfer rate within the nanofluid filled trapezoidal enclosure were found to depend upon various inclination angles, solid volume fractions and Rayleigh numbers [124–126]. It was inferred from earlier studies [124,125] that nanoparticles with the high volume fraction are most effective in enhancing performance of the heat transfer rate. In addition, the structure of the fluid flow and temperature fields within the nanofluid filled trapezoidal enclosure is found to vary with the aspect ratio [126] and Prandtl number [124–126].

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

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D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

7.3. Rhombic/parallelogrammic enclosures  Extensive studies have been reviewed based on the fluid media filled rhombic and parallelogrammic shaped enclosures. The effect of the thermal aspect ratio, inclination angles of the enclosure, Rayleigh number, Prandtl number and Fourier number on the flow field within the rhombic/parallelogrammic enclosure has been investigated in details [127,128,130–135]. Asymmetric flow circulation cells were observed in most of the studies due to the asymmetric geometric configuration of the rhombic/parallelogrammic enclosure. Similar to the triangular and trapezoidal enclosures, the heatlines concept was implemented for the visualization of the heat flow [134,135]. It was inferred that rhombic cavities with the lower inclination angles are found to be useful in the liquid metal processing applications (Pr ¼ 0:015) whereas cavities with the higher inclination angle are efficient in the solar heating applications and thermal processing of chemical solutions and oils (Pr ¼ 1000). Further, the entropy generation studies within fluid filled rhombic/parallelogrammic enclosure conclude that the total entropy generation is found to be low for the lower inclination angles [135].  A number of studies on porous rhombic/parallelogrammic enclosures have been reviewed in details [138–142]. Analysis of heatlines and entropy generation during natural convection in porous rhombic/parallelogrammic cavities has been carried out based on the controlling parameters such as inclination angle [138,140–142], Darcy number [139–142], Rayleigh number [139–142], modified Rayleigh number [138], enclosure gap [139], porosity [139], etc. It was observed that the flow strength and the total heat transfer rate increases with Pr irrespective of Darcy and Rayleigh numbers. Overall, the thermal efficiency within the porous rhombic cavity of lesser inclination angles is concluded to be optimum based on the minimum entropy generation values and greater heat transfer rates [141,142].

7.4. Complex enclosures  A number of works based on natural convection heat transfer within fluid filled complex geometry have been reviewed in details [143–155]. The test studies were carried out for various inclination angles, amplitude-wavelength ratios, undulations on the side walls, irreversibility distribution ratios, aspect ratio, Rayleigh numbers and Prandtl numbers. It was concluded that the presence of the undulation in the walls strongly affects both the local heat transfer rate and flow field as well as thermal field [143–151]. In addition, the flow and thermal characteristics within the fluid filled complex enclosure is found to be highly dependent on amplitude-wavelength ratios for the lesser inclination angles, especially for high Rayleigh numbers [148,151]. The heat transfer rate was found to be higher for the complex enclosures with greater amplitude-wavelength ratio. It was also found that for a particular aspect ratio, the entropy generation is less and heat transfer rate is invariant and independent of Rayleigh number at the conduction regime. On the other hand, the entropy generation and heat transfer rate show an increasing trend with Rayleigh number [153–155].  Natural convection heat transfer in a complex cavity filled with a porous-saturated medium was studied numerically by various authors [56,156–162]. The effect of porosity was studied in details by varying the Darcy–Rayleigh number over a large range. In all the studies, the heat transfer rate was found to be higher for the higher values of Darcy–Rayleigh number as the overall hydraulic resistance within the porous media was con-

siderably lesser for the high Darcy–Rayleigh number. In addition, the flow circulations within the cavities were also found to be intense at the high Darcy–Rayleigh number. The results also illustrated that the increase in the amplitude of the wavy surface and the number of undulations largely enhanced the effective heat transfer rate inside the complex cavity [56,156– 159]. In addition, the fluid flow characteristics within the enclosure also significantly changes and multiple fluid circulations occur for high amplitude-wavelength ratio and higher number of undulations.  A number of works on natural convection heat transfer of nanofluid in various complicated geometries were also presented [86,165–170]. The effect of nanofluids on natural convection is investigated as a function of geometrical and physical parameters and particle fractions such as aspect ratio, Rayleigh number, nanoparticle fraction, effective thermal conductivity ratios, etc. The results illustrate that the heat transfer rate increases with the volume fraction of nanoparticles irrespective of Rayleigh number (Ra). In addition, it is also shown that the heat transfer performance can be optimized via changing the amplitudewave ratio of the enclosure [165–169]. Overall, the results inferred from the various studies provide a useful insight into potential strategies for enhancing the convection heat transfer performance within complicated cavities.

References [1] A. Bejan, Convection Heat Transfer, third ed., Wiley, Hoboken, NJ, 2004. [2] H. Chiang, C. Kleinstreuer, Analysis of passive cooling in a vertical finite channel using a falling liquid-film and buoyancy-induced gas vapour flow, Int. J. Heat Mass Transfer 34 (1991) 2389–2394. [3] P. Payvar, Laminar heat transfer in the oil groove of a wet clutch, Int. J. Heat Mass Transfer 34 (1991) 1791–1798. [4] P.M. Haese, M.D. Teubner, Heat exchange in an attic space, Int. J. Heat Mass Transfer 45 (2002) 4925–4936. [5] K.A. Joudi, I.A. Hussein, A.A. Farhan, Computational model for a prism shaped storage solar collector with a right triangular cross section, Energy Convers. Manage. 45 (2004) 391–409. [6] H. Mistry, Ganapathi-subbu, S. Dey, P. Bishnoi, J.L. Castillo, Modeling of transient natural convection heat transfer in electric ovens, Appl. Therm. Eng. 26 (2006) 2448–2456. [7] A.M.A. Dayem, Experimental and numerical performance of a multi-effect condensation–evaporation solar water distillation system, Energy 31 (2006) 2710–2727. [8] S. Kalaiselvam, M. Veerappan, A.A. Aaron, S. Iniyan, Experimental and analytical investigation of solidification and melting characteristics of PCMs inside cylindrical encapsulation, Int. J. Therm. Sci. 47 (2008) 858–874. [9] S.M. Wang, A. Faghri, T.L. Bergman, A comprehensive numerical model for melting with natural convection, Int. J. Heat Mass Transfer 53 (2010) 1986– 2000. [10] S. Ostrach, Natural convection in enclosures, Advances Heat Transfer, Academic press, New York, 1972, 161–227. [11] S. Ostrach, Natural convection in enclosure, ASME J. Heat Transfer 110 (1988) 1175–1190. [12] C.J. Hoogendoorn, Natural convection in enclosures, Proc. 8th Int. Heat Transfer Conf, vol. I, Hemisphere Publishing Corp., San Francisco, 1986, pp. 111–120. [13] T. Fusegi, J.M. Hyun, Laminar and transitional natural convection in an enclosure with complex and realistic conditions, Int. J. Heat Fluid Flow 15 (1994) 258–268. [14] O. Aydin, A. Unal, T. Ayhan, Natural convection in rectangular enclosures heated from one side and cooled from the ceiling, Int. J. Heat Mass Transfer 42 (1999) 2345–2355. [15] I.E. Sarris, I. Lekakis, N.S. Vlachos, Natural convection in a 2-D enclosure with sinusoidal upper wall temperature, Numer. Heat Transfer A 42 (2002) 513– 530. [16] T. Basak, S. Roy, A.R. Balakrishnan, Effects of thermal boundary conditions on natural convection flows within a square cavity, Int. J. Heat Mass Transfer 49 (2006) 4525–4535. [17] R.S. Kaluri, T. Basak, Analysis of distributed thermal management policy for energy-efficient processing of materials by natural convection, Energy 35 (2010) 5093–5107. [18] Z.H. Kodah, M.A. Jarrah, N.S. Shanshal, Thermal characterization of foam-cane (Quseab) as an insulant material, Energy Convers. Mange. 40 (1999) 349–367. [19] I.T. Togrul, Determination of convective heat transfer coefficient of various crops under open sun drying conditions, Int. Comm. Heat Mass Transfer 30 (2003) 285–294.

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx [20] B. Dhifaoui, S. Ben-Jabrallah, A. Belghith, J.P. Corriou, Experimental study of the dynamic behaviour of a porous medium submitted to a wall heat flux in view of thermal energy storage by sensible heat, Int. J. Therm. Sci. 46 (2007) 1056–1063. [21] M. Javaheri, J. Abedi, H. Hassanzadeh, Linear stability analysis of doublediffusive convection in porous media, with application to geological storage of CO2 , Transp. Porous Media 84 (2010) 441–456. [22] O.V. Trevisan, A. Bejan, Mass and heat-transfer by natural convection in a vertical slot filled with porous medium, Int. J. Heat Mass Transfer 29 (1986) 403–415. [23] J.L. Lage, A. Bejan, The resonance of natural convection in an enclosure heated periodically from the side, Int. J. Heat Mass Transfer 36 (1993) 2027–2038. [24] M. Song, R. Viskanta, Natural convection flow and heat transfer within a rectangular enclosure containing a vertical porous layer, Int. J. Heat Mass Transfer 37 (1994) 2425–2438. [25] G.B. Kim, J.M. Hyun, H.S. Kwak, Buoyant convection in a square cavity partially filled with a heat-generating porous medium, Numer. Heat Transfer A 40 (2001) 601–618. [26] M.A. Hossain, M. Wilson, Natural convection flow in a fluid-saturated porous medium enclosed by non-isothermal walls with heat generation, Int. J. Therm. Sci. 41 (2002) 447–454. [27] J.R. Philip, Free convection at small Rayleigh number in porous cavities of rectangular, elliptical, triangular and other cross-sections, Int. J. Heat Mass Transfer 25 (1982) 1503–1509. [28] T.S. Lee, Computational and experimental studies of convective fluid motion and heat transfer in inclined non-rectangular enclosures, Int. J. Heat Fluid Flow 5 (1984) 29–36. [29] J.M. Hyun, B.S. Choi, Transient natural convection in a parallelogram-shaped enclosure, Int. J. Heat Fluid Flow 11 (1990) 129–134. [30] L. Iyican, Y. Bayazitoglu, L.C. Witte, An analytical study of natural convective heat transfer within trapezoidal enclosure, ASME J. Heat Transfer 102 (1980) 640–647. [31] Y.E. Karyakin, Transient natural convection in prismatic enclosures of arbitrary cross-section, Int. J. Heat Mass Transfer 32 (1989) 1095–1103. [32] Y. Varol, H.F. Oztop, I. Pop, Natural convection in right-angle porous trapezoidal enclosure partially cooled from inclined wall, Int. Commun. Heat Mass Transfer 36 (2009) 6–15. [33] O.M. Kamiyo, D. Angeli, G.S. Barozzi, M.W. Collins, V.O.S. Olunloyo, S.O. Talabi, A comprehensive review of natural convection in triangular enclosures, Appl. Mech. Rev. 63 (2010) 06080. [34] S.C. Saha, M.M.K. Khan, A review of natural convection and heat transfer in attic-shaped space, Energy Build. 43 (2011) 2564–2571. [35] G.D. Mey, M. Wojcik, J. Pilarski, M. Lasota, J. Banaszczyk, B. Vermeersch, A. Napieralski, M.D. Paepe, Chimney effect on natural convection cooling of a transistor mounted on a cooling fin, J. Electron. Packag. 131 (2009) 14501– 14503. [36] Y. Varol, H. Oztop, A comparative numerical study on natural convection in inclined wavy and flat-plate solar collectors, Build. Environ. 43 (2008) 1535– 1544. [37] K.S. Jairaj, S.P. Singh, K. Srikant, A review of solar dryers developed for grape drying, Sol. Energy 83 (2009) 1698–1712. [38] M. Matian, A.J. Marquis, N.P. Brandon, Application of thermal imaging to validate a heat transfer model for polymer electrolyte fuel cells, Int. J. Hydrogen Energy 35 (2010) 12308–12316. [39] A.A. Al-Abidi, S. Mat, K. Sopian, M.Y. Sulaiman, A.T. Mohammad, Numerical study of PCM solidification in a triplex tube heat exchanger with internal and external fins, Int. J. Heat Mass Transfer 61 (2013) 684–695. [40] M.C. Kim, C.K. Choi, D.Y. Yoon, Analysis of the onset of buoyancy-driven convection in a water layer formed by ice melting from below, Int. J. Heat Mass Transfer 51 (2008) 5097–5101. [41] F. Erdogdu, M. Tutar, Velocity and temperature field characteristics of water and air during natural convection heating in cans, J. Food Sci. 76 (2011) 119– 129. [42] H. Salmun, Convection patterns in a triangular domain, Int. J. Heat Mass Transfer 38 (1995) 351–362. [43] D.A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York, 2006. [44] D.B. Ingham, I. Pop, Transport Phenomena in Porous Media, Elsevier, Amsterdam, 1998. [45] R.A. Wooding, Steady state free thermal convection of liquid in a saturated permeable medium, J. Fluid Mech. 2 (1956) 273–285. [46] H.P.G. Darcy, Les Fontaines Publiques de La Ville de Dijon, Victor Dalmont, Paris, 1856. [47] D. Poulikakos, A. Bejan, Numerical study of transient high Rayleigh number convection in an attic-shaped porous layer, ASME J. Heat Transfer 105 (1983) 476–484. [48] Y. Varol, H.F. Oztop, A. Varol, Effects of thin fin on natural convection in porous triangular enclosures, Int. J. Therm. Sci. 46 (2007) 1033–1045. [49] P. Forchheimer, Wasserbewegung durch Borden, Z. Ver. Dtsch. Ing. 45 (1901) 1781–1788. [50] S. Ergun, Fluid flow through packed columns, Chem. Eng. Progr. 48 (1952) 89– 94. [51] H.C. Brinkman, On the permeability of media consisting of closely packed porous particles, Appl. Sci. Res. 1 (1949) 81–86. [52] S. Kakac, R.K. Shah, W. Aung, Handbook of Single-Phase Convective Heat Transfer, John Wiley, New York, 1987.

49

[53] Z.G. Du, E. Bilgen, Natural convection in vertical cavities with internal heat generating porous medium, Wrme-und Stoffbertr 27 (1992) 149–155. [54] M. Bhattacharya, T. Basak, On multiple steady states for natural convection (low Prandtl number fluid) within porous square enclosures: effect of nonuniformity of wall temperatures, Int. J. Heat Mass Transfer 59 (2013) 230–246. [55] K. Vafai, C.L. Tien, Boundary and inertia effects on flow and heat transfer in porous media, Int. J. Heat Mass Transfer 24 (1981) 195–203. [56] X.B. Chen, P. Yu, S.H. Winoto, H.T. Low, Free convection in a porous wavy cavity based on the Darcy–Brinkman–Forchheimer extended model, Numer. Heat Transfer A 52 (2007) 377–397. [57] A.K. Singh, T. Basak, A. Nag, S. Roy, Role of entropy generation on thermal management during natural convection in tilted porous square cavities, J. Taiwan Inst. Chem. Eng. 50 (2015) 153–172. [58] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME Fluids Eng. Div. 231 (1995) 99–103. [59] R. Saidur, K.Y. Leong, H.A. Mohammad, A review on applications and challenges of nanofluids, Renewable Sustainable Energy Rev. 15 (2011) 1646–1668. [60] K.Y. Leong, R. Saidur, S.N. Kazi, A.H. Mamun, Performance investigation of an automotive car radiator operated with nanofluid-based coolants (nanofluid as a coolant in a radiator), Appl. Therm. Eng. 30 (2010) 2685–2692. [61] T. Yousefi, F. Veysi, E. Shojaeizadeh, S. Zinadini, An experimental investigation on the effect of Al2 O3
H2 O nanofluid on the efficiency of flat-plate solar collectors, Renewable Energy 39 (2012) 293–298. [62] M.A. Mansour, M.A.Y. Bakier, Free convection heat transfer in complex-wavywall enclosed cavity filled with nanofluid, Int. Commun. Heat Mass Transfer 44 (2013) 108–115. [63] B. Ghasemi, S.M. Aminossadati, Brownian motion of nanoparticles in a triangular enclosure with natural convection, Int. J. Therm. Sci. 49 (2010) 931–940. [64] S.M. Aminossadati, B. Ghasemi, Enhanced natural convection in an isosceles triangular enclosure filled with a nanofluid, Compu. Math. App. 61 (2011). [65] J. Koo, C. Kleinstreuer, A new thermal conductivity model for nanofluids, J. Nanopart. Res. 6 (2004) 577–588. [66] J. Koo, C. Kleinstreuer, Laminar nanofluid in microheat-sinks, Int. J. Heat Mass Transfer 48 (2005) 2652–2661. [67] E. Abu-Nada, Z. Masoud, H.F. Oztop, A. Campo, Effect of nanofluid variable properties on natural convection in enclosures, Int. J. Therm. Sci. 49 (2010) 479–491. [68] S. Kimura, A. Bejan, The heatline visualization of convective heat transfer, ASME J. Heat Transfer 105 (1983) 916–919. [69] T. Basak, S. Roy, I. Pop, Heat flow analysis for natural convection within trapezoidal enclosures based on heatline concept, Int. J. Heat Mass Transfer 52 (2009) 2471–2483. [70] T. Basak, G. Aravind, S. Roy, Visualization of heat flow due to natural convection within triangular cavities using Bejan’s heatline concept, Int. J. Heat Mass Transfer 52 (2009) 2824–2833. [71] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, FL, 1996. [72] A. Bejan, Method of entropy generation minimization, or modeling and optimization based on combined heat transfer and thermodynamics, Rev. Gen. Therm. 35 (1996) 637–646. [73] A. Bejan, A study of entropy generation in fundamental convective heat transfer, J. Heat Transfer ASME 101 (1979) 718–725. [74] Y. Varol, H.F. Oztop, A. Koca, Entropy production due to free convection in partially heated isosceles triangular enclosures, Appl. Therm. Eng. 28 (2008) 1502–1513. [75] T. Basak, R. Anandalakshmi, P. Gunda, Role of entropy generation during convective thermal processing in right-angled triangular enclosures with various wall heatings, Chem. Eng. Res. Des. 90 (2012) 1779–1799. [76] G.G. Ilis, M. Mobedi, B. Sunden, Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls, Int. Commun. Heat Mass Transfer 35 (2008) 696–703. [77] T. Basak, P. Kumar, R. Anandalakshmi, S. Roy, Analysis of entropy generation minimization during natural convection in trapezoidal enclosures of various angles with linearly heated side wall(s), Ind. Eng. Chem. Res. 51 (2012) 4069– 4089. [78] D.A. Nield, Resolution of a paradox involving viscous dissipation and nonlinear drag in a porous medium, Transp. Porous Media 41 (2000) 349– 357. [79] A.K. Al-Hadhrami, L. Elliott, D.B. Ingham, A new model for viscous dissipation in porous media across a range of permeability values, Transp. Porous Media 53 (2003) 117–122. [80] K. Hooman, H. Gurgenci, Effects of viscous dissipation and boundary conditions on forced convection in a channel occupied by a saturated porous medium, Transp. Porous Media 68 (2007) 301–319. [81] D.A. Nield, A.V. Kuznetsov, An historical and topical note on convection in porous media, J. Heat Transfer ASME 135 (2013) 06120. [82] A.C. Baytas, Entropy generation for natural convection in an inclined porous cavity, Int. J. Heat Mass Transfer 43 (2000) 2089–2099. [83] Y. Varol, H.F. Oztop, I. Pop, Entropy generation due to natural convection in non-uniformly heated porous isosceles triangular enclosures at different positions, Int. J. Heat Mass Transfer 52 (2009) 1193–1205. [84] T. Basak, P. Gunda, R. Anandalakshmi, Analysis of entropy generation during natural convection in porous right-angled triangular cavities with various

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

50

[85]

[86]

[87]

[88]

[89]

[90] [91]

[92]

[93]

[94]

[95]

[96]

[97]

[98]

[99]

[100]

[101]

[102]

[103] [104]

[105]

[106]

[107]

[108]

[109]

[110]

[111] [112]

D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx thermal boundary conditions, Int. J. Heat Mass Transfer 55 (2012) 4521– 4535. P.A.K. Lam, K.A. Prakash, A numerical study on natural convection and entropy generation in a porous enclosure with heat sources, Int. J. Heat Mass Transfer 69 (2014) 390–407. M. Esmaeilpoura, M. Abdollahzadeh, Free convection and entropy generation of nanofluid inside an enclosure with different patterns of vertical wavy walls, Int. J. Therm. Sci. 52 (2012) 127–136. E.H. Ridouane, A. Campo, J.Y. Chang, Natural convection patterns in rightangled triangular cavities with heated vertical sides and cooled hypotenuses, J. Heat Transfer 127 (2005) 1181–1186. H.F. Oztop, Y. Varol, A. Koca, Laminar natural convection heat transfer in a shed roof with or without eave for summer season, Appl. Therm. Eng. 27 (2007) 2252–2265. E.F. Kent, Numerical analysis of laminar natural convection in isosceles triangular enclosures for cold base and hot inclined walls, Mech. Res. Commun. 36 (2009) 497–508. E.F. Kent, E. Asmaz, S. Ozerbay, Laminar natural convection in right triangular enclosures, Heat Mass Transfer 44 (2007) 187–200. G. Saha, Md.T. Islam, S. Saha, Md.Q. Islam, Natural convection in a tilted isosceles triangular enclosure with discrete bottom heating, Thammasat Int. J. Sci. Tech. 12 (2007) 24–35. T. Basak, S. Roy, Ch. Thirumalesha, Finite element analysis of natural convection in a triangular enclosure: effects of various thermal boundary conditions, Chem. Eng. Sci. 62 (2007) 2623–2640. T. Basak, S. Roy, S.K. Babu, A.R. Balakrishnan, Finite element analysis of natural convection flow in a isosceles triangular enclosure due to uniform and non-uniform heating at the side walls, Int. J. Heat Mass Transfer 51 (2008) 4496–4505. A. Koca, H.F. Oztop, Y. Varol, The effects of Prandtl number on natural convection in triangular enclosures with localized heating from below, Int. Commun. Heat Mass Transfer 34 (2007) 511–519. Y. Varol, A. Koca, H.F. Oztop, Natural convection in a triangle enclosure with flush mounted heater on the wall, Int. Commun. Heat Mass Transfer 33 (2006) 951–958. A. Sojoudi, S.C. Saha, F. Xu, Y.T. Gu, Transient air flow and heat transfer due to differential heating on inclined walls and heat source placed on the bottom wall in a partitioned attic shaped space, Energy Build. 113 (2016) 39– 50. R.S. Kaluri, R. Anandalakshmi, T. Basak, Bejan’s heatline analysis of natural convection in right-angled triangular enclosures: effects of aspect-ratio and thermal boundary conditions, Int. J. Therm. Sci. 49 (2010) 1576–1592. T. Basak, G. Aravind, S. Roy, A.R. Balakrishnan, Heatline analysis of heat recovery and thermal transport in materials confined within triangular cavities, Int. J. Heat Mass Transfer 53 (2010) 3615–3628. Y. Varol, H.F. Oztop, A. Varol, Free convection heat transfer and flow field in triangular enclosures filled with porous media, J. Porous Media 11 (2008) 103–115. Y. Varol, H.F. Oztop, I. Pop, Influence of inclination angle on buoyancy-driven convection in triangular enclosure filled with a fluid-saturated porous medium, Heat Mass Transfer 44 (2008) 617–624. T. Basak, S. Roy, S.K. Babu, Natural convection and flow simulation in differentially heated isosceles triangular enclosures filled with porous medium, Chem. Eng. Sci. 63 (2008) 3328–3340. H.F. Oztop, Y. Varol, I. Pop, Investigation of natural convection in triangular enclosure filled with porous media saturated with water near 4  C, Energy Convers. Manage. 50 (2009) 1473–1480. Y. Varol, Natural convection in porous triangular enclosure with a centered conducting body, Int. Commun. Heat Mass Transfer 38 (2011) 368–376. Y. Varol, H.F. Oztop, M. Mobedi, I. Pop, Visualization of natural convection heat transport using heatline method in porous non-isothermally heated triangular cavity, Int. J. Heat Mass Transfer 51 (2008) 5040–5051. T. Basak, S. Roy, D. Ramakrishna, I. Pop, Visualization of heat transport during natural convection within porous triangular cavities via heatline approach, Numer. Heat Transfer A 57 (2010) 431–452. T. Basak, S. Roy, D. Ramakrishna, B.D. Pandey, Analysis of heat recovery and heat transfer within entrapped porous triangular cavities via heatline approach, Int. J. Heat Mass Transfer 53 (2010) 3655–3669. R. Anandalakshmi, R.S. Kaluri, T. Basak, Heatline based thermal management for natural convection within right-angled porous triangular enclosures with various thermal conditions of walls, Energy 36 (2011) 4879–4896. I. Rezaiguia, M. Kadja, R. Mebrouk, N. Belghar, Numerical computation of natural convection in an isosceles triangular cavity with a partially active base and filled with a Cu–water nanofluid, Heat Mass Transfer 49 (2013) 1319–1331. F. Moukalled, M. Darwish, Natural convection in a partitioned trapezoidal cavity heated from the side, Numer. Heat Transfer A 43 (2003) 543– 563. F. Moukalled, M. Darwish, Natural convection in a trapezoidal enclosure heated from the side with a baffle mounted on its upper inclined surface, Heat Transfer Eng. 25 (2004) 80–93. M.E. Arici, B. Sahin, Natural convection heat transfer in a partially divided trapezoidal enclosure, Therm. Sci. 13 (2009) 213–220. A. Silva, E. Fontana, V.C. Mariani, F. Marcondes, Numerical investigation of several physical and geometric parameters in the natural convection into trapezoidal cavities, Int. J. Heat Mass Transfer 55 (2012) 6808–6818.

[113] T. Basak, S. Roy, A. Singh, B.D. Pandey, Natural convection flow simulation for various angles in a trapezoidal enclosure with linearly heated side wall(s), Int. J. Heat Mass Transfer 52 (2009) 4413–4425. [114] T. Basak, D. Ramakrishna, S. Roy, A. Matta, I. Pop, A comprehensive heatline based approach for natural convection flows in trapezoidal enclosures: effect of various walls heating, Int. J. Therm. Sci. 50 (2011) 1385–1404. [115] T. Basak, R. Anandalakshmi, P. Kumar, S. Roy, Entropy generation vs energy flow due to natural convection in a trapezoidal cavity with isothermal and non-isothermal hot bottom wall, Energy 37 (2012) 514–532. [116] D. Ramakrishna, T. Basak, S. Roy, Analysis of heatlines and entropy generation during free convection within trapezoidal cavities, Int. Commun. Heat Mass Transfer 45 (2013) 32–40. [117] A.C. Baytas, I. Pop, Natural convection in a trapezoidal enclosure filled with a porous medium, Int. J. Eng. Sci. 39 (2001) 125–134. [118] B.V.R. Kumar, B. Kumar, Parallel computation of natural convection in trapezoidal porous enclosures, Math. Comput. Simul. 65 (2004) 221–229. [119] Y. Varol, Natural convection for hot materials confined within two entrapped porous trapezoidal cavities, Int. Commun. Heat Mass Transfer 39 (2012) 282– 290. [120] T. Basak, S. Roy, A. Matta, I. Pop, Analysis of heatlines for natural convection within porous trapezoidal enclosures: effect of uniform and non-uniform heating of bottom wall, Int. J. Heat Mass Transfer 53 (2010) 5947–5961. [121] D. Ramakrishna, T. Basak, S. Roy, Heatlines for visualization of heat transport for natural convection within porous trapezoidal enclosures with various wall heating, Numer. Heat Transfer A 63 (2012) 347–372. [122] T. Basak, R. Anandalakshmi, S. Roy, I. Pop, Role of entropy generation on thermal management due to thermal convection in porous trapezoidal enclosures with isothermal and non-isothermal heating of wall, Int. J. Heat Mass Transfer 67 (2013) 810–828. [123] D. Ramakrishna, T. Basak, S. Roy, E. Mamoniat, Analysis of thermal efficiency via analysis of heat flow and entropy generation during natural convection within porous trapezoidal cavities, Int. J. Heat Mass Transfer 77 (2014) 98– 113. [124] H. Saleh, R. Roslan, I. Hashim, Natural convection heat transfer in a nanofluidfilled trapezoidal enclosure, Int. J. Heat Mass Transfer 54 (2011) 194–201. [125] R. Roslan, H. Saleh, I. Hashim, Buoyancy-driven heat transfer in nanofluidfilled trapezoidal enclosure with variable thermal conductivity and viscosity, Numer. Heat Transfer A 60 (2011) 867–882. [126] R. Nasrin, S. Parvin, Investigation of buoyancy-driven flow and heat transfer in a trapezoidal cavity filled with water–Cu nanofluid, Int. Commun. Heat Mass Transfer 39 (2012) 270–274. [127] H. Yuncu, S. Yamac, Laminar natural convective heat transfer in an air-filled parallelogramic cavity, Int. Commun. Heat Mass Transfer 18 (1991) 559–568. [128] F. Moukalled, H. Diab, S. Acharya, Laminar natural convection in a horizontal rhombic annulus, Numer. Heat Transfer A 24 (1993) 89–107. [129] F. Moukalled, S. Acharya, Laminar natural convection heat transfer in an eccentric rhombic annulus, Numer. Heat Transfer A 26 (1994) 551–568. [130] D. Naylor, P.H. Oosthuizen, A numerical study of free convective heat transfer in a parallelogram-shaped enclosure, Int. J. Numer. Method Heat Fluid Flow 4 (1994) 553–559. [131] K.D. Aldridge, H. Yao, Flow features of natural convection in a parallelogrammic enclosure, Int. Commun. Heat Mass Transfer 28 (2001) 923–931. [132] A. Bairi, On the Nusselt number definition adapted to natural convection in parallelogrammic cavities, Appl. Therm. Eng. 28 (2008) 1267–1271. [133] A. Bairi, J.M. Garcia de Maria, N. Laraqi, Transient natural convection in parallelogrammic enclosures with isothermal hot wall. Experimental and numerical study applied to on-board electronics, Appl. Therm. Eng. 30 (2010) 1115–1125. [134] V.A.F. Costa, M.S.A. Oliveira, A.C.M. Sousa, Laminar natural convection in a vertical stack of parallelogrammic partial enclosures with variable geometry, Int. J. Heat Mass Transfer 48 (2005) 779–792. [135] R. Anandalakshmi, T. Basak, Heat flow visualization for natural convection in rhombic enclosures due to isothermal and non-isothermal heating at the bottom wall, Int. J. Heat Mass Transfer 55 (2012) 1325–1342. [136] R. Anandalakshmi, T. Basak, Analysis of entropy generation due to natural convection in rhombic enclosures, Ind. Eng. Chem. Res. 50 (2011) 13169– 13189. [137] R. Anandalakshmi, T. Basak, Natural convection in rhombic enclosures with isothermally heated side or bottom wall: entropy generation analysis, Eur. J. Mech. B Fluids 54 (2015) 27–44. [138] A.C. Baytas, I. Pop, Free convection in oblique enclosures filled with a porous medium, Int. J. Heat Mass Transfer 42 (1999) 1047–1057. [139] F. Moukalled, M. Darwish, Natural convection heat transfer in a porous rhombic annulus, Numer. Heat Transfer A 58 (2010) 101–124. [140] R. Anandalakshmi, T. Basak, Heatline analysis for natural convection within porous rhombic cavities with isothermal/nonisothermal hot bottom wall, Ind. Eng. Chem. Res. 51 (2012) 2113–2132. [141] R. Anandalakshmi, T. Basak, Analysis of natural convection via entropy generation approach in porous rhombic enclosures for various thermal aspect ratios, Int. J. Heat Mass Transfer 64 (2013) 224–244. [142] R. Anandalakshmi, T. Basak, Numerical simulations for the analysis of entropy generation during natural convection in porous rhombic enclosures, Numer. Heat Transfer A 63 (2013) 257–284. [143] Y. Varol, H.F. Oztop, Buoyancy induced heat transfer and fluid flow within a tilted wavy solar collector, Build. Environ. 42 (2007) 2062–2071.

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034

D. Das et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx [144] L. Adjlout, O. Imine, A. Azzi, M. Belkadi, Laminar natural convection in an inclined cavity with a wavy wall, Int. J. Heat Mass Transfer 45 (2002) 2141– 2152. [145] E.H. Ridouane, A. Campo, Free convection performance of circular cavities having two active curved vertical sides and two inactive curved horizontal sides, Appl. Therm. Eng. 26 (2006) 2409–2416. [146] A. Dalal, M.K. Das, Natural convection in a cavity with a wavy wall heated from below and uniformly cooled from the top and both sides, ASME J. Heat Transfer 128 (2006) 717–725. [147] H.F. Oztop, E. Abu-Nada, Y. Varol, A. Chamkha, Natural convection in wavy enclosures with volumetric heat sources, Int. J. Therm. Sci. 50 (2011) 502– 514. [148] P.K. Das, S. Mahmud, Numerical investigation of natural convection within a wavy enclosure, Int. J. Therm. Sci. 42 (2003) 397–406. [149] M. Ashjaee, M. Amiri, J. Rostami, A correlation for free convection heat transfer from vertical wavy surfaces, Heat Mass Transfer 44 (2007) 101–111. [150] M.K. Triveni, R. Panua, Numerical simulation of natural convection in a triangular enclosure with caterpillar (C)-curve shape hot wall, Int. J. Heat Mass Transfer 96 (2016) 535–547. [151] A. Dalal, M.K. Das, Heatline method for the visualization of natural convection in a complicated cavity, Int. J. Heat Mass Transfer 51 (2008) 263–272. [152] P. Biswal, T. Basak, Bejan’s heatlines and numerical visualization of convective heat flow in differentially heated enclosures with concave/convex side walls, Energy 64 (2014) 69–94. [153] S. Mahmud, A.K.M.S. Islam, Laminar free convection and entropy generation inside an inclined wavy enclosure, Int. J. Therm. Sci. 42 (2003) 1003–1012. [154] S. Mahmud, R.A. Fraser, Free convection and entropy generation inside a vertical inphase wavy cavity, Int. Commun. Heat Mass Transfer 31 (2004) 455–466. [155] I. Dagtekin, H.F. Oztop, A. Bahloul, Entropy generation for natural convection in C-shaped enclosures, Int. Commun. Heat Mass Transfer 34 (2007) 502– 510. [156] B.V.R. Kumar, P.V.S.N. Murthy, P. Singh, Free convection heat transfer from an isothermal wavy surface in a porous enclosure, Int. J. Numer. Method Fluids 28 (1998) 633–661. [157] A. Misirlioglu, A.C. Baytas, I. Pop, Free convection in a wavy cavity filled with a porous medium, Int. J. Heat Mass Transfer 48 (2005) 1840–1850. [158] A. Misirlioglu, A.C. Baytas, I. Pop, Natural convection within an inclined wavy enclosure filled with a porous medium, Transport Porous Media 64 (2006) 229–246.

51

[159] K. Khanafer, B. Al-Azmi, A. Marafie, I. Pop, Non-Darcian effects on natural convection heat transfer in a wavy porous enclosure, Int. J. Heat Mass Transfer 52 (2009) 1887–1896. [160] T. Basak, R. Anandalakshmi, P. Biswal, Analysis of Convective heat flow visualization within porous right angled triangular enclosures with a concave/convex hypotenuse, Numer. Heat Transfer A 64 (2013) 621–647. [161] S. Mahmud, R.A. Fraser, Free convection and irreversibility analysis within a circular porous enclosure, Entropy 5 (2003) 358–365. [162] S. Bhardwaj, A. Dalal, Analysis of natural convection heat transfer and entropy generation within porous right-angled triangular enclosure, Int. J. Heat Mass Transfer 65 (2013) 500–513. [163] S. Bhardwaj, A. Dalal, Effect of undulations on the natural convection heat transfer and entropy generation inside a porous right-angled triangular enclosure, Numer. Heat Transfer A 67 (2015) 972–991. [164] S. Bhardwaj, A. Dalal, S. Pati, Influence of wavy wall and non-uniform heating on natural convection heat transfer and entropy generation inside porous complex enclosure, Energy 79 (2015) 467–481. [165] E. Abu-Nada, Z. Masoud, A. Hijazi, Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids, Int. Commun. Heat Mass Transfer 35 (2008) 657–665. [166] O. Abouali, A. Falahatpisheh, Numerical investigation of natural convection of Al2 O3 nanofluid in vertical annuli, Heat Mass Transfer 46 (2009) 15–23. [167] A. Arefmanesh, M. Amini, M. Mahmoodi, M. Najafi, Buoyancy-driven heat transfer analysis in two-square duct annuli filled with a nanofluid, Eur. J. Mech. B Fluids 33 (2012) 95–104. [168] M. Mahmoodi, S.M. Hashemi, Numerical study of natural convection of a nanofluid in C-shaped enclosures, Int. J. Therm. Sci. 55 (2012) 76–89. [169] R. Dehnavi, A. Rezvani, Numerical investigation of natural convection heat transfer of nanofluids in a C shaped cavity, Superlattices Microstruct. 52 (2012) 312–325. [170] C.C. Cho, C.L. Chen, C.K. Chen, Natural convection heat transfer performance in complex-wavy-wall enclosed cavity filled with nanofluid, Int. J. Therm. Sci. 60 (2012) 255–263. [171] M. Saidi, G. Karimi, Free convection cooling in modified L-shape enclosures using copper–water nanofluid, Energy 70 (2014) 251–271. [172] C. C Cho, C.H. Chiu, C.Y. Lai, Natural convection and entropy generation of Al2O3–water nanofluid in an inclined wavy-wall cavity, Int. J. Heat Mass Transfer 97 (2016) 511–520.

Please cite this article in press as: D. Das et al., Studies on natural convection within enclosures of various (non-square) shapes – A review, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.034