Article

AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090

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Al-Qadisiyah Journal for Engineering Sciences

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* Corresponding author.

E-mail address: AmmarAbdulkadhim@mustaqbal-college.edu.iq (Ammar Abdulkadhim)

https://doi.org/10.30772/qjes.v13i2.633

Comprehensive Review of Natural Convection Heat Transfer in Annulus

Complex Enclosures

Ammar Abdulkadhim a*, Khaled Al-Farhany b, Azher M. Abed a, Hasan Sh. Majdi c

a Air conditioning and Refrigeration Techniques Engineering Department, Al-Mustaqbal University College, Babylon, 51001, Iraq
b Department of mechanical engineering, University of Al-Qadisiyah, Al-Qadisiyah, Iraq.
c Department of Chemical Engineering and Petroleum Industries, Al-Mustaqbal University College, Babylon, 51001, Iraq

A R T I C L E I N F O

Article history:

Received 01 March 2020

Received in revised form 26 April 2020

Accepted 30 April 2020

Keywords:

Natural convection

complex enclosures

Triangular

Trapezoidal

Parallelogrammic

Rhombic

Elliptical

A B S T R A C T

The natural convection heat transfer has many applications in engineering like solar collectors, cooling of

electronic equipment, and geothermal engineering. The present work demonstrates the recent publications

in the last ten years in this specific subject for a body located in complex shapes like rhombic, wavy,

trapezoidal, elliptical, and Parallelogrammic enclosure. Many parameters like Ra, Nu, number of

undulations, the position of the inner body had been addressed and discussed to draw the main conclusions

and recommendations. It is worthy to mention that a wavy enclosure has been investigated less than the

other simple enclosure shapes due to its complexity. Besides that entropy generation should be included in

future studies in complex shapes of enclosure as this will helps the researchers to extend their studies. The

inner bodies inside trapezoidal, parallelogrammic enclosure are very limited, and more investigation should

be done. The review concluded for the different shapes of enclosure with the tables that illustrate the major

finding of each study. Finally, the governing equations of the natural convection of enclosure filled with

pure fluid, porous medium, and nanofluid had been addressed.

1. Simple Enclosures

The natural convection heat transfer in simple enclosure shapes as

illustrated in Fig. 1 had been studied numerically by various researchers

due to its importance in energy related applications. Some of these studies

were [1-20]. Ghasemi et al. [2] investigated numerically the influence of

the magnetic field on buoyancy driven fluid flow in a square nanofluid

enclosure filled with Al2O3-water. The results indicated that Rayleigh and

Hartmann number had an opposite influence on heat transfer while the

increasing of nanoparticle loading helps in augmentation of heat transfer.

Another important study focus on using a different model of nanofluid

properties presented by Lai and Yang [3] using Lattice-Boltzmann scheme.

They found that different methods of nanofluid properties affect the Nusselt

number value. Bhuvaneswari et al. [4] computed using a finite volume

approach the natural convection with the magnetic field in a square

enclosure. The major of this study is that the authors applied sinusoidal

temperature distribution to both of the sidewalls and keeping the horizontal

wall adiabatic. The results indicated that increasing Hartmann number

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ABDULKADHIM ET. AL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090 81

reduces the heat transfer. Alam et al. [5] demonstrated the impact of aspect

ratio and heat source power density on fluid flow circulation and heat

transfer of free convection within rectangular enclosure heated and cooled

partially from the vertical walls while the rest of them and others walls

maintained adiabatic. They found that increases the aspect ratio augmented

the heat transfer and it reaches its maximum value at aspect ratio equals to

one. [6, 7] illustrated the double diffusive natural convection in tilted

rectangular enclosure including the impact of a magnetic field and heat

source. They concluded that heat source had a crucial impact on heat

transfer. For more details about the previous works, the reader can be

referring to [8-15]. Wang et al. [16] presented a comparison between the

impact of Al2O3-H2O and Ga-H2O and the radius under different Rayleigh

numbers of heat transfer using a two-phase lattice Boltzmann method. The

authors deduced that Ga-H2O enhances the heat transfer better than Al2O3-

H2O. Al-Farhany and Abdulkadhim [17] examined the conjugate problem

in a square enclosure filled with porous media under various Rayleigh and

Darcy numbers and they obtained that wall thickness effects on the

transformation of the heat mode from convection into conduction which

reduces the heat transfer. Barik and Al-Farhany [21] studied the inflleucne

of inclined baffle in nanofluid/porous square enclosure using COMSOL.

Dutta et al. [18] studied the entropy generation in quadrant porous enclosure

heated sinusoidally from its bottom wall. They concluded that entropy

generation due to fluid friction is dominated at high values of Darcy number

while they noticed the entropy formed due to heat transfer is the major

influencer on low Darcy number. Torki and Etesami [19] experimentally

studied the natural convection of a rectangular enclosure filled with SiO2 at

different nanofluid loading and enclosure inclination angles. They obtained

that the nanofluid did not affect low concentration value and the inclination

angle of a rectangular enclosure had a strong effect. In addition, the Nusselt

number is increased as the inclination angle goes up. Graževičius et al. [20]

studied experimentally and numerically using ANSYS 17.2 of natural

convection for removing heat from the reactor using a passive system.

Also, the natural convection in complex shapes had been reported by

various researchers like [22-25].

Figure 1. Schematic diagram of simple shape of enclosure (square) [2]

Sheikholeslami and Chamkha [22] examined the free convection in a lid-

driven enclosure filled with Fe3O4 with applied magnetic field and wavy

wall. They obtained that increasing magnetic number, Rayleigh number,

and nanofluid volume fraction increases the Nusselt number while

Hartmann number had a reverse impact on Nusselt number. Sheikholeslami

[23] studied the liquid metal due to natural convection in a wavy enclosure

for various values of Rayleigh number, amplitude of wavy wall, and

Hartmann number. It is worthy to mention here that the last parameters

increasing lead to reduce the Nusselt number. Two important studies collect

between MHD and wavy enclosure is presented in [24, 25] and they agreed

with the previous mentioned results in the previous works of various

researchers.

Besides that, various researchers dealt with the different shapes of an inner

body located inside a regular simple enclosure shape like a square or

rectangle that had been presented in Fig. 2. The inner body located within

the enclosure had a wide range of applications like a solar collector, fuel

cell, etc. The researchers among the world interested in understanding the

effect of inner shapes like circular, triangular, elliptical, and wavy inside

different shapes of the enclosure. Other researchers focus on the position of

the inner body and change the direction vertically, horizontally and

longitudinally. These are the main parameters that affect the heat transfer

so that some of these studies are presented by [26-31]. Lee et al. [26] used

an immersed boundary method to examine the changing of an inner

cylinder located within square enclosure horizontally and longitudinally

while another study by [27, 28] illustrates the vertical position on heat

transfer. Ali et al. [29] studied the mixed convection due to rotating inner

circular cylinder within square enclosure filled with air using ANSYS

FLUENT. Roslan et al. [30] studied the heated circular cylinder located

within the cold enclosure. The main important thing in this study is that the

inner circular cylinder had sinusoidal temperature under unsteady

conditions. We summarized the previous publications regarding the natural

convection within simple shape, complex shapes and simple annulus

enclosure in Table 1- 3, respectively.

Finally, the present work concentrates on the inner body located inside the

non-square enclosure.

Figure 2. Schematic diagram of the inner body within simple

enclosure shape [27]

82 ABDULKADHIM ET. AL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090

Table 1 summarized the studies of natural convection inside a simple shape of enclosure

Ref Objective Enclosure shape Software/model used Conclusions Special findings

Ghasemi et al. [2] MHD and nanofluid

effect

Square CVM using SIMPLE

algorithm

Hartmann and Rayleigh

numbers had an inverse

effect on Nusselt number

Nanofluid effect may increases

of decreases Nusselt number

depending upon Rayleigh

number

Lai and Yang [3] Al2O3 nanofluid Square Lattice – Boltzmann

method (LBM)

Nanofluid thermophysical

models effect on the

computation of the Nusselt

number

LBM is recommended for the

practical engineering

applications

Bhuvaneswari et

al. [4]

Sinusoidal temperature

boundary conditions

with MHD effect

Square FVM Increasing Hartmann

number reduces the heat

transfer

The phase deviation of applied

Sinusoidal temperature effect

on Nusselt number

Alam et al [5] Partial cooling/heating

and aspect ratio

Rectangle FEM Nussselt number increases

when the aspect ratio range

from 0.5 – 10 beyond this

value it reduces Nu as it

goes up.

The maximum Nusselt number

is achieved at aspect ratio

equals to one.

Teamah et al. [6] Double diffusive,

Inclinations angle,

MHD, buoyancy ratio

Inclined rectangle CVM SIMPLER

algorithm

Lowest and Highest value

of Sherwood and Nu

numbers were at 75 and 150

Sherwood number is not

affected by heat absorption and

generation.

El Qarnia et al. [9] Phase change due to

melting

Rectangle FVM/FORTRAN Two correlations had been

developed.

The developed model can be

used in phase change material

Nithyadevi et al.

[15]

Effect of numbers of

discrete heater, Prandtl

number and heat

generation

Rectangle FVM Increasing numbers of

heater and Prandtl number

enhance the heat transfer

Increasing heat generation

reduces the heat transfer

Qi at a;. [16] Ra and nanofluid radius Rectangle Two-phase LBM Small radius of nanofluid

can enhance better than the

big size

The augmentation of nanofluid

is better at low Ra number

Al-Farhany and

Abdulkadhim [17]

Conjugate problem in

porous medium

square FEM/COMSOL Increasing Ra and Da

enhance the heat transfer

Increasing the conduction wall

reduces the heat transfer

Dutta et al. [18] Porous media, entropy

generation with non-

uniform bottom wall

temperature

Quadrant FEM Increasing Ra and Da

increase the heat transfer as

mentioned in most of the

publications

When the Darcy number is

low, the heat transfer's entropy

generation is higher while at

the high Da, the entropy

generation due to friction of

fluids is higher

Torki et al. [19] nanofluid, inclined

enclosure

Rectangle Experimental study Increasing Rayleigh number

increases Nu

Inclination effect is higher at

low nanofluid loading

Table 2 summarized the studies of natural convection inside the complex shape of enclosure

Ref Objective Enclosure shape Software/model used Conclusions Special findings

Sheikholeslami

et al. [22]

MHD, lid-driven

cavity, ferro nanofluid

Wavy FEM/FORTRAN Increasing nanofluid

loading, Rayleigh, and

magnetic numbers

increases the heat

transfer while Hartmann

increases to reduce it.

The authors studied the

wavy top wall which is a

little bit make a difference

with the previous

publications

Sheikholeslami

et al. [23]

MHD Wavy CVFEM Increasing Hartmann

number reduces Nu

Hartmann number is highly

effects on fluid flow and

heat transfer

Xiong, et al.

[25]

Nanofluid/porous

layers and MHD

Wavy CVFEM Ha increasing leads to

reduction in the Nusselt

number

Increasing Da leads to an

improvement in the heat

transfer

ABDULKADHIM ET. AL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090 83

Table 3 summarized the studies of natural convection between inner body within regular (simple) enclosure shapes

Ref Objective Enclosure shape Software/model used Conclusions Special findings

Lee et al. [26] Position of inner circular

cylinder in horizontal and

diagonal direction

Circular cylinder

within square

enclosure

FVM Increasing Ra number leads

to enhance the Nusselt

number

It is noted that when the

cylinder becomes closest to the

corner or left walls, the eddies

that located in the direction of

the cylinder will be separated

while one large eddies formed

behind the cylinder

Hussain and

Husssein [27]

Vertical position of the

inner cylinder, constant

heat flux is considered

for the cylinder

Circular cylinder

within square

enclosure

FVM As Ra increases, Nu goes

When the cylinder moves

vertically upward, two inner

cells (eddies) are formed below

it.

Park et al. [28] Same the study of [27] except they applied isotherm hot temperature to the cylinder

[30] and [31] Unsteady case study Wavy temperature

conditions inside a

square cavity

COMSOL Increasing the oscillation of

the heat source leads to

improve the heat transfer.

The frequency of wavy

condition 25π – 30π augments

the heat transfer better

2. Triangular enclosure

This section describes different shapes of an inner body located inside a

triangular enclosure. Schematic diagram of this case study is illustrated

below in Fig. 3. Some of these studies presented by [32-38].

Figure 3. Schematic diagram of the inner body in a triangular

enclosure [32]

Xu et al. [32] examined numerically the natural convection heat transfer

between various shapes of the inner cylinder (circular, square, rhombic, and

triangular) located inside the triangular enclosure tilted for various

inclination angle. The inner cylinder is kept at uniform hot temperature

while the triangular enclosure is cold. The governing equations had been

solved numerically using a finite volume method and validated with the

previously published work along the bottom wall. The results are crucial

and indicate that increasing the Rayleigh numbers breaking down the

symmetry of streamlines contours and concentrates the isotherms contours

to the top between the gaps. Yu et al. [33] examined the influence of various

values of Prandtl number on heat transfer between a circular cylinder

located inside an inclined triangular enclosure. The equations of mass,

energy, and momentum of fluid along with Boussinesq approximations had

been solved using a finite volume scheme. The results indicate that a low

Prandtl number less than 0.7 had no effect of fluid flow intensity and heat

transfer while the inclination angle had strong effect on it. The authors

proposed important empirical equations of Nusselt number in terms of

Prandtl number. Selimefendigil and Öztop [34] examined the mixed

convection between circular cylinder within right-angled triangular

enclosure heated partially from its left vertical wall. The governing

equations had been solved numerically using a finite element method. The

results are important because of increasing Hartmann number reduces both

of entropy generation and heat transfer while increasing nanofluid volume

fraction and speed of rotating circular cylinder increases both of them. Yu

et al. [35] studied the unsteady free convection between circular cylinder

within triangular enclosure under various effects of Grashof number, aspect

ratio (inner diameter), and inclination angle. They simulated this

phenomenon using CFD code ANSYS Fluent 6.3 which is based upon a

finite volume method. They developed a relation between Nusselt number

as a function of Grashof number for different inner cylinder diameters and

inclination angles. Wang et al. [36] examined the mixed convection within

the different sizes of an inner rotating circular cylinder located within a

triangular enclosure and the gap between them was filled with Ethylene

glycol-silicon carbide nanofluid for different Rayleigh numbers. Fluent

CFD code had been used to simulate the whole of this problem. The results

of this paper agreed with the previously published works. Sourtiji et al. [37]

examined the natural fluid flow between a circular cylinder within a

triangular nanofluid enclosure using a control volume based on a finite

element scheme. The nanofluid thermal thermo-physical properties like

viscosity and thermal conductivity had been predicted using Brinkman and

Maxwell-Garnetts. It is obtained that adding void fraction of nanofluid had

a remarkable impact at low Rayleigh numbers. Also, it had been observed

that increasing inner circular cylinder diameter augments the heat transfer

obviously. Another important investigation had been reported in Amrani et

al. [38]. The authors studied the combined effect of radiation as well as free

convective flow for triangular enclosure within rectangular body. As the

most of researchers, finite volume method had been used to simulate this

phenomenon under various Rayleigh numbers and aspect ratios.

3. Trapezoidal

This section summarized the convection heat transfer due to the density

difference between the inner body located within the trapezoidal enclosure

[39-43]. Schematic representation of this case is presented in Fig. 4.

84 ABDULKADHIM ET. AL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090

Figure 4. Schematic diagram of the inner body in a triangular

enclosure [39]

Hussein et al. [39] investigated numerically the mixed convection heat

transfer between the inner circular rotating cylinder immersed in the

trapezoidal enclosure using COMSOL which is based upon the finite

element scheme. The gap between the inner body and the enclosure is

divided into two layers system; the upper layer is filled with Copper-water

nanofluid while the lower layer is consisting of the same nanofluid

immersed in a saturated porous medium. The authors studied the influences

of many dimensionless parameters such as Rayleigh and Darcy numbers,

nanofluid void fraction as well as many geometrical parameters such as

undulations number of a bottom wavy wall, inner body's diameter and

rotational speed and the thickness of the porous layer. The results were

crucial and indicated that the increasing of Rayleigh, Darcy numbers,

nanofluid void fraction, and inner body's diameter rotational leads to

augmentation in the local Nusselt number, and that means enhancement of

the heat transfer. However the behavior of other parameters is inverse

which means as the layer of the porous medium and the number of

undulations goes up, the heat transfer is reduced. Esam et al. [40] examined

the natural convection heat transfer in a trapezoidal enclosure filled with

multilayers using the finite element method. The enclosure is partially

heated from the bottom wall while the top wall is kept at isotherm cold

temperature. The two inclined walls, as well as the inner circular cylinder

and the rest length of the bottom wall, are assumed adiabatic. The upper

layer is filled with Ag-water nanofluid, while the lower layer filled with

porous media saturated with the same nanofluid. The results had been

validated with the previously published works and the agreement was good.

The results indicate that increasing nanofluid, Darcy, and Rayleigh

numbers increases the fluid flow intensity and the heat transfer rate.

However, the behavior of porous layer thickness is completely reversed.

Khan et al. [41] explained the mixed convection heat transfer between the

inner heated circular cylinder rotating counter-clockwise located in a

trapezoidal enclosure. The trapezoidal enclosure is kept adiabatic at its top

and bottom wall while the two inclined walls are kept at cold temperatures.

They made a comparison between the influence of the rotating cylinder and

a motionless cylinder in a square enclosure. Their results confirmed that the

rotating cylinder as well as the inclination angle of the sidewall effect

significantly on the heat transfer. Selimefendigil [42] demonstrated

numerically the natural convection between different shapes of the inner

conductive body within a trapezoidal enclosure filled with different shapes

of nanoparticle (blade, spherical, and cylindrical). The top and bottom walls

are adiabatic while the left and right vertical walls are kept at isotherm hot

and cold temperature respectively. The authors used finite element scheme

to solve the governing equations and the validation seems good. The

parameters of this study were Rayleigh number, thermal conductivity ratio,

solid volume fraction, shapes of the nanoparticle and the inclination angle.

It is worthy to mention that they conclude the shape of nanoparticle change

obviously the heat transfer and the cylindrical shape is recommended for

better Nusselt number and better enhancement in heat transfer. Ahmed et

al. [43] visualized numerically using finite element techniques the heat lines

in a nanofluid trapezoidal enclosure separated by a porous divider. The top

wall is kept cold while a non-uniform temperature profile is applied to the

bottom wall. The side walls are adiabatic. The parameters of this study are

Rayleigh and Darcy number as well as nanofluid volume fraction as

physical parameters. Also, many geometrical parameters had been

investigated such as porous layer thickness and its positions as the authors

moved it vertically upwards and downwards. The major findings were

increasing the thickness of porous divider with the Rayleigh numbers

augments the heat transfer. Also, the position of the porous divider is

significant at low Rayleigh number while at higher Rayleigh number, the

Nusselt number will be at the minimum values when the divider moved

vertically downward.

We summarized the studies of the inner body within triangular and

trapezoidal enclosure in Table 4.

Table 4 summarized the studies of natural convection between inner body within a triangular and trapezoidal enclosure

Ref Objective Enclosure shape Software/

model used

Conclusions Special findings

Xu et al. [32] Laminar free

convection, effect of

Ra, aspect ratio and

inclination angle

Cylinder inside

triangular enclosure

CVM At constant aspect ratio, the

inclination angle and Ra effect

significantly on Nusselt number.

Correlation of Nusselt number as a

function of Ra for each value of

aspect ratio

Yu et al. [31] Effect of Prandtl

number

Cylinder inside

coaxial triangular

FVM Inclination angle had a strong

impact on Nu

Unique effect of low Prandtl

number Nu while when Pr≥0.7 it

does not affect Nu

Selimefendigil and

Öztop [34]

Mixed convection,

MHD, nanofluid,

entropy

Rotating insulated

cylinder inside

triangular enclosure

FEM Increasing nanofluid concentrations

and rotating lead to an increasing in

total entropy and Nusselt number

Hartmann number increasing leads

to a reduction in both the entropy

and Nusselt number

Yu et al. [35] Unsteady natural

convection

Cylinder inside

coaxial triangular

ANSYS

Fluent

Correlations of Nusselt number had

been developed.

Nusselt number history had been

presented

Wang et al. Wang et

al. [36]

Mixed convection,

nanofluid

Cylinder inside

coaxial triangular

ANSYS

Fluent

Increasing Ra and nanofluid

volume fraction improve the Nu

Rotational velocity effect

significantly on Nu

ABDULKADHIM ET. AL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090 85

Sourtiji et al. [37]

Laminar buoyancy

driven flow, nanofluid

Cylinder inside

triangular enclosure

CVFEM Nanoparticle improve the heat as

mentioned in most of the studies

Maxwell–Garnetts (MG) and

Brinkman models had been used to

simulate the nanofluid presence

Amrani et al. [38]

Radiation and natural

convection effect

Rectangle inside

triangular enclosure

FVM Decreasing the aspect ratio and

increasing the Ra number enhances

the heat transfer

Thermal radiation promotes the heat

transfer

Hussein et al. [39] Mixed convection with

multi-layer system

Rotating cylinder in

trapezoidal enclosure

with sinusoidal

bottom wall

FEM The increasing the size and the

rotation speed of the inner cylinder

in addition to increases the Da, R

and nanofluid loading will improve

the heat transfer

Increasing porous thickness and

number of undulation of the bottom

wall reduce the heat transfer

Esam et al. [40] free convection with

multi-layer system

Fixed adiabatic

cylinder within

trapezoidal enclosure

FEM The results indicate that increasing

Ra, Da and nanofluid loading

enhance the heat transfer

Increasing porous thickness reduces

the heat transfer

Khan et al. [41] Mixed convection, air Rotating cylinder in

trapezoidal enclosure

FEM Grashof number for large

inclination angle is very strong

Rotating speed of the inner cylinder

and inclination angle of the

trapezoidal wall effect highly on the

Nusselt number

Selimefendigil [42] Natural convection

with different shapes of

nanoparticles (blades,

spherical and

cylindrical)

Different shapes of

inner body inside

trapezoidal

FEM Effect of Ra, thermal conductivity

ratio, nanofluid loading and shapes

of nanoparticle on Nu had been

discussed

Cylindrical nanoparticle gives better

performance

Ahmed et al. [43] Visualization of

heatlines of free

convection

trapezoidal enclosure

divided by porous

medium partition

FEM Increasing Ra, nanofluid volume

fraction and Darcy number

augments the heat transfer

Porous position equals to 0.5 gives

the better heat transfer

4. Parallelogrammic

This section summarized the convection heat transfer due to the density

difference between the inner body located within the parallelogrammic

enclosure [44-48]. The computational domain of these shapes is inserted in

Fig. 5.

Hussein [44] investigated numerically the influence of the position of

an inner circular cylinder located inside a parallelogrammic enclosure filled

with air using a finite volume method. The inner circular cylinder is kept at

a hot temperature while both vertical walls are cold. The top and bottom

walls are adiabatic. The effect of Rayleigh number, the inclination angle of

the vertical wall, and the inner circular cylinder position had been taken into

account and examined their effect on fluid flow strength and the heat

transfer. It was obtained that the maximum flow strength will be when the

inclination angle is zero i.e., for square enclosure when the cylinder moves

upwards by +0.1. It was also obtained that when the cylinder moves

downward will have greater Nusselt number than moving upwards. Majdi

et al. [45] examined the natural convection between the hot circular cylinder

immersed in a nanofluid parallelogrammic enclosure. The finite element

had been used to solve the governing equations of heat transfer and fluid

flow numerically. The validation was in good agreement with the previous

publishing works. The results indicated that the increase of nanofluid

volume fraction and Rayleigh number enhances the heat transfer especially

if the inner circular cylinder moves vertically downwards until it reaches -

0.1.

Chamkha [46] examined numerically the conjugate (conductive-natural

and forced) convection within the parallelogrammic enclosure separated by

solid partition using the finite volume method. The influence of various

parameters such as Richardson number, the inclination angle of the

enclosure from the cavity left vertical wall as well as the thermal

conductivity ratio. The results were crucial and all of the mentioned

parameters affect the heat transfer rate. Baïri [47] used a finite volume

scheme to simulate free convection under transient conditions within a

parallelogrammic enclosure filled with air. The enclosure is partially heated

from its left vertical wall while cold temperature conditions are applied to

the right wall. The top and bottom walls are kept adiabatic. Hussain et al.

[48] simulated the free convection in a parallelogrammic enclosure

containing a volumetric source under various inclination angles. The

enclosure is heated non-uniformly from its left wall while the right wall is

at isotherm cold temperature. The top and bottom walls are adiabatic.

Figure 5. Schematic diagram of the inner body in a parallelogrammic

enclosure [45]

5. Rhombic

This section summarized the convection heat transfer due to the density

difference between the inner body located within rhombic enclosure [49-

53]. The present case is illustrated in Fig. 6.

Anandalakshmi and Basak [49] examined numerically the entropy

generation and natural convection in a rhombic enclosure filled with

saturated porous medium for different heating situation and inclination

angle. Different uniform temperatures applied on the bottom and top walls

where the bottom wall is warmer. Adiabatic conditions are considered to

86 ABDULKADHIM ET. AL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090

both inclined vertical walls. The finite element scheme had been used to

solve the continuity, momentum, and energy equations. Choi et al. [50]

demonstrated the position of an inner circular cylinder located within the

rhombic enclosure subjected to transient conditions using the immersed

boundary scheme. The simulation of this study was done under various

dimensionless parameters, which are Rayleigh number and inner cylinder

locations, which is changed vertically upwards and downwards. The results

indicated that when the cylinder moves up, two circulations formed below

it, and the Rayleigh number had a significant effect on the maps of

streamlines and isotherms. Another important study was presented by

Hosseinjani and Nikfar [51] focused on the natural convective fluid flow

between two horizontal circular cylinders located within the nanofluid

rhombic enclosure. The impact of symmetry, asymmetry, instability, and

stability of Cu-O nanofluid under various Rayleigh numbers had been

explained. The impact of other parameters had been included such as

diameter and the distance of inner cylinder, nanoparticle void fraction. The

results reported that increasing the distance leads to an increase in transient

asymmetric flow. Dogonchi et al. [53] investigated numerically the natural

convection between the circular cylinder within a partially heated rhombic

enclosure using CVFEM. The influence of nanoparticle shape factor,

nanoparticle volume fraction, and had been examined. It is obtained the

platelet shape had a better heat transfer rate.

The studies of the inner body within Parallelogrammic and Rhombic

enclosure have been summarized in Table 5.

Figure 6. Schematic diagram of the inner body in a Rhombus

enclosure [50]

Table 5 summarized the studies of natural convection between inner body within Parallelogrammic and Rhombic enclosure

Ref Objective Enclosure shape Software/

model used

Conclusions Special findings

Hussein [44] Natural convection in

Parallelogrammic

enclosure for different

vertical locations

Cylinder inside

Parallelogrammic

enclosure filled

with air

FVM Increasing Ra leads to

enhance the heat transfer

and increases the fluid flow

strength. In addition, it is

recommended to incline the

vertical wall 15 for better

fluid strength.

For better fluid flow strength and

heat transfer rate, it is

recommended to move the inner

cylinder towards the bottom wall

vertically

Majdi et al. [45] Natural convection in

Parallelogrammic

enclosure for different

vertical locations

Cylinder inside

Parallelogrammic

enclosure filled

with nanofluid

FEM Ra and nanofluid enhance

the heat transfer and fluid

flow strength

Increasing inclination angle

enhances the heat transfer

Chamkha [46] Mixed convection Parallelogrammic

with solid partition

FVM The Thermal conductivity

ratio, Ri number, inclination

angle and the direction of

movement of the left

vertical wall upward and

downward effect on heat

transfer and fluid dynamics

Mean skin friction coefficient

when the wall moves downward

was higher than if it moves

upward

Baïri [47] Unsteady buoyancy in

buildings

Parallelogrammic

enclosure filled

with air

FVM The influence of Ra number

and the slop of the building

effect on the building

cooling

The authors presented data for

analysis of building including

fluid flow and heat transfer

Hussain et al.

[48]

Free convection, heat

source with non-

uniform left sidewall

Parallelogrammic

enclosure filled

with air

FVM Internal and external

Rayleigh number effect on

heat transfer.

Increases inclination angle in

positive direction leads to a

reduction in fluid flow strength

while when it increases in the

negative direction; the fluid flow

circulation becomes larger.

Anandalakshmi

and Basak [49]

Free convective flow

with entropy

generation with two

different locations

(cases) of the heat

sources

Rhombic filled

with porous

medium

FEM Rhombic with inclination

angle 30 is recommended to

usage as it gives minimum

entropy generation.

However, it gives less heat

transfer enhancement

Case 1 is better in energy

efficiency compared to case 2 as

the latter produces much

irreversibility more than case 1

ABDULKADHIM ET. AL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090 87

Choi et al. [50] Impact of inner

position of circular

cylinder on heat

transfer under

transient state.

Rhombus with

internal cylinder

FVM Increasing Ra number leads

to an augmentation in Nu

It is obtained that minimum Nu

is obtained when the cylinder

moved upward. It is

recommended to move

downward for better heat

transfer

Hosseinjani and

Nikfar [51]

Unsteady free

convection

Two-cylinder

inside rhombic

enclosure

Immersed

boundary

scheme

At low Ra, the nanofluid

had a negligible effect on

Nu. While at high Ra, Nu id

highly effected.

The horizontal distance between

the circles effect on the

instability of the fluid flow

Dogonchi et al.

[53]

Nanoparticle shape

(platelet, cylindrical

and spherical) effect

on heat transfer

Cylinder in

partially thermal

active zone of

rhombus

CVFEM Rayleigh number and

nanofluid increasing helped

in improving Nu

Platelet shape gives better

performance

6. Elliptical

This section summarized the convection heat transfer due to the density

difference between the inner body located within the elliptical enclosure

[54-58] as shown in Fig. 7.

Sheikholeslami et al. [54] examined the natural convection between the

inner elliptical body within a circular cylinder enclosure filled with

nanofluid. The results explained that increasing Rayleigh number,

nanoparticle, and inclination angle leads to an increase in the Nusselt

number. Zhang et al. [55] investigated the natural convection between the

hot elliptical inner body inside the cold square enclosure using a variational

multiscale element scheme. The parameters were the major axis of the inner

ellipse, Rayleigh number, and the inclination angle of the square enclosure.

The results confirmed that inner body size, as well as the angle of

inclination, had a noticeable impact on fluid flow. Sheikholeslami et al. [56]

examined free convection, thermal radiation as well as a magnetic field

between elliptical inner body within elliptical enclosure filled with

nanofluid. Kefayati and Tang [57] examined numerically the inner cylinder

or/and elliptical inner body inside a square enclosure by the lattice

Boltzmann method. Abdulkadhim [58] demonstrated the free convection

heat transfer within the elliptical enclosure with an inner circular cylinder.

The gap was filled with nanofluid. The influence of the magnetic field,

Rayleigh number, heat coefficient had been examined and addressed.

Figure 7. Schematic diagram of the inner body in an elliptical

enclosure [56]

7. Wavy enclosure

This section summarized the convection heat transfer due to the density

difference between the inner body located within wavy enclosure [24, 52,

58-62]. As an illustrative example for this case is indicated in Fig. 8.

Figure 8. Schematic diagram of the inner body in an elliptical

enclosure [62]

One of the interesting investigation that presented by Dogonchi [52] which

concluded the inner rhombic body within wavy enclosure filled with Fe3O4.

Control volume based upon the finite element method had been used in the

simulation. Magnetic field dependent upon new viscosity model is

employed. Many parameters had been included in the study like Rayleigh

and Hartmann number, radiation parameters, aspect ratio, and nanoparticle

shape factor (platelet, cylindrical, and spherical). The results highlighted

that the increase of aspect ratio when the Hartmann number remains

constant will reduce the rate of heat transfer. Jabbar et al. [59] examined

the wavy interface on heat transfer between square enclosure divided into

two layers, nanofluid/porous layer as well as a non-newtonian layer for

various undulation number of wavy and Rayleigh numbers. Hatami and

Safari [60] used the finite element method to solve the free convection

between the circular cylinder inside the wavy nanofluid enclosure. Boulahia

et al. [61] modeled the inner hot and cold cylinders inside the wavy

enclosure. Abdulkadhim et al. [62] illustrated the multilayer system

between the wavy inner cylinder within a wavy enclosure using a finite

element scheme under various inner cylinder location and different

undulations numbers. It can be seen in Table 6 which summarized the

studies of elliptical and wavy enclosure with the internal body.

88 ABDULKADHIM ET. AL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090

Table 6 summarized the studies of natural convection between inner body within elliptical and wavy enclosure

Ref Objective Enclosure shape Software/

model used

Conclusions Special findings

Sheikholeslam

i et al. [54]

Natural convection

with nanofluid

Circular enclosure

with inner elliptical

body

CVFEM Increasing of nanofluid

loading, Rayleigh number

and inclination angle

increases the Nusselt

number

Increasing Ra reduces the

enhancement of heat transfer.

Zhang et al.

[55]

Natural convection Elliptical body

inside tilted square

enclosure

FEM The size of inner ellipse, as

well as Ra, is highly

affected on Nu

The effect of inclination angle is

small on Nu

Sheikholeslam

i et al. [56]

Natural convective

flow, MHD, nanofluid

Elliptical enclosure

with internal

elliptical body

CVFEM Increasing of the inclination

angle of the inner elliptical

body increases the heat

transfer rate

Derivation of formula of Nu

Kefayati and

Tang [57]

Natural convection Circle and elliptical

inner body inside

square enclosure

LBM Increasing the size of

cylinder augments the heat

transfer

As Bingham number goes up, heat

transfer goes down

Abdulkadhim

[58]

Natural convection,

MHD, heat

generation/absorption

and nanofluid

Circular body in a

nanofluid elliptical

enclosure

FEM Hartmann number and Ra

had an inverse effect on Nu

The change of inner cylinder

horizontally effect on the heat

transfer characteristics

Jabbar et al.

[59]

Free convection with

wavy wall

Square enclosure

with wavy wall

FEM Nu decreases as the power

index increases

Thickness layer effect on Nu

Hatami and

Safari [60]

Natural convection

and nanofluid

Internal cylinder

inside wavy

enclosure

FEM The location of inner

cylinder effect on heat

transfer

The central location gives better

heat transfer characteristics

Boulahia et al.

[61]

Natural convection Cylinder inside

wavy enclosure

FVM Ra and nanofluid increasing

leads to increases the heat

transfer

Increasing number of undulations

and reduction in the amplitude of

the wavy surface leads to

augmentation in heat transfer

Abdulkadhim

et al. [62]

Natural convection

with multilayer

system

Wavy internal body

within wavy

enclosure

COMSOL Number of corrugated effect

are small

The inner body position effect on

heat and fluid flow

8. Governing Equation

Finally, it is important to insert the governing equations used in this

specific subject of natural convection heat transfer within enclosure

filled with pure fluid, porous medium, and nanofluid [63].

8.1. Pure fluid

𝜕𝑈

𝜕𝑋
+

𝜕𝑉

𝜕𝑌
= 0 (1)

𝑈
𝜕𝑈

𝜕𝑋
+ 𝑉

𝜕𝑈

𝜕𝑌
= −

𝜕𝑃

𝜕𝑋
+ 𝑃𝑟(

𝜕2𝑈

𝜕𝑋2
+

𝜕2𝑈

𝜕𝑌2
) (2)

𝑈
𝜕𝑉

𝜕𝑋
+ 𝑉

𝜕𝑉

𝜕𝑌
= −

𝜕𝑃

𝜕𝑌
+ 𝑃𝑟 (

𝜕2𝑉

𝜕𝑋2
+

𝜕2𝑉

𝜕𝑌2
) + 𝑅𝑎𝑃𝑟𝜃 (3)

𝑈
𝜕𝜃

𝜕𝑋
+ 𝑉

𝜕𝜃

𝜕𝑌
=

𝜕2𝜃

𝜕𝑋2
+

𝜕2𝜃

𝜕𝑌2
(40

8.2. Porous Medium

𝜕𝑈

𝜕𝑋
+

𝜕𝑉

𝜕𝑌
= 0 (5)

𝜕𝑈

𝜕𝑌
= −

𝐾

𝜇

𝜕2𝑃

𝜕𝑋 𝜕𝑌
(6)

𝜕𝑉

𝜕𝑋
= −

𝐾

𝜇

𝜕2𝑃

𝜕𝑋 𝜕𝑌
+ 𝑅𝑎

𝜕𝑇

𝜕𝑋
(7)

𝑈
𝜕𝜃

𝜕𝑋
+ 𝑉

𝜕𝜃

𝜕𝑌
=

𝜕2𝜃

𝜕𝑋2
+

𝜕2𝜃

𝜕𝑌2
(8)

8.3. Nanofluid

𝜕𝑈

𝜕𝑋
+

𝜕𝑉

𝜕𝑌
= 0 (9)

𝜌𝑛𝑓 (𝑈
𝜕𝑈

𝜕𝑋
+ 𝑉

𝜕𝑈

𝜕𝑌
) = −

𝜕𝑃

𝜕𝑋
+ 𝜇𝑛𝑓 (

𝜕2𝑈

𝜕𝑋2
+

𝜕2𝑈

𝜕𝑌2
) (10)

𝜌𝑛𝑓 (𝑈
𝜕𝑈

𝜕𝑋
+ 𝑉

𝜕𝑈

𝜕𝑌
) = −

𝜕𝑃

𝜕𝑌
+ 𝜇𝑛𝑓 (

𝜕2𝑉

𝜕𝑋2
+

𝜕2𝑉

𝜕𝑌2
) (11)

+𝑔[(1 − 𝜑)(𝜌𝛽)𝑏𝑓 + 𝜑(𝜌𝛽)𝑠](𝑇 − 𝑇𝑐 )

𝑈
𝜕𝜃

𝜕𝑋
+ 𝑉

𝜕𝜃

𝜕𝑌
= 𝛼𝑛𝑓 (

𝜕2𝜃

𝜕𝑋2
+

𝜕2𝜃

𝜕𝑌2
) (12)

9. Conclusion

This paper presents a comprehensive literature review of the most published

papers in the field of natural convection between inner bodies located inside

different complex enclosure shapes. The main conclusions are:

ABDULKADHIM ET. AL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 080–090 89

 The inner bodies inside trapezoidal, parallelogrammic enclosure are

very limited, and more investigation should be done.

 The studies regarding wavy enclosure are limited in a comparison with

other simple shapes of enclosure despite its important applications is

electronic equipment.

 There are limitations in the studies of natural convection between the

inner body located in a wavy enclosure.

 Multi-layers system inside a wavy enclosure is limited as most of the

recent studies focus on nanofluid, porous media filled the enclosure but

there are serious limitations when the nanofluid/porous media filled the

space.

 Dufour and Soret effect on natural flow for the multi-layer system are

not investigated yet in full-details.

ACKNOWLEDGMENT

We would like to thank our institutions (Al-Mustaqbal University College

and Al-Qadisiyah University) for their supports and for giving us the time

for writing and completing the present article review.

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