A Review on the Control Parameters of Natural Convection in Different Shaped Cavities with and without Nanofluid

Abstract

Natural convection in cavities is an interesting subject for many researchers. Especially, in recent years, the number of articles written in this regard has grown enormously. This work provides a review of recent natural convection studies. At first, experimental studies were reviewed and, then, numerical studies were examined. Then, the articles were classified based on effective parameters. In each section, numerical studies were examined the parameters added to the cavity such as magnetic forces, fin, porous media and cavity angles. Moreover, studies on non-rectangular cavities were investigated. Free convection in enclosures depends more on the fluid velocity relative to the forced convection, leading to the opposite effect of some parameters that should essentially enhance rate of heat transfer. Nanoparticle addition, magnetic fields, fins, and porous media may increase forced convection. However, they can reduce free convection due to the reduction in fluid velocity. Thus, these parameters need more precision and sometimes need the optimization of effective parameters.

1. Introduction

Free convection happens in various industries such as solar cells, air, and space, food industry, solar and nuclear collectors, etc. Therefore, it has been studied in different geometries. Cavities are the most important host for these mechanisms of heat transfer. In these cavities, heat transfer occurs in a small space. The heat transfer mechanism in closed cavities is natural due to the absence of an external device to accelerate fluid flow. Heat transfer generally involves three mechanisms, including conduction, convection, and radiation heat transfer. Each of these mechanisms can become sub-branches. One of the branches of convection is the free or natural one. Figure 1 presents various types of heat transfer mechanisms with their sub-branches.

Various applications of cavities in the industry include electronic equipment cooling, solar-related equipment, packaging industries, heat exchangers, power plants, petrochemicals, petroleum industries, and melting metals, etc. The importance of evaluating free convection in cavities, as well as an increased number of studies in this field, means that there is a need for review studies. Free convection can be divided into two subdivisions of internal and external convection. As shown in Figure 2, in the case of internal convection, the fluid is in a closed environment and the temperature difference of the walls causes convection. The right figure indicates free convection and the left one shows forced one.

Another kind of free convection is external flow. In this case, the fluid encounters a plate at a different temperature in an open environment. Consequently, the changing of temperature near the wall, and the creation of difference in the density gradient make free convection. An example of external free convection is shown in Figure 3, which demonstrates different shapes of boundary layers at different wall angles. If some factors such as fan or pump cause fluid motion and its collision to the plane, forced convection happens and if the fluid motion is due to the difference of density because of temperature difference, free convection occurs.

Free convection has many applications in industry or even in human daily life. Two examples of application of free convection are illustrated in Figure 4. The home environment and heating of water due to a heat source are examples of free convection because of the temperature difference in a closed environment. As the temperature gradient increases, the free convection also strengthens and its value increases.

In order to create free convection flow, it is essential to notice the effect of temperature on the fluid density. In other words, a density gradient occurs due to the temperature difference in the environment to create convection. The buoyancy force is created due to the density gradient that can displace the fluid. An approximation is needed to calculate the density difference created in the fluid. Many researchers used the Boussinesq approximation, which relies on the negligible effect of density changes in the momentum equation can be ignored and the effect of density changes is considered only in terms of the fluid motion equation include the gravity acceleration. This approximation can make the momentum equation simpler and make it possible to consider the effect of the density difference. Of course, it is more appropriate to use this approximation for low temperatures. According to the above explanation, it has been shown that free convection is of excessive importance. Free convection within enclosures is an important matter so that it becomes the main focus of this paper.

In these articles, researchers collect and review a series of articles in a particular field, especially articles that have been published in recent years. Given developments regarding cavities, a researcher can understand the development by reading review articles. The current study attempts to present articles on the free convection in cavities in a summarized form.

2. Experimental Studies

A large number of experimentally and numerical studies exist in the literature. Some researchers study heat transfer field and some of them study economy field of heat device [3,4,5,6,7,8,9,10,11]. Karatas and Delbentli [12] included radiation effects in their heat transfer study on a 3-dimensional rectangular cavity. They used air fluid with Rayleigh number (Ra) between 1.6 × 10⁵ and 4.67 × 10⁷, and examined aspect ratio (AR) effect of the cavity on thermal performance. They changed the aspect ratio between 1 and 6, and eventually expressed an equation in terms of Ra number and AR for Nu_ave. Mahmoudinezhad et al. [13] attached a fin and with an angle toward the horizon. They investigated Rayleigh number effect in 1.5 × 10⁵ to 4.5 × 10⁵ range and fin angle of 0 to 90 degrees. They found that the minimum Nusselt number was in the 90-degree fin and the maximum was in 45 degrees. Li and Peterson [14] investigated the nanofluid alumina natural convection. They found that with increasing volumetric percentages of nanoparticles, the heat transfer characteristics raised. Ho et al. [15] examined a suspension containing alumina nanopowder in a cavity and realized that dispersing more nanoparticles was in the range of 1 to 4%. Karatas et al. [16] examined air fluid free convection. Ra number was considered between 4.5 × 10⁵ and 1.13 × 10⁸ and the aspect ratio in the range 1 to 6 on the Nusselt number. They found that when the wall temperature is applied sinusoidally compared to a constant temperature, the Nusselt number decreases. Vasiliev et al. [17] studied turbulent flow considering the Ra number between 2 × 10⁹ and 1.6 × 10¹⁰. Bharti et al. Researchers [18] investigated water heat transfer of for Ra between 7.05 × 10⁵ and 1.84 × 10⁷. Sides had been considered at different temperatures, and horizontal walls had been considered insulated.

Zhang et al. [19,20] examined the turbulent regime of free convection. They examined a three-dimensional rectangular cavity and the influence of Ra on heat transfer. Chen et al. [21] investigated fin geometry effect and its position on heat transfer of airflow by adding a fin to the hot surface. Cordoba et al. [22] also examined air natural convection in the rectangular cavity. Mahdavi et al. [23] examined heat transfer of natural and laminar convections of the air and waters in a 3-D cavity for Ra between of 1.3 × 10⁸ and 1.19 × 10⁹ range. Saxena et al. [24] studied the water-filled open cavity and examined its effect on Nu by resizing the cavity.

Table 1. Summary of experimental investigations performed on the enclosures.

References	Geometry	Type of Fluid	Parameter	Solver	Remark
[12]		air	Ra AR	Experimental	Radiation
[16]		air	Ra AR dp	Experimental	3D
[25]		Fluid Pr = 6.62	Ra Da	FEM Experimental	Entropy Generation Porous Layer
[18]		Water	Ra	Experimental
[21]		air	Ra d	Experimental FDM	Fin 3D
[23]		Water air	Ra	CVFEM Experimental	3D
[26]		air	E td h	Experimental	3D Fin Radiation
[27]		air	Ra S	Experimental	Fin 3D

shows a summary of some experimental investigations performed on the enclosures.

3. Numerical Studies

Due to time spending and the high cost of conducting experiments, the researchers used numerical methods to study the natural convection in the cavities. Knowing thermophysical fluid properties in the cavity is necessary to use numerical methods and solve the governing equations. These properties are expressed by tables or relationships depending on the constituents and the parameters that the fluid depends on.

3.1. Governing Equations

The governing equations assert as follows by assuming laminar, stable flow, incompressible, and with the fluid assumption in a continuous environment. In these equations, the effects of radiation and viscosity loss are neglected. Equation (1) is the mass conservation equation, Equations (2) and (3) are the momentum conservation equation and (4) is the energy conservation Equation (4) (with Boussinesq approximation).

$$\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\frac{\partial \mathrm{v}}{\partial \mathrm{y}}=0$$

(1)

$$\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{u}}{\partial \mathrm{y}}=-\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}+\frac{1}{\mathsf{\rho }}\left(\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\mu }\frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left(\mathsf{\mu }\frac{\partial \mathrm{u}}{\partial \mathrm{y}}\right)\right)$$

(2)

$$\mathrm{u}\frac{\partial \mathrm{v}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{v}}{\partial \mathrm{y}}=-\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{y}}+\frac{1}{\mathsf{\rho }}\left(\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\mu }\frac{\partial \mathrm{v}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left(\mathsf{\mu }\frac{\partial \mathrm{v}}{\partial \mathrm{y}}\right)\right)+\mathrm{g}\mathsf{\beta }\left(\mathrm{T}-T_c\right)$$

(3)

$$\left(\mathrm{u}\frac{\partial \mathrm{T}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{T}}{\partial \mathrm{y}}\right)=\frac{\mathrm{k}}{\rho C_P}\left(\frac{{\partial }^2\mathrm{T}}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathrm{T}}{\partial {\mathrm{y}}^2}\right)$$

(4)

3.2. Thermophysical Properties of Fluids in Cavities

As we need to know the properties of fluid for solving the governing equations, it is essential to study the properties of different fluids in a section. Some researchers, in light of abundance of fluids, such as air and water, examined these two fluids in the cavity. Since Choi [28] used nanofluids for the first time, many researchers have studied the heat transfer by adding these types of fluids to the cavities. Since nanofluids show better thermal performance than simple fluids, in recent years, researchers have been studying this type of fluids. Moreover, some researchers examined non-Newtonian fluids in the cavities. The properties of all three types of fluids are discussed below.

3.2.1. Air and Water

One of the most important fluids used in numerical studies of the cavity is the air or water. Due to their abundance and availability, these fluids are widely used in industries. Hence, many researchers used water or air as the operating fluid in their cavity [29,30,31,32,33,34]. In most numerical studies, the properties of these two fluids were fixed and these properties for temperature and pressure of the room (See

Table 2. Properties of air and water in room temperature and pressure.

	ρ (kg/m3)	Cp (kJ/kg.K)	k (W/m.K)	μ (kg/m.s)
Water [35]	9.971 × 102	4.179	6.13 × 10−1	1 × 10−3
Air [36]	1.169	1.0064	2.607 × 10−2	1.853 × 10−5

).

However, due to the fact that the properties of these two fluids depend on temperature, the researchers also tried to obtain a relationship between the properties of air and water with the temperature. In

Table 3. Properties of water, dry air and saturated fluids.

ρ (kg/m3)	Cp (kJ/kg.K)	k (W/m.K)	μ (kg/m.s)
765.33 +1.8142 × T −0.0035 × T2	28.07 −0.2817 × T +0.00125 × T2 −2.48 × 10−6 × T3 +1.857 × $10-9\times$ T4	−0.5752 + 0.006397 × T −8.151 × $10-6\times$ T2	0.0967 −8.207 × 10−4 × T + 2.344 × 10−6 T2 −2.244 × 10−9 × T3	[23] Water
((3.484 − 1.317) × xv) $\times \frac{{\mathrm{P}}_0}{273.15+\mathrm{T}}$	1.034 −2.8488 × 10−4 × T +7.8168 $\times$10−7 × T2 −4.9707 × 10−10 × T3 +1.077 × $10$−13 × T4	−2.2765 × 10−3 +1.259 × 10−4 × T −1.481 × 10−7 × T2 +1.7355 × 10−10 × T3 −1.0666 × $10$−13 $\times$ T4 +2.476 × $10$−17 × T5	−9.8601 $\times$10−1 +9.0801 $\times$10−2 $\times$ T −1.1763 $\times$ 10−4 $\times$ T2 +1.2349 $\times$10−7 $\times$ T3 −5.797 $\times 10$−11 $\times$ T4	[36] Dry air
1.2933 −5.588 × 10−3 × T +3.860 × 10−5 × T2 −5.253 $\times$10−7 × T3	1.0045 +2.0506 × 10−3 × T −1.6315 × 10−4 × T2 +6.2123 × 10−6 × T3 −8.8304 × 10−8 × T4 +5.071 $\times 10$−11 $\times$ T5	2.4007 $\times$10−2 +7.2784 $\times$10−5 $\times$ T −1.788 $\times$10−2 $\times$ T2 −1.3517 $\times$10−9 $\times$ T3 −3.3224 $\times 10$−11 $\times$ T4	1.7157 $\times$10−5 +4.7224 $\times$10−8 $\times$ T −3.663 $\times$ 10−10 $\times$ T2 +1.873 $\times$10−12 $\times$ T3 −8.050 $\times 10$−14 $\times$ T4	[37] Saturated air

, the properties of the water, dry air, and saturated fluids are provided.

The number of papers reviewed by these two fluids in recent years is less than before (see

Table 4. Some articles used working fluid of air and water in the enclosures.

References	Geometry	Type of Fluid	Parameter	Solver	Remark
[38]		air	Ra ε	FVM	Open Cavity Radiation
[39]		Water	Ra	CVFEM
[40]		air	SC τ β w k	FDM	Thermophoresis Wave Wall
[41]		Air (H2O-CO2)	Ra type of gas	FVM	Radiation Turbulance
[43]		air	Ra d ε	CVFEM	Radiation Open Cavity
[42]		Type of gas	Ra	FDM	Radiation Transient
[44]		air	Ra	FDM	Transition
[45]		air	Ra AR h d	FVM	Radiation Fin
[46]		air	Ra d	FEM	Transient Fin
[47]		air	Ra Ha α	FEM	MHD Fin

), and most researchers are interested in using nanofluids in the cavities. However, Gupta et al. [38] studied natural convection along with the radiation of air fluid in an open cavity. They found that heat transfer near open walls is more than elsewhere. Bhowmick et al. [39] used a triangular cavity and found that the heat transfer rate improved with growing Ra. Grosan et al. [40] examined vibrating wall using the finite differential method. Kogawa et al. [41] included radiation effects in turbulent flow using the finite volume method (FVM). They used water vapor and carbon dioxide fluids as air fluids. Foruzan nia et al. [42] examined natural convection along with transient radiation of gas flux using a finite difference method (FDM). Their cavity consists of several solid walls and two fluid parts.

3.2.2. Nanofluids

With the advent of science and the ever-increasing need of humans for energy, the researchers attempted to use fluids with higher thermal conductivity in the cavities. Therefore, by adding nanofluids instead of air and water, they studied the heat transfer in the heat device [48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68]. The study of nanofluids has grown dramatically in recent years, and articles that have used these fluids are increasing. Their thermal expansion, specific heat capacity, and density are calculated by the following equations:

$${\mathsf{\rho }}_{\mathrm{nf}}=\left(1-\mathsf{\phi }\right){\mathsf{\rho }}_{\mathrm{f}}+{\mathsf{\phi }\mathsf{\rho }}_s$$

(5)

$${\left({\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{nf}}=\left(1-\mathsf{\phi }\right){\left({\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}+\mathsf{\phi }{\left({\mathrm{c}}_{\mathrm{p}}\right)}_s$$

(6)

$${\left({\mathsf{\rho }\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{nf}}=\left(1-\mathsf{\phi }\right){\left({\mathsf{\rho }\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}+\mathsf{\phi }{\left({\mathsf{\rho }\mathrm{c}}_{\mathrm{p}}\right)}_s$$

(7)

$${\mathsf{\beta }}_{\mathrm{nf}}=\left(1-\mathsf{\phi }\right){\mathsf{\beta }}_{\mathrm{f}}+{\mathsf{\phi }\mathsf{\beta }}_s$$

(8)

The viscosity of nanofluids is an important property is measured by performing tests on various nanofluids, volumes, and temperatures [69,70,71,72,73,74,75,76,77,78,79]. A collection of well-known relationships regarding viscosity of nanofluids is briefly listed in

Table 5. Relationships obtained for nanofluid viscosity.

Correlation	Ref
${\mathsf{\mu }}_{\mathrm{eff}}=\left(1+2.5\mathsf{\phi }\right)$	Einstein [80]
${\mathsf{\mu }}_{\mathrm{eff}}=\frac{1}{{\left(1-\mathsf{\phi }\right)}^{2.5}}$	Brinkman [81]
${\mathsf{\mu }}_{\mathrm{eff}}=\left(1+2.5\mathsf{\phi }+6.5{\mathsf{\phi }}^2\right)$	Batchelor [82]
${\mathsf{\mu }}_{\mathrm{eff}}=\left(1+2.5\mathsf{\phi }+25.4{\mathsf{\phi }}^2+\mathrm{O}\left({\mathsf{\phi }}^3\right)\right)$	Lundgren [83]
${\mathsf{\mu }}_{\mathrm{eff}}=\left(1+2.5\mathsf{\phi }+4.5\right)/\left[\left(\mathrm{h}/{\mathrm{d}}_{\mathrm{p}}\right)\left(2+\mathrm{h}/{\mathrm{d}}_{\mathrm{p}}\right){\left(1+\mathrm{h}/{\mathrm{d}}_{\mathrm{p}}\right)}^2\right]$	Graham [84]
${\mathsf{\mu }}_{\mathrm{eff}}=\left(1+2.5\mathsf{\phi }+\left(\frac{125}{64{\mathsf{\phi }}_{\mathrm{p}\max }}\right){\mathsf{\phi }}^2+\dots \right)$	Simha [85]
${\mathsf{\mu }}_{\mathrm{eff}}=\exp \left(\frac{2.5\mathsf{\phi }}{1-\mathrm{k}\mathsf{\phi }}\right)$	Mooney [86]
${\mathsf{\mu }}_{\mathrm{eff}}={\mathsf{\mu }}_{\mathrm{f}}\left(1+\frac{1.25\mathsf{\phi }}{1-\raisebox{1ex}{$\mathsf{\phi }$}\!\left/ \!\raisebox{-1ex}{$0.78$}\right.}\right)$	Eilers [87]
${\mathsf{\mu }}_{\mathrm{eff}}={\mathsf{\mu }}_{\mathrm{f}}\left(1+\frac{1.25\mathsf{\phi }}{1-\mathsf{\phi }}\mathsf{\phi }\right)$	Saito [88]
${\mathsf{\mu }}_{\mathrm{eff}}=\frac{9}{8}\left[\frac{{\left(\mathsf{\phi }/{\mathsf{\phi }}_{\mathrm{m}}\right)}^{1/3}}{1-{\left(\mathsf{\phi }/{\mathsf{\phi }}_{\mathrm{m}}\right)}^{1/3}}\right]$	Frankel and Acrivos [89]

.

The most important reason for using nanofluids is their ability to the thermal conductivity so that thermal conductivity became the most important property. There are several calculation methods have been presented by researchers in order to measure the thermal conductivity of nanofluids. Many researchers are using experimental relationships for the thermal conductivity to express an equation for these fluids [90,91,92,93,94,95,96,97,98,99,100,101,102,103]. Some other researchers are also using theoretical relations, like ones listed in

Table 6. The relationships presented on the thermal conductivity for different nanofluids.

Ref	Correlation
Maxwell [104]	${\mathrm{k}}_{\mathrm{eff}}={\mathrm{k}}_{\mathrm{f}}\frac{{\mathrm{k}}_{\mathrm{p}}+2{\mathrm{k}}_{\mathrm{f}}+2\mathsf{\phi }\left({\mathrm{k}}_{\mathrm{p}}-{\mathrm{k}}_{\mathrm{f}}\right)}{{\mathrm{k}}_{\mathrm{p}}+2{\mathrm{k}}_{\mathrm{f}}-\mathsf{\phi }\left({\mathrm{k}}_{\mathrm{p}}-{\mathrm{k}}_{\mathrm{f}}\right)}$
Wasp [105]	${\mathrm{k}}_{\mathrm{eff}}={\mathrm{k}}_{\mathrm{f}}\frac{{\mathrm{k}}_{\mathrm{p}}+2{\mathrm{k}}_{\mathrm{f}}-2\mathsf{\phi }\left({\mathrm{k}}_{\mathrm{p}}-{\mathrm{k}}_{\mathrm{f}}\right)}{{\mathrm{k}}_{\mathrm{p}}+2{\mathrm{k}}_{\mathrm{f}}+\mathsf{\phi }\left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{p}}\right)}$
Bruggeman [106]	$\mathsf{\phi }\left[\frac{{\mathrm{k}}_{\mathrm{p}}-{\mathrm{k}}_{\mathrm{eff}}}{{\mathrm{k}}_{\mathrm{p}}+2{\mathrm{k}}_{\mathrm{eff}}}\right]+\left(1-\mathsf{\phi }\right)\left[\frac{{\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{eff}}}{{\mathrm{k}}_{\mathrm{f}}+2{\mathrm{k}}_{\mathrm{eff}}}\right]=0$
Hamiltonn and Crosse [107]	${\mathrm{k}}_{\mathrm{eff}}={\mathrm{k}}_{\mathrm{f}}\frac{{\mathrm{k}}_{\mathrm{p}}+\left(\mathrm{n}-1\right){\mathrm{k}}_{\mathrm{f}}+\left(\mathrm{n}-1\right)\mathsf{\phi }\left({\mathrm{k}}_{\mathrm{p}}-{\mathrm{k}}_{\mathrm{f}}\right)}{{\mathrm{k}}_{\mathrm{p}}+\left(\mathrm{n}-1\right){\mathrm{k}}_{\mathrm{f}}-\mathsf{\phi }\left({\mathrm{k}}_{\mathrm{p}}-{\mathrm{k}}_{\mathrm{f}}\right)}$
Koo and Kleinstreuer [108,109]	$$\begin{gathered} {\mathrm{k}}_{\mathrm{eff}}={\mathrm{k}}_{\mathrm{f}}\frac{{\mathrm{k}}_{\mathrm{p}}+2{\mathrm{k}}_{\mathrm{f}}+2\mathsf{\phi }\left({\mathrm{k}}_{\mathrm{p}}-{\mathrm{k}}_{\mathrm{f}}\right)}{{\mathrm{k}}_{\mathrm{p}}+2{\mathrm{k}}_{\mathrm{f}}-\mathsf{\phi }\left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{p}}\right)}+5\times {10}^4{\mathsf{\theta }\mathsf{\rho }}_{\mathrm{f}}{\mathrm{C}}_{\mathrm{pf}}\mathsf{\phi }\mathrm{f}\left(\mathrm{T},\mathsf{\phi }\right)\sqrt{\frac{{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}{{\mathsf{\rho }}_{\mathrm{f}}{\mathrm{d}}_{\mathrm{p}}}} \\ \mathrm{f}\left(\mathrm{T},\mathsf{\phi }\right)=\left(-6.04\mathsf{\phi }+0.4705\right)\mathrm{T}+1722.3\mathsf{\phi }-134.63 \end{gathered}$$
Avsec and Oblak [110]	$k_{\mathrm{eff}}=\frac{k_p+\left(n-1\right)k_f-\left(n-1\right){\alpha }_e\left(k_p-k_f\right)}{k_p+\left(n-1\right)k_f+{\alpha }_e\left(k_p-k_f\right)}$
Jang and Choi [111]	$$\begin{gathered} {\mathrm{k}}_{\mathrm{eff}}={\mathrm{k}}_{\mathrm{f}}\left(1-\mathsf{\phi }\right)+0.01{\mathrm{k}}_{\mathrm{p}}\mathsf{\phi }+18\times {10}^6\frac{{\mathrm{d}}_{\mathrm{f}}}{{\mathrm{d}}_{\mathrm{p}}}{\mathrm{k}}_{\mathrm{f}}{\mathrm{Re}}_{\mathrm{d}}^2{\mathrm{Pr}}_{\mathrm{f}}\mathsf{\phi } \\ {\mathrm{Re}}_{\mathrm{d}}^{}=\frac{{\mathrm{C}}_{\mathrm{RM}}{\mathrm{d}}_{\mathrm{p}}}{\mathsf{\upsilon }}\cdots ,{\mathrm{C}}_{\mathrm{RM}}=\frac{{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}{3{\mathsf{\pi }\mathsf{\mu }}_{\mathrm{f}}{\mathrm{d}}_{\mathrm{p}}{\mathrm{l}}_{\mathrm{f}}} \end{gathered}$$
Pak and Cho [112]	${\mathrm{k}}_{\mathrm{eff}}=1+7.47\phi$
Timofeeva et al. [113]	${\mathrm{k}}_{\mathrm{eff}}={\mathrm{k}}_{\mathrm{f}}\left(1+3\mathsf{\phi }\right)$

, considering Brownian motion, thermophoresis, clustering, to name a few, which are used to estimate nanofluid thermal conductivity.

3.2.3. Non-Newtonian Fluids

Many fluids such as the ones used in the food industry, pharmaceuticals, dye industry, adhesives, etc., whose shear-stress and strain are not linear-dependent. Non-Newtonian fluids can be divided into different categories based on their behavior. Figure 5 illustrates the types of non-Newtonian fluids and their relationship between strain and stress. Non-Newtonian fluids can exhibit different behaviors due to applied strains. Non-Newtonian yield material fluids have a certain amount of stress. In shear-thinning fluids, the stress decreases with the strain rate, while non-Newtonian shear thickening fluids behave inversely. Their stress increases with the strain rate. Figure 5 shows the behavior of these fluids in terms of stress and strain.

The third type of fluid, which some researchers added to the cavities to study heat transfer are non-Newtonian fluids and some models for the stress and strain relationship are presented [114,115,116,117,118,119,120,121,122,123,124,125]. The shear stress tensor of the power-law model, which is widely used for non-Newtonian fluids in numerical studies of cavities, is described as follows:

$${\mathsf{\tau }}_{\mathrm{ij}}=2{\mathsf{\mu }}_{\mathrm{a}}{\mathrm{D}}_{\mathrm{ij}}={\mathsf{\mu }}_{\mathrm{a}}\left(\frac{\partial {\mathrm{u}}_{\mathrm{i}}}{\partial {\mathrm{x}}_{\mathrm{j}}}+\frac{\partial {\mathrm{v}}_{\mathrm{j}}}{\partial {\mathrm{y}}_{\mathrm{i}}}\right)$$

(9)

In (9), D_ij and μ_a are the tensor of shear rate and the apparent viscosity, respectively.

$${\mathsf{\mu }}_{\mathrm{a}}=\mathrm{K}{\left\{2\left[{\left(\frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)}^2+{\left(\frac{\partial \mathrm{v}}{\partial \mathrm{y}}\right)}^2\right]+{\left(\frac{\partial \mathrm{v}}{\partial {\mathrm{x}}_{\mathrm{j}}}+\frac{\partial \mathrm{u}}{\partial {\mathrm{y}}_{\mathrm{i}}}\right)}^2\right\}}^{\frac{\left(\mathrm{n}-1\right)}{2}}$$

(10)

In (10), n and K are the coefficient and power-law index, respectively. Quasi-plastic fluids are those with n greater than unity, dilatant fluids are those with n lower than unity, and Newtonians are those with n equal to unity. For example, Zhou and Bayazitoglu [126] examined a power-law fluid flow by means of the FVM. They found that with the growing n, Nu decreases. Moreover, in a different power-law index, increasing the Rayleigh has a different effect on Nu. Wang et al. [127] used Lattice Boltzmann (LBM) method and examined the sinusoidal thermal boundary condition and power-law model was used for fluid modeling. They found that raising n caused a reduction of Nu.

Table 7. Brief of articles about non-Newtonian fluids within enclosures.

References	Geometry	Type of Fluid	Parameter	Solver	Remark
[126]		Power-law non Newtonian $0.6<\mathrm{n}<1.4$	Ra n Pr rt	FVM	Open Cavity
[127]		Power-law non Newtonian $0.5<\mathrm{n}<1.5$	Ra AR Pr n w dp	LBM	Wave Wall
[128]		Power-law non Newtonian $0.4<\mathrm{n}<1$	Ra AR Da n α	FVM	Open Cavity Porous Layer Angel Cavity Entropy Generation
[129]		Viscoelastic fluid	Ra rd Pr e	FDM	Radiation
[130]		Power-law non Newtonian $0.2<\mathrm{n}<1.8$	Ra Ha n N	LBM	Entropy MHD Fin
[131]		Power-law non Newtonian $0.2<\mathrm{n}<2$	Gr Pr n δ	FEM	Fin
[132]		Power-law non Newtonian $0.7<\mathrm{n}<1.4$	Ra Ha AR n	LBM	MHD

summarizes some of the studies related to convection in the enclosures using non-Newtonian fluids.

4. Effective Parameters on Heat Transfer in Cavities

At the beginning of cavities examination, the researchers examined simple cavities. The geometry of these cavities is either square or rectangular, and the side walls temperature is constant. In these studies, the effective parameters on Nu were low and the influence of Ra on Nu was often investigated. Over time, researchers have tried to improve their studies by adding other effective parameters and changing geometry of the cavities. The following can be considered as simple examples for examining the natural convection in the cavities.

Most of the studies investigated quadratic cavities and the parameters affecting heat transfer include Rayleigh number and thermophysical properties of the operating fluid [49,58,59,60,61,62,63,133,134,135,136,137,138,139,140,141,142,143]. With the advancement of science, researchers also investigated the natural convection in enclosures by adding other imperative parameters such as the magnetic field, fins, porous medium, etc. In the following, the work done with each of the important parameters is important.

4.1. Inclined Cavity

Some researchers examined the effect of this parameter on Nu, temperature, velocity fields by angling the cavity. By giving angle to the cavity, the gravitational force on the cavity also receives an angle. Hence, momentum equations will be as follows:

$$\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{u}}{\partial \mathrm{y}}=-\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}+\frac{1}{\mathsf{\rho }}\left(\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\mu }\frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left(\mathsf{\mu }\frac{\partial \mathrm{u}}{\partial \mathrm{y}}\right)\right)+\mathrm{g}\mathsf{\beta }\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{c}}\right)\sin \mathsf{\gamma }$$

(11)

$$\mathrm{u}\frac{\partial \mathrm{v}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{v}}{\partial \mathrm{y}}=-\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial y}+\frac{1}{\mathsf{\rho }}\left(\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\mu }\frac{\partial \mathrm{v}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left(\mathsf{\mu }\frac{\partial \mathrm{v}}{\partial \mathrm{y}}\right)\right)+\mathrm{g}\mathsf{\beta }\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{c}}\right)\cos \mathsf{\gamma }$$

(12)

In the above equations, γ is the angle of cavity relative to the horizon line. Naturally, due to absence of gravity in the mass and energy conservation equations, no change is made in these two equations. In the study of Goodarzi et al. [144] a constant flux source in the chamber has been studied. They examined the effect of Grashof number on Nu in the range of 400 to 40,000,000. Emami et al. [145] focused on water/copper nanofluid as a two-phase fluid in a cavity. They examined the cavity by adding a porous medium and found that the porous media was beneficial in strong convection, but in poor convection, it may reduce Nu. Raizah et al. [128] studied a porous cavity and found that Nu increased by raising Ra and reducing power-law index.

Kefayati et al. [146,147] reviewed Bingham fluid natural convection through the Finite Difference Lattice Boltzmann (FDLBM) method. They examined the angle effect of the cavity. Sajjadi et al. [148] used the copper/water nanofluid using LBM, they investigated the effect of cavity angles from 0 to 60 degrees. Hosseini et al. [149] used several nanofluids in diagonal cavities. They used nanoparticles, for example, Cu, Ag, Al₂O₃, and TiO₂, in water-base fluid. Sheremet et al. [150] studied alumina/water nanofluid using the FDM in order to the free convection investigation. Ogut [151] examined various angles from 0 to 90 degrees. He used copper oxide, silver oxide, and alumina nanofluids in water base fluid. Abu-nada and Oztop [152] studied influence of cavity angle on Nu by means of the FVM. They used the copper/water nanofluid as an operating fluid and changed the angle of cavity from 0 to 120 degrees. Oztop et al. [59] modified the angle of cavity by studying the heat transfer rate in a cavity with sinusoidal thermal boundary condition by means of the FVM. They changed the angle of cavity from 0 to 90 degrees, and by adding nanofluid of copper and titanium water examined the percentage effects of nanofluids.

Table 8. Summary of some articles studied enclosures with different inclination angles.

References	Geometry	Type of Fluid	Parameter	Solver	Remark
[144]		air	Gr α	LBM	Fin Inclined Cavity
[150]		Al2O3/W $0<\mathsf{\phi }<3\%$	Ra$\mathsf{\phi }$ f α τ	FDM	Inclined Cavity Transient
[147]		nanofluid (water base)	Ra Da d h α	FDM	Thermophoresis Brownian Motion Inclined Cavity Porous Layer
[153]		Al2O3/W $0<\mathsf{\phi }<6\%$	Ra Rd γ φ Ha	FEFVM	Brownian Motion Inclined Cavity Radiation Fin MHD
[154]		Al2O3/W $0<\mathsf{\phi }<6\%$	Ra Rd γ φ Ha	FEFVM	Brownian Motion Inclined Cavity Radiation Entropy MHD
[155]		Al2O3/W $0<\mathsf{\phi }<6\%$	Ra γ φ Ha	FEFVM	Brownian Motion Inclined Cavity Entropy MHD

presents a number of articles studied inclined enclosures.

4.2. Adding a Fin to Cavity

Adding a fin to cavity is one of the most commonly used ways is to increase the heat exchange surfaces. As the heat exchange surface increases, the fluid and the wall are in greater contact with each other, which intensifies the overall heat transfer according to Newton’s law of cooling. Hence, the blades or fins attracted much attention. Adding a blade or a fin to the surface can increase the heat transfer. Of course, another important issue is the ability to pump. By addition of fin, the energy loss in the fluid and the pressure difference increase due to the shear stress. Therefore, it is necessary to be careful in the use of fins in the desired dimensions and proportions. Therefore, many researchers have attempted to optimize the dimensions of blades or fins. Fins can have different shapes and sizes some of which can be seen in Figure 6. Since the fine size is very important in Nu due to the lack of free space, the addition of fin to them has been one of the challenges in discussing the free convection. Excessive fin size can narrow the space in the chamber and reduce the fluid flow rate.

Researchers also examined the effect of fin in the cavities [156,157,158,159,160]. By adding a constant temperature fin, a temperature boundary condition is added to the boundary condition equations. If the fin has a thermal conductivity, in addition to the fin boundary condition, the following relation should also be added to the equations. This relation is the energy equation for the fin.

$$\frac{\partial }{\partial \mathrm{x}}\left(k_{fin}\frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left(k_{fin}\frac{\partial \mathrm{T}}{\partial \mathrm{y}}\right)=0$$

(13)

In some cases, the fin is placed on constant temperature walls or the thermal conductivity, and in some cases, it is placed as a heat source within the cavity. Selimefendigil and Öztop [161] added a barrier to the center of enclosure to evaluate the impact of these barriers on Nu. Sheikholeslami and Sadoughi [162] investigated the effect of four fixed temperature fins in a square enclosure under a magnetic field. Sheikholeslami and Ganji [163] used LBM to examine the fin effect. Miroshnichenko et al. [164] investigated water/alumina nanofluid by the FDM in order to study the free convection. They placed a thermal source in the lower part of the cavity and evaluated the effect of cavity angle on Nu. They concluded that the most Nu appears at 60 angles. Rahimi et al. [165] investigated an L-shaped cavity using LBM method. There was a fixed temperature fin in the cavity. Alsabery et al. [166] also examined the effect of a fin in cavities by means of the FDM.

Siavashi et al. [167] examined the copper-nickel nanofluid while considering two porous fins. They found that by adding a high-Darcy fin, the heat transfer was increased and by adding a low-Darcy fin, the heat transfer was reduced. Alsabery et al. [168] examined the free convection using the FDM. There is a barrier with thermal conductivity within the cavity and, thus, they found that the ratio of thermal conductivity and block size were good controllers for heat transfer. Lugarimi et al. [43] studied the natural and turbulent convection heat by means of the FVM. Inside the cavity, there was a square barrier. Larger fins provide lower Nu values. Alsabery et al. [169] focused on effect of thermophoresis and Brownian motion using the FDM. There was a barrier with a conductivity coefficient in the cavity and with increasing its thermal conductivity in the Ra, Nu increases. Raisi and Arvin [46] examined the heat transfer of transient air using FDM. They put an insulating and flexible rectangular fin inside the cavity and found that by growing the Ra, Nu, and the deformation of the elastic part of the fin increases.

Sheikholeslami et al. [170] examined alumina/water nanofluids Using LBM in a 3-D model under an angled magnetic field. There was a spherical barrier with a warm temperature inside the rectangular three-dimensional cavity. Sobhani et al. [171] included the radiation in a fined cavity. Tighchi et al. [172] did similar research using LBM. Akter and Parvin [47] examined the MHD flow of air using the finite element method (FEM). Increasing Ra and decreasing Ha would enhance Nu. Saeid [158] examined natural and laminar air convection heat transfer in by FVM. The fin position of the constant heat flux in the floor has been studied. Izadi et al. [173] examined the multiwalled carbon nanotube (MWCNT)-Fe₃O₄/water hybrid nanofluid by means of the LBM. The cavity was in the shape of T-inverted and there was a fin with fixed thermal flux on its floor. Shun et al. [174] examined a square cavity with a number of fins inside it. Pordanjani et al. [175] examined two rectangular fins effect on Nu.

Kwak et al. [26] examined the presence of a spherical fin in the cavity. Hoghoughi et al. [176] examined a circular fin cavity. Mun et al. [177] examined presence of four circular fins. Kefayati and Tang [178,179] evaluated the effect of a circular fin in two separate studies. Riahi et al. [180] also examined a rectangular cavity with two circular fins within them. Kefayati and tang [130] studied two circular fins inside it. Hussein et al. [181] examined the effect of a thin fin. Park et al. [182] examined the presence of two circular fins. Cho et al. [183] examined the natural convection with two circular and elliptical fins in a vertical rectangular cavity. Mishra et al. [131] investigated the influence of two circular fins. Verma et al. [27] evaluated adding a number of fins.

Table 9. Summary of some enclosures equipped with fins.

References	Geometry	Type of Fluid	Parameter	Solver	Remark
[164]		Al2O3/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ γ AR	FDM	Open Cavity Fin
[165]		SiO2-TiO2/W-EG $3\%<\mathsf{\phi }<5\%$	Ra $\mathsf{\phi }$ AR	LBM	Entropy Fin
[184]		SiO2/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ K α	FEM	Fin
[185]		Cu/W $0<\mathsf{\phi }<5\%$	Ra $\mathsf{\phi }$	FEM	Fin
[186]		Cu/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ Ha	CVFEM	MHD Fin
[166]		Al2O3/W $0<\mathsf{\phi }<9\%$	Ra $\mathsf{\phi }$ k d	FDM	Entropy Fin
[182]		Fluid Pr = 0.7	Ra α	FVM	Fin
[158]		air	Gr h	CVFEM	Fin
[171]		fluid Pr = 0.71	Ra	LBM	Radiation Fin
[172]		fluid Pr = 0.71	Ra l	LBM	Radiation Fin
[183]		fluid Pr = 0.7	Ra AR	FVM	Fin
[181]		Cu/W $0<\mathsf{\phi }<2\%$	Ra $\mathsf{\phi }$ h ε	FVM	Fin
[180]		Cu/W Ag/W Al2O3/W TiO2/W $0<\mathsf{\phi }<6\%$	Ra $\mathsf{\phi }$ γ	FVM	Fin
[187]		Al2O3/W TiO2/W $0<\mathsf{\phi }<9\%$	Ra $\mathsf{\phi }$ AR	LBM	Fin

presents some articles that used fins in their work.

4.3. Applying a Magnetic Field

The magnetic field is inevitable in many industries because of the electricity near the heat transfer devices works as a flow controller. The presence of a magnetic field in different thermal devices can have different effects. The lid of some appliances, such as ducts or heatsinks, can improve heat transfer or reduce it like enclosures. Hence, the researchers have been studied heat transfer in a magnetic field. Figure 7 shows the types of magnetic field in a tube. It can be seen that a special magnetic field is created by placing different types of magnet in different situation.

Addition of magnetic field leads to the accumulation of nanoparticles. Figure 8 illustrated the effect of the magnetic field on nanoparticles and their accumulation.

The magnetic field leads to the generation of a new force in the nanofluid in addition to Lorentz force. This force can displace the fluid. In some cases, such as free convection in enclosures, it prevents the fluid displacement due to the buoyancy force, as the cavity is sometimes affected by the magnetic field in industry; researchers also examined the influence of a magnetic field on Nu [189,190,191,192,193,194,195,196,197,198]. The researchers have tried to study this issue using various angular fields. Some researchers have entered the field along the horizon and some examined the effect of magnetic field angle by giving angle to the cavity. By applying the angular magnetic field, the momentum equations are as below. Equations of energy and mass conservations, in addition, did not change.

$$\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{u}}{\partial \mathrm{y}}=-\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}+\frac{1}{\mathsf{\rho }}\left(\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\mu }\frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left(\mathsf{\mu }\frac{\partial \mathrm{u}}{\partial \mathrm{y}}\right)\right)-\frac{{\mathsf{\sigma }\mathrm{B}}^2}{\mathsf{\rho }}\left(vsin\mathsf{\Omega }.cos\mathsf{\Omega }-{\mathrm{usin}}^2\mathsf{\Omega }\right)$$

(14)

$$\frac{\partial \mathrm{v}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{v}}{\partial \mathrm{y}}=-\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{y}}+\frac{1}{\mathsf{\rho }}\left(\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\mu }\frac{\partial \mathrm{v}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left(\mathsf{\mu }\frac{\partial \mathrm{v}}{\partial \mathrm{y}}\right)\right)-\frac{\sigma {\mathrm{B}}^2}{\mathsf{\rho }}\left(usin\mathsf{\Omega }.cos\mathsf{\Omega }-{\mathrm{vcos}}^2\mathsf{\Omega }\right)+\mathrm{g}\mathsf{\beta }\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{c}}\right)$$

(15)

In the above equations, Ω, σ, and B are magnetic field angle, electrical coefficient, and strength of magnetic field, respectively. Sajjad et al. [199] studied the effect of the magnetic field on natural convection of copper/water nanofluid and realized that its greatest influence of magnetic field occurred in Rayleigh 10⁴ and the least effect occurred on 10³. Selimefendigil and Oztop [200] did similar work on copper oxide/water and alumina/water nanofluids using the finite element method. There is a conductive wall in the cavity and there are two different nanofluids on both sides. They found that with increasing wall thermal conductivity, the Ra and decreasing Ha increased the heat transfer rate.

Selimefendigil and Oztop [201] worked on carbon nanotube (CNT)/water nanofluids under an angular magnetic field using FEM. There was a conductive obstacle in the middle that separates fluid and nanofluid. They eventually found that by raising the height and the vibrational number of the conduction wall, the Nusselt number decreased by 27 to 32 percent. Kefayati [51] investigated various magnetic fields, Ra, and nanoparticles concentration on Nu, and found that by raising the volumetric percentage of ferromagnetism nanofluids in different Ra, Nu is reduced. Mahmoudi et al. [50] studied the free convection of aluminum oxide nanofluid in a chamber numerically. With a rise in Ra and a decrease of Ha, Nu increases and the magnetic field directly controls the nanoparticles effect. Mejri et al. [202] and Sivaraj and Sheremet [203] did similar works on alumina/water and iron oxide/water nanofluids, respectively. They found that the magnetic field angle is an excellent controller for heat transfer. Ashorynejad and Shahriari [204] investigated natural convection of the hybrid alumina-copper nanofluid under a magnetic field in an open cavity by the LBM method. One of the walls of cavity had a sinusual temperature along with vibration.

They found that Nu augmented by growing Ra and volumetric percentage of nanoparticles, and reducing the Hartmann number. Haq et al. [205] investigated a porous cavity with the finite element method and using vibrating surfaces under a magnetic field. Increasing the vibrations length number and Ra, Nu increases and it decreases by growing the Hartmann number. Javaherdeh and Najjanezami [206] investigated the water flow in a porous medium inside a cavity under a magnetic field using the LBM method. Growing the Darcy number raises Nu. Sheikholeslami and Shehzad [207] examined the natural convection of copper oxide nanofluid as a number in a cavity exposed to constant angular flux. The bottom was kept warm and the top was isolated. They studied the effects of Ra, Darcy, Hartmann, and percent volumes of nanoparticles on heat transfer. Selimefendigil and Oztop [208] investigated a 3-dimensional cavity under magnetic flux using the finite element. There is copper oxide nanofluid in the cavity. They found that growing the nanoparticles concentration and Ra increased Nu. Sheikholeslami and Sadoughi [162] considered the natural convection of the copper oxide nanofluid under a fixed angular magnetic field using LBM. Sides were maintained cool and the bottom and top kept isolated. There were four heaters; two hot, and two cold. They understood that Nu reduced with growing Ha and augmented with growing Ra. Al-Rashed et al. [209] investigated the natural convection of a 3-D cavity containing CNT/water nanofluids under an angular magnetic field in the FVM. They reported that by increasing Ra and declining Ha, the Nusselt number increases. Dogonchi et al. [210] examined the effect of vibrating walls by FVM. Raising the Ra and the nanoparticles concentration intensifies the Nusselt number.

Table 10. Articles studied the enclosures under the influence of the magnetic field.

References	Geometry	Type of Fluid	Parameter	Solver	Remark
[211]		Fe3O4/W $0<\mathsf{\phi }<9\%$	Ra $\mathsf{\phi }$ Ha Da α	FDM	Porous Layer Open Cavity MHD
[199]		Cu/W $0<\mathsf{\phi }<12\%$	Ra Ha $\mathsf{\phi }$	LBM	3D MHD
[200]		CuO/W Al2O3/W $0<\mathsf{\phi }<4\%$	Gr Ha k	FEM	MHD Entropy Brownian Motion
[201]		CNT/W $0<\mathsf{\phi }<3\%$	Ra $\mathsf{\phi }$ Ha α w k	FEM	MHD
[203]		Fe3O4/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ Ha α Ω	FVM
[204]		Cu- Al2O3/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ w Ha	LBM	Wave Wall Open Cavity MHD Hybrid Nanofluid
[210]		Cu/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ w Ha	CVFEM	Wave Wall MHD Brownian Motion
[212]		Cu- Al2O3/W $0<\mathsf{\phi }<4\%$	Ra Ha q $\mathsf{\phi }$ b d	FDM	MHD Hybrid Nanofluid
[208]		CuO/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ Ha h	FEM	3D MHD
[170]		Al2O3/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ Da h	LBM	Porous Layer MHD 3D Brownian Motion Fin
[209]		CNT/W $0<\mathsf{\phi }<5\%$	Ra $\mathsf{\phi }$ Ha α	FVM	MHD 3D
[213]		Al2O3/W $0<\mathsf{\phi }<4\%$	Ra Ha AR d	FVFDM	Entropy MHD
[214]		CuO/W $0<\mathsf{\phi }<4\%$	Ra Ha $\mathsf{\phi }$ m	CVFEM	MHD
[215]		Al2O3/W $0<\mathsf{\phi }<4\%$	Ra Ha $\mathsf{\phi }$ AR α	LBM	Entropy MHD Brownian Motion

presents the papers on the effect of magnetic field in the convection system.

4.4. Add Porous Media to the Cavity

Porous material consists of a solid medium that the fluid can pass through its parts. Passage of fluid through the porous medium can have important influences on the pressure difference and the heat transfer in systems. Figure 9 shows a part of a porous environment in which a fluid flow. This environment can in some cases even act as a fin and transfer the heat. In many cases, the porous environment also acts as a filter of fluid flow. In addition, the porous environment leads to intensification in pressure drop and the pumping power because of an enhancement in the shear stress. Therefore, in cases such as closed enclosures where the fluid velocity is slowed down, it can have a high impact on the heat transfer. It should be considered that the porosity value affects the heat transfer rate.

Since engineers have to use porous media in some industrial and engineering environments, some researchers have explored the effect of adding porous media to the cavities [217]. By adding a porous medium, the Darcy relationship of the momentum equations has variations which can be seen below. Of course, the energy and mass conservation equations remain unchanged [205].

$$\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{u}}{\partial \mathrm{y}}=-\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}+\frac{1}{\mathsf{\rho }}\left(\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\mu }\frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left(\mathsf{\mu }\frac{\partial \mathrm{u}}{\partial \mathrm{y}}\right)\right)-\frac{\mu }{\rho K}\mathrm{u}$$

(16)

$$\mathrm{u}\frac{\partial \mathrm{v}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{v}}{\partial \mathrm{y}}=-\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{y}}+\frac{1}{\mathsf{\rho }}\left(\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\mu }\frac{\partial \mathrm{v}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left(\mathsf{\mu }\frac{\partial \mathrm{v}}{\partial \mathrm{y}}\right)\right)-\frac{\mu }{\rho K}\mathrm{v}+\mathrm{g}\mathsf{\beta }\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{c}}\right)$$

(17)

Darcy’s law has acceptable results for viscous currents. The relationship between the pressure gradient and average volume velocity can be obtained using Forchheimer’s modification of Darcy’s model. This relation can add the term of nonlinear drag to Darcy’s law [218].

$$\frac{\partial \mathrm{p}}{\partial \mathrm{x}}=-\frac{{\mu }_f}{K}{u}_D-\frac{c_f{\rho }_0}{\sqrt{K}}\Vert \sqrt{\left(u_D^2+v_D^2\right)}\Vert u_D$$

(18)

$$\frac{\partial \mathrm{p}}{\partial \mathrm{y}}=-\frac{{\mu }_f}{K}{v}_D-\frac{c_f{\rho }_0}{\sqrt{K}}\Vert \sqrt{\left(u_D^2+v_D^2\right)}\Vert v_D-g\beta {\rho }_0\left(T-T_0\right)$$

(19)

The values of u_D and v_D represent seepage velocity along the x-, y-directions. In the above relation c_f is the dimensionless inertia parameter which can be expressed by Ergun’s [219] model as follows:

$${\mathrm{c}}_{\mathrm{f}}=\frac{1.75}{\sqrt{150{\mathsf{ϵ}}^2}}$$

(20)

where $ϵ$ is the porosity value. In Equations (18) and (19), the first term represents the linear drag produced by Darcy and the second term is the nonlinear drag produced by Forchheimer.

Brinkman [220] improved the Darcy flow model. This relation is valid for high viscous flow (without porous matrix) with high permeability limitation. In this case, the relationship between the pressure gradient and the average volume velocity is obtained as follows:

$$\frac{\partial \mathrm{p}}{\partial \mathrm{x}}=-\frac{{\mu }_f}{K}{u}_D+{\mu }_{\mathrm{eff}}{\nabla }^2{u}_D$$

(21)

$$\frac{\partial \mathrm{p}}{\partial \mathrm{x}}=-\frac{{\mu }_f}{K}{v}_D+{\mu }_{\mathrm{eff}}{\nabla }^2{u}_D$$

(22)

Here μ_eff indicates the fluid effective viscosity. According to Brinkman’s model, energy equations and the momentum will be as follows: [221,222].

$${\mathrm{u}}_1\frac{\partial {\mathrm{u}}_1}{\partial \mathrm{x}}+v_1\frac{\partial {\mathrm{u}}_1}{\partial \mathrm{y}}=-\frac{1}{{\mathsf{\rho }}_0}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}+\frac{1}{{\mathsf{\rho }}_0}\left(\frac{\partial }{\partial \mathrm{x}}\left({\mu }_{\mathrm{eff}}\frac{\partial {\mathrm{u}}_1}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left({\mu }_{\mathrm{eff}}\frac{\partial {\mathrm{u}}_1}{\partial \mathrm{y}}\right)\right)-\frac{\in {\mu }_{\mathrm{eff}}}{{\mathsf{\rho }}_{0}K}{\mathrm{u}}_1$$

(23)

$${\mathrm{u}}_1\frac{\partial v_1}{\partial \mathrm{x}}+v_1\frac{\partial v_1}{\partial \mathrm{y}}=-\frac{1}{{\mathsf{\rho }}_0}\frac{\partial \mathrm{p}}{\partial \mathrm{y}}+\frac{1}{{\mathsf{\rho }}_0}\left(\frac{\partial }{\partial \mathrm{x}}\left({\mu }_{\mathrm{eff}}\frac{\partial v_1}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left({\mu }_{\mathrm{eff}}\frac{\partial v_1}{\partial \mathrm{y}}\right)\right)-\frac{\in {\mu }_{\mathrm{eff}}}{{\mathsf{\rho }}_{0}K}{v}_1+\mathrm{g}\mathsf{\beta }\left(\mathrm{T}-{\mathrm{T}}_0\right)$$

(24)

$$\in \rho c_p\left(\frac{\partial \left({\mathrm{u}}_{1}T\right)}{\partial \mathrm{x}}+\frac{\partial \left(v_{1}T\right)}{\partial \mathrm{y}}\right)=\frac{\partial }{\partial \mathrm{x}}\left({\mathrm{k}}_{\mathrm{eff}}\frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right)+\frac{\partial }{\partial \mathrm{y}}\left({\mathrm{k}}_{\mathrm{eff}}\frac{\partial \mathrm{T}}{\partial \mathrm{y}}\right)$$

(25)

Here k_eff denotes the effective thermal conductivity of the porous media saturated fluid. The relation between Darcy and intrinsic velocities is u_D = $ϵ$ u₁, v_D = $ϵ$ v₁.

Astanina et al. [211] carried out research on a partially-porous trapezoidal enclosure filled by iron oxide nanofluids using the FDM. Miroshnichenko et al. [223] examined the natural convection of aluminum oxide nanofluid in an open cavity. There are two porous walls inside the cavity, which separates the nanofluid in it into two parts. They found that Nu increases by raising the volumetric percentage of nanofluids between the warm wall and the porous medium at low intervals. Sheikholeslami and Seyednezhad [224] examined the natural convection with Fe₃O₄/EG nanofluid radiation in a two-dimensional asymmetric cavity using the finite element method. There is a porous environment in the cavity. They realized that raising Ra and the radiation parameter leads to a raise in Nu. Sheikholeslami et al. [225] studied the natural convection along with radiation in a porous enclosure with FVFEM. Gibanov et al. [226] evaluated the natural convection of iron oxide nanofluids using the FDM in a horizontal magnetic field. They used two different porous media in a rectangular cavity and used the Brinkman extended Darcy model to model porous environments.

Raizah et al. [128] examined an open angular porous cavity with power-law fluids and used the Darcy method for the porous medium. Finally, using the FVM, they solved the governing equations of the cavity. They found that with increasing Ra and decreasing angle of the cavity, the aspect ratio and power-law index would cause the heat transfer rate to increase. Dutta and Biswas [25] studied the natural convection in the finite element method in a circular, quarter circle cavity. There was a porous environment within the cavity. It was realized that with growing the Darcy number, Nu increased. Ahmed et al. [227] studied the porous medium in a conical enclosure by FEM. Dutta et al. [228] examined a quarter of porous enclosure.

Table 11. Summary of some articles related to heat transfer in enclosures saturated with porous media.

References	Geometry	Type of Fluid	Parameter	Solver	Remark
[223]		Al2O3/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ Da AR	FDM	Porous Layer Open Cavity
[224]		Fe3O4/EG $0<\mathsf{\phi }<5\%$	Ra Rd Da $\mathsf{\phi }$	CVFEM	Radition Porous Layer
[228]		fluid Pr = 0.7	Ra Da	FEM	Entropy Porous Layer
[205]		fluid	Ra Ha Da w	FEM	Porous Layer Wave Wall MHD
[206]		Water	Ha Da α	LBM	Porous Layer MHD Phase Devition
[229]		isotropic porous medium	Ra Df Sr Le n	FDM	Soret Dufour Unsteady Double Diffusive Porous Layer
[230]		CuO/W $0<\mathsf{\phi }<8\%$	Ra $\mathsf{\phi }$ d h k ε	FEM	Porous Layer
[231]		Fe3O4/EG $0<\mathsf{\phi }<5\%$	Ra Rd Da V $\mathsf{\phi }$	CVFEM	Radiation Coulomb Forces Porous Layer Electric Field

presents some of the articles in which the porous media is used in the enclosure.

5. Cavity with Two-Phase Approach

In general, both two-phase and single-phase methods can be utilized to simulate nanofluid flow. Many of the simulations employed single-phase fluid inside the enclosure, however, in some papers, especially for nanofluids, the researchers used two-phase analysis for their simulations. In the following, some of two-phase methods and some related articles are introduced. There are several methods for modeling multiphase flow (Figure 10) that will be explained below.

5.1. Eulerian–Eulerian Approach

Different phases are considered as continuous ones in the Eulerian–Eulerian method. The concept of phase volume fraction is presented duo to the fact that phases were not mixed. Concentration is considered as a continuous function, spatially and temporally, and sum of them is equal to unity. Each phase has its conservation equation and obtained separately that have the same structure for all phases [232].

5.1.1. Volume of Fluid Model

The volume of fluid model is a model used to immiscible suspensions simulation when the interfaces among phases are important. In this model, momentum equations are assumed to the fluid and a volume fraction is solved in computational cells to every volume fraction. This approach is used to model the stratified flow, free-surface flow, filling of a reservoir, movement of the liquid surface due to the turbulence, movement of bubbles and transient or steady tracking of the gas-liquid interface. The method is less commonly used for nanofluids within enclosures. It seems that this model alone is not capable to simulate the two-phase of nanofluids [233].

5.1.2. Mixing Model

This model is used for multi-phase fluids. The mixing model uses a hybrid momentum equation so that the mean properties of the different phases are considered in the equation. For dispersed phases, this model uses relative velocity. Of course, if the model does not consider the movement of the dispersed phase, it models the multi-phase flow as a homogeneous medium. Applications of this model are particle-laden and bubbly flows, as well as cyclonic separators. Many researchers used this method to simulate two-phase nanofluid flows in enclosures [234,235,236,237,238,239,240,241,242].

5.1.3. Eulerian Model

This model is the most complex multi-phase model in terms of the number of equations and relationships. This model solves the continuity and momentum equations phase by phase, separately. The coupling of these equations is done by the pressure and inter-phase exchange coefficients. The coupling is based on the type of phases so that it is different for fluid-solid flows from fluid-fluid ones. Although this model has the potential of simulating multiphase nanofluid flows, and even predicts nanoparticle deposition, it has received less attention due to the complexity and high number of equations [243,244,245,246,247].

5.2. Eulerian–Lagrangian Approach

In this model, the fluid phase is considered continuously by solving the time-averaged Navier–Stokes equations. However, the discrete phase is solved by tracking the motion of particles, bubbles, or droplets within the fluid. Energy is exchanged between phases. In fact, the trajectory of each particle is determined in this method. Since the particle trajectory is known, this model is suitable for modeling spray dryers, fuel combustion, and flows containing particles, However, it is not suitable for modeling liquid–liquid mixtures, fluidized bed substrates or any other applications where the secondary phase volume fraction cannot be neglected. Due to its many equations and difficulties, this method has been used less frequently in enclosures and is more applicable for combustion or phase conversion problems [248].

Danjie et al. [249] investigated the heat transfer of copper/water nanofluid using the Lattice–Boltzmann method in a square cavity. By considering the Ra ranging between 10³ and 10⁶ and range of the nanoparticles volume fraction between 1% and 5%, they realized that the two-phase model should be used instead of single-phase homogeneous one due to the creation of slip in the nanofluid when Ra or nanofluid concentration are relatively high. Goodarzi et al. [237] examined a shallow rectangular enclosure filled with CuO using a mixture two-phase method. They carried out their study in the range of Grashof number of 10⁵ to 10¹⁰. They considered the effect of the Brownian motion. A summary of some articles presented in the field of two-phase nanofluids indicated in

Table 12. Summary of some articles related to two-phase nanofluid flow in enclosures.

References	Geometry	Type of Fluid	Parameter	Solver	Remark
[167]		Cu/W $0<\mathsf{\phi }<4\%$	Ra$\mathsf{\phi }$ Da	FVM	Fin Porous Layer Entropy Two-Phase
[176]		Al2O3/W	Nt Nb Nr Ra Da Le Nh d γ	FEM	Wavewall Porous Medium Two-Phase Brownian Motion Fin
[169]		Al2O3/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ k d	FDM	Thermophoresis Brownian Motion Two-Phase Fin
[168]		Al2O3/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$ d k	FDM	Fin Two-Phase
[145]		Cu/W $0<\mathsf{\phi }<3\%$	Ra $\mathsf{\phi }$ Da α	FVM	Two-Phase Porous Layer Inclined Cavity

.

6. Non-Rectangular Cavities

Most scholars have studied rectangular cavities, but in recent years, some researchers have also looked at non-rectangular cavities such as triangular, circular. Rahimi et al. [250] examined SiO₂-TiO₂/W-EG hybrid nanofluid in an H-shaped enclosure using LBM. Bondereva et al. [251] did similar work on a trapezoidal open enclosure by means of the FDM. Bhowmick et al. [252] studied the transient convection with using the FVM in a triangle cavity. Das et al. [253] examined convection in cavities with different geometries such as triangles and square Dogonchi et al. [186] examined the natural convection copper nanofluids under magnetic field in a triangular cavity. There is a semi-circular niche in the lower part. They reported that by growing Ra and decreasing Ha, the thermal characteristics are enhanced. Rashad et al. [212] examined natural convection of the alumina/copper hybrid nanofluid in water using the FDM. The cavity is triangular under the magnetic field. There was a constant thermal flux on the floor of the cavity. They found that the rate of heat transfer raises with increasing the Ra and the part where flux enters the cavity. Aghakhani et al. [132] examined the free convection in power-law model with using a non-Newtonian fluid exposed to a magnetic field in a C-shaped enclosure by means of LBM. They realized that reduction in Nu due to raising the power-law index. Using FVM, Esfe et al. [254] investigated free convection of CNT/water-EG nanofluids in trapezoidal configuration. Esfe et al. [255] studied the T-shaped cavity. There is also

Table 13. Summaries of papers on non-rectangular cavities.

Ref	Geometry
Astanina et al. [211]]	Examined a trapezoidal cavity
Rahimi et al. [165]	L-shaped cavity
Izadi et al. [173]	Inverted T cavity
Dutta and Biswas [25]	Quarter circle cavity
Ahmed et al. [227]	Cone cavity
Lugarini et al. [43]	C-shaped cavity
Zhang et al. [256]	Circular cavity
Guestal et al. [257]	Circular cavity
Armaghani et al. [213]	T-shaped cavity
Enayati et al. [258]	Cylindrical cavity
Snoussi et al. [259]	U-shaped cavity
Ho et al. [44]	Cylindrical cavity
Bhowmick et al. [252]	V-shaped cavity
Mo et al. [187]	U-shaped cavity
Malekpour et al. [260]	I-shaped cavity
Sheikholeslami et al. [231]	Circular sectot between two circles
Almudhaf et al. [229]	Trapezoidal cavity
Amrani et al. [45]	Triangular cavity

, summarizing of studies along with non-rectangular cavities.

Table 14. Summarizing papers along with their figures.

References	Geometry	Type of Fluid	Parameter	Solver	Remark
[250]		SiO2-TiO2/W-EG $3\%<\mathsf{\phi }<5\%$	Ra $\mathsf{\phi }$	LBM	Hybrid Nanofluid Entropy
[261]		CuO/W $0<\mathsf{\phi }<9\%$	Ra $\mathsf{\phi }$ T AR	FDM	Thermophoresis Brownian Motion
[251]		Nanofluid (Water base)	Ra Le Nr Nb Nt	FDM	Transient Thermophoresis Brownian Motion
[262]		CuO/W $2\%<\mathsf{\phi }<5\%$	Ra$\mathsf{\phi }$	FDM
[252]		fluid Pr = 0.71	Ra $\mathsf{\phi }$ AR Pr	FVM	Transient
[263]		Al2O3/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$	LBM	Radiation
[264]		fluid Pr = 7	Ra Da Le n	FVM	3D
[257]		Cu/W TiO2/W $0<\mathsf{\phi }<5\%$	Ra h $\mathsf{\phi }$	FVM	Hybrid Nanofluid
[265]		CuO/W $0<\mathsf{\phi }<4\%$	Ra $\mathsf{\phi }$	LBM	Entropy 3D
[259]		Cu/W Al2O3/W $0<\mathsf{\phi }<1.5\%$	Ra $\mathsf{\phi }$	FVM	3D

summarizes some of the other works on free convection in enclosures by some researchers.

7. Conclusions

Researchers on free convection in cavities have been investigated in the current work. The focus has been on recent articles especially the articles presented in recent years. Experimental studies on the cavities were reviewed. In the next section, the agent fluids in the cavities were investigated including air, water, nanofluids, and non-Newtonian fluids. For each of the fluids, the thermophysical properties have been investigated and their important points were mentioned. Moreover, studies on cavities were investigated in several sections including effective parameters including the magnetic field, the cavity angle, the porous medium, and the fin. In each section, the articles in which the parameters are mentioned are stated. In the last section, papers on non-rectangular compartments have been investigated. This review paper showed that with increasing the Rayleigh number the thermal performance intensifies and the flow field strengths in the enclosure. The magnetic field reduces flow strength in enclosure and can be used as a flow controller. Nanoparticles, in addition, increase the thermal conductivity of fluids; however, they can also increase fluids viscosity. Since the viscosity has a noteworthy effect on convection, this can result in a reduction in thermal characteristics. Therefore, at high nanofluid concentrations, further studies on heat transfer are required. Particularly, there is still no accurate data consistent with the experimental data in two-phase analysis. The evaluation of nanoparticle deposition may be of much interest to researchers.