Numerical analysis of unsteady natural convection from two heated cylinders inside a rhombus enclosure filled with Cu-water nanofluid

International Communications in Heat and Mass Transfer 113 (2020) 104510

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Numerical analysis of unsteady natural convection from two heated cylinders inside a rhombus enclosure filled with Cu-water nanofluid

T

⁎

Ali Akbar Hosseinjania, , Mehdi Nikfarb a b

Department of Mechanical Engineering, Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Unsteady natural convection Nano-fluid Rhombus enclosure Immersed boundary method (IBM)

In this paper, natural convection due to two heated cylinders inside a rhombus enclosure with lower temperature with respect to the cylinders is investigated. Here, Cu-water is the working nanofluid. The numerical algorithm is based on an iterative direct immersed boundary method (IBM). The focus of this article is on stability, instability, symmetry as well as asymmetry of nanofluid flow pattern in three different Rayleigh (Ra) numbers, i.e. 104, 105 and 106. The effects of important parameters such as cylinder diameters, nanoparticle volume fraction and distance between two cylinders are analyzed. The results of this study disclose that at both Ra = 104 and 105, nanofluid flow is symmetric and stable, while at Ra = 106 four different flow regimes including steady-symmetric, steady-asymmetric, unsteady-asymmetric with periodic oscillations and unsteady-asymmetric with irregular oscillations are observed. In addition to geometrical parameters, nanoparticle volume fraction has a considerable impact on the flow regime. Furthermore, the possibility of unsteady-asymmetric flow increases considerably by increasing the distance between two cylinders.

1. Introduction Convection heat transfer caused by hot/cold cylinders in fluid-filled enclosures has received lots of attention because due to its necessity in the studies related to the heat transfer. Myriad cases such as investigating the performance of heat exchangers, chemical reactors, electronic cooling equipment and solar energy collectors can be mentioned for a heated cylinder in cooled fluid-filled enclosures. Both experimental and numerical researches have been performed out to study the effects of fluid features, geometry and boundary conditions on convection heat transfer rate and flow regime in this scope. Fig. 1 shows a few different geometries of heated circular cylinders inside cooled enclosure available in the literature [1–14]. The fluid flow inside a cooled enclosure with single heated circular cylinder has been investigated in lots of studies. The geometrical effects of cavity and Rayleigh (Ra) number on heat transfer intensity have been experimentally and numerically studied by Cesini et al. [3]. They reported that the higher heat transfer rate happened where the distance between heated and cooled walls were minimum. Also, unsteady flow can be seen at Ra = 106. Shu et al. [4] investigated the flow behavior in a cavity encompassing a hot cylinder by a numerical approach called DQ method. They showed that the cylinder locations influences the

flow regime and heat transfer rate. A computational simulation has been done by Roychowdhury et al. [5] to analyze the effect of the geometries, the fluid Prandtl (Pr) number and boundary conditions on the heat transfer rate and flow regime around the heated cylinder inside a cold cavity. Also, Angeli et al. [6] studied the heat transfer and flow field around a horizontal cylinder to present a correlation for Nusselt number. Kim et al. [1] studied 2D unsteady convection heat transfer around a hot cylinder in a cooled cavity via an immersed boundary method (IBM) (Fig. 1A). The influences of vertical position of the cylinder and Ra number have been investigated in this study. An IBM has been applied by Lee et al. [7] to analyze the natural heat convection from a cylinder inside a square cavity at different Ra numbers, i.e. Ra = 103, 104, 105 and 106. Costa and Raimundo [8] investigated the role different parameters such as cylinder diameter and rotational speed on the strength of mixed convection inside a square cavity. Natural convection in cavity with a cylindrical obstacle moving with a linear velocity has been numerically studied by Hussain and Hussein [9]. In this research, the lower and upper walls of the cavity were insulated while side walls both had constant temperature (Fig. 1B). They did a parametric study to understand the role of cylinder position and motion direction as well as the enclosure slop. Hussein [10]analyzed free convection inside a parallelogrammical enclosure filled by air

⁎ Corresponding author at: Department of Mechanical Engineering, Faculty of Industrial and Mechanical Engineering, Qazvin Islamic Azad University, Nokhbegan Blvd., P.O. Box 34185-1416, Qazvin, Iran E-mail addresses: [email protected] (A.A. Hosseinjani), [email protected] (M. Nikfar).

https://doi.org/10.1016/j.icheatmasstransfer.2020.104510

0735-1933/ © 2020 Elsevier Ltd. All rights reserved.

International Communications in Heat and Mass Transfer 113 (2020) 104510

A.A. Hosseinjani and M. Nikfar

∗ → u uB → VΓ → x → x

Nomenclature cp dp df d D → f

→ f → g h h I k κb Ls L2 L0 Nu p p Pr Re S t T T0 → u → u

Specific heat at constant pressure (J kg−1 K−1) diameter of the nanoparticle (m) Diameter of the base fluid molecule (m) Diameter of cylinder (m) Distribution operator

Dimensionless intermediate velocity vector Brownian velocity of the nanoparticle (m s−1) Dimensionless velocity vector on the boundary Position vector (m) Dimensionless position vector

Greek symbols

Force source to impose velocity boundary condition (m s−2)

α β Θ Θ∗ ΘΓ μ ρ ϕ ∇

Dimensionless force source Gravity acceleration vector (m s−2) Heat source to impose thermal boundary condition Dimensionless heat source Integration operator Thermal conductivity (W m−1 K−1) Boltzmann's constant = 1.38066e-23 (JK−1) Side length of enclosure (m) L2 norm Reference length Local Nusselt number Pressure (pa) Dimensionless pressure Prandtl number Reynolds number Distance of cylinders (m) Time (s) Temperature (°K) Reference temperature (°K) Velocity vector (m s−1) Dimensionless velocity vector

Thermal diffusivity (m2 s−1) Coefficient of thermal expansion (oK−1) Dimensionless temperature Dimensionless intermediate temperature Dimensionless temperature on the boundary Viscosity (kg m−1 s−1) Density (kg m−3) Nanofluid volume fraction Gradient operator

Subscripts nf f p fr B H C m Θ

Nanofluid Pure fluid Particle freezing point Brownian Hot cold Momentum Thermal

Nusselt numbers strengthens/decreases as the cylinder becomes close to the upper/lower enclosure walls. Many researchers have analyzed the influence of nano-fluid on convection heat transfer around circular cylinder in an enclosure. A comprehensive review on numerical study of convection heat transfer of nano-fluid have been done by Vankie et al. [17]. Also, a brief review on natural convection of nano-fluid and pure fluid has been done by Oztop et al. [18] for different types of enclosure with localized heating. Izadi et al. [19] published a complete review paper on mixed convection of nano-fluid in various shape of enclosures. Four main classes include rectangular, triangular, trapezoidal and conventional shapes, have been summarized in this paper. Mahian et al. [20] published a review paper and categorized theoretical aspects of recent advanced in nano-fluid modeling. The reported a brief overview of nano-fluid thermo-physical properties. Also, they described all main physical models and numerical modeling methods of nano-fluid and heat transfer such as single-phase and two-phase models [21]. Garoosi and Hoseininejad [22,23] numerically studied the problem of different convection heat transfer regimes in an insulated cavity with a few numbers of heated and cooled obstacles. They observed that the average Nusselt number reduces by changing the position of the cold cylinder from vertical to horizontal mode. Rahmati and Tahery [24] studied laminar free convection of a nano-fluid around a hot object inside a square enclosure via Lattice Boltzmann Method (LBM). They reported that the average Nusselt number enhances by enlarging enclosure length and also falls down by reduction in enclosure width. Sheikholeslami et al. [25] numerically studied natural convection inside an enclosure with wavy edges filled with Cu-water nano-fluid. Their results disclosed that the inclination angle variations have a considerable effect on flow pattern. Arefmanesh et al. [26] using the meshless local Petrov-Galerkin (MLPG) approach simulated nano-fluid (Al2O3) flow and combined convection heat transfer in a square enclosure with lower temperature moving lid at the top, a corrugated wall

containing a heated concentric circular cylinder (Fig. 1E) by resorting to finite volume numerical simulation. He reported that for different values of the slope angle, the average Nusselt numbers on the inner cylinder decrease as the cylinder moves upward, while increasing as the cylinder moves downward. Liao and Lin [11] simulated mixed convection in tilted cavities with a fixed or rotating cylinder by an IBM. The effects of the Rayleigh number, Prandtl number and the cylinder diameter on the flow and heat transfer pattern were investigated. The effect of cavity shape with an internal rotating cylinder at a fixed temperature on heat transfer has been investigated by Shi and Cheng [12]. They revealed that triangular and circular cavities are the best and worst cases for heat transfer respectively. Kang et al. [54] studied the unstable free convection as well as the role of the internal hot cylinder position along a horizontal or diagonal line inside a cooling chamber in the traversing and descending thermal plumes at Ra = 107 (Fig. 1C and D). Park et al. [39] carried out a numerical study on free heat transfer from a heated cylinder inside a tilted cavity. Choi et al. [13] have analyzed the influence of the cylinder vertical position on free convection in a rhombus cavity and observed three different flow regimes, namely steady-symmetric, steady-asymmetric and unsteady-asymmetric flow regimes (Fig. 1F). Two-dimensional computational investigation have been performed by Mun et al. [14] to study the convection heat transfer due to the temperature difference gradient between the cold walls of an inclined square enclosure and an internal heated cylinder at different Prandtl numbers and Rayleigh numbers (Fig. 1G). Khanafer and Aithal [15] have analyzed the characteristics of mixed convection flow and heat transfer in a lid-driven cavity with an inner cylinder (Fig. 1H)). The roles of the Richardson number (Ri), the cylinder radius, its location on the average Nusselt number were all assessed in this study. Chamkha et al. [16] conducted a numerical analysis to understand the behavior of mixed convection inside an enclosure containing a fluid together with porous material and an insulated rotating cylinder (Fig. 1I). They noted that the local and average 2

International Communications in Heat and Mass Transfer 113 (2020) 104510

A.A. Hosseinjani and M. Nikfar

Fig. 1. Some different geometrical domain of hot cylinder in cold enclosure, (A) to (I) a single cylinder, (J) to (L) a pair of cylinder and (M) to (O) four cylinder in enclosure.

in this paper. They observed that the heat transfer rate increases with cylinder rotation in both directions for all nanoparticle types. Selimefendigil and Oztop [30] also studied two adiabatic inner rotating cylinders in 3D cavity filled with nano-fluid. They reported that 38% increases in average Nu number at the highest volume fraction in compare to the pure fluid. Numerical investigation of mixed convection of nano-fluid in a branching channel filled with nano-fluid with a rotating cylinder has been done by Selimefendigil and Oztop [31]. They reported that the average Nu number increases up to 64% for highest volume fraction for lower Richardson number. Sheikholeslami and his coworkers [32–37] also tried to quantify the intensity of a nano-fluid nearby heated cylinders with various shapes in the presence of a magnetic field. Roslan et al. [38] studied heat transfer patterns inside a cavity containing a rotating cylinder. The working fluid in this study was water– Ag, water–Cu, water–Al2O3 or water–TiO2 nano-fluids. They also pointed out that the increase in nanoparticle volume fraction

with higher temperature at the bottom and two side adiabatic walls. Their results showed an enhancement in Nusselt number by rising the concentration of nanoparticles. Nikfar and Mahmoudi [27] investigated the natural convection heat transfer of a nan fluid inside a square enclosure with corrugated side walls via MLPG. They utilized two different models for effective dynamic viscosity. Their observation indicated a significant difference is occurred between the amount of heat transfer by employing different viscosity models. Selimefendigil and Oztop [28] assessed numerically the MHD combined convection heat transfer for a non-fluid inside a cavity with moving lid. The enclosure lower wall was hot while the upper one has a lower temperature. They observed that strengthening Hartman number enhances the heat transfer. Moreover, the heat transfer also increases with increasing nanoparticle presence. Selimefendigil et al. [29] numerically investigated mixed convection in a square enclosure filled with SiO2 nano-fluid. A flexible side wall with a rotating cylinder has been studied 3

International Communications in Heat and Mass Transfer 113 (2020) 104510

A.A. Hosseinjani and M. Nikfar

Fig. 1. (continued)

induced by varying Rayleigh numbers and space between adjacent to obstacles. They also captured a regime transition at Ra = 106. In another study, 2D natural convection inside a square cavity with four heated elliptical obstacles was simulated (Fig. 1O). This parametric study focused on understanding the role of different factors such as Rayleigh number and aspect ratio of the cylinders which have considerable influence on the heat transfer case in this case. All the mentioned studies focus on analyzing the convection heat transfer and fluid behavior with and without nanoparticle around twodimensional cylinders mostly inside square enclosures. In the present paper, we focus on analyzing the nano-fluid behavior rhombus enclosures in terms of stability and instability due to the presence of nanoparticles. To our best knowledge, this problem has not been considered so far. The simulations were performed in three Ra number, i.e. 104, 105 and 106. Also, the effects of geometrical parameters such as the ratio of cylinders diameter to the cavity length, the distance between cylinders are analyzed.

makes the convective heat transfer stronger. Convection heat transfer modeling around a pair of heated cylinders was the object of some other researchers. Park et al. [39] studied the bifurcation and free convection of a pair of cylinders horizontally located inside an enclosure (Fig. 1J). Karimi et al. [40] investigated two horizontal planar cylinders inside a square cavity using a commercial CFD package. They observed both stable and unstable flow regimes. Park et al. [41] studied free convective heat transfer in a square enclosure with two internal cylinders located in different vertical locations (Fig. 1K). Very recently, Ashrafizadeh and Hosseinjani [42] utilized an IBM to perform a parametric study on convective heat transfer from two rotating heated cylinders inside a cold square chamber filled with air as the working fluid (Fig. 1L). The impact of the rotation direction, the Rayleigh and Richardson numbers were investigated in this study. They showed that the Rayleigh and Richardson numbers were the effective parameters Also, the rotation direction was turned out to have crucial impact on generating different flow regimes and can lead to an unstable periodic flow patterns under some conditions. Convection heat transfer modeling in a cold enclosure with four hot obstacles has been the subject of some studies. For instance, Seo et al. [43] studied free convective heat transfer in a cold square enclosure with four heated obstacles at different flow regime (Fig. 1M). They noticed that steady state to the unsteady state transition can happen at a high Rayleigh number (Ra = 106). They also presented a mathematical equation for calculating Nusselt number in the studied cases. Also, Mun et al. [44] numerically studied free convection inside a cavity with four obstacles in the context of a diamond (Fig. 1N). In this case, the temperature gradient between hot obstacles and cooled cavity walls were

2. Nanofluid modeling and governing equations The free convection heat transfer inside an enclosure can be governed by the equations below [45,46]:

4

μnf → (ρβ )nf → → u 1 ∂→ u . ∇→ u = − ∇p + g (T − T0) + f +→ ∇2 u + ρnf ρnf ρnf ∂t

(1)

∂T +→ u . ∇T = αnf ∇2 T + h ∂t

(2)

International Communications in Heat and Mass Transfer 113 (2020) 104510

A.A. Hosseinjani and M. Nikfar

∇→ .u = 0

μ nf

(3)

where A =

Here, → u , p, μ,ρand Tare velocity vector, pressure, viscosity, density and Temperature of the fluid respectively. Also, β is the coefficient of → thermal expansion, g is the gravitational acceleration, T0 is a reference temperature and α is the thermal diffusivity of the fluid. Force source → term f on the right-hand side of Eq. (1) has been used to impose the velocity boundary condition in the IBM formulation. Similar role has been consider for the external heat source h on the right hand side of Eq. (2) to impose the thermal boundary condition in the IBM formulation (For more details on implementing IBM [47,48]). The properties of nano-fluid can be related to the properties of base fluid and nanoparticles as follows [49]:

μf

⎛ (1 − ϕ) + ϕ ρs ⎞ ρf ⎠ ⎝ ⎜

→ βs 1 1 and f and h + ⎛ ϕ ρs + 1⎞ ⎛ (1 − ϕ) ρs + 1⎞ βf ⎠ ⎠ ⎝ (1 − ϕ) ρf ⎝ ϕ ρf terms. Raf and Prf are Rayleigh and Prandtl

, B=

⎟

⎜

⎟

⎜

⎟

are dimensionless source numbers respectively defined as follows.

Prf =

Raf =

νf αf

(14)

→ g βf (TH − TC ) L03 νf αf

(15)

h (non-dimensional thermal source) can be defined as follows:

h=

hL0 2 αf (TH − TC )

ρnf = (1 − ϕ) ρf + φρp

(4)

(ρβ )nf = (1 − ϕ)(ρβ )f + ϕ (ρβ )p

(5)

The no-slip and thermal boundary condition can be expressed as follows:

(ρcp )nf = (1 − ϕ)(ρcp )f + ϕ (ρcp )p

(6)

→ → UB = VΓ

In this research, a homogenous single-phase model has been applied. It has been supposed that the slip between pure fluid and nanoparticle is neglected, the nanoparticle dispresed uniformly through the fluid, and nanoparticles and main liquid are in hydrodynamic and thermal equlibrium. The Corcione formula is used to evaluate the thermal conductivity of nano-fluid as follows [50]: 10

knf

kp T = 1 + 4.4ReB0.4 Pr 0.66 ⎛⎜ ⎞⎟ ⎛⎜ ⎞⎟ kf T ⎝ fr ⎠ ⎝ kf ⎠

ρf uB dp μf

, uB =

(17)

ΘB = ΘΓ (18) → where, VΓ is the non-dimensional boundary velocity and ΘΓ is the nondimensional boundary temperature. The local Nusselt number on the wall surface can be evaluated by the following equation: knf ∂T ⎞ Nu = ⎜⎛− ⎟ ⎝ kf ∂X ⎠ x = 0

0.03

ϕ0.66

(19)

(7) 3. Direct forcing immersed boundary method

where ReB defined as follows:

ReB =

(16)

2κb T πμf dp2

A direct forcing iterative IBM described in [47,51] and [42] is applied here. In accordance with [42] and [47], two operators of I(∅) and D(Φ) are interpolation and distribution function respectively. The → u , f, Θ, and h defined on parameter ∅ stands for variables such as →

(8)

In which subscripts of p and f stand for particle and base fluid. The effective viscosity correlation of nano-fluid obtained by following equation [50].

μnf = μf (1 − 34.87(dp/ df )−0.3ϕ0.66)

Eulerian grid points. Also, Φ is used for the variable stored at the in→ → ternal boundary point (Lagrangian points), i.e. U, F,Θ, and H. At immersed boundary formulation to impose the boundary conditions on the internal boundary point (Eqs. (17) and (18)) the force and heat source term on the right-hand side of Eqs. (11) and (12) have been added. These source terms have been directly calculated by using the non-dimensional governing equations at the direct forcing iterative method. The non-dimensional equations can be discretized as follows:

(9)

The working fluid in this study is Cu-water nano-fluid. The utilized values for the properties of fluid and nanoparticles can be found in Table 1. The following dimensionless variables are introduced to obtain a non-dimensional form for the governing equations:

→ 3 → tαf f L0 pL0 2 → x → → u L0 T − TC → x = , u = ,p= , f = , t = 2, Θ = L0 αf αf 2 L0 TH − TC ρnf α f2 (10) where L0is the length scale. Moreover, TH and TC stand for heated and cold reference temperatures, respectively. Using the introduced dimensionless variables in Eqs. (1)–(3) will results in the non-dimensional equations as follows:

→ ∂→ u +→ u . ∇→ u = −∇p + APr∇2→ u + BRaf Prf Θ + f ∂t

(11)

αnf ∂Θ → + u . ∇Θ = ⎜⎛ ⎟⎞ ∇2 Θ + h ∂t ⎝ αf ⎠

(12)

∇→ .u = 0

(13)

n+1 → u 3 1 3 1 3 n =→ u + δt ⎛ h nm − h nm− 1 − ∇pn + 1 + ∇pn + BRaf Prf 2 2 2 2 ⎝2 →n + 12 1 n 1 n + − BRaf Prf Θ ⎞ + f Θ δt 2 ⎠

(20)

1 3 n 1 n−1 ⎞ + hn + 2 δt. Θn + 1 = Θn + δt ⎛ hΘ − hΘ 2 ⎝2 ⎠

(21)

In Eq. (20), hm contains momentum and diffusion terms, i.e. hm = −(→ u . ∇→ u ) + APr∇2→ u . Similarly, hΘ made of heat convection and αnf → diffusion, i.e. hΘ = −( u . ∇Θ) + α ∇2 Θ. Adams-Bashforth methodis

( ) f

applied to discretize the convection and diffusion terms. The second order central difference has been applied for all other derivatives. Implementing the no-slip boundary condition is formulated as follows:

Table 1 Water and Cu nanoparticles at T = 320 [46].

Cu Water

ρ (kg/m3)

K(W/mK)

Cp(J/kgK)

β × 10−5(K−1)

μ × 10−6(kgm−1s−1)

dp(nm)

8933 993

401 0.628

385 4178

1.67 36.2

– 695

45 0.385

5

International Communications in Heat and Mass Transfer 113 (2020) 104510

A.A. Hosseinjani and M. Nikfar

→ n+1 I(→ u ) = VΓ

(22)

Thermal boundary conditions can be expressed as below:

I(Θn + 1)

(23)

= ΘΓ

n+1 u ) and I(Θn+1) are the non-dimensional In above equations, I(→ fluid velocity and temperature at an internal boundary point ∗ (Lagrangian point) respectively. Θ∗ and → u , as intermediate temperature and velocity, are defined as follows:

3 n 1 n−1 ⎞ Θ∗ = Θn + δt ⎛ hΘ − hΘ 2 ⎠ ⎝2

(24)

3 1 1 1 ∗ n → u =→ u + δt ⎛ h nm − h nm− 1 + ∇pn − BRaf Prf Θn⎞ 2 2 2 ⎠ ⎝2

(25)

By substituting Eqs. (20) and (21) into Eqs. (22) and (23) respectively, employing Eqs. (24) and (25), the following equations can be written:

→n + 1 → 3 3 ∗ n+1 VΓ = I(→ u ) = I ⎛→ u − ∇pn + 1 + BRaf Prf Θn + 1⎞ + I ⎛⎜ f 2 δt⎞⎟ 2 2 ⎝ ⎠ ⎠ ⎝

(

1

)

ΘΓ = I(Θn + 1) = I(Θ∗) + I hn + 2 δt

(26)

Fig. 3. Local Nu number on the outer surface of cylinder.

(27)

impose thermal boundary conditions, the heat sources, hn + 2 δt , at Eulerian grid points can be directly calculated as follows:

1

Eqs. (26) and (27) can be rewritten as follows:

→n + 1 → 3 3 ∗ I ⎛⎜ f 2 δt⎞⎟ = VΓ − I ⎛→ u − ∇pn + 1 + BRaf Prf Θn + 1⎞ 2 2 ⎠ ⎝ ⎝ ⎠

(

1

)

I hn + 2 δt = ΘΓ − I(Θ∗)

1

(28)

By using the distribution operator, f be calculated as follows:

1

) ) = D(Θ

Γ

− I(Θ∗))

(31)

These heat sources at Eulerian points are implemented to impose the thermal boundary condition in the direct forcing iterative IBM. The details of computational algorithm in this study is similar to computational method described in [42,47].

(29)

→n + 12

((

hn + 2 δt = D I hn + 2 δt

at Eulerian grid points can

4. Validation 1 →n + 12 ⎛ →n + ⎞ f δt = D ⎜I ⎜⎛ f 2 δt⎟⎞ ⎟ ⎠⎠ ⎝ ⎝ → 3 3 ∗ = D ⎛ VΓ − I ⎛→ u − ∇Pn + 1 + BRaf Prf Θn + 1⎞ ⎞ 2 2 ⎝ ⎠⎠ ⎝

The numerical method is verified by solving the problem defined in [1] at Ra = 104, 105 and 106. In this test case, the free convective from a single hot circular cylinder in a cold cavity is solved by the described numerical method. Fig. 2 shows the streamline and isotherms obtained by the numerical algorithm. Also, local Nu number on the outer surface of the obstacles and the inner surface of cavity have been compared

(30)

These force terms at Eulerian grid points are implemented to impose the velocity boundary condition in the direct forcing iterative IBM. To

Fig. 2. Streamline and isotherms of a single heated cylinder in a cold enclosure. 6

International Communications in Heat and Mass Transfer 113 (2020) 104510

A.A. Hosseinjani and M. Nikfar

different flow regimes exist inside the cavity atRa = 106. At Ra = 106 and Ls d = 2, the flow field is stable and asymmetric. In other words, however the flow symmetry is disturbed, the flow is still stable and independent of time. At Ra = 106 andLs d = 2.5 and 3.33, the flow is asymmetric and unstable. As it will be discussed in section later in this paper, at Ls d = 2.5, the variations of average Nusselt number on the cylinders with respect to time are periodic, while at Ls d = 3.33, these variations do not follow a periodic trend. At Ra = 104, the flow is conduction-dominated. Therefore, the upward flow field is weak, and two eddies are generated near the upper surface of two cylinders. Also, as the diameter of cylinders increases, two weak eddies are formed in the upper corners of the cavity which is a sign of weak upward flow near the cylinders. In this case, the isotherms are horizontally symmetric in Ls d = 2 , while in Ls = 2.5 and 3.33, the eddies near cylinders vanish and the isotherms d slightly move upward. By increasing Ra to Ra = 105 and developing the role of natural convection in heat transfer, the flow field still remains symmetric and stable. Due to increase in natural convection, the vortexes on the upper side of the cylinders are intensified. As Fig. 10 displays, there are two upward convective flows. The first one is central upward (CU) flow that moves upward in space between two cylinders. The second one is side upward (SU) flow that moves upward in the proximity of the cavity walls. However, there is only one side downward (SD) flow that move close to the cold wall. Indeed, different vortexes are generated because of presence of these upward and downward flows. For instance, inLs d = 2 , because the space between two cylinders is narrow, the flow rate of CU is low. However, two vertexes will be created above the cylinder surface due to the ascending shear flow. There will also be two small lateral vertexes at the corners of the cavity. By increasing the Ls d parameter, the overall flow behavior remains constant, but the vertexes tend to wall. At Ra = 106, the flow pattern is different from what is observed for lower Ra numbers. Under this circumstance, four flow regimes including stable-symmetric, stable-asymmetric, unstable-asymmetric with periodic oscillation and unstable-asymmetric with non-periodic oscillation can occur. As it can be seen in Fig. 9, the flow field is asymmetric and unstable for all the diameters of cylinders. In Ls d = 2 , the unstable-asymmetric flow is periodic, while in two other aspect ratios, the flow is non-periodic. The asymmetric flow happens because of increase in the intensity of CU flow which remarkably disturbs the flow pattern above cylinders. Furthermore, as the space between two cylinders increases, the intensity of CU flow increases which results in asymmetry of the flow above the cylinders and consequently leads to increasing the number of eddies in this region. These eddies are

with Kim et al. [1] study as shown in Figs. 3 and 4. As it can be observed, both results are in good agreement with each other. In this test case, grid dependency study is carried out and it shows that 200 × 200uniform Cartesian grid is appropriate. To determine the accuracy order of the method, in this problem, the time step is set equal to 0.001. L2 norm is defined as follows: N

L2 =

1/2

1⎛ ∑ (|Θcoarser − Θfinest |2 )i ⎞⎟ N ⎜ i=1 ⎝ ⎠

(32)

The above norm is calculated up to 400 × 400. Fig. 5 shows the variations of L2with respect to the grid size. As it can be observed, the numerical method is second order accurate. Two other problems are used to validate the numerical method. To verify the nano-fluid model, heat convection problem in an enclosure filled with Cu-water nano-fluid with hot and cold side walls defined in [52] has been solved at two different Gr numbers, (Gr = 104 and 105). The volume fraction of nano-fluid and Pr number are φ = 0.1 and Pr = 6.8 respectively. Fig. 6 compares the temperature in the midline of enclosure. Good agreement between Khanafer et al. [52] and present results can be observed. Another problem is an inclined enclosure with hot and cold side wall filled with Cu-water nanofluid described in [53]. Fig. 7 shows the averaged Nu number with respect to inclination angle. The nanofluid volume fraction is ϕ = 0.1 and the problem is solved for three different Ra number (Ra = 103, 104 and 105).). Very good agreement can be observed in Fig. 7 with respect to the results reported by Abu-nada and Oztop [53]. 5. Problem definition The computational domain along with boundary conditions is shown in Fig. 8. In this study, the Ls is considered as the specific length and other dimension like S and d are scaled with respect to Ls. The cylinders are kept at a constant temperature (Θh) higher than the temperature of cavity walls (Θc). In this paper, the cylinders are stationary, and the heat transfer occurs only be natural convection. Here, to investigate the effect of cylinders diameter on the nano-fluid pattern inside the enclosure, three ratios of Ls d is considered, i.e. 2, 2.5 and 3.33. First, the effects of diameters of cylinders and their distance on the flow pattern of pure water are analyzed. Then, the effect of nanoparticles at three different volume fractions of 0.5%, 1% and 2% is quantified. The boundary conditions can be described as follows:

→ UB = 0, On the surface of cylinders and inner surface of enclosure (33)

Θh = 1, On the surface of cylinders

(34)

Θc = 0, On the inner surface of enclosure

(35)

6. Results and discussion In this section, the computational results for both pure fluid (water) and Cu-water nano-fluid are presented. For all the test cases, the streamlines, isotherms, local Nusselt number, average Nusselt number as well as the Nusselt number trend over time are investigated. Indeed, the roles of Rayleigh number, geometry and nanoparticle volume fraction on the stability and instability of fluid flow are interpreted. 6.1. Pure fluid Here, the effect of geometry and Ra on the natural convection of pure water inside the enclosure is investigated. Fig. 9 illustrates the streamlines and isotherms at different Ra numbers and ratios of Ls d . The distance between two cylinders (S) is kept constant equal to 0.6Ls. As it can be observed, the fluid field is stable at Ra = 104, 105 while two

Fig. 4. Local Nu number on the inner surface of enlosure. 7

International Communications in Heat and Mass Transfer 113 (2020) 104510

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Fig. 5. L2 norm with respect to the grid size.

Fig. 7. Averaged Nusselt numbers with respect to inclination angles, comparison between the results obtain from presented numerical method and the numerical results reported by Abu-nada & Oztop. [53] at φ = 0.1.

Fig. 6. Temperature on axial midline, comparison between the results obtain from presented numerical method and the numerical results reported by Khanafer et al. [52] at φ = 0.1 and Pr = 6.8.

constantly growing and diminishing. The results show that the generation and dissipation rates of the eddies do not obey any specific rules and their behavior is completely unstable and irregular. To analyze the effect of distance between cylinders (S), Ls d and Ra number are kept 2.5 and 106 respectively, while S changes from 0.45Ls to 0.7Ls. As Fig. 11 shows, at S = 0.45Ls, the CU flow is weak. Under this condition, the flow field is symmetric and stable. By increasing S to 0.5Ls, the CU flow becomes stronger and disturb the symmetry of the flow field. In spite of this, the flow field is still stable. Eventually, at S = 0.7Ls, the CU is strong enough to influence the flow field above the cylinders and generate an unstable asymmetric flow in this area. To summarize, the results of this section illustrate that the CU flow and two SU flows have significant effects on the symmetry and stability of the flow field inside the rhombus cavity. As the space between two cylinders increases, the CU flow is strengthened and leads to instability and asymmetry of the flow field. In contrast, by decreasing the distance between two cylinders, the CU flow is weakened, and the flow pattern tend to remain stable and symmetric.

Fig. 8. Schematic view of computational domain along with boundary conditions.

cavity shown in Fig. 8, here the flow pattern of Cu-water nanofluid at volume fractions (ϕ) of 0.5%, 1% and 2% is investigated. The streamlines and isotherms related to Cu-water nano-fluid at different volume fractions and Ra number for different Ls d ratios are displayed in Fig. 12. As it can be observed, at Ra = 104, the flow is conduction-dominated. Under this condition and at Ls d = 2 , by increasing ϕ, the size of eddy above cylinders decreases which shows that the upward convective flow is being reduced. As we can see, there is no vortex above cylinders at ϕ = 2%. Furthermore, as the nanoparticle volume fraction increases, the isotherms become more symmetric which is an indicator for enhancing conduction in the cavity. The same trend is observed for other Ls d ratios, however the streamlines are inclined towards the upper portion of the enclosure for larger Ls d ratios. By increasing Ra number to 105, the upward convective flow is

6.2. Nanofluid Having analyzed the behavior of pure water inside the rhombus 8

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Fig. 9. Isotherms and streamlines at different Ra numbers and cylinder size for pure water.

one large vortex are generated near the cylinders. The size of these vortexes is changing periodically. By increasing ϕ, the flow field is stabilized such that at ϕ = 1%, the flow field is asymmetric and unstable. In this case, two eddies with different constant sizes appear above the cylinders and remain asymmetric in this region. This happens because increasing ϕ reduces the upward convective flow intensity which results in a stable flow above the cylinders. In ϕ = 2%, the flow is symmetric and stable. In this case, the convective flow is so weak that it cannot disturb the flow symmetry. In Fig. 13, the variations of

intensified, and larger and stronger eddies are generated near the cylinder surfaces. The flow field is symmetric and stable. Isotherms are stretched towards the upper region of the enclosure which shows stronger convective flow. Increasing ϕ causes the flow field to become more conductive and it reduces the severity of the upward convective flow. At Ra = 106, the flow pattern is more complicated. At Ls d = 2 with ϕ = 0.5%, the flow field is asymmetric and periodic unstable similar to what was observed for pure water. In this case, one small vortex and 9

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As a result, the flow symmetry is disturbed, and multiple unstable vortexes are generated inside the enclosure. In ϕ = 0.5%, the flow field is asymmetric, unstable and non-periodic. By increasing ϕ to 1%, the flow is still asymmetric and unstable, but it becomes periodic. Also, in ϕ = 2%, flow is asymmetric and periodic. It should be mentioned that by increasing ϕ, the size of vortexes decreases which confirms that the flow has become weaker. In Fig. 15, the average Nusselt number on the cylinder surface over time is depicted. In ϕ = 0 and 0.5%, no periodic trend is observed, while in ϕ = 1% and 2%, the periodic trend is obvious. It can be concluded that at Ra = 104 and Ra = 105, for the nanoparticle volume fractions, the flow is symmetric and stable. In Ra = 106, considering the values of ϕ and geometrical parameters, four different cases can occur which are summarized in Table 2. In this table, S stands for either steady or symmetric, while U, A, P and NP stand for unsteady/unstable, asymmetric, periodic and non-periodic respectively. At, Ra = 106, Ls d = 2 and ϕ = 1 % ,steady and asymmetric flow can be observed. The logic behind this observation is as follows: at the first instant an upward flow forms. The intensity of this flow leads to an asymmetric flow, which is drawn to one side. It is expected that this flow forms an unsteady flow in the next time increment. However, the formation of a large local vortex (eddy) doesn't let the unsteadiness of this flow to be continued and hence, it keeps its asymmetric state. In fact, this point is the border between the steady and unsteady regimes. That is, the intensity of the central upward flow ruins the symmetry of the flow. In addition, the large local vortex prevents the formation of an unsteady flow and leads to an asymmetric-steady flow. The local Nusselt number on the cavity wall and cylinder surface at different Ls d ratios, Ra numbers and nanoparticle volume fraction are shown in Figs. 16 to 18. In Fig. 16, at Ra = 104(Fig. 16A and B), the flow is conduction dominated and hence local Nusselt number does not change remarkably with changing the nanoparticle volume fraction. In the distribution of local Nusselt number on the cylinder surface, two local maximum exist that locate at the nearest distance between hot cylinders and cold cavity walls because there is the highest temperature gradient in this region. The same trend exists for the local Nusselt number on the enclosure walls. In other words, two maximums of local Nusselt number are observed at the location where hot cylinders are at the closest distance with respect to the cold cavity walls. At Ra = 105, and increasing the intensity of the natural convection, the variations of local Nusselt number on the cylinder surface are significant in the region with the angle of 0° to 120°(0°≤θ ≤ 120°). This region is associated with the locations of two eddies above the cylinders where the eddies' rotation enhances the convection heat transfer rate. By increasing ϕ, the local Nusselt number increases as a result of intensification of temperature gradient inside two vortexes due to the change in their shape. For the enclosure walls, the variations in Nusselt

Fig. 10. Different flow branches in the rhombus enclosure.

average Nusselt number on cylinder surface with respect to time at Ls = 2 is illustrated. As it can be observed, as ϕ increases the flow field d is converted from periodic unstable to stable. In other words, by increasing ϕ, the oscillation amplitude becomes negligible and the flow field approaches to the stable sate. In Ra = 106 and Ls d = 2 , other phenomena are observed. Because of large space between two cylinders, the upward convective flow is strong, and it disturbs the symmetry and stability of the flow field above cylinders. The results indicate that at ϕ = 0.5%, the flow is asymmetric, unstable and periodic. By increasing the volume fraction of nanoparticles to 1%, the flow field remains asymmetric, unstable and periodic, but the vortexes size and the intensity of upward flow both reduce. Finally, at ϕ = 2%, the flow is completely symmetric and stable. The average Nusselt number on the cylinder surface with respect to time atLs d = 2.5 is sketched in Fig. 14. In ϕ = 0, there is no certain trend in Nusselt distribution, while in ϕ = 0.5% and 1% the periodic trend can be observed. In ϕ = 1%, the average Nusselt number is constant and the flow symmetry is maintained. For Ls d = 3.33 and Ra = 106, due to large space between two cylinders, the convective flow between cylinders become more powerful.

Fig. 11. Effect of the distance between cylinders on the flow field at Ra = 106 and 10

Ls

d

= 2.5.

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Fig. 12. Isotherms and streamlines for Cu-water nanofluid at different nanoparticle volume fraction and different Ra numbers in rhombus enclosures with different Ls ratios. d 11

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Fig. 12. (continued)

12

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Fig. 12. (continued)

Fig. 13. Time history of the average Nu number on the cylinder surface at Ls = 2 and Ra = 106. d

Fig. 15. Time history of the average Nu number on the cylinder surface at Ls = 3.33 and Ra = 106. d Table 2 Different flow regimes in rhombus cavity with two rotating cylinders.

Ls

ϕ ϕ ϕ ϕ Ls

ϕ ϕ ϕ ϕ Ls

ϕ ϕ ϕ ϕ

Fig. 14. Time history of the average Nu number on the cylinder surface at Ls = 2.5 and Ra = 106. d

Ra = 104

Ra = 105

Ra = 106

=2 =0 = 0.5% = 1% = 2%

SS SS SS SS

SS SS SS SS

UA(P) UA(P) SA SS

= 2.5 =0 = 0.5% = 1% = 2%

SS SS SS SS

SS SS SS SS

UA(NP) UA(P) UA(P) UA(P)

= 3.33 =0 = 0.5% = 1% = 2%

SS SS SS SS

SS SS SS SS

UA(NP) UA(NP) UA(P) U(P)

d

d

d

In Ra = 106, the local Nusselt numbers dramatically change by changing the nanoparticle volume fraction. As it can be observed from Fig. 16E, the local Nusselt number variations in the region of 0 ° ≤ θ ≤ 150° are more significant. This region is the location of formation of eddies above hot cylinders. Increasing ϕ has a direct effect on the local Nusselt number as it enhances convective heat transfer and

number happens on the AB (s = 0.5 to 1) and DA (s =3.5 to 4). The increase in ϕ which leads to increase in temperature gradient and convective heat transfer rate, is the reason for increasing the local Nusselt number. 13

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Fig. 16. Graph of changes of local Nusselt number on the outer surface of the circular cylinder and the inner surface of the cavity at, Ls d = 2 . (A) outer surface of cylinders at Ra = 104 (B) Cavity walls at Ra = 104 (C) outer surface of cylinders at Ra = 105 (D) Cavity walls at Ra = 105 (E) outer surface of cylinders at Ra = 106(F) Cavity walls at Ra = 106.

14

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Fig. 17. Graph of changes of local Nusselt number on the outer surface of the circular cylinder and the inner surface of the cavity at, Ls d = 2.5 (A) outer surface of cylinders at Ra = 104 (B) Cavity walls at Ra = 104 (C) outer surface of cylinders at Ra = 105 (D) Cavity walls at Ra = 105 (E) outer surface of cylinders at Ra = 106 (F) Cavity walls at Ra = 106.

15

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Fig. 18. Graph of changes of local Nusselt number on the outer surface of the circular cylinder and the inner surface of the cavity at, Ls d = 3.33 (A) outer surface of cylinders at Ra = 104 (B) Cavity walls at Ra = 104 (C) outer surface of cylinders at Ra = 105 (D) Cavity walls at Ra = 105 (E) outer surface of cylinders at Ra = 106 (F) Cavity walls at Ra = 106.

16

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increasing ϕ. The reason for this is that in the region of 250 ° ≤ θ ≤ 350° the ascending current hits the cylinder surface, while in the region0 ° ≤ θ ≤ 150° it flows away from the cylinder surface. In the Ra = 106 the local Nusselt number at the inner surface of the cavity on the two AB and DA walls has its maximum at s = 0 to 0.5 and s = 3.5 to 4, respectively (Fig. 17F). The intense ascending convective flows from these two surfaces make this observation possible. Of course, asymmetric flow disruption has made the local Nusselt number behavior somewhat unexplainable, but in general, it increases the maximum local Nusselt number. At Ls d = 3.33 in all the Ra numbers, the local Nusselt numbers are also presented on the outer surface of the cylinder and the inner surface of the cavity in Fig. 18. In Ra = 104, which is practically conductiondominated, the local Nusselt number changes with ϕ at an angle between θ = 0° to θ = 150°. In this area, most of the flow is due to the formation of vertexes. As the value of ϕ increases, so does the Nusselt number. The reason for this is that increasing ϕ increases the overall conductivity of the fluid and increases the heat transfer rate of the overall conductivity. Two points of maximum Nusselt number occur at angles θ = 160° and θ = 260°, which is the minimum distance between the hot cylinder and the cold walls. Obviously, the highest temperature gradient occurs at these two points. The variations of the local Nusselt number on the inner wall of the cavity surface at the Ra = 104 are also shown in Fig. 18B. Obviously, in this case the maximum Nusselt number is where the distance of the hot cylinder from the cold wall is the least. On the AB and DA walls, the upward fluid flow motion causes the changes of local Nusselt number along the two walls to be most effective. In other words, on the first half of the AB wall (at s = 0 to 0.5) the most local Nusselt number variations can be seen. This figure shows

consequently the eddies structures. In 0 ° ≤ θ ≤ 150° where the upward flow exists, we can also see the variations in local Nusselt number on the cylinder external surface. On the cavity wall (Fig. 16F), on AB (s = 0.5 to 1) and DA (s =3.5 to 4), the most dramatic change in local Nusselt number is observed which is due to formation of eddies on these surfaces. In Fig. 17, the results for Ls d = 2.5 are displayed. In this case, the reduction of cylinders diameter increases the space between hot cylinders and cold cavity walls. Therefore, the temperature gradient is weakened and the maximum local Nusselt number decreases. In Ra = 104, conduction is dominated, but because of large space between cylinder and cavity wall, the upward flow has more freedom to move up. Here, the local Nusselt number variations with respect to nanoparticle volume fraction is more considerable in 0 ° ≤ θ ≤ 150. The same scenario is true for the walls of AB and DA of the enclosure. In Fig. 17C, increasing the Rayleigh number to Ra = 105, enhances the convective flow and increases the effect of ϕ on the local Nusselt number. In Fig. 17D there are two local maxima on the inner walls AB and DA, two of which (s = 0.5 to 1 and s = 3 to 3.5) correspond to the minimum distance between the hot cylinder and the cold wall, and the second two maxima (s = 0 to 0.5, s = 3.5 to 4) is related to the formation of two powerful vertexes above the surface of the cylinders. The most important observation at the Rayleigh number Ra = 106 is shown in Fig. 17E. The creation of local vortexes caused by the asymmetric and scrambled flow has caused the local Nusselt number to behave irregularly. This figure is presented in an instant for all four cases. This figure shows the flow behavior such that in the region of 0 ° ≤ θ ≤ 150°, the local Nusselt number is increased by ϕ but in the region 250 ° ≤ θ ≤ 350°, the local Nusselt number decreases by

Fig. 19. Average Nusselt number (A) on the outer surface of the cylinders (B) on the upper wall of the cavity (C) on the lower wall of the cavity. 17

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that local Nusselt numbers increases with the increase of ϕ. The same trend can be seen on the DA wall at s = 3.5 to 4. By increasing the Rayleigh number to Ra = 105, the local Nusselt number changes will be more dependent on the way the CU and SU convective flows move and how the vertex is created (Fig. 18C). There were actually three local maxima for the local Nusselt number. The first peak occurs at an angle of approximately θ = 100° because of the SU flow. This local maximum should be increased by increasing ϕ. The second local maximum point is created around θ = 180° due to the presence of the lateral vertex. The third maximum occurs at an angle of approximately θ = 300° due to the CU flow. The maximum point created at θ = 300°decreases with increasing value of ϕ which means that the CU flow is weakened. On the inner surface of the cavity, SU flow on the two walls of AB and DA give rise to two local maximums (Fig. 18D). On the AB wall, the first maximum occurs at s = 0 to 0.5 due to the approaching distance between the hot cylinder and the cold wall, and the second maximum at s = 0.5 to 1 is due to SU flow and eddy formation on the cylinder surface. The same is true for the DA wall. In the BC and CD walls only a local maximum due to the low distance between the hot cylinder and the cold wall (extreme local temperature gradient) is visible. In Ra = 106, in Fig. 18E, the disruption of asymmetry causes the amount and location of the Nusselt number on the surface of the cylinders to be time-dependent and variable. However, the most important observation is that between θ = 0° to θ = 180° the value of Nusselt number always increases with increasing ϕ but at the distance between θ = 180° to θ = 360° the local Nusselt number decrease with increasing ϕ. In Fig. 18F, on the inner surface of the cavity on the walls of AB and DA, the maximum of local Nusselt number due to the CU flow is very large, but disturbances of Nusselt number change the location of maximum. Fig. 19 shows the change of the average Nusselt number relative to ϕ in three states of Ls d = 2, 2.5 and 3.33, for Ra = 104,105 and 106 on the cylinder surface, the upper walls (AB and DA) and lower walls (CB and CD) of the enclosure. Generally, the Nusselt number on the cylinder surface increases with increasing Ra number. The graphs show that the variations of the average Nusselt number on the surface of the cylinders at low Ra numbers are negligible. However, with the increase in Ra number, average Nusselt number generally decreases by increasing ϕ. Similar behavior is seen on the upper walls of the cavity (AB and DA). By increasing the Ra number, the Nusselt number increases too. In all the cases, increasing ϕ results in a decrease in the average Nusselt number. In the lower wall of the cavity, this process is reversed. In this case, due to strong upward flows, the temperature gradient near the lower cold wall has practically decreased. Therefore, the reduction of Ra number leads to a decrease in the average Nusselt number. Increasing the Nusselt number by increasing ϕ is visible in this region.

and asymmetric periodic and unstable non-periodic asymmetric regimes may be observed. 5. At Ra = 106, the greater the distance between the two cylinders, the stronger the CU flow and plays an important role in causing unstable and asymmetric flow. In general, the greater the diameter of the cylinders and the smaller the space between the cylinders, the weaker CU flow and the flow stability increases. 6. In Ra = 106, the increase of ϕ generally increases the stability of the flow. For example, a flow that is initially asymmetric and unstable with non-periodic behavior may become asymmetrically unstable with periodic behavior or stable asymmetric flow with increasingϕ. 7. The local and average Nusselt number at low Ra numbers (Ra = 104) is not very much a function of changes of ϕ, but with increasing the Ra number to Ra = 105 and 106, the maximum local Nusselt number on the cylinder surface increases. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relatinships that could have appeared to influence the work reported in this paper. References [1] B.S. Kim, D.S. Lee, M.Y. Ha, H.S. Yoon, A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations, Int. J. Heat Mass Transf. 51 (7–8) (2008) 1888–1906. [2] Y.M. Seo, M.Y. Ha, Y.G. Park, The effect of four elliptical cylinders with different aspect ratios on the natural convection inside a square enclosure, Int. J. Heat Mass Transf. 122 (2018) 491–503. [3] G. Cesini, M. Paroncini, G. Cortella, M. Manzan, Natural convection from a horizontal cylinder in a rectangular cavity, Int. J. Heat Mass Transf. 42 (10) (1999) 1801–1811. [4] C. Shu, H. Xue, Y.D. Zhu, Numerical study of natural convection in an eccentric annulus between a square outer cylinder and a circular inner cylinder using DQ method, Int. J. Heat Mass Transf. 44 (17) (2001) 3321–3333. [5] D.G. Roychowdhury, S.K. Das, T. Sundararajan, Numerical simulation of natural convective heat transfer and fluid flow around a heated cylinder inside an enclosure, Heat Mass Transf. 38 (7–8) (2002) 565–576. [6] D. Angeli, P. Levoni, G.S. Barozzi, Numerical predictions for stable buoyant regimes within a square cavity containing a heated horizontal cylinder, Int. J. Heat Mass Transf. 51 (3–4) (2008) 553–565. [7] J.M. Lee, M.Y. Ha, H.S. Yoon, Natural convection in a square enclosure with a circular cylinder at different horizontal and diagonal locations, Int. J. Heat Mass Transf. 53 (25–26) (2010) 5905–5919. [8] V.A.F. Costa, A.M. Raimundo, Steady mixed convection in a differentially heated square enclosure with an active rotating circular cylinder, Int. J. Heat Mass Transf. 53 (5–6) (2010) 1208–1219. [9] S.H. Hussain, A.K. Hussein, Mixed convection heat transfer in a differentially heated square enclosure with a conductive rotating circular cylinder at different vertical locations, Int. Commun. Heat Mass Transf. 38 (2) (2011) 263–274. [10] A.K. Hussein, Computational analysis of natural convection in a parallelogrammic cavity with a hot concentric circular cylinder moving at different vertical locations, Int. Commun. Heat Mass Transf. 46 (2013) 126–133. [11] C.C. Liao, C.A. Lin, Mixed convection of a heated rotating cylinder in a square enclosure, Int. J. Heat Mass Transf. 72 (2014) 9–22. [12] Y.C. Shih, Y.J. Cheng, The effect of viscous dissipation on heat transfer in cavities of varying shape due to an inner rotating circular cylinder, Numer. Heat Transf. A Appl. 68 (2) (2015) 150–173. [13] C. Choi, S. Jeong, M.Y. Ha, H.S. Yoon, Effect of a circular cylinder’s location on natural convection in a rhombus enclosure, Int. J. Heat Mass Transf. 77 (2014) 60–73. [14] G.S. Mun, J.H. Doo, M.Y. Ha, Thermo-dynamic irreversibility induced by natural convection in square enclosure with inner cylinder. Part-I: effect of tilted angle of enclosure, Int. J. Heat Mass Transf. 97 (2016) 1102–1119. [15] K. Khanafer, S.M. Aithal, Laminar mixed convection flow and heat transfer characteristics in a lid driven cavity with a circular cylinder, Int. J. Heat Mass Transf. 66 (2013) 200–209. [16] A.J. Chamkha, F. Selimefendigil, M.A. Ismael, Mixed convection in a partially layered porous cavity with an inner rotating cylinder, Numer. Heat Transf. A Appl. 69 (6) (2016) 659–675. [17] S.M. Vanaki, P. Ganesan, H.A. Mohammed, Numerical study of convective heat transfer of nano fl uids: a review, Renew. Sust. Energ. Rev. 54 (2016) 1212–1239. [18] H.F. Öztop, P. Estellé, W.M. Yan, K. Al-Salem, J. Orfi, O. Mahian, A brief review of natural convection in enclosures under localized heating with and without nanofluids, Int. Commun. Heat Mass Transf. 60 (2015) 37–44. [19] S. Izadi, T. Armaghani, R. Ghasemiasl, A.J. Chamkha, M. Molana, A comprehensive review on mixed convection of nanofluids in various shapes of enclosures, Powder

7. Conclusion In this paper, we investigate the natural heat transfer of two hot cylinders inside a cold rhombus enclosure filled with Cu-Water nanofluid with different volume fraction of nanoparticles. The results of this study can be summarized as follows. 1. In this problem, four different states of flow may be created around the cylinders: steady and symmetric flow, steady and asymmetric flow, unsteady and asymmetric flow with regular periodic behavior, and unsteady and asymmetric flow with non-periodic behavior. 2. In the Ra = 104 and Ra = 105, the symmetric and steady flow is generally created. 3. The two CU and SU convective flows play the most important role in the flow behavior. The intensity of CU flow decreases with increasing ϕ, but the SU flow is not significantly affected by ϕ. 4. At Ra = 106, depending on the geometry and value of ϕ, all four regimes of steady and symmetric, steady and asymmetric, unstable 18

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