NUMERICAL SIMULATION OF UNSTEADY NATURAL CONVECTION FROM HEATED HORIZONTAL CIRCULAR CYLINDERS IN A SQUARE ENCLOSURE

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1 Numerical Heat Transfer, Part A, 65: , 2014 Copyright # Taylor & Francis Group, LLC ISSN: print= online DOI: / NUMERICAL SIMULATION OF UNSTEADY NATURAL CONVECTION FROM HEATED HORIZONTAL CIRCULAR CYLINDERS IN A SQUARE ENCLOSURE 1. INTRODUCTION Fariborz Karimi 1, Hong Tao Xu 1, Zhiyun Wang 1, Mo Yang 1, and Yuwen Zhang 2 1 School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai, P.R. China 2 Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, Missouri, USA In this article, a 2-D numerical study is carried out to investigate the effect of this interaction on natural convection from two heated horizontal cylinders confined in a square enclosure with isothermal walls at the heat sink temperature. The simulations are carried out for Rayleigh numbers between 10 3 and 10 7, and dimensionless horizontal distances of cylinders between 0.1 and 0.4. The results show that the variation of the area-averaged Nusselt number strongly depends on the distance between the two cylinders if Rayleigh numbers are less than In contrary, the effect of cylinder spacing on heat transfer was found to be nearly negligible when the Rayleigh number is between 10 4 and It is also observed that steady state flow and heat transfer undergo periodical oscillation, and ultimately chaotic oscillation in special positions of cylinders can occur. There are many industrial and environmental applications for the buoyancyinduced flows from heated horizontal cylinders confined in an enclosure, such as space heating, heat exchangers, solar energy collectors, nuclear and chemical reactors, energy storage systems, electronic equipment, and so on [1 5]. A literature survey shows that numerous experimental and numerical studies have been conducted on natural convection in enclosures with the presence of a body with various thermal conditions. These studies focused on natural convection within a square enclosure with either a horizontally or vertically imposed temperature difference or heat flux. Cesini et al. [6] studied the effect of Rayleigh number and the cavity geometry on the heat transfer from a horizontal cylinder enclosed in a square cavity. The temperature distribution in the air and the heat transfer coefficients were measured by Received 12 October 2012; accepted 1 September This project was financially supported by the National Natural Science Funds (no , no ), and the Overseas Collaborative Research Program Grant from National Natural Science Foundation of China (no ). Address correspondence to Mo Yang, School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai , P.R. China. yangm66@gmail.com 715

2 716 F. KARIMI ET AL. NOMENCLATURE c p heat capacity, (Jkg 1 K 1 ) d cylinders diameter, (m) D dimensionless diameter, (¼ d=l) g gravity acceleration, (ms 2 ) h heat transfer coefficient, (W m 2 K 1 ) H enclosure height, (m) L enclosure width, (m) Nu s surface-averaged Nusselt number Nu t time-averaged Nusselt number Nu u local Nusselt number p pressure, (Pa) P dimensionless pressure Pr Prandtl number Ra Rayleigh number s cylinders distance, (m) S dimensionless cylinders distance, (¼ s=l) t time, () T H temperature of heated cylinders, ( C) T C temperature of enclosure walls, ( C) u, v velocity components in x and y directions, (ms 1 ) U, V dimensionless velocity components in X, Y directions U R reference velocity, (ms 1 ) x, y Cartesian coordinates X, Y dimensionless coordinates q density, (kg m 3 ) k conductivity, (W m 1 K 1 ) a thermal diffusivity, (m 2 s 1 ) b thermal expansion coefficient, (K 1 ) n kinematic viscosity, (kg m 1 s 1 ) h dimensionless temperature s dimensionless time Subscripts avg average s surface t time u local a holographic interferometer. Roychowdhury et al. [7] investigated the effects of different thermal boundary conditions of the enclosure wall, Prandtl number, and aspect ratio on the two-dimensional natural convective flow and heat transfer around a heated cylinder in a square enclosure. Dong and Li [8] studied the conjugated natural convection and conduction in the solid wall of a cavity by using vorticity stream function method. Effects of material character, geometrical shape, and Rayleigh number on the heat transfer were investigated. Typical isotherms, streamlines, vorticities, local Nusselt numbers, tangential and radial velocities, and temperature distributions were reported in detail by Saitoh et al. [9]. They attempted to obtaina high-accuracy benchmark solution for the natural convection flow around a horizontal circular cylinder with uniform surface temperature. Angeli et al. [10] numerically investigated the buoyancy-induced flow regimes for a horizontal cylinder centered in a long co-axial square-sectioned cavity filled with air. Atayılmaz and Teke [11] experimentally and numerically studied the natural convective heat transfer from a horizontal cylinder for two different diameters (4.8 mm and 9.45 mm). The average Nusselt number over the cylinder were obtained in the range of 74 < Ra < Kim et al. [12] solved the twodimensional unsteady natural convection between a cold outer square enclosure and a hot inner circular cylinder using the immersed boundary method (IBM). They investigated the effect of the inner cylinder location on the heat transfer and fluid flow with Rayleigh number ranged from 10 3 to Hussain and Hussain [13] studied the effects of vertical cylinder locations and Rayleigh numbers (10 3 to 10 6 ) on fluid flow and heat transfer performance of a two-dimensional steady natural convection problem. They assumed a uniform heat source on the inner circular cylinder in a square enclosure and verified the location of the inner cylinder by

3 NATURAL CONVECTION FROM CYLINDERS IN A SQUARE ENCLOSURE 717 vertically changing it along the centerline of the enclosure from 0.25 L to 0.25 L upward and downward, respectively. De and Dalal [14] studied the natural convection around a tilted heated square cylinder in an enclosure for Rayleigh numbers in the range of 10 3 to 10 6.Theflow and heat transfer for two different thermal boundary conditions (uniform wall temperature and uniform wall heat flux) were studied for different aspect ratios and the locations of the square cylinder. Butler et al. [15] experimentally investigated the natural convection from a heat generating horizontal cylinder enclosed in a square cavity, where a temperature difference existed across its vertical walls for < Ra < and a Prandtl number of Sasaguchi et al. [16] reported the effect of the position of a cooled cylinder in a rectangular cavity on the cooling process of water around the cylinder. Numerical solutions of natural convection induced by a temperature difference between a cold outer square cylinder and a hot inner circular cylinder were carried out by Lee et al. [17]. They used immersed-boundary method (IBM) to model the inner circular cylinder based on the finite volume method to study a two-dimensional natural convection for different Rayleigh umbers in the range of For the case of two cylinders, a numerical study of natural convection of air from two vertically separated horizontal heated cylinders confined in a square enclosure was carried out by Lacroix and Joyeux [18]. The enclosure had vertical walls of finite conductance and horizontal walls at the heat sink temperature. The interaction between convection and conduction in the vertical walls at Rayleigh number from 10 3 to 10 6, dimensionless wall-fluid thermal conductivity ratio in the range of , and dimensionless wall thickness were investigated. Lacroix [19] also studied the natural convection for air around two horizontal heated cylinders confined in a square enclosure cooled from above with two cavity widths and three different top cylinder positions. The local and average Nusselt numbers were determined over the range of Rayleigh numbers from 10 4 to Reymond et al. [20] experimentally investigated the natural convection heat transfer from a pair of vertically aligned horizontal cylinders for different Rayleigh numbers (2 10 6, and ) and for different lengths of cylinder spacing. Their results showed that the fluid flow and heat transfer around the upper heated cylinder were strongly affected by the presence of the heated lower cylinder, because a plume rising from the heated lower cylinder interacted with the heated upper cylinder. Another experimental investigation on the natural convection heat transfer for two parallel horizontal cylinders was carried out by Chae and Chung [21]. They measured the mass transfer rate from the cylinders and obtained the heat transfer rate (Nusselt number) based on the analogy concept by changing various parameters including pitch-to-diameter ratios (P=D) ( ), Prandtl numbers (2,014 8,334), and Rayleigh numbers ( to ). The maximum heat transfer per unit volume is a key issue for heat exchanger design. The heat sources and fluid streams must be well arranged to obtain the maximal heat transfer rate. In this field, the comprehensive knowledge of the effect of tube arrays flow regime on each other and in turn, heat transfer among them, is essential. Another important point is that the natural convection around an array of horizontal cylinders is different from those around a single cylinder because of the mutual interaction of the buoyant plumes generated by the cylinders. In addition to an engineering background, nonlinear phenomena of the problem, including periodical oscillation or chaotic features, are the attractive theoretical aspect in the

4 718 F. KARIMI ET AL. numerical study of unsteady natural convection from two horizontally separated heated cylinders confined in a square enclosure. Only a few of the above cited research considered the time-dependent behavior for the aforementioned case. The objective of this article is to report a numerical study of unsteady natural convection heat transfer from two isothermally heated cylinders centered in a square enclosure with the same wall temperatures at the heat sink. Effects of Rayleigh number and geometrical position of cylinders are studied to reveal the flow and thermal behavior in the enclosure. 2. PHYSICAL MODEL Figure 1 shows a schematic of the physical model considered in the present study. A two-dimensional square enclosure of equal height (H) and width (L) that contains two horizontal oriented cylinders with diameter (d) situated in the middle of enclosure height and separated by a distance (s) will be considered. The width of the cavity, L, serves as the characteristics length scale on which the Rayleigh number is defined. All cavity surfaces are maintained at a constant temperature (T C ), and the uniform temperature on the surfaces of the heated cylinders is T H (T H > T C ). Natural convection occurs due to the temperature gradients that exist across the enclosure. The working fluid is water and the flow is laminar for all cases. The radiation effects are assumed to be negligible. It is also assumed that the thermophysical properties are constant except for the density in the body force term of the y-momentum equation, which follows the Boussinesq approximation. The flow within the cavity is assumed to be unsteady and viscous dissipation and compressibility effects are negligible. The physical models and its boundary conditions with Cartesian coordinates are indicated in Figure 1. The governing equations describing the problem are based on the conservations of mass, momentum, and energy. According to the foregoing assumptions, Figure 1. Schematic diagram of the physical model.

5 NATURAL CONVECTION FROM CYLINDERS IN A SQUARE ENCLOSURE 719 the dimensionless forms of the governing equations are as the following. qu qx þ qv qy ¼ 0 ð1þ qu qs þ U qu qx þ V qu qy ¼ qp qx þ qv qs þ U qv qx þ V qv qy ¼ qp qy þ Pr rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RaPr q2 U þ q2 U qx 2 qy 2 pffiffiffiffiffiffiffiffiffiffiffi Pr RaPr qh qs þ U qh qx þ V qh qy ¼ 1 p ffiffiffiffiffiffiffiffiffiffiffi RaPr where dimensionless variables are defined as follows.! q 2 V qx 2 þ q2 V qy 2 þ h! q 2 h qx 2 þ q2 h qy 2 s ¼ tu R L ; X ¼ x L ; Y ¼ y L ; U ¼ u ; V ¼ v ; u R u R u R ¼ a pffiffiffiffiffiffiffiffiffiffiffi RaPr; P ¼ p ; h ¼ T T C L T H T C qu 2 R The Rayleigh (Ra) and Prandtl numbers (Pr) are defined as follows. Ra ¼ gbl3 ðt H T C Þ ; Pr ¼ n na a Where n, g, and b are the kinematic viscosity, gravitational acceleration, and volume expansion coefficient, respectively. The dimensionless forms of the boundary conditions are specified as follows.. On all solid cavity walls: h ¼ 0; U ¼ V ¼ 0. On the surface of cylinders: h ¼ 1; U ¼ V ¼ 0 Two aspect ratios of the model are defined as follows. ð2þ ð3þ ð4þ ð5þ ð6þ D ¼ d L ; S ¼ s L ð7þ The Nusselt number (Nu) of the cylinder walls is the measure of convective heat transfer at the surfaces. The local Nusselt number on the cylinder surfaces is obtained from the following expression. Nu u ¼ h ud k qh ¼ qn cylinder wall Where h u is the local surface heat transfer coefficient on the cylinder walls, and n is the normal coordinate to the cylinder surface. The average Nusselt number is ð8þ

6 720 F. KARIMI ET AL. calculated by integrating the local Nusselt number along the wall and the areaaveraged Nusselt number for cylinder is defined as follows. Nu s ¼ h avgd k ¼ 1 2p Z 2p 0 Nu u du ð9þ 3. NUMERICAL SOLUTION The unsteady heat transfer predictions were carried out using the finitevolume-method-based commercial CFD package FLUENT. The laminar unsteady equations were solved on structured control volumes using a second-order quadratic upwind interpolation of convective kinematics (QUICK) scheme for the convection terms and by employing the semi-implicit method for pressure-linked equations (SIMPLE) pressure correction algorithm [22]. Transient terms of the equations are discretized by the second order implicit formulation as well. Numerical results were first obtained to check the grid independency. The convergence of the solution to the unsteady state result was obtained when the residual of continuity, X and Y momentum equations are less than For the energy equation, the residual must be less than In order to determine proper spatial resolutions, a grid sensitivity analysis was carried out for the case of two horizontally oriented cylinders with D ¼ S ¼ 0.2 and Ra ¼ 10 4, where the flow regime appeared to be steady state. The nodes arranged in a circular pattern near the cylinders and the grid sizes have been optimized. A total number of 1,704, 2,774, 4,045, 6,861, and 8,681 cells were studied while the time-step was kept constant. The area-averaged Nusselt number of both cylinders was used as an indicator for comparing numerical accuracy among the cases, and its values are reported in Table 1. It is found that the maximum deviation between the two extreme cases is less than 0.15%. When the total number of cells is greater than 6,861, further improvement on mesh resolution would not result in desirable gain in accuracy and the numerical results become nearly independent of the grid size. Therefore, the trade-off between numerical accuracy and computational cost suggests using intermediate mesh size with a total number of 6,861 cells. When the diameter or position of the cylinders changes, the grid points should be refined to obtain accurate results. In this study, the same procedure was performed for different combinations of geometrical parameters. Table 1. Grid independence study at Ra ¼ 10 4, D ¼ 0.2, and S ¼ 0.2 Number of cells Area-weighted average Nusselt number of the cylinders 1, , , , ,

7 NATURAL CONVECTION FROM CYLINDERS IN A SQUARE ENCLOSURE RESULTS AND DISCUSSION Simulations were carried out for various positions of horizontally aligned cylinders at a wide range of Rayleigh numbers from 10 3 to 10 7 in which the conduction dominated mode is transmitted to the laminar free convection region. The influences of Rayleigh number and the distance of cylinderson the flow field and heat transfer are investigated for all cases. The nondimensional diameter of the cylinders is kept constant (D ¼ 0.2) in all studied cases Unsteady Characteristics of Flow and Heat Transfer There are several ways to recognize the route to chaos, including analyzing time history, phase space trajectories, and power spectral method [23]. In this section, effects of the increasing Rayleigh numbers on the route to chaos were analyzed by using time histories of the area-averaged Nusselt number and phase space of velocity at some sample points. Qualitative view of the long-term results from CFD simulations is summarized in Table 2. Different flow regimes are categorized by steady, unsteady periodic oscillation, and unsteady chaotic oscillation. According to thermal and flow patterns, steady ones are those that thermal and flow fields appear to be constant over time. In contrast, regular fluctuation of unsteady results is referred to as unsteady periodical oscillation and the thermal patterns are labeled as unsteady chaotic oscillation in case of complicated non-regular oscillating flows. Time variations of area-averaged Nusselt number Nu s on cylinders with different values of S are presented in Figure 2 for Rayleigh numbers from 10 3 to Generally, the magnitude of Nu s is the minimum at Ra ¼ 10 3 and the maximum at Ra ¼ In addition, as shown in Figure 2, impulses of Nu s at t ¼ 0 declines in a short time period and reaches steady-state in all cases with Ra < 10 7, except for the case of S ¼ 0.2 where oscillation is observed at Ra Figure 2b indicates that both time interval and magnitude of Nusselt number oscillation at Ra ¼ 10 5 are are larger than those of Ra ¼ The sample points of the cavity domain are selected in order to analyze the unsteady characteristics of the problem. The numerical solutions of velocity at selected sample points were obtained at S ¼ 0.2, and different Rayleigh numbers and phase space results are presented in Figure 3. There is no special reason to select Table 2. Characterization of flow regimes S Ra S S S S 10 4 S S S S 10 5 S UP S S 10 6 S UP S S 10 7 UC UC UC UC S: Steady, UP: Unsteady periodical oscillation, and UC: Unsteady chaotic oscillation.

8 722 F. KARIMI ET AL. Figure 2. Time histories of Nu s over the cylinders at different Rayleigh numbers: (a) S¼ 0.1; (b) S¼ 0.2; (c) S¼ 0.3; and (d) S¼ 0.4. a particular point, since the results will be fluctuated periodically at every point in the entire cavity domain if the results are oscillatory. However, some points near boundaries may not be suitable due to small amplitude. As seen in Figure 2b that the time signal is constant for the area-averaged Nusselt number at Ra ¼ 10 4, and the related phase trajectory of velocity at the sample point (0.29, 0.72) is a path that eventually reaches a point for Ra ¼ 10 4, as shown in Figure 3a. When the Rayleigh number increases further, a periodic oscillation occurs at Ra ¼ 10 6 which leads to a closed loop trajectory path in Figure 3b. However at Ra ¼ 10 7, periodicity is lost and chaotic solution finally appears, as shown in Figure 3c. The time-averaged Nusselt number is predicted over a period of time and is calculated by the following expression. Nu t ¼ 1 T Z tþt t Nu s dt ð10þ

9 NATURAL CONVECTION FROM CYLINDERS IN A SQUARE ENCLOSURE 723 Figure 3. Phase space of velocity at the sample points where S ¼ 0.2. (a) Point (0.29, 0.72), Ra ¼ 10 4 ; (b) point (0.3, 0.7), Ra ¼ 10 6 ; and (c) point (0.28, 0.76), Ra ¼ The time-averaged Nusselt numbers of the cases discussed in the previous subsection are calculated by Eq. (10) and shown in Figure 4 based on the variations of horizontal distance between heated cylinders and Rayleigh number. It should be

10 724 F. KARIMI ET AL. Figure 4. Time-averaged Nusselt number of cylinders. (a) Different horizontal distances between heated cylinders; and (b) various values of Rayleigh numbers. noted that calculation of Nu t for the periodic oscillating cases is done for the interval of one oscillation period from time-varied Nusselt number graphs. In addition, we selected two equal periods of time randomly in the situation of non-regular oscillating items that the maximum relative difference of corresponding time-averaged Nusselt numbers were lower than 3%.

11 NATURAL CONVECTION FROM CYLINDERS IN A SQUARE ENCLOSURE 725 Figure 4a reveals that enlarging the cylinder distance leads to a slightly higher Nusselt number at Rayleigh numbers 10 3 and 10 4, but this trend cannot be simply tracked for higher Rayleigh numbers. Moreover, Nu t decreases at first in some cases and then increases again, as can be seen for the corresponding line of Ra ¼ 10 5 in Figure 4a. Figure 4b illustrates that Nu t ascends smoothly from Ra ¼ 10 3 to 10 4 (maximum about 9%), and then sharply increases to its magnitude, around 5.5, at Ra ¼ It can also be seen that various cylinder distances result in different values of time-averaged Nusselt number at Ra ¼ 10 3 and 10 4, but the magnitude of Nu t is almost the same at higher Rayleigh numbers. This means that at a higher Rayleigh number, such as 10 6 and 10 7, the distance between the cylinders does not affect heat transfer characteristics anymore. The detailed reason will be described in the following subsections Flow Features Considering the evolutions of the area-averaged Nusselt number over the cylinder surfaces, convective flow patterns have significant effects on the overall heat transfer. Following the purpose of discovering the reasons of Nu t variation, in this subsection, flow patterns inside the enclosure are presented by means of streamlines. According to prior discussions, two main aspects of the time-averaged Nusselt numbers hould be considered further: 1) Slight increase from Ra ¼ 10 3 to Ra ¼ 10 4, and 2) a Sharp rising trend from Ra ¼ 10 4 to Ra ¼ The streamlines and isotherms of the cavity containing two horizontal heated cylinders at different distances and Rayleigh numbers are presented in Figures. 5 and 6. Generally, the heated lighter fluid moves upward along the hot surface of the inner cylinders until it reaches the cold top wall. Then, the fluid gradually becomes colder and denser while it moves horizontally outward in contact with the cold top wall. Then, the cooled denser fluid descends along the cold side walls of the cavity. Figure 5a shows that the circulating path of the fluid is restricted between the cylinders at S ¼ 0.1 and there are four symmetrical separated circulating vortices at the surrounding space of the heated surfaces. Accordingly, the fluid can easily pass in the space between the cylinders when S becomes larger. At S ¼ 0.4, the fluid should overcome the viscosity resistance that is caused by the space limitation among the cavity vertical walls and cylinders. At Ra ¼ 10 4, natural convection heat transfer becomes more powerful compared with the case of Ra ¼ 10 3 and vortices start to appear or become stronger in the top of the cavity as can be seen in Figure 5b. When the Rayleigh number increases further to 10 5 and 10 6, the vortices at the top of the cylinders become larger, as shown in Figures 5c and 5d. The core of the vortex moves toward the upper part of the cavity and the flow becomes weaker at the lower part. Meanwhile, the flow at the top of the cylinders becomes stronger, which means that fluid flow is dominated by a buoyancy-driven mechanism. At S > 0.1, two more symmetrical rotating vortices are generated between the top of the cylinders and upper horizontal wall of the cavity. In addition, Figures 5c and 5d indicate that the aforementioned rotating vortices become stronger at Ra ¼ 10 6 compared with those at Ra ¼ 10 5 and they forced two outer rotating vortices to a narrow space parallel to the cavity vertical walls. For all of the above described

12 726 F. KARIMI ET AL. Figure 5. Streamlines for different cylinder distances and Rayleigh numbers. (a)ra¼ 10 3 ;(b) Ra¼ 10 4 ;(c) Ra ¼ 10 5 ;(d) Ra¼ 10 6 ; and (e) Ra¼ S ¼ 0.2 (c) and (d) unsteady. (e) All solutions are unsteady and plotted at s ¼ Rayleigh numbers, all streamlines are nearly symmetrical about the vertical centerline. The streamlines are no longer symmetric at Ra ¼ 10 7, as shown in Figure 5e. Moreover, the top inner circulating vortices become stronger compared with the previous ones. Therefore, outer vortices are forced to a narrow space parallel to the vertical walls of the cavity. It is also evident that the circulating fluid flow entirely concentrates at the top part of the cavity, and the fluid in the lower part is almost stationary.

13 NATURAL CONVECTION FROM CYLINDERS IN A SQUARE ENCLOSURE 727 Figure 6. Isotherms for differents cylinder distances and Rayleigh numbers. (a) Ra¼ 10 3 ;(b) Ra¼ 10 4 ; (c) Ra¼ 10 5 ;(d) Ra¼ 10 6 ; and (e) Ra¼ S ¼ 0.2 (c) and (d) unsteady. (e) All solutions are unsteady and plotted at s ¼ Heat Transfer Results Figure 6 shows the isotherms for different cylinders distance and Rayleigh numbers. It is observed in Figures 6a and 6b that the heat transfer in the enclosure is mainly dominated by the conduction mode at Ra ¼ 10 3 and 10 4 ; the isotherms are all parallel to the hot and cold surfaces which fully comply with the rules of conductive heat transfer. Comparing the isotherms in Figure 6a, it can be understood that the heat transfer is enhanced by making cylinders farther from each other since

14 728 F. KARIMI ET AL. Figure 7. Local Nusselt number distribution over the left horizontal cylinder (left column) and right horizontal cylinder (right column) for different Rayleigh numbers. (a) S ¼ 0.1 and (b) S ¼ 0.3; Results of Ra ¼ 10 7 are plotted at s ¼

15 NATURAL CONVECTION FROM CYLINDERS IN A SQUARE ENCLOSURE 729 thermal boundary layers can cover the entire surface of the cylinders at larger S values. The lower side penetration of isothermal lines between the cylinders become more as Figure 6b presents; so the thermal boundary layers further approaches the hot surfaces. In the upper part of the cavity, on the contrary, isotherms become farther apart from cylinders. These two competing procedures affect the overall heat transfer. As seen in Figure 4a, the effect of lower portion is more powerful than the upper one for all S values, which leads to increasing Nu t. The main difference between isotherms in Figures 6b and 6c is that the buoyant plumes appear at Ra ¼ 10 5, especially when S ¼ 0.2 and higher; every cylinder has its own thermal plume directed upward completely. For S ¼ 0.1, Figures 6c and 6d reveal that a small distance of heated cylinders results in an almost merged buoyant plume which is close to the results of one heated cylinders inside the cavity [10, 12]. In cases of larger distance of heated cylinders, i.e., S ¼ 0.2, 0.3, and 0.4, the hot fluid that encounters the top horizontal wall with low temperature becomes cold and falls down between the plumes. The trend of enhanced heat transfer from the cases of Ra ¼ 10 4 to Ra ¼ 10 5 and 10 6 is evident of changing heat transfer mechanism from conduction to convection. Figure 6e indicates that thermal boundary layers become denser, and results in higher heat transfer rate around the cylinders and the cylinders distance have no influence on thermal plumes, except for the case of S ¼ 0.1. More information of the described features can be obtained from the distributions of local Nusselt number over both cylinders, as shown in Figure 7, where cases are chosen for S ¼ 0.1 and 0.3 and Rayleigh numbers 10 3 to 10 4 and 10 6 to The maximum local Nusselt number occurs at the left point of the left cylinder (u ¼ 180 ) and right point of the right cylinder (u ¼ 0 ) for Ra ¼ 10 3 due to the symmetrical shapes of isotherms. However, increasing Rayleigh number to 10 6 and 10 7 results in more intensive natural convective heat transfer and the position of the largest value of Nu u shifts from side points to lower ones, i.e., 240 < u < 300. According to Figures 7a and 7b, the magnitudes of local Nusselt number of both cylinders for Ra ¼ 10 4 are always more than those of Ra ¼ 10 3, except for some limited portion of the cylinders surface; especially the upper quarter parts of the cylinders that faced each other (left one: 0 < u < 90 and right one: 90 < u < 180 ). The local Nusselt number at Ra ¼ 10 7 for both cylinder distances completely prevail over those of Ra ¼ 10 6 in all surface points. Moreover, the distribution of dimensionless heat transfer coefficient over the surface of the left and right cylinders clearly shows that heat transfer is completely asymmetric at Ra ¼ CONCLUSION Numerical simulation of the unsteady natural convection from two heated cylinders confined in a two-dimensional square enclosure with cold isotherm walls was carried out. For the nondimensional cylinders diameter (d) of 0.2, a Rayleigh number had been varied from 10 3 to 10 7 and the effects of cylinders distance were investigated. Development of convective flow and heat transfer was discussed by analyzing the time variation of area-averaged Nusselt number over the walls of heated cylinders and different heat transfer phases including steady state, periodic oscillation, and chaotic oscillation. Moreover, phase space of velocity at sample

16 730 F. KARIMI ET AL. points for various solution phases was observed. Time-averaged Nusselt numbers of cylinders with different horizontal distance and Rayleigh numbers were also studied. Snapshots of streamlines and isotherms for several cases were discussed. Finally, the distributions of local Nusselt number over both heated cylinders for some specific cylinder distance and Rayleigh numbers were monitored. It is concluded that heat transfer mechanism in a cavity with two heated horizontally aligned cylinders is a complex function of Rayleigh number and cylinder distance. The time-averaged Nusselt number has an ascending trend by enlarging cylinder distance at low Rayleigh numbers (Ra < 10 4 ), in which flow and heat transfer are dominated by a conduction mechanism. Increasing the Rayleigh number to 10 5 and more leads to an almost negligible effect of cylinder space on the time-averaged Nusselt number in the range of the spacing investigated. The flow patterns in the cavity at Ra > 10 5 and S > 0.1 show two more circulating vortices at the upper part of the cavity, and the focus of the fluid flow is at the top middle space between the cylinders at Ra ¼ At Rayleigh numbers more than 10 5, thermal plumes of the cylinders combines with each other at low cylinder space (S ¼ 0.1), but every cylinder has a separated upward plume in cylinder distances more than 0.1. It is also observed that steady state flow and heat transfer undergoes periodical oscillation and ultimately chaotic oscillation in special positions of cylinders (S ¼ 0.2), and the flow patterns are unsteady at Ra ¼ 10 7 in all studied cases. REFERENCES 1. B. Sandnes and J. Rekstad, A Photovoltaic=Thermal (PV=T) Collector with a Polymer Absorber Plate: Experimental Study and Analytic Model, Solar Energy, vol. 72, no. 1, pp , H. Chen and J. Chou, Investigation of Natural-Convection Heat Transfer Coefficient on a Vertical Square Fin of Finned-Tube Heat Exchangers, Int. J. of Heat and Mass Transfer, vol. 49, nos , pp , A. Sharma, C. R. Chen, and N. Lan, Solar-Energy Drying Systems: A Review, Renewable and Sustainable Energy Reviews, vol. 13, nos. 6 7, pp , M. Rahman and I. Mulolani, Convective Diffusive Transport with Chemical Reaction in Natural Convection Flows, Theoretical and Computational Fluid Dynamics, vol. 13, pp , M. A. R. Sharif and T. R. Mohammad, Natural Convection in Cavities with Constant Flux Heating at the Bottom Wall and Isothermal Cooling from the Sidewalls, Int. J. of Thermal Sciences, vol. 44, no. 9, pp , G. Cesini, M. Paroncini, G. Cortella, and M. Manzan, Natural Convection from a Horizontal Cylinder in a Rectangular Cavity, Int. J. of Heat and Mass Transfer, vol.42, pp , D. G. Roychowdhury, S. K. Das, and T. Sundararajan, Numerical Simulation of Natural Convective Heat Transfer and Fluid Flow Around a Heated Cylinder Inside an Enclosure, Int. J. of Heat and Mass Transfer, vol. 38, pp , S. F. Dong and Y. T. Li, Conjugate of Natural Convection and Conduction in a Complicated Enclosure, Int. J. of Heat and Mass Transfer, vol. 47, pp , T. Saitoh, T. Sajiki, and K. Maruhara, Benchmark Solutions to Natural Convection Heat Transfer Problem Around a Horizontal Circular Cylinder, Int. J. of Heat and Mass Transfer, vol. 36, pp , 1993.

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