This work presents a numerical investigation on the natural convection heat transfer in a circular enclosure with an internal cylinder at *Ra*=10^{3}-10^{6} in both conduction and convection dominant regimes. The cross-section of the cylinder is of regular polygon geometry with various numbers of edges, including circle, triangle, square, pentagon and hexagon. The polygon cylinders are positioned at two orientations, i.e., the corner-upward and edge-upward orientations where one of the sharp corners or flat edges faces upward. The simulations are performed using our in-house fourth-order finite difference code which is well validated. Our objective is to explore the effects of cylinder geometry and orientation on the thermal and flow characteristics. The results are presented and analysed by the total equivalent conductivity coefficient for the quantitative assessment of the contribution of fluid circulation, the streamlines and thermal fields for the flow pattern and qualitative evaluation of heat transfer performance, and the distributions of local heat transfer rate on the surfaces of cylinder and enclosure. We also perform the first synergy principle analysis on this physical model to identify how the fluid circulation contributes to the heat transfer and its spatial behaviours. Numerical results reveal that the corner-upward orientation generally exhibits better heat transfer performance by permitting the well development of flow above the cylinder and in the top region of the enclosure.

## I. INTRODUCTION

Natural convection in an enclosure is one of the most fundamental and classical physical problems in the fields of fluid mechanics and heat transfer. The enclosure can be heated and cooled on different walls or in different regions on the same wall, or keeps at a different temperature with respect to that of an internal entity. The fluid is driven by the density variation arising from temperature difference, and circulates within the enclosure and results in heat transfer. Similar physical and geometrical configurations are widely used in engineering applications of civil and industrial circumstances, such as indoor energy management, cooling of electronic equipment or shafts in cavities, lubrication of bearing journals and food processing. The physical problem is significant in researches and engineering in that the fluid flow and heat transfer processes are extensively integrated. The fluid is heated by the hot entities which generates buoyancy and drives the fluid circulating within the enclosure; meanwhile, the fluid circulation provides a pathway of convective heat transfer which alters the temperature distributions within the whole enclosure. The most typical configuration is the natural convection in an enclosure with an internal cylinder. The outer enclosure and internal cylinder are kept at low and high temperatures, respectively; the fluid is heated by the cylinder and moves upward till the top of the enclosure, then cooled by the enclosure surface and circulates downward to the bottom of the enclosure.

The natural convection in an enclosure with an internal circular cylinder has been extensively investigated during the past decades and a large number of publications are archived. In the classical work by Kuehn & Goldstein,^{1} the steady-state natural convection in a concentric annulus consist a circular enclosure and an internal circular cylinder has been extensively studied by numerical and experimental approaches for water and air in a wide range of Rayleigh number. They provided the high-fidelity benchmark results for the mean and radial/circumferential distributions of thermal quantities, and good agreement is achieved between the experimental and numerical results. This physical problem was also studied by Liang et al.^{2} using the particle method (moving particle semi-implicit method), and the distributions of characteristic quantities are in good consistence with the experimental results. The geometry of a square enclosure was considered in the work of Kim et al.^{3} in which the effect of vertical position of the cylinder within the enclosure was investigated. The authors observed a multi-cellular flow pattern with up to four vortices depending on the position of the cylinder, and the distribution of local Nusselt number on the surfaces of the cylinder and enclosure are consequently influenced.

Except for the circular geometry of the internal cylinder, natural convection in an enclosure with an internal cylinder of various geometries (other than circular) is also investigated considering its wide applications in engineering circumstances such as heat exchangers, steel-making processes and chemical mixing chambers. The thermal and flow characteristics are greatly determined by the irregular geometry of the cylinder. Ghasemi et al.^{4} considered the cylinder of elliptic geometry to study the effects of Rayleigh number, the inclination and size of the cylinder on the natural convection within the enclosure; the size of the cylinder determines the dominant heat transfer regime by affecting the fluid circulation within the enclosure, and consequently changes the strength of heat transfer between the cylinder and enclosure. Zhang et al.^{5} experimentally explored the natural convection in a circular enclosure where an internal octagonal cylinder with or without slots is placed for Rayleigh number up to around *Ra*=10^{6} using the smoke visualization technique; the correlation of equivalent conductivity coefficient with respect to Rayleigh number was proposed. It was found that the pattern of flow between the cylinder and enclosure is quite similar to that in a circular annulus, while the heat transfer intensity of the slotted cylinder case is higher than that of the unslotted cylinder case due to the fluid stream in and out of the cylinder. The similar geometry of a finite thickness circular ring with slots is studied by Zhang et al.^{6} to explore how the Rayleigh number determines the flow characteristics. Several flow regimes are identified for this specific geometry depending on the Rayleigh number; the flow is steady for *Ra*<4×10^{5}, asymmetric periodic for *Ra*>6×10^{5} and quasi-periodic at *Ra*=6×10^{6}, while the periodicity is lost at even higher *Ra* and the flow is fully non-periodic oscillatory. Sheikholeslami et al.^{7} investigated the geometry of a cylinder with a sinusoidal surface and studied the effect of geometrical parameters, i.e., amplitude and number of undulations of the surface profile, on the thermal and flow characteristics up to *Ra*=10^{6}. The maximum local Nusselt number is observed at the crest of the cylinder and the position of thermal plume on the enclosure; the mean heat transfer rate reaches maximum at the number of undulation *N*=4 and the minimum is observed at *N*=5 for this specific configuration. A similar configuration is studied by Nabavizadeh et al.^{8} to reveal the effects inclination angle, amplitude and number of undulation on the natural convection in a square enclosure. It was found that the three parameters have profound influences on the thermal and flow behaviours. The increase of number of undulation prohibits the fluid circulation in the space between gaps, thus the overall heat transfer rate decreases; however, the increase of amplitude dramatically increases the effective surface area and intensifies the heat transfer.

For triangular cylinder within a circular enclosure, Xu et al.^{9} performed numerical simulations for Rayleigh number up to *Ra*=10^{6} to study the effects of the size and inclination angle of the cylinder on the heat transfer performance. The results revealed that the inclination has negligible effect on the mean heat transfer rate. Multicellular structure appears above the cylinder when one of the flat edges is roughly facing upward, which results in a secondary thermal plume. The same physical configuration is analysed by Yu et al.^{10} to identify the effect of Prandtl number for *Ra*≤10^{6}. Three regimes of mean heat transfer rate are observed as divided by *Pr*=0.7 and *Pr*=7, and the mean heat transfer rate is only slightly affected by the Prandtl number for *Pr*>0.7; the correlations between the mean heat transfer rate and Rayleigh number were also given. The natural convection in a circular enclosure with an internal cylinder of various cross-section geometries was numerically studied by Yuan et al.^{11} by considering radiation heat transfer; the geometry of the cylinder can be cylindrical, elliptical, square and triangular. They found that the radiation improves the local heat transfer rate particularly for high wall temperature condition; the heat transfer on the flat edges of the triangular and square cylinders is enhanced compared with that of the circular cylinder. For a hexagonal cylinder in a square enclosure, El Moutaouakil et al.^{12} concluded that the flow intensity is determined by the size of the cylinder and the Rayleigh number; for a large cylinder, the flow structure and local heat transfer characteristics are greatly affected by the orientation of the cylinder. At high Rayleigh numbers, the overall heat transfer rate is larger when one flat edge of the cylinder faces upward. There are also numerical works on natural convection in an enclosure with multiple cylinders of various geometries^{13–25} and on nanofluid.^{26–37}

It is summarized from the above reviewed literatures that for natural convection of a cylinder of non-circular geometry placed in an enclosure, the thermal and flow characteristics are greatly affected by the cylinder geometry. The overall heat transfer performance of the cylinder-enclosure system is dependent on the position of the cylinder especially in the vertical direction; the local heat transfer rate on the solid walls is determined by the pattern of local flow, i.e., whether the flow is unconfined or locally confined in a small space. In realistic engineering applications, the internal cylinder is normally produced in simple geometries such as circular, triangular or square to reduce the costs of design, manufacture and maintenance and facilitate standardization. However, to the authors’ knowledge, most of existing works primarily studied the natural convection in an enclosure with an internal cylinder of circular geometry, while only quite a few works considered the internal cylinder of non-circular geometry such as those reviewed above. The non-circular cylinders are actually widely used in industrial and civil engineering to meet the specific requirements such as space saving, strength enhancement and vibration control, thus the thermal and flow characteristics for the related configurations should be well analysed and understood. The present study performs a comprehensive numerical investigation on the natural convection in a circular enclosure with an internal cylinder of regular polygon geometry, i.e., triangle, square, pentagon and hexagon. For each of the polygon cylinder, two types of orientation are considered: the corner-upward orientation where one of the sharp corners of the polygon faces upward, and edge-upward orientation where one of the flat edges faces upward and is aligned and opposite to the gravitational direction. The objective is to explore the effects of orientation and geometry (number of edges) of the cylinder on the characteristics of fluid circulation within the enclosure and local heat transfer on the solid walls. The simulations are performed by our in-house fourth-order finite difference code which has been well validated. The results are presented and analysed by the variations of characteristic quantities with the orientation and geometry of the cylinder, including the equivalent conductivity coefficient, temperature and streamfunction fields, and distributions of local heat transfer rate and local velocity in different regions within the enclosure. We also carried out the synergy principle analysis to identify the regions where the convection contributes the most and least significantly on the convective heat transfer.

## II. NUMERICAL DETAILS

### A. Physical model

The geometrical and physical configurations of the problem investigated in this work are schematically given in Fig. 1. The concentric annulus consist a circular enclosure and an internal cylinder of various geometries as regular polygon, i.e., triangle, square, pentagon and hexagon, and the number of edges is denoted as *N*. The circumcircle of the polygon cylinder is represented by the dashed circle in the figure, whose size is fixed at *D*_{i}/*D*_{o}=0.4 in the present study to permit the well circulation of fluid within the enclosure. Since the natural convection is affected by the orientation of the polygon cylinder with respect to the gravitational direction, here we consider two orientations, i.e., the corner-upward orientation where one sharp corner faces upward opposite to the gravitational direction, and the edge-upward orientation in which the top edge is horizontally placed, as respectively shown in Fig. 1(b) and (c). The internal cylinder of circular geometry is also studied for comparison to reveal the effect of number of edges. Both the enclosure and internal cylinder are assumed sufficiently long in the axial direction (perpendicular to the paper) that the end effect leading to three-dimensional flow is ignored. The solid walls are considered isothermal, respectively denoted by the lower temperature *T*_{o} for the enclosure and higher temperature *T*_{i} for the cylinder; this approximation is consistent with the realistic condition that the thermal conductivity of solid material is orders of magnitude that of fluid. The annulus is filled with air (*Pr*=0.71) whose thermophysical properties are constants except for the density variation that follows the Boussinesq approximation; the effects of radiation and viscous dissipation are not considered here. The Rayleigh number under investigation varies in the range *Ra*=10^{3}-10^{6} to cover both the conduction and convection dominant heat transfer regimes.

### B. Numerical methods

The incompressible flow and heat transfer in the annulus follow the two-dimensional continuity, momentum and energy equations in non-dimensional form:^{38}

where *Pr* is the Prandtl number and $Ra=g\beta Ti\u2212ToDo3/\upsilon 2$ is the Rayleigh number. The variables are non-dimensionalized by the reference length *D*_{o}, velocity *u*_{ref} =(*a*/*D*_{o})(*PrRa*)^{1/2}, pressure $pref=\rho uref2$ and time *D*_{o}/*u*_{ref}. The temperature is scaled as *T*=(*T**-*T*_{o})/(*T*_{i}-*T*_{o}) in which *T** is the dimensional temperature. The viscous dissipation term is neglected since it is small in most engineering applications.^{38} In the simulations, no-slip condition is prescribed on the solid walls, and the temperature is *T*=1 for the cylinder surface and *T*=0 for the enclosure surface. The initial solution for the fluid is **u**=0 and *T*=0.

The governing equations (1)–(3) are discretized by an energy conservative fourth-order scheme^{39} on an O-type grid generated in the fluid domain. The discretized equations are solved by our in-house fourth-order finite-difference code based on a semi-implicit fractional step method.^{40} The steady-state solution is obtained by advancing in the physical time from the initial solution. The time advancement in one physical time step consists four sub-steps: the predictor step for the velocity from *u*^{n-1} to *u*^{n-1/2}, the solution of pressure *p*^{n}, the corrector step for velocity from *u*^{n-1/2} to *u*^{n}, and the solution of temperature from *T*^{n-1} to *T*^{n}. The differential equations for the four sub-steps are expressed as follows:

Since the fluid flow and heat transfer achieve steady-state for all configurations and Rayleigh numbers considered here, the time step size would not affect the final solution and is chosen to be around 0.005*D*_{o}/*u*_{ref} for all computations, corresponding to CFL<1. The time integration is performed until the solution satisfies the convergence criterion:

### C. Code validation

The in-house code used in this work has been successfully employed and well validated in our earlier works on steady-state natural convection in an enclosure,^{41} transient or unsteady mixed convection in an enclosure^{42,43} and forced convection flow across cylinders.^{44–49} The code is capable in simulating natural, forced and mixed heat transfer for flow in open or enclosed systems. In this work, we validate the code through the physical problem of natural convection in an annulus consist of two concentric circular cylinders at *Ra*=1.71×10^{6}, which has been experimentally and numerically studied by Kuehn & Goldstein^{1} and detailed quantitative benchmark results are provided. It is noted that the Rayleigh number defined in our work is based on the enclosure diameter, while it is defined based on the gap width between the cylinder and enclosure by Kuehn & Goldstein, thus scaling has to be made for consistent comparison. The Rayleigh number given above corresponds to *Ra*_{L}=5×10^{4} in the experiment^{1} where *L* is the gap width. The mean equivalent conductivity coefficient (*k*_{eq}) obtained under various resolutions is given in Table I. Our results are in good agreement with the benchmark experimental results. The coefficient obtained under various resolutions does not show much difference, i.e., a relative difference less than 0.2% between the results obtained at 256×128 and 512×256 grid, thus the 512×256 grid is deemed sufficient for this simulation and is used in all following simulations.

Source . | Method . | Grid . | _{keq,i}
. | _{keq,o}
. |
---|---|---|---|---|

Kuehn & Goldstein^{1} | 2^{nd}-order FD | 16×19 | 3.024 | 2.973 |

Present | 4^{th}-order FD | 128×64 | 2.972 | 2.981 |

256×128 | 2.961 | 2.967 | ||

512×256 | 2.958 | 2.962 |

Source . | Method . | Grid . | _{keq,i}
. | _{keq,o}
. |
---|---|---|---|---|

Kuehn & Goldstein^{1} | 2^{nd}-order FD | 16×19 | 3.024 | 2.973 |

Present | 4^{th}-order FD | 128×64 | 2.972 | 2.981 |

256×128 | 2.961 | 2.967 | ||

512×256 | 2.958 | 2.962 |

## III. RESULTS AND DISCUSSION

### A. Overall heat transfer rate

For natural convection in an enclosure with internal entities, the overall heat transfer rate generally increases with the Rayleigh number because of the enhanced convective heat transfer arising from intensified fluid circulation. Here we use the mean equivalent conductivity coefficient to quantify the effect of fluid circulation in enhancing the heat transfer which is affected by the geometry and orientation of the polygon cylinder. The quantity is defined as the ratio between the overall heat transfer rate of natural convection case and pure conduction case. Since the conductive heat transfer rate is independent of the orientation of the polygon cylinder, the quantity also reveals the comparative intensity of heat transfer between edge-upward and corner-upward orientations.

Fig. 2 gives the variation of *k*_{eq} with the Rayleigh number for all geometries and orientations of the cylinder. It is noted that *k*_{eq} monotonically increases with the Rayleigh number in the present *Ra*-regime because of the intensified fluid circulation. Based on the magnitude of *k*_{eq}, the convective heat transfer can be roughly categorized into two regimes for all cases investigated. The first is the conduction dominant heat transfer regime roughly at *Ra*≤2×10^{4} as characterized by *k*_{eq}≈1; the fluid circulation is weak and provides only tiny or even negligible contribution to the overall heat transfer performance in addition to conduction. The second is convection dominant heat transfer regime at *Ra*≥5×10^{4} that the contribution from convective flow is pronounced as represented by the substantially growing heat transfer rate. For flow in this regime, we correlate the magnitude of *k*_{eq} with *Ra* roughly as:

as shown in the figure for all cases in the *Ra*-regime considered in this work.

Since *k*_{eq} varies significantly with the geometry and orientation of the cylinder especially in the convection dominant regime, the quantitative results are comprehensively listed in Table II to further investigate the effects of the two factors. For the conduction dominant flow, the magnitude of *k*_{eq} is normally equal to or slightly higher than 1.000. The largest *k*_{eq} is always observed for the corner-upward triangular cylinder (T1) among all cases which is up to 1.120 at *Ra*=2×10^{4}, reflecting a 12% additional heat transfer rate contributed by the weak fluid circulation. The largest *k*_{eq} for the T1 cylinder is attributed to two reasons; the first is the largest fluid domain compared with other geometries which permits the fully development of fluid circulation; the second is that the upward moving fluid around the cylinder is not obstructed by the cylinder surface (e.g., the T2 configuration), thus the circulation is smooth and strong. For the convection dominant flow, the magnitude of *k*_{eq} is always the smallest for the circular cylinder which is attributed to its largest size among all geometries that the fluid is confined within the smallest fluid domain that the circulation is prohibited. For the polygon cylinders, it is obvious that the magnitude of *k*_{eq} is always larger for the corner-upward orientation (T1, S1, P1 and H1) than that of the edge-upward counterpart although the difference is only several percents or even smaller, which is consistent with the conclusions of Xu et al.^{9} for the triangular cylinder. Considering that the influence from spatial resolution is in the order of 0.1%, the difference of *k*_{eq} should not be attributed to the resolution but the intensity of fluid circulation. Comparing the results for all cases, it is found that the difference of *k*_{eq} between the corner-upward and edge-upward orientations is always the largest for the triangular cylinder and the smallest for the hexagon cylinder for all Rayleigh numbers, i.e., the difference monotonically reduces with *N*. Moreover, the difference of *k*_{eq} between the results of two orientations experiences a drastic reduction as *N* increases; the difference is normally 3%-6% for the triangular and square cylinders, while it reduces to a maximum value of 1% for the pentagon and hexagon cylinders which will be discussed in the following sections.

Ra
. | C . | T1 . | T2 . | S1 . | S2 . | P1 . | P2 . | H1 . | H2 . |
---|---|---|---|---|---|---|---|---|---|

10^{3} | 1.000 | 1.000 | 1.000 | 1.001 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

2×10^{3} | 1.000 | 1.000 | 1.000 | 1.001 | 1.001 | 1.000 | 1.001 | 1.000 | 1.000 |

5×10^{3} | 1.002 | 1.007 | 1.006 | 1.005 | 1.005 | 1.003 | 1.004 | 1.002 | 1.002 |

10^{4} | 1.007 | 1.032 | 1.025 | 1.017 | 1.016 | 1.012 | 1.012 | 1.010 | 1.010 |

2×10^{4} | 1.025 | 1.120 | 1.086 | 1.064 | 1.058 | 1.045 | 1.044 | 1.037 | 1.037 |

5×10^{4} | 1.132 | 1.397 | 1.312 | 1.276 | 1.236 | 1.211 | 1.202 | 1.183 | 1.180 |

10^{5} | 1.343 | 1.693 | 1.599 | 1.561 | 1.486 | 1.469 | 1.451 | 1.427 | 1.423 |

2×10^{5} | 1.650 | 2.031 | 1.921 | 1.899 | 1.797 | 1.785 | 1.775 | 1.743 | 1.743 |

5×10^{5} | 2.131 | 2.528 | 2.387 | 2.400 | 2.264 | 2.270 | 2.262 | 2.228 | 2.226 |

10^{6} | 2.547 | 2.940 | 2.798 | 2.830 | 2.661 | 2.693 | 2.668 | 2.649 | 2.634 |

Ra
. | C . | T1 . | T2 . | S1 . | S2 . | P1 . | P2 . | H1 . | H2 . |
---|---|---|---|---|---|---|---|---|---|

10^{3} | 1.000 | 1.000 | 1.000 | 1.001 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

2×10^{3} | 1.000 | 1.000 | 1.000 | 1.001 | 1.001 | 1.000 | 1.001 | 1.000 | 1.000 |

5×10^{3} | 1.002 | 1.007 | 1.006 | 1.005 | 1.005 | 1.003 | 1.004 | 1.002 | 1.002 |

10^{4} | 1.007 | 1.032 | 1.025 | 1.017 | 1.016 | 1.012 | 1.012 | 1.010 | 1.010 |

2×10^{4} | 1.025 | 1.120 | 1.086 | 1.064 | 1.058 | 1.045 | 1.044 | 1.037 | 1.037 |

5×10^{4} | 1.132 | 1.397 | 1.312 | 1.276 | 1.236 | 1.211 | 1.202 | 1.183 | 1.180 |

10^{5} | 1.343 | 1.693 | 1.599 | 1.561 | 1.486 | 1.469 | 1.451 | 1.427 | 1.423 |

2×10^{5} | 1.650 | 2.031 | 1.921 | 1.899 | 1.797 | 1.785 | 1.775 | 1.743 | 1.743 |

5×10^{5} | 2.131 | 2.528 | 2.387 | 2.400 | 2.264 | 2.270 | 2.262 | 2.228 | 2.226 |

10^{6} | 2.547 | 2.940 | 2.798 | 2.830 | 2.661 | 2.693 | 2.668 | 2.649 | 2.634 |

### B. Thermal and flow fields

The effects of geometry and orientation of the cylinder on the thermal and flow characteristics are presented and analysed through the spatial distributions of isotherms and streamlines.

Fig. 3 shows the distributions for the circular cylinder case at various Rayleigh numbers in which the left and right halves of each subfigure present the isolines of streamfunction and temperature, respectively, owing to the symmetric distribution pattern about the vertical centerline (*x*=0). There is always one circulating vortex to both left and right sides of the cylinder whose center moves upward as *Ra* increases, especially from *Ra*=10^{4} to 10^{6} where convection gets pronounced. The intensified fluid circulation is also revealed by the increasing magnitude of streamfunction at the center of the vortex, as denoted in the figure. At *Ra*=10^{3} and 10^{4}, the isotherms present the structure of concentric circles which are parallel with the solid walls, reflecting the conduction dominant heat transfer that the fluid is hardly driven by buoyancy; the isotherms are more clustered around the cylinder due to the higher local heat transfer rate which guarantees the same over heat transfer rate on both entities. At *Ra*=10^{5} and 10^{6}, the fluid circulation is stronger which enhances the heat transfer, thus the isotherms are non-parallel and cluster close to the top of the enclosure and bottom of the cylinder where the local heat transfer is the most pronounced. The isotherms to the left and right sides of the cylinder are almost horizontal at *Ra*=10^{6}, i.e., a typical structure of thermal stratification as the convective heat transfer is significant.

For the polygon cylinders, Fig. 4 gives the structures of isotherms and streamlines at *Ra*=10^{5} and 10^{6} where convection is pronounced. Depending on the geometry, the position of the circulating vortex varies significantly as the flow is partially divided by the sharp corners and squeezed by the flat edges of the cylinder, thus fluid circulation is obstructed. At *Ra*=10^{5}, the vortex center is always above the horizontal centerline (*y*=0) for the corner-upward orientation although it is quite close to the centerline for the pentagon and hexagon geometries because of the short edge length and relatively blunt corners. However, for the edge-upward orientation, the center is around or even below (T2) the horizontal centerline because of the squeezing from the cylinder surface. At a higher Rayleigh number *Ra*=10^{6} where convection intensity is more pronounced, the vortex is stretched approximately in the vertical direction; the vortex center moves upward compared with the cases at *Ra*=10^{5} and is above the horizontal centerline for all cases. Although squeezing from the cylinder edge still exists, the fluid is strongly driven by the buoyancy, and the circulating pattern, in terms of shape and position of the vortex, is less affected by the geometry and orientation of the cylinder compared with the *Ra*=10^{5} cases. It is worth noted that because of the intensified buoyancy which drives the fluid circulating at a higher velocity, the attached flow on the cylinder surface separates at the top corners and forms separation bubbles for the T2, S2 and P2 cylinders, as represented by the isolines of streamfunction with negative magnitude (dashed lines). As *N* increases, the separation bubble rapidly reduces in size and weakens in intensity because of the gradually flattening surface that the discontinuity of curvature is reduced. The separation bubble is not visible for the H2 and all corner-upward cylinders due to the small deflection angle of the flow direction at the sharp corners that the fluid can move smoothly around the corners. Compared with the streamlines, the distribution of isotherms is less affected by the cylinder geometry and orientation. The isotherms cluster at the top of the enclosure and bottom of the cylinder for the corner-upward orientation because of the thermal plume structure, and it is less noticeable as *N* increases.

The formation of the circulating vortex is largely dependent on the geometry and orientation of the cylinder. To further quantitatively evaluate the effects of the two factors on the internal flow, Fig. 5 presents the position of the left circulating vortex with increasing Rayleigh number for all cases in which the position of the vortex center is identified by the local maximum magnitude of streamfunction. The position of the vortex is obviously affected by the cylinder geometry since it is squeezed by the flat edges and partially divided by the corners. It is seen that at a fixed Rayleigh number, the circulating vortex forms in the leftmost region for the circular geometry and the rightmost for the corner-upward triangular geometry (T1). As *N* increases, the center gradually moves away from the cylinder, i.e., in the negative-*x* direction, and gets more close to the enclosure for the edge-upward orientation compared with the corner-upward one. For the circular and most polygon cylinders (T1, S1, S2, P2, H1 and H2), the circulating vortex moves toward the top-right direction with increasing *Ra*, i.e., the center is getting close to the vertical centerline above the cylinder, although the motion can be quite slight in the *x*-direction that the vortex center is almost only lifted up with *Ra* (i.e., S2). Different variation pattern is observed for the T2 and P1 cases that the circulating vortex first moves toward upper-left and then upper-right direction with *Ra* due to the squeezing from the cylinder at the sharp corner. This variation pattern is especially obvious for the T2 configuration since the flat edge of the cylinder is long that the vortex is continuously squeezed as *Ra* increases. The Rayleigh number also greatly affects the formation of circulating vortex. At *Ra*=10^{3}, the vortex forms around *y*=0.0 for the circular, S2 and two hexagon cylinders, at *y*>0.0 for the T1, S1 and P2 cylinders and at *y*<0.0 for the T2 and P1 cylinders. As *Ra* increases, the circulating vortex moves more significantly in space for the corner-upward configurations where the center reaches the position beyond *y*=0.2 at the highest Rayleigh number *Ra*=10^{6}.

### C. Radial distribution of velocity

The heat transfer between the cold enclosure and internal hot cylinder is enhanced by convection through fluid circulation. Since the fluid moves roughly along the *y*-direction as heated by the cylinder and cooled by the enclosure, the *v*-velocity can be used as a good indication of the convection intensity. At *Ra*=10^{6} where convection is pronounced, Fig. 6 gives the distribution of *v*-velocity along the radial direction at several circumferential positions for all geometries and orientations.

The curves at *θ*=0° represent the flow right above the cylinder where the thermal plume structure forms and the fluid moves exactly along the *y*-direction (zero *u*-velocity). For the corner-upward configurations, the magnitude of *v*-velocity is always larger than that of the edge-upward ones which supports the higher heat transfer rate listed in Table I. The magnitude is also larger that of the circular cylinder; considering that the surface area of the circular cylinder is larger than any of the polygon cylinders, this observation reflects that the fluid can be more effectively heated by a polygon positioned in a corner-upward orientation. As *N* increases, the magnitude monotonically decreases due to the reduced fluid domain size in the upper half of the enclosure that the fluid circulation is constrained; the radial position for the maximum magnitude is getting away from the cylinder because the flow is less sufficiently accelerated as it moves along the cylinder surface. For the edge-upward configurations, the magnitude of *v*-velocity is equivalent or smaller than that of the circular cylinder case, and monotonically increases with *N* as a result of the reduced surface area of the top edge of the cylinder where the constraint on the fluid circulation is weakened. It is also noted that the *v*-velocity is negligible in magnitude or even slightly negative in the region adjacent to the cylinder because of the formation of separation bubbles on the top edge (refer Fig. 4). Although the negative *v*-velocity is quite small in magnitude, it is the most visible on the curve of the T2 cylinder but recovers as *N* increases, and we don’t observe the negative *v*-velocity for the H2 cylinder.

The curves for the radial position of *θ*=30° exhibit more complex distribution patterns since the fluid also moves in the *x*-direction and is obstructed by the edges or corners of the cylinder. In the region adjacent to the cylinder where the fluid is heated, the magnitude of *v*-velocity is always the highest, and the difference among the several configurations is notable compare with the region close to the enclosure surface. For the corner-upward orientation, the radial position for the maximum magnitude of *v*-velocity monotonically moves toward the enclosure as *N* increases since the fluid becomes gradually confined by the cylinder. The magnitude is generally the largest for the T1 configuration, while the remained three corner-upward configurations do not have much difference. In the region close to the enclosure, the several curves are almost parallel with each other due to the circulating flow along the enclosure surface; the maximum magnitude monotonically decreases with *N* as the flow weakly circulates as confined by the larger cylinder, in which the large negative *v*-velocity for the T1 configuration also reflects the strong circulation. The velocity magnitude shows only slight difference between the P1 and H1 configurations but is much deviated from that of the circular cylinder case. For the edge-upward orientation, the variation of velocity magnitude with *N* is opposite in trend with that of the corner-upward orientation, i.e., the magnitude monotonically increases with *N* in the region adjacent to the cylinder but decreases close to the enclosure. This is mainly attributed to the increased *N* where the fluid circulation is restricted by a shorter edge around the cylinder that the convection intensity is stronger.

The curves for the radial position of *θ*=60°exhibit different variation patterns. The maximum magnitude of velocity slightly decreases from T1 to S1 but substantially increases as *N* further increases for the corner-upward cylinders. However, it monotonically reduces with *N* for the edge-upward cylinders which is contrary to the variation tendency of curves at *θ*=0° and 30°, while the radial position for the maximum magnitude slightly varies within a small scope *R*/*D*_{o}=[0.20, 0.22]. The magnitude in the region close to the enclosure surface is much larger than that of the *θ*=30° case (strong negative *v*-velocity) and is seldom affected neither by the orientation nor *N*. For curves at the horizontal centerline (*θ*=90°), the velocity distributions are greatly affected by *N* and cylinder orientation since one of the sharp corners may right at this circumferential position, thus the radial position for the maximum magnitude differ a lot depending whether *N* is odd or even. For the corner-upward geometries, it is found that the maximum magnitude does not monotonically vary with *N*, but is the largest for the square (S1) cylinder since its sharp corner is right at the quadrant that the size of gap between the cylinder and enclosure is minimum. The maximum magnitude increases with *N* for the edge-upward cylinders, and the corresponding radial position gradually moves away from the cylinder due to the geometry variation. The curves at *θ*=120° exhibit similar variation patterns. The velocity magnitude for flow below the cylinder (*θ*=150°) is much smaller than those at other circumferential positions. Both the maximum magnitude and its radial position vary in a non-monotonic way with *N*. For the corner-upward geometries, the magnitude is the largest for the S1 cylinder but is the smallest for the T1 cylinder since the bottom edge prohibits the circulation of local flow; however, reverse variation tendency is observed for the edge-upward geometries that the magnitude is the largest for the T2 cylinder but smallest for the S2 cylinder. Since the circulation intensity adjacent to the enclosure surface is weak and the fluid domain size at the *θ*=150° position is large, the magnitude of local recirculating flow is negligible compared with the curves at other circumferential positions.

It is concluded here that both the orientation and number of edges of the cylinder greatly affect the velocity distribution at various circumferential positions. The effects are more pronounced for flow in the region close to the cylinder that the curves for the *v*-velocity distribution vary significantly with the orientation and *N*. The maximum magnitude of the velocity and its radial position does not necessarily vary with the two factors in a monotonic way, but depend on the circumferential position that if the fluid flow is obstructed by the cylinder surface and especially the sharp corners.

### D. Characteristics of local heat transfer

Although the overall heat transfer between the cylinder and enclosure may not be significantly affected by the geometry and orientation of the cylinder (refer Table I), the characteristics of local heat transfer, including the magnitude and distribution on the solid wall and variation with the Rayleigh number, could be potentially affected because the flow is severely obstructed by the flat edges and sharp corners of the cylinder.

Fig. 7 gives the circumferential distribution of local Nusselt number, defined as the temperature gradient in the fluid domain, on the surfaces of the circular cylinder and enclosure. The distributions are given over the right half of the geometry from top (*θ*=0°) to bottom (*θ*=180°) considering the symmetry of the configuration. For heat transfer from fluid to solid wall, the magnitude is regarded positive for the cylinder but negative for the enclosure. Generally, the magnitude of Nusselt number experiences a monotonic variation with the circumferential angle on both solid walls. It is the highest at the bottom of the cylinder and the top of the enclosure where the temperature difference between fluid and solid is the largest, and is the lowest at the top of the cylinder and bottom of the enclosure where the temperature difference is the smallest. The variation amplitude of the curve, i.e., the difference between the maximum and minimum magnitude, increases with the Rayleigh number. At *Ra*=10^{4} where heat transfer is conduction dominant, there is almost no Nusselt number variation on the surfaces of both the cylinder and enclosure since the fluid circulation is rather weak. The variation amplitude increases at *Ra*=10^{5} because of the intensified natural convection; the heat transfer rate at the cylinder bottom is roughly five times that of the cylinder top, while the difference is relatively smaller for the enclosure due to its larger surface area. At *Ra*=10^{6} where the heat transfer is strongly convection dominant, the circumferential variation of Nusselt number is remarkable. The magnitude of Nusselt number monotonically increases from the top to bottom of the cylinder and keeps nearly unchanged roughly for *θ*>120° where the cold fluid impinges on the cylinder and temperature difference is almost constant, and it is the highest at the top of the enclosure where the variation is the most substantial due to the impingement of hot fluid to the local enclosure surface.

For the polygon cylinders, the circumferential distribution of local Nusselt number is presented in Fig. 8. It is noticed that for all cylinders, there is a drastic increase of local Nusselt number at all sharp corners where the magnitude is can be one order higher than that of the flat edge. However, the position with largest Nusselt number does not necessarily occur right at the sharp corners, as exemplified by the curves for the triangle and pentagon cylinders where the maximum magnitude appears on the lateral edges very close to the corner for the corner-upward cylinder and top edge for the edge-upward cylinder. Another noticeable observation is that although the heat transfer rate is relatively larger at higher Rayleigh number, the magnitude is quite dependent on the local geometrical property especially at the corners. The Nusselt number at the top of the cylinder (*θ*=0°) experiences a non-monotonic variation pattern with the Rayleigh number; the local Nusselt number is the highest and lowest at *Ra*=10^{4} and *Ra*=10^{5}, respectively, while the magnitude for *Ra*=10^{6} is medium. For the enclosure, the local Nusselt number is only slightly affected by *N* and orientation in terms of its magnitude, while the distribution pattern and variation with Rayleigh number is the same as that of the circular cylinder case. The magnitude of local Nusselt number is almost constant from the top to the bottom of the enclosure for *Ra*=10^{4} and 10^{5}, and the maximum magnitude at the top of the enclosure does not vary much with *N* and orientation. However, at *Ra*=10^{6} where convection is pronounced, the magnitude of local Nusselt number is relatively larger for the corner-upward cylinder case due to stronger thermal plume structure (see Fig. 4), and the difference is getting smaller as *N* increases, while the magnitude at the top of the enclosure monotonically increases with *N* for the edge-upward cylinders.

### E. Synergy principle analysis

The convection heat transfer within an enclosure is more substantially enhanced at high Rayleigh numbers as discussed above, while the intensity of local heat transfer on the solid walls and in the fluid domain is determined by the pattern of local alignment between the velocity and temperature fields. It is anticipated that the heat transfer is the most enhanced as the velocity vector is aligned with the temperature gradient vector where the fluid circulation could effectively mix the hot and cold fluid and produce the effective heat transfer. Guo et al.^{50} proposed the field synergy (coordination) principle to assess the enhancement by convection heat transfer for single phase flow. The local heat transfer rate contributed by convection can be reflected by the dot product of velocity vector and temperature gradient vector in the form:

where *β* implies the synergy angle between the velocity vector and temperature gradient vector. The field synergy principle states that the velocity and temperature gradient vectors should be synergized with a small included angle to enhance the local heat transfer.

Figure 9 gives the spatial distribution of the synergy coefficient -cos*β* for the convection dominant heat transfer cases, in which -cos*β*=±1.0 represents the alignment of the two vectors that convection is the most effective, and -cos*β*=0.0 for two perpendicular vectors that heat transfer is only realized by conduction and fluid circulation has no contribution. The synergy coefficient is zero at the solid surfaces since the velocity and temperature gradient vectors are always parallel with and perpendicular to the local wall, respectively, thus the heat transfer is realized only by conduction. The synergy coefficient is pronounced right above the cylinder and in the central region of the fluid domain. It is concluded from Fig. 3 and Fig. 4 that in a narrow region above the cylinder, the velocity vector is nearly perfectly aligned and in opposite direction with the temperature gradient, thus the upward motion of the hot fluid from the cylinder surface could effectively heat the fluid in this region. In the central region to the left and right of the cylinder, the velocity vector is aligned with either the same or opposite direction with the temperature gradient, thus heat transfer might be locally significant. It is also noticed that the value of -cos*β* is around -1.0 to the cylinder bottom since the velocity vector is in the same direction with the temperature gradient; however, the magnitude of temperature gradient is only pronounced very close to the cylinder (see the clustered isotherms in Fig. 3 and Fig. 4), the low magnitude of temperature gradient degrades the heat transfer enhancement.^{50}

The distribution of the synergy coefficient at the two Rayleigh numbers is similar in that the magnitude is high in the central region of the enclosure to the left and right of the cylinder; the area of the region for high coefficient is relatively larger at *Ra*=10^{6} that convection effect is significant and contributes more to the heat transfer process. However, there is a zigzag structure for the -cos*β*→1.0 and -cos*β*→-1.0 isolines that the velocity vector can be opposite in direction with the temperature gradient (red lines) even below the horizontal centerline of the enclosure, and the two vectors can be aligned in direction in the upper half of the enclosure (blue dashed lines), which is not observed for the *Ra*=10^{5} cases. Compared with the Rayleigh number, it is noted that the orientation and *N* do not notably affect the coefficient distribution. The difference is mainly clearly observed around the sharp corners of the cylinder especially when there are recirculating bubbles; the convection heat transfer is weakened by the quasi-stationary local fluid flow, and heat transfer is also prohibited on the flat edge of the cylinder.

## IV. CONCLUSIONS

This work performs numerical investigations on the natural convection in a circular enclosure with an inner cylinder of various regular polygon geometries (circle, triangle, square, pentagon and hexagon) at *Ra*=10^{3}-10^{6}. The orientation of the polygon cylinder can be corner-upward or edge-upward with one sharp corner or flat edge faces upward opposite to the gravitational direction. The objective is to explore the effects of orientation and geometry of the cylinder on the thermal and flow characteristics. The main conclusions are as follows:

The heat transfer within the enclosure is mainly dominated by conduction roughly at

*Ra*≤2×10^{4}where*k*_{eq}is only slightly above 1.0, reflecting that the fluid circulation does not contribute much to the heat transfer. The flow is convection dominant for*Ra*>5×10^{4}where*k*_{eq}exponentially grows with the Rayleigh number, and the correlation with*Ra*is proposed.The geometry and orientation of the cylinder determines the flow pattern. For the corner-upward orientation, the attached flow at the cylinder surface is permitted to move upward without constraint from the cylinder edge. However, for the edge-upward orientation, the attached flow subjects a deflection at the corners of the top edge and separates from the surface; the magnitude of

*v*-velocity is lowered in the region close to the top of the cylinder, thus the overall heat transfer rate is relatively smaller than that of the corner-upward orientation.The geometry and orientation of the cylinder do not much affect the distributions of local heat transfer rate at the solid surfaces. The local Nusselt number experiences a drastic increase at or adjacent to the sharp corners for all configurations, while the maximum Nusselt number is not necessarily at the corner. The corner-upward cylinders results in a larger local Nusselt number at the top of the enclosure due to the intensified fluid circulation.

The synergy principle analysis reveals that the fluid circulation intensifies the heat transfer the most significantly in the narrow regions above and below the cylinder, while the contribution is relatively low to the left and right sides of the cylinder which is roughly within the center of circulating vortices.

## ACKNOWLEDGMENTS

This research was funded by Natural Science Foundation of China (51706205) and Joint Project from Natural Science Foundation of China and Liaoning Province (U1608258). W.Z. appreciates the start-up funding provided by Zhejiang Sci-Tech University and Qianjiang Talent Program of Zhejiang Province.

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