Comparison of Natural Convection around a Circular Cylinder with a Square Cylinder Inside a Square Enclosure

Numerical calcu lations are carried out for natural convection induced by a temperature difference between a cold outer square enclosure and a hot inner cy linder with two different geometries (i.e. circular and square). A two-dimensional solution for natural convection is obtained, using the finite volume method for different Rayleigh numbers varying over the range of (103 –105 ). The study goes further to investigate the effect of vertical position of the inner cylinder on the heat transfer and flow field. The location of the inner cylinder is vertically changed along the center-line of the square enclosure. The number, size and form of the vortices strongly depend on the Rayleigh number and the position of the inner cylinder. The results show that for both cylinders, at low Rayleigh numbers of 103 and 104 , the bifurcation from the bicellular vortices to an uni-cellular vortex occurs when an inner cylinder is placed at a certain distance from the center of the enclosure. When Ra = 105 , only a uni-cellu lar vortex is formed in the enclosure irrespective of the position of the inner cylinder. A lso as the obtained total surfaces-averaged Nusselt numbers of the enclosure show, in all cases, at the same Rayleigh number, the rate of heat transfer from the enclosure which the circular cylinder is located inside is better.


Introduction
Natural convection in enclosures is encountered in many engineering systems such as convection in buildings, fluid movement in solar energy co llectors, cooling of electronic circuits, and cooling of nuclear reactors, etc. Because of this wide range of applications, so far many studies have been performed in th is field . Fo r examp le : De and Dalal [1], studied numerically natural convection around a tilted heated square cy linder kept in an enclosu re. St ream funct ionvorticity fo rmu lation of the Nav ier-Stokes equations was solved numerically using fin ite-difference method. Kim et al. [2], perfo rmed numerical simu lation of natural convection induced by a temperature difference between a cold outer square enclosure and a hot inner circu lar cy linder. The imme rsed boundary method (IBM ) was used to model an inner circular cylinder only at AR=0.2 in this paper. Lee et al. [3], carried out fo r n atu ral conv ect ion indu ced by a temperature d ifference between a cold outer square cylinder and a hot inner circu lar cy linder nu merically Ray leigh solution with immersed boundary method(IBM) for different numbers at different horizontal and diagonal locations for only one aspect ratio. The both effects of the inner cylinder location in an enclosure and the buoyancy-induced convection on heat transfer and fluid flo w were investigated. Hussain and Hussein [4], investigated numerically the laminar steady natural convection where a uniform heat source applied on the inner circu lar cylinder enclosed in a square enclosure in which all boundaries are assumed to be isothermal. Hussain and Hussein [5], carried out a nu merical simu lation using a finite volume scheme for a laminar steady mixed convection problem in a t wo-dimensional square enclosure with a rotating circular cylinder enclosed inside it. The effects of various locations and solid-flu id thermal conductivity ratios on the heat transport process were studied in this work. Bararn ia et al. [6], performed a nu merical study of natural convection between a square outer cylinder and a heated elliptic inner cylinder. Lattice Bo ltzmann method (LBM ) had been used to investigate the hydrodynamic and thermal behaviors of the flu id at various vertical positions of the inner cylinder for d ifferent Rayleigh numbers. Xu et al. [7], studied transient natural convective heat transfer of liquid galliu m fro m a heated horizontal circu lar cylinder to its coaxial triangular enclosure numerically by employing the control volume method. Two orientations of the triangular cylinder were investigated in this paper.
The above relevant literature survey shows that even though the fundamental features of natural convection inside enclosures have been identified and analy zed, in most cases, the effects of natural convection around inner cylinders of different geometries have not examined in detail. For example, there is no a comprehensive study that compre the natural convection around the circular cylinder with the square cylinder inside a square enclosure. Because of this, in the present study, different features of the natural convection around the two different geometries of cylinders inside an enclosure is numerically investigated for different Ray leigh numbers and different vert ical locations of inner cylinders.

Physical Model and Assumptions
A schematic of the system considered in the present studyis shown in Fig. 1. The system consists of a square enclosure with sides of length H, where exists a circular cylinder with a radius R(=0.2H) or a square cylinder with diameter 2R. These inner cylinders can move along the vertical centerline of enclosure in the range of -0.2H to 0.2H. The walls of the square enclosure were kept at a constant low temperature o f T c , where as the cylinder was kept at a constant high temperature of T h . The flu id properties are also assumed to be constant, except for the density in the buoyancy term, wh ich follows the Boussinesq approximation. In this paper, two different geometries of inner cylinders (i.e. circular and square) at three values of Rayleigh nu mber (Ra = 10 3 ,10 4 ,10 5 ) are considered. In the simu lations to be reported here the Prandtl number, Pr, has been taken to be 0.7 corresponding to that of air. Since the present study would be a parametric study, the input data are adjusted so that the dimensionless parameters can take our desired values. Accordingly the cold temperature (T c ) was considered 300 K. All propert ies of the working fluid, namely air, which here is assumed to be an ideal gas, were chosen at temperature of 300 K and presented in Table 1.The enclosure length (H) was also considered as an input parameter and in this study, its value was determined 4 cm. It should be noted that the parameter T h is not given as an input parameter and it must be obtained from the following equation : The dimensionless vertical distance δ, which represents the distance from center of the enclosure to the inner cylinder center along the vertical centerline,varies in the range of -0.2 -0.2.

Governing Equations
In order to state the governing equations in a dimensionless form, the following dimensionless variables are defined: (2) Based on the above dimensionless variables, the non-dimensional form of the equations for the conservation of mass, mo mentu m and energy are : Where the Rayleigh, Ra and Prandtl, Pr nu mbers are defined as: The dimensionless boundary conditions are : On the walls of enclosure: U = V = 0 ; θ= 0 (8) On the heated cylinders :U = V = 0 ; θ = 1 (9) In the analysis, the average Nusselt number is defined as: Where n is the normal d irection with respect to the walls and S is distance along the square enclosure.

Numerical Method
The Figures 2 and 3 show algebraic generated grids around a circu lar cylinder and a square clinder inside a square enclosure respectively. These collocated, Circular Cylinder with a Square Cylinder Inside a Square Enclosure non-orthogonal grids based on finite volu me technique have been applied for solution of the non-dimensional equations subjected to the boundary conditions. As it is observed in the Figures 2 and 3, the coarser grids are displayed for clarity. Near the walls of square enclosure and around the inner cylinder where a thin thermal boundary layer forms, a high grid concentration is required to resolve the te mperature distribution accurately.  The diffusion terms in the equations are discretized by a second order central d ifference scheme, while a hybrid scheme is employed to appro ximate the advection terms. In order to link the velocity field and pressure in the momentum equations, the well known SIMPLEC-algorith m is adopted. The set of discretized equations are solved by TDMA line by line method [8].
In order to valid the results of the developed numerical code, a natural convection problem with low temperature for enclosure and high temperature for inner circular cylinder was examined.The wall surface-averaged Nusselt number of enclosure for this test case were co mpared with the benchmark values by Kim et al. [2] as shown in Table 2. The comparison is carried out using the following dimensionless parameters: Pr=0.7, Ra = 10 4 -10 6 and δ=0. A good agreement is observed between the results.
To perform the grid independence study, a case with circular cy linder at δ = 0 and Ra=10 5 is considered. Four different grids, namely, 60×60, 80×80, 100×100 and 120×120 are emp loyed for the numerical calculat ions. The surface-averaged Nusselt number of square enclosure obtained by each kind of grids is shown in Table 3. As it can be seen, the 100×100 grid is sufficiently fine for the numerical calcu lation. The convergence criterion of solutions is satisfied when the normalized residual of all variables (u, v, p and T) goes to 10 −8 .

Results and Discussion
Figs. 4 and 5 show the streamlines and isotherms at Ra=10 4 , 10 5 and different vertical locations for circular and square cylinders respectively. Since the patterns of streamlines and isothrem lines at Ra=10 3 are almost similar to those at Ra=10 4 , the resulted patterns at Ra=10 3 are not presented here for the sake of brevity. But if there is any difference between those, it will be mentioned.
As shown in the figs. 4 and 5, in general, the heated lighter flu id goes upward between the hot surface of the inner cylinders and the vertical symmetry line until it encounters the cold top wall. Then the fluid beco mes gradually colder and denser while it moves horizontally outward in contact with the cold top wall. Consequently, the cooled denser fluid descends along the cold side walls. As shown in figs. 4(a) and 5(a) at δ=0 for Ra = 10 4 , the heat transfer in the enclosure is almost dominated by the conduction mode. The circulat ion of the flow shows two overall rotating symmetric eddies with t wo inner vortices inside each one of the left and right eddies. A careful observation in the figures shows that there is a small difference between inner vort ices around the circular cylinder with those around the square cylinder. This difference is related to the size of these vortices, so that those around the square cylinder are larger than those around the circular cy linder. Because of more space in the enclosure around the square cylinder than the circular cylinder. As the Rayleigh nu mber beco mes 10 5 corresponding to figs. 4(b) and 5(b) for case of δ=0, the convection effect in heat transfer becomes mo re significant and the thermal boundary layer on the surface of the inner cylinder beco mes thinner. Also, a plume starts to appear on the top of the inner cylinder. At this Rayleigh nu mber, the flow filed undergoes a bifurcation where t wo inner vortices merge. The flow at the bottom of the enclosure is very weak co mpared with that at the middle and top regions which suggests stratification effects in the lower regions of the enclosure. At this Ray leigh number (i.e. Ra=10 5 ), similar to pervious cases, two vortices inside the enclosure around the square cylinder are larger than those the around the circular cylinder.
As shown in figs. 4 and 5, When the cylinder moves downward, the size of the lower inner vortex is reduced gradually and the two inner vortices merge into a single vortex atδ = -0.1 fo r circular cylinder and at δ = -0.2 for square cylinder at Ra=10 4 . Because the space between the inner cylinders and the bottom wall o f the enclosure dimin ishes in size. The reason of this difference in δs at Ra= 10 4 is that the lower zone of enclosure around the square cylinder has mo re space thanit around the circular cylinder therefor the vortices have more space for circulation thus those disappear later. For Ra=10 3 , two inner vortices merge into a single vortex at δ = -0.2 for both cylinders. Circular Cylinder with a Square Cylinder Inside a Square Enclosure At Ra=10 5 , when the cylinder moves downward more spaces between the hot inner cylinder and the top cold wall of the enclosure are secured, enhancing the buoyancy induced convection. Thus isotherms move upward and larger plumes exist on the top of the inner cylinder, wh ich increase the thermal gradient on the top of the enclosure. The dominant flow is formed at the upper half of the enclosure, locating the core of the recirculating eddies in the upper half. The stagnant region under the inner cylinder decreases as δ becomes more negative, except for the t wo bottom corners of the enclosure. As shown in the figs. 4(a) and 5(a), When the cylinder moves upward, the bifurcat ion fro m the inner bi-cellular vortices to the uni-cellular vortex occurs at δ= 0.2 for both geometries but for the circular cy linder it is later than that for the case when the circular cy linder moves downward. This is because a stronger convective flow exists in the reg ion between the hot inner cylinder and top wall of the enclosure.
At Ra=10 5 corresponding to the fig. 4(b) and 5(b), When the inner cylinder moves upward, the pattern of isotherms and streamlines is much different fro m that when the inner cylinder moves downward especially for circular cy linder. After the circular cylinder moves upward corresponding to fig. 4(b), isotherms at the upper half of the enclosure are slightly squeezed.When δ= 0.1, the plu me at δ = 0 is d ivide into three plumes. Two upwelling plu mes appear on the top of inner circular cy linder. A third plu me appears above the top of the inner circular cylinder with reverse direction owing to the two secondary vortices newly generated over the upper part of the inner circu lar cylinder. As δ increases further, the reduced space above the top of the inner cylinder confines the vertical mot ion of flow and consequently the heat conduction is predominant over the convective heat transfer locally in this space. Thus, the secondary two vortices over the top of the inner cylinder decreases in size and finally d isappears at δ = 0.2, and accordingly no third plume is found at this δ. For the square cylinder at Ra = 10 5 , when it moves upward, corresponding to fig. 5(b) at δ = 0.1, the plume at δ = 0 is divided into two upwelling plu mes which appear on the top of square cylinder and no third plume appear in here.
Figs. 6(a) and 6(b ) show the distribution of local Nusselt numbers along the cold surfaces of the enclosure for both different geometries of cylinder and different δs at Ra=10 4 and Ra=10 5 respectively. Because the qualitative trend of diagrams for square cylinder is similar to circu lar cylinder and only the values of Nus for circular cylinder is larger than Those for square cylinder at the same δ. Thus in this figures, only the diagrams which is related to circular cylinder is explained. But there is any difference between them, it will be presented.
In the fig. 6(a), when δ=0, the Nu has a maximu m value at point A which is the stagnation point on the topwall of the enclosure. When we move fro m po int A to B the Nu decreases and reaches a local minimu m value close to zero at point B. When we move fro m point B to C the Nu increases, reaches a local maximu m value at s=1where s is the distance fro m point A along the surfaces of the enclosure, and decreases again until it has a local min imu m value at point C. When we move further fro m point C to D, the Nu increases slightly again. When the inner cylinder moves upward, the Nu at the upper half of the enclosure of 0 ≤ s ≤ 1 increases whereas that at the lower half of the enclosure of 1 ≤ s ≤ 2 decrease, compared to those when δ = 0. The variat ion of the Nu values in the region A-B according to the variation of δ is very large whereas that in the reg ion B-C and region C-D is relatively s mall, due to the distribution of isotherms shown in Figs. 4(a) and 5(a).
When the inner cylinder moves downward the distribution of the Nu shows almost symmetric shapes with respect to s=1 compared to when the inner cylinder moves upward. But in the region A-B, the location of the local Nusselt number maxima increases to the bottom wall d irection.
In the fig. 6(b), as it can be seen, the values of the Nus at Ra=10 5 is larger than those at Ra=10 3 and 10 4 , due to the increasing effect of convection. Correspondig to the fig. 6(b), when δ=0.1, there is a difference between two cylinders. For square cylinder, the trend of the distribution of the Nu is fro m a ma ximu m to a min imu m value because no secondary vortices appear on inner cylinder. But for circu lar cy linder, this trend is different due to secondary vortices appeared on the inner cylinder that it is exp lained in the following. For circular cy linder, the secondary eddies are formed on the upper surface of the inner cylinder. Thus the Nu at s = 0 (point A) when δ=0.1 is not the maximu m in the presence of downwelling p lu me unlike to the cases of Ra = 10 3 and 10 4 . When we move fro m the point A to point B at δ=0.1, the Nu increases until it has a maximu m value at the location with the upwelling plu me and then decreases until it has a minimu m value at s = 0.5 (point B). In the region A-B, the value of the Nu when δ=0.1 is smaller than that when δ=0. When δ increases to 0.2, the secondary eddies disappear. As a result, when δ = 0.2, the Nu at s = 0 has a maximu m value and decreases as we move fro m the point A to B until it has a minimu m value at s = 0.5. In the region A-B, the value of the Nu when δ=0.2 is larger than that when δ=0 and increases with increasing δ. When we move fro m point B (s = 0.5) to point C (s = 1.5) at δ > 0, the Nu on the surface of the enclosure increases until it has the maximu m and then keeps decreasing until it has a min imu m value at s = 1.5. The Nu at δ > 0 does not change much and has an almost constant value close to zero, because the region C-D becomes stagnant when the inner cylinder moves upward. When the inner cylinder moves downward at Ra=10 5 , the Nu is different fro m that when the inner cylinder moves upward. Because the secondary vortices do not exist on the upper surface of the inner cylinder, the Nu at δ < 0 has a maximu m value at s = 0 (point A) and decreases as we move fro m point A to point B until it has a minimu m value at s = 0.5 (point B). The Nu in the reg ion of 0.5 ≤ s ≤ 1.5 at δ < 0 has a similar distribution to that at δ > 0. The difference in the value of the Nu in the region of 0.5 ≤ s ≤ 1.5 is not large for all the different values of δ. When we move fro m point C to point D, the Nu at δ < 0 increases with increasing s. The Nu on the bottom wall also increases very rapidly with increasing absolute value of δ at δ < 0 because the gap between the inner cylinder and the bottom wall keeps decreasing and the gradient of isotherms on the bottom wall increases very rapidly when the inner cylinder keeps moving downward.  Fig. 7 shows the total surfaces-averaged Nusselt number of the enclosure, Nu ���� en , as a function of δ at different Rayleigh numbers for both geometries of cylinder. When Ra = 10 3 and 10 4 , Nu ���� en has a parabolic profile with a minimu m value at δ = 0. The value of Nu ���� en at Ra = 10 4 is almost the same as that at Ra = 10 3 . When the Rayleigh number increases to 10 5 , the symmet ry of Nu ���� en is broken and Nu ���� en has a minimu m value at δ = 0.1, because of the presence of the secondary vortices formed on the upper surface of the inner cylinder owing to the rising thermal plumes. In all cases studied, the value of Nu ���� en at δ= 0.2 is smaller than that at δ = -0.2, because the stagnant region at δ= 0.2 beco mes larger than that δ = -0.2. As shown in the fig.  7, in all cases, at the same Rayleigh nu mber, the rate of heat transfer fro m enclosure which the circular cylinder is located inside is better. The value of the Nu at Ra = 10 5 and δ= 0 is almost same for both cylinders. Because the patterns of streamline and isotherm are almost similar but when the cylinders move upward or downward, this similarity becomes less due to difference in patterns of streamline and isotherm especially streamline and also due to difference in space around the cylinders inside enclosure so that more space around the square cylinder cause the themal boundary layer thickness increases and then Nu ���� en reduces. The optimal rate of heat transfer fro m the enclosure (i.e. ma ximu m Nu ���� en ) is at Ra=10 5 for circular cylinder at δ= -0.2. The worst case for rate of heat transfer is at Ra=10 3 for square cylinder at δ= 0.

Conclusions
Free convection heat transfer around a heated cylinder with two d ifferent geometry in an air filled square enclosure investigated numerically. A parametric study was performed and the effects of the Rayleigh number and the position of the heated cylinder on the fluid flow and heat transfer were investigated. The results show that for both cylinders, at low Rayleigh numbers of 10 3 and 10 4 , the bifurcation fro m the bicellular vortices to an uni-cellular vortex occurs when an inner cylinder is placed at a certain distance fro m the center of the enclosure. When Ra = 10 5 , only a uni-cellular vorte x is formed in the enclosure irrespective of the position of the inner cylinder. At these high Rayleigh nu mbers, the effect of the inner cy linder position on fluid flow and heat transfer is significant, especially in the upper half region. Fo r circular cylinder, when Ra=10 5 , the secondary vortices due to the rising thermal p lu me fro m the inner cylinder are present on the upper surface of the inner cylinder.
As the obtained Nu ���� en s show, The optimal rate of heat transfer fro m the enclosure (i.e. maximu m Nu ���� en ) is at Ra=10 5 for circu lar cy linder at δ= -0.2. The worst case for rate of heat transfer is at Ra=10 3 for square cylinder at δ= 0 and also in all cases, at the same Rayleigh number, the rate of heat transfer fro m enclosure which is located inside the circular cylinder is better.