Lattice Boltzmann Simulation of Natural Convection in an Annulus between a Hexagonal Cylinder and a Square Enclosure

Laminar natural convection in a water filled square enclosure containing at its center a horizontal hexagonal cylinder is studied by the lattice Boltzmann method. The hexagonal cylinder is heated while the walls of the cavity are maintained at the same cold temperature. Two orientations are treated, corresponding to two opposite sides of the hexagonal cross-section which are horizontal (case I) or vertical (case II). For each case, the results are presented in terms of streamlines, isotherms, local and average convective heat transfers as a function of the dimensionless size of the hexagonal cylinder cross-section (0.1 ≤ B ≤ 0.4), and the Rayleigh number (103 ≤ Ra ≤ 106).


Introduction
The numerical simulation of natural convection of various fluids in different geometries has become the most widely used approach because of its low cost and the accuracy of its predictions.This is justified by the availability of reliable mathematical models and robust numerical methods that predict the fluid flow and heat transfer characteristics in practically all situations encountered.The lattice Boltzmann method is relatively new compared to the classical approaches based on the Navier-Stokes equations.This technique has been used with a great success to simulate different physical behaviors of magnetohydrodynamic fluids [1,2], inhomogeneous mediums [3,4], phase change materials [5,6], flows with chemical reactions [7], and porous media [8,9].
The lattice Boltzmann method is easier to implement for the numerical simulation of laminar natural flows in the complex geometries.Thus, it has been successfully implemented to predict the flow structures and heat transfers in enclosures of different shapes [10][11][12].But its performances are even more remarkable in the simulation of natural convection in the annulus between a rectangular enclosure and a cylinder of circular cross-section [13][14][15][16][17][18][19][20], square cross-section [21][22][23], or elliptical cross-section [24].However, one finds in the literature many other numerical works that treated the natural convection in such geometries by using the Navier-Stokes equations [25][26][27][28].Generally, the system is heated by the cylinder and cooled by the cavity walls which are maintained at the same temperature.
In these studies, the effects of the shape, size, and position of the heating body on the dynamic and thermal fields in the cavity are analyzed.Thus, Hussain and Hussein [27] studied the effect of the position of a circular cylinder inside a square enclosure.These authors have shown that the position of the horizontal cylinder has an influence on the convective heat transfer which is proportional to the Rayleigh number.Concerning the size effect, it has been studied by Moukalled and Acharya [25] which showed that the average heat transfer is proportional to the cross-section size of the cylinder placed in the center of the cavity.The simultaneous effects of the size and position of a circular cylinder inside a square cavity were analyzed by Ding et al. [13].The results found indicate that these two parameters have a great influence on the dynamic and thermal fields in the annulus between the circular cross-section and the cavity walls.The effect of the cross-section geometry of the heating cylinder on natural convection within a square cavity is studied by Ravnik and Škerget [28].These authors found that a cylinder of elliptical cross-section slightly improves the heat transfer through the walls of the cavity.
The literature review showed that the case of a horizontal hexagonal cylinder inserted in a square enclosure has not been studied.For this purpose, the present work is dedicated to the lattice Boltzmann simulation of two-dimensional natural convection around a horizontal hexagonal cylinder placed in a square cavity filled with water.The heating cylinder is at a hot temperature  ℎ , while the cavity walls are kept at a cold temperature   .The effects of the Rayleigh number, the size and orientation of the hexagonal cylinder on the streamlines, isotherms, and local and average heat transfers are presented and discussed below.

Mathematical Formulation
The system under consideration is a square cavity ( = / = 1), filled with water (Pr = 7), and having four walls kept at a cold temperature   .A cylinder of hexagonal crosssection of side  is placed in the center of the cavity and maintained at a hot temperature  ℎ .The hexagon dimensions are such that 0.1 ≤  = / ≤ 0.4.Two orientations are considered, corresponding to two opposite sides of the hexagonal cross-section which are horizontal (case I) or vertical (case II) (Figure 1).The dimensions of the cavity and the temperature difference ( ℎ −   ) are varied so that 10 3 ≤ Ra = ( ℎ −   ) 3 /] ≤ 10 6 .
The thermophysical properties of water are considered constants, except the density in the buoyancy term where the Boussinesq approximation is adopted.To simulate the fluid flow and heat transfer in the physical domain, the lattice Boltzmann method (LBM) using the two distribution functions  and , respectively, for the dynamic and thermal fields is implemented [29].
For the dynamic field: For the thermal field: The relaxation times for the flow and temperature fields are   and   , respectively.They are related to the kinematic viscosity and thermal diffusivity by Note that the restriction  > 0.5 should be satisfied for both relaxation times to ensure that the kinematic viscosity and thermal diffusivity are positive.The external force appearing in (1) is given by Δ denotes the lattice time step. eq  and  eq  are the equilibrium distribution functions which are calculated with the following relation: ) . ( is the speed of sound.It is related to the velocity of the particles by For both functions  and , we adopt the D2Q9 model which contains nine propagation velocities of particles from a node r to its neighbors r + c i Δ.
For the D2Q9 model, the weighting factor   and the discrete velocities c i are defined as follows: ))  = 6, 7, 8, 9. ( On the active walls, the boundary conditions for   and   are given by The macroscopic quantities such as the density, velocity, and temperature are calculated by Nu  d (11) on the horizontal walls  =  and  =  and inversely on the vertical ones.

Numerical Procedure and Validation
In order to numerically integrate the equations ( 1) and ( 2), a computational mesh of 150 × 150 is used.In fact, it was found that the effect of a finer mesh on the results (Nu  ,  max , and  min ) is negligible.The convergence is considered to be reached when the difference on the velocity and the temperature at each point of the mesh is less than 10 −8 between two successive iterations.
For the computer code validation, the closest studies found in the literature are related to natural convection of air in an annulus between a cold square outer cylinder and a hot circular inner cylinder of radius .The results of our code are compared in Tables 1(a) and 1(b) with those obtained by Moukalled and Acharya [25] and Ding et al. [13] For Ra = 10 6 and / = 0.1, in Figure 2 the dimensionless velocity and temperature profiles at mid-height of the cavity are compared (between  = 0.6 and 1).The figure shows the excellent agreement on the velocity and  temperature of the air (Pr = 0.71) that vary widely, especially near the hot circular wall.3(a) and 3(b), respectively.Generally, the flow structures are symmetrical with respect to the center, the vertical and horizontal centerlines of the cavity.On either side of the vertical centerline, the convective cells have opposite rotations and their number increases with , but their intensities decrease while remaining in general quite low.For  = 0.4, there are four cells that occupy the corners of the cavity.When  decreases, the size of these cells increases which allows those which rotate in the same sense to merge partially ( = 0.2) or totally ( = 0.1).Thus, for  = 0.1, we obtained only two separate cells that surround the cylinder and each of them occupies a vertical half of the cavity.

Results and Discussion
Due to the low intensity of the fluid flow (Ra = 10 3 ), the heat transfer between the cylinder and the cavity walls occurs mainly by conduction.Indeed, Figure 3(b) shows that, for a given , the isotherms are almost symmetrical with respect to the vertical and horizontal median and indicate that the heat transfer is proportional to the hexagon size.For  = 0.4, the isotherms show a good heat transfer in the vicinities of the horizontal walls and the hexagon vertices that are near the vertical walls of the cavity, while for  = 0.1, the isotherms are almost circular with a spacing that increases with distance from the hexagon due to its small size.
Regardless of the size of the heated hexagonal cylinder, the fluid flow becomes much more intense for Ra = 10 6 and the symmetry with respect to the horizontal median is no longer observed (Figure 4(a)).Given that the temperature of the water is higher at the top of the cavity, the flow intensity is more important in this zone.For  = 0.4, we note the appearance of two small counter-rotating cells between the top horizontal walls of the hexagonal cylinder and the cavity.These convective cells are similar to those observed in the natural flows of Rayleigh-Bénard, while two large cells where each surrounds two small cells occupy the space between the cavity vertical walls and the hexagon.The intensity of the upper small cell is higher.For a small size of the hexagon, there are two large counter-rotating cells, each of which occupies a vertical part of the cavity, while in the vicinity of the bottom horizontal wall of the cavity, the fluid is stationary.
The isotherms of Figure 4(b) show that, regardless of the size of the heating body, the convective heat transfer is important over all sides of the hexagon and the top horizontal wall of the cavity.In addition, for  = 0.4, the isotherms show important thermal gradients in the whole domain including the bottom of the cavity.In contrast, for low values of , the increase of the Rayleigh number promotes the thermal exchanges with the top horizontal cold wall and reduces them with the bottom one.Indeed, below the hexagon, the water temperature is practically uniform and equal to that of the cold walls.Note also the formation of a thermal plume when the size of the hexagon is small enough ( = 0.2 or 0.1).

Case II. For this new position of the hexagon (case II)
and different values of , the streamlines and isotherms are presented for Ra = 10 3 in Figures 5(a) and 5(b), respectively.Except for  = 0.4, the flow structures and their intensities have not undergone significant changes compared with case I.However, for  = 0.4, we have for this case only two large counter-rotating cells along the cold vertical walls, each of   which surrounds two small cells of a low intensity.As for the isotherms, they show also here that the heat transfer between the hexagonal cylinder and the walls of the cavity is mainly by conduction.The isotherms of case II can be deduced from those of case I by a simple rotation of 90 ∘ around the center of the cavity.Thus, for  = 0.4, the heat transfer is relatively better in the vicinity of the vertical cold walls and around the middle of the horizontal ones.While for  = 0.1 or 0.2, the isotherms remain circulars, so almost insensitive to the hexagon orientation.
By switching to Ra = 10 6 (Figures 6(a) and 6(b)), for low values of , the influence of the orientation of the hexagon on the streamlines and isotherms is negligible.Only a slight effect on the thermal plume that is located over the top of the hot body is noticeable (Figures 4(a) and 4(b)).However, for  = 0.4, the dynamic and thermal fields have undergone significant changes relative to case I. Thus, the two small cells above the hexagon have undergone considerable reductions in size and intensity.While the most important changes are undergone by the isotherms, this will result in significant differences on local heat transfer along the cold walls of the cavity.

Heat Transfer.
In conduction regime (Ra = 10 3 ), the variations of the local Nusselt numbers along the four cold walls of the cavity are presented in Figure 7 for the two cases and different values of .Regardless of the value of  and the case considered, the evolution of Nu  on two opposite cold walls is almost identical and Nu  on the horizontal (vertical) wall of case II can be deduced from that on the vertical (horizontal) wall of case I.
For small sizes of the hexagon ( = 0.1 or 0.2), the variation of the local Nusselt number is generally independent of the cold wall considered and the orientation of the hot body (case I or II).The maximum and minimum values of Nu  are always obtained, respectively, at the middle ( = 0.5 or  = 0.5) and the ends ( = 0 and 1 or  = 0 and 1) of the considered wall.However, for  = 0.4, Nu  is highly dependent on the hexagon orientation in the cavity.Indeed, in case I, Nu  reached a maximum of 6.5 and 8.5 around the middle of the horizontal and vertical walls, respectively.While in case II, these same maximums of 6.5 and 8.5 are achieved around the middle of the vertical and horizontal walls, respectively.
For Ra = 10 6 and different values of , the variations of Nu  on the horizontal and vertical cold walls are shown in Figures 8(a)-8(d).Firstly it is noted that the variation of Nu  on the top and bottom horizontal walls is no longer the same for all values of .The hexagon orientation (case I or II) has virtually no influence on Nu  along all the cold walls when the hot body size is small enough ( = 0.1 and 0.2).Nu  () is proportional to  on the top cold wall and negligible on the bottom one.Due to the thermal plume shown by the isotherms (Figures 4 and 6), Nu  () is maximum at the middle of the top cold wall.However, the maximum value of Nu  () is reached above the middle of the vertical walls.This behavior is caused by the progressive displacement of the cell centers to the top of the domain when Ra increases.When the hexagon size is large ( = 0.4), its orientation has significant influence on the local heat transfer along the four walls of the cavity.For both cases, Nu  () along the top horizontal wall is maximum at  ∼ 0.3 and 0.7 (Nu ,max ∼ 18.5 (case I) or 17 (case II)) and minimum at  ∼ 0.5 (Nu ,min ∼ 3 (cas I) or 6 (case II)).This behavior is due to the two small Rayleigh-Bénard cells that are formed at the top of the hexagon.Along the bottom cold wall, the heat transfer remains relatively low and Nu  () reaches its maximum at  ∼ 0.5.Concerning the two vertical walls, because of the symmetry, the variation of Nu  () is the same in each case.For case I, the fluid recirculation is reduced to the mid-height of the cavity, which gives a minimum local Nusselt number at  ∼ 0.55 with two peaks on either sides of this position.For case II, two sides of the hexagon are parallel to the vertical walls, resulting in a fairly regular local heat transfer around the middle of the walls.The maximum of Nu  () is located at the top where the intensity of the water flow is higher.
For both cases and different values of , the average Nusselt numbers on all the cold walls of the cavity as a function of the Rayleigh number are presented in Figures 9(a)-9(c).For  = 0.1 and 0.2, Nu  is independent of the hexagon orientation (case I or II) and it is always higher on the top horizontal wall.After a slight decrease between Ra = 10 3 and 10 4 , Nu  on the top horizontal wall increases rapidly with Ra, while on the vertical walls, this growth is moderate.However, on the bottom horizontal wall, Nu  is decreasing due to the displacement of the centers of the convective cells to the top of the cavity.
For  = 0.4, the proximity of the hot sides of the hexagon and the cold walls of the enclosure leads to higher convective heat exchanges.The orientation of the hexagon has an appreciable effect on Nu  of all the cold walls, especially for high Rayleigh numbers.Overall, Nu  is higher in case I and the difference with case II can reach about 20% on the vertical walls if Ra = 10 6 .

Conclusion
The lattice Boltzmann method was used to study the natural convection of water around a hot horizontal hexagonal cylinder inserted in the center of a square enclosure with cold walls.For two orientations of the hexagon (cases I and II), the effects of the size  (0.1 ≤  ≤ 0.4) and the Rayleigh number Ra (10 3 ≤ Ra ≤ 10 6 ) were studied.
For each case, it was found that the flow intensity and the number of convective cells depend on the size of the hexagon and the Rayleigh number.However, the effect of the orientation of the hexagon on the flow structures and heat transfers in the cavity is important only for high values of .The average heat transfers through the walls of the cavity are proportional to  and are more favored in case I, if  and Ra are high.

4. 1 .
Streamlines and Isotherms 4.1.1.Case I.For case I, Ra = 10 3 and different values of  (size of the heating body), the streamlines and isotherms are shown in Figures

Figure 7 :Figure 8 :
Figure 7: Variations of Nu  on cold walls for Ra = 10 3 and different values of : (a) horizontal (case I) or vertical (case II) and (b) vertical (case I) or horizontal (case II).

Figure 9 :
Figure 9: Average Nusselt numbers on the cold walls of cases I and II as a function of Ra: (a)  = 0.4, (b)  = 0.2, and (c)  = 0.1.

Table 1 :
Validation in terms of Nu  and  max for different values of Ra and .
. The good agreement on Nu  (Table 1(a)) and  max (Table 1(b)) may be noted for different values of Ra and .
Side of the hexagon : Dimensionless size of the hexagon c i : Discr etev ecto rv eloci ty   : Sound speed D2Q9: Lattice arrangement   (r, ): Dynamical distribution function :  : Hydrodynamic relaxation parameter   : Thermal relaxation parameter : Stream function   : Weighting factor.