RESULTS AND DISCUSSION

Validation - Forced convection over a heated circular cylinder with an isothermal surface To validate the established immersed boundary model, we first simulate forced convection over a heated circular cylinder placed in an unbounded uniform flow. The geometric set up in the computational domain and the associate physical boundary conditions are shown in Fig. 1. The Dirichlet boundary condition, i.e. a uniform velocity profile, is applied at the inlet boundary and Neumann boundary conditions are applied at lateral and outlet sides. The incoming fluids are cold ( ) whereas the cylinder is considered as a heated body with an isothermal surface ( ). A non-uniform rectangular grid is adopted to discretize the computational domain, with a uniform grid is employed to cover the cylinder.

Simulations have been conducted for a variety of Reynolds numbers based on the cylinder diameter and the free stream velocity. Streamlines, vorticity contours and isotherm contours for simulation at Re = 40 and Re = 100 are shown in Figs. 2 and 3, respectively. It is well known that at low Re, the flow pattern remains

symmetric with a pair of stationary re-circulating vortices behind the cylinder (see Fig. 2). Increasing Re leads to instability of flow structures, so a pair of symmetrical vortices behind cylinder breaks down and the vortex starts to shed up and down alternatively (see Fig. 3). This shedding frequency can be revealed as a

dimensionless parameter, namely Strouhal number. The re-circulation length , drag coefficient , Strouhal number and Nusselt number are compared with some previous works and presented in Table 1. In general, all results obtained by the established model show good agreements with those previous studies.

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Figure 1. Computational domain and boundary conditions.

Table 1: The comparison of average drag coefficients, recirculation length and Strouhal numbers at Re = 40 and 100.

Present study 1.567 2.2192.219 3.32 1.4 0.167 5.08

Figure 2. (a) Streamlines, (b) vorticity contours and (c) isotherms at Re = 40.

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Figure 3. (a) Streamlines, (b) vorticity contours and (c) isotherm contours at Re = 100.

Natural convection in a square enclosure with a heated circular cylinder The established immersed boundary model is further applied to simulations of natural convection in a square enclosure with a heated circular cylinder placed at its center. Due to the temperature difference between the hot cylinder and cold ambient fluid, buoyancy is induced and causes an upward flow. The system consists of a square enclosure of length whereas the diameter of the cylinder is . The walls of the square enclosure are kept at a constant low temperature of while the cylinder is kept at another constant high temperature of . We define , and as the characteristic length, the reference temperature and the wall temperature, respectively. The computational domain as well as the boundarystationary conditions are shown in Fig. 4.

uniform Cartesian grids are employed in the computational domain.

The isotherms and streamline contours for simulations at Ra = , and are presented in Fig. 5. For the case at Ra = , the heat transfer inside the enclosure is mainly dominated by heat conduction. At the higher Ra of , the thermal plume commences to appear on the top of the cylinder due to the buoyancy. The thermal gradient at the upper part of the enclosure is much stronger than the lower one. Consequently, the dominant flow is found in the upper half of the enclosure. It indicates that natural convection plays an important role in flow and thermal field inside the enclosure. As Ra increases to , the heat transfer in the enclosure is mainly by natural convection. The thermal plume strongly impinges on the top of the enclosure to form a thinner thermal boundary layer and enhances the heat transfer.

The surface averaged Nusselt numbers of the cylinder for different Ra are given in Table 2. of the cylinder increases as Ra increases due to the domination of convective heat transfer. To study the effect of fractional values of , we perform simulations for two different types of grids, i.e. A and B as shown in Table 2. The terms A and B indicate that the calculations are performed by ignoring and including fractional values of , respectively. is either 0 or 1 only for A and for B. In terms of our results, it is observed that the difference of calculation between A and B is not significant when simulations are preformed using more grids. Furthermore, we choose uniform grids for all simulations in the following cases.

Table 2: around a circular cylinder placed concentrically inside an enclosure, D = 0.2L. The terms A and B indicate that the calculations are performed by ignoring and including fractional values of , respectively.

will be only either 0 or 1 for A and for B.

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Figure 4. Computational domain and the coordinate system along with boundary conditions.

Figure 5. Isotherms (upper) and streamlines (lower) of natural convection in a square enclosure with a circular cylinder, . The cylinder is at the center of the enclosure.

To further validate the proposed method, we also undertake simulations for natural convection over a heated cylinder placed eccentrically inside a square enclosure. The cylinder has a diameter . Its center is located at from the center of the enclosure. This case is common and has been studied by other scholars. The isotherm and streamline contours at different Ra are presented in Fig. 6 whereas the comparison of the average Nusselt number with previous studies is given in Table 3. The comparisons for both concentric and eccentric cases have shown good agreements between the present method and some previous studies in Tables 2 and 3, respectively.

Table 3: around a circular cylinder placed eccentrically inside an enclosure, D = 0.2L.

Ra = Ra = Ra =

Vega [18] 4.750 7.519 12.531

Pan [19] 4.686 7.454 12.540

Sadat and Couturier [26] 4.699 7.430 12.421 Present study 4.712 7.481 12.523

Figure 6. Isotherms (upper) and streamlines (lower) of natural convection in a square enclosure with a circular cylinder, . The cylinder is not at the center of the enclosure.

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Natural convection in a square enclosure with a moving heated circular cylinder To further demonstrate the capability of the present immersed boundary method in handling fluid-solid interactions with a moving hot object, we perform the simulations of natural convection in a square enclosure with a moving heated circular cylinder. The cylinder has a diameter , where is the length of the square enclosure. The cylinder sways sinusoidally in the horizontal direction with the amplitude of as shown in Fig. 7 and the non-dimensional swaying frequency is equal to 1. The instantaneous displacement of the cylinder from its mean position is given by . The cylinder is at the center of the square enclosure initially.

The simulation is performed till it reaches a periodic phenomenon. The instantaneous streamline and isotherm contours during a cycle are shown in Fig. 8. It should be noted that thermal physical properties of fluid in this case are as same as the case of natural convection at Ra = . It is found that the thermal plume which also occurs in a stationary cylinder case as shown in Fig. 8 sweeps above the cylinder due to the sway of the cylinder. The thermal plume plays a vital role in the heat dispersion from the cylinder as shown in Table 3. It is interesting to know the effect of the cylinder motion on the behavior of the thermal plume and the heat dispersion from the cylinder. In the case of horizontally moving cylinder (Fig. 8), two main eddies due to the thermal plume occupy the entire enclosure. The sizes of those two counter rotating eddies change periodically due to their interaction with the cylinder. In addition to the sway of the cylinder, the vertical motion of the cylinder, which is in the direction of the gravity is also worthy of investigation. Hence, the simulation which considers a cylinder bouncing up and down is undertaken at Ra = . Snapshots of the instantaneous isotherm, streamline and vorticity contours during a cycle are presented in Fig. 9.

Figure 7. Computational domain and the coordinate system along with boundary conditions for mixed-convection in a square enclosure with a moving heated circular cylinder.

Figure 8. Snapshots of isotherms (upper) streamline (middle) vorticity (lower) contours during a cycle, .

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Figure 9. Snapshots of isotherms (upper) streamline (middle) vorticity (lower) contours during a period, .

In addition to swaying and bouncing cylinders, we extend our simulation by combining those motions, i.e. the cylinder moving in a counter-clockwise circular orbit which is concentric with the enclosure and has the radius . Such a movement stirs the flow inside the enclosure in a more complicated way and may further cause more complicated patterns in the natural convection. The description of this problem is given in Fig. 10.

The isotherm, streamline and vorticity contours during a cycle are given in Fig. 11.

Figure 10. Computational domain and the coordinate system along with boundary conditions for mixed-convection in a square enclosure with a heated circular cylinder moving in a counter-clockwise orbit.

Figure 11. Snapshots of isotherms (upper) streamline (middle) vorticity (lower) contours during a period, .

Together with the previous cases, we show the time histories of for those three cases at and presented them in Fig. 12. For the considered amplitude and oscillating frequency of cylinder motion

, it can be observed that for moving objects are lower than the case of the stationary one in Tables 2 and 4. When the hot cylinder starts to move in any direction, it sheds convective cells which are induced by the thermal plume above the heated cylinder. These cells spread heat to the adjacent fluid. After one period, while the convective cells keep growing, the cylinder moves toward its starting position and penetrates the growing hot region, the heat transfer form the hot cylinder to the fluid is consequently reduced. To further observe these phenomena, we perform the simulations at different amplitudes and oscillating frequencies of the cylinder motion. We tabulate the calculation of in Table 4. When the amplitude is very small, e.g. , the cylinder acts as a slightly vibrating object. For all the cases in Table 4, we observe that the highest is found at the smallest amplitude, . On the other hand, smaller oscillating frequencies offer higher due to the slower motion of the cylinder, which spends more time for convective cells spreading heat before the cylinder moves back. It should be noted that all those cases are compared to natural convection at Ra = for the same fluid and thermal physical properties. The flow is more dominated by natural convection than by the forced cylinder motion. Hence, the motion of cylinder does not offer any significant effect to increase heat transfer between the hot cylinder and the fluid.

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Figure 12. Total average Nusselt number of cylinder for different kind of cylinder motion at . The length of the square enclosure is while the diameter of the cylinder is . and are the amplitude and the radius of the cylinder motion, respectively.

Table 4: of a moving circular cylinder at different amplitude and oscillating frequency, D = 0.2L.

Motion

Horizontal 0.1 6.117 6.111 6.062 Horizontal 1.0 6.114 6.083 5.628

Vertical 0.1 6.130 6.125 6.110

Vertical 1.0 6.127 6.103 5.912

Orbital 0.1 6.122 6.120 6.079

Orbital 1.0 6.118 6.085 5.639

CONCLUSIONS

A direct-forcing immersed boundary method is established in this study to explore interactions of fluids and structures and also heat transfer problems. The fraction of solid at each computational cell is determined.

Subsequently, it is involved in the computation of the virtual force and heat source. The virtual force and heat source are added into the momentum and energy equation to estimate the effects of solids on the flow and heat transfer, respectively. The proposed immersed boundary model is validated by a uniform flow past a heated circular cylinder. Good agreements are found in flow characteristics and heat transfer features between the present model and previous studies. Furthermore, the natural convection in a square enclosure where a heated cylinder inside in simulated by the proposed model. The average Nusselt number for the heat cylinder is estimated by the present model and agrees with another previous study. The established immersed boundary model is able to consider a moving solid object. The cylinder inside the square enclosure is forced to move to test the capability of the proposed immersed boundary model. The circular cylinder sways, bounces and moves in a circular orbit in the square enclosure. Influences of the motion of the cylinder on the natural convection inside the enclosure are calculated. Consequently, the proposed direct-forcing immersed boundary method which including the virtual force and heat source is able to simulate heat transfer in a fluid-solid interaction problem.

ACKNOWLEDGEMENTS

This work is supported in part by the National Science Council Taiwan (Grants No. 99-2212-E-011-042 and 96-2115-M-035-001), by National Center for Theoretical Sciences (NCTS) Taiwan, and by Taida Institute for Mathematical Science (TIMS). Authors would like to thank university computing center for providing the computing facility in the university cluster.

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