4. Heat transfer

In this section, the heat transfer characteristics of the semicylinder array are presented for buoyancy assisting and opposing flow.

#### 4.1. Local Nusselt numbers

Figures 12a and b show representative distributions of the local Nusselt number defined in Eq. (8) over the curve length A-B-C-D (body contour of each semicylinder) for Ri ¼ −1 and Ri ¼ 1, respectively. In these figures, the broken and continuous lines correspond to the upstream and downstream semicylinder, respectively. For assisting/opposing buoyancy, when the warm/cold downward flow impinges the front stagnation point of the upstream semicylinder, the temperature gradient is maximum and the local Nusselt number reaches its peak value at point C. Beyond point C, as the warm/cold downward-flowing fluid travels through the front half of the semicylinder along the surface B-C-D, it yields/picks up thermal energy and the local Nusselt number gradually decreases toward points B and D. The cold/ warm upward flow present between both semicylinders impinges the rear of the upstream one, a local maximum is reached at point A and a progressive increase in the local Nusselt number is observed over the curve length B-D. Depending on whether buoyancy assists/ opposes the flow and because of the presence of the recirculation zone within the gap that yields/picks up thermal energy from the wake of the upstream semicylinder, a local minimum of the local Nusselt number is reached at the front stagnation point of the downstream semicylinder. Thus, the local Nusselt number beyond point C gradually increases toward points B and D. As the flow detaches from the tip of the downstream semicylinder (points B and D), the local Nusselt number reaches a local/global maximum for assisting/opposing buoyancy, respectively.

#### 4.2. Overall Nusselt number

Figure 13 shows the time variation of the surface-averaged Nusselt number of both semicylinders with Richardson number. In these figures, the broken and continuous lines correspond to the upstream and downstream semicylinder, respectively. Figure 13 shows how the presence of the upstream semicylinder has a significant effect on the heat transfer characteristics of the downstream semicylinder and lower heat transfer rates are achieved by the latter. For clarity, in the inset of Figure 13, the value of the mean Nusselt number of both semicylinders is plotted in a limited range of the nondimensional time, from τ ¼ 180 to 200. It is worth to mention that the discernible periodic oscillations of the mean Nusselt number of the lower semicylinder are closely related to flow oscillation due to vortex shedding for both cases.

4. Heat transfer

buoyancy assisting and opposing flow.

234 Heat Exchangers– Design, Experiment and Simulation

4.1. Local Nusselt numbers

buoyancy, respectively.

4.2. Overall Nusselt number

shedding for both cases.

In this section, the heat transfer characteristics of the semicylinder array are presented for

Figures 12a and b show representative distributions of the local Nusselt number defined in Eq. (8) over the curve length A-B-C-D (body contour of each semicylinder) for Ri ¼ −1 and Ri ¼ 1, respectively. In these figures, the broken and continuous lines correspond to the upstream and downstream semicylinder, respectively. For assisting/opposing buoyancy, when the warm/cold downward flow impinges the front stagnation point of the upstream semicylinder, the temperature gradient is maximum and the local Nusselt number reaches its peak value at point C. Beyond point C, as the warm/cold downward-flowing fluid travels through the front half of the semicylinder along the surface B-C-D, it yields/picks up thermal energy and the local Nusselt number gradually decreases toward points B and D. The cold/ warm upward flow present between both semicylinders impinges the rear of the upstream one, a local maximum is reached at point A and a progressive increase in the local Nusselt number is observed over the curve length B-D. Depending on whether buoyancy assists/ opposes the flow and because of the presence of the recirculation zone within the gap that yields/picks up thermal energy from the wake of the upstream semicylinder, a local minimum of the local Nusselt number is reached at the front stagnation point of the downstream semicylinder. Thus, the local Nusselt number beyond point C gradually increases toward points B and D. As the flow detaches from the tip of the downstream semicylinder (points B and D), the local Nusselt number reaches a local/global maximum for assisting/opposing

Figure 13 shows the time variation of the surface-averaged Nusselt number of both semicylinders with Richardson number. In these figures, the broken and continuous lines correspond to the upstream and downstream semicylinder, respectively. Figure 13 shows how the presence of the upstream semicylinder has a significant effect on the heat transfer characteristics of the downstream semicylinder and lower heat transfer rates are achieved by the latter. For clarity, in the inset of Figure 13, the value of the mean Nusselt number of both semicylinders is plotted in a limited range of the nondimensional time, from τ ¼ 180 to 200. It is worth to mention that the discernible periodic oscillations of the mean Nusselt number of the lower semicylinder are closely related to flow oscillation due to vortex

Figure 12. Distribution of the local Nusselt number on the surface of each semicylinder versus distance along each semicylinder surface for ReD ¼ 200, BR ¼ 0:2, σ ¼ 3, and Ri ¼ −1 and 1, respectively.

Figure 13. Time-evolution of the overall Nusselt numbers at ReD ¼ 200, BR ¼ 0:2, σ ¼ 3, and Ri ¼ −1 and Ri ¼ 1 for the upstream (broken lines) and downstream (continuous lines) semicylinders, respectively.

#### 5. Conclusions

In this work, numerical simulations have been carried out to study the unsteady flow and heat transfer characteristics around two identical isothermal semicylinders arranged in tandem and confined in a channel. The blockage ratio, Prandtl number and pitch-todiameter are fixed at BR ¼ 0:2, Pr ¼ 7 and σ ¼ 3, respectively. Numerical simulations are performed using the control-volume method on a nonuniform orthogonal Cartesian grid. The immersed-boundary method is employed to identify the semicylinders confined inside the channel. The influence of buoyancy has been assessed on the resulting mean and instantaneous flow, vortex shedding properties, nondimensional oscillation frequencies (Strouhal numbers), phase-space portraits of flow oscillation, thermal fields and local and overall nondimensional heat transfer rates (Nusselt numbers) from each semicylinder. Results show that in this parameter space, the flow patterns reach a time-periodic oscillatory state, the recirculation zone of the upper semicylinder completely fills the space within the gap and vortex shedding from the lower semicylinder occurs. For values of the Richardson number of for Ri ¼ −1 and Ri ¼ 1, steady-state and time periodic oscillations of the mean Nusselt number are observed for the upstream and downstream semicylinder, respectively.

#### Acknowledgements

This research was supported by the Consejo Nacional de Ciencia y Tecnología (CONACYT), Grant No. 167474.

#### Nomenclature


5. Conclusions

236 Heat Exchangers– Design, Experiment and Simulation

semicylinder, respectively.

Acknowledgements

Grant No. 167474.

In this work, numerical simulations have been carried out to study the unsteady flow and heat transfer characteristics around two identical isothermal semicylinders arranged in tandem and confined in a channel. The blockage ratio, Prandtl number and pitch-todiameter are fixed at BR ¼ 0:2, Pr ¼ 7 and σ ¼ 3, respectively. Numerical simulations are performed using the control-volume method on a nonuniform orthogonal Cartesian grid. The immersed-boundary method is employed to identify the semicylinders confined inside the channel. The influence of buoyancy has been assessed on the resulting mean and instantaneous flow, vortex shedding properties, nondimensional oscillation frequencies (Strouhal numbers), phase-space portraits of flow oscillation, thermal fields and local and overall nondimensional heat transfer rates (Nusselt numbers) from each semicylinder. Results show that in this parameter space, the flow patterns reach a time-periodic oscillatory state, the recirculation zone of the upper semicylinder completely fills the space within the gap and vortex shedding from the lower semicylinder occurs. For values of the Richardson number of for Ri ¼ −1 and Ri ¼ 1, steady-state and time periodic oscillations of the mean Nusselt number are observed for the upstream and downstream

Figure 13. Time-evolution of the overall Nusselt numbers at ReD ¼ 200, BR ¼ 0:2, σ ¼ 3, and Ri ¼ −1 and Ri ¼ 1 for the

upstream (broken lines) and downstream (continuous lines) semicylinders, respectively.

This research was supported by the Consejo Nacional de Ciencia y Tecnología (CONACYT),


#### Greek symbols


### Subscripts

