3.3. Response characteristics for opposing flow ðRi ¼ 1Þ

In this section, the response characteristics for opposing flow are presented. Figure 8 shows the nondimensional mean flow values at Ri <sup>¼</sup> <sup>1</sup> <sup>ð</sup>Gr <sup>¼</sup> <sup>4</sup> <sup>×</sup>104 Þ. Clearly, because of the presence of flow reversal, the width of the symmetric recirculation zone present within the gap and at the rear of the downstream semicylinder increases. As a result, the blockage effect is enhanced and the longitudinal velocity component reaches peak values close to the semicylinders. Note how due to secondary flow, both recirculation zones behind each semicylinder have approximately the same size. Also, because of the presence of relatively strong upward flow within the gap, a bridge that reconnects the thermal layers of both semicylinders increases buoyancy strength and vorticity strength reduces.

Figure 9 shows a typical instantaneous flow and thermal pattern at Ri <sup>¼</sup> <sup>1</sup> <sup>ð</sup>Gr <sup>¼</sup> <sup>4</sup> <sup>×</sup>104 Þ, illustrating how the shedding process changes in the presence of flow reversal. Here, the recirculation zone within the gap impinges the forebody of the downstream semicylinder and pairs periodically with the vortices shed by the downstream semicylinder. Note how because of the presence of relatively high upward flow, the downstream semicylinder sheds typical Kármán vortices of relatively large size. The third strip illustrates how vorticity contours become more complex toward the downstream direction. Here, A highlights how wall vorticity merges with downstream vortices with the same sign. The fourth strip shows how the total surface of the downstream semicylinder is completely surrounded by upward flow that produces a thermal plume at the upper stagnation point of the lower hemisphere. As such, heat transfer decreases because of the presence of relatively high temperature fluid within the gap.

Unsteady Mixed Convection from Two Isothermal Semicircular Cylinders in Tandem Arrangement http://dx.doi.org/10.5772/66692 231

Figure 8. Nondimensional mean flow vales at ReD <sup>¼</sup> 200, BR <sup>¼</sup> <sup>0</sup>:2, <sup>σ</sup> <sup>¼</sup> <sup>3</sup>, and Ri <sup>¼</sup> <sup>1</sup> <sup>ð</sup>Gr <sup>¼</sup> <sup>4</sup> <sup>×</sup>10<sup>4</sup>Þ. From left to right: <sup>U</sup> and Vvelocity, Ω vorticity and θ temperature fields, respectively.

3.3. Response characteristics for opposing flow ðRi ¼ 1Þ

230 Heat Exchangers– Design, Experiment and Simulation

several X positions.

nondimensional mean flow values at Ri <sup>¼</sup> <sup>1</sup> <sup>ð</sup>Gr <sup>¼</sup> <sup>4</sup> <sup>×</sup>104

In this section, the response characteristics for opposing flow are presented. Figure 8 shows the

Figure 7. Time variations of the nondimensional longitudinal and transverse velocity components as a function of the nondimensional time at ReD ¼ 200, BR ¼ 0:2, σ ¼ 3, and Ri ¼ 0. The extracted data is obtained at the symmetry plane and

reversal, the width of the symmetric recirculation zone present within the gap and at the rear of the downstream semicylinder increases. As a result, the blockage effect is enhanced and the longitudinal velocity component reaches peak values close to the semicylinders. Note how due to secondary flow, both recirculation zones behind each semicylinder have approximately the same size. Also, because of the presence of relatively strong upward flow within the gap, a bridge that reconnects the thermal

Figure 9 shows a typical instantaneous flow and thermal pattern at Ri <sup>¼</sup> <sup>1</sup> <sup>ð</sup>Gr <sup>¼</sup> <sup>4</sup> <sup>×</sup>104

illustrating how the shedding process changes in the presence of flow reversal. Here, the recirculation zone within the gap impinges the forebody of the downstream semicylinder and pairs periodically with the vortices shed by the downstream semicylinder. Note how because of the presence of relatively high upward flow, the downstream semicylinder sheds typical Kármán vortices of relatively large size. The third strip illustrates how vorticity contours become more complex toward the downstream direction. Here, A highlights how wall vorticity merges with downstream vortices with the same sign. The fourth strip shows how the total surface of the downstream semicylinder is completely surrounded by upward flow that produces a thermal plume at the upper stagnation point of the lower hemisphere. As such, heat transfer decreases

layers of both semicylinders increases buoyancy strength and vorticity strength reduces.

because of the presence of relatively high temperature fluid within the gap.

Þ. Clearly, because of the presence of flow

Þ,

Figure 9. Nondimensional near-wake patterns of instantaneous velocity, vorticity and temperature contours at ReD ¼ 200, BR <sup>¼</sup> <sup>0</sup>:2, <sup>σ</sup> <sup>¼</sup> <sup>3</sup>, and Ri <sup>¼</sup> <sup>1</sup> <sup>ð</sup>Gr <sup>¼</sup> <sup>4</sup> <sup>×</sup>10<sup>4</sup>Þ. From left to right: <sup>U</sup> and <sup>V</sup>velocity, <sup>Ω</sup> vorticity and <sup>θ</sup> temperature fields, respectively.

Figure 10 shows the time variations of the nondimensional longitudinal and transverse velocity components at the symmetry plane and selected longitudinal positions inside the channel. Clearly, time-periodic flow oscillation sets in after an induction time. The inset of the lower left image shows how the recirculation zone within the gap depicts periodic flow oscillation of relatively small amplitude.

Figure 10. Time variations of the nondimensional longitudinal and transverse velocity components as a function of the nondimensional time at ReD <sup>¼</sup> 200, BR <sup>¼</sup> <sup>0</sup>:2, <sup>σ</sup> <sup>¼</sup> <sup>3</sup>, and Ri <sup>¼</sup> <sup>1</sup> <sup>ð</sup>Gr <sup>¼</sup> <sup>4</sup> <sup>×</sup>104Þ. The extracted data is obtained at the symmetry plane and several X positions.

#### 3.4. Strouhal number and phase space plots

The left images in Figure 11 show (from top to bottom) the normalized spectrum of the transverse velocity component as a function of the nondimensional frequency (Strouhal number), St ¼ f D=u<sup>0</sup> for Ri ¼ −1, 0 and 1, respectively.

These images show how for Ri ¼ −1, 0 and 1 and for selected locations within the gap and downstream of the lower semicylinder, there is a sharp peak at St ¼ 0:32111, 0:29448, and 0:22295, respectively, indicating that the wake vortex shedding of both semicylinders is time-periodic and is dominated by a single fundamental frequency. These images exemplify how for the three values of the buoyancy parameter studied, the recirculation zone of the upstream semicylinder locks on to the shedding frequency of the downstream one. In addition, these images show how the Strouhal number decreases for increasing values of the buoyancy parameter. The right images in Figure 11 show the corresponding phasespace relation between the longitudinal and transverse velocity signals after the vortex shedding reaches an established periodicity. The inset of these figures describe the fluctuations at a location of ðX, YÞ¼ð1:5, 0Þ. For all cases, the single orbit with a double loop illustrates how the periodic alternate shedding of vortices takes place at the space within the gap and downstream of the lower semicylinder.

Figure 10 shows the time variations of the nondimensional longitudinal and transverse velocity components at the symmetry plane and selected longitudinal positions inside the channel. Clearly, time-periodic flow oscillation sets in after an induction time. The inset of the lower left image shows how the recirculation zone within the gap depicts periodic flow oscillation of

The left images in Figure 11 show (from top to bottom) the normalized spectrum of the transverse velocity component as a function of the nondimensional frequency (Strouhal num-

Figure 10. Time variations of the nondimensional longitudinal and transverse velocity components as a function of the nondimensional time at ReD <sup>¼</sup> 200, BR <sup>¼</sup> <sup>0</sup>:2, <sup>σ</sup> <sup>¼</sup> <sup>3</sup>, and Ri <sup>¼</sup> <sup>1</sup> <sup>ð</sup>Gr <sup>¼</sup> <sup>4</sup> <sup>×</sup>104Þ. The extracted data is obtained at the

These images show how for Ri ¼ −1, 0 and 1 and for selected locations within the gap and downstream of the lower semicylinder, there is a sharp peak at St ¼ 0:32111, 0:29448, and 0:22295, respectively, indicating that the wake vortex shedding of both semicylinders is time-periodic and is dominated by a single fundamental frequency. These images exemplify how for the three values of the buoyancy parameter studied, the recirculation zone of the upstream semicylinder locks on to the shedding frequency of the downstream one. In addition, these images show how the Strouhal number decreases for increasing values of the buoyancy parameter. The right images in Figure 11 show the corresponding phasespace relation between the longitudinal and transverse velocity signals after the vortex shedding reaches an established periodicity. The inset of these figures describe the fluctuations at a location of ðX, YÞ¼ð1:5, 0Þ. For all cases, the single orbit with a double loop

relatively small amplitude.

232 Heat Exchangers– Design, Experiment and Simulation

3.4. Strouhal number and phase space plots

symmetry plane and several X positions.

ber), St ¼ f D=u<sup>0</sup> for Ri ¼ −1, 0 and 1, respectively.

Figure 11. ReD ¼ 200, BR ¼ 0:2, σ ¼ 3, and Ri ¼ −1, 0 and 1. Left images: Normalized spectrum of the longitudinal and transverse velocities. Right images: Phase-space plot of the longitudinal velocity signal as a function of the transverse velocity signal.
