4 Results And Discussion

As summarized in Table 1, computations were performed for one Reynolds number (10,000), rotation numbers ranging from 0 to 0.28 and inlet coolant-to-wall density ratios Ap/p ranging from 0.122 to 0.40 with two channel orientations of b = 90o and 135o. The Nusselt numbers presented here were normalized with a smooth tube correlation by Dittus-Boelter/McAdams (Rohsenow and Choi [39]) for fully developed turbulent non-rotating tube flow:

Nuo = 0.023 Re08 Pr04

4.1 Velocity and Temperature Fields

Before discussing the detailed computed velocity field, a general conceptual view about the secondary flow patterns induced by angled ribs and rotation is summarized and sketched in

Figure 3.3. The parallel angled ribs in the non-rotating duct (Figure 3.3a) produce symmetric counter rotating vortices that impinge on the top surface. The Coriolis force in the (5 = 90° rotating duct (Figure 3.3b) produces two additional counter-rotating vortices that push the cooler fluid from the core to the trailing surface. For the (5 = 135° rotating duct (Figure 3.3 c), the Coriolis force produces two long vortices parallel to the ribbed surfaces and a third small vortex near the corner of the top-trailing surfaces. The effect of this rotation secondary flow is to combine destructively (opposite directions) with the rib induced secondary flow along the whole leading and trailing surfaces. This is an important concept that will help explain some of the coming flow and heat transfer characteristics.

4.1.1 Smooth Duct.

At two axial stations as defined in Figure 3.1a, Figures 3.4 through 3.6 show the calculated secondary flow vectors and constant temperature contours for the smooth cases as mentioned in Table 1. Note that these axial stations are viewed from upstream of the channel. It can be seen from Figure 3.4a that secondary corner vortices are generated as a result of the Reynolds stress anisotropy. It can be noticed from the corresponding temperature contour plots that the cooler fluid is located in the core region of the channel cross section. Further downstream (Figure 3.4b), the level of the secondary corner vortices is the same and the fluid in the duct core is heated more.

In Figure 3.5, the Coriolis forces produce a cross-stream two vortex flow structure (Figure 3.5a) that pushes the cold fluid from the core toward the trailing surface and then brings it back along the inner and outer surfaces to the leading surface. This means that the thermal boundary layer starts at the trailing surface, grows along the two side surfaces and ends at the leading surface. This results in small temperature gradient near the leading surface (hence lower heat transfer coefficients) and steeper one near the trailing surface (hence higher heat transfer coefficients) as seen from the corresponding contour plot of Figure 3.5a. Moreover, the cooler heavier fluid near the trailing surface will be accelerated by the centrifugal buoyancy force while the hotter lighter fluid near the leading surface will be decelerated to maintain the continuity in the streamwise direction. The Coriolis forces, in the (5 = 135° smooth duct (Figure 3.6a), produce a secondary flow that pushes the cold fluid away from the corner of the leading and top surfaces. This produces two counter rotating vortices with the one near the leading surface stronger than the one near the trailing surface. It can also be noticed that a small vortex is generated at the corner of the top and trailing surfaces. As a result of this secondary flow, the fluid is pushed toward the bottom surface at which part of the secondary flow will move back along the trailing surface while the other part moves along the leading surface such that they meet again at the leading corner. This means that the thermal boundary layer starts at the bottom surface, grows along the trailing and leading surfaces and ends at the leading corner. This can be seen from the corresponding temperature contour plots where high temperature contours are located near the leading corner.

4.1.2 Ribbed Duct.

At several axial stations as defined in Figure 3.2a, Figures 3.7 through 3.10 show the calculated secondary flow vectors and constant temperature contours for the ribbed cases as mentioned in Table 1. Figure 3.7 shows the calculated secondary flow vectors and constant temperature contours for the non-rotating case (case 4). Since the ribs are oriented at a negative 45° angle, the fluid adjacent to the top and ribbed surfaces will reach the ribs first and change direction along the ribbed surfaces toward the bottom surface (Figure 3.7a). It then returns back to the top surface along the centerline of the inclined cross-stream plane. In the same figure, one can also notice the early stages of two symmetric counter-rotating vortices, which become two full symmetric counter-rotating vortices in the midsection of any two ribs (Figure 3.7b). Along the streamwise direction, the size of these two vortices oscillates from the largest in the middle of each inter-rib distance to the smallest on the rib tops (Figure 3.7c). This pattern keeps repeating until the last rib (Figure 3.7d and 7e). The effect of the secondary flow on the temperature field is convecting the cooler fluid from the top surface and along the ribbed surfaces towards the bottom surface. It then moves back to the top surface which results in steep temperature gradients and high heat transfer coefficients on both the top and ribbed surfaces as seen in the corresponding temperature contours.

Figure 3.8 shows the cross-stream velocity vectors and temperature contours for case 5 (Ro = 0.14 and (5 = 90°) at the same planes as in the non-rotating ribbed duct (case 4). As the flow approaches the first rib, this Coriolis force induced secondary flow starts to distort the secondary flow started by the inclined ribs. This effect can be clearly seen by comparing Figures 3.8a through 8e with Figures 3.7a through 3.7e. From this comparison, the following conclusions can be drawn. (1) The magnitude of the Coriolis force induced secondary flow is weaker than the rib induced secondary flow. (2) In the midsections of each of two ribs, the rib induced vortex near the bottom surface is distorted slightly in the midsection of rib 1 and 2 (Figure 3.8b) but this distortion increases as the fluid proceeds downstream the duct (Figure 3.8d). (3) On the ribs (Figure 3.8c), both vortices shrink in size and get distorted only near the bottom. This pattern repeats itself until the last rib (Figure 3.8e). The general effect of the Coriolis force induced secondary flow is to distort the rib-induced vortices. Consequently, the temperature contours are shifted toward the trailing surface, which affects the heat transfer coefficients from both the leading and trailing surfaces as seen from the corresponding temperature contour plot.

Figure 3.9 shows the cross-stream velocity vectors and temperature contours for the low rotation low density ratio b = 135° (case 6) at the same planes as in cases 5 and 6. Comparing Figure 3.9 with Figure 3.8, the following can be noticed. Just before the ribbed section, the rotation induced secondary flow is still dominant as can be seen from comparing Figures 3.9a and 8a. However, from rib 1 on, this low rotation induced secondary flow is dominated by the rib induced secondary flow. A careful comparison between the secondary flow fields of case 6 and case 5 (e.g. Figure 3.9d with Figure 3.8d) shows that there is only minor change in the net effect of the secondary flow fields. This minor change appears more clearly in the temperature field. By comparing the temperature contours in Figure 3.9 with Figure 3.8, we notice that the cooler fluid is pushed back toward the leading surface, reducing the steep temperature gradients on the trailing surface.

As we increase the rotation number and density ratio, the strength of the rotation-induced secondary flow increases and gradually overcomes the rib induced secondary flow (recall Figure 3.3c). By reaching a rotation number of 0.28 and a density ratio of 0.40 as shown in Figure 3.10 (case 9), the rotation-induced secondary flow is found to be dominant over the rib induced secondary flow especially downstream of the channel. This is very clear by comparing the corresponding axial stations in Figures 3.10 and 3.9. This important result has its own consequence on the temperature field and thus the Nusselt number ratio distribution. The rib induced secondary flow is not any more able to drive the secondary flow from the ribs leading side (near the top surface) to the ribs trailing side (near the bottom surface). On the contrary, the rotation induced secondary flow moves the cold fluid from the bottom surface along the ribbed surfaces with the secondary flow along the leading surface is much stronger than the one on the trailing surface. The temperature contours in Figure 3.10 indicate that the cold fluid is moved toward the bottom surface compared to Figure 3.9.

4.2 Detailed Local Heat Transfer Coefficient Distribution 4.2.1 Smooth Duct.

Figure 3.11a shows the Nu/Nuo contour plots on the leading and trailing surfaces for the non-rotating smooth case. The unheated sections were cut to focus on the heated section. The Nusselt number ratios near the beginning of the heated section are high due to the thinner boundary layers. Downstream, they decrease and asymptotically approach the fully developed value. Figure 3.11b and 3.11c show the Nu/Nuo ratio contours on the leading side for the (5 = 90° and 135° rotating cases (Ro = 0.14 and Dp/p = 0.122). Compared to the non-rotating case, the heat transfer in the (5 = 90° case is lower because of the Coriolis force induced secondary flow which pushes the fluid away from the leading surface. For the (5 = 135° case, we notice that the Nu/Nuo ratios are high next to the bottom surface and then decrease toward the top surface. The reason for this is explained in the velocity section where it was mentioned that part of the cold fluid comes back from the bottom surface along the leading surface. This means that the thermal boundary layer grows on the leading surface as the secondary flow moves toward the top surface and thus heat transfer will be high at the bottom surface and then decreases towards the top surface. Figure 3.11d shows the Nu/Nuo ratio contours on the trailing side for the (5 = 90° rotating case (Ro = 0.14 and Dp/p = 0.122). The heat transfer is higher on this surface compared to the non-rotating case. This is again a result of the rotation induced secondary flow that pushes the cold fluid toward the trailing surface (see the velocity section). Figure 3.11e shows the Nu/Nuo ratios contours on the trailing side for the (5 = 135° rotating case (Ro = 0.14 and Dp/p = 0.122). Except for the entry region, the Nu/Nuo ratios are almost constant in the middle portion of the duct.

4.2.2 Ribbed Duct.

For various rotation numbers and density ratios, Figures 3.12 and 3.13 show the local Nusselt number ratio contours of the ribbed leading and trailing surfaces, respectively. The non-rotating case in Figure 3.12a (3.13a for the trailing surface) will be used as a baseline for comparison and discussion. Figures 3.12b through 3.12e (3.13b through 3.13e for the trailing surface) are for (5 = 135° while Figure 3.12f (13f for the trailing surface) is for ( = 90°. The entrance and exit regions were cut to focus on the ribbed heated section. First, the effect of the channel orientation on the Nusselt number ratios is discussed via comparing Figures 3.12b and 3.12f (3.13b and 3.13f for the trailing surface). Second, the effect of increasing the rotation number on the (5 = 135° Nusselt number ratios is discussed via Figures 3.12a through 3.12c (3.13a through 3.13c for thetrailing surface). Third, the effect of increasing the density ratio on the (5 = 135° Nusselt number ratios is discussed via Figures 3.12c through 3.12e (3.13c through 3.13e for trailing surface).

In Figure 3.12a, the highest Nusselt number ratios were obtained on the top of the ribs, and the lower Nusselt number ratios were obtained right before and after the ribs. Between any two ribs, the Nusselt number ratios are highest near the top surface and decrease as we move towards the bottom surface. This is due to the rib induced secondary flow that moves from the top surface (and parallel to the ribbed walls) to the bottom surface.

Effect of channel orientation on the leading and trailing surfaces: For fixed rotation number and density ratio (Ro = 0.14 and Ap/p = 0.122), Figures 3.12b and 3.12f show the Nusselt number ratios contours on the leading side for (5 = 135° and 90°, respectively. Comparing these figures with the non-rotating leading side (Figure 3.12a), it is noticed that the Nusselt number ratios decreased in both cases with the decrease in the (5 = 135° case being the most (a 19% decrease compared to a 10% decrease in the 90° case). Figures 3.13b and 3.13f show the Nusselt number ratios contours on the trailing side for (5 = 135° and 90°, respectively. Comparing these figures with the non-rotating trailing side (Figure 3.13 a), it is noticed that the Nusselt number increased in both cases with the increase in (5 = 135° being the least (a 1% increase compared to a 5% increase in the ( = 90° case). The reason why the Nusselt number ratios in the ( = 135° case decreased more on the leading side and increased less on the trailing side compared to ( = 90° case can be understood in light of the conceptual secondary flow diagram in Figure 3.3. The rotation induced vortex in the ( = 135° configuration move along the full face of the leading or trailing surfaces. However, the rotation induced vortex in the ( = 90° configuration moves along only one half the face of the leading or trailing surfaces. With this in mind, we notice in Figure 3.3 that the two secondary flows produced by rotation and angled ribs for the rotating (5 = 135° duct combine destructively (opposite direction) and thus reduce heat transfer on both the leading surface and the trailing surface. On the other hand, the two secondary flows produced by rotation and angled ribs for the rotating ( = 90° duct combine to (i) constructively (same direction) enhance heat transfer for only one half of each of the leading and trailing surfaces and (ii) destructively (opposite direction) reduce heat transfer for the other half of each of the leading and trailing surfaces.

Effect of increasing the rotation number on the leading surface: In Figure 3.12b, the rotation number is increased to 0.14 while the density ratio is kept fixed at 0.122. As discussed before, this causes the Nusselt number ratios to decrease by 19% compared to the non-rotating case (Figure 3.12a). But when the rotation number was increased to 0.28 (Figure 3.12c), the Nusselt number ratios decreased only by 10% compared to the non-rotating case. Moreover, it is noted that the high Nusselt number ratios regions are shifted to the middle of the ribbed surface. This is because of the rotation induced secondary flow getting stronger and gradually overcomes the rib induced secondary flow.

Effect of increasing the density ratio on the leading surface: In Figure 3.12d, the rotation number is kept fixed at 0.28 while the density ratio is increased to 0.20. It is seen from this figure that the high Nusselt number ratios regions are moved further toward the bottom surface. Increasing the density ratio further to 0.40 (Figure 3.12e), we notice that the high Nusselt number ratios regions are now existing next to the bottom surface with a total decrease of only 4% compared to the non-rotating case.

Effect of increasing the rotation number on the trailing surface: Figure 3.13 shows the same information as in Figure 3.12 but for the trailing surface. Figure 3.13a (Ro = 0.00) will be used as the baseline for comparison and discussion. As discussed before, increasing the rotation number to 0.14 (Figure 3.13b) causes the Nusselt number ratios to increase only by 1%

compared to the non-rotating case. In Figure 3.13c, the rotation number is increased further to 0.28 while the density ration is kept fixed at 0.122. This causes the Nusselt number ratios to increase by 6% compared to the non-rotating case. Also, it is seen from this figure that the high Nusselt number ratios regions are spreading toward the bottom surface.

Effect of increasing the density ratio on the trailing surface: In Figure 3.13d, the rotation number is kept fixed at 0.28 while the density ratio is increased to 0.20. It is seen in this figure that the high Nusselt number ratios regions are pushed slightly more toward the bottom surface. Increasing the density ratio further to 0.40 (Figure 3.13e) causes the Nusselt number ratios to increase by 12% compared to the non-rotating case. It is also seen from this figure that, upstream of the channel, the high Nusselt number ratios are moved toward the bottom surface while downstream they dominate most of the inter-rib regions.

4.3 Spanwise-Averaged Heat Transfer Coefficients and Comparison with Experimental Data 4.3.1 Smooth Duct.

In Figure 3.14, comparisons of the spanwise-averaged Nusselt number ratios (Nu/Nuo) were made with the experimental data of Griffith et al. [14]. In order to compare the effects of the channel orientation on the heat transfer, Figure 3.14 shows the Nusselt number ratios for the three smooth cases: 1, 2 and 3. In this Figure, the inlet coolant-to-wall density ratio was held constant at value of 0.122. The effect of the model orientation can be seen by comparing the b= 135° Nusselt number ratios with the b = 90° ones. It can be seen that the b = 135° Nusselt number ratios are higher on the leading and bottom surfaces, and lower on the trailing and top surfaces. This can be explained in terms of the secondary flow patterns and temperature contours shown in Figures 3.5 and 3.6. For the b = 90° case, the cold fluid reaches the leading surface after it passes over the trailing surface and both of the two side surfaces. On the other hand, the cold fluid in the b = 135° case moves directly to the bottom surface at which it splits and comes back along the leading and trailing surfaces. When the channel orientation was changed from b = 90° to 135°, more cold fluid was flowing to the leading surface while the trailing surface received less cold fluid. This has led to higher heat transfer on the leading surface and lower heat transfer on the trailing surface for the b = 135° case. For the top surface, the lower Nusselt number ratios observed in the b = 135° rotating case can be attributed to the fact that most of the top surface behaves as a leading surface in the sense that the fluid is moving away from this surface. Similarly, the ( = 135° bottom surface behaves as a trailing surface with high heat transfer since it receives the cold fluid directly from the duct core. Comparisons with the experimental values reveal the following: (1) for the non-rotating case, the matching between the experimental and prediction is good on all surfaces, (2) fair agreement on the leading, trailing, top and bottom sides is achieved for the rotating cases of (5 = 90° and 135°.

4.3.2 Ribbed Duct.

Figures 3.15 and 3.16 show the spanwise-averaged and regional-averaged Nusselt number ratios (Nu/Nuo) for the ribbed cases 4 (fi = 90°) and 5 ( = 135°). The rotation number and the inlet coolant-to-wall density ratio were held constant at values of 0.14, and 0.122, respectively. Note that the experimental regional-averaged Nusselt number in Griffith et al. [14] is based on the projected area of each copper plate rather than the true heat transfer surface area which includes the 45° rib-increased area. However, the predicted regional-averaged Nusselt Number is based on the true heat transfer area for the test surfaces with 45° ribs which is 1.25 times the projected area. Therefore, the experimental data were divided by 1.25 to reasonably compare with our regional-averaged Nusselt number, except for the inner and outer surfaces where there were no ribs. The predicted Nusselt number ratios on the leading and trailing surfaces are in good agreement with Griffith et al. [14] data for the non-rotating case (Figure 3.15) while relatively close to the experimental data in the rotating case (Figure 3.16). Downstream of the channel, the predicted Nusselt numbers on the top and bottom surfaces are mildly over-predicted and under-predicted, respectively. This may be partly attributed to the fact that the predicted Nusselt number ratios are based on a uniform wall temperature boundary condition while the experimental ones are based on a uniform wall heat flux boundary condition.

The spanwise-averaged Nusselt number distributions on the leading and trailing surfaces of Figures 3.15 and 3.16 show periodic spikes. The higher spikes which occur on the ribs tops are caused by the flow impingement on the ribs, and the lower spikes (which occur right before and after the ribs) are caused by the flow reattachment between the ribs. The Nusselt number ratios are high in the regions between the ribs. The Nusselt number ratios increase until the last rib, which is similar to the results obtained in Jang's et al. [33] 45°-ribbed square channel and Al-Qahtani's et al. [34] 45°-ribbed rectangular channel (AR = 2). This phenomenon is caused by the rib-induced secondary flow becoming stronger along the duct as discussed in Figure 3.7. The Nusselt number distribution on the top surface of Figures 3.15 and 3.16 shows that it increases all the way to rib 9 as a result of the secondary flow that pushes the cold fluid towards the top surface. For the same reason, the Nusselt number distribution on the bottom surface is decreasing (although mildly) since it receives the heated fluid from the ribbed surfaces.

Figures 3.17 shows the spanwise-averaged Nusselt number ratios (Nu/Nuo) for the (5 = 135° ribbed cases 6 and 9 which presents a comparison between the low-rotation low-density ratio case and the high-rotation high-density ratio cases which is close to the real rotor cooling conditions. The following observations are obtained by comparing Figure 3.17a with 3.17b. (1) It is seen that the Nusselt number ratios on the top surface of case 6 were higher than the ones on the bottom surface due to the rib induced secondary flow which convects the cooler fluid along the ribbed surfaces and then back to the top surface resulting into higher Nusselt number ratios on the top surface. However, in case 9 which represents the high rotation high density ratio range, the situation is reversed where the Nusselt number ratios on the bottom surface are higher than the ones on the top surface. This is a direct result of the rotation induced secondary flow which pushes the cold fluid toward the bottom surface. For the leading and trailing surfaces, the Nusselt number ratios spikes are higher in case 6 compared to case 9, however, the Nusselt number ratios in the inter-rib regions are higher in case 9 compared to case 6. It is found that the overall Nusselt number ratios of case 9 on the leading and trailing surface are higher than those in case 6 by 18% and 11%, respectively.

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