Chapter Heat transfer over protrusions1 In this chapter, heat transfer characteristics and flow structures over protrusions in a turbulent channel flow are systematically investigated by DES numerically. The thermal-hydrodynamic performances are also closely examined, including Nussselt number, friction and performance factor, and the effect of changing the height ratio. Additionally, the distribution of friction factors and Nusselt number are studied with the objective of providing the connectivity, if any, between them and the flow/vortex patterns over the protrusion. 5.1 Configuration of protrusions In this chapter, fluid flows inside a channel with length L, width W and height 2H in the x, z and y direction, respectively (Figure 5.1). For all the cases discussed here, only the lower wall consists of protrusions, while the Part of this chapter has been published as Chen et al. (2012b) 186 upper wall is always flat. The protrusion’s print diameter is a constant at D = 5H, and its height h varies from 5%D to 25%D. Y X Z 2H D h L W Figure 5.1: Channel with protrusions The spherical protrusion with smooth rounded edge (see Figure 5.2) considered in the present chapter is the inverse of the dimple case from Chapter 4. The geometry of protrusion can be described by the following height functions: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ [yi (x, z) + R − h]2 + x2r = R2 , xr < xI ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [yi (x, z) − r]2 + (xr − xe )2 = r2 , xI < xr < xE ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yi (x, z) = 0, xr > xE . In Eq. (5.1), xr = (5.1) (x − xci )2 + (z − zci )2 is the horizontal distance between a protrusion’s surface point to the center axis of protrusion, where (xci , zci ) is the center of the ith protrusion at the plane of the channel floor. 187 r h 2XI d D=2XE R Figure 5.2: Sectional drawing of a single protrusion R, h, d and r are, respectively, the protrusion’s radius of curvature in inner region, height, nominal diameter and rounded edge’s radius. Other parameters are given by the following equations: • Radius of curvature in inner region: R = • External boundary radius: xE = h + d2 , 8h h (2R + 2r − h), • Print diameter: D = 2xE , • Internal region radius: xI = R x . R+r E For the cases in which there are N protrusions on the channel floor, the composite height function is given by the summation of the individual height functions: N Y = −H + yi (x, z), i=1 188 (5.2) where Y = indicates the center plane of the channel, and Y = −H indicates the flat portion of the channel floor where the protrusions reside. The dimpled surface in Chapter 4, which is used to compare with protrusions, has the same dimension with the protrusions except that the dimpled portion is below the flat plate instead of above the flat plate for the protrusion. 5.2 Results and Discussion In this chapter, eight protrusions are closely placed in a staggered pattern on the lower wall of the channel to study the interaction between neighboring protrusions, and the upper wall is smooth and flat (see Figure 5.1). √ As such, a channel with length L = 10 3, width W = 10 and half channel height H = is taken as the main/working computational domain. The grid resolution of current study is 160 × 128 × 96, so the grid size is fairly similar to that of mesh in Table 2.3 albeit slightly better resolution for the protrusion features. In this section, heat transfer and flow structure over protrusions with different height ratios (h/D = 5%, 10%,15%, 20%, 25%) are presented and discussed. 5.2.1 Hydrodynamic and thermal performance The normalized average friction ratio Cf /Cf , Nusselt number ratio N u/N u0 and performance factors (Ga/Ga0 and Gv/Gv0 ) for protrusions with various height ratios are shown in Figure 5.3 and compared with 189 those obtained for dimpled surface taken from Chapter 4. It is observed from Figure 5.3(a) that protrusions with larger height ratio (h/D) produce higher heat transfer rate and friction. It can be also seen that the friction ratio Cf /Cf increases much more rapidly than the Nusselt number ratio N u/N u0 does as the height ratio (h/D) increases. As will be shown in the §5.2.3, the rapid increase of the normalized friction coefficient Cf /Cf is due to the acceleration of main flow with reduced flow cross-sectional area and the flow instability induced by vortices. As shown in Figure 5.3(c), with the increase of height ratio, the performance factors (Ga/Ga0 and Gv/Gv0 ) also show an initial increase. However, as a result of the more rapid rise of the friction factor than Nusselt number, the performance factors very soon reach their asymptotic limit and even start to decrease at larger height ratio (h/D). It can be observed from Figures 5.3(a) and (b) that the trend of hydrodynamic performance of protrusions bears much similarity to that of dimpled surface in terms of trend. Still quantitatively, the protrusions induce much higher friction and heat transfer rate than dimples at the corresponding depth/height ratio (h/D). It is also found from Figure 5.3(c) and (d) that the volume goodness factor Gv/Gv0 for protrusions are much higher than that of dimples at the same depth/height ratio (h/D). Additionally, the area goodness factor ratios Ga/Ga0 for both protrusions and dimples are similar to each other at the same depth/height ratio (h/D). Perhaps based on the Ga/Ga0 criterion, there is the continual rivalry in the application of dimple or protrusion for enhanced heat transfer. The present finding suggests that the optimum depth/height ratio (h/D) to achieve the 190 10 10 Cf /Cf0 Nu/Nu0 Cf/Cf Nu/Nu0 Cf/Cf 0, Nu/Nu0 Cf /Cf0, Nu/Nu0 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 h/D h/D (a) Cf /Cf and N u/N u0 for protrusions (b) Cf /Cf and N u/N u0 for dimples Ga/Ga0 Gv/Gv0 1.5 0.5 Ga/Ga0 Gv/Gv0 2.5 Ga/Ga0, Gv/Gv0 Ga/Ga0, Gv/Gv0 2.5 1.5 0.5 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 h/D h/D (c) performance factors for protrusions (d) performance factors for dimples Figure 5.3: Effect of h/D on Nusselt number, friction coefficient and performance factors: h/D stands for height ratio for protrusion, while it stands for depth ratio for dimple highest volume goodness factor (Gv/Gv0 ) are around 15%–20% for both the protrusions and dimples arrangement. Higher volume goodness factor herein means that implementing protrusions reduces the volume of heat exchanger, although the area goodness factors are comparable which means the surface area of the heat exchanger is still similar. 5.2.2 Distribution of local drag and heat transfer rate The distributions of local average friction factor Cf (comprising the components of time averaged skin friction Sm and form drag F m) and Nusselt number on protrusions with different height ratios are presented and discussed in this section. 191 5.2.2.1 Skin friction The normalized skin friction Sm/Sm0 distribution on protrusions is presented in Figure 5.4. It is shown that two local highest skin friction (red region) are located at the upstream portion of protrusion while the lowest skin friction (blue region) is found around the downstream centerline of protrusion. It can be observed that the skin friction on protrusions with larger height is generally larger than that on protrusions with lower height. The distribution of skin friction factor significantly depends on the height ratio of protrusion h/D. In particular, two local highest skin friction positions are located fairly symmetrically about the streamwise centerline of protrusion when height ratio is low (h/D ≤ 10%). Conversely for h/D ≥ 15%, the value of skin friction distributes asymmetrically about the streamwise of protrusions, especially for the local highest skin friction at the upstream portion. As such, the local highest skin friction as found on one side (which can be on the left or right side, depending on the initial input conditions, see §5.2.2.4) is higher than the other side of upstream portion of protrusion when h/D ≥ 15%. 5.2.2.2 Form drag The normalized form drag F m/Sm0 distribution on protrusions is presented in Figure 5.5. It is shown that the single highest form drag (red region) is located at the upstream portion of protrusion while the lowest form drag (blue region) is found around the downstream centerline of protrusion. It can be observed that the form drag on protrusions with larger 192 0.8 0.6 0.4 0.2 10 10 10 15 10 X X (a) 5% (b) 10% 10 10 10 2.2 1.4 0.6 -0.2 -1 Sm/Sm0 Z 15 10 X X (c) 15% (d) 20% 15 Sm/Sm0 4.4 3.6 2.8 1.2 0.4 -0.4 -1.2 -2 Z 15 Sm/Sm0 2.8 1.2 0.4 -0.4 1.6 1.2 0.8 0.4 -0.4 Sm/Sm0 Z Z Z Sm/Sm0 10 10 15 X (e) 25% Figure 5.4: Normalized friction Sm/Sm0 at different height ratios h/D 193 height is generally higher than that on the protrusions with lower height. It can be also found that the form drag distributes fairly symmetrically about the streamwise centerline of protrusion when height ratio is low (h/D ≤ 10%). Conversely for h/D ≥ 15%, the form drag distributes asymmetrically about the streamwise centerline of protrusions. However, the asymmetry of form drag distribution is less obvious than that for friction drag. Being so, the highest form drag is found at a position slightly offset from the upstream centerline of protrusions (arbitrary offset, either on the left or right side, see §5.2.2.4). 5.2.2.3 Nusselt number To further investigate the influence of protrusions on the heat transfer, the normalized Nusselt number distribution on protrusions is presented in Figure 5.6. It is shown that the highest Nusselt number (red region) is located at the upstream portion of protrusion while the lowest Nusselt number (blue region) is found around the downstream centerline of protrusion. It can be observed that the Nusselt number for the protrusions with larger height is generally higher than that on the counterpart with lower height. It can be also found that Nusselt number distributes fairly symmetrically about the streamwise centerline of protrusion when the height ratio is low (h/D ≤ 10%), and there exists two local highest Nusselt number located on the two sides of centerline of protrusions. Otherwise (h/D ≥ 15%), the Nusselt number distributes asymmetrically about the streamwise centerline of protrusions. Being so, the highest Nusselt number is found on one single side (either on the left or right side, see §5.2.2.4) of upstream portion of 194 10 10 10 15 10 X X (a) 5% (b) 10% 15 Fm/Sm0 27 23 19 15 11 -1 -5 -9 Fm/Sm0 50 40 30 20 10 -10 Z Z Fm/Sm0 11 -1 -3 Z Z Fm/Sm0 2.8 2.2 1.6 0.4 -0.2 -0.8 -1.4 10 10 10 15 10 X X (c) 15% (d) 20% Fm/Sm0 85 70 55 40 25 10 -5 -20 Z 15 10 10 15 X (e) 25% Figure 5.5: Normalized friction F m/Sm0 at different height ratios h/D 195 the protrusion. In addition, the location of the highest Nusselt number generally coincides with the location of both the highest skin friction and form drag. 10 10 10 15 10 X X (a) 5% (b) 10% 10 10 Nu/Nu0 10 9.2 8.4 7.6 6.8 5.2 4.4 3.6 2.8 1.2 10 15 10 X X (c) 15% (d) 20% 15 Nu/Nu0 11 10 Z 15 Nu/Nu0 7.5 6.5 5.5 4.5 3.5 2.5 1.5 4.6 4.2 3.8 3.4 2.6 2.2 1.8 1.4 Z Z Nu/Nu0 Z Z Nu/Nu0 2.2 1.8 1.6 1.4 1.2 0.8 10 10 15 X (e) 25% Figure 5.6: Normalized Nusselt number N u/N u0 at different height ratios h/D 196 5.2.2.4 Effect of initial conditions for protrusions with h/D = 20% It is noticed that the highest skin friction, form drag and Nusselt number are located on one side of protrusions if the height is sufficiently large (h/D ≥ 15%), hence leading to asymmetric distribution. To further verify this finding and to ascertain what affects the location of the highest hydrodynamic and thermal factors, more computational runs were carried out for the flow and heat transfer over protrusion at h/D = 20% with different initial conditions imposed. The results obtained are shown in Figure 5.7. The initial condition is set as follows: ⎧ ⎪ ⎪ ⎪ u = upre [1 + ε N (0, 1)] ⎪ ⎪ ⎪ ⎨ v = ε N (0, 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ w = ε N (0, 1) where ε (5.3) and N (0, 1) is the standard normal distribution. upre is the predicted velocity based on the velocity distribution of turbulent flow over flat channel: upre = ⎧ ⎪ ⎪ ⎨ y+ + < y + < ycritic ⎪ ⎪ ⎩ lny + + C, κ = 0.4 and C = 5.5 κ + y > . (5.4) + ycritic + where ycritic is the height of buffer layer about 15, and a smoothing technology is implemented in this transition region between the viscous sub-layer the log-law region. 197 So the random perturbation component ε N (0, 1) determines the initial condition, which does affect the final result. It is found that the local highest skin friction, form drag and Nusselt number may be found at either side of protrusions subject to the initial conditions. The distributions of hydrodynamic and thermal factors for the original run (shown in earlier sections) and the subsequent run are generally opposite. Additionally, extra runs caried out which are not shown here indicate that the highest localized factors are found at arbitrary side of centerline; these locations are essentially mirror images of each other on the respective side. These imply that the asymmetrical location of the highest Nusselt number generally only depends on the initial condition because all the other parameters are kept the same for these runs. What we have observed can be broadly classified as bifurcation phenomenon, which is inactive or insignificant when protrusion is low and becomes active or important when the protrusion is high. Furthermore, the locations of the highest skin friction, form drag and Nusselt number are found on the same side of the protrusions, thus implying the strong connectivity between them. To understand better the possible mechanisms for such asymmetric distribution of hydrodynamic and thermal factors, the various quantities on streamlines, vortex structures and velocity contours are studied and discussed in the next section. 198 2.2 1.4 0.6 -0.2 -1 10 10 10 15 10 15 X (a) Sm/Sm0 for original run (b) Sm/Sm0 for additional run Fm/Sm0 10 10 10 50 40 30 20 10 -10 Fm/Sm0 Z 50 40 30 20 10 -10 X Z 2.2 1.4 0.6 -0.2 -1 Z Sm/Sm0 15 10 15 X X (c) F m/F m0 for original run (d) F m/F m0 for additional run Nu/Nu0 10 9.2 8.4 7.6 6.8 5.2 4.4 3.6 2.8 1.2 Nu/Nu0 10 9.2 8.4 7.6 6.8 5.2 4.4 3.6 2.8 1.2 Z Z Z Sm/Sm0 10 10 10 15 10 15 X X (e) N u/N u0 for original run (f) N u/N u0 for additional run Figure 5.7: Normalized skin friction Sm/Sm0 , form drag F m/F m0 and Nusselt number N u/N u0 at h/D = 20% 199 5.2.3 Flow structure In order to explore the underlying mechanisms for the asymmetric distribution of hydrodynamic and thermal factors over protrusions with large height ratio, the associated flow structures are studied in some greater details. For simplicity but without loss of generality, only the cases with height ratio h/D = 10% and 20% are compared and shown. 5.2.3.1 Mean flow field In this section, the mean flow field (streamlines) based on time averaged velocity field are investigated. The streamlines in the vicinity of the protrusions (y + = 1.5) for the cases with height ratio h/D = 10% and 20% are compared in Figure 5.8. The fluid bifurcates at the upstream edge of protrusions, and then flows through the valleys between protrusions. Thereafter, fluid starts to recirculate, forming vortex structure behind protrusions. However, these vortex features are symmetric when the height ratio is low, but asymmetric when the height ratio is large. The asymmetric flow and vortex pattern in deep dimples, which was also observed by Kornev et al. (2010), was believed to enhance heat transfer more than symmetric flow in shallow dimple. This may be a cause of higher heat transfer rate on higher protrusions other than blockage effects as introduced by Hwang et al. (2008). According to the Taylor’s hypothesis, the evolution of flow pattern along the mean flow direction can present the temporal evolution of fluid flow. In order to examine what happens to the vortical flow over the 200 Z Z 10 10 10 15 10 X X (a) h/D = 10% (b) h/D = 20% 15 Figure 5.8: Streamlines on y + = 1.5 at different height ratios h/D protrusions, 3-dimensional streamlines are shown in Figure 5.9. For a clearer presentation of the 3D streamlines, the streamlines are colored by vertical position Y . In order to show the whole evolution cycle of fluid flow over protrusions, Figures 5.9(a) and (b) depict the behavior of the fluid before these vortices are generated; on the other hand, Figures 5.9(c) and (d) show the behaviors of fluid after these vortices are generated. Specifically, Figures 5.9(a) and (b) are streamlines traced backwards from the series of markers between the last two row protrusions; on the other hand, Figures 5.9(c) and (d) are streamlines traced forwards from the series of markers between the first two row of protrusions. Similar to planar 2D streamlines above, it is also found in Figure 5.9 that the fluid bifurcates at the upstream edge of protrusions, and then recirculates and is lifted up, hence forming the vortex structure. However, the patterns of the streamlines differ for protrusions with different height ratios both before and after the vortices are generated. Before the vortices are generated, the fluid sweeps down to the valley between the neighboring protrusions from the center plane of channel (Y = 0). The number of the coming flow groups before vortices are generated over 201 the low protrusions is four, but the counterpart over the high protrusions is two. This may be due to the symmetric and asymmetric flows over the low and high protrusions: for symmetric flow pattern over low protrusions, two groups of flow can merge into one flow group inside the valley between protrusions; for asymmetric/inclined flow, only one group of incoming flow can enter the valley between protrusions. However, the angle of sweep of the flow over protrusions with larger height ratio is much larger than that over the low protrusions, hence resulting in a stronger mixing between the fluid in the center region and near-wall region. This partially explains the higher Nusselt number observed for the higher protrusions. After the vortices are generated: (i) for low protrusions (h/D = 10%), the vortices are relatively less intense and symmetric, so they are transported through the valley of next row of protrusions; (ii) for high protrusions (h/D = 20%), the vortices are more intense and asymmetric, so they flow directly downstream and then impinge on one side of the next row of protrusions. As a result of the different behavior of vortices above the protrusions at different height ratios, the Nusselt number distribution is symmetric for low protrusion geometry (h/D = 10%) while asymmetric for high protrusion geometry(h/D = 20%). Furthermore, the positions of the highest Nusselt number for the high protrusion (h/D = 20%) coincide with the impingement between vortex and protrusion wall. There is more intense mixing between the center region of channel and near-wall region, and stronger vortices which impinge the downstream protrusions have led to higher Nusselt number for protrusions with larger height ratio. It is also worthwhile to further investigate the impingement and recir202 Y Y X X Y: -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 Y: Z -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 Z (a) before vortex generation at h/D = (b) before vortex generation at h/D = 10% 20% Y Y X X Y: -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 Y: Z -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 Z (c) after vortex generation at h/D = 10% (d) after vortex generation at h/D = 20% Figure 5.9: 3-D streamlines at different height ratios h/D, the dashed line refers to the streamline tracing markers, the fluid flows from left bottom corner to right top corner culation regions, respectively, in the upstream and downstream portions of protrusions. For simplicity but without loss of generality, only streamlines for the cases with height ratio h/D = 10% and 20% are compared in Figure 5.10. It is found that there is recirculation behind the protrusion around the centerline (Z = 5). It can also be observed that the recirculation for higher protrusion (h/D = 20%) is stronger than that for lower protrusion (h/D = 10%). On the upstream portion of protrusion, the streamlines follows the surface profile of the protrusion on the wind-ward side and over the top and then join the freestream. In addition, the higher protrusion tends to trigger more intense mixing between the center region of channel and near-wall region than the lower protrusion does. In general, the intense 203 mixing flow and recirculation, respectively, on upstream and downstream portions of protrusion can explain the highest Nusselt number located at the upstream portion of protrusion and the lowest Nusselt number located at the downstream portion of protrusion. Y Y -1 -2 -1 10 12 X (a) Z=5 for h/D = 10% -2 X 10 12 (b) Z=5 for h/D = 20% Figure 5.10: Streamlines on X-Y planes for different height ratios h/D 5.2.3.2 Mean velocity contours The asymmetric distribution of Nusselt number for protrusion with height ratio h/D = 20%, however, cannot be easily revealed by the pattern of streamlines alone. Thus, it behooves us to explore and further examine the time averaged velocity in the vicinity of protrusion (y + = 8) which is shown in Figure 5.11. It can be found that the values of both streamwise and vertical velocity (U and V) are symmetric about the centerline of protrusion when height ratio is low (h/D = 10%). Conversely, both U and V are located asymmetrically about the centerline for protrusion with large height ratio (h/D = 20%). In particular, there are two local highest values of U for h/D = 20%, of which one is located a little offset from the centerline and another is located in the valley between two neighboring protrusions. Additionally, there is one single highest value of V for h/D = 20% located 204 a little offset from the centerline. The location of highest values of U and V which is a little offset from centerline coincides with the highest Nusselt number. This is because large magnitude of convection velocity contributes significantly to heat transfer from the wall to the fluid downstream. 10 10 10 15 10 X (a) U for h/D = 10% (b) U for h/D = 20% v 2.6 2.2 1.8 1.4 0.6 0.2 -0.2 10 10 v 2.6 2.2 1.8 1.4 0.6 0.2 -0.2 15 10 X Z Z u 4.5 3.5 2.5 1.5 0.5 -0.5 -1.5 Z Z u -1 15 10 X X (c) V for h/D = 10% (d) V for h/D = 20% 15 Figure 5.11: Velocity contours in vicinity of protrusion (y + = 8) with different height ratios 5.2.3.3 Mean vortical flow structures In this section, the time averaged vorticity field of flow over the protrusion with different height ratios are presented and examined to reveal the mechanism leading to the distribution of skin friction, form drag and Nusselt number as shown in the above sections. Firstly, the streamwise vorticity (ωx ) and streamlines over protrusions are investigated by showing them on the different vertical Z-Y plane slices 205 (these positions are given in Figure 5.12). According to the Taylor’s hypothesis, the evolution of flow pattern through these slices represents the temporal evolution of fluid flow over the protrusions. This is approximately true here since unlike flat plate, the flow is modified by the geometry of the protrusions. The fluid flows through the planes 1, 2, 3, 4, 5, 6, and consequently moves over the second row of protrusions. In doing so, the vorticity and streamlines on these slices can show the evolution of vortex over the protrusions. 1234567 II 1.25 I 2.5 3.75 II Z 6.25 I 7.5 8.75 II 10 11.25 10 15 X Figure 5.12: Slices position over protrusions Before examining the vorticity and streamlines on each Z-Y plane, it is thought expedient to introduce and demarcate the different zones of the Z-Y plane, which are given as follows: 1. Zone I with 1.25 ≤ Z ≤ 3.75 and 6.25 ≤ Z ≤ 8.75. 2. Zone II with ≤ Z ≤ 1.25, 3.75 ≤ Z ≤ 6.25 and 8.75 ≤ Z ≤ 10 206 It shall be noted that Z=0 and 10 are actually the same position for the periodic boundary condition employed in this study. As the fluid passes through planes to consecutively in zone I, broadly it originates from the tail region of the valley between the first row of protrusions and arrives at the front ridge of the second row of protrusion, then consequently flows over the forward-facing ridge and the backward-facing ridge of the second row of protrusions, and finally enters the valley of the third row of protrusions. When the fluid passes through planes to consecutively in zone II, it firstly flows over backward-facing ridge of the first row of protrusions, then flows through the valley between the second row of protrusions, and finally arrives at the upstream or forward-facing ridge of the third row of protrusion. In general, zone I covers the forward-facing and backwardfacing ridge of the protrusion, while zone II covers the valley between two side-by-side protrusions in the spanwise direction. The behavior of fluid respectively in these two zones integrate the whole cycle of fluid flow over the “ridge” and “valley”. The streamwise vorticity (ωx ) and streamlines on different Z-Y plane slices at h/D = 10% are shown in Figure 5.13(a). On each plane, there are generally four pairs of contra-rotating vortices: two pairs located in the zone I (1.25 ≤ Z ≤ 3.75 and 6.25 ≤ Z ≤ 8.75) are characterized as group I; the other two pairs of vortices found in the zone II (0 ≤ Z ≤ 1.25, 3.75 ≤ Z ≤ 6.25 and 8.75 ≤ Z ≤ 10) are characterized as group II. In the zone I, the group I contra-rotating vortices which come from the upstream valley become weaker and weaker as they pass through the planes 1–2. Thereafter, the fluid arrives at the front ridge of the second row of 207 protrusions and is lifted up and forms a new pair of contra-rotating vortices (planes 3–5). Then the fluid flows around the top and backward-facing ridge of the second row of protrusion with rotation leading to a new pair of strong and small vortices very near to the wall (planes 6–7). It can be found that the new pair of vortices generated at backward-facing ridge coexist with the contra-rotating vortices transported from the forward-facing ridge on planes 6–7. In the zone II, the group II contra-rotating vortices, which come from the top of the first row of protrusion, merge with the small but strong vortices which are generated over the backward-facing ridge of the first row of protrusions (planes 1–3). Thereafter, the contra rotating vortex enter the valley between the protrusions in the second row (planes 3–5). Finally, the group II vortices become weak at the upstream rim of the third row of protrusions (planes 5–7). Examining the combined behavior of the contra-rotating vortices in the zones I and II, the flow structures of fluid passing through the ”ridgevalley” topography of interspaced protrusions can be summarized as the following. Firstly, a new pair of contra-rotating vortices is generated on the front ridge of protrusions. Then the vortices merge with the small but strong contra-rotating vortex which are generated over the back ridge of protrusions. Thereafter, the contra-rotating vortices enter the valley between the next row of protrusions and become weaker and weaker till they reach the next “ridge”. Then the flow patterns repeat the “ridgevalley” cycle. It may be noted that the contra-rotating vortices which are generated at the back ridge of protrusions are much stronger than those which are generated at the front ridge of protrusions. The existence of 208 [...]... location of the highest Nusselt number generally coincides with the location of both the highest skin friction and form drag 0 4 6 Nu/Nu0 4.6 4.2 3.8 3.4 3 2.6 2.2 1.8 1.4 1 2 4 Z 2 Z 0 Nu/Nu0 2.2 2 1.8 1.6 1.4 1.2 1 0.8 6 8 8 10 10 0 5 10 15 0 5 10 X (a) 5% (b) 10% 0 4 6 8 10 5 10 Nu/Nu0 10 9.2 8.4 7.6 6.8 6 5. 2 4.4 3.6 2.8 2 1.2 2 4 Z 2 Z 0 Nu/Nu0 7 .5 7 6 .5 6 5. 5 5 4 .5 4 3 .5 3 2 .5 2 1 .5 0 15 X 6 8 10 15. .. geometry of the protrusions The fluid flows through the planes 1, 2, 3, 4, 5, 6, 7 and consequently moves over the second row of protrusions In doing so, the vorticity and streamlines on these slices can show the evolution of vortex over the protrusions 123 456 7 II 0 1. 25 I 2 .5 3. 75 II Z 5 6. 25 I 7 .5 8. 75 II 10 11. 25 0 5 10 15 X Figure 5. 12: Slices position over protrusions Before examining the vorticity and. .. The location of highest values of U and V which is a little offset from centerline coincides with the highest Nusselt number This is because large magnitude of convection velocity contributes significantly to heat transfer from the wall to the fluid downstream 0 4 6 8 u 4 .5 3 .5 2 .5 1 .5 0 .5 -0 .5 -1 .5 2 4 Z 2 Z 0 u 9 8 7 6 5 4 3 2 1 0 -1 6 8 10 10 0 5 10 15 0 5 10 X (a) U for h/D = 10% (b) U for h/D = 20%... 6 5. 2 4.4 3.6 2.8 2 1.2 2 Z 4 6 8 0 Nu/Nu0 10 9.2 8.4 7.6 6.8 6 5. 2 4.4 3.6 2.8 2 1.2 2 4 Z 0 10 6 8 10 0 5 10 15 0 5 10 15 X X (e) N u/N u0 for original run (f) N u/N u0 for additional run Figure 5. 7: Normalized skin friction Sm/Sm0 , form drag F m/F m0 and Nusselt number N u/N u0 at h/D = 20% 199 5. 2.3 Flow structure In order to explore the underlying mechanisms for the asymmetric distribution of. .. 0 5 10 X (c) 15% 15 X (d) 20% 0 Nu/Nu0 11 10 9 8 7 6 5 4 3 2 2 Z 4 6 8 10 0 5 10 15 X (e) 25% Figure 5. 6: Normalized Nusselt number N u/N u0 at different height ratios h/D 196 5. 2.2.4 Effect of initial conditions for protrusions with h/D = 20% It is noticed that the highest skin friction, form drag and Nusselt number are located on one side of protrusions if the height is sufficiently large (h/D ≥ 15% ),... structures and velocity contours are studied and discussed in the next section 198 0 3 2.2 1.4 0.6 -0.2 -1 2 6 Sm/Sm0 3 2.2 1.4 0.6 -0.2 -1 2 4 Z 4 Z 0 Sm/Sm0 6 8 8 10 10 0 5 10 15 0 5 10 15 X X (a) Sm/Sm0 for original run (b) Sm/Sm0 for additional run 0 4 6 Fm/Sm0 50 40 30 20 10 0 -10 2 4 Z 50 40 30 20 10 0 -10 2 Z 0 Fm/Sm0 6 8 8 10 10 0 5 10 15 0 5 10 15 X X (c) F m/F m0 for original run (d) F m/F m0 for. .. cycle of fluid flow over protrusions, Figures 5. 9(a) and (b) depict the behavior of the fluid before these vortices are generated; on the other hand, Figures 5. 9(c) and (d) show the behaviors of fluid after these vortices are generated Specifically, Figures 5. 9(a) and (b) are streamlines traced backwards from the series of markers between the last two row protrusions; on the other hand, Figures 5. 9(c) and. .. and streamlines on each Z-Y plane, it is thought expedient to introduce and demarcate the different zones of the Z-Y plane, which are given as follows: 1 Zone I with 1. 25 ≤ Z ≤ 3. 75 and 6. 25 ≤ Z ≤ 8. 75 2 Zone II with 0 ≤ Z ≤ 1. 25, 3. 75 ≤ Z ≤ 6. 25 and 8. 75 ≤ Z ≤ 10 206 It shall be noted that Z=0 and 10 are actually the same position for the periodic boundary condition employed in this study As the fluid... Z 15 X 6 8 8 10 10 0 5 10 15 0 5 10 X (c) V for h/D = 10% 15 X (d) V for h/D = 20% Figure 5. 11: Velocity contours in vicinity of protrusion (y + = 8) with different height ratios 5. 2.3.3 Mean vortical flow structures In this section, the time averaged vorticity field of flow over the protrusion with different height ratios are presented and examined to reveal the mechanism leading to the distribution of. .. traced forwards from the series of markers between the first two row of protrusions Similar to planar 2D streamlines above, it is also found in Figure 5. 9 that the fluid bifurcates at the upstream edge of protrusions, and then recirculates and is lifted up, hence forming the vortex structure However, the patterns of the streamlines differ for protrusions with different height ratios both before and after . 10% X Z 0 5 10 15 0 2 4 6 8 10 Nu/Nu 0 7. 5 7 6. 5 6 5. 5 5 4. 5 4 3. 5 3 2. 5 2 1. 5 (c) 15% X Z 0 5 10 15 0 2 4 6 8 10 Nu/Nu 0 10 9. 2 8. 4 7. 6 6. 8 6 5. 2 4. 4 3. 6 2. 8 2 1. 2 (d) 20% X Z 0 5 10. 10% X Z 051 0 15 0 2 4 6 8 10 u 4 .5 3 .5 2 .5 1 .5 0 .5 -0 .5 -1 .5 (b) U for h/D = 20% X Z 051 0 15 0 2 4 6 8 10 v 2.6 2.2 1.8 1.4 1 0.6 0.2 -0.2 (c) V for h/D = 10% X Z 051 0 15 0 2 4 6 8 10 v 2.6 2.2 1.8 1.4 1 0.6 0.2 -0.2 (d) V for. 0. 25 0 0 .5 1 1 .5 2 2 .5 3 Ga/Ga 0 Gv/Gv 0 (c) performance factors for protrusions h/D Ga/Ga 0 ,Gv/Gv 0 0 0. 05 0.1 0. 15 0.2 0. 25 0 0 .5 1 1 .5 2 2 .5 3 Ga/Ga 0 Gv/Gv 0 (d) performance factors for dimples Figure