Chapter Heat transfer over asymmetric dimples1 In this chapter, medium deep (h/D ≥ 5%) asymmetric dimples with different depth ratio and skewness are investigated systematically to evaluate the effects of these factors. This study is deemed somewhat unique because the asymmetric dimple studied here is created by skewing the deepest point thus keeping the shape of dimple’s rim as circular, so that the coverage area of dimpled surface remains unchanged. Additionally, multiple dimples instead of single dimple (Isaev et al., 2000b, 2003; Kornev et al., 2010) are arranged in the channel to investigate the interactions between them. The systematic and quantitative thermo-aerodynamic performance factors measured in terms of friction factor, Nusselt number, area and volume goodness factors (Shah and London, 1978) are presented to indicate the trends and optimal configuration of the asymmetric dimple. Furthermore, Part of this chapter has been published as Chen et al. (2012a) 142 the flow and thermal field structures (especially the secondary flow induced by vortex) are investigated to illustrate the possible mechanisms leading to enhanced thermo-aerodynamic performance of the asymmetric dimple visa-vis symmetric dimple. 4.1 Configuration of asymmetric dimples The investigation of flow and heat transfer over a single hemispherical cavity (Terekhov et al., 1995, 1997) showed that sharpness of dimple can double the friction penalty compared to dimple with rounded edge. Thus, dimple with smooth rounded edge are selected in this study for heat transfer enhancement. The symmetric spherical dimple with smooth rounded edge (see Figures 4.1 and 4.3) considered in the present study is described by the following depth 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. (4.1), xr = (4.1) (x − xci )2 + (z − zci )2 is the horizontal distance between a dimple’s surface point to the center axis of dimple, where (xci , zci ) is the center of the ith dimple at the plane of the channel floor. R, h, d and r are respectively the dimple’s radius of curvature in inner region, depth, nominal diameter and rounded edge’s radius. Other parameters are given by the following equations: 143 R D=2XE d h 2XI r Figure 4.1: Sectional drawing of a single dimple • 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 dimples on the channel floor, the composite depth function is given by the summation of the individual depth functions: N Y = −H + yi (x, z), (4.2) i=1 where Y = indicates the center plane of the channel, and Y = −H indicates the flat surface around the dimple (dimple horizon) where the dimples reside. 144 The asymmetric dimple discussed in this study is established by skewing the symmetric spherical dimple in a systematic manner like simple shear deformation (details are shown in Appendix E). Thus the outer boundary of the asymmetric dimple on the channel floor remains circular (i.e. the print diameter is kept unchanged) instead of being changed into other shapes like teardrop or oval (Chyu et al., 1997; Isaev et al., 2000b, 2003). This produces higher coverage ratio of dimpled region than other modified dimples investigated previously (Chyu et al., 1997; Isaev et al., 2000b, 2003), leading reasonably to higher augmentation of heat transfer. Parameter skewness is defined to quantify how much the center of dimple has been shifted in the x and z directions: Dx = Δx , D (4.3) Dz = Δz , D (4.4) where Δx and Δz are respectively the displacements of dimple’s center in x and z direction (see Figure 4.2). More details can be found in Appendix E. √ In the following, a channel with length L = 10 3, width W = 10 and half channel height H = is taken as the main/working computational domain. For all the dimpled cases discussed here, only the lower wall consists of dimples, while the upper wall is always flat. In this study, eight dimples are placed in a staggered pattern on the lower wall of the channel (see Figures 4.3 and 4.4). The flow is driven by the prescribed mean pressure gradient β = −1 in x-direction with frictional Reynolds 145 0.6 0.4 symmetric dimple asymmetric dimple y 0.2 -0.2 dp(-h) -0.4 displacement of the center of dimple -0.6 -3 -2 -1 x Figure 4.2: Dimple’s surface along the streamwise centerline of dimple, the displacement of the deepest point is ΔX = dp(−h), the fluid flows from left to right number Re τ = 180. The grid resolution of current study is 160 × 128 × 96, so the mesh size is fairly similar to that of mesh in Table 2.3 albeit better resolution for the dimple features. 4.2 Configuration of studied cases To explore the asymmetric dimple’s capability to enhance heat transfer efficiency, three different cases are investigated: Case (flat plate), Case (depth ratio h/D = 10%) and Case (depth ratio h/D = 15%). Tables 4.1 and 4.2 summarize the configurations of Case and Case 3, and Figure 4.5 shows the position of deepest point of dimple for different cases. One may note that both Dx and Dz for Case 2C.2* (Table 4.1) and Case 3C.2* (Table 4.2) are zero; the dimple in these cases are called symmetric (or 146 Y X Z W 2H L Flow direction Figure 4.3: Computational domain and dimpled plate general or ordinary spherical) dimple. Since the pressure gradient is normal to the spanwise direction, asymmetric dimples whose spanwise skewness are negative should be considered equivalent to that with positive Dz. Thus only the cases with positive Dz are investigated in this study. The mean bulk velocity Ub is obtained to define Re 2H for each case, whose range is 4, 000 < Re 2H < 6, 000. To maintain uniformity in the interpretation of data, all results are non-dimensionalized by friction velocity in the channel uτ . Each calculation is initiated with initial guess of velocity and temperature fields and allowed to develop under the prescribed pressure gradient and boundary conditions. The global thermo-aerodynamic parameters in terms of friction coefficient ratio Cf /Cf , Nusselt number ratio N u/N u0 , area goodness factor ratio Ga/Ga0 and volume goodness factor ratio Gv/Gv0 are obtained by both temporal and spatial averaging, where the subscript ‘0 ’ denotes the 147 10 d z D 10 15 x Figure 4.4: Arrangement of dimples on bottom channel wall Case D d h Dx Dz 2C.1 0.5 −15% 0% 2C.2* 0.5 0% 0% 2C.3 0.5 15% 0% 2C.4 0.5 17.5% 0% 2S.1 0.5 −15% 15% 2S.2 0.5 0% 15% 2S.3 0.5 15% 15% Table 4.1: Different configurations of dimples for Case at h/D = 10%: ‘C’ stands for the case where dimple’s deepest point is skewed in streamwise centerline (Dz = 0%), ‘S’ stands for the case where dimple’s deepest point is on offset side of centerline (Dz = 15%); * stands for symmetric dimple, cases without * are asymmetric dimples 148 Case D d h Dx Dz 3C.1 0.75 −7.5% 0% 3C.2* 0.75 0% 0% 3C.3 0.75 15% 0% 3S.1 0.75 −7.5% 15% 3S.2 0.75 0% 15% 3S.3 0.75 15% 15% Table 4.2: Different configurations of dimples for Case at h/D = 15%. The meanings of ‘C’, ‘S’ and * are the same as in Table 4.1 2S.1 2S.2 2S.3 centerline Flow 3S.1 3S.2 3S.3 2C.4 centerline Flow 2C.1 2C.2 2C.3 (a) Case 2, h/D = 10% 3C.1 3C.2 3C.3 (b) Case 3, h/D = 15% Figure 4.5: Deepest point of dimple for different configurations numerical results of flat plate case. Note that the numerical results of flat plate has been interpolated to the counterparts at the same Reynolds number Re 2H as the dimpled case by the trend of Eqs. (2.25) and (2.26). All these factor ratios are calculated in order to quantitatively assess the performances such as the pressure loss Cf /Cf and heat transfer capacity N u/N u0 for different heat transfer surfaces, respectively. 149 4.3 4.3.1 Global thermo-aerodynamic performance Symmetric dimple Heat transfer rate and friction of symmetric dimple are studied before asymmetric dimple is investigated. in symmetric dimple. A fair bit of work has been done Summary of the results on heat transfer rate enhancement and pressure loss (see Figure 18 of Ligrani et al., 2003) shows very large variations. For example, the heat transfer rate enhancement reported by different researchers vary between 1.5 and 3.5, and the data on pressure loss have even larger variations (ranging between 1.0 and 5.0). Such wide variation of results can be possibly attributed to different dimple’s geometry, measurement conditions, numerical turbulent models, and even evaluation methods. This observation motivates us to carry out a more systematic study of the performance of symmetric dimples with different depth ratio. To clarify the effects of depth ratio, five different cases (h/D = 0%, 5%, 10%, 15%, 20%; D = 5; d/D = 0.8) are simulated. It should be noted that h/D = 0% denoted smooth flat plate. Friction factor ratio Cf /Cf and heat transfer rate augmentation N u/N u0 are plotted in Figure 4.6(a). It shows that with an increase of depth ratio, N u/N u0 shows higher rate of increase at first (0%≤ h/D ≤10%), then almost reaches asymptotic value of about 1.7–1.9 for h/D ≥10%. The thermal performance obtained are compared with the experimental results in Burgess and Ligrani (2005) as shown in Figure 4.7. 150 It is found that both the trend and level of heat transfer augmentation agree basically well with that of Burgess and Ligrani (2005). Although the trend of approaching an asymptotic limit at high depth ratio found in the present results does not exist in the results in Burgess and Ligrani (2005), such trend is consistent with the findings (Lee et al., 2008) that strong recirculation in deep dimple restricts further heat transfer enhancement. It can also be found that Cf /Cf grows monotonically with the increase of depth ratio. Though quantitatively it does not match well with experimental results because that friction strongly depends on Reynolds number, dimple’s geometry and measurement conditions, its nonlinear increasing trend still qualitatively matches the conclusion obtained by Burgess and Ligrani (2005). When depth ratio h/D increases, Ga/Ga0 and Gv/Gv0 first increase to be followed by a decrease (the critical depth ratio for the change in trend is about 10%). As such the area/volume goodness factors for deep dimple can be lower than that for the shallow dimple (see Figure 4.6(b)). The results indicate that for symmetric dimple, the optimum depth ratio h/D is 10%. 4.3.2 Asymmetric dimple 4.3.2.1 Case (h/D=10%) Figure 4.8 shows the pressure loss and heat transfer capacity for Case (h/D = 10%, Table 4.1). For Case 2C (Dz = 0%), the heat transfer ratio 151 the center line of dimple. Besides, increasing Dx brings about fairly similar effects on the frictional force distribution between the trend for the maximum value found in Figures 4.18(d)–(f) to that in Figures 4.18(a)–(c). 4.4.2.2 Mean characteristics of the form drag Figure 4.19 shows the streamwise component of time-averaged local form drag distribution on the bottom wall for the different sub-cases in Case 3. As is known, positive form drag contributes to increasing the total drag, and conversely negative form drag contributes to reducing the total drag. Figure 4.19(b) demonstrates positive form drag at the upstream portion (region I) and half of the downstream portion (region II) within the dimple, and negative form drag in the downstream rim of dimple (region III). Positive form drag in region I mainly comes from relatively low pressure which is created by the recirculation flow. Concave surface in region II results in centrifugal force, leading to high pressure and positive form drag. Conversely, low pressure and negative form drag in region III are related to convex surface. Recirculation flow is dominant in region I, thus the concave/convex surface effects play less significant role in influencing the pressure distribution and form drag (in other words, only positive form drag exists in region I). As shown by Figure 4.19(a), shifting the center of dimple to the upstream direction (Case 3C.1) makes region I shrink and region III expand compared with the symmetric dimple. Furthermore, form drag in region I increases because of stronger recirculation flow, which is attributed 170 Dz = 15% 11 -4 -3 12 Dz = 0% -1 -1 12 -4 Z -2 -4 12 10 Z Dx = −7.5% -2 -2 10 12 10 X (a) Case 3C.1 (d) Case 3S.1 -8 11 12 X -3 -6 -1 II III Z I Z Dx = 0% -8 -3 -2 6 -8 10 12 X (b) Case 3C.2* 12 (e) Case 3S.2 12 -12 -16 12 10 X -2 -4 -8 Z Z -8 12 -12 12 4 -1 12 -16 12 Dx = 15% 10 12 X 10 12 X (c) Case 3C.3 (f) Case 3S.3 Figure 4.19: Form drag F m/F m0 for Case (h/D = 0.15), the fluid flows from left to right 171 to the steeper slope in this region. Form drag in region II decreases in value while form drag in region III increases (i.e. becoming less negative) because the concave and convex regions become flattened (curvature decreases). Conversely as shown by Figure 4.19(c), shifting the center of dimple to the downstream direction (Case 3C.3) makes region I expand and region III shrink. It is also found that the local form drag in region I is higher compared to the symmetric dimple (Case 3C.2*). Form drag in region II increases in value while form drag in region III decreases (i.e. becoming even more negative) because the concave and convex regions have higher curvature. All these lead to overall higher form drag for Case 3C.3 compared to the symmetric dimple (Case 3C.2*). Figures 4.19(d)–(f) show that contours of the form drag for Case 3S exhibit fairly similar behavior to that described above for Case 3C (Figures 4.19(a)–(c)) except the former are distorted in the spanwise direction, which is due to the asymmetric flow moving towards the deepest point of dimple that is skewed in the spanwise direction. 4.4.2.3 Mean characteristics of the heat transfer coefficients Figure 4.20 shows the time-averaged local Nusselt number N u on the bottom wall for different sub-cases in Case 3. In Figures 4.20(a) and (b) (Cases 3C.1 and 3C.2*), the highest N u is found located on the downstream centerline of dimple in the locality where the free streamlines ‘impinge’ on the wall and fluid ejection occurs. Conversely, Figure 4.20(c) shows that the highest N u is located at both sides away from the centerline of dimple 172 while the deepest point of dimple is skewed in the downstream direction (Case 3C.3). It is also found that the lowest N u is located at the upstream portion of dimple which coincides with the recirculation region. These agree well with the findings of Park et al. (2004). It is observed that the maximum value of N u in Figure 4.20(a) (Case 3C.1) is lower than that in Figure 4.20(b) (symmetric dimple, Case 3C.2*) because the surface is flattened in the downstream rim of dimple. Conversely, maximum value of N u in Figure 4.20(c) (Case 3C.3) is about 25% higher than that in Figure 4.20(b) (symmetric dimple, Case 3C.2*) because the downstream rim has become steeper. Figures 4.20(d)–(f) show that the distributions of N u for Case 3S are similar to those for Case 3C in Figures 4.20(a)–(c) except that they are now distorted in the spanwise direction, which is caused by asymmetric flow moving towards and into the deepest point of dimple. Furthermore, the highest N u is located on the steeper side wall and towards where the center of dimple is skewed. 4.5 Instantaneous characteristics of flow To clarify the mechanisms on why the highest Nusselt number is found located at the downstream rim of dimple and why asymmetric dimple (Case 3C.3) permits higher heat transfer than symmetric dimple (Case 3C.2*), the instantaneous flow field such as velocity and vortex structures are studied in this section. 173 Dz = 0% Dz = 15% 30 25 20 30 20 45 25 15 15 45 10 45 50 25 Z 30 35 40 45 25 25 30 35 40 30 Z 45 10 20 35 Dx = −7.5% 50 15 10 40 45 45 30 40 45 45 50 15 45 15 45 10 15 15 30 25 30 35 10 12 20 45 10 X 12 X (a) Case 3C.1 40 (d) Case 3S.1 30 25 20 45 30 20 25 20 15 45 10 45 50 10 40 55 50 50 30 Z 50 20 Z 25 10 35 50 30 40 50 25 30 35 40 35 40 30 25 45 Dx = 0% 15 55 45 15 45 15 15 15 15 30 25 30 35 10 12 X 12 (e) Case 3S.2 35 30 35 20 45 20 20 25 10 60 50 35 50 45 50 45 35 60 55 20 20 25 30 35 40 50 55 25 30 50 30 35 45 55 20 15 60 Z Z 15 15 10 20 55 Dx = 15% 10 X (b) Case 3C.2* 40 50 20 55 15 15 30 25 20 35 40 10 35 45 20 12 X 10 12 X (c) Case 3C.3 (f) Case 3S.3 Figure 4.20: Nusselt number distribution for Case (h/D = 0.15), the fluid flows from left to right 174 4.5.1 Flow field Similar to the mean velocity field plotted in Figures 4.14 and 4.15 in Section 4.4.1, the instantaneous velocity field is plotted in Figures 4.21 and 4.22 for Cases 3C.2* and 3C.3, respectively, to show flow recirculation and ejection. As can be observed, the ejection in Figures 4.21(b) and 4.22(b) (instantaneous velocity) is stronger and more obvious than that shown in Figures 4.14(b) and 4.15(b) (mean velocity). However, the recirculation in Figure 4.21(a) (instantaneous velocity) is fragmented, and not as smooth and complete as that found in Figure 4.14(a). Undoubtedly, the fact that recirculation region for symmetric dimple (Case 3C.2*, Figure 4.21(a)) is larger in extent than that for Case 3C.3 (Figure 4.21(b)) remains unchanged Y Y as seen previously for the mean flow field streamlines. -1 -2 -1 10 11 -2 12 X 10 11 12 X (a) Case 3C.2* (b) Case 3C.3 Figure 4.21: Instantaneous streamlines patterns on X-Y plane (Z=5), the fluid flows from left to right 4.5.2 Vortex structures In Figures 4.23(a)–(b), the 3-dimensional vortex structures in the vicinity of the bottom wall for Cases 3C.2* (symmetric dimple) and 3C.3 are presented using the method proposed by Jeong and Hussain (1995). The iso-surface 175 v: -5 -4 -3 -2 -1 v: -5 -4 -3 -2 -1 -1 -2 Y Y -1 10 -2 12 X 10 11 12 X (a) Case 3C.2* (b) Case 3C.3 Figure 4.22: Contours of instantaneous vertical velocity v on X-Y plane (Z=5), the fluid flows from left to right of a proper negative λ2 close to zero (say a constant -5 in this study) can be used to identify the ‘targeted’ vortex structures from the rest of complex vortex structures. (Here, λ2 denotes the second largest eigenvalue of the tensor determined by strain rate.) Figures 4.23(a)–(b) show the iso-surface of above mentioned λ2 , and the color refers to the height Y in order to clearly show the position of vortex. The small scale eddies elongated in the streamwise direction are swallowed into and then ejected out from the dimple cavity. Most of these small scale eddies are generated around the deepest point of dimple and ejected into the flow from the downstream rim of dimple, which are then convected into inner region of the channel flow. Figures 4.23(c)–(d) show the contours of λ2 on X-Y plane going through the centerline of dimple, and the color refers to the value of λ2 . For the asymmetric dimple (Case 3C.3), the eddies generated and ejected into the flow are very close to the downstream rim of dimple, which is shown more clearly in Figure 4.23(d) (dark blue region close to downstream rim of dimple). The same occurs for the symmetric dimple (Case 3C.2*) but at lower intensity as depicted in Figure 4.23(c). These small scale eddies 176 are believed to dissipate kinetic energy, generate high friction, and enhance heat transfer, which helps to explain why high friction and Nusselt number both appear at dimple’s downstream rim as shown in Section 4.4.2. Y Y Y: -1.75 -1.5 -1.25 X -1 -0.75 -0.5 -0.25 Y: -1.75 -1.5 -1.25 X Z (a) 3-D vortex for Case 3C.2* -0.75 -0.5 -0.25 (b) 3-D vortex for Case 3C.3 λ2: -6 -5 -4 -3 -2 -1 λ2: -6 -5 -4 -3 -2 -1 -1 -2 Y Y -1 Z -1 10 11 -2 12 X 10 11 12 X (c) Contours of λ2 on Z=5 for Case 3C.2* (d) Contours of λ2 on Z=5 for Case 3C.3 Figure 4.23: Vortex structures 4.6 Turbulent advective heat flux As turbulent advective heat flux v T plays a very significant role in the heat transfer of turbulent flow, it is thus deemed necessary to examine this quantity for Case 3C.2* (symmetric dimple) and Case 3C.3. Figure 4.24 shows the contours of v T for Cases 3C.2* and 3C.3 on different X-Y planes (Z = 5, 5.5. Z is the spanwise coordinate, so Z = is the centerline of dimple, and Z = 5.5 is a little offset from that). The spanwise averaged v T in the flat channel is also shown in Figure 4.24(e) for reference and 177 comparison. The color in Figures 4.24 and 4.24(e) refers to value of v T . An extreme negative v T implies more turbulent advective heat flux is transported from the bottom wall into inner region of the channel. It can be observed that the minima of v T (see blue regions) in these two dimpled channels are much lower than that for the flat channel flow, definitely leading to an enhancement of heat transfer on the dimpled surface. By comparing the left column (Case 3C.2*, symmetric dimple) and the right column (Case 3C.3, asymmetric dimple) in Figure 4.24, several interesting differences can be found. It is noted that the negative v T region (blue region) is located far away from the dimpled wall for Case 3C.2*, while it lies inside the dimple and extends over larger extent for Case 3C.3. Furthermore, there is a large area of positive v T in the recirculation region in the left column (symmetric dimple, Case 3C.2*). Next, Figures 4.24(a)–(b) (Z=5) and Figures 4.24(c)–(d) (Z=5.5) are compared. The minima of v T (blue region) on the plot at Z=5.5 is always lower (darker blue) than that for Z=5 for both the symmetric and asymmetric dimple. In particular, by comparing to Figure 4.24(b), the region of negative v T in Figure 4.24(d) has larger area extent and located nearer to the dimpled wall. This may be due to the above mentioned counter-rotating vortices which are located symmetrically at each side of the downstream rim of dimples. It helps to explain why the highest local Nusselt number is found at both sides away from the centerline of dimple for Case 3C.3. As is known, an extreme negative v T implies more turbulent advec- 178 vt: -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0.002 0.004 0.006 vt: -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 -1 10 11 -2 12 X 10 11 12 (b) Case 3C.3, Z=5 vt: -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0.002 0.004 0.006 vt: -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0.002 0.004 0.006 Y -1 -1 10 11 -2 12 X 10 11 X (c) Case 3C.2*, Z=5.5 (d) Case 3C.3, Z=5.5 vt: -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0.002 0.004 0.006 Y Y X (a) Case 3C.2*, Z=5 -2 0.002 0.004 0.006 -1 -2 Y Y -1 -2 X (e) Spanwise averaged v T for flat plate Figure 4.24: Turbulent advective heat flux v T on X-Y planes 179 12 tive heat flux transported from the bottom wall into inner region of the channel. Thus, it is not surprising that dimpled surface is more efficient for heat transfer than flat surface, and the asymmetric dimple (Case 3C.3) performs even better than the symmetric dimple (Case 3C.2*). 4.7 Turbulent kinetic energy It is worth to investigate the turbulent kinetic energy inside dimples, and to compare the secondary motions in symmetric dimple and asymmetric dimple. Figure 4.25 shows the iso-surface of high turbulent kinetic energy component w = 7. It is found that the secondary motions reflected by high w is concentrated above the downstream rim of dimple, which coincide with the ejection associated with counter rotating vortices described in Ligrani et al. (2001a). In addition,the intensity and coverage area of such secondary motion are strengthened by asymmetric dimple compared to symmetric dimple, which is consistent with the finding in Figure 4.23. Y Y: -1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9 -0.85 -0.8 -0.75 Y X Y: -1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9 -0.85 -0.8 -0.75 Z X Z (a) Case 3C.2* (b) Case 3C.3 Figure 4.25: Iso-surfaces of high w = 7, red dots refer to the location of sampling point over dimples 180 4.8 Spectral analysis of velocity To study the frequency of ejection and secondary motions above the downstream rim of dimples, a sampling point is put in the core region of the high velocity fluctuation w to measure the velocity history. The sampling point is above the dimpled surface at a distance about y + = 6.6. Figure 4.26 shows the power spectral density (PSD) of velocity fluctuation at different angular frequency ω. As a reference, the PSD information of a sampling point in the flat channel with the same distance away from the bottom wall is also presented herein. It is observed that the power of velocity fluctuations above the asymmetric dimple is significantly enhanced at low frequency (ω ≈ 2–5) compared to symmetric dimple. In addition, the PSD of both symmetric and asymmetric dimple are much higher than that for flat channel at almost all frequencies. All these findings are good evidence of ejection and secondary motions at low frequency above the downstream rim of dimple, and show that asymmetric dimple can augment such motions significantly, thus resulting in large heat transfer enhancement. 0.08 flat symmetric dimple asymmetric dimple flat symmetric dimple asymmetric dimple 0.75 PSD for w’ PSD for v’ 0.06 0.04 0.02 0.5 0.25 10 10 10 angular frequency ω 10 10 10 angular frequency ω (a) v’ (b) w’ Figure 4.26: Power spectral density at different angular frequency of velocity fluctuations 181 4.9 Concluding remarks A systematic numerical investigation of heat transfer in turbulent flow between two opposite plates — at the Reynolds number Re 2H (based on bulk velocity and full channel height) ranging from 4,000 to 6,000 and Prandtl number Pr of 0.7 — has been conducted in this study via the detached eddy simulations (DES) model. A newly designed asymmetric dimple is introduced and compared with symmetric dimple in terms of drag, heat transfer rate N u, area goodness factor, and volume goodness factor. Dimples with different depth ratio, skewness and skewing directions (streamwise, spanwise and combination of both) are considered. Furthermore, the mean and instantaneous characteristics of thermo-aerodynamic field are presented to clarify the mechanisms for the enhanced heat transfer associated with the asymmetric dimple. The following conclusions are drawn. 1. The skewing of the center of dimple to the downstream side while maintaining the spherical shaped print diameter is a feasible way to enhance heat transfer with fairly similar pressure loss. Downstream skewness (at say Dx = 15%) increases the Nusselt number by 4% with accompanying drag reduction of 1% as compared to the symmetric dimple when keeping to the same depth ratio of h/D = 10%. When the depth ratio is increased to h/D = 15% such skewness increase further the Nusselt number by 23% with only 8% increase of drag compared to the symmetric dimple. Thus, both the area goodness factor and volume goodness factor have been significantly increased. 182 2. Skewing the center of shallow dimple (h/D < 20%) in the downstream direction provides a more efficient way to enhance heat transfer efficiency than only increasing its depth ratio (h/D ≥ 20%). 3. Skewing the center of dimple to upstream side evidently reduces the drag, but heat transfer decreases more because of a larger region of recirculation. As such, it leads to a reduction of the area goodness factor or volume goodness factor for such asymmetric dimple. 4. Skewing the center of dimple in the spanwise direction generates asymmetric flow structure in near wall region, which was believed to be helpful for heat transfer (Kornev et al., 2010). Although, the heat transfer rate is enhanced, the drag exhibits a larger increase. Consequently, the calculated area goodness factor and volume goodness factor are not as high as those for dimple whose center is only skewed downstream. 5. It is shown that shear drag and heat transfer are mainly induced in the vicinity of the downstream rim of dimple where flow ejection occurs. Thus the skewing of the deepest point of dimple caused the slope of the dimple wall to change and hence offset the drag and heat transfer in this area. In particular, the skewing of the deepest point of dimple downstream produces stronger ejection around the downstream rim of dimple, resulting in further enhancement of Nusselt number. 6. Both the mean and instantaneous velocity fields show that by skewing the dimple’s center downstream, the recirculation region in the upstream portion of dimple is suppressed. As is known, recirculation 183 does not contribute to enhancing the heat transfer. Thus, reducing the recirculation region can lead to increased heat transfer rate. Recirculation region will reduce skin friction and increase the form drag. However, the increment of heat transfer rate is higher than that for drag, thereby contributing to enhancement of thermo-aerodynamic efficiency of the asymmetric dimple. 7. A detailed analysis of the vorticity field reveals that there are streamwise counter-rotating vortex and upwash motion at the downstream portion of the dimple. A noteworthy phenomenon is that the vertical vortical structures observed above at the downstream rim of dimple is associated with fluid ejection which contributes to the heat transfer. Additionally, such stronger vortical structures are observed for the asymmetric dimple with positive streamwise skewness, leading to higher heat transfer as compared to the symmetric dimple. 8. Analysis of the vortex structures based on λ2 shows the mechanisms of enhancing heat transfer by the asymmetric dimple. The fact that most of the small eddies are generated in the downstream portion of dimple and are carried into the inner region of channel by ejections explains the higher local Nusselt number at this region. Furthermore, by skewing the deepest point of dimple downstream makes such eddies flat and closer to the downstream rim wall of dimple, thus leading to higher heat transfer rate than the symmetric dimple. 9. Investigation of the turbulent advective heat flux shows that the maxima of −v T in dimpled channel is higher than that in the flat channel. In dimple whose center is skewed downstream, the maxima 184 of −v T takes on even higher quantity and is located closer to the bottom of the dimpled wall. In summary, the asymmetric dimple with center skewed downstream makes for a more competitive heat exchange surface than symmetric dimple keeping to the same h/D and circular shape print diameter. The better performance of the asymmetric dimple is broadly attributed to its stronger flow ejection, weaker recirculation zone, stronger vortex and eddies, and higher turbulent advective heat flux. Still, more works are required to further optimize other parameters of asymmetric dimple, such as channel height, spacing of dimples, print diameter and rounded edge radius. Besides, future experimental investigations on non-symmetric dimple may be helpful to validate and support the findings herein. 185 [...]... 12 4 -4 Z 2 8 6 4 0 4 4 -2 0 2 7 2 -4 4 4 12 8 10 6 2 Z 4 2 1 0 4 Dx = −7.5% 6 -2 6 4 2 2 2 1 8 -2 8 1 6 10 12 6 8 10 X (a) Case 3C.1 (d) Case 3S.1 6 -8 1 1 8 12 X 8 4 9 -3 8 -6 3 2 6 0 -1 4 4 4 2 6 II III 9 Z 4 3 2 I 8 0 4 2 Z 1 3 Dx = 0% 1 8 -8 -3 8 6 6 2 2 -2 6 4 6 6 3 -8 8 3 4 9 2 2 1 6 1 8 10 12 6 8 X (b) Case 3C.2* 12 (e) Case 3S.2 12 4 8 -12 2 8 -16 12 10 X 0 0 -2 -4 -8 4 0 0 4 2 Z 8 8 4 Z 2 4. .. downstream rim of dimple and why asymmetric dimple (Case 3C.3) permits higher heat transfer than symmetric dimple (Case 3C.2*), the instantaneous flow field such as velocity and vortex structures are studied in this section 173 Dz = 0% Dz = 15% 30 25 20 8 8 30 20 45 25 15 15 45 10 45 50 25 Z 4 30 35 40 45 25 25 30 35 40 30 Z 4 45 10 20 35 Dx = −7.5% 50 6 15 6 10 40 45 45 30 40 45 45 50 15 45 15 45 10 15 6... 2 20 10 12 45 8 6 10 X 12 X (a) Case 3C.1 8 40 (d) Case 3S.1 30 25 20 8 45 30 20 25 20 15 45 10 45 50 10 40 55 50 50 30 Z 50 20 Z 25 10 35 50 30 40 4 50 25 30 35 40 35 40 30 4 25 45 Dx = 0% 6 15 6 55 45 15 45 15 15 15 15 6 30 25 30 35 2 8 2 10 12 6 8 X 12 (e) Case 3S.2 35 30 8 35 20 45 20 20 25 10 60 50 35 50 45 50 45 35 6 6 60 55 20 20 25 30 35 40 50 55 4 25 30 50 30 35 45 55 20 15 60 4 Z Z 15 15... Contours of λ2 on Z=5 for Case 3C.2* (d) Contours of λ2 on Z=5 for Case 3C.3 Figure 4. 23: Vortex structures 4. 6 Turbulent advective heat flux As turbulent advective heat flux v T plays a very significant role in the heat transfer of turbulent flow, it is thus deemed necessary to examine this quantity for Case 3C.2* (symmetric dimple) and Case 3C.3 Figure 4. 24 shows the contours of v T for Cases 3C.2* and 3C.3... 3C.2* 8 40 50 20 55 15 15 30 25 2 20 6 2 35 40 8 10 35 45 20 12 X 6 8 10 12 X (c) Case 3C.3 (f) Case 3S.3 Figure 4. 20: Nusselt number distribution for Case 3 (h/D = 0.15), the fluid flows from left to right 1 74 4.5.1 Flow field Similar to the mean velocity field plotted in Figures 4. 14 and 4. 15 in Section 4. 4.1, the instantaneous velocity field is plotted in Figures 4. 21 and 4. 22 for Cases 3C.2* and 3C.3,... superior thermo-aerodynamic performance in regard to the enhancement of compactness of a heat exchanger; this is true for both h/D = 10% (Case 2) and h/D = 15% (Case 3) By comparing Case 2C.3 or Case 2C .4 in Figures 4. 8 and 4. 9(c) with Case 3C.3 in Figures 4. 10 and 4. 9(d), the most concerned parameter N u/N u0 and Gv/Gv0 for Case 2C.3/2C .4 is higher than those achieved for Case 3C.3 This may suggest... bottom wall for the different sub-cases in Case 3 As is known, positive form drag contributes to increasing the total drag, and conversely negative form drag contributes to reducing the total drag Figure 4. 19(b) demonstrates positive form drag at the upstream portion (region I) and half of the downstream portion (region II) within the dimple, and negative form drag in the downstream rim of dimple (region... characteristics of drag and heat transfer Since Case 3 has higher heat transfer characteristics than Case 2, we shall examine in greater details the various sub-cases for Case 3 (h/D = 15%) 166 u: -0.08 -0. 04 0 0. 04 0.08 0.12 0.16 0.2 0. 24 0.28 0.32 u: -0.08 -0. 04 0.2 0. 24 0.28 0.32 6 Z 6 0. 04 0.08 0.12 0.16 8 Z 8 0 4 4 2 2 6 8 10 12 6 8 10 X (a) Case 3C.2* 12 X (b) Case 3C.3 Figure 4. 16: Contour of longitudinal... Ga/Ga0 for Dz=15% 1.1 Performance ratio Performance ratio 1.1 1 0.9 Ga/Ga0 for Dz=0% Ga/Ga0 for Dz=15% 1 0.9 symmetric dimple 0.8 -0.2 -0.1 0 0.1 0.8 0.2 -0.1 0 Dx 0.1 Dx (a) Ga/Ga0 for Case 2 at h/D = 10% (b) Ga/Ga0 for Case 3 at h/D = 15% 1.6 1.6 1.5 Performance ratio Performance ratio Gv/Gv0 for Dz=0% Gv/Gv0 for Dz=15% symmetric dimple 1 .4 Gv/Gv0 for Dz=0% Gv/Gv0 for Dz=15% 1.3 -0.2 -0.1 0 0.1 1.5 1 .4. .. 3C.2*) except for the Case 3C.1 (Dx = −7.5%, Dx = 0%) Further examination of Figures 4. 9(b) and 4. 9(d) suggests that the area 156 2.8 Cf/Cf0 for Dz=0% Nu/Nu0 for Dz=0% Cf/Cf0 for Dz=15% Nu/Nu0 for Dz=15% 2.7 Performance ratio 2.6 2.5 2 .4 2.3 symmetric dimple 2.2 2.1 2 1.9 1.8 symmetric dimple 1.7 1.6 -0.1 0 0.1 Dx Figure 4. 10: Friction and Nusselt number ratios for Case 3 at h/D = 15% and volume goodness . ratio h/D = 10%) and Case 3 (depth ratio h/D = 15%). Tables 4. 1 and 4. 2 summarize the configurations of Case 2 and Case 3, and Figure 4. 5 shows the position of deepest point of dimple for different. the performances such as the pressure loss C f /C f0 and heat transfer capacity Nu/Nu 0 for different heat transfer surfaces, respectively. 149 4. 3 Global thermo-aerodynamic performance 4. 3.1 Symmetric. dimple 4. 3.2.1 Case 2 (h/D=10%) Figure 4. 8 shows the pressure loss and heat transfer capacity for Case 2 (h/D = 10%, Table 4. 1). For Case 2C (Dz = 0%), the heat transfer ratio 151 depth ratio Performance