Chap chen y

such contra-rotating vortex pair is also supported by the experimental flow visualization in Figure of Ligrani et al. (2001b), though dimples are also installed on the opposite wall of protrusion which may influence the flow structure. The streamwise vorticity (ωx ) and streamlines on different Z-Y plane slices for h/D = 20% are shown in Figure 5.13(b). Different from the flow pattern for h/D = 10%, there are only two pairs of contra-rotating vortices or even two single vortex on each plane. These vortices can also be characterized into two groups pertaining to their locations similar to the above discussion of the vortical flow over protrusions at h/D = 10%: the group I vortex in the zone I (1.25 ≤ Z ≤ 3.75 and 6.25 ≤ Z ≤ 8.75); the group II vortex in the zone II (0 ≤ Z ≤ 1.25, 3.75 ≤ Z ≤ 6.25 and 8.75 ≤ Z ≤ 10). In the zone I, the group I vortex originating from the upstream valley continues to become weaker and weaker and even disappear over the forward-facing ridge and moves towards and over the backwardfacing ridge of the second row of protrusions (planes 1–5). Then the fluid which flows around the ridge of the second row of protrusion starts to rotate and form new pairs of contra-rotating vortices (planes 5–7). It may be noted that these new pairs of contra-rotating vortices are asymmetric. In the zone II, one half (i.e. the vortex with negative ωx ) of the group II contra-rotating vortices becomes more dominating while the other half (i.e. vortex with positive ωx ) diminishes and almost disappears completely (planes 1–3). Thereafter, this vortex with negative ωx enters the valley between the two protrusions on the second row (planes 3–5), and it takes a position closer to the side of protrusion which has resultant higher Nusselt 210 number. Further downstream, the group II vortex becomes weaker and then disappears at the upstream rim of the third row of protrusions (planes 6–7). On the combined behavior of the contra-rotating vortices in zones I and II, the flow structures of the fluid as it passes through the ”ridgevalley” topography of protrusions are summarized as below. New pairs of asymmetric contra-rotating vortices are generated on the back ridge of protrusions, then one half of them (i.e. vortex with negative ωx ) becomes dominant while the other half (i.e. vortex with positive ωx ) subsides and eventually is eliminated. Thereafter, this vortex with negative ωx enters the valley between the next row of protrusions and then becomes weaker and weaker till it reaches the next “ridge-valley” cycle. While the asymmetric vortex with negative ωx remain inside the valley between protrusions, it is closer to one side of protrusion (coincide with the side with the highest Nusselt number). It shall be noted that the contra-rotating vortices which are generated over the front ridge of protrusions with h/D = 10% not exist here (h/D = 20%). All these support the observation in the last section that the vortices are generated at the back ridge of protrusion and impinge on one side of the next row of protrusion, hence resulting in asymmetric friction and heat transfer distributions. 5.2.3.4 Instantaneous vortex structure To examine further the flow pattern inside the channel, the 3-dimensional instantaneous vortex structures in the vicinity of the bottom wall for 211 protrusions with height ratio h/D = 10% and 20% are investigated using the method as proposed by Jeong and Hussain (1995) and shown in Figure 5.14. The iso-surface of a proper negative λ2 close to zero (say at a quantity of −5 set in this study) can be used to identify the vortical structures near the bottom wall from the rest of complex vortical structures. Here, λ2 denotes the second largest eigenvalue of the tensor B wchich has been defined in §1.3.2.2 and can be used to identify the location of vortical structures. Figures 5.14(a)–(b) show the iso-surface of the above-mentioned λ2 , and the color refers to the height Y in order to clearly show the position of vortex. It is found that the small scale eddies are generated at the back ridge of protrusions and are elongated in the streamwise direction. For low protrusions (h/D = 10%), these small scale eddies are transported through the valley between the two protrusions and then impinged on the downstream protrusion on the forward-facing side. For high protrusions (h/D = 20%), the small scale eddies as they pass through the valley indicate a tendency for a portion to move up the slope of adjacent protrusion with some impingement, while the rest follow on through the valley and impinge on the downstream protrusions. The regions where the inclined vortex impinge on the protrusions coincide with the location with the highest Nusselt number. Furthermore, it is also noticed that the high protrusions (h/D = 20%) eject the vortex structures much deeper into the mid-plane of channel than the low protrusions (h/D = 10%), resulting in larger cross stream mixing and hence higher Nu. 212 Y 0.9 0.7 0.5 0.3 0.1 -0.1 -0.3 -0.5 -0.7 -0.9 Y 0.9 0.7 0.5 0.3 0.1 -0.1 -0.3 -0.5 -0.7 -0.9 flow direction flow direction 10 Z 10 Z (a) Vortex for h/D = 10% (b) Vortex for h/D = 20% Figure 5.14: Vortex structure identified by iso-surface of λ2 5.2.4 Turbulent kinetic energy It is worth to investigate the turbulent kinetic energy in the wake of protrusions, and to compare the fluctuating motions over the low protrusion and high protrusion. Figure 5.15 shows the iso-surface of high turbulent kinetic energy k = 3. It is found that the fluctuating motions with high TKE reflected by the iso-surface of k is concentrated in the wake of protrusions, which coincide with the the vortex structures shown in Figure 5.14. In addition, the TKE distributes asymmetrically about the streamwise centerline above the high protrusion (h/D=20%), while symmetrically above the low protrusion (h/D=10%). 5.2.5 Spectral analysis of velocity To study the frequency of asymmetric vortex in the wake of high protrusion, two sampling points are put in the core region of the high TKE as shown in Figure 5.15(b) to measure the velocity history. Two sampling points are 213 Y 0.9 0.7 0.5 0.3 0.1 -0.1 -0.3 -0.5 -0.7 -0.9 Y 0.9 0.7 0.5 0.3 0.1 -0.1 -0.3 -0.5 -0.7 -0.9 10 Z 10 Z (a) h/D=10% (b) h/D=20% Figure 5.15: Iso-surfaces of high turbulent kinetic energy k = 3, red dots refer to the location of sampling point over high protrusion located symmetrically above the centerline of the protrusion to show the non-symmetry of flow more clearly. Figure 5.16 shows the power spectral density (PSD) of velocity fluctuation at different angular frequency ω at these two points above the high protrusion. It is observed that the power of velocity fluctuations of point is significantly higher than that of point at relatively low frequency (ω ≈ 0.9–5). This supports that there is large scale asymmetric vortex in the wake of high protrusion, and shows the frequency of this flow structure which helps heat transfer enhancement at specific side of protrusion in the next row. 1.2 0.4 point point point point PSD for w’ PSD for v’ 0.3 0.8 0.6 0.4 0.2 0.1 0.2 10 10 10 angular frequency ω 10 10 10 angular frequency ω (a) v’ (b) w’ Figure 5.16: Power spectral density at different angular frequency of velocity fluctuations 214 5.3 Concluding remarks In this study, the heat transfer and flow structure of turbulent flow in a channel with single protrusion wall were investigated numerically by Detached Eddy Simulation (DES) method. Thermal and hydrodynamic performances of protrusions were examined in terms of friction, Nusselt number, thermal performance factors (expressed in terms of area/volume goodness ratios) and flow structures. It can be concluded that larger protrusion’s height induces higher friction factor and Nusselt number. This can be associated with the asymmetric flow for the larger height protrusions. However, as the height ratio increases, the friction factor increases more rapidly than the Nusselt number does. As such, the thermal performance factor increases first, and then reaches its asymptotic limit to be followed by decline. It is also found that the highest friction factors and Nusselt number are located at the upstream portion of protrusions, while the lowest friction factors and Nusselt number are found at the downstream portion. This can be attributed to the strong convection and impingement of fluid on the upstream portion of protrusions. In addition, the distribution of friction factor and Nusselt number are symmetric and asymmetric on protrusions when height ratio is low and high, respectively. Further investigation of the flow patterns and vortex structures over the two typical protrusions with different height ratios (h/D = 10% and 20%) reveals that the asymmetric hydraulic-thermal factor distribution at h/D = 20% are related to the asymmetric vortex and flow patterns. It is shown that symmetric vortex 215 and eddies are generate at the back ridge of low protrusions (h/D ≤ 10%). Conversely asymmetric vortex and eddies are generated at the back ridge of high protrusion (h/D ≥ 15%) and then incline to and impinge on one side of the next row of protrusions. The findings in this study provide guidance for possible optimization of protrusions for enhancing heat transfer over them. However, the effects of geometric properties of protrusions (i.e. print diameter and rounding edge radius) which have been initiated in this work can be further studied in the future. The mechanism of activation of asymmetric flow and vortex can be examined further with possibly other analytical tools, catering especially to the study of vortical field. 216 Chapter Overall conclusions and recommendations 6.1 Conclusions In this thesis, a detailed numerical investigation of drag reduction capability of corrugated surface, and heat transfer enhancement of asymmetric dimple and protrusions has been conducted using Detached Eddy Simulation (DES) and Direct Numerical Simulation (DNS) methods. Besides investigating the hydrodynamic and/or thermal performances of these modified surfaces, the flow patterns, drag distribution, heat transfer rate distribution and vortex structures were also carefully examined. These analysis are helpful to understand the mechanisms of drag reduction and heat transfer enhancement associated with drag and heat transfer in the modified channels. The detailed conclusions of each part are summarized as following. 217 6.1.1 Corrugated surface It was verified that corrugated surface with longitudinal wavy grooves) can create similar or higher drag reduction than riblets and other traditional drag reduction devices for turbulent channel flow. It was also found that lower wave number and higher amplitude lead to larger drag reduction. In turbulent channel flow, the numerical results show that corrugated surface can create drag reduction of about 10% when S = and α = 0.25. Furthermore, the theoretical analysis showed that the drag reduction of corrugated channel can approach up to 30–37% at high amplitude (S → 2) and small wave number (α → 0). In-depth examinations demonstrated that the drag reduction is achieved mainly through rearrangement of the bulk velocity distribution, i.e. reducing the bulk term of drag coefficient. Study of turbulent kinetic energy (TKE) and Reynolds stress showed that the corrugated surface rearranges their distributions but does not reduce their volume-averaged intensity. In addition, the investigation of corrugated channel with phase shift also supports the argument that the drag reduction is obtained mainly through rearranging more flow in the wider portion. 6.1.2 Heat transfer over asymmetric dimples The newly designed asymmetric dimple has been compared with symmetric dimple in terms of drag, heat transfer rate N u, area goodness factor, and volume goodness factor. 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 218 dimple. The skewing of the center of dimple to the downstream side while maintaining the circular print diameter is a feasible way to enhance heat transfer with fairly similar pressure loss. Furthermore, 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%). 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. 6.1.3 Heat transfer over protrusions Thermal and hydrodynamic performances of protrusions are examined in terms of friction, Nusselt number, thermal performance factor and flow structures. It shows that larger protrusion’s height induces higher friction factor and Nusselt number. This may be due to asymmetric flow which is activated over higher protrusions. However, when the height ratio increases, friction factor increases more rapidly than Nusselt number does, resulting in the thermal performance factor reaching an asymptotic limit followed by subsequent decline. It is also found that the highest friction factor and Nusselt number are located at the upstream portion of protrusions due to the strong convection and impingement of fluid on the upstream portion of protrusions. Additionally, the distribution of friction factor and Nusselt number are 219 The local maximum velocity and bulk velocity in arbitrary X-Y plane located at z are as following, respectively U0e U0 = =5 log10 uτ e = Reτ e 0.09 0.88 (log10 Reτ e − log10 0.09) 0.88 (D.18) =5.6818 log10 Reτ e + 5.9418 Ube = Ub U0 = − 2.4 u τ e uτ e (D.19) =5.6818 log10 Reτ e + 3.5418 Meanwhile, we also have the local effective friction Reynolds number and friction velocity as Reτ e = Reτ uτ e = u τ S + sin αz 1+ S sin αz 3/2 1/2 As such, we obtained the local bulk velocity in arbitrary X-Y plane located at z Ub = Ub Ub u τ e = uτ uτ e u τ = 5.6818 log10 Reτ 1+ S sin αz 3/2 = 5.6818 log10 Reτ + 8.5227 log10 + S + sin αz + 3.5418 S sin αz 1+ S sin αz 1/2 + 3.5418 1/2 (D.20) 259 D.3.3 Flux—integration of bulk velocity The full channel opening at the location z is 2δ = + S sin αz (D.21) Integrating the bulk velocity in the spanwise direction z leads to the flux Q Q= = uτ 2π α =2 2π 2δ Ub dz uτ 5.6818 log10 Reτ + 8.5227 log10 + S sin ξ + 3.5418 dz 5.6818 log10 Reτ + 8.5227 log10 + 1+ S sin αz 3/2 S + sin αz = α 2π α S sin ξ + 3.5418 3/2 dξ 2π (5.6818 log10 Reτ + 3.5418) = α 1+ 8.5227 log10 + S2 sin ξ [5.6818 log10 Reτ + 3.5418] S + sin ξ 3/2 dξ (D.22) The flux for flat channel can also be obtained in the same way: Q0 = Q0 = uτ 2h 2π α =2 Ub dz uτ (5.6818 log10 Reτ + 3.5418) dz (5.6818 log10 Reτ + 3.5418) 2π = α 260 (D.23) The flux ratio of the corrugated and flat channels at the same imposed friction Reynolds number Reτ can be extracted as: Q = Q0 2π 2π 1+ 8.5227 log10 + S2 sin ξ [5.6818 log10 Reτ + 3.5418] 1+ S sin ξ 3/2 dξ (D.24) D.3.4 Mean bulk velocity The cross section of one spanwise period is AΣ = 2h 2π 4π = α α (D.25) The average bulk velocity for flat channel is Ub flat = Q0 = 5.6818 log10 Reτ + 3.5418 AΣ (D.26) The average bulk velocity for corrugated channel is Ub cor = D.3.5 Q Q Q0 Q = = Ub AΣ Q0 AΣ Q0 flat (D.27) Skin friction coefficient The friction coefficient is defined as Cf = 2τw∗ ρ∗ Ub∗ =2 261 uτ Ub (D.28) So the friction coefficient for flat channel at the given Reynolds number Re2H is Cf = log10 Re2H − 2.4 (D.29) The friction coefficient for corrugated channel is Cf cor = uτ (D.30) Ub cor At the imposed friction Reynolds number, the bulk velocity for flat channel is Ub flat = 5.618 log10 Reτ + 3.5418 where the Reynolds number based on the bulk velocity in the flat channel is Re2H flat = Reτ 0.09 0.88 where Re2H flat is defined as ∗ Ub 2H ∗ = Ub flat Re τ Re 2H flat = flat∗ ν (D.31) Meanwhile, at the same imposed friction Reynolds number, the bulk velocity in the corrugated channel is Ub cor = Q Q Ub flat = (5.6818 log10 Reτ + 3.5418) Q0 Q0 (D.32) where the Reynolds number based on the bulk velocity in the corrugated 262 channel is 0.88 Reτ 0.09 Q Q Re2H flat = Re2H cor = Q0 Q0 (D.33) where Re2H cor is defined as ∗ Ub 2H ∗ Re 2H cor = cor∗ = Ub cor Re τ ν (D.34) The bulk velocity in the flat channel at Reynolds number Re2H = Re2H cor is Ub flat Re2H =Re2H cor =5 log10 Re2H cor − 2.4 =5.6818 log10 Reτ + 3.5418 + log10 (D.35) Q Q0 Thus, the friction coefficient of the corrugated channel and flat channel at the same Reynolds number Re2H = Re2H cor is Cf cor Cf flat = Ub flat Ub cor Re2H =Re2H cor = = Re2H =Re2H cor 5.6818 log10 Reτ + 3.5418 + log10 Q Q0 Q0 Q Q Q0 (D.36) (5.6818 log10 Reτ + 3.5418) 1+ log10 Q Q0 5.6818 log10 Reτ + 3.5418 263 D.4 Reτ = 180 In our simulation, Reτ = 180 is imposed, so the bulk velocity for the flat channel is Ub flat = Q0 = 5.6818 log10 180 + 3.5418 = 16.3558 AΣ where the Reynolds number based on the bulk velocity is Re2H flat = D.4.1 180 0.09 0.88 = 5639 S=0.5 The flux ratio at the given constant friction Reynolds number Reτ = 180 is Q = 1.01883 Q0 (D.37) Thus, the bulk velocity for the corrugated channel is Ub cor = Q Ub = 16.6638 Q0 flat where the Reynolds number based on the bulk velocity in the corrugated channel is Re2H cor = Q = 5745 Re Q0 2H flat 264 So, the bulk velocity in the flat channel at Re2H = Re2H cor = 5744 is Ub flat Re2H =5745 = log10 5745 − 2.4 = 16.3965 Finally, the friction coefficient ratio of the corrugated channel and the flat channel at Re2H = Re2H cor = 5744 is Cf cor Cf flat = Re2H =5745 Ub flat Re2H =5745 Ub cor = 16.3965 16.6638 = 0.96816 (D.38) The drag difference ΔD|α→0 ≈ −3.18% D.4.2 S=1 The flux ratio at the given constant friction Reynolds number Reτ = 180 is Q = 1.07574 Q0 (D.39) Thus, the bulk velocity for the corrugated channel is Ub cor = Q = 17.5946 Ub Q0 flat where the Reynolds number based on the bulk velocity in the corrugated channel is Re2H cor = Q Re = 6066 Q0 2H flat 265 So, the bulk velocity in the flat channel at Re2H = Re2H cor = 6066 is Ub flat Re2H =6066 = log10 6066 − 2.4 = 16.5145 Finally, the friction coefficient ratio of the corrugated channel and the flat channel at Re2H = Re2H cor = 6066 is Cf cor Cf flat = Re2H =6066 Ub flat Re2H =6066 Ub cor = 16.5145 17.5946 = 0.88097 (D.40) The drag difference ΔD|α→0 ≈ −11.90% D.4.3 Validation of results For validation of the theoretical prediction, the numerical results in Figure 3.13(a) are compared with the above results. It can be found that the asymptotic limit of drag difference obtained by the numerical simulation at S = 0.5 and are respectively around 3% and 11%, which matches well with the theoretical one with consideration of nonlinear effect near α → (the fitting line in Figure 3.13(a) is linear). This implies that the theoretical analysis can give good prediction of drag difference of the corrugated channel in turbulent flow when the wave number α is very small. Thus, the theoretical results at Reτ = 180 can be further extended to other Reynolds numbers and even Reτ → ∞. 266 Reτ → ∞ D.5 When the friction Reynolds goes very high, the flux ratio of the corrugated channel and flat channel at such constant given friction Reynolds number is Q Reτ →∞ Q0 lim lim = 2π Reτ →∞ = 2π 2π 2π 8.5227 log10 + S2 sin ξ 1+ [5.6818 log10 Reτ + 3.5418] S + sin ξ 3/2 dξ 3/2 S + sin ξ dξ (D.41) The friction coefficient ratio is lim Reτ →∞ Cf cor Cf flat Q0 Q = lim Reτ →∞ = D.5.1 Q0 Q Re2H =Re2H cor 1+ log10 Q Q0 (D.42) 5.6818 log10 Reτ + 3.5418 S=0.5 The flux ratio at the given constant friction Reynolds number Reτ → ∞ is Q = 1.01175 Reτ →∞ Q0 lim (D.43) Thus, the friction coefficient ratio of the corrugated channel and the 267 flat channel at the same Reynolds number Re2H = Re2H cor is lim Reτ →∞ Cf cor Cf flat = Re2H =Re2H cor Q0 Q = 0.97691 (D.44) The drag difference is ΔD|α→0 ≈ −2.31% D.5.2 S=1 The flux ratio at the given constant friction Reynolds number Reτ → ∞ is Q = 1.04746 Reτ →∞ Q0 lim (D.45) Thus, the friction coefficient ratio of the corrugated channel and the flat channel at the same Reynolds number Re2H = Re2H cor is lim Reτ →∞ Cf cor Cf flat = Re2H =Re2H cor Q0 Q The drag difference is ΔD|α→0 ≈ −8.86% 268 = 0.91143 (D.46) Appendix E Configuration of asymmetric dimple In this part, we discuss the configuration of our asymmetric dimples. The original symmetric dimples can be expressed by ψ (x, y, z) = 0. (E.1) The origin of the coordinate axis-system is located at the center of the symmetric dimple’s edge (Figure E.1). The asymmetric dimple is obtained by simple shear deformation from the symmetric dimple ⎧ ⎪ ⎪ ⎪ x = x + dp · y ⎪ ⎪ ⎪ ⎨ y =y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z = z + dq · y (E.2) where dp denotes how much the dimple is skewed in the negative x269 Y X Z Figure E.1: Coordinate system in 3D rendition of the dimple direction, dq denotes how much the dimple is skewed in the negative zdirection and the depth of dimple is given by y. (As such the center of the symmetric dimple is at −h). The new feature is ⎧ ⎪ ⎪ ψ (x, y, z) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x = x − dp · y ⎪ ⎪ ⎪ y=y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z = z − dq · y ⎧ ⎪ ⎪ ⎪ ψ (x , y , z ) ⎪ ⎪ ⎪ ⎨ ⇒ = ψ (x − dp · y , y , z − dq · y ) . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩= (E.3) The core region of the undeformed (or symmetric) dimple’s geometry is expressed as (y − R + h)2 + (x − xc )2 + (z − zc )2 = R2 . 270 (E.4) The deformed surface of this portion can be calculated as (y − R + h) + (x − xc − dp · y ) + (z − zc − dq · y ) = R2 . (E.5) Expanding and simplifying the equation, we obtain the geometric function of dimple’s core region: + dp2 + dq y − [(R − h) + (x − xc ) dp + (z − zc ) dq] y (E.6) + (R − h) + (x − xc ) + (z − zc ) = 0. The negative root of this binomial equation would be the vertical coordinate y of the dimpled wall. The rounded edge portion of symmetric dimple can be expressed as: ⎧ ⎪ ⎪ ⎨ (y + r)2 + (xr − xE )2 = r2 ⎪ ⎪ ⎩ xr = (E.7) (x − xc ) + (z − zc ) . So the deformed geometry can be written as: ⎧ ⎪ ⎪ ⎨ (y + r)2 + (xr − xE )2 = r2 ⎪ ⎪ ⎩ xr = (E.8) (x − dpy − xc ) + (z − dqz − zc ) . 271 On expanding and simplifying the equation, we obtain Ay + By + Cy + Dy + E = ⎧ ⎪ ⎪ ⎪ A = a2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B = 2ab ⎪ ⎪ ⎪ ⎨ C = 2ac + b2 − dp2 + dq ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D = 2bc + [dp (x − xc ) + dq (z − zc )] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ E = c2 − (x − x )2 + (z − z )2 ⎩ c c where ⎧ + dp2 + dq ⎪ ⎪ ⎪ a = ⎪ ⎪ 2xE ⎪ ⎪ ⎨ r − dp (x − xc ) − dq (z − zc ) b= ⎪ xE ⎪ ⎪ ⎪ ⎪ ⎪ (x − xc ) + (z − zc )2 + x2E ⎪ ⎩c = . 2xE (E.9) (E.10) The biggest real root for this 4th order polynomial equation is the depth of rounded edge region. We introduce a new factor called skewness (Dx and Dz), which is defined as the displacement of the center of dimple respectively in the x-direction and z-direction non-dimensionalized by the print diameter D (Figure 4.2 shows such displacement in x-direction when dp < and dq = 0): Dx = dp × (−h) = −dp × depth ratio, D (E.11a) Dz = dq × (−h) = −dq × depth ratio. D (E.11b) 272 Negative symbol in Eq. (E.11) indicates depth of dimple are negative. Absolute value of skewness cannot be larger than 50%, otherwise the downstream/upstream portion of dimple would be vertical. Thus, dp and dq cannot be more than dpcrtic and dqcritic : dpcritic = dqcritic = 273 D . 2h (E.12) Vita Chen Yu was born on 26 May, 1986 in Nantong, Jiangsu, China. He graduated from the Peking University (PKU), in China, in July 2008 with a Bachelor of Science degree in Theoretical and Applied Mechanics. He joined National University of Singapore in August 2008 in order to study for the degree of Doctor of Philosophy in Mechanical Engineering. 274 [...]... u(2) (±1, z) + S 3 u(3) (±1, z) + · · · yy yy yy yy S 1 ± sin (αz) 2! 2 2 + u(0) (±1, z) + Su(1) (±1, z) + S 2 u(2) (±1, z) + S 3 u(3) (±1, z) + · · · yyy yyy yyy yyy S 1 ± sin (αz) 3! 2 3 +··· = 0 (A.7) which can be rearranged as 1 u(0) (±1, z) + S u(1) (±1, z) ± u(0) (±1, z) sin (αz) 2 y 1 1 +S 2 u(2) (±1, z) ± u(1) (±1, z) sin (αz) + u(0) (±1, z) sin2 (αz) y 2 8 yy 1 1 +S 3 u(3) (±1, z) ± u(2) (±1,... (αz) , z 2 =u (±1, z) + uy (±1, z) ± S sin (αz) 2 S 1 + uyy (±1, z) ± sin (αz) 2! 2 + 1 S uyyy (±1, z) ± sin (αz) 3! 2 2 (A.3) 3 + ··· =0 where uy = ∂u , y uyy = ∂2u y 2 and uyyy = ∂3u y 3 Meanwhile, perturbation theory indicates that the solution u (y, z) can be written in series of : u = u(0) + Su(1) + S 2 u(2) + S 3 u(3) + · · · (A.4) Substituting the series form of u (y, z) i.e Eq A.4 into the... (±1, z) sin2 (αz) y 2 8 yy ± (A.8) 1 (0) u (±1, z) sin3 (αz) 48 yyy +··· = 0 So, the fact that B.C should be satisfied for arbitrary S 1 indicates that u(0) (±1, z) = 0 (A.9) 1 u(1) (±1, z) = ∓ u(0) (±1, z) sin (αz) 2 y (A.10) 240 1 1 u(2) (±1, z) = ∓ u(1) (±1, z) sin (αz) − u(0) (±1, z) sin2 (αz) y 2 8 yy (A.11) 1 1 u(3) (±1, z) = ∓ u(2) (±1, z) sin (αz) − u(1) (±1, z) sin2 (αz) y 2 8 yy 1 (A.12) ∓ u(0)... 48 yyy A.2 Solution A.2.1 Velocity field The 0-order solution u (y, z) which satisfies ⎧ ⎪ ⎪ Δu(0) = −Re τ ⎪ ⎪ ⎪ ⎪ ⎨ (0) ⎪ u (±1, z) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u y, − π = u y, π α α can be written as u(0) = 1 − y2 Re τ 2 (A.13) The higher order solution u(i) (y, z) are symmetric about y = 0, and periodic in z-direction Thus, by variable separation method, the general solution is u(i) (y, z) = C0 + cosh (k y) [C1k... 1 + S sin (αz) , z ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ ⎪ u (−1, z) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u y, − π = u y, π ⎩ α α =0 (B.2) Applying perturbation method, the boundary condition can be rewrite 244 like: u 1+ S sin (αz) , z 2 =u (1, z) + uy (1, z) S sin (αz) 2 S 1 sin (αz) + uyy (1, z) 2! 2 1 S + uyyy (1, z) sin (αz) 3! 2 2 (B.3) 3 + ··· =0 Meanwhile, the solution u (y, z) can be written in series of S: u = u(0) + Su(1) + S 2 u(2)... Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368 (1929), 4775 Doo, J., Yoon, H., Ha, M., 2010 Study on improvement of compactness of a plate heat exchanger using a newly designed primary surface International Journal of Heat and Mass Transfer 53 (25-26), 5733–5746 Eaton, J., 1995 Effects of mean flow three dimensionality on turbulent boundary-layer structure AIAA... turbulent pipe flows by circular-wall oscillation Physics of Fluids 10, 7 Choi, K.-S., 1989 Near-wall structure of a turbulent boundary layer with riblets Journal of fluid mechanics 208, 417–458 Chong, M., Perry, A., Cantwell, B., 1990 A general classification of threedimensional flow fields Physics of Fluids 2, 408–420 Chyu, M., Yu, Y. , Ding, H., Downs, J., Soechting, F., 1997 Concavity 224 enhanced heat... 237 Appendix A Analytical solutions of channel with both corrugated walls A.1 Governing equations The streamwise velocity u satisfies the partial differential equation (P.D.E.) Δu = ∂ 2u ∂ 2u + = −Re τ y 2 ∂z 2 (A.1) with non-slip and periodic B.C.s ⎧ ⎪ ⎪ u ±1 ± S sin (αz) , z ⎨ 2 ⎪ ⎪ ⎩ u y, − π = u y, π α α Applying Taylor series expansion (S 238 =0 (A.2) 1), the boundary condition at y = ±1 ± S 2 sin... 2001 Numerical analysis of the effect of viscosity on the vortex dynamics at laminar separated flow past the dimple on a plane with allowance its asymmetry Journal of Engineering Physics and Thermophysics 74 (2), 339–346 Itoh, M., Tamano, S., Iguchi, R., Yokota, K., Akino, N., Hino, R., Kubo, S., 2006 Turbulent drag reduction by the seal fur surface Physics of Fluids 18, 065102 Iuso, G., Onorato, M.,... Caruelle, B., Ducros, F., 2003 Detached-eddy simulations of attached and detached boundary layers International Journal of Computational Fluid Dynamics 17 (6), 433–451 Ceccio, S., 2010 Friction drag reduction of external flows with bubble and gas injection Annual Review of Fluid Mechanics 42, 183–203 Chen, Y. , Chew, Y T., Khoo, B C., 2010 Turbulent flow manipulation by passive devices In: Proceedings of the . distributes asymmetrically about the streamwise centerline above the high protrusion (h/D=20%), while symmetrically above the low protrusion (h/D=10%). 5.2.5 Spectral analysis of velocity To study the. protrusion located symmetrically above the centerline of the protrusion to show the non-symmetry of flow more clearly. Figure 5.16 shows the power spectral density (PSD) of velocity fluctuation at. mechanism of activation of asymmetric flow and vortex can be examined further with possibly other analytical tools, catering especially to the study of vortical field. 216 Chapter 6 Overall conclusions