Chapter Corrugated surface The interest in this chapter is to examine a novel means—the corrugated surface (longitudinal grooves)—in laminar and turbulent channel flow, and determine the performance and (possible) mechanism(s) for drag reduction. Though Floryan (2011) has investigated drag over corrugated surface in laminar channel flow, the amplitude of corrugated surface studied in this dissertation is much larger than that in his study. The investigation in this chapter is unique since there is no prior work on the performance of macro longitudinal grooves in turbulent channel flow. In this chapter, the geometry of corrugated surface is briefly introduced. The analytical and numerical results of laminar channel flows over corrugated surface are presented in Section 3.2. Thereafter, turbulent channel flow over corrugated surface is investigated in Section 3.3. Both analytical tool using perturbation method (Nayfeh, 1993) and numerical methods are employed in this study. Perturbation method is employed to obtain the analytical solution of laminar channel flow over corrugated 70 surface. For turbulent channel flow, a finite-volume-based parallel Detached Eddy Simulation (DES) code, which requires less computation resources than Direct Numerical Simulation (DNS), is used. However DNS is employed on several selected cases of turbulent flow to check on the accuracy of DES code.) 3.1 Geometry of corrugated channel In this study, fluid flows inside a channel with non-dimensional length L, width W and mean height 2H = in the x, z and y direction, respectively (see Figure 3.1). In addition, all the dimensions have been non-dimensionalized by the reference length scale — half channel height H ∗ . The superscript superscript ∗ ∗ indicates dimensional quantities, quantities without are dimensionless. Note that there are mainly three different cases investigated in this chapter: Case Both walls have grooves (see Figure 3.1(a)). Case Only single wall has groove and the other is flat (see Figure 3.1(b)). Case Both walls are flat. The reduced order geometry model generated by projection of the wall shape onto a Fourier space has been proven to be sufficient to predict the drag coefficient with an acceptable accuracy for laminar channel flow in 71 Floryan (2011). Thus, only sinusoidal grooves are investigated for laminar channel flow in this study. However, the applicability of reduced order geometry model has to be verified for turbulent channel flow first before we focus on sinusoidal grooves in turbulent channel flow. In this section, all the wall shapes tested in this study will be introduced, including sinusoidal, triangular and trapezoidal grooves. 3.1.1 Sinusoidal grooves First, the most important shape—sinusoidal grooves—that only has single mode in Fourier space is introduced. Corrugated channel with sinusoidal grooves can be defined as plate with smooth longitudinal wavy grooves in such a way (see Figures 3.1 and 3.2): • Corrugated bottom wall: y = −1 − • Corrugated top wall: y = + S S sin (αz), sin (αz), where S and α are, respectively, the amplitude and wave number of grooves. Thus the wavelength of grooves in the spanwise direction is λ = 2π/α. Note that the width of channel W in our study is always integral multiples of groove wavelength λ and the top and bottom walls are 180◦ out of phase. 3.1.2 Other groove shapes Some other general wall shapes are also investigated in this chapter to validate the capacity of the reduced order geometry method, especially 72 Y Y X X Z Z 2H W λ λ L S flow direction 2H W L S flow direction (a) (b) Figure 3.1: Channel with (a) two corrugated walls (b) single corrugated wall. Sinusoidal groove is taken as an example here. in turbulent channel flow. Figure 3.2 illustrates two shapes selected for testing, e.g., triangular grooves, trapezoidal grooves together with the sinusoidal grooves. All these grooves were given the same amplitude S and the same effective wave number α in Figure 3.2. Note that the triangular groove can be considered as special trapezoidal groove with b = 0. 0.8 0.6 0.4 Y 0.2 triangular S -0.2 sinusoidal c b a -0.4 -0.6 trapezoidal -0.8 -1 0.2 0.4 0.6 0.8 Z/λ Figure 3.2: Sketches of the grooves used in the analysis: sinusoidal, triangular and trapezoidal 73 3.2 Laminar channel flow The results obtained by Floryan (2011) has demonstrated that the hydrodynamics for most geometry in laminar channel flow can be generally determined by the dominant Fourier mode. Thus, the reduced order geometry method reduces the number of dominant geometric factors to just two for longitudinal grooves in laminar channel flow, i.e., the wave number α, the amplitude S. In this section, analytical results obtained by perturbation method are presented. Parametric investigations of wave number α and amplitude S, and possible physical mechanism(s) of drag reduction in laminar channel flow are demonstrated as the preliminaries for further investigation on turbulent channel flow. Additionally, numerical results for laminar channel flow are also presented in this section for comparison to analytical solution to validate the numerical method’s capability on this problem. 3.2.1 Simplified governing equations Considering the non-slip and periodic boundary conditions applied at the walls and in z-direction, respectively, the laminar channel flow is 2dimensional (uniform in x-direction) and steady (time-independent). As such, we have v = w = 0, ∂u/∂t = ∂u/∂x = 0, and ∂p/∂y = ∂p/∂z = 0. Thus the N-S equations are simplified into the following form: ∂p ∂ 2u ∂ 2u + = Re τ . ∂y ∂z ∂x 74 (3.1) The pressure gradient β = −∂p/∂x is set at a constant 1, so the streamwise velocity u satisfies the partial differential equation (P.D.E.) Δu = ∂ 2u ∂ 2u + = −Re τ ∂y ∂z (3.2) with non-slip B.C. on the bottom and top walls and periodic B.C. in the spanwise direction: ⎧ ⎪ S ⎪ ⎪ ⎨ u ±1 ± sin (αz) , z ⎪ 2π ⎪ ⎪ ⎩ u (y, 0) = u y, α =0 for both corrugated walls (3.3) for single corrugated wall (3.4) or ⎧ S ⎪ ⎪ ⎪ u + sin (αz) , z ⎪ ⎪ ⎪ ⎪ ⎨ u (−1, z) = ⎪ ⎪ ⎪ ⎪ ⎪ 2π ⎪ ⎪ ⎩ u (y, 0) = u y, α =0 Different from the original nonlinear Navier Stokes equations, Eq. 3.2 is linear thus amenable to solution using analytical techniques such as perturbation methods. 3.2.2 Theoretical solution The analytical solutions of laminar flow in channel with both corrugated walls and with single corrugated wall (i.e. velocity and flux) can be derived by perturbation method (see details in Appendix A and B). The flow rate for channel with both corrugated walls in the one wave length λ = 2π/α 75 can be expressed as Q (α, S) ≈ 4πRe τ 3S αS 1+ I1 3α 2α − 3αS (α) , (3.5) where I1 is the first order modified Bessel function of the first kind. Similarly for channel with single corrugated wall, the flow rate is Q (α, S) ≈ αS 4πRe τ 3S 1+ I1 3α 4α − 3αS coth (2α) . 16 (3.6) When the amplitude S tends to zero, corrugated channel becomes flat within the same width λ = 2π/α, in which the flow rate is Q0 = 4πRe τ . 3α (3.7) It can be found that the flow rate ratio between corrugated channel and flat channel Q/Q0 is independent of frictional Reynolds number Re τ , and only depends on wave number and amplitude of longitudinal grooves. Thus the normalized flow rate difference at fixed frictional Reynolds number Re τ is defined as ΔQ (α, S) = Q − Q0 Q = − 1, Q0 Q0 (3.8) which can also be used to evaluate the level of drag reduction. It should be noted that since the pressure gradient is a constant, a positive flow rate difference indicates drag reduction and conversely negative flow rate difference indicates drag increase. The asymptotic limit for flow rate 76 difference can also be obtained as lim ΔQ =0 ⎧ ⎪ ⎪ ⎪ 3S ⎪ ⎪ ⎨ lim ΔQ = α→0 ⎪ ⎪ ⎪ 3S ⎪ ⎪ ⎩ 32 S→0 for both Cases & (3.9) for Case (3.10) for Case 3.2.3 Global performance of drag difference 3.2.3.1 The definition of drag difference To find the quantitative relationship between flow rate increase ΔQ and drag reduction, the normalized drag coefficient difference at fixed Reynolds number Re 2H can be defined as ΔD = Cf − Cf , Cf (3.11) where Cf is the drag coefficient for corrugated channel, Cf is the drag coefficient for flat channel. According to the definitions of drag coefficient Cf (Eq. 2.19), bulk velocity Ub (Eq. 2.21) and Re2H (Eq. 2.27), we have the following relation: Cf = = 2β Ub2 2β Q Re2H AΣ 2Reτ 77 (3.12) where Q = AΣ = =3 Q Q0 Q0 AΣ Reτ = constant Q 4πRe τ /3α Q0 4π/α Q Re τ Q0 Reτ = constant Reτ = constant . Substituting it back to Eq. 3.12, one can obtain the friction coefficient for corrugated surface 2β Cf = Q Re τ Q0 Re2H Reτ = constant 2Reτ Q0 4β = . Q Reτ = constant 3Re2H (3.13) Similarly, we also have the friction coefficient for flat channel as follows: Cf = 4β . 3Re2H (3.14) Thus drag coefficient ratio at fixed Reynolds number Re 2H can be written as Cf Cf = Re2H = constant Q0 Q β, Reτ = constant . (3.15) In other words, drag coefficient ratio at the given Re 2H is inversely proportional to flow rate ratio at given Re τ . With some modifications and simplifications, it can be obtained that ΔD = Q0 ΔQ −1=− . Q ΔQ + 78 (3.16) It is natural to note that the drag difference is independent of Reynolds number Re 2H just like the flow rate does. It can be observed that for given pressure gradient, flow rate increase indicates drag reduction and conversely negative flow rate difference indicates drag increase. Furthermore, higher flow rate increase indicates more drag reduction. Similar to flow rate difference, the normalized drag difference ΔD is also only dependent on wave number α and amplitude S. 3.2.3.2 Theoretical prediction The normalized drag coefficient differences ΔD for corrugated surfaces are plotted in Figures 3.3 and 3.4. It is observed that there exists a critical wave number αc , above which there is drag increase and below which there is drag reduction. Such critical wave number αc for the channel with both corrugated walls is about 1.2, while the critical wave number αc for the channel with only single corrugated wall is about 0.96. Thus the wave number α should be smaller than this critical wave number αc to obtain drag reduction in corrugated channel. It is also observed that lower wave number α produces higher drag reduction and there is an asymptotic limit of drag difference when α approaches zero. Drag reduction for channel with both corrugated walls is about four times of that for channel with only single corrugated wall (it can also be deduced from Eqs. 3.10 and 3.16). Furthermore, higher amplitude S brings more drag reduction when wave number α < αc , while higher amplitude S brings more drag increase when wave number α > αc . Additionally, both drag and flow rate difference go to zero when amplitude S approaches zero as shown in Eq. 3.9. This is 79 9% total skin friction drag reduction. Conversely, when the wave number is large (α > αc , herein is α = and S = 0.5), both the turbulent term and the bulk term increase by 2%–3% compared to a flat surface, resulting in 5% total skin friction drag increase. Overall, it can be concluded that drag reduction of the corrugated surface is achieved through suppressing bulk term of skin friction drag Cf b . Surface Bulk (Cf b /Cf ) Turbulent (Cf t /Cf ) Total (Cf /Cf ) Flat 27.78% 72.22% 100.00% α = 0.5, S = 0.5 25.06% 73.13% 98.18% α = 0.5, S = 21.48% 75.44% 96.92% α = 0.33, S = 20.56% 71.00% 91.56% α = 2, S = 0.5 29.78% 75.59% 105.37% Table 3.5: Ratio of each term of drag coefficient to the total drag coefficient on flat plate It is also found that when the amplitude is fixed, higher wave number produces both higher bulk and turbulent terms of drag coefficient, leading to larger total drag coefficient. Another interesting feature is found by comparing the results of α = 0.5, S = 0.5 and α = 0.5, S = that larger amplitude increases the turbulent term of drag Cf t /Cf , but a greater reduction of the bulk term Cf b /Cf compensates for it, hence resulting in greater total drag reduction. Thus, low wave number α and high amplitude S favour the reduction of total drag, which is similar to the conclusion for laminar flow. To investigate why the effects of wave number and amplitude on total drag of corrugated channel in turbulent flow are similar to those in laminar flow, the key factor of drag reduction—reduction of bulk term of drag Cf b — 126 Surface Cf b /Cf b0 in turbulent flow Cf /Cf in laminar flow Flat 100.00% 100.00% 90.21% 93.32% α = 2, S = 0.5 107.20% 108.36% α = 0.33, S = 74.01% 75.21% α = 0.5, S = 77.32% 78.16% α = 0.5, S = 0.5 Table 3.6: Ratio of each term of drag coefficient on corrugated surface to that on flat plate for laminar flow and turbulent flow are further examined. The total drag ratio can be rewritten as Cf b + Cf t Cf b Cf b0 Cf t Cf t0 Cf = = + , Cf Cf Cf b0 Cf Cf t0 Cf (3.23) which respectively represent the contributions of bulk drag ratio and turbulent drag ratio. Note that the prescribed drag decomposition theory is applicable to laminar flow, and the turbulent term Cf t is zero in laminar flow. So in laminar flow, the ratio of total drag coefficient Cf /Cf equals to the ratio of bulk term of drag coefficient Cf b /Cf b0 . According to the theory of Peet and Sagaut (2009) (see Eqs. 3.19 and 3.21), the ratio of bulk term of drag of corrugated channel to that of flat channel Cf b /Cf b0 at the same Reynolds number can be rewritten as A0 Cf b . = Cf b0 A (3.24) Note that all drag coefficients of both corrugated and flat channels in this study have been interpolated to the same Reynolds number Re. Thus it is obvious to deduce that Cf b /Cf b0 in turbulent flow should be the same 127 as Cf /Cf in laminar flow for corrugated channel with the same geometry; this is because A only depends on the geometry of channel’s cross section. The comparison of these two ratios (Cf b /Cf b0 in turbulent flow and Cf /Cf in laminar flow) is presented in Table 3.6. It shows that apart from the numerical errors the two ratios agree well with each other for corrugated channel with the same geometry. This further supports the conclusion that drag reduction of corrugated channel in turbulent flow is achieved through the same mechanism as in laminar flow. Additionally, Eq. 3.23 also implies that not only Cf b /Cf b0 affects the total drag ratio Cf /Cf , the proportion of bulk term in total drag Cf b0 /Cf and ratio of turbulent term of drag Cf t /Cf t0 are also important factors. The total drag reduction of corrugated surface in turbulent channel flow (3%–10%) is about one third of that in laminar flow (10%–30%) because of low proportion of bulk term of drag in total drag for turbulent flow (i.e. Cf b0 /Cf is about 27%). Furthermore, the unexpected increase of turbulent term of drag (i.e. Cf t /Cf t0 ≥ 1) can partially offset the reduction of bulk term, thus affecting the total drag ratio Cf /Cf . 3.3.6 Theoretical prediction at small wave number When α → 0, the interaction between each portion (including the wide and narrow portions) is weak, each portion acts like a smooth flat channel with different channel heights and constant pressure gradient. The flow behavior is dominated by local conditions: local friction velocity, local friction Reynolds number and local non-dimensional wall distance, etc. 128 Based on this assumption, the flow rate ratio and drag difference has been obtained theoretically in Appendix D. The flow rate ratio of the corrugated channel and flat channel at the constant friction Reynolds number is Q = Q0 2π 2π 8.5227 log10 + S2 sin ξ 1+ [5.6818 log10 Reτ + 3.5418] S + sin ξ 3/2 dξ. (3.25) The skin friction coefficient ratio of the corrugated channel and flat channel at the same full channel height Reynolds number is Cf Cf = Re2H =constant Q0 Q 1+ log10 Q Q0 5.6818 log10 Reτ + 3.5418 . (3.26) This theoretical result has been compared with the numerical results for the corrugated channel with S = 0.5 and S = at Reτ = 180 to validate the theoretical prediction in Appendix D.4.3. Furthermore, this theoretical result can also be extended to other cases which cannot be simulated due to the cost and limitation of the numerical methods. Figure 3.28 shows the theoretical prediction of drag difference ΔD at different Reynolds numbers and amplitude S at very small wave number α. In order to present the effect of Reynolds number on drag reduction clearly, the results of drag difference for the corrugated channel in laminar channel flow is also plotted (results are extracted from Eq. 3.10). It is demonstrated that at arbitrary Reynolds number, the magnitude of the drag reduction increases rapidly and nonlinearly with an increase in the amplitude S. It can also be found that when flow transits from laminar to turbulent, there is a big drop in magnitude of drag reduction. 129 However, in the turbulent region the magnitude of drag reduction ΔD decreases very slowly to the asymptotic limit when Reynolds number increases. Furthermore, the asymptotic level of drag reduction is still very significant when the Reynolds becomes infinitely high. Besides, it can be inferred that in order to maximize drag reduction, the wave number α shall be very small (α → 0) and the amplitude S shall be large (S → 2). For such situations, the drag reduction can go up to 30%–37% in turbulent region, depending on Reynolds number. Though such drag reduction is not as high as that for laminar channel flow (60%), it is still much higher than most other traditional drag reduction methods. 0.1 0.1 ΔD -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 laminar Reτ=180 Reτ=10 10 Reτ=10 Reτ→ ∞ -0.4 -0.5 -0.6 -0.7 0.5 -0.4 -0.5 -0.6 1.5 -0.7 S Figure 3.28: The drag difference ΔD at different Reτ and S when α → Furthermore, it can also be concluded from the analysis that at small wave number, the flux increase and drag reduction induced by corrugated channel is obtained mainly through bulk velocity rearrangement instead of suppression of turbulence motions. 130 3.3.7 Additional cases with phase shift According to the results and discussion above, a hypothesis can be established that the corrugated channel increases flow rate at given pressure gradient because of velocity rearrangement induced by the alternating wide and narrow portions in the cross section. To further validate this hypothesis, it is necessary to investigate additional cases of corrugated channel whose wide and narrow portions are modified or even eliminated. The original corrugated channel is: • Corrugated bottom wall: y = −1 − • Corrugated top wall: y = + S S sin (αz), sin (αz). Shifting the top corrugated wall with respect to the bottom one in such way: • Corrugated bottom wall: y = −1 − • Corrugated top wall: y = + S S sin (αz), sin (αz + ϕ), where the phase shift ϕ will modify the alternating wide and narrow portions of channel. When ϕ = 0, the channel has the biggest wide portion and the smallest narrow portion, which is called ‘converging-diverging channel’; when ϕ = π, the channel becomes ‘sinusoidal channel’ (different from ‘sinusoidal grooves’) where the wide and narrow portions disappear (see Figure 3.30(c)). And it is natural to believe that corrugated channels with ϕ = ϕ0 , ϕ = π − ϕ0 and ϕ = 2π + ϕ0 (ϕ0 is an arbitrary phase shift) 131 will induce the same amount of drag difference because they have actually the same geometry considering the symmetry and periodicity in spanwise direction. Thus it is only necessary to examine the effects of phase shift ϕ in the range of [0, π]. sinusoidal channel 0.02 ΔD -0.02 -0.04 converging-diverging channel ϕ Figure 3.29: Effects of phase shift ϕ on drag coefficient difference ΔD For computation convenience, only corrugated channels with α = 0.5, S = are investigated, and the phase shift ϕ is taken as 0, π/4, π/2, 3π/4 and π. It can be found in Figure 3.29 that as phase shift ϕ increases, corrugated channel induces less drag reduction and even produces drag increase. Thus ‘converging-diverging channel’ (i.e. phase shift ϕ = 0) is the most effective corrugated channel to reduce drag with such fixed wave number α = 0.5 and amplitude S = 1. To discover the mechanisms how phase shift affects drag, the local full 132 channel height of cross section 2δ at arbitrary position z can be written as: 2δ =1 + S S sin (αz + ϕ) − −1 − sin (αz) 2 S [sin (αz + ϕ) + sin (αz)] ϕ ϕ =2 + S cos sin αz + . 2 =2 + In corrugated channel, the height of cross section varies in the range of [2 − cos ϕ S, + cos ϕ S] with the assumption ≤ ϕ ≤ π. Considering that cos (ϕ/2) ranges from to 1, it can be deduced that: 1. In the ‘converging-diverging channel’ (ϕ = 0), the height of the narrow portion reaches its lower limit − S (at z = λ/4), while the height of the wide portion reaches its upper limit + S (at z = 3λ/4). In other words, the wide portion and narrow portion are respectively largest and smallest at given amplitude S, resulting in optimal velocity rearrangement (see Figure 3.30(a), Uwide = 22 and Unarrow = 16) and the largest drag reduction (see Figure 3.29). 2. In ‘sinusoidal channel’ (ϕ = π), the height at arbitrary position of the cross section is constant at 2, indicating no alternating wide and narrow portions. Thus the rearrangement of velocity (alternating high and low speed) in cross section disappears (see Figure 3.30(c), the highest velocity is uniform along the centerline). Furthermore, ‘sinusoidal channel’ has more wetting area compared to flat channel, thus the highest velocity in it (≈ 18) is slightly smaller than that in flat channel (≈ 19), resulting in flow rate decrease and drag coefficient increase. Quantitatively, the normalized wetted area increase of ‘sinu133 soidal channel’ compared to flat channel ΔAw = Aw /Aw0 − is about 1.55%, which is very close to the increase of the total drag coefficient (1.53%). The difference between them may be due to numerical errors. This finding shows that drag increase of ‘sinusoidal channel’ is only caused by wetted area increase, and further supports that drag reduction of ‘converging-diverging channel’ comes from velocity rearrangement. In Figure 3.31, no variation of skin friction drag Sm/Sm0 in spanwise direction can be observed except fluctuations induced by turbulence structures. This is because the highest velocity in centerline of cross section is uniform along the spanwise direction. 3. In the cases between ‘converging-diverging channel’ and ‘sinusoidal channel’ (0 < ϕ < π), the height of the narrow portion is larger than its lower limit − S and the height of the wide portion is lower than its upper limit + S, resulting in weaker velocity rearrangement than original corrugated channel (‘converging-diverging channel’). It subsequently leads to less drag reduction or even drag increase. For instance, when ϕ = π/2 (see Figure 3.30(b)) the height of the wide √ portion (z = λ/8) is + 2S/2 and the height of the narrow portion √ (z = 5λ/8) is − 2S/2. So as shown in Figure 3.30(b), the highest and lowest velocity are located at z = λ/8 and z = 5λ/8, respectively. Furthermore, the highest velocity for ϕ = π/2 (Uwide = 21) is lower than that for ϕ = (Uwide = 22), resulting in less drag reduction than that of ‘converging-diverging channel’. Additionally it is observed in Figure 3.31 that the normalized skin friction drag Sm/Sm0 is higher than in the wide portion (z = λ/8) and is lower than in the narrow 134 portion (5λ/8), which is similar to ‘converging-diverging channel’. It is interesting to note that the positions of highest and lowest Sm/Sm0 for ϕ = π/2 are shifted compared to those for ϕ = 0. Overall, it can be concluded that flow rate increase of corrugated ‘converging-diverging channel’ is achieved through velocity rearrangement induced by the alternating wide and narrow portion in the cross section. If such alternating wide and narrow portions are weakened or are eliminated, the velocity rearrangement will become weak or even disappear, resulting in less flow rate than that of ‘converging-diverging channel’. 1.5 16 12 18 0.5 16 Y 18 Y 20 18 12 -0.5 -1 -2 18 0.2 16 12 18 20 -1 12 16 12 18 18 21 18 -0.5 12 20 21 16 16 20 20 21 22 16 12 -1.5 12 0.5 20 16 18 18 1.5 -1.5 0.4 0.6 0.8 -2 0.2 0.4 Z/λ 0.6 0.8 Z/λ (a) (b) 1.5 12 16 12 0.5 16 Y 18 18 18 18 16 18 -0.5 18 12 -1 -1.5 -2 16 12 0.2 0.4 0.6 0.8 Z/λ (c) Figure 3.30: Streamwise velocity u on Z-Y plane for different phase shift (a) ϕ = 0, (b) ϕ = π/2 and (c) ϕ = π 135 1.4 1.3 1.2 Sm/Sm0 1.1 0.9 0.8 flat ϕ=0 ϕ=π/2 ϕ=π 0.7 0.6 0.5 0.4 0.2 0.4 0.6 0.8 Z/λ Figure 3.31: Shear drag stress on Z-Y plane for different phase shift ϕ at α = 0.5 and S = 3.3.8 Summary It can be concluded that corrugated channel also produces increment of flow rate or drag reduction in turbulent channel flow similar to laminar channel flow. However, quantitatively the magnitude of drag reduction of corrugated channel in turbulent flow is less than that in laminar flow with the same wall shape as frictional drag due to bulk velocity, which is the main contributor to drag reduction, accounts for only about 20–30% of the total drag. When α < αc where αc is about 0.9, there will be drag reduction which intensifies with increasing S because of stronger bulk flow rearrangement. Conversely, the viscous and turbulent interaction between the fluid in the wide and narrow portions weakens such rearrangement, resulting in flow rate decrease when the wave number is high. Furthermore, such interaction is intensified with larger S, amplifying drag increase augmentation. More details of how wave number and amplitude affects 136 the interaction between the fluid in the wide and narrow portions, and how such interaction influence bulk flow rearrangement can be found in §3.2.6. Additionally, instead of suppressing turbulent kinetic energy production and Reynolds stress (e.g. riblets, see Peet and Sagaut, 2009; Choi et al., 1993), the drag reduction achieved by corrugated surface is obtained mainly through rearranging the mean velocity distribution and reducing the bulk term of drag Cf b . This hypothesis has been proven through careful examination of mean velocity, TKE, turbulence structures, drag coefficient decomposition, theoretical prediction and channel with phase shift. Furthermore, in order to maximize the drag reduction in turbulent channel flow, the wave number α shall be very small (α → 0) and the amplitude S shall be large (S → 2). From the inference of theoretical analysis at small wave number, the highest drag reduction of corrugated surface can go up to 30%–37% in turbulent channel flow when α → and S → 2, depending on Reynolds number. 3.4 Concluding remarks A systematic analytical and numerical investigation of the laminar and turbulent channel flow through two opposite corrugated walls has been conducted in this study via the perturbation method and the Detached Eddy Simulations (DES) model respectively. A novel macro-scale corrugated surface (i.e. longitudinal grooves) is introduced and investigated in terms of flow rate, drag coefficient, mean velocity distribution, mean drag distribution, turbulent kinetic energy (TKE), Reynolds stress, turbulence 137 structures and drag decomposition. It has been demonstrated that the reduced order geometry model in laminar flow can also predict the drag coefficient for corrugated surface in turbulent channel flow with sufficient accuracy. The reduced order geometry model adopted in this study is based on the projection of wall shape onto a Fourier space. Thus, only the effects of wave number and amplitudes on drag coefficient are emphasized in this study. The following conclusions are drawn for this chapter (if it is not specified laminar or turbulent flow, it is valid for both of them). 1. The corrugated surface when subjected to the same pressure gradient can permit more flow rate (i.e. drag reduction) when wave number is low (i.e. α < αc ) while it restricts the flow rate (i.e. drag increase) when wave number is high (i.e. α > αc ), where αc is about 1.2 for laminar flow and about 0.8–1 for turbulent flow. 2. Bigger amplitude S increases the flow rate (i.e. drag reduction) for low wave number (i.e. α < αc ), and conversely decreases the flow rate (i.e. drag increase) for high wave number (i.e. α > αc ). 3. The numerical results show that the drag reduction of the investigated corrugated surfaces in turbulent channel flow is about 10% when α = 0.25 and S = 1, which is fairly similar to that of riblets (7%–10%). A reasonable inference of the numerical result is that the corrugated surface with α → and S → can produce drag reduction more than 10%. The theoretical prediction at small wave number has proven this hypothesis (see Conclusion 9). 4. There are two possible mechanisms to explain flow rate increase or 138 drag coefficient reduction obtained by corrugated surface: • To achieve more flow rate at fixed pressure gradient by rearranging the bulk velocity distribution in Y-Z planes, with the largest flux flows through the widest channel opening (see Floryan, 2011); • To reduce bulk term of drag coefficient Cf b by increasing A which is related to the cross section geometry according to the drag decomposition theory (Peet and Sagaut, 2009). 5. Study of turbulent kinetic energy (TKE) and Reynolds stress shows that the corrugated surface rearranges their distributions but does not reduce their volume-averaged intensity. 6. The drag decomposition in turbulent channel flow shows that the drag reduction of corrugated surface in turbulent flow is mainly due to the suppression of the bulk term of drag, while the turbulent term of drag remains approximately the same. 7. The bulk term of drag ratio Cf b /Cf b0 in turbulent flow is theoretically the same as the total drag ratio Cf /Cf in laminar flow for the same corrugated geometry since it depends on the same A. This is also verified by the numerical results. 8. The corrugated surface with the same configuration creates less flow rate increase (drag reduction) in turbulent flow (3%–10%) than it does in laminar flow (10%–30%). As shown in above Conclusion (7), Cf b /Cf b0 are the same for both turbulent flow and laminar flow. Thus the low proportion of bulk term in total drag Cf b0 /Cf (about 27%) 139 in turbulent flow weakens the contribution of Cf b /Cf b0 in total drag, which includes turbulent drag Cf t (about 73% of total drag, see Eq. 3.23). 9. Theoretical prediction of drag reduction at small wave number shows that the effects of Reynolds number on drag reduction in turbulent flow is weak, but the effect of amplitude is very strong. When amplitude is high (S → 2), the drag reduction of the corrugated channel can approach up to 30–37% in turbulent flow and 60% in laminar flow. 10. Shifting top corrugated wall spanwise eliminates the alternating wide and narrow portions of channel in various degree, thus decreasing the drag reduction induced by corrugated channel or even producing drag increase with low wave number and high amplitude (i.e. α = 0.5 and S = 1). When the top corrugated channel wall is shifted to generate a ‘sinusoidal channel’ with uniform gap, it results in drag increase due to the augmentation of wetted area of the channel. In summary, corrugated surface with low wave number and higher amplitude creates drag reduction at similar level to (even higher than) riblets and other traditional drag reduction devices for turbulent channel flow. It shall be noted that riblets can be seemed as longitudinal grooves with very high wave number (α ≈ 160) and low amplitude (S ≈ 0.04) which is outside of the current investigation, and the drag reduction is obtained through stabilizing near wall turbulence. Thus, the conclusion herein about wave number and amplitude effects cannot be simply and directly 140 extrapolated for riblets. Additionally, corrugated surface, being both a passive and macro-scale device, is easier to manufacture and maintain and thus has wider engineering applications. 141 [...]... −1.81% −1 .32 % α = 0.4, S = 1 −6.57% −6.09% α = 0.25, S = 1 −10.17% −9.89% Table 3. 4: Comparison of drag coefficient difference ΔD computed by DES and DNS 3. 3.1.4 Global performance of drag difference The numerical results of ΔD for different corrugated channels are shown in Figure 3. 13, where (a) shows the effects of wave number α, and (b) shows the effect of amplitude S It can be observed in Figure 3. 13( a) that... produces larger drag reduction However, the rate of drag reduction decreases with increasing S which is opposite to that in laminar flow (see Figures 3. 3, 3. 4 and in particular 3. 6) 0 .3 0.04 0.2 S=1 0.02 0.1 ΔD ΔD S=0.5 0 α=0.5 0 -0.02 αc -0.1 -0.04 -0.2 0 0.5 1 1.5 2 0 0.5 1 α (a) 1.5 2 S (b) Figure 3. 13: Drag coefficient difference (a) effects of α (b) effects of S In other wordw, drag reduction of corrugated... the accuracy of analysis of drag reduction mechanisms, the numerical simulation results are used in the following sections of laminar flow, 3. 2.4, 3. 2.5 and 3. 2.6 However, the theoretical prediction results are used in 3. 2 .3. 4 to show the wetted area effect in a wide range of wave number This is acceptable because the general trend of α and S effects on wetted area and the normalized drag difference... typical cases of turbulent channel flow (marked by ‘*’ in Table 3. 2) to verify the accuracy of the results obtained by DES In this section, the capability of corrugated surface for drag reduction in turbulent channel flow is investigated first The effects of amplitude and wave number of longitudinal grooves on drag reduction in turbulent channel flow are also examined Furthermore, in-depth and detailed examination... surface for turbulent channel flow is qualitatively similar to but quantitatively different from that for laminar flow This may be due to the different velocity profiles 1 03 of laminar and turbulent flow (see 3. 3.2) Overall, in order to achieve as much drag reduction as possible, the wave number α and amplitude S shall be low and high enough, respectively 3. 3.1.5 Wetted area effect As shown in the discussion of. .. of drag reduction obtained by corrugated channel 99 3. 3.1 Drag difference In the following, drag difference between the corrugated channel and flat channel will be discussed First, the definition of drag difference will be presented explicitly for turbulent channel flow Then, the reduced order geometry method and DES model will be validated by examining the numerical results Finally, the drag difference for. .. will be discussed 3. 3.1.1 The definition of drag difference Similar to the investigation of corrugated surface in laminar channel flow, the normalized drag coefficient difference ΔD at fixed Reynolds number Re 2H is also used to evaluate the capability of corrugated surface to reduce drag in turbulent channel flow: ΔD = Cf − Cf 0 Cf 0 where Cf stands for corrugated channel and Cf 0 stands for flat channel Note... Figure 3. 9 show that for all the corrugated 88 cases, the fluid in wider portion speeds up while that in the narrower portion slows down For the drag reduction cases in Figure 3. 9(a) and (b), the difference in maximum velocity in the wide and narrow portions of the channel is much larger than that for the drag increasing cases in Figure 3. 9(c) and (d) Their maximum velocity in the wide portion of channel... which there is drag increase and below which there is drag reduction 102 Such critical wave number αc for channel with both corrugated walls in turbulent flow is about 0.8–1 (an estimation because of uncertainty of drag difference) when S = 0.5 and 1, which is fairly similar to the counterpart for laminar flow (αc = 1.2) Figures 3. 13( a) shows that when α > αc higher amplitude S produces higher drag increase,... more drag reduction It is also shown that when wave number α decreases, drag reduction becomes larger When α further decrease and approaches zero, the drag reduction also approaches its asymptotic limit Quantitatively, the highest drag reduction at amplitude S = 0.5 is about 3% –4% when α → 0, while corrugated channel with amplitude S = 1 produces drag reduction up to 10% when α → 0 Figure 3. 13( b) shows . 1 3S 2 32 for Case 2 (3. 10) 3. 2 .3 Global performance of drag difference 3. 2 .3. 1 The definition of drag difference To find the quantitative relationship between flow rate increase ΔQ and drag reduction, . produces higher drag reduction and there is an asymptotic limit of drag difference when α approaches zero. Drag reduction for channel with both corrugated walls is about four times of that for channel. Figure 3. 9(b) case where the maximum velocity of 9.5 is the highest and the drag reduction is the greatest as can be seen in Figure 3. 7. For the drag increasing cases in Figure 3. 9(c) and (d),