3.1. Numerical method

Fully developed turbulent channel flow with wall control is considered using DNS. The results presented are generated by a DNS code (Chung & Talha Reference Chung and Talha2011), based on the fully implicit fractional step method proposed by Kim, Baek & Sung (Reference Kim, Baek and Sung2002), in which a Crank–Nicolson discretisation is used for both the diffusion and convective terms. The Navier–Stokes equations are solved in a Cartesian coordinate system. Using block LU decomposition each velocity equation is split into three one-dimensional problems which are solved using suitable banded matrix algorithms. The pressure is updated via the Poisson equation, taking advantage of the uniform grid discretisation in the streamwise and spanwise directions by performing a combination of a two-dimensional Fourier transform and a tridiagonal matrix inversion. Various versions of the code have been used for DNS and large eddy simulations (LES) (Chung & Talha Reference Chung and Talha2011; Jewkes, Chung & Carpenter Reference Jewkes, Chung and Carpenter2011; Jung & Chung Reference Jung and Chung2012; Chung & Hurst Reference Chung, Hurst, Zhou, Liu, Huang and Hodges2014).

When considering parallelisation using MPI, it is preferable to retain the data from the full domain length in the direction of any one-dimensional algorithms. Therefore, a pencil structure is adopted, dividing the processes into a two-dimensional grid. The code makes use of the 2DECOMP&FFT library developed by Laizet & Li (Reference Laizet and Li2011), which performs optimised transpositions when all the data is required in a particular direction. It utilises a set of halo cells for calculations requiring only local data, to reduce the number of necessary transpositions. Parallel I/O is also used to remove the dependency upon writing data and improve efficiency. Simulations were performed using the local cluster (Minerva) with 6000 Intel Xeon X5650 Westmere-EP 2.66 GHz cores at the Centre for Scientific Computing, University of Warwick. The higher-Reynolds-number simulations at $\mathit{Re}_{{\it\tau}}=1600$ were run on 1024 cores on HECToR and ARCHER (www.archer.ac.uk), the UK National High Performance Computing Facility.

3.2. Simulation parameters

DNS of the turbulent channel flow were performed at four Reynolds numbers $\mathit{Re}_{{\it\tau}}=200$ , 400, 800 and 1600, based on the frictional velocity of the no-control case, $u_{{\it\tau}0}$ , and the channel half-width, $h$ . The simulations were run with a constant mass flow rate by dynamically adjusting the pressure gradient. The length of the computational domain is chosen so that the same wall control parameters, scaled by wall units, can be studied at each of the Reynolds numbers, due to the requirement of an integer number of waves within the domain.

The domain sizes, grid sizes and resolutions used in the main simulations are shown in table 2. Throughout this paper $\text{}^{+}$ represents the wall units based on $u_{{\it\tau}0}$ . The number of grid points was adopted such that the grid spacing was fixed with ${\rm\Delta}x^{+}=10$ and ${\rm\Delta}z^{+}=5$ . A time step size of ${\rm\Delta}t^{+}\approx 0.12$ was used, with time averaging taken over at least $t^{+}=7200$ after the initial transient. All statistics were obtained using plane averaging in the streamwise and spanwise directions as well as time averaging.

Table 2.

Parameters of the channel flow simulations at the four Reynolds numbers studied. Here $L_{x},L_{y}$ and $L_{z}$ are the computational domain size in the $x,y$ and $z$ directions, $N_{x},N_{y}$ and $N_{z}$ are the corresponding grid sizes and $\mathit{Re}=U_{m}h/{\it\nu}$ is the bulk mean Reynolds number.

The current simulations are restricted to forward travelling waves, corresponding to positive values of ${\it\omega}$ and ${\it\kappa}_{x}$ . Using these parameters, the region of maximum drag reduction and the region of drag increase can be investigated. The maximum wall velocity was also fixed at $A^{+}=12$ to aid in comparison with previous work (Quadrio et al. Reference Quadrio, Ricco and Viotti2009; Touber & Leschziner Reference Touber and Leschziner2012). At $\mathit{Re}_{{\it\tau}}=200,400$ and 800, a series of simulations were performed using all combinations of the control parameters ${\it\omega}^{+}$ of $0.01{-}0.18$ and ${\it\kappa}_{x}^{+}$ of $0.002{-}0.016$ . This gives a good view over the region of interest in the ( ${\it\omega},{\it\kappa}_{x}$ ) drag reduction map (Quadrio et al. Reference Quadrio, Ricco and Viotti2009). For the wall oscillation and stationary wave, extra points were chosen in the region of the optimal, ${\it\omega}^{+}=0.06{-}0.1$ and ${\it\lambda}^{+}=1000{-}1250$ . The parameters which achieved the maximum drag reduction for the wall oscillation and stationary wave were also run at $\mathit{Re}_{{\it\tau}}=1600$ . To the best of the authors’ knowledge, this is the highest Reynolds number attempted for flow control DNS using a full domain.

3.3. Grid sensitivity study

First, the effect of the computational domain size was considered through a sensitivity study at $\mathit{Re}_{{\it\tau}}=200$ . Two simulations of larger computational domain sizes were performed with the same grid resolution as in the CH200 case in table 2. The computational domain size was doubled in the streamwise direction ( $L_{x}=32h$ ) and spanwise direction ( $L_{z}=12h$ ), respectively. A comparison of DNS results for the larger domain sizes ( $32h\times 2h\times 6h$ and $16h\times 2h\times 12h$ ) with the baseline case shows that the skin-friction coefficients were not affected by doubling the domain size. The final domain size chosen ( $16h\times 2h\times 6h$ ) is similar to that used by Bernardini et al. (Reference Bernardini, Pirozzoli and Orlandi2014) for channel flow DNS at $\mathit{Re}_{{\it\tau}}=4000$ .

Table 3.

Parameters used in a grid sensitivity study. (a) Four grid resolutions tested for no-control and wall oscillation cases. Here $C_{f}$ values are for no-control cases at $\mathit{Re}_{{\it\tau}}=200$ and ${\mathcal{D}}{\mathcal{R}}$ values are the amount of drag reduction for the optimal wall oscillation case, ${\it\omega}^{+}=0.06$ at $\mathit{Re}_{{\it\tau}}=200,400$ and 800. (b) Various grid spacings tested from the baseline (G1) grid for a further grid sensitivity study. Here $\dagger$ indicates the final grid spacings chosen.

An extensive grid sensitivity study was performed at the lowest Reynolds number ( $\mathit{Re}_{{\it\tau}}=200$ ) due to the computational cost required. Table 3(a) shows samples of grids tested including two very fine grids (G0 and G1). The G0 grid uses $1280\times 380\times 480$ grid points with a total number of 233 million. The G1 grid is the same as the G0 grid except a larger streamwise grid spacing ( ${\rm\Delta}x^{+}=5$ ), and a total number of 117 million grid points were used. The skin-friction coefficient $C_{f}$ values at $\mathit{Re}_{{\it\tau}}=200$ given in table 3(a) are very similar for all cases, and the difference is approximately 1 %. DNS results of flow quantities show little difference between the G0 and G1 grids, so the G1 grid was used as the baseline grid for a further grid sensitivity study. The root-mean-squared (r.m.s.) velocity ( $u_{i,rms}^{+}$ ) fluctuations and the Reynolds shear stress ( $-\overline{uv}$ ) from the G1 grid are shown in figure 1. Also included in the figure are the DNS data of Kim, Moin & Moser (Reference Kim, Moin and Moser1987) at $\mathit{Re}_{{\it\tau}}=180$ . The present DNS results for no-control case exhibit an excellent agreement with the available DNS data.

Figure 1.

Present DNS data for no-control case on the baseline (G1) grid at $\mathit{Re}_{{\it\tau}}=200$ . (a) Velocity fluctuation, $u_{i,rms}^{+}$ , and (b) Reynolds shear stress, $-\overline{uv}$ . Symbols are DNS data of Kim et al. (Reference Kim, Moin and Moser1987) at $\mathit{Re}_{{\it\tau}}=180$ .

The grid resolution was systematically changed only in one direction at a time while grid resolutions in the other two directions remain unchanged from the baseline (G1) grid; ${\rm\Delta}y_{\mathit{min}}^{+}$ and ${\rm\Delta}y_{\mathit{max}}^{+}$ were also changed separately. The grid spacings tested are shown in table 3(b). Various flow properties including the skin-friction coefficient $C_{f}$ were compared with the values from the baseline grid case. From this grid sensitivity study, the G3 grid was chosen based on the accuracy and also computational resources required to create a drag reduction map.

The grid sensitivity study was then extended to higher Reynolds numbers ( $\mathit{Re}_{{\it\tau}}=400$ and 800). The accuracy of the G3 grid has been thoroughly assessed. First, the relation between $Re$ and $\mathit{Re}_{{\it\tau}}$ is compared with other DNS studies (Moser, Kim & Mansour Reference Moser, Kim and Mansour1999; Iwamoto et al. Reference Iwamoto, Suzuki and Kasagi2002; Tanahashi et al. Reference Tanahashi, Kang, Miyamoto, Shiokawa and Miyauchi2004; del Álamo & Jiménez Reference del Álamo and Jiménez2003; del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2004; Hoyas & Jiménez Reference Hoyas and Jiménez2006) in figure 2(a) to examine whether $C_{f}$ is correctly predicted in the present DNS. Also included in figure 2(a) are two correlations proposed by Dean (Reference Dean1978) based on experiments and by Laadhari (Reference Laadhari2007) based on DNS data. $U_{m}^{+}\ (=\sqrt{2/C_{f}})$ values are also compared well with other DNS data (not shown here). The kinetic energy budget terms for all four Reynolds numbers are compared with the DNS data from literature at similar Reynolds numbers (Kim et al. Reference Kim, Moin and Moser1987; Moser et al. Reference Moser, Kim and Mansour1999; del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2004; Hoyas & Jiménez Reference Hoyas and Jiménez2006). The $\mathit{Re}_{{\it\tau}}=200$ and 1600 cases are displayed in figure 2(b). Figure 2 clearly shows that the present DNS results on the final grid are in good agreement with other DNS studies and also with the correlations proposed by Dean (Reference Dean1978) and Laadhari (Reference Laadhari2007).

Figure 2.

Present DNS data for no-control case on the final (G3) grid at four Reynolds numbers considered: (a) $\mathit{Re}_{{\it\tau}}$ versus $Re$ ; (b) turbulence kinetic energy budget terms. Dashed lines are the present DNS data at $\mathit{Re}_{{\it\tau}}=200$ and solid lines are the present DNS data at $\mathit{Re}_{{\it\tau}}=1600$ . Symbols are DNS data of Kim et al. (Reference Kim, Moin and Moser1987) for $\mathit{Re}_{{\it\tau}}=180$ (○) and Hoyas & Jiménez (Reference Hoyas and Jiménez2006) for $\mathit{Re}_{{\it\tau}}=2000$ (♢).

In addition to the no-control case, wall oscillation simulations for the optimal case of ${\it\omega}^{+}=0.06$ were performed at three Reynolds numbers to ascertain the effect of grid resolution on flow control results. Based on the $\mathit{Re}_{{\it\tau}}=200$ results described earlier, only the G2 and G3 grids were considered at high Reynolds numbers as it was found that DNS results are more sensitive to the resolution in the streamwise direction. The difference in ${\mathcal{D}}{\mathcal{R}}$ prediction between the two grids are shown in table 3(a) at $\mathit{Re}_{{\it\tau}}=200$ , 400 and 800. The effect on ${\mathcal{D}}{\mathcal{R}}$ prediction is found to be smaller than ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}=1\,\%$ . A similar trend was observed for the stationary wave case.

4.1. Drag reduction map

Figure 3 shows contours of the drag reduction values achieved at three Reynolds numbers, $\mathit{Re}_{{\it\tau}}=200,400$ and 800, with using 86, 90 and 106 simulations, respectively. The drag reduction map at $\mathit{Re}_{{\it\tau}}=200$ is very similar to that shown in Quadrio et al. (Reference Quadrio, Ricco and Viotti2009). Please note that only forward travelling waves ( $c>0$ ) were considered in this study as this half of the parameter domain contains the optimal travelling wave case while Quadrio et al. (Reference Quadrio, Ricco and Viotti2009) considered both forward and backward travelling waves. In this study, the amount of drag reduction can be defined as ${\mathcal{D}}{\mathcal{R}}=\left(1-(C_{f}/C_{f,0})\right)\times 100\,(\%)$ , where $C_{f,0}$ is the skin-friction coefficient for the no-control case. The maximum drag reduction at $\mathit{Re}_{{\it\tau}}=200$ is approximately ${\mathcal{D}}{\mathcal{R}}_{\mathit{max}}=50$ , and the optimal control parameters are ${\it\omega}^{+}=0.02$ ( $T^{+}=300$ ) and ${\it\kappa}_{x}^{+}=0.008$ ( ${\it\lambda}^{+}=800$ ). It is interesting to note that at first glance, drag reduction maps at the higher Reynolds numbers ( $\mathit{Re}_{{\it\tau}}=400$ and 800) appear to be similar to that at $\mathit{Re}_{{\it\tau}}=200$ , indicating that the overall drag reduction and drag increase characteristics of the travelling wave are not affected significantly by increasing the Reynolds number: the large ${\mathcal{D}}{\mathcal{R}}$ values are observed in the ${\mathcal{D}}{\mathcal{R}}$ peak region around $c_{\mathit{DR}}^{+}\approx 4$ while a drag increase is observed in a narrow region around $c_{\mathit{DI}}^{+}\approx 10$ at all three Reynolds numbers. A line of the minimum $\boldsymbol{{\rm\nabla}}{\mathcal{D}}{\mathcal{R}}$ passing through the ${\mathcal{D}}{\mathcal{R}}_{\mathit{max}}$ location is displayed in the ${\mathcal{D}}{\mathcal{R}}$ map to indicate the location for the ${\mathcal{D}}{\mathcal{R}}$ peak region:

Please note ${\it\kappa}_{x,s}^{+}$ is where the above line intersects the ${\it\omega}^{+}=0$ axis in the ${\mathcal{D}}{\mathcal{R}}$ map.

Figure 3.

Drag reduction map ( ${\it\omega}^{+}$ , ${\it\kappa}_{x}^{+}$ ) for the forward travelling waves at (a) $\mathit{Re}_{{\it\tau}}=200$ , (b) $\mathit{Re}_{{\it\tau}}=400$ and (c) $\mathit{Re}_{{\it\tau}}=800$ . Contour levels are drawn at 5 % intervals. The dark (blue online) indicates a drag reduction and the bright (yellow online) colour indicates a drag increase. White dashed lines indicate the location for the local ${\mathcal{D}}{\mathcal{R}}$ maxima. (d) For ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}={\mathcal{D}}{\mathcal{R}}_{800}-{\mathcal{D}}{\mathcal{R}}_{200}$ , change in drag reduction from $\mathit{Re}_{{\it\tau}}=200$ to 800. The bright (yellow online) colour indicates a negative ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}$ , and the dark (blue online) indicates a positive ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}$ .

While the overall similarity of the drag reduction map is evident at different Reynolds numbers, figure 3 shows that the absolute value of drag change (both drag reduction and drag increase) is decreased when the Reynolds number is increased. The maximum drag reduction at $\mathit{Re}_{{\it\tau}}=400$ is reduced to ${\mathcal{D}}{\mathcal{R}}_{\mathit{max}}=44$ with the optimal control parameters changed slightly. At the optimal control parameters, the difference in the maximum ${\mathcal{D}}{\mathcal{R}}$ achieved at these two Reynolds numbers is quite significant, ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}=-6$ , which means by doubling the Reynolds number from $\mathit{Re}_{{\it\tau}}=200$ , drag reduction has already become roughly 10 % less effective than the lower-Reynolds-number case. The maximum ${\mathcal{D}}{\mathcal{R}}$ decreases further to ${\mathcal{D}}{\mathcal{R}}_{\mathit{max}}=40$ at $\mathit{Re}_{{\it\tau}}=800$ . It is worthwhile to note that the optimal control parameters for $\mathit{Re}_{{\it\tau}}=800$ changes slightly from the lower-Reynolds-number values, and this will be addressed later. The peak region for large ${\mathcal{D}}{\mathcal{R}}$ values also appears to move towards a smaller ${\it\omega}^{+}$ and larger ${\it\kappa}_{x}^{+}$ region. It is observed that the region of the parameter space in which the drag increase is found diminishes as the Reynolds number is increased. There is a valley region in the drag reduction map along $c_{\mathit{DI}}^{+}\approx 10$ where ${\mathcal{D}}{\mathcal{R}}$ is negative. This valley still exists at higher Reynolds numbers, but drag increase in the valley region becomes less pronounced at higher Reynolds numbers, making the valley shallow. Gatti & Quadrio (Reference Gatti and Quadrio2013) suggested that this valley region widens as the Reynolds number increases. Because our primary interest is in the drag reduction region, however, it is not quite clear from the present study whether this valley region widens or not as the Reynolds number increases.

Figure 4.

Drag reduction for the wall oscillation case, ${\it\kappa}_{x}^{+}=0$ . The vertical arrow indicates the location for the maximum ${\mathcal{D}}{\mathcal{R}}$ at each Reynolds number.

Figure 3(d) shows the change in drag reduction values, ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}={\mathcal{D}}{\mathcal{R}}_{800}-{\mathcal{D}}{\mathcal{R}}_{200}$ , to illustrate the effect of increasing the Reynolds number from $\mathit{Re}_{{\it\tau}}=200$ to 800 on ${\mathcal{D}}{\mathcal{R}}$ . With control parameters close to that of the maximum drag reduction, ( ${\it\omega}_{\mathit{opt}}^{+}$ , ${\it\kappa}_{x,\mathit{opt}}^{+}$ ), the change is large and negative with a reduction of up to ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}=-12$ . In the valley region of the drag reduction map, the magnitude of the drag increase also becomes smaller at higher Reynolds numbers corresponding to positive values of ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}$ . For example, in the middle of the drag increase valley ( ${\it\omega}^{+}=0.09$ and ${\it\kappa}_{x}^{+}=0.008$ ), drag increases are ${\mathcal{D}}{\mathcal{R}}=-17$ , $-10$ and $-5$ at $\mathit{Re}_{{\it\tau}}=200$ , 400 and 800, respectively, resulting in a difference in ${\mathcal{D}}{\mathcal{R}}$ between $\mathit{Re}_{{\it\tau}}=200$ and 800 of ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}=12$ .

4.2. Reynolds number effect

The drag reduction results for the wall oscillation ( ${\it\kappa}_{x}^{+}=0$ ) for all the Reynolds numbers are presented in figure 4. The present results shown in figure 4 clearly displays the decrease in the value of ${\mathcal{D}}{\mathcal{R}}$ as the Reynolds number is increased. At all three Reynolds numbers, ${\mathcal{D}}{\mathcal{R}}$ increases initially with ${\it\omega}^{+}$ to attain a maximum ${\mathcal{D}}{\mathcal{R}}$ at ${\it\omega}_{\mathit{opt}}^{+}$ , and then decreases gradually. One interesting feature of the wall oscillation is that the optimal frequency increases slightly from ${\it\omega}_{\mathit{opt}}^{+}=0.06$ ( $T^{+}=100$ ) at $\mathit{Re}_{{\it\tau}}=200$ to ${\it\omega}_{\mathit{opt}}^{+}=0.07$ at $\mathit{Re}_{{\it\tau}}=400$ , and ${\it\omega}_{\mathit{opt}}^{+}=0.08$ at $\mathit{Re}_{{\it\tau}}=800$ although the ${\mathcal{D}}{\mathcal{R}}$ peak becomes broader as the Reynolds number increases, and the ${\mathcal{D}}{\mathcal{R}}_{\mathit{max}}$ is not easily discernible. The drag reduction at ${\it\omega}^{+}=0.06$ ( $T^{+}=100$ ) is calculated up to $\mathit{Re}_{{\it\tau}}=1600$ . At this Reynolds number, the drag reduction has been reduced from ${\mathcal{D}}{\mathcal{R}}=37$ to ${\mathcal{D}}{\mathcal{R}}=22$ , corresponding to a ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}=-15$ decrease from the $\mathit{Re}_{{\it\tau}}=200$ case, which means a 40 % less effectiveness from the low-Reynolds-number result. While the change in ${\mathcal{D}}{\mathcal{R}}$ at the optimal value, ${\it\omega}_{\mathit{opt}}^{+}$ , is greater, there appears to be less of an effect of Reynolds number at larger values of ${\it\omega}^{+}$ (corresponding to smaller $T^{+}$ ) in the drag reduction region. Ricco & Quadrio (Reference Ricco and Quadrio2008) reported a similar trend using results from $\mathit{Re}_{{\it\tau}}=400$ .

Wall oscillation ${\mathcal{D}}{\mathcal{R}}$ data reported in literature have been examined to identify the Reynolds number dependence. Figure 5 shows ${\mathcal{D}}{\mathcal{R}}$ data of Choi et al. (Reference Choi, Xu and Sung2002), Quadrio & Ricco (Reference Quadrio and Ricco2004), Ricco & Quadrio (Reference Ricco and Quadrio2008), Touber & Leschziner (Reference Touber and Leschziner2012) and Gatti & Quadrio (Reference Gatti and Quadrio2013). Some Reynolds number effect is discernible in the data, but there is considerable of scatter between different studies. This is partly because different amplitudes and oscillation frequencies have been used in various studies as summarised in table 1. A clear decrease in ${\mathcal{D}}{\mathcal{R}}$ is shown in Choi et al. (Reference Choi, Xu and Sung2002) (yellow symbols), but their ${\mathcal{D}}{\mathcal{R}}$ values are somewhat smaller than the ${\mathcal{D}}{\mathcal{R}}$ values reported in other studies at the same Reynolds numbers. Quadrio & Ricco (Reference Quadrio and Ricco2004) and Ricco & Quadrio (Reference Ricco and Quadrio2008) (open symbols) display similar trend at $\mathit{Re}_{{\it\tau}}=200$ and 400. A similar trend is also seen in Touber & Leschziner (Reference Touber and Leschziner2012) (orange symbols) at $\mathit{Re}_{{\it\tau}}=200$ , 500 and 1000. Also included are the DNS data of Gatti & Quadrio (Reference Gatti and Quadrio2013) (open symbols) at $\mathit{Re}_{{\it\tau}}=950$ and 2100.

Figure 5.

The ${\mathcal{D}}{\mathcal{R}}$ data for the wall oscillation case, ${\it\kappa}_{x}^{+}=0$ , in literature: Choi et al. (Reference Choi, Xu and Sung2002), small grey (yellow) colour symbols; Quadrio & Ricco (Reference Quadrio and Ricco2004), Ricco & Quadrio (Reference Ricco and Quadrio2008) and Gatti & Quadrio (Reference Gatti and Quadrio2013), open symbols; Touber & Leschziner (Reference Touber and Leschziner2012), grey (orange) colour symbols; boundary layer (Yudhistira & Skote Reference Yudhistira and Skote2011; Skote Reference Skote2012; Lardeau & Leschziner Reference Lardeau and Leschziner2013), black symbols. Present results are shown in dashed lines. The wall oscillation amplitude is $A^{+}=12$ for all cases except for Choi et al. (Reference Choi, Xu and Sung2002), where the data is for $A^{+}=10$ .

The ${\mathcal{D}}{\mathcal{R}}$ values for near-optimal wall oscillation cases are displayed in figure 6. The oscillation frequency used is ${\it\omega}^{+}=0.06$ (or $T^{+}=100$ ) in most studies while different $T^{+}$ values were used in some studies: $T^{+}=125$ is used in Ricco & Quadrio (Reference Ricco and Quadrio2008) and $T^{+}=90$ in Gatti & Quadrio (Reference Gatti and Quadrio2013). All ${\mathcal{D}}{\mathcal{R}}$ data show a clear trend of drag reduction becoming less successful as the Reynolds number increases. The first thing to notice is that there is a large variation in the ${\mathcal{D}}{\mathcal{R}}$ values at $\mathit{Re}_{{\it\tau}}=200$ . The ${\mathcal{D}}{\mathcal{R}}$ values in Choi et al. (Reference Choi, Xu and Sung2002) are particularly low, and this may be partly due to a smaller oscillation amplitude ( $A^{+}=10$ ) used in their simulations. Also the decrease between $\mathit{Re}_{{\it\tau}}=200$ and 400 appears to be larger than in other studies, indicating a strong Reynolds number effect. The ${\mathcal{D}}{\mathcal{R}}$ data of Gatti & Quadrio (Reference Gatti and Quadrio2013) using a small computational domain are also included. Their ${\mathcal{D}}{\mathcal{R}}$ values appear to be significantly larger than the present results even though their ${\mathcal{D}}{\mathcal{R}}$ value for $\mathit{Re}_{{\it\tau}}=200$ is very similar to the ${\mathcal{D}}{\mathcal{R}}$ value from the present study.

Figure 6.

Drag reduction for the near-optimal wall oscillation cases. The oscillation frequency is ${\it\omega}^{+}=0.06$ ( $T^{+}=100$ ), and $A^{+}=12$ . Please note that $T^{+}=125$ is used in Ricco & Quadrio (Reference Ricco and Quadrio2008) and $T^{+}=90$ in Gatti & Quadrio (Reference Gatti and Quadrio2013). Choi et al. (Reference Choi, Xu and Sung2002) used $A^{+}=10$ .

Figure 7.

Drag reduction for the stationary wave case, ${\it\omega}^{+}=0$ . Open symbols are for Quadrio et al. (Reference Quadrio, Ricco and Viotti2009) (○) and Gatti & Quadrio (Reference Gatti and Quadrio2013) (▵).

Figure 8.

Drag reduction for the travelling wave cases. Horizontal lines through the map with (a) ${\it\kappa}_{x}^{+}=0.006$ , (b) ${\it\kappa}_{x}^{+}=0.008$ , (c) ${\it\kappa}_{x}^{+}=0.016$ and (d) a vertical line through the map with ${\it\omega}^{+}=0.02$ . Open symbols in (a) are for Quadrio et al. (Reference Quadrio, Ricco and Viotti2009) (○ for  $\mathit{Re}_{{\it\tau}}=200$ ) and Gatti & Quadrio (Reference Gatti and Quadrio2013) (▵ for  $\mathit{Re}_{{\it\tau}}=950$ and ♢ for $\mathit{Re}_{{\it\tau}}=2100$ ) at ${\it\kappa}_{x}^{+}=0.005$ , and open symbols in (d) are for Quadrio et al. (Reference Quadrio, Ricco and Viotti2009) and Gatti & Quadrio (Reference Gatti and Quadrio2013) at ${\it\omega}^{+}=0.012$ .

The results from the stationary wave case ( ${\it\omega}^{+}=0$ ) are presented in figure 7. Also included are the DNS data of Gatti & Quadrio (Reference Gatti and Quadrio2013). The ${\mathcal{D}}{\mathcal{R}}$ increases initially with ${\it\kappa}_{x}^{+}$ at all Reynolds numbers considered, and the rate of increase is attenuated at high Reynolds numbers. For ${\it\kappa}_{x}^{+}>{\it\kappa}_{x,\mathit{opt}}^{+}$ , ${\mathcal{D}}{\mathcal{R}}$ decreases very gradually, resulting in a broad maximum ${\mathcal{D}}{\mathcal{R}}$ peak at higher Reynolds numbers. It is interesting to note that like the wall oscillation case shown in figure 4, the optimal parameter for the stationary wave also appears to increase with the Reynolds number from ${\it\kappa}_{x,\mathit{opt}}^{+}=0.006$ ( ${\it\lambda}^{+}=1070$ ) at $\mathit{Re}_{{\it\tau}}=200$ although the optimal ${\it\kappa}_{x,\mathit{opt}}^{+}$ peak is very broad at $\mathit{Re}_{{\it\tau}}=800$ . Again, the change in ${\mathcal{D}}{\mathcal{R}}$ is significant near the optimal parameter. At ${\it\kappa}_{x,\mathit{opt}}^{+}=0.006$ , the drag reduction decreases from ${\mathcal{D}}{\mathcal{R}}=48$ at $\mathit{Re}_{{\it\tau}}=200$ to ${\mathcal{D}}{\mathcal{R}}=36$ at $\mathit{Re}_{{\it\tau}}=800$ , corresponding to ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}=-12$ . However, if the wavenumber ${\it\kappa}_{x}^{+}=0.008$ ( ${\it\lambda}^{+}=800$ ) is considered, a drag reduction of ${\mathcal{D}}{\mathcal{R}}=37$ can still be achieved at $\mathit{Re}_{{\it\tau}}=800$ , and even a little higher at ${\it\kappa}_{x}^{+}=0.009$ ( ${\it\lambda}^{+}=700$ ). This means that choosing the optimal parameters determined from a low-Reynolds-number case, and scaling the parameters in wall units will not guarantee the optimal drag reduction at the higher Reynolds number. At $\mathit{Re}_{{\it\tau}}=1600$ , the drag is decreased further to ${\mathcal{D}}{\mathcal{R}}=33$ for ${\it\kappa}_{x}^{+}=0.008$ ( ${\it\lambda}^{+}=800$ ), making it ${\rm\Delta}{\mathcal{D}}{\mathcal{R}}=-15$ from the $\mathit{Re}_{{\it\tau}}=200$ case.

Figure 8 shows the drag reduction for the travelling wave case at Reynolds numbers up to $\mathit{Re}_{{\it\tau}}=800$ . Gatti & Quadrio (Reference Gatti and Quadrio2013) data (open symbols) at similar ${\it\omega}^{+}$ and ${\it\kappa}_{x}^{+}$ are also included for comparison. Looking at horizontal lines through the parameter space with ${\it\kappa}_{x}^{+}=0.006$ ( ${\it\lambda}^{+}=1070$ ), 0.008 ( ${\it\lambda}^{+}=800$ ) and 0.016 ( ${\it\lambda}^{+}=400$ ), ${\mathcal{D}}{\mathcal{R}}$ values are shown in figure 8(a–c). The Reynolds number effect is clearly seen in all ${\it\lambda}^{+}$ figures. Please note that Gatti & Quadrio (Reference Gatti and Quadrio2013) data included in figure 8(a) are at ${\it\kappa}_{x}^{+}=0.005$ . As shown from the ${\mathcal{D}}{\mathcal{R}}$ map in figure 3, the peak region for large ${\mathcal{D}}{\mathcal{R}}$ values intersects the ${\it\omega}^{+}=0$ axis at ${\it\kappa}_{x}^{+}={\it\kappa}_{x,s}^{+}$ . For small ${\it\kappa}_{x}^{+}$ values ( ${\it\kappa}_{x}^{+}<{\it\kappa}_{x,s}^{+}$ ), the stationary wave ( ${\it\omega}^{+}=0$ ) produces the largest ${\mathcal{D}}{\mathcal{R}}$ among all ${\it\omega}^{+}$ cases while for ${\it\kappa}_{x}^{+}>{\it\kappa}_{x,s}^{+}$ , the maximum ${\mathcal{D}}{\mathcal{R}}$ is obtained at ${\it\omega}^{+}=c_{\mathit{DR}}^{+}({\it\kappa}_{x}^{+}-{\it\kappa}_{x,s}^{+})$ . For instance, at $\mathit{Re}_{{\it\tau}}=200$ , the stationary wave is most effective for ${\it\kappa}_{x}^{+}\leqslant 0.004$ . As shown in figure 8(a), the ${\it\kappa}_{x,s}^{+}$ value also increases when the Reynolds number increases. At $\mathit{Re}_{{\it\tau}}=800$ , the stationary wave ( ${\it\omega}^{+}=0$ ) has the largest ${\mathcal{D}}{\mathcal{R}}$ for ${\it\kappa}_{x}^{+}\leqslant 0.006$ .

In figure 8(b), the optimal frequency is found at ${\it\omega}^{+}=0.02$ ( $T^{+}=300$ ) with $\mathit{Re}_{{\it\tau}}=200$ , giving a maximum drag reduction of ${\mathcal{D}}{\mathcal{R}}_{\mathit{max}}=50$ , slightly higher than that of the stationary wave, and this is consistent with Quadrio et al. (Reference Quadrio, Ricco and Viotti2009). At $\mathit{Re}_{{\it\tau}}=800$ , this reduces to ${\mathcal{D}}{\mathcal{R}}=37$ with the same frequency parameter and the ${\mathcal{D}}{\mathcal{R}}$ peak becomes less discernible. Figure 8(a,b) suggest that for the travelling wave, the peak region for large ${\mathcal{D}}{\mathcal{R}}$ moves towards higher ${\it\kappa}_{x}^{+}$ values as $\mathit{Re}_{{\it\tau}}$ increases. It is clear in figure 8(c) that the location for maximum $DR$ at ${\it\kappa}_{x}^{+}=0.016$ moves to a lower ${\it\omega}^{+}$ , again indicating that the peak region for large ${\mathcal{D}}{\mathcal{R}}$ moves with the Reynolds number. This is highlighted in the ${\mathcal{D}}{\mathcal{R}}$ map in figure 3 with white dashed lines.

With fixed ${\it\omega}^{+}=0.02$ ( $T^{+}=300$ ) in figure 8(d), the optimal location also changes from ${\it\kappa}_{x}^{+}=0.008$ ( ${\it\lambda}^{+}=800$ ) at $\mathit{Re}_{{\it\tau}}=200$ to ${\it\kappa}_{x}^{+}=0.01$ ( ${\it\lambda}^{+}=640$ ) at $\mathit{Re}_{{\it\tau}}=800$ , but again the maximum peak becomes very broad. Comparing ${\it\kappa}_{x}^{+}=0.008$ and ${\it\kappa}_{x}^{+}=0.016$ ( ${\it\lambda}^{+}=400$ ) at $\mathit{Re}_{{\it\tau}}=200$ , drag reduction at ${\it\kappa}_{x}^{+}=0.008$ is much more efficient while ${\mathcal{D}}{\mathcal{R}}=43$ for ${\it\kappa}_{x}^{+}=0.016$ , and there is a difference in the measured ${\mathcal{D}}{\mathcal{R}}$ levels of 7 between the two wavenumbers. At $\mathit{Re}_{{\it\tau}}=800$ , however, these two points achieve a similar drag reduction at ${\mathcal{D}}{\mathcal{R}}\approx 37$ , showing the larger reduction in ${\mathcal{D}}{\mathcal{R}}$ at the smaller ${\it\kappa}_{x}^{+}$ . At the new optimal parameter of ${\it\kappa}_{x}^{+}=0.01$ , ${\mathcal{D}}{\mathcal{R}}=40$ is obtained at $\mathit{Re}_{{\it\tau}}=800$ .

4.3. Reynolds number scaling

To explore the Reynolds number effect, the values of the drag reduction at the four Reynolds numbers studied are normalised by the drag reduction achieved in the $\mathit{Re}_{{\it\tau}}=200$ case and plotted against the Reynolds number in figure 9. Here ${\mathcal{D}}{\mathcal{R}}/{\mathcal{D}}{\mathcal{R}}_{200}<1$ means that drag reduction becomes less effective as the Reynolds increases from $\mathit{Re}_{{\it\tau}}=200$ . The results from the wall oscillation are shown in figure 9(a). It clearly shows that drag reduction for the wall oscillation deteriorates at higher Reynolds numbers, but there is a large variation in the rate of decrease in ${\mathcal{D}}{\mathcal{R}}/{\mathcal{D}}{\mathcal{R}}_{200}$ among the wall oscillation cases depending on ${\it\omega}^{+}$ values. The thick line leading to the $\mathit{Re}_{{\it\tau}}=1600$ represents the optimal case ( ${\it\omega}_{\mathit{opt}}^{+}=0.06$ ), indicating the drag reduction becomes much less successful at $\mathit{Re}_{{\it\tau}}=1600$ . The results from the stationary wave simulations are shown in figure 9(b). Again, a decrease in ${\mathcal{D}}{\mathcal{R}}/{\mathcal{D}}{\mathcal{R}}_{200}$ is clearly seen as the Reynolds number increases. The rate of decrease in ${\mathcal{D}}{\mathcal{R}}/{\mathcal{D}}{\mathcal{R}}_{200}$ varies among the cases, and the optimal case ( ${\it\kappa}_{x,\mathit{opt}}^{+}=0.008$ ) shows a better trend than the optimal wall oscillation case. It appears that the wall oscillation suffers more from the Reynolds number effect than the stationary wave. At $\mathit{Re}_{{\it\tau}}=1600$ , the wall oscillation becomes 40 % less effective than the $\mathit{Re}_{{\it\tau}}=200$ case while for the stationary wave, this is less than 30 %. Figure 9 highlights the difference in Reynolds number effect dependent on the control parameters chosen. The Reynolds number dependence is not necessarily limited to the range shown in this figure as this only shows the cases studied. It is possible that the Reynolds number dependence has larger variation than presented.

Figure 9.

Change in drag reduction with $\mathit{Re}_{{\it\tau}}$ scaled by the results from the $\mathit{Re}_{{\it\tau}}=200$ case, ${\mathcal{D}}{\mathcal{R}}/{\mathcal{D}}{\mathcal{R}}_{200}$ . (a) Wall oscillation and (b) stationary wave. A longer dash length corresponds to a larger control parameter; ${\it\omega}^{+}$ for the wall oscillation and ${\it\kappa}_{x}^{+}$ for the stationary wave.

The ${\mathcal{D}}{\mathcal{R}}/{\mathcal{D}}{\mathcal{R}}_{200}$ values for near-optimal cases are displayed in figure 10. Wall oscillation data (Choi et al. Reference Choi, Xu and Sung2002; Quadrio & Ricco Reference Quadrio and Ricco2004; Ricco & Quadrio Reference Ricco and Quadrio2008; Touber & Leschziner Reference Touber and Leschziner2012; Gatti & Quadrio Reference Gatti and Quadrio2013) from figure 6 are included. Also included are the stationary wave data of Gatti & Quadrio (Reference Gatti and Quadrio2013). The Reynolds number effect is clearly seen for all cases in figure 10, with large variations between the data. While Choi et al. (Reference Choi, Xu and Sung2002) display a rapid decrease in ${\mathcal{D}}{\mathcal{R}}$ , Gatti & Quadrio (Reference Gatti and Quadrio2013) suggest rather a moderate decrease in ${\mathcal{D}}{\mathcal{R}}$ for the wall oscillation. It is interesting to note that the present results indicate the Reynolds number effect is much stronger in the wall oscillation than the stationary wave, while the opposite trend was observed in Gatti & Quadrio (Reference Gatti and Quadrio2013) although they considered the ${\it\kappa}_{x}^{+}=0.005$ case.

Figure 10.

Change in drag reduction, ${\mathcal{D}}{\mathcal{R}}/{\mathcal{D}}{\mathcal{R}}_{200}$ , for the wall oscillation case at ${\it\omega}^{+}=0.06$ . Open symbols indicate the optimal stationary wave cases. Present data are for ${\it\kappa}_{x}^{+}=0.008$ , and Gatti & Quadrio (Reference Gatti and Quadrio2013) are for ${\it\kappa}_{x}^{+}=0.005$ . Flow conditions for the wall oscillation are given in figure 6.

In this study, a scaling in the form $\mathit{Re}_{{\it\tau}}^{-{\it\alpha}}$ is calculated from ${\mathcal{D}}{\mathcal{R}}/{\mathcal{D}}{\mathcal{R}}_{200}$ to quantify the effect of the Reynolds number. A large (positive) value of ${\it\alpha}$ corresponds to a large decrease in ${\mathcal{D}}{\mathcal{R}}$ as the Reynolds number increases, and can therefore be interpreted as an unfavourable scaling. The calculated scalings for the wall oscillation and stationary wave control are shown in figure 11 and also summarised in table 4. For the wall oscillation, large values of ${\it\alpha}$ are obtained at lower values of ${\it\omega}^{+}$ with the largest ${\it\alpha}$ at ${\it\omega}^{+}=0.04$ ( $T^{+}=150$ ), emphasising the fact that the drag reduction is severely affected by the Reynolds number effect. The ${\it\alpha}$ value is still quite large near ${\it\omega}^{+}=0.06$ ( $T^{+}=100$ ), the optimal value for $\mathit{Re}_{{\it\tau}}=200$ . Ricco & Quadrio (Reference Ricco and Quadrio2008) also suggested that a stronger effect of Reynolds number occurs for smaller ${\it\omega}^{+}$ (large $T^{+}$ ). The scaling is improved for ${\it\omega}^{+}>0.06$ , partly as the optimal value of ${\it\omega}^{+}$ is increasing with the Reynolds number; at ${\it\omega}^{+}=0.08$ ( $T^{+}=75$ ), the optimal value for $\mathit{Re}_{{\it\tau}}=800$ , the scaling value is ${\it\alpha}\approx 0.17$ . The scaling is further improved at larger ${\it\omega}^{+}$ with ${\it\alpha}\approx 0.1$ . There is no universal ${\it\alpha}$ value which can be applied for a wide range of control parameters. Instead, a large variation in ${\it\alpha}$ values is obtained. The present DNS results demonstrate that this type of scaling does not represent the Reynolds number dependence observed in the spanwise wall forcing flow control. It is interesting to note that Choi et al. (Reference Choi, Xu and Sung2002) used ${\it\alpha}=0.4$ to scale their wall oscillation $S^{+}$ parameter.

Figure 11.

Reynolds number scaling parameter ${\it\alpha}$ for (a) wall oscillation and (b) stationary wave.

Table 4.

The ${\mathcal{D}}{\mathcal{R}}$ scalings for (a) wall oscillation and (b) the stationary wave. Here $\dagger$ indicates the optimal parameter at $\mathit{Re}_{{\it\tau}}=200$ and $\ddagger$ indicates the optimal parameter at $\mathit{Re}_{{\it\tau}}=800$ .

Figure 11(b) and table 4(b) show the Reynolds number scalings for the stationary wave at different ${\it\kappa}_{x}^{+}$ values. Similarly to the wall oscillation case, the ${\mathcal{D}}{\mathcal{R}}$ for the stationary wave case is also seen to be affected more at lower values of ${\it\kappa}_{x}^{+}$ by the Reynolds number, where there are large values of ${\it\alpha}$ in the scaling. Unlike the wall oscillation, ${\it\alpha}$ decreases monotonously with ${\it\kappa}_{x}^{+}$ . This is likely due to the limit on the minimum value of ${\it\kappa}_{x}^{+}$ studied, and is expected to behave similarly to that of wall oscillation. As shown in figure 12(a), the wall oscillation and stationary wave display a similar trend when a relationship of ${\it\omega}^{+}={\it\kappa}_{x}^{+}U_{c}^{+}$ is used to find an equivalent ${\it\omega}^{+}$ value for ${\it\kappa}_{x}^{+}$ , where $U_{c}^{+}=10$ (Viotti et al. Reference Viotti, Quadrio and Luchini2009). The scaling around the region of optimal drag reduction for the stationary wave is better than for the wall oscillation hinting that, at higher Reynolds number, the stationary wave may remain as a more advantageous control method within the control parameters studied.

Figure 12.

Reynolds number scaling parameter ${\it\alpha}$ for (a) wall oscillation and stationary wave and (b) travelling waves ( ${\it\kappa}_{x}^{+}=0.008$ and ${\it\omega}^{+}=0.06$ ). An equivalent ${\it\omega}^{+}$ value (squares, yellow online) is calculated for ${\it\kappa}_{x}^{+}$ using a relationship of ${\it\omega}^{+}={\it\kappa}_{x}^{+}U_{c}^{+}$ , where $U_{c}^{+}=10$ (Viotti et al. Reference Viotti, Quadrio and Luchini2009).

Figure 12(b) and table 5 show the ${\it\alpha}$ values of the scalings calculated for the travelling wave, emphasising the variation of the scaling dependent on the control parameters. Again, an equivalent ${\it\omega}^{+}$ values are calculated for ${\it\kappa}_{x}^{+}$ using ${\it\omega}^{+}={\it\kappa}_{x}^{+}U_{c}^{+}$ . Like the wall oscillation and stationary wave, ${\it\alpha}$ values vary significantly depending on ${\it\omega}^{+}$ and ${\it\kappa}_{x}^{+}$ values. The scaling is better (i.e. smaller ${\it\alpha}$ values) for larger ${\it\omega}^{+}$ and ${\it\kappa}_{x}^{+}$ . The main impact of this is that, although the ${\mathcal{D}}{\mathcal{R}}$ is reducing as the Reynolds number increases, and the scaling is strong at the optimal control parameters, by adjusting the control parameters correctly, the scaling with Reynolds number is improved. This means that, at a higher value of $\mathit{Re}_{{\it\tau}}$ , using the parameters which were not optimal at a low Reynolds number can give a more favourable drag reduction. In the wall oscillation and stationary wave cases, lower values of ${\it\omega}^{+}$ and ${\it\kappa}_{x}^{+}$ have higher ${\it\alpha}$ values, hence the drag reduction deteriorates more rapidly as the Reynolds number increases.

Table 5.

The ${\mathcal{D}}{\mathcal{R}}$ scalings for (a) the horizontal line through the map with ${\it\kappa}_{x}^{+}=0.008$ and (b) the vertical line with ${\it\omega}^{+}=0.02$ . Here $\dagger$ indicates the optimal parameter at $\mathit{Re}_{{\it\tau}}=200$ and $\ddagger$ indicates the optimal parameter at $\mathit{Re}_{{\it\tau}}=800$ .

4.4. Turbulence statistics

Figure 13 shows the initial response of the skin friction to the wall oscillation, normalised by $C_{f,0}$ . The level of oscillation in $C_{f}$ is seen to increase as the Reynolds number is increased while the overall drag reduction deteriorates with the Reynolds number. It has been shown that, when the oscillation is applied for large time periods (corresponding to small ${\it\omega}^{+}$ ), substantial oscillations occur in the skin friction (Jung et al. Reference Jung, Mangiavacchi and Akhavan1992; Touber & Leschziner Reference Touber and Leschziner2012). This is due to the turbulence responding to the wall motion, and an increase of turbulent intensity in the new shear direction. At certain points throughout the oscillation drag is increased, and this can lead to an overall drag increase when ${\it\omega}^{+}$ is sufficiently small (figure 4 when ${\it\omega}^{+}=0.01$ ). Figure 13 suggests that the oscillation in the drag reduction does not scale with wall units, and deteriorates the drag reduction achieved at the optimal parameters as the Reynolds number increases. Another interesting point from this figure is that the rate of the initial decay in $C_{f}$ is very similar at all Reynolds numbers. However, this decay is sustained for longer at the lower Reynolds numbers; for the $\mathit{Re}_{{\it\tau}}=200$ case the initial decrease of $C_{f}$ is sustained until $t^{+}=700$ while the $C_{f}$ minimum is achieved much earlier at $t^{+}=300$ for the $\mathit{Re}_{{\it\tau}}=1600$ case. In global units, the initial decay of $C_{f}$ is even faster at higher Reynolds numbers.

Figure 13.

Initial response of skin friction (normalised by the no-control case) for the wall oscillation with ${\it\omega}^{+}=0.06$ at the four Reynolds numbers. The arrow indicates the increase in Reynolds number: $\mathit{Re}_{{\it\tau}}=200,400,800$ and 1600.

Figure 14.

Stokes layer for oscillation case: (a) ${\it\omega}_{\mathit{opt}}^{+}=0.06$ ( $T^{+}=100$ ) and (b) ${\it\omega}^{+}=0.03$ ( $T^{+}=200$ ). Dashed lines are the laminar Stokes layer solution: $W(y,t)=A\text{e}^{-{\it\eta}}\cos ({\it\omega}t-{\it\eta})$ , here ${\it\eta}=y\sqrt{{\it\omega}/2{\it\nu}}$ .

Figures 14 and 15 show the phase-averaged spanwise velocity profiles for the wall oscillation at two oscillation frequencies, ${\it\omega}^{+}=0.06$ ( $T^{+}=100$ ) and ${\it\omega}^{+}=0.03$ ( $T^{+}=200$ ). Also included are the laminar Stokes flow profiles (Schlichting Reference Schlichting1968). Please note that the laminar Stokes layer thickness is ${\it\delta}^{+}=\sqrt{4{\rm\pi}T^{+}}$ in wall units; ${\it\delta}^{+}=35.4$ for ${\it\omega}_{\mathit{opt}}^{+}=0.06$ ( $T^{+}=100$ ) and ${\it\delta}^{+}=50.1$ for ${\it\omega}^{+}=0.03$ ( $T^{+}=200$ ), and there is no Reynolds number dependence. For the optimal case ( ${\it\omega}_{\mathit{opt}}^{+}=0.06$ ), the spanwise turbulent velocity profiles at ${\it\eta}=1$ , corresponding to $y_{0}^{+}=6$ , are very similar to the Stokes profile at all four Reynolds numbers (figure 15 a), indicating that the turbulent Stokes layer near the wall is not affected by increasing the Reynolds number. This finding is interesting considering the weakened drag reduction observed at higher Reynolds numbers. Further away from the wall ( ${\it\eta}=4.6$ and $2{\rm\pi}$ ) (figure 15 b,c), the spanwise turbulent velocity decays slower than the laminar flow, and the phase difference for different Reynolds numbers is clearly discernible. At ${\it\omega}^{+}=0.03$ ( $T^{+}=200$ ), the spanwise turbulent velocity profile decays much slowly, and this trend was also observed in the previous DNS study at $\mathit{Re}_{{\it\tau}}=200$ (Touber & Leschziner Reference Touber and Leschziner2012). The Reynolds number effect is more evident at the lower ${\it\omega}^{+}$ value, and the spanwise turbulent velocity profile deviates significantly from the Stokes profile in figure 15(d–f). The deviation from the laminar Stokes profile increases at higher Reynolds numbers.

Figure 15.

Phase-averaged spanwise velocity for oscillation case for (a–c) ${\it\omega}_{\mathit{opt}}^{+}=0.06$ ( $T^{+}=100$ ) and (d–f) ${\it\omega}^{+}=0.03$ ( $T^{+}=200$ ). In (a,d) ${\it\eta}=1$ ; in (b,e) ${\it\eta}=4.6$ ; and in (c, f) ${\it\eta}=2{\rm\pi}$ . Dashed lines are the laminar Stokes layer solution.

Figure 16.

Mean velocity ( $U^{+}$ ) profiles at four Reynolds numbers. (a) Wall oscillation for ${\it\omega}_{\mathit{opt}}^{+}=0.06$ and (b) stationary wave cases for ${\it\kappa}_{x,\mathit{opt}}^{+}=0.008$ . Here $u_{{\it\tau}}$ is used in wall units. Black lines represent the flow control case and grey (orange) lines represent the no-control case. The thick grey straight line represents the law of the wall: $u^{+}=(1/{\it\kappa})\ln y^{+}+3.7$ and ${\it\kappa}=0.37$ (Nagib & Chauhan Reference Nagib and Chauhan2008).

The mean velocity profiles are shown in figure 16 for both the wall oscillation and stationary wave cases. The optimal control parameters are considered here: ${\it\omega}_{\mathit{opt}}^{+}=0.06$ ( $T^{+}=100$ ) for the wall oscillation and ${\it\kappa}_{x,\mathit{opt}}^{+}=0.008$ ( ${\it\lambda}^{+}=800$ ) for the stationary wave cases. For the no-control cases, the streamwise mean velocity profiles at four Reynolds numbers agree well with the log-law profile. Also included in the figure is a logarithmic velocity distribution $u^{+}=(1/{\it\kappa})\ln y^{+}+B$ , where ${\it\kappa}=0.37$ and $B=3.7$ (Nagib & Chauhan Reference Nagib and Chauhan2008). When the $u_{{\it\tau}}$ value for each controlled flow is used in wall units, the log-law velocity profiles show a characteristic upward shift (Choi et al. Reference Choi, DeBisschop and Clayton1998; Touber & Leschziner Reference Touber and Leschziner2012), and the magnitude of the shift is roughly proportional to ${\mathcal{D}}{\mathcal{R}}$ . It is interesting to note that the upward shift was much larger for the stationary wave reflecting larger ${\mathcal{D}}{\mathcal{R}}$ values achieved. The upward shift for the wall oscillation is approximately ${\rm\Delta}U^{+}=5.3,3.8,3.1$ and 2.7 at $\mathit{Re}_{{\it\tau}}=200,400,800$ and 1600, respectively while the upward shift is larger for the stationary wave with ${\rm\Delta}U^{+}=7.6,6.2,5.5$ and 5.0 at $\mathit{Re}_{{\it\tau}}=200,400,800$ and 1600, respectively. Since the mass flow rates are constant for the uncontrolled and controlled flows, a large ${\rm\Delta}U^{+}$ means a thicker viscous sublayer in the controlled flow. For the stationary wave, the viscous sublayer is more affected than the wall oscillation case.

Figure 17 shows the velocity fluctuations ( $u_{i,rms}/u_{{\it\tau}0}$ ) and the Reynolds shear stress ( $-\overline{uv}/u_{{\it\tau}_{0}}^{2}$ ) for the stationary wave. The optimal wave number ${\it\kappa}_{x}^{+}=0.008$ is chosen for all four Reynolds numbers studied. When scaled with the friction velocity of the no-control case, $u_{{\it\tau}0}$ , the velocity fluctuations display two distinctive characteristics: first, all three velocity components decrease significantly, and second, the location for the maximum $u_{i,rms}^{+}$ moves away from the wall. A similar trend was also observed in other types of flow control (Choi, Moin & Kim Reference Choi, Moin and Kim1994; Chung & Talha Reference Chung and Talha2011). The decrease in $u_{rms}^{+}$ is much larger at lower Reynolds numbers, consistent with the amount of drag reduction observed earlier. As the Reynolds number increases, the effect on the velocity fluctuations becomes weaker. This can be most clearly seen in $u_{rms}^{+}$ profiles; for the no-control case, the maximum $u_{rms}^{+}$ increases by 0.3 when the Reynolds number increases from $\mathit{Re}_{{\it\tau}}=200$ to 1600, but the difference becomes much larger for the stationary wave with ${\rm\Delta}u_{rms}^{+}=0.45$ , indicating the stationary wave decreases $u_{rms}^{+}$ more significantly at lower Reynolds numbers. The difference between the new location for the maximum $u_{i,rms}^{+}$ and the no-control value is found to be quite large at $\mathit{Re}_{{\it\tau}}=200$ , but this reduces following $y_{u}^{+}\sim \mathit{Re}_{{\it\tau}}^{-0.15}$ as the Reynolds number increases.

The changes in the maximum value of each component of Reynolds stresses are examined at four Reynolds numbers. The reduction in maximum $u_{i,rms}$ , normalised by the maximum $u_{i,rms}$ value for the no-control case is shown in figure 18. All quantities show good correlation with ${\mathcal{D}}{\mathcal{R}}$ : a large ${\mathcal{D}}{\mathcal{R}}$ is accompanied with a large reduction in the maximum value of the Reynolds stresses. Chung & Talha (Reference Chung and Talha2011) observed a similar trend in the opposition control. This finding shows that turbulence fluctuations (e.g.  $u_{rms}$ ) are reduced much less by the flow control at higher Reynolds numbers. Iwamoto et al. (Reference Iwamoto, Fukagata, Kasagi and Suzuki2005) suggested that the degree of turbulence attenuation in the near-wall region is related to the amount of drag reduction achievable. This may explain why drag reduction deteriorates at high Reynolds numbers.

Figure 17.

Reynolds stresses profiles from the stationary wave cases for ${\it\kappa}_{x,\mathit{opt}}^{+}=0.008$ at four Reynolds numbers: (a) $u_{rms}^{+}$ , (b) $v_{rms}^{+}$ , (c) $w_{rms}^{+}$ and (d) $-\overline{uv}$ . Here $u_{{\it\tau}0}$ is used in wall units. Black lines represent the flow control case, and grey (orange) lines represent the no-control case. The inset in (a) is for the location of the maximum $u_{rms}^{+}$ for the stationary wave cases in log–log scale, $y_{u}^{+}\sim \mathit{Re}_{{\it\tau}}^{-0.15}$ .

Figure 18.

Reduction in maximum $u_{i,rms}$ as a function of ${\mathcal{D}}{\mathcal{R}}$ , ${\rm\Delta}u_{i,\mathit{max}}/u_{i,\mathit{max},0}$ . Here ${\rm\Delta}u_{i,\mathit{max}}$ is normalised by the maximum $u_{i,rms}$ value for the no-control case, $u_{i,\mathit{max},0}$ .

Since the drag reduction is directly related to the Reynolds shear stress, $-\overline{uv}$ profiles are analysed further. Fukagata, Iwamoto & Kasagi (Reference Fukagata, Iwamoto and Kasagi2002) showed that the contribution of the Reynolds shear stress to the skin-friction drag is the weighted average of Reynolds shear stress.

where $\mathit{Re}=U_{m}h/{\it\nu}$ . The first term is the laminar flow contribution while the second term represents the turbulent flow contribution. Figure 19(a) shows the $(1-y)(-\overline{uv})$ profile for four Reynolds numbers. The accumulated contribution of the weighted turbulent shear stress from 0 to $y$ is shown in figure 19(b). The Reynolds shear stress in figures 17(d) and 19(a) shows a stronger Reynolds number effect. This is not surprising because the no-control turbulent flow also experiences the Reynolds number effect close to the wall (DeGraaff & Eaton Reference DeGraaff and Eaton2000; Hoyas & Jiménez Reference Hoyas and Jiménez2006).

Figure 19.

(a) Weighted turbulent shear stress, $\left(1-y\right)(-\overline{uv})$ , and (b) accumulated contribution of the weighted turbulent shear stress from 0 to $y$ , $\int _{0}^{y}\left(1-y\right)(-\overline{uv})\text{d}y$ .

In this study, the turbulent flow contribution is further divided into two components: inner and outer regions. The wall-normal location for the maximum of the Reynolds shear stress, $y_{\text{p}}$ , is used

Sreenivasan & Sahay (Reference Sreenivasan, Sahay and Panton1997) argued that the momentum balance in the critical layer around $y_{\text{p}}$ is different from those in the classical inner and outer layers. It is known that the $y_{\text{p}}^{+}$ location increases with the Reynolds number: $y_{\text{p}}^{+}=2\mathit{Re}_{{\it\tau}}^{1/2}$ (Sreenivasan & Sahay Reference Sreenivasan, Sahay and Panton1997; Guala, Hommema & Adrian Reference Guala, Hommema and Adrian2006).

Figure 20(a) shows that for no-control case, the contribution of the outer region increases with the Reynolds number while the contribution of the inner region decreases gradually. As expected, the laminar contribution becomes smaller as the Reynolds number increases. For the stationary wave, the contribution of the outer region is significantly reduced at all Reynolds numbers tested, and the amount of reduction does not appear to be affected by increasing the Reynolds number, indicating a constant drag reduction from the outer region. The contribution of the inner region is roughly constant for the stationary wave. Figure 20(b) displays the contribution of each component to ${\mathcal{D}}{\mathcal{R}}$ . The contribution of the outer region becomes more or less constant, just over ${\mathcal{D}}{\mathcal{R}}\approx 30$ , while the contribution of the inner region diminishes gradually from ${\mathcal{D}}{\mathcal{R}}=20$ at $\mathit{Re}_{{\it\tau}}=200$ . The Reynolds number effect shown in figure 20(b) is related to a weaker drag reduction observed at high Reynolds numbers. This is consistent with Iwamoto et al. (Reference Iwamoto, Fukagata, Kasagi and Suzuki2005). They predicted that the Reynolds number effect on the drag reduction rate is relatively mild (i.e. logarithmic) if the fluctuations in the regions near the wall can completely be damped. This finding suggests an exciting possibility of a finite ${\mathcal{D}}{\mathcal{R}}$ value at a very high Reynolds number, which is very promising for the transport industry, in particular, the air transport, where a typical operating condition is of the order of $\mathit{Re}_{{\it\tau}}=10\,000$ . This is a conjecture at this stage, and a further study with higher-Reynolds-number DNS or experiment will be able to present a clearer picture regarding the Reynolds number effect.

Figure 20.

(a) Skin-friction contribution from laminar component; inner and outer regions. Dashed lines with open symbols are for the no-control cases, and solid lines are for the stationary wave cases. (b) Contribution to ${\mathcal{D}}{\mathcal{R}}$ from inner and outer regions. The total ${\mathcal{D}}{\mathcal{R}}$ values are also included.