3.1. Drag reduction map as a function of frequency and wavelength

Figure 3(a,b) display the maps of ${\rm DR}(\omega ^+,\kappa ^+_x)$ at $Re_\tau = 951$ and $4000$ from the computations listed in table 1. At each $Re_\tau$, we have $56$ DR data points. To generate the maps, we perform bilinear interpolation of our DR data points onto a uniform $20 \times 20$ grid over the parameter space $0 \le \kappa ^+_x \le 0.02$ and $-0.2 \le \omega ^+ \le +0.2$. At $Re_\tau = 951$, the maximum DR of $35.4\,\%$ at $(\omega ^+,\kappa ^+_x) = (0.05,0.021)$ is in close agreement with the DNS of Gatti & Quadrio (Reference Gatti and Quadrio2016) at $Re_\tau \simeq 950$, where the maximum DR was found to be $38.8\,\%$ at $(\omega ^+,\kappa ^+_x) = (0.05, 0.0195)$. At $Re_\tau = 4000$, the maximum DR decreases to $27.5\,\%$ at the same actuation parameters $(\omega ^+,\kappa ^+_x) = (0.05,0.021)$. At each Reynolds number, DR changes more drastically by changing $\omega ^+$ than by changing $\kappa ^+_x$.

Figure 3. (a,b) Maps of DR for $A^+ =12$ at (a) $Re_\tau = 951$ and (b) $Re_\tau = 4000$. The local maximum DR (blue dashed-dotted line) and the local minimum DR (black dashed line) for $\kappa ^+_x > 0$ are indicated for clarity. We label the region on the left-hand side of the blue dashed-dotted line with I, between the blue dashed-dotted line and the black dashed line with II and the right-hand side of the black dashed line with III. (c) Map of the difference in DR between $Re_\tau = 4000$ and $Re_\tau = 951$. (d) Map of the difference in DR between $Re_\tau = 4000$ and GQ's prediction (Gatti & Quadrio Reference Gatti and Quadrio2016) at the same Reynolds number. In plots (a–d) the contour fields and the contour lines show the same quantity. For plots (a,b), the contour lines grow from $-20\,\%$ to $40\,\%$ and, for (c,d), the contour lines grow from $-7\,\%$ to $+7\,\%$.

When $\kappa ^+_x = 0$, there is no travelling wave (plane wall oscillation), and the variation of the DR is symmetric between $\omega ^+ <0$ and $\omega ^+ > 0$. In this case, two equal local maxima (at $\omega ^+ \simeq \pm 0.05$) and a local minimum (at $\omega = 0$) emerge. When $\kappa ^+_x >0$, a travelling wave is generated, and the variation of DR is asymmetric between $\omega ^+ < 0$ and $\omega ^+ > 0$. In this case, at each $\kappa ^+_x$ only one local maximum (blue dashed-dotted curve in figure 3) and one local minimum (black dashed curve in figure 3) appear in the DR. These observations are in agreement with Quadrio et al. (Reference Quadrio, Ricco and Viotti2009) and Gatti & Quadrio (Reference Gatti and Quadrio2016). Overall, within our parameter space, the map of DR consists of three distinct regions. Region I to the left of the local maximum DR (blue dashed-dotted curve) where $\omega ^+ \lesssim 0$ (upstream travelling wave); in this region ${\rm DR}> 0$. Region II represents the crossover from the local maximum to the local minimum DR (between the blue dashed-dotted curve and the black dashed curve). For $\kappa ^+_x \lesssim 0.007$, the local minimum DR is positive, however, for $\kappa ^+_x \gtrsim 0.007$, the local minimum DR becomes negative (hence, a drag increase). Increase in $\kappa ^+_x$ beyond $0.007$ leads to a larger drag increase area, and the local minimum DR becomes more negative; Quadrio et al. (Reference Quadrio, Ricco and Viotti2009) and Gatti & Quadrio (Reference Gatti and Quadrio2016) observe similar trends. Quadrio et al. (Reference Quadrio, Ricco and Viotti2009) find that the local minimum DR follows the line $\omega ^+/\kappa ^+_x \simeq 10$. In other words, a maximum drag increase occurs when the travelling wave speed is about $10u_{\tau _0}$, which is nearly the same as the convective speed of the near-wall flow structures. Similarly, in figure 3(a,b) the black dashed curve that marks the local minimum DR follows $\omega ^+/\kappa ^+_x \simeq 10$. Region III covers the right of the local minimum DR (black dashed curve) where $\omega ^+ > 0$ (downstream travelling wave); in this region an increase in $\omega ^+$ increases the DR.

In figure 3(c) we display the difference in DR as the Reynolds number changes from 4000 to 951. For most of the $(\omega ^+, \kappa ^+_x)$ space, DR is lower at the higher Reynolds number. Only within the range $0.005 \lesssim \kappa ^+_x \lesssim 0.020, +0.1 \lesssim \omega ^+ \lesssim +0.2$ do we observe the opposite trend. This region coincides with the drag-increasing range (${\rm DR} < 0$) with $\omega ^+ > 0$. This observation is consistent with GQ's model (1.5), where ${\rm DR} <0$ (hence, $\Delta B < 0$) predicts an increase in DR as the Reynolds number increases. To make these comparisons more quantitative, in figure 3(d) we show the difference in DR between our results and GQ's model at $Re_\tau = 4000$. To predict DR, the model (1.5) requires $C_{f_0}$ and the value of the log-law shift $\Delta B$ for each set of actuation parameters $(A^+, \kappa ^+_x, \omega ^+)$. For $C_{f_0}$, we use Dean's power-law correlation (Dean Reference Dean1978) that agrees well with the DNS data given by MacDonald et al. (Reference MacDonald, Hutchins and Chung2019). We can obtain $\Delta B$ from a low-Reynolds-number simulation for the same set of $(A^+, \kappa ^+_x, \omega ^+)$ because $\Delta B$ is assumed to be Reynolds number independent. Therefore, we use our results at $Re_\tau = 951$, where for each $(\omega ^+, \kappa ^+_x)$, we find $\Delta B$ from the velocity difference $\Delta U^*$ at $y^* = 200$ (similar to figure 2b). We choose $y^* = 200$ as it is far enough from the wall to fall into the log region, but not too far to fall into the wake region ($y/h \gtrsim 0.3$ according to Pope Reference Pope2000). By having $\Delta B$ at each $(\omega ^+, \kappa ^+_x)$ and having $C_{f_0}$ at $Re_\tau = 4000$, we can reconstruct the DR map based on GQ's model.

Figure 3(d) shows the overall good performance of GQ's model for this range of Reynolds numbers. In region I the difference in DR between LES and GQ's model is less than $2\,\%$, i.e. $|{\rm DR}\%_{LES,Re_\tau = 4000} - {\rm DR}\%_{GQ,Re_\tau = 4000}| \lesssim 2\,\%$. This is a very good agreement considering that DR varies between $15\,\%$ and $28\,\%$ in region I. In regions II and III we observe some slight differences in DR between LES and GQ's model, especially in region II in the drag-increasing range. In this range, the difference in DR between LES and GQ's model reaches $4\,\%$, which is the same order as DR (see figure 3b). In region III, for $\omega ^+ \gtrsim +0.1$ again, we observe good agreement between LES and GQ's model (less than $2\,\%$ difference). In the following sections we investigate the reasons behind the different performance of GQ's model in regions I, II and III related to the changes in the Stokes layer dynamics and the near-wall turbulence in each of these regions.

3.2. Mean velocity profiles

To obtain an overall picture of the mean velocity behaviour in regions I, II and III (see figure 4), we consider the seven runs conducted at $Re_\tau = 4000, A^+ = 12$ and $\kappa ^+_x = 0.007$ for $\omega ^+$ ranging from $-0.2$ to $+0.2$. In figure 4(a) we identify the selected values of $\omega ^+$ (filled squares) on the DR map along with the local maximum DR (blue dashed-dotted line) and the local minimum DR (black dashed line). The corresponding velocity profiles are shown in figure 4(b,c) for $\omega ^+ \le 0$ (upstream travelling waves) up to the local maximum DR (region I) and $\omega ^+ \ge 0$ (downstream travelling waves) beyond the local maximum DR (regions II, III), respectively.

Figure 4. Variation of DR and the mean velocity profiles $U^*$ at $Re_\tau = 4000$, $A^+ = 12$, $\kappa ^+_x = 0.007$ and $-0.2 \le \omega ^+ \le +0.2$. (a) Variation of DR with $\omega ^+$; the inset shows the location of the data points on the DR map. The blue dashed-dotted line and the black dashed line are the local maximum and minimum DR.(b,c) Variation of the $U^*$ profiles with $\omega ^+$ for (b) upstream travelling wave ($\omega ^+ \le 0$) and (c) downstream travelling wave ($\omega ^+ \ge 0$); the profile (—$\square$—) corresponds to the non-actuated case and the profiles with no symbol correspond to the actuated cases. For each profile, the solid line is the resolved portion and the dashed line is the reconstructed portion following Nagib & Chauhan (Reference Nagib and Chauhan2008). For each case, the colour of its $U^*$ profile in (b,c) is consistent with the colour of its DR data point in (a). In (b,c) the inset plots the same profiles in terms of $U^*/y^*$ vs $y^*$. (d,e) Diagnostic function $y^* \,{\rm d}U^*/{{\rm d} y}^*$ for the profiles in (b,c); the inset shows the velocity difference $\Delta U^* = U^*_{act} - U^*_{non{\text {-}}act}$ between each actuated profile $U^*_{act}$ and the non-actuated profile $U^*_{non{\text {-}}act}$.

When $\omega ^+ \le 0$, the log region of the actuated profiles is shortened and shifted above the non-actuated counterpart (figure 4b), corresponding to a positive DR. The shortening of the log region is due to the thickening of the viscous sublayer. We show the viscous sublayer thickening in the inset of figure 4(b), in that the actuated profiles of $U^*/y^*$ are closer to unity for a greater wall distance compared with their non-actuated counterpart. We show the shortening of the log region in figure 4(d) by plotting the diagnostic function $y^* \,{\rm d}U^*/{{\rm d}y}^*$. The log region appears as a plateau with the value of $\kappa ^{-1} \simeq 2.5$. For the non-actuated case, the plateau appears for $100 \lesssim y^* \lesssim 600$. This range is consistent with the DNS of channel flow by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) and Lee & Moser (Reference Lee and Moser2015) at $Re_\tau = 4200$ and $5200$, respectively (see figure 3a in Lee & Moser Reference Lee and Moser2015). For the actuated cases, the plateau is narrowed further (i.e. log region is shortened) as DR increases. We quantify the shift in the log region by plotting $\Delta U^*$ (inset of figure 4d). The magnitude of the shift increases as DR increases. These observations are also reported in the previous turbulent DR studies, including turbulent flow with the spanwise wall oscillation (Di Cicca et al. Reference Di Cicca, Iuso, Spazzini and Onorato2002; Touber & Leschziner Reference Touber and Leschziner2012; Hurst et al. Reference Hurst, Yang and Chung2014), turbulent flow with the streamwise travelling wave (Hurst et al. Reference Hurst, Yang and Chung2014; Gatti & Quadrio Reference Gatti and Quadrio2016), turbulent flow of a polymer solution (Ptasinski et al. Reference Ptasinski, Boersma, Nieuwstadt, Hulsen, Van den Brule and Hunt2003; White & Mungal Reference White and Mungal2008) and turbulent flow over piezoelectrically excited travelling waves (Musgrave & Tarazaga Reference Musgrave and Tarazaga2019). Gatti & Quadrio (Reference Gatti and Quadrio2016) derived their predictive model (1.5) based on similar observations of the velocity profiles in region I, and as a result, GQ's prediction works well in this region (figure 3d). The behaviour of the profiles in region I is consistent with the ISA pathway, where only the inner-scale eddies up to the buffer region are actuated.

In region II we observe a sudden drop in DR as $\omega ^+$ changes from 0 to $+0.1$ (figure 4a), with a corresponding decrease in the logarithmic shift (figure 4c). A distinct feature of region II is the high level of distortion in the $U^*$ profile, which is particularly severe at $\omega ^+ = +0.05$. For this case, the diagnostic function tends towards the plateau $\kappa ^{-1}$, but does not reach it. Similarly, $\Delta U^*$ for this case approaches a plateau of $1.7$ by the resolved height $y^*_{res} \simeq 750$, but does not reach it (inset in figure 4e). This is our most challenging case for computing DR using our approach in § 2.3 (figure 2). For accurate calculation of the DR, $\Delta U^*$ needs to reach a plateau by the resolved height $y^*_{res} \simeq 750$, i.e. the resolved height must fall into the logarithmic region. In Appendix C we deliberately consider this challenging case for a domain size study. We double the domain length and width compared with the medium domain (figure 1b), extending the resolved height to $y^*_{res} \simeq 1500$. The difference in DR is $1.4\,\%$ between the medium domain and the large domain (table 4). Furthermore, the large domain reinforces the approach of $\Delta U^*$ to a plateau of $1.7$ (the inset in figure 18b). To our knowledge, such significant levels of distortion in the $U^*$ profile have not been seen before in previous studies of flows over drag-reducing or drag-increasing surfaces. For example, in rough wall turbulent flows $\Delta U^*$ is almost constant for $y^* \gtrsim 30$ (e.g. figure 6 in Chan et al. Reference Chan, MacDonald, Chung, Hutchins and Ooi2015 or figure 3 in MacDonald et al. Reference MacDonald, Chung, Hutchins, Chan, Ooi and García-Mayoral2017), while in turbulent flows over riblets $\Delta U^*$ is almost constant for $y^* \gtrsim 100$ (e.g. figure 2 in Endrikat et al. Reference Endrikat, Modesti, MacDonald, García-Mayoral, Hutchins and Chung2021b). In §§ 3.4 and 3.5 we discuss the physics behind the highly distorted mean velocity profiles (figure 4c,e).

Table 4. Summary of the LES cases for the domain size study. For all cases, $A^+ = 12$. We consider three cases with $(\kappa ^+_x, \omega ^+) = (0.021, -0.1), (0.021, +0.1), (0.007, +0.05)$. For each case, we perform a medium-domain simulation (figure 1a, $L_x \times L_z \simeq 2.0h \times 0.6 h, y^+_{res} \simeq 1000$), and a large-domain simulation (figure 1b, $L_x \times L_z \simeq 4.0h \times 1.2 h, y^+_{res}\simeq 2000$).

In region III, when $\omega ^+$ increases to $+0.2$ (figure 4c), DR increases to $13\,\%$ and the $U^*$ profile behaves similarly to that seen in region I. A well-defined logarithmic shift appears beyond $y^* \simeq 100$ with viscous sublayer thickening.

3.3. Turbulence statistics

We now assess the behaviour of the Reynolds stress distributions at $Re_\tau =4000$ (figure 5a–d) and the turbulent kinetic energy production $P = -\langle u' v' \rangle _{xzt}\, {\rm d}U/{{\rm d} y}$ (figure 5e, f), where $\langle \cdots \rangle _{xzt}$ denotes averaging over the $xz$ plane and time. We highlight four cases from figure 4 ($A^+ = 12, \ \kappa ^+_x = 0.007$), where we vary $\omega ^+$ from $-0.05$ to $+0.20$. As indicated earlier, we employ a finer grid resolution for these cases to properly resolve the Reynolds stresses (see Appendix B). We plot the profiles scaled by the non-actuated $u_{\tau _0}$ (dashed-dotted lines, figure 5a,c) and by the actuated $u_\tau$ (solid lines, figure 5b,d). Scaling by $u_{\tau _0}$ is comparable to scaling by the bulk velocity $U_b$ (Gatti & Quadrio Reference Gatti and Quadrio2016) because the bulk velocity $U_b$ is the same between the actuated and non-actuated cases. Any difference between the outer-scaled actuated and non-actuated profiles reflects the overall response of turbulence to the wall oscillation (1.1).

Figure 5. Profiles of Reynolds stresses and turbulence production for four cases from figure 4 at $Re_\tau = 4000, A^+ = 12, \kappa ^+_x = 0.007$ and $\omega ^+ = -0.05, 0, +0.05, +0.20$. The insets in (a,b) indicate the considered values of $\omega ^+$ and their DR values. Line and symbol colours are consistent with figure 4. In each panel only the resolved portion of the profiles are shown ($y^+ \lesssim 1000, y^* \lesssim 700$). The black lines with symbols correspond to the non-actuated case. (a,c,e) Plot of the actuated profiles (dashed-dotted lines) scaled by the non-actuated $u_{\tau _0}$ (superscripted with $+$); (b,d, f) plot of the actuated profiles (solid lines) scaled by the actuated $u_\tau$ (superscripted with $*$). (a–d) Reynolds stress profiles for (a,b) the streamwise velocity $\langle u'^2 \rangle _{xzt}$ and (c,d) the spanwise velocity $\langle w'^2 \rangle _{xzt}$. (e, f) Premultiplied production of turbulent kinetic energy.

Scaling by the non-actuated $u_{\tau _0}$ (‘$+$’ superscript), as in figure 5(a,c,e), indicates that the wall oscillation attenuates the $\langle u'^2 \rangle ^+_{xzt}$ levels up to the resolved height $y^+_{res} \simeq 1000$. The cases with the highest DR ($\omega ^+ = -0.05, 0$ in figure 5a) show the highest level of attenuation in their $\langle u'^2 \rangle ^+_{xzt}$. Additionally, for these cases, the inner peak of $\langle u'^2 \rangle ^+_{xzt}$ is farther from the wall. Consistently, the viscous sublayer is thickened and the buffer layer is shifted away from the wall (figure 4b). In contrast to the behaviour of $\langle u'^2 \rangle ^+_{xzt}$, the $\langle w'^2 \rangle ^+_{xzt}$ profiles are amplified near the wall. According to Quadrio & Ricco (Reference Quadrio and Ricco2011) and Touber & Leschziner (Reference Touber and Leschziner2012), this amplification is due to the Stokes layer that forms as a result of the spanwise wall motion. The premultiplied turbulent kinetic energy production $y^+ P^+$ (figure 5e) also displays the attenuation of turbulence that accompanies increasing DR. All these trends are similar to previous studies on spanwise wall oscillation at lower Reynolds numbers (Quadrio & Ricco Reference Quadrio and Ricco2011; Touber & Leschziner Reference Touber and Leschziner2012).

Scaling by the actuated $u_\tau$ (‘$*$’ superscript) is equivalent to inner scaling, which highlights the extent up to which the actuated profiles depart from the non-actuated profile. For $\langle u'^2 \rangle ^*_{xzt}$ and $\langle w'^2 \rangle ^*_{xzt}$ (figure 5b,d), the actuated cases agree with the non-actuated case at distances far from the wall, but near the wall the actuated $\langle u'^2 \rangle ^*_{xzt}$ levels are attenuated, while the $\langle w'^2 \rangle ^*_{xzt}$ levels are amplified. For $\omega ^+ = +0.05$ (in region II), the point where the actuated profiles begin to depart from the non-actuated counterpart occurs at $y^* \simeq 100$, considerably farther than for the other cases $(y^* \lesssim 30)$. The same case yields the strongest level of near-wall amplification for $\langle w'^2 \rangle ^*_{xzt}$ (the red profile in figure 5c,d) and the highest level of distortion in mean velocity (red profile in figure 4c,e).

Regardless of the scaling used, as $\langle w'^2 \rangle _{xzt}$ is amplified near the wall, $\langle u'^2 \rangle _{xzt}$ is attenuated, the viscous sublayer is thickened and DR is increased. This trend occurs in regions I ($\omega ^+ = -0.05, 0$) and III ($\omega ^+ = +0.2$). In region II ($\omega ^+ = +0.05$), however, there is an excessive amplification of $\langle w'^2 \rangle _{xzt}$ near the wall, a thinning of the viscous sublayer and a drop in DR.

3.4. Stokes layer: an important source of ISA

As indicated earlier, the near-wall amplification of $\langle w'^2 \rangle _{xzt}$ is related to the growth of the Stokes layer. We now apply triple decomposition to more precisely uncover how the strength of the Stokes layer modifies the near-wall turbulence, which in turn affects the wall drag. We primarily consider $u_\tau$ scaling, as we are interested in the level of departure from the non-actuated behaviour. In Part 2 we mostly use $u_{\tau _0}$ scaling, as we are interested in studying the overall response of turbulence to the wall actuation. Nevertheless, the conclusions from Parts 1 and 2 are valid regardless of the scaling.

Because the flow is subjected to a harmonic forcing (1.1), the instantaneous flow can be triply decomposed similar to Touber & Leschziner (Reference Touber and Leschziner2012), as in

(3.1a)$$\begin{gather} f(x,y,z,t) = \left\langle f \right\rangle_{xzt}(y) + \underbrace{\tilde{f}(x,y,t) + f''(x,y,z,t)}_{f'(x,y,z,t)}, \end{gather}$$

(3.1b)$$\begin{gather}\tilde{f}(x,y,t) = \frac{1}{N}\sum_{n=0}^{N-1}\left\langle f \right\rangle_z (x,y,t+nT_{osc}) - \left\langle f \right\rangle_{xzt}(y), \end{gather}$$

(3.1c)$$\begin{gather}\left\langle f'^2 \right\rangle_{xzt} = \left\langle \tilde{f}^2 \right\rangle_{xt} + \left\langle f''^2 \right\rangle_{xzt}, \end{gather}$$

where $f$ indicates the quantity of interest, i.e. $u,v$ or $w$. In (3.1a), the total fluctuation $f'$ is decomposed into the harmonic contribution $\tilde {f}$ and the stochastic (turbulent) contribution $f''$. The harmonic contribution $\tilde {f}$ is obtained by phase averaging the spanwise averaged field $\langle f \rangle _z$ in time over the number of periods $N$, and then subtracting the mean vertical profile $\langle f \rangle _{xzt}$. Accordingly, the total Reynolds stress $\langle f'^2 \rangle _{xzt}$ is decomposed into its harmonic component $\langle \tilde {f}^2\rangle _{xt}$ associated with the Stokes layer dynamics and its turbulent (stochastic) component $\langle f''^2\rangle _{xzt}$ (3.1c).

In figure 6 we plot these two components for the cases given in figures 4 and 5 ($A^+ = 12, \kappa ^+_x = 0.007$, $Re_\tau =4000$). For reference, figure 6(a,b) shows the considered $U^*$ profiles (as in figure 4b,c). Figure 6(c,d) displays $\langle u''^2\rangle ^*_{xzt}$, the stochastic component of the streamwise Reynolds stress. By comparing figure 5(b) with figure 6(c,d), we see that $\langle u'^2\rangle ^*_{xzt} \simeq \langle u''^2\rangle ^*_{xzt}$, indicating that the harmonic (Stokes layer) component makes a negligible contribution. For the spanwise velocity, however, the harmonic component $\langle \tilde {w}^2\rangle ^*_{xt}$ contributes significantly to the total spanwise Reynolds stress $\langle w'^2\rangle ^*_{xzt}$ close to the wall (see figure 6e, f). At $y^* \sim {O}(1)$, the harmonic component is about three orders of magnitude larger than the turbulent component, while at $y^* \sim {O}(10)$ they have comparable magnitudes. Figure 6(e, f) indicates that the rate of decay in $\langle \tilde {w}^2 \rangle ^*_{xt}$, hence the protrusion of the Stokes layer, strongly depends on $\omega ^+$. Furthermore, the level of distortion in the $U^*$ profiles (figure 6a,b) strongly depends on the rate of decay in $\langle \tilde {w}^2 \rangle ^*_{xt}$. Interestingly, in region II (figure 6f) the decay rate in $\langle \tilde {w}^2 \rangle ^*_{xt}$ is noticeably slower compared with regions I and III, implying the presence of a more protrusive Stokes layer. Accordingly, the $U^*$ profile in region II is distorted to the highest level. The turbulent stress profiles are also shown in figure 8, where they are accompanied by the turbulent kinetic energy profiles $\langle \mathcal {K} \rangle _{xzt}$, which follow the same trends.

Figure 6. Profiles of velocity statistics at $Re_\tau =4000$ for given cases as in figures 4 and 5 ($A^+ = 12, \kappa ^+_x = 0.007$). Line legends are consistent with figures 4 and 5. In each panel only the resolved portion of the profiles are shown corresponding to $y^* \le 1000$. Plots (a,c,e) correspond to $\omega ^+ \le 0$, and plots (b,d, f) correspond to $\omega ^+ > 0$. (a,b) The $U^*$ profiles; the insets indicate the value of $\omega ^+$ and its DR for each profile. In (c–f) the Reynolds stress profiles are presented in terms of the turbulent component (solid lines) and the harmonic component (dashed lines) following ((3.1a), (3.1b)). (c,d) Turbulent component of the streamwise velocity $\langle u''^2 \rangle ^*_{xzt}$. (e, f) Turbulent component $\langle w''^2 \rangle ^*_{xzt}$ and harmonic component $\langle \tilde {w}^2 \rangle ^*_{xt}$ for the spanwise velocity. On each actuated profile, the cross symbol (+) marks the Stokes layer thickness $\delta ^*_S$, and the bullet symbol ($\bullet$) marks the protrusion height $\ell ^*_{0.01}$ due to the Stokes layer.

To quantify the protrusion of the Stokes layer (figure 6e, f), we calculate two length scales from the spanwise Reynolds stress profiles. The first is the laminar Stokes layer thickness $\delta ^*_S$ that is featured in Stokes’ second problem (Batchelor Reference Batchelor2000). Following Quadrio & Ricco (Reference Quadrio and Ricco2011), we define $\delta ^*_S$ as the height $y^*$ where the amplitude of $\tilde {w}$ decays to $A {\rm e}^{-1}$ (i.e. where $\langle \tilde {w}^2 \rangle ^*_{xt} = {\tfrac {1}{2}}{A^*}^2 {\rm e}^{-2}$). In figure 6 we mark $\delta ^*_S$ on each profile with a cross symbol. The second length scale $\ell ^*_{0.01}$ is new, and it is defined as the height where $\langle \tilde {w}^2 \rangle ^*_{xt} = 0.01$. Our choice for the threshold of $\langle \tilde {w}^2 \rangle ^*_{xt} = 0.01$ is based on the observation that $\langle w''^2 \rangle ^*_{xzt} \sim {O}(1)$ in the buffer and log regions (also reported by Lee & Moser Reference Lee and Moser2015; Baidya et al. Reference Baidya, Philip, Hutchins, Monty and Marusic2021). In other words, we define $\ell ^*_{0.01}$ as the height where the Stokes layer stress $\langle \tilde {w}^2 \rangle ^*_{xt}$ drops to about $1\,\%$ of the spanwise turbulent stress $\langle w''^2 \rangle ^*_{xzt}$. In figure 6 we mark $\ell ^*_{0.01}$ on each profile with a bullet symbol.

The key difference between $\delta ^*_S$ and $\ell ^*_{0.01}$ is that we mark $\delta ^*_S$ where the Stokes layer stress $\langle \tilde {w}^2 \rangle ^*_{xt}$ is a small fraction of its maximum value at the wall ${A^*}^2/2$. Thus, we ignore the background turbulence in this definition. However, we mark $\ell ^*_{0.01}$ where the Stokes layer stress $\langle \tilde {w}^2 \rangle ^*_{xt}$ is a small fraction of the turbulent stress $\langle w''^2 \rangle ^*_{xzt}$, hence considering the background turbulence in this definition. In figure 6(c–f), $\ell ^*_{0.01}$ coincides well with the distance where the actuated $\langle u''^2 \rangle ^*_{xzt}$ and $\langle w''^2 \rangle ^*_{xzt}$ profiles depart from the non-actuated counterpart. However, $\delta ^*_S$ underestimates the actual protrusion by the Stokes layer due to its ignorance of the background turbulence. For instance, for the case with $\omega ^+ = 0$ at $y^* = \delta ^*_S$ (black cross symbol in figure 6e), $\langle \tilde {w}^2 \rangle ^*_{xt} \simeq 8 \langle w''^2 \rangle ^*_{xzt}$, i.e. the Stokes layer is $8$ times stronger than the background turbulence. However, at $y^* = \ell ^*_{0.01}$ (black bullet symbol), $\langle \tilde {w}^2 \rangle ^*_{xt} \simeq 0.01 \langle w''^2 \rangle ^*_{xzt}$, i.e. the Stokes layer is $100$ times weaker than the background turbulence. We propose, therefore, that $\ell ^*_{0.01}$ is a more suitable measure for reflecting the entire penetration of the Stokes layer into the turbulent field.

In regions I and III, the level of protrusion by the Stokes layer $\ell ^*_{0.01}$, as well as the departure height in the $\langle u''^2 \rangle ^*_{xzt}$ and $\langle w''^2 \rangle ^*_{xzt}$ profiles, stay below $20 - 30$ viscous units. As a result, the mean velocity profiles in regions I and III (figure 6a,b) yield a well-defined logarithmic shift beyond $y^* \simeq 100$ with viscous sublayer thickening. However, in region II there is a large increase in $\ell ^*_{0.01}$ and the departure in the $\langle u''^2 \rangle ^*_{xzt}$ and $\langle w''^2 \rangle ^*_{xzt}$ profiles also starts at a larger distance from the wall. For example, for $\omega ^+ = +0.05$ in region II (figure 6d, f), $\ell ^*_{0.01} \simeq 80$, which also closely marks the point where the actuated $\langle u''^2 \rangle ^*_{xzt}$ and $\langle w''^2 \rangle ^*_{xzt}$ profiles depart from their non-actuated counterpart. As a result, the mean velocity profile for $\omega ^+ = +0.05$ in region II (figure 6b) is highly distorted up to $y^* \simeq 200\unicode{x2013}300$.

Furthermore, we can draw a connection between the protrusion height $\ell ^*_{0.01}$ and the level of DR. In figure 7(a,b) we overlay the map of $\ell ^*_{0.01}$ onto the maps of DR and $\delta ^*_S$. In region I (left-hand side of the blue dashed-dotted line), $\ell ^*_{0.01} \lesssim 30$ and $\delta ^*_S \lesssim 7$. In this region, an increase in $\ell ^*_{0.01}$ and $\delta ^*_S$ leads to an increase in DR. For upstream travelling waves ($\omega ^+ < 0$), therefore, the growing protrusion of the Stokes layer has a favourable effect on DR. In contrast, in region II (between the blue dashed-dotted line and the black dashed line) DR drops by increasing $\ell ^*_{0.01}$ and $\delta ^*_S$. Another difference between regions I and II is in the relation between $\ell ^*_{0.01}$ and $\delta ^*_S$. In region I, $\ell ^*_{0.01}$ and $\delta ^*_S$ are proportional to each other with $\ell ^*_{0.01}\approx 4\delta ^*_S$. However, in region II this proportional relation is broken and $\ell ^*_{0.01}$ can reach as high as $8\delta ^*_S$. At each $\kappa ^+_x$, the maximum DR (the blue dashed-dotted line) coincides with the optimal range $20 \lesssim \ell ^*_{0.01} \lesssim 30$ $(5 \lesssim \delta ^*_S \lesssim 7)$. In figure 7(c) we plot DR vs $\ell ^*_{0.01}$ for our simulation cases in regions I and II. Also, following Quadrio & Ricco (Reference Quadrio and Ricco2011) (their figure 9), we plot DR vs $\delta ^*_S$ for the same cases (figure 7d). These plots confirm that the maximum DR coincides with $20 \lesssim \ell ^*_{0.01} \lesssim 30$ and $5 \lesssim \delta ^*_S \lesssim 7$ (shaded in grey). Furthermore, for $\ell ^*_{0.01} \lesssim 20$ ($\delta ^*_S \lesssim 5$), $\ell ^*_{0.01} \simeq 4 \delta ^*_S$ and DR increases linearly with $\ell ^*_{0.01}$ and $\delta ^*_S$ (see the fitting dotted lines in figure 7c,d). Following Quadrio & Ricco (Reference Quadrio and Ricco2011), if we extrapolate the linear fits to ${\rm DR} = 0$, we obtain $\ell ^*_{0.01,{min}}\simeq 5$ and $\delta ^*_{S,{min}}\simeq 1$; these values indicate the minimum limits for DR to occur.

Figure 7. (a) Comparison between the map of DR (contour field) and the protrusion height by the Stokes layer $\ell ^*_{0.01}$ (contour lines) for our considered parameter space at $Re_\tau = 4000$. (b) Comparison between the map of Stokes layer thickness $\delta ^*_S$ (contour field) and $\ell ^*_{0.01}$ (contour lines) for the same cases as in (a). The blue dashed-dotted line and the black dashed line are the local maximum and minimum DR (same as in figure 3b). (c,d) Plot of DR vs $\ell ^*_{0.01}$ and DR vs $\delta ^*_S$, respectively, for the same data as in (a,b); $\kappa ^+_x = 0.00238$ ($\blacklozenge$), $0.004$ ($\blacktriangle$), $0.007$ ($\blacksquare$), $0.010$ ($\blacktriangleright$), $0.012$ ($\blacktriangleleft$), $0.017$ ($\blacktriangledown$), $0.021$ ($\bullet$). At each $\kappa ^+_x$, we plot the cases only in regions I and II (see the map in d), with the maximum DR case highlighted with a green outline. The grey regions in (c,d) $(20 \le \ell ^*_{0.01} \le 30, 5 \le \delta ^*_S \le 7)$ shade the range of maximum DR at each $\kappa ^+_x$. The linear dotted lines in (c,d) fit the data for $\ell ^*_{0.01} \lesssim 20$ (c) and $\delta ^*_S \lesssim 5$ (d). The fitting lines also locate the minimum values for $\ell ^*_{0.01,{min}} \simeq 5$ (c) and $\delta ^*_{S,{min}} \simeq 1$ (d) to achieve DR.

Figure 8. Profiles of the turbulence statistics (a–j) and premultiplied spectrograms (k–t) at $Re_\tau = 4000$ for the non-actuated case and the actuated cases with $A^+ = 12, \kappa ^+_x = 0.007$ and different values of $\omega ^+$ (same cases as in figure 6); $\omega ^+ = -0.05$ (a, f,k,p), $\omega ^+ = 0$ (b,g,l,q), $\omega ^+ = +0.05$ (c,h,m,r), $\omega ^+ = +0.10$ (d,i,n,s) and $\omega ^+ = +0.20$ (e,j,o,t). (a–e) Profiles of turbulent stresses for the streamwise and spanwise velocity compoenents $\left \langle u''^2 \right \rangle _{xzt}, \left \langle w''^2 \right \rangle _{xzt}$. ( f–j) Profiles of turbulent kinetic energy $\left \langle \mathcal {K} \right \rangle _{xzt} = ( \left \langle u''^2 \right \rangle _{xzt} + \left \langle v''^2 \right \rangle _{xzt} + \left \langle w''^2 \right \rangle _{xzt} )/2$. Throughout (a–j), black lines with symbols correspond to the non-actuated case; lines with no symbol correspond to the actuated case scaled by the actuated $u_\tau$ (solid line) and the non-actuated $u_{\tau _o}$ (dashed line). Premultiplied spectrograms for the turbulent part of the streamwise velocity $k^*_z \phi ^*_{u'' u''}$ (k–o) and spanwise velocity $k^*_z \phi ^*_{w'' w''}$ (p–t). The filled contours correspond to the actuated cases and the line contours correspond to the non-actuated case. The contour lines for $k^*_z \phi ^*_{u'' u''}$ (k–o) change from $0.6$ to $4.8$ with an increment of $0.6$, and for $k^*_z \phi ^*_{w'' w''}$ (p–t), change from $0.2$ to $1.8$ with an increment of $0.2$. We locate $\ell ^*_{0.01}$ with a bullet point.

The linear relation between ${\rm DR}, \ell ^*_{0.01}$ and $\delta ^*_S$ is limited to region I. In region II when DR drops, this linear relation is broken. Our observations related to DR vs $\delta ^*_S$ in regions I and II are similar to those by Quadrio & Ricco (Reference Quadrio and Ricco2011). These observations are also applicable to DR vs $\ell ^*_{0.01}$. Quadrio & Ricco (Reference Quadrio and Ricco2011) report the optimal $\delta ^*_S \simeq 6.5$ for the maximum DR, the minimum $\delta ^*_{S,{min}} \simeq 1.0$ for DR to occur, the linear relation between DR and $\delta ^*_S$ in region I and the breaking of this linearity in region II. Given the linear relation in region I, Quadrio & Ricco (Reference Quadrio and Ricco2011) conclude that there exists a unique minimum value of $\delta ^*_S$ to achieve a desired DR. We observe this uniqueness in region I for both $\ell ^*_{0.01}$ (figure 7c) and $\delta ^*_S$ (figure 7d). For instance, to achieve ${\rm DR} \gtrsim 20\,\%$, we require $\delta ^*_S \gtrsim 4.0$, equivalent to $\ell ^*_{0.01} \gtrsim 15$. Quadrio & Ricco (Reference Quadrio and Ricco2011) calculated $\delta ^*_S$ from the laminar $\langle \tilde {w}^2 \rangle ^*_{xt}$ profile based on Stokes layer solution. Here, however, we calculate $\delta ^*_S$ from the actual $\langle \tilde {w}^2 \rangle ^*_{xt}$ profile by phase averaging the simulation data. The laminar Stokes layer solution agrees with the simulation up to region I, where there is a linear relation between ${\rm DR}, \delta ^*_S$ and $\ell ^*_{0.01}$ (figure 11a). In region II, where the linear relation is broken, the Stokes layer solution does not agree with the simulation result (figure 11b). We discuss this in § 3.6.

The variation in DR vs $\ell ^*_{0.01}$ is similar to DR vs $\delta ^*_S$ up to region I, in terms of the linear trends, an optimal thickness for the maximum DR and a minimum thickness for the occurrence of DR. However, in region II when the linear trends are broken, we observe noticeable differences between DR vs $\delta ^*_S$ and DR vs $\ell ^*_{0.01}$. In region II there does not appear to be a consistent relation between DR and $\delta ^*_S$ (the red symbols in figure 7d). In other words, we cannot find a threshold for $\delta ^*_S$ beyond which DR drops. For instance, for the case with $\kappa ^+_x = 0.02, \omega ^+ = +0.2$, DR drops to $-10\,\%$ but $\delta ^*_S \simeq 5$ that is within the optimal range $(5 \lesssim \delta ^*_S \lesssim 7)$. In contrast, there is a much stronger connection between DR and $\ell ^*_{0.01}$, even in region II (figure 7c). For all cases, increasing $\ell ^*_{0.01}$ beyond $30$ decreases DR. As a result, the value of $\ell ^*_{0.01}$ can be used to determine whether we are in region I ($\ell ^*_{0.01} \lesssim 30$) or region II ($\ell ^*_{0.01} \gtrsim 30$). In figure 7(c), at each $\kappa ^+_x$ we can draw a connecting line between the discrete values of DR at different $\omega ^+$ to represent a unique function ${\rm DR}(\ell ^*_{0.01})$. However, we cannot do this for $\delta ^*_S$ (figure 7d).

3.5. Interaction between the Stokes layer and the near-wall turbulence

In a turbulent flow with spanwise wall oscillation, Touber & Leschziner (Reference Touber and Leschziner2012) similarly report that an overly protrusive Stokes layer leads to the degradation of DR. They proposed that the attenuation of $\left \langle u''^2 \right \rangle _{xzt}$ and amplification of $\left \langle w''^2 \right \rangle _{xzt}$ are based on the periodic realignment of the near-wall streaks. To examine this proposal further, we consider energy spectrograms and near-wall flow visualisations (figures 8 and 9). We focus on the same cases as in figure 6, where $Re_\tau = 4000$, $A^+ = 12$ and $\kappa ^+_x = 0.007$.

Figure 9. Profiles of the turbulence statistics (a–d) and visualisation of the near-wall turbulence (e–n) at $Re_\tau = 4000$ for the non-actuated case (i,n) and the actuated cases (e–h, j–m) with $A^+ = 12, \kappa ^+_x = 0.007$ and different values of $\omega ^+$ (same cases as in figures 6 and 8); $\omega ^+ = -0.05$ (a,e,j), $\omega ^+ = 0$ (b, f,k), $\omega ^+ = +0.05$ (c,g,l) and $\omega ^+ = +0.10$ (d,h,m). (a–d) Profiles of turbulent stresses for the streamwise and spanwise velocity components $\left \langle u''^2 \right \rangle ^*_{xzt}, \left \langle w''^2 \right \rangle ^*_{xzt}$ (same profiles as in figure 8a–d); lines correspond to the actuated cases and lines with symbols correspond to the non-actuated case. Visualisation of the near-wall turbulence at (e–i) $y^* = 10$ and (j–n) $y^* = 50$ (located with vertical dashed-dotted lines in a–d). In each plot (e–n) the upper field shows the turbulent streamwise velocity $u''^*$ and the lower field shows the turbulent spanwise velocity $w''^*$. We overlay the spanwise and phase-averaged spanwise velocity $\tilde {w}^*$ (as solid curves) onto the $w''^*$ field.

For the cases with $\ell ^*_{0.01} \lesssim 30$, the streamwise premultiplied spectrograms $k^*_z \phi ^*_{u'' u''}$ (figure 8k,l,o) show the attenuation of $u''^2$ below $y^* \simeq \ell ^*_{0.01}$ (i.e. within the Stokes layer). For these cases, increasing $\ell ^*_{0.01}$ (hence strengthening the Stokes layer) attenuates $u''$ over a wider range of wavelength $\lambda ^*_z$ and height $y^*$ (e.g. compare figure 8k with 8l). At the same time, the energetic peak in $k^*_z \phi ^*_{u'' u''}$ is shifted to a higher $y^*$ and a higher $\lambda ^*_z$. This attenuation is apparent in the visualisations of the instantaneous velocity fields of $u''$ at $y^* = 10$ for the cases with $\ell ^*_{0.01} \lesssim 30$ (figure 9e, f). On the $w''$ fields, we overlay the spanwise and phase averaged $\tilde {w}^*$ (solid black curves) as a measure of the Stokes motion. As $\ell ^*_{0.01}$ increases from $20$ at $\omega ^+ = -0.05$ (figure 9e) to $30$ at $\omega ^+ = 0$ (figure 9f), the Stokes motion becomes stronger and the energy level in the $u''$ field is decreased compared with the non-actuated counterpart (figure 9i). At the same time, the spanwise spacing between the high-speed streaks increases by increasing $\ell ^*_{0.01}$. Overall, for ${\ell ^*_{0.01}} \lesssim 30$, the Stokes layer dampens the level of turbulence within $y^* \lesssim {\ell ^*_{0.01}}$, hence acting favourably towards increasing DR.

For the cases with $\ell ^*_{0.01} > 30$, the Stokes layer is excessively strong and protrusive. As a result, the near-wall flow structures meander, following the Stokes motion. This meandering is observed in the $u''$ and $w''$ fields for the cases with $\ell ^*_{0.01} > 30$ at $y^* = 10$ (figure 9g,h). Even at $y^* = 50$, for the same cases (figure 9l,m), we can see the protrusion of the Stokes motion (solid black curves) and evidence of meandering in the $u''$ fields. This meandering is also evident in the spectrograms. For instance, for $\omega ^+ = +0.05$ with $\ell ^*_{0.01} \simeq 90$ (figure 8m,r), the meandering flow structures at $y^* \simeq 10$ (visualised in figure 9g) manifest as an energetic peak in the $k^*_z \phi ^*_{w'' w''}$ spectrogram (figure 8r) at $(\lambda ^*_z, y^*) \simeq (100, 10)$; this peak coincides with the peak in the $k^*_z \phi ^*_{u'' u''}$ spectrogram (figure 8m). Touber & Leschziner (Reference Touber and Leschziner2012) relate the attenuation of $\langle u''^2 \rangle _{xzt}$ and amplification $\langle w''^2 \rangle _{xzt}$ (e.g. figure 9c,d) to this meandering behaviour and argue that the strong Stokes shear strain periodically re-orients the streaks. As a result, energy is transferred from $u''$ to $w''$, and the anisotropy between $u''$ and $w''$ is reduced. Considering the flow visualisation at $y^* = 10$ for $\omega ^+ = +0.05$ with $\ell ^*_S \simeq 90$ (figure 9g), we see a strong resemblance between the $u''$ and $w''$ fields in terms of the energy level and structure, which support the reduction in anisotropy.

A noticeable difference between the cases with $\ell ^*_{0.01} \lesssim 30$ and those with $\ell ^*_{0.01} > 30$ is the wall distance of the maximum turbulence activity given by the location of the energetic peaks in $k^*_z \phi ^*_{u'' u''}$ and $k^*_z \phi ^*_{w'' w''}$. For the cases with $\ell ^*_{0.01} \lesssim 30$ (figure 8k,l,o), the energetic peak in $k^*_z \phi ^*_{u'' u''}$ is lifted away from the wall to a $y^*$ distance that coincides with $\ell ^*_{0.01}$. However, for the cases with $\ell ^*_{0.01} > 30$, the energetic peaks in $k^*_z \phi ^*_{u'' u''}$ (figure 8m,n) and $k^*_z \phi ^*_{w'' w''}$ (figure 8r,s) instead reside near the wall at $y^* \simeq 10$, well below $\ell ^*_{0.01} \simeq 90$. It appears that when $\ell ^*_{0.01} > 30$, a near-wall cycle of streaks with high turbulence activity is generated within the Stokes layer. Contrast this behaviour to the case when $\ell ^*_{0.01} \lesssim 30$ where the turbulence is damped within the Stokes layer and the cycle of turbulence generation is lifted away from the wall.

Overall, through flow visualisations and spectrograms we could explain the physics behind the trends in DR vs $\ell ^*_{0.01}$ (figure 7). When $\ell ^*_{0.01} \lesssim 30$, turbulence is damped within $y^* \lesssim \ell ^*_{0.01}$. The level of damping increases by increasing $\ell ^*_{0.01}$. As a result, DR increases by increasing $\ell ^*_{0.01}$, with the maximum DR attained when $\ell ^*_{0.01} \simeq 30$. However, when $\ell ^*_{0.01} > 30$, the Stokes layer becomes excessively strong. In this situation a near-wall cycle of turbulence is generated at $y^* \simeq 10$ that meanders following the Stokes motion. As a result, DR drops by increasing $\ell ^*_{0.01}$.

3.6. Power performance analysis

While DR is an important performance parameter for many applications, the efficiency of the flow control effort is often even more important. Here we use the concept of net power saving (NPS):

(3.2)\begin{equation} {\rm NPS} = \frac{P^+_0 - \left( P^++ P^+_{in} \right)}{P^+_0} = {\rm DR} - \frac{P^+_{in}}{P^+_0}, \end{equation}

where $P^+ = ( 1 - {\rm DR} ) U^+_b$ is the pumping power required to drive the flow through the actuated channel, $P^+_0$ is the non-actuated analogue of $P^+$ and $P^+_{in}$ is the input power required to oscillate the wall actuation mechanism (1.1) while neglecting any mechanical losses. A positive NPS indicates that the total power cost of the actuated case is less than the total cost of its non-actuated counterpart. We are also interested in assessing the accuracy of generalized Stokes layer (GSL) theory (Quadrio & Ricco Reference Quadrio and Ricco2011) for estimating $P^+_{in}$. In Part 2 (Chandran et al. Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023) we use this theory to estimate NPS for our experimental data. Here, in Part I, our actuation frequencies fall into the ISA regime. In Part 2 the data fall into both the ISA and OSA regimes.

The input power is given as follows: as first proposed by Baron & Quadrio (Reference Baron and Quadrio1995) for an oscillating plane, and then used by Quadrio & Ricco (Reference Quadrio and Ricco2011), Gatti & Quadrio (Reference Gatti and Quadrio2013) and Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) for a travelling wave,

(3.3)\begin{equation} P^+_{in} = \frac{1}{T^+_{avg} L^+_x L^+_z}\int_{t^+}^{t^+{+} T^+_{avg}} \int_0^{L^+_x} \int_0^{L^+_z} w^+_s \left( \left.\frac{\partial w^+}{\partial y^+}\right|_{y^+{=} 0} \right) \,{{\rm d}\kern0.7pt x}^+ \,{\rm d}z^+ \,{\rm d}t^+, \end{equation}

where all the quantities are normalised by $\nu$ and the non-actuated $u_{\tau _o}$ (hence, superscripted with a cross symbol). In (3.3), $T_{avg}$ is the averaging time, $w_s$ is the instantaneous wall velocity (1.1) and $\partial w^+/\partial y^+ |_{y^+=0}$ is the instantaneous wall-normal gradient of the spanwise velocity at the wall.

In figure 10(a) we present the map of $P^+_{in}/P^+_0$ as computed over our parameter space of $(\omega ^+, \kappa ^+_x)$ at $Re_\tau = 4000$. The map is much more symmetric about $\omega ^+ = 0$ compared with DR (figure 3). We also see that substantially more power is required at higher actuation frequencies. For example, $P^+_{in}/P^+_0\,\%$ can reach up to $100\,\%$ when $\omega ^+ \simeq \pm 0.2$. In region II, between the local maximum and the local minimum DR (between the blue dashed-dotted line and the black dashed line), $P^+_{in}/P^+_0$ decreases to about 30 %–35 %.

Figure 10. (a) Map of $P^+_{in}$ at $Re_\tau = 4000$. The filled contour and line contours show the same quantity.(b) Filled contour is the difference in calculation of $P^+_{in}$ at $Re_\tau = 4000$ between LES and its theoretical estimation from the GSL theory (Quadrio & Ricco Reference Quadrio and Ricco2011); line contours give the Stokes layer protrusion height $\ell ^*_{0.01}$ (same as in figure 7a).

We can use (3.3) only if we have an estimate for $\partial w^+/\partial y^+ |_{y^+=0}$. In most experimental studies, including Part 2 of the present study, this quantity is unavailable and some estimate needs to be made instead. In Part 2 we use GSL theory, which gives the instantaneous spanwise velocity for a laminar flow with wall actuation. That is,

(3.4)$$\begin{align} w^+(x^+,y^+,t^+) &= A^+ \mathcal{R}\left\{ C \exp({{\rm i} (\kappa^+_x x^+{-} \omega^+ t^+)})\vphantom{\frac{1}{2}}\right.\nonumber\\ &\left.\times \mbox{Ai}\left[ {\rm e}^{{\rm \pi} {\rm i}/6}\left( \kappa^+_x [1-{\rm DR} ] \right)^{1/3} \left( y^+{-} \frac{\omega^+}{\kappa^+_x [1-{\rm DR}]} - \frac{{\rm i} \kappa^+_x}{1-{\rm DR}} \right) \right] \right\} \end{align}$$

where $C = \{ \mbox {Ai} [ {\rm i} {\rm e}^{{\rm i} {\rm \pi}/3} ( \kappa ^+_x [ 1-{\rm DR} ] )^{1/3} (\omega ^+/\kappa ^+_x + {\rm i} \kappa ^+_x)/[1-{\rm DR} ] ] \}^{-1}$, Ai is the Airy function of the first kind and $\mathcal {R}\{ \cdots \}$ is the real part of the argument. To use (3.4) for a turbulent flow, one needs to assume that (1) the Stokes layer preserves its laminar structure near the wall, i.e. $\tilde {w}$ is the same in the laminar and turbulent flow; and (2) the turbulent spanwise velocity is negligible near the wall, i.e. $w'' \simeq 0$.

We now compare the results for $P^+_{in}$ using GSL to the results obtained using LES, so as to verify the validity of using GSL estimates in experiments. Figure 10(b) shows the difference between the pumping power obtained from the LES at $Re_\tau = 4000$ ($P^+_{in}/P^+_0\,\%_{{LES}, Re_\tau = 4000}$) and that estimated using GSL theory at the same $Re_\tau$ ($P^+_{in}/P^+_0\,\%_{{GSL}, Re_\tau = 4000}$). Overall, the differences are small, especially in the range $\omega ^+ < 0$ where they differ less than $3\,\%$. Only in region II with $\omega ^+ > 0$ (between the blue dashed-dotted line and the black dashed line) do the differences approach $10\,\%$, especially along the minimum DR line (black dashed line). The overlay of the Stokes layer protrusion height $\ell ^*_{0.01}$ (contour lines) indicates that the region where the power differences are significant coincides with the region where $\ell ^*_{0.01}$ is large. In other words, the error in using GSL theory (laminar Stokes layer assumption) is largest when the Stokes layer is most protrusive.

This behaviour is further substantiated by figure 11, which compares the phase-averaged (harmonic) Reynolds stress profiles $\langle \tilde {w}^2 \rangle ^*_{xt}$ between LES (solid lines) and its laminar solution (3.4) from GSL theory (dashed lines with symbols). We see that in region I with $\omega ^+ \le 0$ (figure 11a), the agreement is reasonably good; we obtain better agreement with $\omega ^+ = -0.05$ ($\ell ^*_{0.01} \simeq 20$) than with $\omega ^+ = 0$ ($\ell ^*_{0.01} \simeq 30$). However, in region II (figure 11b), when $\omega ^+ = +0.05$ ($\ell ^*_{0.01} \simeq 90$) and $\omega ^+ = +0.10$ ($\ell ^*_{0.01} \simeq 80$), we observe significant departures between LES and the GSL theory. For instance, for $\omega ^+ = +0.05$ at $y^* \simeq 20$, $\langle \tilde {w}^2 \rangle ^*_{xz}$ from LES is $2.2$ but from GSL is $0.2$. This is a significant difference considering that the background turbulent stress $\langle w''^2 \rangle ^*_{xzt} \sim {O}(1)$. In region III with $\omega ^+ = +0.20$ where $\ell ^*_{0.01} \simeq 17$, we see a return of the good agreement between LES and GSL theory.

Figure 11. Comparison of the phase-averaged (harmonic) Reynolds stress profiles $\langle \tilde {w}^2 \rangle ^*_{xt}$ between LES (solid lines) and the laminar solution from the GSL theory (dashed lines with symbols) ($A^+ = 12, \ \kappa ^+_x = 0.007$ at $Re_\tau = 4000$). Results are shown for (a) $\omega ^+ = -0.05, 0$ and (b) $\omega ^+ = +0.05, +0.10, +0.20$. The insets plot DR for the selected cases.

Our observations regarding the differences between $P_{in}$ from the simulation and that from the GSL theory (figure 10) are similar to those reported by Quadrio & Ricco (Reference Quadrio and Ricco2011) (their figure 7); they report close agreement between the GSL theory and the turbulence simulation in the drag-decreasing range, but report noticeable differences in the drag-increasing range. They explain this behaviour through the time scale $\mathcal {T}^+ \equiv 2{\rm \pi} /(\omega ^+ - \kappa ^+_x \mathcal {U}^+_w)$, which represents the period of oscillation as observed by the near-wall eddies with the convective speed $\mathcal {U}^+_w \simeq 10$. As discussed in § 3.1, in the drag-increasing range $\omega ^+/\kappa ^+_x \rightarrow \mathcal {U}^+_w$ leading to $\mathcal {T}^+ \rightarrow \infty$. In other words, the spanwise oscillation becomes too slow that close to the wall, the $u$- and $w$-momentum equations are coupled together. However, GSL theory assumes that these equations are decoupled. Here, we add a new explanation based on the protrusion of the Stokes layer. As discussed in § 3.5 (figure 9), in the drag-increasing range the Stokes layer is too protrusive and a near-wall cycle of turbulence is embedded within the Stokes layer. As a result, near the wall, all the terms of the momentum equation (2.1b) are active. However, the GSL theory neglects the advection (nonlinear) terms from the $w$-momentum equation. The departure of the $\langle \tilde {w}^2 \rangle ^*_{xt}$ profiles from the GSL solution (figure 11b) supports the activation of these terms.

We can now explore the NPS. Figure 12(a) demonstrates that for our considered parameter space, NPS is mostly negative. The highest (best) NPS is $0.5\,\%$ at $(\omega ^+, \kappa ^+_x) \simeq (0.05, 0.012)$. In figure 12(b) we plot the map of the difference between NPS from LES at $Re_\tau = 4000$ and its counterpart at $Re_\tau = 951$. If this difference is positive, NPS increases with $Re_\tau$. Over a large portion of our parameter space, the difference is negative, i.e. NPS becomes more negative with $Re_\tau$. However, for a small portion of region I with $\omega ^+ <0$ and $\kappa ^+_x \lesssim 0.0025$, the difference is positive. One experimental case reported by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) falls into this region, with $\omega ^+ = -0.044$ and $\kappa ^+_x = 0.0014$ (see their figure 3a,b). The NPS of this case was negative, but it increased with $Re_\tau$ in accordance with our analysis.

Figure 12. (a) The NPS for LES at $Re_\tau = 4000$. (b) Difference in NPS between LES at $Re_\tau = 4000$ and LES at $Re_\tau = 951$. In plots (a,b) the filled contour and line contours show the same quantity. All the quantities with a ‘+’ superscript are scaled by $\nu$ and the non-actuated $u_{\tau _o}$. The blue dashed-dotted line and black dashed line locate the local maximum and local minimum DR, respectively.

Quadrio et al. (Reference Quadrio, Ricco and Viotti2009), similar to figure 12(a), generate a map of NPS for their travelling wave study at $Re_\tau = 200$ (their figure 5). They report ${\rm NPS}>0$ within the range $0 \lesssim \omega ^+ \lesssim +0.05, 0.002 \lesssim \kappa ^+_x \lesssim 0.025$. This range coincides with the range where ${\rm DR} \gtrsim 40\,\%$. Gatti & Quadrio (Reference Gatti and Quadrio2013) generate a similar map at $Re_\tau = 1000$ (their figure 9). They also observe ${\rm NPS}>0$ within the same range of $(\omega ^+,\kappa ^+_x)$. However, the level of ${\rm NPS} > 0$ is lower at $Re_\tau = 1000$ compared with $Re_\tau = 200$. Considering (3.2), we speculate that the decrease in ${\rm NPS}>0$ from $Re_\tau = 200$ to $1000$ is due to the decrease in DR. We observe a similar trend in figure 12(b). Within the range of $(\omega ^+,\kappa ^+_x)$ where DR is maximum (blue dashed-dotted line), NPS decreases by increasing $Re_\tau$ from $1000$ to $4000$.

Overall, for our considered parameter space, NPS is negative and predominantly decreases with Reynolds number. As discussed in § 1, our parameter space falls into the ISA regime. In Part 2 we conduct experiments with some actuation parameters in the OSA regime, which yield positive values for the NPS that actually increase with Reynolds number.