3.1 Case SP and case ST

First, we determine whether performance for suboptimal control can be maintained, or improved, via the use of a spatial wavenumber filter. In other words, we evaluate performance with control applied only over the spectral region where resolvent analysis predicts favourable effects. To generate resolvent-based predictions, we first need to identify appropriate values for the blowing-and-suction parameter  $\unicode[STIX]{x1D6FC}$ .

Nakashima et al. (Reference Nakashima, Fukagata and Luhar2017) estimated the value of $\unicode[STIX]{x1D6FC}$ using two different approaches for case SP, although both methods assumed the NW cycle to be the dominant flow feature. First, based on known r.m.s. values of the fluctuating spanwise wall shear stress and the magnitude of blowing and suction obtained in previous DNS, this parameter was estimated from (2.4) in inner normalized form as

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{+}\approx \frac{K}{k_{z}}\frac{\unicode[STIX]{x1D719}_{rms}^{+}}{(\unicode[STIX]{x2202}w_{rms}/\unicode[STIX]{x2202}y)_{w}^{+}}\approx \frac{\unicode[STIX]{x1D719}_{rms}^{+}}{(\unicode[STIX]{x2202}w_{rms}/\unicode[STIX]{x2202}y)_{w}^{+}}.\end{eqnarray}$$

The expression above assumes that $K=\sqrt{k_{x}^{2}+k_{z}^{2}}\approx k_{z}$ , which is reasonable since the NW cycle comprises structures that are approximately one order of magnitude longer in the streamwise direction compared to the spanwise direction. Second, Nakashima et al. (Reference Nakashima, Fukagata and Luhar2017) estimated $\unicode[STIX]{x1D6FC}^{+}$ by requiring that $\unicode[STIX]{x1D719}_{\boldsymbol{k}}$ be approximately equal to $v_{\boldsymbol{k}}(y^{+}\approx 10)$ for resolvent modes resembling the NW cycle. More specifically, they considered the wavenumber–frequency combination corresponding to $(\unicode[STIX]{x1D706}_{x}^{+},\unicode[STIX]{x1D706}_{z}^{+},c^{+})\approx (10^{3},10^{2},10)$ as being representative of the NW cycle since the energetic contribution associated with these structures is known to be concentrated around $(\unicode[STIX]{x1D706}_{x}^{+},\unicode[STIX]{x1D706}_{z}^{+})\approx (10^{3},10^{2})$ in spectral space and near the wall-normal location $y^{+}\approx 15$ . This assumption is also consistent with prior studies (e.g. McKeon & Sharma Reference McKeon and Sharma2010; McKeon, Jacobi & Sharma Reference McKeon, Jacobi and Sharma2013; Sharma & McKeon Reference Sharma and McKeon2013; Luhar et al. Reference Luhar, Sharma and Mckeon2015). Both methods led to $\unicode[STIX]{x1D6FC}^{+}\approx 0.5$ for case SP. Since $K/k_{x}\sim O(10)$ for the NW cycle, an estimate similar to that shown in (3.1) obtained using (2.6) suggests that $\unicode[STIX]{x1D6FC}^{+}$ is one order of magnitude larger for suboptimal control based on the streamwise wall shear stress, with $\unicode[STIX]{x1D6FC}^{+}\approx 10$ for case ST.

Figure 2 shows resolvent-based predictions for case SP and case ST with $\unicode[STIX]{x1D6FC}^{+}=0.5$ and $\unicode[STIX]{x1D6FC}^{+}=10$ , respectively. Both panels show the normalized change in squared singular values relative to the uncontrolled case. We consider this measure, which represents the predicted change in energy amplification, to be a surrogate for drag reduction. For each wavenumber combination $(k_{x},k_{z})$ , the ratio of squared singular values is integrated over the range of wave speeds $0\leqslant c\leqslant 1$ , where $c$ is normalized by the centreline velocity (Nakashima et al. Reference Nakashima, Fukagata and Luhar2017). In other words, the predicted ‘drag reduction’ rate across wavenumber space is defined as

(3.2) $$\begin{eqnarray}R_{\unicode[STIX]{x1D70E}}(k_{x},k_{z})=1-\frac{\displaystyle \int _{0}^{1}\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}^{2}(k_{x},k_{z},c)\,\text{d}c}{\displaystyle \int _{0}^{1}\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}^{2}(k_{x},k_{z},c)\,\text{d}c}.\end{eqnarray}$$

The cross markers in figure 2 represent length scales that characterize the NW cycle, $\unicode[STIX]{x1D706}_{x}^{+}\approx 10^{3}$ and $\unicode[STIX]{x1D706}_{z}^{+}\approx 10^{2}$ . Note that the maximum wavelengths shown in this figure correspond to the DNS domain size, $\unicode[STIX]{x1D706}_{x}^{+}\leqslant 4\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}^{+}$ and $\unicode[STIX]{x1D706}_{z}^{+}\leqslant 2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}^{+}$ .

As shown in figure 2(a), resolvent analysis predicts mode suppression (i.e.  $1-(\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0})^{2}>0$ ) for case SP over a wide region of spectral space. Singular values are nearly unchanged in regions where drag reduction is not expected. Based on this prediction, we hypothesize that a spatial wavenumber filter is unlikely to yield a substantial difference in performance. DNS for case SP with control over all streamwise wavelengths yielded approximately 15 % drag reduction in the present study, which is a little smaller than the value reported by Lee et al. (Reference Lee, Kim and Choi1998). This is probably due to the slight increase in Reynolds number. According to Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002), suboptimal control becomes less effective in reducing the skin friction when the friction Reynolds number increases. For instance, drag reduction decreases from 20 % at $Re_{\unicode[STIX]{x1D70F}}=110$ to 15 % at $Re_{\unicode[STIX]{x1D70F}}=150$ . Since Lee et al. (Reference Lee, Kim and Choi1998) obtained ${>}20\,\%$ drag reduction at $Re_{\unicode[STIX]{x1D70F}}=110$ , the slightly lower value obtained here at $Re_{\unicode[STIX]{x1D70F}}=180$ is consistent with prior results.

Figure 2.

Resolvent-based predictions for singular-value suppression at $Re_{\unicode[STIX]{x1D70F}}=180$ for (a) case SP ( $\unicode[STIX]{x1D6FC}^{+}=0.5$ ) and (b) case ST ( $\unicode[STIX]{x1D6FC}^{+}=10$ ).

Figure 2(b) shows resolvent-based predictions for case ST. Unlike case SP, for this case, the effects of control vary significantly across spectral space. For modes with streamwise length scales larger than $\unicode[STIX]{x1D706}_{x}^{+}\approx 200$ , the singular-value ratio is suppressed by 10–20 %. For shorter modes with streamwise length scales smaller than $\unicode[STIX]{x1D706}_{x}^{+}\approx 200$ , the singular-value ratio increases substantially.

In the original simulations, Lee et al. (Reference Lee, Kim and Choi1998) reported that case ST leads to an increase in drag. The model predictions shown in figure 2(b) suggest that drag reduction may be attained if a spatial wavenumber is employed such that control is implemented only over the long-wavelength region. Based on these predictions, we pursued three separate DNS cases, with: (i) control applied only for structures with long streamwise wavelengths, $\unicode[STIX]{x1D706}_{x}^{+}\geqslant 140$ , for which resolvent analysis predicts mode suppression, i.e. the region enclosed by the blue box in figure 2(b); (ii) control applied only for short streamwise wavelengths, $\unicode[STIX]{x1D706}_{x}^{+}<140$ , for which resolvent analysis predicts further amplification, i.e. the region enclosed by the red box in figure 2(b); and (iii) with control applied over all scales, which corresponds to the baseline case ST studied previously. Hereafter, these three cases are termed ST-CL, ST-CS and ST-CA, respectively. Note that in the present study, the term ‘large scales’ is used to denote the NW cycle, not the much larger structures that emerge at higher Reynolds numbers and are multiple outer units in length (e.g. the so-called large-scale and very-large-scale motions (see Smits, McKeon & Marusic Reference Smits, McKeon and Marusic2011)).

Table 2.

Drag reduction rates $R_{D}$ (%) obtained in DNS.

Interestingly, none of these three cases showed a substantial change in drag. As summarized in table 2, DNS for case ST-CL only shows a small amount of drag reduction ( $R_{D}=1.2\,\%$ ) while case ST-CS leads to no perceptible change in drag ( $R_{D}=0.06\,\%$ ). To evaluate why these DNS results contradict resolvent-based predictions, we revisit the blowing-and-suction parameter, $\unicode[STIX]{x1D6FC}^{+}$ , imposed in our DNS. As noted earlier, the value $\unicode[STIX]{x1D6FC}^{+}=10$ was estimated from previous DNS results based on the assumption that the NW cycle is the dominant flow feature close to the wall. Specifically, equation (2.6) was rearranged to yield

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{+}\approx \frac{K}{k_{x}}\frac{\unicode[STIX]{x1D719}_{rms}^{+}}{(\unicode[STIX]{x2202}u_{rms}/\unicode[STIX]{x2202}y)_{w}^{+}}\approx 10\frac{\unicode[STIX]{x1D719}_{rms}^{+}}{(\unicode[STIX]{x2202}u_{rms}/\unicode[STIX]{x2202}y)_{w}^{+}},\end{eqnarray}$$

under the assumption that $K/k_{x}\approx k_{z}/k_{x}=10$ . For case ST-CA, the magnitudes of streamwise wall shear stress and control input were $(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y)_{w}^{+}\approx 0.4$ and $\unicode[STIX]{x1D719}_{rms}^{+}=0.25$ , respectively, in the DNS conducted here. Thus, if the NW cycle is the dominant contributor in the expression above and $K/k_{x}\approx 10$ , the a priori estimate $\unicode[STIX]{x1D6FC}^{+}\sim O(10)$ is reasonable. Indeed, Nakashima et al. (Reference Nakashima, Fukagata and Luhar2017) showed similar arguments to be valid for suboptimal control based on the spanwise wall shear stress (case SP).

To estimate the true value of $\unicode[STIX]{x1D6FC}^{+}$ imposed in the present DNS for case ST-CA, based on the condition that $\unicode[STIX]{x1D719}_{rms}=v_{rms,0}(y^{+}\approx 10)$ , we follow the following procedure. First, we compute the ‘unscaled’ value for the blowing-and-suction magnitude in Fourier space, as

(3.4) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D719}}_{\unicode[STIX]{x1D6FC}}=-\frac{\text{i}k_{x}}{K}\left(\widehat{\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}}\right)_{w}.\end{eqnarray}$$

Next, this unscaled value for the actuation is inverse-Fourier-transformed and its r.m.s. value, $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC},rms}$ , is computed in physical space. Finally, the effective value for $\unicode[STIX]{x1D6FC}$ is estimated from the ratio

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}\approx \frac{v_{rms,0}(y^{+}\approx 10)}{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC},rms}}.\end{eqnarray}$$

Estimates based on this procedure suggest a range, $\unicode[STIX]{x1D6FC}^{+}\in [0.2,1.0]$ , that is much smaller than the assumed value, $\unicode[STIX]{x1D6FC}^{+}=10$ . This indicates that the assumption $K/k_{x}\approx 10$ in (3.3) is inaccurate, and that the NW cycle is not the sole dominant flow feature when it comes to estimating the fluctuating streamwise wall shear stress in turbulent channel flow.

Figure 3.

Resolvent-based predictions for singular-value suppression at $Re_{\unicode[STIX]{x1D70F}}=180$ for case ST with $\unicode[STIX]{x1D6FC}^{+}=0.5$ .

For completeness, figure 3 shows resolvent-based predictions for mode amplification across spectral space under case ST with the modified blowing-and-suction parameter, $\unicode[STIX]{x1D6FC}^{+}=0.5$ . Compared to the predictions shown in figure 2(b) for $\unicode[STIX]{x1D6FC}^{+}=10$ , the predicted changes in mode amplification are much smaller. For instance, the suppression predicted for modes with length scales corresponding to the NW cycle is $R_{\unicode[STIX]{x1D70E}}\approx 0.03$ for $\unicode[STIX]{x1D6FC}^{+}=0.5$ , which is a factor of five lower than the suppression predicted for $\unicode[STIX]{x1D6FC}^{+}=10$ . This suggests that case ST-CL does not lead to substantial drag reduction in DNS due to the relatively weak blowing-and-suction magnitude. Similarly, the lack of substantial mode amplification for $\unicode[STIX]{x1D706}_{x}^{+}<140$ at $\unicode[STIX]{x1D6FC}^{+}=0.5$ also explains why case ST-CS does not lead to a significant increase in drag.

The above discussion suggests that increasing the actuation amplitude for case ST-CL could result in drag reduction, albeit with increased actuation cost. However, additional DNS for case ST-CL with $\unicode[STIX]{x1D719}_{rms}^{+}=0.5$ or $1$ (two or four times the original actuation strength) did not yield substantial increases in drag reduction. Nevertheless, even with this increased actuation, values for the blowing-and-suction parameter were still well below the $\unicode[STIX]{x1D6FC}^{+}\approx 10$ estimate used to generate the predictions shown in figure 2(b).

In sum, suboptimal control for case ST does not lead to observable changes in the present DNS due to the relatively low value of $\unicode[STIX]{x1D6FC}$ ; resolvent analysis only predicts marginal changes in mode amplification for $\unicode[STIX]{x1D6FC}^{+}=0.5$ . In other words, the failure of case ST at reducing drag in the present DNS is caused not just by the detrimental effect on small scales (as hypothesized by Nakashima et al. (Reference Nakashima, Fukagata and Luhar2017)), but also due to the very low actuation amplitude. Note that we also estimated $\unicode[STIX]{x1D6FC}$ for case SP using the procedure described above and again obtained $\unicode[STIX]{x1D6FC}^{+}\sim O(0.1)$ . Thus, it appears that $\unicode[STIX]{x1D6FC}^{+}\sim O(0.1)$ is a robust estimate for suboptimal control in which $\unicode[STIX]{x1D719}_{rms}=v_{rms,0}(y^{+}\approx 10)$ . In the following section, we propose new control laws based on the insights gained here, assuming $\unicode[STIX]{x1D6FC}^{+}=0.5$ for the purposes of resolvent analysis.

3.2 Proposal of new control laws

The DNS results and resolvent-based predictions presented in the previous section suggest the following key insights for suboptimal control based on the streamwise wall shear stress. First, resolvent analysis predicts that control performance is mixed in spectral space. Structures with large streamwise length scales (low $k_{x}$ ) are suppressed by control while shorter structures may be further amplified. Second, with the scaling factor $\unicode[STIX]{x1D6FC}(k_{x}/K)$ in (2.6) and the requirement that $\unicode[STIX]{x1D719}_{rms}=v_{rms,0}(y^{+}\approx 10)$ , the magnitude of blowing and suction is too weak for control to have a substantial effect on the energetic NW cycle.

Based on these observations, we propose new feedback control laws. Specifically, we retain the reliance on streamwise wall shear stress measurements (this is desirable from a practical viewpoint), but change the scaling factor in (2.6) from $\unicode[STIX]{x1D6FC}(k_{x}/K)$ to $\unicode[STIX]{x1D6FC}(k_{z}/K)$ . This effectively increases blowing and suction for longer structures with low streamwise wavenumber such as the NW cycle, and decreases the actuation for shorter structures that are predicted to be further amplified under control per figure 2(b). Actually, replacing $k_{x}$ with $k_{z}$ in the case ST control law leads to a 50–100 % suppression in the actuation amplitude for the spectral region with $\unicode[STIX]{x1D706}_{x}^{+}<140$ and a roughly 100 % increase in actuation amplitude for the larger scales. Two candidate control laws are

(3.6) $$\begin{eqnarray}\displaystyle \text{(case STz)}:\qquad \hat{\unicode[STIX]{x1D719}}=-\unicode[STIX]{x1D6FC}\frac{\text{i}k_{z}}{K}\left(\widehat{\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}}\right)_{w} & & \displaystyle\end{eqnarray}$$

and

(3.7) $$\begin{eqnarray}\displaystyle \text{(case STrz)}:\qquad \hat{\unicode[STIX]{x1D719}}=+\unicode[STIX]{x1D6FC}\frac{\text{i}k_{z}}{K}\left(\widehat{\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}}\right)_{w}. & & \displaystyle\end{eqnarray}$$

Since multiplication by $k_{z}$ in Fourier space denotes a differentiation in the spanwise direction, these new control laws may also be interpreted as modified versions of the discrete formulation identified by Morimoto et al. (Reference Morimoto, Iwamoto, Suzuki and Kasagi2002) using a genetic algorithm,

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D6FC}\left[\left.\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}\right|_{w}(x^{+},z^{+}+20)-\left.\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}\right|_{w}(x^{+},z^{+}-20)\right].\end{eqnarray}$$

Note that, unlike cases SP and ST, the newly proposed laws introduce a spanwise asymmetry depending on the sign of $k_{z}$ . Specifically, the actuation changes for modes with $k_{z}<0$ relative to that for modes with $k_{z}>0$ . Further, case STz has the same control effect on modes with $k_{z}>0$ as case STrz does on modes with $k_{z}<0$ , and vice versa. These spanwise asymmetries in the new control laws are confirmed by the resolvent-based predictions shown in figure 4. Figures 4(a) and 4(b) show predicted singular-value ratios for modes with $k_{z}>0$ for case STz and case STrz, respectively. It has also been confirmed that, for modes with $k_{z}<0$ , the predictions are reversed, i.e. figure 4(a) corresponds to case STrz, while figure 4(b) corresponds to case STz. Of course, if contributions from both oblique modes, $k_{z}>0$ and $k_{z}<0$ , are combined, resolvent analysis predicts identical control effects for cases STz and STrz. Predicted mode suppression for this scenario is shown in figure 4(c).

Figure 4.

Resolvent-based predictions for singular-value suppression at $\unicode[STIX]{x1D6FC}^{+}=0.5$ for: (a) case STz ( $k_{z}>0$ ); (b) case STrz ( $k_{z}>0$ ); and (c) case STz and case STrz when both $k_{z}>0$ and $k_{z}<0$ are considered.

Importantly, figure 4(c) shows that both case STz and case STrz overcome the drawbacks associated with case ST. Namely, singular values are suppressed substantially in the spectral region corresponding to the NW cycle at $\unicode[STIX]{x1D6FC}^{+}=0.5$ , with $R_{\unicode[STIX]{x1D70E}}\approx 0.1$ . Further, in contrast to the predictions shown in figure 2(b), there are very limited detrimental effects on modes with short streamwise length scales.

3.3 Evaluation of new control laws

To test these predictions, we conducted DNS with the new control laws with, and without, spatial filtering. The observed performance for both cases STz and STrz was consistent with predictions made using resolvent analysis. Approximately 10 % drag reduction was obtained without the spatial filtering (CA) and for the case in which control was limited to large streamwise wavelengths (CL). Almost no effect was observed when control was limited to short streamwise wavelengths (CS). Moreover, there were no statistically significant differences between case STz and case STrz. Table 2 provides a summary of the total drag reduction observed in DNS for all the control laws (ST, STz, STrz) and spatial filters (CA, CL, CS) considered here.

To further evaluate the efficacy of the new control laws, the large- and small-scale contributions to total skin friction were quantified using the Fukagata–Iwamoto–Kasagi (FIK) identity (Fukagata et al. Reference Fukagata, Iwamoto and Kasagi2002):

(3.9) $$\begin{eqnarray}\displaystyle C_{f}=\frac{12}{Re_{b}}+12\int _{0}^{1}2(1-y)(-\overline{u^{\prime }v^{\prime }})\,\text{d}y. & & \displaystyle\end{eqnarray}$$

Here, all variables are made dimensionless using the channel half-width, $\unicode[STIX]{x1D6FF}$ , and twice the bulk mean velocity, $2U_{b}$ . The second term in (3.9), which represents the turbulent contribution to the skin friction coefficient, was decomposed into two components in a manner similar to Fukagata, Kobayashi & Kasagi (Reference Fukagata, Kobayashi and Kasagi2010): the large-scale contribution from structures with $\unicode[STIX]{x1D706}_{x}^{+}\geqslant 140$ , and the small-scale contribution from structures with $\unicode[STIX]{x1D706}_{x}^{+}<140$ . This decomposition shows that, in case STz-CL, the large-scale contribution to the total skin friction decreases from 71 % for the uncontrolled case to 61 % (figure 5), which constitutes the majority of the 11 % total drag reduction obtained under cases STz-CA and STz-CL (table 2). Together, these results confirm that the suppression of large-scale structures with $\unicode[STIX]{x1D706}_{x}^{+}\geqslant 140$ is central to the drag reduction obtained under cases STz and STrz, which is consistent with resolvent-based predictions.

Figure 5.

Contribution to friction drag for the uncontrolled case, case ST-CL and case STz-CL.

Figure 6.

Results obtained in DNS. (a) The mean velocity profile. Profiles for the (b) r.m.s. fluctuating streamwise velocity $u_{rms}^{+}$ , (c) wall-normal velocity $v_{rms}^{+}$ , and (d) spanwise velocity $w_{rms}^{+}$ . Black line, uncontrolled case; blue line, case STz-CL; magenta circles, case STz-CA.

Figure 6(a–d) shows the profiles of mean streamwise velocity and r.m.s. fluctuating velocities obtained in DNS. For the uncontrolled case, all the profiles (black lines) are consistent with known features of wall-bounded turbulent flows. The mean velocity profile shows a linear increase in the viscous sublayer and the presence of a logarithmic region further from the wall. The r.m.s. velocity profiles show good agreement with previous spectral DNS results (Moser, Kim & Mansour Reference Moser, Kim and Mansour1999). For the controlled cases, it is clear that cases STz-CA (magenta circles) and STz-CL (blue lines) show nearly identical profiles for these mean turbulence statistics. This essentially means that the new control laws produce the same effect with and without the low-pass spatial filter.

The viscous normalization employed in figure 6 corresponds to the friction velocity for the uncontrolled case. With this in mind, the slight decrease in near-wall $U^{+}$ observed for the controlled cases in figure 6(a) is consistent with a reduction in skin friction, i.e. the decrease in near-wall $U$ represents a decrease in the wall shear stress $\unicode[STIX]{x1D70F}_{w}=\unicode[STIX]{x1D707}\,\text{d}U/\text{d}y$ . In addition, as shown in figure 6(b–d), the r.m.s. values of all velocity components decrease across the channel, which is consistent with a decrease in turbulence intensity. For instance, the peak value of $u_{rms}$ decreases by approximately 10 % for both case STz-CA and case STz-CL. These observations are consistent with previously reported DNS results for effective controls (e.g. Deng et al. Reference Deng, Huang and Xu2016).

Figure 7.

Distribution of the instantaneous control input, $\unicode[STIX]{x1D719}^{+}$ , for case STz-CA at $t^{+}\approx 900$ after control begins.

Figure 7 shows the instantaneous spatial distribution of blowing and suction, $\unicode[STIX]{x1D719}^{+}$ , for case STz-CA at $t^{+}\approx 900$ after control commences. The spatial distribution of $\unicode[STIX]{x1D719}^{+}$ shows a distinct domain-scale periodic pattern (i.e. with $\unicode[STIX]{x1D706}_{x}=L_{x}/2$ and $\unicode[STIX]{x1D706}_{z}=L_{z}$ ) with alternating regions of blowing ( $\unicode[STIX]{x1D719}^{+}>0$ ) and suction ( $\unicode[STIX]{x1D719}^{+}<0$ ) that are tilted in the streamwise–spanwise plane. Similar patterned control inputs have also been reported in other feedback control schemes (Koumoutsakos Reference Koumoutsakos1999; Hasegawa & Kasagi Reference Hasegawa and Kasagi2011), though these previous simulations did not yield oblique structures.

Additional tests indicate that the oblique structures observed here are most likely a numerical artifact arising from the limited size of the computational domain. Specifically, when the computational domain is reduced by a factor of two, the size of the oblique structures is also halved. Moreover, these structures do not appear to play an important role in terms of drag reduction. Statistically, the control effect does not change if the amplitude coefficient for the wavenumber combination corresponding to $(\unicode[STIX]{x1D706}_{x},\unicode[STIX]{x1D706}_{z})=(L_{x}/2,L_{z})$ is artificially set to zero. In any case, the phase speed associated with these patterned inputs is too high ( $c^{+}\simeq 800$ ) to allow for dynamic interaction with the turbulent structures arising in the flow. Although the emergence of these oblique periodic patterns needs to be investigated further, the tests described above indicate that the structures seen in figure 7 do not contribute substantially to drag reduction. An alternative drag reduction mechanism for cases STz and STrz is proposed in § 4.

Figure 8.

Instantaneous distribution of the streamwise velocity fluctuations, ${u^{\prime }}^{+}$ , in the $x$ – $z$ plane at $t^{+}\approx 900$ after control commences: (a,c,e) uncontrolled case; (b,d,f) case STz-CA; (a,b) $y^{+}\approx 3$ ; (c,d) $y^{+}\approx 5$ ; (e,f) $y^{+}\approx 15$ .

Figure 8 shows that the present control input suppresses the formation of streamwise velocity streaks in the near-wall region. As is typical for the NW cycle, the uncontrolled case shows the presence of elongated high- and low-speed streaks aligned in the streamwise direction (figure 8 a,c,e). For case STz, figure 8(b,d) shows that these streaks are damped substantially at $y^{+}\approx 3$ and $y^{+}\approx 5$ . A qualitative comparison between the uncontrolled and controlled cases in figure 8 also suggests that, in the presence of control, the near-wall streaks show an oblique bias. Specifically, there appear to be more streaks inclined in the positive $z$ -direction (from bottom-left to top-right in these panels) compared to the negative $z$ -direction.

To test for this non-equivalence between modes with $k_{z}>0$ and $k_{z}<0$ under case STz, we computed two-dimensional (2D) Fourier transforms for the snapshots shown in figure 8(b,d,f). Fourier transforms for these snapshots confirm that, for $k_{x}>0$ , streamwise velocity fluctuations with $k_{z}<0$ are more energetic than fluctuations with $k_{z}>0$ , i.e. streaks inclined in the positive $z$ -direction are suppressed less under control (data not shown here). Note that this observation is consistent with the asymmetric control effect predicted by resolvent analysis for NW-type modes (see figure 4 a,b). Singular values associated with the NW modes are reduced from $\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}(k_{z}>0)=\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}(k_{z}<0)=3.6$ for the uncontrolled case to $\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}(k_{z}>0)=2.8$ and $\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}(k_{z}<0)=3.2$ for case STz. In other words, control leads to greater suppression for modes with $k_{z}>0$ . The physical mechanism creating this asymmetry in control efficacy for modes with $k_{z}<0$ and $k_{z}>0$ is discussed in § 4.2.

Figure 9.

Premultiplied 2D co-spectra at $y^{+}=5$ : (a–c)  $E_{uu}$ ; (d–f)  $E_{vv}$ ; and (g–i)  $E_{ww}$ . Cases: (a,d,g) uncontrolled case; (b,e,h) case ST-CA; and (c,f,i) case STz-CA.

Figure 9 shows the 2D premultiplied co-spectra for the streamwise ( $E_{uu}$ ), wall-normal ( $E_{vv}$ ) and spanwise velocity ( $E_{ww}$ ) at $y^{+}=5$ obtained from DNS for the uncontrolled case, case ST-CA and case STz-CA. Each co-spectrum is normalized by the maximum value for the uncontrolled case. Figure 9(a,d,g) confirm that the results obtained here for the uncontrolled case are consistent with the spectra observed in previous DNS. The streamwise spectrum $E_{uu}$ shows a clear peak at the wavelength combination corresponding to the NW cycle. Spectra for the wall-normal and spanwise velocity show peaks at slightly lower streamwise wavelengths.

For case STz-CA, spectra for all velocity components show substantial damping, though this is especially apparent for the streamwise component (see figure 9(c,f,i)). The reduction in $E_{uu}$ confirms the suppression of near-wall streaks that was evident in figure 8. Similarly, reduction in $E_{vv}$ and $E_{ww}$ indicates that the quasi-streamwise vortices associated with the NW cycle are also substantially damped. Together, a suppression of spectra for all three velocity components implies a reduction in turbulent kinetic energy, which is directly related to drag reduction. In contrast to the spectra obtained for case STz-CA, spectra for case ST-CA show very little change relative to the uncontrolled case. This is consistent with the marginal reduction in drag obtained for case ST-CA (table 2). Note that the observations for case ST-CA and case STz-CA described above also hold for the case in which control is applied only for long wavelengths (i.e. for cases ST-CL and STz-CL).

To examine the change in $E_{uu}$ ,  $E_{vv}$ and $E_{ww}$ more quantitatively, we integrate these spectra over all wall-normal locations as well as specified regions of spectral space, and compute integrated reduction rates relative to the controlled flow. As an example, the reduction rate for the streamwise component is defined as

(3.10) $$\begin{eqnarray}\displaystyle R_{uu}=1-\frac{\displaystyle \iiint E_{uu,c}(k_{x},y,k_{z})\,\text{d}k_{x}\,\text{d}y\,\text{d}k_{z}}{\displaystyle \iiint E_{uu,0}(k_{x},y,k_{z})\,\text{d}k_{x}\,\text{d}y\,\text{d}k_{z}}. & & \displaystyle\end{eqnarray}$$

Table 3 shows the integrated reduction rates obtained under cases ST-CA and STz-CA for all three components ( $R_{uu},R_{vv},R_{ww}$ ). Note that three different sets of values are shown, corresponding to integrals over (i) all streamwise wavelengths, (ii) only large streamwise wavelengths, $\unicode[STIX]{x1D706}_{x}^{+}\geqslant 140$ , and (iii) only small streamwise wavelengths, $\unicode[STIX]{x1D706}_{x}^{+}<140$ .

For case ST-CA, the reduction rates obtained are small ( ${\lesssim}2\,\%$ ) for all components of velocity and over all length scales. This is consistent with the marginal decrease in average skin friction observed in DNS. Further, the reduction rates are lowest for structures with small streamwise wavelengths, for which resolvent analysis predicts a small increase in mode amplification (figure 3). For case STz-CA, the integrated reduction rates over the long-wavelength region for all components of velocity are approximately 15 %, which is broadly consistent with the resolvent-based predictions for mode amplification presented in figure 4(c). Interestingly, integrated reduction rates over the short-wavelength region ( $\unicode[STIX]{x1D706}_{x}^{+}<140$ ) for all components of velocity are higher, approximately 20 %. Resolvent analysis for case STz does not predict substantial mode suppression over this region, indicating that the observed suppression of these smaller structures is due to nonlinear energy transfer. Specifically, these results suggest that weakening of larger-scale turbulent structures inhibits the cascade of turbulent kinetic energy down to small scales. In the resolvent formalism, these nonlinear effects would result in a reduction of the forcing term for smaller structures. As a result, the forcing normalized analysis of mode suppression considered here is unable to predict a suppression of small-scale turbulence.

Table 3.

Integrated reduction rates for all velocity components obtained in DNS for cases ST-CA and STz-CA ( $\%$ ).