3.1 Flow energy

A total of 48 optimization intervals, corresponding to a time duration of $48T=2880$ , are considered. This is approximately equal to three complete periods of the low-frequency, penetrating inlet mode A (table 1). This time span is long enough for the controlled flow to reach the exit of the computational domain.

The convergence of the cost function ${\mathcal{J}}$ (defined in (2.3)) within each optimization interval is shown in figure 5. Recall that ${\mathcal{J}}$ is defined as the sum of the integral of flow energy over time $T$ plus the cost of actuation. Due to the high computational cost required, the maximum number of iterations in each interval was set to four. As figure 5 clearly demonstrates, there is a large drop of ${\mathcal{J}}$ in the first optimization interval $T$ because the actuation starts from an uncontrolled transitional flow state. After the second interval, ${\mathcal{J}}$ exhibits a repeatable oscillatory pattern, with frequency equal to that of the low-frequency inlet mode A (indeed three cycles can be detected). The shape of the oscillatory pattern is related to the transition activity taking place inside the control region and will be explored in more detail below. Two examples of convergence of the cost function can be seen in the insets of figure 5; both indicate smooth convergence. The reduction of ${\mathcal{J}}$ between the third and fourth iteration is small, so adding further iterations would increase computational cost, without commensurate reduction of the cost function. The low-frequency penetrating mode also modulates the flow, again making more iterations within a single $T$ unnecessary.

Figure 5. Cost function ${\mathcal{J}}$ versus the total number of iterations $n$ . The vertical dashed lines delineate the optimization intervals. The two insets show the convergence of the cost function in the 20th and 32nd interval.

The time evolution of the flow energy $E$ for both uncontrolled and controlled flow is shown in figure 6. Note that in the horizontal axis, time is expressed in terms of the optimization length $T$ (with $T=60$ ). For the uncontrolled flow, three complete periods of the low-frequency mode ( $0{-}16T$ , $16T{-}32T$ and $32T{-}48T$ ) can be detected from the shape of $E$ ; these are delineated by two vertical dash-dot lines. It is clear that the breakdown process that determines $E$ is modulated by the low-frequency penetrating mode A. The first few intervals (the very first interval is exactly the same as in Xiao & Papadakis (Reference Xiao and Papadakis2017)) can be regarded as the initial transient period until the control effect is established after $16T$ . In the current work, we mainly focus on the control effect and the associated control mechanism in time domain $16T{-}48T$ , during which the controlled flow is very different compared to that during the first few intervals.

Figure 6. Evolution of the energy of the flow in the defined sub-region. Solid line, uncontrolled flow; dashed line, controlled flow.

In figure 6, the controlled flow energy is reduced significantly within the first few optimization intervals, and then stays in the range 3000–4500. This is approximately $43\,\%$ of the uncontrolled flow energy. In the first low-frequency period ( $0{-}16T$ ), there is a small phase difference between controlled and uncontrolled flow. This is because when the controller is activated at $t=0$ , the uncontrolled flow is in a state of increasing $E$ . From $t=16T$ onwards, the two energies evolve in phase. Similar flow processes take place in the controlled flow, but with reduced flow energy.

The evolution of $E$ indicates that two local maxima appear within each period. The evolution of the uncontrolled and controlled flow is examined by visualizing the instantaneous $u$ in figure 7 at four successive time instants. The total duration corresponds to one quarter of the period of the low-frequency mode. Figure 7 demonstrates in detail the evolution of one patch of streaks as it convects into the control region. In the uncontrolled flow at $t=16T$ , the head of one patch of high-speed streaks is seen to enter the control slot, while the downstream side is occupied by turbulent flow. Secondary instabilities distort the streaks entering the slot. The front part convects more rapidly than the rear and soon overtakes the downstream turbulent region. The ragged edge of the turbulent region is maintained by turbulent spots that continuously overtake the main turbulent zone. Without this sustenance the turbulent flow would convect out of the numerical domain (Jacobs & Durbin Reference Jacobs and Durbin2001). In figure 6 it can be seen that $E$ is at a local minimum at $t=16T$ . This is due to the region of inactivity between the entering high-speed streaks and the turbulent spot. At $t=19T$ , the actuation region is almost full of turbulent flow, and the flow energy is approaching a local peak value, as expected.

In the controlled flow, at all time instants, the high-speed streaks are still visible and distorted, indicating that the penetrating mode is present, but they have been clearly quenched by the control action, while the turbulent zone has also been significantly affected. Between them, there is an extended region with very low velocities and small spanwise variation in $u$ . In the downstream end, there are some small localized patches of large $u$ but they appear random. This figure indicates that the controller has been very effective in suppressing the energy of the flow. It is very interesting to notice that the control effect has propagated downstream of the control slot. This aspect of control action will be further investigated in § 3.4.

Figure 7. Uncontrolled (a,c,e,g) and controlled (b,d,f,h) instantaneous streamwise velocity in the $x$ – $z$ plane at $y=2$ at four successive time instants: (a,b) $16T$ ; (c,d) $17T$ ; (e,f) $18T$ ; (g,h) $19T$ .

To further understand the control effect, we define $E_{z}(x,t)$ as the energy integrated in the spanwise and wall-normal directions:

(3.1) $$\begin{eqnarray}\displaystyle E_{z}(x,t)=\int _{0}^{y_{lim}}\int _{0}^{L_{z}}[(\boldsymbol{u}(t)-\boldsymbol{U}_{B})\boldsymbol{\cdot }\unicode[STIX]{x1D6FA}(\boldsymbol{x})\boldsymbol{\cdot }(\boldsymbol{u}(t)-\boldsymbol{U}_{B})]\,\text{d}y\,\text{d}z. & & \displaystyle\end{eqnarray}$$

The time-averaged value $\langle E_{z}\rangle _{t}$ is plotted in figure 8 against $x$ . From the definition of $E_{z}(x,t)$ , the area under the curve is equal to the total energy of the flow above the slot. In the uncontrolled case, the energy increases almost linearly as a result of the transition process taking place over the control slot, as described above. In the controlled flow, $\langle E_{z}\rangle _{t}$ is already reduced at $x=1050$ (starting position of the control slot) compared to the uncontrolled flow. This indicates that, in a time-average sense, the actuation effect is already felt upstream of the slot. The controlled $\langle E_{z}\rangle _{t}$ then decreases and reaches a minimum value just before $x=1100$ . After this location, it increases linearly, but with a much smaller rate compared to the uncontrolled flow. This behaviour is in agreement with the $u$ velocity contours of figure 7.

Figure 8. Streamwise variation of $\langle E_{z}\rangle _{t}$ . Solid line, uncontrolled flow; dashed line, controlled flow.

3.2 Optimal control velocity

The spatially (streamwise and spanwise) averaged optimal blowing and suction velocity $\langle v_{w}\rangle _{x,z}$ is shown in figure 9 as a function of time. Only the first 30 optimization intervals are shown; the rest have very similar behaviour. The actuation velocity $\langle v_{w}\rangle _{x,z}$ attains its maximum value at the beginning of each interval and gradually reduces towards the end, which is in agreement with the actuation from control in a single optimization horizon (Xiao & Papadakis Reference Xiao and Papadakis2017). After a short transient period at the beginning of actuation, the value of $\langle v_{w}\rangle _{x,z}$ within each interval stabilizes to an average of $0.685\,\%$ of $U_{\infty }$ . It was demonstrated in figure 6 that the flow energy is reduced dramatically in the first few intervals and then fluctuates around a lower value. The control velocity has a similar behaviour. This suggests that a larger actuation velocity is required to bring down the uncontrolled flow at the beginning of actuation, and subsequently a smaller $v_{w}$ is enough to maintain the flow energy at that lower level.

Figure 9. Spatially averaged actuation velocity against time. Vertical dashed lines demarcate optimization intervals.

In figure 10 the variation of the time- and spanwise-averaged actuation velocity $\langle v_{w}\rangle _{z,t}$ with streamwise distance inside the control slot is plotted. Time averaging here is performed over one period of the low-frequency inlet mode A (i.e. over $16T$ ), starting from four different time instants (mentioned in the figure caption). All four curves collapse reasonably well. This indicates that the spanwise-averaged actuation velocity is repeatable in a time scale equal to the period of the low-frequency inlet mode. We notice in figure 6 that the flow energy is periodically modulated by the slow mode, so it is not surprising that $\langle v_{w}\rangle _{z,t}$ also demonstrates periodic behaviour. Note also that when no constraint on mass is imposed, the control velocity results in a net positive mass flow rate.

It is interesting to examine the streamwise variation of $\langle v_{w}\rangle _{z,t}$ . When the optimization was performed on a single interval (Xiao & Papadakis Reference Xiao and Papadakis2017), $\langle v_{w}\rangle _{z}$ increased monotonically in the streamwise direction and a physical explanation was provided based on the flow pattern and the definition of the objective function. However, when the actuation velocity is averaged over much longer time, there is a local peak at around $1070$ , a local minimum at 1130 and $\langle v_{w}\rangle _{z,t}$ then increases linearly with $x$ .

Figure 10. Time- and spanwise-averaged actuation velocity. The four distributions shown are averaged over one period of low-frequency inlet mode (i.e. $16T$ ) starting from different time instants: solid line, $6T$ ; dashed line, $18T$ ; dash-dot line, $22T$ ; dotted line, $26T$ .

This behaviour is consistent with the instantaneous flow patterns of the controlled flow (right-hand column of figure 7). The peak at $1070$ corresponds to the control action to quench the incoming distorted high-speed streaks. The minimum value corresponds to the low-velocity region (black area), and the linear growth to the suppression of the turbulent patches with large $u$ that appear in the right half of the slot, but they originate upstream (refer to figure 3).

It is interesting to notice that the profiles of both the controlled energy $\langle E_{z}\rangle _{t}$ (figure 8) as well as the actuation $\langle v_{w}\rangle _{z,t}$ (figure 10) reduce due to the control action in the streamwise direction, but then rise again. This behaviour is the result of the competition between the control action and the transition process. If transition to turbulence had not taken place above the control slot, the flow energy as well as $\langle v_{w}\rangle _{z,t}$ would continue to reduce, in a manner similar to that found by Monokrousos et al. (Reference Monokrousos, Brandt, Schlatter and Henningson2008). In their work, the control slot is placed in the upstream laminar flow region and both the flow and control velocity reduce monotonically in the streamwise direction within the slot. This is the expected behaviour if there is no transition.

In the present case, however, the interaction between the elevated negative streaks and the external high-frequency mode that leads to secondary instability starts inside the boundary layer and upstream of the control slot as seen in figure 2. Nolan & Zaki (Reference Nolan and Zaki2013) used laminar–turbulent discrimination techniques and managed to detect the inception and growth of spots. They showed that the highest turbulent spot count appears inside the boundary layer, in the region $y/\unicode[STIX]{x1D6FF}=0.4$ – $0.8$ , where $\unicode[STIX]{x1D6FF}$ is the local boundary layer thickness. The current objective function minimizes the flow energy closer to the wall, below $y/\unicode[STIX]{x1D6FF}_{c}=0.25$ , where $\unicode[STIX]{x1D6FF}_{c}$ is the average boundary layer thickness over the control slot (see also figure 13 a). The spots are then brought towards the wall, and the control velocity in response to this increases along $x$ as seen in figure 10. This top-to-bottom process in relation to the control slot is sketched in figure 3.

In the following two sections, the effect of control action on the time-average flow above and downstream of the slot until the end of the domain is examined. For a more accurate and meaningful comparison, all the mean flow properties for both controlled and uncontrolled flow are obtained by averaging over two periods of low-frequency mode A, starting from $t=16T$ until $t=48T$ . The reason is that at $t=16T$ the effect of actuation has reached the end of the domain (as will be shown later), so the same time window can be used for the flow both above and downstream of the slot.

3.3 Effect of actuation on mean and turbulent quantities above the control slot

3.3.1 Mean velocity profiles

The mean (time- and spanwise-averaged) velocity profiles $\overline{u}$ at three streamwise locations, $x=1075$ , $1130$ and $1250$ , are shown in figure 11. These locations are selected based on the previous analysis of the streamwise variation of the optimal control velocity. They correspond to the peak of actuation close to the inlet, to the trough in the quiescent region between steaks and turbulent flow and to a location inside the fully developed region, respectively.

It can be seen clearly that the control has imparted significant changes in the mean profiles in the three locations. At $x=1075$ , the controlled velocity profile is between the uncontrolled and Blasius profile, while at $x=1130$ , the controlled flow matches well with the Blasius solution. Further downstream at $x=1250$ , the uncontrolled profile is steeper as a result of the enhanced mixing arising from the transition process. The controlled flow velocity has milder slope, but still larger than the Blasius one, indicating that the controlled flow is still disorganized, but with reduced mixing activity. This is in agreement with the findings of the previous section. As already mentioned, the upper boundary of the box in which the objective function is defined is at $y=5$ ; this position is indicated by a horizontal dashed line in figure 11. The optimal controller performs its ‘duty’ as expected; it brings the velocity profile close to the Blasius velocity only below $y=5$ .

Figure 11. Time- and spanwise-averaged velocity profiles at three locations: (a) $x=1075$ ; (b) $x=1130$ ; (c) $x=1250$ . Solid line, uncontrolled flow; dashed line, controlled flow; dotted line, Blasius profile. The horizontal dashed line marks the upper boundary in the wall-normal direction in which the cost function is defined.

The optimal actuation velocity depends on the choice of the objective function. If the location of the upper boundary were set closer to the wall, one would expect that the controlled velocity profile would approach even better the Blasius profile in the near-wall region, leading to lower drag (closer to laminar drag). It is therefore clear that minimizing energy is not expected to lead to drag minimization if the upper boundary is set far from the wall.

The spatial correlation between $v_{w}$ and $u$ , $R_{v_{w},u}^{s}$ , evaluated over the whole control slot is now examined. $R_{v_{w},u}^{s}$ is defined as

(3.2) $$\begin{eqnarray}\displaystyle R_{v_{w},u}^{s}=\frac{\langle (v_{w}-\langle v_{w}\rangle _{x,z})(u-\langle u\rangle _{x,z})\rangle _{x,z}}{[\langle (v_{w}-\langle v_{w}\rangle _{x,z})^{2}\rangle _{x,z}\langle (u-\langle u\rangle _{x,z})^{2}\rangle _{x,z}]^{1/2}}, & & \displaystyle\end{eqnarray}$$

where $\langle \rangle _{x,z}$ denotes average in streamwise and spanwise directions. Figure 12 shows $R_{v_{w},u}^{s}$ averaged over one period of time with $u$ extracted from four wall-normal locations. It can be seen that $R_{v_{w},u}^{s}$ peaks at $y=2.5$ , which means $v_{w}$ is most responsive to the flow in this region. The explanation for this is that the difference between the uncontrolled $\overline{u}$ and Blasius velocity is larger between $y=2$ and $3$ as seen in figure 11. Note also that $v_{w}$ is very weakly correlated with $u$ at $y=6$ . This is not surprising since this location is outside the region where the objective function is defined and the controller does not respond to the flow in this region; therefore $R_{v_{w},u}^{s}$ is very small. Above $y=5$ , the controlled flow profile deviates strongly from the Blasius profile compared to the uncontrolled flow. This again ties with the low correlation between $v_{w}$ and $u$ at $y=6$ .

Figure 12. Time-averaged (over one period) correlation coefficient between control velocity over the whole control region and instantaneous streamwise velocity at various wall-normal locations over the whole control region.

The effect of the control action on the spatial development of the boundary layer is further investigated by examining the boundary layer thickness $\unicode[STIX]{x1D6FF}$ (defined as the location of $0.99U_{\infty }$ ), the Reynolds number based on momentum thickness $Re_{\unicode[STIX]{x1D703}}$ and the shape factor $H$ , which are plotted in figure 13. In the control region, both $\unicode[STIX]{x1D6FF}$ and $Re_{\unicode[STIX]{x1D703}}$ increase, i.e. the boundary layer is thickened by the control action. This is expected since the current controller imparts a positive net mass flow rate and it is known that UB increases the boundary layer thickness (Kametani & Fukagata Reference Kametani and Fukagata2011; Kametani et al. Reference Kametani, Fukagata, Örlü and Schlatter2015; Stroh et al. Reference Stroh, Hasegawa, Schlatter and Frohnapfel2016). Note that $\unicode[STIX]{x1D6FF}$ starts to increase before the control slot is reached, and this is due to the upstream effect of the pressure gradient induced by the actuation, as will be discussed below. The shape factor $H$ is also increased above the control slot, in agreement with the findings from UB (Kametani & Fukagata Reference Kametani and Fukagata2011; Kametani et al. Reference Kametani, Fukagata, Örlü and Schlatter2015).

The effect of control action extends downstream of the control slot. The shape factor $H$ gradually recovers and approaches the value of the uncontrolled flow. At the exit of the domain, $H\approx 1.5$ for both flows. At the moderate values of $Re_{\unicode[STIX]{x1D703}}$ shown in figure 13(b) (less than 1000), $H$ depends on the history of transition (Schlatter & Orlu Reference Schlatter and Orlu2012). Although transition in the present work is due to interacting modes in the free stream (and not due to tripping at the wall), the value at the exit is within the range of values reported in Schlatter & Orlu (Reference Schlatter and Orlu2012). Note also the constant negative shift of the $Re_{\unicode[STIX]{x1D703}}$ plot for the controlled flow downstream of the slot; this is very similar to the shift reported in Stroh et al. (Reference Stroh, Hasegawa, Schlatter and Frohnapfel2016) for UB.

Figure 13. Effect of optimal control on spatial development of the boundary layer: (a) boundary layer thickness $\unicode[STIX]{x1D6FF}$ ; (b) Reynolds number based on momentum thickness $Re_{\unicode[STIX]{x1D703}}$ ; (c) shape factor $H$ . Solid line, uncontrolled flow; dashed line, controlled flow. In (a) the shaded area denotes the region where the cost function is defined.

3.3.2 Pressure distribution

Figure 14 shows the variation of the mean wall pressure. The mean pressure changes significantly near the entrance and the exit of the control slot. The time-average $x$ -momentum equation at the wall can be written as

(3.3) $$\begin{eqnarray}\displaystyle \left.\frac{\unicode[STIX]{x2202}\overline{p}}{\unicode[STIX]{x2202}x}\right|_{w}=\left.\frac{1}{Re}\frac{\unicode[STIX]{x2202}^{2}\overline{u}}{\unicode[STIX]{x2202}y^{2}}\right|_{w}-\left.\overline{v}_{w}\frac{\unicode[STIX]{x2202}\overline{u}}{\unicode[STIX]{x2202}y}\right|_{w}-\left.\overline{v_{w}^{\prime }\frac{\unicode[STIX]{x2202}u^{\prime }}{\unicode[STIX]{x2202}y}}\right|_{w}, & & \displaystyle\end{eqnarray}$$

where the overbar denotes the average in time and the spanwise direction ( $z$ ) and the prime ( $^{\prime }$ ) denotes the fluctuation. Upstream and downstream of the control slot the pressure increases in the streamwise direction, i.e. there is an adverse pressure gradient. This is because at these places $v_{w}=0$ and $\unicode[STIX]{x2202}^{2}\overline{u}/\unicode[STIX]{x2202}y^{2}|_{w}>0$ . The same is reported by Park & Choi (Reference Park and Choi1999), who applied UB to a turbulent boundary layer. The presence of the adverse pressure gradient upstream of the slot explains the upstream effect of the control action mentioned earlier. Above the slot, $\overline{v}_{w}(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y)$ is always positive because $\overline{v}_{w}>0$ as shown in figure 10. Our computations reveal that the third term on the right-hand side of equation (3.3) is much smaller than the second term, and therefore can be neglected. The term $\unicode[STIX]{x2202}^{2}\overline{u}/\unicode[STIX]{x2202}y^{2}|_{w}$ is positive above the control slot and its magnitude is comparable to that of $-\overline{v}_{w}(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y)$ . Therefore the sign of the pressure gradient depends on the relative size of these two terms. In the work of Park & Choi (Reference Park and Choi1999), the viscous term was much smaller as they used relatively large UB velocity; consequently they had a favourable pressure gradient over the entire control slot. In the present case, as seen in figure 14, the pressure gradient is initially favourable, and is followed by a weak adverse pressure gradient, starting from about $x=1100$ , which is the location where $\overline{v}_{w}$ is decreasing to the local minimum (figure 10). After $x=1200$ , the pressure starts to decrease again as $\overline{v}_{w}$ keeps increasing. Near the exit of the domain, there is a small deviation of pressure from $0$ , probably due to the effect of the convective outlet boundary condition.

Figure 14. Variation of mean wall pressure.

3.3.3 Turbulent profiles

In this section, the control effect on second-order turbulent statistics is examined. Figure 15 shows the root mean square (r.m.s.) profiles of the three velocity fluctuations at the same streamwise locations. The variables take physical values (figure 15 a–c), or are expressed in wall units based on local uncontrolled (figure 15 d–f) or case-specific, local friction velocity (figure 15 g–i).

In the uncontrolled flow, the wall-normal location for maximum $u_{rms}$ gradually reduces from $y=1.7$ ( $0.135\unicode[STIX]{x1D6FF}$ ) at $x=1075$ to $y=1.54$ ( $0.07\unicode[STIX]{x1D6FF}$ ). This is due to the fact that the transition process is initiated at the edge of the boundary layer and progressively penetrates inside the boundary layer, as already mentioned. In the controlled case, the values of $u_{rms}$ are reduced close to the wall (within the region where the cost function is defined) at all three locations; the reduction is largest at $x=1075$ . Further away from the wall, however, the values are increased, and more so in the two downstream locations. When normalized by the case-specific (i.e. actual) local friction velocity, $u_{rms}^{+case}$ is increased all along the wall-normal direction, except near the wall at $x=1075$ . The enhanced $u_{rms}^{+case}$ is due to the fact that the friction velocity is decreased, as will be demonstrated later.

The wall-normal turbulence intensity is positive at the wall due to the unsteady actuation velocity. At the first location $x=1075$ , $w_{rms}$ is slightly decreased close to the wall while away from the wall slightly increased, while $v_{rms}$ is only increased at peak. At two downstream locations, $x=1130$ and $x=1250$ , the behaviour of $v_{rms}$ and $w_{rms}$ is different compared to the upstream point. All values increase in the wall-normal direction, which is also found in UB (Sumitani & Kasagi Reference Sumitani and Kasagi1995). The overall effect is that in the right half of the slot turbulence intensity increases, and more so as the downstream end of the slot is approached.

Figure 15. Root mean square of the velocity fluctuations profile at three locations: (a,d,g) $x=1075$ ; (b,e,h) $x=1130$ ; (c,f,i) $x=1250$ . (a–c) Physical variables; (d–f) variables in wall units based on local uncontrolled friction velocity; (g–i) variables in wall units based on case-specific, local friction velocity. Symbols: $\circ$ , $u_{rms}$ ; $\diamond$ , $v_{rms}$ ; $\times$ , $w_{rms}$ . Solid line, uncontrolled flow; dashed line, controlled flow. The dotted line in (a–c) represents the extent of the region in which the cost function is defined.

Differences are also found in the profiles of the Reynolds and viscous shear stresses, shown in figure 16. To facilitate comparison, the variables are expressed in wall units based on the local friction velocity of the uncontrolled flow. At all three locations, the viscous part $\text{d}\overline{u}/\text{d}y^{+nc}$ is decreased close to the wall, which is a direct result of the reduction of the friction velocity. The behaviour of Reynolds stress is different at the first location compared to the other two locations. At $x=1075$ , $-\overline{u^{\prime }v^{\prime }}^{+nc}$ is reduced close to the wall. At $x=1130$ , the controlled $-\overline{u^{\prime }v^{\prime }}^{+nc}$ is very close to that of the uncontrolled flow in the near-wall region, while further downstream at $x=1250$ , it is increased in the controlled case. Away from the wall, $-\overline{u^{\prime }v^{\prime }}^{+nc}$ is increased in all three locations. The behaviour of $-\overline{u^{\prime }v^{\prime }}^{+nc}$ at $x=1075$ , i.e. decreased close to the wall and increased away from the wall, is also observed when using a blowing-only opposition control by Pamiès et al. (Reference Pamiès, Garnier, Merlen and Sagaut2007). On the other hand $-\overline{u^{\prime }v^{\prime }}^{+nc}$ at $x=1130$ and $x=1250$ is similar to those obtained from UB (Pamiès et al. Reference Pamiès, Garnier, Merlen and Sagaut2007; Kametani & Fukagata Reference Kametani and Fukagata2011).

Figure 16. Reynolds stress $-\overline{u^{\prime }v^{\prime }}^{+nc}$ ( $\circ$ ) and viscous shear stress $\text{d}\overline{u}/\text{d}y^{+nc}$ (♢) profiles at three locations: (a) $x=1075$ ; (b) $x=1130$ ; (c) $x=1250$ . Solid line, uncontrolled flow; dashed line, controlled flow. All quantities are non-dimensionalized by the local uncontrolled friction velocity.

3.4 Control effect downstream of the slot

To investigate how the control effect propagates, we divide the distance between the downstream end of the control slot until the end of the computational domain in small segments of length $\unicode[STIX]{x0394}x=50$ . The variation of $E$ with time is computed in each segment (in the wall-normal and spanwise directions the segment dimensions are $0<y<5$ and $0<z<90$ , respectively). The results are displayed in figure 17. Here $E$ is reduced immediately at the first segment $x=1300{-}1350$ and as the control effect propagates, reduction also appears downstream at later time instants. This shows that the control effect does affect the flow downstream of the actuation slot. In fact, the flow energy is reduced right until the end of the domain.

Figure 17. Evolution of energy downstream of the control slot. Each subplot corresponds to a segment of length $\unicode[STIX]{x0394}x=50$ ; the start and end positions are indicated at the top. Solid line, uncontrolled case; dashed line, controlled case.

The propagation of the control effect can be seen in figure 18(a), where the evolution of the energy reduction rate for each segment, defined as $r(t)=(E_{nc}-E_{c})/E_{nc}$ , is plotted. The reduction rate varies from an average of $60\,\%$ in the first segment $x=1300{-}1350$ (closest to the exit of the control slot) to an average of about $30\,\%$ near the end of the domain (furthest from the control slot). In figure 18(b) a space–time diagram of the point at which $|r|\geqslant 3\,\%$ is shown. The linear trend suggests that the actuation effect propagates at an average speed of 0.74 $U_{\infty }$ , which is very close to the convection speed of the streaks upstream as mentioned before.

Figure 18. (a) Energy reduction rate $r=(E_{nc}-E_{c})/E_{nc}$ as a function of time; line colour from dark to light corresponds to increasing starting $x$ of the 10 segments of figure 17. (b) Space–time diagram of the point when $|r|\geqslant 3\,\%$ (indicated by a dashed horizontal line in (a)).

After the control effect has reached the end of the domain and the flow has stabilized, the time-average (between $32T$ and $48T$ ) energy reduction rate downstream of the control slot can be computed. The results are shown in figure 19, and, as expected, the energy reduction rate decreases further downstream. The computational domain is not large enough however to capture the full recovery to the uncontrolled state.

Figure 19. Mean energy reduction rate (averaged from $32T$ to $48T$ ) against distance $x$ downstream of the control slot.

Attention is turned to the velocity profiles. Figure 20 presents the mean velocity profiles at several downstream locations. The change in $\overline{u}$ in the near-wall region ( $y<2$ ) is small compared to the velocity profile above the control slots in figure 11. Above $y=2$ , there is large deviation of the controlled from the uncontrolled flow, but the difference becomes smaller further downstream. This is because the cost function is defined up to $x=1350$ . There is a small difference of the profile slope at the wall, so we expect a small reduction in drag. The controlled $\overline{u}$ is reduced above the control slot (figure 11) and downstream of the control slot (figure 20). The positive actuation introduces additional mass into the system and it is found that the controlled $\overline{v}$ is increased above the control slot to satisfy continuity.

Figure 20. Mean velocity profile. Solid line, uncontrolled flow; dashed line, controlled flow; dotted line, Blasius solution. Streamwise axis shows the downstream location of each profile.

3.5 Effect on skin friction

The time- and spanwise-averaged skin friction profile is shown in figure 21. It can be seen that the controlled skin friction starts to deviate from the uncontrolled line upstream of the control region, which is also observed in other controlled flows (Park & Choi Reference Park and Choi1999; Kim & Sung Reference Kim and Sung2006; Pamiès et al. Reference Pamiès, Garnier, Merlen and Sagaut2007; Kametani et al. Reference Kametani, Fukagata, Örlü and Schlatter2015). This is also in agreement with the mean pressure field shown in figure 14, where $\bar{p}$ at the wall starts to increase from around $x=900$ . The controlled $C_{f,c}$ reduces quickly inside the control slot and reaches the laminar level at around $x=1130$ . Between $x=1130$ and $1200$ , $C_{f,c}$ increases along the streamwise direction, at a rate similar to that of the uncontrolled flow. After $x=1200$ , the controlled $C_{f,c}$ is almost constant until the end of the control region. At this location the actuation velocity $v_{w}$ increases linearly to counteract the transition. The local drag reduction rate $R_{C_{f}}=(C_{f,nc}-C_{f,c})/C_{f,nc}$ is also shown in figure 21(b). Inside the control region, $R_{C_{f}}$ is between $45\,\%$ and $60\,\%$ .

Figure 21. Mean skin friction profiles (a) and corresponding local drag reduction rate $R_{C_{f}}=(C_{f,nc}-C_{f,c})/C_{f,nc}$ (b). Solid line, optimal controlled flow; dashed line, uncontrolled flow; dash-dot line, controlled flow using repeated $v_{w}$ .

Downstream of the control slot, $C_{f,c}$ recovers very quickly, but remains at a lower value than that of the uncontrolled flow up to the end of the domain. In terms of the average boundary layer thickness $\unicode[STIX]{x1D6FF}_{c}$ of the uncontrolled flow, the distance between the end of the control slot and the exit of the computational domain is about $25\unicode[STIX]{x1D6FF}_{c}$ . The average drag reduction rate in this region is $10\,\%$ , with a maximum $R_{C_{f}}$ of $14.5\,\%$ at $7\unicode[STIX]{x1D6FF}_{c}$ and a minimum $R_{C_{f}}$ of $7.5\,\%$ at the exit.

These results have similarities to those of Stroh et al. (Reference Stroh, Hasegawa, Schlatter and Frohnapfel2016). The effect of UB was also extended significantly downstream of the control slot until the exit of the domain, as in our case. This is interesting because in our case turbulence is sustained by instabilities that are initiated inside the boundary layer and not close to the wall.

It has been shown that the optimal control velocity is repeatable in a time scale equivalent to the period of the slow mode (figure 10). The $C_{f,c}$ from optimal control shown in figure 21 (black dashed line) was averaged from $t=16T$ to $48T$ , which is limited by the heavy computational cost for this control method. Although our results are in agreement with those of Stroh et al. (Reference Stroh, Hasegawa, Schlatter and Frohnapfel2016) as mentioned above, we performed an additional check to confirm that the drag reduction observed downstream of the slot is indeed robust, and it is not an artifact of the short time-averaging window. To this end, the temporal and spatial evolution of the optimal control velocity was stored, and then applied repeatedly for $64T$ (equivalent to four periods of the slow mode and two flow-through times).

The evolution of flow energy in the same control box is shown in figure 22. The vertical dotted lines indicate repeated application of $v_{w}$ , i.e. the optimal control velocity from $16T$ to $48T$ was repeated from $48T$ to $80T$ and $80T$ to $112T$ . The figure demonstrates that the repeated application of $v_{w}$ is able to maintain the flow energy at a level very similar to that obtained from piece-wise optimization. The controlled $E$ also has a clearly periodic behaviour over the time, as expected.

The long time-averaged $C_{f,lc}$ from $16T$ to $112T$ (i.e. including both piece-wise optimal control and control using repeated $v_{w}$ profiles) is shown in figure 21 by the dash-dot line. It is somewhat surprising that in the control slot, $C_{f,lc}$ is slightly lower than the $C_{f,c}$ from the receding horizon control. It must be borne in mind however that the objective of the optimal control is to minimize the flow energy, and not $C_{f}$ . Most importantly, the results demonstrate that in the downstream region, the long time-averaged $C_{f,lc}$ remains reduced and has a very similar value to the $C_{f,c}$ from receding horizon control.

Figure 22. Evolution of the flow energy defined in the control box. Black solid line, uncontrolled flow; thick dashed line, optimal controlled flow; thin dashed line, controlled flow using repeated application of $v_{w}$ .

3.6 Correlation between actuation and the flow field

In the present work, the control velocity distribution was obtained solely based on the objective function and the governing equations. In this section we explore the relationship between the optimal actuation and the flow above the slot.

Figure 23 shows the time history of $v_{w}$ and $u$ at $x=1080$ , which is close to the inlet of the control slot. Velocity $u$ is recorded at a distance $y=2.5$ from the wall. The time is from $4T$ to $36T$ , which covers two complete periods of the slow inlet mode. It was shown earlier that at this location the flow contains distorted streaks. The control velocity $v_{w}$ and both uncontrolled and controlled $u$ exhibit periodic behaviour as a function of time and it is clear that they are correlated.

Figure 23. Time history of control velocity $v_{w}(x=1080,z=30)$ (a) and instantaneous streamwise velocity $u(x=1080,y=2.5,z=30)$ (b). Solid line, uncontrolled $u$ ; dashed line, controlled $u$ ; thin horizontal solid line, time averaged value over each optimization interval.

In order to explore quantitatively the relationship between the optimal actuation and the flow above the slot over time, we compute the two-point correlation coefficient between $v_{w}(x,z)$ and a general variable of interest $\unicode[STIX]{x1D719}(x,y,z)$ ; in the present case, $\unicode[STIX]{x1D719}(x,y,z)$ represents velocities $u$ , $v$ . The coefficient is defined as

(3.4) $$\begin{eqnarray}\displaystyle R_{v_{w},\unicode[STIX]{x1D719}}^{t}(x,z;y)=\frac{\langle (v_{w}-\langle v_{w}\rangle _{t})(\unicode[STIX]{x1D719}-v_{w}-\langle \unicode[STIX]{x1D719}\rangle _{t})\rangle _{t}}{[\langle (v_{w}-\langle v_{w}\rangle _{t})^{2}\rangle _{t}\langle (\unicode[STIX]{x1D719}-\langle \unicode[STIX]{x1D719}\rangle _{t})^{2}\rangle _{t}]^{1/2}}. & & \displaystyle\end{eqnarray}$$

The superscript $t$ denotes averaging in time from $4T$ to $36T$ , and the correlation is computed using data from two periods of time, i.e. $32T$ . Note that due to the finite value of $T$ , velocity $v_{w}$ decreases over time within each interval and rapidly increases at the beginning of next interval. On the other hand the velocity field is relatively smooth compared to $v_{w}$ . The correlation coefficients are computed using data from each time instant from $4T$ to $36T$ and also from data averaged within each optimization interval (shown by horizontal line in figure 23).

Before examining the correlation between the control velocity and flow field above it, we first recall in figure 24 the flow properties corresponding to an idealized vortex pair. The largest (smallest) velocity fluctuations $u^{\prime }$ ( $v^{\prime }$ ) at the height of the vortex centre occur between the vortices, while at the centre of the vortex they are zero. Note also that $u^{\prime }$ and $v^{\prime }$ are negatively correlated (leading to ejections and sweeps events). The spanwise variation of the streamwise shear stress $\unicode[STIX]{x1D70F}_{x}$ is consistent with $u^{\prime }$ . On the other hand, the largest $\unicode[STIX]{x1D70F}_{z}$ is directly beneath the vortex centre and therefore is offset from the peak $u^{\prime }$ position by a quarter of the average streak spacing (Naguib, Morrison & Zaki Reference Naguib, Morrison and Zaki2010).

Figure 24. Sketch of a pair of streamwise vortices and corresponding variation in velocity fluctuations ( $u^{\prime },v^{\prime }$ ) at the height of the vortex centre and wall shear stresses $\unicode[STIX]{x1D70F}_{x}^{\prime },\unicode[STIX]{x1D70F}_{z}^{\prime }$ (Naguib et al. Reference Naguib, Morrison and Zaki2010).

Figure 25 shows the contour of $R_{v_{w},u}^{t}$ over the slot for $u$ and $v$ extracted at $y=2.5$ . The correlation was computed using averaged data from $4T$ to $36T$ (correlation with non-averaged data has similar distribution, but reduced value). When averaged data are used, the sample is much smaller (there is one sample per interval, corresponding to the horizontal line of figure 23). It can be seen that for $u$ , there is strong positive correlation over the entire slot; it is only towards the downstream end where some patches with negative correlation appear. Parameter $R_{v_{w},u}^{t}$ has a strong spatial dependence that weakens along the streamwise direction while the correlation of $R_{v_{w},v}^{t}$ remains more uniform over the entire slot. The values of $R_{v_{w},v}^{t}$ are negative in most of the control region, indicative of opposition control.

Figure 25. Contour of correlation between actuation velocity $v_{w}(x,z)$ and averaged data: (a) instantaneous streamwise velocity $u(x,y=2.5,z)$ ; (b) instantaneous wall-normal velocity $v(x,y=4,z)$ .

In order to remove the effect of the transient behaviour at the beginning of each interval due to the instability of the adjoint equations, we recomputed the correlations using the non-averaged data but discarding the first $150$ time steps (equivalent to  $t=12$ ) in each optimization interval. We further process the correlation over the entire slot and produce a (smoothed) normalized histogram of the correlation distribution. Figure 26 shows the histogram of $R_{v_{w},u}^{t}$ for non-averaged data (denoted by ‘all’) and non-averaged data with the first 150 time steps removed (denoted by ‘partial’) for various wall-normal locations. The results show that the transient in $v_{w}$ has small effect on the correlation distribution. There is positive correlation between $v_{w}$ and $u^{\prime }$ at all four locations and $R_{v_{w},u}^{t}$ is larger at $y=2.5$ and $y=4$ than at $y=1$ and $y=5.6$ . This wall-normal dependence is very similar to $R_{v_{w},u}^{s}$ (figure 12). The histogram peaks at average $R_{v_{w},u}^{s}\approx 0.3{-}0.35$ .

Figure 26. Distribution of the correlation coefficient $R_{v_{w},u}^{t}$ between $v_{w}(x,z)$ and $u(x,y,z)$ at different wall-normal locations.

Figure 27 shows the distribution of $R_{v_{w},v}^{t}$ at the same four wall-normal locations. The correlation is negative at three locations, apart from $y=1$ where it is slightly positive. This is because this location is close to the wall and $v^{\prime }$ is strongly affected by the actuation. As for $R_{v_{w},u}^{t}$ , removing the transient data results in slight changes. The histogram peaks at average $R_{v_{w},v}^{s}\approx -0.10$ .

Figure 27. Distribution of the correlation coefficient $R_{v_{w},v}^{t}$ between $v_{w}(x,z)$ and $v(x,y,z)$ at different wall-normal locations.

The standard opposition control (Choi et al. Reference Choi, Moin and Kim1994) imposes the opposite of the wall-normal velocity at a distance away from the wall (detection plane) with the aim of counteracting the up-and-down motion induced by the vortex. In this case the correlation at the detection plane is $R_{v_{w},v}^{s}=-1$ . The predicted negative sign of the $R_{v_{w},v}^{s}$ correlation in our case indicates that opposition control is at play, but the small value of the correlation indicates that it significantly deviates from the standard opposition control. The correlation is positive with the streamwise velocity, so the optimal wall actuation is broadly consistent with the streamwise vortex model shown in figure 24.

Recall that the current actuation has mean positive velocity (figure 10). Figure 28 presents a scatter plot of $v_{w}$ and $u$ from averaged data at one particular location with averaged $R_{v_{w},u}^{t}=0.92$ as an example. A positive $R_{v_{w},u}^{t}$ implies that a larger positive $v_{w}$ is applied to area with positive $u^{\prime }$ , while smaller positive $v_{w}$ is applied to area with negative $u^{\prime }$ . Therefore the actuation is closer to the blowing-only opposition control of Pamiès et al. (Reference Pamiès, Garnier, Merlen and Sagaut2007), where the suction part from the opposition control is removed. Chang et al. (Reference Chang, Collis and Ramakrishnan2002) showed that the effectiveness of standard opposition control reduces as Reynolds number increases in channel flow. Pamiès et al. (Reference Pamiès, Garnier, Merlen and Sagaut2007) demonstrated the blowing-only opposition control can improve drag reduction efficiency when compared to the classic opposition control as well as UB with same mean control velocity. Another similar example was the recent experimental work of Abbassi et al. (Reference Abbassi, Baars, Hutchins and Marusic2017), in which wall-normal jet flow was injected in regions where high-speed streamwise velocity fluctuations were presented. The jet operated in the blowing-only mode.

Figure 28. Scatter plot of time-average (over one interval) $v_{w}$ at $x=1071$ , $z=63$ and $u$ at $x=1071$ , $y=2.5$ , $z=63$ ; at this location $R_{v_{w},u}^{t}=0.92$ .

In § 3.3 it was found that the turbulent profiles (r.m.s. of velocity fluctuations, Reynolds stress) have different behaviour over the control slot. In particular, the controlled flow at $x=1075$ is different from that at $x=1130$ and $x=1275$ . In figure 25 is can be seen that $R_{v_{w},u}^{t}$ is larger in the upstream side of the control slot, especially upstream of $x=1100$ . As discussed above, this indicates the control action is similar to blowing-only opposition control in this region. This is supported by the behaviour of the controlled turbulent profiles at $x=1075$ being very similar to those obtained by blowing-only opposition control (Pamiès et al. Reference Pamiès, Garnier, Merlen and Sagaut2007). The blowing-only opposition control is designed to counteract the sweep events only. Wallace (Reference Wallace2016) reported that close to the wall, for $y^{+}<15$ , the sweep events contribute considerably more to the total Reynolds stress than the ejection events. In figure 16, at $x=1075$ , $-\overline{u^{\prime }v^{\prime }}^{+nc}$ is largely reduced near the wall. There is a very small reduction in $-\overline{u^{\prime }v^{\prime }}^{+nc}$ at $x=1130$ while at $x=1250$ an increase is observed. This behaviour agrees with the spatial distribution of $R_{v_{w},u}^{t}$ .

In the rear part of the control slot when $R_{v_{w},u}^{t}$ is weak, the mean positive actuation indicates the controller might work in a similar way to the UB. This is also proved by the behaviour of turbulent statistics in this region.

The change of the control mechanism over the slot is related to the flow activity in the control region. As shown in § 3.1, near the inlet of the control region, there are mainly distorted streaks entering from upstream and the controller tends to counteract the motion of the streaks (vortices) using blowing-only opposition control. This is effective as the flow energy initially reduces along $x$ (figure 8). As mentioned, the transition still occurs in the controlled flow after $x=1100$ and there are turbulent structures, but with reduced strength. In this region ( $x>1100$ ), the optimal results reveal that the best solution to control the flow is to impose strong positive actuation. Although the controlled flow properties are similar to those obtained from UB, the distribution of optimal $v_{w}$ in this region is not uniform. In Xiao & Papadakis (Reference Xiao and Papadakis2017) it was shown that a variable $v_{w}$ is more efficient in reducing the objective function than UB in a single interval. The same is also demonstrated in the receding horizon control.