6.1. Weakened streamwise vorticity

The QSVs play a crucial role in the near-wall dynamics of the TBL and their variation may be quantitative reflected in the wall-normal profile of $\omega _{x,rms}^ +$ (figure 10), where $\omega _x^ += \partial {W^ + }/\partial {y^ + } - \partial {V^ + }/\partial {z^ + }$ and $\omega _{x,rms}^ += \sqrt {\left( {\sum\nolimits_{i = 1}^{{N_p}} {\omega^{\prime + 2}_x} } \right)/{N_p}}$ (the prime denotes the fluctuating component and N_p (= 3000) is the number of PIV image pairs). The uncertainty of the PIV-measured ω_{x ,rms} results largely from a lack of convergence and the uncertainty of instantaneous ω_x (Benedict & Gould Reference Benedict and Gould1996; Sciacchitano & Wieneke Reference Sciacchitano and Wieneke2016). The standard uncertainty ${s_{{\omega _{x,rms}}}}$ can be calculated through

(6.1)\begin{equation}{s_{{\omega _{x,rms}}}} = \frac{1}{{2\sqrt {\omega _{x,rms}^2 - \overline {\gamma _{{\omega _x}}^2} } }}\sqrt {\omega _{x,rms}^2 + {{\left( {\sqrt 2 {\sigma_{{\gamma_{{\omega_x}}}}}\overline {{\gamma_{{\omega_x}}}} \sqrt {1 + \frac{{\sigma_{{\gamma_{{\omega_x}}}}^2}}{{2{{\overline {{\gamma_{{\omega_x}}}} }^2}}}} } \right)}^2}} \sqrt {\frac{2}{{{N_p}}}} ,\end{equation}

where ${\gamma _{{\omega _x}}}$ is the uncertainty of ω_x of the instantaneous flow, $\overline {{\gamma _{{\omega _x}}}} $ and ${\sigma _{{\gamma _{{\omega _x}}}}}$ are the time-averaged value and the standard deviation of ${\gamma _{{\omega _x}}}$, respectively. The uncertainty of the PIV measurement arises from a variety of sources, such as the density of the seeding particles, interrogation window size, time delay between two successive frames, image distortions and wall reflections (e.g. Raffel et al. Reference Raffel, Willert, Scarano, Kähler, Wereley and Kompenhans2018; Zhang et al. Reference Zhang, Liu, Zhou, To and Tu2018). Following Sciacchitano, Wieneke & Scarano (Reference Sciacchitano, Wieneke and Scarano2013), ${\gamma _{{\omega _x}}}$ is evaluated through the image matching process. The basic concept of this approach is to evaluate the residual distance or particle disparity between the particle image pairs in two successive images (Zhang et al. Reference Zhang, Liu, Zhou, To and Tu2018). Thus determined error bars, i.e. $1.96{s_{{\omega _{x,rms}}}}$ corresponding to a 95 % confidence interval, are shown in figure 10. It is evident that the uncertainty of the PIV-measured ω_{x ,rms} grows considerably for y ⁺ < 10. In the absence of control, the $\omega _{x,rms}^ +$ profile agrees reasonably with Kim, Moin & Moser's (Reference Kim, Moin and Moser1987) DNS data, except the point at y⁺ = 3.5. The discrepancy at this point is ascribed to the poor quality of the PIV images near the wall, due to the reflection of the laser. Once control is applied, $\omega _{x,rms}^ +$ becomes considerably smaller than the uncontrolled flow for y⁺ < 25. The reduced $\omega _{x,rms}^ +$ near the wall was also observed by other DR investigations in the wall-bounded flows. For example, Baron & Quadrio (Reference Baron and Quadrio1996) introduced numerically spanwise wall oscillation into a turbulent channel flow. They achieved a maximum DR of up to 40 % and found that the $\omega _{x,rms}^ +$ near the wall was greatly suppressed. It is also noteworthy that the location of the maximum $\omega _{x,rms}^ +$ moves from y⁺ = 20 to y⁺ = 25 (figure 10). As is well established, this location corresponds to the average position of the centre of QSVs (e.g. Kim et al. Reference Kim, Moin and Moser1987; Kim & Sung Reference Kim and Sung2006), and its shift points to the lift-up of the QSVs due to the blowing jets. Naturally, the sweep motions induced by these QSVs become weaker and less effective in producing high WSS, contributing to the measured DR.

Figure 10. Wall-normal distribution of the r.m.s. value $\omega _{x,rms}^ + $ of the streamwise vorticity fluctuations measured at (x⁺, z⁺) = (33, 0). Control parameters: A ⁺ = 1.42, f ⁺ = 0.42.

6.2. Redistributed energy of turbulent structures

Insight may be gained into the change in the turbulence structures via the power spectral density function E_u of the fluctuating streamwise velocity u acquired with the hot-wire. The value of E_u is calculated from a fast Fourier transform (FFT) algorithm with an FFT window size of 4096. The integration of E_u over the entire frequency range yields the variance of u (Bai et al. Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014). At y⁺ = 3, the manipulated ${f^ + }E_u^ +$ shows a considerable reduction for f⁺ < 0.01 and an increase for f⁺ > 0.01, indicating less energetic large-scale structures and more energetic smaller ones (figure 11). This result is consistent with the observation from smoke-wire flow visualization images (figure 9), where rather stable and small-scale structures occur near the wall in lieu of the natural larger-scale streaky structures. A discernible spike at f⁺ = 0.42 is also observed in figure 11, which results from the applied unsteady blowing. A similar observation is made at y⁺ = 6, though changes are less obvious. The distributions of ${f^ + }E_u^ +$ with and without control at y⁺ = 14 and 38 become undiscernible, if exception is made for the spike at f⁺ = 0.42. The change in ${f^ + }E_u^ +$ near the wall is consistent with the observation of Bai et al. (Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014), who also found similar energy distribution modification, with less energy in larger-scale structure but more energy in smaller-scale structures. The similarity between the two studies is not surprising. In both studies, the streamwise vortex array is generated near the wall and the natural large-scale streaky structures are replaced by smaller-scale motions. In contrast, the present redistribution of ${f^ + }E_u^ +$ is opposite to Iuso et al.'s (Reference Iuso, Onorato, Spazzini and Di Cicca2002) finding, who introduced jet-induced large-scale counter-rotating streamwise vortices in a channel flow, achieving a spanwise-averaged DR of 15 %, and observed an energy increase at low f⁺ but a decrease at high f⁺. The difference may be connected to distinct DR mechanisms involved. In Iuso et al. (Reference Iuso, Onorato, Spazzini and Di Cicca2002), the large-scale manipulation increased both the length and lateral spacing of the near-wall velocity streaks, making them more stable and thus leading to the DR. Therefore, the energies associated with the large-scale structures increased, while those with small-scale structures contracted. However, the present wall-normal jets through the slit array induce small-scale streamwise vortices and inject zero-streamwise-momentum fluid.

Figure 11. Weighted power spectral density functions of the hot-wire-measured fluctuating velocity u at (a) (x ⁺, y ⁺, z ⁺) = (33, 3, 0), (b) (x ⁺, y ⁺, z ⁺) = (33, 6, 0), (c) (x ⁺, y ⁺, z ⁺) = (33, 14, 0) and (d) (x ⁺, y ⁺, z ⁺) = (33, 38, 0). Solid line, uncontrolled TBL; dotted line, controlled TBL, with control parameters A ⁺ = 1.42, f ⁺ = 0.42.

6.3. Dissipation and production

The dissipation $\varepsilon ( = \nu \overline {(\textrm{d}{u_i}/\textrm{d}{x_j})(\textrm{d}{u_i}/\textrm{d}{x_j})} )$ and production rate $- \overline {{u_i}{u_j}} \textrm{(d}\overline {{U_i}} /\textrm{d}{x_j})$ of turbulent kinetic energy (TKE) are two important terms in the budget equation for TKE near the wall in the TBL. Although experimental access to the remaining terms of TKE budget is difficult from the present type of measurements, examining how production and dissipation are altered by control may provide additional physical insight into the DR mechanisms.

The ε involves 12 components or velocity derivatives. Fortunately, among the various derivatives, the term $\overline {{{(\textrm{d}u^+/\textrm{d}y^+)}^2}}$ overwhelms the others and in the unmanipulated case accounts for approximately 80 % of the total ε in the viscous sublayer and 41 %–72 % in the buffer layer for wall-bounded flows (Antonia, Kim & Browne Reference Antonia, Kim and Browne1991). Therefore, it is reasonable to estimate the variation of ε through $\overline {{{(\textrm{d}u^+/\textrm{d}y^+)}^2}} $, as shown in figure 12(a). The standard uncertainty ${s_{\overline {{{(\textrm{d}u/\textrm{d}y)}^2}} }}$ of $\overline {{{(\textrm{d}u/\textrm{d}y)}^2}} $ is calculated by

(6.2)\begin{equation}{s_{\overline {{{(\textrm{d}u/\textrm{d}y)}^2}} }} = \sqrt {{{(\overline {{{(\textrm{d}u/\textrm{d}y)}^2}} )}^2} + {{\left( {\sqrt 2 {\sigma_{{\gamma_{\textrm{d}u/\textrm{d}y}}}}\overline {{\gamma_{\textrm{d}u/\textrm{d}y}}} \sqrt {1 + \frac{{\sigma_{{\gamma_{\textrm{d}u/\textrm{d}y}}}^2}}{{2{{\overline {{\gamma_{\textrm{d}u/\textrm{d}y}}} }^2}}}} } \right)}^2}} \sqrt {\frac{2}{{{N_p}}}} ,\end{equation}

where ${\gamma _{\textrm{d}u/\textrm{d}y}}$ is the measurement uncertainty of du/dy, and $\overline {{\gamma _{\textrm{d}u/\textrm{d}y}}} $ and ${\sigma _{{\gamma _{\textrm{d}u/\textrm{d}y}}}}$ are the time-averaged value and the standard deviation of ${\gamma _{\textrm{d}u/\textrm{d}y}}$, respectively. The error bars in figure 12(a,b) represent $1.96{s_{\overline {{{(\textrm{d}u/\textrm{d}y)}^2}} }}$, corresponding to a 95 % confidence interval. In the absence of control, $\overline {{{(\textrm{d}u/\textrm{d}y)}^2}}$ agrees well with Qiao, Xu & Zhou's (Reference Qiao, Xu and Zhou2019) measurement using parallel hot-wires and reasonably well with Antonia et al.'s (Reference Antonia, Kim and Browne1991) DNS data. Once control is applied, $\overline {{{(\textrm{d}{u^ + }/\textrm{d}{y^ + })}^2}}$ exceeds greatly the uncontrolled flow, especially near the wall, suggesting a substantial increase in dissipation, in agreement with the result of the isotropic estimate. Although these indications only have a qualitative significance, it is interesting to note how the scenario of increased TKE dissipation, suggested by the change in $\overline {{{(\textrm{d}{u^ + }/\textrm{d}{y^ + })}^2}} $, is consistent with stabilized streaky structures and a flow with locally reduced turbulence (figure 9).

Figure 12. Wall-normal variation of (a,b) the dominant component $\overline {{{(\textrm{d}u/\textrm{d}y)}^2}}$ of the mean turbulent energy dissipation, (c,d) production $- \overline {uv} (\textrm{d}\bar{U}/\textrm{d}y)$ measured at (x⁺, z ⁺) = (33, 0). Control parameters: A ⁺ = 1.42, f ⁺ = 0.42. Plots in the left column employ inner scaling, and outer scaling is used for the right column.

The production term $- \overline {{u_i}{u_j}} (\textrm{d}\overline {{U_i}} /\textrm{d}{x_j})$ assumes the simpler expression $- \overline {u^+v^+} (\textrm{d}\bar{U}^+/\textrm{d}y^+)$ in the absence of control (Pope Reference Pope2001), as shown in figure 12. The standard uncertainty ${s_P}$ of the production term $- \overline {uv} (\textrm{d}\bar{U}/\textrm{d}y)$ is calculated through ${s_P} = \sqrt {{{(\overline {uv} )}^2}s_{\textrm{d}\bar{U}/\textrm{d}y}^2 + {{(\textrm{d}\bar{U}/\textrm{d}y)}^2}s_{\overline {uv} }^2} $, where ${s_{\textrm{d}\bar{U}/\textrm{d}y}}$ and ${s_{\overline {uv} }}$ are the standard uncertainties of $\textrm{d}\bar{U}/\textrm{d}y$ and $\overline {uv} $, respectively (Sciacchitano & Wieneke Reference Sciacchitano and Wieneke2016). The error bars plotted in figure 12(c,d) as $1.96{s_P}$ correspond to a 95 % confidence interval. The quantity $- \overline {{u^ + }{v^ + }} (\textrm{d}\overline {{U^ + }} /\textrm{d}{y^ + })$ exhibits a sharp increase in the near-wall region (figure 12c). Again, one should be cautioned that the other terms of the production tensor, difficult to be accurately measured using a two-dimensional PIV, may become non-negligible under control. However, $\overline {{{(\textrm{d}{u^ + }/\textrm{d}{y^ + })}^2}}$ and $- \overline {{u^ + }{v^ + }} (\textrm{d}\overline {{U^ + }} /\textrm{d}{y^ + })$ increase markedly under control, partially due to the altered local inner scales, i.e. the friction velocity u_τ and viscous length scale l_v. When control is applied, u_τ decreases from 0.105 to 0.058 m s⁻¹ and l_v grows from 0.15 to 0.27 mm, resulting in a smaller ${({u_\tau }/{l_v})^2}$.

Naturally, it is of interest to examine the production and dissipation terms normalized by the outer scales, i.e. U_∞ and δ, which are unchanged by control, thus providing a different perspective of modifications. The manipulated $\overline {{{(\textrm{d}{u^\ast }/\textrm{d}{y^\ast })}^2}} $ is again much larger than the uncontrolled flow (figure 12b). The jet-induced small-scale streamwise vortices, which are more dissipative, are probably largely responsible for the increased dissipation (Bai et al. Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014). Under control, $- \overline {{u^\ast }{v^\ast }} (\textrm{d}\overline {{U^\ast }} /\textrm{d}{y^\ast })$ becomes appreciably smaller than that without control for y/δ < 0.017, corresponding to y⁺ < 10, in consistence with the reduced u_mrs/U_∞ in the same region (figure 8d). The reduced $- \overline {{u^\ast }{v^\ast }} (\textrm{d}\overline {{U^\ast }} /\textrm{d}{y^\ast })$ is attributed to the stabilized streaks (figure 9b), which shows unambiguously an interrupted near-wall turbulence regeneration cycle and suppression of the new QSVs. On the other hand, $- \overline {{u^\ast }{v^\ast }} (\textrm{d}\overline {{U^\ast }} /\textrm{d}{y^\ast })$ rises rapidly and exceeds uncontrolled flow for 0.017 < y/δ < 0.1, or 10 < y⁺ < 57; the maximum production climbs by 108 %, although its location moves away from the wall. As the TKE production is closely associated with the QSVs (Wallace Reference Wallace2016), the movement of its maximum is a clear indicator that the QSVs are lifted up. The lifted QSVs grow stronger due to the weakened interaction between the QSVs and near-wall flow (Park & Choi Reference Park and Choi1999), accounting for the climbing production further away from the wall. Accordingly, there is an increase in u_mrs/U_∞ in this region (figure 8d). The reduced outer-scale-normalized production is consistent with the stabilized streaks, as seen from the smoke-wire visualization (figure 9). As such, the DR mechanism is presently discussed based on the outer-scale-normalized results (figure 12b,d).

Overall, the following scenario is proposed. The periodically blowing jets through the streamwise slits generate an array of streamwise vortices near the wall. The vortices, together with the effect of injecting fluid from the wall towards the bulk flow, act to push the QSVs away from the wall (figures 8–12) and work as a barrier between the QSVs and the wall, preventing the sweep motions from reaching the wall, large WSS from being generated and natural near-wall streaky structures from being formed (figure 9b). As a result, the near-wall turbulence regeneration cycle is partially inhibited, causing a reduced TKE production near the wall (figure 12d). Furthermore, the vortices are highly dissipative, leading to a great increase in ε (figure 12b). Meanwhile, the injection of the zero-streamwise-momentum fluid reduces the near-wall streamwise velocity gradient, also contributing to local flow relaminarization (figure 9b) and DR.

6.4. Contributions from different mechanisms to DR

As noted earlier, the injected fluid through slits modifies the TBL in two ways. One is to add zero-streamwise-momentum fluid in the near-wall region, resulting in decelerated velocity in this region and hence contributing to DR. The other is to generate one row of streamwise vortices, which act to suppress the high-speed sweeps (figure 4), thus interrupting the near-wall turbulence generation cycle and also contributing to DR. One important question arises naturally. Which mechanism plays a more important role in reducing drag? Further, can we quantify the two contributions? To address this issue, we conducted two experiments, i.e. the opposition control and the desynchronized control following Abbassi et al. (Reference Abbassi, Baars, Hutchins and Marusic2017). The latter is as an open-loop control, and the injection of jet fluid is desynchronized with the high-speed events.

In the opposition control, the blowing actuator is activated only when high-speed events are detected, preferentially related to sweep motions. A feed-forward control strategy (figure 13) is deployed as described by Qiao, Zhou & Wu (Reference Qiao, Zhou and Wu2017), although their piezo-ceramic actuators are replaced by the present jets. Figure 13 schematically presents 10 actuators and four sensors to form two sensor–actuator groups for the feed-forward opposition control, including the detecting signal U_d, monitoring signal U_m, predicted signal U_p and transfer function G(s), where s is the transform operator. Each sensor–actuator group is composed of an upstream detecting sensor, a downstream monitoring sensor and five slits, and works independently from the other. Following Qiao et al. (Reference Qiao, Zhou and Wu2017) and Qiao, Wu & Zhou (Reference Qiao, Wu and Zhou2018), we used four fixed wall wires, made of tungsten, to measure the streamwise velocity and approximately the WSS. The sensing element of each wall wire is 5 μm or 0.034 wall unit in diameter (d) and 1.2 mm or 8 wall units in length (l), resulting in l/d = 240. The gap between the plate surface and the sensing element is set as 0.5 mm or 3.4 wall units to ensure that the sensing element is immersed in the viscous sublayer (y ⁺ < 5). Two wall wires or detecting sensors are placed 7 mm or 47 wall units upstream of the actuators to measure the velocity signal U_d and detect the incoming large events of WSS. The spanwise separation between the sensors is 10 mm or 67 wall units, i.e. less than the average spanwise separation of 100 wall units between the near-wall low-speed streaks (e.g. Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). The other two wall wires or monitoring sensors are positioned at (x ⁺, y ⁺, z ⁺) = (47, 3.4, ±33) to measure the velocity signal U_m for the estimate of local WSS. The transfer function G(s), developed by Qiao et al. (Reference Qiao, Zhou and Wu2017), is introduced to predict the real-time uncontrolled flow state (U_p) at the leading edge of the actuators from U_d. The transfer function G(s) is determined based on an off-line system identification method, with U_d and U_m as the input and output signals, respectively, and in general G(s) = (c_hs^h + c_{h −1}s^{h− 1} + $\cdots$ + c ₁s ¹ + c ₀)/(sⁿ + k_{n −1}s^{n −1} + $\cdots$ + k ₁s ¹ + k ₀), where [c ₀,…, c_{h −1}, c_h] and [k ₀,…,k_{n −1}] are the undetermined constants and subscripts ‘h’ and ‘n’ are the natural numbers. Applying a prediction error minimization technique, we may determine the optimal constants so that the prediction error $e = 1/{N_G}\sum\nolimits_{i = 1}^{{N_G}} {{{({U_p} - {U_m})}^2}} $ is minimized, where N_G = 5000 is the sampling number of U_m for the determination of G(s). Please refer to Qiao et al. (Reference Qiao, Zhou and Wu2017) and Qiao et al. (Reference Qiao, Wu and Zhou2018) for more details on how to determine the constants. A reasonable prediction is achieved, as demonstrated by Qiao et al. (Reference Qiao, Zhou and Wu2017, Reference Qiao, Wu and Zhou2018). This is because the average length of streaky structures is approximately 1000 wall units in TBL (Robinson Reference Robinson1991), while the distance from the monitoring sensor to the leading edge of the actuators is only 47 wall units. There is a time delay T_c in G(s) associated with the feed-forward opposition control, which is connected to the convection time of the organized structures from the detecting sensors to the actuators; T_c plays a significant role in the control performance. As shown in figure 14, ${\delta _{{\tau _w}}}$ depends on T_c and the optimal T_c occurs at 4.1 ms, where ${\delta _{{\tau _w}}} ={-} 0.39$. However, there is a significant difference between the optimal and expected T_c. Two factors need to be considered in the choice of the expected T_c, i.e. the convection time for the coherent events to travel from the detecting sensors to the trailing edge of the slit array and the response time of the actuators once actuated. The former is 24.0 ms given a convection velocity $\bar{U}_{cn}^ + $ of 11.5 (e.g. Qiao et al. Reference Qiao, Zhou and Wu2017) and the latter is 4.8 ms. Then, the expected time delay is 24.0–4.8 = 19.2 ms, which deviates significantly from the optimal T_c (figure 14). Note that the near-wall high-speed streaks are tilted due to the shear at an angle of approximately 4.7° with respect to the wall (Rebbeck & Choi Reference Rebbeck and Choi2001) and hence their upper parts occur downstream of the lower parts (Lundell & Alfredsson Reference Lundell and Alfredsson2004). That is, the centre of a streak is already far downstream of the detecting sensors when the lower parts are detected. This suggests that the actuators should be activated before the organized structures of concern reach the actuators, which contributes directly to the discrepancy between the optimal and expected time delays.

Figure 13. Schematic diagram of the feed-forward opposition control set-up (units in mm). The detecting and monitoring sensors are placed at (x ⁺, y⁺, z ⁺) = (−193, 3.4, ±33) and (47, 3.4, ±33), capturing the U_d and U_m, respectively. The signal U_p is predicted from U_d based on the transfer function G(s).

Figure 14. Dependence of ${\delta _{{\tau _w}}}$ on the time delay T_c for the feed-forward opposition control. The duty cycle is 350 %.

Figure 15 presents the typical signals of $U_d^ + ,U_m^ + ,U_p^ +$ and the triggering signal U_t for the actuator, all simultaneously obtained, for the feed-forward opposition control. In general, $U_d^ +$ displays a good similarity in large-scale events, both qualitatively and quantitatively, to $U_p^ +$ indicating that the transfer function used is acceptable for the WSS prediction. Once the magnitude of $U_p^ +$, exceeds a threshold of Th ₁ (figure 15c), a high-speed WSS event is detected and U_t jumps from 0 to 1 V, which activates the periodic blowing. The frequency and amplitude (f ⁺, A ⁺) of the blowing jets are chosen to be (0.35, 0.67), corresponding to a DR of 60 % in the open-loop control. Apparently, the threshold Th ₁ for $U_p^ +$ influences directly the duty cycle of U_t (figure 15d) and hence the amount of zero-streamwise-momentum fluid injected, which plays a dominant role in DR. Assume the sweep and ejection events occur equally on the down- and up-draught sides of the QSVs, respectively, which are responsible for producing the high- and low-speed WSS events (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967; Robinson Reference Robinson1991). As such, we choose a threshold, i.e. Th ₁ = 3.2 u_τ such that the duty cycle of U_t (figure 15d) is 50 %. A similar approach was also used by Abbassi et al. (Reference Abbassi, Baars, Hutchins and Marusic2017) who chose a duty cycle of 50 % for their unsteady wall-normal jet injection used to suppress high-speed WSS events. The high-speed WSS events, indicated by arrows in $U_p^ +$, contract significantly in magnitude and duration under control, as shown in $U_m^ +$, accounting for the observed significant DR. Note that the choice of the duty cycle will change the proportion of the contribution to the total DR between the injection of low-streamwise-momentum fluid and controlling large-scale organized events. Take a duty cycle of 60 % for example. The DR, resulting from the latter mechanism, is the same as that when the duty cycle of 50 % is deployed because the high-speed WSS events only account for 50 % of the total time. However, there is an increase for the DR associated with the former mechanism at the duty cycle of 60 %, as compared with 50 %.

Figure 15. (a) U_d from the detecting sensor upstream of the actuator, (b) U_m from the monitoring sensor downstream of the actuator, (c) predicted U_p and (d) actuation signal U_t that drives the actuators via a dSpace control platform. Here, t = 0 s is chosen arbitrarily. Control parameters (f ⁺, A⁺) = (0.35, 0.67). All measurements are taken at U_∞ = 2.4 m s⁻¹.

In the desynchronized control, actuation is not correlated with the organized aspects such as the sweep or ejection events. Hence, the same amount of low-momentum fluid or input energy as the opposition control is injected randomly into the TBL to facilitate a comparison between the two schemes. This is made possible by pre-recording U_t used in the opposition control and then using it to trigger the actuator in a new realization of the flow where the desynchronized control is applied. The same f ⁺ and A ⁺ as the opposition control are used. The triggering signal is obviously unaware of the upstream flow condition and is thus desynchronized with the organized events. The actuation is not directed against the organized structures, and any DR incurred may be ascribed to the injected zero-streamwise-momentum fluid. The assertion is confirmed by the weighted power spectral density function ${f^ + }E_u^ +$ of U_m, as shown in figure 16. Under opposition control, there is an appreciable decrease in ${f^ + }E_u^ +$ for f ⁺ < 0.01 and meanwhile an increase for f ⁺ > 0.015, compared with the uncontrolled case. However, under the desynchronized control, there appears no difference in ${f^ + }E_u^ +$ for f ⁺ < 0.03, and ${f^ + }E_u^ +$ is only slightly larger than the uncontrolled case for f ⁺ > 0.03. The observation indicates an appreciable energy transfer from large- to small-scale structures under opposition control, but this energy transfer is not evident under the desynchronized control.

Figure 16. Weighted power spectral density functions of the hot-wire-measured fluctuating velocity u at (x ⁺, y ⁺, z ⁺) = (47, 3.4, 33) under different control schemes.

The DR is 39 % for the feed-forward opposition control and 30 % for the desynchronized control. The latter results from the injection of low-streamwise-momentum fluid (Abbassi et al. Reference Abbassi, Baars, Hutchins and Marusic2017), while the former also contains the contribution from the blowing-jet-generated streamwise structures, which suppress the large-scale WSS events. This difference of 9 % may be attributed to the blowing-jet-generated streamwise structures and hence a modification of organized structures in the near-wall region. It may be further inferred that the injection of low-streamwise-momentum fluid accounts for 77 % of the total DR, while controlling large-scale organized events is responsible for 23 %.

Some remarks are due on the effect of the slit length. The control performance (e.g. DR and drag recovery length) depends on the penetration depth of actuation, reaching the optimum when this depth is of the order of the viscous sublayer (e.g. Du et al. Reference Du, Symeonidis and Karniadakis2002). In Bai et al. (Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014), the cantilever-supported piezo-ceramic actuator was characterized by the bending displacements from zero at the fixed end to the maximum at the free end, implying a penetration depth range from zero to its maximum. As such, its effective length, corresponding to the most effective or optimum penetration depth, was bound to be greatly shorter than the actuator length. In contrast, the present wall-normal jets are longitudinally uniform. The effective length of the actuator is much larger than that in Bai et al. (Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014), which accounts for the significantly postponed drag recovery. However, this effective length, when changing from 20 mm used in the open-loop control to 22 mm, has little influence on the percentage contributions to DR from the two mechanisms, i.e. the jet-induced highly regularized streamwise vortices and the injection of the zero-streamwise-momentum fluid. This is substantiated by supplementary experiments using the slits of 35 and 50 mm in length, respectively, the latter being 43 % longer than the former. The jets through 50 mm long slits at A ⁺ = 0.91 inject 16 % more of zero-streamwise-momentum fluid into the near-wall region than those of 35 mm long slits at A ⁺ = 1.12. Then, DR may be estimated by 23 % × (1 + 43 %) + 77 % × (1 + 16 %) = 22 % in view of the contributions, 23 % and 77 %, to the total DR from controlling large-scale organized events and the injection of zero-streamwise-momentum fluid, respectively. The measured DRs averaged over x ⁺ = 0–360 are 33 % and 41 % for the slits of 35 and 50 mm in length, respectively. The latter produces 24 % more of the averaged DR than the former, rather close to the estimated 22 %. For the same token, the contributions to DR from the two mechanisms may vary only slightly when DR changes from 60 % to say 70 %.

6.5. Scaling of DR

In this section, the scaling of DR is empirically investigated. Figure 6(a) indicates clearly that ${\delta _{{\tau _w}}}$ depends on A ⁺ and f ⁺, but A ⁺ and f ⁺ are closely coupled. One may surmise that ${\delta _{{\tau _w}}}$ is connected to the energy input per pulse of blowing, which is proportional to A ⁺³/f ⁺. Here, A^{+ 3} is the energy injected into the boundary layer per unit time and f⁺ is the injection number per unit time. Therefore, A^{+ 3}/f⁺ is a direct indicator for the input energy per pulse. After a careful analysis of the experimental data in figure 6(a) along with numerous trial-and-error attempts at least-squares fitting the data, we propose a relationship among ${\delta _{{\tau _w}}}$, A ⁺ and f ⁺, viz.

(6.3)\begin{equation}{\delta _{{\tau _w}}} = \; g\left( {\frac{{{A^{ + 3}}}}{{{f^{+ }}}}} \right)\varPi ({f^ + }),\end{equation}

where g and Π are functions of ${A^{ + 3}}/f{ ^ + }$ and f ⁺, respectively. Equation (6.3) takes into account (i) a coupling between A ⁺ and f ⁺ and (ii) the positive influence of f ⁺ on ${\delta _{{\tau _w}}}$ as evident in figure 6(a). After numerous trial-and-error attempts, Π is determined to be ln(−3f ⁺² + 3.5f ⁺ + 1.6) which fits the data well. Function g in (6.3) is determined by plotting ${\delta _{{\tau _w}}}/\varPi $ against A ⁺³/f ⁺. As shown in figure 17, almost all the data collapse reasonably well with a piecewise least-squares fit by

(6.4a)\begin{gather}\frac{{{\delta _{{\tau _w}}}}}{\varPi } ={-} 0.51 \times {\left( {\frac{{{A^{ + 3}}}}{{f{^ + }}}} \right)^{0.33}},\quad \frac{{{A^{ + 3}}}}{{f^{ + }}} \le 24,\end{gather}

(6.4b)\begin{gather}\frac{{{\delta _{{\tau _w}}}}}{\varPi } ={-} \left( {1.5 - 0.0014 \times \frac{{{A^{ + 3}}}}{{{f^{+ }}}}} \right),\quad \frac{{{A^{ + 3}}}}{{f{ ^ + }}} > 24,\end{gather}

or a single function fit, viz.

(6.5)\begin{align}\frac{{{\delta _{{\tau _w}}}}}{\varPi } = \; {a_1}exp \left[ { - {{\left( {\frac{{{{({A^{ + 3}}/{f^{+ }})}^{0.33}} - {b_1}}}{{{c_1}}}} \right)}^2}} \right] + {a_2}exp \left[ { - {{\left( {\frac{{{{({A^{ + 3}}/{f^{+ }})}^{0.33}} - {b_2}}}{{{c_2}}}} \right)}^2}} \right],\end{align}

where the coefficients a ₁, b ₁, c ₁, a ₂, b ₂ and c ₂ are listed in table 2. The scaling law is obtained and thus valid for a rather limited range of the control parameters, i.e. f⁺ ≤ 0.56 and A⁺ ≤ 2.

Figure 17. Dependence of ${\delta _{{\tau _w}}}/\varPi $ on parameter A ⁺³/f ⁺. The solid and dashed curves correspond to the least-squares fittings to the experimental data, i.e. (6.4) and (6.5), respectively. The circled points are the critical points predicted by (6.5).

Table 2. Coefficients in (6.5).

Some interesting inferences can be made from (6.4) and (6.5). Firstly, recast (6.4a) as ${\delta _{{\tau _w}}} ={-} 0.51\varPi \times ({A^{ + 3}}/f{ ^ + }){\; ^{0.33}}$. Given a fixed A ⁺³/f ⁺, ${\delta _{{\tau _w}}}$ is then directly proportional to Π, suggesting that Π can be understood to be an indicator of the control efficiency. Obviously, as a function of f ⁺, Π increases monotonically with f ⁺, that is, a larger DR can be achieved given a higher f ⁺ under a fixed input energy per pulse. For example, given A ⁺³/f ⁺ = 6.4, ${\delta _{{\tau _w}}}$ is −0.7 for f ⁺ = 0.42, while ${\delta _{{\tau _w}}}$ is −0.6 for f ⁺ = 0.14. Secondly, ${\delta _{{\tau _w}}}/\varPi $ declines parabolically and sharply for A ⁺³/f ⁺ < 24, suggesting a rapid increase in the control efficiency given a small input energy per pulse. In contrast, Bai et al. (Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014) found a linear relation between DR and input energy before reaching a critical point, which is not surprising in view of very different actuators deployed in the two investigations. Nevertheless, beyond the critical point where A ⁺³/f ⁺ = 24, ${\delta _{{\tau _w}}}/\varPi $ rises very slowly and almost linearly, as the input energy per pulse increases further, that is, exceeding a certain level, a further increase in the input energy per pulse affects the control efficiency adversely. Obviously, the higher the input energy per pulse, the larger the blowing penetrates. As the DR depends strongly on the penetration depth, as discussed previously, it is reasonable to infer that the critical level of A ⁺³/f ⁺ is linked to a critical penetration depth. Thirdly, the critical A ⁺ for each specified f ⁺ may be predicted via (6.5). For a given f ⁺, we set $(\partial {\delta _{{\tau _w}}}/\partial {A^ + }){|_{{f^ + }}} = \; 0$ to obtain A ⁺ by using a graphical method (Castillo Reference Castillo1988). For example, given f ⁺ = 0.028, 0.07 and 0.14, the solutions for A⁺ are 0.99, 1.35 and 1.70, respectively, very close to those measured from experiments, cf. those circled points where the gradient of ${\delta _{{\tau _w}}}$varies markedly, as marked by the circles in figure 6(a) and the corresponding ones (also circled) in figure 17. Fourthly, the data points of f ⁺ ≥ 0.28 all populate on the left side of the critical point (A ⁺³/f ⁺ = 24), where ${\delta _{{\tau _w}}}/\varPi $ drops sharply, in figure 17. That is, the control efficiency rises rapidly with increasing input energy per pulse given f ⁺ ≥ 0.28. This is in distinct contrast with Bai et al. (Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014) where ${\delta _{{\tau _w}}}/\varPi $ or the control efficiency does not go up so quickly with increasing input energy (their figure 23). This is not unexpected. Before reaching the critical point, both mechanisms, i.e. zero-streamwise-momentum fluid and externally generated streamwise vortices (§ 6.4), work together presently in favour of DR, leading to a high control efficiency. However, the first mechanism is absent in Bai et al. (Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014). Finally, beyond the critical point, the data are characterized by small f ⁺ (<0.28) and relatively large A ⁺ such as A ⁺ > 0.85 at f ⁺ = 0.028 and the DR due to the second mechanism diminishes rapidly with increasing input energy, as shown in Bai et al. (Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014), which accounts for the present slow rise in ${\delta _{{\tau _w}}}/\varPi $.

Multiplying Π on both left and right sides of (6.5) yields

(6.6)\begin{align}{\delta _{{\tau _w}}} = \left\{ {{a_1}exp \left[ { - {{\left( {\frac{{{{({A^{ + 3}}/{f^ + })}^{0.33}} - {b_1}}}{{{c_1}}}} \right)}^2}} \right] + {a_2}exp \left[ { - {{\left( {\frac{{{{({A^{ + 3}}/{f^ + })}^{0.33}} - {b_2}}}{{{c_2}}}} \right)}^2}} \right]} \right\}\varPi .\end{align}

Then, defining the right side of (6.6) as −ξ, we can achieve a linear function ${\delta _{{\tau _w}}} ={-} \xi $, as shown in figure 18. With this transformation, ${\delta _{{\tau _w}}}$ can be linearly related to a single parameter, which is a combination of both A⁺ and f⁺. It can be found that ${\delta _{{\tau _w}}}$ declines gradually with increasing ξ. The scaling of DR in figure 18 works for almost every data point in present study.

Figure 18. Dependence of ${\delta _{{\tau _w}}}$ on the scaling factor ξ. The solid line is a least-square fit to the data. The circled points are the critical points predicted by (6.5).

6.6. Net energy saving

The efficiency is an important aspect for the proposed DR technique. One would definitely like to know whether a net energy saving is possible and, if yes, under what conditions. Following Kametani & Fukagata (Reference Kametani and Fukagata2011), the net energy saving rate S is defined as

(6.7)\begin{equation}S = \frac{{{P_{{\tau _{w,0}}}} - ({P_{{\tau _{w,ctr}}}} + {w_{in}})}}{{{P_{{\tau _{w,0}}}}}},\end{equation}

where ${P_{{\tau _{w,0}}}}$ and ${P_{{\tau _{w,ctr}}}}$ are the powers that are needed to drive the flow in the absence and presence of control, respectively, and may be calculated as

(6.8)\begin{gather}{P_{{\tau _{w,0}}}} = \int_{{x_1}}^{{x_2}} {\int_{{z_1}}^{{z_2}} {\overline {{\tau _{w,0}}} \,\textrm{d}z\,\textrm{d}x{U_\infty }} } ,\end{gather}

(6.9)\begin{gather}{P_{{\tau _{w,ctr}}}} = \int_{{x_1}}^{{x_2}} {\int_{{z_1}}^{{z_2}} {\overline {{\tau _{w,ctr}}} \,\textrm{d}z\,\textrm{d}x{U_\infty }} } ,\end{gather}

where $\overline {{\tau _{w,0}}} $ and $\overline {{\tau _{w,ctr}}} $ are the time-averaged local WSSs without and with control, respectively, z ₁ = −8.25 mm $(z_1^ += 55)$ and z ₂ = 8.25 mm $(z_2^ += 55)$ correspond to spanwise extent of the streamwise slit array, and x ₁ = 5 mm $(x_1^ += 33)$ and x ₂ = 225 mm $(x_2^ += 1500)$, are the first hot-wire measurement position and the location where the overshoot vanishes in figure 7, respectively. Note that the region over the slit array is not considered in calculation. Following Stroh et al. (Reference SpalartReference Stroh, Frohnapfel, Schlatter and Hasegawa2015), the energy input w_in in (6.7) is given by

(6.10)\begin{equation}{w_{in}} = n\int_0^{{L_z}} {\int_0^{{L_x}} {[|\overline {0.5\rho U_{out}^3} |+ |\overline {\Delta p{U_{out}}} |]\,\textrm{d}x\,\textrm{d}z,} }\end{equation}

where U_out is the jet velocity through the slits, n is the number of the slits, L_z = 0.5 mm and L_x = 20 mm are the width and length of a single slit, respectively. The Δp_w is the pressure difference as flow is blown from the contraction chamber into the TBL, which is measured through pressure taps at contraction chamber wall and x = 5 mm (x⁺ = 33) on the flat plate (§ 2.3). Note that U_out is measured only at the centreline of the slit. The distribution of U_out along the cross section (z direction) of the slit is assumed as a parabolic profile following New, Lim & Luo (Reference New, Lim and Luo2006), given by

(6.11)\begin{equation}{U_{out}}/{U_{out,c}} = 1 - {(2z/{L_z})^2},\end{equation}

The net energy saving is achieved when S > 0, that is, the energy saved is larger than that of input.

The dependence of S on A ⁺ is qualitatively the same for different f⁺ (figure 19). The value of S rises with increasing A ⁺ at first and then drops rapidly when A ⁺ exceeds a critical value. For a given A⁺, S is always large for high f⁺, which is consistent with the fact that the DR is larger at high f⁺ than at low f⁺. Equation (6.7) indicates a dependence of S on both DR and the energy input, both connected to A⁺. For small A⁺, the DR rises rapidly with increasing A⁺ (figure 6a), and S grows gradually. At large A⁺, on the other hand, the rise in DR slows down significantly but the corresponding w_in grows more rapidly. Please refer to (6.10). As such, S drops quickly with further increasing A⁺. Note that the positive S is always achievable given a reasonably small A⁺, irrespective of f⁺. The maximum S of 4.01 % occurs at A⁺ = 0.86 for f⁺ = 0.42, corresponding to ${\delta _{{\tau _w}}} ={-} 49\,\%$ at x⁺ = 33 (figure 6a). From a practical point of view, both positive net energy saving rate and large DR are of great importance. That is, the choice of A⁺ should not be too small to ensure a large DR but cannot be too large to achieve a net energy saving. Given A⁺ = 1.42 for f⁺ = 0.42, the local DR is as high as 70 % and the corresponding net energy saving rate is 2.21 %. Note that the small S results from the fact that S is the spatially averaged results over a large streamwise distance (x⁺ = 33–1500). The energy gain G, defined as the ratio between the energy saved and the energy input, is presented in figure 20. Although S is less than 4 % for the range of A⁺ and f⁺ examined, the maximum G is larger than 10, indicating that the energy saved could be much larger than the energy input.

Figure 19. (a) Dependence of the net energy saving rate S on A⁺ at different f⁺. (b) Zoom-in view of S in the red rectangular region of (a).

Figure 20. Dependence of the energy gain G, defined as the ratio between the energy saved and energy input, on A⁺ at different f⁺.

6.7. Comparison with uniform blowing

Some remarks are due on comparison between periodic and uniform blowings. Firstly, two mechanisms are responsible for DR in case of present periodic blowing, i.e. the injection of zero-streamwise-momentum fluid and the interruption of the near-wall turbulence regeneration cycle. The latter is not directly responsible for DR in case of uniform blowing and it is this mechanism that allowed Bai et al. (Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014) to achieve a local DR of 50 %. These authors deployed one array of piezo-ceramic actuators to produce wall-normal oscillations which formed a spanwise-travelling wave. Yet, this mechanism leads presently to a significantly smaller DR, about 16 % given (A⁺, f⁺) = (1.42, 0.42). This is not surprising. The injection of zero-streamwise-momentum fluid acts to lift up the near-wall coherent structures (Park & Choi Reference Park and Choi1999), resulting in weakened interaction between the periodic-jet-induced streamwise vortices and the natural near-wall turbulent structures, compared with Bai et al.'s case. Secondly, periodic blowing achieves a local DR beyond 70 %, downstream of actuation, which is comparable to what uniform blowing may achieve (e. g. Kametani & Fukagata Reference Kametani and Fukagata2011). It could be inferred from the DR recovery (figure 7) that DR over the slits should be 70 % or more. Thirdly, periodic blowing may improve the control efficiency. As shown in scaling analysis (figure 17), the control efficiency grows with increasing input energy per pulse (A ⁺³/f ⁺) and then drops after reaching the maximum at A ⁺³/f ⁺ ≈ 24 at f⁺ = 0.007–0.14. It is worth mentioning that DR by periodic blowing at higher frequencies (f⁺ ≥ 0.28) behaves essentially like steady blowing (figure 6a) or uniform blowing where DR shoots towards 100 % with increasing A⁺ (Park & Choi Reference Park and Choi1999). Note that the minimum ${\delta _{{\tau _w}}}/\varPi$ at f⁺ = 0.007–0.14 is appreciably lower than at f⁺ ≥ 0.28, suggesting that periodic blowing can achieve a higher control efficiency than steady or uniform blowing. Finally, blowing through one array of slits is practically implementable, while uniform blowing through the wall is extremely difficult, if not impossible, to implement.