Turbulent boundary layer under the control of different schemes

Abstract

This work explores experimentally the control of a turbulent boundary layer over a flat plate based on wall perturbation generated by piezo-ceramic actuators. Different schemes are investigated, including the feed-forward, the feedback, and the combined feed-forward and feedback strategies, with a view to suppressing the near-wall high-speed events and hence reducing skin friction drag. While the strategies may achieve a local maximum drag reduction slightly less than their counterpart of the open-loop control, the corresponding duty cycles are substantially reduced when compared with that of the open-loop control. The results suggest a good potential to cut down the input energy under these control strategies. The fluctuating velocity, spectra, Taylor microscale and mean energy dissipation are measured across the boundary layer with and without control and, based on the measurements, the flow mechanism behind the control is proposed.

1. Introduction

Active control of skin friction drag reduction represents a fascinating challenge to the community of experimental fluid dynamics due to its great importance and direct practical significance in many fields of engineering. Advances have been made over past decades in understanding the near-wall coherent structures and developing predictive methods for turbulent phenomena (e.g. [1–4]), which are crucial for the design of active control schemes to inhibit the turbulent fluctuations and to reduce skin friction drag. The coherent structures in a turbulent boundary layer play an important role in the transport of mass, heat and momentum [5], among which the quasi-streamwise vortices (QSVs) deserve special attention, which make a predominant contribution to the production of the near-wall Reynolds shear stress (e.g. [1,6]). The QSVs are centred roughly at 10 ∼ 50 wall units from the wall in the buffer region, with their typical diameter in the order of 20 ∼ 50 wall units [7,8]. These structures are responsible for inducing relatively high- and low-speed streaks via the sweep and ejection events, respectively, which act to enhance the momentum transport in the near-wall region. The high-speed streaks or events near the wall account largely for a significant increase in the skin friction drag (e.g. [9–11]). Naturally, the skin friction drag may be reduced through suppressing the near-wall QSVs or high-speed events. Du & Karniadakis [12] and Du et al. [13] investigated the effectiveness of a transverse travelling wave, induced by a spanwise force, on the control of a turbulent boundary layer based on direct numerical simulation (DNS) data and observed an almost complete elimination of the high-speed streaks and hence a drag reduction exceeding 50%. Kim & Sung [14] and Xu et al. [15] managed numerically to weaken high-speed streaks and hence to reduce the skin friction drag by as much as 70% by means of unsteady blowing and a localized steady force, respectively. Recently, Bai et al. [16] developed an open-loop control technique for skin friction drag reduction in a fully developed turbulent boundary layer over a flat plate, where a spanwise-aligned array of 12–16 piezo-ceramic (PZT) actuators was deployed to generate a local surface oscillation. A maximum drag reduction by 50% was achieved locally. Correspondingly, the frequency, intensity and duration of bursts were all reduced, along with less energetic large-scale coherent structures and weakened high-momentum sweep motions. However, their control efficiency is very low. One naturally asks a question whether this efficiency can be increased if a closed-loop control scheme is deployed.

Turbulence control can be significantly more effective given a closed-loop control [17]. This was confirmed both numerically and experimentally (e.g. [18,19]). Control action must interact with turbulent fluctuations in a boundary layer, and the random aspects of the fluctuations reduce the effectiveness of an open-loop control. Various linear closed-loop control schemes such as opposition control, phasor control, linear-quadratic regulator and model predictive controller strategies have been numerically developed [20–23]. Based on DNS, Choi et al. [11] imposed blowing and suction at the wall in opposition to the normal velocity component, detected by a sensor placed at 10 wall units above the wall, and suppressed effectively the QSVs, resulting in a drag reduction of up to 25%. Lee et al. [18] argued that this active opposition control strategy was unpractical because the required velocity component above the actuators was in general unavailable and proposed a suboptimal control strategy based solely on wall pressure, achieving numerically a local drag reduction of 22% again based on blowing and suction at the wall. Applying DNS to a turbulent channel flow, Endo et al. [24] deployed a feed-forward control scheme, in which the stress sensors were placed at 12.3 wall units upstream of the deformable wall in order to detect incoming turbulence structures, to attenuate selectively the QSVs and the low-speed meandering streaks. A reduction in skin friction drag by 10% was obtained. Attempts have been made to develop a nonlinear control strategy because linear models do not capture the nonlinear ‘cascade' of energy over a range of length and time scales [5,20]. Hill [25] included a nonlinear term, based on Taylor series expansion, in his optimal control algorism and a feedback scheme, developed partly from physical intuition or experience, achieved numerically a skin friction drag reduction by 15% using blowing/suction at the wall. Lee et al. [26] manipulated the QSVs using blowing/suction in a low Reynolds number turbulent channel flow and developed numerically an adaptive controller based on a neural network, resulting in a skin friction drag reduction by 20%. It is of interest to note that the control performance of the nonlinear control strategies does not outperform the linear. The nonlinear simplified control strategies suffer from inherent limitations in capturing the characteristics or dynamics of the near-wall coherent motions [2,4], which may have reduced their ability to suppress turbulent fluctuations and hence to reduce the skin friction drag.

The experimental demonstration of the closed-loop control of wall turbulence proves to be highly challenging; few attempts have been made physically or experimentally for friction drag reduction in a turbulent boundary layer, perhaps due to difficulties in measuring reliably wall shear stresses and implementing practically complicated control systems. Based on PZT actuators, Jacobson & Reynolds [27] employed a combined feed-forward and feedback control scheme to inhibit the unsteady low/high-speed streaks in a laminar boundary layer based on steady and unsteady actuation created by an array of spanwise suction holes. They reduced drag by 10% and the RMS value of wall shear stress by 80%. Using a spanwise resonant synthetic jet, Rathnasingham et al. [28] developed a combined feed-forward and feedback linear control system and obtained a maximum reduction of 30% in the streamwise velocity fluctuation and 7% in drag at 20 wall units downstream of the actuator array in a turbulent boundary layer. Following Choi et al.'s [11] opposition control, Rebbeck & Choi [29,30] performed a feed-forward control by placing a wall-mounted hot-wire sensor upstream of a pulsed wall-normal jet. The downwash of high-speed fluid during sweep events was successfully suppressed in the near-wall region. In spite of all the efforts, no drag reduction has been physically demonstrated in a turbulent boundary layer under closed-loop control [2].

This work is a continuation of Bai et al.'s [16] investigation and is performed with three objectives. Firstly, develop experimentally different control schemes for the friction drag reduction in a turbulent boundary layer. Secondly, compare the effectiveness of the schemes and develop understanding of flow physics behind the possible differences. Finally, determine the extent to which the control efficiency can be improved on the basis of the open-loop control. Experimental details such as the arrangement for the wall shear sensors and actuators are provided in §2. The control performances for different control schemes are presented in §3–5. The work is concluded in §6.

2. Experimental details

(a). Generation of the turbulent boundary layer

Experiments were conducted in a closed-circuit wind tunnel with a 5.6 m long test section of 0.8 m in width and 1.0 m in height at Shenzhen Campus of Harbin Institute of Technology. A 4.8 m long flat plate with 0.78 m in width and 0.015 m in thickness, rounded at the leading edge, was placed vertically in the test section, slightly inclined with respect to flow and confirmed to be zero in the longitudinal pressure gradient. The flow passage between the flat plate surface and the wind tunnel wall was 0.6 m as shown in figure 1. Flow separation at the leading edge was minimized by tuning the pitch angle of a 0.2 m long tail-end board [31]. The flow was tripped at 0.1 m downstream of the leading edge by two spanwise arrays of screws separated longitudinally by 15 mm and stagger-mounted, as is in Bai et al. [16]. Each screw, 15 mm in height, was separated from its adjacent ones by 15 mm in the spanwise direction.

Figure 1.

Schematic of experimental arrangement (all dimensions given in millimetre).

Measurements were conducted at a free-stream velocity of U_∞ = 2.4 m s⁻¹ and the corresponding free-stream longitudinal turbulence level is 0.4%. The characteristic parameters of the tripped turbulent boundary layer at the test position of 3.2 m downstream of the leading edge of the plate (figure 1) are given in table 1, including the boundary layer disturbance thickness δ₉₉, momentum thickness θ, shape factor H₁₂, momentum-thickness-based Reynolds number Re_θ, averaged wall shear stress ${\overline{\tau }}_{\mathrm{w}}$ (overbar denotes time averaging in this paper), friction velocity $u_{\tau }=\sqrt{{\overline{\tau }}_{\mathrm{w}}/\rho }$ (ρ is the fluid density), length and time characteristic scales ${\delta }_v=\nu /u_{\tau }$ (v is kinematic viscosity) and $t_{\tau }=\nu /u_{\tau }^2$.

Table 1.

Characteristic parameters of the screws-tripped turbulent boundary layer.

U∞ (m s−1)	δ99 (mm)	θ (mm)	H12	Reθ	${\overline{\tau }}_{\mathrm{w}}$(N m−2)	uτ (m s−1)	δv (mm)	tτ (s)
2.4	85	9.2	1.41	1450	0.0134	0.105	0.15	1.42 × 10−3

(b). Actuators

Bai et al. [16] noted that the flapping motion of the PZT actuators array may produce almost as much drag reduction as the transverse wave under the optimum control parameters. Yang [32] investigated the local drag reduction in a turbulent boundary layer under the flapping motion of different PZT actuators, i.e. 8, 6, 4 and 2, and found that the number of actuators has a negligible effect on the drag reduction. As such, we deployed only two PZT actuators, as marked by ‘a₁' and ‘a₂' in figure 1, because this work is focused on the comparison between the control performances of different schemes. Therefore, sensing and the drag reduction observed are local to the point of measurement. The two PZT actuators, each with a dimension of length × width × thickness = 21.5 × 2.0 × 0.33 mm and flush mounted with the plate surface, were presently deployed and placed 3.2 m downstream of the leading edge of the plate (figure 1). The actuators were cantilever-supported on a substrate and were driven by in-phased sinusoidal signals so that their 19.5 mm long active parts generated a flapping motion (figure 1). The average spanwise and longitudinal separations between the near-wall low-speed streaks are in the order of 100 and 1000 wall units, respectively [33]. The spanwise separation of the actuators was 1 mm (figure 1) or 6.7 wall units and the gap between the wall edge and the actuator was nominally 0.025 mm or 0.17 wall unit. The present actuation system was made largely the same as that in Bai et al. [16]. However, compared with Bai et al.'s [16] experiments, there are two differences. Firstly, the length of the present PZT actuators is 0.5 mm shorter. Secondly, the gap between the wall edge and the actuator is 0.025 mm smaller. The peak-to-peak oscillating amplitude $A_{\mathrm{o}}^{+}$ at the actuator tip and frequency $f_{\mathrm{o}}^{+}$ for each actuator were 0.81 ∼ 3.76 and 0.14 ∼ 0.71, respectively, which may be adjusted by changing the driving voltage magnitude and frequency of the sinusoidal wave via a dSPACE control system. The two actuators are controlled synchronously by a single control signal. In this paper, superscript ‘+’ denotes normalization by δ_v, u_τ and/or t_τ. The coordinate system (x, y, z) is defined in figure 1. A maximum tip displacement of less than four wall units for PZT actuator is considered to be aerodynamically smooth at this low Reynolds number flow [16].

(c). Detecting and monitoring the wall shear stress

Following Rebbeck & Choi [29] and Blackwelder & Kaplan [34], we deploy two fixed wall wires, made of a tungsten wire, to capture the streamwise velocity and hence approximately the wall shear stress. One wall wire or detecting sensor was placed 6.2 mm or 41.6 wall units upstream of the actuators to detect the incoming large events of τ_w in the boundary layer. The sensing element of 5 µm in diameter and 1.2 mm in length was placed normally to the mean flow, 0.5 mm or 3.4 wall units away from the wall, as shown in figure 1, and is operated on a constant-temperature circuit (CTA, Dantec Streamline) with a low overheat ratio of 1.5 in order to minimize the wall thermal effect [35]. Another wall wire or monitoring sensor was placed at x⁺ = 14, chosen rather arbitrarily, downstream of the actuators and y⁺ = 3.4 in the viscous sublayer. The output signal U_m from this sensor, approximately proportional to the wall shear stress τ_w based on Newton's law of viscosity, is used to estimate or monitor τ_w.

A single movable calibrated hot-wire was mounted on a computer-controlled three-dimensional traversing mechanism with a spatial resolution of 1 µm, which was used for calibrating the two wall wires. The wall wires were calibrated following the technique proposed by Blackwelder & Kaplan [34]. Briefly, the velocity profile in the near-wall region may be obtained using the calibrated hot-wire. The averaged streamwise velocity at 0.5 mm above the wall is then determined for a number of U_∞ between 0 ∼ 7.5 m s⁻¹, which establishes the relationship between the instantaneous streamwise velocitiy U_d, measured by the detecting sensor, or U_m and the averaged voltage from the wall wire.

It is important to determine the longitudinal separation L⁺ between the wall sensors (figure 1). The near-wall coherent structures may experience the process of formation, evolution and decay, and they interact vigorously with each other from the detecting to monitoring sensor [5]. As such, two factors are considered. Firstly, this separation must be larger than the actuator length, i.e. 144 wall units. Secondly, the variation in the coherent structures through this separation should be adequately small. To ensure this, experiments were conducted to determine the dependence of the two-point correlation function on the separation between these two wall wires, one of which was replaced by a movable wire supported on the three-dimensional traversing mechanism. The averaged convention time τ_c of the near-wall coherent structures from the detecting to monitoring sensor may be estimated from the cross-correlation function (e.g. [36,37]), viz.

$${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}}}(\tau )=\frac{\overline{u_{\mathrm{m}}(t)u_{\mathrm{d}}(t\pm \tau )}}{u_{\mathrm{m},\mathrm{RMS}}{u}_{\mathrm{d},\mathrm{RMS}}},$$ 2.1

where u_m and u_d are the fluctuating components of U_m and U_d, respectively, and subscript ‘RMS’ represents their RMS values. The τ_c corresponds to the maximum ${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}}}$, as illustrated in figure 2. The ${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}}}$ exhibits apparently more spread and less symmetry about its maximum with increasing L⁺, indicating a variation in the convection velocity of the near-wall turbulent structures. Furthermore, the maximum ${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}}}$ or ${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}},\max }$ diminishes, initially rather rapidly, for increasing L⁺ and then appears levelling off for L⁺ > 700 (figure 3a). The decaying ${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}},\max }$ is consistent with previous reports that the coherent structures typically increase in volume due to diffusion of vorticity and their peak vorticity decrease (e.g. [5,38]). The ${\tau }_{\mathrm{c}}^{+}$ and L⁺ are almost perfectly linearly correlated up to L⁺ = 600 (figure 3b), at which ${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}},\max }$ drops to 0.23; the departure from the linearity is appreciable only beyond L⁺ = 600. Johansson et al. [38] noted that the shear layers, i.e. the near-wall organized and deterministic flow structures including wall streaky structures, streamwise vortices etc. could be identified over considerably large distances. Their mean survival time of the shear layers was approximately 50 time units, though some strong shear layers could be followed throughout the entire extent of the time studied (138 time units). Based on their DNS data in a turbulent channel flow, Quadrio & Luchini [39] further suggested that the linearity is indeed associated with a characteristic velocity or the convection velocity $U_{\mathrm{c}}^{+}$ of the most dominant flow structures travelling in the near-wall region (e.g. [40]). Taking $U_{\mathrm{c}}^{+}=11.5$, as estimated from the present experimental data (to be shown later), we find that L⁺ = 600 corresponds to about 52 time units, close to the mean survival time of Johansson et al.'s [38] shear layers. That is, the linearity shown in figure 3b is connected to the average lifespan of the near-wall organized structures. Apparently, given L⁺ < 600, most of passing coherent structures may be captured by both sensors as their major characteristics are largely unchanged, at least not to a great extent.

Figure 2.

Dependence of ${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}}}(\tau )$ on τ in the absence of actuation. U_∞ = 2.4 m s⁻¹.

Figure 3.

(a) Dependence of ${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}},\max }$ on L⁺. (b) Correlation between ${\tau }_{\mathrm{c}}^{+}$ and L⁺. U_∞ = 2.4 m s⁻¹.

L⁺ = 200 is chosen to be the separation between the wall sensors, which corresponds to a reasonably large ${\rho }_{u_{\mathrm{m}}{u}_{\mathrm{d}},\max }(=0.60)$ and τ_c = 24.8 ms, near one-third of the average lifespan of the near-wall organized structures. In the linear region, ${\overline{U}}_{\mathrm{c}}^{+}$ may be estimated to be L/τ_cu_τ = 11.5, within the range of previous reports by, e.g. Krogstad et al. [37] (${\overline{U}}_{\mathrm{c}}^{+}\hspace{0.17em}=13$) and Rathnasingham & Breuer [28] (${\overline{U}}_{\mathrm{c}}^{+}=10.7$). It is desirable from the most effective control point of view that L⁺ should be small enough to ensure a negligible change in the turbulent structures. Note that the upstream wall wire is 6.2 mm or 41.6 wall units from the upstream end of the actuators, determined mainly from the needs of fabrication, that is, there should be adequate separation between the support base of the actuators and the electrical leads of the wall wire (figure 1).

The sampled frequency and duration of U_m and U_d are 2500 Hz and 40 s, respectively, which are adequately large for the converged second-order quantities of the signals, the ensuing experimental uncertainty being less than 2%. U_d and U_m were low-pass-filtered at a cut-off frequency of 1000 Hz and then sent to a real-time control system (dSPACE), which is capable of rapid control prototyping, production code generation and hardware-in-the-loop execution. The output of the controller is then amplified by a dual channel piezo driver amplifier (Trek PZD 700) to activate the PZT actuators. The dSPACE system is compatible with SIMULINK module of Matlab and software of ControlDesk 4.3, where parameters for control schemes can be conveniently tuned online, such as the frequency and amplitude for the sinusoidal wave. The different control schemes are implemented in the dSPACE system. $A_{\mathrm{o}}^{+}$ and $f_{\mathrm{o}}^{+}$ were online-monitored using a Polytec OFV505/5000 laser vibrometer.

3. Physics-based control schemes

(a). Controller design

The present control scheme is the physics-based control, which in essence requires a rather thorough understanding of the underlying flow physics. The performance of physics-based control schemes highly depends on the choice of both physical quantities measured and the control parameters. We have previously developed the physics-based control schemes for jet mixing [41] and for suppressing vortex shedding and vibration of a square cylinder in crossflow [19]. These schemes showed excellent performances. The physics-based controller does not really need any system model because the information (e.g. the excitation frequency and driving voltage) used to drive the actuators is directly based on the flow information (e.g. the predominant frequency and fluctuating velocity magnitude) via say the feedback signal. In general, the actuators are triggered/driven once the targeted event (e.g. a quasi-periodic large-scale organized structure in [19,41]) exceeds a specified level in the physics-based control schemes and, therefore, the actuator transfer function is not needed. In a turbulent boundary layer, the near-wall high-speed events are highly correlated with the significant rise in skin friction drag [9,10]. Therefore, the present control schemes are designed to trigger the actuation once the near-wall high-speed events are detected at the actuator location, with a view to suppressing large skin friction drag. A similar strategy was deployed by Jacobson & Reynolds [27], who inhibited the high-speed streaks created by an array of spanwise suction holes in a laminar boundary layer using PZT actuators. While their laminar boundary layer is characterized by highly repeatable or quasi-periodical events, the present turbulent boundary layer is associated with much more random (not quasi-periodical) coherent structures, that is, it is much more challenging when implementing the physics-based control schemes.

Figure 4 schematically presents the physics-based controller design for the three different schemes, i.e. the feed-forward control, the feedback control, and the combined feed-forward and feedback control. Ideally, the actuators should operate only when the high-speed events just reach the actuator tip. However, it is difficult to detect accurately the events at the actuator tip because the flow near the actuators is influenced by actuation. One way to overcome this problem is to use the upstream streamwise velocity signal to predict the downstream velocity signal. Therefore, a single-input--single-output (SISO) prediction model G(s) is used to predict the streamwise velocity signals near the actuator tip from the detecting sensor signal U_d in the feed-forward control scheme, where c(t) = U_d and y(t) = U_m (t is the time) are the input and output signals, respectively. The details of determining G(s) will be provided later. Another way is to use the monitoring signal U_m to approximate the streamwise velocity near the actuator tip (figure 4). As such, a feedback control scheme is developed where the monitoring sensor has been mounted as closely as possible to the actuators so that U_m is used as the feedback signal. There is no prediction involved in the feedback scheme. The averaged longitudinal size for the high-speed streaks is about 1000 wall units given a low Reynolds number flow [1]. The separation between the monitoring sensor and the actuator tip is only 14 wall units, by far smaller than the high-speed streak length in the near-wall region, implying a minimum evolution of the turbulent structures from the actuators to the monitoring sensor. Therefore, the actuators may timely interact with the high-speed events detected by the monitoring sensor. This strategy is fully consistent with Belson et al.'s [3] suggestion that the feedback control can be effective only when the sensor is near the actuator. To achieve the maximum drag reduction efficiency, we further combine the feed-forward with the feedback control. As the two actuators are controlled synchronously by a single control signal, producing a flapping motion, the feed-forward control scheme is a SISO system. So is the feedback control scheme. However, the combined feed-forward and feedback control scheme is a multiple-input-single-output (MISO) system.

Figure 4.

Schematic diagram of physics-based control plant, including detecting sensor (c(t) or U_d), actuators (a₁, a₂) and downstream control point of monitoring sensor (y(t) or U_m). G(s) is a SISO transfer function for predicting the streamwise velocity signal near the actuator tip, and is determined using an off-line system identification method based on the analysis of one set of U_d and U_m signals. U_p represents the predicted signal from U_d. U_t is the triggering signal. Both a₁ and a₂ are triggered at U_t = 1 V when the near-wall high-speed events are detected at the actuator tip, and are not actuated at U_t = 0 V. (Online version in colour.)

(b). Sensing and forward prediction

The organized structures play an important role in the self-sustaining process of near-wall turbulence, which involves the formation of velocity streaks from the advection of the mean profile by streamwise vortices and the generation of the vortices from the instability of the streaks [42,43]. An autonomous cycle exists in the near-wall region which is independent of the outer flow [43,44]. An interrupt to any part of the self-sustaining process may lead to relaminarization or drag reduction and furthermore the key parts of this process are linear [4]. The development of the coherent structures from the detecting to monitoring sensor could be reflected by the velocity signals, i.e. U_d and U_m, captured from the two sensors. Here, a predetermined transfer function model G(s) is used to obtain U_p near the actuator tip based on U_d.

The development of the transfer function is the key to predict the downstream characteristics of the streamwise velocity fluctuation in a turbulent boundary layer [28]. Based on the U_d and U_m signals measured by the two wall wires in the absence of actuation, we may obtain Y(s) = G(s)C(s) + E(s), where Y(s), C(s) and E(s) represent Laplace transforms of y(t), c(t) and e(t) (e(t) is noise signal), respectively, and s is the transform operator. The G(s) we use in experiments contains two poles and two zeros, which provides us with an acceptable estimation, as will be seen later. The prediction error minimization (PEM) approach is adopted to evaluate the transfer function coefficients. In general, the evaluation algorithm performs two major tasks: parameter initialization and parameter update. The algorithm initializes the estimable parameters using the instrument variable (IV) method [45]. Nonlinear least squares with automatically chosen line search method is then performed to update the parameters, targeting minimizing the weighted prediction error norm. All the identification process mentioned above is realized using the Matlab system identification tool. Eventually, the coefficients of G(s) are determined based on a set of training data c(t) and y(t). Thus, the real-time uncontrolled flow at the actuator tip could be predicted based on this transfer function G(s) and instantaneous U_d.

U_d and U_m simultaneously sampled are used as the training data in the system identification, each record of U_d or U_m consisting of 5000 data points, corresponding to a duration of 2 s. For the purpose of verifying the developed G(s), another set of data are randomly selected from the detecting and monitoring sensor in the absence of actuation. Figure 5 compares signals $U_{\mathrm{m}}^{+}$ and $U_{\mathrm{p}}^{+}$, along with $U_{\mathrm{d}}^{+}$, all simultaneously obtained without actuation, where $U_{\mathrm{p}}^{+}$ denotes the predicted signal from U_m and G(s). The disparity between $U_{\mathrm{d}}^{+}$ and $U_{\mathrm{m}}^{+}$ or $U_{\mathrm{p}}^{+}$ is appreciable; nevertheless, $U_{\mathrm{m}}^{+}$ displays good similarity, both qualitatively and quantitatively, to $U_{\mathrm{p}}^{+}$. The results indicate that the model used is acceptable for the velocity prediction. Departures occur largely in small-scale events that represent the small-scale turbulent structures (e.g. [28]). These departures are not unexpected as the near-wall turbulent structures are nonlinear. Nevertheless, this result points unequivocally to the fact that the presently proposed linear control strategy predicts rather well the near-wall large-scale coherent structures. The goodness of fit between $U_{\mathrm{p}}^{+}$ and $U_{\mathrm{m}}^{+}$, which describes the discrepancy between observed events and those expected under the model of concern, can be determined by the normalized root mean squared error (RMSE) $\text{fit}=1-\parallel U_{\mathrm{m}}^{+}(t)-U_{\mathrm{p}}^{+}(t){\parallel }_{t=0}^{t=T}/\parallel U_{\mathrm{m}}^{+}(t)-{\overline{U_{\mathrm{m}}^{+}(t)}}_{t=0}^{t=T}{\parallel }_{t=0}^{t=T}$, where ||·|| and T denote the Euclidean norm of a vector (i.e. the length of a vector) and the chosen time period, respectively. The fit is a scalar value over a range of (−∞, 100%), where 100% means a perfect fit and −∞ for a poor fit. Several numbers on the poles (no more than five) and zeros (no more than four) have been examined to find a result such that these transfer functions produce a similar fit value approaching 30%. Therefore, a simple G(s) with two poles and two zeros are presently chosen. Additionally, we modify time delay T_c in G(s) slightly, from 13.8 to 22.2 ms (figure 12a), to account for the convection time of the organized structures from the detecting sensor to the actuator tip, with a view to optimizing the control performance.

Figure 5.

Comparison between the detecting, monitoring and predicted signals, $U_{\mathrm{d}}^{+}$ at (x⁺, y⁺, z⁺) = (−186, 3.4, 0), $U_{\mathrm{m}}^{+}$ at (x⁺, y⁺, z⁺) = (14, 3.4, 0) and $U_{\mathrm{p}}^{+}$ at (x⁺, y⁺, z⁺) = (0, 3.4, 0), where t = 0 s is chosen arbitrarily. U_∞ = 2.4 m s⁻¹.

Figure 12.

(a) Dependence of drag reduction ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ on the convection time T_c of the organized structures from the detecting sensor to the actuators. (b) Dependence of cost function Ø on the duty cycle d_c. $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$. U_∞ = 2.4 m s⁻¹.

(c). Implementation details

The flow chart of the feed-forward control is shown in figure 6a. Based on G(s) and U_d, the estimated U_p is used to determine the action of actuators. A real-time moving-average filter with a sampling window of 2 ms is applied on the captured signal U_p to eliminate the random noise. Once the moving-averaged signal $⟨U_{\mathrm{p}}⟩$ exceeds a threshold Th₁, indicating the detection of a high-speed event, the actuators are then triggered to oscillate with the same and specified control parameters of ($\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+}$) for the entire duration of this event. The Th₁ is given by $k_1{\tilde{U}}_{\mathrm{p}}$, where κ₁ is a constant and ${\tilde{U}}_{\mathrm{p}}$ is the short-time-averaged value of the last 2 s of U_p, updated every 3 s, which is adjusted to obtain a compromise between the duty cycle d_c and drag reduction ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}=({({\overline{\tau }}_{\mathrm{w}})}_{\mathrm{on}}-{({\overline{\tau }}_{\mathrm{w}})}_{\mathrm{off}})/{({\overline{\tau }}_{\mathrm{w}})}_{\mathrm{off}},$ where subscripts on and off denote measurements with and without control, respectively. The ${\overline{\tau }}_{\mathrm{w}}$ is given by $\mu \text{d}\overline{U}/dy{|}_{y=0}=\mu {\overline{U}}_{\mathrm{m}}/y_{\mathrm{m}}$, where $y_m^{+}$ is 3.4 (x⁺ = 14, z⁺ = 0) at which ${\overline{U}}_m$ is measured. By the way, the model G(s) determined from the uncontrolled data is still valid under actuation. The actuators mainly affect flow nearby and downstream and their effect is negligible on the upstream flow. Also, the actuators are only 14 wall units (2 mm) from the monitoring sensor. Therefore, the actuation has a negligible influence on the evolution of the near-wall high-speed events from the detecting sensor to the actuator.

Figure 6.

Flow chart for the feed-forward (a) and combined feed-forward and feedback control scheme (b).

Figure 6b schematically shows the flow chart of the feedback control and the combined feed-forward and feedback control scheme. Similarly to the feed-forward control scheme, the real-time captured signal U_m in the feedback control is filtered by a moving-average filter with a sampling window of 2 ms. The actuators are triggered to oscillate if the moving-averaged signal $⟨U_{\mathrm{m}}⟩$ exceeds a threshold $\text{T}{\text{h}}_2=k_2{\tilde{U}}_{\mathrm{m}}$, where the constant κ₂ and ${\tilde{U}}_{\mathrm{m}}$ are determined similarly to κ₁ and ${\tilde{U}}_{\mathrm{p}}$.

To achieve the maximum drag reduction efficiency, we combined the feed-forward with the feedback control to take advantage of both effective linear control algorithm, associated with the feed-forward component, and correction resulting from a threshold-triggering process offered by the feedback component. The combined system is designed such that the actuators are triggered to oscillate once $⟨U_{\mathrm{p}}⟩$ or $⟨U_{\mathrm{m}}⟩$ exceeds $\text{T}{\text{h}}_1=k_1{\overline{U}}_{\mathrm{p}}$ or $\text{T}{\text{h}}_2=k_2{\overline{U}}_{\mathrm{m}}$, respectively.

(d). Response of the actuators

The fast response of the actuators to U_t reflects the ability of actuation-induced flow to interact timely with the high-speed events and is crucial for achieving excellent performance. Figure 7 presents the dependence of the actuation amplitude $A_{\mathrm{o}}^{+}/2$ on U_t. The actuators are triggered to oscillate, with specified control parameters of $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$, at U_t = 1 V and stopped at U_t = 0 V. Note that $A_{\mathrm{o}}^{+}/2$ shows a small overshoot at the onset of U_t = 1 V due to the capacitive nature of this type of PZT actuators and may not come to zero immediately at U_t = 0 V, decreasing relatively slowly due to the inertia of the actuator itself. It takes approximately 4 ms for the actuator at a still state to reach the fully operated state (please refer to the response time in figure 7), which is much smaller than the average convection time of 24.8 ms for the predominant coherent structures to travel from the detecting to monitoring sensor. The result indicates that the feed-forward control may coordinate actuation-induced flow timely to suppress the detected high-speed events and hence to reduce the skin friction drag. However, the 4 ms response time of the actuator may adversely affect the performance of the feedback control scheme where the actuation is triggered by downstream signal U_m.

Figure 7.

Dependence of actuation in $A_{\mathrm{o}}^{+}/2$ on triggering signal U_t, $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$. (Online version in colour.)

It is worth mentioning that the stability is not an issue for the controller of the present physics-based control schemes where the actuators are triggered once the predicted or feedback signal reaches a specified level.

4. Control performance

Four control schemes are investigated and compared in terms of their effectiveness and efficiency, i.e. the open-loop, feed-forward, feedback, and the combined feed-forward and feedback controls.

(a). Open-loop control results

It is important to document the perturbation a single actuator may produce to the flow. This knowledge may facilitate our understanding of flow physics behind the control results. Figure 8 presents $\mathrm{\Delta }{\overline{U}}^{+}={\overline{U}}_{\mathrm{pe}}^{+}-{\overline{U}}^{+}$, where ${\overline{U}}_{\mathrm{pe}}^{+}$ and ${\overline{U}}^{+}$ are the mean streamwise velocities with and without the perturbation generated by one single actuator at the control parameters $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$, measured at U_∞ = 2.4 m s⁻¹ by a single hot-wire movable in the yz plane of x⁺ = 14 in the absence of the wall monitoring sensor. The examined area was located at y⁺ = 2 ∼ 30 and z⁺ = −25 ∼ 25, and the corresponding measurement increment was (Δy⁺, Δz⁺) = (1, 5). The oncoming flow is apparently appreciably disturbed around the actuator, and the most pronouncedly perturbed regions, where $\mathrm{\Delta }{\overline{U}}^{+}<-1$ is approximately centred at (y⁺, z⁺) = (15, ±7). The negative $\mathrm{\Delta }{\overline{U}}^{+}$ indicates the transport of low-speed fluid away from the wall. The $\mathrm{\Delta }{\overline{U}}^{+}$ is slightly positive in the region above the actuator and close to the wall. As expected, the results are very similar to the observation made by Bai et al. [16].

Figure 8.

Iso-contours of $\mathrm{\Delta }{\overline{U}}^{+}$ in the yz plane (x⁺ = 14) perturbed by one single actuator operated at $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$. The contour increment is Δ = 0.5. The actuator is symmetrical about z⁺ = 0, with its sides at z⁺ = ±7, as indicated by the solid rectangle along z⁺. U_∞ = 2.4 m s^−1.

The open-loop control serves as a baseline case in this study and is first performed following Bai et al. [16], who investigated the dependence of drag reduction ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ on $f_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+}$ and phase difference between two adjacent actuators. As they noted from their figure 6a, a high oscillation frequency is beneficial for drag reduction. However, the drag reduction would be levelled-off once $f_{\mathrm{o}}^{+}$ exceeds a critical value, about 0.4 ∼ 0.6 for $A_{\mathrm{o}}^{+}=1.11\sim 2.22$. As a matter of fact, as shown in their fig. 23, once ${(\hspace{0.17em}{f}_{\mathrm{o}}^{+}{A}_{\mathrm{o}}^{+})}^2$, which is physically proportional to the input energy of the produced transverse wave, exceeds about 0.8, the control efficiency may be adversely affected. Therefore, $f_{\mathrm{o}}^{+}$ is presently chosen to be 0.56 after trial-and-error tests. With $f_{\mathrm{o}}^{+}$ preset, we may determine the dependence of ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ on $A_{\mathrm{o}}^{+}$ experimentally.

As shown in figure 9, ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ gradually decreases as $A_{\mathrm{o}}^{+}$ increases from 0.81 to 3.22 and then moves up, which is qualitatively the same as Bai et al. [16] observed. However, their maximum drag reduction is more pronounced, reaching 50% at x⁺ = 17. The maximum local drag reduction is 30%, measured at x⁺ = 14 in our experiment. Furthermore, their maximum occurs at $A_{\mathrm{o}}^{+}\approx 2.0$, appreciably smaller than the present case ($A_{\mathrm{o}}^{+}=3.22$). The difference is not unexpected in view of different flow conditions, albeit slightly, and the actuation system (please refer to §2). It is worth pointing out that the disturbance to the flow or the penetration depth Λ is connected directly to the actuation-induced flow rather than the actuator displacement [16,27]. The optimum Λ should be comparable to the viscous sublayer thickness [13]. The difference in the actuation system may have a rather pronounced impact upon this disturbance flow or Λ and hence the control performance.

Figure 9.

Dependence of ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ on $A_{\mathrm{o}}^{+}(\hspace{0.17em}{f}_{\mathrm{o}}^{+}=0.56)$.The duty cycle is 100%, 50%, 50% and 62% for the open-loop, feed-forward, feedback and combined feed-forward and feedback control schemes, respectively.

It is worth pointing out that the rapid recovery in the friction drag is largely due to the three-dimensional effect. The friction drag does not fully recover until x⁺ ≈ 160 ([16]) when one array of 12 actuators/elements is deployed but is fully recovered presently at x⁺ = 74 (figure 10) when the array is reduced to two elements, whose drag-reduced area shrinks apparently due to the three-dimensional activities of the near-wall flow such as meandering streaky structures (e.g. fig. 10a in [16]). The averaged drag reduction over x⁺ = 14 ∼ 74 is about 14%, as calculated from figure 10.

Figure 10.

Dependence of ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ on x⁺ at z⁺ = 0 under the open-loop control ($f_{\mathrm{o}}^{+}=0.56$, $A_{\mathrm{o}}^{+}=3.22$). U_∞ = 2.4 m s⁻¹.

The control efficiency η_e can be estimated as η_e = P_saved/P_used, where P_saved and P_used are the power saved by drag reduction and consumed by PZT actuators, respectively. As suggested by Bai et al. [16], P_saved can be obtained by, $P_{\mathrm{saved}}=\rho u_{\tau }^2{U}_{\mathrm{\infty }}{L}_{\mathrm{c}}{\int }_{x_{\mathrm{o}}}^{x_{\mathrm{c}}}{\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}\text{d}x,$ where x_c − x_o is the streamwise extent of the affected area downstream of the actuators and L_c is the spanwise affected distance of the two actuators, i.e. L_c = 6 mm or $L_{\mathrm{c}}^{+}=40$, as illustrated in figures 1 and 8. P_saved is estimated to 2.66 × 10⁻⁷ W under the optimal control parameters of $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$. On the other hand, P_used consumed by PZT actuators can be represented as $P_{\mathrm{used}}=6.28\hspace{0.17em}N\hspace{0.17em}{f}_{\mathrm{o}}C\text{tan( }\delta )\hspace{0.17em}{V}_{\mathrm{o}}^2$ ([16]), where V_o is the driven voltage responsible for $A_{\mathrm{o}}^{+}$, C = 20 nF and tan(δ) = 1.7% are the capacitance and dissipation factor of the PZT material, respectively. N = 2 is the total number of actuators. Given f_o = 400 HZ and V_o = 17.2 V for the optimal control parameters, P_used can be obtained as 5.02 × 10⁻⁴ W. As such, η_e = 0.53 × 10⁻³, which is close to that calculated by Bai et al. [16]. Apparently, this is not an efficient control system. However, the present investigation is focused on exploring the extent to which the control efficiency can be improved by the physics-based control, when compared with the open-loop control.

It is worth commenting that there is no flow separation downstream of the actuator tip. Firstly, ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ moves up or its magnitude diminishes as $A_{\mathrm{o}}^{+}$ exceeds its optimum value of 3.22 (figure 9). If the observed drag reduction resulted from flow separation, the magnitude of ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ should continue to increase for larger $A_{\mathrm{o}}^{+}$. Secondly, figure 11 presents the distribution of ${\overline{U}}^{+}$ in the near-wall region with and without control. The linear region of the viscous sublayer is evident over y⁺ = 3 ∼ 6 in the absence of control. This linear region is extended to y⁺ ≈ 12 under control. Bai et al. [16] also noted an extension for the linear region up to y⁺ ≈ 14 under control for a skin friction drag reduction of 35% (x⁺ = 35). This result provides yet another strong evidence for the absence of a separation bubble behind the actuator, for the bubble, if present, would lead to a nonlinear ${\overline{U}}^{+}$ near the wall. The gradual increase in ${\overline{U}}^{+}$ (figure 11), when approaching the wall for y⁺ < 3, is due to the wall thermal effect. The same observation can be made in the absence of control when the actuators are flush with the surface, resulting in no wall blockage. Finally, the shape factors of 1.44 and 1.48 for the boundary layers with and without control, respectively, are below the critical value of 2.0 [46]. This result also indicates that the absence of local flow separation near the actuator tip.

Figure 11.

Distribution of the mean streamwise velocity (a) $\overline{U}/U_{\mathrm{\infty }}$ and (b)$\hspace{0.17em}{\overline{U}}^{+}$ in the near-wall region, measured using a hot-wire placed at (x⁺, z⁺) = (14, 0). $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$. (Online version in colour.)

(b). Feed-forward control results

The parameters $f_{\mathrm{o}}^{+}$ and $A_{\mathrm{o}}^{+}$ are chosen to be 0.56 and 3.22, respectively, where the maximum skin friction drag reduction is achieved for the open-loop control. As discussed earlier, T_c accounts for the convection time of the most dominant coherent structures travelling from the detecting sensor to the PZT actuator tip; its choice plays a significant role in achieving drag reduction and is mainly determined from the tilted near-wall velocity streaks [29] and the response time of actuator, as discussed extensively by Lundell [47] in developing a reactive control system for the control of a flat plate boundary layer. Given ${\overline{U}}_{\mathrm{c}}^{+}\hspace{0.17em}=11.5$, it takes 23.0 ms for the dominant coherent structures to travel from the detecting sensor to the tip of actuators. Given d_c = 50%, the dependence of ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ on T_c is presented in figure 12a. The optimal T_c occurs at 18.6 ms, implying that the actuators are triggered before the coherent structures of concern reach the actuator tip, and the corresponding ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ is −24%.

The difference between the optimal convection time T_c and the travelling time calculated from ${\overline{U}}_{\mathrm{c}}^{+}$ is ascribed to two main effects. Firstly, the near-wall high-speed streaks that occur at near-wall region are tilted due to the shear at an angle of about 4.7° to the wall [29] so that their upper parts are downstream of their lower parts, as shown by Lundell & Alfredsson [48]. Therefore, the centre of a streak, when detected, is already far downstream of the detecting sensor. Secondly, once a PZT actuator is switched on, it takes approximately a response time of 4 ms, directly contributing to a discrepancy between the optimal and expected convection time.

Define the cost function Ø = |d_c/δ_w|, which provides a measure for the input control energy (d_c × 5.02 × 10⁻⁴ W) as per unit energy saved from drag reduction. A decrease in Ø means achieving a larger drag reduction at the expense of less input control energy. Therefore, it is reasonable to use Ø as cost function in the context of the present study. Its minimum can be experimentally determined by varying d_c with $f_{\mathrm{o}}^{+}$ and $A_{\mathrm{o}}^{+}$ fixed, as shown in figure 12b, where d_c = 100% corresponds to the open-loop control scheme. The minimum Ø occurs at d_c = 50%, which is used for the feed-forward control scheme.

Note that the cost function is often defined as the sum of the spatial (temporal) mean input energy and energy consumed by the mean drag. In this investigation, however, the power P_drag ($\rho u_{\tau }^2{U}_{\mathrm{\infty }}{L}_{\mathrm{c}}\hspace{0.17em}(x_{\mathrm{c}}-x_{\mathrm{o}})$) consumed by the mean drag over x⁺ = 14 ∼ 74 is estimated to be 1.90 × 10⁻⁶ W in the absence of control. Under the optimal control parameters of the open-loop control d_c(=100%), P_drag is reduced to 1.63 × 10⁻⁶ W (§4a). Correspondingly, the power P_used, consumed by PZT actuators for achieving drag reduction, is given by $6.28\hspace{0.17em}N\hspace{0.17em}{f}_{\mathrm{o}}C\text{tan( }\delta )\hspace{0.17em}{V}_{\mathrm{o}}^2\times d_{\mathrm{c}}\hspace{0.17em}=5.02\hspace{0.17em}\times d_{\mathrm{c}}\times {10}^{-4}\hspace{0.17em}\text{W}$, where d_c = 0 ∼ 100%. Apparently, this is not an efficient control system where P_drag is (1.63 ∼ 1.90) × 10⁻⁶ W and P_used is (0 ∼ 5.02) × 10⁻⁴ W over d_c = 0 ∼ 100%. P_used is predominant in the sum of P_drag and P_used. However, the minimum of the sum of P_drag and P_used occurs when P_used = 0. Clearly, the often-used definition for the cost function cannot really differentiate the control efficiencies presently when different control schemes are deployed.

With T_c and d_c determined, the ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ achieved by the feed-forward control is investigated under different $A_{\mathrm{o}}^{+}$. The ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ gradually decreases from −5% to −24% as $A_{\mathrm{o}}^{+}$ increases from 0.81 to 3.22 and achieves the minimum of −24% at $A_{\mathrm{o}}^{+}=3.22$ (figure 9). The u_m,RMS of U_m, corresponding to the maximum drag reduction, is reduced from 0.156 to 0.120 m s⁻¹, a drop by 23%, 5% less than that (28%) of open-loop control. Nevertheless, the duty cycle drops to 50% and the minimum Ø is 2.1 at $A_{\mathrm{o}}^{+}=3.22$ (a 37% reduction compared with that of the open-loop control), indicating a substantial improvement in the control efficiency. That is, the skin friction drag has been reduced by suppressing selectively the high-speed events in the near-wall region [16], which are mainly associated with the downwash of quasi-longitudinal vortices that drive high-momentum fluid towards the wall. Similar changes for high-speed events in the near-wall turbulent structures are also experimentally observed by Rebbeck & Choi [29], who deployed an opposition control scheme based on a wall-normal actuation jet.

Figure 13 illustrates signals $U_{\mathrm{d}}^{+}$, $U_{\mathrm{m}}^{+}$ and $U_{\mathrm{p}}^{+}$ captured simultaneously, along with the triggering signal U_t and the oscillating amplitude $A_{\mathrm{o}}^{+}/2$ of the actuator tip measured by a laser vibrometer. Note that $U_{\mathrm{m}}^{+}$ shows more small-scale fluctuations than $U_{\mathrm{d}}^{+}$, which is ascribed to the additional disturbance produced by actuators. In response to rising U_t, $A_{\mathrm{o}}^{+}/2$ shoots up and the corresponding high-speed events in $U_{\mathrm{m}}^{+}$ are thus disrupted, the corresponding magnitude dipping, compared with $U_{\mathrm{d}}^{+}$, while the low-speed events are barely affected.

Figure 13.

Signals for the feed-forward control scheme, where t = 0 s is chosen arbitrarily. $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$. U_∞ = 2.4 m s⁻¹.

(c). Feedback control results

During the feedback control process, κ₂ is adjusted so that d_c is equal to 50% to facilitate the comparison in the control performance with the feed-forward control scheme. The variation in ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ with $A_{\mathrm{o}}^{+}$ (figure 9) is qualitatively the same as the feed-forward control scheme as well as the open-loop control. At $A_{\mathrm{o}}^{+}<2.15$, the feedback control scheme may achieve essentially the same drag reduction as the feed-forward control scheme (figure 9). However, the maximum drag reduction reaches only 20%, 4% less than the feed-forward control scheme, and the minimum Ø is 2.5. The u_m,RMS of U_m reduces from 0.156 to 0.136 m s⁻¹, a drop by 13%, 15% less than its counterpart (28%) in the open-loop control. As shown from $U_{\mathrm{m}}^{+}$ in figure 14a, the high-speed events, e.g. at t = 0.05 s and 0.20 s, may be greatly reduced in magnitude, compared to that in $U_{\mathrm{p}}^{+}$. However, this is not the case for all of the high-speed events. For example, the events at t = 0.44 s and 0.58 s do not seem to be affected to a significant extent. Two factors may be responsible for the less impressive performance than the feed-forward control. Firstly, signal U_m is used to trigger the actuators. As is evident in figure 14a, there are many more small-scale fluctuations in $U_{\mathrm{m}}^{+}$ than $U_{\mathrm{p}}^{+}$, which may originate from the influence of the actuation. These fluctuations may bring about erroneous triggering in U_t, which displays many more triggers of smaller intervals compared to its counterpart (figure 13) for the feed-forward control scheme. As a consequence, the actuators may not work effectively due to their inertia at the start-up and stopping processes, as discussed earlier. Secondly, the time lag, albeit very small, between actuation and the high-speed events makes the performance of the feedback control even worse. The separation of 14 wall units between the actuator tip and the monitoring sensor is responsible for a time delay of 1.7 ms, as calculated from ${\overline{U}}_{\mathrm{c}}$. Yet, the PZT actuators are characterized by a delay in response time by 4 ms (figure 7). Thus, the total time lag of the actuators is 5.7 ms. Nevertheless, the cost function Ø for the maximum drag reduction is 2.5, significantly less than that (3.3) in open-loop control, that is, the feedback control outperforms the open-loop control in efficiency. In the case of a turbulent boundary layer, it is evident from figure 9 that the prediction based on the upstream events in the feed-forward control outperforms that based on the downstream events in the feedback control.

Figure 14.

(a) Signals for the feedback control scheme. (b) Signals for the combined feed-forward and feedback control scheme. t = 0 s is chosen arbitrarily. $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$. U_∞ = 2.4 m s⁻¹.

(d). Combined feed-forward and feedback control results

During experiments, κ₁ is firstly adjusted to a value so that d_c is 50% and then κ₂ is determined experimentally to increase the drag reduction as large as possible, along with keeping a minimum value near 2.1 for cost function Ø, determined from the feed-forward control.

The ${\delta }_{{\hspace{0.17em}}_{{\tau }_{\mathrm{w}}}}$ varies again similarly to other control schemes and is 1 ∼ 4% lower than the feed-forward control over $A_{\mathrm{o}}^{+}=0.81\sim 3.76$ (figure 9). The maximum drag reduction reaches 28%, almost the same as the open-loop control scheme achieves. Compared with the uncontrolled case, u_m,RMS drops by 25%, 3% less than that of the open-loop control. As shown in $U_{\mathrm{m}}^{+}$ (figure 14b), the maximum magnitude of high-speed events at $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$, where the maximum drag reduction is achieved, may drop by 2.5. The duty cycle increases slightly to 62%, corresponding to a minimum Ø of 2.2. The results are encouraging, substantially exceeding the achieved drag reduction of other control systems previously reported, e.g. [27,28].

5. Physical aspects

(a). Change in the velocity spectra

The energy distribution of the fluctuating velocity in the frequency domain is well reflected by the weighted power spectral density function $f^{+}{E}_u^{+}$ of u_m. As shown in figure 15a, there is an appreciable decrease in $f^{+}{E}_u^{+}$ under control, measured at (x⁺, y⁺, z⁺) = (14, 3.4, 0), for f⁺ < 0.02 and meanwhile an increase for f⁺ > 0.03, irrespective of the control schemes. The change of the combined feed-forward and feedback control is largest, especially over relatively low frequencies, even exceeding the open-loop control. This is internally consistent with its lowest u_m,RMS in the control schemes. The observation indicates clearly a shift in energies from low to high frequencies under control, that is, a transfer in energy from large- to small-scale turbulence. This result is similar to Bai et al.'s [16] observation. It may be inferred that the connection between large-scale coherent structures and the wall are weakened or even broken down as a result from suppressing the high-speed or large wall shear stress event. The observation also conforms to a decrease in both intensity and duration of the bursts in the near-wall region under control [16].

Figure 15.

(a) Weighted spectra of the monitoring-sensor-measured fluctuating velocity u_m at (x⁺, y⁺, z⁺) = (14, 3.4, 0). (b) The co-spectra between fluctuating velocities u_d and u_m, measured simultaneously from the detecting sensor at (x⁺, y⁺, z⁺) = (−186, 3.4, 0) and monitoring sensor at (x⁺, y⁺, z⁺) = (14, 3.4, 0), respectively. (Online version in colour.)

The magnitude of the pronounced peak at the imposed f⁺ = 0.56 for the open-loop control is 1.9, 1.6 and 1.4 times that for the feed-forward, feedback, and combined feed-forward and feedback control schemes, respectively. This is consistent with the difference between their duty cycles. Note that this pronounced peak is considerably larger for the feedback control than for the feed-forward control even they have the same duty cycle of 50%. This result is at least partially linked to the fact that U_t triggers more in the case of the feedback control (figure 14a), resulting in more overshoots in $A_{\mathrm{o}}^{+}/2$ of the actuators (figure 13), which contribute to an increase in $f^{+}{E}_u^{+}$ at f⁺ = 0.56.

The co-spectrum $C{o}_{u_{\mathrm{d}}{u}_{\mathrm{m}}}$ and spectral coherence $Co{h}_{u_{\mathrm{d}}{u}_{\mathrm{m}}}(\equiv (C{o}_{u_{\mathrm{m}}{u}_{\mathrm{d}}}^2+Q_{u_{\mathrm{m}}{u}_{\mathrm{d}}}^2)/E_{u_{\mathrm{d}}}{E}_{u_{\mathrm{m}}})$ between u_d and u_m are calculated [49], where $Q_{u_{\mathrm{m}}{u}_{\mathrm{d}}}$ denotes the quadrature spectrum. There is a significant decrease in $C{o}_{u_{\mathrm{d}}{u}_{\mathrm{m}}}$ (figure 15b) and $Co{h}_{u_{\mathrm{d}}{u}_{\mathrm{m}}}$ (figure 16) at f⁺ ≈ 5.1 × 10⁻³. This frequency is close to previous reports on the burst frequency $f_b^{+}$ in the near-wall region by Cantwell [50] ($\hspace{0.17em}{f}_b^{+}\approx 6.6\times {10}^{-3}$, calculated from U_∞/δ₉₉f_b ≈ 6) and Blackwelder et al. [51] ($\hspace{0.17em}{f}_b^{+}\approx 4.0\times {10}^{-3}$). The burst is generated by the breakdown process of low-speed streaks [33,52] and plays a crucial role in the self-sustainable process of turbulence in the near-wall region [2,43]. Presumably, frequency f⁺ ≈ 5.1 × 10⁻³ is linked to the burst frequency. The connection between the weakened burst and friction drag reduction is well established. Choi [53] noted a marked inhibition in the burst signature when the friction drag was reduced by 3% in a turbulent boundary layer over a riblet surface. Choi & Clayton [54] showed that the duration of near-wall burst activities was cut short by nearly one-third of no-control case when the skin friction drag was reduced by 45% over a spanwise oscillating wall. It is plausible that the control may have modified the near-wall coherent structures such that the burst is weakened in intensity and duration, thus stabilizing the velocity streaks and reducing skin friction drag. The result is fully consistent with the reduction in $f^{+}{E}_u^{+}$ over the low frequency range, as shown in figure 15a, suggesting suppressed near-wall turbulent activities. The observation further points to an interruption of the turbulence production cycle, which contributes to the observed drag reduction. The decrease in both $C{o}_{u_{\mathrm{d}}{u}_{\mathrm{m}}}$ and $Co{h}_{u_{\mathrm{d}}{u}_{\mathrm{m}}}$ is smallest of all at f⁺ = 5.1 × 10⁻³ for the feedback control, which is consistent with the lowest skin friction drag reduction (20%) among the control schemes.

Figure 16.

Spectral coherence between fluctuating velocities u_d and u_m, measured simultaneously from the detecting sensor at (x⁺, y⁺, z⁺) = (−186, 3.4, 0) and monitoring sensor at (x⁺, y⁺, z⁺) = (14, 3.4, 0), respectively. (Online version in colour.)

(b). Change in the fluctuating velocity and Taylor microscale

A single calibrated hot-wire was traversed across the boundary layer to measure the fluctuating component u of streamwise velocity U at (x⁺, z⁺) = (14, 0) over y⁺ = 2.5 ∼ 100 with and without control. During measurements, the downstream monitoring sensor was removed in order to eliminate the possible interaction between the wall sensor and the hot-wire. As such, measurements were not performed for the feedback-related controls. In the absence of control, $u_{\mathrm{RMS}}^{+}$ (figure 17a) of u increases initially from 0.9 at y⁺ = 2.5 to about 2.7 at y⁺ = 15, and then diminishes gradually as y⁺ further increases. This result conforms to Purtell et al. [55] and Bai et al.'s [16] observations, thus providing a validation for the present measurement. The $u_{\mathrm{RMS}}^{+}$ drops at y⁺ < 10 but rises for y⁺ = 13 ∼ 20 for both control schemes. There is no appreciable variation for y⁺ > 20, that is, the disturbance introduced by PZT actuators is effectively limited to the viscous and buffer layers, which is fully consistent with the result in figure 8. The appreciably reduced $u_{\mathrm{RMS}}^{+}$ for y⁺ < 10 suggests suppressed turbulence activities, corroborating the significant skin friction drag reduction. It is worth noting that the feed-forward control corresponds to $u_{\mathrm{RMS}}^{+}$ (y⁺ < 10) that falls between those under the open-loop and no control, as does its skin friction drag (figure 9). This is not surprising because the near-wall turbulence activities and skin friction drag are correlated (e.g. [53]). The increased $u_{\mathrm{RMS}}^{+}$ over y⁺ = 13 ∼ 20 in the buffer layer is ascribed to the actuator disturbance, which is largest around y⁺ ≈ 15, as shown in figure 8. This increase was also observed by Bai et al. [16].

Figure 17.

(a) Variation in $u_{\mathrm{RMS}}^{+}$ with y⁺ at (x⁺, z⁺) = (14, 0). (b) Variation in Taylor microscale ${\lambda }_T^{+}$ with y⁺ at (x⁺, z⁺) = (14, 0). Open-loop control: $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$, d_c = 100%; feed-forward control: $(\hspace{0.17em}{f}_{\mathrm{o}}^{+},A_{\mathrm{o}}^{+})=(0.56,\hspace{0.17em}3.22)$, d_c = 50%. (Online version in colour.)

The Taylor microscale, defined by ${\lambda }_{\mathrm{T}}=\sqrt{\overline{2u^2}/\mathrm{\partial }u/\mathrm{\partial }{t}^2}$, has been estimated in order to understand the mechanism of drag reduction. The λ_T is a hybrid scale containing information on both large scales via $\hspace{0.17em}\overline{u^2}$ and small scales via $\overline{{(\mathrm{\partial }u/\mathrm{\partial }t)}^2}$ (e.g. [16]). In the absence of control, the calculated ${\lambda }_{\mathrm{T}}^{+}$ is both qualitatively and quantitatively similar to Metzger's [56] and Bai et al.'s [16] reports with a maximum value of about 10 over y⁺ = 3 ∼ 6 (figure 17b). The ${\lambda }_{\mathrm{T}}^{+}$ drops markedly across the viscous and part of the buffer layers (y⁺ < 20), from about 10 to less than 3 under the feed-forward control and even to about two under the open-loop control. This drop is largely due to an increase in $\hspace{0.17em}\overline{{(\mathrm{\partial }u/\mathrm{\partial }t)}^2}$, with a smaller contribution from the reduced $\overline{u^2}$ (figure 17a). Tardu [57] deployed suction/blowing actuation to manipulate a turbulent boundary layer and suggested that, given a decrease in λ_T by a factor of two, the isotropic dissipation rate could grow by a factor of 12. This suggestion is supported by our mean energy dissipation rate measurement presented below.

(c). Change in the mean energy dissipation rate

A change in the mean energy dissipation rate $\overline{\epsilon }$ under control may provide the crucial information on physical mechanisms behind effective control. The $\overline{\epsilon }$ consists of 12 components. Antonia et al.'s [58] DNS data, obtained by in a channel flow at Re_τ = 395 based on the wall friction velocity and the boundary layer disturbance thickness, indicated that $\overline{{(\mathrm{\Delta }u/\mathrm{\Delta }y)}^2}$ overwhelms the other 11 components, accounting for 41 ∼ 72% of $\overline{\epsilon }$ in the buffer layer and about 80% in the viscous sublayer. Their finding is of close relevance to the present study in view of comparable Re_τ (the present Re_τ of the turbulent boundary layer is 568).

We used a probe consisting of two parallel wires to measure Δu/Δy at (x⁺, z⁺) = (14, 0) across the boundary layer over y⁺ = 6 ∼ 100 with and without control, where y⁺ is measured at the midpoint between the hot-wires. The sensing element, made of Pt-10% Rh, is 2.5 µm in diameter and 0.5 mm in length. Antonia et al. [59] reported based on measurements in a channel flow that $\overline{{(\mathrm{\Delta }u/\mathrm{\Delta }y)}^2}$ could be measured with an adequate accuracy if the separation between the hot-wires was in the range of 2 ∼ 4η, where η is the Kolmogorov length scale. As such, two separations between the wires were examined, i.e. Δy = 0.5 mm and 1.0 mm, corresponding to about 1.7 ∼ 2.2η and 3.4 ∼ 4.4η, respectively, across the viscous and buffer layers (i.e. y⁺ < 30). It has been observed that the measured $\overline{{(\mathrm{\Delta }u/\mathrm{\Delta }y)}^2}$ profile at Δy = 0.5 mm is approximately the same as that at Δy = 1.0 mm, with a maximum departure of 6%, in the absence of control. This is an indicator that, given Δy ≈ 2 ∼ 4η, $\overline{{(\mathrm{\Delta }u/\mathrm{\Delta }y)}^2}$ may be measured presently with an adequate accuracy. The η is estimated from the present hot-wire data using a spectral chart method proposed by Djenidi & Antonia [60], yielding η⁺ = 1.5 ∼ 2 over the range of y⁺ = 5 ∼ 30. The deviation between the presently estimated η and Antonia et al.'s [58] report based on the DNS data obtained in the viscous and buffer layers of a channel flow is no more than 5%. For the open-loop control, η = 0.14 ∼ 0.30 mm or η⁺ = 0.93 ∼ 2, calculated based on the spectral chart method, is obtained across the viscous and buffer layers. Thereby, the data measured using the probe with a separation of Δy = 0.5 mm between the parallel wires are presented in this section so that the condition of Δy ≈ 2 ∼ 4η for the reliable estimate of $\overline{{(\mathrm{\Delta }u/\mathrm{\Delta }y)}^2}$ is met.

The probe was mounted on the computer-controlled traversing mechanism and was traversed across the boundary layer. The cut-off frequency was set at 1000 Hz, which is substantially higher than the Kolmogorov frequency $\overline{U}/2\pi \eta$in the near-wall region. For example, $\overline{U}/2\pi \eta$ is 650 Hz when measured at y⁺ = 10. The sampling frequency and duration were 2500 Hz and 40 s, respectively, which are adequately large to ensure the second-order quantities of the measured signals to be converged to an experimental uncertainty of less than 2%.

In the absence of control, the measured $\overline{{(\mathrm{\Delta }{u}^{+}/\mathrm{\Delta }{y}^{+})}^2}$ (figure 18) exhibits a gradual decrease with increasing y⁺, in reasonably good agreement both qualitatively and quantitatively with Antonia et al.'s [58] DNS data. The small departure is not unexpected in view of different flow conditions (e.g. Re_τ). This agreement provides a validation for the present measurement. Once the control is introduced, $\overline{{(\mathrm{\Delta }{u}^{+}/\mathrm{\Delta }{y}^{+})}^2}$ shows a marked rise near the wall (y⁺ < 30), jumping by three to six times or more in the viscous sub-layer, irrespectively of the control schemes. Note that the viscous sub-layer becomes thicker under control, reaching y⁺ ≈ 12 (figure 11), as noted by Bai et al. [16]. The result indicates a great increase in the mean energy dissipation rate near the wall, which is fully consistent with the corresponding appreciable drag reduction and corroborates Bai et al.'s [16] suggestion that the mean energy dissipation rate grows under control. It seems plausible that the control may have not only interrupted the turbulence generation cycle near the wall but also enhanced the mean energy dissipation rate and subsequently partial relaminarization, thus resulting in a considerable drag reduction. That is, the larger the drag reduction, the larger the mean energy dissipation rate. The drag reduction for the open-loop control is largest, 30%, of all and, therefore, the corresponding peak in $\overline{{(\mathrm{\Delta }{u}^{+}/\mathrm{\Delta }{y}^{+})}^2}$ is the most pronounced. However, this does not necessarily mean that the other schemes such as the feed-forward and feedback control are inefficient. As a matter of fact, the other schemes are more efficient in terms of drag reduction per unit input energy. Take the feed-forward control for example. Compared to the open-loop control, the peak at y⁺ ≈ 8.4 in $\overline{{(\mathrm{\Delta }{u}^{+}/\mathrm{\Delta }{y}^{+})}^2}$ drops by 30% for the feed-forward control and the corresponding drag reduction is only 24%, dropping by 20%. However, the input energy consumed for the feed-forward control is only 50% of that for the open-loop control. That is, giving the same amount of the input energy, the feed-forward control could achieve an energy saving, via drag reduction, by 48%, rather than 30% as in the case of the open-loop control. Nevertheless, it is not clear why the maximum $\overline{{(\mathrm{\Delta }{u}^{+}/\mathrm{\Delta }{y}^{+})}^2}$ occurs at y⁺ ≈ 8.4 under the open-loop control, which certainly contributes to the large drop in u_RMS at the same y⁺ (figure 17a).

Figure 18.

Comparison in $\overline{{(\mathrm{\Delta }{u}^{+}/\mathrm{\Delta }{y}^{+})}^2}$ between measurements at (x⁺, z⁺) = (14, 0) and Antonia et al.'s [58] DNS data obtained in a channel flow. The control parameters are as in figure 17. (Online version in colour.)

6. Conclusion

Different active control schemes are investigated based on spanwise-aligned PZT actuators in a fully developed turbulent boundary layer with a view to suppressing the near-wall high-speed events and hence reducing skin friction drag. The feed-forward control scheme works on predicting, based on a transfer function model, the downstream turbulence evolution and manipulating high-speed friction events. The maximum drag reduction achieved at the measurement point is up to 24%, 6% less compared to that attained from the open-loop control. However, the duty cycle is only 50%, suggesting a substantial drop in the input energy. The effective linear approach hinges on the fact that the well-organized turbulence structures play a crucial role in the self-sustaining process of turbulence in the near-wall region and the key parts of this process are linear [4]. On the other hand, the feedback control works on the assumption that the captured feedback signal is the same as what could be measured at the actuator position and obtains a maximum drag reduction of only 20% given a duty cycle of 50%. Apparently, the prediction based on the downstream evolution exhibits advantages over that on past events in this flow. When the feedback and the feed-forward controls are combined, the maximum drag reduction rises up to 28%, almost the same as that (30%) achieved by the open-loop control. Yet, the corresponding duty cycle increases only slightly to 62%.

The analysis of the flow structures with and without control unveils that there is a shift in energies from low to high frequencies under control, that is, a transfer in energy from large- to small-scale turbulence, regardless of the control schemes. The control further weakens bursts, as is evident in the co-spectra and spectral coherence between the detecting and monitoring velocity signals. The result suggests an interrupted turbulence production cycle. This, along with the reduced u_RMS, a big drop in the Taylor microscale and the greatly enhanced mean energy dissipation rate as indicated in the measured $\overline{{(\mathrm{\Delta }{u}^{+}/\mathrm{\Delta }{y}^{+})}^2}$, may suggest the occurrence of relaminarization in the near-wall turbulence, which is fully consistent with the substantially reduced skin friction drag.

Data accessibility

This article has no additional data.

Authors' contributions

Z.X.Q. performed the experimental measurements and drafted the paper. Z.W. took part in some experiments in physics-based control schemes and revised the relevant texts in the draft. Y.Z. proposed the idea of the study, funded and supervised the project. All authors regularly discussed the progress during the entire work.

Competing interests

We declare we have no competing interests.

Funding

Y.Z. wishes to acknowledge support given to him from NSFC through grant no. 11632006 and from RGC of HKSAR through grant no. PolyU 5319/12E.