Active control of multiscale features in wall-bounded turbulence

This study experimentally investigates the impact of a single piezoelectric (PZT) actuator on a turbulent boundary layer from a statistical viewpoint. The working conditions of the actuator include a range of frequencies and amplitudes. The streamwise velocity signals in the turbulent boundary layer flow are measured downstream of the actuator using a hot-wire anemometer. The mean velocity profiles and other basic parameters are reported. Spectra results obtained by discrete wavelet decomposition indicate that the PZT vibration primarily influences the near-wall region. The turbulent intensities at different scales suggest that the actuator redistributes the near-wall turbulent energy. The skewness and flatness distributions show that the actuator effectively alters the sweep events and reduces intermittency at smaller scales. Moreover, under the impact of the PZT actuator, the symmetry of vibration scales’ velocity signals is promoted and the structural composition appears in an orderly manner. Probability distribution function results indicate that perturbation causes the fluctuations in vibration scales and smaller scales with high intensity and low intermittency. Based on the flatness factor, the bursting process is also detected. The vibrations reduce the relative intensities of the burst events, indicating that the streamwise vortices in the buffer layer experience direct interference due to the PZT control.


Introduction
Turbulence control offers significant advantages in various engineering applications. One of the most important techniques used to improve the efficiency of a working system is to reduce the surface drag caused by turbulent boundary layer (TBL) flow, and understanding TBL flow mechanisms is beneficial for the development of control methods. Quasistreamwise vortices (QSVs) are typical coherent structureinducing [1][2][3] bursting events that dominate the near-wall region, and are associated with most surface friction production [4][5][6]. The ejection process occurs on the updraft side of QSVs, wherein the low-speed fluids in the inner layer are lifted away from the wall to the outer layer with the transfer of the mass, energy, momentum, and vortices [7]. The sweep process occurs on the downdraft side of QSVs, wherein the high-speed fluids move downward to the wall, which is directly related to an increase in the skin friction [8,9]. Thus, QSVs are the main factors that increase the surface friction, so altering the QSVs is a potential approach for achieving drag reduction [10,11].
Numerous passive and active control methods have been developed based on this flow mechanism [12]. As a representative of the passive control methods, Wash and Choi et al. [13][14][15] achieved considerable drag reduction on riblet surfaces by conducting experiments and direct numerical simulations. In comparison with the passive control method, the active control method has wider adaptability to complex flows, and it enhances the control effectiveness [16][17][18][19], which has been confirmed by several experimental and simulation investigations [20][21][22][23][24][25][26][27][28][29]. Bai et al. [2] used a spanwisealigned array of piezoceramic actuators to generate a transverse traveling wave along the wall surface. The actuator can induce a layer of highly regularized streamwise vortices to break the connection between the large-scale coherent structures and the wall, achieving a maximum drag reduction of 50%. Berger et al. [30] obtained a drag reduction of 40% via an open loop-controlled oscillating spanwise Lorentz force that disturbed the semi-equilibrium state between the near-wall streamwise vortices and the wall in channel flow. Zheng et al. [19] applied a single PZT actuator to break the near-wall streamwise vortices and achieved a drag reduction of 27%. Herein, we employed an active strategy based on the aforementioned achievements.
A basic property of turbulence is its multiscale characteristics. However, few studies have investigated the influence of active control on the TBL's multiscale features. We provided a periodic perturbation to the TBL via a PZT actuator. To further observe the modification in the TBL's multiscale characteristics because of the PZT actuator, we conducted hot-wire measurements. This study reports the drag reduction results with a PZT actuator over a wide range of frequencies and amplitudes, and the multiscale properties of the turbulence are analyzed. Finally, the control effect on the bursting process is observed. Figure 1 shows the experimental setup. The experiments were conducted in a closed loop wind tunnel. TBL flow developed along a flat plate, which was mounted vertically in the test section. Pitch angle was finely adjusted to ensure a zero-pressure gradient in the streamwise direction. A tripwire was fixed at x = 80 mm downstream of the leading edge, and a sandpaper band was attached following the tripwire until x = 530 mm . The freestream velocity was U ∞ = 9 m∕s . TBL was fully developed at the streamwise location of x = 1000 mm , wherein the end of the control mechanism and measurement was conducted at x = 1002 mm , as shown in Fig. 1. The thickness of the TBL was approximately = 39.8 mm . Based on U ∞ and momentum thickness, , Reynolds number, Re , was 2183.

Experimental setup and parameters
The PZT actuator was firmly pasted over the plate. The upper surface of the actuator was leveled with the plate's surface. The effective length, thickness, and width of the actuator was 30 , 0.42 , and 3.62 mm , respectively. The actuator comprised a 220-µm-thick PZT-5H material and 200-µm-thick phosphor-copper shim. The basic geometric information and material properties of the PZT actuator are listed in Table 1. The PZT oscillator was excited via a power source, which worked as a cantilever beam. The frequency and amplitude of the actuating voltage were adjustable over a wide range.
We tested three different experimental conditions, wherein the frequencies of actuation were set to 80, 160, and 240 Hz under the same voltage of 100 V. The corresponding amplitudes of the actuator were 0.4 , 0.6 , and 0.2 mm , respectively, which were captured using a high-speed camera. The measuring equipment comprised an IFA-300 with a TSI-1621A-T1.5 miniature boundary layer probe [31,32]. The sensing element

Mean velocity profile
The outer scaled mean velocity profile, scaled using and U ∞ , is linear in the viscous sublayer [34], as shown in Fig. 2a. The profile slope is related to the skin-friction stress w , according to where the air flow density is 1.205 kg/m 3 and kinematic viscosity coefficient v is 1.5 × 10 −5 m 2 /s [35][36][37]. The friction velocity, u , can be obtained as follows where the overbar denotes time averaging. Figure 2b shows the mean velocity profiles normalized by u , and U + = y + is evident for both uncontrolled and controlled cases over y + = 3-5. The local skin-friction reduction is defined as follows where the subscripts on and off denote measurements with and without control, respectively. Table 2 lists the drag reduction results for different cases. It can be seen that case 2 with the PZT frequency of 160 Hz and amplitude of 0.6 mm achieves the maximum drag reduction of 26.4%.

Wavelet energy distribution
The effects of the PZT actuator on multiscale characteristics of the turbulence were investigated by wavelet analysis, which is one of the most widely used multiscale signal processing methods. The definition of this method fits well with the physical characteristics of turbulence [38,39]. Wavelet analysis includes a continuous wavelet transform (CWT) and discrete wavelet transform (DWT). To improve the computing efficiency and avoid redundant information, DWT was employed for data processing [40].
The wavelet transforms of a signal u(t) with scale r at position t are defined as follows [41] where (t) is wavelet function, which satisfies the following admissibility condition  where ̂( k) denotes the Fourier transform of (t) . Equation (5) is the invertibility condition, which is a necessary condition for the signal u(t) to be reconstructed from its wavelet coefficients. Instead of using CWT, the function u(t) can be decomposed into a series of self-similar orthonormal functions obtained via dilatations and translations of a mother function 00 [42]. By choosing separations r k = 2 −k , a set of functions forming the orthonormal basis can be written as follows If u(t) is sampled on a discrete mesh t j , the transformed function can be rewritten in a discrete form as follows where r represents the discretized scale. The wavelet function (r) (i − 2 r j) should obey the following orthogonality condition: Then the coefficients w (r) (i) can be obtained as The reconstructed signal at every scale can be computed as follows In this experiment, DWT in the fluctuating velocity signals was calculated using Daubechies 4 wavelet function. Figure 3 shows the linear relation between the logarithmic characteristic frequency f w r and scale level r. Figure 4 shows the percentage of the reconstructed fluctuations' energy with respect to the total energy, which is a function of the scale levels and wall-normal positions. In Fig. 4a, the uncontrolled case indicates that the inner peak appears at y + = 10-14 at a scale level of 9, corresponding to the most energetic structures in the near-wall turbulence, i.e., the QSVs. Figure 4b-d show the results of the controlled cases. It can be seen that the PZT vibration concentrates the energy at scale levels of 10, 9, and 8, corresponding to the vibration frequencies of 80, 160, and 240 Hz, respectively, which means that the PZT actuator, as expected, has a periodic impact on the downstream flow fields where the hot-wire is measured. As shown, the wall-normal region perturbed by the PZT actuator primarily exists at y + < 40.
According to Wallace et al. [26], typical bursting events, i.e., sweeps and ejections, are closely correlated with each other at a wall-normal position of 15 wall units. In our study, the vibrations affected this position range. The turbulent energy distribution, as shown in Fig. 4, indicates that the impact of the actuator was concentrated in the range y + = 10-15. Therefore, the time series of the original and reconstructed fluctuating signals at y + = 14 in the uncontrolled case and case 2 (100 V and 160 Hz) were compared to investigate the effects of PZT on the scale corresponding to the vibration frequency. In Fig. 5a, b, it can be seen that the original fluctuation u + overlaps better with the reconstructed fluctuation u + r at a scale level of 9, i.e., corresponding to 160 Hz, in the controlled case than that in the uncontrolled case. This implies that the fluctuation components at this dominating scale are coupling to the PZT actuation.

Turbulent intensities
First, some definitions need to be stated for clarity. The vibration scales indicate the scales approximately corresponding to the vibration frequencies 80, 160, and 240 Hz, which are 10, 9, and 8, respectively. Scales with higher characteristic frequencies than the vibration frequencies are defined as smaller scales, whereas those with lower characteristic frequencies are defined as larger scales. Table 3 lists the definitions of the analyzed scalesbelow.
The turbulent intensity distribution is observed at various scales, which is represented by u + rms . As shown in Fig. 6a-

Skewness
The skewness,  near-wall region of y + = 10-15 for all cases. For cases 1 and 2, the skewness in the near-wall region decreases (close to zero) significantly, implying a better symmetry of the PDF for streamwise fluctuations. Then likely reason is that the PZT actuator induces flow structures with a certain frequency and highly fluctuating amplitude, and that the generated structures result in an alteration of the sweep events in the near-wall region. intensity. Combined with the results of the flatness factor, it shows that the PZT disturbance caused fluctuations on the vibration and smaller scales with high intensity and low intermittency.

Flatness
Based on the study reported by Jiang and Zhang [43], bursting process components are the fluctuations that cause the flatness factor (FF) to be larger than 3. The two principal indicators used for our analysis of the data intermittency were the FF and local intermittency measure. FF is defined as follows: This function gives the level of intermittency at scale level r. Therefore, the difference with respect to the Gaussian statistics with an FF equal to 3 can be directly obtained.
The local intermittency measure (LIM) is defined as follows: According to Camussi and Guj [40], LIM is a good qualitative indicator of the intermittency and magnitude of the energy fluctuations. Overall, detecting the bursting process indicates extracting the components in the fluctuations that make FF larger than 3 using these two indicators. The extraction procedure can be summarized as follows. For a certain scale, LIM can be computed as a function of t and threshold level l, can be set for LIM. The initial value of l is set to l 0 = u �2 max ∕u �2 (t) t , where u ′2 max is the maximum value of the fluctuations over the entire time domain at this scale. Then, FF of the fluctuation components u ′ t i ( LIM t i < l = l 0 ) can be calculated. If FF is larger than 3, l is reduced from l 0 by several times at a certain rate. At each time, the FF of the fluctuation components u ′ t i ( LIM t i < l ) is calculated. The detecting process stops when the FF of u ′ t i LIM t i < l drops to 3 and l = l (FF=3) . Finally, the fluctuation components u ′ t j LIM t j > l (FF=3) are extracted as the bursting process. The completion status of the detection process is shown in Fig. 10. The black line denotes the fluctuation at a scale level of 9, whereas red line denotes the bursting processes detected at this scale.
According to this detection method, the statistical results of the bursting process can be studied at all scales. To obtain an overall understanding of the process, we superimposed the energy of the bursting process, E b , and total energy, E , at all scales. Their ratios are obtained in Fig. 11a. E b ∕E , which represents the relative importance of the bursting process, reflecting the fact that the vibration weakens the bursting process and that the case of 100 V and 160 Hz has the best performance. The impact is concentrated at y + < 40 . These results are in agreement with those reported by Zheng et al. [18]. Combined with the drag reduction effect, it appears that the streamwise vortices in the buffer layer experience direct interference due to the PZT actuator, which leads to a breakdown in the burst mechanism in the inner region, thereby reducing the drag. The bursting process includes eject and sweep events with low-and high-speed fluids, respectively. To observe the impact of the vibration more carefully, the sweep and eject events were extracted according to the positive and negative fluctuation signals, respectively. Figure 11b, c show the proportion of eject and sweep event energy, i.e., E e and E s , respectively, relative to E . As can be seen, the relative intensities of the eject and sweep events were both weakened by the vibration in the near-wall region. Even though the energy proportions of the eject and sweep events were different in the original fluctuations, the weakening degrees due to the vibrations were nearly same for the positive (sweep) and negative (eject) fluctuations. According to Fig. 7, the vibrations produced strong fluctuations in the positive and negative symmetries in the near-wall region; therefore, the impacts on the positive and negative signals were similar.

Conclusions
This experiment was conducted considering the Reynolds number of Re = 2183 under three different conditions. The impact of a single PZT actuator on the statistical characteristics in the TBL was investigated using various methods. Statistics were gathered from the streamwise fluctuating velocity signals, measured using a single hot wire in the near-wall region. The major conclusions of the study are as follows.
1. From a multiscale point of view, in the near-wall region, the actuator redistributes the energy into different scales. 2. The changes in the skewness and flatness revealed that the PZT actuator, actuated at a certain frequency by a voltage with a sinusoidal waveform, effectively interferes with the sweep events, reduces the intermittency at smaller scales and on the vibration scales, and enhances the symmetry of the velocity signals at the vibration scales. Furthermore, the structure composition appears ordered. The PDF results indicate that the perturbation caused the fluctuations at the vibration scales and the smaller scales to have high intensity and low intermittency. 3. Based on the change in the flatness, the characteristics of the bursting events were analyzed. The results indi-cate that the vibrations restrain the bursting process for y + < 40 , which indicates that the streamwise vortices in the buffer layer experience direct interference due to the PZT control, leading to a reduction in the drag.