# Active control of multiscale features in wall-bounded turbulence

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## Abstract

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.

## Keywords

Turbulent boundary layer Piezoelectric actuator Drag reduction Discrete wavelet decomposition Bursting process## 1 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. Quasi-streamwise vortices (QSVs) are typical coherent structure-inducing [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 spanwise-aligned 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.

## 2 Experimental setup and parameters

Details of the PZT oscillator materials

Quantity | Value |
---|---|

PZT density | 7.45 × 10 |

Shim density | 8.89 × 10 |

PZT elastic modulus | 7.69 × 10 |

Shim elastic modulus | 11.3 × 10 |

PZT strain constant | − 186 × 10 |

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 \(0.2\;{\text{mm}}\), 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 of the probe was a tungsten wire with a diameter of 4 μm and length 1.25 mm. The calibration was conducted using an air velocity calibrator over the velocity range 0–15 m/s. Sampling rate and low pass cut-off frequency were 100 and 50 kHz, respectively. We obtained time sequences of the streamwise velocity signal at 74 different wall-normal locations. Each sequence comprised 2^{22} velocity samples or approximately 42 s [33].

## 3 Results and discussion

### 3.1 Mean velocity profile

^{3}and kinematic viscosity coefficient \(v\) is 1.5 × 10

^{−5}m

^{2}/s [35, 36, 37]. The friction velocity, \(u_{\tau }\), can be obtained as follows

*y*

^{+}= 3–5. The local skin-friction reduction is defined as follows

Local skin drag reduction

Case | \(U_{\infty } \left( {{\text{m}}/{\text{s}}} \right)\) | \(f\)(Hz) |
| \(u_{\tau } \left( {{\text{m}}/{\text{s}}} \right)\) | \(\Delta_{{\tau_{w} }}\) |
---|---|---|---|---|---|

Uncontrolled | 9 | 0 | 0 | 0.4317 | 0 |

1 | 9 | 80 | 0.4 | 0.3971 | − 15.4% |

2 | 9 | 160 | 0.6 | 0.3693 | − 26.8% |

3 | 9 | 240 | 0.2 | 0.4210 | − 4.9% |

### 3.2 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].

*r*represents the discretized scale. The wavelet function \(\varPsi^{\left( r \right)} \left( {i - 2^{r} j} \right)\) should obey the following orthogonality condition:

*r*.

*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 \(y^{ + } < 40\).

*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.

### 3.3 Turbulent intensities

Definition of the analyzed scales

Case | Parameters | Vibration scale | Smaller scale | Larger scale |
---|---|---|---|---|

1 | 100 V, 80 Hz | 10 | < 10 | > 10 |

2 | 100 V, 160 Hz | 9 | < 9 | > 9 |

3 | 100 V, 240 Hz | 8 | < 8 | > 8 |

*y*

^{+}= 10–15, the turbulent intensities are enhanced by the vibrations on smaller scales. Moreover, in case 2, the vibrations obviously weaken the intensities at larger scales; however, they have only a slight effect at larger scales in cases 1 and 3. Generally, in the near-wall region, the vibrations redistributed the energy to different scales [29].

### 3.4 Skewness

*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.

### 3.5 Flatness

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.

*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 \(l_{0} = u_{\hbox{max} }^{\prime 2} /u^{\prime 2} \left( t \right)_{t}\), where \(u_{\hbox{max} }^{\prime 2}\) is the maximum value of the fluctuations over the entire time domain at this scale. Then, FF of the fluctuation components \(u^{\prime}\left( {t_{i} } \right)\) (\({\text{LIM}}\left( {t_{i} } \right) < l = l_{0}\)) 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 \(u^{\prime}\left( {t_{i} } \right)\) (\({\text{LIM}}\left( {t_{i} } \right) < l\)) is calculated. The detecting process stops when the FF of \(u^{\prime}\left( {t_{i} } \right)\left( {{\text{LIM}}\left( {t_{i} } \right) < l} \right)\) drops to 3 and \(l = l_{{\left( {{\text{FF}} = 3} \right)}}\). Finally, the fluctuation components \(u^{\prime}\left( {t_{j} } \right)\)\(\left( {{\text{LIM}}\left( {t_{j} } \right) > l_{{\left( {{\text{FF}} = 3} \right)}} } \right)\) 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.

## 4 Conclusions

- 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 indicate 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.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 11732010, 11572221, 11872272, U1633109, 11802195) and the National Key R&D Program of the Ministry of Science and Technology, China, on “Green Buildings and Building Industrialization” (Grant 2018YFC0705300).

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