The experiments were conducted in an annular return wind tunnel. The TBL flow was developing along one side of a flat acrylic glass plate mounted vertically in the wind tunnel. Following the right-hand rule, the space coordinate system Oxyz is set with O as the origin at the leading edge of the plate, x as the stream-wise direction, y as the wall-normal direction, and z as the span-wise direction as shown in Fig. 2(a) . By adjusting the pitch angle of the plate, a nearly zero-pressure-gradient in the x direction was achieved. A piece of twisted-pair wire was fixed spanwisely at x = 8 cm, and 4 pieces of No. 240 sandpaper were attached following the wire until x = 53 cm. These measures accelerated the transition of the boundary layer and made us achieve the fully developed TBL flow at about x =1000 mm where the end of the control mechanism was and the measurements were conducted as shown in Fig. 2(b) . ^{[ 25 ]}

Fig. 2.

	Figure Option View Download New Window
	Fig. 2. (a) Schematic of experiment set-up, (b) picture of PZT oscillator and miniature velocity probe, and (c) Euler–Bernoulli beam model of PZT oscillator.

The control mechanism consisted of a cavity with the sizes of 32 mm × 5 mm × 5 mm and a PZT oscillator pasted firmly. The oscillator upper surface was kept flush with the flat plate. The effective length, total thickness, and the width of the oscillator were 30 mm, 0.42 mm, and 3.62 mm. The length and width were designed according to the size of the near-wall stream-wise vortex, a span-wise scale of O (100 ν / u _τ ), and a stream-wise scale of O ( δ ) (both defined later). This unimorph configuration consisted of a piece of 220-μm-thick PZT-5H material and a 200-μm-thick phosphor-copper shim. The thickness of epoxy bond was negligible. To excite the PZT oscillator, we used a power source of YuanFang-GK10005 that supplies AV with adjustable frequency and amplitude in a wide range.

The PZT oscillator was modeled as a cantilever beam ^{[ 26 ]} as shown in Fig. 2(c) . When a voltage was applied between its upper and lower surfaces, the beam bent in the x – y plane. The oscillator changed the wall boundary conditions and introduced interference into the TBL. The assumption of perfect bonding implied that the strain was continuous at the bond interface, and a linear strain distribution in the y direction was adopted.

According to the basic geometrical information and material properties of the PZT oscillator (Table 1 ), we analyze the static deformation and the dynamic vibration by using the Euler–Bernoulli beam theory. For an incompressible air flow, the aerodynamic pressure loading on the oscillator is negligible at low Mach numbers.

Table 1.

Table 1.

Table 1. Details of PZT oscillator materials. . Quantity Value PZT density ρ p /kg·m −3 7.45 × 10 3 Shim density ρ s /kg·m −3 8.89 × 10 3 PZT elastic modulus E p /N·m −2 7.69 × 10 10 Shim elastic modulus E s /N·m −2 11.3 × 10 10 PZT strain constant d 31 /m·V −1 –186 × 10 −12

Table 1. Details of PZT oscillator materials. .

The first-order natural frequency f ₁ is 254 Hz and the direct voltage (DV) gain coefficient H ₀ , i.e., the tip displacement of the PZT oscillator actuated by a voltage of 1 V, is 0.0014 mm/V. These two parameters are both important for the second-order frequency response model which has been verified experimentally. ^{[ 27 ]} The model of AV gain coefficient H ( f ) can be expressed as

Fig. 3.

	Figure Option View Download New Window
	Fig. 3. Frequency responses of PZT oscillator. (a) Amplitude–frequency curve, (b) phase–frequency curve.

Table 2.

Table 2.

Table 2. Drag reduction effect at x = 1002 mm. . V c /V, f c /Hz A t /mm τ w /(N/m 2 ) u τ /(m/s) No control 0 0.2408 0 0.4317 20, 80 0.031 0.2421 0.54% 0.4329 20, 160 0.046 0.2416 0.33% 0.4324 20, 240 0.261 0.2488 3.32% 0.4388 60, 80 0.093 0.2348 –2.49% 0.4263 60, 160 0.139 0.2066 –14.20% 0.3999 60, 240 0.784 0.2400 –0.33% 0.4310 100, 80 0.155 0.2037 –15.41% 0.3971 100, 160 0.232 0.1762 –26.83% 0.3693 100, 240 1.306 0.2290 –4.90% 0.4210

Table 2. Drag reduction effect at x = 1002 mm. .

Under most control conditions, the interference introduced by the PZT oscillator belongs to the category of the Hydrodynamic smooth, because their amplitudes are less than 7 wall units (1 wall unit = ν / u _τ ) (∼0.257 mm). Although the amplitudes are a little bigger than the critical condition of the Hydrodynamic smooth when f _c is set to be 240 Hz, the typical layered structure of the TBL is not altered, owing to the small size of the PZT oscillator compared with the whole flow field.

At x = 1002 mm, i.e., 2 mm downstream the end of the control mechanism, the stream-wise velocity components u in the TBL were accurately measured by the constant temperature anemometry of IFA-300 with a miniature boundary layer probe TSI-1621A-T1.5. The hot wire of this probe is made of tungsten (platinum coated) cylinder with a sensitive length of 1.25 mm and diameter of 4 μm. This probe was specially prepared for velocity measurement, which was located very close to the wall. ^{[ 28 , 29 ]} Before measurement, mean-flow calibration was employed twice and the error was 0.087% based on the fourth-order polynomial curve fitting. We set the sampling rate and low pass cut off frequency to be 100 kHz and 50 kHz, respectively. Time sequences of the stream-wise velocity signals at different wall-normal locations were finally achieved and each sequence consisted of 2 ²² moments in about 42 s.

The free stream velocity U _∞ is 9.0 m/s and the TBL nominal thickness δ is 39.8 mm. Based on U _∞ and the momentum thickness θ , the Reynolds number Re _θ is 2183. The outer scale mean velocity profile (scaled with δ and U _∞ ) is linear in the viscous sub-layer ^{[ 30 , 31 ]} as indicated in Fig. 4(a) . The profile slope is related to skin friction stress τ _w , because

Fig. 4.

	Figure Option View Download New Window
	Fig. 4. Mean velocity profiles of different control conditions in (a) outer and (b) inner scale units.

Table 2 shows the values of relative drag reduction rate under different values of V _c and f _c , and is the skin friction stress of the TBL without control.

Figure 4(b) shows the mean velocity profiles in inner scale units y ⁺ = yu _τ / ν and U ⁺ = U / u _τ . According to the log law U ⁺ = κ ⁻¹ ln y ⁺ + B , B is related to u _τ as the following differential expression:

The integral spatial scale L reflects the spatial dimension of a turbulent structure in large scale with the same order of mean motion, ^{[ 25 ]} and is achieved by the auto-correlation analysis and Taylor’s frozen hypothesis as

Fig. 5.

	Figure Option View Download New Window
	Fig. 5. Profiles of integral special scale L .

In order to detect the most energetic coherent components of TBL flow and reveal the control mechanism, Mallat pyramidal algorithm of the orthonormal discrete wavelet transform is accomplished with db5 wavelet. ^{[ 35 , 36 ]} The wavelet scale index n is linear with respect to logarithmic characteristic frequency , as Fig. 6(a) shows. The curve slope slightly differs from the dyadic property theoretical value of −1. This deviation is caused by the selection of wavelet basis. The intercept is related to the signal sampling frequency. According to the linear logarithmic relationship, the control frequencies of 80, 160, and 240 Hz are corresponding to varied discrete scale n values of 10, 9, and 8 in the multi-scale system of near-wall turbulence.

Fig. 6.

	Figure Option View Download New Window
	Fig. 6. (a) Relationship between n and , (b) distribution of in the plane ( n,y + ) without control.

The values of single-reconstructed velocity fluctuation without control are attained firstly at different values of wall-normal location y ⁺ . Then the energy distribution in dB is shown by Fig. 6(b) . A peak at ( n = 9, y ⁺ = 14) is obvious and it corresponds to the stream-wise vortex in the buffer layer that is located in a frequency band near the frequency of about 147 Hz. These structures contain the most fluctuating energy in the near-wall multi-scale turbulence. ^{[ 37 ]} It is the reason why the controls with a frequency of 160 Hz located in this special frequency band perform best for drag reduction in our experiments.