Frequency Response of Near-Wall Coherent Structures to Localized Periodic Blowing and Suction in Turbulent Boundary Layer

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1 CHIN.PHYS.LETT. Vol. 25, No. 5 (2008) 1738 Frequency Response of Near-Wall Coherent Structures to Localized Periodic Blowing and Suction in Turbulent Boundary Layer LIU Jian-Hua( ), JIANG Nan( ) Department of Mechanical Engineering, Tianjin University, Tianjin (Received 8 January 2008) We experimentally investigate the frequency response of near-wall coherent structures to localized periodic blowing and suction through a spanwise slot in a turbulent boundary layer. Sine wave is applied through a spanwise slot by changing the frequency of periodic disturbance at similar velocities of free stream. The effects of blowing and suction disturbance on energy redistribution, turbulent intensity u + rms over y + and waveforms of phase-averaged velocity during sweeping process are respectively discussed under three frequencies of periodic blowing and suction in near-wall region of turbulent boundary layer, compared with those in a standard turbulent boundary layer. The most effective disturbance frequency is figured out in this system. PACS: De, nb, ld The interests in controlling near-wall turbulence have been activated by progresses in understanding of the coherent structures in wall turbulence flow. Up to date, many attempts have been made to exploit a technique for controlling wall-bounded flow. There are two kinds of controlling methods. One is passive control and the other is active control. The difference between these two kinds of controlling methods is whether the disturbance directly aims at the coherent structures or not. Generally speaking, it is termed as passive control that when the coherent structure arrives at the disturbing position unknown before, the disturbance is imposed on the coherent structures. Passive control is a simple but effective method for wall-bounded turbulence control. Learn from other researchers work, passive control includes the modification of the wall surface by using spanwise oscillating wall, [1] or installing riblet, [2] grooves [3] and complaint coating. [4] All these methods mentioned above have a fatal shortcoming that the disturbance corresponding to geometry of the surface is impossible to be set quantificationally before using it. Periodic blowing and suction deserves much more detailed studies because it provides a simple and easily variable method for wall-bounded turbulence control. The frequency of blowing and suction disturbance is able to be set quantificationally and changed easily, according to researchers needs. Most of the previous studies of local blowing and suction have paid attention on the steady actuation. Compared to numerous experimental and numerical studies on steady blowing and suction actuation, relatively few works have been performed to detect the effect of unsteady blowing and suction on near-wall turbulence. The effects of unsteady blowing on a turbulent boundary layer were demonstrated by Kim et al. [5] by direct numerical simulation (DNS) in Sweeping process plays an important role in turbulence production, transportation and evolution. The skin friction coefficient increases with each successive sweep of turbulent coherent structures [6] and the sweeping process shares much more kinetic energy than ejection process in coherent structures motion [7] at the beginning of the logarithm layer. It is more effective to investigate the effect of periodic blowing and suction on the sweeping process than a burst in coherent structures motion. Discrete wavelets transform (DWT) proposed by the Hungarian mathematician Alfred Haar [8] has been used as an effective tool in turbulent velocity analysis since the end of 20th century. Haar wavelet is also the simplest possible wavelet. The disadvantage of the Haar wavelet is that it is not continuous and therefore not differentiable, as follows: 1, 0 t < 1/2, ψ(t) = 1, 1/2 t < 1, (1) 0, otherwise. In numerical and functional analysis, a DWT is any wavelet transform for which the signals are discretely sampled. The DWT of signal u(t) is calculated by passing it through a series of filters. First the samples are passed through a low pass filter with impulse response g resulting in a convolution: W (a, n) a=1 = (u g)(n) = + k= u(k)g(n k), (2) where a represents the level of wavelet transform and scale of coherent structures; W (a, n) is wavelet coefficient; and n is the number of data points. Supported by the National Natural Science Foundation of China under Grant Nos and To whom correspondence should be addressed. jhliu ning@yahoo.com.cn c 2008 Chinese Physical Society and IOP Publishing Ltd

2 No. 5 LIU Jian-Hua et al The signal is also decomposed simultaneously using a high-pass filter h. The output from high-pass filter gives the detail coefficient and that from the low-pass filter gives the approximation coefficients. It is important that the two filters are related to each other and they are known as a quadrature mirror filter. However, since half the frequencies of the signal have now been removed, half samples can be discarded according to Nyquist s rule. The filter outputs are then downsampled by 2. The decomposition has halved the time resolution since only half of each filter output characterizes the signal. However, each output has the frequency band of the input so the frequency resolution has been doubled. This decomposition is repeated to further increase the frequency resolution and the approximation coefficients decomposed with high and low pass filter and then down-sampled. This is represented as a binary tree with nodes representing a sub-space with different time-frequency localizations. The process is shown in Fig. 1. Here f s is the sampling rate for velocity signals. The highest frequency of signals detected by a sampling rate f s is f s /2 Hz. That is the reason why the frequency band is from 0 to f s /2 Hz shown in the wavelet transform process. Summarized from the wavelet transform, an a-scaled structure contains the velocity signals at the frequency domain [f s /2 a+1, f s /2 a ] Hz. Here t represents time and t = n/f s. in sweeping process is acquired by 1 Na i=1 N u j(a, t i ), a I(a, t i )/I(a) > L&W (a, t i ) > 0, u(a, t) = 1 (4) Na i=1 N u j(a, t i ), a I(a, t i )/I(a) > L&W (a, t i ) < 0, where 1 < j < N c and N c is the number of a-scaled coherent structures at a certain vertical location. The case W (a, t i ) > 0 represents sweeping process of coherent structure motion. The present experiment is conducted in the wooden TULTWT wind tunnel in fluid mechanics lab of Tianjin University, whose test section is 4.5 m in length with a rectangular cross section in height 0.45 m and width 0.35 m. The freestream velocity in the test section ranges from 0.5 m/s to 50.0 m/s, where the primary turbulent level is 0.07%. Two flat steel boards in width 0.35 m, thickness 5 mm and length 1.0 m and 0.9 m, respectively, are fixed horizontally on the centreline, parallel to the freestream. A sandpaper stick on the leading edge of the upwards board reduces the two dimensional wake characteristics of the trip wire. This combination ensures the self-preserving turbulent boundary layer upstream of the location of periodic actuation. A spanwise slot between the upwards board in length 1.0 m and the downstream board in length 0.9 m is 0.5 mm in width. The periodic blowing and suction disturbance is induced through the spanwise slot into a turbulent boundary layer flow. The origin of the coordinate axes is located at the centre of the downstream edge of the spanwise slot. The sketch map of test section is shown in Fig. 2. Fig. 1. Wavelet transform. The sweeping process is abstracted according to the detecting function D(a, t) based on Haar wavelet transform: 1, I(a, t i )/I(a) > L&W (a, t i ) > 0, D(a, t) = 1, I(a, t i )/I(a) > L&W (a, t i ) < 0, 0, otherwise, (3) where I(a, t i ) = W 2 (a, t i ) and I(a, t i ) denotes the intensity of a-scaled structure at time t i, I(a) = Na i=1 I(a, t i) and N a represents the length of a-scaled structure in data points and t i. = i/f s (1 < i < N a ). The ratio I(a, t i )/I(a) is the relative intensity at time t i compared with the mean intensity of a-scaled structure. L is the threshold value. The phase averaged velocity of coherent structures Fig. 2. Sketch map of test section. Longitudinal and vertical velocity signals are obtained in combination with a TSI-IFA300 anemometer and a double slantwise hot-wire probe, which is tungsten filament with a 2.5-mm diameter. There are velocity signals detected in s at a frequency of f s = 50, 000 Hz at a certain vertical location, 0.7 mm off the board surface downstream from the spanwise slot. Momentum Reynolds number is = Uθ/ν = , with the turbulent boundary R eθ

3 1740 LIU Jian-Hua et al. Vol. 25 layer (TBL) thickness δ = 44.1 mm, when the velocity of free stream is m/s in standard TBL. Figure 3 shows the effect of 16 Hz blow and suction disturbance on longitudinal mean velocity profile, maintained until x = 16.8 (84 mm downstream the spanwise slot), which is normalized by the spanwise width of the slot 5 mm. Herein, 16-Hz actuation makes longitudinal mean velocity profile deviate from that of typical turbulent boundary layer obviously at x = 2 (10 mm downstream the spanwise slot), as well as 32- and 64-Hz disturbances, which is the reason why detailed study is carried out at streamwise location x = 2. The velocity profiles at streamwise location x = 2.0 in standard TBL is shown in Fig. 4 as the curve with asterisks. The velocity profiles are also plotted in Fig. 3 with 16-, 32- and 64-Hz blowing and suction disturbed turbulent boundary layers. Compared with the velocity profile for typical TBL, there are significant velocity defects of the velocity profiles for disturbed turbulence in near-wall region y + < 200 and velocity increases in the region 200 < y + < 700. Note that the effect decreases as the actuation frequency increases from 16 Hz to 64 Hz. The influence of blowing and suction under a certain frequency on longitudinal turbulent intensities in TBL is demonstrated by Fig. 5. All the values of u + rms under blowing and suction disturbance are larger than that in the standard TBL at the same vertical location. Herein, the value of u + rms under 16 Hz actuation noticeably exceeds those under the other two frequencies at the same vertical location. Note that the transition point of intensity curve under 16 Hz and 32 Hz actuation are consistent with the transition point in velocity profile, at 150 < y Another peak value of intensity appears in the cases of 16 Hz and 32 Hz. Large turbulent intensity represents the existence of much more coherent structures in this area. Sweep and ejection occur at any vertical location in the turbulent boundary layer but the probability and intensity always differ from one vertical location to another. If ether ejection or sweep is strengthened at a certain vertical location, the turbulent intensity increases definitely at the same location. Summarized from velocity profile in Fig. 4 and turbulent intensity in Fig. 5, longitudinal mean velocity defect and large turbulent intensity exist in the near wall region y + < 150 and longitudinal mean velocity increases and large turbulent intensity appears in the area 200 < y + < 700. As demonstrated by Liu et al., [9] the longitudinal velocity accelerates in stretching process, corresponding to sweeping process of coherent structures motion. Thus the longitudinal mean velocity defect in near wall region is caused by the relatively weaken sweeping process and ejection property is the dominant there. In the region 200 < y + < 700, longitudinal mean velocity increase demonstrates the enhancement of sweeping process and sweeping process is the dominant motion. Learned from the above discussion, it is true that the sweeping process have been pushed away from the wall by the way of weakening them in the near-wall flow by periodic blowing and suction actuation and strengthening them in the area 200 < y + < 700. As well known, large shear force is caused by the strong interaction during sweeping process in near wall region. According to the two ways to achieve drag reduction summarized by Tardu, [10] one is to decrease the intensity of the coherent structures and the other is to weaken their interaction with the near-wall flow by pushing them away from the wall. The sweep significantly weakened by 16- and 32-Hz blowing and suction in near wall region is good for drag reduction in the TBL. Fig. 3. Longitudinal mean velocity profile at streamwise locations x = 0.0 (on the slot), 2.0, 8.0, and Fig. 4. Longitudinal mean velocity U + at different actuation frequencies f = 16, 32 and 64 Hz in the TBL. Fig. 5. Longitudinal turbulent intensities versus wall normal distance, scaled with local shear velocity. The energy redistribution over scale at y + = 27 is shown in Fig. 6. Kinetic energy distribution at y + = 27 in the standard TBL, shown as the curve with asterisks, reaches the peak at the sixth scale. Namely, the maximum energy scale is the sixth scale at this vertical location in the standard TBL. According to DWT, the frequency band of the sixth scale (a = 6) structures is [f s /2 7, f s /2 6 ], namely [390.6, ] Hz.

4 No. 5 LIU Jian-Hua et al The periodic disturbance makes the maximum energy scale shift to the fifth scale (corresponding to frequency band [781.2, ] Hz) for the 64 Hz case and the seventh scale (corresponding to frequency band [195.3, 390.6] Hz) for the 16- and 32-Hz actuations. A large percentage of kinetic energy concentrated on the maximum energy scale denotes that the intensity of coherent structures at the maximum energy scale is strengthened by blowing and suction through the spanwise slot. This characteristic is bad for drag reduction. wide frequency band is drag-reducing effect of 16-Hz blowing and suction disturbance. Compared with the contour in the standard TBL shown in Fig. 7(d), the velocity contour is changed significantly by blowing and suction disturbance. However, the effect differs from one disturbance to another. The temporary duration of coherent structures increases in the situation of periodic disturbance at 16, 32 and 64 Hz, while the duration of sweep events (where the longitudinal fluctuating velocity is positive) increases noticeably at 16-Hz blowing and suction actuation. Moreover, a second ejection event (where the longitudinal fluctuating velocity is negative) occurs after the sweep event in the case of 16-Hz disturbance, as well as 32-Hz disturbance. However, the duration of sweep event under 32-Hz disturbance is much shorter than that in the case of 16-Hz actuation. The amplitude of the longitudinal fluctuating velocity is enlarged by the 16-Hz blowing and suction significantly, because the actuation speed is adversely proportional to the frequency of blowing and suction disturbance. Fig. 6. Energy redistribution over scale under 16, 32 and 64 Hz blowing and suction disturbances at y + = 27, respectively. Here E(a) denotes the kinetic energy occupied by the a-scaled coherent structures and e represents the total kinetic energy shared by all the coherent structures at y + = 27. The contours of conditional phase-averaged longitudinal fluctuating velocity under periodic disturbance are shown in Fig. 7(a) 7(c), corresponding to 16 Hz, 32 Hz and 64 Hz blowing and suction actuation orderly. Plotted in Fig. 7(d) is the contour of conditional phase-averaged longitudinal fluctuating velocity at the same nondimensionalized vertical location y + = 27 in the standard TBL. The horizontal axis represents duration time nondimensionalized by inner scale, and the vertical axis is scale of the coherent structures in the turbulent boundary layer. The velocity contour distributes only in the centre region of t + a plane, which indicates large velocity gradient and strong shear process shown in Fig. 7(d). Seen from Fig. 7, scale range of coherent structures during sweeping event (1st 9th scale) becomes noticeably larger in the 16-Hz blowing and suction case than that in a typical TBL (1st 6th scale). According to the correlation between scale and frequency band, frequency band of sweeping process is [48.83, 25000] Hz in the 16- Hz blowing and suction case and [390.62, 25000] Hz in the typical TBL. Frequency band of fluctuating velocity during sweeping events is widened by induced low frequency events by 16-Hz actuation, as well as in the 32-Hz actuation case. In comparison, frequency band is narrowed by 64 Hz blowing and suction disturbance. In view of Fig. 6, kinetic energy redistribution over a Fig. 7. Contours of conditional phase-averaged longitudinal fluctuating velocity during sweeping events at y + = 27 under (a) 16 Hz, (b) 32 Hz and (c) 64 Hz blow and suction disturbances and (d) in standard TBL. In conclusion, properties of TBL flow have been changed by periodic blowing and suction actuation through a spanwise slot of flat plate. Both the lift of sweeping event and the longer time-duration of sweeping event are drag-reducing effects in TBL, which are significant under 16 Hz blowing and suction actuation. The kinetic energy redistribution over a relatively wide frequency band in the 16-Hz blowing and the suction case is good for drag reduction. However, enhancement of local turbulent intensity or energy concentration on a certain scale prevents from drag reduction. Compared with the effect of blowing and suction actuation under 32 and 64 Hz, the effect of the 16-Hz blowing and suction is much more noticeable.

5 1742 LIU Jian-Hua et al. Vol. 25 References [1] ChoiK S, Debisschop I R, Clayton B R 1998 AIAA J [2] Suzuki Y and Kasagi N 1994 AIAA J [3] ChoiK S 1989 J. Fluid Mech [4] ChoiK S, Yang X, Clayton B R Glover E J, Atlar M, Semenov B N and Kulik V M 1997 Math. Phys. Engin. Sci [5] Kim K Y and Sung H J 2006 J. Fluid Mech [6] Smith A M and Scott S H 1996 J. Neurophysiol [7] Liu J H and Jiang N 2007 Chin. Phys. Lett [8] Haar A zur 1910 Theorie der orthogonalen Funktionensysteme, Mathematische Annalen 69 p 331 [9] Liu J H, Jiang N, Wang Z D and Shu W 2005 Appl. Math. Mech [10] Tardu S F 2001 J. Fluid Mech

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