Development of an array of pressure sensors with PVDF film

Transcription

1 Experiments in Fluids 26 (1999) Springer-Verlag 1999 Development of an array of pressure sensors with PVDF film I. Lee, H. J. Sung 27 Abstract A new type of an array of pressure sensors, composed of PVDF (polyvinylidenefluoride), was devised and evaluated. In order to obtain the system transfer function of the PVDF system, a dynamic calibration was performed utilizing the signal from a 1/8 inch B&K microphone as input. The time history of the unsteady pressure was then reconstructed from the output of the sensor by using this transfer function. The reconstructed pressure signals showed good agreement with the reference signal in both temporal and spectral sense. This sensor was then used to measure the wall pressure fluctuations in a two-dimensional turbulent channel flow. Various statistical moments were obtained from the measurement data set. Comparison of these quantities with the existing studies demonstrated satisfactory agreement. These tests give credence to the relevance and reliability of this sensor for applications in more complicated turbulent flows. List of symbols d diameter of pressure sensor [m] d nondimensional diameter of pressure sensor, du /ν E [ ] expected value f frequency [Hz] G ( f ) cross-spectrum between y(t) and x(t) yx G ( f ) autospectrum of y(t) yy H ( f ) transfer function of PVDF sensor j imaginary unit, 1 k streamwise wavenumber [1/m] 1 k convective wavenumber, ω/u [1/m] c c R (ξ, τ) cross-correlation of pressure with streamwise separation distance ξ and time delay τ [Pa2] pp Re Reynolds number based on channel centerline velocity and boundary layer thickness δ, U δ/ν o Received: 13 August 1996 / Accepted: 26 January 1998 I. Lee, H. J. Sung Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Kusong-dong, Yusong-ku, Taejon, , Korea Correspondence to: H. J. Sung Appreciation is extended to the referees who provided productive and helpful comments. The suggestions of the referees led to improvements in the revised paper. Re Reynolds number based on channel centerline velocity and boundary layer displacement thickness, θ, U θ/ν o R Reynolds number based on friction velocity, u δ/ν U channel centerline velocity [m/s] 0 U convection velocity [m/s] c u friction velocity, τ /ρ [m/s] w x(t) input of linear, time-invariant system x time series of x(t) n X( f ) discrete Fourier transform of x(t) k y (t) output of linear, time-invariant system estimation based on discrete time series with finite record length Greek symbols α decay constant in streamwise direction 1 Γ (ξ, ω) coherence function of pressure δ half width of channel [m] δ* displacement thickness [m] θ boundary layer displacement thickness [m] ν kinematic viscosity [m2/s] ξ streamwise separation distance [m] ρ density [kg/m3] τ time delay [s] τ wall shear stress [Pa] w Φ(ω) autospectrum of pressure [Pa2 s] Φ (ξ, ω) cross-spectrum of pressure [Pa2 s] p Φ (k, ω) streamwise frequency-wavenumber spectrum of pressure [Pa2 ms] p 1 (ξ, ω) phase of cross-spectrum of pressure [rad] p ω circular frequency [rad/s] 1 Introduction Surface pressure fluctuations are closely associated with flow unsteadiness and aerodynamic noise generation in the immediate vicinity of the wall. A knowledge of this quantity is of prime importance in understanding the dynamic behavior of wall turbulent flows and flow noise. One example of a practical application is noise generation caused by the flow over a sonar transducer mounted on a ship or a submarine. Moreover, unsteady pressure data play a fundamental role in the analysis and reduction of the sound radiated from the surface. To predict the acoustic pressure level based on Curle s (1955) solution of the Lighthill equation, an accurate spatial distribution of the wall pressure fluctuations is needed. Much attention to analytical and experimental aspects has been given

2 28 to the construction of a reliable spectral model of pressure fluctuations (Blake 1986; Chase 1991; Corcos 1964). Control of the aforementioned flow noise depends on knowledge of the spatial properties of pressure strength in the vicinity of the surface as a function of position in the flow [5]. Obviously, in order to secure detailed information of pressure behavior in the wavenumber domain, it is advantageous to resolve the spatial and temporal distribution of surface pressure fluctuations by using an array of sensors near the wall. A technique for making such measurements using an array of pressure sensors, which is termed wave-vector filtering, was proposed by Maidanik and Jorgesen (1967). A literature survey reveals that multi-channel applications in measuring the surface pressure fluctuations with a small array of sensors are not numerous. This is attributed largely to difficulties in practical instrumentation; the conventional pressure sensors, such as pressure transducers require considerable space to be installed. Furthermore, when these sensors are utilized, the measuring holes for pinhole microphones are machined onto the surface of the test section, thereby producing a significant distortion of pressure fluctuations in a certain frequency band. Recently, special attention has been given to the utilization of piezoelectric film which can be glued directly to the surface (Nitsche and Mirow 1989). The piezeoelectric film, which is flexible and smooth in nature, gives minimal disturbance to the flow field. Furthermore, by plating a suitable electrode pattern on both sides of the piezoelectric film, an array of pressure sensors with small sensor spacing can be fabricated (Nitsche et al. 1989). However, the qualitative characteristics of this piezoelectric film have not been investigated in sufficient detail. The application of this film in the measurement of fluid flow has been restricted to such qualitative endeavors as the prediction of laminar-turbulent transition or shock detection. The main purpose of this study is to devise and evaluate a new type of an array of pressure sensors, which is composed of piezoelectric film. The frequency response of this sensor was examined by using a reference microphone. In order to compensate for the frequency response of the sensor, a linear modeling of the pressure measurement system was performed, and the transfer function of the sensor was estimated. To validate the reliability of the present array of sensors, in which a signal from one channel can intervene into another channel, a crosstalk measurement was made. It was then deployed to reconstruct the pressure time history from the output of the sensor. Various unsteady pressure signals were generated by using a woofer speaker with a bandwidth of 10 Hz 2 khz. The reconstructed signal was compared wth the output of the reference microphone in the time-domain as well as in the frequency-domain. In order to assess the applicability of this array of sensors, a precise measurement of the surface pressure fluctuations was made in a two-dimensional fully-developed turbulent channel flow, for which the data of well-documented experiments as well as numerical simulation are available (see Table 1). The Reynolds number of the flow, which was defined by the channel half width, was The statistical characteristics obtained from the pressure fluctuation were compared with the existing available data, and the agreement was satisfactory. 2 PVDF array of pressure sensors 2.1 Fabrication of the sensor In the present study, a 28 μm-thick PVDF (polyvinylidenefluoride) film was utilized as the sensing element. PVDF film is suitable for sensing small amplitude pressure fluctuations due to the wide dynamic range and large piezoelectric constant of the transparent semicrystalline polymer film (Nitsche and Mirow 1989). Figure 1a shows the PVDF film used in this study, where 4 10 arrays of circular electrodes are plated. The diameter of these circular electrodes is 3 mm, and the spacing between centers of each electrode is 6 mm. Each electrode serves as an independent sensor and the capacitance of each sensor is about 26.9 pf. The sensing unit, which is displayed in Fig. 1b, consists of the PVDF film and female connectors, which accommodate the electrical wiring between the sensor and the signal processing system. It is noted that the PVDF sensor is mainly responding to pressure fluctuations and is insensitive to surface shear stress fluctuations. This is based on the fact that the present PVDF sensor is laterally clamped by hard glue. For turbulent boundary layer flows, the ratio of wall shear stress fluctuations Fig. 1. PVDF film sensor a plan view of PVDF film; b schematic diagram of sensor unit

3 to mean wall shear stress τ /τ ranges from 0.36 to 0.40 w, w (Alfredsson et al. 1988; Kim et al. 1987). Moreover, p /τn w is about 3. (which will be found later in Fig. 11b). Accordingly, τ /p is about In addition, it is known that the w, piezoelectric constants of the material indicate a shear stress sensitivity less than 10% of the normal stress sensitivity. Therefore, the shear stress component in the transducer output is negligible. A block diagram of the overall pressure measurement system is illustrated in Fig. 2. By employing a 32-channel charge-amplifier system (Kistler Type 5017), 32 unsteady signals can be sampled simultaneously. Prior to A/D conversion, a lowpass filter option with the cutoff frequency of 3 khz was used as an anti-aliasing filter. As has been documented in the literature, noise can interfere with the measurement system by way of the connection lines between various components (Deoblin 1983). In order to alleviate such noise originating from the background fluctuating electromagnetic field, an aluminium foil shield was inserted to wrap around the experimental facilities, thereby reducing 60 Hz noise. Noise from ground loops due to multiple grounds within the circuit was minimized by joining the grounds together. 2.2 Sensor characteristics To determine the frequency characteristics of the present pressure sensor, a plane sound wave ranging from 10 Hz to 2 khz was generated by means of a woofer speaker driven by a white noise generator (B&K type 1207). A schematic diagram of the unsteady pressure calibrator is exhibited in Fig. 3. It is common knowledge that the sound from a woofer speaker may not be entirely white. Thus, precise sound measurements should precede the measurements of the frequency characteristics of the PVDF sensor. Toward this end, a 1/8 inch condenser microphone (B&K type 4138) was employed as a reference sensor. As shown in Fig. 3, the reference microphone and the PVDF sensor were installed inside the calibrator and the unsteady signals from the woofer speaker were measured simultaneously. The diameter of the calibrator is 80 mm and the length is 1 m. This calibrator was designed to get a plane sound wave below about 2 khz. To check the uniformity of sound field in the calibrator, the PVDF sensor was replaced with another B&K microphone. A comparison between two microphones indicated that the nonuniformity is less than 5% below 2 khz. The autospectra from the two sensors are plotted in Fig. 4. Since the frequency response from the 1/8 inch microphone is not reliable below about 10 Hz, no experimental data are presented for frequencies less than 10 Hz. It is seen that the signal-to-noise ratio is very low at low frequencies. A peak noise is detected in the PVDF signal near 60 Hz. For a reliable sensor development, a primary concern is the appropriate remedy to compensate for the PVDF frequency response. 2.3 Linear model of measurement system Figure 5 shows the rms value of the PVDF output (V) plotted against the rms value of the incident sound pressure (Pa). It is evident that a linear relation holds between the pressure and the sensor output. The frequency response of the PVDF sensor was also examined by varying the pressure levels of incident sound, of which autospectrum is almost the same as that of the incident sound shown in Fig. 4. It is revealed that the shape of the autospectrum of the PVDF output is nearly invariant regardless of the incident sound pressure level except in the extremely low frequency band. Hence, a linear model of the relation between the input and the output of the PVDF sensor can be established. 29 Fig. 2. Block diagram of pressure measurement system Fig. 3. Pressure calibrator Fig. 4. Comparison of autospectra from reference microphone output and PVDF output

4 30 Fig. 5. Linearity of PVDF sensor The basic assumption concerning the model is that both the distortion and the phase delay of the reference microphone are identically zero, and the output of the reference microphone can be assumed to be directly proportional to the incident pressure fluctuations. Thus, the proportionality constant corresponds to the sensitivity of the microphone. Based on this assumption, output from the reference microphone can be regarded as input to the PVDF sensor, a linear model stems from outputs from the reference microphone and the PVDF film sensor. The transfer function H ( f ) between the input x(t) and the output y(t), in a linear time-invariant system, can be estimated (Bendat and Piersol 1991); H( f ) E GI yy ( f ) GI yx ( f ) (1) where GI yy ( f ) and GI ( f ) denote the autospectrum of y(t) and yx the cross-spectrum between y(t) and x(t), respectively. E [ ] stands for an expected value and the tilde ( ) represents the estimation based on the discrete time series with finite record length. Here, the input x(t) and the output y(t) represent the reference microphone signal and the PVDF sensor signal, respectively. A typical transfer function of the PVDF sensor is plotted in Fig. 6. It can be easily found that the frequency response of the PVDF sensor is a function of frequency. Thus, as mentioned earlier, a certain compensation method is essential for a reliable measurement with the PVDF sensor. 2.4 Reconstruction of unsteady pressure The reconstruction of real-time pressure from the output signal of PVDF sensor is performed using the following procedure: 1. The discrete-fourier transform (DFT) of the output signal from PVDF sensor is obtained; Y ( f ) 1 2πkn y exp (2) k n j N n 0 Fig. 6. Transfer function of the PVDF sensor a magnitude; b phase 2. The corrected DFT from the PVDF signal X ( f k )is produced; X ( f k ) Y ( f k ) H ( f k ) 3. The inverse-discrete Fourier transform is performed, which yields the dynamically corrected time series, x n ; x n 1 N (3) 1 2πkn X( f ) exp j (4) k N n 0 Figure 7 represents the signal reconstruction in the case of the band-limited white noise input, which has the same conditions as those of Fig. 4. The autospectrum of the reconstructed signal in Fig. 7 shows good agreement with that of the reference signal over most frequency bands. If we compare this output with the preceding figure (Fig. 4), where the large discrepancy is exhibited, the present agreement is quite remarkable. The successful reconstruction in the frequency domain indicates that, by utilizing the present signal reconstruction method, the PVDF sensor is promising in estimating the spectral energy distribution as well as the total energy of the pressure fluctuations. A number of reconstruction tests have been made of the unsteady pressure signals in the time domain with varying frequencies, in particular at low frequency bands. In an effort to appraise the effectiveness of PVDF reconstruction for unsteady pressure signals, comparisons are made between the two signals. Here, the reference signals are generated by a woofer speaker. One of the results is shown in Fig. 8, in which the input is a 10 Hz-sine wave. The phase shift between the two signals is mainly caused by the characteristics of the B&K microphone. Nevertheless, it is still a valid test of signal reconstruction. In Fig. 8, almost all the frequency content, except for the 10 Hz wave, in the original PVDF signal are

5 31 Fig. 9. Crosstalk measurement apparatus Fig. 7. Result of signal reconstruction: white noise input Fig. 8a, b. Result of signal reconstruction: 10 Hz sine wave input. a Before reconstruction; b after reconstruction suppressed, and the phase of the signal is perfectly recovered after reconstruction. This is in support of the usefulness of PVDF sensor as an unsteady pressure sensing device. 2.5 Sensor crosstalk measurement In an array of devices, crosstalk between sensors can be a significant problem. This phenomenon stems from the vibration modes of sensor unit, failure in the shielding of each channel at the connections, etc. The crosstalk should, therefore, be reduced below a certain admissible level; otherwise, the reliability of the measured distribution of pressure fluctuation from this sensor would be low. In this study, a crosstalk measurement apparatus, shown in Fig. 9, was devised. This apparatus consists of a big outer circular cylinder and an inner small circular pipe, which is fitted closely to the cylinder. This circular pipe is filled with sound-absorbing material, and it has two rigid endplates, each of which has a small hole. These holes are connected, with a 2.5 mm diameter, stainless steel pipe. For a crosstalk measurement, a woofer speaker is installed on one side of the inner cylinder and the PVDF sensor unit is attached on the other side. The hole is exactly aligned with one of 40 sensors. The rigid endplates and sound absorbing material inside the cylinder ensure that the sound generated from the woofer speaker can be transmitted only through the stainless steel pipe, which acts as a waveguide. Moreover, the sound through this waveguide is effectively focused onto only one of 40 sensors, because the remaining parts of the PVDF sensor are faced with the rigid signal endplates, i.e., the gap distance between the endplate and sensor surface is kept below 1 mm. The crosstalk can be estimated from the ratio of the output from the non-sound incident sensor to that from the sound incident sensor. Since sound with broadband frequencies is generated by the woofer speaker, the crosstalk is estimated. In the present measurement, the crosstalk value, expressed in decibel scale, is seen to be smaller than 40 db over the frequency range. This amounts to only one percent of the signal from one channel affecting the other channels. Thus, the distribution of wallpressure fluctuations using this sensor can be measured confidently. 3 Channel flow measurements 3.1 The wind tunnel and its flow parameters The wind tunnel used in this study has a 800 mm 80 mm rectangular cross-section, where the length in the streamwise direction is 6000 mm. Air is blown from a centrifugal blower upstream of the test section and flow conditioning devices. In order to reduce the noise generated from the fan, two kinds of silencers were made, which were located between the fan and

6 32 the diffuser: one is an expansion-chamber type silencer for noise reduction in a wide frequency range and the other is the Helmholtz resonator to suppress the noise from the fan blade passing frequency. In Fig. 10, the sound spectra with and without silencers are plotted for comparison. The insertion loss, which is defined by the ratio of the sound pressure with silencers installed to that without any silencers, was measured. In the frequency range below 5 khz, the insertion loss was found to be db with a uniform sound reduction. Hot-wire velocity and turbulence intensity measurements were made for estimating flow parameters, which were characterized by the following values: velocity at the channel centerline U m/s; boundary layer thickness δ 40 mm; Reynolds number Re U 0 δ/ν , Re U 0 θ/ν 2810, shear velocity u m/s; Reynolds number with respect to u and δ, R u δ/ν A comparison with other data was made, which is compiled in Table Root mean square of pressure The wall pressure field contains a wide range of wave numbers or length scales. Since the present pressure sensor has finite dimensions, high wavenumber components become attenuated. Among the various components of wall pressure fluctuations, the viscous length scale is selected, where the nondimensional sensor diameter d du /ν represents the amount of high wavenumber attenuation (Corcos 1963). It is desirable to make the sensor diameter as small as possible for detailed measurements of pressure near the wall. In this study, the nondimensional sensor diameter is measured as d 74. The rms values of pressure p are illustrated in Fig. 11a as a function of d, where the pressures p are normalized by the dynamic pressure q 0 1/2ρU2 0. As seen, the present measured point is located within the boundary of other relevant data (Langeheineken and Dinkelacker 1978; Schewe 1983). The dependence of the wall pressure fluctuations on the Reynolds number R u δ/ν is displayed in Fig. 11b, together with results obtained by various investigations. A comparison of the wall pressure measurements with different investigations reveals that p /τ w depends on the Reynolds number. It can be seen that p increases with increasing R. Table 1. Comparison of flow parameters Case Re Re R d Willmarth and Wooldridge (1962) Blake (1970) Bull and Thomas (1976) Schewe (1983) Farabee and Casarella (1991) Choi and Moin (1990) Present Fig. 11. Root mean square of pressure fluctuations a p against normalized sensor diameter, d ; b p against R Farabee and Casarella (1991) proposed the following relation by integration over the various spectral regions, p τ w ln (R /333) (R 333) (R 333) (5) A closer inspection of Fig. 10b indicates that the present data falls slightly below Eq. (5). As pointed out by Farabee and Casarella (1991), the data scattering in Fig. 11b is attributed to transducer spatial resolution limitations. Fig. 10. Comparison of sound spectra in test section 3.3 Spectral features of wall pressure fluctuations A difficulty that arises in comparing wall pressure measurements from different investigations is the different characteristic velocity and length scales between experiments. Thus, it

7 is meaningless to simply compare the spectra from various studies without any nondimensionalizing procedure. A reliable scaling law is presented which is slightly modified from the proposal by Keith et al. (1992); (a) Scaling by outer variables (centerline velocity, boundary layer displacement thickness) Frequency : ωδ/u 0, spectrum : Φ(ω)/(ρ2δU3 0 ) (b) Scaling by inner variables (friction velocity, viscous length scale) Frequency : ων/u2, spectrum : Φ(ω)u2 /(τ2ν) (c) Scaling by mixed inner and outer variables Frequency : ωδ/u 0, spectrum : Φ(ω)U 0 /(τ2δ). For measuring the spectra, 500 ensemble sets containing 4096 data were used. The random error ε r is given as ε r 1/ n d, where n d is the number of ensemble sets (Bendat and Piersol 1991). The corresponding random error amounts to about 4.5%. The autospectra, nondimensionalized by the outer scaling, are illustrated in Fig. 12a, where the results of Schewe (1983), Choi and Moin (1990), Willmarth and Wooldridge (1962), are included for comparison. Large scatter can be found in data sets. As shown, a complete collapse between data sets is not provided by the outer scaling. It is known that the discrepancy in the high frequency region can be attributed to the differences in transducer spatial resolution and the differences in the low frequency region are caused by Reynolds number effects Keith et al. (1992). The spectra obtained with the inner scaling are displayed in Fig. 12b. Except for the data of Willmarth and Wooldridge (1962), the spectra seem to be collapsed well in the high frequency range. Willmarth and Wooldridge used a large sensor, i.e., d 383 in their experiment. This coarse spatial resolution led to severe attenuation of high frequency components of pressure fluctuations. As pointed out by Keith et al. (1992), to alleviate the Reynolds number dependence from the outer scaling, a correction factor (U 0 /u )4 is used in the mixed scaling. A closer examination of Fig. 12a c reveals that the mixed scaling gives better agreement between data sets in the lower frequency range. The high frequency range data of Choi and Moin (1990) is shown to be less than most values reported experimentally. In general the present pressure measurements follow the other data sets satisfactorily. 3.4 The spatial characteristics of pressure The spatial characteristics of the pressure fluctuations are obtained from the cross spectrum Φ p (ξ, ω), which is defined as Φ (ξ, ω) R (ξ, τ)e j dτ (6) p pp in which R pp (ξ, τ) represents the cross-correlation of wall pressure fluctuations R pp E [ p (x, t) p (x ξ, t τ)]. Here, ξ and τ are the streamwise sensor spacing and time delay, respectively. In general, since the cross-spectrum is a complex quantity, the coherence function Γ (ξ, ω) is employed for the convenience of data presentation Φ Γ (ξ, ω) p [ Φ (ω) Φ (ω)]1/2 p1 p2 In the above, Φ p1 (ω) and Φ p2 (ω) denote the spectra of wall pressure at points x and x ξ, respectively. In this study, the cross spectra and the coherence are obtained in the range 0.15 ξ/δ 5.5. It is known that large scale components of pressure field lose coherence slowly during convection with U c while the small-scale components lose coherence in distances proportional to their wavelengths. Considering this phenomena, the coherence decays with increasing the separation spacing ξ. The convection velocity U c is obtained from the phase of the cross-spectrum Φ p (ξ, ω), i.e., U c (ξ, ω) ωξ/ (ξ, ω). Here, U c varies with ξ and ω. Corcos (1964) assumed that the streamwise coherence decays spatially as a function of the similarity variable ωξ/u c. Thus, the coherence can be expressed in an exponential form, Γ(ξ, ω) exp ( α 1 ωξ/u c ), (8) where α 1 is an empirically determined decay constant, α Figure 13 represents the behavior of the coherence spectra with respect to ωδ/u c when the streamwise separation ξ 0 is fixed. If Corcos similarity scaling given by Eq. (8) holds for all values of the similarity variable, then Γ goes to unity as ω 0. (7) 33 Fig. 12. Comparison of spectra of pressure fluctuations using a outer scaling; b inner scaling; c mixed scaling

8 34 Fig. 13. Streamwise coherence against ωδ/u c Fig. 14. Streamwise coherence against ω 0 ξ/u c However, as ω decreases, the low frequency deviation is present. This deviation from the similarity scaling trend is evidence of a low wavenumber cutoff to the similarity scaling behavior. Clearly the peak in the coherence spectra, which is an approximate measure of the lower limit for which similarity scaling no longer holds, occurs at approximately the same value of ωδ/u c, although there is a mild shift in the value with the spacing ξ/δ. These trends are consistent with the results of Farabee and Casarella (1991), which were measured by flush-mounted pinhole microphones. Next, to provide a more intuitive view of the data, the streamwise separation ξ is varied at a fixed frequency ω 0. The coherence spectra are displayed in Fig. 14 as a function of ω 0 ξ/u c for selected values of the dimensionless frequency (16.3 ωδ/u 195.4). It is seen that each spectrum approaches a value of unity as ξ 0 and the spectra generally collpase to a universal similarity scaling behavior for the higher values of fixed frequency. The exponential scaling law in Eq. (8) using α is also included. However, in the lower frequency range ωδ/u 16.3, the similarity is less satisfactory. For comparison, the values of frequencies and sensor separation at which measurements have been shown are carefully chosen to coincide with those used by Farabee and Casarella (1991). An examination of the coherence data suggests that the wall-pressure fluctuations undergo a spatial decay which can be expressed by a similarity variable ωξ/u c. Another spatial features of the wall pressure fluctuations can be extracted from its streamwise convection. The frequency-wavenumber spectrum Φ(k 1, ω) can be obtained by Fourier transforming the cross-spectrum Φ p (ξ, ω) with respect to space. Blake (1986) provided a formula for the wavenumber spectrum derived from the Corcos model of the cross-spectrum, whose onedimensional version is given as follows: Fig. 15. Frequency streamwise wavenumber spectrum Φ(k 1, ω) Φ(ω)(α 1 k c )2 [(k 1 k c )2 (α 1 k c )2] (9) where k c is the convective wavenumber given as k c ω/u c. According to Eq. (9), the wavenumber spectrum attains a maximum value at k 1 k c, and this represents the convective feature of wall pressure. In Fig. 15, the wavenumber spectrum Fig. 16. Surface plot of frequency streamwise wavenumber spectrum

9 is displayed, which was obtained from the discrete Fourier transform (DFT) of each time history. At a fixed ωδ/u , one-sided wavenumber spectrum Φ p (k 1, ω) was then obtained. The Corcos model spectrum from Eq. (9) is also shown for comparison. In Eq. (9), the mixed variable scaling was employed, and no dependence of U c upon ξ were incorporated. The peak value is clearly seen at k 1 δ 11, which corresponds to the convective wavenumber at ωδ/u , i.e., U 0 /U c 1.61, k c δ ωδ/u c However, the Corcos model overpredicts the spectrum in the subconvective wavenumbers and underpredicts in the superconvective wavenumbers. This finding is consistent with the result of Keith and Abraham (1997). To show the convective characteristics of pressure fluctuations, a surface plot of the frequency-wavenumber spectrum is illustrated in Fig. 16. A distinctive ridge can be found, which shows the convective feature of wall pressure fluctuations. 4 Conclusions A new array of PVDF pressure sensors was devised and its dynamic characteristics were examined. By adopting linear system modeling, the reconstruction of the signal could be achieved satisfactorily. The reconstruction of unsteady pressure signals was also examined and it was found to be very reliable. With the present reconstruction method, the PVDF sensor provides a promising tool for measuring both the temporal and spatial pressure fluctuations at the wall. To validate the present PVDF sensor, a wind tunnel measurement of wall pressure fluctuations in a two-dimensional fully developed channel flow was made, for which the existing data are available for comparison. A variety of statistical quantities were obtained, and by comparison with other results, the usefulness of the PVDF sensor in investigating turbulent wall pressure fluctuations was shown. Spectral features of the spatial pressure fluctuations, such as the streamwise spatial decay and the streamwise convection, were evaluated against data from the literature and the empirical model of Corcos. The agreement was shown to be satisfactory. Deoblin EO (1983) Measurement systems: application and design. 4th ed. New York: McGraw-Hil Farabee TM; Casarella MJ (1991) Spectral features of wall pressure fluctuations beneath turbulent boundary layer. Phys Fluids A3: Keith WL; Hurdis DA; Abraham BM (1992) A comparison of turbulent boundary layer wall-pressure spectra. J Fluids Eng 114: Keith WL; Abraham BM (1997) Effects of convection and decay of turbulence on the wall pressure wavenumber-frequency spectrum. J Fluids Eng 119: Kim J; Moin P; Moser R (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177: Langeheineken T; Dinkelacker A (1978) Fortschritte der Akustik: 391 Duseldorf: VDE-Verlag Maidanik G; Jorgensen DW (1967) Boundary wave-vector filters for the study of the pressure field in a turbulent boundary layer. J Acoust Soc Am 42: Nitsche W; Mirow P (1989) Piezo-electric foils as means of sensing unsteady surface forces. Exp Fluids 7: Nitsche W; Mirow P; Dörpler T (1989) Applications of piezoelectric foils in experimental aerodynamics. Proc Int Congress Instrumentation in Aerospace Simulation Facilities ICIASF 89: Schewe G (1983) On the structure and resolution of wall-pressure fluctuations associated with turbulent boundary-layer flow. J Fluid Mech 134: Sherman CH; Ko SH; Buehler BG (1990) Measurement of the turbulent boundary layer wave-vector spectrum. J Acoust Soc Am 88: Willmarth WW; Wooldridge CE (1962) Measurement of the fluctuating pressure at the wall beneath a thick boundary layer. J Fluid Mech 14: References Alfredsson PH; Johansson AV; Haritonidis JH; Eckelmann H (1988) The fluctuating wall-shear stress and the velocity field in the viscous sublayer. Phys Fluids 31: Bendat JS; Piersol AG (1991) Random data: analysis and measurement procedure. 2nd ed. New York: Wiley & Sons Blake WK (1970) Turbulent boundary-layer wall-pressure fluctuations on smooth and rough walls. J Fluid Mech 44: Blake WK (1986) Mechanics of flow-induced sound and vibration, Chap. 8 New York: Academic Press Bull MK; Thomas ASW (1976) High frequency wall-pressure fluctuations in turbulent boundary layers. Phys Fluids 19: Chase DM (1991) The wave-vector-frequency spectrum of pressure on a smooth plane in turbulent boundary-layer flow at low Mach number. J Acoust Soc Am 90: Choi H; Moin P (1990) On the space-time characteristics of wallpressure fluctuations. Phys Fluids A2: Corcos GM (1963) Resolution of pressure in turbulence. J Acoust Soc Am 35: Corcos GM (1964) The structure of the turbulent pressure field in boundary layer flows. J Fluid Mech 18: Curle N (1955) The influence of solid boundaries upon aerodynamic sound. Proc Roy Soc London Ser A 231: