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Influence of stiffeners on plate vibration and radiated noise excited by turbulent boundary layers
Applied Acoustics 80 (2014) 28–35
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Influence of stiffeners on plate vibration and radiated noise excited by turbulent boundary layers Bilong Liu a,⇑, Hao Zhang a, Zhongchang Qian a, Daoqing Chang a, Qun Yan b, Wenchao Huang b a b
Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, 100190 Beijing, PR China Key Laboratory of Aeroacoustics and Dynamics, Aircraft Strength Research Institute, 710065 Xian, PR China
a r t i c l e
i n f o
Article history: Received 22 July 2013 Received in revised form 2 January 2014 Accepted 14 January 2014 Available online 6 February 2014 Keywords: TBL noise Aircraft noise Plate vibration
a b s t r a c t The influence of stiffeners on plate vibration and noise radiation induced by turbulent boundary layers is investigated by wind tunnel measurements. Plates with and without stiffeners are tested under the flow speed of 60 m/s, 71 m/s and 86 m/s, respectively. The stiffeners are set either perpendicular or parallel to the direction of the free stream. Measured vibration and noise levels are compared with theoretical calculations, where wall pressure cross-spectra are described by the Corcos model. For the plates tested, it is evident that stiffeners perpendicular to the direction of the free stream could increase noise radiation, but have almost no influence on vibration level of plates. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The noise and vibration generated by turbulent boundary layer (TBL) has importance for the design of high speed vehicles. For any type of prediction of noise and vibration levels inside a high speed vehicle, the noise and vibration generated by the TBL must be well described. The noise and vibration of the structure depends on the velocity of the vibrating plates, which in turn is determined by the speed of the vehicle, the geometry, the dimensions and losses, or damping of the plates. Clearly the acoustic properties of the interior system, trim panels, etc., will also influence the interior noise of the vehicle. Much attention has been paid to the prediction of boundary layer induced noise and vibration as discussed in literature [1– 17]. Among these references, Graham [2] proposed a model to predict the noise induced by TBL in an aircraft. Graham successfully developed an analytical expression to evaluate the modal excitation terms. In this way the time to calculate the excitation field was thereby significantly reduced. In another paper, the merit of various models describing the cross power spectral density induced by a flow or turbulent boundary layer across a structure was discussed in some detail by the same author [3]. Han attempted to predict TBL induced noise by energy flow analysis [6,7]. The method has proved to be successful for predicting the response of flat isotropic panels subjected to TBL excitation. However, the noise radiated by the panel was not properly described.
⇑ Corresponding author. Tel.: +86 1082547504. E-mail address: [email protected] (B. Liu). 0003-682X/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apacoust.2014.01.007
The reason is resulting from the inaccuracy of the method to predict the modal averaged radiation efficiency. Recently Maury et al. used wavenumber approach to model the response of panel subjected to acoustic and TBL excitations [8,9]. Finnveden and Birgersson [11–13] introduced a series of numerical method to calculate the vibration of plates induced by boundary layer. While Rocha and Palumbo [15] adopted modal summation method to investigate the sensitivity of sound power radiated by aircraft panels to turbulent boundary layer parameters. The models above are restricted to flat uniform panels without stiffener attachments. Concerning with the influence of stiffeners, Liu et al. [16–18] investigated the effects of ring frames, stringers, damping, and curvature on the noise transmission of aircraft plates under the excitation of acoustic diffuse field and TBL respectively. The results indicate that significant differences resulting from stiffeners and damping could appear when the plates are subjected to acoustic diffuse field and TBL excitations. More specifically, the ring frames have almost no influence on sound transmission through a typical aircraft panel subjected to excitation of an acoustical diffuse field. While the ring frames have significant influences on TBL induced noise radiation according to the prediction. This conclusion, however, is based on numerical prediction. Experimental work together with predictions on the noise radiation of TBL excited plates with and without stiffeners is not available in the literature. The purpose of current work is trying to investigate the influence of stiffeners on the plate vibration and noise radiation under TBL excitations. Different size plates with and without spanwise (perpendicular to the direction of free stream) and streamwise
B. Liu et al. / Applied Acoustics 80 (2014) 28–35
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(parallel to the direction of free stream) stiffeners were tested under the flow speed of 60 m/s, 71 m/s and 86 m/s, respectively. Then the test results were analyzed and compared with theoretical prediction. 2. Measurements A detailed description of measurement set-up and relevant data on TBL spectrum are given in this section. 2.1. Experimental set-up The test was conducted in a subsonic wind tunnel specially designed for acoustic measurement. The diagram for the test set-up is illustrated in Fig. 1. A wind turbine is used to produce volume flow with different velocities. A silencer with the perimeter of about 6 m and the length of about 15 m are mounted downstream and it could eliminate the turbine fan noise effectively, so that pure TBL excitation is obtained. A rectangular aluminum pipe is connected downstream of the silencers. The wall thickness of the pipe is 10 mm, the width and height of the pipe are 170 mm and 370 mm respectively. The test plate is mounted as part of the pipe sidewall. Inside surfaces of the test plate and the pipe wall are at the same plane. A photo of the plate under testing is shown in Fig. 2 as an example. The end of pipe is streamlined to trumpetshaped outlet with absorption liners. Most part of pipe is located in an anechoic room (6.21 7.86 5.05 m, cut off frequency 80 Hz). Two 1/4 in. microphones (B&K, Type 4944) with a separation distance of 1.3 cm were mounted on the pipe wall to monitor the TBL spectrum. An illustration of microphone installation is shown in Fig. 1, where the diameter of the hole in front of microphone is 1 mm. The average sound intensity radiated by the plates is measured by intensity probe (B&K Sound Intensity Probe Kit, Type 3599) with scanning method according to ISO 9614-2 [19]. A 1.0 g accelerometer (PCB, Type 352c68) is randomly attached to the plate by using beeswax to measure vibration level. The vibration is measured by accelerometer repeatedly for sufficient points of the plate and then averaged vibration level is calculated. The measured data is recorded by B&K Pulse, Type 3610. Six aluminum plates, for their parameters listed in Table 1, are tested. All of plates are flat and with thickness of 1.1 mm. The plate I0 has the same the size of the plate I, while attaches two equallyspaced spanwise stiffeners. The stiffeners are connected to the plates by rivets. The plate J0 has the same the size of the plate J, while in the middle of it has one spanwise stiffener attached. The plate J1 has the same the size of the plate J, but has one streamwise stiffener enclosed in the middle. The plate K is the smallest plate and its size corresponds to that of the sub-plate between two
Fig. 2. A photo of the plate with enclosed spanwise stiffeners under testing.
spanwise stiffeners of the plate I. The height and thickness of stiffeners are 50 mm and 1.5 mm respectively. 2.2. The boundary layer thickness The boundary layer thickness, measured by hot wire anemometer, is an important parameter to characterize the turbulence. The measured results are shown in Table 2, where d and d⁄ denotes the boundary layer thickness and the displacement thickness respectively, Uc and Us represent the convection velocity and friction velocity. According to the model proposed by Corcos [20], Uc = 0.6–0.8U1, where U1 is the free-stream velocity. As an example, the velocity profile for a free flow velocity of 71 m/s is also given in Fig. 3. Comparing the results with theory given by Schlichting [21], it is verified that the velocity profile is as in a classical boundary layer. 2.3. Wall pressure power spectral density The point auto spectral density of the wall pressure Upp(Pa2/Hz) was measured using two fixed microphones according to Fig. 1. The measured data with flow speed of 60, 71 and 86 m/s and in the frequency from 100 Hz to 3500 Hz is given in Fig. 4. The measured spectrum levels agree with that in Ref. [22,23]. The wall pressure coherence measured with two microphones with a streamwise length 1.3 cm is given in Fig. 5, and the results calculated according to Corcos’s model [20] is also plotted for a comparison. 3. Measured plate vibration and noise radiation 3.1. Influence of the spanwise stiffeners To perform the measurements, two B&K pulse systems are synchronously connected, which yields totally 12 channels. The fre-
Fig. 1. Schematic of test set-up for plate vibration and sound radiation.
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B. Liu et al. / Applied Acoustics 80 (2014) 28–35
Table 1 Plates used in measurement. Plate no.
Length width cm
Thickness (mm)
Number of stiffeners
I
93 30
1.1
0
I0
93 30
1.1
2
J
62 30
1.1
0
J0
62 30
1.1
1
J1
62 30
1.1
1
K
31 30
1.1
0
Table 2 Estimated flow data from measurements.
Schematic of plate
1
Measurements Corcos' model
0.9
U1
60 (m/s)
71 (m/s)
86 (m/s)
d (mm) d⁄ (mm) Uc (m/s) Us (m/s)
20.5 2.51 42 1.8
20.5 2.48 49.7 2.13
22.5 2.71 60.2 2.58
0.8
Coherence
0.7 0.6 0.5 0.4 0.3 0.2 0.1
1
measured data theory predicted
0.95
0
500
1000
1500
2000
2500
3000
3500
4000
Frequency (Hz) 0.9
Fig. 5. Wall pressure coherence as function of frequency. Solid line is measured coherence and dot line corresponds to calculation according to Corcos’s model. Flow speed 71 m/s.
Uz/U∞
0.85 0.8 0.75 0.7 0.65
0
0.2
0.4
0.6
0.8
1
z/δ Fig. 3. Velocity profile for a free flow of 71 m/s at a distance z from wall scaled with the boundary layer thickness d. Circles are measured velocities and solid line corresponds to a power velocity distribution law [21] with n = 7.2.
86m/s measured 71m/s measured 60m/s measured
-5
dB (ref. 1 Pa2/Hz)
Auto spectral density of wall pressure,
0
-10 -15 -20 -25 -30 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Frequency (Hz) Fig. 4. Measured and predicted wall pressure auto spectral density at flow speed of 60 m/s, 71 m/s and 86 m/s.
quency resolution is 2 Hz and the Hanning window is used. The average number for auto spectrum is set to 150, with 66.7% overlap. The measured data of the plate I for the three different wind speeds is given in Fig. 6, where the measured velocity of the plate in narrow band analysis is shown in velocity auto spectrum in dB (Ref. 1018 m2 s2). The measured noise radiation of the plate is shown in sound intensity level in dB (Ref. 1012 W/m2). It is evident that the velocity level and the radiated sound of the test plate very much depend on the wind speed. An increase of the wind speed by 10% is likely to increase the plate velocity level by 3 dB and the radiated sound level by 3 dB. For the test plates, the measured spectral energy dominates in the frequency range from 100 to 300 Hz and decays with the frequency with a speed of about 10 dB/oct. In practical application such as an aircraft plate in flying condition, however, the spectrum could be much different in comparison with this measured data due to the difference of flow speed and plate parameters. The dominated spectral energy for the aircraft plate could be located at much higher frequency range. Under the flow speed of 60 m/s, 71 m/s and 86 m/s, the measured velocity auto spectrum and the radiated sound intensity of the plates with and without spanwise stiffener attachments are respectively illustrated from Figs. 7–12. The measured date is given in narrow band analysis per Hz and as well in 1/3 octave band analysis. It is interesting to observe that spanwise stiffeners almost have no influence on the vibration level of the plates, while significantly affect the radiated sound of the plates. Below 500 Hz frequency, the spanwise stiffeners influence on the radiated sound intensity is not evident, however, above 500 Hz the radiated sound intensity by the plates with spanwise stiffeners is obviously higher than that of the plates without spanwise stiffeners.
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B. Liu et al. / Applied Acoustics 80 (2014) 28–35
Autospectrum of velocity dB ref.1e-18 m2/s2
86m/s 71m/s 60m/s
110 100 90 80 70 60 50 40 100
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
Radiated sound intensity dB ref.1e-12W/m2
70 60 50 40
140 130 120 110 100 90 80 70 60 50 100
Radiated sound intensity dB ref.1e-12 W/m2
Auto spectrum of velocity dB ref. 1e-18 m2/s2
120
500
1,000
500
1,000
1,500
2,000
2,500
3,000
3,500
2,500
3,000
3,500
(b)
70 60 50 40 30 500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
Plate I Plate I0
Autospectrum of velocity dB ref.1e-18 m2/s2
Fig. 6. Measured data of the plate I at flow speed of 60 m/s, 71 m/s and 86 m/s. (Upper) velocity autospectrum density; (Below) sound intensity. Narrow band analysis in per Hz.
500
1,000
1,500
2,000
2,500
3,000
3,500
(b)
70 60
Radiated sound intensity dB ref.1e-12 W/m2
(a)
80
140 130 120 110 100 90 80 70 60 50 100
Plate I Plate I0
(a)
500
1,000
1,500
2,000
2,500
3,000
80
3,500
(b)
70 60 50 40 30 20 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Frequency (Hz)
50
Fig. 9. Measured data of the plate I and I0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 86 m/s.
40 30 20 100
2,000
Fig. 8. Measured data of the plate I and I0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 71 m/s.
10
140 130 120 110 100 90 80 70 60 50 100
1,500
Frequency (Hz)
Frequency (Hz)
Autospectrum of velocity dB ref.1e-18 m2/s2
(a)
20
0 100
Radiated sound intensity dB ref.1e-12 W/m2
Plate I0
80
20 100
30
Plate I
500
1,000
1,500
2,000
2,500
3,000
3,500
Frequency (Hz) Fig. 7. Measured data of the plate I and I0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 60 m/s.
An example of the measured vibration level with respect to the plate surface and the stiffeners is illustrated in Fig. 13. The measured flexural vibration spectrum of the stiffener is about 10 dB lower than that of plate surface above 500 Hz, which implied that the increased sound radiation of spanwise stiffened plates is not due to the swing vibration of the stiffeners. A comparison of the measured data with respect to plate J and plate K is also illustrated in Fig. 14. It is evident that the plate K (with shorter length) significantly increase the radiated noise in the frequency range from 300 Hz to 1500 Hz, and slightly improve it above 1500 Hz, while the vibration level is almost identical to the plate J above 500 Hz. The measured data partially support the numerical results given in literature [15], where the plate vibration and the noise radiation
of a plate excited by the flow with a speed of 225 m/s are predicted. The prediction in Ref. [15] indicates that the plate with the spanwise stiffeners behaves more like a sub-panel between two stiffeners and may radiate more sound in comparison with that of plate without stiffeners. If compare a reference plate with that of the length doubled in free stream direction, when the flow matches the m-order axial (streamwise) mode of the reference panel at a specific frequency, then the same flow at the same frequency would always match the 2 m-order axial mode of the panel with axial length doubled. At a specific frequency far below the modal critical frequency of the plate, the lower order axial modes always have higher modal radiation efficiencies than that of the higher order axial modes. Therefore, the reference plate is expected to radiate more sound than that of the plate with larger axial length. 3.2. Influence of the streamwise stiffener It is of interest to see how streamwise stiffeners affect sound radiation. The measured velocity auto spectrum and the radiated sound intensity of the plates with and without streamwise
B. Liu et al. / Applied Acoustics 80 (2014) 28–35
Plate J0 Plate J
(a)
500
1,000
1,500
2,000
2,500
3,000
Radiated sound intensity dB ref.1e-12 W/m2
80
3,500
60 50 40 30 20 100
500
1,000
1,500
2,000
2,500
3,000
Plate J Plate J0
(a)
500
1,000
3,500
3,000
3,500
(b)
70 60 50 40 30 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Fig. 12. Measured data of the plate J and J0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 86 m/s.
Plate J Plate J0
(a)
140
plate stiffeners
500
1,000
1,500
2,000
2,500
3,000
80
3,500
(b)
70 60
Autospectrum of velocity dB ref.1e-18 m2/s2
130 120 110 100 90 80
50 70
40 30
60 100
500
1,000
1,500
2,000
2,500
3,000
Frequency (Hz) Fig. 11. Measured data of the plate J and J0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 71 m/s.
stiffeners are given in Figs. 15–17. Similar to the spanwise stiffeners, the streamwise stiffener almost have no influence on the vibration level in the frequency range tested, however, unlike spanwise stiffeners, the radiated sound intensity by plates with and without streamwise stiffener only has slight difference above 500 Hz. 3.3. Comparison with theoretical predictions To validate the results from wind tunnel test, an approach in Ref. [16] is adopted for the prediction of the plate vibration and the radiated sound. In the calculation, the cross power spectral density of the flat-plate boundary layer proposed by Corcos is introduced and shown in below
SPP ðnx ; ny ; xÞ ¼ Upp ðxÞ expðc1 jxnx =U c jÞ expðc3 jxny =U c jÞ expðixnx =U c Þ
500
1,000
ð1Þ
where Upp(x) is the point auto spectrum of the pressure. In the calculation Upp(x) in Eq. (1) is adopted. For a simply supported rectangular plate the eigenfunctions are defined as
1,500
2,000
2,500
3,000
3,500
4,000
Frequency (Hz)
3,500
Fig. 13. Measured velocity auto spectrum of the plate I0 and the stiffener. Flow speed 86 m/s.
Autospectrum of velocity dB ref.1e-18 m2/s2
20 100
Radiated sound intensity dB ref.1e-12 W/m2
Autospectrum of velocity dB ref.1e-18 m2/s2
2,500
Frequency (Hz)
Fig. 10. Measured data of the plate J and J0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 60 m/s.
Radiated sound intensity dB ref.1e-12 W/m2
2,000
80
Frequency (Hz)
140 130 120 110 100 90 80 70 60 50 100
1,500
90
(b)
70
140 130 120 110 100 90 80 70 60 50 100
Radiated sound intensity dB ref.1e-12 W/m2
Autospectrum of velocity dB ref.1e-18 m2/s2
140 130 120 110 100 90 80 70 60 50 100
Autospectrum of velocity dB ref.1e-18 m2/s2
32
140 130 120 110 100 90 80 70 60 50 100
Plate I Plate K
(a)
500
1,000
1,500
2,000
2,500
3,000
90
3,500
(b)
80 70 60 50 40 30 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Frequency (Hz) Fig. 14. Measured data of the plate I and K in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 86 m/s.
33
140 130 120 110 100 90 80 70 60 50 100
Plate J Plate J1
(a)
500
1,000
1,500
2,000
2,500
3,000
80
3,500
(b)
70 60
30 20 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Radiated sound intensity dB ref.1e-12 W/m2
Autospectrum of velocity dB ref.1e-18 m2/s2
Fig. 15. Measured data of the plate J and J1 in 1/3 octave band. (a) Velocity auto spectrum; (b) Sound intensity. Flow speed 60 m/s.
140 130 120 110 100 90 80 70 60 50 100
Plate J Plate J1
(a)
500
1,000
1,500
2,000
2,500
3,000
3,500
(b)
70 60 50 40 30 500
1,000
1,500
2,000
2,500
3,000
3,500
Fig. 16. Measured data of the plate J and J1 in 1/3 octave band. (a) Velocity auto spectrum; (b) Sound intensity. Flow speed 71 m/s.
140 130 120 110 100 90 80 70 60 50 100
Plate J Plate J1
(a)
500
1,000
1,500
2,000
2,500
3,000
90
1 mn 1 mn 1 mn J 1 þ J mn J3 þ J4 ; 2 þ bn am bn am
ð3Þ
where am = mp/a, bn = np/b and
4 ; abl½x2mn ð1 þ jgÞ x2
ð4Þ
8 mn 9 8 9 8 9 1 sin am x sin bn x > J1 > > > > > > > > > > > > > > > > > Z Z mn > > ðb yÞ J mn > 0 0 > > > > sin am x cos bn x > > > > > > > > : 3mn > : : ; ; > ; ða xÞ cos am x sin bn x J4 S pp ðx; y; xÞ cosðxx=U x Þdxdy: e
IðxÞ ¼
x2 X q0 c0 rmn jW mn j2 8
ð5Þ
mn
1 mn 1 mn 1 mn J 1 þ J mn J3 þ J4 ; 2 þ bn am bn am
70 60 50 40
500
1,000
1,500
2,000
2,500
3,000
Example of measured and predicted velocity spectral density and the radiated sound intensity of the plate I is illustrated in Fig. 18 in narrow band analysis. In the calculation the loss factor of the plate is set to 1.5% for a convenience. The predicted and measured velocity and sound radiation show reasonable agreement in the frequency range from 100 Hz to 3500 Hz. The agreement provides solid verification to test data and prediction. As concluded in Ref. [17], the plate with the stiffeners radiates roughly the same level of acoustical power as the sub-panel between two stiffeners. For simplicity, one may compare the predicted data of the plate with different length or width to simulate spanwise or streamwise stiffeners influence. An example of the predicted data of the plates respectively with the size of
3,500
(b)
80
30 100
mn
ð6Þ
80
20 100
jW mn j2
Here l is the mass per unit area of the plate, g is the loss factor and e S pp ðx; y; xÞ cosðxx=U x Þ corresponds to the real part of Spp. The sound intensity radiated by the plate is then
Frequency (Hz)
Autospectrum of velocity dB ref.1e-18 m2/s2
8
40
Frequency (Hz)
Radiated sound intensity dB ref.1e-12 W/m2
x2 X
Svv ðxÞ ¼
W mn ðxÞ ¼
50
ð2Þ
where m and n are integers and the corners of the plate are at (0, 0), (0, a), (a, b) and (b, 0). Following the same approach in Ref. [16], the averaged power spectral density with respect to the velocity of the plate may be calculated according to following expressions
Auto spectrum of velocity dB ref. 1e-18 m2/s2
Radiated sound intensity dB ref.1e-12 W/m2
mpx npy /mn ðrÞ ¼ sin sin ; a b
3,500
Frequency (Hz) Fig. 17. Measured data of the plate J and J1 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 86 m/s.
Radiated sound intensity dB ref.1e-12 W/m2
Autospectrum of velocity dB ref.1e-18 m2/s2
B. Liu et al. / Applied Acoustics 80 (2014) 28–35
130 120 110 100 90 80 70 60 50 40 102
Measured Predicted
103
70 60 50 40 30 20 10 0 102
103
Frequency (Hz) Fig. 18. Measured and predicted velocity auto spectrum and the radiated sound intensity of the plate I. Narrow band analysis in per Hz. Flow speed 86 m/s.
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B. Liu et al. / Applied Acoustics 80 (2014) 28–35
Autospectrum of velocity dB (ref. 1e-18m2/s2)
130 110 100 90 80 70 60 50 102
Radiated Sound Intensity dB (ref.1e-12W/m2)
0.93mx0.3m 0.31mx0.3m
120
103
70 60 50 40 30 20 10 0 102
103
tunnel measurements. For the plates tested, it is evident that spanwise stiffeners may increase noise radiation a few dB above 500 Hz, while have almost no influence on the vibration level of the plates. Unlike the spanwise stiffeners, the streamwise stiffeners have almost no influence both on the vibration level and radiated sound intensity of the plates. The measured and predicted results imply that reducing the length or width for a reference plate has almost no influence for the vibration level of the plate subjected to TBL excitations, while reducing the length (along with the flow direction) of the plate may increase the radiated noise, and reducing the width of the plate has no influence on the radiated sound intensity. These conclusions are tentative if applied to typical aircraft plates, since the experiment condition does not account for the full frame of an aircraft sidewall (skin panel/wall cavity/trim panel) and much higher speed for aircraft during cruise. However, the conclusions imply that ring stiffeners must be considered carefully if TBL-induced noise is of main concerns.
Frequency (Hz)
Autospectrum of velocity dB (ref. 1e-18m2/s2)
Fig. 19. The predicted data of the plates respectively with the size of 0.93 m 0.3 m and 0.31 m 0.3 m.
The authors gratefully acknowledge the financial support from NSFC Grant 10974221 and State Key Development Program for Basic Research of China (973 Program) Grant 2012CB720204.
130
0.62mx0.3m 0.62mx0.15m
120 110
References
100 90 80 70 60 50 102
Radiated Sound Intensity dB (ref.1e-12W/m2)
Acknowledgments
103
70 60 50 40 30 20 10 0 102
103
Frequency (Hz) Fig. 20. The predicted data of the plates respectively with the size of 0.62 m 0.3 m and 0.62 m 0.15 m.
0.93 m 0.3 m and 0.31 m 0.3 m is illustrated in Fig. 19. The other parameters used in calculation is same as the plate I. Similar to the measured plate with and without spanwise stiffeners, reducing the length of the plate almost does not affect the vibration spectrum, however, reducing the length of the plate increases the radiated sound intensity obviously above 500 Hz. The predicted data of the plates respectively with the size of 0.62 m 0.3 m and 0.62 m 0.15 m is illustrated in Fig. 20. The other parameters used in calculation is same as the plate J. Similar to the measured plates with and without streamwise stiffeners, reducing the width of the plate almost does not affect the vibration spectrum and as well the radiated sound intensity in the frequency range of calculation.
4. Conclusions The influence of stiffeners on the plate vibration and noise radiation induced by turbulent boundary layers is investigated by wind
[1] Davies HG. Sound from turbulent boundary layer excited panels. J Acoust Soc Am 1971;49:878–89. [2] Graham WR. Boundary layer induced noise in aircraft, Part I: The flat plate model. J Sound Vib 1996;192(1):101–20. [3] Graham WR. Boundary layer induced noise in aircraft, Part II: The trimmed flat plate model. J Sound Vib 1996;192(1):121–38. [4] Graham WR. A comparison of models for the wavenumber–frequency spectrum of turbulent boundary layer pressures. J Sound Vib 1997;206:541–65. [5] Durant C, Robert G, Filippi PJT, Mattei PO. Vibroacoustic response of a thin cylindrical shell excited by a turbulent internal flow: comparison between numerical prediction and experimentation. J Sound Vib 2000;229(5): 1115–55. [6] Han F, Bernhard RJ, Mongeau LG. Prediction of flow-induced structural vibration and sound radiation using energy flow analysis. J Sound Vib 1999;227:685–709. [7] Han F, Bernhard RJ, Mongeau LG. A model for the vibro-acoustic response of plates excited by complex flows. J Sound Vib 2001;246(5):901–26. [8] Maury C, Gardonio P, Elliott SJ. A wavenumber approach to modelling the response of a randomly excited panel, Part I: General theory. J Sound Vib 2002;252(1):83–113. [9] Maury C, Gardonio P, Elliott SJ. A wavenumber approach to modelling the response of a randomly excited panel, Part II: Application to aircraft panels excited by a turbulent boundary layer. J Sound Vib 2002;252(1): 115–39. [10] Allen M, Vlahopoulos N. Noise generated from a flexible and elastically supported structure subject to turbulent boundary layer flow excitation. Finite Elem Anal Des 2001;37:687–712. [11] Birgersson F, Ferguson NS, Finnveden S. Application of the spectral finite elements method to turbulent boundary layer induced vibration of plates. J Sound Vib 2003;259:873–91. [12] Birgersson F, Finnveden S. A spectral super elements for modeling of plate vibration. Part 2: Turbulence excitation. J Sound Vib 2005;287:315–28. [13] Finnveden S, Birgersson F, Ross U, Kremer T. A model of wall pressure correlation for prediction of turbulence-induced vibration. J Fluids Struct 2005;20:1127–43. [14] Rocha J, Palumbo D. On the sensitivity of sound power radiated by aircraft panels to turbulent boundary layer parameters. J Sound Vib 2012;331:4785–806. [15] Palumbo D. Determining correlation and coherence lengths in turbulent boundary layer flight data. J Sound Vib 2012;331(16):3721–37. [16] Liu B, Feng L, Nilsson AC, Aversano M. Predicted and measured plate velocities induced by turbulent boundary layers. J Sound Vib 2012;331:5309–25. [17] Liu B. Noise radiation of aircraft panels subjected to boundary layer pressure fluctuations. J Sound Vib 2008;314:693–711. [18] Liu B, Feng L, Nilsson AC. Sound transmission through curved aircraft panels with stringer and ring frame attachments. J Sound Vib 2007;300: 949–73.
B. Liu et al. / Applied Acoustics 80 (2014) 28–35 [19] ISO 9614-2. Acoustics-Determination of sound power levels of noise sources using sound intensity – Part 2: Measurement by scanning; 1996. [20] Corcos GM. Resolution of pressure in turbulence. J Acoust Soc Am 1963;35(2):192–9. [21] Schlichting H. Boundary Layer Theory. 7th ed. New York: McGraw-Hill; 1981. 156.
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[22] Keith WL, Hurdis DA, Abraham BM. A comparison of turbulent boundary layer wall-pressure spectura. ASME J Fluids Eng 1992;114(2):338–47. [23] Efimtsov BM. Characteristics of the field of turbulent wall pressure fluctuations at large Reynolds numbers. Soviet Phys Acoust 1982;28(4):289–92.
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Influence of stiffeners on plate vibration and radiated noise excited by turbulent boundary layers Bilong Liu a,⇑, Hao Zhang a, Zhongchang Qian a, Daoqing Chang a, Qun Yan b, Wenchao Huang b a b
Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, 100190 Beijing, PR China Key Laboratory of Aeroacoustics and Dynamics, Aircraft Strength Research Institute, 710065 Xian, PR China
a r t i c l e
i n f o
Article history: Received 22 July 2013 Received in revised form 2 January 2014 Accepted 14 January 2014 Available online 6 February 2014 Keywords: TBL noise Aircraft noise Plate vibration
a b s t r a c t The influence of stiffeners on plate vibration and noise radiation induced by turbulent boundary layers is investigated by wind tunnel measurements. Plates with and without stiffeners are tested under the flow speed of 60 m/s, 71 m/s and 86 m/s, respectively. The stiffeners are set either perpendicular or parallel to the direction of the free stream. Measured vibration and noise levels are compared with theoretical calculations, where wall pressure cross-spectra are described by the Corcos model. For the plates tested, it is evident that stiffeners perpendicular to the direction of the free stream could increase noise radiation, but have almost no influence on vibration level of plates. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The noise and vibration generated by turbulent boundary layer (TBL) has importance for the design of high speed vehicles. For any type of prediction of noise and vibration levels inside a high speed vehicle, the noise and vibration generated by the TBL must be well described. The noise and vibration of the structure depends on the velocity of the vibrating plates, which in turn is determined by the speed of the vehicle, the geometry, the dimensions and losses, or damping of the plates. Clearly the acoustic properties of the interior system, trim panels, etc., will also influence the interior noise of the vehicle. Much attention has been paid to the prediction of boundary layer induced noise and vibration as discussed in literature [1– 17]. Among these references, Graham [2] proposed a model to predict the noise induced by TBL in an aircraft. Graham successfully developed an analytical expression to evaluate the modal excitation terms. In this way the time to calculate the excitation field was thereby significantly reduced. In another paper, the merit of various models describing the cross power spectral density induced by a flow or turbulent boundary layer across a structure was discussed in some detail by the same author [3]. Han attempted to predict TBL induced noise by energy flow analysis [6,7]. The method has proved to be successful for predicting the response of flat isotropic panels subjected to TBL excitation. However, the noise radiated by the panel was not properly described.
⇑ Corresponding author. Tel.: +86 1082547504. E-mail address: [email protected] (B. Liu). 0003-682X/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apacoust.2014.01.007
The reason is resulting from the inaccuracy of the method to predict the modal averaged radiation efficiency. Recently Maury et al. used wavenumber approach to model the response of panel subjected to acoustic and TBL excitations [8,9]. Finnveden and Birgersson [11–13] introduced a series of numerical method to calculate the vibration of plates induced by boundary layer. While Rocha and Palumbo [15] adopted modal summation method to investigate the sensitivity of sound power radiated by aircraft panels to turbulent boundary layer parameters. The models above are restricted to flat uniform panels without stiffener attachments. Concerning with the influence of stiffeners, Liu et al. [16–18] investigated the effects of ring frames, stringers, damping, and curvature on the noise transmission of aircraft plates under the excitation of acoustic diffuse field and TBL respectively. The results indicate that significant differences resulting from stiffeners and damping could appear when the plates are subjected to acoustic diffuse field and TBL excitations. More specifically, the ring frames have almost no influence on sound transmission through a typical aircraft panel subjected to excitation of an acoustical diffuse field. While the ring frames have significant influences on TBL induced noise radiation according to the prediction. This conclusion, however, is based on numerical prediction. Experimental work together with predictions on the noise radiation of TBL excited plates with and without stiffeners is not available in the literature. The purpose of current work is trying to investigate the influence of stiffeners on the plate vibration and noise radiation under TBL excitations. Different size plates with and without spanwise (perpendicular to the direction of free stream) and streamwise
B. Liu et al. / Applied Acoustics 80 (2014) 28–35
29
(parallel to the direction of free stream) stiffeners were tested under the flow speed of 60 m/s, 71 m/s and 86 m/s, respectively. Then the test results were analyzed and compared with theoretical prediction. 2. Measurements A detailed description of measurement set-up and relevant data on TBL spectrum are given in this section. 2.1. Experimental set-up The test was conducted in a subsonic wind tunnel specially designed for acoustic measurement. The diagram for the test set-up is illustrated in Fig. 1. A wind turbine is used to produce volume flow with different velocities. A silencer with the perimeter of about 6 m and the length of about 15 m are mounted downstream and it could eliminate the turbine fan noise effectively, so that pure TBL excitation is obtained. A rectangular aluminum pipe is connected downstream of the silencers. The wall thickness of the pipe is 10 mm, the width and height of the pipe are 170 mm and 370 mm respectively. The test plate is mounted as part of the pipe sidewall. Inside surfaces of the test plate and the pipe wall are at the same plane. A photo of the plate under testing is shown in Fig. 2 as an example. The end of pipe is streamlined to trumpetshaped outlet with absorption liners. Most part of pipe is located in an anechoic room (6.21 7.86 5.05 m, cut off frequency 80 Hz). Two 1/4 in. microphones (B&K, Type 4944) with a separation distance of 1.3 cm were mounted on the pipe wall to monitor the TBL spectrum. An illustration of microphone installation is shown in Fig. 1, where the diameter of the hole in front of microphone is 1 mm. The average sound intensity radiated by the plates is measured by intensity probe (B&K Sound Intensity Probe Kit, Type 3599) with scanning method according to ISO 9614-2 [19]. A 1.0 g accelerometer (PCB, Type 352c68) is randomly attached to the plate by using beeswax to measure vibration level. The vibration is measured by accelerometer repeatedly for sufficient points of the plate and then averaged vibration level is calculated. The measured data is recorded by B&K Pulse, Type 3610. Six aluminum plates, for their parameters listed in Table 1, are tested. All of plates are flat and with thickness of 1.1 mm. The plate I0 has the same the size of the plate I, while attaches two equallyspaced spanwise stiffeners. The stiffeners are connected to the plates by rivets. The plate J0 has the same the size of the plate J, while in the middle of it has one spanwise stiffener attached. The plate J1 has the same the size of the plate J, but has one streamwise stiffener enclosed in the middle. The plate K is the smallest plate and its size corresponds to that of the sub-plate between two
Fig. 2. A photo of the plate with enclosed spanwise stiffeners under testing.
spanwise stiffeners of the plate I. The height and thickness of stiffeners are 50 mm and 1.5 mm respectively. 2.2. The boundary layer thickness The boundary layer thickness, measured by hot wire anemometer, is an important parameter to characterize the turbulence. The measured results are shown in Table 2, where d and d⁄ denotes the boundary layer thickness and the displacement thickness respectively, Uc and Us represent the convection velocity and friction velocity. According to the model proposed by Corcos [20], Uc = 0.6–0.8U1, where U1 is the free-stream velocity. As an example, the velocity profile for a free flow velocity of 71 m/s is also given in Fig. 3. Comparing the results with theory given by Schlichting [21], it is verified that the velocity profile is as in a classical boundary layer. 2.3. Wall pressure power spectral density The point auto spectral density of the wall pressure Upp(Pa2/Hz) was measured using two fixed microphones according to Fig. 1. The measured data with flow speed of 60, 71 and 86 m/s and in the frequency from 100 Hz to 3500 Hz is given in Fig. 4. The measured spectrum levels agree with that in Ref. [22,23]. The wall pressure coherence measured with two microphones with a streamwise length 1.3 cm is given in Fig. 5, and the results calculated according to Corcos’s model [20] is also plotted for a comparison. 3. Measured plate vibration and noise radiation 3.1. Influence of the spanwise stiffeners To perform the measurements, two B&K pulse systems are synchronously connected, which yields totally 12 channels. The fre-
Fig. 1. Schematic of test set-up for plate vibration and sound radiation.
30
B. Liu et al. / Applied Acoustics 80 (2014) 28–35
Table 1 Plates used in measurement. Plate no.
Length width cm
Thickness (mm)
Number of stiffeners
I
93 30
1.1
0
I0
93 30
1.1
2
J
62 30
1.1
0
J0
62 30
1.1
1
J1
62 30
1.1
1
K
31 30
1.1
0
Table 2 Estimated flow data from measurements.
Schematic of plate
1
Measurements Corcos' model
0.9
U1
60 (m/s)
71 (m/s)
86 (m/s)
d (mm) d⁄ (mm) Uc (m/s) Us (m/s)
20.5 2.51 42 1.8
20.5 2.48 49.7 2.13
22.5 2.71 60.2 2.58
0.8
Coherence
0.7 0.6 0.5 0.4 0.3 0.2 0.1
1
measured data theory predicted
0.95
0
500
1000
1500
2000
2500
3000
3500
4000
Frequency (Hz) 0.9
Fig. 5. Wall pressure coherence as function of frequency. Solid line is measured coherence and dot line corresponds to calculation according to Corcos’s model. Flow speed 71 m/s.
Uz/U∞
0.85 0.8 0.75 0.7 0.65
0
0.2
0.4
0.6
0.8
1
z/δ Fig. 3. Velocity profile for a free flow of 71 m/s at a distance z from wall scaled with the boundary layer thickness d. Circles are measured velocities and solid line corresponds to a power velocity distribution law [21] with n = 7.2.
86m/s measured 71m/s measured 60m/s measured
-5
dB (ref. 1 Pa2/Hz)
Auto spectral density of wall pressure,
0
-10 -15 -20 -25 -30 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Frequency (Hz) Fig. 4. Measured and predicted wall pressure auto spectral density at flow speed of 60 m/s, 71 m/s and 86 m/s.
quency resolution is 2 Hz and the Hanning window is used. The average number for auto spectrum is set to 150, with 66.7% overlap. The measured data of the plate I for the three different wind speeds is given in Fig. 6, where the measured velocity of the plate in narrow band analysis is shown in velocity auto spectrum in dB (Ref. 1018 m2 s2). The measured noise radiation of the plate is shown in sound intensity level in dB (Ref. 1012 W/m2). It is evident that the velocity level and the radiated sound of the test plate very much depend on the wind speed. An increase of the wind speed by 10% is likely to increase the plate velocity level by 3 dB and the radiated sound level by 3 dB. For the test plates, the measured spectral energy dominates in the frequency range from 100 to 300 Hz and decays with the frequency with a speed of about 10 dB/oct. In practical application such as an aircraft plate in flying condition, however, the spectrum could be much different in comparison with this measured data due to the difference of flow speed and plate parameters. The dominated spectral energy for the aircraft plate could be located at much higher frequency range. Under the flow speed of 60 m/s, 71 m/s and 86 m/s, the measured velocity auto spectrum and the radiated sound intensity of the plates with and without spanwise stiffener attachments are respectively illustrated from Figs. 7–12. The measured date is given in narrow band analysis per Hz and as well in 1/3 octave band analysis. It is interesting to observe that spanwise stiffeners almost have no influence on the vibration level of the plates, while significantly affect the radiated sound of the plates. Below 500 Hz frequency, the spanwise stiffeners influence on the radiated sound intensity is not evident, however, above 500 Hz the radiated sound intensity by the plates with spanwise stiffeners is obviously higher than that of the plates without spanwise stiffeners.
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B. Liu et al. / Applied Acoustics 80 (2014) 28–35
Autospectrum of velocity dB ref.1e-18 m2/s2
86m/s 71m/s 60m/s
110 100 90 80 70 60 50 40 100
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
Radiated sound intensity dB ref.1e-12W/m2
70 60 50 40
140 130 120 110 100 90 80 70 60 50 100
Radiated sound intensity dB ref.1e-12 W/m2
Auto spectrum of velocity dB ref. 1e-18 m2/s2
120
500
1,000
500
1,000
1,500
2,000
2,500
3,000
3,500
2,500
3,000
3,500
(b)
70 60 50 40 30 500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
Plate I Plate I0
Autospectrum of velocity dB ref.1e-18 m2/s2
Fig. 6. Measured data of the plate I at flow speed of 60 m/s, 71 m/s and 86 m/s. (Upper) velocity autospectrum density; (Below) sound intensity. Narrow band analysis in per Hz.
500
1,000
1,500
2,000
2,500
3,000
3,500
(b)
70 60
Radiated sound intensity dB ref.1e-12 W/m2
(a)
80
140 130 120 110 100 90 80 70 60 50 100
Plate I Plate I0
(a)
500
1,000
1,500
2,000
2,500
3,000
80
3,500
(b)
70 60 50 40 30 20 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Frequency (Hz)
50
Fig. 9. Measured data of the plate I and I0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 86 m/s.
40 30 20 100
2,000
Fig. 8. Measured data of the plate I and I0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 71 m/s.
10
140 130 120 110 100 90 80 70 60 50 100
1,500
Frequency (Hz)
Frequency (Hz)
Autospectrum of velocity dB ref.1e-18 m2/s2
(a)
20
0 100
Radiated sound intensity dB ref.1e-12 W/m2
Plate I0
80
20 100
30
Plate I
500
1,000
1,500
2,000
2,500
3,000
3,500
Frequency (Hz) Fig. 7. Measured data of the plate I and I0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 60 m/s.
An example of the measured vibration level with respect to the plate surface and the stiffeners is illustrated in Fig. 13. The measured flexural vibration spectrum of the stiffener is about 10 dB lower than that of plate surface above 500 Hz, which implied that the increased sound radiation of spanwise stiffened plates is not due to the swing vibration of the stiffeners. A comparison of the measured data with respect to plate J and plate K is also illustrated in Fig. 14. It is evident that the plate K (with shorter length) significantly increase the radiated noise in the frequency range from 300 Hz to 1500 Hz, and slightly improve it above 1500 Hz, while the vibration level is almost identical to the plate J above 500 Hz. The measured data partially support the numerical results given in literature [15], where the plate vibration and the noise radiation
of a plate excited by the flow with a speed of 225 m/s are predicted. The prediction in Ref. [15] indicates that the plate with the spanwise stiffeners behaves more like a sub-panel between two stiffeners and may radiate more sound in comparison with that of plate without stiffeners. If compare a reference plate with that of the length doubled in free stream direction, when the flow matches the m-order axial (streamwise) mode of the reference panel at a specific frequency, then the same flow at the same frequency would always match the 2 m-order axial mode of the panel with axial length doubled. At a specific frequency far below the modal critical frequency of the plate, the lower order axial modes always have higher modal radiation efficiencies than that of the higher order axial modes. Therefore, the reference plate is expected to radiate more sound than that of the plate with larger axial length. 3.2. Influence of the streamwise stiffener It is of interest to see how streamwise stiffeners affect sound radiation. The measured velocity auto spectrum and the radiated sound intensity of the plates with and without streamwise
B. Liu et al. / Applied Acoustics 80 (2014) 28–35
Plate J0 Plate J
(a)
500
1,000
1,500
2,000
2,500
3,000
Radiated sound intensity dB ref.1e-12 W/m2
80
3,500
60 50 40 30 20 100
500
1,000
1,500
2,000
2,500
3,000
Plate J Plate J0
(a)
500
1,000
3,500
3,000
3,500
(b)
70 60 50 40 30 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Fig. 12. Measured data of the plate J and J0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 86 m/s.
Plate J Plate J0
(a)
140
plate stiffeners
500
1,000
1,500
2,000
2,500
3,000
80
3,500
(b)
70 60
Autospectrum of velocity dB ref.1e-18 m2/s2
130 120 110 100 90 80
50 70
40 30
60 100
500
1,000
1,500
2,000
2,500
3,000
Frequency (Hz) Fig. 11. Measured data of the plate J and J0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 71 m/s.
stiffeners are given in Figs. 15–17. Similar to the spanwise stiffeners, the streamwise stiffener almost have no influence on the vibration level in the frequency range tested, however, unlike spanwise stiffeners, the radiated sound intensity by plates with and without streamwise stiffener only has slight difference above 500 Hz. 3.3. Comparison with theoretical predictions To validate the results from wind tunnel test, an approach in Ref. [16] is adopted for the prediction of the plate vibration and the radiated sound. In the calculation, the cross power spectral density of the flat-plate boundary layer proposed by Corcos is introduced and shown in below
SPP ðnx ; ny ; xÞ ¼ Upp ðxÞ expðc1 jxnx =U c jÞ expðc3 jxny =U c jÞ expðixnx =U c Þ
500
1,000
ð1Þ
where Upp(x) is the point auto spectrum of the pressure. In the calculation Upp(x) in Eq. (1) is adopted. For a simply supported rectangular plate the eigenfunctions are defined as
1,500
2,000
2,500
3,000
3,500
4,000
Frequency (Hz)
3,500
Fig. 13. Measured velocity auto spectrum of the plate I0 and the stiffener. Flow speed 86 m/s.
Autospectrum of velocity dB ref.1e-18 m2/s2
20 100
Radiated sound intensity dB ref.1e-12 W/m2
Autospectrum of velocity dB ref.1e-18 m2/s2
2,500
Frequency (Hz)
Fig. 10. Measured data of the plate J and J0 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 60 m/s.
Radiated sound intensity dB ref.1e-12 W/m2
2,000
80
Frequency (Hz)
140 130 120 110 100 90 80 70 60 50 100
1,500
90
(b)
70
140 130 120 110 100 90 80 70 60 50 100
Radiated sound intensity dB ref.1e-12 W/m2
Autospectrum of velocity dB ref.1e-18 m2/s2
140 130 120 110 100 90 80 70 60 50 100
Autospectrum of velocity dB ref.1e-18 m2/s2
32
140 130 120 110 100 90 80 70 60 50 100
Plate I Plate K
(a)
500
1,000
1,500
2,000
2,500
3,000
90
3,500
(b)
80 70 60 50 40 30 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Frequency (Hz) Fig. 14. Measured data of the plate I and K in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 86 m/s.
33
140 130 120 110 100 90 80 70 60 50 100
Plate J Plate J1
(a)
500
1,000
1,500
2,000
2,500
3,000
80
3,500
(b)
70 60
30 20 100
500
1,000
1,500
2,000
2,500
3,000
3,500
Radiated sound intensity dB ref.1e-12 W/m2
Autospectrum of velocity dB ref.1e-18 m2/s2
Fig. 15. Measured data of the plate J and J1 in 1/3 octave band. (a) Velocity auto spectrum; (b) Sound intensity. Flow speed 60 m/s.
140 130 120 110 100 90 80 70 60 50 100
Plate J Plate J1
(a)
500
1,000
1,500
2,000
2,500
3,000
3,500
(b)
70 60 50 40 30 500
1,000
1,500
2,000
2,500
3,000
3,500
Fig. 16. Measured data of the plate J and J1 in 1/3 octave band. (a) Velocity auto spectrum; (b) Sound intensity. Flow speed 71 m/s.
140 130 120 110 100 90 80 70 60 50 100
Plate J Plate J1
(a)
500
1,000
1,500
2,000
2,500
3,000
90
1 mn 1 mn 1 mn J 1 þ J mn J3 þ J4 ; 2 þ bn am bn am
ð3Þ
where am = mp/a, bn = np/b and
4 ; abl½x2mn ð1 þ jgÞ x2
ð4Þ
8 mn 9 8 9 8 9 1 sin am x sin bn x > J1 > > > > > > > > > > > > > > > > > Z Z mn
IðxÞ ¼
x2 X q0 c0 rmn jW mn j2 8
ð5Þ
mn
1 mn 1 mn 1 mn J 1 þ J mn J3 þ J4 ; 2 þ bn am bn am
70 60 50 40
500
1,000
1,500
2,000
2,500
3,000
Example of measured and predicted velocity spectral density and the radiated sound intensity of the plate I is illustrated in Fig. 18 in narrow band analysis. In the calculation the loss factor of the plate is set to 1.5% for a convenience. The predicted and measured velocity and sound radiation show reasonable agreement in the frequency range from 100 Hz to 3500 Hz. The agreement provides solid verification to test data and prediction. As concluded in Ref. [17], the plate with the stiffeners radiates roughly the same level of acoustical power as the sub-panel between two stiffeners. For simplicity, one may compare the predicted data of the plate with different length or width to simulate spanwise or streamwise stiffeners influence. An example of the predicted data of the plates respectively with the size of
3,500
(b)
80
30 100
mn
ð6Þ
80
20 100
jW mn j2
Here l is the mass per unit area of the plate, g is the loss factor and e S pp ðx; y; xÞ cosðxx=U x Þ corresponds to the real part of Spp. The sound intensity radiated by the plate is then
Frequency (Hz)
Autospectrum of velocity dB ref.1e-18 m2/s2
8
40
Frequency (Hz)
Radiated sound intensity dB ref.1e-12 W/m2
x2 X
Svv ðxÞ ¼
W mn ðxÞ ¼
50
ð2Þ
where m and n are integers and the corners of the plate are at (0, 0), (0, a), (a, b) and (b, 0). Following the same approach in Ref. [16], the averaged power spectral density with respect to the velocity of the plate may be calculated according to following expressions
Auto spectrum of velocity dB ref. 1e-18 m2/s2
Radiated sound intensity dB ref.1e-12 W/m2
mpx npy /mn ðrÞ ¼ sin sin ; a b
3,500
Frequency (Hz) Fig. 17. Measured data of the plate J and J1 in 1/3 octave band. (a) Velocity auto spectrum; (b) sound intensity. Flow speed 86 m/s.
Radiated sound intensity dB ref.1e-12 W/m2
Autospectrum of velocity dB ref.1e-18 m2/s2
B. Liu et al. / Applied Acoustics 80 (2014) 28–35
130 120 110 100 90 80 70 60 50 40 102
Measured Predicted
103
70 60 50 40 30 20 10 0 102
103
Frequency (Hz) Fig. 18. Measured and predicted velocity auto spectrum and the radiated sound intensity of the plate I. Narrow band analysis in per Hz. Flow speed 86 m/s.
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B. Liu et al. / Applied Acoustics 80 (2014) 28–35
Autospectrum of velocity dB (ref. 1e-18m2/s2)
130 110 100 90 80 70 60 50 102
Radiated Sound Intensity dB (ref.1e-12W/m2)
0.93mx0.3m 0.31mx0.3m
120
103
70 60 50 40 30 20 10 0 102
103
tunnel measurements. For the plates tested, it is evident that spanwise stiffeners may increase noise radiation a few dB above 500 Hz, while have almost no influence on the vibration level of the plates. Unlike the spanwise stiffeners, the streamwise stiffeners have almost no influence both on the vibration level and radiated sound intensity of the plates. The measured and predicted results imply that reducing the length or width for a reference plate has almost no influence for the vibration level of the plate subjected to TBL excitations, while reducing the length (along with the flow direction) of the plate may increase the radiated noise, and reducing the width of the plate has no influence on the radiated sound intensity. These conclusions are tentative if applied to typical aircraft plates, since the experiment condition does not account for the full frame of an aircraft sidewall (skin panel/wall cavity/trim panel) and much higher speed for aircraft during cruise. However, the conclusions imply that ring stiffeners must be considered carefully if TBL-induced noise is of main concerns.
Frequency (Hz)
Autospectrum of velocity dB (ref. 1e-18m2/s2)
Fig. 19. The predicted data of the plates respectively with the size of 0.93 m 0.3 m and 0.31 m 0.3 m.
The authors gratefully acknowledge the financial support from NSFC Grant 10974221 and State Key Development Program for Basic Research of China (973 Program) Grant 2012CB720204.
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0.62mx0.3m 0.62mx0.15m
120 110
References
100 90 80 70 60 50 102
Radiated Sound Intensity dB (ref.1e-12W/m2)
Acknowledgments
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70 60 50 40 30 20 10 0 102
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Frequency (Hz) Fig. 20. The predicted data of the plates respectively with the size of 0.62 m 0.3 m and 0.62 m 0.15 m.
0.93 m 0.3 m and 0.31 m 0.3 m is illustrated in Fig. 19. The other parameters used in calculation is same as the plate I. Similar to the measured plate with and without spanwise stiffeners, reducing the length of the plate almost does not affect the vibration spectrum, however, reducing the length of the plate increases the radiated sound intensity obviously above 500 Hz. The predicted data of the plates respectively with the size of 0.62 m 0.3 m and 0.62 m 0.15 m is illustrated in Fig. 20. The other parameters used in calculation is same as the plate J. Similar to the measured plates with and without streamwise stiffeners, reducing the width of the plate almost does not affect the vibration spectrum and as well the radiated sound intensity in the frequency range of calculation.
4. Conclusions The influence of stiffeners on the plate vibration and noise radiation induced by turbulent boundary layers is investigated by wind
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