Measurement of an aeroacoustic dipole using a linear microphone array

Measurement of an aeroacoustic dipole using a linear microphone array Peter Jordan and John A. Fitzpatricka) Department of Mechanical Engineering, Trinity College, Dublin 2, Ireland

Jean-Christophe Valie`re LEA/CNRS UMR-6609, University of Poitiers, Poitiers, France

共Received 4 October 2001; accepted for publication 27 November 2001兲 It is shown that the standard beamformer technique is inadequate for both the source location and the measurement of a simple dipole and that this is due to the assumption of monopole propagation in the calculation of the phase weights used to steer the focus of the array. A numerical simulation is used to illustrate the problem and to develop a correction to the signal processing algorithm to account for the dipole propagation characteristic. This is then applied to array measurements for an aeroacoustic dipole produced by a cylinder in a cross flow. The resulting source map and the beamformed spectrum are shown to give a true representation of the source energy and frequency content. A secondary effect of this correction is that the array becomes insensitive to other source types so that in addition to acting as a spatial filter, the array can perform as a source filter. This work also demonstrates how an array measurement can be misinterpreted if applied without consideration of the source mechanism. © 2002 Acoustical Society of America. 关DOI: 10.1121/1.1446052兴 PACS numbers: 43.38.Hz, 43.28.Ra, 43.28.Tc 关MSH兴

I. INTRODUCTION

The use of acoustic beamforming has become increasingly commonplace in experimental aeroacoustics, and microphone arrays have proved a useful tool for source localization in a number of applications. Billingsley and Kinns1 used a microphone array based on the beamforming technique to measure jet noise whereas a polar correlation technique was used by Fisher et al.2 also to measure jet noise. Marcolini and Brooks,3 examined helicopter rotor noise by using an array placed above the rotor and Brooks et al.4 reported on the parameters affecting the response characteristics of an array. The advantages and disadvantages of a phased array have been given by Siller et al.5 who implemented the system for monitoring engine core noise. The performance of an array is usually characterized by the directivity pattern as this defines its spatial sensitivity. The directivity pattern can result in the contamination of measurements from extraneous noise sources, a form of spatial aliasing, and this is a function of both source frequency and microphone array geometry. This has resulted in the development of very large arrays with random spacing so that these effects can be minimized over a range of frequencies.6 The implementation of beamforming for a microphone array normally assumes monopole propagation characteristics in the calculation of the phase weights used to steer the focus of the array. As a consequence, the capacity of the system to identify different source types and provide a measure of the contribution of that source in the far-field is limited in certain circumstances. Since the far-field characteristics of an aeroacoustic system depend on the aerodynamic source mechanisms and the manner by which they interact, it a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

J. Acoust. Soc. Am. 111 (3), March 2002

is important to identify these mechanisms so that their contribution to the acoustic far-field can be assessed. The noise source for many aeroacoustic systems is generated by fluctuating lift forces and these are usually dipole type mechanisms. In this work it is shown numerically and experimentally that the standard beamformer method fails when used to measure a simple dipole source represented by an Aeolian tone generated by cross flow over a circular cylinder. For this, the dipole is generated principally by the fluctuating lift force at the vorticity shedding frequency with very much smaller unsteady drag at twice this frequency as described by Phillips.7 Experimental results reported by Leehey and Hanson8 investigated the effect of Reynolds number and spanwise correlation length on the noise intensity. Their results, which agreed well with predictions based on the analysis of Phillips,7 indicated that the oscillating lift coefficient was the controlling factor in the radiated sound intensity and that this increased sharply with increasing Reynolds number. Thus, identification of coherent sound sources typical of dipole mechanisms is important for practical applications in aeroacoustics. A linear microphone array is used to make measurements of the noise generated by a cylinder in a cross-flow. The failure of the beamformer is manifest in both the location of the source and in the derived spectrum which shows almost no energy above the background levels in a frequency range known to contain a discrete frequency source. This is shown to be due to the assumption of monopole propagation and a correction which accounts for dipole propagation characteristics is developed and applied so that the beamformer can be modified to accurately measure the source. After the correction is applied, the spectrum shows, very clearly, a peak at the source frequency, the level of which is a measure

0001-4966/2002/111(3)/1267/7/$19.00

© 2002 Acoustical Society of America

1267

of the source energy. It is further shown how, due to the source-specific nature of the beamformer modification, higher and lower frequency duct, fan, nozzle and jet noise is eliminated from the spectrum. This work demonstrates how array measurements can be misinterpreted if some effort is not made to account for the physical nature of the source mechanisms. It also demonstrates how, through application of corrections to account for different source types, the array can be used, not only to discriminate spatially between sources, but also to discriminate between different source types. II. AEROACOUSTIC SOURCES

The far-field pressure of monopole, dipole and quadrupole sources have been given by, for example, Powell9 respectively as p M 共 t;x 兲 ⫽

␳ 0 a m2 dU m* , x dt

p D 共 t;x 兲 ⫽

␳ 0 a m2 ycos␽ d 2 U m* , x c dt 2

冉冊冉冊

␳ 0 a 2q a q p Q 共 t;x 兲 ⫽ x 3c

2

d 3 U q,x dt 3

共1兲

,

where ␳ 0 is the air density, a m is the radius of an imaginary sphere of fluid containing the source, and y is the distance between the two poles of the dipole 共assumed small com* is the radial velocity of pared to an acoustic wavelength兲. U m the monopole surface, x is the distance from the source, and c the speed of sound. U q,x is the x component of velocity of the quadrupole surface. Monopole sources are associated with volumetric fluctuations of the local medium and the resultant sound power is proportional to first power of the local Mach number. For aeroacoustics, they arise in propeller or fan noise as described by Ffowcs-Williams and Hawkings.10 Dipoles are associated with fluctuating forces, the resultant sound power varying with the third power of the local Mach number, and are dominant sources for many aeroacoustic systems at moderate Mach numbers. Finally, quadrupole sources, associated with the turbulent distortion of the fluid, generate sound power proportional to the fifth power of the local Mach number and arise mainly in high speed jet noise systems. In terms of far-field propagation, the monopole is the most efficient radiator of sound, followed by the dipole, which is less efficient because of the extra degree of freedom of motion associated with the source mechanism and, then, by the quadrupole which is less efficient again, having yet another degree of freedom of motion. This work concentrates on the dipole source, which tends to generate narrow-band acoustic energy due to unsteady aerodynamic loading in fluid– structure interactions. A. The dipole source

From Eqs. 共1兲 it can be seen that the acoustic field of a dipole is dependent on ␪, where ␪ is the azimuthal angle in the dipole plane. The resulting unsteady pressure fluctuations 1268

J. Acoust. Soc. Am., Vol. 111, No. 3, March 2002

FIG. 1. Polar pressure distribution of a point dipole.

propagate spherically into space as shown in Fig. 1 and there exists a planar region between the two poles, and perpendicular to the dipole axis, where the sound waves combine to interfere destructively as the waves are perfectly out of phase. As an observer moves around the dipole, the cancellation becomes less efficient, until a point is reached 共on the dipole axis兲 where there is no cancellation and the sound pressure observed is a maximum.

III. BEAMFORMING AND THE DIPOLE A. Modeling the dipole

The expression for the acoustic field of a monochromatic dipole can be written as p 共 x, ␽ ,k 兲 ⫽

B cos␽ e jkx , x

共2a兲

where k⫽ ␻ /c is the wavenumber, B is the sound pressure magnitude and ␻ the sound frequency in radians, or, alternatively as p 共 x, ␽ ,t 兲 ⫽

B cos␽ e ⫺ j ␻ t , x

共2b兲

i.e., a source with time dependence proportional to e ⫺ j ␻ t , where j⫽ 冑⫺1. Using this expression, a far-field radiation pattern for a dipole was generated as given in Fig. 1. Using this acoustic field, the response of a linear array of 30 microphones arranged parallel to the dipole axis as shown in Fig. 2 was determined. The frequency of the source was 800 Hz, the array was located 1 m from the source and the microphone spacing was 30 mm. Figure 3 shows ten fully reconstructed temporal signals, where it can be seen that there are variations in both magnitude and phase across the array as would be expected. Jordan et al.: Measurement of an aeroacoustic dipole

microphone-source vectors, r m . The influence of these parameters on the resolution power of an array is significant and has been well documented in the literature.1,2,4,11 Higher frequency sources are more effectively resolved by the array due to a narrowing of the mainlobe with increasing frequency whereas low frequency sources cause a broadening of the mainlobe and are less well resolved. An increase in the overall aperture of an array, defined as the angle subtended between its extremities and the source, results in an improvement in the resolution capacity of the array. Thus, for a fixed aperture, the addition of more microphones 共i.e., reducing the microphone separation兲 results in both an improvement in the dynamic gain of the array, which is defined as the difference between the mainlobe peak and the next highest sidelobe, and its resolution power. C. Results of numerical simulation

FIG. 2. Setup for numerical simulation.

B. Beamforming algorithm

The beamforming operation is defined by11 M

X 共 t, ␻ 兲 ⫽

兺 m⫽1

Wm Y 共 t, ␻ 兲 e ⫺ j ␻ ⌬ m , rm m

共3兲

which, in the frequency domain, is M

X共 f 兲⫽

兺 m⫽1

Wm Y 共 f 兲 e ⫺ j 共 2 ␲ f /c 兲 r m , rm m

共4兲

where W m is the element weighting 共nominally equal to 1兲, r m is the distance from the mth sensor to the focus position, Y m is the Fourier transform for the mth sensor, f is the frequency and c is the speed of sound in air. It can be seen from Eqs. 共3兲 and 共4兲 that the output of the array, as well as being a function of the source frequency, is also a function of the

Using the numerically generated acoustic field, the simulated microphone signals were weighted so as to move the focus position over the linear region AB containing the source as shown in Fig. 2. It is clear from the beamformer response shown in Fig. 4共a兲 that the measured source distribution is in error, with a minimum sound pressure at the source location. By comparison, Fig. 4共b兲 shows the results of a simulation for a point monopole source with the same frequency as the dipole, and it is here readily observed that the beamformer correctly identifies the source. The mechanism responsible for the null measurement at the source location for the dipole is illustrated in Fig. 5, where the phase weighted signals for that focus position are shown. This shows that the signals prior to summation are such that destructive interference will occur, as the first 15 microphone signals are all in phase, with the second 15 out of phase. Thus, the out of phase characteristic of the microphones on either side of the zero-axis results in perfect destructive interference when the entire array of signals is combined. It is clear that, physically, this is due to the spherical wave-fronts being 180 degrees out of phase. This is the distinctive phase signature of the dipole and a modified procedure is required so that beamforming can be used to identify dipole sources which can result in coherent noise generation in aeroacoustic systems. In order to implement such a procedure, the steps required will be as follows : 共1兲 Examine the phase characteristics of the signals from the array for each focus position of interest. 共2兲 Locate a potential dipole source as the position where the phase clearly moves through 180 degrees across the array response. 共3兲 Realign the data to correct for the phase difference. 共4兲 Proceed with the beamforming procedure as normal. IV. EXPERIMENTAL SETUP AND ANALYSIS

FIG. 3. Time domain array data from the simulations. J. Acoust. Soc. Am., Vol. 111, No. 3, March 2002

A series of experiments was conducted to implement the procedures as described above. A cylinder was placed in an open jet flow so that an Aeolian tone was generated representing a realistic aeroacoustic dipole source. The experimental facility of Fig. 6 shows the semi-anechoic environJordan et al.: Measurement of an aeroacoustic dipole

1269

FIG. 5. Array response weighted for focus on source.

range of locations. This range was a 1-m linear region, parallel to the array axis and centered at the source. The weighted frequency domain signals were then summed, for each focus position, in order to give a beamformed FFT. This process was repeated for each consecutive block of 4096 data points, which for a total of 150 000 points results in 36 beamformed data sets. The beamformed spectrum was determined from these in the usual way with 36 ensemble averages. This then represents an estimation of the acoustic energy generated by a source at a given focus position. V. RESULTS

The 4-mm cylinder in a cross-flow of 50 m/s, generated strong Aeolian tones at a frequency of 2560 Hz, and these were clearly audible above all other noise sources. Figure 7

FIG. 4. Beamformer response: 共a兲 dipole source and 共b兲 monopole source.

ment 共effective down to a frequency of 500 Hz兲, the fan and nozzle arrangement and the microphone array. A support structure for the 4-mm-diam cylinder was mounted at the exit from the nozzle which had a maximum velocity of 50 m/s. A linear array consisting of 30 microphones was used to perform measurements with microphone spacing of 35 mm so that frequencies up to 4.9 kHz could be resolved. The array consisted of 30 KE4 Sensheiser electret microphones with a 20–20 000 Hz range and integrated preamplifiers. Data acquisition was performed using two 16-channel Kinetic Systems V200 acquisition cards, mounted on a National Instruments VXI chassis with each card capable of acquiring 16 channels of data simultaneously up to 200 kHz. A controller provided local control of the system and was connected to a PCI card on a PC via an MXI interface. A single acquisition consisted of 150 000 data points acquired at 12.5 kHz. The signals were broken down into blocks of 4096 data points, and each block Fourier transformed. Phase weights were applied to the transforms according to Eq. 共4兲, in order to steer the array focus over a 1270


FIG. 6. Experimental setup. Jordan et al.: Measurement of an aeroacoustic dipole

FIG. 7. Normal beamformer source map.

shows the beamformed spectral levels at the source frequency as a function of focus position. The source is located at 1 m and the results demonstrate clearly the inability of the array using the standard beamformer method either to locate the source or to estimate its amplitude. Even more striking is the result of the beamformed spectrum shown in Fig. 8. Although a strong peak was expected at a frequency of 2560 Hz, the spectrum is dominated by lower frequency fan, duct and nozzle noise, showing only a very weak response over the background level at the source frequency. Figure 9 shows phase distributions across the array, for a range of focus positions. A dipole signature can be identified where the phase moves through 180 degrees across the array and this is marked as a series of circles. This is the dipole source location, and to measure the source using the array, the beamformer must be corrected to allow for the distinctive phase characteristic of the dipole source.

FIG. 9. Phase distributions for beamformer focus positions.

Measurements taken with a beamformer give an average measure of sound power radiated from the source region in the direction of the array. However, this is only true when the

signals are weighted such that those components of the signals representing energy which originated at the focus position are perfectly in phase. Even in the case of a monopole, standard phase weighting will not give perfect phase alignment, due to the nonideal nature of real acoustic sources and their propagation characteristics. For measurement of a dipole source the correction procedure proposed earlier in the article is applied to compensate for the phase misalignment shown in Fig. 9. Once this correction has been applied to give the phase distribution of Fig. 10, summation of the signals will result in constructive interference between components of the signal representative of the dipole. The effect of this phase correction on the source map is shown in Fig. 11 together with the original result from the standard beamformer. This source map now gives a maximum at the source where the ordinary beamformer showed a large attenuation. More dramatic, however, is the change in the beamformed spectrum shown in Fig. 12共a兲 where a peak at 2560 Hz is now clearly evident. To demonstrate the effectiveness of the method in identifying a dipole type source, results were obtained at two other flow velocities and the corrected beam-

FIG. 8. Normal beamformer spectrum.

FIG. 10. Beamformer phase correction.

A. Beamformer correction


Jordan et al.: Measurement of an aeroacoustic dipole

1271

FIG. 11. Source map 共- - : normal; ---: with dipole correction兲.

formed spectra are shown in Figs. 12共b兲 and 共c兲. For these, clear peaks at 2200 and 1850 Hz can be observed whereas the spectra from the uncorrected beamforming approach show the same characteristics for all three velocities. VI. CONCLUSIONS

The use of the standard beamforming method fails to locate a simple aeroacoustic dipole source and results in a spectrum which does not represent the true character of the source. The spectrum generated by the standard beamformer method demonstrates the inability of this technique to identify a discrete frequency dipole source in an aeroacoustic system. The source location and the spectra resulting from the phase correction for measurement of a dipole show that the proposed method works well with experimental data. The results of the corrected spectra are indicative of a number of phenomena. The first is the capacity of the beamformer to identify discrete frequency dipole energy sources in the frequency range 1800–2570 Hz. Secondary effects are the reductions in energy shown in the frequency ranges 0–1500 Hz and at higher frequencies than that of vortex shedding. By phase correction of the beamformer to increase its sensitivity to the dipole source, the array appears to have become less sensitive to other source types, and, so, sound energy which was present in the system as a result of other aeroacoustic phenomena has been attenuated. This indicates that the spectrum generated by the standard beamformer represents the sound energy of sources such as the fan, duct, nozzle and jet 共all of these lie on a horizontal axis containing the dipole, perpendicular to the array axis, and, thus, in the same look direction兲, whereas the modified spectrum represents the dipole energy. Thus, there is potential for the technique to be developed so that it can be used to discriminate between different source types as well as acting as a spatial filter. The results demonstrate the degree to which an array measurement can be in error. Aeroacoustic systems are generally directional, with a large variety of different source mechanisms, all combining to produce the resultant acoustic field. This work constitutes a worst case scenario, where a 1272


FIG. 12. Computed spectra 共- - : normal; --- : with dipole correction兲. 共a兲 Jet velocity 50 m/s 共f⫽2560 Hz兲. 共b兲 Jet velocity 44 m/s 共f⫽2200 Hz兲. 共c兲 Jet velocity 37 m/s 共f⫽1850 Hz兲.

very obvious source is completely missed by the standard technique. This is clearly a consequence of the linear array being aligned normal to the dipole axis. Obviously, if the array was aligned parallel to the dipole axis, it would see an Jordan et al.: Measurement of an aeroacoustic dipole

apparent monopole and the source map would identify this. In practical cases, it is likely that the array axis will be oblique and, therefore, the likely orientation of potential aeroacoustic sources should be considered before experiments are performed using arrays. For many airframe applications, the aeroacoustic sources arise from fluctuating loading forces and these are principally of the dipole type. Further work is required to investigate the application of the proposed method to more complex systems. 1

J. Billingsley and R. Kinns, ‘‘The acoustic telescope,’’ J. Sound Vib. 48, 485–510 共1976兲. 2 M. J. Fisher, M. Harper-Bourne, and S. A. L. Glegg, ‘‘Jet Engine Noise Source Location: The Polar Correlation Technique,’’ J. Sound Vib. 51, 23–54 共1977兲. 3 M. A. Marcolini and T. F. Brooks, ‘‘Rotor Noise Measurement Using a Directional Microphone Array,’’ J. Am. Helicopter Soc. 37, 11–22 共1992兲. 4 T. F. Brooks and W. M. Humphreys, Jr., ‘‘Effect Of Directional Array Size On The Measurement Of Airframe Noise Components,’’ AIAA Paper No.


99-1958, 5th AIAA/CEAS Aeroacoustics Conference, Bellevue, Washington, May 1999. 5 H. A. Siller, F. Arnold, and U. Michel, ‘‘Investigation of Aero-Engine Core Noise using a Phased Microphone Array,’’ AIAA Paper No. 20012269, 7th AIAA/CEAS Aeroacoustics Conference, Maastricht, Netherlands, May 2001. 6 H. Holthusen and H. Smit, ‘‘A New Data Acquisition System for Microphone Array Measurements in Wind Tunnels,’’ AIAA Paper No. 20012169, 7th AIAA/CEAS Aeroacoustics Conference, Maastricht, Netherlands, May 2001. 7 O. M. Phillips, ‘‘The Intensity of Aeolian Tones,’’ J. Fluid Mech. 1, 607– 624 共1956兲. 8 P. Leehey and C. E. Hanson, ‘‘Aeolian Tones Associated with Resonant Vibrations,’’ J. Sound Vib. 13, 465– 483 共1971兲. 9 A. Powell, ‘‘Some Aspects of Aeroacoustics: From Rayleigh Until Today,’’ J. Vibr. Acoust. 112, 145–159 共1990兲. 10 J. E. Ffowcs-Williams and D. L. Hawkings, ‘‘Sound Generation by Turbulence and Surfaces in Arbitrary Motion,’’ Phil. Trans. R. Soc. A, 264, 321–343 共1969兲. 11 D. H. Johnson and D. E. Dudgeon, Array Signal Processing 共Prentice Hall, Englewood Cliffs, NJ, 1993兲.

Jordan et al.: Measurement of an aeroacoustic dipole

1273