circular microphone array in our measurements) are expressed in terms of the pressure field as follows: 16th AIAA/CEAS Aeroacoustics Conference. AIAA 2010- ...
AIAA 2010-4018
16th AIAA/CEAS Aeroacoustics Conference
Correlations of jet noise azimuthal components and their role in source identification Victor Kopiev, Sergey Chernyshev, Georgy Faranosov, Mikhail Zaitsev, and Ivan Belyaev Central Aerohydrodynamics Institute (TsAGI), Acoustic Department, 17 Radio street, 105005 Moscow, Russia In this paper, we study the subsonic cold jet noise using the azimuthal decomposition technique (ADT). The results of measurement of correlations for jet noise azimuthal components are reported. It is shown that the correlations for tone-excited jet strongly differ from those for unexcited jet; for example, an unexpectedly high value of correlation for tone-excited jet noise azimuthal components has been obtained for observation angles close to 90º to the jet axis. This behavior of the correlation renders it quite promising for the jet noise mechanism identification and source localization. An analytical model based on qudrupole source distribution is proposed for description of the correlations for unexited jet noise azimuthal components and its applicability is validated experimentally.
A
I.
Introduction
t present the process of noise generation by turbulent subsonic cold jets is thought to be related with different mechanisms: fine-scale turbulence (Refs. 1 and 2), eigen-oscillations of large-scale vortex structures (Ref. 3), instability waves (Refs. 4 and 5) etc. Identification of these mechanisms in the process of jet noise generation and assessment of their contribution to the total noise is an important task. To achieve this on the basis of the measurements of only far-field total noise directivity is difficult, however. The azimuthal decomposition of the far field noise allows us to obtain more detailed characteristics of the sound sources; the previous studies (Refs. 3 and 6) on modeling the azimuthal components show that the observed experimental data for cold subsonic jet noise can be explained if both large-scale vortex structures (vortex rings) and fine-scale turbulence (modeled as moving point quadrupoles) are accounted for as sound sources, the vortex rings’ contribution to the total noise being about 40%. An excellent collapse of the modeling and experimental curves has been observed, which evinces that this is a plausible framework to model turbulent cold subsonic jet noise. To further validate this model and get new insights into the structure of the sound sources, the analysis has been expanded to include correlation characteristics of the azimuthal components. In Ref. 7 the first results of correlation measurements of azimuthal harmonics for unexcited cold subsonic jets (i.e. the correlation between the simultaneous measurements for the same azimuthal mode at two different points) have been reported that experimentally verify the absence of correlation between different modes, which is what should be expected from the theoretical considerations (azimuthal components are orthogonal). The cross-correlation curve for tone excited jet has characteristic peculiarities, albeit it is generally similar in shape to the unexcited jet curve. These curves demonstrate that the large-scale structure sound radiation is concentrated in the region at the right angle to the jet, because in this region the cross-correlation curve for the tone excited jet is significantly higher than the curve for the unexcited jet. This property of the curve is somewhat unexpected from the general view that large-scale structures in jet radiate in the downstream direction, whereas at the right angle to the jet direction it is so-called fine scale turbulence that radiates. In Ref. 7 the spatial correlations of the azimuthal components have been obtained, but no attempt has been made to propose a theoretical model to explain the observed spatial correlation curves. Such an attempt is made in the present work. The sound sources are modeled as a distribution of moving point quadrupoles. The results of modeling are compared with the results of measurements of cross-correlation function for the far sound field of unexcited cold jet with velocity 120 m/s. The comparison is performed for the zeroth harmonics a0 in two frequency bands 600.
II. Experimental Setup The azimuthal decomposition technique (ADT) applied to aerodynamically generated noise is based on the analysis of the far-field sound power directivity for azimuthal components in narrow frequency bands. The sound field is represented in the form of a Fourier expansion in azimuthal θ-harmonics: P (θ ) = A0 + A1 cosθ + B1 sin θ + A2 cos(2θ ) + B2 sin (2θ ) + ...
(1)
Fourier coefficients An and Bn of the azimuthal expansion (1) on a cylindrical surface of radius Ra (the radius of a circular microphone array in our measurements) are expressed in terms of the pressure field as follows: 1 American Institute of Aeronautics and Astronautics Copyright © 2010 by Victor Kopiev. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
A0 (ω ) =
1 2π
An (ω ) =
1
π
2π
∫ P (θ , ω )dθ , 0
2π
∫ P (θ , ω ) cos(nθ ) dθ ,
Bn (ω ) =
0
1
2π
π
(2)
∫ P (θ , ω ) sin(nθ ) dθ . 0
In ADT the standard procedure of ensemble averaging of squared Fourier coefficients (2) is performed in the first place. Then the obtained mean values (ω) and (ω) are summed up in a given frequency range ω1 < ω < ω2 and normalized by the bandwidth Δω = ω2 − ω1 . Thus one obtains a set of coefficients ω
a02 ( Δω ) =
1 2 A0 A0* (ω ) d ω , Δω ω∫1
an2 ( Δω ) =
1 2 An An* (ω ) d ω , n > 0, 2Δω ω∫1
bn2 ( Δω , z ) =
ω
ω
1 2 Bn Bn* 2Δω ω∫1
( z, ω ) d ω ,
(3)
n > 0,
each of which represents the specific energy of an azimuthal mode in the given frequency band. Particularly, a02 gives an input of the axisymmetric mode to the total noise radiation energy, whereas the values ( a12 + b12 ) and (a22 + b22 ) determine an input of the 1st and 2nd modes, respectively. Note that an2 = bn2 for n ≥ 1 due to the axial symmetry of the statistical characteristics of the jet and its noise; the equality of these modes has been validated experimentally (Refs. 3 and 8). Moreover, the experiments also show that higher-order modes an2 and bn2 ( n ≥ 3 ) are small and can be neglected; thus, only the zeroth, first, and second azimuthal harmonics are considered in the present paper. In this study we investigate an axisymmetric subsonic jet issuing from a conic dc=4cm nozzle at Vjet ≈120m/s. A loudspeaker located at the settling chamber of the test facility rig is used as a longitudinal acoustic excitation source, the acoustic excitation frequency being 2032Hz. The jet noise modal structure is measured in TsAGI acoustic anechoic chamber AK-2 using three circular microphone arrays (Fig. 1). Each array comprises six microphones (Bruel & Kjær’s pre-polarized ¼ inch, type 4935) located on a circle with its center on the jet axis. Arrays 1 and 2 are located in the same cross-section but have different radii, Array 3 is located downstream at the distance L=76cm from the plane of Arrays 1 and 2. The arrays move along the jet axis with the step 5cm from x=-65cm to x=250cm, x=0 corresponding to the nozzle exit plane. It should be clarified that the measurements are not performed while the array system is moving, but only when it has come to a halt.
Figure 1. Three microphone arrays, shifted along the jet axis x. The amplitudes of azimuthal components (see Eq. (1)), measured by a microphone array, can be obtained from the signals si, i=1…6, measured by the microphones of the array, by calculating their linear combinations: 2 American Institute of Aeronautics and Astronautics
ai = ∑ α i j s j
(4)
j
where αij are the known coefficients (Refs. 8 and 9). This is done by the in-house code MATRIX, which has been developed specifically for this purpose and has an advantage over the standard B&K Pulse Time 7789 software in that it performs the transformation of the signals significantly faster. The transformed data are then imported back to PULSE LabShop software for post-processing. In present work, the frequency band 600-1400 Hz is chosen for analysis, so that the band does not include the excitation frequency.
III.
Correlation Measurements
In Fig. 2 are shown the correlation functions R(τ) between azimuthal modes measured by Arrays 2 and 3 for unexcited turbulent jet for the position of Array 2 at 180 cm from the nozzle exit. Although the distance between the arrays is 76 cm, the correlation maximum exceeds 0.8 displaying the strong correlation downstream between the azimuthal components of the noise in the far field. A relatively small value of correlation maximum for a2 seems to be related with smallness of the absolute value of a2 which leads to increase errors of its measurement and calculation of the correlation function. The correlations between different azimuthal modes measured by Array 1 at distance 180 cm from the nozzle edge are shown in Fig. 3, where we can see that different azimuthal components are uncorrelated as should be expected from the orthonormality of azimuthal harmonics. Crosscorre lation(Sig na l 13,Sig na l 7) - Input (Ma g nitud e) Ana lyse Re co rd ed D a ta : Input : Inp ut : FFT Analyzer Zo om1
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Crosscorre lation(Sig na l 15,Sig na l 9) - Input (Ma g nitud e) Ana lyse Re co rd ed D a ta : Input : Inp ut : FFT Analyzer Zo om1
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a1-a1 a2-a2 Figure 2. Cross-correlation of identical azimuthal modes for different arrays, x = 180 cm.
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a1-b1 Figure 3. Cross-correlation of different azimuthal modes for one array, x=180cm. 3 American Institute of Aeronautics and Astronautics
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Figure 4. Unexcited and tone excited jets. For the tone excited jet coherent structures are identical to a vortex ring train. If the jet is acoustically excited, the flow becomes more regular (see Fig. 4) and it can be expected that the correlation characteristics of sound generated by the flow will also change. One can see from Fig. 5 that the cross-correlation curve for tone excited jet has characteristic peculiarities, albeit it is generally similar in shape to the unexcited jet curve. The difference between the jets is clearly demonstrated in Fig. 4, where it is shown that the initial region of tone excited jet consists of periodical vortical structures – vortex rings (moving with about a half of the main jet velocity). According to the analysis performed in Ref. 1, these vortices radiate up to 45% of acoustical power, and the correlation curves from Fig. 5 presents a qualitative confirmation to this conclusion. The curves also demonstrate that the large-scale structure sound radiation is concentrated in the region at the right angle to the jet, because in this region the cross-correlation curve for the tone excited jet is significantly higher than the curve for the unexcited jet. This property of the curve is somewhat unexpected from the general view that large-scale structures in jet radiate in the downstream direction, whereas at the right angle to the jet direction it is so-called fine scale turbulence that radiates. 1,2
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4 American Institute of Aeronautics and Astronautics
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Figure 5. Maximum of cross-correlation value Rmax for azimuthal modes a0, a1, and a2 (Arr2-Arr3) for the tone excited (triangles) and unexcited (squares) jet, 600-1400Hz, V=120m/s. The nozzle edge is at x = 0 cm 5 American Institute of Aeronautics and Astronautics
IV. Analytical Model In this section, the results of modeling are compared with the results of measurements of cross-correlation function for the far sound field of unexcited cold jet with velocity 120 m/s. The comparison is performed for the zeroth harmonics a0 in two frequency bands 600 < f < 1000 Hz and 1000 < f < 1400 Hz. The sound sources are modeled as a distribution of moving point quadrupoles (5) D ij (r , t ) = D0ij A(r ) ξ ( x, t ) ij Here D0 is a quadrupole with components that correspond to a particular azimuthal mode, A(r) is a spatial distribution of the sources, ξ(x,t) is a stationary homogeneous stochastic process, x is directed along the jet axis. For the zeroth azimuthal component the quadrupole components in the model (5) are as follows ⎛ ⎜ ⎜1 0 1 D0ij = ⎜ 0 − ⎜ 2 ⎜ ⎜0 0 ⎝
⎞ ⎟ 0 ⎟ 0 ⎟ ⎟ 1⎟ − ⎟ 2⎠
The sound field generated by the sources (5) is then as follows
p (r , t ) = ∫ G ij (r − r ′, t − t ′) D ij (r ′, t ′) dr ′dt ′
(6)
where the Green’s function for quadrupole sources is
G ij (r, t ) =
3r i r j − δ ij r 2 ⎛ r ∂ r2 ∂2 ⎞ ⎛ r ⎞ ⎜ 1 + + ⎜ c ∂t 3 c 2 ∂ 2t ⎟⎟ δ ⎜⎝ t − c ⎟⎠ r5 ⎠ ⎝
c is the speed of sound. Cross-spectral density of sound field (6) at the points r1 and r2 is
Φ pp (r1 , r2 , ω ) = D0ij D0lk ∫
3rbi rbj − δ ij rb2 3rak ral − δ kl ra2 ⎛ ω ⎞ exp⎜ i (ra − rb )⎟ rb5 ra5 ⎝ c ⎠
⎛ ⎞ ω ω2 ω3 ω4 ⎜⎜1 − i (ra − rb ) − 2 ra2 − 3ra rb + rb2 − 3 (ra − rb )ra rb + 4 ra2 rb2 ⎟⎟ c 3c 3c 9c ⎝ ⎠ A(r ′)A(r ′′) Φξξ (r ′ − r ′′, ω ) dr ′dr ′′
(
Here, ra = r1 − r ′ , rb = r2 − r ′′ ,
τR =τ +
)
1 (ra − rb ), c
(7)
Φ ξξ (r ′ − r ′′, ω ) is a cross-spectral density of the
process ξ(x,t). The cross-correlation function is defined as follows 1∞ R pp (r1 , r2 ,τ ) = ∫ Φ pp (r1 , r2 , ω )exp(i ω τ ) dω
π
0
Maximum of the cross-correlation function
R ABS = max(Abs R pp ( x1 , x 2 , τ ))
(R (x , x , τ ) R (x pp
1
1
pp
2
, x 2 , τ ))
This value for different positions of the microphone arrays with respect to the sources location was measured in experiment and was used for comparison with the modeling. Model (5) can describe the behavior of the sources of different types, depending on the type of the correlation function of the process ξ(x,t). For this paper, the sound sources were modeled as spatially uncorrelated sources with averaged lifetime τ0 moving with velocity V . The cross-correlation function for such kind of sources is as follows ⎛ τ2 ⎞ (8) Rξξ (Δ,τ ) = δ (Δ − Vτ ) exp⎜⎜ − 2 ⎟⎟ 2 τ 0 ⎠ ⎝ 6 American Institute of Aeronautics and Astronautics
where
Δ = x2 − x1
is the distance between the sources. The cross-spectral density of the sources is a Fourier
transform of the cross-correlation function (8). Thus,
⎛ Δ2 ⎞ Δ⎞ 1 ⎛ exp⎜⎜ − 2 2 ⎟⎟ exp⎜ − iω ⎟ V V⎠ ⎝ ⎝ 2τ 0 V ⎠ The sources are supposed to be located at the x-axis with length L along x – coordinate. Thus, the spatial localization is given by a function ⎛ x2 ⎞ A(r ) = exp⎜⎜ − 2 ⎟⎟ δ (r ) ⎝ L ⎠ Parameters of the model were chosen to fit the main peculiarities of the measured cross-correlation function −4 R pp (r1 , r2 ,τ ) in two frequency ranges. These parameters are the following: V = 60 m/s, τ 0 = 3 ⋅ 10 s , L = 0.25 m. Φ ξξ (Δ, ω ) =
Cross-correlation for sound signal in two frequency bands has been calculated for these parameters. A comparison of the measurement data and calculation results is shown in Figs. 7 and 8. The point x = 0 corresponds to the position of the array system where its middle point is located exactly in the plane of the nozzle surface, the middle point of the region of modeled sources was located at the point x = 22 cm. Fig. 7, 8 shows a qualitative agreement of measured and calculated data. The main qualitative difference is the minimum of Rabs at the angle 340 from the jet axis. This minimum is predicted by the model for the frequency range 600 – 1000 Hz but not available in the measurement data. To clarify what determines the shape of the cross-correlation function, calculations for the frequency range 600 – 1000 Hz have been repeated with compressibility excluded from the consideration. The results of calculations are compared in Fig.9 against a background of directivity. The directivity curve has two pairs of clear minima. Each of these minima corresponds to such position of the arrays that one of the arrays is located at a zero of directivity pattern. If one of microphone array (see Fig.1) is located at the zero of the directivity pattern it receives signals both from the positive and negative lobes of the directivity pattern, whereas the second array receives signals only from one lobe of the directivity pattern which results in a significant decrease of correlation of the signals. The considered source description is simplified and takes into account only a few main features. In particular, a comparison of calculated and measured auto-spectra (Fig.10-11) shows that predicted auto-spectrum is too narrow. Besides, the calculated directivity is close to the directivity determined by far-field limit (Fig.12), while there is amplification in the measured directivity pattern in downstream direction, possibly, due to refraction. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -50
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Figure 7. Maximum of cross-correlation, frequency range 600 < f
