The Proper Orthogonal Decomposition (POD) method and the Polar Correlation Technique (PCT) are applied to the analysis of far-field acoustic measurements from a high temperature Mj=1.38axisymmetric jet issuing from a 50.8 mm converging nozzle. The acoustic data is decomposed ineigenfunctions and eigenvalues using POD for a subsequent application of the PCT to determine noisesource locations. The far field acoustic data obtained from a normal jet and with microjet injection at thenozzle exit are examined. For the normal jet, the POD method shows that a single dominant eigenfunctionwith 61% of the energy. With the application of microjet injection, the dominant mode energy issignificantly reduced. The PCT further reveal that the extent and the strength of the source region are reduced by the use of microjet injection.
1. INTRODUCTION The database used in this investigation was
obtained in the High Temperature Supersonic JetFacility at the Fluid Mechanics Research Laboratory ofthe Florida State University. High-pressure microjetsare injected into the primary jet near the nozzle exit to manipulate the dominant source region, which typicallyextends from 5 to 20 diameters from the nozzle exit. Recent results of experiments on a Mj = 0.9 round jetsuggest that the interaction of the microjets with the jetshear layer reduces the turbulence levels in the noiseproducing region of the jet8. It appears that the microjets influence the mean velocity profiles such thatthe peak normalized vorticity in the shear layer is significantly reduced and thus inducing an overallstabilizing effect. Therefore, it is suggested that an alteration in the stability characteristics of the initialshear layer can influence the whole jet exhaust, including its noise field. In the absence of flow fieldmeasurements in the high temperature supersonic jet, we are motivated to use POD and PCT as mathematicaltools to get a better understanding of the noisesuppression mechanisms.
To gain a better understanding of themechanism of noise suppression in a supersonic jet, a Proper Orthogonal Decomposition (POD) analysis together with the Polar Correlation Technique (PCT) are applied to far field microphone data.
The POD method was originally suggested byLumely1 to extract organized large-scale structures from turbulent flows. The method provides a set ofoptimized orthonormal basis functions for an ensembleof data. The most important property of POD is itsoptimality in the sense that it provides the mostefficient way of capturing the dominant features of an infinite dimensional process with only few functions.For this reason POD has been in extensive use toexamine turbulent flows, particularly in the analysis ofDirect Numerical Simulation (DNS) data2-6.
The PCT was proposed by Fisher et. al 7 todetermine noise source location and strength from far field microphones, typically located in a polar array ofequally spaced microphones. In this technique, thefluctuating sources are assumed to be on the jet axis.The source distribution is evaluated by considering thecross-spectral density between the signals of a reference microphone and microphones located in a polar arc. 2. BRIEF REMARKS ON THE POD METHOD
The POD is a basic statistical tool first introduced in the context of fluid mechanics byLumley1, and is also known as the Karhunen-Loeveexpansion. The general aspects of the method and applications may be found in the review of Berkooz et.al 10.
† Don Fuqua Eminent Scholar and Professor, AssociateFellow AIAA ‡ Ph.D. student, Student Member AIAA * D. Moreno is with the Centro de Investigaciones enOptica A.C., Leon, Gto., Mexico.
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9th AIAA/CEAS Aeroacoustics Conference and Exhibit12-14 May 2003, Hilton Head, South Carolina
Following the approach of Lumley1, the dominant structures of the flow can be determined bysolving the following eigenvalue problem: 1
)()()( ),(),()(),,(n
nj
ni
nij xxxxC (8)
The numerical procedure to solve equation (2)can be found in Glauser et. al11. Basically the numericalprocedure consists of the calculation of eigenvalues and eigenfunctions of a matrix of size N, where N is thenumber of microphones (10 for this case) used in theexperiment. Thus, the number of modes obtained is 10. The solution is performed for each frequency, = 2 f,independently, and the resulting matrices are complexHermitian, which means that the matrices are real andsymmetric.
),(),(),,,( )( txdxtxttxxR n (1)
Where R is the two point cross-correlation function,are the eigenfunction and the correspondingeigenvalue. Since time is a homogeneous function, we perform a temporal Fourier transform of R, the twopoint cross correlation function, to obtain the cross spectral matrix
3. THE EXPERIMENT),()(),(),,( )()()( xxdxxxC n
inn
j(2)
Briefly, the experiment was performed in the High Temperature Supersonic Jet Facility at the FluidMechanics Research Laboratory of the Florida State University. The data being examined consists of far-field acoustic measurements that were collected froman axisymmetric jet issuing from a 50.8 mm diameter(D) converging nozzle. The jet had an exit velocity of 755 m/sec and was maintained at a stagnationtemperature of 1033 K. There were 10 B&Kmicrophones that sampled the far field noise, and theywere set up in and arc of radius r = 60D, where D is the diameter of the nozzle exit. The arc covered the polar angle, , range from 90 to 155 degrees, relative to theupstream jet axis. All of the microphones weresimultaneously sampled at 250 kHz and each sample set consisted of 409,600 data points. Krothapalli et al.9provide a more complete description of the experiment.
where C is the cross-spectral density matrix, calculatedin this work by using far field microphone data. The cross-spectral matrix can be represented as:
),(ˆ),(ˆ),,( xFxFxx jiijC (3)
where the symbol * denote complex conjugate and F is the Fourier transform, given as:
dtetxFxF ti),(),(ˆ (4)
The Fourier transform of F can be represented as a sumof eigenfunctions in the following way
4. RESULTS OF THE POD ANALYSIS
1
)( ),()(),(ˆn
nn xxF (5) Solving the eigenvalue problem represented by
equation (2) we get eigenfunctions and eigenvalues. In this case, the eigenfunctions and eigenvalues are spatial-frequency dependent. The higher energy in theeigenvalues represents the dominant feature in the dataset. The total energy for each eigenvalue can be determined as:
where is random and uncorrelated functions of frequency, and can be obtained by projection of theeigenfunction onto individual realizations of the flowfield as:
dffE nn )()()( (9)dxxxF nn ),(),()( )( (6)
The energy contained in each eigenfunction of the PODis depicted in Figure 1 for the normal jet and the jetwith microjet injection. It is clearly depicted that mostof the energy is contained in the first eigenvalue or thefirst mode of the POD. Very little energy is observed at modes higher than about 5. The effect of microjetinjection is to reduce the energy in the dominant modeby almost 67%. It is of interest to determine the directrelationship between the dominant mode and the jetnoise sources.
The spectrum can also be reconstructed at each position by
1
2)()( ),()(),(n
nn xxS (7)
and the cross-spectral density matrix by
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4.1 Shape of the eigenvectors
Due to the loss of temporal phase related to theuse of POD, the corresponding eigenvectors cannot beretrieved in the physical space (x,t) and can be described only in the frequency domain (x, f ).However, for each frequency the organization in the xdirection remains meaningful. Hence, informationrelated with the pressure fluctuations in the jet columncan be obtained for the spectral distribution of (n)
which are plotted in the Figure 3 and 4 for the first four modes. The modulus of , represented by the colorcontours, is shown in the plots. Each eigenfunction isinterpolated by using piecewise cubic Hermiteinterpolating polynomial provided by MatLab. Thiskind of interpolation is used because of the necessity to keep the same shape of the original eigenfunction. In the interpolation, 100 points were used.
Figure 1. Absolute value of the POD energy for normal jet and with microjet injection.
The frequency distributions of the eigenspectra(n)(f) for the first four modes of the POD are shown in
Figure 2. The eigenspectra corresponding to the normaljet shows a series of discrete frequencies, commonlyreferred to as screech tones that are present inunderexpanded jets. The fundamental screech toneoccurs at Strouhal number of 0.24. The fundamentaltone and its first harmonics are present in the first foureigenfunctions. This is because of the nature of the experiment, that is, the microphones acquire thepressure fluctuations from the entire jet column. These eigenspectra resemble closely to the frequency spectra obtained using FFT9. The suppression of screech tones due to microjets is clearly evident. The amplitudereductions at all other frequencies are consistent withthe results of Krothapalli et al9. Figure 3. Shape of the first four eigenfunctions for the normal jet;
a) First mode, b) Second mode, c) Third mode, d) Fourth mode.
The eigenfunctions show a well-definedbehavior for all frequencies. For the first and second modes the eigenvectors have a series of maxima at 135 to 140 degrees. These peaks are attributed to the largescale mixing noise and the crackle that are known topresent in the noise field. These maxima extend up to a Strouhal number of 6. This observation is consistentwith that of Laufer et al12. The remaining two modes 3 and 4 may represent the contributions from othercomponents of noise.
The eigenvectors for the case of microjetinjection are shown in Figure 4. The absence of maximum values at 140 degrees is noticed in the firstmode. From our earlier observations9, it was shown thatmicrojets suppress crackle. Hence, it is logical toidentify these maxima with particular sources of noisein the radiation field. The mixing noise component isstill present in these plots albeit with lesser energy(figure 1). These data also show that the noise is being
Figure 2. Energetic eigenspectra; a) First mode, b) Second mode,c) Third mode, d)Fourth mode.
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shifted towards lower angles suggesting that the noisesources may be confined to regions closer to the nozzleexit.
Figure 4. Shape of the first four eigenfunctions for the jet withmicrojet injection; a) First mode, b) Second mode, c) Third mode,d) Fourth mode.
4.2 Power spectrum reconstruction
The contributions of eigenfunctions to thepower spectrum can be studied by the reconstructionusing equation (7). Figure 5 and 6 show thereconstructed and original power spectrum for thenormal jet and with microjet injection respectively. Theresults were obtained using the same interpolationfunction (Hermite interpolating polynomial) providedby MatLab with the same number of points usedpreviously. By using the first eigenfunction in thereconstruction, it is clear from the figure that most ofthe important details are captured. Also shown in thefigure is the difference between the original andreconstructed power spectra. It is evident that most ofthe low frequency component of the noise is capturedby using the first mode only. Using the first six modesin the reconstruction captures most features of theoriginal power spectrum. With microjet injection, it appears that the contributions of high frequencies to thepower spectrum are minimal. A decrease in the amplitude of the low frequency noise is also observed. These observations are consistent with those reportedby Krothapalli et al9. Hence, we believe that the POD analysis of polar arc microphone data is useful indetermining the effect of control schemes on the far field noise with a better insight than that provided bytraditional analysis methods.
Figure 5. Contribution of eigenfunctions to the power spectrum of the normal jet; a) first mode, b) 1-3 modes and c) 1-6 modes.
5. THE POLAR CORRELATION TECHNIQUE APPLICATION
In the polar correlation technique developedby Fisher et al.7 is aimed at obtaining acoustic sourcesstrengths along the jet axis using two microphones. In this technique, one evaluates the source distribution byconsidering the cross spectral density between thesignals of a reference microphone xo and traversable microphone x.
To compute the source strength, the followingequation obtained from the paper of Fisher et al7 is used.
The spatial accuracy of the results can beimproved by increasing the number of microphonesused to collect the data.
o
ayjo
o
adexxRxpyS o
sin),,(~2
),(),(~ /)sin(1(10)
where
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10
1
)()(2
),()(),(n
onn
o xxp (12)
),(),(),,(~
),,(~xpxp
xxCxxRo
oo
(11)
10
1
2)()(10
1
2
,)()(
10
1
)()()(
),()()()(
),(),()(),,(~
n
nn
no
nn
n
njo
ni
n
o
xx
xxxxR
and p is the power spectra of the reference and traversable microphone, C~ is the cross-spectra density, ao is the speed of sound, is the angular spacingbetween microphones and R~ is called coherence. Inorder to reduce the spacing between microphones toavoid aliasing in the oscillatory term in the integral, thecoherence function has been interpolated by using theHermite interpolating polynomial.
(13)
By making use of equations (12), (13) inequation (10) for each individual eigenfunctions, the source strengths are obtained as shown in Figures 7 and8. In this figure, the acoustical source strengthconsidering the original coherence and the first three modes, are shown. The units of the acoustical sourcestrength are decibels per unit length. The reference microphone is located at 90 degrees with respect theinlet axis which corresponds to = 0 degrees inequation 10. The value of extends up to 65 degrees. In this technique, it is desirable to have a large apertureangle for a good resolution with a small separationbetween microphones to avoid aliasing. The separationbetween microphones used in the present investigationis too wide, but was somewhat overcome by theinterpolation of the data. According to the limitationsimposed by the experimental setup, it is believed thatthe noise sources with frequencies less than 10 kHz are resolved.
Figure 5. Contribution of eigenfunctions to the power spectrum of the jet with microjet injection; a) first mode, b) 1-3 modes and c) 1-6 modes.
According to the equations (7) and (8) we canexpand the coherence function and the reference microphone power spectra in terms of eigenfunctionsas:
Figure 7. Acoustic source strength (decibels per unit length). Normal jet case. a) First eigenfunction, b) second eigenfunctionc) third eigenfunction and d) contribution from all eigen- functions.
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The data in these figures show the source strengths on the jet axis covering the region from thenozzle exit to fifty diameters. The apparent Sourcesupstream of the inlet can be caused by one of threeeffects: 1) leakage from the window side lobes(reduces with aperture) 2) spatial aliasing (microphonestoo widely spaced) and 3) reflections from the rig. In the present set up, the first two effects may beresponsible for the presence apparent upstream sources.Except for the screech tones, most of the noise sources are confined to the region within the first ten diametersfrom the nozzle exit. The frequency range of the strongest sources appears to be between the Strouhalnumber (St = fD/Uj) of 0.2 and 1.0. Higher frequencysources are located closer to the nozzle exit, which is consistent with observations made by many previousinvestigators12.
Figure 8. Acoustic source strength (decibels per unit length). Microjet injection case. a) First eigenfunction, b) second eigenfunction c) third eigenfunction and d) contribution from alleigen- functions.
Figure 8 shows the results of source strengthdistribution for the case of the jet with microjetinjection. Upon comparing the corresponding data ofthe normal jet, it is clear that the amplitude of thesources is decreased. The sources are further confined to the region with in the first five diameters from thenozzle exit. The frequency range of the dominantsources is confined to 0.5 < St < 0.8.
6. CONCLUSION
The use of POD method in the analysis of acoustic data is presented for the characterization of the dominant noise sources of a supersonic hot jet. Thesignals from far-field microphones arranged in a polar
arc are used to determine the eigenfunctions and eigenvalues. The POD method shows that a singledominant eigenfunction contains most of the energy.The data obtained from a jet using the microjetinjection at the nozzle exit is analyzed and the results are compared with the corresponding normal jet. Theenergy in the first mode was found to be reduced byabout 67 % due to microjet injection. An attempt ismade to relate this reduction to the various componentsof the far field noise.
The Polar Correlation Technique was appliedto each individual eigenfuction to determine acoustic source strength along the jet axis. The results showconfinement of noise sources with reduced amplitude toa region close to the nozzle exit by the effect of themicrojets.
We like to thank the Office of Naval Research (Technical monitor: Dr. Gabriel Roy) for the support ofthis work.
7. REFERENCES1 Lumley, J. L., “The Structure of InhomogeneousTurbulent Flows”. In atmospheric turbulence and Radiowave propagations. (Yaglom A. M.; Tatarsky V. I., editors) Nauka, Moscow, pp. 166 (1967).
2Aubry, N., Holmes, P., Lumley, J. L. and Stone, E., “The Dynamics of Coherent Structures in the WallRegion of a Turbulent Boundary Layer”, Journal ofFluid Mechanics, 192, pp. 115-173 (1988).
3Gordeyev, S. and Thomas, F., “Coherent Structure inthe Turbulent Planar Jet. Part.1 Extraction of the ProperOrthogonal Decomposition Eigenmodes and Self-similarity,” Journal of Fluid Mechanics, 414, pp. 145-194 (2000)
Citriniti, J. and George, W., “Reconstruction of theGlobal Velocity Field in the Axisymmetric MixingLayer Utilizing the Proper Orthogonal decomposition,”Journal of Fluid Mechanics, 418, pp. 137-166 (2000).
4Bernero, S. and Fiedler, E., “Application of the particleimage velocimetry and proper orthogonaldecomposition to the study of a jet in a counterflow”,Experiments in Fluids, 29 (7), pp.274-281 (2000).
5Graftieaux L., Michard M., and Grosjean N.,“Combining PIV, POD and vortex identificationalgorithms for the study of unsteady turbulent swirlingflows”, Measurement Science and Technology, 12,pp.1422-29 (2001).
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6 Patte-Rouland B., Lalizel G., Moreau J., and Rouland E., “Flow Analysis of an Annular Jet by Particle Image Velocimetry and Proper Orthogonal Decomposition”, Measurement Science and Technology, 12, pp. 1404-12 (2000).
7 Fisher M., J., Harper-Bourne M., Glegg S. A., “Jet Engine Noise Source Location: The Polar Correlation Technique”, Journal of Sound and Vibration, 51, 23-54(1977).
8 Arakeri, V. H., Krothapalli, A., Siddavaram, V., and Lourenco, L., “On the use of Microjets to Suppress Turbulence in a Mach 0.9 Axisymmetic Jet”, to appear in Journal of Fluid Mechanics, 2003.
9 Krothapalli A., Greska B. and Arakeri V., “High Speed Jet Noise reduction using Microjects”, AIAA paper 2002-2450, June 2002.
10 Berkooz, G., Holmes, P., and Lumley, J. “The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows”. Annual Review Fluid Mechanics. 25,539-575 (1993).
11 Glauser, M. N., Leib S. J., and George W. K., “ Coherent Structures in the Axisymmetric Turbulent Jet Mixing Layer”, Proc 5th Symp. Turbulent Shear Flows, Ithaca, New York, USA (1987).
12 Laufer J., Schlinker R. and Kaplan R. E., “Experiments on Supersonic Jet Noise”, AIAA Journal, 14 (4) , pp. 489-497, (1976).
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