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PAGE 1 AN ASSESSMENT OF TRAILING EDGE NOISE MEASUREMENT TECHNIQUES By CHRISTOPHER JOHN BAHR A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 1 PAGE 2 2010 Christopher John Bahr 2 PAGE 3 To my parents for raising me, my wife Stephanie for supporting me, and God for inspiring me 3 PAGE 4 4 ACKNOWLEDGMENTS Financial support for this work was provided in part by NASA Langley Research Center, monitored by Drs. Mehdi Khorrami and Meel an Choudhari and by the Florida Center for Advanced AeroPropulsion. I a ppreciate the opportunities I have had to interact with Drs. Khorrami and Choudhari, as well as the chance to work with Dr. Florence Hutcheson and Dr. David Lockard. I would like to thank Dr. Louis N. Cattafest a III, my committee chair, for leading me through my development as a researcher. His (thorough) insistence on attention to detail has turned me into a far better experimentalist than I would have been otherwise. I would also like to thank the rest of my committee for the opportun ity to work and consult with them: Dr. Mark Sheplak, Dr. Lawrence Ukeiley, and Dr. Jian Li. I thank all of the Interdisci plinary Microsystems Group for the constant interaction and thoughtful discussions they provide I would particularly like to acknowledge the rest of the work group class of 2003: Vijay Chandrasekha ran, TaiAn Chen, Ben Griffin, Brian Homeijer and Erin Patrick McKnight. It was an honor climbing the ranks w ith you. I owe special thanks to Drew Wetzel, Nik Zawodny, Fei Liu and Tarik Ya rdibi for all the assist ance they have given me with my research. I thank my parents for nurturing my curios ity from an early age, and supporting me through my extended stay in college. I th ank my wife, Stephanie, for her constant encouragement, and for keeping me grounded. Fina lly, I thank God for granting me the gifts He did, and for staying faithful to me through all these years. PAGE 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.........................................................................................................................9 ABSTRACT...................................................................................................................................21 CHAPTER 1 INTRODUCTION................................................................................................................. .23 Trailing Edge Noise................................................................................................................24 Problem Statement...........................................................................................................24 Existing Research............................................................................................................25 Theoretical development..........................................................................................25 Computational development....................................................................................27 Experimental development.......................................................................................28 Research Structure..................................................................................................................31 Existing StateoftheArt.................................................................................................31 Research Objective..........................................................................................................32 Expected Contributions...................................................................................................32 Research Roadmap..........................................................................................................32 2 AEROACOUSTICS THEORETICAL DEVELOPMENT.................................................36 LighthillBased Analysis....................................................................................................... .36 Linearized Hydrodynamics.....................................................................................................41 Howes Analysis.....................................................................................................................42 3 EXPERIMENTS IN TRAILING EDGE NOISE...................................................................46 Paterson, Vogt, Fink & Munch (1973)...................................................................................46 Yu & Joshi (1979).............................................................................................................. ....47 Brooks & Hodgson (1981)......................................................................................................49 Blake & Gershfeld (1988)......................................................................................................51 Brooks, Pope, & Marcolini (1989).........................................................................................52 Hutcheson & Brooks (2002, 2004).........................................................................................54 Examples of Additional Notable Work..................................................................................56 Experimental Body of Work Summary...............................................................................59 4 EXPERIMENTAL METHODS AND SETUP......................................................................61 Experimental Facility UFAFF.............................................................................................61 5 PAGE 6 Airfoil Model..........................................................................................................................62 Model Performance................................................................................................................64 Steady Pressure Behavior................................................................................................64 Unsteady Surface Pressures.............................................................................................67 Hotwire Anemometry......................................................................................................70 Acoustic Measurements.......................................................................................................... 73 CoherenceBased Techniques.........................................................................................73 Coherent output power.............................................................................................74 Threemicrophone method.......................................................................................80 Beamforming Techniques...............................................................................................90 Test Matrix.................................................................................................................... ..........96 5 RESULTS & DISCUSSION................................................................................................138 Installation Effects........................................................................................................... .....138 Comparison of TwoMicrophone Methods..........................................................................142 ThreeMicrophone Method...................................................................................................145 Additional Linear Array Analysis........................................................................................147 Beamforming Results...........................................................................................................1 49 1/3rd Octave Scaling..............................................................................................................156 Mach Number Scaling...................................................................................................156 Angle of Attack Behavior..............................................................................................158 Instrumentation Offset...................................................................................................159 6 CONCLUSIONS & FUTURE WORK................................................................................200 A AIRFOIL COORDINATE DE SIGN AND MEASUREMEN T COMPARISON CODE....207 B ANALYSIS OF THE THREEMICROPHONE METHOD................................................212 Formulation.................................................................................................................... .......212 Simulation & Analysis.......................................................................................................... 213 C EFFECT OF MULTIPLE SOURCES ON COHERENCEBASED ANALYSIS...............231 TwoInput/TwoOutput Analysis.........................................................................................231 Autospectral Scaling......................................................................................................232 CrossSpectral Scaling...................................................................................................235 Coherent Output Power Behavior.................................................................................236 General Monopole MultipleInput/MultipleOutput Behavior.............................................241 Problem Formulation.....................................................................................................241 Simulation..................................................................................................................... .249 Dipole Analysis and Comparison with Experimental Data..................................................252 Summary...............................................................................................................................258 D ARRAY CALIBRATION IN THE PRESENCE OF ECHOES...........................................272 Ideal Case Problem Statement..............................................................................................272 6 PAGE 7 7 Ideal Case Analysis............................................................................................................ ...273 Single Ideal Reflection........................................................................................................ .274 Analysis Options...................................................................................................................279 Impulse Response Analysis...........................................................................................279 Cepstrum Alanysis.........................................................................................................280 Single Ideal Reflecti on: Simulation.....................................................................................284 Real Experimental Conditions..............................................................................................288 Downsampling......................................................................................................................293 Low Frequency Treatment....................................................................................................295 Application to Beamforming Results...................................................................................297 Conclusion............................................................................................................................299 E COMPARISON OF DAS AND DAMAS RESULTS..........................................................344 REFERENCES............................................................................................................................359 BIOGRAPHICAL SKETCH.......................................................................................................368 PAGE 8 LIST OF TABLES Table page 11 Approach noise levels for a Boeing 747400 with Pratt & Whitney Advanced Ducted Propulsor (ADP) engines...................................................................................................34 41 Key UFAFF design characteristics..................................................................................134 42 UF NACA 63215 ModB airfo il profile design coordinates..........................................135 43 UF NACA 63215 ModB chord wise pressure tap locations.........................................136 44 Geometric and correct ed microphone angles...................................................................136 45 UFAFF mediumaperture arra y microphone design coordinates....................................137 46 Test matrix of experiments to study measurement techniques........................................137 51 Integrated array levels at 2,512 Hz for M = 0.17, 0 degree AoA for varying integration region dimensions..........................................................................................198 52 Integrated array levels at 7,600 Hz for M = 0.17, 0 degree AoA for varying integration region dimensions..........................................................................................198 53 Integrated array levels at 20,000 Hz for M = 0.17, 0 degree AoA for varying integration region dimensions..........................................................................................199 B1 Equationunknown scaling for varying channel count....................................................230 8 PAGE 9 LIST OF FIGURES Figure page 11 Schematic of major ai rframe noise sources.......................................................................33 12 Schematic of sources in the vici nity of an example trailing edge......................................33 21 Coordinate system for tr ailing edge noise analysis............................................................45 41 Planform schematic of UFAFF showing th e flow path from the garage entrance (top), through the openje t test section and out through the fan section............................98 42 Isometric schematic of UFAFF..........................................................................................99 43 AWeighted outofflow background noise comparison between different aeroacoustic flow facilities..............................................................................................100 44 Cross section of UF NACA 63215 ModB airfoil..........................................................101 45 Photograph of UF NAC A 63215 ModB airfoil.............................................................101 46 Photograph of UF NACA 63215 inst alled in UFAFF, from below...............................102 47 UF NACA 63215 ModB profile de viation between design and measured coordinates.................................................................................................................... ...103 48 Trailing edge profiles at varying spanwise locations.......................................................104 49 Surface error curve fits for UF NACA 63215 ModB....................................................105 410 XFoil predicted Cp distributions fo r design and modified profiles, M = 0.17...............105 411 Cp for 7.5 geometric AoA (3.14 equivalent lift freestream)......................................106 412 Cp for 5 geometric AoA (2.63 eq uivalent lift freestream).........................................106 413 Cp for 2.5 geometric AoA (1.98 equivalent lift freestream)......................................107 414 Cp for 0 geometric AoA (1.19 equivalent lift freestream)...........................................107 415 Cp for 2.5 geometric AoA (0.65 equivalent lift freestream)........................................108 416 Cp for 5 geometric AoA (0.15 equivalent lift freestream)...........................................108 417 Cp for 7.5 geometric AoA (0.33 equivalent lift freestream).........................................109 418 Cp for 10 geometric AoA (0.76 equivalent lift freestream)..........................................109 9 PAGE 10 419 Installed, equivalentlift AoA scaling with geometric AoA for UF NACA 63215........110 420 Evaluation of spanwise uniform ity at zerodegree geometric AoA.................................110 421 Trailing edge sensor layout for th e NACA 63215 ModB, with orange lines representing ribbon cabling, and all dimensions in inches..............................................111 422 Map of trailing edge array spacings.................................................................................111 423 Schematic of a singlewire, 2axis traverse experiment..................................................112 424 Photograph of hotwire installation for a DU 96W180 airfoil, from below the model, viewing the trailing edge..................................................................................................113 425 Photograph of hotwire installation for a DU 96W180 airfoil from above the model, viewing the trailing edge..................................................................................................114 426 Example hotwire calibration data to be used with Kings Law.......................................115 427 Plot of mean and fluctuating boundary la yer velocity profile in the trailing edge vicinity.............................................................................................................................116 428 Plot of mean and fluctuating wake profile in the trai ling edge vicinity...........................117 429 Schematic of an ideal, spatially lumped line source in a coherent powerbased analysis.............................................................................................................................118 430 Block diagram for a SingleInpu tTwoOutput (SITO) system.......................................119 431 Installation schematic for cohere nt output power measurements....................................119 432 Coherent Output Power compared between two experiment sets...................................120 433 Crosschannel coherence compared between two experiment sets.................................120 434 Coherent Output Power plotted as a function of scaled frequency..................................121 435 Crosschannel phase angle compar ed between two experiment sets...............................121 436 Baseline experimental conf iguration with phased array..................................................122 437 Experimental configurati on with free field microphone lo cated at equivalent array location in free space.......................................................................................................12 3 438 Experimental configuration with array horizontally offset from trailing edge by 0.25 m......................................................................................................................................124 439 Experimental configuration for linear a rray measurements conducted prior to current body of research............................................................................................................... 125 10 PAGE 11 440 Comparison of differing coherenc ebased noise reduction techniques...........................126 441 Effect of shear layer correction on reliability of dipolelike assumption........................126 442 Phase error comparison for linear arra y microphones, referenced to microphone 12.....127 443 Phase error comparison for nonc ontaminated linear array microphones,......................127 444 Comparison of power predictions from different methodologies....................................128 445 Isometric schematic of baseli ne experimental configuration...........................................129 446 Layout of the UFAFF medium apertu re array with acoustic treatment...........................130 447 3 dB beamwidth of the UFAFF medium ap erture array as a function of frequency.......131 448 PSF for mediumaperture array at 1,024 Hz....................................................................131 449 PSF for mediumaperture array at 2,512 Hz....................................................................132 450 PSF for mediumaperture array at 5,008 Hz....................................................................132 451 PSF for mediumaperture array at 7,600 Hz....................................................................133 452 PSF for mediumaperture array at 15,008 Hz..................................................................133 453 PSF for mediumaperture array at 20,000 Hz..................................................................134 51 Model installation effects and backgr ound noise for a Mach number of 0.10................161 52 Model installation effects and backgr ound noise for a Mach number of 0.17................161 53 Installation effects and background nois e for upper mic, Mach number of 0.10............162 54 Installation effects and background nois e for upper mic, Mach number of 0.17............162 55 Cepstrum comparison of installati on effects for a Mach number of 0.10.......................163 56 Cepstrum comparison of installati on effects for a Mach number of 0.17.......................163 57 Cepstrum comparison of installation effects for a Mach number of 0.17, where quefrency has been converted to an equivalent lag distance using the measured speed of sound............................................................................................................................164 58 Cepstrum comparison of installation e ffects for a Mach number of M = 0.10, for free field microphones mounted above the model trailing edge.............................................165 59 Coherent power analysis of inarray B&K 4138 for a Mach number of 0.10.................166 11 PAGE 12 510 Coherent power analysis of inarr ay B&K 4138 for a Mach number of 0.17.................166 511 Coherent power analysis of oppositearray G.R.A.S. 40BE for M = 0.17......................167 512 Coherent power analysis of free field B&K 4939 for a Mach number of 0.10...............167 513 Coherent power analysis of free field B&K 4939 for a Mach number of 0.17...............168 514 Coherent power analysis of freef ield opposing G.R.A.S. 40BE for M = 0.17...............168 515 COP confidence intervals for the free field case for a Mach number of 0.10.................169 516 COP confidence intervals for the free field case for a Mach number of 0.17.................169 517 Ordinary coherence function computed between the upper G.R.A.S. 40BE and lower B&K 4939 trailing edge microphones for the free field case, for two different Mach numbers........................................................................................................................ ....170 518 Uncertainty bounds on installation effect s for COP analysis, Mach number of 0.10......171 519 Uncertainty bounds on installation effect s for COP analysis, Mach number of 0.17......171 520 Comparison of twoand threemicr ophone methods for array case at M = 0.10............172 521 Comparison of twoand threemicr ophone methods for array case at M = 0.17............172 522 Comparison of twoand threemicr ophone methods for free case at M = 0.10..............173 523 Comparison of twoand threemicr ophone methods for free case at M = 0.17..............173 524 Uncertainty bounds for differing me thods for array case at M = 0.10............................174 525 Uncertainty bounds for differing me thods for array case at M = 0.17............................174 526 Uncertainty bounds for differing met hods for free field case at M = 0.10......................175 527 Uncertainty bounds for differing met hods for free field case at M = 0.17......................175 528 Convergence analysis of Monte Carlo uncertainties for free field case of M = 0.10......176 529 Convergence analysis of Monte Carlo uncertainties for free field case of M = 0.17......176 530 Cdf of Monte Carlo results for free field case of M = 0.17 at 2,512 Hz..........................177 531 Cdf of Monte Carlo results for free field case of M = 0.17 at 3,392 Hz..........................177 532 Cdf of Monte Carlo results for free field case of M = 0.17 at 6,000 Hz..........................178 533 Cdf of Monte Carlo results for free field case of M = 0.17 at 12,000 Hz........................178 12 PAGE 13 534 Nominal and mean threemicrophone met hods with confidence intervals, M = 0.10.....179 535 Nominal and mean threemicrophone met hods with confidence intervals, M = 0.17.....179 536 Comparison of covariancebased fitti ng approaches for free field, M = 0.17.................180 537 Variation of Frobenius Norm Method so lution for varying internal iterations...............180 538 Variation of Rank1 Method solution for varying internal iterations..............................181 539 Variation of Maximum Likelihood Method so lution for varying inte rnal iterations.......181 540 Comparison of threemicrophone method and DAS for M = 0.17..................................182 541 Comparison of confidence interval bounds for M = 0.17................................................182 542 Comparison of nominal and Monte Carlo mean solutions for M = 0.17.........................183 543 Beam map of test section at 1,024 Hz, M = 0.17, 0 degree AoA....................................184 544 Cdf of integrated Monte Carlo data at 1,024 Hz, M = 0.17, 0 degree AoA....................184 545 Beam map of test section at 2,512 Hz, M = 0.17, 0 degree AoA....................................185 546 Cdf of integrated Monte Carlo data at 2,512 Hz, M = 0.17, 0 degree AoA....................185 547 Beam map of test section at 5,008 Hz, M = 0.17, 0 degree AoA....................................186 548 Cdf of integrated Monte Carlo data at 5,008 Hz, M = 0.17, 0 degree AoA....................186 549 Beam map of test section at 7,600 Hz, M = 0.17, 0 degree AoA....................................187 550 Cdf of integrated Monte Carlo data at 7,600 Hz, M = 0.17, 0 degree AoA....................187 551 Beam map of test section at 8,800 Hz, M = 0.17, 0 degree AoA....................................188 552 Cdf of integrated Monte Carlo data at 8,800 Hz, M = 0.17, 0 degree AoA....................188 553 Beam map of test section at 15,008 Hz, M = 0.17, 0 degree AoA..................................189 554 Cdf of integrated Monte Carlo da ta at 15,008 Hz, M = 0.17, 0 degree AoA..................189 555 Beam map of test section at 20,000 Hz, M = 0.17, 0 degree AoA..................................190 556 Cdf of integrated Monte Carlo data at 20,000 Hz, M = 0.17, 0 degree AoA..................190 557 1/3rd octave method comparisons for a M ach number of 0.05 and 0 degree AoA..........191 558 1/3rd octave method comparisons for a M ach number of 0.07 and 0 degree AoA..........191 13 PAGE 14 559 1/3rd octave method comparisons for a M ach number of 0.10 and 0 degree AoA..........192 560 1/3rd octave method comparisons for a M ach number of 0.12 and 0 degree AoA..........192 561 1/3rd octave method comparisons for a M ach number of 0.15 and 0 degree AoA..........193 562 1/3rd octave method comparisons for a M ach number of 0.17 and 0 degree AoA..........193 563 1/3rd octave method comparisons for a Mach number of 0.10 at 1.5 degree AoA.........194 564 1/3rd octave method comparisons for a M ach number of 0.10 at 1.5 degree AoA..........194 565 1/3rd octave method comparisons for a Mach number of 0.17 at 1.5 degree AoA.........195 566 1/3rd octave method comparisons for a M ach number of 0.17 at 1.5 degree AoA..........195 567 1/3rd octave method comparisons for M = 0.10, 0 degree AoA with offset array...........196 568 1/3rd octave method comparisons for M = 0.17, 0 degree AoA with offset array...........196 569 Beam map of narrowband CSM at 2,512 Hz for offset array at M = 0.17, AoA = 0......197 570 Beam map of 1/3rd octave CSM at 2,500 Hz for offset array at M = 0.17, AoA = 0......197 61 Acoustic absorption coefficient of sidewalls used in UFAFF.........................................206 B1 Piston in an infinite baffle, directivity pattern for ka = 3.................................................218 B2 Directivity pattern for SNR = 10, us ing nearesttwo microphone selection....................219 B3 Directivity pattern for SNR = 1, us ing nearesttwo microphone selection......................219 B4 Directivity pattern for SNR = 0.1, using nearesttwo microphone selection...................220 B5 Color maps of crossch annel coherence, SNR = 10........................................................220 B6 Color maps of crossch annel coherence, SNR = 1..........................................................221 B7 Color maps of crossch annel coherence, SNR = 0.1.......................................................221 B8 Histogram of power predictions for SNR = 10................................................................222 B9 Histogram of power predictions for SNR = 1..................................................................222 B10 Histogram of power predictions for SNR = 0.1...............................................................223 B11 Directivity from mean powe r prediction method for SNR = 10......................................223 B12 Directivity from mean powe r prediction method for SNR = 1........................................224 14 PAGE 15 B13 Directivity from mean powe r prediction method for SNR = 0.1.....................................224 B14 Directivity for mean power prediction method with outlier rejection, SNR = 10...........225 B15 Directivity for mean power prediction method with outlier rejection, SNR = 1.............225 B16 Directivity for mean power prediction method with outlier rejection, SNR = 0.1..........226 B17 SNR data spread for microphone 10................................................................................226 B18 SNR data spread for microphone 1..................................................................................227 B19 Histogram of estimated powers for microphone 10 and a true SNR of 0.5.....................227 B20 Histogram of estimated powers for microphone 10 and a true SNR of 0.05...................228 B21 Directivity comparison for SNR = 10..............................................................................228 B22 Directivity comparison for SNR = 1................................................................................229 B23 Directivity comparison for SNR = 0.1.............................................................................229 C1 Schematic of TwoInput/TwoOutput system with no additive noise.............................259 C2 Schematic of TwoInput/TwoOutput syst em with incoherent measurement noise........259 C3 TITO situation where, for monopole sour ces, all phase angle differences cancel..........260 C4 Threeobserver MIMO system modeling a trailing edge of incoherent sources.............261 C5 Threeobserver MIMO block diagram for th e analysis of the system shown in Figure C4....................................................................................................................................262 C6 Coherence between microphones 1 and 3 ba sed on the schematic in Figure C4...........263 C7 Threemicrophone prediction for microphone 3 from Figure C4...................................263 C8 Threeobserver MIMO system modeling a trailing edge of incoherent sources.............264 C9 Coherence between microphones 1 and 3 based the schematic in Figure C8................265 C10 Threemicrophone prediction for microphone 3 from Figure C8...................................265 C11 Eightobserver MIMO system modeling a trailing edge of incoherent sources..............266 C12 Coherence function for spanwise microphones based on schematic in Figure C11......267 C13 Threemicrophone predictions for microphones 2 through 4 in Figure C11..................267 C14 Coherence for spanwise microphones, dipole sources located in Figure C11...............268 15 PAGE 16 C15 Threemicrophone solution, microphones 2 through 4 in Figure C11 for dipoles.........269 C16 Ratio of predicted power to true power for microphones in Figure C11 for dipoles.....270 C17 Comparison of experimental coherence with simulated source coherence between microphones 1 and 4 from Figure C11...........................................................................271 D1 Simplified acoustic calibration experi ment with an ideal source field............................300 D2 Block diagram of ideal acoustic calibration.....................................................................301 D3 Acoustic calibration experime nt with a single reflection................................................302 D4 Block diagram of acoustic calibra tion with a single reflection........................................303 D5 True frequency response magn itude of the second microphone......................................303 D6 True frequency response phase angle of the second microphone....................................304 D7 Schroeder Multisine input waveform...............................................................................304 D8 Schroeder Multisine input power spectral density...........................................................305 D9 Microphone 1 power spectral density..............................................................................305 D10 Microphone 2 power spectral density..............................................................................306 D11 Comparison of frequency response estimate magnitudes................................................306 D12 Comparison of frequency re sponse estimate phase angles..............................................307 D13 Impulse response estimate, invers etransformed from FRF estimate..............................307 D14 Cepstrum of impulse response estimate...........................................................................308 D15 Cepstrum of impulse response estimate, showing the full computed record...................308 D16 Relative FRF magnitude error for impulse response windowing....................................309 D17 Relative FRF magnitude e rror for cepstrum windowing.................................................309 D18 FRF phase error for impulse response windowing..........................................................310 D19 FRF phase error for cepstrum windowing.......................................................................310 D20 Schematic of array calibration experimental setup..........................................................311 D21 Multisine used in experiment, meas ured from function generator output.......................312 D22 Power spectral density of function generator output.......................................................312 16 PAGE 17 D23 Power spectral density of reference B&K microphone located at array center...............313 D24 B&K measurement autocorrelation function, normalized to correlation coefficient......313 D25 Frequency response magnitude estimate for electret 1....................................................314 D26 Frequency response phase a ngle estimate for electret 1..................................................314 D27 Frequency response magnitude estimate for electret 63..................................................315 D28 Frequency response phase a ngle estimate for electret 63................................................315 D29 Impulse response estimate between reference microphone and electret 1......................316 D30 Impulse response estimate between reference microphone and electret 63....................316 D31 Cepstrum estimate between refe rence microphone and electret 1...................................317 D32 Cepstrum estimate between refe rence microphone and electret 63.................................317 D33 FRF magnitude estimates for electr et 1 comparing different methods............................318 D34 FRF phase estimates for electret 1 comparing different methods....................................318 D35 FRF magnitude estimates for electr et 63 comparing different methods..........................319 D36 FRF phase estimates for electret 63 comparing diffe rent methods..................................319 D37 FRF magnitude estimates fo r electret 1, 0.25 ms window...............................................320 D38 FRF phase estimates for electret 1, 0.25 ms window......................................................320 D39 FRF magnitude estimates fo r electret 63, 0.25 ms window.............................................321 D40 FRF phase estimates for electret 63, 0.25 ms window....................................................321 D41 Logarithm of the twosided FRF magn itude between B&K and electret 63...................322 D42 Inverse Fourier transform of logarith mic magnitude spectrum from Figure D41..........322 D43 Unwrapped twosided FRF phase angle from B&K to electret 63..................................323 D44 Inverse Fourier transform of imag inary phase angle fr om Figure D43..........................323 D45 Illustration of available FRF data (blue) and desired FRF calibration curve (yellow)....324 D46 Illustration of available FRF data and de sired FRF calibration curve when decimated by 2...................................................................................................................................325 D47 Downsampled logarithm of twosided FRF from B&K microphone to electret 63........326 17 PAGE 18 D48 Cepstrum corresponding to Figure D47.........................................................................326 D49 Unwrapped downsampled phase angle from B&K to electret 63....................................327 D50 Cepstrum corresponding to Figure D49.........................................................................327 D51 Window length effects for original electret 63 FRF magnitude estimate........................328 D52 Window length effects for downsampled electret 63 FRF magnitude estimate..............328 D53 Window length effects for original electret 63 FRF phase estimate................................329 D54 Window length effects for downsampl ed electret 63 FRF phase estimate......................329 D55 Lowfrequency fit effect for original electret 63 FRF magnitude estimate.....................330 D56 Lowfrequency fit effects for downsampl ed electret 63 FRF magnitude estimate..........330 D57 Lowfrequency fit effects for orig inal electret 63 FRF phase estimate...........................331 D58 Lowfrequency fit effects for downsampled electret 63 FRF phase estimate.................331 D59 Electret 43 phase angle estimates....................................................................................332 D60 Plot of phase angles for electrets 1 through 63................................................................332 D61 Phase angles for electrets 1 through 63, a dditional acoustic treat ment is applied...........333 D62 Phase angle estimate for an acousticallytreated case......................................................333 D63 Impulse response windowing magni tude estimate for electret 43...................................334 D64 Downsampled impulse response window ing magnitude estimate for electret 43...........334 D65 Impulse response windowing pha se estimate for electret 43...........................................335 D66 Downsampled impulse response window ing phase estimate for electret 43...................335 D67 Impulse response magnitude estimate with a 50% Tukey window.................................336 D68 Impulse response phase estim ate with a 50% Tukey window.........................................336 D69 Tukeywindowed magnitude estimate for treated calibration experiment......................337 D70 Tukeywindowed phase estimate fo r treated calibration experiment..............................337 D71 Magnitude estimate comparison between untreated and treated experiments.................338 D72 Phase estimate comparison between untreated and treated experiments.........................338 18 PAGE 19 D73 Mediumaperture array calibration experiment beam map at 1120 Hz...........................339 D74 Ideal calibrated response for experiment beam map at 1120 Hz.....................................339 D75 Uncalibrated beam map of offset speaker at 1120 Hz.....................................................340 D76 Calibrated beam map of offset speaker at 1120 Hz.........................................................340 D77 Beam map of offset speaker at 1120 Hz, calibrated with new technique........................341 D78 Uncalibrated beam map of medium aperture array data at 10 kHz.................................341 D79 Calibrated beam map of medium aperture array data at 10 kHz.....................................342 D80 Uncalibrated beam map of medium aperture array data at 20 kHz.................................342 D81 Calibrated beam map of medium aperture array data at 20 kHz.....................................343 E1 Comparison of nomina l DAS and DAMAS outputs........................................................349 E2 Comparison of DAMAS output to DAS uncertainty bounds..........................................349 E3 Reduced scan plane DAMAS solution for 1,024 Hz.......................................................350 E4 Full test section scan plane DAMAS solution for 1,024 Hz............................................350 E5 Reduced scan plane DAMAS solution for 2,512 Hz.......................................................351 E6 Full test section scan plane DAMAS solution for 2,512 Hz............................................351 E7 Reduced scan plane DAMAS solution for 5,008 Hz.......................................................352 E8 Full test section scan plane DAMAS solution for 5,008 Hz............................................352 E9 Reduced scan plane DAMAS solution for 7,600 Hz.......................................................353 E10 Full test section scan plane DAMAS solution for 7,600 Hz............................................353 E11 Reduced scan plane DAMAS solution for 8,800 Hz.......................................................354 E12 Full test section scan plane DAMAS solution for 8,800 Hz............................................354 E13 Reduced scan plane DAMAS solution for 15,008 Hz.....................................................355 E14 Full test section scan plane DAMAS solution for 15,008 Hz..........................................355 E15 Reduced scan plane DAMAS solution for 20,000 Hz.....................................................356 E16 Full test section scan plane DAMAS solution for 20,000 Hz..........................................356 19 PAGE 20 20 E17 Full test section scan plane DAMAS solution for 20,000 Hz with 5,000 iterations........357 E18 Full test section scan plane DAMAS solution for 20,000 Hz with 10,000 iterations......357 E19 1/3rd octave comparison of DAMAS beam ma p regions with DAS and NAFNoise.......358 PAGE 21 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN ASSESSMENT OF TRAILING EDGE NOISE MEASUREMENT TECHNIQUES By Christopher John Bahr May 2010 Chair: Louis N. Cattafesta, III Major: Aerospace Engineering Aircraft noise is a subject of great concer n, both to aircraft designers and community planners. As air travel becomes more prevalen t and populations near ai rports increase, noise generated by aircraft during take off and landing must be addresse d. To this end, much research has gone into quantifying and reducing the strength of major contributors to aircraft noise, such as jet engines, highlift devices and landing gear. However, as these sources are reduced, a new tier of sources grows in relative importance. Trailing edge noise is the next major noise sour ce which needs to be addressed if aircraft designers wish to continue thei r trend of making quieter aircraft with each new generation. This noise source, due to the scatteri ng of pressure fluctuations in the wing boundary layer and near wake by the rear edge of the wing, is not fully understood, even after over thirty years of research. While theoretical models exist, develo ping measurement tools to validate the models is not a trivial task. Experiments performed decad es ago, which were thought to have provided satisfactory data to designers, are now found to have significant flaws in their results due to instrumentation limitations. Modern experiment al methods can readily show the weaknesses in the older analysis techniques, but no comprehensive set of new t ools is currently available for quantifying and analyzin g trailing edge noise. 21 PAGE 22 22 This dissertation seeks to provi de such a tool set by first reviewing the existing body of theoretical and experimental work, and then reviewing the major methodologies available. Existing methodologies are reformulated with additional analysis, and optimal measurement schemes are proposed. A library of acoustic da ta for a NACA 63215 ModB airfoil is obtained, with the intent of benchmarking th is tool set with respect to ol der results using coherent power techniques, and building a library of new ones using both cohe rent power and beamforming techniques. Results show that th e studied techniques predict simila r levels when airfoil noise is the dominant source in the facility. When di stributed background noise sources are dominant, nominal prediction method do not agree, but met hod uncertainties become sufficiently large that exact level estimates are identified as unreliable. PAGE 23 CHAPTER 1 INTRODUCTION Since the inception of commercial flight, excess noise from air travel has been a major concern for communities near airports. Airpor t air traffic has increased significantly with economic growth and improving transport techno logy. This traffic increase, coupled with significant community expansion in the vicinity of airports, ha s lead to the Federal Aviation Administration (FAA) regulating aircraft noise emissions [FAA 2003]. This regulation has become more stringent over time, and has led to significant research interest in the reduction of aircraft noise [Lockard & Lilley 2004; Macaraeg 1998]. Initial research on aircraft noise focused pr imarily on engine noise [Lockard & Lilley 2004]. However, the dramatic reduction achieved [Willshire 2003] has led to the condition where engine noise, at least during approach conditi ons, is of similar intensity to or less than the airframe noise generated by the aircraft [Lockard & Lilley 2004]. Airframe noise is defined as all nonpropulsive noise generate d by an aircraft in flight [Lilley 2001] and does not include engine/airframe interaction [Lockard & Lilley 2004] Major sources of airf rame noise at landing conditions include landing gear and high lift de vices [Willshire 2003]. A schematic of major airframe noise sources on an example aircraft is shown in Figure 11 (contributed by TaiAn Chen). Once those primary sources are addresse d, trailing edge noise from the baseline clean configuration becomes a major c oncern [Lockard & Lilley 2004]. The present research intends to quantify trailing edge noise fro m a particular airfoil model, a NACA 63215 modB [Szelazek & Hicks 1979]. This model profile has been previously evaluated under various flight conditions, both acoustically and aerodynamically [Hutcheson & Brooks 2002; Hutcheson & Brooks 2004]. A more comp lete (with regards to previous research studies) library of analysis tools will be developed and applied to the data collected within this 23 PAGE 24 research study, with the goal of assembling and assessing a set of reliable techniques for the quantification of trailing edge noi se. Attention will be paid to the strengths and weaknesses of various acoustic measurement tools with a focu s on their limiting assumptions in application to wind tunnel experiments. A majo r contribution is expected regarding the determination of the benefits and limitations of most currentlyu sed trailing edge noise measurement methods. This chapter begins with a definition of trailing edge noise, followed by a problem statement for the proposed research. A review of past research is presented. Theoretical work is first described, followed by a brief discussion of computational analysis, and an overview of experimental work. This is followed by a disc ussion of the research objectives, along with expected contributions. Trailing Edge Noise Trailing edge noise is genera lly defined as acoustic radiation induced by the combination of edge scattering of turbulent boundary layer pr essure fluctuations, bl untedge vortex shedding, and any laminar boundary layer instab ilities on the rear edge of a wi ng or airfoil, along with any contributions from downstream wake uns teadiness [Blake & Gershfeld 1988]. Figure 12 is a schematic of these potential noise sources in th e vicinity of a blunt, rounded trailing edge. Trailing edge noise is present in any application involving an airf oil or exposed edge, be it in a flight condition as an aircraft wi ng or helicopter rotor blade, or in wind powe r installations as the turbine blades [Guidati, Barei & Wagner 1996]. In all cases, the physical noise generation mechanisms are the same (although relative power c ontributions may differ), as are the available acoustic analysis tools. Problem Statement Table 11 shows the importance of airframe noise relative to other airc raft noise sources when engine noise has been treated, and why ai rframe noise is an important consideration. 24 PAGE 25 While significant research is being conducted regarding airframe noise components such as landing gear and high lift noise, the reduction in these sources is expected to fall short of anticipated FAA guidelines for acce ptable community noise levels [Lockard & Lilley 2004]. To meet future approach guidelines, current lowen d sources of airframe noise, of which trailing edge noise is one of the most do minant, must be addressed. In the case of wind turbines, trailing edge noise is a dominant source of noise and must be reduced in conjunction with other mechanical sources, such as the turbine gear box [Guidati et al. 1996]. The problem of trailing edge noise has received signifi cant attention in the form of theoretical, computational and experimental development. The stateofthea rt regarding trailing e dge noise analysis has improved significantly since the early years of theoretical studies [Ffowcs Williams & Hall 1969]. However some gaps still exist in th e available research, especially regarding experimental toolsets and thei r potential application to the development and validation of theoretical and computational models. Existing Research Theoretical development Indepth theoretical development of a traili ng edge noise model can be traced back to 1959, with a Lighthill Analogybased model [Powell 1959]. Extensions of this model focused on edgescattering [Ffowcs Williams & Hall 1969] Ffowcs Williams model made use of Lighthills Equation to describe aerodynamic noise scattered by a halfplane [Lighthill 1952]. 22 22 0 222 0011ijijij ijvvpc ctcyy (11) Here, is the thermodynamic density of the fluid, c0 is the isentropic speed of sound, vi is the local flow velocity, and pij is the applied stress tensor. The independent variables are t (time) and yi (space). This equation reformulates the fluid conservation laws of mass, momentum and 25 PAGE 26 energy such that a densitybased wave operator on the left side of the equation exists in balance with equivalent source terms on the right side. These source terms cons ist of a stress tensor containing apparent turbulent st resses (the first source in pare nthesis on the right), physical stresses (the second source term), and a hydrosta tic pressure term for an equivalent atrest acoustic field. Lighthills Equati on is strictly valid when the re gion outside of the source region, where the acoustic waves generated by the source regi on propagate, is isentropic and at rest. No allowance is made for feedback from the acoustic field into the source region. A dimensional analysis of this equation was first performed fo r jets [Lighthill 1954], and the derived scaling laws were used to determine important parameters for jet noise reduction [Lighthill 1962]. This initial evaluation of scaling invol ved a freefield Greens function. Subsequent analysis involved the inclusion of infinite, plane solid boundaries [Curle 1955]. While Curles model accounted for surface pressuredoubling and sourceboundary interaction, it failed to capture scattering effects from a finitesurface edge, which was late r modeled as a semiinfinite half plane [Ffowcs Williams & Hall 1969]. Alternative formulations for trailing edge noi se have been developed involving linearized hydrodynamic equations for flow around an airf oil [Amiet 1975; Amiet 1976; Crighton & Leppington 1971; Crighton 1972]. These formulati ons lack the physical exactness of Lighthillbased solutions but allow more readily for farfield noise prediction based on known turbulent surface pressure statistics in the vici nity of the trailing edge [Amiet 1976]. The aforementioned methodologies were subseq uently unified, along with adhoc source models, as limiting cases of a more robust m odel [Howe 1978]. This analysis technique attempted to resolve the importa nce of the Kutta condition at th e trailing edge and took into account source motion with respect to the trailin g edge. It was later modified to include 26 PAGE 27 thickness and rounding effects [How e 1988] and then transitioned to acoustically compact chord airfoils, where interaction between the leading edge and trailing edge can be analyzed [Howe 1999; Howe 2001]. Compact assumpti ons can be applied when the le ngth scales of interest, in this case the airfoil chord, are significantly sm aller than the acoustic wavelengths of interest. Research involving compactchord ai rfoils was used to modify Amiets original formulation to include this interaction betw een the leading and trailing edges [Roger & Moreau 2005]. Computational development A variety of computational simulations of tr ailing edge noise, with varying degrees of complexity, have been conducted. The simple st modeling techniques involve semiempirical formulations, in which boundary layer parame ters and mean aerodynamic properties [Drela 2001] are computed from thin airf oil theory or vortex panel met hods and then substituted into linearized flow equations [Moriarty 2005; Mori arty, Guidati & Migliore 2005; Parchen 1998]. More advanced models involve computation of the hydrodynamic nearfield using ReynoldsAveraged Navier Stokes (RANS) solvers before s ubstitution into a reformulation of Lighthills acoustic analogy [Khorrami, Berkman & Choudhari 2000; Singer et al. 2000a; Singer, Lockard & Brentner 2000b]. This reformulation is done using the Ffowcs Williams Hawking Equation [Ffowcs Williams & Hawkings 1969]. The fidelity and complexity of near field computations can be increased by using large eddy simulations (LES) [Manoha, Troff & Sagaut 2000; Oberai, Roknaldin & Hughes 2002; Wang & Moin 2000]. Fi nally, direct numerical simulation (DNS) of the hydrodynamic near field is now possible at low Reynolds num bers using modern computing techniques [Sandberg et al. 2007]. DNS is the most accurate of the available computational tools, but is the most computa tionally expensive. Sandbergs work, for example, was limited to a chordbased Reynolds number (Rec) of fifty thousand. This limitation means the turbulence field around the simulated airfoil cannot fully develop. Thus, tu rbulent pressure fluctuation 27 PAGE 28 contribution to trailing edge noise, which is of ma jor interest in fullscale applications, is not fully accounted for using DNS. Experimental development A large body of experimental work has been as sembled to date, is summarized briefly in Table 12 and is discussed in detail in Chapter 3. In one of the earliest works, Paterson et al. collected data from both surface flushmounted microphones and far field microphones [Paterson et al. 1973]. Experiments were conducted in a moderate Reynolds number regime on NACA 0012 and 0018 airfoils. Individual autospectra were evaluated and compared. Fink installed surfacemounted microphones in a splitterplate airf oil and collected data from both these and farfield microphones in several locations in an openjet wind tunnel [Fink 1975]. Data were collected with the airfoil in the center of the te st section as well as mounted at the edge of the openjet inlet in order to isolate trailing edge noise (with onesided flow) from leading edge effects. Yu performed a similar analysis with a wallmounted trailing edge, and also conducted shadowgraph photography of the tu rbulent flow field [Yu & Ta m 1978; Yu & Joshi 1979]. In the first two studies, singlemic rophone autospectra were anal yzed. The latter also used correlation analysis between surface and far field pressure transducers. Schlinker and Amiet made an early attempt at source isolation using a directional microphone. This involved installing a standard omnidirectional microphone at the focal point of an acoustic mirror to provide a directiona l measurement [Schlinker & Amiet 1981]. This setup reduced the influence of background noise on the microphone measurement but was only effective over a limited frequency range. Br ooks and Hodgson took a different approach at roughly the same time, using a coherent output powe r method to evaluate coherent noise sources on a NACA 0012 airfoil [Brooks & Hodgson 1981]. Unsteady surface pressures were measured in this study as well. Subsequent studies began accounting for the effects of model lift 28 PAGE 29 generation on an open jet test section facility [Brooks & Marcolini 1984]. This analysis was extended to multiple trailing edge configurations at several angles of attack [Brooks, Pope & Marcolini 1989]. Blake conducted a study on a differe nt trailing edge model with an installation similar to that of Yu [Blake & Gershfeld 1988]. A large amount of data was collected regarding boundary layer and surface pressure scaling. Extensions of coherent power analysis have been used in more recent aeroacoustic experiments [Bahr et al. 2008; Mendoza, Nance & Ahuja 2008]. Meanwhile, advances were being made in noi se source localization and identification. Multimicrophone phased arrays were designed for broadband measurement applications [Underbrink 1995] and leveraged for aeroacoustic e xperiments in an anechoic, open jet facility [Choudhari et al. 2002; Humphreys et al. 1998]. The Foundational Dutch National Aerospace Laboratory (NLR) used phasedarray technology to study potential sources of noise contamination of openjet wind tunn el facilities with side plates [Oerlemans & Sijtsma 2000]. Phased array techniques were compared to co herent power methods by Hutcheson [Hutcheson & Brooks 2002] and subsequently used to analyze a NACA 63215 mod B model in a manner similar to the earlier Brooks Pope and Marcolini study of a NACA 0012 [Hutcheson & Brooks 2004]. Studies were also conducted in this manner on wind turbine airfoil candidates [Oerlemans 2004]. Studies on trailing edge vortex shedding have been conducted first with coherence based microphone measurements [Kunze et al. 2002] and later with Partic le Image Velocimetry (PIV) [Shannon, Morris & Mueller 2005]. High speed PIV has been used to capture acoustic source behavior in the vicinity of a trailing edge [Schroder et al. 2004]. Laser Doppler Anemometry (LDA) has also been used to capture unsteady flow fields in acoustic studies [McAlpine, Nash & 29 PAGE 30 Lowson 1999; Nash, Lowson & McAlpine 1999]. Aerodynamic loading effects on the trailing edge noise sources have been considered with standard coherence techniques [Moreau & Roger 2005]. Advanced beamforming techniques have been us ed to attempt field analysis on airfoils. Traditional DelayandSum (DAS ) techniques can determine ac oustic source locations, but do not provide reliable acoustic leve ls caused by those sources. Re gional integration of DAS beam maps, with a correction based on an assumed sour ce behavior, is one technique which has been used to determine acoustic levels due to the observed sources [Oerlemans, Broersma & Sijtsma 2007a; Oerlemans, Sijtsma & MendezLopez 2007b]. Eigenvaluedecomposition methods have been considered [Sarradj 2010]. Deconvolution a pproaches have become another popular option in recent years, with codes such as DAMA S [Brooks & Humphreys 2006a], LORE [Ravetta, Burdisso & Ng 2009], CLEANSC [Sijtsma 2007], SCDAMAS and CMF, [Yardibi et al. 2008] and some modified astronomical codes [Ehren fried & Koop 2006]. Source coherence has been crudely accounted for in the DAMASC code [Brooks & Humphreys 2006b], as well as in LORE. This is an advancement over previo us beamforming techniques, as standard beamforming algorithms assume that multiple sources are all statistically uncorrelated monopoles. These deconvolution appro aches attempt to remove the effect of the finite aperture of the phased array to back out the true levels. They have been used in conjunction with traditional beamforming methods to construct hybrid broadband spectra of trailing edge noise [Shannon & Morris 2008]. Time domain extensi ons for broadband noise have been proposed [Dougherty & Podboy 2009]. While coherent sources are being analyzed, the effects of multipole, directional sources on beamforming analysis have yet to be fully characterized. Some modified beamforming procedures have been proposed [Bouchard, Havelock & Bouchard 2009]. 30 PAGE 31 In addition, some wavenumber analysis has been applied to sepa rate higherwavenumber turbulent fluctuations from lowerwavenumber ac oustic fluctuations in a closedwall wind tunnel [Arguillat et al. 2005; Koop & Ehrenfried 2008]. A practical result of the theoretical, comput ational and experimental analyses discussed above was the discovery that airfoil modifications can reduce trailing edge noise. One potential modification, trailing edge serr ations, as discussed by Howe [H owe 1991], has been studied at NLR [Oerlemans et al. 2008]. Detailed, semiempirical code s modeling trailing edge noise have been assembled [Bertsch, Dobrzynski & Guerin 2008; Nark et al. 2008]. Alternative techniques, such as porous airfoil design, have been evaluated [Sarradj & Geyer 2007]. Research Structure Existing StateoftheArt As discussed above, there are ma ny tools available in the analys is of trailing edge noise. While these tools can be sophisticated, the source assumptions involved in each analysis method's implementation can severely limit the validity of results. As will be discussed in Chapter 4, for instance, the Coherent Output Power method suffers from severe noise contamination effects if the source assumption of trailing edge noise, involving dominant dipolelike radiation, is violated, or if the signaltonoi se ratio (SNR) of the measurement is too low. Also, beamforming techniques, which are beco ming extremely popular, suffer from aperture issues, and no standard method exists for determining integration regi ons of a distributed acoustic source. Different deconvolution tec hniques, which seek to overcome standard beamforming aperture issues, ofte n output differing results [Yardibi et al. 2008]. The existing stateoftheart lacks a standard toolbox of m easurements for analysis and comparison between experimental datasets. The cl osest in use is by Brooks, Pope and Marcolini [Brooks et al. 1989], but this study was conducted with an uncambered airfoil, which while 31 PAGE 32 widelystudied is of limited practical use. Also, the data set was constructed entirely with the Coherent Output Power method, and thus suffers fr om the limitations of that technique. A data set which, under identical experi mental conditions, compares a wi de variety of acoustic analysis techniques for trailing e dge noise, is necessary. Research Objective The objective of this dissertation is to pe rform a suite of acoustic measurements on a classic airfoil, which is representative of the ma in element in a modern commercial aircraft, in a single facility. Redundant measurements will be performed with differing techniques and analyzed to attempt to isolat e the individual failings of each methodology. Improved analysis techniques will be developed to minimize noise source interference in an attempt to isolate trailingedgerelated acoustic sources in each experimental set. Expected Contributions This research is expected to provide fundame ntal characterization of different aeroacoustic measurement techniques and their uncertainties. An experimental aeroacoustic analysis tool set will be produced. The data collected will be provided to the airframe noise community as a benchmark data set for their ow n characterization and validation of airfoi ls and facilities. Research Roadmap This dissertation will first revi ew, in greater detail than that provided here, the fundamental theories and models of trailing e dge noise in Chapter 2. In Chap ter 3, a detailed discussion of existing experimental sets will be presented. This will be followed in Chapter 4 by the experimental setup for the measurements in the Un iversity of Florida Aeroacoustic Flow Facility (UFAFF), along with a discussion of the different processing techniques used here and in the literature, with emphasis on core assumptions, SNR effects and processing ambiguities. Appropriate simulation se tup will be discussed in relation to each measurement technique. 32 PAGE 33 UFAFFs basic design and characteri zation will be reviewed. In Ch apter 5, the data set will be presented, followed by discussion, conclusions and future work. Appendices with detailed derivation and analysis of measurement techniques will conclude the document. Figure 11. Schematic of ma jor airframe noise sources. Figure 12. Schematic of sources in th e vicinity of an example trailing edge. 33 PAGE 34 Table 11. Approach noise levels for a Boei ng 747400 with Pratt & Whitn ey Advanced Ducted Propulsor (ADP) engines. Noise Source Effective Perceived Noise (EPN), dB Inlet 93 Aft fan 93 Combustor 83 Turbine 78 Jet 73 Total Airframe 97 Total Aircraft Noise 100 [Golub, Rawls & Russell 2005] 34 PAGE 35 35 Table 12. Summary of experimental research in trailing edge noise. Author(s) Year Measurements Limitations Fink 1975 Single microphone Background noise contamination Yu & Tam, Yu & Joshi 1978, 1979 Flow visualization, correlation analysis Limited inflow data, noise contamination Brooks & Hodgson 1981 Surface pressures, coherent power Coherent power model assumptions Schlinker & Amiet 1981 Directional microphone Limited frequency range Brooks & Marcolini 1984 Coherent power, open tunnel corrections Model assumptions, open tunnel assumptions Blake & Gershfeld 1988 Single microphone, surface pressures, boundary layer data Background noise limitations Brooks, Pope & Marcolini 1989 Coherent power, many model permutations Coherent power model assumptions, no flow measurements Oerlemans & Sjitsma 2000 Arraybased Measurements Source model assumptions, data referenced to BPM 1989 Hutcheson & Brooks 2002 Coherent power, arraybased methods Source model assumptions, data referenced to BPM 1989 Kunze & Lynch 2002 Coherent po wer Source model assumptions Hutcheson & Brooks 2004 Arraybased measurements, many model permutations Source model assumptions, data referenced to BPM 1989, limited flow data Oerlemans 2004 Arraybased measurements Source model assumptions, data referenced to BPM 1989 Schroder & Dierksheide 2004 High speed PIV No correlation to acoustic data Moreau & Roger 2005 Coherent power Coherent power model assumptions Shannon & Morris 2005 PIV, Arraybased measurements Source model assumptions, PIV conducted in different facility from acoustics Oerlemans, Broersma & Sijtsma 2007 Arraybased measurements, field integration Source model assumptions Bahr et al. 2008 Coherent power and arraybased measurements Coherent power assumptions, no attempt at beamforming acoustic level analysis Shannon & Morris 2008 Arraybased measurements Source model assumptions PAGE 36 CHAPTER 2 AEROACOUSTICS THEORETICAL DEVELOPMENT LighthillBased Analysis Lighthillbased analysis techni ques begin with the derivati on of Lighthills aeroacoustic equation. To start, the equations of contin uity and conservation of momentum are stated, respectively [Panton 1996]. 0 V t (21) V VVp t (22) As in Chapter 1, p is the total stress tensor, includi ng pressure and viscous stresses, is the fluid density, and V is the local velocity vector. These tw o equations can be recast in indicial notation, and the source term on the right side of the momentum equation can be moved into the divergence term on the left [Lighthill 1952]. 0i iv tx (23) 0ijij i jvvp v tx (24) An approximate momentum equation for a nearqui escent inviscid flow field is defined in Equation (25) where unindexed p is the thermodynamic pressure. 0i iv p tx (25) The pressure in this equation can be changed to the fluid density by defining an isentropic speed of sound and applying the chain rule. 36 PAGE 37 2 00ii iivv p c txtx (26) The divergence term in Equation (24) can be moved to the rightha nd side of the equation. The gradient term from Equation (26) can then be added to both sides of Equation (24) 2 0 2 0ijijij i iivvpc v c txx (27) The source term of Equation (27) can be recast as a single source tensor. 2 0 ijijij ijTvvpc (28) Here, the total source tensor Tij is equated to a sum of turbulent stresses, pressure stresses, and viscous stresses, minus the equivalent quiescent medium stresses. By taking the time derivative of Equation (23) and subtracting the divergence of Equation (27) the problem can be reformulated into a single equation. 2 22 2 0 2 ij iiijT c txxx x (29) Equation (29) is a more compact restatement of Equation (11). An analog can be drawn between this and a homogeneous wave equati on from linear acoustics [Blackstock 2000]. 22 2 0 2'' 0iic txx (210) By comparison, Lighthills Equation is simply a wave equation with al l the usual nonlinearities of flow lumped into an inhomogeneous source term. This equation is exactly true for a continuum, under the limitation that the far field ac oustic propagation is in a region of quiescent, linear lossless flow. The near field, where th e source term is nonzero, can have complex, turbulent, viscous flow intera ctions. Solutions to Equation (29) generally involve assumptions of boundary conditions and the neglect of differi ng components of the source term to simplify 37 PAGE 38 analysis. Low speed, freefield analysis of a cold turbulent jet exhausting into a quiescent medium was first conducted [Lighthill 1954], and then extended to turbulent eddies convected near a rigid, acoustically compact body [Curle 1955]. Ffowcs Williams and Hall addressed this equatio n in relation to scattering from a semiinfinite half plane [Ffowcs Williams & Hall 1969]. A schematic of their problem is shown in Figure 21 By assuming that in the Lighthill stress tensor, Equation (28), the viscous stresses are negligible (a valid assumpti on in developed turbulent flow fi elds away from boundaries), the second and third terms exactly balance and cancel as the remaining stress es in the second term are only nonzero on the tensor diagon al, just as in the third. This assumption leaves turbulent fluctuations as the on ly source. Equation (29) can be simplified. 2 22 2 0 2ij iiijvv c txxxx (211) This wave equation has a volume source of quadrupole nature [Dowling & Ffowcs Williams 1983]. The effect of edge scattering is addresse d entirely through the selection of the Greens function. By assuming stationary behavior of turbulence in the flow field, a Fourier transform of Equation (211) can be taken with respect to time, reducing it to an inhomogeneous Helmholtz equation. 2 2* 2* ij ii ijvv p kp xxxx (212) Here, k is the acoustic wavenumber, and the su perscript denotes the Fouriertransformed functions of the stated physical quantities. The magnitude of the acoustic wavenumber, k, is related to the circular fre quency of an acoustic wave, through the isentropic speed of sound. 38 PAGE 39 0k c (213) Ffowcs Williams selects an appropriate Greens function for the given equation and boundary condition, and then scales it in two wa ys [Ffowcs Williams & Hall 1969]. First, he addresses eddies in the near field. Assuming characteristic turbulence scales and solving for the far field acoustic intensity, I, defined in Equation (214) with tav as the time interval of averaging and u the instantaneous acoustic velocity magnitude as a function of space and frequency in cylindrical coordinates yields Equation (215) 01avt av I pudt t (214) 2 4242 0 2 3 32 0 0cos 1 sinsin 22 sin ,,; kU Irz cRkr 2 (215) Here, U is the mean flow velocity. The subscripted 0 terms in this equation denote the source location relative to the origin when referencing independent variables, and mean quantities for thermodynamic properties. R is the distance between the observer and source locations 0rr is the characteristic eddy volume, is the normalized turbulence intensity, 0r is the distance from the turbulent eddy center to the halfplane edge, and is the angle of the eddy center referenced from the trailing edge. The selec tion of sine or cosine is dependent on eddy characteristics. This formulation suppresses a te rm regarding the flow a ngle with respect to the half plane. Suppressing directivity terms in Equation (215) scaling 0r as the turbulence characteristic correlation length scale and scaling the normalized turbulence intensity to unity 39 PAGE 40 shows that this edge scattering noise follows a fifthpower law with flow velocity [Ffowcs Williams & Hall 1969]. 52 0 22 0U I cR (216) Equation (216) assumes a characteristic eddy wavenumber, Equation (217) using a characteristic eddy radius, defined as the relative to the characteristic eddy volume Also, the distance from the turbulent eddy center is assumed to be of similar magnitude to the characteristic eddy size. 0U k c (217) For low, subsonic Mach numbers, this fifthpower scaling is more efficient at acoustic radiation than Lighthills analysis of a free jets eighthpo wer scaling [Lighthill 1954], or Curles analysis of the sixthpower scaling of flow past a compact rigid body [Curle 1955]. FfowcsWilliams proceeds to analyze an eddy far from an edge, and shows that th e scaling becomes the same as with a free jet [Ffowcs Williams & Hall 1969]. 82 52 0U I cR (218) The implication from this analysis is that, for low Mach number flows, turbulence near the surface of a halfplane edge is far more effici ent at scattering hydrodynamic pressure fluctuations into a radiating sound field than fl uctuations elsewhere in the flow field. This means that the fluctuations of instability waves in a tran sitional boundary layer, turbulent boundary layer fluctuations, separation vortices, a turbulent near wake of a large airfoil will be significantly louder than fluctuations further upstream on the airfoil surface, or the fr ee fluctuations of the airfoil's turbulent farwake. The analysis also provides some sense of source directivity, as 40 PAGE 41 shown in the angular terms in Equation (215) The problem formulation is limited, however, in that it does not account for convect ion effects on sound propagation. It has also assumed, as stated before, statistically stati onary behavior of the turbulence, and applied a simplistic model to the characteristic frequency c ontent of a turbulent eddy. Linearized Hydrodynamics Another class of analysis of the trailing edge noise problem involves linearizing hydrodynamic equations. This process solves for source and scattered fields using a velocity potential, generally formulated in both the frequency and wavenumber domains. In addition to the assumptions required for the Lighthillbase d formulations, irrotational flow is required throughout the domain of interest to allow for a velocity potential definition. In the preceding section, irrotational flow was only required in the acoustic far field. However, rotationality could be present in the source region. Crighton set down a formulation of edgescatt ered noise using the above assumptions, and recovers the fifthpower scaling of Ffowcs Williams [Crighton & Leppington 1971]. Chase presents a result for the scattered pressure fiel d as a function of the tu rbulent surface pressure fluctuations near the scattering edge. [Chase 1972]. Amiet evalua ted airfoil noise as a function of fluctuating surface pressure [Amiet 1975], and provided the far field pressure autospectrum as a function of surface pressure fluctuations [Amiet 1976]. 2 2 2 0,,0, ,0 2pp z qqbz Gxy ldG c L (219) Here, the far field acoustic autospectrum is subscripted p, while surface pressure fluctuations within a correlation length of the trailing edge are subscripted q. is a term relating to airfoil compactness, lz is the spanwise correlation length of th e characteristic turbulence fluctuations, and is an observer distance scaled to account fo r convection effects. The halfchord of the L 41 PAGE 42 airfoil is b, and the halfspan is d. While this formulation dir ectly relates surface pressure fluctuations to far field pressure fluctuations, the computation of the airfoil compactness term can be involved. This formulation was subsequently corrected for additional leading edge effects [Amiet 1978b; Moreau & Roger 2009; Roger & Moreau 2005]. While these equations are less physic ally exact with regard to the nature of the source field, they do lend themselves better towards experimental validation. The determination of the true turbulent source from Lighthills equation is experimentally difficu lt, as it requires knowledge of the entire turbulent velocity field as a func tion of space and time, requiring simultaneous multidimensional velocity measurements at each point in space, or at least th e simultaneous turbulent correlation behavior within the region. However, unsteady surface pressure measurements in the vicinity of the airfoil tra iling edge are possible using fl ushmounted unsteady pressure transducers, so an approximate surface wavenum berfrequency field can be determined using homogeneous turbulent assumptions. Howes Analysis Howe extended trailing edge noise analysis to include the benefits of su rface pressure measurement with the exactness of Lighthillbased methods [Howe 1978]. This was done by changing the primary acoustic variable from either pressure or density to stagnation enthalpy. Lighthills acoustic analogy can then be reexpressed as in Equation (220) 2 22 2 00 011 1 DDDv Dv B v DtcDtcDt cDt v (220) Here, B is the stagnation enthalpy, is the vorticity vector, and the total derivatives are denoted by D( )/Dt The following approximations and assu mptions are used in the reduction and solution of this equation. 42 PAGE 43 Isentropic flow in the far field Constant temperature flow Taylors Hypothesis applies to turbulence in boundary layer and near wake [Tennekes & Lumley 1972] There is no backreaction from the tr ailing edge to the upstream flow Eddy convection velocity is a f unction of local mean velocity Eddies moving at different convection velocities are uncorrelated Flow fluctuations are approximated as incompressible near the trailing edge Kuttaconditioninduced vorte x shedding is treated as a stationary vortex sheet The mean shear and wake convection velocities are small relative to c0 Howe presents the far field pressure spectru m as a function of surface pressure spectrum, as well as the overall sound pressu re level (OASPL) for a given observer location. The relations are presented both with and without enforcem ent of the Kutta condition for a flyover case. 1 123 222 0 0 2 2 2 3 001s i nc o s 2 2 1cos1 cosV V z I VMM cvVM Ll p r MMM (221) In Equation (221) pI is the far field pressure without Kutta enforcement, c is the integrated, normalized source spectrum, is the root mean square (rms) turbulent fluctuation velocity, MV is the local Mach number, M0 is the free stream Mach number, L is the trailing edge wetted span, is the local surface angle at the tr ailing edge relative to the xaxis (or flatplate chord line) in Figure 21 and MV1 is the xcomponent of the local Mach number. The Kutta condition reduces this radiated intensity by a factor dependent on the difference between the mean eddy velocity and the mean wake velocity. Howe states that this equation reduces to the Ffowcs WilliamsHall solution when free stream and eddy convection effects are neglected. He subsequently presents a generalized model 43 PAGE 44 for projecting surface pressure fluctu ations to the far field. As with the previous sections, Howe shows a nearfifthpower scali ng for trailing edge noise, depe ndent on forward flight Mach number. Howes model also shows agreement with directivity predictions from previous research. The overall agreement between the m odels shows that reduc ing the eddy convection speed and the spanwise eddy corre lation should reduce trailing edge noise radiation. With a measurement of surface pressure spec tra in the vicinity of the airfoil trailing edge, far field noise can be predicted and compared with experiment, a nd several models of trai ling edge noise can be verified. The surface pressure spectra can also be modeled with sufficient mean boundary layer measurements and existing turbulen t boundary layer predictions. Th is is, indirectly, what some existing research attempts in re lating farfield acoustic spectra to mean boundary layer properties [Brooks et al. 1989; Moriarty 2005]. Unfortunately, while the trailing edge noise problem has been solved for ideal cases, and even extended to more complicated geometries such as blunted and serrated trailing edges [Howe 1988; Howe 1991], much research remains. The scaling of empirical models used in design codes still requires experimental validati on. For experimental validation to occur, a comprehensive, consistent set of analysis tool s is required to appropriately compare results between facilities. These tools must reliabl y extract out real contamination effects in experiments while preserving the necessary data for theoretical model validation. While many experimental data sets have b een assembled with far field noise measurements, no consistent method of acquisition and analysis has been selected. This defici ency must be addressed before theoretical model validation can be revisited. The next chapter addresses many of these data sets, discusses which measurement techniques we re used, and what is lacking in each for comparison with other data sets. 44 PAGE 45 45 Figure 21. Coordinate system for trailing edge noise analysis. Adapted from Howes work [Howe 1978]. PAGE 46 CHAPTER 3 EXPERIMENTS IN TRAILING EDGE NOISE Many experimental techniques have been developed for the analys is of trailing edge noise. These techniques vary in complexity from single microphone measurements to phased array methods. The following is a more extensive revi ew of some of the major works present in the literature, as briefly discussed in Chapter 1. Analysis methods will be discussed, including the limitations of the presented experiments. Paterson, Vogt, Fi nk & Munch (1973) One of the earliest experimental works in trailing edge noise was conducted by United Aircraft Research Laboratories (UARL) and Sikorsky Aircraft Division [Paterson et al. 1973]. NACA 0012 and NACA 0018 models we re placed in UARLs Acoustic Research Tunnel; this tunnel was an openjet facility in an anechoic ch amber, set up with sidewalls of unspecified construction to constrain flow over fullspan mode ls to two dimensions. The selected models had chords of 9, and were tested at Reynolds numbers ranging from 8x105 to 2.2x106. Multiple angles of attack were evaluated, and were listed as geometric angles in the literature, with brief mention of required corrections due to open je t streamline curvature as defined by Pope and Harper [Pope & Harper 1966]. Models were instrumented with microphones to measure surface pressure fluctuations. The surface microphones had protective grids remove d, and the diaphragms were flushmounted with the airfoil surface. The microphones were placed at varying chordwise locations, based on the model selection. At least one microphone on each model surface had a variable spanwise installation. The rest were installed at fixed 1/3 span locations. Additional measurements were conducted with a microphone placed in the acoustic fa r field, 7 above the te st section centerline. 46 PAGE 47 Collected data were processed individually, channelbychannel. 1/3rdoctave and 10 Hz binwidth narrowband spectra we re computed using a variable bandpass filter. At lower Reynolds numbers, bandlimited tonal behavior was detected. The tones were seen in wake hotwire data, as well. This was seen to sc ale as a Strouhal number, defined in Equation (31) with twice the laminar boundary la yer thickness as a length scale. Such scaling was noted to imply a wake scale dependency. The tonal behavi or was not present at higher Reynolds numbers, when the boundary layer and wake had transitioned to turbulent behavior. Trips on the airfoil suction side were found to have little effect on the tonal behavi or. Trips forward of 80% chord on the pressure side suppressed the tones. At the highest Reynolds numbers, airfoil noise was indistinguishable from background tunnel noi se, laying the groundwork for a need for measurement and analysis denoising t echniques for wind tunnel testing. Yu & Joshi (1979) In 1979, Yu and Joshi presented a study of trai ling edge noise involving an airfoil in an openjet aeroacoustic facility [Yu & Joshi 1979]. The authors used a pair of NACA 63012 models in their experiments. Both uncambered m odels were identical, with chords of 0.61 m and spans of 0.3 m. One model was instrumented wi th surface pressure transducers, and the other was machined for smoke injection to permit flow visualization. The airfoil trailing edges were machined to razor edges, as specified in th e airfoil design coordina tes [Abbott & Von Doenhoff 1959]. This razor thickness is no t defined. However, if it is assumed that blunt trailing edge shedding occurs at a Strouhal number [Panton 1996], St of approximately 0.2 [Boldman, Brinich & Goldstein 1976], the lowspeed case (30 m/s) would allow a maximum trailing edge thickness of just over 0.5 mm to be above the 10 kHz cutoff of the lowpass filter used in data acquisition. f d St U (31) 47 PAGE 48 Here, d is the length scale of interest, su ch as trailing edge thickness, and U is the mean free stream flow velocity. Given standard caliper resolution, it can be assumed that knife or razor thickness is less than 0.5 mm. The implica tion of this is that the measured noise in the study would exclude any blunt shedding tones. For this experimental set, Reynolds numbers based on model chord of 1.22x106 and 2.21x106 were considered in the surface pressure and acoustic measurements, corresponding to respective flow velocities of 29.7 m/s and 53.9 m/s. For flow visualization, analysis limitations lowered the experiment Reynolds numbers to 2.5x105 and 6.3x105. Here, Reynolds number is defined in Equation (32) [Panton 1996]. Re UL (32) In Equation (32) is the fluid density, L is the length scale of interest, and is the fluid dynamic viscosity. For these cases, the length scale of interest is considered to be the model chord. Boundary layer characteristics and flow scales were measured using flow visualization. Surface pressure fluctuations on the upper and lowe r surfaces were collected simultaneously with far field noise measurements. It should be not ed that the far field noise measurements, when analyzed using as a single microphone, could not discern trailing e dge noise over facility background noise levels. Correlationbased t echniques were used with surface pressure measurements, as well as microphones placed on opposite sides of the trailing edge. A 180 phase shift was seen between the suction and pre ssure sides of the airf oil. Surface pressures measured on the same side of the airfoil as a given far field microphone were seen to be in phase. An attempt was made to measure the effect of the Kutta condition based on phase relations from Howe [Howe 1978], but the measurement data failed to provide the necessary information. It 48 PAGE 49 was hypothesized by the authors that this was due to the surface pressure measurements occurring too far from the trailing ed ge of the model, at 95% chord. Brooks & Hodgson (1981) In 1981, Brooks and Hodgson published an extensive study of trailing edge noise conducted in the Quiet Flow Facility (QFF) at NASA Langley Research Center. A NACA 0012 airfoil model, with a chord of 0.61 m and a span of 0.46 m, was used. The model trailing edge was modified using hardwood extensions. The th ickest case had a trailing edge thickness of 2.5 mm, while the thinnest was defined as sharp. The method used to attach trailing edges lead to a model chord increase of 12.7 mm for the sharp case. It should be not ed that for the listed chord, the true defined trailing edge thickness of a NACA 0012 model is just over 1.5 mm, thus the airfoil used was actually a s lightly blunted NACA 0012. The eff ect of this minor increase in chord was not addressed in the publication, bu t later work [Blake 1986] indicated that an extended model chord shifts the lowfrequenc y cuton where trailing edge noise becomes independent of leading edge scattering and feedb ack effects. All model trailing edge extensions were joined with glossy 0.08 mm thick Teflon tape Surface pressure transducers were placed in an array along the chord and span of the airfoil, in a pattern symmetric along the upper and lower surfaces of the airfoil. The rearmost trans ducers were 2.54 mm, or 0.42% chord, from the trailing edge. It was noted that this distance incr eased as thinning trailing edge extensions were added. An arc of Brel and Kjr (B&K) Type 4133 free field microphones was set around the airfoil midspan. Microphone distances and an gles from the trailing edge were measured geometrically and then modified for openjet wi nd tunnel effects using a shear layer correction method derived by Amiet [Amiet 1978a]. 49 PAGE 50 Data were acquired using a 14 channel Honeywell 9600 FM analog tape recorder. Due to the large number of sensors in the experimental setup, up to five runs were required for each combination of experimental parameters. Aside from trailing edge thickness, parameters varied included mean flow velocity and model angle of attack (AoA). Not all trailing edge thicknesses were used at every AoA. The blunt trailing edge was used at zero degrees five degrees and ten degrees; the sharp trailing edge was used for zer o degrees and five degrees. The remaining trailing edge permutations were onl y evaluated at a zero degree AoA. The impact of both boundary layer tripping and spanwise threedimensional effects on the state of the boundary layer near the trailing edge was evaluated. It was noted that tripping reduced the threedimensional effects caused by the short span of the airfoil and the presence of hardwalled side plates. Accordingly, the bounda ry layer was tripped for all noted studies. Mean flow aerodynamics were analyzed for all run conditions using a moveable Pitotrake and surface mounted Preston tubes. Both meas urements confirmed twodimensionality of the mean flow field. Boundary layer parameters, such as displacement and momentum thickness, were measured, presumably using the Pitotrake, and computed for each run condition. Some doubt was cast on the validity of the computed di splacement thicknesses, so best fit values were instead used for scaling purposes. This was computed by fitting a regression function to the experimental displacement thickness data using traditional known scaling laws. Detailed statistical results of surface pressure measurements were presented in the paper. Correlation and coherence were evaluated in the chord and sp anwise directions, and edge backscatter effects were discussed. That data were evaluated with respect to Corcos turbulent boundary layer model [Corcos 1964]. Through simp lification, surface pr essure data were claimed to agree with theory regarding near fiel d pressure fluctuations near the trailing edge. 50 PAGE 51 As with Yu and Joshis work, it was dete rmined that single microphone measurements were insufficient for determining trailing edge noise in the QFF. A coherent output power method (COP) was discussed in reference to genera l sources. In processing, it was assumed that trailing edge noise behaved in an antisymmetric fashion for the phase angle as had been previously shown. This allowe d the simplification of the COP method and computation of a denoised directivity pattern. The phase delay between microphones placed on opposite sides of the airfoil was computed and compared to theory. This comparison was used to determine the range over which the data were reliable. A phase rela tionship breakdown indica ted the appropriate cutoff frequencies. Spectra were evaluated with regards to potential nondimensional parameters, and overall sound pressure levels ( OASPL) were calculated and fit to curves as functions of free stream velocity. Finally, the measured surface pre ssure fluctuations were used with the existing trailing edge noise models of Howe and Chase to predict radiated trai ling edge noise [Chase 1975; Howe 1978]. Comparison was made with the far field acoustic measurements, and general agreement in spectral trends was seen. Blake & Gershfeld (1988) In 1988, Blake and Gershfeld presented a twopart comprehensive review of research conducted on trailing edge noise [Blake & Gers hfeld 1988]. The first half was a review of previous theoretical and analytic al work. The second half described new research conducted by the authors. Complete information on the ai rfoil selection was not publishe d in the document. A single forwardhalf airfoil was machined such that its afthalf could be swapped out between two trailing edges. One edge was thin, while the other was thickened to capture some vortex shedding effects. The model chord was 40 inches. Trip tape was applied at 5% chord in all 51 PAGE 52 cases. Measurements were conducted in the Anechoic Flow Facility (AFF) of the David Taylor Research Center. Unsteady surface pressures were measured, as in the previous studies. Cross correlations were computed between surface pressures and boundary layer and wake velocities. The experimental setup for the wake measurement was not provided in the publication, but given existing technology at the time, it can be assumed that hotwire anemometry was employed. For far field noise, a pair of B&K Type 4165 halfinch microphones was placed on opposite sides of the trailing edge, as done by Brooks and Hodgson. The acoustic spectra were calculated using the COP method. Directional arrays were consid ered but not used, due to facility limitations, model size and the frequency range of interest. In processing the surface pressure measurements, much attention was paid to the turbulence statistics near the models boundary la yer separation point. Th e airfoil trailing edge sections were selected specifically for this ch aracteristic separation behavior, and detailed correlation and crossspectra were presented and discussed. As with previous work, the surface pressure and far field acoustic data were used to compare experimental results with Howes theory. Also, as with previous work, genera l agreement was seen between experiments and theory. The blunt trailing edge model showed bett er agreement, and it was claimed that this was due to the stronger radiated field. Brooks, Pope, & Marcolini (1989) Brooks, Pope, and Marcolini published a repo rt in 1989 detailing their experimental research on airfoil self noise [Brooks et al. 1989]. The goal of this report was to generate a set of empirical codes for the prediction of airfoil self noise. The codes accounted for various components of airfoil noise such as turbulent boundary layer no ise, separation noise, tip noise and vortex shedding noise. The code produced nois e predictions for input parameters based on 52 PAGE 53 airfoil chord, AoA, tip conditions and flow speed as well as desired observer output locations. The output data were formatted for onethird octave bands. To generate their empirical relations, the authors returned to the NACA 0012 airfoil design, as used by Brooks and Hodgson [Brooks & Hodgson 1981]. However, many 0012 airfoils with chords varying fr om 2.54 cm to 30.48 cm were machined, with sharp trailing edges, defined as less than 0.05 mm thick. There is no documentation on whether the standard NACA 0012 profile was extended to match this sharp condition, thus altering the original chord by a given amount or if the model was sharpened at the specified trailing edge location, and subsequently blended smoothly back along the airfoil body until the true NACA 0012 shape was recovered. No surface pressure instrumentation was used with this data set. Flow measurements were conducted using hot wire anemometry in NASA Langleys QFF. Boundary layer displacement a nd momentum thickness were calcu lated using threedimensional traversing for both singlewire a nd crosswire installations. Boundary layer behavior as a function of model and flight condition was one of the primary desired parameters for the code. Facility test conditions were varied such that the test sect ion flow speed was a maximum at 71.3 m/s, and AoA varied between zero degrees and 25.2 degrees. This experiment leveraged an openjet correc tion for its calculation of model AoA based on previous work [Brooks & Marcolini 1984]. This correction is an atte mpt to account for the deflection of the test sections finite jet by model lift to calculate an equivalent free stream AoA. While this does not account for details in diffe rences of surface pressure distributions, it does find the appropriate equivalentlif t condition for an open jet exhausti ng into an infinite plenum. The correction utilizes a periodic formulation of potential flow vortex sheets to simulate an infinite cascade of airfoils deflecting a freestrea m flow. The original fo rmulation does not allow 53 PAGE 54 for a downstream jet collector turning the flow ag ain, and thus yields questionable results in facilities with finitelength open test sections. Acoustic measurements were conducted with halfinch free field microphones. The microphones were oriented in a manner similar to that of Brooks and Hodgson [Brooks & Hodgson 1981]. Again, the radiat ed sound field was computed us ing the COP method, and shear layer corrections were used. The measured ac oustic field was compared to the boundary layer measurements for each flight condition, and scalin g relations were developed. These empirical relations were converted into a Fortran code appended to the report. This was one of the first attempts at a prediction tool for trailing edge noise. Subsequent c odes have refined it with more robust boundary layer predictions [Moriarty et al. 2005]. Hutcheson & Brooks (2002, 2004) More recently, Hutcheson and Brooks conducted studies of trailing e dge noise, attempting to leverage newer technologies [Hutcheson & Brooks 2002; Hutcheson & Brooks 2004]. These studies were again conducted in NASA Langleys QFF. The primary focus of the first body of work was to compare COP analysis to arrayba sed techniques, while th e second reevaluated trailing edge noise using arra ybased techniques at differe nt AoAs and flight speeds. For these tests, a NACA 63215 ModB airfoil was used. The model had a chord of 16, and a span of 36, and different trailing edge co nfigurations were studied. The baseline design of a NACA 63215 ModB has a shar p trailing edge [Szelazek & Hick s 1979]. As with previous work, the thickness changes altered th e baseline chord, at least for so me cases. In this work, that chord change was addressed and measured. The upstream distance of blending was not discussed. No surface pressure instrumentation was included. Some of the test cases used trip tape; only tripped bounda ry layer experiments were presented. Test section conditi ons varied from Mach numbers of 0.07 to 0.17, corresponding to 54 PAGE 55 Reynolds numbers varying from 6x105 to 1.6x106, respectively. Airfoil geometric AoA was varied between 6.2 degrees and 8.8 degrees, with corrected values spanning a shorter range, depending on the experimental setup. These airfoil corrections di d not account for airfoil camber. Shear layer corrections for acousti c propagation paths were also employed. Acoustic measurements were simultaneous ly acquired from 35 B&K 1/8 microphones placed in the acoustic far field. Two of the mi crophones were situated so as to perform a COP measurement on the trailing edge of the airfoi l, while the other 33 microphones where arranged in a smallaperture array designed for a wide frequency range of opera tion. Its small size allowed for easy traversing for directivity meas urements, but also gave it very poor spatial resolution at lower frequencies of interest. A frequencydependent perfoot weighting technique was used in an attempt to remove this aperture effect and allow for direct comparison with a COPbased microphone measurement. This tech nique was described in detail by Mendoza [Mendoza, Brooks & Humphreys 2002]. This weighting was employed along with diagonal removal, where the crossspectral matrix array diagonal is set to zero to remove the influence of selfnoise on microphones, and background noise s ubtraction, in which the measured acoustic field of an empty test section is subtracted fr om the airfoils acoustic field. These will be discussed in more detail in Chapter 4. With appropriate analysis, it was determined that for most of th e frequency range of interest, arraybased measurements outperforme d the COP method. COP appeared to handle itself better at low frequencies, but its overlap in effectiveness with the SADA measurements was close enough that the second publication was pres ented using array data only. Sound pressure scaling with Mach number and AoA was pr esented in addition to directivity analysis. 55 PAGE 56 General agreement was found with the predic tion codes provided by Brooks, Pope, and Marcolini, except at high frequencies. Examples of Additional Notable Work A body of work involving airfoil self noise was performed in France. Research was conducted at the Laboratoire de Mcanique des Fl uides et Acoustique de lEcole Centrale de Lyon. This facility (ECLLMFA) has an anech oic wind tunnel [Roger & Moreau 2004] and has generated some theoretical st udies evaluating trailing edge noise models [Roger & Moreau 2005]. Their research involves analysis of wind turbine airfoil profiles at lower, transitional Reynolds numbers [Moreau et al. 2003]. Additional experime nts have been conducted to validate a modification of existing theore tical formulations [Moreau & Roger 2009]. Aerodynamic characterization was performed us ing surface pressure transducers. Both mean and fluctuating pressure fields were anal yzed [Moreau & Roger 20 05]. Detailed analysis was performed on surface pressure fluctuations using techniques fr om the previous stated works, as well as more spectral coherence processing allowable with modern computing. Acoustic measurements were performed with microphones placed in the far field. The facility background noise was first measured and then subtracted from the microphone measurement of the model. No additional coherence processing was performed to extract noise from the signal. At Notre Dame, research was conducted involvin g an airfoil trailing e dge similar to that previously used by Blake [Blake & Gershfeld 1988]. The model, a fl at strut with a 0.91 m chord, was placed in Notre Dames An echoic Wind Tunnel (AWT) [Shannon et al. 2005]. A boundary layer trip was applied to the airfoil, which was tested at Reynolds numbers from 1.2x106 to 1.9x106. Surface pressure fluctuations were measured near the trailing edge and far field noise was simultaneously collected with a large aperture array. In a separate closedwalled facility, Particle Image Velocimetry (PIV) was used to ch aracterize the flow field near the model trailing 56 PAGE 57 edge. Observations from the phaselocked PIV measurements and the computed acoustic spectra were qualitatively compared. Noticeable facility based contamination was observed in some of the beamforming results. Subsequent resear ch focused on applying advanced beamforming techniques to remove the contamination of tunnel background noise in trailing edge measurements [Shannon & Morris 2008]. A significant amount of research has been c onducted at NLR in the Netherlands. Some research was targeted at funda mental experimental techniques, involving characterization of sidewall effects on phased array measurements [O erlemans & Sijtsma 2000]. It was shown that significant measurement errors ca n occur if sidewalls are made of a soundhard material. Acousticallytreated sidewalls allow for much closer recovery of true acoustic source levels. Array power measurements have been a nother point of part icular interest. Some research was focused on simulating an id eal trailing edge to generate a frequencydependent correction factor for p eak array output levels in an attempt to evaluate absolute acoustic levels from the trailing edge noise s ource [Oerlemans & Sijtsma 2002]. This method was found to be dependent on trailin g edge spanwise correlation scales. Subsequent research for generalized aircraft sources applied a simplified integrati on method where the summed power region is normalized by the array point spread fu nction within the integration region [Oerlemans et al. 2007a]. A background subtraction technique, where the integrated power spectrum of the empty test section was subtracted from the integrat ed power spectrum of the model, was applied. It was shown through simulation and measurement th at when sidelobes are within the integration region for the simulated point sour ce, but outside of it for the act ual measured source, significant level errors can occur. Cohere nce reduction, due to the presence of free shear layers in openjet facilities, was also shown to cont ribute to underprediction of acousti c levels for larger arrays at 57 PAGE 58 higher test section speeds. This was computed by comparing the array integrated levels to outofflow far field microphones placed in the open jet wind tunnel facility. Additional research studied se veral airfoils as wind turbin e candidates, and compared NACA 0012 data to previous NASA experiments [Oerlemans 2004]. Array measurement integration was conducted with a co rrection factor for line source be havior based on the sizing of the integration region about the midspan of the airfoil models. Rough agreement was seen between the NLR and NASA data sets at some fr equency bands, but the low frequency data saw significant disagreement. Subsequent research conducted on fullscale wind turbines summed the data in the integration region and divided by the array point spread function [Oerlemans & Sijtsma 2002]. Trailing edge serrations were show n to reduce the trailing edge noise signature. As mentioned in the introduction, a large num ber of advanced beamforming techniques have been developed with the intent of impr oving or supplanting the standard delayandsum beamformer. Many of these techniques involve d econvolution, where the array aperture effects, caused by having finite samples within a finite domain of wavenumber space of the acoustic field, corrupt true images and integrated levels of acoustic sources. Most of these methods provide improved spatial resolution at the cost of computation time. Some research has involved more novel expe rimental techniques. Spark photography with a schlieren system, which illuminates density grad ients in a flow, has been used to study the diffraction effects of a sharp trailing edge unde r different flow conditions, when an acoustic wave interacts with an airfoil [Heavens 1978]. This research illumi nated qualitative behavior of edge scattering with and without flow, but did not provide a me thod for computing absolute far field levels. 58 PAGE 59 Experimental Body of Work Summary The discussed research contains a large variety of installations and measurement techniques. From a facility pe rspective, both openand closedtest section wind tunnels have been used. The closed test se ction tunnels suffer from higher background noise, while the open test section facilities require sh ear layer corrections be applied in postproce ssing. Models with and without camber have been evaluated. T hose without camber allow for some degree of problem simplification, since at zero angle of attack the boundary layers on the top and bottom of the model should develop near identically. Th ose with camber are clos er to a true flight profile. Similarly, sharpened trailing edges allow for the isolation of either laminar instability noise, or turbulent broadband scattering, depending on the experiments Reynolds number regime. Models with blunted trai ling edges are more realistic in regards to true flight models. For this body of research, however, the tunnel characteristics and model have already been determined. As discussed in Chapter 4, all experiments are performed in an openjet wind tunnel facility on a cambered model with a blunted trailing edge. The key parameters of study are the experiment types. When comparing to prev ious work, many experiment types must be considered. Acoustically, measurement techniques were so metimes as simple as a single microphone placed in the models far field. This microphone may or may not have been directional. Pressure measurements suffered from exce ssive background noise le vels. To overcome background noise contamination, two primary paths have been taken. One path involves applying coherencebased modeling to the tr ailing edge noise source and using multiple microphones to back out a coherent acoustic fiel d. The other involve s the opposite assumption of source nature, mainly that th e source is composed of a series of incoherent monopoles. This assumption is used to analyze the trailing edge region using beamforming algorithms and 59 PAGE 60 60 spatially reject noise coming from other sources Both of these methods apply significant restrictions to the trailing edge noise source. Coherencebased techniques may break down when the acoustic field is actually ge nerated by a collection of semicohe rent or incoherent sources. Beamforming may fail, or at least lead to inco rrect level measurements, if there is a group of partially coherent sources, or th e sources are not monopole in nature. The experiments conducted in this body of resear ch must be sufficiently broad as to cover older, singlemicrophone techniques, as well as modern beamforming techniques, in successive comparison. The overlap of these data with appropriately computed uncertainties will show which methods, if any, appear to provide c onsistent estimates of trailing edge noise. PAGE 61 CHAPTER 4 EXPERIMENTAL METHODS AND SETUP The planned experiments are discussed with regard to setup. First, the wind tunnel facility characteristics will be summarized. Details on the NACA 63215 ModB model, fabricated by NASA Langley Research Center (LaRC), will follow. Details will include static aerodynamic behavior of the airfoil in the University of Fl orida Aeroacoustic Flow Facility. These sections will be followed by a description of the experime nts conducted, with previouslyacquired sample data shown, demonstrating nominal agreement between the current experimental setup and previous work, as well as framing the expe rimental techniques to be discussed. Experimental Facility UFAFF The University of Florida Aeroacoustic Flow Facility (UFAFF) is an openjet, opencircuit wind tunnel located in the MAEA Building of the University of Floridas Department of Mechanical and Aerospace Engineering. This f acilitys construction details are discussed in detail in the primary designers dissertati on [Mathew 2006] and were summarized in 2005 [Mathew et al. 2005]. The facility was designed for a maximum Mach number of just over 0.2, to simulate approach flight conditions. With this primary c onstraint in mind, the design objectives were set to leverage the existi ng ISO 3745 certified 100 Hz anechoic chamber [Jansson et al. 2002] while maximizing test section Re ynolds number. The upstream portion of the facility was designed to mini mize turbulence intensity, such that leading edge noise effects from turbulence ingestion would not cont aminate airframe noise studies [Guidati et al. 1996]. Downstream sections of the facility were designed to minimize drivesystem noise contamination of the test section. Acoustic fo am sidewalls, 0.15 m thic k S82N manufactured by Reilly Foam, are installed on the two spanwise boundaries of the test section to constrain the openjet flow to a quasitwodimensional condition. The resulting facility characteristics are 61 PAGE 62 listed in Table 41 Figure 41 shows a planform view of UFAFF, and Figure 42 shows an isometric schematic of an example airfoil isolated in the facility test section. Background noise scaling for the facility (AWeighted) is shown in Figure 43 Additional data in the figure are adapted from previous work [Duell et al. 2002]. Airfoil Model A NACA 63215 ModB airfoil wa s selected for use in trailin g edge studies. This was done to allow direct comparison with modern previous research [H utcheson & Brooks 2002; Hutcheson & Brooks 2004]. The model, fabricated by NASA Langley Research Center (LaRC), has a chord of 0.737 m, and a wetted span of the full test section, or 1.12 m. The model is composed of a carbonfiber skin applied to metallic ribs and spars, and is hollow to allow for static pressure tubing, as well as wiring for dynami c pressure transducers. The models full span is 1.83 m, allowing for future swept wing studies. Three hatches are built into the pressure side of the model, allowing for easy access to pressure ports and dynamic pressure transducer installations. Removable panels near the trai ling edge of the model allow for quickswitch access to dynamic pressure transducer arrays. A crosssectional plot of the airfoil profile is shown in Figure 44 A photograph of the model is shown in Figure 45 A photograph of the models underside, installed in UFAFF, is shown in Figure 46 with the locations of an access hatch and a trailing edge panel. As with the model used by Hutcheson and Br ooks, the trailing edge of this model is blunted, in this case to 3.3 mm, geometrically ma tching one of their specific cases. XFoil [Drela 2001] was used to smoothly blend this thickness increase to the maximumthickness point of the airfoil in the design process, thus subtly alte ring the overall airfoil shape. The leadingedge shape which differentiates the ModB from a baseline 63215 [Szelazek & Hicks 1979] was unchanged. Coordinates for this mo dified airfoil sh ape are given in Table 42 and again plotted 62 PAGE 63 in Figure 47 The profile coordinates are listed star ting at the trailing edge of the model, wrapping around the leading edge (designated as the coordinate system origin) along the pressure side of the airfoil, and then back towa rds the trailing edge along the suction side. The trailing edge was rounded w ith fillets on the pressure and suction edges, each with a radius of 1.1 mm. Note that the coordinates provided in Table 42 do not account for skin thickness effects in the vicinity of the trailing edge Some of this thickness issue was taken into account with the coordinate design in XFoil though additional tra iling edge thickening was seen in the quality assurance measurements, as shown in Figure 47 and addressed here. Quality assurance measurements of the model were required before any experiments could be conducted, so in any analysis the true shape of the airfoil would be known, and not just the design profile. This QA was done at NASA LaRC using a coordinate measurement machine (CMM). The airfoil coordinates were mapped at 25%, 50% and 75% span. The locations of all pressure taps were also validated. To properly assess the measured profiles with respect to the design profile, a MATLAB function was written wh ich imported all the measured and design coordinates, performed a leastsquares comparison to account for any coordinatesystem offsets and rotations, and then calculated local normal differences between the measurement and design coordinates. This code is presented in Appendix A. Figure 47 plots both the design and measured coordinates in overlay, and the surfacenormal error from design. The peak absolute error was 0.16% chord and the RMS error was between 0.06% and 0.07% chord, depending on span wise chord profile location. Trailing edge deviation from design was also examined. The trailing edge is plot ted for different spanwise locations in Figure 48 Here, 0% span would be the right side of the model when the observer is upstream of the model, facing downstream. Finally, the code performed a curve fit to the offset profiles and added the resulting smooth 63 PAGE 64 polynomials, one for suctionside and one for pressureside as plotted sidebyside in Figure 49 back onto the original design coordinates. This provided an effective airfoil shape for use in codes such as XFoil and/or FLUENT, to assess any flow differences which may arise from the model surface error. Figure 410 shows the XFoil prediction for both cases at zero angle of attack and Reynolds number of 3.0x106. The data show that manufacturing deviations from the design have a minimal impact on steady aerodynamic behavior. Similarly, little impact on the overall acoustic field is expected from such deviation, but it is r ecorded nonetheless to allow for accurate future comparison. Model Performance Flow behavior around the airfoi l in the facility in which acoustic measurements are performed must also be determined, to provide a complete database for the aeroacoustic dataset. As facilitytofacility differences can lead to significantly different flow behavior around identical models, having a measurement set of flow behavior is necessary for any future comparisons of datasets. Steady Pressure Behavior The mean flow behavior around the model can be easily evaluated by utilizing the pressure taps built into the airfoil s surface. The airfoil has three chordwise sets of pressure taps installed at 25%, 50% and 75% span loca tions, to allow for mapping of the streamwise pressure distribution along the airfoil surface. The distribution of taps along the chordwise profiles was designed using an optimization rout ine to estimate the lift and pr essure drag correctly with 47 pressure taps around the airfoil, designed by Ce sar Moreno. For this model, 23 taps were allocated to the suction side, 23 to the pressure si de, and one at the leading edge stagnation point. Lift simulations were conducted in XFoil. The final design has identical tap distributions on the suction and pressure sides of the mo del. Tap locations are given in Table 43 All three 64 PAGE 65 chordwise profile sets are measured to check fo r spanwise flow uniformity. The model also has spanwise rows of taps installed near its trailing edge on both the suction and pressure side of the airfoil. These will not be utili zed in this experiment set, and will be leveraged in future work regarding quasithreedime nsional sweep effects. Each tap consists of a 0.028 ID metal tube connected to 0.0 40 ID urethane tubing. The narrow tubing was selected to minimize spatial variations in pressure across the tap hole. While this increases the time constant of the tubing and slows its response to pressure changes, it improves measurement resolution u nder primary experiment conditions of interest where the test section speed is constant and a larg e number of averages are taken. Data were acquired using three PSI Netsca nner Pressure Scanners (Model 9116). One Netscanner has a range of 10 H2 O differential, while the other two have 1 PSI differential ranges. All models are accurate to 0.05% full scale. Th ese were connected to a computer running the LabVIEW desktop environment, and data were sampled for each spanwise chord set of taps. Typically, about 500 samples were coll ected over 20 seconds. Uncertainty values of each tap measurement were calculated, and plotted with the pressure coefficient ( Cp) distributions. 21 2p p p C U (41) Equation (41) is valid for incompressible flow cases. All of the experiments conducted in UFAFF for this research satisfy the incompressi ble flow assumption. In this definition of pressure coefficient, is the fluid density, U is the test section velocity upstream of the airfoil, p is the test section static pressure upstream of the model, and p is the local mean surface 65 PAGE 66 pressure at the tap. Dynamic pressure, the to tal denominator term, was measured using a pitotstatic tube, by using the incompressible flow based identity given in Equation (42) 2 01 2Upp (42) Here, 0 p is the flow stagnation pressu re, measured by the stagnation port of the pitot probe. The static ports, similarly, measure p The two ports are connected to a single differential pressure transducer. The numerator in the pressure co efficient calculation was measured by connecting pressure taps to the modules, and then using a common reference pressure of the chamber static pressure for the differential measurement. De nsity for the test section speed computation was calculated from a local weather stations values of pressure and temperature. Pressure distribution deviati on from XFoil prediction was ex pected due to openjet wind tunnel effects, as discussed by Brooks and Ma rcolini [Brooks & Marc olini 1984]. Further deviation from their prediction met hod was also expected, due to the presence of a jet collector at the rear of the test section. An inviscid so lver method which addresse s both the openjet and collector effects on model distributio n has yet to be used to analy ze airfoil behavior in UFAFF. Some preliminary viscous, turbulent CFD analysis of the test section has been conducted using FLUENT, but this proved to be too cumberso me for experiment planning and design. For baseline mean aerodynamic comparison, th e model was installed in the chamber and swept through a range of geometric AoAs from 7.5 to 10, in 2.5 increments. The tunnel test section flow speed was set to 31 m/s, l eading to a chord Reynolds number of 1.5x106. Lift values were computed through pressure distribution integration, and substituted into XFoil to calculate an equivalent lift freestream AoA. Figure 411 through Figure 418 show the data for varying AoAs, compared to the prediction fo r an equivalentlift freestream AoA. 95% confidence interval bars are plotted for the experimental data. 66 PAGE 67 The geometric AoA vs. equivalentlif t AoA of the model is plotted in Figure 419 A linear regression curve is drawn through the da ta, generating the relation given in Equation (43) which has an R2 value of 0.993. 0.2291.37Equiv Geom (43) Based on these measurements, it was determined th at there would be much difficulty attempting to match any highlift configurations for the airfo il model. As such, experiments were limited to 1.5, 0 and 1.5 AoAs. The spanwise uniformity of flow about the model is demonstrated in Figure 420 Unsteady Surface Pressures An array of unsteady pressure transducers was designed and built for insertion into the trailing edge region of the UF NACA 63215 ModB airfoil. Unsteady pressure measurements are of interest for comparison with previous work by many authors [B lake & Gershfeld 1988; Brooks & Hodgson 1981; Yu & Tam 1978]. The surface pressure data allow for detailed analysis of the surface pressure fluctuation st ructure in the vicinity of the trailing edge, evaluating such quantities as correlation length and time scales fo r the streamwise and spanwise directions. These measurements can also be co rrelated with farfield microphone measurements when the tunnel is configured for acoustic data acquisition, as well as hotwire measurements when the tunnel is configured for flow data ac quisition. This correlation data, as well as the unsteady surface pressure spectra, can be used in aforementioned works [Amiet 1976; Howe 1978] for the prediction of the radiated acoustic fi eld. While the analysis of unsteady pressure data in relation to far field acoustic data is beyond the scope of this present study, the data are acquired in conjunction with the pr esent acoustic data for future work. As such, the airfoils unsteady pressure measurement capability is presented here for completeness. 67 PAGE 68 The trailing edge arrays consis t of 14 Kulite LQ1255A unsteady pressure transducers. These sensors have a nominal diaphragm diameter of 0.125, an overall sensor head diameter of 0.16, and are tuned for a 5 psi pressure fl uctuation range. The predicted boundary layer pressure fluctuations of 10% freestream dynamic pr essure fall well within this range. Freestream dynamic pressure is simply calculated as the denominator of Equation (41) Unfortunately, the lack of space available in the tr ailing edge region of the model prevents the use of a reference pressure line. As such, the transducer selection necessitated using an absolute transducer instead of differential. Absolute tran sducers have a stiffer diaphragm, increasing minimum detectible pressure (MDP) fluctuations. The rated natural frequency is 150 kHz. Each transducer is installed from the backside of the airfoil surf ace. The face of the transducer is seated in a cavity which is connected to the surf ace of the airfoil through a pinhole opening. The pinhole has a nominal length l of 0.01 and an area of The cavity has a depth of 0.01, and diameter of 0.13, which translates to a volume V. These dimensions were used to estimate the Helmholtz resonant frequency of the cavity as approximately 16 kHz for standard atmospheric conditions. 20.01" S 21 2 cS f Vl (44) In Equation (44) c is the isentropi c speed of sound. The transducers are arranged in two 7elemen t Lshaped arrays. One array is located on the suction surface of the airfoil, and the other on the trailing edge A layout of the suction side array panel is shown in Figure 421 where the panel boundaries are the solid black lines. The pressure side array is a mirror image of the sucti on side array. The spanwise line of the array is 68 PAGE 69 located approximately 0.64 from the start of the fillet of the rounded trailing edge. The chordwise line of the array is located 3.84 from 50% span of the airfoil. This offset necessary due to the centerspan chord profile of pressure taps. The centertocenter spacing of the transducers at their densestspaced location, the part closest to 50% span and closest to the trailing edge, is the minimum possi ble spacing of 0.16. The next sensors in each direction are located 0.32 away from their neighbors, and th ose after them 0.64 away. This array design leads to a nonredundant population of sensor spacings, maximizing the number of spatial lags available in a crosscorrelation comparison. Th e spacings in each direction are 0.16, 0.32, 0.48, 0.64, 0.96, and 1.12. A plot of these spacings is shown in Figure 422 Essentially, these spacings, or lag distances, are the avai lable crosscorrelation spacings for computing turbulent spatial correlation scales in a homogene ous turbulence field, without making the Taylor frozenfield hypothesis [Tennekes & Lumley 1972] In other words, the crosscorrelation of the turbulence as a function of space is only availabl e at discrete lag vect ors. These discrete coordinates are the data points plotted in the figure. A signalconditioning board is necessary for use with the Kulite sensors, to provide appropriate power and amplification. This board is located outside of the airfoil model. For minimum additional noise, the wires from the Ku lites are run through a grounded shielding tube before exiting the model. Additionally, a thre eaxis PCB accelerometer was mounted inside the model near the trailing edge Kulites, for future analysis of any vibrational coupling into surface pressure fluctuations. Note that only the component normal to the surface of the model was measured, due to data acquisition system channel count considerations. A National Instruments PXI1045 chassis is us ed for all dynamic data acquisition (DAQ). The first slot of the chassis is dedicated to a MXI4 card, which is connected via a fiberoptic 69 PAGE 70 cable to a workstation running La bVIEW. The remaining card slot s in the chassis contain NI PXI4462 DAQ cards, for a total of 68 channels. Ea ch channel has 24bit resolution with 118 dB effective dynamic range. The maximum sampling ra te available perchan nel is 204,800 samples per second. All measurements of linear instrume ntation, such as surface pressure transducers and microphones, are ac coupled with a 3 dB cuton of 3.4 Hz. Nonlinear measurement schemes such as hotwire data acquisition ar e not ac coupled. The DAQ cards automatically select the appropriate antialiasing filter characte ristics for the operation of their sigmadelta A/D converters. Hotwire Anemometry Constanttemperature hotwire anemometry (CTA) was selected for inflow velocity measurements in the vicinity of the trailing ed ge. While the technique suffers from directional ambiguity for singlewire probes, it provides high spatial and temporal resolution. The resulting mean and fluctuating velocity data can be used in conjunction with steady and/or unsteady pressure measurements to characterize the mode l behavior in a given facility, and allow for crossfacility comparison of results. A hotwire works on the principle of forced co nvection, using a heat transfer model to determine flow velocity. For constanttemperatu re anemometry, a closedloop controller is used to hold the wire at a constant resist ance, and thus a constant temperature [Bruun 1995]. wT0 n wQTTABU (45) Equation (45) known as Kings Law, relates the total heat flux, Q to the difference in temperature between the hotwire and the flow, Tw T0, and the local flow velocity normal to the axis of the wire, nominally the local streamwise flow speed U in a shear flow. A B and n are calibration constants. By assuming that heat flux is related to the square of bridge voltage E a 70 PAGE 71 directly measureable quantity, a calibration relation can be constructed, as shown in Equation (46) 2 0 n wE ABU TT (46) A Dantec 55P15 boundary layer hotwire prob e is attached to a 55H21 probe holder, connected to a 90N10 streamline CTA controller frame. The probe holder is enclosed in a NACA 0012 airfoil extrusion with a 4 chord, to minimize vibration and probe arm drag for centertestsection measurements. This airfoil ex trusion is connected to a traverse above the wind tunnel test section. A 2axis Velmex st epper traverse was used for vertical and axial movement of probes. A schematic of a singlewire installation with the 2axis traverse is shown in Figure 423 Photographs of a similar installation in UFAFF featuring a DU 96W180 airfoil are shown in Figure 424 and Figure 425 from below and above the model, respectively. The hotwire system is calibrate d in the tunnel free stream using a pitotstatic probe. Calibration is conducted relative to a Heise HQS 1 015 H2O module installed in an ST2H chassis. Both the pitotstatic probe and the hot wire anemometer are place d in the core flow of the wind tunnel test section, and the wind tunnel operated at many di screte test section velocities. Test section fluid temperature is also measured using a Heise system. Wire temperature is output from the hotwire bridge controller, and dccoupl ed voltage fluctuations are measured using a PXI4462 card. These data are fit to Equation (46) using a leastsquares method in MATLAB, to determine the appropriate calibration constant s. For preliminary hotwire measurements, the DAQ system was set to acquire data at a rate of 32,768 samples per second. The range was set for zero to five volts. An example calibration curve is given in Figure 426 The coefficients determined through the leastsquares fit can th en be used, in conjun ction with the mean temperature and fluctuating voltage measuremen ts, to determine the fluctuating velocity 71 PAGE 72 magnitude observed by the wire. It should be noted that the effective sensitivity of the hotwire is driven by the temperature difference between the ambient fluid and the wire, so operating at a large temperature difference is preferable. In these cases the wire temperature was approximately 1.8 times the absolute mean temperature of the fluid. A single hotwire boundary layer survey was condu cted at the centerline of the trailing edge of the NACA 63215 modB model, for a Mach number of M = 0.17. This was done right off of the beginning of the trailing edge fillet of the model. The steady and fluctuating components of the measurement were separated and evaluated. The survey was conducted by bringing the hotwire to just off the surface of the airfoil (< 0.001 m), and traversing upwards towards the freestream. The mean and fluctuating profiles fo r the trailing edge boundary layer are shown in Figure 427 The coordinate system shown is not wallnormal, but the trailing edge angle at the rear end of the airfoil is very small, so th e plot should approximately match a wallnormal profile. This experiment was repeated just behind th e airfoil (< 0.001 m behind the trailing edge), and the full wake profile was measured. Results are shown in Figure 428 Comparing the two profiles reveals potential wake thi nning as the shear layer leaves th e trailing edge of the airfoil. Integral parameters for the boundary layer profil e were computed using a leastsquares fit to Muskers boundary layer velocity profile [Muske r 1979], as at the time of measurement probe vibration was such that the nearwall region of the boundary layer could not be measured. The boundary layer thickness was estimated with this method to be 0.018 m, with a displacement thickness of 0.0036 m and a momentum thickness of 0.0024, leading to a shape factor of 1.5. As an aside, since the data were collect ed a new traversing system and hotwire installation hardware were purchased for use in the facility to reduce this vibration problem. 72 PAGE 73 Acoustic Measurements To determine the acoustic character of an airfoil, farfield microphone measurements must also be performed. Unfortunatel y, the noise source of interest in this case, the airfoils trailing edge, is only one of many sources a singl e microphone may measure. A brief, noncomprehensive list of other potential contributors to a single microphone is presented. Leading edge noise Edge noise from the lip of the front of the test section Shear layer noise Shear layercollector interaction noise from the rear of the test section Scrubbing noise from the test section acoustic walls Noise from unsteady modelwall interactions Local hydrodynamic pressure fluc tuations over th e microphone face Electronic noise Due to the potential presence of all these cont aminants, multiple microphone measurements must be made within a single experiment set. The data from these microphones can be processed in numerous ways. Both coherencebased techniques and basic array techniques will be considered. CoherenceBased Techniques Coherencebased techniques make few assumptio ns on the nature of the source, aside from the linearity of signal behavior between one or multiple sources. However, implicit within the use of such techniques is the assumption that ther e is either a single, dom inant, coherent source; or that the multiple coherent, incoherent or partially coherent sources generating the measured field can be spatially lumped relative to the meas urement domain. In essence, this means that the characteristic source dimensions are significantly smaller than the length scales of the measurement. A schematic of this situation showing a line source which would meet this criterion is given in Figure 429 where the length scale of the source, l, is much smaller than the distance D from the source to the observers, shown as microphones. The consequences of this 73 PAGE 74 assumption are discussed in more detail in A ppendix C. Coherencebased techniques rely primarily on statistical analysis and averaging to remove any noise sources which are incoherent between multiple measurement microphones, and focus on a coherent field which may be generated by multiple coherent or incoherent sources. Such techniques should be very effective at removing line noise and local hydrodynamic contamination, but will have difficulty separating the effects of multiple acoustic sources whic h are correlated and common to all microphones. Coherent output power The simplest coherence technique to util ize involves a twomicrophone measurement. This technique has been used extensively in previous research [Brooks & Hodgson 1981; Hutcheson & Brooks 2002]. While the technique can be used in conjunction with a dipole source model for trailing edge noise, as dem onstrated by Hutcheson and Brooks, the basic coherent power method can be derived without one. To start, some simple spectral relations will be defined [Bendat & Piersol 2000]. *1 2lim,,yy k k TGfEYfTYfT T (47) *1 2lim,,xy k k TGfEXfTYfT T (48) 2 2xy xyxy xy xxyy xxyyGf GfGf f GfGfGfGf (49) Equation (47) defines the onesided autospectral density function of a signal. Here, Y is the Fourier Transform of a bloc k of acquired time series y Y* is the complex conjugate of the original transformed signal, T is the length of the data block in units of time, k is the individual block number, and f is the frequency of interest. This equation is valid for all frequencies aside from the zerofrequency and Nyqui st frequency. For those data points, the coefficient of 2 74 PAGE 75 becomes unity. Equation (48) defines the cross spectral density of a pair of signals, which gives a relationship between signals x and y This relationship is normalized to a zerotounity coefficient in Equation (49) where 2 is defined as the ordinary coherence function relating signals x and y Note that for subsequent equations, frequency dependence is implied but not stated. Figure 430 shows a SingleInput/TwoOutput (SITO) system. Here, X is the source, and is treated as unmeasureable. U is the source signal after it has propagated from the source to a microphone, and is considered the signal of interest. N is uncorrelated measurement noise, either due to electronic line noise, or a physical mechanism exciting a sensor. Y is the sensor output. Equations (410) through (413) provide the mathematical iden tities associated with a SITO system. ,1,iiiiiYHXNUNi 2 (410) 2 *ii ii iiiiiiyyixxnniixxnnuunGHGGHHGGGG n (411) 1212yyuuGG (412) 12 12 11222 ** 2 1212 ** 11221uu xx xx uu uuuu xx xxG HHGHHG GGHHGHHG (413) By assuming that the noise terms Ni between the two channels ar e perfectly incoherent, the crossspectrum of the measurement collapses to the autospectrum of the source passing through two unique transfer functions, as shown in Equation (412) This allows for the definition of the source coherence between the two channels as unity in Equation (413) This can be leveraged to define the Coherent Output Power (COP) for a given channel in Equation (414) 75 PAGE 76 12 12 12 22 11 1211 22 22 2222 222 22 2 1 2 1 2211 11 1uu yy uu uu uu xx yyyy nn yyuunn uuG GG GG HG COPG G GGG SNRSNR G (414) Here, it is shown that the Coherent Output Powe r at microphone 1 is proportional to the coherent source strength measured by the channel, but is biased low by the signalt onoise ratio (SNR) of the second microphone. A similar equation can be constructed for microphone 2. Thus, if both microphones have significant noise contamination, the Coherent Output Power method will underpredict the true source levels at each microphone [Bahr et al. 2008]. If the two microphones are placed at equal but opposite locations above and below a trailing edge, then some additional simplifica tions can be made [Brooks & Hodgson 1981]. The transfer function for the trailing edge noise propagation can be tr eated as having a distancebased attenuation factor and a distancebased time lag, of the form shown in Equation (415) Here the sign of the pi term in the phase angle is depe ndent on whether the microphone is referenced as located above or below the model trailing edge. 2ijk iiHHrer (415) For two microphones placed at equal and opposite sides of a trailing edge, then Equation (416) holds. 12 12 12* 1122 ****** 12122112 **** 12 122 2 22yy yy yyGEHXNHXN T GEHHXXHXNHXNNN T GHHEXXHEXN TT 212 HEXN T 122 ENN T 122 22 12mmjkrjkr j yy xxm m mxxGHHGHreHreHrGe (416) 76 PAGE 77 Therefore, if the true trailing edge noise signal is simply considered to be the magnitude of the source signal at a given distance from the trailing edge, then Equation (417) provides a direct solution. 12 11222 yy mxxuuuuGHrGGG (417) This formulation ideally provide s a noiseless estimate of traili ng edge noise without bias, but, unlike the generalized coherent output power prev iously defined, is sensitive to any positional uncertainties in microphone placement, and requ ires a shear layer correction to account for refraction effects on the signal pr opagation path. Note that if the shear layer corrections are identical for the upper and lower microphones, e.g. if openjet test secti on flow about the model was behaving in a symmetric manner due to neglig ible lift, then the sh ear layer correction will cancel between the two microphones. The first acoustic measurements performe d on the UF NACA 63215 ModB involved a coherent power analysis. For these experiment s, a pair of Brel & Kjr Model 4939A011 microphones with Type 2670 preamplifiers and a Type 2829 power supply was used. These were connected to a PXI4462 DAQ board, and instal led in the facility at opposing sides of the model trailing edge, as shown in Figure 431 Note that the displayed coordinate system is that used for constructing trailing edge noise directivity plots, and is si milar to that shown in previous chapters, Figure 21 as the upper or lower side of the m odel in the previous discussion was arbitrary. This coordinate system, however, is different from the one used in beamforming analysis. The microphones were covered with foam windscreens to reduce local hydrodynamic pressure fluctuations, and ca librated with a B&K Pistonphone Type 4228 at a single frequency. It was assumed that, given the measurement ba ndwidth of the microphones used (~100 kHz) the instrumentation would have nearflat magnitude and phase response for the majority of the 77 PAGE 78 measured acoustic bandwidth (10 kHz), so only a singletone calibration was necessary. Data were acquired at 65,536 samples per second. Spectral blocks were 4096 points long, resulting in 16 Hz binwidth FFTs, with 3000 blocks collected and averaged. Results were compared to a data set provided by Dr. F. V. Hutcheson of NASA LaRC. Dr. Hutcheson kindly provided the crossspectral matrix (CSM) for a data set most closely matching UFs trailing edge configuration, operated at a 1.2 AoA and a Mach number of 0.17. The UF data were collected using a 0 AoA at a Mach number of 0.17. The comparison, plotted in Figure 432 shows agreement for frequencies with moderate coherence, shown in Figure 433 Note the slight offset in peak location is due to a small difference in trailing edge thicknesses t between models, as Figure 48 shows the UF models trailing edge is sli ghtly undersized from design. This can be accounted for as approximately a 10% difference, and the data collapse with thicknessbased Strouhal scaling, shown in Figure 434 Level comparisons are al so approximate, as the NASA data binwidth is 17.45 Hz. No shear layer co rrections have been applied to this data. Of key interest in this comparison is the phase relationship between the two microphones. In an ideal measurement, the coherent output power method should see a phase offset of radians from the first microphone to the second. If the microphones are not quite equally spaced, there should be a linear phase shift as a function of frequenc y, as shown above in the crossspectral magnitude formulation. The crosschan nel phase relationship between the first and second microphones in the UF and NASA data set is plotted in Figure 435 As shown here, there is a linear phase relati onship between the two microphones in both cases below 5 kHz, indicating that the source assump tions may be correct at these fr equencies. However, above 5 kHz there is a total br eakdown of the phase relationship, corresponding to the previously seen breakdown in coherence. This appears to be indicative that either the trailing edge noise source 78 PAGE 79 is not strong above this frequency, or the nature of the contaminating noise is such that the coherent power method cannot separate it from the tr ailing edge noise source. All experiments for this body of work were or ganized such that a coherent output power solution could be calculated as a reference value while other data were collected. The aeroacoustic array plate used, discussed subseq uently, was designed such that a B&K 4138 1/8 pressure field microphone could be installed at the array center for comparison between array output and direct microphone measurement. A second microphone, a G.R.A.S. 40BE, was present on the opposing side of the airfoil for re ference, along with several other microphones. Both microphones were placed a nominal 1 m dist ance from the trailing edge of the array. A schematic of this setup is shown in Figure 436 For comparison, experiments were repeated with a B&K 4939 free field microphone in pla ce of the array plate, located where the 4138 microphone was located when the plate was present, as shown in Figure 437 A third configuration involved offsetting the phased arra y downstream of the airfoil by 0.25 m, as shown in Figure 438 For all of these experiments, 30 seconds of data were acquired continuously at 102,400 samples per second to allow for maximum flex ibility in postprocessing. All time series data were stored as 32bit floating point binary files (littleendian). Uncertainty calculations with this method are straightforward. Both are available from Bendat & Piersol [Bendat & Piersol 2000]. For the generalized coherent output power, the normalized standard deviation is given by Equation (418) 12 122 1 var 2vv yy vv yydG COP G n (418) Here, nd is the effective number of spectral block aver ages. This is computed by finding the total number of spectral averages from overlap an alysis and correcting for the selected window 79 PAGE 80 function, as computed from Equation (419) where is the total number of nonoverlapped blocks available, is the overlap fraction, here 0.75, and 0.52 is a datadependency correction factor computed for a hanning window at 75% overlap [Cattafesta 2010]. blocksNr 1 0.521 1blocks dN nfloor r (419) Note that while the true probability density f unction follows a Chisquared distribution, for a large number of degrees of freedom (spectral bloc k averages in this case) this converges to a Gaussian function and computing a 95% confidence interval is t hus straightforward. For the case with modeled, ideal micr ophone locations, the normalized st andard deviation is given by Equation (420) For coherent power methods, bias er ror in the narrowband spectral estimates will not be considered, but should be negligible for these situations based on scaling [Bendat & Piersol 2000]. 12 1221yy dyyG n (420) Threemicrophone method The SITO system developed in the previous section can be extended to a generalized SingleInput/MultipleOutput system with a minimum of threemicrophones. The solution for this threemicrophone system is developed and ca st into a readily solvable set of equations [Bendat & Piersol 2000]. 80 PAGE 81 1213 11 23 1223 22 13 1323 33 12yyyy uu yy yyyy uu yy yyyy uu yyGG G G GG G G GG G G (421) Equation Set (421) provides a basic statement of the result s, but suffers dramatically when the crossspectral magnitude in the denominator of an equation is small. This formulation is recast into a more stable solution in the first part of Appendix B, where positive power for the predictions is enforced. This is summarized starting with Equation (422) and continuing through Equation (424) 1 1,1iiiiiiiiyyuunnuu iGGGGi SNR ,2,3 (422) 23 1213 13 1223 12 13232 22 1 2 22 2 2 22 31 1 1 1 1 1yy yyyy yy yyyy yy yyyySNR SNR SNR (423) ,1,2, 1 1ii iiyy uu iG G SNR 3 i (424) The threemicrophone method has the advantage of completely removing incoherent noise contamination, regardless of differences in m odeled propagation paths between the different channels as was required for the coherent output power method, a lthough the source must still be spatially lumped. This is done by modeling each channels obser ved noise in addition to the 81 PAGE 82 desired signal, as shown in Equation (423) where the signaltonoise ratios, SNR, are evaluated. As with all other coherencebased techniques, the method may have diffi culties resolving true physical noise sources when the sources become spat ially distributed with partial coherence, as is so often the case in real measurements. This is addressed in more detail in Appendix C. The threemicrophone method can be applie d when more than threemicrophones are present in a measurement setup by selecting subs ets of microphone triads. However, ambiguities exist regarding which microphone combinations provide the best resu lts. This is pertinent to trailing edge noise in that a sk ewed local prediction of acoustic levels may lead to erroneous directivity results, in addition to providing false overall sound pressure level information. One approach, discussed in Appendix B, involves building a statistical table of all possible threemicrophone predictions. An alternative is to move to a technique which uses more than threemicrophones in each individual solution. As discussed in Appendix B, the number of equations quickly outpaces the number of unknowns when more than threemicrophones are used in a single solution, as the number of unknowns scales linearly with the number of mi crophones while the number of equations scales quadratically. This scaling is shown in Table B1 with both formulae and examples. One method for dealing with this is a leastsquares solution [Bahr et al. 2008]. This method performs a minimization fit for signal and noise estimate s of each channel. This is done by rescaling Equations (411) and (413) while leveraging Equation (412) The resulting functions for minimization are given in Equations (425) and (426) where v is used in place of u 11ii iiiiyy i vvnnG e GG (425) 82 PAGE 83 2 21ij ij iijjyy vvvvG e GG (426) While preliminary results using this techni que are included, the computation cost is sufficiently high that it will not be applied to the main body of this research. As a byproduct of this work, more efficient techniques have been conceived using covariancebased approaches based upon a single source assumption[Du et al. 2010]. The first of these methods minimizes the Frobenius Norm of the difference between the crossspectral matrix (CSM) of the microphone signals and the modeled signal and noise matr ices. The second method leverages the Rank1 behavior of the signal matrix to model the di fference between the CSM and the noise data as Rank1. The third method, a Maximum Likelihood method, assumes independent and identically distributed random Gaussian proce sses and minimizes a loglikelihood function. These codes are applied to the data collected in this body of research. Preliminary experiments involving the thre emicrophone method and leastsquares method were conducted using a line array of microphones in UFAFF. A schematic of this installation is shown in Figure 439 These runs used a single Brel & Kjr Type 4939 inch microphone located at 90 from the traili ng edge of the airfoil, opposite an 11element linear array composed of Panasonic WM61A omnidire ctional electret microphones. Note that the microphones are powered from the PXI4462 cards th rough a constantcurrent 4 mA out put. As shown, the linear array was placed 1.13 m below the model chord li ne, with microphone 5 located directly below the model trailing edge at 90. The array elem ent spacing was 0.178 m, covering the physical angles between 58 and 133. This setup wa s subsequently invert ed, and directivity measurements were also conducted on the suction side of the airfoil. During the preliminary experiment phase, the electret microphones were calib rated in a planewave tube referenced to a 83 PAGE 84 B&K 4939 microphone up to 6 kHz. While this calibration did not cover their remaining bandwidth of interest, 6 kHz to 20 kHz as audio microphones, the ca libration does cover the frequency range where trailing edge noise is observed to be dominant, below 5 kHz. Linear array measurements were collected as contiguous time series for each channel. The sampling rate was set to 32,768 samples per second, and 30 seconds of data were collected for each channel. For spectral estim ation, block lengths were designated as 2048 samples, resulting in a 16 Hz narrowband bin width. The data bl ocks were processed with 75% overlap and a Hanning window, resulting in 996 effective averag es and 3.2% normalized autospectral random uncertainty. In these preliminary experiments, the methods were applied to linea r array measurements to estimate the trailingedge noise source direc tivity pattern, after application of a shear layer correction to compute the correct sourcemi crophone angle relati on[Amiet 1978a; Sanford 2008]. Table 44 lists the geometric microphone installation angles relati ve to the trailing edge downstream ray, and their correspon ding corrected angles in a frees tream of infinite extent with Mach number M For the computation, the shear layer was located 0.42 m from the microphones, with a test section Mach number of 0.17. The spectra, processed with varying techniques, were integrated, normalized by th e corrected 90 measurement, converted to (unweighted) overall levels (dB), and compared to Howes solution [Howe 1978]. Normalization divides Equation (221) by all terms held constant in the experimental set, leaving the directivity function given in Equation (427) 2 2 10 2 3 2 90deg2sin 2 ,10log 1cos1 cosr rVp dB p MMM r (427) 84 PAGE 85 Here, r is the shear layer corrected angle and MV is the eddy convection Mach number, assumed to be approximately 60% of the meanf low Mach number [Brooks & Hodgson 1981]. The spectral integration is performed using the scaling provided by Blake [Blake 1986] for the appropriate cuton frequencie s of trailing edge noise. The lower bounds of the overall sound pressure level integration of each microphones pow er spectral density can be established using the trailing edge boundary layer thickness via Equation (428) 0.6 U (428) This gives the lowest frequency of interest fo r trailing edge noise, while the upper bounds are dictated by the facility background noise. Equation (428) gives a characteristic frequency estimate from a boundary layer parameter such that fluctuations of wavelength on the order of or larger than the boundary layer thickness in length scale will not be noticeably scattered from the trailing edge. This is actually only one of thr ee presented by Blake, the others being stated in Equations (429) and (430) These two are an airfoil compact ness statement based on the airfoil chord C and an argument of leading edge independence from trailing edge noise radiation. The leading edge independence condition is computed to be approximately 250 Hz for M = 0.17. Of The boundary layer condition is approximately 310 Hz, while the compactness condition is about 465 Hz. The leading edge independence conditi on is selected as it provides the largest bandwidth (and thus the greatest possible error) in the analysis. 02 C c (429) 10 2 C U (430) 85 PAGE 86 For multiplemicrophone measurements, the phase between each microphone in the linear array and the Brel & Kjr microphone placed opposite the model is extracted from the crossspectral matrix. Then the deviation of the phase, adjusted by the shear layer correc tion, from that of a dipole at the trailing edge is computed. The upper integration bound is determined by the frequency where this phase relationship degrades to random behavior, which coincides with the frequency where the ordinary cohe rence function trends to zero. The results from these preliminary experi ments compare the differing coherent power methods and allow for planning of the main experi mental body of the research. The COP, threemicrophone and leastsquares methods are applied to the microphone 5 data in Figure 440. Note that microphone 12 is the opposing B&K 4939 from Figure 439 and is the reference microphone used for coherent power. This mi crophone, along with microphone 4 in the linear array, is included in the processing for the 3microphone method. The least squares method is computed using all of the microphones except mi crophones 1 and 2 closest to the collector. These microphones are neglected because when the linear array was placed on the pressure side of the model, microphones 1 and 2 were in a dow nwash region generated by the models lift. This downwash region generated an exce ssive amount of hydrodynamic fluctuation contamination in the microphone measurements. These microphones were not influenced when the linear array was positioned on the suction side of the model. Figure 441 shows a comparison of the correcte d, measured phase angle between microphones 5 and 12 compared to a theoretical prediction for a dipole located at the trailing edge (without a scattering correction). This presentation shows that for low frequencies, the corrected data possess the expected dipolelik e behavior. The near ze ro coherence at high frequencies results in random pha se angle behavior. Note the necessity of the shear layer 86 PAGE 87 correction in this analysis due to the unequal spacing of the microphone locations from the trailing edge. The deviations of the linear array phase plots from dipole behavior are shown in Figure 442 and Figure 443 with Blakes cuton criteria. The plotted data are the difference, in radians, between the corrected cro ssspectral phase and that predicted by theory, with Blakes predicted cuton frequency for trailing edge noise plo tted [Blake 1986]. As shown, aside from microphones 1 and 2, the phase behavior is in good agreement from the computed lower bounds up to 4 kHz. Note that the data points near the maximum and minimum phase values Figure 443 actually phasewrap back onto the main band of the plot. The directivity plots are shown in Figure 444 for integrated levels from 250 to 4000 Hz. For most cases, as long as the measurement wa s well away from the model downwash, even the raw autospectral integration provided good agre ement with the theory, although without crossspectral information an upper integration bound coul d not be set. All of the coherent power methods are in good agreement with theory. With the exception of 75, a ll of the methods differ from each other by no more than ~1 dB, while the maximum deviation from theory is ~2 dB. This preliminary data set serves several purpos es in the layout of subsequent experiments for the research. One of the first is that it s hows that integrated direc tivity levels, while in good agreement between the different analysis techniqu es, do a poor job at highlighting the detailed differences of each method. As such, directivity in the twodimensional sense, will not be a parameter of interest in the fi nal research. Second, the data show that all of these methods predict different levels. Above a certain threshold, when the predicted dipolelike phase behavior breaks down, all of the methods have significant disagreement. As such, uncertainties 87 PAGE 88 are necessary to see if, when accounted for, th ey can demonstrate agreement between different methods. Also, it should be noted that so far none of these experiments have addressed threedimensional and multisource field effects in terms of coherent power analysis. Based on the discussion in Appendix C, when a distributed so urce region composed of incoherent sources is observed by a set of microphones, the coherenc e measurements between microphones can trend to zero, even though each microphone could be obse rving an identical set of sources with no incoherent noise contamination. This behavior is observed to be a f unction of experimental geometry, and is associated with a near field measurement effect. In essence, if the source field is not spatially lumped, an average cancella tion effect can occur depending on source and observer orientation. The conclusion of the appendi x is that this effect can be mitigated, at least in the case of a source like trailing edge noise where the geometry is known to be symmetric, but placing the microphones such that they are located on the source lines plane of symmetry. For the nominal threemicrophone analysis in this body of work, the G.R.A.S. 40BE downstream of the one used for twomicrophone solutions is used for all experimental configurations. For additional analysis options in future work G.R.A.S. 40BE microphones are placed along the airfoils upper span in addition to those placed chordwise in Figure 436 Figure 437 and Figure 438. Figure 445 expands upon the previous experimental configuration figures by showing the orientation of this upper array of microphones. Note that by assuming equivalent shear layer behavior above and below the model, as the upper and lower microphones ar e located equidistant from the model trailing edge in Figure 437 as previously discussed no shear layer correction should be necessary for the cohe rent power method from Equation (417) This assumption can be considered valid for lowlift conditions, as this will lead to minimal test section jet deflection. 88 PAGE 89 Given the equivalentlift Ao A formulation from Equation (43), this assumption should be valid for the planned angles of attack. The additional upper array microphones are placed based on overall test section dimensions and data acquisition constraint s. 7 G.R.A.S. microphones are the maximum allowable with a 68channel data acquisition syst em. The phased array requires 45 channels for the array electrets plus an additional cha nnel for the center reference microphone. Surfacemounted Kulites require another 14 channels. The model accelerometer requires a single channel, leaving 7 channe ls for the upper array. Uncertainty estimates are necessa ry for comparison between techniques, as just discussed. While they were relatively simple for twomicrophone methods, with three or more microphones they are not as straightforward. The absolu te value operating on the square root terms in Equation (423) makes a simple classic uncertainty propagation difficult without expanding approximate derivatives. Since the uncertainty will be large for the predicted lowcoherence frequency bins, such linear expansions may be in error [Yardibi et al. 2010a]. As such, the simplest solution is to conduc t a set of Monte Carlo simula tions for the multimicrophone solution, where the inputs to th e code are perturbed by a Gaussi an random input of standard deviation determined by the random uncertainty of each input component. For example, with the threemicrophone methods the inputs consist of the raw autospectra and ordinary coherence functions. Bendat & Piersol [Bendat & Piersol 2000] give the normalized autospectral random error repeated in Equation (431) 1iiyy dG n (431) The uncertainty of the ordinary coherence function is given in Equation (432) where frequency dependence is implied and suppressed. 89 PAGE 90 2 221 ,ij ij ijyy yy yydi n j (432) More advanced coherent power techniques may le verage the full complexvalued cross spectral matrix (CSM). For these methods, the diagonal of the cross spectral matrix is perturbed using Equation (431) The offdiagonal terms must be separated into their re al and imaginary, or coand quadspectral, components, as in Equation (433) 1,ijij ijyyyy yyGCQi j (433) The cospectrum normalized standard deviation is given in Equation (434) and the quadspectrum normalized standard deviation in Equation (435) again from Bendat & Piersol [Bendat & Piersol 2000]. 22, 2iijjijij ij ijyyyyyyyy yy yydGGCQ Ci Cn j (434) 22, 2iijjijij ij ijyyyyyyyy yy yydGGQC Qi Qn j (435) Beamforming Techniques Traditional beamforming techni ques involve the assumption of a source field consisting of a sum of incoherent monopoles. When this assu mption is made, beamforming can be used to estimate the locations of acoustic so urces, as well as the field level at the array center due to each source. This technique has been ap plied often to analysis of traili ng edge noise experimental data with various modifications [Brooks & Hu mphreys 2006a; Hutcheson & Brooks 2002; Shannon et al. 2005]. For an openjet facil ity, shear layer corrections ar e applied to each microphones signal prior to the application of beamforming algorithms [Amiet 1978a]. This is done using a precomputed array of data for known microphone and scanning grid [Sanford 2008]. Classic 90 PAGE 91 delayandsum (DAS) methods can be used to c onstruct source maps, but these are sensitive to contamination from local flow noise [Humphreys et al. 1998; Mosher 1996] When these methods are cast in the frequency domain, one commonlyused method for dealing with contamination is diagonal removal (DR) from th e CSM. By setting al l autospectral diagonal terms to zero, the system noise as listed in Equation (411) goes to zero. However, the coherent source contribution to the diagona l is also eliminated. This ap proach can significantly improve the resulting source maps but can cause erroneou s, negative level calcu lations over distributed source regions [Hutcheson & Brooks 2004]. These nega tive power levels can be dealt with in an integrated level code by setting them equa l to zero in the power summation [Oerlemans et al. 2007a]. Diagonal removal also assumes that noi se sources are uncorrelated between microphone channels, such that contamination is constrained to the cross spect ral matrix diagonal. However, if nonacoustic correlated noise is introduced, such as coherent flow structures passing over the array surface, significant errors may still be present in the results. To further reduce contamination, the cr oss spectral matrix element, Equation (412) can be modified to Equation (436) ijijijyyuunnGGG (436) Here, it is no longer assumed that noise contamin ation sources are incoherent with each other, although it is still assumed that th e noise sources are not coherent with the acoustic sources of interest. One method employed in previous literature [Humphreys et al. 1998] uses background noise subtraction. With backgr ound noise subtraction, the measured crossspectral matrix of an empty test section with flow is measured and as sumed to be the coherent noise term in Equation (436) Then, the test is repeated with the mode l in place and is measured. The difference is assumed to be the desired coherent acoustic sources of interest with the model present. In other 91 PAGE 92 words, the coherent noise term is assumed to be the same with and without the model installed. However, this approach can lead to unexp ected results because th e installed model can significantly change the flow patterns in the ch amber, leading to new noise sources. Based on previous research in UFAFF, diagonal removal will be used, but background subtraction will not be used, as background subtraction has been obser ved to induce clearly er roneous behavior in beam maps from the facility [Bahr et al. 2008]. The power at a grid point in a scanning plan e for the standard delay and sum beamformer in the frequency domain is given in Equation (437) [Dougherty 2002; Yardibi et al. 2010a; Yardibi et al. 2010c]. 21H llPa M lGa (437) Here, l is the index of the scanning point, M is the number of mi crophones in the array, G is the CSM, is the steering vector defined in Equation a(438) and H a is the conjugate transpose of the steering vector. ,1 ,,1 ,01 ,...,lT jkr jkr lll M larere r l M (438) The distance is from the array center to the l th scanning point, from each microphone to the scanning point, ,0lr,lmr T the nonconjugate transpose, k the acoustic wavenumber for the frequency of interest as defined in Equation (213) and 1 j (as opposed to an index as used earlier in this chapter). If diagonal removal is used, the denominator in Equation (437) is changed to 2 M M to account for the reduced number of microphones in the final summation. Integrated levels are computed using Equation (439) [Oerlemans et al. 2007a; Yardibi et al. 2010c]. 92 PAGE 93 l lL L lLP P PSFl (439) L is the integration region of interest, and PSF(l) is the array response, or pointspread function, to an ideal centered source at scanning location l Essentially, the summed DAS prediction within the integration region is normalized by th e summed point spread function. This is a dramatically simplified expression of a true d econvolution, and suffers from several pitfalls depending on the behavior of the pointspread function in the integr ation region [Oerlemans et al. 2007a]. Array aperture affects ca n still be important ev en with this integr ation normalization. However, the computational expense of this method is much less than more advanced algorithms, and the method has a recent form ulation for overall uncertainty [Yardibi et al. 2010a]. Due to computational expense and the inte rest in uncertainty, th e experiments presented in this work will be formulated and analyzed only in the context of the standard DAS beamformer, with acknowledgement of the methods inherent weaknesses. A zeroredundancy spiral aperture array was designed using the tools designed by Underbrink [Underbrink 1995], and fabricated for us e in the UFAFF. The physical array consists of a 0.5 m diameter, thick aluminum pl ate with 45 flushmounted Panasonic WM61A microphones. The disk edges are tr eated with acoustic foam to minimize scattering effects. A photograph of this array is shown in Figure 446 A solid plate is selected over free field installation of the microphones on r ods due to microphone packaging, installation and traversing simplicity. This array was selected over a pr eviouslydesigned, 1.82 m diameter 63 element array for two reasons. First, the na ture of the free jet test section flow field is such that having a larger array located only 1 m from the mode l would induce significan t flow over the array surface. Second, a smaller array provides the oppor tunity to offset the array from its baseline location, and study how small or moderate shifts in array position can affect the relationship 93 PAGE 94 between an integrated array level and a coheren cebased method. Additionally, previous internal research has shown that large arrays will underpredict acoustic levels from directional sources such as dipoles. The 3 dB beam width of the array is plotted in Figure 447 Note that part of the cost of a smaller, more flexible array arises in its complete inability to discern source direction below approximately 650 Hz. Coordinates for th is medium aperture array are listed in Table 45 The installation uncertainty of th e microphone locations in the plat e is on the order of 0.001. A selection of individualfrequency pointspread functions of the array with contours of 5 dB, 10 dB and 15 dB are shown in Figure 448 through Figure 453 The selected frequencies correspond to frequencies analyzed in detail in Ch apter 5. Note that at 20,000 Hz, the arrays sidelobes grow beyond the 10 dB level. For phased array measurements, the phase re sponse of individual microphones within the array is important, and can play a significant ro le in the uncertainty of output spectra [Yardibi et al. 2010a]. As such, array calibration is extrem ely important over the entire bandwidth of interest of the array. Unfortunately, experience has shown the basic group array calibration technique [Dougherty 2002] to be very sensitive to imperfections in the anechoic environment, as well as partial scattering off of the array e dges, even with acoustic edge treatment. Group array calibration can correct for errors in steering vectors du e to microphone response, as well as those due to microphone positional uncertainties As mentioned previously, the microphone positional uncertainties in a platebased array ar e minimal. Therefore, if the microphones have proper frequency response calibrations over their entire range of operation, the group calibration procedure should no longer be necessary. This assumes small differences in temperature and ambient pressure between individua l calibration and experimental c onditions play a small role in microphone responses, but such an assumption is always inherent in an experiment unless 94 PAGE 95 calibration occurs immediately before or after data acquisition. An ins itu method of individual array microphone calibration refe renced to the arraycentered B&K 4138 microphone is given in detail in Appendix D. The summary of this cal ibration result is that the array microphones have been calibrated for use from 512 Hz to 20 kH z, assuming that the B&K microphone has flat response within this bandwidth. Array measurements were collected simultaneous ly with all other channels, as previously discussed, at a sampling rate of 102,400 sample s per second for 30 seconds. CSMs were built using 6400point discrete Fourier tr ansforms (DFTs) for a 16 Hz bi nwidth. The data blocks were processed with 75% overlap usi ng a Hanning window function, lead ing to 996 effective averages with a normalized autospectral ra ndom uncertainty of 3.2%. Diagonal removal is used in all processing. The integration regi on is set to a 0.4 m x 1.06 m box centered on the trailing edge of the model in the z = 1 plane of the array coordinate system shown in dn Figure 445 The xdimension of the region is chosen to balance computational costs with capturing the trailing edge noise lobe at as low of a frequency as possible, while still attempting to reject noise from the front and back of the test section (setting a ma ximum dimension of 0.5 m). The ydimension of the integration region is chosen to encompass almost the entire model wetted span, aside from a small region on each side of the model to reject some of the possible sidewall noise. Note that this means the entire point spread function 3 dB lobe isnt contained with in the ydimensions of the integration region below 1,392 Hz, and within the xdimensions until 3,312 Hz. Shear layer corrections are computed for the entire region in a 0.02 m x 0.02 m dense grid. This same grid density is used in the integration procedure. The existing uncertainty code for a dela yandsum beamformer conducts Monte Carlo trials based on uncertainties in the beamformer input [Yardibi et al. 2010a]. Simple mechanical 95 PAGE 96 uncertainties include the microphone locations in the array plate, already estimated at 0.001, as well as uncertainty in the distance from the array to the source scan plane, which is considered as the measurement uncertainty of distances in the wind tunnel facility, 1/16, as well as the thickness of the model trailing edge, 0.13, to acc ount for the ambiguity of the reference location on the trailing edge for measurement relative to the true noise source plane. These inputs to the code are all drawn from Gaussi an random number generators, w ith the aforementioned values used as the input standard devi ation. Temperature uncertainty is computed based on insitu measurements during testing, and again is treated as an independe nt Gaussian random variable. The final input uncertainties are due to the microphone frequency response functions. Rather than attempt to compute individual response uncer tainties for each microphone this is treated as a batch problem, where the magnitude and phase responses of all of the microphones in the array, computed in Appendix D, are used to compute the overall system variance, and the subsequent standard deviations are used to perturb microphone be havior again using a Gaussian random number generator. Test Matrix The final test matrix for the dissertation expe riment set can include significant tunnel speed variations, minor angle of attack changes, and so me minor instrumentation offsets. The goal of these perturbations on a baseline case isnt to exhaustively qu antify the trailing edge noise signature of this particular m odel, but to provide enough variati on to the signal to cover a large spread of potential inputs to the analysis techniques. The overa ll test matrix is given in Table 46 The test section Mach number is swept from the tunnel minimum of 0.05, to the maximum with the NACA 63215 modB installe d of 0.19. The angle of a ttack is perturbed to 1.5. The array is offset 0.25 m behind the model from its baseline location. This moderate offset is selected for a minor directiv ity change, without moving the array too close to the known 96 PAGE 97 contamination region of the test section jet collector. This o ffset also maintains a microphone directly opposite the array center. Finally, the array is remove d and replaced with a single B&K 4939 microphone to study the plat e installation effects. Th ese conditions should provide sufficient data to make some statements regardin g the agreement of differ ent trailing edge noise measurement techniques. 97 PAGE 98 Figure 41. Planform schematic of UFAFF showing the flow pa th from the garage entrance (top), through the openje t test section and out through the fan section. 98 PAGE 99 Figure 42. Isometric schematic of UFAFF. Note that the silenc er boxes next to the contraction have been removed and blocked off with sealed panels and acoustic wedges for all experiments conducted for the main body of this research. 99 PAGE 100 Figure 43. AWeighted out offlow background noise co mparison between different aeroacoustic flow facilities. NonUF data are adapted from Duell et al. [Duell et al. 2002]. Note that the UF data were collected only 1 m from the test section cente rline. Most other facilities measurements occurr ed much further away, leading to lower acoustic levels from models in addi tion to lower background noise levels. 100 PAGE 101 Figure 44. Cross section of UF NACA 63215 ModB airfoil. Figure 45. Photograph of UF NACA 63215 ModB airfoil. 101 PAGE 102 Figure 46. Photograph of UF NACA 63215 installed in UFAFF, from below. Trip tape is visible on the model leading edge. Note the large access panel in the center of the model, and the smaller dynamic pressure tr ansducer panel near the trailing edge. 102 PAGE 103 Figure 47. UF NACA 63215 ModB profile deviation between design and measured coordinates. 103 PAGE 104 Figure 48. Trailing edge profiles at varyi ng spanwise locations. 104 PAGE 105 Figure 49. Surface error curve fits for UF NACA 63215 ModB. Figure 410. XFoil predicted Cp distributions for design and modified profiles, M = 0.17. 105 PAGE 106 Figure 411. Cp for 7.5 geometric AoA (3.14 equivalent lift freestream). Figure 412. Cp for 5 geometric AoA (2.63 equivalent lift freestream). 106 PAGE 107 Figure 413. Cp for 2.5 geometric AoA (1.98 equivalent lift freestream). Figure 414. Cp for 0 geometric AoA (1.19 equivalent lift freestream). 107 PAGE 108 Figure 415. Cp for 2.5 geometric AoA (0.65 equivalent lift freestream). Figure 416. Cp for 5 geometric AoA (0.15 equivalent lift freestream). 108 PAGE 109 Figure 417. Cp for 7.5 geometric AoA (0.33 equivalent lift freestream). Figure 418. Cp for 10 geometric AoA (0.76 equivalent lift freestream). 109 PAGE 110 Figure 419. Installed, equiva lentlift AoA scaling with ge ometric AoA for UF NACA 63215. Figure 420. Evaluation of spanwise uniformity at zerodegree geometric AoA. 110 PAGE 111 Figure 421. Trailing edge se nsor layout for the NACA 63215 ModB, with orange lines representing ribbon cabling, and all dimensions in inches. Note the vertical spacings are equal to horizontal ones. Distance from the trailing edge is given in the text. Figure 422. Map of trai ling edge array spacings. 111 PAGE 112 Figure 423. Schematic of a singlewire, 2axis traverse experiment. 112 PAGE 113 Figure 424. Photograph of hot wire installation for a DU 96W180 airfoil, from below the model, viewing the trailing edge. 113 PAGE 114 Figure 425. Photograph of hot wire installation for a DU 96W180 airfoil from above the model, viewing the trailing edge. 114 PAGE 115 Figure 426. Example hotwire calibration data to be used with Kings Law. 115 PAGE 116 Figure 427. Plot of mean and fluctuating boundary layer velocity profile in the trailing edge vicinity. Note that the yaxis does not orig inate on the airfoil su rface. An offset of approximately 5x104 m is present. 116 PAGE 117 Figure 428. Plot of mean and fluctuating wake profile in the trailing edge vicinity. 117 PAGE 118 Figure 429. Schematic of an ideal, spatially lumped line source in a coherent powerbased analysis. 118 PAGE 119 Figure 430. Block diagram for a Singl eInputTwoOutput (SITO) system. Figure 431. Installation schematic fo r coherent output power measurements. 119 PAGE 120 Figure 432. Coherent Ou tput Power compared between two experiment sets. Figure 433. Crosschannel coherence co mpared between two experiment sets. 120 PAGE 121 Figure 434. Coherent Out put Power plotted as a func tion of scaled frequency. Figure 435. Crosschannel phase angle compared between two experiment sets. 121 PAGE 122 Figure 436. Baseline experimental configuration with phased array. 122 PAGE 123 Figure 437. Experimental config uration with free field microphone located at equivalent array location in free space. 123 PAGE 124 Figure 438. Experimental conf iguration with array horizontally offset from trailing edge by 0.25 m. 124 PAGE 125 Figure 439. Experimental configuration for linear array measurements conducted prior to current body of research. 125 PAGE 126 Figure 440. Comparison of differing coherencebased noise reduction techniques. Figure 441. Effect of shear layer correc tion on reliability of dipolelike assumption. 126 PAGE 127 Figure 442. Phase error comparison for linear array microphones, referenced to microphone 12. Figure 443. Phase error comparison for noncontaminated linear array microphones, 127 PAGE 128 Figure 444. Comparison of power predictions from differe nt methodologies. 128 PAGE 129 Figure 445. Isometric schematic of ba seline experimental configuration. 129 PAGE 130 Figure 446. Layout of the UFAFF medium aperture array with acoustic treatment. 130 PAGE 131 Figure 447. 3 dB beamwidth of the UFAFF medium aperture arra y as a function of frequency. Figure 448. PSF for mediuma perture array at 1,024 Hz. 131 PAGE 132 Figure 449. PSF for mediuma perture array at 2,512 Hz. Figure 450. PSF for mediuma perture array at 5,008 Hz. 132 PAGE 133 Figure 451. PSF for mediuma perture array at 7,600 Hz. Figure 452. PSF for mediuma perture array at 15,008 Hz. 133 PAGE 134 Figure 453. PSF for mediuma perture array at 20,000 Hz. Table 41. Key UFAFF design characteristics. Facility Characteristic Design Value Maximum test section velocity 76 m/s Test section height 0.74 m Test section width 1.12 m Test section length 1.83 m Maximum Reynolds number (based on model chord of 2/3rd test section width, or 0.75 m) 3.7x106 Measured turbulence intensity <0.05% 134 PAGE 135 Table 42. UF NACA 63215 ModB airfoil profile design coordinates. x, m y, m x, m y, m x, m y, m x, m y, m x, m y, m 0.7374 0.0013 0.5845 0.0247 0.2784 0.0634 0.0005 0.0025 0.3876 0.0399 0.7364 0.0015 0.5808 0.0254 0.2583 0.0638 0.0006 0.0030 0.4060 0.0378 0.7322 0.0021 0.5747 0.0265 0.2404 0.0639 0.0008 0.0034 0.4244 0.0355 0.7265 0.0029 0.5711 0.0271 0.2218 0.0640 0.0016 0.0050 0.4428 0.0332 0.7230 0.0033 0.5650 0.0282 0.2034 0.0641 0.0023 0.0063 0.4613 0.0307 0.7172 0.0041 0.5614 0.0289 0.1850 0.0640 0.0031 0.0073 0.4797 0.0280 0.7137 0.0046 0.5552 0.0300 0.1665 0.0637 0.0039 0.0082 0.5086 0.0236 0.7081 0.0054 0.5516 0.0307 0.1481 0.0631 0.0077 0.0117 0.5124 0.0230 0.7045 0.0059 0.5455 0.0318 0.1297 0.0623 0.0151 0.0162 0.5315 0.0200 0.6997 0.0065 0.5419 0.0324 0.1112 0.0609 0.0226 0.0196 0.5352 0.0194 0.6954 0.0071 0.5358 0.0335 0.0928 0.0588 0.0300 0.0224 0.5543 0.0165 0.6897 0.0078 0.5322 0.0342 0.0743 0.0559 0.0374 0.0249 0.5582 0.0160 0.6862 0.0083 0.5261 0.0352 0.0669 0.0543 0.0448 0.0271 0.5772 0.0131 0.6791 0.0092 0.5226 0.0359 0.0595 0.0525 0.0522 0.0291 0.5811 0.0126 0.6768 0.0095 0.5165 0.0369 0.0521 0.0503 0.0595 0.0308 0.6002 0.0099 0.6696 0.0106 0.5129 0.0375 0.0447 0.0479 0.0669 0.0325 0.6040 0.0094 0.6674 0.0109 0.5069 0.0385 0.0373 0.0448 0.0743 0.0339 0.6231 0.0069 0.6615 0.0118 0.4981 0.0400 0.0299 0.0412 0.0927 0.0372 0.6270 0.0065 0.6579 0.0123 0.4797 0.0431 0.0224 0.0367 0.1112 0.0399 0.6461 0.0043 0.6520 0.0133 0.4613 0.0460 0.0150 0.0306 0.1296 0.0422 0.6494 0.0040 0.6484 0.0138 0.4429 0.0488 0.0075 0.0221 0.1480 0.0441 0.6499 0.0039 0.6425 0.0148 0.4245 0.0513 0.0038 0.0155 0.1665 0.0457 0.6691 0.0021 0.6388 0.0154 0.4060 0.0537 0.0030 0.0137 0.1849 0.0469 0.6724 0.0019 0.6329 0.0164 0.3920 0.0554 0.0023 0.0118 0.2033 0.0478 0.6729 0.0018 0.6292 0.0170 0.3866 0.0560 0.0015 0.0094 0.2217 0.0484 0.6922 0.0008 0.6233 0.0180 0.3778 0.0570 0.0008 0.0064 0.2402 0.0486 0.6955 0.0007 0.6197 0.0186 0.3692 0.0579 0.0006 0.0057 0.2586 0.0485 0.6960 0.0007 0.6136 0.0196 0.3590 0.0589 0.0005 0.0048 0.2770 0.0481 0.7154 0.0010 0.6100 0.0203 0.3508 0.0596 0.0003 0.0039 0.2955 0.0474 0.7187 0.0011 0.6039 0.0213 0.3412 0.0604 0.0002 0.0026 0.3139 0.0464 0.7193 0.0011 0.6003 0.0219 0.3323 0.0610 0.0000 0.0000 0.3323 0.0452 0.7347 0.0013 0.5943 0.0230 0.3139 0.0622 0.0002 0.0010 0.3507 0.0436 0.7352 0.0013 0.5906 0.0236 0.2955 0.0630 0.0003 0.0018 0.3692 0.0418 0.7374 0.0013 135 PAGE 136 Table 43. UF NACA 63215 ModB chord wise pressure tap locations. Tap number Tap distance from leading edge, m 1 0.0000 2 0.0037 3 0.0074 4 0.0147 5 0.0221 6 0.0313 7 0.0405 8 0.0516 9 0.0663 10 0.0884 11 0.1105 12 0.1363 13 0.1694 14 0.2099 15 0.2541 16 0.3094 17 0.3683 18 0.4420 19 0.5156 20 0.5525 21 0.5893 22 0.6261 23 0.6629 24 0.6998 Table 44. Geometric and corrected microphone angles. Microphone number Geometric angle from trailing edge, degrees Corrected angle from trailing edge ( M = 0.17), degrees Geometric distance from trailing edge, m Corrected distance from trailing edge ( M = 0.17), m 1 57.7 59.2 1.34 1.34 2 64.7 67.1 1.25 1.25 3 72.5 75.6 1.19 1.19 4 81.0 84.6 1.14 1.15 5 90.0 93.6 1.13 1.13 6 99.0 102.3 1.14 1.15 7 107.5 110.1 1.19 1.19 8 115.3 117.0 1.25 1.25 9 122.3 122.8 1.34 1.34 10 128.3 127.6 1.44 1.44 11 133.4 131.5 1.56 1.56 136 PAGE 137 137 Table 45. UFAFF medi umaperture array microphone design coordinates. x, in. y, in. x, in. y, in. x, in. y, in. 0.7310 0.0000 1.7023 0.8207 1.7268 3.8568 0.5600 0.4699 1.8316 0.4655 3.802 1.8445 0.1269 0.7199 1.1039 1.5339 4.0981 1.0309 0.3655 0.6331 2.7990 1.6970 2.4767 3.4239 0.6869 0.2500 3.2350 0.4992 0.3036 4.2149 0.6969 0.2500 2.1572 2.4618 2.9418 3.0336 0.3655 0.6331 0.0701 3.2725 5.1648 4.1624 0.1269 0.7199 2.0498 2.5520 1.2809 6.5084 0.5600 0.4699 3.2106 0.6373 3.2023 5.8091 0.1404 1.8846 2.8691 1.5755 6.1871 2.3916 1.3189 1.3535 1.1852 3.0512 6.2769 2.1449 1.8803 0.1890 1.0534 3.0991 3.4297 5.6778 1.5619 1.0639 4.2035 0.4329 1.0223 6.5540 0.5127 1.8190 3.4984 2.3703 4.9960 4.3635 0.7765 1.7229 1.1563 4.0645 6.6319 0.1313 Table 46. Test matrix of experi ments to study measurement techniques. Array Location Mach number range Geometric Angles of Attack 1 m below trailing edge 0.05 to 0.19 in 0.01 increments 1.5, 0, 1.5 1 m below trailing edge and 0.25 m downstream 0.05 to 0.19 in 0.01 increments 0 Removed, B&K 4939 1 m below trailing edge 0.05 to 0.19 in 0.01 increments 0 PAGE 138 CHAPTER 5 RESULTS & DISCUSSION This chapter will discuss the results of the experiments from the test matrix proposed in Table 46 First, the data will be evaluated and compared between experimental conditions to check for installation effects of the microphone phased array. This is followed by evaluation of the twoand threemicrophone methods available. Analysis will focus on the comparison of results between different inst allation techniques, and the di fferent ways the twomicrophone solution can be computed. Uncerta inties will be used to show where different methods agree. Preliminary results from covariancebased approaches will be shown and discussed. Beamforming results will then be computed and compared to previous techniques. Finally, 1/3rdoctave band result scaling will be shown for diff erent Mach numbers and angles of attack, and compared to existing empirical solution codes [Moriarty 2005]. Installation Effects The first data to be analyzed are autospectra l data. The necessity of various processing techniques can be quickly verified by co mparing the raw measurement autospectra to background noise measurements, collected in prev ious work using G.R. A.S. 40BE microphones. Four cases are selected for this evaluation. Two cases are for the pha sed array installation, Figure 436 and two for the freefiel d microphone installation, Figure 437 The zerodegree AoA case will be considered, as the freefield mi crophone data were not collected for any other angles of attack. Mach numbers of 0.10 and 0.17 are selected for comparison. These data are plotted in Figure 51 and Figure 52 Background noise data for the facility at a nearequivalent microphone location are only available from prev ious work in 5 m/s increments, so the background noise data plotted are at a slightly higher speed of 35 m/ s for the airfoil experiments 34.4 m/s at Mach 0.10. For the Mach 0.17 data which was equivalent to 58.5 m/s, the two 138 PAGE 139 closest background noise data sets are 55 m/s a nd 60 m/s. Since the experimental data fall between these two values, both noise curves are plotted. The background noise measured is also from a microphone 5 closer to the test section than the data collected with th e model installed. Background noise measurements were conduc ted using G.R.A.S. 40BE free field microphones in free space. Two things are immediately evident from these plots. First, the models noise is only above the facility background noise for a portion of the audible spectrum. Second, there is a dramatic difference in observed signal from the freefield microphone to the arrayinstalled microphone. In an attempt to separate the local e ffects of the array instal lation, such as acoustic scattering and/or flow over the array face, from any global installation effects, these same plots are generated in Figure 53 and Figure 54 for the upper central microphone from Figure 436 and Figure 437 Here, background noise measurements are available at the same speeds, but 7 closer to the test section, so background noise comparisons are agai n qualitative. It is clear from these comparisons that the offset in the arra y microphone measurement is a local effect, while some of the ripple may be global, at least at the lower Mach number of M = 0.10. The term global is applied here as inst alling the phased array on one side of the model affects a free field microphone measurement on the opposing side. The ripple in the microphone spectra must be addressed next. Ri pple in an acoustic spectrum is a characteristic of a reflected acoustic signal. A reflection, as derived in detail in Appendix D, can appear as a ripple in autospectral levels. One wa y to identify the time scales of a reflection is to look at the cepstrum magnit ude of a signal [Randall & Hee 1981]. While the details of this analysis are again shown in Appe ndix D, in essence an in verse Fourier transform 139 PAGE 140 of the logarithm of a power spectrums magnitude will accentuate the time scales associated with any ripple in an autospectrum. The cepstrum magnitudes, or gamnitudes, of the array and freefield microphone autospectra are plotted in Figure 55 and Figure 56 as a function of quefrency. Quefrency can be thought of as lag time, like in correlation analys is, and has dimensions of time. The ripple in the array data manifests as a fluctuation in gamnitude at around a quefrency of 5.5 6.5 milliseconds. This feature appears to be Mach number independent. Multiplying quefrency by the speed of sound and replotting the data for a Mach number of 0.17 can give an indication of the propagation distance of this re flective feature. In this case, the reflected signal appears to have travelled an additional distance of appr oximately two meters, beyond the initial one meter transit from the airfoil tr ailing edge, as shown in Figure 57 The most likely candidate for a 2 m reflection based on the geometry illustrated in Figure 436 would be an acoustic signal which reflects off of the array face and back up to the model (1 m), and then reflects off of the model and back down to the array (1 m). The ripple in the G.R.A.S. 40BE signal in Figure 53 can be analyzed in a similar manner. The cepstrum plots for the arrayand fr ee field comparison si gnals are shown in Figure 58 Again, there is a significant difference between the signals at a quefrency of 6 ms, although the comparison is not as clean as in the previous plots. As these data were acquired simultaneously with the data from the B&K cepst rum analysis, again th is corresponds to an additional distance traveled of approximately 2 m. Again Figure 436 can be used to illustrate the situation. Here, a wave can be pictured leaving th e trailing edge and traveling one meter before arriving at both the array plate and the upper center free field microphone A reflection then occurs off of the array plate, and travels two additional meters befo re arriving at the upper microphone. Again, the 140 PAGE 141 array plate is the source of the observed reflec tion, which would explai n why the ripple isnt present in the free field autospectrum plotted in Figure 53 for the experiment al configuration from Figure 437 Note that the ripple is not noticeably present in the G.R.A.S. autospectra in Figure 54 One potential hypothesis as to why this is the case could be due to shear layer refraction and convective effects. These are sign ificantly stronger at M = 0.17 as compared to M = 0.10, and would contribute to the reflected signal from the array being convected downstream of the upper microphone. Also, multiple transitions of the acoustic signal through multiple shear layers could break down the corr elation of the reflection with the original signal. Such correlation is required for reflections to cause ripple in autospectra. Having tentatively identified the ripple featur e in the spectra, seve ral things can now be said about it. First, this feature is likely due to a model noise source, and is independent from tunnel background noise. Second, th is feature should be expected for all frequencies for which the trailing edge is the source and may be a good indicator that the airf oil trailing edge is significantly contributing to a given bandwidth. The only frequencies which would be unaffected by this effect would be those which are too low (i.e. wa velength too large) to interact with the 20 diameter aluminum disk used for the phased array, and such frequencies are well below the scope of interest of this study, due to the limitations of the acoustic arrays calibration bandwidth. Such frequencies also fall below the uppe r values of Blakes criteria for trailing edge noise [Blake 1986]. Also, as a coherent reflection of the airfoil noise source, this ripple feature should be preserved for all analys is techniques, since none of the present techniques considered can isolate a signal which is an image of a base signal at a specific time delay. Finally, the difference in propagation behavior between th e upper and lower microphones when the array is present will invalidate the second, noiseless model of the coherent output power method. This 141 PAGE 142 can be said because Equation (417) is dependent on the assumption from Equation (415) that the propagation path model from the trailing edge to each microphone is of equal magnitude but opposite phase above and below th e model. If one of the microphones is experiencing a reflection which the other does not observe, or observes in a redu ced fashion, this assumption cannot hold. Equation (414) can still be used as it makes no such assumption, but as shown the results will suffer a bias from SNR issues. Comparison of TwoMicrophone Methods The twomicrophone techniques from Equation (414) and Equation (417) are now evaluated for the two expe rimental conditions from Figure 436 and Figure 437 As just discussed, the crossspectral magnitude e xpression of the coherent power, Equation (417) is expected to be invalid for measur ements with the array plate instal led in the tunnel. Nonetheless, this method is compared to the appropriate cohe rent power method for that installation, Equation (414) to evaluate any significant differences. Bo th techniques are valid for the free field microphone installation. However, they should no t give the same results for acoustic signals except under conditions where the measurement is noiseless, as the general coherence method is biased low by the line noise of the opposing microphone, while the crossspectral magnitude is not. Finally, the noisecontaminated coherent power technique must be evaluated between the cases with the phased array plate, Figure 436 and those with the equivalentlylocated free field microphone, Figure 437 to see how if this method is approx imately installationindependent. In all subsequent figures, the general coherent pow er case from Equation (414) will be labeled as COP, General while the trailing edge noise specific coherent pow er case from Equation (417) will be labeled as COP, Dipole Assumption. Coherent power results for Mach 0. 10 and Mach 0.17 are plotted in Figure 59 and Figure 510 for the arraybased microphone case of Figure 436 As plotted, both cases appear to agree 142 PAGE 143 within the frequency range of the shedding peak, wh ich is clearly visible in each plot. However, upon further consideration it is clear that this cannot be the case, as in an ideal measurement the general COP method should underpre dict the dipolespecific COP method by the noise power in the measurement. For the two techniques to agre e, there would be negligible measurement noise in the second microphone. However, the offset in the raw autospectrum indicates a significant amount of incoherent noise in the first microphones autospectrum, and as Figure 511 shows, the difference between the second microphones au tospectrum and general COP solution would indicate a significant amount of incoherent noise in that micr ophones measurement as well. Notably the dipoleassumption COP predicts higher levels than the autospectrum for the second microphone in this case. As the microphone autospectrum should be a sum of signal and noise powers, both of which are nonnegative terms, it should not be possible for a signal prediction to be higher than the microphones autospectrum. This is indicative of a breakdown in the assumptions in this method. As such, the agreement of the two methods in the B&K 4138 prediction for a limited bandwidth is suspect, and may be coincidental. Coherent power results for Mach 0. 10 and Mach 0.17 are plotted in Figure 512 and Figure 513 for the free field microphone case of Figure 437 These appear to be more inline with the mathematics of each method, where the coherence multiplication method should appear similar to the crossspectral magnitude method, but with lo wer levels due to the noise level bias. Note that no shear layer corr ection has been applied to the dipo leassumption COP case, as for these low lift conditions, with these specific microphone locations, the shear layer corrections should cancel for this method, as prev iously discussed. The predic tions for the opposing G.R.A.S. 40BE microphone are shown in Figure 514 and show a more appropria te expected behavior for both methods in regards to the microphone autosp ectrum. The two methods have very similar 143 PAGE 144 spectral shape up to a frequency of about 3 kHz for the Mach 0.10 case an d 5 kHz for the Mach 0.17 case, before degrading to what appears to be noise. In an attempt to determine if the character of the methods changes at these fr equencies, the 95% confidence bounds of the methods, computed from Equation (418) and Equation (420) will be plotted and checked for overlap, shown in Figure 515 and Figure 516 It is clear from th ese plots that above the aforementioned frequencies, the measurements are the same within the shown confidence intervals as the blue and red regi ons overlap to a purple shade, wh ereas at lower frequencies they are distinct. Also at higher frequencies the si ze of the confidence intervals grows dramatically, quickly scaling from being one or two dB wide to on the order of tens of decibels. As these techniques are coherencebased, the ordi nary coherence between the upper and lower microphones is an obvious culprit, appearing in the denominator of the corresponding equations, Equation (418) and Equation (420) Figure 517 is a plot of the c oherence between the two microphones for these two cases, and confirms this behavior. Finally, the twomicrophone prediction between the array case and free field case must be compared. This is only done for the genera l COP formulation, as the dipole assumption formulation has just been shown to be erroneous for the array case. Figure 518 and Figure 519 plot the results for Mach numb ers of 0.10 and 0.17. As shown, the results of the array installation and free field insta llation are distin ctly different at fr equencies of nonzero coherence. The measured coherent field is dram atically different, as would be expected for an installation which generates additional scattering a nd reflection. These data illustrate that care must be taken when comparing results using di fferent experimental se tups, as significantly different physical effects can be observed. 144 PAGE 145 ThreeMicrophone Method Two microphone methods have been shown to be affected by the installation of the phased array. The dipolebased coherent power method provide s erroneous results when the array plate is present. The general coherent power method should function properly in both cases, but insufficient information is present in the solu tion to validate the output, since the method is affected by SNR. The behavior of both methods appears consistent for the free field microphone case, but again insufficient information is present to validate the relative behavior, since a noise power estimate is required to check the offset from the one method to the next. The threemicrophone method provides an estimate of the SNR, so it is the next method analyzed. Figure 520 and Figure 521 show the nominal threemicrophone si gnal estimate as compared to the general coherent power results fo r the installed array cases from Figure 436 at Mach numbers of 0.10 and 0.17. Figure 522 and Figure 523 show the same plots, as well as dipole assumptionbased coherent power result, for the free field installation from Figure 437 for Mach numbers of 0.10 and 0.17. As expected, for the arraybased cases of Figure 436 the threemicrophone method falls somewhere between the raw microphone autospectr um and two microphone pr ediction. Plotting the uncertainties for these two Mach numbers in Figure 524 and Figure 525 shows that, at least for the nonzero coherence frequency region, the threemicrophone method is truly distinct from the coherent power method. For the free fi eld microphone cases, the threemicrophone method should match the crossspectral magnitude me thod for an ideal source. However, as Figure 526 shows, the results are close but distinct for al most the entire coherent measurement region. Specifically, the coherent power method finds a solution between 13 dB below the threemicrophone method below 1 kHz. Above 1.3 kHz, the coherent power method is off from the threemicrophone method by about a decibel until th e uncertainty regions become large near 2 145 PAGE 146 kHz. Based on Equation (417) and Equation (424) both methods provide a noiseless estimate of the coherent acoustic strength For them to differ, the assu mptions inherent in one or both methods must be faulty. For instance, if the dom inant source for a given fr equency is not located at the trailing edge of the model, such a fiel d could generate misleadi ng results with the dipole assumptionbased coherent power method. Alte rnatively, as discusse d in Appendix C, the source field could be su fficiently distributed such that bo th the threemicrophone method and the dipole assumptionbased coherent power meth od both fail. The agreement between both methods when the airfoil trailing edge is the dominant source is reinforced in Figure 527 where the two methods completely match in the vicinity of the shedding peak of the airfoil trailing edge. Such a comparison could be a good indica tor of the dominance of trailing edge noise within a given bandwidth. If the crossspectral magnitude a nd threemicrophone methods agree within uncertainty bounds, a domin ant acoustic source is likely located in the vicinity of the airfoil trailing edge. As discussed in Chapter 4, Monte Carlo analys is was used to compute uncertainties for the threemicrophone method. This was done using a Gaussian random number generator to perturb the coherence of each microphone combination, using Equation (432) to determine the appropriate variance and subseque nt standard deviation of the Gaussian random variable. A positivity constraint was assigned in the loop such that when low coherence levels with large variance were perturbed, the resu ltant coherence could not be nega tive. Instead it was set to an infinitesimallysmall positive value. The behavior of these Monte Carlo simulations must be evaluated. The uncertainty bounds plotted so far have been fo r 1,000 iterations of the Monte Carlo simulation. To check convergence, 10,000 iterations were run and the confidence interval compared to the 1,000 iteration case. This is shown in Figure 528 and Figure 529 for the free 146 PAGE 147 field cases. For almost ever y frequency bin the data appear converged. The cumulative distribution function (cdf) of the da ta is evaluated at several fre quencies in the overall spectrum of the free field Mach number 0.17 case. The frequency 2,512 Hz is selected in Figure 530 as this appears to be the shedding peak of the airfoil. Here, the data appear Gaussian in nature with a skewness of 0.168 and a kurtosis of 2.95. The frequency 3,392 Hz is selected in Figure 531 as this frequency is where the spread in the uncer tainty region becomes large. The data appear nominally Gaussian at first glan ce, but the nonzero offset at ze ro power present at the beginning of the cdf is nonphysical. This is due to the nonnegative zero constraint placed on coherence perturbations in the Monte Carlo analysis. Indeed nonGaussian behavior is identified via the skewness of 0.492 and a kurtosis of 3.69. Next, the frequency 6,000 Hz is shown in Figure 532 and 12,000 Hz in Figure 533 These two frequencies are fully in to the large uncertainty region. Both show that a significant number of the predic tions are forced to zero power. Also, a bimodal shape appears in the data, more pronounced in th e 6,000 Hz data than the 12,000 Hz data. Both conditions are dramatically far from Gaussian data behavior, w ith 6,000 Hz having a skewness of 3.37 and a kurtosis of 12.6, and 12,000 Hz having a skewness of 5.80 and a kurtosis of 36.2. Finally, the 95% confidence interval is plot ted under both the nominal solution and the mean of the Monte Carlo trials in Figure 534 and Figure 535 It appears that the nominal and mean value diverge as the Monte Carlo trial distribution becomes bim odal, with the mean solution predicting higher than the nominal. This could be an additional breakpoint beyond which the results are considered unreliable. Previous methods of checking for nearzero coherence and a breakdown in regular phase behavior showed similar break frequencies. Additional Linear Array Analysis The techniques described by Du et al. [Du et al. 2010] are also applied to the data collected, using the data set of the upper array of G.R.A.S. 40BE microphones and the inplate 147 PAGE 148 B&K 4138 microphone as shown in Figure 436 The nominal results for the methods for the default condition of 20 internal iterations are plotted in Figure 536 and compared to threemicrophone uncertainty levels for the array plat e experimental setup. The threemicrophone method is used for comparison as it also solves for both signal and noise powers. These methods estimate the same parameters for higher channel counts. As shown in th e plot, the methods all come extremely close to the 95% confidence in terval of the threemicrophone method near the shedding peak of the airfoil, such that within their confidence intervals they may match. However, discrepancies are seen in most of the re st of the signal bandwidth. This may indicate that the 20 internal iterations to each algorithm are insufficient for this data set, so a brief convergence study was conducted for each method, su ch that 50 and 100 iter ations were also allowed. This becomes important for the uncertainty analysis, as a 1000iteration Monte Carlo analysis with 20 internal itera tions is manageable, and has be en conducted for the single case shown in Figure 536 The solution time scaled to unmanag eable levels for the scope of this research with internal iteration counts of 50 and 100. Figure 537 shows the solution variation with increasing internal iteration count for the Frobenius Norm Method. As shown, for the en tire range for which the solution appears wellbehaved, the solution output is independent of itera tion count. This solutio n could be considered converged. However, in the upper frequency range of this case, the so lution is dependent on iteration count. Figure 538 shows the convergence behavior of the Rank1 Method. Aside from the shedding peak, these data within the expect ed model acoustic range ma y not be converged at 20 iterations, or even 50 or 100 iter ations. Interestingly, the data for higher frequencies, where most of the coherent power met hods discussed so far appear to fail and have severe uncertainty issues, appears converged and wellbehaved. Ho wever, the physical meaning of this result, 148 PAGE 149 which is in disagreement with all previous methods, is not fully understood, and thus quantitative uncertainty comparison will be avoided for the time being. Finally, Figure 539 shows the convergence behavior for the Maximum Likelihood Method. This method again shows convergence in the vicinity of the trailing edge sh edding peak, but nowhere else. As none of the methods appear fully converged, the single availa ble Monte Carlo trial will not be presented in this work. It should be noted that all of thes e methods assume a single coherent source between all of the microphones, as is assumed for twoand threemicrophone methods. The number of iterations for convergence could be an indicator of the nature of the acoustic field, and again if the trailing edge is acting as a single, domi nant source within a gi ven bandwidth. For frequencies where a given method has difficulty with convergence, there may be an indication that the source field strongly vi olates the methods assumptions. Beamforming Results Phased array data were collected for the entire test matrix and analyzed using a frequencydomain DelayandSum (DAS) beamformer, as discussed previously. However, due to computational expense, only one Mo nte Carlo analysis of an inte grated spectrum was conducted. The zerodegree angle of attack case at a Mach number of 0.17 was selected. Figure 540 shows the results of the nominal beamformer inputs and threemicrophone data in comparison to the B&K autospectrum from the array installation of Figure 436 The two methods are in closest agreement near the models shedding peak, and ha ve predictions within 1 dB of each other. They have the greatest difference in prediction at approximately 5 kHz, where they differ by over 10 dB. Based on the behavior of the predictions in relation to the microphone autospectrum, this is the region where the coherent acoustic source becomes weak in relation to the incoherent noise. Figure 541 shows the uncertainty bounds from the Monte Carlo trials in comparison to the threemicrophone method uncertainty bounds. Aside from the shedding peak, the data are 149 PAGE 150 not in agreement for the lower frequencies of th e measurement bounds. This could be due to the aperture effects of this smaller array at lower fr equencies, the presence of sources which violate the assumptions of the threemicrophone method, and even the incoherent source assumption of the beamformer. Based on the unc ertainty bounds of the analysis methods, there is overlap and thus agreement at higher frequencies. However, this is due in large part to the uncertainty bounds on the threemicrophone method results spanning such a great range of power predictions. Several features of the uncertainty bounds in the beamformer out put are striking and warrant further evaluation. The hump structure in the uncertainty bounds at higher frequencies is one such feature. Also, the trend of the beamfo rmer at frequencies slightly above the shedding peak warrants investigation, as it is significantly diffe rent from the threemicrophone behavior. Another, as shown in Figure 542 is the trending of the nominal integrated level with the Monte Carlo mean. With beamforming, spatial informa tion is available, so at each investigation frequency selected a full test section beam ma p will be generated, in addition to cumulative distribution function estimates. The first frequency to be evaluated is 1,024 Hz This is a frequency where the array has a large main lobe, approximately 1.6 m based on Figure 447 so the majority of noise reduction will occur from the diagonal removal operation on the CSM. In effect, because the weighting of the steering vectors from Equation (438) will be small at very low frequencies for a small array, at this frequency the DAS output could be consid ered the mean of the sum of the offdiagonal powers of the CSM. The beam map of these data, with approp riate shear layer corrections applied, is shown in Figure 543 and the Monte Carlo cdf in Figure 544 In this beam map, as in all subsequent ones, the space occupied by the m odel is indicated with a shaded rectangle. A 150 PAGE 151 blackbordered box is drawn as the boundary of the integration region. Flow through the test section is from right to left, so the integration box is drawn ar ound the model trailing edge. For this case, allowing for the large array response region, it would appear that the dominant noise source is coming from behind the model. Given the size of the arrays main response lobe at this frequency, it would be difficult to blindly stat e whether the dominant acoustic source is due to the models wake region or UFAFFs jet collector leading to the diffuser. However, based on the comparison of autospectra in Figure 52 between experimental configurations and facility background noise, it is evident that this acoustic source is not solely due to background jet collector effects. The skewness of the Monte Carlo trial da ta is 0.237, and the kurtosis 3.33. The second frequency evaluated is the shedding peak as noted in the free microphone cases previously at 2,512 Hz. The beam map is shown in Figure 545 and the cdf in Figure 546 As expected, the airfoil trailing edge is the domin ant noise source in the beam map. The jet collector appears to have some slight contri bution to the overall meas ured noise at this frequency, and very little leadi ng edge or inlet contri bution is noticeable. The data skewness is 0.186, and its kurtosis 3.19, showing Gaussianlik e behavior as at the previous frequency. The third frequency, 5,008 Hz is evalua ted next, with its beam map shown in Figure 547 and cdf in Figure 548 This frequency was selected as it is near the point on the integrated spectra where the mean and nominal level predic tions begin to diverge, and the uncertainty bounds increase. From the beam map, it is clea r that at this frequenc y sidewall boundary layer noise and jet collector noise are the two dominant sources. Al so, while in the previous two frequencies the nominal power pred iction fell below the median and mean predictions, here it is nearly the same. With a skewness of 0.338 and kur tosis of 3.59, the data begin to diverge more from expected Gaussian behavior. Also notable is the fact that at this frequency the results are 151 PAGE 152 very different from those predicted by the th reemicrophone method, even when accounting for uncertainty bounds. This would be consistent with the lack of dominance of the trailing edge noise source. At 7,600 Hz, the uncertainty bounds span nearly 25 dB, and the nominal integrated value is dramatically higher than the Monte Carlo mean. The beam map of this frequency is shown in Figure 549 and its cdf in Figure 550 The sidewall noise appears to be the dominant noise source, and collector noise has reduced below the plotted thres hold for much of the collector area. A strong noise source is no ticeable on the uppe r leading edge of the ai rfoil. This may be due to a leak in the sidewalls, which is highly possible given the in stallation methodology. While it could also be related to the sidewa ll boundary layer rolling up as a horseshoe vortex around the airfoil leading edge, sim ilar behavior should be seen on the lower leadi ng edge in the plot if that can be a strong noise source. Stat istically, the data show a strong bimodal behavior, where the nominal integrated leve l falls near the peak of the second, smaller mode, while the mean falls towards the end of the first, larger pe ak. The skewness and kurtosis of the data are of course significantly off from Gaussian va lues, at 0.904 and 2.18, respectively. At 8,800 Hz, the spread of the uncertainty re gion has decreased dramatically from 7,600 Hz. The beam map and cdf of this frequency are shown in Figure 551 and Figure 552 Again, the sidewalls appear as the dominant noise source. Again, the distribution appears bimodal, with the nominal integrated level falling on the sec ond, smaller peak. The spread between the peaks is reduced from the previous frequency. The da ta skewness is 0.692, and its kurtosis is 2.38. At 15,008 Hz, the data spread is again large. The beam map is shown in Figure 553 and the cdf in Figure 554 The sidewall boundary layers no lo nger appear to be dominant noise sources. The leading edge source is still strong here, as are some other, unidentifiable noise 152 PAGE 153 sources near the rear of the test section. Within the integration re gion, little is visi ble. The data spread behaves similarly to the previous bimodal cases, with a skewness of 0.946 and kurtosis of 2.78. Finally, at 20,000 Hz, the Monte Carlo mean surpasses the nominal integration by a significant amount. The beam map is plotted in Figure 555 and the cdf in Figure 556 The same noise sources are visible here as were visible in the 15,008 Hz map, although additional sources appear within the field. These may be si delobe images of true noise sources. The data spread appears to have returned to singlem ode behavior, but with a skewness of 0.482 and kurtosis of 2.50 is significantly deviated from Gaussian. The nominal condition falls on the lower tail of the distribution. Based on the Monte Carlo trials of the data, it appears that the behavior of the Monte Carlo results appear as tight, nearGau ssian distributions for frequencie s where the airfoil trailing edge is dominant. This also appears to be the case fo r frequencies where airfoil trailing edge noise is a significant contributor, as in th e frequencies below the airfoil shedding peak. When the sidewall noise is dominant, the distributi on has strong bimodal behavior. Th is could be inherent in the noise mechanism of the sidewall boundary layers, or it could be due to the sidewall noise passing in or out of the integration region in a manner th at is dependent on the Monte Carlo parameters. It would appear that when the sidewalls are no lo nger a significant contribution, either aperture effects or the beamforming noise floor are the do minant sources within the integration region. At these frequencies, the distribu tion of the Monte Carlo data agai n changes character. In future work, detailed uncertainty analysis of the entire beamforming region at all of these frequencies is warranted. Due to computational expense, such analysis is beyond the scope of the current work. 153 PAGE 154 In evaluating DAS integrated le vels, a brief discussi on should be dedicated to the effect of varying integration bounds. The nominal bounds, 0.4 m x 1.06 m centered on the airfoil trailing edge, are selected to provide so me rejection of sidewall noise while capturing the majority of the trailing edge shedding peak behavi or. Computational expense is al so a consideration. Several frequencies of a single case, Mach number of 0.17 at 0 degree angle of attack, are evaluated with varying integration regions to s ee what role this effect may pl ay in computing power levels. The first frequency evaluated is the airfoil shedding peak at 2,512 Hz. The integrated results for different integr ation bounds are presented in Table 51 It is evident from these data that varying the integration region in the xdirection has little eff ect on computed acoustic levels. Varying the ydimensions shifts the power by a pproximately 2.4 to 2.5 dB. A solution on this order of magnitude might be expected if the tr ailing edge consisted of a line of incoherent dipoles. Including the edges of the span will have a noticeable power contribution. However, doubling the number of sources in the integration region, by doub ling the length of trailing edge included in the region, will not double the predic ted power as the additional sources may be further from the array. The next frequency evaluated is 7,600 Hz. Th is frequency is selected as it has little observable trailing edge noise in its nominal beam map in Figure 549 and appears dominated by sidewall noise and the model s leading edge source. The di fferent power predictions are listed in Table 52 The results show that DAS is predicting zero power for the airfoil centerspan region, as every integration region which completely excludes the sidewall noise by having a small ydimension is computed as negativelyin finite on a decibel scale. As the integration region begins to include sidewall sources the power prediction climbs dramatically. Similarly, increasing the xdimension of the integration region substantially increases the power prediction. 154 PAGE 155 This behavior may contribute to the large uncerta inty bounds of this frequency. In the Monte Carlo simulation, perturbing th e DAS inputs will alter the amount of sidewall noise in the integration region. Varying the integration region, similarly, a lters the amount of sidewall noise contributing to the overall integrated level. The final frequency considered is 20,000 Hz. Th is frequency is selected as the only major observable noise sources are due to the leading ed ge and some reartest section sources, shown in the nominal beam map in Figure 555 The results of this analysis are listed in Table 53 Trends in these data are more di fficult to evaluate. For most cases a small increase in predicted power occurs by increasing the overall ydimension of the integration region. The xdimension slightly increases the prediction when going from a small to medium size, but has no effect when going from medium to large. Notably, for an xdimension of 1.2 m, when the ydimension increases from 0.4 m to 0.8 m a power decrease o ccurs. Intuitively this may make little sense as integration is a summation of source power within a region, but recall from Equation (439) that the integration is normalized by the arrays PSF within the integration regi on. If additional sources are not introduced, but th e PSFs summation increases due to the inclusion of additional sidewalls, the predicted power within a region will d ecrease. If a true source is present within an integration region, the inclusi on of the PSFs sidelobe should be accompanied by the inclusion of the sources sidelobe such that the summation is ideally balanced and no additional power is predicted. A power reduction may indicate that the integrated source power is actually due to sidelobes of sources outside of th e integration region. This reinfo rces the suspicion regarding the visible acoustic sources at this fr equency as mentioned previously. 155 PAGE 156 1/3rd Octave Scaling Mach Number Scaling Spectral scaling for the coherent power, th reemicrophone, and beamforming methods is now shown for various Mach numbe rs. These are shown in 1/3rd octave levels, to allow direct comparison with one of the free software tools for airfoil noise analys is, NAFNoise [Moriarty 2005]. The NAFNoise inputs are com puted using the equivalentlift AoA of the airfoil, XFoil for boundary layer properties, and the BPM method for turbulent boundary layer noise calculations, which were measur ed using the dipole assumpti onbased coherent power method applied to data from uncambered airfoils [Brooks et al. 1989]. Based on the beam maps from the previous section, it appears that leading edge noise can be neglected for these computations. This would make sense given the low test se ction turbulence intensity of the facility. 1/3rd octave plots are synthesized for the coherent power a nd threemicrophone methods by summing the appropriate bins of the narrowband sp ectra. The plotted data are all for the arrayinstalled cases, not for the free field microphone cases. 1/3rd octave plots for the beam forming data are computed both this way, and by summin g the appropriate bandwid th of CSMs prior to beamforming, and then beamforming at the center frequency of each band. While data for Mach numbers of 0.05 through 0. 19 in 0.01 increments were collected, for the sake of conciseness and due to computational time limitations a subset of the data is presented for Mach numbers of 0.05, 0.07, 0.10, 0.12, 0.15 and 0.17. The data are shown in Figure 557 through Figure 562 The lower frequency bounds of trailing edge noise based on the previouslydiscussed criteria defined by Blake [Blake 1986] in general fall below the minimum frequency bound of the plots. An uppe r frequency bound is drawn as a black dashed line over the data, and is located by determin ing the frequency breakdown of the coherence between the upper and lower microphones, specifica lly where the ordinary coherence function 156 PAGE 157 drops below 1%. Essentially this is the uppe r bound of the valid region of coherent power methods. Immediately evident is the fact that for the most part all methods are in agreement with NAFNoise regarding the location of the sh edding peak of the airfoil for higher Mach numbers. For lower Mach numbers, the coheren cebased methods appear to capture the rolloff after the shedding peak, but both be amforming methods miss it. This is sensible, as the shedding peak is sufficiently low frequency at lower Mach numbers that the array has no ability to reject extraneous acoustic sources. For the most part, the two different array computation methods are the same aside from end effects of the summing process in the first and last frequency bins. However, at the condition of 0.12 Mach number, they are dramatically different at higher frequencies. It is not immediatel y clear why this might occur only fo r this case, but as this is the bandwidth where sidewall noise woul d appear to dominate it may be that some selfrelationship of the sidewall noise within the CSM is the culprit. Also noticeable is that at M ach numbers of 0.05, 0.07 and 0.17 all of the methods except the coherent power method come extremely close to matching the NAFNoise output, and even coherent power is close at Mach 0.17. This is puzzling, as it should not be the case. NAFNoise uses prediction codes formulated by Brooks et al. in 1989 [Brooks et al. 1989], using the dipole assumptionbased coherent power technique. The measurements conducted in QFF were performed using freefield microphones, so the obse rved airfoil behavior, as shown previously, should be dramatically different. Specifically, al l of the phased array data shown here are from the experimental conf iguration shown in Figure 436 but the BPM data are from a configuration similar to that in Figure 437 Comparing arrayinstalled to fr ee field data in narrowband spectra was shown to have up to a 10 dB difference at the shedding peak ear lier in this chapter, as shown in Figure 51 Figure 52 Such results would indicate th at NAFNoise tends to overpredict 157 PAGE 158 shedding noise strength. This conclusion is consistent with the rest of the 1/3rd octave bands presented in this chapter. While at the highest Mach number most of th e analysis techniques ca tch the shape of the airfoil shedding peak, none of th em match the low or high freque ncy behavior of the prediction code. Based on the beam maps, this can be explained by considering the facility background noise sources. UFAFF background noise appears to have significant contributions from the jet collector at low frequencies. QFF, on the other hand, has no jet collector as the wind tunnel exhausts into a large room, thus the calibration data collected fo r the BPM study did not have to deal with lower frequency contam ination. A crosscorrelation ed iting technique was used in the BPM data for the reduction of facility background noi se. This could also play a significant role in the behavior of the code at high frequencies, although the levels are in sufficient agreement with the coherent power analysis that lowcoherence uncertainties could account for the behavior. As an additional consideration, the Q FF data were collected us ing rigid sideplates to constrain the flow to a nominally twodimensiona l field, instead of acoustic foam as used in UFAFF. This could cause stan ding wave patterns across the test section and l ead to altered acoustic measurements [Oerlemans & Sijtsma 2 000]. Additionally, the boundary layer behavior over the flat plates coul d be dramatically different over these solid sidewalls as compared to the behavior over porous acoustic foam, which would have an undetermined effect on the sidewall noise sources behavior. Angle of Attack Behavior The data for AoA variation are shown in Figure 563 through Figure 566 The data are for geometric AoAs of 1.5 and 1.5, which correspond to XFoil pred icted equivalentlift AoAs of 1.715 and 1.025. 0 geometric AoA corresponds to 1.37 equivalent lift. Again for brevitys sake and due to computational expens e, only data for Mach numbers of 0.10 and 0.17 158 PAGE 159 are reduced and plotted. As shown, the cohere ncebased methods show slight variation above the shedding peak as a function of AoA, as co mpared to the NAFNoise prediction. However, little other difference exists for this small perturbation in AoA. Instrumentation Offset The results for traversing th e array 0.25 m further downstrea m along the test section axis, as illustrated in Figure 438 are shown in Figure 567 and Figure 568 The data are in much poorer agreement with the NAFNoise output, and actually appears to fa il at detecting the shedding peak. It was suspected that there was sufficient flow over the face of the array that even the gain from beamforming with diagonal removal was insufficient to overcome flow noise contamination. This suspicion is reinforced by the behavior of the coherence between the B&K 4138 microphone and the opposing G. R.A.S. 40BE, which was sufficiently low, < 1% through the entire bandwidth of interest that no upper bound could be determ ined for the acoustic field at a Mach number of 0.10. As such, no upper bound is drawn on the plot. For the data at M = 0.17, only one frequency bin, 2688 Hz, is above 1% coherence. This bin has a coherence of 1.2%. Because of this behavior no upper bound is applied to the M = 0.17 data either. The array data are examined at several condi tions to confirm suspicions regarding flow contamination. One condition, the airfoil sheddi ng peak of 2,512 Hz at a Mach number of 0.17, is shown in Figure 569 with the appropriate 1/3rd octave plot at 2,500 Hz is shown in Figure 570. In both of the cases, some noise is visible in the vi cinity of the airfoil, but the appropriately positioned integration region fails to capture it, a nd the overlay of the model position shows that this noise source doesnt r eally appear to be trailing edge as it was with the centered array. This mass of apparent noise may be due to flow over the array, or it could be caused by invalidation of the shear layer correction method. Dependi ng exactly on how the test section shear layer interacts with the jet collector, which is now much closer to the array, the planar shear layer 159 PAGE 160 assumption likely breaks down as th e flow field becomes a complicated, viscous region. If this is the case, beamforming would no longer be capable of extracti ng usable model information, and the dramatically higher microphone selfn oise would make acous tic extraction through coherent power method s extremely difficult. 160 PAGE 161 Figure 51. Model installati on effects and background noise for a Mach number of 0.10. Figure 52. Model installati on effects and background noise for a Mach number of 0.17. 161 PAGE 162 Figure 53. Installation effect s and background noise for upper mic, Mach number of 0.10. Figure 54. Installation effect s and background noise for upper mic, Mach number of 0.17. 162 PAGE 163 Figure 55. Cepstrum compar ison of installation effects for a Mach number of 0.10. Figure 56. Cepstrum compar ison of installation effects for a Mach number of 0.17. 163 PAGE 164 Figure 57. Cepstrum comparison of installation effects for a Mach number of 0.17, where quefrency has been converted to an equiva lent lag distance using the measured speed of sound. 164 PAGE 165 Figure 58. Cepstrum comparison of installation effects for a Mach number of M = 0.10, for free field microphones mounted above the model trailing edge. 165 PAGE 166 Figure 59. Coherent power analysis of inarray B&K 4138 for a Mach number of 0.10. Figure 510. Coherent power analysis of in array B&K 4138 for a Mach number of 0.17. 166 PAGE 167 Figure 511. Coherent power analysis of oppositearray G.R.A.S. 40BE for M = 0.17. Figure 512. Coherent power analysis of fr ee field B&K 4939 for a Mach number of 0.10. 167 PAGE 168 Figure 513. Coherent power analysis of fr ee field B&K 4939 for a Mach number of 0.17. Figure 514. Coherent power analysis of fr eefield opposing G.R.A.S. 40BE for M = 0.17. 168 PAGE 169 Figure 515. COP confidence intervals for the free field case for a M ach number of 0.10. Figure 516. COP confidence intervals for the free field case for a M ach number of 0.17. 169 PAGE 170 Figure 517. Ordinary coherence function co mputed between the upper G.R.A.S. 40BE and lower B&K 4939 trailing edge microphones for the freefield case, for two different Mach numbers. 170 PAGE 171 Figure 518. Uncertainty bounds on installation effects for COP an alysis, Mach number of 0.10. Figure 519. Uncertainty bounds on installation effects for COP an alysis, Mach number of 0.17. 171 PAGE 172 Figure 520. Comparison of twoand threemicrophone methods for array case at M = 0.10. Figure 521. Comparison of twoand threemicrophone methods for array case at M = 0.17. 172 PAGE 173 Figure 522. Comparison of twoand threemicrophone methods for free case at M = 0.10. Figure 523. Comparison of twoand threemicrophone methods for free case at M = 0.17. 173 PAGE 174 Figure 524. Uncertainty bounds for differi ng methods for array case at M = 0.10. Figure 525. Uncertainty bounds for differi ng methods for array case at M = 0.17. 174 PAGE 175 Figure 526. Uncertainty bounds for differing methods for free field case at M = 0.10. Figure 527. Uncertainty bounds for differing methods for free field case at M = 0.17. 175 PAGE 176 Figure 528. Convergence analysis of Monte Carlo uncertainties for free field case of M = 0.10. Figure 529. Convergence analysis of Monte Carlo uncertainties for free field case of M = 0.17. 176 PAGE 177 Figure 530. Cdf of Monte Carlo results for free field case of M = 0.17 at 2,512 Hz. Figure 531. Cdf of Monte Carlo results for free field case of M = 0.17 at 3,392 Hz. 177 PAGE 178 Figure 532. Cdf of Monte Carlo results for free field case of M = 0.17 at 6,000 Hz. Figure 533. Cdf of Monte Carlo results for free field case of M = 0.17 at 12,000 Hz. 178 PAGE 179 Figure 534. Nominal and mean threemicrophone methods with confidence intervals, M = 0.10. Figure 535. Nominal and mean threemicrophone methods with confidence intervals, M = 0.17. 179 PAGE 180 Figure 536. Comparison of covariancebased f itting approaches for free field, M = 0.17. Figure 537. Variation of Frobe nius Norm Method solution for varying internal iterations. 180 PAGE 181 Figure 538. Variation of Rank1 Method solution for va rying internal iterations. Figure 539. Variation of Maxi mum Likelihood Method solution for varying internal iterations. 181 PAGE 182 Figure 540. Comparison of threemi crophone method and DAS for M = 0.17. Figure 541. Comparison of confid ence interval bounds for M = 0.17. 182 PAGE 183 Figure 542. Comparison of nominal and M onte Carlo mean solutions for M = 0.17. 183 PAGE 184 Figure 543. Beam map of test section at 1,024 Hz, M = 0.17, 0 degree AoA. Figure 544. Cdf of integrated Monte Carl o data at 1,024 Hz, M = 0.17, 0 degree AoA. 184 PAGE 185 Figure 545. Beam map of test section at 2,512 Hz, M = 0.17, 0 degree AoA. Figure 546. Cdf of integrated Monte Carl o data at 2,512 Hz, M = 0.17, 0 degree AoA. 185 PAGE 186 Figure 547. Beam map of test section at 5,008 Hz, M = 0.17, 0 degree AoA. Figure 548. Cdf of integrated Monte Carl o data at 5,008 Hz, M = 0.17, 0 degree AoA. 186 PAGE 187 Figure 549. Beam map of test section at 7,600 Hz, M = 0.17, 0 degree AoA. Figure 550. Cdf of integrated Monte Carl o data at 7,600 Hz, M = 0.17, 0 degree AoA. 187 PAGE 188 Figure 551. Beam map of test section at 8,800 Hz, M = 0.17, 0 degree AoA. Figure 552. Cdf of integrated Monte Carl o data at 8,800 Hz, M = 0.17, 0 degree AoA. 188 PAGE 189 Figure 553. Beam map of test section at 15,008 Hz M = 0.17, 0 degree AoA. Figure 554. Cdf of integrated Monte Carl o data at 15,008 Hz, M = 0.17, 0 degree AoA. 189 PAGE 190 Figure 555. Beam map of test section at 20,000 Hz M = 0.17, 0 degree AoA. Figure 556. Cdf of integrated Monte Carl o data at 20,000 Hz, M = 0.17, 0 degree AoA. 190 PAGE 191 Figure 557. 1/3rd octave method comparisons for a Mach number of 0.05 and 0 degree AoA. Figure 558. 1/3rd octave method comparisons for a Mach number of 0.07 and 0 degree AoA. 191 PAGE 192 Figure 559. 1/3rd octave method comparisons for a Mach number of 0.10 and 0 degree AoA. Figure 560. 1/3rd octave method comparisons for a Mach number of 0.12 and 0 degree AoA. 192 PAGE 193 Figure 561. 1/3rd octave method comparisons for a Mach number of 0.15 and 0 degree AoA. Figure 562. 1/3rd octave method comparisons for a Mach number of 0.17 and 0 degree AoA. 193 PAGE 194 Figure 563. 1/3rd octave method comparisons for a Mach number of 0.10 at 1.5 degree AoA. Figure 564. 1/3rd octave method comparisons for a Mach number of 0.10 at 1.5 degree AoA. 194 PAGE 195 Figure 565. 1/3rd octave method comparisons for a Mach number of 0.17 at 1.5 degree AoA. Figure 566. 1/3rd octave method comparisons for a Mach number of 0.17 at 1.5 degree AoA. 195 PAGE 196 Figure 567. 1/3rd octave method comparisons for M = 0.10, 0 degree AoA with offset array. Figure 568. 1/3rd octave method comparisons for M = 0.17, 0 degree AoA with offset array. 196 PAGE 197 Figure 569. Beam map of narro wband CSM at 2,512 Hz for offset array at M = 0.17, AoA = 0. Figure 570. Beam map of 1/3rd octave CSM at 2,500 Hz for offs et array at M = 0.17, AoA = 0. 197 PAGE 198 198 Table 51. Integrated arra y levels at 2,512 Hz for M = 0.17, 0 degree AoA for varying integration region dimensions. The nomina l integration region, spanning from x = 0.2 m to 0.2 m and y = 0.53 m to 0.53 m and shown in Figure 545 with the frequencys beam map, is listed first. xboundary, min (m) xboundary, max (m) yboundary, min (m) yboundary, max (m) SPL (dB ref 20 Pa, 16 Hz binwidth) 0.2 0.2 0.53 0.53 60.2 0.2 0.2 0.2 0.2 58.1 0.2 0.2 0.4 0.4 59.5 0.2 0.2 0.6 0.6 60.5 0.4 0.4 0.2 0.2 57.9 0.4 0.4 0.4 0.4 59.3 0.4 0.4 0.6 0.6 60.4 0.6 0.6 0.2 0.2 58.1 0.6 0.6 0.4 0.4 59.5 0.6 0.6 0.6 0.6 60.5 Table 52. Integrated arra y levels at 7,600 Hz for M = 0.17, 0 degree AoA for varying integration region dimensions. The nomina l integration region, spanning from x = 0.2 to 0.2 m and y = 0.53 m to 0.53 m and shown in Figure 549 with the frequencys beam map, is listed first. xboundary, min (m) xboundary, max (m) yboundary, min (m) yboundary, max (m) SPL (dB ref 20 Pa, 16 Hz binwidth) 0.2 0.2 0.53 0.53 42.5 0.2 0.2 0.2 0.2 Inf 0.2 0.2 0.4 0.4 34.7 0.2 0.2 0.6 0.6 44.4 0.4 0.4 0.2 0.2 Inf 0.4 0.4 0.4 0.4 38.0 0.4 0.4 0.6 0.6 48.0 0.6 0.6 0.2 0.2 Inf 0.6 0.6 0.4 0.4 40.2 0.6 0.6 0.6 0.6 50.1 PAGE 199 Table 53. Integrated arra y levels at 20,000 Hz for M = 0.17, 0 degree AoA for varying integration region dimensions. The nomina l integration region, spanning from x = 0.2 to 0.2 m and y = 0.53 m to 0.53 m and shown in Figure 555 with the frequencys beam map, is listed first. xboundary, min (m) xboundary, max (m) yboundary, min (m) yboundary, max (m) SPL (dB ref 20 Pa, 16 Hz binwidth) 0.2 0.2 0.53 0.53 40.4 0.2 0.2 0.2 0.2 38.4 0.2 0.2 0.4 0.4 38.9 0.2 0.2 0.6 0.6 40.6 0.4 0.4 0.2 0.2 40.0 0.4 0.4 0.4 0.4 40.1 0.4 0.4 0.6 0.6 42.0 0.6 0.6 0.2 0.2 40.0 0.6 0.6 0.4 0.4 38.4 0.6 0.6 0.6 0.6 41.9 199 PAGE 200 CHAPTER 6 CONCLUSIONS & FUTURE WORK The measurement of acoustic sources in wind tunnel testing is a co mmon task, and one in which many tools are available. Autospectral an alysis is of course the simplest but has no mechanism for decoupling the acoustic field of inte rest from measurement noise. Basic analysis tools, such as coherent powe r analysis formulated in tw o different ways in Equation (414) and Equation (417) provide an estimate of the coherent sign al field, but are eith er biased in power by measurement noise in the first case, or highly specific in thei r assumptions and application in the second. A slight increase in measurement complexity yields the threemicrophone method, Equation (424) which provides estimates of both signa l and noise strength. However, the assumptions inherent in this method limit its applicability in multisource measurements, as discussed in Appendix C. Gene ralized analysis techniques such as covariancebased fitting methods may provide more robust estimates for a single dominant noise source when more than three microphones are used in an experimental co nfiguration, but additiona l research is required to determine convergence parameters for the met hods when sources are distributed, as well as how to best leverage the methods in experiment al setup. Beamforming methods, such as the frequencydomain DAS b eamformer in Equation (437) provide the most information, but at the cost of experimental and processing complexity. Also, different algorith ms can yield different results. The data shown in the previous chapte r make one thing abundantly clear: measurement of trailing edge noise is not a straightforward task. The meth ods which are so often blindly used in acoustic analysis can have tremendous systematic uncertainties in their output, and only through careful consideration of these uncertainties and the p hysical setup can results be compared. 200 PAGE 201 In regards to the specifics of trailing edge noise measurement, cohe rencebased techniques appear to operate well on the ac tual airfoil trailing edge noise source, within their limiting assumptions given the number of microphones used but suffer severe penalties in frequency ranges where noise appears to come from di stributed background sources. Unfortunately, beamforming is required to assess the presen ce of other potential noise sources. Beamforming methods can provide more information and improve d results, but even they suffer from large uncertainty in output depending on measurement conditions. Standard beamforming methods also have no way of removing low frequency noise as the array resolution becomes poor. Also, high frequency aperture effects ma y be an issue. It is only through a combined effort of examining all the possible analysis methods that some understanding of the noise source behavior can be gained. The overall data acquired in this body of work appear to be in qualitative agreement with previous work. This is not necessarily due to the quality of the current or previous results, but to the large measurement uncertainties over so much of the bandwidth of interest. At the airfoil shedding peak and below, most uncertainty bounds are distinct and conclusions can be drawn about when and why some methods differ, for instance generalized coherent power underpredicting acoustic levels from beamforming or the threemicrophone method due to limitations in its formulation. However, at hi gher frequencies where th e background noise is not well behaved, uncertainties become large and a ll methods overlap solely because the confidence intervals occupy such a large lo garithmic power domain. Base d on the levels seen, it would appear that the airfoil is not generating appreciable noise wi thin this higher bandwidth. However, the background noise could just be loud enough to mask an interesting noise feature of the airfoil. 201 PAGE 202 From the present work, it appe ars that arraybased measurements provide the best, most wellbehaved results in determining trailing edge noise. The lower frequency bound on the arraybased methods is unfortunately a str ong limitation given the pr edicted bandwidth of trailing edge noise. To properly re ject contaminating noise at low frequencies, a larger array is necessary. Unfortunately the cost of using a larg er array is that sidelobe contamination becomes a concern at lower frequencies. While existi ng deconvolution approaches appear and warrant further investigation, as show n in Appendix E, the computati onal expense involved makes them prohibitive for routine use. As a preliminary recommendation, it appears that a hybrid approach could be used. Here, for frequencies where the ar ray spatial resolution is poor relative to the size of the wind tunnel test secti on, Appendix C would suggest th at the threemicrophone method would perform well at extracti ng the acoustic field of the measurement, although it would not provide information regarding the nature of the acoustic source. Array output could be considered once the array beamwidth becomes small enough to start rejecting acoustic energy from extraneous sources in the facility. It should be noted that, base d on Appendix C, the validity of coherent power methods is strongly dependent on the spanwi se behavior of the trailing edge. For a large number of incoherent sources along the tr ailing edge span, which based on the discussion in Chapter 2 could physically model the turbulent flow field in the vicinity of the trailing edge, spanwise microphone coherence breaks down more quickly as a function of reduced sourcetosource spacing. This indicates that a reduction in span wise correlation scales, for example due to the introduction of trailing edge se rrations [Howe 1991] would have an adverse effect on any spanwise coherence measurements. As the tr ailing edge noise source no longer exists along a single, centered line with resp ect to the measurement microphones, even a centerspan coherence 202 PAGE 203 measurement may underestimate acoustic levels at higher frequencies as long as the measurement microphones are near the model. Specifically, a degradation in ordinary coherence levels could occur which would cause coherent powerbased prediction methods to provide underestimates of the true acoustic field. For an ideal trailing e dge noise measurement with these t echniques, a facility should have noise sources far removed from the airfoil noise source, and measurement microphones far removed from the model. The coherencereducti on effects of Appendix C would be for the most part negated under these condi tions. While UFAFF is limite d in the overall measurement domain, it may be useful to consider removing the jet collector and forward part of the facility diffuser. This acoustic source appears to cont aminate a large portion of the lower frequency spectra, but moving the collector to the rear of UFAFFs an echoic chamber will reduce the maximum flow speed. Nonetheless, the noise re duction benefits may be worthwhile. Regarding sidewall noise, some additional investigation of th e sidewalls is warranted as beam maps indicate they are a major noise source from near 5 kHz to somewhere below 15 kHz. The acoustic absorption coefficient of the sidewalls used in the facility is plotted in Figure 61 Clearly, the absorption is nonideal at frequencies well with in the predicted bandwidth of this models trailing edge. In addition, th e porous nature of the sidewall s may allow for flow through the surface, introducing additional noise. A hybrid si dewall fabrication should be examined, where an acoustically treated thin su rface is used, backed with a bul k acoustic absorber with better absorption properties. This wall should be bounde d with impermeable mate rial to prevent large scale flow through the wall, and the f acility background noise reexamined. Regarding instrumentation for this idealized setup, it has clearl y been shown that a mediumsized platebased array contaminates the test section acoustic field beyond acceptable 203 PAGE 204 levels. This array design was selected based on convenience of installation for the WM61a microphones, as previous work with them in on freefield rod installati ons proved difficult. Also, the platebased array minimized the n eed for steering vector corrections, based on positional uncertainty of microphones. However, additional edgescattering issues were introduced and complicated the group microphone calibration proce ss, as discussed in Appendix D. For an ideal measurement, individuallycalibrated freefield microphones would be used, installed on acoustically treated rods. Unfortunately, these will suffer selfscattering issues at higher frequencies which are directivity dependen t, but for a model with a predicted bandwidth below 5 kHz such scattering would not be an issue. As long as these microphones could be accurately located in free space relative to the model, beamfo rming could be conducted without contaminating the model noise spectra with reflec tions. Additionally, larger arrays with nested subarray designs should be used. The larger array will localize noise sources and specify overall integration bounds, while the smaller ar ray will conduct the true source integration without suffering from source directiv ity effects. Currently such a design is infeasible in UFAFF due to DAQ system channel count limitations but should be evaluated for future work. Future work must leverage source behavior in the analysis of these measurement techniques. The data acquired here were co llected in conjunction with surface pressure fluctuations in the vicinity of the trailing edge, on both sides of the model. These pressure fluctuations, once processed appropr iately, can be used with sour ce model techniques to attempt to match far field noise predictions with measurem ents. Also, the surface pressure data must be collected in conjunction with inflow interroga tion techniques such as twopoint hotwire correlations, PIV, and/or LDV to gain a better un derstanding of how the ove rall flow field in the vicinity of the airfoil trailing edge relate s to the surface pressure fluctuations. This current body 204 PAGE 205 of research demonstrated that trailing edge noi se is a difficult measurement, and there can be tremendous ambiguity in acoustic results. The ne xt step in trailing edge noise analysis will require more source information to leverage in processing results. The next step in acoustic analysis is less clear. Fast deconvolution approaches must be investigated. A preliminary comparison between DAS and DAMAS is provi ded in Appendix E. SCDAMAS [Yardibi et al. 2008] should also be investigated and compared to baseline DAMAS results. DAMAS is still limited by the source assumptions used in DAS, but a more generalized, semicoherent source model for beamforming, such as DAMASC or MACS [Yardibi et al. 2010b] could be applied to the data to determine if any coherent source fiel ds are present, or if the results are driven by incoherent trailing edge (or side wall, diffuser etc.) sources. Th ese methods could be used in conjunction with surface pressure and flow correlation data to de termine over what bandwidth an aeroacoustic noise source is largely correlated, lending itself to cohere nt power analysis, and over what bandwidth it is largely uncorrelate d, lending itself to conventional beamforming approaches. It is only with this combination of source and field information that some of the measurement ambiguity present in aeroac oustic wind tunnel testing can be reduced. 205 PAGE 206 206 Figure 61. Acoustic absorption coeffi cient of sidewalls used in UFAFF. PAGE 207 APPENDIX A AIRFOIL COORDINATE DESIGN AND MEASUREMENT COMPARISON CODE function [s,e,MSE,t1n] = AFSNE(X,XM,x0,y0,theta) % AFE is a function to compute th e surfacenormal error of a final, % fabricated airfoil to its initial design coordinates. % This function is of the form % % [s,e,MSE,t1n] = AFSNE(X,XM,x0,y0,theta) % % Where X is a twocolumn matrix of the design airfoil coordinates, and XM % is the measured coordinates of the final airfoil, in the same format. % These matrices should be formatted as XFoil coordinates, where they % begin at the suctionside traili ng edge, wrap around the airfoil suction % side, around the leadi ng edge, and back to th e pressureside trailing % edge. Files output from FoilXPort.m are of the correct format. t1n has % been added as an exported quantity to allow for foil design % modifications based on curvefitted offset tables. t1n is the local % normal slope to the surface (in radians). % % x0 and y0 are x and y offsets to a pply to the measured coordinates, in % case there is a coordinate syst em offset from the design values. % Similarly, theta is a rotation applied to the measured coordinates, to % correct for rotational disparities, in radians. % % The output s is the surface distance along the airfoil from the % suctionside trailing edge, based on a linear approximation. e is the % error at each s location, e(s). MSE is the meansquare of e. % % Note that this code will assume that both the airfoil design coordinates % and the measured surface profile ar e reasonably smooth, and that they % are already positioned close to the final overlay. This only works % for a 2D case. If the trailing edge is significantly different between % the two profiles, the calculated error will be erroneous. Also, an % alternate form of this code, AF MSE, is written which only outputs the % meansquare error, MSE, for use with fmincon and other optimizing % software. % % Example: Given a design airfoil c oordinate set specified by the matrix % UF_Design, and measured coordinates of the final airfoil, UF_QA, both in % inches, a known offset of .05 inches in the xdi rection, .07 inches in % the ydirection and 1/60th of a ra dian (3 degrees) is applied. The % usage would be % % [s,e,MSE] = AFSNE(UF_D esign,UF_QA,.05,.07,1/60) % 207 PAGE 208 % to get the normal e rror around the airfoil. %% This cell block implements the desired offsets to XM. % Implement offsets XMo(:,1) = XM(:,1) x0; XMo(:,2) = XM(:,2) y0; % Implement rotation matrix G = [cos(theta) sin(theta); sin(theta) cos(theta)]; XMq = XMo*G; %% Calculate the slope of the design profile at all supplied coordinates % Central difference is used for a ll points except the beginning and end, % which use respective forwar d and backward differences. M = length(X(:,1)); t1a = zeros(1,M); t1a(1) = atan2(X(2,2)X (1,2),X(2,1)X(1,1)); for i = 2:(M1); t1a(i) = atan2(X(i+1,2 )X(i1,2),X(i+1,1)X(i1,1)); end; t1a(M) = atan2(X(M,2)X(M1,2),X(M,1)X(M1,1)); t1 = unwrap(t1a); % Calculate the slope of the line normal to the design airfoil surface at % each point. t1n = t1 pi/2; % Trim the trailing edge of the meas ured profile, if necessary. Apply a % rotation to the surface using th e surface normal angle to find which % points are beyond the surface profile. Retain the points just before % and after the profil e for interpolation. [Ytemp,Imin] = min(XMq(:,1)); XTtemp(:,1) = XMq(:,1) X(1,1); XTtemp(:,2) = XMq(:,2) X(1,2); RTtemp = [cos(t1n(1)) sin(t1n(1 )); sin(t1n(1)) cos(t1n(1))]; XTR = XTtemp*RTtemp; jj = 1; for i = 2:Imin; if XTR(i1,2) < 0; if XTR(i,2) > 0; jj = i1; 208 PAGE 209 end; end; end; XTtemp2(:,1) = XMq(:,1) X(M,1); XTtemp2(:,2) = XMq(:,2) X(M,2); RTtemp2 = [cos(t1n(M)) sin(t1n(M )); sin(t1n(M )) cos(t1n(M))]; XTR2 = XTtemp2*RTtemp2; kk = length(XMq(:,1)); for i = (Imin+1):length(XMq(:,1)); if XTR2(i,2) > 0; if XTR2(i1,2) < 0; kk = i; end; end; end; j = 0; for i = jj:kk; j = j + 1; XMp(j,:) = XMq(i,:); end; N = length(XMp(:,1)); % Calculate the streamwise distance s = zeros(1,M); smp = zeros(1,N); s(1) = 0; smp(1) = 0; for i = 2:M; s(i) = s(i1) + sqrt((X(i,2)X(i1,2))^2 + (X(i,1)X(i1,1))^2); end; for i = 2:N; smp(i) = smp(i1) + sqrt((XMp(i,2)XMp(i1,2))^2 ... + (XMp(i,1)XMp(i1,1))^2); end; %% Locate the two QA points closest to each design point, calculate the % line between those two points, and then the intersection point between % the line and the normal of the design point. Use this point to % calculate the local no rmal error. Note that for debugging, A and B are % indexed. If need arises, these values could be l ooponly. Once B is % determined as the closest point, angles are used to enforce A and B % occurring on opposite sides of the surface normal vector. A = zeros(M,2); B = zeros(M,2); As = zeros(1,M); 209 PAGE 210 Bs = zeros(1,M); J = zeros(1,M); for i = 1:M; A(i,:) = XMp(1,:); As(i) = smp(1); B(i,:) = XMp(2,:); Bs(i) = smp(2); for j = 2:N; if (sqrt((X(i,1)XM p(j,1))^2+(X(i,2)XMp(j,2))^2) < ... sqrt(( X(i,1)B(i,1))^2+(X(i,2)B(i,2))^2)); B(i,:) = XMp(j,:); Bs(i) = smp(j); A(i,:) = XMp(j1,:); As(i) = smp(j1); J(i) = j; end; end; end % To catch the points in erro r, apply a translation to each % local set, and then a rotation to set the surface normal to an angle of % zero. The OXB angle should have an opposite sign from the OXA angle. % If not, shift A and B forward by one point each. At = zeros(M,2); Bt = zeros(M,2); Ats = zeros(M,2); Bts = zeros(M,2); OXA = zeros(1,M); OXB = zeros(1,M); for i = 2:M1; At(i,:) = A(i,:) X(i,:); Bt(i,:) = B(i,:) X(i,:); rt = [cos(t1n(i)) sin(t1n( i)); sin(t1n(i)) cos(t1n(i))]; Ats(i,:) = At(i,:)*rt; Bts(i,:) = Bt(i,:)*rt; OXA(i) = atan2(Ats(i,2),Ats(i,1)); OXB(i) = atan2(Bts(i,2),Bts(i,1)); if J(i) < N1; if sign(OXA(i)) == sign(OXB(i)); A(i,:) = B(i,:); As(i) = Bs(i); B(i,:) = XMp(J(i)+1,:); Bs(i) = smp(J(i)+1); end; end; 210 PAGE 211 211 end; tb = zeros(1,M); theta2 = zeros(1,M); dl = zeros(1,M); for i = 1:M; tb(i) = atan2(B(i,2)A(i,2),B(i,1)A(i,1)); theta2(i) = atan2(A(i,2)X(i,2),A(i,1)X(i,1)); dl(i) = sqrt((X(i,1)A(i,1))^2+(X(i,2)A(i,2))^2); end; tb = unwrap(tb); theta2 = unwrap(theta2); phi = 2*pi tb (pi theta2); theta3 = t1n theta2; psi = pi theta3 phi; e = dl.*sin(phi)./sin(psi); MSE = mean(e.*e); PAGE 212 APPENDIX B ANALYSIS OF THE THREEMICROPHONE METHOD Formulation The system shown in Figure 430 is extended to more cohe rent outputs from a single source, contaminated by extraneous, incoherent lin e noise. The autospectral density of a signal is defined in Equation (411) but it should be noted that this is strictly only true when source and noise are incoherent, as the cross te rms between the two shown in Equation (B1) cancel out through averaging. ****1 2limiiyy iiiiiiii TGEUUUNNUN T N (B1) Similarly, the relation sh own for the crossspectral density given in Equation (412) only holds if the noise inputs are incoherent both with the source signal and with each other, cancelling the additional terms shown in Equation (B2). ****1 2limijyy ijijijij TGEUUUNNUN TN (B2) For m microphones, the autospectr al density of microphone i is reformulated in Equation (B3) 1 1iiiiiiiiyyuunnuu iGGGG SNR (B3) This, in conjunction with Equation (412) and Equation (413) allows for a restructuring of the problem in terms of measured coherence and channel SNRs. 22 211 11 1 1 1 1ij ij ij jj ii iijjyy yy uu ij yy yy yyyy i jGG SNRSNR G G GG SNR SNR 1 (B4) 211 11ijyy i jSNRSNR 1 (B5) 212 PAGE 213 Equation (B3) and Equation (B5) comprise a system with m + ( m 1)/2 equations for 2 m unknowns ( m autospectral densities and m SNRs). As summarized in Table B1 for two microphones, the number of unknowns (4) exceeds the number of equations, so the COP method must be used. When threemicrophones are us ed, an exact solution is possible. For m >3 microphones, the number of equations exceeds th e number of unknowns, and other options, such as a leastsquares constrained solution or covariancebased appr oaches, must be considered. Alternatively, the th reemicrophone method can still be applied, by evaluating different combinations of microphones within the data set. Simulation & Analysis A piston in an infinite baffle is simulated with a ka of 3, for nonuniform directivity pattern without true nulls [Blackstock 2000]. This model is selected due to its simple, analytic nature. The overall situation is similar to trailing edge noise measurements conducted in UFAFF, where microphones are not expected to be present in directivity nulls, but signifi cant power variation may occur over a line array of transducer s. The baseline pattern is shown in Figure B1 along with 19 sampling points selected to simulate microphone measurements. The piston is sized such that th e frequency of interest is 2048 Hz. The Sampling rate is set to 65,536 samples per second for 60 seconds, and indi vidual block length set to 4096, for a total of 3837 blocks, or 1995 effective averages with a hanning window and 75% overlap. Note that in subsequent analyses, even the zeronoise case was processed using a hanning window, despite the fact that the blocks were sized correctly fo r no leakage in processing a pure sinusoid. This was done to preserve peak behavi or between the zero noise case and the finite SNR cases. The microphones are placed in 10 degree increments from 90 degrees to 90 degrees around the field. Noise power was generated by using a Gaussian ra ndom generator with unity standard deviation to simulate a microphone measurement at each channel. A crossspectral matrix was generated 213 PAGE 214 for the noise signal, and then the autospectral no ise powers were used to determine the baseline signaltonoise ratio. The cros sspectral matrix (CSM) was then scaled to generate a desired signaltonoise ratio, and added to th e clean simulation of the acoustic CSM. Figure B2 Figure B3 and Figure B4 show the predicted directivity pattern using the threemicrophone method for several signaltonoise ratios (SNRs). The pred icted directivity is compared to the true directivity pattern, the di rectivity computed by a processed, clean signal autospectrum, and the directivity computed usin g the pure autospectral power. The two nearest microphones to the microphone of interest are select ed for the coherent power prediction. The plots show that with a su fficient number of averages, the nearestmicrophone method successfully recovers the directivity for a SignaltoNoise Ratio (SNR) of 10, and gets the overall shape correct for an SNR of 1. The method fails for an SNR of 0.1. It should be noted that simulations were run for shorter periods of time and with fewer averages, and resulted in output directivity patterns that were unresolvable for an SNR of 1. The SNR is defined for the frequency bin of interest only as this is a tonal simulation, as opposed to bandintegration for broadband noise as in true measurements in th e chamber. This was done to reduce the overall simulation complexity, since the threemicr ophone method is formulated for frequencybyfrequency analysis, so if the threemicrophone met hod can fail in a tonal situ ation, it can fail in a broadband one. Multiple references [Bendat & Piersol 2000; Chung 1977] discuss the threemicrophone method and derive it, showing that the solution is exact for incoherent noise, regardless of input signaltonoise ratio. One of th e assumptions of the solution must be violated for it to fail in predicting the true signal pow er. The first, and easiest to evaluate with this simulation, is that of uncorrelated noise contamination. 214 PAGE 215 The noise added with Matlabs random number generator is not completely uncorrelated between channels. Even a small value of noise cohere nce can contaminate th e method. It would appear that the contamination becomes a true problem when the coherence of the overall measured signal approaches the magnitude of the coherence of the contaminating noise. Figure B5 Figure B6 and Figure B7 show color maps of the crosssp ectral matrix coherence at the input frequency of interest, as compared to co lor maps of the simulated noise CSM. As the diagonal of the coherence matrix w ould be identically equal to unity, it is set to zero to increase the resolution of the color map for the rest of th e matrix. Not shown is the crossspectral matrix of the input signal without noise, which wa s verified to be universally unity. Given that the threemicrophone method begins to break down at lower signaltonoise ratios when selecting the n earest two microphones for each refe rence microphone, the next step in evaluating the method is to determine if a ny combination of microphones successfully recover the true power. To do this, a histogram is cons tructed of the power prediction of all possible microphone combinations for the midarray microphone and compared to the true signal power. Figure B8 Figure B9 and Figure B10 show these histograms for varying SNRs. The data spread gets large as the SNR becomes low, so the likelihood of an individual microphone combination being in error becomes high. However, for the most part, the data are still clustered around the true power value. A modified approach is taken to the computation of the th reemicrophone power estimate. All possible combinations of microphones are used and then the average power from all the estimates is computed for each microphone. The di rectivity pattern for this technique is shown in Figure B11 Figure B12 and Figure B13 The data for an SNR of 1 have improved, but SNR = 0.1 still s hows poor results. 215 PAGE 216 Based on the histograms, it is evident that the low SNR conditions have significant outliers present in the power predictions An outlier rejection method based on the modified ThompsonTau technique [Wheeler & Ganji 1996] can be a pplied to the power predictions, and the mean power for each microphone recalculated. These m odified directivity predictions are shown in Figure B14 Figure B15 and Figure B16 An SNR of 1 is clearly resolved, and even in the case of SNR of 0.1, the overall directivity shap e begins to emerge, at least enough to allow for qualitative discussion. A technique is required to determine when the mean threemicrophone method is trustworthy when the true noise data are unknown. The predicte d SNR ratio for the simulation can be plotted with respect to the true input SNR for microphone 10, and evaluated both without and with outlier rejection. This is shown in Figure B17 Without outlier rejection, significant estimate spread and breakdown is seen below true SNR values of 1. Once outlier rejection is applied, the mean value does not deviate signifi cantly from the true power on a log scale until approaching an SNR of 0.25. Tw o data characteristics are evident in this regime. First, the prediction spread becomes greater than an order of magnitude. S econd, the lowpower tail of the prediction cluster spreads further and further from the central cluster. These can be used as warning indications as to when mean thre emicrophone method predictions become suspect. For the entirety of the previous discussion, the microphone on the main lobe has been used as the reference microphone. This microphones stat istics would be considered the best case for the simulation, as it received th e strongest output from the sound s ource. One of the worstcase microphones, microphone 1 located at 90, is br iefly analyzed as another example case of outlier rejection. As shown in Figure B18 this method again begins to falter from the true power at SNRs below 0.5. However, for the most part the data trend ho lds near the true power 216 PAGE 217 until below SNRs of 0.05. Here, the data spread again passes an order of magnitude, and tails in the logscale SNR prediction become signi ficant even after outlier rejection. It should be noted that this SNR of 0.05 corresponds to the same case where microphone 10 has an SNR of 1, so a conservative appli cation of this technique may involve generating logarithmic tables of SNR predictions for ev ery microphone in the measurement system, and checking to see if any of them fail the potential criteria discussed above. If so, that microphone should be discarded from directivity analysis. Regarding the difference in SNR reliability of Microphone 1 vs. Microphone 10, it can be hypothesized that this is due to the availability of other highSNR microphones in the measurement. Microphone 10 may only be reliable down to an SNR of 0.5 to 0.3 because it has the highest true signal available within the directivity lobe, and therefore has no other stronger reference mi crophones for denoising. Microphone 1 may be reliable down to 0.05 because it can referen ce to the stronger microphones on the main body of the lobe. As the data rejection scheme is shown on a log scale, it may be helpful to evaluate the histograms of rejected data linearly to get a sense of the computed di stribution both with and without an outlier rejection scheme. Figure B19 and Figure B20 show histograms of the data without and with point re jection, to assess the behavior of th e scheme. As shown, the rejection methodology does little to eliminate lowend power estimates, but handles overestimated powers well. Finally, the threemicrophone method is co mpared against itself, using the nearest neighbor selection vs. using the mean power techni que with outlier rejecti on, as well as a twomicrophone coherent output power method, where the coherent power is just taken as the crosschannel coherence multiplied by th e autospectrum of the channel of interest. For the two217 PAGE 218 microphone method, the nearest adjacent microphone channel is used and the coherent output power is computed using the ge neral formulation of Equation (414) Figure B21 Figure B22 and Figure B23 show the directivity comparis on for varying true SNR values. As is shown on the plots, the meanmethod with outlier rejecti on for threemicrophones has the best performance, in th eory coming with greater computat ional cost. However, the threemicrophone method still runs in a trivial amount of time, especially when compared to beamforming algorithms, on a modern workstation using Matlab. As such, when a large number of source observation locations are available for a simple source unde r noisy conditions, the mean threemicrophone method with outlier rejecti on should be considered. As sources become more complicated, however, this method will fall vi ctim to the same effects as other coherencebased techniques. Appendix C goes into more deta il regarding the effect of more complicated source fields on coherencebased analysis. Figure B1. Piston in an infinite baffle, directivity pattern for ka = 3. 218 PAGE 219 Figure B2. Directivity patte rn for SNR = 10, using near esttwo microphone selection. Figure B3. Directivity patte rn for SNR = 1, using near esttwo microphone selection. 219 PAGE 220 Figure B4. Directivity patte rn for SNR = 0.1, using near esttwo microphone selection. Figure B5. Color maps of cro sschannel coherence, SNR = 10. 220 PAGE 221 Figure B6. Color maps of cro sschannel coherence, SNR = 1. Figure B7. Color maps of cro sschannel coherence, SNR = 0.1. 221 PAGE 222 Figure B8. Histogram of pow er predictions for SNR = 10. Figure B9. Histogram of pow er predictions for SNR = 1. 222 PAGE 223 Figure B10. Histogram of power predictions for SNR = 0.1. Figure B11. Directivity from mean power prediction method for SNR = 10. 223 PAGE 224 Figure B12. Directivity from mean power prediction method for SNR = 1. Figure B13. Directivity from mean power prediction method for SNR = 0.1. 224 PAGE 225 Figure B14. Directivity for m ean power prediction method with outlier rejection, SNR = 10. Figure B15. Directivity for m ean power prediction method with outlier rejection, SNR = 1. 225 PAGE 226 Figure B16. Directivity for m ean power prediction method with outlier rejection, SNR = 0.1. Figure B17. SNR data spread for microphone 10. 226 PAGE 227 Figure B18. SNR data spread for microphone 1. Figure B19. Histogram of estimated power s for microphone 10 and a true SNR of 0.5. 227 PAGE 228 Figure B20. Histogram of estimated power s for microphone 10 and a true SNR of 0.05. Figure B21. Directivity comparison for SNR = 10. 228 PAGE 229 Figure B22. Directivity comparison for SNR = 1. Figure B23. Directivity comparison for SNR = 0.1. 229 PAGE 230 230 Table B1. Equationunknown scal ing for varying channel count Number of Microphones, m Number of Unknowns, 2 m Number of Equations, ( m2+ m )/2 Solution Method 2 4 3 COP 3 6 6 Exact 4 8 10 LeastSquares Constrained, Microphone Subsets 22 44 253 LeastSquares Constrained, Microphone Subsets PAGE 231 APPENDIX C EFFECT OF MULTIPLE SOURCES ON COHERENCEBASED ANALYSIS For the most part in this body of work, the ma thematics of coherencebased analysis have been discussed in terms of a single cohe rent source, as shown schematically in Figure 430 for two output observations. As aeroacoustic source s can have complicated natures and distributed behavior, it is of interest to s ee how multiple coherent sources a ffect the output of these methods. TwoInput/TwoOutput Analysis A block diagram for a twoinput two output (TITO) system with no additive incoherent noise is shown in Figure C1 In this figure, the input acousti c sources as functions of time are denoted by 1 x t and 2 x t. The propagation path from the first source to the first observer is denoted as a functi on of frequency by 11Hf. The path from the first source to the second observer is denoted by This convention is maintained throughout this work, where the path from the i th source to the j th observer will be 12Hf ijHf. Similarly, the signal at the first observer due to the first source is denoted as a function of time by 1 1ut, and the signal at the first observer due to the s econd source is denoted as 2 1ut. In general, the signal at the j th observer due to the i th source will be i jut. Finally, the total measured signal at the first observer is and that at the second observer is 1yt 2yt. The only data which would be experimentally available is the measured data at the observers. All analysis in this discussion will be c onducted in the frequency domain. The Fourier transform of each source and observer signal can be defined using Equation (C1) where T is used consistently as in Chapter 4 as the data block length. 2 0 T jft X fxtxte dt (C1) 231 PAGE 232 Discretization effects will not be considered. Autoand crossspectral densities are computed using the expressions in Equation (47) and Equation (48) and the ordinary coherence function is computed from Equation (49) At this stage of analysis, the coherence between the sources, 122 xx f is undetermined. From systems analysis [Bendat & Piersol 2000], the propagation path or impulse response function woul d be convolved in the time doma in, so the frequencyresponse function, the Fourier transform of the impulse response function, is multiplied in the frequency domain as in Equation (C2) i ji jiUfHfXf (C2) For all subsequent equations, fre quencydependence is suppressed. Autospectral Scaling The autospectral density of the signal at obs erver 1, suppressing the block length limit to show the estimate instead of the true value, is shown in Equation (C3) 11 11 11 11221221 11 11111111* *1 21 2 11 1111 1*11*22*12*2 11111111 1*1 1*2 2*1 2*2 11 11 11 1122 2 2222yy yy yy yy uuuuuuuuGEYYEUUUU TT GEUUUUUUUU T GEUUEUUEUUEUU TTTT GGGGG (C3) The observed autospectral density is the sum of autospectral dens ities of the sources and their crossspectral densities. These cr ossspectral densities can be reexpressed as phasors, shown in Equation (C4) where 12 11uu would be the phase angle betw een sources 1 and 2 observed at observer 1. 12 11 1212 1111 21 11 2121 11 11 uu uuj uuuu j uuuuGGe GGe (C4) 232 PAGE 233 For real signals i x t and j y t, the identities in Equation (C5) and Equation (C6) hold. 12 21 11 11uuuuGG (C5) 12 21 11 11uuuu (C6) The coherence between the two obs erved signals can be expressed as the coherence between the two sources. 12 12122112 11 1111 1111 12 11 112211221122 1111111111112 2uu uuuuuuuu uu uuuuuuuuuuuuG GGGG GGGGGG (C7) 11 11 11 11 22 22 22 112 1*1 ** 11 1111111111 11 2 2*2 ** 11 2122122121 2122 22xx xx uu x xx uuGEUUEHXHXHHGHG TT GEUUEHXHXHHGHG TT x (C8) 12 12 11 21 21 111*2 ** 11 1112121121 2*1 ** 11 212111211122 22 x x uu x x uuGEUUEHXHXHHG TT GEUUEHXHXHHG TT (C9) 12 11** 21211111 2uuHHHH 2112** 11112121xxxxGG HHHH 12 12 1122 11222 2xx x x xxxx xxxxG GG GG (C10) This in turn can be used to reexpress the crossspectral density magnitude in Equation (C11) 12 1122 12 11 11112xx uu uuuuGG G (C11) Substituting Equation (C11) into Equation (C5) while substituting Equation (C5) and Equation (C6) into Equation (C4) and finally substituting Equation (C4) into Equation (C3) yields Equation (C12) 12 12 11 11 1122 1122 11 12 1111 11112uu uujj yy xx uuuu uuuuGGGeeGG (C12) 233 PAGE 234 This equation can be modified using Eulers Formula in Equation (C13) yielding a final form shown in Equation (C14) 12 11 12 12 11 11 12 11 12 12 11 11cossin cossinuu uuj uu uu j uu uuej ej (C13) 1122 11 1111 12 12 11 11 cossinyy uuuu uu uuGGG j 12 12 11 11cossinuu uuj 1122 12 1111 1122 12 1122 11 12 1111 11 11112 22cosxx uuuu yy xx uuuu uu uuuuGG GGG GG (C14) In essence, the power measured by observer 1 is equal to the sum of the individual source contributions and the coherent power between the two sources. Limiting cases of behavior can be evaluated. If the sources are pe rfectly coherent, with and in phase, meaning that the phase angle 1221xx12 110uu Equation (C14) limits to Equation (C15) 1122 1122 11 1111 11112yy uuuu uuuuGGGGG (C15) This can be further simplified by assuming the sources have equal power contribution at the observer location, as in Equation (C16) 111111 11 11224yyuuuu uuuuGGGGG (C16) Here, the coherence reinforces the signal, such that two equa l inputs quadruple the measured power of a single source. This is sensible since a doubling of signal level would yield a quadrupling of power for a single selfcoherent signa l. If instead the sources are assumed to be perfectly coherent, equal power and out of phase with a phase angle of 12 11uu Equation (C14) reduces to Equation (C17) 234 PAGE 235 111111 1122yyuuuu uuGGGG 0 (C17) Coherent, out of phase sources (where the phase angle between the sources is ) of equal power cancel perfectly. Finally, if the sources are assumed to be incoherent such that 1220xx then Equation (C14) simplifies to Equation (C18) 1122 11 1111yy uuuuGGG (C18) The autospectral density measured at observer 1 is simply the sum of the individual source power spectral densities propagated to the observer. CrossSpectral Scaling The crossspectral density between meas urements at observers 1 and 2 from Figure C1 is given in Equation (C19) 12 12 12 11122122 12 12121212 11 12 1* *1 21 2 12 1122 1*11*22*12*2 12121212 1*1 1*2 2*1 2*2 12 12 12 1222 2 2222yy yy yy yy uuuuuuuu yy uGEYYEUUUU TT GEUUUUUUUU T GEUUEUUEUUEUU TTTT GGGGG GG 11 12 21 22 12 12 12 12 12 21 22 21 21 21 2uu uu uu uujjjj uu uu uu ueGeGeGe (C19) Leveraging similar techniques as th ose used in the autospectral sc aling allows for the use of the coherence between the sources to simplify the crosssource terms. 12 1122 12 12 11222 xx uu uuuuGGG (C20) 21 2211 12 12 11222xx uu uuuuGG G (C21) Equation (413) can be used for the remaining two terms. 11 1111 12 1122uuuuuuGGG (C22) 235 PAGE 236 22 2222 12 1122uuuuuuGGG (C23) Without knowledge of the propagation path behavior and the relative phases of the sources, the phase angles within Equation (C19) cannot be simplified. The overall expression for the crossspectral density is restated in terms of indivi dual autospectral densitie s and phase angles in Equation (C24) 11 22 12 12 1111 2222 12 1122 1122 12 21 12 12 1122 2211 12 12 1122 112222 uu uu uu uujj yy uuuu uuuu j xx xx uuuu uuuuGGGeGGe GGeGGe j (C24) The first and second terms can be interpreted as the individual source cont ributions to the cross spectrum, while the third and fourth terms are based on source coherence, and can behave similarly to the crosssource terms in the autosp ectral analysis for limiting cases of unity and zero coherence. Coherent Output Power Behavior The fundamental contributions of each source to the overall, noiseless measured values have just been discussed. Now the effects of th ese contributions to an analysis on a noisy signal must be evaluated. Figure C2 shows the modified setup. Without additional source information or more advanced analysis methods, there is no way to decouple th e two sources using a coherent output power analysis. The method is simply used to remove the individual channel noise terms, and and analyze the acoustic field wh ich is coherent between the two microphones, denoted as and 1nt2nt1wt 2wt. In the pressure domain these add linearly such that the resultant desired parameters of interest are given in Equation (C25) and Equation (C26) 12 111wtutut (C25) 12 222wtutut (C26) 236 PAGE 237 Unfortunately, analysis must occur in terms of power in the frequency domain, so the additive behavior is coherencedepende nt, as demonstrated in prev ious sections. The acoustic autospectral density is thus given in Equation (C27) and Equation (C28) and is the key parameter desired using c oherencebased analysis. 1122 12 1122 11 12 1111 11 111122cosww xx uuuu uu uuuuGGG GG (C27) 1122 12 1122 22 12 2222 22 222222cosww xx uuuu uu uuuuGGG GG (C28) As the noise terms are uncorrelated, the measured autospectral densities are simply the acoustic autospectral densities plus the individual noise autospectral densities, shown in Equation (C29) and Equation (C30) 111111yywwnnGGG (C29) 222222yywwnnGGG (C30) Based on Equation (412) the crossspectral density can be expressed in Equation (C31) where is simply equivalent to Equation 12wwG(C24) 1212yywwGG (C31) It must now be determined if Equation (414) and Equation (417) behave as expected under twosource conditions. To analyze Equation (414) the behavior of the ordina ry coherence function between observers 122yy must be evaluated. As it is evident that this will contain a significant number of terms, the general coherent output po wer can be reexpressed in Equation (C32) using Equation (C30) and Equation (C31) 12 12 1211 22 222222 2 1 yy ww yyyy yywwnnGG COPG GGG (C32) 237 PAGE 238 This equation must then be evaluate d to see if the identity in Equation (C33) which would yield the coherent power at observer 1, biased low by the opposing observers SNR as shown in Equation (414) holds. Note that for equations where the statement is to be validated, is replaced with to indicate an equivalency check is necessary. ? 12 11 11 22 2222 222 2? 1 1 1ww ww ww nn wwnn wwG GG G GG SNR G (C33) This can be evaluated by substituting from Equation (C24) and Equation (C30) 12 12 12 12 11 12 2222 22 22 221122 22 2 21 ?? 1 1ww ww ww ww ww ww wwnn ww ww wwwwGG GG G GGGGGG SNR 1 (C34) The formulation becomes a test of the cohere nce function when two sources are present. 12 1212 12 11221122 11 22 12 21 12 12 12 12 1111 2222 1122 2211 12 12 1122 1122 1122 1122 1122 12 112 12 1111 11 112 2 22 22cosuu uu uu uuww wwww ww wwwwwwww jjj xx xx uuuu uuuu uuuu uuuu xx uuuu uu uuuG GG GGGG GGeGGeGGeGGe GG GG j 2 11 11 22 12 21 12 12 12 12 1111 2222 1122 2211 12 12 1122 1122 1122 1122 1122 12 1122 12 2222 22 222222 22cosuu uu uu uuu jj xx xx uuuu uuuu uuuu uuuu xx uuuu uu uuuuGGejGGeGGeGGe GG GG j (C35) In general, this is intractable without making simplifying assumptions. For the simplest case, the sources can be treated as perfectly in coherent. In such a case, Equation (C35) can be simplified to Equation (C36) 11 22 11 22 12 12 12 12 1111 2222 1111 2222 1122 1122 1122 1122 12 11221122 11112222 11112222 11112 1212 11221122 11222 11222cosuu uu uu uu uuuujjj uuuu uuuu uuuu uuuu ww uuuuuuuu uuuuuuuu uuuuuGGeGGeGGeGGe GGGG GGGG GGG j 222 1122 1111222211222211 1122112211221122uuu uuuuuuuuuuuuuuuuG GGGGGGGG (C36) In general, the cosine term in the numerator of this equation will reduce the coherence of the signal. Therefore, for incoherent sources, the general COP method will not hold. However, for 238 PAGE 239 further simplification, the cosine term can be ev aluated at unity. This is equivalent to the schematic in Figure C3 where the phase angle between observers for source 1 is equal to the phase angle between obs ervers for source 2. 11112222 11112222 11221122 11221122 12 1111222211222211 112211221122112222uuuuuuuu uuuuuuuu ww uuuuuuuuuuuuuuuuGGGGGGGG GGGGGGGG (C37) Finally, if all of the s ource powers at the observer locations ar e equal, the coherence function in Equation (C37) can be reduced one last time. 122 2 22 4 1 4uuuuuuuuuuuuuuuu uu ww uuuuuuuuuuuuuuuu uuGGGGGGGG G GGGGGGGG G (C38) This shows that for perfectly incoherent source s of equal power with eq ual distance to observer locations, the general coherent power method will behave as expected. Specifically, with proper microphone placement, if trailing edge noise were considered a line of incoherent sources, the general coherent power method woul d still function as expected. However, if the geometry of the source arrangement becomes more complicated such that the phase angle difference term returns, the coherence function will underpredict from what is expected. Also, when source coherence is introduced, additiona l complications occur. This would indicate that the method would handle distributed source fields with comp licated coherence, such as boundary layer noise on the sidewalls, poorly. This procedure can also be applied to the dipoleassumption based coherent power method of Equation (417) as shown in Equation (C39) 12 11?wwwwGG (C39) 239 PAGE 240 12 11 22 12 12 1111 2222 1122 1122 12 21 12 12 1122 2211 12 12 1122 1122 1122 12 1122 12 1111 11 111122 2 ? 2cosuu uu uu uuww jj uuuu uuuu j xx xx uuuu uuuu xx uuuu uu uuuuG GGeGGe GGeGGe GG GG j (C40) Again, the solution to this is complicated. Again, it can be simplified by assuming incoherent sources. 11 22 12 12 1111 2222 1122 12 1122 1122 1111?uu uujj ww uuuu uuuu uuuuGGGeGGeGG (C41) As a dipole assumption is already applied to the data, the phase angles in Equation (C41) can be evaluated. 1122 1212uuuu (C42) 11 22 12 121uu uujjee (C43) Again, if equal source powers are assumed, the equation properly reduces to the assumed behavior, as shown in Equation (C44) 122ww uuuuuuuuuuuuuuGGGGGGG G (C44) As with the general coherent power method, the dipole assu mptionbased coherent power method should behave as expected under the same conditions set for the general coherent power method, in addition to the assumption that the sources are incoherent dipoles, again schematically shown in Figure C3 if the sources are considered as dipoles instead of monopoles. It will be shown in subsequent discussion that in coherent dipole sources will behave similarly to incoherent monopoles regarding co herent power analysis. For ge neral distributed, partiallycoherent source fields, the crossspectral magnitude will not properly predict the microphone autospectra due to the phasor summation. 240 PAGE 241 General Monopole MultipleInput /MultipleOutput Behavior The TITO analysis presented previously s hows that depending on the source nature and arrangement with respect to a measurement, a coherencebased analysis method may predict incorrect levels due to source coherence underprediction. Unde r certain limiting circumstances, however, the methods may predict the correct acou stic levels. Validati on of these methods under a more generalized condition is necessary to determine if underprediction is likely in trailing edge noise measurements. Problem Formulation Generalized equation scaling as conducted in the previous sec tion would be difficult and is beyond the scope of the discussion in this appendix. Instead, a specific physical problem will be formulated in terms of a multipleinput/multipleoutput (MIMO) system and then analyzed. Trailing edge noise is a func tion of, among other things, the spanwise coherence of the boundary layer turbulence as shown in Equation (219) and Equation (221) If it is assumed that this correlation length scale is small relative to the wetted span of an airfoil, the trailing edge noise source can be simplified, for the purposes of this discussion, to be a line of uncorrelated sources with a spacing of this corre lation length scale. A schematic of this idealized source setup is shown in Figure C4 while the associated block diagram is shown in Figure C5 For this stage of discussion, all sources are treated as monopoles. Observers ar e considered to be idealized microphones. This threemicrophone se tup is analogous to the spanwise microphone setup shown in Figure 445 The equation for the pressure field generated by a harmonic monopole operating at a single freq uency, adapted from Blackstock [Blackstock 2000], is given in Equation (C45) 0000',,Re cos 44qjtkr qAA prtqe tkr rr (C45) 241 PAGE 242 Here, p denotes the acoustic pressure fluctuation at location r from the source at time t. The source is fluctuating at 0 radians per second, with an associated wavenumber defined via Equation 0k(213) with as the acoustic speed of sound in meters per second. A is the acoustic source strength, here treated as an arbitrary constant set to unity. 0cq is the randomized source phase. To construct a set of in coherent sources, a random variab le is necessary such that the average crosspower between the sources is zero. This phase randomization is introduced for such a purpose. The mathematics of this will be evident in the subseque nt discussion. This phase angle in reality would likel y be some function of the tur bulent correlation time scale, but for this discussion will be treated as a function of da ta block number q. The analysis is treated such that a continuous data set is available which is broken up into Q total blocks of data, each T seconds long. The timedomain expression from Equation (C45) can be converted into a frequencydomain expression of the monopole sound field th rough a finite Fourier transform in Equation (C46) 0 00 0 0000 0 0000 0 00'',,',, ',,cos 4 coscossinsin 4 coscossinsincossin 4 cos coscos 4T jt T jt q T jt qq T qq qpPrqprtqedt A Prq tkredt r A tkrtkredt r A tkrtkrtjt r A kr tt r dt 00 00 0000 00cos cossin sin sincossin sinsinTT q TT qqdtjkr ttdt kr ttdtjkr ttdt (C46) 242 PAGE 243 Orthogonality can be applied such th at the function is zero for all except 0 (and 0 but this analysis only deals with singlesided spectra, so only positive frequencies will be discussed, and power will be doubled where appropriate). 22 00000 00 00 0 00 00',,coscossinsin 4 11 cos sin2sin sin2 42 4 2 4TT qq TT qqA Prq kr tdtjkr tdt r At t kr tjkr t r 0 (C47) As periodicity is assumed over the block length T, Equation (C47) reduces to Equation (C48) 0000',,cos sin 42 28qjkr qqAT T AT Prq krjkre rr (C48) From this equation, the output at microphone 1 due to source m is expressed in Equation (C49) Microphone 2s output is expressed in Equation (C50) and Microphone 3s in Equation (C51) Note that here the subscript of has been updated to reflect its dependence on source as well as block number. 22 010 22, 8mqjkmxh mAT Uq e mxh (C49) 22 020 22, 8mqjkmxh mAT Uq e mxh (C50) 22 030 22, 8mqjkymxh mAT Uq e ymxh (C51) The total pressure field for each microphone is thus expressed in Equation (C52) Equation (C53) and Equation (C54) 22 010 10 22, 8mqjkmxh M mMATe Yq Nq mxh (C52) 243 PAGE 244 22 020 20 22, 8mqjkmxh M mMATe Yq N mxh q (C53) 22 030 30 22, 8mqjkymxh M mMATe Yq N ymxh q (C54) Autoand crossspectral densities will now be considered. The autospectral density of source m at microphone 1 is expressed in Equation (C55) Here, each blocks phase angle term identically cancels. 11 22 0* 01 1 222 2 8mm mqmm uu jkmxhGEUU T AT e QT mxh 22 01 22 8mqQ q jkmxhAT e mxh 2 22 2 32 AT mxh (C55) Microphones 2 and 3 are construc ted similarly in Equation (C56) and Equation (C57) 222 0 2 232mmuuAT G mxh 2 (C56) 332 0 2 232mmuuAT G ymxh 2 (C57) The crossspectral density between two sources at a given micr ophone is expressed by Equation (C58) 244 PAGE 245 11 22 22 2222* 01 1 22 1 22 22 2 8 8 32mn mq nqmn uu Q jkmxh q jknxh jkmxhnxhGEUU T AT e QT mxh AT e nxh ATe Q 2222 2 1mqnqQ j qe mxhnxh (C58) The variables mq and nq are defined as uncorrelated random variables, so the phasor term mqnqje should vary randomly between 1 and 1 on the real axis, and j and j on the imaginary axis. For a sufficiently large number of blocks Q the summation term should approach zero, validating this method of defining incoherent monopole sources, which should have crossspectra which are identically zero, as given in Equation (C59) The same holds for the crossspectra between individual s ources at microphones 2 and 3. 11 22 33000mn mn mnuu uu uuGGG0 (C59) The autospectral density of each microphone signal can now be constructed (assuming noise is uncorrelated with input). 22 0 11 22 0 11 110 22 1 11 22 2 22 22 8 8 1 32mq nq mmQ MM jkmxh yy qmMnM jknxh M nn uu mM mAT Ge QT mxh AT eN N nxh AT GG mxh 111111M nnwwnn MGGG (C60) 245 PAGE 246 Equation (C60) simplifies as such because it was shown in Equation (C58) and Equation (C59) that summation terms are zero for mn The autospectral densities for microphones 2 and 3 are given in Equation (C61) and Equation (C62) 22 22 222222 222 0 22 21 32mmMM yy nn nnwwnn uu mM mMAT GG G mxh G G G (C61) 33 33 33 33 33332 0 22 21 32 mmMM yy nn nn uu mM mM wwnnAT GG ymxh GG G G (C62) As expected for incoherent sources, the overall so urce power field is equiva lent to the sum of the individual power contri butions of each source. Crossspectral densities must now be considered. The crossspectral density between the contribution of source m at microphone 1 and source m at microphone 2 is derived in Equation (C63) 12 22 0* 01 2 222 2 8mm mqmm uu jkmxhGEUU T AT e QT mxh 22 01 22 8mqQ q jkmxhAT e mxh 2 22 2 32 AT mxh (C63) As shown previously, for equalpower monopoles w ith equidistant observers the crossspectral density will reduce to the auto spectral density. However, microphones 1 and 3 and microphones 2 and 3 do not form equidistant pairs. The fo rmulation for these crossspectral densities, computed the same way as that between microphones 1 and 2, is more complicated. 246 PAGE 247 2222 0 132 0 222 232mmjkmxhymxh uuATe G mxhymxh 2 (C64) 2222 0 232 0 222 232mmjkmxhymxh uuATe G mxhymxh 2 (C65) In Equation (C64) and Equation (C65) a phase offset can exist in each of the crossspectral density terms. As with the autospectral dens ity computations, any cr osssource/crossobserver crossspectral densities will be equal to zero. As such, the overall crossspectral density between each observer pair is computed in Equation (C66) Equation (C67) and Equation (C68) 12 12 120 22 21 32mmMM yy ww uu mM mMAT G mxh G G (C66) 2222 0 13 13 132 0 2 222232mmjkmxhymxh MM yy ww uu mM mMATe GG mxhymxh G (C67) 2222 0 23 23 232 0 2 222232mmjkmxhymxh MM yy ww uu mM mMATe GG mxhymxh G (C68) As has been demonstrated fo r a single source in Equation (413) the coherence of a single sources signal across two microphones is unity. 12 13 23222 001mm mm mmuu uu uu0 (C69) The total crosschannel cohe rence between microphones 1 an d 2 is given in Equation (C70) 247 PAGE 248 12 12 11 22 11 22 12 12 11 112 2 2 0 22 22 2232 32 32 32 32 32 32mm nn pp mm rr nn pM uu mM yy MM nn nn uu uu nM pM MM uu uu mM rM M nn uu u nMAT G AT AT GGGG AT AT GG AT AT GGG 22 22pM nn u pMG (C70) In the limiting condition of no incoherent line noise, this reduces to Equation (C71) 1222 22 22 2 0 22 22 2211 32 32 1 11 32 32MM mM rM yy MM nM pMAT AT mxh rxh AT AT nxh pxh (C71) The coherence between microphones 1 and 2 in the absence of incoherent line noise reduces to unity, indicating that these two microphones will properly predict local coherent source power when using coherencebased analysis methods. This is consiste nt with the TITO system shown in the previous section. However, when one of the microphones, microphone 3, is offset, the scaling is more difficult. For noiseless conditions, the coherence is given in Equation (C72) 13 13 13 13 13 1133 11 33* 2 22 0mm rr nn ppMM uu uu ww mM rM yy ww MM wwww uu uu nM pMGG G GG GG (C72) The numerator term can be expanded as in Equation (C73) 248 PAGE 249 2222 13 2222 0 22 042 2 4 2222 2222 42 41024 1024jkmxhymxh M yy mM jkrxhyrxh M rM jkmxhAT e G mxhymxh e rxhyrxh AT e 22 2222 02222 2222 ymxh MM mMrM jkrxhyrxhmxhymxh e rxhyrxh (C73) Given the nature of the problem, no assumption can be made with regard to the scaling of m x or y to h so no further simplifi cation may be possible. Simulation Instead, an example will be used similar to an experimental setup in UFAFF (neglecting flow and shear layer effects). Here, the micr ophone distance from the trailing edge will be h = 1 m, and the microphone spacing will be y = 0.178 m. This horizontal spacing is slightly less than that shown in Figure 445 but has been used in previous wo rk. As the correlation scales for the model are currently unknown, a correlation le ngth scale/source spacing will be selected as x = 0.02 m, as this is the grid density used in beamforming analysis. The model wetted span of 44 gives a value for M of 28 (rounding up) for a total of 57 sources. The source strength A will be taken as unity, and the block length will be T = 0.0625 sec as is often used in UFAFF spectral processing. Solving for the coherence between microphones 1 and 3 as a function of frequency yields the curve shown in Figure C6 As previouslydiscussed, cohe rent power methods are dependent on the noiseless source field of interest having a coherence of unity. This plotted coherence 249 PAGE 250 function shows that if microphones are oriented along the span of a line of sources, and one of those microphones, microphone 3, is not centered along that line, a dram atic reduction in the coherence function occurs as frequency increases. This is due to the phasor summation in the crossspectral magnitude term of Equation (C73) As such, both twomicrophone coherent power methods, Equation (414) depending on the coherence function and Equation (417) depending on the crossspectral magnitude, w ould dramatically underpredict the true autospectral levels at higher frequencies, as based on Equation (C55) and Equation (C60) the autospectral density levels of th e sources are frequencyindependent. Based on Equation (423) the effect this would have on threemicrophone predictions is less straightforward. If microphones 1 and 2 are observing the same field, which would be true for being positioned on equal, opposite sides of a line of incoherent monopoles as in Figure C4 their coherence in a noiseless measurement will be unity as shown in Equation (C71) Also, as both microphones are observing the same signal, the coherence between microphones 1 and 3 will be the same as that between microphones 2 and 3. This would indicate that when computing the SNR values for microphones 1 and 2, the error from the coherence underprediction would ideally divide out, although realis tically the nulls in the visibl e hump structure could lead to dividebyzero issues. Only microphone 3s SNR prediction would suffer, as is shown in Figure C7 where microphone 3s true autospectral dens ity is compared to its threemicrophone prediction. This means that under ideal cond itions, as long as the first two microphones are placed carefully with respect to a known source field, a third microphone can have suboptimal source location and still allow for good SNR estimates of the first two. In the case of trailing edge noise, the third microphone can have some o ffset as long as the first two are centered on opposite sides of the trailing edge. The dipolelike behavior of the trailing edge source is not 250 PAGE 251 expected to have a significant effect on this behavior beyond what is seen with the monopoles, as these computations are dependent on the crossspectral magnitude, so the phase difference between the upper and lower su rfaces should not contribute. Having analyzed the effects of offsetting an observer along the source line span, the observer can now be offset dept hwise. This is analogous to the actual threemicrophone method used in this dissertation, wh ere the third microphone is sli ghtly downstream from the upper microphone, as shown in Figure 437 The source schematic for this setup is shown in Figure C8 The autospectral and crossspectral densities for each sources contribution to the third microphone must be slightly updated based on the new geometry. 332 0 22 232mmuuAT G mxzh 2 (C74) 22222 13 232 00 2222 232mm mmjkmxhmxzh uu uuATe GG mxhmxzh 2 (C75) The updated source contributions from Equation (C74) and Equation (C75) can be substituted into Equation (C72) and the coherence function recomputed using the same values as previously selected, except with y z The resultant cohere nce plot is shown in Figure C9 This case shows far improved behavior, with less than 5% coherence loss by 20 kHz. Centering the third microphone along the span of sources and locating it slightly offset in the z direction has dramatically improved the performance of the coherence pred iction, and in this case a dividebyzero condition is no longer of concern in the threemicrophone method. The corresponding power prediction for microphone 3 is shown in Figure C10 This result would also indicate that small positional errors in microphone 1 and microphone 2 placement would 251 PAGE 252 have a small effect on coherence breakdown based on multiple sources, as long as the microphones remain in the plane which bisects the line of incoherent sources. In an effort to evaluate the severity of th is spanwise effect, an additional simulation is conducted using a set of eight microphones, as shown in Figure C11 This spanwise arrangement of microphones, with the same spaci ng of 0.178 m as used previously, mirrors a previouslyacquired data set in UFAFF involving a DU 96W180 airfoil. All previous formulations for the monopole field are used with the appropriate substitutions of 2 y and 3 y for y depending on the microphone location. The resu ltant coherence plots for several of the microphones are shown in Figure C12 The local autospectral density predictions are shown in Figure C13 as computed using the threemicrophone method referencing microphones 1 and 5. As shown, for typical length scal es within UFAFF, increasing span wise offset of a measurement microphone causes increased coherence breakdown due to phasor cancel lation in the crossspectral magnitude. An increase is noted at high frequencies for 122 yy with a corresponding increase in predicted power The behavior may be part of some largescale spectral periodicity, but this cannot be know n without further investigation, which is not conducted at this time. 22uuGDipole Analysis and Comparison with Experimental Data While it was previously assumed that a line of incoherent dipoles would behave similarly to a line of incoherent monopoles, this a ssumption should be verified. The timedomain expression for a harmonic dipole [Dowling & Ff owcs Williams 1983] is adapted in Equation (C76) 000 2cos1 ',,,Re 4qjtkrjk f prtq e rr (C76) 252 PAGE 253 Note the additional angular dependence Also note that this angle is 2 offset from that used in Figure 21 distinguishing these ideali zed dipoles from the true trailing edge coordinate system. The dipole strength is f The timedomain expression can be expanded, as in Equation (C77) 0 00 00 2 000000 00 00 22cos1 ',,,Re cos sin 4 cos sin cos Re 4 cos sin qq qq qqjk f prtq tkrjtkr rr jktkrktkr f rr tkrjtkr rr 000 00 2coscos cossin 44qqftkrf ktkr rr (C77) As with the monopole analysis, this expressi on can be converted to the frequency domain through a finite Fourier transform in Equation (C78) 00 0 000 0 0000 0 000cos cos ',,, 4 sin cos1 coscossinsin 4 sincosT q jt T jt q T jt qqtkr f Prq edt rr ktkredt f tkrtkre rr ktkr dt 00 0 0000 0 00000 0cossin cos1 coscossinsin 4 cossin sincoscossin T jt qq T qq T qqtkredt f tkrtkr rr tjtdt ktkrtkr cossin tjtdt (C78) 253 PAGE 254 Again, orthogonality cancels the integrals between sines and cosi nes, as well as those where 0 frequency mismatches, leaving Equation (C79) 0 2 00 0 0 22 000 0 00 2 00 0 0 0 0cos cos ',,, cos 4 sin sin cos sin sin cos cos cos 1 sin2 42 4T q TT q q T q qkr f Prq tdt rr jkr tdtjkkr tdt r kkrtdt kr ft rr 0 0 0 0 0 0 00 0 0 0 00 0 0 0sin 1 sin2 24 1 cos sin2 24 1 sin sin2 24 cos 4T T q T q T qt jkr t t r t jkkr t t kkr t f 000 0000 0 00 0 2cos sin 22 cos sin 22 cos cos sin 422 cos 1 8qqq qq qq jkrTkrjTkr rrr jkTkrkTkr jkT fT krjkr rr fT jkre r (C79) The output at microphone 1 due to source m can thus be defined in Equation (C80) by substituting Equation (C81) 254 PAGE 255 22 0 22 022 22 10 0 22 22 0 3 22 2,1 8 1 8mq mqjkmxh m jkmxhh fT mxh Uq jkmxhe mxh fTh jkmxhe mxh (C80) 2cosh mxh 2 (C81) The autospectral density contribution of each source is thus given in Equation (C82) 11 22 0 22 0* 01 1 22 0 3 12 2 2 22 0 3 22 22 1 2 8 1 8mm mq mqmm uu Q jkmxh q jkmxhGEUU T fThjkmxh e QT mxh fThjkmxh e mxh 22 0 22 022 3 22 2 22 0 1 22 0 32 1 1mq mqQ jkmxh q jkmxhfTh Qmxh jkmxhe jkmxhe 22 222 0 3 22 21 32fThkmxh mxh (C82) If the original, more generali zed coordinate system is de sired, the output at microphone i due to source m can instead be stated in Equation (C83) 255 PAGE 256 000 2cos ,1 8m imqm jkr i mm i m ifT Uq jkre r i (C83) Here, m i is the angle from source m to the ith microphone, and is the distance from source m to the ith microphone. The corresponding autospectra l density contribution is given in Equation m ir(C84) using the same methodology as shown in Equation (C82) 2 222 0 0 4 2cos1 32mm iimm ii uu m ifT kr G r (C84) Similarly, the crossspectr al density between microphones i and j due to source m is given in Equation (C85) 0 000 2 1 0 2 2 22 2cos 2 1 8 cos 1 8 coscos 32 m imq mm ij m jmqm Q jkr i m i uu m q i m jkr j m j m j mm ij mm ijfT Gj k r QT r fT jkre r fT rr e 0 02 000 2 22 2 2 0001 cos cos 64 1mm ij mm ijjkrr mmmm ijij mmmm ijij mm ij jkrr mmmm ijijjkrjkrkrre fT rr jkrjkrkrre (C85) Equation (C84) and Equation (C85) can be substituted into noiseless, generalized forms of Equation (C60) and Equation (C66) to construct the total autospectral and crossspectral density at the i th microphone and between the i th and j th microphones, given respectively in Equation (C86) and Equation (C87) 256 PAGE 257 0mm ii ii iiM yy ww uu mMGGG G (C86) 0mm ij ij ijM yy ww uu mMGG (C87) With these formulae, identical analysis can be conducted as with the monopole sources for a constant dipole strength f of unity, with the source and microphone locations defined in Figure C11. The resulting coherence functions are shown in Figure C14 and look similar to the monopole behavior. The corres ponding threemicrophone power predictions are shown in Figure C15 Note that the powerlaw based increase s hown for the true autospectral densities of the microphone measurements is due to the term in the numerator of the autospectral and crossspectral density solutions. Note that as the true autospectral density scales with the square of frequency, the bounds of the oscillatory beha vior in the threemicrophone prediction appear reasonably flat. A comparison of the ratio of th e predicted power to th e true power, shown in 2 0kr Figure C16 appears similar to the monopole behavior. The rolloff follows a roughly 6 dB per octave line. The highfrequenc y increase in predicted power for microphone 2 is present with the dipole sources, as well. Finally, a comparison is made between monopol e simulations, dipole simulations, and real experimental data for a DU 96W180 airfoil at 0 degree AoA a nd a Reynolds number of 1e6. The microphone locations for the DU 96 match those from Figure C11 where the trailing edge of the airfoil is colocated with the line running through the acoustic source centers. The microphone spacing is 0.178 m. The DU96 airfoil is oriented and installed similarly to the NACA 63215 model shown in Figure 437 but the spanwise microphone array is below the model in the test section instead of above it. The real data were acqui red at 65,536 samples per second using UFAFFs PXI chassis and proces sed with a Hanning window. The coherence 257 PAGE 258 between microphones 1 and 4 is shown in Figure C17 While the data do not overlap perfectly, the shape and behavior of the experimental data is very close to the simulated data, indicating that the mechanisms driving the experimental resu lts could definitely be those derived in this appendix. One open question for future work fro m this plot is whether an inverse problem formulation can be applied to experimental data in an attempt to determine the source fields correlation length scale. Such a discussi on is beyond the scope of this appendix. Summary Derivations of coherencebased analysis have been conducted for generalized sources in a TITO system. These derivations showed that under specific conditions, which match the idealized installation conditions for twomicrophone measurements in previous work [Brooks & Hodgson 1981; Hutcheson & Brooks 2002], the cohere nt power methods behave as expected. These ideal installation conditions are defined su ch that the microphones ar e at a models center span and directly above and below the models trailing edge. Similarly, the threemicrophone method is shown to, under ideal inst allation conditions, pr operly extract the c oherent power field for centered microphones. Simulations for incohere nt lines of sources agree with derivations. However, when microphones are no longer ideall y centered or equallyspaced with respect to source fields, it appears cancellation effects in crossspectral density source summations force coherent power methods, both twoand threemicrophone, to dr amatically underpredict true coherent field levels. This cancellation structure is observed in both simulated and experimental data. Such results make even measurements conducted with ideal insta llation conditions suspect, as while the installation locations are ideal with re spect to the airfoil trailing edge noise source, they are not ideal with respect to other background noise sources Cancellation effects may be present when the facility backgr ound noise is similar to or grea ter than trailing edge noise in level, leading to unreliable power estimates fr om coherencebased techniques for distributed 258 PAGE 259 source regions. The structure of this coherence breakdown is observed to be highly dependent on both the source and observer locations, and ma y be related to compactness conditions where the acoustic wavenumber k is small with respect to the measurement dimensions. This compactness condition would induce an effective upper frequency limit on coherencebased measurement techniques. Figure C1. Schematic of TwoInput/Two Output system with no additive noise. Figure C2. Schematic of TwoInput/TwoOutput system with incoherent measurement noise. 259 PAGE 260 Figure C3. TITO situation where, for monopol e sources, all phase angle differences cancel. 260 PAGE 261 Figure C4. Threeobserver MIMO system mode ling a trailing edge of incoherent sources. Here, the third observer is offset span wise along the simulated trailing edge. 261 PAGE 262 Figure C5. Threeobserver MIMO block diagram for the analys is of the system shown in Figure C4. 262 PAGE 263 Figure C6. Coherence between microphone s 1 and 3 based on the schematic in Figure C4 Figure C7. Threemicrophone prediction for microphone 3 from Figure C4 263 PAGE 264 Figure C8. Threeobserver MIMO system mode ling a trailing edge of incoherent sources. Here, the third observer is offs et depthwise into the page. 264 PAGE 265 Figure C9. Coherence between microphones 1 and 3 based the schematic in Figure C8 Figure C10. Threemicrophone prediction for microphone 3 from Figure C8 265 PAGE 266 Figure C11. Eightobserver MIMO system modeling a trailing edge of incoherent sources. Here, the lower observers are offset spanwi se along the simulated trailing edge from the central, fifth observer. 266 PAGE 267 Figure C12. Coherence function for sp anwise microphones based on schematic in Figure C11 Figure C13. Threemicrophone predic tions for microphones 2 through 4 in Figure C11 267 PAGE 268 Figure C14. Coherence for spanwise microphones, dipole sources located in Figure C11 268 PAGE 269 Figure C15. Threemicrophone solution, microphones 2 through 4 in Figure C11 for dipoles. 269 PAGE 270 Figure C16. Ratio of predicted pow er to true power for microphones in Figure C11 for dipoles. 270 PAGE 271 271 Figure C17. Comparison of expe rimental coherence with simula ted source coherence between microphones 1 and 4 from Figure C11 PAGE 272 APPENDIX D ARRAY CALIBRATION IN THE PRESENCE OF ECHOES In the use of aeroacoustic arrays, array calibra tion is a necessity if one desires to extract quantified information from beam maps. This calibration may compensate for steering vector errors due to positional uncertainty of the microphones, and it may substitute for individual magnitude and phase calibration of each ar ray microphone when an appropriate reference microphone is provided [Dougherty 2002]. Howeve r, all microphones in the array are required to see the same calibration signal. If they see an additional sour ce which is incoherent with the calibration signal, the beamforming algorithms may still correct the phase angles of the steering vectors, but the magnitude of the reference microphone will be in error. If the additional sources are coherent, e.g. from reflections of the calib ration signal, the magnitude and phase calibrations will be in error. Due to the difficulty of insitu, reflectionfr ee calibration in some aeroacoustic facilities, an analysis technique which allo ws for the detection and filtering of these contaminating signals would be helpful, at least for postprocessing ca libration data which are found to be in error, when rerunning calibrati on experiments is infeasible or impossible. Ideal Case Problem Statement For initial discussion, an a rray will be treated as a simp le twomicrophone measurement, rather than a large channel count multiarm logarith mic spiral array, as addressed later. In an ideal case, these two microphones are at known locations from a monopole source, in an environment with a preciselymeasured speed of sound. The first microphone is treated as a reference microphone with an ideal response. The second microphone has an unknown response function. The objective of this idealized experiment is to determine the second microphones frequency response function. A schematic of this experiment is shown in Figure D1 A block diagram of the signal path is shown in Figure D2 where x t is the ideal source output, 1Hf 272 PAGE 273 and are the ideal propagation frequency respons e functions as subsequently defined, is the unknown microphone response, and 2HffmH 1 y t and 2 y t are the final measured signals from microphones 1 and 2. Once the s econd microphones respons e function is known, data collected could be corrected by divi ding by the response functions magnitude, and subtracting its phase shift. Ideal Case Analysis Assuming ideal farfield monopole behavior, the observed ideal acoustic pressure field can be defined in Equation (D1) [Blackstock 2000], a generalized version of that defined in Equation (C45) 04 r xt c r prt (D1) As in prior analysis, is the isentropic speed of sound. The equivalent frequencydomain expression for a harmonic source is adapted from Equation 0c (C48) in Equation (D2) 4jkr X fe f r Pr (D2) The wavenumber k is defined in Equation (213) The pressure field at each microphone can thus be formulated in Equation (D3) and Equation (D4) 11Yf14jkrXfe r (D3) 22Yf24jkr mHfXfe r (D4) Although the measurement is noiseless and so a direct calculation would work, formulation for an optimal Wiener filter is still used for consistency in discussion. All power spectral 273 PAGE 274 densities are given as twosided, as subsequent spectral analysis and modification are computed on twosided functions. However, the factor of two in the denominator due to the twosided nature of Fourier transforms of sines and cosines is omitted. For ideal functions, the autoand crossspectr al densities can be expressed in Equation (D5) through Equation (D8) *1xxSfXX T (D5) 11 11* 11 2 22 1 11 16 4jkrjkr xx yyS XXee SfYY Tr Tr (D6) 22 222 ** 22 2 22 2 21 16 4jkrjkr mxx mm yyHS HHXXee SfYY Tr Tr (D7) 12 12 12* 12 2 12121 4416jkrr jkrjkr mm x yyHXXeeHSe SfYY TTrrrr x (D8) The relationship between the two microphone signals can be leveraged to calculate the frequency response function of microphone 2, as shown in Equation (D9) and Equation (D10) 12 12 11mxx yy yy yyHS S Hf S 122 116jkrre r 2 216 r 2 1r xxS 121 2 jkrr mr He r (D9) 21 122 1 jkrr my yr HfHe r (D10) Single Ideal Reflection Figure D3 and Figure D4 schematically show the effect of adding an ideal reflective surface to the calibration experi ment. The autocorrelation of microphone 1 will be computed and transformed into its autospectral density to demonstrate the effect of adding a reflection to a 274 PAGE 275 general signal. Then, assuming broadband, timeharmonic monopole behavior, the remaining power spectral densities will be computed. The signal at microphone 1 can now be expresse d as the sum of the two signals, where the ideal reflecting surface is treated as creating an image source of the monopole, such that the reflected wave is treated as tr avelling a total distance (and unde rgoing spherical spreading) of This is given in Equation 21r (D11) 11 21 0 1 11 2144 rr xtxt cc yt rr 0 (D11) The autocorrelation of a stationa ry signal is defined in Equation (D12) [Bendat & Piersol 2000]. 01 limT xx TR xtxtdtExtxt T (D12) Here, the expected value operator occurs as a time average of the product of the delayed signal to the baseline signal. The auto correlation of the first microphone is computed in Equation (D13) 1111 21 11 21 0000 11 21 11 214444yyrrrr xtxtxt xt cccc RE rrrr (D13) Assuming a sufficiently long record, a change of variables can be applied, defining 11 0' r tt c 275 PAGE 276 112111 2111 00 11 21 11 21 2111 0 22 2 11 1121'' '' 4444 '' '' 16 16yyrr rr xt xt cc xt xt RE rrrr rr xtxt c xtxt EE rr r 2111 2111 2111 00 22 2111 21'''' 16 16 rr rrrr xtxtxtxt cc EE rr r0 2c (D14) Again applying a change of variables within individual expected value operations and solving, the final form of the autocorrelation is constructed in Equation (D15) 112111 2111 222 1121 1121 0 01111 16yy xx xx xxrrrr RRRR rrrrc c (D15) The autospectral density of a signal is the Fourier transform of its autocorrelation, shown in Equation (D16) [Bendat & Piersol 2000]. 2 jf xx xxSfRed (D16) Using this identity, the autospec tral density of the first micro phone is constructed in Equation (D17) 112 222 1121 22 2111 2111 1121 0 0111 16 1 jf yy xx jf jf xx xxSf Red rr rr rr RedRe rr c c d (D17) The integration variable in the second and third terms can be changed, giving Equation (D18) 276 PAGE 277 11 2111 2111 002 222 1121 2' 2'' 1121 222 1121111 16 1 ''' '' 111 16jf yy xx rr rr jf jf cc xx xxSf Red rr RedRed rr rr 2111 2111 0022 1121 2111 222 11211121 01 1112 cos2 16rr rr jf jf cc xx xx xxSfeeSf rr rr fSf rrrrc (D18) In addition to the magnitude offset from the reflection, the autospectr al density shows an oscillatory behavior in the ma gnitude response with a period dependent on the speed of sound and difference in ray lengths, regardless of the na ture of the stationary waveform. If a harmonic source waveform is assumed, this can also be shown through frequencydo main construction of the power spectral de nsities in Equation (D19) through Equation (D22) 11 211 112144jkrjkree Yf Xf rr (D19) 12 222 122244jkrjkr mee YfHf Xf rr (D20) 12 22 12 22 222 22 12221222 2 2212 222 122212221 4444 112 cos 16jkrjkr jkrjkr yy m xx m xxeeee SfYYH S Trrr H krkrS rrrr r (D21) 11 21 12 22 12 1112 1122 2112 2122* 12 2 11211222 2 11121122211221221 16 16jkrjkrjkrjkr m yy xx jkrrjkrrjkrrjkrr m x xH eeee SfYY S Trrrr H eeee S rrrrrrrr (D22) 277 PAGE 278 As shown, autospectra and crossspectra are co ntaminated by real and complex oscillatory behavior, dependent on lag times. While and (the direct path le ngths) are well known for this calibration problem, the reflected path lengths may not be. Equation 11r12r (D23) and Equation (D24) show the results of attempting to solv e for the second microphones response function. 1112 1122 2112 2122 12 12 111112112221122122 2111 22 11211121112 cosjkrrjkrrjkrrjkrr m yy yy yyeeee H rrrrrrrr S Hf S krkr rrrr (D23) 12 1112 1122 2112 21222111 22 11211121 1112112221122122112 cosmy y jkrrjkrrjkrrjkrrkrkr rrrr HfH eeee rrrrrrrr (D24) Without the reflected path lengths, the microphone response cannot be determined, as the reflection adds terms to both the magnitude a nd phase of the computed frequency response function. Attempting to solve for the microphone response using the known direct distances will yield the erroneous estimate of the response in Equation (D25) 1211 12 1112 1122 2112 2122 1211 122212 11 1112112221122122 12 11 2111 22 11211121 12 11 22 112 cos 1 jkrr m yy jkrrjkrrjkrrjkrr m jkrr jkrr mr HfHe r eeee H rrrrrrrr r e r krkr rrrr rere r H 2111 121121221112 21 2122 2 1111 2111 2 212112cosjkrr jkrrjkrrrree rr r rr krkr rr (D25) 278 PAGE 279 Analysis Options Filtering in the frequency domain cannot solve this problem, as the contaminating terms occupy the same bandwidth as the function of inte rest. One potential option is to window the autocorrelation function and cros scorrelation function, as the refl ections occur at different time delays from the original function occurrence. If the original function is approximately white noise, its autocorrelation is a de lta function, so correlation windowi ng should be straightforward. However, in many situations the signal is bandlimited, so the autocorrelation has a finite duration. If this duration overlaps the time de lay of the reflection, the reflection cannot be windowed without significantly altering the baseline functions correla tion functions, and subsequently its spectral density functions. On e option is to inverse transform the computed frequency response, edit the reflections out of it, and forward transform. Another is cepstral analysis (or alanysys, following cepst ral nomenclature) [Randall & Hee 1981]. Impulse Response Analysis The second microphones impulse response functi on referenced to the first microphone can be constructed by computing the inverse Fourie r transform of the frequency response function from Equation (D23) as shown in Equation (D26) 12 12 1112 1122 2112 2122 0001 0 2222 1112 1122 2112 2122 2111 22 0 11211121 112 cos2 yy yy m rr rr rr rr jf jf jf jf ccchtHf ht eeee rrrrrrrr rr f c rrrr jfedfd 0c (D26) 279 PAGE 280 This effective impulse response function is si mply a convolution of the microphones impulse response with a series of delta functions at each combination of lag times due to the contaminating reflections. Assuming the microphone s frequency response function is broad, its impulse response function should be sharp, allowing for a boxcar window to be placed around the primary peak in the impulse response function, corresponding to the time given in Equation (D27) 1211 0rr t c (D27) This windowed impulse response function can be forwardtransformed into the frequency domain. It can then be analyzed as in the ideal, noreflection case to solve for the second microphones frequency response function. Cepstrum Alanysis Computing a cepstrum of a signal is a form of homomorphic analysis where signals which are convolved or multiplied in some domains are tr ansformed to be additive in another [Randall & Hee 1981]. For example, a signal which is convolved in the time domain is shown in Equation (D28) 0 tythxtd (D28) This convolution becomes multiplication in the frequency domain, shown in Equation (D29) 0 tYfythxtdHfXf (D29) If the logarithm of each side is taken, the product of the input and frequency response function becomes a sum. While these two signals may stil l overlap in the frequency domain, the inverse Fourier transform of this logarithm can be take n. Depending on the nature of these functions, 280 PAGE 281 they may be easily separable in this new quefrency domain. This is demonstrated in Equation (D30) and Equation (D31) loglogloglog YHXH X (D30) 111logloglog YH X (D31) Note that strictly speaking, there are two types of cepstral alanysis. In the first, the autopower spectrum is converted to a logarithmic scale and i nversetransformed. This can be edited and transformed back into the frequency domain, but as it only operates on real power spectra, no crosspower spectra can be evaluated, prev enting its use in this study. A second method exists where the complex logarithm of each term is computed, given in Equation (D32) [Randall & Hee 1981]. lnlnlnj X XeXj (D32) When a function is analytically known, this is straightforward. However, when dealing with discrete data, a phaseunwrapping method is required else the inverse transform of this function will be nonsense. A phaseunwrapping/wrapping scheme is implemented in the MATLAB function cceps and its inverse icceps. Note also that when computing an inverse Four ier transform of a finite, discrete data set, aliasing occurs in the cepstral computation. This is due to the nonlinearity introduced by the logarithmic operation generating harmonics of th e base spectral content of the signal, well beyond the Nyquist sampling requireme nts of the inverse transform. This can lead to erratic behavior of the signal in the quefrency domain and must be kept in mind when filtering (or liftering). This can be a voided through a polezero factoriz ation method of a wellbehaved spectrum, also available in MATLAB, but the factorization method is dramatically more expensive from a computational perspective. 281 PAGE 282 The second ideal case, with a single reflection, is restated in logarithmic form in Equation (D33) 112111 22 11211121 0112 lnlnln cos2 2ln4yy xxrr SS f rrrrc (D33) This equation consists of three terms. The first is the logarithmic autospectral density of the input function, and the last is a constant. The ce ntral term is the natural logarithm of an offset cosine function. The cepstrum of the sp ectrum can then be computed in Equation (D34) 11 1112 2 2111 22 11211121 0 2ln ln ln 112 ln cos2 2ln4jf jf yy yy xx jf jfSSedfSedf rr2 f edf rrrrc edf (D34) The last term in the cepstrum is the inverse Fourier transform of a constant. 22ln4 2ln4jfedf (D35) If it is assumed that the input au tospectral density is true white noise, its cepstrum also becomes the inverse transform of a constant, as given in Equation (D36) 2ln lnjf xx xxSedfS (D36) The remaining term can be expressed as the series expansion of the natural logarithm [Abramowitz & Stegun 1965]. 2311 ln111... 23 zzzz (D37) The expansion in Equation (D37) assumes that 11 z and 0 z which for illustrative purposes can be assumed to be satisfied with appr opriate choice of ray lengths. This expansion can then be shown in Equation (D38) 282 PAGE 283 2111 2111 22 22 11211121 0 11211121 0 2 2111 22 11211121 0 2 11112 112 ln cos2 cos2 1 1112 cos2 1 2 111 3 rr rr ff rrrrcrrrrc rr f rrrrc rr 3 2111 2 211121 02 cos2 1... rr f rr c (D38) This expansion acts as a constant offset summed with an infinite set of harmonics. These harmonics share a fundamental periodicity in the frequency domain, given in Equation (D39) 0 0 2111c f rr (D39) When the inverse transform of this term is computed, it will manifest as a series of delta functions. One will occur at a quefrency of zer o, corresponding to the constant offset. The remaining delta functions will occur at quefrencies which are harmonics (or rahmonics), given in Equation (D40) 2111 0 0rr c (D40) Therefore, the influence of the reflection, as ide from the offset at zero quefrency, can be removed by multiplying the power cepstrum by zero for quefrencies above a certain cutoff of interest. The cepstrum computa tion has effectively separated the envelope of the input power spectrum from the influence of the reflections. The current discussion holds for a white noise input, but if the input were white noise a correlation window would al so prove satisfactory. This ap plication must again involve the frequency response function, where the input powe r spectral term has been removed, shown in Equation (D41) 283 PAGE 284 1112 1122 2112 2122 121112112221122122 2111 22 11211121lnlnln 112 ln cosjkrrjkrrjkrrjkrr yy meeee HH rrrrrrrr krkr rrrr (D41) Here, the cepstrum of the function will consist of the cepstrum of the reflections and the cepstrum of the second microphones response. Th e cepstrum of the reflection terms will again consist of delta functions (or in the case of the imaginary terms, delta distributions) located mostly at higher quefrencies for the dimensions of a typical aeroacoustic facility. The cepstrum of the second microphones respons e function should be concentrated at lower quefrencies for a given measurement bandwidth, modeling it as a canonical secondorder system, for example. For such a case, most of the influence of the re flections should be rem ovable through liftering, where certain quefrency values are set to zero by multiplying the cepstrum by a window function, and inversetransforming the frequency response function. Using the alternative of the frequency response between the source and seco nd microphone, this can be expressed in Equation (D42) 12 22 21222lnlnln ln4jkrjkr xy mee HH rr (D42) Note that in subsequent usage, the complex cep strum will be used whenever crossspectral or crosscorrelation term s are of interest. Single Ideal Reflection: Simulation A simple simulation is now performed, using an input typical of that used in the University of Florida Aeroacoustic Flow Facility (UFAFF) for calibration, and typical reflection length scales and speeds of sound. The second microphone is simulated as an underdamped, canonical second order system with a corner frequency of 16,000 Hz and a passband magnitude of unity. 284 PAGE 285 The damping ratio is set to 0.5 for a moderate re sonance peak. While this is not necessarily how a good measurement microphone would perform, it a dds an interesting feature to the frequency response function (or FRF) which the filtering tec hniques must resolve. The total response and phase angle of the microphone frequenc y response are given in Equation (D43) and Equation (D44) respectively [Cattafesta 2010]. Th e magnitude response is shown in Figure D5 The phase angle is shown in Figure D6 2 2 21 12j m ccHf e ff ff (D43) 1 22 tan 1c cf f f f f k (D44) The input waveform is a Schroeder multisine, which is a periodic broadband signal defined in Equation (D45) through Equation (D47) [Simon & Schoukens 2000]. 1cos2F kk kxtAft (D45) 0 k f kf (D46) 1kkk F (D47) Here, the fundamental frequency is selected as 16 Hz. The waveform is designed to have uniform spectral content from 304 Hz ( k = 19) to 24,304 Hz ( k = 1519), simulating the passband of the speaker system used in UFAFF for sy stems calibration. Waveform magnitude is normalized to a peaktopeak level of unity. The waveform is plotted in Figure D7 Residual noise with a power magnitude of below the passband signal is added to the remaining 1210 285 PAGE 286 signal bandwidth to improve performance of the cepstral computations, as problems were noted when zeros occurred on the unit circle of a data sequences Ztransform. The power spectral density of the multisine is plotted in Figure D8 The distance from the first microphone to the source is set to 1 m, simulating a typical array distance from a source in th e facility test section. The distance from the first microphone to the second is 0.559 m, to match the dist ance from the center microphone to the outermost microphone in the facilitys large aperture phased array. The remaining distances are given in Equation (D48) through Equation (D50) when the reflecting plane is treated as a test section sidewall, 0.559 m horizontally from th e source (opposite the second microphone). 22 121.5591.146 m r (D48) 22 212.5.5591.500 m r (D49) 221.952 m r (D50) The speed of sound is set to 345 m/s. Data ar e simulated as acquired at 102,400 samples per second. A single block of data spanning 1/16th of a second (6400 points) is analyzed. To simulate an ideal delay of the periodic input, th e records are circularly shifted by an appropriate amount at each microphone. The resultant power spectral densities (or PSDs) of microphones 1 and 2 are shown in Figure D9 and Figure D10 The corresponding frequency response function magnitude and phase estimates for the microphone are shown in Figure D11 and Figure D12 compared to their true values. As seen in Figure D13 and Figure D14 in both forms of analysis the reflections are easily separable from the initial signal. Here, gamnitude is simply the cepstral term for magnitude, and is a dimensionless logarithmic quantity. Note that in the impulseresponse function, reflections appear as offset and scaled forms of the initial peak, as would be expected 286 PAGE 287 when the microphone impulse response is convolv ed with a delta sequence at the reflection times. However, in the cepstral plot, while th e initial peak resembles the impulse response function, the secondary reflection peaks are delta functions, since the cepstrum of the microphone impulse response is added to the refl ection delta sequence inst ead of convolved with it. For both methods, a boxcar window is applied to the primary peak of the plot. Several sizes of boxcar window are applie d, although all are sufficiently sm all to cut out any reflection terms. Note that while the impulse response is onesided, and thus onl y the beginning of the sequence need be passed, the cepstrum has some twosided characteristics. As such, the cepstral window must be shifted to pass the end of the cepstral sequence, as shown in Figure D15 For simplicity in these simulations, the same shifted window will be applied to both the cepstrum and the impulse response function. In true analys is applications a more optimum method may be desired. Window sizes of 1.2 ms, 1.6 ms, 2.0 ms, 2.3 ms and 2.7 ms are used. Both methods come sufficiently close to pred icting the frequency response function that plotting direct overlays will not give a sense of their accuracy. The relative magnitude error of the two methods for different window sizes is plotted in Figure D16 and Figure D17 The phase error is plotted in Figure D18 and Figure D19 Overall, the impulse response windowing appears to perform better than the cepstrum wi ndowing, although for the small window case the tail of the base impulse response function was cli pped, leading to worse performance. It would appear that more care is necessa ry in applying the window to impulse response functions than to cepstral functions. Instead of centering the wi ndow at zero time, a boxcar with some small sample lead should be applied near the p eak of the primary impulse response peak. 287 PAGE 288 Real Experimental Conditions The simulated data show that both im pulse response windowing and cepstrum windowing/liftering have the capacity to remove echoes from a signal and solve for an unknown frequency response function. This study must now be extended to real experimental conditions. First, the major components of a real experiment will be shown schematically, and the problem formulated. Then, acquired data will be analyzed. Figure D20 shows a schematic of the experimental setup used in array calibration. A Schroeder multisine is generated by a function gene rator, run through an amplifier to a speaker. The speaker is mounted to an exponential horn, which is designed to minimize internal reflections while reducing the e ffective source area, such that a compact monopole behavior can be expected over the frequency range of operati on of the speaker [Blackstock 2000]. Note that this horn still has some internal reflections whic h will manifest in the output signal. Also, some small nonlinearities may exist in the amplifier and speaker output when the power is set high enough for desired signal strength. The effective acoustic source at the exit of th e hornspeaker assembly has three displayed paths to a given microphone. If the microphone is not centered, it may experience up to five paths. The direct path is assumed to follow simp le spherical spreading, as discussed in previous sections. However, the sidewall reflections and edge scatter are nonideal, and undergo a frequency dependent magnitude and phase shif t. Depending on the setup, these may be attenuated by additional sidewall padding and an acoustic foam edge treatment applied to the edges of the array disk, as shown in Figure 446 Note that the shown array is a 45element array. The array used in this sections analysis is UFAFFs LA MDA array, a larger 63element array [Bahr et al. 2008]. The sidewall padding, for th e most part, removes the sidewall reflections, but the array edge treatment does not completely mitigate edge scattering. 288 PAGE 289 A B&K 4939 1/8 condenser microphone is located at the center of the array plate as the ideal microphone reference to the array microphone s, which are uncalibrated Panasonic WM61a microphones. The signal at the B& K microphone is given in Equation (D51) 11 21 311 11 2144jkr jkr jkr wall edge sxxee YfHHeHG rr (D51) The second term in the brackets is simply the FRF of the wall material for a given oblique incidence multiplied by the spherical spreading function for the reflected image source. The third term cannot be treated as a spherical image source due to the more complex nature of edge scattering, but it should have a dominant compon ent due to propagation from the speaker to the edge to the microphone, The amplifierspeakerhorn response 31r s H accounts for both the direct frequency response of the system, as well as any internal reflections from the horn end to the speaker and back to free space. Nonlinearities in this term as mentioned above would make quantitative analysis of it ques tionable, but the crosscorrelati ons and impulse response estimates involving it can still give a good id ea of the system time delays. Similarly, for any electret microphone in the system, the signal measured is given in Equation (D52) 12 12 2 31 321212 12 12 1 444 mmm mmmm jkr jkr jkr jkr jkr wallm wallm edgem edgem mmm sxxYfH eee HHHeHe rrr HG (D52) As mentioned previously, in the more genera l case separate reflections will pass by the microphone from each sidewall and each surface wave. The signal used in this particular case is a modified Schroeder multisine with a bandwidth from 7,248 Hz to 14,480 Hz, in 16 Hz steps. As th is is slightly less than an octave band, any nonlinearities in the system will be evident from harmonics, sum frequencies and different 289 PAGE 290 frequencies outside of the signa l passband. The measured output from the function generator (before amplification) is shown in Figure D21 with its power spectral density in Figure D22 While some frequency content is present outside of the measurement bandwidth, due to the D/A of the arbitrary waveform editor, it is over four orders of magnitude below the band of interest and so can be safely neglected. Note that th e plotted PSDs are segments of twosided spectra, and are thus offset by a factor of two from a true onesided PSD. The power spectral density of the re ference B&K microphone is plotted in Figure D23 Some small system nonlinearity is present, as the next octave band above the function generator output octave has a relatively flatband shape, ap proximately two orders of magnitude reduced in level. At such a reduced level this may not play much of a roll in frequency response estimates of the amplifierspeakerhorn system, so af ter finding the microphone response functions it would be possible to go back and solve for the amplifierspeakerhorn FRF. However, such a task is beyond the scope of this work. Previously, it was mentioned that when suffi cient bandwidth is present in a signal, its autocorrelation function could be direc tly edited to remove reflections. Figure D24 presents the autocorrelation function of the B&K refere nce microphone. While the multipath peaks are visible in the signal for the appropr iate delays (1.4 ms for the first sidewall, 3 ms for the first internal horn reflection and 3.7 ms for the edge reflection), there is no distinct zero region between the primary autocorrelation peak and th ese reflections. Editing the autocorrelations would likely remove important parts of the directinc idence signal. Note that this data are for the worstcase scenario, with untreated side walls and untreated array plate edges. Figure D25 and Figure D26 show the magnitude and phase estimates for an electret close to the center of the array (0.107 m) denoted here as electret 1. Figure D27 and Figure D28 290 PAGE 291 show the magnitude and phase estimates for an electret far from the center of the array (0.559 m), labeled as electret 63. These numbers ar e assigned based on the microphones array index. Both sets of data show strong contaminati on from reflections. The impulse response and cepstrum of the impulse response between the reference microphone and each of the electret microphones must be examined to see if filtering or lifte ring is possible. The functions are computed fo r both microphones 1 and 63. Reflection time scales are clearly evident in the im pulse response plots of Figure D29 and Figure D30 while by inspection there are only small ripples present in the cepstral plots of Figure D31 and Figure D32. However, the as cepstrum is logarithmic in s cale, it may contain more information in a small ripple than is directly evident. A .window is applied to the data which passe s all data 1 ms after and 1 ms circularly before the peak of each function. The data ar e then reverse transformed and analyzed. The magnitude and phase estimates of microphone 1 are shown in Figure D33 and Figure D34 respectively. Microphone 63s estimates are shown in Figure D35 and Figure D36 For both microphones, the magnitude response is significantly smoothed while trending along a line of expected behavior. However, cep stral liftering appears to do a worse job of restoring the phase behavior for microphone 63 than impulse response filtering. Window length is evaluated by filtering/lifte ring again with a 0.25 ms window ahead and behind the peak of the impulse response and cepstra sequences. The results for electret 1 are plotted in Figure D37 and Figure D38 The results for electret 63 are plotted in Figure D39 and Figure D40 For the smaller window, the magnitude response again looks reasonable. However, the cepstrum liftering performs very poorly with phase estimate. At first glance it would appear that impulse response filtering is the best option. However, it should be noted that 291 PAGE 292 as the filtering window is decreased in size, the impulse response function approaches a delta function, which would have a frequency response magnitude of one and phase shift of zero. Since the bandwidth of interest appears to be reasonably flat for the microphone, an overfiltered impulse response could still appear to be capturing the system dynamics. Furthermore, the algorithm used to compute the si gnal cepstrum utilizes a phaseunw rapping technique. Since the signal bandwidth is undefined below 7,248 Hz, the phaseunwrap will be unwrapping random noise phase angles, dramatically altering th e behavior of the cepstrum computation. To get a better sense of the behavior of im pulse response filtering, a larger bandwidth measurement is necessary so the rolloff region of the electret microphones can be captured. For cepstrum analysis, a larger bandwidth signal is al so required to fill in the lower frequencies and check if phase unwrapping is the dominant probl em. Two options are available for broadband analysis. The first experiment conducted used a signal with multisine components ranging from 304 Hz to 28,816 Hz. This measurement is the easi est to analyze, but may suffer signaltonoise issues related to the maximum system output, as well as nonlinear contamination of upper frequencies from lower ones. The second option is to patch together the seven octave bands run in separate experiments. This overlap process would not be entirely accurate when evaluating the autospectra of a give n microphone, since the sp eaker output may not be the same from run to run due to amplifier instability or ambient cond itions. However, as the physical setup was completely unaltered as the bands were shifted, the mictomic frequency response functions should be the same. If they are the same, then a hybrid, broadband FRF can be patched together from the seven runs spanning 304 Hz to 28,816 Hz. This should capture the entire range of operation of the BMS speaker, and provide enough bandwidth to check both the cepstral phase 292 PAGE 293 issues, by filling out the lower frequencies of th e measurement, and the reliability of impulse response filtering, by measuring beyond the cutoff frequency of the electret microphones. For brevitys sake, the broadband analysis, cond ucted identically to the octave band above, is not presented. The results s howed similar behavior to the oc tave band analysis. Based on the questions raised above, a more re liable processing technique is necessary. For large arrays, visual inspection of each microphone frequency response function and manual editing of the appropriate filters will prove infeasible, and automation requires a consistently reliable processing technique. Downsampling For cepstra, one of the easiest things to evaluate is the cont ribution of individual components of the FRF to the cepstrum. Reca lling that the complex cepstrum of a frequency response function is the sum of the magnitude and phase components of the FRF, the inverse transform of the two components ar e additive and thus can be ev aluated individually. In this analysis, the broadband data are used. Note that unless otherwise stated, logarithms are base e As Figure D41 shows, when signal levels are low, the estimate of the frequency response function breaks down. This observation is reinforced in Figure D43 with the sudden breakdown of a clear phase relations hip at frequencies below and above the speaker output band. The second plot in the phase figures is the mi nimumdelayshift signal, which is simply the unwrapped phase angle with a cons tant integer delay applied. This minimizes the imaginary contribution to the zeroquefrency bin, which can interfere with signal r econstruction if only the real component of cepstra is taken, as it would be since a true cepstrum s hould be nominally real. These plots also show that the phase data have the largest cont ribution to lowquefrencies from Figure D44 while the magnitude cepstrum is best used for identification of reflections, which are visible as ripples in Figure D42 293 PAGE 294 Figure D45 is a schematic of the existing data situ ation. While the desired region does not actually extend to the dc compone nt of the FRF curve, the verylow frequency behavior is necessary to avoid fluctuations in the phase of the calibration curve in the lower bands of interest. Removal of highfreque ncy components is more straightforward, and will be discussed first. The most obvious solution is to apply an idea l lowpass filter to the data. While this may also lead to irregular behavior in the impulse response function of the filter, as the timedomain representation of an ideal lowpass filter is noncausal, the data may still be usable from a reflectionremoval standpoint a nd will thus be evaluated in subsequent discussion. More troubling is the effect this will have on cepstral analysis. Zeroing out the higherfrequency data will result in negatively infinite data when the lo garithm is taken, leading to singular behavior in the inverse Fourier tran sform applied to construct a cepstru m. While the magnitude can be filtered after the logarithm is taken and the phase filtered separately, a more direct solution would be to downsample the data. If the freque ncy response curves are bandpass filtered at +//2 on the unit circle, and then the resulting data expanded to encompass the entire unit circle, the data have effectively been downsampled by a factor of two. In essence, the timedomain sampling rate has been cut from 102,400 samples per second to 51,200 samples per second. The unit circle schematic of this is shown in Figure D46 Figure D47 through Figure D50 show the magnitude and phase results for the downsampled data. Downsampling the data has re moved sufficient fluctuation from the cepstral plots that reflections ar e far more evident in both the ma gnitude and phase components of the cepstrum. From here, the data can be boxcar wi ndowed as previously, and transformed back into the frequency domain. Comparisons between the full dataset liftering and downsampled liftering 294 PAGE 295 are shown in Figure D51 through Figure D54 While the magnitude response in both the original data and the downsamp led data appears to trend reasonably well with the original estimate, with some mild lowfrequency undersh oot, downsampling clearly improves the data fit for phase angle estimates. However, the ripple in the phase angle is still sufficient to yield questionable calibration results. Low Frequency Treatment To improve the performance of the data fit, the lower frequencies for which no data were collected will now be modeled. The base fre quency used in lowband signals was 304 Hz, but judging from the plotted phase response, the sp eaker output may have been insufficient for calibration below 512 Hz. By fitting magnitude and phase data to these frequencies, it may be possible to reduce the ripple in the phase esti mate and improve the lowfrequency magnitude estimate. Fitting data to the magnitude response is simp le. As data are only missing for the 0 Hz to 496 Hz bins, it can be assumed that the microphone passes these signals with little difficulty. While in reality the microphone likely has a lowfrequency cuton, this frequency should be sufficiently low that the first nonzero bin, 16 Hz, will be unaffected. For the sake of data fitting, 0 Hz will be treated as part of the pass band. Qualitatively, a value can be assigned by assigning these low frequency bins a magnitude response e qual to the average magnitude of the 304 Hz to 10 kHz bins, which should be reasonably close to the true microphone response for a measurement microphone with 20 kHz bandwidth. Phas e fitting can be a bit more difficult. As a first attempt, a regression line will be drawn th rough the unwrapped phase angle from 10 kHz to 10 kHz, neglecting the 496 Hz to 496 Hz bins. This should project a rough estimate of the phase behavior for these low fre quency bins, while maintaining an identically zero phase offset at 0 Hz. 295 PAGE 296 The liftered signals with the curve fit low frequencies are plotted in Figure D55 through Figure D58 The plots show an improvement in th e passband response trend of the magnitude estimate, but little change for the phase. Note th e jump at high frequencies in the phase behavior is due to phase angle wrap, which was unwrapped in previous plots. Downsampling the signal improves phase angle estimates. Fitting the low frequency data improves the magnitude estimate. However, the ripple in phase angle prediction is still a problem. As this could be a microphonespecific issu e, another microphone is selected for evaluation. The downsampled phase es timate for electret 43 is shown in Figure D59 Here, a severe error in the phase angle is present. It appears that th e reflection contri bution to phase angle fluctuations is sufficiently large that it causes a phase sh ift of greater than 180 degrees, which tricks phase unwrapping algorithms. While a larger unwrap tolera nce could be applied, this would have implications at other locations in the phase spectrum. This observation leads to questioning how many microphones su ffer from this issue. Figure D60 shows the unwrapped angles for all 63 electret microphones, before lag time correction from the B&K microphone is applied. As shown, most of the microphones undergo huge phase shifts in the 1000 Hz to 1200 Hz band, which are on the order of 360 degrees. That is, the jump is sufficiently large that there is actually no jump at all when wrapped phases ar e plotted. This evidently causes severe issues with cepstrumbased methods wh en predicting phase response. The phase angles for a case where there is e dge treatment on the array and the test section sidewalls have additional, wedge based absorption are shown in Figure D61 As is evident, the phase angle shift is not as severe. Upon further inspection, however, an attempt at a phase angle estimate proves fruitless, as seen in Figure D62 While manual tweaking of windows and phase shifts may allow for cepstral liftering, for blind calibration it appears that impulse response 296 PAGE 297 windowing may prove to be both the simpler and safer choice. The impulseresponsewindowed signals for the case of untreated edges and untre ated sidewalls for electret 43 is shown in Figure D63 through Figure D66 Clearly, impulse response windowing does not suffer from the same problems as cepstral methods, as no phase wrapping is required. Correct window size selection is still important for minimum ripple in the fr equency response estimates. Also, downsampling appears to reduce the performance of an impulse response window method. In an effort to improve performance and reduce the sensitivit y to window size selection, a 50 %Tukey window is applied with the results plotted in Figure D67 and Figure D68 The Tukey window, which consists of a flat pass region which transitions to a flat stop region thr ough a blended, raised cosine section, is selected to perfectly pass the main por tion of the impulse response. It appears to allow longer segments of the impulse response to be used, as shown, before significant ripple is introduced into the FRF estimate. After some minor experimentation, a Tukey wi ndow of 0.4 ms (+/0.2 ms around the peak of the impulse response function), is selected. This is used for both treated and untreated experiments. The treated result s for electret 43 are shown in Figure D69 and Figure D70 and show far less dependence on window size than the untreated case. A comparison between 0.4 ms window cases with treated and untreated conditions is shown in Figure D71 and Figure D72 While there is some variation in magnitude estimates, the phase estimates match remarkably well. Application to Beamforming Results The need for this calibration t echnique is now demonstrated. Figure D73 shows the standard delayandsum beam map of a calibratio n experiment with the mediumaperture array from Figure 446 at 1120 Hz. Figure D74 is the same data set after application of an eigenvaluebased group calibration technique, designed to compensate for steering vector errors 297 PAGE 298 [Dougherty 2002]. In both figures, the acoustic source is located at the cent er of the region plot, centered within the black integration box. The data are contaminated by nonideal reflections located in the upper and lower portions of the sc an region, as no other acoustic sources were present in this experiment. While inspection of the original beam ma p may obviously indicate these data are unsuitable for a group calibra tion technique, if only a few frequencies are contaminated like this and a broadfrequency range, narrowband calibration is desired, this calibration method could easily be applie d without knowing it is in error. When the source is offset by 0.5 m in the posi tive xdirection, the unc alibrated data appear to locate the source, although the source is highly skew, in Figure D75 However, when the calibration which corrected Figure D73 to Figure D74 is applied, Figure D76 is the result. As expected by applying a grossly inaccurate calib ration, the new beam map is nonsense as it completely misses the true source and predicts a large source region in empty space. In contrast, when the new calibration technique is applied to the data, the result in Figure D77 indicates that the true source behavior is preserved although the distribution is still smeared. The calibration technique is now evaluated at frequencies where the data do not appear corrupted by coherent reflection effects. 10 kHz and 20 kHz are well be yond the absorption cutons for sidewall foam used in UFAFF, so both should so reasonable behavior even for uncalibrated cases assuming the microphones do not require dramatic calibration corrections. The uncalibrated beam maps are shown in Figure D78 and Figure D80 The source field appears wellbehaved. The source shape is not perfectly circular like a true monopole, but it is close. The center of the beam maps predicted source is slightly offset from the nominal source location, but this offset is within experimental uncertainty of the horns location in free space. The associated calibrated beam maps are shown in Figure D79 and Figure D81 The shape of 298 PAGE 299 the higherpower beam contours in these plots may be slightly mo re circular, but such judgment is mainly qualitative in nature. The physical acoustic source may be slightly skew due to irregularities in the horn or speaker, so there is no proof that the cal ibration has improved or worsened the shape of the acoustic field. What is evident is the dramatic increase in predicted power at 20 kHz. Throughout this appendix it has been clear that the magnitude and phase rolloff of the electret microphones begin well below 20 kHz. The calibration procedure has, upon initial investigation, compensated for the indivi dual sensor responses in the beam map power estimate. While this is not proof that the calibra tion technique is truly qu antitatively correct, it does provide evidence that it is behaving as intended. Conclusion Impulse response filtering appears to be a vi able method for calibrating array microphones in a semiechoic environment. Under certain conditions, cepstral methods may work. However, one of the primary strengths of cepstral met hods, the ability to alge braically separate a spectrums envelope from reflections and othe r convolved phenomena, appears to be wasted when dealing with frequencyresponse functions, since the timedomain equivalent of these are sharply separated from their reflections. If spectral filtering were of interest, where the autocorrelation of a signal overlap s with its reflection, a cepstra l liftering method would likely be a better alternative to timedomain windowing of a signal. Impulse response filtering has b een applied to an array measurement contaminated with coherent reflections, and has demonstrated th e ability to preserve source location from a reflective calibration data set, in which a standa rd array calibration fails. It also appears to appropriately compensate for sensor re sponse rolloff at higher frequencies. 299 PAGE 300 Figure D1. Simplified acoustic calibration experiment with an ideal source field. 300 PAGE 301 Figure D2. Block diagram of ideal acoustic calibration. 301 PAGE 302 Figure D3. Acoustic calibration expe riment with a single reflection. 302 PAGE 303 Figure D4. Block diagram of acoustic calibration with a single reflection. Figure D5. True frequency response magnitude of the second microphone. 303 PAGE 304 Figure D6. True frequency response phase angle of the second microphone. Figure D7. Schroeder Mu ltisine input waveform. 304 PAGE 305 Figure D8. Schroeder Multisine input power spectral density. Figure D9. Microphone 1 power spectral density. 305 PAGE 306 Figure D10. Microphone 2 power spectral density. Figure D11. Comparison of frequency response estimate magnitudes. 306 PAGE 307 Figure D12. Comparison of frequenc y response estimate phase angles. Figure D13. Impulse response estimate, inversetransformed from FRF estimate. 307 PAGE 308 Figure D14. Cepstrum of impulse response estimate. Figure D15. Cepstrum of impulse response estimate, showing the full computed record. 308 PAGE 309 Figure D16. Relative FRF magnitude error for impulse response windowing. Figure D17. Relative FRF magnit ude error for cepstrum windowing. 309 PAGE 310 Figure D18. FRF phase error for impulse response windowing. Figure D19. FRF phase error for cepstrum windowing. 310 PAGE 311 Figure D20. Schematic of array calibration experimental setup. 311 PAGE 312 Figure D21. Multisine used in experiment measured from function generator output. Figure D22. Power spectral density of function generator output. 312 PAGE 313 Figure D23. Power spectral de nsity of reference B&K microphone located at array center. Figure D24. B&K measurement autocorrelation f unction, normalized to correlation coefficient. 313 PAGE 314 Figure D25. Frequency response ma gnitude estimate for electret 1. Figure D26. Frequency response phas e angle estimate for electret 1. 314 PAGE 315 Figure D27. Frequency response ma gnitude estimate for electret 63. Figure D28. Frequency response phas e angle estimate for electret 63. 315 PAGE 316 Figure D29. Impulse response estimate be tween reference microphone and electret 1. Figure D30. Impulse response estimate be tween reference microphone and electret 63. 316 PAGE 317 Figure D31. Cepstrum estimate betw een reference microphone and electret 1. Figure D32. Cepstrum estimate between reference microphone and electret 63. 317 PAGE 318 Figure D33. FRF magnitude estimates for electret 1 comparing different methods. Figure D34. FRF phase estimates for electret 1 comparing different methods. 318 PAGE 319 Figure D35. FRF magnitude estimates for electret 63 comparing different methods. Figure D36. FRF phase estimates for el ectret 63 comparing different methods. 319 PAGE 320 Figure D37. FRF magnitude estimat es for electret 1, 0.25 ms window. Figure D38. FRF phase estimates for electret 1, 0.25 ms window. 320 PAGE 321 Figure D39. FRF magnitude estimat es for electret 63, 0.25 ms window. Figure D40. FRF phase estimates for electret 63, 0.25 ms window. 321 PAGE 322 Figure D41. Logarithm of the twosided FRF magnitude between B&K and electret 63. Figure D42. Inverse Fourie r transform of logarithmi c magnitude spectrum from Figure D41 322 PAGE 323 Figure D43. Unwrapped twosided FRF phase angle from B&K to electret 63. Figure D44. Inverse Fourier transf orm of imaginary phase angle from Figure D43 323 PAGE 324 Figure D45. Illustration of available FRF data (blue) and desired FRF calibration curve (yellow). 324 PAGE 325 Figure D46. Illustration of available FRF data and desired FRF calibration curve when decimated by 2. 325 PAGE 326 Figure D47. Downsampled logarithm of twosided FRF from B&K microphone to electret 63. Figure D48. Cepstrum corresponding to Figure D47 326 PAGE 327 Figure D49. Unwrapped downsampled phase angle from B&K to electret 63. Figure D50. Cepstrum corresponding to Figure D49 327 PAGE 328 Figure D51. Window length effects for orig inal electret 63 FRF magnitude estimate. Figure D52. Window length eff ects for downsampled electret 63 FRF magnitude estimate. 328 PAGE 329 Figure D53. Window length effects for or iginal electret 63 FRF phase estimate. Figure D54. Window length effects for downs ampled electret 63 FRF phase estimate. 329 PAGE 330 Figure D55. Lowfrequency fit effect for or iginal electret 63 FRF magnitude estimate. Figure D56. Lowfrequency fit effects for dow nsampled electret 63 FRF magnitude estimate. 330 PAGE 331 Figure D57. Lowfrequency fit effects fo r original electret 63 FRF phase estimate. Figure D58. Lowfrequency fit effects for downsampled electret 63 FRF phase estimate. 331 PAGE 332 Figure D59. Electret 43 phase angle estimates. Figure D60. Plot of phase a ngles for electrets 1 through 63. 332 PAGE 333 Figure D61. Phase angles for electrets 1 through 63, additional acoustic treatment is applied. Figure D62. Phase angle estimate for an acousticallytreated case. 333 PAGE 334 Figure D63. Impulse response windowi ng magnitude estimate for electret 43. Figure D64. Downsampled impulse response windowing magnitude estimate for electret 43. 334 PAGE 335 Figure D65. Impulse response window ing phase estimate for electret 43. Figure D66. Downsampled impulse respons e windowing phase estimate for electret 43. 335 PAGE 336 Figure D67. Impulse response magnitude estimate with a 50% Tukey window. Figure D68. Impulse response phase estimate with a 50% Tukey window. 336 PAGE 337 Figure D69. Tukeywindowed magnitude esti mate for treated calibration experiment. Figure D70. Tukeywindowed phase estima te for treated calibration experiment. 337 PAGE 338 Figure D71. Magnitude estimate comparison be tween untreated and treated experiments. Figure D72. Phase estimate comparison betw een untreated and treated experiments. 338 PAGE 339 Figure D73. Mediumaperture array calib ration experiment beam map at 1120 Hz. Figure D74. Ideal calibra ted response for experime nt beam map at 1120 Hz. 339 PAGE 340 Figure D75. Uncalibrated beam ma p of offset speaker at 1120 Hz. Figure D76. Calibrated beam map of offset speaker at 1120 Hz. 340 PAGE 341 Figure D77. Beam map of offset speaker at 1120 Hz, calibrated with new technique. Figure D78. Uncalibrated beam map of me dium aperture array data at 10 kHz. 341 PAGE 342 Figure D79. Calibrated beam map of medium aperture array data at 10 kHz. Figure D80. Uncalibrated beam map of me dium aperture array data at 20 kHz. 342 PAGE 343 343 Figure D81. Calibrated b eam map of medium apertu re array data at 20 kHz. PAGE 344 APPENDIX E COMPARISON OF DAS AND DAMAS RESULTS A single experimental case is processed with a deconvolution algor ithm, DAMAS [Brooks & Humphreys 2006a]. This method uses an itera tive, and computationall y expensive, procedure to attempt to remove the effects of an arrays point spread function from a beam map. While expensive, it does see broad use as a comparator for DAS results, as well as other deconvolution algorithms. DAMAS is used with the same cas e for which a delayandsum integrated power uncertainty estimate is computed, that bei ng the NACA 63215 ModB model at a Mach number of 0.17 and 0 degree AoA. The initial scan plane se lected is the same for that used with the DAS integration, having an extent of 0.4 m in the xaxis and 1.06 m in the yaxis, with 0.02 m steps in each direction. This was discussed previously as involving several tradeoffs of computation time and noise rejection, balanced around adm itting the majority of the trailing edge noise signature. This admittance is demonstrated at the shedding peak frequency of 2,512 Hz in Figure 545 as the integration regions black border bounds the majority of the trailing edge noise source. DAMAS is run for 1,000 iterations. The c onvergence behavior of this iteration count is discussed towards the end of this appendix. Nominal integrated spectra from the two algor ithms are shown in comparison to the arraycentered B&K 4138 microphone in Figure E1 As shown, for most of the measurement bandwidth the DAMAS solution is several decibels above the DAS solution, until just above 10 kHz when the two converge. A comparison of the DAMAS solution to the DAS 95% confidence region is shown in Figure E2 to see if the uncertainty in the DAS prediction may actually overlap with the deconvolution prediction. Th is plot shows that below 3 kHz the DAMAS solution is just outside of the uncertainty bounds of DAS. Between 3 kHz and 10 kHz the DAMAS prediction is a bit higher, and above 10 kHz DAMAS falls within the DAS bounds. 344 PAGE 345 However, the DAMAS data warrant further investigation. DAS integrated levels, with appropriate grid resolution, ar e only dependent on the integra tion bounds of the computation. Any scanning region grid points outside of the integration bounds have no effect on the integrated DAS solution. This is not so with DAMAS. As DAMAS is an inverse problem, the entire scan plane is used in the deconvoluti on process and the predicted levels within the integration region are very much dependent on the overall scan plane solution. While the integration region is selected for consistencys sake with DAS results, the current scan plane, which consists of only a fraction of the wind tu nnel test section and as shown in Chapter 5 clearly excludes major noise sources, is selected for computational expediency. For this grid size, a single integrated spectrum from 512 Hz to 20 kHz required slightly over 24 hours on a modern workstation operating at 3.33 GHz in seri al mode. For a grid encompassing the entire test section, the computational time would increase by a factor of approximately 32, as DAMAS computation time scales as the squa re of the number of grid points. While a narrowband integrated spectrum could not be constructed to evaluate scan plane effects with DAMAS, individual frequency bins could be checked. A series of seven frequencies is evaluated. These frequencies are the same fo r which DAS beam maps ar e available in Chapter 5. For the DAMAS figures, the reduced inte gration region plots will have a dotted line representing the integration region, as well as the total beam map area. The full integration region plots will have a dotted line representing the integration region, and another dotted line representing the overall beam map area. This st yle is different from the previous DAS beam maps due to the pointlike nature of the DAM AS predictions and data visibility issues. The first frequency to be evaluated is 1, 024 Hz, which has a DAS beam map plotted in Figure 543 The limitedregion DAMAS beam map is shown in Figure E3 while the full test 345 PAGE 346 section beam map is shown in Figure E4 In the smaller scan plane, most of the predicted sources lie on the very edge of the plane boundaries while in the larger scan plane case sparse sources are present on the airfoi l trailing edge, as well as some near the rear boundary. The larger scan plane level prediction is well below the DAS integrated level. It should be noted that physically neither of these beam maps resemble the DAS solution. Given the arrays 3dB beamwidth as plotted in Figure 447 the full test section scan plane may still be too small to properly resolve sources using DAMAS. The second frequency evaluated is the sheddi ng peak as noted in the free microphone cases previously at 2,512 Hz, with a DAS beam map in Figure 545 The reduced and full DAMAS scan plane beam maps are shown respectively in Figure E5 and Figure E6 These cases differ from each other by 1 dB, with the full scan plan e predicting the lower valu e. This large scan plane data is only 0.1 dB off from the nominal DAS solution. In both cases, the trailing edge is the dominant visible noise source, with only a fe w sources falling outside of the integration bounds in the full scan plane plot. As with so many other analysis methods, when the trailing edge is the dominant noise sour ce the methods tend to be well be haved and predict similar levels. The third frequency, 5,008 Hz is evaluated next, with its DAS beam map shown in Figure 547. The reduced DAMAS beam map is shown in Figure E7 while the full test section one is shown in Figure E8 The reduced scan plane shows very few sources away from the integration region borders, while the full scan plane shows no significant sources within the integration region at all. This is consistent with the DAS beam map, which indicates diffuser and sidewall noise are the dominant sources at this frequency. The reduced scan plane beam map appears to attempt to resolve them by pushing its observed sour ces to the edge of the region. The full test section beam map appropriately locates them, a nd predicts an SPL significantly below DAS as 346 PAGE 347 DAMAS is removing the lobe width effects from the sidewall and diffuser noise, preventing the noise from contaminating the integration region. The frequency 7,600 Hz, with a DAS beam map shown in Figure 549 shows similar behavior to 5,008 Hz. The reduced and full DAMAS scan planes are shown in Figure E9 and Figure E10 Again, the reduced scan plane detects mo st sources on the very edge of its scan plane, and results in levels hi gher than those estimated with DAS. Again, the full scan plane DAMAS beam map shows that the sources are not within the integration region, and predicts a power lower than DAS as it removes the lobe e ffects from sidewall noise sources. The same behavior is seen at 8,800 Hz. The DAS beam map is given in Figure 551 The DAMAS reduced and full solutions are shown in Figure E11 and Figure E12 15,008 Hz and 20,000 Hz array predictions both appear to be driven by the leading edgesidewall noise source, as is clearly visible in the DAS beam maps of Figure 553 and Figure 555 In both of these cases, the DAS solution appe ars as if it may be contaminated by sidelobe effects of this noise source. In both of th ese cases, the reduced DAMAS solution shown in Figure E13 and Figure E15 interprets these possible sidelobes as major noise sources. This is likely because the main lobes of the noise sour ces are outside of the DAMAS solution region, and thus deconvolution fails to account for that beha vior of the arrays point spread function. As the full test section DAMAS beam maps of Figure E14 and Figure E16 show, these noise sources actually have little impact on the in tegration region when proper deconvolution occurs, and the resultant noise prediction is dramatically below the DAS output. Briefly, the convergence of the DAMAS solutions will be checked. A ll of the cases shown were analyzed with 1,000 internal DAMAS itera tions. A single case, the 20,000 Hz full test section beam map, was selected and run for 5,000 and 10,000 iterations. The 5,000 iteration case 347 PAGE 348 is shown in Figure E17 and the 10,000 iteration case is shown in Figure E18 These figures show, in comparison to the 1,000 iteration solution of Figure E16 that nominal convergence is achieved as the beam maps appear graphically to be the same, and the integrated levels match to within the displaye d 0.1 dB precision. Finally, a 1/3rd octave band analysis is conducted for the small and large beam map regions with DAMAS, and compared to the results from DAS and NAFNoise. The plot of these results is shown in Figure E19 As with the narrowband integrat ion, DAMAS with a small beam map tends to predict levels slightly above DAS. Ho wever, DAMAS with a full test section beam map predicts significantly lower than DAS for a larg e portion of the measurement bandwidth. While DAMAS still fails to reject much of the low frequency noise suspected to come from the diffuser, much of the higherfreque ncy contamination is eliminated.. The DAMAS algorithm appears to be able to morefinely resolve the acoustic sources in this data set than the standard DAS algorithm, and predicts significantly lower acoustic levels than DAS for frequencies where th e airfoil trailing edge is not the dominant acoustic source, and background noise sources are physic ally located outside of the defined integration region. However, the required scan plane for the algorithm encompasses the entire test section, and a full integrated spectrum using such a domain is com putationally infeasible. Using a reduced scan plane in an attempt to get a solution in a reduced amount of time leads to beam maps where incorrect acoustic fields are constr ucted, leading to erroneous integrated levels. If such a tradeoff must be made, a DASbased solution provides a more physically reasonable acoustic field, with the benefits of shorter computation tim e and existing uncertainty analysis codes. Alternative codes, such as SC DAMAS, may provide a faster d econvolution alternative [Yardibi et al. 2008]. 348 PAGE 349 Figure E1. Comparison of no minal DAS and DAMAS outputs. Figure E2. Comparison of DAMAS output to DAS uncertainty bounds. 349 PAGE 350 Figure E3. Reduced scan plane DAMAS solution for 1,024 Hz. Figure E4. Full test section scan plane DAMAS solution for 1,024 Hz. 350 PAGE 351 Figure E5. Reduced scan plane DAMAS solution for 2,512 Hz. Figure E6. Full test section scan plane DAMAS solution for 2,512 Hz. 351 PAGE 352 Figure E7. Reduced scan plane DAMAS solution for 5,008 Hz. Figure E8. Full test section scan plane DAMAS solution for 5,008 Hz. 352 PAGE 353 Figure E9. Reduced scan plane DAMAS solution for 7,600 Hz. Figure E10. Full test section scan plane DAMAS solution for 7,600 Hz. 353 PAGE 354 Figure E11. Reduced scan plane DAMAS solution for 8,800 Hz. Figure E12. Full test section scan plane DAMAS solution for 8,800 Hz. 354 PAGE 355 Figure E13. 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PAGE 368 BIOGRAPHICAL SKETCH Chris Bahr was born and raised in Orlando, Florida. He attended the International Baccalaureate program at Winter Park High School, and graduated as a National Merit Scholar in the spring of 1999. He began attending the University of Florida in the fall of 1999, and received his Bachelor of Scie nces degree in aerospace engineer ing in the spring of 2003. Along the way, he began working for Dr. Lou Cattafest a as an undergraduate researcher. Swayed by the stories of fame and glory coming from gra duate students, Chris applied and was accepted to UFs graduate Mechanical and Aerospace Engi neering program in the fall of 2003. He continued on with Dr. Cattafesta for seven years as a graduate student. During that time he met Stephanie Mecca, and married her in the summer of 2009. Upon completion of his degree in the spring of 2010, Chris will begin a postdoctoral position, working for Dr. Cattafesta. 368 