ThePennsylvaniaStateUniversity TheGraduateSchool THEPREDICTIONOF TONALANDBROADBANDSLATNOISE AThesisin AerospaceEngineering by AnuragAgarwal c 2004AnuragAgarwal (cid:13) Submitted inPartialFul(cid:2)llment oftheRequirements for theDegreeof Doctor ofPhilosophy May 2004 Thethesisof AnuragAgarwalwasreviewedandapproved by thefollowing: (cid:3) PhilipJ.Morris Boeing/A.D.WelliverProfessor ofAerospaceEngineering ThesisAdviser,Chairof Committee LyleL.Long Professor of AerospaceEngineering KennethS.Brentner Associate Professor ofAerospaceEngineering Victor W.Sparrow Associate Professor ofAcoustics DennisK.McLaughlin Professor of AerospaceEngineering Headof theDepartmentof AerospaceEngineering Signatures areon (cid:2)leintheGraduateSchool. (cid:3) Abstract Noise from high-lift devices such as slats and (cid:3)aps can contribute signi(cid:2)cantly to the overall aircraft sound pressure levels, particularly during approach. The acoustic spectrum of the noise radiated from slats exhibits two distinct features. There is a high-frequency tonal noise component, and a high-energy broadband component ranging from low to mid-frequencies. The objective of this thesis is topredictboththetonalandthebroadbandslatnoise. Anaeroacousticwhistling mechanism is proposed to predict the tonal noise generation. When the vortex sheddingfrequencyattheblunttrailingedgeoftheslatcomesclosetooneofthe normalmodesofthegapbetweentheslatandthemainelement,anintensetonal noise is produced. The normal modes are calculated based on the geometry of the wing. The vortex shedding frequency is predicted based on a linear stability analysis of the slat’s wake region. An e(cid:14)cient and robust scheme is developed by which the stability calculation can be performed by a modular algorithm in a relativelyquicktime. Thebroadbandnoiseispredictedusingatwo-stepprocess. First the noise sources are modeled based on the local turbulence information. Then, the sound from these sources is propagated by assuming that the (cid:3)ow past the wing is uniform. A Boundary Element Method is developed to (cid:2)nd the Green’s function for wave propagation in a moving medium in the presence of thewing. Thenoise inthefar(cid:2)eldisthenpredicted byforming aconvolution of the Green’s function with the modeled sources. Finally, a technique is presented to account for nonuniform (cid:3)ow around the wing. This requires a solution of the linearizedEulerEquations. However,theseequationssupportacousticaswellas instability waves. Theinstability waves cancompletely overwhelmtheacoustic- wave solution. Thus it is imperative for an accurate noise-prediction scheme to suppress the unwanted instability waves. A detailed mathematical analysis is presented that demonstrates that the instability wave solution is suppressed if the governing equations are solved in the frequency domain. The main focus of this thesis is in the development of numerical schemes and models, and then their use to explore the physics of noise generation, and the prediction of noise iii radiation, from slats. iv Table of Contents ListofFigures viii ListofTables xi Acknowledgments xii 1 Introduction 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Background andmotivation . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Tonalnoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Broadbandnoise . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Tonalnoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 Broadbandnoise . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Roadmaptothe restof thethesis . . . . . . . . . . . . . . . . . . . . 12 2 Tonal slatnoise prediction 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Basicequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Localstability analysis (cid:150)Numericalprocedure . . . . . . . . . . . . 17 2.4 Wakecharacteristics (cid:150)Localandglobal dynamics . . . . . . . . . . 18 2.5 Aeroacoustic whistle . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Parametricvariations . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6.1 Slatgap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6.2 Machnumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6.3 Reynoldsnumber . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Broadbandslatnoise 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 v 3.2 Green’s function for the convected wave equation in the presence ofasolidbody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Boundaryelementformulation . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Numericalimplementation . . . . . . . . . . . . . . . . . . . . 37 3.4 Examplesof theapplicationof BEM . . . . . . . . . . . . . . . . . . 40 3.4.1 Simplyconnected domains . . . . . . . . . . . . . . . . . . . . 41 3.4.2 Multiconnected domains . . . . . . . . . . . . . . . . . . . . . 41 3.5 Noiseprediction scheme . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4 Thecalculation ofsound propagationinnonuniform(cid:3)ows 54 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Generaltheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2.1 Space-timeresponse: Solution to theinitial-valueproblem . . 57 4.2.2 Unstableresponse . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.3 Assumedtime-harmonic response . . . . . . . . . . . . . . . . 61 4.3 Aone-dimensionalexample . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.1 Space-timeresponse . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.2 Assumedtime-harmonic response . . . . . . . . . . . . . . . . 63 4.4 Soundpropagation through atwo-dimensional jet . . . . . . . . . . 63 4.4.1 Space-timesolution . . . . . . . . . . . . . . . . . . . . . . . . 65 4.4.2 Assumedtime-harmonic response . . . . . . . . . . . . . . . . 65 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 Conclusions andfuturework 72 A Numericalcomputationoflinearstability offree-shear(cid:3)ows 75 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.2 Dispersionrelation solver . . . . . . . . . . . . . . . . . . . . . . . . 77 A.2.1 Contour integration technique . . . . . . . . . . . . . . . . . . 78 A.3 Absoluteandconvective instability solver . . . . . . . . . . . . . . . 83 A.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 B Free space Green’s function: time-harmonic point source in a uniform mean(cid:3)ow 91 C Coupling oftwo waves(cid:150)Analytical solution 93 C.1 Space-timeGreen’sfunction . . . . . . . . . . . . . . . . . . . . . . . 93 C.2 Assumedtime-harmonic, constant-frequency response . . . . . . . 96 vi D Analytical solution toLilley’sequation 98 Bibliography 105 vii List of Figures 1.1 A comparison of the noise spectra from the various wing compo- nentsofamodelMD11aircraft (from Guo etal. [2]). . . . . . . . . 3 1.2 Three-element high-lift system. a) Three-dimensional model, b) cross-sectional view(from Khorrami et al.[3]). . . . . . . . . . . . . 5 1.3 Acoustic spectrum basedon1/12thoctave binswitharray focused on slat region: con(cid:2)guration angle of attack is 10 deg, Reynolds number is 7.2 million, and Mach number is 0.2 (from Singer et al. [4]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Acoustic spectrum basedon1/12thoctave binswitharray focused on slat region: con(cid:2)guration angle of attack is 9 deg, Reynolds numberis7.2million,andMachnumberis0.2;(cid:151)(cid:151),lowfrequency microphone array and (cid:150) (cid:150) (cid:150), high frequency microphone array (from Khorrami etal. [10]). . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Space-time evolutions G(x;t) of a pulse introduced at the origin. The pulse evolution is: absolutely unstable in (a), convectively unstablein(b),andstablein(c)(From Bers[19]). . . . . . . . . . . . 15 2.2 Mean (u) velocity pro(cid:2)les at di(cid:11)erent downstream locations for the 30 deg. slat de(cid:3)ection case: a) progressively more deep, b) progressively moreshallow, wake. . . . . . . . . . . . . . . . . . . . 19 2.3 Mean (u) velocity pro(cid:2)les at di(cid:11)erent downstream locations for the 20 deg. slat de(cid:3)ection case: a) progressively more deep, b) progressively moreshallow, wake. . . . . . . . . . . . . . . . . . . . 20 2.4 The real and imaginary part of the absolute frequency for the slat wakeat30deg. slatde(cid:3)ection. . . . . . . . . . . . . . . . . . . . . . 21 2.5 The real and imaginary part of the absolute frequency for the slat wakeat20deg. slatde(cid:3)ection. . . . . . . . . . . . . . . . . . . . . . 22 2.6 Feedbackprocess leadingtoresonance neartheslattrailing edge. . 26 3.1 The coordinate systems: (x;y) (cid:150) Cartesian; ((cid:24);(cid:17)) (cid:150) body-(cid:2)tted or- thogonal curvilinear. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 viii 3.2 Surfacedeformation such thatinthelimit (cid:15) 0,x x(cid:0) . . . . . . . 36 ! ! 3.3 Surfacediscretization forthe BEM. . . . . . . . . . . . . . . . . . . . 38 3.4 Schematicfor acoustic scattering from acylinder. . . . . . . . . . . . 41 3.5 Sounddirectivitypatternsforthescatteringofacousticwavesfrom a line source at (20,0) by a cylinder centered at (0,0). The source frequencyis200rad/s: (cid:151)(cid:151)(cid:150),BEMsolution; ^,analytical solution. 42 3.6 Comparison of sound directivity patterns for the scattering of acoustic waves from a line source by a cylinder. (cid:150) (cid:150) (cid:150) (cid:150), M=0; (cid:151)(cid:151)(cid:150),M=0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 Multiconnected domain(cid:10) = (cid:10) (cid:10) . . . . . . . . . . . . . . . . . . 43 1 2 [ 3.8 Schematicfor acoustic scattering from two cylinders. . . . . . . . . 44 3.9 Rmspressureforthescatteringofacousticwavesfromalinesource bytwo cylinders. (cid:151)(cid:151),BEMsolution; ,analytical solution. . . . . 44 (cid:5) 3.10 Surfacede(cid:2)nition ofthewing geometry for theBEM. . . . . . . . . 48 3.11 Thedirectivity pattern at aradius of 1.82m from apoint source of frequency2000Hzintheslat-cove region. . . . . . . . . . . . . . . . 49 3.12 Contourplotoftheturbulentkineticenergyinthevicinityoftheslat. 50 3.13 Comparisonwithexperimentaldataofthepredictedspectralden- sity at 270 deg. to the downstream direction:(cid:151)(cid:151), experiment, Khorrami et al. [10]; (cid:144), prediction with the wing; (cid:150) (cid:150) (cid:150), prediction without thewing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.14 Comparison of noise spectra at di(cid:11)erent observer locations. The points represent predictions with the wing: +, (cid:18) = 310 deg.; (cid:144), (cid:18) = 270 deg.; (cid:0), (cid:18) = 210 deg. The lines represent predictions without the wing: (cid:151)(cid:151)(cid:150), (cid:18) = 270 deg.; (cid:150) (cid:150) (cid:150), (cid:18) = 310 deg., , (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:18) = 210deg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.15 The e(cid:11)ect of mean (cid:3)ow on the noise spectra at 270 deg:+, M = 0:2 prediction and (cid:144), M = 0 prediction, with wing; , M = 0:2 (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) prediction and(cid:150)(cid:150)(cid:150),M = 0prediction, without wing. . . . . . . . . 53 4.1 LoweringoftheinverseFourier-contourfromF F inconjunction 0 ! withtheinverseLaplacecontourfromL L . TheinverseLaplace- 0 ! contours lie above the respective !(k) maps to maintain causality. Theresidueat ! dominates thetime-asymptotic response. . . . . . 59 o 4.2 Comparisonof theanalyticalsolution ( )along thesideline y = 15 (cid:5) withnumericalsolution ((cid:151)(cid:151))obtainedby aspace-time solver. . . 65 4.3 Comparisonof theanalyticalsolution ( )along thesideline y = 15 (cid:5) with numerical solution ((cid:151)(cid:151)) obtained by a direct, frequency- domainsolver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ix A.1 An integration contour avoiding the singularities in the complex y-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 A.2 A contour integration path, consisting of piece-wise continuous line-segments,for Eq.(A.3). . . . . . . . . . . . . . . . . . . . . . . . 80 A.3 Lowering of the contours in the k-plane (a) and their respective mapsinthe!-plane(b)inorder to uncover thecusp(k ;! ). . . . . 85 o o A.4 The ratio of group to phase velocity as a function of the velocity ratio R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 A.5 Locusof branchpoints y duringthe k !mappingprocedure . . 88 c ! D.1 Locationoftheinverse-Fouriercontour(F)andthesteepest-descent paths (S’s) in the k-plane. Also shown are the locations of the i branch-points ( ) and branch-cuts (hatched lines) of (cid:23), and the (cid:5) eigenvaluesof Eq.(D.3)(marked +). . . . . . . . . . . . . . . . . . . 100 D.2 Location of the critical point (y ) and the branch-cut for the loga- c rithmicsingularity inEq.(D.3). . . . . . . . . . . . . . . . . . . . . . 104 x