PhD thesis of Anurag Agarwal

121 Pages·2012·1.37 MB·English

by  Anurag Agarwal

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