Fuselage boundary-layer refraction of fan tones radiated from an installed turbofan aero-engine

39 Pages·2017·1.99 MB·English

Fuselage boundary-layer refraction of fan tones
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radiated from an installed turbofan aero-enginea)
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James Gaffney and Alan McAlpineb)
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InstituteofSoundandVibrationResearch,UniversityofSouthampton,Southampton,SO171BJ,UK
Michael J. Kingan
DepartmentofMechanicalEngineering,UniversityofAuckland,Auckland,NewZealand
February 6, 2017
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a)Partofthisworkwaspresentedin“Soundradiationoffantonesfromaninstalledturbofanaero-engine: fuselage
boundary-layerrefractioneffects”,Proceedingsofthe22ndAIAA/CEASAeroacousticsconference,Paperno. AIAA-
2016-2878,Lyon,France,30May–1June2016.
b)Authortowhomcorrespondenceshouldbeaddressed. Electronicmail: [email protected]
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Abstract
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Adistributedsourcemodeltopredictfantonenoiselevelsofaninstalledturbofanaero-engine
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is extended to include the refraction effects caused by the fuselage boundary layer. The model
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is a simple representation of an installed turbofan, where fan tones are represented in terms
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of spinning modes radiated from a semi-infinite circular duct, and the aircraft’s fuselage is
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represented by an infinitely long, rigid cylinder. The distributed source is a disc, formed by
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integratinginfinitesimalvolumesourceslocatedontheintakeducttermination. Thecylinderis
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located adjacent to the disc. There is uniform axial flow, aligned with the axis of the cylinder,
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everywhereexceptclosetothecylinderwherethereisaconstantthicknessboundarylayer. The
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aimistopredictthenear-fieldacousticpressure,andinparticulartopredictthepressureonthe
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cylindrical fuselage which is relevant to assess cabin noise. Thus no far-field approximations
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areincludedinthemodelling. Theeffectoftheboundarylayerisquantifiedbycalculatingthe
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area-averaged meansquare pressure over thecylinder’s surface withand without the boundary
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layer included in the prediction model. The sound propagation through the boundary layer is
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calculatedbysolvingthePridmore-Brownequation. Resultsfromthetheoreticalmethodshow
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that the boundary layer has a significant effect on the predicted sound pressure levels on the
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cylindrical fuselage, owing to sound radiation of fan tones from an installed turbofan aero-
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engine.
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I. Introduction
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Organisations such as the International Civil Aviation Organisation (ICAO) and the American Fed-
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eral Aviation Administration (FAA) oversee noise certification of civil aircraft, including setting
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stringent targets for the reduction of noise. To aid meeting these requirements, practical and reli-
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able noise prediction methods are an essential part of the research and development toolkit. This
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paperpresentsanewtheoreticalmethodtopredictthesoundradiationoffantonesfromaninstalled
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turbofan aero-engine, focussing on the prediction of the fuselage sound pressure levels (SPL), and
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howtheselevelsareaffectedbyrefractioneffectscausedbythefuselageboundarylayer.
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Noise from aero-engines has a major impact on communities around airports as well as the pas-
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sengers and crew aboard the aircraft. At take-off and cruise conditions the aero-engines are the
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dominant source of noise from civil aircraft. A turbofan engine represents a very complex noise
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source,andassuchthenoisegenerationandradiationmechanismsaredifficulttomodelaccurately.
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Thecomplexityiscompoundedbythepresenceoftheairframe. Consequently,turbofannoisemod-
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els often predict the sound radiated into free space. However, the radiated noise is affected by the
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whole aircraft, including the fuselage, wings, high-lift devices, landing gear etc. How the radiated
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acousticfieldisaffectedbyinteractionwiththeairframeisreferredtoasinstallationacoustics.
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This paper will focus on prediction of the sound pressure levels on an aircraft fuselage. The
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key aim is to develop a simple installation acoustics method based on a canonical problem: how
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to predict the pressure field due to a source located near an infinite, rigid cylinder. The basis of
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the method is a classical problem: acoustic scattering by a cylinder. The analytical solution for
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the acoustic field due to a point source located adjacent to an infinite cylinder is given in the text
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by Bowman, Senior and Uslenghi (Chapter 2, §2.5.2, pp. 126–127) [1]. Simplifications in the
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geometryandthemeanflowmustbemadetouseanalyticaltechniques,butthesedonotnecessarily
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curtail the usefulness of the solutions. In this case, the aircraft’s fuselage is represented by the
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cylinder. Also, except close to the fuselage, the flow is assumed to be uniform and its direction is
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alignedwiththeaxisofthecylinder.
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Of principal interest in this current work is the effect of the fuselage boundary layer. One of the
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earliest examples of work in this area is by Hanson [2] who studied the problem of a point source
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over a rigid flat plate with a boundary layer. Subsequently, Hanson and Magliozzi [3] examined
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the more advanced and realistic problem of an open-rotor source located adjacent to a cylindrical
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fuselage, and with the inclusion of the fuselage boundary layer. This type of model also has been
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investigated by Lu [4], who utilized a point source model. Most recently Brouwer [5] studied the
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effects of a boundary layer on the near side of a cylinder for an open-rotor type source. The model
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wasextendedtothefar-fieldwheretherewasamodesteffectonthefree-fieldpredictionscausedby
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theboundarylayer.
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Historically, these type of theoretical methods for fuselage installation acoustics have focused
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on open-rotor noise sources. More recently, the procedure given by Hanson and Magliozzi was
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followed by McAlpine, Gaffney and Kingan [6] to develop a scattering model for a realistic tur-
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bofan source located adjacent to an infinitely long, rigid cylindrical fuselage. A disc source was
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formulated so that the source distribution was representative of sound radiation for a ducted fan
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noise source. The source is a disc formed by integrating infinitesimal volume sources located on
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the duct termination. The source strength (volume velocity) is proportional to the axial component
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of the particle velocity at the duct termination, following the Kirchhoff approximation. Originally
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Tyler and Sofrin [7] used this approach to construct a Rayleigh integral in terms of a non-uniform
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velocity distribution across the duct termination. The particle velocity is given by the well-known
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‘spinning’ mode solution for sound propagation in a cylindrical waveguide. Thus, tones generated
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by a ducted fan are modelled in terms of spinning modes propagating inside the intake duct. This
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approach neglects diffraction effects at the duct lip, but Hocter [8] has shown that these are largely
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insignificant for sound radiation up to approximately 70 degrees relative to the axis of the duct.
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Accordingly,thescatteringmodelisonlyvalidforsoundradiationintheforwardarc.
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This paper extends recent work published in Ref. [6]. The key advancement is the inclusion of
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theeffectofboundary-layerrefractioninthemodelling. Thecylindricalfuselageisinthenear-field
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of the source. Thus it is not valid to use a far-field approximation of the sound field radiated by the
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source, a` la the far-field solution of the Rayleigh integral for a circular disc source. In Ref. [6] the
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ducted fan noise source was incorporated into a cylinder scattering model with uniform mean flow
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everywhere. In this paper, the same method for a ducted fan noise source is extended to include
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refractioneffectscausedbyaboundarylayerofconstantthicknessonthecylindricalfuselage.
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II. Theory
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A. Fuselage scattering
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The acoustic pressure field owing to a fan tone radiated from an intake duct located adjacent to
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an infinitely long, rigid cylindrical fuselage was derived in McAlpine, Gaffney and Kingan [6]. A
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sketch of the problem set-up is shown in Figure 1; also shown is some of the nomenclature used in
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the solution, and the Cartesian reference frames for the source (x,y,z) and the cylinder (x¯,y¯,z¯).
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¯
Theequivalentcylindricalpolarcoordinatereferenceframesare(r,φ,z)forthesourceand(r¯,φ,z¯)
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forthecylinder. Thereisasubsonicuniformmeanflow,MachnumberM ,directedinthenegative
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z¯-direction. The total pressure field (sum of the incident and scattered fields) is calculated in the
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Figure1: Sketchofthecylindricalfuselage(radiusa )andthediscsource(radiusa). The
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centrelineofthecylinderisalignedwiththez¯-axis. Thediscsourceislocatedintheplanez¯= 0,
andthedistancebetweenthecentreofthediscandthecentreofthecylinderisb. Alsoshownisthe
edgeofthefuselageboundary-layer(thicknessδ).
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forwardarcz¯> 0.
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(cid:0) ¯ (cid:1) (cid:0) ¯ (cid:1)
From Ref. [6], the time-harmonic total pressure p r¯,φ,z¯,t = pˆ r¯,φ,z¯ exp{iω t}, is ex-
90 t t 0
pressed in terms of a Fourier series, and each Fourier harmonic is itself expressed in terms of an
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inverseFouriertransform. TheresultfromRef.[6]canbewritteninnon-dimensionalformas
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pˆ (cid:0)r¯,φ¯,z¯(cid:1) = 1 (cid:88)∞ (cid:26)(cid:90) ∞ (cid:0)p +p (cid:1)e−ikzz¯dk (cid:27)e−inφ¯, (1)
t (2π)2 (cid:101)in (cid:101)sn z
n=−∞ −∞
wheretheincidentfield
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p (r¯,k ) = π2ξ P (−1)l+ne−i(l−n)β(k +k M )Ψ H(2) (Γ b)J (Γ r¯), (2)
(cid:101)in z lq lq 0 z ∞ lq l−n 0 n 0
andthescatteredfield
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H(2) (Γ b)J(cid:48) (Γ )
p (r¯,k ) = −π2ξ P (−1)l+ne−i(l−n)β(k +k M )Ψ l−n 0 n 0 H(2)(Γ r¯), (3)
(cid:101)sn z lq lq 0 z ∞ lq H(2)(cid:48)(Γ ) n 0
n 0
where · denotes variables which have been Fourier transformed. The Bessel function of the first
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kind of order (cid:96) is denoted by J , and the Hankel function of the second kind of order (cid:96) is denoted
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by H(2). The (cid:48) denotes differentiation with respect to the function’s argument. Full details are in
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McAlpine,GaffneyandKingan[6].
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In Equations (1–3), all the variables have been made non-dimensional by taking the reference
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length scale equal to the radius of the cylinder, a , the reference velocity equal to the speed of
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sound,c ,andthereferencedensityequaltotheambientdensity,ρ . Thenthepressureisscaledby
101 0 0
ρ c2.
102 0 0
The fan tone source is modelled by the time-harmonic spinning mode (l,q), where l denotes the
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azimuthal order and q denotes the radial order. The mode has amplitude P , wavenumber k and
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angularfrequencyω . Thedispersionrelationshipforthemodeis
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k 2 +κ2 = (cid:0)k +k M (cid:1)2 , (4)
zlq lq 0 zlq ∞
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andthemodeamplitudecoefficient
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k
zlq
ξ = . (5)
lq (cid:0) (cid:1)
k +k M
0 zlq ∞
ThetermΨ isfoundbyintegrationoverthediscsource. Thisintegralcanbeevaluatedexactly.
107 lq
Fornon-plane-waveexcitation,forr > a
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
 (κ2Γ−0aΓ2)Jl(κlqa)J(cid:48)l(Γ0a) ifΓ0 (cid:54)= κlq
Ψ = lq 0 , (6)
lq
(cid:16) (cid:17)
 21 a2 − κl22 J2l(κlqa) ifΓ0 = κlq
lq
andforplane-waveexcitation
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
Ψ =  −Γa0J(cid:48)0(Γ0a) ifΓ0 (cid:54)= 0 . (7)
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
 21a2 ifΓ0 = 0
SimilartoEquation(4),the‘radial’wavenumberoftheradiatedfield,Γ ,isgivenby
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Γ2 = (k +k M )2 −k 2 , (8)
0 0 z ∞ z
andequalszeroat
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k k
k− = − 0 and k+ = 0 . (9)
z 1+M z 1−M
∞ ∞
FromEquations(8)and(9),thewavenumberΓ isdefinedas
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(cid:112) (cid:112) (cid:112)
Γ = −i 1−M2 −i(k −k+) i(k −k−), (10)
0 ∞ z z z z
on taking the principal value of the square root. In the range k− < k < k+, the wavenumber Γ
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is real and positive, and corresponds to wavenumbers which radiate to the far field. Outside of this
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(cid:112)
range, Γ = −iγ where γ = k2 −(k +k M )2 > 0. This wavenumber range corresponds to
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an evanescent field. Accordingly, the integration of the inverse Fourier transform in Equation (1) is
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splitintothreepartsasshowninTable1. Moreover,thedefinitionofΓ givenbyEquation(10)also
117 0
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Limits
Label Γ
0
lower higher
I −∞ k− −iγ
1 z 0
(cid:112)
I k− k+ (k +M k )2 −k2
2 z z 0 ∞ z z
I k+ ∞ −iγ
3 z 0
Table1: BreakdownoftheinverseFouriertransformintegralinEquation(1). Thefullintegralis
givenbyI +I +I . ThevalueofthewavenumberΓ ineachregionisspecified.
1 2 3 0
prescribestheappropriatelocationofthebranchcutsfrompointsk− andk+ suchthatontakingthe
118 z z
inverseFouriertransformtheradiationconditionswillbecorrect.
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Measured relative to the axis of the intake duct (z), as the source Helmholtz number k a in-
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creases, the mode and propagation angles also increase. There are several ways that can be used to
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predicttheradiationangleoftheprincipallobe. Oneestimateisgivenbythemode(orphase)angle,
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whichcanbecalculateddirectlyfromthedispersionrelationship(4). Rice[9]derivedanalternative
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expressionfortheradiationangle,viz.
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(cid:112) (cid:26) 1−1/ζ2 (cid:27)1/2
cos θ = 1−M2 , (11)
∞ 1−M2 (1−1/ζ2)
∞
whereζ isthecut-offratio
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k
0
ζ = . (12)
(cid:112)
k 1−M2
zlq ∞
Thisresultiscommonlyusedtopredicttheprincipalradiationangleinthepresenceofflow.
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Alternatively the radiation angle can be found numerically by direct computation of the direc-
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tivity function. For the disc source, the radiated pressure field, expressed in the source coordinates
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(r,φ,z),isgivenby
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ξ P (cid:90) ∞
pˆ (r,φ,z) = lq lq (k +k M )Ψ H(2)(Γ r)e−ikzz dk e−ilφ , (13)
i 4 0 z ∞ lq l 0 z
−∞
atfieldpointsr > a. Equation(13)istakenfromMcAlpine,GaffneyandMcAlpine([6],Eq. (20)).
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Thelocationonthecylinder’ssurfacewheretheSPLisatamaximumcanbeestimatedutilizing
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the radiation angle of the principal lobe and simple geometry. An example of this is shown in
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Figure 2. Numerical evaluation of the radiation angle provides the closest estimate to the location
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of the maximum SPL on the cylinder. This is because the cylinder is located in the near field,
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whereastheanalyticformulasforthemodeandpropagationanglesarevalidinthefarfield.
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Figure2: NormalisedtotalSPLonthesurfaceofthecylinder. Thelocationofthemaximumvalue
oftheSPLisshownbythebluedot. Predictionsofthispointobtainedusing(i)mode(phase)
angle,(ii)radiationangle(11),or(iii)numericalevaluationofthenear-field(intheabsenceofthe
cylinder),arealsoshown. Key: (i)magentadot;(ii)blackdot;(iii)greendot. Otherrelevant
parametersinthisexampleare(l,q) = (2,1),k a = 5,b = 3andM = 0.5.
0 ∞
B. Fuselage scattering including boundary-layer refraction
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The technique to include the boundary layer in the method is outlined in this section. A sketch of
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the cylindrical fuselage, including the constant-thickness boundary layer is shown in Figure 1. The
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