Computational Aeroacoustic Prediction of Discrete-Frequency Noise
Generated by a Rotor-Stator Interaction
S. Sawyer∗ M. Nallasamy† R. Hixon‡
small Mechanical Engineering QSS Group, Inc. Mechanical EngineeringUniversity of Akron Mail Stop 500-QSS University of ToledoAkron, OH 44325 NASA Glenn Research Center Toledo, OH 43606
Accurate prediction of turbomachinery noise is a continuing concern for aircraft
engine designers. Although significant reductions in the noise generation have been
achieved, future noise reductions will rely on a fundamental understanding of the noise
generation process. The discrete-frequency noise generated by a rotor-stator interaction
is computed by solving the fully nonlinear Euler equations in the time domain in two-
dimensions. The acoustic response of the stator is determined simultaneously for the first
three harmonics of the convected vortical gust of the rotor. The spatial mode generation,
propagation and decay characteristics are predicted by assuming the acoustic field away
from the stator can be represented as a uniform flow with small harmonic perturbations
superimposed. The computed field is then decomposed using a joint temporal-spatial
transform to determine the wave amplitudes as a function of rotor harmonic and spa-
tial mode order. The frequency and spatial mode order of computed acoustic field was
consistent with linear theory. Further, the propagation of the generated modes was also
correctly predicted. Although, some dispersion at short wavelengths was apparent. The
upstream going waves propagated from the domain without reflection from the inflow
boundary. However, reflections from the outflow boundary were noticed. The amplitude
of the reflected wave was approximately of 5% of the incident wave.
Introduction
Aircraft noise is a significant environmental concern.In fact, NASA under its “Global Civil Aviation pil-lar” has made a commitment to reduce noise by 10dB within a decade and 20 dB within the next twodecades. Achieving NASA’s long-term 20-decibel ob-jective for noise reduction will, in most cases, containobjectionable aircraft noise within the airport bound-aries (55 Day Night Level contour), freeing airports ofmost noise restraints. Aircraft noise has dropped dra-matically in the last 30 years with the development ofhigh bypass ratio engines. As the overall noise level hasbeen reduced, fan noise has become more prominentin the engine noise signature. Thus, accurate enginenoise predictions will rely heavily on the proper mod-eling of turbomachinery noise sources.
R. W. Dyson. Published by the American Institute of Aeronauticsand Astronautics, Inc. with permission.
broadband noise floor with discrete-frequency tones atthe multiples of rotor blade pass frequency superim-posed. For subsonic fans, the discrete-frequency tonesare usually 10−15 dB above the broadband level. Thediscrete-frequency tones provide only a minor contri-bution to the overall noise level but are the main sourceof irritating noise at take-off and landing when noisegeneration is most important. Thus, broadband anddiscrete-frequency noise play equally significant roles.
The discrete-frequency tones are generated by pe-riodic interactions between rotating and non-rotatingblade rows. Namely, the fan exit guide vane is subjectto the potential field and viscous wake from the fanrotor. The impact of the rotor potential field is mit-igated by maximizing the axial spacing between thefan rotor and the exit guide vane. Note that potentialflow interactions can become of concern if the axialspacing between blade rows is small or the flow Machnumber is high. The viscous wake of the rotor persistsover considerable axial distances, and is the primarysource of excitation seen by the exit guide vane.
The noise generation process is not well understood.The fan exit guide vane operates in an extremely com-plex flow field generated by highly loaded, arbitrarily
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American Institute of Aeronautics and Astronautics Paper 2003–1917
9th AIAA/CEAS Aeroacoustics Conference and Exhibit12-14 May 2003, Hilton Head, South Carolina
shaped airfoils. These complexities change the convec-tion characteristics of the gust and make calculation ofthe generated noise difficult. The complexity of boththe flow and geometry necessitates the solution of theEuler equations on a body-fitted grid. It is expectedhowever that in the far-field away from the fan exitguide vane the generated sound propagates in a man-ner that is consistent with classical linear theory. Thispaper presents the analysis of the noise generated bya rotor-stator interaction in terms of classical lineartheory.
Rotor-Stator Interaction
The stator is modeled herein as an isolated bladerow that is excited by the convected vortical gust ofthe rotor. Thus, the acoustic response of the statoris uncoupled from the rotor and can be independentlycalculated. The acoustic waves generated by this in-teraction propagate or exponentially decay in the ductupstream and downstream of the stator. The propa-gation or decay of an acoustic wave can be predictedassuming the acoustic pressure and unsteady velocityare small unsteady perturbations on a uniform meanflow. In a two-dimensional representation of this prob-lem, a cylindrical slice is taken from the annulus and“unwrapped” to form a two-dimensional cascade of air-foils, Fig. 1.
For a fan stage with NR rotor blades and NS statorvanes, the generated spatial modes must satisfy
m = nNR − lNS (1)
where n is the rotor harmonic and l = 0,±1,±2, ... isan arbitrary integer.
Computational Technique
The Euler equations in Cartesian coordinates are
∂Q
∂t+
∂E
∂x+
∂F
∂y= 0 (2)
where
Q = [ρ, ρu, ρv, ρw, E] (3)
E = [ρu, ρu2 + p, ρuv, ρuw, u(E + p)] (4)
F = [ρv, ρuv, ρv2 + p, ρvw, v(E + p)] (5)
and
p = (γ − 1)
(
E −1
2ρ(u2 + v2)
)
(6)
The chain-rule is applied to obtain the Euler equationsin curvilinear coordinates.
∂Q
∂τ+
(
∂ξ
∂x,∂η
∂y
)
�
[(
∂E
∂ξ,∂E
∂η
)
+
(
∂F
∂ξ,∂F
∂η
)]
= 0 (7)
The spatial derivatives of Equation 7 are approx-imated using the prefactored sixth-order compactscheme and explicit boundary stencils of Hixon.3
Equation 7 is then integrated in time using a low stor-age fourth-order nonlinear extension of Hu’s 5-6 LowDispersion and Dissipation Runge-Kutta scheme.2 Fi-nally, a 10th order explicit filter is used at everystage of the Runge-Kutta solver to dissipate unre-solved waves.
On solid boundaries such as the blade surface, themomentum normal to the surface is set to zero at eachRunge-Kutta stage ensuring the tangency of the flowwith the surface. At the inflow and outflow planes,Giles boundary conditions are used.
Giles Boundary Conditions
Giles non-reflecting boundary conditions are ob-tained using characteristic theory.5 Four character-istics are identified that correspond to upstream anddownstream going acoustic waves, a convected vortic-ity wave, and a convected entropy wave. The char-acteristics are determined using time derivative valuesfrom the interior equations. The characteristics aredefined as
c1
c2
c3
c4
=
−c2∞ 0 0 1
0 0 ρc∞ 00 ρc∞ 0 10 −ρc∞ 0 1
δρδuδvδp
(8)
where ρc∞ is the characteristic impedance of thefreestream. These four characteristics correspond toentropic, vortical, downstream going acoustic andupstream going acoustic waves, respectively. In-flow boundary conditions are created by prescribingthe downstream going characteristics c1, c2 and c3
and leaving the upstream going characteristic c4 un-changed. In a similar manner, outflow boundary con-ditions are created by prescribing the upstream goingcharacteristic c4 and leaving the downstream goingcharacteristics c1, c2 and c3 unchanged.
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American Institute of Aeronautics and Astronautics Paper 2003–1917
Fourth order, two-dimensional unsteady inflowboundary conditions are obtained by setting
∂c1
∂t= −v
∂c1
∂y
∂c2
∂t= −v
∂c2
∂y− 1
2 (c − u)∂c3
∂y− 1
2 (c + u)∂c4
∂y
∂c3
∂t= − 1
2 (c − u)∂c2
∂y− v
∂c3
∂y
(9)
Second order, two-dimensional unsteady outflowboundary conditions are obtained by setting
∂c4
∂t+ u
∂c2
∂y+ v
∂c4
∂y= 0 (10)
The time derivatives of the flow variables are thendetermined by
∂ρ
∂t=
1
2c2∞
[
−2∂c1
∂t+
∂c3
∂t+
∂c4
∂t
]
∂u
∂t=
1
2ρc2∞
[
∂c3
∂t−
∂c4
∂t
]
∂v
∂t=
1
ρc∞
[
∂c2
∂t
]
∂p
∂t=
1
2
[
∂c3
∂t+
∂c4
∂t
]
(11)
Linear Theory
Away from the stator vane passage, the unsteadyvelocity and acoustic fields are represented as smallperturbations superimposed on a uniform mean flow.4
Thus, the linearized Euler equations govern the flowin this region.
Dρ
Dt+ ρ∞∇ · ~v = 0 (12)
ρ∞D~v
Dt+ ∇p = 0 (13)
where ρ∞ is the freestream density, ρ is the pertur-bation density, ~v = (u, v) is the perturbation velocity,
p is the acoustic pressure, DDt
= ∂∂t
+ (~V · ∇), and ~Vis the freestream velocity. The assumed form of thesolution is
uvp
=
uvp
ei(kxx+kyy+ωt) (14)
where kx and ky are the wave numbers in the axialand tangential directions, ω is the frequency of theexcitation, and the overbar quantities are the pertur-bation amplitudes. The spatial mode order is relatedto the tangential wave number ky = − 2π
NSSm where
S is the pitch spacing. Substitution of Equation 14into Equations 12-13 gives a homogeneous linear sys-tem of equations. All nontrivial solutions to these
equations must have a determinant equal to zero. Innon-dimensional terms, the determinant is
[(k + kxMcosα + Mkysinα)2 − (k2x + k2
y)]
(k + kxcosα + kysinα) = 0 (15)
where the reduced frequency k = ωC/V , M is thefreestream Mach number, and α is the absolute flowangle. Equation 15 describes two types of waves. Thefirst is a vortical wave convected with the mean flow,and the second is a set of acoustic waves which travelupstream and downstream from the cascade. The axialwavenumber can be determined for a given reducedfrequency and the known tangential wave number thatsatisfies rotor-stator periodicity requirements.
The vortical waves have an axial wave number givenby
kx = −k + kysinα
Mcosα
The vortical waves have vorticity
ξ = ∇× ~v = (ky v − kxu)eiωt
that is simply convected with the mean flow and noacoustic part.
Acoustic wave solutions have axial wave numbersgiven by
kx =M2cosα(k + kysinα)
1 − M2cos2α±
√
k2M2 + 2kkyMsinα − (1 − M2)k2y
1 − M2cos2α(16)
All acoustic waves have vorticity ξ ≡ 0. The behaviorof an acoustic wave is determined by the argument ofthe square root.
• k2M2 + 2kkyMsinα− (1 − M2)k2y > 0, there are
two real wave numbers which describe waves thatpropagate upstream and downstream in the duct.
• k2M2 + 2kkyMsinα− (1−M2)k2y < 0, there is a
complex conjugate pair of wave numbers describ-ing waves which must decay exponentially awayfrom the cascade.
• k2M2 + 2kkyMsinα− (1 − M2)k2y = 0, there are
real repeated wave numbers which describe a res-onance condition.
Acoustic waves which decay in the duct are called “cut-off”, and waves which propagate are called “cut-on”.
An acoustic wave will propagate when the spatialmode order falls in the range m1 < m < m2 where m1and m2 are given by
m1,2 =NSS
2π
kM
1 − M2
[
M sinα ±
√
1 − M2 cos α2]
(17)
Applying spatial and temporal Fourier transforms tothe computed acoustic pressure upstream and down-stream of the stator, the decomposed acoustic field
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American Institute of Aeronautics and Astronautics Paper 2003–1917
becomes a function of frequency, spatial mode orderand axial location. The propagation or decay charac-teristics can then be examined to determine the natureof the response and the amplitude of the propagatingacoustic waves.
Joint Spatial-Temporal Transform
In a rotor-stator interaction, the response is not onlyharmonic in time, but also periodic in the tangentialdirection y, and Fourier transforms in time and spaceare applied. The acoustic pressure, computed Nt timesat position (x, y) over the interval T = Nt∆t where∆t is the time between samples, can be decomposedusing a joint Fourier transform of the acoustic pres-sure p(x, y, t). The joint transform gives the acousticpressure as a function of axial location, frequency andspatial mode order P (x, m, n). The joint Fourier trans-form is
P (x,m, n) =2
NNt
N−1∑
l=0
Nt−1∑
j=0
plj(x) e−2πi(nj/Nt+ml/N) (18)
where plj(x) is p(x, yl, tj), n = 0, ..., Nt/2, the fre-quency fn = n/T , N is the number of points in thetangential direction y, and −N/2 < m < N/2 is thespatial mode order. Note that only integer values of mare allowed (corresponding with an integer number ofperiods around the annulus). Also keeping only pos-itive frequencies implies the rotor and its convectedgust are rotating with the rotor shaft in the positivedirection.
In summary, the computed acoustic pressure can bedecomposed in time and space to determine the am-plitude of the acoustic waves as a function of rotorharmonic n, spatial mode order m and axial locationx. The behavior of the decomposed acoustic field willthen be compared with the behavior of the waves pre-dicted by classical linear theory where only certainspatial modes are generated by the rotor-stator inter-action and the propagation or exponential decay of thegenerated modes can be predicted.
Results
The interaction of a rotor and stator in an annularduct generates acoustic waves at the multiples of therotor blade pass frequency. The acoustic waves in theannulus are naturally periodic in the circumferentialdirection. The spatial mode order is the measure ofthe circumferential periodicity and gives the number ofperiods over circumference of the annulus. For a rotorand stator with NR rotor blades and NS stator vanes isis possible to predict the spatial mode order of the gen-erated waves. Further, the axial propagation, decay orresonance of the generated waves can also be predictedassuming the acoustic wave is a small perturbation ona uniform flow. The linear theory characteristics ofthe response will be used to support the veracity ofthe computed solution.
The stator cascade under consideration has NS = 27stator vanes and is excited by the wakes of NR = 11rotor blades.6 The computational grid for one passage,shown in Fig. 2, contains 9 blocks and 12,700 pointsper passage. The cascade has a pitch spacing S/C =2/3. The mean inflow and outflow conditions are givenby
inflow:
Pi = 1Ti = 1
αi = 36o
outflow: po/Pi = 0.92 (19)
where Pi is the non-dimensional stagnation pressureat the inlet, Ti is the non-dimensional stagnation tem-perature at the inlet, αi is the absolute flow angle atthe inlet, and po is the non-dimensional static pressureat the exit.
The stator is excited by the convected gust of therotor at the first three multiples of the rotor blade passfrequency. The rotor wake is steady in the rotatingframe and appears as a velocity deficit in the relativeflow direction. In the stationary frame, the rotor wakesare harmonic in time and periodic in the tangentialdirection. The vortical gust at the inflow plane is givenby
~ug = {a1cos[kyy − ωt]+
a2cos[2(kyy − ωt) + φ2)+
a3cos[3(kyy − ωt) + φ3]}eβ (20)
where eβ = (cosβ,−sinβ) is the unit vector in thedirection of the relative flow, the relative flow angleβ = 50o. For the case under consideration here, theexcitation frequency ω = 3π/4 and the tangential wavenumber ky = 11π/9. The amplitude of the gust is
a1 = 7 × 10−3
a2 = 3 × 10−3 φ2 = −7π/5
a3 = 4 × 10−4 φ3 = −π/2
Fig. 3 shows computed the acoustic pressure at aninstant in time corresponding to the beginning of theperiod (t/T = 0) for a portion of the 27 vane cascade.The tangential periodicity in the solution is evident.The maximum non-dimensional acoustic pressure inthe computational domain pmax = 0.03 or 163 dB oc-curs at the leading edge. Away from the cascade theacoustic pressure is on the order of 122 dB. The timeevolution of this field is decomposed using the jointspatial-temporal transform to determine the acousticpressure as a function of rotor harmonic n, spatialmode order m and axial location in the duct x. Thetransform is accomplished by saving 64 snapshots overthe course of two periods of excitation. A tempo-ral Fourier transform is applied to transform fromthe time to the frequency domain where only posi-tive frequencies are kept (i.e. n = 0, 1/2, 1, 3/2, ..., 32).
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American Institute of Aeronautics and Astronautics Paper 2003–1917
X
Y
-1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
Fig. 2 Computational grid for one of the 27 statorvane passages.
Response is only expected at n = 1, 2and3. To ac-complish the spatial transform rectangular regions areextracted upstream and downstream of the cascade.The acoustic pressure is now known as a function ofposition (x, y) and harmonic n. The spatial Fouriertransform is applied to remove the dependence on y.The computed acoustic field can now be represented asa function axial position x for a given harmonic n andspatial mode order m, but before showing the com-puted results the linear theory analysis will be appliedto predict (1) the generated spatial modes and (2) thecut-off spatial mode order for propagation/decay of thewave in the duct.
It is expected the rotor-stator interaction produceacoustic waves at the frequencies it is excited (n =1, 2, 3) and no others. Fig. 4 shows the time historyand the amplitude of the pressure spectrum at the in-flow plane (x, y) = (−1.5, 0). Clearly, the computedacoustic response is behaving in a linear manner with-out any evidence of nonharmonic response.
The spatial modes genererated at the first three
Time (Periods)
Pre
ssur
e
0 0.5 1 1.5 20.8372
0.8373
0.8374
0.8375
0.8376
0.8377
0.8378
0.8379
a) Pressure time history.
Rotor Harmonic
Pre
ssu
reA
mp
litud
e
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160
5E-05
0.0001
0.00015
0.0002
0.00025
b) Amplitude of the pressure spectrum.
Fig. 4 Pressure history and spectrum at (x, y) =(−1.5, 0).
rotor harmonics determined using Equation 1 m =nNR − lNS, Table 1. The computed results now de-composed in time and space are shown at the inflowplane (x = −1.5) for all three harmonics, Fig. 5. Thecomputed pressure exists only at the predicted spatialmode orders with no modal “spillover”.
Now the range of spatial mode which will propa-gate in the duct can be determined by application ofEquation 17, Table 2. The change in the mean flowvariables upstream and downstream of the cascadechanges significantly the range of propagating spatialmodes. Note that the acoustic response is “cut-off” atBPF with no propagating waves. At 2BPF, there isone wave m = −5 propagating upstream and down-stream in the duct. At 3BPF, there are two wavesm = −21 and 6 propagating in the upstream duct, but
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American Institute of Aeronautics and Astronautics Paper 2003–1917
Spatial mode order (m)
Pre
ssur
eA
mpl
itud
e
-97 -70 -43 -16 11 38 65 920
1E-05
2E-05
3E-05
4E-05
5E-05
a) BPF
Spatial mode order (m)
Pre
ssur
eA
mpl
itud
e
-86 -59 -32 -5 22 49 760
5E-05
0.0001
0.00015
0.0002
0.00025
b) 2BPF
Spatial mode order (m)
Pre
ssur
eA
mpl
itud
e
-75 -48 -21 6 33 60 870
5E-06
1E-05
1.5E-05
2E-05
2.5E-05
3E-05
3.5E-05
c) 3BPF
Fig. 5 Modal content of the pressure.
only one m = 6 propagating in the downstream duct.Note in the downstream duct the wave m = −21 at3BPF is very near the linear theory resonance condi-tion.
Finally, the linear theory analysis gives the abilityto examine the axial behavior of the generated modesin the duct. Several things are expected (1) decayingwaves decay exponentially away from the cascade (2)propagating waves propagate with constant amplitudeand (3) the generated waves exit the domain withoutreflection from the inflow or outflow boundaries.
The axial wavelengths and wavenumbers of thepropagating waves can be analytically predicted us-
Inflow OutflowAbs flow angle 35.99 Abs flow angle -0.16Mach Number 0.364 Mach Number 0.269Reduced Freq 6.47 Reduced Freq 8.76
Table 3 Wavelengths and wavenumbers of the gen-erated modes.
ing Equation 15. Here the change in the mean flowvariables upstream and downstream of the cascadecauses only small changes in the wavelengths of thegenerated modes, Table 3. Note numerical disper-sion in the computed results will cause the computedwavenumbers/wavelengths to differ from these analyt-ically predicted values.
The behavior of the generated modes at each mul-tiple of BPF will be examined in turn. The acousticresponse at BPF is shown in Fig. 6. As predicted, thegenerated waves decay in the direction away from thecascade.
Fig. 7 shows a series of plots related the propa-gation or decay of the modes generated at 2BPF inthe region upstream of the cascade. Fig. 7(a) showsthe amplitude of the generated waves as a functionof axial location x. The m = −5 mode propagatesat nearly constant amplitude (2.45E − 04, 121.3 dB)while all other generated waves decay in the directionaway from the cascade. Fig. 7(b) shows the m = −5mode with the nearly linear phase distribution of anupstream traveling wave. Fig. 7(c) shows the complexamplitude of the m = −5 mode. A perfectly propagat-ing upstream going wave would appear on Fig. 7(c) asa circle. Finally, Fig. 7(d) shows some evidence of dis-persion in the solution. The analytical wave numberis roughly 87% of the numerical value.
The 2BPF acoustic response downstream of thecascade is shown in Fig. 8. The propagating wavem = −5 shows signs of a reflection from the down-stream boundary, Fig. 8(a). The amplitude of thedownstream going wave is (2.57E− 04, 121.3 dB), andthe reflection is approximately 6% of the incident wave.The complex amplitude of the wave, Fig. 8(c), showsa noncircular shape with irregular angular phase dif-
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American Institute of Aeronautics and Astronautics Paper 2003–1917
X
Pre
ssur
eA
mpl
itud
e
-1.5 -1.2 -0.9 -0.6
5E-05
0.0001
0.00015
0.0002
0.00025
0.0003
0.00035
0.0004
0.00045
0.0005
0.00055
0.0006
0.00065
0.0007
0.00075
0.0008
m = -16m = 11m = 38
a) Pressure Amplitude at BPF, inflow.
X
Pre
ssur
eA
mpl
itud
e
0.6 0.9 1.2 1.5
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.001
0.0011
0.0012
m = -16m = 11m = 38
b) Pressure Amplitude at BPF, outflow.
Fig. 6 Acoustic waves at BPF (all waves “cut-off”).
ference from point to point. The modulation in am-plitude and irregular phase shift are characteristics ofan acoustic field with a reflected wave superimposedon the downstream going incident wave. The variationin the point-to-point phase difference is also manifestin the variation of wavenumber, Fig. 8(d). Note thatwith the longer wavelength of the downstream goingwave the numerical wavenumber is much closer to theanalytical value.
At 3BPF, there are two modes m = −21 and 6 prop-agating upstream at the inflow, Fig. 9. Again as inthe 2BPF response, there is little evidence of reflec-tion from the inflow boundary. The wave amplitudesare nearly constant. The amplitude of the m = 6 mode(3.63E−05, 104.7 dB) is the higher than the amplitudeof the m = −21 mode (2.62E − 05, 101.9 dB). Eachof the propagating waves has a numerical wavenum-ber which is slightly lower than the analytical value,Fig. 9(d).
The behavior of the generated modes downstreamof the cascade for 3BPF is shown in Fig. 10. The largeamplitude of the m = −21 mode is evidence of its nearresonance condition, Fig. 10(a). Evidence of reflectionof the propagating mode m = 6 is also seen. Theamplitude of the incident wave is (1.99E−05, 99.5 dB)with a reflected wave amplitude of approximately 5%.
In summary, the amplitudes of the upstream anddownstream propagating waves generated by therotor-stator interaction are shown in Fig. 11(a) andFig. 11(b), respectively. In the upstream duct, thewaves propagate with little apparent reflection withthe inflow boundary. However in the downstream duct,reflections are apparent at the outflow boundary.
Conclusions
Aircraft noise is is significant environmental con-cern. The proper prediction of turbomachinery noisesources will play an important role in meet ever morestringent noise requirements. This paper presentedrotor-stator generated discrete-frequency noise predic-tions. Although the fully nonlinear Euler equations aresolved, the generated noise was expected to have manycharacteristics which could be predicted and comparedwith a linear theory analysis where the acoustic field isa small harmonic perturbation on a uniform freestreamflow. From the linear theory analysis, the frequency,the generated modes and the propagation or decayof the generated modes can be predicted and com-pared with the computed values. The interaction noisewas “cut-off” at BPF, but propagating modes were“cut-on” at 2BPF and 3BPF. The following commentssummarize the results:
• the computed acoustic field existed only at thefrequencies of excitation
• the modal content of the acoustic field was con-sistent with linear theory
• the propagation or decay of the generated modeswas consistent with linear theory
• some dispersion of the generated waves was appar-ent. The dispersion error decreased as expectedfor waves with longer wavelengths.
• the waves propagating upstream from the cas-cade showed little sign of reflection with the inflowboundary
• the wave propagating downstream from the cas-cade showed signs of reflection with a reflected
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American Institute of Aeronautics and Astronautics Paper 2003–1917
X
Pre
ssur
eA
mpl
itud
e
-1.5 -1.2 -0.9 -0.62.52E-09
5.00E-05
1.00E-04
1.50E-04
2.00E-04
2.50E-04
m = -32m = -5m = 22
a) Pressure Amplitude at 2BPF inflow.
X
Pre
ssur
eP
hase
(Rad
ians
)
-1.5 -1.2 -0.9 -0.6-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
b) Pressure Phase at 2BPF (m = −5) inflow.
Real P
Imag
P
-0.0002 -0.0001 0 1E-04 0.0002-0.00025
-0.0002
-0.00015
-0.0001
-5E-05
0
5E-05
1E-04
0.00015
0.0002
0.00025
c) Complex Pressure Amplitude at 2BPF (m = −5) in-flow.
X
Axi
alW
aven
um
ber
-1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.60
1
2
3
4
5
6
7
8
d) Axial wavenumber for m = −5 at 2BPF inflow.
Fig. 7 Acoustic waves at 2BPF (m = −5 propagating).
wave amplitude roughly 5% that of the incidentwave
Although it is not possible to accurately predict theamplitude of the propagating waves using linear the-ory, it would be impossible to compute the correct fieldif each of the above conditions were not met. It isnoted that further computations are required to verifythe amplitude of the noise predictions.
Acknowledgments
This work was supported by the NASA Glenn Re-search Center under the Quiet Aircraft TechnologyProgram (NAG3-2537). The authors are thankful forthe technical correspondence and support of Dr. EdEnvia. The authors would also like to recognize Dr.Christopher Miller who was responsible for the setup,
maintenance and administration of the Beowulf clusterused for these computations.
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American Institute of Aeronautics and Astronautics Paper 2003–1917
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