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frequencies where several radial modes are able to propagate along the duct, depending on the spinning order
number and hub to tip ratio. Cut-on cut-off and/or cut-off cut-on modal transitions could also occur in the inlet and
aft fan duct.
Nayfeh, et al.1
presented an extensive review on acoustic propagation inside aircraft engine duct systems, up to
1975. In the last three decades the acoustic propagation inside ducts has been exhaustively studied. A more recent
review was given by Eversman2
, published in 1991. Syed, et al.3
derived an analytical solution of sound propagation
in a straight annular fan exhaust duct with axially segmented treatment for the uniform flow and the radially sheared
flow cases. Nayfeh, et al.4
used the method of multiple scales for sound propagation in a slowly variable area duct.
Eversman5
developed a FE model for duct propagation and radiation for axially symmetric duct in non-uniform
flows. More recently, Richards, et al.6
used the linearized Euler equations (LEE) to determine the near field
propagation using a higher order temporal and spatial scheme. Far field directivity is predicted using an integral
solution of the FW-H equations.
The aim of this paper is to accomplish three main objectives. The first objective is to validate the far field
radiation code and compare prediction against available experimental data for both hard wall and acoustically
treated configurations. The reason for this validation is to check the implementation of the acoustic treatment
boundary condition which is based on the full Myers7
boundary condition. The second objective is to carry out a
comprehensive study to investigate modal scattering inside the duct and its impact on the far field directivity. The
work by Ovenden, Eversman and Rienstra8
will be extended to include the soft wall boundary effect on mode cut-on
cut-off or cut-off cut-on transitions. The third objective is to investigate the important parameters that affect the far
field directivity patterns such as inlet lip thickness. Acoustic measured data and numerical results obtained by other
investigators will be used whenever possible to validate numerical results.
II. Problem Formulation
A. Governing equations
For an inviscid, non-heat conducting compressible, isentropic flow, the governing equations in non dimensional
form can be written as
Continuity: ( ) 0~~
~
=•+
t
(1)
Momentum: 0~~~
~
~ =+•+ p
t
(2)
Equation of State: ~~ =p (3)
The problem is non-dimensionalised using the duct radius at the fan face R, as a reference length, the free stream
density, , and speed of sound, c . For a potential flow, the velocity can be written in terms of a velocity potential
as follows:
( ) ( )tXtX ,,~ rr
= (4)
The governing equations are solved by splitting the velocity potential into a first order perturbation acoustic
velocity potential superimposed on a steady velocity potential field, so that
),()(),( tXXtX
rrr
o += (5)
and
),()(),(~ tXXtX
rrr
o += (6)
where the o subscript denotes the mean flow field variables and the unsubscripted ones represent the acoustic
perturbation variables. By substitution of Eqs. (4) – (6) into Eqs. (1) – (3), yields
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Steady Flow:
( ) 0=• oo (7)
E
C
=+
12
1 2
2
o (8)
12
= ooC (9)
Acoustic Perturbation:
( ) ( ) ( ) 0
2
=
•+
•+•
oC
i
i (10)
( )•+= ip (11)
2
oCp = (12)
where the harmonic dependence of the acoustic quantities are defined in terms of , the non dimensional frequency
of the acoustic source. Because the acoustic perturbation equations are dependent on the mean flow variables, the set
of Eqs. (7) – (9) must be solved first to obtain the mean flow before starting solving the acoustic perturbation
problem. The same mesh used for obtaining the mean flow is used for the acoustic calculations. The Eversman’s
radiation code solves Eq. (7) for the velocity potential, assuming a constant density field while the modified code
solves the compressible set of Eqs. (7) – (9) using an iterative scheme.
B. Boundary conditions
1. The radiation code
The computational domain used in the Eversman’s radiation code is shown in Fig. 1a. The computational mesh
is divided into three regions. The first region is contained inside the inlet and extends up to the highlight circle. The
second region extends from outside the inlet to a certain boundary, B, from the inlet, where the flow is nearly
uniform. The third region extends from B to the outer boundary. Regions I and II use a conventional finite elements
whereas region III uses mapped infinite wave envelope elements. For the mean flow calculations, the normal
velocity component vanishes at the center body and at the outer wall. The mean flow is defined completely by the
fan face and the free stream Mach numbers, Mf and M ; respectively. Using a mass flux balance between the inflow
and outer flow the complete set of equations can be solved by forcing the velocity potential to a constant value at a
single point inside the domain. For the acoustic perturbation problem, Eqs. (10)-(12), the normal component of the
particle velocity vanishes at the center body and at the outer wall for hard wall surfaces and the full Meyers
boundary condition is used for treated boundaries9
. The source is assumed to propagate from left to right. For the
inlet duct, the sound wave propagate against the mean flow whereas in the fan exit the wave propagates with the
mean flow. For a specific spinning mode number, the complex modal amplitudes of the right running incident waves
are specified at the source plane. The left running reflected waves are calculated as part of the solution. The
radiation code uses mapped infinite wave envelope elements at far field which allow propagation in only one
direction simulating a non reflection radiation condition. More details about the boundary conditions implemented in
the radiation code is given in Ref. 10.
2. The modified propagation code
The computational domain used in the modified code is shown in Fig. 1b. The boundary conditions at the fan
face, inner nacelle, and center body surfaces are the same as the radiation code. The mean flow velocity potential
takes a constant value at the termination boundary8
. For the acoustic perturbation equations, a non reflecting
termination boundary condition is used which allows only right propagating waves and the left running waves are
forced to vanish at that boundary.
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III. Numerical Solution
Both the steady flow and the acoustic perturbation equations are solved using the standard Galerkin FE
formulation. The computational domain is divided into a finite mesh which contains at least 5 quadratic elements per
effective wavelength. The effect of the mean flow on the acoustic wavelength is taken into consideration. The same
grid is used for both mean flow calculations and acoustic solution. Although the far field radiation problem contains
the duct as part of the solution it was decided, for computational efficiency to develop a modified propagation code
with a non reflecting termination boundary to study the modal scattering inside the duct due to geometry, steady
flow and the acoustic treatment boundaries. Both the inner nacelle and center body surfaces could be treated. The
source boundary allows reflected waves from inside the duct. The right running incident waves are input at the
source plane while the left running reflecting waves are obtained as part of the solution.
IV. Results and Discussion
The Eversman radiation code is first validated against measured far field data for both hard wall and treated
inlets. The JT15D11
nacelle, used in the validation case, is shown in Fig. 2. Both hard wall and treated nacelle
configurations are shown. The acoustic source is generated by inlet flow distortion rotor interaction. The flow
distortion is created by the wakes of 41 rods in the inlet. The number of fan blades is 28. At a non dimensional
frequency of 15.4 only one radial mode, (-13,1), first radial, is propagating. Predicted and measured directivity
pattern of that mode for the hard wall case is shown in Fig. 3a. Good agreement between prediction and
measurement is noticed. For the treated case, the specific acoustic impedance is Z = 2.272 + 0.5i, which is the same
value reported in Ref. 11. The validation between predicted and measured directivity for treated nacelle is still in
good agreement. There is about 1-2 dB discrepancy in the peak sound pressure level. This could be attributed to the
uncertainty in the given value of the impedance. The sound pressure level contours for the JT15D inlet for both hard
and treated nacelles are shown in Fig. 4, which shows the attenuated levels with the presence of the acoustic
treatment.
Table 1. The four cases considered in the inlet modal scattering.
Case No. Acoustic Source Duct Boundary Inlet Mean Flow
1 m = 21, n = 4, = 41.0 Hard Wall No Flow
2 m = 21, n = 4, = 41.0 Hard Wall Mf = 0.5
3 m = 21, n = 4, = 41.0 Treated Wall No Flow
4 m = 21, n = 4, = 41.0 Treated Wall Mf = 0.5
The modified propagation code is used next for inlet and fan exit modal scattering due to duct geometry, mean
flow field, and acoustic treatment boundaries. For the inlet modal scattering problem, four cases are considered as
given in Table 1. The computational domains for the four inlet configurations are shown in Fig. 5. The first
configuration is hard wall inlet with no mean flow. The second configuration is hard wall inlet with Mf = 0.5. The
third configuration is treated inlet wall with no mean flow. The fourth configuration is treated inlet wall with
Mf=0.5. Streamlines and Mach number contours for the inlet with Mf = 0.5 are given in Fig. 6. Notice that the flow
goes from right to left, i.e. inlet flow. The acoustic source used in this case is the same as the one used in Ref. 8, m =
21, n = 4, and = 41.0. This case was chosen for comparison with the results obtained from Ref. 8. The sound
pressure level contours for the four different cases are shown in Fig. 7. Case 1 shows the (21,4) mode will encounter
cut-on cut-off at about xt = 1.5 similar to Ref. 8. Case 2 shows that adding an inlet flow with Mf = 0.5 to the hard
wall inlet will eliminate the cut-on cut-off inside the duct and the (21,4) mode will continue to propagate inside the
duct and radiates to the far field. Case 3 shows some modal scattering and sound pressure level attenuation inside
the duct. Although the (21,4) mode will continue to cut-off at about the same station as the hard wall case, the
acoustic treatment will scatter the mode to other radial orders as shown in Fig. 7c. Case 4 shows that by adding an
inlet flow with Mf = 0.5 to the treated duct, the (21,4) mode will continue to propagate but with more attenuated
levels than case 2.
The outer wall sound pressure level for the same mode is shown in Fig. 8 for the four cases. Case 1 shows a
strong standing wave pattern inside the inlet duct due to the interaction between incident and reflected acoustic
modes. Again the (21,4) mode is cut off for x 1.5 and the sound pressure level rises just before the mode cuts-off
and then decays rapidly in the duct. Introducing an inlet flow, the (21,4) mode will continue to propagate. As
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mentioned previously, the acoustic treatment will enhance modal scattering in the duct and will attenuate the sound
pressure level during the propagation phase of the mode. Due to modal scattering, other modes will continue to
propagate in the duct contributing to the total acoustic pressure. Adding an inlet flow with the treated duct will again
reduce the standing wave pattern in the duct as in case 4. The far field sound pressure level directivity for the four
cases are obtained from the radiation codes and shown in Fig. 9. The directivity patterns show that both the mean
flow and the acoustic treatment have significant effect on the far field sound pressure levels.
Table 2. The four cases considered in the fan exit modal scattering.
Case No. Acoustic Source Duct Boundary Fan Exit Mean Flow
1 m = 20, n = 6, = 52.0 Hard Wall No Flow
2 m = 20, n = 6, = 52.0 Hard Wall Mf = 0.5
3 m = 20, n = 6, = 52.0 Treated Wall No Flow
4 m = 20, n = 6, = 52.0 Treated Wall Mf = 0.5
In the fan exit propagation problem, the sound waves propagate with the flow. Similar to the inlet, four cases are
considered as given in Table 2. The computational domains for the four cases are shown in Fig. 10. The mean flow
streamlines and Mach number contours are shown in Fig. 11 for Mf = 0.5. Due to the duct axial variation, the mean
flow expands to higher Mach numbers than in the inlet case. The acoustic source selected is m = 20, n = 6, and =
52.0. The sound pressure level contours for the (20,6) mode for the four cases are given in Fig. 12. The (20,6) mode,
which is propagating mode at the source plane, will encounter cut-on cut-off transition at xt = 0.9. Case 2 shows that
by adding fan exit flow the mode will continue to propagate but with reduced levels than in case 1. Case 3 shows
that the acoustic treatment will attenuate the sound pressure levels inside the duct and the axial station of the cut-on
cut-off transition is slightly moved upstream of the hard wall case. Adding fan flow, Mf = 0.5, to the treated duct
will increase the sound pressure levels inside the duct but the (20,6) mode will continue to cut-off. The outer wall
sound pressure levels of the (20,6) mode, for the four cases, are shown in Fig. 13. As we noticed in the inlet cases,
the standing wave pattern generated in the duct due to the interaction between incident and reflected waves are
reduced by the presence of the flow. Also, the acoustic treatment attenuates the sound pressure levels inside the duct.
It is interesting to note that the sound pressure level for case 2 is nearly flat.
Table 3. The three cases considered in the inlet lip study.
Case No. Acoustic Mode Frequency Cut-off Ratio
1 m = 1, n = 1 = 5.0 3.35
2 m = -13, n = 1 = 15.4 1.03
3 m = -13, n = 1 = 25.0 1.67
During static engine tests, a bell mouth inlet is used to simulate the flight conditions around the lip region. The
inlet lips used with those inlets are usually different from those of the flight inlet lips. The acquired static engine
data are extrapolated to flight conditions for noise certification purposes. One of the objectives of the current study
is to investigate the effect of inlet lip on noise radiation .
Two different inlets are considered as shown in Fig. 14. Both inlets have the same duct length and radius. The
only difference between the two inlets is the lip thickness, inlet 1 has a thicker lip than inlet 2. The radiation code is
used in this problem. Three different cases are considered as given in Table 3. The first case is m = 1, n = 1, = 5.0,
the mode is well above cut-on mode. The second case is m = -13, n = 1, = 15.4, the mode is barely cut-on mode.
The third case is m = -13, n = 1, = 25.0, the mode is above cut-on. Those three cases are selected to represent
acoustic sources at low, mid, and high frequencies with different cut-off ratios. The fan face Mach number, Mf, is
0.175. The far field directivity patterns for all cases and for the two inlets, are shown in Fig. 15. The directivity
patterns of the (1,1) mode at = 5.0 are given in Fig. 15a. There is a slight difference between the two patterns at
directivity angles above 50o
, the directivity angle is measured from the forward arc center line. The directivity
patterns of the (-13,1) mode at = 15.4 are shown in Fig. 15b. Since this mode is barely cut-on, there is a significant
difference between the directivity patterns of the two inlets. Also there is a shift in the peak directivity angle. The
peak angle of the thin lip is about 55o
, while for the thick lip is 50o
. The directivity patterns of the (-13,1) mode at
= 25.0 are given in Fig. 15c. Since the (-13,1) mode at = 25.0 is well cut-on, there is a slight difference between
the two lips at lower angles but there is appreciable difference at larger angles. The sound pressure level contours
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for the two inlets, for the (-13,1) mode at = 15.4, are given in Fig. 16. The inlet with thinner lip direct the sound
waves to higher directivity angles and the peak angle shifts to larger angle.
V. Conclusions
The Eversman radiation code has been validated with measured data for both hard wall and treated ducts. The
comparison between predicted and measured data shows good agreement. A modified version of the Eversman’s
code has been used to study the cut-on cut-off and cut-off cut-on modal transition inside the duct. The work of
Ovenden, et al. has been extended to include the acoustic treatment and flow for inlets and fan exhausts. The flow
effect in the inlet noise propagation is to delay the cut-on cut-off transition. Adding mean flow to the duct reduces
the standing wave pattern generated due to the interaction between incident and reflected waves. The acoustic
treatment enhances the modal scattering inside the duct and attenuate the sound pressure levels of propagating
modes. The optimal acoustic nacelle design is to maximize the cut-on cut-off occurrence in both inlet and fan ducts.
The noise directivity from static inlet lip could be different from flight inlet lip for modes which are close to cut-on.
The prediction could be used to provide correction factors for static-to-flight extrapolation process.
References
1
A.H. Nayfeh, J.E. Kaiser, and D.P. Telionis, “Acoustics of aircraft engine duct systems,” AIAA J., Vol. 13, no. 2, 1975, pp.
130-153.
2
W. Eversman, “Aeroacoustics of flight vehicles: Theory and practice,” Vol. 2: Noise control, pp. 101-163, NASA reference
publication 1258, 1991.
3
A.A. Syed, R.E. Motsinger, G.H. Fiske, M.C. Joshi, R.E. Kraft, “ Turbofan aft duct suppressor study,” NASA CR 175067,
July, 1983.
4
A.H. Nayfeh, J.E. Kaiser, R.L. Marshall and C.J. Hurst, “A comparison of experiment and theory for sound propagation in
variable area ducts,” Journal of Sound and Vibration, 1980, 71(2), 241-259.
5
W. Eversman and I. Danda Roy, “Ducted fan acoustic radiation including the effects of nonuniform mean flow and acoustic
treatment,” 15th
AIAA Aeroacoustics Conference, AIAA 93-4424.
6
S.K. Richards, X. Zhang and X.X. Chen, “Acoustic radiation computation from an engine inlet with aerodynamic flow
field,” 42nd
AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2004-0848.
7
M.K. Myers, “On the acoustic boundary condition in the presence of flow,” J. of sound and vibration, 71, 429-434, 1980.
8
N.C. Ovenden, W. Eversman and S. W. Rienstra, “Cut-on cut-off transition in flow ducts: comparing multiple-scales and
finite-element solutions,” Paper 10th
AIAA/CEAS Aeroacoustics Conference, AIAA 2004-2945.
9
W. Eversman, “The Boundary Condition at an Impedance Wall in a Nonuniform Duct With Compressible Potential Mean
Flow,” J. of Sound and Vibrations, 110, 2001, 41-47.
10
W. Eversman and O. Okunbor, “Aft Fan Duct Acoustic Radiation,” J. of Sound and Vibrations, 213, No. 2, 1998, 235-257.
11
K.J. Baumrister and S.J. Horowitz, “Finite element-integral simulation of static and flight fan noise radiation from the
JT15D turbofan engine,” NASA TM 82936.
12
C.L. Morfey, “Acoustic energy in non-uniform flows,” Journal of sound and Vibration, 1971, 14(2), 159-170.
13
E.C. Rice and M. F. Heidmann, “Modal propagation angles in a cylindrical duct with flow and their relation to sound
radiation,” 17th
Aeroacoustics Meeting, 1079, Paper 79-0183.
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(a) Domain of the radiation code (b) Domain of the modified code
Figure 1. Computational domains for both the radiation and the modified code.
(a) Hard wall nacelle (b) Treated wall nacelle
Figure 2. JT15D nacelle used in the validation case, both hard wall and acoustically treated duct are
shown with the specific acoustic impedance given.
(a) Hard wall nacelle (b) Treated wall nacelle
Figure 3. Far field directivity pattern of a single radial mode (-13,1) propagating in the JT15D inlet at =
15.4, solid line is prediction and symbols are data.
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8
(a) Hard wall nacelle (b) Treated wall nacelle
Figure 4. Sound pressure level contours for the JT15D inlet for hard and treated wall configurations.
Figure 5. Computational domain for four different inlet configurations a) Hard wall inlet, no flow b)
Hard wall inlet, Mf = 0.5 c) Treated inlet, no flow d) Treated inlet, Mf = 0.5.
(a) Mean flow streamlines (b) Mach number contours
Figure 6. Mean flow streamlines and Mach number contours for the inlet case with Mf = 0.5.
(d) Treated Nacelle, Mf = 0.5(c) Treated Nacelle, no flow
(a) Hard wall, no flow (b) Hard wall, Mf = 0.5
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9
Figure 8. Outer wall sound pressure level for m = 21, n = 4, = 41.0 a) Hard wall inlet, no flow b) Hard
wall inlet, Mf = 0.5 c) Treated inlet, no flow d) Treated inlet, Mf = 0.5.
Figure 7. Sound pressure level contours for m = 21, n = 4, = 41.0 a) Hard wall inlet, no flow b) Hard
wall inlet, Mf = 0.5 c) Treated inlet, no flow d) Treated inlet, Mf = 0.5.
(a) Hard wall, no flow (b) Hard wall, Mf = 0.5
(c) Treated wall, no flow (d) Treated wall, Mf = 0.5
(a) Hard wall, no flow (b) Hard wall, Mf = 0.5
(c) Treated wall, no flow (d) Treated wall, Mf = 0.5
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Figure 9. Far field directivity patterns for the four inlet cases m = 21, n = 4, = 41.0.
Figure 10. Computational domain for four different fan duct configurations a) Hard wall fan duct, no flow
b) Hard wall fan duct, Mf = 0.5 c) Treated fan duct, no flow d) Treated fan duct, Mf = 0.5.
(a) Mean flow streamlines (b) Mach number contours
Figure 11. Mean flow streamlines and Mach number contours for the fan exit case with Mf = 0.5.
(a) Hard wall, no flow (b) Hard wall, Mf = 0.5
(c) Treated wall, no flow (d) Treated wall, Mf = 0.5
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Figure 12. Sound pressure level contours for m=20, n=6, =52.0 a) Hard wall fan duct, no flow b) Hard
wall fan duct, Mf = 0.5 c) Treated fan duct, no flow d) Treated fan duct, Mf = 0.5.
Figure 13. Outer wall sound pressure levels for m=20, n=6, =52.0 a) Hard wall fan duct, no flow b)
Hard wall fan duct, Mf = 0.5 c) Treated fan duct, no flow d) Treated fan duct, Mf = 0.5.
(a) Hard wall, no flow (b) Hard wall, Mf = 0.5
(c) Treated wall, no flow (d) Treated wall, Mf = 0.5
(a) Hard wall, no flow (b) Hard wall, Mf = 0.5
(c) Treated wall, no flow (d) Treated wall, Mf = 0.5
(a) Hard wall, no flow (b) Hard wall, Mf = 0.5
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Figure 14. Inlet nacelle with two different lip thicknesses.
Figure 16. Sound pressure level contours for the two inlets m = -13, n = 1, = 15.4.
Figure 15. Far field directivity patterns for a single radial propagating mode for the two inlets, Mf = 0.175.
(a) m =1, n=1, = 5.0 (b) m = -13, n = 1, = 15.4 (c) m = -13, n = 1, = 25.0
(a) Thick Lip (b) Thin Lip
(a) Thick Lip (b) Thin Lip
