Comparison of Mode-Matching and Two

Comparison of Mode-Matching and Two-Port
Formulations for Acoustic Impedance Eduction
of Liners Under Grazing Flow
Augusto Amador Medeiros
Vibrations and Acoustics Laboratory, Federal University of Santa Catarina, Florianópolis, Brazil
Zargos Masson
Vibrations and Acoustics Laboratory, Federal University of Santa Catarina, Florianópolis, Brazil
Pablo Giordani Serrano
Vibrations and Acoustics Laboratory, Federal University of Santa Catarina, Florianópolis, Brazil
Danillo Cafaldo dos Reis
EMBRAER S.A., São José dos Campos, Brazil
Julio Apolinário Cordioli
Vibrations and Acoustics Laboratory, Federal University of Santa Catarina, Florianópolis, Brazil
Abstract
Measuring acoustic impedance of liners used in aircraft engines has become a point of interest in the
last decades, especially in the presence of grazing flow, similar to what occurs under their operational
conditions. Different indirect methodologies have been developed by independent research groups to
assess this problem, such as the Mode-Matching technique and the Two-Port method. In this paper,
these two indirect techniques are implemented and then validated by means of numerical simulations
using the Finite Element Method in both no-flow and grazing flow conditions of up to 0.3 Mach
numbers. The two methods are then compared based on their accuracy, computational cost, required
inputs and limitations. Finally, both methods are evaluated with test data measured in a new grazing
flow impedance eduction test rig designed and built by the authors.
1. Introduction
The rapid growth of the aircraft transportation sector
in the last decades, coupled with the urbanization of
airports’ surrounding areas in most cities, has led to
stricter regulations regarding noise levels radiated by
aircraft, specially during take-off and approach conditions. As a consequence, aircraft manufactures have
focused their attention in identifying and reducing the
main noise sources of aircrafts, specially at take-off
and landing [1].
One of the main sources of noise in an aircraft is
its engine, which holds specially true during previously cited conditions. Because of that, noise treatment of aircraft engines has become a point of interest for manufactures. Since engine noise is generally dominated by tonal components, associated with
blade-passage frequencies, one efficient way of noise
(c) European Acoustics Association
treatment for this application are tunable narrow frequency acoustic treatments known as liners. Their
typical form is a layer of honeycomb material, on top
of a rigid plate, and below a perforated plate. That
configuration can be seen as a single-degree of freedom system, providing good attenuation in a narrow
frequency band, which can be tuned to be around a
specific blade-passage frequency [2].
A very important property of an acoustic liner is
the acoustic impedance, specially under grazing flows,
similar to what is seen under operational conditions.
The determination of impedance under grazing flow
is a non-trivial task, and efforts have been carried out
by multiple research groups around the world in the
last decades to develop a methodology, resulting in
many indirect techniques [3, 4, 5].
The so-called "impedance eduction techniques" usually consist in measuring the acoustic field in a duct
where a sample of liner material is subject to a grazing flow. An analytical or numerical model is used
to calculate the acoustic field for a given impedance
value and the results are compared to the measured
l
FORUM ACUSTICUM 2014
7-12 September, Krakow
Methods for Impedance Eduction Under Grazing Flow
z
data. The impedance is then varied until the calculated acoustic field matches the measured data. z
b
y
b
The mainx differences between the multiple techz
niques available are the way the acoustic field in the
presence of the liner material is calculated, and the
optimization process adopted.
h
y
x
In this paper, two different impedance eduction
methods were implemented, validated using numerical simulation results in both no-flow and grazing flow
conditions of up to 0.3 Mach numbers, and compared
based on accuracy, computational cost, required inputs and limitations.
The first technique, hereafter called Two-Port
Method (TPM) [6, 7] uses a formulation based on twoport matrices to describe a lined section of a duct. An
analytic transfer matrix is used in a cost function for
an optimization problem from which the longitudinal wavenumbers in the lined duct are found. These
wavenumbers are then used to directly calculate the
unknown impedance.
The second technique, hereafter called the ModeMatching Method (MMM) [8, 9] uses the modematching approach to calculate the amplitude of each
mode propagating in a lined duct for an expected
impedance value. Those modes and their corresponding amplitudes and wavenumbers are then used to
calculate the acoustic field in the duct, which is compared to the measured data in a cost function. The
expected impedance is then varied in an optimization
process until the calculated acoustic field converges to
the measured data.
In the last section of this paper, both methods will
be used to calculate the impedance of a liner sample
in no-flow and Mach 0.25 conditions, using test data
from a new test rig designed and built by the authors.
l
z
Figure 1. Rectangular duct with height h and width b,
whose wall at x = b has an impedance Zwx along a section
of length l.
2.1. Sound Propagation in a Duct with Uniform Mean Flow
In a duct with uniform mean flow in the axial direction, the convected wave equation [10] for linear
acoustics is given by
∇2 p −
fc =
c0
(1 − M 2 )1/2 ,
2h
(2)
where M is Mach number of the uniform mean flow in
the duct, and h could be replaced by b for the modes
in the other direction. The acoustic field in each of
the three ducts n, pn , is calculated by the summation
of the Q modes in the duct, whose mode shapes are
given by Φ(x, y), so that
pn =
Both the TPM and the MMM assume a duct with
the geometry shown in Figure 1, whose walls are rigid
except for a section of length l where an impedance
Zwx is applied to the wall at x = b. The duct can be
seen as having three different sections: an inlet duct
(1), then a lined duct, which represents the region
with the liner in the experimental configuration (2),
followed by a outlet duct (3). Both the inlet and outlet
ducts have hard walls. Uniform flow is assumed in all
sections.
(1)
where D/Dt is the material derivative. Each solution
to the wave equation represents a mode, that propagates only above its cut-on frequency fc , given by
2. Impedance Eduction Techniques
The two techniques cited in the introduction are very
similar in essence, so this section will be used to describe some of the theory shared by both methods.
After that, each method will be presented in more
detail separately.
1 D2 p
= 0,
c20 Dt2
Q
X
q=1
(q)
(q)
(q)
ani Φni e−jkzni z +
Q
X
(q)
(q) jkznr z
a(q)
(3)
nr Φnr e
q=1
where the indexes i and r represent the incident and
reflected waves which propagate respectively in the
z+ and z− directions, q is the index of the mode,
in a crescent order of its cut-off frequency, a is the
(q)
amplitude of each mode, and kzn is the wavenumber
in the z direction for the q-th mode, that satisfies the
dispersion relation [10]
kx2 + ky2 + kz2 = (k0 ± M kz )2 ,
(4)
where k0 is the wavenumber ω/c0 , c0 being the speed
of sound in the fluid. The wavenumbers in the crosssectional directions, kx and ky , are implicit in the
mode shapes Φ(x, y). An omitted ejωt harmonic time
FORUM ACUSTICUM 2014
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Methods for Impedance Eduction Under Grazing Flow
dependence is assumed in the solution of equation 3
and all equations derived from it.
Using the hard-wall boundary condition at all walls
in ducts 1 and 3, the wave numbers and the mode
shapes can be calculated. The wavenumbers in the x
and y directions for the q-th mode are
and the reflection coefficient at its exit. These quantities can be computed using the well-known TwoMicrophone Method on each hard-wall section. A
more general procedure for determining these quantities using multiple microphones at each side of the
lined section is presented here, which has the advantage of helping suppress random errors like aerodynamic fluctuations in test data, as seen in references
[13] and [14].
Assuming that all higher order modes have decayed,
only plane waves propagate in the hard ducts. Therefore the acoustic field, equation 3, can then be rewritten to include only propagating waves in the form
(q)
(q)
kxi = kxr
= π(q − 1)/b, and
(q)
(q)
kyi = kyr
= π(q − 1)/h,
(5)
with the mode shapes having a cosine form for symmetric modes and a sine form for anti-symmetric
modes, in each direction [9].
An important hypothesis made by both the TPM
and the MMM is that only plane waves propagate
in the section with hard walls, i.e., the maximum frequency under analysis should be below the lowest cuton frequency in the duct, given by equation 2. That
means that if only plane waves are incident in duct
1, higher-order modes generated at the interface with
duct 2 will decay exponentially with axial distance.
The same happens with modes generated at the interface of duct 3 to duct 2.
In the lined section, the boundary conditions are
symmetric in the y direction, and, similarly to what
occurs in the hard-wall sections, the wave numbers
can be found from equation 5. Below the first cut-on
(1)
(1)
frequency in the y direction, ky2i = ky2r = 0, and from
the dispersion relation, equation 4, the wave numbers
in the x direction are
q
(q)
(q)
(q)
kx2i = (k0 − M kz2i )2 − (kz2i )2 , and
(6)
q
(q)
(q)
(q)
kx2r = (k0 + M kz2r )2 − (kz2r )2 .
Also, by taking the expression for the acoustic field in
the duct, equation 3, and applying the Myers boundary condition [11] for the impedance at x = b, the
following equation is derived:
!
(q) 2
k0
kz2i
(q)
Zwx = jZ0 (q) 1 − M
cot(kx2i b), (7)
k
0
k
x2i
where Z0 is the characteristic impedance of the fluid,
given by the product of its density and speed of sound.
The same equation could be written in terms of the reflected wave numbers (index r) by changing the sign of
the Mach number M inside the parenthesis [7]. Equation 7 is used by both the TPM and the MMM in
slightly different but equivalent forms [8].
2.2. Multiple-Microphone Wave Decomposition
As will be seen later, both methods rely on some form
of the Two-Microphone Method [12]. The TPM requires acoustic pressure and velocity on both sides
of the lined duct, and the MMM takes as input the
amplitude of the incident wave in the lined section
(1)
(1)
(1)
jkzr z
p(z) = pi (z)+pr (z) = ai e−jkzi z +a(1)
.(8)
r e
In this case, the wave numbers in directions x and y
are zero, and the dispersion relation, equation 4, can
(1)
be solved for kz , so that
kz(1) =
k0
,
1±M
(9)
(1)
where the plus sign is used for the incident wave (kzi )
(1)
and the negative, for the reflected wave (kzr ). The
velocity in the z-direction in a given position z can be
calculated [15] from:
!
(1)
(1)
(1)
ai −jk(1) z ar jkzr
z
uz (z) =
e zi −
e
,
(10)
Z0
Z0
or, by comparison with equation 8:
uz (z) =
1
1
pi (z) −
pr (z).
Z0
Z0
(11)
From equation 8, it is possible write the pressure
at a position z = z1 as a function of the pressure at
another position z = z2 as
p(z1 ) =pi (z1 ) + pr (z1 )
(1)
=pi (z2 )e−jkzi
(z1 −z2 )
pr (z2 )e
(12)
+
(1)
jkzr
(z1 −z2 )
.
If the acoustic pressures in two positions z1 and
z2 are known, one could use equation 12 to form a
system of 2 equations and 2 unknowns, the pressure
amplitudes at, for instance, z1 , pi (z1 ) and pr (z1 ). This
system could also be extended to an overdetermined
system of N measurements in different positions, resulting in:


p(z )

 2 


† p(z3 ) 
pi (z1 )
= A
,
.
pr (z1 )

 .. 



p(zN )

e
(1)
−jkzi (z2 −z1 )
(1)
ejkzr (z2 −z1 )

(1)
 −jk(1) (z3 −z1 ) jkzr
(z3 −z1 ) 
 e zi

e

..
..



.
.
where [A] = 

(1)
e−jkzi
(zN −z1 )
(1)
ejkzr (zN −z1 )
(13)
,
FORUM ACUSTICUM 2014
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Methods for Impedance Eduction Under Grazing Flow
and the † represents the Moore-Penrose pseudoinverse [16]. With equation 13, it is possible to calculate the incident and reflected waves at any axial
position within a hard duct, given that the pressure
is known on at least two positions.
2.3.1. Hard-soft Wall Transition
2.3. Two-Port Method
In the Two-Port Method, it is assumed that there is
only a single, dominant mode propagating throughout
the whole duct, although its exact mode shape might
change depending on which region it is propagating
in. In this case, equation 3 could be rephrased for the
lined section, duct 2, as
p2 = a2i Φ2i e−jkz2i z + a2r Φ2r ejkz2r z .
(14)
Applying the Linearized Momentum Equation to
equation 14, the acoustic velocity distribution can be
derived [7] as:
1
1
−jkz2i z
−jkz2r z
Φ2i e
−
Φ2r e
, (15)
uz2 =
Zi
Zr
where Zi e Zr are defined as
k0 − M kz2i
, and
Zi = Z0
kz2i
k0 + M kz2r
Zr = Z0
.
kz2r
Santana [7] suggests that the effects of the hard-soft
wall transition to the present method be considered
through the addition of a new transfer matrix, [Ttr ],
which represents an infinitesimal transition element
before and after the lined section, as shown in Figure
2.
𝑝1𝑜𝑢𝑡 𝑝2𝑖𝑛
𝛿
(16)
Once the pressure and velocity fields along the lined
section are known, the incident pressure at the exit of
the lined section (index 2out) can be written in terms
of the pressure at its inlet (index 2in), in a similar
fashion to what was done for the hard ducts in section
2.2. The relations between pressures and velocities on
both ends of the lined section can be written as a twoport system represented by a transfer matrix of the
form [6]
p2out
p2in
= T
, where [T ] =
u2in
u2out
" + −jk l − jk l + − −jkz2i l ikz2r l # (17)
Z Z (e
−e
)
z2i +Z e z2r
Z e
Z + +Z −
e−jkz2i l −ejkz2r l
Z + +Z −
As outlined above, the TPM assumes that only one
mode propagates throughout the entire duct. This can
be approximately true in most of the duct length, if
the frequency considered is below the first cut-on frequency. However, although the higher order modes
generated by scattering in the hard-soft-wall transitions decay with distance, they are still present in
their vicinities, and the transfer matrix approach does
not take this into account. Additionally, there’s data
suggesting that the physical phenomena in these transitions in the presence of flow are reasonably complex
[17, 18, 19].
Z + +Z −
Z − e−jkz2i l +Z + ejkz2r l
Z + +Z −
This is the main equation of the TPM. It presents a
system of two equations and two unknowns (kz2i and
kz2r ) that when solved provides the wave numbers
in the axial direction inside the lined duct. Applying
these wave numbers to equation 6, the wavenumbers
in the x direction can be calculated, and then used to
calculate the unknown impedance, Zwx , using equation 7. The method implies that pressure and velocity
before and after the lined section need to be known
beforehand, what can be achieved using the technique
presented in section 2.2. This assumes that the pressure and velocity at the beggining of duct 2 equals
pressure and velocity at the exit of duct 1, and the
same is assumed at the interface of ducts 2 and 3. This
assumption will be discussed in the following section.
T𝑡𝑟
𝑝2𝑜𝑢𝑡 𝑝3𝑖𝑛
l
𝛿
T
−1
T𝑡𝑟
Lined Duct
𝑢1𝑜𝑢𝑡 𝑢2𝑖𝑛
𝑢2𝑜𝑢𝑡 𝑢3𝑖𝑛
Figure 2. Schematic representation of the two-port matrix
representation of the inifinitesimal transition element.
The matrix [Ttr ] relates the pressures and velocities before and after the transition region. The effects
on both sides of the lined section are considered symmetric, so that the transfer matrix on the exit is the
inverse matrix of [Ttr ]. It can be shown that the pressure and velocity at the beggining of the third duct
can then be related to the pressure and velocity at the
end of the first duct by:
p3in
u3in
−1
= [Ttr ][T ][Ttr ]
p1out
.
u1out
(18)
At first, no assumption is made about the form of
the transfer matrix, [Ttr ], and 4 new unknowns (the
elements of [Ttr ]) are included in the analysis, additionally to the two wave numbers in the z direction
from equation 17. Since there are 6 variables to be
determined, and the system given by equation 18 is
actually a system of 2 equations, two additional independent measurements have to be made, which can be
achieved by means of the two-source or two-load techniques [20]. The final system consists of 6 equations
and 6 variables, which can be solved, for instance, via
an optimization procedure.
FORUM ACUSTICUM 2014
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Methods for Impedance Eduction Under Grazing Flow
2.4. Mode-Matching Method
the exit reflection coefficient Re = a3r /a3i , which is
zero for all q > 1 since the only reflected mode is the
plane-wave mode, as per equation 21. Both inputs can
be easily computed using the procedure outlined in
section 2.2. It is also required to know the wave numbers for each mode, in each direction, in each duct.
In the hard sections, they are easily computed from
equations 5 and 9. In the lined section, they have to
be computed from an expected impedance value by
solving together equations 6 and 7.
From solving the forementioned system of equations
for an expected impedance value, the modal amplitudes are found, and the acoustic field can be computed at any position in the ducts with equations 19
to 21. It makes sense then to use the microphones
(1)
(q)
positions already used to compute a1i and Re to
compare the calculated acoustic field to the measured
one. From that, a cost function is built, and by minimizing it the unknown impedance can be found.
A more detailed derivation and the full system of
equations can be seen in references [8] and [9].
(q)
The TPM uses a two-port representation of the lined
section to describe how a propagating plane wave
would be affected by the liner. To achieve that, it was
assumed that the waves in the vicinity of the hardsoft-wall transition were plane. This can lead to errors, since the impedance discontinuity scatters the
incident waves into higher order modes [19]. Below
the plane-wave cut-off frequency, these modes decay
exponentially with distance, but ideally they should
still be taken into account when coupling the acoustic
fields of the hard ducts with the lined section.
The MMM solves this problem by taking into account as many modes as necessary in each duct. It
still assumes that the only mode propagating towards
both sides of the lined section is a plane wave mode,
such that the acoustic fields in ducts 1 to 3 can be
written as:
(1)
(1)
(1)
p1 = a1i Φ1 e−jkz1i z +
Q
X
(q)
(q)
(q)
a1r Φ1 ejkz1r z ,
(19)
q=1
p2 =
Q
X
(q)
(q)
(q)
a2i Φ2i e−jkz2i z +
q=1
p3 =
Q
X
Q
X
(q)
(q)
(q)
a2r Φ2r ejkz2r (z−l) ,(20)
q=1
(q)
(q)
(q)
(1)
(1)
(1)
a3i Φ3 e−jkz3i (z−l) +a3r Φ3 ejkz3r (z−l) .(21)
q=1
In equation 19 the summation only occurs for the reflected modes, since the only incident mode is the
plane-wave mode (q = 1). Analogously, in equation
21 the summation only occurs for the incident modes,
since the only reflected mode (which propagates in
z−, i.e., towards duct 2) is the plane-wave mode.
The MMM then assumes continuity of pressure and
axial velocity at the interface of duct 1 with the lined
section (z = 0), and then at the interface of the lined
section with duct 3 (z = l), so that:
p1 (x, y, 0) = p2 (x, y, 0),
(22)
p2 (x, y, l) = p2 (x, y, l),
(23)
uz1 (x, y, 0) = uz2 (x, y, 0),
(24)
uz2 (x, y, l) = uz3 (x, y, l).
(25)
and
Elnady [9] then applies the boundary conditions
(equations 22 to 25), to end up with a system of 4Q
equations and 4Q unknowns, the model amplitudes
(q)
(q)
(q)
(q)
a1r , a2i , a2r and a3i , for q from the first mode to
the Q-th mode.
The required inputs to the system of equations are
(1)
the incident plane-wave amplitude in duct 1, a1i , and
(q)
(q)
3. Numerical Validation
Both the TPM and MMM were implemented in MATLAB, using the built-in fsolve optimization function
with the Levenberg-Marquardt algorithm and mostly
default options [21].
The generation of input data for both methods was
done using numerical models based on the the Finite Element (FE) method [22]. The FE model followed approximately the geometry of the grazing flow
impedance eduction test rig built at the Vibration and
Acoustics Laboratory of Federal University of Santa
Catarina by the authors [1]. The main parameters of
the test rig are:
• Duct cross-section of 0.04 by 0.10 m,
• Test section for liner samples 0.20 m, covering the
entire duct height (0.10 m), and
• 4 microphones before and after the lined section,
each group symmetrically positioned in the wall opposite to the liner sample, non-uniformly spaced,
with the closest at 0.28 m from the lined section
and the furthest away 0.59 m from it.
The mesh, seen in Figure 3, was built with 20 elements per wavelength of the highest frequency of
interest (2500 Hz), taking into account the highest
simulated flow speed (102.9 m/s, or Mach 0.3). The
acoustic mesh was created with TETRA elements, using linear interpolation functions. The software chosen
for solving the model was the FFT ACTRAN 13 [23]
because it allows inclusion of uniform flow effects in
the solution of the FE method, required for a complete validation of the TPM and MMM.
Besides the impedance boundary condition on the
region of the liner sample (green region in Figure 3),
an arbitrary impedance was applied to the duct exit
FORUM ACUSTICUM 2014
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Methods for Impedance Eduction Under Grazing Flow
Normalized Impedance
TPM, 280 seconds, in a 3.2 GHz quad-core Core i7
with 16 Gb of RAM.
Figure 3. FEM model used on the validation of the methods. The green elements represent the liner region. In red,
both the inlet and outlet faces. The dark points are the
microphone positions.
5
0
−5
Relative Error (%)
Normalized Impedance
−10
4
3
2
0
0
0
500
500
Normalized Impedance
Relative Error (%)
Relative Error (%)
2
5
5
Normalized Impedance
1
0
00
1000
1500
2000
2500
Frequency [Hz]
Reference, Real
Figure 5.
no-flow
validation
results.
Reference,
Real
Reference,
Real numericalReference,
Imaginary
−5 MMM
Reference,
RealImaginary
Reference,
−5
Reference,
Reference
(Real
Part)
Reference
Imaginary
Educed,
Real (Imaginary
Reference,
Imaginary
Educed,
Real
Educed,(Real
Real Part)
Educed,
Imaginary
Part)
Educed
Educed
(Imaginary
Part).
Educed, Real
Educed, Imaginary
−10 Educed, Imaginary
Educed, Imaginary
4−10
8
3
6
3.2. Flow
speed of Mach 0.2
2
4
For the 1second validation case, a uniform mean flow
2
of Mach number
0.2, or 68.6 m/s, was included in the
acoustic0 FE
simulation.
The results
can
0 0
500
1000
1500
2000be seen
2500 in
0
500 Frequency
1000 [Hz]
1500
2000
2500
1000
1500
2000
2500
1000Frequency
1500[Hz] 2000
2500 Frequency [Hz]
Frequency [Hz]
Relative Error (%)
Relative Error (%)
Relative Error (%)
Normalized Impedance
Normalized Impedance
The first validation case is a simple, no-flow case. Figures 4 and 5 show the educed complex impedance
for
5
5
the TPM and the MMM, respectively. In each
figure,
the reference impedance curve is also plotted in the
0air charsame graph. All values are normalized by the
0
acteristic impedance. Below each figure, the relative
error (calculated as the percentage difference−5between
the reference value and the educed one) is−5shown. Additionally, each impedance result will show a blue ver−10 since it
tical line in the plane-wave cut-off frequency,
−10 4
is theoretically the limit of application for both
meth8
ods.
3
6
It can be seen from the results that both
meth2
ods converged to almost the imposed impedance
in
4
the no-flow case, with almost identical error1 curves.
2
The MMM solve time was around 30 seconds
and the
0
3
500
0
Relative Error (%)
3.1. No-flow Case
Relative Error (%)
Normalized Impedance
Normalized Impedance
Relative Error (%)
Relative Error (%)
Normalized Impedance
Normalized Impedance
5
(in this case, the air characteristic impedance).
At
1
5
5
5
the inlet face, a modal duct boundary condition
[23]
0
was created to impose a plane-wave incident mode
00
500
1000
1500
2000
2500
0
Frequency [Hz]
in the duct as an excitation and also to make0 it non0
Reference, Real
reflecting (anechoic). For the TPM, in order to obtain
Figure 4.
no-flow
results.
Reference,
Real
Reference,
Realnumerical validation
Reference,
Imaginary
−5 TPM
linearly independent measurements, two other
simuReference,
RealImaginary
Reference,
−5
Reference,
Reference
(Real
Part)
Reference
Imaginary
−5
Educed,
Real (Imaginary
Reference,
Imaginary
lations were carried on for each flow speed:
−5 first, the
Educed,
Real
Educed,(Real
Real Part)
Educed,
Imaginary
Part)
Educed
Educed
(Imaginary
Part).
Educed, Real
Educed, Imaginary
exit impedance was doubled (two-load technique), an
−10 Educed, Imaginary
Educed,
Imaginary
−10 (modal
then the boundary conditions switched places
4−10
−10 4
8
duct on the outlet and impedance on the inlet - two8
3
source technique).
6
3
6
2
The impedance curves imposed in the liner
region
4
2
were obtained using the Extended Helmholtz
Res5 1
4
2
onator model [24], with parameters chosen in1order to
0
obtain a curve that resembled typical liner 2impedance
0 0
500
1000
1500
2000
2500
0
0
values.
0
500 Frequency
1000 [Hz]
1500
2000
2500
500
1000
1500
2000
2500
0 0
500
1000Frequency
1500[Hz] 2000
2500 Frequency [Hz]
In all MMM results in this paper, only 0one mode
Frequency [Hz]
will be taken into account in the solution. Although
−5
this article will not get into detail on this matter, increasing the number of modes only improve
−10
marginally the results, at small number of frequen4
cies [9].
Methods for Impedance Eduction Under Grazing Flow
Figures 6 and 7 for the TPM and the MMM, respectively.
Again, the results are very similar. In this case,
however, there is a slight deviation of the results in
lower frequencies with the MMM displaying larger erros below 700 Hz. Overall, the relative error is sligthly
higher in the MMM when compared to the TPM, but
both results are still acceptable, below 6% over the
whole frequency range.
Solve time was 51 seconds for the MMM and 304
seconds for the TPM.
The third and last validation case was carried out with
a uniform mean flow of 102.9 m/s, or Mach 0.3. The
results are shown in Figures 8 and 9 for the TPM and
the MMM, respectively.
Normalized Impedance
5
3.3. Flow speed of Mach 0.3
0
−5
3
2
1
00
500
Normalized Impedance
Normalized Impedance
Normalized Impedance
5
5
5
1000
1500
2000
2500
0
0
Frequency [Hz]
0
Reference, Real
Figure 7.
MMM
numerical
validation
results
for
M = 0.2.
Reference,
Real
Reference,
Real
Reference,
Imaginary
−5
Reference,
RealImaginary
Reference,
−5
Reference,
Reference
(Real
Part)
Reference
Imaginary
−5
Educed,
Real (Imaginary
Reference,
Imaginary
−5
Educed,
Real
Educed,(Real
Real Part)
Educed,
Imaginary
Part)
Educed
Educed
(Imaginary
Part).
Educed, Real
Educed, Imaginary
−10 Educed, Imaginary
Educed, Imaginary
−10
4−10
−10 4
8
8
3
6
3
6
2
4
2
4
1
2
1
2
0
lative Error (%)
elative Error (%)
10
4
2 5
Relative Error (%)
−5
4
0
5
5
Normalized Impedance
1
0
1000
1500
2000
2500
Frequency [Hz]
Reference, Real
Figure 8.
numerical
results
forReal
M = 0.3.
Reference,
Reference,
Real validation
Reference,
Imaginary
−5 TPM
Reference,
RealImaginary
Reference,
Imaginary
−5
Reference,
Reference
(Real
Part)
Reference
(
Imaginary
Educed, Real
Reference,
Imaginary
Educed,
Real
Educed,(Real
Real Part)
Educed,
Imaginary
Educed
Educed
(Imaginary
Part).
Part)
Educed, Real
Educed, Imaginary
−10 Educed, Imaginary
Educed, Imaginary
4−10
8
3
6
2
4
5 1
2
0
0 0
500
1000
1500
2000
2500
0
500 Frequency
1000 [Hz]
1500
2000
2500
10000
1500
2000
2500
1000Frequency
1500[Hz] 2000
2500 Frequency [Hz]
Frequency [Hz]
500
0
Relative Error (%)
Relative Error (%)
00
−5
−10
8
elative Error (%)
0
Relative Error (%)
Normalized Impedance
−10
6
5
Relative Error (%)
Normalized Impedance
Normalized Impedance
Normalized Impedance
−5
2
6
4
5
2
5
Normalized Impedance
500
500
3
0
00
1000
1500
2000
2500
Frequency [Hz]
Reference, Real
Figure 9.
numerical
resultsImaginary
for
M = 0.3.
Reference,
Real
Reference,
Real validation
Reference,
−5MMM
Reference,
RealImaginary
Reference,
−5
Reference,
Reference
(Real
Part)
Reference
Imaginary
Educed,
Real (Imaginary
Reference,
Imaginary
Educed,
Real
Educed,(Real
Real Part)
Educed,
Imaginary
Part)
Educed
Educed
(Imaginary
Part).
Educed, Real
Educed, Imaginary
−10 Educed, Imaginary
Educed, Imaginary
4−10
8
3
6
2
4
1
2
0
Relative Error (%)
0
−5
500
0
elative Error (%)
0
5
Normalized Impedance
1
500
5
Relative Error (%)
Relative Error (%)
2
00
Relative Error (%)
3
5
Relative Error (%)
10
4
5
1
Normalized Impedance
Relative Error (%)
Normalized Impedance
−5
2
1000
1500
2000
2500
0
0
Frequency [Hz]
0
Reference, Real
Figure 6.
TPM
numerical
validation
results
for
M = 0.2.
Reference,
Real
Reference, Imaginary
−5 Reference, Real
Reference,
RealImaginary
−5
Reference,
−5
Reference,
Reference
(Real
Part)
Reference
Imaginary
Educed,
Real (Imaginary
−5
Reference,
Imaginary
Educed,
Real
Educed,(Real
Real Part)
Educed,
Imaginary
Part)
Educed
Educed
(Imaginary
Part).
Educed,
Real
Educed,
Imaginary
−10 Educed, Imaginary
Educed, Imaginary
−10
4−10
−10 4
8
8
3
3
6
6
2
2
4
4
5 1
1
2
2
0
0 0
500
1000
1500
2000 0 2500
0 0
0
500 Frequency
1000 [Hz]
1500
2000
2500500
10000
1500
2000
2500
0
500
1000Frequency
1500[Hz] 2000
2500 Frequency [Hz]
Frequency [Hz]
Normalized Impedance
0
3
0
0
−10
4
−10
4
5
5
Relative Error (%)
Normalized Impedance
Normalized Impedance
FORUM ACUSTICUM 2014
7-12 September, Krakow
FORUM ACUSTICUM 2014
7-12 September, Krakow
Methods for Impedance Eduction Under Grazing Flow
These results repeat the trends seen in the previous case. Again the TPM achieves errors below 4% in
the entire frequency range, reaching their peak around
1600 Hz. For the first time, though, they are also
around 4% in the first analyzed frequencies, similarly
to the MMM results, which this time reach around
8% in the lowest frequencies, the biggest difference in
all test cases.
The MMM solution took 38 seconds this time, versus around 310 for the TPM.
Both methods made the assumption that the frequency range was below the plane-wave cut-off frequency, so that only plane waves propagate in the
hard sections of the duct. However, upon testing these
methods with FE method generated data, both had
good agreement even above the first transverse mode
cut-on frequency, which might suggest that the acoustic field is still dominated by the plane-wave mode,
and the assumption is approximately valid.
The TPM and the MMM showed good accuracy
and reasonably insignificant solution times in all test
cases, where more then a 100 frequencies were solved.
However, the TPM was still around ten times slower
then the MMM in the current implementation, but
its results, at least in the numerical validation, were
slightly better.
In the test cases solved, both methods seemed to
find reasonable solutions, which resemble typical liner
impedance curves.
It was also interesting that worse results were found
in lower frequencies for the higher mach-number test
data. It is believed that such oscillation might be due
to high vibration levels in the test rig affecting the
measurements. More research is needed to better understand how to improve the measurements and the
signal-to-noise ratio in this cases, because of the high
sound pressure level needed to overcome the aerodynamic noise from the high speed flow.
4. Data From New Impedance Eduction Test Rig
A new grazing flow impedance eduction test rig was
designed and built by the authors at the Vibration
and Acoustics Laboratory of Federal University of
Santa Catarina. Its test section geometry is similar
to the numerical model described in section 3, but
only 3 microphones were used in the measurements
(same positions of the microphones closer to the lined
section).
The measured sample is a typical single degree of
freedom honeycomb/perforated plate liner, and measurements were made in no-flow condition and up to
Mach 0.25. Data was acquired from 100 Hz to 3000
Hz, in steps of 20 Hz, using tonal excitation. More
details on the test rig, acquisition system and specifications of the measurements can be seen in reference
[1].
The first result, shown in Figure 10, is for the noflow condition. Satisfactory agreement is seen, specially from 500 Hz and up. There is some slight oscillation around the plane-wave cut-off frequency, but
the curves match again above 2200 Hz.
The second test case, for target Mach 0.25 (actual
Mach 0.24), shows more dispersion and oscillation in
the results obtained by both methods, specially below
500 Hz. Above 1000 Hz both methods show satisfactory agreement.
5. Conclusions
In this paper, two techniques for acoustic impedance
eduction of liners under grazing flow conditions were
described, implemented and validated by means of numerical simulation using the Finite Element Method.
The first method, called Two-Port Method (TPM),
used a straight-forward two-port representation of a
lined duct to find the wave numbers in it and calculate the acoustic impedance. The second, called ModeMatching Method (MMM), used the mode-matching
technique to couple the acoustic fields in the hard and
lined sections of the duct and find the modal amplitudes in each duct for a given impedance value,
which was iterated on until the calculated acoustic
field matched a measured (or simulated) one.
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Methods for Impedance Eduction Under Grazing Flow
FORUM ACUSTICUM 2014
7-12 September, Krakow
Methods for Impedance Eduction Under Grazing Flow
5
Normalized Impedance
0
−5
−10
TPM, Real Part
TPM, Imaginary Part
MMM, Real Part
MMM, Imaginary Part
−15
0
500
1000
1500
Frequency [Hz]
2000
2500
3000
Figure 10. No-flow impedance eduction from test data: comparison between MMM and TPM. Vertical blue line represents
the plane-wave cut-off frequency.
10
Normalized Impedance
5
0
−5
−10
TPM, Real Part
TPM, Imaginary Part
MMM, Real Part
MMM, Imaginary Part
−15
0
500
1000
1500
Frequency [Hz]
2000
2500
3000
Figure 11. Impedance eduction from test data at Mach 0.24: comparison between MMM and TPM. Vertical blue line
represents the plane-wave cut-off frequency.

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