23rd AIAA/CEAS Aeroacoustics Conference : On the Effect ... - AIAA ARC

On the effect of boundary conditions on impedance eduction results André M. N. Spillere∗ and Julio A. Cordioli† Acoustics and Vibration Laboratory Federal University of Santa Catarina, 88040-900, Florian´ opolis, Brazil

Hans Bodén‡ MWL, Aeronautical and Vehicle Engineering, KTH, S-100 44 Stockholm, Sweden

The acoustic impedance determination of liners remains a challenging task. Recent works have shown that different experimental procedures may lead to different impedance results. Since the acoustic impedance is a key parameter related to the absorption coefficient, it must be correctly measured. The experimental techniques can be divided in two categories: in-situ and inverse methods. In general, the latter assume, in the presence of a mean flow, an infinitely small boundary layer over the liner surface i.e. the so called Ingard-Myers boundary condition. More recent boundary conditions have been proposed that include a small but finite boundary layer. In this paper, the Brambley boundary condition is implemented for a mode matching method, whereas a shear flow profile is considered in a straightforward method. The educed impedances for different liners and flow velocities are compared and the results are discussed.

Nomenclature (x, y, z) Spatial coordinates A Product of the mode amplitude by its mode shape a Mode amplitude b Duct width c0 Speed of sound e Cost function f Frequency h Duct height √ i Imaginary unit −1 J Total number of microphones j Microphone index k0 Free-field wave number kx Transverse wave number in x-direction ky Transverse wave number in y-direction kz Axial wave number L Liner length M Flow Mach number N Total number of acoustic modes n Mode index p Pressure t Time ∗ M.Sc

Student, Federal University of Santa Catarina, AIAA Student Member Professor, Federal University of Santa Catarina ‡ Professor, KTH Aeronautical and Vehicle Engineering † Associate

1 of 18 American Institute of Aeronautics and Astronautics

U v X Z

Mean flow velocity Acoustic particle velocity Non-dimensional duct width Liner impedance

Subscripts avg max i r

Average Maximum Downstream propagating wave Upstream propagating wave

Symbols δI0 , δI1 Brambley coefficients ∆z Distance between microphones δ Boundary layer thickness µ Auxiliary axial wave number ω Angular frequency 2πf ψ Mode shape ρ0 Air density

I.

Introduction

he characterization of acoustic liners by means of a locally reacting wall impedance has been employed T since the earliest works on this topic. This approach has several advantages: i) it is possible to predict the liner impedance based on semi-empirical models given the liner geometry and operating conditions, ii) 1–4

the optimum impedance of ducts i.e. the impedance that gives the highest modal decay rate is known,5, 6 and therefore the liner can be designed to achieve this impedance, iii) several experimental techniques are available to evaluate the liner impedance, for instance in situ techniques7 and impedance eduction methods.8–10 Each of these advantages has also its drawbacks, most of them inherent to the non-linear nature of the liner and its environment. For instance, in aeronautical applications the liner is subject to grazing flow and high sound pressure level (SPL), and both of them have shown to modify the liner impedance experimentally11–14 and numerically.15 Although there is a general agreement that these conditions modify the acoustic impedance, it remains a challenge to correctly quantify it. One of the main issues lies on the experimental techniques, and therefore on assumptions such as i) locally reacting liners and ii) acoustic particle displacement continuity across a thin vortex sheet over the liner surface (known as the Ingard-Myers boundary condition).16, 17 The latter is usually assumed in the aforementioned impedance eduction methods, and for the sake of clarity they will be further explained. First, a liner sample is placed in a duct. The acoustic field is measured at certain points, for instance along the duct walls. Then, a numerical or analytical model simulates the acoustic field in the duct for a given wall impedance guess. Finally, an iterative routine is used to find the impedance that matches the experimental and the simulated acoustic fields. This can be achieved by minimizing the difference between experimental and numerical pressure, or reflection and transmission coefficients, for example. The underlying assumptions on the acoustic field simulation are the key difference between the eduction methods. Of major concern are the flow assumptions, such as uniform flow with the Ingard-Myers boundary condition at the wall, or shear flow and no-slip condition at the wall. To the author’s knowledge, most of them make use of the IngardMyers boundary condition, which is necessary to account for the grazing flow effects. A brief comment about in situ techniques: a transfer function between the pressure at the perforated plate and backing is used to find the acoustic impedance, thus no grazing flow effect is explicitly or mathematically defined. The Ingard-Myers boundary condition was the first attempt to handle grazing flow effects by collapsing the boundary layer. However, over the past years, this assumption has shown to lead to time domain instabilities18 and differences between the measured impedance using downstream and upstream propagating waves.12–14 The first issue is related to numerical simulations, whereas the latter has several implications, such as erroneous calibration of semi-empirical models and incorrect impedance used in aeroacoustic simulations. One could argue that such numerical models also use the Ingard-Myers boundary condition, and therefore the errors from experimental and numerical data would be mutually cancelled. However, this analysis is outside the scope of this work. Some attempts have been made to improve this boundary condition. 2 of 18 American Institute of Aeronautics and Astronautics

Rienstra19 and Brambley20 included a small but finite boundary layer thickness in the mathematical formulation, whereas Renou and Aurégan12 introduced a factor in the classical Ingard-Myers boundary condition. Such representations have improved experimental results12, 13 and model accuracy when compared to an exact solution of the boundary layer.21 The aim of this paper is to compare the more recent boundary conditions, such as the proposal of Brambley,20 to the Ingard-Myers boundary condition, in the context of impedance eduction techniques, more specifically in the mode matching method (MMM).8 The modified Ingard-Myers boundary condition, as proposed by Renou and Aurégan,12 makes use of upstream and downstream results. However, differences between both results may be inherent to the liner behaviour in the presence of flow,22 and thus this approach is not considered here. Rienstra and Darau boundary condition19 is asymptotically equivalent to Brambley’s and thus it has been excluded from this analysis. Another impedance eduction method proposed in the recent years is the straightforward method (SFM).9 It receives this name because no iteration is required. The acoustic field along the lined section is approximated by a series of complex exponentials, and the Prony’s method is used to extract the axial wave numbers. Then, the flow can be assumed uniform, and by means of the Ingard-Myers boundary condition the wall impedance is found.9 In the case of shear flow assumption, the Pridmore-Brown equation23 has to be numerically integrated. Since the flow has zero velocity at the wall (no-slip condition), the impedance boundary condition without flow is used.24 Thus, the results from both approaches are also compared. This paper is organized as follows. Section II gives a brief explanation of the impedance eduction techniques and how the boundary conditions are applied. In Section III the boundary conditions are presented and the necessary equations for the MMM and SFM are derived. Section IV describes the experimental set-up and Section V shows the main results.

II.

Impedance eduction techniques

This section briefly presents the MMM and SFM. A detailed derivation can be found on the original papers.8, 9, 24, 25 A.

Mode matching method

Fig. (1) shows the reference geometry, where two hard wall sections (1 and 3) and a lined section (2) compose the duct of dimensions b and h, respectively the width and height of the duct. The liner can be seen as a wall impedance Z of length L at x = 0. The convected wave equation, assuming uniform flow in the z+ direction, is given by 1 D2 p = 0, (1) ∇2 p − 2 c0 Dt2 where D/Dt is the material derivative and ∇2 is the Laplace operator. The solution to the wave equation at each section is given by the sum of N modes of amplitude a(n) and mode shape ψ (n) (x, y) (harmonic time dependence eiωt is omitted) p(x, y, z) =

N X

(n)

(n)

(n)

ai ψi (x, y)e−ikzi

n=1

z

+

N X

(n)

(n) ikzr z a(n) , r ψr (x, y)e

(2)

n=1

where i and r denote the incident and reflected waves propagating respectively in the z+ and z− directions, n is the index of the mode, and kz is the axial wave number, which satisfies the dispersion relation 2

kx2 + ky2 + kz2 = (k0 ± M kz ) ,

(3)

where k0 = ω/c0 is the free-field wave number, M = U/c0 is the mean flow Mach number and U is the mean flow velocity in the z direction. It is assumed that only plane waves are propagating towards the lined section, depicted in Fig. (2), which results in the followings acoustic fields at each section: (1)

(1)

p1 = a1i e−ikz1i z +

N X

(n)

(n)

(n)

a1r ψ1r eikz1r z ,

n=1


(4)

b Z

(1) h

(2) y

(3) x z L

Figure 1. Rectangular duct with width b and height h. At x = 0 in section (2) the wall has an impedance Z and length L.

p2 =

N X

(n)

(n)

(n)

a2i ψ2i e−ikz2i z +

n=1

p3 =

N X

N X

(n)

(n)

(n)

a2r ψ2r eikz2r (z−L) ,

(5)

n=1 (n)

(n)

(n)

(1)

(1)

a3i ψ3i e−ikz3i (z−L) + a3r eikz3r (z−L) .

(6)

n=1

The acoustic field at sections (1) and (3) has to satisfy the hard wall boundary conditions. At x = 0 in section (2) an appropriate boundary condition has to be selected and this is further discussed in Section III. The next step is to satisfy the continuity of pressure and axial particle velocity at the interfaces (z = 0 and z = L, and thus the name mode matching). By using the orthogonality between modes it is possible to end (n) (n) (n) (n) up with a system of 4N equations and 4N unknowns: the modal amplitudes a1r , a21 , a2r and a3i . The system of equations is solved for an impedance guess. Using the calculated modal amplitudes, the acoustic field is computed at positions which correspond to the microphones. From that, a cost function is built based on the experimental pressure pj,exp at the j-esim microphone, and the semi-analytical pressure given by the MMM, pj,MMM , e(Z, f ) =

2 J X pj,exp (f ) − pj,MMM (Z, f ) pj,exp (f )

j=1

,

(7)

By minimizing it the liner impedance can be found. Notice that the cost function is solved independently for each frequency f . In the present work, the Matlab fsolve minimizer with the Levenberg-Marquardt algorithm was used,26 with mostly default options. Upstream Loudspeaker

(1)

(q)

(1)

p1r

p1i Flow

Microphones

(2)

x

Downstream Loudspeaker

(3)

Microphones

(q)

(q)

p2r

p2i

z

Liner sample

(q)

(1)

p3i

p3r

Microphones

Figure 2. Schematic top view of the test rig with hard wall sections (1) and (3) and lined section (2). The loudspeakers are placed upstream and downstream to the liner. The microphones used in the mode matching method are placed in sections (1) and (3), whereas the microphones for the straightforward method are placed along the liner in section (2).


B.

Straightforward method

Differently from the MMM, the SFM is a direct impedance eduction technique i.e. no iteration nor minimization is required. The pressure at the wall opposed to the liner can be rewritten from Eq. (2) as a sum of complex exponentials 2N X (n) p= A(n) eµ z , (8) n=1 (n)

where A(q) is the product of the mode amplitude by its mode shape at the duct wall, and µ(n) = (−1)n ikz . Since both upstream and downstream propagating waves are considered, the total number of modes is 2N . If the microphones are equally spaced, Prony’s method can be employed to extract the axial wave numbers. Thus, Eq. (8) is rewritten as 2N X (n) pj = A(n) ejµ j∆z , (9) n=1

for j = 1, ..., J and ∆z the distance between consecutive microphones positioned according to Fig. (2). The procedure is to convert such a non-linear problem into a linear least-square problem by means of Prony’s method, which gives the mode amplitudes A(n) and axial wave numbers µ(n) . Since spurious solutions may be obtained, a systematic procedure must be employed to eliminate non-physical results.9, 24 Once the axial wave number is known, it can be related to the transverse wave number kx , and therefore to the impedance Z by means of an appropriate boundary condition, for instance the Ingard-Myers boundary condition.16, 17 This is further discussed in Section III.

III.

Boundary conditions

In this section the boundary conditions at the lined section are discussed. The hard wall boundary condition in the x direction at x = b is given by ∂p = 0. (10) ∂x x=b At x = 0 the presence of the liner requires an appropriate boundary condition that relates acoustic particle velocity and pressure as a function of the wall impedance Z. The Ingard-Myers boundary condition is widely used, but more recent boundary conditions have been proposed to improve the accuracy of the physics near the wall. Brambley20 boundary condition is investigated for the MMM, whereas an exact solution based on the Pridmore-Brown equation23 is analysed for the SFM.24 A.

Ingard-Myers

The Ingard-Myers boundary condition17 collapses the boundary layer by assuming acoustic particle displacement continuity across a vortex sheet over the liner surface, such that at x = 0 ∂ 1 −v = iω + U p (11) iωZ ∂z Introducing into Euler equation in x-direction, ∂p ∂ = −ρ0 iω + U v, ∂x ∂z

(12)

assuming a pressure field as given by Eq. (2) and applying the hard wall boundary condition given by Eq. (10), the following relation is found kx tan (kx b) =

i 2 (k − M kz ) , kZ

where Z is the normalized wall impedance.


(13)

B.

Brambley

Brambley20 proposed an alternative boundary condition by introducing a small but finite boundary layer δ. Using an asymptotic expansion and retaining the leading order terms the following equation is derived:20 k − M kz δI1 kz2 i 2 −Z p (14) −v Z − (k − M kz ) δI0 = k k i (k − M kz ) where the coefficients δI0 and δI1 are given by Z δ 2 (k − M0 (x)kz ) δI0 = 1− (15) 2 dx (k − M kz ) 0 Z δ 2 (k − M kz ) δI1 = 1− (16) 2 dx (k − M0 (x)kz ) 0 The former may be interpreted as a correction to the impedance as seen by the acoustic field in the presence of a uniform velocity profile, turning into an ”effective” impedance. The latter is responsible for the wellposedness of the boundary condition. In both equations it was assumed a constant density across the boundary layer. Repeating the procedure shown for the Ingard-Myers boundary condition, the following relation is derived 2 i (k − M kz ) − kZδI1 kz2 , (17) kx tan (kx b) = 2 kZ − i (k − M kz ) δI0 Notice that, if δI0 = δI1 = 0, the Ingard-Myers boundary condition is recovered. Assuming a constant density ρ0 (y) = 1 and a linear velocity profile u0 (y) = M y/δ, then δI0 =

δM kz (2M kz − 3k) 3 (M kz − k)

2

,

δI1 = δM

kz k

(18)

Gabard has shown a little effect on changing the boundary layer profile, i.e. quadratic, power law, etc, on the absorption coefficient of the surface,21 and therefore the linear profile was chosen for the sake of simplicity. A boundary layer thickness of 25% relative to half of the duct height was chosen, representing the strongly sheared flow seen in experimental test rigs. Although relatively thick, kδ is small for most of the frequency range under analysis, which is a basic assumption in Brambley’s derivation. C.

Exact solution

The previous boundary conditions are based on uniform flow assumption. However, test rigs are usually of small duct cross section, and therefore the flow is strongly sheared. A better approach is to include the flow profile in the wave propagation. In this case, the governing equation is the Pridmore-Brown equation,23 i d2 p 2kz dM0 dp h 2 2 + (k − M (x)k ) − k + (19) 0 z z p = 0, dx2 k − M0 (x)kz dx dx where the flow profile M0 (x) can be selected based on curve fitting the experimental data. This formulation is relatively easy to include in the SFM and the following description is given by Jing.24 Once the axial wave number is known by means of the Prony’s method, the Pridmore-Brown equation can be numerically integrated. Firstly, it is rewritten as a pair of first order differential equations,  dF   =G  dX (20) 2kz dM dG   =− G − (k − M0 (X)kz )2 − kz2 F  dX k − M0 (X)kz dX where X = x/b and F = p. The boundary conditions are given as follows. At the hard wall, the acoustic particle velocity is zero, thus G(1) = 0, whereas the pressure is an arbitrary constant, for instance F (1) = 1. At the lined wall, the no slip boundary condition is applied, ik F (0) (21) Z Eq. (20) is numerically integrated by means of a fourth-order Runge-Kutta scheme, and thus the unknown impedance can be found. G(0) =


IV.

Experimental setup

The experimental apparatus follows the geometrical specifications of the analytical formulation i.e. a rectangular duct, with a liner sample on one wall, and hard wall sections before and after the lined section. A.

Test rig

The test rig built at the Federal University of Santa Catarina (UFSC) follows the schematic representation from Fig. (2). It is composed of five interchangeable sections to accommodate different test configurations. For instance, the source section, where the speakers are connected, can be positioned upstream or downstream to the liner sample. As a drawback, the ventilation system has to be restarted, which may lead to slightly different flow velocities when comparing upstream and downstream results. A total of 8 speakers can be connected to a single duct section in the test rig, resulting in a SPL over 140 dB at the lined section. However, in this work all the measurements were done using a single speaker with pure single tones and never exceeding 130 dB. The cross-section of the test section is b = 0.04 m by h = 0.10 m, which results in a no-flow cut-on frequency of 1700 Hz for the first transverse mode. The microphones are positioned at the nodal line of this mode, which corresponds to half of the duct height. Therefore, the effect of the first higher order mode is not captured. As the source frequency approaches the second transverse mode cut-on frequency (3400 Hz), the error in the impedance eduction techniques increases.27 The frequency range under analysis is therefore limited from 500 to 3000 Hz. A total of 8 microphones are positioned in the hard wall sections (1) and (3), flush to the duct walls, according to the MMM formulation. For SFM, the same 8 microphones are repositioned at the wall opposed to the liner. A list of the microphone positions for each impedance eduction technique is given in Table (1), assuming z = 0 at the beginning of the lined section. The acoustic sources can be placed upstream or downstream to the liner, leading to downstream and upstream propagating waves, respectively. An amplification system is responsible to achieve sound pressure levels (SPL) up to 130 dB. This level was chosen to avoid non-linear effects on the liner impedance. A ventilation system is used to generate a uniform flow at the duct inlet, which is able to achieve an average Mach number of 0.30. Table 1. Microphone positions in the test rig. Notice that all microphones are position at half of the duct height i.e. y = 0.05 m. The coordinate system follows Fig. (2).

Method

Location

Upstream MMM Downstream

SFM

Along liner

Microphone 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

x [m] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04

y [m] 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

z [m] -0.59 -0.42 -0.33 -0.28 0.48 0.53 0.62 0.79 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17

Excitation signal generation, microphone signal acquisition, and controlling of the flow velocity, are executed by a custom software implemented in Labview. Three arbitrary velocities were chosen to perform the measurements, Mach 0.28 being the maximum velocity in this work. The uniform flow velocity used in 7 of 18 American Institute of Aeronautics and Astronautics

the impedance eduction algorithms is an average Mach number calculated from the acoustic measurements at the microphones using a custom routine.28 Thus, the shear flow has a centreline Mach number slightly higher than the average Mach number. B.

Test samples

Three liner samples were available for measurement at UFSC. The main geometrical parameters of each sample are given in Table (2). The liner sample is 0.20 m long, covering the entire duct height. All of them are typical single degree of freedom locally reacting liners used in aeronautical applications. Liner C is covered by a wire mesh and is also called linear liner because it is more flow-insensitive than the traditional perforated liner (A and B), which has the holes exposed to the grazing flow. The open area is defined as the ratio between the area of the holes and total face-sheet area. The values given in Table (2) are the effective open area, since the honeycomb cells can partially block the holes in the assembly process. Table 2. Liner samples used in the measurements.

Test sample Type Hole diameter [mm] Face-sheet thickness [mm] Cavity depth [mm] Open area [%]

C.

Liner A Perforated SDOF 1.0 0.65 19.0 5.18

Liner B Perforated SDOF 2.0 0.8 19.0 8.63

Liner C Linear SDOF – – 30.0 –

Flow profile

In order to correctly assess the effect of the boundary conditions from the previous section, the flow profile in the test rig has to be further examined. Three arbitrary flow velocities are shown in Fig. (3). The experimental results are based on measurements using a Pitot tube at half of the duct height i.e. at y = 0.05 m, and at half length of the liner i.e. z = 0.10 m. To perform the measurements, the lined section was replaced by a hard wall section with a small hole at the top to insert the Pitot tube. The uniform value is given by the average of the measurements points not only in the x-direction, but also in the y-direction, which results in a total of 29 measurement points. The analytical profile is given by an equation based on the power law, (1/7) M (X) = Mmax [−4X (X − 1)] , (22) where 0 ≤ X ≤ 1. It has been observed that the ratio between the average and maximum Mach number Mavg /Mmax is approximately 0.9 at any flow velocity. This analytical expression has been chosen due to the good agreement with experimental results, particularly at the first three measurement points. This choice of profile has its drawbacks. For instance, its derivative to X, necessary to the PridmoreBrown equation, is given by dM Mmax (−6/7) = [−4X (X − 1)] (−8X + 4) (23) dX 7 Thus, as X approaches zero, dM/dX tends to infinity. In terms of numerical implementations, the PridmoreBrown equation and its boundary conditions have to be evaluated at small but not zero distance, otherwise the second case in Eq. (20) cannot be evaluated. Physically, an infinite derivative would also mean an infinite friction velocity at the wall. A more appropriate approach is to divide the profile in the viscous sublayer and the outer turbulent layer. However, no experimental information is available regarding the flow profile near the wall, and therefore any modelling assumption would find no experimental data for comparison. Nevertheless, the power law provides a good approximation of the actual flow and it is much more representative than the uniform flow assumption.

V.

Results

This section is divided into two parts. The first shows the results for the MMM using the classical IngardMyers boundary condition16, 17 and the Brambley boundary condition.20 Results for both downstream and 8 of 18 American Institute of Aeronautics and Astronautics

Non-dimensional duct width [−]

1 Analytical Uniform Experimental

0,8 0,6 0,4 0,2 0

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

Mach number [−] Figure 3. Flow profiles at different flow velocities. The analytical profile ( ) is given by Eq. (22), the uniform profile ( ) is an average value of the cross section and the experimental points ( ) were measured using a Pitot tube.

upstream propagating waves are shown since there is a noticeable difference between the results. The second part is dedicated to the SFM using the classical Ingard-Myers boundary condition16, 17 and the exact solution given by the Pridmore-Brown equation.23 Again, both downstream and upstream propagating waves are analysed. A.

Mode matching method

The MMM results are available in the frequency range from 0.5 kHz to 3.0 kHz. On the following, the results for each test sample are discussed. 1.

Liner A

In general, the attenuation from 0.5 kHz to 1.0 kHz is very low, and the MMM finds difficulty to correctly educe the impedance. This is aggravated for downstream waves in the presence of flow, when the attenuation is even lower. Therefore, the results are very unstable in this frequency range, specially at M = 0.28 and downstream waves. Regarding the effect of the boundary conditions, Fig. (4) shows some interesting trends. At M = 0.10 both the resistance and reactance are very similar regardless the boundary condition and the wave orientation. As flow velocity increases, assuming the Ingard-Myers boundary condition, the resistance becomes significantly higher for the upstream propagating wave when compared to the downstream propagating wave. A difference between both cases is also seen in the reactance, the upstream result being lower than the downstream result. When the Brambley boundary condition is applied, the difference between the curves is reduced. The reactance is related to the cell cavity, so it is expected that upstream and downstream propagating waves have a small influence on the reactance. In other words, the wave orientation should not affect the reactance, at least in the frequency range here considered. On the other side, the resistance shows a slight difference between both cases. Since the acoustic wave is subject to vortex shedding and turbulence generated at the perforated plate, the trend in the resistance may be different for upstream and downstream propagating waves. 2.

Liner B

In contrast to liner A, where a good signal-to-noise ratio was achieved for most of the frequency range, liner B has a very high attenuation between 2.2 kHz and 2.7 kHz, such that the microphones after the liner (or before, depending on the source location) are measuring flow-induced noise rather than an attenuated sound wave. As a consequence, the results are oscillating around this frequency in Fig. (5), but nevertheless the same trends from liner A are observed. The Brambley boundary condition increases the downstream


5

2 Myers - Downstream Myers - Upstream Brambley - Downstream Brambley - Upstream

1 0 Reactance [−]

Resistance [−]

Mach 0.10

4 3 2

−1 −2 −3 −4

1

−5 0

0,5

1

1,5

2

2,5

−6

3

1,5

2

2,5

3

0,5

1

1,5

2

2,5

3

0,5

1

1,5

2

2,5

3

1

4

0 Reactance [−]

Resistance [−]

1

2

5

Mach 0.22

0,5

3 2

−1 −2 −3 −4

1

−5 0

0,5

1

1,5

2

2,5

−6

3

2

5

1 0 Reactance [−]

Resistance [−]

Mach 0.28

4 3 2

−1 −2 −3 −4

1

−5 0

0,5

1

1,5

2

2,5

3

−6

Frequency [kHz]

Frequency [kHz]

Figure 4. Impedance eduction result for liner A at different flow velocities using the mode matching method and uniform flow assumption on the following cases: Myers boundary condition and downstream propagating wave ( ), Myers boundary condition and upstream propagating wave ( ), Brambley boundary condition and downstream propagating wave ( ) and Brambley boundary condition and upstream propagating wave ( ).


resistance and decreases the upstream resistance. However, both results are similar even by applying the Ingard-Myers boundary condition, and thus there is no noticeable change. This is not the case for the reactance. The Brambley boundary condition results in a smaller difference between the curves for upstream and downstream propagating waves, although a collapse is not achieved as in the case of liner A. 3.

Liner C

Differently from the previous test samples, liner C is a linear SDOF, and a more flow-insensitive impedance is expected. In fact, even at low flow velocities, for instance at M = 0.10, the resistance is relatively high when compared to traditional perforated plate liners. As flow velocity increases, the results remain very similar, with exception of the resistance in the case of upstream propagating wave, as seen in Fig. (6). Once again, the use of Brambley boundary condition barely changed the downstream results. However, the resistance in the upstream propagating wave is considerably closer to the downstream propagating case. In contrast to the perforated plate liners, the reactance curves were not affected, even in the low frequency range of 0.5 kHz to 1.0 kHz. A general trend observed not only in liner C, but also in liners A and B, is that the impedance using the Ingard-Myers and Brambley boundary conditions are virtually the same after 2.0 kHz. It is not clear although whether it is a physical phenomenon or a limitation of the Brambley boundary condition, which is valid under the assumption that kδ 1. In our test cases, at 3.0 kHz, kδ ≈ 0.28, and therefore the boundary condition has no effect on the educed impedance when compared to the Ingard-Myers condition. Thus, a better approach would be to include the shear flow profile in the impedance eduction technique. B.

Straightforward method

The results using the SFM are available from 1.0 kHz to 3.0 kHz. On the following, the results for each test sample are discussed. 1.

Liner A

Fig. (7) shows the same trend observed when applying the Brambley boundary condition in the MMM. Under the assumption of uniform flow, the educed impedance from upstream and downstream measurements shows a significant difference, specially in the frequency range from 1.0 kHz to 1.5 kHz and flow velocity greater than Mach 0.20. By considering the flow profile, a better agreement between upstream and downstream curves is obtained, mainly in the reactance (although some differences are still present). An interesting trend is observed for the resistance. The downstream results are almost constant in the frequency range under analysis, and the upstream results show a decrease with frequency, regardless the flow profile assumption. Therefore, this behaviour may be related to the underlying physics of the liner, and not to flaws in the impedance eduction technique. 2.

Liner B

As stated in the previous section, liner B shows a very high attenuation, an just as the MMM results, the SFM results are also affected due to flow-induced noise measurements. As a consequence, the number of microphones used in the post-processing is reduced, and unexpected oscillations in the curves from Fig. (8) are observed, which could be related to spurious results from Prony’s method. Regarding the modifications in the impedance eduction, the resistance is slightly affected by the shear flow assumption, but the biggest difference is seen for the reactance, where the upstream and downstream curves are very close for low Mach number. 3.

Liner C

Liner C results are shown in Fig. (9). The trend is opposite to the previous liners. The results for upstream and downstream propagating waves are closer assuming uniform flow than shear flow. Up to M = 0.10 no noticeable change is observed. At M = 0.22, the resistance from upstream and downstream measurements assuming uniform flow are the same around 2.5 kHz, but different and even with opposed trend when assuming shear flow (downstream result increases with frequency, whereas upstream result decreases with frequency). The same observation is valid at M = 0.28, and it is in disagreement with the results using the MMM. Thus, this deviation should be further investigated. 11 of 18 American Institute of Aeronautics and Astronautics

5


1 0 Reactance [−]

Resistance [−]

Mach 0.10

4 3 2

−1 −2 −3 −4

1

−5 0

0,5

1

1,5

2

2,5

−6

3

1,5

2

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3

0,5

1

1,5

2

2,5

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4

0 Reactance [−]

Resistance [−]

1

2

5

Mach 0.20

0,5

3 2

−1 −2 −3 −4

1

−5 0

0,5

1

1,5

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1 0 Reactance [−]

Resistance [−]

Mach 0.25

4 3 2

−1 −2 −3 −4

1

−5 0

0,5

1

1,5

2

2,5

3

−6

Frequency [kHz]

Frequency [kHz]

Figure 5. Impedance eduction result for liner B at different flow velocities using the mode matching method and uniform flow assumption on the following cases: Myers boundary condition and downstream propagating wave ( ), Myers boundary condition and upstream propagating wave ( ), Brambley boundary condition and downstream propagating wave ( ) and Brambley boundary condition and upstream propagating wave ( ).


5


1 0 Reactance [−]

Resistance [−]

Mach 0.10

4 3 2

−1 −2 −3 −4

1

−5 0

0,5

1

1,5

2

2,5

−6

3

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0 Reactance [−]

Resistance [−]

1

2

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Mach 0.22

0,5

3 2

−1 −2 −3 −4

1

−5 0

0,5

1

1,5

2

2,5

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3

2

5

1 0 Reactance [−]

Resistance [−]

Mach 0.28

4 3 2

−1 −2 −3 −4

1

−5 0

0,5

1

1,5

2

2,5

3

−6

Frequency [kHz]

Frequency [kHz]

Figure 6. Impedance eduction result for liner C at different flow velocities using the mode matching method and uniform flow assumption on the following cases: Myers boundary condition and downstream propagating wave ( ), Myers boundary condition and upstream propagating wave ( ), Brambley boundary condition and downstream propagating wave ( ) and Brambley boundary condition and upstream propagating wave ( ).


5

2 UF (Myers) - Downstream UF (Myers) - Upstream SF (exact) - Downstream SF (exact) - Upstream

3

1 0 Reactance [−]

Resistance [−]

Mach 0.10

4

2

−1 −2 −3 −4

1

−5 0

1

1,5

2

2,5

−6

3

2

2,5

3

1

1,5

2

2,5

3

1

1,5

2

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3

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4

0 Reactance [−]

Resistance [−]

1,5

2

5

Mach 0.22

1

3 2

−1 −2 −3 −4

1

−5 0

1

1,5

2

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1 0 Reactance [−]

Resistance [−]

Mach 0.28

4 3 2

−1 −2 −3 −4

1

−5 0

1

1,5

2

2,5

3

−6

Frequency [kHz]

Frequency [kHz]

Figure 7. Impedance eduction result for liner A at different flow velocities using the straightforward method on the following cases: uniform flow (Myers boundary condition) and downstream propagating wave ( ), uniform flow (Myers boundary condition) and upstream propagating wave ( ), shear flow (exact solution) and downstream propagating wave ( ) and shear flow (exact solution) and upstream propagating wave ( ).


5


3

1 0 Reactance [−]

Resistance [−]

Mach 0.10

4

2

−1 −2 −3 −4

1

−5 0

1

1,5

2

2,5

−6

3

2

2,5

3

1

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0 Reactance [−]

Resistance [−]

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2

5

Mach 0.20

1

3 2

−1 −2 −3 −4

1

−5 0

1

1,5

2

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1 0 Reactance [−]

Resistance [−]

Mach 0.25

4 3 2

−1 −2 −3 −4

1

−5 0

1

1,5

2

2,5

3

−6

Frequency [kHz]

Frequency [kHz]

Figure 8. Impedance eduction result for liner B at different flow velocities using the straightforward method on the following cases: uniform flow (Myers boundary condition) and downstream propagating wave ( ), uniform flow (Myers boundary condition) and upstream propagating wave ( ), shear flow (exact solution) and downstream propagating wave ( ) and shear flow (exact solution) and upstream propagating wave ( ).


5


3

1 0 Reactance [−]

Resistance [−]

Mach 0.10

4

2

−1 −2 −3 −4

1

−5 0

1

1,5

2

2,5

−6

3

2

2,5

3

1

1,5

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2,5

3

1

1,5

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4

0 Reactance [−]

Resistance [−]

1,5

2

5

Mach 0.22

1

3 2

−1 −2 −3 −4

1

−5 0

1

1,5

2

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−6

3

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5

1 0 Reactance [−]

Resistance [−]

Mach 0.28

4 3 2

−1 −2 −3 −4

1

−5 0

1

1,5

2

2,5

3

−6

Frequency [kHz]

Frequency [kHz]

Figure 9. Impedance eduction result for liner C at different flow velocities using the straightforward method on the following cases: uniform flow (Myers boundary condition) and downstream propagating wave ( ), uniform flow (Myers boundary condition) and upstream propagating wave ( ), shear flow (exact solution) and downstream propagating wave ( ) and shear flow (exact solution) and upstream propagating wave ( ).


VI.

Conclusions

In this work, two different boundary conditions were implement in impedance eduction techniques. In the MMM, where uniform flow is assumed, the Ingard-Myers boundary condition was substituted by the Brambley boundary condition. The former assumes an infinitesimal vortex sheet over the lined surface, whereas the latter considers a small but finite boundary layer, leading to a ”modified” impedance as seen by the acoustic field. The SFM is based on Prony’s method to extract the axial wave numbers, and therefore any flow profile can be considered when integrating the Pridmore-Brown equation. Uniform and shear flows were investigated, the former assuming the Ingard-Myers boundary condition, and the latter assuming a no-slip condition at the wall. The modification to Brambley’s boundary condition in the MMM leads to a very consistent trend in the educed resistance. When the acoustic wave is propagating downstream, the resistance increases, whereas in the upstream case the resistance decreases. This is valid for the frequency range under analysis (from 0.5 kHz to 3.0 kHz) and for three different liners. The educed reactance from both wave directions are in good agreement for the perforated plate liners, eliminating the deviation seen when assuming Ingard-Myers boundary condition. Since the reactance is related to the cell cavity, one should not expect differences in the educed impedance regarding the wave orientation. In the SFM, the shear flow assumption resulted in very similar educed reactances for upstream and downstream propagating waves. The resistance, however, shows a distinct behaviour in frequency for upstream and downstream propagating waves, regardless of the flow profile considered. The trends observed in the resistance are in agreement with the MMM results, except for liner C, and this should be further investigated. Overall, the educed impedance is affected when more realistic flow profiles or boundary conditions in test rigs are considered. However, a mismatch is still present regarding upstream and downstream propagating waves. Future works should investigate whether this is a modelling failure in the impedance eduction technique, or a consequence of different acoustic-flow interactions on the perforated plate under grazing flow.

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