Comparison of the effect of flow direction on liner

Comparison of the effect of flow direction on liner impedance using different measurement methods Hans Bod´en∗ MWL, Aeronautical and Vehicle Engineering, KTH, S-100 44 Stockholm, Sweden

Julio A. Cordioli†, Andr´e M. N. Spillere‡and Pablo G. Serrano§ Acoustics and Vibration Laboratory Federal University of Santa Catarina, 88040-900, Florian´ opolis, Brazil

There is an ongoing scientific discussion about the effect of flow on experimental techniques for determination of acoustic liner impedance. This paper contributes to this continuing effort to gain confidence in results obtained under different acoustical and flow excitation configurations. A majority of the test rigs for determination of liner impedance including the effect of mean flow use plane wave excitation on the upstream side of the liner, but some studies have compared the results for downstream acoustic excitation. Especially for the so-called inverse impedance eduction techniques, it has been reported that different flow directions compared to the acoustic excitation can provide different educed impedances. It is still an open question if this results are due to the application of the Ingard-Myers boundary condition, to other errors or flaws in the measurements or a characteristic of the liner itself. This paper revisit some previous published results and compares the result obtained by means of inverse impedance eduction techniques, which in general adopt the Ingard-Myers boundary condition, and in-situ impedance measurements, which do not require the definition of a boundary condition. It is seen that discrepancies between downstream and upstream measurement can be observed in both approaches, and a discussion on such behavior is presented.

I.

Introduction

erforated liners are used for noise control of aircraft engines as well as for other vehicles and machines. P Their properties and noise reduction depends on the mean flow field and other parameters such as temperature and acoustic excitation level. Therefore, there has been a large interest in developing techniques to measure the acoustic impedance of liners under grazing flow conditions.1–31 A number of different techniques for extracting the liner impedance from measurement have been developed and the dominating ones, at least in terms of numbers of publications, are the so-called inverse impedance eduction techniques.1–27 In order to gain confidence in the results, which may depend on both the test rig used and on the impedance education method, some comparative studies have been performed.1, 10, 11, 18, 20, 27 The in-situ impedance measurement technique,28 in which the liner is instrumented, has also been successfully used to measure the liner impedance.27, 29, 30 The present paper continues the effort from Ref. 24, where measurement data with both upstream and downstream acoustic excitation has been collected and compared, by also considering results obtained using the in-situ method. Data for five different liners obtained in different test rigs using different impedance eduction techniques were compared in,24 and it was concluded that systematically different results were obtained for upstream and downstream excitation. Similar results have also recently been obtained by numerical simulation.32 Renou and Auregan19 were the first to point this out and attributed it to a failure of ∗ Professor,

KTH Aeronautical and Vehicle Engineering Professor, Federal University of Santa Catarina ‡ M.Sc Student, Federal University of Santa Catarina, Non AIAA Member § PhD Student, University of Southampton, Non AIAA Member † Associate

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the Ingard-Myers boundary condition,33 which is applied together with plug flow conditions in the impedance eduction techniques. The in-situ impedance measurement technique does not use the Ingard-Myers boundary condition. Therefore, if the application of this boundary condition is the reason for the discrepancies they should not be seen when using the in-situ technique. In order to address some of these questions, a summary of the experimental data considered in24 is presented, followed by a review of some recently publish results in order to propose hypotheses for the discrepancies observed between upstream and downstream. Finally, experimental data comparing impedance eduction techniques and the in-situ method are shown and the results discussed in view of the hypotheses previously established.

II.

Experimental techniques for impedance determination and test setups

In this section, the test rigs, liner samples and impedance eduction methods used in24 are briefly described, together with the in-situ method. A.

Impedance eduction techniques

In Ref. 24, results obtained using a number of different test rigs and impedance eduction methods were reviewed and discussed. Figure 1 shows an example of one such test rig. Two impedance eduction techniques were considered in Ref. 24: the Mode-Matching Method (MMM) and the Straightforward Method (SFM). Both techniques use microphones in the test duct containing the liner sample to characterize the sound field. Inverse techniques involving models of the acoustic field in the lined test section are then used to estimate the liner surface impedance. The SFM can be described as a simplified version of the single mode method (another method used in Ref. 11), where the Prony method23 is used to obtain the axial wave number taking as input the pressure measured at the wall opposite to the liner at positions regularly spaced. The impedance is then obtained by using the Ingard-Myers boundary condition and the dispersion relation for the transverse wave number. The MMM describes the acoustic field by means of the Convected Helmholtz Equation (CHE) with plug flow and Ingard-Myers boundary condition at the lined wall. Details of the method can be found in Refs. 13 and 14. It assumes that plane waves are incident on the lined section from the hard wall ducts, and mode matching is used to assure continuity of acoustic pressure and axial particle acoustic velocity between the lined and the hard wall duct sections. A cost function is built based on the difference between the measured and calculated pressures at the microphone positions in the hard wall ducts on both sides of the lined section, and the liner impedance is obtained by minimizing it.

Figure 1. Sketch of a liner test setup at KTH.20

Table 1 shows a summary of the liner specifications and impedance eduction techniques used to obtain the data discussed in.24 It should be noted that in Ref. 11 NASA conducted a comparison study of four different impedance eduction methods using the NASA Langley GFIT test rig. In24 only the results for the straightforward method were considered. B.

In-Situ impedance measurement technique

The in-situ measurement technique was introduced by Dean28 builds on estimating the acoustic particle velocity using the pressure difference over the perforated top sheet. Figure 2 shows a sketch of the principle 2 of 11 American Institute of Aeronautics and Astronautics

Table 1. Summary of the data sets analyzed in.24

Data set Hole diameter [mm] Plate thickness [mm] Cavity depth [mm] Open area [%] Length/width [mm] Flow Mach numbers Eduction method

NASA11 1.0 0.65 38.0 8.7 610 x 50.8 0.3, 0.5 SFM

Renou19 1.3 1.2 37.5 10.5 200 x 100 0.2 SFM

KTH120, 21 1.0 1.0 40.0 1.5 558 x 63 0.1, 0.2 MMM

KTH222 0.75 0.6 19.0 16.3 56 x 28 0.12, 0.24 MMM

UFSC124 1.0 0.65 19.0 5.18 200 x 100 0.25 SFM/MMM

UFSC224 2.0 0.8 19.0 8.63 200 x 100 0.25 SFM/MMM

for the in-situ impedance measurement setup. There is one surface pressure transducer (Mic1) which gives the acoustic pressure at the liner surface. The other pressure transducer (Mic 2) is placed at the back plate, but, since it is assumed that the sound field in the cavity is known from theory, the pressure just inside the top sheet can be calculated. The main considerations of the technique area that the backing plate are: (i) fully reflective and (ii) that only standing plane waves exist in the cavity. In this case, the relation between the pressure at the any point in cavity P for a given incident pressure P0 is given by P = 2P0 eiωt cos(ky)

(1)

where y is the distance from the bottom of the cell. From the momentum equation it is possible to obtain the particle velocity as 2P0 iωt u = −i sin(ky) e (2) ρc If continuity of acoustic particle across the perforate is assumed, the particle velocity at the liner surface may be written as PB iωt uf = i sin(kL) e (3) ρc where PB = 2P0 is the acoustic pressure at the bottom of the cell. The normalized impedance is given by the ratio between acoustic pressure and particle velocity at the liner surface Pf , uf

(4)

−i Pf iφ e , sin(kL) Pb

(5)

Z= so, substituting, Z=

In practical terms, (Pf /Pb )eiφ is the Frequency Response Function Hpb between Mic1 and Mic2, so Z = −i

Hpb sin(kL)

(6)

The in-situ technique implicitly assumes that the liner is locally reactive, but there are no assumptions regarding the flow outside the liner. Therefore, it does not rely on the definition of a boundary conditions such as Ingard-Myers boundary condition since it will not attempt to model the acoustic field as in the impedance eduction techniques.

III. A.

Previous results

Data set results from Ref.24

In Ref. 24, an attempt was made to analyze the trend in impedance results for both upstream and downstream configurations obtained using different impedance eduction methods and in different test rigs. In general, it has been observed that downstream results tend to display higher values at lower frequency and a 3 of 11 American Institute of Aeronautics and Astronautics

Pf

PB

Figure 2. Sketch of an in-situ impedance measurement setup.20

more pronounced decay with frequency. An example of such behavior can be seen in Figure 3. Here, results for M = 0.25 using two different eduction techniques are presented: the mode matching method (MMM) and the straightforward method (SFM). It can be seen that the downstream values are slightly higher at lower frequencies than the upstream results, and converging at higher frequencies. Although the results diverge at lower frequencies between eduction methods, the downstream/upstream trend can be noted in both methods.

Figure 3. Educed impedance results using MMM (straight line) and SFM (dotted) for the UFSC1 data set: black - upstream excitation, red - downstream excitation. (a) and (b) Results for mean Mach number M = 0.25. (c) and (d) Local Mach number Mδ = 0.165.

In order to allow a better comparison between the data sets described in Table 1, Boden et al.24 normalized the resistances by their respective percentage of open areas and the frequency by the liner resonance frequency. Figure 4 shows the new curves for downstream and upstream conditions. Only the resistance is considered since most discrepancies in the results between upstream and downstream excitation were observed. Even though other parameters are left outside the normalization applied (such as Mach number and hole geometry), the procedure seems to result in a good collapse of the curves for upstream excitation, with the exception of UFSC data set 2 at low frequency and KTH data set 2 at high frequencies. In general, the resistance under upstream condition shows an almost frequency-independent behavior. This agrees with some semi-empirical models for the acoustic impedance of liners.34, 35 Since these models have been adjusted based on experimental measurements, it may be that they were optimized for upstream excitation, without regarding the possible difference between upstream and downstream conditions. A very good collapse of the curves can be observed for the resistance obtained with downstream excitation. In this case, a clear frequency-dependent behavior can be noted, with an almost constant decrease of the resistance with frequency. Such trend is also observed in some liner impedance semi-empirical models,36, 37 suggesting that these models were derived from experimental data obtained under downstream condition. Overall, the data sets show a clear difference between educed liner resistance for upstream and downstream conditions. This supports the analysis carried out in the previous sections and reinforce the need of alternatives for the use of the Ingard-Myers boundary conditions in impedance eduction techniques.


Normalized Resistance

0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

2.5

3

2.5

3

(a) Upstream excitation

Normalized Resistance

0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

Normalized Frequency (b) Downstream excitation Figure 4. Normalized impedance multiplied by the percentage of open area: black - KTH1 at M = 0.2, gray - KTH2 at M = 0.24, red - UFSC1 at M = 0.25, blue - UFSC 2 at M = 0.25, green - NASA at M = 0.3, cyan Renou and Aur´ egan at M = 0.2. (a) Upstream excitation (b) Downstream excitation24

B.

Boundary conditions and velocity profile

In a companion paper,38 the authors compare the classical Ingard-Myers boundary conditions with more recently proposed boundary conditions, such as the one presented by Brambley,39 when applied in impedance eduction techniques. In the study, both the MMM and SFM techniques are used, with the assumptions of uniform and shear flow and also the numerical solution of the Pridmore-Brown equation40 in the case of the SFM. The flow profiles considered in the analysis are show in Figure 5, together with experimental data from the test rig. Figure 6 compare the results for the educed impedance at M = 0.28 by the MMM considering both Ingard-Myers boundary condition (uniform flow) and Brambley boundary condition with a boundary layer thickness of 25 % of the duct radius. The discrepancies between downstream and upstream results are clearly visible for Ingard-Myers boundary condition, but when the Brambley boundary condition is applied, the difference between the curves is reduced. These results suggest that Brambley boundary condition seems to be an improvement over Ingard-Myers boundary condition, but it still fails to collapse the curves. A similar trend is seen in Figure 7, where results obtained by means of the SFM considering both the Ingard-Myers boundary condition and a numerical solution of the Pridmore-Brown equation are compared. Again, the use of Ingard-Myers boundary condition leads to large discrepancies between downstream and upstream cases. The implementation of the Pridmore-Brown equation seems to improve the agreement at lower frequencies, but with a cost of larger discrepancies at higher frequencies. A more detailed discussion on these results is presented in the next section.


Figure 5. Different low profiles considered in Ref.38 at different flow velocities. 5

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

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

0,5

1

1,5

2

2,5

3

Frequency [kHz]

Frequency [kHz]

Figure 6. Impedance eduction results of Ref.38 for liner A using the mode matching method and uniform flow.

C.

Locally reactive assumption

More recently, Dai and Auregan32 have studied the locally reactive assumption by comparing two different representations of the liner acoustic behavior under shear flow. Both approaches are summarized in Figure 8. In the first approach, duct and liner are considered as a periodic system and the solution for a single cell (highlighted in Figure 8.a) by means of the Periodic Theory (also known as Floquet-Bloch theorem) allows the determination of the wavenumber content at the duct axial direction. The second approach uses the classical assumption that the liner is a homogeneous surface that can be represented by a locally reactive impedance in order to obtain the axial wave numbers. The comparison between axial wave numbers obtained using each approach in the case of no flow show a perfect agreement. In other words, the liner considered in the analysis can be correctly represented by means of a locally reactive impedance in the no flow condition. In the case with a shear flow (M = 0.3), a difference scenario is observed, and the equivalent impedance computed from the Bloch wave numbers (calculated using the Periodic Theory) depends on the acoustic propagation direction (upstream or downstream) as can be seen in Figure 9. It was then concluded that a single impedance of not capable of representing the liner in this case. A discussion on the impact of these results on the traditional view of liner behavior is presented in the next section.


5

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

3

1 0 Reactance [−]

Resistance [−]

Mach 0.28

4

2

−1 −2 −3 −4

1

−5 0

1

1,5

2

2,5

3

−6

1

1,5

2

2,5

3

Frequency [kHz]

Frequency [kHz]

Figure 7. Impedance eduction results of Ref.38 for liner A using the straightforward method on the following cases: uniform flow - UF (Myers boundary condition) and shear flow - SF (exact solution - Pridmore-Brown equation)

Figure 8. (a) Sketch of the periodic representation of a liner with periodic cell considered in the calculations highlighted; and (b) liner representation by means of continuous system with equivalent impedance.32

IV. A.

Results for in-situ technique and discussion

In-situ versus impedance eduction techniques

As mentioned above, the in-situ technique provides an estimative of the liner local impedance based solely on a frequency response function (FRF) between pressure at the liner surface and at the back plate. Therefore, no assumptions is made regarding how the acoustic field is affect by the liner (locally reactive assumption) or about the flow beyond the liner surface (uniform or shear flow). As a consequence, the analysis of the impedance obtained from the in-situ measurement for upstream or downstream configuration can provide a interesting insight on the reasons for the discrepancies observed between these configurations. The same sample used to obtain the data set UFSC1 (Table 1) was instrumented to perform in-situ measurements. Details of the instrumentation are given in Ref. 41. The FRF between top surface microphone and back plate microphone was measured for different flow speeds and for both upstream and downstream configuration when an acoustic field of averaged 130 dB SPL was applied. From the measured FRFs, the liner impedance was calculated using Eq. 6. A correction for the different percentage of open area (POA) in the instrumented cell has also been applied.41 Figures 10, 11 and 12 compare the impedance results for M = 0.10, M = 0.22 and M = 0.28, respectively, obtained using the in-situ technique for both upstream and downstream configuration. The results are also compared with data obtained using the SFM method, whose input data (sound pressure measured at the 7 of 11 American Institute of Aeronautics and Astronautics

+ Figure 9. Real and imaginary parts of the liner equivalent impedance for upstream kB0 conditions and down− stream kB0 conditions for Mach number M = 3 and m = 15.32

opposed wall) was measured during the same test campaign. Overall, the agreement observed between the in-situ and the SFM is similar to what has been previously reported in the literature.27, 41 The discrepancies between upstream and downstream results are clearly visible on the SFM data, but are considerably reduced on the in-situ results. Nevertheless, the in-situ downstream results are consistently higher that the upstream values for all flow speeds considered. This suggest that the liner reactive and dissipative mechanism for a single cell may be influence by the propagation direction. 3

2 1

2.5

0 -1

Reactance [-]

Resistance [-]

2

1.5

1

-2 -3 -4 -5 In situ - Upstream source In situ - Downstream source SFM - Myers (uniform) - Upstream source SFM - Myers (uniform) - Downstream source

-6

0.5

-7 0 500

1000

1500

2000

2500

3000

-8 500

1000

1500

Frequency [Hz]

2000

2500

3000

Frequency [Hz]

Figure 10. In-situ versus SFM - M = 0.10

B.

On the flow direction discrepancies

Different hypotheses have been proposed in the literature to explain the discrepancies observed between liner impedance obtained experimentally or numerically under upstream and downstream conditions. These hypotheses may be summarized as follows: 1. A shear flow can be represented by a uniform flow; 2. The liner can be modeled as locally-reactive surface; 3. The liner reactive and dissipative mechanisms are independent of the wave propagation direction. The application of Ingard-Myers boundary condition assumes that all three hypotheses are valid. However, the results presented in Section III.B show that the consideration of a shear flow does reduce the upstream/downstream discrepancies. Indeed, the use of Ingard-Myers boundary conditions in impedance 8 of 11 American Institute of Aeronautics and Astronautics

3

2 1

2.5

0 -1

Reactance [-]

Resistance [-]

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-8 500

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eduction techniques which rely in data from test rigs with long ducts can lead to significant errors due to the strong shear layer seen in such test rigs. Both Brambley boundary condition and the numerical solution of Pridmore-Brown equation seem to be improvements over the Ingard-Myers boundary condition, but are not the full answer to the upstream/downstream discrepancies. The analysis performed by Dai and Auregan32 and summarize in Section III.C show that a impedance model may not be sufficient to properly represent the liner. Even when fully representing the shear layer, an equivalent impedance was unable to model the liner for both downstream and upstream conditions, suggesting that second hypotheses above is not always valid and a more general model of the liner may be necessary. Nonetheless, the analysis carried out did not take into account some important parameters such as viscosity, turbulence, and non-linear effects. However, some considerations about such effects may be drawn from the analysis presented in Section IV.A. The comparison of impedance results obtained by means of the in-situ technique for both upstream and downstream conditions presented in Section IV.A pointed to small upstream/downstream discrepancies. The differences observed were seen at all Mach numbers, with downstream data always slightly higher than upstream results, specially at lower frequencies. The results suggest that the hypothesis (3) may also not be always valid, since the other two hypothesis are not assumed in the in-situ technique, leaving only the


third hypothesis to explain the small differences. In some way, such discrepancies should be expected in view of the interaction between the cell and the flow dynamics. As show in previous studies,42 the vortex shedding present at the holes is likely to be seen by the acoustic field in different ways depending on the wave propagation. Such interaction are likely to become even stronger at high SPL and lead to higher upstream/downstream discrepancies.

V.

Conclusions

A reviews of previous studies, both experimental and numerical, on educed liner impedance under different flow directions have been presented, together with new data comparing a impedance eduction technique and the in-situ method. The analysis has shown that the discrepancies usually seem on educed impedance data for downstream and upstream conditions may not be caused by a single effect. Indeed, the analysis suggests that the violation of three hypotheses commonly assumed in these methods are actually responsible for the discrepancies. The hypothesis are: (i) a uniform flow condition can be assumed, (ii) the liner is locally reactive, and (iii) the liner dynamic behavior is independent of the flow direction. Although the analysis carried above suggest that the current models used to represent the liner are not capable of fully modeling its physics, such models have been used in the past to design liners for real applications with relative success. In this sense, the present study suggests that these models can continue to be used with care depending on the conditions under analysis, but improved models may be necessary if an optimal design is desired.

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