Compact asymmetric treatments for perfect sound absorption in venti- lated problems Jean Boulvert 1 Laboratoire d’Acoustique de l’Université du Mans (LAUM) Institut d’Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France Vicent Romero García Laboratoire d’Acoustique de l’Université du Mans (LAUM) Institut d’Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France Gwénaël Gabard Laboratoire d’Acoustique de l’Université du Mans (LAUM) Institut d’Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France Jean-Philippe Groby Laboratoire d’Acoustique de l’Université du Mans (LAUM) Institut d’Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France

ABSTRACT This presentation introduces compact asymmetric treatments for sound absorption, i.e., simultaneous cancellation of reflection and transmission, in a lined duct of constant cross section. The treatments are composed of side by side and sub-wavelength resonators, they are thus compact which facilitates their practical use. The simplest asymmetric treatment is formed by a pair of detuned resonators that can lead to a perfect monochromatic sound absorption. Furthermore, the combination of multiple pairs of resonators generates multiple low quality factor absorption peaks inducing an absorption plateau. Two compact asymmetric treatments are presented: one is composed of quarter-wavelength porous resonators to prove their ability to achieve perfect sub-wavelength absorption in a duct prob- lem, and the other is composed of Helmholtz resonators and is mounted on the walls of a large cross- section duct. In both cases, the evanescent coupling between the resonators has an important impact on the treatments behavior and is thus taken into account during their optimization. The experimental results are found in good agreement with the theory and, for each treatment, a mean absorption coefficient of 99% over a large target and subwavelength frequency range is observed.

1. INTRODUCTION

This work deals with the unidirectional absorption of sound in ducts by means of asymmetric and flush mounted treatments. The term “unidirectional absorption” refers to the cancellation of the sound wave from the incident side, both in terms of reflection and transmission. Infinite ducts of constant section and flush mounted treatments are considered so that the response of the system only depends on the acoustic properties of the treatment and of the cross section of the duct. The frequency range

1 jean.boulert@univ-lemans.fr

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of interest is below the first cut-off frequency of the duct allowing only plane waves to propagate. Under these conditions, if the treatment is only composed of a single point resonator, its absorption coefficient, 𝛼 , cannot exceed 0.5 [1]. Asymmetric treatments overcome this limitation as they can target perfect absorption from the incident side, 𝛼 + = 1 [1]–[6]. Their response depends on the inci- dent side of the sound wave and are composed of at least two different resonators or identical reso- nators laid out by groups. In both cases, the resonators or group of resonators are spaced in the axial direction of the duct. This work provides simple and generic explanations of the operating principle of flush mounted asymmetric absorbers. In addition, a particular attention is paid to the compacity of the absorbers, both in terms of height and length, using sub-wavelength and side-by-side resonators, respectively. Finally, two absorbers are designed, optimized, and tested experimentally. The first one is composed of quarter-wavelength porous resonators to showcase their sub-wavelength properties in a duct prob- lem, and the other is composed of Helmholtz resonators and is mounted to the walls of a wide duct to explore the specificities of such type of ducts. No mean air flow is considered inside the duct except during the experimental testing of the first absorber. 2. SIMPLE AND GENERIC EXPLANATIONS OF FLUSH MOUNTED ASYMMETRIC ASBORBERS

This section presents the prediction methods used in this work and their lessons learned.

2.1. Prediction of the system acoustic behavior Transfer Matrix Method The Transfer Matrix Method (TMM) provides a 1D model of a duct lined by an absorber (see Fig. 1(a))[6] leading to an analytic description of its acoustic behavior. Only plane waves are con- sidered, the absorbers are formed of point resonators, i.e., with no axial extent, and the possible eva- nescent coupling between the resonators is therefore neglected. The TMM does not account for all the complexity of the problem at hand but still captures its key features. This method is thus the one used to design most of the asymmetric absorbers presented in the literature. We use it to derive ana- lytic expressions of optimal impedances of the asymmetric absorbers’ resonators. Mode Matching Technique The Mode Matching Technique (MMT) provides a 3D model of the system (see Fig. 1(b)) and con- siders the potential coupling between the resonators. The latter are accounted for by their surface impedance. This model is more accurate than the TMM but not allows to derive analytic rules. It is used to evaluate the impacts of the evanescent coupling between the resonators and to optimize the two presented absorbers.

Finite Element Method Finally, a Finite Element Method (FEM) is used to validate the MMT predictions. A 3D model of the system, the full geometry of the resonators and the potential evanescent coupling between them are considered.

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a b

Figure 1: Schematic representation of an absorber composed of two flush resonators.

ncident ave

(a) TMM. (b) MMT.

2.2. Resonators optimal impedance An absorber composed of two different resonators, axially spaced by a distance 𝛿, is considered for monochromatic perfect absorption at 𝑓 0 : 𝛼 + ሺ𝑓 0 ሻ= 1 (see Fig. 1(a)). It is analyzed by means of the TMM. The downstream (i.e., farthest from the incident wave) resonator blocks the wave propagation by forming a soft wall inside the duct. To do so, its surface impedance must be such that

𝑍 2 ሺ𝑓 0 ሻ

𝑆 2 𝑆 0 , (1)

𝑍 0 ≪

where 𝑍 2 is the surface impedance of the downstream resonator, 𝑍 0 is the characteristic impedance of the air, 𝑆 2 is the section of the downstream resonator connected to the duct (e.g., the cross section of the neck if the resonator is a Helmholtz resonator), and 𝑆 0 is the cross section of the duct. To obtain a perfect soft wall, i.e., 𝑇ሺ𝑓 0 ሻ= 0 , the resonator should be lossless such that at its resonance fre- quency, its impedance is null: 𝑍 2 ሺ𝑓 0 ሻ= 0 . In practice, resonators have intrinsic losses. The maximum attenuation is reached at the resonance frequency of the resonators and depends on Reሺ𝑍 2 ሻ, 𝑆 2 , and 𝑆 0 . This way, the efficiency of a given resonator to block the wave propagation is proportional the duct cross section. The downstream resonator blocks the wave by reflecting it back. To avoid that this backward wave propagates in the upstream direction away from the absorber, the upstream resonator impedance matches the system by having an impedance 𝑍 1 such that

𝑍 1 ሺ𝑓 0 ሻ

𝑆 1 𝑆 0 ሺsin 2 ሺ𝑘 0 𝛿ሻ−i sinሺ𝑘 0 𝛿ሻሻ , 𝑘 0 𝛿≠0 mod 𝜋 (2)

𝑍 0 =

2𝜋𝑓 0

where 𝑆 1 is the section of the upstream resonator connected to the duct and 𝑘 0 =

𝑐0 the wave num- ber of the air at 𝑓 0 . The impedance match can thus be perfect, 𝑅 + ሺ𝑓 0 ሻ= 0 , as the optimal impedance is not null. This simple equation synthetizes all the acoustic features of the problem. For a given non dimensional value of 𝑘 0 𝛿 , the wider the duct, the higher the optimal resistance, Reሺ𝑍 1 ሻ, and losses of the upstream resonator. In addition, the analysis of the 𝑘 0 𝛿 dependance indicates that the upstream resonator resistance and intrinsic losses are minimal (resp. maximal) for 𝑘 0 𝛿 close to 0 mod 𝜋 (resp. 𝜋

2 mod 𝜋ሻ and that it resonates at a higher (resp. lower) frequency than the downstream resonator for

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𝜋

𝜋

2 [ mod 𝜋 (resp. 𝑘 0 𝛿∈ ]

2 ; 𝜋[ mod 𝜋 ). The two resonators resonate at the same frequency

𝑘 0 𝛿∈ ]0;

𝜋

for 𝑘 0 𝛿=

2 mod 𝜋 .

2.3. Broadband absorption To increase the frequency range of the absorber, more than two resonators can be used [6]. To broaden the frequency range of quasi-null transmission, multiple resonators can be used and put either at the same position [5] or at different axial locations [7]. They all create a soft wall around their resonance frequency and at their axial position. The number of required resonators to tackle a given frequency range depends on 𝑆 0 and on their losses. In a similar fashion, multiple resonators can be used to obtain an impedance match of the system at multiple frequencies. The larger the number of resonators, the harder to explain the impact of each resonator on the behavior of the absorber. This is because all resonators have an impact on the reflection and transmission coefficients of the system at all frequen- cies.

2.4. Evanescent coupling The potential evanescent coupling between the resonators is neglected when the TMM is used to explain the behavior of asymmetric absorbers and to optimize them. However, this coupling can have strong effects on the system. In particular, the capacity of two identical and axially spaced resonators to block the wave propagation can be strongly decreased when the resonators couple. A MMT anal- ysis we performed on such system indicates that the coupling effects cannot be neglected when the resonators are close ( 𝛿< ~5 cm), when the duct is wide ( > ~5 ∗5 cm²), when the resonators are weakly damped, and when the resonators are tuned at similar frequencies. The impacts of the evanescent coupling can be problematic when designing compact absorbers, i.e., absorbers composed of side-by-side resonators and when designing absorbers for wide ducts. To sof- ten the evanescent coupling effects, the resonators should be relatively damped which is contradictory with Eq. (1). This way, the right balance between low losses leading to low resistance and high losses leading to less coupling should be found. In addition, when the absorber is designed for broadband absorption, its constitutive resonators should not be arranged in a monotonic cascade of resonance frequencies. 3. OPTIMIZED COMPACT ABSORBERS

This section introduces two compact and optimized absorbers for perfect absorption in sub-wave- length and wide target frequency ranges.

3.1. Compact and porous absorber for narrow duct This first absorber is composed of folded quarter wavelength resonators (white elements in Fig. 2(c)) filled with air or with a porous material (red elements in Fig. 2(c)) [2], [3]. The resonators are folded to resonate at a sub-wavelength frequency. The filling porous material is a micro-lattice. Its lattice constant (governing the pore size) can easily be adjusted. This way, tunning the folding of the reso- nators and the pore size of their filling porous medium allows to adjust the resonance frequency and the losses of the resonators independently [7].

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The absorber is composed of 12 different resonators along the axial duct direction. They are repeated 3 times along the lateral dimension of the duct to cover its bottom face. The height, width, and length of the absorber are 3 cm, 5 cm, and 20 cm, respectively. The width and height of the duct cross section are 5 cm and 4 cm. This way, the open area ratio of the system is

4∗5

4∗5+3∗5 = 57% . The absorber is optimized for perfect absorption in 𝑓∈[1250; 2150] Hz with no mean air flow. It is 3D printed and tested experimentally without mean air flow and with an air flow either propagating in the same direction than the incident sound wave (M0.24) or in the opposite direction (M-0.25), as presented in Fig. 2(a,b). Without air flow, the measure and FEM predictions are in good correlation and a mean absorption of 99% for 𝑓∈[1225; 2120] Hz and a Transmission Loss (TL= −20 logሺȁ𝑇ȁሻ ) larger than 19 dB are measured. The air flow mainly impacts the transmission coefficient of the system with an expected increase (resp. decrease) of the TL for backward (resp. forward) propagating air flow. The porous absorber is compact and subwavelength as its thickness corresponds to 1/9.1 of the wave- length in air at 1225 Hz.

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Figure 2: Compact and porous absorber optimized for a narrow duct. (a) Absorption coefficient. (b)

Transmission Loss (TL). (c) Picture of the absorber mounted in a sample holder.

3.2. Compact absorber for wide duct The second absorber is composed of 16 different Helmholtz resonators along the axial duct direction. They are repeated 8 times along the lateral dimension of the duct to cover its bottom face (see Fig. 3(c)). The height, width, and length of the absorber are 5 cm, 15 cm, and 30 cm, respectively. The width and height of the duct cross section are 15 cm and 14.8 cm. This way, the open area ratio of

250000800000850050000A000 2500508000008500509000000 — 25005065000000500 Frequency (Hz) Frequency (Hz)

14.8∗15

the system is

14.8∗15+5∗15 = 75% . The absorber is optimized for perfect absorption in 𝑓∈[700; 800] Hz with no mean air flow. It is 3D printed and tested experimentally only without mean air flow as the wide duct test bench available during this work could not generate a mean air flow. Its absorption coefficient and TL are presented in Fig. 3(b). The experimental results are in good correlation with the MMT and FEM predictions while being much different than the TMM predictions. This is be- cause, conversely to the TMM, the MMT and FEM consider the potential evanescent coupling effects. Experimentally, a mean absorption of 99% for 𝑓∈[700; 800] Hz and a TL larger than 20 dB are observed. The absorber is compact and subwavelength as its thickness corresponds to 1/9.8 of the wavelength in air at 700 Hz.

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Figure 3: Compact absorber optimized for a wide duct. (a) Absorption coefficient. (b) Transmission

Loss (TL). (c) Picture of the absorber. 4. CONCLUSION

A flush mounted absorber for perfect monochromatic absorption composed of two spaced resonators has been studied by mean of a 1D model leading to simple and generic explanation of their operating principle. Broadband absorption can be obtained by using more resonators and the roles of the reso- nators become more intricate. In addition, the potential evanescent coupling between resonators has been presented. This coupling can have problematic impacts when perfect absorption is targeted in wide ducts and with compact absorbers. To prove that this problem can be overcome, two compact

a

absorbers are designed, optimized, and tested numerically. They both present a mean absorption co- efficient of 99% over a target frequency range. The first absorber is porous. It highlights that porous materials can have a subwavelength behavior not only at normal incidence but also in ducts problems. The second absorber is composed of sub-wavelength Helmholtz resonators and is flush mounted to a wall of a wide duct leading to a very high open area ratio of 75%. 5. ACKNOWLEDGEMENTS

The authors acknowledge the financial support from the ANR industrial chair MACIA (ANR-16- CHIN-0002). They also acknowledge Safran Aircraft Engines and the Natural Sciences and Engi- neering Research Council of Canada (NSERC) for supporting and funding this research. 6. REFERENCES

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