Acoustic damping performance of coupled Helmholtz resonators with a sharable perforated sidewall in the presence of grazing flow

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Acoustic damping performance of coupled Helmholtz resonators with a sharable perforated sidewall in the presence of grazing flow Weichen Pan, Xiao Xu, Jun Li, Yiheng Guan ∗ School of Energy and Power Engineering, Jiangsu University of Science and Technology, Mengxi Rd 2, Zhenjiang city, Jiangsu Province, China

a r t i c l e

i n f o

Article history: Received 13 August 2019 Received in revised form 5 October 2019 Accepted 14 November 2019 Available online xxxx Keywords: Aeroacoustics Helmholtz resonators Transmission loss Thermoacoustics

a b s t r a c t Helmholtz resonators are widely used in gas turbines and aeroengines, because of its simple structure and high noise damping. To broaden its effective frequency range, two or more Helmholtz resonators could be applied. In this work, systematic studies are conducted to evaluate the aeroacoustic damping performance of two coupled Helmholtz resonators with a sharable perforated sidewall in the presence of a grazing flow. For this, 2D numerical model of a duct with two Helmholtz resonators implemented is developed via solving linearized Navier-Stokes equations in frequency domain. The model is validated first by comparing with the experimental data available in the literature. It is then modified to examine the effects of 1) the perforated orifice width Dx with respect to the back-cavity width Dr , 2) the mean grazing flow Mach number Ma, 3) the flow direction: forward and reverse. It is found that unlike conventional uncoupled Helmholtz resonators, the coupled ones are associated with three or more damping peaks. The local transmission peaks depend strongly on Dx /Dr , Ma and the flow direction in terms of the magnitude and the resonant frequencies. Furthermore, increasing Dx /Dr leads to the secondary peak being shifted to a higher frequency by approximately 100 Hz. However, increasing the grazing flow Mach number gives rise to deteriorated noise damping performance in terms of the local maximum transmission losses by about 10 dB. Finally, the classical theoretical formula ω2 = c 2 S/VLeff fails in predicting the resonant frequencies of the coupled resonators in presence of the grazing flow. The present study help on optimizing the design of coupled multiple Helmholtz resonators in application. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Gas turbine engines and aero engine produce loud noise. There are two typical and dominant noise sources associated with the engines. One is the combustion noise and the other is aerodynamic noise. Combustion noise is caused by a sudden decrease of density in the combustion chamber. Combustion process is acting like a monopole sound source. Aerodynamic noise [1,2] is generated due to the flow motion such as flow separation and turbulence of the working medium. Increasing noise emission requirements means that effective noise dampers need to be designed and applied. Helmholtz resonators [3,4] are widely used as noise-damping devices to suppress combustion and aerodynamic noise. Due to its simple structure and low manufacture and maintenance cost, they are widely applied in engine systems as a passive control actuators [5–7]. Helmholtz resonators are generally composing of a connecting neck and a large back/resonant cavity. It’s shape likes

*

Corresponding author. E-mail addresses: [email protected] (W. Pan), [email protected] (Y. Guan). https://doi.org/10.1016/j.ast.2019.105573 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

a beer bottle. When sound waves propagate in the engine passing through the neck, part of the incident sound will be reflected back upstream due to the impedance change [8]. Meanwhile, the acoustic disturbance at the neck will push the air in and out of the back cavity. Thus resonance occurs. This mechanism will convert acoustical energy into heat and finally be dissipated by the working medium. Helmholtz resonators are successfully demonstrated to be applied to mitigate self-excited thermoacoustic oscillations [9–11]. However, the effective frequency range of Helmholtz resonators is typically narrow. In order to broaden its damping frequency range, two or more Helmholtz resonators could be implemented [12]. There are two typical implementation configurations of multiple Helmholtz resonators. One is series-connected configuration [13,14]. Intensive researchers have already been conducted on Helmholtz resonators and coupled ones. A single resonator [15] was studied by considering the interaction with the modal resonance of an enclosure. It was found that regardless of the magnitude of the impedance of the resonator, the performance of the resonator was affected. Howard and his co-workers [16,17] used a Helmholtz resonator with optimized mesh and parameters to tune the mass damper. Then he combined the previous work on the

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rocket payload fairing for noise reduction, and using the modal coupling theory. As far as Helmholtz resonators are applied for noise reduction, both experimental and numerical studies were conducted in Ref. [18] on the unsteady airflow and noises in the interior of the vehicle. Fleming et al. [19] first introduced a new technique to control the resonant sound field by using a loudspeaker. Cheng et al. [20] then studied parallel coupled resonant metamaterials. They found that the frequency bandwidth is closely related to the number of resonators. The application of thin films was proposed in Ref. [21]. It was found that the presence of the film can decrease the vibration range. The other implementation configuration is parallel-connected Helmholtz resonators. Transmission loss of a parallel coupled Helmholtz resonator network was theoretically studied by using the Green’s function method [22]. The geometric shape effect of the parallel coupled Helmholtz resonators was then experimentally studied in Ref. [25]. It was found that the flexible sidewall motion introduces additional transmission-loss peaks at the nonresonant frequency of the cavities [27]. Broadening the frequency range of Helmholtz resonators was attempted in the literature [23, 26,28]. For example, the approaches include reducing the length of the connecting neck or increasing the diameter of the neck, periodically applying the resonators, and/or tuning the sharable sidewall. The parallel-connected resonators with different geometric shapes was analyzed in Ref. [24] in terms of acoustic impedance and transmission loss. Griffin et al. [29] conducted a more detailed study of the acoustic attenuation characteristics of coupled Helmholtz resonators and found that mechanically coupled resonators can provide a wider effective frequency bandwidth. This feature can be used to design a more damping effective noise damper. Tang et al. [30] used a simplified physical model to study the Helmholtz resonator transmission loss. In addition, when maximizing the resonator damping performance [31], the length of the neck of the resonator could be optimized [32,33]. In this paper, a two-dimensional numerical model of two coupled Helmholtz resonators with a sharable perforated sidewall is developed in the presence of a grazing flow. The configuration of the coupled Helmholtz resonator is quite similar to Ref. [25], except that there is a perforated aperture on the sharable sidewall. By solving linearized Navier–Stokes equations in frequency domain, aeroacoustic damping performances of the couple Helmholtz resonators are evaluated in terms of its transmission loss and power absorption coefficient. This is described in Sect. 2. The model is first validated by comparing with the results available in Ref. [33] first. It is then applied to examine the effects of the mean grazing flow Mach number and its flow direction (i.e. forward and reverse flow direction). Further study on the perforated aperture size Dx /Dr is then conducted. These results are discussed in Sect. 3. Key findings are summarized in Sect. 4. 2. Description of the 2D model and its validation The 2D model of a rectangular pipe with two parallel coupled Helmholtz resonators being flushed mounted is developed in this work. Fig. 1 shows the schematic drawing of the modelled duct and the parallel-coupled resonators with a sharable perforated sidewall. The perforated aperture width and the cavity width are denoted by D x and Dr respectively. The ratio of Dx /Dr could be set to 0%, 13.5%, 27.0%, 40.5% or 100%. There is a mean flow through the duct. The Mach number denoted by Ma could be varied from 0 to 0.1. The working air could either move forward and in the reverse direction, as shown schematically in Fig. 1(a) and (b) respectively. The dimensions and geometry of the Helmholtz resonators are summarized in Table 1.

Fig. 1. Schematics of the modelled parallel-coupled Helmholtz resonators with a sharable perforated sidewall in the presence of a mean grazing flow denoted by the Mach number Ma ; (a) forward flow, (b) reversed flow.

Table 1 Geometric dimensions of the modelled parallel-coupled Helmholtz resonators. Parameters

Values

Parameters

Values

Parameters

Values

Lu Lv Lw Dd Dn1,2

0.464 m 0.208 m 0.513 m 13.6 cm 5.0 cm

Ln Dr Vr1 Vr2 Ma

8.0 cm 7.4 cm 0.00068 m3 0.00113 m3 0→0.1

ρ0

1.2 kg/m3 101325 Pa 150–600 Hz 297 K 0.2 cm

p0

ω/2π T0 Dt

By following the previous study [32], the linearized Navier– Stokes equations in frequency domain is numerically solved. The governing equations include 1) Mass conservation,

Dρ Dt

+ ρ ∇ · u = 0.

(1)

2) Momentum conservation

ρ

Du Dt

=∇ ·Σ +ρ f.

(2)

The left hand side Eq. (2) represents the momentum change rate per unit volume of air, and the right side represents the surface and mass forces respectively converted to air per unit volume. The stress and strain rate, or the relationship between the stress tensor and the strain tensor, is usually called the constitutive equation, which is Σ = − pI − 23 μ I ∇ · u + 2μ S . Eq. (2) can be reformulated into:

ρ

Du Dt

2

= −∇ p − ∇(μ∇ · u ) + 2∇ · (μS) + ρ . 3

(3)

3) Energy conservation

ρ

De Dt

= − p ∇ · u + ∇ · (k∇ T ) + Φ

(4)

The left hand side of Eq. (4) denotes the internal energy change rate per unit volume of air, and the right side represents the compression work power, heat-transfer power, and work power

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of the viscous stress during fluid deformation when the air volume changes. Here, Φ is called the dissipation function and can ∂u be expressed as Φ = τi j ∂ x i . In a Cartesian coordinate system, Φ = j

∂u j 2 ∂ xi ) , where

λ( ∂∂uxk )2 + μ2 ( ∂∂ ux ij + k

λ = − 23 μ. The working medium-

air is assumed to be perfect gas and follow the thermodynamic state equations and the relationships such as, p = p (ρ , T ) and e = e (ρ , T ). Here p is pressure, ρ is density, T is temperature, u denotes velocity, μ is dynamic viscosity, and τi j is the viscous stress tensor. Since the present model involves a mean flow present in the duct. An appropriate turbulence model should be selected. There are generally three types. The first is that the mixed-length model. A typical model is the k − ε two-equation model. It is based on the assumption of eddy viscosity coefficient, called eddy viscosity (EV) model. Secondly, the assumption of the eddy viscosity coefficient can be abandoned. Reynolds stress is directly solved from the transport equation of the Reynolds stress itself. This is the Reynolds stress (RS) model. Finally, large-eddy simulation (LES) is a compromise between the direct numerical simulation and general turbulence models. For simplicity, we apply the two-equation k − ε turbulence model. The eddy viscosity coefficient νt is determined by the turbulent energy k and the turbulent energy dissipation rate ε.

νt = C μ

k2

ε

(5)

,

where C μ is the model coefficient. It is set to 0.09. Turbulence energy and dissipation: k and ε are determined by the respective transport equations as given as

 ∂k + P − ε, Ck +ν ε ∂ xi    ∂ ∂ε ∂ε k2 ε ε2 ∂ε + C ε1 P − C ε2 , + ui = Cε +ν ∂t ∂ xi ∂ xi ε ∂ xi k k ∂ ∂k ∂k + ui = ∂t ∂ xi ∂ xi



k2



(6) (7)

where P = + = 0.1, C ε = 0.08, C ε1 = 1.42, and C ε2 = 1.91. The numerical mesh as shown in Fig. 2 is unstructured. Finer mesh is applied near the resonator neck and the sharable perforated sidewall. Mesh independence study is conducted first by comparing 3 different number of meshes (i.e. 960641, 1428519 and 1968164 cells). A total of 1.428519 million meshes is chosen. This mesh selection is based on the comparison between the numerical results and the experimental data available in the literature [33]. The comparison between the numerical and experimental results is shown in Figs. 3 and 4 in terms of transmission loss, as Dx /Dr = 0. Here, the transmission loss TL is defined as

   p i (ω)   ,  T L ≡ 20 log10  p (ω) 

prediction of the resonant frequency in radians by using the classical formula:

ω12,2 =

∂ uk ∂ u i ∂ xi ) ∂ xk , C k

νt ( ∂∂ uxki

Fig. 2. Photos of the unstructured mesh of a 2D modelled Helmholtz resonators coupled with a sharable perforated sidewall. (a) mesh number N = 1428519 and (b) mesh number N = 42420.

(8)

t

where || p i (ω)|| denotes the incident sound pressure amplitude and || pt (ω)|| is the transmitted sound pressure amplitude. Since the frequency range is low, only plane waves are propagating within the duct. It is assumed that both the inlet and the outlet are acoustic open, i.e. p  = 0 Pa. This is consistent with the experimental setup boundary conditions as described in Ref. [33]. In the experimental measurement of the plane wave propagating in the duct. TL are determined experimentally by using a two-microphone technique. It can be seen from Fig. 3 that the present simulations results qualitatively agree with the experimental ones, as the Mach number is set to 4 different values as the same as the experimental flow conditions. Further comparison is done with the theoretical

c 2 S 1,2

( V 1,2 ) L e f f ,1,2

(9)

where c represents the speed of sound, S 1,2 denote the crosssectional areas of the two resonator necks, V 1 and V 2 represent the volume of the two resonant cavities, and L eff,12 denote the effective lengths of the two resonator necks [33]. Comparing the theoretical predictions and the numerical results reveals that there is a good agreement. Further transmission loss evaluation is performed, as the Mach number is increased up to Ma = 0.1 This is shown in Fig. 4. It can be seen that as the Mach number is non-negligible (Ma > 0.01), the transmission loss peaks and the resonant frequencies depend strongly on the grazing flow Mach number. The maximum transmission loss is found to be decreased by more than 10 dB, as Ma is increased from 0 to 0.1. The resonant frequencies are found to be reduced slightly by about 10%. 3. Results and discussion 3.1. Preliminary study on the effect of Dx /Dr = 100% and grazing flow direction The coupled Helmholtz resonators model is preliminarily evaluated, when Dx /Dr = 100%, i.e. there is no sharable sidewall between the resonators [34–36]. Fig. 5 shows the variation of the transmission loss (TL) with the forcing frequency, as the grazing flow Mach number is set to 4 different values. It can be seen from Fig. 5 that increasing the grazing flow Mach number leads to decreased maximum transmission losses (TLmax ). This finding is consistent with that obtained in Fig. 4. Comparing Fig. 5 with Fig. 4 reveals that the resonant frequencies corresponding to the local transmission loss peaks are increased. This is most likely due to the fact that the sharable sidewall is removed and the resonant

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Fig. 3. Comparison of transmission loss performances between the present numerical simulation, theoretical, and experimental results available in Ref. [33], as Dx /Dr = 0%.

Fig. 4. Variation of the transmission loss with the forcing frequency under Ma = 0, 0.03, 0.07, and 0.1, as Dx /Dr = 0%.

volume of each resonator is varied. As the Mach number is increased to 0.1 (the green dash curve in Fig. 5), transmission loss is negative between 260 and 290 Hz. This means that the resonators are generating sound, which is also known as ‘whistling. Interested readers can refer to Refs. [37,38] for more details. The whistling phenomenon is known to be related to the instability of the shear flow, associated with a hydrodynamic feedback or with an acoustic

Fig. 5. Variation in the transmission loss with forcing frequency, as Dx /Dr , =100%, i.e. in the absence of a sharable sidewall, as the grazing flow Mach number is set to Ma = 0, 0.03, 0.07, and 0.1. (For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.)

feedback. Hydrodynamic feedback occurs when the vortices generated along the shear layer reach an area where the steady velocity exhibits a gradient. Feedback can happen as well in reverberating acoustic conditions. In both cases, the feedback velocity fluctuation modulates the vorticity in the shear layers and the energy is transferred from the main flow to self-sustained oscillations.

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Fig. 6. Variation in the transmission loss the forcing frequency, as Dx /Dr = 100%, and Ma = 0, 0.03, 0.07, and 0.1 and the mean grazing flow direction is either forward or reversed.

Further studies are conducted to examine the grazing flow direction effect, as Dx /Dr = 100% and the Mach number is set to the same value. This is shown in Fig. 6. It can be seen that when Ma ≤ 0.03, the flow direction does not lead to any transmission loss performance. However, as Ma > 0.07 (see Fig. 6(c) and (d)), the flow direction is found to play an important role on affecting the maximum transmission loss performance and the effective frequency range. Whistling is not observed, as the flow is sucked (reversal flow) into the duct. This finding is not applicable to the configuration of the forward flow. As Dx /Dr = 0%, the grazing flow direction effect is illustrated in Fig. 7. It can be seen that when Ma ≤ 0.07, the forward flow direction is beneficial to improve the noise damping performance [39] in comparison with the reversed direction flow. The two local transmission loss peaks correspond to the separated resonator resonances. However, as the grazing flow Mach number is increased to 0.1, reversing the flow direction leads to ‘whistling’ corresponding to the negative transmission loss. For a given flow direction, the increase of Mach gives rise to deteriorated transmission loss performances. As Dx /Dr = 27%, the grazing flow direction effect is illustrated in Fig. 8. It can be seen that in general, the reversed flow direction is beneficial to improve the noise damping performance in comparison with the forward direction flow. The two local transmission loss peaks are observed. They are more widely separated in frequency spectra comparing with Fig. 7, ‘whistling’ effect [37, 38] is observed over a certain frequency range, as Ma ≥ 0.07. For a given flow direction, increasing Mach number Ma gives rise to deteriorated transmission loss performances. This is consistent with the previous studies and Table 2 summarizing the two local TLmax with the grazing flow Mach number being set to 4 different values.

Table 2 Summary of the local TLmax , as the grazing flow Mach number is varied. Grazing flow

Flow direction

Mach number Ma

Forward TLmax1 (dB)

TLmax2 (dB)

TLmax1 (dB)

TLmax2 (dB)

Ma = 0 Ma = 0.03 Ma = 0.07 Ma = 0.1

32.93 28.27 14.63 12.23

11.12 9.33 8.15 7.15

36.15 28.24 13.79 23.14

23.93 16.14 13.14 7.7

Reverse

3.2. Effect of Dx /Dr and Ma for a given Dx Now we consider the two Helmholtz resonators coupled with a perforated sharable sidewall. The width of the perforated orifice is set to 3 different values, i.e. Dx /Dr = 13.5%, 27%, or 40.5% in the presence of a mean grazing flow. This is shown in Fig. 9. It can be seen that for a given Mach number, the perforated orifice width Dx /Dr affects the 2nd transmission loss peaks strongly, in terms of the resonant frequency. The frequency of the secondary peak could be changed by approximately 35%, as the orifice width Dx /Dr is increased from 13.5% to 40.5% and Ma≤0.07. As Dx/Dr is increased and Ma = const, the 2nd resonant frequency is shift to higher value. This is due to the resonant frequency ω2 = c 2 S/VLeff . Since Dx/Dr is increase, the cross-sectional area is increased and thus the 2nd resonant frequency. Finally comparing Fig. 9(c) and (d) reveals that there may be an optimum orifice width Dx /Dr . Fig. 10 shows the performances of the two coupled Helmholtz resonators with a perforated sharable sidewall, as Ma is set to 4 different values. For a given Dx /Dr , the transmission loss is maximized in the absence of the grazing flow, i.e. Ma = 0. As the grazing flow Mach number is increased for a given Dx /Dr , the transmission loss performance is found to be deteriorated. Fig. 11

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Fig. 7. Variation in the transmission loss with the forcing frequency, as Dx /Dr = 0, Ma = 0, 0.03, 0.07, and 0.1 and the mean grazing flow direction is set to either forward or reversed.

Fig. 8. Variation in the transmission loss the forcing frequency, as Dx /Dr = 27%, and Ma = 0, 0.03, 0.07, and 0.1 and the mean grazing flow direction is either forward or reversed.

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Fig. 9. Variation of the transmission loss with the forcing frequency, as Ma is set to 4 different values and the flow direction is forward and Dx /Dr is set to 3 different values.

Fig. 10. Variation in the transmission loss with forcing frequency, as the forward mean grazing flow Ma is set to 0, 0.03, 0.07, and 0.1. (a) Dx /Dr = 13.5%, (b) Dx /Dr = 27%, (c) Dx /Dr = 40.5%.

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two or more resonators are coupled, or when the structure of resonators are modified, 2D/3D numerical predictions [40–45] are needed, especially in the presence of a grazing flow. In general, the present research shed lights on the design of effective acoustic resonators [46–54]. These acoustic dampers have great potential to be applied to attenuate self-excited thermoacoustic oscillations [55–63]. 4. Conclusions The present work considers 2D numerically modelling of a duct with two coupled Helmholtz resonators implemented in the presence of a mean grazing flow. These two resonators share the same perforated sidewall. The effects of (1) the grazing flow Mach number Ma, (2) its direction, and (3) the perforated aperture size Dx /Dr of the sharable sidewall are evaluated one at a time. The kind findings are summarized as

• For two coupled Helmholtz resonators, two and three even Fig. 11. Variation of the local maximum of transmission loss peaks with the grazing flow Mach number and the perforated orifice width ratio of Dx /Dr .

• • • •

more transmission loss peaks could be generated, as the grazing flow Mach number is varied. As the grazing flow Mach number is increased, the transmission loss performance is deteriorated in general. The grazing flow direction is found to not only affect the local transmission loss peaks in terms of the magnitude and the resonator frequencies but also lead to ‘whistling’ phenomenon. The size Dx /Dr of the perforated orifice in the sharable sidewall is found to strongly influence the maximum transmission losses. There may be an optimum size of the perforated orifice. The classical formula ω2 = c 2 S/V L eff used to predict resonant frequencies could not be applied to determine the two coupled resonators, especially in the presence of the perforated orifice and the non-negligible grazing flow.

In general, the present work sheds light on the effective design of two or more coupled Helmholtz resonators in the presence of non-negligible grazing flow. Declaration of competing interest The authors declare that there is no conflict of interest. Fig. 12. Comparison of the resonant frequencies ω1 and ω2 corresponding to the domain 2 local peaks with the theoretical ones, as the width ratio Dx /Dr = 0% and Dx /Dr = 100% and the Mach number is varied from 0 to 0.1.

summarizes the local maximum of transmission loss peaks varied with the grazing flow Mach number and the orifice width Dx /Dr . It is obvious that the maximum transmission losses depend strongly on the grazing flow Mach number and the orifice width Dx /Dr . As the grazing flow Mach number is increased, the TLmax is generally decreased. Only when the Mach number is increased to and above 0.07, TLmax may be decreased as Ma is increased further. However, the difference is less than 3 dB and is negligible. The difference may be due to the flow-acoustics-structure interaction at high Mach number. Fig. 12 illustrates the comparison of the resonant frequencies corresponding to the dominant 2 local peaks, as Dx /Dr = 0% and Dx /Dr = 100% and the Mach number is varied. It can be seen that the resonant frequencies ω1 and ω2 are dramatically changed as Dx /Dr = 0% is changed to 100%. Both the dominant and secondary peaks are shifted to higher frequencies. These resonant frequencies cannot be predicted by using the classical formula ω2 = c 2 S/VLeff by comparing with the theoretical predicted values. Here, c is the sound speed, V, S and Leff are the resonator’s volume, neck crosssectional area and the effective neck length. This reveals that when

Acknowledgements The authors would like to acknowledge the financial support of Jiangsu University of Science and Technology and the National Natural Science Foundation of China with grant No. 51506080. References [1] H. Zhao, Z. Lu, Y. Guan, Z. Liu, Effect of extended necks on transmission loss performances of Helmholtz resonators in presence of a grazing flow, Aerosp. Sci. Technol. 77 (2018) 228–234. [2] G. Wu, Z. Lu, X. Xu, W. Pan, Numerical investigation of aeroacoustics damping performance of a Helmholtz resonator: effects of geometry, grazing and bias flow, Aerosp. Sci. Technol. 86 (2019) 191–203. [3] J.Y. Chung, D.A. Blase, Transfer function method of measuring in-duct acoustic properties. I. Theory, J. Acoust. Soc. Am. 68 (1980) 907–913. ¯ [4] M.L. Munjal, Advances in the acoustics of flow ducts and mufflers, Sadhan a¯ 15 (1990) 57–72. [5] Iain D.J. Dupère, Ann P. Dowling, The use of Helmholtz resonators in a practical combustor, J. Eng. Gas Turbines Power 127 (2005) 268–275. [6] D. Zhao, C. A’Barrow, A.S. Morgans, J. Carrotte, Acoustic damping of a Helmholtz resonator with an oscillating volume, AIAA J. 47 (2009) 1672–1679. [7] C. Li, D. Zhao, N. Han, J. Li, Parametric measurements of the effect of in-duct orifice edge shape on its noise damping performance, J. Sound Vib. 384 (2016) 130–145. [8] Sjoerd W. Rienstra, Impedance models in time domain, including the extended Helmholtz resonator model, in: AIAA Aeroacoustics Conference, 2006.

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[9] Dan Zhao, A.S. Morgans, Tuned passive control of combustion instabilities using multiple Helmholtz resonators, J. Sound Vib. 320 (2009) 744–757. [10] J.M. Debedout, R.J. Bernhard, L. Mongeau, M.A. Franchek, Adaptive-passive noise control with self-tuning Helmholtz resonators, J. Sound Vib. 202 (1997) 109–123. [11] D.L. Gysling, G.S. Copeland, D.C. McCormick, W.M. Proscia, Combustion system damping augmentation with Helmholtz resonators, J. Eng. Gas Turbines Power 122 (2000) 269–279. [12] U. Zalluhoglu, N. Olgac, A study of Helmholtz resonators to stabilize thermoacoustically driven pressure oscillations, J. Acoust. Soc. Am. 139 (4) (2016) 1–11. [13] K.T. Chen, Y.H. Chen, K.Y. Lin, C.C. Weng, The improvement on the transmission loss of a duct by adding Helmholtz resonators, Appl. Acoust. 54 (1998) 71–82. [14] Dan Zhao, X.Y. Li, A review of acoustic dampers applied to combustion chambers in aerospace industry, Prog. Aerosp. Sci. 74 (2015) 114–130. [15] F.J. Fahy, C. Schofield, A note on the interaction between a Helmholtz resonator and an acoustic mode of an enclosure, J. Sound Vib. 72 (1980) 365–378. [16] Carl Q. Howard, Rick Morgans, Colin H. Hansen, Anthony C. Zander, A tool for the optimisation of vibro-acoustic systems using a parallel genetic algorithm and a distributed computing network, Noise Control Eng. J. 53 (2005) 256–267. [17] C.Q. Howard, C.H. Hansen, A.C. Zander, Noise reduction of a rocket payload fairing using tuned vibration absorbers with translational and rotational DOFs, in: Proceedings of ACOUSTICS, 9-11 Nov. 2005, Busselton, Western Autralia, in: Aust. Acoust. Soc., 2005. [18] B. Crouse, S. Senthooran, D. Freed, G. Balasubramanian, M. Gleason, Experimental and numerical investigation of a flow-induced cavity resonance with application to automobile buffeting, in: AIAA/CEAS Aeroacoustics Conference, 2006. [19] A.J. Fleming, D. Niederberger, S.O.R. Moheimani, M. Morari, Control of resonant acoustic sound fields by electrical shunting of a loudspeaker, IEEE Trans. Control Syst. Technol. 15 (2007) 689–703. [20] Y. Cheng, J.Y. Xu, X.J. Liu, Broad forbidden bands in parallel-coupled locally resonant ultrasonic metamaterials, Appl. Phys. Lett. 92 (2008). [21] Umar Farooq, Shahin S. Nudehi, A nonlinear acoustic resonator, in: Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 1, 2008. [22] D. Guan, Dan Zhao, J. Li, J. Li, Evaluations of acoustic damping performances of double-layer in-duct perforated plates at low Mach and Helmholtz number, J. Acoust. Soc. Am. 146 (5) (Nov. 2019), https://doi.org/10.1121/1.5134063. [23] Yu Hanren, Wen Guilin, Chen Zhifu, Wang Yanguang, Analysis of acoustical characteristics of coupled intake resonators, Constr. Eng. Des. 19 (2012) 289–293. [24] Matthew F. Calton, Scott D. Sommerfeldt, Modeling of acoustic resonators and resonator arrays for use in passive noise control, J. Acoust. Soc. Am. 137 (2015) 2340. [25] Chenzhen Ji, Dan Zhao, Xinyan Li, Mya Win Yin, Jing Li, Geometry effect investigation on parallel-coupled Helmholtz resonators for duct noise damping, in: AIAA/CEAS Aeroacoustics Conference, 2016. [26] Guang-Sheng Liu, Yao-Yin Peng, Ming-Hao Liu, Xin-Ye Zou, Jian-Chun Cheng, Broadband acoustic energy harvesting metasurface with coupled Helmholtz resonators, Appl. Phys. Lett. 113 (2018). [27] Dan Zhao, Transmission loss analysis of a parallel-coupled Helmholtz resonator network, AIAA J. 50 (2012). [28] Z.B. Wang, Y.S. Choy, Tunable parallel barriers using Helmholtz resonator, J. Sound Vib. 443 (2019) 109–123. [29] Steve Griffin, A. Lane Steven, Steve Huybrechts, Coupled Helmholtz resonators for acoustic attenuation, J. Vib. Acoust. 1 (2001) 11–17. [30] S.K. Tang, On Helmholtz resonators with tapered necks, J. Sound Vib. 279 (2005) 1085–1096. [31] Z. Zhang, D. Zhao, N. Han, S. Wang, J. Li, Control of combustion instability with a tunable Helmholtz resonator, Aerosp. Sci. Technol. 41 (2015) 55–62. [32] C. Cai, C.M. Mak, X. Shi, An extended neck versus a spiral neck of the Helmholtz resonator, Appl. Acoust. 115 (2017) 74–80. [33] A. Selamet, L. Lee, Helmholtz resonator with extended neck, J. Acoust. Soc. Am. 113 (4) (2003) 1975–1985. [34] C.H. Sohn, J.H. Park, A comparative study on acoustic damping induced by half-wave, quarter-wave and Helmholtz resonators, Aerosp. Sci. Technol. 15 (8) (2011) 606–614. [35] R.A. Prydz, L.S. Wirt, H.L. Kuntz, Transmission loss of a multilayer panel with internal tuned Helmholtz resonators, J. Acoust. Soc. Am. 87 (1990) 1597–1602. [36] F. Ghanadi, M. Arjomandi, B. Cazzolato, A. Zander, Effectiveness of flow-excited Helmholtz resonator on turbulence structures in streamwise and spanwise directions, in: International Symposium on Turbulence, 2015.

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[37] R. Lacombe, P. Moussou, Y. Auregan, Whistling of an orifice in a reverberating duct at low Mach number, J. Acoust. Soc. Am. 130 (2011) 2662–2672. [38] P. Testud, Y. Auregan, P. Moussou, A. Hirschberg, The whistling potentiality of an orifice in a confined flow using an energetic criterion, J. Sound Vib. 325 (2009) 769–780. [39] X.Q. Zhou, D.Y. Yu, Acoustic energy absorption dissipation characteristics of Helmholtz resonator enhanced and broadened by acoustic black hole, Aerosp. Sci. Technol. 81 (2018) 237–248. [40] A. Selamet, N.S. Dickey, J.M. Novak, Theoretical, computational and experimental investigation of Helmholtz resonators with fixed volume: lumped versus distributed analysis, J. Sound Vib. 187 (2) (1995) 358–367. [41] A. Selamet, P.M. Radavich, N.S. Dickey, J.M. Novak, Circular concentric Helmholtz resonators, J. Acoust. Soc. Am. 101 (1) (1997) 41–51. [42] E. Selamet, A. Selamet, A. Iqbal, H. Kim, Effect of Flow on Helmholtz Resonator Acoustics: A Three-Dimensional Computational Study vs. Experiments, SAE Technical Paper 2011-01-1521, 2011. [43] V. Bellucci, P. Flohr, C.O. Paschereit, F. Magni, On the Use of Helmholtz Resonators for Damping Acoustic Pulsations in Industrial Gas Turbines, Technical Report 2001-GT-0039, ASME TURBO EXPO 2001, June 4–7, 2001. [44] D.L. Gysling, G.S. Copeland, D.C. McCormick, W.M. Prosnia, Combustion system damping augmentation with Helmholtz resonators, ASME J. Eng. Gas Turbines Power 122 (2000) 269–274. [45] I.D.J. Dupere, A.P. Dowling, The use of Helmholtz resonators in a practical combustor, J. Eng. Gas Turbines Power 127 (2005) 268–275. [46] D. Zhao, E. Gutmark, Mitigating self-excited flame pulsating and thermoacoustic oscillations using perforated liners, Sci. Bull. 64 (13) (2019) 941–952. [47] D. Zhao, B. Wang, C.Z. Ji, Geometric shapes effect of in-duct perforated orifices on aeroacoustics damping performances at low Helmholtz and Strouhal number, J. Acoust. Soc. Am. 145 (4) (2019) 2126–2136. [48] L. Li, D. Zhao, Prediction of stability behaviours of longitudinal and circumferential eigenmodes in a choked thermoacoustic combustor, Aerosp. Sci. Technol. 46 (2015) 12–21. [49] D. Zhao, Y. Guan, A. Reinecke, Characterizing hydrogen-fuelled pulsating combustion on thermodynamic properties of a combustor, Commun. Phys. 2 (2019) 44, https://doi.org/10.1038/s42005-019-0142-8. [50] X. Li, D. Zhao, J. Li, Y. Xu, Experimental investigation of anti-sound approach in damping self-sustained thermoacoustic oscillations, J. Appl. Phys. 114 (20) (2013) 204903. [51] S. Li, D. Zhao, C. Ji, J. Li, Combustion instabilities in a bifurcating tube: openand closed-loop measurements, AIAA J. 52 (11) (2014) 2513–2523. [52] C. Ji, D. Zhao, Two-dimensional lattice Boltzmann investigation of sound absorption of perforated orifices with different geometric shapes, Aerosp. Sci. Technol. 39 (2014) 40–47. [53] D. Zhao, L. Ang, C. Ji, Numerical and experimental investigation of the acoustic damping effect of single-layer perforated liners with joint bias-grazing flow, J. Sound Vib. 342 (2015) 152–167. [54] D. Zhao, A.S. Morgans, A.P. Dowling, Tuned passive control of acoustic damping of perforated liners, AIAA J. 49 (2011) 725–734. [55] Z. Zhong, D. Zhao, Time-domain characterization of the acoustic damping of a perforated liner with bias flow, J. Acoust. Soc. Am. 132 (2012) 271–281. [56] D. Guan, D. Zhao, J. Li, Y. Chen, J. Li, Aeroacoustic damping performance studies on off-axial double-layer in-duct orifices at low Mach and Helmholtz number, Appl. Acoust. 156 (2019) 46–55. [57] X. Li, D. Zhao, B. Shi, Coherence resonance and stochastic bifurcation behaviors of simplified standing-wave thermoacoustic systems, J. Acoust. Soc. Am. 145 (2019) 692–702. [58] D. Zhao, J. Li, Feedback control of combustion instabilities using a Helmholtz resonator with an oscillating volume, Combust. Sci. Technol. 184 (2012) 694–716. [59] D. Zhao, Z. Lu, H. Zhao, X. Li, B. Wang, P. Liu, A review of active control approaches in stabilizing combustion systems in aerospace industry, Prog. Aerosp. Sci. 97 (2018) 35–60. [60] D. Zhao, E. Gutmark, L.P.H. de Goey, A review of cavity-based trapped vortex, ultra-compact, high-g inter-turbine combustors, Prog. Energy Combust. Sci. 66 (2018) 42–82. [61] D. Zhao, C. Ji, J. Li, L. Ang, Experimental comparison of noise dissipation effects of single- and double-layer acoustic liners, Appl. Acoust. 141 (2018) 281–292. [62] D. Zhao, Y. Sun, S. Ni, C. Ji, D. Sun, Experimental and theoretical studies of aeroacoustics damping performance of a bias-flow perforated orifice, Appl. Acoust. 145 (2019) 328–338. [63] W.M. Jungowski, G. Grabitz, Self-sustained oscillation of a jet impinging upon a Helmholtz resonator, J. Fluid Mech. 179 (1987) 77–103.