HAL Id: hal-02161528
https://hal.archives-ouvertes.fr/hal-02161528
Preprint submitted on 20 Jun 2019
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Porous Material
Rémi Roncen
To cite this version:
Rémi Roncen. Acoustic Nonlinearity of a Perforated Plate Backed by a Porous Material. 2019.
�hal-02161528�
Material
R´emi Roncen
ONERA / Department of Multi-Physics for Energetics, Toulouse University, F-31055, Toulouse, France
Abstract
An experimental investigation is carried out to study the acoustic nonlinearity of units made of a perforated facing backed by a porous-filled cavity. Porous media of different natures and microstructural properties are placed inside the cavity, topped by perforated plates of different properties. Impedance measurements are performed with and without porous materials. The addition of a porous inside the cavity reduces the nonlinear ratio of the perforated plate by an approximate factor of two. The proposed rationale for this decrease is that vortex-shedding is impeded in the cavity by viscous effects within the porous material, flattening the velocity profile at the perforation discontinuity and smoothing the outflow.
Keywords: acoustic liner; nonlinearity; sound level; porous material; perforated plate
1 Introduction
One of the main challenges in aeronautics remains the reduction of noise, in environments where materials are subject to high sound pressure levels (SPL) and a grazing flow. The current state of the art in passive acoustic absorption relies on the use of liners, acoustic units consisting of perforated sheets backed by a cavity and behaving as a resonator. Very efficient near a resonance frequency, current acoustic liners lack broadband absorption properties. In addition, the acoustic behavior of such resonators has been shown to change as a function of the SPL it is exposed to [1–
7]. As the SPL increases, the strong flow motion inside the liner perforations triggers nonlinear vortex-shedding at the outflow side of the perforated sheet [6, 7]. The vortex are convected away and ultimately converted into heat by molecular dissipation. This can result in either additional acoustic absorption, since the vortices add dissipation, or loss of efficiency, since the phenomenon also acoustically rigidifies the liner, thus progressively impeding waves from entering as the SPL increases. The design of liners currently relies on the prior knowledge of both the incident acoustic field and of the liner behavior when subject to a high SPL.
Porous materials used in acoustic absorption applications, such as foams or fiber materials, usually have a high absorption coefficient at higher frequencies [8]. In addition, porous media display a more constant behavior when subject to an SPL increase, when compared to a classical liner. The combination of a perforated sheet and a porous material has been fairly well studied in the past [9–12], and indeed showed an increase in broadband efficiency. However, studies on the acoustic nonlinearity caused by an SPL increase are scarce in the literature for this type of acoustic units [13–15]. A porous material is a dissipative medium, that also displays nonlinear properties [16–18], albeit of lower intensity than a perforated plate’s, but higher than that of an air cavity. It is thus not immediately obvious what would be the influence of the introduction of such a material inside the liner cavity on the SPL nonlinearity. Either one of three scenarios seem viable:
1
1. Due to the nonlinear nature of the porous material, the overall nonlinearity of the acoustic unit might increase in the presence of a porous material. This is the conclusion that can be drawn in Ref. [14], even though the author mostly discussed the nonlinearity in a hydrodynamic sense, not an acoustic one. The author showed that a nonlinear interference occurred between the perforated plate and the porous material, that would increase the hydrodynamic flow resistance of the unit. The model developed in Ref. [14] accounts for this interference and seems to be able to capture well the influence of the presence of a porous material on the nonlinearity.
2. In Ref. [15], an equivalent fluid model representing the nonlinear behavior of a perforated plate was used in conjunction with the Atalla & Sgard model [12] to represent the acoustic behavior of a porous material topped by a perforated sheet. No special attention was given in the new modeling at the interface between the perforated sheet and the porous material, and of possible nonlinear interactions. However, the model seemed perfectly capable of repre- senting the experimental results (impedance measurements) at high SPL. This suggests that integrating a porous material inside the liner cavity might have no significant effect on the global nonlinearity of the liner.
3. Since a porous material is a dissipative resistive media, it is possible that vortex-shedding could be inhibited in a porous filled cavity, thus reducing the overall non-linearity. This theory was not made explicit elsewhere in the literature before, to the best of the author’s knowledge.
However, a related argument was considered for hydrodynamic configurations [19–21], with the main results being reported in De Lemos’ book [22, Sec. 11.5]. It was shown that the recirculation zone after a backward-facing step configuration would be drastically shortened, sometimes even canceled, in the presence of a porous material at the step discontinuity. This demonstrated the capacity of porous media to impede the mechanism leading to the conversion from a detached boundary layer to rotational energy. Due to the dissipative nature of the porous material, the velocity profile at the discontinuity tends to flatten faster than in air, which lessens the chance of a detached flow occurring. This gives credit to the possibility that the acoustic nonlinearity of liners with porous-filled cavities could be reduced by a certain amount, due to vortex-shedding not occurring in the cavity side anymore (or at a lower intensity). This hypothesis has not been evaluated before, and was the primary motivation for this study (part of which was performed during the author’s PhD thesis [23, Chap. 8]).
The main purpose of this work is to experimentally quantify the acoustic nonlinear behavior of liners made of a nonlinear perforated sheet and a porous-filled cavity, and to evaluate the validity of the above hypotheses. A theoretical background is first given in Section 2. The measurement method and materials used are presented in Section 3. In Section 4, results of the measurement campaign are analyzed: impedance tube measurements at different SPL for different combinations of perforated sheets and porous materials are investigated to extract their nonlinear features. Conclusions are laid out in Section 5.
2 Theoretical background
In this section, elements regarding the acoustics of liners are given. In the following, the convention is e+jωt, where ω = 2πf is the angular frequency, with f the frequency in Hz. An acoustic unit can be characterized by its normalized surface impedance Zs(ω), defined as a pressure to normal velocity ratio
Zs(ω) = 1 ρfcf
p
v·n, (1)
with ρf and cf the fluid density and celerity, wherep is the acoustic pressure,v is the particular acoustic velocity andnis the normal to the surface directed toward the material. The impedance can be seen as a homogenized boundary condition, representing the macroscopic response of an acoustic system at normal incidence. It is used in boundary conditions to aggregate the local complexities of the unsteady flow near and inside the liner, generalizing the rigid wall (Zs =∞) or atmospheric (Zs = 1) boundary conditions. Since the impedance is a complex-valued quantity, it is usually decomposed into its real (resistance R) and imaginary (reactance χ ) parts: Zs(ω) = R(ω) + jχ(ω).
Impedance models of a perforated plate are usually based on the fluid mechanics approach of Crandall [24], and gave way to a vast literature on the subject [4, 25–27]. Different physical phenomena are taken into account in these models. The general consensus among the acoustic community for the linear acoustic dissipation mechanism is as follows [8, Chap. 9]. The real part of the impedance (resistance) is controlled by viscous effects occurring in the viscous boundary layer inside the perforation, and by the viscous friction near the orifice aperture caused by a distortion of the acoustic flow. The reactance is related to inertial effects, induced by the oscillation of an air mass within the perforation and by the cavity. Air distortion and radiation at the perforation extremities also account for an additional inertia, usually taken into account in the modeling via a plate thickness correction. In the following, the linear contribution of resistance and reactance to the impedance are writtenRLand χL, respectively.
In the nonlinear case, after a certain SPL is reached, an additional dissipation starts to oc- cur [2]. While research on the subject is still ongoing in the acoustic community [5, 7, 28], the current consensus is that vortex-shedding is triggered at the outflow side of the perforation, thus converting the acoustic energy into a rotational one, that eventually gets dissipated by viscous effects. This phenomenon mainly increases the acoustic resistance of the material, which can either improve or deteriorate the acoustic unit’s performance (at normal incidence, the optimal impedance isZs= 1).
This nonlinear phenomenon is commonly represented in the modeling by additive relationships as
Rtot(ω) =RL(ω) +RN L(ω),
χtot(ω) =χL(ω) +χN L(ω), (2)
where the tot subscript stands for total and the N L subscript for nonlinear. To characterize the non linearity of a liner, one mainly requires the knowledge ofRN L, since the reactance usually does not change much as the SPL increases [4, 29].
3 Measurements and materials
In this section, the experimental set-up is introduced, and the characterization results of the dif- ferent porous materials and perforated sheets are presented.
3.1 Experimental set-up
An impedance tube is used to measure the normal incidence acoustic impedance of a multilayer liner placed above a rigid backing using the two microphone method [30], as depicted in Figure 1.
The liner is made of a perforated plate (PP) and a porous material. A reference case is also performed where the porous material is removed, thus representing a classical liner configuration.
A loudspeaker is used to generate a standing wave pattern in the tube, at all frequencies of interest (white noise). Due to the dimensions of the tube, the lowest frequency accessible to analysis is
≈200 Hz, while the cut-off frequency is around 6000 Hz. The SPL inside the tube is then varied
from 100 dB to a maximum of 140 dB. A white noise is used in all experiments, and the SPL is calculated by integrating the noise spectrum between the aforementioned frequency limits, averaged at three different microphone locations. Using the pressure measurements at different axial locations
pi
pr
Porous material
Rigid backing mic. 1
mic. 2
PP
Figure 1: Impedance tube set-up. PP: perforated plate.
in the tube, the standing wave field pattern can be decomposed into an incident and a reflected field [30]. This allows the immediate calculation of the reflection coefficient ˜β, which gives access to the surface impedance Zs as
Zs= 1 + ˜β
1−β˜. (3)
3.2 Materials characterization
Four different rigid porous materials (with thicknessesLreported in Table 1) are used in this work, along with six different perforated plates. The porous microstructural properties are identified prior to their inclusion inside the liner cavities. The plate properties are measured optically.
When the solid phase of the porous can be assumed rigid and the frame motion can be neglected, one usually resorts to using an equivalent fluid model where only the slow compressional wave propagation in the fluid phase of the porous material is considered. Due to its extensive use in the literature, the Johnson-Champoux-Allard-Lafarge (JCAL) equivalent fluid model [31–33] is used in this work. The model inputs (intrinsic microstructure properties) are: the material volume porosity φ, the tortuosity α∞, the static viscous permeability k0 (unit m2), the viscous length Λ (unit m, [31]), the thermal length Λ0 (unit m, [32]) and the static thermal permeabilityk00 (unit m2, [33]). The characterization approach used in this work is an inverse method, that relies on an optimization procedure relating the intrinsic parameters (model inputs) to an observed quantity (i.e. the impedance).
Since impedance tube measurements are readily available in most labs, inverse methods based on the measurement of a surface impedance are now widely spread [34–38]. Here, the strategy of Zielinski [36] is followed. The method consists in measuring the surface impedance of a porous sam- ple in different configurations, where air gaps of different thicknesses are placed behind the porous material. The acoustic propagation inside the air gap being known, multiplying the measurements allows for a more precise identification. Using the JCAL equivalent fluid model, one can then relate the porous material intrinsic properties (JCAL model inputs) to the measured impedance. A dif- ferential evolution algorithm [39] is then used to minimize the difference between the model output and the measured impedance, which yield its minimum for parameter values that are considered to be the true ones. Identification results are given in Table 1, for materials M1 through M4. Material M1 consists of a pack of sphere, hence the low porosity and high tortuosity. Materials M2 and M3 are open cells melamine-like foams, of high porosity and low tortuosity, differing by the pore scale size, which results in material M2 being less resistive than material M3. Material M4 is a ceramic foam of lower porosity that possibly had a behavior too different from that of an equivalent fluid (double porosity, heterogeneities), since its identification was almost impossible. No coherent values were found for the thermal parameters Λ0 and k00 (Λ0, k00 → ∞, as in free space, thus suggesting that thermal effects inside the pores can be neglected). While the different materials show a wide
range of porosity and tortuosity, they seem to share a static viscous permeability that falls into a narrow range, compared to what can be achieved in porous media (k0 ∈[1e−11,1e−7]m2). This might be detrimental for the generalization of this work, but overall the materials in this range of static viscous permeabilities are typical of what can be encountered in practice in noise absorption applications.
Table 1: Material intrinsic properties, identified using the inverse method in Ref. [36].
Parameters φ α∞ k0 (10−9m2) Λ (µm) Λ0/Λ k00/k0 L (mm)
M1 0.47 2.0 1.81 195 1.6 1 24
M2 0.99 1.05 4.84 202 1.01 1.16 24
M3 0.99 1.0 1.55 109 2.3 1 39
M4 0.8 1.15 2.15 79 − – 46
Six different perforated plates are used in this work, whose properties are displayed in Table 2 (withφp the plate porosity,rp the perforation radius andethe plate thickness). Note that PP1’ is made by adding tape on PP1 in order to hide perforations to reduce the apparent porosity of the plate.
Table 2: Perforated plates properties.
Plates φp rp(mm) e(mm)
PP1 0.05 0.15 0.8
PP1’ 0.013 0.15 0.8
PP2 0.017 0.20 0.5
PP3 0.022 0.5 1.0
PP4 0.025 0.25 1.0
PP5 0.015 0.3 0.8
Using 6 perforated plates, 4 different porous media and a reference with an air cavity (of thickness ≈ 4 cm) results in 30 configurations. For each configuration, impedance measurements are carried out at different total SPL accounting for both the incident and reflected waves inside the impedance tube, ranging from 100 dB to 140 dB by increments of ≈ 5 dB. The impedance of such multilayer units is then measured as the SPL increases, and compared to each reference case with an air cavity. This endeavor is pursued in Section 4.
4 Results and analysis
Measurements consist of a set of impedance values (resistance and reactance) from 200 Hz to 6000 Hz, at different SPL. Due to the different natures of the porous samples, and due to their different thicknesses, directly comparing the evolution of the impedance at a given frequency is difficult, if relevant at all. The frequency axis is normalized by a single value of the resonance frequency for each configuration, as was done in Ref. [29]. The resonance occurs when the reac- tance becomes zero and is increasing. Since liners behave as resonators, and since engineers are mostly interested in the frequency range around the resonance, it was deemed the most appropriate quantity.
4.1 Evolution of the resistance with the SPL
In order to assess the nonlinearity degree of a configuration from the impedance curves, the following procedure is taken. At a given frequency (normalized by the resonance frequencyfr), the nonlinear contribution to the impedance ZNL is measured by subtracting the total measured impedanceZtot
by the linear impedance valueZL measured at the lowest SPL :
ZNL =Ztot−ZL. (4)
Since the reactance changes as the SPL increases, the resonance frequency also varies. In this work, the normalizing resonance frequency remains the one evaluated at the lowest SPL measure- ment of a given configuration.
4.1.1 Individual nonlinearity
Before tackling the issue of the SPL nonlinearity on acoustic units made of a perforated plate and a porous material, each element was tested separately in the impedance tube to evaluate its initial nonlinear behavior (6 perforated plates backed by an air cavity, and 4 porous materials). While a white noise is used in the impedance tube experiment, only the resonance frequency is used in the following analysis. Results are displayed in Fig. 2, where the delta of resistance ∆R=Rtot−RL is displayed. At a low SPL value, ∆R≈0 for all configurations, which allows for an easier comparison between the materials.
(a) Perforated plates (b) Porous media
Figure 2: Increase of the resistance as a function of the SPL
The plates of lowest porosities display the highest increase in resistance, while porous media in general have a very low nonlinearity relative to the SPL. However, the increase in resistance for porous media is still present, and will have to be taken into account in the remainder of the study, to decouple the nonlinearity associated to the perforated plate from the total nonlinearity of the acoustic unit.
4.1.2 Nonlinear ratio for the resistance
A new parameter ξR is introduced, representing the frequency dependent ratio between nonlinear regimes with and without a porous material inside the cavity. This nonlinear ratio is calculated by a least-mean square minimization, as
ξR= arg min
ξ
RNL,0−ξ 1 K
X
k∈K
RNL,k
2
, (5)
whereRNL,0 is the reference nonlinear resistance with an air cavity andK= [1,2,3,4] is the index array of the different porous media. It follows thatξRis representative of the average ratio between the reference plate nonlinearity, and the plate nonlinearity when backed by a porous-filled cavity.
Note that since the SPL values between each configuration do not exactly match, an interpo- lation is first conducted (thusRNL,0 is of sizeN = 9 different SPL values used for the interpolation between 100 and 140 dB). RNL,k is obtained by taking the measured nonlinear resistance Rtot,k
of the perforated plate with a porous-filled cavity, and by subtracting the nonlinear resistance RNL porous,k of the porous material alone (shown in Fig. 2). As a result, only the nonlinearity of the perforated plate is evaluated:
RNL,k =Rtot,k−RNL porous,k. (6)
At the exit of the perforation, the local flow velocity is important, which means that the nonlinearity of the porous material could be locally increased. However, due to the low nonlinearity level of the porous media, this effect is neglected in the following.
To better illustrate the meaning of ξR, a simple case is represented in Fig. 3. The perforated plate PP1 is considered, along with porous material M3 (k= 3 in Eq. 5). Only a single high SPL value (≈140 dB) is considered, so as to makeRNL,0 of sizeN = 1. Due to the linear nature of M3, one hasRNL,3≈Rtot,3, and Eq. 5 simplifies to
ξR≈RNL,3/RNL,0= 0.55 (7)
at the resonance frequency. While the reactance changes a little, its variation is much less than the resistance. As a result, it is hereafter not further studied. One notes that despite a linear resistance that is more important due to the addition of a resistive porous material inside the cavity, the total resistance at high SPL is lower than the reference value. It is a somewhat conterintuitive result that introducing a porous material inside an aircavity would reduce the total resistance of the material in certain conditions.
0.9 0.95 1 1.05 1.1
0.5 1 1.5 2
RNL,0 RNL,3
(a) Resistance
0.9 0.95 1 1.05 1.1
-1 -0.5 0 0.5
PP1+air 100dB PP1+air 140dB PP1+M3 100dB PP1+M3 140dB
(b) Reactance
Figure 3: Comparison of impedances at low and high SPL
The minimization in Eq. 5 is done at every normalized frequency of interest, extending from 0.4fr to 1.6frfor each perforated plate, considering all the porous media simultaneously to average the results. Frequencies outside these bounds extend beyond the impedance tube limitations for certain configurations and are thus discarded. Results obtained for the six different perforated plates are displayed in Fig. 4.
A nonlinearity ratio of 0.5 ±0.1 is obtained for the resistance. This tends to show that nonlinear effects related to the perforated plate, caused by vortex-shedding at the outflow side, are divided by a factor of ≈2 when a porous material is present. An explanation to this phenomenon can be found by an analogy with Assato’s work [19–21]. The velocity profile at the perforation discontinuity on the cavity side (sudden expansion) is likely flattened by the viscous effects within
the porous medium, which impedes the formation of a vortex-shedding mechanism (in Assato’s work, it is a recirculation past a backward-step configuration that is hindered). On the other side of the perforated plate, the vortex-shedding mechanism can occur without being hindered by the porous material, as it would with an air cavity.
0.4 0.6 0.8 1 1.2 1.4 1.6
0 0.2 0.4 0.6 0.8 1
PP1 PP1' PP2 PP3 PP4 PP5
Figure 4: Evolution of the nonlinearity resistance ratioξRas a function of the normalized frequency 4.1.3 On the presence of an airgap
An additional experiment is conducted to evaluate whether the nonlinearity decrease is due to the removal of nonlinear effects occurring in the vicinity of the perforations, in the cavity side.
Only considering the perforated plate PP1, an airgap of ≈2 mm is placed between the perforated plate and the porous materials. The previous analysis is repeated for this configuration, and the nonlinearity ratio is displayed in Fig. 5. The value ofξR is consistently closer to 1, indicating that nonlinear effects are reverting to their reference level, obtained with a perforated plate and an air cavity.
0.4 0.6 0.8 1 1.2 1.4 1.6
0 0.2 0.4 0.6 0.8 1
PP1+2mm airgap+foam PP1+ foam
Figure 5: Influence of an airgap on the nonlinearity resistance ratioξR
This confirms that the nonlinearity reduction happening in the presence of a porous material in the cavity is located in the vicinity of the perforations. This shows the validity of the third
scenario in the Introduction of this work: porous materials pertain to blocking the vortex-shedding mechanism in the cavity, thus reducing the global nonlinearity of a multilayer liner.
5 Conclusion
Liners made of a perforated plate and a cavity are widely spread acoustic units. Their impedance is known to change as a function of the sound pressure level they are exposed to. An experimental study is conducted with different perforated plates and porous materials placed inside the liner cav- ity. The nonlinearity of each configuration is evaluated by measuring the increase of the resistance, as a function of the sound pressure level.
Liners having a porous-filled cavity are shown to consistently display a nonlinearity that is halved, compared to a reference case with an air cavity, at frequencies ranging from 0.4 to 1.6 times the resonance frequency of the acoustic unit. An additional experiment is performed where a small air gap is present between the perforated plate and the porous material, showing that the nonlinearity level tends to revert back to its reference level in the presence of an air gap. This further shows that the nonlinearity reduction happens in the vicinity of the perforations. The rationale for such a decrease is that the velocity profile at the perforation discontinuity is flattened by the viscous dissipation within the porous material, thus hindering the shedding of a vortex in the cavity, that are responsible for approximately half the nonlinear increase in resistance.
Different remarks can be made regarding these findings:
• It is sufficient, when considering a porous material behind a perforated plate, to update a nonlinear model for the perforated plate by multiplying the nonlinear contribution by a factor of ξR= 0.5. For the case of Crandall’s model [24], this gives
Rtot =RL+
ξR
z}|{1 2 ·
Crandall’s term
z }| { 4
3πρf1−φ2P
φPCD2 |v0|, (8)
where CD is a discharge factor coefficient, depending on the perforation shape, ρf is the ambient fluid density,φP is the plate perforation rate andv0is the particle velocity associated to a given SPL, outside the perforation.
• The nonlinearity reduction is localized close to the perforations. One could then obtain a decrease in the nonlinear resistance by adding only a thin layer of porous material in contact with the perforated plate. This could prove useful in certain applications where the cavity cannot be entirely filled with porous media. In addition, this would limit the additional (linear) resistance caused by the presence of a porous material within the cavity, which can reduce the effectiveness of a liner at certain frequencies.
Acknowledgment
Some early results of this study were obtained during the author’s PhD thesis, which was partly founded by the French Occitanie Region. The author would like to thank D. Sebbane for her help in setting up the experiment, as well as E. Piot and F. Simon for their guidance during the completion of the author’s PhD.
References
[1] L. Sivian, Acoustic impedance of small orifices, The Journal of the Acoustical Society of America 7 (2) (1935) 94–101. doi:10.1121/1.1915795.
[2] U. Ing˚ard, S. Labate, Acoustic circulation effects and the nonlinear impedance of orifices, The Journal of the Acoustical Society of America 22 (2) (1950) 211–218. doi:10.1121/1.1906591.
[3] U. Ingard, H. Ising, Acoustic nonlinearity of an orifice, The journal of the Acoustical Society of America 42 (1) (1967) 6–17. doi:10.1121/1.1910576.
[4] T. H. Melling, The acoustic impendance of perforates at medium and high sound pressure levels, J.
Sound Vib. 29 (1) (1973) 1–65. doi:10.1016/S0022-460X(73)80125-7.
[5] X. Jing, X. Sun, Sound-excited flow and acoustic nonlinearity at an orifice, Physics of Fluids 14 (1) (2002) 268–276. doi:10.1063/1.1423934.
[6] C. K. Tam, H. Ju, M. G. Jones, W. R. Watson, T. L. Parrott, A computational and experimental study of slit resonators, Journal of Sound and Vibration 284 (3-5) (2005) 947–984. doi:10.1016/j.
jsv.2004.07.013.
[7] Q. Zhang, D. J. Bodony, Numerical investigation and modelling of acoustically excited flow through a circular orifice backed by a hexagonal cavity, Journal of Fluid Mechanics 693 (2012) 367–401. doi:
10.1017/jfm.2011.537.
[8] J. Allard, N. Atalla, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials 2e, John Wiley & Sons, New York, 2009. doi:10.1002/9780470747339.
[9] W. Davern, Perforated facings backed with porous materials as sound absorbers-an experimental study, Applied Acoustics 10 (2) (1977) 85–112. doi:10.1016/0003-682X(77)90019-6.
[10] R. Kirby, A. Cummings, The impedance of perforated plates subjected to grazing gas flow and backed by porous media, J. Sound Vib. 217 (4) (1998) 619–636. doi:10.1006/jsvi.1998.1811.
[11] W.-H. Chen, F.-C. Lee, D.-M. Chiang, On the acoustic absorption of porous materials with different surface shapes and perforated plates, Journal of sound and vibration 237 (2) (2000) 337–355. doi:
10.1006/jsvi.2000.3029.
[12] N. Atalla, F. Sgard, Modeling of perforated plates and screens using rigid frame porous models, Journal of sound and vibration 303 (1-2) (2007) 195–208. doi:10.1016/j.jsv.2007.01.012.
[13] R. Tayong, T. Dupont, P. Leclaire, On the variations of acoustic absorption peak with particle velocity in micro-perforated panels at high level of excitation, The Journal of the Acoustical Society of America 127 (5) (2010) 2875–2882. doi:10.1121/1.3372714.
[14] F. Peng, Sound absorption of a porous material with a perforated facing at high sound pressure levels, Journal of Sound and Vibration 425 (2018) 1–20. doi:10.1016/j.jsv.2018.03.028.
[15] Z. Laly, N. Atalla, S.-A. Meslioui, Acoustical modeling of micro-perforated panel at high sound pressure levels using equivalent fluid approach, Journal of Sound and Vibrationdoi:10.1016/j.jsv.2017.09.
011.
[16] H. L. Kuntz, D. T. Blackstock, Attenuation of intense sinusoidal waves in air-saturated, bulk porous materials, The journal of the Acoustical Society of America 81 (6) (1987) 1723–1731. doi:10.1121/1.
394787.
[17] D. K. Wilson, J. D. Mcintosh, R. F. Lambert, Forchheimer-type nonlinearities for high-intensity prop- agation of pure tones in air-saturated porous media, The Journal of the Acoustical Society of America 84 (1) (1988) 350–359. doi:10.1121/1.396937.
[18] J. D. McIntosh, R. F. Lambert, Nonlinear wave propagation through rigid porous materials. i: Nonlinear parametrization and numerical solutions, The Journal of the Acoustical Society of America 88 (4) (1990) 1939–1949. doi:10.1121/1.400217.
[19] M. Assato, M. de Lemos, Numerical treatment and applications of a nonlinear eddy viscosity model for high and low reynolds formulations, Proceedings of the 2nd ETT-Brazilian School on Transition and Turbulence (2000) 11–15.
[20] M. Assato, M. de Lemos, Numerical simulation of turbulent flow through axisymmetric stenosis using linear and non-linear eddy-viscosity models, in: Proceedings of the 16th Brazilian Congress of Mechan- ical Engineering, Uberlˆandia/MG, Brazil, 2001.
[21] M. Assato, M. H. Pedras, M. J. De Lemos, Numerical solution of turbulent channel flow past a backward- facing step with a porous insert using linear and nonlinear k-εmodels, Journal of Porous Media 8 (1).
doi:10.1615/JPorMedia.v8.i1.20.
[22] M. J. De Lemos, Turbulence in porous media: modeling and applications, 2nd Edition, Elsevier, 2012.
doi:10.1016/C2011-0-06981-8.
[23] R. Roncen, Mod´elisation et identification par inf´erence bay´esienne de mat´eriaux poreux acoustiques en a´eronautique (modelling and bayesian inference identification of acoustic porous materials in aeronau- tics), Ph.D. thesis, Universit´e de Toulouse (2018).
URLhttps://hal.archives-ouvertes.fr/tel-02004624
[24] I. B. Crandall, Theory of vibrating systems and sound, D. Van Nostrand Company, 1954.
[25] E. Rice, A model for the pressure excitation spectrum and acoustic impedance of sound absorbers in the presence of grazing flow, in: Aeroacoustics Conference, 1973, p. 995. doi:10.2514/6.1973-995.
[26] D.-Y. Maa, Potential of microperforated panel absorber, the Journal of the Acoustical Society of America 104 (5) (1998) 2861–2866. doi:10.1121/1.423870.
[27] T. Elnady, H. Boden, On semi-empirical liner impedance modeling with grazing flow, in: 9th AIAA/CEAS Aeroacoustics Conference and Exhibit, 2003, p. 3304.
[28] C. K. W. Tam, K. A. Kurbatskii, Microfluid dynamics and acoustics of resonant liners, AIAA journal 38 (8) (2000) 1331–1339. doi:10.2514/2.1132.
[29] M. Meissner, The influence of acoustic nonlinearity on absorption properties of helmholtz resonators.
part ii. experiment, Archives of Acoustics 25 (2) (2000) 179–190.
[30] J. Y. Chung, D. A. Blaser, Transfer function method of measuring in-duct acoustic properties. I. Theory, J. Acoust. Soc. Am. 68 (3) (1980) 907–913. doi:10.1121/1.384778.
[31] D. L. Johnson, J. Koplik, R. Dashen, Theory of dynamic permeability and tortuosity in fluid-saturated porous media, J. Fluid Mech. 176 (1987) 379–402. doi:10.1017/S0022112087000727.
[32] Y. Champoux, J.-F. Allard, Dynamic tortuosity and bulk modulus in air-saturated porous media, J.
Appl. Acoust. 70 (4) (1991) 1975–1979. doi:10.1063/1.349482.
[33] D. Lafarge, P. Lemarinier, J. F. Allard, V. Tarnow, Dynamic compressibility of air in porous structures at audible frequencies, J. Acoust. Soc. Am. 102 (4) (1997) 1995–2006. doi:10.1121/1.419690.
[34] Y. Atalla, R. Panneton, Inverse acoustical characterization of open cell porous media using impedance tube measurements, Canadian Acoustics 33 (1) (2005) 11–24.
[35] P. Cobo, F. Sim´on, A comparison of impedance models for the inverse estimation of the non-acoustical parameters of granular absorbers, J. Appl. Acoust. 104 (2016) 119–126. doi:10.1016/j.apacoust.
2015.11.006.
[36] T. G. Zieli´nski, Normalized inverse characterization of sound absorbing rigid porous media, J. Acoust.
Soc. Am. 137 (6) (2015) 3232–3243. doi:10.1121/1.4919806.
[37] J.-D. Chazot, E. Zhang, J. Antoni, Acoustical and mechanical characterization of poroelastic materials using a bayesian approach, J. Acoust. Soc. Am. 131 (6) (2012) 4584–4595. doi:10.1121/1.3699236.
[38] M. Niskanen, J.-P. Groby, A. Duclos, O. Dazel, J. Le Roux, N. Poulain, T. Huttunen, T. L¨ahivaara, Deterministic and statistical characterization of rigid frame porous materials from impedance tube measurements, J. Acoust. Soc. Am. 142 (4) (2017) 2407–2418. doi:10.1121/1.5008742.
[39] R. Storn, K. Price, Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim. 11 (4) (1997) 341–359. doi:10.1023/A:1008202821328.
