© The Author(s), Published by EDP Sciences, 2024

1 Introduction

In this work, we investigate how the sound absorption coefficient of perforated foams is affected by increasing sound pressure levels (SPL). While common porous materials such as glass wool do not display strong non-linear features, perforated structures are known to exhibit non-linear regimes [1]. This effect limits the success of perforated plates, screens, and foams in the case of sound levels higher than 100 dB SPL, since a fluctuation of the ambient noise level can influence the efficiency of the absorbers.

Classical porous materials are usually modeled using an equivalent fluid approach, which itself is derived from a multiple-scale homogenization approach [2]. More particularly, the Johnson-Champoux-Allard (JCA) model [3–5] provides the equivalent mass density ρ and bulk modulus B of such equivalent fluids for rigid-frame porous materials. Both these quantities are complex-valued and frequency-dependent. The JCA model encapsulates the complete dynamics of wave propagation and dissipation into 5 parameters, which emerge from the local fields of temperature and particle velocity within the porous medium. The JCA parameters are: the open porosity ϕ and thermal characteristic length Λ′, which are both purely geometric quantities; the high-frequency tortuosity τ^∞ and characteristic viscous length Λ which both result from the non-viscous fluid flow in the high-frequency limit; and finally the visco-static permeability, K⁰, which describes the particle velocity fields in the low-frequency limit ω→0. A complete overview of the relation between micro- and macroscopic flow, even for complex geometries, and the numerical determination of the transport parameters is given in [6]. It illustrates also how the effective mass density depends on visco-inertial effects, represented by K⁰, τ^∞, and Λ, while the bulk modulus is related to the thermo-acoustic effects through Λ′.

The JCA parameters can be retrieved experimentally (an overview is given in [4], paragraphs 5.2.2 and 5.3.5), using standardized porosimetry or flow resistivity methods, using ultrasound waves to measure the refraction index and therefrom derive the tortuosity, or using challenging two-gas measurements for the characteristic lengths. To assist the design of efficient absorbers, the transport parameters can also be predicted by flow simulations in microscopic representative volume elements (RVE) of the porous material under consideration [6, 7]. Perforated surfaces take a special place in the study of sound absorbers, since they can achieve high, but narrow-band, acoustic absorption while keeping the entire system relatively thin compared to materials with open pores. Although predictive models for the acoustic absorption by perforated plates are well studied [8–11], the case of perforated foams is less commonly investigated [12, 13]. The presence of small perforations in low-density foams can lead to low-frequency sound absorption, for a material thickness smaller than 1/10 of an acoustic wavelength [14]. It is well known that the sound absorption potential of perforated systems depends on the SPL because of non-linear changes in the thermal and viscous losses [9, 15–17]. Several publications describe how to include SPL-dependent effects into models for perforated plates and porous media, both semi-empirical [18] and analytical [19, 20].

For the case of perforated foams, the dynamic flow under harmonic loading is affected by edge effects, such as flow separation close to the perforation, but also by the finite extent of the pores which might further inhibit the fluid flow. The pressure-dependent volume flow was initially studied in geophysical settings, showing that the linear dependency between pressure and flow rate, as described by Darcy [21], breaks down when the flow speed reaches a certain level [22, 23]. Since the JCA model requires this (asymptotic) visco-static flow resistivity, the inlet pressure can be taken into account as the driving parameter for the foam’s non-linear absorption behavior. We will show that this simulated value is sufficient to explain the SPL-dependent maximum absorption coefficient. The work by Laly et al. [19] predicts a change of the frequency-dependent tortuosity as well, but this effect is not captured by the JCA model. Including the static tortuosity, as proposed by Pride, could yield similar results.

This work aims to assess the variation of the JCA parameters with respect to SPL, which in practice only affects the static flow resistivity (or viscous permeability). The other four parameters are not affected by a change in driving force. The increase of air-flow resistivity σ as a function of the applied pressure gradient can be directly measured according to ISO standard 9053-1. Furthermore, impedance tube measurements are performed with increasing SPL in order to obtain the absorption curves of the perforated foam. An analysis on linear and non-linear behavior for two different base foams is carried out, in order to characterize the model’s dependence on SPL.

This paper is organized as follows: first the experimental methods for flow-resistivity and acoustic impedance are described in Section 2. The effect of an increasing pressure gradient on the simulation of the static flow resistivity is shown in Section 3. Then the non-linear effects on sound absorption are presented in Section 4. Finally, the discussion and outlook is given in Section 5.

2 Materials & methods

2.1 Perforated mineral foams

The base material is a rigid gypsum foam produced by the company de Cavis. The gypsum foams are prepared from calcium sulfate hemihydrate, mixing water, a foam formation powder (FFP), a blowing agent, and a catalyst to trigger the blowing reaction. The FFP [24] plays a key role in the process. It comprises partially hydrophobized particles [25] that form a particle shell [26] at the air-water interface during expansion and yield wet foams of unprecedented stability. Particle-stabilization is a prerequisite to create gradient-free, homogeneous foams with tailored pore size. The porosity is empirically determined by the blowing agent concentration, while the pore size can be adjusted by varying the amount of FFP and blowing agent. This leads to highly porous (ϕ_tot > 90%), low-density (<250 kg/m³), closed-cell foams. By varying the foaming and stabilizing agent concentrations, the total porosity, pore size, and wall thickness can be tuned to desired values.

For the work at hand, foams with a median pore diameter of 2 mm (material F2) and 3 mm (material F3) and wall thickness of respectively 0.2 mm and 0.15 mm were investigated. The mass densities of the two materials are 182 kg/m³ and 124 kg/m³. The total porosity of the materials is calculated from the mass density of the resulting foams, and is 92.1% for F2 and 94.6% for F3. A sample of material F3 is shown in Figure 1(a). However, the pores in the hardened foam are predominantly closed, apart from a small amount of micro-cracks and the inherent micro-porosity of gypsum. The flow resistivity of the produced foams is in the order of 10⁶ Pa.s.m⁻², which results in a negligible acoustic absorption.

Figure 1 (a) Sample of perforated gypsum foam F3, cast in a gypsum ring for better measurement repeatability. (b) Optical microscope image showing the perforations relative to the pore size. (c) Experimental setup for flow resistivity measurements. (d) Experimental setup for the absorption measurement using the two-microphone method.

A fraction of the pores is interconnected by a square array of perforations (visible in Fig. 1(b), an optical microscope image made by a Dino-Lite Edge 3.0 microscope) with a diameter d = 0.5 mm–smaller than the pore size–and a distance L = 5 mm between perforations, which is larger than the pore size. We have shown before that this foam treatment leads to low-frequency absorption due to a high value of the dynamic tortuosity [14, 27].

2.2 Flow resistivity measurement

According to standard ISO 9053-1, we measure the pressure drop across the sample for a series of chosen fluid flow values. A digital mass flow meter (Bronkhorst EL_Flow) controls the air inlet, and the pressure difference between inlet and outlet is measured by a Rotronic differential pressure sensor. The thickness of the sample is 25 mm, and the diameter 100 mm. To avoid leakage around the sample when placed in the sample holder, the foam is cast in a gypsum ring, which is sealed with vaseline and pressed between the two parts of the holder as shown in Figure 1(c). This approach ensures repeatable measurements.

According to the standard, the pressure gradient should be measured for increasing flow rates, measurements that should be extrapolated to sufficiently low pressure drops to ensure the linear Darcy resistance. From this data, the visco-static permeability K⁰ and air-flow resistivity σ are directly derived using Darcy’s law [21]:

$$\mathbf{v}=-\frac{K^0}{\eta }\nabla p,$$(1)

where the volume force due to gravity is neglected, v is the particle velocity, p the acoustic pressure, and η the dynamic viscosity of air. The static air-flow resistivity is then given by σ = η/K⁰.

From the measurements at different pressure-velocity couples (and enforcing the zero-pressure/zero-velocity point according to the standard), the linear visco-static permeability of the medium is obtained for a flow speed of $\left|{\mathbf{v}}_{\mathrm{ref}}\right|=0.5 \mathrm{mm}.{\mathrm{s}}^{-1}$. However, the same setup allows us to measure the pressure-dependent permeability and flow resistivity of the sample. With the used setup, flows up to 5 l/min can be generated, which yields a pressure drop of 31.0 Pa for sample F2 and 20.2 Pa through sample F3. At the lower limit, a pressure drop as low as 0.5 Pa can be reliably measured.

2.3 Impedance tube measurement

The perforated foam is tested in an impedance tube under plane wave excitation, as described in the standard ISO 10534-2. The test rig is the impedance tube Type-4206 provided by Brüel and Kjaer¹ with a diameter of 100 mm, and a frequency measurement range of 50–1600 Hz. The tube is mounted vertically and the gypsum ring of the sample is placed on its edge, backed by a heavy plate as shown in Figure 1(d). We can thus assume that the foam’s surface is entirely exposed to the impinging plane sound wave, and is rigidly backed by the metal mass. Sealing the gypsum ring with vaseline avoids leakage around the sample, a typical problem when hard samples are placed inside the tube [28]. The acquisition of pressure signals is made using two 1/4 inch microphones type B&K 4958 and a data acquisition system NI USB-4431. The TubeCell software² is used for the measurement and absorption calculation, and allows to change the equivalent SPL inside the tube as a measurement parameter. A photograph of the impedance tube setup is shown in Figure 1(d).

3 Simulation of the flow resistivity

In total, five non-acoustical parameters have to be derived to use the JCA model for the prediction of the equivalent fluid properties of a porous material. We use microscopic flow simulations in a well chosen RVE according to [14]. Since the foam starts from a liquid phase, we choose the Kelvin cell as the pore geometry, to allow a periodic arrangement. The RVE consists of exactly one cell in the thickness, and has an in-plane size equal to the perforation distance L. At the center of the RVE, the perforation connects several parts of Kelvin cells, to allow for a fluid flow. It is shown in Figure 2(b). The results in [14] show that most of the JCA parameters are sufficiently well predicted to anticipate the frequency of the maximum absorption peak.

Figure 2 Geometry of two different RVEs for the flow simulation with 5 unit cells (a) and 1 unit cell (b) through the thickness. The perforations highlighted in green shows the connection between an entire Kelvin cell and two parts of neighboring cells. The flow resistivity values (c) show a convergence from 5 unit cells, for two different materials and two different inlet pressures.

In the assumption of a rigid matrix, the open porosity and thermal characteristic length, which are purely dependent on the geometry, do not change for increasing sound pressure levels. The high-frequency parameters, tortuosity and viscous characteristic length, are calculated based on an equivalent electrical model, and they are therefore also independent of the sound pressure level. To calculate the remaining parameter, the non-linear flow resistivity, the RVE must be altered. In the linear case, the flow resistivity is independent of the number of cells through the thickness of the model. If the volume flow becomes pressure dependent however, the retrieved flow resistivity, which depends on multiple narrow perforations in series, is altered the thickness of the RVE. We performed calculations with an increasing number of periods through the thickness, up to 5 in total (the situation shown in Fig. 2(a)). For more than 5 unit cells, the flow resistivity does not change more than 5% so this amount is deemed to be a good compromise between model size and accuracy. The results of this convergence study are shown in Figure 2(c).

The flow simulation is performed in Ansys Fluent 2022R2. A pressure inlet, varying from 0.01 Pa to 100 Pa, is defined on the top surface, and the laminar flow velocity is calculated using no-slip boundary conditions v = 0 on all walls. The volume mesh has an element size of 10 μm, with a refinement to 5 μm on the perforation edges. A 3-element inflation layer ensures a stable converging simulation for all pressure levels. This leads to a total of 2.05 × 10⁶, elements, and 8.26 × 10⁶ model nodes. The model converges when the relative error for the three velocity components changes less than 1 × 10⁻⁶, which requires typically between 150 and 220 iterations. The total calculation time amounts to up to 20 min per simulated case.

The flow resistivity is then calculated from Darcy’s law, taking the average total flow speed in the RVE as calculated by the software. Even for the highest inlet pressure, introducing a turbulent flow model (K-ε with standard Fluent settings) does not lead to significantly different results from the faster laminar-flow simulations, with a difference less than 1% which is supposed to be within the numerical error. Therefore, laminar models are deemed to be sufficient, which avoids fine tuning of several turbulent-flow parameters.

4 Experimental and numerical results

4.1 Flow resistivity

Due to sensor limitations, the flow resistivity can be measured for volume flows between 0.1 and 5 l/min. This leads to a measured pressure difference ranging from 0.25 Pa to 30 Pa, depending on the measured material. The measured flow resistivities are shown in Figure 3. It is clear that the flow resistivity of both materials remains constant up to a pressure difference of around 2.0 Pa, which is equivalent to a sound pressure level of 100 dB SPL. Above this value, the resistivity increases rapidly.

Figure 3 Modeled (red dashed line) and experimental (blue line) flow resistivity values for material F2 (a) and F3 (b). The RVE model captures the low-pressure value but overestimates the nonlinear increase of σ.

In the model, the inlet pressure is varied from 0.01 Pa to 100 Pa and the resulting volume flow is calculated. Darcy’s equation then yields the resistivity value for this geometry. Although the low-pressure limit is very similar to the measured results, its nonlinear increase as shown in Figure 3 is more pronounced than in the measurements. The resistivity increases already at 0.3 Pa (85 dB SPL), and the increase is much faster. The reason is most likely that microcracks and micropores in the cell walls lower the flow resistance, and increase the linear pressure regime.

The model also gives qualitative insight in the reason for the increase in resistivity as illustrated in Figure 4. It can be seen that at low input pressures, the flow follows a smooth path within the Kelvin cell. If the pressure increases, the faster flow hits the opposite wall, which leads to flow inhibitions and a higher resistivity.

Figure 4 Model of the laminar flow for an inlet pressure (from left to right) of 0.001 Pa, 1 Pa, and 100 Pa. At higher flow velocities, the flow is distorted by the cell wall.

4.2 Normal incidence sound absorption

According to the JCA model, only the mass density of the equivalent properties is affected by a change of the flow resistivity, the bulk modulus remains the same. In [14] we showed that the JCA parameters can be calculated using microscopic flow models. For material F2, these values are ϕ = 0.45, σ = 70, 170, and 300 kPa.s.m⁻², τ^∞ = 10.9, Λ = 50 μm, and Λ′ = 530 μm. For material F3, they are ϕ = 0.45, σ = 45, 112, and 200 kPa.s.m⁻², τ^∞ = 6.8, Λ = 45 μm, and Λ′ = 800 μm. These values are used to predict the equivalent fluid properties and sound absorption curves for increasing flow resistivity values. Figure 5 shows that mainly the imaginary part of the equivalent fluid mass density is affected by the pressure-dependent change in air-flow resistivity. However, despite a fairly small variation of this value in the 500–1000 Hz range, the sound absorption drops considerably.

Figure 5 Equivalent fluid properties of material F3 according to the JCA model for increasing flow resistivity values. Solid lines show the imaginary part of the quantities, dashed lines the real part. The experimental values for σ at the desired dynamic pressure are used.

The sound absorption is measured in the impedance tube with a rigid backing at increasing driving levels of the loudspeaker. The TubeCell software shows the SPL during the measurement at the two loudspeaker positions, and these levels are used for the analysis of the absorption curves. Sound levels between 80 and 130 dB SPL can be achieved. The experimental values are shown in Figure 6, and follow the trends of the models. A sub-wavelength absorption peak (the wavelength/thickness ratio is equal to 18 at the absorption peak located at 750 Hz for material F3 and equal to 27 at the absorption peak located at 500 Hz for material F2) is clearly reduced at higher SPLs.

Figure 6 Modeled (dashed lines) and measured (solid lines) sound absorption curves for increasing SPL for material F2 (a) and F3 (b). In the modeled curves, only the value for flow resistivity is changed in the JCA model. The squares show the maximum of the modeled absorption curves.

The measured and predicted sound absorption spectra match reasonably well for material F3 as seen in Figure 6(a). The sound absorption is slightly overestimated at higher frequencies, and underestimated at lower frequencies, but the absorption peak is well captured. For material F2 (Fig. 6(b)), the JCA model combined with the microscopic flow predictions only correctly captures the peak absorption. The measured absorption curve is higher than predicted. There are several reasons for this discrepancy. First, the wall thickness varies more in material F2 than F3, which has a major effect on the flow calculations. It is known that a wider statistical distribution of pore and wall properties can improve the sound absorption. Second, the large aspect ratio between pore and window size is known to limit the validity of the JCA model. However, since this model is still able to predict the peak absorption, we decided not to investigate more complex homogenization models.

5 Conclusions

Perforated mineral foams have shown their ability to provide strong acoustic absorption in sub-wavelength regimes, making these structures viable for thin low-frequency absorbers. Porous materials are typically well described by multiple-scale homogenization theory and by the JCA theory. This allows to depict the mineral foam as an equivalent fluid having complex and frequency-dependent properties. However, the response of micro-perforated structures is known to display non-linear behavior at high SPLs. In this work we investigated the non-linearity in perforated porous materials and establish its connection to changes in effective properties of the equivalent fluid, and more specifically the change in flow resistivity, to explain changes in the acoustic absorption. Models show how the flow through perforated Kelvin cells is affected by the presence of pore walls, leading to a nonlinear pressure-flow velocity relationship, and thus an increase in flow resistivity. The models show a low-pressure asymptotic behavior of the flow resistivity, and a steady increase with inlet pressure, which is in line with nonlinear Forchheimer flow through porous media. Measurements show good agreement in the linear regime, but at higher inlet pressures the flow resistivity is lower than predicted. A probable reason for this is the presence of cracks and micropores in the cell walls, which lowers the air flow resistance compared to closed walls. The JCA model is able to predict the frequency and amplitude of the peak absorption. However, the model is not accurate enough to capture the frequency-dependent absorption curve at low and high frequencies. Once again, the additional sound absorption by the microporous walls cannot be adequately included. A possibility to achieve better agreement between models and experiments is the use of more advanced models such as the Pride correction at low frequencies, or to integrate the multiscale porosity. The practical implications coming out of this work are mainly found in the tunability of the absorption curve for certain high-SPL applications. In theory, the flow resistivity can be lowered by increasing the perforation diameter or the amount of perforations per unit area, so that its value can be optimized for a chosen SPL. However, a change in the perforation pattern also affects the other JCA parameters and finally the absorption curve, so that the optimization is not trivial.

Acknowledgments

The authors gratefully acknowledge U. Gonzenbach, P. Sabet, and P. Sturzenegger from de Cavis AG. This work is jointly funded by Innosuisse project 56633.1 and de Cavis AG.

Conflicts of interest

The authors declare no conflicts of interest.

Data availability statement

The data are available from the corresponding author on request.

References

All Figures

	Figure 1 (a) Sample of perforated gypsum foam F3, cast in a gypsum ring for better measurement repeatability. (b) Optical microscope image showing the perforations relative to the pore size. (c) Experimental setup for flow resistivity measurements. (d) Experimental setup for the absorption measurement using the two-microphone method.
In the text

	Figure 2 Geometry of two different RVEs for the flow simulation with 5 unit cells (a) and 1 unit cell (b) through the thickness. The perforations highlighted in green shows the connection between an entire Kelvin cell and two parts of neighboring cells. The flow resistivity values (c) show a convergence from 5 unit cells, for two different materials and two different inlet pressures.
In the text

	Figure 3 Modeled (red dashed line) and experimental (blue line) flow resistivity values for material F2 (a) and F3 (b). The RVE model captures the low-pressure value but overestimates the nonlinear increase of σ.
In the text

	Figure 4 Model of the laminar flow for an inlet pressure (from left to right) of 0.001 Pa, 1 Pa, and 100 Pa. At higher flow velocities, the flow is distorted by the cell wall.
In the text

	Figure 5 Equivalent fluid properties of material F3 according to the JCA model for increasing flow resistivity values. Solid lines show the imaginary part of the quantities, dashed lines the real part. The experimental values for σ at the desired dynamic pressure are used.
In the text

	Figure 6 Modeled (dashed lines) and measured (solid lines) sound absorption curves for increasing SPL for material F2 (a) and F3 (b). In the modeled curves, only the value for flow resistivity is changed in the JCA model. The squares show the maximum of the modeled absorption curves.
In the text