Air-saturated Porous Materials Research Articles

Characterizing high-frequency wave interaction in rigid porous bilayer materials: Insights from acoustic parameter sensitivity analysis

This study aims to investigate the sensitivity of high-frequency acoustic parameters in rigid air-saturated porous bilayer materials using the equivalent fluid theory. The interaction between the fluid and solid phases of the porous medium incorporates visco-inertial and thermal exchange, characterized by two functions: the dynamic tortuosity α(ω) proposed by Johnson etal. and the dynamic compressibility β(ω) proposed by Allard, refined by Sadouki for the low-frequency domain of ultrasound. Various parameters, including porosity, tortuosity, viscous characteristic length, thermal characteristic length, as well as viscous and thermal shape factors, are examined in this study. A 10% variation in these parameters is considered to assess their impact on the amplitudes of transmitted waves. The findings of this study contribute to a deeper understanding of high-frequency wave behavior in porous bilayer materials. The investigation sheds light on the sensitivity of the parameters and provides valuable insights for the design and optimization of such materials in engineering, technology, and applied sciences.

Estimating the airflow resistivity of porous materials in an impedance tube using an electroacoustic technique

Airflow resistivity is an essential parameter for characterizing air-saturated porous sound-absorbing materials theoretically and selecting sound-absorbing materials in practice. Although standardized methods can determine this non-acoustic parameter in the laboratory, many indirect alternative methods have been proposed to measure it. One of them is the technique presented in the 1980s by Ingard and Dear using a standing wave tube, a loudspeaker, and two microphones. This paper suggests an electroacoustic procedure based on a modification of the Ingard and Dear setup. Equations are derived through the transfer matrix method. After a simple calibration, the airflow resistivity of a material sample is indirectly estimated from the total electric impedance measured at the loudspeaker input connection terminals. Thus, implementing the proposed method is straightforward and inexpensive, since microphones and complex instrumentation are unnecessary. The method is tested by comparing measured values of the airflow resistivity of different material samples with those obtained through the Ingar and Dear approach and the ISO standardized method. Reasonably good agreement is observed, confirming the validity of the electroacoustic method.

Experimental characterization of air-saturated porous material via low-frequencies ultrasonic transmitted waves

An improved acoustic method is proposed for measuring the tortuosity and the viscous and thermal characteristic lengths of air-saturated porous materials via low-frequency ultrasonic transmitted waves (70 kHz–110 kHz). The equivalent fluid model is considered. The interaction between the saturated fluid and the structure is taken into account in two frequency response factors: the dynamic tortuosity of the medium and the dynamic compressibility of the air which are described by their high-frequency expansion in powers of the viscous and thermal skin depth, either limiting the expansion to the first two leading terms or the first three leading terms with the introduction of two new viscous and thermal shape factors. These two new factors play an important role in stabilizing and improving the inverted parameters of the tortuosity and the viscous and thermal characteristic lengths in the frequency band considered. The inverse problem is solved numerically in the frequency domain by minimizing the spectrum of simulated and experimental transmitted signals. Optimized parameters of tortuosity, viscous and thermal characteristic lengths, and viscous and thermal shape factors are obtained simultaneously. The tests are carried out with three samples of plastic foam. The results obtained by the two procedures are discussed and compared to those given in the literature.

Estimation of all six parameters of Johnson-Champoux-Allard-Lafarge model for acoustical porous materials from impedance tube measurements.

The open porosity of air-saturated acoustical porous materials is estimated from the low-frequency or high-frequency asymptotes of the real part of the dynamic bulk modulus. Combining this technique with the estimation of the static air-flow resistivity from the low-frequency asymptote of the imaginary part of the dynamic mass density and the analytical inversions of the remaining parameters from the dynamic mass density and bulk modulus [methods introduced by Panneton and Olny, J. Acoust. Soc. Am. 119(4), 2027-2040 (2006) and Olny and Panneton 123(2), 814-824 (2008)], this work estimates all six parameters of a Johnson-Champoux-Allard-Lafarge model from impedance tube measurements. A classical two-microphone impedance tube, as well as three- or four-microphone tubes, can be used for these measurements and estimations. Examples of applications and limits of the method are presented and a tool to estimate the open porosity and the static air-flow resistivity is made available online.

An electroacoustic method for measuring airflow resistivity of porous sound-absorbing materials

In this paper, a method for measuring the airflow resistivity of air-saturated porous sound-absorbing materials is presented. The method is based on a modification of the previous device developed by Dragonetti et al. The approach used in the present work involves a cavity and a Helmholtz resonator that are coupled through a loudspeaker so that the complete system behaves as a fourth-order symmetrical band-pass loudspeaker system. After a straightforward calibration, the airflow resistivity of a material sample is indirectly estimated from the direct measurement of the total electric impedance at the loudspeaker connection terminals. In this way, the use of microphones is not necessary, which makes its implementation very simple and inexpensive. Experimental results obtained with the present method agree well with those obtained through a standardized method as long as the values of the material’s airflow resistance are not too high.

Spatial Laplace transform for complex wavenumber recovery and its application to the analysis of attenuation in acoustic systems

We present a method for the recovery of complex wavenumber information via spatial Laplace transforms of spatiotemporal wave propagation measurements. The method aids in the analysis of acoustic attenuation phenomena and is applied in three different scenarios: (i) Lamb-like modes in air-saturated porous materials in the low kHz regime, where the method enables the recovery of viscoelastic parameters; (ii) Lamb modes in a Duralumin plate in the MHz regime, where the method demonstrates the effect of leakage on the splitting of the forward S1 and backward S2 modes around the Zero-Group Velocity point; and (iii) surface acoustic waves in a two-dimensional microscale granular crystal adhered to a substrate near 100 MHz, where the method reveals the complex wavenumbers for an out-of-plane translational and two in-plane translational-rotational resonances. This method provides physical insight into each system and serves as a unique tool for analyzing spatiotemporal measurements of propagating waves.

Experimental determination of the viscous flow permeability of porous materials by measuring reflected low frequency acoustic waves

An acoustic reflectivity method is proposed for measuring the permeability or flow resistivity of air-saturated porous materials. In this method, a simplified expression of the reflection coefficient is derived in the Darcy's regime (low frequency range), which does not depend on frequency and porosity. Numerical simulations show that the reflection coefficient of a porous material can be approximated by its simplified expression obtained from its Taylor development to the first order. This approximation is good especially for resistive materials (of low permeability) and for the lower frequencies. The permeability is reconstructed by solving the inverse problem using waves reflected by plastic foam samples, at different frequency bandwidths in the Darcy regime. The proposed method has the advantage of being simple compared to the conventional methods that use experimental reflected data, and is complementary to the transmissivity method, which is more adapted to low resistive materials (high permeability).

Design of sound absorbing metamaterials by periodically embedding three-dimensional resonant or non-resonant inclusions in rigidly backed porous plate

Air saturated porous materials, namely, foams and wools, are often used as sound absorbing materials. Nevertheless, they suffer from a lack of absorption efficiency at low frequencies, which is inherent to their absorption mechanisms (viscous and thermal losses), even when used as optimized multilayer or graded porous materials. These last decades, several solutions have been proposed to avoid this problem. Among them, metaporous materials consist in exciting modes trapping the energy between the periodic rigid inclusions embedded in the porous plate and the rigid backing or in the inclusions themselves. The absorption coefficient of different foams is enhanced both in the viscous and inertial regimes by periodically embedding 3D inclusions, possibly resonant, i.e., air filled Helmholtz resonators. This enhancement is due to different mode excitation: a Helmholtz resonance in the viscous regime and a trap mode in the inertial regime. In particular, a large absorption coefficient is reached for wavelengths in the air 27 times larger than the sample thickness. The absorption amplitude and bandwidth is then enlarged by removing porous material in front of the neck, enabling a lower impedance radiation, and by adjusting the resonance frequencies of the Helmholtz resonator.

On the influence of frequency-dependent elastic properties in vibro-acoustic modelling of porous materials under structural excitation

The aspects related to modelling the frequency dependence of the elastic properties of air-saturated porous materials have been largely neglected in the past for several reasons. For acoustic excitation of porous materials, the material behaviour can be quite well represented by models where the properties of the solid frame have little influence. Only recently has the importance of the dynamic moduli of the frame come into focus. This is related to a growing interest in the material behaviour due to structural excitation. Two aspects stand out in connection with the elastic-dynamic behaviour. The first is related to methods for the characterisation of the dynamic moduli of porous materials. The second is a perceived lack of numerical methods able to model the complex material behaviour under structural excitation, in particular at higher frequencies. In the current paper, experimental data from a panel under structural excitation, coated with a porous material, are presented. In an attempt to correlate the experimental data to numerical predictions, it is found that the measured quasi-static material parameters do not suffice for an accurate prediction of the measured results. The elastic material parameters are then estimated by correlating the numerical prediction to the experimental data, following the physical behaviour predicted by the augmented Hooke׳s law. The change in material behaviour due to the frequency-dependent properties is illustrated in terms of the propagation of the slow wave and the shear wave in the porous material.

Measuring static viscous permeability of porous absorbing materials.

Conventional acoustical methods for measuring the permeability or flow resistivity of a porous material require a priori estimation of the porosity. In this work, an acoustical method is presented in which a simplified expression (independent of both the frequency and porosity) for the transmitted waves at the Darcy's regime (low frequency range) is derived, and used for the inverse determination of both the viscous static permeability (or flow resistivity) and the thickness of air-saturated porous materials. The inverse problem is solved based on the least-square numerical method using experimental transmitted waves in time domain. Tests are performed using industrial plastic foams. Experimental and numerical validation results of this method are presented, which show the advantage of measuring the viscous permeability and thickness of a porous slab, without the required prior knowledge of the porosity, but by simply using the transmitted waves.

Simultaneous determination of porosity, tortuosity, viscous and thermal characteristic lengths of rigid porous materials

We present an improved method for the characterization of air-saturated porous materials by simultaneous measurement of porosity, tortuosity, viscous, and thermal characteristic lengths via ultrasonic transmission only. The proposed method is based on a temporal model of the direct and inverse scattering problem for the transient ultrasonic waves in a homogeneous isotropic slab of rigid porous material. The advantage of the proposed method is that the four parameters are determined simultaneously using just transmitted experimental waves from a porous material saturated by one gas (air). In addition, no relationship is assumed between the two characteristic lengths.

Numerical and analytical solutions for sound propagation and absorption in porous media at high sound pressure levels

On the basis of the work of Wilson et al. [J. Acoust. Soc. Am. 84, 350-359 (1988)], a more exact numerical approach was constructed for predicting the nonlinear sound propagation and absorption properties of rigid porous media at high sound pressure levels. The numerical solution was validated by the experimental results for sintered fibrous porous steel samples and its predictions were compared with the numerical solution of Wilson et al. An approximate analytical solution was further put forward for the normalized surface acoustic admittance of rigid air-saturated porous materials with infinite thickness, based on the wave perturbation method developed by Lambert and McIntosh [J. Acoust. Soc. Am. 88, 1950-1959 (1990)]. Comparisons were made with the numerical results.

Acoustic properties of air-saturated porous materials containing dead-end porosity

This study examines the acoustic properties of materials with complex micro-geometry containing partially open or dead-end (DE) porosity. One of these kinds of materials can be obtained from dissolving salt grains embedded in a solid metal matrix with the help of water. The solid matrix is obtained after the metal, in liquid form, has invaded the granular material formed by the salt particles at negative pressure and high temperature, and after cooling and solidification of the metal. Comparisons between theoretical and experimental results show that the classical Johnson-Champoux-Allard model does not quite accurately predict the acoustic behavior. These results suggest that the assumptions of the Biot theory may not all be fulfilled and that cavity resonators and dead ends can be present in the material. The first part of the study proposes a simple model to account for this geometry. Based upon this model, two acoustic transfer matrices are developed: one for non-symmetric and one for symmetric dead-end porous elements. It is thought that this model can be used to study the acoustic absorption and sound transmission properties of the type of material previously described. In the second part, a series of simplified samples are proposed and tested with a three-microphone impedance tube to validate the exposed model. Finally, the third part compares the predictions of the exposed model to the impedance tube results on a real aluminum foam sample containing dead-end pores. These first results are encouraging and show that this simple model also provides a good prediction for these materials with more complicated microstructure.

Ultrasonic characterization of air-saturated double-layered porous media in time domain

This paper concerns the ultrasonic characterization of air-saturated double-layered porous materials by solving the inverse problem using experimental reflected signals at normal incidence. The double-layered porous media consist of two slabs of homogeneous isotropic porous materials with a rigid frame. The ultrasonic propagation in double-layered porous material is modeled using a temporal model in which the inertial effects are described by the tortuosity. The viscous and thermal losses of the medium are described by two susceptibility kernels which depend on the viscous and thermal characteristic lengths. The sensitivity of porosity, tortuosity, and viscous characteristic length of each layer is studied showing their effect on the reflected interface waveforms. The inverse problem is solved numerically by the least-squares method. Five parameters are inverted: porosity and tortuosity of the two layers and the viscous characteristic length of the first layer. The minimization of the discrepancy between experimental and theoretical data is made in the time domain. The inverse problem is shown to be well posed and its solution to be unique. Experimental results for waves reflected by the interfaces of the double-layered porous material are given and compared with theoretical predictions.

Ultrasonic characterization of porous absorbing materials: Inverse problem

This paper concerns the ultrasonic characterization of air-saturated porous materials by solving the inverse problem using experimental data. It is generally easy to solve the inverse problem via transmitted waves, obtaining optimized values of tortuosity, viscous and thermal characteristic lengths, but this is not the case for the porosity because of its weak sensitivity in the transmitted mode. The reflection mode is an alternative to the transmission mode, in that it gives a good estimation of porosity and tortuosity by processing the data relative to measurements of the wave reflected by the first interface. The viscous and thermal characteristic lengths cannot be determined via the first interface reflection. The wave reflected by the second interface can be experimentally detected only for the weakly resistive porous materials. In this case, the characteristic lengths can be estimated. But for common air-saturated porous materials, the second reflection is very damped and its experimental detection is difficult. We propose in this paper to solve the inverse problem numerically by the least-squares method, using both reflected and transmitted experimental data. We determine simultaneously all the physical parameters intervening in the propagation. The minimization between experiment and theory is made in the time domain. The inverse problem is well posed, and its solution is unique. As with the classic ultrasonic approach for characterizing porous material saturated with one gas, the characteristic lengths are estimated by assuming a given ratio between them. Tests are performed using industrial plastic foams. Experimental and numerical results, and prospects are discussed.

Numerical Analysis of Grashof and Darcy Number Effects on Dissipative Natural Convection Boundary Layers in a Micropolar Fluid-Saturated Geological Porous Medium

The thermo-micropolar non-Newtonian theory is used to formulate a transport model for combined free convection heat and species transfer through a micropolar-fluid saturated Darcian porous medium. Temperatures in the medium are assumed to be high enough for viscous heating effects to be significant. The influence of thermal Grashof number, species Grashof number and Darcian porous number on the momentum, angular momentum, temperature and concentration flow fields are studied using the finite element method. Temperature is observed to be reduced with a rise in the thermal Grashof number. Species transfer is seen to be also decreased with increasing species Grashof number. Micro-rotation values are also decreased with both thermal and species Grashof numbers near the stretching surface and also depressed with a rise in the Darcy number in the near-field regime. The flow field is accelerated with a rise in Darcy number as indicated by the increase in translational velocities. Our computations are relevant to for example the diffusion of hydrogen gas in air-saturated porous materials, pollutant release in a geophysical regime etc.

Reflection and transmission at normal incidence onto air-saturated porous materials and direct measurements based on parametric demodulated ultrasonic waves

The present work is related to the characterization of air-saturated porous media by using parametric demodulated ultrasonic waves. One uses two different powerful ultrasonic emitters working either at 47 kHz or at 162 kHz which are electronically amplitude modulated over the 200 Hz–4 kHz or 2 kHz–40 kHz bandwidths respectively. The demodulation process takes place in air, due to its nonlinearity enabling to generate audio range acoustical waves or alternatively low frequency ultrasonic waves which can be used to characterize porous materials in the reflection configuration at normal incidence. Some appropriate theoretical calculations are introduced for three configurations of interest, i.e. a porous slab, a porous layer mounted onto a rigid plate, and a porous half space, in the case of the equivalent-fluid model. Comparisons between theoretical modeling and experimental data are provided and prospective industrial applications are discussed.

Application of nonlinearly demodulated acoustic signals for the measurement of the acoustical coefficient of reflection for air saturated porous materials

The present Note describes work related to the measurement of the coefficient of reflection in automotive felt materials, by using a mixed ultrasonic/audio range technique. Powerful 162 kHz ultrasonic waves are amplitude modulated in the audio range. By applying appropriate procedures borrowed from underwater nonlinear ultrasonic methods (the so-called parametric antennae), one produces low frequency (i.e. in the 5–30 kHz range) acoustical waves which are generated in the pulse echo mode by short bursts. The coefficient of reflection of various felt materials are measured, and the results are compared to the standard ‘fluid-equivalent’ model which describes the propagation of acoustic waves in poroelastic air-saturated materials. To cite this article: M. Saeid et al., C. R. Mecanique 332 (2004). La présente Note décrit un travail relatif à la mesure du coefficient de réflexion pour des feutres de l'industrie automobile, en utilisant une technique mixte dans le domaine ultrasons/audio. Des ondes ultrasonores de puissance à 162 kHz sont modulées en amplitude dans le domaine audio. En utilisant des procédures appropriées empruntées aux méthodes de l'acoustique sous-marine non linéaire (c'est-à-dire les antennes paramétriques), des ondes acoustiques de basse fréquence (dans la bande 5–30 kHz) sont produites dans le régime d'échos impulsionnels avec des paquets d'onde courts. Le coefficient de réflexion de divers matériaux en feutre sont mesurés, et les résultats sont comparés avec le modèle standard de « fluide-équivalent » qui décrit la propagation des ondes acoustiques dans des matériaux poroélastiques saturés par l'air. Pour citer cet article : M. Saeid et al., C. R. Mecanique 332 (2004).

Application of nonlinearly demodulated acoustic signals for the measurement of the acoustical coefficient of reflection for air saturated porous materials

The present Note describes work related to the measurement of the coefficient of reflection in automotive felt materials, by using a mixed ultrasonic/audio range technique. Powerful 162 kHz ultrasonic waves are amplitude modulated in the audio range. By applying appropriate procedures borrowed from underwater nonlinear ultrasonic methods (the so-called parametric antennae), one produces low frequency (i.e. in the 5–30 kHz range) acoustical waves which are generated in the pulse echo mode by short bursts. The coefficient of reflection of various felt materials are measured, and the results are compared to the standard ‘fluid-equivalent’ model which describes the propagation of acoustic waves in poroelastic air-saturated materials. To cite this article: M. Saeid et al., C. R. Mecanique 332 (2004).

Porosity measurement by comparison of air volumes

A method for measuring the volume of a solid with arbitrary shape and the open porosity (volume fraction of the open pores) of air-saturated porous materials is proposed. The experimental principle is inspired from Beranek’s method [J. Acoust. Soc. Am. 13, 248 (1942)] based on the application of the ideal gas law to the gas surrounding the solid. The originality of the proposed method lies in the comparison of the volume of a measurement chamber containing the sample with that of an empty reference chamber. During a measurement, the reduction of volume in the measurement chamber due to the introduction of the solid is compensated by increasing the volume of a piston connected to the measurement chamber. An important aspect of this experimental system is that the influence of temperature can be considerably reduced as nearly the same variations of temperature apply simultaneously to the measurement and the reference chambers. A preliminary calibration experiment without sample in the measurement chamber is carried out in order to equalize the volumes of the two chambers. The calibration can be performed once and for all, thus minimizing the number of operations in the measurement process. The porosities of several materials are measured and compared to reference values. The precision and possible future improvements are discussed.