Rigid-frame Porous Materials Research Articles

Acoustic characterization of porous windscreens for sound pressure and particle velocity sensors

Acoustic measurements of ground vehicles, wind turbines or aircrafts are often performed outdoors in presence of wind. Air movement around an acoustic sensor may significantly disturb the measured signals due to flow-induced vibrations and turbulence. This is specially relevant for pressure-gradient and particle-velocity based acoustic transducers which are more sensitive to these perturbations. To attenuate this effect and improve measurement quality, the sensors are typically covered with windscreens made of porous materials. However, designing an appropriate windscreen poses a challenging task, demanding a deep understanding of sound transmission characteristics. This study works towards quantifying acoustic properties of rigid-frame porous materials through in-situ measurements and fitting a parametric fluid-equivalent model to find optimal model parameters. The parameters are subsequently investigated by comparing windscreen insertion loss through finite-element numerical simulations and measurements on a sound intensity PU probe.

Time-domain simulations of sound propagation in a flow duct with extended-reacting liners

A general methodology to simulate acoustic propagation in ducts with extended-reacting liners in the time domain is presented, including a generic perforated sheet on the air-material interface. The Linearized Euler Equations (LEE) with a mean flow profile are solved in the duct and the linearized equations on an equivalent fluid are solved in the liner material. The auxiliary differential equation method (ADE) is used to prevent the computation of convolution integrals, and leads to a formulation compatible with high-order numerical schemes. The methodology is illustrated for the case of liners consisting of rigid-frame porous materials and a prototype of locally-resonant acoustic metamaterials. A one-dimensional (1D) test case is first used to validate the algorithm and assess the numerical error. The numerical order of the algorithm is the expected one, independently of the interface, as long as the number of poles retained in the partial fraction expansions involved in the formulation is high enough. The algorithm is then applied to a realistic two-dimensional (2D) configuration in a duct with flow, and is used to illustrate the restrained validity of the locally-reacting approximation. Finally, the impact of the flow Mach number on the acoustic performance of porous and metamaterial extended-reacting liners is briefly assessed.

Pole identification method to extract the equivalent fluid characteristics of general sound-absorbing materials

A method is presented to characterize general sound-absorbing materials through a pole-based identification of the equivalent fluid. This is accomplished by 1) determining the extended equivalent fluid of the material sample through the transfer function method (TFM), 2) identification of the acoustic response of the material through the poles of the extended effective density and compressibility, and 3) build the effective density and compressibility from the poles associated to the local acoustic response. Real pole pairs describe a dissipative medium (or equivalently an over-damped resonating medium), which is the natural behavior of rigid-frame porous materials, while complex-conjugate pole pairs describe a locally-resonant medium typical of metamaterials. Complex-conjugate poles associated to elastic resonances of the sample are discarded. We test the method for several non-conventional porous materials. In general, a better fit to the measured surface impedance is obtained than with an acoustics-based identification to the Johnson-Champoux-Allard-Pride-Lafarge model (JCAPL), and the method appears also to be robust to errors of the TFM.

Two-dimensional finite-difference time-domain analysis of sound propagation in rigid-frame porous material based on equivalent fluid model

An equivalent fluid model based finite-difference time-domain algorithm is proposed to simulate the frequency characteristic of rigid-frame porous material in 2-dimensional sound field. Two study cases are simulated at oblique incidence of sound, and the calculated surface impedance agree well with the theoretical value in broad frequency range. The errors are also discussed when simulating a porous material with shape with the staircase approximation. As an example, sound propagation in a porous material with triangular shape is simulated using the algorithm.

An equivalent fluid model based finite-difference time-domain algorithm for sound propagation in porous material with rigid frame.

An equivalent fluid model based finite-difference time-domain algorithm is proposed to simulate the frequency characteristic of porous material with rigid frame. The rigid-frame porous material is replaced by an equivalent fluid characterized by the effective density and the effective bulk modulus, which reflect the viscous effects and thermal effects in the porous material, respectively. The effective density and the effective bulk modulus are frequency-dependent complex values, which are designed in the form of infinite impulse response filters in the frequency domain. The Z transform theory is used to discretize the frequency-domain wave equations. The accuracy of the proposed algorithm is preliminarily validated by good agreement between the calculated and measured normal-incidence sound absorption coefficients of two samples composed of different multiple-layered materials in a wide frequency range.

Deterministic and statistical characterization of rigid frame porous materials from impedance tube measurements.

A method to characterize macroscopically homogeneous rigid frame porous media from impedance tube measurements by deterministic and statistical inversion is presented. Equivalent density and bulk modulus of the samples are reconstructed with the scattering matrix formalism, and are then linked to its physical parameters via the Johnson-Champoux-Allard-Lafarge model. The model includes six parameters, namely the porosity, tortuosity, viscous and characteristic lengths, and static flow and thermal permeabilities. The parameters are estimated from the measurements in two ways. The first one is a deterministic procedure that finds the model parameters by minimizing a cost function in the least squares sense. The second approach is based on statistical inversion. It can be used to assess the validity of the least squares estimate, but also presents several advantages since it provides valuable information on the uncertainty and correlation between the parameters. Five porous samples with a range of pore properties are tested, and the pore parameter estimates given by the proposed inversion processes are compared to those given by other characterization methods. Joint parameter distributions are shown to demonstrate the correlations. Results show that the proposed methods find reliable parameter and uncertainty estimates to the six pore parameters quickly with minimal user input.

Acoustics of multiscale sorptive porous materials

This paper investigates sound propagation in multiscale rigid-frame porous materials that support mass transfer processes, such as sorption and different types of diffusion, in addition to the usual visco-thermo-inertial interactions. The two-scale asymptotic method of homogenization for periodic media is successively used to derive the macroscopic equations describing sound propagation through the material. This allowed us to conclude that the macroscopic mass balance is significantly modified by sorption, inter-scale (micro- to/from nanopore scales) mass diffusion, and inter-scale (pore to/from micro- and nanopore scales) pressure diffusion. This modification is accounted for by the dynamic compressibility of the effective saturating fluid that presents atypical properties that lead to slower speed of sound and higher sound attenuation, particularly at low frequencies. In contrast, it is shown that the physical processes occurring at the micro-nano-scale do not affect the macroscopic fluid flow through the material. The developed theory is exemplified by introducing an analytical model for multiscale sorptive granular materials, which is experimentally validated by comparing its predictions with acoustic measurements on granular activated carbons. Furthermore, we provide empirical evidence supporting an alternative method for measuring sorption and mass diffusion properties of multiscale sorptive materials using sound waves.

Acoustics of sorptive porous materials

Sound propagation in conventional rigid-frame porous materials is affected by the viscosity of the saturating gas and the heat transfer between the gas and the solid frame. This paper investigates the acoustical properties of hierarchical porous materials that support mass transfer processes, such as sorption and different types of diffusion, in addition to the previously mentioned effects. The two-scale asymptotic expansion method of homogenisation for periodic media is used to derive the macroscopic acoustic description. This description allowed to conclude that, at the leading order, neither sorption nor diffusion affects the macroscopic fluid flow through the material. However, the effective compressibility is significantly modified by these physical phenomena. Specifically, the real part of the low-frequency bulk modulus can take values much smaller than those found in conventional materials while its imaginary part is increased around the characteristic frequency associated with diffusion. As a consequence, the sound waves are slowed down and more attenuated at low frequencies. The results are exemplified for sorptive porous, fibrous, and granular materials and the influence of their microstructural descriptors on their effective acoustical properties is discussed.

Characterization of porous material and a sound suppression system using a shock tube

This work describes the design and construction of an experimental apparatus for testing shock waves and a sound suppression system for student learning. The shock tube will be used to characterize both porous media and the suppression system response to a non-linear shockwave. The effects of a plane shock wave will be used to determine acoustic properties, such as transmission, reflection, and absorption coefficients, of rigid frame porous materials. The suppression system is comprised of 20 conical chambers that cause a loss of energy in the shock wave by the expansion and acceleration of the flow. The sound pressure level from the shock is in excess of 140 dB re 20 μPa. Material properties and the performance of the designed suppression system will be presented.

Impedance tube experiments as a tool to teach acoustics of porous media

As a practical matter, sound propagation in many everyday situations includes interactions with porous materials. Such materials include textiles, foams, gravel, grass-covered ground, asphalt, and trees. Modeling sound propagation in complex environments, such as passenger cabins or urban spaces, with porous materials like these is essential to understanding realistic acoustics problems. The theory behind the sound absorption in porous material is not typically included in introductory acoustics courses. This is, at least in part, attributable to the mathematical complexity of the topic. Despite this barrier, acoustic properties of porous material can be easily introduced to typical undergraduate students using simple experiments to demonstrate the acoustic effects of different materials. This presentation describes acoustic property measurements of rigid-frame porous material with straight cylindrical pores as well as foam and soil included in an “Introduction to Acoustics” class at the Catholic University of America. The apparatus is composed of a LabVIEW measurement interface, a pair of microphones, and a simple aluminum impedance tube. The modular design allows students to adapt the software or hardware for other experiments. These hands-on experiences are intended to impart a conceptual understanding of the theory of sound absorption and the acoustic performance of sound absorbing (porous) media.

Using simple shape three-dimensional rigid inclusions to enhance porous layer absorption.

The absorption properties of a metaporous material made of non-resonant simple shape three-dimensional rigid inclusions (cube, cylinder, sphere, cone, and ring torus) embedded in a rigidly backed rigid-frame porous material are studied. A nearly total absorption can be obtained for a frequency lower than the quarter-wavelength resonance frequency due to the excitation of a trapped mode. To be correctly excited, this mode requires a filling fraction larger in three-dimensions than in two-dimensions for purely convex (cube, cylinder, sphere, and cone) shapes. At long wavelengths compared to the spatial period, a cube is found to be the best purely convex inclusion shape to embed in a cubic unit cell, while the embedment of a sphere or a cone cannot lead to an optimal absorption for some porous material properties and dimensions of the unit cell. At a fixed position of purely convex shape inclusion barycenter, the absorption coefficient only depends on the filling fraction and does not depend on the shape below the Bragg frequency arising from the interaction between the inclusion and its image with respect to the rigid backing. The influence of the incidence angle and of the material properties, namely, the flow resistivity is also shown. The results of the modeling are validated experimentally in the case of cubic and cylindrical inclusions.

Generalized equation for transient-wave propagation in continuous inhomogeneous rigid-frame porous materials at low frequencies

This paper provides a temporal model for the propagation of transient acoustic waves in continuous inhomogeneous isotropic porous material having a rigid frame at low frequency range. A temporal equivalent fluid model, in which the acoustic wave propagates only in the fluid saturating the material, is considered. In this model, the inertial effects are described by the inhomogeneous inertial factor [A. N. Norris, J. Wave Mat. Interact. 1, 365-380 (1986)]. The viscous and thermal losses of the medium are described by two inhomogeneous susceptibility kernels which depend on the viscous and thermal permeabilities. The medium is one-dimensional and its physical parameters (porosity, inertial factor, viscous, and thermal permeabilities) are depth dependent. A generalized wave propagation equation in continuous inhomogeneous material is established and discussed.

Microstructure, transport, and acoustic properties of open-cell foam samples: Experiments and three-dimensional numerical simulations

This article explores the applicability of numerical homogenization techniques for analyzing transport properties in real foam samples, mostly open-cell, to understand long-wavelength acoustics of rigid-frame air-saturated porous media on the basis of microstructural parameters. Experimental characterization of porosity and permeability of real foam samples are used to provide the scaling of a polyhedral unit-cell. The Stokes, Laplace, and diffusion-controlled reaction equations are numerically solved in such media by a finite element method in three-dimensions; an estimation of the materials’ transport parameters is derived from these solution fields. The frequency-dependent visco-inertial and thermal response functions governing the long-wavelength acoustic wave propagation in rigid-frame porous materials are then determined from generic approximate but robust models and compared to standing wave tube measurements. With no adjustable constant, the predicted quantities were found to be in acceptable agreement with multi-scale experimental data and further analyzed in light of scanning electron micrograph observations and critical path considerations.

Importance-based sampling for porous material physical parameter estimation

Bayesian inference is applied to the problem of determining the physical parameters of a rigid-frame porous material. Such materials may be characterized by the parameters of flow resistivity, porosity, and tortuosity. The Biot-Allard and Attenborough models predict the propagation of sound in rigid-frame porous materials by modeling the characteristic impedance and propagation coefficient of the material in terms of these physical parameters and a shape factor. In this work, model-based Bayesian analysis is formulated to estimate the values of these parameters from experimental measurements of the characteristic acoustic impedance or propagation coefficient of a material under test. Importance sampling and related methods are used in the parameter estimation procedure as Monte Carlo approaches to characterize the posterior probability distribution which is used to represent the likelihood of parameter values. To increase the efficiency of the sampling, adaptive strategies are employed to extend the class...

Analytical method for the ultrasonic characterization of homogeneous rigid porous materials from transmitted and reflected coefficients

A frequency domain method dedicated to the analytic recovery of the four relevant parameters of macroscopically homogeneous rigid frame porous materials, e.g., plastic foams, at the high frequency range of the Johnson-Champoux-Allard model is developed and presented. The reconstructions appeal to experimental data concerning time domain measurements of the ultrasonic fields reflected and transmitted by a plate of the material at normal incidence. The effective density and bulk modulus of the material are first reconstructed from the frequency domain reflection and transmission coefficients. From the latter, the porosity, tortuosity, and thermal and viscous characteristic lengths are recovered. In a sense, the method presented herein is quite similar in the ultrasonic range, but also quite complementary, to the method developed by Panneton and Olny [J. Acoust. Soc. Am. 119, 2027-2040 (2006); 123, 814-824 (2008)] at low frequency, which appeal to experimental data measured in an impedance tube.

Bayesian parameter estimation of porous materials.

This work proposes a new application of Bayesian analysis to determine the physical parameters of a rigid frame porous material from the measurement of the material’s acoustic impedance. Bayesian analysis allows for numerical estimation of physical parameters, their uncertainties, and inter-relationship of the parameters from the measured complex impedance. Complex impedance data are simulated based on measurements of a sample material to test the Bayesian formulation. Using synthesized data allows for investigations of different frequency resolutions and different levels of noise on the algorithm. Results obtained from testing the parameter estimation algorithm with synthesized data will be utilized in applying the algorithm to measured complex impedance data.

Time domain formulation of the equivalent fluid model for rigid porous media

A set of equations has been derived which corresponds to the time domain formulation of the equivalent fluid model. It models the propagation of an acoustic pulse in rigid frame porous material accounting for both viscous and thermal effects. It has been shown analytically and confirmed numerically that the equations can be reduced to those published previously in the limit of long and short duration pulses. Numerical solutions have been found for different pulse durations and the results have been compared with other time domain models.

On the Theory of Sound Transfer in Rigid-Frame Porous Materials

The modeling of the macroscopic propagation of sound inside porous media is usually based on first order acoustical quantities. Single or multiple layers are treated as an equivalent impedance network which can be also solved with a transfer matrix formulation. In this work the one dimensional, time averaged, macroscopic sound transfer process in rigid-frame porous media is reinterpreted with the concepts developed in the field of sound intensity. An algebraic derivation shows that the active intensity is proportional to the spatial gradient of the sound energy density, so that the two quantities obey a diffusion equation. The name sound conductivity is suggested for the related coefficient, and the expressions of the sinks of sound energy and of the sources of reactivity in the material are formalized. Moreover a differential formulation for the energy densities inside the material is developed and solved analytically.

Reconstruction of material properties profiles in one-dimensional macroscopically inhomogeneous rigid frame porous media in the frequency domain

The present paper deals with the inverse scattering problem involving macroscopically inhomogeneous rigid frame porous media. It consists of the recovery, from acoustic measurements, of the profiles of spatially varying material parameters by means of an optimization approach. The resolution is based on the modeling of acoustic wave propagation in macroscopically inhomogeneous rigid frame porous materials, which was recently derived from the generalized Biot's theory. In practice, the inverse problem is solved by minimizing an objective function defined in the least-square sense by the comparison of the calculated reflection (and transmission) coefficient(s) with the measured or synthetic one(s), affected or not by additive Gaussian noise. From an initial guess, the profiles of the x-dependent material parameters are reconstructed iteratively with the help of a standard conjugate gradient method. The convergence rate of the latter and the accuracy of the reconstructions are improved by the availability of an analytical gradient.

Generalized hyperbolic fractional equation for transient-wave propagation in layered rigid-frame porous materials

This paper provides a temporal model for the propagation of transient ultrasonic waves in a layered isotropic porous material having a rigid frame. A temporal equivalent fluid model is considered, in which the acoustic wave propagates only in the fluid saturating the material. In this model, the inertial effects are described by the layered tortuosity and the viscous and thermal losses of the medium are described by two layered susceptibility kernels which depend on the viscous and thermal characteristic lengths. The medium is one dimensional and its physical parameters (porosity, tortuosity, and characteristics lengths) are depth dependent. A generalized hyperbolic fractional equation for transient sound wave propagation in layered material is established.