Acoustic characterization of air-saturated porous materials by solving the inverse problem.

Transcription

1 Acoustic characterization of air-saturated porous materials by solving the inverse problem. Z.E.A. Fellah 1, A. Berbiche 1,2, M. Felllah 2, E. Ogam 1 and C. Depollier 3 1 Laboratoire de Mécanique et d Acoustique, Centrale Marseille, Aix-Marseille University, 4 impasse Nikolas Tesla CS 40006, Marseille Cedex 13, France. 2 Laboratoire de Physique Théorique, Faculté de Physique, USTHB, BP 32 El Alia, Bab Ezzouar 16111, Algérie. 3 LUNAM Universite du Maine. UMR CNRS 6613 Laboratoire d Acoustique de l Universite du Maine UFR STS, Avenue O. Messiaen Le Mans CEDEX 09 France. ABSTRACT We present in this contribution, several experimental methods for the acoustic characterization of air-saturated porous materials. The equivalent fluid model is considered, in which the acoustic wave propagates in the saturating fluid of the porous material. The principle of these methods is the detection of experimental waves reflected and transmitted by the porous material. The high and low frequency domains are studied. The expressions of the reflection and transmission coefficients are established in time and frequency domains. The sensitivity of the non-acoustic parameters - porosity, tortuosity, viscous and thermal characteristic lengths, viscous and thermal permeabilities is studied, showing their effects on the transmitted and reflected waveforms. It is shown that some parameters are more sensitive to the transmission mode, and others to the reflection. The inverse problem is solved numerically using experimental data of waveforms in the time domain for these parameters. Tests are performed using industrial plastic foams. The inverted values of non-acoustic parameters are close to those measured by classical methods. The results of the experimental and numerical validation of this method are presented and compared with theoretical predictions and a very good agreement has been found. 1. INTRODUCTION The acoustic and ultrasonic characterization of air-saturated porous materials as plastic foam, fibrous or granular materials is of great interest for a wide range of industrial applications automotive, aeronautic industries, building trade[1, 2]. According to the frequency domain, the physical parameters describing the acoustic propagation in porous materials are different. In the high frequency domains, the main important parameters are: tortuosity, viscous and thermal characteristic lengths [3]. In the low frequency range, the parameters intervening in the acoustic propagation are: the static viscous and thermal permeabilities [4], and the inertial factor low frequency tortuosity. In addition to these parameters, the porosity φ is a key parameter playing an important role for all frequencies. The high and low frequency ranges [3, 4], are defined by comparing the viscous and thermal skin thicknesses δ = 2η/ωρ 1/2 and δ = 2η/ωρP r 1/2 with the radius of the pores r ρ is the density of the saturating fluid; ω the pulsation frequency; P r the Prandtl number. In the asymptotic domain high frequencies, the skin thicknesses become narrower and the viscous effects are concentrated in a small volume near the frame δ r and δ r. Acoustic characterization of materials is often achieved by measuring the attenuation coefficient and phase velocity in the frequency domain [1] or by solving the direct and inverse problems directly in the time domain [5 16]. The attractive feature of a time domain based approach is that the analysis is naturally limited by the finite duration of ultrasonic pressures and is consequently the most appropriate approach for the transient signal. The objective of this contribution is to propose some theoretical and experimental methods for the acoustic characterization of air-saturated porus materials. The direct and inverse scattering problems are solved in time domain using experimental reflected and transmitted signals. The physical parameters of the porous medium are recovered by solving 1 Fellah@lma.cnrs-mrs.fr

2 the inverse problem at the asymptotic domain corresponding to the high frequency range ultrasound, and at the viscous domain low frequency range. 2. THEORY In the acoustics of porous materials, one distinguishes two situations according to whether the frame is moving or not. In the first case, the dynamics of the waves due to the coupling between the solid skeleton and the fluid is well described by the Biot theory [17 19]. In air-saturated porous media, the vibrations of the solid frame can often be neglected in absence of direct contact with the sound source, so that the waves can be considered to propagate only in fluid. This case is described by the equivalent-fluid model [1, 3, 4] which is a particular case of the Biot model [17 19], in which fluid-structure interactions are taken into account in two frequency response factors: dynamic tortuosity of the medium αω given by Johnson et al [3], and the dynamic compressibility of the air in the porous material βω given by Allard et al [1]. In the frequency domain, these factors multiply the density of the fluid and its compressibility respectively and represent the deviation from the behavior of the fluid in free space as the frequency increases. The expressions of the dynamic tortuosity and compressibility are given by [1, 3, 4]: αω = α M2 jx j x where x = ωα ρ f and M = 8k 0α σφ φλ. 1 2 βω = γ γ 1/ [ 1 1 j x 1 M ] 2 j x where x = ωρ fk 0P r ηφ and M = 8k 0 φλ 2. the parameter k 0 introduced by Lafarge [4] called thermal permeability by analogy to the viscous permeability. Consider a homogeneous porous material that occupies the region 0 x L. A sound pulse impinges normally on the medium. It generates an acoustic pressure field p and an acoustic velocity field v within the material. The acoustic fields satisfy the following equivalent-fluid macroscopic equations along the x axis[1, 3, 4]: ραωjωv = p x, βω jωp = v K a x, 2 where, j 2 = 1, ρ is the saturating fluid density and K a is the compressibility modulus of the fluid. For a slice of porous material occupying the domain 0 x L, the incident and scattered fields are connected by reflection and transmission operators in time domain. They are defined by [19]: t p r x, t = Rτp i t τ + x dτ, 3 c 0 p t x, t = 0 t 0 T τp i t τ L c x L dτ. 4 c 0 The following interface conditions are assumed acoustic pressure and flow are continuous: p0 +, t = p0, t, pl, t = pl +, t v0, t = φv0 +, t, vl +, t = φvl, t 5 φ is the porosity of the material. The expression, in frequency domain, of the transmission coefficient T ω of a slab of porous material is given by [9, 10, 14]: T ω = 2Y ω 2Y ω coth kωl Y 2 ω sinh kωl, 6 where: βω ραωβω Y ω = φ, and kω = ω, αω K a

3 3. HIGH FREQUENCY DOMAIN INTER-NOISE 2016 In the high frequency domain, the viscous effects are concentrated in a small volume near the frame and the compression/dilatation cycle is faster than the heat transfer between the air and the structure, and it is a good approximation to consider that the compression is adiabatic [3] αω = α 1 2 Λ η jωρ f 1/2, ω. 7 where j 2 = 1, η is the dynamic viscosity of the fluid, ω the angular frequency, ρ f the density of the fluid, [1] βω = 1 2γ 1 Λ η P r ρ f 1/2 1 1/2, ω, 8 jω where P r is the number of Prandt and γ the adiabatic constant. The general expressions of R and T for a normal incidence are given by see Ref. [9] φ + α Rt = φ + α T t = 4φ α α + φ 2 n 0 φ α φ + α φ α n 0 φ + α 2n [ G t, 2n L G t, 2n + 2 L ], c c 2n G t + L c 0, 2n + 1 L c where G is the Green function of the medium [19]. These expressions take into account the multiple n reflections within the porous material. Given the high attenuation of acoustic waves in air-saturated porous media, multiple reflections are negligible. For reflections at the interfaces x = 0 and x = L, the expressions of the operators of reflection and transmission are simplified Ref. [9] as follows: Rt = T t = α φ α + φ δt 4φ α α φ 4φ α φ + α G t + L 2 c, L c G α + φ 3., 9 t, 2L c. 10 The first term on the second member of the equation 9: α φ/ α + φ δt is equivalent to the instantaneous reflection response of the porous material. This term corresponds to the wave reflected by the first interface x = 0. It depends only on the porosity and tortuosity of the material. The reflected wave to the first interface has the advantage not to be dispersive, but simply attenuated. This shows that it is possible to measure the porosity and tortuosity of the porous material by measuring just the first reflected wave. The second term of equation 9: 4φ α α φ α +φ G t, 2L 3 c corresponds to the reflection by the second interface x = L. This term depends on the Green function of the medium that describes the propagation and scattering of the acoustic wave having made a round trip in the slab of porous material. Green s function depends on the tortuosity, and viscous and thermal characteristic lengths Λ and Λ material, but does not depend on the porosity. Experimentally, this second contribution to the debate can not be measured for low-resistive materials, because the acoustic signal is very attenuated. 3.1 Inversion of Transmitted Experimental waves The inverse problem [9] is to find the parameters α, φ, Λ and Λ which minimize numerically the discrepancy function Uα, φ, Λ, Λ = i=n i=1 pt expx, t i p t x, t i 2, wherein p t expx, t i i=1,2,...n is the discrete set of values of the experimental transmitted signal and p t x, t i i=1,2,...n the discrete set of values of the simulated transmitted signal predicted from Eq. 4. The inverse problem is solved numerically by the least-square method. For its iterative solution, we used the simplex search method Nedler Mead [20] which does not require numerical or analytic gradients. Experiments are performed in air using a broadband Ultran NCT202 transducer with a central frequency of 190 khz in air and a bandwith of 6 db extending from

4 Figure 1. Experimental set-up of the ultrasonic measurements. Table 1. Physical characteristics of ample M1 obtained by classical methods Material Thicknesses cm φ α Λµm Λ µ.m Sample M1 0.8± ± ± ± 1 60± 3 to 230 khz. Pulses of 400V are provided by a 5058PR Panametrics pulser/receiver. The received signals are filtered above 1MHz to avoid high-frequency noise. Electronic interference is eliminated by 1000 acquisition averages. The experimental setup is shown in Fig. 1. Consider the plastic foam sample M1 having the physical characteristic given in Table 1. After solving the inverse problem simultaneously for the porosity φ, tortuosity α, viscous and thermal characteristic lengths Λ and Λ for the sample M1 we obtain the results given in the Table 2. It can be seen that the values of the inverted parameters are close to those obtained by Table 2. Physical characteristics of the sample M1 obtained by solving the inverse problem. Material φ α Λµm Λ µ.m Sample M1 0.87± ± ± ± 0.5 conventional methods [1, 21 23]. The variation of the minimization function U with the porosity, tortuosity, viscous characteristic length, and the ratio between Λ and Λ are given in Figures 2 and 3. A comparison between an experimental transmitted signal and simulated transmitted signal for the optimized values of φ, α, Λ and Λ is given in figure 4. The reader can see the small difference between experimental et predicted transmitted signals, which leads us to conclude that the optimized values of the physical parameters are correct. 3.2 Inversion of reflected Experimental waves The reflection coefficient in time for an oblique incidence is given by [6] Rt = rt + Rt, 11 where where rt = 1 E 4E1 E δt and Rt = 1 + E 1 + E F t, 2L 3 c. 12 E = φ 1 sin2 θ α α cos θ

5 Figure 2: leftexperimental incident signal solid line and experimental transmitted signal dashed line. Right:Variation of the minimization function U with porosity and tortuosity Figure 3: Left Variation of the cost function U with the viscous characteristic length Λ and the ratio Λ /Λ. Right Comparison between the experimental transmitted signal black dashed line and the simulated transmitted signals red line using the reconstructed values of φ, α, Λ and Λ. θ is the incident angle shown in Fig. 4 left The simulated reflected signal is computed using Eqs. 3 and 11. Figure 4: Left Geometry of the problem. Right Experimental set-up of the ultrasonic measurements in reflected mode. P.G: pulse generator, H. F. F-P. A: high frequency filtering -preamplifier, D. O: digital oscilloscope, C: computer, T: transducer, S: sample

6 Experiments are carried out in air with two broadband Ultran NCT202 transducers having a 190kHz central frequency in air and a bandwith at 6dB extending from 150kHz to 230kHz. A goniometer used in optic has been employed for the positioning of the transducers. Pulses of 400V are provided by a 5052PR Panametrics pulser/receiver. The received signals are amplified up to 90dB and filtered above 1MHz to avoid high frequency noise. Electronic perturbations are removed by 1000 acquisition averages. The experimental setup is shown in Fig. 4 right. The distance between the transducers and the samples is 20cm.In this section, we measure the tortuosity and the porosity knowing the reflection coefficient at the first interface for different values of the incidence angle θ. The expression of the reflection coefficient at the first interface is given by [6] rt, θ = α cos θ φ α sin 2 θ α cos θ + φ δt. 14 α sin 2 θ For two values of the incidence angle θ 1 and θ 2, it is easy to calculate the expression of the tortuosity function of the reflection coefficients r 1 = rt θ1 and r 2 = rt θ2 corresponding to the angles θ 1 and θ 2 respectively α = 1 r21+r 1 cos θ r 21 r 1 cos θ1 sin 2 θ 1 sin 2 θ r21+r 1 cos θ 2 1+r 21 r 1 cos θ1 Knowing the value of the tortuosity, we deduce the expression of the porosity function of θ i and r i by the expression φ = α 1 r i cos θ i 1 + r i α sin 2 θ i, i = 1, Consider a plastic foam ample M2 having the flowing characteristics: Table 4. gives the experimental data Table 3. Physical characteristics of ample M2 obtained by classical methods Material Thicknesses cm φ α Λµm Λ µ.m Sample M ± ± of the reflection coefficient for different values of the incidence angle θ. The values of the tortuosity and Table 4: Experimental data of the reflection coefficient for different values of the incidence angle θ foam M2 θ deg r porosity are calculated using of Eqs. 15 and 16 for each pair of incidence angles, we obtain the values given in Table 5. Fig. 5 left shows the experimental data of the reflection coefficient for different values of Table 5: Physical characteristics of ample M2 obtained by solving the inverse problem in reflection mode. Material Thicknesses cm φ α Λµm Λ µ.m Sample M ± ± the incident angle θ, and the simulation of the variation of the reflection coefficient with the incident angle

7 θ for the values of the tortuosity α = 1.07 and porosity φ = The values of porosity and tortuosity of the plastic foam M2 given by the classical methods are given in Table 4. A small difference is observed between the values of the porosity and tortuosity measured using this method and the other classical methods [1, 21 23]. Fig. 5 right shows the comparison between the simulated reflected signal at the first interface calculated for α = 1.07 and φ = 0.97, and the experimental reflected signal for θ = 17deg. The difference Figure 5: Left Simulation of the variation of the reflection coefficient solid line with the incident angle θ and experimental data of the reflection coefficient star for the material M2. Right Experimental set-up of the ultrasonic measurements in reflected mode. P.G: pulse generator, H. F. F-P. A: high frequency filtering -pre-amplifier, D. O: digital oscilloscope, C: computer, T: transducer, S: sample. between the simulated reflected signal and experimental reflected signal is relatively weak which leads to the conclusion that the values of the tortuosity and porosity obtained are satisfactory. The reflected wave by the second interface of the porous slab is difficult to observe experimentally except in certain cases studied in our previous work for non-resistive materials [7]. The sensitivity of viscous and thermal characteristic lengths is very low making the determination of these lengths difficult if not impossible reflective mode. However, this study showed that the determination of porosity and tortuosity is fairly easy in reflection. 4. LOW FREQUENCY DOMAIN In the low frequency domain [1, 4, 8, 10, 12, 12 14], the viscous effects are important in all the pore volume, and the compression dilatation cycle in the porous material is slow enough to favor the thermal interactions between fluid and structure. At the same time the temperature of the frame is practically unchanged by the passage of the sound wave because of the high value of its specific heat: the frame acts as a thermostat. In addition, the thermal conductivity of the solid is high, and the excess heat is immediately evacuated by the solid, which therefore remains at the same temperature during the compression dilatation cycle [14] αω = ηφ + α 0 + 2α 4 k 3 0 ρ jω jωρk 0 ηλ 4 φ 3 p3 βω = γ + γ 1k 0P r ρ ηφ jω α 0γ 1k 0 2 P 2 r ρ 2 ω η 2 φ 2 In these equations, k 0 is the static permeability, p is a geometrical parameter introduced by Pride et al [24] and revisited by Lafarge [4], γ is the adiabatic constant, P r the Prandtl number, α 0 is the thermal tortuosity introduced by Lafarge [4], this parameter is the thermal counterpart of the inertial Norris [25] parameter α 0. In this domain of frequency, the expression of the transmission coefficient is given by [14]: t T t = DτGt τ + L/c, L/cdτ, 19 0 Dt = 2 1 u 2 π t 3/2 2t 1 u u2 ρk 0 γπ u exp du, =, 4t η

8 where G is the Green function [26] of the porous material: Gx, t = 1 x b 1 exp x2 π c 4 b t3/2 16c 2 b t F ζ, t = ℵζ, s = π π exp s t ss + 1/2b ℵζ, sds, [ cos 2b ζ s y ] ss + 1/2b ζ 2 x2 c 2 b b = Bc 2, c = Ac 2, = 1 + 4b c. x c b F ζ, t dζ, ζ 2 x2 b c 2 yd y 1 y 2, In the very low frequency range: αω = ηφ, ρ f k 0 jω ω 0, 20 βω = γ, ω In this range of frequencies, the expressions of the transmission and reflection coefficients are given by [10, 12]: T ω = Rω = 2C 1 jω 2C 1 jω cosh LC2 jω C 2 1 jω sinh LC 2 jω, 22 1 C 2 1ω sinh LC 2 jω 2C 1 jω cosh LC2 jω C 2 1 jω sinh LC 2 jω, 23 where C 1 = γρk 0 φ γηφ, C 2 = 24 η K a k 0 At very low frequencies [8, 13], the viscous permeability is the only predominant parameter in sound attenuation, it is unnecessary to consider the effect of other parameters heat permeability and inertial parameter that have an effect for frequencies well higher. Experiments are performed in a guide pipe[8, 10, 12, 12 14], having a diameter of 5 cm. The experimental set up is given in Fig. 6 left. For measuring the static thermal permeability and the inertial factor, a length of 3 m of the pipe is enough, since the frequencies used in the experiment are higher than 1 khz. However, for measuring the viscous permeability and flow resistivity[8, 13], a length of 50 m can be used. In this case, it is not important to keep the pipe straight, it can be rolled in order to save space without perturbations on experimental signals the cut-off frequency of the tube f c 4kHz. The difference between the tube Kundt [1] and the proposed method of this long tube is that the Kundt tube is suitable for Stationary signals, however, in this method, transient signals are privilege as pulses or burst, it is more easy to target frequency ranges. In addition it is easier to use different frequency ranges. The principle of this tube is used to work under similar conditions as ultrasound. A sound source Driver unit "Brand" constituted by loudspeaker Realistic is used. Bursts are provided by synthesized function generator Standford Research Systems model DS345-30MHz. The signals are amplified and filtered using model SR 650-Dual channel filter, Standford Research Systems. The signals incident and transmitted are measured using the same microphone Bruel&Kjaer, 4190 in the same position in the tube. The incident signal is measured without porous sample, however, the transmitted signal is measured with the porous sample. Consider a cylindrical sample of plastic foam M3 of diameter 5 cm and thickness L = 4.15cm, the physical characteristics of the sample M3 are given in Table 6: After solving the inverse problem numerically for the thermal permeability k 0 and the inertial factor α 0, we find the following optimized values given in Table 7. Here again the Difference between inverted values of the physical parameters thermal permeability and inertial factor are close to those given by classical methods [1, 4]. A similar characterization has been developed in the very low frequency range using the expressions 22 and 23 of the transmission and reflection

9 Table 6. Physical characteristics of plastic foam sample M3 INTER-NOISE 2016 Materiel φ k m 2 k m 2 Λ α α 0 α 0 Sample M ± ± µm 1.20 α 0 /α 2.15 ± 0.15 Table 7. Inverted values of plastic foam sample M3 Materiel k m 2 α 0 Sample M ± Figure 6: Left Experimental setup of acoustic measurements. Right Minima of the inverse problem. coefficients for the determination of the static viscous permeability [8, 10, 12, 13]. This study shows that it is possible to characterize a porous material saturated with air in different frequency regimes, using an appropriate models and by determining the parameters in the high and low frequency domains, using experimental data reflected waves and transmitted. The inverse problem is solved by minimizing the gap between theoretical and experimental waves. It is interesting to note that the reflected and transmitted waves do not give the same settings in some cases as in the high frequency domain. 5. CONCLUSION Different methods are proposed in this paper for the acoustic characterization of air-saturated porous materials in the high ultrasound and low frequency ranges. The materials included in this study are the plastics foam samples and fibrous materials used frequently for passive control and sound insulation. The principle of characterization is based on solving the inverse problem using experimental data transmitted and reflected waves by a slab of porous material. According to the frequency range studied high and low frequency domains, the inverted physical parameters are different. This study shows that the reflection and transmission modes are a complementary approaches for the acoustic characterization. REFERENCES [1] J. F. Allard and N. Atalla. Propagation of sound in porous media: Modeling sound absorbing materials. Wiley Chichester UK, 1:1 284, [2] K. Attenborough. Acoustical characteristics of porous materials. Physical Reports, 82: ,

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