Sound absorption of porous composites with heated impervious inclusions

Gabriel Núñez 1

Universidad Austral de Chile (UACh) Instituto de Acústica, Valdivia, Chile

Rodolfo Venegas 2

Universidad Austral de Chile (UACh) Instituto de Acústica, Valdivia, Chile

Claude Boutin 3

Université de Lyon Ecole Nationale des Travaux Publics de l’Etat (ENTPE), Lyon, France

ABSTRACT This paper investigates the sound absorption properties of porous composites with solid impervious heated inclusions, i.e. heat sources, which allow the temperature tuning of the porous constituents. By means of the two-scale asymptotic method of homogenisation, the wave equation in the composites with heated impervious inclusions is derived, allowing the material to be modelled as an equivalent fluid with e ff ective density and compressibility. Analytical and semi-analytical models for the acoustical properties of porous composites with inclusions of di ff erent geometries are developed, which are also numerically validated using the finite element method. This work demonstrates that adjusting the temperature of the porous constituents can provide an alternative way of controlling the absorptive properties of hard-backed porous composite layers, and highlights the dependence of the pressure di ff usion phenomenon on temperature.

1 gabriel.nunez@alumnos.uach.cl

2 rodolfo.venegas@uach.cl

3 claude.boutin@entpe.fr

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

1. INTRODUCTION

Sound absorption of porous composites with solid impervious heated inclusions, i.e. heat sources, is investigated in this paper. These materials are constituted by a matrix made of a permeable material that contains one or more inclusions that can be made of either resistive or impervious materials, and can have di ff erent geometry or shape. Using the two-scale method of homogenisation [1], which allows to model heterogeneous media as equivalent fluids with atypical e ff ective parameters, a macroscopic description of sound propagation through these materials is derived. Analytical models for the e ff ective parameters of these composites are developed, which are also employed to exemplify the temperature-dependence of both the e ff ective properties of these materials, as well on the sound absorption coe ffi cient of hard-backed rigid-frame porous composites. Conventional permeable materials take an important role in the field of acoustics due to their capability of dissipating acoustic energy. If the solid frame of a conventional permeable material is considered rigid because it is either much heavier or sti ff er than the fluid saturating the pores, this capability exists due to the viscosity of the fluid and the thermal exchanges occurring between the fluid and the solid frame. Moreover, these e ff ects are quantified by modelling these materials as equivalent fluids with frequency-dependant complex-valued e ff ective properties such as dynamic viscous permeability [2] and compressibility [3], respectively. Since the acoustical properties of these materials have been extensively studied [4], their limitations are also accounted for. An alternative to these materials are porous composites which, due to the permeability contrast in their constituents, present two or more local acoustic pressure fields, where the pressure field in the matrix is constant at the local scale while the pressure in the inclusions fluctuates locally. The acoustic properties of porous composites have been thoroughly studied [5,6], including the development of a macroscopic description of acoustic wave propagation derived through homogenisation for composites with a single resistive inclusion per unit cell [5], and polydisperse heterogeneous porous composites [6]. These works, however, have not considered the influence of the temperature of the saturating fluid on the sound absorption properties of rigid-frame porous composites. Moreover, the dependence of the pressure di ff usion phenomenon on the temperature of the air inside the inclusions has not been investigated. Consequently, in this paper a rigorous description of sound propagation through porous composites with solid impervious heated inclusions is developed through the two-scale asymptotic method of homogenisation. The key contribution of this paper is to demonstrate the e ff ect of temperature in both the sound absorption properties of the composites, as well as in the pressure di ff usion phenomena occurring locally.

2. THEORY

2.1. Geometry and local description A generic geometry of a periodic rigid-frame porous composite with heated impervious inclusions saturated by air is shown in Figure 1, where L represents the macroscopic characteristic length and l the period of the material. The representative elementary volume (REV) of the composite is denoted by Ω and comprises a permeable matrix, indicated by Ω p , with N r highly resistive porous inclusions Ω i , indexed by i = 1 , ..., N r , and N s totally impervious solid inclusions Ω s j , where j = 1 , ..., N s , which act as slowly varying, with respect to the acoustic characteristic time, heat sources in the material. Moreover, all the inclusions can have di ff erent shape or size, and each of the resistive inclusions can have an impervious inclusion, i.e. a heat source, embedded inside of them. Then, the volume fraction of the matrix is given by ϕ p = Ω p / Ω while those of the i -th resistive inclusion and j -th impervious inclusion are ϕ i = Ω i / Ω and ϕ sj = Ω sj / Ω , respectively. The inclusions are completely surrounded by the matrix and their boundaries are denoted by Γ i for the resistive inclusions and Γ s j for the impervious inclusions. For the propagation of sound in the material, a long wavelength condition is assumed (i.e. λ ≫ l ,

Figure 1: Sketch of the scales and representative elementary volume (REV) of a generic porous composite with solid impervious heated inclusions.

where λ is the sound wavelength related to L as L = λ/ 2 π ). This separation of scales allows to define a small parameter ε = l / L which is used in the homogenisation process. Every constituent of the composite is modelled as an equivalent fluid with e ff ective parameters, i.e. the dynamic viscous permeability tensor k and the dynamic compressibility C , indexed by either p or i . Then, the key conditions to be satisfied by the constituents are that the matrix is much more permeable than the inclusions, i.e. | k p | ≫| k i | , and that the e ff ective compressibilites are comparable, i.e. | C p | = O ( | C i | ). Also, due to the slowly varying characteristics of the heat sources, it is considered that the temperature is constant in both the matrix and the porous inclusions, with the possibility of keeping each at a di ff erent temperature. Porous composites which exhibit local temperature gradients in its constituents are beyond the scope of this work, as it is also the case of porous composites in which macroscopic temperature gradients develop. The local description of sound propagation in the composites is given by the e ff ective equation of conservation of mass (Equation 1) and the dynamic Darcy’s law (Equation 2) for both the matrix and the inclusions, ∇· V u + j ω p u C u ( ω ) = 0 in Ω u , (1)

V u = − k u ( ω )

η · ∇ p u in Ω u , (2)

where V u and p u are the Darcy velocity and pressure, respectively. Likewise, ω is the angular frequency, η is the dynamic viscosity of air, and u = p , i . These equations are coupled via boundary conditions representing the continuity of mass and pressure (Equation 3) in each of the interfaces between the matrix and the resistive inclusions, and of zero normal velocity for the solid inclusions (Equation 4), i.e V p · n = V i · n & p p = p i on Γ i , (3)

V p · n = 0 on Γ s j . (4)

2.2. Two-scale asymptotic homogenisation method By means of the two-scale asymptotic method of homogenisation [1], the macroscopic description of sound propagation through porous composites with heated impervious inclusions is derived from the local description presented in the previous section. The main steps of this method as follows: analysis of the local physics, rescaling of the local description, expressing the variables as a series expansion in terms of ε , identification of boundary-value problems, and derivation of the macroscopic description. For the case of porous composites with heated impervious inclusions, the homogenisation process is analogous to the derivation of the macroscopic description for sound propagation in polydisperse heterogeneous porous composites, carried out in [6]. Thus, the reader is referred to [6] for the details of the derivation.

2.3. Macroscopic description The macroscopic equations that govern sound propagation in porous composites with heated impervious inclusions are the macroscopic mass balance equation (Equation 5) and the dynamic Darcy’s law (Equation 6), i.e. ∇· V + j ω p C ( ω ) = 0 , (5)

V = − k ( ω )

η · ∇ p . (6)

The e ff ective parameters associated with the above e ff ective equations are the e ff ective compressibility C ( ω ) and dynamic viscous permeability tensor k ( ω ) of the composite. The former is calculated as

N X

i = 1 ϕ i C i ( ω ) F i ( ω ) , (7)

C ( ω ) = ϕ p C p ( ω ) +

with F i ( ω ) = 1 − j ω B i ( ω ) /ϕ i D i , where the pressure di ff usivity is given by D i = K i /η C i for an inclusion made from an isotropic porous material, and the pressure di ff usion functions B i ( ω ) are calculated from the solution of a pressure di ff usion problem [7, 8] formulated in the inclusions. It is also relevant to define the pressure di ff usion characteristic frequencies of the inclusions, which are given by ω bi = ϕ i D 0 i / B 0 i , where B 0 i = B i ( ω = 0) and D 0 i = K 0 i /η C 0 i correspond to the static values of the pressure di ff usion function and the pressure di ff usivity of the i -th porous inclusion, respectively. Since the pressure di ff usivity depends on the e ff ective parameters of the inclusion material, these characteristic frequencies can be tuned by varying the temperature of the fluid saturating the porous inclusions, thereby modifying η . The dynamic viscous permeability tensor is calculated as

k = ϕ p k p · α − 1 ∞ , (8)

where α ∞ represents the tortuosity tensor [9] induced by the presence of both the resistive and impervious inclusions in the permeable matrix. From now on, macroscopically isotropic composites are considered, which means that the dynamic viscous permeability and tortuosity tensors can be written as k = K I and α ∞ = α ∞ I , where I is the second rank identity tensor.

3. RESULTS

In this section, analytical models for composites with heated impervious inclusions of cylindrical or spherical shapes are presented in order to exemplify the theory developed in Section 2. Also, the developed upscaled theory is validated through computational experiments using the finite element method.

The analytical models used to obtain the pressure di ff usion of porous composites with heated impervious inclusions of canonical shape, i.e. cylindrical or spherical inclusions, were developed in [6] (see Equations 27 and 28). The inclusions can either be impervious or be made of resistive porous materials. Also, the resistive inclusions can have an impervious inclusion of the same shape embedded inside of them (see image insert of Figures 2a and 2c). Several examples of porous composites with heated impervious inclusions are now presented in order to illustrate the theory by analysing the frequency behaviour of the e ff ective parameters of the composites, particularly the dynamic viscous permeability and e ff ective compressibility, at di ff erent temperatures. Also, the e ff ect of temperature over the characteristic frequencies of the material will be studied. Additionally, theoretical predictions of the sound absorption coe ffi cient of layers of composites with heated inclusions at di ff erent temperatures are compared with results obtained from numerical experiments in order to validate the developed theory.

3.1. Matrix and inclusion at the same temperature This example considers a composite, referred to as PCH1, with a squared REV of side b = 20 mm made of a fibrous material modelled as an array of solid cylinders of radius a p = 30 µ m and porosity φ p = 0 . 97 [10]. The unit cell contains one resistive inclusion made of a granular material ( a i = 50 µ m and φ i = 0 . 45 [11,12]) of radius r i = 8 mm, placed at its centre. The resistive inclusion contains an impervious inclusion of radius r s = 2 . 5 mm located at its centre and which acts as the heat source in the material. The volume fractions of the matrix and the inclusions are ϕ p = ϕ i = 0 . 5. In this first example, the temperature in both the matrix and the resistive inclusion is constant and is determined by the temperature of the impervious inclusion. Figures 2a and 2b show the real and negative of the imaginary parts of the normalised dynamic viscous permeability of PCH1 at di ff erent temperatures, respectively. The plot shows that increasing the temperature of the saturating fluid significantly alters the behaviour of the permeability. Specifically, both the transition region in the real part and the peak in the negative of the imaginary part are shifted upwards in frequency when the temperature increases. It is also important to note that the low and high frequency behaviour of the real part is not altered by temperature. The real and negative of the imaginary parts of the normalised e ff ective compressibility of PCH1 are shown in Figures 2c and 2d. At ambient temperature, i.e. 20 ◦ C, the pressure di ff usion phenomenon in the inclusion induces a transition region in the real part as well as a peak in − Im { C } located at the pressure di ff usion characteristic frequency. When the temperature of the air in the material is increased, both the transition region and the peak are shifted to a lower frequency. This is consistent with Figure 2e, which demonstrates that increasing the temperature in the composite leads to a lower pressure di ff usion characteristic frequency. Moreover, this figure shows that the thermal characteristic frequency of the material increases with temperature until reaching a peak (at approximately 350 ◦ C) and then decreasing again. This behaviour explains the low frequency behaviour of − Im { C } . Figure 2e also shows that the rise in temperature increases the value of the viscous characteristic frequency, which is consistent with the analysis of Figures 2a and 2b. It is also of importance to mention that the low frequency behaviour of Re { C } is not altered by the change in temperature. Lastly, Figure 2f shows the sound absorption coe ffi cient of a hard-backed 6-cm layer of PCH1. It is clear that the change in temperature has a significant influence on the absorption of the material. As the temperature increases, the absorption peak of the material is located at a higher frequency, also flattening the absorption curve. This behaviour is due to the combined e ff ect of classical visco- thermal dissipation together with the pressure di ff usion phenomenon and the tortuosity induced by the inclusions. As detailed in the analysis of Figures 2a–2e, these e ff ects are dependant on the temperature of the air saturating the material. Furthermore, at very low frequencies the absorption curves tend to similar values, which is explained by the fact that the asymptotic low frequency values of both the dynamic viscous permeability and e ff ective compressibility are not altered by temperature. In

addition, a good agreement is found between the numerical results and the theoretical predictions, which leads to conclude that the developed theory correctly captures the acoustical behaviour of porous composites with heated impervious inclusions.

(a) (b)

(c) (d)

(e) (f)

Figure 2: (a) Real part and (b) negative of the imaginary part of the dynamic viscous permeability ( K / K 0 ) of PCH1 at di ff erent temperatures. (c) Real part and (d) negative of the imaginary part of the e ff ective compressibility ( C P 0 ) of PCH1 at di ff erent temperatures. (e) Characteristic frequencies of the material: Continuous red line: pressure di ff usion characteristic frequency; Continuous green line: Thermal characteristic frequency of the matrix material; Discontinuous blue line: Thermal characteristic frequency of the porous inclusion material; Dotted orange line: Viscous characteristic frequency of the matrix material. (f) Sound absorption coe ffi cient of PCH1 at di ff erent temperatures.

3.2. Matrix and inclusion at di ff erent temperatures: varying the temperature of the inclusion To further illustrate the behaviour of the e ff ective parameters, an example of a composite is introduced, called PCH2, where the temperature in the inclusion is di ff erent to that in the matrix. Then, it is considered that the matrix is at ambient temperature while in the inclusion the temperature is determined by the heat source. The geometry and parameters of the material as the same as those of PCH1. Moreover, as the dynamic viscous permeability of the composites is determined solely by that of the matrix, the behaviour of this parameter will not be altered in this example, and thus will be omitted from the analysis. Figures 3a and 3b show the real and negative of the imaginary parts of the e ff ective compressibility of PCH2, respectively. The main di ff erence to the previous example is that Re { C } is not largely a ff ected by the temperature of the air inside the inclusion for frequencies under 350 Hz. This is also true for − Im { C } but for frequencies under 100 Hz. Then, only altering the temperature inside the inclusion does not have a noticeable e ff ect on the e ff ective parameters of the material at lower frequencies. The main di ff erences appear at around the pressure di ff usion characteristic frequency. Since this frequency decreases with temperature, as shown in Figure 3c (and in Figure 2c), the transition region in Re { C } and the peak in − Im { C } will be located at a lower frequency when the temperature inside the inclusion increases. Thus it is possible to obtain a quasi constant value of the e ff ective compressibility by altering the temperature of the air inside the inclusion. Moreover, Figure 3c clearly shows that the thermal characteristic frequency of the matrix is not altered, due to the temperature of the air inside the matrix not being altered. Figure 3d shows the sound absorption coe ffi cient of a hard-backed 6-cm layer of PCH2. The absorption of the material is not significantly altered by temperature for frequencies under 1000 Hz, which is consistent with the analysis of the e ff ective compressibility. The main di ff erences appear over 1000 Hz, where it can be observed that a higher temperature leads to a small increase in the sound absorption coe ffi cient. This is coincidently the frequency range where there are more noticeable di ff erences in the e ff ective compressibility due to the change in temperature. Additionally, a good agreement is found between the numerical and theoretical results, further showing the validity of the developed theory.

3.3. Matrix and inclusion at di ff erent temperatures: varying the temperature of the matrix The last example considers a composite, referred to as PCH3, whose REV comprises a rectangular porous matrix of sides b = 2 cm and 2 b with one resistive inclusion of radius r i = 8 mm and one impervious inclusion with radius r s = r i (see image insert in Figure 4a). Both the matrix and the resistive inclusion are made of the same fibrous and granular materials as the previous two examples, respectively. The volume fractions of the matrix and the inclusions are ϕ p = 0 . 5, ϕ i = ϕ s = 0 . 25. In this example, it is considered that the heat source, i.e. the impervious inclusion, only alters the temperature of the air saturating the porous matrix, and that the temperature inside the inclusion is constant and corresponds to ambient temperature. This design may be di ffi cult to achieve in practice unless the inclusion is thermally isolated without a ff ecting the macroscopic pressure forcing it. Nonetheless, the purpose of this example is to further illustrate the behaviour of porous composites with heated impervious inclusions. The real and negative of the imaginary parts of the e ff ective compressibility of PCH3 are shown in Figures 4a and 4b. Contrary to the second example (see Figures 3a and 3b), the high frequency behaviour of the compressibility is not altered by the change in the temperature of the air inside the porous matrix. The e ff ects of increasing the temperature are more noticeable in from the low frequency regime up until the pressure di ff usion characteristic frequency. This is due to the altered e ff ective compressibility of the matrix due to the higher temperature. Since the temperature inside the inclusion is constant, the pressure di ff usion phenomenon is not altered, and thus the frequency location of the transition region and peak in Re { C } and − Im { C } , respectively, is not changed. This is further confirmed by Figure 4c, where it can be observed that the pressure di ff usion characteristic

(a) (b)

(c) (d)

Figure 3: (a) Real part and (b) negative of the imaginary part of the e ff ective compressibility ( C P 0 ) of PCH2 with the inclusion at di ff erent temperatures. (c) Characteristic frequencies of the material: Continuous red line: pressure di ff usion characteristic frequency; Dotted green line: Thermal characteristic frequency of the matrix material; Discontinuous blue line: Thermal characteristic frequency of the porous inclusion material. (d) Sound absorption coe ffi cient of PCH2 with the inclusion at di ff erent temperatures. For simplicity, the plot only shows the numerical results of the composite with a temperature of 500 ◦ C.

frequency is independent of the temperature inside the matrix. Finally, Figure 4d shows the sound absorption coe ffi cient of a 8-cm hard-backed layer of PCH3. The plot shows that the absorption peak is shifted towards a higher frequency when the temperature in the matrix is increased. Moreover, the absorption curve is flattened for higher temperatures, which is similar to the behaviour of the sound absorption coe ffi cient of PCH1 (see Figure 2f). Also, as was the case for the two previous examples, a good agreement is found between the theoretical predictions and the results of computational experiments.

4. CONCLUSIONS

Sound propagation through porous composites with heated impervious inclusions was investigated in this paper A general theory of acoustic wave propagation was developed through the two-scale asymptotic method of homogenisation. The theory was successfully validated by comparing theoretical predictions of sound absorption of composites with heated inclusions with the results of finite-element computational experiments. This theory showed that the temperature of the air saturating both the matrix and the inclusions of the material has a significant e ff ect on both the

(a) (b)

(c) (d)

Figure 4: (a) Real part and (b) negative of the imaginary part of the e ff ective compressibility ( C P 0 ) of PCH3 with the matrix at di ff erent temperatures. (c) Characteristic frequencies of the material: Continuous red line: pressure di ff usion characteristic frequency; Dotted green line: Thermal characteristic frequency of the matrix material; Discontinuous blue line: Thermal characteristic frequency of the porous inclusion material. (d) Sound absorption coe ffi cient of PCH3 with the matrix at di ff erent temperatures.

pressure di ff usion phenomenon as well as the absorptive properties of hard-backed layers of these composites. This in contrast with the behaviour of conventional porous media, where the e ff ect of temperature is only substantial around the thermal and viscous characteristic frequencies of the material, and does not significantly alter the low and high frequency asymptotic values of the e ff ective parameters. The main conclusion of this paper is that the developed theory correctly captures the acoustic behaviour of porous composites with heated impervious inclusions Moreover, the upscaled model also provides accurate predictions of the sound absorption coe ffi cient of such materials when compared with computational experiments. The theory clearly shows that the mechanisms responsible for sound dissipation in composites, i.e. classical visco-thermal dissipation and pressure di ff usion phenomena in the inclusions, are significantly dependent on the temperature of the air saturating the material, and that altering said temperature appears as an e ff ective way of tuning the sound absorption of composites. In summary, this paper provides a rigorous theoretical framework for sound propagation in composites with heated impervious inclusions, and the results presented here can be of use for the development on new materials with tunable acoustic properties.

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