Mathematical Modeling of Porous Medium for Sound Absorption Simulations II: Wave Propagation and Interface Conditions

The application of porous medium has a myriad of applications in different industries: automotive, aerospace, civil (commercial, residential), environmental noise control, and biomedical. In the past, design questions involving porous material were addressed with seat-of-the-pants decisions that led to multiple/iterative prototypes and experiments that were costly and time consuming. The objective, in this series of publications pertaining to porous medium, is to establish tools that will lead to effective and accurate simulations involving porous medium. In this third installment of this series the focus is on establishing the constitutive equations using tensors and then applying Transfer Matrix Method (TMM) to calculate diffuse field Transmission Loss (TL) across structures that comprises of layers of different porous medium. The constitutive equations are obtained by relating information regarding the micro-structure make-up to macro level properties. In order to apply the TMM, the equations for wave propagation across different mediums need to be developed and in turn represent these propagation properties in a matrix format. Additionally, the boundary condition between each layer type is defined in order to ensure numerical stability. The author’s current research effort is running simulations for the automotive industry to predict NVH environments. Therefore, TL calculations pertaining to the materials that are utilized in the interior of automobiles are used, in this paper, as a test bed for the developed analytical tools. Case in point, the TL for a multi-layered material consisting of one panel and two different layers of foam is calculated and compared to experimental data. Future publication goals will be to apply these tools in the biomedical field; an example will be to model and run siHow to cite this paper: Teagle-Hernandez, A., Ohtmer, O. and Nguyen, D. (2019) Mathematical Modeling of Porous Medium for Sound Absorption Simulations II: Wave Propagation and Interface Conditions. Journal of Applied Mathematics and Physics, 7, 2780-2795. https://doi.org/10.4236/jamp.2019.711191 Received: October 14, 2019 Accepted: November 10, 2019 Published: November 13, 2019 A. Teagle-Hernandez et al. DOI: 10.4236/jamp.2019.711191 2781 Journal of Applied Mathematics and Physics mulations of different organs like the liver and lungs that are porous in nature.


Introduction
This paper is the third installment from a series of publications pertaining to modeling of porous medium. The first paper, Teagle et al. [1], derived a coupled set of fluid/structure equations for a porous medium applying asymptotic and homogenization techniques. In [1] it is shown that there are mainly 3 modes of energy transformation: 1) The first mode is through the connection between the micro and macro structural framework of the porous skeleton, 2) The second is via the viscous boundary layer, and 3) The third interaction is through thermal (entropy) boundary layer. The combination of the viscous boundary layer and how tortuous the porous material is, results in the encapsulation of the fluid medium. This, in turn, changes the apparent mass of the structural medium. Details pertaining to these encapsulating phenomena can be found in the work by Johnson, et al. [2]. In [2] the concepts of tortuosity, viscous length, and viscous permeability are explained. The thermal interchange is described in Teagle, et al. [3]. In [3], the mathematical description of the thermal energy dissipated by the thermal boundary layer is explained along with the relaxation process. This thermal exchange changes the acoustic bulk modulus of the porous medium and thus the speed of sound inside the porous layer. These thermal phenomena are then represented by the parameters of thermal length and thermal permeability. This paper is a continuation of [1]. The equations of motion derived in the aforementioned publication will be presented in a form that is used for calculation of TL for porous material used in the automobile industry. The combination of 1) the change of density due to the viscous effect and 2) the change of acoustic bulk modulus due the thermal exchange will be applied. Additionally, a matrix representation of the wave propagation (for both forward and reflected propagating wave) that incorporates the aforementioned viscous and thermal effects is developed. Each porous layer makes contact with another type of porous medium, elastic panel, or air. This study will establish the correct boundary conditions between mediums in order to run numerical simulations that lead to stable and unique solutions. In order to translate these boundary conditions to the numerical model, interface matrices are developed. In continuation, the global matrix that represents the multi-layered material is assembled and applying the definition of impedance the diffuse field TL is calculated. In the TMM formulation it is assumed that each layer is of infinite extent. A correction applying Green's function technique is applied.

Basic Tensor Calculus and Notation
Tensor calculus and concepts from differential geometry are used extensively in this paper. This notation affords a level of abstraction that leads to an efficient explanation of the stresses, strains, and their relationships. Here, the usual notation ( ) h are scaling factors that satisfy the following is the components of the metric tensor ij g . The Einstein summation convention is used in which covariant index followed by the identical contravariant index is implicitly summed over, thus contracting the order of the tensor. In this paper, if 2 tensors of different order are shown next to each other, this contraction rule is followed. To obtain expression for strains, the gradient of the first term of the asymptotic expansion of the displacement vector, ( ) o u x , is required. The expression for the components of the tensor of order 2 is the following and the expression for strain is given as ( ) The elastic constant, Thus, the physical component of the force field in the jth direction is given by The use of orthogonal coordinates results in A.

Fluid-Structure Interaction: Dynamic Equations
This paper applies the definitions and results obtained in [1]. In that publication, the fluid/structure interaction equations are derived via asymptotic and homogenization techniques. A slight modification is applied to the averaged relative displacement of the fluid with respect to the skeleton, ( ) w x . In this paper this equation is represented as In [1], Equations (54) and (55) are derived and represent the structural stress, o τ  , and internal fluid pressure, o p . These are rewritten below as Equations (9) and (10) [ ] [ ] where , bi ii C C and ib C are elastic constant tensor (fourth rank) per their explanations given in [1].
κ is the interstitial fluid bulk modulus, details on its derivation can be found in Teagle et al. [4]. Additionally, in [4] Subtracting Equation (13) from (12c) and substituting the definition of ( ) , w x y , Equation (8), the following expression for the structural portion is ob- Setting and applying the definition for 12 ρ , an efficient representation of Equation (13) is ( ) Substituting for o p , in Equation (9), the expression in (10), and in turn taking the divergence the dynamical equation for the structural portion of the foam can be expressed as In order to derive (17), the definition for β , Equation (12b), was used. When 0 M = , it is easy to verify that λ and μ are new Lame' constants of the elastic portion when the porous material is drained.
Additionally, taking the gradient of Equation (10) followed by a multiplication by -ϕ , the following expression dealing with pressure is obtained Combining Equations (14) and (17), the dynamic equations pertaining to the structural/skeleton portion of the porous medium is obtained Similarly, combining Equations (16) and (18) and by applying the coupling definition of Q, Equation (20) becomes ( ) A.

Wave Participation Factor: Eigenvalue Problem
In this section, the study adapts the concepts from Brouard, et al. [5]. The motion of the structural/skeleton and the fluid portion are described by introducing the potential scalar functions φ and Ψ and the potential vector functions, G and H , such that Here, the basic definition from mechanics is used where the gradient represent portion of the displacement vector that is purely dilatational or in compression.
The curl of H and G represent any shear motion associated with the displacement. Substitute these definitions into Equations (21) and (22) [ Recognize that the ∇ operator commutes with 2 ∇ i.e.
for any general vector ζ .
Using these relationships, Equations (25) and (26) Gathering terms corresponding to the gradient operator results in These equations will give the formulation to solve for the compression waves.
The eigenvalues produce the complex wave number for the compression waves and they will have the following expression Two sets of eigenvectors are generated known as the participation factor. Substituting the eigenvectors into Equation (31) ones gets These results indicate that there exist two compression waves traveling the porous medium, one is fast and the other one is slower. i µ is important since the number will indicate which wave, whether acoustic or solid, has the most contribution at that particular frequency. For the shear wave we accumulate all terms in Equation (27) and (28)

Wave Propagation: Porous Medium
This portion will analyze wave propagation in a semi infinite porous medium.
The disturbance wave is traveling down at an angle of incidence of θ in the general xz plane and will impinge the porous medium at the z = 0 level. A portion of this wave will be reflected and some of it will be transmitted to travel through the porous medium until it impinges the next layer, at z = L, of different impedance. A portion of the wave will again be reflected back into the porous medium and the rest will be transmitted to the next layer.
Since there exist 3 wave types, 2 compression (dilatation) waves and 1 shear wave, and since each wave type has 2 waves (forward moving and the reflected wave) there will be 6 variables. These variables are: The continuity of these variables at the interface between 2 layers will become the boundary conditions that the traveling wave has to satisfy. Using the definition of the participation factor above, the potential functions can be expressed as and Figure 1. Porous medium element: Thickness L, angle of incidence θ.
The six variables can be written as a function of the potential functions and using the above notation: The following three equations are essential to define the stress components Applying the corresponding expressions in (44)

T A A A A A A A A A A A
The idea behind a transfer matrix is to relate the 6 variable, V(z), at z = L to the conditions at z = 0, i.e. relate the A matrix with respect V(L) The conditions at z = 0 can be written  is known as the transfer matrix (58) Deresiewicz, et al. [6], under the context and language set forth in this paper.

Interface Conditions: Uniqueness
The kinetic energy, per unit volume for the two phase system is expressed as 12 22 ρ , 12 ρ , and 22 ρ are 11 ρ , 12 ρ , 22 ρ void the portion with tortuosity (without viscous effect). The dynamic equations in Equation (17)  : : The following tensorial relationship holds Combining Equations (61, 62) and the divergence theorem the expression for power is obtained The portion inside the volume integral represents the traction and inertial work applied to the skeleton and acoustic medium respectively. Equations (21) and (22) are written in shorthand form term is the viscous force term due to the interstitial fluid, Β is a viscous transfer function replicating the viscous effects. When (64) In the common portion c S , For the non-intersecting boundaries (S 1 and S 2 ) the boundary conditions will have to be given. At the S c boundary, continuity is required across the interface to be able to maintain uniqueness. If the skeleton phase is to remain in contact with each other and to maintain the principle of conservation of mass of the acoustic medium it is continuity of the normal relative velocity of the acoustic medium with respect to the skeleton, i.e.
The continuity of n n u w +   was applied to obtain Equation (75).

Results and Conclusion
A Simple application of TMM is applied to the simple layer configuration shown in Figure 2. The figure shows a multi layer system where there is a plate that is glued to foam 1 which in turn is connected to foam 2. The parameters pertaining to plate 1, foam 1, and foam 2 are listed in Table 1. This simulation also applied the following parameters that are not listed in the Equations (9) and (10)    Analyzing Equations (17) and (18)   ("meas") results and also to the empirical formulas of Delaney and Bazley ("DB") [10] [11]. The graph shows that the calculated results come within 0.7 dB.