Fighting noise pollution is a high order importance for modern societies, especially with the development of the aeronautics, automobile and railway industries. To minimize the noise generated in systems such as aircraft turbojet engines, fans, compressors and automobile exhaust mufflers, several studies have been carried out in recent years to identify noise sources and accurately model the propagation environments of acoustic waves. In most of these systems, before radiating outwards, the noise is channeled by guides of more or less complex geometries (elbows, change of section, presence of obstacles) which can contain fluids whose movement can be more or less rapid depending on the applications (ventilation duct, exhaust lines) and more or less complex (turbulent, laminar, pulsed, etc.). In order to avoid certain phenomena linked to end effects (reflection, radiation at the exit of the guides), the guides are often considered infinite.

In this study, our interest is the acoustic propagation in infinite axisymmetric cylindrical lined guides without flow. The acoustic sources are assumed to be known and we are interested in their radiation in guided environments, neglecting the coupling between acoustic and aerodynamic phenomena which rather concerns the Lighthill analogy [1] [2] . The study of acoustic propagation in guides is an open subject on which several works have been carried out. In most of these works [3] [4] [5] , 2D straight guides with or without acoustic treatment on the walls are been considered. Acoustic propagation in these guides is governed by the Helmholtz wave equation. We adopted this equation here because our method of numerical resolution of acoustic propagation problems is based on a modal representation with the acoustic pressure as a scalar unknown. Furthermore, a significant part of our work is devoted to the study of acoustic wave attenuation and absorption. The absorbent material taken into account in this study is of complex local impedance type which is introduced into the wall boundary conditions.

Studying the acoustic propagation in an infinite guide by the finite element method requires truncating the infinite domain by artificial boundaries on which non-reflection conditions must be written. Thus, the most important part of our work concerns the implementation of transparent boundary conditions. A technique often used is the boundary element method (BEM). However, the BEM method requires a Green’s function and is better suited for wave propagation problems in infinite space and a homogeneous medium. Another transparent boundary condition called perfectly matched layers (PML) was developed by Béranger [6] for transient problems in electromagnetism. However, PML can be unstable in certain, cases particularly in the presence of inverse modes. On the other hand, PML cannot be produced from physical absorption models and the solution obtained in the layers has no physical reality. In ducted propagation, for a solution using the finite element method, an efficient transparent boundary condition is to consider a boundary condition resulting from a development based on the guide eigenfunctions. The transparent boundary condition of the Dirichlet-to-Neuman (DtN) operator type [7] [8] [9] [10] [11] fitted into this perspective. It’s an operator based on a modal decomposition of the acoustic pressure. For a rigid wall duct, the transverse modes are orthogonal in the sense of the usual scalar product; it is then easy to explain the DtN operator. For lined guides, some difficulties occurred. Redon et al. [12] in their study for plane problems showed that the eigenvalue problem is no longer self-adjoint for a lined guide even without flow. The DtN operator has been nevertheless established by exploiting the biorthogonality relation existing between the modes [13] . In this study, their works are generalized to explain the DtN operator for axisymmetric problems and we studied the convergence error as a function of the number of modes. Searching modes allowed us in particular to examine the dispersion curves and asymptotic behavior of these modes. It also helped us to establish an orthogonality relation between modes, essential for writing the DtN operator as modal decomposition which the determination of several coefficients (modal and normalization coefficients of transverse modes) is required.

In the following, the problem statement including the equation of acoustic propagation and the appropriate boundary conditions is presented in Section 2. The eigenvalue problem is described in Section 3 including the asymptotic behavior of the modes and their attenuation in the guide. The orthogonality relation between modes is established in Section 4. Then in Section 5, the essential steps leading to implementing the DtN operator for the lined guide are explained and the main numerical results of this study are shown in Section 6.

The space domain is an infinite axisymmetric cylindrical guide with radius R and axis z represented in Figure 1 such that Ω ∞ = { − ∞ < z < + ∞ , 0 < r < R } . Its wall Г_Z (r = R) is covered with an absorbent material characterized by an impedance Z ∈ ℂ . The equation of the acoustic propagation in the domain is established from the equations of fluid mechanics in the general framework of the thermodynamics of continuous media. We adopted the classic hypotheses of linear acoustics. Viscous constraints and thermal conduction phenomena are assumed to be negligible from an acoustic point of view. All the physical quantities used are functions of Euler variables. The contained fluid in the duct was at rest.

In cylindrical coordinates, the equation of acoustic propagation is given by:

Δ p − 1 c 0 2 d 2 p d t 2 = 0 (1)

where p designates the acoustic pressure, c₀ is sound celerity. The operator ∆ is the Laplacian in cylindrical coordinates. The boundary condition on the duct axis Г₀ is a homogeneous Neumann condition [14] :

∂ p ∂ n = 0 (2)

The boundary condition at the wall Г_Z is related to the impedance Z by:

∂ p ∂ n = i k p Z (3)

For a harmonic regime ( e − i ω t ) without flow, Equation (1) becomes:

∂ 2 p ∂ r 2 + 1 r ∂ p ∂ r + 1 r ² ∂ 2 p ∂ θ 2 + ∂ 2 p ∂ z 2 + k 2 p = 0 (4)

With k = ω/c₀ the wave number. A search for the solution to Equation (4) by the variable separation method while taking into account a convergent solution on the duct axis (r = 0) led us to a general solution of the form:

p ( r , θ , z ) = a A z J m ( α r ) ( A θ e i m θ + B θ e − i m θ ) e i β z (5)

where a , A z , A θ and B θ are modal coefficients. J m is a Bessel function of m order, α is a wave number and β is the propagation constant along the z axis. α and β verify the following dispersion equation:

α 2 = k 2 − β 2 (6)

Solution of Equation (5) being a harmonic function of the azimuthal variable θ with m the angular mode order and the guide presenting a symmetry of revolution, then the study can therefore be reduced to the plane (r, z). So, the function J m can be limited to the 0 order, φ ( r ) = a J 0 ( α 0 n r ) . In order to solve this problem by the finite element method, it is necessary to truncate the infinite domain. We therefore chose two artificial boundaries (Σ− and Σ+) on which transparent boundary conditions should be imposed (see Figure 2).

The boundary conditions on Σ− and Σ+ are both given by:

∂ p ∂ n = − T Z ± ( p ) (7)

where T Z ± ( p ) represents the DtN operators. To determine these DtN operators the perfect knowledge of the transverse modes of the lined guide and therefore the eigenvalue problem and all essential coefficients are required. This task is carried out in Sections 3 and 4.

In this section, we are looking for the acoustic transverse modes of the guide. As shown previously, considering only axisymmetric solutions (i.e. m = 0, p ( r , z ) = φ ( r ) e i β z ), the transverse modes φ ( r ) are solutions of:

d 2 φ d r 2 + 1 r d φ d r + ( k 2 − β 2 ) φ = 0           Ω (8)

d φ d r = i k φ Z           Γ Z (9)

d φ d r = 0           Γ 0 (10)

where φ ( r ) = a J 0 ( α r ) . The transverse wave numbers α are solutions of the following transcendental equation which depends on the impedance Z:

− α J ′ 0 ( α R ) = i k J 0 ( α R ) Z → − α n J 1 ( α n R ) J 0 ( α n R ) = i k Z (11)

Note: Although the impedance Z of materials is often a function of frequency in physical problems, we considered it constant for simplification. In this study, we determined α n numerically by the Newton-Raphson algorithm. However, the Newton-Raphson method requires initial values of α n which are calculated by the finite element method applied in the cross section of the guide. The propagation constants β n are determined using the following dispersion equation: α n 2 + β n 2 = k 2 . Figure 3 shows the curves of β n in the complex plane without flow for k = 5 respectively for a rigid guide and a guide with an absorbing wall of impedance Z = 1.5 − 1.5i. There are two situations for the rigid guide:

· If k 2 > α n 2 , the propagation constant β n is pure real (see Figure 3(a)):

β n = ± k 2 − α n 2 (12)

The mode of order n is propagative in the increasing z direction: β n = β n + (respectively in the decreasing z direction: β n = β n − ). In particular, there is a finite number of propagative modes and the firt mode such as: β 1 = k is always propagative.

· If k 2 < α n 2 , then the propagation constant is pure imaginary (see Figure 3(a)):

β n = ± i α n 2 − k 2 (13)

In this case, the mode of order n for β n = β n + (respectively for β n = β n − ) is evanescent in the increasing z direction (in the decreasing z direction). There are

an infinite number of evanescent modes. For k = α n , we define the cut-off frequency of the mode of order n (n > 1) by: f c n = α n c 0 2 π .

For the lined guide, the eigenvalues α n are complex (see Table 1), all modes are attenuated. However for each wave number k (here k = 5), the imaginary part of β n became higher than its real part beyond a certain order n (here n = 2). The mode is then strongly attenuated. This corresponds to a pseudo-cutting phenomenon (Figure 3). The propagative modes and the evanescent modes respectively present a weak imaginary part and a weak real part unlike the rigid guide case where β n was either pure real or pure imaginary.

Finally, it should be noted that for both hard-wall guide and the lined guide without flow, the eigenvalues α n are identical in the two directions of propagation (i.e. α n + = α n − = α n ), when the propagation constants β n are opposite ( β n + = − β n − )

as shown in Figure 3.

Knowing the impact of the absorbent on the propagative modes (first modes) and the higher order modes (evanescent modes), an orthogonality relationship between these modes must be defined. Indeed, to write a transparent boundary condition based on a DtN operator, it is necessary to calculate the modal coefficients appearing in the modal decomposition of the acoustic pressure. This is done using an orthogonality relation. This relationship is classic in the case of a rigid guide since the eigenvalue problem is self-adjoint. With absorbent, a different situation arises because the boundary condition on the wall takes into account the impedance Z. In their work for a rectangular guide, Redon et al. [12] showed that the modes are no longer orthogonal in the sense of the usual scalar product because the eigenvalue problem is no longer self-adjoint [14] . To be able to determine our modal coefficients, we use the biorthogonality relation existing between the modes for non-self-adjoint problems [13] . It is defined by:

( φ n , φ m ) Σ * = ∫ Σ φ n φ m r d r = 0       for     m ≠ n (17)

where φ n ( r ) is a transverse mode of order n, φ n ( r ) = a n J 0 ( α n r ) . Its norm is calculated according to the relation in Equation (17). According to Abramowitz and Stegun [16] , it is given by:

( φ n , φ n ) Σ * = ∫ 0 R φ n φ n r d r = 1 = a n 2 R 2 2 [ J 0 2 ( α n R ) + J 1 2 ( α n R ) ] (18)

The normalization coefficients a n are therefore given by:

a n = 1 R 2 J 0 2 ( α n R ) + J 1 2 ( α n R ) (19)

This normalization is only possible if:

J 0 2 ( α n R ) + J 1 2 ( α n R ) ≠ 0 (20)

The discrete values α n c roots of Equation (20) are associated with critical values of the impedance Z n c which had already been discussed in the 2D case [17] . According to Equation (11), we define a function:

S ( α n ) = α n J 1 ( α n R ) J 0 ( α n R ) + i k Z (21)

Having as derivative: S ′ ( α n ) = α n R J 0 2 ( α n R ) [ J 0 2 ( α n R ) + J 1 2 ( α n R ) ] . This implies that S ( α n c ) = S ′ ( α n c ) = 0 . The critical eigenvalues α n c are therefore double

roots of Equations (11) and (20). To determine the critical impedances for a given wave number k, we first calculate the roots of Equation (20) and deduce Z n c using the transcendental Equation (11). The search for critical impedances is more complicated here than the case of a 2D rectangular guide, because the modes are expressed by Bessel functions. The Bessel functions J₀ and J₁ being continuous on ℝ, we take h_c a complex function and α n R = x + i y . We must therefore resolve:

h c ( x , y ) = J 0 2 ( x + i y ) + J 1 2 ( x + i y ) = 0 (22)

The functions J₀ and J₁ are known even and odd respectively, but their squares are even. This means that the sum of the squares of J₀ and J₁ is also even. So, if the couple (x, y) is a solution of Equation (22), then the couples (−x, y), (x, −y) and (−x, −y) are also solutions of Equation (22). The study can therefore be limited to x ≥ 0 and y ≥ 0. To determine the zeros of Equation (22), we chose x and y such that the function h_c is close to 0. These values of x and y are then refined by the Newton-Raphson method to find the exact values for which the function h_c vanishes. The critical impedances Z n c are then calculated using the transcendental relation in Equation (11):

Z n c = − i k J 0 ( α n c R ) α n c J 1 ( α n c R ) (23)

Table 2 indicates the real and imaginary parts of α n c and the critical impedances calculated for a wave number k = 5 and for the following intervals of x and y chosen so that the function h_c is close to 0: x ∈ [ 0 , 25 ] and y ∈ [ 0 , 5 ] . For these intervals of x and y, we obtained seven decreasing critical impedances (see Figure 9).

These impedance values make impossible the normalization of the transverse modes φ n , so Z is chosen in such we can be able to calculate the coefficients α n and subsequently be able to determine the operators DtN.

As seen previously in Figure 2, to allow discretization by the finite element method, the domain is truncated by artificial boundaries Σ− and Σ+. The boundary conditions on these boundaries are given by the DtN operators already mentioned

in Equation (7): ∂ p ∂ n = − T Z ± ( p ) . The DtN operator being based on a modal

decomposition of the acoustic pressure on the boundaries Σ± which truncate the physical domain, the biorthogonality relation in Equation (17) which made it possible to normalize the modes of the guide will also make it possible to determine the modal coefficients resulting of this decomposition. In front of Σ+, we got this:

∀ z ≥ L ,     p ( r , z ) = ∑ n = 0 + ∞     A n + φ n ( r ) e i β n + ( z − L ) (24)

In particular on z = L, it became:

p ( r , z = L ) = ∑ n = 0 + ∞     A n + φ n ( r ) ;     0 ≤ r ≤ R (25)

A product of p and a transverse mode φ n ( r ) using the biorthogonality relation Equation (17), led to the expressions of the modal coefficients A n + :

A n + = ( p , φ n ) Σ * (26)

By replacing A n + in Equation (25) and making a derivation about z, we deduced the operator DtN on Σ+:

∂ p ∂ z ( r , L ) = − T Z + ( p ) = ∑ n = 0 + ∞     i β n + ( p , φ n ) Σ * φ n ( r ) → T Z + ( p ) = − ∑ n = 0 + ∞     i β n + ( p , φ n ) Σ * φ n ( r ) (27)

By similar development, we found on Σ−:

T Z − ( p ) = ∑ n = 0 + ∞     i β n + ( p , φ n ) Σ * φ n ( r ) (28)

The determination of the DtN operators completes all the boundary conditions of the equation of the acoustic propagation in the lined guide. We can therefore carry out numerical simulations for some practical examples.

In the following, numerical results obtained by the finite element method of acoustic propagation problems in an axisymmetric cylindrical lined guide, will be presented. The absorbing material is taken to be a complex impedance Z = 1.5 (1 − i) and the fluid is at rest.

In this study, we proposed a new method for the modeling by the finite element method of a transparent boundary condition in acoustic propagation problems for infinite axisymmetric cylindrical guides with a wall lined with a locally reacting absorbent material. The transparent boundary condition is carried out with DtN operators which are determined from a decomposition of the acoustic pressure on the guide eigemodes. For a rigid wall, the DtN operator is easy to carry out, but became complex for liner guides. In the case of a lined guide without flow, the modes are no longer orthogonal in the sense of the usual scalar product but there exists a biorthogonality relation which makes it possible to compute the DtN operators. The method developed in this work is successfully applied to a straight axisymmetric lined guide by imposing a mode on one of the artificial boundaries of the truncated guide. The results are in good agreement with the analytical solutions. We then applied the method for a non-uniform axisymmetric lined guide without flow. For this complex case, the method has proven its effectiveness and the results compared to those of PML layers are in very good agreement.

The authors declare no conflicts of interest regarding the publication of this paper.

Ouedraogo, B. and Redon, E. (2024) Study of Axisymmetric Infinite Guide Lined with Locally Reacting Material without Flow Using DtN Operators. Open Journal of Applied Sciences, 14, 572-588. https://doi.org/10.4236/ojapps.2024.142041