A mixed finite element model has been derived for the acoustic analysis of perforated dissipative silencers including several effects simultaneously: (1) High temperature and thermal gradients in the central duct and the outer absorbent material; (2) A perforated passage carrying non-uniform axial mean flow. For such a combination, the properties of sound propagation media and flow are inhomogeneous and vary with position. The material of the outer chamber can be modelled by its complex equivalent acoustic properties, which completely determine the propagation of sound waves in the air contained in the absorbent medium. Temperature gradients introduce variations in these properties that can be evaluated through a heterogeneous temperature-dependent resistivity in combination with material models obtained at room temperature. A pressure-based wave equation for stationary medium is then used with the equivalent density and speed of sound of the absorbent material varying as functions of the spatial coordinates. Regarding the central air passage, a wave equation in terms of acoustic velocity potential can be used to model the non-uniform moving medium since the presence of temperature variations introduce not only heterogeneous acoustic properties of the air but also a gradient in the mean flow velocity. The acoustic connection between the central passage and the outer chamber is given by the acoustic impedance of the perforated duct. This impedance depends on the heterogeneous properties of the absorbent material and the non-uniform mean flow, leading to a spatial variation of the acoustic coupling and also to additional convective terms in the governing equations. The results presented show the influence of temperature, thermal gradients and mean flow on the transmission loss of automotive silencers. It has been found that high temperature and thermal-induced heterogeneity can have a significant influence on the acoustic attenuation of an automotive silencer and so should be included in theoretical models. In some particular configurations it may be relatively accurate to approximate the temperature field by using a uniform profile with an average value, specially for low resistivity materials. It has been shown, however, that this is not always possible and attenuation overestimation is likely to be predicted, mainly for high radial thermal gradients and high material flow resistivities, if the temperature distribution is not taken into account.

A finite element approach is proposed for the acoustic analysis of automotive silencers including a perforated duct with uniform axial mean flow and an outer chamber with heterogeneous absorbent material. This material can be characterized by means of its equivalent acoustic properties, considered coordinate-dependent via the introduction of a heterogeneous bulk density, and the corresponding material airflow resistivity variations. An approach has been implemented to solve the pressure wave equation for a nonmoving heterogeneous medium, associated with the problem of sound propagation in the outer chamber. On the other hand, the governing equation in the central duct has been solved in terms of the acoustic velocity potential considering the presence of a moving medium. The coupling between both regions and the corresponding acoustic fields has been carried out by means of a perforated duct and its acoustic impedance, adapted here to include absorbent material heterogeneities and mean flow effects simultaneously. It has been found that bulk density heterogeneities have a considerable influence on the silencer transmission loss.

This work presents a mathematical approach based on the point collocation technique to compute the transmission loss of perforated dissipative silencers with transversal temperature gradients and mean flow. Three-dimensional wave propagation is considered in silencer geometries with arbitrary, but axially uniform, cross section. To reduce the computational requirements of a full multidimensional finite element calculation, a method is developed combining axial and transversal solutions of the wave equation. First, the finite element method is employed in a two-dimensional problem to extract the eigenvalues and associated eigenvectors for the silencer cross section. Mean flow as well as transversal temperature gradients and the corresponding thermal-induced material heterogeneities are included in the model. In addition, an axially uniform temperature field is taken into account, its value being the inlet/outlet average. A point collocation technique is then used to match the acoustic fields (pressure and axial acoustic velocity) at the geometric discontinuities between the silencer chamber and the inlet and outlet pipes. Transmission loss predictions are compared favorably with a general three-dimensional finite element approach, offering a reduction in the computational effort.

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This work presents an efficient numerical approach based on the combination of the mode matching technique and the finite element method (FEM) to model the sound propagation in silencers containing granular material and to evaluate their acoustic performance through the computation of transmission loss (TL). The methodology takes into account the presence of three-dimensional (3D) waves and the corresponding higher order modes, while reducing the computational expenditure of a full 3D FE calculation. First, the wavenumbers and transversal pressure modes associated with the silencer cross section are obtained by means of a two-dimensional FE eigenvalue problem, which allows the consideration of arbitrary transversal geometries and material heterogeneities. The numerical approach considers the possibility of using different filling levels of granular material, giving rise to cross sections with abrupt changes of properties located not only in the usual central perforated passage, but also in the transition between air and material, that involves a significant change in porosity. After solving the eigenvalue problem, the acoustic fields (acoustic pressure and axial velocity) are coupled at geometric discontinuities between ducts through the compatibility conditions to obtain the complete solution of the wave equation and the acoustic performance (TL). The granular material is analysed as a potential alternative to the traditional dissipative silencers incorporating fibrous absorbent materials.Sound propagation in granular materials can be modelled through acoustic equivalent properties, such as complex and frequency dependent density and speed of sound. TL results computed by means of the numerical approach proposed here show good agreement with full 3D FE calculations and experimental measurements. As expected, the numerical mode matching outperforms the computational expenditure of the full 3D FE approach. Different configurations have been studied to determine the influence on the TL of several parameters such as the size of the material grains, the filling level of the chamber, the granular material porosity and the geometry of the silencer cross section.