Tuned optimization of extended reacting acoustic liners

A finite element Galerkin formulation was employed to study the optimum attenuation and reflection characteristics of acoustic waves propagating in tw0 dimensional straight ducts with extended reacting absorbing walls without a mean flow. The reflection and transmission of acoustic energy at the entrance and the exit of the duct were determined by coupling the finite element solutions in the absorbing portion of the duct to the eigenfunctions of an infinite, uniform, hard wall duct. In the frequency range where the duct height and acoustic wave length are nearly equal, power attenuation contours were examined to determine conditions for minimizing acoustic transmission through the duct. The extended reacting liners were found to significantly minimize the reflective characteristics of the duct rather than to increase absorption in the liner. Thus, extended reacting wall liner properties can be theoretically chosen to yield large impedance mismatches in an open straight duct

Acoustic propagation in curved ducts with extended reacting wall treatment

A finite-element Galerkin formulation was employed to study the attenuation of acoustic waves propagating in two-dimensional S-curved ducts with absorbing walls without a mean flow. The reflection and transmission at the entrance and the exit of a curved duct were determined by coupling the finite-element solutions in the curved duct to the eigenfunctions of an infinite, uniform, hard wall duct. In the frequency range where the duct height and acoustic wave length are nearly equal, the effects of duct length, curvature (duct offset) and absorber thickness were examined. For a given offset in the curved duct, the length of the S-duct was found to significantly affect both the absorptive and reflective characteristics of the duct. A means of reducing the number of elements in the absorber region was also presented. In addition, for a curved duct, power attenuation contours were examined to determine conditions for maximum acoustic power absorption. Again, wall curvature was found to significantly effect the optimization process

Galerkin finite difference Laplacian operators on isolated unstructured triangular meshes by linear combinations

The Galerkin weighted residual technique using linear triangular weight functions is employed to develop finite difference formulae in Cartesian coordinates for the Laplacian operator on isolated unstructured triangular grids. The weighted residual coefficients associated with the weak formulation of the Laplacian operator along with linear combinations of the residual equations are used to develop the algorithm. The algorithm was tested for a wide variety of unstructured meshes and found to give satisfactory results

Effect of surface deposits on electromagnetic waves propagating in uniform ducts

A finite-element Galerkin formulation was used to study the effect of material surface deposits on the reflective characteristics of straight uniform ducts with PEC (perfectly electric conducting) walls. Over a wide frequency range, the effect of both single and multiple surface deposits on the duct reflection coefficient were examined. The power reflection coefficient was found to be significantly increased by the addition of deposits on the wall

Discretization formulas for unstructured grids

The Galerkin weighted residual technique using linear triangular weight functions is employed to develop finite difference formula in cartesian coordinates for the Laplacian operator, first derivative operators and the function for unstructured triangular grids. The weighted residual coefficients associated with the weak formulation of the Laplacian operator are shown to agree with the Taylor series approach on a global average. In addition, a simple algorithm is presented to determine the Voronoi (finite difference) area of an unstructured grid

Modal element method for scattering of sound by absorbing bodies

The modal element method for acoustic scattering from 2-D body is presented. The body may be acoustically soft (absorbing) or hard (reflecting). The infinite computational region is divided into two subdomains - the bounded finite element domain, which is characterized by complicated geometry and/or variable material properties, and the surrounding unbounded homogeneous domain. The acoustic pressure field is represented approximately in the finite element domain by a finite element solution, and is represented analytically by an eigenfunction expansion in the homogeneous domain. The two solutions are coupled by the continuity of pressure and velocity across the interface between the two subdomains. Also, for hard bodies, a compact modal ring grid system is introduced for which computing requirements are drastically reduced. Analysis for 2-D scattering from solid and coated (acoustically treated) bodies is presented, and several simple numerical examples are discussed. In addition, criteria are presented for determining the number of modes to accurately resolve the scattered pressure field from a solid cylinder as a function of the frequency of the incoming wave and the radius of the cylinder

A finite element model for wave propagation in an inhomogeneous material including experimental validation

A finite element model was developed to solve for the acoustic pressure field in a nonhomogeneous region. The derivations from the governing equations assumed that the material properties could vary with position resulting in a nonhomogeneous variable property two-dimensional wave equation. This eliminated the necessity of finding the boundary conditions between the different materials. For a two media region consisting of part air (in the duct) and part bulk absorber (in the wall), a model was used to describe the bulk absorber properties in two directions. An experiment to verify the numerical theory was conducted in a rectangular duct with no flow and absorbing material mounted on one wall. Changes in the sound field, consisting of planar waves was measured on the wall opposite the absorbing material. As a function of distance along the duct, fairly good agreement was found in the standing wave pattern upstream of the absorber and in the decay of pressure level opposite the absorber

Parametric study of electromagnetic waves propagating in absorbing curved S ducts

A finite-element Galerkin formulation has been developed to study attenuation of transverse magnetic (TM) waves propagating in two-dimensional S-curved ducts with absorbing walls. In the frequency range where the duct diameter and electromagnetic wave length are nearly equal, the effect of duct length, curvature (duct offset), and absorber wall thickness was examined. For a given offset in the curved duct, the length of the S-duct was found to significantly affect both the absorptive and reflective characteristics of the duct. For a straight and a curved duct with perfect electric conductor terminations, power attenuation contours were examined to determine electromagnetic wall properties associated with maximum input signal absorption. Offset of the S-duct was found to significantly affect the value of the wall permittivity associated with the optimal attenuation of the incident electromagnetic wave

Finite element analysis of electromagnetic propagation in an absorbing wave guide

Wave guides play a significant role in microwave space communication systems. The attenuation per unit length of the guide depends on its construction and design frequency range. A finite element Galerkin formulation has been developed to study TM electromagnetic propagation in complex two-dimensional absorbing wave guides. The analysis models the electromagnetic absorptive characteristics of a general wave guide which could be used to determine wall losses or simulate resistive terminations fitted into the ends of a guide. It is believed that the general conclusions drawn by using this simpler two-dimensional geometry will be fundamentally the same for other geometries

Modal element method for potential flow in non-uniform ducts: Combining closed form analysis with CFD

An analytical procedure is presented, called the modal element method, that combines numerical grid based algorithms with eigenfunction expansions developed by separation of variables. A modal element method is presented for solving potential flow in a channel with two-dimensional cylindrical like obstacles. The infinite computational region is divided into three subdomains; the bounded finite element domain, which is characterized by the cylindrical obstacle and the surrounding unbounded uniform channel entrance and exit domains. The velocity potential is represented approximately in the grid based domain by a finite element solution and is represented analytically by an eigenfunction expansion in the uniform semi-infinite entrance and exit domains. The calculated flow fields are in excellent agreement with exact analytical solutions. By eliminating the grid surrounding the obstacle, the modal element method reduces the numerical grid size, employs a more precise far field boundary condition, as well as giving theoretical insight to the interaction of the obstacle with the mean flow. Although the analysis focuses on a specific geometry, the formulation is general and can be applied to a variety of problems as seen by a comparison to companion theories in aeroacoustics and electromagnetics