Calculations of modes in circumferentially nonuniform lined ducts

Abstract

This paper proposes a computationally efficient method of determining eigenfunctions and eigenvalues of acoustic modes propagating in circular lined ducts with zero or uniform flow. Linings with circumferentially nonuniform impedance, as found in nacelle acoustics, are the focus. The method of solution adapts in two important respects--the presence of flow and the imposition of impedance boundary conditions--the series expansion method first proposed by Pagneux et al. [J. Acoust. Soc. Am. 110, 1307-1314 (2001)] to calculate the eigenvalues of Lamb modes. The inclusion of flow, and a corresponding different method of solution leading to an improved convergence of eigenvalues (O(N(-5)), N is the truncation of radial basis of expansion), is the important new feature as compared to the previous adaptation by Bi et al. [J. Acoust. Soc. Am. 122, 280-291 (2007)]. As a result, it becomes possible to safely determine and in a simple manner the eigenvalues and eigenfunctions of circumferentially nonuniform lined ducts in the presence of uniform flow, up to relatively high frequencies (e.g., 30<kR<80). All required eigenvalues are calculated at a time and no initial values guesses are needed. Results are presented and the method is discussed.

Eigenmodes of lined flow ducts with rigid splices

“…The infinite azimuthal Fourier sum that is numerically truncated at N modes is shown in §5.2 to lead to an eigenvalue error that decays exponentially for increasing N . The comparable numerical scheme of Bi (2008) involves a doubly infinite sum of radial and azimuthal modes, which, if both are truncated at the same limit N 1/2 (as suggested by Bi for azimuthally nonuniform ducts) leads to N terms in the truncated sum and eigenvalue errors that decay at the rate O(N 5/2 ) (see Bi 2008, figure 3). The finite-element calculation of duct modes performed by Wright (2006) was reported to have taken approximately 19 minutes.…”

“…Both of these studies posed an expansion of the eigenfunction as a sum of solutions of the Helmholtz equation for the duct but, unlike the multimodal expansion of Pagneux et al (1996), these solutions were not required to satisfy the boundary conditions but were required to all have the same axial wavenumber; a similar technique, although derived in a different manner using a Green's function approach, is used here, and results in a singly infinite (Fourier) modal sum. In contrast, Bi (2008) tackled the eigenmode problem in an infinite spliced lined duct with uniform flow using a type of spectral method consisting of a modal decomposition in terms of hard-walled duct modes augmented by a pressure-release duct mode to aid faster convergence, leading to a doubly infinite modal sum. The spliced eigenmode problem has also been considered numerically by Wright (2006) and Gabard & Astley (2008), who both used two-dimensional finite elements to numerically calculate the eigenmodes within the lined section of a duct and then match these to the rigidwalled modes of the infinite rigid duct at either end of the lined section; both of these studies used a uniform mean flow.…”

Computation of dispersion relations for axially symmetric guided waves in cylindrical structures by means of a spectral decomposition method

“…The method was adapted for use in aeroacoustic applications and extended to bends [33] and lined ducts [34][35][36]. In [37] the calculation of wave numbers in nonuniform lined ducts was adapted to the presence of uniform flow and subsequently improved with respect to the convergence rate of the eigenvalues while in [38] the convergence of pressure fields calculated in two-dimensional or axially symmetric geometries has been improved by supplementing the base of Laplace operator eigenfunctions by a boundary mode designed to encapsulate the less convergent part of the series. To the best of our knowledge no attempts have been made so far to extend the use of the spectral decomposition method to the calculation of dispersion relations in solid waveguides other than plates.…”

Discontinuous Galerkin method for the computation of acoustic modes in lined flow ducts with rigid splices

“…In addition, small areas of the lining may be replaced by hard patches for maintenance reasons. These splices may have a significant effect on the radiated noise and therefore the infinite duct model presenting lining non-uniformities or discontinuities has been already the object of many studies [2][3][4][5][6] by means of various methods (analytical or numerical). Except in specific cases with thin splices (see [6]), the effect of the splices is generally to reduce the damping, as it would be guessed since the lined area is smaller.…”

“…The previous investigations of circumferentially non-uniformly lined duct have been conducted by means of finite element method with [4] and without flow [2] or by means of multimodal method [3,5] in the presence of uniform flow. In these studies, the equations are written under the Pridmore-Brown form and therefore the only considered variable is the acoustic pressure.…”