An improved multimodal method for sound propagation in nonuniform lined ducts

Abstract

An efficient method is proposed for modeling time harmonic acoustic propagation in a nonuniform lined duct without flow. The lining impedance is axially segmented uniform, but varies circumferentially. The sound pressure is expanded in term of rigid duct modes and an additional function that carries the information about the impedance boundary. The rigid duct modes and the additional function are known a priori so that calculations of the true liner modes, which are difficult, are avoided. By matching the pressure and axial velocity at the interface between different uniform segments, scattering matrices are obtained for each individual segment; these are then combined to construct a global scattering matrix for multiple segments. The present method is an improvement of the multimodal propagation method, developed in a previous paper [Bi et al., J. Sound Vib. 289, 1091-1111 (2006)]. The radial rate of convergence is improved from O(n(-2)), where n is the radial mode indices, to O(n(-4)). It is numerically shown that using the present method, acoustic propagation in the nonuniform lined intake of an aeroengine can be calculated by a personal computer for dimensionless frequency K up to 80, approaching the third blade passing frequency of turbofan noise.

Improved multimodal admittance method in varying cross section waveguides

“…k 2 −1 is found negative (assuming of course that the truncation includes all the propagative modes), k −1 is imaginary, representing an evanescent part of the wavefield. As N is increased, it is more and more evanescent, as can be seen in the explicit expression obtained from (2.8) 10) which implies that k 2 −1 ∝ −N 2 for large N (for details, see [15]). Coupled mode equations (2.7) with the boundary mode (as well as equations (2.8) and (2.9)) have exactly the same form as the coupled mode equations without the boundary mode [6].…”

“…As any method used to solve the coupled mode equations, the multimodal admittance method has a rate of convergence with respect to the number of local transverse modes in the series expansion of the pressure at each location on the axis of the waveguide. In the literature, several propositions have been made to improve this rate of convergence [7][8][9][10][11]. Similar to the classical attachment mode used in structural mechanics [12][13][14], all these techniques use a boundary mode which is not a local transverse mode but that encapsulates the less convergent part of the series.…”

“…The choice of the supplementary (N + 1)th transverse function is the key point of our improved method. As in [7,10,11,15], it is chosen to absorb the less convergent part of the series representing the pressure, but the novelty is that we take it to be orthogonal to the first N local transverse modes. When projecting the wave equation on these (N + 1) transverse functions, this latter property ensures that the coupled mode equations take the following classical form: the identity matrix appears in front of the higher order differential terms, and more importantly the boundary mode appears as an evanescent mode.…”

“…It is based on an enriched modal expansion of the pressure, as in [7,10,11,15]. The pressure is projected on N + 1 transverse functions associated with N + 1 components.…”

Improved multimodal method for the acoustic propagation in waveguides with a wall impedance and a uniform flow

“…The presence of the flow seems to make the presence of these small-scale structures more penalizing. [41]). Nevertheless, it is of interest to consider the limit of vanishing transition region with ε → 0 (incidentally, note that realistic discontinuous admittances coincide rather to this limit).…”