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

Abstract

We present an efficient multimodal method to describe the acoustic propagation in the presence of a uniform flow in a waveguide with locally a wall impedance treatment. The method relies on a variational formulation of the problem, which allows to derive a multimodal formulation within a rigorous mathematical framework, notably to properly account for the boundary conditions on the walls (being locally the Myers condition and the Neumann condition otherwise). Also, the method uses an enriched basis with respect to the usual cosine basis, able to absorb the less converging part of the modal series and thus, to improve the convergence of the method. Using the cosine basis, the modal method has a low convergence, 1/N, with N the order of truncation. Using the enriched basis, the improvement in the convergence is shown to depend on the Mach number, from 1/N 5 to roughly 1/N 1.5 for M = 0 to M close to unity. The case of a continuously varying wall impedance is considered, and we discuss the limiting case of piecewise constant impedance, which defines pressure edge conditions at the impedance discontinuities.

Exact semi-separation of variables in waveguides with non-planar boundaries

“…The origin of the CMT goes back to early 50s, first appearing in connection with electromagnetic [2], and acoustic [3] waveguides of varying cross sections (horns). Since then, the CMT has been applied to the study of almost all types of elongated waveguides( 5 ): microwave and optical waveguides (sometimes under the name generalized telegraphist's equations) [4], [5], [6]; duct acoustics (horns) [7], [8], [9]; duct acoustics with mean flow [10]; hydroacoustics [1] (Sec. 7.1), [11], [12], [13], [14]; seismology and geophysics [15], [16]; water waves [17], [ The local vertical eigenfunctions n Z , as obtained by the reference waveguide, satisfy boundary conditions which, in general, are incompatible with the boundary conditions of the field ( , )…”

“…The origin of the CMT goes back to early 1950 s, first appearing in connection with electromagnetic [2] and acoustic [3] waveguides of varying cross sections (horns). Since then, the CMT has been applied to the study of almost all types of elongated waveguides 2 : microwave and optical waveguides (sometimes under the name generalized telegraphist's equations) [4][5][6]; duct acoustics (horns) [7][8][9]; duct acoustics with mean flow [10]; elastic waveguides [11][12][13]; hydroacoustics [1,Sec. 7.1], [14][15][16][17]; seismology and geophysics [18,19]; water waves [20][21][22][23][24][25][26][27][28][29][30][31]; hydroelasticity [26].…”

Propagation of acoustic perturbations in non-uniform ducts with non-uniform mean flow using eigen analysis in general curvilinear coordinate systems

“…The current method is a more natural approach and removes one of the sources of error discussed in the above studies (the convergence of the solution series to a true duct solution). The reasons given by Mercier and Maurel [13] for the latter choice are twofold: firstly that it is not proven that a mathematically complete series solution exists in the general case, and secondly that "the notion of adapted transverse functions becomes fuzzy" in the case of continually varying impedance. The first issue is not relevant to the current work, as the problem is first discretised, and the eigenmodes found form a complete basis with respect to the discretised problem.…”

Homogenization of the vibro–acoustic transmission on periodically perforated elastic plates with arrays of resonators

“…Up to our knowledge, the acoustic transmission in a nonstationary fluid flowing through a periodically perforated interfaces has not been treated by the homogenization so far in the published literature. Numerical aspects of the acoustic waves in a uniform flow were reported in [7] where the applied Lorentz transformation yields the Helmholtz equation. Another study [8] employed the Galbrun equation.…”