Abstract
Cut‐on cut‐off transition of acoustic modes in hard‐walled ducts with irrotational mean flow is well understood for Helmholtz numbers of order unity, where previous analyzes have shown that the incident mode undergoes a total reflection with a phase shift of π/2. Finite‐element simulations of this phenomenon, however, appear to indicate the possibility of energy scattering into neighboring modes at a moderately large Helmholtz number, which can then propagate beyond the transition point. In this work, we attempt to explain and predict such scattering phenomena in slowly varying aeroengine ducts using a multiple‐scale approach. It is found that, for sufficiently high frequencies, two mechanisms exist whereby energy can be scattered into neighboring modes by an incident propagating mode. One mechanism occurs only when a mean flow is present in the duct and induces scattering at significantly lower frequencies than the other mechanism that remains present even without a mean flow. An efficient system of coupled ordinary differential equations is developed to obtain the corresponding transmitted and reflected amplitudes of the scattered modes as well as the overall acoustic pressure field. Moreover, the theory appears to demonstrate that some interaction and exchange of energy between the acoustic and mean flow fields occur during scattering.
Highlights
The propagation of unsteady disturbances in ducts of slowly-varying geometry, such as those typical of an aeroengine, can be successfully modelled using a multiple scales approach
A recent comparison paper [8] contains finite-element simulations of cut-on cut-off transition that appear to indicate the possibility of energy scattering into neighbouring modes at large Helmholtz numbers
A brief argument put forward by the authors of the paper was based on the smaller separation of neighbouring eigenvalues at high frequency and it was conjectured that scattering may occur at non-dimensional frequencies of the order of ε−2, where ε ≪ 1 is the so-called slowly-varying parameter
Summary
The propagation of unsteady disturbances in ducts of slowly-varying geometry, such as those typical of an aeroengine, can be successfully modelled using a multiple scales approach. A brief argument put forward by the authors of the paper was based on the smaller separation of neighbouring eigenvalues at high frequency and it was conjectured that scattering may occur at non-dimensional frequencies of the order of ε−2, where ε ≪ 1 is the so-called slowly-varying parameter (a typical feature of the nondimensional duct geometry variation as described below). We attempt to explain and predict such observed scattering phenomena in slowly varying aeroengine ducts using a multiple-scales approach. The straightforward application of the theory to ducts of both circular and annular cross-section is described towards the end of the paper
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