Cut-on Cut-off Transition in Flow Ducts: Comparing Multiple-scales and Finite-element Solutions

Flow and geometry induced scattering of high frequency acoustic duct modes.

“…2/3 as before and, as a result, the non-parallel terms above become comparable in the inner region when ω ∼ ϵ −2 ; this agrees exactly with the initial approximate estimate obtained previously [8]. Hence for the case ω = O(ϵ −2 ), substituting our expression for the inner solution (14) into (13) leads to…”

“…Terms 1 and 2 alone lead to Airy's equation for ω ∼ 1 when X − X t ∼ ϵ 2/3 but we can conclude from our scalings that at least one of the non-parallel terms 3 and 4 driven by the crosswise mean velocity become comparable for frequencies of O(ϵ −1/2 ) or higher-a scaling that is significantly lower than the previous estimate obtained [8]. Note that these additional terms require a mean flow to exist and their magnitude is determined by both the strength of the mean flow, assuming…”

“…In this paper, we attempt to explain and predict such observed scattering phenomena in slowly varying aeroengine ducts using a multiple-scales approach. Our analysis appears to show firstly that the order of magnitude estimate obtained previously [8] of the nondimensional frequency required for modal scattering to occur appears overestimated when a mean flow is present within the duct. Furthermore we show that, for sufficiently high frequencies, two mechanisms exist whereby energy can be scattered into neighbouring modes by an incident propagating mode undergoing cut-on cut-off transition.…”

“…Significantly for typical aeroengine applications, a similar partial reflection also occurs in lined ducts of finite impedance [12]. However, 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. Out of the seven benchmark cases presented in that comparison paper, modal scattering was observed in the high-frequency cases with a high cross-sectional eigenvalue for the incident mode, and where a mean-flow is present within the duct.…”

“…The multiple-scales approach has a number of distinct advantages over full numerical methods as it is ideally suited to handle higher frequencies and the computational complexity is only marginally more than calculating the eigenmodes inside a straight parallel duct. The accuracy and usefulness of the multiple scales approach has been validated against finite-element methods [6] for realistic aeroengine configurations and acoustic frequencies [7,8].…”