Sound transmission in strongly-curved slowly-varying cylindrical and annular lined ducts with flow

In this paper we consider the propagation of acoustic waves along a curved hollow or annular duct with lined walls. The curvature of the duct centreline and the wall radii vary slowly along the duct, allowing application of an asymptotic multiple scales analysis. This generalises Rienstra's analysis of a straight duct of varying cross-sectional radius. The result of the analysis is that the modal wavenumbers and mode shapes are determined locally as modes of a torus with the same local curvature, while the amplitude of the modes evolves as the mode propagates along the duct. The duct modes are found numerically at each axial location using a pseudo-spectral method. Unlike the case of a straight duct, there is a fundamental asymmetry between upstream and downstream propagating modes, with some mode shapes tending to be concentrated on either the inside or outside of the bend depending on the direction of propagation. The interaction between the presence of wall lining and curvature is investigated in particular; for instance, in a representative case it is found that the curvature causes the first few acoustic modes to be more heavily damped by the duct boundary than would be expected for a straight duct. Analytical progress can be made in the limit of very high mode order, in which case well-known 'whispering gallery' modes, localised close to the wall, can be identified.

Generation and scattering of sound in vortical flow

We present solutions to scattering problems for unsteady disturbances to a mean swirling flow in an annular duct with a rigid 'splitter'. This situation has application to rotor-stator interaction noise in aeroengines, where the flow downstream of the fan is swirling and bifurcates into the by-pass duct and the engine core. We also consider the trailing edge extension of this problem. Inviscid mean flow in a cylindrical annulus is considered, with both axial and swirling (azimuthal) velocity components. The presence of vorticity in the mean flow couples the acoustic and vorticity modes of irrotational flow. Instead we have one combined spectrum of acoustic-vorticity waves in which the 'sonic' and 'nearly-convected' modes are fully coupled. In addition to the aeroacoustics application the results offer insight into the behaviour of these acoustic-vorticity waves, and the precise nature of the coupling between the two types of mode. Two regimes are discussed in which progress has been made, one for a specialised mean flow, uniform axial flow and rigid body swirl, and a second regime in which the frequency is assumed large, valid for any axisymmetric mean flow. The Wiener-Hopf technique is used to solve the scattering problems mathematically, and we present numerical evaluations of these solutions. Several new effects are seen to arise due to the mean vorticity, in particular the generation of sound at a trailing edge due to the scattering of a nearly convected disturbance, in contrast to the way a convected gust silently passes a trailing edge in uniform mean flow.