a1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
In this paper we consider the propagation of acoustic waves on top of an inviscid steady flow along a curved hollow or annular duct with hard or lined walls. The curvature of the duct centreline (which is not restricted to being small) and the wall radii vary slowly along the duct, allowing application of an asymptotic multiple-scales analysis. 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 explicitly at each axial location using a pseudospectral numerical method.
Unlike the case of a straight duct carrying uniform flow, 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, curvature and steady-flow Mach number. The interaction between the presence of wall lining and curvature is also significant; 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.
Using ray theory, we suggest explanations of these features. For the lowest azimuthal-order modes, three distinct regimes are found in which the modes are localized in different parts of the duct cross-section. This phenomenon is explained by a balance between whispering-gallery effects along the duct and refraction by the steady flow. At the opposite extreme, strongly spinning modes are investigated, and are seen to be due to a different whispering-gallery effect across the duct cross-section.
(Received May 01 2007)
(Revised September 03 2007)