Propagation of unsteady disturbances in a slowly varying duct with mean swirling flow

The propagation of unsteady disturbances in a slowly varying cylindrical duct carrying mean swirling flow is described. A consistent multiple-scales solution for the mean flow and disturbance is derived, and the effect of finite-impedance boundaries on the propagation of disturbances in mean swirling flow is also addressed. Two degrees of mean swirl are considered: first the case when the swirl velocity is of the same order as the axial velocity, which is applicable to turbomachinery flow behind a rotor stage; secondly a small swirl approximation, where the swirl velocity is of the same order as the axial slope of the duct walls, which is relevant to the flow downstream of the stator in a turbofan engine duct. The presence of mean vorticity couples the acoustic and vorticity equations and the associated eigenvalue problem is not self-adjoint as it is for irrotational mean flow. In order to obtain a secularity condition, which determines the amplitude variation along the duct, an adjoint solution for the coupled system of equations is derived. The solution breaks down at a turning point where a mode changes from cut on to cut off. Analysis in this region shows that the amplitude here is governed by a form of Airy's equation, and that the effect of swirl is to introduce a small shift in the location of the turning point. The reflection coefficient at this corrected turning point is shown to be exp (i pi /2). The evolution of axial wavenumbers and cross-sectionally averaged amplitudes along the duct are calculated and comparisons made between the cases of zero mean swirl, small mean swirl and O(l) mean swirl. In a hard-walled duct it is found that small mean swirl only affects the phase of the amplitude, but O(l) mean swirl produces a much larger amplitude variation along the duct compared with a non-swirling mean flow. In a duct with finite-impedance walls, mean swirl has a large damping effect when the modes are co-rotating with the swirl. If the modes are counter-rotating then an upstream-propagating mode can be amplified compared to the no-swirl case, but a downstream-propagating mode remains more damped.