On the Acoustics of Unbounded and Ducted Vortex Flows

Abstract

The propagation of sound is considered in a potential cylindrical vortex, with superimposed axial flow, by means of explicit analytical solutions. The sound waves are sinusoidal in time, and in the axial and azimuthal directions; the convected wave equation leads to a radial dependence specified by an ordinary second-order dierential equation, with two singularities, at the origin and at infinity. Both singularities are irregular, implying that the acoustic fields have an essential singularity. In the neighborhood of the vortex axis, the essential singularity of the acoustic field is specified by an exponential of the integrated Doppler shift; using the latter as a factor, the acoustic fields are specified by asymptotic expansions in ascending powers of the radius. In the neighborhood of the point at infinity, where the tangential mean flow velocity vanishes, the leading terms are outward or inward propagating cylindrical waves; these factors multiply asymptotic expansions in descending powers of the radius. The two pairs of solutions, around the vortex axis and the point-atinfinity are valid in all space or overlapping regions, as far as the asymptotic expansions can be calculated. The case of an annular nozzle, with uniform axial flow, and potential swirl is used as an example; the eigenvalues are obtained for rigid wall boundary conditions and the corresponding eigenfunctions are plotted.