On swirl mach number effects on acoustic-vortical waves

Abstract

The propagation of sound in a uniform flow with rigid body swirl has been considered in the approximation of low swirl Mach number [Formula: see text],where the swirl Mach number [Formula: see text] is specified by the angular velocity [Formula: see text], stagnation sound speed [Formula: see text] and radial distance r; in this case the mass density and sound speed in the mean flow are constant, and the convected wave equation is solved in terms of Bessel functions. In this paper the restriction on swirl Mach number is relaxed from [Formula: see text] to [Formula: see text], thus keeping O ([Formula: see text]) terms in the mean flow, mass density and sound speed, that become non-uniform; the acoustic vortical wave equation is no longer the convected wave equation, and is extended to this case and solved in terms of generalized Bessel functions. The radial dependence to second order in the swirl Mach number is specified by a generalized Bessel differential equation for decoupled acoustic or vortical modes and for acoustic-vortical waves to first order in the swirl Mach number. The solutions are obtained: (i) for finite radius in terms of generalized Bessel and Neumann functions, that determine the radial wavenumbers, natural frequencies and normal eigenfunctions for cylindrical or annular ducts with rigid or impedance wall boundary conditions; (ii) asymptotically for large radius in terms of generalized Hankel functions, specifying the growth of amplitude with radius either as a power law or as an exponential of one-half of the square of the swirl Mach number. Thus the compressible uniform mean flow with rigid body rotation is spatially unstable in the radial direction; it is also unstable in time for cut-on modes with real axial wavenumbers and cut-off modes with imaginary axial wavenumbers. Compared with acoustic waves the acoustic-vortical waves have more modes, some with complex rather than real eigenvalues leading to instabilities.