On the extension of cylindrical acoustic waves to acoustic-vortical-entropy waves in a flow with rigid body swirl

Abstract

The noise of jet and rocket engines involves the coupling of sound to swirling flows and to heat exchanges leading in the more complex cases of triple interactions to acoustic-vortical-entropy (AVE) waves. The present paper presents the derivation of the AVE equation for axisymmetric linear non-dissipative, compressible perturbations of a non-homentropic, swirling mean flow, with constant axial velocity and constant angular velocity for a perfect gas with constant density. The axisymmetric AVE wave equation is obtained for the radial velocity perturbation, specifying its radial dependence for any frequency and axial wavenumber. The AVE wave equation in the case of zero axial wavenumber, corresponding to cylindrical AVE waves, has no singularities for finite radius, including the sonic radius, where the isothermal Mach number for the swirl velocity is unity. The exact solution of the AVE wave equation for the fundamental axisymmetric mode with zero axial wavenumber is obtained without any restriction on frequency, as series expansions of Gaussian hypergeometric type: (i) covering the whole flow region; (ii) specifying the wave field at the sonic radius; (iii) specifying near-axis and asymptotic scaling for small and large radius. Using polarization relations among wave variables specifies exactly and allows the plotting of the perturbations of: (i,ii) the radial and azimuthal velocity; (iii,iv) pressure and mass density; (v,vi) entropy and temperature. Thus the extension of cylindrical acoustic waves, that are specified by Bessel functions, to cylindrical acoustic-vortical-entropy waves, is specified by Gaussian hypergeometric functions.