A wave propagating in a non-uniform flow can have a critical layer where it is absorbed, amplified, reflected, or converted to another mode, possibly exchanging energy with the mean flow. Two examples are the propagation of: (i) fan noise in the shear flow in the air inlet of a jet engine; (ii) turbine noise in the swirling flow in the jet exhaust. Both situations (i) and (ii) are included by considering wave propagation in an axisymmetric isentropic non-homentopic mean flow allowing for the simultaneous existence of shear, swirl, temperature and density gradients. This corresponds to coupled acoustic-vortical modes that have both continuous and discrete spectra. It is shown that a critical layer exists where the Doppler shifted frequency vanishes. A stability condition is obtained for the continuous spectrum for all frequencies, axial and azimuthal wavenumbers generalizing previous results from the homentropic to the isentropic non-isentropic case. The wave fields are calculated and plotted for a case of non-rigid body rotation, namely angular velocity of swirl proportional to the radius; this demonstrates the mode conversion between acoustic and vortical waves across the critical layer where the waves are absorbed because the pressure spectrum vanishes.
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