A Perfectly Matched Layer (PML) for aeroacoustic problems using Galbrun's equation in the presence of an axial and a swirling steady mean flow is investigated in a cylindrical coordinates system. This equation is based on an Eulerian-Lagrangian description and leads to a wave equation written only in terms of the Lagrangian perturbation of the displacement. Galbrun's equation is solved by a mixed pressure-displacement Finite Element Method (FEM). To avoid instabilities in the presence of mean flow, a geometric transformation is presented. The validity and efficiency of the proposed PML formulation are established through comparisons with analytical, semi-analytical model based on Pridmore-Brown equation (extended to an axial and a swirling mean flow) and with multiple-scale models. The interest of the formulation is shown through an example of aeroacoustic radiation.