We study the diffraction in time harmonic regime of an acoustic wave by a rigid plate in the presence of a uniform flow in a duct. Contrary to prior analytical studies, using Wiener-Hopf techniques and thus restricted to semi-infinite plates, we use a, finite elements method which allows us to deal with plates of finite length. To take into account irrotational perturbations induced by the trailing edge of the plate, a potential formulation requires the introduction of a vortex sheet behind the plate. The key point of the method is to get access at the singular coefficient of the velocity potential near the trailing edge, in order to cancel it using the so-called Kutta-Joukowski condition. This approach leads to an efficient finite elements method, and numerical computations are presented: we show the amplitude of the vortex sheet versus the Mach number and the plate length and the dissipated acoustic power versus the Mach number and the frequency. This method is extended to the case of two aligned plates to analyze the influence of the choice of the boundary condition on the downstream plate which interacts with a vortex sheet. © IMACS.