Time harmonic acoustic scattering in presence of a shear flow and a Myers impedance condition

Noise reduction of aircraft engines can be achieved by a well-suited internal coating of the nacelle, which is generally modelized by the Myers impedance condition. Therefore there is a need of a numerical method for acoustics in presence of a complex flow and treated boundaries. We consider the time harmonic acoustic radiation in a confined flow in presence of treated boundaries. The case of a potential flow with Myers condition leads to a scalar formulation which is tractable by a standard finite element method.^1-3 On the other hand, Galbrun's equation^4? seems to be well-suited for handling non potential flows.^{5, 6} For a uniform flow we present a scalar problem in presence of PMLs (Perfectly Matched Layers). Well-posedness is proved which ensures the convergence of the finite element discretization. To extend to a shear flow, the vectorial Galbrun's equation is used and we show that Myers condition is natural and easy to incorporate in Galbrun's framework. We explain why the proof of the well-posedness is not so easy than in the scalar case and has not been achieved, even for a uniform flow. Finally the di±culty is solved by enriching the Myers condition.