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 ∞ow and treated boundaries. We consider the time harmonic acoustic radiation in a conflned ∞ow in presence of treated boundaries. The case of a potential ∞ow with Myers condition leads to a scalar formulation which is tractable by a standard flnite element method. 1{3 On the other hand, Galbrun’s equation 4 seems to be well-suited for handling non potential ∞ows. 5,6 For a uniform ∞ow we present a scalar problem in presence of PMLs (Perfectly Matched Layers). Well-posedness is proved which ensures the convergence of the flnite element discretization. To extend to a shear ∞ow, 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 ∞ow. Finally the di‐culty is solved by enriching the Myers condition.