Transparent boundary condition for acoustic propagation in lined guide with mean flow

A finite element analysis of acoustic radiation in an infinite lined guide with mean flow is studied. In order to bound the domain, transparent boundary conditions are introduced by means of a Dirichlet to Neumann (DtN) operator based on a modal decomposition. This decomposition is easy to carry out in a hard-walled guide. With absorbing lining, many difficulties occur even without mean flow. Because the eigenvalue problem is no longer selfadjoint, acoustic modes are not orthogonal with respect to the L2-dot product. However, an orthogonality relation exists which permit to write the modal decomposition. For a lined guide with uniform mean flow, orthogonality relation doesn't exist but a new dot product allows us to define the DtN operator. We consider first the case of an infinite rectangular two-dimensional lined guide with uniform mean flow to present the methodology. Then, the extension to the axisymmetric cylindrical problem is presented.