The studies on the interactions between flow and porous materials located in wall ducts have been initiated in 2001. Some of these studies investigate the use of the Multimodal Method in lined flow ducts. The other studies are related to the prediction and the measurement of wall impedances in the presence of flow. A recent results is the experimentally and numerically proven failure of the classical boundary condition used in the calculations of lined ducts with flow. This condition known as the Ingard-Myers condition implies the continuity of the acoustic displacement in the vicinity of an acoustic treatment with flow. The condition which we determined experimentally is located midway between the displacement continuity and the velocity continuity. Some of the studies on the acoustic liners involves the use of acoustic metamaterials (see Figure). A first work was done as part of a Post-Doc funded by the REBECCA contract (B. Nennig) on porous with inclusions. This subject was continued as part of the European project FlowAirS (L. Xiong thesis). Another explored subject is the use of quasi-trapped modes (Fano resonance) to reduce the transmission of acoustic treatment. These studies have led us to study the effect on the flow of a sound wave slowed by an acoustic treatment. In this context we have considered the properties of the scattering matrix when a change in the speed of sound occurs and we studied the analogy of this kind of propagation with gravitational waves.

Thin slow sound material used in flow ducts as a liner

The problem is to understand the effect of hydrodynamic modes in the vicinity of an impedance discontinuity in the wall and to study the influence of system parameters on the amplification and on the wavelength hydrodynamic modes. This should determine when these modes can have a significant influence in mitigation calculations. Such an effect was measured with PIV at LEA (Poitiers) as part of a collaboration initiated in the COMATEC contract. Visualizations (see Figure) show the appearance of coherent structures above the material. This work has been completed with a theoretical study of the instability mechanisms. An additional experimental study on the porous materials was conducted LAUM which is now equipped with PIV. The instability already seen for locally reacting materials also appears above porous materials.

PIV measurements over a liner with flow. Coherent structures are clearly visible.

The modal methods for the propagation of guided waves are based on the discretization on a mode basis of the transverse components of the field (perpendicular to the preferred direction of propagation). This base is typically that of the modes in a uniform waveguide, so that the solution of the modal method in the trivial case of a waveguide without perturbation is known and exact. In the general case, the multimodal equations and the numerical schemes to solve them should describe couplings that take place in the non-uniform regions, given the radiation and source conditions at the ends [Pagneux2010]. These methods, now widely investigated and applied, have proved their numerical efficiency. Efficiency meaning notably energy conservation and reciprocity, and a rapid convergence with respect the number of modes. Recent developments of the multimodal methods in waveguides have concerned (i) the use of geometrical transformations and an improved formulation that significantly fasten the convergence of the multimodal method in waveguides with complex geometry or boundary conditions [Maureletal2014], (ii) a formulation for waveguides filled with an heterogeneous medium, (iii) one-way approximations of the coupled-mode propagation in slowly varying waveguides.

[Pagneux2010] V. Pagneux, J. Comput. Math. Appl. **234**, 1834 (2010).

[Maureletal2014] A. Maurel, J.-F. Mercier, V. Pagneux, Proc. R. Soc. A **470**, 20130448 (2014).