Optimal Disturbance in a Flow Duct with a Lined Wall

Acoustic lining is widely used to reduce sound. However, for specific liners and under particular flow conditions, some experiments have shown that a convective hydrodynamic instability may grow on the liner and is likely to lead to a sound amplification. In this paper, such a phenomenon is studied through a direct optimal growth analysis of the linearized Euler equations in a two-dimensional flow duct with the bottom wall which is partly lined. A Discontinuous Galerkin scheme is used for spatial discretization, which proved to be quite efficient in handling the acoustic impedance discontinuity at the interface between the lined and hard wall. The optimal perturbation exhibits a spatial amplification similar to the one predicted by a local stability analysis. Moreoever, we can observe a mechanism of conversion of the hydrodynamic surface mode, which appears above the lined wall, into a high amplitude acoustic wave which travels downstream of the liner.