Study of the interaction between an unstable boundary-layer flow and a flexible surface. Application to the prediction of the hydrodynamic noise on a sonar antenna.

This work, supported by Thales Underwater Systems through a Cifre thesis grant, and by DCNS, aims at improving the prediction of the hydrodynamic self-noise of a sonar antenna, due to the pressure fluctuations in the boundary-layer flow that develops along the dome. Noise estimations are generally based on semi-empirical models, that take only partially into account the dome flexibility. The present study readdresses the issue of the hydrodynamic noise, in the simplified archetype case of an unstable boundary layer flow along an elastic flat plate.

The first part of the study consists in the direct numerical simulation of a two-dimensional boundary layer flow, characterized by a highly supercritical Reynolds number, over an elastic clamped plate. The incompressible Navier-Stokes equations are solved by means of a time-dependent mapping, associated to a mixed finite differences – Chebyshev collocation spatial discretization. A fractional step method enables a full coupling between the plate model and the fluid system. A velocity forcing is introduced in the latter, at unstable frequencies and high amplitudes, in order to give rise to convective and non-linearly saturated instabilities, which interact with the plate motions. The latter vibrates around an initial bent state resulting from the coupling with the non perturbed flow. Plates of various materials and lengths are tested, to characterize the vibrations, in terms of levels and spatial structures, as well as their retro-action on the flow instabilities, depending on the values of the plate natural frequencies. For the cases considered in the present investigation, we observe that the wall pressure spectrum possesses additional modal components, with very low wavenumbers and relatively high frequencies, which may give rise to increased radiation.

A post-treatment is implemented to estimate the noise generated by the boundary layer velocity fluctuations. The radiated pressure in the uniform flow domain is evaluated from the simulation data in the framework of the Lighthill's analogy. The calculations are performed in the spectral domain, using an appropriate Green function whose expression takes into account the presence of the elastic plate. In this approach, the plate vibrations induce at some frequencies, including the plate natural frequencies, higher radiated pressure levels than in the rigid case, by favouring larger spatial structures. Additional models, derived from the same formulation, but taking into account the boundary layer compressibility in the computation of the vibratory contributions, highlight a clear increase in the radiated pressure levels in a wide frequency range around the plate natural frequencies, when the latter are distinct from the forcing frequencies, as well as the benefit from considering a full coupling in comparison to classical approaches based on a weak coupling assumption.

Finally, the fluid-structure system is investigated experimentally using a hydrodynamic tunnel setup, in order to measure the vibrations of an elastic clamped plate triggered by a transitional or turbulent boundary layer flow, as well as the radiated noise in the cavity beneath the plate, which is equipped with a hydrophone. The measurements, acquired using laser vibrometry and laser Doppler velocimetry, are detailed and analysed.