## Section: New Results

### Aeroacoustics

#### Time-harmonic acoustic scattering in a vortical flow

Participants : Antoine Bensalah, Patrick Joly, Jean-François Mercier.

This activity is done in the framework of the PhD of Antoine Bensalah, in partnership with Airbus Group. We study the time-harmonic acoustic radiation in a fluid in a general flow which is not curl free, but has restricted vortical areas. The objective is to take into account the complicated coupling between acoustics and hydrodynamics. The Galbrun approach developed previously in 2D is too expensive in terms of degrees of freedom for 3D simulations. As an alternative, we propose to consider instead the Goldstein equations, which are vectorial only in the vortical areas and remain scalar elsewhere.

We have proved that the Goldstein equations are well-posed in a domain $\Omega $ for a potential flow, or for a vortical flow if the flow is $\Omega $-filling (each point of $\Omega $ is reached by a streamline coming from the inflow boundary in a finite time). A non $\Omega $-filling flow corresponds to the presence or recirculations areas and we have shown that, for such flows, some of the closed streamlines can be resonant. To study deeper this phenomenon, we focused on the case of a rotating flow in an annular geometry. We proved that outside the set of resonance frequencies, the radiation problem is well-posed. Work is under progress to determine the solution on a resonant streamline.

#### Propagation of solitons through Helmholtz resonators

Participant : Jean-François Mercier.

With Bruno Lombard (Laboratoire de Mécanique et Acoustique of Marseille), we studied the propagation of an acoustic solitary wave in a 1D waveguide connected to a periodic array of Helmholtz resonators. Starting from a model of the literature, obtained by approximations, our goal was to provide a numerical modeling, which validates (or not) the underlying model and the assumptions. The model consists of two coupled equations evolution: a nonlinear PDE describing acoustic waves (similar to the Burgers equation), and a linear ODE describing oscillations in the Helmholtz resonators. We have developed a numerical method based on two main ingredients: a diffusive representation of fractional derivatives and a splitting method applied to the evolution equations. The numerical scheme has been validated by comparison with exact solutions. The properties of non-linear solutions have been investigated numerically.

In collaboration with O. Richoux of the LAUM, this work has been extended, comparing to experimental results. Adjustments had to be made, the attenuation of the numerical model being weaker than that observed experimentally. To remedy this, we have incorporated some attenuation mechanisms that we had neglected. One consequence of these additions is that a more sophisticated numerical method had to be developed. A good agreement has been found with experimental results.