Section: New Results

Acoustics and aeroacoustics

High-order absorbing boundary conditions with corner treatment for high-frequency acoustic scattering

Participant : Axel Modave.

This work is done in collaboration with C. Geuzaine (University of Liège) and X. Antoine (IECL & EPI SPHINX)

We address the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high-frequency acoustic scattering problems. While high-order absorbing boundary conditions (HABCs) are accurate for smooth fictitious boundaries, the precision of the solution drops in the presence of corners if no specific treatment is applied. We analyze two strategies to preserve the accuracy of Padé-type HABCs at corners: first by using compatibility relations (derived for right angle corners) and second by regularizing the boundary at the corner. We show that the former strategy is well-adapted to right corners and efficient for nearly-right corners, while the later is better for very obtuse corners. Numerical results are proposed to analyze and compare the approaches for two-and three-dimensional problems.

Time-harmonic acoustic scattering in a vortical flow

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

We study the time-harmonic acoustic radiation in a fluid in flow. To go beyond the convected Helmholtz equation, only adapted to potential flows, we use Goldstein's equations, coupling exactly the acoustic waves to the hydrodynamic field. We have studied the hydrodynamic part of Goldstein equations, corresponding to a generalized time-harmonic transport equation and we have investigated its well-posedness. The result has been established under the assumption of a domain-filling flow, which in 2D is simply equivalent to a flow that does not vanish. The approach relies on the method of characteristics, which leads to the resolution of the transport equation along the streamlines and on general results of functional analysis. The theoretical results have been illustrated with numerical results obtained with a SUPG Finite Element scheme.

In complement we have developed a new model for Goldstein's equations in which the description of the hydrodynamic phenomena is simplified. The model, initially developed for a carrier flow of low Mach number $M$, is proved theoretically to remain accurate for moderate Mach numbers, associated to a low error bounded by $M^2$. Numerical experiments confirm the $M^2$ law and the good quality of the model for flows of non-small Mach numbers.