CERFACS CERFACS Scientific Activity Report Jan. 2010 – Dec. 2011

FINITE ELEMENT SIMULATION OF ACOUSTIC SCATTERING IN A SUBSONIC FLOW
One main advantage of the Galbrun’s model is that the unknown is well adapted to write boundary
conditions. Indeed boundary conditions are naturally expressed with the displacement rather the pressure or
the velocity. In particular, in presence of lined walls, it is simple to write a condition on the displacement,
defined by divu = ikZu where Z is the dimensionless impedance. Now , in this case, the proof of wellposedness
is not straightforward with this boundary condition even if the flow is uniform. Some work has
been done in order to clarify this situation. We have proposed an other boundary condition depending on a
small parameter β. We have proved that in this case, the new problem is well-posed and when β tends to
zero, the boundary condition tends to the condition of lined walls. Numerical results are in progress [see
Figure 4.3).
FIG. 4.3: Real part of the displacement component for a value Z.
[12] P. AZERAD, Analyse des équations de Navier-Stokes en bassin peu profond et de l’équation de
transport, Thèse de l’Université de Neuchâtel (Suisse), 1996.
[13] P.B. BOCHEV AND M.D. GUNZBURGER, Least-Squares Finite Element Methods, Springer, Applied
Mathematical Sciences, Vol. 166.
[14] A. S. Bonnet-Ben Dhia, E. M. Duclairoir, G. Legendre and J. F. Mercier, “Time-harmonic acoustic
propagation in the presence of a shear flow”, J. of Comp. and App. Math., 2007.
[15] A. ERN AND J.-L. GUERMOND, Theory and Practice of Finite Elements, Springer, Applied
Mathematical Sciences, Vol. 159.
4.3 A new model called the Goldstein-Visser model
Let us come back to the potential case. Contrary to the Galbrun case, only a scalar equation has to be solved,
which is very attractive. But this potential equation is only available in specific case in particular when the
flow is irrotational, so when the coupling between acoustic and hydrodynamic effects are neglected. We
have extended this approach in more general cases. We have proposed to write an augmented equation
by introducing a new variable ξ. This latter is directly linked to the vorticity of the flow and for example
is equal to zero when the flow is irrotational. Again, this variable is obtained through a time harmonic
64 Jan. 2010 – Dec. 2011