Coupled phenomena for waves in fluids and solids


POEMS has strong links with the french community of acoustics and in this framework we are interested in the acoustic propagation in complicated environments which couple phenomena propagating at very different speeds. A first example is the case of a moving fluid (aeroacoustics), in which acoustic waves and hydrodynamic waves (vorticity) propagate at very different speeds. A second example is given by an elastic solid, coupling naturally pressure and shear waves, which leads in the degenerated case of a nearly incompressible material to very different speeds of the two waves.

Aeroacoustics via Goldstein's equations

The aeroacoustic project concerns the simulation of time-harmonic acoustic propagation in presence of a stationary flow (called the mean flow). The difficulty is twofold because, in addition to the usual difficulties of acoustics, there are also difficulties arising from hydrodynamics phenomena, induced by the flow and coupled with the acoustic. This subject has been introduced in the activities of POEMS at the beginning of the 2000's. Aeroacoustics is very close to applications, mainly the reduction of aircraft noise and therefore has been the object of a long term collaboration with Airbus Group Innovations and more recently with Naval Group.

As a model for aeroacoustics, rather than the linearized Euler equations, we have chosen the less known Goldstein's equations, better suited in our opinion for both theory and numerics. The unknowns are the velocity potential \(\varphi\) which satisfies a convected Helmholtz equation and a vectorial hydrodynamic unknown \(\boldsymbol{\xi}\) which satisfies a non-classical harmonic transport equation. Both equations are coupled.

Mathematical analysis and numerical modeling
\(\Re e(\varphi)\) in a general flow

First has been studied the generalized transport equation (assuming that the potential is known). For the theory, we concentrated ourselves on flows without recirculation. We have shown that such flows are non-resonant : the transport problem is well posed and has good stability properties. The approach is based on the method of characteristics, which leads to the resolution of ODE's along streamlines.

Numerically we developed a particular finite element scheme of SUPG type that allowed us to obtain numerical results illustrating the theory.

In a second part, we studied the full model, i. e. the coupling of the transport equation with the convected Helmholtz equation. The analysis is based on the elimination of \(\boldsymbol{\xi}\) and on the Fredholhm's alternative for the reduced problem in \(\varphi\). Well-posedness is obtained under a smallness assumption about the vorticity of the mean flow. Numerically, we couple standard Lagrange finite elements for the convected Helmholtz equation with the above mentioned SUPG method for the transport equation.

Simplified low Mach number model
Velocity potential with the exact hydrodynamic phenomena (left) and with the approximated model (right)

In addition but for a slow flow, we have studied the acoustic radiation in a 1D varying flow in a straight guide. From Goldstein's equations, we developed a new model in which the description of hydrodynamic phenomena is simplified. We proved that this model, initially developed for a mean flow of low Mach number \(M\), is ultimately very precise, associated with a low error in \(M^2\).

Coupling with elastic waves

Solving linear elastodynamics equations using potentials

It is difficult to simulate the elastic propagation of waves in living soft tissues and commercial codes used to fail because standard displacement based formulations are highly penalized by the fact that shear waves propagate much more slowly than pressure waves. A remedy is the Helmholtz decomposition of the displacement field which allows the decoupling of the two dynamics giving the hope to build space and time discretizations adapted to each type of wave. Then using finite element methods, the challenge was to treat in a stable way boundaries and interfaces, where the two waves are recoupled. Straightforward approaches lead to severe instability phenomena and we built an original stabilized mixed formulation of the problem, based on a clever choice of Lagrange multipliers living on boundaries and interfaces.

Resonances of an elastic plate in a flow
Stable and unstable areas in the \((c,U)\) plane, theoretically (stable=blue areas) and numerically (stable=below the black squares)

We took into account the coupling of elastic waves with pressure waves when studying the acoustic resonances of an elastic beam placed in a fluid in uniform flow. While the resonance frequencies of analogous configurations are generally real, the originality of this study is that the combined effects of the elasticity of the beam and of a flow can create complex frequencies corresponding to instabilities (similar to the instabilities of so-called float or flag membranes). We theoretically studied the conditions ensuring stability as a function of three parameters, the velocity of the flow \(U\) and the ratios of the densities \(\rho\) and of the velocities of sound \(c\) of the plate and of the fluid. A volume in the parameter space is defined explicitly, in which no instability can develop.