## Section: New Results

### Perfectly Matched Layers for the Shallow Water equations

Participants : Hélène Barucq, Julien Diaz, Roland Martin, Carlos Couder, Mounir Tlemcani [ Assistant Professor, University of Oran, Algeria ] .

In the past few years ago, substantial progress has been made in the development of the PML technique for the Euler equations, starting with the studies for cases with constant mean flows, followed by extensions to cases with non-uniform mean flows. Most recently, applications of PML to linearized Navier-Stokes equations and non-linear Navier-Stokes equations have been discussed in [69] , [68] , [79] . Although the PML technique itself is relatively simple when it is viewed as a complex change of variables in the frequency domain, it is important to note that, for the PML technique to yield stable absorbing boundary conditions, the phase and group velocities of the physical waves supported by the governing equations must be consistent and in the same direction.

For applications of full non-linear Navier-Stokes equations, we use an optimized perfectly matched layer (PML) technique that has proven to be efficient in elastodynamics to absorb surface waves as well as body waves with non grazing incidence [71] . It is possible to use this unsplit convolutional PML (CPML) for staggered and finite difference integration schemes to improve the computational efficiency reducing the number of computational arrays and therefore the memory storage for the flux domain. We applied this first to non-linear Shallow water equations in presence or not of Coriolis forces and friction forces, and also to subsonic and supersonic directed flows for industrial contexts of interest.

Possible applications of the Shallow water equations with PML are for massive avalanches like turbidites, pollutant contaminants travelling in coastal regions (Gulf of Mexico) with many chemical species, avalanches and tsunamis, ocean-earth interactions etc.... We restricted successfully our results obtained for Navier-Stokes to directed chanellized shallow water flows.

During this year, we studied air critical ejector diffuser simulations. The ejectors are commonly used to extract gases in the petroleum industry where it is not possible to use an electric bomb due the explosion risk because the gases are flammable. In this work we develop a numerical code to investigate the unsteady supersonic flow in the ejector diffuser to have an efficient tool that allows modeling different diffusers designs. The model is developed using curvilinear coordinates transformation to adapt the ejector design to a regular computational plane where a finite differences scheme could be applied, and for control the outflow conditions we use a convolutional perfectly matched layer (CPML) technique. The CPML has demonstrated its convenience because the time evolutions of damping mechanisms do not need to be split and only the space derivatives of velocities need to be stored at each time step reducing the number of computational array used in the numerical code. In this context of a directed flow, the results obtained shows that the perfectly matched layer (CPML) technique can absorb efficiently the out-going supersonic flux at the outlet condition and absorbing zone is reflection less.

In the same time, we have worked on the stabilization of a PML for the linearized Shallow Water equations.Indeed, the PML technique induces some instabilities when applied to Euler equations [69] or to shallow water equations. Much works have been devoted to the stabilization of the PML for linearized Euler equations [68] , [79] and we have developed a stable PML for the linearized shallow water equations with a Coriolis term and a uniform mean flow. Our method follows an idea proposed by Nataf [79] for linearized Euler equations and is based on the splitting of the vorticity waves from the advective and entropy waves. Then since the propagation of vorticity and entropy waves is governed by a classical transport equation, these waves can be easily absorbed by a transparent and exact condition at the end of the layer and the PML condition is applied to the advective waves only. We have used a transformation proposed by Hu [68] before applying a classical PML technique to avoid an exponential growth of the waves and numerical experiments [20] show both the accuracy of the condition and its long-time stability. The main results have been presented at the peer-reviewed conference Waves 2009.