## Section: Scientific Foundations

### Numerical schemes for fluid mechanics

Participants : Rémi Abgrall, Marc Duruflé, Mario Ricchiuto.

A large number of industrial problems can be translated into fluid mechanics ones. They may be coupled with one or more physical models. An example is provided by aeroelastic problems, which have been studied in details by other INRIA teams. Another example is given by flows in pipelines where the fluid (a mixture of air–water–gas) does not have well-known physical properties. One may also consider problems in aeroacoustics, which become more and more important in everyday life. In some occasions, one needs specific numerical tools because fluids have exotic equation of states, or because the amount of computation becomes huge, as for unsteady flows. Another situation where specific tools are needed is when one is interested in very specific quantities, such as the lift and drag of an airfoil, a situation where commercial tools can only provide a very crude answer.

It is a fact that there are many commercial codes. They allow users to compute many flow realizations, but the quality of the results is far from being optimal in many cases. Moreover, the numerical tools of these codes are often not the most recent ones. An example is the noise generated by vortices crossing through a shock wave. It is, up to our knowledge, even out of reach of the most recent technologies because the numerical resources that would necessitate such simulations are tremendous ! In the same spirit, the simulation of a 3D compressible mixing layer in a complex geometry is also out of reach because very different temporal and physical scales need to be captured. Consequently, we need to invent specific algorithms for that purpose.

In order to reach efficient simulation of complex physical problems, we are working on some fundamental aspects of the numerical analysis of non linear hyperbolic problems. Our goal is to develop schemes that can adapt to modern computer architectures. More precisely, we are working on a class of numerical schemes specifically tuned for unstructured and hybrid meshes. They have the most possible compact stencil that is compatible with the expected order of accuracy. The order of accuracy typically ranges from two to four. Since the stencil is compact, the implementation on parallel machines becomes simple. The price to pay is that the scheme is necessarily implicit, though some progress have been made recently so that this is not anymore a constraint. We are also interested in Discontinuous Galerkin type schemes. However, the compactness of the scheme enables us to use the high performance parallel linear algebra tools developed by the team for the lowest order version of these schemes. The high order versions of these schemes, which are still under development, will lead to new scientific problems at the border between numerical analysis and computer science. In parallel to these fundamental aspects, we also work on adapting more classical numerical tools to complex physical problems such as those encountered in interface flows, turbulent or multiphase flows.

Within a few years, we expect to be able to consider the physical problems which are now difficult to compute thanks to the know-how coming from our research on compact distribution schemes and the daily discussions with specialists in computer science and scientific computing. These problems range from aeroacoustic to multiphysic problems, such as the ones mentioned above. We also have interest in solving compressible MHD problems in relation with the ITER project. Because of the existence of a magnetic field and the type of solutions we are seeking for, this leads to additional scientific challenges. Our research work about numerical algorithms has led to software FluidBox which is described in section 5.2 . This work is supported by the EU-Strep ADIGMA, various research contracts and in part by the ANR-CIS ASTER project (see section 4.3 also), and also by the ERC grant ADDECCO.