## Section: Scientific Foundations

### Numerical schemes for fluid mechanics

Participants : Rémi Abgrall, Marc Duruflé, Mario Ricchiuto, Pietro Congedo.

A large number of industrial problems involve fluid
mechanics. They may involve the coupling of 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
to take into account *e.g.* a fluids' exotic equation of state, or because the
amount of required computational resources becomes huge, as in unsteady flows. Another situation
where specific tools are needed is when one is interested in very specific physical quantities, such as
*e.g.* 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
simulate a lot of different flow types. The quality of the results is however
far from optimal in many cases. Moreover, the numerical technology implemented in these codes
is often not the most recent. To give a few examples, consider the noise generated
by wake vortices in supersonic flows (external aerodynamics/aeroacoustics),
or the direct simulation of a 3D compressible mixing layer in a complex geometry (as in combustion chambers).
Up to our knowledge, due to the very different temporal
and physical scales need to be captured,
a direct simulation of these phenomena is
not in the reach of the most recent technologies because the numerical
resources required are currently unavailable !
*We need to invent specific algorithms for this purpose.*

In order to efficiently simulate these 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*, known in literature as
Residual Distribution schemes, *specifically
tailored to unstructured and hybrid meshes*. They have the most possible compact
stencil that is compatible with the expected order of accuracy.
This *accuracy is at least of second order, and it can go up to
fourth order in practical applications.*
Since the stencil is compact, the implementation on parallel machines becomes
simple. These schemes are very flexible in nature, which is so far one of the most importat advantage
over other techniques. This feature has allowed us to adapt the schemes to the requirements of different
physical situations (*e.g.* different formulations allow either en efficient explicit
time advancement for problems involving small time-scales, or a fully implicit space-time
variant which is unconditionally stable and allows to handle stiff problems
where only the large time scales are relevant). This flexibility has also enabled
to devise a variant using the same data structure of the popular Discontinuous Galerkin
schemes, which are also part of our scientific focus.

The compactness of the second order version of the schemes enables us to use efficiently the high performance parallel linear algebra tools developed by the team. However, the high order versions of these schemes, which are under development, require modifications to these tools taking into account the nature of the data structure used to reach higher orders of accuracy. This leads 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, material science.

Within a few years, we expect to be able to deal with physical problems out of today's reach, such as aeroacoustics, unsteady aerodynamics, and compressible MHD (in relation with the ITER project). This will be achieved by means of a multi-disciplinary effort involving our research on compact distribution schemes, the parallel advances in algebraic solvers and partitioners, and the strong interactions with specialists in computer science and scientific computing.

Another topic of interest is the quantification of uncertainties in non linear problems. In many applications, the physical model is not known accurately. A typical example is the one of turbulent flows where, for a given turbulent model which depends on many coefficients, the coefficients themselves are not know accurately. A similar situation occur for real gas or multiphase flows where the equation of state form suffer from uncertainties. The dependency of the model with respect to these uncertainties can be studied by propagation of chaos techniques such as those developped during the recent years via polynomial chaos techniques. Different implementations exists, depending whether the method is intrusive or not. The accuracy of these methods is still a matter of research, as well how they can handle an as large as possible number of uncertainties or their versatility with respect to the structure of the random variable pdfs.

Our research in numerical algorithms has led to the development of the
`RealfluiDS ` platform which is described in
section
5.2 . This work is
supported by the EU-Strep IDIHOM, various research contracts and in
part by the ANR-CIS ASTER project (see
section
4.3 also), and also by the
ERC grant ADDECCO.