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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.


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