Section: Research Program

Numerical schemes for fluid mechanics

Participants : Luca Arpaia, Héloïse Beaugendre, Pietro Marco Congedo, Cécile Dobrzynski, Andrea Filippini, Maria Kazolea, Luc Mieussens, Mario Ricchiuto, Maria Giovanna Rodio.

A large number of engineering 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, and there are even more exotic situations. In some occasions, one needs specific numerical tools to take into account e.g. a fluids' exotic equation of state, or a the influence of small flow scales in a macro-/meso-scopic flow model, etc. Efficient schemes are needed in unsteady flows where the amount of required computational resources becomes huge. 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, or the boundary of the area flooded by a Tsunami.

In these situations, commercial tools can only provide a crude answer. These codes, while allowing users to simulate a lot of different flow types, and “always” providing an answer, often give results of poor quality. This is mainly due to their general purpose character, and on the fact that the numerical technology implemented in these codes is 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 that 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.

Our goal is to develop more accurate and more efficient schemes that can adapt to modern computer architectures, and allow the efficient simulation of complex real life flows.

We develop 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 any order of accuracy, even though fourth order is considered for 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 important 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, geophysical flows, and material science. A particular attention has been devoted to the implementation of complex thermodynamic models permitting to simulate several classes of fluids and to take into account real-gas effects and some exotic phenomenon, such as rarefaction shock waves.

Within these applications, a strong effort has been made in developing more predictive tools for both multiphase compressible flows and non-hydrostatic free surface flows.

Concerning multiphase flows, several advancements have been performed, i.e. considering a more complete systems of equations including viscosity, working on the thermodynamic modelling of complex fluids, and developing stochastic methods for uncertainty quantification in compressible flows. Concerning depth averaged free surface flow modelling, on one hand we have shown the advantages of the use of the compact schemes we develop for hydrostatic shallow water models. On the other, we have shown ho to extend our approach to non-hydrostatic Boussinesq modelling, including wave dispersion, and wave breaking effects.

We expect to be able to demonstrate the potential of our developments on applications ranging from the the reproduction of the complex multidimensional interactions between tidal waves and estuaries, to the unsteady aerodynamics and aeroacoustics associated to both external and internal compressible flows, and the behaviour of complex materials. This will be achieved by means of a multi-disciplinary effort involving our research on residual discretizations schemes, the parallel advances in algebraic solvers and partitioners, and the strong interactions with specialists in computer science, scientific computing, physics, mechanics, and mathematical modeling.

Concerning the software platforms, our research in numerical algorithms has led to the development of the RealfluiDS platform which is described in section  5.3 , and to the SLOWS (Shallow-water fLOWS) code for free surface flows, described in sections  5.10 . Simultaneously, we have contributed to the advancement of the new, object oriented, parallel finite elements library AeroSol , described in section 5.1 , which is destined to replace the existing codes and become the team's CFD kernel. Concerning uncertainty quantification and robust optimization, we are developing the platform RobUQ .

New software developments are under way in the field of complex materials modeling and multiphase flows with heat and mass transfer. Concerning the materials modelling, these developments are performed in the code in the solver COCA (CodeOxydationCompositesAutocicatrisants) for the simulation of the self-healing process in composite materials. These developments will be described in section  5.2 . Concerning the numerical simulation of multiphase flows, we have developed the code sDEM , which is one of rare code, permitting to simulate metastable states with a complex thermodynamics and considering uncertainty quantification techniques.

Funding and external collaborations. This work is supported by several sources including the last of the ADDECCO ERC grant, the FP7 STORM, the ANR UFO and the PIA project TANDEM . Important contributions to these activities are given by our external collaborators, and in particular R. Abgrall (Universit$\ddot{\text{a}}$t Z$\ddot{\text{u}}$rich), P. Bonneton (UMR EPOC Bordeaux), G. Vignoles (LCTS lab Bordeaux), and D. De Santis (via the associated team AQUARIUS ).