Section: Scientific Foundations

Simulations of Complex Fluid Flows

Conservation Laws

A major issue in the numerical approximation of systems of conservation laws is the preservation of singularities (shocks, contact discontinuities...). Indeed, the derivatives of the solutions usually blow-up in finite time. The numerical scheme should be able to reproduce this phenomenon with accuracy, i.e. with a minimum number of points, by capturing the profile of the singularity (discontinuity), and by propagating it with the correct velocity. The scheme should also be able to give some insight on the interactions between the possible singularities. Quite recently, new anti-diffusive strategies have been introduced, and successfully used on fluid mechanics problems. We focus on multidimensional situations, as well as on boundary value problems. Since a complete theory is not yet available, the numerical analysis of some prototype systems of conservation laws is a good starting point to understand multi-dimensional problems. In particular, a good understanding of the linear case is necessary. This is not achieved yet on the numerical point of view on general meshes. This question is particularly relevant in industrial codes, where one has to solve coupled systems of PDEs involving a complex coupling of different numerical methods, which implies we will have to deal with unstructured meshes. Thus, deriving non-dissipative numerical schemes for transport equations on general meshes is an important issue. Furthermore, transport phenomena are the major reason why a numerical diffusion appears in the simulation of nonlinear hyperbolic conservation laws and contact discontinuities are more subject to this than shocks because of the compressivity of shock waves (this is another reason why we focus at first on linear models).

The next step is to combine non-dissipation with nonlinear stability. An example of such a combination of preservation of sharp shocks and entropy inequalities has been recently proposed for scalar equations and is still at study. It has also been partially done in dimension one for Euler equations.

Of course, there are plenty of applications for the development of such explicit methods for conservation laws. We are particularly interested in simulation of macroscopic models of radiative hydrodynamics, as mentioned above. Another field of application is concerned with polyphasic flows and it is worth specifying that certain numerical methods designed by F. Lagoutière are already used in codes at the CEA for that purpose. We also wish to apply these methods for coagulation-fragmentation problems and for PDEs modelling the growth of tumoral cells; concerning these applications, the capture of the large time state is a particularly important question.

Control in Fluid Mechanics

Flow control techniques are widely used to improve the performances of planes or vehicles, or to drive some internal flows arising for example in combustion chambers. Indeed, they can sensibly reduce energy consumption, noise disturbances, or prevent the flow from undesirable behaviors.

Recently, open and closed active flow control were carried out in order to study the flow behavior over a backward-facing step in a transitional regime. It was done either by a global frequency destabilization at the entry of the domain, or by a local blowing or suction through the lower and upper parts of the step by the use of small jets ( [54] , E. Creusé, A. Giovannini (IMFT Toulouse) and I. Mortazavi (MC2 INRIA EPI, Bordeaux)). The numerical computations were based on a vortex-in-cell method. Such controls were shown to be efficient in reducing the average recirculation length value, the global flow energy, as well as the global flow enstrophy. We have now in mind to apply such a strategy on cavity-stent flows, in order to study the effect of passive and/or active control on the average emptying time of the cavity, corresponding to a lot of possible industrial or health applications (combustion, blood circulation in arteries,...).

Passive as well as active control were also performed on the "Ahmed body geometry", which can be considered as a first approximation of a vehicle profile. This work was carried out in collaboration with the EPI INRIA MC2 team in Bordeaux (C.H. Bruneau, I. Mortazavi and D. Depeyras), as well as with Renault car industry (P. Gillieron). We recently combined active and passive control strategies in order to reach efficient results, especially concerning the drag coefficient, for two and three dimensional simulations [46] . We are now interested in the same kind of study, but for a ${25}^{\circ }$ rear-window configuration, for which the 3D-effects are very important and have to be considered in the numerical simulations.

In another field of applications, a work was performed with the TEMPO Laboratory of Valenciennes. The objective of this collaboration was to study the pressure wave generated by high-speed trains entering tunnels in order to improve the shape of the tunnel sections.

Numerical Methods for Viscous Flows

Numerical investigations are very useful to check the behavior of systems of equations which modeling very complicate dynamics. In order to simulate the motion of mixtures of immiscible fluids having different densities, a recent contribution of the team was to develop an hybrid Finite Element / Finite Volume scheme for the resolution of the variable density 2D incompressible Navier-Stokes equations. The main points of this work were to ensure the consistency of the new method [49] as well as its stability for high density ratios [47] . In order to answer to these questions, we have developed respectively a MATLAB code and a C++ code. In the following of this work, we now have in mind the following objectives :

• To allow the corresponding MATLAB code distribution, in order to promote some further collaborations with other researchers in the domain, and to make the comparison of our results with alternative numerical methods possible. For this objective, a graphics interface was developed as well as some post-processing tools and an accurate documentation. This was the object of the Manuel Bernard internship in the SIMPAF team (Mars 2011 – Sept. 2011). For exemple, the code was already used to study the influence of new strategies for updating LU factors of existing preconditioners. In [48] C. Calgaro et al. address the problem of computing preconditioners for solving linear systems of equations with a sequence of slowly varying matrices. This problem arises in many important applications, for example in computational fluid dynamics, when the equations change only slightly possibly in some parts of the domain.

• To generalize the stability results obtained in [47] for the scalar transport equation to the full 2D Euler system. The target is now to ensure a positivity principle for vertex-based finite volume methods, allowing to simulate some cases involving in particular very low density values density (near vacuum), while maintaining a sufficient accuracy. This work is being developed by Yohan Penel during his post-doc position in the SIMPAF team (Nov. 2010 – Dec. 2011), in collaboration with C. Calgaro, E. Creusé and T. Goudon;

• To modify the already existing C++ code to treat certain more general hydrodynamic models arising in combustion theory, as well as models describing mixing of compressible fluids arising for instance when describing the transport of pollutants. The interesting thing is that this kind of model can be derived by a completely different approach through a kinetic model. Besides, this model presents interesting features, since it is not clear at all whether solutions can be globally defined without smallness assumptions on the data. Then, a numerical investigation is very useful to check what the actual behavior of the system is. Accordingly, our program is two-fold. On the one hand, we develop the density dependent Navier-Stokes code, where the incompressibility condition is replaced by a non standard condition on the velocity field. In particular, if the closure model is the Fick's law, one obtain the so called Kazhikhov-Smagulov model. The first phenomena we try to reproduce are the powder-snow avalanches. The influence of the characteristic Froude number on the front progression is also investigate. On the other hand, we wish to extend our kinetic asymptotic-based schemes to such problems. This work is being developed in collaboration between C. Calgaro, E. Creusé and T. Goudon (INRIA Sophia Antipolis, team COFFEE).

• In the case of the PhD of M. Ezzoug, co-advised by C. Calgaro and E. Zahrouni (Monastir University, Tunisie), to study numerically and theoretically the influence of a specific stress tensor, introduced for the first time by Korteweg.

• Finally, to prove the convergence of the numerical scheme in order to ensure the theoretically performance of the method. This work started very recently by a collaboration between C. Calgaro, E. Creusé and E. Zahrouni (Monastir University, Tunisie).