## Section: Scientific Foundations

Keywords : Conservation Laws, Anti-Diffusive Schemes, Viscous Flows, Control, Turbulence, Finite elements methods (finite element method, finite element method, finite elements method), Finite volume methods.

### Simulations of Complex Fluid Flows

#### Conservation Laws

A major issue in the numerical analysis 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

Nowadays, passive control techniques are widely used to improve the performances of planes or vehicles. In particular these devices can sensibly reduce energy consumption or noise disturbances. However, new improvements can be obtained through an active control of the flow, which means by activating mechanical devices. This is a very promising theme.

The first results are concerned with the control of the 2D compressible Navier-Stokes equations over a dihedral plane. The technical device consists in a small hole which allows to suck or to inject some fluid in the flow, depending on the pressure measured at another point. This improves the aerodynamics performance of the dihedral. Variants are possible, for instance by considering several such devices and taking into account the local properties of the flow.

Another work is concerned with simulation of the control of low Reynolds number flows (laminar regimes) over a backward facing step by imposing pulsed inlet velocities. Such a flow can be considered as a toy-model for the modelling of combustion phenomena. The goal is to understand and control vortex formations, by making the frequency and amplitude of the incoming fluid vary.

Recently, previous results on the step were generalized to the transitional regime, with a work of E. Creusé, A. Giovannini (IMFT Toulouse) and I. Mortazavi (EPI INRIA MC2, Bordeaux). The nonstationarity property of the uncontrolled flow allows to use some closed-loop control strategies. The control process is either a global one, by imposing a pulsed inlet velocity like for the laminar case, or a local one by the use of two horizontal jets located on the vertical side of the step.

Our current objective is now to apply such techniques on the "Ahmed body geometry", which can be considered as a first approximation of a vehicle profile. This work is performed in collaboration between E. Creusé and C.H. Bruneau (EPI INRIA MC2, Bordeaux) in the context of the research and innovation program on terrestrial transports supported by the ANR and the ADEME, leaded by Renault and PSA and managed by Jean-Luc Aider (ESPCI Paris). We have in mind to combine active and passive control strategies in order to reach efficient results, especially concerning the drag coefficient, on two and three dimensional simulations. Another important point consists in deriving more sophisticated control laws, using either adaptive or optimal control processes.

#### Numerical Methods for Viscous Flows

In the large scale computations of fluid flows, several different numerical quantities appear that are associated to different eddies, structures or scales (in space as well as in time). An important challenge in the modelling of turbulence and of the energy transfer for dissipative equations (such as Navier-Stokes equations, reaction-diffusion equations) is to describe or to model, for the long time behavior, the interaction between large and small scales. They are associated to slow and fast wavelengths respectively. The multiscale method consists in modelling this interaction on numerical grounds for dissipative evolution equations. In Finite Elements and Finite Differences discretizations the scales do not appear naturally as in spectral approximations, their construction is obtained by using a recursive change of variables operating on nested grids; the nodal unknowns (Y) of the coarse grids are unchanged (they are of the order of magnitude of the physical solution) and those of the fine grids are replaced by proper error interpolation, namely the incremental unknowns (Z); the magnitude of the Zs is then "small". This allows to make a separation of the eddies in space (presence of nodal and incremental quantities) but also in frequency since the incremental unknowns are supported by the fine grids which capture the high frequencies while the nodal unknowns are defined on coarse grids which can represent only slow modes. Note that this approach differs from the LES model that proposes to split the flow into a mean value and a fluctuation component, the latter having small moments but not necessarily a small magnitude. This change of variable defines also a hierarchical preconditioner. It is well known that the (semi)explicit time marching schemes have their stability region limited by the high modes, so a way to enhance the stability is to tread numerically the scales (Y and Z) in a different manner. The inconsistency carried by the new scheme acts only on small quantities allowing for efficient and accurate schemes for the long time integration of the equations. We develop and apply this approach to the numerical simulation of Navier-Stokes equations in highly non stationary regimes. In this framework of numerical methods, we focus on the domain decomposition method together with multiscale method for solving incompressible bidimensional NSE; the stabilized explicit time marching schemes are also studied.

The already written code can be used to treat certain low Mach number 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 will develop a density dependent Navier-Stokes code, in 2D, the incompressibility condition being replaced by a non standard condition on the velocity field. The numerical strategy we use mixes a Finite Element method for computing the velocity field to a Finite Volume approach to evaluate the density. As a by-product, the code should be able to compute a solution of the 2D incompressible Navier-Stokes system, with variable density. On the other hand, we wish to extend our kinetic asymptotic-based schemes to such problems.

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