# thierry Goudon - Université Côte d'Azur, Inria, CNRS, LJAD - Keywords: fluid mechanics, complex flows, mixture, numerical simulation

The development of numerical methods for fluid mechanics is still a very active field of research.

We have developped numerical methods for the simulation of complex flows characterized by

1/ the influence of density gradients: the flow is non homogeneous, and the numerical challenge is precisely to capture the density variations;

2/ a constraint on the velocity field.

We can split the activity by distinguishing

a/ the simulation of the Incompressible Navier--Stokes system with variable density,

b/ the modeling of mixture flows, that makes unusual constraints appear.

A common feature

of the different numerical strategies we develop is to work, in a convenient way, on staggered grids.

Surprisingly enough, there are not many studies on the incompressible Navier-Stokes equations with variable density,

compared to the huge literature on the constant density case. The difficulty is related to the fact that the system mixes equations

of different types, roughly speaking a hyperbolic conservation law for the mass conservation

coupled to an equation of parabolic type for the momentum equation; the whole being

coupled to the divergence free constraint.

We have proposed a new numerical method based on an

hybrid Finite Volume/Finite Element scheme where both the mass conservation and

the momentum equation are treated by a well adapted method. The point consists in

defining a suitable footbridge between the two schemes.

We have also developped an approach based entirely on the

Finite Volume framework (DDFV schemes).

In any cases, the basic requirement which guides our construction of

numerical fluxes

consists in preserving the homogeneous solutions.

These approaches

give a lot of freedom concerning the meshes

and they allow the

use of non structured meshes if necessary, opening perspectives to complex geometries, front

detection and mesh refinements.

We are also interested

in the modeling of mixtures: in such situations, there is a constraint that relates the

divergence of the velocity field and derivatives of the density.

This is the case for the Kazhikov-Smagulov system, where the constraint is intended to describe diffusion fluxes at the interfaces between the

constituents of the fluid, and a similar difficulty appears in zero-Mach flows.

We have adapted our approaches

paying a specific attention to regimes relevant for the simulation of powder-snow avalanches.

Furthermore, we have developped original schemes based on staggered discretization for conservation laws.

The scheme has been designed to deal with the Euler equations.

The adaptation for mixture flows

permits us to preserve exactly the mass conservation and to justify for the discrete models all

identities that appear for the continuous model. Our work also brings out delicate stability issues

related to the close--packing modeling, which imposes a threshold on the particle volume fraction.