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

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

Contribution title: Numerical methods for complex flows

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.