A Model for Two Coupled Fluids: Numerical Approximation by Finite Element

Speaker: Dr. Driss Yakoubi

Affiliation: Universite de Laval

This talk focuses on numerical analysis and simulation of two models in fluid mechanics.

The first one deals with a coupled two-fluid RANS turbulence model, which couples the steady Navier-Stokes Equations (NSE) with the equation for the turbulent kinetic energy (TKE). The link includes the eddy viscosities, the boundary condition on the interface and the source term in the energy equation, which is only in $L^1$ and then presents a high complexity. We change the initial system to a new variational system, whose eddy viscosities and source term are regularized by convolution. We perform a full finite element discretization of an iterative linearization procedure and prove its convergence to the continuous scheme for large enough eddy viscosities.

The second part concerns two immiscible Newtonian fluids where the surface tension at the fluid interface has to be accounted for. The surface tension is modeled as a Continuum Surface Force (CSF). The CSF model allows us to treat the dynamic boundary condition at the interface implicitly. The main difficulty in this problem is that, of course, each fluid flows through a time-dependent domain. I will investigate the existence of a solution in the non-realistic case where the velocity satisfies homogeneous boundary conditions. Next, I will propose a discretization of it by the characteristics method in time and standard conforming finite elements in space.