A discretization of phase mass balance in fractional step algorithms for the drift-flux model
Résumé
We address in this paper a parabolic equation used to model the phases mass balance in two-phase flows, which differs from the mass balance for chemical species in compressible multi-component flows by the addition of a non-linear term of the form $\dive \rho \phi(y) \, u_r$, where $y$ is the unknown mass fraction, $\rho$ stands for the density, $\phi(\cdot)$ is a regular function such that $\phi(0)=\phi(1)=0$ and $u_r$ is a (non-necessarily divergence free) velocity field. We propose a finite-volume scheme for the numerical approximation of this equation, with a discretization of the non-linear term based on monotone flux functions \cite{eym-00-fin}. Under the classical assumption \cite{lar-91-how} that the discretization of the convection operator must be such that it vanishes for constant $y$, we prove the existence and uniqueness of the solution, together with the fact that it remains within its physical bounds, \ie\ within the interval $[0,1]$. Then this scheme is combined with a pressure correction method to obtain a semi-implicit fractional-step scheme for the so-called drift-flux model. To satisfy the above-mentioned assumption, a specific time-stepping algorithm with particular approximations for the density terms is developed. Numerical tests are performed to assess the convergence and stability properties of this scheme.
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