1.10. OpenFOAM and driftFluxFoam

These are notes on driftFluxFoam (Brennan 2001).

1.10.1. Two-fluids Model Conservation of Mass

\[\begin{split}{{\partial } \over {\partial t}}(\alpha_{n} \rho_n) + \nabla \cdot (\alpha_{n} \rho_n \vec{u}_{n}) = 0 \\\end{split}\] Conservation of Momentum

\[{{\partial } \over {\partial t}} (\alpha_{n} \rho_n \vec{u}_{n}) + \nabla \cdot ( \alpha_{n} \rho_n \vec{u}_{n} \vec{u}_{n}) = - {{\alpha_{n}}} \nabla {p_n} + \nabla \cdot ( \alpha_{n} \vec{\vec{\tau}}_{n} ) + \alpha_{n} \rho_n \vec{g} + \vec{\vec{M}}\]

1.10.2. Mixture Model

  • Consider a mixture, not two phases: one continuity equation, one momentum equation and one convective diffuction equation for the dispersed phase

  • Increased efficiency: reduces number of equations from 4 to 3

  • Increased robustness: interphase momentum transfer term disappears (as momentum equations are added and the terms are equal and opposite)

  • Physically realistic: main slip between phases is gravitational settling, the settling velocity

  • New concepts: mixture density, centre of mass of mixture velocity, and diffusion velocities with respect to centre of mass of mixture velocity Mixture Density

\[\rho_m = \alpha_1 \rho_1 + \alpha_2 \rho_2\] Centre of Mass of Mixture Velocity

\[u_m = {{\alpha_1 \rho_1 u_1 + \alpha_2 \rho_2 u_2} \over {\rho_m}}\] Diffusion Velocities wrt Centre of Mass of Mixture Velocity

\[u_{1m} = u_1 - u_m\]
\[u_{2m} = u_2 - u_m\] Mixture Conservation of Mass

The above concepts can be used to derive a new continuity equation for the mixture model:

\[{{\partial \rho_m} \over {\partial t}} + \nabla \cdot (\rho_m \vec{u}_m) = 0\]