Implicitly coupled phase fraction equations for the Eulerian multi-fluid model

In this work, the implementation, verification and validation of an implicitly coupled solution procedure for the phase fraction equations in the Eulerian multi-fluid model are presented. The model is implemented within the foam-extendtoolbox, a community-driven fork of OpenFOAM. The implicitly coupled system for an arbitrary number of phases is based on the modified formulation of the phase fraction equation. This formulation takes advantage of the mixture divergence-free velocity and the cross-coupling with the remaining phase fraction equations via the non-linear relative velocity term. The linearised and implicitly coupled phase-fraction equations are solved simultaneously within a single block matrix. The model is tested for a bubbly air-water upward flow which forms a mixing layer inside a square duct. In the first test, the mesh verification analysis is performed on structured grids with different levels of refinement. The second test investigates the influence of the number of bubble phases on the flow solution for the same flow conditions. In the third test, the implemented model is validated against experimental data from the literature. The last test compares the performance of the implemented implicitly coupled solution procedure for the phase fraction equations against the standard segregated implementation. The proposed method shows good agreement with experimental data, and has proven to be consistent both in terms of the number of phases and grid refinement. Furthermore, the method improved the convergence of the solution for flows at higher bubble phase fraction.