A new numerical algorithm based on the ALE formulation has been developed for a fully coupled solution of the free-surface problems. The mesh deformation is handled by solving the linear elasticity equations. At the interface, the mesh motion in the normal direction is related to interface normal velocity component meanwhile the linear elasticity equation is solved in tangential direction. The surface tension is computed as a tangential force at the interface using high-order polynomials. The pressure and viscous forces are treated as discontinuous functions at the interface while satisfying the jump conditions.

The rising bubble benchmark problem given by Hysing and Turek, 2007  (C. Guventurk).

The present Arbitrary-Lagrangian approach which combines the advantages of both Lagrangian and Eulerian methods
has been extended to three-dimensions through the TUBITAK 217M358 project. A special attension is given to satisfiy the discrete geometric conservation law (DGCL) for the kinematic boundary condition in order to conserve the total mass for each species at machine precision in tree-dimensions.  The interface normal vector is computed using the mean weighted by sine and edge reciprocal (MWSELR) approach (Max, 1999) and the surface tension force is treated to be a tangent force at the interface. The present approach significantly reduce the parasitic currents. The surface tension force is also treated semi-implicitly. The resulting algebraic equations are solved in a fully coupled manner and a more advanced block proconditioner has been implement to allow large-scale parallel multiphase simulations.

The three-dimensional  rising bubble benchmark problem given by Andelsberger et al., 2014  (C. Guventurk).


The three-dimensional  rising Taylor bubble problem given by Bugg and Saad., 2002  (C. Guventurk).