An Arbitrary Lagrangian Eulerian Formulation with Exact Mass Conservation for the Numerical Simulation of a Three-Dimensional Rising Bubble Problem

An arbitrary Lagrangian Eulerian (ALE) framework has been proposed to solve incompressible multiphase fluid flow problems with exact mass conservation in three-dimensions. The incompressible Navier-Stokes equations are discretized over unstructured moving hexahedral meshes using the div-stable face-centered finite volume method where the continuity equation is satisfied exactly within each element. The pressure field is treated to be discontinuous across the fluid-fluid interface with the discontinuous treatment of density and viscosity. The surface tension term is considered as a force tangent to the interface. For the application of the interface kinematic boundary condition due to the normal displacement of interface, a special attention is given to satisfy both local and global discrete geometric conservation law (DGCL) in order to conserve the total mass of both species at machine precision. The resulting algebraic equations are solved in a fully coupled (monolithic) manner and a one-level restricted additive Schwarz preconditioner with a block-incomplete factorization is utilized within each partitioned sub-domain. The proposed method is validated by simulating a classical single rising bubble in a viscous fluid due to buoyancy in three-dimensions.