Hybrid Finite Element and Finite Volume Methods for Two Immiscible Fluid Flows

We have successfully extended our implicit hybrid finite element/volume solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered finite volume method. The pressure Poisson equation is solved by the node-based Galerkin finite element method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid-interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrixfree finite volume method as the one used for momentum equations is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm which enforces the conservation of the mass for each fluid.