A monolithic finite element approach for interface capturing in multiphase flow problems

Abstract

Multiphase flows are fundamental to diverse industrial applications, from inkjet technology and droplet formation to bubble dynamics in chemical reactors. Despite their prevalence, the numerical simulation of interfacial dynamics remains a significant challenge due to the stiff non-linearities introduced by surface tension and the geometric complexities of moving boundaries. Traditional numerical frameworks often struggle with restrictive capillary time step constraints, the need for frequent interface reinitialization, and the inaccuracies associated with explicit curvature estimation.

This thesis presents a robust, monolithic finite element solver designed for the simultaneous calculation of velocity, pressure, and interface position in incompressible multiphase flows. Building upon curvature free level set and indicator-based material function formulations, the proposed method eliminates the need for explicit calculation of interface normals and curvature. This approach bypasses the traditional capillary time step restriction and removes the necessity for separate redistancing or reinitialization procedures, which are integrated directly into the non-linear formulation.

The governing equations are discretized using the high-order stable FEM pair Q_2⁄(P_1^disc ) for velocity and pressure, while a Q_2approximation is employed for the interface scalar fields. The resulting non-linear system is resolved using a discrete Newton solver with a divided difference evaluation of the Jacobian matrices. To handle the resulting saddle-point problems, a suitable linear solver is implemented within the FeatFlow software framework.

The accuracy, stability, and mass conservation properties of the solver are rigorously validated against established test cases and benchmark, including the static bubble, the oscillating bubble, and the rising bubble cases. The results demonstrate that the monolithic coupling provides an efficient and robust framework capable of handling high surface tension coefficients without compromising interface integrity, offering a significant advancement over decoupled or explicit interfacial solvers.