MOOSE-based finite element framework for mass-conserving two-phase flow simulations on adaptive grids using the diffuse interface approach and a Lagrange multiplier

A numerical framework capable of simulating incompressible laminar two-phase flows has been developed within the Multiphysics Object-Oriented Simulation Environment (MOOSE). The fully-coupled and fully-implicit-in-time methodology relies on the continuous Galerkin finite element discretization of the coupled Cahn-Hilliard Navier-Stokes (CHNS) equations. Despite the computational advantages of adaptive mesh refinement (AMR), mass-conserving interpolation schemes do not exist for transferring the solution to a newly adapted mesh. This paper introduces a new time-dependent scalar Lagrange multiplier to ensure mass conservation on adaptive grids while efficiently handling the interpolation errors involved in mesh coarsening. To assess the accuracy of the numerical implementation, several two-phase flow benchmark problems have been studied and validated against reference solutions. The comparisons demonstrate the accuracy of the code and the overall methodology. The proposed method can be effectively applied to any 2D, 2D axisymmetric and 3D complex immiscible two-phase flows, leveraging conservative AMR without compromising conservation principles.