An Energy Stable Nonlinear Incompressible Multi-Phase Flow Formulation
Jan Nordström, Arnaud. G. Malan
TL;DR
This work addresses energy stability for incompressible multi-phase flow in the volume-of-fluid setting by reformulating the one-fluid VOF equations into a skew-symmetric, density-based system with scaled variables, enabling a well-defined energy rate $dE/dt$. The authors derive a continuous energy energy-rate expression and design boundary conditions (inflow-outflow with nonlinear energy-based BCs and solid-wall with SAT penalties) that bound the energy by external data, yielding strong energy stability. They also outline a straightforward nonlinear SBP-based discretization that preserves the energy structure at the semi-discrete level, indicating a path to provably energy-stable schemes for VOF-based simulations. The methodology provides a rigorous framework for energy bounds and practical numbers for boundary treatment, with potential impact on reliable nonlinear simulations of incompressible multi-phase flows.
Abstract
We show that a reformulation of the governing equations for incompressible multi-phase flow in the volume of fluid setting leads to a well defined energy rate. Weak nonlinear inflow-outflow and solid wall boundary conditions complement the development and lead to an energy estimate in terms of external data. The new formulation combined with summation-by-parts operators lead to provably nonlinear energy stability.