Overview
This study addresses the computational challenges associated with simulating variable-density Cahn–Hilliard–Navier–Stokes (CHNS) flows on finite element meshes that undergo changes between accepted time levels. Specifically, it focuses on scenarios involving fixed-topology mesh motion or topology-changing remeshing. A core challenge arises when discrete spaces vary over time, preventing the direct transfer of phase, kinetic, and pressure histories, which are typically measured in distinct discrete structures, by a single operator.
Research Context
The Cahn–Hilliard–Navier–Stokes equations model complex fluid dynamics involving phase separation and fluid motion. Accurate and stable numerical discretizations of these equations are critical, particularly when dealing with evolving computational domains. Traditional multistep schemes often rely on the assumption of a static discrete structure, which is violated in applications requiring adaptive meshing or fluid-structure interaction where mesh topology or node positions change over time. The history terms in multistep schemes—which carry information from previous time steps—become problematic when the underlying discrete representation changes.
Approach
The researchers developed decoupled backward Euler (BE) and second-order backward differentiation formula (BDF2) schemes designed to accommodate variable-density CHNS discretizations on evolving finite element meshes. The methodology integrates exact physical cross-mesh pairings with history representations. These representations are specifically crafted to be compatible with the corresponding phase-energy, kinetic-energy, and pressure-gradient storages within the changing discrete structures. This compatibility is crucial for accurately transferring historical data across different mesh configurations.
Key elements of the approach include:
- The phase update component, which simultaneously determines an Abels–Garcke–Grün-consistent mass flux. This flux is subsequently utilized in the momentum transport equations.
- A scalar capillary-exchange equation implemented to separate the phase and fluid solve steps. This separation is achieved while ensuring the discrete energy exchange is maintained.
The resulting field subproblems derived from these schemes are linear. The scalar capillary-exchange equation specifically yields a unique positive solution. A significant characteristic of these schemes is their ability to satisfy modified energy balances without imposing a time-step restriction, provided certain admissibility assumptions are met.
Findings
Numerical experiments conducted using the developed schemes confirmed several key properties and performance metrics:
- Temporal Convergence: The schemes demonstrated second-order temporal convergence under both types of mesh updates considered: fixed-topology motion and topology-changing remeshing.
- Mass Conservation: Phase-mass conservation was observed across the simulations.
- Energy Decay: In unforced test cases, the schemes exhibited modified-energy decay, indicating stability.
- Dynamic Behavior: The schemes accurately reproduced comparable Rayleigh–Taylor and rising-bubble dynamics, suggesting their applicability to standard fluid instability and multiphase flow problems.
The linearity of the field subproblems and the unique positive solution for the scalar capillary-exchange equation contribute to the practical solvability and robustness of the proposed methods. The satisfaction of modified energy balances without time-step restriction, under specified admissibility assumptions, is a critical finding for computational efficiency and stability in challenging variable-density multiphase flow simulations.
Why This Matters
The development of energy-stable finite element schemes for variable-density CHNS flows on evolving meshes addresses a significant computational challenge in simulating dynamic multiphase systems. The ability to handle mesh changes, whether through motion or remeshing, while maintaining numerical stability and accuracy (second-order temporal convergence, mass conservation, modified-energy decay) removes a common limitation in adaptive simulation strategies. The schemes' linearity and lack of time-step restrictions, under admissibility, enhance their utility for efficient and reliable large-scale simulations in fields where variable-density multiphase flows and evolving geometries are critical, such as fluid dynamics, materials science, and biomedical engineering.