cfdmfFTFoam: A front-tracking solver for multiphase flows on general unstructured grids in OpenFOAM

This paper introduces cfdmfFTFoam, a novel OpenFOAM-based Front-Tracking solver designed for multiphase flows on general unstructured grids, which addresses the scarcity of open-source FTM tools by integrating advanced algorithms for front mesh communication, advection, and correction to enable accurate and efficient simulations in both serial and parallel environments.

The Gist

Imagine you are trying to simulate how a drop of oil moves through water, or how a bubble rises in a glass of soda. In the world of computer simulations, this is a notoriously difficult puzzle. You have two different "worlds" trying to talk to each other: the fluid (the water and oil) and the interface (the invisible skin separating them).

For a long time, computer scientists had to choose between two bad options:

1. The "Blurry Photo" approach: They treated the boundary as a fuzzy line. It was easy to calculate, but the oil and water would slowly mix together in the computer, losing their sharp edges.
2. The "Rigid Grid" approach: They forced the simulation to happen on a perfect, square grid (like graph paper). This was accurate, but it couldn't handle complex shapes or real-world engineering problems where the space isn't a perfect box.

Enter cfdmfFTFoam. Think of this new software as a high-tech, shape-shifting GPS system that can navigate any terrain, no matter how messy.

Here is how it works, broken down into simple analogies:

1. The Two Teams: The Grid and the Dancers

Imagine the fluid (water/oil) is a giant, invisible grid of tiles covering the floor. This is the "Eulerian grid." It stays mostly still, just measuring the speed and pressure of the fluid passing over it.

Now, imagine the boundary between the oil and water is a team of dancers holding hands in a circle. This is the "Front." Unlike the grid, the dancers can move anywhere, stretch, twist, and spin. They don't care about the tiles on the floor; they just follow the music (the fluid flow).

The Problem: The dancers need to tell the tiles, "Hey, we are here, and we are pushing you!" and the tiles need to tell the dancers, "The floor is slippery here, move faster!"

2. The Magic Translator (RKPM)

In older software, the dancers and the tiles spoke different languages, or the grid was too rigid to understand the dancers' complex moves.

cfdmfFTFoam introduces a brilliant translator called RKPM. Imagine the dancers are throwing invisible "sticky notes" (data) onto the floor tiles. Even if a dancer is standing between four tiles, this translator calculates exactly how much of the dancer's "push" belongs to each tile. This allows the simulation to work on unstructured grids—meaning the floor tiles can be triangles, hexagons, or weird shapes, just like a real, messy engineering part.

3. The "Stretchy" Problem (Volume Correction)

There's a catch. When dancers spin and stretch, sometimes they accidentally lose a little bit of their "hand-holding." In physics terms, the drop of oil might shrink or grow slightly just because of math errors.

To fix this, the software has a Volume Correction feature. Think of it like a smart elastic band. If the dancers notice the circle is getting slightly too small or too big, the software gently nudges every dancer inward or outward by a tiny, calculated amount to restore the perfect size. It's like a self-correcting hug that ensures the oil drop never accidentally evaporates or expands in the computer.

4. The Shape-Shifting Mesh (Remeshing)

As the dancers spin, the circle they form might get too stretched out (long and skinny) or too squished. If the shapes get too weird, the math breaks.

The software has a Remeshing feature. Imagine the dancers have a magical ability to instantly swap partners. If a part of the circle gets too stretched, they quickly grab a new neighbor to form a perfect triangle again. If a part gets too crowded, they let go of a partner to make room. This keeps the "dance floor" looking neat and healthy, no matter how wild the dance gets.

5. Why This Matters

Before this tool, if you wanted to simulate a complex shape (like a car part or a human heart valve) with oil and water, you were stuck. You either had to use a blurry approximation or a rigid grid that couldn't fit the shape.

cfdmfFTFoam is the first open-source tool that lets scientists:

• Use any shape of grid (unstructured).
• Keep the boundary between fluids crisp and sharp (no blurring).
• Run these simulations on supercomputers (parallel processing) to solve big problems fast.

The Bottom Line

This paper introduces a new, free software tool that acts like a super-accurate, shape-shifting camera for fluid dynamics. It allows engineers and scientists to watch exactly how bubbles, drops, and liquids interact in complex, real-world environments without the "blur" or "rigid grid" limitations of the past. It's a major step forward for designing better engines, medical devices, and understanding nature's fluids.

Technical Summary

1. Problem Statement

Multiphase flow simulations involving complex interface phenomena (e.g., droplet collision, bubble rise, spray breakup) require Fully Resolved Simulations (FRS). While interface capturing methods like Volume-of-Fluid (VOF) and Level-Set (LS) are widely available in open-source CFD tools, they suffer from numerical diffusion, leading to interface smearing and inaccurate curvature calculations.

The Front-Tracking Method (FTM) offers a superior alternative by explicitly tracking the interface using a Lagrangian surface mesh, providing sharp interface resolution and accurate curvature computation. However, existing open-source FTM solvers (e.g., PARIS, FronTier++, TrioCFD) are largely restricted to structured Cartesian grids. Furthermore, hybrid approaches (like lentFoam) exist for unstructured grids but are not "pure" FTM solvers, relying on Eulerian fields for interfacial forces. There is a significant gap in the availability of an open-source, pure FTM solver capable of running on general unstructured grids within a parallel computing environment.

2. Methodology

The paper introduces cfdmfFTFoam, a new solver integrated into the OpenFOAM v9 CFD framework. The methodology involves the following key technical components:

• Hybrid Architecture: The solver couples an Eulerian volumetric grid (for solving Navier-Stokes equations) with a Lagrangian surface mesh (the "front") to track the interface.
• Integration of Ftc3D: The core FTM logic is adapted from the Ftc3D code (originally developed by Tryggvason et al. for structured grids) and ported to OpenFOAM's unstructured grid topology.
• Key Algorithms Implemented:
  1. Front-to-Field Communication: To transfer data (e.g., surface tension) from the Lagrangian front to the Eulerian grid, the Reproducing Kernel Particle Method (RKPM) is implemented. This ensures conservation of zeroth- and first-order moments (forces and torques) and handles general unstructured cell connectivity.
  2. Volume Correction (VC2): Since FTM is not inherently mass-conserving, a volume correction algorithm is employed. It adjusts vertex positions along the normal vector by solving a cubic equation to minimize volume error.
  3. Remeshing: A dynamic remeshing strategy controls mesh quality and resolution, including:
     • Refining: Subdividing elements that exceed maximum edge length or aspect ratio thresholds.
     • Coarsening: Merging elements that become too small.
     • Smoothing (Undulation Removal): Algorithms like TSUR3D and VCS are used to remove numerical noise while preserving physical curvature.
  4. Surface Tension Computation: A direct element-based method is used to calculate interfacial forces on triangular elements.
  5. Indicator Function Construction: Two methods are provided: solving a Poisson's Equation (for smooth gradients) and the Closest Point Transform (CPT) (geometric approach).
  6. Front Advection: The Lagrangian points are advected using OpenFOAM's built-in particle tracking capabilities, utilizing trilinear interpolation and explicit Euler integration.

3. Key Contributions

• First Pure FTM on Unstructured Grids: This work presents the first open-access, open-source pure FTM solver capable of handling general unstructured grids (hexahedral, polyhedral, tetrahedral, etc.) in a parallel domain-decomposed environment.
• OpenFOAM Integration: Successfully bridged the gap between the legacy Ftc3D Fortran code and the modern C++ OpenFOAM ecosystem, enabling the use of OpenFOAM's advanced meshing, parallelization, and linear solvers.
• Comprehensive Algorithm Suite: The package includes a full suite of FTM sub-algorithms (volume correction, RKPM communication, multiple remeshing strategies) that were previously unavailable in a unified open-source tool for unstructured grids.
• Code Availability: The solver is released under the GPLv3 license, with source code available on GitHub and the CPC Library, facilitating community development and research.

4. Results and Validation

The solver was validated against five standard multiphase flow benchmarks:

1. 3D Deformation Benchmark: A droplet subjected to a divergence-free analytical velocity field.

   • Result: The FTM solver achieved an advection error of 1%, significantly outperforming VOF-MULES (41%) and VOF-isoAdvector (36%). It successfully avoided unphysical interface rupture seen in VOF methods.
   • Convergence: Grid convergence studies showed an apparent order of convergence of ~2 and a Grid Convergence Index (GCI) below 1%.

2. Stagnant Droplet Benchmark: A static droplet used to test surface tension and parasitic currents.

   • Result: The solver achieved a pressure error of 0.046% and a parasitic current Capillary number of $2.1 \times 10^{-3}$, vastly superior to the VOF-based interFoam (12.6% error).

3. Droplet in Poiseuille Flow (Stokes Regime):

   • Result: The solver accurately predicted terminal drop shapes on various grid topologies (hexahedral, polyhedral, tetrahedral).
   • Scalability: Parallel efficiency tests showed the solver scales reasonably well, achieving near-linear speedup on up to 4 processors, comparable to standard OpenFOAM solvers.

4. Taylor Droplet Rising in a Tube:

   • Result: Validated against experimental data. The solver successfully captured the terminal shape and rise dynamics. The study highlighted the importance of tuning the undulation removal frequency to prevent front self-intersection during rapid acceleration.

5. Free Rising Bubble (High Inertia/Density Ratio):

   • Result: Simulated a bubble with a density ratio of ~1100. While the terminal velocity was accurate, the deformation was slightly underpredicted. The study demonstrated the sensitivity of the solution to the RKPM kernel radius ($\alpha$) and the necessity of Tikhonov regularization to prevent matrix singularity at small kernel radii.

5. Significance

The cfdmfFTFoam solver fills a critical void in the computational fluid dynamics community. By enabling pure Front-Tracking on general unstructured grids, it allows researchers to:

• Simulate complex geometries that are difficult or impossible to mesh with structured Cartesian grids.
• Achieve higher accuracy in interface curvature and topology evolution compared to VOF/Level-Set methods.
• Leverage the robust parallel infrastructure of OpenFOAM for large-scale simulations.

The paper serves as both a software release and a methodological guide, providing a foundation for future research into contact-line dynamics, topology changes (coalescence/breakup), and hybrid FTM strategies. The open-source nature of the project encourages further algorithmic improvements and broader adoption in academic and industrial multiphase flow research.