A Volume of Fluid Immersed Boundary Method for Industrial Polymer Mixing

This paper presents a novel block-coupled Volume of Fluid Immersed Boundary (BC-VOF-IB) solver implemented in OpenFOAM that overcomes numerical instabilities caused by high viscosity contrasts to accurately simulate free-surface polymer mixing in partially filled industrial extruders.

The Gist

Imagine you are trying to mix a giant vat of thick, sticky honey with air inside a spinning machine. This is essentially what happens in industrial polymer mixing, where companies like Pirelli need to blend melted plastic with additives to make tires, medical devices, or car parts. The goal is to get everything perfectly mixed so the final product is strong and uniform.

However, simulating this process on a computer is a nightmare for mathematicians and engineers. Here is why, and how this paper solves it, using simple analogies:

The Problem: The "Thick Honey vs. Thin Air" Struggle

In these machines, you have two very different fluids:

1. Polymer Melt: Extremely thick, sticky, and slow-moving (like cold honey).
2. Air: Very thin and fast-moving.

When you try to simulate how these two interact inside a machine with spinning screws, standard computer programs get confused. It's like trying to calculate the movement of a snail and a race car on the same track using the same set of rules. The computer tries to take tiny, tiny steps to keep the "snail" (the thick plastic) from moving too fast, which makes the simulation incredibly slow—sometimes taking days to finish a few seconds of real-time mixing.

Furthermore, the machines have complex, spinning parts (screws) that move inside a fixed container. Traditionally, to simulate this, you have to build a digital mesh (a grid of tiny boxes) that perfectly wraps around the spinning screws. As the screws spin, this grid has to constantly reshape itself, which is like trying to knit a sweater while the person wearing it is running a marathon. It's messy, difficult, and prone to errors.

The Solution: A New "Smart Grid" and a "Team Approach"

The authors of this paper developed a new way to run these simulations using a software called OpenFOAM. They combined two powerful techniques:

1. The Immersed Boundary Method (The "Ghost Wall" Trick)
Instead of reshaping the grid to fit the spinning screws, they kept the grid fixed and rigid (like a solid block of ice). They then told the computer, "Hey, there is a spinning screw inside this block of ice."

• The Analogy: Imagine a swimming pool with a fixed grid of tiles on the bottom. Instead of moving the tiles to fit a swimmer, you just tell the water, "Don't go through the swimmer." The computer uses math to create a "ghost wall" around the screw, forcing the fluid to flow around it without ever needing to rebuild the grid. This makes handling complex, moving shapes much easier.

2. The Volume of Fluid (VOF) Method (The "Tracking Paint" Trick)
To see where the thick plastic ends and the air begins, they use a "paint" that fills the cells.

• The Analogy: Imagine the computer grid is a 3D checkerboard. Some squares are 100% plastic, some are 100% air, and some are a mix. The computer tracks how much "plastic paint" is in each square to see the surface of the liquid.

3. The Block-Coupled Scheme (The "Team Huddle")
This is the most important breakthrough. In standard simulations, the computer solves for the speed of the fluid in the X, Y, and Z directions one by one, like three people taking turns talking. When the fluid is super thick (like polymer), this "taking turns" approach causes the simulation to crash or slow down to a crawl because the thick fluid couples all directions together tightly.

The authors changed this to a Block-Coupled approach.

• The Analogy: Instead of three people taking turns, they all huddle up and solve the problem together at the exact same time. By treating the movement in all directions as one giant, interconnected team, the computer can handle the massive difference between the thick plastic and thin air without getting stuck.

The Results: From Hours to Minutes

The team tested their new method on two scenarios:

1. A Dog-Bone Shaped Channel: A test case where thick plastic is injected into a narrow, twisting channel.

   • Old Way: The standard computer program crashed or took 7 hours to simulate a few seconds because it was forced to take tiny steps.
   • New Way: Their new "Team Huddle" method finished the same job in just 16 minutes and didn't crash, even when the plastic got extremely thick.

2. Real Industrial Machines: They simulated real-world single-screw and twin-screw extruders (the machines used to make plastic pellets).

   • They successfully showed how the plastic fills the machine, how the pressure builds up, and how the air gets pushed out.
   • They proved that their "Ghost Wall" method works just as well as the old, difficult method of reshaping the grid, but much faster and easier to set up.

What's Next?

The paper concludes that this is a major step forward for the industry. It bridges the gap between academic math and real factory needs. However, the authors note that their current model assumes the temperature stays the same (isothermal). In reality, mixing plastic generates heat, which changes how thick the plastic is. Adding temperature effects and more complex "stretchy" plastic behaviors are the next steps for future research.

In short: They built a faster, more stable way to watch thick plastic mix with air in spinning machines on a computer, turning a process that used to take hours into one that takes minutes, without needing to rebuild the digital world every time a screw spins.

Technical Summary

Technical Summary: A Volume of Fluid Immersed Boundary Method for Industrial Polymer Mixing

Problem Statement
The numerical simulation of polymer mixing processes presents significant challenges due to the complex interplay of non-Newtonian rheology, multi-phase free-surface dynamics, and moving complex geometries. Industrial mixers, such as single-screw (SSE) and twin-screw extruders (TSE), often operate under partial filling conditions where highly viscous polymer melts coexist with air. Simulating these scenarios requires capturing the interface between phases while handling rotating solid boundaries.

Standard Computational Fluid Dynamics (CFD) approaches face two primary hurdles in this context:

1. Geometric Complexity: Generating body-fitted (conforming) meshes for rotating screws with complex geometries is computationally expensive and often requires dynamic mesh techniques that can lead to mesh distortion.
2. Numerical Instability: The extreme viscosity contrast between polymer melts and air (often exceeding $10^5$ to $10^8$) causes severe numerical instabilities in standard segregated Volume of Fluid (VOF) solvers. Specifically, the explicit treatment of the transposed velocity gradient in the viscous diffusion term imposes severe time-step constraints ($\Delta t \propto h^2$), making simulations of high-viscosity flows computationally prohibitive or unstable.

Methodology
This work proposes a novel numerical framework integrating a Volume of Fluid (VOF) interface-capturing approach with a non-conforming Immersed Boundary (IB) method, implemented within the OpenFOAM Finite Volume library.

• Immersed Boundary (IB) Approach: The method utilizes a fixed background grid where solid boundaries (e.g., rotating screws) are represented by triangulated STL surfaces. A Weighted Least Squares (WLS) interpolation operator is employed to enforce boundary conditions (Dirichlet for velocity, Neumann for volume fraction) on cells intersected by the immersed boundary. This eliminates the need for conforming mesh generation and allows for the handling of moving contact lines along complex, rotating geometries.
• VOF Formulation: The interface is tracked by solving a transport equation for the phase volume fraction ($\alpha$). The method employs the MULES (Multidimensional Universal Limiter with Explicit Solution) algorithm to maintain interface sharpness and boundedness, including a compressive velocity term to counteract numerical diffusion.
• Block-Coupled (BC) Scheme: To address the stability issues arising from high viscosity contrasts, the authors introduce a block-coupled scheme for the momentum equation. Unlike standard segregated solvers that treat velocity components sequentially, the BC approach treats the full viscous diffusion term $\nabla \cdot (\mu(\nabla \mathbf{u} + \nabla^\top \mathbf{u}))$ implicitly. This involves coupling the three velocity components within a single linear system. The discretization of the tangential gradient term utilizes a Finite Area (FA) method combined with WLS interpolation on extended stencils (involving point-neighbors rather than just face-neighbors), adapted from elasticity applications [11].
• Solution Algorithm: The solver employs a modified PIMPLE algorithm (combining SIMPLE and PISO) to handle the pressure-velocity coupling. The IB constraints are integrated into the algebraic system, modifying the matrix coefficients and source terms to satisfy boundary conditions on the immersed surfaces while maintaining mass conservation.

Key Contributions

1. Development of BC-VOF-IB Solvers: The paper introduces the UCInterIbFoam solver, which combines the non-conforming IB method with the block-coupled VOF approach. This is the first implementation of such a coupled method within the OpenFOAM-10 framework for two-phase flows.
2. Stability Enhancement via Block-Coupling: The work demonstrates that the fully implicit treatment of viscous diffusion significantly relaxes time-step stability constraints. This allows for the simulation of high-viscosity polymer flows with viscosity ratios up to $10^8$ without the simulation crashing due to Courant number violations.
3. Handling Moving Contact Lines: The framework successfully addresses the geometric and algorithmic difficulties of moving contact lines where the two-phase interface intersects immersed boundaries, a critical requirement for partially filled mixing devices.

Numerical Results
The proposed methods were validated and tested on two categories of problems:

• Benchmark Case (Dog-bone Injection Molding):

  • A comparison between segregated (interFoam), block-coupled conforming (UCInterFoam), and block-coupled non-conforming (UCInterIbFoam) solvers was performed on a high-viscosity Power-law fluid injection case.
  • The standard segregated solver failed to produce stable results within reasonable timeframes due to severe time-step restrictions ($\Delta t \approx 10^{-6}$ s).
  • The block-coupled solvers achieved stable solutions with time steps up to $10^{-3}$ s, reducing computational time from 7 hours to 16 minutes (conforming) and 58 minutes (non-conforming).
  • The non-conforming IB solution showed qualitative agreement with the conforming solution, with minor discrepancies in viscosity peaks near geometric restrictions, attributed to the non-conforming nature of the grid.

• Industrial Applications (SSE and TSE):

  • Simulations were conducted on simplified Single-Screw Extruders (SSE) and realistic Twin-Screw Extruders (TSE) under both fully filled and partially filled (starve-fed) conditions.
  • The solvers successfully captured the evolution of the free surface, filling ratios, and pressure fields.
  • Results aligned with physical expectations: fully filled scenarios ($Q_{in} > Q_0$) exhibited negative pressure gradients from inlet to outlet, while partially filled scenarios ($Q_{in} < Q_0$) showed pressure increases near the outlet.
  • The IB method proved effective in handling the complex intermeshing geometry of the TSE without requiring dynamic mesh regeneration.

Significance and Claims
The paper claims that the proposed BC-VOF-IB framework represents a meaningful step toward bridging the gap between academic CFD research and industrial polymer processing demands. By overcoming the numerical instabilities associated with high viscosity contrasts and eliminating the need for complex body-fitted mesh generation for rotating geometries, the method enables physically consistent predictions of velocity and pressure fields in industrially relevant, partially filled mixing devices.

The authors acknowledge that the current work is limited to isothermal processes and generalized Newtonian rheology. They state that future developments must include thermal effects (energy equation) and viscoelastic constitutive models to fully capture the physics of industrial polymer mixing. However, the current framework establishes a rigorous foundation for these extensions and provides a robust tool for simulating free-surface flows in complex, moving geometries.