💻 computer science

Adaptation of the hybrid fictitious domain-immersed boundary method for Reynolds-averaged turbulence modeling

This paper presents an open-source adaptation of the hybrid fictitious domain-immersed boundary method within OpenFOAM to enable robust, remeshing-free Reynolds-averaged Navier-Stokes (RANS) simulations for shape and topology optimization, demonstrating accuracy comparable to standard body-fitted CFD across various benchmarks and flow conditions.

Original authors: Lucie Kubíčková, Martin Isoz

Published 2026-06-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lucie Kubíčková, Martin Isoz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an engineer trying to design the perfect shape for a car part, a pipe, or a wing. To do this, you need to simulate how air or water flows around it. Traditionally, this is like trying to draw a perfect map of a winding river by cutting out a piece of paper that matches the river's exact shape. Every time you change the river's shape (even a little bit), you have to cut out a brand new piece of paper and redraw the whole map. This is slow, expensive, and frustrating.

This paper introduces a new, smarter way to do this simulation, called openHFDIBRANS. Here is how it works, using simple analogies:

1. The "Ghost Wall" Trick (Immersed Boundary)

Instead of cutting a new piece of paper (mesh) for every new shape, the researchers use a giant, simple grid (like a checkerboard) that covers the whole area. They don't change the grid. Instead, they use a "magic marker" to tell the computer: "Hey, these specific squares are solid walls, and these are empty air."

If you want to change the shape of the object, you just move the "magic marker" on the grid. You don't have to rebuild the whole grid. It's like playing with a digital sandbox where you can reshape a castle instantly without rebuilding the entire sand table.

2. The "Traffic Cop" Problem (Turbulence)

The tricky part is that fluids (like air or water) get messy and turbulent, especially near walls. In the real world, the air right next to a wall moves very slowly, while the air a little further away moves fast. This is called the "boundary layer."

Usually, computers need a very detailed, high-resolution map right next to the wall to see this. But since our "magic marker" grid is simple and doesn't hug the wall perfectly, the computer can't see those tiny details.

The authors solved this by adding a "Traffic Cop" (Wall Functions). Instead of trying to see every tiny swirl of air near the wall, the Traffic Cop uses a set of rules (mathematical formulas) to guess what the air is doing based on how fast it's moving a little further away. It's like a police officer at a busy intersection who doesn't need to see every single car to know the traffic flow is heavy; they just use a rule of thumb to manage the flow.

3. The "Tightrope" Challenge (Stability)

When you mix this "Ghost Wall" trick with the "Traffic Cop" rules, things can get unstable. It's like trying to balance a broom on your finger while someone keeps pushing it. The computer might get confused, the numbers might go crazy, and the simulation crashes.

The authors built a special "Safety Harness" into their software. They made sure the computer checks the rules before it tries to calculate the next step, and they added a "correction step" at the end of every calculation to make sure the numbers stay on the tightrope. This ensures the simulation doesn't fall off the edge, even when the shapes are weird or the speeds are very high.

4. Did It Work? (The Results)

The team tested their new tool on several classic challenges:

Water in a pipe: They tested it from very slow flow to extremely fast flow. The results matched the traditional, slow, "cut-the-paper" method almost perfectly.
A step in a channel: They simulated water flowing over a sudden drop. The new tool predicted where the water would swirl and settle just as well as the old method.
A car shape (Ahmed body) and an airplane wing (NACA profile): They tested how air flows around these shapes. The new tool predicted the drag (air resistance) and lift (upward force) very accurately.

The Catch:
In some cases, the new tool slightly "overestimated" the size of the wake (the messy, swirling air left behind an object). It's like the tool thought the object was slightly bigger than it really was, leaving a slightly larger trail of turbulence. However, the authors say this is a predictable behavior and doesn't ruin the results.

Why Does This Matter?

The main goal of this work is Optimization. Because this new tool doesn't require rebuilding the grid every time the shape changes, it is perfect for Topological Optimization. This is a process where a computer tries thousands of different shapes automatically to find the absolute best one (e.g., the lightest bridge or the most aerodynamic car part).

The authors have made their code open-source (free for anyone to use) and built it on a popular platform called OpenFOAM. They claim it is robust enough to be used right now for designing real-world engineering components, like sensors or molds for casting metal, without needing to be a math genius to set it up.

In short: They built a faster, more flexible way to simulate fluid flow that works great for designing shapes, provided you accept a tiny, predictable error in the "tail" of the flow.

Technical Summary: Adaptation of the Hybrid Fictitious Domain-Immersed Boundary Method for RANS Modeling

Problem Statement
Engineering applications frequently require shape or topology optimization (TO) of fluid components, where computational fluid dynamics (CFD) serves as the basis for cost function evaluation. A significant bottleneck in CFD-based TO frameworks is the necessity for frequent remeshing as geometries change. While Immersed Boundary Methods (IBMs) offer a solution by utilizing simple, non-conforming meshes and representing geometry via scalar indicator fields, their integration with Reynolds-averaged Navier-Stokes (RANS) turbulence models remains challenging. The primary difficulties involve modeling the boundary layer without anisotropic mesh refinement, preventing spurious oscillations in pressure and wall shear stress at high Reynolds numbers, and ensuring a tight coupling between momentum and turbulence variables to avoid instability. Furthermore, existing IBM-RANS solvers are often not open-source or lack standard CFD practices.

Methodology
The authors propose openHFDIBRANS, an open-source solver built upon the OpenFOAM library that adapts the Hybrid Fictitious Domain-Immersed Boundary (HFDIB) method for steady-state, incompressible RANS simulations. The methodology integrates the following key components:

Governing Equations: The solver addresses the steady-state RANS equations coupled with four two-equation turbulence models: $k$ - $\epsilon$ , $k$ - $\omega$ , $k$ - $\omega$ SST, and realizable $k$ - $\epsilon$ . The equations are modified by a phase indicator field $\lambda$ to restrict solutions to the fluid domain and include source terms ( $f_{ib}$ and $S_{ib}$ ) to enforce boundary conditions.
Masking and Cell Classification: The computational domain is divided into in-solid cells ( $\Omega^h_{ic}$ ), boundary cells ( $\Omega^h_{bc}$ ), and free-stream cells ( $\Omega^h_{fc}$ ) based on the $\lambda$ field. Masking fields ( $\alpha_u, \alpha_k, \alpha_\omega$ ) limit the application of source terms to relevant regions.
Wall Modeling and Boundary Treatment: To handle the boundary layer without resolving it, the method employs wall functions.
- Friction Velocity: To ensure stability and tight coupling between momentum and turbulence, the friction velocity ( $u_\tau$ ) is calculated using turbulence variables rather than local velocity profiles.
- Imposed Fields: In boundary cells where the boundary layer is unresolved ( $y^+_{ib} > y^+_{lam}$ ), velocity is not reconstructed via interpolation but corrected solely by turbulent viscosity. In resolved regions ( $y^+_{ib} \leq y^+_{lam}$ ), polynomial interpolation is used to impose velocity and turbulence kinetic energy ( $k$ ) based on wall functions.
- Turbulence Variables: Specific source terms are not added for the specific dissipation rate ( $\omega$ ) because it diverges near walls. Instead, $\omega$ and the production of $k$ ( $P_k$ ) are directly assigned in affected cells by manipulating the discretization matrix, effectively enforcing Dirichlet-like conditions.
Solution Strategy: The solver utilizes an extended SIMPLE algorithm. Crucially, before the momentum predictor, imposed fields are calculated. After the momentum equation is solved, a correction step ( $u = u + \alpha_u(u_{ib} - u)$ ) is applied to enforce the boundary conditions directly. This step is vital for convergence in steady-state simulations where under-relaxation is required. The incompressibility condition is enforced only in the fluid region to avoid conflicts with the IBM forcing.

Key Contributions

Open-Source Implementation: The work provides a fully open-source implementation of an IBM-RANS solver within the widely used OpenFOAM framework, incorporating standard CFD practices.
Stability-Oriented Design: The authors address common IBM-RANS instabilities by evaluating friction velocity via turbulence variables and enforcing a tight connection between momentum and turbulence transport equations.
Robustness for Optimization: The method is specifically designed for geometry optimization applications, demonstrating robustness across various Reynolds numbers ( $10^1$ to $10^6$ ) and complex geometries without the need for remeshing.
Comprehensive Benchmarking: The solver is validated against standard body-fitted simulations (simpleFoam) and experimental data across six scenarios: 2D/3D pipe flow, backward-facing step, flow over a cylinder, NACA-0009 airfoil, and the Ahmed body.

Results
The performance of openHFDIBRANS was evaluated against standard body-fitted CFD (simpleFoam) and experimental data:

Accuracy: The solver produced results consistent with simpleFoam across a wide range of Reynolds numbers and different turbulence models. $L_2$ norm errors for velocity, turbulent viscosity, and Reynolds stress were generally low, though slightly higher in regions with abrupt changes in turbulent viscosity (e.g., $k$ - $\omega$ SST and realizable $k$ - $\epsilon$ models).
Sensitivity: The method showed satisfactory robustness regarding the position of the solid-fluid interface relative to the mesh. While alignment with mesh faces improved accuracy, the solver remained functional even with misaligned interfaces.
Drag and Lift: For the cylinder and NACA-0009 airfoil, the solver captured flow patterns and predicted lift and drag coefficients with good agreement to reference data.
Limitations: In specific cases (e.g., backward-facing step and NACA-0009), the solver tended to slightly overestimate the size of the wake region behind the solid objects. Additionally, like other RAS models, it could not resolve the drag crisis for the cylinder, which requires more advanced two-layer models.

Significance and Claims
The paper claims that openHFDIBRANS is a viable tool for engineering applications requiring geometry or topology optimization. The authors emphasize that while the method may exhibit minor overestimation of wake sizes, its behavior is predictable and consistent across different geometries. This consistency is identified as the most critical factor for optimization frameworks. The work positions the solver as a ready-to-use alternative to standard body-fitted methods for TO, eliminating the computational cost of frequent remeshing while maintaining sufficient accuracy for engineering design. Future applications cited by the authors include the topology optimization of biosensors and die-casting molds.