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arxiv: 2606.06135 · v1 · pith:SWWCB2QWnew · submitted 2026-06-04 · 💻 cs.CE

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

Lucie Kub\'i\v{c}kov\'a , Martin Isoz This is my paper

Pith reviewed 2026-06-27 23:07 UTC · model grok-4.3

classification 💻 cs.CE
keywords immersed boundary methodfictitious domainRANS turbulence modelingSIMPLE algorithmwall functionstopology optimizationOpenFOAM
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The pith

A hybrid fictitious domain-immersed boundary method adapted for RANS turbulence models produces results consistent with body-fitted CFD across Reynolds numbers from 10 to 10^6.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper adapts the hybrid fictitious domain-immersed boundary method to solve Reynolds-averaged Navier-Stokes equations with wall functions. This removes the need for repeated remeshing when using CFD inside shape or topology optimization loops. The key step is an IB-aware steady-state solver built around the SIMPLE algorithm inside OpenFOAM. The authors demonstrate that the approach matches standard body-fitted results for the most common two-equation RANS models on several benchmarks. They further test robustness on a NACA profile at changing angles of attack to support use with arbitrary geometries.

Core claim

The hybrid fictitious domain-immersed boundary forcing terms and wall-function treatment can be combined with the steady SIMPLE algorithm to produce an IB-aware RANS solver whose outputs remain consistent with body-fitted CFD for two-equation models, Reynolds numbers spanning five orders of magnitude, and both canonical and general geometries.

What carries the argument

Hybrid fictitious domain-immersed boundary forcing terms with wall-function treatment inside the steady SIMPLE algorithm for RANS equations.

If this is right

  • CFD-based topology optimization can proceed without remeshing at each design iteration.
  • The method applies directly to the standard two-equation RANS closures over the full practical Reynolds-number range.
  • Results remain consistent on both simple benchmarks and more general shapes such as airfoils at varying incidence.
  • The implementation is open-source and already integrated with an existing CFD library.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same forcing structure could be tested on unsteady or higher-fidelity turbulence closures.
  • Computational savings would be largest in optimization loops that require dozens or hundreds of flow evaluations.
  • Accuracy on very thin or highly curved immersed surfaces remains an open practical question.

Load-bearing premise

The immersed-boundary forcing and wall-function treatment stay stable and accurate for arbitrary immersed geometries when the flow is solved with the steady SIMPLE algorithm.

What would settle it

A benchmark case with an arbitrary immersed shape at Reynolds number 10^6 where the immersed-boundary RANS solution deviates measurably from an equivalent body-fitted solution in skin friction or separation location.

Figures

Figures reproduced from arXiv: 2606.06135 by Lucie Kub\'i\v{c}kov\'a, Martin Isoz.

Figure 1
Figure 1. Figure 1: (a) Discrete spatial domain Ωh. (b) Overlay of a solid body over the domain from (a) with highlighted subdomains from (5). (c) Resulting λ field. (d) Cell groups constructed based on the λ field. To construct the masking fields αu and αk, the solid bodies need to be projected on the spatial domain Ω. For the description, we will focus solely 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of y⊥ for a boundary cell Ωh P in different situations with Ωh N as its in-solid neighbor. In all cases, orange indicates the solid surface position and r the normal direction to the surface. (a) Situation where λP ∈ (0, 0.5). (b) Situation where λP = 0 and λN < 1.0. (c) Situation where λP = 0, λN = 1.0 and cells Ωh P and Ωh N share one vertex. (d) Same situation as in (c) but Ωh P and Ωh N s… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of yeff and y⊥ for three cases of solid surface position. In each case, the bottom cell is a boundary cell and the top an in-solid cell. (a) Boundary cell is intersected by the solid surface. (b) The solid surface goes in between the cell pair. (c) In-solid cell is intersected. 2.5. Calculation of imposed fields The imposed fields uib, kib, ωib, and Pib are calculated differently for differ￾ent … view at source ↗
Figure 4
Figure 4. Figure 4: Construction of the interpolation polynomial for a boundary cell [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of the SIMPLE algorithm with additional steps required in openHFDI [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Computational domains and meshes used for simulations of 2D pipe flow. Dashed [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Simulation of 2D pipe flow with different [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Simulation of 2D pipe flow with Re = 106 and different turbulence models. Data sampling and normalization was done as in [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Qualitative comparison of νt fields from the 2D simulation of pipe flow run with different turbulence models. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Investigated variations of solid-fluid interface position. On the left side, the position [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Simulation of 2D pipe flow with different variations in the solid-fluid interface [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Isometric (left) and cross-sectional (right) view onto computational domains and meshes used for simulation of 3D pipe flow. Only halves of each domain are depicted. Domains used for simpleFoam are colored in grey and for openHFDIBRANS by λ field. and openHFDIBRANS is given in [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Simulation of 3D pipe flow in domains from Figure [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Backward facing step benchmark. (a) Computational domain used by simpleFoam [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Backward facing step benchmark. Profiles were sampled along lines given in [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Zoomed view onto flow fields in the backward facing step benchmark. (a-b) Fields [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Evolution of the (a) wall shear stress τw and (b) skin friction coefficient Cf sampled along the bottom wall in the backward facing step benchmark. In (b), the x range is adjusted to match the available experimental data from [30]. HFDIBRANS agree nicely. Also, the agreement with experiment is acceptable given that the models do not operate with resolved boundary layer since y + is kept in the logarithmic… view at source ↗
Figure 18
Figure 18. Figure 18: Flow over a cylindrical obstacle. (a) Domain used by simpleFoam with indicated boundaries. (b) Domain and λ field used by openHFDIBRANS. 3.4. 2D flow over a cylindrical obstacle To evaluate the openHFDIBRANS ability to estimate drag coefficients, a test scenario with flow over a cylindrical obstacle was prepared. The utilized 27 [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Evolution of the drag coefficient Cd for a cylindrical obstacle in flow. The evolution of the drag coefficient with respect to Re is given in [PITH_FULL_IMAGE:figures/full_fig_p028_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Flow over a NACA-0009 airfoil. (a) Computational domain used by simpleFoam with indicated boundaries and visualization of the relation between the inlet velocity uin and the angle of attack γ. (b) Domain and λ field used by openHFDIBRANS. Black highlights the data sampling line. 3.5. Flow over a NACA-0009 airfoil As the next test scenario, the simulation of flow over a NACA-0009 airfoil was chosen. A view… view at source ↗
Figure 21
Figure 21. Figure 21: Simulation of flow over a NACA-0009 airfoil. Results from simpleFoam (left) and [PITH_FULL_IMAGE:figures/full_fig_p030_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Fields profiles from simulation of flow over a NACA-0009 airfoil with different [PITH_FULL_IMAGE:figures/full_fig_p031_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Evolution of the (a) drag and (b) lift coefficients for NACA-0009 with respect to the angle of attack. calculate the drag and lift coefficients. The coefficients were calculated as Cd = 2 ρ Aref ∥uin∥ 2 Z Γsf ftot · vd dS, Cℓ = 2 ρ Aref ∥uin∥ 2 Z Γsf ftot · vℓ dS vd = (cos(γ),sin(γ), 0)T, vℓ = (− sin(γ), cos(γ), 0)T , (20) where the calculation of the total force ftot is given in (19). The reference area … view at source ↗
Figure 24
Figure 24. Figure 24: (a) Visualization of the Ahmed body. (b) Representation of the Ahmed body [PITH_FULL_IMAGE:figures/full_fig_p033_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Vertical and horizontal slices through flow fields from the simulation of flow over [PITH_FULL_IMAGE:figures/full_fig_p035_25.png] view at source ↗
read the original abstract

Engineering practice often calls for shape or topology optimization (TO) of fluid defining components, while the ever-increasing computing power allows the optimized cost functions to be based on computational fluid dynamics (CFD). However, a common bottleneck in CFD-based TO frameworks is the requirement for frequent remeshing. In order to alleviate this bottleneck, we propose an adaptation of an immersed boundary (IB) method variant, the hybrid fictitious domain-immersed boundary method, to leverage Reynolds-averaged Navier-Stokes (RANS) equations and wall function. The main contribution of the present work lies in the design and open-source implementation of the IB-aware steady-state solution of the RANS equations via the SIMPLE algorithm in the OpenFOAM library. For the most common two-equation RANS models, Reynolds numbers from $10^1$ to $10^6$, and several benchmarks, such as flow over a backwards facing step or an Ahmed body, the framework gives results consistent with the standard body-fitted CFD. Furthermore, given the intended application in TO, special emphasis is placed on the robustness and applicability of the approach to general geometries, which is tested on a NACA profile under various angles of attack.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript adapts the hybrid fictitious domain-immersed boundary (IB) method to Reynolds-averaged Navier-Stokes (RANS) turbulence modeling with wall functions. The central contribution is an open-source implementation of an IB-aware steady-state RANS solver using the SIMPLE algorithm within OpenFOAM. The authors claim that, for standard two-equation RANS models and Reynolds numbers from 10^1 to 10^6, the approach produces results consistent with body-fitted CFD on benchmarks including the backwards-facing step, Ahmed body, and NACA airfoil at varying angles of attack, with emphasis on robustness for general geometries to enable topology optimization without remeshing.

Significance. If the consistency and robustness claims hold with quantitative support, the work removes a key practical obstacle (repeated remeshing) in CFD-driven shape and topology optimization of fluid components. The OpenFOAM implementation and focus on steady SIMPLE coupling for RANS are practical strengths that could facilitate adoption in engineering workflows.

major comments (2)
  1. [Abstract] Abstract: the headline claim that 'the framework gives results consistent with the standard body-fitted CFD' for the listed benchmarks and Re range is load-bearing, yet no quantitative error metrics, coefficient comparisons, grid-convergence indices, or residual histories are supplied to allow verification of that consistency.
  2. [Abstract] Abstract (robustness paragraph): the extension to 'general geometries' required for topology optimization rests on the unshown stability and accuracy of the hybrid fictitious-domain IB forcing terms plus wall-function treatment when the immersed surface is non-smooth or multiply connected and the pressure-velocity coupling uses the steady SIMPLE algorithm; only the smooth NACA profile is mentioned as a test case.
minor comments (1)
  1. [Abstract] Abstract: the notation '$10^1$ to $10^6$' for Reynolds number could be clarified to indicate whether transitional regimes (where RANS applicability is limited) are included.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and have made revisions to strengthen the presentation of our results and claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline claim that 'the framework gives results consistent with the standard body-fitted CFD' for the listed benchmarks and Re range is load-bearing, yet no quantitative error metrics, coefficient comparisons, grid-convergence indices, or residual histories are supplied to allow verification of that consistency.

    Authors: We agree that the abstract claim would be strengthened by quantitative support. The full manuscript presents visual comparisons and qualitative agreement in the results section for the backwards-facing step, Ahmed body, and NACA airfoil cases across the stated Re range. To address the concern directly, we will revise the abstract to include specific quantitative indicators such as relative errors in drag/lift coefficients and key flow quantities (e.g., reattachment length) where they are reported in the body of the paper. revision: yes

  2. Referee: [Abstract] Abstract (robustness paragraph): the extension to 'general geometries' required for topology optimization rests on the unshown stability and accuracy of the hybrid fictitious-domain IB forcing terms plus wall-function treatment when the immersed surface is non-smooth or multiply connected and the pressure-velocity coupling uses the steady SIMPLE algorithm; only the smooth NACA profile is mentioned as a test case.

    Authors: The robustness paragraph highlights the NACA profile at varying angles of attack because it directly tests the method under changing flow conditions relevant to optimization. However, the backwards-facing step and Ahmed body benchmarks explicitly include sharp edges, corners, and non-smooth surfaces, which exercise the IB forcing and wall-function treatment on non-smooth geometries under the steady SIMPLE algorithm. We will revise the abstract to explicitly reference these cases as supporting evidence for applicability to general (including non-smooth) geometries. revision: yes

Circularity Check

0 steps flagged

No circularity: implementation paper validated on external benchmarks

full rationale

The paper describes an adaptation of the hybrid fictitious domain-immersed boundary method to RANS equations with wall functions, implemented via the steady SIMPLE algorithm in OpenFOAM. Its central claim is empirical consistency with body-fitted CFD results across listed benchmarks (backwards-facing step, Ahmed body, NACA profile) for common two-equation models and Re 10^1–10^6. No derivation chain is present; the work consists of code-level modifications and direct numerical comparisons against independent external solvers. No equations reduce to fitted inputs by construction, no uniqueness theorems are imported via self-citation, and no ansatz or renaming is smuggled in. The extension to general geometries is asserted via the NACA tests but remains an empirical claim, not a self-referential derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that standard RANS turbulence models and wall functions can be combined with the existing fictitious-domain forcing without new calibration; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard two-equation RANS closures and wall functions remain valid when the near-wall treatment is replaced by immersed-boundary forcing.
    Invoked by the statement that results are consistent with body-fitted CFD for common RANS models.

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