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AI CFD Scientist: Toward Open-Ended Computational Fluid Dynamics Discovery with Physics-Aware AI Agents

Nithin Somasekharan, Rabi Pathak, Manushri Dhanakoti, Tingwen Zhang, Ling Yue, Andy Zhu, Shaowu Pan

arXiv:2605.06607v3physics.flu-dyncs.AI

TL;DR

High-fidelity CFD challenges AI scientists because solver completion does not ensure physical validity, while key failures may appear only in flow-field imagery. AI CFD Scientist integrates literature-grounded ideation, validated execution, source-code modification, vision-based verification, and figure-grounded writing, discovering a Spalart–Allmaras correction that reduces lower-wall Cf RMSE against DNS by 7.89%.

  • Problem

    AI scientists for CFD need physical-validity gates because solver completion can mask wrong geometries, missing flow features, and degenerate outputs invisible to logs.

  • Method

    AI CFD Scientist couples literature-grounded ideation, validated execution, source-code modification, vision-based flow-field verification, and figure-grounded writing in an OpenFOAM workflow.

  • Results

    Across five tasks, the system supports regular experimentation, custom-model compilation, and open-ended discovery, achieving a 7.89% lower-wall Cf RMSE reduction against DNS in one discovery task.

  • Takeaways & Limitations

    The framework supports an inspectable CFD discovery loop that converts validated computational experiments into defensible scientific claims.

  • Takeaways & Limitations

    Results use only GPT-5.5, cross-framework comparisons rely on manual expert artifact reading, and the framework is supervised assistance rather than unattended publication.

Abstract

from arXiv · show

Recent LLM-based agents have closed substantial portions of the scientific discovery loop in software-only machine-learning research, in chemistry, and in biology. Extending the same loop to high-fidelity physical simulators is harder, because solver completion does not imply physical validity and many failure modes appear only in field-level imagery rather than in solver logs. We present AI CFD Scientist, an open-source AI scientist for computational fluid dynamics (CFD) that, to our knowledge, is the first to span literature-grounded ideation, validated execution, vision-based physics verification, source-code modification, and figure-grounded writing within a single inspectable workflow. Three coupled pathways cover parameter sweeps within a fixed solver, case-local C++ library compilation for new physical models, and open-ended hypothesis search against a reference comparator, all running on OpenFOAM through Foam-Agent. At the center of the framework is a vision-language physics-verification gate that inspects rendered flow fields before any result is accepted, rerun, or written into a manuscript. On five tasks under a shared GPT-5.5 backbone, AI CFD Scientist autonomously discovers a Spalart-Allmaras runtime correction that reduces lower-wall Cf RMSE against DNS by 7.89% on the periodic hill at Reh=5600; under matched LLM cost, two strong general AI-scientist baselines (ARIS, DeepScientist) execute partial CFD workflows but lack the domain-specific validity gates needed to convert runs into defensible scientific claims; and a controlled planted-failure ablation shows that the vision-language gate detects 14 of 16 silent failures missed by solver-level checks. Code, prompts, and run artifacts are released at https://github.com/csml-rpi/cfd-scientist.

1 Introduction

AI CFD Scientist addresses the difficulty of making high-fidelity CFD simulations scientifically defensible by integrating ideation, validated execution, vision-based physics verification, source-code modification, and figure-grounded writing. Across five tasks, it demonstrates open-ended model discovery while showing that domain-specific validity gates are necessary beyond solver-level checks.

  • Motivation: High-fidelity physical simulators remain an underexplored frontier for AI-scientist systems because discovery requires deeper simulator interaction than text-mediated tool use.Prior agents have closed substantial portions of the discovery loop in software-only machine learning, chemistry, and biology.
  • CFD challenge: CFD solver completion does not establish physical validity, because clean runs can still contain incorrect geometry, missing flow features, or degenerate output invisible to solver logs.Validity gates such as mesh independence and reference-data alignment must be confirmed before scientific claims.
  • Research gap: Generic AI-scientist frameworks lack physical-validity gates, while CFD-specific agents stop short of an end-to-end workflow for defensible scientific claims.CFD research variables can also require case-local C++ source modification rather than configuration changes.
  • Framework: AI CFD Scientist spans literature-grounded ideation, validated execution, vision-based physics verification, source-code modification, and figure-grounded writing in one inspectable OpenFOAM workflow.Its three coupled pathways cover parameter sweeps within a fixed solver, case-local C++ model-library compilation, and open-ended discovery.
  • Results: 7.89% lower-wall Cf RMSE reduction against DNS was achieved by an autonomously discovered Spalart–Allmaras runtime correction on the periodic hill at Reh=5600.The result comes from five tasks under a shared GPT-5.5 backbone; ARIS and DeepScientist executed only partial CFD workflows under matched LLM cost.

2 Related Work

Related work spans autonomous physical experimentation, LLM-based software research, and increasingly structured CFD/OpenFOAM agents. Recent systems provide strong software-science baselines and CFD-specific orchestration, but the literature reflects distinct domain and workflow emphases.

  • Robot scientists and autonomous laboratories: Robot Scientist systems demonstrated end-to-end hypothesis generation and physical experimentation, while Eureqa automated equation discovery from data.Self-driving laboratories later combined robotic experimentation with Bayesian-optimization planners for chemistry, materials, and biology.
  • LLM-based AI-scientist frameworks: LLM-based frameworks now automate software research through end-to-end paper generation, multi-agent collaboration, inter-paper memory, critique, iterative refinement, and tool use.Examples include The AI Scientist, AI Scientist-v2, Agent Laboratory, AgentRxiv, AI co-scientist, CycleResearcher, AI-Researcher, and Zochi.
  • LLM-based AI-scientist frameworks: DeepScientist and ARIS are the two most recent strong baselines used for head-to-head comparison, while domain-specific systems target chemistry and biology.Bohrium–SciMaster, AstaBench, PaperBench, and MLR-Bench evaluate artifact quality in machine-learning research workflows.
  • CFD- and OpenFOAM-specific agents: CFD/OpenFOAM agents progressed from Python case manipulation and prompt assistance toward structured orchestration, case authoring, retrieval, reviewer loops, chat, multi-agent decomposition, and physics constraints.Named systems include PythonFOAM, foamlib, FoamPilot, AutoCFD, OpenFOAMGPT, MetaOpenFOAM, Foam-Agent, ChatCFD, CFDagent, SwarmFoam, PhyNiKCE, CFD-copilot, turbulence.ai, and FlamePilot.

3 CFD Scientist

AI CFD Scientist structures CFD discovery as expert-guided execution pathways with checkpointed or modular orchestration and structured artifacts exchanged across agents. Its workflow combines mesh-independence checks, vision-language physics verification, gate-driven reruns, and figure-grounded manuscript writing.

  • Architecture: The framework encodes CFD discovery through expert-written prompts, guidelines, and execution pathways rather than a generic chat loop.It provides a checkpointed LangGraph workflow and a modular skills-based implementation.
  • Architecture: Agents exchange structured artifacts including study JSON, requirements, source-edit plans, run directories, figure manifests, interpretation JSON, and manuscript drafts.These artifacts support orchestration across the workflow’s stages.
  • Three pathways: Three pathways organize experimentation, including regular studies that run CFD simulations without modifying simulator source code.The regular pathway uses literature-aware ideation, novelty filtering, structured study specification, and validation.
  • Mesh-independence gate: A 5% threshold flags mesh-comparison quantities of interest for Richardson/GCI escalation after baseline and refined cases are compared.The refined mesh applies approximately 10% near-wall and 5% bulk refinement while preserving topology, blocking, and meshing method.
  • VLM physics-verification gate: The VLM gate quality-checks rendered figures, verifies expected flow features against experiment requirements, and drives reruns and writing decisions.Separate VLM calls first assess figure readability and then inspect accepted figures for physical consistency.

4 Experiments: AI CFD Scientistwith GPT-5.5

Across five CFD tasks, AI CFD Scientist combined regular experimentation, source-code modification, and open-ended discovery with vision-based physics verification. It recovered physically plausible behavior, validated custom models, discovered an SA correction reducing lower-wall Cf RMSE by 7.89%, and detected 14 of 16 planted failures.

  • Task coverage: Five tasks spanned BFS turbulence sensitivity, jet/plume Reynolds-number sweeps, custom viscosity, a custom Spalart–Allmaras modifier, and open-ended SA discovery.The first two used regular experimentation; the next two used code modification, while T5 used open-ended discovery.
  • T1 — BFS turbulence sensitivity: The VLM flagged a BFS reattachment-extractor sign/origin error and rejected a k–ε output as inconsistent with separated-flow physics.SST and SA produced the most plausible recirculation topology in streamlines.
  • T2 — Jet/plume Re sweep: Seven jet cases recovered centreline velocity scaling from 0.09 to 0.60 m/s as Re increased from 60 to 600, while high-Re oscillations emerged and one case was flagged anomalous.All cases ran end-to-end on identical 35,156-cell meshes.
  • T3 — Custom viscosity: The custom viscosity library compiled on the first attempt, and its n=1 law reproduced the analytic Newtonian baseline within 0.5% while six cases reached steady state.Centreline velocity varied approximately 3.8% across the sweep, from 1.4542 to 1.5231 m/s.
  • T5 — Open-ended SA discovery: 7.89%: the best discovered SA model reduced lower-wall Cf RMSE against DNS from 0.004297 for baseline SA to 0.003958.The model was found after 44 discovery iterations and delivered as a coded fvModels block.
  • VLM physics-verification ablation: 14/16 planted failures were detected, including all missing-deliverable, wrong-magnitude, and broken-postprocessing cases; only 2/4 truncated runs were flagged.Incomplete simulations could appear visually complete when edited endTime values obscured convergence insufficiency.

5 Cross-Framework Comparison: AI CFD Scientist vs. ARIS vs. DeepScientist

Under a shared GPT-5.5 backbone, AI CFD Scientist is compared with ARIS and DeepScientist on T1–T4 using manual, artifact-based evaluation, revealing that CFD-specific validity gates and fuller scientific pipelines distinguish defensible outputs from runnable simulations.

  • Evaluation setup: AI CFD Scientist, ARIS, and DeepScientist were compared on T1–T4 under the same GPT-5.5 backbone using archived artifacts, logs, libraries, figures, and reports.T5 was excluded because neither baseline supports open-ended source-level discovery.
  • Scientific validity: ARIS and DeepScientist often execute simulations and produce clean trends, but lack CFD-specific gates for determining whether those trends are scientifically supported.On T1 and T2, they report closure rankings or St(Re) correlations despite missing mesh or reference-data evidence.
  • Scientific validity: AI CFD Scientist records an unresolved verdict when evidence is incomplete instead of converting a runnable case into a scientific claim.This conservative behavior contrasts with baseline outputs that report conclusions despite missing supporting evidence.
  • Custom-model workflows: All three frameworks compiled case-local custom models on T3 and T4, so compilation capability—not pipeline completeness—did not distinguish them.ARIS and DeepScientist each ran one custom variant against a reference and reported a markdown summary.
  • Custom-model workflows: AI CFD Scientist additionally ran an APG=0 control, produced a DNS overlay against the reference, and emitted a figure-grounded LaTeX draft.These steps extend beyond the custom-variant runs and markdown summaries produced by ARIS and DeepScientist.

6 Conclusion · A Input topic for T1-T5

AI CFD Scientist is presented as an open-source CFD AI scientist that connects natural-language topics to manuscript drafts through literature-grounded ideation, validated execution, physics verification, code modification, and figure-grounded writing. The conclusion bounds these results to GPT-5.5, expert manual evaluation, and supervised scientific assistance, while the accompanying topics define five CFD studies spanning sensitivity analysis, parameter sweeps, custom models, and open-ended discovery.

  • 6 Conclusion: AI CFD Scientist closes the discovery loop from a natural-language topic to a manuscript draft in an open-source CFD workflow.The framework is described as integrating ideation, execution, verification, source modification, reference-data alignment, and figure-grounded writing.
  • 6 Conclusion: The framework integrates literature-grounded ideation, novelty filtering, mesh-independence gating, source-level modification, VLM physics verification, reference-data alignment, and figure-grounded writing.
  • 6 Conclusion: All reported numbers use GPT-5.5 (Codex), while cross-framework comparison relies on manual expert artifact reading rather than an automated CFD-paper rubric.Additional LLM sweeps and baselines are deferred for cost, and Table 5 reflects expert artifact reading.
  • 6 Conclusion: The framework is supervised scientific assistance rather than unattended publication.
  • A Input topic for T1-T5: T1 studies turbulence-model sensitivity on the backward-facing step at Re_h = 25,400 using four RANS closures on one mesh-independence-passed mesh.The closures are k–ε, k–ω SST, Spalart–Allmaras, and a Reynolds-stress model, with identical inlet profiles, boundary conditions, time controls, and post-processing.
  • A Input topic for T1-T5: T2 performs seven logarithmically spaced transient cases over Re ∈ [60, 600] to characterize centreline U_x scaling and dominant oscillation frequency.The study requires statistically converged statistics, centreline profiles, representative contour snapshots, and per-case dominant-frequency estimates.
  • A Input topic for T1-T5: T3 implements a case-local power-law viscosity ν_eff = k|γ̇|^(n−1) and validates it against a Newtonian reference before sweeping five operating points.The Newtonian case uses n = 1 and must reproduce the built-in solution within numerical tolerance; the sweep varies (k, n, ∇p).

B Per-Task Experiment Matrices and Quantitative Results … D Cross-Framework Evidence Ledger

The appendix reports per-task CFD experiment configurations, quantitative outcomes, the 44-iteration open-ended Spalart–Allmaras discovery trajectory, and artifact-level cross-framework evidence. The finalist reduces Cf RMSE against DNS by 7.89% and is characterized by a physically targeted quadRecTail source deployed as a coded fvModels block.

  • B Per-Task Experiment Matrices and Quantitative Results: The T1–T4 matrices define task-specific CFD conditions, including Reh=25,400 for the backward-facing step, a 35,156-cell jet mesh, a generalized-Newtonian channel model, and a Reh=10,595 periodic hill.The T2 spectral metrics are unresolved because of a cross-experiment metadata-parser failure; T3 uses a case-local compiled library, and T4 compares Cf RMSE against DNS over matched wall points.
  • B.5 T5 — Open-ended SA discovery (overview): 7.89% reduction: finalist iter_044_quadRecFine12 lowers Cf RMSE versus DNS from 0.004297 for baseline SA to 0.003958 at Reh=5600.The full discovery trajectory contains 44 iterations.
  • C.1 Discovery objective and reference: The discovery objective minimizes lower-wall Cf RMSE over 99 wall samples against an exact-match DNS reference, rather than separation or reattachment error.Baseline separation and reattachment estimates are x/h=0.269 and 7.753, versus DNS values 0.191 and 4.726.
  • C.2 Iteration trajectory: The OED trajectory scores each iteration by Cf RMSE versus DNS, with REVISE marking worsened scores and PROCEED marking improved scores whose gates were accepted.The baseline SA score is Cf RMSE 0.004297.
  • C.3 Discovered quadRecTail model: form and coefficients: The discovered quadRecTail model adds an implicit source to the SA eν equation through four terms: recovery production, outlet-overshoot suppression, upstream secondary production, and tail damping.The terms are Grec, Gsink, Gsrc, and Gtail, respectively, and their coefficients are archived in Table 11.
  • C.4 Deployment as a coded fvModels block: The finalist is delivered as a coded fvModels runtime block that assembles K from Gaussian patches and adds it implicitly with fvm::Sp(K, eqn.psi()).The artifact uses the nuTilda field and requires no separate compilation.
  • D Cross-Framework Evidence Ledger: Under matched GPT-5.5 on the four standard tasks, the cross-framework ledger evaluates evidence directly from each framework’s run archive.The T1 and T2 ledgers document backward-facing-step sensitivity and jet/plume Reynolds-number sweeps.
  • D Cross-Framework Evidence Ledger: All three frameworks generated and compiled case-local C++ libraries for T3 and executed case-local custom SA libraries for T4, while differing in validation breadth and manuscript support.T4 differentiators include APG=0 control-case validation, DNS overlay rendering, and manuscript output.

E What AI CFD ScientistDid Well, Per Task

Table 16 consolidates AI CFD Scientist’s per-task strengths, with each row grounded in a specific archived artifact from the workflow.

  • Per-task strengths: Table 16 summarizes the system’s strengths separately for each task.The table consolidates the per-task strengths described in Section 4.1.
  • Per-task strengths: Each table row is grounded in a specific archived artifact.The artifacts include study JSON, requirement files, run directories, VLM judgments, figure manifests, source-code libraries, and manuscript fragments.
  • Per-task strengths: The evidence spans planning, execution, verification, implementation, and scientific writing artifacts.These categories are represented by the archived files and judgments listed for the table rows.

F What AI CFD ScientistDoes Not Yet Do Well · G Failure-Mode Taxonomy and Detection Gates · H Architectural Details: Agent Inventory and State Schema

The framework’s remaining weaknesses concentrate in cross-experiment post-processing, while its taxonomy assigns distinct detection and recovery stages to different CFD failure modes. Its architecture documents agents and a redundantly checkpointed LangGraph state through human- and machine-readable handoffs.

  • F What AI CFD ScientistDoes Not Yet Do Well: Most residual failures occur in cross-experiment post-processing rather than solver execution, custom-model compilation, or the VLM gate.Recorded limitations include parser fragility, reattachment-extraction sign conventions, and spectral-metadata reconstruction.
  • F What AI CFD ScientistDoes Not Yet Do Well: The framework records concrete limitations conservatively instead of concealing them.These limitations are summarized in Table 17, with each residual issue tied to the framework’s recorded results.
  • G Failure-Mode Taxonomy and Detection Gates: CFD automation failures are separated into distinct axes because different failure types require different detection gates.Table 18 formalizes the failure-mode taxonomy used by the framework.
  • G Failure-Mode Taxonomy and Detection Gates: Detection occurs at the stage where each failure becomes observable rather than through a single executable/non-executable decision.Each taxonomy class is detected at a different stage and triggers a different recovery action.
  • G Failure-Mode Taxonomy and Detection Gates: The VLM physics gate addresses evidential failures that solver logs and validators cannot observe.This gate exists because some failures appear in physical evidence rather than in execution-level checks.
  • H Architectural Details: Agent Inventory and State Schema: The appendix documents the agents implementing the framework’s pathways and the shared LangGraph state object.Table 19 lists each agent’s primary inputs, outputs, and functional role.
  • H Architectural Details: Agent Inventory and State Schema: Every agent handoff is both human-readable and machine-readable, while the state is checkpointed and intentionally redundant.The requirement records what should be run, the case directory records what was actually run, and figures expose result validity.

I LLM Cost: Token Usage and USD per Framework · J VLM Physics-Verification Gate: Planted-Failure Ablation · J.1 Setup: 4 categories × 4 flows + 4 controls

Under matched GPT-5.5, the three frameworks complete four CFD experiments at comparable dollar cost, while a controlled 20-case planted-failure ablation isolates the VLM gate’s sensitivity using deterministic ground truth. The cost accounting excludes the 19-call VLM sweep, whose total cost is below $1.

  • I LLM Cost: Token Usage and USD per Framework: Production end-to-end costs are logged for four standard CFD experiments under matched GPT-5.5 using shared accounting middleware.Each run directory contains llm_token_usage.json, with provider_usage reporting where available.
  • I LLM Cost: Token Usage and USD per Framework: Costs use $1.25 per 1M uncached input tokens, $0.125 per 1M cached-input tokens, and $10.00 per 1M output tokens.AI CFD Scientist has zero cached input by construction, whereas ARIS and DeepScientist use cache-replay volumes.
  • I LLM Cost: Token Usage and USD per Framework: $41.19, $41.92, and $37.94 are the respective costs for AI CFD Scientist, ARIS, and DeepScientist across four CFD experiments.The comparable envelope is $38–$42, supporting a level-playing-field cost comparison.
  • I LLM Cost: Token Usage and USD per Framework: The VLM-ablation sweep is excluded from these production costs because its 19 single-shot vision calls cost below $1 in total.The reported framework costs cover only production end-to-end CFD runs in Sections 4 and 5.
  • J VLM Physics-Verification Gate: Planted-Failure Ablation: The retrospective found the VLM gate caught 7/21 silent failures, adding a 33% catch rate beyond Foam-Agent’s crash detection.These failures were from four production GPT-5.5 runs that had already passed the solver-level reviewer loop.
  • J.1 Setup: 4 categories × 4 flows + 4 controls: The ablation contains 16 planted-failure cases and 4 unperturbed clean controls from jet, bfs, hill, and chan production cases.Each read-copied template receives one file-system-level perturbation from a four-bucket failure taxonomy.
  • J.1 Setup: 4 categories × 4 flows + 4 controls: Post-hoc perturbation makes each requirement–case-state pair deterministically FAIL or OK, isolating verifier sensitivity from solver-side rerun noise.Only the perturbation changes from the production-passed case, providing unambiguous ground truth.

J.2 Results … 3 Methods

AI CFD Scientist’s ablation shows that vision-based verification catches most silent CFD failures while exposing a convergence blind spot, and its methods evaluate a bounded Spalart–Allmaras production correction on a mesh-independent periodic-hill benchmark.

  • J.2 Results: 87.5% overall recall (14/16; F1 = 82.4%) was achieved, with 100% recall on missing_deliverable, wrong_magnitude_metric, and broken_postprocessing but 50% on convergence_not_settled.The two missed convergence cases had endTime edited to match truncated states, making figures appear complete.
  • J.2 Results: 76.6 s mean wall-clock per case (64–89 s) required one LLM call per case, about an order of magnitude cheaper than the production interpret.py loop.The full sweep used 19 calls and took approximately 24 minutes, versus 10–15 minutes and 2–11 calls per case for interpret.py.
  • J.3 What the ablation tells us: 100% recall (12/12) covered missing deliverables, wrong-magnitude metrics, and broken postprocessing that solver-level review misses after clean completion.Convergence-not-settled remains a blind spot; a deterministic residual-plateau or QoI-drift detector is proposed before the VLM call.
  • J.3 What the ablation tells us: Per-flow recall of 3–4 of 4 across jet, BFS, hill, and channel was statistically indistinguishable, indicating geometry-independent failure detection.All four clean controls were flagged REVISE, yielding 77.8% overall precision because the ablation rendered only a small interpret-mode figure subset.
  • 1 Introduction: The periodic-hill benchmark targets separated and reattached boundary layers at Reh = 5,600, where standard SA can overproduce eddy viscosity in rotation-dominated regions and suppress recirculation.The study evaluates a modified SA model against DNS-informed skin-friction behavior on the canonical periodic-hill geometry.
  • 2 Related Work: Prior approaches include local production-term corrections, field-inversion and machine-learning corrections, and hybrid RANS–LES methods with higher computational cost.Physics-informed neural networks augmented with SA equations were reported to improve periodic-hill mean-velocity reconstruction error by up to 73%.
  • 3.3 Custom SA Model with pMult Production Modifier: The custom model applies a bounded multiplier pMult to SA production as a function of R = S/Ω, reducing production below Rref in rotation-dominated recirculation and compiled as a case-local runtime library.The parameterization uses Rref, β, and clamp bounds [pMin, pMax], with production levels approaching or exceeding standard SA above Rref.

4 Results · 5 Discussion · 6 Conclusion

Across the results and discussion, the pMult modifier deepens predicted separation monotonically with β while remaining unable to reproduce the DNS valley-floor Cf topology; the study concludes that the baseline mesh is adequate and identifies verified Rref and wall-normal comparisons as priorities for future work.

  • 4.1 Overview of Predicted Velocity Fields: The pMult modification progressively deepens the hill-lee recirculation zone with increasing β, while Rref = 1.0 resembles baseline SA and the tight clamp resembles the design point.The velocity-field comparison covers all six cases at t = 5,000.
  • 4.2 Baseline SA Behaviour and DNS Comparison: RANS peak |Cf| values of 0.028–0.038 occur near x/h ≈8.5, whereas DNS valley-recirculation values of order 10^-3 occur over x/h ≈0.12–4.57 and are not directly comparable.The distinct streamwise stations and mechanisms must be preserved when interpreting the RANS–DNS comparison.
  • 4.2 Baseline SA Behaviour and DNS Comparison: The baseline SA misses the DNS negative-Cf valley region, while pMult suppresses production in the rotation-dominated valley and directionally improves separation prediction without matching the DNS bubble.Baseline Cf remains positive or near-zero across the valley, whereas modified cases can generate shorter, downstream closed bubbles.
  • 4.3 Streamwise Velocity and Separation Structure: 7.7%: Umax decreases from 0.03283 m/s (case 1, baseline SA) to 0.03032 m/s (case 4, β = 9), while UCL x,min decreases from 0.02260 to 0.02119 m/s, indicating deeper separation.The centreline velocity spread increases from 0.00401 m/s (case 1) to 0.00670 m/s (case 4).
  • 4.4 Design-Point Custom SA Performance: For case 2, peak |Cf| increases from 0.02809 (case 1) to 0.03543 (case 2), a 26 % increase, and a closed bubble appears between x/h = 5.754 and x/h = 6.232.The custom SA also reduces valley-region turbulent working variable and eddy viscosity through pMult suppression.
  • 4.5.1 Sensitivity to β: β = 3 to 9 advances separation onset from x/h = 7.621 to x/h = 4.618 and increases downstream-face |Cf| from 0.03240 to 0.03751, while sensitivity diminishes at high β.Cases 3 and 6 lack reattachment zero-crossings within x/h ∈[0, 9], whereas cases 2 and 4 reattach at x/h = 6.232 and 6.662.
  • 4.5.2 Sensitivity to Rref: Rref = 1.0 effectively deactivates pMult, yielding Umax = 0.03259 versus baseline 0.03283 (−0.7 %) and fields nearly indistinguishable from baseline, while its xsep/h = 1.502 and xreat/h = 1.896 pair is physically suspect.The anomalous pair is attributed to numerical noise near the hill crest rather than a physical separation event.
  • 4.5.3 Sensitivity to Clamp Bounds; 4.6 Near-Wall Resolution; 4.7 Failure Cases and Negative Results; 6 Conclusion: The clamp range [0.10, 3.0] is effectively non-binding relative to [0.05, 5.0], while the 44,749-cell baseline mesh produces QoIs varying by less than 2.5 % after 18.6 % enrichment and Reτ = 339.3 fully converged.Unverified wider-clamp and additional parameter attempts were excluded because of configuration propagation errors; future work should complete the Rref sweep below 0.82 and add full Cf and wall-normal comparisons.

A Reproducibility Details · B Claim–Evidence Table · Claim Evidence

The study documents a reproducible OpenFOAM periodic-hill setup with a case-local custom Spalart–Allmaras library, then links parameter variations to validated changes in separation, velocity, turbulence, and skin friction. Evidence shows β dominates behavior, Rref = 1.0 disables pMult, and clamp bounds are non-binding.

  • A Reproducibility Details: All simulations use standard-release OpenFOAM, while the custom SA model is compiled as a case-local shared library without modifying system sources.This isolates the implementation to customModels/.
  • A Reproducibility Details: The α = 1.0 periodic hill uses a multi-block structured blockMesh with near-wall grading ratio 1815, 44,749 cells, and 61,000 points.Periodic streamwise boundaries, no-slip walls, and a fixed pressure gradient maintain Ub = 0.028 m/s with ν = 5 × 10−6 m2/s.
  • A Reproducibility Details: PISO coupling, second-order upwind convection, pseudo-steady integration, and 5,000 time units define the numerical protocol for every run.Wall shear stress and y+ are extracted at t = 5,000, with cross-case fields and figures generated using PyVista and Matplotlib.
  • Claim Evidence: Baseline SA predicts no valley recirculation, with Case 1 Cf near-zero for x/h = 1–7 and a single zero-crossing at x/h = 7.742.Relative to baseline UCL, pMult deepens separation and decreases x,min 5.0 % in case 2 versus case 1.
  • Claim Evidence: β is the dominant control parameter, with Umax monotonically decreasing from 0.03283 to 0.03032 as β increases from 3 to 9.The same progression shows increasingly deep ˜ν suppression.
  • Claim Evidence: Rref = 1.0 deactivates pMult: Case 5 reaches Umax = 0.03259 versus baseline 0.03283 (−0.7 %) and has near-identical Ux and ˜ν fields.This behavior is visually and quantitatively close to baseline.
  • Claim Evidence: Design-case clamp bounds are non-binding, with case 6 versus case 2 giving ∆Umax = +0.6 % and ∆peak |Cf| = −2.8 %.The two cases are visually indistinguishable in overview.
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