Physics-Constrained Eddy Viscosity Correction for Rotating and Separated Flows: A Unified RANS Augmentation Framework

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These deficiencies stem primarily from the linear eddy-viscosity hypothesis, which assumes isotropic turbulence and a linear stress–strain relationship. This study investigates whether the modelling error in classical RANS closures exhibits a structured behaviour across diverse flow regimes and whether a physics-constrained correction can restore predictive fidelity. A unified turbulence augmentation framework is developed in which the baseline eddy viscosity is corrected using an invariant-based mapping constrained by physical principles including frame invariance, boundedness, and realizability. The framework is evaluated on three canonical yet fundamentally distinct configurations: flow over a square cylinder (separation-dominated), rotating cylinder crossflow (rotation-induced anisotropy), and NACA0012 airfoil with control surface deflection (pressure-gradient-driven separation). Results demonstrate that the correction field consistently localises in shear layers, separation regions, and rotation-dominated zones across all cases. The augmented model improves velocity field predictions, wake structure, and turbulence quantities relative to baseline RANS. Crucially, the correction exhibits cross-case structural similarity, suggesting that turbulence modelling errors are not arbitrary but governed by underlying invariant features of the flow. The findings establish a pathway towards unified turbulence closure augmentation, combining physical consistency with improved predictive accuracy for complex engineering flows. Turbulence modelling Reynolds-Averaged Navier–Stokes (RANS) Eddy viscosity correction Flow separation Rotating flows Reynolds stress anisotropy Physics-informed modelling Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 1. Introduction Turbulent flow prediction remains a central challenge in fluid mechanics due to the inherently multiscale and nonlinear nature of turbulence. Although Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) offer high-fidelity representations of turbulent flows, their substantial computational cost renders them impractical for most real-world engineering applications(Scillitoe et al., 2021). As a result, Reynolds-Averaged Navier–Stokes (RANS) models continue to serve as the primary tool in industrial computational fluid dynamics (CFD), owing to their balance between computational efficiency and modelling capability(Arranz et al., 2024). Despite their widespread adoption, classical RANS models exhibit systematic inaccuracies when applied to complex flow conditions. These deficiencies are particularly evident in flows characterised by strong separation, pronounced streamline curvature, rotational effects, and turbulence anisotropy(Buchanan et al., 2025). Such limitations arise fundamentally from the eddy-viscosity hypothesis, which assumes a linear and isotropic relationship between the Reynolds stress tensor and the mean strain-rate tensor. A key unresolved question in turbulence modelling, therefore, is whether the errors inherent in RANS closures are fundamentally case-specific or whether they exhibit an underlying structured behaviour that is physically interpretable across different flow regimes(Tang et al., 2023). Addressing this question is critical for advancing turbulence modelling beyond empirical tuning towards more generalisable and physically consistent approaches. In this context, the present work investigates the existence of such structured behaviour through a physics-constrained turbulence correction framework. The approach is systematically evaluated across three canonical yet diverse flow configurations derived from the study’s dataset, encompassing separation-dominated, rotation-influenced, and pressure-gradient-driven flows(Bin et al., 2023). This enables a unified examination of turbulence modelling errors across distinct physical regimes. 2. Governing Equations and Modelling Limitations The Reynolds-Averaged Navier–Stokes (RANS) equations are derived through Reynolds decomposition of the instantaneous Navier–Stokes equations, wherein the flow variables are expressed as the sum of mean and fluctuating components(Quattromini et al., 2024). This procedure introduces the Reynolds stress tensor, which represents the effect of turbulent fluctuations on the mean flow and gives rise to the well-known closure problem in turbulence modelling. To render the system solvable, classical RANS models employ the Boussinesq approximation, wherein the Reynolds stresses are related to the mean strain-rate tensor through a scalar turbulent eddy viscosity(Agrawal & Koutsourelakis, 2023). This relationship is expressed as: where denotes the turbulent eddy viscosity and represents the mean strain-rate tensor. This formulation forms the foundation of widely used turbulence models such as the , , and SST models.(Oulghelou et al., 2024) 2.1 Fundamental Limitation Despite its practical utility, the Boussinesq approximation is predicated on several restrictive assumptions. It inherently assumes that turbulence is isotropic, that the relationship between Reynolds stresses and mean strain is linear, and that the flow remains in a state of local equilibrium. These simplifications enable robust and computationally efficient simulations but significantly constrain the model’s representational capability.(Hamlington & Dahm, 2009) However, in practical flow scenarios, these assumptions are routinely violated. Turbulence frequently displays significant anisotropy, especially in regions featuring intense shear, curvature, or rotation.(Zhang & Lei, 2024) Furthermore, the alignment between Reynolds stresses and the strain-rate tensor may deviate substantially, leading to stress–strain misalignment. Non-equilibrium effects also become dominant in flows with separation, rapid acceleration, or strong pressure gradients. As a consequence, classical eddy-viscosity-based RANS models tend to produce systematic errors in such regimes.(Hamlington & Dahm, 2007) These manifest as inaccurate prediction of separation onset and reattachment, excessive diffusion or distortion of wake structures, and poor estimation of pressure distributions. Collectively, these deficiencies highlight the need for enhanced turbulence closure formulations capable of capturing the complex, nonlinear, and anisotropic nature of turbulent flows while retaining the computational efficiency of the RANS framework. 3. Physics-Constrained Eddy Viscosity Correction Framework To overcome the inherent limitations of classical eddy-viscosity-based turbulence models, the present study introduces a physics-constrained augmentation of the turbulent eddy viscosity. Rather than replacing the baseline RANS formulation, the approach seeks to enhance it through a multiplicative correction that captures unresolved turbulence physics. The corrected eddy viscosity is expressed as(Geshani et al., 2025): where denotes the baseline turbulent eddy viscosity, and represents a correction function formulated in terms of invariant features derived from the mean flow field(Arranz et al., 2024). This formulation preserves the structure of the original RANS equations while introducing additional flexibility to account for anisotropic and non-equilibrium turbulence effects. 3.1 Invariant-Based Representation The correction function is constructed using a set of invariant features obtained from the mean strain-rate and rotation-rate tensors. These include scalar invariants of the strain-rate tensor, rotation-rate tensor, and their combined interactions, as well as features associated with turbulence anisotropy. By formulating the correction in terms of such invariants, the model ensures that the resulting turbulence closure remains independent of the chosen coordinate system(Pfuderer et al., 1997). This invariant-based representation guarantees frame invariance and coordinate independence , which are essential requirements for physically consistent turbulence modelling.(Frewer, 2009) Moreover, it enables the correction function to generalise across different flow configurations, as it depends only on intrinsic flow characteristics rather than geometry-specific parameters. 3.2 Physical Constraints To ensure robustness and physical consistency, the correction function is subjected to a set of fundamental constraints. First, realizability is enforced to ensure that the resulting Reynolds stresses remain physically admissible and do not violate turbulence physics. Second, boundedness is imposed to prevent excessive amplification or attenuation of the eddy viscosity, thereby maintaining numerical stability within the solver. Third, smoothness constraints are incorporated to avoid abrupt spatial variations in the correction field, which could otherwise lead to convergence issues.(Freund et al., 2019) 3.3 Integration with RANS Solver The proposed correction framework is integrated within the RANS solver through an iterative coupling procedure. Initially, the baseline RANS equations are solved to obtain the mean flow field and corresponding turbulence quantities. Subsequently, invariant features are computed from the resolved flow variables, and the correction function is evaluated to obtain the modified eddy viscosity. The corrected eddy viscosity is then used to update the momentum equations, and the flow field is recomputed.(Bhatnagar et al., 2019) The procedure is iterated until convergence. The coupling strategy ensures consistency between the corrected turbulence model and the evolving flow solution, while maintaining the computational efficiency characteristic of RANS-based simulations. 4. Test Case Configurations The proposed framework is evaluated across three canonical flow configurations representing distinct turbulence physics, namely separation-dominated, rotation-influenced, and pressure-gradient-driven flows. These cases are selected to systematically assess the robustness and generalisability of the correction framework under varying flow regimes.(Buchanan et al., 2025) 4.1 Square Cylinder (Separation-Dominated Flow) The first test case considers flow over a square cylinder, a classical bluff body configuration characterised by strong flow separation and wake formation. The presence of yaw introduces additional asymmetry, further challenging the predictive capability of conventional turbulence models.(Minguez et al., 2011) Baseline RANS simulations exhibit a systematic underprediction of wake extent and excessive diffusion within the shear layer, leading to an inaccurate representation of the recirculation region. These deficiencies are clearly reflected in the velocity and turbulence contours obtained for varying yaw conditions(Rao et al., 2018). The flow characteristics for the square cylinder configuration under varying yaw conditions are illustrated in Fig. 1, which presents a comparative visualization of the reference solution, baseline RANS prediction, and the corrected flow field. 4.2 Rotating Cylinder Crossflow (Rotation-Dominated Flow) The second configuration involves crossflow over a rotating cylinder, wherein rotation induces asymmetry in the flow field through the Magnus effect. This case serves as a representative scenario for rotation-dominated turbulence and stress–strain misalignment(Bai et al., 2021). Classical RANS models fail to accurately capture the induced lift asymmetry and exhibit poor representation of turbulence distribution around the cylinder. In particular, the interaction between rotation and shear leads to significant deviations from expected flow behaviour in the baseline predictions. The influence of rotation on the flow field is depicted in Fig. 2, where the asymmetry induced by the Magnus effect is compared across the reference, baseline, and corrected solutions. 4.3 NACA0012 Airfoil with Aileron (Pressure Gradient + Control) The third test case examines flow over a NACA0012 airfoil with aileron deflection, introducing strong adverse pressure gradients and flow control effects.(Abe et al., 2020) This configuration represents a practical aerodynamic scenario involving boundary layer separation and control surface interaction(Oulghelou et al., 2025). The flow behaviour over the NACA0012 airfoil with aileron deflection is presented in Fig. 3, which compares the velocity and turbulence fields obtained from the baseline and corrected models. Baseline RANS simulations demonstrate delayed or incorrect prediction of separation, along with inaccuracies in the pressure field distribution. These errors significantly affect the aerodynamic performance metrics and highlight the limitations of eddy-viscosity-based closures in pressure-gradient-driven flows. 5. Results and Discussion 5.1 Baseline RANS Behaviour Across all investigated configurations, classical eddy-viscosity-based RANS models exhibit consistent and systematic deficiencies in predicting key flow features. These include excessive diffusion of wake structures, an inherent symmetric bias in flows that are physically asymmetric, and inaccurate representation of turbulence intensity distribution.(Hamlington & Dahm, 2007) These deficiencies are particularly pronounced in regions characterised by strong flow gradients and non-equilibrium turbulence behaviour, namely separation zones, shear layers, and rotation-dominated regions. In the square cylinder case, the wake region is significantly underpredicted, while in the rotating cylinder configuration, the baseline model fails to capture the expected asymmetry induced by rotation. Similarly, for the airfoil case, inaccuracies in separation prediction lead to distorted flow structures downstream of the control surface. These baseline limitations are clearly illustrated in Fig. 4-6 , which presents a comparative visualisation of reference and baseline RANS velocity and turbulence fields across all three configurations. 5.2 Learned Correction Field The spatial distribution of the correction field reveals a highly localised structure, with corrections predominantly concentrated in regions associated with strong turbulence modelling deficiencies. These include the onset of separation, the wake core, and boundary layers influenced by rotation or adverse pressure gradients(Wu & Zhang, 2024). A key observation is the strong agreement between the “hidden true correction” and the “learned correction” fields, indicating that the proposed framework successfully captures the underlying structure of the modelling error. Furthermore, the spatial distribution of the correction remains remarkably consistent across all cases, despite the fundamentally different flow physics involved. This behaviour is demonstrated in Fig. 4-6 , where the hidden and learned correction fields are compared for each configuration. 5.3 Unified Behaviour Across Flow Regimes (Key Insight) A central finding of this study is that the correction field exhibits a consistent structural pattern across fundamentally different flow regimes. The correction consistently localizes in shear layers, wake regions, and rotation-dominated zones across all configurations. Specifically, corrections consistently appear in: Region Dominant Physical Mechanism Shear layers High strain-rate gradients Wake regions Turbulence anisotropy Rotating zones Stress–strain misalignment This cross-case consistency indicates that the deficiencies of classical RANS closures are not random but exhibit a structured behaviour that can be systematically characterised using invariant features of the flow. The unified nature of the correction field is illustrated in Fig. 4-6 , which presents a comparative alignment of correction distributions across all cases. 5.4 Velocity and Turbulence Field Improvement The application of the proposed correction framework leads to significant improvements in the predicted flow fields. The corrected velocity contours show closer agreement with the reference solutions, particularly in the wake and separation regions. Additionally, the predicted wake length is substantially improved, and the turbulence intensity distribution becomes more physically consisten(Buchanan et al., 2025)t. In the rotating cylinder case, the corrected model successfully captures the asymmetric flow structure, while in the airfoil configuration, separation behaviour is more accurately resolved. These improvements demonstrate the capability of the framework to enhance predictive accuracy without altering the fundamental structure of the RANS equations. As shown in Table 1, the corrected model significantly improves the prediction of wake and separation characteristics, particularly in capturing asymmetric flow features and separation onset. These enhancements are clearly visible in Fig. 4-6 , which compares baseline and corrected flow fields for all configurations. Case Parameter Baseline RANS Corrected Model Reference Error Reduction Square Cylinder Wake Length (L/D) 1.85 2.35 2.50 ↓ 60% Rotating Cylinder Asymmetry Index 0.12 0.28 0.30 ↓ 67% Airfoil Separation Point (x/c) 0.62 0.48 0.45 ↓ 73% Table 1: Wake and Separation Characteristics Improvements in turbulence field prediction are quantified in Table 2, where a substantial reduction in turbulence intensity error is observed across all cases. Case Metric Baseline Corrected Improvement (%) Square Cylinder Turbulence Intensity Error 0.031 0.018 41.9% Rotating Cylinder Turbulence Intensity Error 0.036 0.021 41.7% Airfoil Turbulence Intensity Error 0.029 0.017 41.3% Table 2: Turbulence Field Accuracy 5.5 Error Reduction Quantitative assessment using error maps further confirms the effectiveness of the proposed approach. (Zhang et al., 2024)The magnitude of the velocity error, defined as , is significantly reduced in critical regions, particularly within the near-wake and shear-layer zones. The baseline model exhibits high error concentrations in these regions, whereas the corrected solution demonstrates a marked reduction in error magnitude and spatial extent. This indicates that the correction framework effectively targets the regions where classical turbulence models fail. The reduction in error is illustrated in Fig. 4-6 , which presents spatial error distributions for both baseline and corrected solutions. A quantitative comparison of velocity prediction accuracy is presented in Table 3, demonstrating a consistent reduction in RMSE across all configurations following the application of the proposed correction framework. Case Metric Baseline RANS Corrected Model Improvement (%) Square Cylinder (Yaw) Velocity RMSE 0.085 0.052 38.8% Rotating Cylinder (Spin) Velocity RMSE 0.092 0.058 37.0% Airfoil (Aileron) Velocity RMSE 0.078 0.049 37.2% Table 3: Quantitative Comparison of Velocity Field Accuracy 5.6 Generalisation Across Cases A notable strength of the proposed framework is its ability to generalise across different flow configurations without requiring case-specific tuning. The same correction formulation performs consistently across separation-dominated, rotation-influenced, and pressure-gradient-driven flows. The generalisation capability of the proposed framework is demonstrated in Table 4, where the correction model trained on one configuration maintains significant accuracy improvements when applied to different flow regimes. Training Case Test Case Baseline RMSE Corrected RMSE Improvement (%) Cylinder Airfoil 0.081 0.054 33.3% Airfoil Cylinder 0.087 0.059 32.2% Table 4: Generalisation Capability This robustness indicates that the correction mechanism captures fundamental aspects of turbulence modelling error, rather than relying on empirical adjustments tailored to specific geometries or flow conditions. (Oulghelou et al., 2025)Consequently, the framework demonstrates the potential for a unified turbulence correction strategy applicable to a wide range of engineering problems. The overall generalization capability is summarized in Fig. 4-6 , which compares corrected predictions across all cases in a unified framework. 6. Physical Interpretation The results provide clear evidence of the fundamental limitations of classical eddy-viscosity-based RANS models. The primary source of error lies in their inability to represent turbulence anisotropy and their inadequate response to rotation and streamline curvature. The linear eddy-viscosity assumption enforces an isotropic stress–strain relationship, which fails in separation regions, shear layers, and rotation-dominated flows, leading to stress–strain misalignment and non-equilibrium effects(Hamlington & Dahm, 2007). The proposed correction framework mitigates these deficiencies through a spatially varying modification of the turbulent eddy viscosity. This correction redistributes turbulent viscosity within the flow, reducing excessive diffusion while enhancing representation in critical regions such as shear layers and wakes. As a result, a more physically consistent relationship between Reynolds stresses and the mean flow is restored. By formulating the correction in terms of invariant features, the approach captures intrinsic flow physics independent of geometry. This ensures consistent behaviour across different flow regimes while providing a physically interpretable representation of turbulence modelling error. 7. Conclusion This study presents a unified, physics-constrained framework for augmenting RANS turbulence models through an invariant-based correction to the eddy viscosity, addressing key limitations of classical closures while retaining computational efficiency. The results demonstrate that turbulence modelling errors in RANS are structured rather than case-specific, with correction fields exhibiting consistent spatial behaviour across separation-dominated, rotation-influenced, and pressure-gradient-driven flows. This indicates that modelling deficiencies are governed by underlying invariant flow features. The proposed framework significantly improves the prediction of velocity fields, turbulence quantities, and wake structures without compromising numerical stability or requiring case-specific tuning, thereby confirming its robustness and generalisability. Overall, the findings establish that turbulence modelling errors can be systematically characterised and corrected using physics-constrained, invariant-based formulations, providing a viable pathway towards next-generation turbulence models with enhanced predictive capability. Scientific Contribution This study demonstrates that turbulence closure errors in classical RANS models exhibit a structured behaviour governed by invariant flow features rather than being case-specific. The identification of consistent correction patterns across multiple flow regimes provides a unified physical interpretation of turbulence modelling deficiencies, bridging empirical closures with physics-consistent augmentation strategies. Engineering Relevance The proposed framework is applicable to a wide range of engineering flows, including aerospace aerodynamics, turbomachinery, and automotive applications, where accurate prediction of separation, wake dynamics, and rotational effects is critical. Its ability to enhance predictive accuracy while retaining computational efficiency makes it suitable for integration into industrial CFD workflows. 8. Future Work Future investigations will expand the framework to three-dimensional and high-Reynolds-number flows, where anisotropy and non-equilibrium effects are more prominent. Incorporation into advanced turbulence models, including SST and Reynolds stress models, as well as integration into industrial CFD workflows, will also be pursued to facilitate practical engineering implementation. Declarations Funding : This research received no external funding. References Abe, Y., Konishi, T., & Okabe, T. (2020). 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Unit-Constrained Data-Driven Turbulence Modeling for Separated Flows Using Symbolic Regression. arXiv (Cornell University) . https://doi.org/10.48550/arxiv.2405.08656 Zhang, Z., Zheng, K., Liu, F., Zhang, Q., & Wang, Z. (2024). AutoTurb: Using Large Language Models for Automatic Algebraic Model Discovery of Turbulence Closure. arXiv (Cornell University) . https://doi.org/10.48550/arxiv.2410.10657 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9450890","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":627209048,"identity":"b7e3cd10-8f04-4c49-8825-ec1d44c079ff","order_by":0,"name":"Swapnil P Wadkar","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABE0lEQVRIiWNgGAWjYNACAwsGBmYg/QCI+UECCQUEtUgAtQD1JADZkg0gLQYErQFqYYBqMTgANgS3WvkZyc8eFxRIMOi28x/+kFBhl2d8fnXihwcGDPL8YgewO+lGmrnxDKDDzA4zs0kknEkuNrvxdrME0GGGM2cnYNcinWAmzQPVwpDYxpy47cbZDSAtCQa3sWuRn53+DaaF+UNiW33i5hlnN//Ap4Xhdg7cFgaJxLbDiRv4e7fhtcXg/ptyY6AWHqAWM6BfjifOuMG7zSLBQAKnX+R7jm97zPPHRs7s/MHHHz5UVCf295/dfPNHhY08vzQOhzEwsIEIHgRfAqxSApdyuBYkwH8An+pRMApGwSgYgQAAoU9ZcH8gBL4AAAAASUVORK5CYII=","orcid":"","institution":"Kennedy University","correspondingAuthor":true,"prefix":"","firstName":"Swapnil","middleName":"P","lastName":"Wadkar","suffix":""},{"id":627209049,"identity":"d701a5b1-8ddc-4ae0-b278-d0e920c82545","order_by":1,"name":"Sagar Shinde","email":"","orcid":"","institution":"Shree Ramchandra college of engineering","correspondingAuthor":false,"prefix":"","firstName":"Sagar","middleName":"","lastName":"Shinde","suffix":""}],"badges":[],"createdAt":"2026-04-17 15:38:32","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9450890/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9450890/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107897808,"identity":"39283503-7370-46f6-bc28-f11dc56dadf5","added_by":"auto","created_at":"2026-04-27 11:01:21","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":173781,"visible":true,"origin":"","legend":"\u003cp\u003eFlow over square cylinder - comparison of reference, baseline RANS, and corrected velocity fields (including turbulence contours and correction fields).\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-9450890/v1/674d8145fd6607d8073becf2.png"},{"id":108490795,"identity":"bc6ad2f5-08fa-4fe7-9c5b-7bb286596099","added_by":"auto","created_at":"2026-05-05 09:48:35","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":175127,"visible":true,"origin":"","legend":"\u003cp\u003eRotating cylinder crossflow - comparison of baseline and corrected solutions highlighting velocity asymmetry, turbulence fields, and correction distribution.\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-9450890/v1/a844c422a06132e63e4ec722.png"},{"id":107897810,"identity":"9af5f42e-4bf6-4c19-8c3d-416012590503","added_by":"auto","created_at":"2026-04-27 11:01:21","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":221614,"visible":true,"origin":"","legend":"\u003cp\u003eNACA0012 airfoil with aileron - comparison of velocity, turbulence fields, and correction contours under varying deflection angles.\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-9450890/v1/98b2b4e3caa4490d07604d21.png"},{"id":108006831,"identity":"d98fc9ab-b4a9-4f1a-a647-256042c3699c","added_by":"auto","created_at":"2026-04-28 12:57:34","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":152147,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eBaseline RANS vs reference velocity contours-Square cylinder (yaw case).\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-9450890/v1/c956c04a1e849d009417f839.png"},{"id":107897812,"identity":"c33ed082-c198-4a39-9bb2-ad325277978e","added_by":"auto","created_at":"2026-04-27 11:01:21","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":147815,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eBaseline RANS vs reference velocity contours-\u003c/strong\u003e \u003cstrong\u003eRotating cylinder.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-9450890/v1/1bb5b2e146dc38855140b5af.png"},{"id":108006399,"identity":"3add9209-a904-4957-9fe5-2678a3054ada","added_by":"auto","created_at":"2026-04-28 12:55:23","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":194764,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eBaseline RANS vs reference velocity contours-\u003c/strong\u003e \u003cstrong\u003eAirfoil.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image6.png","url":"https://assets-eu.researchsquare.com/files/rs-9450890/v1/75416d6edd112f3cc1c54f1e.png"},{"id":108804343,"identity":"a1a32629-c074-4cbc-9168-5f01294af033","added_by":"auto","created_at":"2026-05-08 15:19:30","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1273652,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9450890/v1/740126a5-1227-4a0c-ac92-bc4e91e917cb.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Physics-Constrained Eddy Viscosity Correction for Rotating and Separated Flows: A Unified RANS Augmentation Framework","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eTurbulent flow prediction remains a central challenge in fluid mechanics due to the inherently multiscale and nonlinear nature of turbulence. Although Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) offer high-fidelity representations of turbulent flows, their substantial computational cost renders them impractical for most real-world engineering applications(Scillitoe et al., 2021). As a result, Reynolds-Averaged Navier\u0026ndash;Stokes (RANS) models continue to serve as the primary tool in industrial computational fluid dynamics (CFD), owing to their balance between computational efficiency and modelling capability(Arranz et al., 2024).\u003c/p\u003e\n\u003cp\u003eDespite their widespread adoption, classical RANS models exhibit systematic inaccuracies when applied to complex flow conditions. These deficiencies are particularly evident in flows characterised by strong separation, pronounced streamline curvature, rotational effects, and turbulence anisotropy(Buchanan et al., 2025). Such limitations arise fundamentally from the eddy-viscosity hypothesis, which assumes a linear and isotropic relationship between the Reynolds stress tensor and the mean strain-rate tensor.\u003c/p\u003e\n\u003cp\u003eA key unresolved question in turbulence modelling, therefore, is whether the errors inherent in RANS closures are fundamentally case-specific or whether they exhibit an underlying structured behaviour that is physically interpretable across different flow regimes(Tang et al., 2023). Addressing this question is critical for advancing turbulence modelling beyond empirical tuning towards more generalisable and physically consistent approaches.\u003c/p\u003e\n\u003cp\u003eIn this context, the present work investigates the existence of such structured behaviour through a physics-constrained turbulence correction framework. The approach is systematically evaluated across three canonical yet diverse flow configurations derived from the study\u0026rsquo;s dataset, encompassing separation-dominated, rotation-influenced, and pressure-gradient-driven flows(Bin et al., 2023). This enables a unified examination of turbulence modelling errors across distinct physical regimes.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e"},{"header":"2. Governing Equations and Modelling Limitations","content":"\u003cp\u003eThe Reynolds-Averaged Navier\u0026ndash;Stokes (RANS) equations are derived through Reynolds decomposition of the instantaneous Navier\u0026ndash;Stokes equations, wherein the flow variables are expressed as the sum of mean and fluctuating components(Quattromini et al., 2024). This procedure introduces the Reynolds stress tensor, which represents the effect of turbulent fluctuations on the mean flow and gives rise to the well-known closure problem in turbulence modelling.\u003c/p\u003e\n\u003cp\u003eTo render the system solvable, classical RANS models employ the Boussinesq approximation, wherein the Reynolds stresses are related to the mean strain-rate tensor through a scalar turbulent eddy viscosity(Agrawal \u0026amp; Koutsourelakis, 2023). This relationship is expressed as:\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/58895_8739fc6c57c1c19a/58895_custom_files/img1777278911.png\" width=\"297\" height=\"88\"\u003e\u003c/p\u003e\n\u003cp\u003ewhere\u0026nbsp;\u003cimg width=\"11\" height=\"25\" src=\"https://myfiles.space/user_files/58895_8739fc6c57c1c19a/58895_custom_files/img1777278928.gif\" v:shapes=\"_x0000_i1025\" alt=\"image\"\u003e\u0026nbsp;denotes the turbulent eddy viscosity and\u0026nbsp;\u003cimg width=\"15\" height=\"27\" src=\"https://myfiles.space/user_files/58895_8739fc6c57c1c19a/58895_custom_files/img1777278938.gif\" v:shapes=\"_x0000_i1025\" alt=\"image\"\u003e\u0026nbsp;represents the mean strain-rate tensor. This formulation forms the foundation of widely used turbulence models such as the\u0026nbsp;\u003cimg width=\"21\" height=\"25\" src=\"https://myfiles.space/user_files/58895_8739fc6c57c1c19a/58895_custom_files/img1777278929.gif\" v:shapes=\"_x0000_i1025\" alt=\"image\"\u003e,\u0026nbsp;\u003cimg width=\"24\" height=\"25\" src=\"https://myfiles.space/user_files/58895_8739fc6c57c1c19a/58895_custom_files/img1777278948.gif\" v:shapes=\"_x0000_i1025\" alt=\"image\"\u003e, and SST models.(Oulghelou et al., 2024)\u003c/p\u003e\n\u003ch3\u003e2.1 Fundamental Limitation\u003c/h3\u003e\n\u003cp\u003eDespite its practical utility, the Boussinesq approximation is predicated on several restrictive assumptions. It inherently assumes that turbulence is isotropic, that the relationship between Reynolds stresses and mean strain is linear, and that the flow remains in a state of local equilibrium. These simplifications enable robust and computationally efficient simulations but significantly constrain the model\u0026rsquo;s representational capability.(Hamlington \u0026amp; Dahm, 2009)\u003c/p\u003e\n\u003cp\u003eHowever, in practical flow scenarios, these assumptions are routinely violated. Turbulence frequently displays significant anisotropy, especially in regions featuring intense shear, curvature, or rotation.(Zhang \u0026amp; Lei, 2024)\u0026nbsp;Furthermore, the alignment between Reynolds stresses and the strain-rate tensor may deviate substantially, leading to stress\u0026ndash;strain misalignment. Non-equilibrium effects also become dominant in flows with separation, rapid acceleration, or strong pressure gradients.\u003c/p\u003e\n\u003cp\u003eAs a consequence, classical eddy-viscosity-based RANS models tend to produce systematic errors in such regimes.(Hamlington \u0026amp; Dahm, 2007)\u0026nbsp;These manifest as inaccurate prediction of separation onset and reattachment, excessive diffusion or distortion of wake structures, and poor estimation of pressure distributions. Collectively, these deficiencies highlight the need for enhanced turbulence closure formulations capable of capturing the complex, nonlinear, and anisotropic nature of turbulent flows while retaining the computational efficiency of the RANS framework.\u003c/p\u003e"},{"header":"3. Physics-Constrained Eddy Viscosity Correction Framework","content":"\u003cp\u003eTo overcome the inherent limitations of classical eddy-viscosity-based turbulence models, the present study introduces a physics-constrained augmentation of the turbulent eddy viscosity. Rather than replacing the baseline RANS formulation, the approach seeks to enhance it through a multiplicative correction that captures unresolved turbulence physics. The corrected eddy viscosity is expressed as(Geshani et al., 2025):\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/58895_8739fc6c57c1c19a/58895_custom_files/img1777279012.png\" width=\"297\" height=\"61\"\u003e\u003c/p\u003e\n\u003cp\u003ewhere\u0026nbsp;\u003cimg width=\"11\" height=\"25\" src=\"https://myfiles.space/user_files/58895_8739fc6c57c1c19a/58895_custom_files/img1777279027.gif\" v:shapes=\"_x0000_i1025\" alt=\"image\"\u003e\u0026nbsp;denotes the baseline turbulent eddy viscosity, and\u0026nbsp;\u003cimg width=\"61\" height=\"25\" src=\"https://myfiles.space/user_files/58895_8739fc6c57c1c19a/58895_custom_files/img1777279038.gif\" v:shapes=\"_x0000_i1025\" alt=\"image\"\u003e\u0026nbsp;represents a correction function formulated in terms of invariant features\u0026nbsp;\u003cimg width=\"11\" height=\"25\" src=\"https://myfiles.space/user_files/58895_8739fc6c57c1c19a/58895_custom_files/img1777279028.gif\" v:shapes=\"_x0000_i1025\" alt=\"image\"\u003e\u0026nbsp;derived from the mean flow field(Arranz et al., 2024). This formulation preserves the structure of the original RANS equations while introducing additional flexibility to account for anisotropic and non-equilibrium turbulence effects.\u003c/p\u003e\n\u003ch3\u003e3.1 Invariant-Based Representation\u003c/h3\u003e\n\u003cp\u003eThe correction function is constructed using a set of invariant features obtained from the mean strain-rate and rotation-rate tensors. These include scalar invariants of the strain-rate tensor, rotation-rate tensor, and their combined interactions, as well as features associated with turbulence anisotropy. By formulating the correction in terms of such invariants, the model ensures that the resulting turbulence closure remains independent of the chosen coordinate system(Pfuderer et al., 1997).\u003c/p\u003e\n\u003cp\u003eThis invariant-based representation guarantees \u003cstrong\u003eframe invariance\u003c/strong\u003e and \u003cstrong\u003ecoordinate independence\u003c/strong\u003e, which are essential requirements for physically consistent turbulence modelling.(Frewer, 2009)\u0026nbsp;Moreover, it enables the correction function to generalise across different flow configurations, as it depends only on intrinsic flow characteristics rather than geometry-specific parameters.\u003c/p\u003e\n\u003ch3\u003e3.2 Physical Constraints\u003c/h3\u003e\n\u003cp\u003eTo ensure robustness and physical consistency, the correction function is subjected to a set of fundamental constraints. First, \u003cstrong\u003erealizability\u003c/strong\u003e is enforced to ensure that the resulting Reynolds stresses remain physically admissible and do not violate turbulence physics. Second, \u003cstrong\u003eboundedness\u003c/strong\u003e is imposed to prevent excessive amplification or attenuation of the eddy viscosity, thereby maintaining numerical stability within the solver. Third, \u003cstrong\u003esmoothness\u003c/strong\u003e constraints are incorporated to avoid abrupt spatial variations in the correction field, which could otherwise lead to convergence issues.(Freund et al., 2019)\u003c/p\u003e\n\u003ch3\u003e3.3 Integration with RANS Solver\u003c/h3\u003e\n\u003cp\u003eThe proposed correction framework is integrated within the RANS solver through an iterative coupling procedure. Initially, the baseline RANS equations are solved to obtain the mean flow field and corresponding turbulence quantities. Subsequently, invariant features are computed from the resolved flow variables, and the correction function is evaluated to obtain the modified eddy viscosity. The corrected eddy viscosity is then used to update the momentum equations, and the flow field is recomputed.(Bhatnagar et al., 2019)\u003c/p\u003e\n\u003cp\u003eThe procedure is iterated until convergence. The coupling strategy ensures consistency between the corrected turbulence model and the evolving flow solution, while maintaining the computational efficiency characteristic of RANS-based simulations.\u003c/p\u003e"},{"header":"4. Test Case Configurations","content":"\u003cp\u003eThe proposed framework is evaluated across three canonical flow configurations representing distinct turbulence physics, namely separation-dominated, rotation-influenced, and pressure-gradient-driven flows. These cases are selected to systematically assess the robustness and generalisability of the correction framework under varying flow regimes.(Buchanan et al., 2025)\u003c/p\u003e\n\u003ch3\u003e4.1 Square Cylinder (Separation-Dominated Flow)\u003c/h3\u003e\n\u003cp\u003eThe first test case considers flow over a square cylinder, a classical bluff body configuration characterised by strong flow separation and wake formation. The presence of yaw introduces additional asymmetry, further challenging the predictive capability of conventional turbulence models.(Minguez et al., 2011)\u003c/p\u003e\n\u003cp\u003eBaseline RANS simulations exhibit a systematic underprediction of wake extent and excessive diffusion within the shear layer, leading to an inaccurate representation of the recirculation region. These deficiencies are clearly reflected in the velocity and turbulence contours obtained for varying yaw conditions(Rao et al., 2018). The flow characteristics for the square cylinder configuration under varying yaw conditions are illustrated in Fig. 1, which presents a comparative visualization of the reference solution, baseline RANS prediction, and the corrected flow field.\u003c/p\u003e\n\u003ch3\u003e4.2 Rotating Cylinder Crossflow (Rotation-Dominated Flow)\u003c/h3\u003e\n\u003cp\u003eThe second configuration involves crossflow over a rotating cylinder, wherein rotation induces asymmetry in the flow field through the Magnus effect. This case serves as a representative scenario for rotation-dominated turbulence and stress\u0026ndash;strain misalignment(Bai et al., 2021).\u003c/p\u003e\n\u003cp\u003eClassical RANS models fail to accurately capture the induced lift asymmetry and exhibit poor representation of turbulence distribution around the cylinder. In particular, the interaction between rotation and shear leads to significant deviations from expected flow behaviour in the baseline predictions. The influence of rotation on the flow field is depicted in Fig. 2, where the asymmetry induced by the Magnus effect is compared across the reference, baseline, and corrected solutions.\u003c/p\u003e\n\u003ch2\u003e4.3 NACA0012 Airfoil with Aileron (Pressure Gradient + Control)\u003c/h2\u003e\n\u003cp\u003eThe third test case examines flow over a NACA0012 airfoil with aileron deflection, introducing strong adverse pressure gradients and flow control effects.(Abe et al., 2020)\u0026nbsp;This configuration represents a practical aerodynamic scenario involving boundary layer separation and control surface interaction(Oulghelou et al., 2025). The flow behaviour over the NACA0012 airfoil with aileron deflection is presented in Fig. 3, which compares the velocity and turbulence fields obtained from the baseline and corrected models.\u003c/p\u003e\n\u003cp\u003eBaseline RANS simulations demonstrate delayed or incorrect prediction of separation, along with inaccuracies in the pressure field distribution. These errors significantly affect the aerodynamic performance metrics and highlight the limitations of eddy-viscosity-based closures in pressure-gradient-driven flows.\u003c/p\u003e"},{"header":"5. Results and Discussion","content":"\u003ch3\u003e5.1 Baseline RANS Behaviour\u003c/h3\u003e\n\u003cp\u003eAcross all investigated configurations, classical eddy-viscosity-based RANS models exhibit consistent and systematic deficiencies in predicting key flow features. These include excessive diffusion of wake structures, an inherent symmetric bias in flows that are physically asymmetric, and inaccurate representation of turbulence intensity distribution.(Hamlington \u0026amp; Dahm, 2007)\u003c/p\u003e\n\u003cp\u003eThese deficiencies are particularly pronounced in regions characterised by strong flow gradients and non-equilibrium turbulence behaviour, namely separation zones, shear layers, and rotation-dominated regions. In the square cylinder case, the wake region is significantly underpredicted, while in the rotating cylinder configuration, the baseline model fails to capture the expected asymmetry induced by rotation. Similarly, for the airfoil case, inaccuracies in separation prediction lead to distorted flow structures downstream of the control surface.\u003c/p\u003e\n\u003cp\u003eThese baseline limitations are clearly illustrated in \u003cstrong\u003eFig. 4-6\u003c/strong\u003e, which presents a comparative visualisation of reference and baseline RANS velocity and turbulence fields across all three configurations.\u003c/p\u003e\n\u003ch3\u003e5.2 Learned Correction Field\u003c/h3\u003e\n\u003cp\u003eThe spatial distribution of the correction field reveals a highly localised structure, with corrections predominantly concentrated in regions associated with strong turbulence modelling deficiencies. These include the onset of separation, the wake core, and boundary layers influenced by rotation or adverse pressure gradients(Wu \u0026amp; Zhang, 2024).\u003c/p\u003e\n\u003cp\u003eA key observation is the strong agreement between the \u0026ldquo;hidden true correction\u0026rdquo; and the \u0026ldquo;learned correction\u0026rdquo; fields, indicating that the proposed framework successfully captures the underlying structure of the modelling error. Furthermore, the spatial distribution of the correction remains remarkably consistent across all cases, despite the fundamentally different flow physics involved.\u003c/p\u003e\n\u003cp\u003eThis behaviour is demonstrated in \u003cstrong\u003eFig. 4-6\u003c/strong\u003e, where the hidden and learned correction fields are compared for each configuration.\u003c/p\u003e\n\u003ch3\u003e5.3 Unified Behaviour Across Flow Regimes (Key Insight)\u003c/h3\u003e\n\u003cp\u003eA central finding of this study is that the correction field exhibits a consistent structural pattern across fundamentally different flow regimes. The correction consistently localizes in shear layers, wake regions, and rotation-dominated zones across all configurations.\u003c/p\u003e\n\u003cp\u003eSpecifically, corrections consistently appear in:\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"667\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eRegion\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eDominant Physical Mechanism\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eShear layers\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eHigh strain-rate gradients\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eWake regions\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eTurbulence anisotropy\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eRotating zones\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eStress\u0026ndash;strain misalignment\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eThis cross-case consistency indicates that the deficiencies of classical RANS closures are not random but exhibit a structured behaviour that can be systematically characterised using invariant features of the flow. The unified nature of the correction field is illustrated in \u003cstrong\u003eFig. 4-6\u003c/strong\u003e, which presents a comparative alignment of correction distributions across all cases.\u003c/p\u003e\n\u003ch3\u003e5.4 Velocity and Turbulence Field Improvement\u003c/h3\u003e\n\u003cp\u003eThe application of the proposed correction framework leads to significant improvements in the predicted flow fields. The corrected velocity contours show closer agreement with the reference solutions, particularly in the wake and separation regions. Additionally, the predicted wake length is substantially improved, and the turbulence intensity distribution becomes more physically consisten(Buchanan et al., 2025)t.\u003c/p\u003e\n\u003cp\u003eIn the rotating cylinder case, the corrected model successfully captures the asymmetric flow structure, while in the airfoil configuration, separation behaviour is more accurately resolved. These improvements demonstrate the capability of the framework to enhance predictive accuracy without altering the fundamental structure of the RANS equations. As shown in Table 1, the corrected model significantly improves the prediction of wake and separation characteristics, particularly in capturing asymmetric flow features and separation onset.\u003c/p\u003e\n\u003cp\u003eThese enhancements are clearly visible in \u003cstrong\u003eFig. 4-6\u003c/strong\u003e, which compares baseline and corrected flow fields for all configurations.\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"667\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eCase\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eParameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eBaseline RANS\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eCorrected Model\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eReference\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eError Reduction\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eSquare Cylinder\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eWake Length (L/D)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e\u0026darr; 60%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eRotating Cylinder\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eAsymmetry Index\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e\u0026darr; 67%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eAirfoil\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eSeparation Point (x/c)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e\u0026darr; 73%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cstrong\u003eTable 1: Wake and Separation Characteristics\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eImprovements in turbulence field prediction are quantified in Table 2, where a substantial reduction in turbulence intensity error is observed across all cases.\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"667\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eCase\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eMetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eBaseline\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eCorrected\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eImprovement (%)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eSquare Cylinder\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eTurbulence Intensity Error\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.031\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.018\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e41.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eRotating Cylinder\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eTurbulence Intensity Error\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.036\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.021\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e41.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eAirfoil\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eTurbulence Intensity Error\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.029\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.017\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e41.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cstrong\u003eTable 2: Turbulence Field Accuracy\u003c/strong\u003e\u003c/p\u003e\n\u003ch3\u003e5.5 Error Reduction\u003c/h3\u003e\n\u003cp\u003eQuantitative assessment using error maps further confirms the effectiveness of the proposed approach. (Zhang et al., 2024)The magnitude of the velocity error, defined as , is significantly reduced in critical regions, particularly within the near-wake and shear-layer zones.\u003c/p\u003e\n\u003cp\u003eThe baseline model exhibits high error concentrations in these regions, whereas the corrected solution demonstrates a marked reduction in error magnitude and spatial extent. This indicates that the correction framework effectively targets the regions where classical turbulence models fail.\u003c/p\u003e\n\u003cp\u003eThe reduction in error is illustrated in \u003cstrong\u003eFig. 4-6\u003c/strong\u003e, which presents spatial error distributions for both baseline and corrected solutions. A quantitative comparison of velocity prediction accuracy is presented in Table 3, demonstrating a consistent reduction in RMSE across all configurations following the application of the proposed correction framework.\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"667\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eCase\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eMetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eBaseline RANS\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eCorrected Model\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eImprovement (%)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eSquare Cylinder (Yaw)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eVelocity RMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.085\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.052\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e38.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eRotating Cylinder (Spin)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eVelocity RMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.092\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.058\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e37.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eAirfoil (Aileron)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eVelocity RMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.078\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.049\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e37.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cstrong\u003eTable 3: Quantitative Comparison of Velocity Field Accuracy\u003c/strong\u003e\u003c/p\u003e\n\u003ch3\u003e5.6 Generalisation Across Cases\u003c/h3\u003e\n\u003cp\u003eA notable strength of the proposed framework is its ability to generalise across different flow configurations without requiring case-specific tuning. The same correction formulation performs consistently across separation-dominated, rotation-influenced, and pressure-gradient-driven flows. The generalisation capability of the proposed framework is demonstrated in Table 4, where the correction model trained on one configuration maintains significant accuracy improvements when applied to different flow regimes.\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"667\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eTraining Case\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eTest Case\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eBaseline RMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eCorrected RMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eImprovement (%)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eCylinder\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eAirfoil\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.081\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.054\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e33.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eAirfoil\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eCylinder\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.087\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.059\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e32.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cstrong\u003eTable 4: Generalisation Capability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis robustness indicates that the correction mechanism captures fundamental aspects of turbulence modelling error, rather than relying on empirical adjustments tailored to specific geometries or flow conditions.\u0026nbsp;(Oulghelou et al., 2025)Consequently, the framework demonstrates the potential for a unified turbulence correction strategy applicable to a wide range of engineering problems.\u003c/p\u003e\n\u003cp\u003eThe overall generalization capability is summarized in \u003cstrong\u003eFig. 4-6\u003c/strong\u003e, which compares corrected predictions across all cases in a unified framework.\u003c/p\u003e"},{"header":"6. Physical Interpretation","content":"\u003cp\u003eThe results provide clear evidence of the fundamental limitations of classical eddy-viscosity-based RANS models. The primary source of error lies in their inability to represent turbulence anisotropy and their inadequate response to rotation and streamline curvature. The linear eddy-viscosity assumption enforces an isotropic stress\u0026ndash;strain relationship, which fails in separation regions, shear layers, and rotation-dominated flows, leading to stress\u0026ndash;strain misalignment and non-equilibrium effects(Hamlington \u0026amp; Dahm, 2007).\u003c/p\u003e\n\u003cp\u003eThe proposed correction framework mitigates these deficiencies through a spatially varying modification of the turbulent eddy viscosity. This correction redistributes turbulent viscosity within the flow, reducing excessive diffusion while enhancing representation in critical regions such as shear layers and wakes. As a result, a more physically consistent relationship between Reynolds stresses and the mean flow is restored.\u003c/p\u003e\n\u003cp\u003eBy formulating the correction in terms of invariant features, the approach captures intrinsic flow physics independent of geometry. This ensures consistent behaviour across different flow regimes while providing a physically interpretable representation of turbulence modelling error.\u003c/p\u003e"},{"header":"7. Conclusion","content":"\u003cp\u003eThis study presents a unified, physics-constrained framework for augmenting RANS turbulence models through an invariant-based correction to the eddy viscosity, addressing key limitations of classical closures while retaining computational efficiency. The results demonstrate that turbulence modelling errors in RANS are structured rather than case-specific, with correction fields exhibiting consistent spatial behaviour across separation-dominated, rotation-influenced, and pressure-gradient-driven flows. This indicates that modelling deficiencies are governed by underlying invariant flow features.\u003c/p\u003e\n\u003cp\u003eThe proposed framework significantly improves the prediction of velocity fields, turbulence quantities, and wake structures without compromising numerical stability or requiring case-specific tuning, thereby confirming its robustness and generalisability.\u003c/p\u003e\n\u003cp\u003eOverall, the findings establish that turbulence modelling errors can be systematically characterised and corrected using physics-constrained, invariant-based formulations, providing a viable pathway towards next-generation turbulence models with enhanced predictive capability.\u003c/p\u003e\n\u003ch3\u003eScientific Contribution\u003c/h3\u003e\n\u003cp\u003eThis study demonstrates that turbulence closure errors in classical RANS models exhibit a structured behaviour governed by invariant flow features rather than being case-specific. The identification of consistent correction patterns across multiple flow regimes provides a unified physical interpretation of turbulence modelling deficiencies, bridging empirical closures with physics-consistent augmentation strategies.\u003c/p\u003e\n\u003ch3\u003eEngineering Relevance\u003c/h3\u003e\n\u003cp\u003eThe proposed framework is applicable to a wide range of engineering flows, including aerospace aerodynamics, turbomachinery, and automotive applications, where accurate prediction of separation, wake dynamics, and rotational effects is critical. Its ability to enhance predictive accuracy while retaining computational efficiency makes it suitable for integration into industrial CFD workflows.\u003c/p\u003e"},{"header":"8. Future Work","content":"\u003cp\u003eFuture investigations will expand the framework to three-dimensional and high-Reynolds-number flows, where anisotropy and non-equilibrium effects are more prominent. Incorporation into advanced turbulence models, including SST and Reynolds stress models, as well as integration into industrial CFD workflows, will also be pursued to facilitate practical engineering implementation.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e: This research received no external funding.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eAbe, Y., Konishi, T., \u0026amp; Okabe, T. (2020). Numerical Simulation of Wake Deflection Control around NACA0012 Airfoil Using Active Morphing Flaps. \u003cem\u003eJournal of Flow Control Measurement \u0026amp;amp Visualization\u003c/em\u003e, \u003cem\u003e8\u003c/em\u003e(3), 121. https://doi.org/10.4236/jfcmv.2020.83007\u003c/li\u003e\n \u003cli\u003eAgrawal, A., \u0026amp; Koutsourelakis, P. (2023). A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertainty. \u003cem\u003earXiv (Cornell University)\u003c/em\u003e. https://doi.org/10.48550/arxiv.2307.02432\u003c/li\u003e\n \u003cli\u003eArranz, G., Ling, Y., Costa, S., Goc, K., \u0026amp; Lozano-Dur\u0026aacute;n, A. (2024). Building-block flow model for computational fluids. \u003cem\u003earXiv (Cornell University)\u003c/em\u003e. https://doi.org/10.48550/arxiv.2403.09000\u003c/li\u003e\n \u003cli\u003eBai, X., Ji, C., Grant, P., Phillips, N., Oza, U., Avital, E., \u0026amp; Williams, J. (2021). Turbulent flow simulation of a single-blade Magnus rotor. \u003cem\u003eAdvances in Aerodynamics\u003c/em\u003e, \u003cem\u003e3\u003c/em\u003e(1). https://doi.org/10.1186/s42774-021-00068-9\u003c/li\u003e\n \u003cli\u003eBhatnagar, S., Afshar, Y., Pan, S., Duraisamy, K., \u0026amp; Kaushik, S. (2019). Prediction of aerodynamic flow fields using convolutional neural networks. \u003cem\u003eComputational Mechanics\u003c/em\u003e, \u003cem\u003e64\u003c/em\u003e(2), 525. https://doi.org/10.1007/s00466-019-01740-0\u003c/li\u003e\n \u003cli\u003eBin, Y., Huang, G., Kunz, R. F., \u0026amp; Yang, X. I. A. (2023). Constrained Recalibration of Reynolds-Averaged Navier\u0026ndash;Stokes Models. \u003cem\u003eAIAA Journal\u003c/em\u003e, \u003cem\u003e62\u003c/em\u003e(4), 1434. https://doi.org/10.2514/1.j063407\u003c/li\u003e\n \u003cli\u003eBuchanan, T., Lăcătuş, M., West, A., \u0026amp; Dwight, R. P. (2025). Data-Driven RANS Closures Using a Relative Importance Term Analysis \u0026nbsp; Based Classifier for 2D and 3D Separated Flows. \u003cem\u003earXiv (Cornell University)\u003c/em\u003e. https://doi.org/10.48550/arxiv.2504.06758\u003c/li\u003e\n \u003cli\u003eFreund, J. B., MacArt, J. F., \u0026amp; Sirignano, J. (2019). 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A New Physically-Based Fully-Realizable Nonequilibrium Reynolds Stress Closure for Turbulence RANS Modeling. \u003cem\u003eDeep Blue (University of Michigan)\u003c/em\u003e. https://hdl.handle.net/2027.42/76281\u003c/li\u003e\n \u003cli\u003eHamlington, P. E., \u0026amp; Dahm, W. J. A. (2009). Reynolds Stress Closure Including Nonlocal and Nonequilibrium Effects in Turbulent Flows. \u003cem\u003eDeep Blue (University of Michigan)\u003c/em\u003e. https://hdl.handle.net/2027.42/77039\u003c/li\u003e\n \u003cli\u003eMinguez, M., Brun, C., Pasquetti, R., \u0026amp; Serre, \u0026Eacute;. (2011). Experimental and high-order LES analysis of the flow in near-wall region of a square cylinder. \u003cem\u003eInternational Journal of Heat and Fluid Flow\u003c/em\u003e, \u003cem\u003e32\u003c/em\u003e(3), 558. https://doi.org/10.1016/j.ijheatfluidflow.2011.03.009\u003c/li\u003e\n \u003cli\u003eOulghelou, M., Cherroud, S., Merle, X., \u0026amp; Cinnella, P. (2024). 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AutoTurb: Using Large Language Models for Automatic Algebraic Model \u0026nbsp; Discovery of Turbulence Closure. \u003cem\u003earXiv (Cornell University)\u003c/em\u003e. https://doi.org/10.48550/arxiv.2410.10657\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Turbulence modelling, Reynolds-Averaged Navier–Stokes (RANS), Eddy viscosity correction, Flow separation, Rotating flows, Reynolds stress anisotropy, Physics-informed modelling","lastPublishedDoi":"10.21203/rs.3.rs-9450890/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9450890/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eReynolds-Averaged Navier–Stokes (RANS) turbulence models remain the industrial standard for simulating high-Reynolds-number flows; however, their predictive capability deteriorates in flows characterised by separation, curvature, and rotation. These deficiencies stem primarily from the linear eddy-viscosity hypothesis, which assumes isotropic turbulence and a linear stress–strain relationship. This study investigates whether the modelling error in classical RANS closures exhibits a structured behaviour across diverse flow regimes and whether a physics-constrained correction can restore predictive fidelity.\u003c/p\u003e\n\u003cp\u003eA unified turbulence augmentation framework is developed in which the baseline eddy viscosity is corrected using an invariant-based mapping constrained by physical principles including frame invariance, boundedness, and realizability. The framework is evaluated on three canonical yet fundamentally distinct configurations: flow over a square cylinder (separation-dominated), rotating cylinder crossflow (rotation-induced anisotropy), and NACA0012 airfoil with control surface deflection (pressure-gradient-driven separation).\u003c/p\u003e\n\u003cp\u003eResults demonstrate that the correction field consistently localises in shear layers, separation regions, and rotation-dominated zones across all cases. The augmented model improves velocity field predictions, wake structure, and turbulence quantities relative to baseline RANS. Crucially, the correction exhibits cross-case structural similarity, suggesting that turbulence modelling errors are not arbitrary but governed by underlying invariant features of the flow.\u003c/p\u003e\n\u003cp\u003eThe findings establish a pathway towards unified turbulence closure augmentation, combining physical consistency with improved predictive accuracy for complex engineering flows.\u003c/p\u003e","manuscriptTitle":"Physics-Constrained Eddy Viscosity Correction for Rotating and Separated Flows: A Unified RANS Augmentation Framework","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-27 11:01:05","doi":"10.21203/rs.3.rs-9450890/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"8b6c2c08-bd2a-440d-99b5-324ac4b1830d","owner":[],"postedDate":"April 27th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-04-27T11:01:05+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-27 11:01:05","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9450890","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9450890","identity":"rs-9450890","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}