Flow-induced vibrations in wall-bounded flows remain a major challenge in many engineering applications. While fully predictive turbulence-induced vibration simulations generally require scale-resolving approaches, unsteady Reynolds-averaged Navier–Stokes (URANS) methods are employed in practice due to their favorable computational cost. However, their predictive reliability is limited by uncertainty in turbulence-model parameters, structural model–form error, and deficiencies in representing turbulent fluctuations. This work presents a surrogate-based Bayesian calibration framework for the SST – model, incorporating anisotropic pressure-fluctuation reconstruction to quantify and reduce model uncertainty. Global Sobol’ sensitivity analysis, surrogate modeling, and Bayesian inference are combined to identify influential parameters and assimilate high-fidelity reference data. The framework is applied to turbulent channel flow, turbulent annular flow, and a blunt-end cantilevered-rod configuration. Across all cases, the specific dissipation-rate coefficient and the turbulent kinetic-energy dissipation coefficient consistently dominate variability in mean-flow quantities and reconstructed fluctuation statistics. Bayesian calibration constrains these parameters to consistent regions of the admissible space, while less influential coefficients remain weakly informed. Gaussian process regression and polynomial chaos expansion surrogates yield nearly identical posterior and predictive distributions, demonstrating robustness to surrogate choice. Remaining discrepancies with reference data are attributed primarily to structural turbulence model limitations rather than parametric uncertainty.

This study presents a novel approach for enhancing Reynolds-averaged Navier-Stokes (RANS) turbulence modeling through the application of a Relative Importance Term Analysis (RITA) methodology to develop a new zonally-augmented k−ω SST model. Traditional Linear Eddy Viscosity Models often struggle with separated flows. Our approach introduces a physics-based binary classifier that systematically identifies separated shear layers requiring correction by analyzing the relative magnitudes of terms in the turbulent kinetic energy equation. Using symbolic regression, we develop compact correction terms for Reynolds stress anisotropy and turbulent kinetic energy production. Trained on 2D configurations, our model demonstrates significant improvements in predicting separation dynamics while maintaining baseline performance in fully attached flows. Generalization tests on Ahmed body and Faith hill 3D configurations confirm robust transferability, establishing an effective methodology for targeted enhancement of RANS predictions in separated flows.

This study presents a compact data-driven Reynolds-averaged Navier-Stokes (RANS) model for wind turbine wake prediction, built as an enhancement of the standard - formulation. Several candidate models were discovered using the symbolic regression framework Sparse Regression of Turbulent Stress Anisotropy (SpaRTA), trained on a single Large Eddy Simulation (LES) dataset of a standalone wind turbine. The leading model was selected by prioritizing simplicity while maintaining reasonable accuracy, resulting in a novel linear eddy viscosity model. This selected leading model reduces eddy viscosity in high-shear regions—particularly in the wake—to limit turbulence mixing and delay wake recovery. This addresses a common shortcoming of the standard - model, which tends to overpredict mixing, leading to unrealistically fast wake recovery. Moreover, the formulation of the leading model closely resembles that of the established -- model. Consistent with this resemblance, the leading and -- models show nearly identical performance in predicting velocity fields and power output, but they differ in their predictions of turbulent kinetic energy. In addition, the generalization capability of the leading model was assessed using three unseen six-turbine configurations with varying spacing and alignment. Despite being trained solely on a standalone turbine case, the model produced results comparable to LES data. These findings demonstrate that data-driven methods can yield interpretable, physically consistent RANS models that are competitive with traditional modeling approaches while maintaining simplicity and achieving generalizability.

We consider the task of calibrating turbulence models against reference data, with particular reference to the estimation of parameters of Reynolds-averaged Navier-Stokes closure models. Traditional calibration methods against canonical flows are discussed, and we introduce the framework of Bayesian probability to generalize this basic approach. A major benefit of Bayes is that we obtain uncertainty information indicating the level of confidence in the calibration results; which we can use to subsequently estimate the uncertainty in predictions due to the model. Numerical methods for solving the challenging computational problems involved are discussed. Finally we apply the same Bayesian framework to the data-assimilation problem of indirectly identifying turbulence anisotropy fields given experimental data pressure data.

We present a data-driven approach to Reynolds-averaged Navier-Stokes (RANS) turbulence closure modelling in magnetohydrodynamic (MHD) flows. In these flows the magnetic field interacting with the conductive fluid induces unconventional turbulence states such as quasi two-dimensional (2D) turbulence, and turbulence suppression, which are poorly represented by standard Boussinesq models. Our data-driven approach uses time-averaged Large Eddy Simulation (LES) data of annular pipe flows, at different Hartmann numbers, to derive corrections for the - SST model. Correction fields are obtained by injecting time averaged LES fields into the MHD RANS equations, and examining the remaining residuals. The correction to the Reynolds-stress anisotropy is approximated with a modified Tensor Basis Neural Network (TBNN). We extend the generalised eddy hypothesis with a traceless antisymmetric tensor representation of the Lorentz force to obtain MHD flow features, thus keeping Galilean and frame invariance while including MHD effects in the turbulence model. The resulting data-driven models are shown to reduce errors in the mean flow, and to generalise to annular flow cases with different Hartmann numbers from those of the training cases.

Reliable prediction of aviation’s environmental impact, including the effect of nitrogen oxides on ozone, is vital for effective mitigation against its contribution to global warming. Estimating this climate impact however, in terms of the short-term ozone instantaneous radiative forcing, requires computationally-expensive chemistry-climate model simulations that limit practical applications such as climate-optimised planning. Existing surrogates neglect the large uncertainties in their predictions due to unknown environmental conditions and missing features. Relative to these surrogates, we propose a high-accuracy probabilistic surrogate that not only provides mean predictions but also quantifies heteroscedastic uncertainties in climate impact estimates. Our model is trained on one of the most comprehensive chemistry-climate model datasets for aviation-induced nitrogen oxide impacts on ozone. Leveraging feature selection techniques, we identify essential predictors that are readily available from weather forecasts to facilitate the implementation therein. We show that our surrogate model is more accurate than homoscedastic models and easily outperforms existing linear surrogates. We then predict the climate impact of a frequently-flown flight in the European Union, and discuss limitations of our approach.

Accurately predicting wind turbine wake effects is essential for optimizing wind-farm performance and minimizing maintenance costs. This study explores the applicability of the Sparse Regression of Turbulent Stress Anisotropy (SpaRTA) framework to develop a simple yet robust Reynolds-averaged Navier-Stokes (RANS) model for wake prediction in wind energy contexts. The framework introduces two correction terms into two-equation models, with k - ϵ model being utilized in the current study. One correction term resembles the residual of the Turbulent Kinetic Energy (TKE) equation, and the other corrects the deviatoric part of the Reynolds Stress Tensor (RST). The terms are calculated from high-fidelity measurement or simulation data, and symbolic regression is used to determine the model for these terms. In this study, Large Eddy Simulation (LES) data from a single turbine is used as the training dataset, and a sample pre-selection process is employed to discover a correction model efficiently. The derived model incorporates two terms based on Pope's basis tensors and their invariants. The expression of the obtained model shows that it functions as a modification to the constant Cμ in the k - ϵ model. The model is evaluated by comparing its predicted velocity and TKE fields with the LES data used for the training. The model showed satisfactory performance in predicting both fields. Additionally, its generalizability is evaluated by testing it against a more complex six-turbine unseen case. The results indicate that the model effectively captures the velocity field and power output, but it tends to overpredict TKE, especially in the wake region.

Purpose: This paper aims to improve Reynolds-averaged Navier Stokes (RANS) turbulence models using a data-driven approach based on machine learning (ML). A special focus is put on determining the optimal input features used for the ML model. Design/methodology/approach: The field inversion and machine learning (FIML) approach is applied to the negative Spalart-Allmaras turbulence model for transonic flows over an airfoil where shock-induced separation occurs. Findings: Optimal input features and an ML model are developed, which improve the existing negative Spalart-Allmaras turbulence model with respect to shock-induced flow separation. Originality/value: A comprehensive workflow is demonstrated that yields insights on which input features and which ML model should be used in the context of the FIML approach.

The state-of-the-art in wind-farm flow-physics modeling is Large Eddy Simulation (LES) which makes accurate predictions of most relevant physics, but requires extensive computational resources. The next-fidelity model types are Reynolds-Averaged Navier–Stokes (RANS) which are two orders of magnitude cheaper, but resolve only mean quantities and model the effect of turbulence. They often fail to accurately predict key effects, such as the wake recovery rate. Custom RANS closures designed for wind-farm wakes exist, but so far do not generalize well: there is substantial room for improvement. In this article we present the first steps towards a systematic data-driven approach to deriving new RANS models in the wind-energy setting. Time-averaged LES data is used as ground-truth, and we first derive optimal corrective fields for the turbulence anisotropy tensor and turbulence kinetic energy (t.k.e.) production. These fields, when injected into the RANS equations (with a baseline k–ɛ model) reproduce the LES mean-quantities. Next we build a custom RANS closure from these corrective fields, using a deterministic symbolic regression method to infer algebraic correction as a function of the (resolved) mean-flow. The result is a new RANS closure, customized to the training data. The potential of the approach is demonstrated under neutral atmospheric conditions for multi-turbine constellations at wind-tunnel scale. The results show significantly improved predictions compared to the baseline closure, for both mean velocity and the t.k.e. fields.

In fluid dynamics, constitutive models are often used to describe the unresolved turbulence and to close the Reynolds averaged Navier–Stokes (RANS) equations. Traditional PDE-based constitutive models are usually too rigid to calibrate with a large set of high-fidelity data. Moreover, commonly used turbulence models are based on the weak equilibrium assumption, which cannot adequately capture the nonlocal physics of turbulence. In this work, we propose using a vector-cloud neural network (VCNN) to learn the nonlocal constitutive model, which maps a regional mean flow field to the local turbulence quantities without solving the transport PDEs. The network is strictly invariant to coordinate translation, rotation, and uniform motion, as well as ordering of the input points. The VCNN-based nonlocal constitutive model is trained and evaluated on flows over a family of parameterized periodic hills. Numerical results demonstrate its predictive capability on target turbulence quantities of turbulent kinetic energy k and dissipation ɛ. More importantly, we investigate the robustness and stability of the method by coupling the trained model back to RANS solver. The solver shows good convergence with the simulated velocity field comparable to that based on k–ɛ model when starting from a reasonable initial condition. This study, as a proof of concept, highlights the feasibility of using a nonlocal, frame-independent, neural network-based constitutive model to close the RANS equations, paving the way for the further emulation of the Reynolds stress transport models.

Data-driven Reynolds-averaged Navier–Stokes (RANS) turbulence closures are increasing seen as a viable alternative to general-purpose RANS closures, when LES reference data is available—also in wind-energy. Parsimonious closures with few, simple terms have advantages in terms of stability, interpret-ability, and execution speed. However experience suggests that closure model corrections need be made only in limited regions—e.g. in the near-wake of wind turbines and not in the majority of the flow. A parsimonious model therefore must find a middle ground between precise corrections in the wake, and zero corrections elsewhere. We attempt to resolve this impasse by introducing a classifier to identify regions needing correction, and only fit and apply our model correction there. We observe that such classifier-based models are significantly simpler (with fewer terms) than models without a classifier, and have similar accuracy, but are more prone to instability. We apply our framework to three flows consisting of multiple wind-turbines in neutral conditions with interacting wakes.

Multi-fidelity optimization methods promise a high-fidelity optimum at a cost only slightly greater than a low-fidelity optimization. This promise is seldom achieved in practice, due to the requirement that low- and high-fidelity models correlate well. In this article, we propose an efficient bi-fidelity shape optimization method for turbulent fluid-flow applications with Large-Eddy Simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) as the high- and low-fidelity models within a hierarchical-Kriging surrogate modelling framework. Since the LES–RANS correlation is often poor, we use the full LES flow-field at a single point in the design space to derive a custom-tailored RANS closure model that reproduces the LES at that point. This is achieved with machine-learning techniques, specifically sparse regression to obtain high corrections of the turbulence anisotropy tensor and the production of turbulence kinetic energy as functions of the RANS mean-flow. The LES–RANS correlation is dramatically improved throughout the design-space. We demonstrate the effectivity and efficiency of our method in a proof-of-concept shape optimization of the well-known periodic-hill case. Standard RANS models perform poorly in this case, whereas our method converges to the LES-optimum with only two LES samples.

This paper presents a novel approach for correcting wind-tunnel wall interference in the nonlinear flow regime, that is, in the presence of phenomena such as flow separation and shocks. The methodology uses a gradient-based optimization to minimize the difference between experimental measurements and a Favre-averaged Navier–Stokes (FANS) simulation. The aim is to exploit the high-fidelity experimental data to correct turbulence-modeling errors in the FANS simulations, as well as to use the accurate angle of attack and Mach number from the FANS simulations to correct the in-tunnel flow conditions. The optimization is carried out directly in free air, thus avoiding the requirement to mesh the wind-tunnel walls and/or to model the ventilated-wall boundary condition. A byproduct of this method is the availability of flow information everywhere around the test object, which augments and complements the experimental data. The methodology is tested on two-dimensional and three-dimensional flow cases, demonstrating a significant improvement in the agreement between experimental and numerical data.