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.

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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.

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The use of load surrogates in offshore wind turbine site assessment has gained attention as a way to speed up the lengthy and costly siting process. We propose a novel probabilistic approach using mixture density networks to map 10 min average site conditions to the corresponding load statistics. The probabilistic framework allows for the modeling of the uncertainty in the loads as a response to the stochastic inflow conditions. We train the data-driven model on the OpenFAST simulations of the IEA 10 MW reference wind turbine (IEA-10MW-RWT) and compare the predictions to the widely used Gaussian process regression. We show that mixture density networks can recover the accurate mean response in all load channels with values for the coefficient of determination (R2) greater than 0.95 on the test dataset. Mixture density networks completely outperform Gaussian process regression in predicting the quantiles, showing an excellent agreement with the reference. We compare onshore and offshore sites for training to conclude the need for a more extensive training dataset in offshore cases due to the larger feature space and more noise in the data.@en

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.

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While efforts have been made to curb CO2 emissions from aviation, the more uncertain non-CO2 effects that contribute about two-thirds to the warming in terms of radiative forcing (RF), still require attention. The most important non-CO2 effects include persistent line-shaped contrails, contrail-induced cirrus clouds and nitrogen oxide (NOx) emissions that alter the ozone (O3) and methane (CH4) concentrations, both of which are greenhouse gases, and the emission of water vapour (H2O). The climate impact of these non-CO2 effects depends on emission location and prevailing weather situation; thus, it can potentially be reduced by advantageous re-routing of flights using Climate Change Functions (CCFs), which are a measure for the climate effect of a locally confined aviation emission. CCFs are calculated using a modelling chain starting from the instantaneous RF (iRF) measured at the tropopause that results from aviation emissions. However, the iRF is a product of computationally intensive chemistry-climate model (EMAC) simulations and is currently restricted to a limited number of days and only to the North Atlantic Flight Corridor. This makes it impossible to run EMAC on an operational basis for global flight planning. A step in this direction lead to a surrogate model called algorithmic Climate Change Functions (aCCFs), derived by regressing CCFs (training data) against 2 or 3 local atmospheric variables at the time of emission (features) with simple regression techniques and are applicable only in parts of the Northern hemisphere. It was found that in the specific case of O3 aCCFs, which provide a reasonable first estimate for the short-term impact of aviation NOx on O3 warming using temperature and geopotential as features, can be vastly improved [1]. There is aleatoric uncertainty in the full-order model (EMAC), stemming from unknown sources (missing features) and randomness in the known features, which can introduce heteroscedasticity in the data. Deterministic surrogates (e.g. aCCFs) only predict point estimates of the conditional average, thereby providing an incomplete picture of the stochastic response. Thus, the goal of this research is to build a new surrogate model for iRF, which is achieved by : 1. Expanding the geographical coverage of iRF (training data) by running EMAC simulations in more regions (North & South America, Eurasia, Africa and Australasia) at multiple cruise flight altitudes, 2. Following an objective approach to selecting atmospheric variables (feature selection) and considering the importance of local as well as non-local effects, 3. Regressing the iRF against selected atmospheric variables using supervised machine learning techniques such as homoscedastic and heteroscedastic Gaussian process regression. We present a new surrogate model that predicts iRF of aviation NOx-O3 effects on a regular basis with confidence levels, which not only improves our scientific understanding of NOx-O3 effects, but also increases the potential of global climate-optimised flight planning.@en

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.

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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.

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Heteroscedastic Gaussian process regression, based on the concept of chained Gaussian processes, is used to build surrogates to predict site-specific loads on an offshore wind turbine. Stochasticity in the inflow turbulence and irregular waves results in load responses that are best represented as random variables rather than deterministic values. Moreover, the effect of these stochastic sources on the loads depends strongly on the mean environmental conditions - for instance, at low mean wind speeds, inflow turbulence produces much less variability in loads than at high wind speeds. Statistically, this is known as heteroscedasticity. Deterministic and most stochastic surrogates do not account for the heteroscedastic noise, giving an incomplete and potentially misleading picture of the structural response. In this paper, we draw on the recent advancements in statistical inference to train a heteroscedastic surrogate model on a noisy database to predict the conditional pdf of the response. The model is informed via 10-minute load statistics of the IEA-10MW-RWT subject to both aero- and hydrodynamic loads, simulated with OpenFAST. Its performance is assessed against the standard Gaussian process regression. The predicted mean is similar in both models, but the heteroscedastic surrogate approximates the large-scale variance of the responses significantly better.@en

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.

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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.

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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.

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In this short note we apply the recently proposed data-driven RANS closure modelling framework of Schmelzer et al.(2020) to fully three-dimensional, high Reynolds number flows: namely wall-mounted cubes and cuboids at Re=40,000, and a cylinder at Re=140,000. For each flow, a new RANS closure is generated using sparse symbolic regression based on LES or DES reference data. This new model is implemented in a CFD solver, and subsequently applied to prediction of the other flows. We see consistent improvements compared to the baseline k−ω SST model in predictions of mean-velocity in complete flow domain.

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Currently, the state of the art in wind farm flow physics modeling are Large Eddy Simulations (LES) which resolve a large part of the spectra of the turbulent fluctuations. But this type of model requires extensive computational resources. One wind speed and direction simulation of the Lillgrund wind farm can take between 160k and 3000k processor hours depending on how the turbines are modeled [1, 2]. The next-fidelity model types are Reynolds-Averaged Navier-Stokes (RANS) models which resolve only the mean quantities and model the effect of turbulence fluctuations. These models require about two orders of magnitude less computational time, but generally do not produce accurate predictions of the mean flow field. Proposed modifications made to these models so far do not generalize well and there is room for improvement. Hence, we present the first steps towards using a data-driven approach to aid in deriving new RANS models that generalize well to different turbine types, varying atmospheric stability, and farm layouts. To do so, time-averaged LES data is used to derive corrections to existing RANS models. The approach uses a deterministic symbolic regression method to infer algebraic correction terms to the RANS turbulence transport equations. Optimal correction terms to the RANS equations are derived using a frozen approach where time-averaged flow fields from LES are injected into the RANS equations. The potential of the approach is demonstrated under neutral conditions for multi-turbine constellations at wind-tunnel scale. The results show promise, but more work is necessary to realize the full potential of the approach.

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Simulating wind-turbines in Reynolds-averaged Navier-Stokes (RANS) codes is highly challenging, at least partly due to the importance of turbulence anisotropy in the evolution of the wake. We present a preliminary investigation into the role of anisotropy in RANS simulations of vertical-axis turbines, by comparison with LES. Firstly an LES data-set serving as our ground-truth is generated, and verified against previously published works. This data-set provides raw turbulence anisotropy fields for several turbine configurations. This anisotropy is injected into RANS simulations of identical configurations to determine the extent to which it influences (i) the production of turbulence kinetic energy, (ii) the turbulence momentum forcing, and finally (iii) the mean-flow. In all these quantities we observe the anisotropy has a surprisingly limited effect, and is certainly not the leading-order error in Boussinesq RANS for these cases. Nevertheless we go on to show that it is feasible to predict anisotropy fields for unseen configurations based only on the mean-flow, by using a tensorized version of random-forest regression.

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A novel machine learning algorithm is presented, serving as a data-driven turbulence modeling tool for Reynolds Averaged Navier-Stokes (RANS) simulations. This machine learning algorithm, called the Tensor Basis Random Forest (TBRF), is used to predict the Reynolds-stress anisotropy tensor, while guaranteeing Galilean invariance by making use of a tensor basis. By modifying a random forest algorithm to accept such a tensor basis, a robust, easy to implement, and easy to train algorithm is created. The algorithm is trained on several flow cases using DNS/LES data, and used to predict the Reynolds stress anisotropy tensor for new, unseen flows. The resulting predictions of turbulence anisotropy are used as a turbulence model within a custom RANS solver. Stabilization of this solver is necessary, and is achieved by a continuation method and a modified k-equation. Results are compared to the neural network approach of Ling et al. [29]. Results show that the TBRF algorithm is able to accurately predict the anisotropy tensor for various flow cases, with realizable predictions close to the DNS/LES reference data. Corresponding mean flows for a square duct flow case and a backward facing step flow case show good agreement with DNS and experimental data-sets. Overall, these results are seen as a next step towards improved data-driven modelling of turbulence. This creates an opportunity to generate custom turbulence closures for specific classes of flows, limited only by the availability of LES/DNS data.

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A multilevel Monte Carlo (MLMC) method for quantifying model-form uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS) simulations is presented. Two, high-dimensional, stochastic extensions of the RANS equations are considered to demonstrate the applicability of the MLMC method. The first approach is based on global perturbation of the baseline eddy viscosity field using a lognormal random field. A more general second extension is considered based on the work of [Xiao et al. (2017)], where the entire Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For two fundamental flows, we show that the MLMC method based on a hierarchy of meshes is asymptotically faster than plain Monte Carlo. Additionally, we demonstrate that for some flows an optimal multilevel estimator can be obtained for which the cost scales with the same order as a single CFD solve on the finest grid level.

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In this work recent advancements are presented in utilising deterministic symbolic regression to infer algebraic models for turbulent stress-strain relation with sparsity-promoting regression techniques. The goal is to build a functional expression from a set of candidate functions in order to represent the target data most accurately. Targets are the coefficients of a polynomial tensor basis, which are identified from high-fidelity data using regularised least-square regression. The method successfully identified a correction term for the benchmark test case of flow over periodic hills in 2D at Reh = 10595.@en

An anti-fairing is a concave deformation of the wall around a wing-body junction that can decrease the aerodynamic drag through the activation of a propulsive force generated by the interaction of the curved concave shape and the high-pressure region in proximity of the wing leading-edge. Although this mechanism is well understood, the dynamics of the interaction between the anti-fairing and the junction flow remain largely unexplored. This work brings together all the numerical and experimental studies of the anti-fairing to investigate its effect on turbulent quantities and the robustness of its design to changes to the incoming flow parameters, and to estimate the drag change with respect to a normal wing/flat-plate configuration. It is found that the interaction of the streamwise pressure gradient generated by the anti-fairing with the incoming boundary layer substantially reduces the shear responsible for viscous drag. Furthermore, no significant influence of the incoming boundary layer thickness on the anti-fairing performance is observed. However, a direct drag measurement with a force balance casts some doubts on the possibility to achieve large drag reductions.

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Downwind wind turbine blades are subjected to tower wake forcing at every rotation, which can lead to structural fatigue. Accurate characterisation of the unsteady aeroelastic forces in the blade design phase requires detailed representation of the aerodynamics, leading to computationally expensive simulation codes, which lead to intractable uncertainty analysis and Bayesian updating. In this paper, a framework is developed to tackle this problem. Full, detailed aeroelastic model of an experimental wind turbine system based on 3-D Reynolds-averaged Navier-Stokes is developed, considering all structural components including nacelle and tower. This model is validated against experimental measurements of rotating blades, and a detailed aeroelastic characterisation is presented. Aerodynamic forces from prescribed forced-motion simulations are used to train a time-domain autoregressive with exogenous input (ARX) model with a localised forcing term, which provides accurate and cheap aeroelastic forces. Employing ARX, prior uncertainties in the structural and rotational parameters of the wind turbine are introduced and propagated to obtain probabilistic estimates of the aeroelastic characteristics. Finally, the experimental validation data are used in a Bayesian framework to update the structural and rotational parameters of the system and thereby reduce uncertainty in the aeroelastic characteristics.

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