A short note on turbulence characteristics in wind-turbine wakes

Analytical wake models need formulations to mimic the impact of wind turbines on turbulence level in the wake region. Several correlations can be found in the literature for this purpose, one of which is the formula proposed in A. Crespo, J. Hernandez, Turbulence characteristics in wind-turbine wakes, Journal of Wind Engineering and Industrial Aerodynamics 61 (1) (1996) 71 - 85, which relates the added turbulence to the induction factor of the turbine, ambient turbulence intensity, and normalized distance from the rotor through an equation with one coefficient and three exponents for the effective parameters. Misuse of this formula with an incorrect exponent for the ambient turbulence intensity is propagating in the literature. In this note, we implement the original and the incorrect formulation of turbine-induced added turbulence in a Gaussian wake model to quantify its impact by studying the Horns Rev 1 wind farm. The results reveal that the turbulence intensity and the normalized power of the waked turbines predicted by the wake model with the correct and the incorrect implementation of turbine-induced added turbulence correlation have a difference equal to 1.94% and 3.53%, respectively, for an ambient turbulence intensity of 7.7%. For an ambient turbulence intensity of 4%, these discrepancies grow to 2.7% and 4.95%.Comment: 4 pages, 2 figure

An extended k−εk-\varepsilon model for wake-flow simulation of wind farms

The Reynolds-averaged Navier-Stokes approach coupled with the standard $k-\varepsilon$ model is widely utilized for wind-energy applications. However, it has been shown that the standard $k-\varepsilon$ model overestimates the turbulence intensity in the wake region and, consequently, overpredicts the power output of the waked turbines. This study focuses on the development of an extended $k-\varepsilon$ model by incorporating an additional term in the turbulent kinetic energy equation. This term accounts for the influence of turbine-induced forces, and its formulation is derived through an analytical approach. To assess the effectiveness of the proposed model, we begin by analyzing the evolution of normalized velocity deficit and turbulence intensity in the wake region, and the normalized power of the waked turbines. This investigation involves a comparison of the predictions against results from large-eddy simulations in three validation cases with different layouts. We then simulate a wind farm consisting of 30 wind turbines and conduct a comparative analysis between the model-predicted normalized streamwise velocity and wind-tunnel measurements. Finally, to conclude our assessment of the proposed model, we apply it to the operational wind farm of Horns Rev 1 and evaluate the obtained normalized power with the results from large-eddy simulations. The comparisons and validations conducted in this study prove the superior performance of the extended $k-\varepsilon$ model compared to the standard version.Comment: 14 pages, 14 figure

Frozen propagation of Reynolds force vector from high-fidelity data into Reynolds-averaged simulations of secondary flows

Successful propagation of information from high-fidelity sources (i.e., direct numerical simulations and large-eddy simulations) into Reynolds-averaged Navier-Stokes (RANS) equations plays an important role in the emerging field of data-driven RANS modeling. Small errors carried in high-fidelity data can propagate amplified errors into the mean flow field, and higher Reynolds numbers worsen the error propagation. In this study, we compare a series of propagation methods for two cases of Prandtl's secondary flows of the second kind: square-duct flow at a low Reynolds number and roughness-induced secondary flow at a very high Reynolds number. We show that frozen treatments result in less error propagation than the implicit treatment of Reynolds stress tensor (RST), and for cases with very high Reynolds numbers, explicit and implicit treatments are not recommended. Inspired by the obtained results, we introduce the frozen treatment to the propagation of Reynolds force vector (RFV), which leads to less error propagation. Specifically, for both cases at low and high Reynolds numbers, propagation of RFV results in one order of magnitude lower error compared to RST propagation. In the frozen treatment method, three different eddy-viscosity models are used to evaluate the effect of turbulent diffusion on error propagation. We show that, regardless of the baseline model, the frozen treatment of RFV results in less error propagation. We combined one extra correction term for turbulent kinetic energy with the frozen treatment of RFV, which makes our propagation technique capable of reproducing both velocity and turbulent kinetic energy fields similar to high-fidelity data

Physics-guided machine learning for wind-farm power prediction: Toward interpretability and generalizability

With the increasing amount of available data from simulations and experiments, research for the development of data-driven models for wind-farm power prediction has increased significantly. While the data-driven models can successfully predict the power of a wind farm with similar characteristics as those in the training ensemble, they generally do not have a high degree of flexibility for extrapolation to unseen cases in contrast to the physics-based models. In this paper, we focus on data-driven models with improved interpretability and generalizability levels that can predict the performance of turbines in wind farms. To prepare the datasets, several cases are defined based on the layouts of operational wind farms, and massive computational fluid dynamics simulations are performed. The extreme gradient boosting algorithm is used afterward to build models, which have turbine-level geometric inputs in combination with the efficiency from physics-based models as the features. After training, to analyze the models' capability in generalization, their predictions for the unseen cases with different operating conditions, inflow turbulence levels, and wind-farm layouts are compared to the Park model and an empirical-analytical Gaussian wake model. Results show that the physics-guided machine-learning models outperform both physics-based models showing a high degree of generalizability, and the machine is not sensitive to the choice of the physics-based guide model

An Analytical Model for the Effect of Vertical Wind Veer on Wind Turbine Wakes

In this study, an analytical wake model for predicting the mean velocity field downstream of a wind turbine under veering incoming wind is systematically derived and validated. The new model, which is an extended version of the one introduced by Bastankhah and Porté-Agel, is based upon the application of mass conservation and momentum theorem and considering a skewed Gaussian distribution for the wake velocity deficit. Particularly, using a skewed (instead of axisymmetric) Gaussian shape allows accounting for the lateral shear in the incoming wind induced by the Coriolis force. This analytical wake model requires only the wake expansion rate as an input parameter to predict the mean wake flow downstream. The performance of the proposed model is assessed using the large-eddy simulation (LES) data of a full-scale wind turbine wake under the stably stratified condition. The results show that the proposed model is capable of predicting the skewed structure of the wake downwind of the turbine, and its prediction for the wake velocity deficit is in good agreement with the high-fidelity simulation data

POD-mode-augmented wall model and its applications to flows at non-equilibrium conditions

Insights gained from modal analysis are invoked for predictive large-eddy simulation (LES) wall modeling. Specifically, we augment the law of the wall (LoW) by an additional mode based on a one-dimensional proper orthogonal decomposition (POD) applied to a 2D turbulent channel. The constructed wall model contains two modes, i.e., the LoW mode and the POD-based mode, and the model matches with the LES at two, instead of one, off-wall locations. To show that the proposed model captures non-equilibrium effects, we perform a-priori and a-posteriori tests in the context of both equilibrium and non-equilibrium flows. The a-priori tests show that the proposed wall model captures extreme wall-shear stress events better than the equilibrium wall model. The model also captures non-equilibrium effects due to adverse pressure gradients. The a-posteriori tests show that the wall model captures the rapid decrease and the initial decrease of the streamwise wall-shear stress in channels subjected to suddenly imposed adverse and transverse pressure gradients, respectively, both of which are missed by currently available wall models. These results show promise in applying modal analysis for turbulence wall modeling. In particular, the results show that employing multiple modes helps in the modeling of non-equilibrium flows