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Abstract Despite substantial growth in wind energy technology in recent decades, aerodynamic modeling of wind turbines relies on momentum models derived in the late 19th and early 20th centuries, which are well-known to break down under flow regimes in which wind turbines often operate. This gap in theoretical modeling for rotors that are misaligned with the inflow and also for high-thrust rotors has resulted in the development of numerous empirical corrections which are widely applied in textbooks, research articles, and open-source and industry design codes. This work reports a Unified Momentum Model to efficiently predict power production, thrust force, and wake dynamics of rotors under arbitrary inflow angles and thrust coefficients without empirical corrections. The Unified Momentum Model is additionally coupled with a blade element model to enable blade element momentum modeling predictions of wind turbines in high thrust coefficient and yaw misaligned states without using corrections for these states. This Unified Momentum Model can form a new basis for wind turbine modeling, design, and control tools from first principles and may enable further development of innovations necessary for increased wind production and reliability to respond to 21st century climate change challenges. 

As turbines continue to grow in hub height and rotor diameter and wind farms grow larger, consideration of stratified atmospheric boundary layer (ABL) processes in wind power models becomes increasingly important. Atmospheric stratification can considerably alter the boundary layer structure and flow characteristics through buoyant forcing. Variations in buoyancy, and corresponding ABL stability, in both space and time impact ABL wind speed shear, wind direction shear, boundary layer height, turbulence kinetic energy, and turbulence intensity. In addition, the presence of stratification will result in a direct buoyant forcing within the wake region. These ABL mechanisms affect turbine power production, the momentum and kinetic energy deficit wakes generated by turbines, and the turbulent mixing and kinetic energy entrainment in wind farms. Presently, state-of-practice engineering models of mean wake momentum utilize highly empirical turbulence models that do not explicitly account for ABL stability. Models also often neglect the interaction between the wake momentum deficit and the turbulence kinetic energy added by the wake, which depends on stratification. In this work, we develop a turbulence model that models the wake-added turbulence kinetic energy, and we couple it with a wake model based on the parabolized Reynolds-averaged Navier–Stokes equations. Comparing the model predictions to large eddy simulations across stabilities (Obukhov lengths) and surface roughness lengths, we find lower prediction error in both power production and the wake velocity field across the ABL conditions and error metrics investigated. 

Wind turbines operate in the atmospheric boundary layer (ABL), where Coriolis effects are present. As wind turbines with larger rotor diameters are deployed, the wake structures that they create in the ABL also increase in length. Contemporary utility-scale wind turbines operate at rotor diameter-based Rossby numbers, the non-dimensional ratio between inertial and Coriolis forces, of$$\mathcal {O}(100)$$where Coriolis effects become increasingly relevant. Coriolis forces provide a direct forcing on the wake, but also affect the ABL base flow, which indirectly influences wake evolution. These effects may constructively or destructively interfere because both the magnitude and sign of the direct and indirect Coriolis effects depend on the Rossby number, turbulence and buoyancy effects in the ABL. Using large eddy simulations, we investigate wake evolution over a wide range of Rossby numbers relevant to offshore wind turbines. Through an analysis of the streamwise and lateral momentum budgets, we show that Coriolis effects have a small impact on the wake recovery rate, but Coriolis effects induce significant wake deflections which can be parsed into two regimes. For high Rossby numbers (weak Coriolis forcing), wakes deflect clockwise in the northern hemisphere. By contrast, for low Rossby numbers (strong Coriolis forcing), wakes deflect anti-clockwise. Decreasing the Rossby number results in increasingly anti-clockwise wake deflections. The transition point between clockwise and anti-clockwise deflection depends on the direct Coriolis forcing, pressure gradients and turbulent fluxes in the wake. At a Rossby number of 125, Coriolis deflections are comparable to wake deflections induced by$${\sim} 20^{\circ }$$of yaw misalignment. 

To achieve decarbonization targets, wind turbines are growing in hub height and rotor diameter, and they are being deployed in new locations with diverse atmospheric conditions not previously seen, such as offshore. Physics-based analytical wake models commonly used for design and control of wind farms simplify atmospheric boundary layer (ABL) and wake physics to achieve computational efficiency. This is accomplished primarily through a simplified model form that neglects certain flow processes, such as atmospheric stability, and through the parametrization of ABL and wake turbulence through a wake spreading rate. In this study, we systematically analyze the physical mechanisms that govern momentum and turbulence within a wind turbine wake in the stratified ABL. We use large-eddy simulation and analysis of the streamwise momentum deficit and wake-added turbulence kinetic energy (TKE) budgets to study wind turbine wakes under neutral and stable conditions. To parse the turbulence in the wake from the turbulent, incident ABL flow, we decompose the flow into the base ABL flow and the deficit flow produced by the presence of a turbine. We then analyze the decomposed flow field budgets to study the effects of changing stability on the streamwise momentum deficit and wake-added TKE. The results demonstrate that stability changes the relative balance of turbulence and advection for both the streamwise momentum deficit and wake-added TKE primarily through the nonlinear interactions of the base flow with the deficit flow. The stable cases are most affected by increased shear and veer in the base flow and the neutral case is most affected by the increased ambient turbulence intensity. These differences in the base flow that arise from stratification are relatively more important than the buoyancy forcing terms in the wake-added TKE budget. The wake-added TKE depends on the ABL stability. An existing wake-added TKE model that neglects the effects of ABL stability yields $15–25\%$error compared to large-eddy simulation, with errors that are higher in stable conditions than neutral. These results motivate future research to develop fast-running models of wake-added TKE that account for stability effects. Published by the American Physical Society2024 

Combined wake steering and induction control is a promising strategy for increasing collective wind farm power production over standard turbine control. However, computationally efficient models for predicting optimal control set points still need to be tested against high-fidelity simulations, particularly in regimes of high rotor thrust. In this study, large eddy simulations (LES) are used to investigate a two-turbine array using actuator disk modeling in conventionally neutral atmospheric conditions. The thrust coefficient and yaw-misalignment angle are independently prescribed to the upwind turbine in each simulation while downwind turbine operation is fixed. Analyzing the LES velocity fields shows that near-wake length decreases and wake recovery rate increases with increasing thrust. We model the wake behavior with a physics-based near-wake and induction model coupled with a Gaussian far-wake model. The near-wake model predicts the turbine thrust and power depending on the wake steering and induction control set point. The initial wake velocities predicted by the near-wake model are validated against LES data, and a calibrated far-wake model is used to estimate the power maximizing control set point and power gain. Both model-predicted and LES optimal set points exhibit increases in yaw angle and thrust coefficient for the leading turbine relative to standard control. The model-optimal set point predicts a power gain of 18.1% while the LES optimal set point results in a power gain of 20.7%. In contrast, using a tuned cosine model for the power-yaw relationship results in a set point with a lower magnitude of yaw, a thrust coefficient lower than in standard control, and predicts a power gain of 13.7%. Using the physics-based, model-predicted set points in LES results in a power within 1.5% of optimal, showing potential for joint yaw-induction control as a method for predictably increasing wind farm power output. 

Wind speed and direction variations across the rotor affect power production. As utility‐scale turbines extend higher into the atmospheric boundary layer (ABL) with larger rotor diameters and hub heights, they increasingly encounter more complex wind speed and direction variations. We assess three models for power production that account for wind speed and direction shear. Two are based on actuator disc representations, and the third is a blade element representation. We also evaluate the predictions from a standard power curve model that has no knowledge of wind shear. The predictions from each model, driven by wind profile measurements from a profiling LiDAR, are compared to concurrent power measurements from an adjacent utility‐scale wind turbine. In the field measurements of the utility‐scale turbine, discrete combinations of speed and direction shear induce changes in power production of −19% to +34% relative to the turbine power curve for a given hub height wind speed. Positive speed shear generally corresponds to over‐performance and increasing magnitudes of direction shear to greater under‐performance, relative to the power curve. Overall, the blade element model produces both higher correlation and lower error relative to the other models, but its quantitative accuracy depends on induction and controller sub‐models. To further assess the influence of complex, non‐monotonic wind profiles, we also drive the models with best‐fit power law wind speed profiles and linear wind direction profiles. These idealized inputs produce qualitative and quantitative differences in power predictions from each model, demonstrating that time‐varying, non‐monotonic wind shear affects wind power production. 

Wind turbine control via concurrent yaw misalignment and axial induction control has demonstrated potential for improving wind farm power output and mitigating structural loads. However, the complex aerodynamic interplay between these two effects requires deeper investigation. This study presents a modified blade element momentum (BEM) model that matches rotor-averaged quantities to an actuator disk model of yawed rotor induction, enabling analysis of joint yaw-induction control using realistic turbine control inputs. The BEM approach reveals that common torque control strategies such as K−Ω^2 exhibit sub-optimal performance under yawed conditions. Notably, the power-yaw and thrust-yaw sensitivities vary significantly depending on the chosen control strategy, contrary to common modeling assumptions. In the context of wind farm control, employing induction control which minimizes the thrust coefficient proves most effective at reducing wake strength for a given power output across all yaw angles. Results indicate that while yaw control deflects wakes effectively, induction control more directly influences wake velocity magnitude, underscoring their complementary effects. This study advances a fundamental understanding of turbine aerodynamic responses in yawed operation and sets the stage for modeling joint yaw and induction control in wind farms. 

Wind farm design generally relies on the use of historical data and analytical wake models to predict farm quantities, such as annual energy production (AEP). Uncertainty in input wind data that drive these predictions can translate to significant uncertainty in output quantities. We examine two sources of uncertainty stemming from the level of description of the relevant meteorological variables and the source of the data. The former comes from a standard practice of simplifying the representation of the wind conditions in wake models, such as AEP estimates based on averaged turbulence intensity (TI), as opposed to instantaneous. Uncertainty from the data source arises from practical considerations related to the high cost of in situ measurements, especially for offshore wind farms. Instead, numerical weather prediction (NWP) modeling can be used to characterize the more exact location of the proposed site, with the trade-off of an imperfect model form. In the present work, both sources of input uncertainty are analyzed through a study of the site of the future Vineyard Wind 1 offshore wind farm. This site is analyzed using wind data from LiDAR measurements located 25 km from the farm and NWP data located within the farm. Error and uncertainty from the TI and data sources are quantified through forward analysis using an analytical wake model. We find that the impact of TI error on AEP predictions is negligible, while data source uncertainty results in 0.4%–3.7% uncertainty over feasible candidate hub heights for offshore wind farms, which can exceed interannual variability. 

Collective wind farm flow control, where wind turbines are operated in an individually suboptimal strategy to benefit the aggregate farm, has demonstrated potential to reduce wake interactions and increase farm energy production. However, existing wake models used for flow control often estimate the thrust and power of yaw-misaligned turbines using simplified empirical expressions that require expensive calibration data and do not extrapolate accurately between turbine models. The thrust, wake velocity deficit, wake deflection and power of a yawed wind turbine depend on its induced velocity. Here, we extend classical one-dimensional momentum theory to model the induction of a yaw-misaligned actuator disk. Analytical expressions for the induction, thrust, initial wake velocities and power are developed as a function of the yaw angle ( $$\gamma$$ ) and thrust coefficient. The analytical model is validated against large eddy simulations of a yawed actuator disk. Because the induction depends on the yaw and thrust coefficient, the power generated by a yawed actuator disk will always be greater than a $$\cos ^3(\gamma )$$ model suggests. The power lost due to yaw misalignment depends on the thrust coefficient. An analytical expression for the thrust coefficient that maximizes power, depending on the yaw, is developed and validated. Finally, using the developed induction model as an initial condition for a turbulent far-wake model, we demonstrate how combining wake steering and thrust (induction) control can increase array power, compared to either independent steering or induction control, due to the joint dependence of the induction on the thrust coefficient and yaw angle.