To meet the 2050 net-zero carbon emissions target in the United States, up to a 30-fold increase in wind capacity is required. Similarly, global net-zero targets are estimated to require an 11-fold increase in wind power. Despite substantial growth in wind energy technology in recent decades, aerodynamic modeling of wind turbines relies on models derived in the late 19th century, which are well-known to break down under flow regimes in which wind turbines often operate. To overcome issues associated with this breakdown, all current predictions of wind turbine forces, power, and wakes that are used to drive contemporary design and control protocols are based on empirically-fit formulas. Eliminating this reliance on empirical modeling unlocks opportunities to develop breakthrough innovations for wind power that were previously only accessible through trial-and-error. The new theory, model, and validation in this study enables predictions of turbine power, loads, and wakes across operating regimes without empirical corrections for the first time, enabling improved turbine and farm design and control for the next generation of on and offshore farms without empiricism.
Schematic illustrating the rotor thrust coefficient variations with rotor-normal induction across various operational scenarios (propeller state, windmill state, and turbulent wake state) for a yaw-aligned actuator disk. Model predictions are shown using classical one-dimensional momentum modeling, Glauert’s empirical relation, and the Unified Momentum Model introduced in this study.
A new, Unified Momentum Theory
This study returns to first-principles of rotor aerodynamics to derive a new, analytical, Unified Momentum model to predict power production, thrust force, and wake dynamics of rotors under arbitrary inflow angles and thrust coefficients without empirical corrections for the first time. The actuator disk representation of a rotor, and the associated classical one-dimensional momentum theory, is the starting point for any textbook on aerodynamics with applications to wind turbines, propellers, helicopters, drones, hydrokinetic turbines, and more. This classical theory results in the Betz limit model which predicts that the maximum power extraction for a wind or hydrokinetic turbine is 59.3% of incident flow power. However, this classical theory results in a well-known breakdown outside of a very narrow range of rotor operation, or if there is any misalignment between the inflow and the rotor, which is ubiquitous in practice due to the constant variations in wind direction in the turbulent atmosphere. This gap in theoretical modeling has prompted the development of numerous empirical corrections, which have found widespread application in textbooks, research articles, and open-source and industry design codes.
This study establishes a new starting point for understanding and modeling rotor aerodynamics by developing an analytical aerodynamic model that is validated against high-fidelity large eddy simulation computational fluid dynamics. The model unifies rotor aerodynamics by removing assumptions made in the classical theory, introducing a novel model for the pressure deficit in the wake under arbitrary thrust coefficients, and modeling the transverse velocities when the rotor is misaligned. These are critical quantities for accurate predictions of rotor aerodynamics. The model predicts rotor characteristics, such as thrust and power. It also predicts wake characteristics, the velocity deficit associated with turbine power extraction, that are critical for modeling, designing, and controlling arrays of rotors, such as those within wind farm configurations (Howland et al. Nature Energy (2022)). The validated model can replace classical one-dimensional momentum theory in open-source wind energy design and control tools.
The model also results in a new first-principles prediction for the maximum efficiency of a wind turbine, replacing the widely used Betz limit (Betz (1920)). For turbines that are aligned with the incident wind flow, the new maximum power extraction is on the order of one percent higher than the classical Betz limit, and it is achieved at an operational setpoint that is higher than the setting to operate at the classical Betz limit. This new theoretical limit stems from the ability to extract energy from the static pressure, in addition to the incoming flow velocity. Going further, the model also generalizes the prediction of the maximum power of a turbine to settings where the turbine is misaligned with the inflow, yielding a new theoretical maximum in power extraction for this misaligned state that the classical momentum theory cannot predict.
Next generation design and control for wind turbines and farms
Momentum theory provides the basis for aerodynamic modeling of wind power. These predictions guide wind turbine design and control in the form of `blade element momentum’ modeling, which combines momentum theory predictions for the wind velocity at the rotor with blade element modeling predictions of the lift and drag forces on the wind turbine blades. Momentum theory is also the basis for wind farm flow models (wake models) used to design and operate wind farms. Wind turbines extract kinetic energy from the incident flow, reducing the available power for downwind turbines that operate in the wake region of an upwind turbine. Recent research has uncovered that operating wind turbines collectively within a wind farm by changing the thrust force or the alignment of certain turbines with the incident wind can increase wind farm energy production. Yet all of these applications, from wind turbine design to wind farm flow control, rely on momentum theory predictions that break down in high thrust or misaligned operational states. Therefore, all of these applications have relied on empiricism to guide decision-making.
The Unified Momentum model provides a new basis for wind turbine and wind farm modeling that can guide design and control for wind turbines and farms. The Unified model provides insights out-of-the-box, identifying that turbines that are misaligned with the incident inflow should increase their thrust coefficient to maximize power production. This has a direct impact on wind farm flow control methods, which have historically controlled the yaw misalignment or the thrust coefficient separately. The Unified model enables physics-based predictions across yaw and thrust regimes, and the results suggest that yaw and thrust control should be jointly optimized.
Opportunities for similar insights based on the Unified Momentum model exist in wind turbine and wind farm design. This Unified Momentum model provides a new basis to address the analysis and modeling of unsteady aerodynamics associated with dynamic control or floating motion for an offshore turbine, along with wind speed and direction shear associated with wind flow in the turbulent atmosphere.  Â
More details of this work can be found in our article published in Nature Communications
References:
1. Howland, Michael F., Jesús Bas Quesada, Juan José Pena MartÃnez, Felipe Palou Larrañaga, Neeraj Yadav, Jasvipul S. Chawla, Varun Sivaram, and John O. Dabiri. “Collective wind farm operation based on a predictive model increases utility-scale energy production.” Nature Energy 7, no. 9 (2022): 818-827. https://doi.org/10.1038/s41560-022-01085-8
2. Betz A. Das maximum der theoretisch möglichen ausnützung des windes durch windmotoren. Sci. Res. 26, 307–309 (1920).