A newly developed model successfully simulates airflow around rotors, even under severe conditions. This groundbreaking model of rotor aerodynamics has the potential to enhance the design of turbine blades and wind farms, as well as improve the operation of wind turbines.
The design of propeller and wind turbine blades is based on aerodynamic principles that were first mathematically articulated over a century ago. However, engineers have long noted that these formulas are not universally applicable. To address this issue, they often resort to ad hoc “correction factors” derived from real-world observations.
For the first time, researchers at MIT have created a complete, physics-based model that accurately reflects airflow around rotors even under extreme situations, such as high-speed operations or when blades are set at particular angles. This model could revolutionize rotor design and the configuration and management of wind farms. These findings are detailed in the journal Nature Communications, in an open-access paper authored by MIT postdoctoral researcher Jaime Liew, Ph.D. student Kirby Heck, and Michael Howland, the Esther and Harold E. Edgerton Assistant Professor of Civil and Environmental Engineering.
“We have developed a new theory for rotor aerodynamics,” Howland explains. This theory can be employed to calculate the forces, flow velocities, and power generated by a rotor, whether it’s harnessing energy from the air, like a wind turbine, or contributing energy to the flow, as seen in ship or airplane propellers. “This theory is applicable in both scenarios,” he adds.
Given its foundational mathematical nature, some aspects of this new understanding can be utilized immediately. For instance, wind farm operators need to continually adjust various factors, including each turbine’s orientation, rotation speed, and blade angle, to optimize power output while ensuring safety. The new model offers a straightforward and efficient approach for real-time adjustments.
“What excites us is its immediate and significant potential for impact throughout the wind power value chain,” Howland says.
Momentum Modeling
Previously, the theory governing rotor interaction with fluid environments—air, water, and more—was established in the late 19th century, known as momentum theory. This enables engineers to start with a specific rotor design and configuration to determine the maximum power that can be generated by that rotor — or, in the case of a propeller, the power required to produce a particular thrust.
Equations found in momentum theory are typically the first subject covered in wind energy textbooks and the initial topic discussed in Howland’s wind power classes. This theory has roots in the work of physicist Albert Betz, who, in 1920, calculated the maximum energy that could be theoretically extracted from wind, known as the Betz limit, which is 59.3 percent of the incoming wind’s kinetic energy.
However, researchers soon discovered that momentum theory tends to fail dramatically under high forces associated with increased blade rotation speeds or varied blade angles, as Howland notes. The theory inaccurately predicts not only the magnitude but also the direction of thrust force changes at higher rotation speeds or different angles; the theory indicates that thrust should decrease after a certain point, while experiments have shown it continues to rise. “Thus, it’s not just quantitatively flawed, but qualitatively incorrect,” Howland clarifies.
This theory also struggles when there’s any misalignment between a rotor and airflow, which Howland describes as “common” in wind farms, where turbines constantly adjust to shifting wind directions. In a previous study conducted in 2022, Howland and his team discovered that slightly misaligning certain turbines against the incoming airflow can greatly enhance the overall power production of the wind farm by minimizing wake disturbances affecting downstream turbines.
Historically, when engineering blade designs, arranging wind turbine layouts, or managing day-to-day turbine operations, engineers have depended on improvised adjustments to the original formulas, relying on wind tunnel tests and practical experience without theoretical backing.
To develop this new model, the team closely examined the interaction between airflow and turbines through detailed computational simulations of aerodynamics. They found that previous assumptions—such as a rapid return to normal pressure just downstream of the rotor—were increasingly imprecise as thrust forces grew. Howland notes, “This assumption becomes less valid as thrust continues to increase.”
Remarkably, this inaccuracy arises very near the Betz limit that predicts optimum turbine performance, effectively falling within the ideal operational range for turbines. “We have Betz’s prediction of optimal turbine operation, and within just 10 percent of this set point that supposedly maximizes power, the original theory fails dramatically,” Howland states.
Through their simulations, the researchers also discovered a method to correct the original model’s traditional reliance on one-dimensional modeling that presumes perfect rotor alignment with airflow. They achieved this by applying fundamental equations initially designed for predicting lift on three-dimensional wings in aerospace contexts.
They formulated their new model—termed the unified momentum model—through theoretical analysis and subsequently validated it using computational fluid dynamics. They plan further validation through wind tunnel and field tests in unpublished follow-up work.
Breakthrough Understanding
A notable result from this new formula is its adjustment of the Betz limit calculation, indicating that slightly more power can be extracted than the previous formula suggested. While this change is minimal—amounting to a few percent—it’s significant that the new theory modifies the long-standing Betz limit, which has been a guiding principle for a century. “Now that we have a new theory, it allows us to revise the Betz limit, making it immediately applicable,” Howland emphasizes. The new model also demonstrates methods to enhance power output from turbines misaligned with airflow, an aspect the Betz limit does not address.
The implications for controlling both individual turbines and turbine arrays can be applied without needing alterations to the current hardware in place at wind farms. This has already been seen in previous work by Howland and his team two years ago that analyzed wake interactions between turbines, relying on existing empirical formulas.
“This advancement naturally extends our prior research on optimizing utility-scale wind farms,” he remarks, acknowledging how their prior analyses highlighted the weaknesses of existing methods for assessing forces and forecasting power from wind turbines. “Empirical modeling alone wasn’t sufficient,” he concludes.
In a wind farm, each turbine extracts energy from nearby turbines due to wake effects. Therefore, precise wake modeling is crucial for both the design and daily functioning of a wind farm, enabling operators to determine the optimal angles and speeds for each turbine in real time.
Previously, Howland notes, operators, manufacturers, and turbine blade designers lacked a theoretical foundation to predict how much a turbine’s output would change with specific alterations, such as its wind angle, without resorting to empirical adjustments. “This gap existed because the theory was missing. Our work provides a direct theory that articulates how to operate a wind turbine for maximum power output without empirical corrections,” he adds.
Because of the similar fluid dynamics, this model is also relevant to propellers in aircraft and marine vessels, as well as hydrokinetic turbines like those for tidal or river currents. This aspect wasn’t the focus of the current research, but is inherently included in the theoretical framework, Howland notes.
The new theory is available as a set of mathematical equations that users can integrate into their own software, or as an open-source software package that can be freely accessed on GitHub. “This engineering model is designed for speedy tools to support rapid prototyping, control, and optimization,” Howland explains. “Our modeling aims to shift the wind energy research field towards aggressive development of wind capacity and reliability needed to combat climate change.”
This research received funding from the National Science Foundation and Siemens Gamesa Renewable Energy.
