Engineers at MIT have developed a comprehensive, physics-based model that accurately represents the airflow around rotors, even under extreme conditions, such as when the blades are operating at high forces and speeds, or are angled in certain directions. The model could improve the way in which rotors are designed, but also the way wind farms are laid out and operated.

The blades of propellers and wind turbines are designed based on aerodynamics principles that were first described mathematically more than a century ago. But engineers have long realised that these formulas don’t work in every situation. To compensate, they have added ad hoc ‘correction factors’ based on empirical observations. The new research does away with these correction factors by comprehensively modelling airflow.

‘We’ve developed a new theory for the aerodynamics of rotors,’ Howland said. This theory can be used to determine the forces, flow velocities and power of a rotor, whether that rotor is extracting energy from the airflow, as in a wind turbine, or applying energy to the flow, as in a ship or airplane propeller. ‘The theory works in both directions.’

Because the new understanding is a fundamental mathematical model, some of its implications could potentially be applied right away. For example, operators of wind farms must constantly adjust a variety of parameters, including the orientation of each turbine as well as its rotation speed and the angle of its blades, in order to maximise power output while maintaining safety margins. The new model can provide a simple, speedy way of optimszing those factors in real time.

‘This is what we’re so excited about, is that it has immediate and direct potential for impact across the value chain of wind power,’ Howland said.

Known as momentum theory, the previous model of how rotors interact with their fluid environment – air, water, or otherwise – was initially developed late in the 19th century. With this theory, engineers can start with a given rotor design and configuration, and determine the maximum amount of power that can be derived from that rotor – or, conversely, if it’s a propeller, how much power is needed to generate a given amount of propulsive force.

Momentum theory equations ‘are the first thing you would read about in a wind energy textbook, and are the first thing that I talk about in my classes when I teach about wind power’, Howland said. From that theory, physicist Albert Betz calculated in 1920 the maximum amount of energy that could theoretically be extracted from wind. Known as the Betz limit, this amount is 59.3 per cent of the kinetic energy of the incoming wind.

But just a few years later, others found that the momentum theory broke down ‘in a pretty dramatic way’ at higher forces that correspond to faster blade rotation speeds or different blade angles, Howland said. It fails to predict not only the amount, but even the direction of changes in thrust force at higher rotation speeds or different blade angles. Where the theory said the force should start going down above a certain rotation speed or blade angle, experiments show the opposite – that the force continues to increase. ‘So, it’s not just quantitatively wrong, it’s qualitatively wrong,’ Howland said.

The theory also breaks down when there is any misalignment between the rotor and the airflow, which Howland said is ‘ubiquitous’ on wind farms, where turbines are constantly adjusting to changes in wind directions. In fact, in an earlier paper in 2022, Howland and his team found that deliberately misaligning some turbines slightly relative to the incoming airflow within a wind farm significantly improves the overall power output of the wind farm by reducing wake disturbances to the downstream turbines.

In the past, when designing the profile of rotor blades, the layout of wind turbines in a farm, or the day-to-day operation of wind turbines, engineers have relied on ad hoc adjustments added to the original mathematical formulas, based on some wind tunnel tests and experience with operating wind farms, but with no theoretical underpinnings.

To derive the new model, the team analysed the interaction of airflow and turbines using detailed computational modelling of the aerodynamics. They found that, for example, the original model had assumed that a drop in air pressure immediately behind the rotor would rapidly return to normal ambient pressure just a short way downstream. But it turns out, Howland said, that as the thrust force keeps increasing, ‘that assumption is increasingly inaccurate.’

And the inaccuracy occurs very close to the point of the Betz limit – and therefore is just at the desired operating regime for the turbines. ‘So, we have Betz’s prediction of where we should operate turbines, and within ten per cent of that operational set point that we think maximizes power, the theory completely deteriorates and doesn’t work,’ Howland said.

The researchers derived their new model, which they call a unified momentum model, based on theoretical analysis, and then validated it using computational fluid dynamics modelling. In follow-up work, they are doing further validation using wind tunnel and field tests.

Until now, Howland said, operators of wind farms, manufacturers and turbine-blade designers had no way to predict how much the power output of a turbine would be affected by a given change such as its angle to the wind without using empirical corrections. ‘That’s because there was no theory for it. So, that’s what we worked on here. Our theory can directly tell you, without any empirical corrections, for the first time, how you should actually operate a wind turbine to maximise its power,’ he said.

The research has been published in *Nature Communications*.