Penn State engineering graduate student Divya Tyagi (right) presents her work on a century-old math problem to her adviser, Sven Schmitz, a faculty member in the College of Engineering. Credit: Penn State / Kevin Sliman.
Penn State engineering graduate student Divya Tyagi (right) shows her work on a century-old math problem to her adviser, Sven Schmitz, a faculty member in the College of Engineering. [Credit: Penn State / Kevin Sliman.]
In 1926, British aerodynamicist Hermann Glauert introduced an equation that shaped a century of wind turbine development — a third-order polynomial that determines the optimum axial induction factor. Yet, at Penn State nearly one hundred years later, Divya Tyagi, an engineering student revisited and improved this classical result by deriving missing analytical approaches for rotor thrust and blade loading.
The results were published in Wind Energy Science.
Tyagi’s work fundamentally expands Glauert’s classical wind turbine optimization theory in a few ways. First, she developed an elegant calculus of variations methodology to the power coefficient optimization problem, recovering Glauert’s results through a more elegant mathematical approach. Second, she appears to have derived the first exact analytical integrals for thrust and bending moment coefficients, thus filling a gap in rotor disk theory. Third, her mathematical analysis revealed asymptotic behaviors: both power and bending moment coefficients converge to the theoretical Betz limit as tip speed ratio approaches infinity, while thrust and bending moment coefficients maintain non-zero values at zero tip speed ratio.
Mathematical refinements to Glauert’s rotor disk model
In simple terms, Glauert determined how an idealized wind turbine (with an infinite number of blades) could extract the maximum power from the wind for a given tip-speed ratio (blade speed to wind speed ratio). His approach gave the maximum power coefficient (fraction of wind energy captured) as a function of tip-speed ratio, and the corresponding “optimum” distributions of axial and angular induction factors along the disk. Yet Glauert’s derivation relied on simplifying assumptions. That is, assuming there would be a constant pressure jump across the rotor and that all kinetic energy for wake rotation comes from the freestream. Glauert also focused exclusively on power, not on the total load on the rotor. Glauert did not explicitly derive the overall thrust force or bending moment on the rotor blades in his optimum solution, leaving those aspects unaddressed.
If you have your arms spread out and someone presses on your palm, you have to resist that movement. We call that the downwind thrust force and the root bending moment, and wind turbines must withstand that, too. You need to understand how large the total load is, which Glauert did not do.
–Schmitz
Under the guidance of Penn State’s Sven Schmitz, Tyagi applied the calculus of variations to re-derive the optimal induction distributions. This mathematically elegant approach matches Glauert’s original optimality conditions but does so with fewer assumptions and more transparency. Tyagi’s solution recovers Glauert’s optimum for axial and angular induction and opens the door to new derivations.
Impact on wind turbine design and performance
Tyagi’s refinements promise practical benefits for wind turbine design. They could lead to a more complete expression of the power coefficient that helps designers pinpoint tip-speed ratios and improve efficiency, even if the gains are incremental. Her calculus-of-variations approach confirms the ideal loading distribution, providing analytical targets for blade twist, chord, and angle of attack. Tyagi’s model also lets engineers estimate overall thrust and bending moments, aiding blade and hub design.
The real impact will be on the next generation of wind turbines using the new knowledge that has been unveiled.
–Schmitz
Tyagi’s work began as her Schreyer Honors College undergraduate thesis, for which she was awarded the Anthony E. Wolk Award, presented to the aerospace engineering senior with the best thesis. “I would spend about 10 to 15 hours a week between the problem, writing the thesis and on research. It took a long time because it was so math intensive,” Tyagi was quoted as saying in a press release. “But I feel really proud now, seeing all the work I’ve done.” Professor Schmitz’s noted that she was the fourth student he had challenged with looking at the Glauert problem, but “the only one who took it on.”
Now pursuing her master’s degree in aerospace engineering, Tyagi has shifted her focus to computational fluid dynamics simulations that analyze airflow around helicopter rotors. Her current U.S. Navy-supported research examines how ship airwake interacts with helicopters attempting to land on deck, with the goal of improving flight simulation and pilot safety. Tyagi sees her mathematical contributions as steps toward enhancing wind energy production while reducing costs: “Improving the power coefficient of a large wind turbine by just 1% has significant impacts on the energy production of a turbine… potentially powering an entire neighborhood.”