Indian Origin Student Divya Tyagi Solves Century-Old Math Problem, Paves Way for Advancements in Wind Turbine Design

Reported By Amit Singh
Pennsylvania, USA, 19th March 2025: Indian origin student Divya Tyagi, a graduate student in aerospace engineering at Penn State University, has achieved a remarkable breakthrough by refining a century-old math problem, opening new avenues in wind turbine design and aerodynamics research. Tyagi’s work revisits and simplifies a problem first posed by British aerodynamicist Hermann Glauert, enhancing its applicability in modern engineering.

Tyagi’s research, which began as an undergraduate project for her Schreyer Honors College thesis, introduces an addendum to Glauert’s original work. This addendum determines the optimal aerodynamic performance of a wind turbine by solving for ideal flow conditions that maximize power output. Her findings were published in Wind Energy Science and have since garnered attention in the academic and engineering communities.

A New Perspective on a Historic Problem

Glauert’s original problem, formulated in the early 20th century, focused exclusively on the maximum attainable power coefficient — a measure of how efficiently a wind turbine converts wind energy into electricity. However, according to Tyagi’s adviser, Sven Schmitz, the Boeing/A.D. Welliver Professor in the Department of Aerospace Engineering at Penn State, Glauert’s model did not account for crucial factors like the total force and moment coefficients acting on the rotor — the spinning unit with attached blades — or how turbine blades bend under wind pressure.

“If you have your arms spread out and someone presses on your palm, you have to resist that movement,” Schmitz explained. “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.”

Tyagi’s work fills this gap by using the calculus of variations, a mathematical method used for constrained optimization problems, to create a more comprehensive model. Her approach simplifies the problem while offering new insights into turbine performance under various conditions.

Real-World Impact on Wind Energy

The implications of Tyagi’s work extend far beyond the classroom. By improving the understanding of aerodynamic performance, her research can directly contribute to the development of more efficient wind turbines. Even a 1% improvement in a turbine’s power coefficient could significantly increase energy output, potentially powering entire neighborhoods and driving down energy costs.

“Improving the power coefficient of a large wind turbine by just 1% has significant impacts on the energy production of a turbine,” Tyagi said. “That improvement also translates to the other coefficients we derived relations for.”

Schmitz emphasized the long-term impact of Tyagi’s contribution, noting that her elegant solution could influence the next generation of wind turbine designs and become a staple in aerodynamics coursework worldwide.

“The real impact will be on the next generation of wind turbines using the new knowledge that has been unveiled,” Schmitz said. “As for Divya’s elegant solution, I think it will find its way into the classrooms, across the country and around the world.”

Recognition and Academic Journey

In recognition of her groundbreaking research, Tyagi received the Anthony E. Wolk Award, presented to the senior aerospace engineering student who develops the best thesis in the field.

Tyagi completed her bachelor’s degree in aerospace engineering with a perfect 4.0 GPA and is currently pursuing her master’s degree at Penn State, focusing on computational fluid dynamics simulations. Her current research analyzes airflow around a helicopter rotor in complex environments, such as when landing on a ship deck. This U.S. Navy-supported project aims to improve flight simulation accuracy and enhance pilot safety by better understanding the dynamic interactions between airflows around ships and helicopters.

“The goal is to integrate that with the complex flow around a ship to see how the ship airwake interacts with a helicopter trying to land on its deck,” Tyagi said.

A Journey of Perseverance

Reflecting on the process, Tyagi described her undergraduate research as challenging but rewarding. She spent 10 to 15 hours per week working on the problem, writing her thesis, and conducting research. The math-intensive nature of the problem demanded a high level of perseverance, but her dedication ultimately led to a significant breakthrough.

Schmitz, who had contemplated Glauert’s problem for decades, praised Tyagi’s tenacity.

“When I thought about the Glauert problem, I thought steps were missing and it was very complicated,” Schmitz said. “There had to be an easier way to do it. That’s when Divya came in. She was the fourth student I challenged with looking at it, and she was the only one who took it on. Her work is truly impressive.”

Tyagi’s achievement not only honors the legacy of Hermann Glauert but also propels wind energy research forward, potentially contributing to a greener, more energy-efficient future.