Elucidating Fluid-Structure Interactions: Studying the Propulsion of Flexible Foils


Natural flyers and swimmers can provide essential new knowledge for the design of more efficient propulsion systems for both aerial and aquatic vehicles. A key feature of many natural flyers, such as birds and insects, and swimmers, such as fish, is the use of flexible wings or fins to navigate their environment and the deformation of these flexible structures in response to fluid dynamics plays a critical role in enhancing propulsive efficiency and maneuverability. However, trying to understand this complex interplay between the flexible structures and the surrounding fluid environment can be challenging because the interaction involves complex dynamics where the properties of the fluid and the structural characteristics of the wing or fin are deeply intertwined. Moreover, this dynamic coupling can significantly affect propulsion performance, and several factors such as thrust generation, energy efficiency, and overall agility play a major role. Furthermore, the non-steady nature of the fluid environment and the non-linear responses of flexible structures under fluid forces add another layer of challenge. To address these challenges, theoretical modeling of such systems is often complicated and requires accurate simulation of the fluid-structure interactions and simplified models which can be beneficial because it is less complex, often need to approximate the behavior of these systems without losing the essence of the underlying physics.

To this end, new study published in Journal of Fluid Mechanics and led by Assistant Professor Feng Du and Professor Jianghao Wu from the Beihang University proposed a simplified analytical model based on elastokinetics and linear potential flow theory. Their study clarified the kinematics and propulsion performance of a flexible thin foil which serves as an idealized representation of a wing or fin and pitches in flow. The new model considers the dynamic forces involved, including the inertial forces of the foil and the non-steady fluid pressures, to determine the average deformation angle of the foil. Professors Feng Du and Jianghao Wu tested the predictions of their analytical model concerning the propulsion performance of a flexible thin foil, an idealized proxy for natural flexible fins and wings, when exposed to fluid dynamics. They investigated several prototypes of thin foils with differing levels of flexibility, which allowed them to simulate varying degrees of natural fin and wing flexibility. These foils were excited by pitching motion at the leading edge and then subjected to controlled flow conditions. With the adjustment in the speed of the flow and the frequency of the foil’s pitching motion, their modelling replicated the non-steady fluid environments these natural systems would encounter. The authors observed that foils with optimized flexibility and properly tuned pitching frequencies demonstrated enhanced propulsive efficiency and thrust generation compared to their rigid counterparts. Specifically, the analytical formulations showed that the resonance conditions where the natural frequency of the foil’s vibration matched the frequency of the pitching motion significantly amplified the propulsion performance. This result highlights the importance of the flow-structure interaction, where the right combination of material properties and motion frequency could exploit natural fluid dynamics to maximize propulsion efficiency. Moreover, the researchers showed that the phase angle between the deformation of the foil and its pitching motion critically influenced the propulsion efficiency. Optimal phase alignment, where the deformation of the foil was synchronized with the fluid forces acting upon it, led to reduced energy expenditure for a given amount of thrust. This optimal synchronization resulted in a reduction of the drag forces acting against the motion of the foil, thereby improving the overall efficiency.

In conclusion, the study findings of Professors Feng Du and Jianghao Wu has far-reaching significance and practical implications specially in bio-inspired engineering, robotics, and fluid dynamics. The development of a simplified analytical model that accurately predicts the propulsion performance of flexible thin foils in fluid environments, bridges a crucial gap in our understanding of fluid-structure interactions and can provide valuable knowledge for optimal design parameters for flexible propelling systems. This knowledge can be directly applied to the development of more efficient and adaptable underwater and aerial vehicles. These vehicles can mimic the natural efficiency of fish fins and bird wings, leading to improvements in energy efficiency, maneuverability, and speed. Moreover, the study can inform the design of bio-inspired robots used in various applications, including environmental monitoring, search and rescue operations, and autonomous exploration. Robots equipped with flexible propulsion systems can navigate complex fluid environments more effectively, performing tasks with greater efficiency and less energy consumption. The new model can be applied to improve the performance of marine and aeronautical vehicles and ships and submarines could benefit from flexible hull designs that reduce drag and enhance propulsion efficiency and also aircraft could incorporate flexible wing elements to improve aerodynamic performance. Indeed, the simplified analytical model can be considered a new useful tool for understanding the complex interactions between flexible structures and fluid flows.

Elucidating Fluid-Structure Interactions: Studying the Propulsion of Flexible Foils - Advances in Engineering Elucidating Fluid-Structure Interactions: Studying the Propulsion of Flexible Foils - Advances in Engineering

About the author

Feng Du is an Assistant Professor in the School of Transportation Science and Engineering, Beihang University. He completed his PhD at Peking University in 2015 studying Solid mechanics. He worked as a postdoctoral fellow and research assistant professor at the Department of Mechanical and Industrial Engineering, Northeastern University, from 2016 to 2018. In 2019, he joined Beihang University as an Assistant Professor. His research interests include Fluid-structure interaction of flapping wing, Surface and Interface mechanics of soft materials, Ice mechanics and friction.

About the author

Professor Wu Jianghao serves as the head of the insect flight and micro aerial vehicles lab, as well as the director of the Bionic Unmanned System Program at the Interdisciplinary Center of the Ministry of Education, Beihang University, and the director of the Autonomous Unmanned System Program under the Industry-University-Research Integration Platform of Artificial Intelligence developed by the National Development and Reform Commission. His research focuses on the mechanisms of insect flight and its application in the design of bionic micro aerial vehicles. He has led over 30 scientific research projects, including national key research and development plans and projects funded by the National Natural Science Foundation of China. He has published over 100 papers in journals such as Progress in Aerospace Sciences, Journal of Fluid Mechanics, and AIAA Journal, and has been granted over 30 national invention patents. Professor Wu has received the Second Prize of the National Natural Science Award and the First Prize of Natural Science Award from the Chinese Society of Aeronautics and Astronautics. He has also been included in the 2023 Stanford’s List of World’s Top 2% Scientists.

About the author

Jiefei Li works as Head of Comprehensive Department of Earthquake Forecasting and Network Technology Support. He received the B.S. degree in applied geophysics and engineering seismology from Institute of Disaster Prevention in 1997 and the M.S. degree in land resource management from Hunan Normal University in 2008. His work involves earthquake monitoring, forecasting and warning, earthquake information networks and security, and construction of basic platforms.

About the author

Yongxian Zhang works as Professor and Head in the Research Division of Short-term Earthquake Forecasting, Institute of Earthquake Forecasting, China Earthquake Administration, Deputy chairman of the APEC Cooperation for Earthquake Science, and Executive Director Boarder member of China Seismological Society. She visited UC Davis of USA as a senior visiting scholar in 2012. Her study interests are modeling of seismogenic processes, seismicity patterns, earthquake anomaly mechanisms, synthetic methods of earthquake forecasting, and predictability testing. She has published more than 100 papers and 8 works. She won several national awards in China such as National Excellent Scientist from China Association for Science and Technology, Special allowance of the State Council of China, etc.

About the author

Xuhui Shen serves as Professor in the State Key Lab of Solar Activity and Space Weather, National Space Science Center, Chinese Academy of Sciences since 2022. He worked separately in the Institute of earthquake Forecasting and the Institute of Crustal Dynamics of China Earthquake Administration and took the duties of the Head Scientist and Deputy-Designer-in-general of the CSES mission in the last 20 years. He received his Master degree separately in 1990 and PhD in 1996 in the Institute of Geology, China Earthquake Administration. His main research area includes space geophysics, remote sensing application in natural hazard mitigation and lithosphere-atmosphere-ionosphere coupling.


Du F, Wu J. Analytical results for pitching kinematics and propulsion performance of flexible foil. Journal of Fluid Mechanics. 2024;979:A5. doi:10.1017/jfm.2023.1028

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