Modeling and Simulation of Interacting Wings: Collective Dynamics in Inertial Fluid Flows

This project aims to understand how collections of flapping wings propel and interact through a fluid medium, such as water or air. Many living and non-living objects propel themselves by moving a set of propulsors (wings, fins, or a tail) that are attached to a main body. While there has been substantial progress in understanding how microscopic objects (such as bacteria) interact through slow fluid flows, little is known about how macroscopic objects (like fish) are affected by the relatively fast flows they generate. The new mathematical models constructed in this project will be used to assess how fluid flows affect the speed, energy consumption and stability of a collection of flapping wings traveling in different formations. The insights gained from this research have the potential to impact the biology and engineering communities. Specifically, recent experiments have suggested that fish and birds may travel in orderly formations in order to save energy, but the role of fluid flows in mediating schooling and flocking remains poorly understood. On the engineering side, the design of autonomous underwater vehicles is often inspired by the flapping mechanics of fish, and this project could inform new design and control principles. The project will train undergraduate and graduate students in mathematical modeling and physical applied mathematics, giving them tools that can be broadly applied to problems arising in the natural sciences and engineering. Moreover, the beautiful displays exhibited by animal collectives can readily be appreciated by the public, making the project suitable for outreach initiatives conducted by the university.

Mathematical models of varying degrees of fidelity and complexity will be constructed and analyzed to probe different aspects of the wings' behavior. A key challenge is that flapping wings generate long-lived fluid flows in the form of complex vortical structures. This leads to temporally nonlocal hydrodynamic interactions between the constituents, wherein the forces between wings at a given time depend on their past positions and velocities. A nonlinear discrete-time delay-difference map, which explicitly models the wings' shed vortices, has been validated against recent experiments and will be used to assess the dependence of a formation's speed, energy consumption and stability on geometric parameters. Conformal mapping and integral equation techniques will be used to represent two- and three-dimensional flows, respectively, and to assess the influence of wing shape. Furthermore, a continuum PDE theory for a dense collective of wings in a high Reynolds number flow will be systematically derived from the aforementioned models. Traveling wave solutions of the continuum theory will be characterized through numerical simulations, and the emergence of positional and orientational order will be assessed. The goals are to determine the extent to which hydrodynamic interactions alone can account for the spatiotemporally complex formations observed in dense collectives of flapping objects, and to uncover novel emergent phenomena that have been conjectured to arise in systems influenced by inertial fluid flows.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.