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

该项目旨在了解拍打翅膀的集合如何在流体介质（如水或空气）中推进和相互作用。许多生物和非生物物体通过移动附着在主体上的一组推进器（翅膀、鳍或尾巴）来推进自己。虽然在理解微观物体（如细菌）如何通过缓慢的流体流动相互作用方面取得了实质性进展，但对于宏观物体（如鱼）如何受到它们产生的相对较快的流体的影响，我们知之甚少。在这个项目中建立的新的数学模型将用于评估流体流动如何影响在不同地层中飞行的一组扑翼的速度、能量消耗和稳定性。从这项研究中获得的见解有可能影响生物学和工程界。具体来说，最近的实验表明，鱼类和鸟类可能为了节省能量而以有序的队形行进，但流体流动在调节鱼群和群集中的作用仍然知之甚少。在工程方面，自主水下航行器的设计经常受到鱼类拍打机制的启发，该项目可以为新的设计和控制原理提供信息。该项目将对本科生和研究生进行数学建模和物理应用数学方面的培训，为他们提供可以广泛应用于自然科学和工程问题的工具。此外，动物集体展示的美丽展品很容易被公众欣赏，这使得该项目适合大学开展的推广活动。将构建和分析不同保真度和复杂程度的数学模型，以探索机翼行为的不同方面。一个关键的挑战是，扇动的翅膀会以复杂的旋涡结构的形式产生长期的流体流动。这就导致了组成部分之间的时间非局部流体动力相互作用，其中机翼之间的力在给定时间取决于它们过去的位置和速度。一个非线性离散时滞差分图，明确地模拟了机翼的脱落旋涡，已经在最近的实验中得到了验证，并将用于评估编队速度、能量消耗和稳定性对几何参数的依赖。保角映射和积分方程技术将分别用于表示二维和三维流动，并评估机翼形状的影响。此外，将从上述模型中系统地推导出高雷诺数流动中密集翼群的连续偏微分方程理论。通过数值模拟对连续介质理论的行波解进行表征，并对位置和方向顺序的出现进行评估。目标是确定流体动力相互作用在多大程度上可以单独解释在密集的扑动物体集体中观察到的时空复杂的形成，并揭示被推测在受惯性流体流动影响的系统中出现的新出现现象。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.