The investigator will study five fundamental and interrelated problems dealing with the motions of flapping appendages in inviscid fluids. The first project is a study of the large-amplitude flapping of a rigid plate which sheds vortex sheets into an oncoming fluid. This study will build on the method developed by the proposer for computing the coupled motions of vortex sheets and a flexible flag to consider leading-edge vortices and their interactions with a body and with trailing-edge vortices. The second problem is an improved understanding of how to approximate the spatial complexity of vortex sheets computationally, in settings of vortex-vortex and vortex-body interactions. These approximations will consist of truncations of the multipole expansion of the velocity field induced by vortex sheets. The third project is a study of the optimal flexibility for propulsion, in simple situations modelling two-dimensional motions of fish fins. The work ranges from analysis of linearized solutions to computational solutions of the full system of nonlinear singular integrodifferential equations. The fourth problem considers the mechanics of flexible fins in more detail, through a constrained optimization of a fin ray model. The constraints will be chosen with both the relevant scientific questions and the well-posedness of the optimization problem in mind. The fifth project considers how to control and stabilize the fin ray using our understanding of the mechanics underlying actual fish fins. This effort requires finding optimal target trajectories when the fin stiffness is varied periodically in time, and stabilizing the target trajectories against undesired excitations from body-vortex coupling.
The project will develop methods to answer questions of broad biological interest about swimming with fish fins and other flexible locomotory structures. The project will also improve the quantitative understanding of strategies for propulsion which can be readily assimilated into theoretical engineering fields concerning how solids and fluids interact. Applications to propulsion technology include improving the efficiency and maneuverability of submersible vehicles.
