Topics in Unsteady Vortical Flows in Three Dimensions

New York University, New York NY

Investigators

Abstract

The investigator studies two related but different aspecta of vortex dynamics in three space dimensions. The first project concerns the global regularity of Euler flows in three dimensions. This important outstanding problem of applied analysis is attacked by the investigator and colleagues with the aim of showing that, under appropriate conditions of flow topology, global regularity is obtained. In addition an estimate of the time rate of growth of the maximum vorticity should be accessible. The approach utilizes maximization of the growth of vorticity and a variational construction of an artificial structure (the "cocoon") near the vorticity maximum. The second project concerns analysis together with numerical and physical experiments of modes of hovering in flapping flight. An oscillating flow facility already in use provides the starting point for investigations of the vortical flow in the vicinity of small flapping hoverers. The investigator and the research staff of the CIMS Applied Mathematics Laboratory develop analytical and numerical models for the study of hovering flight, motivated by the current interest in micro-aerodynamic vehicles. The project deals with two related problems involving the difficult problem of analysis of three-dimensional fields of fluid motion and the complex eddying motions that occur in many important time-dependent flows. The first problem that the investigator and his colleagues study is the theoretical question of "blowup" of solutions of Euler's equations, wherein the flow develops a rough or singular structure after a finite time. Although much effort has been devoted to trying to find such behavior in fluid flows, the question remains open. The contention the investigator explores is that in fact for most flows, especially those for which the "knottedness" of the vorticity is sufficiently simple, blowup does not occur. This regularity property of the flows can be established in some simple cases, and methods under development by the investigator promise to be applicable to a larger class, whose size needs to be determined. The study has important implications for global analysis of Euler flows, and perhaps also for the numerical computation of these flows. The second project is different in emphasis and brings into play a combination of analysis with numerical and physical experiments to study the hovering flight of small flapping flyers of the kind envisioned in the context of micro-aerodynamic flight. This work aims to understand the eddying motions produced by certain kinds of flapping structures, and the mechanism of lift production in various cases of hovering flight. The work has possible applications to the engineering design of extremely small air vehicles, as well as a bearing on the understanding of insect flight.