CAREER: Non-Uniqueness in Inviscid Flow and Algebraic Vortex Spirals

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

A ubiquitous feature in fluid flow is emergence of vortex sheets: surfaces across which tangential velocity changes sharply, while pressure and normal velocity are continuous. This shearing motion causes the Kelvin-Helmholtz instability and other poorly understood phenomena. Vortex sheets are produced by sharp corners of solid objects moving in a fluid, such as wings of accelerating aircraft or by interacting shock waves in compressible flow. Vortex sheets generally end in a spiral. Since computer simulation of vortex spirals is notoriously unstable, it is especially important investigating them with analytical tools from the theory of differential and integral equations. The project is aimed in particular at understanding and construction of certain classical spirals, followed by a generalization to compressible and viscous flow models. Recent numerical experiments by the principal investigator and analytical results by others suggest that the Euler equation, a mathematical model of fluid flow that neglects viscosity and heat conduction, may suffer from non-uniqueness: the initial state of the fluid does not determine the state at later times unambiguously. This project will put these unclear results into a more satisfactory state by finding rigorous proofs for flows that have simpler structure and are physically relevant, namely Pullin's separated spirals. The non-uniqueness phenomenon surfaces not only in theory, but also in computer simulation and in physical experiments which are crucial for designing aircraft and other sensitive applications. Understanding non-uniqueness is important because it indicates that the theoretical predictions are less reliable than previously thought, with potentially catastrophic consequences. Broader impacts of this project include combining education with hands-on research in order to attract students, especially at the critical advanced undergraduate stage, to the fluid dynamics research. Classroom innovations and several undergraduate projects on computer simulation of compressible fluids, with emphasis on extensive independent study instruction, are planned; PhD students will be trained.