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

职业：无粘流和代数涡旋的非唯一性

• 批准号: 1054115
• 负责人: Volker Elling
• 金额: $ 42.08万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-05-01 至 2017-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1054115&HistoricalAwards=false
• 关键词: CAREER; Non; Uniqueness; Inviscid; Flow

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.

流体流动中一个普遍存在的特征是涡面的出现：在涡面上切向速度急剧变化，而压力和法向速度是连续的。这种剪切运动导致开尔文-亥姆霍兹不稳定性和其他知之甚少的现象。涡面是由在流体中运动的固体物体的尖角产生的，例如加速飞行器的机翼或可压缩流中相互作用的冲击波。涡面通常以螺旋形结束。由于涡旋螺旋线的计算机模拟是众所周知的不稳定，因此用微分和积分方程理论的分析工具来研究它们是特别重要的。该项目的主要目的是理解和构建某些经典螺旋，然后推广到可压缩和粘性流动模型。主要研究者最近的数值实验和其他人的分析结果表明，欧拉方程，一种忽略粘性和热传导的流体流动数学模型，可能会受到非唯一性的影响：流体的初始状态并不明确地确定随后的状态。这个项目将把这些不清楚的结果到一个更令人满意的状态，通过寻找严格的证明，具有简单的结构和物理相关的流动，即普林的分离螺旋。非唯一性现象不仅在理论上，而且在计算机模拟和物理实验中也是如此，这对于设计飞行器和其他敏感应用至关重要。理解非唯一性很重要，因为它表明理论预测比以前认为的更不可靠，可能会带来灾难性的后果。该项目的更广泛的影响包括将教育与实践研究相结合，以吸引学生，特别是在关键的高年级本科阶段，进行流体动力学研究。课堂创新和几个本科生项目的计算机模拟可压缩流体，重点是广泛的独立学习指导，计划;博士生将接受培训。