Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics

流体力学一些基本模型中的极端和奇异行为的系统搜索

• 批准号: 1515161
• 负责人: Charles Doering
• 金额: $ 45.49万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1515161&HistoricalAwards=false
• 关键词: Systematic; Search; Extreme; Singular; Behavior

The investigator develops and applies effective mathematical analysis and scientific computation tools to systematically search for extreme behavior in some of the fundamental equations of physical fluid mechanics. The goal is to derive precise predictions of physically significant quantities from first principles. The work capitalizes on recent developments to implement ideas from optimal control theory and the calculus of variations to compute fluid flows achieving maximal mixing, optimal transport, or other extreme dissipation. Transport, mixing, and dissipation are among the most fundamental features of fluid flows and are of foundational significance for important applications ranging from microfluidics engineering to modeling in climate science and astrophysics. The control and optimization techniques adopted here constitute a new and unified computationally aided analysis approach to these problems. This project directly involves advanced training for graduate students and postdoctoral researchers.This project utilizes methods of modern applied mathematics and scientific computation. Mathematical measures of mixing introduced by the investigator and collaborators are utilized in optimal control analyses of the advection and advection-diffusion equations in order to place absolute limits on passive tracer mixing by incompressible flows, and to illuminate key features of particularly effective stirring strategies. Computational control and applied analysis are employed to construct incompressible fluid flows optimizing transport between impenetrable surfaces and produce new transport bounds for buoyancy-driven Rayleigh-Benard convection and the outstanding problem of turbulent convection. Optimal control techniques are developed and deployed to determine maximal enstrophy production in the incompressible three-dimensional Navier-Stokes equations over finite time intervals. Extremal solutions provide new insight into fully nonlinear vorticity amplification in unforced flows, and this component of the project is a novel and promising framework for the study of one of the signal challenges for 21st century applied mathematics: the regularity question for the 3D Navier-Stokes equations.

研究人员开发并应用有效的数学分析和科学计算工具来系统地寻找物理流体力学的一些基本方程中的极端行为。 目标是根据第一原理得出物理重要量的精确预测。 这项工作利用最新的进展来实现最优控制理论和变分计算的思想，以计算实现最大混合、最优传输或其他极端耗散的流体流量。 传输、混合和耗散是流体流动最基本的特征，对于从微流体工程到气候科学和天体物理学建模等重要应用具有基础意义。 这里采用的控制和优化技术构成了解决这些问题的一种新的、统一的计算辅助分析方法。 该项目直接涉及研究生和博士后研究人员的高级培训。该项目利用现代应用数学和科学计算的方法。 研究者和合作者引入的混合数学测量用于平流和平流扩散方程的最优控制分析，以便对不可压缩流的被动示踪剂混合施加绝对限制，并阐明特别有效的搅拌策略的关键特征。 采用计算控制和应用分析来构造不可压缩流体流动，优化不可穿透表面之间的传输，并为浮力驱动的瑞利-贝纳德对流和湍流对流的突出问题产生新的传输边界。 开发和部署最优控制技术来确定有限时间间隔内不可压缩三维纳维-斯托克斯方程中的最大熵产生。 极值解为非受迫流中的完全非线性涡度放大提供了新的见解，该项目的这个组成部分是一个新颖且有前途的框架，用于研究 21 世纪应用数学的信号挑战之一：3D 纳维-斯托克斯的正则性问题方程。