Studies in Mathematical Physics: Advection, Convection and Turbulent Transport

数学物理研究：平流、对流和湍流传输

• 批准号: 0555324
• 负责人: Charles Doering
• 金额: $ 29.51万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555324&HistoricalAwards=false
• 关键词: Studies; Mathematical; Physics; Advection; Convection

This research project in mathematical physics and applied analysis is a study of qualitative and quantitative properties of solutions of the partial differential equations of fluid mechanics including the Navier-Stokes equations. The Navier-Stokes equations constitute the basic mathematical model of fluid flow and are believed to contain turbulence among their solutions. Turbulent transport and mixing have important applications in, and implications for, many areas of applied physical sciences and engineering and present a number of outstanding challenges for theoretical physics and applied mathematics. The investigations will be carried out utilizing modern applied analysis, computation and numerical simulation with graduate students and postdoctoral researchers working under the direction of Principal Investigator Charles R. Doering at the University of Michigan. The project has three major components: . Mathematical methods previously developed by principal investigator and collaborators for the study of turbulent transport in the Navier-Stokes and related equations will be extended and applied to the advection-diffusion equation and turbulent mixing. This analysis will place limits on mixing efficiencies for passive scalar fields in terms of bulk and statistical features of the applied flows, and indicate key features of particularly efficient or inefficient stirring strategies. . Theoretical and mathematical issues in thermal convection will be studied via rigorous analysis and direct numerical simulation. Modern enhancements of the analytical techniques pioneered by the principal investigator will be developed and applied to open problems including homogeneous convection, Rayleigh-Benard convection with free-slip boundaries, infinite Prandtl number models and flows driven by internal heating with applications in geophysics. . The turbulent energy cascade and enstrophy generation will be investigated for solutions of the incompressible Navier-Stokes equations. Variational approaches capable of bounding turbulence driven by time-independent body-forces will be extended and applied to time-dependent and broadband (fractal) forcing. Work in progress will continute to determine maximum enstrophy generating flow-field configurations, how they are related to structures observed in fully developed turbulence, and their role in the development of singularities. With regard to the intellectual merit of this activity, knowledge gained from this project will contribute to fundamental understandings of mathematical models in fluid dynamics that are of direct relevance to many branches of applied science and engineering. In the long term this research wll aid the development of practical techniques for simulation, prediction and control of physical processes with applications ranging from meteorology to materials manufacturing. With regard to this activity's even broader impacts, there are several significant advanced training aspects to this project: it provides frontier dissertation research opportunities for graduate students in Michigan's Ph.D. program in Applied & Interdisciplinary Mathematics and support and guidance for postdoctoral researchers at the University of Michigan. This research also involves collaborations and interactions with investigators, including graduate students and postdoctoral researchers, from other institutions.

该数学物理和应用分析研究项目是对流体力学偏微分方程（包括纳维-斯托克斯方程）解的定性和定量性质的研究。纳维-斯托克斯方程构成了流体流动的基本数学模型，并且被认为在其解中包含湍流。湍流传输和混合在应用物理科学和工程的许多领域具有重要的应用和影响，并对理论物理和应用数学提出了许多突出的挑战。调查将利用现代应用分析、计算和数值模拟进行，研究生和博士后研究人员将在密歇根大学首席研究员 Charles R. Doering 的指导下开展工作。该项目由三个主要部分组成： .先前由主要研究者和合作者开发的用于研究纳维-斯托克斯湍流输运及相关方程的数学方法将被扩展并应用于平流扩散方程和湍流混合。该分析将在所应用流的体积和统计特征方面对被动标量场的混合效率进行限制，并指出特别有效或低效的搅拌策略的关键特征。 。将通过严格的分析和直接数值模拟来研究热对流的理论和数学问题。由首席研究员开创的分析技术的现代增强将被开发并应用于开放问题，包括均匀对流、具有自由滑移边界的瑞利-贝纳德对流、无限普朗特数模型以及由内部加热驱动的流动及其在地球物理学中的应用。 。将研究湍流能量级联和熵生成，以求解不可压缩纳维-斯托克斯方程。能够限制由与时间无关的体积力驱动的湍流的变分方法将被扩展并应用于与时间相关的宽带（分形）强迫。正在进行的工作将继续确定最大熵生成流场配置、它们与在充分发展的湍流中观察到的结构的关系，以及它们在奇点发展中的作用。关于这项活动的智力价值，从该项目中获得的知识将有助于对流体动力学数学模型的基本理解，这些模型与应用科学和工程的许多分支直接相关。从长远来看，这项研究将有助于开发模拟、预测和控制物理过程的实用技术，其应用范围涵盖从气象学到材料制造。就这项活动更广泛的影响而言，该项目有几个重要的高级培训方面：它为密歇根大学博士研究生提供了前沿论文研究机会。应用和跨学科数学项目以及对密歇根大学博士后研究人员的支持和指导。这项研究还涉及与其他机构的研究人员（包括研究生和博士后研究人员）的合作和互动。