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.

这项数学物理学和应用分析的研究项目是对包括Navier-Stokes方程在内的流体力学局部微分方程的定性和定量特性的研究。 Navier-Stokes方程构成流体流的基本数学模型，并且被认为包含其溶液中的湍流。动荡的运输和混合在许多应用物理科学和工程领域都具有重要的应用，并影响了许多领域，并为理论物理和应用数学带来了许多出色的挑战。该调查将通过现代应用分析，计算和数值模拟进行，并在密歇根大学首席研究员Charles R. Doering的指导下与研究生和博士后研究人员进行。该项目有三个主要组成部分：。先前由首席研究者和合作者开发的数学方法将扩展并应用于对流扩散方程和湍流混合，将扩展到Navier-Stokes和相关方程中的湍流传输研究。该分析将根据施加流的批量和统计特征来混合被动标量场的效率限制，并指示特别有效或效率低下的搅拌策略的关键特征。 。热对流中的理论和数学问题将通过严格的分析和直接数值模拟进行研究。将开发由主要研究者开创的分析技术的现代增强，并将其应用于开放问题，包括均质对流，瑞利 - 贝纳德对流，具有自由滑移边界，无限的Prandtl数字模型和由内部暖气在地球物理学中应用的内部供暖而驱动的流动。 。将研究不可压缩的Navier-Stokes方程解决方案的湍流能量级联反应和发电。能够限制由时间无关的身体驱动的湍流的变异方法将扩展并应用于时间依赖性和宽带（分形）强迫。正在进行中的工作将继续确定最大的腹膜生成流场构型，它们与在完全发育的湍流中观察到的结构的关系以及它们在奇异性发展中的作用。关于这项活动的智力优点，从该项目中获得的知识将有助于对流体动力学中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实用技术，以模拟，预测和控制物理过程，其应用从气象到材料制造等。关于这项活动的更大影响，该项目有几个重要的高级培训方面：它为密歇根州博士学位的研究生提供了前沿论文研究机会。密歇根大学应用和跨学科数学和支持和指导的计划。这项研究还涉及其他机构的合作和互动，包括研究生和博士后研究人员。