Applied Analysis of the Navier-Stokes and Related Equations

纳维-斯托克斯及相关方程的应用分析

• 批准号: 0244859
• 负责人: Charles Doering
• 金额: $ 26.7万
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244859&HistoricalAwards=false
• 关键词: Applied; Analysis; Navier; Stokes; Related

This is a proposal for fundamental research in mathematical physics and applied mathematics fo-cused on the challenges presented by the incompressible Navier-Stokes and related equations ofuid mechanics. The Navier-Stokes equations constitute the basic mathematical model of uidow, and are believed to contain turbulent dynamics among their solutions. Turbulence in uidmechanics remains one of the outstanding challenges for theoretical physics and applied mathe-matics with important applications in, and implications for, many areas in the physical sciencesand engineering. The work in this project will be carried out via modern applied analysis andnumerical computation and simulation by the principal investigator (PI), Prof. Charles R. Doeringof the University of Michigan, and graduate students performing doctoral dissertation work. Thisproject has three specific objectives:For one, a mathematical technique for deriving rigorous bounds on turbulent dissipation anddrag, which has come to be known as the \background method," will be developed and expandedto new areas including magnetohydrodynamics, drag-reducing polymer ows, and ows over roughboundaries. The background method was introduced by the PI and his collaborator a decade agofor the Navier-Stokes equations, and since then it has been developed and applied by the PI, hisstudents, and many other researchers to a number of fundamental shear ow and thermal convectionproblems. One particular goal of this project will be to explore applications to a wider variety ofproblems of scientific interest.Another focus of this project is to continue the ongoing investigation of theoretical and mathe-matical issues in the analysis of thermal convection models where the background method is capableof putting limits on the heat transfer rate. Problems of concern here include laminar and turbulentconvection with free-slip boundaries, xed-ux convection, ows driven by internal heating, andinfinite Prandtl number models inspired by applications in geophysics.In a third direction of research, power consumption and enstrophy generation will be studiedfor forced ows and free ows in the absence of rigid boundaries. The PI and collaborators haverecently developed a new approach for the analysis of turbulence driven by time-independent body-forces, and it is proposed to extend the results to time-dependent forces. A distinct problem forunforced ows is to solve a variational problem for the maximum enstrophy-generating configurationand study how it relates to structures observed in fully developed turbulence or the potentialdevelopment of singularities.With regard to the intellectual merit of this activity, knowledge gained from this project willfurther our understanding of some basic mathematical models in uid dynamics of direct relevanceto many branches of engineering and applied science. In the long term, this kind of mathematicalresearch could help the development of practical techniques for the prediction and/or control ofphysical processes ranging from meteorology to materials manufacturing.And with regard to this activity's broader impacts, there are several significant advanced train-ing aspects to the project. For one, it provides research support and opportunities for graduatestudents within the University of Michigan's new Ph.D. program in Applied & InterdisciplinaryMathematics. Moreover, this project also involves other investigators|including graduate studentsand postdoctoral researchers from the University of Michigan as well as other institutions|whowill collaborate in the research.

这是数学物理和应用数学基础研究的提案，重点关注不可压缩纳维-斯托克斯和流体力学相关方程所带来的挑战。纳维-斯托克斯方程构成了 uidow 的基本数学模型，并且被认为在其解中包含湍流动力学。流体力学中的湍流仍然是理论物理和应用数学面临的突出挑战之一，在物理科学和工程的许多领域具有重要的应用和影响。该项目的工作将由首席研究员（PI）、密歇根大学的 Charles R. Doering 教授和研究生通过现代应用分析、数值计算和模拟进行。该项目有三个具体目标：首先，将开发一种用于导出湍流耗散和阻力严格界限的数学技术，该技术被称为“背景方法”，并将其扩展到新领域，包括磁流体动力学、减阻聚合物流、背景方法是由 PI 和他的合作者在十年前针对纳维-斯托克斯方程引入的，从那时起它就被开发和应用。 PI、他的学生和许多其他研究人员致力于解决一些基本的剪切流和热对流问题。该项目的一个特定目标是探索其在更广泛的科学问题中的应用。该项目的另一个重点是继续正在进行的研究。热对流模型分析中的理论和数学问题，其中背景方法能够限制传热速率，这里关注的问题包括层流和湍流对流。自由滑移边界、xed-ux 对流、内部加热驱动的流动以及受地球物理学应用启发的无限普朗特数模型。在第三个研究方向中，将研究在没有严格的界限。 PI 和合作者最近开发了一种新方法来分析由时间无关的物体力驱动的湍流，并建议将结果扩展到时间相关的力。受迫流的一个独特问题是解决最大熵生成构型的变分问题，并研究它与在充分发展的湍流中观察到的结构或奇点的潜在发展如何相关。关于这项活动的智力价值，从该项目中获得的知识将进一步加深我们对与工程和应用科学的许多分支直接相关的流体动力学中的一些基本数学模型的理解。从长远来看，这种数学研究可以帮助开发预测和/或控制从气象到材料制造等物理过程的实用技术。就这项活动的更广泛影响而言，有几个重要的高级培训方面到项目。其一，它为密歇根大学新博士项目的研究生提供研究支持和机会。应用与跨学科数学课程。此外，该项目还涉及其他研究人员，包括来自密歇根大学以及其他机构的研究生和博士后研究人员，他们将参与这项研究。