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.

本文针对流体力学中不可压缩的Navier-Stokes方程和相关方程提出的挑战，对数学物理和应用数学的基础研究提出了建议。Navier-Stokes方程构成了uidow的基本数学模型，其解被认为包含了湍流动力学。流体力学中的湍流仍然是理论物理和应用数学的突出挑战之一，在物理科学和工程的许多领域都有重要的应用和影响。这个项目的工作将通过现代应用分析、数值计算和模拟来进行，由密歇根大学的首席研究员Charles R.Doering教授和正在进行博士论文工作的研究生进行。该项目有三个具体目标：首先，将开发一种用于推导湍流耗散和阻力的严格界限的数学技术，即众所周知的背景方法，并将其扩展到新的领域，包括磁流体动力学、减阻聚合物OWS和粗边界上的OWS。背景方法十年前由PI和他的合作者提出，用于求解Navier-Stokes方程。从那时起，PI和他的学生以及其他许多研究人员发展并应用于许多基本剪切流动和热对流问题。这个项目的一个特别目标将是探索在更广泛的科学兴趣问题上的应用。这个项目的另一个重点是继续正在进行的对热对流模型分析中的理论和数学问题的研究，其中背景方法能够对热传递速率施加限制。这里关注的问题包括具有自由滑移边界的层流和湍流对流，X-UX对流，内部加热驱动的OWS，以及受地球物理应用启发的无限Prandtl数模型。在第三个研究方向，将研究强迫OWS和没有刚性边界的自由OWS的功率消耗和拟能产生。PI和合作者最近发展了一种新的方法来分析与时间无关的体力驱动的湍流，并建议将结果推广到与时间相关的力。非受迫OWS的一个独特问题是解决最大位能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之间的关系。考虑到这一活动的智力价值，从这个项目中获得的知识将加深我们对UID动力学中一些基本数学模型的理解，这些模型与工程学和应用科学的许多分支直接相关。从长远来看，这种数学研究可以帮助开发预测和/或控制从气象学到材料制造的物理过程的实用技术。至于这项活动的更广泛影响，该项目有几个重要的高级培训方面。首先，它为密歇根大学新开设的应用与跨学科数学博士项目的毕业生提供研究支持和机会。此外，该项目还涉及其他研究人员，包括密歇根大学的研究生和博士后研究人员以及其他机构，他们将合作进行这项研究。