Studies in Mathematical Physics: Advection, Convection and Turbulent Transport

本研究方向为数学物理与应用分析，研究流体力学偏微分方程（包括Navier-Stokes方程）解的定性与定量性质。Navier-Stokes方程构成了流体流动的基本数学模型，并被认为在其解中包含湍流。湍流输运和混合在应用物理科学和工程的许多领域都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。在密歇根大学首席研究员Charles R. Doering的指导下，研究生和博士后研究人员将利用现代应用分析、计算和数值模拟进行研究。该项目有三个主要组成部分：。先前由首席研究员和合作者开发的用于研究Navier-Stokes和相关方程中的湍流输运的数学方法将扩展并应用于平流-扩散方程和湍流混合。这种分析将限制被动标量场在体积和应用流的统计特征方面的混合效率，并指出特别有效或低效的搅拌策略的关键特征。热对流的理论和数学问题将通过严格的分析和直接的数值模拟来研究。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，自由滑移边界的瑞利-贝纳德对流，无限普朗特数模型和内部加热驱动的流动，以及在地球物理学中的应用。本文将研究不可压缩Navier-Stokes方程的湍流能量级联和熵的产生。能够由时间无关的身体力驱动的边界湍流的变分方法将被扩展并应用于时间相关和宽带（分形）强迫。正在进行的工作将继续确定产生最大熵的流场构型，它们与完全发展的湍流中观察到的结构之间的关系，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，这项研究将有助于物理过程模拟、预测和控制的实用技术的发展，其应用范围从气象学到材料制造。关于这项活动的更广泛的影响，有几个重要的高级培训方面的项目：它提供前沿的论文研究机会，在密歇根大学的应用和跨学科数学博士项目的研究生和支持和指导博士后研究人员在密歇根大学。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。

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.