Studies in Mathematical Physics: Advection, Convection and Turbulent Transport

本课题研究流体力学中的Navier-Stokes方程等偏微分方程的解的定性和定量性质。后者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，并对数学物理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，与研究生进行博士研究和博士后研究人员在PI的指导下工作。该项目有三个主要组成部分：平流：由PI和合作者开发的数学方法将应用于对流扩散方程和湍流混合。这种分析将绝对限制被动标量场的搅拌流的体积和统计特征方面的扩散增强，并指出特别有效或低效的搅拌策略。对最佳搅拌策略进行新的探索，并对湍流的混合效果进行研究。对流：将使用严格的分析和数值模拟来研究热对流的理论问题。将研究由固定热通量和固定温度条件维持的对流湍流之间的差异。PI的分析技术将被开发和应用于表面张力驱动的convention.Energy耗散和拟能生产：工作将继续确定如何最大拟能产生流场配置与充分发展的湍流中观察到的结构。变分方法推导的先验界限的能量耗散率的复杂和湍流将被扩展到相关的地球物理和天体物理的应用程序的流动configuration.Knowledge从这个项目中获得的将有助于我们在流体动力学的数学模型的基本理解，直接相关的应用物理科学和工程。关于这项活动在教育方面的广泛影响，它为密歇根大学的博士生提供了前沿论文研究机会，并为博士后研究人员提供了支持。这项研究还涉及与世界各地机构的研究人员的广泛合作和互动。从长远来看，这项研究将有助于开发从航空到天体物理学，从气象学到材料制造等应用的实用技术。

This project is a study of qualitative and quantitative properties of solutions of the partial differential equations of fluid mechanics including the Navier-Stokes equations. The latter 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 many areas of applied physical sciences and engineering and present a number of outstanding challenges for mathematical physics. The investigations will be carried out utilizing modern applied analysis, computation, and numerical simulation in collaboration with graduate students performing doctoral research and postdoctoral researchers working under the direction of the PI. The project has three major components.Advection: Mathematical methods developed by the PI and collaborators will be applied to the advection-diffusion equation and turbulent mixing. This analysis will place absolute limits on diffusive enhancements for passive scalar fields in terms of bulk and statistical features of the stirring flows, and indicate particularly efficient or inefficient stirring strategies. New searches for optimal stirring strategies will be undertaken, and the mixing effectiveness of turbulence will be investigated. Convection: Theoretical issues in thermal convection will be studied using rigorous analysis and numerical simulation. Differences between convective turbulence sustained by fixed heat flux and fixed temperature conditions will be investigated. The analytical techniques of the PI will be developed and applied to surface tension driven convection.Energy dissipation and enstrophy production: Work will continue to determine how maximum enstrophy generating flow-field configurations are related to structures observed in fully developed turbulence. Variational approaches for the derivation of a priori bounds on energy dissipation rates for complex and turbulent flows will be extended to flow configurations relevant to geophysical and astrophysical applications.Knowledge gained from this project will contribute to our fundamental understanding of mathematical models in fluid dynamics, of direct relevance in the applied physical sciences and engineering. With regard to this activitys broader impacts in education, it provides frontier dissertation research opportunities for doctoral students and support for postdoctoral researchers at the University of Michigan. This research also involves extensive collaborations and interactions with investigators from institutions worldwide. In the long term this research will aid the development of practical techniques for applications ranging from aeronautics to astrophysics, and meteorology to materials manufacturing.