Studies in Mathematical Physics: Advection, Convection and Turbulent Transport

本课题是研究流体力学偏微分方程（包括Navier-Stokes方程）解的定性和定量性质。后者构成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，并对数学物理提出了许多突出的挑战。研究将利用现代应用分析、计算和数值模拟，与PI指导下的博士研究生和博士后研究人员合作进行。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平流-扩散方程和湍流混合。该分析将根据搅拌流的体积和统计特征对被动标量场的扩散增强施加绝对限制，并指出特别有效或低效的搅拌策略。将进行新的最优搅拌策略的搜索，并研究湍流的混合效果。对流：热对流的理论问题将采用严格的分析和数值模拟进行研究。将研究在固定热通量和固定温度条件下持续的对流湍流的差异。PI的分析技术将得到发展，并应用于表面张力驱动的对流。能量耗散和熵产生：工作将继续确定产生最大熵的流场构型如何与充分发展的湍流中观察到的结构相关。用于推导复杂和湍流的能量耗散率先验边界的变分方法将扩展到与地球物理和天体物理应用相关的流动构型。从这个项目中获得的知识将有助于我们对流体动力学数学模型的基本理解，与应用物理科学和工程直接相关。鉴于该活动对教育的广泛影响，它为密歇根大学的博士生提供前沿论文研究机会，并为博士后研究人员提供支持。这项研究还涉及与来自世界各地机构的研究人员的广泛合作和互动。从长远来看，这项研究将有助于从航空到天体物理学，从气象学到材料制造等应用领域的实用技术的发展。

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.