Studies in Mathematical Physics: Advection, Convection and Turbulent Transport

这项研究计划致力于开发和应用严格的分析和计算方法来解决流体力学和湍流中的一些长期存在的问题，目的是为平流、平流-扩散、Navier-Stokes和相关的偏微分方程组的解得出可靠的重要物理量的数学估计。它们在应用物理科学和工程中有重要的应用，包括天气预报和气候模拟。该项目有三个主要组成部分：平流：主要研究人员和合作者提出的数学混合措施将用于研究平流和平流扩散方程的解，作为层流和湍流混合的模型。分析将根据搅拌流动的体积和/或统计特征对被动示踪剂的混合施加绝对限制，并将指出特别有效的搅拌的关键特征。将对最佳搅拌策略进行新的探索，并将研究湍流的混合效果。对流：将通过分析和直接数值模拟来研究热对流问题。通过层流的渐近分析和计算以及湍流的高瑞利数模拟，将检验经典的二维Rayleigh-Benard对流模型中新的严格传热极限的敏锐性。我们将利用Boussinesq近似下的温度方程的最大值原理，对三维对流进行新的估计。能量耗散和拟能产生：将启动一个确定有限时间间隔内三维N-S方程的最大拟能产生的主要新程序。数学和计算技术的背景下最大复古生产的二维纳威-斯托克斯方程将得到发展。对于当前分析方法失效的简单流动装置，将寻求新的方法来确定Navier-Stokes方程解中体积和时间平均湍流能量耗散率的绝对极限。更广泛的影响：这些项目适合密歇根大学的博士生和博士后研究人员。首席调查员的研究通常涉及与研究生、博士后、初级教员和美国及其他地方机构不同部门的杰出高级研究人员的合作。这些互动促进了结果的广泛传播，刺激和激励了新的调查，并促进了跨学科、跨机构和跨国界的数学方法的转移。首席调查员还积极参与有组织的努力，鼓励和加强妇女和任职人数不足群体的成员参与物理和数学教育和研究。

This research program is focused on the development and application of rigorous analytical and computational approaches to some longstanding problems in fluid dynamics and turbulence with the goal of deriving reliable mathematical estimates of physically important quantities for solutions of the advection, advection-diffusion, and Navier-Stokes and related systems of partial differential equations. These have important applications in the applied physical sciences and engineering, including weather prediction and climate modeling. The project has three major components: Advection: Mathematical mixing measures introduced by the principal investigator and collaborators will be applied to study solutions of the advection and advection-diffusion equations as models of laminar and turbulent mixing. Analysis will place absolute limits on mixing for passive tracers in terms of bulk and/or statistical features of the stirring flows, and it will indicate key features of particularly efficient stirring. New searches for optimal stirring strategies will be undertaken, and the mixing effectiveness of turbulence will be investigated. Convection: Issues in thermal convection will be studied via analysis and direct numerical simulation. The sharpness of new rigorous limits on heat transport in the classical two-dimensional model of Rayleigh-Benard convection will be tested via asymptotic analysis and computation of laminar flows and high Rayleigh number simulations of turbulent flows. New estimates for three-dimensional convection will be pursued exploiting the maximum principle for the temperature equation in the Boussinesq approximation. Energy dissipation and enstrophy production: A major new program to determine maximal enstrophy production in the three-dimensional Navier-Stokes equations over finite time intervals will be initiated. Mathematical and computational techniques in the context of maximal palinstrophy production in the two dimensional Navier-Stokes equations will be developed. New methods for determining absolute limits on the bulk and time averaged turbulent energy dissipation rate in solutions of the Navier-Stokes equations will be sought for simple flow setups where current analysis methods fail. Broader impacts: These projects are suitable for doctoral students and postdoctoral researchers at the University of Michigan. The Principal Investigator's research routinely involves collaborations with graduate students, postdocs, junior faculty, and distinguished senior researchers in a variety of different departments at institutions across the United States and beyond. These interactions foster broad dissemination of results, stimulate and motivate new investigations, and promote transfer of mathematical methods across disciplinary, institutional, and national boundaries. The Principal Investigator is also actively engaged in organized efforts to encourage and enhance the participation of women and members of under-represented groups in physics and mathematics education and research.