Studies in Mathematical Physics: Advection, Convection and Turbulent Transport

本研究计划的重点是发展和应用严格的分析和计算方法来解决流体动力学和湍流中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系统的解提供可靠的物理重要量的数学估计。这些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建模。该项目有三个主要组成部分：平流：由首席研究员和合作者引入的数学混合测量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。分析将根据搅拌流的体积和/或统计特征对被动示踪剂的混合提出绝对限制，并将指出特别有效搅拌的关键特征。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：热对流的问题将通过分析和直接数值模拟来研究。本文将通过层流的渐近分析和计算以及湍流的高瑞利数模拟来检验经典二维瑞利-贝纳德对流模型中热传递新严格限制的尖锐性。利用Boussinesq近似中温度方程的最大值原理，对三维对流进行新的估计。能量耗散和熵产生：一个主要的新程序，以确定在有限时间间隔内三维纳维-斯托克斯方程的最大熵产生将被启动。将发展二维Navier-Stokes方程中最大回弹产生的数学和计算技术。对于当前分析方法失效的简单流动装置，将寻求确定Navier-Stokes方程解中体积和时间平均湍流能量耗散率绝对极限的新方法。更广泛的影响：这些项目适用于密歇根大学的博士生和博士后研究人员。首席研究员的研究通常涉及与研究生、博士后、初级教师和杰出的高级研究人员合作，这些研究人员来自美国和其他国家的各个机构的不同部门。这些互动促进了结果的广泛传播，刺激和激励了新的研究，并促进了数学方法跨学科、机构和国家边界的转移。首席研究员还积极参与有组织的努力，鼓励和加强妇女和代表性不足群体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.