Studies in Mathematical Physics: Advection, Convection and Turbulent Transport

该研究计划的重点是开发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的一些长期存在的问题，目的是推导出对流，对流扩散和Navier-Stokes方程以及相关偏微分方程系统的物理重要量的可靠数学估计。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：主要研究者和合作者介绍的数学混合措施将用于研究平流和对流扩散方程的解，作为层流和湍流混合的模型。分析将在搅拌流的体积和/或统计特征方面对被动示踪剂的混合施加绝对限制，并且它将指示特别有效搅拌的关键特征。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：将通过分析和直接数值模拟研究热对流问题。通过层流的渐近分析和计算以及湍流的高Rayleigh数模拟，将测试Rayleigh-Benard对流经典二维模型中热传输的新严格限制的尖锐性。三维对流的新估计将采用Boussinesq近似中温度方程的最大值原理。能量耗散和涡度拟能产生：将启动一个主要的新方案，以确定三维纳维尔-斯托克斯方程在有限时间间隔内的最大涡度拟能产生。数学和计算技术的背景下，最大的回文生产的二维Navier-Stokes方程将被开发。新的方法来确定的体积和时间平均湍流能量耗散率的Navier-Stokes方程的解决方案的绝对限制，将寻求简单的流动设置，目前的分析方法失败。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在各种不同的部门在美国和超越机构的合作。这些互动促进了结果的广泛传播，刺激和激励新的调查，并促进跨学科，机构和国家边界的数学方法的转移。首席研究员还积极参与有组织的努力，以鼓励和加强妇女和代表性不足的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.