Collaborative Research: Analysis of incompressible high Reynolds number flows

该项目的目的是在美国和巴西两组科学家之间开展国际合作研究，对湍流不可压缩流体进行数学分析和建模。要研究的题目是：二级流体的小粘度状态；二维Euler方程线性扰动弱解的唯一性螺旋对称三维Navier-Stokes方程的消失粘度极限；寻找具有异常耗散的二维和三维Navier-Stokes方程的不变测度结构或平稳统计解的假设；具有物理边界域内流动的能量级联搜索具有大间隙双峰强迫的二维叶栅。湍流是流体运动中的一种常见现象，其中宏观量（速度、压力、温度等）与流动的全局参数不再具有确定性关系。在实际应用中，如在地球物理建模和工业过程中的混合中，对大雷诺数湍流的直接数值模拟，即使是最先进的计算机能力也无法实现。因此，迫切需要对这个具有挑战性的问题进行分析，通过开发严格的数学和统计工具来研究它，并对这些工具进行计算测试。虽然湍流每天都在发生，但我们在许多方面的理解仍然不足。量化小尺度对大尺度动力学的影响是现代多尺度科学的基础。该项目的目标是使紊流的预测分析研究成为可能。这项研究将影响广泛的应用，从地球物理建模，如海洋中污染物的扩散，到生物和工业建模，如聚合物材料的设计。该项目将巩固主要研究人员与其巴西同行之间业已建立的合作努力，并可能导致新的合作，特别是初级研究人员之间的合作。科学家之间的国际合作是全球市场经济竞争力的关键。四家美国学术机构和三家巴西学术机构参与了该项目。计划中的两个讲习班进一步强调了该项目的国际层面。至少六名博士生和博士后的培训和监督也通过计划的暑期学校和科学研讨会实现。来自美国的学生和博士后将前往巴西参加研讨会和暑期学校，并与美国和巴西的研究人员进行互动。本项目由国际科学与工程办公室美洲项目共同资助。

The purpose of this project is to conduct international collaborative research, between two teams of scientists from the USA and Brazil, in the mathematical analysis and modeling of turbulent incompressible fluids. The topics to be investigated are: small viscosity regime of second grade fluids; uniqueness of weak solutions for certain linear perturbations of the two-dimensional Euler equations; the vanishing viscosity limit of the three-dimensional Navier-Stokes equations with helical symmetry; the search for hypotheses on the structure of invariant measures or stationary statistical solutions of the two and three-dimensional Navier-Stokes equations with anomalous dissipation; the search for energy cascade for flows in domains with physical boundaries; two-dimensional cascades with large gap bimodal forcing. Turbulence is a common phenomenon in fluid motion, in which macroscopic quantities (velocity, pressure, temperatures, etc.) no longer have a deterministic relation with global parameters of the flow. Direct numerical simulations of turbulent flows at large Reynolds numbers, that occur in practical applications, such as in geophysical modeling and mixing in industrial processes, are out of reach even for the state-of-the-art computer power. Therefore, there is an urgent need to pursue this challenging problem analytically, by developing rigorous mathematical and statistical tools to investigate it, and to test these tools computationally. While turbulence is an everyday occurrence, our understanding is still lacking in many aspects. Quantifying the effect of small scales on the dynamics of large scales is fundamental in modern multiscale science. The goal of the project is to enable a predictive analytical study of turbulent flows. This study will impact wide-ranging applications, from geophysical modeling, such as dispersion of pollutants in the ocean, to biological and industrial modeling, such as design of polymeric materials. The project will consolidate the well-established collaborative efforts of the principal investigators with their Brazilian counterparts, and may lead to new collaboration, especially among the junior research personnel. International collaboration among scientists is a key to economic competitiveness in global markets. Four US and three Brazilian academic institutions are involved in the project. The international dimension of the project is further emphasized through two planned workshops. Training and supervision of at least six Ph.D. students and postdoctoral fellows is also achieved through planned summer schools and scientific workshops. Students and postdoctoral fellows from the US will travel to Brazil to participate in the workshops and the summer schools and interact with the US and Brazilian researchers.This project is co-funded with the Americas Program of the Office of International Science and Engineering.