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CAREER: Nonlinear stability mechanisms and boundary layer singularities in fluid flows

职业：流体流动中的非线性稳定机制和边界层奇点

基本信息

批准号：
1652134
负责人：
Vlad Vicol
金额：
$ 42万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2019-01-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1652134&HistoricalAwards=false
关键词：
CAREER Nonlinear stability mechanisms boundary

项目摘要

The project concerns the mathematical analysis of fluid flows in the vicinity of physical boundaries, a fundamental problem in fluid dynamics. The realization that drag forces exerted on a body moving through air (or water) take place on a thin layer around the body lead to the development of boundary layer theory. The practical importance of boundary layer theory in aerodynamics and hydrodynamics is immense: computing the drag on an object and the energy dissipation rate nearby, with implications for fuel efficiency; understanding the problem of wing stall, the dependence of the lift on the angle of attack; and even the enhancement of heat transfer near solid walls. Mathematically, these questions concern the behavior of solutions to the Navier-Stokes equations in the vanishing viscosity limit, and the stability of the ensuing boundary layers. This project will develop new analytical tools to study the validity of the vanishing viscosity limit, the formation of singularities in boundary layers, and to explore the nonlinear stability mechanisms by which these lead to dynamic boundary layer separation. This theoretical information will lead to more accurate reduced models and finer predictions about real fluid flows. The project will train undergraduate, graduate, and post-graduate researchers, in modern research problems in applied mathematics and the tools to study them.This project aims to develop new mathematical tools that will further our understanding of the vanishing viscosity limit: the question whether solutions of the incompressible Navier-Stokes equations converge to solutions of the incompressible Euler equations as the viscosity approaches zero, in the presence of a characteristic physical boundary. For fixed external parameters, the vanishing viscosity limit is equivalent to the infinite Reynolds number limit, and thus this problem is of vital importance to the study of the onset of turbulence in fluid flows. The PI and collaborators will address the emergence of singularities in the Prandtl boundary layer equations by establishing more robust finite time blow-up scenarios. The stability of nearly laminar boundary layers will be studied via a hypocoercive analysis of the Prandtl system and of higher order models. Finally, the PI will develop new energy methods that intertwine Eulerian and Lagrangian approaches to prove the vanishing viscosity limit in high regularity regimes. The goal is to develop nonlinear, solution-adapted methods. Graduate students and post-doctoral fellows will be mentored and included in the research activities.

该项目涉及物理边界附近流体流动的数学分析，这是流体动力学的一个基本问题。边界层理论的发展是由于人们认识到物体在空气（或水）中运动时受到的阻力作用在物体周围的薄层上。边界层理论在空气动力学和流体动力学中的实际重要性是巨大的：计算物体上的阻力和附近的能量耗散率，对燃料效率的影响;理解机翼失速问题，升力对迎角的依赖性;甚至加强固体壁附近的传热。在数学上，这些问题涉及的行为的解决方案的Navier-Stokes方程在消失的粘度限制，以及随后的边界层的稳定性。本项目将开发新的分析工具，以研究粘性极限消失的有效性、边界层中奇点的形成，并探索导致动态边界层分离的非线性稳定机制。这一理论信息将导致更准确的简化模型和更精细的预测有关真实的流体流动。该项目将培训本科生、研究生和研究生研究人员，了解应用数学中的现代研究问题及其研究工具。该项目旨在开发新的数学工具，以加深我们对消失粘度极限的理解：当粘性趋于零时，不可压缩Navier-Stokes方程的解是否收敛于不可压缩Euler方程的解，存在一个特定的物理边界。对于固定的外参数，粘性极限的消失等价于雷诺数极限的无穷大，因此这一问题对于研究流体流动中湍流的起始是至关重要的。PI和合作者将通过建立更鲁棒的有限时间爆破场景来解决普朗特边界层方程中奇点的出现。近层流边界层的稳定性将通过普朗特系统和高阶模型的次强制分析来研究。最后，PI将开发新的能量方法，这是欧拉和拉格朗日方法之间的矛盾，以证明在高规则性区域中消失的粘性极限。我们的目标是开发非线性，解决方案适应的方法。研究生和博士后研究员将得到指导并参与研究活动。

项目成果

期刊论文数量（6）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

On Singularity Formation in a Hele-Shaw Model

Hele-Shaw 模型中奇点的形成

DOI：
10.1007/s00220-018-3241-6
发表时间：
2018
期刊：
Communications in Mathematical Physics
影响因子：
2.4
作者：
Constantin, Peter;Elgindi, Tarek;Nguyen, Huy;Vicol, Vlad
通讯作者：
Vicol, Vlad

Vorticity Measures and the Inviscid Limit

涡量测量和无粘极限

DOI：
10.1007/s00205-019-01398-1
发表时间：
2019
期刊：
Archive for Rational Mechanics and Analysis
影响因子：
2.5
作者：
Constantin, Peter;Lopes Filho, Milton C.;Nussenzveig Lopes, Helena J.;Vicol, Vlad
通讯作者：
Vicol, Vlad

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit