A Stochastic Langrangian approach to non-linear transport equations

非线性输运方程的随机朗格朗日方法

• 批准号: 0966947
• 负责人: Gautam Iyer
• 金额: $ 3.49万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0966947&HistoricalAwards=false
• 关键词: Stochastic; Langrangian; approach; non; linear

This area of mathematics approaches viscous transport equations in terms of a noisy inverse flow map. This is a natural generalization of the method of characteristics (used for inviscid problems) to viscous problems by adding noise to particle trajectories and averaging. The main focus of this research is to apply these techniques to the Navier-Stokes and Boussinesq equations. The Navier-Stokes equations can be elegantly formulated in the above framework using the inviscid Webber formula, which essentially represents the Navier-Stokes equations as the average of the Euler equations plus Brownian motion. We have a wide variety of applications in mind: A numerical (stochastic) method to compute solutions to the Navier-Stokes equations, a study of the effect of obstacles and boundaries on viscous fluid flows, turbulence models for incompressible fluids, analytical estimates and weak solutions, and front propagation in the Boussinesq equations.This research is based on the framework in which a viscous incompressible fluid can be explicitly represented as the average of an inviscid fluid plus Brownian motion. A study of several aspects of fluid dynamics in this framework is proposed. One important application is the implementation of a numerical method to compute fluid flows in turbulent settings (for instance air flow around an airplane wing, or in a jet engine). A related application is to use this method to find a "turbulence model" for fluid flows: namely, the velocity field of the fluid is expressed as a slowly varying "mean field" (which is easy and inexpensive to compute), plus a rapid fluctuating "noisy" part, which is modeled using stochastic methods. Another application is to study boundary layer effects on the interior flow in the inviscid limit. A study of burning and flame propagation is also proposed.

该数学领域根据噪声逆流图来处理粘性输运方程。这是通过向粒子轨迹添加噪声并求平均来将特征方法（用于无粘性问题）自然推广到粘性问题。这项研究的主要重点是将这些技术应用于纳维-斯托克斯和布辛涅斯克方程。纳维-斯托克斯方程可以使用无粘性韦伯公式在上述框架中优雅地表述，该公式本质上将纳维-斯托克斯方程表示为欧拉方程加上布朗运动的平均值。我们考虑到了各种各样的应用：计算纳维-斯托克斯方程解的数值（随机）方法、障碍物和边界对粘性流体流动的影响的研究、不可压缩流体的湍流模型、分析估计和弱流体模型。这项研究基于这样一个框架，其中粘性不可压缩流体可以明确地表示为无粘性流体加上布朗运动的平均值。提出了在此框架中对流体动力学的几个方面进行研究。一个重要的应用是实施一种数值方法来计算湍流环境中的流体流动（例如飞机机翼周围或喷气发动机中的气流）。一个相关的应用是使用这种方法来找到流体流动的“湍流模型”：即，流体的速度场表示为缓慢变化的“平均场”（计算起来很容易且成本低廉），加上快速变化的“平均场”。波动的“噪声”部分，使用随机方法建模。另一个应用是研究无粘极限条件下边界层对内部流动的影响。还提出了燃烧和火焰传播的研究。