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.

该数学领域以嘈杂的逆流图接近粘性传输方程。这是通过向粒子轨迹和平均添加噪声来对特征方法（用于无粘性问题）的自然概括。这项研究的主要重点是将这些技术应用于Navier-Stokes和Boussinesq方程。 Navier-Stokes方程可以在上述框架中使用Indiscid Webber公式在上述框架中进行优雅配制，该公式基本上代表Navier-Stokes方程为Euler方程和Brownian Motion的平均值。我们有各种各样的应用：一种数值（随机）方法，用于计算Navier-Stokes方程的解决方案，研究障碍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，不可压缩的流体，分析性估计和弱解决方案，以及对BOUSSINESS的范围的流动性。明确表示无粘性流体和布朗尼运动的平均值。提出了该框架中流体动力学的几个方面的研究。一种重要的应用是实施数值方法来计算湍流设置中的流体流动（例如，飞机机翼围绕飞机或喷气发动机中的空气流）。相关的应用是使用这种方法来找到流体流的“湍流模型”：即，流体的速度场表示为一种缓慢变化的“平均场”（易于计算，易于计算），加上快速波动的“ Noisy”部分，该部分是使用随机方法建模的。另一个应用是研究边界层对无粘性极限内内部流动的影响。还提出了对燃烧和火焰传播的研究。