The Lagrangian Averaged Navier-Stokes Equations with Applications to Turbulence Modeling

拉格朗日平均纳维-斯托克斯方程及其在湍流建模中的应用

• 批准号: 0105004
• 负责人: Steve Shkoller
• 金额: $ 9.46万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0105004&HistoricalAwards=false
• 关键词: Lagrangian; Averaged; Navier; Stokes; Equations

0105004ShkollerThis three year research effort is founded upon the author's recent development of a novel Lagrangian averaging procedure for the Euler and Navier-Stokes equations of fluid dynamics. The method is based on expressing the exact Lagrangian fluid flow as a composition of a smooth deterministic volume-preserving Lagrangian flow and a rough stochastic flow consisting of volume-preserving near-identity transformations. This decomposition is asymptotically expanded about the identity, substituted into the variational principle, and then averaged. The resulting deterministic system of equations is termed the Lagrangian averaged Navier-Stokes (LANS) equations in the presence of viscosity, and the Lagrangian averaged Euler (LAE) equations in the ideal case when viscosity is absent. Both the LAE and LANS models are parameterized by a small spatial scale alpha, and are derived in such a fashion as to accurately reproduce the dynamics of the Euler and Navier-Stokes equations at spatial scales larger than alpha, while averaging (or homogenizing) the fluid motion at scales smaller than alpha. Unlike current approaches such as Reynolds averaged Navier-Stokes (RANS) or Large Eddie Simulation (LES) models which add artificial dissipation to the system to remove subgrid scales, the LAE/LANS equations preserve the underlying structure of the inviscid dynamics, namely, energy, helicity, and circulation, by instead using a geometric, nonlinear dispersive mechanism. As a result, our LANS model, unlike RANS or LES, does not artificially suppress intermittency, a fundamental feature of fluid turbulence. The resulting system is a set of dynamically coupled partial differential equations for the mean velocity field and covariance tensor. This system will be the backbone of a massive analytic and computational assault on the modeling and understanding of fluid turbulence.Although heavily studied by numerous researchers for over a century, incompressible fluid turbulence elusively remains one of the last great challenges of modern scientific exploration. Its understanding is of paramount importance in a wide range of engineering and physical applications, ranging from the design of airplanes and automobiles, to daily weather forecasts and global climate prediction. Roughly speaking, a flow becomes turbulent when all of the spatial scales in the fluid are activated, or in other words, when the fluid is moving so chaotically, as to create smaller and ever smaller vortices. In such a flow regime, the trajectory of each fluid particle appears unpredictable, yet the challenge is to derive a mathematical set of equations which can describe this unpredictable motion. About 150 years ago, the Navier-Stokes equations were introduced for this very purpose, and although it is now generally accepted that these equations do indeed provide a remarkable physical model of reality, it remains a mathematical mystery as to whether or not unique solutions to these equations exist for all time. Moreover, even numerical approximations of these equations on the world's fastest supercomputers are incapable of modeling the small-scale structures and patterns which are formed in a turbulent regime -- the computer simply runs out of memory long before it can simulate the prohibitively small vortical motion. The LANS model, described above, is intended to alleviate these fundamental difficulties, and make the computational simulation of turbulent flows feasible.

0105004SHKOLLERTHIS这三年的研究工作是在作者最近开发的，用于为Euler和Navier-Stokes方程式的流体动力学方程式开发。 该方法基于表达精确的拉格朗日流体流量作为平滑确定性量的拉格朗日流的组成，以及由容量具有近近置转换组成的粗糙随机流。 这种分解是渐近地扩展了围绕身份的扩展，被替换为变分原理，然后平均。 在粘度的存在下，所得的方程式确定性系统称为Lagrangian平均Navier-Stokes（LAN）方程，而在不存在粘度时，在理想情况下，Lagrangian平均欧拉（LAE）方程在理想情况下。 LAE和LAN模型均通过小空间尺度α进行了参数化，并且以一种方式得出的方式可以准确地在比Alpha的空间尺度上准确地重现Euler和Navier-Stokes方程的动力学，而平均（或平均（或匀浆））流体运动小于Alpha的流体运动。 与当前的方法不同，诸如Reynolds平均Navier-Stokes（RANS）或大型Eddie模拟（LES）模型，这些模型添加了人造耗散以消除亚网格尺度，LAE/LANS方程可维护不粘性动态的基本结构，即使用，能量，螺旋，螺旋，螺旋和循环，而不是使用一个几何形式，而不是循环。 结果，与Rans或LE不同，我们的LAN模型不会人为地抑制间歇性，这是流体湍流的基本特征。所得系统是一组平均速度场和协方差张量的动态耦合部分微分方程。 该系统将是对流体湍流的建模和理解的大规模分析和计算攻击的骨干。尽管许多研究人员在一个世纪以来一直在大量研究中进行了大量研究，但不可压缩的流体湍流仍然是现代科学探索的最后一个巨大挑战之一。 在广泛的工程和物理应用中，其理解至关重要，从飞机和汽车的设计到日常天气预报和全球气候预测。 粗略地说，当流体中的所有空间尺度都被激活时，或者换句话说，流体在混乱的过程中移动时，流动变得湍流，以创造出较小且越来越小的涡旋。 在这样的流程中，每个流体粒子的轨迹看起来不可预测，但挑战是得出一组数学方程，可以描述这种不可预测的运动。 大约150年前，为此目的引入了Navier-Stokes方程，尽管现在人们普遍认为，这些方程确实确实提供了一个非凡的现实物理模型，但仍然存在数学上的谜，即是否一直存在这些方程式的独特解决方案。 此外，即使是世界上最快的超级计算机上这些方程式的数值近似值也无法建模在湍流方向上形成的小规模结构和模式 - 计算机很早就可以模拟了过度的小型涡流动作。 上述LANS模型旨在减轻这些基本困难，并使湍流的计算模拟可行。