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

0105004 Shkoller这三年的研究工作是建立在作者最近开发的一种新的拉格朗日平均程序的欧拉和Navier-Stokes方程的流体动力学。 该方法是基于表达的精确拉格朗日流体流作为一个组成的一个光滑的确定性体积保持拉格朗日流和一个粗糙的随机流组成的体积保持近恒等变换。 这个分解是渐近展开的身份，代入变分原理，然后平均。 由此产生的确定性系统的方程被称为拉格朗日平均Navier-Stokes（局域网）方程的存在下的粘性，和拉格朗日平均欧拉（LAE）方程在理想情况下，当粘性不存在。 LAE和局域网模型都由小的空间尺度α参数化，并且以这样的方式导出，以便在大于α的空间尺度上准确地再现欧拉和纳维-斯托克斯方程的动力学，同时在小于α的尺度上平均（或平均）流体运动。 与目前的方法，如雷诺平均的Navier-Stokes（RANS）或大埃迪模拟（LES）模型，增加人工耗散的系统，以消除亚网格尺度，LAE/局域网方程保留的基本结构的无粘动力学，即能量，螺旋度，和环流，而不是使用一个几何，非线性色散机制。 因此，我们的局域网模型，不像RANS或LES，不人为地抑制湍流，流体湍流的基本特征。由此产生的系统是一组动态耦合的平均速度场和协方差张量的偏微分方程。 该系统将成为对流体湍流建模和理解的大规模分析和计算攻击的支柱。尽管世纪以来许多研究人员对不可压缩流体湍流进行了大量研究，但不可压缩流体湍流仍然是现代科学探索的最后一个重大挑战之一。 它的理解在广泛的工程和物理应用中至关重要，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，当流体如此混乱地移动时，流动变得湍流，从而产生越来越小的漩涡。 在这样的流动状态下，每个流体粒子的轨迹似乎是不可预测的，但挑战是导出一组数学方程，可以描述这种不可预测的运动。 大约150年前，Navier-Stokes方程就是为了这个目的而引入的，尽管现在人们普遍认为这些方程确实提供了一个了不起的现实物理模型，但这些方程是否存在唯一的解仍然是一个数学之谜。 此外，即使在世界上最快的超级计算机上对这些方程进行数值近似，也无法模拟在湍流状态下形成的小尺度结构和模式-计算机在能够模拟非常小的旋涡运动之前很久就耗尽了内存。 上述局域网模型旨在减轻这些基本困难，并使湍流的计算模拟可行。

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.