Velocity Gradient Statistics in Fully Developed Turbulence - Statistical Evolution Equations, DNS Investigations and Implications for Reduced Models

充分发展的小尺度水动力湍流可以用速度场的导数来全面表征，速度场的导数定义了速度梯度张量。在这个项目中，这些小尺度特征的统计特性是在精确统计方程的框架内研究的，这些方程控制着速度梯度张量的概率密度函数的演化。由于流体运动方程的非线性和非局部特性，当只考虑有限数量的空间点时，所得的统计方程是不闭合的。在项目中，由于这个封闭问题而丢失的信息将由直接数值模拟的数据提供。特别地，将研究局部自放大、耗散和非局部压力贡献的影响。目标是更深入地了解这些动态贡献对观察到的小尺度统计的影响。除了描述所研究的统计量的形状和演变外，研究结果将有助于为低维随机模型制定改进的闭包。这样，就在基础湍流研究和应用建模（例如风能转换领域）之间建立了自然联系。

The small scales of fully developed hydrodynamic turbulence can be comprehensively characterized in terms of the derivatives of the velocity field, which define the velocity gradient tensor. In this project, the statistical properties of these small-scale features are studied within the framework of exact statistical equations governing the evolution of the probability density functions of the velocity gradient tensor. Due to the nonlinear and nonlocal character of the equations of fluid motion, the resulting statistical equations appear unclosed when only a finite number of spatial points is considered. Within the project, the information missing due to this closure problem will be contributed by data from direct numerical simulations. In particular, the influence of the local self-amplification, dissipation and nonlocal pressure contributions will be studied. The goal is to gain a deeper understanding of the impact of these dynamical contributions on the observed small-scale statistics. Apart from characterizing the shape and evolution of the studied statistical quantities, the results will help to formulate improved closures for low-dimensional stochastic models. By this, a natural connection between fundamental turbulence research and applied modeling, e.g. in the field of wind energy conversion, is established.