Lagrangian aspects of turbulent superstructures: numerical analysis of long-term dynamics and transport properties

湍流流体流动通常表现出大规模的、长寿命的相干结构，称为湍流超结构。在湍流剪切流动中，上层结构的特征是沿流速度波动的细长区域，而在湍流对流中，上层结构的特征是时间平均速度或温度场的模式。虽然在欧拉参考系中可以清楚地观察到，但这些上层结构的起源和动力学以及它们对湍流性质和输运过程的影响还远远没有被理解。为此，需要在欧拉规范之外对湍流上层结构进行数学描述。在第一个资助期，基于轨迹的方法已经开发出来，用于研究拉格朗日参照系中的相干行为，使用来自动力系统和遍历理论以及数据科学的概念。在项目的第二阶段，我们将继续开发和应用计算方法来近似和表征三维湍流中的拉格朗日超结构。我们的重点将放在(i)上层结构的长期动力学分析，包括分岔的预测，（ii）尝试系统地理解欧拉和拉格朗日上层结构之间的关系，以及（iii）表征它们的输运性质。在优先计划内与我们的跨学科项目伙伴合作，我们将研究瑞利-巴萨纳德对流和湍流剪切流中湍流上层结构的拉格朗日方面。

Turbulent fluid flows often exhibit large-scale, long-lived coherent structures, termed turbulent superstructures. In turbulent shear flows superstructures are characterized by elongated regions of streamwise velocity fluctuations, whereas in turbulent convection they are identified as patterns in the time-averaged velocity or temperature fields. While clearly observable in the Eulerian frame of reference, the origin and dynamics of these superstructures and their impact on turbulent flow properties and transport processes are far from being understood. To this end, there is a need for mathematical descriptions of turbulent superstructures beyond the Eulerian specifications.In the first funding period, trajectory-based methods have been developed for the study of coherent behavior in the Lagrangian frame of reference, using concepts from dynamical systems and ergodic theory as well as data science. In the second phase of the project, we will continue to develop and apply computational approaches to approximate and characterize Lagrangian superstructures in 3D turbulent flows. Our focus will be on (i) the analysis of the long-term dynamics of superstructures including the prediction of bifurcations, (ii) attempting a systematic understanding of the relation between Eulerian and Lagrangian superstructures, and (iii) a characterization of their transport properties. In cooperation with our interdisciplinary project partners within the Priority Programme, we will study the Lagrangian aspects of turbulent superstructures both in Rayleigh-Bénard convection and in turbulent shear flows.