Multi-scale geometry of Lagrangian and vortex-surface fields in turbulence

研究人员和他的同事们应用基于曲波的多尺度几何（CBMSG）方法来识别，表征和分类的尺度相关的几何形状内的三维，不断发展的拉格朗日标量和涡面标量场在几个不同的湍流。CBMSG方法首先将给定场分解为尺度相关的分量场.然后，从每个字段中提取结构的等值面和每个结构的几何形状的特征在于由一组有限的几何参数获得的联合概率分布的表面形状指数和曲率的时刻的函数。这允许在可视化空间中将每个尺度下的结构集合的图形描绘为点云。该空间内的点的密度提供了关于嵌入在原始场中的尺度依赖结构的统计几何形状的信息。研究的一个重要组成部分是发展方法，用于数值模拟拉格朗日标量场的时间演变，因为它们被湍流速度场对流，变形和拉伸。这是通过一种新的粒子反向跟踪方法来实现的，该方法直接构造时间向后或反向拉格朗日映射。 具体而言，研究内容包括均匀湍流的高分辨率拉格朗日标量场演化的几何学研究，槽道湍流的欧拉场和拉格朗日场的统计几何学研究，以及无粘和粘性流体流的涡面场跟踪方法的发展。自然界中物体的形状或几何学往往在其行为中起着至关重要的作用。 鸟类或昆虫翅膀的形状，生物细胞或复杂分子的详细形状，或树木的组织成一个大树干，较小的树枝和树叶，都与它们的特殊功能有关。这种几何形状可以是变化的，而不是静止的;例如，包括水或其他流体运动的“漩涡”长期以来一直被认为具有可重复的形状，如在云的形成中，在破碎的海浪中以及在火箭排气或海底漏油的波涛和折叠形状中所看到的。通常，这种复杂的自然几何形状不能简单地用一个简单的形状来理解，而必须理解为相互关联但不同形式的混合物。 树和云就是很好的例子。 本研究的目的是开发定量的，统计的方法来表征复杂的三维几何形状的涡流，包括湍流的流体流动。 所采用的方法和技术来源于现代计算应用数学。直接的实际应用是，研究结果将为发展先进的计算方法提供信息，用于在广泛的工业和环境设置中对湍流进行数值预测，包括污染物和气候建模。虽然本发明的焦点在于包括流体动力学湍流的涡流的几何形状，但是所开发的方法原则上对任何上述说明性示例具有更普遍的适用性。这项研究预计将导致在其他领域的科学和工程的可视化和减少复杂的“有机”几何嵌入在巨大的数据库仍然是一个具有挑战性的问题的影响。

The investigator and his colleagues apply a curvelet-based, multi-scale geometric (CBMSG) methodology to the identification, characterization and classification of scale-dependent geometry within three-dimensional, evolving Lagrangian-scalar and vortex- surface scalar fields in several differing turbulent flows. The CBMSG methodology first decomposes the given field into scale- dependent component fields. Structures from each field are then extracted by iso-surfacing and the geometry of each structure is characterized by a finite set of geometrical parameters obtained as functions of moments of the joint probability distributions of the surface shape-index and curvedness. This allows pictorial depiction of the set of structures at each scale as a cloud of points in a visualization space. The density of points within this space provides information on the statistical geometry of scale-dependent structures embedded within the original field. An important component of the research is the development of methodologies for numerically simulating the time-wise evolution of Lagrangian scalar fields as they are convected, deformed and stretched by turbulent velocity fields. This is achieved using a novel particle backward-tracking method that constructs directly the time-backward or reverse Lagrangian map. Specifically, the research includes study of the geometry of high-resolution, Lagrangian-scalar-field evolution for homogeneous turbulence, the statistical geometry of both Eulerian and Lagrangian fields for turbulent channel flows and the development of a methodology for tracking vortex-surface fields for inviscid and viscous fluid flow.The shapes or geometry of objects in nature often plays a crucial role in their behavior. The form of a bird or insect wing, the detailed shape of a biological cell or of a complex molecule or the organization of a tree into a large trunk, smaller branches and leaves are all related to their special function. This geometry can be changing and not static; for example the ``eddies'' comprising water or other fluid motion have long been recognized to have repeatable shapes as can seen in cloud formation, in breaking sea waves and in the billowing and folding shapes of a rocket exhaust or an oil leak from the sea floor. Typically, this complex natural geometry cannot be easily perceived in terms of a single simple shape but must be understood as an amalgam of interconnected but different forms. Trees and clouds are good examples. The aim of this research is to develop quantitative, statistical methods for characterizing the complex three-dimensional geometry of the eddies that comprise turbulent fluid flow. The methods and techniquesused come from modern computational applied mathematics. The immediate practical application is that the results will inform the development of advanced computational methods for the numerical prediction of turbulent fluid flows in a wide range of industrial and environmental settings, including pollutant and climate modeling. While the present focus is on the geometry of eddies that comprise fluid-dynamic turbulence, the methods developed have more general applicability, in principle to any of the above illustrative examples. This research is expected to lead to impact in other areas of science and engineering where the visualization and reduction of complex ``organic'' geometry embedded within huge data bases remains a challenging problem.