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

研究者和他的同事们将基于曲线的多尺度几何（CBMSG）方法应用于几种不同湍流中三维、不断演变的拉格朗日标量场和涡旋表面标量场中尺度相关几何的识别、表征和分类。CBMSG方法首先将给定的域分解为与尺度相关的分量域。然后通过等曲面提取每个场的结构，并通过表面形状指数和曲率度联合概率分布矩的函数获得有限几何参数集来表征每个结构的几何特征。这允许在可视化空间中以点云的形式对每个尺度的结构集进行图形化描述。该空间内的点密度提供了原始域内嵌入的尺度相关结构的统计几何信息。该研究的一个重要组成部分是发展数值模拟拉格朗日标量场在湍流速度场的对流、变形和拉伸作用下随时间演变的方法。这是通过一种新的粒子反向跟踪方法来实现的，该方法直接构建了时间向后或反向拉格朗日映射。具体来说，研究包括高分辨率几何的研究，均匀湍流的拉格朗日-标量场演化，湍流通道流动的欧拉场和拉格朗日场的统计几何，以及跟踪无粘和粘性流体流动的涡表面场的方法的发展。自然界中物体的形状或几何形状往往在它们的行为中起着至关重要的作用。鸟或昆虫翅膀的形状，生物细胞或复杂分子的详细形状，树木形成大树干、小树枝和树叶的组织都与它们的特殊功能有关。这种几何形状可以改变，而不是静态的；例如，由水或其他流体运动组成的“漩涡”长期以来一直被认为具有可重复的形状，如云的形成、破碎的海浪、火箭排气或海底漏油的翻腾和折叠形状。通常，这种复杂的自然几何形状不能简单地理解为单一的简单形状，而必须理解为相互联系但不同形式的混合体。树和云就是很好的例子。本研究的目的是发展定量的统计方法，以表征包含湍流流体流动的漩涡的复杂三维几何形状。所使用的方法和技术来自现代计算应用数学。直接的实际应用是，这些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流体流动进行数值预测，包括污染物和气候建模。虽然目前的重点是包含流体动力学湍流的涡流的几何形状，但所开发的方法原则上对上述任何说明性示例都具有更普遍的适用性。这项研究预计将对科学和工程的其他领域产生影响，在这些领域，嵌入在庞大数据库中的复杂“有机”几何图形的可视化和简化仍然是一个具有挑战性的问题。

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.