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

这位研究人员和他的同事们应用基于曲线的多尺度几何(CBMSG)方法来识别、表征和分类几种不同湍流中三维、演变的拉格朗日标量场和涡面标量场中依赖尺度的几何图形。CBMSG方法首先将给定场分解成尺度相关的分量场。然后通过等值面提取每个区域中的结构，每个结构的几何特征由有限的几何参数集表征，这些几何参数是曲面形状指数和曲率的联合概率分布的矩的函数。这允许将每种比例的结构集图形化地描述为可视化空间中的点云。该空间内的点密度提供了嵌入在原始场中的尺度相关结构的统计几何信息。这项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、变形和拉伸过程中随时间演变的方法。这是使用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反向或反向拉格朗日映射。具体地说，这项研究包括研究均匀湍流的高分辨率拉格朗日标量场的几何演化，湍流槽道流动的欧拉场和拉格朗日场的统计几何，以及无粘和粘性流体流动的涡面场跟踪方法的发展。鸟类或昆虫翅膀的形状，生物细胞或复杂分子的详细形状，或将树木组织成大树干、小树枝和树叶，都与它们的特殊功能有关。这种几何形状可以是变化的，也不是静态的；例如，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，如云的形成、海浪的破碎以及火箭尾气或海底漏油的翻滚和折叠形状。通常，这种复杂的自然几何形状不容易被理解为单一的简单形状，但必须理解为相互关联但不同形式的混合体。树和云就是很好的例子。这项研究的目的是开发定量的统计方法来描述构成湍流流动的涡旋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，以在广泛的工业和环境环境中进行湍流流动的数值预测，包括污染物和气候模拟。虽然目前的重点是组成流体动力湍流的涡旋的几何形状，但所开发的方法在原则上对上述任何说明性例子都具有更广泛的适用性。这项研究预计将在科学和工程的其他领域产生影响，在这些领域，嵌入巨大数据库的复杂“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.