Multi-scale, Geometrical Study of Eddy-structure in Turbulence

研究者和他的学生致力于开发一种新的计算数学框架，用于识别和表征湍流中的涡流结构。一般方法包括对湍流体积数据集进行多尺度分析，然后推导出感兴趣的结构及其几何特征。通过曲波变换进行多尺度分析。结构的导出是通过对不同尺度的体数据集等等高线来实现的。几何特征是基于形状指数和弯曲度的概率密度函数，就面积覆盖而言，与每个结构相关联。这允许结构集的全局特征，以及研究和比较包含在这个集合中的相关结构组。研究人员已经开发出了基本框架，该框架已经经过了基于再现合成“虚拟湍流”几何特征的验证测试。目前的工作重点是研究周期立方体内512^3直接数值模拟得到的标量湍流的几何结构。为了更好地适应湍流数据库的特性，在提取和分类层面上进一步完善和改进这一框架的工作正在进行中。该方法也被应用于K. Horiuti（日本名古屋）提供的相同湍流图像的256^3,512^3和1024^3数据集（以三种不同的分辨率）。所研究的湍流场包括标量耗散和由速度梯度衍生的量，如涡度，速度梯度张量的不变量，这些量的函数确定局部，涡度占主导地位的区域，以及压力的函数。该方法也将应用于非均匀湍流场，如湍流通道流。从莱昂纳多·达·芬奇（Leonardo Da Vinci）创作的漩涡流体流动的详细图像开始，到现在增强的计算机图形学和自然现象可视化的时代，人们对湍流流体流动的自然几何特征和理解一直很着迷。但是，尽管进行了深入的研究，湍流漩涡的结构和形态仍然难以捉摸。更好地理解这种结构，既可以阐明自然界的一个深刻奥秘，同时也可以为改进湍流流体流动的预测模型的发展提供坚实的基础，这些模型可以应用于许多不同的科学和工程领域，从银河系尺度到太阳系动力学、恒星形成和恒星内部动力学、太阳风、地球气候模型、环境流体动力学和工业和工程应用。本研究的动机是最近高保真数据库的可用性，这些数据库代表了通过密集的计算机模拟获得的非常详细和真实的湍流流场。研究者和他的学生将计算机科学领域的新模式识别技术与基于“多尺度”分析的新应用数学方法相结合，来研究这些数据。这项工作的意义在于，它将提供一种新的方法来分析极大的湍流流体流动数据场的潜在几何结构和内容。在这项研究中编制的计算机代码将通过论文和档案期刊的出版物公开提供。这应该允许潜在用户将在这项工作中开发的建模方法应用于从数值模拟和实验中获得的大型数据库。除了流体流动之外，还可以应用于任何一组连续油田。

The investigator and his students work to develop a new computational-mathematical framework for the identification and characterization of eddy-like structures in turbulence. The general methodology consists of a multi-scale analysis of a turbulence volume data set followed by the eduction of structures of interest and their geometrical characterization. The multi-scale analysis is performed through the curvelet transform.The eduction of structures is achieved by isocontouring the volume data sets for different scales. The geometrical characterization is based on the probability density functions of shape index and curvedness, in terms of area coverage,associated with each structure. This allows a global characterization of the set of structures as well as the study and comparison of relevant groups of structures contained within this set. The investigator has already developed the basic framework which has been subjected to validation testing based on reproducing the geometrical character of a synthetic ``virtual turbulence''. The work is presently focused on studying the geometrical structure of scalar turbulence obtained from 512^3 direct-numerical simulation in a periodic cube. Further development is in progress to refine and improve this framework at the extraction and classification levels in order to better adapt it to the properties of turbulence data bases. The methodology is also being applied to the 256^3, 512^3 and 1024^3 data sets of the same turbulence image (at three different resolutions) provided by K. Horiuti (Nagoya Japan). Turbulent fields under study include the scalar dissipation and quantities derivable from velocity gradients such as vorticity, invariants of the velocity-gradient tensor, functions of these quantities that identify local, vorticity-dominant regions, and functionals of the pressure. The methodology will also be applied to non-homogeneous turbulent fields such as those obtained from turbulent channel flow. From the time of Leonardo Da Vinci, who crafted detailed images of eddying fluid flow, the to present era of enhanced computer graphics and the visualization of natural phenomena, there has been an ongoing fascination with both characterizing and understanding the natural geometry of turbulent fluid flow.But despite intense study, the structure and morphology of turbulent eddies remains elusive. A better understanding of this structure should both elucidate one of nature's profound mysteries and at the same time provide a firm basis for the development of improved predictive models for turbulent fluid flow for application to many diverse areas of science and engineering ranging from the galactic scale, through solar-system dynamics, star formation and stellar interior dynamics, the solar wind, climate modeling of planet earth, to environmental fluid dynamics and industrial and engineering applications. The present research is motivated by the recent availability of high-fidelity data bases representing very detailed and realistic turbulent flow fields obtained from intensive computer simulation.The investigator and his students combine novel pattern-recognition techniques from the field of computer science with new applied-mathematical methods based on ``multi-scale'' analysis, to study these data. The significance of this work is that it will provide a new methodology for analysing the underlying geometrical structure and content of extremely large, turbulent fluid-flow data fields.The computer codes developed in this research will be made openly available, with documentation through publications in thesis dissertations and in archival journals.This should allow potential users to apply the modeling methodologies developed in this work to large data bases obtained from both numerical-simulation and experiment. Applications beyond fluid flows, to any set of continuous fields, are envisioned.