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

California Institute Of Technology, Pasadena CA

Investigators

Abstract

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 techniques used 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.