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.
