Time‐Dependent 2‐D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures

Abstract

This paper presents an approach to a time-dependent variant of the concept of vector field topology for 2-D vector fields. Vector field topology is defined for steady vector fields and aims at discriminating the domain of a vector field into regions of qualitatively different behaviour. The presented approach represents a generalization for saddle-type critical points and their separatrices to unsteady vector fields based on generalized streak lines, with the classical vector field topology as its special case for steady vector fields. The concept is closely related to that of Lagrangian coherent structures obtained as ridges in the finite-time Lyapunov exponent field. The proposed approach is evaluated on both 2-D time-dependent synthetic and vector fields from computational fluid dynamics.

Time Line Cell Tracking for the Approximation of Lagrangian Coherent Structures with Subgrid Accuracy

“…In contrast, time lines crossing forward LCS and integrated in forward direction are subject to strong deformation (due to separation and/or shear). At the same time, Sadlo and Weiskopf [SW10] have shown that time lines seeded exactly on a saddle‐type LCS collapse to a single point. For the same reason, backward time lines have to align with dominant forward LCS structures in this case: If the (collapsed) saddle line is enclosed by time lines at time step t 0 , the same time lines have to enclose this structure at any other time step , since both structures are guaranteed to be material structures, i.e.…”

“…In literature, the relation of streak lines (and streak surfaces) to LCS has been discussed in detail by Sadlo et al . [SW10] and Üffinger et al . [USE12].…”

The State of the Art in Topology‐Based Visualization of Unsteady Flow

“…Sadlo and Weiskopf [SW10] present an approach to time‐dependent 2D VFT based on generalized streak lines , i.e., streak lines with a moving instead of a fixed seed point. This allows them to give a generalized definition of saddle‐type critical points for unsteady flow.…”

Lagrangian Coherent Structures with Guaranteed Material Separation

“…Shadden et al [SLM05] have shown that ridges of FTLE are approximate material structures, i.e., they converge to material structures for increasing integration times. This fact was used in [SW10] to extract topology‐like structures and in [LM10] to accelerate the FTLE computation in 2D flows. Also in the visualization community, different approaches have been proposed to increase performance, accuracy and usefulness of FTLE as a visualization tool [SP09, GLT*09, GGTH07, SP07, SRP11, KPH*09].…”