## Beads Problem: Streak Line Vector Fields

We provide streak line vector fields for the 2D time-dependent Beads Problem flow. These vector fields have been computed as described in the paper Streak Lines as Tangent Curves of a Derived Vector Field and used there for visualizing and analyzing streak lines.

Wiebel et al.1 reported of a biofluid dynamic model where neither classic visualization methods such as LIC or path lines, nor feature extraction methods such as vector field topology or FTLE where able to detect an apparent attractor in the flow, i.e., a point in the flow where particles aggregate. Wiebel et al. used particle density to extract the attractor. An analytic variant of this flow has been formulated by Ronny Peikert:2

$$\vv(x, y, t) = \begin{pmatrix} -(y - \frac{1}{3} \sin(t)) - (x - \frac{1}{3} \cos(t))\\ (x - \frac{1}{3} \cos(t)) - (y - \frac{1}{3} \sin(t)) \end{pmatrix}$$

Since this is a time-periodic flow, the attractor can be found using the method of Shi et al.3; it is a path line with the following parametric description: $\xx(t) = \frac{1}{3} \left(\sin(t) + \cos(t), -\cos(t) + \sin(t)\right)^T$. We computed the streak line vector field for this flow and extracted cores of swirling streak lines from it. Both, streak line vector field and cores, are available for download below.

1 A. Wiebel, R. Chan, C. Wolf, A. Robitzki, A. Stevens, and G. Scheuermann. Topological Flow Structures in a Mathematical Model for Rotation-Mediated Cell Aggregation. In Proc. Topo-In-Vis 2009, Snowbird, Utah, U.S.A., 2009.

2 R. Peikert, 2009. private communication.

3 K. Shi, H. Theisel, T. Weinkauf, H. Hauser, H.-C. Hege, and H.-P. Seidel. Path Line Oriented Topology for Periodic 2D Time-Dependent Vector Fields. In Proc. Eurographics / IEEE VGTC Symposium on Visualization (EuroVis), pages 139-146, Lisbon, Portugal, 2006.

## General Remarks about Streak Line Vector Fields

Given a time-dependent flow field, its streak lines can be described as the tangent curves of the corresponding streak line vector field. Hence, a simple tangent curve integration can be used to obtain the streak lines of the original flow. This novel mathematical description opens the gates to a number of visualization and analysis tools that have been developed in our community, but were previously only available for stream and path lines. Not surprisingly, streak lines and surfaces can be computed almost instantly with a streak line vector field. Furthermore, it allows to extend known feature extraction and analysis tools to work with streak lines.

A 2D time-dependent flow field has two components u,v and is defined in space-time (x,y,t), i.e., in a 3D domain. Its corresponding streak line vector field is defined in a 4D domain (x,y,t,τ). It has 4 components where two of them are constant.

More information, including details on how to use such fields for computing streak lines, can be found in the paper: Streak Lines as Tangent Curves of a Derived Vector Field.

## Visualizations

### Cores of Swirling Streak Lines

We computed the cores of swirling streak lines using the streak line vector field and found the following: with decreasing τ, the streak line cores converge to the attractor. Note that streak line cores of a 2D time-dependent flow are 4D surfaces. Hence, we have to intersect them along τ to get a line in space-time.

In other words, our new method is able to detect the attractor reliably in contrast to various other feature extraction methods. The video shows the streak line core together with the attractor, as well as the cores of swirling stream and path lines. The latter two methods are clearly off the attractor, i.e., do not detect it. We omit the FTLE field since it is, interestingly, constant for this flow and does not reveal any features at all.

We believe that this is a very promising result, since the Beads problem is considered to be one of the major test cases for a successful approach to an unsteady flow topology. However, it has to be left to future investigations whether or not this extends to other challenging flows and streak line cores can serve as a basis for an unsteady flow topology.

More information can be found in the paper: Streak Lines as Tangent Curves of a Derived Vector Field.

## Technical Details

The data set describes a 4D streak line vector field of a 2D time-dependent flow and therefore it consists of two non-constant components w1,w2 and two constant components 0 and -1. Only the non-constant components are saved in the files.

### Resolution

Two different version are provided:

• A normal resolution version which is very well suited to compute and visualize streak lines. It is given on a 4D uniform grid with the following specifications:

• Grid: 32 x 32 x 101 x 141 (number of grid points in x,y,t,τ-direction)
• Bounding Box: [-2, 2] x [-2, 2] x [0, π] x [-π, π] (extents in x,y,t,τ-direction)
• A high resolution version which has been used to extract the streak line cores as shown in the paper. It is given on a 4D uniform grid with the following specifications:

• Grid: 50 x 50 x 101 x 100 (number of grid points in x,y,t,τ-direction)
• Bounding Box: [-0.75, 0.75] x [-0.75, 0.75] x [9, 15] x [0, -9] (extents in x,y,t,τ-direction)

### Data Format

Each time step is written as a single file in AmiraMesh format, i.e., a file represents a 3D (x,y,τ)-subspace. This makes it very easy to apply time-unaware visualization techniques to individual time steps. In particular, streak lines can already be computed using a single time step of a streak line vector field.

The cores of swirling stream, path, and streak lines are provided as well as the ground truth for the attractor. These files are written as SeedingLineSets. Note that the core of swirling streak lines is a surface in 4D. It is given here as a series of lines living in space-time and their τ-values are attached as data values.

## How to Acknowledge

You are free to use these data sets as long as you give proper acknowledgement. Please use a LaTeX snippet similar to the following:

The streak line vector fields as well as the extracted core lines
are courtesy of Weinkauf and Theisel \cite{weinkauf10c}.


with the following BibTeX entries:

@ARTICLE{weinkauf10c,
author = {T.~Weinkauf and H.~Theisel},
title = {Streak Lines as Tangent Curves of a Derived Vector Field},
journal = {IEEE Transactions on Visualization and Computer Graphics (Proceedings Visualization 2010)},
year = {2010},
volume = {16},
pages = {1225--1234},
number = {6},
month = {November - December},
abstract = {Characteristic curves of vector fields include stream, path, and streak
lines. Stream and path lines can be obtained by a simple vector field
integration of an autonomous ODE system, i.e., they can be described
as tangent curves of a vector field. This facilitates their mathematical
analysis including the extraction of core lines around which stream
or path lines exhibit swirling motion, or the computation of their
curvature for every point in the domain without actually integrating
them. Such a description of streak lines is not yet available, which
excludes them from most of the feature extraction and analysis tools
that have been developed in our community. In this paper, we develop
the first description of streak lines as tangent curves of a derived
vector field -- the streak line vector field -- and show how it can
be computed from the spatial and temporal gradients of the flow map,
i.e., a dense path line integration is required. We demonstrate the
high accuracy of our approach by comparing it to solutions where
the ground truth is analytically known and to solutions where the
ground truth has been obtained using the classic streak line computation.
Furthermore, we apply a number of feature extraction and analysis
tools to the new streak line vector field including the extraction
of cores of swirling streak lines and the computation of streak line
curvature fields. These first applications foreshadow the large variety
of possible future research directions based on our new mathematical
description of streak lines.},
keywords = {unsteady flow visualization, streak lines, streak surfaces, feature extraction},
url = {http://tinoweinkauf.net/}
}