## 2D Flow Around a Cylinder: Streak Line Vector Fields

We provide streak line vector fields for a 2D time-dependent flow around a cylinder. 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.

## 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

### Exploration of Streak Lines and Surfaces

Our approach lends itself to a new interaction metaphor: given a spatio-temporal seeding location, the so-defined streak line can be computed more or less instantly, whereas the classic approach requires to develop the streak line over a one-parameter family of streak lines. Hence, our approach allows for a faster exploration of the space of streak lines. The figure below shows 5000 forward and backward integrated streak lines of the 2D time-dependent flow behind a cylinder. They have been computed in under a second in the τ-interval [0, ± 2.5]. The pre-computation time for the streak line vector field is 52 seconds. To compute the same set of streak lines with the classic approach requires 167 minutes. This clearly demonstrates one of the advantages of integrating streak lines in the streak line vector field.

5000 forward (red) and backward (blue) integrated streak lines of a time step in the 2D time-dependent flow around a cylinder. The streak lines are rendered such that they become less opaque the closer they get to their seeding position at τ = 0.

The following figure shows a LIC visualization of the first two components of the streak line vector field at different τ-values for the cylinder data set. These are its two non-constant components. Note that the LIC does not show streak lines, but rather the local direction of the streak lines at a given τ. One can think of the depicted lines as being built up of instantaneous streaklets. The vortices of the von Kármán vortex street are revealed by this: longer integrated parts of a streak line roll up in the vortices of this flow. Hence, the streak lines have a stronger rotational behavior for larger τ-values, which is encoded in the non-constant components of the streak line vector field and exposed in this LIC visualization.

With increasing τ from left to right, the first two components of the streak line vector field visualized using LIC reveal the von Kármán vortex street of the cylinder data set.

### Feature Extraction and Analysis for Streak Lines

The description of streak lines as tangent curves of a derived vector field allows us to apply feature extraction and analysis tools to streak lines that were previously only available for stream and path lines. The following figure shows volume renderings of the velocity magnitude and the curvature of streak lines. These scalar quantities can be computed just by considering the derivatives of the streak line vector field. Integrating the streak lines themselves is not required. Both examples exhibit the rolling up of the vortices in the von Kármán vortex street.

Derived scalar fields of the streak lines of the 2D time-dependent flow around a cylinder (gray tube). Shown is the (x,y,τ)-domain for a fixed time step. Two parts of the domain are not defined (front-left and back-right), since the path lines left the space-time domain during the computation of the streak line vector field. Left: Velocity magnitude of the streak lines between τ = (-3,3). Right: Curvature of the streak lines between τ = (-1.5,1.5).

## 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 time-resolved version corresponding to a period of vortex shedding in the von Kármán vortex street. It is given on a 4D uniform grid with the following specifications:

• Grid: 400 x 80 x 37 x 501 (number of grid points in x,y,t,τ-direction)
• Bounding Box: [1, 5] x [-0.4, 0.4] x [18, 18.6] x [-2.5, 2.5] (extents in x,y,t,τ-direction)
• A high resolution version of a single time step (t = 19). A single time step suffices to compute streak lines and derived quantities for that particular time step. It is given on a 3D uniform grid with the following specifications:

• Grid: 750 x 146 x 801 (number of grid points in x,y,τ-direction)
• Bounding Box: [0, 7] x [-0.487, 0.487] x [-2.8, 2.8] (extents in x,y,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.

## 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:

These streak line vector fields are courtesy of Weinkauf and Theisel \cite{weinkauf10c}.
The underlying original flow field has been simulated
using the Free Software \emph{Gerris Flow Solver} \cite{gerrisflowsolver}.


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/}
}

@ARTICLE{gerrisflowsolver,
author = {S. Popinet},
title = {Free Computational Fluid Dynamics},
journal = {ClusterWorld},
year = {2004},
volume = {2},
number = {6},
url = {http://gfs.sf.net/}
}