## 3D Flow Around a Confined Square Cylinder: Streak Line Vector Field

We provide a time step of the streak line vector field for a 3D time-dependent flow around a confined square cylinder. It has been computed as described in the paper *Streak Lines as Tangent Curves of a Derived Vector Field* and used there for visualizing streak surfaces.

The underlying original flow has been obtained from a direct numerical Navier Stokes simulation by Simone Camarri and Maria-Vittoria Salvetti (University of Pisa), Marcelo Buffoni (Politecnico of Torino), and Angelo Iollo (University of Bordeaux I).

## 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 3D time-dependent flow field has three components u,v,w and is defined in space-time (x,y,z,t), i.e., in a 4D domain. Its corresponding streak line vector field is defined in a 5D domain (x,y,z,t,τ). It has 5 components where two of them are constant. A single time step of such a streak line vector field lives in the 4D domain (x,y,z,τ) and its t-component can be discarded, i.e., it can be treated as a 4D vector field.

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

## Visualizations

Here we show two streak surfaces in the flow around the square cylinder. They have been integrated in forward and backward direction from the depicted seeding lines. We computed them using the **streak line vector field**, which describes streak lines of the original flow as tangent curves. This allows us to compute the streak surfaces as stream surfaces in the derived streak line vector field. The user is able to manipulate the seeding line and gets almost instant feedback in the form of a fully developed streak surface. This way, our new approach provides an interesting, orthogonal alternative to the classic variant for computing streak surfaces: whereas the classic computation scheme focuses on showing the evolution of streak surfaces, the streak line vector field provides quasi-instant results at any given time step. Note that computing a streak surface as a stream surface in the streak line vector field is considerably faster, since one only has to check the front line for adaptivity, whereas the classic approach for streak surfaces requires to check the whole surface for adaptivity.

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 time step of a 5D streak line vector field of a 3D time-dependent flow. We treat this time step as a 4D vector field: it consists of three non-constant components w_{1},w_{2},w_{3} and the constant τ-component -1. Only the non-constant components are saved in the files.

The chosen time step is t = 100.099.

### Resolution

The data set is given on a 4D uniform grid with the following specifications:

**Grid:**205 x 80 x 60 x 204 (number of grid points in x,y,z,τ-direction)**Bounding Box:**[-0.5, 20] x [-4, 4] x [0, 6] x [-60.0985, 61.9015] (extents in x,y,z,τ-direction)

### Data Format

Each τ-step is written as a single file in AmiraMesh format. The accompanying file `SquareCylinder.t100.099.fileseries` lists the files in the correct order from τ_{min} to τ_{max}.

### Additional Files

The surface of the cylinder itself is provided together with the underlying original flow field.

## How to Acknowledge

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

```
This streak line vector field is courtesy of Weinkauf and Theisel \cite{weinkauf10c}.
The underlying original flow field has been simulated by Camarri et al.\ \cite{camarri05}.
```

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/}
}
@INPROCEEDINGS{camarri05,
author = {S.~Camarri and M.-V.~Salvetti and M.~Buffoni and A.~Iollo},
title = {Simulation of the three-dimensional flow around a square cylinder between parallel walls at moderate {Reynolds} numbers},
booktitle = {{XVII Congresso di Meccanica Teorica ed Applicata}},
year = {2005}
}
```