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Next: Rendering Up: Visualizing characteristic trajectories Previous: Introduction


A thread of streamlets

To come up with a useful technique of locally enhanced stream lines, we propose a model for the generation of a thread of streamlets. Near a stream line of interest  $\mathcal{T}_\mathbf{s}$ (the base trajectory) many short streamlets are placed. Thereby a continuous representation of the system's behavior in the vicinity of the base trajectory is approximated.
  
Figure 7.1: Relation between streamlet density ( $\mathit{no}$), streamlet integration length ( $\mathit{len}$), and streamlet instantiation interval ( $\mathit{dt}$).
\framebox[\textwidth]{
\centerline{\includegraphics[scale=1.2]{figs/model-a.eps}}
}

Using constant flow as a reference model - stream lines are straight lines in this case - the thread of streamlets  $\left\{\mathcal{T}_{\mathbf{s}_i}\right\}$ is defined as follows: Any cross-section perpendicular to base trajectory  $\mathcal{T}_\mathbf{s}$ is pierced by a constant number ( $\mathit{no}$) of streamlets. Using integration time t as parameterization of the base trajectory ( $\mathcal{T}_\mathbf{s}(0)=\mathbf{s}=\textnormal{seed point of }\mathcal{T}_\mathbf{s}$), streamlets  $\mathcal{T}_{\mathbf{s}_i}$ are instantiated at time  $t_i=i\!\cdot\!\mathit{dt}$ and integrated over the time interval  $\left[i\cdot\mathit{dt}\pm\mathit{len}/2\right]$. See Fig. 7.1 for an illustration of the relationship between $\mathit{no}$, $\mathit{dt}$, and $\mathit{len}$, i.e.,  $\mathit{dt}=\mathit{len}/\mathit{no}$. Seed points  $\mathcal{T}_{\mathbf{s}_i}(t_i)\!=\!\mathcal{T}_{\mathbf{s}_i}(i\!\cdot\!\mathit{dt})$ of newly instantiated streamlets are randomly chosen within a perpendicular cross-section through $\mathcal{T}_\mathbf{s}(i\cdot\mathit{dt})$ (reference location on the base trajectory) corresponding to a probability distribution function (PDF)  $d(\alpha ,r)$ (see Eq. 7.1 and Fig. 7.2).

 \begin{displaymath}d(\alpha,r) = \left\{ \begin{array}{rl}
D & \textrm{ if } 0<...
...f } qR<r\le{}R \\
0 & \textrm{ if } R<r
\end{array} \right.
\end{displaymath} (7.1)

PDF  $d(\alpha ,r)$ is defined by parameters R (the maximal distance between $\mathcal{T}_\mathbf{s}(t_i)$ and $\mathcal{T}_{\mathbf{s}_i}(t_i)$) and q$\in$[0,1). The latter parameter is used to define PDF d as a truncated cone. This shape provides the fade-out characteristic of the streamlet placement procedure with respect to the distance from  $\mathcal{T}_\mathbf{s}$. To guarantee that d is a PDF, $\int{}\!d(\alpha,r)\,\mathrm{d}\alpha\,\mathrm{d}r$ must equal to 1, i.e., the volume of the truncated cone must be 1. This constraint can be expressed as specification for parameter D:

\begin{displaymath}D = \frac{3}{(1+q+q^2)R^2\pi}
\end{displaymath}

In other words, Computing a thread of streamlets for the reference model ( $\dot{\mathbf{x}}$= const.), a bunch of line segments (streamlets  $\left\{\mathcal{T}_{\mathbf{s}_i}\right\}$) of equal spatial length ( $\mathit{len}\!\cdot\!\left\vert\dot{\mathbf{x}}\right\vert$) is generated. It this case of constant flow the streamlets are parallel to the base trajectory which is a straight line itself. The seed points of the streamlets, i.e., $\left\{\mathbf{s}_i\right\}$= $\left\{\mathcal{T}_{\mathbf{s}_i}(t_i)\right\}$, are determined according to the PDF  $d(\alpha ,r)$. For any time t the cross-section perpendicular through $\mathcal{T}_\mathbf{s}(t)$ is pierced by exactly $\mathit{no}$= $\mathit{len}/\mathit{dt}$ streamlets.
  
Figure 7.2: Probability density function  $d(\alpha ,r)$ for the instantiation of streamlets based on a perpendicular cross-section through the base trajectory.
\framebox[\textwidth]{
\centerline{\includegraphics[scale=1]{figs/d-2.eps}}
}

Applying this model to real (usually non-constant) flow data, local flow characteristics are visualized through the following variations from the constant flow reference setup:


  
Figure 7.3: Visualizing the flow near a linear node repellor in 3D:  (a) eigenvectors and eigenvalues,  (b) characteristic trajectories plus threads of streamlets.
\framebox[\textwidth]{
\begin{tabular*}{.93\linewidth}{@{}@{\extracolsep{\fill}...
...ht=61mm]{pics/realfp.2e.ps}
\\ {\small{}(a)}
& {\small{}(b)}
\end{tabular*} }

Taking a linear node repellor with eigenvalues 1, 10, and 100 as example, the flow characteristics in the vicinity of this critical point can be visualized in different ways (see Fig. 7.3). Using threads of streamlets for a visualization of the characteristic trajectories - those which are aligned with the eigenvectors of the critical point's Jacobian matrix - a dense and intuitive representation of the 3D flow near the critical point is generated. Through the threads of streamlets (Fig. 7.3(b)) the flow next to the characteristic trajectories is visualized. A purely abstract notation (Fig. 7.3(a)) encodes the eigenvectors of the Jacobian matrix and the magnitudes of the associated eigenvalues. No direct information about the vicinity of the characteristic trajectories is provided.


next up previous contents
Next: Rendering Up: Visualizing characteristic trajectories Previous: Introduction
Helwig Löffelmann, November 1998,
mailto:helwig@cg.tuwien.ac.at.