Investigation of some of the clusters in Fig. 2. The top row shows the actual selections, while the bottom row gives the associated path lines in their 3D context (color coding according to temporal evolution from yellow to red). For the discussion of the figures, we refer to the main text (Sec. 4.2). 

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... data sets. The full output of the statistical evaluation can found in the supplemental material. A remark on time- vs. arc length-parametrization We observe that the arc length-parametrized data set allows a representation with the same number of factors, or fewer, following Kaiser’s criterion or the scree plot test. With the 100% proportion criterion, no clear trend is observable. It is worthwhile noticing that an arc length parametrization represents the geometry of the path line more faith- fully, but lacks information on the dynamics (uniform speed with respect to arc length!). Hence, the trend to be expressible by fewer factors may actually originate for that fact that this representation causes an information loss. On the other hand, we see that the shape descriptors inv perform well under both parametrization. Hence, we may conclude that the geometry-wise advantage of an arc length parametrized data set is too small to outweighed the possible risk of information loss. After we determined both dimensionality and representative attributes, we now demonstrate how an interactive visual flow analysis based on our findings can look like. First, we describe the framework used. Then we analyze two different data sets. Both data sets have been investigated in previously published work, which allows us to assess the results we achieve. The framework used for this paper is the SimVis software [4]. This software is an interactive visual analysis environment, tailored to meet the special requirements of computational fluid dynamics. Apart from multiple linked views, consisting of different information visualization views (e.g., histograms, scatter plots, parallel coordi- nated), the system provides a passive 3D view for focus+context visualization of the flow domain. Besides this, the frame work offers the opportunity to derive new flow attributes on the fly. For further details we refer to Doleisch’s paper on the SimVis software [4] and the references therein. One of the views, that makes the framework especially useful in the context of path line attributes, is the curve view [15, 20]. The curve view is used to display large families of function graphs at once (cf., e.g., Fig. 7) plotting the function values against time. Lines are selected by brushing a certain value range for a specific time step. Functions with multiple components can be analyzed component-wise. This data set has been investigated by Lež et al. [17]. Their paper is not targeting the question of which attributes to choose for a interactive flow analysis, however, the authors suggest several attribute combinations that they found useful for the case study. These attribute combinations are start to end distance and path line length (arc length), maximum velocity and mean velocity along path line, and maximal curvature and maximal torsion. All those attributes are constant along the path lines. As also mentioned in the original paper by Lež et al., one of the goals in the design of exhaust manifolds is the decrease of flow resistance (so-called back pressure). Hence, the detection of path lines/particles causing back pressure is a natural task in this context. In order to make the visual analysis based on the different attribute sets comparable, we identify an analytically defined set of path lines that we try then try to retrieve using both the original variable combinations and the here proposed attribute set. We restrict the analysis to particles seeded in the middle tube and its imminent vicinity. We identify particles possibly causing back pressure as those which 1) move upstream (i.e. max t ( pos x ( t 0 ) − pos x ( t )) > 0, x denoting the axis aligned with the stream and assuming the stream to have positive sign) and 2) are upstream from the middle pipe at some point in time (i.e. max t ( pipeboundary x − pos x ( t )) > 0, with pipeboundary x being the position on the x-axis where the inflow pipe is connected to the outlet and under the same assumptions as before). Hence, the path lines in question are those where both parameters are positive. See Fig. 1 for an overview over the path lines identified and their seeding positions. Obviously, an ad hoc analytic definition of interesting path lines is only possible in rela- tively clear and intuitive situation as this. We use this for the sake of comparability only. First, we investigate the attribute combination start to end distance and path line length. We have preselected particles in the middle pipe and its immediate vicinity. In Fig. 2 we see a scatter plot of the two attributes. The red dots represent the path lines to investigate, the yellow dots give the context (i.e. the remaining path lines). In the scatter plots opacity scaling according to point density is used. In their paper, Lež et al. suggest investigating "unusual clusters", and we can visually identify several of them. We select those clusters one after the other and monitor the path lines associated to them (Fig. 3). We see that none of the visually distinguishable main clusters gives satisfactory results: on the one hand the clusters in Fig. 3(a) and Fig. 3(b) describe the same path line behavior, the cluster in Fig. 3(c) contains both path lines we are interested in (left branch) as well as path lines that seem not to be associated with back pressure (lighter path lines in the right branch). Further refinement of the query could help this, but no visual clues on how to do this are present in the scatter plot. The next attribute combination investigated is maximum velocity and mean velocity along the path line. Fig. 4 shows a scatter plot of these two variables, the colors have the same meaning as before. In this case the visual detection of unusual clusters is harder. The most apparent abnormality seems to be the high share of path lines in question in the center of scatter plot. As Fig. 6(a) shows these path lines are indeed associated with the behavior we want to track. However, we systematically miss out on path lines seeded in a specific region (marked up with the circle). Finally, we investigate the combination of maximum curvature and maximum torsion along the path line (see Fig. 5 for the respective scatter plot). Here, no clusters are visible. This means we would have to rely on thresholding. This thresholding gives, however, not the desired results, as seen in Fig. 6(b). Choosing a higher threshold refines the selection, but it fails to discriminate different types of flow behavior (Fig. 6(c)). As a summary, we conclude that, following the state-of-the-art approach as described by Lež et al., we could find only a part of the path lines targeted. Now we use the attribute set suggested by our statistical analysis. As remarked earlier, all of these attributes are time series. Hence, we make extensive use of the curve view. First we look at the stream- wise position (in the same sense as used earlier). As in the first investigation, we select particles that originate from the middle pipe and its vicinity (Fig. 7(a) top). In order to cause back pressure, particles have to still be in the pipe, at the next stroke of the engine. Hence, we discharge ("not-selection") particles that are in the outlet at the time step the next stroke occurs (selection in Fig. 7(b) top). In the top of Fig. 7(c) we see the path lines corresponding to this selections. The particles that move "upstream" exhibit the same pattern as the once found by the analytic definition. However, our selection is, at the current point, still containing a number of path lines with clearly different (so to say "correct") flow behavior. Hence, we move to a different attribute to refine our selection. In the bottom of Fig. 7(a) we see the time series for the second quadratic statistical invariant (in the following: inv2). We see (at least) two clearly distinguishable patterns: path lines with a medium-high value of inv2 in the beginning of the time series, and others with a rather low value. We select the ones with the higher values and see (cf. Fig. 7(c) bottom) that now nearly all path lines exhibit the expected behavior. A small number of path lines is not of the expected type, representing particles being sucked in the rightmost tube. In fact, also the selected time series of inv2 have two branches (upper and lower, see Fig.7(b) bottom). As Fig.8 shows the two branches are indeed associated to the different types of path line behavior present. Comparing the seed points of the path lines found by our analysis to the seed points of the reference path lines, we see a clear correspondence of the two sets (Fig. 9, in contrast to the situation in Fig. 6(a) bottom). We see that the interactive flow analysis of the data set based on our suggestions is able to find the targeted path lines. In addition, the process is intuitive in the sense that different flow behavior is reflected by clearly distinguishable clusters in the attributes. We discussed our results with a domain expert, who confirmed the expressiveness of our results. This data set has been investigated by Pobitzer et al. [23] in the context of finite-time Lyapunov exponents (FTLE). One of the interesting features is a separation structure in front of the obstacle, separating particles passing at the two sides of the obstacle. Another structure is a recirculation zone in front of the obstacle. Due to its definition, the FTLE approach is not suitable to investigate the internal structure of the recirculation. We therefore investigate this question by means of interactive flow analysis of the attribute set proposed earlier in this paper. Since we want to target recirculation behavior, we preselect par- ticles seeded upstream from the obstacle. Since the recirculating particles do not pass the obstacle, we can assume that the distance from start to end is not too big. Hence, we exclude the path line cluster associated to high distances from our analysis, using a "not"- selection (Fig. ...