Delocalized Unsteady Vortex Region Detectors

Raphael Fuchs1,2, Ronald Peikert3, Filip Sadlo3, Bilal Alsallakh2, Meister Eduard Groller1

1: Institute of Computer Graphics and Algorithms, TU Vienna, Austria2: VRVis Research Center for Virtual Reality and Visualization, Vienna, Austria

3: Computer Graphics Laboratory, ETH Zurich, SwitzerlandEmail: {raphael, groeller}@cg.tuwien.ac.at

{peikert,sadlo}@inf.ethz.chalsallakh@VRVis.at

Abstract

In this paper we discuss generalizations of instan-taneous, local vortex criteria. We incorporate in-formation on spatial context and temporal devel-opment into the detection process. The presentedmethod is generic in so far that it can extend anygiven Eulerian criterion to take the Lagrangian ap-proach into account. Furthermore, we present a vi-sual aid to understand and steer the feature extrac-tion process. We show that the delocalized detectorsare able to distinguish between connected vorticesand help understanding regions of multiple inter-acting vortex structures. The delocalized detectorsextract smoother structures and reduce noise in thevortex detection result.

1 Introduction

Recent research in the field of Lagrangian coherentstructures [8, 9, 17, 18] suggests that we need torefine our approach to understanding fluid behav-ior. Even though the local information has shownto be highly valuable when trying to understandthe nature of turbulent fluid movements, we needto look further and find ways to include informa-tion on temporal development and particle move-ment into the analysis. In this paper we show thatvortex feature extraction can retain the knowledgethat we have on local properties of the flow and stillinclude the Lagrangian perspective into the analy-sis.

The Lagrangian approach is based on taking thetrajectories of particles into account for analysis.We can think of the detectors presented in this pa-per as criteria where local detector responses are ac-cumulated along trajectories to achieve both spatial

and temporal coherency. The Lagrangian approachintroduces new questions into the analysis. Sincethe result of the Lagrangian vortex feature detectoris dependent on the length of the particle trajecto-ries1 analyzed, we get an additional parameter withsignificant impact on the results of the analysis. Weneed a way to control the length of the trajectorythat contributes to the vortex detector response. Re-cent publications have suggested this as an impor-tant open research question [18, 8, 7]. We presentan approach which allows to control this parameternon-uniformly using an interactive analysis view.

A problem mentioned by several publicationsdealing with Lagrangian coherent structures andparticle trajectories in general [6, 16, 19] is the factthat particle trajectories can quickly leave the sim-ulation domain (e.g., through an outlet). In thiscase we do not have enough information available togive a good accumulated detector response. The ap-proach of delocalized Eulerian detectors gives threeanswers to the problem of short trajectories: firstly,local criteria have been demonstrated to give reli-able results on their own, thus we are less depen-dent on having long trajectories available to gener-ate good results. Secondly, we allow to include theupstream information by using backwards integra-tion into the analysis to compensate for short par-ticle trajectories in forward time. And thirdly, bytaking the proportion of the unknown region intoaccount, the lack of information due to extremelyshort trajectories is included into the detection re-sult.The contributions of this paper are as follows:

• The extension (delocalization) of Eulerian vor-

1since trajectories are streamlines for steady data and pathlinesfor unsteady data we will speak of trajectories when the differenceis not relevant

VMV 2008 O. Deussen, D. Keim, D. Saupe (Editors)

tex criteria to extract coherent structures whichimprove on the features detected using λ2, Qor the swirl criterion.

• A method to interactively control the crucialintegration length parameter.

• An extension of trajectories to include up-stream information to deal with the problemof short particle trajectories.

• Comparison and evaluation of the results ac-cording to numerical issues, smoothness andseparation of vortices.

In the next section we discuss related work. Thethird section presents the basic concepts of delocal-ized feature detectors and a 2D view of particle tra-jectories to analyze the local detector responses. Inthe fourth section we present evaluation results.

2 Related Work

For an overview of feature based flow visualizationwe refer to Post et al. [13].

The finite time Lyapunov exponent (FTLE) canbe used to measure separation of trajectories intime-dependent flows. Sadlo and Peikert [17] ex-tract ridges from 3D FTLE. Garth et al. [7] presenta method for the direct visualization of 2D FTLEinformation. Recent work shows how height ridgesof the FTLE field [16] and and direct visualizations[7] can be computed efficiently.

Cucitore et al. [4] review Eulerian detectors (Q,λ2, swirl, and others) and suggest a non-local mea-sure of swirl based on trajectories to extract vor-tices. Jiang et al. [10] search for trajectories rotat-ing about a common axis to verify the existence ofa vortex, while Sadarjoen and Post [15] computecurvature centers of trajectories. In earlier workLugt [12] requires a vortex to be a portion of thefluid moving around a common axis. As an in-dicator for such a structure he proposes closed orspiralling pathlines. Haller [9] describes vorticesthrough the stability of manifold structures whichare related to fluid trajectories. The Mz criterion[9] can be considered as an accumulation of a lo-cal measure based on the strain acceleration ten-sor along a trajectory. A single instability at onecell introduces noise into all trajectories throughthat cell. In recent work Haller [9] (see also Sah-ner et al. [18]) therefore adds up all time stepsalong the trajectory at which the particle is classi-fied to belong to a vortex. This can lead to over-

representation of the downstream situation, and wewill discuss how controlling both the locality of thecriterion and the weighting along the trajectory canimprove the accuracy of the feature extraction pro-cess. The discussed approaches are similar in thatthey compute trajectories and then evaluate a mea-sure of coherence from the shape or the relation ofendpoints of trajectories and are therefore depen-dent on the additional parameter of integration time.The presented method is also related to line inte-gral convolution (LIC) and similar methods, wherea flow visualization is produced by convoluting anoise texture along streamlines. Cabral and Leedom[3] present LIC for a dense visualization of two-dimensional flow fields. LIC-related methods areused to compute direct visualizations and are there-fore less concerned with understanding and control-ling the specific details of the results for a singlepixel of the final image. An extended formulationfor unsteady flow fields was published by Shen andKao [20].Salzbrunn et al.[19] introduce the concept ofboolean pathline predicates, to select pathlines ofinterest. They do not compute additional attributesand their approach is not related to vortex detection.Shi et al. [21] present the concept of a pathline at-tribute data set that is computed from the originalflow data set. The computed information consistsof scalar properties such as the average particle ve-locity, Euclidean distance to start, curvature or ve-locity. Recently Shi et al. [22] have suggest pathline integral convolution of kinetic energy and mo-mentum to get insights into the dynamical processesof the flow. They show that they can extract vor-tex structures which are almost as good as λ2 butcannot claim to improve on λ2 vortex extraction.This may result from not convolving the Euleriandetectors themselves and from using very long tra-jectories for accumulation, since the problem of in-tegration length specification is not tackled in re-cent work. In this paper we present a method thatimproves the vortex detection results.

3 Delocalized Vortex Detectors

In the following subsection we discuss the non-localextensions of the Eulerian detectors. In the secondsubsection we present the line view which helps tosteer the extraction. See Figure 1 for an illustrationof the delocalized vortex region detectors approach.

particle starting position

evaluation positions

particle trajectory

detector response color coded

trajectories + Eulerian detectors occlusion-free visualization accumulation along pathlets

particle starting position

(a) (b) (c)

sorting criterion(key length)

time

λ2-10k

+10k

λ2-10k

+10k

Figure 1: We illustrate the suggested approach. (a) By analyzing Eulerian quantities along trajectorieswe can improve the extraction of vortices which we consider as coherent structures and limit the effectof numerical issues. (b) Using a simple two-dimensional representation of the detector responses alongtrajectories we can understand and steer the locality of the accumulation. (c) The resulting method promotescoherency in space and time by accumulating information along a trajectory of a fluid particle for each point.

path length

local detector response

backward integrationboundary

forward integrationboundary

particle start position

unknownregion

backward trajectoryend position

forward trajectoryend position

detector threshold

key length

t ft b

weighting function

positive response

negative response

unknown

inflection point

w(t)

t b t f

Figure 2: Accumulation of local detector values along a particle trajectory. The local values are weightedaccording to their distance from the seeding point of the trajectory.

3.1 New Criterion

For accumulation along a trajectory we need an Eu-lerian vortex detector E(x,J(x,t),u(x,t)) → [0,1]where x ∈ D ⊂ R

3 is a point inside the simulationdomain D, u(x,t) is the velocity at point x at timet and J(x,t) is the Jacobian. See Figure 1 (a) foran illustration. The criteria we have found to bene-fit most from delocalization are λ2, Q, the swirlingstrength criterion and vorticity magnitude.A pathline can be expressed as

p(t +Δt) = p(t)+∫ t+Δt

tu(p(s),s)ds

where p(t) is the position of the particle at timet, p(t + Δt) is the new position after time Δt andu(p(t),t) is the velocity of the particle at position

p(t) at time t. The Runge-Kutta method (RK4) canbe used for numeric integration of these pathlines[14]. Given a vector field u we call pt0,x0 the path-line starting at point x0 and time t0.Now we can define the Eulerian detector responsefor a pathline pt0,x0 as

E(pt0,x0 ,t)= E(pt0,x0(t),J(pt0 ,x0(t),t),u(pt0,x0(t),t))

if pt0,x0(t) ∈ D.For a pathline we define the two maximal integra-

tion length parameters t f and tb as the maximal timethis pathline remains inside the simulation domainD during forward (resp. backward) integration.In contrast to unsteady flow LIC, where a colorpixel value is advected through the flow field, in thethe context of computing a vortex detector value it

makes sense to include the backward direction: thevorticity of a position is not only dependent on itsfuture contribution to a vortex, but also on its pastdevelopments (we can think of a particle inside theborder of a strong vortex region to be justly assigneda high vorticity value).

The delocalized version of the Eulerian detectorat position x0 and time t0 finally is

E(pt0,x0 ,tb,t f ) =

∫ min(t f ,t f )max(tb,tb)

w(s) ·E(pt0,x0 ,s)ds∫ t ftb w(s)ds

(1)with tb < t0 < t f and w(x) a weighting function.Good parameters for forward and backward integra-tion time, t f and tb allow the delocalized detectorimprove on the local information. In case the tra-jectory leaves the domain before the selected inte-gration times are met (t f > t f or tb > tb) we can ac-cumulate the requested information only partially.Weighting the result with the integral of w(x) overthe complete selection [tb,t f ] decreases the delo-calized detector result and incorporates the uncer-tainty resulting from short trajectories. The formal-ism does not change for steady and unsteady flows,since for steady flows the definition of a pathlinecoincides with the definition for a streamline.

The weighting function should give sufficientcontrol over the accumulation and produce pre-dictable results for the user. The first option is lin-ear weighting where the weight for a position onthe trajectory is given by the difference in phys-ical time from particle release time t0. That isw(t) := 1− (t − t0)/(t f − t0) for t ≥ t0 and w(t) :=(t0 − t)/(t0 − tb) for t < t0 with (tb < t0 < t f ). Thesecond option is a an accumulation using a Gaus-sian filter w(x) = 1/(σ

√2π) · e−0.5(t−t0)2/σ . See

Figure 2 for an illustration. The line view presentedin the next section allows the user to determine andspecify the relevant parameters, i.e. σ , t f and tb.Using Gaussian weighting, the influence of a sam-pling point quickly becomes very small after the in-flection point is reached, thus σ can be used to con-trol the locality of the criterion. For brevity we willwrite Eσ

t for a delocalized detector using Gaussianaccumulation when t f = tb = t. Units are secondsfor t, and σ has unit of ’average cell size times me-ters’.

So far we have not discussed how the integrationparameters tb and t f can be chosen appropriately.This will be the topic of the following subsection.

3.2 Line View

The purpose of this view is to visualize the com-puted trajectories in a 2D view as straight lines.This gives more space to convey visual informa-tion and enables easier selection and brushing oper-ations. The delocalized criteria are robust and onlyin complex flow regions a single threshold does notperform well. In this case it is necessary to use mul-tiple thresholds, which are difficult to define usingthe occluded 3D trajectory rendering. By evaluatingthe distribution of local detector values in combina-tion with selective 3D visualization of the relevantstreamlines it becomes possible to select a few suit-able integration length parameters to separate inter-acting vortices.

The engineers are interested in the relationshipsbetween the fluid cells of the mesh, therefore weseed one trajectory per grid cell. Each trajectory isvisualized simply by placing its segments succes-sively on a straight horizontal line. The resultinghorizontal lines are spaced vertically so that they fillthe available viewing space. Our main interest forthe line view is to observe how the vortex classifierresponse is distributed along the trajectories. Theview works in coordination with the other views inthe visualization framework to allow filtering rele-vant trajectories. See Figure 3 for an illustration.While selecting integration length parameters in theline view, the currently relevant trajectories are ren-dered as lines in the 3D view, conveying the spatialinformation for these trajectories. Additionally, adegree of interest (DOI) can be specified by brush-ing in other views, thus assigning a DOI value to thesample points in the data set. Lines with zero DOIat their starting positions are filtered out.

Good sorting of the lines is crucial so that trajec-tories belonging to the same structure are orderedclosely together. For that purpose the view offers arange of sorting and filtering criteria:

• Key length: is the maximal time interval in-cluding t0 inside which the particle remainsinside a region of positive local detector re-sponse without interruption (see Figure 2).

• Line length: lines are sorted or filtered accord-ing to their length

• Delocalized response: after selection the linescan be reordered according to their delocalizeddetector response

Line fusion: line fusion is necessary when morelines are currently in focus than there are pix-

Cells Ts Type Grid ROI Lines Acc. Int.

T-Junc. 30 K 100 incomp. struc. all 137 MB 0.1 sec 1 minCool. J. 1538 K 1 (steady) incomp. unstruc. 95 K 650 MB 2 sec 5 min

2-Stroke 1156 K 91 comp. unstruc. 81 K 570 MB 2 sec 4 minRankine 262 K 1 synth. struc. all 1.6 GB 3 sec 3 min

Table 1: Comparison of the datasets evaluated in the application study. We have evaluated a simulationof a pulsating T-Junction, a Cooling Jacket, a 2-stroke engine and the synthetic rankine vortex model.The region of interest (ROI) showing complex vortical behavior was always much smaller than the wholedataset. (Abbreviations: Ts - time steps, ROI - cells in region of interest, Acc.- accumulation of delocalizeddetector, Int. - integration of trajectories)

(a) (b)

λ2

|u|

}tftb

λ2-10k

+10k λ2

key length (c)

Figure 3: Linking the Line View. (a) Attribute selection on the multivariate simulation data set allows tofilter the data points of interest. Only the lines seeded at the selected points are rendered in the line view.(b) The remaining lines are displayed in 2D and the user can specify integration parameters by drawingline segments onto the view. (c) The view is linked to the 3D rendering. While selecting the forward andbackwards integration length parameters in the 2D view the trajectories below the tip of the cursor arerendered in the 3D view.

els available on the screen. We employ post-classification, by first combining the detector re-sponses for line segments, and then assigning colorto the resulting line by means of a transfer function.To combine a group of lines into one line, we keepadvancing a vertical scan-line, until all segments aredrawn onto a storage texture.

Integration length specification: the selectionranges for the lines can be defined interactively bydrawing two curves on the view. This way, for eachgroup of lines (after fusion) the user can specify theparameters t f and tb. When Gaussian weighting isselected, the interaction allows to select the loca-tion of the inflection point. It is sufficient to selectthe ranges for very large groups of lines and onlywhen this approach fails it is necessary to zoom inand perform a more elaborate selection. With goodsorting parameters the delocalization is robust, andall figures in this paper were made without zoom-ing.

Using linking and brushing the line view allowsto select appropriate integration length parametersfor different regimes of the flow, which is necessaryto separate interacting structures. Furthermore, theselective rendering of 3D trajectories using linkingbetween the line view and the 3D view can serve asa useful analysis tool by itself.

4 Evaluation

In this section we discuss vortex feature detectionresults. Table 1 gives an overview of the evaluatedcases. To be able to evaluate at which point higherthresholds start to remove important parts of the fea-ture we include vortex core lines computed with theapproach of Sujudi and Haimes [23] for the steadycooling jacket data set and its extension for time de-pendent flows [5]. These vortex core line detectorscan produce spurious or shifted solutions as well,but for the strongest and largest features they are a

good measure for comparison with the extracted re-gions.

4.1 Cooling Jacket

Q >1000 Q > 1000

inflow

Region of Interest

ductsout

A AB D

CB

15 cm

0.8

16 cm

43 cm

c E

DE

0.1~

Figure 4: The cooling jacket data set. (top)Overview of the geometry and region of interest.(bottom) We compare the detector results for Q anddelocalized Q using σ = 0.8 and 0.1 sec. integrationtime: large parts of the feature A are removed by theinstantaneous Q criterion, while feature C becomesmore localized and connected to the larger region,which is better in the light of the results obtainedusing λ2.

The first data set contains a steady simulationgrid of fluid moving through a cooling jacket (seeFigures 4 and 5). In this section we extend theanalysis of turbulence inside the cooling jacket per-formed in a previous publication [2]. The coolingjacket in focus (see Figure 4 (top)) is designed fora four cylinder engine. The shape of the coolingjacket is designed to provide optimal temperaturefor the engine. Between the cylinder head and thecylinder block lies the cooling jacket gasket. It con-sists of a series of small holes that serve as conduitsbetween the block and head. These ducts can bequite small relative to the overall geometry but arevery important because they are used to control themotion of fluid flow through the cooling jacket. We

focus on a section of the jacket close to the inlet,where the flow is fast and the gasket causes strongturbulence.

In Figure 4 (bottom) we compare the Q criterionand the delocalized Q criterion. We set the thresh-old to 1000 in order to remove the large amount oftiny structures, such that three larger structures be-come distinguishable. We can see that inside thelargest structure (A) half of the length of the coreline is removed and large holes appear. While thetwo other structures (C+D) contain the same coreline features, the delocalized regions appear muchsmoother and are disconnected.

In Figure 5 we compare the λ2 vortex detectorat thresholds 1000 and 5000 with the delocalizedversion of λ2. The top row shows a side view of thesituation, where we can see turbulent regions appearbehind the gaskets. At a threshold level of 1000 allthe relevant vortex core lines are present, but the re-sulting iso-surface is difficult to understand and wehave a single connected region. At this level the fea-ture A is still present, but if we want to separate thedifferent structures, we have to set a higher thresh-old where the core line is no longer fully inside theselected domain. If we look at the correspondingdelocalized λ2 regions, we can see that the featureA remains intact. We can also see that the core lineat position C hints towards the assumption that thisis a rather important feature which is correctly con-nected to the large region (A). Also the regions (B)and (C) are disconnected. By comparing the resultsof the delocalized versions of the Q and the λ2 cri-terion we can make another finding: the delocal-ized results are extremely similar for both criteria,even though the iso-surfaces of the instantaneousversions bear little semblance.

4.2 Two-stroke Engine

The second dataset is a high-performance two-stroke engine dataset, which contains the simulationresults for injection and combustion of fuel duringone crank revolution.

In Figure 6 we can see the results of the vortexdetection in the Eulerian and the Lagrangian frame.Here we have only one extremely strong main fea-ture and therefore a single coreline is detected. Forthe main feature (A) the results for the Eulerianand the delocalized detector are similar. But for asmaller structure (B) we have a similar result as forthe cooling jacket: to keep B intact we have to se-

λ2 < 1000 < 5000delocalized λ2λ2 λ2Gauss0.80.8

< 5000< 5000

A

B

C

DE

AB

C

D

E

A

B

C

D E

AB

C

D

E

A

B

C

DE

AB

C

D

E

A

B

C

D E

AB

CD

E

(a) (b) (c) (d)

0.1

~

Figure 5: The delocalized λ2 outperforms λ2 regarding feature separation and noise suppression. (a) Thedelocalized version of λ2 extracts three non-connected regions which contain the strongest and longestvortex core lines. Small features and noise are removed. (b) At a threshold of λ2 < 1000 we get a singleconnected region and all the vortex core line features remain intact. We also get a lot of small or weakfeatures which we are not interested in. (c) At a level of λ2 < 5000 different (still connected) featuresappear, still a lot of noise is extracted and vortex A breaks in two parts. (d) Smoothing λ2 using a Gaussiankernel removes noise, but also feature E.

lect a low threshold (swirl > 0) and the feature (B)is difficult to recognize. (b) At a threshold wherethe features become distinguishable, the feature (B)is split in two components.

4.3 T-Junction

The t-junction data set is a small unsteady simula-tion of pulsating flow through a t-junction. In thecenter of the domain, behind an obstacle, a smallvortex appears. Du to the good temporal resolutionand relative simplicity of the situation we can dis-cuss how the delocalization process allows to con-centrate on different types of features. We can ob-serve the development of four features in the dataset. These are (A) two longitudinal vortices behindthe inlet, (B) a transversal vortex created by the pul-sating inflow boundary condition, (C) a vortex ap-pearing behind the obstacle and (D) a region of tur-bulence at the outlet. In (a) we can see that it ispossible to select a threshold to separate the struc-tures (A) and (B). In (b) we use streamlines to depictthe shape of the feature (C) a the current moment.We can see that the feature has the same height asthe obstacle. From this we can conclude that in (a)the threshold necessary to separate the features (A)and (B), removes too much of feature (C). In (c)

we can see that the delocalized version of λ2 allowsto separate the vortex behind the obstacle and stillvisualize the full transversal vortex (B). By select-ing specific integration length parameters for eachof the now disconnected regions we have also dese-lected the turbulent region at the outlet.

4.4 Filter Properties

noise 1% 5% 10%

λ2 0.007 5.734 12.988MF(λ2) 0.007 0.019 7.813

Gauss(λ2) 0.872 1.1 4.185

λ20.8

0 0 3.002

Table 2: Numerical evaluation of noise in the Rank-ine vortex model. The table shows the error rates ofthe classification results of the local detector (λ2),after application of a median filter (MF(λ2)), afterapplication of a Gaussian filter (σ = 0.8), and theresults for the delocalized λ2 detector (σ = 0.8).

To test numerical stability we use a simple syn-thetic solution so that we can know where the vor-tex has to be detected. A simple model for a vortexis given by the combination of a rigid-body rota-

(a) (b) (c)

AB B BA A

Figure 6: Two-stroke engine data set. We compare the result of the swirling strength criterion and thedelocalized swirling strength criterion. (a) At a low threshold (swirl > 0) the two features are not distin-guishable. (b) Searching for a better threshold we find that in order to get a good separation between thetwo features we have to select a value at which feature B breaks in two components (swirl > 10000). Avisualization including streamlines seeded at the gap shows that the two components belong together. (c)The delocalized delta detector with σ = 0.8 and tf = tb = 0.01 allows to visualize both features intact.

tion within a core, and a decay of angular velocityoutside [1]. The Rankine vortex model can be de-scribed by

uθ ={

ω · rR ,r ≤ R

ω · Rr ,r > R

ur = 0 uz = u,

where R is the radius of the vortex, u controls ax-ial velocity and ω controls the maximal tangentialvelocity. The model has a long history in mete-orological studies of tropical cyclones and can beconsidered a good approximation of measured data.This is also an example much in favor of the localdetectors since they all have 100% correct classi-fication in the absence of noise. Nevertheless thedelocalized vortex detectors outperform their Eule-rian counterparts consistently. From an image pro-cessing viewpoint one can consider the presenteddetector as a special case of an isotropic filter. Toshow that the reasoning behind the convolution ac-tually improves the detection results, we compareour results to the error rates after the application ofa median and a Gaussian filtering kernel.In Table 2 we can see the results of the numericalstudy. Noise was added to each cell using a linearcombination of random noise vectors for each cellni, j,k at sample position (i, j,k) in the regular gridand the original flow value vi, j,k such that a noiselevel of p% is computed as vi, j,k +(p/100) · ni, j,k .We use σ = 0.8 for accumulation of the delocal-ized detector values. Changing the velocity vector

locally will affect the estimated gradient of all thesurrounding cells. Isotropic filtering therefore can-not deal with noise as well as the delocalized vortexdetectors. The error of the delocalized vortex de-tectors at a noise level of 10% of the original signalstems from the fact that we have a very sharp vor-tex boundary in the model such that a small devia-tion from the correct trajectory can already degradethe performance. A second reason is that the tra-jectories at the corners of the rectangular domainhave very short integration times and quickly leavethe simulation domain. It is quite unnatural to havesuch sharp vortex boundaries and also the large per-centage of short streamlines is to the disadvantageof the delocalized detector. Even though the de-localized detector outperforms the other methods.The high error of the Gaussian filter stems from thefact that for low error rates it blurs the errors andcan actually increase error.

5 Implementation Details

In the context of our application we know that theengineers are interested in understanding the rela-tionships between the computational cells of thesimulation grid. Therefore we seed one path- orstreamline per cell to be able to compute a delocal-ized detector response for each cell. During inter-action the main computational cost lies in the line

(a) (b) (c)

in

out

obstacle

A

BC D

λ2 < -1

delocalized λ2 < -1

obstacle

0.8

1

vortex behindobstacle

λ2-1

+1

Figure 7: Comparison of λ2 and the delocalized version of λ2. (a) Behind the inlet two small longitudinalvortices appear (A). When setting a threshold to separate the transversal vortex (B) from the longitudinalvortices, the third vortex (C) behind the obstacle almost vanishes. (b) Streamlines show that the vortexC extends to the full height of the obstacle. (c) Using the delocalized λ2 detector we can focus on thetransversal vortex and still select the full vortex C while deselecting A and D.

fusion approach and our prototype can compute thefinal texture for small regions of interest with upto 100K cells at interactive rates. A more efficientimplementation could be several magnitudes faster.Trajectories are computed off line and stored in anadditional data set, which takes several hours for thefull two-stroke data set, but for a specific region ofinterest at a selected time step only small subset ofthese lines has to be computed.

Conclusion and Discussion

The main drawback of the presented method is thatthe detected results are no longer objective in thesense that each engineer will come to exactly thesame vortex detection results. The exact location ofthe vortex boundary is dependent on the specifica-tion of integration length parameters. These differ-ences are typically small and as long as we do nothave a general definition for what a vortex is, thisfuzzyness can be considered appropriate. The sec-ond disadvantage of the presented approach is thetime consumed by integrating trajectories throughthe fluid. We used a rather inefficient implementa-tion where the timings cannot be considered repre-sentative, and GPU-based implementations are re-ported to compute trajectories at nearly interactiverates [11]. Another drawback is that interaction isoften a necessity. Using bad integration length pa-rameters, the results are more blurred and worsethan λ2 regions, even though the delocalized detec-tors have shown to be very robust in our experience.The approach to use a single trajectory length pa-

rameter as seen in the results presented in Figures4, 5, 6 and 7 only works with carefully selected re-gions of interest and even then the line view is nec-essary to find good parameters for σ and t = tf = tb.

An obvious idea for estimating good integrationlength parameters automatically is to search forminima of detector response along the trajectory.This way we could hope to find the boundary ofthe vortex region without interaction. This has pro-duced mixed results for the data sets evaluated inthis paper. A criterion for good integration lengthparameters based on physical principles indepen-dent of user interaction would further improve thedelocalized detectors.

In this paper we have proposed delocalized vor-tex region detectors. With little interaction to de-termine reasonable parameters, the delocalized vor-tex detectors improve the feature extraction process.We have also discussed how the ability to controlthe range of integration improves the expressivenessof the detectors over their local counterparts. Thedelocalized detectors are a combination of the Eu-lerian and the Lagrangian approach to vortex regionextraction. The basic message here is that the Eu-lerian and the Lagrangian are not different alterna-tives to vortex extraction, opposite to each other, butthat they can be combined to one technique sharingthe benefits of both. The good local vortex detectionperformance of the Eulerian criteria and the globalinformation of the Lagrangian view point combineto generate well separated and smooth detection re-sults.

Acknowledgements

This work has been funded by the FWF PVGproject supported by the Austrian Science Fund(FWF) under grant no. P18547-N04. The 2-strokeengine dataset is courtesy of the ”Institut fur Ver-brennungskraftmaschinen und Thermodynamik” atthe Technical University of Graz. The coolingjacket CFD simulation dataset is courtesy of AVLList GmbH, Graz, Austria.

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