*Corresponding author. Present address: Manchester Visual-ization Centre, Manchester Computing, University of Manches-ter, Oxford Road, Manchester M13 9PL, UK. Tel.: #44-161-275-7040; fax: #44-161-275-6071.

E-mail addresses: ari.sadarjoen@mcc.ac.uk (I. Ari Sadarjoen),frits.post@cs.tudelft.nl (F.H. Post).

Computers & Graphics 24 (2000) 333}341

Data Visualization

Detection, quanti"cation, and tracking of vortices usingstreamline geometry

I. Ari Sadarjoen*, Frits H. Post

Faculty of Information Technology and Systems, Delft University of Technology, Zuidplantsoen 4, 2628 BZ Delft, Netherlands

Abstract

We present two techniques for vortex detection in 2D velocity "elds, based on the macroscopic geometric properties ofstreamlines. The methods do not depend on the local #ow patterns at a single point used in many other vortex detectiontechniques. Both methods begin by covering the full domain with a large number of streamlines, and select the curveswith circular or looping geometry. The "rst method uses local cumulations of curvature centers which may indicate thatmany streamlines are circling around a cluster of closely spaced center points. The second method detects loopingpatterns in streamlines by looking at the cumulative changes of direction, as represented by the winding angle. Thesecond method is very e!ective in detecting weak vortices, as it does not depend on velocity magnitude but only on thepattern. The methods can be used for quanti"cation of vortices using numerical attributes which are suitable for featuretracking in time dependent #ows. We present results of the methods with stationary and time-dependent CFD datasets. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Flow visualization; Vortex detection; Feature extraction

1. Introduction

In many areas of science and engineering, vortices areimportant #ow features, both from a theoretical anda practical viewpoint, both in science and engineeringpractice. Since the underlying physics of vortices are notcompletely understood yet, detection and visualization ofvortices is still an important topic. Some recent publica-tions in this context are [1,2].

Traditionally, vortex detection methods have beenbased on physical quantities, such as pressure, vorticity,and helicity [3,4]. These are typically evaluated locally,or derived from gradient quantities evaluated in in"ni-

tesimal regions. Unfortunately, these methods often failto "nd weaker vortices.

In this paper, we present two di!erent vortex detectiontechniques, based on geometric properties of streamlines.The second technique is also used as a basis for quanti"-cation and tracking of vortices.

Both methods start by globally calculating a largenumber of streamlines. The "rst method then determinesthe curvature centers for many points on the streamlines.For a perfectly circular streamline, the curvature centersof these points accumulate at one pint. Assuming a vor-tical region mainly consists of (nearly) circular stream-lines, we should be able to "nd vortex cores fromaccumulations of streamline curvature centers.

The second method does not use curvature centers,but directly looks at the geometry of the streamlines, bydetermining their winding-angle. This is the cumulativeangle the streamline has &tuned around', and is conse-quently $2p for a full turn, p for a half turn, etc. Anadvantage of this method is that the streamline need notbe perfectly circular, but may wiggle back and forth a bit,as long as the overall rotation direction remains the

0097-8493/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 9 7 - 8 4 9 3 ( 0 0 ) 0 0 0 2 9 - 7

Fig. 2. Curvature center points are accumulated into a new grid,resulting in the curvature center density (CCD) scalar "eld.

Fig. 1. (a) Circular streamline with coinciding curvature centersC and (b) non-circular streamline with scattered curvaturecenters.

same. Then we cluster the selected streamlines, in orderto identify individual vortices.

Next, we quantify the vortices by calculating numericalattributes of the streamline clusters. These attributes areused for two purposes: iconic vizualization and tracking.We use abstract ellipse or ellipsoid icons to visualize theshape and orientation of the vortices. We perform track-ing of vortices in time-dependent datasets, by consideringmultiple timesteps of a time-dependent dataset, and fol-lowing the positions and sizes of the detected vortices. Inaddition, we have applied the technique in a new casestudy.

The structure of this paper is as follows. In Section 2,we give an overview of related work. Then, we describeout geometric methods for vortex detection: the curva-ture center method in Section 3, and the winding-anglemethod in Section 4. In Section 5, we show some resultsof applying our methods to CFD simulations.

2. Related work

The "rst class of vortex detection methods is typicallybased on point samples of physical quantities. The quan-tities involved are usually pressure, velocity, quantitiesderived from the velocity vector, or quantities derivedfrom the velocity gradient tensor. All of these quantitiesare based on the assumption either that vortices areregions with a high amount of rotation, or that thereexists a pressure minimum at vortex cores. Banks andSinger [3] and Roth and Peikert [4] have surveyeda number of quantities, and concluded that they often failto capture all vortices. An important cause is that vor-tices are regional features, but these criteria are strictlybased on point samples, or "rst-order approximations inin"nitesimal regions. Recently, Roth and Peikert [5],recognizing the de"ciencies of "rst-order approxima-tions, proposed a higher-order method which is also ableto detect bent vortices.

The second class of vortex detection methods is geo-metric, i.e. based on geometric properties of streamlines.De Leeuw and Post [6] describe and interactive way todetect vortices using a box-shaped probe in which samplepoints are taken. For all the sample points in the box,a number of properties were calculated, including thecenter of curvature of the streamline through the samplepoint. When the box contained a vortex, the centers ofcurvature would accumulate near a point, otherwise theywould be scattered.

Portela [7] has developed a formal mathematicalframework for de"ning vortices in 2D, which corres-ponds to the intuitive notion of swirling motion arounda central set of points. To de"ne a central set of points, heproposed so-called Jordan structures; to de"ne swirlingmotion, he used the winding-angle concept known fromdi!erential geometry.

In [8], we applied two geometric techniques to severalhydrodynamic cases. The "rst technique used curvaturecenters to "nd vortex cores, the second used a simpli"edwinding-angle concept.

In [9], we extended the second technique to a tech-nique for automatic feature extraction, characterizing thefeatures by calculating a set of quantitative attributes,such as position, size, and rotation speed and direction.This was done by clustering the selected streamlines anddetermining numerical attributes of the vortices. This hasthe advantage that vortices can be described by a smallset of attributes, which naturally causes a dramatic datareduction.

The present paper shows new results obtained withthis technique, by applying it to a sequence of time-stepsof an existing case, and in a completely new case.

3. The curvature center method

The curvature center method tries to detect vortices in2D by sampling the "eld at many points, typically at allgrid nodes. For each sample point, the center of curvatureis determined, which is the center of the osculating circleof the streamline through that point [10]. In vorticalregions of the "eld, the centers of curvature should accu-mulate at a point, as in Fig. 1a. The samples taken on thisperfectly circular streamline all project to the same centerof curvature. In non-vortical regions of the "eld, thecenters of curvature will be scattered, as in Fig. 1b.

In this way, a set of curvature center points is obtained,which are accumulated into a new grid, as illustrated inFig. 2. The number of curvature centers in each cellconstitutes a new scalar "eld which we call the curvaturecenter density (CCD) "eld.

334 I.A. Sadarjoen, F.H. Post / Computers & Graphics 24 (2000) 333}341

Fig. 4. In circular #ow (a), there is a peak in the CCD "eld (b). In elliptic #ow (c), the peak in the CCD "eld is spread out (d).

Fig. 3. Paci"c Ocean with global streamlines and a white height"eld of the curvature center density.

Fig. 3 shows an example of a CCD "eld. The 2D dataset originates from a numerical #ow simulation of thePaci"c Ocean, which models the west coast of NorthAmerica [11]. The grid used is a rectilinear 2D grid of117]84 nodes, at each of which the velocity has beencalculated. The "gure shows streamlines released fromevery grid node. The CCD "eld has been rendered asa white height "eld. Thresholding has been applied toselect only the highest peaks of the "eld: CCD'0.8CCD

.!9.

This method works, but has the same limitations astraditional point-based detection methods. There aresome false and some missing peaks. Some of the falsepeaks may be "ltered out by thresholding or "ltering.Also, supersampling may be applied to get more samplesper grid cell [12]. An important cause of these problemsare the streamlines which are not perfectly circular, butelliptical or elongated; this is often due to interactionbetween adjacent vortices, or the e!ect of the shear com-ponent in the #ow. The e!ect is shown in Fig. 4: inperfectly circular #ow (see Fig. 4a), there is a clear peak inthe CCD "eld (see Fig. 4b). However, in slightly elliptic#ow (see Fig. 4c), the peak is &spread out' (see Fig. 4d).This causes many missing peaks, and possibly also somefalse peaks.

4. The winding-angle method

Another geometric method for detecting vortices in2D, inspired by Portela [7], builds upon the intuitiveidea of a swirling pattern around a central set of points.The method tries to detect vortices by selecting loopingstreamlines and then clustering them. Selection is per-formed using a simpli"ed winding-angle criterion anda distance criterion.

Let Sibe a 2D streamline, consisting of points P

i,jand

line segments (Pi,j

, Pi,j`1

), and letL(A, B, C) denote theangle between line segments AB and BC. Then, the wind-ing-angle au,i

of streamline Siis de"ned as the cumulative

I.A. Sadarjoen, F.H. Post / Computers & Graphics 24 (2000) 333}341 335

Fig. 5. The winding-angle aw

is the sum of the angles betweenthe edges.

change of direction of the streamline segments:

au,i"

N~1+j/1

L(Pi,j~1

, Pi,j

, Pi,j`1

) (1)

(see Fig. 5). We use signed angles, with positive rotationfor a counterclockwise-rotating curve, and negativerotation for a clockwise-rotating curve. Obviously,au,i

"$2p for a fully closed curve; lower values may beused to "nd winding streamlines which do not make a fullrevolution.

The selection process tries to "nd the streamlines thatbelong to a vortex by using two criteria: (1) the winding-angle of a streamline should be k ) 2p, with k*1, and(2) the distance between the starting and "nal point ofthe streamline should be relatively close.

We have extended the work described in [8] froma visual, qualitative selection technique, to a more quant-itative feature extraction technique. We now use theselected streamlines for automated vortex extraction andfor determining numerical vortex attributes. This is donein two stages: clustering and quantixcation.

The purpose of clustering is to group those streamlinestogether which belong to the same vortex. Rather thanclustering streamlines, it is easier to cluster points. To thisend, each streamline is mapped to a point by determiningthe center point, or geometric mean, of all sample pointson the streamline. These center points are then clusteredas follows. The "rst cluster is formed by the "rst point.For each subsequent point, it is determined which pre-vious cluster lies closest. If the point is not within a prede-termined radius of all the existing clusters, it constitutesa new cluster. In this way, the selected streamlines arecombined into a distinct number of groups. Streamlinesof the same group are considered to be part of the samevortex.

Once the streamlines have been clustered, quantixca-tion of the vortices is performed by calculating numericattributes of the corresponding streamline clusters. Weapproximate the shape of the vortices by ellipses. Fittingan ellipse to a set of points is done by calculating statist-ical attributes, such as mean, variance, and covariance, of

the points [13]. In addition, we calculate speci"c vortexattributes, such as rotation direction and angular velo-city. We denote the number of points on a streamlineSias DS

iD, a cluster of streamlines as C

k"MS

k,1, S

k,2,2N,

where Sk,l

is streamline dl in cluster dk, the number ofstreamlines in that cluster as DC

kD, and all the points on all

the streamlines in that cluster as t(Ck). Now, we can

calculate the following attributes for each vortex:

f streamline center: SMi"(1/DS

iD)+ @Si @

j/1Pi,j

.

f cluster center: CMk"(1/DC

kD)+ @Ck @

l/1(SM

k,l).

f cluster covariance: Mk"cov(t(C

k)).

f ellipse axis lengths: jk"eig(M

k).

f ellipse axis directions: dk"eigvec(M

k).

f vortex rotation direction: dk"sign(a

w,k).

f vortex angular velocity: uk"(1/DC

kD*t)+ @Ck @

l/1aw,l

.

The vortices can be visualized by mapping their at-tributes to icons: the "rst three statistical attributes areused to calculate the axis lengths and directions of anellipse which approximates the size and orientation ofa vortex. The rotation direction of a vortex is visualizedby small arrows. Finally, the angular velocity of a vortexis visualized by adding wheel spokes to the ellipse, thenumber of which is made proportional to the angularvelocity: fast rotation is suggested by many spokes, slowrotation by few.

5. Results

5.1. Currents in the North-Atlantic Ocean

The "rst example uses a data set of a simulation per-formed at the Hadley Centre for Climate Research andPrediction, the UK Meteorological O$ce. One of thegoals of the simulation was to predict the e!ects of airpollutant emissions to global warming and ocean currents.

The model is de"ned on a curvilinear grid of288]143]20 nodes spanning the globe with a resolu-tion of 1.253 longitude and latitude. From this grid, weselected the part covering the North-Atlantic ocean.

The simulation spans a period between the year 1860and 2099, with one time step per year, but we use onlyone time step (1999). At each node, the simulation cal-culated three velocity components, the velocity magni-tude, and temperature.

Fig. 6 shows the result of applying the winding-anglemethod. The global #ow patterns in the data set arevisualized by the grey streamlines released from everygrid node in a horizontal grid slice at the center of thegrid (sliced10 out of 20). Selected streamlines are drawnin black. The ellipse icons visualize the approximate sizeand shape of the vortices, with the ellipse axes drawn indashed lines. Arrows indicate the rotation direction ofthe vortices. The number of spokes indicate the strength

336 I.A. Sadarjoen, F.H. Post / Computers & Graphics 24 (2000) 333}341

Fig. 7. Atlantic Ocean with streamlines and vortices approximated by ellipses. Red and blue ellipses rotate in opposite directions.

Fig. 6. Vortices in the North-Atlantic Ocean. The number ofspokes is proportional to u.

Table 1Some numerical attributes of the vortices in the North-AltanticOcean

Number of clusters 14Number of CW vortices 12Number of CCW vortices 2Min. radius (km) 37.127Max. radius (km) 920.914Min. u(s~1) 0.042872Max. u(s~1) 0.706842

of the vortices: the higher the number of spokes, the fasterthe rotation.

It can be seen that this method captures all vorticesconsisting of rotational streamlines, including elongated

and weak ones. An impression of the vortex strength(rotation speed) is immediately visible for the number ofspokes.

Fig. 7 shows a color visualization, where, the topogra-phy is much clearer due to the use of a texture map of theearth. Instead of ellipses, here ellipsoids are used, whichdo not show the angular velocity, but their color showsthe rotation direction of the vortices: red indicates clock-wise rotation, and blue counterclockwise rotation.

Once the vortices have found, numerical attributes aredetermined for each of them. Table 1 shows some of theresults. Notice the di!erences between the largest and thesmallest vortex (approx. factor 25), and between the fas-test and the slowest one (approx. factor 15). There does

I.A. Sadarjoen, F.H. Post / Computers & Graphics 24 (2000) 333}341 337

Table 2Statistics of the vortices in the #ow past a tapered cylinder

Time-step No. vortices max jCW

max jCCW

xCCW

yCCW

12110 1 0.7559 * 1.2009 0.272412130 1 0.8299 * 1.2961 0.246912150 2 0.8812 0.2514 1.4277 0.217512170 2 0.8700 0.4236 1.5792 0.177312190 2 0.8714 0.4231 1.7627 0.1403

not seem to be any correlation between the size and therotation speed of the vortices.

5.2. Flow past a tapered cylinder

The second example uses a data set of a simulationperformed at NASA-Ames Research Center which con-cerns a laminar #ow past a tapered cylinder [14]. Thistapered cylinder has a variable radius depending on thez-coordinate, which in#uences the vortex-shedding fre-quency at that height. The grid used is a structured,cylindrical grid with 64]64]32 nodes, each of whichcontains density, x, y, z-momentum, and stagnation. Thetime-dependent simulation has many time-steps, fromwhich we use the ones at t"12110212190.

We have applied the winding-angle method forextracting vortices from one horizontal slice (z"20)in "ve di!erent time steps, to achieve a simple formof temporal vortex tracking. Table 2 shows numericalstatistics of these vortices, where max j

CWis the

maximum axis length of the clockwise vortex, maxjCCW

of the counterclockwise vortex. xCCW

and yCCW

arethe x, y-coordinates of the geometric centers of thestreamlines, which corresponds roughly to the centers ofthe vortices. It can be seen that for t"12110, 12130,there is only one clock-wise vortex, while for t"12150,12170, 12190, there is one clockwise and one counter-clockwise vortex. The birth of a new counterclockwisevortex between t"12130 and t"12150 is an event whichcan be detected automatically using an event detectiontechnique [15].

Fig. 8 visualizes the vortices for these time steps. The#ow goes from left to right, past the cylinder which isdrawn as a semi-circle on the left. Again, the global #owpattern is shown by grey streamlines, the selected stream-lines are drawn in black, and the vortices are approxi-mated by ellipse icons. Spokes indicate the rotationspeed, and arrow heads the rotation direction.

Fig. 9 shows a color visualization of the slice at z"20.Five ellipsoids are shown, at t"12110, 12130,2, 12190.All ellipsoids but the frontmost have been rendered insemi-transparent white, the frontmost ellipsoid has beenrendered in red (yellow on the inside). These "ve ellipsesshow the trajectory of the clockwise vortex.

The grid slice has been colored with pressure. Lowvalues are supposed to indicate the presence of vortices.However, in this example, the lowest (blue) values forp are observed behind rather than in front of the cylinder,without any obvious vortices. Therefore, in this example,our winding-angle criterion turns out to be better thanpressure.

5.3. Performance and comparison

The curvature center density (CCD) method is fasterthan the winding-angle method. The method consists ofthree algorithmic components: velocity gradient deter-mination, curvature center determination, and cumulat-ion of the centers into a grid. The "rst component is themost computationally intensive one, requiring the calcu-lation of "nite di!erences to approximate velocity gradi-ents of the velocity vectors at all the grid points in a gridlayer. The second and third components are less com-putationally intensive, as they only involve vector addi-tions and summations.

The winding-angle method is more computationallyintensive. The method consists of four algorithmic com-ponents: streamline calculation, winding-angle calcu-lation, clustering, and quanti"cation. The "rst two aremost computationally intensive. Since the accuracy of thewinding-angle method increases with the number ofstreamlines, generally a very large number of streamlinesis calculated in the "rst step. Calculation of the windingangles is then performed for each of the line segments ofthe same number of streamlines. The following clusteringand quanti"cation components are less computationallyintensive, as they are only performed on a much smallernumber of clusters.

Taking into consideration both performance andqualitative aspects, we can state the following advantagesand disadvantages of both methods: the CCD methodhas the advantage of giving quicker results, but the disad-vantages of often not "nding weak and non-circularvortices. The winding-angle method requires more execu-tion time, but is qualitatively superior, because it cap-tures more vortices, and allows for quanti"cation.

338 I.A. Sadarjoen, F.H. Post / Computers & Graphics 24 (2000) 333}341

Fig. 8. Flow past a tapered cylinder; di!erent time-steps show di!erent vortices.

6. Conclusions and future work

We have described two geometric vortex detectionmethods. The curvature center method has limitationsdue to incomplete separation of rotation and shear com-ponents of the #ow. The winding-angle method is e!ec-

tive in "nding both strong and weak vortices. Anotherimportant advantage is that it also allows for quanti"ca-tion, which leads to data reduction.

Future work includes incorporating critical points inthe winding-angle method, to trace streamlines only inthe neighborhood of critical points, rather than globally

I.A. Sadarjoen, F.H. Post / Computers & Graphics 24 (2000) 333}341 339

Fig. 9. Flow past a tapered cylinder, a grid slice colored with pressure, and subsequent positions of a vortex indicated by semi-transparent ellipsoids.

in the entire "eld. Another useful application of thenumerical attributes of vortices would be to performspatial matching and temporal tracking of vortices.Matching and connecting ellipses found in adjacentx/y/z-slices allows us to "nd 3D vortices, as long as theyproject reasonably well to x/y/z-slices.

Temporal tracking is currently done simply by ap-plying the extraction algorithm to each time step inde-pendently, but we intend to do this using an automatedtracking algorithm as described by Reinders et al. [15].

Acknowledgements

We thank Freek Reinders of Delft University of Tech-nology for his e!orts in preprocessing the tapered cylin-der data. The North-Atlantic data set is courtesy ofthe Hadley Centre for Climate Research and Prediction,the U.K. Meterological O$ce. We thank Paul Lever ofthe Manchester Visualization Centre for his help withthis data set.

References

[1] Pagendarm HG, Henne B, RuK tten M. Detecting vorticalphenomena in vector data by medium-scale correlation.

In: Proceedings of Visualization '99. New York: ACMPress, 1999. p. 409}12.

[2] Peikert R, Roth M. The parallel vectors operator* a vec-tor "eld Visualization primitive. In: Ebert D, Gross M,Hamann B, editors. Proceedings of Visualization '99.New York: ACM Press, 1999. p. 263}70.

[3] Banks DC, Singer BA. A predictor-corrector technique forvisualizing unsteady #ow. IEEE Transactions on Visualiz-ation and Computer Graphics, 1995;1(2):151}63.

[4] Roth M, Peikert R. Flow visualization for turbomachinerydesign. In: Yagel R, Nielson GM, editors. Proceedings ofvisualization '96. Los Alamitos, CA: IEEE Computer So-ciety Press, 1996. p. 381}4.

[5] Roth M, Peikert R. A higher-order method for "ndingvortex core lines. In: Ebert D, Hagen H, Rushmeier H,editors. Proceedings of Visualization '98. New York: ACMPress 1998. p. 143}50.

[6] de Leeuw WC, Post FH. A statistical view on vector "elds.In: GoK bel M, MuK ller H, Urban B, editors. Visualization inscienti"c computing. Eurographics, Wien: Springer, 1995.p. 53}62.

[7] Portela, LM. On the identi"cation and classi"cation ofvortices. Ph.D. thesis, Stanford University, School ofMechanical Engineering, 1997.

[8] Sadarjoen IA, Post FH, Ma B, Banks DC, Pagendarm HG.Selective visualization of vortices in hydrodynamic #ows.In: Ebert D, Hagen H, Rushmeier H, editors. Proceedings ofVisualization '98. New York: ACM Press, 1998. p. 419}23.

340 I.A. Sadarjoen, F.H. Post / Computers & Graphics 24 (2000) 333}341

[9] Sadarjoen IA, Post FH. Geometric methods for vortexextraction. In: GroK ller E, Ribarsky W, LoK !elmann H,editors. Data Visualization '99, Proceedings of the JointEurographics IEEE TCCG Symposium on Visualization.Wien: Springer, 1999. p. 53}62.

[10] Farin G. Curves and surfaces for computer aided geomet-ric design. New York: Academic Press, 1990.

[11] Zhu ZF, Moorhead RJ. Extracting and visualizing oceaneddies in time-varying #ow "elds. Proceedings of theSeventh International Conference on Flow Visualization.WA: Seattle, 1995.

[12] Sadarjoen, IA. Extraction and visualization of geometriesin #uid #ow "elds. Ph.D. thesis, Delft University of Tech-nology, 1999.

[13] van Walsum T, Post FH, Silver D, Post FJ. Featureextraction and iconic visualization. IEEE Transactionson Visualization and Computer Graphics 1996;2(2):111}9.

[14] Jespersen DC, Levit C. Numerical simulation of #ow pasta tapered cylinder. Technical Report AIAA 91-0751,NASA Ames Research Center, Reno, NV, 1991. 29thAIAA Aerospace Sciences Meeting and Exhibit.

[15] Reinders KFJ, Post FH, Spoelder HJW. Attribute-basedfeature tracking. In: GroK ller E, Ribarsky W, LoK !elmann H,editors. Data Visualization '99, Proceedings of the JointEurographics IEEE TCCG Symposium on Visualization.Wien: Springer, 1999. p. 63}72.

I.A. Sadarjoen, F.H. Post / Computers & Graphics 24 (2000) 333}341 341