American Institute of Aeronautics and Astronautics1

Extraction of vortical flow features in a turbomachinerysimulation

Shuang ZhangAnsys/Fluent Inc., Lebanon, NH, 03766, USA

Stefano Mereu 1

Ansys/Fluent Inc., Evanston, IL, 60201, USA

Jin Yan and Dipankar ChoudhuryAnsys/Fluent Inc., Lebanon, NH, 03766, USA

[Abstract] In this paper, we use flow feature extraction to study a CFD application from theturbomachinery industry. We show that for this RANS, nearly incompressible simulation, our neweigen helicity density scheme outperforms the widely accepted λ2 scheme in identifying vortex regions.A new vortex coreline extraction algorithm is introduced, which extracts accurately the vortexcorelines of the horse-shoe vortices in this case. This new coreline algorithm, combined with the eigenhelicity vortex region scheme, shows promise in reducing the false positives that are affecting existingvortex coreline algorithms. We analyze the potential impact of the automatic feature extractionschemes on turbomachinery flow analysis.

Nomenclature∇u = velocity gradient tensorω = vorticity vectorΩ = vorticity tensorλ2 = second invariant of the symmetric tensor ||S||2+||Ω||2

∆ = discriminant of ∇ue0 = real eigen vector of ∇u when ∇u is not symmetrice1, e2 = eigenvectors corresponding to the complex conjugate eigenvalues of ∇u when ∇u is not symmetricHe = eigen helicity densityS = rate of strain tensoru = velocity vector

I. IntroductionHE application of computational fluid dynamics in turbomachinery design and analysis is of growingimportance. It has reached a stage where integral quantities, such as lift and drag coefficients, are not sufficient

for an effective study. Details of the three-dimensional flow fields are necessary to answer questions such as the 3Dblade geometry design, the film cooling holes design, the efficiency, and the impact of unsteady phenomena.

3D flow solutions are large-scale in storage, complex with unstructured grid, vector valued, and often transient.The visualization and analysis of these data present significant challenges to engineers, who usually have limitedexposure to scientific visualization. A certain level of automation is highly desirable in order to relieve the engineersfrom computer science jargons, and to allow them to focus on the most important information in a time-efficientmanner.

Qualitative visual observations are useful but not sufficient when coming to a design assessment. Visuallyobserved results should be made quantifiable in order to answer design questions. For example, an understanding ofvortices is very important to make appropriate design changes. Traditional visualization techniques could indicatethe existence of vortices, and show qualitatively where they are. However, an analysis on the impact of the vorticesto a design demands answers to questions like how strong the vortices are, where exactly the vortices are in

1 Development Director, Ansys/Fluent Evanston Office, 1007 Church St. Suite 250, Evanston IL 60201 USA.

T

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Copyright © 2007 by Ansys Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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comparison to the turbo blade, etc. These questions cannot easily be answered by the existing data analysiscapability.

Flow feature extraction has shown promises in meeting the challenges in automation and quantification. Inaddition to its parameter free nature in visualizing the data, the extracted flow feature can be further quantified,modeled, and even used to steer the solution process1,2.

Feature extraction, rooted from pattern recognition technology in signal processing community, has grown intoan important sub-division in visualization science3. The technology is still maturing, but attempts are being made tofocus the application of feature extraction in an engineering production environment4.

In this paper, we study the vortex features in a turbomachinery application. We extend our new vortexidentification scheme, eigen helicity density5, to analyze the results of a CFD simulation for turbomachinery. Weshow that our generic, Galilean invariant scheme, outperforms the widely used λ2 (ref 6) scheme in educing thevortex regions. We further introduce a vortex coreline identification algorithm which significantly reduces the falsepositives that are affecting existing coreline algorithms7. We demonstrate the application of the feature extractiontechnology in turbomachinery analysis, and provide an outlook of its potential in general data analysis andvisualization.

The test case that we have chosen to examine is the Durham test, which is a popular validation case used byEuropean Research Community on Flow Turbulence and Combustion (ERCOFTAC, a special interest group inturbomachinery), denoted as case F1 hereafter. Case F1 is a simulation of 3D steady flow in a linear turbine cascade.The geometry of the cascade is shown in Figure 1. In the simulation, only the flow over one blade is computed,while symmetry boundary conditions are enforced to model the cascade.

II. New vortex region identification scheme: eigen helicity density

A. Review of vortex region identification schemesVortex identification has been a topic of interests to both fluid dynamics and scientific visualization

communities. In this section, we will briefly review the schemes used in the fluid community. Ref. 5 has anoverview of traditional schemes based on pressure, vorticity, helicity, second invariant and discriminant.

One of the main questions that most traditional methods fail to address is the direction of the rotation of a vortex.The alignment of two vectors related to the flow field has a dominant role in answering the question. Attention hasbeen mainly centered around the relationships between the velocity vector (u), vorticity vector (ω), and realeigenvector of the ∇u (e0). Mostly explored combinations are velocity projected to vorticity (helicity)9, and velocityprojected to the real eigenvector of ∇u (ref. 7). Velocity is the Galilean variant component of these choices. Anatural choice replacement for Galilean Invariance is a vector associated with the eigen system defined by ∇u.

When analyzing this eigen system, the symmetric tensor derived from ∇u has been popular choice because itallows to work with real eigenvalues and eigenvectors. Upon examining two strain-rate based eigen alignmentmodels10,11, we conclude that eigenvectors of the strain-rate tensor S might not align with the direction of the vortextubes, consistent with the observations of ref. 12.

We do not think that the alignment scheme based on a symmetric derivative of ∇u is appropriate, becausevorticity production also contributes to the generation of strain. Instead, the full ∇u tensor should be considered.

For a general 3x3 tensor such as ∇u, only one real eigenvalue is guaranteed. The other two eigenvalues could beeither real or complex conjugates. The anti-symmetric part of ∇u not only determines the strength of the rotation(which equals to the imaginary part of the complex conjugate eigenvalues), but also contributes to the set-up of the

Figure 1. Geometry and setup of Durham test case (case F1).

Validation against experiments andverification of different turbulence models,plus some general discussions on CFDanalysis in turbomachinery design, can befound in ref. 8. In this study, we choose oneparticular simulation data as we are focusingon generic vortex extraction algorithm.

The simulation is performed in Fluent6.0with Reynolds stress transport turbulencemodel.

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plane of rotation. Additionally, the Cartesian space defined by the three eigenvectors of ∇u could be non-orthogonal, which may be the case for highly skewed vortices subject to significant amount of unsteady,compressibility, and/or baroclinic effects.

Based on the above analysis, ref. 5 proposed the following new scheme that presents a different alignmentscheme: eigen helicity density, or the He scheme:

≤∆>∆×

=0,0

0,

when

whennH swirl

e

ω

where ∆ is the discriminant of ∇u (ref. 13), nswirl = ereal x eimag = -(e1xe2) i/2; ereal = (e1+e2)/2; eimag = -(e1-e2) i/2, e1

and e2 are the two eigenvectors corresponding to the complex conjugate eigenvalues of ∇u. Note e0, the eigenvectorcorresponding to the real eigenvalue, is not necessarily aligned with nswirl.

In ref. 5, we compare the He scheme with various existing vortex region identification schemes: pressure,vorticity magnitude, Q, helicity and λ2. We show that for the compressible baroclinic Richtmyer-Meshkov flow, He

gives promising results.In this paper, we apply the He scheme to case F1, and compare the result with other schemes, specifically with

the state of art scheme λ2, which has been successfully applied to incompressible simulation results.

B. Application of the He scheme to identifying vortex regions in turbomachinery flowsFor the turbine blade mounted on wall, shown in Figure 2a, a famous horse-shoe vortical structure exists. In

turbomachinery industry, the upper “leg” of the horse shoe is called passage vortex. It is noteworthy that the portionof the lower “leg” below the blade is mixed with the boundary shear layer. An established vortical flow is onlyvisible off the trailing edge of the blade. Hence this visible portion of the lower “leg” is labeled tip vortex.

Figure 2a shows the result of the He scheme applied to case F1. We can see that He correctly identifies both“legs”. He also provides an accurate detection around the leading edge of the blade, where the horse-shoe vortexemerges. A number of regions are also educed with no clear vortical flow associations. A viable explanation mightrelate to the RANS nature of the simulation, where vortical flows below certain scales are averaged. Vortex sheetsmight appear as shear layers, while the eigen system of ∇u still shows the characteristics of vortices. Theverification of this hypothesis is left to future work.

The accurate detection of He method can be further appreciated when comparing its results against a number ofother popular vortex region identification schemes used by fluid scientists and engineers.

The results from the λ2 scheme6, the most widely accepted and applied vortex identification scheme, are shownin Figure 2b. λ2 is the second largest eigenvalue of the symmetric tensor Ω2+S2, where Ω and S are the vorticity andstrain-rate tensors, respectively the anti-symmetric and symmetric part of ∇u. It is clear that λ2=0 isosurfaces arenoisier when compared with results from Figure 2a.

We have shown that for highly-skewed vortex, i.e., flow fields with substantial unsteady, compressibility, and/orbaroclinic effects, the λ2 scheme has some fundamental limitations. For instance, this is the case for Richmyer-Meshkov flow examined in ref. 5. It is however somewhat surprising that, for the weakly compressible case studiedin this paper, the results obtained via the He method are significantly better than the λ2 results. We note that it ispossible to get cleaner result by adjusting λ2 isovalue in a trial-and-error manner, although this diminishes one of themajor advantages of λ2, i.e., the threshold-free property, as we will examine in the next section.

Ref. 10 proposed λ+ scheme, a variant of λ2. λ+ is defined as the eigenvalue of tensor Ω2+S2 which yieldsmaximum alignment between the corresponding eigen vector and vorticity vector. λ+ equals to λ2 in some regions,but not always. Since λ+ is a relatively new scheme, we are not aware of any previous application to practicalengineering simulation data. Figure 2c indicates that the results are marginally better than λ2. Some noisy smallregions are eliminated. However, major deficiencies similar to those of Figure 2b still exist.

Figure 2d shows the vortical structure educed by another popular scheme: Q, which is the second invariant of ∇u(ref. 18). Despite the noisy results comparing to He, the shear layer surrounding the wing is also incorrectly educedby Q. We also observed in our study that for vortices, Q value is usually fairly big. Small positive values of Q areusually associated with noise, invalidating the correspondence between Q ≥ 0 and vortex regions.

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C. ConclusionBased upon our study of vortex region identification schemes applied to the F1 case, the following conclusions

can be drawn:(a). In educing the two-leg horse-shoe vortical structure for visualization purposes, He scheme shows the most

accurate results. It is especially noteworthy that the results from the λ2 scheme appear somewhat poor, since thisscheme has been validated in many incompressible cases. This calls for a more careful examination before applyingλ2 as a generic vortex region identification scheme.

(b). Some region schemes are computationally more expensive than others. For examples, He and λ2 require tosolve the eigen system of ∇u, while Q does not. This is an important issue to keep in mind when efficiency andinteractivity are important.

(c). The threshold dependency is an issue that He scheme does not fully address. Although schemes such as λ2

and Q attempted to resolve the problem, by translating the definition of a vortex region to a fixed threshold of ascalar (e.g., λ2 ≤ 0 and Q≥ 0), in practice they are not working very well. This is because both schemes are based on∇u, a numerically sensitive quantity especially in unstructured grid. Numerical noise almost always requires a trialand error threshold. Furthermore, from physics standpoint, these schemes are based on assumptions on the flow fieldwhich might not be valid, as discussed above and in ref. 5.

In the next section we will discuss a different approach that removes the threshold dependency and has asignificant impact on the automatic post-processing of the results.

WALL

INFLOW (a)

(c)

(b)

(d)

Passage vortex

Tip vortex

Figure 2. Comparison of different vortex region identification schemes.(a). He; (b) λ2; (c) λ+; (d) Q.

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III. Vortex coreline algorithm and resultsA. Review of vortex coreline extraction schemes

In addition to the fluid dynamics literature on vortex identification, a parallel research effort is undertaken by thecomputer science community devoted to flow visualization. These research activities are mainly focused ondetermining the center of rotation of a vortex, i.e., the vortex coreline. A review in chronological order of theliterature related to this topic is given below.

The first literature implying a vortex coreline algorithm is from ref. 9. When applying the helicity as a vortexregion identification scheme, grid points with normalized helicity density ±1 are used to seed streamlines. Theauthors claim that these streamlines are vortex corelines. Two major issues about this approach are identified by ref.26. First, normalized helicity with ±1 might not be lying exactly on the grid points. Furthermore, they might notexist at all while the vortical structure are still seen, such as in cross flow separation vortices. Secondly, there areexamples where the integrated streamline deviates from the center of rotation. Ref. 26 further examined thecondition that the streamlines with zero curvature (center of rotation) will coincide with maximum normalizedhelicity line: extreme of velocity magnitude, vorticity magnitude, pressure and density all should occur on the line,which is seldom the case in practice.

Later works in the literature turned their attention to the topology theory of a vector field. Ref. 16 defined vortexcoreline by integrating curves from critical point (u=0), when ∇u has one real and two complex eigenvalues. Invector field topology, this type of critical points is classified as spiral saddle. The integration algorithm steps a smalldistance from the critical point, along the direction of the eigenvector corresponding to the real eigenvalue. The useof eigenvector, instead of velocity vector, is an important merit of this approach, which is incorrectly interpreted bysome later literature. Despite its sound approach and implementation in FAST visualization package, this algorithmhas not gained enough popularity.

Another vortex coreline algorithm based on vector field topology was introduced by ref. 7. Similar to ref. 16,eigenvalues and eigenvectors of the ∇u are used. The eigen vector corresponding to the real eigenvalue is projectedto the velocity vector, to yield a “reduced velocity” field. Critical points of this field define the loci of the vortexcoreline. The authors compare vortex coreline extraction results between their method and the method from ref. 16and claim to achieve more accurate results. However, in the regions where vortices are clearly identified, the resultsfrom ref. 16 produces more coherent lines, while in the regions where the two sets of results differ ref. 7's algorithmproduces noisy segments and it is not easy to draw a definite conclusion. One problem that has been correctlypointed out about ref. 16’s algorithm is convergence: integrating streamlines of eigenvector field might not convergeto the coreline, because sometimes it start spiraling. Our experience shows that there are two potential factors thatmight cause this problem. The integration parameters are more sensitive when the spirals (imaginary part of thecomplex eigenvalue) are not overwhelmingly stronger than the real eigenvalue. Or, more seriously, the assumptionthat the vortex coreline is the integral line of eigenvector might be wrong. Note this assumption is made based uponthe linearization of the flow field, i.e, the vector field topology is determined by ∇u, but not by higher order terms.

Since both papers are using ref. 16’s eigen solver, we would categorize ref. 7's corelines algorithm as a variant ofref. 16’s. Another interesting variant is ref. 19, where the authors exercised Sperner’s lemma in combinatorialtopology to implement the procedure of locating projected critical point. Only the cells that contain the vortexcoreline are identified in this study.

Driven by its commercialization by major CFD postprocessors Fieldview, Ensight Gold, and Tecplot, ref. 7’salgorithm has gained great exposure in the engineering simulation community. Refinements have also beensuggested, such as connecting line segments, and replacing velocity vector with convection vector (forcomputational efficiency and robustness), or with vorticity to enforce Galilean Invariant property23. Note this lastsuggestion is associated with helicity based method by ref. 9. In addition to a few application case studies17,20,21, thismethod has also been integrated into a finite element solver for guiding mesh refinement around the vortex core15.

Ref. 14 assume that vortex corelines are streamlines of the vorticity field. During the vorticity streamlineintegration, the pressure minimum in the plane perpendicular to the vortex coreline is used in the predictor-correctorprocedure.

Ref. 24 used the maximum line of vorticity field to extract a vortex coreline. The maximum line of a vector fieldis defined as the locus of points where the vector magnitude is maximum in a plane perpendicular to the vector.

A milestone work in vortex coreline identification is the parallel vector framework23. This work extracted thecommon components from all vortex coreline algorithms mentioned above, that is, the vortex coreline usuallyinvolves two vectors that are parallel to each other. All the existing algorithms could be reformulated with the samemathematical representation, which improves efficiency, accuracy, and software reuse. For example, ref. 7’s reduced

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velocity is equivalent to finding points where the acceleration a is parallel to the velocity v, or to finding points ofzero curvature. The acceleration vector a is defined as a=Du/Dt = ∇u • u for steady flows.

An additional contribution of Roth’s thesis is a high order scheme, which is motivated by the difficulties inapplying ref. 7’s algorithm to turbo-machinery flows, where weak, curved secondary vortices dominate. Ref. 25attempted an algorithm for finding vortex corelines using 2nd order derivative of velocity, also based on their parallelvector framework. The success however is limited due to numerical difficulties when evaluating high-orderderivatives of u.

Connections have been made between the helicity density and the streamline curvature26, but no attempt hasbeen made to analyze the relationships of the parallel vector vortex coreline detection framework with the vortexregion boundary identification schemes.

B. Advantages of vortex coreline over vortex regionThe determination of a vortex coreline has several advantages over region boundary surface based vortex

methods, which are summarized below and also clearly illustrated in Figure 3.1. Coreline is able to isolate vortices close to each-other. Due to its local nature, it is capable of identifying

secondary vortices. Region based schemes usually produce vortex regions that are lumped together, for example, thetwo vortex tubes in Figure 3a. Weaker vortices could be completely omitted if they are enclosed within a biggervortical structure.

2. Coreline can be quantified easily and consistently. The precise location and direction of the vortices providegreat power to engineering analysis.

3. Coreline relieves the user from manually finding a threshold with trial-and-error, hence provides the potentialfor automation. Given the physical ambiguity in the vortex definition, and the numerical inaccuracies associatedwith the definition of the appropriate scalar, region based approaches always require the definition of an appropriatethreshold. Coreline, on the contrary, has the best conceptual clarity. Whenever a vortex is resolved, a center line ofrotation should exist. No manual inputs are required for the definition of these corelines.

4. With vortex coreline, an accurate construction of the vortex structure, including the boundary surface of thevortex region, is possible, but not vice-versa: geometry center line of a vortex boundary does not necessarilyconverge to a vortex coreline.

Obviously, these advantages do not come without a cost. As we reviewed in the previous section, corelinealgorithms are more complex and computationally intensive compared to region based approach.

Figure 3. Advantages of vortex coreline over vortex region schemes. (a). Vortex surface identified bytypical vortex region based schemes. Note the two vortex tubes are lumped into one surface, whilethere is no way to tell the configuration of each tube, and the direction of rotation. Furthermore, thesurface will look different at another threshold, sometimes dramatically even with a small step. (b).Vortex corelines of the two tubes, which clearly indicates the two separate vortices. The direction ofrotation is indicated with colors.

C. vortex coreline algorithmIn this section, we propose a new coreline algorithm, which includes some of the advantages of the region based

schemes. The algorithm is focused on improving the quality of the vortex coreline, and also on providing a level ofautomation suitable to engineering analysis.

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Many of the methods reviewed in section III.A are facing one common issue: the presence of false positives, i.e.the fact that there are non-negligible amount of locus points being extracted, which are not associated with vortices.This is due to the fact that critical point evaluation (where velocity vector vanish) has to be done tolerantly, as in anyother numerical zero evaluation. In reducing the false positives, some scalar based post-filtering (filter afterextraction) mechanisms are introduced in order to filter extracted vortex corelines. As these post-filters increase theclarity, the coherency and completeness of the extraction is sacrificed. These post filters in many cases also filter outportion of real vortex corelines. In the example shown in figure 4a, the continuous lines become segmented after thepost filter.

In this paper, we propose that the filtering should be done both before and after applying the vortex corelineextraction algorithm. The pre-filtering criteria should be based on the vortex region schemes coming from fluidresearch as discussed in section II. Specifically, pre-filtering has a number of advantages,

(a). The areas that produce false positives are usually filtered out by a good vortex region indicator.(b). Region identification schemes are mostly field calculations, which are less computational intensive than

coreline algorithms. A pre-filter hence greatly improves the computational efficiency.Ref. 4 developed a concept of “domain reduction” before applying vortex coreline algorithms, with the main

focus on the second benefit pointed out above.In this work, we introduce the use of the He scheme as a pre-filter, in order to ensure a good starting point for the

vortex coreline algorithm. We also implement a new vortex coreline extraction algorithm, based on parallel vectorframework23.

The different steps of the algorithm are described in the section below. The algorithm can be applied to analyzesimulation results from any general CFD code regardless of the type of grid (structured or unstructured) or thediscretization type (finite volumes or finite elements). In addition to solution data (velocity and pressure), gradientsof velocity are assumed to part of the output from the CFD code. It is important to point out that solver-consistentinterpolation and gradient evaluation are crucial aspects in determining the quality of the coreline.

Pre-filtering.The addition of a pre-filtering step is an important point when comparing the algorithm to others existing in the

literature. The pre-filtering check involves a loop on all the internal faces in the grid. Before searching for a solution,a check is performed in order to determine if a solution is likely to be present in the face. Faces that fail the checkare simply skipped from the remainder of the computation.

This not only improves the efficiency, but also significantly reduces the noise and the false positive pointsassociated with the noise. This is beneficial for later steps such as reducing line segments and easier connection.

The pre-filtering check involves the use of a scalar field. The choice of the scalar field is dictated by threefactors:

- Ease and efficiency of computation- Possibility to use normalized values around a bounded value.- Galilean invariant.

In our tests, we found that the use of the discriminant is an appropriate choice since it gives a good balancebetween efficiency of computation and accuracy of results. The use of the normalized eigen helicity density, withthreshold value of 0.9, is recommended when the most accurate results are desired. The disadvantage of this scalar isthat it requires the eigen solution to be computed for the whole computational domain, resulting in significantcomputational effort.

Other possible choices include λ2, λ+, Q, and helicity. However helicity is not Galiean invariant, neither it isbounded, hence it is impossible to select a threshold that is applicable to every situation.

Parallel Vector builderThe Parallel Vector builder refers to the operation between a pair of vector fields that can be seen as the

fundamental building block for computing different line-type features of vector and scalar fields23. Ourimplementation involves a loop over the internal faces of the computational domain that have passed the pre-filtering check. For each face, a Newton-Raphson iteration is performed in order to determine if a solution, wheretwo vectors are parallel, exists within the face. Faces that meet these criteria are added to a list of possible solutionpoints.

Different choices for the search of parallel vectors are possible. We found that parallel vector builder does notprovide reliable solutions when high order derivatives of a solution vector are used. This is not an inherent limitationof the algorithm, but it is due to the difficulty in computing high order derivatives from general CFD solutions. Forthis reason, we provide an implementation of the parallel vector for convection alignment and vorticity alignment.

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a. Convection AlignmentIn this implementation, vortex corelines are defined as the 3D curves where velocity vector (u) is parallel to

convection vector (u•∇u). Heuristically, this is equivalent to stating that u is parallel to e0 (which is the only realeigenvecotr of ∇u). Ref. 23 showed that u//e0 is a sufficient condition for u//(u•∇u), but there is no mathematicalproof that this is also a necessary condition. In practical applications, discrepancies are observed: usually u//e0 is asubset of u//( u•∇u), which might due to the numerical errors in the eigenvector evaluation.

Nevertheless, this equivalency avoids the expensive eigenvalue calculation in the original algorithm7. When theflow is steady, this definition is also equivalent to loci of zero curvature points in the flow. Note this is notnecessarily the streamline with minimum curvature.

b. Vorticity AlignmentVorticity alignment is the algorithm that has been used for the results obtained in this paper.The Vorticity Alignment implementation finds vortex corelines defined as loci where velocity vector is parallel

to vorticity vector. Theoretically, this is equivalent to finding the ridge line of the normalized helicity field. Note thenormalization implicitly involved in the algorithm results in discrepancies between the vortical structure identifiedin helicity and this algorithm. The normalization usually lifts numerical noise (where one of the two vectors areextremely small, which gives unity (high) normalized helicity, but not necessarily high helicity).

As stated above, the Parallel Vector algorithm operates on two vectors (that are assumed to be defined on eachnode of a face) and finds if there is a point where the two vectors are parallel.

Starting from the nodal values of the two vectors, locating the points where these two vectors are parallel isequivalent to minimizing the vector valued function

f = v x w = 0

A Newton-Raphson iteration is performed on each face (regardless of face geometry) in order to look for asolution within the face. During each iteration, bilinear interpolation is used to locate the position and compute theobjective function value.

It is important to note that a solution can be obtained only using a relatively loose tolerance (~10-3) asconvergence criteria. An ‘exact’ zero evaluation is not obtainable from a numerical discretization. A strictertolerance will not produce more accurate results. Instead, strict tolerance may yield no result.

Post-filteringThe first step in the post-filtering algorithm is the computation of the rotation strength based on vector field

topology theory, for each solution point. The points with zero rotation strength solution are discarded.The second step is a filter based upon a scalar quantity. All the quantities used for pre-filtering could be used for

post-filtering to further reduce the false positives of the extracted point-set., although it is suggested that the quantityfor post-filtering should be different from one used in the pre-filter. Our preferred choice is the eigen helicitydensity, since the post-filtering computation is performed on a small subset of points. Eigenvalues and eigenvectorsare already computed as part of the rotation strength calculation, therefore the computation of eigen helicity densityonly adds a small computational effort.

ConnectionThe ‘connection’ step refers to the construction of the geometric coreline based upon the filtered data built in the

parallel vector step. The data set consists of a number of points corresponding to the faces that are candidates forbeing part of the vortex coreline. Thanks to the pre and post filtering, most of the cells end up with points on two oftheir faces, which implies a uniquely defined path for the vortex coreline. This path can simply built by connectingtwo faces within a cell and repeating the process for the adjacent cells until a face that is part of the solution isencountered.

This is an advantage over ref. 7's algorithm, where each cell produced a segment, but the segments are notconnected.

It is also important to note that the connection step inherently allows the determination of multiple corelines, ifpresent in the solution. In our implementation, multiple corelines are treated as separate surfaces that can beindividually visualized for post-processing operations. An additional filter based upon the ‘length’ of the corelinecan also be built in order to improve the clarity of the results. For instance, corelines that are made of a very smallnumber of faces (e.g. 3) can be removed from the solution.

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D. ResultsIn Figure 4 we compare the results obtained with our algorithm and the results from existing coreline

implementations. The comparison is done against two different flavors of the MIT algorithm based upon ref. 7'swork, which is available in many commercial post-processing packages, hence widely used.

Figure 4 Comparison of vortex coreline algorithms. (a) Vortex corelines extracted by ref. 7 in one ofcommercial post-processing package. (b) Vortex coreline extracted by ref. 20. (c). Vortex corelinesextracted by the algorithms presented in this paper. Note the view is rotated to expose better the end ofthe two legs. A total pressure loss plane contour plot corresponds to ref. 8 is shown for verification. Eachpoints on the coreline are wrapped with a sphere, with its radius corresponds to the rotation strength,defined as the imaginary part of the complex eigenvalues of ∇u. (d). Streamlines seeded around thevortex corelines show the spiral flows around the vortex corelines. A colormap corresponds to totalpressure loss is applied to the streamlines.

Figure 4a shows the vortex corelines extracted by one of the commercial postprocessors using ref. 7's algorithm.In the parallel vector framework, this algorithm is based on the alignment of velocity vector with the realeigenvector of ∇u, if ∇u has only one real eigenvector. It is clear to see the false positives and the discontinuouslines. ref. 20 suggested that a replacement of velocity with vorticity might yield an improvement of the vortexcoreline in terms of solution completeness, i.e., longer vortex corelines. Figure 4b, produced by employing thisvorticity based alignment schemes, shows the opposite in this application. The result is even noisier with many falsepositives, and the tip vortex is almost completely gone.

(c)

(a) (b)

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Figure 4c and 4d show the results obtained with the proposed algorithm. Two continuous vortex corelines areclearly extracted. In Figure 4c, we placed a plane cut at the measured station where a comparison the total pressureloss with experimental data is available8. The accurate capturing of the center of rotation, for these two closelyadjacent vortices at this station, is remarkable.

In addition to offering more accurate results, our implementation of the algorithm has also been focused toprovide significant usability improvements. The extracted vortex corelines are represented as separate line segmentsthat can be selected to perform further operations such as pathlines seeding, color coding, and reduced orderrepresentation based on the selected coreline segments, as in Figure 4c.

In Figure 4c we show the points that form the vortex coreline wrapped by spheres, whose radius corresponds tothe rotation strength of the vortex coreline at that point. Rotation strength is defined as the imaginary part of thecomplex eigenvalue of ∇u, a quantity that is related to vorticity magnitude.

In Figure 4d, an array of streamline is released around the vortex corelines and integrated in both forward andbackward directions. In addition to showing the spiral flow patterns (which verifies the accuracy of the corelinecomputation), these streamlines are also colored by total pressure loss.

Figure 5 shows examples of the quantitative results that can be obtained from the extracted vortex corelines.

Figure 5 Quantification of case F1: the two legs of the horse-shoe vortex, red – passage vortex; blue –corner vortex. Y direction points counter-streamwise direction, i.e, these plots start from the end of thechannel, approaching the blade along the legs. (a). Y coordinate versus rotation strength. (b). Ycoordinate versus distance above the wall (Z coordinate).

In Figure 5a, we plot rotation strength against the streamwise coordinate Y. Interesting trends are observed forboth legs. The passage vortex (blue curve), it is gaining strength during the initial roll-up phase from the wall shearlayer (Note that it is negative). Peak strength is reached above the blade, and we see a dissipation of the strength ofthe passage vortex. When approaching the trailing edge of the blade, a sudden change of the dissipation rate isobserved. At the same time, we see the emergence of the corner vortex, which is gaining strength off the bladebottom shear layer. The correlation of the growing corner vortex and speed-weakening of the passage vortex isinteresting and remains to be studied in further detail in the future. When the corner vortex gained its peak strength,the two legs display remarkable similarity in both the absolute value of the strength and the dissipation rate.

In Figure 5b, we plot Y coordinate versus Z coordinate, which is the distance of the vortex coreline above thewall. This distance is an important parameter when designing the turbo blade cascade.

E. ConclusionIn this section, we reviewed the main literature in the past decade on vortex coreline extraction. Combining the

advantages of vortex coreline and vortex region identification schemes, we designed and implemented a new vortexcoreline algorithm, which yields good results comparing with existing state of art algorithms and can be used as aneffective tool for engineering analysis of the flow simulation.

(a) (b)

American Institute of Aeronautics and Astronautics11

IV. Closing remarks

The continuous increase in computing power allows CFD simulation to be run with increasing larger meshes. Atthe same time, the lack of capabilities for interpreting the huge amount of information produced by these simulationsdemands innovations in data-post processing and visualization techniques. The main challenge is to relieve thelaborious task of manual exploration of the areas of interest, and to translate quickly and accurately the large amountof data coming from the simulation into engineering information. One key element in meeting this challenge isextensive use of feature based flow analysis, i.e. extract automatically region of interests, or features, tracking thesefeatures in time and quantify them in engineering useful terms. We have shown how extracting vortex feature can beused to achieve this goal. Thus, the purpose of the work described in this paper has been two-fold:(a) Investigate advances and refinement in the vortex coreline algorithm.

Following a review of the region and coreline based identification schemes, we have proposed an algorithm thatcombines the advantages of the two methods and we have shown that it outperforms existing algorithms for aturbomachinery application. We have also further tested, in the context of an incompressible flow, the region basedHe scheme that had previously been tested in a compressible, baroclinic flow5.(b) Provide an implementation that can be easily used in engineering applications.

We have provided implementation through a software architecture that is designed to allow full access to theextracted vortices, thus allowing accurate quantification without the need of trial-error or manual procedures. Thisstep has the potential to push the technology beyond visualization to further allow flow control, optimization, 3Dsimulation/experiment data assimilation and sensitivity analysis.

Acknowledgments

The authors acknowledge the support from many colleagues at Fluent. In particular, S.Z. would like to thank thevaluable discussions with Paul Felix, Vahe Haroutunian, Chris Hill, and Greg Stuckert.

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