a new parameter in vortex identification and visualization

A NEW PARAMETER IN VORTEX IDENTIFICATION AND VISUALIZATION:SYMMETRY OF VORTICAL FLOW

Katsuyuki NakayamaDepartment of Mechanical Engineering

Aichi Institute of TechnologyToyota, Aichi 470-0392

JapanEmail: [email protected]

Yasumasa OhiraGraduate School of EngineeringAichi Institute of TechnologyToyota, Aichi 470-0392


Shoko YamadaGraduate School of EngineeringAichi Institute of TechnologyToyota, Aichi 470-0392


ABSTRACTThe present investigation introduces a parameter to repre-

sent symmetry of a vortex. An important feature of a vortex to beidenti�ed in many �uid engineering �elds is its stability derivedfrom the �ow kinematics, which is associated with the symmetryof its vortical �ow. Although many vortex de�nitions and identi�-cation methods have been proposed in terms of various physicalaspects of a vortex, such symmetry property or its identi�cationis lacked. We focus on the �ow geometry speci�ed by the ve-locity gradient tensor, and show that the feature of its complexeigenvectors derives the symmetry of the vortical �ow in the swirlplane. This property is invariant, and extracts the high symmetryvortex or vortical region irrespective of the intensity of the vor-tex. It does not require additional calculation except the complexeigenvectors, but it brings useful information of the �ow stateand skewness of vortices.

INTRODUCTIONVortices are associated with many �uid problems or engi-

neering �elds such as aerofoil, wind turbine or turbine blade,power plants, and structures. In these engineering �elds, iden-ti�cation of vortices and their stability are important subjects tobe analyzed.

On the other hand, the universal de�nition of a vortex hasnot been established, and many vortex de�nitions have been pro-posed corresponding to physical aspects of the characteristics ofa vortex, i.e. invariant swirling �ow motion speci�ed by the

velocity gradient tensor ∇v (the ∆-de�nition [1]), the positiveLaplacian of the pressure associated with the vorticity that ex-ceeds the irrotational strain (the Q-de�nition [2]), the Hessian ofthe speci�c pressure (discarding the unsteady straining and vis-cous effects) to identify the pressure minimum region inducedby vortical �ow motion (the λ2-de�nition [3]), the Hessian of thepressure and swirling �ow in the pressure minimum plane [4],the vorticity [5,6], the helicity [7], eigen-helicity density de�nedby the complex eigenvectors of ∇v and the vorticity [8], as wellas other characteristics [9�14]. In the analysis of vortex identi�-cation, it might be appropriate to apply a vortex de�nition whichis suitable to detect a vortex with speci�c physical characteristicsthat we attach importance to.

One of the important points in the vortex identi�cation inengineering �elds is to detect stable vortices. A vortex has char-acteristics that its �ow geometry (topology) brings stability tomaintain the �ow state itself. And the symmetry of vortical �owin a vortex is associated with this stability of the vortex [15]. TheBurgers vortex [16] is an axisymmetric vortex that maintains asteady state. The Rosenhead vortex [17] and a vortex model byVatistas [18] are also examples of axisymmetric vortices. Fur-thermore, the identi�cation of vortical �ow with vortex modelsis required in engineering �elds such as wind turbine technolo-gies [19]. In this identi�cation, analysis of the symmetry of thevortical �ow is indispensable. However, in spite of many de�-nitions focusing on various physical characteristics of a vortex,such property specifying the symmetry is lacking.

The eigenvalues of ∇v have been used for the classi�cation

Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014

November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-39859

1 Copyright © 2014 by ASME

of �ow kinematics (geometry) [1]. The complex eigenvalues arealso associated with popular vortex de�nition such as the Q- andλ2- de�nition, and applied in other vortex identi�cation methods[4,13,14,20�22]. Nakayama investigated the detail �ow geome-try derived from ∇v, in which the vortical �ow in the swirl planeis decomposed into the radial and azimuthal velocities and thesymmetry properties for respective velocity components are de-rived [23]. It is also shown that the complex eigenvalues are dif�-cult to specify the vortical �ow geometry precisely and uniquely,and that an additional property for the symmetry is necessary.

These symmetry properties for the radial and azimuthal ve-locities are based on an invariant in terms of the complex eigen-vectors, and this invariant itself represents a symmetry of thevortical �ow geometry around a point. It enables to analyze thesymmetry of the local vortical �ow in the vortical region, andto identify the high symmetry region or high symmetric vorticesirrespective of the intensity of a vortex. This identi�cation withthe symmetry is also ef�cient in the vortex identi�cation withinvortex models.

Hereafter we show an invariant derived from ∇v, that repre-sents a symmetry of vortical �ow. This invariant is necessary tospecify the vortical �ow geometry uniquely, and it gives a newaspect of vortical �ow analysis with respect to the skewness or�ow state of vortices.

A NEW PARAMETER OF SYMMETRY OF VORTICALFLOWFlow geometry speci�ed by the velocity gradient ten-sor

First we summarize the �ow geometry derived from ∇v. Weconsider the velocity �eld in an inertia coordinate system, xi (i=1;2;3), where a point to be considered is the origin and moveswith its velocity, i.e., an reference frame with this point. ThenTaylor expansion of the velocity vi around the considered point(origin), neglecting the second and higher order terms, gives thefollowing differential (autonomous) equation with respect to xi:

dxidt=

∂vi∂x j

x j; (1)

where summation notation is applied. We consider incompress-ible �uids, i.e., ∂vi=∂xi = 0, then the algebraic condition that theeigenequationΦ(λ ) have conjugate complex solutions is derivedas the ∆ de�nition in the inequality: ∆ � (Q=3)3+(R=2)2 > 0,where Q and R represent the second and third invariants, re-spectively [1]. Then the solution trajectory of Eqn. (1) can beexpressed in terms the eigenvalues εi and the eigenvectors ξ

(i)

(i= 1;2;3), i.e.,

~x=3

∑i=1cieεit~ξ (i); (2)

where ci (i = 1;2;3) are constants. If ∇v has a pair of complexconjugate eigenvalues ε1;ε2 = εR� iψ (ψ > 0) (the upper opera-tor of � concerns the �rst variable of the left-hand side), where iis the imaginary unit, and one real eigenvalue ε3 = εa, we denotethe complex eigenvectors of ε1 and ε2 as ξ

(1);ξ (2) = ~ξp� i~ηpand that of ε3 as ξ

(3) = ξ axis. Then the solution trajectory ofEqn. (1) can be represented as

~x= 2c01eεRtf~ξp cos(ψt+ c02)�~ηp sin(ψt+ c02)g+ c03eεatξ axis; (3)

where c0i (i = 1;2;3) are constants [23]. Equation (3) indicatesthat the trajectory (�ow) swirls with an angular velocity ψ inthe plane de�ned by ~ξp and ~ηp, hereafter referred to as the swirlplane P . It then converges towards or diverges from the con-sidered point, and proceeds along a vortical axis ξ axis. The signof εR classi�es a convergent (in�owing) or divergent (out�ow-ing) vortex behavior. We note that Zhou et al. used the imag-inary part ψ as a parameter in the vortex identi�cation coiningthe term �swirling strength� [20]. Nakayama showed that ψ isequal to the geometrical average of the intensity of azimuthal ve-locity (eigenvalues of the quadratic form of azimuthal velocity)in a plane (e.g. P) in the case that ∇v has a pair of complex con-jugate eigenvalues, and de�ne this property as �swirlity� [23].

A property specifying symmetry of vortical �owThe complex eigenvectors have an important feature that

the directions and lengths of real and imaginary vectors, i.e.,~ξp and ~ηp, depend mutually. ~ξp and ~ηp are speci�ed using theeigenequations of ∇v (= A);

A(~ξp� i~ηp) = (εR� iψ)(~ξp� i~ηp): (4)

This eigenequation with respect to real and imaginary partsyields

(A� εRE)~ξp =�ψ~ηp; (5)

(A� εRE)~ηp = ψ~ξp: (6)

Equations (5) and (6) derives

(A� εRE)2~ζ =�ψ2~ζ ; (7)


FIGURE 1. FLOWGEOMETRY INP DERIVED FROM∇vWITHSAME COMPLEX EIGENVALUES BUT DIFFERENT r, WHEREεR =�1 AND ψ = 2. (i) r = 1=4, (ii) r = 1=2, AND (iii) r = 5=6.

where ~ζ = ~ξp or ~ηp. Then ~ξp and ~ηp are the eigenvectors of thematrix (A� εRE)2 with respect to the degenerated eigenvalue�ψ2. This indicates that the directions of ~ξp and~ηp are arbitraryvectors in a speci�c plane (the eigenplane Q corresponding tothe eigenvalue �ψ2 of (A� εRE)2). We can specify that ~ξp?~ηp(~ξp;~ηp 2Q), and the directions of ~ξp and ~ηp are given from thisorthogonal condition and Eqn. (5) or (6):

~ζ (A� εRE)~ζ = 0: (8)

Thus ~ξp and ~ηp is speci�ed from the above condition. Then therelative length of these vectors, i.e., the ratio

r =j~ξpjj~ηpj

orj~ηpjj~ξpj

(0< r � 1); (9)

is speci�ed from Eqns. (5) and (6). We note that j~ξpj 6= j~ηpj ingeneral.

r is a property to specify vortical �ow geometry, which hasa different role from that of the complex eigenvalues. Figure 1shows the vortical �ow geometry speci�ed by ∇v, i.e., the com-plex eigenvalues and r. In spite of the same complex eigenvalues,the geometries differ according to r. As r changes from 0 to 1,deformed vortical �ow geometry changes to be symmetric.

We note that r is independent of the complex eigenvalues εRand ψ , and is invariant associated with the complex eigenvec-tors. In addition, the vortical �ow geometry is not determineduniquely without r as shown in Fig. 1. This property r speci�esnot only the vortical �ow geometry with respect to a consideredpoint, but also the degree of the symmetry. This symmetry isassociated with stability of a vortex in terms of �ow kinematics.We note that r becomes 0 when a vortex disappears and that rdevelops from 0 when a vortex is generated (the eigenvalues of∇v change from real numbers to complex numbers). Then r has

characteristics that, in a vortical region of a �nite scale vortex, rbecomes high in the core of a vortical region and increases as thevortex develops. The analysis of r gives useful information withrespect to skewness, situation, and stability of the vortical regionin terms of �ow kinematics in an instantaneous velocity �eld.

NUMERICAL SCHEMEThe integration of the analysis of the symmetry into vor-

tex identi�cation is simple because the popular vortex de�ni-tions for the identi�cation of a vortex, e.g., the ∆-, Q- and λ2-de�nitions and other corresponding de�nitions, are associatedwith ∇v. Then this detail analysis of vortex identi�cation withr is possible by including the calculation of the complex eigen-vectors.

The algorithm of this analysis in vortex identi�cation with∇v is described as follows:

1. Calculate ∇v of a considered point.2. If ∇v has conjugate complex eigenvalues, calculate theeigenvalues (εR, ψ).

3. Specify the complex eigenvectors (~ξp, ~ηp) of ∇v so that~ξp?~ηp. The directions of ~ξp and ~ηp are calculated fromEqns. (7) and (8), and r is speci�ed from Eqn. (5) or (6).

4. Calculate properties corresponding to the applied vortex de-�nition.

In vortex identi�cation associated with ∇v, the above thirditem is the only additional calculation.

NUMERICAL ANALYSISWe show an example in a direct numerical simulation of

decaying isotropic homogeneous turbulence analyzed with thepseudo spectral method in the region (2π)3 composed of 2563nodes with wave number jkij < 85 (i = 1;2;3) of wavenum-ber vector ~k = (k1;k2;k3) and time step ∆t = 0:001. Theinitial velocity �eld is given by a energy spectrum E(k) =(k=kp)4 expf�2(k=kp)2g=2 (k= j~kj, kp= 4) with random phasesof wavenumber vectors, where the Taylor Reynolds numberReλ � 154, the Taylor microscale λT � 0:58, the Kolmogorovlength η � 0:021, and the eddy turnover time te � 0:064.Figure 2 shows a vortical region, i.e., the contour of non-dimensional swirlity φ = 2 where φ is equal to ψ divided by theroot mean square value of it. In Fig. 2, The non-dimensional timet 0 (the time divided by the initial eddy turnover time) is 124:1, af-ter the peak of the enstrophy.

We concentrate on a vortex to analyze its characteristics ofvortical structure, and focus on a vortex indicated by the arrow inFig. 3 which is an enlarged �gure of Fig. 2. Figure 4 shows thecontours of φ = 1:5;2 and r = 0:85 in the vortical region of thevortex indicated in Fig. 3. The contour φ = 2 is inside of that


FIGURE 2. VORTICAL REGION IN DECAYING ISOTROPICTURBULENCE (t 0 = 124:1), WHERE φ = 2.

FIGURE 3. AN ENLARGED PART OF SUBREGION IN FIG. 2AND A VORTEX TO BE ANALYZED IN TERMS OF φ AND r DIS-TRIBUTIONS (INDICATED BY AN ARROW).

FIGURE 4. CONTOURS OF φ AND r IN THE VORTEX INDI-CATED IN FIG. 3, WHERE φ = 1:5;2 AND r = 0:85.

of φ = 1:5, and, a high symmetry region, the contour r = 0:85,is inside of that of φ = 2. If we increase the value of r in thecontour, its contour moves into the core region of the vortex.Fig. 5 shows the contours of φ = 2 and r = 0:9, in which thecontour of r = 0:9 lies inside of that of φ = 2. Figure 6 shows

FIGURE 5. CONTOURS OF φ AND r IN THE VORTEX INDI-CATED IN FIG. 3, WHERE φ = 2 AND r = 0:9.

FIGURE 6. CONTOURS OF φ AND r IN THE VORTEX INDI-CATED IN FIG. 3, WHERE φ = 2:7 AND r = 0:9.

that most of part of the contour of r = 0:9 is still inside of thatof φ = 2:7. These �gures indicate that high r region is locatedin the core region of the vortex where φ is high. As for a vortexwith a �nite scale, the swirlity and symmetry is higher in its coreregion.

On the other hand, each vortex has each degree of the sym-metry in an instantaneous velocity �eld, irrespective of the inten-sity of the vortex. Figure 7 shows a contour (distribution) of φ

in a plane parallel to the swirl planeP of a strong vortex whichhas a local maximum point of φ (φ � 3) indicated by an arrowin Fig. 7 (t 0 = 46:5). Figure 8 shows a contour of r in the planein Fig. 7, which indicates that the distribution of r does not havea sharp peak and that r � 0:61 at this local maximum point ofφ . The local vortical �ow of this vortex is obtained by subtract-ing the velocity �eld by the velocity components at this point, asshown in Fig. 9.

Although this vortex has a strong swirling, i.e., high swirlingintensity of the swirlity, the vortical �ow geometry is elliptic andvertically long shaped by a shear �ow (in the vertical direction)in the right and left sides of the vortex in Fig. 9. The symmetryproperty r � 0:61 suggests this situation of the geometry.

We focus on a weak vortex in a similar way. Figure 10 showsa contour (distribution) of φ in a plane parallel toP of a vortex


FIGURE 7. CONTOUR OF φ IN A PLANE PARALLEL TOP OFA STRONG VORTEXWITH A LOCALMAXIMUM φ (φ � 3) INDI-CATED BY AN ARROW.

FIGURE 8. CONTOUR OF r IN THE PLANE SPECIFIED INFIG. 7 (STRONG VORTEX).

which has a local maximum point of φ (φ � 0:95) indicated byan arrow. Figure 11 shows a contour of r in the plane in Fig. 10,which indicates that r � 0:77 at the local maximum point of φ .Figure 12 shows the local vortical �ow of this vortex, where thevelocity �eld is subtracted by the velocity components at thispoint. In spite that this vortex is weak (low φ ) and small, thisvortical �ow has a high symmetry and is stable in terms of its�ow kinematics.

This analysis of the symmetry enables to investigate tempo-ral processes of the development or decay of a vortex where notonly the strength of swirling but also the symmetry of vortical�ow changes. Then the analysis is effective in the precise obser-vation of the vortex phenomena. It is also useful in the vortexidenti�cation within vortex models in experimental or numericalanalysis data, in order to identify the �ow symmetries (axisym-metry) speci�ed in the models.

FIGURE 9. VELOCITY DISTRIBUTION OF THE STRONG VOR-TEX SPECIFIED IN FIG. 7.

FIGURE 10. CONTOUR OF φ IN A PLANE PARALLEL TO P

OF A WEAK VORTEX WITH A LOCAL MAXIMUM φ (φ � 1) IN-DICATED BY AN ARROW.

CONCLUSIONA parameter, which is derived from complex eigenvectors

of the velocity gradient tensor, is introduced to identify the sym-metry of local vortical �ow in vortex identi�cation. It enablesto identify the symmetry of vortical �ow without investigatingthe velocity distribution, and vortices with speci�ed (high) sym-metry irrespective of the intensity of swirling. An analysis in adecaying isotropic homogeneous turbulence shows that a vortexhas a characteristic that the symmetry is high in the core region,similar to the swirlity.

The present symmetry property and its analysis may be ef-fective for observing the detail vortical �ow phenomena or de-velopment of a vortex.

This analysis requires simple additional calculation, butbring about further detail information of vortices.


FIGURE 11. CONTOUR OF r IN THE PLANE SPECIFIED INFIG. 10 (WEAK VORTEX).

FIGURE 12. VELOCITY DISTRIBUTION OF THE WEAK VOR-TEX SPECIFIED IN FIG. 10.

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a new parameter in vortex identification and visualization

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