A Liutex based definition and identification of vortex core center lines *

Yi-sheng Gao, Jian-ming Liu, , Yi-fei Yu, Chaoqun Liu

1. Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019, USA 2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Abstract: Six core issues for vortex definition and identification concern with (1) the absolute strength, (2) the relative strength, (3) the rotational axis, (4) the vortex core center, (5) the vortex core size, and (6) the vortex boundary (Liu C. 2019). However, most of the currently popular vortex identification methods, including the Q criterion, the 2λ criterion and the ciλ criterion etc., are Eulerian local region-type vortex identification criteria and can only approximately identify the vortex boundary by somewhat arbitrary threshold. On the other hand, the existing Eulerian local line-type methods, which seek to extract line-type features such as vortex core line, are not entirely satisfactory since most of these methods are based on vorticity or pressure minimum that will fail in many cases.The key issue is the lack of a reasonable mathematical definition for vortex core center. To address this issue, a Liutex (previously named Rortex) based definition of vortex core center is proposed in this paper. The vortex core center, also called vortex rotation axis line here, is defined as a line where the Liutex magnitude gradient vector is aligned with the Liutex vector, which mathematically implies that the cross product of the Liutex magnitude gradient vector and the Liutex vector on the line is equal to zero. Based on this definition, a novel three-step method for extracting vortex rotation axis lines is presented. Two test cases, namely the Burgers vortex and hairpin vortices, are examined to justify the proposed method. The results demonstrate that the proposed method can successfully identify vortex rotation axis lines without any user-specified threshold, so that the proposed method is very straightforward, robust and efficient.

Key words: Vortex definition, vortex identification, flow visualization, Liutex, Liutex magnitude gradient, vortex core lines

Introduction

Vortices are ubiquitous in nature and can be easily observed from smoke rings to wingtip vortices and from tropical cyclones to even galaxies[1]. In fluid mechanics, it is well recognized that the ubiquity of multi-scale vortical structures, more formally referred to as coherent structures[2-4], is one of the most prominent characteristics of turbulent flows and these spatially coherent and temporally evolving structures serve a crucial role in the generation and evolution of turbulence[5-7]. Several important vortical structures, including hairpin vortices[8-10]and quasistreamwise vortices[3,11-12], etc., have been identified and intensively studied. Surprisingly, although vortices can be readily observed and intuitively regarded as the rotational or swirling motion of fluids,an unambiguous and generally accepted definition of a vortex has yet to be achieved (and the definition of coherent structures is somewhat ambiguous as well).Since vortices can be considered as elementary structures of organized motion[9]or building blocks for turbulent flows[13], the lack of a rigorous vortex definition is possibly one of the major obstacles to thoroughly understand the mechanism of turbulence generation and sustenance[14].

During the last several decades, a variety of vortex identification methods have been proposed to attempt to answer the deceptively complicated question of the vortex definition. One of the most intuitive idea to define a vortex is that closed or spiraling streamlines or pathlines imply the existence of a vortex. One such example is a definition given by Lugt for steady motions[15]: Any mass of fluid moving around a common axis constitutes a vortex.Mathematically, such motion can be described by closed or spiraling streamlines (or pathlines) if a reference frame exists for which the flow field becomes steady. Robinson[3]also introduces a definition based on instantaneous streamlines which exhibit a roughly circular or spiral pattern. However,these seemingly reasonable definitions will be plagued with a devastating shortcoming: streamlines are not invariant under the Galilean transformation. Another common candidate for vortex definition is the vorticity. The vorticity is mathematically well-defined(the curl of the velocity), so a vortex is commonly associated with a vortex filament/tube[16]or a finite volume of vorticity[17]. While vorticity-based methods seem to be straightforward, they will run into serious problems in viscous flows, especially in turbulence.The vorticity itself cannot distinguish a real rotational motion from a shear layer and thus the association between regions of strong vorticity and actual vortices can be rather weak in the turbulent boundary layer[18].In many wall-bounded flows such as the Blasius boundary layer, the magnitude of the vorticity in the near-wall region can be relatively large compared to its surroundings but does not induce any rotational motion. And the maximum vorticity magnitude does not necessarily imply the vortex core. Actually, the vorticity magnitude can be considerably decreased along the vorticity lines entering the vortex ring in the flat plate boundary layer flow[19]. Furthermore, it is not unusual that the local vorticity vector is not aligned with the vortex core direction in turbulent wall-bounded flows, especially in the near-wall regions. Gao et al.[20]point out that the vorticity can be somewhat misaligned with the vortex core direction in turbulent wall-bounded flows. Pirozzoli et al.[21]also show the differences between the local vorticity direction and the vortex core orientation in a supersonic turbulent boundary layer.

The issues of streamline-based and vorticitybased methods prompt the development of Eulerian local region-type vortex identification criteria[22].These local region-type methods are based on the concept that a point is inside a vortex if a prescribed criterion is met in that point. Overall, most of the currently popular Eulerian local region-type vortex identification methods are based on the local velocity gradient tensor. More specifically, these criteria are exclusively determined by the eigenvalues or invariants of the velocity gradient tensor and thereby can be classified as eigenvalue-based criteria[23]. For example, in the Q criterion, Q denotes a measure of the vorticity magnitude in excess of the strain rate magnitude, which is equal to the second invariant of the velocity gradient tensor for incompressible flows[24]. The Δ criterion identifies a vortex as a region where the velocity gradient tensor has complex eigenvalues by the discriminant of the characteristic equation[25]. Theciλ criterion can be considered as an extension of the Δ criterion and uses the (positive)imaginary part of the complex eigenvalue to determine the swirling strength[26]. The2λ criterion defines a vortex as a connected region with two negative eigenvalues of the symmetric tensor of(S, W represent the symmetric and the antisymmetric parts of the velocity gradient tensor,respectively)[27]. Usually,2λ cannot be expressed in terms of the eigenvalues of the velocity gradient tensor. But when the eigenvectors are orthonormal,2λ can be exclusively determined by the eigenvalues[28]. These eigenvalue-based Eulerian local regiontype methods have several advantages: (1) they are Galilean invariant in contrast with closed or spiraling streamlines or pathlines, (2) they can discriminate against shear layers, (3) they only depend on the local flow property and can be easy to implement.Nevertheless, the existing eigenvalue-based criteria are not entirely satisfactory. One critical shortcoming of these criteria is the case-related threshold. It is crucial to determine an appropriate threshold, since different thresholds will present different vortical structures. But no one knows what an appropriate threshold is in advance and even sometimes there exists no appropriate threshold at all[13]. It has been found that “vortex breakdown” will be exposed with the use of a large threshold for the2λ criterion,while will not be observed with a small threshold even if the same DNS data are examined[29]. Due to the sensitivity to the threshold change, it is difficult to make a judgement if the identified vortical structures are complete or not. To avoid the usage of case-related thresholds, relative-value based methods are favored, such as the Omega method proposed by Liu et al.[30-31]. The Omega method (Ω） is originated from an idea that the vortex is a connected region where the vorticity overtakes the deformation and thus Ω can be defined as a ratio of vorticity tensor squared norm over the sum of vorticity tensor squared norm and deformation tensor squared norm. The Omega method is robust to moderate threshold change and capable to capture both strong and weak vortices simultaneously while most of eigenvalue-based methods are sensitive to threshold change. Another obvious drawback of the existing criteria is the inadequacy of identifying the rotation axis. Since the existing eigenvalue-based criteria are scalar-valued criteria, no information about the rotation or swirling axis will be obtained. In addition, eigenvalue-based criteria can be severely contaminated by shearing[23,32-33].As pointed out by Gao and Liu[23], the existing eigenvalue-based criteria will suffer from this problem as they are associated with the imaginary part of the complex eigenvalues.

In our previous work, a novel eigenvector based Liutex method (previously named Rortex) was proposed to remedy the situation[23,34]. One of the most salient features of Liutex is that Liutex is a systematical definition of the local fluid rotation,including the scalar, vector and tensor interpretations.The scalar version or the magnitude of Liutex represents the local rotational strength (angular velocity). The direction of the Liutex vector, which is determined by the real eigenvector of the velocity gradient tensor, represents the local rotation axis,consistent with the analysis of the instantaneous solution trajectories govern by first-order ordinary differential equations[25-26]. The tensor form of Liutex,rather than the vorticity tensor, represents the real rotational part of the velocity gradient tensor, which can be used for the decomposition of the velocity gradient tensor[35]. Meanwhile, Liutex can eliminate the contamination due to shearing and hence can accurately quantify the local rotational strength[23].Moreover, compared to the current eigenvalue-based criteria, not only the Liutex iso-surface, but also Liutex vector field and Liutex lines can be adopted to visualize and investigate the vortical structures.Nonetheless, a user-specified threshold is still required if the Liutex iso-surface is used. For this problem, a combination of the ideas of the Omega method and Liutex would be a possible solution[36].

Though Eulerian local region-type methods are widely used for vortex identification due to the convenience in applications, Eulerian local line-type methods may be preferred to identify line-type features like vortex core lines or vortex skeletons.Several vortex core line detection algorithms are proposed, based on the vorticity or helicity. Strawn et al. propose an algorithm to collect points that have local maximum vorticity magnitude in a plane normal to the vorticity vector to trace the vortex skeleton[37].Similarly, Banks and Singer develop a two-step predictor-corrector scheme to obtain a series of points that approximate a vortex skeleton[38]. Starting from a seed point, the vorticity direction is first examined to predict the new position of the vortex skeleton and then the candidate location is corrected to the pressure minimum in the plane perpendicular to the vorticity vector. Levy et al.[39]connect points of maximum helicity density to locate the vortex core lines. Based on the concept of eigen helicity density, Zhang and Choudhury[40]propose a Galilean invariant scheme to identify vortex tubes and extract vortex core lines in a simulated Richtmyer-Meshkov flow. Additionally, the concept of pressure minimum is also widely applied to identify vortex core lines. Miura and Kida[41]formulate a pseudo-pressure expression to search for sectional local pressure minima and then connect the points of pressure minima to construct the axial lines.This method is further improved by imposing the additional swirling constraint[42]. Linnick and Rist extract vortex core lines by connecting points of local minimum of2λ on the plane normal to the gradient of2λ[43]. As the real eigenvector corresponding to the real eigenvalue indicates the direction of the local swirl axis, Sujudi and Haimes develop an algorithm for identifying the center of swirling flow by finding points where the velocity projected on the plane normal to the real eigenvector is equal to zero[44]. Roth analyzes many different line-type methods and introduce a unifying framework for identifying line-type features in terms of a “parallel vectors”operator[45]. Extensive overview of the currently available vortex identification methods, including both Eulerian local region-type and line-type methods,can be found in Refs. [22, 46].

Six core issues for vortex definition and identification concern with (1) the absolute strength,(2) the relative strength, (3) the rotation axis, (4) the vortex core center, (5) the vortex core size, and (6) the vortex boundary[13]. As already stated, most of Eulerian local region-type vortex identification criteria are based on a scalar quantity, so can only approximately identify the vortex boundary by somewhat arbitrary threshold. However, even for identifying the vortex boundary, these methods are sensitive to threshold change. If the threshold is too small, weak vortices may be captured, but strong vortices will be smeared. If the threshold is too large, weak vortices will be almost excluded[13]. On the other hand, the current line-type methods, which aim to extract line-type features such as vortex core lines, are not very successful since most of these methods are based on the vorticity or pressure minimum which will fail in many cases. The key issue of the existing methods is the lack of a reasonable mathematical definition for the vortex core center. In this paper, a Liutex based mathematical definition for vortex core center, also called vortex rotation axis line, is proposed. Motivated by the characteristic of the Liutex magnitude gradient lines and the continuity of the Liutex lines inside vortices, the vortex rotation axis line is defined as a line where the Liutex magnitude gradient vector is aligned with the direction of the Liutex vector, which mathematically implies that the cross product of the Liutex magnitude gradient vector and the Liutex vector on the line is equal to zero. And a novel three-step method for extracting vortex rotation axis lines is presented. Despite of a preliminary manual process, the proposed method is straightforward,robust and efficient.

1. Liutex vector and Liutex magnitude gradient vector

Liutex is a systematical definition of the local fluid rotation[23,34-35,47]. According to critical point theory[25], if the velocity gradient tensor has complex conjugate eigenvalues, the instantaneous streamline pattern presents a local swirling motion around the direction of the real eigenvector. Therefore, when the velocity gradient tensor ?v has complex eigenvalues, the direction r of the Liutex vector, which represents the local rotation axis, is defined as the real eigenvector of the local velocity gradient tensor and can be written as

whererλ is the real eigenvalue. Since the normalized eigenvector is only unique up to a±sign, a second condition is imposed[47], which reads

where ω represents the vorticity vector. The magnitude of the Liutex vector represents the local rotational strength (angular velocity). In the original definition, the magnitude is determined on the plane perpendicular to r, involving somewhat cumbersome coordinate rotation[23,34]. Recently, Wang et al.[47]introduce a simple and explicit formula to substantially simplify the calculation. This explicit formula for the Liutex magnitude can be expressed as

Intuitively, it is expected that the vortex rotation axis line should consist of some kind of local extreme points located on the plane perpendicular to the vortex rotation axis line. Since the gradient of a physical quantity is commonly used to find the local extreme point and the Liutex magnitude can represent the accurate local rotational strength, it is natural to choose the gradient of the Liutex magnitude for identifying vortex rotation axis lines.

Definition 1: The Liutex magnitude gradient vector is defined as the gradient of the Liutex magnitude, which reads

2. The Liutex based definition of vortex rotation axis line

However, the Liutex magnitude gradient lines obtained by the straightforward integration of the Liutex gradient vector do not always present distinct and continuous lines. Instead, the integration often results in discontinuous segments, as shown in Fig. 1.Actually, the usage of the gradient of other common scalar quantities such as Q ,ciλ ,2λ or even Ω will also fail owing to the ubiquity of stationary points(zero gradient) in the whole flow fields and one example of Ω gradient lines is shown in Fig. 2. To seek local extreme points, another possible candidate can be derived from the concept of “parallel vectors”operator[45], which is given by

Fig. 1 (Color online) Liutex magnitude gradient lines obtained by the straightforward integration of the Liutex magnitude gradient vector

where F denotes a scalar quantity like2λ used in Ref. [43], H the Hessian matrix of F and2η the second eigenvalue of H (using ascending order).Equation (6) indicates the vectors H??Fand ?F are parallel to each other. It is easy to show that ?F is the eigenvector of the Hessian matrix H and the points satisfying Eq. (6) are local extrema points ofon the plane perpendicular to the direction of. But it has been pointed out in Ref. [45] that the solutions of Eq. (6) are actually loci of zero curvature.Worse still, the numerical noise will dramatically undermine the accuracy even if high order methods are applied for the calculation of the Hessian matrix H[43], making this definition impractical.

Fig. 2 (Color online) Ω gradient lines obtained by the straightforward integration of the gradient of Ω

Fortunately, the definition of Liutex not only involves the magnitude, but also the direction. The Liutex lines obtained by the integration of the Liutex vector are continuous inside the vortex region (with zero threshold) and have been successfully used to demonstrate the skeleton of hairpin vortices[23,34].Hence, we can combine the Liutex lines and the Liutex magnitude gradient lines to extract vortex rotation axis lines. The Liutex vector is defined at every point inside the vortex where the velocity gradient has complex eigenvalues but has different direction from the Liutex magnitude gradient. It is expected that the Liutex vector is aligned with the direction of the Liutex magnitude gradient only on the vortex rotation axis line. The physical meaning of the vortex rotation axis line is the concentration of Liutex magnitude gradient lines as the local Liutex maxima on the plane perpendicular to the vortex rotation axis line, and, meanwhile, the concentration line is a special Liutex line where the Liutex vector is aligned with the Liutex magnitude gradient, which reasonably leads to a new and unique mathematical definition of the vortex rotation axis line.

Definition 2: The vortex core center or vortex rotation axis line is defined as a line consisting of points which satisfy the condition

where r represents the direction of the Liutex vector.(It should be noted that every point in the pure rigid body rotation satisfies Eq. (7), so no unique vortex rotation axis line will be obtained. But this type of fluid motion occurs infrequently in real flows.) Eq. (7)implies that the angle between the Liutex magnitude gradient vector ?R and the Liutex vector r is equal to zero. Theoretically, the vortex rotation axis lines can be obtained by connecting all points satisfying Eq. (7) in a vortex region. However, the computational error made it difficult to solve Eq. (7)directly since the computational error could make large noises around zero points of Eq. (7), leading to discontinuous or non-unique zero-point distribution.In practice, we can find a start point in the vortex which satisfies Eq. (7) and then integrate the Liutex vector passing through this point to obtain a Liutex line which would be the vortex rotation axis line or a very close approximation. Based on the observation that all of the Liutex magnitude gradient lines will converge to a concentration line which can be considered as a vortex rotation axis line, the start point can be determined by picking up a point on the concentration line. For two-dimensional flows, the vortex rotation axis line degrades to a point and the rotation axis always points out of the 2-D plane. In this case, Eq. (7) becomes

which is also valid. Thus, Definition 2 is a reasonable definition for the vortex rotation axis line.

Based on this definition, a three-step manual method for extracting vortex rotation axis lines is introduced:

Step (1) Set a slice which shows the contour of Ω or the Liutex magnitude.

Step (2) Determine the concentration line of some Liutex magnitude gradient lines passing through the slice (a few Liutex magnitude gradient lines or even one line will be enough for each individual vortex).

Step (3) Find the intersection point of the concentration line and the slice and create a Liutex line passing through the intersection point. The resulting Liutex line is a vortex rotation axis line or a very close approximation.

Usually only one intersection point is needed for any individual vortex. In addition, no user-specified threshold is required and any vortex core, no matter strong or weak, can be identified. Consequently, the current method is very straightforward, robust and efficient.

3. Test cases

First, the Burger vortex is examined. The Burger vortex is an exact steady solution of the Navier-Stokes equation and has been widely applied for modelling fine scales of turbulence. The velocity components in the cylindrical coordinates for the Burger vortex are given by

where Γ represents the circulation, ξ the axisymmetric strain rate and ν the kinematic viscosity. The Reynolds number can be defined as

Hence, the velocity components in the Cartesian coordinate system will be written as

Accordingly, the analytical expression of the Liutex magnitude can be easily obtained as

where

For the existence of Liutex, ζ should be larger than zero, which yields a non-dimensional vortex radius size of, consistent with the result of Ref. [28]. The direction of the Liutex vector is always along the positive z- direction inside the vortex. According to Eqs. (16), (18), the Liutex magnitude has a maximum along the z- axis and the Liutex magnitude gradient vector is perpendicular to the Liutex vector elsewhere inside the vortex.Therefore, there exists only one vortex rotation axis line satisfying the condition imposed by Eq. (7),namely the Liutex line along the z- axis. Figure 3 shows the vortex rotation axis line and several Liutex magnitude gradient lines on the xy- plane. Figure 4 demonstrates that streamlines exhibit a spiral pattern around the vortex rotation axis line, indicating the proposed definition of vortex rotation axis line is valid for the Burgers vortex. Although the vortex rotation axis line can be analytically obtained in this case, we can also adopt the three-step method to identify the vortex rotation axis line. First, an arbitrary slice is set in the flow field. Here a z- slice is set because it is convenient to determine the Liutex magnitude gradient line on the slice. And then, two Liutex magnitude gradient lines are determined on the slice.These Liutex magnitude gradient lines point to the concentration point located on the center of the slice.So, the Liutex line, passing the concentration point and located on the z- axis, is the vortex rotation axis line. The whole process for extracting the vortex rotation axis line is shown in Fig. 5.

Fig. 3 (Color online) Vortex rotation axis line and Liutex magnitude gradient lines for the Burgers vortex

Fig. 4 (Color online) Vortex rotation axis line (red color) and streamlines (black color) for the Burgers vortex

In the following, the proposed method is applied to extract vortex rotation axis lines for hairpin vortices.

The data is obtained from a direct numerical simulation (DNS) of the late transition of flat plate boundary layer[5]. The simulation is performed with about 60 million grid points and over 4×105timesteps at a freestream Mach number of 0.5. For the detailed computational setting, one can refer to Ref. [5].

Fig. 5 (Color online) Three-step method for extracting the vortex rotation axis lines of the Burgers vortex

Fig. 6 (Color online) Liutex magnitude gradient lines for hairpin vortices

Fig. 7 (Color online) Vortex rotation axis line (black color) on which Liutex magnitude gradient lines concentrate

Fig. 8 (Color online) The concentration of the Liutex magnitude gradient lines and the x -slice

Figure 6 illustrates the Liutex magnitude gradient lines that are obtained from the integration of the Liutex magnitude gradient vector. It can be found that although the Liutex magnitude gradient lines usually present discontinuous segments, these lines concentrate in a special line which can be considered as a vortex rotation axis line and the Liutex magnitude gradient vector is aligned with the local Liutex vector on this line, as shown in Fig. 7. Figure 8 demonstrates the concentration of the Liutex magnitude gradient lines is locally maximum on the plane normal to the direction of the concentrationline(Noteax-sliceisused here to approximate the exact plane due to the convenience in practice), which means that the vortex rotation axis line is a local maximum on the plane normal to the vortex rotation axis line as well, as shown in Fig. 9. Figure 10 indicates when the Liutex magnitude gradient line is approaching to the vortex rotation axis line (the concentration line), the Liutex magnitude gradient line will be gradually aligned with the Liutex line, justifying that the vortex rotation axis line is also a Liutex line. The process for extracting vortex rotation axis lines is shown in Fig. 11. Slices with the contour of Liutex magnitude are set in the flow field and several Liutex magnitude gradient lines are obtained, as shown in Fig. 11(a). And then, the intersection points of the Liutex magnitude gradient lines and the slices are determined, from which the Liutex lines are constructed. The resulting Liutex lines are actually the vortex rotation axis lines, as shown in Figs. 11(b), 11(c). No user-specified threshold is required and almost all the vortices, whether strong or weak, can be clearly identified. Figure 12 demonstrates that the Liutex magnitude varies along the vortex rotation axis lines, which means the rotational strength is quite different along the vortex rotation axis line. This would suggest that the usage of isosurfaces may not be appropriate for the identification of vortical structures in addition to improperly large thresholds which would exclude all vortices below the threshold. Figure 13 shows the vortex rotation axis lines and the streamlines projected on the x- slice,indicating that the streamlines rotate around the vortex rotation axis line.

Fig. 9 (Color online) Vortex rotation axis lines and x- slice

Fig. 10 (Color online) Liutex magnitude gradient lines (red)aligned with Liutex line (black)

Fig. 11 (Color online) Three-step method for extracting vortex rotation axis lines of hairpin vortices

4. Conclusion

In this paper, a Liutex based definition of vortex rotation axis line is presented, which requires that the Liutex magnitude gradient vector is aligned with the Liutex vector. Mathematically, this implies that the cross product of the Liutex vector and the Liutex magnitude gradient vector on the vortex rotation axis line is equal to zero. Based on this definition, a novel three-step method for extracting vortex rotation axis lines is introduced and verified by the Burgers vortex and hairpin vortices. The results demonstrate that the present method can accurately extract vortex rotation axis lines without any user-specified threshold, so that the present method is straightforward, robust and efficient. Since the results in this paper are obtained by a preliminary manual method, we will develop an algorithm to automatically identify the complete vortex rotation axis lines in the near future.

Fig. 12 (Color online) The vortex rotation axis lines with Liutex magnitude (rotational strength)

Fig. 13 (Color online) The vortex rotation axis lines and streamlines projected on the x -slice

Acknowledgements

This work was supported by the Department of Mathematics at University of Texas at Arlington and AFOSR (Grant No. MURI FA9559-16-1-0364), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No.18KJA110001) and the Visiting Scholar Scholarship of the China Scholarship Council (Grant No.201808320079). The authors are grateful to Texas Advanced Computational Center (TACC) for providing computation hours. This work is accomplished by using code DNSUTA developed by Dr. Chaoqun Liu at the University of Texas at Arlington.