Most of the current Eulerian vortex identification criteria, including the Q criterion and the λci criterion, are exclusively determined by the eigenvalues of the velocity gradient tensor or the related invariants and thereby can be regarded as eigenvalue-based criteria. However, these criteria will be plagued with two shortcomings: (1) these criteria fail to identify the swirl axis or orientation; (2) these criteria are prone to severe contamination by shearing. To address these issues, a new vector named Rortex which represents the local fluid rotation was proposed in our previous work. In this paper, an alternative eigenvector-based definition of Rortex is introduced. The direction of Rortex, which represents the possible axis of the local rotation, is determined by the real eigenvector of the velocity gradient tensor. And then the rotational strength obtained in the plane perpendicular to the possible axis is used to define the magnitude of Rortex. This new equivalent definition allows a much more efficient implementation. Furthermore, a systematic interpretation of scalar, vector, and tensor versions of Rortex is presented. By relying on the tensor interpretation, the velocity gradient tensor is decomposed to a rigid rotation part and a non-rotational part including shearing, stretching, and compression, different from the traditional symmetric and anti-symmetric tensor decomposition. It can be observed that shearing always manifests its effect on the imaginary part of the complex eigenvalues and consequently contaminates eigenvalue-based criteria, while Rortex can exclude the shearing contamination and accurately quantify the local rotational strength. In addition, in contrast to eigenvalue-based criteria, not only the iso-surface of Rortex but also the Rortex vectors and the Rortex lines can be applied to investigate vortical structures. Several comparative studies on simple examples and realistic flows are studied to confirm the superiority of Rortex.

Vortical structures, also referred to as coherent turbulent structures,1–4 are generally acknowledged as one of the most salient characteristics of turbulent flows and occupy a pivotal role in turbulence generation and sustenance. Several important coherent structures have been identified, including hairpin vortices in wall-bounded turbulence,5–8 quasi-streamwise vortices,4,9,10 vortex braids in turbulent shear layers,11,12 etc. Naturally, the ubiquity and significance of such spatially coherent, temporally evolving vortical motions in transitional and turbulent flows necessitate an unambiguous and rigorous definition of vortex for the comprehensive and thorough investigation of these sophisticated phenomena. Unfortunately, although vortex has been intensively studied for more than one hundred years, a sound and universally-accepted definition of vortex is yet to be achieved in fluid mechanics,13–15 which is possibly one of the chief obstacles causing considerable confusions in visualizing and understanding vortical structures.16–18

The classic definition of vortex is associated with vorticity which possesses a clear mathematical definition, namely, the curl of the velocity vector. As early as 1858, Helmholtz first considered a vorticity tube with infinitesimal cross section as a vortex filament,19 which was followed by Lamb to simply call a vortex filament as a vortex in his classic monograph.20 Since vorticity is well-defined, vorticity dynamics has been systematically developed for the generation and evolution of vorticity and applied in the study of vortical-flow stability and vortical structures in transitional and turbulent flows.14,21,22 However, the use of vorticity will run into severe difficulties in viscous flows, especially in turbulence: (1) vorticity is unable to distinguish between a real vortical region and a shear layer region; (2) it has been noticed by several researchers that the local vorticity vector is not always aligned with the direction of vortical structures in turbulent wall-bounded flows, especially at locations close to the wall;23–25 (3) the maximum vorticity does not necessarily occur in the central region of vortical structures.26,27

The problems of vorticity for the identification and visualization of vortical structures in turbulence motivate the rapid development of vortex identification methods, including the Q criterion,28 the $Δ$ criterion,29,30,15 the λci criterion,24,31 the λ2 criterion,32 the Ω method,33 etc. These Eulerian vortex identification criteria are based on the analysis of the velocity gradient tensor. More specifically, most of these criteria are exclusively determined by the eigenvalues of the velocity gradient tensor or the related invariants and thereby can be regarded as eigenvalue-based criteria. (Ω and λ2 cannot be expressed in terms of the eigenvalues of the velocity gradient tensor and are not investigated in this paper). Despite the widespread use, these eigenvalue-based criteria are not always satisfactory. One obvious drawback is the inadequacy of identifying the swirl axis or orientation. The existing eigenvalue-based criteria are scalar-valued criteria and are thus unable to identify the swirl axis or orientation. Another shortcoming is the contamination by shearing. Recently, the λci criterion has been found to be seriously contaminated by shearing motion.34,35 In fact, as described below, other eigenvalue-based criteria will suffer from the same problem, as long as the criterion is associated with the complex eigenvalues. This issue prompts Kolář to formulate a triple decomposition from which the residual vorticity can be obtained after the extraction of an effective pure shearing motion and represents a direct and accurate measure of the pure rigid-body rotation of a fluid element.36 However, the triple decomposition requires a basic reference frame to be first determined. Searching for the basic reference frame in 3D cases will result in an expensive optimization problem for every point in the flow field, which limits the applicability of the method. And the triple decomposition has not yet been thoroughly investigated for 3D cases. Hence, Kolář et al. introduced the concept of the average corotation of line segments near a point to reduce the computational overhead.37 In addition to the widely used Eulerian vortex identification methods, some objective Lagrangian vortex identification methods have been developed to study the vortex structures involved in the rotating reference frame.3,38 For an extensive overview of the currently available vortex identification methods, one can refer to review papers by Epps13 and Zhang et al.39

To address the above-mentioned issues of the existing eigenvalue-based criteria, a new vector quantity, which is called the vortex vector or Rortex, was proposed and investigated in our previous studies.40,41 In this paper, an alternative eigenvector-based definition of Rortex is presented. The direction of Rortex, which represents the possible axis of the local rotation, is determined by the real eigenvector of the velocity gradient tensor. And then, the rotational strength determined in the plane perpendicular to the possible axis is used to define the magnitude of Rortex. The rotational strength of Rortex is equivalent to Kolář’s residual vorticity in 2D cases, but Kolář’s triple decomposition has yet to be fully studied in 3D cases and thus the result is unclear. The main distinguishing feature of Rortex is that Rortex is eigenvector-based and the magnitude (rotational strength) is strongly relevant to the direction of the real eigenvector. Although Gao et al. used the real eigenvector to indicate the orientation about which the flow swirls, they chose the imaginary part of the complex eigenvalues as the swirling strength and the swirling strength was determined independent of the choice of the orientation.23 The present eigenvector-based definition is mathematically equivalent to our previous one but it significantly improves the computational efficiency. Furthermore, a complete and systematic interpretation of scalar, vector, and tensor versions of Rortex is presented to provide a unified and clear characterization of the local fluid rotation. The scalar represents the local rotational strength, the vector offers the local swirl axis, and the tensor extracts the local rigidly rotational part of the velocity gradient tenor. Especially, the tensor interpretation brings a new decomposition of the velocity gradient tensor to investigate the analytical relations between Rortex and eigenvalue-based criteria. The velocity gradient tensor in a special reference frame is examined to indicate that shearing always manifests its effect on the imaginary part of the complex eigenvalues and consequently contaminates eigenvalue-based criteria. By contrast, Rortex can exclude the shearing contamination and accurately quantify the local rotational strength. A comprehensive comparison of Rortex with the Q criterion and the λci criterion on several simple examples and realistic flows is carried out to confirm the superiority of Rortex.

The remainder of the paper is organized as follows. In Sec. II, our previous definition of Rortex is revisited, followed by an eigenvector-based definition, and the new implementation is also provided. The systematic interpretation of scalar, vector, and tensor versions of Rortex and the analytical comparison of Rortex and eigenvalue-based criteria are elaborated in Sec. III. Several comparative studies on simple examples are carried out in Sec. IV. Section V shows the comparison of Rortex and eigenvalue-based criteria on the direct numerical simulation (DNS) data of the boundary layer transition over a flat plate. The conclusions are summarized in Sec. VI.

To reasonably define a vortex vector or Rortex, we propose the following principles:

1. Local. Although a vortex is regarded as a non-local flow motion, the presence of viscosity in real flows leads to the continuity of the kinematic features of the flow field,31 and numerous studies have suggested that the cores of vortical structures in turbulent flows are well localized in space.32 Moreover, critical-point concepts based on local kinematics of the flow field have successfully provided a general description of the 3D steady and unsteady flow pattern.15,42 And non-locality commonly implies much more complexity in computation.

2. Galilean Invariant. It means that the definition is the same in all inertial frames. This principle is followed by many Eulerian vortex identification criteria.28,31,32 Objectivity it may be preferred when involved in a more general motion of the reference frame,43,44 but it is beyond the subject of the present study.

3. Unique. The description of the local rigidly rotation must be accurate and unique. It requires the exclusion of the contamination by shearing.

4. Systematical. The definition will contain a scalar version which is the strength of the rigid rotation, a vector version which provides both the rotation axis and rotation strength, and a tensor version which represents the rigid rotation part of the velocity gradient tensor.

Based on the above principles, the concept of the local fluid rotation and a vector named vortex vector or Rortex which represents the local fluid rotation was introduced in our previous work.40,41 The direction of Rortex is determined by the direction of the local rotation axis Z, and the magnitude of Rortex is defined by the rotational strength of the local fluid rotation, which is determined in the XY plane perpendicular to the Z-axis. If U, V, and W are velocity components along the X, Y, and Z axes, respectively, the matrix representation of the velocity gradient tensor in the XYZ-frame can be written as

$∇V→=∂U∂X∂U∂Y0∂V∂X∂V∂Y0∂W∂X∂W∂Y∂W∂Z.$
(1)

Generally, the z-axis in the original xyz-frame is not parallel to the Z-axis, so the velocity gradient tensor in the origin xyz-frame

$∇v→=∂u∂x∂u∂y∂u∂z∂v∂x∂v∂y∂v∂z∂w∂x∂w∂y∂w∂z$
(2)

cannot fulfill Eq. (1). Thus, a coordinate transformation is required to rotate the z-axis to the Z-axis. There exists a corresponding transformation between $∇V→$ and $∇v→$,

$∇V→=Q∇v→Q−1,$
(3)

where Q is a rotation matrix and

$Q−1=QT.$
(4)

In Ref. 41, the existence of the local rotation axis Z was proved through real Schur decomposition.45 The direction of the local rotation axis Z can be obtained by solving a nonlinear system of equations through the Newton-iterative method40 or by a fast algorithm based on real Schur decomposition.41

If the direction of the Z-axis in the xyz-frame is given by $r→=rx,ry,rzT$,

$r→=QT001$
(5)

and

$Qr→=001$
(6)

represents the direction of the local rotation axis Z in the XYZ-frame.

Once the local rotation axis Z is obtained, the rotation strength is determined in the XY plane perpendicular to the local rotation axis Z. This can be achieved by a second coordinate rotation in the XY plane. When the XYZ-frame is rotated around the Z-axis by an angle θ, the velocity gradient tensor will become

$∇V→θ=P∇V→P−1,$
(7)

where P is the rotation matrix around the Z-axis and can be written as

$P=cos⁡θsin⁡θ0−sin⁡θcos⁡θ0001,$
(8)
$P−1=PT=cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001.$
(9)

So, we have

$∂U∂Y|θ=α⁡sin2θ+φ−β,$
(10a)
$∂V∂X|θ=α⁡sin2θ+φ+β,$
(10b)
$∂U∂X|θ=α⁡cos2θ+φ+12(∂U∂X+∂V∂Y),$
(10c)
$∂V∂Y|θ=−α⁡cos2θ+φ+12(∂U∂X+∂V∂Y),$
(10d)

where

$α=12∂V∂Y−∂U∂X2+∂V∂X+∂U∂Y2,$
(11)
$β=12∂V∂X−∂U∂Y,$
(12)
$φ=arctan∂V∂X+∂U∂Y∂V∂Y−∂U∂X,∂V∂Y−∂U∂X≠0π2,∂V∂Y−∂U∂X=0,∂V∂X+∂U∂Y>0−π2,∂V∂Y−∂U∂X=0,∂V∂X+∂U∂Y<0$
(13)

(note: If $∂V∂Y−∂U∂X=0, ∂V∂X+∂U∂Y=0$, $∂V∂X=β$, $∂U∂Y=−β$ for any θ, thus φ is not needed).

The criterion to determine the existence of local fluid rotation in the XY plane is

$gZmin=β2−α2>0,$
(14)

and $∂V∂X$ can be regarded as the angular velocity of the fluid at the azimuth angle θ relative to the point

$ωθ=∂V∂X|θ=α⁡sin2θ+φ+β.$
(15)

Since ωθ will change with the change in the azimuth angle θ, the fluid-rotational angular velocity in the XY plane is defined as the absolute minimum of Eq. (15),

$ωrot=ωθmin=β−α,gZmin>00,gZmin≤0.$
(16)

Here, we assume β > 0. If β < 0, we can rotate the local rotation axis to the opposite direction to make β positive. The local rotation strength (the magnitude of Rortex) is defined as twice the fluid-rotational angular velocity

$R=2(β−α),gZmin>00,gZmin≤0.$
(17)

The factor 2 is related to using 1/2 in the expression for the 2-D vorticity tensor component. It should be noted that Eq. (17) is equivalent to Kolář’s residual vorticity in 2D cases.36,37 But Kolář’s triple decomposition (TDM) is yet to be thoroughly examined for general 3D cases. A numerical comparison of Rortex and the TDM is presented in  Appendix A to indicate the superiority of Rortex over the TDM.

Our previous work provides a physical description of Rortex, but the relation between Rortex and the eigenvalues of the velocity gradient tensor and eigenvalue-based criteria is unclear. It motivates the present study.

Definition 1.

A local rotation axis is defined as the direction of $r→$ where $dv→=αdr→$.

This definition means that in the direction of the local rotation axis, there is no cross-velocity gradient. For example, if the z-axis is the rotation axis in a reference frame, the velocity can only increase or decrease along the z-axis, which means only dw ≠ 0, but du = 0, and dv = 0. Accordingly, we can obtain the following theorem:

Theorem 1.

The direction of the local rotation axis is the real eigenvector of the velocity gradient tensor $∇v→$.

Proof: If $r→=rx,ry,rzT$ represents the direction of the local rotation axis, we have $dv→=αdr→$. On the other hand, from the definition of the velocity gradient tensor,
$dv→=∇v→⋅dr→.$
(18)
Therefore,
$∇v→⋅dr→=αdr→$
(19)
and
$∇v→⋅r→=αr→,$
(20)
which means $r→$ is the real eigenvector of $∇v→$ and α is the real eigenvalue.
The alternative definition of the direction of Rortex is equivalent to our previous one. If $r→$ is the real eigenvector of the velocity gradient tensor $∇v→$, then we have
$∇v→⋅r→=αr→.$
(21)
Under the coordinate rotation Q which rotates the z-axis to the direction of $r→$, it can be written as
$Q∇v→QTQr→=αQr→.$
(22)
According to Eq. (3), we can find
$∇V→⋅Qr→=αQr→.$
(23)
According to Eq. (6), $Qr→$ is the direction of the local rotation in the XYZ-frame.
Conversely, the vector $001$is the real eigenvector of $∇V→$ in the XYZ-frame since
$∇V→⋅001=∂U∂X∂U∂Y0∂V∂X∂V∂Y0∂W∂X∂W∂Y∂W∂Z001=∂W∂Z001.$
(24)
If we rotate the XYZ frame back to the origin xyz-frame, we have
$QT∇V→QQT001=QT∂W∂Z001,$
(25)
$∇v→⋅QT001=∂W∂ZQT001.$
(26)
Through Eq. (5), $r→=QT001$ represents the real eigenvector of the velocity gradient tensor $∇v→$ in the xyz-frame.

The definition of the rotational strength is the same as the previous one. It is determined in the plane perpendicular to the direction of the real eigenvector by Eq. (17).

It should be noted that when the velocity gradient tensor has three real eigenvalues, there exist more than one real eigenvector, which implies the existence of multiple possible axes. However, according to our previous work,41 when there exist three real eigenvalues λ1, λ2, and λ3, $∇V→$ will become a lower triangular matrix which can be written as
$∇V→=λ100∂V∂Xλ20∂W∂X∂W∂Yλ3.$
(27)
In this case, we have
$α=12λ2−λ12+∂V∂X2,$
(28)
$β=12∂V∂X.$
(29)
Since αβ, the rotation strength R given by Eq. (17) will be equal to zero. Therefore, Rortex is a zero vector in this case, which is consistent to our definition. Non-zero Rortex exists only if the velocity gradient tensor has one real eigenvalue and two complex eigenvalues. So, Rortex is equivalent to the Δ criterion and the λci criterion when a zero threshold is applied.

In Ref. 41, we use real Schur decomposition to prove the existence of the (possible) local rotation axis. But the uniqueness is not mentioned. Through the above eigenvector-based definition, the existence and uniqueness of the (possible) local rotation axis can be immediately proved from the existence and uniqueness (up to sign) of the normalized real eigenvector of the velocity gradient tensor when there exists a pair of complex eigenvalues.

By relying on the eigenvector-based definition, the use the Newton-iterative method or real Schur decomposition, applied in our previous work,40,41 can be avoided, making a significantly simplified implementation. The complete calculation procedure consists of the following steps:

1. Compute the velocity gradient tensor $∇v→$ in the xyz-frame.

2. Calculate the real eigenvalue λr of the velocity gradient tensor $∇v→$ when the complex eigenvalues exist (the analytical expression is provided in  Appendix B).

3. Calculate the (normalized) real eigenvector $r→=rx,ry,rzT$ corresponding to the real eigenvalue λr (the analytical expression is provided in  Appendix C);

4. Calculate the rotation matrix Q* using Rodrigues’ rotation formula,46

$Q*=cosϕ+rx2(1−cosϕ)rxry1−cosϕ−rzsinϕrxrz1−cosϕ+rysinϕryrx1−cosϕ+rzsinϕcosϕ+ry2(1−cosϕ)ryrz1−cosϕ−rxsinϕrzrx1−cosϕ−rysinϕrzry1−cosϕ+rxsinϕcosϕ+rz2(1−cosϕ),$
(30)
$ϕ=acos(c),$
(31)
$c=001⋅r→.$
(32)
1. Obtain the velocity gradient tensor $∇V→$ in the XYZ frame via

$∇V→=Q*T∇v→Q*.$
(33)
1. Calculate α and β using Eqs. (11) and (12);

2. Obtain R according to the signs of α2β2. (Here, we assume β > 0. If not the case, we can rotate the local rotation axis $r→$ to the opposite direction to make β positive. In addition, R is an invariant in the XY plane, so the calculation of the rotation matrix P can be avoided,)

$R=2(β−α), α2−β2<00, α2−β2≥0.$
(34)
1. Compute Rortex $R→$ via

$R→=Rr→.$
(35)

Although Rortex is defined as a vector, we can propose a tensor interpretation of Rortex. When 2θ + φ = π/2 (assume β > 0), the velocity gradient tensor given by Eq. (7) becomes

$∇V→θmin=12∂U∂X+∂V∂Y−β−α0β+α12∂U∂X+∂V∂Y0∂W∂Xθmin∂W∂Yθmin∂W∂Zθmin.$
(36)

If λcr represents the real part of the complex eigenvalues, λci represents the imaginary part of the complex eigenvalues, and λr represents the real eigenvalue, we can obtain

$λr=∂W∂Zθmin,$
(37)
$λcr=12∂U∂X+∂V∂Y,$
(38)
$λci=β2−α2.$
(39)

Equation (36) can be decomposed into two parts,

$∇V→θmin=λcr−ϕ0ϕ+ελcr0ξηλr=R+S,$
(40)
$R=0−ϕ0ϕ00000,$
(41)
$S=λcr00ελcr0ξηλr=000ε00ξη0+λcr000λcr000λr,$
(42)

where ϕ = βα = R/2, ε = 2α, $ξ=∂W∂Xθmin$, $η=∂W∂Yθmin$. Since the local rotational strength (magnitude) can be regarded as the scalar version of Rortex and the direction of the local rotational axis with the magnitude can be regarded as the vector version, Eq. (41) can be regarded as the tensor interpretation of Rortex which exactly represents the local rigidly rotational part of the velocity gradient tensor and is consistent with the scalar and vector interpretations of Rortex. Equation (42) contains the pure shearing and the stretching or compressing parts of the velocity gradient tensor. Because S has three real eigenvalues (multiple λcr and λr), S itself implies no local rotation. Although the decomposition given by Eq. (40) is similar to Kolář’s triple decomposition, Kolář’s method is applied in the basic reference frame which remains unclear in 3D cases, while our decomposition is a clear explicit expression which is obtained in a special coordinate frame determined by the orientation of the real eigenvector and the plane rotation given by Eq. (7). Additionally, Eq. (40) also sheds light on an analytical relation between Rortex and eigenvalues

$λci=ϕ(ϕ+ε).$
(43)

This expression will be applied in the following to examine the relations between Rortex and eigenvalue-based criteria.

As stated earlier, most of the popular Eulerian vortex identification methods are based on the analysis of the velocity gradient tensor $∇v→$. More specifically, these methods are exclusively dependent on the eigenvalues of the velocity gradient tensor or the related invariants. Assume that λ1, λ2, and λ3 are three eigenvalues. The characteristic equation can be written as

$λ3+Pλ2+Qλ+R̃=0,$
(44)

where

$P=−λ1+λ2+λ3=−tr∇v→,$
(45)
$Q=λ1λ2+λ2λ3+λ3λ1=−12tr∇v→2−tr(∇v→)2,$
(46)
$R̃=−λ1λ2λ3=−det∇v→,$
(47)

and P, Q, and $R̃$ are three invariants. For incompressible flow, according to the continuous equation, we have P = 0.

Here we consider two representatives of eigenvalue-based criteria, namely, the Q criterion and the λci criterion.

#### 1. Q criterion

The Q criterion is one of the most popular vortex identification method proposed by Hunt et al.28 It identifies vortices of incompressible flow as fluid regions with positive second invariants, i.e., Q > 0. Meanwhile, a second condition requires the pressure in the vortical regions to be lower than the ambient pressure, despite being often omitted in practice. Q is a measure of the vorticity magnitude in excess of the strain-rate magnitude, which can be expressed as

$Q=12Ω2−S2,$
(48)

where S and Ω are the symmetric and antisymmetric parts of the velocity gradient tensor, respectively,

$S=12∇v→+∇v→T=∂u∂x12∂u∂y+∂v∂x12∂u∂z+∂w∂x12∂v∂x+∂u∂y∂v∂y12∂v∂z+∂w∂x12∂w∂x+∂u∂z12∂w∂y+∂v∂z∂w∂z,$
(49)
$Ω=12∇v→−∇v→T=012∂u∂y−∂v∂x12∂u∂z−∂w∂x12∂v∂x−∂u∂y012∂v∂z−∂w∂x12∂w∂x−∂u∂z12∂w∂y−∂v∂z0,$
(50)

and $∥⋅∥2$ represents the Frobenius norm.

#### 2. λci criterion

The λci criterion is an extension of the Δ criterion and is identical to the Δ criterion when zero threshold is applied.24 When the velocity gradient tensor $∇v→$ has two complex eigenvalues, the local time-frozen streamlines exhibit a swirling flow pattern.15 In this case, the eigen decomposition of $∇v→$ will give

$∇v→=v→rv→crv→ciλr000λcrλci0−λciλcrv→rv→crv→ci−1.$
(51)

Here, $λr,v→r$ is the real eigenpair and $λcr±λci,v→cr±v→ci$ is the complex conjugate eigenpair. In the local curvilinear coordinate system (c1, c2, c3) spanned by the eigenvector $v→r,v→cr,v→ci$, the instantaneous streamlines are the same as pathlines and can be written as

$c1t=c10eλrt,$
(52a)
$c2t=c20cos(λcit)+c30sinλciteλcrt,$
(52b)
$c3t=c30cosλcit−c20sinλciteλcrt,$
(52c)

where t represents the time-like parameter and the constants c1(0), c2(0), and c3(0) are determined by the initial conditions. From Eqs. (52b) and (52c), the period of the orbit of a fluid particle is 2π/λci, so the imaginary part of the complex value λci is called swirling strength.

The analytical relation between ϕ and λci has been given by Eq. (43). The analytical relation between ϕ and Q can be obtained as

$Q=λ1λ2+λ2λ3+λ3λ1=λcr+iλciλcr−iλci+λcr−iλciλr+λrλcr+iλci=λcr2+λci2+2λcrλr=λcr2+ϕϕ+ε+2λcrλr.$
(53)

Since λci and Q are eigenvalue-based, the same eigenvalues always yield the same values of λci and Q. By contrast, Rortex cannot be exclusively determined by eigenvalues. Assume that two velocity gradient tensors $∇v→A$ and $∇v→B$ have the same eigenvalues λcr + ci, λcrci, and λr but different real eigenvectors. Through appropriate rotation matrices $QA$ and $PA$, we can obtain $∇V→θminA$ as

$∇V→θminA=PAQA∇v→A(QA)T(PA)T=λcr−ϕA0ϕA+εAλcr0ξAηAλr=0−ϕA0ϕA00000+λcr00εAλcr0ξAηAλr.$
(54)

Similarly, through appropriate rotation matrices $QB$ and $PB$, we have

$∇V→θminB=PBQB∇v→B(QB)T(PB)T=λcr−ϕB0ϕB+εBλcr0ξBηBλr=0−ϕB0ϕB00000+λcr00εBλcr0ξBηBλr.$
(55)

Since the eigenvalues are identical, we have

$QA=QB,$
(56)
$λciA=λciB$
(57)

and the following conditions

$ϕAϕA+εA=λci2,$
(58)
$ϕBϕB+εB=λci2.$
(59)

However, there is no further relation of ϕA and ϕB, since four unknowns, i.e., ϕA, ϕB, εA, εB cannot be uniquely determined by Eqs. (58) and (59). Therefore, in general, the rotational strength ϕAϕB. Consider a specific case. Two matrices

$∇V→θminA=1−20210ξAηA2=0−20200000+100010ξAηA2,$
(60)
$∇V→θminB=1−10410ξBηB2=0−10100000+100310ξBηB2$
(61)

have the same eigenvalues 1 + 2i, 1 − 2i, and 2. Certainly, we have $QA=QB=9$ and $λciA=λciB=2$. But the rotational strengths are quite different: ϕA = 2 and ϕB = 1.

From Eq. (43), we can find that the shearing effect ε always exists in the imaginary part of the complex eigenvalues. Therefore, as long as eigenvalue-based criteria are dependent on the complex eigenvalues, they will be inevitably contaminated by shearing. Equations (43) and (53) indicate the shearing effect on λci and Q, respectively. The investigation of this contamination in some simple examples and realistic flows will be given in Sec. IV and V, respectively.

First, we consider 2D rigid rotation. The velocity in the polar coordinate system can be expressed as

$vr=ωrvθ=0.$
(62)

Here, ω is a constant and represents the angular velocity. We assume ω > 0, which means that the flow field is rotating in counterclockwise order. Then, the velocity in the Cartesian coordinate system will be written as

$u=−ωyv=ωx.$
(63)

In this simple case, we can analytically express Rortex, Q, and λci as

$R=2ω,$
(64)
$Q=ω2,$
(65)
$λci=ω.$
(66)

It can be found that Rortex is exactly equal to vorticity.

Now consider the superposition of a prograde shearing motion, which is given by

$u=0v=σx, σ>0.$
(67)

σ > 0 implies that the shearing motion is consistent with the clockwise rigid rotation. The velocity becomes

$u=−ωyv=(ω+σ)x.$
(68)

It can be easily verified that Eq. (68) fulfills 2D vorticity equations.

According to Eq. (40), the velocity gradient tensor is decomposed to

$0−ω0ω+σ00000=0−ω0ω00000+000σ00000,$
(69)

which exactly presents the rigidly rotational part and the shearing part. The explicit expressions of Rortex, Q, and λci are given by

$R=2ω,$
(70)
$Q=ω(ω+σ),$
(71)
$λci=ω(ω+σ).$
(72)

It is expected that in this case the rigidly rotational part of fluids should not be affected by the shear motion. Only Rortex remains the same as the no-shearing case and provides the precise rigidly rotational strength as expected, whereas Q and λci are altered by the shearing effect σ. Obviously, the stronger shearing will result in the larger alteration of Q and λci, as shown in Fig. 1. In Fig. 1, the shearing effect σ is normalized by the angular velocity ω. With the increase of the shearing effect σ, Q and λci both indicate significant deviations from the values in the no-shearing case, which implies that these criteria are prone to contamination by shearing and cannot reasonably represent the local rotation. By contrast, Rortex excludes the shearing effect and remains exactly twice of the angular velocity ω as the no-shearing case.

FIG. 1.

R (Rortex), Q, and λci as functions of σ/ω for 2D rigid rotation superposed by a prograde shearing motion.

FIG. 1.

R (Rortex), Q, and λci as functions of σ/ω for 2D rigid rotation superposed by a prograde shearing motion.

Close modal

Here we examine the Burgers vortex. This vortex has been widely used for modeling fine scales of turbulence. The Burgers vortex is an exact steady solution of the Navier–Stokes equation, where the radial viscous diffusion of vorticity is dynamically balanced by vortex stretching due to an axisymmetric strain. The velocity components in the cylindrical coordinates for a Burgers vortex can be written as

$vr=−ξr,$
(73a)
$vθ=Γ2πr1−e−r2ξ2ν,$
(73b)
$vz=2ξz,$
(73c)

where Γ is the circulation, ξ is the axisymmetric strain rate, and ν is the kinematic viscosity. The Reynolds number for the vortex can be defined as Re = Γ/(2πν). The velocity in the Cartesian coordinate system will be written as

$u=−ξx−Γ2πr21−e−r2ξ2νy,$
(74a)
$v=−ξy+Γ2πr21−e−r2ξ2νx,$
(74b)
$w=2ξz.$
(74c)

The analytical expressions of Rortex, Q, and λci are given by

$R=2Reξζ,$
(75)
$Q=ξ2Re2ζζ+ε−3,$
(76)
$λci=Reξζ(ζ+ε),$
(77)

where $r̃=rξ/ν$ and

$ζ=1r̃21+r̃2e−r̃22−1,$
$ε=2r̃21−1+r̃22e−r̃22.$

Since Rortex and λci are equivalent to the Δ criterion with a zero threshold, the existence conditions of Rortex and λci are identical, namely, ζ > 0, which yields a non-dimensional vortex size of $r̃0$ = 1.5852, consistent with the result of Ref. 31.

Equations (76) and (77) indicate that the shearing part ε will affect Q and λci. To investigate this shearing effect, we consider the superposition of a shearing motion (with an appropriate external force term to fulfill the Navier-Stokes equations), which is given by

$u=−CReξr̃02yv=0w=0,$
(78)

where C is a user-specified constant. We choose Re = 10 and ξ = 1. Figures 2 and 3 demonstrate the iso-contours of Rortex, λci, and Q in the xy plane for the Burgers vortex superposed with the shearing motion when C is set to 1 and 10, respectively. It can be obviously seen that the increase of the shearing motion slightly modifies the distribution of Rortex, but the rotational strength near the central part nearly remains constant. On the other hand, the distribution of λci and Q is significantly disturbed. The value of λci near the central part is increased from 6 to 14 and the value of Q near the center is increased from 40 to 180. This significant deviation demonstrates that these two criteria are prone to the contamination by shearing and the λci criterion is not a reliable measure of the local swirling strength at least when high shear strain exists.

FIG. 2.

Iso-contours of Rortex (a), λci (b), and Q (c) in the xy plane for the Burgers vortex superposed with the shearing motion (C = 1).

FIG. 2.

Iso-contours of Rortex (a), λci (b), and Q (c) in the xy plane for the Burgers vortex superposed with the shearing motion (C = 1).

Close modal
FIG. 3.

Iso-contours of Rortex (a), λci (b), and Q (c) in the xy plane for the Burgers vortex superposed with the shearing motion (C = 10).

FIG. 3.

Iso-contours of Rortex (a), λci (b), and Q (c) in the xy plane for the Burgers vortex superposed with the shearing motion (C = 10).

Close modal

A 2-D Blasius-profile shear layer excited with a perturbation to induce the formation and merging of vortices is studied here. The Reynolds number, which is based on the displacement thickness, is 1000. The inflow Mach number is 0.5. The 2-D flow field is simulated by a DNS code called DNSUTA with a local refined grid which consists of 256 × 512 nodes in the streamwise and the normal direction, respectively.7 A sixth-order compact scheme is applied in the spatial discretization and a third-order total variation diminishing (TVD) Runge-Kutta scheme is adopted for time integration.

Figure 4 demonstrates the evolution of the shear layer and vortices. At the beginning, there exists a large amount of vorticity distribution in the shear layer, as shown in Fig. 4(a). In the following instants, the initial instability induces the roll-up of the shear layer, and two clockwise vortices are generated. And then, these vortices approach each other, as shown in Fig. 4(d). The vortices shown in Figs. 4(b)–4(d) are identified by the contour lines of Rortex (black line) with zero thresholds. In this case, Rortex can clearly capture the boundaries of the vortices. Two points A and B as shown in Fig. 5 are used to investigate the shearing effect. The values of λci (2D version),47 Q (2D version),47 the shearing components ε, and Rortex strengths are presented in Table I. It can be found that with the large decrease of the shearing component, λci is reduced by 16% and Q is reduced by 31%, while Rortex strength remains almost the same. Accordingly, the λci criterion and the Q criterion are sensitive to the shearing effect but Rortex is not.

FIG. 4.

The temporal evolutions of the shear layer and vortices.

FIG. 4.

The temporal evolutions of the shear layer and vortices.

Close modal
FIG. 5.

Contours of λci (a), Q (b), shearing component ε (c), and Rortex strength R (d).

FIG. 5.

Contours of λci (a), Q (b), shearing component ε (c), and Rortex strength R (d).

Close modal
TABLE I.

Comparison of λci, Q, shearing component ε, and Rortex strength R for points A and B.

PointλciQεR
0.1688 0.028 49 0.2445 0.1724
0.1410 0.019 79 0.1487 0.1701
PointλciQεR
0.1688 0.028 49 0.2445 0.1724
0.1410 0.019 79 0.1487 0.1701

The Sullivan vortex is an exact solution to the Navier-Stokes equations for a three-dimensional axisymmetric two-celled vortex.48 The two-celled vortex has an inner cell in which air flow descends from above and flows outward to meet a separate airflow that is converging radially. The mathematical form of the Sullivan Vortex is

$vr=−ar+6νr1−e−ar22ν,$
(79a)
$vθ=Γ2πrHar22νH∞,$
(79b)
$vz=2az1−3e−ar22ν,$
(79c)

where

$Hx=∫0xe−t+3∫0t1−e−τ/τdτdt.$
(80)

In this case, we set a = 1, Γ = 10, and ν = 0.001 to illustrate the local rotational axis. Figure 6 shows the Rortex vector lines on the iso-surface which represent the local rotational axis. It can be seen that the local axis given by Rortex is consistent with the global rotation axis, that is, the z axis, which means that the direction of Rortex is physically reasonable.

FIG. 6.

Rortex vector lines for Sullivan vortex.

FIG. 6.

Rortex vector lines for Sullivan vortex.

Close modal

Here we use the DNS data of late boundary layer transition on a flat plate to compare Rortex with Q and λci. The DNS data are generated by the DNS code DNSUTA.7 A sixth-order compact scheme is applied in the streamwise and normal directions. In the spanwise direction where periodic conditions are applied, the pseudo-spectral method is used. In order to eliminate the spurious numerical oscillations caused by central difference schemes, an implicit sixth-order compact filter is applied to the primitive variables after a specified number of time steps. The simulation was performed with near 60 × 106 grid points and over 400 000 time steps at a free stream Mach number of 0.5. For the detailed case setup, please refer to Ref. 7.

Although all methods illustrate the similar iso-surfaces of vortical structures as shown in Fig. 7, the values of λci and Q can be found contaminated by shearing. Examine three points A, B, and C on the Rortex and λci iso-surfaces of the quasi-streamwise vortical structure, as shown in Fig. 8. Point A is located on both the Rortex and λci iso-surfaces, B is located on the Rortex iso-surface only, and C is located on the λci iso-surface only. The corresponding velocity gradient tensors of A, B, and C are given by Eq. (81). The eigenvalues, the magnitudes of Rortex, and shearing components are provided in Table II. From Table II, we can find that A and B possess the same local rotational strength with different eigenvalues, while A and C have the same imaginary value of the complex eigenvalues but different local rotational strength. The shearing parts are so strong that the λci criterion will be seriously contaminated. Especially for point C, the shearing component ε = 0.81 is significantly larger than the actual local rotation strength R = 0.018, making point C being mistaken for a point with large swirling strength by the λci criterion. From Fig. 8, it can be found that point B which has a strong rotation (R = 0.06) is missed by the λci criterion, but point C which contains a weak rotation (R = 0.018) is mis-identified by the λci criterion. The Q criterion as shown in Fig. 9 will indicate a similar result of contamination, so the detailed analysis is omitted here,

$∇v→A=0.06220.2150.560−0.007 330.003 120.0562−0.0121−0.0514−0.0664,$
(81a)
$∇v→B=0.04980.2110.361−0.001 09−0.009 860.0697−0.006 25−0.0417−0.0406,$
(81b)
$∇v→C=0.0372−0.000 8970.8180.000 0640.02810.000 12−0.01240.000 250−0.0674.$
(81c)
FIG. 7.

Iso-surfaces of Rortex (a), λci (b), and Q (c) for late boundary layer transition.

FIG. 7.

Iso-surfaces of Rortex (a), λci (b), and Q (c) for late boundary layer transition.

Close modal
FIG. 8.

Iso-surfaces of Rortex and λci for the quasi-streamwise vortical structure.

FIG. 8.

Iso-surfaces of Rortex and λci for the quasi-streamwise vortical structure.

Close modal
TABLE II.

Comparison of eigenvalues (λr, λcr, and λci), Rortex strengths (R), and shearing components (ε) for points A, B, and C.

ABC
λr 0.019 7 0.016 2 0.0281
λcr −0.001 04 −0.008 43 −0.0151
λci 0.086 0.059 0.086
0.06 0.06 0.018
ε 0.218 0.086 6 0.81
ABC
λr 0.019 7 0.016 2 0.0281
λcr −0.001 04 −0.008 43 −0.0151
λci 0.086 0.059 0.086
0.06 0.06 0.018
ε 0.218 0.086 6 0.81
FIG. 9.

Iso-surfaces of Rortex and Q for the quasi-streamwise vortical structure.

FIG. 9.

Iso-surfaces of Rortex and Q for the quasi-streamwise vortical structure.

Close modal

Since Rortex is a vector quantity, we can visualize the local rotation axis of the vortex structures. Figures 10 and 11 demonstrate that the Rortex vector is actually tangent to the iso-surface of Rortex. Assume that a point P is located on the iso-surface and a point P* is on the direction of Rortex vector at P, as shown in Fig. 12. According to Definition 1, when P* limits toward P, only the velocity along the local rotation axis Z can change. Correspondingly, only the component along the local rotation axis Z of the velocity gradient tensor can change. So, the component of the velocity gradient tensor in the XY plane will not change, which means that P* will be located on the same iso-surface in the limit and Rortex vector is tangent to the iso-surface of Rortex at point P.

FIG. 10.

Rortex vector on the leg part of the vortical structure.

FIG. 10.

Rortex vector on the leg part of the vortical structure.

Close modal
FIG. 11.

Rortex vector on the vortex ring.

FIG. 11.

Rortex vector on the vortex ring.

Close modal
FIG. 12.

Illustration of the Rortex vector at point P.

FIG. 12.

Illustration of the Rortex vector at point P.

Close modal

Figure 13 shows the structures of Rortex lines and Fig. 14 demonstrates vorticity lines. Both pass the same seed points. As can be seen, vorticity lines can only represent the ring part of the hairpin vortex. By contrast, Rortex lines can provide a skeleton of the whole hairpin vortex. It is expected that Rortex lines will offer a new perspective to analyze the vortical structures.

FIG. 13.

Rortex lines for hairpin vortex.

FIG. 13.

Rortex lines for hairpin vortex.

Close modal
FIG. 14.

Vorticity lines for hairpin vortex.

FIG. 14.

Vorticity lines for hairpin vortex.

Close modal

For large data sets, the eigenvector-based definition brings remarkable improvement to the computational efficiency. In our earliest implementation, the direction of Rortex was obtained by solving a nonlinear system of equations through the Newton-iterative method.40 In Ref. 41, a fast algorithm based on real Schur decomposition was proposed to reduce the computational cost. The real Schur decomposition is performed using a standard numerical linear algebra library LAPACK.49 Table III illustrates the computational times of our previous and present methods. The computational times of the λci criterion and the Q criterion are presented as well. All the calculations are run on a MacBook Pro (Late 2013) laptop with 2.0 GHz CPU and 8 GB memory. It can be observed that the computational time of the present method is reduced by one order of magnitude compared to our previous methods and comparable to that of the λci criterion.

TABLE III.

Comparison of computational times of different methods for the DNS data of late boundary layer transition on a flat plate.

Real Schur
MethodNewton-iterativedecompositionλciQPresent
Time (s) 264 120 7.5 0.5 11.3
Real Schur
MethodNewton-iterativedecompositionλciQPresent
Time (s) 264 120 7.5 0.5 11.3

In the present study, an alternative eigenvector-based definition of Rortex is introduced. A systematic interpretation of scalar, vector, and tensor versions of Rortex is presented to provide a unified characterization of the local fluid rotation. Several conclusions are summarized as follows:

1. The real eigenvector of the velocity gradient tensor is used to determine the direction of Rortex, which represents the possible axis of the local fluid rotation, and the rotational strength obtained in the plane perpendicular to the possible local axis is defined as the magnitude of Rortex.

2. Eigenvalue-based criteria are exclusively determined by the eigenvalues of the velocity gradient tensor. If two points have the same eigenvalues, they are located on the same iso-surface. But Rortex cannot be exclusively determined by the eigenvalues. Even if two points have the same eigenvalues, the magnitudes of Rortex are generally different.

3. The existing eigenvalue-based methods can be seriously contaminated by shearing. Since shearing always manifests its effect on the imaginary part of the complex eigenvalues, any criterion associated with the complex eigenvalues will be prone to contamination by shear. While Rortex eliminates the contamination and thus can accurately quantify the local rotational strength.

4. Rortex can identify the local rotational axis and provide the precise local rotational strength, thereby can reasonably represent the local rigidly rotation of fluids.

5. In contrast to eigenvalue-based criteria, as a vector quantity, not only the iso-surface of Rortex but also the Rortex vector field and Rortex lines can be used to visualize and investigate vortical structures.

6. A new velocity gradient tensor decomposition is proposed. The velocity gradient tensor is decomposed to a rigid rotation part and a non-rotational part including shearing, stretching, and compression, different from the traditional symmetric and anti-symmetric tensor decomposition.

7. Since both the local rotation axis and magnitude of Rortex are uniquely determined by the velocity gradient tensor without any dynamics involved, Rortex is a mathematical definition of fluid kinematics.

8. Our new implementation to calculate Rortex dramatically improves the computational efficiency. The calculation time of the present method is reduced by one order of magnitude compared to our previous methods and comparable to that of the λci criterion.

This work was supported by the Department of Mathematics at the University of Texas at Arlington and AFOSR Grant No. MURI FA9559-16-1-0364. The authors are grateful to the Texas Advanced Computing Center (TACC) for providing computation hours. This work was accomplished by using Code DNSUTA, which was released by Dr. Chaoqun Liu at the University of Texas at Arlington in 2009. The name of Rortex is credited to the discussion with many colleagues in the WeChat groups.

A numerical comparison of Rortex and Kolář’s triple decomposition (TDM) is performed for the DNS data of late boundary layer transition on a flat plate. The case setup is identical to the DNS example in Sec. V. Since the explicit expression for the basic reference frame (BRF) is unavailable, thorough search is required to determine the BRF. The angular step size is set to 4° according to the recommendation given in Ref. 48. The iso-surfaces of Rortex and the residual vorticity, as shown in Fig. 15, indicate a resemblance of the vortical structures identified by two methods. However, the TDM costs several orders of magnitude more time compared to Rortex, as shown in Table IV, which means a severe limitation for the practical applicability of the TDM.

FIG. 15.

Iso-surfaces of Rortex (a) and residual vorticity (b) for late boundary layer transition on a flat plate.

FIG. 15.

Iso-surfaces of Rortex (a) and residual vorticity (b) for late boundary layer transition on a flat plate.

Close modal
TABLE IV.

Comparison of computational times of Rortex and TDM for DNS data of late boundary layer transition on a flat plate.

MethodRortexTDM
Time 12 s About 18 h
MethodRortexTDM
Time 12 s About 18 h

In this appendix, an analytical solution for the eigenvalues of the velocity gradient tensor is presented. Let A be a matrix representation of the velocity gradient tensor in the original xyz-frame

$A=∂u∂x∂u∂y∂u∂z∂v∂x∂v∂y∂v∂z∂w∂x∂w∂y∂w∂z$
(B1)

and λ be the eigenvalue. The characteristic equation of the matrix A is given by

$λ3+Pλ2+Qλ+R̃=0,$
(B2)

where

$P=−λ1+λ2+λ3=−tr(A),$
(B3)
$Q=λ1λ2+λ2λ3+λ3λ1=−12trA2−tr(A)2,$
(B4)
$R̃=−λ1λ2λ3=−det(A).$
(B5)

Here, tr represents the trace of the matrix and det represents the determinant. The cubic Eq. (B2) can be solved by a robust algorithm to minimize the roundoff error.50 Here we are only concerned about the case of the existence of two complex roots as the existence of three real roots implies no local rotation. First, we compute

$S≡P2−3Q9,$
(B6)
$T≡2P3−9PQ+27R̃54.$
(B7)

If T2 > S3, the cubic equation has two complex roots. By computing

$A=−sgn(T)T+T2−S31/3,$
(B8)
$B=S/A (A=0)0 (A=0),$
(B9)

where sgn is the sign function, the three roots can be written as

$λ1=−12A+B−P3+i32A−B,$
(B10)
$λ2=−12A+B−P3−i32A−B,$
(B11)
$λ3=A+B−P3.$
(B12)

Because A and B are both real, λ1 and λ2 are the complex eigenvalues and λ3 is the real eigenvalue.

Here, we derive the analytical expression for the normalized real eigenvector $r→$ corresponding to the real eigenvalue λr. Also, we focus on the case of the existence of two complex eigenvalues and one real eigenvalue. In this case, the normalized real eigenvector is unique (up to sign). Assuming that A is a matrix representation of the velocity gradient tensor and $r→*=rx*,ry*,rz*T$ represents an unnormalized eigenvector corresponding to λr, we can obtain the following equation:

$Ar→*=λrr→*.$
(C1)

Equation (C1) can be rewritten as

$∂u∂x−λr∂u∂y∂u∂z∂v∂x∂v∂y−λr∂v∂z∂w∂x∂w∂y∂w∂z−λrrx*ry*rz*=0.$
(C2)

By checking three first minors

$Δx=∂v∂y−λr∂v∂z∂w∂y∂w∂z−λr,Δy=∂u∂x−λr∂u∂z∂w∂x∂w∂z−λr,Δz=∂u∂x−λr∂u∂y∂v∂x∂v∂y−λr,$
(C3)

we can find the maximum absolute value

$Δmax=maxΔx,Δy,Δz$
(C4)

(note: not all the minors will be equal to zero, thus $Δmax>0$. Otherwise, we will arrive at a contradiction that the normalized real eigenvector is nonunique, or the real eigenvector is a zero vector).

If $Δmax=Δx$, we can set

$rx*=1.$
(C5)

By solving

$∂v∂y−λr∂v∂z∂w∂y∂w∂z−λrry*rz*=−∂v∂x−∂w∂x,$
(C6)

we obtain the other two components of $r→*$ as

$ry*=−∂w∂z−λr∂v∂x+∂v∂z∂w∂x∂v∂y−λr∂w∂z−λr−∂v∂z∂w∂y,$
(C7)
$rz*=∂w∂y∂v∂x−∂v∂y−λr∂w∂x∂v∂y−λr∂w∂z−λr−∂v∂z∂w∂y.$
(C8)

Similarly, if $Δmax=Δy$, we have

$ry*=1,$
(C9)
$rx*=−∂w∂z−λr∂u∂y+∂u∂z∂w∂y∂u∂x−λr∂w∂z−λr−∂u∂z∂w∂x,$
(C10)
$rz*=∂w∂x∂u∂y−∂u∂x−λr∂w∂y∂u∂x−λr∂w∂z−λr−∂u∂z∂w∂x.$
(C11)

In the case of $Δmax=Δz$, we have

$rz*=1,$
(C12)
$rx*=−∂v∂y−λr∂u∂z+∂u∂y∂v∂z∂u∂x−λr∂v∂y−λr−∂u∂y∂v∂x,$
(C13)
$ry*=∂v∂x∂u∂z−∂u∂x−λr∂v∂z∂u∂x−λr∂v∂y−λr−∂u∂y∂v∂x,$
(C14)

and the normalized real eigenvector $r→$ will be

$r→=r→*/r→*.$
(C15)
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