Stretching and Shearing Contamination Analysis for Liutex ...
Transcript of Stretching and Shearing Contamination Analysis for Liutex ...
Stretching and Shearing Contamination Analysis forLiutex and Other Vortex Identi๏ฟฝcation MethodsPhushpa Shrestha
University of Texas at ArlingtonCharles Nottage
University of Texas at ArlingtonYifei Yu
University of Texas at ArlingtonOscar Alvarez
University of Texas at ArlingtonChaoqun Liu ( [email protected] )
University of Texas at Arlington https://orcid.org/0000-0002-7204-1974
Research
Keywords: Liutex, vorticity, vortex, Omega method, transition ๏ฟฝow, velocity gradient tensor, principalcoordinate, principal tensor, Liutex tensor decomposition.
Posted Date: October 7th, 2020
DOI: https://doi.org/10.21203/rs.3.rs-86116/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License
Version of Record: A version of this preprint was published on March 12th, 2021. See the publishedversion at https://doi.org/10.1186/s42774-020-00060-9.
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Stretching and Shearing Contamination Analysis for Liutex and
Other Vortex Identification Methods
Pushpa Shrestha, Charles Nottage, Yifei Yu, Oscar Alvarez, and Chaoqun Liu1
Department of Mathematics, The University of Texas at Arlington, Arlington, Texas 76013, USA
Abstract: The newly developed vortex-identification method, Liutex, has provided a new systematic
description of the local fluid rotation, which includes scalar, vector, and tensor forms. However, the advantages
of Liutex over the other widely used vortex-identification methods such as ๐, โ, ๐2, and ๐๐๐ have not been
realized. These traditional methods count on shearing and stretching as a part of vortex strength. But, in the real
flow, shearing and stretching do not contribute to fluid rotation. In this paper, the decomposition of the velocity
gradient tensor is conducted in the Principal Coordinate for uniqueness. Then the contamination effects of
stretching and shearing of the traditional methods are investigated and compared with the Liutex method in
terms of mathematical analysis and numerical calculations. The results show that the Liutex method is the only
method that is not affected by either stretching or shear, as it represents only the local fluid rigid rotation. These
results provide supporting evidence that Liutex is the superior method over others.
Keywords: Liutex, vorticity, vortex, Omega method, transition flow, velocity gradient tensor, principal
coordinate, principal tensor, Liutex tensor decomposition.
1. Introduction
A vortex is recognized as the rotational motion of fluids. Within the last several decades, a lot of vortex
identification methods have been developed to track the vortical structure in a fluid flow; however, we still lack
unambiguous and universally accepted vortex identification criteria. This is the impediment that has caused a lot
of confusion and misunderstandings in turbulence research (Liu et al., 2014)[1]. In most research papers and
textbooks, vorticity tube/filament is regarded as a vortex and the magnitude of the vorticity as the local
1 Corresponding author: Chaoqun Liu, Email: [email protected]
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rotational strength. Many researchers widely acknowledged the concept of vortex defined as the vorticity
concentration and other vorticity-based methods (Helmholtz, H., 1858; Saffma, P., 1990)[2-3] as the vorticity
vector was believed to offer a mathematical definition of fluid rotational motion. Zhou and Antonia (1993)[4]
utilized the spatially phased correlated vorticity to characterize large-scale and organized structures in the
cylinder wake. However, problems arose while applying in viscous flows, particularly in turbulent flows. In
turbulent viscous flows, the rotation strength is minimal near the wall, where shear stress is dominant. Many
researchers in the literature have supported this claim, indicating the inadequacies of vorticity-based methods.
Epps (2017)[5] observed that vorticity could not distinguish a vortical region with rotational motions from a
strong shear layer. Robinson (1991) [6] also uncovered the fact that the regions of strong vorticity and actual
vortices are weakly related. These vorticity-based methods are the first generation (1G) of vortex identification
methods as per Liu et al. (2019)[7], while scalar based schemes such as โ, Q, ๐2, ๐ci, and ฮฉ are known as second-
generation (2G) criteria.
Attempting to remedy the problems of vorticity based methods, 2G vortex identification criteria such as โ (Chong and Perry,1990) [8, 9], Q (Hunt et al., 1988)[10], ๐2 (Jeong, J., and Hussain, F., 1995)[11], ๐ci (Zhou et al.,
1999)[12], ฮฉ (Liu et al., 2016; Dong et al., 2018)[13-14], etc., have been introduced and extensively applied in
visualizing vortex structures over the past three decades. Although the 2G- schemes have been able to judge the
presence of the local rotational motion, these methods need an adjustment of threshold to visualize the iso-
surface plot accurately. This poses a problem as it is difficult to adjust the threshold that will define the
boundary of vortical structures in a particular case (Liu et al., 2016; Zhang et al., 2018)[13,15]. Moreover, the ๐
and ๐2 methods are restrictive for incompressible flows due to their incompressibility assumption. These and
other issues of 1G and 2G vortex identification methods prompted the development of a new vortex
identification method, the Liutex method (Liu et al., 2018; Gao et al., 2018)[16-17], also known as a third-
generation vortex identification scheme.
According to Chong and Perry (1990) [8], if the velocity gradient tensor of a three-dimensional flow field has a
pair of complex conjugate eigenvalues and a real eigenvalue, then the instantaneous streamline pattern presents
a local swirling motion. Liu et al. (2019)[7] acknowledged that these streamline patterns are the direct result of
the real eigenvector of the velocity gradient tensor. In fact, this real eigenvector is the axis of fluid rotation
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which is also known as the direction of Liutex. Unlike 2G schemes, the Liutex method is a novel eigenvector-
based method that is local, accurate, unique, and systematic. Furthermore, the systematical definition of Liutex
is given as a vector which has its corresponding scalar and tensor forms[17]. The vector form of Liutex gives the
direction of the local fluid rotation, while the magnitude of Liutex represents the rotational strength of a fluid
rotation. Also, the Liutex vector is Galilean invariant (Haller, 2005; Wang et al., 2018; Liu et al., 2019)[18-20].
According to Liu et al. [7], the Liutex method, Liutex-Omega method (Liu et al., 2018)[16], Liutex Core Line
method, and other Liutex-based methods (Dong, 2019; Liu et al., 2019)[30-32] are the third generation (3G) of
vortex identification methods. The Liutex method is discussed briefly in section 2.6.
After the introduction of the Liutex method in 2018, many doors of investigations of fluid rotations and related
topics have opened. Based on Liutex definition, Gao and Liu proposed a velocity gradient tensor decomposition
(Gao and Liu, 2018)[24] in the rotated coordinate system, which was somewhat lengthy and cumbersome. So,
later in the same year, they gave explicit expression for the Rortex/Liutex tensor and velocity gradient tensor
decomposition (Gao and Liu, 2018)[25] without coordinate rotation. Recently, Yu and Liu proposed a new
unique definition of Principal Coordinate and conducted a unique Principal Tensor decomposition under the
Principal coordinate (Yu and Liu, 2020)[29]. In this paper, the concept of Principal Coordinate and Principal
Tensor decomposition is revisited briefly in section 3. Also, the issue of shear and stretching contamination of
1G and several 2G vortex identification methods are investigated in terms of mathematical analysis and
numerical calculations. Our investigation involves calculating the velocity gradient tensor at a point inside the
numerical domain (flow field). Then, a unique Principal Tensor decomposition is performed in the Principal
Coordinate. In other words, the velocity gradient tensor is uniquely decomposed into its rigid rotation part,
stretching part, and shearing part in the Principal Coordinate. From our investigation, it is found that 1G and
several 2G vortex identification methods misinterpret shearing and stretching as a constituent of vortical
structure, which is not true. Also, these methods have dimensional problems at different levels, which is
discussed in section 4 in detail.
The organization of this paper is as follows: In Section 2, some traditional vortex identification methods are
revisited including vorticity tube/filaments, โ, ๐, ๐2 ๐๐๐ ๐๐๐ methods. Section 3 describes the Liutex-based
Principal Tensor Decomposition in the unique Principal Coordinate. Based on the unique Principal Tensor
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Decomposition in the Principal Coordinate, the theoretical contamination analysis of the vorticity, โ, ๐, ๐2 ,๐๐๐ ๐๐๐ Liutex schemes are examined and presented in Section 4. Numerical experiments and their results can
be found in Sections 5 and 6, respectively, and some concluding remarks are made in the last section.
2. Review of 1G and 2G Vortex Identification Methods.
2.1 Vorticity-based method
Since Helmholtz introduced the concept of vorticity tube/filament in 1858[2], many researchers have believed
that vortices consist of small vorticity tubes which are called vortex filaments, and the magnitude of vorticity
gives the vortex strength. The vorticity vector is mathematically defined as the curl of the velocity. i.e.
Vorticity = โ ร ๐ = | ๐ ๐ ๐๐๐๐ฅ ๐๐๐ฆ ๐๐๐ง๐ข ๐ฃ ๐ค|= ๐ (๐๐ค๐๐ฆ โ ๐๐ฃ๐๐ง) โ ๐ (๐๐ค๐๐ฅ โ ๐๐ข๐๐ง) + ๐(๐๐ฃ๐๐ฅ โ ๐๐ข๐๐ฆ) (1)
Helmholtz also defined vortex lines and vortex filaments based on the vorticity vector. According to Helmholtz,
vortex lines are drawn through the fluid mass so that their direction at every point coincides with the direction
of the momentary axis of rotation of the water particles lying on it. He also defined vortex filament as the
portions of the fluid mass cut out from it by way of constructing corresponding vortex lines through all points of
circumference of an infinitely small surface element. Another vorticity-based method to define a vortical
structure is vortex-tube. Lamb[21] defines vortex-tube as: โIf through every point of a small closed curve we
draw the corresponding vortex-lie, we mark out a tube known as vortex-tube.โ
2.2 ๐ซ Method
The โ method defines a vortex to be the region where the velocity gradient tensor ๐ต๏ฟฝโโ๏ฟฝ has a pair of complex
conjugate eigenvalues and a real eigenvalue (Chong et al., 1990)[8]. If ๐1, ๐2 ๐๐๐ ๐3 are the eigenvalues of the
3ร 3 matrix of ๐ต๏ฟฝโโ๏ฟฝ , then the characteristic equation can be written as
๐3 + ๐ผ1๐2 + ๐ผ2๐ + ๐ผ3 = 0 (2)
where, ๐ผ1, ๐ผ2, and ๐ผ3 are the first, second, and third invariants of the characteristic equation (2) and given by
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๐ผ1 = โ(๐1 + ๐2 + ๐3) = โ๐ก๐(๐ต๏ฟฝโโ๏ฟฝ ) (3)
๐ผ2 = ๐1๐2 + ๐2๐3 + ๐3๐1 = โ 12 [tr(๐ต๏ฟฝโโ๏ฟฝ 2) โ tr(๐ต๏ฟฝโโ๏ฟฝ )2] (4)
๐ผ3 = โ๐1๐2๐3 = โdet(๐ต๏ฟฝโโ๏ฟฝ ) (5)
The discriminant of the characteristic equation (2) is given by
๐ฅ = (๏ฟฝฬ๏ฟฝ3)3 + (๏ฟฝฬ๏ฟฝ2)2 (6)
where ๏ฟฝฬ๏ฟฝ = ๐ผ2 โ 13 ๐ผ12 and ๏ฟฝฬ๏ฟฝ = ๐ผ3 + 227 ๐ผ13 โ 13 ๐ผ1๐ผ2.
For incompressible flow, the first invariant ๐ผ1 = 0, which results in equation (6) becoming โ = (๐ผ23)3 + (๐ผ32)2. If ๐ฅ โค0, all three eigenvalues of ๐ต๏ฟฝโโ๏ฟฝ are real, but if โ > 0, there exists one real and two conjugate complex
eigenvalues. The latter means that the point is inside a vortex region. Although the ๐ฅ-method can capture the
vortex region successfully, it is susceptible to the threshold value, which is man-made and arbitrary in general.
2.3 ๐ธ Method
Proposed by Hunt et al. (1988)[10], the ๐ method is one of the most popular methods used to visualize the vortex
structure. Although ๐ is the second invariant in the eigenvalue equation (2) given above, the value of ๐ can also
be calculated as half of the difference of squares of the Frobenius norm of the vorticity tensor and strain-rate
tensor. i.e.,
๐ = 12 (โ๐ฉโ๐น2 โ โ๐จโ๐น2) (7)
where A and B are the symmetric and anti-symmetric part of the velocity gradient tensor, respectively.
๐จ = 12 (๐ต๏ฟฝโโ๏ฟฝ + ๐ต๏ฟฝโโ๏ฟฝ ๐ป) = [ ๐๐ข๐๐ฅ 12 (๐๐ข๐๐ฆ + ๐๐ฃ๐๐ฅ) 12 (๐๐ข๐๐ง + ๐๐ค๐๐ฅ)12 (๐๐ฃ๐๐ฅ + ๐๐ข๐๐ฆ) ๐๐ฃ๐๐ฆ 12 (๐๐ฃ๐๐ง + ๐๐ค๐๐ฆ)12 (๐๐ค๐๐ฅ + ๐๐ข๐๐ง) 12 (๐๐ค๐๐ฆ + ๐๐ฃ๐๐ง) ๐๐ค๐๐ง ]
(8)
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๐ฉ = 12 (๐ต๏ฟฝโโ๏ฟฝ โ ๐ต๏ฟฝโโ๏ฟฝ ๐ป) = [ 0 12 (๐๐ข๐๐ฆ โ ๐๐ฃ๐๐ฅ) 12 (๐๐ข๐๐ง โ ๐๐ค๐๐ฅ)12 (๐๐ฃ๐๐ฅ โ ๐๐ข๐๐ฆ) 0 12 (๐๐ฃ๐๐ง โ ๐๐ค๐๐ฆ)12 (๐๐ค๐๐ฅ โ ๐๐ข๐๐ง) 12 (๐๐ค๐๐ฆ โ ๐๐ฃ๐๐ง) 0 ]
(9)
The ๐ method considers that a vortex occurs in the region where ๐ > 0. Using the ๐ method, it is easy to track
the vortical structure by iso-surface plotting. However, ๐ is scalar-valued, and a proper threshold, which in
general does not exist, is required to visualize the vortex region. Also, there exists an inconsistency between
the โ method and ๐ method as (๏ฟฝฬ๏ฟฝ2)2 is always positive, and even if ๐ < 0, it is still possible that ฮ could be
positive.
2.4 ๐๐๐ Criterion
The ๐๐๐ criterion (Zhou et al., 1999; Chakraborty et al., 2005)[12, 22] uses the imaginary part of the complex
eigenvalues of the velocity gradient tensor to visualize the vortex structure. It is based on the idea that the local
time-frozen streamlines exhibit a rotational flow pattern when ๐ต๏ฟฝโโ๏ฟฝ has a pair of complex conjugate eigenvalues.
In this case, the tensor transformation of ๐ต๏ฟฝโโ๏ฟฝ is given by
๐ต๏ฟฝโโ๏ฟฝ = [๐ฃ ๐ ๐ฃ ๐๐ ๐ฃ ๐๐] [๐๐ 0 00 ๐๐๐ ๐๐๐0 โ๐๐๐ ๐๐๐] [๐ฃ ๐ ๐ฃ ๐๐ ๐ฃ ๐๐]โ1 , (10)
where ๐๐ is the real eigenvalue with the corresponding real eigenvector ๐ฃ ๐ and the pair of complex conjugate
eigenvalues are ๐๐๐ ยฑ ๐๐๐๐ with corresponding eigenvectors ๐ฃ ๐๐ ยฑ ๐ ๐ฃ ๐๐ . In this case, in the local curvilinear
system (๐1, ๐2, ๐3) spanned by the eigenvector (๐ฃ ๐ , ๐ฃ ๐๐ , ๐ฃ ๐๐) , the instantaneous streamlines exhibit a spiral
motion. The equations of such streamlines can be written as:
๐1(๐ก) = ๐1(0)๐๐๐๐ก (11)
๐2(๐ก) = [๐2(0) cos(๐๐๐๐ก) + ๐3(0)๐ ๐๐(๐๐๐๐ก)]๐๐๐๐๐ก (12)
๐3(๐ก) = [๐3(0) cos(๐๐๐๐ก) โ ๐2(0)๐ ๐๐(๐๐๐๐ก)]๐๐๐๐๐ก (13)
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Despite being used by many researchers to track the vortical structure, the strength of this swirling motion was
improperly quantified by ๐๐๐, because ๐๐๐ cannot be the pure rotational strength. ๐๐๐ is also characterized as a
scalar-valued criterion (Liu et al., 2019)[7]. Note that Eq. (10) represents a similar transformation (not an
orthogonal transformation) that cannot keep vorticity constant.
2.5 ๐๐ criterion
The ๐2-criterion is calculated based on the observation that in a vortical region, pressure tends to be the lowest
on the axis of a swirling motion of fluid particles. This happens because the centrifugal force is balanced by the
pressure force (the cyclostrophic balance). This method is valid only in a steady inviscid and incompressible
flow (Jeong, J., and Hussain, F., 1995)[11]. However, this assumption fails to accurately identify vortices under
compressible, unsteady, and viscous conditions. By neglecting these unsteady and viscous effects, the
symmetric part S of the gradient of the incompressible NavierโStokes equation can be expressed as:
๐บ = ๐จ2 +๐ฉ2 = โ ๐(๐๐)๐ (14)
where p is the pressure and equation (14) is a representation of the pressure Hessian matrix
((๐ต(๐ต๐))๐๐ = ๐2๐๐๐ฅ๐๐๐ฆ๐. To capture the region of local pressure minimum in a plane perpendicular to the vortex
core line, Jeong & Hussain defined the vortex core as a connected region with two positive eigenvalues of the
pressure Hessian matrix, i.e., a connected region with two negative eigenvalues of the symmetric tensor S. If ๐๐1, ๐๐2 & ๐๐3 are three real eigenvalues of the symmetric tensor S and when setting them in order in such a way
that ๐๐1 โฅ ๐๐2 โฅ ๐๐3, there must be ๐๐2 < 0 as two eigenvalues are negative, which confirms the existence of a
vortex. In general, ๐๐2 cannot be expressed in terms of eigenvalues of the velocity gradient tensor; however, in
some special cases when eigenvectors are orthonormal, ๐๐2 can be exclusively determined by eigenvalues of the
velocity gradient tensor. Vortex structure can be visualized as iso-surface by selecting a proper threshold of ๐๐2
(Liu et al., 2019)[7]. The relation between the eigenvalues of the symmetric tensor A2+B2 and second invariant Q
is given by:
๐ = โ 12 ๐ก๐(๐จ2 +๐ฉ2) = โ 12 (๐๐1 + ๐๐2 + ๐๐3) (15)
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While the Q-criterion measures the excess of vorticity rate over the strain rate magnitude in all directions, the ๐2
criterion looks for this excess only on a specific plane (Jeong, J., and Hussain, F., 1995)[11]. For comparative
studies of different schemes, another article, โon the relationship between vortex identifications methodsโ by
Chakraborty & Balachandar (2005) [22] is also recommended. There is yet another article by Liu et al., โThird
generation of vortex identification methodsโ, which reviews the second generation of vortex identification
methods. These 2G methods are based on the eigenvalues of ๐ต๏ฟฝโโ๏ฟฝ . Afterward, we are presented with the novel
third generation of vortex identification methods[7].
2.6 Liutex
Definition 1: Liutex is a vector defined as ๏ฟฝโ๏ฟฝ = ๐ ๐ , where R is the magnitude of Liutex, and ๐ is the direction of
Liutex.
According to Ref. [23], ๐ is the real eigenvector of the velocity gradient tensor and the explicit formula of R is
๐ = ฯโโ โ ๐ โ โ(ฯโโ โ ๐ )2 โ 4๐๐๐2 (16)
Dr. Liu classified the Liutex method as the third generation of vortex identification method, while the Q method, ๐๐๐ criterion and ๐2 criterion are all classified as second-generation methods since they are all scalar and
eigenvalue related. Liutex is a vector which overcomes the drawbacks of the scalar methods, e.g., it can give the
direction of rotation and represents the rotational strength accurately. Liutex is finally found as a mathematical
definition of fluid rotation or vortex.
3. Principal Coordinates and Principal Decomposition.
Definition 2: The Principal Coordinate at a point is a coordinate that satisfies (Yu et al., 2020)[29]:
1. Its Z-axis is parallel to the ๐ (direction of Liutex)
2. The velocity gradient tensor under this coordinate is in the form of:
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๐๐ฝ = [ ๐๐๐ ๐๐๐๐ 0๐๐๐๐ ๐๐๐ 0๐๐๐๐ ๐๐๐๐ ๐๐]
, (17)
where r and cr
are the real eigenvalue and real part of the conjugate complex eigenvalue pair of the velocity
gradient tensor, respectively, for rotation points.
3. ๐๐๐๐ < 0 and |๐๐๐๐| < |๐๐๐๐|
The following two theorems are valid in the Principal Coordinate.
Theorem 1: Under the Principal Coordinate, ๐๐๐๐ = โ ๐ 2, where R is the magnitude of Liutex.
Proof: For an arbitrary velocity gradient tensor๐๐, there exists a rotation matrix ๐ธ๐ that aligns the Z-axis of
the new frame XYZ with the Liutex direction ๐ after rotation.
๐๐ฝ = ๐ธ๐๐๐๐ธ๐๐ป = [ ๐๐๐๐ ๐๐๐๐ 0๐๐๐๐ ๐๐๐๐ 0๐๐๐๐ ๐๐๐๐ ๐๐๐๐ ]
(18)
Where xyz is the original frame and XYZ is the new frame after ๐ธ๐-rotation.
Applying a second rotation ๐ท๐ around the Z-axis (P rotation), we have the Principal Coordinate. Under the
Principal Coordinate, the corresponding velocity gradient tensor ๐๐ฝ๐ฝ can be written as
๐๐ฝ๐ฝ = ๐ท๐๐๐ฝ๐ท๐๐ป = [ ๐๐๐๐|๐ ๐๐๐๐|๐ 0๐๐๐๐|๐ ๐๐๐๐|๐ 0๐๐๐๐ |๐ ๐๐๐๐ |๐ ๐๐๐๐ |๐]
, (19)
where the rotation matrix ๐ท๐ is given by
๐ท๐ = [ ๐๐๐ ๐ ๐ ๐๐๐ 0โ๐ ๐๐๐ ๐๐๐ ๐ 00 0 1] (20)
and,
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๐๐๐๐|๐ = ๐ผ sin(2๐ + ๐) โ ๐ฝ (21)
๐๐๐๐|๐ = ๐ผ sin(2๐ + ๐) + ๐ฝ (22)
๐๐๐๐|๐ = โ๐ผ cos(2๐ + ๐) + 12 (๐๐๐๐ + ๐๐๐๐) (23)
๐๐๐๐|๐ = ๐ผ cos(2๐ + ๐) + 12 (๐๐๐๐ + ๐๐๐๐) (24)
๐ผ = 12โ(๐๐ฃ๐๐ฆ โ ๐๐ข๐๐ฅ)2 + (๐๐ฃ๐๐ฅ + ๐๐ข๐๐ฆ)2 (25)
๐ฝ = 12 (๐๐ฃ๐๐ฅ โ ๐๐ข๐๐ฆ) (26)
๐ ={ ๐๐๐๐ก๐๐ (๐๐ฃ๐๐ฅ+๐๐ข๐๐ฆ๐๐ฃ๐๐ฆโ๐๐ข๐๐ฅ), ๐๐ฃ๐๐ฆ โ ๐๐ข๐๐ฅ โ 0 ๐2 , ๐๐ฃ๐๐ฆ โ ๐๐ข๐๐ฅ = 0, ๐๐ฃ๐๐ฅ + ๐๐ข๐๐ฆ > 0 โ ๐2 , ๐๐ฃ๐๐ฆ โ ๐๐ข๐๐ฅ = 0, ๐๐ฃ๐๐ฅ + ๐๐ข๐๐ฆ < 0 (27)
Then, the Liutex magnitude/rotational strength is defined as (Liu et al., 2019)[7]
๐ = { 2(|๐ฝ| โ ๐ผ), ๐ฝ2 > ๐ผ2 0, ๐๐กโ๐๐๐ค๐๐ ๐ (28)
From part (2) in Def. 2,
๐๐๐๐|๐ = ๐๐๐๐|๐ (29)
thus,
cos(2๐ + ๐) = 0 (30)
Which implies sin(2๐ + ๐) = 1 ๐๐ โ 1. This leads us to the following four cases.
Case 1: ๐ฝ > 0 and sin(2๐ + ๐) = 1
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๐๐๐๐|๐ = ๐ผ โ ๐ฝ = โ 12๐ (31)
Case 2: ๐ฝ > 0 and sin(2๐ + ๐) = โ1
๐๐๐๐|๐ = โ๐ผ โ ๐ฝ (32)
๐๐๐๐|๐ = โ๐ผ + ๐ฝ (33)
However,
|โ๐ผ โ ๐ฝ| > |โ๐ผ + ๐ฝ| (34)
This contradicts part 3 in Def. 2, therefore, sin(2๐ + ๐) โ โ1.
Case 3: ๐ฝ < 0 and sin(2๐ + ๐) = 1
๐๐๐๐|๐ = ๐ผ โ ๐ฝ (35)
๐๐๐๐|๐ = ๐ผ + ๐ฝ (36)
However,
|๐ผ โ ๐ฝ| > |๐ผ + ๐ฝ| (37)
This contradicts part 3 in Def. 2, therefore, sin(2๐ + ๐) โ 1.
Case 4: ๐ฝ < 0 and sin(2๐ + ๐) = โ1
๐๐๐๐|๐ = ๐ผ โ ๐ฝ = โ 12๐ (38)
Theorem 2: For an arbitrary velocity gradient tensor, the Principal Coordinate is unique.
Proof:
From the proof of theorem 1, if ๐ฝ > 0, then cos(2๐ + ๐) and sin(2๐ + ๐) must be 0 and 1, respectively. Thus, 2๐ + ๐ = ๐2 โ ๐ = 12 (๐2 โ ๐) . Similarly, if ๐ฝ < 0 , then cos(2๐ + ๐) and sin(2๐ + ๐) must be 0 and -1,
12
respectively. So, 2๐ + ๐ = 3๐2 โ ๐ = 12 (3๐2 โ ๐). Therefore, the angle of rotation around the Z-axis is unique,
and so is the Principal Coordinate.
Definition 3: The Principal Decomposition is the decomposition of velocity gradient tensor under the Principal
Coordinate i.e.
๐๐ฝ = [ ๐๐๐ โ ๐ 2 0๐ 2 + ๐ ๐๐๐ 0๐ ๐ ๐๐] = [0 โ ๐ 2 0๐ 2 0 00 0 0] + [0 0 0๐ 0 0๐ ๐ 0] + [๐๐๐ 0 00 ๐๐๐ 00 0 ๐๐] = ๐ด + ๐ต + ๐ถ (39)
Here A represents the rotation part, B represents the shear part, and C represents the stretching part.
Unlike traditional Cauchy-Stokes decomposition, The Principal Decomposition correctly and uniquely
decomposes the velocity gradient tensor into its rotation part, shear part, and stretching part. This decomposition
expresses that the shear and stretching contamination analysis can only be uniquely done in the Principal
Coordinate[29].
4. Theoretical Contamination Analysis
The velocity gradient tensor in the Principal Coordinate is:
๐ต๏ฟฝโโ๏ฟฝ = [ ๐๐๐ โ 12๐ 012๐ + ๐ ๐๐๐ 0๐ ๐ ๐๐ ] (40)
Then the Principal Decomposition is given by
๐ต๏ฟฝโโ๏ฟฝ = [ ๐๐๐ โ 12๐ 012๐ + ๐ ๐๐๐ 0๐ ๐ ๐๐ ] = [0 โ ๐ 2 0๐ 2 0 00 0 0] + [0 0 0๐ 0 0๐ ๐ 0] + [๐๐๐ 0 00 ๐๐๐ 00 0 ๐๐] = ๐ด + ๐ต + ๐ถ (41)
A is the rotation part, B is the shear part and C is the stretching part
13
For the first time in the literature, we are defining the several vortex identification schemes in the Principal
Coordinate, which will allow us to analyze the contamination effects on these traditional vortex identification
methods.
4.1 Contamination of vorticity
Plugging in the gradient components from (40) into the equation (1) gives us the vorticity vector as ๐ = (๐, โ๐, ๐ + ๐)๐ (42)
and its magnitude is โ๐โ = โ๐2 + ๐2 + (๐ + ๐)2 (43)
From the equations (42) and (43), it can be concluded that a vorticity vector does not only represent
rotation but also claims shearing and stretching components to be a part of the vortical structure, which is
clearly contaminated by shears.
4.2 Contamination of Q method
First, we perform the Cauchy-Stokes decomposition on the Principal Coordinate. Then, the scalar magnitude of
the Q method can be calculated based on the definition provided by equation (7).
๐ต๏ฟฝโโ๏ฟฝ = [ ๐๐๐ โ 12๐ 012๐ + ๐ ๐๐๐ 0๐ ๐ ๐๐ ] = [ ๐๐๐ 12 ๐ 12 ๐12 ๐ ๐๐๐ 12 ๐12 ๐ 12 ๐ ๐๐ ]
+ [ 0 โ 12๐ โ 12 ๐ โ 12 ๐12๐ + 12 ๐ 0 โ 12 ๐12 ๐ 12 ๐ 0 ]
= ๐จ๐ธ +๐ฉ๐ธ (44)
๐ = 12 (โ๐ฉ๐ธโ๐น2 โ โ๐จ๐ธโ๐น2)
= 12 [2 (๐ 2 + ๐2) + 2 (๐2)2 + 2 (๐2)2] โ 12 [2 ๐๐๐2 + ๐๐2 + 2 (๐2)2 + 2 (๐2)2 + 2 (๐2)2] = (๐ 2)2 + 12๐ โ ๐ โ ๐๐๐2 โ 12 ๐๐2 (45)
In the expression of Q above, there are not only R (the magnitude of rotation) but also ๐, ๐๐๐ and ๐๐ , which are
components of either the shear part or stretching part. Therefore, the value of Q is undoubtedly contaminated by
shear and stretching. Also, ๐ contains an R2 term, indicating dimensional distortion of a fluid rotation.
14
4.3 ๐๐จ๐ง๐ญ๐๐ฆ๐ข๐ง๐๐ญ๐ข๐จ๐ง ๐จ๐ ๐๐๐ Criterion
The characteristic equation of velocity gradient tensor given by (40) is
(๐ โ ๐๐) [(๐ โ ๐๐๐)2 + ๐ 2 (๐ 2 + ๐)] = 0 (46)
Thus, the eigenvalues are
๐1 = ๐๐, ๐2 = ๐๐๐ + ๐โ๐ /2(๐ /2 + ๐), ๐3 = ๐๐๐ โ ๐โ๐ /2(๐ /2 + ๐) Since rotation is orthogonal, eigenvalues are the same as the original velocity gradient tensor,
๐2 = ๐๐๐ + ๐โ๐ /2(๐ /2 + ๐) = ๐๐๐ + ๐๐๐๐ ๐3 = ๐๐๐ โ ๐โ๐ /2(๐ /2 + ๐) = ๐๐๐ โ ๐๐๐๐
Therefore, we have
๐ 2 (๐ 2 + ๐) = ๐๐๐2 (47)
Thus,
๐๐๐ = โ๐ 2 (๐ 2 + ๐) (48)
Despite being dimensionally correct, the expression of ๐๐๐ has ๐, which is a component of the shear part and
thus is contaminated by shear.
4.4 ๐๐จ๐ง๐ญ๐๐ฆ๐ข๐ง๐๐ญ๐ข๐จ๐ง ๐จ๐ โ method
In section 4.2, the three roots of the characteristic equation are
๐1 = ๐๐, ๐2 = ๐๐๐ + ๐โ๐ /2(๐ /2 + ๐), ๐3 = ๐๐๐ โ ๐โ๐ /2(๐ /2 + ๐) Plug their values into (2),(3), and (4)
15
๐ผ1 = โ(๐1 + ๐2 + ๐3) = โ๐๐ โ 2๐๐๐ (49)
๐ผ2 = ๐1๐2 + ๐2๐3 + ๐3๐1 = 2๐๐๐๐๐ + ๐๐๐2 + ๐ 2 (๐ 2 + ๐) (50)
๐ผ3 = โ๐1๐2๐3 = โ๐๐ [๐๐๐2 + ๐ 2 (๐ 2 + ๐)] (51)
๏ฟฝฬ๏ฟฝ = ๐ผ2 โ 13 ๐ผ12 = โ 13 (๐๐๐ โ ๐๐)2 + ๐ 2 (๐ 2 + ๐) (52)
๏ฟฝฬ๏ฟฝ = ๐ผ3 + 227 ๐ผ13 โ 13 ๐ผ1๐ผ2 = 227 (๐๐๐ โ ๐๐)3 + 23 (๐๐๐ โ ๐๐) ๐ 2 (๐ 2 + ๐) (53)
Then, the expression for โ can be written as
๐ฅ = (๏ฟฝฬ๏ฟฝ3)3 + (๏ฟฝฬ๏ฟฝ2)2 = 1243 [9 (๐ 2)3 (๐ 2 + ๐)3 โ 6(๐ 2)2 (๐ 2 + ๐)2 (๐๐๐ โ ๐๐)2 + 5๐ 2 (๐ 2 + ๐) (๐๐๐ โ ๐๐)4] (54)
The expression of ๐ฅ includes ๐, ๐๐ and ๐๐๐, indicating the contamination of ๐ฅ method by shear and stretching.
Moreover, ๐ฅ contains the sixth power of ๐ , which confirms that it cannot accurately denote rotation strength.
Next, we investigate the contamination of these schemes with the help of a numerical example.
5. Vortex Example: Burgers Vortex
In this test case, Burgers vortex is examined to justify the comparison of the effects of shearing and
stretching/compression on different criteria. The Burgers vortex is an exact steady solution of the Navier-Stokes
equation that can be used to model fine scales of turbulence (Gao et al., 2019)[27]. The Burgers vortex forms
when an inward radial flow concentrates and spins around the symmetric axis and the flow moves out in both
directions along the z-axis (Webster, D. R., and Young, D. L., 2015)[28]. The velocity gradient tensor is obtained
from the Burgers vortex.
The velocity components of Burgers vortex in the cylindrical coordinate system are given by:
๐ฃ๐ = โ๐๐ (55)
๐ฃ๐ = ๐ค2๐ฑ๐ (1 โ ๐โ๐2๐2๐ ) (56)
16
๐ฃ๐ง = 2๐๐ง (57)
where ๐ค represents the circulation, ๐ the axisymmetric strain rate, ๐ the kinematic viscosity, and ๐ is the
distance of the chosen point from the centerline in the Burgers vortex.
For post-processing, the velocity components are converted into the Cartesians coordinate system given by:
๐ข = โ๐๐ฅ โ ๐ค2๐ฑ๐2 (1 โ ๐โ๐2๐2๐ ) ๐ฆ (58)
๐ฃ = โ๐๐ฆ + ๐ค2๐ฑ๐2 (1 โ ๐โ๐2๐2๐ ) ๐ฅ (59)
๐ค = 2๐๐ง (60)
The existence of the vortex structure is highly contingent on the selection of parameters. For the proper vortex
structure visualization, we take ๐ = 0.042, ๐ค = 1.45 and ๐ = 0.01 . The calculation domain is taken with
50 ร 20 ร 20 grid points with a step size of 0.5. The vortex strength is strong in the core and becomes weak
when moving away from the center. The streamlines of the Burgers vortex exhibit a spiral pattern around the
vortex rotation axis line which is demonstrated in Figures 1a and 1b, which show that flows enter from a radial
direction and stretches outward spinning around the axis.
17
(a)Top View (b) Side View
Fig 1. (Color online) Streamlines of Burger vortex from top and side view with Liutex magnitude, which
depicts the rotational strength of the fluid particles.
6. Numerical Contamination Analysis
The velocity gradient tensor in the Principal Coordinate is:
๐๐ฝ = [ ๐๐ข๐๐ฅ ๐๐ข๐๐ฆ ๐๐ข๐๐ง๐๐ฃ๐๐ฅ ๐๐ฃ๐๐ฆ ๐๐ฃ๐๐ง๐๐ค๐๐ฅ ๐๐ค๐๐ฆ ๐๐ค๐๐ง ]
= [ ๐๐๐ โ 12๐ 012๐ + ๐ ๐๐๐ 0๐ ๐ ๐๐ ] (61)
where ๐ข , ๐ฃ and ๐ค are the three components of the velocity along the ๐ฅ , ๐ฆ, and ๐ง directions in Cartesian
Coordinate.
Since our purpose is to demonstrate the shear and stretching contaminations of different criteria numerically, we
add shear and stretching components separately and calculate how different criteria respond.
6.1 Adding Shear components
The matrix corresponding to shear is in the form of:
๐๐ฝ๐๐๐๐๐ = [ 0 0 0๐๐ 0 0๐๐ ๐๐ 0] (62)
The subscript โ๐โ refers to the โaddingโ of shearing components.
The new velocity gradient tensor in the Principal Coordinates after adding shear is
๐๐ฝ๐ = [ ๐๐๐ โ 12๐ 012๐ + ๐ + ๐๐ ๐๐๐ 0๐ + ๐๐ ๐ + ๐๐ ๐๐] (63)
18
Under the Principal Coordinate, the local rotation axis is the z-axis, so ๐๐ (value of ๐๐ค๐๐ฅ component) and ๐๐
(value of ๐๐ค๐๐ฆ component) are not in the rotation plane. Thus, these components will not influence the rotation
strength. However, ๐๐ is in the rotation plane, and thus affecting rotation strength. By the Liutex definition, the
magnitude of rotation strength is min๐ {|๐๐ข๐๐ฆ|๐ , |๐๐ฃ๐๐ฅ|๐}. |๐๐ข๐๐ฆ|๐ and |๐๐ฃ๐๐ฅ|๐ are the minimum absolute values of ๐๐ข๐๐ฆ
and ๐๐ฃ๐๐ฅ respectively, when we rotate the coordinate ๐ angle anti-clockwise along the z-axis.
Proposition 1: The rotation strength does not change if ๐๐ โฅ โ๐; the rotation strength changes if ๐๐ < โ๐.
Proof:
If ๐๐ โฅ โ๐, then |12๐ + ๐ + ๐๐| โฅ |โ 12๐ |, which shows that the Liutex magnitude is still ๐ which is defined as
twice the angular speed of the rigid rotation.
Remark: Physically, ๐๐ < 0 means adding a shear against the rotation direction and ๐๐ < โ๐ indicates that
shear is strong to the extent that it can affect rotation strength. So, keeping the rotation strength same, the shear
matrix we add should satisfy the following requirement: ๐๐ โฅ โ๐.
6.2 Adding Stretching components
The matrix corresponding to stretching is of the form
๐๐ฝ๐๐๐๐๐๐๐๐๐๐ = [๐ผ๐ 0 00 ๐ฝ๐ 00 0 ๐พ๐] (64)
The subscript โ๐โ means โaddingโ the stretching components.
The new velocity gradient tensor in the Principal Coordinate after adding stretching is
๐๐ฝ๐ = [๐๐๐ + ๐ผ๐ โ 12๐ 012๐ + ๐ ๐๐๐ + ๐ฝ๐ 0๐ ๐ ๐๐ + ๐พ๐ ] (65)
Proposition 2: The rotation strength does not change if ๐ผ๐ = ๐ฝ๐.
19
Proof: Based on theorem 1, the rotation strength of ๐๐ฝ๐ is still ๐ . Therefore, to keep the rotation strength
unchanged, the added ๐๐ฝ๐๐๐๐๐๐๐๐๐๐ should satisfy ๐ผ๐ = ๐ฝ๐.
6.3 Stretching Contamination Analysis
The following procedure was implemented to depict the reaction of the vortex identification schemes over the
change in stretching effects. Firstly, we select a point in Burgers vortex whose velocity gradient tensor is:
๐ต๏ฟฝโโ๏ฟฝ ๐ฉ = [โ0.0419999994 โ0.0711665452 0.00000000000.0711665452 โ0.0419999994 0.00000000000.0000000000 0.0000000000 0.0839999988] (66)
Then for convenience, the stretching matrix ๐๐ฝ๐๐๐๐๐๐๐๐๐๐ is added and the values of ๐, ๐ฅ, ๐2, ๐๐๐, and Liutex
are recorded where ๐๐ฝ๐๐๐๐๐๐๐๐๐๐ is given by (note that it satisfies the requirement of adding stretching):
๐๐ฝ๐๐๐๐๐๐๐๐๐๐ = [0.02 0 00 0.02 00 0 โ0.04] (67)
The sum of diagonal elements of ๐๐ฝ๐๐๐๐๐๐๐๐๐๐ is equal to zero, so it satisfies the continuity equation of
incompressible flow. The stretching effect given by ๐๐ฝ๐๐๐๐๐๐๐๐๐๐ is increased repeatedly, step by step, and the
values of every scheme are recorded. The results are presented in the following graph where the x-axis
represents the relative stretching rate (the ratio of stretching component and vorticity magnitude), and the y-axis
gives the corresponding values of different vortex identification methods.
20
Fig. 2. Line graphs depicting the effect of stretching on different vortex identification methods.
The change in stretching components result in a change in ๐, ๐ฅ, and ๐2. On the other hand, ๐๐๐ and L are not
changed, indicating that ๐๐๐ and L are not contaminated by the stretching effect, as shown in Fig. 2. Further
observation of Fig. 2 reveals that the increase in the stretching effect on ๐ decreases its value after a certain
point. The Q values ultimately become negative, indicating a non-vortical structure. However, other criteria
indicate there is a vortex structure. So, it can be concluded that ๐ may conflict with other criteria and violate the ๐ฅ-rule. Theoretically, this is understandable as in Eq. (6), ๐ฅ could be positive even if ๐ is negative provided that
the square of the third variant is large enough. The computational results shown in Fig. 2 coincide with the
theoretical analysis given by Eqs. (45), (48) and (54).
6.4 Shear Contamination Analysis
A similar procedure to the stretching effect is implemented for the graphic representation of the shearing effect
on the vortex identification schemes. Again, for our convenience, the shearing matrix ๐๐ฝ๐๐๐๐๐ is defined as:
๐๐ฝ๐๐๐๐๐ = [ 0 0 00.02 0 00.02 0.02 0] (68)
๐๐ฝ๐๐๐๐๐ is added to ๐ป๏ฟฝโ๏ฟฝ ๐ต a few times, and then the corresponding values of the different criteria are recorded.
The results are presented in the following graph, where the x-axis represents the relative shear rate, and the y-
axis gives the corresponding values of different vortex identification methods.
21
Fig. 3. Line graphs depicting the effect of shear on different vortex identification methods.
The relative shear rate is the ratio of the shearing component over vorticity magnitude. More precisely, it is the
ratio of shear over vorticity at any point in our numerical domain. The reason we are dividing shear by vorticity
is to refrain shear from getting too large. Fig. 3 indicates that ๐, ๐ฅ, ๐2, and ๐๐๐ are all affected by the change in
the shearing component, while Liutex has remained unaffected by shear. It can be concluded from Fig. 2 and
Fig. 3 that ๐, ๐ฅ, ๐2, and ๐๐๐ are affected by either shear, stretching, or both at different levels, whereas the
Liutex method is unaffected by both.
Since the values of ๐ฅ are very small, ๐ฅ looks to be consistently zero because ๐ฅ has sixth power of R which is
small; however, if we use zoom in, it can be observed that the ๐ฅ values are increasing. The ๐ฅ - plots are given in
Fig. 4 below:
22
Fig. 4. Stretching and shearing effect on ๐ฅ method of vortex identification.
7. Conclusion
As the theoretical relation between Liutex and several 2G vortex identification methods are given, and a
numerical example of Burger vortex is implemented, the following conclusions can be summarized.
1) Mathematical relations between Liutex and the 2G vortex identification methods, such as ๐, ๐ฅ, ๐2 and ๐๐๐ methods are derived mathematically, which clearly show the second generation methods are severely
contaminated by shearing, stretching, or by both. On the other hand, Liutex is the exact mathematical
definition of fluid rotation or vortex.
2) The velocity gradient tensor in the Principal Coordinate can be uniquely decomposed into its rigid
rotational part, stretching/compression part, and shearing part. This decomposition is known as the Liutex
decomposition.
3) The ๐ method conflicts with ๐ฅ, ๐2 , and ๐๐๐ methods, meaning that at some points, Q may mistreat the
vortex point as a non-rotational point. Evidence of this can be seen in the graphical representations (see
Figure 2).
4) According to Table 1, ๐, ๐ฅ, and ๐2 are stretch sensitive, while ๐, ๐ฅ, ๐2, and ๐๐๐ are affected by the shearing
factor. The numerical results from the Burgers vortex coincide with the theoretical analysis. However,
Liutex remained the same and is not affected. Therefore, we can conclude ๐, ๐2, and ๐๐๐ methods may not
23
appropriately represent vortex strength, while Liutex is the only one that correctly represents fluid rotation
or vortex.
Methods ๐ ๐ฅ ๐2 ๐๐๐ L
Contamination by
stretching
Yes Yes Yes No No
Contamination by
shearing
Yes Yes Yes Yes No
Table 1 Contamination by stretching and shearing on different criteria.
8. Declarations
8.1 Availability of data and materials
The data and materials that support the findings of this study are available from the corresponding author
upon reasonable request.
8.2 Competing Interest
The authors declare that they have no competing interests.
8.3 Funding
There is no funding to support this project.
8.3. Authorsโ Contributions
PS performed the overall analysis of stretching and shearing effects of different vortex identification
methods and was responsible for writing this manuscript. YY helped in proofreading and mathematical
discussion about Principal Coordinates and Principal Tensor Decomposition. OA helped in coding to get
the streamlines of Burger vortex. CN helped in sketching the graph as well as in English grammar. CL is
the corresponding author of this paper who gave the original ideas and instructions in every step of the
research and checked and modified the paper for more than 10 times.
8.5 Acknowledgement
24
The authors are grateful to the Mathematical Department of University of Texas at Arlington for housing
the research team. The code โLiutex-UTAโ is released by Prof. Chaoqun Liu in 2019. The authors thank
TACC (Texas Advanced Computational Center) for providing the CPU hours.
8.6 Authorโs information
PS, YY, OA and CN are all PhD students of the Mathematics Department of UTA and CL is the
Distinguished Professor and Director of the Center for Numerical Simulation and Modeling (CNSM) of
UTA
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Figures
Figure 1
(Color online) Streamlines of Burger vortex from top and side view with Liutex magnitude, which depictsthe rotational strength of the ๏ฟฝuid particles.
Figure 2
Line graphs depicting the effect of stretching on different vortex identi๏ฟฝcation methods.
Figure 3
Line graphs depicting the effect of shear on different vortex identi๏ฟฝcation methods.
Figure 4
Stretching and shearing effect on ฮ method of vortex identi๏ฟฝcation.