Yifei Yu, Pushpa Shrestha, Charles Nottage, Chaoqun Liu

Department of Mathematics, University of Texas at Arlington, Arlington 76019, USA

Abstract: Helmholtz velocity decomposition and Cauchy-Stokes tensor decomposition have been widely accepted as the foundation of fluid kinematics for a long time.However, there are some problems with these decompositions which cannot be ignored.Firstly,Cauchy-Stokes decomposition itself is not Galilean invariant which means under different coordinates, the stretching (compression)and deformation are quite different.Another problem is that the anti-symmetric part of the velocity gradient tensor is not the proper quantity to represent fluid rotation.To show these two drawbacks, two counterexamples are given in this paper.Then “principal coordinate” and “principal decomposition” are introduced to solve the problems of Helmholtz decomposition.An easy way is given to find the Principal decomposition which has the property of Galilean invariance.

Key words: Velocity decomposition, Helmholtz, Cauchy-Stokes, Liutex, principal coordinate

Introduction

Turbulence is still a mystery that has yet to be fully understood, although scientists have been intensively working on this topic for centuries.Features of turbulence, including the instability,irregular motion, and hardness to measure, make it so confusing that there is even not a universally accepted definition of turbulence, along without the delicate and quantitative analysis.However, turbulence plays a salient role in real life, in the design of aircraft and analyzing the blood motion in vessels to avoid blockage for example.The behavior of the fluid motion is coupled with translation, rotation, stretching,and shear deformation.So, a natural idea to reduce the complexity of fluid motion is to decompose the velocity gradient tensor of fluid.

Helmholtz velocity decomposition, which is equivalent to a tensor form called Cauchy-Stokes decomposition, has been accepted for a long time since Helmholtz first proposed his theory in 1858[1]It is so well-known that most textbooks in fluid dynamics consider Helmholtz velocity decomposition as the fundamental theory of fluid kinematics.Cauchy-Stokes decomposition, which is corresponding to Helmholtz theory, decomposes the velocity gradient tensor into two parts with a symmetric part and anti-symmetric part.

whereAis the symmetric part,Bis the antisymmetric part.

The anti-symmetric partBcorresponding to vorticity ?Vis what people used to consider as rotation of fluids under the influence of the fact that vorticity represents angular speed for rigid body rotation in solid mechanics.Admittedly,Cauchy-Stokes decomposition is correct in mathematics, but, however, the doubt of the correctness of physical meanings of Cauchy-Stokes decomposition has arisen after thoughtful thinking.Scientists keep questioning whether vorticity can represent rotation of fluids within the past 40 years.Robinson[2]pointed out that the associate region can have strong vorticity but its actual vortices are quite weak.Also, Wang et al.[3]found that the vorticity magnitude is much smaller inside the vortex region than the neighboring area outside the vortex from a DNS research.Many researchers devoted themselves to finding a better physical quantity which can better represent vortex.Hunt et al.[4]proposed the popularQmethod in 1988.Based on the critical point theorem, Chong et al.[5]suggested using the existence of complex eigen values as a criterion to determine if there is a vortex in the region, which is known as Δ criterion.Zhou et al.[6]further extended Δ criterion intoλcicriterion.In Ref.[7], Liu classified the above vortex identification methods as the first (vorticity) and second (others)generation methods and introduced the Liutex method which can be categorized as the third generation.So far, more and more evidences[8]show Liutex is the best one among all these vortex identification methods.In this paper, Liutex is used as the right way to detect vortex.

Galilean invariance is a property first described by Galilean[9]in 1632, indicating whether a quantity changes under different coordinates.Any objective quantity that does not rely on the choice of coordinates must be Galilean invariant.Unfortunately,Cauchy-Stokes decomposition is not Galilean invariant as shown in the following content of this paper

1.Basic form of fluid motion

First, we want to make it clear which part of the velocity gradient represents rigid body rotation as shown in Fig.1, stretching, shear deformation, and symmetric deformation.For simplicity, we consider a 2-D velocity gradient tensor

Fig.1 Movement of velocity gradient corresponding to rigid rotation

Definition 1.1The velocity gradient corresponds to rigid body rotation if

Suppose the velocity at O is 0 and ?u/?y=-?v/?x=-ω, the coordinates at any point on the circle could be expressed as (cosθ,sin)θand the velocity at that point is

The magnitude of the velocity is

For each point on the circle, the direction of the velocity is tangent to the circle and the magnitude of velocity isω.Therefore, the movement is a rigid body rotation.

Definition 1.2The velocity gradient corresponds to stretching (as shown in Fig.2) if

and at least one of ?u/?xand ?v/?yis non-zero.

Fig.2 Movement of velocity gradient tensor corresponding to stretching

Suppose velocity at O is 0, ?u/?x=0 and?v/?y=c.Then,va=acandvb=-ac.After a small time step, the control volume will become longer.Therefore, this movement is stretching.

Definition 1.3The velocity gradient corresponds to shear deformation if one of ?u/?yand ?v/?xis 0 and the other one is non-zero, and

Suppose ?u/?y=c＞0,?v/?x=0 and velo-city at O is 0, thenuA=ac,uB=-acandv=0 in the whole domain.After a small time step, the shape of the control volume will become the right part of Fig.3.Therefore, it is a shear deformation.

Fig.3 Movement of velocity gradient tensor corresponding to shear deformation

Definition 1.4The velocity gradient corresponds to symmetric deformation if

Suppose the velocity at O is 0,?u/?y=?v/?x=c.As shown in Fig.4, the velocity distributes linearly along the two legs and the behavior is symmetric deformations.Also, symmetric deformation could be considered as a combination of two shear deformations.One is ?u/?y=cand?v/?x=0, the other is ?v/?x=cand ?u/?y=0.

Fig.4 (Color online) Movement of velocity gradient tensor corresponding to symmetric deformation

2.Problems with Cauchy-Stokes decomposition

2.1 Vorticity cannot represent fluid rotation

To illustrate that vorticity is not a proper way to represent rotation, we consider the following 2-D case as shown in Fig.5.

Hereais a positive constant,uandvare respectively thex-component andy-component of the velocity.

Fig.5 2-D Couette flow

The velocity gradient tensor is

The Cauchy-Stokes decomposition is

This decomposition is correct in mathematics.However, its physical meaning is questionable.From Definition 1.3, (13) represents a shear deformation that contains no rotation, but Cauchy-Stokes decomposition (14) results in a rotational part that does not exist.Now, we are going to analyze this case in detail to see where this artificial rotation comes from.In the original matrix, ?u/?y=2a.This “2a”is divided into “a” and “a”, then distributed to matrixAand matrixB.Similarly, we split?v/?x=0into “-a” and “a”, and distribute them to matrixAand matrixBas shown in Fig.6.

Fig.6 (Color online) Distribution of Cauchy-Stokes decomposition

Let’s look at “ ?v/?x=0”.The ?v/?xcomponent does not have anything in the original matrix, but we are forcing it to give “-a” to matrix B because a rigid body rotation matrix requires?u/?y=-?v/?xin Definition.1.1.Also, it distributes an artificial “a” to matrixAfor balance.We think this process is artificially making something from nothing which causes a man-made rotation in Cauchy-Stokes decomposition.To avoid this situation,we create a decompose criterion described as follows:

(1) Ifp＞0, then a decomposition of ap=∑vimust satisfy 0≤vi≤p.

(2) Ifp＜0, then a decomposition of ap=∑vimust satisfyP≤vi≤ 0.

(3) Ifp=0, then a decomposition of ap=∑vimust satisfyvi=0.

Based on the decompose criterion, the decomposition of ?v/?x=0must be 0.From Definition 1.1, in a rotation matrix, ?u/?y=-?v/?xand ?u/?x=?v/?y=0.So, we can deduce that ?u/?y=0.Then the rotational part of (13) is, indicating there is no rotation which is consistent with the physical fact.The Cauchy-Stokes tensor decomposition is found to violate the above criteria.

So, to find the real rotational part in the decomposition of velocity gradient tensor, instead ofB=0.5(?V-?VT), it seems that we should choose the minimum of.Otherwise the decompose criterion is not satisfied.It is reasonable to defineBas

However, there is another problem,are changing when the coordinate changes.If the given coordinate is rotated with angleθcounter-clockwise toOx′y′, the rotation matrix is

The velocity gradient tensor under the new coordinate should be

So, the cap of decomposition based on decompose criterion is different under different coordinates.Figure 7 shows the 2-D laminar flow in different coordinates.In Definition 1.1, we knowais angular speed.It is impossible for two different angular speeds to exist.The reason is the rotation matrix could be contaminated by shear deformation matrix, because they have a similar pattern of rotation matrix.We need to distinguish rotation and shear deformation.Since angular speed is fixed at a time point, velocity gradient tensor should be capable of decomposing out the angular speed rotation matrix under any coordinate.The maximum angular speed that the velocity gradient tensor can provide depends on.Consequently, we need to chooseorto make sure that under any coordinates

Fig.7 2-D laminar flow in different coordinates

2.2 Galilean invariant

Not satisfying Galilean invariant is another problem of Cauchy-Stokes decomposition that cannot be ignored.If the decomposition is different under different coordinates, then which one is correct? Let’s observe a 3-D example.

Performing Cauchy-Stokes decomposition, we can get:

whereA0is the symmetric deformation part,B0is the rotation part,C0is the stretching part.

where 1, 2, ∞ representL1,L2,L∞norms respectively.

Let the rotation matrices be:

After rotation, the velocity gradient tensor under the new coordinates is

Performing Cauchy-Stokes decomposition

It is not difficult to notice that the norm of rotation,symmetric deformation and stretching matrices are all different, except for, under different coordinates.A question arises, under which coordinates does the Cauchy-Stokes tensor decomposition give the right stretching (compression) and deformation? We hypothesis that there must be a unique coordinate that can give a unique tensor decomposition for stretching(compression), deformation, shear, and rotation.We call this coordinate the “principal coordinate”.Under principal coordinate,.Performing velocity gradient decomposition under this coordinate system can easily, clearly, and correctly find the rotation, stretching, and shear deformation.The definition of principal coordinate is given in the following section.

3.Principal coordinates

Definition 3.1Principal coordinate at a point is a coordinate that satisfies and itsz-axis is parallel to ther(direction of Liutex[10]).

The velocity gradient tensor under this coordinate is in the form of

whereλr,λcrare real eigenvalue and real part of the conjugate complex eigenvalue pair of the velocity gradient tensor respectively for rotation points and assuming

Noting that rotation by matricesP1andP2is an orthogonal transformation, the eigenvalues of the new velocity gradient tensor in the Principal Coordinates are the same as the original one.Note that the requirements of the Principal coordinate include two zero elements at the upper right corner and the two same upper diagonal elements which areλcr.These requirements sufficiently determine the rotational metricsP1andP2, and then determine the principal coordinates uniquely.

From part (1) of the definition “z-axis is parallel to ther”, we can find one axis of the Principal Coordinates.As a result, the possible coordinate could only rotate around thez-axis.Then rotate the coordinate in a plane parallel to Liutex until ?U/?Xand ?V/?Ycomponents under this coordinate are bothλcr, because, in this condition, the value of?U/?Ycomponent is exactly.

Definition 3.2Liutex[10]is a vector whose magnitude is twice of the angular speed of fluids and its direction is swirl axis.It can be expressed as

whereRmeans Liutex vector,Rmeans the magnitude ofR, andris a normalized vector that represents the local rotation axis direction.

There are only 2 eigenvalue situations of a 3×3 matrix: one real eigenvalue and two conjugate complex eigenvalues (rotational points) or three real eigenvalues (non-rotational points).Since there is no rotation for 3 real eigenvalues situation whereR=0 according to the Liutex definition, we only consider the situation of one real eigenvalue and two conjugate complex eigenvalues in this paper.Denote the real eigenvalue and two conjugate complex eigenvalues asλr,λcr+iλciandλcr-iλci.

According to Ref.[11],ris the normalized eigenvector corresponding to the real eigenvalue of?vand the explicit way to calculate the magnitude ofRis

Whereωis vorticity,ris the direction of Liutex andλciis the imaginary part of the complex eigenvalue of ?v.The detailed definition and illustration of Liutex can be found in Ref.[10].

Then we introduce the decomposition under principal coordinate, called “principal decomposition”.From the definition of “principal coordinate”, we know that the velocity gradient tensor is

where ?U/?Y=-R/2.Also, letε=?V/?X-R/2,?W/?X=ξand ?W/?Y=η, then the tensor can be expressed as

The principal decomposition is

whereArepresents rotation part,Brepresents shear deformation, andCrepresents stretching deformation.This decomposition exactly corresponds with the definition of the matrices of rotation, shear,and stretching.Consequently the physical meaning of this decomposition is clear and correct.It is emphasized that the Principal Tensor Decomposition is unique and Galilean invariant.

It seems that we need to do the rotation of coordinate twice to find the decomposition based on the definition.Firstly, rotatez-axis torbyP1and then rotate around thez-axis to make sure?U/?X=?V/?Y=λcrbyP2.However, noting that Liutex is Galilean invariant[12], there is a quick way to figure out the decomposition.

The characteristic equation of ?Vis

Thus, the eigenvalues are:

Since rotation is orthogonal, eigenvalues are the same as the original velocity gradient tensor

Therefore, we have

whereωis vorticity,ωzis the third component of vorticity.

SolvingRandεfrom Eqs.(56) and (57), we can get:

This is also the proof of the explicit formula of Liutex magnitude.So, indeed, there is no need to do any coordinate rotation to find these elements except forξandη.

Also, we can do principal decomposition in the original coordinates on vorticity[13](this decomposition is under the original coordinate, not principal coordinate)

whereRrepresents rotation andSrepresents shear deformation which is non-rotational.

From the decomposition of vorticity, we observe that vorticity is not only rotation but also partially shear.For a rigid body, it is assumed that there is no deformation so that vorticity can precisely be rotation.However, for fluids, deformation cannot be ignored,and thus, using vorticity to identify vortex does not work for fluid in general.In many cases, vorticity mainly represents shear, but not rotation.

After defining “principal coordinate” and“principal decomposition”, let us analyze a point in a DNS case of the boundary layer transition[14]shown in Fig.8.

Fig.8 (Color online) Selected point in a DNS case of boundary layer transition

The velocity gradient tensor at the circled point under the original coordinate is

Performing Cauchy-Stokes decomposition, we get:

HereA0,B0andC0are the symmetric deformation, rotation, and stretching parts in Cauchy-Stokes decomposition respectively.Ris the magnitude of Liutex.

Next, we rotate the coordinate randomly, for example, rotate π/6 around thex-axis and π/4 around thez-axis then we can get the new velocity gradient tensor

Performing Cauchy-Stokes decomposition, we get:

HereA1,B1andC1are symmetric deformation,rotation, and stretching part in Cauchy-Stokes decomposition respectively.Ris the magnitude of Liutex.

The velocity gradient tensor under principal coordinate is

Similarly, perform Cauchy-Stokes decomposition.

Again, the Cauchy-Stokes decomposition is not Galilean invariant, as onlyand none of the other norms of Cauchy-Stokes decomposition components are equal.The magnitude of the stretching and deformation parts are different under different coordinates.The magnitudes of vorticity and Liutex remain the same, but, however, as discussed above, only Liutex can represent fluid rotation.

The principal decomposition is:

HereA,BandCare rotation, shear deformation,and stretching part, respectively.

Since principal decomposition is defined under a particular coordinate which is called “principal coordinate”, this decomposition is unique and Galilean invariant.Additionally, ?V2(1,1) and?V2(2,2) are exactly the real part of the complex eigenvalues of?V0, corresponding to the “principal coordinate” definition.

To show more evidence, we analyze another point shown in Fig.9.

Fig.9 (Color online) Another selected point in a DNS case of boundary layer transition

The velocity gradient tensor at the circled point under the original coordinate is

The Cauchy-Stokes decompositions under the original coordinate are:

The new velocity gradient tensor and Cauchy-Stokes decomposition under a new coordinate that rotates π/6 aroundx-axis and π/4 aroundz-axis from the original coordinate are:

The velocity gradient tensor and Cauchy-Stokes decomposition under principal coordinate are:

4.Conclusions

This paper proposes a so-called “principal coordinates” and develops a so-called “principal decomposition” of a velocity gradient tensor.Both are unique.Following conclusions can be made:

(1) Vorticity is not vortex.From the Principal Decomposition of vorticity, we observed that vorticity is actually a rigid rotation plus shear.For rigid body,shear deformation is assumed to be zero.Only in such a special case, vorticity can be considered as a measurement of rotation.

(2) Liutex is a physical quantity to accurately measure the fluid rotation.A number of vortex identification methods based on Liutex have been reported including vortex identification and vortex core line detection in Refs.[15-16].

(3) Cauchy-Stokes decomposition itself is not Galilean invariant, so it is uncertain under which coordinates we can get the right physics like rotation,shear, deformation and stretch (compression).

(4) A new and unique coordinate called“Principal Coordinate” is defined based on the velocity gradient tensor in this paper.Under this coordinate, we can decompose the fluid motion uniquely to the rotation, shearing, and stretching parts corresponding to the real physics.

(5) Principal Decomposition not only has a clear and correct physical meaning but also overcomes the Galilean variant drawback of Cauchy-Stokes Decomposition which also mistreats anti-symmetric tensor as fluid rotation.It could become a foundation of fluid kinematics for further analysis in fluid dynamics.

Data availability

The data that supports the findings of this study are available from the corresponding author upon reasonable request.