Mathematical foundation of turbulence generation-From symmetric to asymmetric Liutex *

Jian-ming Liu , Yue Deng, Yi-sheng Gao, Sita Charkrit , Chaoqun Liu

1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

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

3. Department of Physics, University of Texas at Arlington, Arlington 76019, Texas, USA

Abstract: Vortices have been regarded as the building blocks and muscles of turbulence for a long time. To better describe and analyze vortices or vortical structures, recently a new physical quantity called Liutex (previously named Rortex) is introduced to present the rigid rotation part of fluid motion (Liu et al. 2018). Since turbulence is closely related to the vortex, it can be postulated that there exists no turbulence without Liutex. According to direct numerical simulations (DNS) and experiments, forest of hairpin vortices has been found in transitional and low Reynolds number turbulent flows, while one-leg vortices are predominant in full developed turbulent flows. This paper demonstrates that the hairpin vortex is unstable. The hairpin vortex will be weakened or lose one leg by the shear and Liutex interaction, based on the Liutex definition and mathematical analysis without any physical assumptions. The asymmetry of the vortex is caused by the interaction of symmetric shear and symmetric Liutex since the smaller element of a pair of vorticity elements determines the rotational strength. For a 2-D fluid rotation, if a disturbance shear effects the larger element, the rotation strength will not be changed, but if the disturbance shear effects the smaller element, the rotation strength will be immediately changed due to the definition of the Liutex strength. For a rigid rotation, if the shearing part of the vorticity and Liutex present the same directions, e.g., clockwise, the Liutex strength will not be changed. If the shearing part of the vorticity and Liutex present different directions, e.g., one clockwise and another counterclockwise, the Liutex strength will be weakened.Consequently, the hairpin vortex could lose the symmetry and even deform to a one-leg vortex. The one-leg vortex cannot keep balance, and the chaotic motion and flow fluctuation are doomed. This is considered as the mathematical foundation of turbulence formation. The DNS results of boundary layer transition are used to justify this theory.

Key words: Turbulence generation, mathematical principle, Liutex/Rortex, asymmetry

Turbulence is generally acknowledged as one of the most complex phenomena in nature[1-2]. In 1883,Reynolds revealed the complex flow pattern through his famous round tube experiment. During the next over one hundred years, a large number of scientists engaged in turbulence research and solved a large number of engineering problems. However, due to the complexity of turbulence, the universal mechanism of turbulence generation is yet to be fully understood.This leads to various turbulence generation theories.Richardson described vortex chains generated by large vortex breakdown[3]. But from many direct numerical simulations (DNS), the vortex chains are never observed. Kolmogorov accepted the idea of the Richardson energy cascade and vortex breakdown and thought that the large eddies passed energy to small eddies through vortex breakdown, then continue to smaller scale, until the Kolmogorov's smallest scale[4].But according to the most accurate experimental equipment, no one can confirm that turbulence is caused by vortex breakdown. Currently, no mathematical principle has been found to explain the generation of turbulence, especially the occurrence and evolution of asymmetry structures in the boundary layer flow transition process. This paper attempts to provide a mathematical principle of turbulence generation based on the definition of the Liutex vector.

Turbulence is commonly observed in everyday phenomena and most realistic engineering flows[1-2].Richard Feynman has described turbulence as the most important unsolved problem in classical physics[3]. Turbulent flow is very irregular, diffusive,dissipative and chaotic. A vortex is the building block and muscle of turbulence since turbulence consists of variety of vortices of different sizes and rotational strengths[4]. Without vortices, there would be no turbulence. Without asymmetric vortices, there would be no turbulence.

Many vortex identification methods were developed during the past decades[4-9]. Recently, a new physical quantity called Liutex has been proposed to represent the rigid rotation of fluid motion[10-11].Liutex is a vector uniquely defined by R=Rr. The direction is defined as the real eigenvector of the velocity gradient tensor ?u and its magnitude R is defined as the angular speed of rigid rotation, i.e.,andwhere r is the direction of the Liutex vector[12-13], ω the vorticity andciλ the imaginary part of the complex eigenvalue of ?u and ω?r＞0. From our previous work, the RS decomposition is written as

here the magnitude of R is 2min,A simple shear tensor can be described asfor 2-D. Let us add the disturbance shear to the channel flow, we will getAlthough R is zero at the beginning, the first shear does not generate Liutex,but the second one does. This clearly shows shear may or may not generate rotation depending on the shear direction. If both disturbance shear stresses increase the larger element of the base shear/Liutex, there is no change in the rotation strength. If the disturbance shear decreases the smaller element of the base shear/Liutex, the rotation strength will be reduced. Let us consider the interaction of a shear and a rigid rotation in which the tensor of the rigid rotation iswith positive φ, s.

In this paragraph, we will give a justification of definition for magnitude of Liutex. A correct definition must be valid for both 2-D and 3-D cases. First,a 2-D laminar channel flow (Fig. 1(a)) is used to justify the definition of the magnitude of Liutex. The exact solution is, v=0 and hence,,and.In order to define the magnitude of Liutex which should measure the strength of the rigid rotation or angular speed, we have several choices like the maximum, the minimum, or the average. Since there is no rotation, it is apparently inappropriate to pick the maximum or the average which is actually equivalent to the vorticity component. Since there is no rotation in the laminar channel flow,=2min{0,, therefore taking the minimum is appropriate to describe the rotation strength. The boundary layer solution on a flat plate or Blasius solution (Fig. 1(b)) will lead the same conclusion that Liutex magnitude should be R=2min, which means that there is no rotation in a laminar boundary layer. Furthermore, if a rigid rotation (Fig. 1(c)) is considered, we have u= ω y,v =-ω x, R=2ω where ω is the exact angular speed. As a result, the shear stress can be presented by/2=0 and there is no energy dissipation in the rigid rotation.and the shear is. By adding them together, we will haveAssuming ω is positive, we will have two totally different cases: (1)If s is positive, R remains constant no matter how larges is, according to the definition of R=2min. (2) If s is negative, we will have several possibilities: 1) If | s|＜ ω then

Fig. 1 Simple flows

Let us take a look at the interaction of shear and Liutex in 3-D flows. The tensor formula could be:

and

The conclusion should be the same as in 2-D case. If s is positive (the shear and rotation have the same directions), the magnitude of Liutex R will not change at all, which means that the shear cannot change the strength of rotation. On the other hand, if s is negative (the shear and rotation have the opposite directions), the magnitude of Liutex or the strength of rotation will decrease or even disappear.

For the interaction of shear and non-rigid rotational vortex, which is very common in a boundary layer, we should have

and assume both φ ,1s are positive. The same conclusion will be achieved just like what we discussed above for the 2-D case.

Fig. 2 (Color online) Symmetric Liutex

Let us consider the interaction of a shear and a hairpin vortex. Both are symmetric. Assume the direction of shear is clockwise which we think s is positive, and the hairpin has two counter-rotating legs with the right leg clockwise and the left leg counterclockwise. Interacted with a clockwise shear,the right leg will keep the same rotation strength or R but the left leg may be weakened (become thinner)or even disappear. So, the original symmetric hairpin will become an asymmetric hairpin or even a one-leg vortex. One example with the symmetric Liutex is shown in Fig. 2. Moreover, Fig. 3 shows the one-leg vortex ring appears in the upper level of the boundary layer and Fig. 4 depicts a secondary vortex ring with one strong leg and one weak leg in the lower boundary layer.

Fig. 3 (Color online) One-leg vortex or asymmetric vortices in the upper boundary layer (Liutex iso-surfaces colored by shear magnitude)

As confirmed by DNS and experiments, there are forest of hairpin vortices in the flow transition and early stage of turbulence, but the hairpin vortex could be deformed or degenerated in the lower boundary layer where the viscosity is large or in fully developed turbulence zones due to the shear interaction with legs(Fig. 3). Note that the condition for the deformation or degeneration of symmetric hairpin is the existence of symmetric shears. That is the reason why in the inviscid flow region, the hairpin vortex keeps symmetric for a long time but the hairpin vortex in the lower boundary layer could quickly lose one leg. The only condition is the existence of fluctuated shear. If the shear moves in a clockwise motion, the clockwise vortex leg will not be affected. However, the counterclockwise vortex leg will be weakened or even disappeared. The asymmetric shear-Liutex interaction will cause the asymmetries of the hairpin vortices and further generate more to two asymmetric legs with one strong and one weak or even a one-leg vortex.These could happen on the top of hairpin vortices (see Fig. 3) or secondary vortices located in the lower boundary layers (see Fig. 4).

Fig. 4 (Color online) One-leg vortex in the second level vortex rings near the wall (Liutex iso-surfaces)

As confirmed by both DNS and experiment,there are many one-leg vortices inside the lower boundary layer and fully turbulence. The one-leg vortex cannot keep static as the nature of Liutex,which keeps rigid rotation. The asymmetric Liutex will keep swinging, producing asymmetry with fluctuating, swinging, shaking and chaos. As we addressed early, there is no turbulence if we have no Liutex and no asymmetric Liutex. However, shear is always in the boundary layer, especially in the lower boundary layer and hairpin vortices always appear in the flow transition and early turbulence (see Fig. 5).Unfortunately, the interaction of the hairpin vortex and shear will cause non-symmetry due to the nature of shear and vortex interaction. Therefore, asymmetry,the one-leg vortex, shaking of asymmetric vortices,and chaos are doomed. In other words, turbulence is doomed and that is the nature.

According to the above discussion, the following conclusions can be achieved: (1) Liutex can represent the rigid fluid rotation. (2) The strength of Liutex is determined by a minimum element in a 2-D plane,described by. (3) Shear will not change the rotation strength if shear and Liutex present the same directions, but shear may reduce the rotation strength if they present opposite directions and the shear magnitude is larger than the original shear contained in the original vortex. (4) The symmetric hairpin vortex may lose its symmetry when it interacts with symmetric shear. (5) A one-leg hairpin vortex can be weakened or disappeared due to the shear-Liutex interaction. Therefore, a hairpin vortex is unstable in boundary layer flows. (6)One-leg or asymmetric vortices are shaking, swinging,chaotic, and then cause turbulence. (7) The nature of Liutex magnitude definition (smaller element of a pair)and interaction of shear and Liutex is the mathematical foundation of turbulence generation,therefore, the symmetry loss and chaos are doomed.

Fig. 5 (Color online) Liutex iso-surfaces colored by shear magnitude

Acknowledgments

This work was supported by the Department of Mathematics of University of Texas at Arlington.The research was partly supported by the Visiting

Scholar Scholarship of the China Scholarship Council(Grant No. 201808320079). The author is thankful to Dr. Lian-di Zhou for beneficial discussions on vortex and turbulence. The authors are grateful to

Texas Advanced Computational Center (TACC) for providing computation hours. This work is accomplished by using code DNSUTA developed by Dr. Chaoqun Liu at the University of Texas at Arlington.