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# Mathematical foundation of turbulence generation—From symmetric to asymmetric Liutex

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## 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

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## Copyright information

© China Ship Scientific Research Center 2019

## Authors and Affiliations

• Jian-ming Liu
• 1
• 2
• Yue Deng
• 3
• Yi-sheng Gao
• 2
• Sita Charkrit
• 2
• Chaoqun Liu
• 2
Email author
1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouChina
2. 2.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA
3. 3.Department of PhysicsUniversity of Texas at ArlingtonArlingtonUSA