[Paper Review] Letter: Mathematical Foundation of Turbulence Generation-Symmetric to Asymmetric Liutex/Rortex

Jianming Liu, Yue Deng+3 more|arXiv (Cornell University)|May 8, 2019

Fluid Dynamics and Turbulent Flows9 references4 citations

TL;DR

This paper establishes a mathematical foundation for turbulence generation by demonstrating that symmetric hairpin vortices become asymmetric due to shear-Liutex interactions, leading to vortex leg loss and chaotic flow. Using the Liutex (Rortex) definition—which isolates rigid rotation—mathematical analysis shows that asymmetric shear forces weaken Liutex strength when directions oppose, destabilizing vortices and initiating turbulence without physical assumptions.

ABSTRACT

Vortex has been considered as the building block and muscle of turbulence for long time. A new physical quantity called Liutex (previously named Rortex) has been defined as the rigid rotation part of fluid motion. From DNS and experiment, forest of hairpin vortices has been found in flow transition and low Reynolds turbulence, but more and more one leg vortices appear in fully developed turbulence. This letter shows hairpin vortex is unstable. Hairpin vortex will weaken or lose one leg by the shear and vortex interaction. This conclusion is made by Liutex definition and mathematical analysis without any physical assumptions. The asymmetry of vortex is caused by the interaction of symmetric shear and symmetric Liutex due to the definition of Liutex, which considers the smaller element of a pair of vorticity elements as 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 immediate changed due to the Liutex strength definition. For a rigid rotation, if both vorticity of the shear and Liutex have the same directions, e.g., clockwise, the Liutex strength will not be changed. If the vorticity of the shear and Liutex have different directions, e.g., one clockwise and one counterclockwise, the Liutex strength will be weakened. Then, the hairpin vortex could lose the symmetry and even deform to a one leg vortex. The one leg vortex cannot keep balanced, and the chaotic motion and flow fluctuation are doomed. This is considered as the mathematical foundation of turbulence formation. This theory has been checked and proved by our DNS results of boundary layer transition.

Motivation & Objective

• To establish a mathematical basis for turbulence formation rooted in vortex asymmetry.
• To explain the transition from symmetric hairpin vortices to asymmetric one-leg vortices in turbulent flows.
• To demonstrate that vortex asymmetry arises from the interaction between symmetric shear and Liutex (Rortex) under the Liutex definition.
• To show that this asymmetry is a direct consequence of the Liutex strength definition, not empirical assumptions.
• To validate the mechanism via direct numerical simulation (DNS) of boundary layer transition.

Proposed method

• Defining Liutex (Rortex) as the rigid rotation component of fluid motion, based on the smaller of a pair of vorticity elements.
• Applying mathematical analysis to assess how shear forces affect Liutex strength depending on the relative direction of shear vorticity and Liutex.
• Using the Liutex strength definition to determine whether shear forces preserve or weaken rotational strength.
• Analyzing 2D rigid rotation cases: when shear and Liutex have same direction (e.g., both clockwise), Liutex strength remains unchanged; when opposite, it weakens.
• Modeling vortex deformation under asymmetric shear, leading to loss of one vortex leg.
• Validating the theoretical predictions with DNS results of boundary layer transition.

Experimental results

Research questions

• RQ1 How does the Liutex definition explain the destabilization of symmetric hairpin vortices?
• RQ2 What role does the directional alignment of shear vorticity and Liutex play in vortex strength preservation or weakening?
• RQ3 Why do hairpin vortices evolve into one-leg vortices in fully developed turbulence?
• RQ4 Can vortex asymmetry and subsequent turbulence formation be derived purely from mathematical analysis without physical assumptions?
• RQ5 What is the mathematical mechanism behind the transition from symmetric to asymmetric vortex structures?

Key findings

• Hairpin vortices are inherently unstable due to shear-Liutex interactions, leading to loss of one vortex leg.
• When shear vorticity and Liutex have opposite directions, Liutex strength is immediately weakened, breaking symmetry.
• If shear affects the smaller vorticity element (as defined by Liutex), rotational strength changes significantly; if it affects the larger, no change occurs.
• The asymmetry arises from the Liutex definition’s focus on the smaller vorticity element, making the system sensitive to directional shear.
• One-leg vortices cannot maintain balance, leading to chaotic motion and flow fluctuations—hallmarks of turbulence.
• The DNS results of boundary layer transition confirm the theoretical prediction of vortex asymmetry and turbulence onset.