Mathematical foundation of turbulence generation—From symmetric to asymmetric Liutex

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 [C. Liu et al., "Rortex-A new vortex vector definition and vorticity tensor and vector decompositions", Phys. Fluids 30, 035103 (2018)]. Since turbulence is closely related to vortex/rotation, it is concluded that there is no turbulence without Liutex and there is no turbulence without asymmetric Liutex. 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.