Liutex core line and POD analysis on hairpin vortex formation in natural flow transition

In this study,the new method of the vortex core line based on Liutex definition,also known as Liutex core line,is applied to support the hypothesis that the vortex ring is not a part of theΛ-vortex and the formation of the ring-like vortex is formed separately from theΛ-vortex.The proper orthogonal decomposition(POD)is also applied to analyze the Kelvin-VHelmholtz(K-H)instability happening in hairpin ring areas of the flow transition on the flat plate to understand the mechanism of the ring-like vortex formation.The new vortex identification method named modified Liutex-Omega method is efficiently used to visualize and observe the shapes of vortex structures in 3-D.The streamwise vortex structure characteristics can be found in POD mode one as the mean flow.The other POD modes are in stremwise and spanwise structures and have the fluctuation motions,which are induced by K-H instability.Moreover,the result shows that POD modes are in pairs and share the same characteristics such as amplitudes,mode shapes,and time evolutions.The vortex core and POD results confirm that theΛ-vortex is not self-deformed to a hairpin vortex,but the hairpin vortex is formed by the K-H instability during the development of Lambda vortex to hairpin vortex in the boundary layer flow transition.

Mathematical foundation of turbulence generation-From symmetric to asymmetric Liutex

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.