Statistical Analysis for Liutex Growth in Flow Transition

In computational fluid dynamics, many researchers and textbooks accepted that vorticity is vortex. However, this is a misunderstanding of the tensors derived from the Cauchy-Stokes decomposition of the velocity gradient tensor. It was believed that the symmetric tensor $${\varvec{A}}$$ and antisymmetric tensor $${\varvec{B}}$$ (vorticity tensor) represented stretching/compression and rotation, respectively. Decomposing the vorticity tensor yields $${\varvec{R}}$$ (rotation part) and $${\varvec{S}}$$ (antisymmetric shear deformation part). Liutex, on the other hand, represents rigid rotation and the Liutex magnitude represents twice the angular speed. We analyze three flow areas in boundary-layer transition: laminar, transitional, and turbulent. In laminar flow, there is no vortex structure. In transitional flow, the formation of hairpin vortex rings will begin. Finally, in turbulent flow, many vortex rings have formed. In this paper, a DNS simulation of boundary transition is conducted, then statistical analysis is performed on the recorded results for Liutex, shear, and vorticity. The resulting values for Liutex followed the proper growth trend, starting at zero in laminar flow and steadily increasing through the transitional and turbulent flows. On the other hand, the vorticity values were much greater and remained consistent with little change throughout the flow transition periods. The analysis also revealed that the shear component negatively relates with Liutex, i.e., as Liutex increases, shear decreases. Since shear substantially impacts the vorticity value where it can be misrepresented as rotation in laminar flow, vorticity, in general, should not be considered vortex.