Mathematical Definition of Vortex Boundary and Boundary Classification Based on Topological Type

Recently, new progress has been made in vortex recognition, and the definition of Liutex (Rortex) based on eigenvectors has established a relation between rotation axis and eigenvectors of velocity gradient tensor. Based on this relation, the geometric and physical meanings of Liutex magnitude and the concept of rigid-rotation of vortex are revisited. Considering the simple shear, geometric meaning of Liutex can be explained through geometric relations in this paper, believing that the dominant quantity rotation described by Liutex implies rotational strength without non-circular symmetry. Moreover, The mathematical condition of vortex boundary is given: the set of points with multiple roots in the characteristic equation of velocity gradient tensor in a flow field. In this way, the topological structure of critical point theory is applied to vortex boundary. According to whether the velocity gradient tensor can be diagonalized, there are shear boundary and non-shear boundary, while according to the positive, negative and zero of the double root, there would be stable boundary, unstable boundary and degenerate boundary. Under different decompositions, we can analyze the superposition of fluid deformation behavior more clearly by mapping image space, which can be used to compare Helmholtz decomposition with Liutex velocity gradient decomposition.