Local Streamline Pattern and Topological Index of an Isotropic Point in a 2D Velocity Field

In fluid mechanics, most studies on flow structure analysis are simply based on the velocity gradient, which only involves the linear part of the velocity field and does not focus on the isotropic point. In this paper, we are concerned with a general polynomial velocity field with a nonzero linear part and study its streamline pattern around an isotropic point, i.e., the local streamline pattern (LSP). A complete classification of LSPs in two-dimensional (2D) velocity fields is established. By proposing a novel formulation of qualitative equivalence, namely, the invariance under spatiotemporal transformations, we first introduce the quasi-real Schur form to classify the linear part of velocity fields. Then, for a nonlinear velocity field, the topological type of its LSP is either completely determined by the linear part when the determinant of the velocity gradient at the isotropic point is nonzero or controlled by both linear and nonlinear parts when the determinant of the velocity gradient vanishes at the isotropic point. Four new topological types of LSPs through detailed sector analysis are identified. Finally, we propose a direct method for calculating the index of the isotropic point, which also serves as a fundamental topological property of LSPs. These results do challenge the conventional linear analysis paradigm that simply neglects the contribution of the nonlinear part of the velocity field to the streamline pattern.