Spiral streamline pattern around a critical point: Its dual directivity and effective characterization by right eigen representation

Abstract

In recent years, vortex recognition method based on velocity gradient has rapidly become a research hotspot, accompanied by a wide range of applications. As a linear velocity field with the origin as its critical point uniquely corresponds to the velocity gradient, its streamline pattern (SP), especially the spiral streamline structure when the velocity gradient has a couple of complex eigenvalues, is studied using the right eigen representation based on the real Schur form. Compared with the left eigen representation, the right eigen representation of velocity gradient is seldom concerned. In this paper, the right eigen representation is carried out in terms of the spectral representation, and its relation with the left eigen representation is also derived. For two-dimensional case, the SP classification is listed and the typical streamlines are illustrated. After detailed investigation, it is shown that the parameters extracted from the right eigen representation would be more suitable to describe the geometrical features of the spiral streamline pattern around the critical point, and the dual directivity of spiral streamline structure is clarified. Some discussions and an illustrative example from the direct numerical simulation (DNS) data are presented.