Kinematic vorticity decomposition for generic two-dimensional flow

The present paper begins with evaluating the matrix representation of the velocity gradient tensor A by applying the method of differential geometry to the intrinsic streamline triad. Then, a direction-dependent kinematic decomposition of the vorticity is proposed for a generic two-dimensional (2D) flow, where the corresponding expressions are derived for the two elementary vorticity modes (R(t),s(t)), physically accounting for the orbital rotation (as a specific form of the rigid rotation mode) and the shear under the field description. We show that (R(t),s(t)) are expressible in terms of the angular velocities of a pair of orthogonal material line (surface) elements, naturally resulting in a physically intuitive representation of A. For the first time, the vorticity constituents (RN,sN) in the established invariant vorticity decomposition, previously deduced from the real Schur form of A at the vortex-structure level, are reformulated based on the direction-dependent variables (R(t),s(t)). It is proved that the disparity between the magnitudes of sN and s(t) is solely attributed to the potential flow, being determined by the difference between the relative stretching rates along the tangential and principal normal directions of a streamline. Next, a thorough analysis is performed for all the possible cases of a generic 2D flow by which the physical understanding of the distinct vorticity modes is, thus, deepened. Finally, the significance of the theoretical results are further illustrated by analyzing the velocity fields of the viscous flow inside a lid-driven cavity and the intriguing coherent structures in the Jupiter's Great Red Spot (GRS) extracted from the Galileo cloud images. The analysis demonstrates the generality of the proposed kinematic vorticity decomposition and its superiority in capturing the details of flow structures. The present exposition could be valuable in separating the elementary vorticity modes at different levels of description, as well as understanding their kinematic evolution mechanisms in a wide range of vortical-flow phenomena.