Evaluation of vortex criteria by virtue of the quadruple decomposition of velocity gradient tensor

Abstract

Vortices play a crucial role in fluid dynamics, which is closely related to fluid diffusion mixing, force, heat, and noise. Five widely-used vortex identification criteria, i.e. the -criterion, Q-criterion, -criterion, ci-criterion, and 2-criterion are analyzed, and four of them are compared with each other based on the velocity-gradient-tensor decomposition method. A new quadruple decomposition method (QDM) is introduced for the first time, so far as we know, to decompose fluid motions into four fundamental components: dilatation, axial deformation along the principal axes of the strain-range sensor, planar motion, and pure shearing. This method helps make the kinematic implications of the four vortex identification criteria more clear. It is found that the mean rotation of fluid elements always contains the pure shearing motion. Non-zero mean rotation does not guarantee the existence of the spiraling streamlines, e.g. in a typically parallel shear flow. A positive Q value indicates the strength of the pure rotation of a fluid element in the 2D complex eigenvalue plane on top of the axial deformation, which however is a sufficient but not a necessary condition for the existence of pure rotation. The -criterion can correctly tell the existence of pure rotation, but cannot accurately determine its strength. The ci-value represents the absolute strength of the pure rotation, which is the combined effect of the canonical rotation in the complex eigenvector plane and the pure shearing. The proposed QDM enables us to achieve a deeper understanding of vortices and motions in fluid dynamics.