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Based on the analysis of the velocity gradient tensor, we investigate in thispaper the physical interpretation and limitations of four vortex criteria:$\omega$, $Q$, $\varDelta$ and $\lambda_{ci}$, and reveal the actual physicalmeaning of vortex patterns which are usually illustrated by level sets ofvarious vortex criteria. A quadruple decomposition based on the normality ofthe velocity gradient tensor is proposed for the first time, which resolves themotion of a fluid element into dilation, axial stretch along the normal frame,in-plane distortion, and simple shear, in order to clarify the kinematicalinterpretation of various vortex criteria. The mean rotation characterized bythe vorticity $\omega$ always consists of simple shear; the $Q$-criterion canreflect the strength of net rotation within the invariant plane relative to theaxial stretch of a fluid element, and it is a sufficient but unnecessarycondition for the existence of net rotation; the $\varDelta$-criterion canexactly identify the existence of net rotation, but it is not the strength ofnet rotation; in the case that net rotation exists, $\lambda_{ci}$ characterizeits absolute strength. net rotation is the total effect of the normal rotationwithin the invariant plane and the simple shear, where the former is the mostbasic rotation. The newly introduced quadruple decomposition can improve ourunderstanding of vortices in fluids and their motions.

Evaluation of Vortex Criteria by Virtue of the Quadruple Decomposition of Velocity Gradient Tensor