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Velocity gradient is the basis of many vortex recognition methods, such as Qcriterion, $\Delta$ criterion, $\lambda_{2}$ criterion, $\lambda_{ci}$criterion and $\Omega$ criterion, etc.. Except the $\lambda_{ci}$ criterion,all these criterions recognize vortices by designing various invariants, basedon the Helmholtz decomposition that decomposes velocity gradient into strainrate and spin. In recent years, the intuition of 'no vortex in straight flows'has promoted people to analyze the vortex state directly from the velocitygradient, in which vortex can be distinguished from the situation that thevelocity gradient has couple complex eigenvalues. A specious viewpoint to adoptthe simple shear as an independent flow mode was emphasized by many authors,among them, Kolar proposed the triple decomposition of motion by extracting aso-called effective pure shearing motion; Li et al. introduced the so-calledquaternion decomposition of velocity gradient and proposed the concept of eigenrotation; Liu et al. further mined the characteristic information of velocitygradient and put forward an effective algorithm of Liutex, and then developedthe vortex recognition method. However, there is another explanation for theincreasingly clear representation of velocity gradient, that is the localstreamline pattern based on critical-point theory. In this paper, the tensorialexpressions of the right/left real Schur forms of velocity gradient areclarified from the characteristic problem of velocity gradient. The relationsbetween the involved parameters are derived and numerically verified. Comparingwith the geometrical features of local streamline pattern, we confirm that theparameters in the right eigen-representation based on the right real Schur formof velocity gradient have good meanings to reveal the local streamline pattern.Some illustrative examples from the DNS data are presented.

No vortex in straight flows -- on the eigen-representations of velocity gradient