Physical meaning of vorticity based on the RS decomposition and an explicit formula for the Liutex vector

In the present study, the physical meaning of vorticity is revisited based on the RS decomposition proposed by Liu et al. in the framework of Liutex (previously named Rortex), a vortex vector field with information of both rotation axis and swirling strength [ C. Liu et al., "Rortex-A new vortex vector definition and vorticity tensor and vector decomposition", Phys. Fluids 30, 035103 (2018)]. It is demonstrated that the vorticity in the direction of rotational axis is twice the spatial mean angular velocity in the small neighborhood around the considered point while the imaginary part of the complex eigenvalue (λ_ci) of the velocity gradient tensor (if exist) is the pseudo-time average angular velocity of a trajectory moving circularly or spirally around the axis. In addition, an explicit expression of the Liutex vector in terms of the eigenvalues and eigenvectors of velocity gradient is obtained for the first time from above understanding, which can further, though mildly, accelerate the calculation and give more physical comprehension of the Liutex vector.