Explicit expressions for Rortex tensor and velocity gradient tensor decomposition

Abstract

The introduction of Rortex provides a new perspective to investigate the local properties of vortical structures in transitional and turbulent flows, as Rortex offers a new and systematic description of the local fluid rotation, including scalar, vector and tensor forms. Unfortunately, the previous definition of Rortex is not straightforward, which requires the explicit calculation of somewhat cumbersome coordinate rotation. In this letter, a new explicit tensor form of Rortex and the relevant explicit velocity gradient tensor decomposition are presented, based on an explicit formula of the Rortex vector. The explicit tensor form represents the real local rotational part of the velocity gradient tensor in the original coordinate system. The explicit calculation of coordinate rotations can be totally avoided, which indicates an important improvement of Rortex based velocity gradient tensor decomposition.