A theory and analysis that a vortex makes the vorticity lines vortical through vortex stretching*

The present study shows that the vortex stretching in vortical region with swirling flow gives an effect to swirl vorticity lines or vector, with the formulation of the stretching with respect to the vortex space that is the specific Galilei invariant coordinate system associated with the swirl plane of the local flow. A numerical analysis of bundle of vorticity lines in core region of a vortex clarifies this geometrical phenomenon in the direct numerical simulation of a homogeneous isotropic turbulence. The swirlity andsourcity that represent the unidirectionality and intensity of respective azimuthal andradial flowsin terms of the local flow, and the local axis geometry theory are applied to specify detail geometrical characteristic of the vorticity lines, in the swirl plane of a vortical core region. It shows that the characteristic of the non-azimuthal component of the vorticity lines or vector specifies the swirling feature of the vorticity lines with the stretching, and that the intensity and sign of the sourcity of local vorticity field (vorticity flow) are associated with it. The vortical flow geometry and this particular feature of swirling of the vorticity lines are combined with the vortex stretching in the vortical region.