A mechanism of formation of vortical structure with interaction between vortical flow and bundle of vorticity lines derived from local geometry

The present study clarifies that vortical flow and bundle of vorticity lines in vortical region have an interaction and that it forms the vortical structure. The detail geometrical theory of local flow geometry given by gradient tensor of a vector field derives a universal representation of an asymmetric vortical flow, where the radial and azimuthal components are given by specific quadratic forms and periodic with wavenumber 2. Then both inflow and outflow exist midway between long and short axes of elliptic vortical flow. A Galilei invariant coordinate system associated with these axes and the swirl plane can be defined as vortex space, and formulation of the vortex stretching in the space reveals that it gives a nonlinear effect and rotates the vorticity vector in its vector field. On the other hand, the representation of the velocity gradient tensor in the vortex space indicates that the gradient of the axial flow in the plane is zero, and then its distribution is given by its Hessians in the swirl plane. A mathematical insight shows that an axial flow with a uniform direction is associated with the swirling of the vorticity lines, and eigenvalues of the azimuthal component of the lines have an intense correlation. Thus, the vortical flow swirls the vorticity lines, and then the swirling lines generate the uniform axial flow, which forms an important aspect of the vortical structure. A vortical analysis in homogeneous isotropic turbulence with direct numerical simulation shows these interactive characteristics.