I want to know that if a vortical region exists, does it mean the turbulence kinetic energy has higher values in that region? (the relation between tubulence kinetic energy and vorticity)
Well, first a vorticity field does not imply automatically a vortical structures. For example the laminar Poiseulle solution has vorticity without vortical structures.
The relevant part of the vorticity equation is in the stretching term that is responsible to redistribute the energy between vortical structures of different scales.