I will cite the two definitions from  as they are presented in V. Levin's translation  and then explain them in terms relevant to this work.
What these two definitions add up to, if you ignore the details needed to convince a mathematician that the world does exist, is that the probability distribution for the relative velocity of two particles in the fluid (G) only depends on the distance between the particles. - Definition 1 removes the dependence of location and definition 2 removes the dependence of relative direction.
Kolmogorov then argues that if you are looking at the system at length scales, which are far from both the length at which energy is pumped into the system, and from the length scale at which the energy dissipates from the system as heat due to viscosity, the only parameter needed to describe the probability distribution of the relative velocity field in a homogeneous isotropic turbulent fluid is the average energy dissipation.
With this hypothesis of similarity, a dimension analysis of the
available parameters (, , and R)
results in the following scaling relation