The basic phenomenology of turbulence can be recovered
from a simple dimensional analysis of Navier-Stokes equations,
using the image of the *turbulent cascade*
proposed by Richardson [9].

The kinetic energy is supposed to be injected by an external forcing which sustains the motion of large scale eddies. This structures are deformed and stretched by the fluid dynamics, until they break into smaller eddies, and the process is repeated such that energy is transported to smaller and smaller structures. Finally at small scales the kinetic energy is dissipated by the viscosity of the fluid. The whole process of transport of energy from the large scale of injection to the small dissipative scale, through the hierarchy of eddies is known as turbulent cascade. It is worthwhile to remember that the eddies must not be thought as real vortices, but just as a metaphoric description of the triadic interaction between modes which has been formally presented in the previous section.

A dimensional analysis of the different terms of Navier-Stokes equation provides an estimate for the time required to transfer energy from an eddy of size to smaller eddies , where is the rms velocity fluctuation on the scale , and the time required to dissipate the energy contained in the same eddy by the viscous term: .

Three different range of scales can thus be identified:

**Injective range**- which corresponds to the large scales where the forcing injects the energy.
**Inertial range**- where the time required for energy transfer is shorter than the dissipative time and the energy is thus conserved and transported to smaller scales.
**Dissipative range**- where the energy dissipation overcomes the transfer and the cascade is stopped.

The hypothesis of a statistically steady state for the turbulent cascade
requires a constant energy flux in the inertial range,
i.e. a constant rate of energy transfer that must be equal to the energy
dissipation rate :

The border between the inertial and dissipative range
is identified by the *Kolmogorov scale* , where the
dissipative and transfer times are equal
: