A brief overview

Laminar smooth flows are found in well controlled situations. In the nature, laminar flows are exceptions, the predominant state is the turbulent one, where the flow presents a very irregular and complicated motion both in space and time. An usual way to characterize a flow, is by the Reynolds number $Re$, defined as

\begin{align} Re=\frac{UL}{\nu} , \end{align}

where $U$ is the mean velocity, $L$ is the integral length scale and $\nu$ is the kinematic viscosity. The degree of irregularity of the flows increases as $Re$ increases, and this irregularity can be seen in a broad range of scales, from the smallest scales of motion $\eta$ to the largest scales $L$.

For $Re \rightarrow \infty$, one could expect statistical universality, at least in a specific range far less than the scales ($L$), where kinetic energy enters into the system, and much larger than the dissipative scales ($\eta$). These ideas were postulated by A. N. Kolmogorov in 1941, when he introduced the concept of locally isotropic turbulence and the use of the velocity differences ($v_r=u(x+r)-u(x)$) statistics through structure functions ($S_p(r)\equiv \left<v_r^p \right>_x$). Kolmogorov's original work is commonly stated as two universal equations, valid for the inertial subrange $\eta \ll r\ll L$ and is frequently referred as K41:
the two-thirds law:

\begin{align} $S_2(r)=C_k\,\epsilon^{2/3}\,r^{2/3}$, \end{align}

and the four-fifties law:

\begin{align} $S_3(r)=-\frac{4}{5} \,\epsilon\, r$, \end{align}

where$C_k$ is an universal constant and $\epsilon$ is the mean energy dissipation.

With these equations, it is possible to obtain a law for higher statistics moments:

\begin{align} $S_p(r)=C_p\,\epsilon^{p/3}\,r^{p/3}\propto r^{\zeta_p}=r^{p/3}$. \end{align}

The linear scaling behaviour $\zeta_p=p/3$ is a characteristic of a self-similar process. Which means that the transformation $f(r)\rightarrow f(\lambda r)$ modify the process by a pre-factor $\lambda^h$, that is $f(\lambda r) =\lambda^h f(r)$. Accoding to the Kolmgorov predictions in 1941, $h=1/3$. Owing to the Wiener-Kintchine theorem, the $\zeta_p=p/3$ scaling corresponds to the $k^{-5/3}$ spectrum.

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