Turbulence

Overview

Laminar flows only occurs in well controlled situations; in the nature, the favourite state is the turbulent one. Turbulence seems to be behave as a permanent complex state and not as a transient or a complicated state. A usual way to characterize a flow, is by the Reynolds number $Re$, define as

(1)
\begin{align} Re=\frac{UL}{\nu} . \end{align}

For $Re \rightarrow \infty$, one could expect statistical universality, at least in a specific range far less than the scales ($L$), where kinetic energy enter into the system, and much larger than the dissipative characteristic scales ($\eta$). These ideas were postulated by A. N. Kolmogorov in 1941, when 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 frequently referred as K41:
the two-thirds law:

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

and the four-fifties law:

(3)
\begin{align} $S_3(r)=-\frac{4}{5} \,\epsilon\, r$, \end{align}

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

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

(4)
\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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