Kolmogorov K41

The naive picture drawn above can be formally stated within the $K41$ theory in term of the scaling properties of the Structure functions:

$$S_p(\ell) \equiv \langle (\delta u_{\ell})^p \rangle$$ (1.46)

which are defined as the moments of the distribution of longitudinal velocity increments $\delta u_{\ell}({\mbox{\boldmath$x$}}) \equiv [{\mbox{\boldmath$u$}}({\mbox{\b... ...{\mbox{\boldmath$u$}}({\mbox{\boldmath$\ell$}})] \cdot {\mbox{\boldmath$\ell$}}$.

The longitudinal velocity increments are easily achievable in experiments, e.g with hot wire anemometry. Let's suppose to have a velocity field ${\mbox{\boldmath$u$}}$ which can be decomposed in a mean flow ${\mbox{\boldmath$U$}} = (U,0,0)$ and a turbulent fluctuating part $u$' = $u$ - $U$ whose intensity is assumed to be small compared with the mean flow $\langle \vert{\mbox{\boldmath$u$}}'\vert^2 \rangle^{1/2} \ll U$. By putting an hot wire perpendicular to the mean flow, let's say in the $z$ direction, and measuring its resistance which is reduced because of the cooling due to the flow, it is possible to obtain the time series of the velocity integrated in the direction of the wire, i.e.

$$u_N = [ (u'_x + U)^2 + {u'_y}^2 ]^{1/2} = U [ 1 + \frac{u'_x}{U} + O({\frac{u'^2}{U^2}})]$$ (1.47)

where it has been supposed that amplitudes of fluctuations in the two direction perpendicular to the wire are of the same order $u'_y \sim u'_x$. Within Taylor's hypothesis, i.e. assuming that the turbulent velocity field for short time delay $\tau$ is almost frozen and it is simply transported through the wire by the fast mean flow ${\mbox{\boldmath$U$}}$, the time series $u'_x(t)$ can be considered as a space series

$$u'_x(x,t+\tau) = u'_x(x - U/\tau,t)$$ (1.48)

allowing to obtain the longitudinal structure function.

The basic assumption of the Kolmogorov theory is the Similarity Hypothesis. Kolmogorov's hypothesis assumes that if the inertial range is large enough, the influence of the large scale forcing and the small scale viscous dissipation can be neglected, and the scale invariance of Navier-Stokes equation in the inviscid limit:

$$\begin{array}{ll} t,{\mbox{\boldmath$r$}},{\mbox{\boldmath$u... ...$u$}} & \lambda \in \mathbb{R}_+ , h \in \mathbb{R}\end{array}$$ (1.49)

is recovered by the turbulent velocity field in a statistical sense. The velocity fluctuations on scale $\ell$ within the inertial range are supposed to be self-similar

$$\delta u_{\lambda \ell} \sim \lambda^h \delta u_{\ell}$$ (1.50)

i.e. their probability distribution are supposed to be identical once they are rescaled according to the scaling exponent $h$, and the structure functions in the limit $Re \to \infty$ are expected to display a power law behavior

$$S_p(\ell) = \langle (\delta u_{\ell})^p \rangle \sim \ell^{hp}$$ (1.51)

Starting from the Karman-Howarth-Monin relation [1] Kolmogorov derived an exact result for the third order structure function, the famous

Kolmogorov's four-fifths law

In the limit of infinite Reynolds number the third order (longitudinal) structure function of homogeneous isotropic turbulence, evaluated for increments $\ell$ small compared to the integral scale, is given in terms of the mean energy dissipation per unit mass $\epsilon$ by

$$S_3(\ell) \equiv \langle (\delta u_{\ell})^3 \rangle = - {4 \over 5} \epsilon \ell$$ (1.52)

The four-fifths law allows to fix the value of the scaling exponent $h=1/3$ and together with the scaling hypothesis for the structure functions leads to the Kolmogorov scaling law:

$$S_p(\ell) \sim C_p \epsilon^{p/3} \ell^{p/3}$$ (1.53)

and to the Kolmogorov energy spectrum

$$E(k) \equiv 2 \pi k \langle \vert\hat{\mbox{\boldmath$u$}}({... ...\boldmath$k$}})\vert^2 \rangle = C \epsilon^{2/3} k^{5/3} \;.$$ (1.54)

A Kolmogorov energy spectrum has been observed in many different physical situations, from the experiments in tidal channel [10] which gave the first confirmations of Kolmogorov's theory, to recent measurements in wind-tunnel experiments[11].

Figure 1.1: Kolmogorov's energy spectrum in the S1 wind tunnel ONERA [11]