# K41 Kolmogorov scaling relation and structure functions (turbulence)

1. Jan 16, 2013

### nicholasmr

Hi there.

I am having trouble interpreting the Kolmogorov K41 scaling relation for homogeneous and isotropic turbulence:

$S_{p}(l)$ = <$\delta u(l)^{p}$> = <$|u(r+l) - u(r)|^{p}$> $\propto$ ($\epsilon l$)$^{p/3}$

where $l$ is the length of displacement between two points under consideration in the flow, $\epsilon$ is the mean energy dissipation, and $u(r)$ is the velocity at point $r$ in the flow.

My question is:
Since we assume the flow to be homogeneous and isotropic in a statistically averaged sense (fully developed turbulence), how come <$\delta u(l)^{p}$> is not zero? If the turbulent flow is homogeneous and isotropic on average, then the (averaged) flow in the two point should be similar? I cannot see the $l$-dependence in my mental picture of this situation?

Regards Nicholas.