K41 Kolmogorov scaling relation and structure functions (turbulence)

[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.

 

[LightHero]

The Kolmogorov K41 scaling relation for homogeneous and isotropic turbulence does not assume that the flow between two points is similar. Instead, it assumes that the average velocity difference between two points is equal to the product of the mean energy dissipation and the length of displacement between the two points. This is due to the fact that turbulent flows are chaotic and unpredictable and so the velocity differences between two nearby points can vary greatly. This is why the K41 scaling relation is useful in predicting the behavior of homogeneous and isotropic turbulence.