# Turbulence and Structure Functions

1. Jun 21, 2014

### joshmccraney

Hey PF!

I was wondering if any of you were familiar with structure functions and how they relate to turbulence? Do you know of any good articles to research?

Structure functions may be defined as $$S_{ij}(\vec{R}) = \mathbb{E} \Big[ \big( u_i(\vec{x}+\vec{R}) - u_i(\vec{x}) \big) \big( u_j(\vec{x}+\vec{R}) - u_j(\vec{x}) \big) \Big]$$

where $u_i$ is the $ith$ component of velocity, $\vec{R}$ is a small displacement in the $\vec{x}$ direction, and $\mathbb{E}$ is the mean. a mean makes sense here because we are averaging dozens of realized velocities.

let me know if i need to be more in depth!

Thanks!

Josh

2. Jun 27, 2014

### Greg Bernhardt

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

3. Jun 29, 2014

### joshmccraney

Thanks for the interest, Greg!

Nothing too new. I'm researching journals like crazy trying to find some good information about it. I've done in depth readings on covariance, since the structure functions look something like those. They also look like a derivative (the numerator) but I'm not sure what these things mean (the structure functions that is).

The papers I read talk simply about them as though they are "well understood" and it seems that K45 (Kolmogerav 1945) used them, but I can't see why.

To be honest, the math is so simple that I thought the significance would be easy to intuit...obviously not.