Decay of correlations in turbulent flows Rich literature exists that discusses the decay of correlations in the velocities between two points in space and time:
$ R(r,\tau) = \mathbb{E}\left[ v({\bf x},t) v({\bf x} + r {\bf e}, t + \tau)\right]$
where, ${\bf e}$ is the stream-wise direction, $r$ is some scalar distance and $\mathbb{E}$ represents an ensemble average. 
There are models that describe this correlation as having elliptic contours
in space-time. My question is how fast do these correlations decay?
I understand that in case of these elliptic models, the aspect ratio and tilt of the ellipse depend on the flow region (the paper I linked to above discusses turbulent shear flow) but I cannot grasp quite how fast these 
correlations decay and how this decay is related to the turbulence regions.
Intuitively, the decay should be faster in the outer layer and slower in the boundary layer but how much slower? Are there analytical/numerical results that suggest the form of decay in different regions (I am particularly interested in finding out where, if at all, the decay would be exponential)?
Thank you very much for your time! 
 A: Suppose you plot the contours of constant correlation (which are ellipses, according to the article), where each contour is labeled by its value. Then the value of the contour decreases as you move away from the  space-time origin, in any direction. Also this method of plotting iso-correlation contours shows how one can get the value of velocity correlation in space-time, say $R(r,\tau)=\langle u(x,t)u(x+r,t+\tau) \rangle$, from velocity correlation in space alone, $R(r_c,0)=\langle u(x,t)u(x+r_c,t) \rangle$, where $r_c$ is the value of radial separation at which the specified contour for $R(r,\tau)$ meets the $r$-axis in $r-t$ plane (see fig-1 in the article you have cited). Therefore it suffices to speak of velocity correlation in space alone.
If you define the normalized velocity correlation in space: $R'(r)=\frac{\langle u(x)u(x+r) \rangle}{u_{rms}(x)u_{rms}(x+r)}$, which would become equal to $\frac{\langle u(x)u(x+r) \rangle}{u_{rms}^2(x)}$ if the turbulence were statistically homogeneous, then correlation length scale is defined as (see Fluid dynamics by Kundu & Cohen, in particular the chapter on turbulence):
\begin{align}
L_x=\int_{0}^{\infty}dr~R'(r)
\end{align}
This gives you a length scale over which the correlation goes to zero, and is called "integral length scale". A similar definition for integral time scale can be given using velocity correlation function over time. As far as elliptic contours in $r-t$ plane are concerned, what this means is that, as you move away from the origin contour-value decreases at such a rate that it comes close to zero on that contour whose intercept on $r$-axis is $\sim L_x$. Anyway I think what you need is an approximation for the function $R'(r)$, and such are indeed given in the book Structure of Turbulent Shear Flows by Townsend in the very first chapter. As far as I can remember, an exponentially decaying function for $R'(r)$ seems to be a good fit for turbulent shear flows. As to how the function would change from boundary layer to bulk of the turbulent flow, I have no idea.
