  By analyzing the Karman-Howarth equation for filtered velocity fields in
turbulent flows, we show that the two-point correlation between filtered
strain-rate and subfilter stress tensors plays a central role in the evolution
of filtered-velocity correlation functions. Two-point correlations-based {\it
statistical priori tests} thus enable rigorous and physically meaningful
studies of turbulence models. Using data from direct numerical simulations of
isotropic and channel flow turbulence we show that local eddy viscosity models
fail to exhibit the long tails observed in the real subfilter stress-strain
rate correlation functions. Stronger non-local correlations may be achieved by
defining the eddy-viscosity model based on fractional gradients of order
$0<\alpha<1$ rather than the classical gradient corresponding to $\alpha=1$.
Analyses of such correlation functions are presented for various orders of the
fractional gradient operators. It is found that in isotropic turbulence
fractional derivative order $\alpha \sim 0.5$ yields best results, while for
channel flow $\alpha \sim 0.2$ yields better results for the correlations in
the streamwise direction, even well into the core channel region. In the
spanwise direction, channel flow results show significantly more local
interactions. The overall results confirm strong non-locality in the
interactions between subfilter stresses and resolved-scale fluid deformation
rates, but with non-trivial directional dependencies in non-isotropic flows.
