LAUSR

Abstract By analysing the Karman–Howarth equation for filtered-velocity fields in turbulent flows, we show that the two-point correlation between the filtered strain-rate and subfilter stress tensors plays a central role in the evolution of filtered-velocity correlation functions. Two-point correlation-based statistical a 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 $0<\alpha <1$ (where $\alpha$ is the fractional gradient order) 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. Hence, non-local operators thus exhibit interesting modelling capabilities and potential for large-eddy simulations although more developments are required, both on the theoretical and computational implementation fronts.