Two-point stress-strain-rate correlation structure and non-local eddy viscosity in turbulent flows

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 similar to 0.5 yields best results, while for channel flow alpha similar to 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.

Automated summary

By analyzing the Karman-Howarth equation for filtered-velocity fields in turbulent flows, it is shown that the two-point correlation between the filtered strain-rate and subfilter stress tensors is central in the evolution of filtered-velocity correlation functions. Statistical a priori tests based on two-point correlations enable rigorous and physically meaningful studies of turbulence models. It is found that using fractional gradient orders rather than classical gradients in eddy-viscosity models can lead to stronger non-local correlations.