Use of Lagrangian statistics for the analysis of the scale separation hypothesis in turbulent channel flow

Turbulence models often involve Reynolds averaging, with a closure providing the Reynolds stress tensor as function of mean velocity gradients, through a turbulence constitutive equation. The main limitation of this linear closure is that it rests on an analogy with kinetic theory. For this analogy to be valid there has to be a scale separation between the mean velocity variations and the turbulent Lagrangian free path whose mean value is the turbulent mixing length. The aim of this work is to better understand this hypothesis from a microscopic point of view. Therefore, fluid elements are tracked in a turbulent channel flow. The flow is resolved by direct numerical simulation (DNS). Statistics on particle trajectories ending on a certain distance y0 from the wall are computed, leading to estimations of the turbulent mixing length scale and the Knudsen number. Comparing the computed values to the Knudsen number in the case of scale separation, we may know in which region of the flow and to what extent the turbulence constitutive equation is not verified. Finally, a new non-local formulation for predicting the Reynolds stress is proposed.