Use of Lagrangian Statistics for the Direct Analysis of the Turbulent Constitutive Equation

Abstract

Turbulence models often involve Reynolds averaging, with a closure providing the Reynolds stress $\overline{u'v'}$ as function of mean velocity gradient dū/dy, through a turbulence constitutive equation (Eq. 1). The main limitation of this linear closure is that it rests on an analogy with the kinetic theory. For this analogy to be valid there has to be scale separation. 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 are computed leading to estimations of the turbulent mixing length scale and the Knudsen number. Comparing the computed values to the values in the case of scale separation we may know where and to what extent Eq. (1) is not verified. Finally, a new non-local formulation for predicting the Reynolds stress is proposed.