Direct Calculation of the Eddy Viscosity Operator in Turbulent Channel Flow at Re$_τ$=180

Abstract: This study aims to quantify how turbulence in a channel flow mixes momentum in the mean sense. We applied the macroscopic forcing method (Mani and Park, Physical Review Fluids, 2021, p.054607) to direct numerical simulation (DNS) of a turbulent channel flow at Re_tau=180 using two different forcing strategies that are designed to separately assess the anisotropy and nonlocality of momentum mixing. In the first strategy, the leading term of the Kramers-Moyal expansion of the eddy viscosity is quantified, revealing all 81 tensorial coefficients that essentially characterize the local-limit eddy viscosity. The results indicate: (1) the eddy viscosity has significant anisotropy, (2) Reynolds stresses are generated by both the mean strain rate and mean rotation rate tensors associated with the momentum field, and (3) the local-limit eddy viscosity generates asymmetric Reynolds stress tensors. In the second strategy, the eddy viscosity is quantified as an integration kernel revealing the nonlocal influence of the mean momentum gradient at each wall-normal coordinate on all nine components of the Reynolds stresses over the channel width. Our results indicate that while the shear component of the Reynolds stress is reasonably controlled by the local mean gradients, other components of the Reynolds stress are highly nonlocal. These results provide an understanding of anisotropy and nonlocality requirements for closure modeling of momentum transport in wall-bounded turbulent flows.