  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.
