Nonlocal analysis of the Reynolds stress in turbulent shear flow

An exact expression for the Reynolds stress is derived using the response function for the velocity fluctuation. The nonlocal eddy viscosity in the expression represents a contribution to the Reynolds stress by the mean velocity gradient at remote points in space and time. A direct numerical simulation of channel flow is conducted to validate the nonlocal expression. The transport equations for the velocity and the response function are numerically solved to evaluate the nonlocal eddy viscosity; it is shown that the nonlocal expression is accurate for both normal and shear stresses. A local expression for the Reynolds stress is also evaluated and reveals that the local approximation is not accurate enough near the wall. Analysis using the nonlocal expression is shown to be useful for obtaining a better understanding of turbulent shear flow.