Closing the loop: an explicit calculation of the nonlinearity in the resolvent formulation of wall turbulence

The resolvent-based analysis of wall turbulence (McKeon & Sharma 2010) is extended with an explicit treatment of the non-linearity in the Navier-Stokes equations. The equivalent of triadic interaction in the wall-normal direction is described, and it is found that the resulting forcing has a phase shift of π/2 in terms associated with symmetric spatial directions. Explicit forms for the nonlinearity in fully-developed pipe and channel flows, i.e. in cylindrical and Cartesian coordinate systems.