Spatio-temporal quantification of triadic contributions to the turbulent velocities in channel flow

To help clarify the role of the advective nonlinearity in turbulent wall flows, we analyze data from the direct numerical simulation (DNS) of an incompressible turbulent channel flow at a friction Reynolds number of 550 (Flores and Jimenez, JFM 2006). After performing a Reynolds decomposition, the quadratic fluctuation-fluctuation term in the Navier-Stokes equations is treated as a non-linear forcing to the associated linear operator. This forcing is studied in the Fourier domain, in which it becomes a convolution sum over all triadically compatible wavenumber-frequency triplets. The linear resolvent operators (McKeon and Sharma, JFM 2010) are applied to each pair of interacting triplets, and the resulting contributions to the corresponding velocity fields are computed. The results show the importance of interactions involving large streamwise scales. Furthermore, the regions of non-linear interactions permitted under the quasi-linear (QL) and generalized quasi-linear (GQL) reductions (Marston et al., PRL, 2016) are shown to be significant contributors to the velocity fields, providing a possible reason for the success of QL and GQL simulations.