Spatio-temporal characterization of non-linear forcing in turbulence

The quadratic convection term in the incompressible Navier-Stokes equations is considered as a non-linear forcing to the linear operator, and it is studied in the Fourier domain through the analysis of interactions between triadically compatible wavenumber-frequency triplets. Interaction coefficients are proposed to quantify the contribution to the forcing by each pair of triplets and are computed using data from direct numerical simulations of a turbulent channel at $Re_{\tau} \approx 550$. The coefficients show the importance of interactions involving streamwise large scales. The regions of non-linear interactions permitted under the quasi-linear (QL) and generalized quasi-linear (GQL) assumptions are shown to be significant contributors to the forcing and increasing the number of GQL-large scales is shown to monotonically increase the total forcing captured, providing a possible reason for the success of QL and GQL simulations. The tools presented are expected to be useful for improving modeling of the nonlinearity, especially in QL, GQL, and resolvent analyses, and understanding the amplitude modulation mechanism relating large-scale fluctuations to the modulation of near-wall structures.